FRACTURE OF POLYMERS, COMPOSITES AND ADHESIVES II
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The Behaviour of Short Fatigue Cracks Edited by K.J. Miller and E.R. de los Rios The Fracture Mechanics of Welds Edited by J.G. Blauel and K.-H. Schwalbe Biaxial and Multiaxial Fatigue Edited by M.W. Brown and K.J. Miller The Assessment of Cracked Components by Fracture Mechanics Edited by L.H. Larsson Tielding. Damage, and Failure of Anisotropic Solids Edited by J.R Boehler High Temperature Fracture Mechanisms and Mechanics Edited by R Bensussan and J.R Mascarell Environment Assisted Fatigue Edited by R Scott and R.A. Cottis Fracture Mechanics Verification by Large Scale Testing Edited by K. Kussmaul Defect Assessment in Components Fundamentals and Applications Edited by J.G. Blauel and K.-H. Schwalbe Fatigue under Biaxial and Multiaxial Loading Edited by K. Kussmaul, D.L. McDiarmid and D.F. Socie Mechanics and Mechanisms of Damage in Composites and Multi-Materials Edited by D. Baptiste High Temperature Structural Design Edited by L.H. Larsson Short Fatigue Cracks Edited by K.J. Miller and E.R. de los Rios Mixed-Mode Fatigue and Fracture Edited by H.R Rossmanith and K.J. Miller Behaviour of Defects at High Temperatures Edited by R.A. Ainsworth and R.R Skelton Fatigue Design Edited by J. Solin, G. Marquis, A. Siljander and S. Sipila Mis-Matching of Welds Edited by K.-H. Schwalbe and M. Ko^ak Fretting Fatigue Edited by R.B. Waterhouse and T.C. Lindley Impact of Dynamic Fracture of Polymers and Composites Edited by J.G. Williams and A. Pavan Evaluating Material Properties by Dynamic Testing Edited by E. van Walle Multiaxial Fatigue & Design Edited by A. Pineau, G. Gailletaud and T.C. Lindley Fatigue Design of Components. ISBN 008-043318-9 Edited by G. Marquis and J. Solin Fatigue Design and Reliability. ISBN 008-043329-4 Edited by G. Marquis and J. Solin Minimum Reinforcement in Concrete Members. ISBN 008-043022-8 Edited by Alberto Carpinteri Multiaxial Fatigue and Fracture. ISBN 008-043336-7 Edited by E. Macha, W. B.edkowski and T. Lagoda Fracture Mechanics: Applications and Challenges. ISBN 008-043699-4 Edited by M. Fuentes, M. Elices, A. Martin-Meizoso and J.M. Martinez-Esnaola Fracture of Polymers, Composites and Adhesives. ISBN 008-043710-9 Edited by J.G. Williams and A. Pavan Fracture Mechanics Testing Methods for Polymers Adhesives and Composites. ISBN 008-043689-7 Edited by D.R. Moore, A. Pavan and J.G. Williams Temperature-Fatigue Interaction. ISBN 008-043982-9 Edited by L. Remy and J. Petit From Charpy to Present Impact Testing. ISBN 008-043970-5 Edited by D. Franfois and A. Pineau Biaxal/Multiaxial Fatigue and Fracture ISBN 008-044129-7 Edited by A. Carpinteri, M. de Freitas and A. Spagnoli
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l^RACTURE OF POLYMERS, COMPOSITES AND ADHESIVES II
Editors: B.R.K. Blackman A. Pavan J.G. Williams
ESIS Publication 32 This volume contains 47 peer-reviewed papers selected from those presented at the 3rd ESIS TC4 conference, "Fracture of Polymers, Composites and Adhesives" held in Les Diablerets, Switzerland, 15-18th September 2002.
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2003
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CONFERENCE COMMITTEE Dr Ph. Beguelin, EPFL, Lausanne (Switzerland), Dr B.R.K. Blackman, Imperial College London (UK), Dr W. Bohme, IWM, Freiburg (Germany), Dr A J. Brunner, EMPA, Duebendorf (Switzerland), Dr L. Castellani, ENICHEM, Mantova (Italy), Dr A. Cervenka, UMIST, Manchester (UK), Dr D.R. Moore, ICI, Wilton (UK), Professor A. Pavan, Politecnico di Milano (Italy), Dr F. Ramsteiner, BASF, Ludwigshafen (Germany), Dr RE. Reed, University of Twente (The Netherlands), Professor J.G. Williams (Chairman), Imperial College London (UK)
Elsevier Internet Homepage - http://www.elsevier.com Consult the Elsevier homepage for full catalogue information on all books, journals and electronic products and services. Elsevier Titles of Related Interest CARPINTERI Minimum Reinforcement in Concrete Members. ISBN: 008-043022-8
MOORE ETAL. Fracture Mechanics Testing Methods for Polymers, Adhesives and Composites. ISBN: 008-043689-7
CARPINTERI ETAL. Biaxial/Multiaxial Fatigue and Fracture. ISBN: 008-044129-7
MURAKAMI Metal Fatigue Effects of Small Defects and Nonmetallic Inclusions ISBN: 008-044064-9
FRANCOIS and PINEAU From Charpy to Present Impact Testing. ISBN: 008-043970-5
RAVICHANDRAN ET AL. Small Fatigue Cracks: Mechanics, Mechanisms & Applications. ISBN: 008-043011-2
FUENTESETAL. Fracture Mechanics: Applications and Challenges. ISBN: 008-043699-4
REMY and PETIT Temperature-Fatigue Interaction. ISBN: 008-043982-9
JONES Failure Analysis Case Studies II. ISBN: 008-043959-4
TANAKA & DULIKRAVICH Inverse Problems in Engineering Mechanics II. ISBN: 008-043693-5
MACHAETAL. Multiaxial Fatigue and Fracture. ISBN: 008-043336-7
VOYIADJISETAL. Damage Mechanics in Engineering Materials. ISBN: 008-043322-7
MARQUIS & SOLIN Fatigue Design of Components. ISBN: 008-043318-9
VOYIADJIS & KATTAN Advances in Damage Mechanics: Metals and Metal Matrix Composites. ISBN: 008-043601-3
MARQUIS & SOLIN Fatigue Design and Reliability, ISBN: 008-043329-4
WILLIAMS & PAVAN Fracture of Polymers, Composites and Adhesives. ISBN: 008-043710-9
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CONTENTS
Foreword
xi
1. Polymers LI. Slow Crack Growth and Fatigue Micromechanisms of Slow Crack Growth in Polyethylene C.J. G. Plummer, A. Goldberg and A. Ghanem
3
Analysis of the Fracture Behavior of Amorphous Semi-Aromatic Polyamides B, Brule, L. Monnerie and J.L Halary
15
Experimental Analysis of Glassy Polymers Fracture Using a Double Notch Four Point-Bending Method N. Saad, C. Olagnon, R. Estevez and J. Chevalier
27
Toughening Effect in Highly Filled Polypropylene Through Multi-Scale Particle Size G. Orange and Y. Bomal
39
Experimental and Theoretical Investigation of the Contact Fatigue Behaviour of an Epoxy Polymer Under Small Amplitude Sliding Micro-Motions M. C Dubourg and A. Chateauminois
51
1.2. Essential Work of Fracture Experimental Study of Rubber-Toughening of PET A'; Billon and J.-R Meyer
65
Essential Work of Fracture of Injection Moulded Samples of PET and PET/PC Blends J.J. Sanchez, 0.0. Santana, A. Gordillo, M.Ll. Maspoch and A.B. Martinez
77
Rate and Temperature Effects on the Plane Stress Essential Work of Fracture in Semicrystalline PET A. Pegoretti and T. Ricco
89
1.3. Environmental Stress Cracking Effects of Detergent on Crack Initiation and Propagation in Polyethylenes M. Rink, R. Frassine, P. Mariani and G. Carianni
103
The Environmental Stress Cracking of a PBT/PBA Co-Poly(ester ester) KB. Kuipers, A.C. Riemslag, R.F.M. Lange, M. Janssen, R. Marissen, K. Dijkstra and A. Bakker
115
1.4. Rate Effects A New Way for Polymer Characterisation Using a Combined Approach LEFM-Plastic Zone Corrected LEFM C Grein, Ph. Beguelin and H.-H. Kausch
129
Effects of Constraint on the Traction-Separation Behaviour of Polyethylene S.K.M. Ting, J.G. Williams and A. Ivankovic
143
Micromechanical Modelling of Rate and Temperature Dependent Fracture of Glassy Polymers R. Estevez, S. Basu and E. Van der Giessen
155
Cohesive Properties of a Crystalline Polymer Craze Under Impact Extension P. Leevers, S. Hazra and L Wang
167
Laboratory Test for Measuring Resistance to Rapid Crack Propagation A. Biirgel, T. Kobayashi and D.A. Shockey
175
Rate Dependent Fracture Toughness of Plastics Z. Major and R. W Lang
187
Numerical Determination of the Energy Calibration Function g^ for High Rate Charpy Impact Tests A. Rager, J.G. Williams and A. Ivankouic
199
The Three Dimensional Stress Fields at the Dynamic Crack Tip Associated with the Crack Branching in PMMA M. Watanabe A Drop Tower Method for High Rate Fracture Toughness Testing of Polymers /. Horsfall, C.H. Watson and C.G. Chilese The Strain Rate Dependence of Deformation and Fracture Behaviour of Acrylonitrile-Butadiene-Styrene (ABS) Copolymer in Impact Test W-S. Lee and H-L Lin
207
221
231
Elastic and Viscoelastic Fracture Analysis of Cracks in Polymer Encapsulations a Wittier, P. Sprafke and B. Michel
241
Modelling the Drop Impact Behaviour of Fluid-Filled Polyethylene Containers A. Karac and A. luankouic
253
Inverse Method for the Analysis of Instrumented Impact Tests of Polymers V. Pettarin, P. Frontini and G. Eligabe
265
2. Adhesive Joints Fracture Mechanics Tests to Characterize Bonded Glass/Epoxy Composites: Application to Strength Prediction in Structural Assemblies P. Davies and J. Sargent
279
On the Mode II Loading of Adhesive Joints B.R.K. Blackman, A.J. Kinloch and M. Paraschi
293
Cohesive Failure Characterisation of Wood Adhesive Joints Loaded in Shear E Simon and G. Valentin
305
Rate Dependent Fracture Behaviour of Adhesively Bonded Joints /. Georgiou, A. luankouic, A.J. Kinloch and V Tropsa
317
Experimental Characterization of Carbon-Fiber/Concrete Adhesive Interface for Retrofitting of Concrete Bridge Structures T. Kusaka, H. Yagi, H. Namiki and N. Horikawa
329
The Determination of Adhesive Fracture Toughness for Laminates by the Use of Different Test Geometry and Consideration of Plastic Energy Correction Factors D.R. Moore and J. G. Williams
341
Fracture Toughness of a Laminated Composite S. Kao-Walter, P. Stdhle and R. Hdgglund
355
Combinatorial Edge Delamination Test for Thin Film Adhesion - Concept, Procedure, Results M.Y.M. Chiang, J. He, R. Song, A. Karim, W.L Wu andEJ. Amis
365
Bond Parameters Affecting Failure of Co-Cured Single and Double Lap Joints Subjected to Static and Dynamic Tensile Loads K.C. Shin and J J. Lee
373
3. Composites 3.1. Short Fibre Composites Fracture Mechanisms in Short Fibre Polymer Composites: The Influence of External Variables on Critical Fibre Angle S. Fara and A. Pavan Fracture Behaviour of Short Glass Fibre-Reinforced Rubber-Toughened Nylon Composites M. Gomina, L. Pinot, R. Moreau and E. Nakache
387 399
3.2. Laminates Comparison of Interlaminar Fracture Toughness Between CFRP and ALFRP Laminates with Common Epoxy Matrix at 77K in LN2 M. Hojo, S. Matsuda, B. Fiedler, K. Amundsen, M. Tanaka and S. Ochiai
421
Delamination Fracture in Cross-Ply Laminates: What Can be Learned from Experiment? A.J. Brunner and B.R.K. Blackman
433
Fracture Toughness of Angle Ply Laminates M.R. Piggott and W Zhang
445
Strain Energy Release Rate for Crack Tip Delaminations in Angle-Ply Continuous Fibre Reinforced Composite Laminates C. Soutis and M. Kashtalyan
455
The Effect of Residual Stress on Transverse Cracking in Cross-Ply Carbon-Polyetherimide Laminates Under Bending L.L. Warnet, R. Akkerman and P.E. Reed
465
3.3. Z-Pinned Laminates and Bridging Analysis Deducing Bridging Stresses and Damage from GIQ Tests on Fibre Composites A.J. Brunner, B.R.K. Blackman and JG. Williams
479
Z-Pin Bridging Force in Composite Delamination K-Y. Liu, W. Yan and Y-M. Mai
491
Effects of Mesostructure on Crack Growth Control Characteristics in Z-Pinned Laminates D.D.R. Cartie, A.J. Brunner and I.K. Partridge
503
Fracture Toughness and Bridging Law of 3D Woven Composites V Tamuzs and S. Tarasous
515
3.4. Modelling and Lifetime Prediction 3D Modelling of Impact Failure in Sandwich Structures C Yu, M. Ortiz and A.J. Rosakis
527
Interfacial Stress Concentrations Near Free Edges and Cracks by the Boundary Finite Element Method J. Lindemann and W. Becker
539
Stability of J-Controlled Cracks in Pipes J. Lellep
549
Alternative Fatigue Formulations for Variable Amplimde Loading of Fibre Composites for Wind Turbine Rotor Blades R.PL. Nijssen and D.R. V. van Delft
563
Author Index
575
Keyword Index
577
FOREWORD It is my great pleasure to introduce the proceedings of the ESIS TC4 conference, "Fracture of Polymers, Composites and Adhesives", which was held in the mountain resort of Les Diablerets, Switzerland between 15-18* September 2002. This was the third conference organised by TC4 and, as on the two previous occasions, it reflects the main activities of the committee which are focussed on developing fracture mechanics test methods for polymers, adhesive joints and composites. For polymers, the essential work of fracture has remained a very popular technique for films and the characterisation of fracture behaviour under cyclic or high rate loading continues to receive much interest. This is reflected in these proceedings. For adhesive joints, we present papers applying fracture mechanics methods to analyse joints loaded under shear and high rate test conditions and also on the use of fracture mechanics to measure adhesion and predict structural strength.
For
composites, the intense research activity in the areas of angle ply delamination, third direction reinforcement and modelling of bridging and damage has ensured that these areas are well represented.
I hope you enjoy these proceedings and are inspired to both contribute, and attend, the 4''' ESIS TC4 conference in September 2005. Bamber Blackman
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1. POLYMERS 1.1 Slow Crack Growth and Fatigue
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
MICROMECHANISMS OF SLOW CRACK GROWTH IN POLYETHYLENE CHRISTOPHER J.G. PLUMMER,^ ANNE GOLDBERG,^ ANTOINE GHANEM^ ^ Laboratoire de Technologic dcs Composites ct Polymercs (LTC), Institut dcs Materiaux, Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Switzerland. ^ Solvay Polyolefins Europe, 310 Rue de Ransbeek, 1120 Brussels, Belgium ^ Solvay Research and Technology Centre, 310 Rue de Ransbeek, 1120 Brussels, Belgium
ABSTRACT Notched specimens of polyethylene (PE) have been subjected to constant and cycUc loading at 80 °C, and their microdeformation behaviour investigated by OM, SEM and TEM. Under constant loading, a transition from full ligament yielding to slow crack growth (SCG) was observed as the stress intensity factor, K, decreased, reflected by a macroscopic ductile-brittle transition with decreasing applied load. SCG was characterized by formation of a wedgeshaped crack tip deformation zone, whose internal structure became progressively finer as K decreased further. The behaviour of relatively SCG resistant third and second generation grades of PE in the low K limit was inferred from TEM of specimens subjected to accelerated testing in IgepaF^ to be breakdown of diffuse zones of interlamellar voiding rather than development of a mature fibrillar structure. This latter failure mode gave smooth fracture surfaces, similar to those observed under conditions, which have been linked to a transition from discontinuous (stick-slip) crack growth to continuous crack growth with decreasing peak K. TEM again indicated the smooth fracture surfaces obtained under low level cyclic loading conditions to be associated with breakdown of regions of interlamellar voiding, suggesting the micromechanisms of failure to be similar in both types of accelerated test. On the other hand, at higher K, crack advance under dynamic loading led to more extensive fibrillar retraction than for static loading. Keywords: polyethylene, microdeformation, slow crack growth, fatigue, TEM, cavitation
C.J.G. PLUMMER, A. GOLDBERG AND A. GHANEM INTRODUCTION The relatively low glass transition temperature, Tg, and melting point, Tm, of polyethylene (PE) lead to globally ductile behaviour between room temperature and Tm. However, its response to long term low level loading remains of major concern, especially in hostile environments, owing to slow crack growth (SCG). Although the stress intensity factor, K, associated with flaws under such conditions is initially less than the critical stress intensity for crack initiation, ^c, derived from short term tests, sub-critical SCG results in a gradual increase in the effective flaw size, and catastrophic failure after a time ^b- At high stresses, cr, where ductile failure dominates, crvs. /b is relatively flat, reflecting the weak dependence of the flow stress Cy on the deformation rate. At intermediate to low cr, however, where SCG dominates, the dependence of cron ^b is much stronger, making extrapolation from short term data difficult. SCG is generally attributed to disentanglement; given sufficient mobility (and time), individual chains escape their entanglement constraints by moving along their own contour, a process associated with the transition from rubbery to viscous behaviour with increasing T and/or decreasing deformation rate in glassy polymers. Although chain folding alters local conformations during solidification of semicrystalline polymers such as PE, the topology of the melt remains essentially unchanged. Entanglement may therefore be considered to be trapped by the crystalline lamellae for Tra> T » Tg, in which case disentanglement is essentially opposed by resistance to chain slip through the lamellae. It is therefore hindered by high molar mass, M, chain branching (or bulky side groups) and high levels of crystallinity. The sensitivity of SCG in PE to M is well estabUshed and has been cited in support of disentanglement mediated SCG under a wide range of conditions [1-4]. SCG is also reduced in the presence of branched chains, suggesting that a combination of high M and branching should give optimum SCG resistance [5]. Moreover, the presence of lateral groups (typically from methyl to octyl) strongly influences crystalline order in PE and decreases the lamellar thickness, resuhing in increased molecular connectivity between the lamellae, which may further improve SCG resistance provided the lamellae remain thick enough to anchor the chains effectively [1, 6-8]. "First generation" high density PE (HDPE), introduced in the 1950's, typically contains a relatively low branch density of 3-7 ethyl groups per 1000 carbon atoms, as opposed to 20-30 ethyl or butyl groups in the earlier low density PE (LDPE), which is too compliant for many engineering applications. Improved SCG performance, associated with increased crack-tip plasticity [9], was subsequently obtained using "second generation" medium density branched ethylene copolymers (MDPE), but at the expense of reduced stiffness and poor bulk creep properties. In the more recent "third generation" short-chainbranched HDPE copolymers, the branched units are concentrated in relatively long chains. The resistance to disentanglement of these long branched chains and their influence on the crystalline morphology lead to excellent SCG resistance, with the advantage that the overall degree of crystallinity and hence the bulk creep properties remain comparable with those of the first generation grades [5, 10]. Since failure times in these grades may be extremely long under service conditions, pre-screening for applications typically involves accelerated fracture mechanics testing of notched specimens (Fig. 1), under either controlled load or controlled K conditions. SCG may be accelerated by testing at high temperatures, and room temperature lifetimes inferred by extrapolation, bearing in mind possible deviations from Arhennius behaviour [9, 11]. However, it is often necessary to accelerate SCG fiirther using cyclic loading conditions and/or a surfactant [12]. Non-ionic surfactants such as IgepaF'^ (nonyl phenol ether glycol) are particularly effective for reducing the tb of SCG resistant grades under low level loading
Micromechanisms of Slow Crack Growth in Polyethylene
5
without affecting their ranking with respect to other grades tested under the same conditions, although it is not clear how the failure mechanisms are affected [13]. The aim of the present work has therefore been to examine in detail the micromechanisms of failure associated with different types of PE subjected to accelerated testing in order to ascertain the extent to which they fit into a single consistent overall framework.
(a) /^^a
Fig. 1 (a) Cylindrical notched bar (CNB) and (b)full notch creep test (FNCT) specimen geometries.
^^> <
•
•*
D
•
V\
EXPERIMENTAL All mechanical tests were performed in tension at 80 °C on representative first, second and third generation PEs with comparable weight average molar masses, described in more detail elsewhere [14]. Cylindrical circumferentially notched bars (CNB) were used for testing in air (Fig. 1(a)). Circumferential notching promotes plane strain in the remaining ligament so that necking tends to be suppressed in favour of SCG. SCO tests can therefore be performed at relatively high loads, allowing reduction of the overall test times. Static tests were carried out using a screw driven tensile test apparatus in force control mode, after tempering for one hour at 80 °C. The crack length, a, used to calculate the initial values of the stress intensity, ^^i [15], and the ligament stress, a, was determined from the fracture surface after each test. Dynamic tests were also performed on CNB specimens using a hydraulic tensile test apparatus under either controlled force or controlled A^ conditions, with adjustment of the load as a function of crack-tip advance by video feedback [16, 17]. Accelerated tests in Igepal made use of the full notched creep test (FNCT) geometry (Fig. 1(b)) [18]. The specimens were loaded using a commercial creep loading apparatus (IPT, Germany) in 2% Igepal CO-630 in demineralized water after tempering for 1 hour at 80 °C. Although numerical calculations indicate K to be heterogeneous in the FNCT specimens, its minimum value, equal to K\ in the CNB specimens for given a and Z), is found to control SCG under these conditions [18,19]. After the tests, the fracture surfaces were trimmed and embedded in either epoxy or polymethylmethacrylate (PMMA). The epoxy was cured at room temperature and the MMA polymerized overnight at 40 °C and post-cured at 80 °C for 24 hours. Positive or negative staining of the HDPE was observed for both types of embedding medium, depending on the depth at which the sections were taken and the degree of penetration of internal voids by the resin and/or the stain. An ultramicrotome (Reichert-Jung Ultracut E) was used for sectioning. Semi-thin sections (about 10 pm) were first removed for optical microscopy (OM). The
C.J.G. PLUMMER, A. GOLDBERG AND A. GHANEM specimen was then exposed to RUO4 vapour overnight and 50 to 100 nm thick sections prepared at room temperature with a diamond knife. Transmission electron microscopy (TEM) was carried out using a Philips EM430 (accelerating voltage 300 kV), or a Philips CM20 (200 kV). Fracture surfaces were observed by scanning electron microscopy (SEM) (Philips XL-30, equipped with a field emission gun) at about 2 kV.
W
0.51
0.45 0.4 0.35 §^
0.3
g
0.25
••^ 0
0
^
0 d
.-, 0.2
0-
0.15-
o.r 0.05-
1000
10000
100000 time [s]
1000000
1000
10000
100000 time [s]
1000000
Fig. 2 (a) (5i plotted against it in CNB specimens of the first generation HOPE (squares) and the third generation HDPE (circles), under constant tensile loading; (b) K, plotted against Xb in the same specimens. The open and filled symbols are for notch depths of 2 and 4 mm respectively. Specimen
FNCTl
FNCT2
FNCT3
FNCT4
CNBl
Material
3^*^ generation HDPE
y^ generation HDPE
2°^ generation MDPE
regeneration HDPE
3'** generation HDPE
Test
static
static
static
static
dynamic (IHz)
CNB 2 2>^^ generation HDPE dynamic (IHz)
CifMPaJ
3.0
3.0
3.0
3.1
KifAdPam'^^J
0.12
0.12
0.12
0.13
h(h)
2523
1175
641
10
109
130
Fracture surf.
smooth
smooth
smooth/ductile
ductile
smooth/ductile
smooth/ductile
dAKIdN= 10"^ CTmax^ 5 M P a MPam^^^/cyc, {K^,-0.251 ^ 0 = 0.1 R = 0.\ 391-485
Cycles to fail
467'805
Table 1. Selected results from accelerated testing (all specimens machined from compression mouldings with the exception ofFNCT2, which was from an extruded pipe; R = (SmJcma))RESULTS Results from constant load testing are given in Fig. 2 for specimens of the first and third generation HDPE in air, illustrating the sharp drop in the effective critical stress intensity with
Micromechanisms of Slow Crack Growth in Polyethylene
t\, associated with the onset of SCG, and the improved SCG resistance of the third generation HOPE [14]. The data in Fig. 2 confirmed failure to be controlled by Ki rather than a. They were also broadly consistent with the widely reported scaling a~a^ (K/KoY during steady state discontinuous SCG, with 7» 4 for a wide range of PEs [20]. Table 1 gives representative results from accelerated tests. In the static tests, the ^b were significantly reduced for a given Ki, but the SCG performance ranking expected for the different grades was respected, i.e. third generation > second generation > first generation. Microdeformation Static tests. Short-term failure in notched specimens at high K{ was generally associated with yielding across the whole of the load-bearing ligament (sometimes accompanied by coarse cavitation in the specimen interior). As K decreased, the transition to SCG was marked by the formation of a wedge-shaped fibrillar crack-tip damage zone at the notch tip, although ductile necking tended to persist in the final stages of failure. This follows from the weak dependence of c^ on a in this geometry, and the fact that at large a, d increases more rapidly with alD than K [20-23]. For K immediately below its value at the transition from necking to SCG, the deformed ligaments separating cavities in the crack tip damage zone were relatively wide (Fig. 3(a)) and their intemal structure (Fig. 3(b)) closely resembled that in macroscopic necks [24]. Indeed, "fibrillar" is somewhat of a misnomer here, since the cavities were isolated in the early stages of deformation, and SEM revealed coarse ribbon-like structures on the fracture surface. The structure in the damage zone nevertheless became progressively finer as K decreased, with fibril diameters in the peripheral regions of fracture surfaces corresponding to the lowest K{ in Fig. 2(b) approaching the dominant lamellar thicknesses, and indeed continuity has been observed between the fibrils and dominant lamellae in the undeformed polymer [14]. The draw ratio was roughly constant within the damage zone under these conditions, as shown in Figure 4 for the third generation HDPE, suggesting a surface drawing mechanism to mediate fibril extension, although there remained significant interconnectivity between individual fibrils. At still lower K, the mode of crack propagation was inferred from accelerated testing in Igepal of the SCG resistant third generation HDPE to change to one of localized interlamellar cavitation, lamellar cleavage and crack propagation via breakdown of interlamellar ligaments, as illustrated in Fig. 5. The corresponding regions of the fracture surface appeared relatively featureless under SEM, as has been reported elsewhere for low level loading conditions [14, 25], The presence of extensive interlamellar cavitation behind the main crack front was taken to reflect the stabiHty of the interlamellar fibrils during the early stages of deformation, breakdown intervening much later as a result of disentanglement, assumed to be relatively sensitive to loading time. The first generation HDPE showed coarse fibrillation and substantial necking under these conditions and a reduced lifetime (FNCT4, Table 1),fi*omwhich it is inferred that the interlamellar material broke down more readily in this grade, consistent with its limited degree of chain branching (coarse fibrillation was also observed in the third generation HDPE tested in Igepal, but only at higher K\). Finally, the second generation MDPE showed intermediate microdeformation behaviour, as illustrated in Fig. 6, with cavitation and fibrillation on the scale of the lamellae close to the notch tip (Fig. 6(a)), but coarser fibrillar texture and extensive macroscopic cavitation in the central part of the fracture surface (Fig. 6(b)). The sinuous fibrillar textures visible in Fig. 6(b) are assumed to have resulted from recoil and partial collapse of the damage zone structure, reflecting rapid crack growth towards the end of the test, limited stress relaxation and hence relatively high true stresses in the fibrils immediately prior to failure.
C.J.G. PLUMMER, A. GOLDBERG AND A. GHANEM
* ¥ ll-'iK
200 nm Fig. 3 The crack tip deformation zone in the first generation HDPE tested with K, = 03 MPam^^^ in air: (a) low magnification TEM image of the deformation zone tip (the dark areas of over-stained epoxy correspond to voids); (b) detail of the internal structure of the region marked (i) in (a).
Fig. 4 TEM of fibrillar deformation in the third generation HDPE tested with K, = 0.26 MPam^^^ in air (tensile axis horizontal).
Micromechanisms of Slow Crack Growth in Polyethylene
Fig. 5 TEM of a section through part of the fracture surface of the third generation HDPE tested with K/ = 0.12 MPam^^^ in Igepal (Table 1, FNCTl): (a) overview; (b) detail of regions of interlamellar cavitation behind the fracture surface.
Fig. 6 TEM of a section through part of the fracture surface of the second generation MDPE tested with K, = 0.12 MPam^^^ in Igepal (Table 1, FNCT3): (a) deformation close to the notch tip; (b) coarse fibrillation near the centre of the fracture surface.
10
CJ.G. PLUMMER, A. GOLDBERG AND A. GHANEM
Cyclic tests. Qualitatively similar SCG behaviour to that described above is seen in testing accelerated by use of cyclic loading conditions, namely discontinuous crack growth accompanied by extensive fibrillation at relatively high K, and quasi-continuous crack growth with a relatively featureless fracture surface at relatively low K [26]. Thus, the fracture surfaces of specimens CNBl and CNB2 in Table 1 (Fig.7) showed a progressive increase in fracture surface roughness with crack advance, which was associated with an increase in X^max in each case (the asymmetry of the fracture surfaces reflects the difficulty in precisely controlling loading conditions during crack advance, leading to slight variations in K). In both CNBl and CNB2, OM of transverse sections indicated relatively little deformation at the notch tip and in the outer regions of the fracture surface. Closer to the sample centre, however, sub-surface deformation was more widespread, particularly in CNB2, with crazelike deformation zones growing out of the plane of the fracture surface. The central regions of the fracture surface of CNBl showed a "tufted" structure locally, as well as larger scale concentric features, assumed to reflect discontinuous crack propagation. Plastic necking was more extensive in CNB2, since in this case most of the crack extension took place in the final stages of fracture, accompanied by a very steep increase in K.
Fig. 7 Reflected light OM offlracture surfaces and transmitted light OM of transverse thin sections taken from specimens of the third generation HDPE subjected to cyclic loading: (a) controlled K test (Table 1, CNBl); (b) controlled load test (Table 1, CNB2). Deformation close to the notch tip closely resembled that in FNCTl, FNCT2 and FNCT3, as illustrated by the TEM image in Fig. 8, with predominantly interlamellar deformation up to failure. Details of the structure associated with the "tufts" in Fig. 7(a) are shown in Fig. 9(a) and Fig. 10. A highly voided fibrillar structure similar to that observed in the creep specimens at intermediate K was observed at the base of these tufts, again with
Micromechanisms of Slow Crack Growth in Polyethylene
continuity between the fibrils and the dominant lamellae in the undeformed polymer (Fig. 10(b)). However, the fibrils showed extensive retraction leading to a correspondingly dense, collapsed structure with undulating fibrillar trajectories, suggesting high elastic strains and hence high true stresses at the instant of fibril breakdown. This contrasted with fibrillar structures obtained under static loading, where, with the exception of those in the central regions of the fracture surfaces (cf Fig. 6(b)), the fibrils remained almost fully extended after failure (cf Fig. 4, for example). Moreover, in the regions separating the tufts in CNBl, the fibrils were elongated parallel to the plane of the fracture surface. This implies much higher stresses to be present within the fibrillar zones than the surface drawing stress [27], assumed to be of the order of cjy. \m nm
Fig. 8 TEMofa thin section from close to the notch tip in the specimen in Fig. 7(b) (Table 1, CNB2), showing interlamellar deformation (top left) and lamellar cleavage..
Fig. 9 SEM micrographs of the intermediate fibrillar regions of the fracture surface in (a) Fig. 7(a) and (b) Fig. 7(b).
12
CJ.G. PLUMMER, A. GOLDBERG AND A. GHANEM
0.5 [im ^ Fig. 10 TEM of a section through a tuft such as in Fig. 9(a) showing partially collapsed fibrillar structure at the base of the tuft and (b) the interface between the deformed material and the undeformed material.
DISCUSSION The overall response of PE to low level loading may be considered to reflect (a) its structural heterogeneity and (b) the presence of a number of competing irreversible microdeformation mechanisms, namely interlamellar cavitation/fibrillation, breakdown of interlamellar fibrils, local lamellar breakdown (cleavage, block slip) and large scale co-operative lamellar deformation (shear yielding). Interlamellar voiding is generally first observed to occur in regions in which the dominant lamellar trajectories are nearly perpendicular to the tensile axis, although lamellar cleavage also gives rise to localized deformation in regions where the lamellae are nearly parallel to the tensile axis (as in Fig. 8, for example). At stress levels well below c^ ((Jy is insensitive to the duration of the test), and given a relatively high critical stress for short term disentanglement, interlamellar voiding is initially the dominant stress relaxation mechanism, and lamellar deformation occurs at a more local scale to give fine fibrillar textures [14]. Indeed, as shown schematically in Fig. 11, at sufficiently low K, i.e. low local stresses, disentanglement may lead directly to interlamellar crack propagation. Although this latter failure mode is observed to occur in the third and second generation grades in both static tests in Igepal and in dynamic tests, it is not accessible to static tests in air in the available timeframe, and has not so far been observed in the first generation HDPE (so that it is not clear that the resistance to breakdown of interlamellar fibrils by disentanglement is sufficient to allow formation of stable regions of interlamellar cavitation under any conditions in this material). Although the dynamic tests were restricted to the third generation grade, the micromechanisms of deformation appeared to show at least qualitative similarity with those observed for static loading in Igepal. Certainly, the effects of M, comonomer content and branch distribution have been shown elsewhere to be broadly similar in the two types of test.
Micromechanisms
of Slow Crack Growth in
13
Polyethylene
so that although the absolute failure times are significantly reduced at any given temperature, rankings based on fatigue crack propagation generally correlate well with those obtained in static tests [9, 11, 28-30]. However, significant compression during each cycle is found to reduce the fatigue crack resistance in materials with high degrees of branching, to the extent that rankings in terms of SCG resistance may be reversed with respect to those established from static loading or under limited compression [31]. The fracture mode has also been observed to change from discontinuous to continuous crack growth as R increases at fixed ^max [11], suggesting that damage sustained by the craze during the compressive part of the cycle may contribute to fibril breakdown in tension [31]. Although it is not possible to verify this from the present results, it is clear that the final stages of fracture within fibrillar regions of the fracture surfaces are significantly different in the static and dynamic tests.
Crack Propagation
Formation of large cavities, homogeneous shear of intervening ligaments Formation of relatively coarse fibrillar structures
Formation of relatively fine fibrillar structures
Decreasing K
Fig. 11 Sketch of the trends in microdeformation behaviour accompanying SCG as a function ofK in PE subject to low level loading. CONCLUSIONS High load-short term failure in notched PE specimens generally initiates by yielding across the whole of the load bearing ligament. The transition to SCG as the load is reduced, is characterized by the appearance of a localized fibrillar damage zone at the notch tip, and the scale of the fibrillar decreases substantially as the load is reduced further. At sufficiently low loads, it is also possible to induce a transition to interlamellar failure in specimens of relatively SCG resistant second and third generation materials by accelerated testing in Igepal. This is argued to reflect the relative stability of interlamellar fibrils in these materials during the early stages of deformation, and results from dynamic testing suggest that this mode of failure is not unique to specimens tested in Igepal. On the other hand, the local stress states
14
CJ.G. PLUMMER, A. GOLDBERG AND A. GHANEM
associated with final failure of the fibrillar zones appear very different under cyclic loading, presumably contributing to the significant shifts in the time and temperature regimes associated with the different modes of failure with respect to those observed for static loading. REFERENCES [I] [2] [3] [4] [5] [6] [7] [8] [9] [10] [II] [12] [13] [14] [15] [16] [17] [18]
Huang, Y. and Kinloch, AJ. (1992) Polymer 33, 1331 Huang, Y. and Brown, N. (1991) / Polym. ScL - Polym, Phys. Edn. 29, 129 Brown, N., Lu, X.C., Huang, Y.L. and Qian, R.Z. (1991) Makromol Chem. Makromol Symp. 41, 55 Brown, N., Lu X., Huang, Y., Harrison, LP. and Ishikawa, N. (1992) Plast. Rubber & Comp. Processing and Applications 17, 255 Bohm, L.L., Enderle, H.F. and Fleissner, M. (1992) Adv. Mater. 4,231 Lu, X., Wang, Q. and Brown, N. (1988) J. Mater. Sci. 23, 643 Bubeck, R.A. and Baker, H.M. (1982) Polymer 23,1680 Channell, A.D. and Glutton, E.Q. (1992) Polymer 33,4108 Parsons, M., Stepanov, E.V., Hiltner, A. and Baer, E. (2000) J. Mater. ScL 36, 5747 Hubert, L., David, L., Seguela, R., Vigier, G., Degoulet, G. and Germain, Y. (2001) Polymer 42, 8425 Parsons, M., Stepanov, E.V., Hiltner, A. and Baer, E. (2000) J. Mater. Sci. 35, 2659 Ward, A.L., Lu, X., Huang, Y. and Brown, N. (1991) Polymer 32,2127 Fleissner, M. (1998) Polym. Eng. & Sci. 38, 330 Plummer, G.J.G., Goldberg, A. and Ghanem, A. (2001) Polymer 42, 9551 Brown, W.F. and Crawley, J.E. (1966) AST STP 410, 15 G'Sell, C , Hiver, J.M., Dahoun, A. and Souahi, A. (1992) J. Mater Sci. 11, 1 Favier, V., Giroud, T., Strijko, E., Hiver, J.M., G'Sell, C , Hellinckx, S. and Goldberg, A. (2002) Po/ywer 43,1375 Nishio, N., Imura, S., Yashura, M. and Nagatani, F. (1985) Proc 9th Plastic Fuel Gas Pipe Symp. 29 Goldberg, A. and Hellinckx. S^. (2000) Internal Report, Solvav Polvolefins Europe &
[ 19] Solvav [20] Chan, M.K.V. and Williams, J.G. (1983) Polymer 24, 234 [21] Richard, K., Diedrich, G. and Gaube, E. (1959) Kunststoffe 49, 616 [22] Lu, X.C. and Brown, N. (1997) Polymer 38, 5749 [23] Brown, N. and Lu, X. (1995) Polymer 36, 543 [24] Lagaron, J.M., Capaccio, C , Rose, L.J. and Kip, B.J. (2000) J. Appl. Polym. Sci. 77, 283 [25] G'Sell, C, Favier, V., Giroud, T., Hiver, J.M., Goldberg, A. and Hellinckx, S. (2000) Proc. 11th International Conference on Deformation, Yield and Fracture of Polymers, Cambridge, U.K., 10-18th April, p73 [26] G'Sell, C , Dahoun, A. (1994) Mater. Sci. Eng. A175, 183 [27] Brown, H.R. (1991) Macromolecules 24, 2752 [28] Shah, A., Stepanov, E.V., Klein, M., Hiltner, A. and Baer, E. (1998) J. Mater. Sci. 33, 3313 [29] Shah, A., Stepanov, E.V., Capaccio, G., Hiltner, A. and Baer, E. (1998) J. Polym. Sci. Part B - Polym. Phys. 36, 2355 [30] Parsons, M., Stepanov, E.V., Hiltner, A. and Baer, E. (1999) J. Mater. Sci. 34, 3315 [31] Harcup, J.P., Duckett, R.A. and Ward, LM. (2000) Polym. Eng. & Sci. 40, 635
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
15
ANALYSIS OF THE FRACTURE BEHAVIOR OF AMORPHOUS SEMI-AROMATIC POLYAMIDES B. BRULE, L. MONNERIE and JL. HALARY Ecole Superieure de Physique et Chimie Industrielles de la Ville de Paris, Lahoratoire PCSM (UMR 7615), F-75231 Paris cedex 05, France ABSTRACT Three-point bending experiments were performed on two series of amorphous semi-aromatic polyamides, so-called SAPA-A and SAPA-R. Thick samples, carefully notched with a razor blade, were examined at temperatures ranging from -100°C up to the vicinity of the main mechanical relaxation temperature, Ta. The values of the critical energy release rate, Gic, and of the critical strain intensity factor, Kic, were extracted from the load-displacement curves. Fracture behavior was shown to depend on several factors including yield stress value, polymer molecular weight, density of entanglement and molecular mobility. Yield stress governs the formation of the plastic zone, whereas the other factors affect its resistance to crack propagation. In the low temperature range, fracture behavior is molecular weight independent. As crack propagation is related to chain scission in the plastic zone, molecular mobility is the critical parameter, and toughness is increased for the samples including terephthalic units, which are likely to provoke (3 cooperative motions. In the high temperature range, as crack propagation results from chain disentanglements, molecular weight and density of entanglement are the more critical parameters. This line of reasoning was also useftil to explain why the SAPA-Rs are significantly tougher than their SAPA-A homologues. KEYWORDS Fracture, semi-aromatic polyamides, plastic zone, entanglements, molecular mobility, relaxations. INTRODUCTION In recent years, the relaxation and yield behavior of amorphous semi-aromatic polyamides has been the subject of a detailed analysis at the molecular level [1-6]. Two series of materials were investigated, so-called SAPA-R and SAPA-A (Table 1). In the SAPA-R series, the chemical structure is based on isophthalic or terephthalic acid and 2-methyl 1,5pentanediamine. In the SAPA-A series, the chemical formulae include isophthalic or terephthalic acid residues, diamino dimethylcyclohexylmethane residues, and lactam-12 sequences.
16
B. BRIJLK, L. MONNKRIEANDJ.L
HALARY
Table 1. Chemical formulae and nomenclature of the materials under study Chemical formula
Series
Nomenclature
XT y
1
H
SAPA-R
HO
-N-CH-pi^(CI|),-N-C
0 XT
1-x^
O H
SAPA-A
-(C-(CH2)„-NV
II I r-\ / - A "i^ C-NH^^CH^HTVNCH3
R-I
0
R-T0.5I0.5
0-5
R-T0.7I0.3
^'^
A-1.8I
0 1.; 1 1. 0 1 0.7 1
A-1.8T A-II A-IT0.7I0.3
Convincing connections were evidenced between chemical structure, molecular motions and plastic deformation. By the way, the role played by the p secondary relaxation motions, and especially by those presenting a cooperative character, was emphasized [1,6], as previously reported for other polymers [7,8]. In a second step, optical and transmission electron microscopy were used to investigate the microdeformation mechanisms in thin films of SAPA-A and SAPA-R [9]. In the chosen range of test temperatures (between -120 °C and the principal mechanical relaxation temperature, Ta), three successive microdeformation mechanisms were identified: chain scission crazing (CSC) at the lowest temperatures, formation of shear deformation zones (SDZs) at intermediate temperatures and chain disentanglement crazing (CDC) at the highest temperatures. The critical stress for SDZ formation was identified with the experimental yield stress, ay, whereas the critical stresses for CSC and CDC were derived from model expressions, accounting for the molecular weight between entanglements, Mg, the monomeric friction coefficient and the plastic flow stress, apf. Variations in the transition temperatures among the different polymers were attributed to differences in the temperature dependence of the yield stress, and hence to variations in chain mobility. The purpose of the present study was to go one step further, by analyzing the fracture behavior of bulky SAPA-A and SAPA-R samples in plane strain conditions.
EXPERIMENTAL Materials With the exception of A-II, which was a commercial product purchased from EMS Co, the SAPA-A were supplied by Atofina. The SAPA-R were kindly synthesized for this work [1] by Rhodia. As already mentionned [2,6], all these polymers are amorphous. The main physical characteristics of the materials are listed in Table 2. The weight average molecular weight, Mw, and the number average molecular weight, Mn, were determined by size exclusion chromatography in benzyl alcohol at 120 °C. The principal mechanical relaxation temperature, Ta, was deduced from viscoelastic measurements performed in the vicinity of the glass transition using a MTS-831 testing system. Ta was defined as the temperature at which the loss modulus E" passed through a maximum at a frequency of 1 Hz. Ta did not vary with the molecular weight in the available range. The molecular weight
Analysis of the Fracture Behavior of Amorphous Semi-Aromatic Polyamides between entanglements, Me, was deduced from melt viscoelasticity measurements performed using a Rheometrics-RDA II rheometer in the parallel plate geometry. To this end, the rubbery plateau modulus, GN°, was taken to be the value of G' at the frequency corresponding to the minimum of tan 6 in the plateau region. Values of Mg and of the density of entanglements, Vg, were derived from GN° using the well-known relation: GN° = p R T Me' ^ = ve R T NA ^ (1) where p is the density of the polymer, determined at 25°C, and NA is the Avogadro number. Table 2. Physical characteristics of the materials
Mw
Mn
(g.mol-^) 22000 23000 39000 21000 26000 23000 32000 18000 23000 22000
(g.mol-^) 8500 9000 12000 8000 10000 8500 11000 6000 6500 5000
Sample A-1.81 A-1.8T(23) A-1.8T(39) A-1I(21) A-1I(26) A-lTo.7lo.3(23) A-lTo.7lo.3(32)
R-I R-T0.5I0.5 R-T0.7I0.3
Ta {°C) 130 137 137 161 161 171 171 141 145 147
P , (kg.m-^) 1042 1042 1042 1055 1055 1057 1057 1194 1196 1196
Me
10-^Se
(g-mol-^) 2700 3000 3000 2800 2800 3100 3050 2750 2900 2950
(m-^)
2.3 2.1 2.1
2.25 2.25
2.0 2.0 2.6 2.5 2.45
Three-point bending experiments Plane strain fracture tests in mode I were performed on three point bending samples, whose critical dimensions (Fig. 1) satisfy the criteria of the ISO standard 13586-1 [10].
1
p re-crack notch
\\
.
W = 2B
\ B
A
4W
A
Fig. 1. Schematic drawing of a sample for fracture test. The dimensions a, W, and B appear in equations (1) to (5). These samples present a sharp pre-crack, formed with a fresh razor blade at the base of a machined notch. Both length of the pre-crack and sharpness of the crack tip (Fig. 2) were adjusted using a falling weight apparatus especially designed to operate the blades in reproducible conditions [1]. The samples were then loaded in a MTS 810 testing machine at temperatures ranging from - 80°C to 140°C and a constant cross-head speed of 1 mm.min-^ The critical stress intensity factor for the initiation of crack growth, Kic,[ll] was calculated from:
17
B. BRULE, L MONNERIE AND J.L HALARY K =f_Lma where B and W represent sample thickness and height, respectively, Pmax is the maximum load recorded during the test, and f is a geometrical factor, corresponding to:
^"^^W)
,. _ . , . (\+7 aVi
. .3/2 av
(3)
where a is the cumulated length of notch plus pre-crack.
Fig. 2. Optical micrograph showing the pre-crack tip in a sample A-1 To? 10.3. The critical energy release rate, Gic,[ll] was also determined experimentally by using the relation: U. where Uj is the area under the curve until the load reaches Pmax, and O is a geometrical factor equal to: _0 + 18.64 (5) 0 = d© 0 , appearing in equation (5) should be calculated from the equation: 0=
\6iV^ r ^^'
-I8.9-33.7174+79.616{'^1
s5l 112.952(A) +84.815(A) -25.672(A) | (6)
Kic and Gic values were taken as the average of at least three concordant measurements on samples whose pre-crack quality had been previously checked by optical microscopy. Error bars on the data were evaluated as being ± 0.1 MPa.m and ± 0.1 kJ.m" , respectively. For the experiments reported in the present article, the geometrical characteristics appearing in Fig. 1 were set as: B = 6 mm and 0.45 < a/W < 0.55. Some measurements performed on thicker samples (B = 10 mm) did not reveal any change in Kjc and Gk values.
Analysis of the Fracture Behavior of Amorphous Semi-Aromatic Polyamides
19
Three-point bending experiments (SENB) are often suspected to be less rigourous for Kic and Gic determination than the compact tension experiments (CT) [11]. A few measurements were carried out on the SAPAs using this latter geometry, which is penalized by the complex and time-consuming machining of the samples (see reference [1] for more details on sample geometrical characteristics and equations). As shown in Table 3, no significant difference in Kic or Gic was observed as a function of test geometry. This result was regarded here as a validation of the SENB measurements. Table 3. Comparison of SENB and CT data for A-lTo.7lo.3(32)
80°C
CT 2.6 3.7 2.7 4.1
SENB 2.5 3.5 2.8 4.3
Kic (MPa.m^^^) Gic (kJ.m-^) Kic (MPa.m^^^) Gic (kJ.m-^)
25°C
Sample preparation Before use, all the polymer pellets were dried at Ta +20 K for at least 48 h. Then, sheets of controlled thickness B were firstly compression molded under vacuum at Ta + 50 K and then cooled down slowly through the glass transition region. Next, samples of dimensions suitable for the SENB tests were cut from the sheets with a diamond saw, machined to produce the notch, and annealed once more at Ta +20 K for 72 h. The purpose of this thermal treatment is double: first, it allows the elimination of the residual stresses induced by sample pressing and machining; and secondly, it permits removal of eventual moisture which is known to affect dramatically the SAP A mechanical properties [12]. The pre-crack was produced in the samples just before testing. RESULTS AND DISCUSSION The values of Kic and Gic, determined at various temperatures for all the SAPA samples, are given in Tables 4 and 5, respectively. Data analysis will be carried out in three successive stages. First, some general features will be extracted. Then, emphasis will be put on the sensitivity of K^ and Gic to polymer molecular weight. And finally, the influence on toughness of the details of polymer chemical structure will be discussed. Table 4. Average values of K^ (MPa.m^^"^) -80°C -60°C -40°C -20°C 0°C
A-1.8I A-1.8T(23) A-1.8T(39) A-1I(21) A-1I(26) A-lTo.7lo.3(23) A-lTo.7lo.3(32) R-I R-T0.5I0.5 R-T0.7I0..3
2.6 2.8 2.7 2.7 2.8 2.95 2.95 3.6 3.7 3.95
2.55 2.75 2.7 2.8 2.65 2.85 2.85 3.25 3.55 3.8
2.4 2.7 2.7 2.7 2.6 2.8 2.8 3.3 3.5 3.75
2.45 2.65 2.6 2.35 2.35 2.6 2.6 3.45 3.8 3.7
2.2 2.6 2.8 2.3 2.3 2.55 2.5 3.6 3.95 3.65
20°C 40°C 60°C 80°C 100°C 120°C 140°C
2.3 2.45 3.0 2.4 2.35 2.45 2.5 3.8 4.0 3.8
2.6 2.5 3.25 2.3 2.6 2.4 2.55 3.75 3.9 3.4
2.5 2.4 3.3 2.4 2.5 2.2 2.6 3.6 3.6 3.35
2.2 2.0 3.1 2.2 2.4 2.0 2.8 3.5 3.4 3.4
1.7 1.9 2.9 2.05 2.15 1.6 2.6 3.55 3.3 3.2
1.9 1.9 1.3 2.35
1.7 1.8 0.9 1.95
20
B. BRULE, L MONNERIE AND J.L. HALARY
Table 5. Average values of Gic (kJ.m') -SOX -60°C -40°C -20°C
3.0 3.4 3.4 2.9 2.9 A-lTo.7lo.3(23) 3.5 A-lTo.7lo.3(32) 3.5 R-I 3.1 3.3 R-T0.5I0.5 3.8 R-T0.7I0..3
2.85 3.6 3.5 3.1 2.8 3.4 3.4 2.9 3.2 3.6
A-1.8I A-1.8T(23) A-1.8T(39) A-1I(21) A-1I(26)
2.6 3.4 3.3 2.9 2.75 3.3 3.2 3.0 3.25 3.4
2.6 3.3 3.6 2.6 2.4 3.0 3.0 3.2 4.0 3.3
o°c
20°C 40°C 60°C 80°C 100°C 120°C 140°C
2.45 3.4 3.8 2.4 2.6 2.9 2.8 3.6 4.05 3.2
2.75 3.0 5.8 2.6 2.8 2.9 3.05 3.9 4.2 3.7
3.3 3.6 6.8 2.5 3.2 2.8 3.6 4.0 4.3 3.2
3.2 2.9 7.0 2.75 3.6 2.6 3.9 3.95 4.1 3.5
2.6 2.6 6.4 2.6 3.3 2.1 4.3 3.75 4.2 3.8
1.75 2.3 6.0 2.4 3.1 1.6 4.1 3.8 3.9 4.0
2.0 2.4 1.1 3.4
1.7 1.7 0.6 2.6
Introductory observations Generally speaking, Tables 4 and 5 show unambiguously that all the SAPAs under study present a remarkable resistance to fracture in the glassy state, irrespective of temperature, chain molecular weight and details of chemical structure. In this respect, their performance resemble very much that of other polymers, such as bisphenol-A polycarbonate, which also present phenyl rings in their main chain repeat unit. Obviously, this common characteristic can be explained by considering the high entanglement density, Vg, of all these materials. Consider, at this early stage of the discussion, the average values of 2.8 and 3.6 MPa.m'^"^ as representative of Kic for the SAPA-A and SAPA-R series, respectively, over an extended low temperature range. As shown in Fig. 3, it turns out that these values nicely position on the plot of Kic versus Ve, drawn some years ago by Wu [13] for various amorphous and semicrystalline polymers. The physical meaning of such a connection has been ascertained on the basis of theoretical models of chain scission, as initially proposed by Brown [14] and then by Kramer's group [15,16]. 6
n—
r
~~r~
1
%^
5
^
e
r 4
d
^
^
•
^
"
'
"
^
'
~
•
-•--
--
f
c
£ 3 --
-
h a
1 ~'\ 0
b 1
()
1
1
2 10 2 ^ M
1
1
3
4
• ' )
Fig. 3. Critical strain intensity factor versus entanglement density for various polymers (filled squares: data taken from [13]; open squares: this study), a: polystyrene; b: poly(methyl methacrylate); c: poly(vinyl chloride); d: polyamide 6; e: polyoxymethylene; f: bisphenol-A polycarbonate; g: poly(ethylene terephthalate); h: SAPA-A series; i: SAPA-R series.
Analysis of the Fracture Behavior of Amorphous Semi-Aromatic Polyamides
21
Secondly, considering chain molecular weight influence on toughness, the data of Tables 4 and 5 reveal a complicated behavior, in the sense that Kic and Gic look like roughly independent of chain length at low temperature, but chain length dependent at higher temperature. The low temperature observation is well understandable from the abundant literature available on the toughness - molecular weight relationship for amorphous polymers such as poly(methyl methacrylate), polystyrene, or bisphenol-A polycarbonate [17-22]. It is recognized, indeed, that room temperature toughness strongly increases with increasing Mw up to a critical value around 7 or 10 Mg, and then stabilizes at a constant level. Owing to the characteristics of the SAPA samples (Table 2), it would just be normal that the lower M^ samples exhibit the plateau values in that case. On the other hand, no information was found in the literature to account for the high temperature observation. This point will be discussed later on in the article. And finally, it is worth recalling that Kic and Gic are two quantities which are related to each other, in plane strain conditions, by the well-known relationship [11]: Gic=Kic'(l-v')E-^ (7) where v and E are the Poisson ratio and the tensile modulus of the material, respectively. As Gic varies as the square of K^, possible changes in Gic for a given sample as a fiinction of test temperature are usually more pronounced than the relevant changes in K^, and hence, more likely to be discussed with confidence. It is reasonable to assume that the term (1 - v^) in equation (7) remains fairly constant. On the other hand, the temperature dependence of E may differ significantly from one SAPA to the other. For this reason, one may find some samples which present about the same value of Gk but differ in K^. Such is, for example, the case of A-II and A-1.81 in the low temperature range: larger values of K^ for A-II than for A-1.81 are associated to higher values of tensile modulus at any given temperature [5,6]. Influence of polymer molecular weight on toughness Figure 4 illustrates, on the example of A-IT0.7I0.3, the influence of temperature and molecular weight on Kic and G^. 3
4.5
-w
-a-^-^
2.5 I i
A ; A A
4 3.5 ^
2
3
•^-wt
22.5
i.1.5 1.5 1 0.5 -100
1 J
-50
L
0 50 100 Temperature (°C)
150
0.5 -100
-50
0 50 100 Temperature (°C)
150
Fig.4. Kic (left) and Gic (right) versus temperature for the samples A-lTo.7lo.3(23) (squares) and A-lTo.7lo.3(32) (triangles).
22
B. BRULE, L MONNERIE AND J.L HALARY
Two important features show up: 1) the independence of the fracture characteristics on Mw over a broad low temperature range, say up to about 0°C. Over this range, K^ and Gic exhibit a slight decrease with increasing temperature; and 2) the occurrence of Mw influence at higher temperature. Whereas K^ and Gic present a continuous decrease with increasing temperature for the lower Mw sample, these quantities go through a maximum for the higher Mw sample. The same trends are actually observed by drawing from the experimental data (Tables 4 and 5) the plots (not shown) of Kic and Gic versus temperature for the samples A-1.8T and A-II. They are expected to hold also for A-1.81 and the SAPA-Rs, samples for which one molecular weight only was available (Table 2). Conventionally, the plots of Kic and Gic versus temperature for the samples of sufficiently high Mw, such as A-lTo.7lo.3(32), can be divided in three successive temperature ranges, so-called a, b, and c: toughness slightly decreases with increasing temperature over the range a, and then it increases over the range b before it finally decreases again over the range c, on the upper temperature side. Tentative justification of these regimes is based on the usual description of the fracture in glassy polymers [11], which is governed by the characteristics of the Dugdale/Barenblatt zone formed at the crack tip, including the yield stress value and the stability of the craze(s) formed at the crack tip, as evidenced by electron microscopy (Fig. 5).
Fig. 5. Electron micrograph showing the deformation micromechanisms of A-lTo.7lo.3(32) in the vicinity of the crack tip at low temperature. It is likely that fracture of craze fibrils mainly results from chain scission over the temperature range a, in agreement with the observation of CSC during the deformation of thin films of the same samples [9]. As the strength required to break a chemical bond is temperature independent and that CSC stress presents a weak decrease with increasing temperature, toughness would decrease weakly with temperature, as actually observed. Progressive disappearence of chain scission at the benefit of chain disentanglement mechanism over the range b [9] would lead to stronger and more stable craze fibrils, as made from longer chains, and hence responsible for the increase in toughness. Finally, the opposite trend over the range c, also characterized by CDC, would result from the increase in molecular mobility at the approach of Ta. It has been reported, indeed, that disentanglement phenomena, detrimental here to fibril stability, are favored by chain mobility [23]. According to this description, one can justify the toughness-temperature profiles of the low Mw samples (such as A-lTo.7lo.3(23) in Fig. 4) by the disappearence of the range b. As the chains become short, no stabilization of the craze fibrils may result from the replacement of chain scission by chain disentanglement in that case. Thus, from a qualitative viewpoint, the above analysis yields a comprehensive description of the temperature dependence of toughness. One should mention, however, that it remains a crude approach, which may ignore some peculiar features. An example of that is the huge increase in toughness (from 3.3 up to 7.0 kJ.m'^ in Table 5) which is observed for A-1.8T(39) from the range a to the range b, and which is followed by a modest decline over the range c (down to 6.0 kJ.m'^). This behavior, unusual from a quantitative viewpoint, has been shown to result from a crack deviation around the very stable plastic zone [1].
Analysis of the Fracture Behavior of Amorphous Semi-Aromatic Polyamides
23
Influence ofpolymer chemical structure on toughness A careful inspection of Tables 4 and 5 shows that changes, even limited, in the details of SAPA chemical structure may affect noticeably their fracture behavior. Although it has been carried out [1], a systematic study of their effects falls out of the scope of this article, because too many parameters are concerned. They include yield stress, p relaxation characteristics, tensile modulus, entanglement density, Mw / Me ratio, and also, in the high temperature range, the gap (Ta-T) between Ta and the test temperature T. Instead, emphasis will just be put here on two features whose explanation is quite simple. The first one deals with the role of the entanglements in SAP A-A and SAPA-R samples, hiformation has already be given for the low temperature fracture behavior (Fig. 3). A ftirther effect can be shown in the higher temperature range by comparing the plots of Kic and Gic versus (Ta-T) for the samples R-To.5lo.5(23) and A-lTo.7lo.3(23), which present the same Mw value (Fig. 6). Whereas the latter does not exhibit any range b, as above discussed, the former presents both ranges b and c as the result of a certain fibril stability in the chain disentanglement crazes when chain mobility is not too high, i.e. for large values of (Ta-T). Obviously, the reason for this difference is the value of the entanglement density, Ve, which is larger for R-To.5lo.5(23) than for A-lTo.7lo.3(23) (2.5x10^^ against 2.0x10^^ m'^ in Table 2). 4,5
"1
4
. v.;
3,5
1
r
r
--v--^-: V
v,^v
4
a
(£2,5
2h
4,5
-D:--
^3,5
I 3
en
0^2,5
.a.-.:n.aa.o-^;
U
_L 100 150 Ta-T (K)
250
1,5
2h-n-
1 J
0,5 0
50
\
I
100 150 Ta-T (K)
L
200
1,5 250
0
50
200
Fig. 6. Kic (left) and Gic (right) versus (Ta-T) for the samples A-lTo.7lo.3(23) (squares) and R-To.5lo.5(23) (triangles). The second feature concerns the replacement of isophthalic units by terephthalic units (Table 1), whose effects on toughness are especially clear in the low temperature range. As shown in Fig. 7, this change in chemical structure leads systematically to tougher materials at -40°C, irrespective of the series of SAPA under consideration (A-1, A-1.8, and R-). At this temperature, neither Mw (which does not affect the toughness characteristics) nor the yield stress value (which is roughly the same for all the samples [6,9]) can explain the observations. Changes in entanglement density are also unlikely to be invoked: their effects would go in the wrong direction since the most entangled chains are the isophthalic ones (Table 2). Therefore, the unique factor suitable for explaining the observed results seems to be the molecular mobility, whose characteristics at this temperature are governed by the p relaxation processes [3-5]. These (3 motions are known to present a marked cooperative character in the
24
B. BRULE, L MONNERIE AND J.L. HALARY
terephthalic samples as the result of Tr-flips of the para-disubstituted phenyl rings, but to remain isolated in the isophthalic samples which cannot undergo 7r-flip motions. This peculiarity has been recently reported to affect both yielding behavior [6] and nature of deformation micromechanisms [9] of these samples. Accounting also for the reports [24-26] pointing out that the molecular processes responsible for the formation of the plastic zone at the crack tip would imply (3 relaxation motions, one may suggest that the extra contribution of cooperative P motions in terephthalic materials leads to the stabilization of the plastic zone formed by CSC in that case.
£ 3.5 1
3h2.5
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Fig. 7. Dependence of Kic at -40°C (left) and d c at -40°C (right) on the fraction of terephthalic units for the series A-1.8 (circles), A-1 (triangles) and R (squares). CONCLUSIONS Consideration of series of polymers in which the chemical structure is modified very progressively proves to be a useful tool for the understanding of polymer toughness on the molecular scale. As compared to previous papers on the same topics, this approach aims to give a deeper insight into the respective influence of the different parameters known to govern toughness in amorphous glassy polymers, hi this respect, some novel results are provided, including in particular the influence of test temperature on the sensitivity of fracture characteristics to chain molecular weight. In the perspective of practical applications, one should bear in mind the potential of SAP As as tough amorphous polymers. ACKNOWLEDGMENTS Thanks are due to Atofma for their interest in this study and for the thesis grant provided to one of us (B.B.). The authors would also like to thank Prof H.H. Kausch (E.P.F.L., Switzerland) and Prof E.J. Kramer (Cornell Univ., USA) for stimulating discussions. REFERENCES 1. 2. 3.
Brule, B. (1999). Thesis, University Pierre and Marie Curie (Paris 6), France. Choe, S., Brule, B., Bisconti, L., Halary, J.L. and Monnerie, L. (1999) J. Polym. ScL, Polym. Phys. Edn 37, 1131. Beaume, F., Laupretre, F., Monnerie, L., Maxwell, A. and Davies, G.R. (2000) Polymer^!, 2611.
Analysis of the Fracture Behavior of Amorphous Semi-Aromatic Polyamides
4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
25
Beaume, F., Laupretre, F. and Monnerie, L. (2000) Polymer 41, 2989. Beaume, F., Brule, B., Halary, J.L., Laupretre, F. and Monnerie, L. (2000) Polymer 41,5451. Brule, B., Halary, J.L. and Monnerie, L. (2001) Polymer 42, 9073. Tordjeman, P., Teze, L., Halary, J.L. and Monnerie, L. (1997) Polym. Eng. ScL 37, 1621. Teze, L., Halary, J.L., Monnerie, L. and Canova, L. (1999) Polymer 40, 971. Brule, B., Kausch, H.H., Monnerie, L., Plummer, C.J.G. and Halary, J.L. (2003) Polymer 44, 1181. ISO 13586-1. Determination of fracture toughness (Gjc and KjJ for plastics. An LEFM approach. Kinloch, A.J. and Young, R.J. (1983). Fracture Behaviour of Polymers. Appl. Sci. Pub., London. Park, Y., Ko, J., Ahn, T. and Choe, S. (1997) J. Polym. Sci., Polym. Phys. Edn 35, 807. Wu, S. (1992) Polym. Eng. Sci. 32, 823. Brown, H.R. (1991) Macromolecules 24, 2752. Hui, C.Y., Ruina, A., Creton, C. and Kramer, E.J. (1992) Macromolecules 25, 3948. Sha, Y., Hui, C.Y., Ruina, A. and Kramer, E.J. (1995) Macromolecules 28, 2450. Berry, J.P. (1964) J. Polym. Sci., Part A 2,4069. Kusy, R.P. and Turner, D.T. (1974) Polymer 15, 395. Kusy, R.P. and Turner, D.T. (1976) J. Mater. Sci. 11, 1475. Prentice, P. (1983) Polymer 24, 344. Pitman, G.L., Ward, LM. and Duckett, R.A. (1978) J. Mater. Sci. 13, 2092. Pitman, G.L. and Ward, LM. (1979) Polymer 20, 895. Plummer, C.J.G. and Donald, A.M. (1991) Polymer 32, 409. Schirrer, R. and Goett, C. (1981) J. Mater. Sci. 16, 2563. Trassaert, P. and Schirrer, R. (1983) /. Mater. Sci. 18, 3004. Doll, W. and Konczol, L. (1990) Adv. Polym. Sci. 91/92.
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
27
EXPERIMENTAL ANALYSIS OF GLASSY POLYMERS FRACTURE USING A DOUBLE NOTCH FOUR POINT-BENDING METHOD
N. SAAD, C. OLAGNON, R. ESTEVEZ, J. CHEVALIER GEMPPMINSA Lyon, France ABSTRACT A twin notch specimen under four point bending is designed to analyse the mechanisms and the properties of glassy polymers fracture. The two notches are submitted to an identical bending moment so that one will fail and provides a measure of the toughness while the other serves as a snap-shot of the strain fields prior to unstable crack propagation. The evolution of the toughness with the loading rate and the influence of the notch radius is analyzed for both PMMA and PC. KEYWORDS Polymer fracture, plasticity, crazing, toughness INTRODUCTION Failure of amorphous polymers in the glassy state results from the competition between shear yielding a nd crazing. W hen c razing c an b e s uppressed, for i nstance u nder c ompression, t he bulk material shows a localized plastic deformation through shear bands related to soflening upon yielding followed by progressive strain hardening as the deformation continues. Crazing involves also some localized plasticity [1], albeit at a smaller scale, and precedes crack propagation. After initiation for a critical stress state, the crazes widen by the growth of fibrils which break down for a critical width resulting in the nucleation of a crack. In a numerical study [2] featuring a viscoplastic model for shear yielding and a viscoplastic cohesive zone for crazing, it was demonstrated that the competition between shear yielding and crazing is governed by the time scales involved in each mechanism. This competition together with the condition for craze fibrils breakdown and crack propagation determines the level of toughness and governs the ductile to brittle transition with increasing loading rate. A ductile response is related to the development of some plasticity in the bulk prior to crack propagation while a brittle response corresponds to the development of crazing only, the bulk remaining elastic.
28
N. SAAD, C. OLAGNON, R. ESTEVEZANDJ.
CHEVALIER
Although the failure properties for metals have been widely investigated, for which standard experimentations are available, attention to glassy polymers has been focused later on. Due to their intrinsic softening response, no analytical results are available for such materials. For a properly designed specimen and configuration test, linear elastic conditions need to be ensured while non linear deviations have to remain confined at a small scale. Under these conditions, classical linear elastic solution can be used for the analysis of failure and the estimation of the toughness. The characterization of the fracture features has then required the development of appropriate test configurations and specific preparation rules of the specimen, and related standards are now emerging. The present experimental study is connected to a recent modeling of crazing [2] within a cohesive surface methodology which incorporates the three stages of initiation, widening and breakdown of crazes; and this mechanism is assumed to precede failure. The motivation of the present experimental study is to define a protocol to calibrate the parameters involved in this theoretical description and attention is devoted here to the craze widening. This mechanism is viscoplastic so that the energy dissipated by crazing for the nucleation of a crack is time dependent and the parameters involved in the kinetics of craze widening are related to the toughness. Therefore, the evolution of the toughness with the loading rate is one of the key features for the calibration of the parameters used in the description of craze widening. A complete calibration requires additional experiments to analyze craze initiation and craze fibril breakdown which are out of the scope of the present work. A twin notch four point-bending configuration is developed to analyse fracture as shown in Fig. la. The two notches are on the same side of the specimen and located in the region in which a constant bending moment prevails. The sample is designed to fulfil both small scale yielding and plane strain conditions. As the load increases, crack propagation takes place at one of the two notches. The critical stress intensity factor is then estimated. The remaining notch is used as a snap-shot of the strain fields prior to crack propagation. Its observation under crossed poralizers indicates whether or not shear yielding has developed prior to crack propagation and serves to verify the condition for small scale yielding. An example of such test is reported in Fig. lb for two glassy polymers.
P/2 p W
1^
1
n P/2
11A M : II' A1 V
1
Si
^
Fig. la : Twin notch configuration under bending
Fig. lb : samples of glassy polymers after testing,
Experimental Analysis of Glassy Polymers Fracture
29
The first part of the paper is devoted to the design of the notched specimen and the analysis of the resulting stress intensity factor from a finite element calculation. The second part deals with the practical preparation of the samples. In a third part, the evolution of the toughness and the analysis of the crack tip fields with the loading rate are presented for PMMA and PC. The influence of the notch radius is also considered. DESIGN OF THE TWIN NOTCH SPECIMEN The dimensions of the specimen need to meet theoretical and practical size requirements. From a dimensional point of view, the classical size criterion for plain strain and small scale yielding conditions needs to be verified [3]:
a,B,(W-a)>2.5
K ic 'y
N2
0) ;
The parameters a and W are the crack length and the specimen width indicated in Fig.la, B is the thickness and Gy is the yield stress of the material when no crazing takes place as in compression. A compromise between these requirements and practical aspects implies that the size of the specimen remains not too large. Thus, we fix the thickness to 10 mm to ensure plane strain conditions while Si=90 mm and S2= 40mm. Several geometries are considered and indicated in table 1. The geometry of PI follows the recommendations for mode I under pure bending according to Tada et al. [4] (S2/2W larger than two in Fig. la) which are close to that of the TC4 committee of ESIS [3]. For the specimen Gl, the ratio S2/W is more compact and the conditions of mode I and pure bending are verified numerically in the sequel as well as for the configuration with the twin notches G2. For a single notch specimen under pure bending and an elastic isotropic material, the stress intensity factor is [4] Kj =aoV7ia'F(a) , (2) where a = a/W , F(a) = 1.122~1.40a + 7.33a^-13.08a"^+14.0a'* and the reference stress ao(P) = 3P(Si - S2)/2W^B with P the applied load. Configuration
1
.
iPi
1
1
Gl 1
1 i I 1
W(mm) B(mm)
10 10
20 10
20 10
a (mm)
5
10
10
Tab.l Dimensions characteristics of the three configurations
G2
30
N. SAAD, a OLAGNON, R. ESTEVEZANDJ.
CHEVALIER
First, the elastic stress distributions of the un-notched specimens are obtained from a finite element analysis. For the PI un-notched specimen, the discrepancy between the finite element and the analytical result is very small (about 0.01%), thus validating the finite element calculation in terms of accuracy through the meshing and the type of element used. Therefore a similar calculation is conducted on the Gl un-notched specimen where the span to height ratio is smaller. The mismatch on the maximum stresses at the bottom and at the top of the beam between the finite element calculations and the analytical solution is 0.74% in tension and 0.79% in compression (and remains constant upon further mesh refinement). This estimation of the stress distribution is then used for the following evaluation of the stress intensity factor. We use the weight function method to evaluate the stress intensity factor. Basically, it consists in a superposition problem conducted in two steps: the estimation of the stress distribution along a fictive crack in an un-notched configuration and the knowledge of a displacement solution along the crack of a similar problem involving an edge crack under mode I. The weight function method [5] for the calculation of any the stress intensity factor involves: r^ n I— Ki=faoV7ia
uu with
r faCT(x/a)m(a,x/a) , f=J—^^ .— -dx ,
,.. (3)
where a is the crack length and x the coordinate along the crack. The term a(x/a) is the stress distribution along the fictive crack in the un-notched specimen and CTO(P) is a reference stress taken as the remote stress related to the applied load P. The definition of the weight function m(a,x) is: m(A,X)=^'^^-^^^-"^"\ (4) Kref(a) da where E'=E for plane stress and E'=E/(l-v^) for plane strain, Kref is the stress intensity factor for the same load and the reference edge crack problem. The function Uref is a known reference elastic displacement along the crack which is differentiated with respect to the crack length. Approximated and simplified expressions of the weight functions are found in Wu and Carlsson [5] for the edge crack problem. It suffices here to indicate the expression we used for the calculation of the function fin Eq. (3):
f
5
1 jfH^ip,.(l-x/ay-^'^dx. ; ;-'"
(5)
i=l
where the five coefficients pi derive from a series expansion of the elastic displacement and are given in [5]. The calculation in Eq. (5) involves the longitudinal component axx(y) along the domain [0,a] of the stress distribution from the un-notched specimen which has been approximated with a multi-linear function to further simplify the calculation in (5). Then, the S.I.F. derives from the expression of Eq. (3). From the analytical solution of a(x) for pure bending, we compared the SDF estimated with the weight function method [5] with that of Tada et al.'s handbook [4] in Eq. (2). The discrepancy
31
Experimental Analysis of Glassy Polymers Fracture
is about 1%, so that the weight function methodology can be adopted. The SIF has therefore been calculated for the Gl configuration at different positions x of the crack (Fig.2), i.e. for different shifts from the centre. The deviation of the calculated SIF from the pure bending solution remains very small (< 2%), even for cracks strongly off-centre. Since the experimental errors in measurements of the toughness are generally higher, this allows the use of the simple expression of Ki (Eq. 2) for the configuration Gl. In the configuration Gl and the position of the crack corresponding to G2, the domain dominated by the stress singularity is smaller than the ligament and the off-centre abscissa of the crack so that no overlap between the singularities from the two notches is assumed. This suggests that the SIF calculation of Gl can be used for G2. However, we verify experimentally that the toughness is similar with both configurations Gl and G2 for an elastic material. 2.0
1.0 \
f. 0.0 I
2x/S2 = 0 2x/S2 = 0.25 2x/S2 = 0.5 2x/S2 = 0.75
^ •1.0 \
-2.0
0.2
0.3
0.4 aAV
0.5
0.6
Fig.2. Comparison of the SIF of a single notch under pure bending (Ki^) and that from finite element calculation of the unotched Gl configuration (Ki). The different x values represent increasing off centre, normalised by the inner span.
A^. SAAD, C OLAGNON, R. ESTEVEZANDJ. CHEVALIER
32
EXPERIMENTAL Materials The fracture process is investigated for two glassy polymers: polymethyl methacrylate (PMMA) and polycarbonate (PC) which are generally thought to show a brittle and a ductile response respectively and thus selected to illustrate the method. These materials consist of commercial sheets (from Goodfellow) of 10 mm thickness which ensures plane strain conditions for both materials. Caution about plane strain conditions concerns primarily PC which is prone to develop plasticity and a 10 mm thickness appears reasonable according to analysis of the influence of the thickness on its toughness found in [6, 7]. In order to check experimentally the size criterion given in Eq. (1), we analysed the evolution of the yield stress ay with strain rate for both materials. Compression tests were carried out with cylinders of 10 mm height and 8 mm diameter. An Instron tensile/compression test machine was used with prescribed clamp speeds of 6.10'"^ - 60 mm/min, , resulting in initial strain rates of lO'Vs lO'Vs. The resulting yield stress varied from 60 to 130 MPa for PMMA and 50 to 65 MPa for PC.
Micrometric thrust
J
Razor blade (fixed)
15=
±Moving platei
-> Sample (transverse motion)
Fig. 3. Device for machining automatically a sharp notch with a razor blade. Notch preparation A first notch of 250 micrometers radius at the tip was mill cut with a rotary cutter. In order to prevent heating while machining, specimens were cooled with fresh compressed air during cutting. A sharp notch was fiirther introduced at the tip of the first notch with a razor blade. The displacement of the razor blade was controlled by a micrometric thrust so that slow and careful control of the blade advance could be monitored. Figure 3 shows the device used to machine the sharp notches automatically in order to improve reproducibility. Examples of the blunt and sharp notches are shovm in Fig.4. From the first blunt notch of 250 jim radius (Fig. 4a), the machined sharp notches for PMMA (Fig. 4b) and for PC (Fig. 4c) are similar and their crack tip is about few micrometers. The comparison between Figs. (4b) and (4c) suggests that some plasticity is induced by the machining in PC and not in PMMA.
Experimental Analysis of Glassy Polymers Fracture
33
(a) (b) (c) Fig. 4. Observations of the notch tip obtained by (a) milling a slit in the material, (b) razor blade for PMMA and (c) razor blade for PC This is confirmed from an analysis under crossed polarizers reported in Fig. 5. The observations in Fig. 5 correspond to a region close to the center of the specimen wddth, where the extension of these non linear effects is larger. A non linear zone is also observed when focusing at the surface but its extension is smaller. This indicates that some initial stresses are induced in the preparation of PC while these effects are negligible for PMMA.
(a) (b) Fig. 5. Photoelasticity of PC for (a) 250 jam notch radius and (b) a sharp notch radius.
Bending test Four point bending tests are used to investigate the evolution of the toughness with the loading rate for the two materials (see Fig.l and table 1). We used an Listron servohydraulic tensile test machine in which a force rate was prescribed from 12N/mn to 5200N/mn. We choose to represent the influence of the loading rate with the variable Kj which is derived from Eq. 3. The stress rateCTQis then involved and estimated from the prescribed force rate. This variable K J is preferred to the prescribed force rate to provide data for the material fracture under mode
N. SAAD, C OLAGNON, R. ESTEVEZANDJ. CHEVALIER
34
I from the present investigation and to allow comparisons with data reported from other devices and with an accompanying numerical investigation [2]. However, Kj is actually directly proportional to the force rate. The specimens were tested at room temperature, in air and for loading rates from 10"^ MPa Vm/s to 0.2 MPa Vm/s for PC and to 0.4 MPa Vm/s for PMMA. The load at fracture was used to calculate the toughness Kic from Eq. (2). RESULTS Influence of the configuration on the toughness For PMMA, the evolution of the toughness with the loading rate is reported in Fig.6. We notice that all the configurations (PI, Gl and G2) provide the same estimation of Kic. The observations of the unbroken notch under crossed polarizers do not show any birefiingence so that the material response remains i sotropic and elastic until failure. S ince the experimental toughness is similar for all configurations and the material is elastic, the similar toughness observed for the three configurations is in agreement with the calculation presented in the design section where a similar SIF was predicted between PI and Gl. Moreover, this justifies experimentally that the expression of Ki from Eq. (2) can be used for both Gl and especially the twin notch specimen G2. The toughness of PMMA is observed to increase slightly with increasing loading rate. For this material, only crazing takes place so that the variation of the toughness with the loading rate reflects the influence of the time dependent craze mechanism in the failure process. OG2 PMMA DG! PMMA OP1 PMMA • G 2 PC BGI PC • P I PC 6.00 5.00 ^4.00
I
•••
•
f
t
•
2.00
0.00 0.0001
0.001
^
o^o e
O
1.00
0.01
B
0.1
Fig. 6. Evolution of toughness versus loading rate for PMMA and PC for the three configurations Pi, Gi and G2 and sharp notches
Experimental Analysis of Glassy Polymers Fracture
35
The corresponding evolution for PC is also reported in Fig. 6. The level of toughness is observed to be about three to four times larger than that of PMMA and shows more scattering. The results are dependent on the configuration. The highest values of the toughness are observed for PI. By taking the toughness approximately to 4.5MPavm/s, the ligament (W-a) in PI is of 5 mm, the size criterion of Eq. (1) would be fulfilled for a yield stress about 100 MPa which is unrealistic for PC. Therefore, the value of Kic obtained with the configuration PI is questionable and overestimated. For the configurations Gl and G2, by taking a mean value of the toughness to 3.8MPaVm/s, the size requirements from the criterion (1) involve a yield stress larger than 60 MPa. The strain rate at the crack tip is difficult to estimate due to the stress and strain concentration but one can reasonably consider that its value is larger than the strain rate used for the compression tests. Therefore, a yield stress about or larger than 60 MPa is likely to be observed at the crack tip so that the size criterion is fulfilled for these geometries. The estimation of the toughness for Gl and G2 were observed to be similar and most of the experiments have been conducted with the twin configuration G2. The evolution of the toughness with the loading rate shows some scattering, however. The preparation of the sharp notch resulted in similar geometries so that the scatter is unlikely due to variations of the notch size. However, it probably originates in initial stresses subsequent to the notch machining.
(a) (b) (c) Fig.7. Deformation zone of the remaining crack tip for PC with (a) Ki =10"^MPaVm/s ,(b) Kj =10-^MPaVii5^/s and(c) Kj =10"^MPaVm/s . The snap-shots prior to crack propagation of the crack tip region (Fig.7) show stress induced birefringence for all the loading rates considered. Along the crack symmetry plane, a craze appears at the crack tip while localized plasticity has developed out of the crack plane. Therefore, the measured toughness does not involve only the energy dissipated during crazing but accounts for energy dissipation due to plasticity.
N. SAAD, C. OLAGNON, R. ESTEVEZANDJ. CHEVALIER
36
Influence of the notch tip radius on the fracture toughness The crack tip radius recommended for the ASTM standard is 250 ^m (O.Olin). The machining is thought to be easier and expected to introduce less plasticity than the razor blade procedure so that the results are expected to be more reproducible. We investigated the influence of this notch radius on the toughness and the strain fields around the crack tip with our twin configuration. These samples will be designed by GO (having the same dimensions as G2) and the related toughness is compared with that of 02 specimens with sharp notches. -GOPC # 6 2 PC XGO PMMA OG2 PMMA
: S. 3
0.0001
X
X
0.001
0.01
I
i
•t
0.1
Fig. 8 Toughness versus loading rate for blunted and sharp twin notch configuration. The evolution of the toughness with the loading rate is reported for both notch types in Fig. 8. For PMMA, the toughness of the blunt notches specimens GO is about two to three times higher than for the sharp notch specimens 02. For the configuration GO, we notice a drop in the toughness over Ki=10-^ MPa. vm/s. For loading rates smaller that this value, the observations of the crack tip region under crossed polarizers reported in Fig. 9 show a non linear region (Figs. 9a-b) while the material appears elastic for higher loading rates (Fig. 9c). Although PMMA is generally thought to be brittle and to remain primarily elastic, a ductile to brittle transition can be evidenced when blunt notches are used for these low loading rates.
Experimental Analysis of Glassy Polymers Fracture
37
(a) (b) (c) Fig.9 Non linear zone around the notch tip for GO PMMA at different loading rates: (a) Kj =10"^MPaVm/s ,(b) Ki =10"^MPaVm/s and(c) Kj =10"^MPaVm/s . For PC, we also reported in Fig. 8 the corresponding comparison for the toughness resulting from blunt (GO) and sharp (G2) notch specimens. The toughness of the blunt cracks configurations is higher than the corresponding value for sharp cracks and remains at a constant level with increasing loading rates. The observations of the crack tips under crossed polarizers (Fig. 10) show that shear yielding has developed for all loading rates. The craze is observed at the tip of the plastic zone and not at the crack tip as demonstrated in Estevez et al. [2], with this location coinciding with that of maximum hydrostatic stress^ where the shear bands intersect. The scatter in the results is smaller than for the sharp cracks configurations G2 but not negligible.
(a) (b) (c) Fig. 10 Non linear zone (shear bands) around the notch tip for GO PC at different loading rates: (a) Ki =10~^MPaVm/s ,(b) Kj =10"^MPaVm/s and(c) Kj =10"^MPaVm/s
DISCUSSION The twin notch configuration presented here allows for an analysis of polymers fracture at two scales: the toughness is measured at the macroscopic level while at a micro scale, the deformation fields at the onset of crack propagation can be observed. The observations at the latter scale provide additional information about the fracture process than the usual analysis of the fracture surface or the analysis of the crack path. For the glassy polymers investigated here.
38
N. SAAD, C OLAGNON, R. ESTEVEZAND J. CHEVALIER
the twin notch configuration indicates whether or not plasticity accompanies failure of the material by crazing, for the loading rate under consideration. We compared the estimation of the toughness from the twin configuration with blunt (crack tip of 250 micrometers) and sharp cracks, with a crack tip of few micrometers. The sharp cracks are machined automatically so that its geometry is reproducible. For a brittle glassy polymer like PMMA, the results for both sharp and blunted cracks are reproducible and show negligible scattering. The sharp cracks give a lower estimation of the toughness but this is not only related to a notch root effect. In the case of a blunt crack with notch radius of 250 micrometers, stress induced birefringence related to plasticity is observed for loading rate smaller than Kj = 10~^MPaVm/s while the response is fully elastic for larger values. Thus, a brittle to ductile transition is observed for these low loading rates. Therefore, machining a sharp notch is recommended to avoid non linear effects for the loading rate domain considered here. For PC, which is more ductile, the configuration with sharp notches generates noticeable scattering in the measure of the toughness. The machining of the sharp crack introduces some stresses which affects the failure process and related measure of the toughness. The scatter is significantly reduced when blunt cracks are used but then, plasticity is enhanced and is observed for the whole domain of loading rates. Thus, the toughness is not representative of the failure process by crazing. To analyse crazing only, the preparation of the sharp notch could be improved by performing the notching at a lower temperature in order to suppress any plasticity and the development of initial stresses. This is currently under investigation. This work is a contribution to the definition of an experimental protocol which aims in identifying the parameters involved in a description of crazing within a cohesive surface methodology. The results obtained for PMMA are valuable for the calibration to perform in connection to the numerical work of Estevez et al. [2]. The method of preparation needs to be improved for more ductile material in order to characterize the failure by crazing only. While restricted here to glassy polymers, such configuration test may be extended to other polymeric materials such as polymer blends and could help in determining the mechanism involved before unstable crack propagation. REFEFIENCES 1. Kramer H.H. and Berger L.L. (1990). Advances in Polymer Science. Springer, Berlin. 2. Estevez R., Tijssens M.G.A. and Van Der Giessen E. (2000) J. Mech. Phys. Sol 48,2585. 3. WilHams J.G., Moore D.R. and Pavan. A. (2001). Fracture Mechanics Testing Methods for Polymers Adhesives and Composites. Elsevier, Oxford. 4. Tada H., Paris P.C. and Irwin O.K. (2000). The Stress Analysis of Cracks Handbook. Professional Engineering Publishing, London. 5. Wu X.R. and Carlsson A.J. (1991). Weight Functions and Stress Intensity Factor Solutions. Pergamon Press, Oxford. 6. Parvin M. and Williams J.G. (1975) Int. J. Fracture, 11, 963. 7. Liberg J.P.F. and Gaymans R.J. (2002) Polyme, 43, 3767
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
39
TOUGHENING EFFECT IN HIGHLY FILLED POLYPROPYLENE THROUGH MULTI-SCALE PARTICLE SIZE
G. ORANGE, Y. BOMAL RHODIA RECHERCHES, Centre de Recherches d'Aubervilliers CRA, 93308 Aubervilliers, France
ABSTRACT The fracture toughness of a semi-crystalUne polymer (PP) filled with mineral submicron and micron scale particles is investigated according to the J-integral method: determination of the crack initiation energy (J^), and the crack propagation resistance dJ/d(Aa). Ultrafme mineral particles as precipitated silica (aggregate particle size < 0.1 |im) strongly affect the viscoelastic properties of the PP matrix, with an increase of the elastic modulus. Mineral particles with surface treatment or larger diameter size (> 1 |im), lead to an improvement of the PP fracture toughness by induced local debonding mechanism. Debonding with void growth is the main mechanism to allow dissipation energy in these mineral filled materials: debonding process is controlled by the local stress at particle/matrix interface and the adhesion energy. A synergy effect is obtained by association of calcium carbonate CaCOs (> 1 |Lim) and ultrafme silica Si02 particles (< 0.1 |Lim) : the behaviour of PP/ 2% Si02/ 5% CaC03 composites is considerably improved, with both high fracture toughness value and a high elastic modulus. The combination of ultrafine and larger mineral particles leads to an increase of both the process zone size ahead of the main propagating crack and energy density dissipated in this process zone. The dispersion of multi-scale mineral particles is a new way to develop significantly improved mechanical behaviour of polymers, specially elastic modulus with high fracture toughness. KEYWORDS Polypropylene, fracture toughness, elastic modulus, filler particle, crack resistance, debonding, cavitation, process zone, silica, calcium carbonate. INTRODUCTION Mechanical properties of semi-crystalline thermoplastics polymers can be improved by incorporating various modifier particles with different physical properties [1]. Particulate mineral fillers generally enhance the stiffness but reduce the fracture strength and toughness, while toughening rubbery inclusions reduce stiffness [2, 3]. However, it is possible to improve
40
G. ORANGE AND Y. BOMAL
the fracture behaviour of polymers by using extremely fine and well-dispersed mineral particles, together with specific surface treatment of the particles or chemical modification of the matrix for optimising adhesion [4]. But up to now the role of stiff filler characteristics on the fracture toughness of polymer composites has remained unclear [5-7]. With mineral filler, a toughening effect is generally observed when adhesion is limited: the matrix-particle debonding takes place of the elastomeric particles cavitation, which results in modification of the stress state in the material with an extended plasticity in the matrix (shear yielding). This explanation is denied by Bartczack and Maratoglu [8,9] who consider that the good ihipact resistance of dispersoi'd reinforced semi-crystalline polymers comes from the crystalline structure modification. According to Bartczack, it is the presence of fine oriented lamella in the vicinity of particles which leads to a local diminution of yield stress. However, debonding phenomena are observed at matrix/particle interface under mechanical stress, and according to other results, the co-existence of both phenomena (modification of crystalline structure + decohesion) are needed to improve shock resistance [10]. Similar impact resistance values are observed in case of hybrid material such as HPDE with EOR nodules and CaCOs particles [8]. If the shock resistance improvement is due to multiple crazes from particles, the physical characteristics of the nodules have no effect in reinforcement efficiency : only a good dispersion and a minimum of adhesion are required [11]. Another advantage of such reinforcement with mineral particles through a decohesion mechanism is the good stability of the polymer shock resistance with temperature: in the temperature range above matrix vitreous transition, there is no ductile-brittle transition temperature. Without any particle-matrix decohesion, rigid particles have just an effect in the modification of the crack front geometry. An increase of the critical stress intensity factor is possible through crack front blunting for low volume fraction, and crack pinning for high volume fraction [12]. Some recent results were obtained with hybrid mixes, with both rigid and elastomeric reinforcement particles in order to combine shock resistance and good stiffness [13-15]. Polypropylene homopolymer (PP) is a widely used thermoplastic material, despite its brittle behaviour at either low temperature or high loading rates, hnprovement in the fracture toughness of PP can be achieved by either modifying the crystalline structure, or addition of a second phase material [16]. The toughening effect and mechanisms of different second phase materials such as stiff fibres, soft rubbery inclusions (EPR, EPDM), and some mineral fillers have been analysed. Recent developments concern the effect of hybrid system consisting of rigid and rubbery inclusions. hi this paper, the toughness of multi-scale mineral filled polypropylene (PP) was studied via the J-integral method: crack initiation and propagation energy. Homogeneous dispersion of the particles in the polymer matrix is a prerequisite, and specific techniques are needed in case of ultrafine (nano) particles as silica. Specific attention will be paid to the fracture resistance of PP/Si02/CaC03 multi-scale composites measured using the J-integral approach, and especially to quantify the different fracture mechanics properties. EXPERIMENTAL PROCEDURE Materials and Sample Preparation Isostatic polypropylene homopolymer, with a weight-average molecular weight Mw of 345 kDa, a melting point of about 160°C and a density of 0.902, was used as matrix material: ELTEX
Toughening Effect in Highly Filled Polypropylene through Multi-Scale Particle Size
41
PHVOOIP (SOLVAY). The resin was supplied in powder form to obtain good dispersion with mineral particles during the compounding process. Different minerals were used as filler : natural calcite CaCOs (OMYA : 90, 95T), precipitated CaCOs (RHONE POULENC : Calofort U, Calofort S), and amorphous precipitated Si02 (RHODIA : Tixosil 365, Tixosil NM61). Granulometry of natural CaCOs is micronic, with a mean grain size of about 2 fim (0.5 < d < 10 |im); in the case of chemically synthesised minerals, particles are very fine with a mean grain size of 0.2 |Lim (0.05 < d < 2 jim) and even less for silica if well desagglomerated (d < 0.1 \\XVL). A stearic acid treatment (around 50% of particle surface covered) was used to improve dispersion of natural CaCOs (Omyalite 95T), and precipitated CaC03 (Calofort S) fillers into the matrix. Silica aggregate size is very small (around fifty nanometers), but effective measured size in the polymer matrix after compounding may be much higher, because of the presence of some agglomerates. A specific process was used to limit agglomerates. Compounds with 2% to 10% (vol.) filler and an antioxidant additive (Irganox N225 0.2wt%) were prepared with a twin screw extruder (ZSK), at a temperature of 190°C. In case of silica, a first compound was prepared with a high shear mixer (Brabender) before extrusion, to obtain a good dispersion. Specimens were cut out from the 4 mm thick plates obtained from compound granulates by hot pressing (180°C, 360 bars). With hot pressing, any skin effect was avoided to assure homogeneous materials. The PP microstructure modifications obtained in presence of the different fillers were analysed: degree of crystallisation X^ (from DSC analysis), and spherulite size Dg measured from microscopy observations. Mechanical Properties Elastic modulus and yield stress were measured by compression tests on 10 x 5 x 4 mm^ prisms cut out from hot pressed plates. The J-integral measurements (J-Aa curve) were performed from tensile tests on CT specimens of 48 x 40 x 4 mm^ size, with a closed loop controlled machine (Schenk-Trebel). The starting crack was made using a razor blade (notch radius < 50 \xm) up to a depth ao of 10 mm. As can be seen on Fig. 1, the filler characteristics have a considerable effect on the material behaviour : the crack opening displacement is increased from 1 to more than 4 mm as a fiinction of the filler size. The single specimen technique was used to determine the value of the energy J versus crack growth Aa. The crack growth resistance curve is derived from experimental data, according to a power law : J = b (Aa)^ (where b and c are constants). Specimen dimensions B, W agree with the Non Linear Fracture Mechanics criteria , and J data can be considered as reliable : B, W-ao > 25 x J^/ay [17]. Experimental data are independent of specimen geometry (B, W); the scatter in fracture criteria results is very small. Two fracture
42
G. ORANGE AND Y. BOMAL
criteria were calculated from the J-Aa curve : the critical energy J^ = J0.2 and the slope dJ/d(Aa) at Aa = 0.2 mm (ESIS-TC4 protocol for J-crack growth resistance curve tests on plastics). 0.25 ^--
W = 40
2 ^msize CaCOj
Vi/",
B=4
0.2 nm size CaCOj
ao«20
0
1
2
Crack Opening
3 (mm)
Figure 1 : CT specimen (PP-10% CaCOs): typical load-crack opening curve : i) sub-micronic precipitated CaC03 ii) micronic natural CaCOs From the elastic modulus E, the yield stress Gy and the slope of the J resistance curve, it is possible to calculate the tearing modulus T^^ [18]. : T = ^m
E y
dJ d(Aa)
The J.Tm parameter can be used as a criterion of material resistance to crack propagation [19]. RESULTS AND DISCUSSION The content of amorphous phase and the small size of spherulites lead to an improvement of the fracture toughness of Polypropylene [16]. Li presence of mineral filler, the particle surface chemistry can induce some specific microstructural characteristics of the PP matrix parameters such as degree of crystallisation, spherulite size, and p phase content (a/p ratio) [16]. The mechanical properties of filled PP have to be analysed as a fiinction of the filler itself (content, particle size and particle-matrix bonding) but also through microstructural modifications of the polymer matrix induced by the filler. With CaC03, the spherulite size is significantly reduced (Ds = 10-15 jim) and the particle surface chemistry induces some specific microstructural characteristics of the PP matrix; small size surface treated CaC03 particles promote formation of the P phase. Without surface treatment, CaC03 has a nucleating effect : the degree of crystallisation is increased by about 20% (Xc = 65%).
43
Toughening Effect in Highly Filled Polypropylene through Multi-Scale Particle Size
Ultrafine Si02 particles have no significant effect on the PP microstructure: the degree of crystalhsation is constant, and the spheruUte size is sHghtly reduced. This could be explained by the amorphous structure of Si02, and the size of the particles (10^ to 10^ times smaller than Ds). Table 1. Microstructural characteristics of unfilled and filled PP specimens. PP (neat) + 10%CalofortU + 10%CalofortS + 10%Omya (90, 95T) + 4% Silica (Tixosil365)
Xc (%) 55 65 58 57 55
Dj
5 im)
p/(a+p) 0 0 12 6 0
50 10 15 15 35
The effect of crystallinity on the PP fracture behaviour was observed from tests on the neat polymer, by using different crystallisation temperatures and annealing treatment: spherulite sizes range from 20 |Lim to 80 |im, and crystallinity X^, from 64% to 75% [20 -21]. As the crystallinity is increased, the elastic modulus is enhanced and the toughness (both critical energy JQ 2 and propagation energy) is considerably reduced : a ductile to brittle transition is observed at X^ > 70% This is coherent with results from Ouedemi [22]. The different results for untreated fillers (Omyalite 90, Calofort U, Silica Tixosil 365 and Tixosil NM61), and surface treated fillers (Omyalite 95T, Calofort S) are reported in Table 2. Table 2. Mechanical properties of specimen : effect of mineral size/surface chemistry. hicorporated filler in Polypropylene Unfilled :PP (neat) Calofort U (10%) Omyalite 90 (10%) Calofort S (10%) Omyalite 95T (10%) Silica Tixosil 365 (4%)
Silica Tixosil NM61 (4%)
Young modulus (MPa) 1250 1950 1650 1600 1675 1600
Yield stress (MPa) 27 35 28 27.5 29 30
J0.2 (kJ/m2)
(dJ/dAa)o.2 (10^ kJ/m3)
T^m
14 10 18 20 18.5 6
25* 38 20 40 6
42* 4 80 42 85 10
590* 40 1440 850 1575 60
1675
32
8
4.5
7
56
in
JT (kJ/m2)
(*) Under test conditions (23°C, quasi-static), polypropylene has a semi-brittle behaviour, to a crack extent of 1 mm, and is brittle at larger crack growth.
The considerable increase of elastic modulus with low amount of ultrafine amorphous silica Si02 (< 0.1 \XVCL) shows the nanoparticles to be well dispersed. It cannot be explained by classical models (Kemer, Nielsen): we have to take into account that a part of the polymer matrix is occluded in the aggregates. It can also be explained by adsorption of the polymer on the surface of the silica. Silica-PP adhesion is high, and so the molecular mobility is reduced: this effect is all the more important as the surface area is high (> 150 m^/g). This effect has been observed on elastomeric materials, where polymer adsorption on silica control the modulus [23].
G. ORANGE AND Y. BOMAL
44
In the case of calcium carbonate (CaC03), the effect on modulus is due to both adhesion between filler and matrix, and increase in the degree of crystallisation (X^ : 55% -> 65%). A similar effect is observed on the yield stress. The J-R curves (J - Aa) obtained with the different CaC03 fillers are plotted in Fig. 2. Different mechanical behaviour can be observed according to filler characteristics, from quasi-brittle PP with a very limited critical energy (crack initiation) and crack propagation resistance dJ/dAa (non surface treated small size CaC03 : 0.2 )Lim /Calofort U) to quasi-ductile PP with a considerable increase of both the critical energy Jo.2 ^^^ the crack propagation energy level dJ/dAa (large size natural CaCOs : > 2 |am /Omyalite 95T). With surface treatment, small size CaCOs (Calofort S) leads to some ductility.
0,25
0,5
0,75 Aa (mm)
1
1,25
1,5
Fig. 2 : J energy as a function of crack growth Aa for 10% CaC03 filled Polypropylene (23°C) From these results, it is clearly shown that particle size is one of the major parameters controlling the CaCOs/PP composite fracture toughness. Surface treatment contributes to better particle dispersion, and also to lower particle - matrix interaction [24]. Li the case of large size CaCOs (natural CaC03), a whitened damage zone appears in front of the main crack which is larger than the plastic zone : 'process zone'. This process zone corresponds to a high density of dissipated energy through void growth which leads to the main toughening effect in these materials [25]. Above 10% vol., there is no increase of fracture parameters.
Toughening Effect in Highly Filled Polypropylene through Multi-Scale Particle Size
45
With ultrafme Si02 filler, there is no toughening effect, even at high filler content up to 5% vol. Both the critical energy Jo.2 and the crack propagation resistance dJ/dAa are reduced compared to unfilled PP, leading to a very stiff and brittle material. The J-R curves (J - Aa) are plotted in Fig. 3. We can observe similar behaviour with the two different silica grades (2% vol.).
40 35 30 £25
•
e.20 15
•
•
•
AA
10
• PP X
X
X
X
A 0.1 micron (T365)
X
x O , 1 micron (NM160)
5 0 ^ 0,25
0,5
0,75 Aa ( m m )
1
1,25
1,5
Fig. 3 : J energy as a function of crack growth Aa for 2% Si02 filled Polypropylene (23 °C) The brittleness of PP is enhanced with nanosize Si02 : lower value of critical energy 10,2? and reduced crack propagation resistance (dJ/dAa). Silica particles do not induce any modification of the stress state in the material, and so no extended plasticity in the matrix. Nanosized silica particles can be considered as a modifier of polymer chain displacement, and not as a reinforcement filler. There is adsorption of PP on silica surface and consequent reduction of molecular mobility with a large increase of elastic modulus. We do not observe any 'process zone', and mechanisms as particle/matrix decohesion as well as crack pinning or blunting are not effective. The crazing process does not induce large-scale energy dissipation in pure Polypropylene; so the cavitation / dewetting process is the main toughening mechanism in CaCOs filled Polypropylene. The basic micro-mechanisms depend on the material properties and on the loading conditions. Under uni-axial tension, the effect of particles is mainly to be stress concentration sites. During crack propagation, the stress field is tri-axial: debonding with void growth accommodates the volume expansion of the material at the crack front. The stress re-distribution in the matrix
46
G. ORANGE AND Y. BOMAL
results in a plane stress transition which allows extensive plasticity within a larger zone. Microstructural observations show that the developing damage in the process zone as the crack propagates through the material corresponds to the formation of a large density of cavities localised around dispersed mineral particles. The process zone size can be increased up to 0.8 mm, with large (2 |im) CaCOs particles. A schematic crack tip profile with damage process ahead of the crack is shown in Fig. 4: localised plastic and cavitation zones. Plastic zone
Cavitation zone
Figure 4 : Schematic crack tip profile, with plastic and process zones. The stress distribution analysis shows that maximum stress concentration develops in the radial direction at the pole of the particle, and shear yielding is initiated at around 45° on the surface of rigid particles. Debonding occurs at the pole of the particle, and extends to a critical angle [27]. In case of total adhesion, debonding does not occur and there is cavitation in the matrix, at some distance from the particle pole (not at the interface). The debonding stress ai) can be evaluated from the balance between energy necessary to create new surfaces and the change in elastic energy : a where Gr is a thermal stress, G is the interface energy, E is the local elastic modulus, (^ is a constant and d is the particle size [28,29]. Decohesion and void growth are controlled by the debonding stress at the particle/matrix interface: it only occurs when the local debonding stress is lower than the fracture stress (crazing stress) of the matrix itself G is correlated with the local toughness at the interface: it is dependent on the particle/matrix adhesion energy. Li connection with this debonding process, the matrix material between the voids deforms more easily to achieve shear yielding. The dependence of the debonding stress on adhesion and particle size can be used to explain the observed yield and fracture stress in mineral particle filled polymer [29]. hi the case of not treated precipitated CaC03 or Si02, the debonding stress dj) is higher than the local fracture stress (5J of the matrix (crazing). An increase of the elastic modulus (i.e. nucleating effect) can contribute to an increase of the debonding stress. The debonding stress can be reduced by surface treatment (decrease of surface free energy), or by using large size particles (natural
Toughening Effect in Highly Filled Polypropylene through Multi-Scale Particle Size
47
CaC03) : GD < (sj. The schematic sequence is shown in Fig. 5 : stress concentration locahsed at particle pole (micronic particle), to induce either local matrix fracture (crazing) or interface decohesion (cavitation). The modification of polymer deformation mechanisms in the localised area between mineral particles is another toughening contribution, hi our case, the ligament thickness A, is about the same size as the mean particle diameter: X, = d/((7r/6Vf)^^^-l) where d is the mean particle size and Vf the vol. fraction [26]. However, the contribution of this mechanism to the toughness is mainly to crack initiation critical value (Jo.2)» ^^^ ^^t to the crack propagation resistance dJ/dAa. Local fracture of PP matrix ('crazing') influeni Local stress concentration
micronic particle (CaC03)
ij Void : local decohesion filler /matrix
m
iltrafine particle (Si02) micronic particle (CaC03)
+P I'igure 5 : Schematic sequence of local deformation, around filler particle. From previous results, we can state that to improve the damage behaviour of a semi-cristalline polymer as polypropylene it is necessary to enhance the decohesion effect (Jc » ) together with a high level of adhesion (E » ) . With toughening effect obtained by matrix/particle interface debonding, the particle geometry is an important parameter. Debonding must be initiated at a stress close to local matrix yield stress. The stress level which leads to debonding is correlated to both shape and interface properties. One way is to develop specific filler as core-shell particles, where the shell is brittle enough to promote decohesion. Another possibility is to use a hybrid material, with an association of particles of different physical properties or different sizes [30]. Kim and Michler have observed the relationship between morphology and strain micromechanisms in cases of both rigid and elastomeric filler: growth of voids, by cavitation or debonding [7,31]. Oshyman has reported a transition, at a certain fraction of filler, correlated to the evolution from macroscopic homogeneous strain to micromechanisms such as crazes. It is in fact a transition between independent mode and correlated mode of strain micromechanims [32].
48
G. ORANGE AND Y. BOMAL
The incorporation of 2% nanometric Si02 (T365) with 5% micrometric CaC03 (95T) in PP is a way to obtain high elastic modulus (contribution of Si02 particles), and good toughness (contribution of CaC03 particles). Compounds were made from 2 to 4% Si02 (< 0.1 jam) with 5% surface treated natural CaC03 (2|Lim). Specimen were prepared through hot pressing from compound granulates. Table 3. Mechanical properties of specimen: effect of multi-scale size fillers. Young Yield (dJ/dAa)o 2 Jo.2 modulus stress (kJ/m2) (103 kJ/m3) (MPa) (MPa) 1250 27 14 25 32 1650 8 10 1700 32 6 7.5 1475 25.5 20 40 1700 28 37 20 1650 33 70 30
hicorporated filler in Polypropylene Unfilled PP (neat) Silica T365 (2%) Silica T365 (4%) CaC03 95T (5%) Si02 (2%) + CaC03 (5%) Si02 (4%) + CaC03 (5%)
T^m
TT (kJ/m2)
42 15 12 85 80 100
620 120 75 1575 1600 3000
The polymer filled by 2% Si02 with 5% CaC03 effectively presents a good stiffness and improved fracture toughness: this is the consequence of the Si02 and CaC03 contributions. If the Si02 content is increased (from 2% to 4%), the elastic modulus remains quite constant: there is a plateau effect of the volume of polymer controlled by silica. But we observe a considerable improvement of fracture toughness parameters, Jo.2 and dJ/dAa when we increase the content of silica. 80 70 ^
60
^
E 50 ^
^
40
"^
30
^
U
20 ^
10
k
^
4 #
A ^
^
• X
m X
•
<^
^ w X
^
m X
•
•
•
•
i
i
1
#
' 0,25
X
4
•
• —
0,5
0,75
Aa (mm)
I
—
•
• •
X
•
X
• PP • 5% CaC03 (Omya 9 iT) A2%Si02(T365) • 4%Si02 (T365) x 2 % S i 0 2 + 5% CaC03 X 4 % Si02 + 5 % C a C 0 3 1
1,25
1.5
Figure 6 : J energy as a function of crack growth Aa for multi-scale Si02-CaC03 filled Polypropylene (23°C)
Toughening Effect in Highly Filled Polypropylene through Multi-Scale Particle Size
49
Results are shown on Fig. 6: improvement of both critical energy Jo,2 arid the crack propagation resistance through dJ/dAa is observed. The hybrid micrometric/nanometric filled polymer has significantly improved properties, both modulus and toughness, if we compare it for instance to surface treated natural CaC03 (95T) as a filler. The process zone size, developed ahead of the main propagating crack, is extended up to 2-4 mm. The high value of J.Tm parameter is the consequence of a high energy, dissipative process in such a material. It seems interesting to consider micronic CaC03 particles (2 |Lim) as a toughening filler to improve fracture behaviour through decohesion and submicronic Si02 particles (< 0,1 |Lim) as a polymer modifier to increase elastic modulus through polymer adsorption and occluded polymer formation. The synergy effect which is observed on composition as 4% Si02 with 5% CaCOs can be explained as an expansion of the decohesion zone around CaC03 particles due to the presence of a high density of well dispersed ultrafine particles. Because of the high density of Si02 particles, there is a modification of the stress field around the large CaCOs particles. Crazes are developed under the local stress, which contributes to increase of the energy dissipation density in the process zone. CONCLUSION Experimental fracture toughness values were determined (J-Aa curves) on PP/CaC03 and PP/Si02 composites. The toughening effect depends on the particle size and the adhesion between particle and matrix: best results (increase of both critical energy Jo,2 and crack propagation resistance dJ/dAa) are obtained with large particles of 2 jum size (10% volume fraction). The dissipation of energy as the crack propagates through the material is due to a debonding and crack growth process, localized within the process zone in front of the main crack. Low volume fractions (2% to 4%) of ultrafine Si02 particles improve the material stiffness, but to the detriment of fracture toughness. The development of a criterion for debonding mechanism, which assumes that the debonding stress is proportional to the strength of adhesion and depends on the particle size of the filler, might explain the experimental observations on toughness. By association of mineral particles with different size (0,1 jim, and 2 |im), a synergy effect can be developed. These results show that it is possible to develop thermoplastic materials with a high modulus and also good fracture toughness properties by using mineral fillers.
50
G. ORANGE AND Y. BOMAL
REFERENCES 1. Bucknall C.B.(1977), Toughened Plastics, Applied Science Publ. (London). 2. Kinloch A. J., Young R.J. (1983), Fracture Behaviour of Polymers, Applied Science Publ. (London). 3. Wu S.A., /. Appl Polym. Sci. 35 (1988), 549-561. 4. Jancar J., Dibenedetto A.T., Dianselmo A., Polym. Eng. Sci. 33 (1993), 559-563. 5. Muratoglu O.K., Argon A.S., Cohen R.E., Weinberg M., Polymer 36 (1995), 92L 6. Bartczak Z., Argon A.S., Cohen R.E., Weinberg M., Polymer 40 (1999), 2347-2365, 7. G.M. Kim, G.H. Michler, Polymer 39 (1998), 5699-5703. 8. Bartczak Z., Argon A.S., Cohen R.E., Weinberg M., Polymer 40 (1999), Part. I 2331-2346, Part, n 2347-2365. 9. Muratoglu O.K., Argon A.S., Cohen R.E., Polymer 36 (1995), 921-930. 10. Suwanprateeb J., J. ofAppl. Polymer Sci., 75 (2000), 1503-1513. 11. Ou Y., Yang F., Yu Z.Z., J. of Polymer Sci., 36 (1998), 789-795. 12. Phillips M.A., Pritchard G., Abou-Torabi A., Polymer and Polymer Composites, 3 (1995), 71-77. 13. Wu J., Mai Y-W., J. ofMater. Sci. 28 (1993), 6167-6177. 14. Tarn W.Y., Cheung Y.H., Li R.K, J. ofMater. Sci. 35 (2000), 1525-1533. 15. Fu Q., Wang G., Shen J., J. ofAppl. Polymer Sci., 49 (1993), 673-677. 16. Karger-Kocsis J. (1995), Polypropylene : Structure, Blends and Composites, Chapman and Hall (London). 17. Broek D.(1991), Elementary Engineering Fracture Mechanics, 4th ed. Kluwer Academic (Dordrecht). 18. Grellman W., Seidler S., J. of Polym. Eng 11(1992) 71-101. 19. Seidler S., Grellmann W., Impact and Dynamic Fracture of Polymers and Composites, ESIS publ. 19, (1995), 171-179. 20. Labour T., Gauthier C, Seguela R., Vigier G., Bomal Y., Orange G.,. Polymer , 42 (2001), 7127-7135. 21. Orange G., Fracture of Polymers, Composites and Adhesives, ESIS publ. 27, (2000), 247257. 22. Ouedemi M., Philips P.J., /. of Polym. Science Part.B 33 (1995), 313-322. 23. Ladouce L., Bomal Y., Flandin L., Labarre D., ACS, Rubber Division Meeting, Dallas (USA), april 2000 (to be published in Rub. Chem. Technol.). 24. Bomal Y., Godard P., Polym. Eng Sci., 36 (1996), 237. 25. Donming L., Wenge Z., Zongneng Q, J. Mater. Sci. 29 (1994), 3754-3758. 26. Wu S., J. ofAppl. Polymer Science 35 (1988), 549-561. 27. Goodier J.N., J. Appl. Mech. (Trans ASME) 55 (1933), 39-44. 28. Gent A.N., J. Mater. Sci. 15, 2884-2888, 1980. 29. Pukansky B., Voros G., Composite Interfaces 1 (1993), 411-427. 30. Patent FR0088634. (June 2000). 31. Kim G.M., Michler G.H., Polymer 39 (1998), 5689-5697. 32. Dubnikova I.L., Muravin D.K., Oschyman V.G., Polym. Eng. Sci. 37 (1997), 1301-1313.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
51
EXPERIMENTAL AND THEORETICAL INVESTIGATION OF THE CONTACT FATIGUE BEHAVIOUR OF AN EPOXY POLYMER UNDER SMALL AMPLITUDE SLIDING MICRO-MOTIONS M.C. DUBOURG^ and A. CHATEAUMINOIS^ 1 - Laboratoire de Mecanique des Contacts, UMR 5514, INSA de Lyon, 69622 Villeurbanne, France 2 - Laboratoire de Physico-Chimie Structural et Macromoleculaire, UMR 7615, ESPCI, 75005, Paris, France ABSTRACT The cracking behaviour of a DGEBA/IPD epoxy material contacting a rigid glass sphere has been investigated under small amplitude cyclic micro-motions, i.e. fretting loading. Under predominantly elastic contact conditions, the initial damage within the epoxy counterface was found to be associated with the propagation of two main cracks at the edge of the contact and along two symmetrical locations with respect to the sliding direction. In situ visualisation of the contact interface also revealed a transition from fatigue to brittle propagation as the crack length increased. A theoretical contact mechanics modelling indicated that the cracks were initiated under predominantly mode I opening conditions and along an initial orientation close to the experimental one. A subsequent theoretical investigation of the crack response was carried out, which took into account the frictional response of the crack faces under the cyclic contact loading. The results demonstrated that, using the known bulk toughness of the epoxy, the calculated transition from fatigue to brittle failure was consistent with the experimental observations. KEYWORDS Contact fatigue, epoxy, fretting, nucleation, propagation, toughness INTRODUCTION Various semi-quantitative macroscopic fatigue wear models have been largely derived in order to account for the wear resistance of polymers substrates sliding against rigid counterfaces which are insufficiently rough for the elastic limit of the polymer to be exceeded during micro-asperity deformation [1-3]. These theoretical approaches were focused on the prediction of the wear resistance of tribological systems, but little attention was paid to the
52
M C DUBORG AND A. CHA TEA IJMINOIS
identification of the fatigue wear damage mechanisms and their relationships with the polymer bulk mechanical properties. Direct experimental evidence of contact fatigue processes, however, are scarce: propagating cracks in polymers are difficult to detect on a micro-asperity scale because of the elastic recovery. As a result, the exact nature of the contact damage micro-mechanisms (brittle or fatigue failure) and the manner they relate to the toughness or fatigue properties remain largely unestablished. In a previous investigation [4], we have demonstrated that macroscopic contacts between glassy polymers and glass counterfaces, can be used as model 'single-asperity' contacts simulating, at an observable scale, the damage micro-mechanisms which may be induced at the micro-asperity level in real contacts between rough surfaces. Under small amplitude cyclic tangential micromotions (i.e. fretting), specific contact loading conditions can be selected which ensure that superficial fatigue cracking of the polymer substrate is the main induced damage. Using the in situ contact visualisation, these failure processes can be monitored under a well-controlled contact stress environment, which is an essential prerequisite for any attempt to correlate the contact damage to the bulk materials properties. Although the complex deformation conditions in the contact zone do not necessarily realise those which are addressed in 'conventional' mechanical testing, the relatively mild strain and strain rates encountered under fretting conditions make it realistic to establish some relationships between the observed cohesive wear damage processes and the bulk failure properties. In the present investigation, the experimental contact fatigue behaviour of an epoxy resin contacting a rigid glass sphere has been analysed in the light of a theoretical contact mechanics approach focused on the prediction of the location and the initial orientation of the cracks induced under the cyclic frictional loading. A modelling of the crack propagation stages was also carried out in order to investigate crack arrest mechanisms and possible transitions from fatigue to brittle response. MATERIALS AND EXPERIMENTAL TECHNIQUES Materials The polymer under investigation was an epoxy thermoset obtained by curing a stoichiometric mixture of diglycidyl ether of bisphenol (DGEBA) and isophoron diamine (IPD). The DGEBA and IPD monomers were supplied by CIBA (LY556, Switzerland) and by Hiills (Germany), respectively. After degassing under vacuum, the reactive mixture was poured into 30x30x5 mm^ PTFE moulds and cured into an oven 2 h 30 at 140°C followed by 15 mn at 165°C. The physical and mechanical properties of the resuking material are reported in Table I. During crosslinking, one of the specimen's faces was directly exposed to air. The resulting mirror-like surface was subsequently used in the fretting tests. All the specimens were kept in a desiccator before use. A smooth (r.m.s. roughness less than 2 nm) glass hemisphere with a 48 mm radius was used as a counterface. It was obtained by polishing cubic specimens 10x10x12 mm^ of E glass (Vetrotex International, Chambery). Before use, the glass specimens were cleaned using ethanol and heated at 630°C for 1 hour in order to remove any organic contamination which could result from the polishing procedure.
Experimental and Theoretical Investigation of the Contact Fatigue Behaviour of an Epoxy Polymer
53
Table I Mechanical properties of the DGEBA/IPD epoxy network. (a) Quasi-static Young's modulus measured by Hertzian indentation; (b) Data taken from ref [5]; (c) Measured by Dynamic Mechanical Thermal Analysis (D.M.T.A) at 1 Hz (Ta is taken as the temperature of the maximum in tan 5); (d) ay" and (5^ are the yield stress under uniaxial and plane strain compression, respectively, for an equivalent strain rate of SxlO""^ s"* (see ref [6] for details on the experimental procedure). E (GPa)"
Kic MPa.m 1/2 b
Ta (°C)'
tan 8 (25°C)'
2.8
1.20
165
0.005
ay" (MPa) ^ ay^ (MPpa) 107
135
Fretting tests The fretting-wear tests were carried out using a modified tension-compression hydraulic device (Fig. 1). The flat epoxy specimens were rubbed against the glass spherical caps under a constant applied normal load, P = 100 N. An oscillating tangential displacement in the range ±10 \ym to ± 60 |Lim was imposed to the contacting specimens. The frequency of the associated triangular shaped signal was set to 1 Hz. During the fretting tests, the tangential load, Q, and the relative displacement, 8, were continuously recorded in order to obtain the so-called Q = f(8) fretting loops. The relative displacement was measured using a high precision extensometer located close to the contact in order to minimize the effects of machine compliance. These data were used as an input in a real-time feedback loop which ensured the constancy of the amplitude of the imposed relative displacement throughout the test. In situ visualization of the contact interface was carried out through the glass counterface in a light transmission mode using a microscope device and a CCD camera. For the contact conditions under consideration, the measured diameter of the contact area, 2a, was 2.1 mm, which yielded an average contact pressure pm = 32 MPa.
Fig. 1 Schematic description of the fretting device. (1) spherical glass counterface, (2) epoxy specimen, (3) tangential load transducer, (4,5) normal load transducers, (6) extensometer, (7) microscope and CCD camera.
54
MC. DUBORG AND A. CHATEA UMINOIS
RESULTS AND DISCUSSION Determination of the contact conditions Fretting loading is associated with complex contact conditions which are characterized by the occurrence of either partial slip conditions or gross slip conditions, depending on the contact loading (normal load, imposed displacement), the elastic properties of the contacting bodies and the frictional response of the contact interface (coefficient of friction). Under partial slip conditions, the contact area is characterised by the existence of a central adhesive zone (no slip) surrounded by an external micro-slip annulus. In contrast, micro-slip is distributed over the whole contact area under gross slip conditions. The precise identification of these contact conditions has been shown to be crucial for the analysis of the associated contact damage and the determination of the necessary boundary conditions in any contact mechanics modelling [7, 8]. In a previous investigation [9], the fretting conditions of the contact under investigation have been analysed as a function of the normal load, the imposed displacement and the number of fretting cycles. On the basis of these results, the present investigation will be focused on the analysis of the cracking mechanisms which occur within the gross slip regime, i.e. for displacement amplitudes greater than ± 40 ^m under a 100 N normal load. A first stage in the analysis consisted in determining the elastic or plastic nature of the contact loading using the known yield properties of the bulk DGEBA/IPD system. The latter were established experimentally assuming that they obey a modified Von Mises criterion taking into account the effect of the hydrostatic pressure. This criterion may be written as: T^OCt = T^OCtO + CtP
(1)
where Xoct and P are the octahedral shear stress and the hydrostatic pressure respectively, which can be expressed, in terms of the principal stresses as : ^oct =-^Fi -^2f+{^^ 1 P = -^(^1+^2+^3)
-^3? +(^2 -^3)^1
(2)
(3)
In equation (1), Xocto corresponds to the shear yield stress under zero pressure and a is a pressure coefficient, which quantifies the yield stress sensitivity to pressure. Such a yield criterion has previously been shown to hold for epoxy resins under a wide range of pressure, temperature and strain rate conditions [10, 11]. The two parameters, Xocto and a were found to be 44 MPa and 0.173 respectively from the uniaxial and plane strain compression results reported in table I. The distribution of octahedral shear stress and pressure within the contact has been calculated from the explicit theoretical expressions of the contact stresses beneath a rigid sliding sphere which were derived by Hamilton [12]. As reported in reference [9], the coefficient of friction, ^, increases slightly during the early stages of the fretting process. The maximum, stabilised, value (ii= 1.2) was taken into account for the present calculation. The calculated stress profiles are reported in Figure 2, at the surface of the epoxy specimen and along a direction orthogonal to the sliding direction, where the maximum values of the octahedral shear stress are likely to occur. These results indicate that the contact remains loaded elastically during sliding, except within a narrow region located at the edge of the contact. It must, however, be noted that the yield stress criterion was established, for practical reasons, at an equivalent
Experimental and Theoretical Investigation of the Contact Fatigue Behaviour of an Epoxy Polymer
55
Strain rate of 5x10"^ s•^ while the strain rates within the epoxy surface layer are in the order of lO''^ s'^ under fretting conditions. Accordingly, the values of the octahedral shear stress at the onset of yield are probably underestimated. In addition to the limited viscoelastic response of the epoxy material at the considered frequency and temperature (tan 5 = 0.005 at 25°C and 1 Hz, table I), this analysis supports the validity of a global elastic description of the contact stress environment. 60
40 (0 Q.
20
(0 (0 O CO
^
-20
-40
-1
0
x/a Fig.2 Profiles of (a) the limit octahedral shear stress at yield, (b) the octahedral shear stress and (c) the hydrostatic pressure at the surface of the epoxy specimen and along the sliding direction (a is the radius of the contact area). In situ analysis of the initial damage mechanisms A typical example for the development of a contact fatigue crack network is shown in Figure 3. In situ visualisation showed that, after about 300 fretting cycles, two cracks nucleated at the edge of the contact and at two approximately symmetrical locations along the sliding direction. During about 100 fretting cycles, crack propagation occurred in progressive, fatigue like manner. When a critical length (between 200 jam and 400 |am) was reached, brittle propagation occurred. During this final brittle propagation stage, a strong decrease in the contact tangential stiffness, K, was observed. As indicated in Figure 3, K was measured from the initial linear slope of the fretting loop, i.e. during the incipient stages of the tangential loading. Under such conditions, the contact stiffness is essentially a function of the contact diameter and the elastic shear response of the contacting bodies [13]. As no significant change in the size of the contact area was noted during crack propagation, the strong drop in K can thus be attributed to the reduced resistance of the epoxy body to tangential displacement, due to the crack opening processes induced by the tangential loading. This was supported hy postmortem microscope observation of specimens cross sections (Fig. 3), which showed that the depth of the two main cracks at the edge of the contact (up to 900 ^m, i.e. the order of magnitude of the contact radius)) allowed significant crack opening mechanisms. The monitoring of the contact stiffness therefore suggests that substantial changes in the contact stress field are induced during the propagation of the two main cracks. The initial growth
M C DUBORG AND A. CHA TEA UMINOIS
56
direction of these cracks was found to be about 11° with respect to the normal to the surface. It did not change very appreciably with the propagation depth.
1 mm
Sliding Direction
Contact cross-section
Fig. 3 Crack initiation and propagation processes in the gross-slip regime. Pictures (except bottom right) are taken from in situ visualisation of the contact. The arrows indicate the location of the cracks. As the number of fretting cycles was increased, secondary cracks developed close to the initial cracks, but they resulted in a more limited and progressive decrease in the contact stiffness. After the propagation of the initial main cracks, their neighbouring areas are unloaded as a consequence of crack opening during the tangential loading. These processes can account for the limited propagation of the secondary cracks.
Experimental and Theoretical Investigation of the Contact Fatigue Behaviour of an Epoxy Polymer
57
Theoretical analysis of the location and orientation of cracks In a first stage, the theoretical analysis of the cracking processes was directed toward the prediction of the location of the cracks and their initial orientation with respect to the surface. This analysis was carried out using a two steps procedure. At first, the contact problem was solved as a unilateral contact problem obeying a Coulomb's friction law. By means of a Kalker's algorithm [14], the contact area, the contact pressure distribution and the internal stresses were determined under a gross-slip contact condition. The magnitude of the tangential load was determined from the experimental value of the coefficient of friction within the gross slip regime {\i= 1.2). In a second stage, the initial crack propagation processes were analysed in the meridian plane of the contact, y = 0 (Fig. 4), where the cracks first initiate. The crack initiation mechanisms have been considered theoretically in the light of a simple dislocation dipole model which was initially introduced for metallic materials [15, 16]. Parameters based on the amplitude of the shear stress, Xm, along a particular direction and the amplitude of the tensile stress, am, perpendicular to this direction, have been considered to derive the crack initiation criterion. In order to take into account the strong stress gradient close to the contact interface, Xm and am have been averaged over a finite length (4 inm) fi*om the polymer surface. As detailed in a previous paper using aluminium alloys [17], the choice of these parameters can be justified from well established physical arguments based on the growth of dislocations from the surface of the crystalline materials. Within the context of this study, the parameters based on Xm and am will be used without underlying assumption regarding the nature of the microscopic processes involved in crack initiation within the glassy amorphous epoxy. As a first approach, they will just be used as mechanical parameters which allows discriminating between predominant mode I and mode II crack propagation driving forces. 1 "P
1
^m = — 2 < ^ n t
"p ^ 1
km=
1 "p —2tTnn Hp 1
Fig. 4 Theoretical modelling of crack initiation. Calculation of the average shear (Xm) and tensile (am) stresses as a function of crack orientation, a, with respect to the contact plane. For various discrete steps of the cyclic tangential loading, the values of Xm and am have been calculated for different orientations, a, with respect to the normal to the surface and for different locations (x,z) within the meridian plane. In Figure 5a, the maximum value, Aam*, of the effective amplitude of the average tensile stress on the surface (z = 0) of the epoxy specimen has been reported as a function of the orientation. Aam* is defined as the amplitude QiihQ positive values of the average tensile stress (am> 0), which are assumed to represent the driving force regarding the opening of tensile cracks. In the same figure, the orientation, a*, of the plane corresponding to the maximum amplitude of the effective average tensile stress has also been reported. Due to the loading symmetry and for the sake of clarity, only the results corresponding to one half of the contact have been represented. The results shows that the maximum amplitude of am occurs at the edge of the contact (x/a = -1) and along an
M.C DUBORGANDA. CHATEAUMINOIS
58
orientation, a* = 7°, which is very close to the experimental initial crack propagation direction (11°). The calculated value of a* was found to be roughly independent on the depth up to 50 |xm, while the amplitude of Aam* decreased from about 70 MPa to 60 MPa within the same range (Fig. 5b). Sliding amplitude
80 jw Q.
(a)
30 25
60 f
20 B
£
* E 40 f
15 2 .
<
10
20
5
-1.4
-1.2
-1
80
*~«-*-* - , _
^
60
0.
1,
(b) «
<
^
x/a
-0.8
-0.6
P - o^ O H X M : )
p-o-o-5-8=8=0-o:^^_^ - c / ^ ^ ^ ^ ^ ^ ^^• - • - •^- ^^^
0 •0.4
\
10
o 0)
40 -]
E 20
x/a = -1 1
1
1
1
0.01
0.02
0.03
0.04
~ 0.05
z/a Fig. 5 Maximum amplitude of the average effective tensile stress, Aam*, and orientation, a*, of the corresponding plane as a function of (a) the location within the contact interface; z = 0 (b) the depth, z, for x/a = -1. The calculations also showed that, at the edge of the contact, the maximum amplitude of the shear stress, Axm, was minimised along the direction corresponding to the maximum tensile stress amplitude (Fig. 6). The combined analysis of Aam* and Aim therefore establishes that the main cracks which nucleate close to the contact edge correspond to predominantly tensile (mode I) fatigue cracks. In addition, the distribution of Aam* within the contact plane can interestingly be considered (Fig. 7). The maximum amplitude of the tensile stresses is located within two croissant shaped areas oriented perpendicular to the sliding direction, which corresponds to the regions were crack initiation was observed experimentally for the various tests carried out under gross slip condition.
Experimental and Theoretical Investigation of the Contact Fatigue Behaviour of an Epoxy Polymer
59
x/a=-1
Q. E
15 10 5
0
2
4
6
8
10
12
14
Orientation a (°)
Fig. 6 Change in the maximum amplitude of the shear stress, Aim, at the edge of the contact (x/a = -1, z = 0) as a function of the orientation, a.
Sliding direction
1 2.0
1 -1.2
1 -0.4
1 0.4
1 1.2
r 2.0
x/a
13
25
38
50
63
AcT„* (MPa)
Fig.7 Distribution of the maximum amplitude of the tensile stress, Aam*, in the contact plane (z = 0). The dotted circle delimits the contour of the contact area. Analysis of crack propagation The second part of the investigation was directed toward the understanding of the crack propagation stages. Two questions were especially addressed: (i) the determination of the transition from fatigue to brittle fracture (ii) the determination of the crack arrest conditions. Within the frame of linear elastic fracture mechanics, such an analysis requires the
60
M C. DUBORG AND A. CHA TEA UMINOIS
determination of the stress intensity factors at the crack tip, which are dependent on the contact conditions at the crack interface. It is therefore of importance to determine whether the cracks faces are in contact or not and whether they slide with respect to each other. This analysis was undertaken within the frame of a general model developed by Dubourg and Villechaise [18-20]. This theoretical two-dimensional linear elastic approach of muhiple frictional contact fatigue cracks allows us to determine the stress and displacements fields in cracked solids and the associated stress intensity factors Ki and Kn. The model is based upon a modified dislocation theory and on the resolution of the contact problem between crack faces as a unilateral contact with fiiction. Stress and displacement fields are given by superimposing the individual responses of the uncracked solid and of the cracks to the contact loading in a manner that satisfies the boundary conditions along the faces of the cracks. The continuum stress field within the uncracked solid is obtained numerically from the abovementioned Kalker's algorithm. The crack response is associated with displacement discontinuities along its faces, opening and slip, which generate stresses. These displacement zones are modelled using continuous distributions of dislocations. For more details on the model, the reader should refer to refs [18-20]. The analysis was carried out in the contact meridian plane (y = 0) where a single straight crack located at x/a = -1 and oriented to 7° with respect to the normal to the surface was considered. This simplified description of the crack geometry was justified by the observation that their orientations did not change greatly as a function of depth. As a first order approximation, the unknown coefficient of fiiction between the crack faces was assumed to be equal to 0.6. The crack behaviour and the associated stress intensity factors during the various discrete steps of the cyclic tangential loading are represented in Figure 8. During the alternate tangential loading, opening (steps 1 to 3) and closing (steps 4 to 6) mechanisms of the crack were simulated in accordance with the in situ observation of the crack dynamics. Interestingly, the simulation also predicts the occurrence of some sliding at the tip of the closed crack (steps 5 and 6), which demonstrates a posteriori the necessity of taking into account the tribological behaviour of the crack faces. The calculation of Ki and Kn also clearly demonstrates the complex non-proportional loading of the crack tip, which complicates any quantitative comparison of the stress intensity factors with the bulk toughness data obtained under pure mode I or mode II conditions. In Figure 9 the maximum calculated values of Ki and Kn have been reported as a fimction of the crack length. As the maximum values of Ki and Kn do not necessarily occur at the same time during the cyclic loading, it is not obvious how to quantify the level of mode mixity from these data. It can, however, be noted that Ki is the preponderant stress intensity factor over the whole range of crack lengths, which confirms the predominantly tensile nature of the observed cracks. As the crack length is increased, the maximum value of Ki goes up to values slightly higher than the experimental fracture toughness (Kic = 1.2 MPa vcv', ref [5]). Accordingly, the simulation predicts a transition from fatigue to brittle crack propagation, which is consistent with the above reported in situ observations. Taking into account the experimental toughness, this transition should occur for a crack depth close to 200 jam. The corresponding experimental value was not available, but it was observed that the length of the cracks at the surface of the epoxy specimens at the onset of brittle propagation (between 200 jLim and 400 jiim) was of the same order of magnitude. For crack lengths greater than 250 jum, a progressive decrease in the maximum value of Ki is calculated. For a crack length of about 400 jim, this value is lowered below Kjc, which corresponds to crack arrest conditions for brittle propagation. The observed maximum crack lengths at the end of the fretting tests were, however, about twice the predicted critical length for crack arrest. This discrepancy could be attributed to the occurrence of some additional sub-critical fatigue propagation after the brittle propagation stage, to the effects of mode mixity or to an insufficient description of the
Experimental and Theoretical Investigation of the Contact Fatigue Behaviour of an Epoxy Polymer
61
frictional behaviour of the crack faces, which could affect the calculation of stress intensity factors. 1
2
3
4
1.4 1.2
K,
A W
5
6
^i^
f
A w
1 Q.
1^
V^~~*^
0.8
\
0.6
^^^—^
\
\
0.4 0.2
K,,
X \
0 1
2
3
4
•
%
5
6
Loading step
Fig. 8 Theoretical description of the crack behaviour and calculated values of Ki and Kn during the discrete steps of a cyclic tangential loading (crack depth : 350 |am).
100
200
300
400
500
600
700
800
900
1000
Crack length (^m)
Fig. 9 Maximum calculated values of Ki and Kn as a function of crack depth.(the dotted line corresponds to the experimental bulk fracture toughness).
62
M C. DUBORG AND A. CHA TEA UMINOIS
CONCLUSION The cyclic damage behaviour of a DGEBA/IPD epoxy resin contacting a smooth glass hemisphere under fretting conditions was found to be associated with the nucleation and the propagation of two deep cracks at the edge of the contact. From a theoretical modelling of the elastic contact stresses, it was possible to ascribe these cracking mechanisms to predominantly mode I opening mechanisms. A theoretical analysis of the crack behaviour under the contact induced stress field also allowed determination of the changes in the stress intensity factors, Ki and Kn, as a function of the crack length. Despite the non-proportional nature of the mixed-mode crack tip loading, the comparison of the calculated Ki values to the experimental toughness properties of the epoxy allowed a reasonable description of the observed transition from fatigue to brittle propagation as the crack length increased. This investigation therefore supports the validity of a fi*acture mechanics approach of contact fatigue processes within glassy amorphous polymers. ACKNOWLEDGEMENT A. Chateaimiinois wishes to thank C. Gauthier (GEMPPM, INSA de Lyon, France) for her help in measuring the yield properties of the epoxy system.
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Jain, V. K. and Bahadur, S. (1980) Wear 60, 237. Jain, V. K. and Bahadur, S. (1982) Wear 79, 241. Lancaster, J. K. (1990) Wear 141, 159. Chateauminois, A., Kharrat, M. and Krichen, A. (2000). Analysis of Fretting Damage in Polymers by Means of Fretting Maps, ASTM STP 1367, pp.325-366, Chandrasekaran, V. and Elliott, C. B., Eds, West Conshohocken. Urbaczewski-Espuche, E., Galy, J., Gerard, J. F., Pascault, J. P. and Sautereau, H. (1991) Pol. Eng Sci. 31, 1572. Quinson, R., Perez, J., Rink, M. and Pavan, A. (1997) J. Mat. Sci. 32,1371. Vincent, L. (1994) Materials andfretting, in ESIS 18, Mech Eng Pub, pp.323-337, London. Fouvry, S., Kapsa, P. and Vincent, L. (1996) Wear 200,186. Kharrat, M., Chateauminois, A. and Krichen, A. (l999).Trib Tran 42, 377. Lesser, A. J. and Kody, R. S. (1997) J. Polym. Sci. B: Polym. Phys. 35,1611. Kody, R. S. and Lesser, A. J. (1997) J. Mat. Sci. 32, 5637. Hamilton, G. M. (1983) Proc. Inst. Mech. Eng 197C, 53. Johnson, K. L. (1985). Contact Mechanics. Cambridge University Press, Cambridge. Kalker, J. J. (1990). Three dimensional elastic bodies in rolling contact. Kluwer Academic Publishers, Dordrecht. Tanaka, K. and Mura, T. (1981) j : Appl. Mech. 47, 111. Yamashita, N. and Mura, T. (1983) Wear 91, 235. Lamacq, V. and Dubourg, M. C. (1999) Fatigue & Fracture of Eng. Mat. and Struct. 22, 535. Dubourg, M. C. and Villechaise, B. (1992) ASMEJ. Trib. 114, 462. Dubourg, M. C. and Villechaise, B. (1989) Eur. J. Mech. A 8, 309. Dubourg, M. C. and Villechaise, B. (1992) ASME J. Trib. 114,455.
1.2 Essential Work of Fracture
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65
EXPERIMENTAL STUDY OF RUBBER-TOUGHENING OF PET N. BILLON* and J-P. MEYER ** ^Centre de Mise en Forme des Materiawc (CEMEF), Ecole des Mines de Paris, UMR CNRS 7635, BP 207, F-06904 Sophia-Antipolis Cedex **Rohm and Haas Company, European Laboratories, 371, rue Beethoven, F-06560 SophiaAntipolis, France
ABSTRACT This work deals with impact toughening of semi-crystalUne and amorphous polyethylene terephtalate by the addition of core-shell elastomer particles. Two types of particles, differing by the reactivity of their shell, are compared. Specimen are analysed using a combination of various experimental techniques. Essential work of fracture allows discriminating energy dissipation processes. Izod tests allow to estimate the efficiency of the toughening. TEM observations allow to characterise dispersion and damage of the particles. Real time light backscattering and tensile tests allow comparing the level and the moment of occurrence of cavitation. Provided that loading conditions are highly tri-axial, the particles are able to significantly toughen PET, especially in the case where a reactive shell is used. Semicrystalline PET appears to be more demanding as far as the modifier-matrix interface is concerned. Comparing results of the mechanical characterisations and damage analysis allows proposing a scheme for toughening process. KEY WORDS Toughness, modifier, polyethylene terephthalate, core-shell, essential work of fracture INTRODUCTION Aside textile and packaging applications the use of PET (Poly(ethylene Terephthalate) for structural applications is rather limited compared to equivalent polymers such as polyamides. Two main reasons can be given. Firstly, the high sensitivity of PET toward hydrolysis and its slow crystallisation kinetics constrain its processing. Secondly, its low glass transition temperature constrains its use if amorphous, whereas its weak impact resistance if semicrystalline constrains its use when crystallised. The industrial objective of this work deals with the latter of these points: increasing the impact resistance of semi-crystalline PET.
66
K BILLON AND J.-P, MEYER
As PET has the abihty to be obtained either in the amorphous state or not an additional scientific goal exists that is: studying the interrelations between the microstructure of semicrystalline polymers and the efficiency of rubber toughening [1]. To achieve these points several complementary techniques, including essential work of fracture, are combined. EXPERIMENTAL Material and processing Core-shell toughening; General description The addition of a distributed rubber phase is known to be a good solution to toughen polymers [2, 3]. As far as PET is concerned, the rubber phase could be introduced by co-polymerising a rubber-like segment within the PET chain. Unfortunately, crystalline phase should be disturbed or even avoided in the resulting copolymer. Blending is than a better solution to combine toughening and crystallisation. However, correct dispersion, which is known to be a key issue for toughening [3], is sometimes difficult to control during processing even when compatibility of the rubber phase has been improved (e.g., via epoxy modification [4, 5]). Consequently, core-shell technology appears to be a more interesting route. Core-shell nodules result from emulsion polymerisation and are composed of a spherical core, including a rubber phase, on which a rigid shell has been grafted. They were first developed to toughen PMMA but can successfully be used for other polymers [6]. This technique enables to control the diameter (lower than one micron) and the shape of the rubber phase [7, 8]. Rigid shell ensures the compatibility with the matrix and can, in addition, be modified to promote chemical bounding with the matrix [9]. This is the route we aim at exploring. Materials The matrix is a high IV grade PET supplied by Eastman (12822, average mass: M^ = 36500g/mol) recommended for semi-crystalline applications. Two different core-shell particles of similar chemical make up (Butyl acrylate core, Polymethyl methacrylate shell) having approximately the same diameter (400 nm) and the same core to shell ratio (80 % core in mass) are used. From the two particles, one has epoxy reactive groups on its shell. Epoxy groups can promote chemical bounding with PET ends of chain. In consequence, this type of particles is expected to lead to strong matrix / nodule interface. It is referred to as the reactive particle (R). The non-grafted particle is referred to as the non-reactive particle (NR). Processing Processing of samples involved two steps, being blending and injection moulding, respectively. To prevent hydrolysis phenomenon (and the resulting decrease in the molecular weight) PET resin was cautiously dried (150°C, during at least 4hours) prior to any of these two steps. For the same reason, processing conditions were suited according to the state of the art [10]. Nonblended PET experienced the same thermo-mechanical history to account for possible degradation in the polymer.
67
Experimental Study of Rubber-Toughening of PET
The blends (PET plus core-shell elastomers) were compounded using a co-rotating twin screw extruder. The contains of particules ranged from 7 to 21 % in mass. Test bars were injection moulded in a second step, hi order to produce compound of different levels of crystallisation, the mould wall temperature was varied. Truly amorphous mouldings could not be obtained. However, for mould temperature of 5°C only a small amount (less than 8 %) of crystalline phase was observed. These mouldings will be considered as amorphous, since their continuous phase is amorphous. Conversely, a wall temperature of 145°C made it possible to reach the maximum level of crystallisation (approximately 30 % in mass). Finally, no evidence for a significant crosslinking effect due to reactive nodules was found. Experimental protocol Specimen were analysed, without additional drying or conditioning, using a combination of various experimental techniques: • impact tests to evidence efficiency of toughening, • tensile tests associated with 3D extensometer and real time light backscattering analysis to point out the appearance of cavities and the kinetics of their development, • essential work of fracture to discriminate energy dissipation processes, • electron microscopy to locate damage and to depict microstructure of blends. EFFICIENCY OF THE NODULES Non toughened semi-crystalline PET is a very brittle polymer whatever the loading conditions are: i.e., un-notched and notched tensile tests, dart test (impact of a hemispherical striker against a clamped plaque) and izod test (Fig. 1. to Fig.4.). Amorphous PET exhibits a more ductile behaviour except when notched. In such a case even amorphous PET is a brittle material at room temperature (Fig. 3. and 5.). In presence of nodules, ultimate properties in tension for non-notched samples remain unchanged. The only visible effect is a decrease in young modulus, E, and yield stress due to the rubber content. In example, young modulus of blends, E, obeys Kemer model [11]: l + ABO E = E^ ^ Ol-B¥
(1)
V
where Eo is the elastic modulus of the matrix. For their part. A, B and \|/ (Eq. (2)) are parameters that only depend on the elastic modulus of the rubber phase, E2, on the Poisson coefficient of the matrix, v and on a packing factor, (|)ni (given in table 1.):
^^1-5^
8-lOv'
(E2
Eo
E2
A
-E^ + A Eo
m
A^. BILLON AND J.-P. MEYER
68
Conversely, nodules can significantly increase the amount of energy dissipated during deformation of notched specimens (Fig. 5., 6.). They contribute to decrease brittle to ductile transition temperature. Non-reactive nodule can toughen amorphous PET to a certain extend, being nevertheless always less efficient than reactive one. On the contrary, only the reactive nodules exhibit a certain level of efficiency in semi-crystalline PET, provided that concentration is at least 21 %.
0.1 (a)
1
0.4
\
0.6 (b)
r
0.8
Fig. 1. Tensile properties of blends at room temperature and at 50 mm/min (i.e., initially 0.02 s"^). Comparison between semi-crystalline (a) and amorphous PET (b) for different amounts of core-shell: (X) Non-blended PET, (O) PET with 14 % of NR, (•) PET with 14 % of R, (U) PET with 21 % of NR, (•) PET with 21 % of R. Force (kN)
Fig. 2. Impact properties of PET at room temperature. Comparison between semi-crystalline (a) and amorphous PET (b). Results are obtained using a falling mass apparatus (initial energy of 147 J, initial velocity of 3.6 ms"\ striker and plaque diameter of 25 mm and 40 mm, respectively).
69
Experimental Study of Rubber-Toughening of PET
Table 1. Kemer's parameters for semi-crystalline and amorphous PET. Eo
Semi-crystalline Amorphous gQ Stress (MPa)
(MPa) 3570 2240
E2
(MPa) 1.2 1.2
V
m
0.35 0.35
0.64 0.64
Stress (MPa)
Fig. 3. Tensile properties of notched blends at constant strain-rate (0.005 s"^) and room temperature. Comparison between semi-crystalline (a) and amorphous PET (b) for different notches: (X) Non-blended non notched PET, (+) Non-notched blend with 21 % of R, (O) Nonblended PET and 1 mm-radius notch, (•) Non-blended PET and 0.25 mm-radius notch, (D) PET blended with 21 % of R and 1 mm-radius notch, (•) PET blended with 21 % of R and 0.25 mm-radius notch.
Fig. 4. Dimensions of the samples used for Izod and essential work of rupture measurements (ASTM D 256).
N. BILLON AND J.-P. MEYER
70
Normalized energy (J/m)
Normalized energy (J/m) 10^ 800-
6oa 4oa
60
80
100
10
Temperature (°C) (a)
15 20 25 Temperature (°C)
(b)
Fig. 5. Impact behaviour of PET and blends. Comparison between semi-crystalline (a) and amorphous (b) PET. Energy at rupture vs. temperature in the case of the non-blended PET (1) and blends with 14 % of NR (2), 14 % of R (3), 21 % of NR (4) and 21 % of R (5).
"Ductile" rupture (%)
"Ductile" rupture (%)
Temperature (°C) 1—
20 (a)
(b)
Fig. 6. Impact properties of PET and blends. Percentage of ductile fractures as a function of temperature. Comparison between semi-crystalHne (a) and amorphous PET (b). Results are obtained using a falhng mass apparatus (initial energy of 147 J, initial velocity of 3.6 ms"\ striker diameter of 25 mm and plaque diameter of 40 mm). At this point it can be concluded that core-shell nodules can toughen PET when loading conditions induce multiaxial stress fields (i.e., notched samples or dart tests). Precise physical phenomena involved have to be identified. This is done combining several techniques: • Essential work of fracture. • Tensile tests in association with 3D-video extensometer and real time light backscattering technique. • Transmission electron microscopy.
Experimental Study of Rubber- Toughening of PET
71
ESSENTIAL WORK OF FRACTURE This technique were used because of its simpHcity and its abihty to quaUtatively decompose the energy at rupture into its two intrinsic components: energy for creating new surfaces (essential work of fracture, We) and "plastic" work, or more seemingly dissipated energy due to the deformation of the bulk. This concept, developed by Brobeg [12], is often used for ductile polymers [13-15] instead of more sophisticated approaches such Rice integral [16]. Basic assumption is that plastic deformation (or damage) occurs in a confined zone in front of the crack. Li that zone a significant amount of energy is dissipated before crack can "produce" new surfaces. The energy required for the formation of new surfaces is assumed to be related to the intrinsic energy for rupture [17], Wg. Additional energy, Wp, is characteristic for the deformation process prior to rupture. From a practical point of view the technique consists in mechanical tests performed on precracked samples having variable crack lengths. Tests are conducted up to rupture. The total energy at rupture, Wf, is deduced from this measurements as a fiinction of the crack length, or more precisely asftinctionof the ligament length, 1. If one assume We to be proportional to the surface to be produced and dissipation phenomenon to be confined in a small volume in front of the crack, then: Wf =Wetl + (}Wptl2
(3)
where t is the thickness of the sample, We is the essential energy for rupture and Wp the specific "plastic" work. According to that description, the plot of Wf / ti vs. 1 should result in a linear evolution whose slope is p Wp and whose origin is Wg. In our case, Izod test bars (Fig. 4.) were used. Crack was initiated using new razor blades and a rubber hammer following previous work [18]. Crack length was deduced from post mortem observations using a microscope (precision of 0.01 mm). Ligament lengths were chosen higher than 3 t, as recommended, to ensure confinement of plastic deformation [13, 17]. Tests were three points bending tests performed at 5 mm/min and room temperature. Results are depicted in Fig. 7. and table 2. The four materials exhibit the expected linear evolution. Concerning non-blended PET, it appears that no energy, Wp, is dissipated before rupture. When nodules are added a certain amount of energy is dissipated additionally to the essential work of rupture ((} Wp higher than 0). This is the trace of the appearance of a new phenomenon occurring in the material during crack propagation. As this phenomenon is able to dissipate energy it toughens PET. The phenomena induced by reactive and non-reactive nodules seems to lead to equivalent efficiency in amorphous PET, whereas they are significantly different in the case of semi-crystalline PET. Hence, one can conclude that the efficiency of nodules is related to the appearance of a new deformation process, whereas rupture energy remains constant.
72
A^. BILLON AND J.-P. MEYER
jw/tlO^/irf)
^ Jt^^
40 20
•l .
i
t •
' V
1 4
_i <^
^iM
6
4 10
(a) Fig. 7. Essential work of fracture. Comparison between semi-crystalline (a) and amorphous PET (b). ( • ) Non blended PET; (•) PET with 21 % of NR; (A)PET with 21 % of R. Table 2. Essential work of fracture; Values for the parameters.
Semi-crystalline PET Amorphous PET Semi-crystalline PET with 21 % of NR Semi-crystalline PET with 21 % of R Amorphous PET with 21 % of NR Amorphous PET with 21 % of R
(kJ/mQ 10.4 9.8 6.6 14.1 6.9 7.3
Pwp (kJ/m) 0 0 2.4 5.7 5.9 6.0
DAMAGE PROCESS ANALYSIS Videometric extensometry [19] Videometric extensometer made it possible to evaluate the volume change during deformation as it gives access to the strains in the three directions of the space. In our case, these evaluations clearly point out an increase in volume (most presumably related to a cavitation process) during tensile deformation for all modified products even under uniaxial loading conditions. This increase in volume can reach 15 %. In the same evaluations, the unmodified PET, either amorphous or semi-crystalline, shows little or no volume change. From these experiments, it is possible to conclude that both amorphous and semi-crystalline PET break according to a non-cavitational process (i.e. shear yielding). Conversely, the modified products exhibit an additional cavitation process, whose importance could depend on the nodule. That is, lower in case of a reactive surface (<10 %): i.e., the more efficient the nodule is the lower the volume change is. Transmission electron microscopy [19, 20] First difference between the two considered particles is their respective ability to be dispersed in the PET. The reactive modifier is uniformly distributed to the individual emulsion particles (400 nm) whereas the non-reactive particle leads to clusters of several particles (Fig. 8.). This is
Experimental Study of Rubber- Toughening of PET
73
observed in both amorphous and semi-crystalHne polymer. This difference in the distribution results from the blending step and is never modified by crystallization nor additional processing.
NR R Fig. 8. TEM observations of semi-crystalline PET containing 21 % of core-shell particles. hi the case of semi-crystalline PET, comparing the TEM photographs and the measured spherulite sizes, it can be assumed that the individual reactive particles should be distributed in within the spherulitic structure. Concerning the non-reactive one it is highly probable, knowing the small size of the semi-crystalline microstructure, that the modifier clusters remain outside the spherulites. Finally, crystalline microstructure of PET is not significantly modified by the blending if one except a small nucleating effect of the non-reactive additive that can be observed in laboratory conditions. However, this nucleating effect is not due to the nodule itself [20] as no transcrystallisation can be observed on particles. Additionally, this nucleating effect does not lead to significant evolution in mean spherulite diameter in injection moulded parts. The post mortem observation of fracture surfaces of Izod bars confirms cavitational mode in the modified products. The close observation of the TEM pictures offers a fiirther insight: while cavities are sometimes visible inside the reactive modifier particles (R), the cavities in the blends containing non-reactive particles (NR) are always located at the modifier / matrix interface. That is, the "low" volume change in the case of reactive nodules could be related to nodules cavitation, whereas the "higher" volume change in the case of non-reactive nodules can be related to de-bounding of clusters. Real time light backscattering Light scattering experiments were used [21, 22] to investigate the cavitation kinetics in blends during tensile tests. This technique is based on the fact that voids in the material act as optical scatters. The analysis of the coherent light backscattering pattern of the material allows the measurement of the mean free transport light path, L*, which is related to the mean distance between scatters, that is voids in our case.
74
N. BILLON AND J.-P. MEYER
These experiments confirm that scatters (presumably voids) appear during deformation as L* decreases (Fig. 9.). Moreover, they emphasise the difference in the cavitation processes encountered in the modified amorphous and semi-cristalHne PET. hi the case of semicristalHne blends the onset of cavitation is quasi-simultaneous to the yielding of the material (Fig. 9a). Conversely, for the amorphous materials, cavitation only appears after yielding of the matrix (Fig. 9b). 150
60
100-
T
40 (2
/i\
\ A\ AlW
50 T
/ 1 ' X
^1
20
»8,1 « " 5; K 4 X 100
50 Time (s)
(a) 60
ACi -,
X
w illi,
X
AH -
ou •
/
sn -
/ ^
•"t'-H^-K K_^
X
0-
(\
TTTr
50
40 '
- 1
^ T T 20
c :< X k 100
Time (s)
(b) Fig. 9. Light meanfi-eepath, L*, and stress vs. time during a tensile experiment at a constant strain rate of 5.10'^ s"\ Comparison between semi-cyistalline (a) and amorphous PET (b) both containing 21% of non-reactive modifier. CONCLUSION hi this work it was possible to demonstrate the efficiency of the core-shell nodules to toughen PET, especially when reactive shell is used and in particular for tri-axial loading conditions (notched tests). Semi-crystalline PET appears to be more demanding as far as the modifiermatrix interface is concerned, hi that case, only a reactive modifier is able to provide a step change improvement in the energy absorbed during impact. Essential work offi-acturemeasurements clearly show that this efficiency always corresponds to a significant increase in the apparent "plastic" work, whereas essential work of rupture remains low. This observation suggests that toughening results fi*om an additional process induced by the particles.
Experimental Study of Rubber-Toughening of PET
75
Volume variation during deformation and light backscatering allow to evidence that this process is a cavitation process inside or around the particles. Comparing all the results of the mechanical characterisation and the analysis of the damage one can suggests that: • Toughening results from the formation of voids in the material. These voids locally disturb stress field in the material in such a way that damage (or plastic deformation) occurs. This leads to energy dissipation. • For toughening to be optimal cavitation has to occur before yielding of the matrix and inside the nodule. • This early rubber cavitation is not necessary for an efficient toughening of amorphous PET because of its lower sensitivity to triaxial stress states (compared to semi-crystalline PET). • For semi-crystalline PET, rubber-matrix de-cohesion does not lead an efficient toughening. • Efficient toughening of semi-crystalline PET requires a well adhered and dispersed elastomer leading to an early cavitation (quasi-simultaneous to the overall yielding) of the nodules themselves.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
Meyer, J.P. (1999). PhD Tesis, Ecole des Mines de Paris, France Bucknall,, C. B. (1977). Toughened plastics. Applied science pubhshers Ltd, London. Schirrer, R. and Gehant, S. (1997) Polym. Eng. and ScL, 37, 10. European patent appUcation EPA 89312834 European patent EP 309 575 HavriUak, S., Cruz, C.A.and Slavin, S.E. (1996) Polym. Eng Sci. 36, 2327. Vasquez, F., Cartier, H., Landfester, K., Hu, G.H., Pith, T. and Lambla, M. (1994) Polym. for Adv. Technol. 6, 309. Gardon, J.L. (1968) J. Polym. ScL, Polym. Phys. Ed. 6, 643. Jha, A. and Bhowmick, A.K. (1997) Polymer 38, 4337. Nagae, S., Nagura, K.,Yamane, N., Miyake, K. and Inoue, K. (1996) Proc. 12*^ Int. Conf. of The Polymer Processing Society, PPS-12 , Sorento, Italy. Kemer, E.H. {\956)Proc. Phys. Soc. B69, 808. Broberg, K.B. (1971) J. Mech. Phys. Solids, 19, 407. Saleemi, A.S. and Nairn, J.A. (1990) Polym. Eng. Sci. 30, 211. Mai, Y.W. and Powell, P. (1991) J. Polym. Sci., Polym. Phys. Ed. 29, 785. Chan, W.Y.F. and WilUams, J.G. (1994) Polymer 35,1666. Rice, J.R. (1968) J. Appl. Mech. 68, 379. Wu, J. and Mai, Y.W. (1996) Polym. Eng Sci. 36, 2275. Maazouz, A. (1996). PhD Thesis, INS A Lyon, France. Meyer, J.P., Schirrer, R. and Billon, N. (1999) Proc. if^ Int. Conf of The Polymer Processing Society, PPS-15 , 's Hertogenbosch, The Netherlands, ID17. Meyer, J.P. and Billon, N. (1999) Proc. IndESAFORM Conf on Materials Forming Guimaraes, Portugal, 155. Schirrer, R., Lenke, R., and Boudouaz, (1997) J. Polym. Eng. and Sci. 37,10. Gehant, S. and Schirrer, R (1999) J. ofAppl. Polym. Sci., Polym Phys. Ed., 37,113,
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
77
ESSENTIAL WORK OF FRACTURE OF INJECTION MOULDED SAMPLES OF PET AND PET/PC BLENDS J.J. SANCHEZ, 0.0. SANTANA^, A. GORJDILLO, M.Ll. MASPOCH and A.B. MARTINEZ Centre Catala del Plastic CCP. Universitat Politecnica de Catalunya UPC. c/Colom, 114. 08222-Terrassa (Barcelona), Spain
ABSTRACT The fracture behaviour at low strain rates of "amorphous" injection moulded PoIy(EthyleneTerephthalate) (PET) samples (< 6 % crystallinity) and its blends with Bisphenol A Polycarbonate (PC) were studied by the Essential Work of Fracture (EWF) method and the Ferrer-Balas partitioning methodology. The effect of specimen orientation, testing rate and PC content were investigated. In PET samples, at the same testing rate, WQ is slightly lower while /?Wp is higher, when crack propagation occurs in the melt flow direction, as previously reported. Additionally, a decreasing trend in We with testing rate was observed, which seems to be related to the restriction of some deformation processes at the propagation stage as the partitioning method reveals. In the case of PET/PC blends, where evidence of transesterification during melt blending have been verified, as the PC content increases, lower values of the EWF parameters were obtained for transverse-to-flow crack propagation. This trend, unlike the parallel-to-flow situation, seems to be dictated by deformation processes involved at crack initiation. In general, while at lower PC content the fracture behaviour is mainly controlled by transesterification effects, at higher content the oriented and stratified morphology which is developed seems to control the fracture behaviour. KEYWORDS: PET, PET/PC Blends, Essential Work of Fracture (EWF), Injection Mould, Orientation INTRODUCTION In those technical applications where Polyethyleneterephalate (PET) with low crystallinity is used, there has been much interest in the toughness, thermal and hydrolytic stability, and the reduction of crystallization. Thus, blends of PET with other thermoplastics polyesters and related materials have been extensively studied, specially those with Bisphenol A Polycarbonate (PC). For these systems it is well known that their melt blending involves interchange reactions, i.e. transesterification among others, giving rise to block and/or graft copolymers that promote the miscibility between the components and reduce the PET's crystallization capacity [1-12]. Although many papers on PET/PC blends have been published.
78
JJ.SANCHEZETAL
few of them make a detailed study of mechanical [1,4-7,10,12] and fracture [1-3] properties as a function of blend composition, degree of transesterification and correlations between deformation mechanisms with morphological features. The aim of this paper is to show some results on the fracture behaviour at low strain rates of injection moulded PET/PC blends applying the Essential Work of Fracture (EWF) theory and the partitioning methodology, in terms of blend composition and orientation. Additionally, the effect of crosshead rate and orientation on low crystalline injected moulded PET samples will be discussed. Essential Work of Fracture (EWF) This analysis was developed on the basis of Broberg's idea and has been extensively used to evaluate the fracture behaviour of ductile polymeric systems [13-22]. It suggests that the total work to break a ductile body (^f) can be divided into the essential work (We) consumed in an Inner Process Zone (IPZ) and associated with the actual crack propagation and deformation processes involved close to it, and the non-essential work {Wne) done in the Outer Plastic Zone (OPZ) surrounding the former, where large deformations during propagation are distributed. The latter, generally plastic deformation, is dependent on geometry, loading type and process zone size. Due to their geometric sample dependence, the following relationships have been considered [13, 14]: Wv=W,+
l^ne = We B L + Wp p BL^
WF
= We + PWp L
(1)
where We is the specific essential work offracture per unit of ligament area, considered as a material property for a given thickness and Wp the specific non-essential work of fracture per unit volume of deformed material, y^ is a plastic zone shape factor, B is the specimen thickness and L the ligament length. The experimental procedure, mathematical treatment and a results validation of this analysis are described elsewhere and compiled in the EUROPEAN STRUCTURAL INTEGRITY SOCIETY'S (ESIS) protocol [22]. The fact that during EWF tests
materials such as PP, and PET display a Load-displacement (P-d) trace as shown in Fig. 1, have motivated the development of additional methodologies based on partitioning the curves that have led to the separation of energetic contributions for the initiation and propagation of the crack [16, 19]. Full yielding of ligament lenght
k
p
max
1
/ OS
o hJ
/
^
Full yielding of ligament lenght
^max
Necking
1 \ ^ Necking
Crack Propagation ^
/
Onset of Crack ^ \ Propagation \^
o
h^
w^ (a)
\ w
Displacement, d
j j
Crack Propagation
1 Onset of CracK^-^ 1 Propagation \^
(*A
/ Displacement, d
Fig. 1. Schematic representation of partitioning methodologies proposed by (a) Karger-Kocsis et a/.[16] and (b) Ferrer-Balas et. aL[\9].
Essential Work of Fracture of Injection Moulded Samples of PET and PET/PC Blends
79
Fig.l shows how the partitioning is performed. Karger-Kocsis et al [16] proposed that Wf could be divided into two components. One associated with the ligament yielding (Wy) and the other with the necking and subsequent fracture (Wn) (Fig.la). Ferrer-Balas et al. [19] proposed a redefinition of these terms by considering that the global work is divided into that corresponding to the crack initiation (Wi) and that with crack propagation (Wn) (Fig. lb). The difference with the former is the stored elastic energy during initiation is dissipated at the propagation stage. In this way, Wi involves all the irreversible initiation processes, local yielding, local necking and blunting while Wn includes the crack propagation and generalized necking around the ligament. Both methodologies make use of eq.l and its related mathematical treatment for each work contribution, giving rise to two essential work components: We,y and We,n in Karger-Kocsis's methodology and w^ and Wg" in the FerrerBalas analysis and two non-essential terms; pyWp,y and y^nWp,n, 0w^^ and y^W", respectively.
EXPERIMENTAL Materials, blends and specimen preparation: Blends were prepared using a PET copolymer based on isopthalic acid (« 2.3% mol), with IV= 0.80 dl/g, and a Bisphenol A PC with MVR (300°C/1200) - 26 cmVlO min. Melt blending was performed using a COLLiN ZK-25 corotating twin screw extruder with a L/D= 36 and a screw diameter of 25 mm. Prior to extrusion both polymers were dried, PET at 140°C for 4 h using a PlOVAN T30IX dehumidifying dryer hopper, and PC in an oven at 130°C for 16 h. After the drying period the pellets were mixed quickly in the solid phase and put into the drying hopper at 120°C. A constant extrusion profile was set at 270°C and a screw speed of 125 rpm was used. Nitrogen, as an inert atmosphere, and vacuum in the last heating zone were applied. The PC amounts added were 5, 10, 20 and 30% w/w, and the following nomenclature will be used: PC##, where ## is the nominal PC content. Prior to injection in the mould, PET and blended pellets were re-crystallized in an oven at 130°C for 16 hr, in order to avoid agglomerations in the hopper, and dried in the PIOVAN hopper at 140°C for 4 hr connected to a MATEU & SOLE Meteor 70/20 injection moulding machine with L/D= 20 and D = 26 mm. The injection moulding was performed using a temperature profile ranging from 290 to 300°C and at a pressure of between 120-140 bar. The mold temperature was set at 18°C for PET and blends and 50°C for PC. In these conditions, square plaques (2 x 100 x 100 mm) and tensile dumbbell specimens (ISO 3167) were obtained. Tensile and fracture properties: Young's modulus (F), engineering yield stress (ay), engineering flow (necking stabilization) stress (af) and strain at yield (Cy) were determined from tensile tests (ISO 527), at room temperature and several crosshead rates (Re) (2 to 50 mm/min), using an universal testing machine equipped with a video-extensometer. The EWF tests were performed using the whole square plaques in a Deep Double Edge NotchedTension (DDENT) configuration, with a distance between grips (Z) of 50 mm and at room temperature. Notches were made at 0° (P) and 90° (T) with respect to the melt flow filling direction using a circular saw and sharpened with a fresh razor blade just before testing. For each condition 18 specimens were evaluated with the ligament length (L) ranging between 5 and 32 mm following the ESIS protocol recommendations. The effect of Re was evaluated in PET samples at 2, 10, 25 and 50 min/min, while the blends were tested at 10 mm/min. In order to increase the accuracy of the calculations, L values and the total height of necking zone (H) were measured after testing with a travelling binocular microscope.
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J.J. SANCHEZ ETAL
Morphological and fractographic inspection of the materials were performed on cryogenic fractured surfaces taken from the central region of tested and untested DDENT specimens, using a JEOL JSM6400 Scanning Electronic microscope (SEM). RESULTS AND DISCUSSION Morphological and mechanical characterization Prior to discussing mechanical and fracture aspects some observations dealing with Differential Scanning Calorimetry {DSC) results (not shown here), will be considered. Firstly, reduced crystallization ability and crystal perfection in the blends compared to PET were obtained which was shown by the dramatic reduction in degree of crystallization during slow cooling {Xc) (up to 77% ), and the Melting Temperature {Tm) depression, TC for PC05 and up to lOT for PC30. Additionally, the ''cold-crystallization'' {c-c) phenomenon, commonly observed on highly "amorphous" PET samples during the heating scan, shows an important depression with PC content as the low Xc-c and increase of cold-crystallizaton temperature {Tc-c) revealed. All these observations let us to affirm that, for the lower PC content and the melt blending conditions selected, some degree of transesterification had occurred, a fact confirmed by additional FTIR analysis. According to the DSC and DMTA results, for up to PCIO, only one Tg signal {= 81°C) corresponding to the PET-phase was detected. For PC20 and PC30 two Tgs were observed (at 84°C for the PET rich phase, and the other at =138*'C, for the PC rich phase) making evident the two phase nature of these systems. Compared to pure PC {Tg = 140°C), the PC phase of the blends undergoes a depression, indicating that there is some interaction between phases. Direct evidence of this two phase situation was obtained from some SEM observations carried out on the central region of untested specimen at 0° (P) and 90° (T) to the flow direction during mould filling (Fig 2). The low PC-phase size (
Essential Work of Fracture of Injection Moulded Samples ofPET and PET/PC Blends81 Table 1. Mechanical properties and Crystallinity degree {X^ evaluated by DSC E Material Xc (%) Re Cy Gy Qf/ay CJf (mm/min) [GPa]* [MPa]* [MPa]* [%]* Dumbbell TET 53.6 0.46 2 2.39 25.1 3.20 0.44 54.9 2.40 24.1 3.38 10 9.0 56.7 0.41 25 2.40 23.3 3.50 57.3 2.41 23.0 0.40 50 3.60 56.2 0.44 25.0 6.8 PC05 10 2.47 3.43 26.3 0.46 2.51 6.1 PCIO 10 57.6 3.50 29.7 6.7 PC20 59.7 0.50 2.59 10 3.56 58.0 0.54 31.6 6.5 PC30 10 2.63 3.63 59.2 45.0 10 2.35 5.94 0.76 PC -
Xc (%) Plaques 6.6 6.3 5.4 4.3 3.3
--
* Standard deviations represent less than 4 % of the mean.
(b) (d) Fig. 2. SEM micrograph of cryogenic fracture surfaces taken at: T (upper) and P (lower) crack propagation. {a,b) PCIO, (c,d) PC20 and (e,f) PC30. Essential Work ofFracture (EWF) analysis The EWF tests were named with a code indicating material, crosshead rate and orientation, in that order. In this way, PC20-10T (or P) results correspond to tests carried out on PC20 blends tested at 10 mm/min in 90° (or 0°) crack propagation with respect to the melt flow direction. It is important to mention that even the fact that tensile parameters were determined on thicker (4 mm) dumbbell specimens could invalidate their use in EWF validations due to some morphological and crystallinity differences, some SEM observations and DSC measurements on plaques and dumbbell test specimens verified that these microstructural features are quite similar on both specimen geometries (Table 1).
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Fig. 3. (a) Schematic representation of phase distribution for PC20 and PC30 blends, (b) Flow filling profile ("Fountain flow" pattern) obtained after short injections. Extensional flow field arrowed. Fracture behaviour. Depending on Crosshead rate {Re), ligament length (L), blend composition, and orientation, there were 5 types of fracture observed, associated with features on the Load-displacement {P-d) traces (Fig. 4) with the help of real time recorded videos: (a) "Post-yielding" (PY) (Fig. 4(a)). The crack initiates its stable propagation after complete yielding of the ligament. This behaviour was shown by PET-2T/2P, PET-lOT/lOP, PET-25T and SOT for L < 15.7 and 12.2 mm, respectively on all blends in the 90° configuration (T) as well as PC05-10P and PC 10-1 OP. {b) "Sequential post-yielding plus tearing" (sPY-T) (Fig. 4(6)). Only observed on PET-25T and PET-50T for L > 15.7 and 12.2 mm, respectively. During crack propagation a tear point and partial ligament breaking were observed. This behaviour is considered as the transition between the former type of fracture and "ductile instability". (c) "Ductile instability" (DI) (Fig. 4(c)). Characterized by a small-scale (confined to the crack path) plastic post-yielding deformation and unstable crack propagation, causing a brittle fracture. For these tests, even though the maximum net stress (amax) follows the expected trend with L, the application of EWF analysis could be invalid. This behaviour was observed on PET-25P, PET-50P, PC20-10P for L > 15-17 mm and PC30-10P. {d) "Post-yielding and fibrillation" (PY-F) (Fig. 4(^0). This was observed on PC20-10P at L < 17 mm. Once the ligament yielded a simultaneous crack propagation and fibrillation followed by tearing was observed. Although this instability restricts the use of EWF methodology, the wp-L plots show a very good linear fit, supporting its use for comparative purposes. (e)"Rapid crack propagation" (RC) (Fig. 4(e)). Observed exclusively on PC for both orientations. Crack propagation begins before the whole ligament yields. Although the EWF application is restricted its use gave parameters and trends in close agreement with those reported by Hashemi et al. [15] (we= 27 to 35 k j W and pWp= 3.46 to 3.01 kjW) (Table 2). EWF determination. After the P-d traces were checked for self-similarity (satisfied by all the systems with post-yielding behaviour and even for PC), wp-L valid points were selected according to the mean value of Gmax (<7m) criteria: 0.9Gm > amax > l-lam [22], and plotted as shown in Fig. 5. In order to increase the accuracy of parameter determination, verification of the upper limit of valid ligamet length (Lmax) was made by the plastic zone size estimation (2rp), using the following relationship:
Essential Work of Fracture of Injection Moulded Samples of PET and PET/PC Blends
0
2
4 6 8 Displacement, d [mm]
10
12
Fig. 4. Typical P-d traces and fracture behaviour: {a) post-yielding, {b) Sequential postyielding and tearing, (c) Ductile instability, {d) Post-yielding and fibrillation, and (e) rapid crack propagation, (a): the plastic zone geometry (diamond shaped) used for y^determinations.
2rp = k(EwjGy )
(2)
with k =\/K for circular geometry or k = n/S for the linear case. Regardless of the geometry considered, values between 12 and 15 mm were obtained, clearly lower than W/3 (=33.3 mm). However, it has been reported that a very good linear fit could be obtained beyond these calculated values, supported by the self-similarity of P-^ traces [16,20,22]. Thus, Lmax, in most of the cases, was set at 28 mm, where good linear fitting was observed. Re and orientation effects on PET. It has been proposed that, if PET samples with the same molecular features are considered, WQ could be affected by the microstructure and aggregation state, ranking its behaviour as [17]: strong oriented semi-crystalline (bi-oriented) < amorphous < semi-crystalline. Taking into account that some molecular characteristics of these PET (comonomer type and content) could be different from PET used in the actual work, our results seem to agree with this ranking. The We values (Table 2) lie between those reported by Chang et al. [21] (53.6 kjW, amorphous) and Hashemi et al. [17] (62.5 kjW, semi-crystalline and oriented), and higher than those from Karger-Kocsis et al. [17] (45-47 kjW, bi-oriented), all of them determined at 2 mm/min. As can be seen, We values for O^' configuration (P) are lower than those for 90° (T), and both of them decrease further with testing rate {Re) (Table 2). These trends could be associated with the semi-crystalline, partially oriented microstructure and its strain-induced crystallization during the necking process in addition to the viscoelastic nature of the network entanglement. Evidence of this strain-induced crystallization was obtained by complementary DSC measurements of the deformed zone after testing, where important decreases in the cold crystallization process compared to untested samples were observed, indicating an increase of crystalline fraction prior to the heating scan.
83
84
J.J. SANCHEZ ETAL 1
.{a)
1
1
—T
1
1
1
1 y.y u
390 -
320
j
^
3
^
-
•
5 250 ^
• (1) 2 mm/min 0 (2) 10 mm/min A (3) 25 mm/min A (4) 50 mm/min
180 110 1
0
•
1
•
1
\ •
•
7 14 21 28 35 Ligament Length, L [mm]
0
7 14 21 28 35 Ligament Length , L [mm]
Fig. 5. WF VS. L pots for 90° tests in (a) PET at different Re and (b) PET/PC blends at 10 mm/min (Coefficient regressions > 0.9898 ). (Y-intercept was omitted for simplification) Table 2. EWF and partitioning parameters for PET at different crosshead rate (Re) and crack orientation (90" and 0°) with respect to flow filling. Re [mm/min]: Parameters We[kJ/m'] /?xlOO Wp [MJ/m^] wj [kJ/m^] y^Wp^ [MJ/m^] We"[kJ/m^]
2 90° 59 ± 4 12.6 ± 0 . 2 9.2 ± 0.2 137 ± 6 20 ± 1 1.2±0.1 39 ± 4
10 90° 51±5 11.8±0.2 8.6 ± 0 . 2 137 ± 5 22 ± 1 1.3±0.1 29 ± 5
25^ 90° 40 ± 4 11.6 ± 0 . 4 8.3 ± 0 . 3 140 ± 1 0 20 ± 2 1.9 ± 0 . 2 20 ± 4
50^ 90° 42 ± 4 13.8±0.6 8.0 ± 0 . 9 173 ± 2 0 23 ± 1 1.8 ± 0 . 2 19±4
2 0° 56 ± 3 12.6 ± 0 . 3 9.0 ± 0 . 2 140 ± 8 13±2 2.1 ± 0 . 1 43 ± 6
10 0° 42 ± 3 12.4 ± 0 . 2 8.8 ± 0 . 2 141 ± 5 19±2 2.2 ± 0 . 1 22 ± 3
y^Vp"[MJ/m^]
11.4 ± 0 . 2
10.5 ± 0 . 2
9.7 ± 0 . 4
12.0 ± 0 . 5
10.5 ± 0 . 3
10.2 ± 0 . 2
flwp [MJ/m^]
Fitting based on: (7) 10 points and (2) 8 points. Applying the partitioning methodology proposed by Ferrer-Balas et al.[19], it was observed that while the We^ component (associated to cracking initiation, local necking, and crack tip blunting) remains constant, Wg" (associated with stable crack propagation) shows an important decrease for both orientations, following the same trend as the global Wg. Thus, the w^ variation seems to be governed by the local events associated with the propagation stage of the crack, i.e. additional local yielding in the Inner Process Zone (IPZ). These trends could be explained by the crystalline entities (possible orientation in the moulding stage) acting as preferential sites for the newer ones induced by deformation (kinetically favoured by the increase on Re) giving an autonucleation effect, increasing the restriction to large deformation (plastic flow and network deformation) once the ligament has yielded, reducing crack stability during propagation causing the transition to "Ductile Instability" observed at higher testing rates. Some additional dichroic FTIR analysis was made to support this hypothesis. Analysing the fiw^ values shows a slight decrease for both orientations with Re. No clear information has been found dealing with the actual trend of this parameter; while
Essential Work of Fracture of Injection Moulded Samples of PET and PET/PC Blends for unoriented and bioriented PET samples [20, 21] no change has been observed, for amorphous CoPET [16] and filled bioriented PET [21] an increase is reported. Based on the height of necked zone (H) in the valid L range, and approaching a "diamond shape" geometry (H=2;fiL) (Fig. 4a), y^factors were calculated, and were in close agreement with those reported for similar materials [18,21]. The slight decrease observed indicates that a reduction on the Outer Plastic Zone (OPZ) size is taking place supporting the restricted necking hypothesis due to preferential orientation of the entities. Taking into account these geometry factors, Wpvalues estimated for both orientations were almost independent of test speed. The increase at 50 mm/min could be considered as an artefact due to the low number of valid wp-L points considered in the fitting. Effect of PC content and orientation in PET/PC blends. As can be seen in Table 3 and Fig. 6, We in the 90° configuration (T-CP) shows the expected decreasing trend according to the "Additive Mixing Law" {AMI). For 0° evaluations (P-CP) a mixed behaviour, on the basis of AML, could be observed; an increase up to 10 % w/w of PC added, reducing the anisotropy observed (difference between 90° and 0° determinations) on pure PET samples. As the PC content increased a sudden drop was observed, raising the degree of anisotropy. For lower PC content (PC05 and PCIO), the reduction of the crystallization ability and the loss of preferential crystalline orientation added to the lower capability of strain- induced crystallization, are seen to be playing the main role in the fracture behaviour. Table 3. EWF and Ferrer-Balas' partitioning parameters for PET/PC at 10 mm/min.
Wp [MJ/m^] We^ [ k J W ] pVp^ [MJ/m^] w}^ [kJ/m^]
51±5 11.8±0.2 8.6 ± 0 . 2 137 ± 5 22 ± 1 1.3±0.1 29 ± 5
Blends PCIO PC20 PC30 T-•CP (90^ crack propagation) 46 ± 4 41±4 47 ± 3 43 ± 3 11.8±0.2 11.5±0.2 10.6 ± 0 . 2 10.3 ± 0 . 2 9.3 ±0.2 9.1 ± 0 . 2 9.0 ± 0 . 2 8.6 ± 0 . 2 127 ± 5 123 ± 2 128 ± 5 113±5 14±1 18±3 15±1 12±2 1.6 ±0.2 1.8±0.1 2.1 ± 0 . 1 2.2 ± 0 . 1 28 ± 4 31±3 29 ± 3 29 ± 3
31±1 3.1 ± 0 . 1 4.3 ± 0 . 2 73 ± 4 27 ± 2 1.5±0.1 4±1
p"wp" [ M J W ]
10.5 ± 0 . 2
10.2 ±0.2
8.1 ± 0 . 1
1.5±0.1
We" [kJ/m^]
42 ± 3 12.4 ± 0 . 2 8.8 ± 0 . 2 141 ± 5 19±2 2.2 ± 0 . 1 22 ± 3
P-CP (0" crack 48 ± 3 47 ± 1 11.9±0.1 10.1 ± 0 . 2 8.8 ±0.4 7.9 ±0.3 128 ± 7 136±6 20 ± 1 24 ± 3 1.6 ± 0 . 1 1.6 ±0.1 27 ± 3 24 ± 3
35 ± 1 2.4 ±0.1 2.9 ± 0 . 2 83 ± 6 24 ± 1 1.6±0.1 12±1
p"wp"[MjW]
10.2 ± 0 . 2
10.3 ±0.1
-
Parameters
We [ k j W ] Pwp[MjW]
pxlOO
We[kjW] pWp[MjW]
PxlOO Wp [MJ/m^] We^ [ k j W ]
pV[MjW]
PET
PC05
9.7 ± 0 . 3
8.5 ± 0 . 2
8.5 ±0.1 propagation) 23 ± 3 8.1 ± 0 . 2 2.9 ± 0 . 4 280 ± 40 13±2 2.4 ± 0 . 2 9±3 5.7 ± 0 . 2
PC
0.80 ±0.03
85
JJ. SANCHEZ ETAL However, the sudden change observed as PC content increases (PC20 and PC30) could mean that morphology aspects rather than the consequences of transesterification reactions govern the fracture process. Taking into account that PET reaches the yield condition before PC (Cy-PEi < Cy.pc, see Table 1), in addition to the apparent low strength interphase and the heterogeneous oriented stratified morphology, PC-phase cavitations could be favoured in the constant stress loading condition, i.e. 0° crack propagation test configuration (see Fig. 3a), increasing the instability during crack propagation. The non-essential term {J3w^) shows a smooth decrease with PC content for both orientations following closely the AMI trend (Fig. 6a). However, when their /^-values are considered and Wp is calculated, this trend seems to be disrupted for PC20 at the 0° configuration (P-CP) (Fig. 6b). However, it must recalled the that fibrillar breaking nature of this blend could give rise to uncertainties in fiw^ and/or /? determinations. Even so, the decreasing trends seem to be related to the progressivly lower homogeneous necking capability of the blends as the PC content increases, inferred from the trend on Qf/ay (Table 1) and observed by fractographic inspection (Fig. 7).
60 1
1
1
1
1
1
'
1
'T
• 0
50 _ 40 L1 Y"^ J
" 1 1 60
j J4 0 ^
^~~T-~ ^^^^
§ 30 ^^20
10 r^~5~*~
(a)
30'g \
^
0L
20 K
\ 10 1
1
1
DJ\J\
PW-TU p 1 Bw-PU 50 p II
i-UL. i
1 1
20 40 60 80 100 PC content (% w/w)
0
1
*
1
300 -
\
_ 250
^B
Ap
»
1
»
1
•
1
•
1
•
W-TI
D
w"- P " p 1
^
^ 140 — 120
100 80 60
(b)
' * -*
20 40 60 80 100 PC content (% w/w)
Fig. 6. (a) EWF parmeters and (b) specific non-essential work of fracture, Wp, for PET/PC blends on 90° (T) and 0° (P) configuration. Solid lines represent the "Additive Mixing Law".
1 mm Fig. 7. SEM micrograph of tested specimen at P-CP showing the "necking degree" (n) at the ligament region, (a) PET; (b) PC 10 and (c) PC20.
Essential Work of Fracture of Injection Moulded Samples of PET and PET/PC Blends
3
10 15 20 25 30 35 40 45 50 55 60
w [kJ/m ] Fig. 8. Essential components {WQ and Wg") from the partitioning as a function of global essential work of fracture (we) for PET/PC blends. Solid and dashed lines represent the trends followed. In order to elucidate which of the events during fracture (initiation or propagation) govern the global behaviour, the partitioning components {w^ and Wg") were plotted as a function of global We (fig. 8). As can be seen, for T-CP (90°) configuration, a steeper trend was obtained between Wg and w^ , implying that the irreversible initiation processes at the crack tip (local yielding, local necking and blunting-initiation), seem to dictate the global energy consumption during fracture. In the P-CP (0° configuration) case, even though Wg seems to have a dependence on both terms, the steeper one with Wg" allows us to say that the local events associated with the propagation stage of the crack, i.e. additional local yielding in the inner process zone (IPZ), are controlling the fracture energy consumption. CONCLUSIONS Once again it is observed that, although some test conditions could promote the violation of the EWF theory pre-requisites, its application could be possible if self-similarity of P-^i traces and extensive ligament yielding are observed. For the injection moulding conditions used, a crosshead rate and orientation dependence of Wg is shown on highly amorphous PET samples. This dependence could be related, to a first approximation, to the oriention favoured straininduced crystallizations phenomena that reduces the necking capability, and thus the crack propagation stability is affected. The same effect (restricted necking) could be observed as the PC phase is added to the system. In the blends, with up to 10% w/w of PC added, transesterification seems to play the major role, destroying the anisotropy effect of the partial oriented crystalline entities observed in pure PET samples. At higher PC content, the two phase nature of the blends and the oriented and stratified morphology govern the fracture process, increasing the anisotropy, lowering Wg and pWp values for parallel-to-flow cracks compared to transverse-to flow crack propagation. The partitioning analysis can be used as an useful tool in order to determine which of the fracture stages (initiation or propagation) is the main one. Thus, for amorphous PET samples all the events involved in crack propagation seem to control the fracture process. As the PC phase is added, this situation changes, and orientafion effects can be observed. For crack propagation in the parallel-to-flow direction (0° evaluations) the
87
JJ.SANCHEZETAL propagation stage rather than the initiation events is the controlling factor. It is the reverse for 90° evaluations, where the initiation process involved at the crack tip is the controlling features of the whole fracture behaviour. ACKNOWLEDGEMENTS The authors are grateful to CICYT (Spain) for the financial support to the project MAT-1112 in which this work is involved. J.J. Sanchez wishes to thank to AECI (Spain) and FUNDAYACUCHO (Venezuela) for the pre-doctoral grant and additional financial support. Special thanks to Dr. A. Miiller and Dra. M. L. Amal from GPUSB (Venezuela) for the use of DMTA equipment, and to Prof J. G. Williams and Dr. Blackmann for the support received for this publication. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
Liao, Z-L. and Chang, F-H. (1994) J. Appl. Polym Sci. 52, 1115. Abu-Isa, LA., Jaynes, C.B. and O'Gara, J.F. (1996) J. Appl. Polym. Sci. 59, 1957. Wu, J., Xue, P. and Mai, Y-W. (2000) Polym. Eng. Sci. 40, 786. Garcia, M., Eguiazabal, J.I. and Nazabal, J. (2001) J. Appl. Polym. Sci. 81, 121. Pesetskii, S.S., Jurkowski, B. and Koval, V.N. (2202) J. Appl. Polym. Sci. 84, 1277. Murff, S.R., Barlow, J.W. and Paul, D.R. (1984) J. Appl. Polym. Sci. 29, 3231. Chen, X-Y. and Birley, A.W. (1985) Br. Polym. J. 17, 347. Porter, R.S. and Wang, L-H. (1992) Polymer 33, 2019. Berti, C, Bonora, V., Pilati, F. and Fiorini, M. (1992) Makromol. Chem. 193, 1665. Abis, L., Braglia, R., Camurati, I., Merlo, E., Natarajan, K.M., Elwood, D. and Mylonakis S.G. (1994) J. Appl. Polym. Sci. 52, 1431. Zhang, G.Y., Ma, J.W., Cui, B.X., Luo, X.L. and Ma, D.Z. (2001) Macromol. Chem. Phys. 202, 604. Kong, Y. and Hay J.N. (2002) Polymer 43, 1805. Cotterell, B. and Reddel, J.K. (1977) Int. J. Fract. 13: 267. Mai, Y-W. and Cotterell, B. (1986) Int. J. Fract. 32, 105. Hashemi, S. and Paton, C. A. (1992) / Mat. Sci. 27, 2279; (1993) ibid. 28, 6178; (1997) ibid. 32, 1563; (2000) ibid 35, 5851. Karger-Kocsis, J., Czigany, T. and Moskala, E.J. (1997) Polymer 38, 4587; (1998) Ibid 39, 3939; (2000) ibid, 41, 6301. Karger-Kocsis, J. and Czigany, T. (1996) Polymer, 37, 2433; and (2000), Polym. Eng. Sci. 40, 1809. Maspoch, M.LL, Santana, O.O., Grando, J., Ferrer, D. and Martinez, A.B. (1997) Polym. Bull. 39, 249; (1991) ibid 39,511. Ferrer-Balas, D., Maspoch, M.Ll, Martinez, A.B. and Santana, 0.0. (1999) Polym. Bull. 42,101. Maspoch, M.L., Henauk, V., Ferrer-Balas, D., Velasco, J.I. and Santana, 0.0. (2000) Polym. Test. 19, 559. Chan, W.Y.F. and Williams, J.G. (1994) Polymer 35, 1666. Glutton, E.Q. (2001). In: Fracture Mechanics Testing Methods for Polymers, Adhesives and Composites, pp. 177-195, Moore, D.R., Pavan, A. and Williams, J.G. (Eds.), ESIS Publication 28, Elsevier Science Ltd., Amsterdam.
Fracture of Polymers, Composites and Adhesives 11 B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
89
RATE AND TEMPERATURE EFFECTS ON THE PLANE STRESS ESSENTIAL WORK OF FRACTURE IN SEMICRYSTALLINE PET A. PEGORETTI Department of Materials Engineering, University of Trento, via Mesiano 77, 38050 Trento, Italy T. RICCO Department of Chemistry and Physics for Engineering and Materials, University of Brescia, via Valotti 9, 25123 Brescia, Italy
ABSTRACT The present work is aimed at studying the strain rate and temperature effects on the fracture behaviour of a semicrystalline poly(ethylene-terephthalate) (PET) film. The evaluation of toughness under plane stress conditions was performed by the Essential Work of Fracture (EWF) method. EWF experiments were carried out varying both temperature (from 0°C to 70°C) and displacement rate (from 1 mm/min up to 6-10'^ mm/min). The moderately high displacement rate of 6-10'^ mm/min (i.e. Im/s) was reached by an instrumented impact pendulum in the tensile configuration. Results showed that specific essential work of fracture, Wg, was scarcely affected by both temperature and displacement rates at least up to 5*10^ mm/min. Under impact conditions a marked increase of Wg was measured. As recently proposed by several authors, the specific total work of fracture, Wf, was partitioned into two terms, one representing the specific work for yielding up to the onset of fracture, Wy, and another term related to the specific work for subsequent necking and tearing, Wj^^. Both terms showed a linear relationships as a function of ligament length, from which the specific essential (Wgy , Wgnt) ^iid non-essential (Pwpy , Pwp j^^) related work components were obtained. Applying this partitioning procedure, it was found that the specific essential terms related to yielding (Wg y) and necking/tearing (Wg ^t) were significantly rate and temperature dependent, with opposite trend as a function of these variables. This behaviour seemed to reflect the viscoelastic character of the material thus suggesting the possibility to apply a timetemperature reduction approach for the construction of master curves for both the yielding and necking/tearing related components of the specific essential work of fracture. KEYWORDS Fracture toughness, essential work of fracture, plane stress, viscoelasticity, thin films, poly(ethylene-terephthalate).
90
A. PEGORETTI AND T. RICCO
INTRODUCTION The viscoelastic nature of polymers generally determines rate and temperature dependence of their mechanical properties. At low strain levels, i.e. in a linear regime, this dependence is well described by intrinsic material properties defined within constitutive viscoelastic laws [1]. At high strains, in presence of failure processes, such as yielding or fracture, it is more difficult to establish a constitutive behaviour as well as to define material properties able to intrinsically characterise the failure process and its possible viscoelastic features. One approach widely used to characterize the fracture behaviour of ductile materials is the essential work of fracture (EWF) method, first developed by Cottrell, Reddel and Mai [2, 3] for metals, following an idea originally proposed by Broberg [4]. In the EWF method the fracture toughness is defined in the framework of an elasto-plastic fracture mechanics approach [5], for notched samples whose ultimate failure is preceded by extensive yielding and slow crack growth. The energy associated with fracture is partinioned into two parts. One term is specific to the fracture of the material and consequently is a material parameter. The second term is related to the plastic deformation processes in the region outside the fracture process zone. After the notation proposed by Cotterel et al. [2], the specific work expended in the fracture process zone is usually called "specific essential work of fracture" (w^) while the work dissipated in the plastic zone surrounding the crack is called "specific non-essential work of fracture" (Wp). The essential work of fracture has been shown to be a material property for a given specimens thickness and independent of the specimen geometry [6]. Rate [7-18] and temperature [16-26] effects on the EWF parameters have been investigated by many authors on several different polymeric materials, like PET [7,18,22,24], PBT [12,21,25] PBT/PC blend [8], PEN [1L 26], amorphous copolyester [9], iPP [10,23], sPP [15], UHMWPE [13], LDPE, LLDPE and LDPE/LLDPE blends [19], ABS [13], POM [14] uPVC [16], and PC [17, 20], The results are not indicating a clear and general trend showing that rate and temperature effects on the fracture parameters strictly depend on the material under investigation. Aim of this paper is to investigate if the rate and temperature effects on the fracture parameters obtained by the EWF approach under plane stress condition are in some way related to the viscoelastic nature of the selected material (semicrystalline PET). THEORETICAL BACKGROUND In the EWF approach it is assumed that the total work of fracture, Wf, is the sum of two energy terms: Wf=We + Wp
(1)
where Wg (called essential work offracture) is the energy consumed in the fracture process zone, where the actual fracture occurs, and Wp (called non-essential work of fracture) is the energy dissipated in the outer plastic region, where a number of energy dissipation mechanisms may occur. By assuming that Wg is proportional to the ligament area, and Wp is proportional to the volume of the outer plastic region, the following specific terms can be defined: W w, = —^; ^ BL
w„ P
W ^ ^BL^
(2)
Rate and Temperature Effects on the Plane Stress
91
where Wg is the specific essential work of fracture, and Wp is the specific non-essential work of fracture, B is the specimen thickness, L is the ligament length and p is a plastic zone shape factor depending on the geometry of the specimen and the crack. By combining the specific work terms reported in Eq. (2) into Eq. (1) the following relationship for the total work of fracture is obtained: W^ = w^BL + jSw BL^
(3)
which can be rewritten as: W Wo = — - = w + /Sw L f BL ^ P
(4)
where Wf is the specific total work of fracture. As suggested by equation (4), the parameters Wg and Pwp can be evaluated by linearly extrapolating the experimental data of Wf versus L and considering the intercept of the regression line at L=0 (i.e. Wg) and its slope (i.e. PWp). A test protocol for EWF testing and data reduction has been assessed by ESIS TC4 group in order to ensure a certain reproducibility of results [5]. Many authors such as Karger-Kocsis et al [9,27], Hashemi et al [16,20,21,24] and FerrerBalas et al. [10], recently proposed that the total work of fracture Wf can be partitioned into two components: i) the work Wy for yielding of the ligament region; ii) the work W^t for necking and subsequent tearing of the ligament region: Wf=Wy + W„t
(5)
This energy partition is usually done by considering Wy as the energy under the loaddisplacement curve up to the maximum load, and W^^ as the energy from the maximum load up to final fracture [9,10,16,20,21,24,27]. Similarly to Eq. (4), the authors cited above expressed the variation of Wy and Wj^^ with the ligament length as
^y = ? r ^ ' ^ ^ ' y ^ ^ y ' ^ P ' y ^ '
"^^^ =-577^ "^^'^t + z^nt Wp,ntL ; (6)
where Wg y and Wg j^^ represent the yielding and the necking/tearing components of the specific essential work of fracture, respectively, and PyWp y and PntWp,nt ^^^ ^^e yielding and the necking/tearing related parts of the specific non-essential work of fracture, respectively. By considering Eq. (5) and Eq. (6) it is evident that the following relationships can be written: Wg = Wg y + Wg nt ;
PWp = PyWp y + p^tWp ^t
(7)
92
A. PEGORETTIAND T. RICCO
EXPERIMENTAL PROCEDURE This study was conducted on a commercial semicrystalline PET film (Mylar® by DuPont) of nominal thickness 52 jim. Rectangular coupons having width of 50 mm (24 mm for impact test) and length of 90 mm (grip distance 50 mm) were cut such that their longitudinal axis was parallel to the machine direction of the extruded film. Coupons were then razor notched to obtain double edge notched tension specimens with ligament length in the range from 5 to 20 mm. Tensile tests at low to intermediate displacement rates (i.e. 1, 10, 100 and 500 mm/min) were performed by an Instron tensile tester model 4502 equipped with a 1 kN load cell. Temperature in the range from 0°C to 70°C was controlled by an Instron thermostatic chamber model 3119. Tests at the moderately high displacement rate of 60000 mm/min (1 m/s) were carried out at room temperature by an instrumented CEAST impact pendulum in the tensile configuration. Load data were collected at a sampling time of 30 fxs and load v^ displacement curves were provided directly by the CEAST software (DAS 4000 Extended Win Acquisition System Ver. 3.30). RESULTS AND DISCUSSION Figure 1 demonstrates the self-similarity of load-displacement curves of DENT specimens of various ligament lengths tested at 1 mm/min at 23 °C. It is worth noting that, for all the experimental condition explored in this work, although the load-displacement curves are indicating an extensive crack tip yielding, it was not possible to visually detect the full extent of yielding before crack growth. In fact, the plastically deformed PET had the same refractive index as the bulk material, as previously observed in a recent study regarding the non-elastic deformations of this material [28]. Nevertheless the applicability of the EWF method on similar semicrystalline PETs is reported on some recent publications [7,18,22,29].
L = 14.34 mm
O L = 7,65 mm
0.5
1
1.5
displacement (mm) Fig. 1. Load-displacement diagram for DENT specimens at various ligament lengths tested at a cross-head displacement rate of 1 mm/min and at a temperature of 23 °C.
Rate and Temperature Effects on the Plane Stress
93
The effects of temperature and displacement rate on the shape of typical load-displacement curves of DENT samples of a given ligament length are illustrated in Fig. 2.
a)
b)
10 mm/min; L = 7.1 ±0.2 mm
T = 23 °C; L = 12.3 ± 0.5 mm 00
-
80
-
60 O
O
40
-
//
Yk
y \
(impact test)
20 '\
0.5
1
1
1
\
0.5
1.5
\
1 1 1 1 , 1 1 1 1 1 , 1 1 1 1 1 1 i l v 1 1 1 i l 11
1
1.5
2
2.5
displacement (mm)
displacement (mm)
Fig. 2. Effect of temperature a) and displacement rate b) on the load-displacement curves of DENT specimens. From Fig. 2a it is evident that as the test temperature raises from 0 up to 70 °C the resulting load-displacement curves display a decrease of the maximum load and an increase of the elongation at failure. On the other hand, as reported in Fig. 2b, when the displacement rate increases the load-displacement curves evidence an increase of the maximum load and, quite surprisingly, also the elongation at break increases. It is worth noting, that this rather unexpected rate effect on the load-displacement curves of notched samples has been already reported by Karger-Kocsis and co-workers for both biaxially oriented filled PET [7] and amorphous copolyester [9], and by Plummer et al for polyoxymethylene tested at high temperature [14]. 160
T = 23°C; V = 1 m/s = 60000 mm/min
0
0.5
1
1.5
2
2.5
3
3.5
displacement (mm) Fig. 3. Load displacement curves of DENT specimens tested under tensile impact conditions.
A. PEGORETTIAND T. RICCO
94
Results of tensile tests under impact conditions (displacement rate of Im/s) on DENT samples are reported in Fig. 3 that clearly indicates how the self-similarity of load-displacement curves is still maintained even at such moderately high testing rate. Moreover, the overall shape of the experimental curves do not consistently differs from those obtained at lower speed (see Fig. 1). Data reduction in accordance to the ESIS protocol for the EWF method [5] yielded the plots reported in Fig. 4. As can be seen, at various temperatures and testing speeds, the plots of the total specific work of fracture (Wf) against the ligament length are essentially linear over the whole ligament length range. A relatively higher dispersion can be observed for the data obtained under impact conditions.
b)
10 mm/min
T = 23 "C
500 -D —{^ -•
400
- V = 1 mm/min - V = 10 mm/min V = 100 mnfi/min V = 500 mm/min V = 60000 mm/min
B 300 ^
200 100
4
8
12
16
0
5
10
15
ligament (mm) ligament (mm) Fig. 4. Effect of temperature a) and displacement rate b) on the total specific work of fracture (Wf) versus ligament length curves. The values of the specific essential (w^) and non-essential (pWp) work of fracture parameters obtained at various temperatures and displacement rates are summarized in Fig. 5 and 6. 50
V^ -
40 ^
30 k
^
20
16
v =10 mm/min
-*
f
14
- - ^ ---^
—>•
k
12 "CD
1
10
<-.
• •
10
. f- —
1
20
,
,
/ <
1
40
,
<
temperature (°C)
60
80
Fig. 5. Temperature dependence of specific essential (fiill circles; Wg) and non-essential (open circles; Pw^) work of fracture parameters.
Rate and Temperature Effects on the Plane Stress 80 70
25
T = 23 °C
impact rate. 20
60 ^
95
•CD
50 15
^-
30 10
20 10 0
"@ 0
1
2
3
4
5
log V (mm/min) Fig. 6. Rate dependence of specific essential (full circles; Wg) and non-essential (open circles; |3wp) work of fracture parameters. It can be seen from Fig. 5 that as the temperature rises the specific essential work of fracture term (w^) slightly decreases from 41.6 kJ/m^ at 0°C to 35.6 kJ/m^ at 70°C. On the other hand, the specific non-essential parameter (|3wp) steadily increases with temperature from 4.7 MJ/m^ to 10.9 MJ/m^. It is important to observe that the temperature range explored in this work is below the glass transition temperature (Tg) of the material. In fact, by considering the loss factor peak, a Tg value of about 105 °C can be estimated by dynamic mechanical test performed at a frequency of 1 Hz [28]. This trend of EWF parameters, at temperatures lower than glass transition, is in quite good agreement with data reported in recent works by Arkhireyeva and Hashemi [18,22] referred to a similar semicrystalline PET film (Melinex® by DuPont). From the data reported in Fig. 6, it can be seen that the specific essential work of fracture term is practically insensitive to displacement rate up to 500 mm/min (being its value slightly oscillating around 40 kJ/m^) and that a marked increase (up to 63.7 kJ/m^) occurs only under impact conditions. The specific non-essential parameter gradually increases with loading rate from 5.2 MJ/m^ to 18.5 MJ/m^. Again, this behaviour is consistent with previous literature data obtained by Arkhireyeva and Hashemi [18] for a similar material over a much narrow displacement rate interval (from 2 to 50 mm/min). At this point is interesting to consider the energy partition procedure described above that allows a distinction between the energy required for the yielding of the ligament region and for the subsequent necking and tearing processes. This energy partitioning procedure is schematically depicted in Fig. 7, where yielding (Wy) and necking/tearing (W^t) related components of the total work of fracture are indicated. Analysis of this components as suggested by Eq. 6 yields linear relationships of the specific Wy and Wj^t terms as a function of ligament length (here not reported for shortness). From the intercept and the slopes of the linear regression lines, the terms Wg y and Wg ^^ can be obtained, that represent the yielding and the necking/tearing related components of the specific essential
96
A. PEGORETTIAND T. RICCO
work of fracture, respectively. These terms are reported in Fig. 8 as a function of displacement rate and at various temperatures.
1
2.5
1.5
displacement (mm) Fig. 7. Typical load-displacement curve of a DENT specimen with indication of the total work partition into yielding related (Wy) and necking/tearing (W^t) related components. As reported in Fig. 8, it is quite clear that the specific essential terms w^ y and Wg j^^ are significantly rate and temperature dependent, with opposite trend as a function of these variables. Generally speaking, it is interesting to observe that the yielding related component of the essential work of fracture (w^ y), represented in Fig. 8a, is increasing with strain rate while the necking/tearing related term (w^ ^t)? reported in Fig. 8b, tends to decrease. This behaviour seems to be not only peculiar of PET since it has been observed by the authors on various polymeric films [30], like LLDPE, PP, PA6, and rubber toughened PA66.
a)
b)
25
-^ T = 0 °C - e — T = 23 "C -{i— T = 50 °C
80 20 h
6
-e— T = -70 ''c
60 M
40 20
15
fi 10
s--^ 1
2
3
log V (mm/min)
4
1
2
3
4
log V (mm/min)
Fig. 8. Effect of displacement rate on the yielding (w^ y) and the necking/tearing (Wg ^t) related parts of the specific essential work of fracture at various temperatures.
Rate and Temperature Effects on the Plane Stress
97
On the other hand, the temperature has an opposite effect on these terms since w^ y decreases while Wg m increases as temperature increases. The general trend of the data reported in Fig. 8 is suggesting the applicability of an empirical time-temperature reduction approach that has been already successftiUy applied to interpret the viscoelastic nature of crack propagation in polymers [31-36]. Master curves for the yielding and the necking/tearing related parts of the specific essential work of fracture, both referred to a temperature of 23 °C, are reported in Fig. 9 and 10, respectively. The master curves for the Wg y and Wg nt components, have been obtained by horizontally shifting the data of Fig. 8 to best superposition with respect to the data obtained at 23°C.
\-
80
T
[
a
L
40
= 23 °C
V O A D
r
60
>% 6
ref
T = 0°C T = 23 °C T = 50 °C T = 70 °C
T
,
'•f
^^'
r
1
^
. -® ' ' '(D
20
~i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1, , 1 , , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i l
-
3
-
2
-
1
0
1
2
3
4
5
log [V a^] (mm/min) Fig. 9. Master curve for the yielding related part (Wg y) of the specific essential work of fracture. 25
20
T
=23°C
cp---i--.t
V O A
a
*(P
15
M
ref
T T T T
= 0°C = 23 °C = 50 °C = 70 °C
Q)
.
CP
10
6
I
I
I
I
I
I
I
I
I
I
- 1 0
I
I
I
I
I
I
I
I
I
1 2
I I
3
4
log [V a ] (mm/min) Fig. 10. Master curve for the necking/tearing related part (Wg ^t) of the specific essential work of fracture.
A. PEGORETTIAND T. RICCO
98
These shift factors are reported in Fig. 11 as a function of the inverse of the absolute temperature. The temperature dependence of these quantities it clearly follows an Arrhenius type equation in the form: AE log a X = RT
(8)
where a^ is the rate-temperature horizontal shift factor, AE is an activation energy for the process, and R is the gas constant (8.31
). The linear regression of the data reported in molK Fig. 11 is indicating activation energy values practically coincident for the yielding (AE = 57.5 kJ kJ ) and the necking/tearing (AE = 57.7 ) processes, respectively. mol mol
t
0P
O
-4 h
2.8
3.2
3.4
3.6
3.8
1000/T (K-^) Fig. 11. Temperature dependence of the shift factors for the master curves of the yielding (triangles, Wg y) and the necking/tearing (circles, w^ ,^t) related parts of the specific essential work of fracture. CONCLUSIONS Rate and temperature effects on the fracture behaviour of DENT samples of semicrystalline PET film have been investigated by the EWF approach. Results showed that the specific essential work of fracture term, Wg, was scarcely affected by both temperature (in the range from 0 to 70 °C) and displacement rates (up to 5-10^ mm/min). A marked increase of the specific essential work of fracture parameter occurred only under impact conditions (displacement rate of 1 m/s). On the other hand, the specific non-essential parameter, |3Wp, gradually increased with temperature and loading rate over the whole experimental range. By partitionig the total specific work of fracture energy into a yielding related (Wy) and a necking/tearing (Wj^^) related component, the specific terms Wg y and Wg j^^ can be obtained, that
Rate and Temperature Effects on the Plane Stress
99
represent the yielding and the necking/tearing related components of the specific essential work of fracture, respectively. These terms resulted to be significantly rate and temperature dependent, with opposite trends as a function of these variables. This behaviour seemed to reflect the viscoelastic character of the material thus suggesting the applicability of a timetemperature reduction approach that allowed the construction of master curves for both the yielding and necking/tearing related components of the specific essential work of fracture. ACKNOWLEDGEMENT The authors thank Mr A.Casagranda for his contribution to the experimental work. REFERENCES 1. Ferry, J.D. (1961). Viscoelastic Properties of Polymers. John Wiley & Sons. New York. 2. Cotterell, B., Reddel, J.K. (1977) Int J. Fract. 13, 267. 3. Mai, Y.W., Cotterell, B. (1986) Int. J. Fract 32, 105. 4. Broberg, K.B. (1968) Int J. Fract 4, 11. 5. Clutton, E. (2001). In: Fracture Mechanics Testing Methods for Polymers, Adhesives and Composites, pp. 177-195, Moore, D.R., Pavan, A., Williams, J.G. (Eds). Elsevier, Oxford. 6. Wu, J., Mai, Y.W. (1996) Polym. Eng. & Sci. 36, 2275. 7. Karger-Kocsis, J., Czigany, T. (1996) Polymer 37, 2433. 8. Hashemi, S. (1997) Polym. Eng & Sci. 37, 912. 9. Karger-Kocsis, J., Czigany, T., Moskala, E.J. (1998) Polymer 39, 3939. 10. Ferrer-Balas, D., Maspoch, M.L., Martinez, A.B., Santana, 0.0. (1999) Polym. Bull. 42, 101. 11. Karger-Kocsis, J., Czigany, T. (2000) Polym. Eng & Sci. 40, 1809. 12. Hashemi, S. (2000) Polym. Eng & Sci. 40, 132. 13. Ching, E.C.Y., Poon, W.K.Y., Li, R.K.Y., Mai, Y.-W. (2000) Polym. Eng & Sci. 40, 2558. 14. Plummer, C.J.G., Scaramuzzino, P., Steinberger, R., Lang, R.W., Kausch, H.-H. (2000) Polym. Eng. & Sci. 40, 985. 15. Karger-Kocsis, J., Barany, T. (2002) Polym. Eng & Sci. 42, 1410. 16. Arkhireyeva, A., Hashemi, S., O'Brien, M. (1999) J. ofMater. Sci. 34, 5961. 17. Hashemi, S. (2000) J. ofMater. Sci. 35, 5851. 18. Arkhireyeva, A., Hashemi, S. (2001) Plast Rubber & Compos. 30, 337 19. Casellas, J.J., Frontini, P.M., Carella, J.M. (1999) J. Appl. Polym. Sci. 74, 781. 20. Hashemi, S., WiUiams, J.G. (2000) Plast Rubber &. Compos. 29, 294. 21. Hashemi, S. (2000) Polym. Eng &. Sci. 40, 1435. 22. Arkhireyeva, A., Hashemi, S. (2001) Plast Rubber & Compos. 30, 125. 23. Ferrer-Balas, D., Maspoch, M.L., Martinez, A.B., Ching, E., Li, R.K.Y., Mai, Y.-W. (2001) Polymer 42, 2665. 24. Arkhireyeva, A., Hashemi, S. (2002) J. of Mater. Set 37, 3675. 25. Hashemi, S. (2002) Polymer 43, 4041. 26. Arkhireyeva, A., Hashemi, S. (2002) Polymer 43, 289. 27. Karger-Kocsis, J., Czigany, T., Moskala, E.J. Polymer (1997) 38 4587. 28. Pegoretti, A., Guardini, A., Migharesi, C, Ricco, T. Polymer 41 (2000) 1857. 29. Maspoch, M.L., Henault, V., Ferrer-Balas, D., Velasco, J.L, Santana, 0.0. (2000) Polymer Testing 19, 559. 30. Pegoretti, A., Bertoldi, E., Ricco, T. in preparation.
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31. 32. 33. 34. 35. 36.
A. PEGORETTIAND T. RICCO
Knauss, W.G. (1989) In: Proceedings oflCF 7 Conference, Advances in Fracture Research, pp. 2683-2711, Salama, K. et al. (Eds). Vol. 4 Houston Texas. Popelar, S.F., Popelar, C.H., Kenner, V.H. (1991) Int. J. Fract. 50, 115. Pfel, M.C., Kenner, V.H., Popelar, C.H. (1993) Eng. Fract. Mech. 44, 91. Frassine, R., Rink, M., Leggio, A., Pavan, A. (1996) Int. J. Fract. 81, 55. Mariani, P., Frassine, R., Rink, M., Pavan, A. (1996) Polym. Eng. <Sc Sci. 36, 2758. Pegoretti, A., Ricco, T. (2001) /. ofMater. Sci. 36, 4637.
1.3 Environmental Stress Cracking
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
103
EFFECTS OF DETERGENT ON CRACK INITIATION AND PROPAGATION IN POLYETHYLENES M. RINK and R. FRASSINE Politecnico di Milano - Piazza L. da Vinci 32, 20133 Milano, Italy P. MARIANI and G. CARIANNI Polimeri Europa - Via lannozzi 1, 20097 San Donato Milanese, Italy ABSTRACT It is well known that polyethylene, although tough when stressed in air, becomes brittle in the presence of some specific substances, among which are non-ionic detergentwater solutions. In this work two polyethylenes, obtained with two different catalytic systems, were considered. Fracture behaviour was examined in air and in the presence of a stress-cracking agent (a water/Antarox CO-630 solution) at 50°C, for both polyethylenes. The effect of detergent concentration in water was then investigated for one of the two materials. Fracture tests were performed on single edge notched specimens, in four point bending under constant load (creep). The resuhs show that the fracture behaviour of the two materials is qualitatively very similar, although the fracture resistance levels are different. In the presence of the stress cracking agent it was observed that there is a critical value of the applied stress intensity factor, different for the two materials, above which the detergent does not seem to have any effect on fracture initiation time nor on crack propagation rate. Below this value, however, the environmental stresscracking agent accelerates fracture initiation. The concentration of the detergent in water on fracture behaviour affects both crack initiation and propagation, the stronger effect occurring for a value around 50%. The results are discussed in terms of material and stress-cracking agent characteristics, and their interactions. KEYWORDS Environmental stress-cracking, polyethylene, detergent, creep, molecular mass INTRODUCTION It is well known that some polymeric materials, although tough when stressed in air, become brittle in presence of some specific substances, commonly called
104
M.RINKETAL
environmental stress cracking (ESC) agents. In particular, polyethylene reduces its fracture resistance when kept in contact with non-ionic detergent-water solutions. The subject of the environmental fracture of polyethylene has been extensively studied in the past 30 years, and some general theories have been proposed to explain failure mechanisms. Lustiger [1,2] described the material behaviour at the molecular scale, recognizing the fundamental role of the so-called "tie molecules" (molecules having both ends trapped inside adjacent crystalline phases) in determining the fracture resistance of the material when ESC agents are present, assuming that the effectiveness of the physical entaglements in transferring the stress across the crystallites is severely reduced by the plasticizing effect of the ESC agent. Tonyali [3] proposed a more refined mechanism for ESC, based on the observation that non-ionic water-detergent solutions tends to form a micellar structure around small hydrocarbon molecules that may be present as impurities. Such structures are easily absorbed at the material surface, and hydrocarbon molecules rapidly diffuse inside the damaged material at a crack tip again producing a plasticizing effect. This theory was used to explain some effects observed at varying concentration of the detergent. These models identify the key structural parameters controlling ESC. The most important are: • • • •
molecular mass and molecular mass distribution; comonomer content; degree of crystallinity; lamellae orientation and thickness.
Molecular theories are fundamental in assisting material producers to develop more ESC-resistant materials. However, where load-bearing structures are concerned, an engineering approach to ESC capable of setting up suitable design criteria is required. This should result in the safety of the structure being at an acceptable level with reasonable confidence. Such criteria make use of fracture mechanics and its critical stress intensity factor paramenter, Kic. Several authors (Ward et al. [4], Brown et al. [5] and Willams [6] among others) observed that ESC agents produce a "transition" in the time-dependent fracture behaviour of polyethylene, which is not observed in the absence of aggressive environments. Such a transition occurs for times larger than a critical time, varying with material and testing conditions, at which a significant reduction of the fracture toughness of the material is observed. This effect was attributed to a diffusioncontrolled plasticization mechanism acting in the damage zone ahead of the crack tip. A model proposed by Williams [7] predicts that, depending on competition between detergent diffusion and material failure, three different stages may be observed at varying crack speed: for the lowest crack speeds, the effect of the environment is fully exploited giving rise to a complete plasticization of the material at the crack tip and producing the lowest crack resistances values. On the opposite side, for the highest crack speeds no difference in the fracture resistance values is expected as compared with results in air or other non-aggressive environments, since the diffusion speed is much slower than the crack speed. For intermediate crack speeds a "transition" between the two mechanisms occurs, which is expected to produce the stronger rate dependence for the critical stress intensity factor.
Effects of Detergent on Crack Initiation and Propagation in Polyethylenes
105
In this work, the ESC resistance of two high-density polyethylenes obtained using different catalytic systems was compared at varying detergent concentration, in order to obtain some insights into the role played by the molecular structure of the material. EXPERIMENTAL DETAILS Materials Two HDPEs showing different ESC behaviour were kindly supplied by Polimeri Europa SpA (Italy). ESC testing results on notched samples (Bell test) together with some other general physical and mechanical properties are given in Table 1. The polymer synthesis was conducted on a mixture of ethylene with about 1.5% hexene comonomer in the reactor, which resulted in about 1% by weight comonomer in the polymer chain. Different catalytic systems were adopted to obtain different molecular mass distributions (Fig.l). The polymer fraction having higher molecular mass in the case of HDPE-2 is expected to produce thicker and/or more regular crystallites as compared to HDPE-1. The stress-cracking agent used for ESC testing in the present work was a water solution of the industrial detergent nonylphenoxy poly(ethyleneoxy)ethanol (Antarox® CO-630). Concentration was varied between 2% and 100%. Table 1. Physical and mechanical properties of the two polyethyllenes examined Property Density (kg/dm^) MFI2.16kg(g/10') MFI21.6kg(g/10') Yield stress (MPa) Break stress (MPa) Break strain (%) Modulus (GPa) ESCR Bell test (h) IZOD impact (J/m)
HDPE-1 0.9540 0.20 21.1 27.4 32.3 1100 1.05 65 130
HDPE-2 0.9533 0.20 18.6 27.6 27.6 890 1.20 >600 170
Normative ASTMD1505
Conditions 23°C
ASTMD1238
190°C
ASTM D638 ASTMD1693 ASTM D256
23°C, 50 mm/min 23°C, 1 mm/min 50°C 23°C
ESC testing The testing geometry is shown in Fig. 2. Specimens were prepared from compressionmoulded rectangular plates of 170 x 200 mm, which were sawn into 80 x 25 mm samples, about 10 mm thick. The specimens were side-grooved on both faces along the centerline path parallel to the shorter side of the specimen, thus reducing the thickness at the fracture plane to about 70% of the original plate thickness.
M
106
RINKETAL
HDPE-1
4
3
HDPE-2
5
6
Log (molecular mass) Fig. 1. Molecular mass distribution of the two poly ethylenes examined. A sharp starting notch was machined by sliding a razor bade into the material to a relative depth, a/W, varying from 0.45 to 0.6. For testing, specimens were supported on two parallel rollers spaced 55 mm apart, and the load was applied by slowly releasing, via a pneumatic device a dead load ranging from 4 to 18.5 kg. The latter was connected to two upper rollers, which were symmetrically positioned over the notch 18.3 mm apart. The specimen deflection was recorded during the test using a displacement transducer connected to the load-points. displacement transducer
iff 173 I
I
/
, L/3 I
side-grooves I ligament
y.
W
• ^ notch
Fig. 2: Specimen configuration and four-point bending testing geometry. For environmental testing, specimens were wrapped inside a semi-rigid multilayer plastic bag filled with detergent solution. To allow the specimen to bend freely during the test, a folded section was introduced into the central portion of the bag (Fig. 3). Testing was conducted at 50°C and 70°C in air and at 50°C in detergent solution respectively.
Effects of Detergent on Crack Initiation and Propagation in Polyethylenes
107
detergent-filled semi-rigid bag
Fig. 3: Environmental testing arrangement. DATA ANALYSIS Given the applied load and the specimen geometry, the applied stress intensity factor, Kic, can easily be calculated using the following equation [8]: K,=Y
PL
(1)
in which Y is a shape factor, P the applied load, L the support span, B the specimen thickness, W the specimen width and a the crack length. The crack initiation time, ti, for any given applied Kic can easily be identified based on the load-point displacement recorded during the test. The displacement curves obtained from two specimens having blunt and sharp notches are shown in Fig.4. Since the blunt-notched specimen undergoes the same viscoelastic deformation as the sharp-notched specimen, but without crack propagation, the crack initiation time can be identified as the time at which the two curves diverge. Determination of the subsequent crack growth is a somewhat more difficult task: direct optical observation was not possible, due to the wrapping bag. An indirect method based on compliance analysis was therefore adopted. For linear elastic materials, deflection at the load-point of a four-point bending beam is given by two different contributions, accounting for both bending and shear deformation. In the case of notched specimens, a third term accounting for crack length arises. The resulting analytical expression for the specimen compliance C is as follows:
W
EBW
r + - (2 + v) + 0 1 TW l/2 O V / T WI / 81
I — Y^ d{—) TI/ ^ TI/^ W W
J
(2)
in which E and V are the elastic modulus and Poisson's ratio of the material respectively.
M
108
£ E
RINKETAL
/
2 CRACK INmATION
1
E u
"(/IE
1
BLUMT NOTCH
.
0
1
•
1
1
.1
1
.
L
.
1
.
2
1
"
•
3
time (h) Fig. 4: Comparison of the displacement curves obtained on two specimens differing for the notch sharpness (HDPE-1; 23°C;13 kg). Deformation curves in Fig. 4 clearly show that materials cannot be considered linearly elastic, and therefore Eq. 2 cannot be used directly in this form. However, by subtracting from the sharp-notched specimen curve the flexural and shear deformation contributions given by the blunt-notched specimen curve, a curve accounting for the crack length contribution to the specimen compliance can be derived from the data of Fig. 4. Eq. 2 can then be rewritten as follows:
\w] VJ
EBW^ i W
W
(3)
in which Cb is the blunt-notched specimen compliance. Since the compliance curve C (a) is known from experiments, inversion of Eq. 3 gives a straightforward method for calculating the crack length during the propagation stage, provided that the time dependence of the modulus E is known. For comparison purposes, calculations were made using both time-independent and time-dependent moduli. The latter was obtained from the slope on the displacement vs. time curve of the blunt-notched specimen using the conventional linear elastic formula for bending. It was found that, at least within the experimental conditions and materials examined, the two solutions were in agreement within 1%. In the reminder of the work, therefore, the modulus of the material was assumed to be time-independent, which greatly simplifies data reduction. Fig.5 shows the crack length vs. time curve obtained from calculations based on data reported in Fig. 4.
Effects of Detergent on Crack Initiation and Propagation in Polyethylenes
109
0.7
0.65 0.6
0.55 0.5
Fig. 5: Crack length vs. time curve obtained from data of Fig. 4 (line) and data obtained from multispecimen procedure (dots). The crack length values obtained from the compliance method were finally compared with crack length measurements derived from a multispecimen procedure, in which a set of identical specimens was subjected to the same applied stress intensity factor (Kic = 0.48 MPa m^ ^) for different times and then broken open in liquid nitrogen, and excellent agreement was found. RESULTS Non-aggressive environment The values of crack initiation time and crack speed obtained in air at 50° and 70°C on the two materials examined at varying applied stress intensity factor are shown in Figs. 6 and 7 respectively. HDPE-2 crack resistance to initiation appears to be higher than that of HDPE-1 at both temperatures, which may be attributed to the different failure resistance of the materials inside the failure zone at the crack tip. Moreover, the stress intensity factor values decrease for increasing initiation time on both materials, showing a linear trend in the log-log plot. This indicates that a power-law dependence of the stress intensity factor on initiation time is obeyed, which is probably to be attributed to the viscoelastic behaviour of the material surrounding the failure zone. The fact that the slope of the lines (i.e. the exponent of the power-law) is approximately constant at varying material and temperature also indicates that no significant differences in the deformation behaviour are apparent under these conditions. The fracture resistance of the two materials under crack propagation shows essentially the same features as for initiation. The only significant difference is that an increasing trend of the stress intensity factor vs. crack speed data is observed, which in turn was expected based from the viscoelastic behaviour of the materials examined.
M. RINKETAL
no
HDPE-2 (50*>C)
1 0.8 (0
a.
5o
l-HDPE-2 (70°C)
0.6 0.4
a
0.2 0.1
HDPE-1 (50*>C)
HDPE-1 (TO'^G)
0.1
1
10
100
1000
initiation time (h) Fig. 6: Stress intensity factor vs. initiation time data obtained from the two materials in air at two different temperatures '
' Hbp&:i "/""
W
J i f l K ' ' ' HDPE-1
1.1 h
J^
0.6 HDPE-2 ^^jM (50°C),'' j^HJ u iZ" 0.4
^
HDPE-2 ,$^*.JiF
0.2
1 it'iiiMiiK^i I I I mil
10-^ lO"'
I1
10"^ 10-^ 10-^ 10°
10^
crack speed (|im/s) Fig. 7: Stress intensity factor vs. crack speed data obtained from the two materials in air at two different temperatures Detergent ESC testing has been conducted at 50°C only, since at 70°C the detergent was immiscible in water even at very low concentrations. Fig. 8 shows crack initiation times obtained in a 10% detergent/water solution at varying applied stress intensity factor for the two materials examined (filled points). For comparison purposes, data obtained in air at the same temperature are also reported (empty points). For sufficiently high values of the applied stress intensity factor, a linear trend which is quite close to that obtained in the non-aggressive environment is observed.
Effects of Detergent on Crack Initiation and Propagation in Polyethylenes
111
HDPE-2 (10% detergent) HDPE-1 (10% detergent)
HDPE-2 (no detergent)
HDPE-1 (no detergent)
1
10
100
1000
initiation time (h) Fig. 8: Stress intensity factor vs. initiation time data obtained in aggressive environment at 50°C for the two materials (filled points). Data obtained in air are also reported (empty points). However, a critical value is found for the applied stress intensity factor, K ic, below which the environmental stress-cracking agent accelerates fracture initiation: below this 'transition" value, the slope of the stress intensity factor vs. initiation time plot is higher but still similar for the two materials. Fig.s 9 and 10 show results obtained for the propagation stage on HDPE-1 and HDPE2 respectively.
10"^
10-^
10-^
crack speed (|Lim/s) Fig. 9: Stress intensity factor vs. crack speed data obtained for HDPE-1 in aggressive environment at 10% concentration of the detergent
112
M.RINKETAL
^'' no detergent (fitting).
1,1
10 Q. u
1,0 0,8 detergent
0,6 0,4 0,2
10-^
10-^
10-^
10'^
10'
crack speed (iim/s) Fig. 10: Stress intensity factor vs. crack speed data obtained for HDPE-2 in aggressive environment at 10% concentration of the detergent Again, a transition is observed for a certain value of the applied stress intensity factor, below which the crack propagation resistance is much lower than that measured in air. As for initiation, however, the ranking of the two materials in the ESC region remains essentially unchanged. Detergent concentration effects Testing was conducted at 50°C on HDPE-1 with different concentrations of the detergent in the water solution for an applied stress intensity factor of 0.35 MPa m^^^. Initiation times and crack speeds obtained are shown in Figs. 11 and 12 respectively. Results show that, at least for the stress intensity factor value applied, the more pronounced effect of the ESC agent occurs for a detergent concentration of about 40% by volume. DISCUSSION Results obtained in air at varying temperature indicate that the fracture mechanism is very similar for the two materials, the main difference consisting in the stress intensity factor value required to initiate fracture at a given time and to propagate it at a given speed. The same behaviour is observed in an aggressive environment, where the ranking of the two materials is maintained even below the critical value for the diffusion-controlled "transition" in the fracture behaviour. This observation is in qualitative agreement with the ranking of the two materials given by ESCR data presented in Tab. 1, and it is probably to be attributed to the influence of the tie molecules, which is higher for HDPE-2 due to its larger fraction of molecules having high molecular mass.
Effects of Detergent on Crack Initiation and Propagation in Polyethylenes
0
20
40
60
80
113
100
detergent concentration (%) Fig. 11: Initiation time at varying detergent concentration for HDPE-1
0
20
40
60
80
100
detergent concentration (%) Fig. 12: Crack speed at varying detergent concentration for HDPE-1 Initiation time and propagation rate turned out to be greatly accelerated in aggressive environment, provided that sufficient time was given to the detergent to diffuse inside the craze zone ahead of the crack tip. However, even under diffusion-controlled conditions, the time dependence of the fracture process is essentially the same for the two materials. This indicates that the chemical structure of the macromolecules probably dominates over possible time-dependent deformation mechanisms associated with molecular mass distribution. This is not surprising, since higher molecular mass molecules are expected to increase the fracture resistance by increasing the tie molecules, but not to affect the entanglement density in the amorphous phase between crystallites, which governs the time-dependence of the deformation before fracture occurs.
114
M. JUNKETAL
Finally, as for the effect of detergent concentration in the water solution for HDPE-1 al 50°C, an decrease in the fracture resistance was observed for increasing detergent concentration until about 40%. This may be attributed to micellar structures that are formed in the solution around small hydrocarbon impurities, as suggested by Tonyali [3]. These structures are easily absorbed by the fibrillated material inside the process zone ahead of the crack tip, resulting in plasticization of the material and subsequent significant acceleration of the fracture process. Above the limiting concentration of about 40%, however, the fracture resistance was observed to increase for increasing detergent concentration. This opposite trend observed at higher concentrations may be attributed to a decrease of the number of micellar structures, due to the rise in viscosit\ of the solution or to some limiting concentration effects. The increased viscosity ma\ also reduce the diffusion rate at the crack tip. Another possible explanation is thai material plasticization may induce excessive blunting at the crack tip, which in turn may imply crack branching. None of these possible mechanisms could be assessed in the present work, and interpretation of the results calls for further investigation. CONCLUSIONS A straightforward experimental procedure for determining ESCR in polymers using fracture mechanics has been set-up. Comparison between the results obtained in air and in detergent shows that aggressive environment affects the crack resistance only below a certain "critical" value, K ic. This is probably to be attributed to a diffusion-controlled plasticization mechanism, which requires times larger than a certain "critical" time (for crack initiation) or crack speeds lower than a certain "critical" speed (for crack propagation) to be activated. This assumption is confirmed by literature data. A significant concentration effect has been evidenced which could not be explained simply by detergent diffusion. A possible interpretation comes from a model of micellae formation around hydrocarbon particles that may be present into the water solution. REFERENCES [1] Lustiger, A. (1986) in "Failure of Plastics" Brostow and Corneliussen Eds., Hanser, New York. [2] Lustiger, A. and Corneliussen, R.D. (1987) J. Mater. Sci. 22, 2470. [3] Tonyali, K., Rogers, C.E. and Brown, H.R. (1989) Polymer 28, 1472. [4] Ward, A.L., Lu, X., Huang, Y. and Brown, N. (1991) Polymer 32, 12. [5] Lu, X., Qian, R. and Brown, N. (1991) J. Mater. Sci. 26,917. [6] Williams, J.G. and Marshall, G:P. (1975) Proc. Roy. Soc. Lond. A342, 55. [7] Chan, M.K.V. and Williams, J.G. (1983) Polymer 24,234. [8] Brown, W.F. and Srawley, J.E. (1969) "Plane Strain Crack Toughness Testing of High Strength Metallic Materials" ASTM-STP 410.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
115
THE ENVIRONMENTAL STRESS CRACKING OF A PBT/PBA CO-POLY(ESTER ESTER) N.B. Kuipers^*\ A.C. Riemslag^\ R.F.M. Lange^\ M. Janssen^\ R. Marissen^\ K. Dijkstra^\ A. Bakker^\ Department of Materials Science, Delft University of Technology, Rotterdamseweg 137,2628 AL Delft, The Netherlands nancy .kuipers @ tnw.tudelft.nl ^ Department of Aerospace Engineering, Delft University of Technology, The Netherlands ^ DSM Research, Polymeric Construction Materials, P.O. Box 18, 6160 MD Geleen, The Netherlands ABSTRACT Preliminary research has indicated that a PBT/PBA co-poly(ester ester) is susceptible to environmental stress cracking in water and in phosphoric acid solution, in both cases at 80°C. Time-to-Failure creep experiments were initiated to obtain quantitative data. These tests were performed in water and phosphoric acid solutions (pH = 1.6) at 80°C with notched tensile specimens under constant load (ranging from 0.6-7 MPa). The results have shown that the phosphoric acid solution decreases the lifetime when compared to tests done in water. Both environments decrease the lifetime tremendously when compared to creep tests in air. Comparing the results with the influence of that only hydrolysis on its own has on the degradation of mechanical properties has showed that the results of the Time-toFailure creep experiments cannot be explained exclusively on the basis of hydrolysis. This confirms the conclusion drawn in our preliminary experiments that the PBT/PBA co-poly(ester ester) is susceptible to stress cracking in both water and phosphoric solution at 80°C. The influence of water on the environmental stress cracking of the PBT/PBA copoly(ester ester) can be of a physical and /or chemical nature. The influence of acid, however, must be chemical because just having a small quantity of acid (11.4 g/1) in water will not influence the physical properties but it will decrease the time to failure up to a factor of 10 when compared to pure water. KEYWORDS Environmental Stress Cracking (ESC), Thermoplastic Elastomer (TPE), Creep, Timeto-Failure, Hydrolysis, Co-poly(ester ester), Poly(butylene terephthalate), Poly(butylene adipate).
116
N.B. KUIPERS ETAL
INTRODUCTION The Environmental Stress Cracking (ESC) of polymer materials can be can give rise to the sudden and unpredicted failure of constructions. The chance of ESC occuring can easily be overlooked because neither loading nor the environment factor in isolation can be termed damaging to the material, but the two in combination is fatal. Knowledge of ESC mechanisms is important if one is to succesfully prevent premature failure and economic loss. In metal alloys the combination of stress and environment can also lead to premature failures, indicated as Stress Corrosion Cracking, SCC [1]. The influence of the environment on SCC is generally of a chemical nature; a chemical reaction occurs between the metal and the environment. Most of the research published on the ESC of polymers focuses on ESC in which the environment influences the material only physically [2-8]. In such cases the mechanism of ESC is studied and models are established for ESC prediction [9]. These models for physical ESC are based predominantly on the solubility parameters of the considered polymer/environment combination. In other words, ESC is mainly a consequence of polymer softening, i.e. it is a reduction of the interaction between the polymer chains that lowers the yield stress. The aim of this paper is to consider ESC under not only the physical but also the chemical influence of the environment. In this case the chemical reactions (for example resulting in chain scission) are important and models based on solubility parameters are not valid. Hardly any literature is available on this subject. Moskala [10] reported chemical ESC of poly (ethylene terephthalate) (PET) in an aqueous sodium hydroxide (NaOH) solution. A discontinuous crack growth with an increased crack growth rate was found upon increasing NaOH concentration. However, one major drawback is that the effect of hydrolysis without external stress was not considered. Hydrolysis on its own can potentially lead to premature failures, whether or not the material is in a stressed or unstressed condition. This paper proves the (partly) chemical ESC of a PBT/PBA copoly(ester ester) in water and in phosphoric acid solution, by showing that there is a mutual intensification of the destructive effect of loading and environment. Therefore the separated and combined influences of loads and environments will be compared so that hydrolysis can be distinguished from ESC. Furthermore, the influence of water will be compared with the influence of phosphoric acid solution, both with and without load. This last comparison leads toflieconclusion that the ESC of PBT/PB A copoly(ester ester) in phosphoric acid solution is mainly of a chemical nature. The influence of the load on the time to failure and the failure mechanism will be shown. The failure mechanisms are interpreted through SEM images of fracture surfaces.
The Environmental Stress Cracking of a PBT/PBA Co-poly(ester ester)
117
MATERIAL, HYDROLYSIS AND SPECIMENS Material and hydrolysis Copoly(ester ester)s belong to the family of thermoplastic elastomers (TPEs) and consist in general of thermo-reversible hard and elastic soft domains [11]. The copoly(ester ester) used here consists of 60% poly(butylene terephthalate), 35% poly(butylene adipate) and 5% 4,4'-methylenebis(phenyl isocyanate), and shows domain sizes of about 20 nm [12]. The material possesses a rubber plateau between the glass transition temperature of the mixed amorphous PB A/PBT phase (the PET phase is semi-crystalline) at about -30°C and the melting point of the PET at about 220°C. Due to the vulnerability of the amorphous PEA/PET soft domains towards water attack [13] the PET/PEA copoly(ester ester) is used here to study the existence of ESC of a chemical rather than a physical nature. For the sake of clarity it should be emphasized that no additives have been used in the copoly(ester ester) described here. It is a well known fact that water can hydrolyse polyesters, especially in the presence of acid and at elevated temperatures. To distinguish hydrolysis from ESC, the separate influence (i.e. without external stress) of hydrolysis on the mechanical properties is determined by measuring the fracture stress as a function of the exposition time. The specimens (for dimensions: see next paragraph) are immersed in water or phosphoric acid solution (pH=1.6), both at 80°C, for a certain time and this is followed by a tensile test (after drying) to measure the fracture stress. The specimens are tensile tested at a speed of 500 mm/min, according to ASTM standard D412-98a [14], on an Instron 5500R tensile testing machine with a load cell of 1 kN. Figure 1 shows the (nominal) fracture stress of the notched specimen as a function of the exposition time in water or phosphoric acid solution, both at 80°C.
10
s
50% of initial fracture stress
5
—•— water -o A
acid air
o
0
276 240 hours 120 Exposition time (hours)
Fig. 1: Nominal fracture stress of notched specimens as a function of pre-exposition time for water and a phosphoric acid solution (pH = 1.6) at 80°C. The time to hydrolysis, taken as the time corresponding to a 50% reduction in the fracture stress, is 276 (+/-12) hours for both water and phosphoric acid solution
118
KB. KUIPERS ETAL
(pH = 1.6). An increased hydrolysis rate of the copoly(ester ester) in phosphoric acid when compared to water may be expected but that was not observed, possibly due to the autocatalytic (a catalyst is produced during the chemical reaction) nature of ester hydrolysis. Monitoring the absorption of water and a phosphoric acid solution both at 80°C via the increase in weight shows that after 8 hours the maximum weight is reached in both fluids. This indicates the saturation of the specimen with the fluid. So after 8 hours the specimen is vulnerable to hydrolysis over the total cross-section. Specimens The PBT/PBA copoly(ester ester) was injection molded into dogbone tensile specimens. The cross-section in the tapered area of the specimen is 4 mm x 10 mm. The gauge length is 50 nrni. The middle of the specimens is notched at both sides by inserting a razor blade (American safety single edged blades, thickness 1.0 mm). The razor blade is inserted by means of a mechanical testing machine at low speed (0.20 mm/min) to minimize the introduction of internal stresses in the specimen. For every notch a new razor blade is used to make sure that every notch has the same sharpness. This notching procedure is carried out according to the notch method used for polyethylene specimens, ASTM F1473 [15]. The notches (2.5 nun) reduce the cross-section of the specimen by 50% (from 40 mm^ to 20 nun^). ESC EXPERIMENTS Time-to-failure (TTF) tests on notched specimens are used to characterize environmental stress cracking as a function of load level and environment. The TTF is the total period of time needed for a crack to initiate and grow before final failure occurs. Usually -for a polymer in air- the stress as a function of the logarithm of the TTF is a knee-shaped curve with different slopes before and after the ductile-brittle transition. An environment may have little or no effect when the stress is high and the failure time is short, while there might be a significant reduction in TTF at lower stresses accompanied by longer fracture times. Therefore the duration of the test must be long enough to ensure that this transition (from creep conditions without significant environmental influence to ESC) can be detected. The notching of specimens and the testing at elevated temperatures may shift this transition to shorter times [2]. The results of the TTF experiments are of course dependent on specimen geometry and thus do not produce an intrinsic material property. Therefore comparisons between TTF results will only be valid in experiments that use identical specimens. However, TTF experiments are shown to discriminate, to a high resolution, the ESC resistance of polymer/fluid combinations [16]. The applied nominal stress, i.e. the load divided by the remote unnotched crosssection of 40 nun^, is 0.65 - 7 MPa. Measurements are performed in air, demineralised water and phosphoric acid solution (pH=1.6). All tests are performed at
The Environmental Stress Cracking of a PBT/PBA Co-poly(ester ester) 119 SC'C. A raised temperature is used to accelerate the test and to also reflect the service conditions that PBT/PBA copoly(ester ester)s normally experience. The experimental set-up is depicted in Fig. 2.
motor
c
QTUn IZZZHB %
timer top side of oven upper grip (movable) specimen glass container
signal
y umi\ IBG
IZZ^S-
medium lower grip (fixed) bottom of oven level indicator constant load
Fig. 2: A schematic view of the set-up for time-to-failure tests in a medium (water or phosphoric acid solution) in an oven. The clamped specimen is placed in a glass container and the container is filled with the medium (water or phosphoric acid solution). The upper medium level is above the upper grip. A flexible top covers the container to minimize vaporization of the medium. The container with the clamped specimen and medium is placed in an oven. During the test a motor corrects the position of the upper grip to compensate for specimen elongation. A timer monitors the time to failure.
120
N.B. KUIPERSETAL
RESULTS AND DISCUSSION Results of ESC experiments The results of the ESC experiments obtained with the notched specimen in air, water and phosphoric acid solution are depicted in Fig. 3. The TTF is plotted against the nominal stress, and the dashed lines at 8 and 276 hours denote the respective saturation and the time to hydrolysis (50% chemical degradation; see "Material and Hydrolysis"). The time to hydrolysis and the TTF cannot be directiy compared because they are the result of two totally different test methods, i.e. a tensile test and a constant-load test. However the fracture stress as a function of the exposition time shows a sharp decline around the time to hydrolysis, while there is hardly any influence at shorter exposition times. Therefore a significant influence of only hydrolysis on the TTF can only be expected around and after the time to hydrolysis.
10 100 Time to fracture [hours]
10000
Fig. 3: The nominal stress versus the time to failure in air, water and a phosphoric acid solution (pH = 1.6) at 80°C. Note that the local stress is much higher due to the notch. "2" denotes that there are two measurements at the given points, which overlap.
The tests in air with loads ranging from 0.65 to 3.75 MPa are stopped before failure after 6620 hours (no longer within the range of Fig. 3) because of the long duration of the tests.
The Environmental Stress Cracking of a PBT/PBA Co-poly(ester ester)
121
ESC and the influence of environment on JTF The influence of water and the phosphoric acid solution becomes clearly demonstrated when one compares the TTF results obtained in air, water and a phosphoric acid solution,. At low stresses (0.6 MPa), hydrolysis, i.e. chemical bulk degradation of the copoly(ester ester) by water and the phosphoric acid solution, is responsible for the failure. However, at higher stress levels, the significant difference in TTF between the aqueous environment and air points to the existence of ESC. The reduced lifetime in the phosphoric acid solution compared to water suggests an increased effect of acid. Physical ESC At the high stress levels, 6 to 7 MPa, there is almost no difference between the TTF in water and phosphoric acid solution. There is, however, a significant difference in TTF between the aqueous environment and air. At 6 MPa the TTF in the fluids is less than 5% of the TTF in air. This is a considerable difference but it corresponds with only a small (less than 1 MPa) decrease in tensile strength. The faster ductile failure in water/phosphoric acid solution is thought to be due to swelling, and thus making it a physical (ESC) process. The failure times in water and phosphoric acid solution are comparable. Apparently, the physical factor is the same in both water and phosphoric acid solution, as already indicated by the similar saturation times and degradation times (Fig. 1). Chemical ESC At lower stress levels, 3-5 MPa, the mean TTF is lowered by 25-40% in the phosphoric acid solution compared to water although the physical influence of both the water and phosphoric acid solutions is about the same. This is a first indication of the presence of chemical ESC, but in this region the scatter is considerable. A definite proof of the existence of chemical ESC is obtained at lower stress levels, especially from 1.6 to 2.5 MPa, where the results show less scatter and where the acid clearly decreases the TTF compared to water. The influence of acid decreases as the load decreases: from 2.5 to 0.6 MPa the ratio of TTF in water and acid decreases from about 10 to 1.25. A load of 0.6 MPa results in a TTF of 315 hours in water and 246 hours in phosphoric acid solution. These times are comparable to the 276 hours after which the failure stress is halved due to hydrolysis so we conclude that the failures obtained at this low stress of 0.6 MPa are only due to bulk degradation. To sunmiarize, the hydrolysis tests show that without load the degradation in phosphoric acid solution is comparable to that in water (Fig. 1). Clearly, the influence of acid is larger if the specimen is loaded (Fig. 3, acid decreases the mean TTF by a factor 10 to 1.25 compared to water for nominal stresses below 6 MPa). This, in combination with the fact that the physical influence of water and phosphoric acid solution is the same (Fig. 3 at 6 and 7 MPa), proves that the environmental stress cracking in phosphoric acid solution has a strong chemical nature.
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N.B. KUIPERS ETAL
Dependence ofTTF on applied stress Figure 3 can be divided into different regions based on the relation between the TTF and the applied nominal stress. These regions are listed in table 1. The failure appearance is also listed in this table and will be further discussed at the end of this chapter. I'able 1: Three different ESC regions and failure mechanisms.
Uepm ', :.;\ I II
m
[MPa] 0.6 - 2.5
3-4 5-7
Fimk^;^^i^m^^^ tBrittle Brittle with blunting Ductile
The average results of regions I and III are represented by a linear fit in Fig. 3. The middle section with an apparent lifetime inversion is roughly matched with a curved line through the data, connecting the ends of both other fits. Regions I and III resemble the well-known knee-shaped curves [2] of thermoplastic polymers in air with different slopes at high (region III) and low stress levels (region I). The short failure times at high stress levels are commonly ascribed to the mechanism of ductile creep failure of the complete ligament. The long failure times at low stress levels are commonly related to the mechanism of crack initiation, subsequent stable crack growth and final ductile failure of the remaining ligament. In most cases, the process of stable crack growth does not involve macroscopic deformations, resulting in the general classification of "brittle" failure. This is in accordance with the failure mechanisms found (see Fig. 5). A deviation from the well-known knee-shaped curves [2] is found in region II where higher stress levels result in longer failure times. For example, all four measurements at 5 MPa in water are higher (TTF 93-316 hours) than those at 3.5 MPa (TTF 68-90 hours). We identify this as "hfetime inversion". Lu and Brown found a similar lifetime inversion after the transition from ductile to brittle failure in the creep tests of notched polyethylene copolymer specimens [17]. Their explanation for the observed lifetime inversion is that there is a blunting of the notch due to creep, which reduces the stress concentration so that crack growth is impeded. This is also a possible explanation for the lifetime inversion shown in Fig. 3. An investigation of the fracture surfaces using the Scanning Electron Microscope (SEM) reveals relatively large fibrils at the notch tip of specimens in the "lifetime inversion region" (region II) while these relatively large fibrils are not present in region I. For example Fig. 4 is a SEM image that shows relatively large (up to about 1 mm) fibrils at the notch tip of a specimen tested in water at 3.5 MPa (within the lifetime inversion region) while Fig. 5, a SEM image of the fracture surface of an ESC specimen in phosphoric acid solution at 2.2 MPa (within region I) does not show any plastic deformation.
The Environmental Stress Cracking of a PBT/PBA Co-poly(ester ester)
123
Fig. 4: Fracture surface of an ESC specimen in water at 3.5 MPa. The arrow indicates the crack growth direction.
Fig. 5: Fracture surface of an ESC specimen tested in phosphoric acid solution at 2.2 MPa. Crack growth from the right to the left side of the photo. The arrows indicate the end of the first five bands. The fibrils (as seen in Fig. 4) are the remains of blunting at the notch tip. The formation of fibrils is also an indication of work hardening due to orientation. Strain induced crystallization may even further enhance this work hardening. The competition between crack growth and blunting-related mechanisms that impede crack growth is a plausible explanation for the lifetime inversion in region II.
124
N.B. KUIPERSETAL
Discontinuous crack growth Bands are visible at the brittle fracture surfaces (see for example Fig. 4 and Fig. 5). This is a strong indication that there is discontinuous crack growth. Possible explanations for discontinuous crack growth are the forming of crazes created and broken one after another and/or the stronger influence of the medium within the yielded zone leading to fracture of (part of) this zone. The band size at 2.2 MPa is approximately 0.14 mm, which could correspond with crazes/yielded zones of the same length or longer if they are only partly broken. The Dugdale strip yield model [18] gives an appropriate description of crazes. According to this model, the length of the yielded strip (rp) is given by:
p-81—I
0)
with Gys = yield strength = 9.8 MPa (at 80°C) for this copoly(ester ester) and Ki is the stress intensity factor (18) under tensile loading, which for a double notched tensile specimen is: Kj = Cc^m
(2)
with: a = crack length, a = nominal stress and C = geometry factor (18): .122-1.122[-|)-0.820[-|] +3.768(~] -3.04o[-|] (3)
with: W = specimen width. Ki = 0.23 MPaVm according to equation 2 for a = 2.2MPa and notch size = a = 2.5 mm. This results in a yielded strip size of 0.22 nmi according to equation 3. This value, of namely 0.14 mm, is much higher than the measured band size. Possible explanations for this discrepancy are the fact that the material is not behaving in a linear elastic manner and/or that only a part of the craze/yielded zone breaks at once. The rapid failure as seen in water and acid solution in regions I and II are due to chemical ESC, as discussed before. A plausible explanation of chemical ESC is chain scission caused by the combined action of the high mechanical stress at the (notch) crack tip and chemical attack of the chains due to local hydrolysis.
The Environmental Stress Cracking of a PBT/PBA Co-poly(ester ester)
125
CONCLUSIONS The environmental stress cracking of a PBT/PBA copoly(ester ester) in water and phosphoric acid solution (pH = 1.6) at 80°C is demonstrated by comparing the combined influence of mechanical load and environment with the separate individual influences. The ESC in phosphoric acid is both physical and chemical. The physical component is dominant at relatively high loads of 6-7 MPa which corresponds with short failure times of up to 3 hours. The chemical component is dominant when the failure times are long because of lower loads. The phosphoric acid does not change the physical ESC behaviour significantly when compared to water, but it does accelerate the chemical ESC. This leads to a decrease of the mean time to failure from a factor of 1.25 at 0.6 MPa to a factor of 10 at 5 MPa compared to water. The time-to-failure (TTF) curve as a function of applied nominal stress can be divided into three regions with different relations between TTF and applied stress: I. 0.6 - 2.5 MPa, TTF increases as expected as stress decreases; linear correlation between stress and the logarithm of TTF; II. 3 - 4 MPa, "lifetime inversion area"; large scatter in measurements; TTF increases as stress increases due to blunting and orientation; III 5 - 7 MPa, TTF increases as expected with decreasing stress; linear correlation between stress and the logarithm of TTF; slope of linear fit smaller than in region I. To sum up then, it is demonstrated that environmental stress cracking is not necessarily a physical process, as is generally assumed, but that it can also be a chemical process. ACKNOWLEGDEMENTS The authors would like to acknowledge the fruitful discussions with many colleagues at DSM Research and thank the following persons for their experimental work: A.C. LambrechtSy M.H,E. Janssen, M.J.M. Groote Schaarsberg, C. Louter, K. Verkaik and DJ.L Rijsdijk.
126
REFERENCES
N.B.KUIPERSETAL
1. Craig, B. (1987) Environmentally induced cracking, in Metals Handbook 13 Corrosion, ASM International pi45. 2. Wright, D. (1996) Environmental Stress Cracking of Plastics, Rapra, Shropshire, UK. 3. Breen, J. (1993) Environmental stress cracking of PVC and PVC-CPE, part I, Crazing, J. of Materials Science 28, 3769-3776. 4. Breen, J. (1994) Environmental stress cracking of PVC and PVC-CPE, part II, Failure mechanisms, J. of Materials Science 29, 39-46. 5. Breen, J. (1995) Environmental stress cracking of PVC and PVC-CPE, part III, Crack growth, J. of Materials Science 30, 5833-5840. 6. Yang, A.C.M., Jou, E.C.Y., Chang, Y.L. and Jou, J.H. (1995) The solvent-induced cracking in glassy polymer coatings by atomic force microscopy. Mat. Chem. and Physics 42, 220-224. 7. Arnold, J.C. (1995) The influence of liquid uptake on environmental stress cracking of glassy polymers. Mat. Sc. andEng. A197,119-124. 8. Arnold, J.C, Li, J. and Isaac D.H. (1996) The effects of pre-immersion in hostile environments on the ESC behaviour of urethane-acrylic polymers, J. of Mat. Processing Technology 56, 126-135. 9. Hansen, CM. and Just, L. (2001) Prediction of Environmental Stress Cracking in Plastics with Hansen Solubility Parameters, Ind. Eng. Chem. Res. 40, 21-25. 10. Moskala, E.J. (1998) A fracture mechanics approach to environmental stress cracking in poly(ethyleneterephtalate). Polymer 39-3, 675-680. 11. Adams, R.K., Hoeschele, G.K. and Witsiepe W.K. (1996) in Thermoplastic Elastomers; Holden, G., Legge, N.R., Quirk, R. and Schroeder, H.E., Hanser Munich pl91. 12. van Berkel, R.W.M., Borggreve, R.J.M., van der Sluis, CL. and Werumeus Buning, G.H. (1997) in Handbook of Thermoplastics, Olabisi, O., Marcel Dekker, NYp397. 13. March, J. (1985) Advanced Organic Chemistry; reactions, mechanisms and structure, Wiley-Interscience, New York p292. 14. Standard Test Method for Vulcanized Rubber and Thermoplastic ElastomersTension, ASTM standard D412-98a 15. Standard Test Method for Notch Tensile Test to Measure the Resistance to Slow Crack Growth of Polyethylene Pipes and Resins, ASTM F1473-01. 16. Hough M.C and Wright, D.C (1996) Two new test methods for assessing environmental stress cracking of amorphous thermoplastics. Polymer Testing 15, 407-421. 17. Lu, X. and Brown, N. (1990) The ductile-brittle transition in a polyethylene copolymer, /. of Mat. Sc. 25, 29-34. 18. Janssen, M., Zuidema, J., Ewalds, H.L. and Wanhill, R.J.H. (2002) Fracture Mechanics, DUP, Delft, Netherlands. For further information on the Environmental Stress Cracking of a PBT/PB A copoly (ester ester): it is expected that a second article will be published in "Polymer Engineering & Science" in the year 2003.
1.4 Rate Effects
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
129
A NEW WAY FOR POLYMER CHARACTERISATION USING A COMBINED APPROACH LEFM - PLASTIC ZONE CORRECTED LEFM *C. Grein, ^Ph. Beguelin, ^H.-H. Kausch * Borealis GmbH, St.-Peter StraBe 25, A-4021 Linz, Austria ^ Laboratoire de Polymeres, DMX, Ecole Polytechnique Federale de Lausanne, 1015 Ecublens, Switzerland
ABSTRACT Toughness quantification of polymers is still an open issue, since no unique fracture mechanics method is able to provide intrinsic crack propagation resistance values over the whole range of material behaviours from brittle to ductile. A sensitive way to characterise the mechanical performance of these materials, exemplified with rubber modified isotactic polypropylenes, is to determine their ductile-brittle transitions as key-factor for material ranking, A conventional Linear Elastic Fracture Mechanics (LEFM) is often used for data reduction. Although being strictly reserved for brittle fracture assessment, it constitutes a guaranty for continuity of analysis over several decades of test speeds. Nevertheless, material descriptors such as K, the stress intensity factor, provided by this method are geometry dependent in the ductile range. To overcome this difficulty, an original approach, which takes into account the high amount of plasticity in tough polymers, is proposed. It consists of determining experimentally the size of the plastic zone developed at a crack tip using specimens of different ligament lengths. It was shown to provide intrinsic values over a large range of test conditions, except near the ductilebrittle transitions. A combined approach "conventional LEFM-plastic zone corrected LEFM" is therefore necessary to apprehend accurately the fracture behaviour of the studied blends.
KEYWORDS: iPP/EPR blends, ductile-brittle transition, toughness evaluation, LEFM, plastic zone correction, intrinsic parameters
130
C. GREIN, PH. BEGUELINANDH.-H. KAUSCH
INTRODUCTION Toughness assessment of ductile polymers is still a matter of debate. A sensitive way to characterise the mechanical performance of these materials, and to rank them, is to determine their ductile-brittle transitions. Test speed can thus be varied over several decades of test speed, while keeping the temperature constant, or a wide range of temperature can be scanned in controlled steps at given velocity. In the first case, the higher the speed at which the tough-tobrittle transition occurred, the better the grade in terms of fracture resistance. In the latter case, the lower the temperature at which the brittle-to-ductile transition occurred, the more suited the material for impact applications. Additionnally, raw data are often analysed using a single fracture mechanics approach. Although being a guaranty for analysis continuity, it doesn't provide intrinsic crack propagation resistance values over the whole range of material behaviours, since each approacli has a limited field of application: Linear Elastic Fracture Mechanics (LEFM) is de\ oted to brittle failure modes, J-integral and Essential Work of Fracture to more ductile ones. In this study, we will show how to combine conventional LEFM with plastic zone corrected LEFM, a method which takes into account the high amount of plasticity in tough polymers, to get (i) indisputable material ranking and (ii) geometry independent materials descriptors as well in the ductile as in the brittle range. To clarify the purpose of this paper, the analysis will be restricted to the stress intensity factor, K, although the proposed plastic zone correction works also with the energy release rate, G. Our approach will be exemplified with two ethylene-propylene rubber toughened isotactic polypropylene blends (iPP/EPR). EXPERIMENTALS MATERIALS The main features of both studied iPP/EPR blends are summarised in Table 1. They differ mainly by their rubber content, iPP/EPR-1 having been obtained by dilution with neat matrix of grade iPP/EPR-2. The amount of rubbery phase was provided by the material supplier (Atofina). Mn and M^ of the xylene insoluble fractions (corresponding roughly to the matrix) were evaluated by Gel Permeation Chromatography (GPC). The degree of crystallinity was measured by Differential Scanning Calorimetry (DSC) assuming that a 100% crystalline iPP has a melt enthalpy of about 207 J/g (in the second heat of+10/-10/+10 K/min scans). Details about the morphological features, the mechanical performances and the microdeformation mechanics of the investigated grades can be found in Ref. [1] (the grades which corresponds to iPP/EPR-1 is Ddu, to iPP/EPR-2 is D).
iPP/EPR-l iPP/EPR-2
Rubber content M„-matrix Mw-matrix MWD-matrix rwt%i [kg/mol] fkg/mol] r-1 15 46.5 237 5.1 30 38 240 6.3 Table 1. Some characteristics of the investigated materials.
Crystallinity
r%i 35 34
131
A New Way for Polymer Characterisation
EXPERIMENTAL SET-UP The mechanical tests were carried out on Compact Tension (CT) specimens (24 x 24 x 4 mm^). The crack length, a, was varied between 10 and 18 mm, corresponding to a variation of a/W of between 0.3 and 0.7 (with W the length of the CT sample). The specimens were deformed in mode I using a servohydraulic high-speed tensile test machine at crosshead speeds between 0.0001 and 10 m/s. All or part of the test speed range was investigated. At speed higher than 0.1 m/s, a damper was used to reduce the dynamic acceleration of the sample. It should therefore be recalled that the results reflect in this speed range material properties at high loading rate rather than the behaviour associated with dynamic impact testing. The displacement was measured optically by detecting the position of a laser beam emitted from an optical fibre located at the moving part of the tested specimens [2,3]. TYPICAL FORCE-DISPLACEMENT CURVES Typical force-displacement (F-d) are shown in Fig. l(i). With decreasing speed (or increasing temperature) four material behaviours were observed: (i) a brittle behaviour. The F-d curves were linear elastic assessing for deformation mechanisms not having been initiated before failure. The fracture surface were very smooth and mirror-like. At the microscopic scale, they are believed to be associated with the development of a single crack. (ii) a semi-brittle behaviour with pronounced non-linearity prior to unstable crack propagation. Fracture occurred before the critical stress to flow (yield stress) could be reached. The fracture surfaces were rough, probably composed of several planes of macrocracks. Stress-whitening, accounting for changes in the refractive index of the material (and thus to voiding in form of matrix crazing or particle cavitation), was not visible; (iii) a semi-ductile behaviour with initiation and partial development of the damage mechanisms before unstable fracture. Near the crack tip, the fracture surface was fully whitened accounting for the limited stable growth of the crack (the extent of whitening decreased with increasing test speed); far from the crack tip, it was rough like in the semi-brittle case; (iv) a ductile behaviour characterised by stable crack propagation and entire stresswhitening of the fracture surface.
250] F I N ] /
Izi
15o|
1
lOOl
nFlN]
lOlFlN]
50lF[Nl
(i)
200| /
'^
501
O'l
0
1
2
Displacement (mm)
3
0
1
2
Displacement [mm]
3
0
2
4
Displacement [mm]
6
0
5
10
15
Displacement [mm]
Fig. 1. Force-displacement curves corresponding to elementary materiaPs behaviours: (i) brittle, (ii) semi- brittle, (iii) semi- ductile and (iv) ductile.
20
132
C GREIN, PH. BEGUELINAND H.-H. KAUSCH
DATA REDUCTION AND ANALYSIS LEFM was used for data reduction. The classical approach and an original method for estimating the size of the plastic zone developed at the crack tip are presented latter. The fracture toughness, K, which describes the intensity of the stress field at a crack tip was chosen to characterise the studied materials. ASSESSMENT OF DUCTILE-BRITTLE TRANSITIONS USING THE LEFM APPROACH DETERMINATION OF K The stress intensity factor in mode I, Ki, was calculated from: K,=f(a/W)
F
(eq. 1)
B4W
with Ki = Kic, the critical stress intensity factor, when the LEFM fracture criteria are satisfied and K = Kimax when the global linearity criteria of LEFM is macroscopically not satisfied. Fmax is the maximun of force on the force-displacement curves; B, the thickness of the specimen; W, its width; f(a/W) a function depending on the specimen characteristics, available in tables for all standard geometries. A UNIQUE DUCTILE-BRITTLE TRANSITION All experiments in this section were performed with a constant sample geometry of a/W = 0.5. As obvious from the section "typical force-displacement curves'\ it was difficult to define unambiguously a single brittle-ductile transition: four elementary materials behaviours involve three distinct ductile-brittle transitions! K , „ , , [MPa.mi/2]
i i
5 4 3 ]
g
i
2
S I D iPP/EPR-1
1
0.00001 0.0001
•
0.001
0.01
iPP/EPR-2
100
V [m/s] Fig. 2. Evolution of the apparent toughness, Kimax, with the logarithm of the test speed, v, at room temperature for iPP/EPR-1 and iPP/EPR-2. The arrows indicate the test speed at which the ductile-brittle transitions occur.
Selecting the "ductile - semi-ductile" transition as ductile-brittle transition was attempting but not realistic: it lead admittedly to substancial reduction of the global fracture energy, but it is known to be highly geometry. Chosing between the "semi-ductile - semi-brittle" transition or
133
A New Way for Polymer Characterisation
the "semi-brittle - brittle" transition was not possible because both of them could lead to a lowering of Kimax, the fracture toughness. We therefore prefer to define the transition in terms of the evolution of Kimax as a function of the test speed: it was taken to occur at the speed where Kimax passed its maximum. It reflects therefore a decrease in the initiation of plastic deformation. As obvious from Fig. 2, the ductile-brittle transition of iPP-EPR-1 occurs at about 0.6 m/s, that of iPP/EPR-2 at about 7 m/s. As expected, the grade with the highest modifier fraction was that with the highest toughness. LIMITATIONS OF THE CONVENTIONAL LEFM APPROACH The LEFM approach have been conceived for brittle materials exhibiting unstable crack propagation. The obtained K-values are considered to be intrinsic properties when [4,5]: (i) they are independent of the crack length of the specimen; (ii) the specimens have been tested in plane strain conditions to assure the most conservative values for the parameters which describe the resistance to propagation. In practice, one considers this condition satisfied if: B,a,W-a>2,5\
(eq. 2)
with B, the thickness of the specimen; W, the width of the specimen; a, the ligament length; and Gy, the yield stress obtained for comparable specific times than Kic. Both criteria are exemplified in Table 2 and 3 for iPP/EPR-1 tested at room temperature. Table 2 shows (i) to be violated when the mode of failure is ductile (i.e at 0.001 m/s), whereas it remains valid, as expected, in case of brittle fracture (i.e 6 m/s). Table 3 highlights that plane stress conditions prevail roughly up to speeds higher than one decade of test speed tthan the ductile-brittle transition. 0.7 0.4 0.5 0.6 0.3 [-] 3.1 2.75 2.5 [MPa.m''^] 3.7 3.45 2.4 2.6 [MPa.m'T 2.6 2.6 2.5 Table 2. Evolution of the apparent toughness, Kimax, versus a geometry factor, a/W, for iPP/EPR-1 tested at 0.001 m/s and 6 m/s at room temperature.
aAV Kimax-0.001 m/s Kimax-6lIl/S
[m/s] 1 3 0.0001 0.001 0.1 10 0.01 3.4 2.7 3.0 3.5 2 [MPa.m'''] 3.25 2.8 41 45 24.5 26 28.3 32.2 50 [MPa] CTv 17.2 9 32.7 33.3 33.0 29.5 [mm] 4 Bmin Table 3. Evaluation of the plane strain thickness, Bmin, calculated after equation 2. If Bmin > 4 mm (the thickness of the tested samples), it doesn't fulfil the plane strain criterion. Material iPP/EPR-1 tested at room temperature. test speed Kimax
Although material ranking was successful and corresponded to application tests using a conventional LEFM approach, ways to get geometry independent material descriptors have to be found. We will describe in next section the method we used to achieve it.
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C GREIN, PH. BEGUELINANDH.-H. KAUSCH
DETERMINATION OF INTRINSIC TOUGHNESS VALUES USING A PLASTIC ZONE CORRECTED LEFM ORIGINAL APPROACH SOME ELEMENTS OF THEORY [4] On the contrary to the basic premise of the conventionnal LEFM approach, a polymer is never completely elastic. It doesn't describe the distribution of stress field of equation (3), sketched in Fig. 3(i) and characterised by the presence of an infinite stress infi*ontof the crack tip, consequence of its 1/Vr singularity. •K,
LiJiO)
(eq. 3)
with Gij, the stress field; Ki, the initial stress intensity factor in mode I; r, the distance from the crack tip; fy, a proportionality factor.
(ii)
^37
^ Singularity
Fig. 3. (i) Normal stress towards the fracture plane in mode I for an elastic material; (ii) redistribution of stress by development of a plastic zone in front of a crack tip according to Irwin.
Indeed the radius in front of the crack tip is not zero and inelastic deformation mechanisms, such as local plasticity or crazes, can be initiated at the crack tip. This is particularly true in heterogeneous polymers such as those investigated here, in which very small size inclusions initiate local plasticity in interparticle domains, even when the external stress applied is well below the yield stress of the neat matrix. When this zone is small enough and does not disturb the global elastic behaviour of the structure, the stress intensity factors can be corrected. Among all small-scale yielding corrections, two are particularly famous: that of Dugdale-Barenblatt [6] and that of Irwin [7]. In this paper, we will concentrate on the latter. In the Irwin's approach, plasticity is assumed to develop when the material exceeds locally its yield stress, ay. The stress in front of the crack tip is therefore truncated at ay as highlighted in Fig. 3(ii). In order to take into account the stored energy in the elastic part of the material, a plastic zone of radius rp, must be added to the crack length, a, which becomes: a^^ =a + r„
(eq. 4)
A New Way for Polymer Characterisation
135
The energy conservation is thus assured and the stress redistributed in front of the crack tip according to more realistic physical principles. The toughness values can be recalculated with: (eq. 5)
^eff=4^effh^[^
with A,(aeff), a factor depending on the geometry of the specimen, and in first approximation plastic radii values estimated at: 1 '''
IK
1
in plane stress (eq. 6), and,
''"
in plane strain (eq. 7).
6JC
PROPOSED PLASTIC ZONE CORRECTION OF LEFM According to equation 1, K can be assessed graphically by plotting Fmax vesrus (D^W)/f for different crack lengths. In case of elastic linear brittle behaviour the straight line described by the experimental points runs through the origin (see Fig. 4(i)) and its slope gives Kimax (= Kic) [8-10]. In case of ductile behaviour where stress whitening occurs at the crack tip (Fig. 4(ii)), Fmax and (BVW)/f exhibit also a linear correlation. This latter, however, does not pass through (0,0) accounting for the development of a plastic zone in front of the crack tip. We propose to deduce numerically the size of this plastic zone, rp (defined as in equation 4), in such a way that the straight line describing all data is forced through the origin. The related slope gives thus the effective toughness, Keff, and is a geometry-independent quantity. Practically the crack length, a, is replaced by a + rp in the proportionnality factor f; f(a/W) becomes thus f((a+rp)/W) and the value of rp is then adjusted iteratively until the straight line passes through (0,0). Furthermore, the values of rp found experimentally express the radius of the equivalent Irwin damaged zone. Physically, rather than the strict physical size of the plastic zone, the values calculated express the equivalent size of undamaged material affecting the global compliance of the specimen. (ii)
tF ^ max
BW"2/f
[/ r
. . • • ^
^'^Imax
BW"2/f
Fig. 4. Experimental determination (i) of Kimax in case of linear elastic material behaviour; (ii) of K^ff by correction with the radius of the plastic zone, present at the vicinity of the crack tip; agrap designs the crack length used for graphical determination of K.
An example of a plastic zone correction is given in Fig. 5(i) for iPP/EPR-1 tested at 0.001 m/s and room temperature. The excellent coefficient of linear regression (r^ > 0.99) of both couples fo data is a guaranty of the reliability of the applied Irwin-like proposed correction. The effective toughness, Keff, is higher than both the toughness measured by varying a/W without applying the plastic zone correction (Fig. 5(i), Kimax = 4.13 MPa.m^^^) and the mean value of
C GREIN, PH. BEGUELINAND K-H. KAUSCH
136
Kimax (Fig. 5(ii), Kimax- 3 MPa.m ) directly calculated from equation 1 for a/W = 0.5. The fracture resistance potentialities of the investigated material were therefore underestimated with the uncorrected values.
K [MPa.m 1/21
(ii)
t-rr*
K.eff
r„ = 2,28 mm K,Imax
•
-
^ r„ = 0 mm
BWi/2/f[10^iii^^]
0.2
0.4
0.6
aAVH
0.8
Fig. 5. (i) Graphical determination of the size of the plastic zone, (ii) Evolution of the toughness, K, with a geometry factor aAV before correction (Kimax) and after correction (Keff) with the size of the plastic zone. Material iPP/EPR-1 tested at 0.001 m/s and room temperature.
SIZE OF PLASTIC ZONE, EFFECTIVE TOUGHNESS AND RUBBER CONTENT The size of the plastic zone, rp, of the investigated materials is, for the studied materials, rather independent of the rubber content as long as they remain ductile. This fact is valuable for variations of both test speed (Table 4) or temperature (Table 5). Whereas under unstable crack growth, rp lowers and tends progressively to 0, it valuer 2 mm for stable propagatio n of the damage. Since both grades have roughly the same matrix, rp seems to reflect the intrinsic ductility of the continuous iPP phase once the damage mechanisms have been initiated in the early stages of the deformation. test speed
0.001 0.01 0.1 1 [m/s] 2.28 ±0.40 2.1 ±0.32 2.07 ±0.29 0.74 ±0.27 0.58 ± 0.47 [mm] r„-iPP/EPR-2 2.10 ±0.32 1.97 ±0.25 1.97 ±0.20 1.95 ±0.25 1.8 ±0.35 [mm] Table 4. Size of the plastic zone, rp, as a function of the test speed at room temperature for iPP/EPR-1 and iPP/EPR-2. In grey, unstable crack propagation. -30 -5 23 [°C] 60 temperature 1.66 ±0.52 2.31 ±0.32 2.28 ±0.40 2.28 ± 0.32 [mm] r„- iPP/EPR-1 1.78 ±0.22 1.89 ±0.27 2.10 ±0.32 2.13 ±0.26 [mm] r„- iPP/EPR-2 Table 5. Size of the plastic zone, rp, as a function of the temperature at room temperature for iPP/EPR-1 and iPP/EPR-2.
The evolution of the effective toughness, Keff (deduced graphically), towards the apparent toughness, Kimax (calculated for aAV = 0.5 using equation 1), over 5 decades of test speed - at room temperature - and from -30 to 60°C - at 0.001 m/s, i.e. in the ductile range for both grades - can be summarised as follows according to Fig. 6 and 7:
A New Way for Polymer Characterisation
137
the ductile-brittle transition is marked by a decrease of both Kimax and Keff (Fig. 6, case ofiPP/EPR-1); Keff is always higher than Kimax- In the ductile range, Kimax/Keff ^ 0.70 ± 0.03 for both grades whatever the test conditions, hi case of unstable crack propagation, Kimax ~ Keff; The values of Keff(iPP/EPR-l)/Keff(iPP/EPR-2) for given test conditions are close to those of Kiniax(iPP/EPR-l)/Kiniax(iPP/EPR-2) when both grades exhibit the same macroscopic behaviour. In other words, Kimax is a semi-quantitative toughness parameter, whereas Keff provides a quantitative description of the fracture resistance.
K [MPa.mi iPP/EPR-2
iPP/EPR-1
rti di
°K n K
^
[tl
rh
0.001
0.01
0.1
1
6
0.001
0.01
0.1
^
1
6
V [m/s] Fig. 6. Evolution of the apparent toughness, Kimax, and the effective toughness, Kgff, for iPP/EPR-1 and iPP/EPR-2 tested over a wide range of test speeds at room temperature.
-30'^C
-5°C
23°C
60T
-30°C
T[°C]
-5°C
23X
60°C
Fig. 7. Evolution of the apparent toughness, Kimax, and the effective toughness, Keff, for iPP/EPR-1 and iPP/EPR-2 tested over a wide range of temperature at 0.001 m/s (i.e. in the ductile range for both grades).
138
C. GREIN, PH. BEGUELINANDH.-H. KAUSCH
NECESSITY OF A MIXED APPROACH The possibility to get geometry independent parameters constitutes a master tnunp to characterise properly a ductile polymer. However, whereas far from the ductile-brittle transition the evaluation of Keff is achieved easily, it is more challenging closer to it. The difficulties to determine reliable effective toughness values in this latter case and thus precise ductile-brittle transitions will be illustrated with grade iPP/EPR-1 tested at room temperature. TRANSITIONS WHICH ARE DEPENDENT ON THE LIGAMENT LENGTH Tested with a/W = 0.5 and using the conventional LEFM approach for data reduction, iPP/EPR-1 had its ductile-brittle transition at about 0.6 m/s. Further fracture tests were carried out with a/W varying between 0.3 and 0.7 at 0.001, 0.01, 0.1, 0.4, 0.7, 1 and 6 m/s. The corrected LEFM method was applied to raw data. Partial results are available in Table 4 and Fig. 6. Plastic zone sizes and effective fracture resistances could be assessed in an incontestable manner up to 0.1 m/s (included) where iPP/EPR-1 was ductile and above 1 m/s (included) where it exhibited unstable crack propagation for all a/W. At speeds of 0.4 and 0.7 m/s, it showed, however, not a single macroscopic behaviour: the crack grew in an unstable wa}^ (either semi-brittle or semi-ductile behaviour) for 0.3 < a/W < 0.5; in a stable way for a/W > 0.5. The evolution of Kimax (calculated according to equation 1) over the test speeds for different ligament lengths is shown in Fig. 8. Due to the limited number of points near the maximum of Kimax (taken as the ductile-brittle transition), transition zones (covering 0.3 decades of speeds) were considered. It appeared that the shorter the ligament length (i.e. the higher a/W) was, the lower was the speed at which the ductile-brittle transition occurred.
0.0001
0.001
0.01
0.1 V [m/s]
Fig. 8. Apparent toughness, Kimax, plotted against the test speed, v, for different ligament length. The shadowed regions correspond to the transition zones. Material iPP/EPR-1 tested at room temperature.
TRANSITION ZONE: UNCERTAINTY ABOUT SOME VALUES OBTAINED WITH THE CORRECTED LEFM APPROACH Because of the dependence of the ductile-brittle transition with the crack length, it would be advantageous to define it in terms of rp or Keff. Their evolution over the investigated range of test speeds is given in Fig. 9. To define an unequivocal ductile-brittle transition is, however, ambitious: does it occur at 0. 4 m/s, the maximum of Keff, or at 0.7 m/s, the inflection in rp?
A New Way for Polymer Characterisation
139
By analogy with what have been done with the conventional LEFM approach, the first solution seems to be the most appropriate. This point is, however, ambiguous: two values of rp and therefore of Keff could not have been determined with certainty: those in the transition zone at 0.4 and 0.7 m/s as direct consequence of the unstable-stable transition occurring in each series. We have indeed noticed that for an a/W, where both macroscopic behaviours have been observed, the values of Fmax were 5 to 10% higher under unstable crack propagation than for stable fracture growth. Table 6 shows the different values of rp and Keff obtained: (i) by considering all the raw data (stable + unstable crack growth); (ii) by correcting the Fmax values of about 7% to "align" them with the preponderant mode of failure in a series; (iii) by suppressing the tests which do not correspond to the predominant macroscopic behaviour in a series.
r [mm],K,„3jMPa.mi/2]
0.0001
0.001
0.01 0.1 V [m/s] Fig. 9, Size of the plastic zone, rp, and effective fracture resistance, Keff, plotted against the logarithm of the test speed. Grey zone: transition zone, incertainty about the values rp and Keff. Material iPP/EPR-1 tested at room temperature. Case (iii) Case (ii) Case (i) 1.55 ±0.30 1.77 ±0.26 2.03 ± 0.29 [mm] 5.23 ±0.21 4.66 ±0.31 4.93 ±0.19 [MPa.m''^] Keff Table 6. Illustration of the variations of the size of the plastic zone, rp, and the effective toughness, Keff, as a function of the conditions of determination (i), (ii) and (iii) defined in the text above. Material iPP/EPR-1 tested at 0.4 m/s and room temperature. TD
0.3
None of the methods for data reduction was satisfactory. For (i) the slope of the Fmax-BVW/f became (artificially) sharper, since the weight of higher Fmax for an unstable crack behaviour was reported to low a/W (i.e. high BVw/f); the calculated rp and Keff were therefore overestimated. Methods (ii) and (iii), which imply a correction of correction, restrict the a/W interval used to estimate rp. It would be of minor importance if the rp were totally independent
140
C GREIN, PH. BEGUELINAND H.-H. KAUSCH
of the ligament length. Table 7 shows that it was nevertheless not the case. Consequently, the speed at which the ductile-brittle transition occurs, can not be assessed by a precision over 0.5 decades of test speed with the plastic zone corrected LEFM approach. The speculation about the rp values near the transition is not acceptable for a fine tuning ranking of various materials. WHY TO USE A MIXED MODE APPROACH? A mixed mode approach would consist: (i) of getting precisely the speed at which the ductile-brittle transitions occur using the conventional LEFM approch with one ligament length (i.e one a/W) as it has been done in section "A unique ductile-brittle transition"; (ii) and of determining in a targetted way intrinsic toughness values for modelisation purpose. Indeed, as obvious from both exemples given in Fig. 2, the transition could thus be determined accurately within 0.1-0.2 decades of test speeds with few samples in a relative short time frame. Moreover, as the apparent values (Kimax) are always lower than the effective parameters (Keff), none of the material descriptor would be overestimated. In addition, since Kimax-values have been shown to provide a semi-quantitative evaluation (in terms of test speed or temperature) of fracture resistance parameters, a coherent material comparison would be possible over the whole investigated range. This remark remains true as long as the grades have similar rp. For iPP grades, it should be checked (and considered with more caution) when materials exhibit different particle and matrix melt flow rates, or different crystalline structures. It should also be investigated in detail when different polymer families (ABS versus HIPS or rubber modified iPP) are compared. CONCLUSION A new approach, which takes into account the high amount of plasticity of tough polymers, has been proposed to assess for material toughness. It consists of determining experimentally the size of the plastic zone developed at a crack tip using specimens of different ligament lengths. The effective parameters Keff were derived from the slopes of the plots Fmax over BW*^^/f(a€fi/W) with aeff = ao + rp in such a way that the plastic zone radii, rp, were obtained numerically using an iteration procedure in such a manner all data fall on a line through the origin. This approach has been shown to provide intrinsic values over a large range of test conditions, except near the ductile-brittle transition. This was all the more a pity that this latter is a key-parameter in the toughness characterisation of polymers. It was therefore suggested to combine the plastic zone corrected LEFM with a more conventional approach, which aims to determine precisely the speed at which the ductile-brittle transition occurs at fixed ligament length (typically a/W = 0.5). Since, both of the approaches provide complementary information about mechanical performances, they are thought to give a complete picture of the fracture behaviour of any given material.
A New Way for Polymer Characterisation
141
REFERENCES [1] C. Grein, P. B6guelin, C.J.G. Plummer, H.-H. Kausch, L. T6z6, Y. Germain, in Fracture of Polymers, Composites and Adhesives, J.G. Williams and A. Pavan Editors, ESIS Pulication 27, Elsevier Science, UK, 319333, (2000). [2] P. B^guelin and H.-H. Kausch, in Impact and Dynamic Fracture of Polymers and Composites, J.G. Williams and A. Pavan Editors, Mechanical Engineering Publications, UK, 3-19, (1995). [3] P. B^guelin, C. Fond and H.-H. Kausch, International Journal of Fracture (1998), 89,85. [4] T. L. Anderson, Fracture Mechanics - fundamentals and applications, CRC Press (USA), (1995). [5] ESIS Technical Committee 4, A Linear Elastic Fracture Mechanics (LEFM) Standardfor Determining Kc and Gc Testing Protocol (1990). [6] D. S. Dugdale, Journal of Physics of Solids (1962), 8, 100. [7] G. I. Barenblatt, Advances in Applied Mechanics (1962), 7, 55. [8] P. L. Fernando and J. G. Williams, Polymer Engineering and Science (1980), 20(3), 215. [9] J. M. Hodkinson, A. Savadori and J. G. Williams, Journal of Materials Science (1983), 18,2319. [10] J. I. Velasco, J. A. De Saja and A. B. Martinez, Fatigue and Fracture of Engineering Materials and Structure (1997), 20, 659.
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
143
EFFECTS OF CONSTRAINT ON THE TRACTION-SEPARATION BEHAVIOUR OF POLYETHYLENE
S.K.M. TING, J.G. WILLIAMS and A. IVANKOVIC Department of Mechanical Engineering, Imperial College London, Exhibition Road, London, SW7 2BX, United Kingdom
ABSTRACT Previous studies have shown that the formation and failure of the craze structure ahead of the crack tip is the precursor to fracture in polyethylene (PE). A knowledge of the craze development and its structure should lead to an understanding of the crack growth behaviour. However, to date there have been very few studies of the craze behaviour from its initiation and growth to eventual breakdown. In the present study, the traction-separation (TS) curve was used to describe and quantify the crazing process utilising a circumferentially notched tensile specimen. Here, the effects of constraint on the TS curves of two grades of PE, a pipe grade and a high density grade, have been investigated experimentally over a wide range of loading speeds. The level of constraint was varied by changing the notch depth and the specimen geometry. The results show that the TS behaviour changes with varying degrees of constraint and loading rate, indicating that the initiation, evolution and subsequent failure of the craze are dependent on these parameters. These data will provide a framework for future modelling work on predicting crack growth in PE. KEYWORDS polyethylene, craze, fracture, constraint, rate, traction-separation. INTRODUCTION The increasing use of polyethylene (PE) for water and gas distribution has resulted in extensive investigation into its behaviour under various loading and environmental conditions. One of the fracture modes that is associated with pipe failure is slow crack growth (SCG) [1].
144
S.K.M. TING, J.G. WILLIAMS AND A. IVANKOVIC
It is well documented [2-4] that the precursor to fracture in PE is the failure of the craze structure ahead of the crack tip during SCG. The formation of the craze and the mechanism that leads to craze breakdown have been described frequently. The craze nucleation is characterised by the formation of a highly localised zone ahead of the crack tip which consists of multiple voids. Their growth and subsequent coalescence leads to the formation of a fibrous structure. Depending on the stability of the craze structure, the craze may widen by drawing material from the craze-bulk interface into the craze fibrils and eventually rupture at the midribs, or fail at the craze-bulk interface with little or no signs of material fibrillation [5]. The rate dependent fracture behaviour and ductile crack growth characteristics of tough PE are well known [6-8]. Since the fracture process is governed by crazing, it is possible to evaluate the fracture behaviour by measuring the behaviour of the craze formed in front of the crack tip. By measuring the craze behaviour under different rate conditions, Pandya and Williams [9] were able to quantify the long term fracture properties of different grades of PE. Ivankovic et. al [10] modelled the crack growth of PE and concluded that in order to achieve accurate predictions both rate and constraint effects must be accounted for. Therefore, an accurate assessment of the craze behaviour at various rates and constraint levels is required to elucidate the ductile crack growth mechanisms in PE. Although some work has been performed to investigate the effects of loading rate on the deformation and failure mechanisms [11], the role of the crack-tip constraint on the fracture behaviour has yet to be examined extensively. The present experiments were designed to measure in-situ the development of the damage zone (craze) under different constraint conditions over a wide range of loading speeds. Here, the localised fracture process is represented in terms of a traction separation relation on the plane of fracture. The aim was to show that the formation and subsequent failure of the damage zone was dependent on the constraint level, in addition to the applied rate. MATERIALS USED AND SPECIMEN PREPARATION The physical characteristics of the two grades of PE studied are given in Table 1. PESO is a copolymer and is one of the toughest grades of PE commercially available whereas HDPE is a brittle homopolymer. The selection of these materials forms a basis for comparing the TS behaviour since each grade is known to exhibit a different resistance to crack growth. Table 1. The properties of the PE studied. Density Branch density PE type (per 1000 C) (kg/m-') 4.5 Pipe grade PE 940 (PESO) 954 0 High density PE (HDPE)
Mw (g/mol) 185 000
Mn (g/mol) 14 000
Melt Row Index (g/10 min) 0.2
355 000
24 000
2.1
Test specimens were cut from compression moulded sheet in the form of rectangular bars with the following dimensions (a) 10 x 10 x 110 mm, (b) 16 xl6 x 110 mm and (c) 20 x 20 X 110 mm (Figure 1). A lathe was used to produce the symmetrical notch using a single point cutting tool of tip radius < 20 |Lim.
Effects of Constraint on the Traction-Separation Behaviour of Polyethylene
Not to scale
145
^'^
Figure 1. Dimension of the circumferentially notched tensile specimen. EXPERIMENTAL PROCEDURE Experiments were performed under constant speed conditions on the Instron testing machine over the range of 0.005 mm/min to 50 mm/min. A standard 10 kN load cell was used to measure the load while the extension was measured by a video extensometer. The traction was calculated based on the original ligament area while separation was measured from a 5 mm gauge length across the notch opening. The experimental set-up is shown schematically in Figure 2.
5 mm gauge length
Figure 2. Experiment set-up. RESULTS AND DISCUSSION Two different procedures were used to vary the constraint level in the specimen hgament. In the first, the specimen thickness is kept fixed while the notch depth, i.e. the Hgament to bulk area (LBA) ratio, is varied as illustrated in Figure 3(a). In the second procedure, the LBA ratio was fixed and the specimen thickness varied (Figure 3 (b)). 20 mm X 20 mm
10 % LBA ratio
20 mm x 20 mm
30 % LBA ratio
(a)
20 mm x 20
50 % LBA ratio
lOmmxIOmm
'^
16mmx16
20 mm X 20 mm
10 % LBA ratio
10 % LBA ratio
111
10 % LBA ratio
(b)
Figure 3. An illustration of the variation of constraint level by changing (a) notch depth (LBA ratio) and (b) specimen thickness. Notch depth effects (LBA ratio) Figure 4 shows the TS curves along with the associated damage evolution of PESO at 0.05 mm/min. The specimen thickness was 20 mm and it can be seen from Figure 4(b) that the
146
S.K.M. TING, J.G. WILLIAMS AND A. IVANKOVIC
stiffness (initial slope) and the maximum stress reduce with increasing LBA ratio. On the other hand, the separation distance increases with increasing LBA ratio resulting in increased separation energy (the area under the curve). According to Figure 4(a), the differences in separation distance and hence energy seen in Figure 4(b), can be attributed to the different damage mechanisms as the LBA ratio varies. When the LBA ratio was 10%, craze fibril rupture was evident across the entire ligament area. A somewhat similar observation was seen at 30% LBA ratio with an increased amount of material drawn into the craze. However, as the LBA ratio increases to 50% (notch depth decreases), the deformation became very ductile and there was a continuous drawing of material from the bulk into the craze until failure occurred. Specimen thickness — 20 mm
LBA ratio (%)
Damage evolution
(a) (b) Figure 4. Comparison of the (a) damage evolution and (b) the TS behaviour of PESO for different LBA ratios at 0.05 mm/min. Figures 5(a) and 5(b) show the damage evolution and the corresponding measured TS curve of PESO at 50 mm/min. In Figure 5(a), the damage evolution is similar and is characterised by the abrupt rupture of the craze structure formed across the ligament area for all LBA ratios. Figure 5(b) shows that the separation distance increases with increasing LBA ratio, but the increase is much smaller than at 0.05 mm/min. Specimen thickness — 20 mm
LBA ratio (%)
H H 50% LBA ratio 1
X^
—
30% LBA ratio
- ^ 1 0 % LBA ratio |
Damage evolution
(a) (b) Figure 5. Comparison of the (a) damage evolution and (b) the TS behaviour of PESO for different LBA ratios at 50 mm/min.
Effects of Constraint on the Traction-Separation Behaviour of Polyethylene
147
Figure 6(a) compares the damage evolution of HDPE for different LBA ratios at 0.05 mm/min. It can be seen that at 10% LBA ratio, the craze formed around the notch fails abruptly with no evident of fibrillation. On the other hand, at higher LBA ratio, the craze formed breaks gradually from the notch perimeter inwards. As a result, the craze becomes highly localised in the ligament center with evidence of fibrils drawing before failure. The TS curves in Figure 6(b) capture quantitatively the differences in the deformation behaviour. As can be seen in Figure 6(b), the TS curve of 10% LBA ratio shows a lower separation distance and a higher breaking stress than that of the 30% and 50% LBA ratio curves. By contrast, the higher LBA ratio curves displayed large separation distances and consequently high separation energy. Specimen thickness — 20 mm
LBA ratio (%)
- • - 5 0 % LBA ratio —
30% LBA ratio
= ^ 1 0 % LBA ratio
I/'A
Damage evolution
(a) (b) Figure 6. Comparison of the (a) damage evolution and (b) the TS behaviour of HDPE for different LBA ratios at 0.05 mm/min. Figure 7 shows the TS curves along with the associated damage evolution of HDPE at 50 mm/min. As the loading speed increases, the failure becomes more brittle regardless of the LBA ratio as seen in Figure 7(a). According to Figure 7(b), there is little difference in the separation distance between each curve. By comparison, the separation distance increases with the LBA ratio as seen in Figure 5 for PESO. Specimen thickness — 20 mm
LBA ratio (%)
/z ^x^ ^^^
- • - 5 0 % LBA ratio i — 30% LBA ratio - ^ 1 0 % LBA ratio |
rr
Damage evolution
(a) (b) Figure 7. Comparison of the (a) damage evolution and (b) the TS behaviour of HDPE for different LBA ratios at 50 mm/min.
148
S.KM. TING, J.G. WILLIAMS AND A. IVANKOVIC
Figures 8 and 9 show the fracture surfaces of the PE80 and the HDPE specimens respectively. As can be seen from the figures, the fracture surface changes as LBA ratio and test speed vary. The change was most apparent for increasing LBA ratio at low speed. As shown in Figure 8 the fracture surface of PE80 showed a transition from a craze fibrils failure at low LBA ratio and at high test speed to one which exhibited gross drawing and yielding at high LBA ratio and at low test speed. By comparison, HDPE showed a transition to localised materials fibrillation as seen in Figure 9. LBA ratio (%)
A
0.005
0.05 0.5 5 Test speed (mm/min)
Figure 8. Comparison of the fracture surface of PE80 for different LBA ratios and test speeds with a 20 mm specimen thickness. LBA ratio (%)
4
50
30 •"•
J:'": •
Pi^rf'':;'- l
^f!H"-r-\'
W^^:V''' '
10 0.005
0.05
0.5
5
50
Test speed (mm/min) Figure 9. Comparison of the fracture surface of HDPE for different LBA ratios and test speeds with a 20 mm specimen thickness.
Effects of Constraint on the Traction-Separation Behaviour of Polyethylene
149
Size effects Figures 10(a) and 10(b) compare the TS behaviour between different specimen thickness of PESO and HDPE respectively at 0.01 mm/min. The LB A ratio was 10%. According to Figure 10(a), there is little to distinguish between the TS behaviour of 10 mm and 16 mm specimen thickness. However, the difference in the TS behaviour becomes obvious when compared with the curve of 20 mm specimen thickness: the gradient of the pre-peak curve is smaller while the separation distance is larger than that of specimen with 10 mm and 16 mm thickness. The trend is similar in HDPE as seen in Figure 10(b). The results seem to suggest that the constraint found in the specimen with 20 mm thickness was lower than that of specimen with 10 mm and 16 mm thickness, while there is not much to differentiate in the constraint between the latter.
(a) (b) Figure 10. Size effects on the TS behaviour of (a) PESO and (b) HDPE at (a) 0.01 mm/min. Examination of the fracture surfaces of PESO specimens in Figure 11 reveals that the fracture is characterised by the craze fibrils failure regardless of the specimen thickness and loading rate. This was quite different from the fracture characteristics examined in Figure S when the LBA ratio was varied. Specimen thickness (mm)
A
20
16-1
10
0.1 1 Test speed (mm/min) Figure 11. Comparison of the fracture surface of PESO for different specimen thickness and test speeds with a 10% LBA ratio. 0.01
150
S.KM. TING,J.G. MLLIAMSANDA.
IVANKOVIC
Figure 12(a) shows the damage evolution of HOPE at 0.1 mm/min while the resultant TS behaviour is shown in Figure 12(b). The trend of the TS curves in Figure 12(b) resembles that seen in Figure 10(b) where low constraint level in the 20 mm curve has been found. As can be seen in Figure 12(a), the limited extentability of the craze resulted in a small postpeak extension (Figure 12(b)). Specimen thickness (mm)
20-1
10 %LBA ratio
\' ,m:
Damage evolution
(a) (b) Figure 12. Comparison of the (a) damage evolution and (b) the TS behaviour of HDPE for different specimen thickness at 0.1 mm/min. Figure 13 compares the fracture appearances of HDPE specimens for different speeds and different specimen thickness. One trend that emerged from the study is the closeness in the appearance between the different specimen thickness. As expected, the evidence suggests that the variation in constraint is not considerable for fixed LB A ratio specimens of different thickness. A similar deduction can be drawn upon for PESO, as seen in Figure 11. Specimen thickness (mm)
A
0.01
0.1
1
Test speed (mm/min) Figure 13. Comparison of the fracture surface of HDPE for different specimen thickness and test speeds with a 10% LB A ratio.
Effects of Constraint on the Traction-Separation Behaviour of Polyethylene
1^ 1
Comparison of the TS behaviour between PESO and HDPE Figure 14 compares the TS behaviour between PESO and HDPE at 0.5 mm/min with a 20 mm specimen thickness and 50% LBA ratio. It can be seen from Figure 14(a) that the deformation mechanism exhibited by each grade is different. As the damage progresses, PESO showed extensive blunting of the notch tip with considerable amount of material being drawn into the craze. Eventually, the failure occurred homogenously across the weakest section of the craze structure (i.e. the middle). By comparison, there was less evidence of tip blunting in HDPE. Instead, the craze formed near the tip vicinity has failed and damage progressed in a craze-crack growth like manner until fracture occurs near the ligament center. According to Figure 14(b), the shape of the TS curve for PESO is broader than that of HDPE due to its ability to sustain the load after the peak and a larger separation distance. On the other hand, the shape of HDPE curve is narrower with a higher peak stress and the slope of the post-peak curve is steeper. It becomes obvious that the manner in which the energy was dissipated via the deformation mechanism influences the overall separation energy as seen in Figure 14(b). Clearly, the deformation mechanism (drawing and subsequent crack tip blunting) of PESO was able to achieve a higher energy of separation through a large separation distance even though the peak stress was lower than that of HDPE. PE type
PESO
HDPE
Damage evolution
(a) (b) Figure 14. Comparison of the (a) damage evolution and (b) the TS behaviour between PESO and HDPE with a 20 mm specimen thickness and 50% LBA ratio at 0.5 mm/min. Crack tip constraint The crack tip constraint factor, M, defined as the ratio of the craze stress a^., over uniaxial yield stress, ay provides a basis for comparing the degree of constraint in various geometries. M=
(1)
Cc is defined as the maximum stress obtained from the measured TS curve whereas Gy is measured from standard tensile tests using dumbbell specimen of 70 mm gauge length and 1.5 mm thickness. The data obtained from tensile tests at different speeds for PESO and HDPE are shown in Table 2. First, the data of a^ and Gy as a function of test speed are considered. A plot of a^ and Gy against the test speed for PESO and HDPE are shown in Figures 15 and 16 respectively. As can be seen in Figure 15, a^ and Cy increase with the logarithm of test speed and both showed an almost linear rise in the stress value. By comparing the a^ values between, for example, different specimen thickness with a 10% LBA ratio at 10 mm/min, the variation in the stress is negligible (Figure 15). A similar observation was also found in specimens with
1^2
S.KM. TING, J.G. WILLIAMS AND A. IVANKOVIC
30% and 50% LBA ratios. The results indicate that the measured a^ is likely to be a material property in that it is independent of the specimen size. The same observation is also seen in Figure 16 for HOPE. Table 2. Uniaxial yield stress of PESO and HOPE measured under different Speed (mm/min) 0.05 0.1 0.5 1 5 0.005 0.01 13.2 12.4 13.5 15.1 16.3 PE80 10.8 10.1 (MPa) 16.2 18.2 19.4 13.2 15 21.6 HOPE 12.3 (MPa)
speeds. 10 17.9
50 21.2
24.2
27.5
40
30
a 1
a. 7 20
^ 2 0 x 2 0 mm 50% LBA ratio 420 x20 mm 30% LBA ratio
10
iss 20 X 20 mm10% LBA ratio <>10x10mm 50% LBA ratio ^ l O x 10 mm 30% LBA ratio Q 10x10 mm 10% LBA ratio # Yield stress
0 0.001
0.01
0.1 1 Test Speed (mm/min)
10
100
Figure 15. Variation of <3^ and Gy with test speed of PE80. 50
40
i I
S. 30
A
I
m
?
4
S 20
.^ 20 X 20 mm-50% LBA ratio 4 20 X 20 mm-30% LBA ratio » 20 X 20 mm-10% LBA ratio
10
0 1 0 X 10 mm-50% LBA ratio /.^ 10 X 10 mm-30% LBA ratio c 10x10 mm-10% LBA ratio • Yield stress
0 0.001
0.01
0.1
1
10
Test Speed (mm/min)
Figure 16. Variation of a^ and Gy with test speed of HDPE.
100
Effects of Constraint on the Traction-Separation Behaviour of Polyethylene
153
Figures 17 and 18 show the computed M against test speed for PESO and HDPE respectively. One trend common to both materials is the decrease in both M and in the variation of M between the different specimens as the test speed increased. At a speed of 0.005 mm/min, specimen with 10mm thickness and 10% LB A ratio has the highest constraint factor (M), whereas constraint is the lowest for specimen with 20mm thickness and 50% LBA ratio. Therefore, as M increases with decreasing rate, one might expect a highly constrained failure normally associated with internal voiding and cavitation mechanisms. On the other hand, a low M at low rate would suggest a ductile drawing mechanisms linked to gross yielding. This is supported by the findings given in Figures 4 and 6. At high loading rate (> 5mm/min), the M value from different conditions converges. The results given in Figures 17 and 18 suggest that differences in deformation observed due to different M at low speed become indistinguishable at high loading rate. The analyses performed in Figures 5 and 7 confirmed this point showing that the damage evolution is alike irrespective of the LBA ratios for both of the PE grades. 2.5
2.0
; 1.5
; 1.0
f- 20 X 20 mm-50% LBA ratio 4 20 X 20 mm-30% LBA ratio m 20 X 20 mm-10% LBA ratio
0.5
^ e ^ 10x10 mm-50% LBA ratio -T?r 10x10 mm-30% LBA ratio ~ Q - 10x10 mm-10% LBA ratio
0.0 0.001
0.01
^^—
0.1
10
Test Speed (mm/min)
100
Figure 17. Variation of M with test speed of PE80.
2.5
" " * -^'''^^====^.,^_g^ 2.0
1.5
h
^
4
^T^'""^-'^
" *^"#^^^^3
1.0
f.
20 X 20 mm-50% LBA ratio
4
20 X 20 mm-30% LBA ratio
5
20x20 mm-10% LBA ratio
1
— ^ 10x10 mm-50% LBA ratio
0.5
—^
10x10 mm-30% LBA ratio
—H— 10x10 mm-10% LBA ratio 0 0
0.001
^
0.01
L
^
0.1
I
I
I
.
n i |
1
Test Speed (mm/min)
Figure 18. Variation of M with test speed of HDPE.
^
100
154
S.K.M. TING, J. G. WILLIAMS AND A. IVANKOVIC
CONCLUSION The present study showed that the variation in the constraint condition affects the TS curve/craze behaviour. This was captured by the shape of the TS curve being broader at the low constraint condition (i.e. at high LBA ratio/shallow notch depth) due to the higher break separation value. In addition, the study has identified the different damage and failure mechanism which explained the distinctly different shape of TS curves exhibited by each grade and associated separation energy values. Furthermore, at a given applied rate it was revealed that the measured a^ remains relatively constant even though the specimen thickness was varied, suggesting that a^ is the property of the material though a function of M. In conclusion, the study has showed that the crack growth behaviour in PE is constraint dependent, in addition to rate and both effects should be taken into account in predicting the failure of these materials.
ACKNOWLEDGEMENT The authors are grateful to the financial support provided by BP Chemical Ltd (currently BP-Solvay) for the project.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Beech S.H., Ferguson C.R. and Clutton E.Q. (2001) in the Conference Proceedings of Plastics Pipes XI, 401. Bhattacharya S.K. and Brown N., (1985) /. Mafl Set, 20, 2767. Lu X., Qian R. and Brown N. (1991) /. Mafl Sci, 26, 881. Duan D.M. and Williams J.G. (1998) X Mat'l. Sci., 33, 625. Lagaron J.M., Capaccio G., Rose L.J. and Kip B.J. (2000), /. App. Polymer Sci., 77, 283. Pandya K.C. (2000), PhD thesis, Imperial College London, United Kingdom. Brooks N., Duckett R.A. and Ward I.M. (1992), Polymer, 33, 1872. Plummer C.J.G., Goldberg A. and Ghanem A. (2001), Polymer, 42, 9551. Pandya K.C. and Williams J.G. (2000) Polym. Eng. Sci., 40, 1765. Ivankovic A., Pandya K.C, and Wilhams J.G. (2002) Eng. Fract. Mech., in press. Pandya K.C, Clutton E.Q. and Rose L.J., in preparation.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
155
MICROMECHANICAL MODELLING OF RATE AND TEMPERATURE DEPENDENT FRACTURE OF GLASSY POLYMERS R. ESTEVEZ^, S. BASU* and E. VAN DER GIESSEN^ ° GEMPPM/INSA delyon, 20 Av Albert Einstein, 69621 Villeurbanne Cedex, France ^ University ofGroningen, Department of Applied Physics/Micromechanics of Materials, Nijenborgh 4, 9749 AG, Groningen, The Netherlands
ABSTRACT This paper is concerned with the fracture of glassy polymers at high loading rates, where thermal effects due to self-heating of the material can become important. A coupled thermomechanical, small-scale yielding calculation is performed that incorporates a recent cohesive zone model for crazing while a viscoplastic model is used to describe shear yielding. When crazing takes place, this mechanism is identified as the major heat source for the temperature increase while the conversion of bulk viscoplasticity to heat appears to be negligible. At sufficiently high remote loading rates, the glass transition temperature can be reached next to the craze but the hot zone is too small to promote thermal blunting by large plastic deformations near the crack tip. Since crack growth is caused by craze breakdown, the evolution of the toughness with increasing loading rate is primarily governed by the craze properties. Their rate dependence can partially explain the brittle to ductile transition. KEYWORDS Polymer fracture, thermal effects, crazing, plasticity INTRODUCTION The evolution of the toughness of glassy polymers with increasing loading rate shows a ductile to brittie transition followed by a second transition from britde to ductile at sufficientiy high loading rates. On the basis of the observation that failure of amorphous polymers in the glassy state occurs by the competition between shear yielding and crazing, it has been shown in [1] that the time scales involved in each mechanism primarily govern this competition. For loading rates where isothermal conditions prevail, the first transition from ductile to brittie results from this competition. The second transition from brittie to ductile is accompanied by a temperature rise at the propagating crack tip of about hundreds of Kelvins [2] with traces of material decomposition on the fracture surface. Since shear yielding is a viscoplastic process and crazing involves also some viscoplasticity [3], albeit at a smaller length scale, part of the second transition from brittie to ductile is thought to result in enhanced plastic dissipation caused by the temperature increase. For instance, Williams and Hodgkinson [4] suggested that the temperature increase promotes extensive plasticity at the crack tip resulting in the increase of the toughness. Localized plasticity within a strip zone along the propagating crack was also invoked by Fuller et al. [2] to interpret the toughness increase with increasing loading rate.
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R. ESTEVEZ, S. BASUAND E. VAN DER GIESSEN
A previous study [1] used a small-scale yielding analysis featuring a viscoplastic model for shear yielding and a cohesive surface model for crazing to explore the first ductile to brittle transition. Whereas the above-referenced work was restricted to isothermal conditions, the present thermo-mechanical investigation incorporates the dissipation related to viscoplasticity in the bulk material as well as that involved in the crazing process. The idea is that the resulting local temperature rise affects their competition responses via the temperature dependence of both plasticity and crazing. Therefore, the aim of the present numerical study is to gain insight into the key features involved in the second transition from brittle to ductile with increasing loading rate. CONSTITUTIVE LAW The constitutive model used to describe large plastic deformations of glassy polymers involves a separate formulation for temperatures above and below the glass transition Tg, since the underlying deformation mechanisms are different. In either regime, the formulation is based on the decomposition of the rate of deformation into an elastic part D^ and a plastic part DP so that Z) = D^ + Z)P. By assuming an isotropic yield stress, the isochoric plastic strain rate D^ is given by the flow rule Z)P^-|-a',
(1)
where the equivalent plastic strain rate ^f = '^(T, T,p) ^ is a function of temperature T and pressure /?, the effective stress deviator & and the equivalent driving stress T = sJXjW' -^ will be specified subsequently. The elastic part of the deformation is formulated through the hypoelastic law o=jCeD^-Cacf/,
(2)
V
where a is the Jaumann rate of the Cauchy stress, L^ the usual fourth-order isotropic elastic modulus tensor and / the second-order symmetric identity tensor. The coefficients C and a^ are the bulk modulus and the coefficient of cubic thermal expansion, respectively. For r < Tg, the viscoplastic model used here accounts for intrinsic softening upon yielding followed by progressive orientational hardening. Rate dependent flow is taken to be governed by Argon's formulation [5] of the equivalent plastic strain rate
f = yoexp
•^'-a)
5/6
for T < To,
(3)
where % A and the athermal shear stress ^o are material parameters. Boyce et al. [6] have suggested a modification of (3) to account for intrinsic softening by substituting SQ with s which evolves from the initial value SQ to a steady state value Sss according to i = /i(l ^Superscript 'p' for plastic.
Micromechanical Modelling of Rate and Temperature Dependent Fracture of Glossy Polymers
157
s/s^^)'f, with h controlling the rate of softening. The yield stress of glassy polymers is also pressure dependent and this is incorporated by using j + a/? in (3) with a a pressuresensitivity coefficient. The increase in hardening observed for increasing plastic deformation is incorporated through the definition of a back stress tensor b originating from the stretch of the molecular chains and also dependent on the network chain density n. This is described in detail by Wu and Van der Giessen [7]. Thus, the deviatoric part a' of the driving stress o — g —^ is used to specify the equivalent shear stress i in the flow rule (1) according to For T >Tg, most studies of the mechanical response found in the literature focus on the description of the molten state [8] due to its practical importance while no constitutive law for glassy polymers in the rubbery state is available. The mechanical response of the molten material is non-Newtonian for most polymers and described by x = rjf", where r| and m are material parameters. We will assume that this non-Newtonian response prevails as soon as Tg is exceeded. Hence, within the same framework as used below Tg, the equivalent plastic strain rate is taken as
f=(-j
forr>rg.
(4)
The deformation is assumed to result in chain slippage but some chain entanglement will influence the mechanical response. This feature is assumed to be lumped into the parameters ri and m so that no back stress contribution appears above Tg. Therefore, the driving stress reduces to G = O and the equivalent shear stress t in (4) is that of the Cauchy stress O. For T < Tg, the back stress £> is a function of the plastic stretch and depends on the entanglement density n. The latter is taken to depend on temperature as well, according to the suggestion in [11] that n{T) =B- D&xp{-Ea/RT). (5) In the above expression, Ea is the dissociation energy, R is the gas constant, while B and D are material parameters. Such evolution of the entanglement density is used to model a reduction of the hardening with temperature. Consistent with the above-mentioned assumption that the effects of chain entanglement when T > Tg are considered to be lumped into r| and m, we shall here employ the side condition that n{Tg) = 0. With this, it is seen that the parameters in (5) need to satisfy B/D = exp{-EjRTg). CRAZE MODELLING Motivated by the Kramer and Berger [3] description of the crazing process, Tijssens et al. [9] proposed a viscoplastic crazing model within theframeworkof a cohesive zone methodology. The traction-separation law proposed in [9] comprises three parts corresponding to initiation, thickening and breakdown of the craze. A craze is supposed to initiate by a stress and temperature controlled mechanism and the initiation criterion is motivated by Stemstein*s [10] classical work and is taken as a n > a m - ^ + 7—,
(6)
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R. ESTEVEZ, S. BASUAND E. VANDER GIESSEN
under plane strain conditions. Here, Gn and Cm correspond to the normal and mean stresses at the plane of initiation. Furthermore, the coefficients A^ and 5^ are assumed to be dependent on temperature according to an Arrhenius relation [9]. Since craze thickening involves an intense viscoplastic activity within the active zone [3], the process is taken to be time and temperature dependent. The craze thickening rate A^ is given by [1, 9]
with Ao, A^ and a^ being material parameters. Craze breakdown is supposed to occur when the craze thickness attains a value of A^^^ This parameter is of major importance since it governs the onset of crack propagation due to craze breakdown. Within a cohesive zone representation of a craze, heat generated during the viscous craze thickening process is accounted for by defining a heat flux along the surfaces of the cohesive zone as discussed in the next section. PROBLEM FORMULATION In this work, a small-scale yielding model of a stationary crack is assumed, witii the remote boundary being subjected to a mode I loading at a constant loading rate ki. The crack initially has a blunt tip with a radius n. Crack growth by crazing is allowed to occur only along the plane of the crack fe = 0). Shear yielding in the bulk of the material is incorporated through the constitutive law presented in the previous section. The definition of the plastic strain rate Z>P together with the expression for the driving stress c specifies the energy dissipation rate per unit volume d : DP = y/lvf. The energy balance in the material can then be written as pcpf = /:V2r + a:£>P,
(8)
with k the isotropic heat conductivity in accordance with Fourier's law and Cp the specific heat. It is important to realize however, that there is a second source of energy dissipation, which results from the fibrillation process during craze thickening, as described above. Per unit of area, the dissipation amounts to CnAJ and represents a heat flux q = /:grad T into the system through the surface of the cohesive zone. If crazing has not initiated, there is no heat flux across the symmetry plane X2 = 0. Once crazing has nucleated, craze thickening takes place and the above heat flux across the craze surfaces is considered. The energy balance (8) is subject to the following boundary conditions on the cohesive surfaces : on X2 = 0, without a craze
dx2
1 / \ 1 *u ^ \ ^CJn /S^/k on JC2 = ±5An(jci), along the craze surfaces
where An is the thickness along the cohesive zone ahead of the crack.
(9)
Micromechanical Modelling of Rate and Temperature Dependent Fracture of Glossy Polymers 159
Fig. 1. Response of the bulk material for uniaxial tension at room temperature under adiabatic conditions. RESULTS The present study aims at giving insight into the processes involved in glassy polymer fracture. The material parameters used are representative for styrene-acrylonitrile (SAN), and are the same as in [1]. The parameters m and r| used for the material description above Tg are taken to be representative of most polymers in the molten state [8] with m = 0.4 and r| = 0.35Pas. With the parameters used for the bulk material, the corresponding stress-strain response for a tensile test at room temperature and under adiabatic conditions is shown in Fig. 1. Before the glass transition is reached, the intrinsic response governed by (3) prevails while it switches to the melt description of (4) above Tg. At relatively low strain rates of 10^ s~^ to 10^ s~^ the material significantly loses stress carrying capacity when the glass transition is reached. For the higher strain rate of 10^ s~^ the response with the melt formulation results in a level of stress comparable to that of the solid. The parameters for the craze description are also borrowed from this numerical study; those of case 8 in [1] were observed to capture well the first transition from ductile to brittle and are used here. At sufficiently low loading rates, the process can be considered to be isothermal and as discussed previously in [1], crazing can initiate from the root of the notch or in the bulk of the material. In the latter case, the initiation occurs since shear bands around the tip (see [13]) enhance the local hydrostatic stress and thus trigger crazing. In this paper we will consider higher loading rates, Ki = 300 to 3000 MPa^in/s, where crazing in this material will initiate from the notch root. At these high strain rates, the effective yield strength of the material is so high that no significant shear yielding takes place.
R. ESTEVEZ, S. BASUANDE.
160
VANDER GIESSEN
1
T(K) 393 383 373 363 353 343 333 323 313 303 293
0.5 w-y^'r-%y/!f>';Vv^'
0|
vyf5,»j'K.-^v.?;-,; •f^;-ii•;s;5i;•c^'•:W;:f .*??
mmmBStmmsm
-0.5 -1
(a)
(b)
Fig. 2. Temperature distributions at the onset of craze fibril breakdown, for (a) Ki = 300 MPaVm/s and (b) ^i = 3000 MPavS/s. Figure 2 shows the temperature distribution for two different loading rates at the moment that breakdown of fibrils starts. At these critical values Kfy the cracks propagate in an unstable manner. Indeed, below these loading rates, the temperature rise is negligible when crack propagation takes place. The temperature distributions in Fig. 2 show that a noticeable variation is located along the faces of the craze, with heat dissipated during craze thickening diffusing into the material. No temperature rise appears at the craze tip, where further craze initiation occurs. Therefore, even though the craze initiation criterion [10] is temperature dependent, initiation is not enhanced from heat generated by the crazing process itself. Figure 3 shows the craze growth resistance curves for the above loading rates together with that for Ki = 30 MPa\/m/s from [1] for which isothermal conditions prevail. As the loading rate increases, Kf^ remains constant. A toughening caused by temperature effects is not observed, even when local temperature increases are significant as tiie loading rate increases. The temperature distribution during crack propagation is shown in Fig. 4, and corresponds to the circle in the resistance curves in Fig. 3. During crack advance, the heat continues to diffuse along the normal to the craze surfaces but the extension of the hot zone is confined to a size comparable to that of the craze opening. The maximum temperature increase is located at the crack-craze interface, where the craze thickening and related heat flux into the bulk is larger. At this location, the temperature reaches the glass transition Tg but plasticity is not enhanced in the bulk which remains primarily elastic during crack propagation.
Micromechanical Modelling of Rate and Temperature Dependent Fracture of Glossy Polymers 161
K,/(So«f)
K, = 30MP«.ni'"/8 .
k, = 300MP«Lm'^/s k , = 3000MPa.ni*'*/8
Length (cnze -i- crack) / r,
Fig. 3. Craze-crack resistance curves for isothermal conditions with Aj = 30 MPaVin/s and temperature dependent stress-displacement fields for Ki = 300 MPa^/iin/s and Ki = 3000 MPaVin/s. The triangle indicates the initiation of crazing while the square corresponds to the onset of unstable crack propagation, defining Kf\
i/'^'
T(K) 393 383 373 363 353 343 333 323 313 303 293
0.5|0 -0.5 h -x/r
(b) Fig. 4. Temperature distribution during crack propagation at the instant shown in Fig. 3 by the circle for (a) Ki = 300 MPav/m/sec and (b) i^i = 3000 MPaVm/s.
162
R. ESTEVEZ, S. BASUAND E. VANDER GIESSEN
T(K) 393 383 373 363 353 343 333 323 313 303 293
(a)
(b)
Fig. 5. Temperature distribution prior to crack propagation when a temperature-dependent craze critical opening A^^ is accounted for with (a) Ki = 300 M P a y ^ / s and (b) Ki = 3000 MPax/m/s.
K,/(soi;")
Temperature dependent A^'
. Constant A^ "
o!0
-J
L
Length (craze + crack) / r, -I
I
I
I
I
I
Fig. 6. Craze-crack resistance curves for a constant and a temperature dependent craze critical opening with Ki = 3000 MPaVm/s.
Micromechanical Modelling of Rate and Temperature Dependent Fracture of Glossy Polymers 163
As reported by Doll and Konczol [14], the critical craze thickness AJ^^ is observed experimentally to be temperature dependent in some cases. The mechanism responsible for this variation is not yet clearly identified, however. To get some feeling for its influence, we have considered the case where AJ^^ varies linearly from its value at room temperature to twice that value at Tg. The corresponding temperature distributions for the same loading rates as before are shown in Fig. 5. As the loading rate increases, the temperature rise increases as well and a higher AJ^^ is observed. This has a direct influence on the resistance curves, yielding a higher K{^ as A^^^ increases. This is demonstrated in Fig. 6 for Ki = 3000 MPa-v/m/s. This difference in Kf' is about 20%. DISCUSSION In the vicinity of the crack tip, the local temperatures prior to unstable crack propagation show a noticeable increase for loading rates higher than Ki = 300 MPa^/m/s. The heat generated by the craze thickening results in a hot zone around the craze surfaces, the extent of which is comparable to that of the craze thickness. In an investigation of thermal effects during crack propagation in PMMA, Fuller et al. [2] estimated the dimension of the hot zone normal to the crack path to be about one to three micrometers. The maximum thickness of a craze in PMMA has been estimated to be about two to three micrometer by Doll and Konczol [14]. Albeit from two separate experiments, the size of the hot zone normal to the crack path and the critical opening observed for PMMA are of the same order of magnitude. The corresponding quantities resulting from our calculations appear to be at least consistent with these observations. In addition, we observe that craze viscoplasticity is the major heat source during fracture once crazing has initiated, since the bulk remains primarily elastic. A complete and necessary long parametric study is not presented here; we therefore prefer to present results based on some average material parameters for glassy polymers. Those for Styrene-acrylonitrile are taken for the bulk since its viscoplastic response with no crazing lies in between that of PMMA and polycarbonate, which are commonly regarded as typical brittle and ductile polymers, respectively. The thermal properties do not vary much for standard glassy polymers and mean values are used. The craze parameters are borrowed from a previous investigation [1] because this set was shown to capture the ductile to brittle transition which occurs at low loading rates. From this we have deduced that the response of standard amorphous polymers in which crazing is observed is reasonably accounted for. Proper identification of the material parameters involved in the present model is in progress; this should then allow for a quantitative comparison with the model predictions. The temperature increase during crack propagation reaches the glass transition temperature at the location of the crack-craze transition, where plastic dissipation caused by the craze thickening is maximum. However, this remains confined to a small volume around the crack-craze surfaces (see Fig. 4) so that no plasticity on a larger scale is promoted. Therefore, thermal blunting due to enhanced plasticity from the temperature increase is not seen. Thus, as long as the crack propagates by a single craze, the present calculations indicate that thermal blunting alone cannot be invoked as the explanation of the brittle to ductile transition
164
R. ESTEVEZ, S. BASUAND E. VAN DER GIESSEN
observed at sufficiently high loading rates. However, a temperature dependent critical craze opening may give a tendency in the correct direction. Since the cohesive surface used to describe crazing in glassy polymers is now implemented in a thermomechanical framework, the methodology has some similarities to the thermal decohesion model of Leevers [12] which is based on a modified strip model. A crack is made of two surfaces partly bridged by highly stretch material, which looks similar to a craze but with a much larger thickness. During craze thickening, our description comes out with a normal stress at the craze surfaces [1] which is approximately constant while this is an assumption in Leevers' formulation. Leevers' thermal decohesion model assumes that heat generated by the plastic work in the strip diffuses into the surrounding bulk and that decohesion of the crack faces will happen when a melt temperature is reached over a critical length. This critical length is identified as the molecular chain contour. Therefore, a ductile to brittle transition is predicted for impact loading conditions due to thermal effects reducing the stress carrying capacity of the material. Our description does not estimate the toughness on the basis of a purely thermal effect, but relates it to craze breakdown through a critical craze opening. The temperature is expected here to influence the kinetics of shear yielding and crazing and their competition since both mechanisms are taken to be viscoplastic. Therefore, even if a similar methodology is adopted in both formulations, the mechanisms involved in the models' parameters are different. The thermal decohesion model is restricted to dynamic loading rates while our formulation is not. The present study is restricted to a quasi-static formulation and focusses on the beginning of crack propagation. Therefore, our attention has been devoted to exploring possible causes for the onset of the brittle to ductile transition for such loading conditions. Therefore, arguments based on inertia effects are not invoked, nor is crack branching at high crack speeds [15]. One of the limitations in the present analysis is the craze initiation criterion (6), which is assumed to be time independent and governed only by the definition of a critical stress state. As we are concerned with loading rates for which the time to failure is on the order of microseconds, the relevance of a time-independent craze criterion is questionable: craze nucleation involves a cavitation process by local plastic expansion from pre-existing heterogeneities (dust particles, micro-voids) [16] in the bulk material to form a micro-hole according to a nucleation time. For a stress much lower than that for yielding, the growth of these craze nuclei occurs by a meniscus instability mechanism with a propagation speed determined in [17]. For stresses near yielding, the meniscus instability is not observed and pore formation is likely to govern craze initiation [17]. At any rate, the step of craze initiation shows a characteristic time scale which is bypassed in the formulation (6) adopted here. If the time for craze initiation is accounted for, another time scale is involved in the competition between crazing and shear yielding since it determines whether crazing takes place or not. Therefore, a switch from crazing to shear yielding may provide a possible interpretation of the transition from brittle to ductile at high loading rates. This point will be examined in a forthcoming analysis.
Micromechanical Modelling of Rate and Temperature Dependent Fracture of Glossy Polymers 165 REFERENCES 1. Estevez R., Tijssens M.G.A., Van der Giessen E., (2000), / Mech. Phys. Solids 48 2585-2617. 2. Fuller K.N.G., Fox RG., Field J.E., (1975), Proc. R. Sac, Land,, A341, 537-557. 3. Kramer, H.H., Berger, L.L., (1990% Adv. Polym. Sc„ 91/92,1-68. 4. Williams J.G., Hodgkinson J.M., (1981), Proc. R. Sac. Land. A 375, 231-248. 5. Argon, A.S., (1973), Phil. Mag., 28, 839-865. 6. Boyce, M.C., Parks, D.M., Argon, A.S., (1988), Mech. Mater, 7,15-33. 7. Wu RD., Van der Giessen E., (1993), / Mech. Phys. Solids, 41, 427^51. 8. Pearson J.E.A., (1985), Mechanics of polymer processing, Elsevier Applied Science Publishers. 9. Tijssens, M.G.A., Van der Giessen, E., Sluys L.J., (2000), Mech. Mat., 32,19-35. 10. Stemstein, S.S., Ongchin, L., (1969), Polymer preprints, 10(2), 1117-1124. 11. Raha, S. and Bowden, RB., (1972), Polymer, 13,174-183 . 12. Leevers RS., (1995), Int. J. Fracture, 73,109-127 13. Lai, J., Van der Giessen, E., (1997), Mech. Mater. 25,183-197. 14. Doll, W., Konczol, L., {\990), Adv. Polym. Sc. (91-92), 138-214. 15. CotterellB., (1965), Appl. Materials Research, 4, 227-232. 16. Argon, A.S., Hannoosh, J.G., (1977), Phil. Mag., 36(5), 1195-1216. 17. Argon, A.S., Salama, M.M., (1977), Phil. Mag., 36 (5), 1217-1234.
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hracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
167
COHESIVE PROPERTIES of a CRYSTALLINE POLYMER CRAZE under IMPACT EXTENSION P Leevers, S Hazra and L Wang Department of Mechanical Engineering, Imperial College London, London SW7 2BX, UK
ABSTRACT In a thermoplastic which undergoes stable craze fibril drawing, rapid thickening of the craze layer will cause highly localised adiabatic heating. A thermomechanical analysis, predicated on the existence of a crack-tip craze, has previously allowed both impact fracture resistance and resistance to rapid crack propagation to be predicted for crystalline thermoplastics. If the crack-tip craze is small, i.e. the cohesive stress is high, this prediction can be made from bulk properties only, without knowing the cohesive stress. However, these materials often have a high toughness to yield stress ratio, the craze grows to a substantial proportion of specimen size, and its properties do affect the prediction. A more rigorous numerical model has been formulated, but it requires data for both fibril stretch ratio and cohesive stress. A modified Full Notch Creep test has been developed to measure the cohesive properties of a planar, craze layer, under thickening rates of up to 2 m/s. Displacement is imposed directly on the notch surface by a rigid insert within the moulded specimen. Both the cohesive stress and the lifetime under rapid extension are measured. For the high density polyethylene tested these correlate well via the adiabatic decohesion model.
KEY WORDS Cohesive stress, decohesion, craze, impact, adiabatic, polyethylene.
INTRODUCTION The cohesive zone approach to fracture mechanics reduces the fracture resistance properties of a material to a traction-separation law. This 'law' relates, a^, the normal cohesive stress which resists the parallel separation of two internal planes which were initially very close together, to the current increase r] in their separation. The fracture resistance G^ (and hence the fracture toughness KJ is simply given by the area under the o^{r]) curve up to 6^. The simplest form of traction separation law assumes o^{r]) to be constant up to a critical maximum separation 6^, and this was developed analytically into a fracture model by Dugdale
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P. LEEVERS, S. HAZRA AND L WANG
[1] and Barenblatt [2]. More complex laws demand more complex methods [3], but the computational methods which implement them are not, at least, required to represent either singular stresses or an infinitely sharp crack tip. The cohesive zone approach offers a particular advantages for polymers. It is particularly appropriate for describing a planar craze zone [4], so that what is known about craze mechanics can be applied to model fracture and to express G^ in terms of more basic material properties. On the other hand, since polymers do not separate at cleavage planes, there is mass transport across planes near that of fracture and this can make cohesive surfaces difficult to define, particularly in short-fibre composites undergoing pull-out. A further advantage appears at the continuum level, i.e. the level at which a component might be stress-analysed for engineering design or failure analysis. Since a cohesive zone can initiate under finite stress, strength calculations could be made for an unflawed component without invoking the 'initial flaw' concept of conventional LEFM In some tough, crystalline thermoplastics such as PE, a craze develops readily on loading at the tip of a sharp crack. These materials tend to be less fracture resistant under impact loading or at high crack speed than they are at normal testing rates. In several previous papers [5-9], we have attributed this high-rate brittle behaviour to localised adiabatic melting as a crack-tip craze is forced to thicken rapidly. A model based on this mechanism has been used to make quantitative predictions for impact fracture resistance and for resistance to rapid crack propagation. In early papers it was shown that if the craze remains small, the predictions can be expressed as closed-form equations for toughness (K^ or GJ in which the cohesive stress (i.e. the craze stress) does not appear. In practice, this small-craze assumption is seldom valid for the tough plastics of greatest practical interest, tested in the small Charpy-like specimens most commonly used. In this case, we need to know traction-separation law or at least the average craze stress. The critical displacement need not be measured since this is what the model predicts; if it can be measured, then the predictions can be tested. In this paper we first outline a simple numerical model (described in detail elsewhere [10]) to simulate the conduction and convection processes which accompany adiabatic heating at a cohesive zone surface during rapid thickening. This model is capable of much greater generality and realism than the linear, analytical model previously used. Secondly we describe an experimental method for measuring the cohesive stress and the lifetime of a craze when it is extended uniaxially, uniformly and steadily at a constant high rate. Finally, we present results for a modified high-density pipe grade PE and compare the measured craze lifetimes with those predicted via the numerical model from the measured cohesive stress.
ADIABATIC HEATING and DECOHESION According to the adiabatic decohesion model, impact crack initiation and rapid crack propagation phenomena in a craze-forming polymer are governed by a limiting thermal criterion. For crystalline polymers, this criterion is the formation, at the cohesive surface, of a melt layer equal in thickness to the length s^ of an extended chain (Fig. 1). Other criteria — e.g. a local critical temperature, or a critical degree of degradation by a temperature-time activated reaction — might be suitable for other classes of polymer. In any case, this model emphasises the overwhelming importance of thermal effects, which arise from the low thermal diffusivity of polymers.
Cohesive Properties of a Crystalline Polymer Craze under Impact Extension
169
Previous work pursued the model analytically, for a linearly elastic [5] (or, later, non-linearly elastic [6]) material with constant thermal properties. The analytical model explained several measured fracture properties of thermoplastics: the magnitude of impact fracture toughness and its dependence on impact speed [7] and the absolute magnitude of resistance to rapid crack propagation [8]. Recent results have shown that the impact fracture properties of some amorphous and crosslinked polymers show the same rate dependence [11]. However, the thermal properties of polymers are far from independent of temperature. In particular, the latent heat of fusion and other endotherms (e.g. the transformation peak in /3PP), could have significant effects. The model has recently been re-formulated using a general finite-difference numerical model of the local thermomechanical process [10]. This model accounts not only for one-dimensional conduction but also for convection through the active craze surface cohesive layer (Fig. 1), for a material of non-uniform thermal properties and arbitrary cohesive stress. The governing conduction and convection equations are solved explicitly, through time, along the fibril axis (i.e. the vertical direction in Fig. 1) using a onedimensional finite-difference method. The outcome of the computation is an axial temperature distribution which can be used to implement any chosen thermal decohesion criterion. For the present we consider only PE, and continue to use the criterion which has already proven to work well for it: attainment of the melt temperature throughout a layer of thickness equal to one extended chain length. As Fig. 1 shows, the numerical method drives the adiabatic heat source forward as conduction raises adjacent bulk material to the melt temperature. decohesion
time Critical thickness (structural dimension)
—•! Bulk convection
Melt layer Adiabatic source intensity = cohesive stress X drawing speed
Cohesive stress Fig. 1. Convection of material into a stable craze fibril through an active deformation layer. Since the new model accounts for convection into the craze from the bulk material, it is sensitive to the craze fibril stretch ratio, i.e. the craze density. For the Lauterwasser-Kramer type craze assumed, a constant craze fibril stretch ratio Ap corresponds to a constant craze density p/Ap, where p is the bulk density (Fig. 1). If Ap is low and the craze density exceeds some critical value, adiabatic heating is overwhelmed by convection of *cold' bulk material into the craze and the cohesive layer cannot fail thermally (although the craze fibrils themselves might do so).
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The craze density of a hot-drawn craze is not known, but its maximum value can be estimated from an equivalent monodisperse molecular weight, M, on the simple assumption that randomly conformed polymer chains of n backbone bonds (i.e. n = M/14 for PE, where M is the segment molecular weight) become fully extended into nearly perfect fibrils. Hence
where for PE C«, = 6.7. Similarly, the structural dimension s^ can be taken as the extended chain length 5 = — X 0.127 nm. 14 Increasing M therefore increases Ap and decreases convective loss from the adiabatic heat source. However, increasing M also increases the energy which this source must supply to melt the thicker active layer. The nett effect of an 'equivalent' monodisperse molecular weight parameter M is small; indeed, there is a value at which the failure time determining passes through a maximum. For PE, this value is close to the weight average value M^ for the grade we have tested: 310 kg/mol. Taking M = M^ leads us to assume a maximum fibril extension of 57.5 and a structural dimension of 2.8 |im. However, the thermal decohesion time predicted for a craze layer which thickens at constant high rate still depends strongly on the cohesive stress. We now describe a method by which this parameter can be measured directly, and by which the impact lifetime of the craze can be measured and compared to the predictions of the model. Moulded specimen Square section 10 mm across flats
3
44 mm diameter steel loading washer (moulded-ln)
Notch plane diameter 5 mm Lead-in angle 45°
Fig. 2. The full notch impact test specimen geometry.
THE FULL NOTCH IMPACT TEST The objective of this test method is to measure the cohesive stress and the time to failure of a crystalline polymer craze layer under rapid, uniform extension. The method is an impact variant of the Full Notch Creep test used by Fleissner [12], Duan and Williams [13], Pandya and Williams [14] and others. The specimen (Fig. 2), a square-section tensile bar, is injection moulded. At the mid-plane of the gauge length a sharp, deep circumferential notch reduces the cross-section to about one fifth of its original area. This notch plane is formed by a mouldedin, hardened steel washer. Specimens were injection moulded at 210°C into a warm (100°C) mould and air cooled to 40°C using a hold pressure of 45-50 bar.
171
Cohesive Properties of a Crystalline Polymer Craze under Impact Extension
The test arrangement is shown diagramatically in Fig. 3. The upper end of the specimen is screwed into a strain-gauged PMMA Hopkinson bar of 19 mm diameter and 1.5 m length, calibrated for short-time response. Pilot tests had shown that the craze lifetime under impact lay within the expected uniaxial stress wave return time of 1.2 ms. Strain gauge signal
V
Phase 2: Falling-weight impact
PMMA Hopkinson bar
J Guide bush
Phase 1:1 Slow, wedge-loaded extension
^ ^
Lost motion linkage
Fig. 3. Full Notch Impact Test arrangement (schematic), showing a statically extended specimen about to undergo impact extension. The lower end of the specimen is screwed into a light shackle which can be driven downwards, through a wedge, a lead-screw and a lost motion linkage, by a stepper motor. During Phase I of each test, the specimen is extended slowly by this motor. The notched plane cavitates and thickens to form a primary craze [15], and the static load eventually rolls over a maximum. This phase corresponds to a conventional full-notched tension test. While the final quasi-static displacement is maintained, Phase 2 of the test begins with release of a 14 kg falling weight from a controlled height. In the original version of this impact test [15] the falling weight struck an anvil attached to the lower shackle. Using the new niethod, the weight strikes a moulded-in washer whose surfaces define the notch profile shape (Fig. 2), so that it almost directly displaces the craze surface. The lost-motion linkage allows the specimen shackle, pushed downwards rapidly by the lower half of the specimen, to slide down through its guide bush and decouple from the slow-extension linkage. The craze thickening rate is thus almost instantaneously increased to the impact speed, which is maintained until the craze fails.
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P. LEEVERS, S. HAZRA AND L WANG
RESULTS Tests were carried out at 23°C on a pipe-grade (PEIOO), modified high density polyethylene, at speeds between 0.2 and 2 m/s. The impact speed was calculated from the initial drop-weight height and load/time traces were recorded using a Nicolet high-rate transient recorder system. 1
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Fig. 4. Load/time trace from a typical test. Viscoelastic delay of strain recovery in the PMMA Hopkinson bar is compensated for by dynamic calibration. A load/time trace from a test at 0.3 m/s is shown in Fig. 4. When the striker hits the specimen loading washer the load rises linearly (though with some noise) as the existing static craze is reloaded. The more or less well defined plateau which follows is interpreted as craze drawing and it is from this plateau that the cohesive stress and the decohesion time are estimated, as shown. Figure 4 also illustrates the effect of viscoelastic creep in the PMMA Hopkinson bar: about 25% of the strain accumulated during creep loading is not recovered during the sudden load drop on decohesion. This effect was factored out of the results by dynamic calibration. Further along the trace (not shown) are clear reflections from the end of the Hopkinson bar, whose period provides reassurance that the data quoted below were dynamically 'clean'. As far as the craze drawing lifetime (Fig. 4) was concerned, estimation was certainly rather more subjective. Nevertheless, two series of tests interpreted by two different experimenters produced similar results, their estimates were supported independently by their supervisor, and neither experimenter knew of the values which would be predicted by the model. For this material M^ = 310k; density, specific enthalpy and thermal conductivity were known as functions of temperature; and the craze stress measured using full notch impact tests was in the range 20-30 MPa. Figure 5 compares the measured decohesion times to those predicted by the model, plotted as trend lines for two constant values of cohesive stress — 20 and 50 MPa — and two values of effective molecular weight (which has only a secondary effect).
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DISCUSSION and CONCLUSIONS Because the events which it deals with are transient and relatively inaccessible, it is very difficult to estimate how realistic the adiabatic decohesion model is. Ultimately it can only be judged by how well it succeeds in predicting high-rate decohesion and fracture behaviour. The new numerical model offers a flexible method for predicting decohesion under a wide range of cohesive surface displacement v^. time histories — of which the present situation represents the simplest possible — and for a variety of thermo-mechanical material properties. For conditions of constant craze thickening rate, agreement between prediction and experiment is already very encouraging, particularly at higher speeds. There are still refinements to be made to the experimental method, and drop-weight striker speeds of less than 0.5 m/s will remain inherently difficult to control. Nevertheless, the Full Notch Impact test provides a promising direct method for the measurement of cohesive properties in tough polymers.
ACKNOWLEDGEMENTS The authors would like to express special appreciation for the work of undergraduate students David Eagle and Giles Lunn (of Imperial College London) and Neda Tozija (St. Kiril and Metodij University, Skopje, visiting under the lASTE scheme) for their development and testing work on full notch impact testing. This work was supported by EPSRC Grant GR/R 15788/01.
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REFERENCES 1.
Dugdale, D. S. (1960) 7. Mechs. Phys. Solids 8, 100
2.
Barenblatt, G. I. (1962). In: Advances in Applied Mechanics, vol. 7, pp. 55-129, Academic Press, New York)
3.
De Borst, R. (2001) Int. J. Num. Methods. Engng. 52, 63.
4.
Hine, P. J., Duckett, R. A. and Ward, I. M. (1981), Polymer 22, 1745
5.
Leevers, P. S. (1995) Int. J. Fracture 73, 109.
6.
Leevers, P. S. & Morgan, R.E. (1995) Engng. Fracture Mechs. 52, 999.
7.
Leevers, P. S. (1996) Polymer Eng. Sci. 36 2296.
8.
Greenshields, C. J. and Leevers, P. S. (1996) Int. J. Fracture 79, 85-95
9.
P. S. Leevers, S. Hazra & S. Hillmansen (2000) Plastics Rubbers Comps. 29, 460
10.
P. S. Leevers, Adiabatic decohesion mechanisms of fracture in thermoplastics. In preparation.
11.
Rager, A. (2003). PhD Thesis, University of London, UK.
12.
Fleissner, M., Polymer Eng. Sci. (1998) 38, 330
13.
Duan, D-M. and Williams, J. G., /. Mater. Sci. Sci. (1998) 33, 325
14.
Pandya, K.C. and Williams, J. G., Polymer Eng. Sci. (2000) 40, 1765
15.
Hazra, S. K. (2000) PhD Thesis, University of London, UK.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
175
LABORATORY TEST FOR MEASURING RESISTANCE TO RAPID CRACK PROPAGATION
ANDREAS BURGEL, TAKAO KOBAYASHI, AND DONALD A. SHOCKEY Center for Fracture Physics SRI International, Menlo Park, CA ABSTRACT Whether a pressurized structure will leak or burst when suddenly cracked depends on the material's resistance to rapid crack propagation and its ability to arrest a crack. Thus, the governing material property, namely the stress intensity-crack velocity relationship, is necessary input for designing pressurized pipes and vessels and establishing safe operating conditions. Unfortunately, this material property is not easily measured, and reUable data do not generally exist. This paper describes and presents results from rapid fracture experiments on polymethyl methacrylate (PMMA) compact tension specimens and analyzes pubHshed data from other specimen geometries. After a certain propagation distance, the stress intensity at a fast moving crack front. KID, is governed by the initial stress intensity, KQ, and the dynamic response of the specimen. A plot relating KID/KQ to normalized crack length, aAV, unifies the RCP test data and defines a master curve for the specific specimen geometry. The results suggest a practical and simple method for measuring toughnesses associated with rapidly running and arresting cracks, and hence a route for managing catastrophic failure of pressurized structures. KEYWORDS Rapid crack propagation, dynamic fracture toughness, crack arrest, polyethylene pipes, pressure vessels
THE RAPID CRACK PROPAGATION (RCP) PROBLEM The sudden introduction of a crack in a pressurized pipe or vessel can result in two widely different failure behaviors with widely different consequences. If the crack does not
176
A. BURGEL, T. KOBAYASHIANDD.A. SHOCKEY
propagate, the pressurizing medium can leak slowly, but if the crack rapidly propagates and the structure bursts, the sudden release of energy, destruction of the structure, and fragmentation can be costly and jeopardize safety. Therefore, structures must be designed to resist fast fracture and arrest a rapidly propagating crack. Consider the following scenario. A backhoe impacts a buried gas pipeline and produces a crack in the pipe wall. Under the tensile hoop stresses in the wall resultingfromthe gas pressure, the crack begins to propagate in the pipe axis direction. As the gas leaksfromthe crack, the local pressure begins to fall and hence the driving force on the crack falls. If the pressure decreases quickly to a level less than that needed to drive the crack, the crack will arrest after running a short distance. If the crack propagates too rapidly to allow adequate depressurization, the crack will continue to propagate. Thus, whether a crack will run a great distance or quickly arrest depends on the relative rates of crack propagation and depressurization. Prediction of the outcome requires a computer code that compares the gas dynamics and thefracturedynamics. The framework for such RCP prediction has been developed [1], but an important task remains before RCP management can be implemented: A fundamentally sound, independent test for measuring the material property governing RCP and crack arrest. During the 1970s, a concentrated effort was made under the sponsorship of the U.S. Nuclear Regulatory Commission and the Electric Power Research Institute to understand and develop tools for managing rapid crack propagation and crack arrest [2, 3]. In this effort, methods such as optical caustics and dynamic photoelasticity were used to measure directly the instantaneous stress intensity at the tip of a rapidly propagating and arresting crack. The relationship between crack tip stress intensity and crack velocity (Figure 1) was established as the material property governing crack behavior in a structure. This relationship was later confirmed by Zehnder and Rosakis [4].
A Fundamental Material Property
a
Fig. 1. Relationship between crack tip stress intensity and crack velocity—a fimdamental material property, KiD(a).
Laboratory Test for Measuring Resistance to Rapid Crack Propagation
111
This relationship is not easily measured for most engineering materials, because the caustic, photoelastic, and strain gauge methods are either difficult or impossible to apply (formation of a large plastic zone at a crack tip or no photoelastic sensitivity). As a result, the relationship for such materials as the polyethylenes used for gas pipelines is not yet known. Thus a practical experimental method is needed to characterize this relationship. AN RCP MATERIAL PROPERTY TEST Key questions addressed by the efforts of the 1970s were whether the relationship between crack tip stress intensity and crack velocity was independent of specimen geometry, and whether crack arrest toughness could be assessed with a static analysis. Kalthoff et al. [5] were first to compare the dynamic stress intensity during crack propagation with the static equilibrium stress intensity factor. Testing double cantilever beam specimens under conditions that were initiallyfixedgrip, they showed the crack tip stress intensity dropped sharply when the crack suddenly began to propagate, and then remained relatively constant as the crack rapidly grew (Figure 2), crossing the falling static
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Fig. 2. Variation of crack tip stress intensity with crack length for rapidly propagating cracks in Araldite B epoxy resin [5].
178
A. BURGEL, T. KOBAYASHI AND DA. SHOCKEY
equilibrium curve. The dynamic stress intensity then decreased gradually, but remained above the static curve until the crack arrested,finallyreaching the static value after several oscillations. Kalthoff s results show that at early stages of rapid crack propagation, significantly less energy is delivered to the crack tip than would be predicted by staticfracturemechanics, but at later stages more energy than predicted is delivered. Thesefindingsillustrate the nonequilibrium energyflowat a rapidly moving crackfrontand show that the crack interacts with the specimen and its surroundings. Thus, the energy transfer at a moving crack tip is complex, but if energyflowcan be analyzed and quantified, an experimental procedure for measuring Ki£)(a) can be developed. In our attempt to develop a practical test method for measuring the resistance of a material to rapid crack propagation, we performed crack propagation experiments using PMMA compact tension specimens and looked for ways to simplify the procedure and unify the results. EXPERIMENTS Specimen Design Considerations and Experimental Setup A specimen geometry was sought that would encourage the rapidly propagating crack to remain on a plane normal to the applied load, produce a range of crack speeds in a single test, and simplify the interaction of reflected stress waves with the propagating crack tip. We considered double-cantilever-beam, compact tension, and single-edge notched specimens, and with the help of computational simulations, selected the specimen shown in Figure 3. IT
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Laboratory Test for Measuring Resistance to Rapid Crack Propagation
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Specimens 254mm wide and 366mm high were machined from polymethylmethacrylate (PMMA) plates 12.7mm thick. A 71mm-long starter slot was cut into the edge at midheight and blunted by drilling a 2.0mm-diameter hole at the slot tip. Holes for the loading pins were drilled 45mm. from the specimen edge and 91mm from the slot. This geometry provides a K-field which decreases with crack length, and thus was expected to result in a range in crack velocity during a single test. The large height was chosen to encourage the crack path to remain along the median plane. The blunted slot tip allowed the specimen to store enough energy to drive the crack at high speeds. Lesser quantities of stored energy and hence lower velocities could be achieved by drawing a sharp knife blade along the slot surface to reduce the bluntness effect. Figure 4 shows the specimen in the tensile loading machine. A standard load cell and LVDT recorded the load and specimen displacement. During the crack propagation event an electro-optical displacement transducer measured the pin displacements at 400 kHz, and strain gauges along the crack path measured the strain histories. A ladder gauge recorded crack front position vs. time and provided a measure of crack velocity complementing that derivable from the strain gauges. The dynamic stress intensity during propagation was determined by analyzing the records from six strain gauges located along the expected crack path and equally spaced at either 12.7 or 19.1mm., at the same positions as the rungs on the ladder gauges.
Fig. 4. Experimental arrangement showing loading configuration and instrumentation
A. BURGEL, T. KOBAYASHIAND D.A. SHOCKEY
180
In affixing the strain gauges to the specimens, care was taken to eUminate unwanted influences of higher terms in the stress distribution (series) and plastic zone effects. Strain gauges were appHed 3.18mm. from the corresponding crack tip positions along a line 61.7 degrees from horizontal; the gauge axis was at an angle of 66.8° from horizontal. Thus, the distance from the crack tip was much larger than the strain gauge length (0.79mm) and large with respect to the plastic zone radius (0.025mm). The positions and aUgnments of the strain gauges were chosen according to the technique proposed by Dally and Sanford for measuring mode-I stress intensity factors [6], although the strain gauges were mounted closer to the crack path than recommended to increase the sensitivity of the measurements. Relating Strain Gauge Signals to Crack Tip Stress Intensity To obtain stress intensity values from the strain gauge signals, a static calibration technique was used. Subcritical loads were applied to specimens containing sawn-in notches of various lengths with strain gauges affixed in the same configuration as in the rapid crack propagation experiments. The static stress intensity was computed from the load and notch length and correlated with the peak strains measured by the strain gauges. The procedure was carried out for pin positions of 25.4 mm and 63.5 mm from the specimen edge. The peak strain from the strain gauge normalized to the stress intensity factor was plotted against the relative position of the crack tip to the strain gauge location (Figure 5).
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Laboratory Test for Measuring Resistance to Rapid Crack Propagation
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All data points from all strain gauges and both pin positions followed the same characteristic curve and confirmed a unique relationship between the peak strain from the gauge record and the stress intensity factor. In addition, the peak strain values measured by the gauges provided a consistent and accurate measure of the crack tip position. Hence crack velocities could be determined from the signal times recorded by the oscilloscopes. Results Four experiments were performed. Loads ranging from 3270 to 4270N produced KQ'S from 1.86 to 2.40MPa Vm and velocities ranging from 158 to 271 m/s. Figure 6 shows typical strain gauge records. In each test the strains measured by the gauge nearest the notch tip were considerably lower than those measured by the other gauges. This is beUeved to be the effect of the blunt notch tip. The remaining five gauges recorded smoothly decreasing peak strains, and the dynamic stress intensity factors were determined at these crack lengths using a procedure that eliminates influences of creep.
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A. BURGEL, T. KOBAYASHIAND D.A. SHOCKEY
The four experiments produced crack velocities ranging from about 271 m/s to 158 m/s. In any given experiment, the crack velocity, as measured by the ladder gauges and confirmed by the strain gauge signals, decreased by as much as about 35% over a crack extension of about 75 mm. Figure 8 shows the variation of stress intensity with crack velocity using data from all experiments. The results are in accord with the crack arrest toughness (0.99MPa Vm) measured by Takahashi and Arakawa [7]. The stress intensity versus crack velocity curve has a shape similar to that shown in Figure 1.
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Laboratory Test for Measuring Resistance to Rapid Crack Propagation
183
INTERPRETATION OF RESULTS The results shown in Figure 8 provide the required data for computing the dynamic crack behavior: how fast and what distance the crack runs and when and where it arrests. However, the measurements of dynamic stress intensity require special expertise and are costly and time consuming. When different batches of material need to be routinely evaluated, such as polyethylene material for piping, such tests are impractical. A way was sought to simplify the procedure. The data from the four experiments in Figure 7 show a similar trend, suggesting the results can be unified by normalization. When normalized by the initial stress intensity, KQ, and the width of the specimen, W, the data points lie on a single curve, as shown in Figure 9a. The same normalization procedure applied to Kalthoff s data from DCB specimens also collapsed the data onto a single curve above a certain crack length. Figure 9b. Thus, a characteristic curve for each specimen geometry describes the dynamic crack behavior in the latter half of the specimen. Furthermore, after the crack has extended roughly half the specimen width (aAV = 0.4 to 0.6), the dynamic stress intensity decreases in a well-behaved manner. Thus, in the latter portion of the specimen the normalized curve may be useful as a master curve that permits KID vs. crack velocity to be determined quickly and easily.
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The procedure for measuring RCP resistance would be as follows. Once the master curve for a given specimen size and shape is estabUshed, for instance by the method used here, RCP tests are performed on specimens of the same geometry, but with no instrumentation other than a ladder gauge. From the crack length vs. time measurements, the crack velocity is calculated at each gauge point, and the KID for a specific crack length is obtained by referring to the master curve, then plotted as a function of the crack velocity to obtain the RCP material property (Figure 1). Several tests with specimens having different notch bluntnesses may be required to map out the KID VS. crack velocity curve sufficiently. DISCUSSION The early undulation of the dynamic curves in Figure 7 about the equivalent static curves attest to energy exchange between the propagating crack tip and the surrounding material in the specimen. Der et al. [8] also observed an initial (although much smaller) drop in stress intensity in their SEN specimens. However, the stress intensity behavior during the subsequent rapid crack propagation differed markedly from that in Figure 7 and from Kalthoff s result. The optofoUower records showed that loading point displacement remained constant during that phase of the experiment when the crack passed the strain gauges. This implies that no energy flowed out of the specimen during the time data were taken and thus that the propagation event was intrinsically controlled. One way to deal with energy balance and energy flow associated with rapid crack growth is to consider system compliance. Irwin [9] used compliance to derive the relationship between the stress intensity factor and the energy release rate, and Okamura et al. [10] used compliance to extendfi-acturemechanics to structural analysis. They showed that the crack arrest toughness of brittle materials can be assessed by considering the load variation and the stress intensity changes induced by changes in the specimen compliance during crack propagation (as long as the crack velocity is not rapid compared with elastic wave speeds). However, their evaluations were based on the equilibrium state; dynamic effects and nonequilibrium energy flow were not considered. We intend to pursue a compliance approach to analyze and describe the stress intensity vs. crack length behavior in RCP experiments. Specimen geometry has an important effect on stress intensity and crack history. Long, slender DCB-type specimens experience larger load drops and longer periods of constant K behavior than the CT specimen geometry. Taller, narrower SEN-type specimens have much smaller load drops and the dynamic stress intensity may not reach the static value until the crack approaches the specimen boundary. Thus considerable leeway exists for tailoring the specimen geometry to obtain the desired crack propagation behavior. The most suitable specimen design for a given application is best achieved with the assistance of finite element analysis and parametric studies. The significant simplification of the master curve concept notwithstanding, the method still requires expensive experiments to generate the master curve. Thus, an ability to generate the curve with fewer experiments would be welcome. Future work will attempt to gain an understanding of the specimen mechanics and material rate effects and to achieve the capability to predict the steady state portion of the KID VS. a curve.
Laboratory Test for Measuring Resistance to Rapid Crack Propagation
CONCLUSIONS 1. Normalizing dynamic stress intensity, KID, data with initial stress intensity, KQ, and plotting this quantity versus crack length unifies the data from RCP experiments and defines a master curve after a certain crack extension. 2. The steady-state portion of the RCP master curve lies above the static equilibrium curve. 3. Normalized data from DCB specimens reported in the literature confirm the CT specimen results reported here, and show that master curves are geometry-specific. 4. A slowly loaded specimen instrumented with only a load cell and a ladder crack velocity gauge forms the basis of a practical test method for determining the material property governing RCP and crack arrest. ACKNOWLEDGMENTS Financial support for this work was provided by the Deutsche Forschungsgemeinschaft and SRI International. We are grateful to Dr. M. Mamoun of the Gas Technology Institute for bringing this problem to our attention and for early discussions. We also thank our SRI colleague, K. Stepelton, for skillfully and meticulously applying strain gauges. REFERENCES 1. Kanninen, M.F., et al, (1997). Design and Technical Reference to Mitigate Rapid Crack Propagation in Polyethylene Pipes for Gas Distribution. Southwest Research Institute Final Report (Contract 5088-271-1822) to the Gas Research Institute. 2. Hahn, G.T. and Kanninen, M.F. (Eds.) (1977). Fast Fracture and Crack Arrest, ASTMSTP 627, American Society for Testing and Materials. 3. Hahn, G.T. and Kanninen, M.F. (Eds.) (1980). Crack Arrest Methodology and Applications, ASTMSTP 711, American Society for Testing and Materials. 4. Zehnder, A.T. and Rosakis, A. J. (1990). Dynamic Fracture Initiation and Propagation in 4340 Steel under Impact Loading, Intemational Journal of Fracture, 43, pp. 271-285. 5. Kalthoff, J.F., Beinert, J. and Winkler, S. (1977). Measurements of Dynamic Stress Intensity Factors for Fast Running and Arresting Cracks in Double-Cantilever-Beam Specimens. In: Fast Fracture and Crack Arrest, ASTMSTP 627, pp. 161-176, Hahn, G.T. and Kanninen, M.F. (Eds.). American Society for Testing and Materials. 6. Dally, J.W. and Sanford, R.J. (1987). Strain-Gage Methods for Measuring the Opening-Mode Stress-Intensity Factor, Kj. Experimental Mechanics 27 4, pp. 381-388. 7. Takahashi, K. and Arakawa, K. A parameter influential in dynamic fracture. In: Dynamic Fracture, pp. 8-19, Homma, H. and Kanto, Y. (Eds.). Proceedings of OJI Intemational Seminar on Dynamic Fracture, Toyohashi, Japan. 8. Der, V.K., Holloway, D.C. and Kobayashi, T. (1978). Techniques for Dynamic Fracture Toughness Measurements. Prepared for the National Science Foundation by Photomechanics Laboratory, Mechanical Engineering Department, University of Maryland, College Park Campus. 9. Irwin, G.R. (1956). Onset of Fast Crack Propagation in High Strength Steel and Aluminum Alloys, Sagamore Research Conference Proceedings 2, pp. 289-305. 10. Okamura, H., Watanabe, K. and Takano, T. (1973). Applications of the Compliance Concept in Fracture Mechanics. Progress in Flaw Growth and Fracture Toughness Testing, ASTMSTP 536, American Society for Testing and Materials, pp. 423-438.
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
187
RATE DEPENDENT FRACTURE TOUGHNESS OF PLASTICS Z. Major, R.W. Lang Institute of Polymer Technology, JOANNEUM RESEARCH Forschungsges.m.b.H Institute of Materials Science and Testing of Plastics, University of Leoben, A-8700 Leoben, Austria ABSTRACT To characterize the rate dependent fracture behavior of various engineering polymers, instrumented impact tests were performed with bending. and_ tensile type specimens in the testing rate range of 10"^ m/s up to 8 m/s. Load-tin^- signals were recorded using an instrumented striker and a fixture equipped with a piezoelectric load cell and strain gages, respectively. Furthermore, the time-to-fracture, tf, was detected with different strain gage types applied to the specimen side surfaces in the vicinity of the crack tip. The data reduction to determine rate dependent fracture toughness values was carried out according to different procedures (conventional force based analysis and "dynamic key curve" method) taking specific local crack tip loading rates into account. In the quasi-brittle failure regime, good agreement was found between the fracture toughness values determined using various specimen and loading configuratons and by force based and dynamic data reduction schemes. Also, there is a clear tendency for a decrease in fracture toughness with increasing impact rate for all materials investigated. However, the rate sensitivity of fracture toughness values strongly depends on the specific polymer type.
KEYWORDS Fracture toughness, rate dependence, bending and tensile type specimens, force-based analysis and dynamic data reduction, engineering polymers. INTRODUCTION For many engineering applications, impact fracture behavior is of prime practical importance. While impact properties of plastics are usually characterized in terms of notched or un-notched impact fracture energies, there has been an increasing tendency to also apply fracture mechanics techniques over the last decade [1, 2 and 3]. For quasi-brittle fracture, a linear elastic fracture mechanics (LEFM) approach with a force based analysis (FBA) is frequently applied to determine fracture toughness values at moderate loading rates. However, in high rate fracture testing, several problems are encountered due to dynamic effects (inertia effects, wave propagation, etc.) which may completely overshadow the true mechanical response of the material to be characterized [4, 5]. While, the control of dynamic effects for
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Z MAJOR AND R. W. LANG
plastics at impact rates up to 1 m/s frequently makes use of mechanical damping in the load transmission by placing a soft pad between the striker and the specimen, for intermediate impact rates above 1 m/s to 10 m/s, a dynamic technique referred to as dynamic key curve method (DKC) has been proposed [6, 7]. The objectives of this paper are (1) to compare force based and dynamic tests methods and data reduction schemes for both bending and tensile type fracture specimens, (2) to define the requirements and limitations for the applicability of FBA and DKC methods, and (3) to determine fracture toughness values of several engineering polymers over a wide range of loading rates (up to 7 decades). EXPERIMENTAL Materials For this study two amorphous engineering polymers, commercial grade poly(carbonate) (PC) (LEXAN 9030, GE Plastics, NL) and commercial grade poly(vinylchloride) (PVC) (SICODEX (grey), EVC, I), and two semi-crystalline engineering polymers, commercial grade poly(oxymethylene) (POM) (Hostafrom C2552, Hoechst AG, D) and P-nucleated poly(propylene) homopolymer (p"^-PP(H)) (Borealis G.m.b.H, A) were used. The materials PC, PVC and POM were supplied as extruded sheets with a nominal thickness of 10 mm, p^-PP(H) was compression molded to plaques with a thickness of 10 and 15 mm, respectively. Specimen configurations and Test Procedures To study the effects of specimen geometry and configuration on the dynamic specimen response and fracture toughness, plane-sided (ps) and side-grooved (sg) single edge notched bending (SENB) specimens (W=2B, a/W=0.5; specimen width, W and specimen thickness, B), Charpy-type specimens with dimensions of 10x10x55/40 (in mm), compact type (C(T)) specimens (W=40 mm, B=10-15 mm, a/W=0.5) and cracked round-bar (CRB) specimens (diameter, D=12 mm, a/W=0.5) were machined from the sheets and plaques and were subsequently notched and razor blade pre-cracked (see Fig. 1). Schematic illustrations of the bending type specimen test set-up (main parts: striker, fixture with anvils, specimen (instrumented for higher testing rates) are shown in Fig. 2. While an identical striker was used in all bending test set-ups, the anvil geometry and distance in the 2 test set-ups is different (anvil radius of 1 mm in the Charpy fixture; anvil radius of 5 mm in the SENB fixture). For tensile type fracture specimens 2 test set-ups were realized, to accommodate either the pin-loaded C(T), SENT and DENT specimens (Fig. 3(a)) or the grip loaded CRB specimens (Fig. 3(b)). While conventional grips were used for testing rates up to 0.1 m/s, a modified C(T) or CRB fixture was used with a slack adapter at higher testing rates (for more details see reference [4]). Fracture tests were performed using a high-rate servohydraulic test system (MTS 831.59 Polymer Test System, MTS Systems Corp., MN, USA). The striker for the bending type specimens and the fixtures for the tensile type specimens were equipped with transducer type strain gages (WK-05-125AD-350, MM, USA) and a piezoelectric load washer (Kistler 9041 A, Kistler AG, CH). The load-point displacement associated with the striker movement was determined from an LVDT signal of the piston. For impact rates above 1 m/s some specimens were also instrumented. In several test series, strain gages (CEA-06-32UW-120, same supplier as before) were applied in the vicinity of the crack tip.
Rate Dependent Fracture Toughness of Plastics
Charpy specimen
SENB specimen
1 r"
55
189
"^^^ 10-15
90-140
'Od:
W
£ » ^ [-•.CO
o6
C(T) specimen
50
10-* 15
41 ai2
CRB specimen
43 1
j
%a
^ ^
-L^I
28
_ . CN
)
20
Fig. 1. Bending and tensile specimen types and geometries for monotonic fracture tests.
Instrumented striker
V)
specimen instrumentation
1
Charpy type fixture TZ 4
1^*"^^
Fig. 2. Test set-up for Charpy-type fracture specimens.
Charpy specimen
Z MAJOR AND R. W. lANG
190
slack adapter
C(T) specimen
rT\
r\
^
V
moving loading pin
a
damping jacket
load cell C(T) fixture
K777A i?L damping jacket
CRB specimen
upper grip
r]
load ce
lower grip
(b) Fig. 3. Test set-up for tensile-type fracture specimens; (a) test set-up for pin-loaded specimens (C(T), SENT and DENT), (b) test set-up for grip loaded specimen (CRB).
Rate Dependent Fracture Toughness of Plastics
191
Data Reduction Quasi-static fracture toughness values, Kc were calculated according to equation (1):
^^-'^'^vwi BW
(1)
where W is the specimen width, Y(aAV) is the LEFM geometry factor and Fp is the peak force value obtained in the force-time curves (reasons for choosing the peak force, Fp, instead of FQ are described elsewhere [3, 4]). The geometry factors for SENB and C(T) specimens were taken from [7] and for the CRB specimen from [8, 9]. The dynamic fracture toughness, K
K, =
7"
^^vM''"{t = tr)
(2)
41/l/'/< 7 + V
where E is the elastic modulus of the specimen, S is the span length, Y is the geometry factor, Cs is the dimensionless compliance, Cs is the specimen compliance. Cm is the machine compliance, vo is the effective testing rate, tf is the time-to-fracture and k ^(cit/W) is the "dynamic key curve" (with ci the wave propagation velocity and t the time). At higher loading rates the load signal is not of sufficient quality to directly determine dK/dt values. Hence, to better compare the results of various specimen configurations the local loading rate defined as Kc/tf (secant method) was used as the relevant rate parameter. Moreover, the dynamic method also yields directly Kc/tf values. RESULTS AND DISCUSSION Typical examples of load-time traces of pre-cracked Charpy specimens at impact rates from 0.1 up to 3.3 m/s are shown in Fig. 4a for PVC. While the recorded non-damped signal up to about 0.5 m/s is of sufficient quality to directly determine the fracture force, Fp, significant force oscillations are visible on the signals for higher loading rates. PVC
PVC Charpy specimen, a/W=0.3 undamped
Charpy specimen 3.3 m/s
1
1 <^
,
''
,
t — 1
^ '
i
time t, ms
(a) (b) Fig. 4. (a) Effect of testing rate on the external load signal; (b) comparison of the external load signal and the instrumented specimen signal at high testing rates (>1 m/s) for PVC.
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Z MAJOR AND R. W. LANG
The control of dynamic effects at impact rates up to 1 m/s (in some instances somewhat higher) frequently makes use of mechanical damping in the load transmission by placing a soft pad (elastomer or grease) between the striker tup and the specimen [3,5]. Above about 1 m/s inertia effects overshadow the true mechanical response of the specimen. Due to such dynamic effects, the applicability of FBA is limited to loading rates up to about 1 to 2 m/s for bending type fracture specimens. Typical traces of striker signals (Ustdker) and specimen strain gage signals (Usg) also for PVC Charpy specimens but for a higher loading rate (3.7 m/s) are shown in Fig 4b. Detailed experiments with numerous materials and measurement techniques have shown [4; 7] that crack tip strain gage signals may be used to determine accurate time-to-fracture values, tf. Based on the results for tf, k ^" values were determined to be equal to 1 for most tests performed up to 8 m/s, except for PVC at test rates above 5 m/s, and for PC and for POM at test rates of 8 m/s, in which cases k^^ was calculated according to the definition in [7]. Examples of load-displacement curves for the tensile loading mode specimens of the C(T) and CRB type are shown in Fig. 5 for POM and for PC, respectively for testing rates of 10'^ m/s up to 6 m/s. Typical traces of external load cell signals (load, F) and specimen strain gage signals (Usg) also for same specimens a higher loading rate (7 m/s) are shown in Fig. 6. POM C(T)-ps specimen 0.01 to 7 rrVs
displacement s, mm
(a) (b) Fig. 5. Effect of testing rate on the external load signal for tensile-type specimens; (a) C(T) specimen for POM, (b) CRB specimen for PC intemat load . 00
2500-
2000-
1500-
' N
PC CRB, a/W=0.5 7 m/s
0-
D
1/
1000-
500-
\1
-0.2
>
external l o a d /
_,
^
(a) (b) Fig. 6. Comparison of the external load signal and the instrumented specimen signal at high testing rates (>1 m/s) for tensile-type specimens; (a) C(T) specimen for POM, (b) CRB specimen for PC.
Rate Dependent Fracture Toughness of Plastics
193
In contrast to the results with the Charpy bending type specimens in Fig. 4. the loaddisplacement results of the tensile load specimens in Fig. 5. apparently are of sufficient quality to determine fracture force values even at loading speeds above 1 m/s. As described elsewhere [11], the improved signal quality of the tensile mode C(T) and CRB specimens over bending type specimens is a result of the higher specimen and lower contact stiffness associated with the former specimen configurations as well as the more effective damping situation in the loading unit. Moreover, as a rule, the striker or other external load signal was considered of sufficient quality for data reduction whenever the shape of the external signal was similar to the instrumented specimen signal and the test duration indicated by each of these signals was approximately equal (i.e., difference of less than 10 %).
1x10-3
Charpy specimen a/W=0.3
2
D -g
0)
•?, t
1x10"'
iS 6
•
1x10""
A
•
PC PVC POM
•
p'-PP(H)
••»• « &
10"'
10°
testing rate
10'
v, m/s
Fig. 7. Rate dependence of time-to-fracture values, tf, based on crack tip strain gage measurements for all materials investigated. 3500
0.40
Q.
LU (0 _3 3 TJ O
E o E (0 c >» •a
3000
2500
2000 1E-5
yjH—I
1E-4
I miMl—•
1E-3
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I —
0.01
0.1
I—r-TTTTm,—•
1
10
••••••'I
100
1000
t i m e t, s
Fig. 8. Time dependence of dynamic modulus (E*) and Poisson's ratio (v*) for POM.
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Z MAJOR AND R. W. LANG
The essential parameters for the calculation of IQ are the time-to-fracture, tf, and the material modulus, E. Values for tf were determined via crack tip strain gage signals according to a procedure described in [5, 6]. The testing rate dependence of tf values are shown in Fig. 7. for materials investigated. Adequate values for the rate dependent modulus E (as well as for the rate dependent Poisson's ratio which enters into the proper definition of k^^ in Eq. 2) of the various engineering polymers used in this study, were determined experimentally [4]. A:i example of data is shown for POM in Fig. 8.
Q.
PC Charpy specimen, aA/V=0.3
(0 (0
^
c O) 3
• 4
K^ force based analysis K DKC method
2 3 ** O
f
(0
10'
1 m/s
10'
10^
II
10^
loading rate dK/dt, MPa-m^'^s""
(a)
POM Charpy specimen, a/\N=0.3
re
Q.
•
! •
tf) tf) 0)
k
c
O) 3
o
K^, force based analysis method K„ DKC method
^
0
!^
1
io'
io'
i(
loading rate dK/dt, MPa.m'''s'
(b) Fig. 9: Loading rate dependence of quasi-static fracture toughness values, Kc^^'^, determined by an FBA method and dynamic fracture toughness values, Kd, determined by the DKC method for Charpy type fracture specimens of; (a) PC and (b) POM.
Rate Dependent Fracture Toughness of Plastics A further purpose of this investigation was to check whether there is a continuous fracture toughness testing rate dependence when comparing force based quasi-static fracture toughness values of the low testing rate regime and dynamic fracture toughness values based on the DKC method in the high testing rate regime. Hence, loading rate dependence of quasi-static fracture toughness values, Kc, determined by an FBA method and dynamic fracture toughness values, Kd, determined by DKC method for Charpy type fracture specimens are shown in Fig. 9 for PC and POM. Good correspondence between FBA fracture toughness data and DKC based dynamic fracture toughness data is observed in an intermediate rate regime with a continuous match of rate dependent fracture data. Loading rate dependent, fracture toughness values ("apparent" Kc^^^ and Kd) for the four materials investigated are shown in Figs. 10 to 13. Each of these figures includes fracture data generated with various specimen configurations using the appropriate force based and dynamic data reduction method, respectively. The local loading rate, dK/dt, used in these diagrams as proper rate parameter was determined as Kc/tf. In all diagrams the onset of ranges for quasibrittle failure (based on load-displacement traces) and for valid Kic determination according to relevant standards [8,9] using the appropriate rate dependent yield stress values is indicated. In the loading rate regime of quasi-brittle fracture, good agreement of the fracture toughness values can be seen for all materials. That is, for a given material grade bending and tensile type specimens yield equivalent Kc values. Moreover, in the high loading rate regime, good correspondence was found between Kc (FBA method) and Kd (DKC method) values, thus corroborating the applicability of the DKC method for engineering polymers if appropriate rate dependent material properties (elastic modulus, Poisson's ratio) are used. As expected, in the ductile failure regime and in the ductile-brittle transition regime of the loading rate scale, where LEFM methods are no longer applicable, large differences in Kc^^^ values were obtained for various specimen configurations, as is shown in Fig. 13 for p^-PP(H). Nevertheless, in this case too, the apparent fracture toughness values, Kc^^^, of the different specimen types converge at in the quasi-brittle fracture regime at high loading rates. Finally, the results also show that the rate dependence of fracture toughness data over the investigated rate regime strongly depends on the specific polymer type. While for PVC and POM a continuous decrease in Kc and K^ values with increasing loading rate is observed, PC and (3"*"PP(H) are apparently much less rate dependent. CONCLUSIONS Based on the investigations performed covering 7 decades of loading rates, the following conclusions may be drawn with regard to the determination of rate dependent fracture toughness values of engineering polymers in the quasi-brittle failure regime: — LEFM methods are applicable to characterize the rate dependent fracture behavior of engineering polymers in the regime of quasi-brittle failure, yielding material specific fracture toughness values independent of specimen configuration. — When applying an appropriate damping procedure, FBA methods can be effectively used up to 2 m/s for bending type and up to 8 m/s for tensile type fracture specimens. — At even higher loading rates, for which no valid force-time signal for applying an FBA data reduction scheme can be recorded, the DKC-method may be used to determine dynamic fracture toughness values, Kd. This requires an adequate technique to measure the time-to-
195
196
Z MAJOR AND R. W. LANG
fracture (e.g., specimens instrumented with crack tip strain gages) and the characterization of proper rate dependent values for modulus and Poisson's ratio.
Q.
force based analysis, K^ • SENB specimen, aW=0.5 # Cliarpy specimen, aA/V=0.33 A C(T) specimen, a/W=0.5 ^ CRB specimen, aA/V=0.5 dynamic analysis (DKC), K^ !3i Charpy specimen, a/W=0.33
PC
(/) Q) C
valid K,^
O) 3 O
J
iS
quasi-brittle fracture mnr10-^
o
TTTTTTIl
10-^
r
TTTT1|—
10^
10°
10^
10^
10"
10=
10°
local loading rate dK/dt, MPa.m s" Fig. 10. Effect of local loading rate on fracture toughness of PC using various specimen configurations and data reduction schemes.
E 0.
10
force based analysis, K^ • SENB-PS specimen, a/W=0.5 # Charpy-PS specimen, a/W=0.3 A C(T)-PS specimen, a/W=0.5 dynamic analyis (DKC), K^ i2i Charpy-PS specimen, aA/V=0.3
PVC
AB (0 0)
valid K
• ; :
c
O) 3
o
ductile fracture
quasi-brittle fracture
o
rrrmi
10"'
10'
loading rate
10^
dK/dt,
mri—
10^
10'
MPam^V
Fig. 11. Effect of local loading rate on fracture toughness of PVC using various specimen configurations and data reduction schemes.
197
Rate Dependent Fracture Toughness of Plastics
force based analysis, K^ SENB-PS specimen, a/W=0.5 • Charpy-PS specimen, aAA/=0.3 O Charpy-SG specimen, aAA/=0.3 A C(T)-PS specimen, a/W=0.5 dynamic analyis (DKC) Charpy-PS, K^ specimen, a/W=0.3
•
OS Q.
10-
POM
ia
Q. "^ Q.
valid K,.
(0
°
0)
c
1• « . A
o
^
O) 3
o
^
ia
quasi-brittle fracture rTTT]
TTTTTT
o
1—I
10°
10"'
10"^
I 11 I I I ]
1—I
10'
I I iiii|
I—I
r I iiii|—
10'
rTTT]
10'
1
I
10'
I
10'
local loading rate dK/dt, MPa.m s' Fig. 12. Effect of local loading rate on fracture toughness of POM using various specimen configurations and data reduction schemes.
• • D O gi
plane sided specimens side-grooved specimens DKC method valid K
5
(/) 0 C U) 3 O
4-]
KJDKC)
3-] 2-1 1
ductile """I
10"^
10-'
mni
10'
quasi-brittle fracture
transition
n
rTTTT|
10°
10'
1
rTTTT—
10'
local loading rate dK/dt,
rTTTT,—•
10'
I
m m |
10'
10'
MPam s'
Fig. 13. Effect of local loading rate on fracture toughness of (3'^-PP(H) using various specimen configurations and data reduction schemes.
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Z MAJOR AND R. W. LANG
— Over equivalent local loading rate ranges, experiments with tensile type specimens using an FBA data reduction and experiments with bending type specimens using the DKC da a analysis method yield equivalent fracture toughness values. — In the regime of quasi-brittle failure, there is a clear tendency for a decrease in fractuie toughness with increasing impact rate for all materials investigated. However, the ra':e sensitivity of fracture toughness values strongly depends on the specific polymer type. REFERENCES 1. Williams, J.G. (1984). Fracture Mechanics of Polymers, Ellis Horwood Series in Engineering Science, Chichester. 2. Leevers, P.S. and Douglas, M. (1999). In: Limitations of Test Methods for Plastics, ASTM STP 1369, J.S. Pesaro, (Ed.) ASTM, West Conshohocken, PA. 3. Beguelin, Ph. and Kausch, H.H. (1995). In: ESIS 19 Impact and Dynamic Fracture of Polymers and Composites, J.G. Williams and A. Pavan (Eds.) Mechanical Engineering Publ, London, pp. 93-102. 4. Major, Z. (2002). PhD Thesis, University of Leoben. 5. Pavan A. (1998). In Fracture from Defects, pp. 1363-1368., Brown. M.W., de los Rios, E.R. and Miller, K.J., (Eds)., EMAS, ECF 12, Sheffield. 6. Bohme, W. (1998). Application of Dynamic Key Curves (DKC) on the determination of the Impact Fracture Toughness, Kw of Plastics at High Rates of Loading „> 1 m/s", FIIWM, Freiburg. 7. Bohme, W. (1995). In ESIS 19 Impact and Dynamic Fracture of Polymers and Composites, J.G. Williams and A. Pavan (Eds.) Mechanical Engineering Publications, London, pp. 93102. 8. ESIS TC4 (1990). A Linear Elastic Fracture Mechanics Standard for Determining Kc and Gc for Plastics, Testing Protocol - March. 9. ESIS TC4, (1997). A Linear Elastic Fracture Mechanics Standard for Determining Kc and Gc for Plastics, Appendix 3 - High Rate Testing (draft 9). 10. Scibetta, M., Chaouadi, R. and Van Walle, E. (2000) Int. J. of Fracture 104: 145-168. 11. Major, Z., Lang, R.W. (1997) J. Phys IV France 7, C3-1005.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
199
NUMERICAL DETERMINATION OF THE ENERGY CALIBRATION FUNCTION gd FOR HIGH RATE CHARPY IMPACT TESTS A. RAGER, J. G. WILLIAMS, A. IVANKOVIC Department of Mechanical Engineering, Imperial College, University of London, SW7 2BXLondon, United Kingdom ABSTRACT The static load-based procedure for the evaluation of G is not valid at high rates because of dynamic effects. For high rate impact tests (>lm/s) on polymers, timing devices are used to determine the fracture time. The times to fracture are then used to fmd a static value of G from the displacement at fracture and the specimen compliance. Finally, this static value is corrected for dynamic effects to give the true value of G. The dynamic correction function to do this, gd, must be calculated, since there is no direct experimental method available for its determination. In this work, a fmite volume procedure with the crack tip closure integral method is applied for this purpose. The sensitivity of gd to the contact stiffness is investigated. The contact stiffness is varied by changing the elastic modulus of the striker and a significant change of the dynamic correction function is observed. Due to the nonlinear dependence on the contact stiffness the dynamic correction function also changes with impact velocity. Results from the crack tip closure integral are compared with the J integral method and close agreement is found between them. KEYWORDS Energy release rate, impact test, Charpy, crack closure, fmite volume, J integral. INTRODUCTION In recent years there has been growing interest in determining the fracture properties of polymers at high rates. For the critical quasi-static energy release rate, Gc, a standardised test procedure has been developed within ESIS TC4 [1]. This procedure uses the loaddisplacement curve of the fracture test to calculate Gc. The area under the curve is calculated and from this area the energy release rate may be found. If the test velocity is increased, dynamic effects become dominant and hence invalidate the load-based procedure. At moderate speeds a layer of damping material in between the striker and the specimen may be used to reduce the oscillations in the load signal. Following this damping approach a test procedure has been developed within ESIS TC4 [2]. This allows the determination of G at test speeds up to Im/s. For impact velocities above Im/s the load signal cannot be used in most cases as inertia effects and oscillations dominate. One way of obtaining G in this case is to
200
A. RAGER, J. G. WILLIAMS AND A. IVANKOVIC
measure time to fracture, tf. With tf and the constant impact velocity, V, the applitd displacement can be calculated. From the applied displacement and the stiffness of the specimen-striker system an equivalent load and hence a quasi-static energy release rate Gqs can be calculated. This approach has been used previously i.e. in [3] and [4]. This energy relea-.e rate must be corrected for dynamic effects, using a dynamic correction function, gd, which has been introduced for this purpose [4] with
(1)
g . = ^
where Gd is the dynamic energy release rate. In order to correct for the dynamic effects, this dynamic correction function must be determined and one way of doing this is with a numerical analysis, which is presented here. In order to cancel out the effect of the specimen size and different wave propagation velocities in the specimen a nondimensional time C,t
(2)
X=—
W
is used. W is the specimen width, t is the time and ci is the longitudinal wave speed in the specimen. This parameter was introduced by Bohme [5,6] for a dynamic correction function for the stress intensity factor. THE DETERMINATION OF G Two different procedures were applied for the calculation of G. The first procedure is the crack tip closure integral method. The crack tip closure integral was first introduced by Irwin [7]. Irwin's idea was that if a crack extends by a small amount, Aa, the strain energy released in the process is equal to the work required to close the crack back to its original length. Rybicki and Kanninen [8] used the crack tip closure integral method to calculate the stress intensity factor in a finite element program. The crack closure integral can be evaluated to givQ the mode I energy release rate
where Gyyt is the stress at the crack tip and Vt is half of the crack opening displacement. A finite volume (FV) interpretation of Equation (3) is given in Fig. 1. The advantage of this method is the simplicity of the calculation and the mesh insensitivity [8]. The second method used in this study is the J integral. Since the work of Eshelby [9] and Rice [10] the J integral has been used extensively in numerical routines for the evaluation of G, which is equal to J for elastic materials. For a stationary crack under dynamic loading it is possible to obtain a path independent far field integral [11]
^
du^^
r.V
' ^1 J
/ = j C/«, - a,n^ ^
8 u du^
\dY + jP^^dA.
(4)
Numerical Determination of the Energy Calibration Function g^
201
This integral includes an integration over the area AQ. FQ is a curve surrounding the crack tip (Fig. 2), beginning on one crack face, ending on the other face. Ao is the area enclosed by F, crack face
crack path
Fig. L The crack tip closure method.
Fig. 2. J-integral paths.
Fo and the crack faces. The curve F is shrunk onto the crack tip to give J. ni is a unit vector that is normal to F or Fo and that points away from the crack tip. ni is the component of ni into xi direction. U is the strain energy density and u is the displacement vector. The material density is p. The advantage of a far-field integral such as Equation (4) is that J can be evaluated along any curve surrounding the crack tip and the singularity at the crack tip, which is problematic in numerical models, can be avoided. B W
Fig. 3. The simple model.
FINITE VOLUME ANALYSIS The calculations for this paper were performed with the FV method, which is particularly suitable for dynamic, non-linear and large problems for several reasons: • It uses mass, momentum and energy conservation laws in their original integral form, which makes the method attractively simple, yet conservative. It lends itself to a segregated solution algorithm, thereby offering extremely efficient memory management, since the equations are linearized and sets of equations for each dependent variable are decoupled. • Equations are solved sequentially using an iterative solver. The technique is inherently suited for solving non-linear problems, where non-linearity arises either from material behaviour, geometry or boundary conditions. The routines developed are currently incorporated within a commercial package called 'FOAM' (Field Operation And Manipulation [12]), which is a C++ library of FV discretisation routines of continuum mechanics problems.
A. RAGER, J.G. WILLIAMS AND A. IVANKOVIC
202
V= 1m/s B
K—^
W
K# it 1
P
B
Fig. 4. The model including contact effects. Two different FV models of the same problem were generated. The first is a simple model neglecting contact effects and the second is a model, which includes a contact procedure. The numerical results were calculated for a specimen used in [5,6] with the following dimensions: Width, W=0.1m, length L=0.55m (LAV=5.5), span S=0.4m (S/W-4), notch depth a=0.03m (aAV-0.3) and thickness B=0.01m. In the simple model an impact velocity of Im/s is applied to one boundary face on the upper side of the specimen (Fig. 3) where the striker hits the specimen. For the anvil one boundary face is fixed and bouncing is allowed at both striker and anvil. A mesh size of 17600 cells is used and only half of the beam is modelled because of symmetry. Dynamic correction function g^, for a/W=0.3 and L/W=5.5, S/W=4 Contact influence
1,4 -
d? 1-2 •2 §
1.0
| o = -
8
1 0.6 Q
// \
^
/^
\ V/\/^/^^ //
\^-^
^^— ^O—
steel striker, steel anvil Cell faces displaced
0.40.2 00 < Nondimensional time x
Fig. 5. Comparison between simple model and model including contact. The second FV model includes a more accurate calculation of contact effects. A newly developed contact procedure is used in this work, which is based on implicit, and therefore very accurate, updating of the contact parameters: i.e. contact surfaces and forces. This
Numerical Determination of the Energy Calibration Function gd
203
procedure was used for both contacts at the striker and the anvil. The striker radius was 0.008m and the anvil radius was 0.01m. The mechanical properties of steel (E=210GPa, p=7800kgW, v=0.3) were used for anvil and striker, except for calculations where the striker stiffness was varied. The mechanical properties of epoxy (Araldite B) were used for the specimen (E=3.38GPa, p=1216kg/m^ v=0.33). Dynamic correction function g^, fora/W=0.3 and L/W=5.5, S/W=4 Striker stiffness influence
Nondimensional time x
Fig. 6. Striker stiffness influence on gd.
Dynamic correction function g^j for a/W=0.3 and L/W=5.5, S/W=4 Impact velocity influence
Nondimensional time x
Fig. 7. Impact velocity influence on gd. A constant displacement rate of Im/s was applied to the striker (Fig. 4). A locally refined FV mesh with 25875 cells was generated. Five levels of refinement on the specimen side and four levels on the striker/anvil side were used to have a sufficient number of similar sized cells in contact. At an average load there were around 20 cells in contact at the striker/specimen contact and around 10 cells for the specimen/anvil contact.
A. RAGER, J.G. WILLIAMS AND A. IVANKOVIC
204
RESULTS AND DISCUSSION The results in Fig. 5 -7 have been calculated with the crack tip closure method. In Figure 5 the influence of the contact procedure can be seen. The dynamic correction function gd rises more steeply initially for the simple model. The reason is that the contact stiffness resulting from the simple model is higher than the contact stiffness predicted by the code with the contact J integral path
00 H(t)
I t t t
t I t I , 24 mm ,
1
I
Big path Intermediate path
E o
1
E
Small path
104 mm
1 1 1 1 1 1 1 1
a=12 mm 52 mm
Fig. 8. Center cracked panel and paths for J integral evaluation. procedure due to a larger contact area in the latter case. The influence of contact stiffness is quite significant, which means that contact effects cannot be neglected. Therefore the effect of the contact stiffness on gd was further investigated by varying the elastic modulus of the striker. Fig. 6 shows the influence of the striker stiffness on the dynamic correction function gd. Not surprisingly the initial slope of the gd curve reduces with decreasing striker stiffness and furthermore the shape of the curve changes. The curve for E=3.4GPa is particularly interesting. This curve was obtained for a epoxy striker and specimen. Computations with a steel specimen and a steel striker actually showed virtually the same curve as for the epoxyepoxy combination. If the same material is used for specimen and striker, the shape of the gd curve changes drastically and it would lead to wrong results if the gd curve for a steel striker and epoxy specimen were to be applied in this case. Dynamic J integral versus crack tip closure for fine mesh
Dynamic J integral and cracl^ tip closure for coarse mesh a)
b)
SOT
*
40
3.0
20 -
1.0-
\0
' °*° °
A ^ < ^ o • . o D »
Crack tip closure Small path Big path Intermediate path
Crack tip closure Big path Intermediate path Small path
1 .OE-5
time [s]
Fig. 9. Results for the center cracked panel from J integral and crack tip closure. Fig. 7 shows the influence of the impact velocity on the calculations with the contact procedure. The impact velocity affects gd because of the nonlinear dependence of the contact stiffness on displacement. Due to increasing loads and therefore increasing contact stiffness the initial part of the curve, which is of most practical importance for high rate tests, rises more steeply with increasing velocity. As there were some doubts about the validity of the crack closure integral in the dynamic case, a J-integral expression from Equation (4) was implemented in the FV program. Because it was found that the J-integral is mesh sensitive and requires a very fine mesh, a test case of a center
Numerical Determination of the Energy Calibration Function gj
205
cracked panel was set up. The panel can be seen in Fig. 8, together with three different paths for the J integral evaluation. A quarter of the plate was modelled for symmetry reasons and a load step function ao-H(t), where ao=lMPa and H(t) is the Heavyside function, was applied. A coarse mesh with 260 cells and a fine mesh with 4160 cells were used. For the coarse mesh the J integral shows path dependence, but this is due to the sensitivity of the J integral method to spatial discretization (Fig. 9a). The converged results for the fine mesh from different paths agree (Fig. 9b). The results indicate that the J integral expression from Equafion (4) is path independent in the case of a dynamically loaded stationary crack. The J integral and crack Up closure yield the same results for a converged solution. The crack closure integral method is however less mesh sensitive and due to its simplicity easier to implement in a computer program. CONCLUSIONS The results of this study show that there is a significant influence of the contact stiffness on the dynamic correction function gd. A different striker stiffness will affect the shape of the curve. Furthermore, it was shown that there is also a velocity influence due to the nonlinearity of the contact stiffness. Two different methods for the calculation of gd were compared. The J integral gives values, which agree well with the crack tip closure method. The mesh sensitivity of the J integral is higher than the crack tip closure method, making it more expensive to use. Crack tip closure seems to be the most straightforward method for the calculation of G, at least for the finite volume method. REFERENCES 1. ESIS TC4 (1990). Testing Protocol. A Linear Elastic Fracture Mechanics Standard for Determining Kc and Gc for Plastics. 2. ESIS TC4 (1997). Testing Protocol. A Linear Elastic Fracture Mechanics Standard for Determining Kc and Gc for Plastics at High Loading Rates. 3. Williams, J.G., Adams, G. C. (1987) Int. J. Fracture 33, 209. 4. Williams, J.G., Tropsa, V., MacGillivray, H., Rager, A. (2001) Int. J. Fracture 107, 259. 5. Bohme, W. (1985) PhD Thesis, TH Darmstadt, Germany. 6. Bohme, W. (1995). In: Impact and Dynamic Fracture of Polymers and Composites, ESIS 19, pp 59-71, Mechanical Engineering Publications, London. 7. Irwin, G. R. (1958). In: Handbuch der Physik 6, p 551, Springer, Berlin. 8. Rybicki, E. F., Kanninen, M. F. (1977) Eng. Fracture Mech. 9, 931. 9. Eshelby, J. D. (1956). In: Solid State Physics, Vol. Ill, pp 79-144, Academic Press, New York. 10. Rice, J. R. (1968) y. appl. Mech. 35, 376. 11. Nakamura, T., Shih, C. F., Freund, L. B. (1984) Int. J. Fracture 27, 229. 12. www.nabla.co.uk.
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
207
THE THREE DIMENSIONAL STRESS FIELDS AT THE DYNAMIC CRACK TIP ASSOCIATED WITH THE CRACK BRANCHING IN PMMA
MASAAKI WATANABE Faculty of Engineering , Kinki University, Takaya Higashi-Hiroshima 739-2116, Japan ABSTRACT In order to investigate the cause of the microcrack branching instability observed by Fineberg, Gross, Marder and Swinney (1992, Phys. Rev. B45, 5146), we have calculated the stress fields at the tip of dynamic mode-I crack associated with the sound waves which form the standing waves across the elastic plate of PMMA (polymethyl-methacrylate) and modulate the stress intensity factor, Kj, of the crack as, 2 KjCff^ cos((ot - p ^x) cos p^z, where C)"^ is the amplitude of modulation. The quantity, o), is the angular frequency of the waves. The quantities, p^ and p^, are the wavenumbers in x and z-direction, respectively. The crack propagates in x-direction along the plane, y = 0. The boundary condition for the singular stress field, a^^ = 0, at the plate surfaces, z = ±>v/ 2, determines the quantity, p^, by the equation, cos P„w/2 = 0. The boundary conditions at the plate surfaces for the 1-st order stress fields, a^^^ = a^^^ = 0, give the following equation, 2 Pn^)['^sm(n + 0.5)jr = 0, where n is the integer and r, the distance from the crack tip. Combining this equation with the normalization condition of the modulation amplitude, 2 W = 1, we find that the interference patterns of the stress fields at the crack tip associated with the standing waves across the plate are enhanced by the order of magnitude depending on the choice of the mode numbers of the standing waves. The possible effect of this result on the microcrack branching instability is discussed. KEYWORDS Brittle fracture, dynamic crack branching, crack wave interaction INTRODUCTION Although much work has been done on the dynamic crack propagation in brittle materials for many years, the mechanism that governs the dynamics of a crack is not well understood [1,2]. Recently, the problems associated with the dynamic brittle fracture have attracted much attention in physics community. Fineberg, Gross, Marder and Swinney [3] performed a refined experiment of mode-I dynamic crack propagation in brittle material, PMMA (polymethyl-methacrylate). Improving the resolution of the crack velocity measurements, they have discovered the existence of a critical velocity, v^ = 0.36c^, beyond which the velocity of
208
M WATANABE
the crack begins to oscillate, where c^ is the Rayleigh wave speed. They have also found that the velocity of the crack, v, agree with the velocity obtained from the following equation.
v=c^{\-aja).
(1)
where a is the length of the crack at time t and a^, the initial crack length. They found that Eq. (1) corresponds within 1% to the velocity predicted by Freund's theory [4]. Taking the derivative of Eq. (1) with respect to time, one can easily find that the acceleration of the crack takes the maximum value at v/ c^ = 1/3, as shown in Fig. 1. The velocity, v/ c^ = 1/3, is very close to the critical velocity. Thus the acceleration of the crack could play an important role in determining the critical velocity. In order to take account of the effect of the acceleration of the crack we calculate the 1-st order stress fields, which is generated by the singular stress fields at the tip of the crack. d V
dr c
Fig. 1 The acceleration of the crack vs. the normaUzed crack velocity. The normalized time, r, is defined as r=-Cj^tlaQ, Another important feature of the experiment [3] is that the experiment is designed to finish before the sound waves reflected from the plate boundary interact with the propagating crack. The reason why they avoid the interaction between the reflected waves and the crack is that the waves strongly interact with the crack and change the dynamic behavior of the crack. Based on this fact we now ask the question, "What about the effect of the sound waves reflected from the plate surfaces and propagate with the crack by forming the standing waves across the plate. Do these waves change the dynamics of the crack?" The frequency of the sound waves which correspond to that of the standing waves across the plate are certainly detected when the branched microcrack grows for both of the materials, PMMA and glass [5] although the details of the frequency spectrum depend on the material. The similar effect for the standing waves across the plate to that for the reflected sound waves from the boundary, i.e., strong influence on the dynamics of the crack, is naturally expected to occur. We then assume that the stress intensity factor of the crack is modulated by these standing waves in the following. 2 KiC^I"^ cos{a)t - p^x) cosp^z,
(2)
where the quantity, Cj"^, is the modulation amplitude of the stress intensity factor, Kj, of the crack with the normalization condition, 2 Q^"^ = 1- We will calculate the stress fields up to the 1-st order at the tip of the crack, which is modulated by the sound waves. We then try to see whether the acceleration of the crack and the standing waves across the plate could have any effect on the microcrack branching instability.
209
The Three Dimensional Stress Fields at the Dynamic Crack Tip
THE BASIC EQUATIONS We consider a homogenous and isotropic elastic solid occupying three dimensional region. The equation of motion without body force is given as iJL = cfV(V• M) - q V XV X M ,
(3)
where the quantity, u, is the displacement vector field. The quantities, c^ and q , are the propagation velocities of the longitudinal and transverse sound waves, respectively. The displacement vector field, u, can be divided into the longitudinal and transverse components as w = M^ + M^ , where these quantities are determined by the following equations, VxM^ = 0, § - q V « , = 0, dt
(4a)
V-M, = 0,
(5a)
^.cyu,=0. at
(4b) (5b)
We introduce the translating coordinate systems {x^,y^ for u^, and (^pjj) ^^^ ^t ^i^^ i^s origin at the tip of the moving crack. The x^ axis is aligned with the initial crack plane, y = 0. These coordinates are defined as x^= X- a{t),
y. = a , j ;
a. = ^ 1 - v^/c^^,
/ = 1,2
(6)
where a{t) is the position of the crack tip. The z axis is perpendicular to the plate surfaces, which are located at z = ±w 12. The velocity of the crack is defined as v =dal dt. The displacement fields are expressed as Ui = ^^"\x,,y,,t)tx^[-i{(ot-p^x-
p^z)],
(7a)
u, = 2«/"'(^.,>'2,0exp[-/(cor- ftjr -/S„z)],
(7b)
n
n
where (o is the angular frequency of the sound wave. The quantities, p^ and p^, are their wavenumbers in x and z direction, respectively. The wavenumber, p^, will be determined by the boundary conditions at the plate surfaces. Substituting Eqs. (7a) and (7b) into Eqs. (5a) and (5b), respectively, we find .^ 1 .^.. ^ 2. dvdU,^"^ 2v y, da, d'-uj"^ 2v d'-uj"^ V\u,^^^{2i{wv-p^c',)-—}-^-^^^-^-f-, ! , a^c, dt dx, a^c, a, dt dx^dy, a,q atdx, V]u, = ^^^Kow-p^cl)-—}-^-^^-^-f-—f-, ; , a^q dt dx, a^c^ a^ dt ox^dy^ a^c^ dtdx,
(8a)
(8b)
where the quantities, V\ and V^, are defined as Vi=TT + TT, ox, dy,
(9a)
^l=TT + TT' dx, dy^
Eqs. (4) and (8) are the basic equations of the following analysis.
(9b)
210
M WATANABE
PERTURBATION ANALYSIS The lowest (0-th) order displacement vector fields, u\^^'^"^ and M,^^^'^"^ , are proportional to the square root of the distance from the crack-tip. They are given by the solutions of Laplace equations and their components are expressed as C'^"^ = Arl'^-cosiO, 12),
(10a)
^(o),(n) ^ ^^^112 ^^^^Q^ / 2),
MJ'^'^"^
(1 la)
= A^rl'\m{ej2),
(10b)
M^;^^' "^ = Brl"- sm{e^ / 2),
(lib)
where the quantities, A, Ay, B^ andB will be determined in the following analysis. Tiie quantities, {r^,6.), (i = 1,2) are defined and related with the polar coordinate (r,d) as, r. = ^xl + y^ ,
tand.=y./x^,
r = ^x^ +y^ ,
\3ne = y/x^,
tan0. = a. tan0,
^i=^lfiiO),
(L2a)
^(0) = (1-v'sin'0/cf)"'^^
(12b)
We note here that z-component of the 0-th order displacement field does not exist. The zcomponent of the displacement fields are the solutions of the Laplace equation. They are the 1-st order quantities which are proportional to r^^^ and given as ^iiun) ^ Dj-^^'\os(3dJ2),
(13a)
u[l^'^"^ = Ej^'\os(30J2).
(13b)
Another components of the 1-st order displacement fields are given as follows, ^n),(n) ^ r[^^{D,cosidJ2) + D^cos(3dJ2) +D^cos(5dJ2)},
(14a)
^a).(.) ^ r^'\D^,sm(dJ2)
(14b)
+ D^,sm(3dJ2) +D^,sm(5ej2)},
^a),(n) ^ ^3/2^^^^^^^^^^ /2) + E^^cos(302 /2) + E^^ cos(5a, /2)}, ^(i),(n) ^ r^'\E,sin(dJ2)
(15a) (15b)
+ E^sm(3d^ /2)+E^sm(5ej2)}.
The coefficients which appear in Eqs. (14a), for example, can be determined by substituting Eq. (14a) into left hand side (LHS) of the following equation, ,<„,<„, ^ V^lx
lijayy-p,cl)-dvldt ~
22
a<>'"> o
Ivy, da, a^<'"" 32
T.
o ^
2v a^<'-"" 2 2
^ .^
»
V ^W
which is derived from Eq. (8a). We can not determine the coefficient, D^, from Eq. (16) since the term proportional to D^ in Eq. (14a) is the solution of the homogeneous equation. Substituting the expressions of the quantities, uf^'^"^ dind uf^'^"^, respectively defined by Eqs. (10), and (13a), (14), into Eqs. (4a), we find the following relations, A^ = -a^A,
(17a)
D^=2ip^AI3,
(17b)
D^, = a,(D,-SD,),
(18a)
^,2 = « i A .
(18b)
D^^ = 2a,(ip^A + 2D^-SD^-3DJ2)I3.
(18c)
211
The Three Dimensional Stress Fields at the Dynamic Crack Tip
Substituting expressions of the quantities, u\^^'^"^ and uf^'^"\ defined by Eqs. (11) and (13b), (15), respectively, into Eq. (4b), we find the following relations. (19)
B^ = -a,B, ^.1
E^^=2a^(ip^B-2E,
(20b)
E^^ = a^E^,
(20a)
= a^{E^-^E^),
+SE^
(20c)
-3EJ2)/3.
In this section we have disregarded the suffix («) for such coefficients as A, B, D. and E., for simplicity. These quantities, for example, should be read as A-^A^"\
B-^B^"\
D,^ E^"\
(21)
E.^ E^\
The 0-th order stress and displacement fields The stress fields can be obtained by substituting the displacement fields into the Hooke's law, and then making use of the boundary conditions at both surfaces of the crack and the plate. Making use of the boundary condition for the 0-th order stress, a^^^^ = 0 at 0 = ±jr , and the definition of the stress intensity factor in terms of the stress a^^y\ we obtain the following equations,
(22a)
j^^^.J^A-\
M!l^M2(v)^
l + a^
(22b)
v2;r
2 ^(v) = - — ^ ^ ^ ^ ^ ^ ,
(23)
where the quantity, fn, is the shear modulus. We define such normalized stress, a^^\ and displacement field, u^^^, respectively, as ^ r ~ < \
(24a)
«(o) = ± . ^ « ( o > .
(24b)
The normalized 0-th order stress and displacement fields are given in the following, • ^ ^ = 2<^/"'{(l+2af - a ' ) / . ( 0 ) c o s ^ - ^ ^ ^ / , ( 0 ) c o s % c o s ( c o ? - 6 , x ) c o s f i 7, (25a) cr*°*
d
- f - = I2a,d-;'{f,{e)sin^F(v) n 2 ^=lC';\-(Ual)Md)cos^ r{V)
n
e
f,{e)sin-^}cos{wt 2
ftx)cos/S„z,
+ ^^f,(d)cos^}cosiwt-p^x)cosp„z,
Z i + O f j
TTZ = 2 ^ ; ^ - ( l - -^)A{e)cos-^cos(wt-p^x)cosp^z F(v) n q q 2
^
,
(25b)
(25c)
(25d)
212
M WATANABE
- ^ = 2 2 C r { — 5 - c o s ^ - ^ ^ ^ — c o s ^ } c o s ( « ) r - /3 x)cos)S„z,
(26a)
We note here that the boundary condition, a^^ = 0, at the plate surfaces, z = ±>v/ 2, determines the wavenumber, P^, as P^ =(2w+l);r/w,
(27)
where the quantity, n, is the integer. The 1-st order stress and displacement fields The method of deriving the stress and displacement fields is not difficult, however, the algebra involved in deriving the 1-st order fields is involved. We then show only the important steps in deriving these fields, which are slightly different from the conventional method [6]. Deriving the 1-st order stress fields, we obtain the following relations from the boundary conditions, o^^^ = 0, and o^^l = 0, at the crack surfaces, 6 =±ji, respectively as
= 0.5(3 - 5al)E^ + 0.5(3 + \9al)E^ - a^iD^ + 5D^ - 3DJ, E^ = -Aip^a^a^AI[3{\ + a^)].
(28a) (28b)
We define the 1-st order quantities, Kf^, as 2i^ftf^'^"^cos(co^-jS,jc)cosj3„z=
lim ^Ijilro^^^,
n
(29)
'
in which the modulation amplitude, Cf^^"^, of the 1-st order quantities, A'f ^, is included. The quantity, Kj^^, corresponds to the stress intensity factor in the 0-th order stress field, which is the natural extension from the 0-th order to the 1-st order fields. The functional dependence of the quantity, Kj^^, on the velocity and the length of the crack must be derived separately. Substituting the expressions of the quantities, a^^^^ in the right hand side (RHS) of Eq. (29) and using Eq. (28a), we find the following equation, 1.5D;, - /s:f^C^/^'^"V(V)/{J2^^) - ip^A(l- al)/(I + a^) - (D^ +5D^)I2 = 2F{v){ip^A{2 -al-q/cl)
+ {3- a\ -2c\ /c\)D^ + 2{3al -1 + 2cf / c^)D^
-2a^[{l- al)E,^-2{\ + 3a\)E^y{\ + al)}.
(30)
We define such normalized quantities, o^J^, D^ and ^ ™ = ^ a S ,
(31a)
D,=^^D„
(31b)
S-=^^^<'.
(3lc)
213
The Three Dimensional Stress Fields at the Dynamic Crack Tip We obtain various normalized coefficients by solving Eqs. (8), which are shown as follows, C^"^ a^
wvr q
v^ 2a^c^
d 2
2v dF F(v) dv (32b)
D^ = ^ < ^ ; \
a ^ a + a^)
q cf6 ^^ c^
c^ 4v^
v^
2v^
(1+a^)C2
2a^c^
a^c^
2v dF^^ F{v) dv
where the quantities, d and e^, are defined as follows,
^ = 4f.
(34a)
^^=^F^-
(34b)
In the following, we show the normalized 1-st order stress and displacement fields, - 2 ^ = l[-^{-^[{iP,rC"'
+A - 4A)cos^ + 2 A c o s ^ ]
+(1 + lal - a'){(-2/)S^rC)"^ - 30, + 16D2)cos^ + ( - ^ - 60^)00^—^g,
8/aijg,rv'C;"^
-^cos—^}
(5 - 3al)E, + (3 +19«,')£, - 2a,{D, + SD,).
^
+(£;-4£2)cos—^-f^cos—^} 3Ar/i o 2 2,cos(0i/2) Aa^a^oosiO^ll)^^ ^ .^ + —^{(1 + 2nt - al) :/ I 11 '}]cxp[-i(wt
« .. « - /3,^)]cos p„z , (35a)
^(v)
C2
n f,{0) r^.4cf
^
^ 2.;:.
q l+«2;;. 2
c^
. «2 rr 8/aiferv^C}"^
r
i
l
f2(d)
—
997"^"
q 3a
l+al~
10,
2
2
2
(5 -3a2^)£; + (3 + 19a|)£2 - 2a,(D, + 5 A ) . '7
(l + a;ycl , z^
. 7^ .
2
l + a^ 30^
~
10^ ^
I COS
^2
2
- ( E -4£,)cos--^4-£:,cos—^} ^ ^ 2 2 3 A . .. 2 . c o s ( a / 2 ) Aa.a.cos{O.I2\^ , ., ^ , ^ +-^{-(1+^2) :' +, S ' ;,' ' } ] e x p H M - j 8 , ^ ) ] c o s j 3 , z , (35b) 2 /i(0) 1+^2 /2(^)
214
M WATANABE
T f r = ( T ^ - 1)l7777[{4/iS,''(l + cxl)Cf
+ 2(1 + 3al)b,
- 3 2 a , ^ ^ + — A}cos - ^
+{-2-Di + 4(1 +3ai )/^}cos—^ - —D2COS—-\txp[-i{wt -
p^x)]cos(\z, (35c)
- ^ = 2 [ - ^ { 2 A s i n ^ +(A - 4 A ) s i n ^ - A s i n ^ } 1 r r l + « 2 ;; /2(0)^^ 2 '
3 - 1 3 a 2 ;; 2 -
-
^~ ^
+[ ^ £ ; + 2 ( l - 3 a , X ] s i n ^
^9 - ^ 2 ^
E,sm^}
^ ~ s i n ( a / 2 ) sin(0,/2) +3aA{'/^, + ;; ^}]exp[-/(a;^ - p,x)]cosp„z, /l(c^)
^
=^ a ^ l P . ^ ^ ^ ^ 1
J7(1)
R
~
^^>cosM-^..)sm^„z, n
"^
f^iO)
(350
R
^ 2 S a f t f f l _a,^ ^ ^ ^ (<•; -7)£; . a 3 - 3 „ ; ) f , 3 (l+a^) l+a^ 2(1+ a,^) ;; 5a. ~ cos(3a/2) 2aia^ cos(30W2) , 2
(35d)
/2(f^)
l + a2
3g, 2
/2(0)
(36a)
fliSy^
^3{\^alfc\ 3(1+a,^) 2a,(A+5A). . 30, ~ 50.
2
~ sin(3a/2) -«A( ,3,^, ^(0) ^z^
4^^
F(v)
3n
+ — i — i — r - ^ ^ l Sin—^ + E^ cos—^} 3(l+a') 2 ' 2^ 2 sin(30,/2),^ . . . . r ' , 3 ' O]exp[-z(co^-j3,^)]cosj3,z, l + a2 /2(0)
.„) cos(30i/2) /i(0)
2aia2Cos(302/2)^ l + a,
^
o x . ,
(36b) .0^ x
AC^)
The boundary conditions, o^^l = 0 and o^J^ = 0 , at the plate surfaces, z= ±wl 2, give the following equation, 2 Prr^\"^ sin( w + 0.5);r = 0 ,
(37)
The Three Dimensional Stress Fields at the Dynamic Crack Tip
215
since z dependent terms can be separated from other terms, which can be easily seen in Eqs. (35e) and (350DISCUSSION We examine the effect of the standing waves across the plate on "the three dimensional stress fields" at the crack-tip. We consider those sound waves which propagate with the crack, i.e., (o = p^v. In this case the quantity, a^^^, given by Eq. (25b), for example, can be written as, ^(0)
Q
r\V)
L
Q
--f- = 2a,{/,(0)sm^-A(0)sin^}2C;">cosi3„z.
(38)
An
Eq. (38) consists of the well known two dimensional singular stress field, o^y /^(^)|2D » multiplied by the factor, 2 Q^"^cos)5„z, which makes the field "three dimensional". Thus the n
Stress field is modulated by the standing waves across the plate and nothing new would have happened if we only consider the 0-th order stress fields. By taking account of the 1-st order stress fields, we find the additional equation (37) from which we determine the modulation amplitude, Cf^, which is normalized by the following equation.
2cr' = i.
(39)
When the two standing waves with mode numbers, n^ and n^, are present, the amplitudes, Cf^^ and Cf^^ are obtained from Eqs. (37) and (39). These amplitudes are given in Eqs. (40). The quantity, 2 ^/"^ cos ^^z, is explicitly given by Eq. (41) and shown in Fig. 2. P^ rsm{n^ +0.5)jt
(40a)
P^rsin(w2 + 0.5)ji -p^rsm(n^ + 0.5)Ji P^ rsin(nj +0.5)jr
(40b)
P^jsin(n^ + 0.5) ji - p^^rsin{n^ + 0.5)ji
0.5
Fig. 2 The interference pattern of the standing waves across the plate is shown for two modes with mode numbers, n^ =10 and /I2 = 12.
z/w
216
M WATANABE
2C;^cos)3^ = — n
p„^rsin(n2 + 0.5);r - p rsm(n^ + 0.5)Jt
^~-.
(41)
The interference pattern of the standing waves across the plate for the case, n^ =10 and n^ =12 shown in Fig. 2 shows the enhancement of the amplitude by the order of magnitude although the various different patterns can be obtained by changing the combination of the integers, n^ and «2. If the standing wave of our interest is the shear wave, the dispersion relation of this wave in a homogeneous material is given as
(42)
co'^4(pl+pl).
When the shear waves propagate through the elastic layer, or the elastic plate and reach the steady state, the type of the wave, SH wave for example, and it's dispersion relation are determined by the boundary conditions at the plate surfaces [7]. We have assumed that the sound waves modulate the stress fields at the tip of the crack, and then solved the wave equations with the boundary conditions at the surfaces of the crack and the plate. If the analysis is extended to derive the higher order fields and the dispersion relation of the wave is then obtained, such a wave do exist in the steady state. In this case we could confirm the existence of such "new wave" associated with the crack. Much algebra is required to obtain the higher order fields, however, it is not difficult to see the structure of the fields with the boundary condition at the plate surfaces. We find the boundary conditions at the plate surfaces for the second order stress fields are satisfied by the factor, cos p^z, in the similar manner to Eq. (25d). The boundary condition for the 3rd order stress fields, which is proportional to r^ ^, Oy^ = 0, at z = ±w / 2, for example, is given by the following equation,
I ^ . < > U - a / ^ - ^ ^ " > : - " \ ...f^^. .{Pl^Pl)cl-(o^ 6a\c\ 1
2a, \^a\
_ ^
^ ^ ^ . s i n O l ^
sin(^2/2) ^ '"\h{e)f
^
V 1 da.. 2 2a. sin(110W2)
+77T(^T
-TT),
\
,'
^ ' 5 '}sm(n+0.5);r = 0.
(43)
512 a^c^a^ dt \^a{ Vh{^n The whole expression is much more complicated and this equation is only small part of it. It is easy to see, however, that Eq. (43) can not be separated in such form, i.e., the product of two terms, as in Eq. (38). We then find that the angular frequency, 0), depends on the angle Q, which is irrelevant. From these argument we find that the sound wave which is assumed in this paper do not exist in the steady state condition since the dispersion relation of the wave cannot be obtained. When the sound wave generated from the branched microcrack or reflected from the boundary of the plate interact with the crack, the singular stress fields at the tip of the crack are modulated in such form as given by Eqs. (25), which are proportional to cos(co? - P^x). On the other hand the 1-st order fields consist of two terms which are proportional either to cos(ft>? - p^x) or sin(co^ - p^x). The latter term, for example, comes from the real part of such term, -^^/jS,rC}''>cos^exp[-/(cor-)3,^)]cosiS„z, / i ( " ) ^2
(44)
^
as given by the first term in RHS of Eq. (35a). When the condition, w = j3^v, with x =vt.
The Three Dimensional Stress Fields at the Dynamic Crack Tip
111
are satisfied we find no contribution from Eq. (44) since sin(co^ - p^x) = 0. When the sound waves modulate the stress fields of the propagating crack they could temporarily enhance the amplitude of the stress fields as shown in Fig. 2, for example, however, they will be scattered away since they cannot remain to be the coherent wave as discussed above. In order to calculate the stress fields at the tip of the crack numerically, which are given by Eqs. (25) and (35), we use the following model of the stress intensity factor [2], (45a)
Kj = k{v)a-Jjta,
(45b)
k(y) =
where the quantity, a, is the applied constant stress. Let us consider the case, Cj / C2 = 3 for the numerical analysis. In this case we find, c^ = 0.92c2 and h = 0.95. Making use of Eqs. (1) and (45), the parameters defined by Eqs. (34a) and (34b) are, respectively, calculated as 3 a
0.91c^ c^
vr 1 dk{v) ^ c] k{v) dt
(46a)
V la
(46b)
The quantity, v/ a, in RHS of Eq. (46b), for example, can be easily calculated from Eq. (1) as, V a
I Cp-v
dv
(47)
dt
Choosing the parameter values, r I a =0.1, c^ = 2.76 x 10 [m/s], which corresponds to the experiment of Fineberg, et. al. [3], we have numerically calculated the parameter, e^, in Fig. 3. The similar graph for the parameter, d, can be obtained by changing the vertical scale of Fig. 1 if the appropriate numerical values are substituted in RHS of Eq. (46a) . Since the numerical value of these parameters are rather small, the contribution from the 1-st order stress fields is expected to be small as well.
0.002 0.0015 0.001 0.0005
0.0005
tr?^^..,^^^ 0.4
0.6
0.8
/
1
vlc^
-0.001 0.0015
Fig. 3 The parameter, s^, defined by Eq. (46b) versus the crack velocity for the case, c^ /cl = 3 and r /a = 0.2. The hoop stress at the tip of the crack is defined as. OQQ = a^^sin^^ + o^^ cos^ 6 - a^sin 16
(48)
Making use of the stress fields given by Eqs. (25) and (35), we have numerically calculated the hoop stress at the center of the plate, zl w = 0, when the wave propagates with the crack, i.e..
218
M WATANABE
CO = p^v. In this case the hoop stress, shown in Fig. 4, becomes identical to the corresponding two dimensional one which can be confirmed by substituting the following three equations cos{(x)t - p^x) = 1, sin(a)t - p^x) = 0 and cos p^z = 1, into Eqs. (25) and (35) or by sim ply looking into the original assumption expressed by Eq. (2). The shape of the hoop stress is not much different from that of Yoffe [8] except the stress for the higher crack velocity. We find that the maximum of the hoop stress at 0 ?i 0 for the crack velocity vl c^"^ 0.6 disappears in Fig. 4, unlike that of Yoffe [8] when the 1-st order stress fields are included.
+ b(1) 1
ee
^"^-x^^ vlc2 = 0.8
0.8 0.6 0.4
N ^ V/C2=0 VIC2 = 0-6 ^ t > x ^
0.2 0.5
1 1 . 5 2
V
2^r-^^]x^0(rad.)
0.2 Fig. 4 The normalized hoop stress at z/w=0 for various values of the crack velocity, v/ C2 = 0,0.6,0.8, versus the angle, 0. cl/cl = 3, r la = 0.2. In order to visualize the effect of the standing waves across the plate, the three dimensional plot of the normalized hoop stress, a^Q + aQQ , is shown for the identical case of Fig. 2 at the fixed crack velocity, vl 0^ = 0.8.
zlw
3
0
0(rad.) Fig. 5 The three dimensional plot of the normalized hoop stress, OQQ + CTQQ , versus the angle, 6 (rad.) and z/w at the crack velocity, v/c2 = 0.8. «! = 1 0 , ^2 = 12, cllc\ = 3 , r / a = 0.2.
The Three Dimensional Stress Fields at the Dynamic Crack Tip
219
Before we conclude we should mention the following. Firstly the wave discussed in this paper is not related with the crack front wave, which is proved to exist theoretically [9] and experimentally [10], since we did not consider any effect associated with the shape of the crack front. Secondly we have numerically calculated the hoop stress at the center of the plate, ^ee + ^^86 y ^^^ shown in Fig. 4, which is identical to the hoop stress of the two dimensional crack. In this figure no hoop stress maximum at 0 ?s 0 is found up to the crack velocity, v/ C2 = 0.8 unlike the result of Yoffe [8], in which the quantity, a^^Q , at the tip of the two dimensional crack is calculated. In other words we find that the crack keeps to propagate along the plane of the crack, i.e., 6 =0, even though the crack velocity exceeds the critical velocity, v/ C2 = 0.6, for branching predicted by Yoffe [8], when the 1-st order stress fields are included provided the crack propagates in the direction along which the hoop stress at the tip of the crack takes the maximum value. We now discuss the experiments performed by Fineberg, et.al. [3, 5, 10, 11]. When the velocity of the crack exceeds the critical velocity, v^ = 036cj^, for PMMA, the microcrack branches [3], whose width is of the order of w/10, for example, is generated [11]. It is clear from this result that the three dimensional stress fields at the tip of the crack must be used to analyze the microcrack branching instability. Besides, it is experimentally known that the branched microcrack emits sound waves as it grows and the time scale of the frequency spectrum of this wave corresponds to the time scale obtained from the sound wave velocity divided by the plate thickness although the details of the frequency spectrum for PMMA and glass are different [5]. The authors of this experiment concluded that a theoretical explanation of the microcrack branching instability must predict both of the critical velocity, v^, and the characteristic frequency of the observed oscillation. Although it is not easy to fulfill this requirement, the sound wave generated by the branched microcrack seems to form the standing waves across the plate and interact with the propagating crack. In this paper we have calculated the three dimensional stress fields at the tip of the crack associated with the standing waves across the plate, assuming the waves modulate the stress intensity factor of the crack. Thus the present analysis takes account of these important experimental observations for the microcrack branching instability described above. We are unable, however, to clarify the direct cause of this instability but we have pointed out that the transient interference pattern of the standing waves could enhance the stress fields at the tip of the crack, as shown in Fig. 5, and possibly change the dynamics of the propagating crack.
REFERENCES 1.
Kanninen, M.F. and Popelar, C.H. (1985). Advanced Fracture Mechanics Oxford University Press 2. Freund, L.B. (1990) Dynamic Fracture Mechanics Cambridge University Press 3. Fineberg,J., Gross, S.P., Marder, M., and Swinney, H.L., (1992) Phys. Rev. B45, 5146 4. Freund, L.B. (1972), J. Mech. Phys. of Solids, 20, 129 5. Gross, S.P., Fineberg, J., Marder, M., McCormick, W.D. and Swinney, H.L. (1993) Phys. Rev. Letters 71, 3162 6. Freund, L.B. and Rosakis, A.J.(1992), J. Mech. Phys. of Solids, 40, 699 7. Achenbach, J.D. (1984), Wave Propagation in Elastic Solids, North Holland 8. Yoffe, E.H. (1951), Philos. Mag. 42, 739 9. Ramanathan, S. and Fisher, D.S. (1997), Phys. Rev. Letters 79, 877 10. Sharon, E., Cohen, G. and Fineberg, J. (2002). Phys. Rev. Letters 88, 085503-1 11. Sharon, E. and Fineberg, J. (1998). Philos. Mag. B78, 243
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
221
A DROP TOWER METHOD FOR HIGH RATE FRACTURE TOUGHNESS TESTING OF POLYMERS I. HORSFALL, C.H. WATSON, C.G. CHILESE Engineering Systems Department, Cranfield University, RMCS Shrivenham, UK
ABSTRACT This paper describes the use of an instrumented drop tower to perform high strain rate fracture testing of polymers using a method previously described by Williams [1] and Rager [2]. This is being investigated by Technical Committee 4 of the European Structural Integrity Society (ESIS) as a possible future test standard. The method utilises a high rate test on a single edge notched bend test (SENB) specimen, from which fracture toughness parameters are determined by analysis based on time to failure. Tests were conducted on three polymers PVC, PE and PMMA at impact speeds in the range Ims'^ to 16ms'^ Contact stiffness was measured at similar impact speeds using a similar geometry striker against fully supported specimens. Dynamic elastic moduli were also measured in compression tests by longitudinal compression using similar geometry but un notched samples from the same materials batch. Tests were also monitored by high-speed video in order to provide crack tip velocity data and to confirm the nature of the fracture process. Finally a comparison is made between the dynamic key curve analysis and mass spring model analysis for to determine dynamic fracture toughness. KEYWORDS Dynamic properties, fracture toughness, fracture energy, dynamic modulus, dynamic testing, impact tests, drop tower, high strain rate. INTRODUCTION High rate fracture parameters are important in the assessment of structural integrity, particularly in applications where there is a risk of rapid crack propagation. Linear elastic fracture mechanics provide a method for assessing fracture parameters under quasi-static conditions but under impact conditions various dynamic effects make both experimental measurement and analysis difficult.
222
/. HORSFALL, C.H. WATSON AND C.G. CHlLESE
For slow impacts at speeds of up to Ims"^ tests can be conducted in a relatively straightforward manner using a single edged-notched bend specimen. Damping pads between the striker and the specimen limit transient effects and the dynamic fracture toughness can be determined by measuring the load to crack initiation [3,4]. However, at higher speeds the time to crack initiation is comparable with the time taken by the stress waves to travel across the test piece. The influence of the resulting oscillations in the load signal make it impossible to accurately define the initiation load value. It is possible to overcome some of these problems by using an analysis based upon time to failure, rather than specimen load, and by applying a dynamic correction in order to account for the some of the transient effects. One method has been proposed by B5hme [5] based on investigations with epoxy specimens and with the use of caustics. The basic assumption of this method is that the crack tip loading history Kid(t) can be separated into a quasi-static part, Kst(t), and a dynamic correction function, k^^^(t):
(1)
K,,{t)^K,^{t)xk'^{t) The first term can be easily calculated by an analytical equation:
ExSxri^xVxtf 4 x f F ^ x r X 1 + -2—J—
\ where E S Y a W tf Cs* Cm B
c;
Specimen modulus Support span Srawley's relationship [6] Crack length Specimen width Time to fi'acture Bucci's formula for dimensionless specimen compliance [7] Contact compliance Specimen breadth
The second term k^^"(t) is a dynamic correction function that was determined in a model experiment by the evaluation of caustics. These result in a set of dynamic correction functions, which in a normalised form are called Dynamic Key Curves (DKC). A dynamic key curve has been established for a specimen having a relative initial crack length a/W = 0.30 ± 0.02, relative specimen length L/W = 5.50 ± 0.10, and relative support span S/W = 4.0-4.2. For the range of tf addressed in this paper (tf<9.2 W/Ci where Ci is the longitudinal wave velocity in the specimen) the dynamic key curve is given by the following approximation
fe^" =-0.9096+ 0.8176 x f - l ^ V 0.1005 x f _ i ^ l
+ 0.003765 x f j ^ l
(3)
This method has to be considered as an engineering approach, with the accuracy currently estimated to be about ±10%.
A Drop Tower Method for High Rate Fracture Toughness Testing of Polymers
223
An alternative approach based on a mass spring model (MSM) has been proposed by Williams [1]. The test system is represented by a lumped mass model with the contact stiffness between the striker and specimen being ki, this acts on an equivalent mass m which is 17/35 of the specimen mass, and the specimen stiffness is ki. A vital factor is the contact stiffness ki, which controls the dynamics of the system. In reality, this factor is not linear but for a case of the contact of a finite cylinder on plane used here, it can be approximated as linear. From this it is possible to calculate the natural frequency of the system o where
Eli
(4)
o> = ,[^l-^ \ m And Kid(t) can be expressed as
Kj)-K,{t)x
^ sm((ot)' wt
(5)
giving the appropriate value of the fracture toughness for t = tf This approach does not address the higher frequency effects so an improved mass spring model has been proposed by Williams et al [8]. The calculation is somewhat cumbersome and for practical purposes has to be solved using a mathematics package such as MATHCAD. This improved model is used in this paper to calculate both the dynamic fracture toughness Kd and the dynamic fracture energy Ga. It should be noted that both the MSM and DKC analyses require that the time to failure tf is accurately measured and that the modulus (and by implication the longitudinal wave speed) of the specimen is known. In addition to this the MSM analysis requires that the contact stiffness be known. HIGH RATE FRACTURE TESTS Impact tests were carried out on a Rosand Precision Ltd IFW8 instrumented accelerated drop tower. This uses a rubber bungee to accelerate a drop mass prior to impact with the specimen. The striker is equipped with a piezoelectric load cell, which is positioned as close as is practical to its tip. Figure 1 shows the effect of load cell position on the measured striker load during an impact test on a PMMA specimen at 2ms"^ The initial design of striker used in early tests had the load cell positioned 100mm behind the tip of the steel striker giving a mass in front of the load cell of 1 lOg. The output of this test shows considerable oscillation imposed on the load output both before and after the peak load is reached. The best practical arrangement was a titanium alloy striker weighing only 20g with the load cell 15mm from its tip. Even in this case there is significant periodicity in the load data and it is not clear whether this represents a true measure of the specimen load. Tests were performed on SENB specimens of geometry 10mm x 10mm x 55mm. These were mechanically notched and then a pre-crack was introduced such that the total length (a) produced an a/W ratio of 0.3±0.02. For the PE and PVC specimens the pre-crack was introduced by razor cutting. For the PMMA specimens a razor was tapped into the notch until a pre-crack was propagated by the wedging action of the blade.
224
/. HORSFALL, C.H. WATSON AND C.G. CHILESE 1400 —Tup to gauge 100mm 110g
I I
Load cell
--Tup to gauge 15mm 20g
1008
0.0P02
(c)
-600Time (s)
Fig 1. Initial (a) and final (b) striker designs with (c) the respective force output for a 2ms"^ impact on a PMMA specimen. Specimens were sputter coated with gold palladium alloy along their top edge to provide an electrical contact for detection of the striker impact. A 2mm length strain gauge was bonded to the specimen immediately adjacent to the tip of the pre-crack so that the crack tip stress field could be measured to determine the crack initiation time. A small voltage was applied to the contact strip in order that the start of the loading period could be accurately determined. The time to failure tf was then taken as the interval between the contact occurring and the time of the maximum strain detected by the crack tip strain gauge. The test layout is illustrated in figure 2. Following the impact test the actual crack length was measured on the broken specimen halves at five locations, equally spaced across the fracture surface; the average of these values was used in the analysis.
Fig 2. Typical fracture test setup with a PMMA specimen, showing contact strip connection, crack tip strain gauge, striker load cell and high speed video equipment.
Fig 3. Layout of the test for the contact stiffness test with an un-notched PE specimen.
225
A Drop Tower Method for High Rate Fracture Toughness Testing of Polymers
CONTACT STIFFNESS AND MODULUS In order to calculate the fracture properties from the SENB test it is necessary either to know or derive the stiffness of the specimen and the contact stiffness between the striker and the specimen. For this work the contact stiffness was measured directly by striking a fully supported and un-notched specimen fig 3. Striker force and displacement were measured and used to determine the contact stiffness. Test were conducted at an impact velocity of 0.5ms , and the contact stiffness was 3.7MPa for the PE, lOMPa for the PMMA and 7.69MPa for the PVC. Elastic modulus was measured by longitudinal compression tests using a screw driven test machine to achieve strain rates between lO'^s"^ and 10"^s"^ and a drop tower to test at strain rates lO^s"^- 10^s"\ Specimen load was measured using each machines load cell whilst strain was measure using a pair of strain gauges on opposing sides of the specimen. The same strain gauge amplifier was used in all tests, this being a Fylde FE-H359-TA strain gauge amplifier which has a frequency response in excess of 500kHz. RESULTS The results of the quasi static and dynamic modulus tests are shown in fig 5. It can be seen that the dynamic values are generally greater than the quasi-static values, which is as expected but there is also a decreasing trend amongst the dynamic values which is not expected. Unfortunately the dynamic modulus test suffers from similar transient effects as the force based fracture test with the load signal having a large periodic variations. At the lowest dynamic test speed the force signal undergoes typically 10 oscillations, at the intermediate value typically 5 oscillations and at the highest velocity only 2 or 3 oscillations. As a consequence the accuracy of the highest velocity tests are is doubtful. As the trend in the data is unclear it was decided not to use this data for further analysis but to use fixed reference values of elastic moduli namely PMMA 3.1GPa, PVC 3.15GPa and PE 1.05GPa.
6.00-
- A-PMMA -•-PVC
,...A"'
5.00-
. -A A
-^-HDPE
....-•* ' * ' * '
Q.
3 3
4.003.00-
"
2.00-
^.,=^-^...----^^-
,
, 0.001
0.01
«
•
-
-
1.00 -
0.1
^
•
^ 0
om1
10
100
strain Rate (s'"*)
Fig 4. Typical arrangement for the dynamic modulus test.
Fig 5. Quasi-static and dynamic modulus results for each of the three materials.
1000
/. HORSFALL, C.H. WATSON AND C.G. CHILESE
226
Figure 6 shows the output from the load crack tip strain gauge and contact strip for a 2ms"^ test on PVC. The time to failure is measured from the start of the fall in contact strip voltage until the peak in crack tip gauge signal, in this case tf is 203fis. Contact is also witnessed by a spike in the striker load cell output signal probably due to electrical interference as the contact ground is via the sheath of the load cell connection cable. It can be seen that there are large oscillations in all three signals with the contact strip signal indicating a loss of contact twice during the test. The three oscillations are of similar period but out of phase as might be expected as each is measured at a different location within the specimen and striker.
«
•
—
—
-^ ^
, tr If
^ • '^
7/ \
1
—load —strain ' • contact
UJ -0.3
-0.26
1-0.2
-0.15
-0.1
\
!
-0.05
-
1
—
%. ,JyKL£ *
'
•
-
0.05
0
Time [ms]
Fig 6. Load cell, crack tip gauge and contact strip output for a 1.85ms test on PVC Figure 7 shows a similar output for a 15ms"^ test on PVC. The measured tf is now 25ias which is significantly less than the first oscillation period for the 2ms"^ test. It is also apparent from the data that there is a significant delay between striker contact and the start of either crack tip or striker loading. It can be seen that the peaks in load cell and crack tip gauges are not coincident even for the lower velocity test. ~
.4
— « i ^
tf
. *
^
w
— load —strain "' - contact
1
v,^
k r
-o.to^
-WTT
^Z^^^^^. 0
0.01
0.02
^ \MH 0.03
v aWV 0.04
Time [ms]
Fig 7. Load cell, crack tip gauge and contact strip output for a 15.3 ms"^ test on PVC.
A Drop Tower Method for High Rate Fracture Toughness Testing of Polymers
111
Data from the SENB fracture tests were used to determine Kj value for the three materials using the MSM [8] and DKC [3] methods, the MSM method was also used to determine Ga values as shown infigs8-10. 3.5 _
A—PMMAMSMGd 3-1 • --PMMAMSMKd ^-> ' PMMA DKC Kd
\ 5
I 2.5 5
¥ 2
I
+3
i
2 tS {5 0.5 0 6 8 10 Impact rate [ms"'']
16
Fig 8. Dynamicfi-acturetoughness (from MSM and DKC) andfi*actureenergy for PMMA 3.5 3
— PVCMSMGd PVCDKCKd • - PVC M S M Kd
+5
% 2.5
E
3
Q.
«
2
§
1.5
+3 g
+ 2% 2
2 "0.5
A
LL
+1
A" 0
0 10 Impact rate [ms'"*]
15
20
Fig 9. Dynamic fracture toughness (MSM and DKC analysis) and fracture energy for PVC. 3.5 „
3
d 2.5 t4
Q.
I 2
3 %
I 1.5 I '
2 ^ 2
2 "0.5
1
0 6 8 10 Impact rate [ms'"']
12
14
16
0
Fig 10. Dynamic fracture toughness (MSM and DKC analysis) and fracture energy for PE.
/. HORSFALL, C.H. WATSON AND C.G. CHILESE
228
The measured Gd and Kd values are within the range of expected values for static properties. For all tests there was good agreement between MSM and DKC analysis at impact velocities of up to lOms"^ but the results tended to diverge at higher impact velocity. The PVC and PMMA data was quite similar in both absolute values and trends. The DKC results showed less tendency for change across the range of impact speeds whilst the MSM analysis tended to show increasing Kd at the higher impact speeds. It is not clear whether these trends are real and what mechanism might account for them. It is also apparent particularly in the lOms"^ tests that the DKC analysis tends to reproduce the scatter in measured tf between similar speed tests whilst the MSM analysis reduces or eliminates this scatter. The PE data showed a decrease in all parameters to a minimum at approximately lOms"^ with a slight rise towards 16ms"\ This is in agreement with other published data and is thought to indicate a reducing trend up to the point where crack tip thermal softening becomes significant and produces and increasing fracture resistance. Examination of fracture surfaces and high speed video footage indicated brittle failure in all cases for the PVC and PMMA. High-speed video observation of the tests showed typical crack tip velocities of approximately 70ms"^ in PVC and 55ms"^ for the PMMA specimens and did not vary as a function of impact velocity. These crack velocities are much lower than those reported in the literature which may indicate that an initial fast crack is quickly decelerating as it propagates through the specimen this might be expected as the frame period of the video of 25|bis provides a relatively low time resolution. For the PE samples, it was evident from the high speed video observation that the failure was of a ductile nature. The PE specimens showed a crack tip velocity that varied with and was approximately equal to the velocity of the striker, as would be expected for a ductile failure mechanism, it is also apparent from the fracture surfaces that there are significant shear lips which would hide the fracture propagation in the centre of the specimen. The presence of significant crazing near the pre-crack tip and along the specimen sides (fig 11) suggests that there is a significant ductile component to the crack propagation at all but the highest impact speeds. Tip of starter crack
Direction of fracture propagation 1ms-'
2ms-^
4ms-^
8ms'^
16ms-^
Fig 11. Fracture surfaces of PE specimens showing the variation of shear lip size, crack tip crazing and crack arrest marks as a function of impact velocity.
CONCLUSIONS A drop tower has been shown to be capable of performing high rate SENB tests and the associated tests such as elastic modulus and contact stiffness. Fracture toughness was assessed
A Drop Tower Method for High Rate Fracture Toughness Testing of Polymers
229
using the modified mass spring model and the DKC method which showed broadly similar results although the DKC method generally showed less variation in Kd with test speed. It might be expected that the large increase in Ka values for PMMA and PVC at higher test speed might produce markedly different fracture surfaces. The lack of any such differences tends to indicate that the relatively constant Ka values obtained by the DKC analysis are more likely to be correct. There is also evidence that the PE specimens failed in a partially ductile mode at the lower impact speeds with brittle fracture only becoming dominant above 10ms'\ This accounts for at least part of the decrease in Kd between test speeds of 2ms'^ and 5ms"^ ACKNOWLEDGEMENTS The author would like to acknowledge help of Prof J.G.Williams and Mr A. Rager for their help in analysis and sample preparation. REFERENCES 1 Williams, J.G. (1986) Int. J. Fracture 33,47. 2. Rager, A. (1999). Diploma Thesis, Imperial College, London. 3. Pavan, A. (1998). In: ECF12 - Fracture from Defects, pp. 1363-1368, Brown, M. W., de los Rios E. R., Miller. K .J. (Eds). EMAS, Sheffield. 4. ISO 17281:2002 ''Plastics- Determination of fracture toughness (Gic andKjc) at moderately high loading rates (1 m/s/\ 5. Bohme, W. (1995). In: Impact and dynamic fracture of polymers and composites, ESIS Publication 19, pp. 59-71,Williams, J.G. and Pavan, A. (Eds). Mechanical Engineering Publications, London. 6. Srawley, J.E. (1976) Int. J. Fracture 12,475. 7. Bucci, R.J, Paris, P.C, Landes,J.D, Rice, J.R. (1973). ASTM STP 514, ASTM Philadelphia, 40. 8. Williams, J. G., Tropsa, V., MacGillivray, H., Rager, A. (2001) Int. J. Fracture 107,259.
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
231
THE STRAIN RATE DEPENDENCE OF DEFORMATION AND FRACTURE BEHAVIOUR OF ACRYLONITRILE-BUTADIENE-STYRENE (ABS) COPOLYMER IN IMPACT TEST WOEI-SHYAN LEE and HUANG-LONG LIN Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC
ABSTRACT The impact properties and fracture characteristics of aciylonitride-butadiene-styrene (ABS) copolymer are investigated by using a servo-hydraulic machine and a compressive split-Hopkinson bar at room temperature over a strain-rate range of 10'^ s"^ to 4x10^ s"^ The effects of strain rate on the stress-strain response, Young's modulus, strain-rate sensitivity and thermal activation volume are evaluated. Scanning electron micrographs (SEM) of the fractured specimens are presented to illustrate the damage initiation and fracture mechanisms. It is found that the deformation and fracture behaviour of ABS are very sensitive to the applied strain rate. There is a significant increase in Young's modulus, flow stress and yield strength as the strain rate increases. By contrast, increasing strain rates lead to decreasing fracture strains. It is also found that the strain-rate sensitivity increases with an increasing range of strain rate, but decreases with strain in large strain-rate ranges due to the development of deformation heat. An inverse phenomenon is observed for activation volume. SEM examination reveals that damage is a complex process and occurs initially at two separate sites: 1) as microvoids and microcracks radiating from the center of the specimen; 2) as microcracks parallel to the specimen axis, starting at the equatorial midline of the cylindrical wall. Rapid microcracking as well as microvoid formation, growth and coalescence lead to final failure. The damaged specimens twist in a catastrophic manner as high strain-rate impact loadings are imposed. KEYWORDS ABS, strain-rate, activation volume, dynamic fracture, cracking INTRODUCTION Acrylonitrile-butadiene-styrene (ABS) copolymer, an important rubber-toughened thermoplastic, is widely used owing to its favorable cost/performance ratio. The advantages of ABS include its luster and resistance to impact [1-2]. ABS is therefore used in various instruments and structures for small elemental parts as well as large structural ones. In spite of
232
W.-S. LEE AND H.-L LIN
extensive ABS application, its critical mechanical behaviours such as plastic deformation and ductilefracturehave not yet been fully analyzed, especially in high strain-rate impact loading conditions. To assess the suitability of ABS for impacted-structure applications, it is essential to obtain dynamic mechanical properties and fracture characteristics corresponding to high strain rates for this material. It is well known that polymeric materials are very sensitive to strain rate and temperature. Some experimental data of polymeric behaviour at high strain rate have been reported in the literature [3-7]. These results indicate a significant change in mechanical response as the rate of strain is increased. With regard to the thermal effects during deformation, Buckley et al. [8] provided impact deformation data obtained for three glassy polymers and showed die specimen temperature to increase by between 20 and 30°C during compression impact tests on all three materials. Swallowe et al. [9] concluded from their impact tests on a range of polymers that very high temperatures (up to 700°C) can be obtained in materials which undergo catastrophic failure, and that the temperatures obtained are related to the material's thermal and mechanical properties. The effect of strain rate and heat developed during deformation on the stress-strain curve of plastics was studied by Chou et al. [4], who pointed out that the temperature rise developed during deformation cannot be neglected in determining the dynamic response of those tested materials. On the basis of these previous investigations, it is clear that both strain rate and temperature affect the flow response of polymers, and that these effects are often coupled during deformation and fracture studies. Unfortimately, information on the deformation and performance of ABS under high strain-rate deformation is generally unspecified. Thus, the aim of this paper is to characterize the impact deformation andfracturebehaviour of ABS at room temperature under strain rates ranging from 10"^ s"^ to 4x10^ s'^ by using a servo-hydraulic machine and a compressive split-Hopkinson bar. The changes in flow response as a function of strain rate are explored. Fracture features are presented to illustrate the eflfect of dynamic impact on damage initiation andfracturemechanisms. EXPERIMENTAL PROCEDURES Commercial grade ABS manufactured by Chi Mei Corporation of Taiwan, was used as-received and it contained 20% polybutadiene rubber. The average density, heat distortion temperature and mold shrinkage were 1.04 g/cm^, 110°C and 0.4%, respectively. The tested material was fabricated in a parallepipedic shape of 12x 12x 150 mm^ by the standard injection molding technology, at 230 C and a mold temperature of 50 °C. The impact specimens were machined on a lathe from the injection-molded ABS square bars. Their configuration was a cylinder of 8 mm diameter and 8 mm length. The surfaces of the cylinders were carefiiUy polished prior to testing to remove surface defects, and lubricated with M0S2 grease to reduce fiictional losses during impact. Quasi-static compression tests, with strain rates rangmg from 10"^ s'^ to 10"^ s"\ were performed by using an Instron universal testing machine. By recording the applied load and resulting displacement, the corresponding quasi-static stress-strain relationship can be established. The impact tests, with high strain rate from 10^ s"^ to 4x10^ s'\ were carried out in a compressive spUt Hopkinson apparatus [10], which consists of an incident bar, a transmission bar and a striker bar. Both the incident and transmission bars were madefrom12
The Strain Rate Dependence of Deformation and Fracture Behaviour ofABS
233
mm diameter 6061-T6 aluminum bars. The specimen was placed between the incident and transmission bars. The striker bar impacted the incident bar and generated a compressive loading pulse which traveled along the incident bar towards the specimen. Because of the variation of mechanical impedence when the wave passed through the specimen, this incident wave, Sj, was partly reflected (e^), and partly transmitted ( s j . From a one-dimensional theory of elastic wave propagation, the strain, stress and strain rate in the specimen are calculatedfromthe following expressions, respectively [11,12]:
'=^i>.
Bs(t)=^-^
S,(t)dt
a3(t) = E(A)e^(t) A. -2r es(t) = ~T^er(t)
(1)
(2)
(3)
where e^ (t), Cg (t) and 8^ (t) are the strain, stress and strain rate, respectively, in the specimen and they vaiy with time. E and Co are Young's modulus and the elastic wave velocity in the bars, respectively, while L is the initial length of the specimen and A/As is the area ratio between the bars and the specimen. After impact testing, thefracturedsurfaces were coated with a thin layer of gold prior to SEM examination. Observation of the fracture features was performed using a JEOL JX A-840 scanning electron microscope operated at 20 kV. RESULTS AND DISCUSSION Stress-strain relations The true stress and strain curves at various strain rates for ABS copolymer in comprossion at room temperature are shown in Fig. 1. In the strain-rate range being studied, ABS exhibits a ductile mechanical behaviour for all strain rates. At high-strain rate deformation, the stress-strain curve exhibits a yield point, followed by a drop in stress imtil fracture occurs. The flow softening becomes more obvious as the strain rate increases. In contrast, when the specimen is deformed at strain rates lower than 10'^ s"\ the yield stress is followed by strain softening. At higher strains, typical strain hardening is observed. It is found from Fig. 1 that ABS flow behaviour is strain rate dependent and is more rate sensitive at high strain rates. Table 1 lists ABS impact properties as afimctionof strain rate. As the stram rate increases, the yield stress ay (maximimi a in the experimental a-e curve) and Young's modulus both increase, and the strain atfracture8f decreases. A similar increasing trend with the strain rate can be found for the stress softening amplitude ^ssAC'^^j^'^pf)- The data of Table 1 also show that a lower stress softening amplitude corresponds to a higher fracture strain. This behaviour is similar to that in glassy polymer observed by Tordjeman et al.[13]. On the other hand, for strain rates lower than 10'^ s"\ the plastic flow stress c^^ (minimum a in the experimental a-e curve after the yield point) as well as the stress hardening amplitude
W.-S. LEEANDH.-L. LIN
234
^SHA (stress increases from a^^ up to the stress at true strain of 0.85) increase with increasing strain rate. Table 1. Impact properties of ABS deformed at various strain rates. Strain Rate, s"^ 10-^ 10-2
10-^ 10^ 2x 10^ 3x 10^ 4x 10^
'
E, GPa 1.2 1.3 1.4 2.6 3.1 3.6 4.0
MPa 43 48 52 98 108 113 120
1 '
1
120
'
1 ' 1 ' — • — 4x10's-»
-e— sxio's-' ^^^^
40
30
20 h
-\
\
L
1
0.2
± J 0.4
10's-' 10-2s-» lO-'s-i 1 1
0.8
^SHA>
MPa 4 5 6 31 32 34 35
MPa 3 4 5 -
1
1
1
4xlOJs-> 3x103 s-' 2x103 s-i 1x103 s-i
1
Cf
0.9 0.8 0.8 0.7
1
1
1
J^ / ^ ^
j
lOh
I^^^^^S-JT^ i^ l ii ^ *i ^ j•L j *y ^ —A— —0— —•— 1 1 0.6
^SSA»
i
A
-^^-' 2x10's-' — e — 1x10's-i
80
^
MPa 39 43 46 -
-
\ 1.
1 1 ...J
0
1
i 1 1 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 True Strain
True Strain
Fig. 2. Variations of temperature rise as function of plastic strain and strain rate.
Fig. 1. Typical true stress-strain curves at room temperature for various strain rates.
Since ABS copolymer has a large deformability associated with a poor thermal conductivity, the total work done on a specimen during the loading process may convert to deformation heat, which affects ABS flow behaviour, particularly at a high strain-rate loading. Thus, the effect of temperature rise during deformation must be taken into consideration. From the true stress-strain curves of Fig. 1, the temperature rise in a specimen can be calculated as a function of strain for each strain-rate level by an integral form of AT(8) =:-ipCy
fads
Jo
(4)
where a is the true stress, s is the true strain, p is the material density (1.04 g/cm ), c^ is the specific heat capacity at constant volume (plastic flow is essentially isochoric) and
The Strain Rate Dependence of Deformation and Fracture Behaviour ofABS
235
r| = 0.6 according to the work of Buckley et al. [8]. It should be noted that during initial loading (i.e., up to the maximum load), the main mechanism of deformation in the specimen is elastic straining accompanied by a small amount of plastic flow. Therefore, the heating effect prior to the maximum load is likely negligible. Figure 2 shows the calculated temperature rise for ABS as a function of strain and strain rate afler yielding. It is evident that temperature rise in the specimens increases markedly with both the strain and strain rate. At very high strains and strain rates, temperature rises up to 25 °C are expected. Our calculated results in the temperature rise of deformed specimen are consistent with that in glassy polymers measured by Buckley et al. [8] under impact strain rates between 10"^ and 5x 10^ s'\ Influence of strain rate on the strength The strain-rate dependence of flow stress can be described directly by plotting the flow stress against strain rate. By taking the temperature rise of specimens into account, the effect of strain rate on strength is best illustrated in a plot of o/T as afimctionof logs, as shown in Fig. 3. It appears there are two distmct regions corresponding to different strain-rate sensitivities. In the low strain-rate range, the ratio of flow stress to deformation temperature increases only gradually with the strain rate and can be approximately represented by a linear fimction of the logarithm of stram rate. However, when the strain rate rises above 10^ s"^ (i.e. the transition strain rate, 8^^), there is a sharp change in behaviour as evidenced by a sudden increase in slope of the ratio of flow stress to deformation temperature with the strain rate. This suggests that ABS is more strain rate sensitive at high strain rates. In addition, under low strain rate conditions, the ratio of flow stress to deformation temperature at afixedtrue strain of 0.6 is higher than that obtained at 0.4 which indicates that a strain hardening effect occurred at large strains. Moreover, it can be noted that for strain rates lower than 10'^ s'\ the strain-rate sensitivity is almost strain independent, while for strain rates higher than 10^ s \ a decreasing strain-rate sensitivity with increasing plastic strain is observed. 0.5
n"™"rT™T iiiiii^ iiiii^ 1111 Ilium limn iiiw|
\
0.4 L
1
•
yeild strain
O
e=0.2
2
o
6=0.4 8 = 0.6
JJU I f
A
iiiw
_ ]
0.3 h
\
//^
^ 0.1 ....J u i d .
t^
\
1
10"' 10"' 10"'iitid 10"'iiiid 10°mid lO' mid lO'inid lo' niidl lO'iiiJ lO' Strain Rate ( s ^
Fig. 3. Ratio of flow stress a to deformation temperature T as fimction of log strain rate under yield strain and different fixed plastic strains. Actually, the strain-rate sensitivity is often used as a useful basis for examining different aspects of deformation. From the unpact deformation standpoint, the value of strain-rate
236
W.-S. LEE AND H.-L LIN
sensitivity can be used to evaluate the relative augmentation of a material's strength as the strain rate is increased. In this study, the strain-rate sensitivity p, as defined by the slope of the flow stress versus the log strain rate relationship, is determined by the Fig. 3 in the following form: p=(aa/ahi8X,
(5)
According to the definition of strain rate sensitivity mentioned above, the values of strain rate sensitivity in both low and dynamic stram rate regions are calculated and listed in Table 2. It is found that dynamic strain rate sensitivity decreases with plastic strain, but low strain rate sensitivity is independent of plastic strain. Furthermore, the dynamic strain rate sensitivity is generally higher than low strain rate sensitivity for each true strain level. Table 2 Comparison of the static and dynamic strain rate sensitivity obtained at different plastic strain levels. Strain rate sensitivity
Plastic strain
(MPa)
yield
0.2
0.4
0.6
10-^ S-' - 10^ S-' 10^ s-^-4x10^ s-^
1.5
1.5
1.5
1.4
15.8
13.2
10.8
8.3
Estimation of thermal activation volume The onset of plastic deformation in ABS copolymer can be seen as a thermally and stress activated process. In thatfi-amework,the Eyring equation can be written in terms of strain rate and temperature as 8 = 8oexp[-(AH-v*a)/kbT]
(6)
where EQ is constant, AH is the activation energy for an activated-rate process, a is the ture stress, v* is the activation volume, k^ is the Boltzmann constant and T is absolute temperature. Takmg logs and rearranging gives AH . , 8 , r T r + ln(—)
(7)
In this form, the Eyring equation shows that the activation volume for a given single flow process can be obtained directly by plotting the flow stress against the logarithm of strain rate at constant temperature, provided that the observed strain rate is due entirely to that flow process. This version of the Eyring equation is implied in the expression of v* = k J ( a i n 8 / a a ) T
(8)
237
The Strain Rate Dependence of Deformation and Fracture Behaviour ofABS
Since the slopes of line fits in Fig. 3 (where a/T is plotted against In 8) are equal to kb/v*, the activation volume thus can be evaluated directlyfi-omthe equation (8) by using those slopes. Table 3 hsts the values of activation volume in both low and dynamic strain rate regions for ABS copolymer. As observed, a decrease of the activation volume with the plastic strain appears in large strain-rate range. However, the activation volume is mdependent of plastic strain for specimens deformed in low strain rate range. It is also foimd that in the ranges of strain rate studied, an increase of strain-rate range leads to a decrease of activation volume when a specimen is deformed at afixedtrue strain. In fact, the variations of v* with strain rate indicate that the relation between activation energy and stress changes with strain rate. Since the stress dependence of the activation energy is determined by the microscopic processes controlling plastic deformation, it follows that different mechanisms control the deformation over the range of strain rates. Hence, when v* is calculated on the basis of strain rates, it represents an average of activation volumes characteristics of different microscopic mechanisms. As demonstrated by Seguela et al. [14] and Haussy et al. [15], the plastic flow of glassy polymers varies strongly with thermal activation volume. This phenomenon impUes that different deformation modes and molecular mechanisms dominated the plastic flow behaviour under various temperature and strain rate conditions. Table 3 Comparison of the static and dynamic activation volume obtained at different plastic strain levels. Plastic strain Activation volume (nm^)
yield
0.2
0.4
0.6
10-^ s^ - 10^ s"^
2.0
2.0
2.0
2.1
10^ s-^-4x10^ s-^
0.19
0.23
0.28
0.36
Fractographic observations From fractographic analysis, it appears that the fracture characteristics of ABS copolymer depend strongly on the applied strain rate. In the case of quasi-static loading, nofracturewas found in the specimens. This could be due to the fact that ABS is already a tough material containing a soft component (polybutadiene). Under high strain-rate impact conditions, however, cracking and formation of microvoids become important. The damage initiates simultaneously in both the equatorial midline of the cylindrical wall and the central region of a specimen. The initiated microcracks and microvoids then propagate and coalesce into different fracture types depending on the imposed loading rate. Figure 4(a) shows a scanning electron micrograph of an ABS specimen deformed at a strain rate of 10^ s"^ It is evident that intensely localized microvoids and microcracks formed and radiatedfromthe center region of the specimen. The mainfractureevents during impact seem to be microvoid formation, void coalescence and subsequent ductile tearing of the ligament between voids. This can be seen more clearly m Fig. 4(b) which corresponds to the intensely localizedfracturezone in Fig.4(a) at a higher magnification. It should be noted that at this stram-rate level, only a slight microcracking appeared in the equatorial plane of the specunen, without any linkage to the central radialfractureregion. Figure 4(c) shows a typical fracture surface of afracturedspecimen after it had undergone
238
W.-S. LEEANDH.-L LIN
Fig. 4. Fracture surface micrographs of specimen deformed at: (a) strain rate of 10 s ; (b) high magnification of (a); (c) strain rate of 3x10^ s'^; (d) high magnification of (c).
Fig. 5 Fracture surface micrograph of specimen deformed at strain rate of 4xl0^s'^ deformation at a strain rate of 3x10 s" . Massive plastic deformation and catastrophic fracture occurred at the central region of the specimen due to the enhancement of applied stress. As expected, extensive microcracking at the equatorial plane of the specimen had linked to the
The Strain Rate Dependence of Deformation and Fracture Behaviour ofABS
239
central radialfractureregion, leading to breakage of the specimen along the loading axis. The fracture was dominated by cracking and tearing mechanisms. Consequently, the fracture surface exhibited the well-known parabolic markings and the convexity was oriented in the propagation sense, as shown in Fig. 4(d). Comparison of Fig. 4(d) with Fig. 4(b) shows that the fracture surface was more planar as the strain rate increased. This indicates a loss in ductility, which can be confirmed by thefracturestrain values reported in Fig. 1. Finally, at the highest tested strain rate of 4x10^ s"\ Fig. 5, a large proportion of thefracturesurface is covered with viscous slide facets owing to the high level of deformation heating. The fracture mechanism was similar to that observed at a strain rate of 3x10^ s"\ but the fracture morphology was quite different. Signs of ductile tearing associated with an intensive twisting along the specimen axis are clearly observed. Such twisting resulted from a spring-back of fractured material after an extreme impact loading. CONCLUSIONS The influence of strain rate on the impact deformation and fracture behaviour of ABS copolymer has been studied. The results show that the stress-strain response of ABS is very sensitive to the strain rate. The yield stress. Young's modulus and stress softening amplitude, GssA, increase with increasing strain rate. However, a decrease of the dynamicfracturestrain with the strain rate is observed, although nofractureof spechnen occurred under quasi-static loading conditions. Thefracturestrain is also related to agg^. A low Ggg^ indicates a more ductile behaviour. ABS exhibits a bilinear relationship between the ratio of flow stress to deformation temperature and log strain rate and the strain-rate sensitivity increases at strain rates about 10^ s"\ The observed linearity of the dependence of flow stress on strain rate obeys the Eyring theory. In addition, the activation volume decreases with increasing range of strain rate, but due to the dynamic softening effect, an increase of activation volume with increasing strain is foimd at high loading rates. SEM analysis reveals that damage initiates shnultaneously from both equatorial plane and central region of a specimen. Rapid microcrack and microvoid growth and connection lead to fmal fracture. The damage specimens twist m a catastrophic manner as high strain-rate impact loading is imposed. REFERENCES 1. Dear, J.P. (2000) Polym. Testing 19, 569. 2. Chen, C.C. and Sauer, J.A. (1990) J. Appl Polym. Scl 40, 503. 3. Tay, T.E., Ang, H.G. and Shim, V.P.W. (1995) Compos. Struct. 33, 201. 4. Chou, S.C, Robertson, K.D. and Rainey, J.H. (1973) Exp. Mech. 13,422. 5. Ciferri, A. and Ward, I.M. (1979). Ultra-High Modulus Polymers. Applied Science Publishers LTD, England. 6. Ward, I.M. and Hadley, D.W. (1993). An Introduction to the Mechanical Properties of Solid Polymers. John Wiley & Sons, New York. 7. Nielsen, L.E. and Landel, R.F. (1994). Mechanical Properties of Polymers and Composites, Marcel Dekker, Inc., New York. 8. Buckley, C.P., Harding, J., Hou, J.P., Ruiz, C. and Trojanowski, A. (2001) y. Mech. Phys. Solids. 49,1517.
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9. Swallowe, G.M., Field, J.E. and Horn, L.A. (1986) J. Mat. Scl 21,4089. 10. Lindholm, U.S. (1964) J. Mech Phys. Sol 12, 317. 11. Nicholas, T. (1982). In: Impact Dynamics, pp. lll-'i^l, Zukas, J. (Eds). John Wiley & Sons, New York. 12. FoUansbee, P.S. (1985). In: Metals Handbook, 9'^ ed., Vol 8, pp. 190-207. 13. Tordjeman, P., T6z6, J., Halary, J.L. and Monnerie, L. (1997) Polym. Eng. Scl 37, 1621. 14. Seguela, R., Staniek, E. Escaig, B. andFillon, B. (1999)/ Appl Polym. Scl 71, 1873. 15. Haussy, J., Cavrot, J.P., Escaig, B. and Lefebvre, J.M. (1980)/ Poly Scl 18, 311.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
241
ELASTIC AND VISCOELASTIC FRACTURE ANALYSIS OF CRACKS IN POLYMER ENCAPSULATIONS
O. WITTLER and P. SPRAFKE Department of Plastics Engineering, Corporate Research and Development, Robert Bosch GmbH, D-71332 Waiblingen, Germany B. MICHEL Department of Mechanical Reliability and Micro Materials, Fraunhofer IZM, D-13355 Berlin, Germany ABSTRACT Cracks inside polymeric packaging materials, which are used for protection against environmental impacts and isolation of electronic components, can lead to failure of the whole system. Therefore the understanding of crack initiation and propagation becomes vital for the design of reUable microsystems. In this paper the loading situation of cracks inside the polymer is analysed considering an encapsulated metal structure. The analysis is based on finite element simulations of thermally induced stresses, where rate-dependent material behaviour is being taken into account. Different fracture criteria like Schapeiy's Work of Fracture, the modified virtual crack closure integral (MCCI), the path independent J-Integral, and the stress intensity factor are being compared and evaluated with respect to their applicabiUty to thermomechanical loading situations (cooling). A good agreement with experimental results is achieved, when the actual load Ki, loading speed dKj/dt and temperature is used to describe critical loading situations. The obtained results are applied to the analysis of a demonstrator. Elastic and viscoelastic simulations are compared and effects of loading rate are investigated. Thus a methodology is being illustrated which enables the consideration of rate dependent material behaviour in a fracture mechanical analysis under thermomechanical loading. KEYWORDS Fracture of polymers, viscoelasticity, crack growth initiation, finite element simulation, crack closure integral, thermomechanical reliability, rate dependence
242
O. WITTIER, P. SPRAFKE AND B. MICHEL
INTRODUCTION The application of electronic components under varying temperature conditions leads to mechanical stresses, as materials with different thermal expansion are combined. This can cause crack initiation and growth inside the encapsulation and failure of the whole component. Therefore the avoidance of such cracks plays an important role during design and testhig. Finite elements simulations of thermomechanical stresses can lead to an understanding of failure modes. For a good benefit of such simulations, models need to be developed, where reality and model coincide to a sufficient degree. The main concern of this paper is going to be the time dependent stress-strain material behaviour and the fi-acture behaviour of the encapsulation material where afilledepoxy is used as an example. Many publications show that a good correlation of thermomechanical simulations and measured deformations is obtained, when a viscoelastic material model is applied for the description of stress-strain-behaviour [1,2]. The difference between elastic and viscoelastic model is not generally negligibly small, either [3]. Now the question arises, what kind of fracture concept needs to be applied for such simulations. The critical parameters of such a concept should be measurable as well as applicable to a thermomechanical loading situation, which is the situation, where loading of electronic components can become critical. Also, it should be possible to calculate the fracture parameter in a finite element model. The stress intensity factor appears to be quite appropriate, as it is used in many analyses of viscoelastic crack growth, which were reviewed by Bradley et al. [4]. To explain the rate dependence of Kic the approach of Frassine et al. showed on different thermosets that the Work of Fracture, which was introduced by Schapeiy [5], is constant for different loading rates [6,7]. But this concept was not directly implemented in finite element simulations. Here modified crack closure integrals [8] or generaUsed domain integrals [9,10] find their field of application. To our knowledge a combined simulative and experimental investigation has not yet been undertaken. Also an evaluation of these presently available concepts with respect to thermomechanical loads needs further investigation. Therefore this paper starts with a short theoretical introduction of these concepts. Their validity in thermomechanical loads is evaluated and analysed. On a simplified application example it is shown what benefit can be expected from a viscoelastic fracture analysis with respect to the simpler pure elastic simulation. THEORY Schapeiy *s Work of Fracture One of the most comprehensive theoretical works about viscoelastic fracture is provided by Schapeiy [5]. Here the crack is formed by a process zone which is surrounded by a viscoelastic material. The process zone is assumed to be negligibly small, leading to a relation of the stress intensity factor Kj and the work per unit area ^done on the zone: (1) Wf = \C{t -f)^f at C is the so-called plane-strain viscoelastic creep compUance. It is connected to the relaxation modulus E(t) and Poisson's ratio v(t) by
Elastic and Viscoelastic Fracture Analysis of Cracks in Polymer Encapsulations
243
sE (2) where the bar denotes the Laplace transformation of a fiinction of time and s is the independent variable in the Laplace room. Later publications restrict the Poisson's ratio to be constant in time [11,12]. In many cases this assumption may hold, but if temperature changes are considered with the time-temperature superposition principle, this assumption is no longer valid and the older formulation needs to be used. Also later work introduces correspondence principles and a path independent Jrhitegral, which gives the energy release rate for the reference elastic body [12]. Many current finite element tools can evaluate the 3-dimensional solution of a thermomechanically loaded viscoelastic model in a nonlinear analysis. Therefore the reference elastic body is not needed in this case and the J^-Integral is not discussed in this paper.
Modified crack closure integral To evaluate crack load in finite element simulations, a virtual crack extension was used by Dubois et al. [9]. Xiong et al. [8] applied a modified crack closure integral to analyse viscoelastic effects on interfacial delamination. Both approaches lead to the same equations, which can be generalised for the 3D case: G *^ = lim fcr„ (Aa - r)/SM^ {r)dr 0 Aa
G *jj = lim f(Ti2 (Aa - r)Au^ {r)dr
(3)
0 Aa
G *jjj = lim f(723 {Aa - r)Au^ {r)dr "^
0
where Aa is the amount of crack advance in xi-direction. A local coordinate system corresponding to figure 1 is assumed and G* = G*j +G*jj +G*jjj is the energy dissipated at an unit step of crack propagation, which can be separated into three modes. An asterix is added to G to denote the calculation method. In the elastic case it equals the classical energy release rate. The stresses cry are given for (p =0 and the relative displacements Aui = ui^-uf are given through the displacements of the upper crack face ut {(p= ^) and the lower crack face u,' i
(4)
ut-±K^P^.ut=±Kr^^,ut.-±K^^IIThey are related to the stress intensity factors Ka by
K':\t)=]c^{t-t')^t' 0
(5)
^*
for a=I,II,III and the viscoelastic creep CompUance Ca(t), which is given for a = / a n d planestrain situations in equation (2). Following Dubois et al. [9] the crack tip parameters KJ^^ and the stress intensity factors Ka can be related to G* with
G„''=\KX:'
(6)
244
O. WITTIER, P. SPRAFKEAND B. MICHEL
for a=I,II and ///. Although G* as well as the work offractureintroduced by Schapery denote the energy requh-ed to form a crack extension, a mathematical equality has not been shown. The following numerical analysis shows similarity between both approaches (figure 2). Especially for the case of mode I crack growth and an instantaneous and constant load K/o, an exact equality of both approaches occurs: G*(t) = W^it) = ^Cit)K,,
(7)
To evaluate G* in finite element analyses it is well established to modify the crack closure integral (3) in a way introduced by Rybicki and Kanninen [13]. For 8-noded 2nd order elements a formulation is takenfrom[14] and extended to the 3D case [19].
Fig. 1. Local coordinate system at the crack tip Generalized J-Integral Another approach to estimate the energy flux into the crack tip, which is used for viscoelastic [10] as well as elastic-plastic [15] crack growth, can be pursued by the appUcation of generalised path-independent integrals. It can be deducedfroman energy balance of the region around the crack tip [16]. If effects of inertia are being neglected, the energy flux dEp to the crack tip per imit crack propagation da and unit thickness B can be calculated by ^* = - & = I(^*"^ -^,«u)^^- j(^*,i -^ij^i,n)dA Bda ^ ;J with the strain energy density U* including dissipative and elastic parts by, U*='\a,d8^
(8)
(9)
0
where the areay4 is surrounded by the contour F, One basic assumption of this approach is that crack growth occurs self-similar. If the stresses are a single valued function of strain and the strain energy density has the form of dU* then the area integral equals zero and equality of J and 7* is obtained. Generally this is obtained for elastic and non-linear elastic materials. If dissipative effects occur the equality is only valid for monotonic loading inside the integration areay4. Comparison of different approaches For a comparison of the different approaches the energy release rate is calculated for a CTSpecimen (W=20mm, B= 6mni, a/W = 0.5), which is loaded in Is to a maximum load of 100 N {Ki = 36 MPa mm^^^) and unloaded in 1 s after 50 s. A simplified and artificial viscoelastic
Elastic and Viscoelastic Fracture Analysis of Cracks in Polymer Encapsulations
245
model is used for the description of stress-strain-behaviour, with relaxation times r; = 0.1 s and r2 = 1 s. It is described by a Prony-Series, r ^(0 = ^.+Z,.,^,exp / (11) where the coefficients Ei and E2 both equal 1500 MPa and Eoo equals 500 MPa. The Poisson's ratio is assumed to be constant with v= 0.3. The Work of Fracture ^ i s calculated accordmg to the approach of Schapery (equation (1)), while the crack closure integral for G* is calculated from the results of a finite element analysis and the formula from Rybicki and Kanninen [13]. The J-Integral is obtainedfrom2D plane strain finite element analyses and standard procedures of the commercially available finite element solvers ABAQUS and ANSYS. Results are displayed infigure2. During loading (time < 1 s) no difference between G* and J can be observed, while an obvious but small difference occurs in comparison to the approach of Schapeiy Wj, because equation (1) and (6) with (5) are not generally the same. For the time t becoming much larger than the relaxation times ti (t» Ti) equation (7) becomes valid and G* equals Wf. Fortiiistime scale the material properties are no longer time dependent and G* and ^become equal to the elastic solution of the energy release rate with IE^. This is not tiie case for the J-Integral calculation. Both solutions calculated in ANSYS and ABAQUS differ from this solution for ^-^00. This approach is therefore not as suitable as the crack closure integral for such a loading case. Especially after unloading the J-Integral becomes path dependent and does not approach zero, because condition (10) is no longer given. But when the CT-Specimen is free of external forces, also the crack tip stresses are zero and no energy is released in a virtual crack propagation. This contradiction is a shortcoming of the solution for J and in the same way of the work of fracture Wf, which was not deduced for the case of unloading. Overall it can be concluded that the energy release rate calculated via a modified crack closure integral (G*) appears to be the best suitable approach for a viscoelastic crack analysis. Further tests will have to prove, if this is also the case for thermomechanical loading.
O. WITTIER, P. SPRAFKEANDB. MICHEL
246
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Elastic and Viscoelastic Fracture Analysis of Cracks in Polymer Encapsulations
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Thermomechanical Loading In the case of a thermomechanical loading case the situation is quite different to the isothermal case described above and the question arises which critical values are valid for the assessment of risk of crack growth in real loading situations of electronic components. From the results above two hypotheses can be derived. Hypothesis A: Risk offracturecan be estimated by the energy dissipation rate, with a risk of crack growth above G*c,min = 0.17 N/mm. Hypothesis B: Risk of fracture can be estimated by the critical stress intensity factor Kic, dependent on the actual loading Ki, the actual loading speed dK/dt and the actual temperature T, To evaluate the Hypotheses, CT-specimens were loaded thermomechanically by an increasmg force with decreasing temperature untilfractureoccured. The temperature range from the start of loading to the temperature atfractureis given in figure 5, where the time scale of cooling was controled by the cooling time of the testing chamber with power off. The loading Ki(t) and temperaturefimctionsT(t) dxt used as input data for finite element simulations, where G* is calculated at the time of crack initiation and a uniform temperature distribution was assumed, with an integrated form of reduced time. Results are shown infigure5.
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Temperature Range [°C] Fig. 5. Results for G*c calculated from different thermomechanical loading experiments m comparison with the critical value of G*c,mm = 0.17 N/mm and with an indication of the cooling rates For most experiments, hypothesis A appears to be an appropriate approach to evaluate a risk of failure except when the temperature T(t) sweeps across the glass transition temperature, which is about 143°C for the measurement of thermal expansion. In this case crack growth occurs far above the assumed critical value, which is especially the case during cooling from 170°C to
Elastic and Viscoelastic Fracture Analysis of Cracks in Polymer Encapsulations
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110°C but also for cooling from 150°C to 30°C. Besides the fact that a vaUdity of the constitutive model is questionable for this temperature range, an additional and important reason for this difference can be found in the fact that for a calculation of G* only self-similar crack growth is assumed, which is also the case for J and T*. This assumption does not coincide with reality, because crack growth is obviously not self-similar for these loading cases, as can be observed in figure 6. Large crack deformations with wide crack openings occur above the glass transition temperature. When crack growth initiates below glass transition temperature, the material is less deformable and crack growth occurs with a smaller crack tip opening. This behaviour is at least qualitatively inherent in the linear viscoelastic material model, as simulations on crack propagation after thermomechanical loading show [19]. Therefore, the outlined theoretical approaches cannot estimate correctly the required energy for crack propagation under thermomechanical loading.
Fig. 6. Images of a loaded crack at 155°C (a) and at 107°C after crack growth initiation (b) Isothennal Measurement
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250
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given for all cases with a maximum error of 16 % in one case. A better correlation than for hypothesis A can be found for hypothesis B in temperature ranges (170->110°C and 150 -> 30 °C), where errors of ± 10 % occur. Therefore, the analysis of iJie thermomechanically loaded demonstrator will be based on Hypothesis B and the results shown in figure 7. APPLICATION EXAMPLE To exemplify the application of a rate dependentfracturemechanical analysis a demonstrator is chosen where a metal core is being encapsulated. The upper and lower surfaces are left free from encapsulation material to give the possibility to observe possible cracks visually. A circular crack with a length of 0.5 mm is inserted in the model as indicated in figure 8. The typical loading situation of such a component occurs at first after curing, when it is cooled from 130°C to room temperature. After a few days it is tested in temperature shocks from 150 °C to - 40 °C. In a viscoelastic analysis the whole loading history needs to be taken into account. Preliminary simulations show that the situation at 150 °C can be assimied to be stress free. Here, the material is above glass transition temperature, where the Young's modulus is more than one order of magnitude smaller than at room temperature. Therefore, 150 °C is used as the stress free temperature in the simulation of the shock experiment, where ANSYS 5.7 is used as a solver and Kj is obtamed indirectly through equation (6) after calculation of G* and KI^K It has to be noted that G* is not used with a direct physical meaning, but as a quantit> to relate Ki and K/^\ The solution of the elastic material model is compared to the solution of the viscoelastic material model, where the component is assumed to cool down in 10 s and for the second case in 45 min. For this first approach also a uniform temperature distribution is assumed. In figure 9, results for Ki are compared to the critical values from figure 7. Comparing the elastic shnulation to the viscoelastic simulations, it can be seen that crack load is about 15 to 40 % larger for the elastic simulation depending on the loading speed in the viscoelastic simulation. A reduced loading speed leads to a smaller crack load, while fracture toughness (Kic) is reduced at the same time. Consequently for both loading speeds Kic is about 30 to 40 % larger than the corresponding crack load. Thus for the investigated loading conditions the rate dependent mechanical behaviour does not lead to a rate dependent criticality, which is the ratio offracturetoughness to crack load. This dependence might rather be due to other effects like thermomechanical stresses, which are the result of a temperature gradient. As such a gradient is not considered in this analysis, the evaluation offracturecriteria under thermal shock might be interesting forfixtureinvestigations. Nevertheless it needs to be mentioned, that this simulation and evaluation of the results only considers effects of loading rate for a single loading event. For example for long time storing or cyclic loading, fiirther research needs to be done.
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Elastic and Viscoelastic Fracture Analysis of Cracks in Polymer Encapsulations
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CONCLUSION The analysis shows that differences of viscoelastic and elastic simulations can not always be neglected as they reach up to 40 % for the thermomechanically loaded example case. An elastic simulation gives a worst-case estimation for the crack load, when compared to a viscoelastic simulation. But when results are compared to the critical fracture toughness, also the time dependence of fracture properties needs to be considered. It is shown that the parameters comparable to the introduced Modified Crack Closure Integral (MCCI) with G* and path-independent integrals like J may be applicable in isothermal and rate-dependent loading conditions. However they do not correctly describe the energy release rate for the thermomechanical loading regime, because the inherent assumption of self similar crack growth is shown to be invalid. This can lead to large errors when the temperature sweeps across the glass transition temperature during thermomechanical load. The presented concept, that considers the actual load Ku actual loading speed dKildt and actual temperature, leads to a better correspondence to the verification experiments. With these results a demonstrator representing an encapsulated electronic component is analysed. For this analysis a reduction of loading speed does not reduce the criticality of the crack, as the reduction in crack load is accompanied by an increase of fracture toughness. Therefore the loading rate dependence of reliability, which is generally inherent to thermal shock experiments, might rather be due to other effects than viscoelastic mechanical behaviour and should be the focus of fiiture investigations.
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REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
Zhu, J., Zou, D. and Liu, S. (1998) Journal of electronic packaging 120, 160. Sonner, M., Vogel, D., Sprafke, P., Michel, B. and Reichl, H. (2001) Materialsweek 2001,1-4 October, Munich. Wittier, O., Sprafke, P., Auersperg, J., Michel, B. and Reichl, H. (2001) In: Proceedings: 1st Int. IEEE Conference on Polymers andAdhesives in Microelectronics and Photonics, pp. 203-208, IEEE, Piscataway, NJ. Bradley, W., Cantwell, W. J., Kausch, H. H. {199%) Mechanics of time-dependent materials 1, 241. Schapeiy, R. A.(1975) Int. J. Fract. 11, 141, 369, 549. Frassine, R., Rink, M. and Pavan A. (1995) In: Impact and Dynamic Fracture of Polymers and Composites, ESIS 19, pp. 103-111, Williams, J. G. and Pavan, A., Mechanical Engineering Publications, London. Frassine, R., Rink, M. and Pavan, A. (1996) Int. J. Fract. 81, 55. Xiong, Z. and Tay, A. A. O. (2000) In: Proceedings: 50th Electronic Components and Technology Conference, pp. 1326-1331, IEEE, Piscataway, NJ. Dubois, F., Chazal, C. and Petit, C. (1999) Journal of theoretical and applied mechanics 2,207. Dubois, F., Chazal, C. and Petit, C. (1999) Mechanics of time-dependent materials 2, 269. Schapeiy, R. A. (1984) Int. J Fract. 25, 195. Schapeiy, R. A. (1990) Int. J Fract. 42, 189. Rybicki, F. F. and Kanninen, M. F. (1977) Eng Fract. Meek 9, 931. Dattaguru, B., Venkatesha, K. S., Ramamurthy, T. S. and Buchholz, F. G. (1994) Eng Frac.Mech. 49,451, Brust, F. W., Nishioka, T., Atluri, S. N. and Nakagaki, M. (1985) Eng Fract. Mech. 22,1079. Will, P. (1988) Integralkriterien und ihre Anwendung in der Bruchmechanik, VDI Reihe 18, vol. 56, VDI-Verlag, Dusseldorf. Wittier, O., Sprafke, P., Walter, H., Gollhardt, A., Vogel, D., Michel, B.(2000) hi: Proceedings: Materials Week 2000, URL: http://www.materialsweek.org/ proceedings/, 8 pages, Werkstoffwoche-Partnerschaft, Frankfijrt, Germany. Walter, H., BierOgel, C, Grellmann, W., Fedtke, M., Michel, B. (2000) In: Proceedings: 3rd International Micro Materials Conference, pp. 537-540, ddp goldenbogen, Dresden, Germany. Wittier, O. (to be pubUshed) PhD Thesis, Technische Universitat Berlin, Germany.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
253
MODELLING THE DROP IMPACT BEHAVIOUR OF FLUID-FILLED POLYETHYLENE CONTAINERS
A. KARAC and A. IVANKOVIC Imperial College London, Mechanical Engineering Department, London SW72BX, UK
ABSTRACT Drop impact resistance of fluid-filled plastic containers is of considerable concern to containers manufacturers as well as distribution industries using the containers for transportation of various liquids. This is due to potential failure of the containers following the drop impact and subsequent spillage of the transported liquid, and consequent safety and economical issues. In this work, a series of drop impact experiments is conducted on water filled bottles made of blow moulded high-density polyethylene (HOPE). During experiments, pressure and strain histories are recorded at various positions. The experiments are then simulated numerically. This problem falls into a category of strongly coupled fluid-structure interaction (FSI) problems due to comparable stiffnesses of the container and its liquid content. Hence, accurate prediction of containers' behaviour requires a liquid-container interaction model. Here, a two-system FSI model based on the Finite Volume Method is employed, and a good agreement is found between measured and predicted pressure and strain histories. To obtain fracture properties of HDPE conventional essential work of fracture tests are performed. Two grades of blow-moulded HDPE are tested at different test speeds. The main aim of these tests is to estimate traction-separation (cohesive zone) properties of the materials. In future work, these will be combined with the fluid-structure interaction model to provide a powerful tool for predicting the complex behaviour and potential failure of fluid-filled containers under drop impact.
KEYWORDS drop impact, plastic containers, fluid-structure interaction. Finite Volume Method (FVM), essential work of fracture, traction-separation.
A. KARAGANDA. IVANKOVIC
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INTRODUCTION Different approaches can be used to measure or evaluate the drop impact resistance of blowmoulded containers: standard drop test procedures (e.g. ASTM D2463-95), theoretical predictions (e.g. water-hammer or mass-spring theory [1]), numerical simulations [2,3], etc. The standard procedures provide a critical drop height above which a particular container will fail, by using a statistical approach. Containers of different shapes, sizes and material properties must be tested individually, making this approach very expensive in design optimisation, although very quick and useful in controlling the manufacturing process. On the other hand, application of the analytical predictions, i.e. pressure propagation, pressure distribution, etc., is constrained to simple geometry and simple (i.e. linear) material behaviour. Thus, a properly validated numerical model is a useful tool to assist and accelerate product development, providing it includes an appropriate FSI model coupled with a failure model of the container material.
EXPERIMENTAL PROCEDURE In order to study the behaviour of drop impact containers and to validate the numerical procedures, a series of drop impact experiments was conducted using a specially designed rig (Fig. 1). The rig was an assembly of two aluminium end-caps held together with three steel connection bars, and the bottle specimen with the cylindrical cross section was placed in-between. The rig was manufactured to house two types of specimen: a bottle-shaped specimen without the original base and a bottle with its base as originally manufactured. Both specimens were of the same size (diameter/) = 84 mm, thickness t = 1.5 mm) and type, the first having the base cut off and the remaining bottle being fixed to the lower cap by a worm driven hose clip. The reason for removing the original base was twofold. Firstly, the influence of the base shape and type (rigid and flat as opposed to flexible and complex base shape) on the pressure and strain distributions in the bottle could be examined. Secondly, a numerical simulation of the bottle with flat and rigid base was much easier to perform and validate than that of a real bottle, for which additional complex issues, such as contact has to be accounted for. Upper cap - ^ Connection bars 1/ Rubber 0-ring Strain gauges 2 1
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The specimens were filled with water to a level of 125 mm above the base and were instrumented with two strain gauges positioned at 25 and 50 mm (occasionally 80 mm) fi'om the base, to record the deformation of the wall. In addition, pressure transducers (one placed
Modelling the Drop Impact Behaviour of Fluid-Filled Polyethylene Containers
255
in a central axial holder, and two in the lower cap for bottles without their original base) were used to measure pressure histories (see Fig. 1). The instrumented rig was dropped onto a concrete floor from a given height. To ensure square landing of the rig a set of three U-profile aluminium guides were used. Signals from strain gauges and pressure transducers were recorded simultaneously.
MODELLING AND RESULTS To simulate drop impact of fluid-filled containers a two-system fluid-structure interaction (FSI) model, based on FVM, was employed [4]. Here, the fluid and solid parts of the solution domain have separate meshes, but there is a common interface between them. The solid and fluid models were combined within a single code FOAM (Field Operation And Manipulation, [5]). The systems of equations were solved for each mesh, and interface conditions were exchanged: tractions from fluid to solid and velocity from solid to the fluid. Both meshes remain fixed during the calculations. Small-strain analysis was performed for the solid and mesh distortions were neglected. An Eulerian frame of reference was used for the fluid, and the information about the motion of the neighbouring solid wall was passed to the fluid boundary via wall interface velocity. The scheme uses two sets of 'inner' iterations; one for each mesh, and implicit coupling in time was achieved through a series of 'outer' iterations, which solves the total system to convergence within each time step [4]. Due to the axisymmetric nature of the problem, only a section of both domains was considered in the analysis (see Fig.2). Two different cases were investigated: (i) the problem with flat rigid base (Fig. 2-left), thus simulating the drop impact of the bottle without its original base, and (ii) the problem with different flexible base shapes (flat and curved, as shown in Fig.2-right) to investigate the base-shape effect. In the latter case, the bottle was allowed to bounce after the impact. Dimensions of the domains corresponded to the actual bottle dimensions (diameter D = M mm, thickness t = 1.5 mm, 7/= 125 mm). The total number of cells for the solid domain was 150 for the first and 162 for the second case, whereas for the fluid domain 1000 and 642 cells were used, respectively. Material properties for the fluid and solid were as follows: • solid (HDPE) - modulus of elasticity E= \3 GPa; density p = 948 kg/m^; Poisson's ratio v=0.35, • fluid - density p = 998.2 kg/m^; dynamic viscosity r] = 0.001 Pas. flat flexible base Solid domain Fluid-structure interface
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Only results corresponding to the drop height of 0.4 m are presented here. This gives the initial impact speed of 2.8 m/s. In the first case the base surface for both domains was suddenly fixed. Namely, displacements and velocities at the base surface were set to zero with dp/dz = 0, where p is pressure and z is the axial direction. The solid top surface was assumed to be a plane of symmetry, whereas the rest of the boundaries were traction free. The pressure at the fluid top surface was set to atmospheric, simulating the free surface, with a zero gradient boundary condition for velocity. In the second case, the real base was modelled in order to simulate impact of the bottle with the rigid floor. Bouncing of the bottle was achieved by disallowing tensile stresses anywhere on the base surface, i.e. by replacing them with stress-free boundary conditions. The time step in all simulations was set to 1 jis, and the total running time was 5 ms. Fig.3 presents numerical predictions of the strain histories in the bottle at two different positions - 25 and 50 mm from the base. As soon as the bottle hits the floor, a compressive pressure wave is generated and starts to travel towards the top of the bottle, deforming the bottle wall. As expected, the position nearest to the base (25 mm) is first reached by the pressure wave, followed by the position further away, i.e. pressure wave propagation can be observed by the time delay between strain histories at different positions. The process is very similar to the water-hammer phenomena in pipelines, and is characterised by a sinusoid-like wave. The high frequency oscillations superimposed on the main signal are due to natural oscillation of the bottle.
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Similar behaviour, pressure wave propagation characterised by the time delay between different positions, can also be seen in experimental results (Fig.4), which is opposed to the observations made by Reed et. al [1]. Superimposed oscillations with higher frequency are also present in the trace. However, the longer low-pressure period can be seen in the experimental results just after 2 ms. This is due to cavitation that takes place during the experiment. In the numerical simulation, cavitation is not modelled, and pressure can drop below absolute zero. Figure 5 shows the influence of the base shape on the strain history in the bottle wall 50 mm from the base, as obtained from numerical simulations. It can be seen that the pressure rise rate is lower for the bottle with original curved base. This is caused by a gradual stoppage of the water column resulting in an increased pressure rise time due to deformation of the base, as opposed to a sudden stop and pressure rise in case of the bottle with flat rigid base. The
Modelling the Drop Impact Behaviour of Fluid-Filled Polyethylene Containers
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shape of the first loading pulse is triangular, and slightly smaller in magnitude. The subsequent oscillations are much shorter and smaller in magnitude since the bottle has bounced. Similar behaviour and good resemblance can be observed in the experimental results, as shown in the Fig.6. However, the numerical simulation somewhat overestimates the strain magnitude. This is probably due to the assumptions that the wall thickness is constant and the landing is perfectly square, which is unlikely in reality.
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3D FLUID-STRUCTURE-FRACTURE MODEL The full 3D fluid-structure-fracture (FSF) model has been first developed to simulate rapid crack propagation in plastic pipes [6], and is adopted in the present work. Apart from fluidsolid coupling issues described elsewhere [3,4,7], there are two main issues that require special care in order to develop predictive model of failures of plastic containers: • description of the fracture process (see Fig. 7a) • passing the information about the crack opening of the container wall to the fluid domain (see Fig. 7b). A fracture process is described by employing a local failure criterion. Here, a Cohesive Zone Model (CZM) or local traction-separation law is used. It gives a relationship bet>\een tractions holding the separating surfaces and the separation displacement between them. Crack initiation and subsequent growth can be determined directly in terms of CZM parameters: the strength of cohesion tc, critical separation displacement 4 , and the area Go under the traction-separation curve representing the fracture toughness (see Fig. 7a where a Dugdale type CZM is shown for simplicity). With regard to the second issue, difficulties were experienced in coupling the fracturing solid with the contained fluid [6]. As the crack propagates and the solid opens up, a special interpolation procedure was developed to pass this information across the interface to the fluid. This is because the crack-gap appeared creating the escape route for the fluid, which was no longer fully contained within the pipe (Fig. 7b). In order to accurately capture the geometry of the crack and its influence on the flow field irrespective of the resolution of the solid-fluid interface, three possible modes of interaction between fluid surface and fracturing solid were considered: i) Fluid cell-face fully covered with the solid, ii) Fluid cell-face fully uncovered, and iii) Fluid cell-face partly covered. Coupling of the first two modes was straightforward. The third one was treated as a combination of the covered and uncovered part, each providing an appropriate contribution to the cell balance through a proportion of fixed-value (for covered part) andfixed-gradient(uncovered part) boundary conditions. This proportion was determined by calculating the (un)covered fraction of the cell area (Fig. 7b). On the other hand, passing the pressure values from the fluid to the solid was reasonably straightforward as all solid cell-faces on the interface were always fully covered by the fluid, and standard pressure interpolation was sufficient. crack
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Modelling the Drop Impact Behaviour of Fluid-Filled Polyethylene Containers
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FRACTURE PROPERTIES OF CONTAINER MATERIAL Since the containers are made of thin blow-moulded material, it was decided to use conventional essential work of fracture (EWF) tests to obtain the plane stress fracture properties. More details about the EWF theory and its applications can be found in the ESIS protocol [8], and other publications [9-12]. In the present work, two different grades of high-density polyethylene (BP Solvay Rigidex HM5411EA and HD5502XA) have been tested at different tests speeds. The specimens were nominally 1.6 mm thick and 40 mm wide, with an 80 mm gauge length. Five different ligament sizes were used (varying from 6.5 to 13.5 mm). The test speed was varied from 1 mm/s to 100 mm/s in order to obtain the fracture properties at rates comparable to those during drop impact testing. These speeds are however significantly larger than that proposed by the protocol [8]. Figure 8 shows a set of load-displacement curves for HM5411EA tested at 1 mm/s. Following the EWF procedure, the plot of the specific work of fracture, Wf vs. ligament length, / is produced (Fig.9). It can be seen that linear approximation fits the data very well. From the intercept between the fitted line and the >^-axes, the value of 25.18 kJ/m^ is obtained for the essential work of fracture. This value represents fracture toughness under plane stress conditions. The slope of the linear fit represents the plastic work dissipation factor, PiWp, where jS is a shape factor associated with the shape and size of the plastic zone, and Wp is the plastic work dissipation per unit volume of material. The values of piWp for all cases are given in Table 1. 700,
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It has recently been suggested [13] that EWF results can be used to obtain the CZM parameters, based on the self-similarity of the load-displacement curves. Two different 'scaling' techniques can be used to extract the CZM curves, one that is based on the maximum separation displacement and another based on displacement at peak load. In the first approach, which is employed in this work, graphs of Ojn vs. I (maximum stress vs. ligament length) and d^ vs. I (separation displacement vs. ligament length) are plotted. Applying a linear fit to the set of maximum stress data (Fig. 10) and separation distance data (Fig. 11), the maximum stress and the maximum separation at zero ligament length are obtained. These two values are assumed to represent the parameters of the CZM curve: cohesive strength and maximum separation displacement, respectively. The next step is to convert load-displacement (Fig. 8) into stress-displacement curves by dividing the load with the ligament cross-sectional area. These curves are then scaled into a
260
A. KARAGANDA. IVANKOVIC
curve that should represent the traction-separation law at zero ligament length, using equations:
& —Q
225
H— 0
E15 3
E
1
Maximum stress (1= 0) =28.12 MPa
Liganient length, mm
Ligament length, mm
Fig.lO a„-l plot for HM541 lEA at 1 mm/s
ay(0) = ayiD d^iO). 6,(1)
Fig. 11 6„-/ plot for HM541 lEA at 1 mm/s
oJl-0)
(1)
f^m(0
(2)
«5m(0
Obviously, all the scaled curves have the same maximum stress and separation displacement. Having in mind the self-similar nature of load-displacement curves, all the curves are expected to scale into one curve. This is shown in Fig. 12, and this unique curve is thought to represent the material traction-separation law. The area under the curve (average of the all curves) represents the cohesive zone fracture toughness. A value of 20.35 kJ/m^ is obtained, which is 20% lower than the one obtained by the conventional EWF protocol. The reason can be found in the slight scatter of the maximum stress and separation displacement data. Namely, the small changes in data distribution and deviation from the linear fit can cause significant changes in linear fit parameters (intersection with y-axes and slope). Results from 20 mm/s tests on HM541 lEA are considered next. The Wf-l plot is shown in Fig.l3, whereas 0^-1 and d^-l graphs are presented in Fig.l4 and Fig.15, respectively. It can be seen that linear fit is not as accurate as for 1 mm/s tests, due to significant data scatter. This could be due to the fact that longer ligaments are not fully yielded prior to the crack growth, which is shovm by the tendency in Wf-l plot to plateau at longer ligament lengths (Fig. 13). Also, large scatter in measurement of ^ , which is due to variations in fibril extension between the specimens, causes noticeable scatter in fracture energies.
Displacement, mm
Fig. 12 Traction separation low for HM541 lEA at 1 mm/s
Ligament length, mm
Fig. 13 wrl plot for HM541 lEA at 20 mm/s
Modelling the Drop Impact Behaviour of Fluid-Filled Polyethylene Containers
261
Maximum stress distribution shows linear dependence on ligament size (Fig. 14), whereas separation displacement distribution has a similar pattern to that ofwf-l graph. The value of We obtained by the EWF test was 14.53 kJ/m^, and the one obtained using maximum separation displacement is 26.6 kJ/m^. The difference is probably due to overestimated separation displacement at the zero ligament length shown in Fig 15.
Ligament length, mm
Ligament length, mm Fig. 15 dm-l plot for HM5411EA at 20 mm/s
Fig. 14 Om-l plot for HM541 l E A at 20 mm/s
According to the EWF protocol, all data which do not lie within a 95% confidence region from the linear fit in the Wf-l plot should be excluded from the analysis, and the procedure repeated. This rule was not strictly followed in the current work, as a substantial amount of data would have to be excluded. This was not thought to cause major difficulties since the CZM curves obtained from EWF tests are only intended to provide an initial guess for the numerical simulations of EWF tests where CZM parameters will be finally calibrated. Once CZM parameters are fixed for a given speed and over a wide range of ligament lengths, such calibrated parameters will be used to represent the local traction-separation law of the material at a given speed, and will be embedded in the numerical model for predicting failures of the containers. The second grade, HD5502XA, was tested at 10 mm/s and 100 mm/s, using 0.8 mm and 1.6 mm thick specimens. Figure 16 presents a set of load-displacement curves for 0.8 mm thick specimens tested at 10 mm/s. The self-similarity of the curves can still be observed, but the variation in the separation distance is much more pronounced. This causes large scatter not only in the separation displacement plot (Fig. 19) but also in specific work of fracture data (Fig. 17), while o^-l plot is largely unaffected and reasonably linear (Fig. 18). CM
^
400
140
r
g> ^20
Z 300.
i
Iff^
\
\
Increas 9 In ligament size
Z. 80 o t 60
£ 200.
i ^
100. 0.
^1.11 III L^lll
'-'- *-
Displacement, mm Fig.l6 F-d plot for HD5502XA at 10 mm/s
W
40
Essential work of fracture = 57.6 kJ/m^
Ligament length, mm Fig.l7 Wfl plot for HD5502XA at 10 mm/s
Consequently, a very high value of essential work of fracture is obtained, We = 57.6 kJ/m^, whereas the expected value is around 20 kJ/m^. The maximum separation approach for
262
A. KARAGANDA. IVANKOVIC
extracting CZM curves also becomes very inconclusive since it depends on the separation displacement data.
t Separation displacement (i = 0) = 5.4 mm
Ligament lengtli, mm
Ligament lengtli, mm
Fig.l8 a„,'l plot for HD5502XA at 10 mm/s
Fig.l9 dr„-l plot for HD5502XA at 10 mni/s
A further set of experiments was conducted on HD5502XA at 100 mm/s loading rate. Unlike in the previous case where testing speed was 10 mm/s, load-displacement curves (Fig.20) are much more repeatable, having only a small difference at the tails of the curves. Also, much smaller separation distances were recorded at 100 mm/s.
Displacement, mm Fig.20 F-6 plot for HD5502XA at 100 mm/s
Ligament length, mm Fig.21 wrl plot for HD5502XA at 100 mm/s
The Wfl plot shows a very good data grouping (Fig. 21). Linear fit seems to be reasonable, although some deviation can be seen at longer ligaments. The essential work is 15.97 kJ/m^, which is also reasonable. Good linearity of the 0^-1 data and d^-l data is demonstrated in Fig. 22 and Fig. 23, respectively. One can argue that at higher loading rates, adiabatic heating may play an important role in material separation giving much smaller but also less scattered separation values, while at intermediate rates the variation in material microstructure between the specimens significantly affects the separation.
Ligament length, mm
Fig.22 a„,-l plot for HD5502XA at 100 mm/s
Ligament length, mm
Fig.23 dm-l plot for HD5502XA at 100 mm/s
Modelling the Drop Impact Behaviour of Fluid-Filled Polyethylene Containers
263
Summary of the main testing parameters and the results for both grades are presented in Table 1. Table 1. Various data for EWF tests on HM541 lEA and HD5502XA Material: Thickness, mm Test speed, mm/s We, kJ/m^ o^„ MPa 6imx, mm Gc,kJ/mHl51 Pwp, MJ/m^
HM5411EA 1.6 1.6
HD5502XA 1.6
0.8
1
1
20
10
100
10
100
25.18 28.12 1.61 20.35 10.04
14.53 33.24 1.8 26.6 4.34
57.6 33.9 5.4 5.25
15.97 36.3 0.87 15.05 3.83
23.58 35.78 1.35 6.2
14.58 36.94 0.41 6.4 3.33
CONCLUSIONS This paper presents the combined experimental/numerical investigation of the behaviour of fluid-filled plastic containers subjected to drop impact. Drop impact experiments were conducted on original and modified bottles. During the test, strain and pressure histories were recorded at various positions. Tests were simulated numerically using the two-system FSI model. Both solid and fluid domains remain fixed during the calculations, i.e. a smallstrain analysis was performed for the solid while an Eulerianfi-ameof reference was used for the fluid. This procedure was found to be simple, stable and efficient. Numerical results agreed well with experimental data, demonstrating the capability of the code to cope with this complexfluid-structureinteraction problem. The procedure presented the skeleton for the development of a general, predictive fluidstructure-fracture procedure that will be applied to predict failures of fluid-filled containers under drop impact. The missing constituent required by the model is the traction-separation data for the materials considered. In order to calibrate the CZM parameters, combined experimental/numerical work employing the essential work of fracture and FV simulations has been conducted. The EWF results were obtained for two HDPE grades at various test speeds. In addition to conventional EWF resuhs, a special scaling analysis was performed to obtain CZM parameters. Some uncertainties due to the large scatter in measured separation distances were experienced, in particular at intermediate test speeds around 10 mm/s. However, the CZM data will only be used as the initial guess values in the numerical simulations of EWF tests. These simulations are designed for accurate calibration of the CZM parameters. Once the parameters are calibrated, they will be embedded in the predictive FSF model of the drop impact tests. ACKNOWLEDGEMENTS The authors would like to thank British Petroleum pic for their financial support.
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A. KARAC AND A. IVANKOVIC
REFERENCES 1.
P.E. Reed, G. Breedveld, B.C. Lim, Simulation of the Drop Impact for Moulded Thermoplastic Containers, Int. J. Imp. Eng., 24, (2000), 133-153.
2.
A. Karac, A. Ivankovic, Behaviour of Fluid-Filled Plastic Containers under Drop Impact, in Proc. Int. Conf on Computational Engineering Sciences ICES2K, LA, USA (2000).
3.
A. Karac, A. Ivankovic, Drop Impact of Fluid-Filled Plastic Containers: Finite Volume Method for Coupled Fluid-Structure-Fracture Problems, in Proc. Fifth World Congress on Computational Mechanics WCCM F, Vienna, Austria (2002).
4.
C.J. Greenshields, H.G. Weller, A. Ivankovic, The Finite Volume Method for Coupled Fluid Flow and Stress Analysis, Computer Modeling and Simulation in Engineering, 4, (1999), 213218.
5.
www.nabla.co.uk
6.
A. Ivankovic, H. Jasak, A. Karac, V. Tropsa, The Prediction of Dynamic Fracture of Plastic Pipes., in Proc. 10th ACME Conference on Computational Mechanics in Engineering, Swansea (2001), 173-176.
7.
A. Ivankovic, A. Karac, E. Dendrinos, K. Parker, Towards Early Diagnosis of Atherosclerosis: The Finite Volume methods for Fluid-Structure Interaction, Biorheology 39 (2002), 401-407.
8.
E. Glutton, Fracture Mechanics Testing Methods for Polymers, Adhesives and Composites, Essential Work of Fracture, ESIS Publication 28, 2001.
9.
B. Cotterell, J.K. Reddel, The Essential Work of Plane Stress Ductile Fracture, Int. J. Fract., 13, (1977), 267-277.
10. A.S.Saleemi, J.A. Nairn, The Plain-Strain Essential Work of Fracture as Measure of the Fracture Toughness of Ductile Polymers, Polym. Eng. Sci., 30 (1990), 211-218. 11. J. Wu, Y. Mai., The Essential Work of Fracture Concept for Toughness Measurement of Ductile Polymers. Polym. Eng. Sci., 36 (1996), 2275-2288. 12. W.Y.F. Chan and J. G. Williams, Determination of the Fracture Toughness of Polymeric Films by the Essential Work Method, Polymer, 35 (1994), 8. 13. Essential Work of Fracture and Cohesive Zone Fracture Toughness, Testing protocol prepared for ESIS TC4 by E. Glutton and revised by D.R. Moore (July 2002).
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
265
INVERSE METHOD FOR THE ANALYSIS OF INSTRUMENTED IMPACT TESTS OF POLYMERS VALERIA PETTARIN*, PATRICIA FRONTINI, AND GUILLERMO ELI^ABE Institute of Materials Science and Technology (INTEMA), University of Mar del Plata and National Research Council (CONICET) Av. J.B. Justo 4302, 7600, Mar del Plata, Argentina.
ABSTRACT Impact testing has become an important technique to determine the parameters associated with dynamic fracture of polymeric materials. These parameters are commonly calculated from the experimentally measured load versus time curves. However, these curves are not what theoretically should be used for this purpose, because the measured load is not equal to the load exerted on the tested specimen, load from which the mechanical performance of the material must be evaluated. The recorded load is corrupted by the other forces acting during the experimental run, which depend in part on the characteristics of the tester and in part on the properties and geometry of the tested material. In order to extract from the corrupted load the useful information, a simple model composed of springs, point masses, and viscoelastic elements is used. The model is employed to formulate an inverse problem from which the load on the specimen is obtained using the recorded load. The methodology is tested using simulated as well as experimental curves of different polymeric materials such as polypropylene homopolymer, mid-density polyethylene, and rubber toughened polymethylmetacrylate. The simulated curves demonstrate the validity of the inverse technique applied. The experimental curves confirm the methodology in a real situation. KEY WORDS Instrumented impact tests, three-point bending, bending force, analogical model, inverse problem. INTRODUCTION Since the advent of high speed recording equipment, impact testing has become a more useful technique than it was before, to test the most severe load conditions to which a material can be subjected. From impact testing the fracture resistance of a material can be infered if proper interpretation of the collected data is performed. In fact, fracture resistance parameters are directly related to the bending force exerted on the tested specimen. However, the force registered by the testing instrument is not actually the bending force but is the one applied on the striker, where the transducer is mounted. The relationship between the recorded force and
266
V. PETTARIN, P. FRONTINI AND G. ELIQABE
the one needed for the determination of the fracture parameters is not straightforward. This is due, in large part, to the very comphcated nature of the dynamic phenomena involved during the time in which the tested specimen interacts with the striker and finally breaks. This fact has imposed a limitation to the direct interpretation of load-time records, and has led several authors to propose dynamic models with different degrees of complexity with the aim to extract the actual force applied on the tested material from the measured one. One of the first models available is that proposed in a paper by Williams [1], from which several other studies have been initiated. This model has been challenged in a paper by Zanichelli et al. [2]. These authors have presented a detailed study of the first moments of the impact testing event and proposed a model that is based on experimental evidence that shows that: i) at the beginning the tested specimen does not interact with the support, ii) the mass initially involved is not the equivalent total mass of the specimen but only a part of it initially in contact with the striker, and iii) the stiffness that really plays a role at the beginning is a local one also related to the contact area. Later, Marur et al. [3], using auxiliary measurements, have validated experimentally a complete model similar to those proposed by the authors mentioned before. More recently, Pavan and Draghi [4] have developed a more complete model than the ones already available and verified it for the case in which the specimen is tested without using supports. The need for models has been envisioned from two angles: a) as a tool to improve the understanding of the dynamic phenomena involved in impact testing, and b) as a way to connect the remote measurement (force exerted on the striker) to the sought measurement (force exerted on the material), with the purpose of extracting the latter from the former. This latter approach was taken in the past by Cain [5] who used frequency domain techniques to filter the load-time records out of spurious oscillations associated with the dynamics of impact testing. This author used a model of the same characteristics as the ones mentioned before, but no analytical development was performed. The model was just used to numerically estimate the type of filters that could be used to clean the recorded signals of unwanted oscillations. The work presented here will consider in detail the problem of recovering the bending force acting in a three point bend test under impact loading, using the recorded force exerted in the striker. The methodology will be developed based on the model described by Pavan and Draghi [4]. The recovery of the bending force acting in the specimen from impact force measurements will be reported for different polymeric materials. The usual differential equations that describe the behaviour of the mass-spring-dashpot configuration were transformed into a discrete model. In this form, the problem of obtaining the bending force acting in the specimen from impact measurements becomes an algebraic inverse problem. It will be shown that the solution of this problem is not easy due to the small errors present in the measurements, which appear greatly amplified in the solution. In order to obtain useful results the problem will be regularized using the Phillips-Tikonov technique [6]. The parameters of the model needed to apply this methodology will be independently calculated through single determinations. Simulated and experimental load-time records will be processed for different polymeric materials. The resuhs will demonstrate that this methodology yields adequate recoveries of the bending force. MATHEMATICAL MODEL As mentioned in the introduction, different models have been proposed in the literature to describe impact testing. Some of them are accurate but rather complex, others are more simple but less accurate. Based on the needs of the application proposed here, the model developed by Pavan and Draghi [4] will be used in all what follows. Figure 1 illustrates the model. This
Inverse Method for the Analysis of Instrumented Impact Tests of Polymers
267
model describes in a simple mamier the main effects that take place when the striker of a falling weight impact machine hits a sample in three point bending mode. The model consists of a series of point masses, springs and viscoelastic elements, connected together in a form in which the first contact between the striker and the specimen, the motion of the portion of the specimen first in contact with the striker, and the motion of the remaining part of the specimen, are taken into account. This model has the advantage that, despite the inclusion of all the relevant dynamic effects, it keeps the simplicity of being linear and one-dimensional. The model equations are easily derived and are the following: m,2^, =K^,t-z„^)-kXz„, -z„J-rX2„, -2,J (1) ^.w4,,, = kf, (z,,^^ - z^^^^) + r^ (4^^ - z^^^^) - k^z^^^^ (3) where simple and double dot denote first and second derivative respectively. The meaning of the variables and parameters involved in the equations is given in Figure 1. The striker is assumed as a large mass that moves at constant speed VQ. Its tup or nose is represented by a separate unit with mass ntt and stiffness kt. Evidence of the need of representing the tup in this form are the oscillations observed when the tup is out of contact with the specimen. The force that is experimentally measured is the one acting in the tup. Ideally the spring represented by kt accounts for the stiffness of the gauging device. Thus the force sensed in the experiment is given by: P.=k,[Vot-zJ (4) The contact force exerted on the tup by the specimen dynamically balances this force. The tup/specimen contact is modelled using a Kelvin-Voight element having stiffness kc and damping coefficient r^. Thus the contact force is given by: Pc = K (z„,, - z„„) + r, (i„, - z„^) (5)
P, Pr
~i
Pr
PH
V ^
7\
TT
PH
Pn
^
Fig. 1. Schematic representation of the model used to analyse the actual tests, and configuration of the test.
268
V. PETTARIN, P. FRONTINl AND G. ELICJABE
Experimentally it has been observed that the tup may loose contact with the specimen [4, 7]. This event can be included in the model equations when they are solved. In fact this can be done by stating that the contact force, Pc, is non-negative. Initially, contact between tup and specimen is assumed and the equations solved with proper boundary conditions. As soon as Pc becomes negative, integration is stopped and, the model and boundary conditions updated. This is done by removing the Kelvin-Voigt element and setting the new boundary conditions equal to the speeds and positions of the point masses of the different elements. In this form Eq (1) runs now independently of Eqs (2-3). This situation is kept until the two point masses become in contact again. The procedure is repeated as many times as Pc becomes negative. To model the flexural dynamics of the test specimen, two masses and a Kelvin-Voigt element are used. The first mass, nisc, represents the inertia of the central part of the specimen and it is also the mass first involved in the local interaction at the contact point. The second mass, rrisw, represents the inertia of the wings of the specimen. It is important to notice that under the assumptions of this model, Pc does not represent the force responsible for the flexural deformation of the specimen. Under dynamical conditions Pc acts only locally producing mainly indentation. The actual force acting in the specimen and related to its bending is: Pb = K (^«,.. - ^m,.,) + h (4,, - 4,,,) (6) The bending force, Pb, is the one that ideally should be determined from the impact test. What is proposed in this work is to treat Pb as an unknown function, and try to infer it from what is actually measured; i.e. Pf In order to follow this approach not all the equations of the model are needed. With this in mind, the model to be used is reduced to Eqs (1,2) with the last two terms of Eq (2) replaced by Pt as per Eq (6). In this form, from all the parameters originally involved in the model (nine), only kt, nit, h, Vc and rrisc need to be determined to complete the model. If these five parameters are determined independently, a model that relates Pb and Pt would be available. Thus Pt can be certainly calculated from Pb using Eqs (1,2). As stated before this is not the problem of interest, what is sought is to estimate Pb from Pt, something that in principle is not obvious. Mathematical Model in Matrix Form In order to extract Pb, from Pt and the model, a possible strategy is first to transform the differential model into an integral one, and then discretize it to obtain a set of linear algebraic equations. These equations in matrix form can be, in principle, easily inverted to estimate Pb. The first step is to obtain the transfer function between Pt and Pb- First Eqs (1-2) are expressed in terms of Pt and Pb as follows: fnj,n^^ = K{z„,^ -z^^J + r,{z„^^ -z^J-P, Applying Laplace transform to Eqs (7,8) the following transfer function is obtained: PXs) = H,(s)S(s) + H,(s)P,(s) _
(8) (9)
where the upper bar indicates Laplace transformed variable. S(s) is the transform of the Dirac S(t) function and the transfer functions of Eq (9) are given by: //, (s) = -m,,Vk, (r,s + k^ )/a(s) (10) H,(s) = -k,{r^s + kJ/a{s) (11) with a{s) = m^m^^/ + r^.(m^ + m^Js^ + (k^X^ + ^.c) + ^i^sc)^~ + ^t^c^ + ^iK (12) In time domain Eq (9) can be written as
Inverse Method for the Analysis of Instrumented Impact Tests of Polymers
P,{t)= \d{T)h,{t-T)dT+ 0
\PXr)h,{t-T)dT
269
(13)
0
with h,{t) = L-'{H,{s)} (14) h,{t) = L-'{H,{s)} (15) where L'^ indicates inverse Laplace transform. Eq (13) can be discretized using any quadrature formula and written in algebraic form as follows: p, =h + Ap, (16) Here p/ is a vector containing the values of Pt{t) at the discretization times, h is a vector containing the values of the first integral in Eq (13) at the discretization times, A is a matrix result of the quadrature process used in the second integral in Eq (13), and p^ is a vector containing the unknown values of Ptit) at the discretization times. EXPERIMENTAL DETAILS Instrumented Impact Tests Experiments were conducted on different commercial polymeric materials. Polypropylene homopolymer (PP), mid-density polyethylene (MDPE), and rubber toughened polymethylmetacrylate (RT-PMMA), kindly supplied by Petroquimica Cuyo SAIC, Siderca, and Ineos Acrylics, respectively. Pellets of the materials were compression moulded into 10 mm thick plaques. Rectangular bars for fracture experiments were cut and then machined to reach the fmal dimensions and improve edge surface finishing. Sharp notches were introduced by scalpel-sliding a razor blade having an on-edge tip radius of 13 jim. The specimen thickness, B, and the span to depth ratio, S/W, were always kept equal to W/2 and 4 respectively. The notch-depth to specimen-width ratio was varied from 0.1 to 0.9 in every case. Pre-crack length, ao, was determined postmortem or after test from the fracture surface using a Profile Projector with a magnification of 20x. Pre-cracked specimens were tested in three point bending (mode I) at room temperature and at VQ =\ m/s using a falling weight type machine Fractovis 6789 by Ceast. Determination of the Model Parameters The striker stiffness (k^. The stiffness of the striker may be obtained by making it to hit a highly rigid surface, such as steel [4]. Under these conditions, the model of Fig. 1 can be well approximated during the first moments of the impact, by the simple configuration of Fig. 2. The condition imposed before, i.e. the constant speed of the large mass M of the striker is removed now and replaced by an inifial speed, Vo at the moment of impact. The solufion of this model in terms of the force applied on the sensing device is: sin jk, Mt P.=VA-^^(17) k^ M The derivative of P/ vs. t at /=0 gives Vokt. Thus the slope of the recorded load-time curve at t=0 (see Fig. 3) together with the known speed at impact, gives the value of kt. This parameter, which depends only on the machine, was obtained at a speed of 0.5 m/s. With this value of kt. Mean also be calculated by fitting Eq (17) to the experimental register, which will be used in the calculus of the tup/specimen contact coefficients.
270
V. PETTARIN, P. FRONTINIAND G. ELIQABE
V{0)=Vo
tp.
i V
Fig. 2. Schematic representation of the model and test configuration used to estimate kt aid M
5000
0.0
0.1
0.2
0.8
1.0
1.2
f(ms)
Fig. 3. Rebound test on steel placed on a flat rigid surface: recorded force Pt (-0-) and linear fit to the initial part of P^ (—). The tup equivalent mass (m^. The mass of the tup can be obtained when it freely oscillates after the completion of the impact test [4] or when no sample is used. From the load-time curve recorded after fracture or without sample, the frequency of the free oscillations, co = ^k^ /m, , can be obtained. With the value of kt already available, mt can be determined. The tup/specimen contact stiffness and damping coefficient (kc and Vf), The tup/specimen contact stiffness, k^ and damping coefficient, r^ can be determined by performing an additional rebound test in which the specimen is tested laid on a flat rigid surface [4]. In this case the proposed model is reduced to that shown in Fig. 4. Again, and because this is also a rebound test, the assumption of constant speed is replaced by an initial speed at t=0. In this particular set up only the two contact parameters are unknown. These will be estimated for different materials by fitting the model simplified as in Fig. 4, to the initial parts of the rebound test registers in which the speed of the striker remains fairly constant, more precisely up to a time when the speed reduced 10% of its initial value. This is a condition imposed by tfie fact that the damping coefficient is rate dependent. Contact parameters were estimated for three materials: polypropylene homopolymer (PP), mid-density polyethylene (MDPE), and rubber
Inverse Methodfor the Analysis of Instrumented Impact Tests of Polymers
271
toughened polymethylmetacrylate (RT-PMMA). In Fig. 5 the model fit to the experimental points is shown for PP as an example. V{0)=Vo
1' T
V
Fig. 4. Schematic representation of the model and configuration test used to estimate kc and rsc. 800
600
400
200
0.00 0.05 0.10
2.00
2.50
f(ms)
Fig. 5. Rebound test on PP placed on a flat rigid surface: recorded force Pt (-a-) and model fit to the initial part ofPt (—). The value of the contact mass, rrisc, depends not only on the material but also on the geometry of the sample. For this reason, it is highly desirable to obtain its value from the register acquired when the actual sample is tested. Therefore, it must be kept in mind that in Eq (16) the elements of h and A are functions of rrisc and could be more precisely written as \i{msc) and A(m^c). In the next section it will be explained how the value of nisc can be estimated at the same time as the main unknown, PbANALYSIS: THE INVERSE PROBLEM To obtain the bending force, Pb, from the force exerted on the tup, Pt, take Eq (16), that relates P/ and PA, and assume for simplicity that the value of nisc is known. The solution of this
272
V. PETTARIN, P. FRONTINIAND G. ELK^ABE
equation for the general case in which the number of experimental determinations of Pt. i.e. m, is larger than the number of elements of Pb, i.e. n, is given, in principle, by the least squares solution of an over-determined system of linear equations p, =(A^A)-'A^(p^ - h ) = (A'^'Ar'Ay (18) Although this solution appears to be straightforward, it is well documented in the literature [6,8-10] that small errors in p/ (i.e., quadrature and experimental errors) result in large errors in Pb The amplification of errors occurs independently of the fact that the inverse of (A^A) can be calculated exactly, and it is a direct consequence of the near singularity of the matrix A (if m=n), or more generally (if m>n) of its near incomplete rank. However, by constraining the least-squares solution by means of a penalty function, approximate useful solutions can be obtained [11]. This implies to extend the original least squares problem to mm
+r^(pj )
(19)
where ^(p^) is a scalar function that measures the correlation or smoothness of p/,, and ;^is a nonnegative parameter that can be varied to emphasize more or less one of the terms of the objective functional given by the previous equation. Note that we have included the value of msc as unknown that makes the inverse problem to be solved, non-linear. If / is set to 0, the equation reduces to the previous case, a solution that generally exhibits large oscillations. On the other hand, when ;A->CO the minimization leads to a perfectly smooth solution judged by the measure of ^(p^) but totally independent of pt and, therefore, useless. Clearly, intermediate values of /, that balance the amount of fitting to the data, p/, against the amount of smoothness of pb, are the ones expected to produce acceptable solutions to the original problem. Several functions can be chosen to establish the desired correlation level or the smoothness of p^. An interesting class of functions can be formulated by using a quadratic form of the vector pb with the advantage that they yield in part an analytical solution to the minimization problem. For example,
^(P/,)==E(2A-A,_,-A,J' plHp,
(20)
where matrix H: : K ' K is given by 0 1 -2 1 1 1
-4
0
6
H=
(21) -4
6 - 4
1 - 4
1
5 - 2
0 . . 0 1 -2 1 Thus, it can be shown that the solution to the minimization problem of Eq (19) is given by[12]: p, = [A'imJAimJ+far^'i'nJplimJ (22) where the value of msc is the one that minimizes the single variable equation obtained after replacing Eq (22) in Eq (19), i.e.: ^(m,,) = |p;(m,,)-A(m,,)pj' +x^(pj (23) To compute y several methods have been proposed. In this work, the Generalized Cross Validation (GCV) technique will be preferred. GCV is explained in detail elsewhere [13]. The value of /computed by means of GCV is the one that minimizes the following function:
Inverse Method for the Analysis of Instrumented Impact Tests of Polymers
Yiy)-
l(i-z)p;|
273
(24)
^raceyV - Z)| where Z = A(A^A+;H)A^ and I is the identity matrix. It is clear that for each value oinisc there is a different value of /that minimizes Eq (24) and then Eq (23) must be minimized under this condition. RESULTS Verification of the Methodology: Application to a Simulated Load-Time Curve In order to test the proposed methodology before using it with real data, a synthetic experiment is generated using the complete model of Fig. 1 with the parameters obtained by other authors for RT-PMMA [14] (a/^=0.5, A:,=110 MPa.m, ^,=3.94 MPa.m, ^^=0.459 MPa.m, ^«=11.58 MPa.m, mt=29.5 g, msc=6.5 g, rc=62 Ns/m, r^=3.2 Ns/m, V=l m/s). The simulated register of P/ generated using all the parameters is plotted in Fig. 6. The "real" Pb is also plotted to compare the curves estimated at different values of /. We follow the proposed methodology and then assume that only the values of kt, mt, kc, and re are known as a result of preliminary tests. Equations (22) and (23) are then used to estimate Pt and rrisc- The results are also plotted in Fig. 6 for different values of y. 300
3001
Fig. 6. Simulation example using a set of parameters taken from the literature [14]: recorded force P( (-O^, actual bending force Pb (-•-) and (a) estimated Pb with /=10"^^ (—) and estimated Pb with x=10"^^ ( ); (b) estimated Pb with /=10''^( -^) and estimated Pb with r=\0'^\—) It can be seen in Fig. 6.a that for very small /(lO""^^) the recovered Pb is very oscillatory and it differs completely from the "real" one. The resulting line shows actually the superposition of two oscillations having periods of approx. 0.2 and 0.008 ms. The oscillation with the smaller period shows an amplitude increasing with time. This latter oscillation is a consequence of numerical instability [9] and its frequency is related to the discretization of the experimental curve. This resuh validates the prevention taken at the moment of inverting Eq (17); such small
K PETTARIN, P. FRONTINI AND G. ELIC^ABE
274
Y is in practice equivalent to have inverted Eq (17) without any precaution; i.e. without regularization. When a value ofy=W^^ was used the estimated Pb follows quite well the "real" curve (Fig. 6.b). This ;K makes the estimated value of nisc be exactly the "real" one. These results theoretically validate the proposed methodology and alert on the attention that must be put on the selection of the regularization parameter y, because it affects not only the estimated curves with, sometimes, obvious spurious oscillations, but also the parameter w^c vvith no simply observable effects.
Application to Real Load-Time Curves The estimated tup parameters are kt= 72.6 MPa.m, M= 5.317 kg and mt= 0.189 kg. The tup/specimen contact parameters estimated as stated in previous sections are listed in Table I. Three typical load-time curves, with different notch-depth to specimen-width ratio, a/W, of PP, MDPE and RTPMMA are plotted in fig. 7, 8 and 9 respectively. Also, the recovered sending force is plotted together with the recorded force, for all cases. The estimated values of the mass of the central part of the specimen, rrisc, are listed in Table II. The values ofPb and rrisc obtained are in agreement with those expected, being the former a smoothed version of the force records and the latter in harmony with the total mass of the tested specimens.
Table I. Tup/specimen contact parameters PP Material 4.447 kc (MPa.m) 209 Vc (Ns/m)
MDPE 0.237 15
RTPMMA 0.5403 469
Table II. Estimated masses of the central part of the specimen for all tested materials PP Material MDPE RTPMMA 0.34 a/W 0.26 0.48 0.2 0.4 0.5 0.57 0.16 0.23 2 1.5 2.5 0.5 0.5 0.5 9 7.5 9.5 rrisc (g)
(£i)
400
30O
300
M
k ^_1 JAi
2-2D0
100
QO
(c)
400
Q2
f(tTB)
Q4
Q6
-J^\
100
/x/
QO
02
04 f(rTB)
06
QO
Q2
—,
Q4
1
Q6
f(nB)
Fig. 7: Load-time signals for PP, measured Pt (-D-) and recovered Pb (—). a) a/^=0.26; b) alW=Q3A\ and c) alW=^A%.
Inverse Method for the Analysis of Instrumented Impact Tests of Polymers
275
ZKXJ
(c)
eoo -2:
yi. J^ '.V-
^300
QO Q5 1.0 1.5 20 25
00 05 1.0 1.5 20 25
0 >M^', , , \' '^ 00 05 1.0 1.5 20 25
f(rTB)
f(nfB)
f(rTB)
Fig. 8. Load-time signals for PE, measured Pt (-O^ and recovered Pb (—). a) alW=02\ b) fl/^=0.4;andc)fl/^=0.5.
h ^^
8D0
eoo
eoo g^oo
h
Ini
QO Q2 04 06 08 1.0 f(tTB)
2D
w
80O
eoo tl
S400
Q.
2D0
(b)
80O
r
\f\
QO Q2 04 06 08 1.0 t(nB)
21) 0
A
/)C
iV
'\fV
00 02 04 06 08 1.0 f(nB)
Fig. 9. Load-time signals for RT-PMMA, measured Pt (-D- ) and recovered Pb (—). a) fl/«^=0.16; b) ^/^=0.23; and c) alW=Q.51. CONCLUSIONS A comprehensive methodology to process load-time registers from impact tests in three point bending mode capable of extracting the effects of inertial loading and resonance in the strikersample-support system which are superimposed over the true impact response has been proposed. This methodology is based on a model available from the literature [4]. A non-linear inverse problem in which the unknowns are the bending force and the contact mass must be solved. The solution of the resulting inverse problem requires the calculation of a regularization parameter. This parameter is automatically calculated for each run based only on the model an the experimental register. As far as the authors know this is the first work in which inversion techniques are used to solve this problem. The proposed method has the following characteristics: a) Independent calibration is only needed for the parameters related to the testing machine {kt and m^ and the two related to the
276
V. PETTARIN, P. FRONTINI AND G. ELIQABE
material/machine contact {kc and r^), the fifth parameter, nisc, related to both material and geometry of the tested specimen is estimated during the actual test. This fact reduces the number of independent calibration experiments to three: 1) one to characterize the stiff less of the tup (A:/), 2) one to characterize the mass of the tup (mi), and 3) one for each new material to be tested {kc and r^), regardless of crack length, span, and rigidity of the supports, b) The tests designed to determine the parameters that must be known in advance (kt, rrit, kc and r^), are strictly based on the original model [4] and estimated under similar dynamic conditions as the actual tests. Compared to other methods used to extract the bending force from impact tests, it can be noted that using the present approach several drawbacks existing with the other methodologies are removed: a) the increase of fracture time and addition of nonlinearities characteristic of mechanical damping; b) the lack of foundation of methods used to numerically smooth the experimental registers; c) the additional cost incurred when samples are instrumented. Finally, when real load-time curves of pre-cracked specimens are processed, the critical fracture parameters deduced result in good agreement with the ISO protocol [15]. ACKNOWLEDGEMENT The authors would like to thank the National Research Council of Argentina (CONICET), ANPCYT (PICT 14-07247), the University of Mar del Plata and CEAST for financial support. REFERENCES [1] [2] [3] [4]
Williams, J. G. and Adams, G. C. (1987) Int. J. Fracture 33, 209 Zanichelli, C, Rink, M., Pavan, A. and Ricco, T. (1990) Potym. Eng. Scl 30, 1117 Marur, P. R., Simha, K. R. Y. and Nair, P. S. (1994) Int. J. Fracture 68, 201 Pavan, A. and Draghi, S. (2000) In: Fracture of Polymers, Composites and Adhesives ESIS Publication 27, pp. 347-361, J. G.Williams and A. Pavan (Eds). Elsevier, The Netherlands [5] Cain, P.J. (1987) In: Instrumented Impact Testing of Plastics and Composite Materials, pp. 81-102, S.L. Kessler, G.C. Adams, S.B. Driscoll, and D.R. Ireland (Eds). American Society for Testing and Materials, Philadelphia. [6] Phillips, D. L. (1962) J. Assoc. Comput. Mack 9, 84 [7] Kalthoff, J. F. (1985) Int. J. Fracture 27, 277 [8] Twomey, S. (1963) J. Assoc. Comput. Mack 10, 97 [9] Twomey, S. (1977) Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements. Elsevier, New York. [10]Bertero, M., De Mol, C. and Viano, G. A. (1980) In: Inverse Scattering Problems in Optics; Topics in Current Physics, pp. 161, Bakes, H. P. (Ed). Springer Verlag, New York. [ll]Eli9abe, G. E. and Garcia Rubio, L. H. (1990) In: Polymer Characterization. Physical Property, Spectroscopic, and Chromatographic Methods, pp. 83-104, C. Craver and T. Provder (Eds). Advances in Chemistry Series, 227. [12] Frontini, G. L. and Eli9abe G. E. (2000) J. Chemometrics 14, 51 [13] Golub, G. H., Heath, M. and Wahba, G. (1979) Technometrics 21, 215 [14] Draghi, S. (1996) Thesis, Politecnico di Milano, Italy. [15]IS0/DIS 17281: "Plastics - Determination of fracture toughness (Gic and Kjc) at moderately high loading rates (1 m/s)" (2002).
2. ADHESIVE JOINTS
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
279
FRACTURE MECHANICS TESTS TO CHARACTERIZE BONDED GLASS/EPOXY COMPOSITES: APPLICATION TO STRENGTH PREDICTION IN STRUCTURAL ASSEMBLIES PETER DAVIES, JEFFREY SARGENT Materials & Structures group, IFREMER, 29280 Plouzane, France BAE Systems, Advanced Technology-Centre Sowerby, Bristol, UK ABSTRACT This paper presents results from a study of assemblies composed of glass fibre reinforced epoxy composites. First, tests performed to produce mixed mode fracture envelopes are presented. Then results from tests on lap shear and L-stiffener specimens are given. These enabled failure mechanisms to be examined in more detail using an image analysis technique to quantify local strain fields. Finally the application of a fracture-mechanics-based analysis to predict the failure loads of top-hat stiffeners with and without implanted bond-line defects is described. Correlation between test results and predictions is reasonable, but special attention is needed to account for size effects and micro-structural variations induced by the assembly process. Keywords. Fracture envelope, Stiffener, Top-hat, Debonding, Image analysis. INTRODUCTION Adhesive bonding is finding increasing applications in marine and aerospace composite structures. The weight savings associated with this form of assembly can be considerable, but designing to optimise strength requires great care. The overall aim of the present work is to develop reliable methods to predict the failure strength of composite assemblies. There have traditionally been two approaches available for strength analysis, either stress-based or fracture mechanics based, though several variants exist according to the adherend and adhesive properties [1-3]. The stress-based approach involves performing a stress analysis of the structure, either analytically, for simple assemblies, or numerically by finite elements for more complex structures, and then comparing stresses or strains to failure criteria for the different constituents (adherend, adhesive and if possible interface). The difficulties with this approach in the past have been the analysis of thin bond-line regions and tackling the stress singularities at joint extremities. With the increasing computer power available today the former is now less of a problem, but the latter remains a major limitation. It should be noted also that sources of reliable strength data to apply in failure criteria have always been scarce, particularly when the adherends are composites. At first sight a fracture mechanics approach which enables crack tip fields to be treated looks very attractive for this type of problem, and indeed it has been employed in the description of delaminations in orthotropic materials with some success (refs). However its application to composite assemblies has been limited so far, partly due to a severe shortage of fracture data to use in a failure criterion but also because it
280
P. DA VIES AND J. SARGENT
is necessary to assume the presence of a crack. The question is then raised, where should this crack be introduced and how long should it be ? Some work has been performed to address this question, notably by Femlund and Spelt [4,5] and encouraging results have been presented for a range of bonded metal joints. It should be noted that these two approaches are not the only ones attracting interest at present. Other recent developments have included the use of damage mechanics models [6] and the development of cohesive zone models [7,81. A first aim in the present study was to establish whether the approach proposed by Femlund & Spelt, which will be described in more detail below, could be applied to bonded composite joints. A second objective was to evaluate the benefits to be gained from a fracture mechanics characterisation when industrial structural assemblies are considered, both qualitatively in material selection, and quantitatively when failure predictions are required. The paper is presented in three parts. First, the tests employed to determine the mixed mode fracture envelope of a glass fibre reinforced epoxy composite adhesively bonded with either a brittle or a ductile adhesive are briefly described. These include mode I (DCB), and mixed mode (MMB) with various mixed mode (I/II) ratios. In the second part of the paper different structural joints will be discussed. These include single and double lap shear and Lspecimens. In a recent European thematic network lap shear and double lap shear composite joints were tested, and predictions of failure load were made by different academic and industrial partners [9,10]. It was apparent that considerable differences existed between different analytical predictions and FE analyses, and correlation with tests proved complex. In particular, the progressive damage development in assemblies bonded with a ductile adhesive was not treated adequately. A more detailed study of damage mechanisms was therefore undertaken, using image analysis combined with microscopy to examine the crack tip strain fields and measure adherend displacements. This is described below and correlation is made between predicted displacements and failure loads, based on the mixed mode envelope determined previously, and measured values. In the final part of the paper the extension of the fracture mechanics approach to the prediction of the pull-off of top hat stiffeners on sandwich panels is presented. Composite tophat stiffeners have been studied by several workers as they are often used in ship structures. Smith [11] gives an overview of structural connections in marine structures and describes experimental results. Shenoi and colleagues have described failure mechanisms under pull-off and flexural loading and compared failure loads with predictions from finite element analysis [12,13]. For their geometry and materials they predict delamination of the overlaminate under tensile pull-off loads, using a failure criterion based on a critical value of through-thickness stress. They also examined strain energy release rates, but for the overlaminate not the bonded interface [13], In the present work we have concentrated on the latter. Experimental details and initial results are given and the influence of implanted bond-line defects is discussed. MATERIALS The results described in this paper were all obtained from tests on E-glass reinforced composite materials produced by hand lay-up. This is the manufacturing route most frequently used for marine structures. For the majority of the tests reported here the E-glass fibres were either quasi-unidirectional (250 g/m^ with 1 g/m^ of polyester fibres bonded in the 90° direction to keep the UD fibres in place) or stitched quadriaxial (0/45/907-45° 1034 g/m^) cloths. The same uniaxial ply is used in both cloths. The resin is based on DGEBA epoxy (SRI 500) with an amine hardener (2505) from Sicomin, France. All epoxy specimens were post-cured at 90°C for 6 hours. Some results are also shown for a woven glass (0/90° 500 g/m^) reinforced isophthalic polyester for comparison, as this is the traditional marine
Fracture Mechanics Tests to Characterize Bonded Glass/Epoxy Composites
281
material. Fibre fraction of the quasi-unidirectional panels was measured by bum-off to be 57±1% by weight, that of the quadriaxial panels was 58 ±1% (this is around 37% by volume). Starter films for crack initiation were 8 micron thick polypropylene. The bonded assemblies were produced using two adhesives. The first, which will be discussed in more detail here, is the matrix epoxy resin employed either to bond the composite adherends (in the case of lap shear and L specimens) or to overlaminate onto a balsa sandwich or monolithic composite base in the case of top hat specimens. The second is a more ductile epoxy resin (Redux 420), used to bond the composite adherends. Figure 1 gives an indication of the tensile behaviour and fracture toughness of these resins.
— Brittle epoxy
Gc 300 J/m^
—Tough epoxy
Gc1400J/m2 0
2
4 6 Tensile Strain, %
8
10
Figure 1. Tensile stress-strain behaviour. Indicative Gc values from tests on cast brittle resin, and on tough epoxy as adhesive on steel substrates. FRACTURE MECHANICS APPROACH TO PREDICT FAILURE LOAD In order to determine the failure load of a structure under a given loading the approach is essentially as follows: First determine the maximum strain energy release rate in the structure for the load of interest. This may be achieved either by applying analytical expressions, when the geometry is simple, or by finite element methods using virtual crack closure for more complex structures. This gives values of Gi and Gn (or Jj and Jn). to be compared to a fracture criterion, which has been determined by a separate series of tests on the same materials. If the loads of interest are well defined only a small part of the mixed mode failure envelope may be needed, but more generally the complete mixed mode envelope is required. Figure 2 shows this procedure schematically for a double lap shear specimen. 1. Analysis of specimen, Gi:Gii
Figure 2. Schematic illustration of procedure to determine failure load, Gc=Gi + Gii, (non-linearity (NL) values shown).
282
P. DA VIES AND J. SARGENT
DETERMINATION OF MIXED MODE FRACTURE ENVELOPE The use of fracture mechanics tests to characterise structural adhesives was proposed over 30 years ago by RipHng, Mostovoy and colleagues, e.g. [14]. Interlaminar fracture testing of composites and adhesively bonded composite assemblies has been reviewed recently [15]. The loading of interest for the structures described in the present paper are in the mode I region so the test configurations used were the double cantilever beam (DCB) for pure mode I and the mixed mode MMB for mode I/mode II. The determination of the mixed mode fracture envelope for the composite and adhesively bonded specimens has been described elsewhere [16,17]: the envelope shovm in Figure 2 comes from that work. Several initiation criteria can be defined, based on values taken from the load-displacement plots (e.g. non-linearity, maximum load), acoustic emission parameters or visual crack observations. It seems reasonable to apply these values to predict the behaviour of the same materials in the form of small assemblies such as lap shear specimens, produced by bonding composite adherends with the matrix epoxy resin, provided the adhesive bond-line thickness is similar to the interlaminar layer thickness and that the same criterion is applied. This exercise will be described in the following section. ANALYSIS OF TEST SPECIMENS A first step in the validation of this approach is to test simple specimens under controlled conditions and to compare predictions with measured failure load values. First lap shear geometries were examined, then an L-geometry was studied in more detail. The bond-line in these small specimens was very similar to that in the quasi-unidirectional fracture specimens as the small dimensions allow panels to be pressed uniformly after assembly (which is not the case for industrial top-hat stiffeners). Lap shear specimens Femlund and Spelt [5] used large deformation beam theory (permitting modelling of nonlinear geometric behaviour) together with the J formulation of the critical strain energy release rate to derive fracture load predictions for cracked lap-shear joints. This had the merit that they were accessible, and could be applied relatively easily to a range of different geometries based on different adherend lengths and thickness. The solutions assumed that the adhesive layer was sufficiently thin such that the global deformation of the joint was determined entirely by adherend material. Assuming no influence of adhesive is probably reasonable given long free adherend lengths. However, if the adherends were short, giving rise to a short effective crack length, or if the adhesive layer was thick, then the adhesive would make a significant contribution to the global deformation, and the analysis would be approximate. In addition, it should be noted that the analysis did not deal explicitly with the consequences of any non-linear plasfic behaviour of the adherends (e.g. as a resuh of damage or yielding) or adhesive. In spite of these limitations, however, the authors noted good agreement between theory and experiment for a rubber toughened adhesive system with very thick (12.7 mm) aluminium adherends and long free adherend lengths (-140 - 260 mm). In order to examine the application of this approach to composite assemblies, tests were performed on single and double lap shear specimens as shown in Table 1. Test results were compared with predictions based on reference [5] ^ Figure 3 shows representative results for both adhesives. NB Equation 25 in reference [5] is incorrect. The correct equation, used in tlie analysis here, is: -DiC3>.i^cosh(>.iLi) = -D2>.2^{C7COsh(?.2Li) + C8sinh(>.2Li)}
283
Fracture Mechanics Tests to Characterize Bonded Glass/Epoxy Composites
Single lap shear SLS
Double lap shear, DLS
Specimen Note two acoustic emission transducers on each specimen
^B^lT^P^^ 3/3/3 & 3/6/3 10,20,30 SR1500, Redux 420 No fillet
3,6 10,20,30 SR1500, Redux 420 With & without fillet
Thickness (mm) Overlaps (mm) Adhesive Joint end geometry
j
Table 1. Lap shear tests performed Redux 420
z
"S 12000
i
SR1500
Test no fillet
SI
Model
Test no fillet
Test with fillet
Model
Figure 3. Test-prediction comparisons for 3mm adherend single lap shear specimens. The correlation is quite good for the SRI 500 resin, while for the more ductile adhesive resin the predictions overestimate the measured failure loads. However, in the latter case an extensive damage zone develops before final failure and the non-linear elastic fracture model is no longer appropriate. It is interesting to note however, that when a fillet is left at the end of the overlap the test values are much closer to the predictions. Strain mapping and deformation analysis of "L " type joints. The "L" type joint used here and shown in Figure 4 represent a common generic geometric element of structures used in modem aircraft designs.
Figure 4. Schematic diagram showing the loading and geometry for the "L" type specimen. These could be found, for example, in "zed" section stringers or parts of rib-elements in aircraft wings. In-situ testing of small, approximately 2mm wide sections, of such structures was performed on an optical microscope using a "Minimaf miniature materials tester [18]
284
P. DAVIESANDJ. SARGENT
with the loading arrangement shown schematically in Figure 4. Image analysis [19] was then used to spatially correlate surface features from images taken in the initial unloaded state, with the same features in the loaded state, to derive strain field maps within the adhesive bond-line and to make measurements of adherend displacements. Figure 5 shows an example of a typical vector displacement map for the detailed inset region of the above specimen.
Figure 5. Vector displacement map for the inset region of the specimen from Figure 4. Heavy dashed white lines show adherend outline. Lower adherend approximately 1.6 mm thick. It was estimated that displacements could be obtained with an accuracy of better than 1/20^^ of a pixel, giving an equivalent accuracy of approximately 0.2JLI when using a 2.5x microscope objective, and strains could be measured with an accuracy of approximately 0.1%. Figure 6(a) and (b) shows, respectively, maps of the tensile (^Uy/5y) and shear {dwjdy +dnyldx) strain components recorded for the specimen from Figure 5 with an intact fillet and 1.6 mm thick SRI 500 composite adherends under a load P3 of approximately 20 N/mm.
0.02
•g O.OIH 0.0 I
(a) {dviyldy)
(b) {dwjdy +auy/ax)
Figure 6a, b. Strain maps for the specimen with a fillet from Figure 5. Load P3 = 20N/inm. Examination of these images showed significant strain developed throughout the whole fillet with maximum strain values of approximately 1% for the tensile (Syy ) and shear (Cxy = {dwxidy +d\Jiyldx)l2 ) components at a region located within the fillet and adjacent to its free surface. Specimens without a fillet were also tested in which starter cracks had been deliberately introduced into the bond line. Figure 7 (a) and (b) shows, respectively, the tensile and shear components of strain for an example of such a specimen with a crack of length 0.5mm.
Fracture Mechanics Tests to Characterize Bonded Glass/Epoxy Composites
285
0.02
c
•g 0.01
I
00
0.0 I
(a) {dwyldy)
(b) {d\xjdy
+d\Xyldx)
Figure 7. Strain maps for specimen without a fillet and with a 0.5 mm crack (effective crack length "a" = 4.8 mm). Load P3 = lON/mm. This specimen, which failed at a load P3 of 11.7N/mm, is shown at a load before failure of approximately lON/mm. Inspection of Figure 7 shows that whilst the bulk of the adhesive shows tensile strains of approximately 0.1%, regions of tensile and shear maxima apparently followed the outline of the interfacial region between adherend and adhesive, with some regions indicating strains in excess of 2-3%. It should also be noted however, that a careful inspection of the vector displacement map showed that these large apparent strains were in fact due to a discontinuity in the displacement vectors as a result of sub-critical damage accumulation ahead of the nominal crack front. Figure 8 shows a plot of the adherend displacements (Uy) as a function of "x" for the specimen from Figure 7. '^
1....LJ i L
5r
r
X
^
1
xifeiais Fflajc bnXejnt
"^"^x;^
^ i „
idhen ToV'<
1
c
^ ocd
'^
U-l(|)
^
X-J....
i"^ .440W eri.^ 4dh<j renldttt ±:x :t:: :±:
"x"iDixel s
1400
Figure 8. Transverse adherend displacements Uy for the specimen from Figure 7. The nominal crack front was located at an "x" pixel position of 450, equivalent to an effective crack length "a" of approximately 4.8 mm from the loading point P. The dotted line shows the net w(x) displacement difference between the adherends. Beam theory modelling of "L " type adherend deflection Beam theory methods offer a potentially simple method for modelling the deformation of the adherends and also for the prediction of failure loads using linear elastic fracture mechanics. These methods usually make use of beam on elastic foundation models in order to describe the deformation of the adherends, from which the strain energy release rates can then be calculated. An analysis of the transverse beam deflection w(x) was undertaken here by approximating the lower adherend as a single beam resting on an elastic foundation of modulus "k" comprising a variable adhesive bond thickness and fixed rigid upper adherend of constant thickness. The accuracy of such an approximation was determined by comparing the predicted beam displacement with that measured experimentally (shown in Figure 9).
286
P. DAVIESANDJ. SARGENT
10.0 Approximate crack location
Effective crack length "a" (mm)
Figure 9. Comparison between predicted and experimental measurements of net beam displacement (w(x)) for the specimen from Figure 5. Formally, the differential equation relating the transverse deflection of a simple beam of width "b", thickness "h" is given by: ,4
-^p-H4l^w(x) := 0 where: X^ = 3.k/Ex.b.h^ dx'
Ex = modulus of adherend in "x" direction (25.6GPa), k = foundation modulus, with a general solution of the form: w(x) := C.cos(Xx)cosh(^x) + C.cos(lx)sinh(^x) + C.sin(\x)cosh(lx) + C.'sinh(^x)sin(^x)
This follows analysis given by Sargent and Wilson [20] whereby the adherend was approximated as a partly free beam and a beam partly supported on the adhesive. The partly supported beam was divided into five discrete sections, each of which rested on a Winkler elastic foundation with a foundation modulus which was representative of the adhesive thickness under that particular part of the beam, together with a contribution from the deforming beam and fixed beam. A Winkler type foundation can be imagined as a continuous array of independent linear springs which simulates the transverse elasticity of the uncracked region. By matching boundary conditions at each beam element junction this resulted in a set of six simultaneous differential equations which were solved numerically. Figure 10 shows the resulting stress distribution calculated using this approach. It may be noted that the stepwise changes in stress reflect the 5 discrete foundation modulus sections used in the analysis. 80-
5.8
9.8 6.8 7. Effective crack length "a" (mm)
Figure 10. Approximate stress distribution (ay) for "L" type specimen predicted for load P3 of lON/mm. Predictions based on 5 stepwise adhesive thicknesses from 0.8 mm to 0.08 mm at a = 4.8 mm, 5.6 mm, 6 mm and 6.4 mm. See text for details.
Fracture Mechanics Tests to Characterize Bonded Glass/Epoxy Composites
287
Reference to Figure 9 shows that the predicted displacement is underestimated using this procedure. Using a modulus of 2.8 GPa for the adhesive in combination with the stress derived from Figure 10 implied predicted tensile strain levels adjacent to the crack front (a = 4.8 mm) of approximately 1%. These may be compared with the measured values of approximately 0.1% for the bulk adhesive strains shown in Figure 8. It is likely that these discrepancies between theoretical and experimental strain values arise from the exclusion of the damaged regions, which were noted in Figure 7. Failure load prediction of'L" type joints If the "L" type specimen is considered to act as a single deforming beam which can be modelled using simple linear elastic beam theory, and that both damage accumulation and adhesive shear deformation may be ignored, then a simple first approximation to estimating failure loads may be derived by considering that the failure load is controlled simply by a critical mode I strain energy release rate. Making the further assumption that a single average adhesive bond line thickness may be used, Sargent and Wilson [20] shows that the failure load prediction (P) for the single beam arm of the specimen with the current geometry may be found from: 6 P -a
Ex
1-H
r» uO-25
b^-h^-Ex
Bh
-a
B^.h^-^
where:
8Gxy
B:= 3
Exb/
where: Gxy = shear modulus of adherend (2.5GPa), a = effective crack length (distance between load point "P" and crack front shown in Figure 4). Noting that the specimen is statically determinate, then the failure load P3 is related to P from Figure 4 via P = P3/ (1 + L1/L2). Figure 11 shows a comparison between the predicted (P3) and experimental failure loads as a function of the effective crack length. In general, failure loads are underestimated using this procedure, though there is some scatter in results. 20 18 16 14 12 o 10-1
• Experimental pointsj -»- Predictions
8-1
6
4-] 2-| 0 3.5
4.0
4.5
5.0
5.5
6.0
Effective crack length "a" (mm)
Figure 11. Comparison between predicted and experimental failure loads (P3) for "L" type specimen as a function of effective crack length. Using Gic = 240 J/m^. APPLICATION TO A MARINE TOP-HAT STRUCTURE The results above suggest that it may be possible to apply fracture mechanics data to determine failure loads of more complex structures, provided that (i) the adhesives used are not too ductile, (ii) bondline thickness is known and controlled, (iii) non-linear behaviour due to adherend and interface damage is limited, and (iv) the specimens employed to determine
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P. DAVIESANDJ. SARGENT
the fracture envelope are representative of the real structure. In order to examine this two series of top hat stiffeners were manufactured. These are typical of real stiffeners in naval superstructures. The first series used a woven glass fibre reinforced polyester representing current shipbuilding practice. The second series was based on the glass/epoxy material. For both materials, in addition to the standard stiffener additional specimens were prepared with starter cracks (8 micron thick polypropylene film) of different lengths (20, 40 and 60 mm) placed at the stiffener/base interface during manufacture. The test set-ups used are shown in Figure 12. When the base is a stiff balsa sandwich material the loading is pure mode I, Figure 12(a). As the base becomes thinner there is a mode II component. Figure 12(b), so a simple tensile pull-off allows the I/II mode mix to be varied. In the limited space available here only results from the former will be given.
Figure 12. Top hat stiffener pull-off test set-up (a) mode I, (b) mixed mode I/II. Two digital cameras interfaced with a PC and a load input were used to record the crack length and opening displacement continuously. An image analysis programme then enabled crack length versus applied force and crack opening displacement plots to be generated. Two acoustic emission transducers were placed on the stiffeners and interfaced to a second PC. The load and machine cross-head displacement were recorded on a third PC. The specimens without implanted defects will be considered first, it is apparent from the load-displacement plots that replacing the woven glass/polyester by a QX/epoxy material results in a very significant increase in pull-offload. Figure 13. Pull-off tests, no defect 8000 QX/Epoxy
5
10
15
20
displacement, mm
Figure 13. Load-displacement plots, top-hats without defects.
Fracture Mechanics Tests to Characterize Bonded Glass/Epoxy Composites
289
However it should be emphasised that many factors affect pull-off load including geometry, fillet material, and surface preparation. The top-hats employed here were produced to allow the existing fracture data to be applied, they are certainly not optimised. When specimens with implanted defects are tested there is a very large drop in pull-off load for both materials. Examples of load and crack length versus time are shown in Figure 14. For the shortest defect the crack propagation is unstable and asymmetric but for the longer defects stable, symmetrical propagation was observed. QX/Epoxy pull-off Nominal ao = 40mm
QX/epoxy pull-off, Nominal ao = 20mm
Figure 14. Load and crack length versus time plots top-hats with defects. Transverse pull-off tests induce mainly mode I loading, provided the base panel is sufficiently rigid. Finite element analyses have been performed to look at this geometry in more detail, and will be reported elsewhere, but here a simple analytical beam theory expression is used to predict the pull-off failure load [21]: F=2 [
4B'h'EGjc 21a'
]
The results from 12 pull-off tests on QX/epoxy specimens with implanted defects are shown in Figure 15. Both measured and predicted values are shown. Different criteria may be used to compare top hat pull-off and fracture test values. These include various acoustic emission parameters (first acoustic events, first events above a certain amplitude), visual or image analysis parameters or values on the load-displacement plots. Several criteria have been examined, here non-linear values are shown (Gic = 240 J/m^, the lower, dashed line). QX/Epoxy top hat pull-off (E=15 GPa, t=5mm) 4000 1 M" 3000
I
c o c w o
2000
1
1
A \\ \\ \\ \\
• NL experimental points - - Gc = 240 J/m^
^%."
Gc = 330J/m2
1000 H 04 0
1
0.02
0.04
0.06
0.08
Defect length, m
Figure 15. Test results and predictions, QX/epoxy top-hats on balsa sandwich.
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P. DA VIES AND J. SARGENT
FURTHER CONSIDERATIONS While the results from this first attempt to use fracture data from standard specimens to predict failure loads are quite encouraging there are a number of aspects which have been neglected and these must be examined in more detail if the work is to be extended in the future. Size effects One point which should be addressed is size effects. This is important, not only because dimensions of marine structures can be very large and the hand lay-up fabrication can lead to rather heterogeneous materials, but also because the reinforcement repeat unit size may be tens of millimetres. Larger specimens were therefore produced, 50 mm wide and 350 mm long rather than the standard 20 mm by 140 mm dimensions tested previously. These large specimens were manufactured from exactly the same materials as the top hat stiffeners, i.e. four layers of quadriaxial glass cloth each side of the starter film for a total specimen thickness of 8 mm. The layers were arranged symmetrically about the specimen mid-plane, with the starter film placed between two 0° layers, i.e. exactly the same interface as that in the specimens used previously. A special scaled-up MMB fixture was developed to test them under mixed mode loading. Here only the mode I results will be presented, in the following section. Manufacturing effects If the production assembly process influences the material structure, compared to that found in the quasi-unidirectional specimens used to obtain the fracture envelope, then there is no reason for predictions to correlate with measurements. Two manufacturing procedures were studied, the first was a continuous lay-up in which a panel was produced with no pause during manufacture. This corresponds to the composite adherend fabrication. The second procedure was in two steps. Half the panel thickness (4 layers) was impregnated. Fabrication was then stopped, a peel ply layer was applied and the panel was left overnight. The following day the peel ply was removed and the four remaining layers were laminated. This second procedure corresponds more closely to real manufacturing of top hat stiffeners by over-lamination, where large panels are produced first and stiffeners are added later. It can result in poor bonding and/or a resin rich layer at the interface between the stiffener and base plate. When specimens of both materials were tested there was no significant difference in initiation \alues for the epoxy composite but the polyester material showed lower values for the material manufactured with a delay. Thus while tests on continuously-produced specimens indicate very similar values for the polyester and the epoxy when the real fabrication process is used the former are only half those of the latter and this is reflected in the pull-offloads.
Continuous With delay
Woven/Polyester 50 X 350mm2 247 (9%) 177(6%)
QX/Epoxy 50 X 350 mm^ 292 (28%) 331 (19%)
UD/Epoxy 20 X 140 mm^ 242(11%) -
Table 2. Initiation values of Gic at non-linearity, J/m^ (coefficients of variation in brackets) The values measured on the QX/epoxy with delay are also higher than those for the UD material. Predictions with this higher value are shown on Figure 15, but are not sufficient to explain the under-estimation of the pull-offloads in the top hats with longer defects.
Fracture Mechanics Tests to Characterize Bonded Glass/Epoxy Composites
291
Another aspect to be considered is the difficulty in producing curved structures with the same fibre content as flat laboratory panels. This effect is shown in Figure 16, at the comer the laminate thickness is larger than at the flat section and fibre content is rather lower. This will affect the bending stiffness of the arm and the predicted failure load. This figure also shows the fillet, which is critical to initiation in the specimens without implanted defects. It is well known that fillets can significantly alter the load path in lap shear joints and increase the failure loads (see [1] and Figure 3 for example). If a fracture mechanics approach is to be applied this effect must be considered. Some recent studies on stress intensity factors for such cases may allow this to be addressed [22].
Figure 16. Detail of top hat stiffener crack tip Ductile adhesives While the approach described above works reasonably well for the polyester and epoxy matrix resins which do not show extensive ductility there is clearly a need for an alternative approach when very ductile structural adhesives are applied. The development of extensive damage zones can be treated more efficiently using damage mechanics models and these are now being examined. 3-D structures Finally, it should be emphasised that the objective of this work is to optimise design of real three-dimensional structures such as stiffened panels. While the top hat section is part of this structure debonding is generally seen to initiate at the stiffener ends under complex (mixed mode) loading and stiffener/panel debonding is not the unique failure mechanism. The top hat pull-off test is still of some interest, as mixed mode loading (I/II) can be studied by varying the plate thickness of the lower panel (Figure 12b). However, tests on real structural elements are important if the correct failure mechanisms are to be modelled and some of these are being examined in a current project (EUCLID RTF 3.21, [23]). CONCLUSION Fracture mechanics characterisation tests have been performed to determine the mixed mode fracture envelope of an epoxy bonded glass/epoxy composite. Analysis of lap shear, and Lstiffener geometries has shown that for this relatively brittle adhesive reasonable first estimations of failure loads can be obtained for both cracked and uncracked specimens. An image analysis technique has been developed which enables failure mechanisms to be
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P. DA VIES AND 1 SARGENT
analysed in more detail. Damage development and accumulation have been observed before final fracture, and must be taken into account if better predictions are required. The extension of the predictions to an industrial top hat stiffener shows that while the fracture mechanics properties do mirror the structural behaviour many geometrical and manufacturing details must be taken into account if reliable predictions are to be made even for this simple geometry. Further work is concentrating on more detailed damage modelling and more complex loading cases. REFERENCES [I] Adams RD, Comyn J, Wake WC, Structural adhesives joints in engineering, Chapman & Hall, 2"^^ edition, 1997. [2] Tong L, Steven GP, Analysis and design of structural bonded joints, Kluwer Academic, 1999. [3] Curley A J, Hadavinia H, Kinloch AJ, Taylor AC, Predicting the service life of adhesively bonded joints. Int. J. Fracture, 103, 2000, 41-69. [4] Papini M, Femlund G & Spelt JK, Effect of crack growth mechanism on the prediction of fracture load of adhesive joints, Comp. Sci. & Tech., 52, 1994, 561. [5] Papini M, Femlund G & Spelt JK, The effect of geometry on the fracture of adhesive joints, Int. J. Adhesion & Adhesives, Vol 14, 1, 1994, 5. [6] Allix O, Leveque D, Perret L, Identification and forecast of delamination in composite laminates by an interlaminar interface model, Comp. Sci & Tech., 58, 1998, 671-678. [7] Tvergaard V, Hutchinson JW, The relation between crack growth resistance and fracture process parameters in elastic-plastic solids, J. Mech & Physics of solids, 40, 6, 1992, 1377-97. [8] Chandra N, Li H, Shet C, Ghonem H, Some issues in the application of cohesive zone models for metal-ceramic interfaces. Int. J. Solids & Structures 39, 2002, 2827-55. [9] Davies et al. Failure of bonded glass/epoxy composite joints, Proc. Structural Adhesives in Engineering (SAE6), Bristol, June 2001. [10] DOGMA thematic network project website http://www.dogma.org.uk/ [II] Smith CS, Design of Marine structures in composite materials, Elsevier Applied Science, 1990, section 4.6. [12] Dodkins AR, Shenoi RA, Hawkins GL, Design of joints and attachments of FRP ships' structures. Marine Structures, 1994, 7, 365-98. [13] Phillips HJ, Shenoi RA, Moss CE, Damage mechanics of top-hat stiffeners used in FRP ship construction. Marine Structures, 12, 1999, 1-19. [14] Ripling EJ, Mostovoy S, Corten HT, Fracture mechanics: a tool for evaluating structural adhesives, J. Adhesion, 3, 1971, 107-123. [15] Moore DR, Pavan A, Williams JG, eds.. Fracture Mechanics Testing methods for Polymers, Adhesives and Composites, ESIS Publication 28, 2001, Elsevier. [16] Ducept F, Gamby D, Davies P, A mixed mode failure criterion from tests on symmetric and asymmetric specimens Comp. Sci & Technology, 59, 1999, p609-619. [17] Ducept F, Davies P, Gamby D Mixed mode failure criteria for a glass/epoxy composite and an adhesively bonded composite/composite joint. Int. Journal of Adhesion & Adhesives, 20, 3, 2000 p233-244. [18] "MINIMAT" miniature materials tester, Polymer Laboratories, UK. [19] "microDAC", Fraunhofer, IZM, Gustav-Meyer-Allee 25, D-13355 Berlin. [20] J P Sargent and Q Wilson, "Prediction of "Zed" section stringer pull-offloads". Accepted for publication in Int. J. Adhesion and Adhesives, 2002. [21] Hashemi S, Kinloch AJ, Williams G, Mixed mode fracture in fiber-polymer composite laminates, ASTMSTP 1110, ed O'Brien TK, 1991, 143-168. [22] Wang CH, Rose LRF, Compact solutions for the comer singularity in bonded lap joints, Int J. Adhesion & adhesives, 20, 2000, 145-154. [23] EUCLID RTP 3.21, website http://research.dnv.com/euclid_rtp3.21/
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
293
ON THE MODE II LOADING OF ADHESIVE JOINTS B.R.K. BLACKMAN, AJ. KINLOCH & M. PARASCHI Department of Mechanical Engineering Imperial College London Exhibition Road London SW7 2BX. UK. ABSTRACT This paper reports the results of research studies on measuring the mode II fracture toughness, Giic, of structural adhesive joints via a Linear Elastic Fracture Mechanics (LEFM) test method. Adhesive joints were manufactured using carbon-fibre reinforced plastic substrates and these were bonded with one of two commercial structural epoxy adhesives. Mode II loading was achieved using the end-loaded split test geometry and the specimen dimensions were controlled to ensure the conditions of LEFM were not violated. Data analysis was achieved by the use of corrected beam theory and experimental compliances approaches. Values of Guc were plotted against the measured crack length to yield the apparent mode II resistance curves (i.e. the mode II R-curves). These curves revealed a characteristic shape that required detailed interpretation. Uncertainties in the measured crack length values were identified as the most likely cause of the disagreement between the values of Guc deduced via corrected beam theory and experimental compliance approaches. It is suggested that these uncertainties are due to the problem of misinterpreting microcracking as the main crack growing. KEYWORDS Adhesive joints, mode II, cohesive failure, R-curve, microcracking INTRODUCTION Whilst test methods to measure the mode I fracture resistance of polymer fibre-composites [1] and structural adhesive joints [2,3] have now proceeded to full standards, the efforts to standardise a mode II method have been beset by a number of problems. Firstly, there exist a number of competing tests: e.g. the end-loaded split (ELS), the end-notched flexure (ENF), the four point end-notched flexure (4-ENF) and the stabilised end-notched flexure (S-ENF) have all been proposed and have been subjected to various inter-laboratory evaluations. However, the results of these programmes have always revealed a very large scatter in the values of Guc measured e.g. [4], and this scatter has been variously attributed to friction
294
B.R.K. BLACKMAN, A.J. KINLOCHANDM. PARASCHI
effects, problems with specimen calibration, difficulty in measuring crack length and microcracking. Indeed, such problems will influence the mode II testing of both polyme • fibre-composites and adhesive joints, however, it has been far from clear which of thes'.; issues are the most significant. Early attempts to quantify the frictional effects during the mode II ELS testing of polymer fibre-composites [5] gave promising results, suggesting that friction could account for a;; much as 20% of the measured mode II fracture energy in a carbon-fibre PEEK composite and as much as 30% in a carbon-fibre epoxy composite. However, repeating these analyses for structural adhesive joints has so far been less successful. The results would suggest that whilst friction does indeed play a role in elevating the measured G\\Q values and hence the Rcurves in the ELS test, it is not the major factor responsible for the very strongly rising values of Giic seen in the present work. The results presented here provide some insight into thtnature of the mode II R-curves measured during ELS testing of structural adhesive joints and demonstrate that the fracture toughness variation calculated is consistent with the development and propagation of microcracks within the adhesive layer. EXPERIMENTAL Joint manufacture The adhesive joints tested in the present work were manufactured using 2mm thick carbon fibre-composite substrates (T300/924 from Hexcel, UK). The substrates were thoroughly dried in an oven immediately prior to bonding to remove any pre-bond moisture from the beams. The substrates were abraded and solvent wiped prior to bonding. Joints were then formed by bonding with either an aerospace epoxy-film adhesive (API26 from 3M Inc.) or with a general purpose epoxy-paste adhesive (ESPl 10 from Permabond Pic). A release film of PTFE was inserted into the bondline at one end to create a crack starter. The joints were cured according to the adhesive manufacturer's instructions. The bondline thickness was controlled during manufacture and was 0.08+0.04 mm for the epoxy-film adhesive and 0.4+0.05 mm for the epoxy-paste adhesive. The sides of the joints were machined to remove excess adhesive and white paint was applied to the joint sides to facilitate crack length measurement.
Fig. 1. The end-loaded split (ELS) test fixture and adhesive joint specimen.
295
On the Mode II Loading ofAdhesive Joints
Fracture testing The joints were precracked in mode I, prior to mode II testing such that the crack grew 2 to 3 mm away from the end of the insert film. This was to improve the stability of the mode II test and create a sharp, natural crack. The specimens were then clamped in the ELS test fixture with a free length, L, of about 125mm. Tests were run in displacement control at a rate of Imm/min. The crack was monitored at the edge of the specimen using a travelling microscope with a magnification X7. This follows the recommendations of the ESIS mode II ELS test protocol for composites [7]. The crack was grown until it reached 10 mm from clamp point. The load, cross-head displacement and crack length were recorded for each increment of crack growth. The test was then stopped and final unloading was performed at 5mm/min. On final unloading, the load-displacement trace returned to the origin, confirming that no permanent plastic deformation of the substrate arms had occurred during the test. The arms of the specimen were then examined for any signs of permanent deformation, before being broken open to assess the locus of joint failure. The compliance of the test fixture was measured by clamping a very stiff calibration specimen in place of the test specimen, and loading the system up to the loads expected during the tests. The system compliance measured was IxlO'"^ mm/N. The displacement values measured for the fracture test were then corrected for system compliance effects. Some additional tests were performed using a more powerful microscope connected to a CCD camera to monitor the crack propagation. ANALYSIS The tests have been analysed to determine the mode II adhesive fracture energy, Guc, using both corrected beam theory and the experimental compliance approaches. The corrected beam theory used was that derived in [6] for the fracture of fibre-composite specimens and now embodied in the ESIS TC4 protocol for ELS testing [7]. The earlier studies on mode I loading [2, 3] have demonstrated that, provided the adhesive layer is relatively thin, it can be neglected in a global energy balance approach for the determination of G, as followed here. The analysis requires the load, P, the displacement, 5, and crack length, a, to be measured at a number of crack length increments. Also required is the beam width, b, the beam height, h and the known flexural modulus of the beam, E. This may be measured in an independent modulus test, or taken as quoted by the manufacturer if a standard grade of material is used. To correct beam theory for the effects of beam root rotation and transverse shear, a correction factor, A\\, is required where A\\ is usually determined from the mode I correction. In this scheme, A\\ =Ziih and Xu is deduced from Zvr^-^^X\ [^J- The value ofxi ^^Y be determined from a mode I test or may be calculated from the elastic properties of the substrate. Corrected beam theory (CBT) may thus be expressed as:
where F is a large displacement correction factor defined in [6]. Beam theory may also be used to eliminate the modulus in eqn. (1) and such an approach leads to a second expression for Giic, called here corrected beam theory with displacement (CBTD): . 9P5 i^ + ^II? '^^^~ 2b{a + Ajj) 3{a + Ajjf +{L + 2Ajf
^ N
(2)
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B.R.K. BLACKMAN, A J. KINLOCH AND M. PARASCHI
where L is the free length of the specimen i.e. between the load-Hne and the clamp poin^, A\=x\h and // is a correction factor to account for the presence of the bonded on end-blocks and was defined in [6]. From this procedure, the flexural modulus of the beam, Ef, may b^ expressed as: p(3(a + A / / ) ^ + ( l + 2A/)^) ^f-^^ ^^^ ;^ ^ ^ ^-A^ 2hh^5
(3)
Thus, corrected beam theory can be used to deduce the value of Ef from the mode II fracture test, and this allows an important cross-check on the results to be performed, as the value can be compared to the known, or independently measured value, E, as used in eqn (1). Also the value of £/so deduced should be independent of crack length, a. If a compliance calibration of the form C=Co+mfl^ is assumed as suggested in the ESIS TC4 protocol [7], where C is the compliance i.e. {5IP) and Co and m are constants, then G\\c may be determined from the experimental compliance method (ECM) as:
G , c - ' - ^ - F
(4)
where m is determined from the slope of a plot of C versus a^. Such a procedure allows a cross check to be made with the corrected beam theory values of Guc- The results of the ELS tests on the adhesive joints are now presented and discussed. RESULTS & DISCUSSION All crack propagation was cohesive in the adhesive layer and was largely stable. Fig. 2 shows a typical force-displacement trace for a joint bonded with the epoxy-film adhesive. In this test, the initial precrack length, Up, was 74mm and the free length, L, was 125mm. The experimentally measured crack lengths have been indicated on Fig. 2. The crack initiation values determined via (a) deviation from initial linearity (NL), (b) visually detected (VIS) and (c) 5% offset of initial compliance (5%) are all shown. Fig. 3 shows a typical forcedisplacement trace for a joint bonded with the epoxy-paste adhesive. Here the ap=65mm and again L= 125mm. The experimentally measured values of crack length for initiation and propagation are again shown on the diagram. The data in Figs. 2 and 3 were analysed as described above to yield values of Guc as a function of measured crack length, i.e. the R-curves were computed. In addition, the back calculated values of substrate modulus were compared to the known values of the flexural modulus of the arms. The R-curves are shown in Figs. 4 and 5 for the epoxy-film and epoxypaste adhesives respectively. Each curve shows the values of Guc calculated via the three analysis approached outlined above, i.e. the CBT approach (eqn. 1), the CBTD approach (eqn. 2) and the ECM approach (eqn. 3). The general pattern was for Guc to rise strongly following crack initiation and then fmally reach a plateau of almost constant Guc- The initial apparent decrease in Guc following initiation is an artefact caused by the quite extensive apparent crack growth prior to the 5% offset initiation value. As the 5% initiation value of Guc was plotted
On the Mode II Loading ofAdhesive Joints
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10
15 20 25 30 35 Displacement [mm] Fig. 2 Typical force-displacement trace for a CFRP joint bonded with the epoxy-film adhesive. (Crack length, a, values in mm.) 400
20 30 Displacement [mm]
50
Fig. 3 Typical force-displacement trace for a CFRP joint bonded with the epoxy-paste adhesive. (Craclc length, a, values in mm.)
B.R.K. BLACKMAN, A J. KINWCHANDM. PARASCHI
298 4000 3500 3000 _
2500 y 2000
O
1500 1000
o n e (CBT) ^— GIIC (CBTD)
500
»— o n e (ECM)
0
I
70
80
I
90 100 Measured crack length [mm]
I
I
I
I
I
110
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Fig. 4 Giic versus measured a for the CFRP joints bonded with the epoxy-film adhesive. 7000 6000
5000
I
4000
^
3000
2000
-^— GIIC (CBT) -^^— GIIC (CBTD)
1000
-<— GIIC (ECM)
0
60
70
80 90 Measured crack length [mm]
100
110
Fig. 5 GIIC versus measured a for the CFRP joints bonded with the epoxy-paste adhesive.
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On the Mode II Loading ofAdhesive Joints
at the initial crack length, ^p, rather than at the visually assessed crack length at this instant, there is an apparent decrease in Gnc following this point. It is seen that the different analysis schemes return somewhat different R-curves for each adhesive, and the trend was that the highest values were deduced by using the CBT analysis and the lowest by the ECM analysis, with the CBTD analysis giving results that lie inbetween. 170
'
'
'
1 1 ' —1
1
1
1 1
-
160
-
150
_ -
-
• • o
•
•
o
•
•
•
•
_ -
0
«
\^
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•o
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130
-
120
-
110 100
^ 60
0
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o
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•
-
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• o 1
i
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1 1_j
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1
1
1
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1
1
1
epoxy-film adhesive epoxy-paste adhesive 1
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1
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Measured crack length [mm] Fig. 6. Values of back-calculated modulus for the joints as a function of measured crack length. (Note that the known value of theflexuralmodulus, E, of the substrate was 126GPa) The accuracy of the CBT and CBTD equations can be assessed by determining the values of the back-calculated modulus, Ef, of the substrate arms. These values are shown in Fig. 6 for both sets of data. It is clear from these results that the back-calculated values of Ef are about 19% higher than the known values for the substrate when bonding with the epoxy-film adhesive and about 11% higher than the known values when bonding with the epoxy-paste adhesive. This discrepancy is most likely to be the result of the uncertainties in measuring the crack length. Fracture in mode II occurs by the development and coalescence of microcracks [9, 10] and microcracking was observed and photographed experimentally in the present work [11]. Fig. 7 shows the microcracking observed at the side of the specimen for a joint bonded with the epoxy-film adhesive. The top photograph shows this at a magnification of X80 and the lower photograph at XI80. For this joint, the cracking extends across the entire height of the adhesive layer. Clearly, measuring crack length with a high degree of certainty when microcracking is occurring is a difficult task and one which is yet to be addressed in the mode II test protocols. Whilst it would be obviously desirable to have a more accurate method to measure crack lengths experimentally in mode II, such techniques are currently not readily
100
B.H.K. BLACKMAN, A.J. KINLOCH ANDM. PARASCHI
available, and whilst crack gauges can work well in mode I, they would be very unreliabl ; when loaded in shear.
If one assumes that the measured crack length is likely to be only approximate in the presence of such extensive microcracking, the use of beam theory expressions requiring this parameter will also be subject to this uncertainty. However, corrected beam theory can be used to determine an effective crack length, a,, via the measured compliance value and eqn (3) ma:y be rearranged to give an effective crack length thus: (a )
Fig. 7. Photographs of microcracking in the adhesive layer for a joint bonded with the epoxy. film adhesive. Magnification: (a) X80 (b) x180. (The vertical black lines in (a) are drawn Imin apart).
x
-A,,
where dl=,ylh and &=X[[Iz hence a value for the effective crack length, a,, can be determined by using the measured compliance, 6/P, and the known modulus, E , via eqn ( 5 ) . The use of eqn ( 5 ) to determine the crack length has a number of implications. Firstly, if this value of at is used to correct eqns (I) and (2) in place of a,then clearly these equations are no longcr independent as they now both rely on the measured compliance and the known, independent flexural modulus, E . It was found that calculating Grrcin this way returns very similar values to those obtained from an experimental compliance analysis method, i.e. eqn (4). Secondly, if eqn (3) is similarly corrected, then obviously E=EF However, the most important
On the Mode II Loading ofAdhesive Joints
301
implication is that the uncertainties in measuring crack length are now unimportant because the effective crack length, QQ, is calculated. Thus, if we calculate Ue on a point-by-point basis, then an insight into the likely measured crack length uncertainties can be obtained. By following this procedure, the difference between the effective crack length, ag, and the measured crack length, a, can be determined, and expressed as a percentage for each value of the measured crack length. These data are shown in Fig. 8 for four repeat specimens bonded with the epoxy-film adhesive. These values are deduced from {(Ue -a)/ae)xlOO%. \
^1
r
^
\
^ \ ^
r
•
0
O
•
-n—1
^—^—^—
Specimen 1 Specimen 2
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Specimen 3
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Specimen 4
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;
•
n
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110
120
Fig. 8. Discrepancy between the effective and measured crack lengths (expressed as a %) for joints bonded with the epoxy-film adhesive plotted as a function of measured crack length. These data suggest that the measured crack length values, a, are on average between 10% and 20% greater than the effective crack length values, Qe, calculated from the compliance. When these more reliable crack length values are used in the corrected beam theory analysis, then a much better agreement with the ECM analysis method is observed. This is shown in Fig. 9 for a joint bonded with the epoxy-paste adhesive (Fig. 9 is plotted from the same test as used for Fig. 5). However, the results still indicate a strongly rising R-curve effect as the microcracking develops and then a plateau to this curve, presumably after the microcracking reaches a steady-state condition. The values of G\\c for initiation (via the NL definition) and propagation (via the plateau value) for both adhesives have been compared to corresponding values measured in mode I, i.e. from double cantilever beam tests, and these results are shown in Table 1. Table I summarises the results for a large number of repeat tests for each joint type [11]. The results show that, for mode I, values of G\c for crack initiation are very similar to the G\c values for propagation, i.e. very flat R-curves are measured in mode I for both adhesive joints. However, for mode II, crack initiation occurs at a similar value of GQ as for mode I, but in mode II a very strongly rising R-curve was observed, with the plateau value of G\\c being significantly higher than the initiation value. For the joints bonded with the epoxy-film
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B.R.K. BLACKMAN, A J. KINLOCHANDM. PARASCHI
adhesive, the plateau value of Guc was about two and a half times higher than the initiation value and for the joints bonded with the epoxy-paste adhesive the plateau value of Guc was about five times higher than the initiation. 7000
' I ' ' ' • I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I '
- « — GIIC (CBT, but with a values used in place of a) - e — GIIC (ECM)
6000 5000 ^S
4000
^^
3000 2000 1000
60
65
70
75
80
85
90
95
100
Measured crack length [mm] Fig. 9 Giic versus measured a for the CFRP joints bonded with the epoxy-paste adhesive.
(Values of Guc deduced via eqn 4 (ECM) and via eqn 2 (CBTD) with the measured crack length, a, replaced with the effective crack length, a^.)
The elevation of the R-curve may depend entirely on the increased fracture surface area generated when crack propagation is via the initiation and growth of microcracks formed at 45 degrees to the original crack plane. If the microcracks extend across the entire height of the adhesive layer (as was observed in Fig. 7 for the joints bonded with the epoxy-film adhesive) then the enhancement in fracture surface area for a 45 degree microcrack can be approximated hy42{hid) where h is the bondline thickness and d is the spacing between microcracks. For the epoxy-film adhesive, /z=0.08mm and the microcrack spacing can bo measured from the photographs to be c/=0.047mm. Hence, the enhancement in surface area for this case is 2.4 times which is very close to the 2.5 times enhancement in Guc observed. Photographs for the epoxy-paste adhesive are not currently available however, it is obvious that the thicker bondlines used for these joints (/z=0.4mm) would lead to a greater enhancement in the fracture surface area and this is in agreement with the steeper R-curves measured for these joints. Of course, more detailed photographs of the development and propagation of the microcracks in both joints are desirable to more fully understand this phenomenon.
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Table 1. Comparison of G\c and G\\c values at initiation and for propagation in test joints tested. Mode I (DCB) Joint Mode II (ELS) Gic(init)max/5% Giic (init) NL Gic (Prop) GIIC (Plateau) 1449+113 CFRP/AF126 1488+156 1351±481 3428+758 903+74 945+28 CFRP/ESPllO 755±301 4119+577 1 Note: Cic values deduced via corrected beam theory [2,3]5 G\\c values deduced via corrected beam theory using effective crack lengths, Ug.
CONCLUSIONS Adhesive joints were manufactured using CFRP substrates bonded with either an epoxy-film or an epoxy-paste adhesive. These were tested in mode II using the ELS test geometry. Values of Gnc were deduced using two forms of corrected beam theory and an experimental compliance approach. Back-calculated values of substrate modulus were also deduced from the mode II fracture tests using the corrected beam theory. Substantial crack growth was observed prior to the maximum load point in the ELS tests. The R-curves showed strong rising effects following crack initiation until an almost constant plateau value of Gnc was obtained. The agreement between the different analysis schemes was quite poor, with the CRT yielding the highest values and the ECM yielding the lowest values. In addition, the values of the back-calculated substrate modulus were between 10-20% higher than the known values. Photographs of the specimen side revealed extensive microcracking and this made it very difficult to accurately determine the crack length by visual techniques. Crack length values were also determined via compliance measurements and this approach indicated that the measured crack lengths were typically 10-20% greater than the effective values via eqn (5). The use of effective crack length values in the analysis improved the accuracy and led to substantially better agreement between the corrected beam theory and experimental compliance approaches, as would be expected. However, even with the more accurate values of crack length, substantial rising R-curves were recorded for all the joints. The values at crack initiation and from the plateau were compared to the mode I values of Gic- Whilst crack initiation occurs at approximately the same value of Gc in the joints for both mode I and mode II, the propagation values are substantially greater in mode II than mode I. It is suggested that the extra fracture surface area generated by microcracking ahead of the main crack in the mode II ELS tests is the main cause of the strongly rising R-curves shown here. ACKNOWLEDGEMENTS We wish to thank the Engineering and Physical Sciences Research Council (EPSRC) for an Advanced Research Fellowship (AF/992781) and a Platform Grant, and the National Physical Laboratory (NPL) for funding the PhD. studentship of Marion Paraschi. Also, we wish to thank Professor J.G. Williams for valuable discussions on the ideas presented here.
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REFERENCES 1. 2. 3. 4.
5. 6. 7.
8. 9. 10. 11.
ISO, Standard test method for mode I interlaminar fracture toughness, Gjc, of unidirectional fibre-reinforced polymer matrix composites. ISO 15024: 2001. Blackman, B.R.K., H. Hadavinia, AJ. Kinloch, M. Paraschi and J.G. Williams, The calculation of adhesive fracture energies in mode I: revisiting the tapered double cantilever beam (TDCB) test. Engineering Fracture Mechanics 2003. 70:p. 233-248. BSI, Determination of the mode I adhesive fracture energy, Gjc, of structural adhesives using the double cantilever beam (DCB) and tapered double cantilever beam (TDCB) specimens. 2001. BS 7991. Davies, P., G.D. Sims, B.R.K. Blackman, A.J. Brunner, K. Kageyama, M. Hojo, K. Tanaka, G.B. Murri, C.Q. Rousseau, B. Gieseke, and R. Martin, Comparison of test configurations for determination of mode II interlaminar fracture toughness results from international collaborative test programme. Plastics, Rubber and Composites, 1999. 28(9): p. 432-437. Blackman, B.R.K. and J.G. Williams. On the mode II testing of carbon-fibre polymer composites, in ECF12 Fracture from Defects. 1998. Sheffield, UK: EMAS publishing. Hashemi, S., A.J. Kinloch, and J.G. Williams, The analysis of interlaminar fracture in uniaxial fibre-polymer composites. Proceeding of the Royal Society London, 1990. A427: p. 173-199. Davies, P., B.R.K. Blackman, and A.J. Brunner, Mode II delamination, in Fracture mechanics testing methods for polymers adhesives and composites, D.R. Moore, A. Pavan, and J.G. Williams, Editors. 2001, Elsevier: Amsterdam, London, New York. p. 307-334. Wang, Y. and J.G. Williams, Corrections for mode IIfracture toughness specimens of composite materials. Composites Science and Technology, 1992. 43: p. 251-256. O'Brien, T.K., Composite interlaminar shear fracture toughness, Gnc-' Shear measurement or sheer myth? in Composite Materials: Fatigue and Fracture 1. 1998 ASTM STP 1330 p.3-18. Lee, S.M., Mode II delamination failure mechanisms of polymer matrix composites. Journal of Materials Science, 1997. 32: p. 1287-1295. Paraschi, M., A fracture mechanics approach to the failure of adhesive joints, PhD thesis. Department of Mechanical Engineering. 2002, Imperial College, University of London: London.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
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COHESIVE FAILURE CHARACTERISATION OF WOOD ADHESIVE JOINTS LOADED IN SHEAR F. SIMON ^'^ G. VALENTIN ^ ^ Laboratoire de Mecanique des Materiaux C.T.B.A. Pole CONSTRUCTION, BP 227, 33028 Bordeaux Cedex, France' ^ Laboratoire de Rheologie du Bois de Bordeaux, Unite Mixte 5103 CNRS / INRA / Universite Bordeaux I, Domaine de THermitage, 33610 Cestas Gazinet, France. ABSTRACT The damage and fracture of wood adhesive joints subjected to shear loading was experimentally investigated, with efforts focusing on cohesive cracking of the joint. Experiments were performed on one specimen type especially designed to give stable crack propagation in the joint; i.e. the modified Tapered End Notched specimen in Flexure. Loaddeflection curves allowed an evaluation of the energy involved in fracture and the critical strain energy release rate (Gc) was measured for elastic crack propagation. The fracture energy (Gf) included damage and non-linear phenomena during the cohesive crack initiation, both of which appeared to be very important parameters in crack stability. Crack propagation during testing was also observed with a high magnification video camera, and compared to fractographic observations to provide a better understanding of the fracture mechanisms. It was determined that fracture surface morphology may be correlated to the variation of the calculated energy G. Moreover, numerical calculations have been performed for a bulk adhesive specimen and wooden adhesive joints with a finite element code (CASTEM 2000) assuming that the bond line displays a softening behaviour (Smeared Crack Approach). The predictions of the non linearity of the load displacement relation and the crack path of the model are in fair agreement with the experimental results for the bulk adhesive specimen. In the case of wooden adhesive joints calculations give qualitative results on the damage occurring in the joint for a cohesive failure. KEYWORDS Fracture Mechanics, Wood adhesive joints, shear loading, cohesive failure. INTRODUCTION In bonded joints, a damage area, the size of which is often linked to the joint thickness and the adhesive properties, may appear ahead of the crack tip before fracture [1, 2, 3]. In the case of wooden adhesive joints, the overall behaviour of the structure may be affected by the presence
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and the size of the fracture process zone [4]. Moreover, for brittle or glassy polymers, the failure of the joint is related to the stress-induced growth and breakdown of crazes. Failure models show that cross-tie fibrils can have a profound effect on the failure mechanisms of a craze as they can transfer stress between the broken and unbroken fibrils [5, 6, 7]. In the case of timber finger-joints, bondline brittleness and the defects present may have a great influence of the strength of the joint [8]. Due to the vast differences in fracture energies encountered during crack extension, it is important to distinguish three debonding modes in the failure of wooden bonded joints. The extension of a debonding crack into a wooden adhesively-bonded assembly may he considered "adhesive" (i.e. along one of the two adhesive-substrate interfaces), "cohesive" (i.e. inside the thickness of the adhesive layer), or in a poor cohesion area very close to one of the two adhesive-substrate interfaces, leading to the "interphase" notion [9]. Although the third case of failure occurs very close to one of the two interfaces, the adhesive/adherend interface is in fact coated everywhere with very thin pieces of adhesive resin. Thus, it may be addressed as another type of cohesive failure. In the case of wooden joints, a poor cohesion zone or interphase of this type may be linked to the migration of low molecular weight components (for example wood resin) in the interfacial area. The present study is a part of a research project dealing with the mechanical behaviour of wooden adhesive joints generally used in laminated wood products such as glued-laminated timber (glulam). The experiments described in this paper were carried out in order to gain a clearer understanding of the damage and fracture mechanisms of a wood adhesive joint loaded in shear. By promoting cohesive failure in the bulk of the joint it was possible to understand the mechanical behaviour of the polymer as a thin layer, before and during its failure. Specifically, the cohesive failure analysed was that which occurred in the bulk and in the poor cohesion zone of the bond line. An experimental process based on non-linear fracture mechanics assumptions was developed to produce such a failure mode for wooden adhesi\'e joints. Then, a numerical model was developed to describe the experimental results and the crack propagation path. TESTING METHODS Material and specimen Previous observations made using photoelasticity [5, 10] on a specimen especially designed to give stable crack propagation in shear have allowed the indirect identification of the damaged area in the joint during cohesive failure. In this study, fracture tests were performed using a high magnification video camera and compared to previous fractographic observations [10] lo understand the fracture mechanisms occurring in the joint during loading. The specimen tyj)e especially designed to give stable crack propagation in solid wood or in the joint [12] is the modified Tapered End Notched specimen in Flexure (mTENF) (Fig. 1), where the loss of stiffness when the crack grows is compensated by increasing the total height. A damage zone exists ahead of the crack tip before the crack propagation in quasi-brittle materials such as wood [13], and the adhesive chosen for the present study. In this damage zone, non-linear phenomena (micro-cracks, porosity) are responsible for energy dissipation and stress redistribution before and during the crack propagation [13]. The mechanical behaviour of the materials and the geometrical particularities of the Tapered End Notched specimen provide it with good crack stability for different modes of loading (tension or shear). Tests were made with a polyurethane resin used in the wood industry (Young's Modulus 1()0 MPa) displaying a quasi-brittle behaviour, and substrates made from Norway Spruce (14^'/o
Cohesive Failure Characterisation of Wood Adhesive Joints
307
moisture content). Clear wood is used for manufacturing adherends which contains no defects such as knots, grain deviation and resin dimples to limit the natural scattering of mechanical properties. The surfaces of the adherends to be bonded together were first slightly ground, degreased with Toluene-Ethanol 1-2 and dried. Subsequently, the bond thickness of the mTENF adhesive joints was controlled by inserting PTFE strip of the desired shape and thickness between the two adherends [14]. Observations of adhesive penetration into wood cells showed that pressure applied during curing is not only carried by the PTFE strip and allows adhesive penetration in the wood. The bonded joint contained an initial pre-crack of length a, introduced at the centre of the bond line with the PTFE strip using a razor blade. The crack tip was chevron shaped with a 90° angle to give better crack initiation in shear to avoid brittle failure [12]. The wood grain was oriented converging to give a small angle to the joint (Fig. 1.) to promote crack propagation in the bulk of the joint [15]. Calculations made using Finite Element Methods [16] on this specimen predicted good crack stability in shear for such an adhesive joint and all the bond thicknesses from 0.25 to 1 mm.
Wood substrate
^
Adhesive joint
^
Annual growth ring
Fig. 1: mTENF testing device in three point bending. Tests were made in three point bending at a constant cross head speed (2 mm/min for loading and 4 mm/min for unloading). During testing, midpoint deflection was measured with two nocontact laser extensometers (Fig. 1.) and joint damage was observed and recorded by a high magnification video camera device [16]. As a result of the load applied in bending before failure, residual displacements occurred, which were limited using both laser extensometers (Fig. 1.). Only joint thickness results equal to 0.25mm are presented in this paper.
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Evaluation of fracture energy Non-linear fracture mechanics assumptions and fracture energy evaluations are discussed i i detail in Refs. [10, 11]. In these studies, two types of fracture energy were evaluated. Firstl)', the critical energy release rate Gc was calculated from the energy dissipated during an elastic crack propagation of length &r. Secondly, fracture energy Gf was also evaluated, taking into account all the phenomena occurring at the crack tip during crack initiation as well as the crack propagation. The modified bonded Tapered End Notched specimen in Flexure was developed for the measurement of fracture energy from the load-deflection curve as: Gf=\lA^PdS (1). where A is the new crack area, P the load and 5 the load displacement. Then, by measuring A on each fractured specimen, fracture energy may be evaluated (Fig. 2.) with: G, = UciA and G^ = Uf j A (2). aa
_i
1
1
1
1
a«+c
1
a^i+c+da,
1
1£
w ^ — I — I — I — I — I — I — I
^
r ^ — I
1 — I — I — I — I — I — h S
Fig. 2: Fracture energy determination. COHESIVE FRACTURE TESTS RESULTS Mechanical results As presented in the introduction, two types of cohesive failure in shear were considered in the experiments. In the first case, the cohesive failure occurred in the bulk of the adhesive layer, while in the second case, failure occurred very close to the upper interface. The first failure case produced a highly rough cracked surface (Fig. 3(a).) associated with a large damage level throughout the thickness of the bond line. In the second case, however, fracture surfaces appeared mostly quite smooth and typical of brittle failure (Fig. 3(b).). In this case also, the crack grew without any deviation, very close to the upper interface. Experimental load deflection curves (Fig. 3.) illustrate the large difference in crack propagation observed in each case. A difference in stiffness between both bonded specimens is observed and results from either a difference in the bond line quality or from interfacial conditions. For both specimens, adherends were made from the same sample of wood. Both wood substrates contained no apparent defects and had the same longitudinal Young's modulus (14500 MPa). Both also had the same growth characteristics (oven dry specific density, annual growth rings), and as a consequence very close values of transverse and shear modulus adjacent to the bond line. Thus, any difference in stiffness is likely to be due to
Cohesive Failure Characterisation of Wood Adhesive Joints
309
interface conditions. Interfaces certainly contain contaminant elements, included in the joint during specimen fabrication; they create a poor cohesion zone in the joint responsible for the second crack path (Fig. 3(b).). Further inspection of the load deflection curves (Fig. 3.) reveals that in the first case, crack grovvth occurs very slowly at a high load level, after a large non-linear part. In the second case, crack propagation occurs at a lower load level and displays significantly different behaviour, with a slight increase in slope gradient following crack initiation. The third curve is presented as a reference for a solid wood specimen, to demonstrate that for both cases of cohesive crack propagation, load and deflection recorded at the crack initiation reach a higher level than that found for solid wood.
2
3
4
5
Load Deflection (mm)
Fig. 3: Adhesive cracked surfaces observed in shear and experimental load deflection curves: (a) high roughness surface, (b) low roughness surface, (c) solid wood. Differences in macroscopic behaviour of wood bonded assemblies immediately prior to failure (in the non-linear sections of the experimental curves) have been studied with loadingunloading cycles before crack propagation, to identify if such non-linear behaviour arises from damage or plastic deformation in the joint, or a combination of these two phenomena. Results show that a residual displacement remains in the joint following testing, possibly as a result of frictional forces acting along the bond line, or plastic deformations in the joint or in the adherends. Frictional forces acting along the bond line result from bending load, but are limited by inserting in the initial notch two very thin PTFE strips (whose thickness is equal to the joint thickness). During unloading, slope decreases shown in the experimental curves indicate that damage phenomena also exist. The second cohesive crack path was often observed in our testing, contrary to the initial test. It seems that Norway Spruce bonding involves such an interphase creation and, as a consequence, the existence of such a poor cohesion area close to the interface must be taken in account in the prediction of the specimen failure. The difference between the two crack paths clearly appears when analysing the results of the fracture tests (Table 1). Table 1. Summary of experimental cohesive failure results in shear. Case a
Caseb
Solid Wood
Giic(J/m^)
2100 + 450
1120 ±300
675 ±150
Giif(J/m')
2500 ± 550
1550 ±300
800 ± 200
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F. SIMON AND G. VALENTIN
Firstly, for different samples of wood, it is observed that the dispersion of longitudin.il Young's Modulus of the substrate reaches 20%. This is the common scattering record for mechanical properties of natural materials such as wood. For crack growth in the poor cohesion area of the joint, the average fracture energy is evaluated as Gnf at 1550±300 J/m^ The critical energy release rate Gnc equals 1120±300 J/m^. The main difficulty found in obtaining critical values was taking into account the constant increase of load during propagation when identifying the real crack starting point. The results of the second crack propagation path show that Gnf is averaged at 2700±550 J/ml This value depends heavily on the degree of damage to the cracked surface, but is significantly higher than for the other propagation path. As a reference, fracture energ>' Guf for a specimen made with solid wood is equal to 8001200 J/m^. High magnification video camera observations Fracture tests were conducted in conjunction with monitoring using a high magnification video camera device, which observed damage of the joint surface during testing. At first, observations were made during loading-unloading cycles to identify phenomena occurring in the joint during the non-linear part of the load-deflection curve[16]. Differences in the joint aspect appeared with the non-linear behaviour and a damage zone, growing with shear loading increase, containing micro-cracks and void dimples was clearly observed. Although the observations taken here are useful in understanding failure process occurring in the joint, it is important to realise that all parameters were actually observed on only one side of the joint. For a same load level experimental loading-unloading cycle curves showed an increase in the compliance without any crack propagation, a behaviour synonymous with damage phenomena Both specimens whose cracked surfaces and experimental load deflection curves appear Fig. 3, show, in Fig. 4 different types of damage.
Crack propagation
F=2589N Fig. 4: Damage visualisation of the bond line during loading. In Fig. 4(a). corresponding to the very rough surface, the bond line appears damaged across its entire thickness. A micro crack oriented to 45° appears ahead of the crack tip, and the crack path is very forked. Such a micro-crack orientation may indicate that the fracture process of the bond line is mainly controlled by the maximum opening constraint a ee. Distortion of the vertical references indicates that the joint is not yet fully cracked. Fig. 4(b). shows that the crack growth occurs very close to the upper interface, according a very straight path without any deep damage, unlike the case shown in Fig. 4(a). The bond line also keeps its initial colour. In this case, vertical lines are totally broken with the joint. Localisation of
Cohesive Failure Characterisation of Wood Adhesive Joints
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the real crack tip for cohesive failure is very difficult with such a testing device because cohesion may persist (fibrils bridging) between the crack lips when the joint has failed. Moreover, only one side of the joint is observed and the crack generally grows in the bond line thickness in a thumb nail shape. Fig. 5. shows an alternation between both failure paths which certainly added to the natural dispersion of mechanical fracture results. Crack propagation
Fig. 5: Mixed cohesive crack propagation. The evaluated fracture energy which is directly influenced by the failure path, may be scattered by the proportion of cohesive failure in the bulk of the bond line or in the interphase. The fracture energy in this case reaches 2525 J/m^. Other damage characteristics observed during cohesive failure appear Fig. 6. These observations, made at an higher magnification than previous observations, show micro-cracks growing, coalescence of micro-voids, fibrils bridging in the adhesive and also between the adhesive layer and the wood substrate. Fibrils bridging between wood adherends and the adhesive layer indicates that substrates are also damaged. As a consequence, non-linear behaviour of the wood adhesive assembly is also as a result of damage in the wood. Local whitening of the bond line indicates plastic yielding of the PU resin. Fig. 6. shows the great tendency of the crack to deviate along one of the adhesive-adherend interphases. As a consequence, it involves wavy crack propagation between the two interfaces already observed in the literature for steel or aluminium substrates [17, 18] Crack propagation
5 mm
Fig. 6: Damage parameters observed during shear loading. Recording crack propagation during bending loading has provided the observation that damage of the bond line is mainly made in shear. Fig. 6. shows the adhesive joint just after crack propagation and allows the measurement of a local opening displacement between both crack lips (measured equal to 0.15 mm). Subsequently, crack growth is made under mixed mode, perhaps induced by the local roughness of the failed surface. All the fractographic observations of the failed surfaces made [10] confirm the existence of damage parameters observed. Thus, damage parameters observed on the side of the joint during loading also appear in the plane of fracture. Different authors [19, 20] have shown for epoxy resins that fracture toughness of the joint was affected by energy dissipation mechanisms around the crack tip in the adhesive material, which are related to the morphology of the fracture surfaces. When the crack is located in the mid-plane of the adhesive layer (cohesive failure).
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the fracture process is largely controlled by the plastic deformation and damage zone around the crack tip. For the joint thickness studied here (0.25 mm), wood adherends restrict the formation of the plastic zone and lower the fracture energy reached for thicker joints [19]. Moreover, in case of the interphase crack propagation, the closeness of wood adherends restricts the adhesive plastic deformation and increases the constraint to reach a brittle failure with no real damage of the joint. Consequently, the fracture energy is still lower [20], NUMERICAL SIMULATIONS Because of the high scattering of experimental resuhs and the great difficulty in reaching the fully cohesive failure of wooden adhesive joints, a nimierical analysis has been made to give a better knowledge of their mechanical behaviour for various parameters (adhesive used, joint thickness, loading mode, e t c . ) . For the PU resin tested previously in shear, such an analysis has been made on two steps; first simulations have been made on bulk adhesive specimen to determine the mechanical behaviour of the resin and the numerical results obtained have been implanted in the FE code CASTEM 2000 [21] for the mTENF bonded specimen loaded by shear. Bulk adhesive specimen. Fracture tests on notched bulk adhesive specimen have been made in three point bending to observe the real mechanical behaviour of the resin [16]. The Young's modulus is estimated to be equal to 96±10 MPa and fracture energy measured reaches 3900 ± 600 j W and the PU resin displayed softening behaviour.
-^^^fc
Fig. 7: Experimental testing device of bulk adhesive specimen. As a consequence, the numerical model chosen for computations was the fictitious crack model extended to include the effect of crack shearing, for better simulations of wooden joints in shear loading. Thus, it is possible to establish a smeared version of the fictitious crack model for quasi brittle materials such as concrete and rocks, and by extension, to wood oi bonded wood. All assumptions and numerical parameters are presented and discussed in details in Ref [22]. Specimen meshes include a radial mesh to evaluate the Rice integral J per integration over a crown of elements surrounding the crack tip [23]. Numerical results obtained (Fig. 8) for the well-suited parameters of the model ( Young's modulus 100 MPa, Poisson's ratio 0.4, limit in tension 10 MPa, fracture energy as defined in the model, 950 J/m^, slip modulus 0.2 MPa) [16] show a softening behaviour of the PU resin. For different initial crack lengths, numerical simulations allow to give pseudo crack propagation load deflection curve (automatic remeshing of the crack tip was impossible because of fracture energ> calculation).
Cohesive Failure Characterisation of Wood Adhesive Joints
O.OOE+OO
2.00E-03
4.00E-03
6.00E-03
8.00E-03
1.00E-02
1.20E-02
313
1.40E-02
Deflection (mm)
Fig. 8: Numerical crack extension of bulk adhesive specimen. Moreover, the decrease of the stress field ahead of the crack tip is explained by the development of a damage zone with load increase (Fig. 9). Loading is made prior to reaching the experimental fracture energy evaluated by the Rice Integral (as a crack stop criterion). Nevertheless, the mesh distortion at the crack tip does not allow to increase more loading because Ottosen model is not usually applied on notched beams.
L axx (Pa) ayy (Pa) Fig. 9: Numerical stress field calculated at the crack tip with Ottosen model. Application to wood adhesive joints The mTENF bonded specimen mesh requires a large number of elements to describe precisely the behaviour of the adhesive layer. It is, indeed, necessary to describe precisely the adhesive layer (for which the thickness is weak with respect to the size of the specimen (from 0.1 to 1 mm of adhesive for 50 mm of total height of the specimen)), and to model finely the zone of the crack tip, since the later calculations concern the stress and strain fields of this area.
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Fig. 10: mTENF bonded specimen loaded in three point bending. Various calculations were carried out in plane stress taking into account that wood is ;in orthotropic material, the adhesive layer is isotropic with the same numerical parameters than in the first part of the numerical computations, except for the fracture energy (as defined in the model) in shear which is equal to 1500 J/m^. For this case, we introduced unilateral conditions as well on the lips of the crack as on the notch in order to simulate the conditions given experimentally by the PTFE strip ( to avoid any inter-penetration of the elements). Shear loading is applied as in the experimental conditions at constant cross head displacement rate in the middle of the specimen resting to two supports located at 30 mm of the extremities. The crack is initiated in the middle of the adhesive layer. For load and deflection applied equivalent to the experimental ultimate conditions before failure, the J integral evaluated on a crown of elements surrounding the crack tip is lower than the experimental fracture energy evaluated. This significant difference comes from the fact that wood substrates are damaged, in the area very close to the interfaces. As a consequence, the non linear behaviour obtained numerically is not pronounced enough to describe the experiments. Applying non linear assumptions only on the adhesive layer does not describe the real behaviour of the wood bonded specimen. We must take into account damage in wood but the Ottosen model cannot be used in orthotropic materials. Nevertheless, the simulations confirm the experimental observations made with the high magnification video camera, as we can see on Fig. 11.
Fig. 11: Non linear deformations in the wood adhesive joint loaded in shear at the critical experimental conditions. We can clearly observe on Fig. 11 the crack deviation from the middle of the adhesive layer to the upper interface. Moreover, the distance between crack lips increases with loading to reach a mixed mode crack propagation, as observed in the experimental part. As a consequence, if it remains a poor cohesion area close to the interface, the crack will naturally fork to this area for a brittle failure.
Cohesive Failure Characterisation of Wood Adhesive Joints
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CONCLUSION Applying non-linear fracture mechanics to wood bonded joints appears to be a reliable method for dimensioning adhesive joints in timber engineering. Nevertheless, natural variability of mechanical properties of wood associated with one of the bond lines induces large scatter in experimental results. Cohesive failure of wooden adhesive joints may be a method to study mechanical properties of the adhesive as a thin layer, but remains very difficult to obtain experimentally on wooden joints. Observations made on wooden joints (high magnification video camera, SEM) show the importance of damage and non-linear phenomena during failure. For the PU resin used in this study, fracture energy is mainly dependant on cohesive crack path. The interphase creates a poor cohesion zone in the joint where fracture surfaces were characteristically brittle associated to the close proximity of wood adherends which limits development of the damage zone. As a consequence, crack propagation in the interphase lowers fracture energy. Moreover, for mid-plane propagation in the joint, plastic and damage phenomena control the crack propagation occurring at higher level of fracture energy. It has been shown that crack propagation was locally made under mode I loading and subsequently, fracture energy for real cohesive failure reaches values from bulk adhesive specimens loaded in Mode I. The close proximity of wood substrates induced by the small joint thickness (0.25 mm) limits the fracture energy obtained. The same bonding conditions can lead to different failure modes (cohesive, adhesive or deviated into wood) for wooden adhesive joints, that must be carefully accounted for when designing a structure. Fracture energy may be of very different magnitude between all crack paths. Moreover, in case of Norway Spruce substrate bonded by the PU resin and loaded by shear, crack propagation occurring in the interphase is the most frequent path observed during testing, certainly because of the interfacial conditions coming from such a resinous substrate. DSC analysis of the bond line shows a total adhesive polymerisation for all specimens. Numerical simulations give qualitative answers to a few questions but need to apply non linear and damage model to wood adherends. Due to cellular constitution of wood, knowledge of mechanical behaviour of wood cells filled by the adhesive penetration will be the main difficulty in such a computation.
Acknowledgements We would like to thank the Regional Council of Aquitaine and the National Building Federation for their financial support for this project. REFERENCES 1. 2. 3. 4. 5. 6. 7.
Chai, H. (1992). Int. J. of Fracture 58, 223. Chai, H. (1995). Acta metallurgica materialia 43, 163. Sheppard, A., Kelly, D., Tong, L., (1998), Int. J. of Adhesion andAdhesives 18, 385. Wemersson, H., (1994), Fracture characterization of wood adhesive joints. Report TVSM-1006, Lund University, Division of Structural Mechanics, Lund, Sweden, Simon, F., Morel, S., Valentin, G. (1997). In: Proceedings of the Euromech Colloquium 358, Mechanical behaviour of adhesive joints, analysis, testing and design, PluraUs, Paris, pp. 341-351. Brown, H.R., (1991), Macromolecules 24, 2752. Xiao, F., Curtin, W.A., (1995), Macromolecules, 28, 1654.
316
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
F. SIMON AND G. VALENTIN
Serrano, E., Gustafsson, PJ., (1999), Int. J. of Adhesion andAdhesives 19 (1), 9-17. Bikerman, JJ., (1968), The Science of Adhesive Joints, Acad. Press, New York. Simon, F., Valentin, G. (2000). In Fracture of Polymers, Composites and Adhesives, ESIS PubUcation 27, pp 285-296. Simon, F., Valentin, G. (2000), French Journal of Timber Engineering 5, 45. Aicher, S., Bostrom, L., Gierl, M., Kretschmann, D., Valentin, G. (1997), Determination of fracture energy of wood in Mode II, RJLEM TCI33 Report, SP Report 1997:13. Morel, S., Schmittbuhl, J., Bouchaud, E., Valentin, G., (2000), Physical Revievv Letters 85, N° 8, 1678. Sener, J.Y., Delannay, F., (2001), Int. J. of Adhesion andAdhesives 21 339. Koutski, J.S., Mijovic, J.A., (1979), Wood Science 11, 3, 164. Simon, F. (2001). Endommagement et Rupture des joints colles sollicites en traction ou cisaillement. Application au collage du bois. PhD Thesis, University Bordeaux I. Akisanya, A.R., Fleck, N.A. (1992), Int. J. of Fracture 55, 29. Chai, H, (1987), Int. J. of Fracture 32, 211. Chai, H, (1986), 7* ASTM Symp. On Composites Materials, Testing and Design, ASTM STP 893, J.M. Whitney Eds. Daghyani, H.R., Ye, L., Mai, Y.W., (1995), J. of Adhesion, 53, 149. Millard A., (1993), Castem 2000: Guide d'utilisation. Rapport DMT/93-006, CEA, Direction des Reacteurs Nucleaires, Departement de Mecanique et Technologic Ottosen N.S., Dahlblom O., (1990) J. of Engineering Mechanics 116, 1, pp 55-76. Brochard J., Suo X.Z., (1994),. Formulation de la methode G-6 et description de la programmation dans Castem 2000. Rapport DMT/94-640, Direction des Reacteurs Nucleaires, Departement de Mecanique et Technologic.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
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RATE DEPENDENT FRACTURE BEHAVIOUR OF ADHESIVELY BONDED JOINTS
I. GEORGIOU, A. IVANKOVIC, A. J. KINLOCH AND V. TROPSA Imperial College London, Department of Mechanical Engineering, South Kensington campus, London, SW7 2AZX, UK ABSTRACT Standard small-scale peel tests, such as the impact wedge-peel (IWP-ISO 11343) and Tpeel tests are often employed to analyse the fracture behaviour of structural adhesives. The current work aims to examine the behaviour of adhesively bonded joints under various loading rates by conducting a series of peel tests and simulating the results from such tests numerically. In all tests, thin sheets of aluminium alloy substrates were bonded together using 'XD4600' and 'XD1493' structural epoxy-adhesives. Numerical modelling of the tests was conducted using the Finite Volume (FV) method. For this purpose, transient, 3D, procedures were developed including a newly developed contact model and a cohesive zone (CZ) model as a local failure criterion. The CZ model was defined using two materials parameters, the adhesive fracture energy, Gc, and the maximum cohesive stress, dm. In order to measure the adhesive fracture energy tapered double cantilever beam (TDCB) tests were performed, whereas the value of am was estimated from the stress-strain curves at corresponding rates and taken to be the ultimate tensile strength (UTS). Numerical analysis of the tests was conducted in order to calibrate the traction separation curves at various rates, and to examine the stick-slip crack behaviour which appeared in TDCB specimens with 'XD4600' adhesive tested at high rates. The calibrated CZ curves were then used in the prediction of failures in the IWP specimens. Work on modelling T-peel tests is currently in progress. KEYWORDS Adhesive, cohesive zone model, finite volume method, impact wedge peel test, TDCB. INTRODUCTION Due to the environmental impact of motor vehicles, automotive manufacturing companies are being encouraged to produce lighter and more fuel-efficient vehicles. Vehicle structure offers an appropriate scope for potential weight saving, with the body-in-white (BIW)
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providing the largest contribution through the use of new technologies or new lightweight materials. Note that BIW is a conventional name adopted by automotive engineers to represent the body/chassis structure along with any exterior skin panels such as bonnet, boot lids, etc. Recent developments have revealed that up to 30% reduction of the total weight of the car can be achieved by substituting steel with aluminium alloy [1]. However, the substitution of one material by another may give rise to various problems, and one of them is the method of joining. The current research will investigate the application of adhesive bonding as a possible solution to the above challenge. An important feature of any vehicle structure is its behaviour under impact loading. Adhesives for automotive applications are based on structural toughened-epoxy polymers that exhibit viscoplastic deformations, and therefore a strain-rate sensitivity. Under impact loading, adhesive joints may fail in a brittle manner. As a consequence, the surrounding body panel material would be unable to undergo substantial plastic deformation, which would result in a low dissipated energy at impact. Hence, the reliability of the adhesively bonded joints, when they are used in safety-critical areas, is of primary importance to the automotive industry. The present work aims to investigate the mechanical and fracture behaviour of adhesively bonded joints under various loading rates. EXPERIMENTAL PROCEDURES Materials tested Substrates. Two commercial aluminium alloys (see Table 1), received as hot-rolled sheets (from ALCAN), were the main body-panel materials used in the current work. The thickness of the sheets varied from 1 to 3 mm for the 5754-0 alloy and from 1 to 2 mm for the 6111-T6 alloy. For the tapered double-cantilever beam (TDCB) tests, where the substrates should remain within the elastic region, a high yield strength alloy, 2014, was used throughout. The specimens were prepared and pre-treated prior to bonding using the procedures proposed by Blackman et. al. [2]. The chemical compositions of the alloys employed are given in Table 1. Table 1. Chemical composition (wt%) of substrates used in the present work Mn Fe Cu Alloy Si Mg Cr Zn 0.4 0.4 0.5 0.1 0.2 5754-0 2.6-3.6 0.3 0.5-0.9 0.15-0.45 0.5-1.0 0.1 6111-T4 0.7-1.1 0.4 0.15 3.9-5.0 0.40-1.20 0.2-0.8 0.1 2014 0.5-0.9 0.5 0.25 Notes: (1) Balance is aluminium. (2) 2014 also contains 0.1% Nickel
Ti 0.15 0.10 0.15
Adhesives. Two single part, rubber toughened hot cured structural epoxy adhesives were used throughout the current research. Both adhesives were provided by Dow Automotive, Europe. The glass transition temperature, Tg, as well as the curing cycle employed for each adhesive are given in Table 2. Mechanical Testing Uniaxial tensile tests were conducted to obtain the basic mechanical properties of the substrate materials and adhesives for a wide range of test rates. For testing of the substrate materials dog-bone specimens of 30 mm gauge length were machined from the sheets according to the ASTM (E8m-89b) standard [3]. Prior to testing, aluminium specimens were heat treated for 30 minutes at 180°C, simulating the adhesive curing process. A
319
Rate Dependent Fracture Behaviour ofAdhesively Bonded Joints standard dumb-bell shaped specimen of 30 mm gauge length was employed for the testing of the adhesives according to ASTM D 638-72 [4]. Specimens were machined following the EN ISO 2818:1996 standard [5] from a bulk 4 mm thick plate, and prepared according to the BS ISO 15166-1:1998 standard [6]. Table 2. Adhesives used in the current research Adhesive
Form
Cure Temperature [°C]
Cure time [min]
T, [°C]
'XD4600' Single-part 150-190* 15 113 'XD1493' Single-part 180 30 91 (*) 'XD4600' adhesive requires a heat curing at two different temperatures indicated above for the time of 15 min. at each. Tapered Double Cantilever Beam (TDCB) Tests The TDCB test configuration was employed to determine the adhesive fracture energy, Gc. Tests were conducted at different crosshead speeds between 10'^ and 1 m/s. A schematic of the test specimen is illustrated in Fig. 1(a). Due to symmetry only half of the specimen is considered in the numerical analysis, Fig. 1(b). The profile of the arms is machined such that the rate of compliance increases linearly with the crack length and hence the derivative of the compliance with crack length remains constant. The beams are contoured to the profile described by Eqn. 1 [2], where h is the height of the beam, a is the crack length and m is a constant {m = 2000 m"' in the present work).
H
The thickness of the TDCB specimens (5 = 10 mm) is sufficient to ensure plain strain conditions. It should be noted that during the test the arms remain within their elastic limit. Therefore, from simple beam theory [7], and by the use of linear elastic fracture mechanics, the strain energy release rate of the adhesive can be obtained using Eqn. 2, where P is the load at failure and Es is the substrate modulus. The calculated adhesive fracture energy was employed in the simulation of the TDCB and impact wedge-peel (IWP) tests. G . = ^
(2)
Impact Wedge Peel (IWP) Tests To study the fracture behaviour of the selected structural adhesive joints, impact wedgepeel tests (ISO 11343 [9]) were also conducted using a high-performance servo-hydraulic Instron machine. The test speed was varied between 0.4 and 11 m/s to investigate the effect of the rate on the cleavage force. A schematic representation of the impact wedge peel (IWP) test specimen is shown in Fig. 1(c), with its numerical representation given in Fig. 1(d). A wedge is pulled through the adhesive joint, which is shaped like a tuning fork. The substrates, cut out from aluminium sheets (1,2 and 3 mm thick), were prepared prior to bonding and bonded together over a length of 30 mm. The excess of adhesive that was present at the ' V formed by substrates was carefully removed before curing to keep a minimum bead of adhesive in this region. High-speed photography was also employed with a number of tests to show clearly the characteristic events during impact, i.e. wedge/specimen contact, crack initiation and propagation, bouncing of the arms, etc. A Kodak HS 4540 camera [10] fitted with a 135 mm Nikon lens was focused on the stationary wedge at a slight angle, in order to have a clear view of the crack propagation.
320
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Substrates 20 mm
Fig. 1: Schematic of the: (a) TDCB test specimen; (b) FV representation of the TDCB geometry; (c) IWP test specimen; (d) FV representation of the IWP geometry. A bond line thickness of 0.4 mm is used for all test specimens. T-peel Tests A series of T-peel tests was conducted at 1, 5 and 50 mm/min in order to investigate the effect of the test rate on the average peeling force. A schematic of the test geometry is illustrated in Fig. 2. Symmetrical 90° peel test specimens were fabricated using rectangular substrate (20±0.25 mm x 300±1 mm) cut from aluminium sheets of different thickness and bonded by structural adhesives. From the measured peel forces, and by employing an analytical elastic-plastic approach, the value of adhesive fracture energy Gc was measured [8].
Adhesive layer thickness - 0.25 mm
Fig. 2: Schematic of the T-peel test geometry. FINITE VOLUME METHODOLOGY A fmite volume (FV) approach was used for the numerical simulation of the experiments [11]. The model was implemented using the 'FOAM' package, i.e. a C-H- library for
Rate Dependent Fracture Behaviour ofAdhesively Bonded Joints
321
continuum mechanics [12]. Schematics of the FV representation of the TDCB and IWP test specimens can be seen in Fig. 1(b) and (d) respectively. In the numerical analysis, the equation governing linear momentum is solved while Hooke's law was used to describe the behaviour of both bulk adhesive and substrate materials. This approach is only suitable for modelling linear elastic TDCB tests, and the finite strain plasticity is required for modelling of the T-peel and the IWP tests. Indeed, our current work is concerned with the peel tests. The CZ model was employed to describe the traction-separation behaviour of the adhesive material along the prospective crack path. The CZ model represents a local failure mechanism of the material. In the present work, only mode I failure was considered due to symmetry conditions, and the normal separation distance was related to the normal tractions via a Dugdale-like curve (Fig. 3). Particular care was taken to achieve mesh independent results. (a)
Numerical crack tip position
(b)
FV mesii
m. 4
S
Crack path Fig. 3: (a) Schematic of the FV CZ model; (b) Dugdale traction-separation curve used in the analysis. Early in the process the cohesive surfaces behave as the surrounding bulk material. After the stress level reaches its maximum value, (Jm, the material is irreversibly damaged. Since no direct measure of (Jm was available, this value was assumed to be equal to the ultimate tensile strength (UTS) of the adhesive. However, as it will be shown in Section 'Calibration of CZ model parameter (Jm \ the results are not very sensitive to the magnitude of this parameter. A further increase in the external work inflicts further damage on the material and, eventually, the separation distance reaches its critical value 5w, i.e. the critical crack opening displacement. At this stage the material is fractured and the stresses on the fracture surfaces drop to zero, i.e. they no longer transfer the load across the surfaces. The work done by the cohesive forces is equal to fracture energy, Gc, of the adhesive material, which was obtained directly from the TDCB tests. The materials' mechanical properties used for the numerical simulations are shown in Tables 3 and 4. RESULTS Mechanical Properties Families of tensile stress-strain curves have been generated for strain rates in the range of 10'^ - 10 s'^ at 23°C, for both the epoxy adhesives. These are illustrated in Fig. 4 (a) and (b). The tensile properties were found to increase progressively with the increasing the rate. Calculated mechanical properties are summarised in Table 3. The properties of the aluminium alloys are not significantly affected by the rates considered and may be regarded as rate independent [13]. The mechanical properties of the aluminium alloys used in the current research are summarised in Table 4.
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0.06 0.08 0.10 0.12 0.03 0.06 Strain Strain Fig. 4: Tensile stress/strain data for the epoxy adhesives at specified average strain rates, at 23°C: (a) 'XD1493'; (b) 'XD4600'. 0.02
0.04
Table 3: Tensile properties of the 'XD1493' and the 'XD4600' epoxy adhesives at specified average strain rates, at 23 °C. (Typical number of test replicates was four). 'XD1493' 'XD4600' Strain rate UTS [MPa] Modulus [GPa] UTS [MPa] Modulus [GPa] 10-^ 10-^ 10-^ 10-^ 10-^ 10-^ 10'
33.6 ±0.6 36.6 ±0.9 40.5 ±0.5 42.6 ±0.9 47.7 ± 0.6 49.1 ±2.3 57.0 ±1.3
1.61 ±0.02 1.80 ±0.02 1.89 ±0.03 1.99 ±0.06 2.17±0.14 2.25 ± 0.22 2.38 ±0.14
59.8 ±1.5 64.1 ±0.4 69.6 ±1.3 74.1 ±1.0 80.9 ± 0.8 83.2 ±1.3 88.3 ±0.6
3.25 ±0.07 3.41 ±0.09 3.46 ±0.17 3.76 ±0.13 4.07 ± 0.47 4.43 ±0.37 4.56 ±0.51
Table 4: Tensile properties of the aluminium alloys employed in the current research obtained at a constant strain rate of 10""* s'\ at 23°C. (Typical number of test replicates was four); Alloys 5754-0 6111-T4 2014 Modulus - Es [GPa] Yield stress - a [MPa] Poisson's Ratio - v
65.3 ±0.9 98.5 ±1.0 0.33 ±0.005
69.8 ±1.1 281.5 ±0.4 0.33 ± 0.008
71.7±1.0 0.33 ± 0.006
Calibration of CZ model parameter am As mentioned earlier, the Gc value required to define the CZ model is obtained from TDCB tests. The remaining parameter (Jm is chosen as the UTS, and was extracted from the stressstrain curves at the corresponding rates. This was an arbitrary choice, since the level of the constraint near the crack tip is higher than that in uniaxial tensile tests used to obtain the stress-strain curves. Therefore, a sensitivity study on this parameter was performed. For illustration purposes, a numerical analysis carried out on TDCB test specimens bonded with the two adhesives under investigation is shown in this section. The value of dm was varied from 20 to 80 MPa and numerical predictions of load versus time were compared against the experimental results. Fig. 5 shows a comparison of the FV and experimental results for different (Jm values for TDCB tests performed at 0.1 mm/min. The best fit dm value should be able to predict correctly both the experimental force and crack history. (Note that the latter was found to be less sensitive to changes of the cohesive strength.)
323
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Fig. 5: Comparison of FV and experimental TDCB results obtained at 0.1 mm/min for different Gm values: (a) Force versus time response of a TDCB specimen bonded with 'XD1493' adhesive, Gc = 5037 J W ; (b) Crack history of a TDCB specimen bonded with ^XD1493' adhesive, Gc = 5037 J/m ; (c) Force versus time response of a TDCB specimen bonded with ^XD4600' adhesive, Gc = 3043 j W ; (d) Crack history of a TDCB specimen bonded with ^XD4600' adhesive, Gc = 3043 J/ml It can be seen that for a range of am values close to the UTS values of the two adhesives obtained experimentally at the corresponding strain-rates, i.e. 10'^ strains/s, the predicted and experimental results are in very good agreement. The low sensitivity of results to the variation in am assures accurate numerical predictions as long as am is within an acceptable realistic level, i.e. close to the UTS value. A different calibration procedure applicable for quasi-static situations is presented in [14]. TDCB Tests In the low rate tests the crack propagation force, as well as the crack speed, were found to be approximately constant (see Figs. 6(a) and 6(b)), giving a constant value of the adhesive fracture energy with the length of the propagating crack, i.e. no 'R-curve' effects were observed. Initially, the load increased linearly with displacement without affecting the crack length. Fig. 6 (a). Once the critical cohesive stress is reached, a damage zone forms in front of the crack tip (see Fig. 7), resulting in the deviation of the load/displacement curve from a straight line. Fig. 6(a). When the damage zone is increased to the extent where the critical separation distance is reached, the crack starts propagating at a constant speed. Fig. 6(b). Although the profiles of the arms are machined such that the compliance increases linearly with the crack length, the crack propagated in a transient manner soon after the crack front approached the last quarter of the TDCB specimen. Such behaviour was observed in the
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324
experiments and was also predicted by the numerical model. The numerical and experimental results are in very good agreement, both qualitatively and quantitatively, see Figs. 6(a) and 6(b). The choice of am being equal to UTS seems to be satisfactory, as both load-time and crack-length-time data are predicted very well numerically. At higher test rates of 0.5 and 1 m/s, significant changes in the fracture behaviour of the TDCB joints were observed: • Impact loading conditions cause an early crack initiation or crack tip damage resulting in a decrease of the specimen compliance, as can be seen from the early change in the slope of the force/displacement trace compared to the numerical FV predictions (Fig. 8(a)). • Dynamic effects occur causing oscillations in the force signal (Fig. 8(a)). • The type of failure for the 'XD4600' adhesive was found to alter with increasing test rate. Whereas all 'XD1493' adhesive TDCB joints failed in a stable continuous manner, TDCB specimens bonded with the 'XD4600' adhesive showed a transition to stick-slip behaviour at these increased test rates (Fig. 8). It is argued that this is probably due to the strain rate dependent fracture toughness of this adhesive. • The crack propagation force was also found to increase with increasing test rate. As can be seen from Fig. 8, this was more pronounced in the case of the 'XD1493' adhesive, where a 25% increase in the propagation force value was measured compared to Imm/min test presented in Fig. 6(a). (a) 25001
(b) 300r
2000
1500 lOOOh
4 5 6 7 Displacement [mm]
8
9
10 11
0
10 20 30 40 50 60 70 Time [min]
100 110
Fig. 6: TDCB results of tests performed at 1 mm/min, at 23°C. Comparison of FV and experimental results: (a) Load/Displacement curves; (b) Crack length histories. CZM parameters: 'XD1493' - am = 33.6 MPa, Gc = 5037 J/m^; 'XD4600' - am = 59.8 MPa, Gc = 3043 J/ml Displacements magnified by a factor of 5 for clarity
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Fig. 7: Normal stress component %. [Pa] at time T = 75 min after the beginning of the test in the TDCB FV simulation.
Rate Dependent Fracture Behaviour ofAdhesively Bonded Joints (a) 3001
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FVXD1493 Experimental XD1493 FVXCH600 Experimental XD4600
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Fig. 8: TDCB results of tests performed at 1 m/s, at 23°C. Comparison of FV and experimental results: (a) Load/Displacement curves; (b) Crack length histories. CZM parameters: 'XD1493' - Gm = 54.3 MPa, Gc = 7465 J/m^; 'XD4600' - (Tm = 86.4, Gc = 2981 J/ml Numerical results for the high rate TDCB tests are generally in a close agreement with experiments, giving good predictions of the force and crack histories. However, in cases where stick-slip behaviour was observed, e.g. for 'XD4600' adhesive, a rate independent CZ model has not produced satisfactory agreement with experiment, i.e. stick-slip behaviour was not possible to predict for any chosen set of CZ parameters. Results are limited to a quantitative estimation of the average force and crack speed values, (Figs. 8(a) and 8(b)). In order to overcome this problem a crack velocity dependent CZ model was considered. From the analysis of the numerical TDCB results, the crack speed was related to the strain rate values in the process region. Values of dm were then extrapolated from available stress-strain data, see Table 3, at the corresponding test rates. Fracture energy values were calculated analytically using the upper and lower bounds of the saw-tooth force versus time trace, assuming that the minimum Gc value corresponds to the highest crack speed and vice versa. Also, a linear relationship between CZ parameters and the crack speed was assumed, see Fig. 9. A comparison between the numerical prediction, with rate dependent CZ parameters, and the experimental results for a TDCB test performed at 0.5 m/s is illustrated in Fig. 10. Although the agreement is not perfect, the theoretical results clearly predict stick-slip behaviour. (a) 32001
10
12 14
Crack velocity [m/s]
4 5 6 7 Displacement [mm]
8
9
10 11
experimental Fig. 9: Velocity dependence of cohesive parameters Fig. Fig. 10: Comparison of FV and experimenta results for a TDCB specimen tested at 0.5 m/s
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T-Peel Tests Typical force versus time traces obtained from T-Peel tests are shown in Fig. 11. As can be seen, the peel load reaches a nearly constant value for both substrate materials, with some minor fluctuations superimposed on the results. These values have been used in an analytical model to calculate the adhesive facture energy, Gc [8]. For all the tests performed the crack propagated cohesively in the adhesive layer. The peel load was found to depend on the alloy type and on the thickness of the substrates, since most of the energy during the test is dissipated by plastic deformation of the arms. Numerical FV work is in progress.
1 mm thick 5754 A. A. 0.9 mm thick 6111 A. A. J
^
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Fig. 11: Force versus displacement response of T-Peel specimens bonded with 'XD4600' adhesive and tested at 5 mm/min, 23 °C. IWP Tests From the tests performed, the crack was always found to propagate cohesively through the adhesive layer of the IWP specimens. The crack propagated either in a transient or in a quasi-static manner, depending on the substrate thickness. Typical IWP force versus time traces for both of these propagation regimes are shown in Fig. 12(a). In the case of quasistatic crack growth (1 mm thick specimens) the presence of two distinct regions, an initial high-peak region followed by a 'plateau' region, can be easily detected. From a careful examination of the high-speed photography results, it can be speculated that the main causes for these initial peaks are dynamic effects, which arise from (a) the initial contact between the wedge and the specimen, and (b) crack initiation from the bead of adhesive that was formed in the ' V of the specimen. This was further examined with a series of tests performed on specimens with a pre-crack. Specimens with a PTFE tape of 10 mm length inserted in the adhesive, experienced significant reduction of the initial peak (see Fig. 12(b)). In the case of the quasi-static crack growth, the crack speed, within the 'plateau' region, was found to be equal to the test rate. In the case of the transient crack growth (2 mm thick specimens), again the initial region of the force versus time response consists of one or more peaks, due to the same reasons as before. However, this time rapid crack propagation takes place, and the absence of the 'plateau' region can be clearly seen in Fig. 12(a).
Rate Dependent Fracture Behaviour ofAdhesively Bonded Joints (a) 25(X)r " ^ ° — Quasi-Static Crack Growth (1mm thick specimen 57540) 2250 Z h Transient Crack Growth (2mm thick specimen-6111-T4) ~ 1
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4
6
8 10 Time [ms]
Fig. 12: Force versus time response from IWP specimens showing stable and unstable crack growth (a) Standard IWP specimen; (b) IWP specimen with a pre-crack. The reason for this behaviour is that in the case of the thicker and, therefore, stiffer substrate materials, a larger amount of energy was elastically stored in the substrates prior to crack initiation. Hence, after the onset of the crack growth, the rate of energy release would be much higher than that required for quasi-static crack growth, resulting in the transient crack propagation. Here, the crack velocities were much higher than the test rate. Analysis of the high-speed photographs has shown a crack velocity of 26 m/s for a test conducted at 2 m/s (see Fig. 13). Careful examination of the failed specimens revealed that the main characteristics of quasi-static crack growth were the large plastic deformation of the substrate arms and the associated high-energy dissipation. In contrast, specimens that exhibited transient crack growth showed no plastic deformation and, as would be expected, very low energy dissipation.
Fig. 13: High-speed photography of an IWP test which exhibited transient crack growth. Aluminium 6111-T4 series bonded with 'XD4600' adhesive and tested at 2 m/s, 23°C. The inter-frame time was 222 |is. A preliminary numerical simulation of the IWP tests was performed assuming linear-elastic behaviour for both the adhesive and the substrates. Two different geometries (i.e. 1 and 2 mm thick substrates) were modelled at an impact speed of 2 m/s. The model used the traction-separation obtained from the TDCB numerical results. Although the analysis was limited to elastic materials, the numerical observations were qualitatively in a good agreement with the experiments. The numerical simulations predicted that in the case of thin specimens the crack was driven by the wedge at the test rate, whereas in the case of thick substrates the crack was predicted to propagate in a transient manner. The calculated crack speed was also in qualitative in agreement with that measured from the high-speed photography.
328
/. GEORGIOUETAL
CONCLUSIONS Experimental results obtained from various tests have demonstrated the effects of loading rates on both the basic mechanical and fracture properties of adhesive joints. The 'XD1493' adhesive joints showed an increase in the fracture toughness from about 5 kJ/m^ at 1 mm/min loading rate, to about 7.5 kJ/m^ at 1 m/s. On the other hand, the average value of Gc did not change considerably with test rate for the 'XD4600' adhesive joints, but stickslip behaviour was observed at rates above 0.5 m/s. A general method of calibrating CZ parameters was established. It is based on Gc values calculated from the TDCB results, while the value of dm was chosen being equivalent to the UTS from the stress-strain curves at the appropriate rate. The 'fine-tuning' of this parameter was achieved by numerically fitting the measured load-displacement curves and the crack length histories. A novel rate dependent CZ model was developed and successfully applied for predicting the stick-slip behaviour of the 'XD4600' TDCB joints. Numerical predictions of the IWP tests, where the CZ parameters from TDCB tests were employed, were very encouraging. Further work on modelling the IWP and T-peel tests, including large strain plasticity, is in progress. The work will help to clarify the issue of transferability of the CZ parameters between the TDCB and the peel geometries, which is still an issue of some debate. ACKNOWLEDGEMENTS The authors wish to acknowledge funding and support from the following companies and organizations: ALCAN Int., DERA, EPSRC and FORD Motor Company. They would also like to thank Mr. Aleksandar Karac for his contribution in the development of the numerical procedure for predicting stick-slip behaviour. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
W. S. Miller, L. Zhuang, J. Bottems, A. J. Wittebrood, P. De Smet, A. Haszler and A. Vieregge, J. Mater. Sci. and Eng. A280 (2000) 37 B.R. K. Blackman, A. J. Kinloch, A. C. Taylor and Y. Wang, J. Mater. Sci. 35 (2000) 1867 American Society for Testing and Materials, ASTM E 8M - 89b (1986) American Society for Testing and Materials, ASTM D 638 - 72 (1972) European Standard EN ISO 2818. Plastics, CEN European Committee for Standardisation (1999) International Standards Organisation, ISO 15166-1 (1998) S. Mostovoy, P. B. Crosley and E. J. Ripling, J. Materials 2 (1967) 661 I. Georgiou, H. Hadavinia, A. Ivankovic, A. J. Kinloch, V. Tropsa and J. G. Williams, J. Adhesion, in press International Standards Organisation, ISO 11343 (ISO, Geneva, 1993) EPSRC. Web page Address: www.eip.rl.ac.uk A. Ivankovic, Computer Modelling and Simulation in Engineering 4 (1999) 227 H. G. Weller, G. Tabor, H. Jasak and C. Fureby, Computers in Physics 12 (1999) 620 The Aluminium Association, Publication, AT6 (1998) T. Ferracin, C. Landis, F. Delannay and T. Pardoen, Proc. 10^^ International Conf on Fracture - ICFIO, Hawaii, USA, Dec. 2001.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
329
EXPERIMENTAL CHARACTERIZATION OF CARBON-FIBER/CONCRETE ADHESIVE INTERFACE FOR RETROFITTING OF CONCRETE BRIDGE STRUCTURES T. KUSAKA*, H. YAGI*, H. NAMIKI** and N. HORIKAWA*** * Department of Mechanical Engineering, Ritsumeikan University, 1-1-1 Noji-Higashi, Kusatsu 525-8577, Japan ** Kyobashi Construction Corporation, 2-2-21 Shigino-Nishi, Joto-ku, Osaka 536-0014, Japan *** New Energy and Industrial Technology Development Organization, 1-1-1 Noji-Higashi, Kusatsu 525-8577, Japan ABSTRACT The modes I+II fracture behavior of the adhesive interface between concrete and carbon fiber sheets was investigated on the basis of the Hnear fracture mechanics. The mixed mode disk specimen was proposed to evaluate the mixed mode energy release rate of the adhesive interface. The evaluation formula for the energy release rate was also derived and its validity was studied on the basis of the results of finite element analysis. The numerical results indicated that the theoretical formula was not valid for evaluating the absolute value of the energy release rate though it could well describe the dimensional tendency between the energy release rate and geometric parameters of the specimen. Hence, it was concluded that the combination of finite element analysis and theoretical prediction was necessary to evaluate the energy release rate with high degree of accuracy. The mixed mode fracture toughness test of an acrylic resin adhesive interface was carried out by applying the present method. The mode II fracture toughness was more than twice as high as the mode I fracture toughness. However, those values were much lower than the fracture toughness of the adhesive resin itself, because the locus of failure was microscopically not in the interface but in the concrete. The mixed mode fracture toughness followed the linear fracture criterion; the principle of superposition was valid for the present specimen. KEYWORDS Concrete structure. Retrofitting, Carbon fiber sheet, Adhesive interface, Mixed mode. Fracture toughness. Fracture criterion
INTRODUCTION Many civil structures made of concrete need to be repaired or reinforced because of the deterioration of materials, increase of traffic volume, damage by accidental force, and
330
T. KUSAKA ETAL 150
• RC beam without CF shxts 100
3
RC beam with tensioned CF sheets Debonding of CF sheets
50
a'0 (a) Damage by Hanshin-Awaji earthquake
5 lb iV ' ^ 20 Displacement S, mm (a) Load-displacement relations
RC beam t
Tension
t
t
t
t
t Bonding f
|
f
f
CF sheets
i
f
f
Tension
^ Compression, (b) Retrofitting of RC beam Fig. 1. Schematic drawing of retrofitting method for concrete beams with tensioned carbon fiber sheets.
(b) Debonding of carbon fiber sheets Fig. 2. Experimental results of 4-point bending test for concrete beams with and without bonding of carbon fiber sheets.
so on. Especially, strong and urgent needs for reinforcement of highway and railway concrete bridges have arisen in Japan since the Hanshin-Awaji earthquake occurred in Kobe in 1995. In addition, the development of industries related to renovation and maintenance of civil structures has been politically encouraged in Japan since 1999 [1]. Retrofitting using CF (Carbon Fiber) sheets has been considered as one of the most promising methods for reinforcement of RC (Reinforced Concrete) bridge structures [25]. The method has many more advantages in construction cost and efficiency than the conventional methods, and some trials have been made to improve the efficiency oi' reinforcement further in recent years. Above all, the method using tensioned carbon fiber sheets, as shown in Fig. 1, is paid the most attention in the field of reinforcement of concrete bridge structures [6-9]. This method can gain the durability for the dead load remarkably by introducing compressive stress to the tensioned side of the structure. However, the efficiency of reinforcement is often limited by the debonding of carbon fiber sheets, as shown in Fig. 2 [6]. Hence, the evaluation and improvement of strength of the adhesive interface between concrete and carbon fiber sheets is vitally iniportani for this kind of reinforcement. However, the strength of the adhesive interface has been usually characterized on the basis of averaged stress without enough consideration on stress concentration at the end of adhesive region in this kind of engineering field [10] Especially, very few works are available on the effect of mode mixture, though the debonding of carbon fiber sheets usually occurs under modes I+II loading condition [11].
Experimental Characterization of Carbon-Fiber/Concrete Adhesive Interface
331
Load
Steel jigs
. >
Idiicrete blocks
Carbon fiber sheets (a) Finite element model
Load (a) Mixed mode disk specimen Adhesive region
Carbon fiber sheets
(b) Detail of the adhesive region ^. ^ e u ^' A ' € 4U A A Fig. 3. Schematic drawing of the mixed mode .. , . disk specimen.
XTTTX
(b) Detail of the crack tip region Fig- 4. Finite element model of the mixed '^ mode disk specimen; A^ = 0.3125 mm. ^
In the present work, a novel experimental method was proposed to evaluate the modes I+II fracture toughness of the adhesive interface between concrete and carbon fiber sheets. The validity of the evaluation formula was studied on the basis of the results of finite element analysis. The mixed mode fi*acture toughness and fracture criterion of the adhesive interface were also studied on the basis of the results of fracture toughness test using the proposed method. MIXED MODE DISK SPECIMEN Specimen
configuration
The MMD (Mixed Mode Disk) specimen was proposed to investigate the modes I+II fracture behavior of the adhesive interface between concrete and carbon fiber sheets. This specimen, which consists of two concrete blocks inserted carbon fiber sheets and steel jigs, is basically similar to the Arcan specimen, as shown in Fig. 3 [12]. However, the evaluation formula for the Arcan specimen can not be directly applied to the MMD specimen owing to the difference of geometries and the existence of carbon fiber sheets. For the present specimen, the adhesive region was set to be smaller than the face of
332
T. KUSAKA ETAL
concrete blocks to and II components varied by changing condition; ^ = 90°
prevent the to the total the loading corresponds
gross fracture of concrete blocks. The ratio of modes I energy release rate, G|/(G,+ Gjj), Gji/(G|-hG,|), can be angle, 0; 0 = 0° corresponds to the pure mode II loading to the pure mode I loading condition [12].
Data reduction scheme The evaluation formula of the mixed mode energy release rate, G, for the MMD specimen can be obtained by modifying that for the Arcan specimen on the basis of some geometric considerations [13,14];
where P is the load applied to the specimen. The terms of /j and /jj correspond to the modes I and II energy release rates, Gj, Gjj, respectively. Hence, the fraction of modes I and II components are given by G,/G = f^sm^O/{f?sm^O+f^cos^O),
(2)
GjG = f^,cos^O/{f^sm^O+f^cos^O),
(3)
where 0 is the loading angle, a, fi, y and x ^^ respectively the geometric and material parameters defined by a = L/L^,
P=B/B^,
Y = a/L,
X = EJE^,
(4)
where IQ, B^ and EQ are the length, width and Young's modulus of the concrete blocks, respectively. L and B are the length and width of the adhesive region, respectively, a is the crack length of the specimen. £Q, which is approximated to be 30 GPa, is the Young's modulus of a general concrete, /j and /jj are the correction factors defined by /,-1.12-0.231/+10.5/2-21.2/^ + 30.3/,
(5)
/„ = (1.12-0.561/+0.085/ + 0.180/)/(l - / ) ^ / ^
(6)
The validity of the above formula is studied in the following section. FINITE ELEMENT ANALYSIS Finite element model Finite element analysis was carried out to study the validity of the evaluation formula derived in the previous section. The MMD specimen was discretized with 2-dimensional plane strain or 3-dimensional solid elements according to the purpose of the analysis, as shown in Fig. 4. The finite element code, MARC 2001, was used for the analysis. The dimensions of the specimen was varied to investigate their effects on the evaluation of the energy release rate, G, as a = 0.22-1.00 {L = 20-90 mm for LQ = 90 mm), /? = 0.20-1.00 (J5 = 4-20 mm for BQ = 20 mm) and / = 0.38-0.61 (a = 35-55 mm for Z = 90 mm). The Young's modulus of concrete blocks was also varied as 2^ = 0.67-1.33
Experimental Characterization of Carbon-Fiber/Concrete Adhesive Interface
333
{EQ = 20-40 GPa for £^Q = 30 GPa). The cases with and without carbon fiber sheets were simulated to clarify the effect of their existence. The longitdinal Young's modulus of carbon fiber sheets, E^, was 250 GPa. The mixed mode energy release rate, G, were calculated by the following equations based on the modified crack closure method [15];
where the first and second terms correspond to the modes I and II energy release rates, Gp Gjj, respectively. / ^ and / ^ are the x and y components of nodal force of the node A. (WQ^ — w^.^) and (WQ — u^) are the x and y components of relative displacement between nodes B and C. Aa is the length of the crack tip elements. Effects of the geometries of the specimen Figures 5-7 shows the effects of the geometric parameters, a, /?, /, on the evaluation of the mixed mode energy release rate, G, which was calculated by Eqn (7) and normalized by {P/BQf{7rF)/{EQLQ), where F = f^sirp- 0 ^f^^cos^ 0. All the results are for the case without carbon fiber sheets. As shown in Figs. 5-7, the energy release rate, G, was proportional to \/a, \/j3 and /. These results were totally consistent with the prediction by Eqn (1) on their dimension. However, the absolute values of G predicted by Eqn (1) were different from those obtained by the finite element analysis, as suggested by the fact that the values of GLQEQI{P/BQY/{nF) depended on the loading angle, 0\ all the plots should be on the same lines without depending on the loading angle, 6, if the energy release rate, G, could be calculated by Eqn (1). In other words, the correction factors, /,, / „ , would not be exact enough for the present specimen. Effects of the Young's modulus of the concrete Figure 8 shows the effects of the material parameter, x^ ^^ the evaluation of the mixed mode energy release rate, G, which was calculated by Eqn (7) and normalized by (P/5o)2(;rF)/(^oZo), where F = f^^ir?-6^f^^cos^0. All the results are for the case without carbon fiber sheets. As shown in Fig. 8, the energy release rate, G, was proportional to X/x- This result was consistent with the prediction by Eqn (1) on its dimension. However, the absolute values of G predicted by Eqn (1) were different from those obtained by the finite element analysis, as similar to the results shown in the previous section. Especially, the results for ^ = 0 differed much from the others. This might be the consequence of less stiffness caused by the constricted parts of the steel jigs. Effects of the existence of the carbon fiber sheets Figure 9 shows the relationship between the loading angle, 0, and the mixed mode energy release rate, G, normalized by P^. The open and solid circles with broken lines
334
T. KUSAKA ETAL 10'
\-
1
1
1
1
1
1
1
10^
1-
E
~i
^
I
I
\
I
r~nj
e = 30-90° Q (9=15°o
^ ^ = 45-90° Q <9 = 30° 8 ^ = 15°o. r 6' = 0° o
^
\
Sr
lO^t
P
<9 = 0 ° o
^
-1
of 10^
-1
1
tt^
/^i
Eqn (I)
n
o 10
10-
Eqn (1)
1
^
CC = L/LQ
Fig. 5. Effect of the geometric 1 1 1parameter, i 1 1 1 a, on evaluation of the mixed mode energy release rate, G; without carbon fiber sheets, >^=:1.00, / = 0.50, j = 1 . 0 0 . 10'
I
I
I
10'
I TT.
^
OQ
fiq
10 10-
(9 = 15-90° I 8 ' -.oP^ ^ = 0° o o9^ Eqn (1) ^-^^^ |1 V/-" 1 J
I
y = a/L
I
I
I
I
I I I
P = BIB,
10"
Fig. 6. Effect of the geometric parameter, p, on evaluation of the mixed mode energy release rate, G; without carbon fiber sheets, cr=1.00, / = 0.50, ;}r=1.00.
^ 10"b
J
^^lo-
10"
"T
I
I
I I I I II
I
I
I
I I I 14-1
Eqn (1)
\
8
10^
^^ eJ6> = 15-90°
q
^6> = 0°
I I
10"
Fig. 7. Effect of the geometric parameter, y. on evaluation of the mixed mode energy release rate, G; without carbon fiber sheets, a =1.00, y? = 1.00, / = 1 . 0 0 .
'«lo-
J
I I I I MM
J
I I\| I III
10'
Fig. 8. Effect of the material parameter, j , on evaluation of the mixed mode energy release rate, G; without carbon fiber sheets, a =1.00, y^ = 1.00, / = 0 . 5 0 .
represent the finite element results for the case with bonding of carbon fiber sheets on upper and lower sides of the crack, respectively. The solid triangles with broken line represent the finite element results for the case without bonding of carbon fiber sheets. The open triangles with solid line represents the theoretical prediction by Eqn (1). All the results are for a = 0.44, J3 = 0.50, 7 = 0.50 and ^ c = ^^ ^P^As shovm in Fig. 9, the theoretical prediction did not agree with the finite element results, which corresponds to the tendency discussed in the previous section. The finite elements results with and without bonding of carbon fiber sheets did not also agree each
Experimental Characterization of Carbon-Fiber/Concrete Adhesive Interface
335
0.15
+
5^ O.lOh
A^^ ^/ 1> 0.05
•
O o
ts
/
(U 73 O
2
30 60 Loading angle 9, deg
90
30 60 Loading angle 6, deg
90
—•-—Numerical with CF sheets (upper)
—•^—Numerical with CF sheets (upper)
—a—Numerical with CF sheets (lower)
—o-Numerical with CF sheets (lower)
—^ — Numerical without CF sheet
—Ar — Numerical without CF sheet
—A— Theoretical without CF sheets
—^— Theoretical without CF sheets
Fig. 9. Effect of the bonding of carbon fiber sheets on evaluation of the energy release rate, G; a = 0.44, p = 0.50, / = 0.50, £•(. = 30 GPa, E^ = 250 GPa.
Fig. 10. Effect of the bonding of carbon fiber sheets on evaluation of the mode ratio, G,/(Gi+Gi,); a = 0.44, y^=:0.50, / = 0.50, E^ = 30 GPa, E^ = 250 GPa.
other. This result suggested that the bonding of carbon fiber sheets affected the evaluation of the energy release rate, G. However, the position of the carbon fiber sheets had little effects on the evaluation of the energy release rate, G. Figure 10 shovs^s the relationship between the loading angle, 0, and the mode ratio, G|/(Gj + Gjj). The open and solid circles with broken lines represent the finite element results for the case with bonding of carbon fiber sheets on upper and lower sides of the crack, respectively. The solid triangles with broken line represent the finite element results for the case without bonding of carbon fiber sheets. The open triangles with solid line represents the theoretical prediction by Eqn (1). All the results are for a = 0.44, /3 = 0.50, r = 0.50 and E^ = 30 GPa. As shown in Fig. 10, the theoretical prediction somewhat different from finite element results. The finite elements results with and without bonding of carbon fiber sheets did not agree each other. The position of the carbon fiber sheets also affected the evaluation of the mode ratio, Gj/(Gj + Gjj). Strictly speaking, the oscillation terms arising near the interface of different materials should be considered in the above discussion [16]. However, the purpose of the present work is to macroscopically characterize the fracture behavior of the adhesive interface for the design of civil structures. Hence, the inhomogenity of the specimen was neglected for the evaluation of the macroscopic fracture toughness.
336
T. KUSAKA ETAL Table 1. Propenties and construction of materials. Water (kg/m^) 157
Cement (kg/m^) 302
Sand (kg/m^) 756 Poisson's ratio
Young's modulus E^ (GPa) 29
0.2
Gravel (kg/m^) 1077
W/C (%) 52
Compressive strength a^ (MPa) 30
(a) Concrete Young's modulus E^ (GPa) 245
Tensile strength C7s (MPa)
3400
Nominal thickness /g (mm) 0.17
(b) Carbon fiber sheet (UT70-30,, Toray) Young's modulus E^ (GPa) 4.8
Tensile strength ox (MPa) 34
Shear strength zx (MPa) 13
(c) Adhesive resin (DK-530, Denka)
Summarizing the results shown in Figs. 5-10, it can be concluded that the combination of finite element analysis and theoretical prediction was necessary to evaluate the energy release rate, G, with high degree of accuracy; the mixed mode energy release rate, G, should be firstly obtained by finite element results, and then it should be corrected for the geometric parameters, a, p, y, and the material parameter, x^ ^f Qdic\i specimen using the theoretical relation given by Eqn (1). In the following section, the mixed mode fracture toughness, G^, was determined by using the result represented by the open circles in Figs. 9 and 10 as master curves together with the correction based on Eqn (1).
FRACTURE TOUGHNESS TEST Materials and specimen Table 1 shows the specifications of the concrete, carbon fiber sheet and adhesive resin investigated in the present work. The concrete blocks of 30 x 20 x 90 mm were made of normal grade portland cement and cured in water for 4 weeks. The carbon fiber sheet (UT70-30, Toray) was a unidirectional cloth bound with a few lateral fibers. The adhesive resin (DK-530, Denka) was a modified acrylic resin of two component system. Two concrete blocks were carefully bonded by inserting a carbon fiber sheet with the adhesive resin and cured at room temperature for 48 hours as the manufacturer's recommendation. A thin releasing film was inserted between the concrete blocks and carbon fiber sheet to introduce an artificial crack and to make an adhesive region of 20 x 10 mm. The blocks were then bonded to the steel jigs with the same adhesive resin, as shown in Fig. 11. A liquid type primer was used to gain the adhesiveness of the interface between the concrete and carbon fiber sheet.
337 Experimental Characterization of Carbon-Fiber/Concrete Adhesive Interface
Fig. 11. External view of mixed mode disk specimen.
Fig. 12. Experimental apparatus of the mixed mode fracture toughness test.
Experimental procedure The mixed mode fracture toughness tests were carried out on a universal testing machine at loading rate of 1 mm/min. The loading angle was varied to study the criterion of mixed mode fracture as (9 = 0°, 15°, 30°, 90° (G,/G = 0.01, 0.31, 0.55, 0.95). The load applied to the specimen was measured by a load cell of 5 kN. The displacement of the specimen was approximately measured from the movement of the crosshead by a displacement sensor of strain gage type. The analog outputs were converted and stored in a computer as digital data. Load-displacement relation Figure 13 shows a typical result of fracture toughness test. The abscissa shows the displacement of crosshead. The ordinate shows the load applied to the specimen. As shown in Fig. 13, the fracture behavior was unstable and rapid. The load-displacement relation had a linearity up to the maximum point of load, though the relation was not linear owing to the plays of the steel jigs and testing machine during the early stage of loading. Hence, the critical point of onset of the crack growth, P^., was assumed to be the maximum point of load, imaxv Fracture morphology Figure 14 shows the fracture surfaces of the specimen. The initial crack tip was on the center of the specimen. The artificial crack and adhesive region were respectively the right and left halves of the specimen in the photographs.
T. KUSAKA ETAL
338
l.U
-1
1
1
1
1
71
'max
0.8
s
oT O
'
1
^
\
= /'c
J
0.6
J
/
0.4 0.2
n
// / / f / / / / / / / / /
f
\
1
0.1
0.2
1
0.3
J
0.4
L.
0.5
Displacement 5^ mm Fig. 13. Load-displacement relation of the mixed mode fracture toughness test; <9 = 15°.
Fig. 14. Macroscopic fracture morphology of the mixed mode disk specimen.
As shown in Fig. 14, a thin concrete layer always remained on the adhesive region; the crack path was inside the concrete blocks. Hence, the experimental results did not strictly indicate the properties of the adhesive interface. Flowever, this behavior totally agreed with the actual fracture behavior of the RC beam reinforced with carbon fiber sheets [6]. This behavior was also consistent with the finite element results, where the principal stress in the concrete blocks partially exceeded their tensile strength. The above results suggest that the macroscopic strength of the adhesive interface can not be improved only by improving the strength of the adhesive resin, but the toughening of the interphase region including the concrete itself is the most necessary and important. Recently, the manufacturer of the present adhesive resin has developed a permeable primer strengthening the concrete near the interface. The authors are now preparing the experiment of the specimen using the primer. Fracture
criterion
Figure 15 shows the diagram of the mixed mode fracture toughness, G. The abscissa shows the mode I component of the mixed mode energy release rate, Gj. The ordinate shows the mode II component of the mixed mode energy release rate, Gjj. Each plot is the average value of 5-10 test data. As shown in Fig. 15, the mode II fracture toughness, Gjj^^, was more than twice as high as the mode I fracture toughness, G^Q. However, the difference was much smaller than that of the other materials such as polymer matrix composite materials. This would be the consequence that the microscopic fracture of the present specimen was not pure mode II owing to the discrepancy between the macroscopic crack face and microscopic crack path even though the mode II loading was applied macroscopically, as suggested by the results shown in Fig. 14.
Experimental Characterization of Carbon-Fiber/Concrete Adhesive Interface 40 ^-
-I—I—I—I—I—
339
' ' I ' '
b—G„p = 36 J/m^ G, + G,T = const.
G,c = 14 'j/m2
5 10 15 20 Mode I component G,, J/m^ Fig. 15. Mixed mode fracture toughness of the adhesive interface between concrete and carbon fiber sheets; each plot is the average value of 5-10 test data.
In addition, the fracture toughness, G, of the present specimen was much lower than that of the adhesive resin itself. This would be also the consequence that the microscopic fracture occurred in the concrete blocks. However, the mixed mode fracture toughness, G, approximately coincided with the linear fracture criterion, Gj + Gj, = const., [17]. This result suggests that the principle of superposition was valid for the present specimen, which is very important conclusion for the structural design. CONCLUSIONS In order to apply the characterization and structural design of the concrete bridge structures retrofitted with tensioned carbon fiber sheets, the mixed mode disk specimen was proposed and applied to the mixed mode fracture toughness tests. The following conclusions were derived from the results and discussions; • The theoretical formula was not valid for evaluating the absolute value of the energy release rate, though it could well describe the dimensional tendency between the energy release rate and geometric parameters of the specimen. • Hence, The combination of finite element analysis and theoretical prediction was necessary to evaluate the energy release rate with high degree of accuracy. • The mixed mode fracture toughness of the acrylic resin adhesive interface was obtained by applying the proposed method. The mode II fracture toughness was more than twice as high as the mode I fracture toughness. • However, the fracture occurred microscopically in the concrete blocks. Consequently, the apparent fracture toughness of the adhesive interface was much lower than the adhesive resin itself. • The mixed mode fracture toughness of the adhesive interface followed the linear fracture criterion. This result suggests that the principle of superposition was valid
340
T. KUSAKA ETAL for the present specimen, which is very important conclusion for the structural design.
ACKNOWLEDGEMENTS The authors would like to acknowledge K. Suzukawa of Toray and S. Kato of Denka for supplying the materials used in the present work. REFERENCES 1. Administrative Program for Regeneration of the Construction Industries, (1999), Ministry of Construction, Japan. 2. Shahawy, M.A., Arockiasmy, M., Beitelman, T. and Sowrirajan, R. (1996), Composite: Part B, 27, 225-233. 3. Meier, U. and Kaiser, H. (1991), Proceedings of the ASCE Specialty Conference on Advanced Composite Materials in Civil Engineering Structures, 224-232. 4. Wu, Z.S., Yin, J. and Niu, H.D. (2001), Proceedings of the International Conference on FRP Composites in Civil Engineering, 12-15. 5. Murahasi, H., Kimura, K., Katsumata, H., Kakuda, A. and Tanigaki, M. (2000), Repairing and Reinforcing Method for Civil Structures with Continuous Fibers, Rikogakusha, Tokyo. 6. Horikawa, N., Kusaka, T, Takagi, N., Inoue, M. and Namiki, H. (2002), Proceedings of the Fifth China-Japan-US Joint Conference on Composites, 81-86. 7. Meier, U., Deuring, H., Meier, H. and Schwegler, G. (1993), Proceedings of the First International Conference on FRP Reinforcement for Concrete Structures, 432^34. 8. Wu, Z.S., Matsuzaki, T, Yokoyama, K. and Kanda, T. (1999), Proceedings of the Fourth International Conference on FRP Reinforcement for Concrete Structures, 291-296. 9. Stoecklin, I. and Meier, U. (2002), Proceedings of the First International Conference on Bridge Maintenance, Safety and Management, 1-8. 10. Sugiyama, H. and Morikawa, H. (2001), Proceedings of the JSMS Symposium on Repair, Reinforcement and Upgrade of Concrete Structures, 219-226. 11. Kagayama, K., Kimpara, I. and Esaki, K. (1995), Proceedings of the Tenth International Conference on Composite Materials, 597-604. 12. Jurf, RA. and Pipes, R.B. (1982), J Compos. Mater, 16, 386-348. 13. Murakami, Y. (1987), Stress Intensity Factors Handbook, Volume 2, Pargamon, Tokyo. 14. Broek, D. (1986), Elementary Engineering Fracture Mechanics, Kluwer, Dordrecht. 15. Rybicki, E.F. and Kanninen, M.F. (1977), Eng Fract. Meek, 9, 931-938. 16. Erdogan, R (1965), J Appl Meek, 32, 403-411. 17. Reeder, R.J. (1993), Composite Materials: Testing and Design, ASTM STP 1206, 303-322.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
341
THE DETERMINATION OF ADHESIVE FRACTURE TOUGHNESS FOR LAMINATES BY THE USE OF DIFFERENT TEST GEOMETRY AND CONSIDERATION OF PLASTIC ENERGY CORRECTION FACTORS DR MOORE, J G WILLIAMS Imperial College, Exhibition Road, London SW7 2AZ, UK
ABSTRACT Fixed arm peel and T-peel test procedures are used to measure peel strength for flexible laminates. Analysis of the contributions from elastic and plastic deformations of the peel arms during these tests enables the energy contribution from plastic effects to be subtracted from the energy required to peel the laminate. In this way, the adhesive fracture toughness is determined. Seven independent laboratories have conducted these procedures (through the ESIS TC4 group) on a flexible polypropylene laminate system. Experiments for fixed arm peel have been conducted at a common peel angle (90°) and also for multiple angles (60°-160®) in order to determine the adhesive fracture toughness. These laboratories also conducted measurements with a T-peel geometry. Similar values were obtained for the adhesive fracture toughness demonstrating that with careful thought to the various energy contributions in the peel tests that a geometry independent value can be derived. The corrections for plastic deformation have been further analysed for metallic substrate laminates, using these two test geometries. A method is suggested for monitoring the correction. It is believed that corrections up to about 70% can be accommodated by the test protocols. Moreover, by investigating the temperature dependence of adhesive fracture toughness for the metallic substrate system, even larger plastic corrections may be tackled (greater than 80%). However, the accuracy of the ensuing results becomes more doubtful. Keywords Adhesion, laminates, fracture mechanics, adhesive fracture toughness, polymers, metal substrates, T-peel, fixed arm peel
342
D.R. MOORE AND J. G. WILLIAMS
INTRODUCTION Peel strength is commonly used in order to monitor adhesive strength. This relates to adhesives in many applications and a number of test methods have been used, includmg fixed arm peel [1], T-peel [2], climbing drum peel [3] and floating roller peel [4]. As industrial tests, the measured peel strength is used synonymously with adhesive strength. However, there are a number of other energy absorbing mechanisms associated with the peel process (e.g. viscoelastic and plastic deformations) that frustrate accuracy through such a simple link. A number of workers [1,5] have demonstrated how plastic deformation can be subtracted from a peel strength in order to approximate to a true adhesive fracture toughness. In this current work we hope to utilise some of these methods in order to achieve three aims: (i) To demonstrate that a geometry independent adhesive fracture toughness can be achieved if proper account is made of plastic deformation. To this end, we adopt two test geometries, namely a fixed arm peel and a T-peel. (ii) To investigate the extent of plastic energy correction through defining its contribution and by commenting on how much can be tolerated, (iii) By illustrating a practical adhesion problem where high plastic correction is present but where a fracture mechanics approach can assist in a pragmatic sense. ADHESIVE FRACTURE GEOMETRIES
TOUGHNESS
USING
DIFFERENT
TEST
Procedures and Laminate System. The purpose of this stage of the work was to determine adhesive fracture toughness on a common laminate system by the use of two different test geometries. Fixed arm and Tpeel test procedures are fully documented in an ESIS publication [6]. This reference provides a full description of the analytical procedures and indicates a source for the associated software for conducting some of the calculations. In order to determine the adhesive fracture toughness two experiments are required in each procedure. First, a peel test is conducted on the laminates i.e. either a fixed arm or a T-peel. Second, a tensile stress-strain plot is required for the peel arm(s). The peel test provides a measure of the peel strength (force/specimen width), whilst the tensile test provides values of elastic (E'l) and plastic (E2) modulus together with a value for yield strain (sy). This is shown in Figure 1. These data are then used to determine peel toughness with elastic (GA^^) and plastic (GA^^) corrections, (eb refers to elastic bending and db refer to dissipated bending) The adhesive fracture toughness (GA) is the difference between these values. In the case of the T-peel procedure, since there are two peel arms, the adhesive fracture toughness is the sum of these differences for each of the peel arms. Experiments were conducted by seven laboratories that were participants from ESIS Technical Committee 4:-
The Determination of Adhesive Fracture Toughness for Laminates
343
ICI pic, UK (D R Moore, R S Hardy), DuPont S.A., Switzerland (S Ducret), Imperial College, London (J G Williams, B Blackman), ATO-DLO & University of Twente, Holland (H Bos, P E Reed). BASF, Germany (F Ramsteiner), Politecnico de Milano, Italy (A Pavan), Insa de Lyon, France (A A Roche). Stress^
E^
i
^
!
/ 1
1 1 J •1 i
i
\
i
\
I
f t i f r 1 1
; ! ;
: I
; ;
%
Strain
Figure 1 Tensile stress-strain plot for peel arm showing elastic modulus (Ei), plastic modulus (E2) and yield strain. The same laminate system was used by all laboratories, namely a laminar structure with five layers based on polypropylene (PP), an adhesive, an ethylene vinyl alcohol (EVOH) layer another adhesive and another polypropylene layer This is designated PP/adh/EVOH/adh/PP and was used in both fixed arm peel tests and T-peel tests. Peel specimen were 15 mm in width and a notional 100 mm in length. The peel arms were PP/adh (with a thickness of 51^m) and EVOH/adh/PP (with a thickness of TS^im). Although there is no rigorous value for "modulus" for such multi-layered arms, we have obtained a tensile "modulus" value for each arm and assumed the materials of the arms to be uniform for the purposes of our calculations. Peel tests were conducted on universal testing machines at 23°C at a test speed of 10 mm/min. In the fixed arm peel test there was a requirement that the peel fixture was attached to the test machine through a friction-free linear bearing system. This enabled the direction of force on the peel arm to remain vertical whilst the peel fracture was propagating. A number of peel angles were used in the fixed arm test, but all laboratories included a configuration where the peel angle was 90^. A tensile test on the peel arm is used to obtain the parameters of "elastic" modulus, "plastic" modulus and yield strain. In this test, it has been necessary to use an extensometer for measuring strain at small magnitudes (i.e. up to about 2%) in order to obtain sufficient accuracy in the determination of Ei. It is also important to continue the tensile test to fracture, in order to define enough of the plastic region for an accurate
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D.R. MOORE AND J.G. WILLIAMS
determination of E2. During the T-peel test the two peel angles are measured as shown in Figure 6. Results from a Fixed Arm Peel Test. All of the participating laboratories conducted 90^ fixed arm peel tests; their results are shown in Figure 2. Three adhesive fracture toughness values are included in this plot. First, with elastic corrections only (GA^^), second with plastic correction (GA^^) and finally the difference (GA= GA^^ -GA'^^). The GA^^ values are directly related to peel strength and it is observed that laboratory 3 did not show agreement with other results. There is no apparent reason for this. The adhesive fracture toughness value (GA) al:so shows some scatter between laboratories. Laboratory 3 have a relatively low value (not surprisingly since their GA^^ value was low)) but Laboratory 4 shows a high value. The value for GA depends on all aspects of measurement i.e. peel strength and parameters derived from the tensile stress-strain plots. In this sense Laboratory 4 deviatedfi"omthe procedures used by all other laboratories. They determined a "plastic" modulus at a much lower value of tensile strain and therefore their value for E2 was much higher than that for other laboratories. This is illustrated in Figure 3 where the ratio E2/E1 is plotted. It is suspected, therefore, that the value of GA from laboratory 4 is too large. 600 500 h 400 h
5 ^ 300 U
• Gaeb o Gdb ^ Ga
200 h 100 h
Figure 2 Fixed arm peel results with a peel angle of 90° from all the participating laboratories.
The Determination of Adhesive Fracture Toughness for Laminates
3
345
4
Laboratory Figure 3 Ratio E2/E1 from data derived from the tensile tests on the peel arm PP/adh. for all the participating laboratories. The average value of GA from the data for 90° fixed arm peel is 214 J/m^, with a standard deviation of 46 J/m^. This excludes the value obtained from Laboratory 4 for the reasons given. Five of the laboratories conducted fixed arm peel tests for a range of peel angles. The characteristics of these results are illustrated in Figure 4 from Laboratory 2.
•
800
•
•
600
0
• • CO
o
E
400
0
200
n
0 A
^
J
1
40
80
•
120
• Gaeb o Gdb
0
A Ga
•
•
160
Peel angle degrees
Figure 4 Fixed arm peel data from Laboratory 2 at five different peel angles for the PP laminate system.
346
D.R. MOORE AND J.G. WILLIAMS
The important features of Figure 4 are the dependence of both GA^^ and G^^ on peel angle. This suggests that these parameters cannot be objective material (or laminate) properties. However, GA can be seen to be independent of peel angle and hence this quantity is considered to be an objective measurement of the adhesive fracture toughness. The GA values for all five laboratories are shown in Figure 5.
• Lab 2 • Lab 3 • Lab 5 ss Lab 6 + Lab7
E (0
O
0
20
40
60
80
100 120 140 160 180
Peel angle (degrees) Figure 5 Adhesive fracture toughness as a function of peel angle for 5 laboratories for fixed arm peel tests on PP laminate system. There is some scatter in these adhesive fracture toughness data. However, there are no reasons for excluding any of the results. The mean value is 206 J/m^ with a standard deviation of 42 J/m^. With consideration to the overall level of scatter, this gives good agreement with the results for the data at a peel angle of 90"" (214 j W ) . Results from a T-peel Test. There are two peel arms to consider in the T-peel test. These arms have slightly different values of stiffness, although similar moduli. Therefore, the peel angles are not 90** (which would be the case for equal stiffness arms) and as a consequence there may be a difference in peel strength dependent on the specimen configuration. The two possibilities are shown in Figure 6 and are designated configurations A and B. Configuration A has the stiffer peel arm (peel arm 2...PP/adh/EV0H) at the bottom of tho arrangement.
The Determination of Adhesive Fracture Toughness for Laminates
347
Peel arm 1 PP/adh
^
--'vJ
Configuration A
Peel arm PP/adh/EVOH
Configuration B
Figure 6 The two configuartions for T-peel where peel arm 2 is stiffer than peel arm 1 for the PP laminate system. Analysis of the data follows a similar route to that for the fixed arm procedure. However, for the T-peel there are adhesive fracture toughness values to determine for each peel arm, (GA) and (GA)^. The adhesive fracture toughness for the laminate (GA) is then the sum of these two values. Table 1 summarises all results from six laboratories. Almost all of the laboratories conducted tests with both configurations A and B. In all cases, the peel angles (0 and (j)) were not equal to 90° because the stiffness of the arms was not equal. The measured smaller peel angle ((|)A and 9B where A and B relate to configuration) is quoted in the table (in the form of the range of angles). It is clear that the smaller angle is less in the case of configuration B i.e. when the stiffer arm is at the top (where both peel angles [(9 + (t))A and (9 + ^)B\ add up to 180^). It is probable that this is due to the larger weight of the top peel arm in configuration B, due to its greater thickness. Therefore the difference in the size of the smaller peel angle is a gravity effect in the test and has implications for the length of the tail and how it might be supported (or not). This was not specified in our experiments but in general the peel tails were imsupported. The average of the adhesive fracture toughness values from all of the labortories is as follows:
D.R. MOORE AND J.G. WILLIAMS
348
Configuration A 190 j W Configuration B 188 J/m^ It would therefore seem that the tail effect is either not relevant or was fortuitouidy similar in the tests conducted by all laboratories. Laboratory
1 3 4 5 6
Configuration A Average GA Small peel (JW) angle ^ 40-64 183 50-61 179 56-70 191 50-64 211 60 184
Number of tests 4 2 2 2 1
Configuration B Average Small peel Number of angle "^ tests GA(J/m') 191 173 203 184
44-49 40-44 43-47 36
3 2
2
1
1
Table 1 Summary of T-peel results from 5 laboratories using both test configurations. It is concluded that the adhesive fracture toughness is independent of configuration. However, the peel strength and peel angles do depend on configuration. Comparison of Adhesive Fracture Toughness by Different Test Geometries There is good agreement for adhesive fracture toughness for the two different test geometries. The fixed arm peel gave two values; for 90** peel angles a value of 214 J/m^ and for multiple angles a value of 206 J/m^. These are in good agreement with the overall average fi'om the T-peel tests, namely 189 J/m^. It is probable that these values agree within experimental scatter. An alternative way to analyse the effect of test geometry is to restrict analysis of data to specific laboratories (as opposed to pooling all the results). Therefore, four laboratories provided pairs of adhesive fracture toughness values obtained by both test procedures. Configuration A was used for the T-peel. For each test geometry values were averaged and results were plotted as shown in Figure 7. A line is drawn in Figure 7 equating to equal value for adhesive fi*acture toughness fi'om each procedure. The data are a reasonable fit to this line indicating that the adhesive toughness is geometry independent. There has often been a suspicion that in the fixed arm peel test it is possible to experience tenting as the laminate is slightly pulled away from the base plate. This would introduce an unaccounted energy term and therefore the adhesive fracture toughness value would be too large. This process can indeed occur, and Laboratory 7 cited some observations of tenting. However, from the overall experience, it is apparent that for the work reported by the seven laboratories that the laminate is adequately stuck to the base plate. It is important to ensure that this happens with regularity. However, the procedure of seeking agreement between a fixed arm and T peel procedure for adhesive fracture toughness can confirm this.
349
The Determination of Adhesive Fracture Toughness for Laminates
50
100
150
200
250
Ga F Peel J/m2 Figure 7 Comparison of adhesive fracture toughness by T-peel and fixed arm peel for data from 4 specific laboratories. Each laboratory has a pair of averaged adhesive fracture toughness values for both tests.
INVESTIGATION OF ENERGY CORRECTION FACTORS DETERMINATION OF ADHESIVE FRACTURE TOUGHNESS.
IN
THE
Preamble The determination of adhesive fracture toughness involves the measurement of a toughness term with elastic deformation corrections and subtracting from this energy term a contribution due to plastic deformation. Therefore, the relative size of the plastic deformation term will govern the accuracy of the adhesive fracture toughness. The correction factor can be determined from this relative size. This is achieved by first defining a "quality coefficient" (q.c) and then by determining the correction: q.c:
{Q:'-Q^'')
(1)
Correction = {[1-qc] x 100}%
(2)
i.e. Correction = K ' - ^ ; x 100}%
(3)
350
D.R. MOORE AND J. G. WILLIAMS
G^^ can never be zero but Ga"^^ can be zero when there is no plastic deformation in the bending or tension of the peel arm. Therefore the qc term will have a maximum value of 1 (no correction) and a minimum value above zero (maximum correction). For example, if q.c. = 0.20 then 80% of the input energy to pull the laminate apart is due to plastic deformation and has to be subtracted from the total input energy in order to calculate the adhesive fracture toughness. Naturally, the smaller the correction (the larger q.c), the more credible the peel data. In practical terms, one hopes for a correction less than about 70% i.e. a q.c value above 0.30. However, such values are merely pragmatic. (These considerations apply equally to the T-peel analysis but can only be determined for each peel arm separately). Results for Polypropylene Laminates The quality coefficient associated with the PP laminate system was about 0.50 for the lower peel angles in Figure 4 and about 0.30 for the higher peel angles. These relate to corrections in the range 50% to 70%. With consideration to the errors associated with measurement and the design of equipment aimed at minimising frictional effects, there is a belief that the adhesive fracture toughness values determined from these data can be considered to be sufficiently accurate. This is further reflected in the comments on comparative results from the two test geometries. Therefore, corrections of 50% are considered to be acceptable and values up to 70% might also be satisfactory. However, as the correction becomes larger, then more doubt is associated with the data. Results from Metallic Laminates Adhesives in electrical applications involve laminates with metal substrates and often have to withstand high temperatures. The laminates are often fairly thin and reasonably flexible; therefore fixed arm and T-peel procedures can be helpfiil in measuring their adhesive strength. For example, a BMI system was bound to copper foil of thickness 25 |Lim. Some fixed arm (where the base plate formed a bond with aluminium) and T-peel results for this system are summarised in Table 2 on 15 mm width specimens of length 150 mm. It is observed from the results in Table 2 that again a common value of adhesive fracture toughness is obtained from either geometry, to within experimental error. The plastic corrections for the data in Table 2 are high. They are 77% and 85% for the Tpeel results and 81% for the fixed arm peel. In addition, the peel forces were low (of the order of 1 N) necessitating the use of a special air bearing system developed for these types of laminates. Consequently, there cannot be a high credibility associated with the adhesive strength measurements for these laminates.
77?^ Determination of Adhesive Fracture Toughness for Laminates
T-Peel Peel Peel Arm2 Arm 1 'GA*(J/m^) GA^'a/m^) Individual arm Overall G A ( J W )
123.7 104.5 19.2
20.0 15.5 4.5 23.7
351
Fixed arm peel (70° peel angle)
116 94 22.0
Table 2 Peel results at 23°C for copper laminates from fixed arm and T-peel tests. Nevertheless, such practical systems need to be investigated in terms of their adhesive strength and in particular in terms of their temperature dependence. TEMPERATURE DEPENDENCE OF ADHESIVE LAMINATES: A PRACTICAL PROBLEM. Despite the difficulties associated with the BMI/copper laminates (as just discussed), an understanding of their adhesion characteristics remains important. In particular there is an interest in the relationship between adhesive fracture toughness and temperature. This can be approached by use of either test geometry. The fixed arm peel procedure can be conducted at different test temperatures. The tensile stress-strain properties of the peel arm can also be measured at these temperatures and adhesive fracture toughness calculated in the usual manner and plotted against temperature. This can be a timeconsuming process that can be overcome by use of a T-peel procedure operating as a temperature scan. The T-peel specimen is tested in a servo-hydraulic Instron whilst at the same time the temperature of the specimen environment is increased. The measurement of the peel force can then be determined as a function of temperature. With knowledge of the tensile stress-strain behaviour of the peel arms (which in the case of copper was independent of temperature up to 100°C) the peel force can be converted into the adhesive fracture toughness, by use of the protocol for conducting a T-peel test (6). One critical aspect of our experimental procedure is to ensure that the rate of peel is commensurate with the Tpeel specimen accommodating the rise in temperature. We selected a test speed that achieved 1 mm of peel whilst the temperature rose by 6^C. Temperature on the specimen was measured at 4 different positions as well as at another position in the environmental chamber. In addition, the use of a digitally controlled servo-hydraulic machine provided the appropriate control of test speed during the test. Our selection of test parameters was pragmatic and further investigations of whether steady state conditions have been established for the peel process would be worthwhile. The adhesive fracture toughness versus temperature results for the peel scan and those based on isothermal T and fixed arm peel are as shown in Figure 8. The agreement between the two methods is encouragingly good, particularly in consideration of the low values for adhesive fracture toughness, the low quality coefficient for the data (about 0.2)
D.R. MOORE AND J.G. WILLIAMS
352
and the small measured peel forces. (The peel force was 1 N at 23°C but reduced to 0.2 N at lOO^'C and then reduced further to 0.03 N at 150T). Consequently, this method is approaching its limit of accuracy for these particular laminates. However, it certainly establishes the economy in conducting a T-peel scan procedure where, with a minimal test time, the complete adhesive fracture toughness -temperature scan can be obtained on a single specimen. 30 25
I
20 CM
I 15 (0
O 10
A <— Scan 1 •
Scan 2
- ^ \ - ^
60
100
160
200
Temperature (degrees C) • T-Peel
• FPeel
Scani—Scan2
Figure 8 Adhesive fracture toughness versus temperature for BMI/copper laminate by Tpeel scan and fixed arm peel procedures Although the temperature scan technique provides an efficient method for monitoring the temperature dependence of adhesive fracture toughness, the data quality remains an issue. We have discussed this for the results at 23^C but the issue increases with rising temperature. However, it is important to recognise that merely reverting to the measurement of peel strength versus temperature would not overcome any of these issues. The accuracy in the method will be higher if the adhesive fracture toughness of the laminate is larger and if the correction factors are not so large.
The Determination of Adhesive Fracture Toughness for Laminates
353
CONCLUDING COMMENTS The determination of adhesive fracture toughness for a polypropylene laminate system has shown that different test geometries (fixed arm and T peel) can produce a common value. In following particular test protocols (6) many of the experimental and analytical difficulties can be overcome. The major correction that is necessary in the determination of the adhesive fracture toughness is the energy associated with plastic deformation. This correction can be quite large and a means of monitoring its level has been suggested. It has not been possible to rigorously define the maximum level of correction that can lead to sensible values for adhesive fracture toughness, although corrections up to about 70% seem to provide adequately accurate values for adhesive fracture toughness. Examples of laminates where the corrections are above 80% have been shown. Even at such levels, the procedures for determination of adhesive fracture toughness provide probably the best approach currently available. However, further investigation of what is an acceptable level of correction is required. This is under investigation by the authors. ACKNOWLEDGEMENT The authors acknowledge the experimental contribution of R S Hardy, ICI pic. REFERENCES 1 Kinloch, A. J., Lau, C.C. and J G Williams (1994), Int. J, Fract 66,45-70 2 ASTM D 1876-95 9 (1995) Standard Test Methodfor Peel Resistance ofAdhesives (T-Peel) 3 ASTM D 1781-93 (1993) Standard Test Methodfor Climbing Drum peel for Adhesives 4 ASTM D 3167-97 (1997) Standard Test Methodfor Floating Roller Peel Resistance of Adhesives 5 Moore, D.R. and Williams, J.G., (2000) Fracture ofPolymers, Composites and Adhesives ed Pavan, A., and Williams, J.G. ESIS Pub 27, Elsevier, ISBN 80437109, 231-245 6 Moore, D.R., Pavan, A. and Williams, J.G. (2001) Fracture Mechanics Testing Methods for Polymers, Adhesives and Composites ESIS pub 28, Elsevier, ISBN 80436897,203-223
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
355
FRACTURE TOUGHNESS OF A LAMINATED COMPOSITE Sharon Kao-Walter, Dept of Mechanical Engineering, Blekinge Institute of Technology, Karlskrona, Sweden, e-mail:[email protected] Per Stahle, Div. of Solid Mechanics, Malmo University, Malmo, Sweden, e-mail: [email protected] Rickard Hagglund, SCA Research AB, Sundsvall, Sweden, e-mail: [email protected] ABSTRACT The fracture toughness of a polymer-metal laminate composite is obtained by mechanical testing of a specimen containing a pre-crack. The laminate is a material used for packaging. It consists of a thin aluminium foil and a polymer coating. A centre cracked panel test geometry is used. Each of the layers forming the laminate is also tested separately. The result is compared with the measured fracture strength of the individual layers. It is observed that the load carrying capacity increases dramatically for the laminate. At the strain when peak load is reached for the laminate only aluminium is expected to carry any substantial load because of the low stiffness of the LDPE. However, the strength of the laminate is almost twice the strength of the aluminium foil. The reason seems to be that the aluminium forces the polymer to absorb large quantities of energy at small nominal strain. The toughness compares well with the accumulated toughness of all involved layers. Possible fracture of the interface between the layers is discussed. KEYWORDS Laminate, aluminium foil, polymer, crack, fracture toughness INTRODUCTION Liquid food packages are often made of packaging materials consisting of different material layers to fulfil several requirements of the package. It is very important to ensure that every layer maintains its function during the forming, filling and transportation processes. As an example here a liquid food packaging material is considered. This is a laminate consisting of LDPE (Low Density Polyethylene) and an aluminium foil (Al-foil). Several studies of different mechanical properties of these materials have been performed [1-5]. It was found in [2] that aluminium foil and LDPE laminated together provide significantly higher stress and strain at fracture as compared with the simplified analytical prediction. Related works can also be found in [6], [7], [8] where notched tensile strength, fracture toughness as well as fatigue resistance for fiber/aluminium composite laminate were studied. The purpose of this work is to study the fracture toughness of a laminated material in relation to the adhesion between the layers. Load and extension were measured for a two-layer laminate specimen with a pre-crack as well as for the individual layers of the laminate. The same specimen geometry was used in all tests. For comparison, measurements were also done for the laminate without any adhesion between the layers.
356
S. KAO-WALTER, P. STAHLEANDR.
HAGGLUND
MATERIAL Laminate in this work consists of Al-foil and Low Density Polyethylene (LDPE). Fully annealed AA1200 Al-foil and LDPE with the product name LD270 is used. Load versus extension were measured for the following materials: Case 1: Al-foil with the thickness 8.98 |Lim. Case 2: LDPE with the thickness 27.30 |Lim. Case 3: Al-foil / LDPE - Al-foil coated by LDPE and the total thickness is 36.28 jLim. Case 4: Al-foil // LDPE - Al-foil and LDPE joined together without adhesion between the layers and the total thickness is 36.28 |Lim. Here "/" applies to two material layers bonded together and "//" applies to two layers put together without any adhesion in between. For case 3, pieces of Al-foil were cut from a roll of fabricated material as shown in Fig. 1(a). The laminate in case 3 was then prepared in a Haake film extruder with a 36 fxm Polyester (PET) as carrier (see Fig. 1(b)). The foil was mounted on the PET carrier while LDPE was extruded and coated on Al-foil at a melting temperature of 278 °C. A nip with the pressure 202 bar was used to press the layers together. The laminated specimen for case 3 was than cut from roll including PET/Al-foil/LDPE as shown in Fig. 1(c). After producing the material for case 3, LDPE was continuously extruded on the PET carrier under the same conditions but without the Al-foil. By peeling off the PET carrier, the LDPE produced here was used for making the specimen for case 2 and case 4. And the specimen of case 1 is taken from the Al-foil roll of the same direction as the other cases.
Extruder
I'
Melted LDPE
LDPE/Al-foil/PET
* PET carrier
Al-foil
(b) Fig.l. Schematic description of specimen preparing.
(a)
Fracture Toughness of a Laminated Composite
357
EXPERIMENTAL METHOD Centre cracked panels as shown in Fig.2(aj are used for evaluating the fracture toughness of the laminated composite and components of it. Pre-fabricated cracks are manually cut, using a razor blade, to a total slit length of 2a = 45 mm. The width and gauge length of the specimens was 2W= 95 mm and 2/i = 230 mm, respectively. A pair of wide clamps is utilised, see Fig.2(b). The tests are made in a MTS Universal Testing Machine. The upper clamp is attached to a 2.5 kN load cell as well as a crosshead in the MTSmachine. Since the specimen is mounted vertically, the clamps are equipped with needles to facilitate a correct positioning. After the positioning of a sample the upper and lower clamps are closed and the pressure is applied by tightening four equally spaced quick-acting locking nuts along the front of each clamp, see Fig.2(b). Locking pins at the centre of the front jaws keep the clamps in an open position during mounting. The largest sample that can be accommodated is 420 mm wide. Specimens are tested by traversing the upper crosshead up to move the sample under increasing tension w at a constant crosshead speed of 9.2 mm/min. The software TestWorks is used to control the load frame and also to record data. During testing both displacement between the crossheads and load is monitored and recorded. All tests are run until the entire cross section has fractured.
K/2
\ >
7h la
>i
vfl (aj. Centre cracked panel.
(b). Set-up for fracture mechanical testing of laminated composites. The specimen shown here is case 4. Fig. 2. Specimen and experiment set up.
358
S. KAO-WALTER, P. STAHLE AND R. HAGGLUND
ANALYTICAL APPROACH ELASTIC BEHAVIOUR OF THE SPECIMEN The force elongating the specimen may be separated into that of the unbroken specimen, Po, and the reduction, Pc, due to the presence of a crack. The former is calculated as follows: ,
p . - ^
(1)
h and the latter is obtained from the energy, Uc, released during the cutting of a crack [6]:
(2)
U,=^JlKfda' . The stress intensity factor for the crack is given by uF ^i=^=
(3)
(Ka/W).
This gives the energy i/W
1 „ u^EWt ""^^ U,=i-uP,= ^'^"' ](/>{TyiT. 2 4/1(1-v') 0
(4)
Using ^ given in [9] the integral of (4) is found to be 0.08 for a/W= 0.47. With W= 47.5 mm and h=ll5 mm (1), (3) and (4) give X. pX. = p-p
..
=
t^EWt
^
1+
a/W
J-
2(1-v^)
mT)dT
r
(5)
giving P =0.92i^^. h
(6) ^
Thus, the stiffness of specimen decreases only 8% as compared with a corresponding unbroken specimen. CRACK TIP DRIVING FORCE G According to the theory of fracture mechanics, a crack tip driving force G is defined as the rate of change in potential energy per crack area and per unit of length of crack front [10, 11]. It is assumed that the critical driving force, Gc, of a crack in a single layer is constant and independent of whether the layer is bonded to other layers or not. Assuming that no delaminating occurs during the growth of a crack in the laminate and that energy is not dissipated in the material, the crack tip driving force for the laminate is the accuraulated driving force for all layers, i.e.
359
Fracture Toughness of a Laminated Composite
(7)
G = ^^l^^^ h +h
Here the assumption that G = Gc as crack growth criterion is examined. This criterion is valid if the crack grows in an approximate steady state. The load at onset of crack growth is usually regarded to define the fracture toughness. For materials with considerable toughening the incipient growth of the crack leads to increased fracture toughness. At small scale yielding the maximum toughness may be of interest but in the present analyses the yielding is considerable. There is a difficulty to define onset of crack growth since the observation is that the crack grows in the Al-foil until the crack traverses the entire specimen before there is any substantial crack growth in the LDPE layer. PEAK LOAD Fm AND DISSIPATED ENERGY U For practical use the maximum load carry capacity may be limiting the reliability of the packaging structure. Therefore it is interesting to compare peak load both for each material separately and as a laminate. It is also interesting to examine the energy dissipated before breaking is reached. The dissipated energy, U, is found from the load displacement curve, P(S) as follows
"^r
dP
0
^^1\dS<0
S'
(8)
The second term represents the recovered elastic energy stored in the specimen. At the point marked with triangles (see curves from case 3 and case 4 in Fig.4) the crack has grown in the Al-foil and traversed the entire specimen whereas the crack in LDPE in most cases did not grow at all. The average strain at peak load is still in the elastic regime for the LDPE material. A first assumption would therefore be that the entire energy is due to the fracture of the aluminium and that the LDPE does not contribute to the fracture toughness. The motivation for the interest in only the dissipated part of the energy is that the elastic part may be recovered which means that accumulated energy, e.g. a Shockwave may temporarily overcome the barrier that the elastic energy represents. If this results in crack advance the recovered energy is restored which may lead to rapid crack growth. RESULTS AND DISCUSSION CALIBRATION OF LOAD CELL A load cell of 2.5 kN has to be used because of the relatively large weight of the clamp. Prior to testing, the 2.5 kN load cell was calibrated and accuracy was investigated. The result for the comparison with 100 N load cells is shown in Fig. 3. It can be seen that the deviation is around
360
S. KAO-WALTER, P. STAHLEANDR.
HAGGLUND
Dev. of measured value from 2.5 kN load cell
0,08-
? c o .2
•a
0,060,040,02-
A 1
P 4 j % l ^ ^ t 1 f i i i i A 4.A4A A ^ A.4
0 -0,02 ^) -0,04 -
10
?0
30
1
4b
4 Load (N)
Fig.3. Difference in % between measurements of a 100 N load cell and the 2.5 kN load cell that is used in the experiment.
0.03% for loads larger than 5 N and in the interval 1 N to 5 N the error is less than 0.08%. MEASUREMENTS Measurements are made on a centre cracked panel as described above. The specimen is loaded via the clamps attached to the ends of the specimen. The test speed is 9.2 mm/min that gives a constant strain rate of 4% per min according to ASTM standard [12]. For each curve, five specimens are tested in room temperature. The averaged curves for the four materials are displayed in Fig.4. This figure shows the force versus extension diagram for the single layers together with the laminate. For each case, the specimen was extended until the crack growth through the entire specimen and break to two peaces. For case 3 (Al-foil coated by a LDPE layer), it can be found that the maximum load value is the highest of all four cases. It was observed that the crack grows only in the Al-foil layer until the curve comes to the point that marked with a triangle where the crack in Al-foil traverses the entire specimen. Similar observations have been made for case 4 in the beginning of the curve. After passing the triangular point where crack in Al-foil traverses the entire specimen, the curve follows the curve of case 2 very well since there is only a single layer of LDPE remaining. By experiment results maximum force F^, dissipated energy before the onset of fracture Am, total dissipated energy Atot as well as estimated stiffness dF/d^ in the elastic region are calculated and shown in Table 1. From these values and the diagrams in Fig.4, the results can be analyzed and discussed in the following.
Fracture Toughness of a Laminated Composite
361
- ^ K - l.Al-foil
-•-
T3 10
2.LDPE
— 1 — S-AVLDPE 4.Ay/LDPE
25 7
extension (mm)
fbj Fig. 4. Load versus extension for centre cracked panels. Diagram in faj shows the resuhsfor one of the 5 tests from easel to case 4. The standard deviation of the peak load is shown on the curve for each case. Picture above was taken when the load for case 3 reaches the peak load. Diagram in (h) shows the detail result at extension from zero to 5 mm.
The Stiffness dF/de may be used together with (6) to predict the modulus of elasticity. The estimated modulus of elasticity is 9.3 GPa for Al-foil and 92.9 MPa for LDPE. The modulus of elasticity for aluminum is much smaller than expected (34 GPa in [2]). On the other hand, the modulus of elasticity for LDPE is rather close to the value from the earlier measured result (136 MPa in [2]). The measured displacements in the elastic region are therefore very uncertain and the stiffness results cannot be regarded as reliable.
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S. KAO-WALTER, P. STAHLEANDR.
HAGGLUND
Table 1. Calculated values. F^ is the maximum force, A^ is the area under the curve from zero to maximum force. Atot is the total area under each curve ofFig.4. dF/de is the stiffness in elastic region Max and min refers to the largest and the smallest values obtained among the five tests.
1. Al-foil 2. LDPE 3. Al/LDPE 4. A1//LDPE 14.2 10.0 22.5 12.9 10.3 15.8 23.5 16.3 -* m max F 12.0 20.2 9.8 10.2 '• m min 114 32.7 8.1 6.3 Am [N mm] 118 8.9 37.6 7.7 ^mmax 111 6.3 26.7 5.7 ^m min 257 247 238 13.6 Atot [N mm] A 264 273 331 15.5 ^^tot max 252 191 253 10.1 ^tot min 0.97 31.1 dF/de [N/mm] 25.6 25.3 1.0 45.4 42.4 39.4 dF/de max 0.9 23.2 14.1 15.0 dF/de min Cases Fm[N]
However for case 4, the peak load was found to be the sUghtly larger than peak load for the Alfoil alone. The peak load 14.2 N may be compared with the peak load of aluminum 12.9 N, the elongation at peak load of aluminum is 0.84 mm and the estimated stiffness in the elastic regime of LDPE is 0.97 N/mm. The latter gives an expected peak load of 12.9 + 0.84 x 0.97 = 13.7 N which is within 5% of the measured result (14.2 N). For case 3, the laminate of LDPE and Al-foil joined together display a much higher peak load and larger extension at peak load. The value of this peak load of 22.5 N was found to be very close to the summation of peak load of Al-foil (12.9 N) and peak load of LDPE (ION). During the test, small scale delamination between the layers was observed. The area under the load deflection curv^, i.e., the energy required, is observed to be very large (32.7 N mm) compared with the energy required to break the Al-foil (13.6 N mm) and stored in the LDPE at the corresponding strain (3.0 N mm). Direct visual inspection of Fig. 4 shows that, an astonishingly much larger amount of energy is consumed in the laminated material at moderate tension. The reason for this could be that the extension of LDPE requires multiple fracture of the Al-foil or delaminating. Multiple fracture of the Al-foil would consume a large amount of energy. Examination of the total energy consumed during complete fracture of the specimens for case 1, 2, 3 and 4 reveal that almost the same total energy is consumed at complete fracture irrespective of if the layers are bonded together or not. Here, energy required to break Al-foil is 13.6 N mm and 257 N mm for the LDPE layer. This may be compared with the energy, 238 N mm, required to break the laminate (case 3) and 247 N mm, required to break both layers of material in case 4. It is believed that less energy is consumed during the fracture of the LJDPE in the laminated case because the straining of the LDPE is concentrated to the thin gap that is defined by the broken Al-foil. The energy for onset the fracture in the Al-foil (Am = 6.3 N mm) is certainly consumed already when the laminate has reached its peak load (compare to ^^Ai/LDPE _ 22 7 N mm). However, unexpectedly also the energy to break the laminate is almost entirely consumed at small straining of the specimen. The reason for this has to be sought in the mechanics of the fracture process region.
Fracture Toughness of a Laminated Composite
363
The almost equal energies in case 4 and 3 rules out the hypothesis of presence of additional dissipative processes in the laminate as an explanation for the much higher toughness of the laminate. One observes also that more energy is consumed at small strains and, hence, less at larger strains. Further the peak load for the laminate is almost the same as the sum of the peak load for the Al-foil and the peak load for the LDPE layer. This suggests that the fracture processes distribute strain so that peak load occurs simultaneously in both materials. This is anticipated for a laminate with little delamination. The assumption is that both materials reach peak stress in a small process region in the vicinity of the crack tip. CONCLUSIONS An experimental investigation was performed on a laminate and separately on the individual components of the laminate. Peak load, energy dissipation at onset of fracture and at fast fracture were investigated. Fracture toughness cannot easily be defined because of very large scale yielding. Therefore an alternative toughness measurement was proposed. The following observations were made: - Onset of crack growth was observed to occur approximately at peak load. In the laminate fracture of LDPE occurred after completed failure of the Al-foil. -Peak load for the laminate is almost the same as the sum of the peak load for the Al-foil and the peak load for the LDPE layer. This suggests that both materials reach peak stress in a small region in the vicinity of the crack tip. -The total energy needed to complete fracture of the ligament in case 3 and case 4 are almost equal. The result rules out the hypothesis of significant additional dissipative processes in the laminate as an explanation for the much higher toughness of the laminate. -The energy required before onset of fracture is unexpectedly large and around five times larger than for the separate Al-foil layer. The last observation indicates a very important enhancement of the fracture toughness of the laminate that calls for further investigations. The finding is expected to be exploited in future composition of a broad range of laminates. ACKNOWLEDGEMENT This work was granted by Tetra Pak R&D AB, Tetra Pak Carton Ambient AB, SCA Research AB and the Swedish KKs-Foundation. The authors would also like to thank Mr. K. Asplund and Mr. E.M. Mfoumou for assisting during the experimental work.
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REFERENCES [I] [2]
[3 ] [4] [5] [6]
[7]
[8] [9] [10] [II] [12]
Kao-Walter, S. and Stable, P. (2001) "In situ SEM study of fracture of an ultra thin Alfoil - modelling of the fracture process", SPIE 3rd ICEM proceeding. Kao-Walter, S., Dahlstrom, J. Karlsson,T. and Magnusson, A. (2002) "A study of relation between mecbanical properties and adbesion level in a laminated packaging material", accepted by Mech. of Comp. Mat., Vol.39, Issue Nr.5-6, 2003. Lau, C. C. (1993) "A Fracture Mechanics Approach to the Adhesion of Packaging Laminates", Doc. Thesis, Imperial College of Science, UK. Tryding, J. (1996) "In Plane Fracture of Paper", Doc. Thesis, Division of Structural Mechanics, Lund Institute of Technology, Sweden. Kao-Walter, S. and Stable, P. (2001) "Mechanical and Fracture Properties of Thin Alfoil". Research report, Blekinge Inst, of Tech., 2001:09. Macheret, J. and Bucci, R.J. (1993) "A crack Growth Resistance Curve Approach to Fiber/Metal Laminate Fracture Toughness Evaluation", Eng. Frac. Mech. Vol.45, N0.6, pp.729-739. Gregory, M.A. and Roebroels, G.H.J.J. (1991) "Fiber/metal laminates: a solution to weight, strength and fatigue problems", 30^^ Ann. Conf. of Metallurgical Society of CIM, Ottawa, Canada. Wells, J.K. and Beaumont, P.W.R. (1987) "The prediction of R-curves and notched tensile strength for composite laminates", J. Mater. Sci. 22, 1457-1468. Isida, M.(1971) "Effect of Width and Length on Stress Intensity Factors of Internally Cracked Plates Under Various Boundary Conditions", Int. J. Fract. Mech. Vol.7, p.301 Anderson, T. (1995) "Fracture Mechanics: fundamentals and applications", 2^^ ed. CRC Press, USA. Irwin, G.R.(1948). "Fracture Dynamics" Fracture of Metals, American Society for metals, Cleveland. ASTM (1991) "Standard Test Methods for Tensile Properties of Thin Plastic Sheeting", D882-91.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
355
COMBINATORIAL EDGE DELAMINATION TEST FOR THIN FILM ADHESION CONCEPT, PROCEDURE, RESULTS Martin Y.M. Chiang, Jianmei He^ Rui Song^, Alamgir Karim, Wen-li Wu, Eric. J. Amis Polymers Division National Institute of Standards and Technology Gaithersburg, MD 20899 USA
ABSTRACT A high-throughput combinatorial approach to edge delamination test is proposed to map the failure of adhesion as a function of both temperature and film thickness in a single step. In this approach, a single specimen of a thin film bonded to a substrate with orthogonal thickness and temperature gradients is subdivided into separate samples. This approach can be adopted to measure the adhesion for films with thickness in the sub-micron range by the addition of an overlayer. Requirements for valid testing results from a mechanistic viewpoint are analyzed using three-dimensional computational fracture mechanics. An initial test result is presented to demonstrate the feasibility of the approach. Keywords: Combinatorial approach, adhesion, interfacial debonding, thin film, edge delamination, fracture mechanics, finite element
^ Current Address:
National Space Development Agency of Japan (NASDA) Tsukuba Space Center, 2-1-1 Sengen, Tsukuba-shi Ibaraki-ken, 305-8505 Japan
^ Current Address:
Institute of Chemistry, the Chinese Academy of Sciences, Beijing, China
M.YM. CHIANGETAL
366 INTRODUCTION
The objective of this study is to develop a combinatorial (or multivariant) approach based on edge delamination to investigate the adhesion between a thin film and a substrate. This technique is expected to provide information about the interface integrity and is not a substitute for a fundamental adhesion measurement (in this study, the adhesion means the debonding energy or the fracture toughness) [1-3]. More importantly, the proposed combinatorial approach can also be extended to measure the adhesion for films with thickness in the sub-micron range, where the measurement of the adhesion for such thin films is very difficult. The application of a combinatorial approach, which originally aimed at speeding synthesis and screening of large composition libraries for drugs and functional materials, has enabled researchers to quickly evaluate how variables influence chemical and physical properties of materials and rapidly screen for optimal material properties [4-7]. In this study, a three-dimensional finite element analysis was developed to evaluate the feasibility of the combinatorial approach outlined below.
Apply Cooling Initial Debonding at Free Edge
Free Edge
Further Debonding at Critical Temperature
Edge Delamination
Fig. 1 A schematic of the free edge effect and edge-delamination test. During the cooling of a bi-material film/substrate system with an initial interfacial crack at a stress-free edge (Fig. 1), a further crack extension (debonding) ^'^' ^ooooooooooooooooo u along the interface will occur at a critical temperature due to the stress concentration near the crack tip. The edge delamination test is based on this debonding mechanism Fail S -20 using the thermal stress generated during the cooling to cause separation of the film from the substrate. Accordingly, the 0 100 200 300 adhesion (or adhesive strength) between Coating Thickness (|a,m) the film coating and the substrate can be deduced (e.g., [8]). By repeating the test Fig. 2 A schematic of the failure map of for numerous samples with different film a film on a substrate as a function of thicknesses, a failure map as a function of temperature and coat thickness. temperature and film thickness can be constructed [9]. This failure map provides a tool to assess the reliability limit of the coating as a function of multiple independent variables. For industry, to construct a comparable failure map to screen numerous new formulations and specialty materials is time-consuming. In this study, we propose to combine the important variables (temperature and thickness) in a single experiment to map the interfacial failure of the film. Subsequently, the value of the adhesion of the film to substrate
Combinatorial Edge Delamination Test for Thin Film Adhesion
367
can be deduced from the failure map if the internal stress-temperature relationship of the film is known. Essential details in the test design and requirements for valid testing results will be described in the next section of the paper. Numerical results and discussion, as well as an initial test result will be presented in the following sections, and some conclusions will be drawn based on the results. For more information on the combinatorial delamination test and experimental procedure, readers may refer to our forthcoming publications [7,10,11].
COMBINATORIAL EDGE DELAMINATION TEST In the proposed experiment, a film is coated onto a relatively rigid substrate in such a way that the film has a thickness gradient in one direction (Fig. 3a). The film is scribed in a form of a square grid pattern to form an array of individual edge delamination samples on the substrate (Fig. 3b). The cut penetrates some distance into the substrate also. The edges are at 90° to the interface of the film/substrate. The depth (d) and width (w) of the cut are the design parameters that need to be optimized and will be discussed later (Fig. 3c). Due to the existence of residual biaxial stresses during the solidification of the film and the stress-free edges (a) Bonded Film after dicing in a bi-material With Initial Crack system, stress concentrations Temperature arise at the interface near the free edges. These stress concentrations are sufficient to create small initial interfacial flaws at the film/substrate boundary. This is the wellknown free-edge effect that is unique to bi-material systems [12-15]. Coupled with an interface having finite adhesion strength, these initial flaws are the nucleation sites for film interfacial debonding after a substrate further loading. To introduce wedge the further loading, the specimen is cooled with a (C) temperature gradient applied in the direction orthogonal to the Fig. 3 A schematic of the combinatorial approach to the thickness gradient (Fig. 3a). edge delamination test: the multivariant specimen with film Interfacial debonding events thickness and temperature gradients, and final failure map will be observed for those (a); a square pattern array of individual edge delamination samples having critical stresses samples on the substrate (b); the cutting depth, d, and width, that depend on the combination w(c). of local temperature and film thickness. Consequently, a failure map as a function of temperature and film thickness can be constructed with one step, as shown in Fig. 3a. In principle, if the adhesion of a film to a substrate is independent of temperature, the adhesion
368
M.Y.M. CHIANGETAL
can be deduced from this failure map as long as the thermo-mechanical property (the stresstemperature relation) of the test film is well characterized [9,16]. Sometimes, especicilly for large film thickness, the residual stresses (or internal energy) resulting from the solidification of the test film on a substrate (the film preparation step) could be large enough to cause premature failure of the film or interface before further cooling in the edge delamination test [9]. Conversely, it could be the case that a film has such a strong bond with the substrate that the stress concentration generated during the cooling process is insufficient to induce debonding. In either case, by adjusting the film thickness and the upper and lower limits of temperature, one can ensure that the debonding condition will be met. To measure the adhesion for films with thickness in sub-micron region, instead of coating a test film with a thickness gradient (Fig. 3a), a test film with a very small constant thickness is coated onto the substrate. Then, an overcoating layer (stress-generating layer) with a thickness gradient is deposited on the top of the thin test film (Fig. 4). The rest of the experimental procedures are identical with the original one. The thickness of the overcoating layer needs to be much larger than that of the test film such that the debonding Temperature^ energy contributed from the test film during the thermal Stress generating cooling can be layer neglected, and only the overcoating layer serves as the stressTest film generating layer. One assumption in this modified approach is that the bonding Fig. 4 A schematic of the combinatorial approach to the modified strength between the edge delamination test: the multivariant specimen with constant film test thin film and the thickness, overcoating layer thickness and temperature gradients. overcoating layer is much higher than the bond between the test film and substrate. In this case, the stress-temperature relation of overcoating is only needed to calculate the adhesion. Consequently, one may use this modified combinatorial approach to obtain the critical bond energy for the thin film in the sub-micron range. It is worthwhile to note that once a well-characterized overcoating layer has been chosen, it can be used as a standard overcoating layer for different test films as long as good adhesion exists between the overcoating and the test film. For the test results to be valid in this combinatorial edge delamination test, the stress state at the crack tip in each individual square sample must be independent of interfacial crack length. This requirement arises because the initial interfacial crack length in each sample cannot be well controlled. A second condition for validity of the test is that there must be no stress interaction among the separate edge delamination samples within one combinatorial specimen. Thus, two issues that must be determined to ensure valid test results are: the minimum initial crack length such that the stress states are independent of crack length (Fig. la), and the required cutting depth and width ("d" and "w" in Fig. 3c) so that the stress interaction
Combinatorial Eldge Delamination Test for Thin Film Adhesion
369
among separate edge delamination samples is negligible. A three-dimensional stress analysis (finite element method) and computational fracture mechanics were used to provide answers to these two important questions. RESULTS AND DISCUSSION The commercial finite element program, Abaqus [17], was used to calculate the stress distribution in an edge delamination sample. A fully three-dimensional model of the combinatorial edge delamination specimen was constructed for the finite element analyses (FEA). For clarity, some of the FEA results and schematics are presented as two-dimensional configurations in this paper (e.g.. Fig. 1). The film and substrate were assumed to be linearly elastic. The ratio of the film stiffness to the substrate stiffness was assumed to be 1/100 to reflect the relative rigidity of the substrate. This ratio also represents a typical organic overcoating on silicon substrate. The Poisson's ratios of the film and the substrate were assumed to be the same. The ratio of the , • • • coefficient of thermal expansion • (CTE) of the film to the substrate • was assumed to be 10. An initial • crack was introduced along • 11 Steady State^ • ^ 2.2 film/substrate interfaces to mimic 0) • the initial flaws; the length of this N initial crack was varied to • determine the region where the O crack-tip stress is independent of • • crack length. The adhesion between the film and the substrate is assumed to be temperature independent. Normalized Crack Length (a/h^) Fig. 5 shows the variation Fig. 5 The variation of the stress normal to the of the stress normal to the interface at the crack tip with the initial crack length. interface at the corner of the The stress is normalized by the applied stress. sample as a function of the initial crack length. This normal stress is the driving force for interfacial debonding. The corner is where two interfacial cracks meet in the edge delamination sample, so the stress concentration is somewhat higher, and the cracks tend to propagate from the corner inward. The results in the figure indicate that the stress at the corner achieves nearly a steady state if the crack length is larger than 4 % of the film thickness. Thus, once the initial crack lengths of the individual edge delamination samples in the proposed combinatorial specimen are more than 4 % of the film thickness, the stress states would be only a function of temperature and film thickness. This requirement is not a significant barrier for using the proposed combinatorial approach since the initial debonding caused by the free-edge effect —
,,,,
1—
,
1—
,
— , —
,,,,...,
_,—
' Certain commercial computer code is identified in this paper in order to specify adequately the analysis procedure. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology (NIST) nor does it imply that they are necessarily the best available for the purpose.
M.YM
370
CHIANGETAL
after dicing is typically longer than 4 % of the film thickness. Chemical etching could also be used to obtain a suitable initial crack length if necessary. Once the dicing penetrates into the substrate to form the array of individual edge delamination samples, it would create a wedge near 90° as shown in Fig. 3c. This wedge will induce a stress concentration during the thermal cooling process due to the existence of a geometric discontinuity. The stress concentration could interfere with the stress state at the film/substrate interface. Also, the stress concentration at different wedges can interact with one another if the cutting width (Fig. 3c) is not large enough. This interaction could also translate into the interface and compound the stress states at the interface. Therefore, in order i o make the stress-state at the interface in each sample independent, the geometry parameters (w and d in Fig. 3c) have to be optimized. Fig. 6 shows the variation of the normal stress (a) at the corner with cutting w idth (w) as a function of cutting depth (d). The results clearly demonstrate that when the cutting depth is greater than or equal to 50 % of the film thickness, the normal stress at the comer is independent of cutting width. Experimentally, one would like to have the cutting width as small as possible in order to accumulate many samples on a single combinatorial specimen. Therefore, the results suggest that the cutting J depth is a critical geometric parameter, which should be greater c« 26 \ than half of the film thickness. In the Co - • - d / h f = 0.1 1 figure, the deviation of the normal . ^ f ^ d/hf = 0.3 J N 24 stress from a steady state (for d/hf = 13 - • — d/hf > 0.5 1 0.1 or 0.3) implies that there is some o influence on the stress state at the interface when the adjacent samples are close together. The difference in 1 1 1 , , , the magnitude of the steady-state 3 stress is evidence of the effect of the w/h. stress concentration from the cutting wedge on the interfacial stress state. Fig. 6 The variation of the normal stress at the ^ ^ • • ~ ~ "
'^22
corner with the cutting width (w) as a function of the cutting depth (d).
(b)
Liquid Nitrogen Thickness Gradient 40 |im - 220 |im
Fig. 7 The initial test result of the combinatorial edge delamination test for PMMA adhesion to silicon substrate.
Based on these results for the geometric requirements, we prepared a combinatorial specimen using silicon as the substrate, PMMA as the test film and commercial epoxy as the overcoating layer. The test film thickness was 10 nm. The overcoating layer thickness varied from 40 |im to 220 |im (Fig. 7 (a), the standard uncertainty is 5 |im). The contrast in the photograph of the figure is due to the reflected light. Next, one side of the specimen, was dipped nto the liquid nitrogen (Fig. 7(b)) for 15 min to form a temperature gradient from -
Combinatorial Edge Delamination Testfor Thin Film Adhesion
371
180 °C to -120 °C (the standard uncertainty is 2 °C). Finally, interfacial debonding for those edge delamination samples having critical stresses can be observed by eye (Fig. 7(c)). CONCLUSIONS A three-dimensional finite element modeling with fracture mechanics has been carried out to demonstrate the feasibility and design the experimental protocol for the combinatorial edge delamination test for thin film adhesion measurement. By combining variables that are important and readily controllable in practice (temperature and film thickness), the effect of stress concentration on the debonding of the film from the substrate is spatially varied in one experiment. Consequently, the failure map of the adhesion as a function of both film thickness and temperature can be constructed in a single step. This map of adhesion reliability can be used to determine the critical bond energy of the thin film in sub-micron thickness range. The approach is expected to provide accurate results because of its larger sampling space. Necessary geometry parameters affecting debonding at the film/substrate interface are defined, and the validity of this combinatorial approach is successfully demonstrated in this study. An initial test result indicates this approach is very promising. ACKNOWLEDGMENTS The authors gratefully acknowledge Dr. Carl Schultheisz for valuable suggestions during the preparation of this manuscript. REFERENCES: [I] Mittal, K.L. (1980) Pure and Applied Chemistry, 52, 1295. [2] Mittal, K.L. (1987) J. of Adhesion Sci. and Tech., 1, 247. [3] Buchwalter, L. P. (2000) /. of Adhesion, 72, 269. [4] Zhao, J. C. (2001), Advanced Engineering Materials, 3, 143. [5] Amis, E. J., Sehgal, A., Meredith, J. C, Karim, A. (2001) Abstracts ofpapers of the American Chemical Society 221:70-BTEC, Part 2. [6] Amis, E. J. (2001) Abstracts of papers oftheAmer. Chem. Soc. 222:339-Poly, Part 2. [7] Chiang, M.Y.M., Wu, W. L., He, J., Amis, E.J. (2003) Thin Solid Films, in press. [8] Farris, R. J. and Bauer, C. L. (1988) J. of Adhesion, 26, 293. [9] Shaffer, E. O., Townsend, P.H. and Im J., (1997) ULSIXII, MRS. [10] Chiang, M.Y.M., Song R., Crosby, A. J., Karim, A. and Amis, E. J., *' The combinatorial approach to the thin film adhesion as a function of film thickness and surface energy," in preparation. [II] Song R., Chiang, M.Y.M., Crosby, A. J., Karim, A. and Amis, E. J., " The combinatorial Measurement on the Adhesion of PMMA Thin Film in Nano Range," in preparation. [12] Bogy, D. B. (1968) J. of Applied Mechanics, 35, 460. [13] Ting, T. C. T. and Chou, S. C. (1981) Int, J. Solids Structures 17, 1057. [14] Stolarski, H. K. and Chiang, M.Y.M. (1989) Int. J. Solids Structures 25, 75. [15] Ting, T. C. T. (1996). Anisotropic Elasticity - Theory and Applications, Oxford University Press, Oxford. [16] Thouless, M. D., Cao, H. C. and Mataga, P. A. (1989) J. of Materials Science, 24, 1406. [17] ABAQUS (2000). Finite Element Analysis Code and Theory (Standard and CAE), Version 6.2, Hibbitt, Karlsson & Sorensen, Inc., RI, USA.
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
373
BOND PARAMETERS AFFECTING FAILURE OF CO-CURED SINGLE AND DOUBLE LAP JOINTS SUBJECTED TO STATIC AND DYNAMIC TENSILE LOADS
K. C. SHIN and J. J. LEE
Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Kusong-dong, Yusong-gu, Taejon-shi 305-701, Korea
ABSTRACT Generally, static and dynamic joint strengths of a lap joint are dependent on the surface roughness. Because the composite laminate has a different stiffness with respect to the stacking angle, it is also important to consider the dependence of the strength of a lap joint on the stacking sequence of the composite laminate. And, manufacturing pressure in the autoclave during the bonding process affects the wetting behavior of the resin onto the surface of the steel adherend. In this paper, bond parameters, namely surface roughness, stacking sequence, and manufacturing pressure, affecting failure of the co-cured single and double lap joints with steel and carbon fiber-epoxy composite adherends were investigated through static and dynamic tensile tests. Systematic failure mechanisms of the co-cured single and double lap joints were explained. Tensile load bearing capacities of the co-cured single and double lap joints were calculated using stress distributions obtained through finite element analysis considering residual thermal stresses and then compared to those of the experimental results. KEYWORDS Co-cured lap joints, surface roughness, stacking sequence, manufacturing pressure, tensile load bearing capacity, fatigue characteristics.
INTRODUCTION Carbon fiber-epoxy composite materials have been used in advanced engineering structures such as spacecraft, aircraft, automobile transmission shafts and robot structures because of their high specific stiffiiess and high specific strength [1, 2]. Since conventional metal alloys are To whom correspondence should be addressed. Phone: (82) (42) 8693033; Fax: (82) (42) 8693210; E-mail: [email protected]
K.C. SHIN AND J J. LEE
374
most widely used in engineering structures compared to advanced composite materials, the joining of polymeric composite materials to metal alloy is very useful for the manufacturiag of various engineering structures. Generally, the efficiency of the composite structures is largely dependent on the joint used rather than the structure itself [3,4]. The co-cured joining method, which is regarded as an adhesively bonded joining method, is an efficient joining technique because both the curing and joining processes for the composite structures can be achieved simultaneously [5]. Despite several advantages of a co-cured joining method, only a few related studies on the co-cured joint are available [5-15]. Generally, static and dynamic joint strengths of a lap joint are dependent on the surface roughness [16-18]. Because the composite laminate has a different stiffness with respect to the stacking angle, it is also important to consider the dependence of the strength of a lap joint on the stacking sequence of the composite laminate [19]. Manufacturing pressure in the autoclave during bonding process also affects the wetting behavior of the resin onto the surface of the steel adherend.
J
J
T
Temperature Curve
/
7
Pressure Curve
1.0 MPa
J /
Pressure Curve 0.7 MPa
|7
Pressure Curve
)0
\
\
1
\
o".4MP'a
150
.
1
Vl
200
Time ( min )
Fig. 1. Cure cycle for the manufacturing process of the co-cured joint.
Fig. 2. Photograph of the co-cured lap joint specimens, (a) Co-cured single lap joint; (b) co-cured double lap joint.
In this paper, the manufacturing process of the co-cured lap joint with steel and carbon fiberepoxy composite adherends was introduced and specimens of the co-cured single and double lap joints of the plate type were fabricated and tested under the static and dynamic tensile loads. Bond parameters, namely surface roughness, stacking sequence, and manufacturing pressure, affecting failure of the co-cured single and double lap joints with steel and carbon fiber-epoxy composite adherends were investigated through static and dynamic tensile tests. Based on the test results, systematic failure mechanisms of the co-cured single and double lap joints were explained. Static tensile load bearing capacities of the co-cured single and double lap joints were calculated using stress distributions obtained through a finite element analysis considering residual thermal stress and then compared to those of the experimental results. SPECIMEN FABRICATION AND EXPERIMENTAL PROCEDURE The steel adherend should be carefully machined by a grinding machine and surface treatment of the steel adherend using different sand papers was performed to improve the joint strength. After the abrasion of the steel adherend, contamination caused by dust, an oxide layer or oil on the adherend was eliminated by cleaning with acetone. After cleaning the bond area, the composite adherend, which was fabricated by stacking with composite prepregs, was immediately bonded to the steel adherend to prevent any oxide layers forming on the steel adherend. Before curing completely the co-cured joint in the autoclave, the uncured composite prepreg adherend is bonded lightly (i.e. pre-bonded) to the steel adherend to maintain position.
375
Bond Parameters Affecting Failure of Co-Cured Single and Double Lap Joints
using the resin on the uncured composite prepreg adherend which will be completely cured in the autoclave during undergoing high temperature. Resin in the uncured composite prepreg adherend is viscous and plays a role of an adhesive during curing process. Then, the uncured co-cured joint is completely cured under 0.7 MPa pressure, using the manufacturer's recommended cure cycle in an autoclave. Co-cured joints should be cured without a resin bleeder and peel ply to prevent the excess resin from bleeding because the excess resin plays the role of an adhesive. Figure 1 shows the cure cycle for the co-cured lap joints and table 1 shows the material properties of the carbonfiber-epoxycomposite (USN 150) produced by SK Chemicals (Suwon, Korea). After the curing and bonding processes, the co-cured joint should be finished using various abrasive sandpapers to obtain a better joint strength by eliminating sharp edges. A complete co-cured joint is composed of two adherends and a resin layer of about 5 to 25 [im thickness. We measured the thickness of the resin layer with the variation of the manufacturing pressure through a microscope of 2,000 magnifications. In the case of the co-cured double lap joint specimen, a Teflon block was used to prevent steel adherends from bonding to each other. Figures 2 and 3 show co-cured single and double lap joint specimens whose shape and dimensions were determined on a basis of ASTM D3165 and D3528, respectively [20]. Table 1. Material properties of the carbonfiber-epoxycomposite material (USN 150). 130 1800 St' (MPa) El (GPa) 8 -1400 Sc^ (MPa) E2, E3 (GPa) 61 4.5 St', St' (MPa) G12, Gi3 (GPa) 3.6 -130 Sc^ Sc' (MPa) G23 (GPa) VI2, V13
0.28
V23
0.49
S23(MPa)
ai(io-Vn)
-0.9
Ply Thickness (mm)
Si2,Si3(MPa)
85 40 0.15
27 1.56 a2,a3(10-VD) Density (g/cm^) : Unidirectional carbon fiber prepreg manufactured by Sunkyung Industry Co., Suwon, Korea.
In the case of the static tensile load test, the average surface roughness of the steel adherend of the co-cured single and double lap joint specimens were 0.2, 0.3, 0.7, 1.2, and 1.7 |im. The surface roughness was measured using a portable, self-contained instrument for the measurement of surface texture (Surtronic 3+, manufactured by Rank Taylor-Hobson Limited). The stacking sequence of the carbonfiber-epoxycomposite adherend in the co-cured single and double lap joints was {[±0]4s}s and [±0]4s (0 = 0, 15, 30, 45°), respectively. Manufacturing pressure in the autoclave during curing process of the co-cured lap joint specimens was 0.4, 0.7, and 1.0 MPa. Bond length of the specimens under the static tensile test was 30 mm. The cocured lap joint specimens were tested with a 100 kN materials testing system (MTS). The cross-head displacement rate used in this experiment was 1.27 mm/min. In the case of the dynamic tensile test, co-cured single lap joint specimens selected in this paper were of three types: A-type with
[0]32T stacking
sequence and 1.2 [im surface roughness, B-type
with [0]32T stacking sequence and 0.3 fim surface roughness, and C-type with {[±45]4s}s stacking sequence and 1.2 [im surface roughness. Co-cured double lap joint specimens were also of three types: A-type with
[0]I6T
stacking sequence and 1.2 [im surface roughness, B-type
K.C. SHIN AND J J. LEE
376
with [0]i6T stacking sequence and 0.3 [im surface roughness, and C-type with [±45]4s stacking sequence and 1.2 |ini surface roughness. The bond length of the specimens under cyclic tensile test was 20mm. Cyclic tensile tests were performed under the condition of stress ratio R = OA and a loading frequency / = 5 Hz. Cyclic tensile loads applied to the co-cured single lap joint specimens were 30%, 40%, 50%, 60%, and 70% of the tensile load bearing capacity obtained from the static tensile load test.
n
^
f=f^'
xm Fig. 3. Shape and dimensions of the co-cured lap joint specimens, (a) Co-cured single lap joint and (b) co-cured double lap joint.
Fig. 4. Photograph of the typical failed surfaces on the steel adherend of the cocured lap joints, (a) Co-cured single lap joint and (b) co-cured double lap joint.
1 J
6®
B
a
A
^
6 0
Manufacturing Pr..>ur a : 0 . 4 M P a
O
Manufacturing Praxur a : 0 . 7 M P a
A
Manufacturing Pr...ur a : 1 . 0 M P a
Surface Roughness ( \ i m )
D
n 0 A
M . ufaeturing Praaaur. : .4 M P . Ma ufacturing Praaaura : .7MPa Ma ufaeturing Praaaura : .OMPa
Surface Roughness (\x.m )
(a) (b) Fig. 5. Tensile load bearing capacities of the co-cured lap joints with respect to the surface roughness between steel and composite adherends. (a) Co-cured single lap joint and (b) cocured double lap joint. EXPERIMENTAL RESULTS OF THE CO-CURED LAP JOINTS UNDER STATIC TENSILE LOADS Figure 4 shows typical failure surfaces obtained from tensile tests of the co-cured single and double lap joint specimens. In the case of the co-cured single lap joint, as the surface preparation on the steel adherend is better, a greater amount of carbon fibers and epoxy resin is attached to the steel adherend. Failure mechanism is a partial cohesive failure mode at the 1^* ply of the composite adherend. In contrast with the co-cured single lap joint, failure mechanism of the co-cured double lap joint is the partial cohesive failure or interlaminar delamination failure at the 1^^ ply of the composite adherend because interfacial out-of-plane peel stress
111
Bond Parameters Affecting Failure of Co-Cured Single and Double Lap Joints
hardly occurs in the co-cured double lap joint due to its symmetrical configuration and the mechanical interlocking effect occurs at the rough interface between the resin layer and the steel adherend. Figure 5 shows tensile load bearing capacities of the co-cured single and double lap joints with respect to the surface roughness between steel and composite adherends. Surface roughness at the interface between steel and composite adherends only slightly affects the tensile load bearing capacity of the co-cured single lap joint because of out-of-plane peel stress caused by the unsymmetrical configuration. In the case of the co-cured double lap joint, tensile load bearing capacity increases on the whole as the surface roughness increases, but the extent of the increment decreases when the surface roughness is larger than 1.2 [im. Co-cured double lap joints have a symmetrical configuration, which can induce a mechanical interlocking effect between steel and composite adherends. ] 1
Q
\
A
B
1
A
A
D
i i
^ I a
M.nur.cturinB Pr.>.ur
§ H
B O
: 0.4 M P .
M . n u •during Pr.ssur
: 0.4 M P .
O M.nu acturing Pr.ssut
: 0.7 M P .
O
Manufacturing Pt«s
M.nu •ctu.ing Pressur
: 1.0 M P .
A
M.nutacturing P r u . u r
D A
h
: 1.0 M P .
Stacking Angle (^•)
Stacking Angle ( i ' )
(a) (b) Fig. 6. Tensile load bearing capacities of the co-cured lap joints with respect to the stacking sequences of the composite adherend. (a) Co-cured single lap joint and (b) co-cured double lap joint.
? 0
Q. 4000.
o
a> ^ 2000-
-• % <»
f
§
^
\
B
n o
stacking Angle : 0 i*
A
Stacking Angle : 30 *•
O
Stacking Angle : 45 ^'
Stacking Angle : 15 ^*
Stacking Angle: 0 ° Stacking Angle: 15Stacking Angle: 3 0 ' Stacking Angle: 45 *
n.
Manufacturing Pressure (MPa)
Manufacturing Pressure (MPa)
(a) (b) Fig. 7. Tensile load bearing capacities of the co-cured lap joints with respect to the manufacturing pressure in the autoclave during bonding or curing process, (a) Co-cured single lap joint and (b) co-cured double lap joint. Figure 6 shows tensile load bearing capacities of the co-cured single and double lap joints with respect to the stacking sequence of the composite adherend. Tensile load bearing capacity of the co-cured single lap joint increases as the stacking angle of the composite adherend increases. Although mechanical interlocking surely occurs in the single lap joint, its efficiency is very small compared with that of the co-cured double lap joint because of the peel stress at the
378
K.C. SHIN AND J.J. LEE
interface comer of the joint. The difference in stiffness between the adherends increases as the stacking angle increases. Then, the composite adherend have a reduction in strength in loading direction. However, the larger difference in stiffness between the adherends makes the composite adherend be flexible and have lower transverse tensile stress (or peel stress). Therefore, it is important to regulate the stacking angle of the composite adherend. In contrast with the co-cured single lap joint, tensile load bearing capacity of the co-oured double lap joint decreases as the stacking angle of the composite adherend increases. Since the co-cured double lap joint has symmetrical configuration, it is important to consider the out-ofplane shear stress rather than the out-of-plane peel stress in designing the joint. Figure 7 shows tensile load bearing capacities of the co-cured single and double lap joints with respect to the manufacturing pressure in the autoclave during the bonding process. Tensile load bearing capacity of the two joints increases on the whole, but the extent of the increment decreases as the manufacturing pressure increases. Thickness of the resin layer can be controlled by regulating the manufacturing pressure in the autoclave. In general, the joint strength is dependent on the thickness of the adhesive layer because of several reasons such as the manufacturing thermal stresses. FINITE ELEMENT ANALYSIS AND EVALUATION OF THE TENSILE LOAD BEARING CAPACITY OF THE CO-CURED LAP JOINTS In this study, residual thermal stresses were also considered because co-cured lap joints generally undergo temperature drop (from 1200 to 20D) during the curing process. The stress distributions in the co-cured single and double lap joints were analyzed using ABAQUS .'\8 to be commercial finite element analysis software [21].
Interface
Fig. 8. Coordinate system of the co-cured single and double lap joints for calculating ;tress distributions in the ply-axis. The co-cured single and double lap joints were modeled as a three-dimensional solid structure. The resin layer was ignored in this analysis because the average thickness of the resin layer between two adherends was about 10 [im. Therefore, the co-cured single and double lap joints were assumed perfectly bonded at the interface between steel and composite adherends. The Teflon block between two adherends in the co-cured double lap joint was ignored in this analysis because its modulus (0.6 GPa) is very small and its surface is so slippery that it was not attached to the two adherends. Since the co-cured double lap joint had a quarterly symmetric configuration, just a quarter of the co-cured double lap joint was modeled. Figure 8 shows the coordinate system for the co-cured lap joints. Figure 9 shows finite element meshes and boundary conditions of the co-cured single and
Bond Parameters Affecting Failure of Co-Cured Single and Double Lap Joints
379
double lap joints. The element used for the steel adherend was a 20-node, three-dimensional isotropic solid element, and the element used for the composite adherend was a 20-node, threedimensional orthotropic solid element. The number of elements used in co-cured single and double lap joint was 13,760 and 16,500, respectively. When observing the failure mechanism of the co-cured single and double lap joints, it is important to consider out-of-plane transverse and shear stresses at the interface between steel and composite adherends.
(a) 1 Synmetnc 3>
^ . (b)
Fig. 9. Finite element meshes and boundary conditions of the two joints, (a) Co-cured single lap joint and (b) co-cured double lap joint. 125-, ^
125 Tensile load = 3 kN
100 J
Tensile load = 5 kN 1
Tensile load = 3 kN
100
Tensile load = 7 kN
75
Thermal load
50-1
Tensile load = 5kN Tensile load = 7 kN Thermal load
,t
^
-25-^ -50 •] 5
10 X (distance, mm)
(a) (b) Fig. 10. Interfacial out-of-plane transverse stress distribution, a^, of the co-cured single lap joint along the interface between steel and composite adherends. (a) [0]32T stacking sequence and (b) {[±45]4s}s stacking sequence. Tensile load = 3 kN
150i. ^
100-
\
Thermal load
50-
^^^^^
^•-••.-:-.
f
-50-
f
\
Tensile load = 7 kN
0)
1
150-
Tensile load = 5 kN
^ X (distance, mm)
15
20
1
Tensile load = 7 kN
100-
Thermal load
0-50-
10
!
Tensile load = 3 kN Tensile load = 5 kN
J
'^>^_ ^ 10
15
20
X (distance, mm)
(a) (b) Fig. 11. Interfacial out-of-plane shear stress distribution, a^, of the co-cured single lap joint
K.C SHIN AND J J. LEE
380
along the interface between steel and composite adherends. (a) [0]32T stacking sequence ar d (b) {[±45]4s}s stacking sequence. Figures 10 and 11 show interfacial out-of-plane transverse and shear stress distributions of cocured single lap joints with [0]32T and {[±45]4s}s stacking sequences along the interface between steel and composite adherends, respectively. Interfacial out-of-plane peel stress level of the co-cured single lap joint with [0]32T stacking sequence is larger than that of the co-cured single lap joint with {[±45]4s}s stacking sequence. In the case of interfacial out-of-plane shear stress distribution, the total interfacial out-of-plane shear stress level of the co-cured single lap joint with {[±45]4s}s stacking sequence is similar to that of the co-cured single lap joint with [0]32T stacking sequence. Therefore, interfacial out-of-plane transverse stress plays an important role in failure of the co-cured single lap joint.
r
Tensile load = 12 kN Tensile load = 18 kN Tensile load = 24 kN Thennal load
«
0.0 J
1
/^
^f'
—
\ Tensile load = 12 kN Tensile load = 18 kN Tensile load = 24 kN Thermal load
(0 -5.0x10'c
E
X (mm)
x(mm)
(a)
(b)
Fig. 12. Interfacial out-of-plane transverse stress distribution, a^^, of the co-cured double lap joint along the interface between steel and composite adherends. (a) [0]I6T stacking sequence and (b) [±45]4s stacking sequence.
^
1.0x10'
.£
-5.0x10
r
-i.oxio'
E
-Tensile load = 12kN • Tensile load = 18 kN - Tensile load = 24 kN - Thennal load
- Tensile load = 12kN • Tensile load = 18kN - Tensile load = 24kN - Themnal load
x(mm)
(a)
x(mm)
(b)
Fig. 13. Interfacial out-of-plane shear stress distribution, a^, of the co-cured double lap joint along the interface between steel and composite adherends. (a) [0]I6T stacking sequence and (b) [±45]4s stacking sequence. Figures 12 and 13 show interfacial out-of-plane transverse and shear stress distributions of the co-cured double lap joints with [0]I6T and [±45]4s stacking sequences along the interfaces between steel and composite adherends, respectively. It is important to consider interfacial outof-plane shear stress rather than interfacial out-of-plane transverse stress because of the compressive stress distribution due to the symmetric configuration of the co-cured double lap
381
Bond Parameters Affecting Failure of Co-Cured Single and Double Lap Joints
joint. Total interfacial out-of-plane shear stress of the co-cured double lap joint with [0]I6T stacking sequence is smaller than that of the co-cured double lap joint with [±45]4s stacking sequence along the interface. Based on the failure mechanisms and stress distributions at the interface between steel and composite adherends of the co-cured single and double lap joints, tensile load bearing capacities of the two joints were evaluated. Since failure started at the edge of the interface between steel and composite adherends, it is important to consider the failure criterion using interfacial out-of-plane stress distributions at the interface. Three-dimensional Tsai-Wu and Yedelamination failure criteria were used to predict partial cohesive failure or interlaminar delamination failure in the co-cured single and double lap joints. The three-dimensional Tsai-Wu failure criterion can be expressed as follows [22]: FI=Fp,,+Fp,,+Fp,,+F,f5'^ +F,pJ +F,p,' +FjjJ +F,p' +Fjj,' +2F,p,fy,,+2F,p,p,,+2F,p,p,,
(1)
Where a-j denotes stress components referred to the principal materials coordinates. F. and F.j can be expressed as follows [23]:
/r=_L+_L, F,=±J^, S^^ S^^
S^^ S^^
•^U "^l 1
^^22 "^22
1 F =——
F,^±^,
O33
S^^
^^33 "^33 '
1 1 F =—— F =——
^^23^
"^13
F —-J-JlJL
/J\
^^
^^12 '
F __V
F __±_}LJL
22^33
Where S. denotes strengths of the composite material. In order to avoid failure, FI of Equation (1) must be less than 1; failure is predicted when FI is > 1. Based on the threedimensional Tsai-Wu failure criterion, tensile load bearing capacities of the co-cured single and double lap joints were calculated. Ye-delamination failure criterion can be expressed as follows: W h e n 0-33 > 0 r-
0"33
5. When G
\2
r-
_^ £ l 3
/ - <1
\1
^ 1 =1
5.
(3)
<0 2
r
2 =1
(4)
v^-w
Where S^ and S,^ are out-of-plane peel and shear strengths between steel and composite adherends, respectively. S,p is out-of-plane shear strength in the plane (i.e. zy-plane) perpendicular to the fiber direction. All the stress components are defined as the averaged stresses over a distance from the free edge, c. Table 2 shows various values used to calculate Ye-delamination failure indices. The distance c was defined as one-ply thickness and Sj^ = S,p was assumed [23]. Figure 14 compares the tensile load bearing capacity of the co-cured single lap joint evaluated
K.C. SHIN AND J.J. LEE
382
through the two failure criteria with experimental results. Analytical results evaluated through the Ye-delamination failure criterion were in good agreement with experimental results, although analytical results evaluated on {[±30]4s}s and {[±45]4s}s stacking sequences were slightly higher. This is caused by the failure mechanism, cohesive failure mode at the interface between steel and composite adherends, of the co-cured single lap joint. Table 2. The distance c and interfacial properties of the interface between the steel and composite adherends. 0.15 40 160
c *(mn)
Interfacial peel strength (MPa) Interfacial shear strength (MPa) : The distance that is defined as one-ply thickness. : Estimated value.
- Tsai-Wu failure criterion -Ye-delamination criterion Experimental results
Stacking angle (deg)
Fig. 14. Comparison of tensile load bearing capacities of co-cured single lap joint (fabricated under 0.7 MPa manufacturing pressure) calculated from the two failure criteria with experimental results.
-D—Tsal-Wu failure criterion -O—Ye-delamination criterion •
Experimental results
Stacking angle (deg)
Fig. 15. Comparison of tensile load bearing capacities of the co-cured double lap joint (fabricated under 0.7 MPa manufacturing pressure) calculated from the two failure criteria with experimental results.
Figure 15 compares the tensile load bearing capacity of the co-cured double lap joint calculated from the two failure criteria with experimental results. The tensile load bearing capacity of the co-cured double lap joint evaluated through the three dimensional Tsai-Wu failure criterion was in good agreement with the experimental results because of its failure mechanism, interlaminar delamination failure at the 1^^ ply of the composite adherend. EXPERIMENTAL RESULTS OF THE CO-CURED LAP JOINTS UNDER DYNAMIC TENSILE LOADS Failure mechanism of the co-cured single lap joint under cyclic tensile loads was a partial cohesive failure in the resin layer. Figure 16 shows the relationship between the normalized load range and the number of cycles to failure of the co-cured single lap joint obtained from cyclic tensile tests. The B- and C-type joints would have good fatigue characteristics under the high cyclic tensile load range and under the low cyclic tensile load range, respectively. However, since these results were based on the cyclic load range, we should also consider the magnitude of the static tensile load bearing capacity, Ps, of the joint. Failure mechanism of the co-cured double lap joint under cyclic tensile loads was a partial cohesive failure in the thin adhesive layer. Figure 17 shows the relationship betweea the
383
Bond Parameters Affecting Failure of Co-Cured Single and Double Lap Joints
normalized load range and the number of cycles to failure of the co-cured double lap joint obtained from cyclic tensile tests. The C-type joint has better fatigue characteristics than A- and B-type joints. The A- and B-type joints have similar fatigue characteristic although they have different surface roughness between steel and composite adherends. This means that we should also consider the magnitude of the static tensile load bearing capacity, Ps, of the joint because these results were based on the cyclic load range. 1.00 n
0.8-|
D
A
OAQk
D
o
0.75-
o
A
ID OMO A an
s
A
D O * O 0.50-
8
*2
•§ 0.4"S
°
OJC
D
z
r-—r-r-T-r-rTTT—^-r- —
1 —
Il
1-
Number of cycles to failure, N,
Fig. 16. Relationship between the normalized load range and the number of cycles to failure of the co-cured single lap joint.
CO
A
• i i ^
(:»=• 0.25-
Ah* 0.2-
DiCD
a cno
4^
*«
(TypeA):P3=4200N (Type B): P, = 3180 N (Type C): P^ = 4980 N
o
O
• (Type A): P,= 14000 N 0 (Type B): P, = 10500 N * (Type C):Pg= 11200 N
•=»
0.00Number of cycles to failure, N,
Fig. 17. Relationship between the normalized load range and the number of cycles to failure of the co-cured double lap joint.
CONCLUSIONS In this study, co-cured single and double lap joint specimens with steel and carbon fiber-epoxy composite adherends were fabricated and tested under the static and dynamic tensile loads with respect to bond parameters, namely surface roughness, stacking sequence, and manufacturing pressure. Using the stress distributions, obtained through finite element analysis considering residual thermal stress, in the interface of the co-cured lap joints, static tensile load bearing capacities were calculated and compared to those of the experimental results. From the experimental and analytical investigations, the following conclusions were derived: (1) Surface roughness at the interface between steel and composite adherends little affects the tensile load bearing capacity of the co-cured single lap joint because of out-of-plane peel stress caused by the unsymmetrical configuration. In the case of the co-cured double lap joint, however, the tensile load bearing capacity was slightly affected by the surface roughness because of its symmetrical joint configuration, which can induce a mechanical interlocking effect between steel and composite adherends. (2) It is important to consider the stacking sequence of the composite adherend of the co-cured single and double lap joints because the out-of-plane peel stress can be reduced through possible regulation of the stiffness difference between steel and composite adherends. (3) Manufacturing pressure in the autoclave during curing process affect a little tensile load bearing capacities of co-cured single and double lap joints. (4) Analytical results evaluated through the Ye-delamination failure criterion were in good agreement with static experimental results of the co-cured single lap joint. This is caused by the failure mechanism, cohesive failure mode at the interface between steel and composite adherends, of the co-cured single lap joint. The tensile load bearing capacity of the co-cured double lap joint evaluated through the three dimensional Tsai-Wu failure criterion was in good agreement with the experimental results because of its failure mechanism, interlaminar
384
K.C. SHIN AND J.J. LEE
delamination failure at the 1^^ ply of the composite adherend. (5) In the case of the co-cured single lap joint, the B- and C-type joints would have good fatigue characteristics under the high cyclic tensile load range and under the low cyclic tensile load range, respectively. In case of the co-cured double lap joint, the C-type joint has better fatigue characteristics than A- and B-type joints. Surface roughness of the steel adherend and the stacking sequence of the composite adherend were important bond parameters in design of the co-cured single and double lap joints under the cyclic tensile loads. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
Mallick P. K. (1988). Fiber-Reinforced Composites, Marcel Dekker, New York. Vinson J.R. and Sierakowski R. L. (1987). The Behavior of Structures Composed of Composite Materials, Martinus Nijhoff, Dordrecht. Hart-Smith L. J. (1987) Composites, T. J. Reinhart (Ed.), pp. 479-495, ASM International, Ohio. Messier R. W. (1993). Joining of Advanced Materials, Butterworth-Heinemann, Stoneham, MA. Shin, K. C , Lee, J. J. and Lee, D. G. (2000). J. Adhesion Sci. Technol. 14, 123. Shin, K. C. and Lee, J. J. (2000). J. Adhesion Sci. Technol. 14, 1539. Shin, K. C. and Lee, J. J. (2000). J. Adhesion Sci. Technol. 14, 1691. Jones, R., Chiu, W. K., and Paul, J. (1993). Composite Struct. 25, 201. He, S. and Rao, M. D. (1994). J. Composite Mater. 28, 112. Olivier, P. and Cottu, J. P. (1998). Composites Sci. Technol. 58, 645. Kim, H. S., Lee, S. J. and Lee, D. G. (1995). Composite Struct. 32, 593. Choi, J. H. and Lee, D. G. (1997). J. Composite Mater. 31, 1381. Lee, S. W., Lee, D. G. and Jeong, K. S. (1997). J. Composite Mater. 31, 2188. Cho, D. H., Lee, D. G. and Choi, J. H. (1997). Composite Struct. 38, 309. Cho, D. H. and Lee, D. G. (1998). J. Composite Mater. 32, 1221. Crane, L. W., Hamermesh, C. L. and Maus, L. (1976). SAMPE J. 12, 6. Parker, B. M. and Waghome, R. M. (1982). Composites 13, 280. Lee, D. G., Kim, K. S. and Im, Y. T. (1991). J. Adhesion 35, 39. Renton, J. W. and Vinson, J. R. (1975) Eng. Fracture Mech. 7, 41. (1995). Annual Book of ASTM Standards (General Products, Chemical Specialties, and End Use of Products), Vol. 15, American Society for Testing and Materials, Philadelphia, PA. (1998). ABAQUS/Standard User's Manual Hibbitt, Karisson & Sorensen, Pawtucket. RI. Reddy, H. N. (1997). Mechanics of Laminated Composite Plates: Theory and Analysis, CRC Press, Inc., Boca Raton, Florida. Ye, L. (1988) Composites Sci. Technol. 33, 257. Burden R. L. and Faires J. D. (1989). Numerical Analysis. PWS-KENT Publishing Company, Boston.
3. COMPOSITES 3.1 Short Fibre Composites
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
387
FRACTURE MECHANISMS IN SHORT FIBRE POLYMER COMPOSITES: THE INFLUENCE OF EXTERNAL VARIABLES ON CRITICAL FIBRE ANGLE S. FARA and A. PAVAN Dipartimento di Chimica, Materiali e Ingegneria Chimica "Giulio Natta' Politecnico di Milano, Piazza Leonardo da Vinci 32,1-20133 Milano, Italia ABSTRACT The influence of external variables, i.e. testing and environmental conditions such as rate of loading, temperature and moisture, on fracture mechanisms and hence on fracture toughness in short fibre polymer composites is investigated as a function of fibre orientation with respect to the fracture plane. Unidirectionally oriented materials with different polymer matrix (polyamide 6.6 and polyoxymethylene) and different glass fibre content (15wt% and 30wt%) are examined. The critical stress intensity factor, KQ, at fracture initiation was measured by single-edge notched three-point bending tests carried out at low (lOmm/min) and high (Im/s) load point displacement rate under different environmental conditions (temperature and moisture). All systems examined show the same fundamental result found in previous works: the critical stress intensity factor, Kc, bears a bi-linear relationship with the factor characterising fibre orientation, with different slopes over different ranges of the orientation factor, suggesting a transition between different fracture mechanisms at a "critical" angle. The variation of Kc with fibre orientation and the transition between the two regimes at a "critical" angle depend on external variables as well as on the constitution of the material. KEYWORDS Short fibre composites, fibre orientation, fracture toughness, fracture mechanisms, fibre pullout, fibre debonding, critical fibre angle. INTRODUCTION Short fibre polymer composites are being increasingly used as engineering materials because they provide mechanical properties superior to neat polymers and can be processed easily by the same fabrication methods, e.g. injection moulding. The mechanical properties of these materials are dependent on a complex combination of several internal variables, such as type of matrix, fibre-matrix interface, fibre content, fibre dimensions, fibre orientation, and external
388
S. FARA AND A. PA VAN
variables, i.e. environmental and testing conditions. Fracture resistance makes no exception. It is thus necessary to understand the relationship between fracture toughness and all these variables if the best performance is to be exploited out of these materials. The relationship between fracture toughness and some characteristics of the constituent fibres, such as fibre-matrix compatibility, fibre content, fibre diameter and fibre length distribution, has been investigated in recent years [1]. While some authors considered, in relation to the fracture toughness, only a gross microstructural characterisation [2, 3], Friedrich [4, 5] and Karger-Kocsis [1] pointed out the need for a more detailed microstructural characterisation of short fibre polymer composites, including consideration of fibre orientation distribution. They assume that fracture toughness is linearly dependent on a "reinforcing effectiveness parameter", R, which encompasses the effects of all microstructural variables and is directly related to the geometrical arrangement of the fibres through the plate thickness. More recently, however, other authors [6, 7] have questioned this assumption on the basis of the observation that different failure mechanisms occur with different fibre length and orientation [1, 8, 9]. Hence the need of distinguishing fracture mechanisms and relating fracture toughness to the effectively operating fracture mechanisms. It is well known that a critical fibre length /c exists, and sets a line of demarcation between two different fracture mechanisms: fibre pull-out is the prevailing fracture mechanism when fibre length is smaller than the critical fibre length (//c) [1, 8]. On the other hand, the influence of fibre orientation on fracture mechanisms has been less studied, maybe because fibre orientation characterisation of these materials is laborious and time consuming. Recently, considering the simplest case of unidirectional short fibre composites, it was found that fracture toughness bears a bi-linear relationship with the factor characterising fibre orientation, with different slopes over different ranges of the orientation factor [7]. This is suggestive of a transition between different fracture mechanisms. At low values of the orientation factor, crack propagation is likely to occur mainly via fibre debonding and matrix fracture, whereas at high values of the fibre orientation factor, fibre breakage and pull-out likely are the dominating mechanisms. This entails that besides a critical fibre length, also a "critical" fibre angle should be considered [10, 11,12]. To confirm these results and gain a deeper insight of the failure phenomena involved, in the present work the effects of loading rate and environmental conditions (temperature and moisture) on the above-said transition were investigated. To best detect the transition, materials with nearly unidirectionally oriented fibres were used and fracture was induced to occur on crack planes having varying orientation with respect to fibre direction. The investigation spanned over a range materials having different polymer matrix (polyamide 6.6 and polyoxymethylene) and different glass fibre content (15wt% and 30wt%). EXPERIMENTAL Materials Plates of polyamide 6.6 (PA6.6) with 30wt% content of glass fibre (G30) and plates of polyoxymethylene (POM) with 15wt% and 30wt% content of glass fibre (G15 and G3()) were prepared by hot compression moulding of thin extruded rods (~2-3mm in diameter) laid parallel in the mould cavity. The technique relies on the findings that the orientation of the fibres in these rods is highly unidirectional and such an orientation is maintained by the hot
389
Fracture Mechanisms in Short Fibre Polymer Composites
compression moulding because of the limited flow occurring in this process [7, 10, 11]. The rods were first cut and kiln dried at 80°C for 90h in the case of polyamide 6.6, and 16h in the case of polyoxymethylene. Then, they were accurately laid down, side by side, in a rectangular mould cavity (170x200mm). Rods of polyamide 6.6 were compression moulded at 290°C for '-20min, rods of polyoxymethylene at 200°C for ~15min. Plates of different thicknesses, in the range 5-12mm, were prepared for the present study. Microstructural characterisation The microstructure of all mouldings was characterised by determining fibre content (weight and volume fraction), average fibre dimensions (diameter and length), fibre length distribution and fibre orientation distribution. Fibre content and dimensions were regarded as uniform throughout the plate thickness. They were measured on the average of three small samples taken from each plate. In order to isolate the fibre component the polymeric matrix was burnt off in a muffle furnace at 700°C. From the residue weight the fibre weight fraction was determined. The fibre aggregates were then immersed in acetone to ease fibre disentanglement, and a sample containing at least 2000 fibres for each specimen was examined under an optical microscope equipped with image analysis facilities to measure fibre dimensions. Fibre orientation was checked at six different in-plane points of the plate. At each point, the through-thickness orientation distribution was determined by sectioning the plate and taking pictures of the section surface at 20-30 different through-thickness positions on the section plane, each picture or frame containing at least 200 fibres. The orientation of each single fibre was determined by measuring the length of the axes of the elliptical section of the fibre and the average orientation of the fibres appearing within one frame was described using the AdvaniTucker tensor [13]. This tensor is defined as:
"y
where a-^x is an orientation tensor component (or orientation factor), p-^ is the component, with respect to the /-axis, of the unit vector p aligned with the k-th fibre (Fig. la), /^ is of the same fibre length and F^ is a weighting factor to convert a surface measurement into a volumetric property and it is related to the projection of the fibre length l^ on the axis n normal to the section plane (Fig. lb): ^k=7-^
(2)
The procedure for characterising fibre orientation consists essentially of: sectioning the moulded plate with a plane perpendicular to the extruded rods direction (plane 2^-3^ in Fig. 2a); polishing the section surface using metallographic techniques; measuring the length of the axes of the elliptical cross-section of each fibre by means of an especially developed image analysis software [14]; calculating the orientation factors using the Advani-Tucker tensor via Eq. (1); determining the distribution of the orientation factors through the thickness of the plate.
390
S. FARA AND A. PAVAN
a)
b)
Fig. 1. a) Unit vector p; b) Fibre length projection, a)
.30^3
b)
1^ extruded rods direction "Extruded rods direction
applied stress direction
Fig. 2. Reference systems on: a) material plate; b) fracture test specimen. Fracture testing Mode I fracture tests were performed in three-point bending (SE(B)) at a low (lOmm/min) and a high (Im/s) loading rate according to the testing protocols that were developed by the European Structural Integrity Society - Technical Committee 4 (ESIS TC4) to determine fracture toughness in plastics by a linear elastic fracture mechanics approach, which have become now international standards [15,16]. In order to assess the relationship among fibre orientation and fracture toughness, fracture specimens were cut at varying angles a between the nominal fibre direction (i.e. the direction 1° of the extruded rods) and the direction normal to the notch plane (the applied stress direction 1) as shown in Fig. 2b. Specimen dimensions were: width W=3B, span S=4W, initial notch depth ao=0.5W and the specimen thickness B was the plate thickness. Notching was made by sliding and pressing a truncated razor blade - scalpel wise - by means of a milling cutter (made by Ceast) producing fast alternating strokes. Sharpness of the notches was checked by observation under an optical microscope: notch root radius was constantly --ISiim. Fracture tests were carried out using an Instron 1125 testing machine for the low loading rate (lOmm/min) and an instrumented impact pendulum Resil 25 by Ceast for the high loading rate (Im/s) tests at room temperature. Using an instrumented falling-weight tower Fractovis by Ceast equipped with a thermostatic chamber, fracture tests at high loading rate (Im/s) were performed also at 80°C and -70°C on specimens of PA6.6 G30 and POM G30 respectively. Loading rates and environmental conditions are summarised in Table 1. Fracture testing at high loading rates presents special problems because of the presence of dynamic effects: the inertial forces caused by the acceleration imparted to the specimen produce vibrations in the test system, oscillations in the recorded signal and forces on the test
Fracture Mechanisms in Short Fibre Polymer Composites
391
specimen which are different from the forces sensed by the test fixture. At speeds approaching Im/s the dynamic effects are significant but can still be controlled by damping the impact as suggested in [16]: a soft pad is placed where the tup strikes the specimen so as to cushion the impact and reduce the inertial effects by reducing the "contact stiffness". A layer of plasticine of 0.1mm and 0.2mm thickness turned out to be optimal for the reinforced polyamide 6.6 and the reinforced polyoxymethylene respectively. Fracture toughness, expressed in term of the critical stress intensity factor Kc, was determined at fracture initiation and that point was identified from the load diagram. Although linearity requirements as set out in the protocol were generally satisfied, the identification of the point of fracture initiation from the load diagram was often uncertain because of the difficulty to distinguish small, residual fluctuations in the load signal due to dynamic effects from a pop-in effect. When in doubt about that the test was discarded. Conditioning. All specimens were kiln dried at 100°C for 16h just before testing. Some specimens of PA6.6 G30 were immersed in distilled water at 23°C for 96h, before testing them at 23°C and 80°C at Im/s, in order to assess the moisture effects. Table 1. Loading rates and environmental conditions. Material Polyamide 6.6 G30 Polyoxymethylene G15 Polyoxymethylene G30
Loading rate
Temperature
Moisture
lOmm/min Im/s lOmm/min Im/s lOmm/min Im/s
23°C 23°C, 80°C 23°C 23°C 23°C -70°C, 23°C
dry saturated dry dry dry dry
Fractography After fracture, scanning electron microscopy (SEM) was used to qualitatively analyse the fracture surfaces so as to assess the fracture mechanisms involved. RESULTS AND DISCUSSION Microstructure Average fibre content (weight and volume %, Wf and Vf) and dimensions (number and mass average length, /„ and /m, and diameter, d) are given in Table 2 for the different materials investigated. There was no difference among samples taken at different in-plane points of the same plate. The two reinforced polyoxymethylenes have fibres with the same diameter but slightly different average lengths, probably because of fibre breaking occurring in the course of the extrusion process which cuts down fibre length the more the larger fibre content is. Comparison of the fibre length distributions of the two POM materials as represented by the IJln ratio leads to the same conclusion.
392
S. FARA AND A. PAVAN
Table 2. Average fibre content and dimensional characteristics. /n
Vf
Material
% 28.8 14.9 29.9
Polyamide 6.6 G30 Polyoxymethylene G15 Polyoxymethylene G30
% 15.4 8.8 19.1
|im 417 144 122
|xm 544 197 172
Uk
d
1.30 1.37 1.41
11 12 12
Fibre orientation turned out to be highly unidirectional (a?i~l, a22~a33--0) and uniform through the thickness B as expected considering the preparation technique used: Figure 3 (a and b) shows an example. By comparing Fig. 3a and 3b it can be observed that an increase in fibre content from 15 wt% to 30 wt% yields values of the orientation factor aJi closer to 1 and the values of 0^2 closer to 0, i.e. better unidirectionality. A value of 2L% very close to zero was obtained in any case, i.e. fibre orientation is always substantially planar. a)
b)
1.0
2 0.8
i2 0.8
%
% >2o.6 C
POM G15
>2o.6 C o
0.0
POM G30
o
[)(.X.)C.X iX .K X .y X jf-if-iH^^^
-B/2
0
X )OC X )C X X X Iffrk
B/2
Through-thickness relative position
Through-thickness relative position
Fig. 3. Fibre orientation factors al: a) POM 015, B= 5 mm; b) POM G30, B= 5 mm. Fracture toughness andfracture mechanisms Figure 4 (a and b) shows an example of the loading curves recorded in fracture tests performed at 23°C at low (lOmm/min) and high (Im/s) loading rate on specimens of PA6.6 G30 cut at different angles a. The loading curves show different shapes on varying fibre orientation (angle a). In particular, at lOmm/min, the degree of non-linearity prior to the attainment of the maximum load decreases, the peaks become sharper and the unstable fracture occurs earlier as the fibre orientation angle a is increased. Non-linearity can be attributed to mechanisms of fibre pullout while unstable fracture can be associated with fibre debonding. It is worth noting that the loading curves, observed in the a range comprised between 30° and 60°, are non-monotonic before the maximum, i.e. kind of pop-in occurs. The variation of the loading curve shape with fibre orientation (angle a) is less regular in Im/s fracture tests and the identification of the point of fracture initiation from the load diagram was often problematic. Load-point displacements at fracture initiation in 1 m/s tests appear to be larger than in low rate tests. This "apparent" result is not unexpected, in view of the damping technique used in the impact tests which increases the compliance of the test system initially.
Fracture Mechanisms in Short Fibre Polymer Composites
a)
393
b) PA6.6G30
500
g
500 . PA6.6G30
10mm/min
400
y—^
g
Im/s
400
MM
11 1
•O300
^ 300 o
(0
o
—1
-J
200
^ ' ' y / ^ \ \
200
\
100
100 0
1 Jin
,A"\ c)
0.2
^^'''^•^'^^' 71?—^— 0.4
0.6
Displacement (mm)
0.8
0
1
./>5^ \
^^^v^ 0.5
C
\
1
1
\
K-JLI 1.5
Displacement (mm)
2
2.
Fig. 4. Loading curves of PA6.6 G30 recorded in fracture tests on specimens cut at angle a varying by intervals of 15° in the range from 0° to 90°. Loading rate: a) lOmm/min; b) Im/s. Test temperature: 23°C. b) 5.0 POM G30
4.5
o^g^ •
1 m s" ^
"E 4.0
^
. . . • . • • •
CO
% 3.5 o
^ 3.0
-1^^^
lOmmmin'"'
: 1
X
2.5 0.2 0.4 0.6 0.8 Orientation factor a,.
0.2
0.4
J_
0.6
Orientation factor a^
0.8
c) 5.0 4.5
:
P0MG15
"E4.0
•1
1 ms
(0
I 3.5 o ^ 3.0 2.5
......I-0 fi
8^
..I- •
.I--:'
XX
X
ig
1 1
p
8 -^ 8^1. lOmmmin'""
8 1
.
. *i
I
0.6 0.2 0.4 0.8 Orientation factor k
.
.
.
1
Fig. 5. Critical stress intensity factor Kc versus fibre orientation factor an: tests at 23°C, lOmm/min (o) and Im/s (x), on dry specimens of: a) PA6.6 G30; b) POM G30; c) POM G15.
394
S. FARA AND A. PAVAN
a)
b) 7.0
POM G30 6.0
§ - ^
0
2l6__—
-70**Co^ 5.0
q^""'^^
^.""^
4.0 3.0 0.2
0.4
0.6
Orientation factor a..
O
•
., .-r
A
••
0.8
0.2
•::::-|
%
23°C
\ 0.4
0.6
Orientation factor a,.
0.8
Fig. 6. Critical stress intensity factor Kc versus fibre orientation factor an: tests at Im/s, on dry specimens of: a) PA6.6 G30 at 23°C (x) and 80^C (o); b) POM G30 at 23°C (x) and -70X (o).
0.2
0.4
0.6
0.8
Orientation factor a^^
Fig. 7. Critical stress intensity factor Kc versus fibre orientation factor an: tests at 23^C and Im/s on dry (x; and moist (o) specimens of PA6.6 G30. The critical stress intensity factor, Kc, obtained at different loading rates and environmental conditions was then reported as a function of fibre orientation. It was expected that the most evident relationship with the considered mechanical property would have been shown by the orientation factor in the applied stress direction (direction 1 in Fig. 2b). Therefore the orientation factor ajj previously measured on the plane 2^-3° was transformed into the orientation factor a^ defined with respect to the applied stress direction 1 (Fig. 2 a and b) by a coordinate axis rotation of an angle a. Further details on such data handling can be found in ref [7, 13]. It is worth reminding here that a^ and a go opposite: when a increases from 0 to 90 degrees a^ decreases from 1 to zero. The critical stress intensity factor, Kc, obtained from specimens of PA6.6 G30, POM G30 and POM G15 tested under different testing and environmental conditions is reported as a function of the fibre orientation factor ai i in Figures 5 (a, b and c), 6 (a and b) and 7. All systems examined here show the same fundamental result observed previously [10, 11, 12], i.e. the critical stress intensity factor Kc bears a bi-linear relationship with the fibre orientation factor ail, with different slopes over different ranges of the orientation factor. A ''knee" between the two linear branches of the Kc versus an diagram appears at a "critical" value of the orientation factor, (aiOc. The existence of this discontinuity is also suggested by the observation of the crack growth
Fracture Mechanisms in Short Fibre Polymer Composites
395
direction. In general the crack proceeds in the notch plane in specimens with aii>(aii)c, whereas it suddenly deviates from the plane of the original notch to follow the prevailing direction of the fibres when aii<(aii)c [11]. The sharp variation in toughness and the sudden change in crack growth direction at (aiOc were interpreted as a transition between different fracture mechanisms. This interpretation is supported by the scanning electron microscope analysis of the fracture surfaces. Test specimens with aii<(aii)c show smooth fracture surfaces displaying clean looking fibres. Test specimens with aii>(aii)c show rough fracture surfaces where fibre pulled-out and voids left by pulled-out fibres are evident. Figure 8 (a and b) shows an example. As the orientation factor increases, the prevailing failure mechanism changes from fibre debonding (and matrix fracture) to fibre pull-out. Fibre breakage has never been observed in this work: this result can be justified by the fact that fibres were shorter than their critical length in all materials examined here. a)
b)
Fig. 8. Scanning electron micrographs at low and high magnification (left and right respectively) of fracture surfaces of POM G30 at two extremes of the orientation factor range: a) aii~0; b) a n - l . Test conditions: 23°C, lOmm/min. The slopes of the two linear branches in the Kc-an diagram and the critical value of the fibre orientation factor (aii)c vary from system to system and with the external variables. Just as the critical fibre length, they depend on the combination of internal variables (such as type of matrix, fibre content, fibre dimensions, fibre-matrix interface) and external variables, i.e. testing and environmental conditions. We can explain the influence of testing and environmental conditions by examining, for sake of simplicity, two extreme cases: aii=0 and aii=l. With reference to the aii=0 case (see the intercepts on the left-hand side vertical axis in Figs. 5 to 7) the increase in fracture toughness with decreasing loading rate (Fig. 5a and 5b),
396
S. FARA AND A. PA VAN
increasing temperature (Fig. 6a) and increasing content of water (a plasticizer for PA6.6) (Fig. 7) can be justified as due to the corresponding decrease in yield stress of the matrix. If its yield stress decreases, the volume of yielded material at the tip of the crack can be expected to increase and so fracture resistance. At the same time as the matrix yield stress decreases, the failure locus moves from the fibre matrix interface to the core of the matrix as it is seen, for example, in Figure 9 (a and b) where the fracture surfaces observed on specimens of PA6.6 G30 with aii~0 tested at different temperatures are shown. The fracture surface of the specimen tested at 23°C (Fig. 9a) shows some clean looking fibres and cavities left by debonded fibres, indicating that fracture occurred via debonding in this case. By contrast the fracture surface of the specimen tested at 80°C (Fig. 9b) shows fibres having scraps of the matrix polymer on them, indicating that fracture occurred mainly inside the matrix. a)
b)
Fig. 9. Scanning electron micrographs of fracture surfaces of PA6.6 G30 with an-O tested at 1 m/s and different temperatures: a) 23°C; b) 80°C. A different trend was found with POM G15 tested at different loading rates (Fig. 5c) and POM G30 tested at different temperatures (Fig. 6b). Firstly, with POM G15 (Fig. 5c), toughness is seen to increase with increasing loading rate. This finding can be explained as the result of two different effects: an increase of matrix yield stress along with an increase of the strength of the bond between fibre and matrix. If the yield stress increases, the volume of yielded material at the tip of the crack can be expected to decrease and so fracture toughness, but if the fibrematrix strength increases, fracture toughness can be expected to increase too. In either case the failure mechanism is fibre debonding. Comparing results obtained with POM G30 (Fig. 5b) and POM G15 (Fig.5c) at the same loading rate shows that fracture toughness increases with increasing fibre content, as expected, at 10 mm/min, but fracture toughness decreases with increasing fibre content in impact tests. Also this result can be explained only by a particular combination of two different effects. As to POM G30 tested at different temperatures. Fig. 6b shows that toughness increases with decreasing temperature. This may be due to higher thermal stresses set in on cooling the material down to -70°C. In view of the sign of the difference in thermal expansion coefficient between polymer matrix and glass fibres, the thermal stress on the fibres is compressive: this provides a strengthening of the bond between fibre and matrix and therefore an increase in toughness. With reference to the aii=l case (see the intercepts on the right-hand side vertical axis in Figs. 5 to 7) the increase in fracture toughness with increasing loading rate (Fig. 5c) and decreasing temperature (Fig. 6b) can be explained as an effect of the increase in pull-out force, which may occur with either weak or strong fibre-matrix interface. If the fibre-matrix interface is weak, the pull-out force increases, as the loading rate increases.
Fracture Mechanisms in Short Fibre Polymer Composites
397
because the rate of fibre pull-out, and so wear and friction, increase. As temperature is lowered the pull-out force increases because of the increase in thermal stresses acting on the fibres and hence increase in friction. If the fibre-matrix interface is strong, the pull-out force increases as the loading rate increases or the temperature decreases, because of the increase in shear strength of the matrix. In the former case (weak fibre-matrix interface) fracture surfaces can be expected to show clean looking pulled-out fibres, as found in POM G15 specimens (Fig. 10a), while in the latter case (strong fibre-matrix interface) fracture surfaces can be expected to show "dirty" looking pulled-out fibres, bearing scraps of the matrix polymer on them, as found in PA 6.6 G30 specimens (Fig. 10b).
Fig. 10. Scanning electron micrographs of fracture surfaces of: a) POM G15; b) PA6.6 G30, with aii-'l. Test conditions: 23°C, Im/s. Also the increase in fracture toughness with increasing water content in PA6.6 G30 (Fig. 7) can be explained as due to an increase in pull-out force, accompanied by a weak fibre-matrix interface. If this is the case the pull-out force increases because of the swelling of the matrix which may increase the compression exerted on the fibre and hence friction. A strong fibre-matrix interface, on the other hand, would set off this interpretation. As a matter of fact water content would reduce the shear strength of the matrix and thus fracture toughness. At any rate, a strong interface can become weak in the presence of water. CONCLUSIONS All systems examined in this work showed the same fundamental result observed in previous works [10, 11, 12], i.e. the critical stress intensity factor, Kc, bears a bi-linear relationship with the fibre orientation factor an, with different slopes over different ranges of the orientation factor. A "knee" between the two linear branches of the Kc versus an diagram pinpoints a "critical" value of the orientation factor, (aii)c. This sharp variation in toughness has been interpreted as a transition between different fracture mechanisms. Matrix fracture and fibre debonding are the prevailing fracture mechanisms at low values of the fibre orientation factor, while at high values the main fracture mechanism is fibre pull-out. Thus, besides a critical fibre length, also a "critical" fibre angle should be considered. The slopes of the two linear branches in the Kc-an relationship and the critical value of the fibre orientation factor (aii)c vary from system to system and with the external variables. Just as the critical fibre length, they depend on a combination of internal variables (such as type of
398
S. FARA AND A. PA VAN
matrix, fibre content, fibre dimensions, fibre-matrix interface) and external variables, i.e. loading rate, temperature and water content. The influence of loading rate, temperature and water content is different for the different mechanisms. ACKNOWLEDGEMENTS Thanks are due to CEAST, Turin, for assistance with the instrumentation of the impact test apparatuses. The PA6.6 G30 material in the form of extruded rods was kindly supplied b> Radici NovaCips, Villa d'Ogna, Italy and the POM G15 and POM G30 samples in the ibrm of extruded rods were kindly supplied by Rhodia, Ceriano Laghetto, Italy. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
16.
Karger-Kocsis, J. (1989). In: Application of Fracture Mechanics to Composite Materials, Ch. 6, Friedrich, K. (Ed.), Elsevier, Amsterdam. Hashemi, S. and Mugan, J. (1993) Journal ofMaterials Science 28, 3983. Huang, D.D. (1995) Polymer Composites 16, 10. Friedrich, K., Carlsson, L.A., Gillespie, J.W. Jr., Karger-Kocsis, J. (1991/ In: Thermoplastic Composite Materials, Ch. 8, Carlsson, L.A. (Ed.). Elsevier, Amsterdam. Friedrich, K. (1985) Composite Science and Technology 11,43. Choi, N.S. and Takahashi, K. (1996) Journal ofMaterials Science 31, 731. Lumini, F. and Pavan., A. (1998) Plastics, Rubber and Composites Processing and Application 11,240. Carling, M.J. and Williams, J.G. (1990) Polymer Composites 11, 307. Lapique, F., Gaarder, R.H. and Larsen, A. (2001) Journal of Reinforced Plastic and Composites 20, 744. Fara, S. and Pavan, A. (2000). In: Advances in Mechanical Behaviour, Plasticity and Damage, pp. 195-200, Miannay, D., Costa, P., Fran9ois, D. and Pineau, A. (Ed.s). Vol.1, Elsevier, Amsterdam. Fara, S. and Pavan, A. (2001). la* Proc. 6th Int. I Conf on Deformation and Fracture of Composites, pp. 123-132, Manchester, UK, 4-5 April 2001. Fara, S. and Pavan, A. "Fracture toughness in short fibre composites: analysis of fracture mechanisms in relation to fibre orientation in unidirectional materials ' to be submitted. Advani, S.G. and Tucker, C.L. (1987) Journal ofReology 31, 751. Piccardi, M., Software for image analysis especially developed for the present work. ISO 13586:2000. Plastics - Determination of fracture toughness (Gjc andKjc) - Linear elastic fracture mechanics (LEFM) approach. See also: Williams, J.G. (2001). In: Fracture Mechanics Testing Methods for Polymers, Adhesives and Composites, ESIS Publ. 28, pp. 11-26, , Moore, D.R., Pavan, A. and Williams, J.G. (Eds). Elsevier, Amsterdam. ISO 17281:2002 Plastics - Determination of fracture toughness (Gjc and Kjc) at moderately high loading rates (1 m/s). See also: Pavan, A. In: Fracture Mechanics Testing Methods for Polymers, Adnesives and Composites, ESIS Publ. 28, pp. 27-58, Moore, D.R., Pavan, A. and Williams, J.G. (Eds). Elsevier, Amsterdam.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
399
FRACTURE BEHAVIOUR OF SHORT GLASS FIBRE-REINFORCED RUBBER-TOUGHENED NYLON COMPOSITES
M. GOMINA*, L. PINOT*, R. MOREAU^ E. NAKACHE° * Equipe Structure et Comportement Thermomecanique des Materiaux (ESCTM) du CRISMAT, UMR 6508 CNRS, ENSICAEN 6 Bd du Marechal Juin, 14050 Caen Cedex 04, France. ^ Institut Superieur de Plasturgie dAlen^on (ISPA), Site Universitaire de Montfoulon, 61250 Damigny, France. ° Laboratoire de Chimie Moleculaire et Thio-organique UMR CNRS 6507, 6 Bd du Marechal Juin, 14050 Caen Cedex 04, France. Abstract: The effects of the amount of rubber, the concentration of fibres and the state of the fibre/matrix interface upon the mechanical behaviour of short glass fibre-reinforced rubbertoughened nylon 6 ternary blends are described. First, tensile tests were carried out on different intermediate materials and then on the ternary blends to derive the stress-strain curves and document the damage mechanisms. Fracture toughness tests were implemented on compact tension specimens and the results were correlated to fractographic observations and acoustic emission analysis to assess the role of the different constituents. Keywords : rubber-toughened thermoplastic ; glass fibre ; ternary blends ; mechanical tests; fracture toughness; J-integral; image analysis; fractographic observations; acoustic emission. INTRODUCTION Fibre-reinforced thermoplastics, such as glass fibre-reinforced nylon, are increasingly used as substitute for wood, metallic or ceramic materials in applications for which high strength and temperature are not decisive parameters. They are being incorporated into components in the automotive, naval, sport and leisure industries due to the potential of reduced production costs of lighter parts and recycling possibilities. When thermoplastics are reinforced with short fibres, both stiffness and strength may be increased, but these improvements are accompanied by a reduction of the ultimate strain for high concentrations of fibres [1-5]. This brittle behaviour is incompatible with growing engineering applications in which the parts are subjected to cyclic loadings or impacts, as in automotive under-the-hood applications. Thus, very tough nylon thermoplastics with enhanced creep and impact resistances have been produced by the introduction of a rubbery phase to the
400
M GOMINA ETAL
base material [6-12]. Up to now, little effort was directed towards the mechanical behaviour of the ternary blends obtained by association of short fibres to these ductile matrices [13-18]. The purpose of this investigation is to determine the effects of the rubber, the fibre concentration and the state of fibre/matrix interface upon the damage and fracture behaviour of short glass fibre-reinforced nylon-based materials. EXPERIMENTAL PROCEDURE Materials In this study, the following materials were used : - granules of nylon 6, Grillon A28GM Natur type, and nylon 6 blended with 8 and :?.0 wt% rubber, supplied by EMS Chemie (Switzerland). The rubber was a maleic-anhydride (MA) modified Ethylene-Propylene. The supplier claimed that 95% of the succenic anhydride grafted on the rubber had reacted with the nylon. - E-glass fibres of two types, A and B, supplied by Vetrotex (Chambery, France). The main physical and mechanical characteristics are : 10 jum in diameter, initial length 4.5 mm. Young's modulus 73 GPa, tensile strength 3.4 GPa and ultimate strain 4.5%. Both coatings contained a y-amino-propyltriethoxysilane (y-APS) as a coupling agent, but for type B fibre the composition of the polyurethane was designed to prevent direct contact between the fibre and the rubber. Rectangular plates of dimensions 90x60x5.7 mm and dumbbell-shaped specimens according to the ISO 527-2/1A standard were prepared by injection molding the three different granules: 100 wt% nylon 6 (GM matrix), 92 wt% nylon /8 wt% rubber (DZ matrix) and 80 wt% nylon /20 wt% rubber (NZ matrix). The fibrous composites were blended by first compounding the different granules with four concentrations of type A or B fibres (1, 10, 20 and 30 wt%)) on a twin screw co-rotating Clextral type extruder (rotor speed 77 rpm; barrel temperature 265°C) prior to injection molding on a KM 90/340B type injection press (mold temperature 70°C).The mould fill direction was parallel to the largest dimension of the specimens. In this paper, all the blends and composites are designated by the type of matrix (G for the neat nylon, D for the 8 wt % rubber-modified nylon and N for the 20 wt % rubber-modified nylon), the concentration of fibres and the type of fibre/matrix interface (A or B). As an example, a material designed DIOB is a ternary blend made of DZ matrix and 10 wt% of type B fibres. After drying the specimens for 24 hours at 100°C, they were stored in plastic bags inside a desiccator. In comparison with freshly injection moulded samples, the moisture content in the specimens ready for mechanical testing is about 2 wt%. All the mechanical tests were conducted in an environmental chamber in controlled conditions: a temperature of 20°C under a continuous argon flow. For the morphological analysis of the binary blends (materials DZ and NZ), the samples were cryofractured in liquid nitrogen and the rubber phase was dissolved after exposure at 140°C for 6 hours to xylene vapour. After the chemical treatment, the surfaces were gold-sputtered for SEM observation.
Fracture Behaviour of Short Glass Fibre-Reinforced Rubber- Toughened Nylon Composites Morphological analysis of the blends The SEM micrographs of the xylene treated surfaces of rupture were transformed into binary images to allow the selection of the desired phase for the analysis: the dark phase (1) corresponds to the rubber and the complementary white phase (0) is the nylon. These binary images can then be subjected to measurements, by means of home-made software, with the aim of determining different micro-structural parameters [19]. The rubber particles were characterized in terms of: (i) (ii) (iii)
the specific surface of the particle/matrix interface by unit volume of the matrix, Sv(E/P). This affords information on the fineness of the rubber dispersion; the mean free path within the rubber phase, Lpm(E); the distribution of the maximum free path within the individual particles, F(l). These data are related to the size distribution of the particles.
Mechanical tests Uniaxial tensile tests were carried out to determine the stress-strain curves and document the damage growth on a computer-controlled Instron model 8516 servo hydraulic testing machine operating at a strain rate of 5% min"\ The macroscopic tensile yield stress was considered equal to the maximum stress on the loading curve. The Young's modulus was determined as the plateau value of a plot of the secant modulus as a function of the strain. Fracture toughness tests were conducted with ASTM E561 type compact tension specimens (CT) machined from the moulded plaques (W=40 mm). The initial notch, parallel to the mould fill direction, was prepared by first machining a saw-cut slot which was then sharpened with a razor-blade (pre-crack tip radius «20jLtm,). A Schenck RMC type universal testing machine operating at a crosshead speed of 3 mm min'^ and equipped with a 10 kN capacity load cell was used. The loadline displacement was measured between the loading pins by using a linear variable differential transducer. Damage accumulation in the specimen under load was monitored by mean of an acoustic emission set (Dunegan 3000) with a total gain of 80 dB, while damage growth was documented by in situ observations of the crack path by using an optical travelling microscope. Once a specimen had been loaded to the desired point Pj (si, ai) on the loading curve, it was unloaded and then broken in liquid nitrogen for the direct measurement of the crack extension, Aa, using a graduated scale on an eye-piece. Fracture surfaces were then gold-sputtered for observafions with a Philips FEG XL30 type scanning electron microscope (SEM). Linear Elastic Fracture Mechanics (LEFM) describes the brittle behaviour of a material in term of the critical value of the stress intensity factor at the crack tip, KQ, at the onset of propagation at a critical load value Pc:
where B is the specimen thickness, a is the crack length, W is the specimen width, and f(aAV) is a geometric factor : /(<^)=7^^^[0-886+4.64(2-13.32a2+14.726^3_5 6^4] (l-aj
(2)
401
402
M GOMINA ETAL
where a = a/W is the relative crack length. KQ is the provisional plane-strain fracture toughness, Kic, and it will be treated as Kic if several criteria on the specimen dimensions and loading curve are fulfilled:
(0
B,aand(W-a)>2.5(— \C7y
and
<1.10
(3')
^50.
where Oy is the yield stress and P50/0 is the load corresponding to a 5% offset on the loaddisplacement curve. Conversely to elastic materials, during crack propagation in inelastic materials the potential energy stored in the region adjacent to the crack path is not totally used up to create new crack surfaces: an amount of the released energy is required to deform the material plastically. T le crack growth in these elastic-plastic materials is defined by an energy per unit area (J) necessary to extend a stable propagating crack. The J-integral was originally defined as a pathindependent line integral around the crack sides [20] which can be expressed in term of energy
^~
(4)
Bda
where U is the potential energy of the loaded body (the area under the load-loadliae displacement curve). Practically, JR values were calculated from the areas UTI associated to incremental crack extensions Aai using equation (4): J.^
t z
Q.
(4')
BAa, Sharp notch
1/
>^
/}// V / / / / / ' 3
//// 11////
/T^\ " /^
//// if J //•//-^-^^^^ ^ 4
/ % ^
ductile 2UT4
I/rA^-^-y^^-\
"* B(a -a )
brlttfa
h(mm)
Fig. 1 : Evaluation of the elastic-plastic crack propagation energy rate, J, from incremental crack growth measurements.
The representation of the JR values as a function of the crack extensions Aai defines the resistance curve J-Aa.
Fracture Behaviour of Short Glass Fibre-Reinforced Rubber-Toughened Nylon Composites For materials which undergo large scale elastic-plastic deformation, macroscopic crack propagation is preceded by an amount of crack tip blunting. The macroscopic crack initiation point is thus not clearly defined. Many standards address this question: - ASTM E813-87 [21] and 89 [22] define the crack initiation point by the intersection of the power law curve fitted to the valid J-Aa points with an offset line (0.2 mm offset) parallel to the blunting line of J=2Aaay; - ASTM E813-97 [23] advocates a critical point at the intersection of the regression line fitted to the valid J-Aa points with a vertical line of abscissa 0.2 mm; - Narissa and Takemori [24] advocates a critical point defined by reading the ordinate intercept by the J-Aa plot. Thus, there is not yet a unique Jic evaluation procedure. In this study, the crack growth resistance curves were analysed on the ASTM E813-81 standard basis [25], which considers only the experimental points lying between two offset lines drawn parallel to the pre-crack blunting line J=2Aaay. The first offset is 0.6% of the initial ligament (W-ao) and the second is 6% (Fig.2). A linear regression line is drawn to the points lying within these lines, the intersection of which with the crack blunting line defines the onset of crack propagation and the provisional fracture toughness value, Jic. Like the LEFM fracture toughness Kic, a valid Jic may be obtained whenever
B,{^-a),W>
25 ' ^ l a' n d
•"
d{^a)
<' cr^,
(5)
>k Crack bkjntiimj ^ 2 a rAa-0,006b) line / f^ / ~5 /
JIC
' /
0
'
J : ^ : ^ ^ ^ :
/ ii^r*^^
f
'f/ f
-'
/
J = 2a/Aa-0,06b) '
/ ^
Aa (mm)
^
Fig. 2: Determination of Jic using the ASTM E813-81 standard [24]. Jic is the critical value of the near-tip quasi-static fracture parameter J, i.e. the crack initiation toughness. As for ductile materials a stable crack growth can occur at J values several times the value Jic, the non-dimensional tearing modulus TR was proposed for defining the crack growth toughness [26] : _E_dJ_ (6) cjy
da
where dJ/da is the slope of the J-Aa resistance curve in the stable crack growth regime.
403
M GOMINA ETAL
404
RESULTS AND DISCUSSION Morphology of the blends Typical SEM micrographs of cryogenically fractured surfaces show an homogeneous distribution of the rubber particles for both DZ and NZ materials (Fig. 3). In Table I are reported the specific surface of the nylon/particle interface and the mean free path within the rubber phase. It appears that the rubber particles are by far finer in the NZ blend even though the number of particles is underestimated for this material due to the presence of very small particles and the poor contrast between the rubber and the nylon. "*^^f «i(i -t
#*
f „^ M'"-^^^^,m^-i^'M'•mmW' ^*-^ M^,*
m^^
Fig. 3: SEM micrographs of cryogenically fractured materials (a) DZ blend; (b) NZ blend. Table I: Morphological parameters determined on digitalized SEM micrographs by using the image analysis technique. Amount of rubber Wt% Vol% 8(DZ) 10.1 (DZ) 20 (NZ) 24.3 (NZ)
Parameters Vv(E) 16.9 8.0
Lpm(E) 0.91 0.23
Mechanical tests Tensile tests Role of the rubber in the binary blends Figure 4 shows the tensile stress-strain curves of the different rubber-toughened nylon 6 materials. The main features are: (i) no acoustic emission is recorded up to catastrophic failure. This means that no acoustic emission is associated to the cavitation of the rubber particles, or the emission lies outside the sensitive band width of the transducer (25-625 kHz). (ii) as the amount of rubber is increased, the stiffness decreases as does the macroscopic yield point. In fact only the yield stress decays due to the presence of rubber, the yield strain being unaltered.
Fracture Behaviour of Short Glass Fibre-Reinforced Rubber-Toughened Nylon Composites 405 Thus, more likely, cavitation of the rubber particles has a limited effect upon yielding of these materials. Yielding may occur due to the stress concentration around the particles since the magnitude of this mechanism is higher when the density and the fineness of the particles are also high.
GM (PA6)
^ / " D Z (PA6+8wt% rubber) NZ (PA6+20wt% rubber)
2
4
Strain (%)
Fig. 4 : Monotonic tensile stress-strain curves associated to the different matrices. Glass fibre-reinforced nylon 6 : effects of the fibre concentration and the state of fibre/matrix interface. Depending on the injection moulding conditions, the thickness of the moulding and its geometry, pronounced differences may be noted in the degree of the fibre orientation and in the sizes of the layers across the thickness. Generally, for thin injection-moulded plaques three layers occur. As an example, figure 5a is a schematic representation of the structure observed for nylon reinforced with 30 wt% of type A fibres. In terms of a simple core-skin model, the 30 wt% fibre-reinforced nylon sample represented in figure 5b and 5 c consists of a core covering 24% of the total thickness where the fibres are perpendicular to the melt flow direction whereas in the skin regions the fibres are parallel to the flow direction. Thus in our specimens the skin dominates the fibre orientation. Figures 6a and 6b show the stress-strain curves for the different blends of nylon with type A or B fibres, respectively. Generally, for a given amount of fibre, composites with type A interface exhibit a higher Young's modulus, strength and yield stress. Further investigation on the interfacial shear stress in these materials have pointed to a stronger fibre/matrix interface when type A fibres are associated with the nylon [27]. Moreover, for both types of interface a pronounced embrittelement is observed for the lower densities of reinforcement i.e. for fibre concentration < 20 wt%. The correlation of the quasi-static loading curves, the acoustic emission counts and SEM observations reveals three successive damage mechanisms as loading proceeds: (i) creation of micro voids at fibre ends and generalized plastic deformation in the matrix at fibre ends; (ii) propagation of interfacial cracks issued from the fibre ends; (iii) formation of a macrocrack by fracture of the brittle matrix lying between neighbouring, but non interacting, fibre ends. Different authors have reported detailed observations of these mechanims [28, 29].
M. GOMINA ETAL
406
Skin
Core
Skin
Mould fill direction
® (a)
Fig. 5: (a) Schematic representation of the fibres arrangement across the thickness of the 30 wt% fibre-reinforced nylon, the melt filling direction is indicated; view of the fibre orientation in (b) the skin and (c) the core regions.
160 ^
^•'30%
Type A fibres
120 /,-•
10%
8 0
S 40
ol r ()
\ 2
1 1 4 6 Strain (%)
(a)
1 1 8
4
6
Strain (%) (b)
Fig. 6: Influence of the concentration of fibres and state of fibre/matrix interface on the uniaxial tensile stress-strain curves of fibre-reinforced nylon composites : (a) type A interface; (b) type B interface. The semi-ductile behaviour observed with 30 wt% of fibres (Fig. 7) is explained by the superposition of different damage mechanisms: (i) the overlapping of the plastic deformation zones of adjacent fibres; (ii) generation of deformation bands in the small matrix layers
Fracture Behaviour of Short Glass Fibre-Reinforced Rubber-Toughened Nylon Composites between neighbouring fibres; (iii) intensive shear in these bands giving rise to an amount of stable macrocrack growth prior to catastrophic failure.
Fig. 7: SEM micrograph showing the ductile behaviour of the nylon reinforced with 30 wt% of type A fibres. Glass fibre-reinforced rubber-toughened nylon 6 Conversely for fibre-reinforced nylon composites, the stress-strain curves of type A fibrereinforced rubber-toughened nylon exhibit a plastic elongation plateau, the extent of which decreases with the amount of type A fibres (Fig. 8). Moreover, the only benefits of increasing the concentration of type A fibres are higher Young's modulus and macroscopic yield stress. This clearly shows an embrittlement effect due to the type A interface. In figure 8a it appears that the association of 1 wt% of type A fibres to the rubber-toughened nylon deteriorates the stress-strain curve as compared to the un-reinforced matrix, which means that a weak load transfer occurs as the type A fibres are not strongly bonded to the matrix. In contrast, for the same amount of type B fibres, the loading curve is not lowered and for higher amounts of type B fibres the stress-strain curves are noticeably enhanced (Fig. 8b); thus type B fibres are more strongly bonded to the rubber-toughened nylon.
Q- 80
/" yy"
o S 40
Type A fibres
0% — 1% — 10% 20% • —30%
ir"
"•^•^'^^•-Trsr.-i-n
* Non broken at8= 18%
5
10
15
Strain (%)
(a) Fig. 8: Influence of the concentration of fibres and the state of fibre/matrix interface on stressstrain curves for glass fibre-reinforced DZ materials.
407
M GOMINA ETAL
408
Fracture toughness Neat nylon 6 Figure 9a shows a typical load-loadline displacement curve of a nylon compact tension specimen. The slight deviation from linear elastic behaviour prior to fracture relates to the presence of a plastic deformation zone confined near the pre-crack tip (Fig. 9b). This stable crack propagation domain is adjoining a wide hackle zone (rapid crack growth domain) characteristic of a brittle fracture. The mean critical stress intensity factor value of 4.5 MPaVm obtained for five identical specimens overestimated the fracture toughness: a minimum thickness of 14 mm is required for the validity criteria advocated in Equation 3.
(b) Stable propagation = instable crack propagation Notch
m
H\J\J
A
PR
300 ^200
i
100 PA6
n/
1 1
1
h(mm)
2
Fig. 9 : Load-loadline displacement curve (a) and SEM micrograph of the surface of rupture (b) of a nylon 6 compact tension specimen.
Fibre-reinforced nylon 6 materials Due to the fibres arrangement across the thickness, these blends can be regarded as layered composite materials [30], with a degree of anisotropy within each of the layers. Therefore, different damage mechanisms will operate simultaneously during the stable propagation of a macrocrack. Many workers have investigated the damage mechanisms and the fracture parameters of injection-moulded short fibre-reinforced unfilled thermoplastics with emphasis on the structural anisotropy (fibre orientation, alignment and length distribution) [4, 13, 31-35]. The results show that the improved fracture toughness relatively to the neat matrix is essentially due to the combination of fibre pull-out and an enhanced load bearing capacity [32]. For the fibre-reinforced nylon composites under study, three main trends of the load-loadline displacement curves were observed (Table II) and documented using optical and scanning electron microscopy (Figure 10), depending on the fibres concentration and the fibres spatial distribution at the crack front:
Fracture Behaviour of Short Glass Fibre-Reinforced Rubber-Toughened Nylon Composites -
-
At low densities of fibres, fracture proceeds by brittle failure or debonding of the fibres: a fully linear elastic loading curve is observed (type 1 loading curve in Table II). At intermediate concentrations of fibres (10 and 20 wt%), fracture initiates first by the formation of a frontal damage zone (microvoids at fibre ends, fibre decohesion, microrupture of the matrix) and then by a stable growth of microcracks and their junction into a macrocrack which propagates catastrophically (types 2 and 2' loading curves in Table II, associated with Figures lOA and IOC, lOD respectively). At higher concentrations of fibres or at intermediate concentrations when a few fibres around the crack tip are orientated perpendicular to the notch plane, the loading curve increases linearly up to a maximum load Pi as the load is transferred onto the fibres at the crack front and a process zone develops. Fracture of the fibres lying normal to the notch plane resuhs in unstable crack propagation until it is arrested by a packet of fibres favourably orientated; then the applied load must be increased to create a new frontal process zone. Therefore the successive unstable crack extensions result in a saw-tooth like loading curve behaviour (types 3 and 3' loading curves in Table II, associated with Figures lOB and lOE , lOF respectively).
The fracture toughness values associated to the different reinforced-nylon composites are plotted in figure 11a as a function of the concentration of fibres and the type of fibre/matrix interface. The introduction of a small amount of both types of fibres (1 wt%) results in a drastic reduction of the fracture toughness, relatively to the unfilled nylon. Pecorini and Hertzberg [36] relate this difference to fracture of the fibres when the matrix and the interfacial bonds strengths are high. Others [31] explain it by a reduction of the contribution of the ductile matrix to the fracture energy by a transition from a plane stress condition in the thickness of the neat resin to a nearly plane strain condition in the thickness of the composite with the low fibre concentration. Observation of the general trends in figures 6a and 6b invalidates the former hypothesis because the material GIB with the weakest interface would exhibit a higher toughness value than GIA. Above 1 wt% of fibres, for both coatings Kic increases steeply with the amount of reinforcement, the higher values being obtained with the type A fibres which are strongly bonded to the nylon. As the fibres concentration is increased, higher stress levels are needed to develop the frontal plastic zone. This trend was analysed in terms of the mean distance between the fibre ends, d, [5, 37] and the results confirm a transition from brittle to ductile behaviour when d is lower than about 60 \xm which is six times the fibre diameter (Fig. 1 lb). Fibres A Fibres B
-•Q-
\
^ K
^
1
0 10 20 30 Concentration offibres(wt%) (a)
0
1
50
100
150
Distance between fibre ends (|jm)
(b)
Fig. 11: Evolution of the fracture toughness Kjc as a function of the fibre concentration (a) and the distance between fibre ends (b).
409
M GOMINA ETAL
410
Table II : Schematic of load-loadline displacement curves associated to the fibre-reinforced nylon materials, and the related damage and fracture mechanisms (see also Fig. 10). Loaclmg cm'\=^GS IV1iit€iiay P-flh)
Neat Nylon (GM) GIA GIB
Damage at the crack front ^p Brittle fiiacture by fliies; deb 0 ndiitg at the crack front Semi-b little fracture ' Instaible propagatiaii
Frontal damage zone growth iqp to Pi
GIDA G20B
G20B
G1QB G21H
a: tnicrovoids b: plastic defM , =c: f/m deb ending / d : matrix microniptJ
Stable growth and junction of ;/ microcracks 19 to
Damaging at the crackfinont19 to Pi Instable crack propagation at Pi due to fibres rupture at the ccrack front
G30A G30B
f^aquetsi'^ \o\ fibres ) ew damage zone
Fracture Behaviour of Short Glass Fibre-Reinforced Rubber-Toughened Nylon Composites
Brittle re I fracture
Crack tip biuntingy Frontal process zone | j^—m
^1 ^
r.te
• w^^mt^
Front of the notch ""''''""^ MF J®^/. ' ^ j M
Instable extension i stable crack propagation
Tiny secondary damage zone Front of the notch
Fig. 10: SEM micrographs of the frontal process zones typical to the fibre-reinforced PA6 materials. The type of loading curve in Table II associated to a specific fracture surface morphology is designated by a number: (A) GlOA, type 2 loading curve; (B) G20A, type 3 loading curve; (C,D) G20B, type 2' loading curve; (E,F) G30A, type 3' loading curve.
411
412
M GOMINA ETAL
Rubber-toughened PA6 The monotonic load-loadline displacement curves obtained on DZ (92wt% nylon / 8wt'^ rubber) and NZ (80wt% nylon / 20wt% rubber) materials clearly show an elastic-plast c behaviour characterized by stable crack propagation (Fig 12a). Due to the difference in tl e rubber concentration, the curve associated to the DZ material is noticeably wider than the orie corresponding to the NZ. Conversely, the JR resistance curve of the NZ material is wider and rises steeper than the one of the DZ (Fig. 12b). Table III summarizes the critical J values associated to crack initiation for the three matrices GM, DZ and NZ; also given are tl e minimum specimen sizes required for valid toughness measurement. 150
Crack initiation 5 mm/min T = 20°C
100 L
-^
50
DZ (8% EPR) NZ (20% EPR) i 5 10 15 h(mm)
J. 2
0 DZ (8% EPR) --#--NZ(20%EPR) 4 Aa (mm)
(b)
(a)
Fig. 12 : Load-loadline displacement curves (a) and J-Aa resistance curves (b) obtained on DZ and NZ materials. Table III : Critical J values and validity of fracture toughness derivation for the three matrices GM (neat nylon), DZ and NZ. Material
Qy tension.
Jlc
l^Jmin ~ VVjmin
(MPa)
(kJ/m^)
(mm)
74,4
6,0 ^Gc
X
DZ (8wt% EPR)
56,2
9,6 (valid)
4,30-8,60
NZ (20wt% EPR)
40,7
PA-6 (GM)
31,1 (non valid)
19,1-38,2
The increase of the Jic values with the amount of rubber illustrates the efficiency of the main damage mechanisms responsible of the high plastic deformation in the matrix, i.e. the stress concentration around the rubber particles and their cavitation. The SEM micrographs in figure 13 show the regular morphology with layers of matrix highly stretched in the direction of crack propagation, which can be regarded as cavitation shear bands. The morphology of the surface of rupture of the NZ material is much finer than the one of the DZ, in relation to the meaQ rubber particle size.
Fracture Behaviour of Short Glass Fibre-Reinforced Rubber-Toughened Nylon Composites
(a)
(b)
Fig. 13 : SEM micrographs of the surfaces of rupture of the DZ (a) and NZ (b) materials. Ternary blends Whatever the concentration of fibres and the state of fibre/matrix interface, stable crack propagation is obtained up to total failure. With type A interface, increasing the fibres concentration results in a loss of resilience by simultaneous reduction of the peak load and the ultimate displacement. In figure 14a are plotted the loading curves associated to the DZ and NZ matrices reinforced using 30 wt% of fibres. In the case of the DZ matrix, the enhancement of the crack growth resistance when using type B fibres is due to the strongly bonded fibre/matrix interface. For the NZ matrix, the compatibility of the type B fibres results in an appreciable rise of the peak load; the displacement at break being controlled by the ductility of the matrix. The J-Aa resistance curves determined from the plots in figure 14a are shown in figures 14b . Association of type A fibres to the rubber-toughened nylon leads to rather flat J-Aa curves as compared to the behaviour induced by type B fibres. The higher crack initiation toughness of material N30A compared to D30A is most likely due to a fretting effect as the NZ matrix is softer. The wider J-Aa curves are obtained when type B fibres are associated with the NZ matrix; which illustrates the beneficial influence of the rubber and a strong fibre/matrix interface. 26 20
QT
->
10
^,'''o
5 0
)
( (a)
<>P-^'N30B
o
15
1 +•
2
Aa N30A
(mm)
J
3
4
(b)
Fig.l4:Loading curves (a) and associated J-Aa resistance curves (b) for different ternary blends.
413
M GOMINA ETAL
414
For both types of interface, the crack initiation toughness Kic decreases monotonically with the concentration of fibres towards a plateau value of about 4.2 MPaVm comparable to the ne;it nylon (Fig. 15). This trend is different from the resuhs reported by Pecorini [36 ] who noticed that the toughness of a rubber-modified nylon 6,6 increases with the amount of short gla^ s fibres, but less that the toughness of the fibre-reinforced neat nylon. They explain the trend in the evolution of the fracture toughness as a function of the concentration of fibres by a competition between two contributions: toughness increases as the strengh rises with the concentration of fibres while their opposition to the matrix stretching at the crack front acts to reduce the frontal plastic zone size. Therefore, the results shown in figure 15 mean that in these materials a drastic reduction of the plastic zone occurs when the fibre concentration is above 10 wt%, probably because the DZ matrix (8 wt% rubber-toughened nylon) is not ductile enough. Furthermore, for a given concentration of fibres, the toughness values associated to type A fibres are always lower. This trend is related to the microscopic features observed on tie surfaces of rupture and correlated to acoustic emission analysis. SEM micrographs of tie surfaces of rupture (Fig. 16a and 16b) clearly show long pull-out lengths of the type A fibres with very smooth surfaces whereas type B fibres are covered with matrix. The cohesive rupture of the matrix reinforced with type B fibres is consistent with the occurrence of a unique population of acoustic emission events (Fig. 16a) whereas two populations are observed with type A fibres (Fig. 16b) associated to fibre decohesion and sliding mechanisms.
•m Fibres A -•-Fibres B 6
5\ -
24
•
7
-i
0
10
20
1
30
Concentration of fibres (wt%)
Fig. 15: Evolution of the fracture toughness Kjc as a function of the concentration of fibres in the DZ matrix.
Fracture Behaviour of Short Glass Fibre-Reinforced Rubber-Toughened Nylon Composites
(a)
(b) D30B
^Si
o a> U V
<£) In
/5 «
A /
J
J
J
(b')
1
"i
\\ \
_ i ij
j
Amplitude (dB)
Amplitude (dB)
Fig. 16: SEM micrographs of the surface of rupture of ternary blends with type A fibres (a) or type B fibres (b) and the associated acoustic emission distribution curves (a') or (b'). The tearing modulus continuously decreases as the fraction of fibres rises (Fig. 17a). That means once the crack is initiated, its propagation is easiest when the fibre concentration is high: the J-Aa resistance curves become more and more flat (Fig. 17b). O •
Fibres A 1 Fibres B
O 1
h
^
•
_j
t
i-J
0 10 20 30 Concentration of fibres (wt%)
1 1
1
0.5
1 Aa (mm)
(a) (b) Fig. 17 : Evolution of the tearing modulus (a) and the J-Aa resistance curves as a function of the concentration of fibres in the DZ matrix. The characteristic values of the crack initiation and propagation parameters (Jic, Kic and TR) are listed in Table IV for the different ternary blends. In all cases the requirements for valid fracture toughness measurement were fulfilled. These results confirm that type B fibres are better suited as reinforcement for rubber-toughened nylon materials. But the marked reduction
415
M GOMINA ETAL
416
of the process zone size as a function of the fibres concentration (Fig. 18) indicates that a higher amount of rubber must be used for the materials to be tough enough for the forseen applications. Table IV : Critical values of crack initiation and propagation parameters for the different ternary blends.
Material
TR*
Jic (=Gic)
Bjmin J vVjmin
Kic
(kJ/m^)
(mm)
(MPam-^^^)
D1A
1113
3.64; 7.28
5.52
6.94
D1B
11.96
3.97 ; 7.95
5.85
7.19
D10A
3.20
0.95; 1.90
3.69
5.81
D10B
6.00
1.75; 3.49
4.60
3.91
D20A
2.85
0.78; 1.56
4,16
5.87
D20B
3.38
0.97; 1.94
4.18
3.07
D30A
1.97
0.48; 0.96
4.05
2.14
D30B
3.25
0.80; 1.61
4.21
1.56
N30A
0.83
0.27 ; 0.55
2.35
3.18
N30B
4.18
1.46; 2.92
5.33
8.60
0
5
10
15
20
25
30
1
35
Concentration of fibres (wt%)
Fig. 18: Evolution of the frontal process zone size as a function of the fibres concentration for the 8 wt% rubber-toughened composites.
Fracture Behaviour of Short Glass Fibre-Reinforced Rubber-Toughened Nylon Composites CONCLUSIONS This paper was aimed at determining the influence of the amount of rubber, the concentration of fibres and the state of fibre/matrix interface on the mechanical behaviour of ternary blends. The results indicate that the stronger the fibre/matrix interface the higher the physical and mechanical characteristics. The presence of the rubber considerably enhances the crack initiation toughness, especially at the lower fibre concentrations, and increases noticeably the reliability owing to the stable crack propagation thus induced. - Addition of brittle fibres to a rubber-toughened nylon 6 provides a higher stiffness but brittleness increases with the concentration of fibres whatever the state of the fibre/matrix interface. Further improvements of the mechanical properties may be obtained with higher amounts of rubber, but then problems may arise due to the lack of workability of the blends. ACKNOWLEDGEMENTS We thank Vetrotex (Centre of Chambery, France) and EMS Chemie (Switzerland) for the materials supplied. This work is part of a research program financially supported by the Conseil Regional de Basse Normandie in the frame of the "Reseau Polymere, Plasturgie" du Grand Bassin Sud-Parisien.
REFERENCES [I] [2] [3] [4] [5] [6]
Bader M.G. and Bowyer W.H., J. Phys. D. 5 (1972) 2215. Curtis P.T., Bader M.G., Bailly J.E., J. Mater. ScL 13 (1978) 377. Nair S.V., Shiao M.L., Garrett P.D., ANTEC'93, 2552. Hashemi S., Mugan J., J. Mater. Sci. 28 (1993) 3983. Shiao M.L., Nair S.V., Garrett P.D., Pollard R.E., J. Mater. Sci. 29 (1994) 1739. Cimmino S., D'Orazio L., Greco R., Maglio G., Malinconico M., Mancarella C , Martuscelh E., Palumbo R. and Ragosta G., Polym. Eng. Sci. 24 (1984) 48. [7] Huang D.D., Wood B.A. and Flexman E.A. In Advanced Materials, Vol. 10, No 15, WILLEY-VCH Verlag Gmbh, D-69469 Weinheim. [8] Bucknall C.B., Heather P.S., Lazzeri A., /. Mater. Sci. 16 (1989) 2255. [9] Hashemi S., Williams J.G., J. Mater. Sci. 26 (1991) 621. [10] Oostenbrink A.J. and Caymans R.J., Polymer, 33 (1992) 3086. [II] Ghidoni D., Fasulo G.C., Cecchele D., Merlotti M., Sterzi G., Nocci R., J. Mater. Sci. 28 (1993)4119. [12] Keskkula H. and Paul D.R (1994). In Nylon Plastics handbook, Kohan M. (Ed), Carl Hans Verlag, Munich. [13] Leach D.C., Moore D.R. In Proceedings ofTEQC.IPC Conf (Sept 1983), Guilford, p. 330. [14] Bailey R.S. and Bader M.G. In Proceedings of 5'^ Int. Conf on Comp. Mater. (1985) 947. [15] Dijktra K., van der Wall A., Caymans R.J., J. Mater. Sci. 29 (1994) 3489. [16] Nair S.V., Shiao M.L., Garrett P.D., J. Mater. Sci. 11 (1992) 1085.
417
418
M.GOMINAETAL
[17] Gaymans RJ., Oostenbrink A.J., A.C.M. van Bennekom, Klaren J.E., Plastic Institute of London, Conf. Preprints on deformation and fracture of composites, April 1991, paper 23. [18] Nair S.V., Subramaniam A., Goettler L.A., J. Mater. Sci. 32 (1997) 5347. [19] Vansse O., (2000) Ph. D. Thesis, University of Caen, France. [20] Rice J.R., J. Appl Mech. ASME 35 (1968) 379. [21] ASTM E813-87, Standard Test Method for Jjc, a Measure of Fracture Toughness, 1987 Annual Book of ASTM standards. Part 10 (American Society for Testing and Materials, Philadelphia, Pennsylvania, p.968 [22] ASTM E813-89, Standard Test Method for Jjc, a Measure of Fracture Toughness, 1989 Annual Book of ASTM standards. Part 10 (American Society for Testing and Materials, Philadelphia, Pennsylvania, p.700. [23] ASTM E813-97, Standard Test Method for Jjc, a Measure of Fracture Toughness, 1997 Annual Book of ASTM standards. Part 10 (American Society for Testing and Materials, Philadelphia, Pennsylvania, p.802. [24] Narisawa I., Takemori M.T., Polymer Engineering and Science, 29 (1989) 671. [25] ASTM E813-81, Standard Test Method for Jjc, a Measure of Fracture Toughness, 1981 Annual Book of ASTM standards, Part 10 (American Society for Testing and Materials, Philadelphia, Pennsylvania, p.810. [26] Paris P.C, Tada H., Zahoor A. and Ernst H. In Elastic-Plastic Fracture, ASTMSTP 668 (1979), J.D. Landes, J.A. Begley and W.G. Clark (Eds), 5. [27] Pinot L., (2001) Ph. D. Thesis, Caen University, France. [28] Sato N., Kurauchi T., Sato S., Kamigaito O., J. Compos. Mater. 22 (1988) 850. [29] Choi N.S., Takahashi K., J. Mater. Sci. 33 (1998) 2357. [30] Wyzgoski M.G., Novak G.E., J. Mater. Sci. 26 (1991) 6314. [31] Malzhan J.C, Friedrich K., J. Mater. Sci. 3 (1984) 861. [32] Friedrich K, Compos. Sci. Technol. 22 (1985) 43. [33] Voss H., Friedrich K, J. Mater. Sci. 5 (1986) 569. [34] Akay M., O'Regan D.F. & Bailey R.S., Compos. Sci. Technol. 55 (1995) 109. [35] Ulrych F., Sova M., Vokrouhlecky and Turcic B., Polymer Composites, June 1993, 14 (3) 229. [36] Pecorini T.J. and Hertzberg R.W., Polymer Composites, June 1994,15 (3) 174. [37] Shiao M.L., Nair S.V., Garrett P.D., Polymer 35(2) (1994) 306.
3.2 Laminates
This Page Intentionally Left Blank
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
421
COMPARISON OFINTERLAMINAR FRACTURE TOUGHNESS BETWEEN CFRP AND ALFRP LAMINATES WITH COMMON EPOXY MATRIX AT 77K IN LN, M. HOJO*, S. MATSUDA**, B. FIEDLER***, K. AMUNDSEN*, M. TANAKA* and S. OCHIAI**** * Department of Mechanical Engineering, **** International Innovation Center, Kyoto University, Kyoto 606-8501, Japan ** Department of Chemical Engineering, Himeji Institute of Technology, Himeji 671-2201, Japan *** Polymer and Composite Section, Technical University Hamburg-Harburg, D21073 Hamburg, Germany ABSTRACT Mode I and n interlaminarfracturetoughness was investigated with unidirectional carbonfiber(CF)/ epoxy and aluminafiber(ALF)/epoxy laminates. Acommon bisphenol Atype epoxy matrix was used to highlight the influence of thefiberphysical and geometrical properties. The tests were carried out at 77K in liquid nitrogen (77K-LN2), and the results were compared with those at room temperature in air (RTair). The mode Ifracturetoughness of the neat epoxy matrix was also evaluated both in 77K-LN2 ^ ^ RT-air for comparison. Thefracturetoughness of the neat resin increased by 1.8 times by changing the test environmentfromRT-ak to 77K-LN2- ^^^ increase was directly translated into the increase of the mode Ifracturetoughness values ofALF/epoxy where the toughness in 77K-LN2 was 1.8 times higher than that in RT-air. This environmental effect for CF/epoxy laminates was smaller and the toughness in 77K-LN2 was only 1.3 times higher than that in RT-air. On the other hand, the mode II interlaminar fracture toughness values of both CF and ALF/epoxy laminates were insensitive to the test environment. The ratios of thefracturetoughness of ALF/epoxy to those of CF/epoxy were about two for both laminates in 77K-LN2 and RT-air. The mechanism of thefiberand environmental effects were discussed from the mesoscopic view points on the bases of microscopic observation. KEYWORDS Carbonfiber.Aluminafiber,Epoxy matrix, Interlaminarfracturetoughness, Liquid nitrogen INTRODUCTION Although twenty years have passed since the importance of the interlaminar strength was recognized [1,2], the individual contribution of mesoscopic structure (fiber, resin, interface and their geometrical arrangement) still remains to be understood [3,4]. One of the reasons for this delay is the difficulties in separating the effect of only one factor on the complicatedfractureprocess of composite laminates. Carbonfiber(CF)/epoxy laminates have mainly been applied to high performance structures where the superior specific strength and stiffness are requked, and the size and properties of fiber and resin differ only slightly within the limited range. Recently, aluminafiber(ALF)/epoxy laminates have been selected for cryogenic application because the thermal conductivity of ALF/epoxyfromcryogenic temperature
422
M.HOJOETAL
(4K) to room temperature (RT) is lower than that of any other structural materials [5]. One of the representative applications is the thermal insulating load support system for the superconducting magnet of the magnetic levitation trains [6]. Carbon fiber (CF)/epoxy composites have also been tested ai^ liquid hydrogen tanks for space vehicles [7], Since laminated structures are used for these applications, the evaluation of interlaminar strength at cryogenic environment is essentialfromthe view point of structural integrity. The evaluation of interlaminar strength for CF/epoxy and ALF/epoxy laminates at cryogenic temperature gives an interesting comparisonsfrommesomechanical view points. The comparison between CF and ALF indicates the effect of fiber diameter, fiber transverse modulus etc., and the comparison between RT and cryogenic temperature depicts the effect of the change in resin properties and the thermal residual stress. Thus, these results possibly separate the contributions of mesoscopic components on interlaminarfracturetoughness both under mode I and n loadings. In the present study, mode I and n interlaminarfracturetoughness was investigated with unidirectional CF/epoxy and ALF/epoxy laminates at 77K in liquid nitrogen (JllL-U^^, and the resuhs were compared with those at room temperature in laboratory air (RT-air). liie epoxy matrix was conmion for both CF and ALF/epoxy laminates to separate the effects of the fiber properties and diameter clearl). EXPERIMENTALPROCEDURE Materials and specimens Laminates used in this study were madefromCF (Toray, T300, PAN based high strength type, ^1 \jm), ALF (Sumitomo Chemical, Altex, 85%Y-Al203,15%Si02, ^^^ M^"^) ^ ^ bisphenol Atype epoxy (EpoxyH, curing temperature: 393 K). For T300/EpoxyH, 16-ply unidirectional laminates with nominal thickness of 3 mm were molded in autoclaves. For Altex/EpoxyH, 24-ply unidirectional laminates with a nominal thickness of 3 mm were molded. The volumefractionof fiber was measured as 50% for both laminates using the combustion method [8]. Plaques of neat EpoxyH (100x100x10 mm) were also molded for the measurement of the resin toughness. Elastic moduli of the neat resin and fibers aie summarized in Table 1. The glass transition temperatures (T) of the neat EpoxyH and both laminates, measured using a differential scanning calorimeter (DSC, Rigaku, Thermo Plus DSC 8230), are shown in Table 2. The value for the neat EpoxyH is the average of three tests, and those for composite laminates are the average of two tests. The agreement of T for the neat resin and laminates indicates that the microstructures of the matrix resin are similar. Double cantilever beam (DCB) and 4-point end notched flexure (4ENF) tests [9,10] were used for the evaluation of interlaminarfracturetoughness of both laminates under mode I and n loadings, respectively, both in RT-air and 77K-LN2' Conventional 3-point ENF (3ENF) tests were carried out only for Altex/EpoxyH laminates at RT. Figures 1 and 2 show examples of the DCB and 4ENF specimens. The width and the span length of the specimens used in 77K-LN2 were smaller than those for standiird test methods (width, B=20 to 25 mm, span, 2L=100 mm) [9,11] because of limitations of the space for the tests in liquid nitrogen. The details of the specimen size are summarized in Table 3. For T300/EpoxyH laminates, the width of the specimens in RT-air was made the same as that in 77K-LN2.4-point ENF Table 1. Elastic moduU of neat resin and 5
Mass density (g/cm )
El
Elastic moduli (GPa)
Ej
Poisson's ratio
Vi?,
Gl2
EpoxyH T300 1.2 1.76 3.2 235 3.2 9.1 1.2 7.2 0.39 0.28
fiber. Altex 3.30 210 210 81 0.29
"^f'^ 2. Glassfransitiontemperature of neat resm and lammate Glass transition Laminates temperature, T„ EpoxyH T300/EpoxyH Altex/EpoxyH
389 396 395
423
Comparison of Interlaminar Fracture Toughness between CFRP andALFRP Laminates
. Polyimide film , Precrack
Polyimide film
ME:
Precrack of
30
-Q.
J^
TJ 1 8 .
TJ
50
Fig 1. Double cantilever beam specimen (Dimensions are in mm).
Fig 2. Four point end notched flexure specimen (Dimensions are in mm).
Table 3. Sunmiary of type and dimension of specimens. Mode Laminates RT-air 77K-LN2 Width, B (mm) 10 10 ^''^^'^ T300/EpoxyH Length (mm) 150 90 Width, B (mm) 20 13 Altex/EpoxyH Length (mm) 170 100 Method 4ENF Mode II 4ENF T300/EpoxyH Width, B (mm) 10 10 100/60 50/30 Span fmm) Method 3ENF 4ENF Altex/EpoxyH Width, B (mm) 20 10 100 50/30 Span fmm) Mode I ao = 25 mm Mode II ao = 25 mm for 3ENF, support span = 100 mm ao = 35 mm for 4ENF, support span = 100 mm ao = 18 mm for 4ENF, support span = 50 mm * Support span/Loading nose span tests were introduced in order to stabilize the crack growth under mode n loading without controlling the crack shear displacement [12]. This was necessary because it is difficult to measure the crack shear displacement in 77K-LN2. Starter slits were introduced into the specimens by inserting 12.5jxm thick polyimidefilmduring molding. The surface and edge of thefilmwere coated with a mold release agent before molding. Mode I precracks of lengths from 1 to 5 mm were introduced into the specimens by clamping the specimens across the entire width approximately at the end of the starter slits and then manually wedging open the specimens. Two tests were carried out for each test condition. Compact (C(T)) specimens of 25x24x10 mm were cut from EpoxyH plaques. The loading pin holes were drilled using HSS tools. The specimens were heat-treated at 413 K (about 20 K above the glass transition temperature) for 1 hr followed by slow cooling at the rate of 1 K/min to reduce the residual stress during curing. Precracks of 1 mm were introduced into the C(T) specimens by tapping razor blades. Figure 3 shows the apparatus used for tapping. Four tests were carried out for each test environment. Basic mechanical properties Bending moduli of the neat resin and the laminates in thefiberdirection were measured using three point bending tests in RT-au* and 77K-LN2- Th^ specimens used for neat resin were 4 nrni thick, 5 mm wide
424
M.HOJOETAL
Apparatus for tapping
C(T) specimen
Fig. 3. C(T) specimen and tapping apparatus for initial crack.
Fig. 4. Photo of loading apparatus for mode II tests in77K-LN..
and 80 mm long. For the bending tests of laminates in fiber direction, uncracked parts of the ENF specimens were used. The three point bending tests were carried out with the support span of 50 mm and the cross head speed of 0.5 min/min. The testing machine compliance was taken into account for the calculation of the bending modulus. The bending strength in the transverse dkection of the laminates were also measured using three point bending tests. Small specimens cut from ENF specimens (width=5 mm, length=20 nmi, nominal thickness=3 nmi) were used for tests in RT-air and 77K-LN2- The support span was 16 nmi, and the cross head speed was 0.2 mm/min. Five tests were carried out for each condition. Fracture toughness tests The tests were carried out in a computer-controlled servohydraulic testing system (Shimadzu 4880,9.8 kN). Load cells of 490 N and 2.45 kN capacity were attached for the tests under mode I and II loadings, respectively. A cage or gate type load frame was designed for tests in 77K-LN2 as shown in Fig. 4. A special loading apparatus with universal joints were used for the tests both under mode I and n loadings to avoid unequal crack growth in the width direction [9,13,14]. All tests at 77K were carried out in liquid nitrogen. The laminate specimens installed in the loading apparatus werefirstplaced about 5 mm above the liquid nitrogen surface followed by dipping deeply in liquid nitrogen for about 30 minutes. Tests were started after the liquid nitrogen had stopped boiling. Since the thickness of the neat resin C(T) specimen is large, the neat resin fracture toughness tests were started after dipping the specimen for about 80 minutes. The cross head speed was controlled to be 0.5 mm/min in C(T) and DCB tests [15], and 0.2 mm/min in 4-point ENF tests [10]. The crack shear displacement was controlled to be 0.01 mm/min in 3-point ENF tests [9,12]. Ilie energy release rate, Q for C(T) specimens were calculated from the stress intensity factor, K, using G=(l-v2)K2/E where v is the Poisson's ratio and E is the elastic modulus of the neat resin. The energy release rate for DCB tests under mode I loading was calculated using the modified compliance calibration method [9,15]. That for ENF tests under mode II loading (both 3- and 4-point ENF) was calculated using experimentally obtained compliance calibration curves for each specimen to avoid the effect of the specimen thickness scatter [14,16]. Bending moduli of laminates were also used for these calculations. The initial values of thefracturetouglmess (Gj^(NL), Gjj^(NL)) were evaluated using the nonlinear point which were determined by the point of deviation from linearity in the initial load-displacement curve. EXPERIMENTAL RESULTS AND DISCUSSION Effects of temperature on modulus and strength Table 4 shows the summary of the bending modulus in RT-air and 77K-LN2- ^ ^ modulus of neat EpoxyH in 77K-LN2 was more than twice higher than that at RT. Change of the matrix modulus affected the shear moduli of laminates, G^^, calculated by the Halpin-Tsai equation. Since the shear deformation of laminates contributes to the bending deformation, the bending moduli infiberdirection, Ej^ is often lower than the tensile moduli in thefiberdirection, Ej. The estimated Ej based on E^^ and the consider-
Comparison of Interlaminar Fracture Toughness between CFRP andALFRP Laminates
425
Table 4. Bending modulus of neat resin and laminates. EpoxyH Laminates T300/EpoxyH Altex/EpoxyH RT-air 77K-LN. RT-air 77K-LN. RT-air 77K-LN. 1.2 2.2 1.5 Mass density (g/cm^) Volume fraction of fiber, Vf (%) 49.5 49.6 108 100 3.2 6.8 Measured 91 99 Eib Elastic moduli G12 3.65 1.4 2.4 Halpin-Tsai 4.0 6.8 5.8 (GPa) El* 119 117 103 108 121 119 107 108 Calculated based on E15 and consideration of shear deformation ** Calculated based on fiber and matrix moduli and rule of mixture Table 5. Bending strength in transverse direction. Material Temperature T300/EpoxyH Altex/EpoxyH
Bending strength in transverse direction (MPa) RT-air 77K-LN2 88 150 110 260
ation of the shear deformation agreed well with the elastic moduli in fiber direction calculated by the rule of mixture, Ej^^^^. Thus, these facts indicate reliability of the measured values, and the importance of the contribution of the shear deformation components. The E^^ values obtained were used for the calculation of the energy release rate under mode n loading. The bending strength of composite laminates in transverse direction is presented in Table 5. The strength of Altex/EpoxyH was 1.3 times (RT-air) and 1.7 times (77K-LN2) higher than that of T300/EpoxyH. The strength increase by changing the test environment from RT-air to 77K-LN2 is 1.7 times for T300/ EpoxyH and 2.3 times for Altex/Epoxy. Often the coefficient of thermal expansion of the fiber is one order lower than those of epoxy matrix. Thus, the anisotropic modulus of carbonfiberindicated in Table 1 gives lower compressive radial thermal residual stresses at the fiber/matrix interface [14,17]. Resin strength has been suggested to increase by lowering the test temperatures [18,19]. These possibly contribute to the increase of the bending strength in transverse direction for both laminates. The difference of the transverse strength at the same temperature indicates that the apparent interfacial strength of Altex/ Epoxy is higher than that of T300/EpoxyH both in RT-air and 77K-LN2. Fracture toughness of neat resin The load-displacement curves for C(T) tests of the neat EpoxyH were almost linear until thefinalunstable fracture. Thefi-acturetoughness value in 77K-LN2 ^ ^ ^^^ ^1^ ^ ^ ^^^^ ^ RT-air was 120 J/m^. Thus the toughness increased by 1.8 times by changing the test environment from RT-air to 77K-LN. Brown and co-workers have found that amorphous polymers crazed in 77K-LN2' ^^^ ^^^ ^ ^ helium or vacuum at about 78K [20-22]. They have also reported that the stress-strain behavior of all polymers, amorphous and crystalline, is affected by Nj at low temperatures [22]. Kneifel has reported that the fracture toughness of epoxy in 77K-LN2 i^ higher than that in RT-air and 5K, and that the reason for this is the reduced notch effect by plastic deformation [23]. Then, the increase of the fracture toughness of the neat EpoxyH in this study is probably caused by the similar effect. Mode I interlaminar fracture toughness Figure 5 indicates the relationship between the load and crack opening displacement for both T300/ EpoxyH and Altex/EpoxyH laminates in 77K-LN2- Stick-slip behavior was observed for both laminates, and the crack growth was partially unstable. This behavior was only observed in 77K-LN2' ^ ^ the crack growth in RT-air was stable with the smooth load-displacement relations. The crack length was measured using traveling microscopes during the tests, and the obtained relationship between the specimen compliance and the crack length was used for the calculation of the energy release rate for the
M.HOJOETAL
426
^^T300/EpoxyH, Altex/EpoxyH Mode I. 77K-LN,
50 I • ' \ ^ ' • ' I ' ' ' I • • ' I ' ' ' I • ' ' r
.^^VM, _D
T300/EpoxyH
25 I
-Altex/EpoxyH
0
2 4 6 8 10 12 Crack opening displacement, 6 (mm)
0.8 1.0 1.2 1.4 Compliance, (BC)^'
1.6
1.8
Fig. 5. Relationship between load and crack opening Fig. 6. Relationship between normalized crack displacement for mode I interlaminar fracture tough- length and compliance for DCB specimen in ness tests in T300/EpoxyH and Altex/EpoxyH lami- T300/EpoxyH laminates in 77K-LN2nates in 77K-LN2.
10 20 30 40 50 60 Increment of crack length, Aa (mm) Fig. 7. Effects of fiber type (T300 and Altex) and test environment (RT-air and 77K-LN2) on the relation between mode I interlaminar fracture toughness and increment of crack length in fiber/EpoxyH laminates. tests in RT-air. For the tests in 77K-LN2» the stick-slip behavior explained above left the traces of crack fronts on thefracturesurface shown by white lines [24]. These traces indicated the stable crack growth region during the stick-slip corresponding to the maximum points of each serration pattern. Then, the crack length was measured from these white traces on thefracturesurfaces. Figure 6 shows two ex-
Comparison of Interlaminar Fracture Toughness between CFRP and ALFRP Laminates
All
Table 6. Summary of mode I toughness values for neat resin and laminates. EpoxyH Material T300/EpoxyH Altex/EpoxyH Temperature RT-air 77K-LN. 77K/RT| RT-air 77K-LN, llYJKVi RT-air 77K-LNo 77K/RT Gic(J/mV 120 210 1.8 120 160 1.3 210 370 1.8 250 460 1.8 160 210 1.3 Gis(J/mO 1.2 1.2 1.3 1.3 * NL for laminates, Pmax for neat resin G^: Average of plateau region of GJR amples of the relationship between the crack length, a, normalized by the thickness, 2h, and the compliance, C, normalized by the specimen width, B in 77K-LN2 ^^r T300/EpoxyH laminates. The fit of the data to the straight lines is quite good, and this relation for each specimen was used for the calculation of the energy release rate in 77K-LN2. These maximum points of each serration pattem were used for the calculation of the propagation values of the energy release rate, Gjj^, because these points correspond to the stable crack growth. The relation between the mode I fracture toughness and the increment of the crack length (R-curve) for both composite laminates are presented in Fig. 7. In thisfigure,the open and solid marks indicate the results in RT-air and 77K-LN2' ^ ^ ciidQ and square marks indicate the results for T300/EpoxyH and Altex/EpoxyH laminates, respectively. The data points at Aa=0 correspond to the initial values of the fracture toughness, Gj^, at the nonlinear point {GJNV}). The toughness values increased slightly from the initial values, Gj^, and then levelled off where Aa is larger than 2 to 5 mm. The average of the plateau region of Gjj^ was calculated as Gj^ when Aa is larger than 10 to 20 mm. Table 6 summarizes the averaged Gj^ and Gj^ values for the neat resin and composite laminates. The ratios of Gj^ to G^^ were 1.2 to 1.3. These ratios were rather small, and insensitive to laminates and environments. On the other hand, the ratios of the toughness values in 77K-LN2 to those in RT-air showed clear dependency of the laminates. This ratio for Altex/EpoxyH was 1.8, while that for T300/EpoxyH was only 1.3. It is interesting to note that this ratio for Altex/EpoxyH and that for the neat EpoxyH are the same. The comparison between Gj^ of the composite lammates and the neat resin gave the result that Gj^ of T300/EpoxyH was similar to that of the neat EpoxyH. On the other hand, Gj^ ofAltex/EpoxyH was much higher than that of the neat EpoxyH. The reason for this difference will be discussed later in the microscopic observation. Mode II interlaminar fracture toughness Figure 8 indicates the relationship between load and load-line displacement for both composite laminates in 77K-LN2 using 4-point ENF specimens. For the case of T300/EpoxyH, the crack growth was stable, and only one unstable growth was observed in the final stage of the test. The stick-slip behavior similar to that under mode I was observed for Altex/Epoxy laminates. The crack growth in RT-air was stable as that under mode I loading for both laminates. For the tests in RT-air, the crack length was measured using traveling microscopes during the tests, and the relation obtained between the specimen compliance and the crack length was used for the calculation of the energy release rate. For the tests in 77K-LN2' the crack length was measured from the white traces of the stick-slip behavior on the fracture surfaces. Figure 9 shows examples of the relationship between the compliance parameter, EjjjBC(2hy, and the crack length, a, where Ej^ is the bending modulus infiberdirection, B is the specimen width, and 2h is the thickness of the specimen. The dashed line indicates the relationship for T300/EpoxyH obtained using the simple beam theory [14]. Although the number of data points was not e n o u ^ for T300/ EpoxyH because of rather stable crack growth, and the scatter of the data points for Altex/EpoxyH was larger than that under mode I, thefitof the data to the straight line is acceptable for the calculation of the energy release rate. For the case of the stick-slip growth, the maximum points of each serration pattem were used for the calculation of the propagation values of the energy release rate, G^^, because these points correspond to stable crack growth. The relation between the mode n fracture toughness and the increment of the crack length (R-curve) for both composite laminates are shown in Fig. 10. Only one test was successfully carried out for Altex/ EpoxyH in 77K-LN2. The initial increase of the toughness is much larger than that under mode I loading. Thus, the toughness values reach plateau values when Aa was larger than 2 to 5 mm. The average of the
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M.HOJOETAL
J300/EpoxyH, Altex/EpoxyH Mode M.77K-LN, (xlO^TSOO/EpoxyH, Altex/EpoxyH Mode II, 77K-LN. I ' ' • • I 1200 max^
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Fig. 10. Effects of fiber type (T300 and Altex) and test environment (RT-air and 77K-LN2) on the relation between mode n interlaminarfracturetoughness and increment of crack length in fiber^poxyH laminates. plateau region of Gjjj^ was calculated as Gjj^ where Aa is larger than 2 to 6 nmi. Table 7 sunmiarizes the averaged Gjj^ and G^^ values for the composite laminates. The ratios of Gjj^ to Gjj^ were 1.2 to 1.9, and those in 77K-LN2 were lower than those in RT-air. The ratios of GHc in 77K-LN2 to that in RT-air (1.2 for T300/EpoxyH and 1.6 for Altex/EpoxyH) were similar to those under mode I loading. On the other hand, the propagation values were insensitive to the test envnonment for both laminates.
Comparison of Interlaminar Fracture Toughness between CFRP andALFRP Laminates
429
Table 7. Summary of mode n toughness values for composite laminates. T300/EpoxyH Material Altex/EpoxyH Temperature RT-air 77K-LN9 77K/RT RT-air 77K-LN9 77K/RTI 440 530 1.2 680 1100 1.6 Gnc(J/mV 710 720 1.0 1300 1300 1.0 Gns(J/mO 1.6 1.4 1.9 1.2 Gns/Gflc * NL Gns: Average of plateau region of GnR Table 8. Ratio of toughness value for Altex/EpoxyH to that forTSOO/EpoxyH
RT-air 77K-LN2
(Altex/EpoxyH)/(T300/EpoxyH) Mode I Mode II Gfc Gis Gnc Gns 1.8 1.6 1.5 1.8 2.3 2.2 2.1 1.8
Table 8 compares the ratios of the toughness values for Altex/EpoxyH to those forT300/EpoxyH under both mode I and n loadings at RT and 77K. Although the temperature effect on the interlaminar fracture toughness values much depends on the kind of fiber (T300 or Altex) and the fracture mode (mode I or n), the ratios of the toughness values ofAltex/EpoxyH to those of T300/EpoxyH under the same loading mode and temperature were about two and the range is rather limited (1.5 to 2.3). Micrographic observation and mechanism consideration Figure 11 shows the micrographs of the transverse section of both laminates at a prepreg interface which is not affected by the starterfihn.The diameter of Altex is about twice as much as that of T300, and this difference is directly transferred to the difference of the thickness of the resin rich region at the prepreg interface. The crude estimation of the thickness of the resin rich layer for T300/EpoxyH is 10 jim, and that for Altex/EpoxyH is 20 ^m. Figure 12 compares the fracture surfaces of T300/EpoxyH and Altex/EpoxyH laminates in RT-air and 77K-LN2 both under mode I and n loadings. Arrows indicate the crack growth direction. The fracture surfaces under mode I loading indicate a clear effect of fibers. For T300/EpoxyH, the areal ratio of interfacial fracture is 70%(RT-air) to 50%(77K-LN2). On the other hand, about 70%(RT-air) to 100%(77K-LN2) of the surface is covered with resin for Altex/Epoxy laminates. This is probably the reason why the increase of the toughness by lowering the temperature for Altex/Epoxy was the same as that for neat EpoxyH. The increase of the resin toughness was not directly translated into the increase of the interlaminar fracture toughness for T300/EpoxyH owing to the interfacialfracture.The lower compressive radial thermal residual stress is probably responsible for this interfacialfracture[14]. Since
(a)T300/EpoxyH (b) Altex/EpoxyH Fig. 11. Optical micrographs of transverse section for both composite laminates.
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(b) Altex/EpoxyH under mode I
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Comparison of Interlaminar Fracture Toughness between CFRP andALFRP Laminates
431
resin-rich fracture surface is observed for Altex/EpoxyH, the difference between the toughness for Altex/ EpoxyH and that for the neat EpoxyH looks contradictory at first sight. This fact explains clearly the importance of the mesoscopic effect in the interlaminarfracturetoughness of composite laminates. Complicated crack path affected by the geometrical arrangement offibers,contribution offiberbridging etc. are possible cause for the increase of the toughness for Altex/EpoxyHfromthat for the neat EpoxyH. The reason for the ratio of the toughness of Altex/EpoxyH to '1300/EpoxyH is more complicated. As shown in Table 5, the ratio of the transverse strength of Altex/EpoxyH to T300/EpoxyH is lower than that for the fracture toughness. However, the linear increase of the fracture toughness with the increase of the thickness of the resinrichregion is reported when the thickness is lower than about 100 jmi [25]. The damage zone at the crack tip is often arrested byfibersand this size was correlated to the change of the toughness [26,27]. Thus, similar mechanisms contributed to the difference between the toughness of Altex/EpoxyH and T300/EpoxyH. Fracture surfaces under mode n loading are also shown in Fig. 12. Microcracking normal to the principal stress or formation of hackles which are typical for the mode n interlaminarfracturewas observed [28]. The hackle size in 77K-LN2 i^ i^iuch smaller than that in RT-air. This means that the resin volume involved in the formation of hackles is much smaller in 77K-LN2. On the other hand, the toughness of the neat EpoxyH in 77K-LN2 i^ ^^^^ ^^ much as that in RT-air. Then, the increase of the resin toughness and the decrease of the resin volume involved in hackle formation, i.e. the decrease of resin shear plastic deformation may compensate for each other. This is probably the reason for the insensitiveness of the propagation values of the mode nfracturetoughness to the testing environment. The initial values can be related to the onset of microcracks. Since these microcracks are located perpendicular to the principal tensile stress, the deformation mode is locally mode I, and the initial values are affected by the increase of the resin toughness by changing the environmentfromRT-air to 77K-LN2- The reason for the ratio of the toughness ofAltex/EpoxyH to T300/EpoxyH is rather simple under mode n loading. Since the whole resinrichregion at the prepreg interface is involved into deformation andfracture,the increase of volume of resinrichregion was directly translated to the increase of the interlaminarfracturetoughness under mode n loading. The difference of the size of the hackle pattem indicates this fact. The areal ratio of the interfacial fracture is 80%(RT-air) to 50%(77K-LN2) for T300/EpoxyH. This for Altex/EpoxyH is 50%(RT-air) to 40%(77K-LN2)- Since the mode n interlaminarfracturetoughness is sensitive to the interfacial strength, this fact also contributes to the difference between the laminates [29]. CONCLUSIONS Mode I and n interlaminarfracturetoughness was investigated with unidkectional CF/epoxy and ALF/ epoxy laminates in RT-air and 77K-LN2- ^ ^ " ^ ^ ^ ^ epoxy matrix was used for both laminates. Mode Ifracturetoughness of the neat resin was also measured for comparison. The results are summarized as follows: (1) The mode Ifracturetoughness of the neat resin increased by 1.8 times changing the environment from RT-air to 77K-LN2. (2) The ratio of the mode I interlaminarfracturetoughness in 77K-LN2 ^^ ^^^^ ^^ RT-air was 1.8 for ALF/epoxy laminates. This ratio was decreased to 1.2 to 1.3 for CF/epoxy laminates. Thus, the increase of the matrix toughness was directly translated into the increase of the interlaminarfracturetoughness for ALF/epoxy laminates. The effect was smaller for CF/epoxy laminates. Fractographic observation indicated the contribution of resin is larger for ALF/epoxy laminates. Dominant interfacial fracture concealed the effect of resin for CF/epoxy laminates. (3) Propagation values of thefracturetoughness under mode n loading were insensitive to test environment for both CF/epoxy and ALF/epoxy laminates. Only initial values increased by changing the environmentfromRT-air to 77K-LN2- This amount of increase was similar to that under mode I loading. Deformation andfractureof hackles are responsible for this insensitivity.
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(4) The ratio of the toughness for ALF/epoxy to that for CF/epoxy was about two, and this ratio remained constant without respect to thefracturemode and test environment. This ratio was correlated to geometrical factors such as the thickness of the resin rich region at the prepreg interface. ACKNOWLEDGEMENTS The authors would like to thank Messrs. S. Machida, T Kawada and T. Ando for their help in carrying out experiments. REFERENCES 1. 2. 3. 4.
O'Brien, T.K. (19^2)ASTMSTP 775,140. (1983) Standard Tests for Toughened Resin Composites, NASA RP1092, Ochiai, S. and Hojo, M. (1994) Composite Interfaces 2,365. Hojo, M., Matsuda, S. and Ochiai, S. (1997) Proc. of the International Conference on Fatigue of Composites, 15. 5. Takeno M., Nishijima, S., Okada, T, Fujioka, K., Tsuchida, Y. and Kurokawa, Y. (1985) Advances in Cryogenic Engineering 32, 217. 6 Aoyama H., Hattori, T, Iwasa, M., Nemoto, T, Sonobe, T, Takizawa, T, Watanabe, H., Fukushi, K., Ichinose, M. and Terai, M. (1996) Trans. Japan Society ofMechanical Engineers, Sen A 62., 1519. 7. Aoki, T, Ishikawa, T, Kumazawa, H. and Morino, Y. (1999) Proc. ICCM12, CD-ROM. 8. (1991) JIS K7075. 9. (1996) JIS K7086. 10. Martin, R.H., Elms, T. and Bowron, S. (1998) Proc. of the 4th European Conference on Composites: Testing & Standardization, 161, The Chamelon Press. 11. (2000) ISO 15024. 12. Kageyama, K., Kikuchi, M. and Yanagisawa, N. (1991)ASTM STP 1110, 210. 13. Hojo, M., Ochiai, S., Gustafson, C.-G. andTanaka, K. (1994) Engineering Fracture Mechanics 49, 35. 14. Hojo, M., Matsuda, S., Fiedler, B., Kawada, T, Moriya, K., Ochiai, S. and Aoyama, H. (2002) InternationalJ. of Fatigue, 24,109. 15. Hojo, M., Kageyama, K. and Tanaka, K. (1995) Composites 26, 243. 16. Matsuda, S., Hojo, M. and Ochiai, S. (1999) JSME InternationalJ., Sen A 42,421. 17. Fiedler, B., Hojo, M., Ochiai, S., Schulte, K. and Ochi, M. (2001) Composites Science and Technology 61, 95. 18. Kinloch, A.J. and Young, R.J. (1983) Fracture Behaviour of Polymers, 288, Elsevier. 19. McCrum, N.G, Buckley, C.R and Bucknall, C.B. (1997) Principles of Polymer Engineering, 2nd ed., 191, Oxford University Press. 20. Parrish, M.F. and Brown, N. (1972) Nature (Physical Science) 237,122. 21. Brown, N. and Parrish, M.F. (1972)/. of Polymer Science: Polymer Science Edition 10,777. 22. Brown, N. (1973)/. of Polymer Science: Polymer Physics Edition 11, 2099. 23. Kneifel, B. (1979) In: Nonmetallic Materials and Composites at Low Temperatures, 123, Clark, A.F., Reed, R.R and Hartwig, G (Eds.) Plenum Press. 24. Matsuda, S., Hojo, M., Ochiai, S., Moriya, K. and Aoyama, H. (1999) Trans. Japan Society of Mechanical Engineers, Ser.A 65,2411. 25. Sela, N., Ishai, O. and Banks-Sills, L. (1989) Composites 20,257. 26. Kinloch, A.J. and Shaw., S.J. (1981)/. Adhesion 12,59. 27. Ozdil, F. and Carlsson, L.A. (1992) Engineering Fracture Mechanics 41,645. 28. Bradley, W.L. (1989) IR: Application of Fracture Mechanics to Composite Materials, 169,, Friedrich, K. (Ed.) Elsevier. 29. Hojo, M., Ochiai, S., Tsujioka, N., Kotaki, M., Hamada, H. and Maekawa, Z. (1995) Proc. COMP'95,3, Paipetis, S.A. and Youtsos, A.G (Eds.) University of Patras, Greece.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
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DELAMINATION FRACTURE IN CROSS-PLY LAMINATES: WHAT CAN BE LEARNED FROM EXPERIMENT? A.J. BRUNNER^ and B.R.K. BLACKMAN* ^ EMPA, Polymers/Composites Laboratory Ueberlandstrasse 129, CH-8600 Duebendorf, Switzerland * Department of Mechanical Engineering, Imperial College London Exhibition Road, London SW7 2BX, UK
ABSTRACT With the initial aim of evaluating the applicability of the ISO 15024 standard for fracture toughness testing for non-unidirectional composite laminates, ESIS TC4 has conducted a number of round robin studies on cross-ply laminates with different stacking sequences. We report here the results of two test laboratories from the S^'^ round robin activity in which double cantilever beam (DCB) specimens made from unidirectional (0°/0°-interface) composite laminates were compared to DCB-specimens with 0790°- and 90790° interfaces. In the analysis, correlation with observations from the delamination growth (e.g., deviation from the mid-plane) and post-test (e.g., fracture surface) inspection was attempted. The results are compared with some results in the literature. KEYWORDS Cross-ply laminates, Mode I delamination fracture, Composite, R-curve. INTRODUCTION The development of a standard test procedure for Mode I delamination fracture of unidirectional fibre-reinforced composites [1] leads naturally to the question of whether this procedure is also applicable to composites with other lay-ups. Since delamination fracture in multi-directional composites was known to show a wide range of complicated phenomena [2] that make a consistent, quantitative analysis of the data virtually impossible, cross-ply laminates with alternating 0°- and 90°-layers were considered a first step towards testing of engineering laminates without necessarily presenting the frill complexity of multi-directional laminates. First test results on carbon-fibre reinforced epoxy (CFRP) cross-ply laminates (T300/970, symmetric and non-symmetric lay-up) had been promising [3, 4] in spite of reports indicating that at least some crack-branching seemed inevitable [5]. The present paper
A.J. BRUNNERANDB.R.K. BLACKMAN
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reports the results of the 3"^^ round robin conducted within the European Structural Integrit} Society (ESIS) Technical Committee 4 using cross-ply laminates with another carbon fibrereinforced epoxy (IM7/977-2). The emphasis in the present work is on selected approaches for the analysis and on the interpretation of the test results based on observations from the tests, complemented by post-test analysis of the CFRP-specimens. EXPERIMENTAL DETAILS The laminates were provided by Cytec Fiberite Pic, the details are summarised in Table 1. Specimens were cut and prepared at Imperial College, the nominal width was 20 mm and the nominal length 175 mm. The round robin participants were asked to perform the tests according to the Draft International Standard version of the ISO standard test foi unidirectional fibre-reinforced laminates [1]. This implies two load cycles, one from the insert starter film, the second from the Mode-I pre-crack after unloading from the first cycle. The procedure leaves only choices of the cross-head speed between 1 and 5 mm/min and the option of using a travelling microscope or following delamination propagation by eye. Laboratory 1 used a cross-head speed of 1 mm/min and a travelling microscope, laboratory 2 used 2 mm/min and observation by eye. Table 1: CFRP laminate types (all IM7/977-2) and nominal dimensions Specimen Material and thickness of thickness insert starter foil [|Lim] [mm] 53 4 FEP * /12.5 [0°]24 unidirectional 53 4 FEP / 12.5 [0790°]6s symmetric 53 4 FEP/12.5 [0°/90°]i2 non-symmetric FEP = tetra-fluor-ethylene-hexa-fluor-propylene copolymer Lay-up
Total insert length 1mm]
PHENOMENOLOGY AND OBSERVATIONS Observations during the test, as reported by all round robin participants included: (1) Repeated unstable delamination growth (stick-slip) in most specimens with symmetric [0790°]6s cross-ply lay-up, occurring throughout the test. (2) Some unstable delamination growth (stick-slip) in most specimens with non-symmetric [0790°] 12 cross-ply lay-up, mostly occurring at longer delamination lengths. (3) Delamination propagation was stable in all specimens with unidirectional lay-up (4) Deviation from the mid-plane during delamination propagation in the cross-ply lay-up, specifically jumping from one adjacent unidirectional layer to that on the other side (symmetric) at regular delamination length intervals, resulting in a "saw-tooth"-type of fracture surface (Fig. 1). (5) Some fibre-bridging by fibre bundles from the 90°-plies perpendicular to the direction of delamination propagation in most specimens for the symmetric and the non-symmetric lay-up, varying in amount from one specimen to the other (Fig. 1) as well as fibre bridging in the unidirectional lay-up. The amount in the unidirectional specimens, ii' quantified, was low or medium.
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(6) Unstable delamination growth (stick-slip) has been observed to coincide with "jumps" from one side of the specimen mid-plane to the other during "saw-tooth" delamination propagation in the cross-ply laminates. (7) No observations of crack-branching or multiple delamination propagation (common in multidirectional lay-ups [2, 5]) in either type of cross-ply laminate have been reported.
Fig. 1: Regular "saw-tooth"-type fracture surface during delamination propagation in (left) non-symmetric lay-up [0°/90°]i2 and (right) symmetric lay-up [0790°]6s, (top row) low and (bottom row) high amount of fibre-bridging (photographs courtesy of D.D.R. Cartie). The visual inspection after the test showed relatively smooth, featureless fracture surfaces for the unidirectional lay-up (Fig. 2). Both symmetric ([0790°]6s) and non-symmetric lay-up ([0790°] 12) yielded two different types of fracture surface. Some of the specimens yielded a rather rough saw-tooth pattern consistent with that shown in Fig. 1. The ridges and troughs are, in general, evenly spaced but do not run strictly parallel to each other (Fig. 2). Other specimens showed, in the beginning, near the insert film the "saw-tooth" pattern with a gradual transition to an increasing area with a smooth surface (Fig. 3). At least three specimens with symmetric and two with non-symmetric lay-up showed this transition initiating on the specimen edge near the starter film. In some specimens, this transition started at a later stage, usually inside the fracture surface, not at the edge. DATA ANALYSIS AND RESULTS An analysis of photographs (e.g., Fig. 1) of the "saw-tooth" pattern yields an average "wavelength" of about 1.22 mm and an "amplitude" of about 0.16 mm for the non-symmetric layup [0790°]i2. The values for the symmetric lay-up [0790°]6s are 2.56 mm and 0.38 mm. The average specimen thickness of close to 4 mm (with some thickness variation) yields an average ply-thickness of 0.17 mm. The "saw-tooth"-amplitude, therefore correspond to one and two plies, respectively.
436
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[0124-4
[0°/90l6s-4
Fig. 2: Photographs of the fracture surface of one specimen with unidirectional (left) and of one with symmetric lay-up (right) after the test, specimen width 20 mm each.
Fig. 3: Photographs of the fracture surface of two types of specimens with non-symmetric lay-up after the test, specimen number is indicated beside the lay-up. Note the shorter wavelength of the topography compared with the symmetric lay-up (Figure 2), specimen width 20 mm each.
A first comparison of the three specimen types ([0°]24, [0790°]6s, and [0790""] 12) is shown in Fig. 4. The load-displacement values used in the analysis are plotted for all specimens tested in one laboratory (testing from the Mode-I pre-crack). Both cross-ply laminates show larger displacements for comparable delamination lengths, much more scatter and somewhat higher but comparable maximum load values compared with the unidirectional lay-up. There is no clear difference between the cross-ply laminate types, except that the symmetric lay-up yields the lowest load values and hence the largest scatter. In the data from the other laboratory, this is reversed, i.e., the non-symmetric lay-up showing the larger scatter. Corrected Beam Theory (CBT) values of initiation from the Mode-I pre-crack according to [1] are summarised in Tables 2-4 for the two laboratories. The values in Table 2 can be compared with the average values for the unidirectional lay-up reported by a third participating laboratory in [4] for initiation from the insert 273 J/m^ (standard deviation 2.1%), initiation from the Mode-I pre-crack 304 J/m^ (2.8%), and maximum propagation values of 369 J/m^ (1.9%). The sign convention for the A-value is explained in the notes to the tables.
Delamination Fracture in Cross-Ply Laminates
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Displacement [mm]
Fig. 4: Load-displacement values for the three laminate types tested in one laboratory (light grey symbols [0°]24, black symbols [0°/90°]6s, dark grey symbols [0790°] n), filled/open symbols indicate specimens with a single/mixed topography of the fracture surface (open symbols are connected by dashed lines to guide the eye). Table 2: Data for unidirectional lay-up (S.D. = standard deviation) Calculated Maximum Interlaminar A [mm] *** Gic [J/m'l £-modulus propagation MAX/5% delamination [GPa] [J/m^] offset (CBT) * length [mm] 162 359 309 8.3 68.5 155 7.1 365 316 67.0 154 341 324 6.7 65.0 3.5 136 350 332 62.0 137 4.9 358 333 65.0 6.1 149 355 323 12 / 7.8% 10 / 3.2% 1.9/30.3% 9 / 2.6% 175 362 304** 5.3 60.0 159 332 6.7 285** 60.0 150 3.6 306 290** 60.0 1 2/3 145 325 2/4 280 -1.5'^ 60.0 4.4 160 338 2/5 337 60.0 158 299 5.0 333 Average 1.3/26.4% 11/7.3% S.D. 23 / 7.7% 20/6.1% (*) Corrected Beam Theory analysis [1], loading from insert yielded average MAX/5% of (286 ± 24) J/m^, standard deviation of 8.4% for laboratory 1 and of (286 ± 63) J/m^, standard deviation of 21.9% for laboratory 2 (**) Visual initiation (VIS) instead of MAX/5% (***) Note sign reversal convention, positive A-values are from the negative intercept, negative values from a positive intercept of the x-axis C)fronitip of insert, including pre-crack lengths of 4-5 mm (^) from positive X-axis intercept, not included in average Lab. / Specimen No. 1/1 1/2 1/3 1/4 1/5 Average S.D. 2/1 2/2
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438
A.J. BRUNNERANDB.R.K. BLACKMAN
Table 3: Data for symmetric lay-up [0790°]6s (S.D. = standard deviation) Calculated Maximum Interlaminar delamination £-modulus propagation length [mm] *** [GPal [J/m^] 4.6 64.0 68 793 94 9.7 55.0 1'548 10.2 64.5 ' 93 ri49 r2io 951 15.0 66.5 117 1728 1 1/4 44 -11.9^ 59.5 1/5 1'178 ri78 9.9 989 83 1'291 Average 185/18.7% 4.2/42.8% 28 / 33.7% 362 / 28.0% S.D. 0.9 759 66 70.0 1'565 2/1 5.1 709 60.0 76 2/2 1'196 667 -4.8^ 82 63.0 2/3 roo9 -4.2^ 2/4 537 90 60.0 r434 742 2.0 85 2/5 60.0 1'143 683 2.7 Average 80 1'269 89/13.0% 2.2 / 80.7% 226/17.8% 9/11.6% S.D. (*) Corrected Beam Theory analysis [1], loading from insert yielded average MAX/5% of (415 ± 75) J/m^ standard deviation of 18.1% for laboratory 1 and for laboratory 2 MAX of (607 ± 85) J/m^ standard deviation of 14.0% and VIS of (278 ± 38) J/m^ standard deviation of 13.8% (**) Note sign reversal convention, positive A-values are from the negative intercept, negative values from a positive intercept of the x-axis (***) from tip of insert, including pre-crack lengths of 4-5 mm C) from positive X-axis intercept, not included in average Lab./ Specimen No. 1/1 1/2 1/3
Gic [J/m^] MAX/5% offset (CBT) * 722 943
A [mm] **
Table 4: Data for non-symmetric lay-up [0790°]n (S.D. = standard deviation) Interlaminar Lab./ A [mm] ** Calculated £ - Maximum GIC [J/m'l modulus propagation Specimen delamination MAX/5% [GPa] No. [J/m^l length [mm] *** offset (CBT) * 3.7 455 66 r442 66.0 1/1 6.1 362 77 r232 1/2 66.0 2.5 61 71.0 r297 1/3 4.4 566 68 1/4 r206 60.5 2.1 438 67 1'462 68.0 1/5 3.7 68 455 1'328 Average 1.6/42.6% 84/18.5% 6 / 8.5% S.D. 118/8.9% -3.5^ 705 77 r314 60.0 2/1 -2.0 "^ 74 794 915 60.0 2/2 724 66 1'314 60.0 2/3 -o.r 1.2 64 819 864 65.0 2/4 -3.6^ 76 713 60.0 2/5 roo9 1.2 71 751 1'083 Average 6 / 8.4% 217/20.0% 52 / 6.9% S.D. (*) Corrected Beam Theory analysis [1], loading from insert yielded average VIS of (320 ± 112) J/ra", standard deviation of 34.9% for laboratory 1 and of (246± 52) J/m^, standard deviation of 21.0% f( r laboratory 2 (**) Note sign reversal convention, positive A-values are from the negative intercep;, negative values from di positive intercept of the x-axis (***) from tip of insert, including pre-crack: lengths of 4-5 mm C)frompositive x-axis intercept, not included in average
Delamination Fracture in Cross-Ply Laminates
439
The values in Table 4 can be compared with values reported in [4], their average values for the non-symmetric lay-up for initiation from the insert were 317 J/m^ (5.7%), initiation values from the Mode-I pre-crack were 518 J/m^ (4.1%), and the maximum propagation values were r295 j W (14.5%). Tables 2-4 show that, for the unidirectional lay-up, averages of initiation and maximum propagation values do agree with each other and with the data presented in [4]. The R-curves are rather flat, with an increase between about 30 and 60 J/m^ (10-20%) over the total delamination length (Fig. 5). For the cross-ply laminates, however, this is not the case. The average initiation values for the symmetric lay-up differ by more than one standard deviation (values from [4] are not available) while for the non-symmetric lay-up, laboratory 1 and [4] agree within 15%, while laboratory 2 obtains larger average values. The averages of the maximum propagation do agree within one standard deviation and within about 20% are the same for both types of cross-ply laminates. These differences will be further explored in the discussion. 2500
2000
E
1500
'^^"^^;,c,
1000 500
20
40
60
80
100
120
140
Delamination length [mm]
Fig. 5: R-curves for unidirectional [0°]24 (light grey symbols), symmetric [0°/90°]6s (black symbols) and non-symmetric lay-up [0790°] 12 (dark grey symbols), tested in one laboratory, filled/open symbols indicate specimens with a single/mixed fracture surface topography (open symbols are connected by dashed lines to guide the eye). The R-curves calculated according to [1] for all test data shown in Fig. 4 are shown in Fig. 5. If no distinction between the different types of fracture surface topography is made, it is evident that the unidirectional lay-up yields rather "flat" R-curves with very small scatter among the 5 specimens. Both symmetric and non-symmetric lay-up, however show large scatter with relatively steep R-curves. Some of these turn "flat" at Gic-values between 1*500 and 1700 J/m^, the symmetric lay-up yielding the steeper rise and slightly higher maximum values. Others, after the initial increase, drop with increasing delamination length, in two cases even below the average value of the unidirectional material (about 350 J/m^).
440
A J. BRUNNER AND B.R.K. BLACKMAN
DISCUSSION The "saw-tooth" fracture surface can be interpreted as a sequence of interlaminar and intralaminar delaminations oscillating back and forth at regular intervals [4]. The amplitud(i of the "saw-tooth" pattern determined from photographs is consistent with the assumption that it is equal to the thickness of the centre 90°-ply of the cross-ply specimens. For the symmetric lay-up with two 90°-plies at the centre, the "wave-length" of the "saw-tooth" pattern is more than doubled (factor around 2.1) compared with the non-symmetric lay-up. The steady-state delamination in the cross-ply specimens is oscillating between two 0°-plies on either side of the centre 90°-ply with a wavelength that seems to depend on ply thickness. The saw-tooth pattern indicates that interlaminar delamination propagation in both cross-ply lay-ups occurs mostly outside the mid-plane, between the unidirectional layer closest to the mid-plane and that separated by the adjacent 90°-ply. If the delamination reaches the unidirectional layer at a point in its course, it can be redirected from saw-tooth to straight delamination propagation in one of the unidirectional plies, at least in part of the specimen width. The cause for the initiation of the transition into the unidirectional ply has not been identified. It could be due to local microscopic inhomogeneities or mesoscopic features that act as sites of stress concentration, e.g., such as regions of high local fibre density. Statistically, the chance for the occurrence of this transition seems to be relatively high, around 50%. Which factors govern the course of delamination propagation after the transition are even less clear. Beside the type of fracture surface shown in Fig. 3, where the portion of delamination occurring in the unidirectional plies is increasing with delamination length, there are examples of intermittent propagation in the unidirectional ply as well as examples of constant width of the delaminated area within the unidirectional ply. If, as in the fracture surface type shown in Fig. 3 on the left hand side, the width of the featureless delamination is increasing with delamination length this is also reflected in the Rcurve (Fig. 5). There, values are dropping from relatively high "cross-ply"-values to a value typical of the unidirectional laminate. This could be a hint that energy release rates determined for the cross-ply laminates using the analysis developed for the unidirectional material are valid in the sense that relative proportions are conserved. The different bending stiffness of the unidirectional, symmetric, non-symmetric lay-up is shown by independent three-point bending E-modulus measurements to decrease in that order from about 145 GPa to 73 GPa and 62 GPa. The same trend, with somewhat larger values is seen in the back-calculated E-moduli (Tables 2-4). This is also apparent from the larger displacements (at similar delamination lengths) of the cross-ply lay-ups compared with the unidirectional material. There is a considerable difference between initiation values from the insert starter film determined from the maximum load/5% offset (MAX/5%) and the non-linear or visual point (NL/VIS) for the two cross-ply laminates (NL/VIS-values not reported in the Tables since they were not determined for all specimens). The NL/VIS-values, when determined, are comparable to initiation from the insert for the unidirectional material (determined as MAX/5%- or NL/VIS-values). The first few mm of delamination propagation in the cross-ply laminates usually occur in the mid-plane of the beam, but deviating from it before the maximum load value is reached. The difference between the NL/VIS- and MAX/5%)-values quite likely reflects this transition from mid-plane to "saw-tooth" pattern.
Delamination Fracture in Cross-Ply Laminates
441
According to the requirements specified in the ISO standard [1] deviation of the delamination propagation from mid-plane invalidates the test. In that sense, any data analysis of the crossply laminates, therefore, will yield invalid results. This point was discussed extensively in [4] where the authors also used the Finite-Element method to supplement their analysis and concluded that the corrected beam theory data reduction scheme seemed to remain applicable and that the non-symmetric cross-ply material yielded apparently valid fracture toughness data, even though these were probably affected by transverse cracking. It could be argued that even in the unidirectional lay-up the delamination is not strictly running along the mid-plane, at least on a microscopic scale (Fig. 6). Deviations are on the order of 0.05 mm. In the case of the cross-ply material, the maximum deviation is limited by the distance between the 0°-plies, i.e. about 0.17 and 0.38 mm, respectively, as long as the delamination does not further deviate into the unidirectional plies. This is about 3 and 7 times, respectively, more than in the unidirectional laminates. "Sufficiently small", limited deviations will not affect the data analysis in a way that renders the values meaningless. In the following, it is proposed that the validity of cross-ply data can be defined by requiring the delamination not to deviate into the adjacent unidirectional plies. This criterion could easily be verified by inspection of the fracture surface. Deviations of the delamination from the mid-plane imply mixed-mode conditions, i.e., a deviation from a pure Mode I opening load. The degree of mode mixity will depend on the relative stiffness of the two unequal beams. In the cross-ply laminates, this stiffness difference will be "small", as long as the unidirectional plies that essentially provide the axial stiffness are equal in both beams. This is a rational justification for a validity criterion for cross-ply laminates that allows deviations from the mid-plane, as long as the delamination does not show a transition into the unidirectional plies.
Fig. 6: Delamination in a specimen with unidirectional lay-up as seen through the travelling microscope. On this scale, slight deviations from the mid-plane are observed, the black vertical lines are each 1 mm apart. The Gic-values of those specimens with a single type of fracture surface topography, i.e., those that do not show any deviation of the delamination into the adjacent unidirectional plies are plotted versus delamination length in Fig. 7. Data shown are from both, laboratory 1 and 2 (Tables 2-4). The R-curves for the unidirectional lay-up agree well for both laboratories. The non-symmetric (light grey symbols) and, in particular, the symmetric lay-up (dark grey symbols) show a large scatter, for both, the data sets (2 each) from each laboratory and the inter-laboratory comparison. It is clear, however, that the R-curves for both lay-up types rise
A J. BRUNNERANDB.R.K. BLACKMAN
442
considerably above those for the unidirectional lay-up. Comparing the R-curves for both types of cross-ply laminates, the difference seems to be within the scatter bands. The drop in the R-curves seen for some specimens in Fig. 5 relates to the change in fracture surface topography. If the delamination deviates into the adjacent unidirectional (0°) plies, the Gic • values drop to those comparable for the unidirectional material. The onset of the drop in GK correlates well with the observed position of the change in fracture surface topography. In principle, this can already be recognised in the load-displacement plots (comparing Fig. 4 and 5). It is also clearly seen as a change in the slope of plots of the cube-root of the normalised compliance (C/N)^^^ versus the delamination length a. This yields erroneous values of A when all data points are fitted to a single straight line. 2500
• 2000
^
1500
^ 1000
500
1
^ •v • • 20
40
60
80
100
120
140
Delamination length [mm]
Fig. 7: Plot of Gic-values versus delamination length for specimens tested at two laboratories all showing homogeneous, single type topography of the fracture surface, i.e., featureless for unidirectional lay-up [0°]24 (light grey symbols), saw-tooth with higher "wavelength" for symmetric lay-up [0790°]6s (black symbols) and lower for non-symmetric lay-up [0790°] 12 (dark grey symbols), respectively (the open symbols and those connected by dashed lines to guide the eye indicate those from the second laboratory). Fig. 7 shows that the values from the second laboratory tend to be lower than those from the first for both types of cross-ply lay-up, while those for the unidirectional lay-up agree fairly well. The scatter still seen in the R-curves for the cross-ply laminates with a single fracture surface topography (Fig. 7) can probably, at least in part, be attributed to different amounts of fibre-bridging (compare Fig. 1). Another factor is micro-cracking in front of the delamination that may make accurate determination of the delamination length difficult. This would also offer an explanation for the steep rise seen in the R-curves of those specimens for which the delamination does not deviate into the unidirectional plies. This is discussed in detail in [6]. "Small" (local and short-term) deviations of the delamination into the unidirectional plies not recognised in the visual inspection of the fracture surfaces might also contribute to the scatter by temporarily reducing Gic. Finally, the oscillating interlaminar - intralaminar type of delamination propagation could also account for some of the observed scatter. The analysi > presented in [4] concludes that the intralaminar G is considerably smaller than the
Delamination Fracture in Cross-Ply Laminates
443
interlaminar G, both Mode-I dominated, the difference is estimated to be as much as 30%. The recorded load-displacement traces show that a fair number of delamination length readings have been taken at arrest following stick-slip [1], i.e., quite likely before the transition from intralaminar to interlaminar delamination propagation. These data points would show as "low" values on the R-curve. A point-by-point data analysis, removing obvious arrest points (Fig. 8) reduced the scatter, both in-laboratory and between laboratories. Statistically, the data, maybe, reveal a distinct "ranking" based of the two cross-ply lay-ups, with the symmetric lay-up tending to slightly higher values than the non-symmetric. This could probably be explained by the lower total fracture surface per unit specimen length formed by the latter. 2500
2000
• ^ 1500 E 1000
1
•
• •
^
500
^"»
i
20
40
60
80
100
120
140
Delamination length [mm]
Fig. 8: Plot of Gic-values versus delamination length for specimens tested at two laboratories same as Fig. 7 but with obvious arrest points removed; unidirectional lay-up [0°]24 (light grey symbols), regular saw-tooth with lower and higher "wavelength" for symmetric lay-up [0790°]6s (black symbols) and non-symmetric lay-up [0790°] 12 (dark grey symbols), respectively (the open symbols and those connected by dashed lines to guide the eye indicate those from the second laboratory). The comparison of the different initiation points in the three laminate types (Tables 2-4) raises the question which definition shall be used for initiation in the cross-ply laminates. Since visual initiation (VIS-point) and probably also non-linearity of the load-displacement plot (NL-point) yield values similar to initiation values in the corresponding unidirectional laminate, the maximum load or 5% offset in compliance (MAX/5%-point) seems to reflect the higher delamination resistance of cross-ply compared with unidirectional laminates better. Further analysis of additional data from the 3" round robin may allow a better assessment of this question. The trends seen in the present analysis seem to support the conclusion that, if the type of fracture is considered, a meaningful relative ranking of cross-ply lay-ups (symmetric or nonsymmetric) with respect to a unidirectional lay-up of the same material can be achieved. The load-displacement plots and R-curves show that cross-ply materials will yield a larger scatter but, if effects from changing fracture surfaces are recognised and those specimens are
444
A J. BRUNNER AND B.R.K. BLACKMAN
eliminated from the analysis, it is clear that cross-ply laminates will yield much steeper Rcurves than the corresponding unidirectional lay-up. In order to determine quantitative data, a larger number of specimens may have to be tested. Based on load-displacement plots and fracture surface analysis, about 2-3 cross-ply specimens out of 5 per type, i.e., about 50%. seem to yield the "saw-tooth" pattern up to the maximum delamination length without deviating into the 0°-plies. It is hence estimated that 10 to 12 specimens per lay-up may have to be tested for a comparison with reasonable statistics (at least 5 valid specimens per material type). SUMMARY/CONCLUSIONS Comparative round robin testing of a symmetric and a non-symmetric cross-ply (0790°) layup following the standard test procedure for unidirectional laminates has been performed. The cross-ply laminates show oscillatory interlaminar and intralaminar delamination resulting in a regular saw-tooth pattern of the crack. Two basic types of fracture surface are identified in the cross-ply laminates, one with a ridge-trough-pattem, the other with transitions to a flat, featureless pattern. It is argued that the former fracture surface type yields Gic-values and Rcurves (Gic plotted versus delamination length) that allow at least a relative ranking of the delamination resistance of the cross-ply laminates compared with unidirectional laminates. ACKNOWLEDGEMENTS The authors wish to thank Cytec Fiberite Pic and Dr. D.R. Moore (ICI Pic) for the supply of materials and Mr Christian Murphy (Imperial College London) and Dr D.D.R. Cartie (Cranfield University) for the contribution of test data and photographs. Comments by Dr. M. Barbezat (EMPA) are also gratefully acknowledged.
REFERENCES [1] [2] [3]
[4] [5] [6]
ISO 15024 (2001) "Fibre-reinforced plastic composites - Determination of Mode I interlaminar fracture toughness, Gic, for unidirectionally reinforced materials. International Organisation for Standardisation". Choi, N.S., Kinloch, A.J., Williams, J.G., (1999) "Delamination Fracture of Multidirectional Carbon-Fiber/Epoxy Composites under Mode I, Mode II and Mixed Mode I/II Loading", J. Comp. Mat. 33, No.l, pp. 73-100. Blackman, B.R.K., Brunner, A.J., (1998) "Mode I Fracture Toughness Testing of FibreReinforced Polymer Composites: Unidirectional versus Cross-ply Lay-up", Proceedings 12^^ European Conference on Fracture ECF-12: Vol III, Fracture from Defects, EMAS Publishing, pp. 1471-1476. de Morais, A.B., de Moura, M.F., Marques, A.T., de Castro, P.T., (2002) "Mode-I interlaminar fracture of carbon/epoxy cross-ply composites", Composites Sci. d: Technol 62, pp. 679-686. La Saponara, V., Kardomateas, G.A., (2001) "Crack branching in cross-ply composites: an experimental study". Compos. Struct. 53, pp. 333-344. Brunner A.J., Blackman, B.R.K., Wilhams, J.G. "Deducing Bridging Stresses and Damage from Gic Test on Fibre Composites" (at press). Proceedings 3^^ ESIS TC4 Conference on Fracture of Polymers, Composites and Adhesives, Elsevier.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
445
Fracture Toughness of Angle Ply Laminates M. R. Piggott and W. Zhang Chemical Engineering, University of Toronto, Toronto M5S 3E5, Canada
Abstract Angle ply laminates have a very high resistance to through thickness fracture and special methods are needed for their toughness testing. Thus the standard ASTM El922 test is not suitable because few of the fibres break during the test, and the crack does not develop in a self-similar mode, as required by the standard. Because of this, a special wide, double edge notched sample, has been developed. It had a width of 43mm and a gauge length of 20mm. Some samples were coated with photoelastic material and photoelastic patterns were recorded during testing. Balanced angle ply laminates were used with angles of 15, 30, 45, 60, and 75 degrees. Quasi isotropic carbon-epoxy laminates were also tested. The angle ply laminates were very tough. Thus the 15 degree angle ply materials had fracture toughnesses of 130 (carbonepoxy) and 110 MPaVm (glass-epoxy). The corresponding works of fracture were about 1.2 MJm"^ in both cases. By contrast the value for the quasi isotropic laminate was only 50 MPaVm, i.e. a work of fracture of 54 kJm'^. Angle ply laminates are also very strong and stiff, but this, too, can only revealed by testing wide samples. The notched strength tests showed that the 30° carbon-epoxy balanced angle ply laminate was 40% stronger than the quasi isotropic one, as well as being seven times as tough. Once these excellent properties are more widely appreciated we can expect to see much greater use of angle ply laminates in aerospace and other critical applications. KEYWORDS Fracture toughness, composites, angle ply laminates, new toughness test method
1.
Introduction
The testing of laminates for strength, stiffness and toughness is standardized using the same philosophy that holds for metals and polymers, i.e., the use of long and narrow specimens. This is a very reasonable approach for isotropic materials, as it minimizes the effects of stress concentrations.
M.R. PIGGOTTAND W. ZHANG
446
However, the long, narrow coupon has been shown to give misleading resv Its with angle ply laminates [1], has been publicly criticized [2], and the ASTM standard for tensile strength and stiffness - ASTM D3039 - has been modified to take account of this ])roblem, dubbed "edge softening". The softening arises because the off-axis fibres emerge i'rom the edges of the coupon, and in severe situations, are virtually unstressed. This accounts for the low values obtained in Tsai's early work [3]. It has unfortunately led to the almost total avoidance of angle ply laminates by the composites industry. The ASTM standard for through-thickness strength - D6415 - has the same problem [4], and our recent experiments show that the same is true for the laminate through-thickness fracture toughness test - ASTM El922 [5]. Here, few of the fibres are broken in the test and the crack does not proceed in a self-similar mode as required by the standard. Thus we have been obliged to devise a different test configuration for angle ply laminates. This paper describes the test, and the results obtained therefrom.
2.
Experimental Method
Wide and short samples of [±(p]s angle ply laminates were notched at both edges and tensile tested. Some of these samples had a photoelastic resin glued to one side, and the patterns recorded during the test. The prepregs used were S glass-epoxy (T4S 216 F584) from Hexcel and carbon-epoxy (BMS CLIGR 190) provided by Boeing Canada. Samples, 150x150 mm^, were moulded according to the suppliers' instructions. Thus the glass prepreg was heated at 2°Cmin" to 180°C, held there for 150min and then allowed to cool, all the time under a pressure of 1.6MPa. The carbon prepregs were treated the same way, except that the hold time was only 120min.
<— l±15Is
2
4
6
6
8
1^75), Displacement (mm) (a)
l±75|, Displacement (mm) (b)
Fig. 1. Force-distance plots for notched laminates: (a) carbon and (b) glass.
8
Fracture Toughness of Angle Ply Laminates
447
Each moulding was then sHt to make three test coupons, each 43mm wide. These were notched to various depths, on both edges, across the middle, using a band saw. This gave a notch width of about 1mm. The samples were then end-tabbed with aluminium 2mm thick, leaving a gauge length of 20mm, with the notches on the centre line. The testing was carried out using an MTS servohydraulic machine with hydraulic grips, into which the coupons just fitted. The crosshead speed was held constant at 2mmmin"^ during the test. There were three to five samples for each test, and the error bars indicate ± one standard deviation. (In the absence of an error bar the variation is smaller than the symbol.) The photoelastic polymer used was PS-1, from Measurements Group It was 2.5mm thick, and had a coating fringe value of 7.57mm/m. It was bonded with PC-1 adhesive from the same supplier. It was monitored during the test using a lab-built circular reflecting polariscope, and the patterns were recorded using a digital camera with the cross head movement temporarily stopped.
3.
Experimental Results
Fig. 1 shows the force-distance plots for the carbon, (a), and the glass, (b) laminates with a notch depth of 5mm (for all samples except cp = 15° glass, which had 4mm notches). It can be seen that the (p = 15° and 30° laminates were stressed to failure without any significant plasticity effects being evident. The 45° coupons, on the other hand, had much quasi-stable deformation before failure, while the 60° and 75° laminates broke very easily.
Carbon
1.0
S. o
0-8
c 9
J^ \ f
0.4
^
I0,+45,90,-45]s
0.2
15
30
—r45
— I —
60
75
Ply Angle (Degrees)
Fig. 2. Notched strengths of notched carbon and glass fibre laminates vs. laminate angle; notch depth 5mm, except for glass for (p = 15°, which had 4mm notches. The corresponding strengths, shown in fig. 2, are not the maximum strengths. Here the notch depth was 5mm. When strength is plotted vs. notch depth it generally increases
M.R. PIGGOTTAND W. ZHANG
448
monotonically; see fig. 3. This is because the fibres emerging from the edges, the "disabled fibres", are increasingly stressed as notch depth increases.
0.6 H (0 Q.
a
c (0
0.2
- 1 —
10
15
20
Notch Depth (mm)
Fig. 3. Strength vs. notch depth for [±45]s carbon and glass fibre laminates. Nevertheless, only the cp =15° and 30° showed significant amounts of fibre breakage; all the other samples failed by some sort of fibre shearing process. Table 1 lists the fractions of fibres broken during the failure process. Table 1. Fractions of fibres broken.
System
—>
Notch depth —> Laminate Quasi isotropic [±15]s [±30]s
carbon-epoxy
glass-epoxy
5mm 10mm 15mm
5mm 10mm 15mm
70% 55% 55% 100% 100% 100% 80% 85% 90%
no laminate tested 70% 90% 100%* 5% 20% 70%
[±45]s, [±60]s and [±75]s carbon and glass fibre laminates: no fibres broken at any notch depth. * 13mm notch depth
The maximum strengths can be quite high, fig. 4. What is particularly notable is that the carbon fibre angle ply with 9=30° is 40% stronger than the quasi isotropic laminate.
Fracture Toughness of Angle Ply Laminates
449
The quasi isotropic sample gives the photoelastic contours to be expected of an isotropic material; the isochromes follow the lines of maximum principal stress difference in a characteristic pattern; see Fig. 5a. i
Carbon
c B in E 3
E
I0,+45,90,-451,
s
Ply Angle (Degrees)
Fig. 4. Maximum strengths of angle ply laminates. This is not true for the angle ply laminates; here the fibre direction can have a striking effect. This is most apparent w^hen (p=45°. At lOOMPa, w^ith a notch depth of only 2.5mm, the fibres show themselves in a very obvious way, as can be seen in Fig. 5b. The ±45° isochromes are visible at all stress levels up to failure, and are still visible when the stress is released.
0.24 GPa, 5mm Quasi Isotropic
0.10 GPa, 2.5mm [±45]s
0.72 GPa, 2.5mm [±15°]s
0.21 GPa, 12.5mm [iBO'^ls
(a) (b) (c) (d) Fig. 5 Photoelastic patterns, from left to right: quasi isotropic laminate; 45°; 15° and 30° angle ply laminates; stresses (GPa) and notch sizes (mm) as indicated.
M.R. PIGGOTTAND W. ZHANG
450 150
l-rr-l
100 i
CL
c
50 H
I0,+45,90,-45|,^ *J
o
10
—I
10
15
Notch Depth (mm)
Fig. 6. Fracture toughnesses of carbon fibre laminates, with laminate angles indicated on the curves. The (p=15° coupons gave only a single lobe at each notch at this notch depth, Fig. 5c, and for (p=30° we see elongated 30° lobes, consistent with the cp = 45° case. Fig. 5d. (Not all the photoelastic tests were successful, so we cannot, at this time, compare all the different angles at the same notch depth.) At (()=60° and 75° the isochromes follow the fibres from the notch tip, as for(p=45°.
^^
30 i
Q.
s10 c n
20
i2 10
1±751, —1—
10
15
Notch Depth (mm)
Fig. 7. Fracture toughnesses of carbon fibre laminates, with laminate angles indicated on the
Fracture Toughness of Angle Ply Laminates
4.
451
Discussion
Assuming the laminates have a property that we can identify as fracture toughness, it can be calculated from: K^ = (F/Wt)<(na)f(a/W)
(1)
Here F is the breaking force, W is the width, and t is the thickness of the coupon, a is the notch depth and Kx is the fracture toughness. Wis 43mm Sindf(aAV) was obtained from J. G. Williams [6]. (Table 6.1 was used, and values were obtained by interpolation.) Note that the subscript for K indicates the direction, and is the same as that adopted in a recent text on fibre composites [7].
E
(0 Q.
100
in
c
JZ O) 3
i2 £
50
3
u
—I—
10
15
Notch Depth (mm)
Fig. 8. Apparent fracture toughnesses of glass fibre laminates, with laminate angles indicated on the curves. The fracture toughness should be independent of notch depth. Figs. 6-9 show that this is approximately the case for a > 5mm, for (p = 15°and 30° for carbon-epoxy, 60° and 75° for carbon-epoxy and glass-epoxy, and for the quasi isotropic laminate, but not for the lower angle glass-epoxies or the 45° carbon-epoxy. The behaviour of the carbon fibre laminates shown in Fig. 6 can probably be explained on the basis of linear elastic fracture mechanics at notch lengths of 10-15mm. (The lower values at the shorter notch lengths can probably be accounted for by the edge softening effect.) Most of the fibres were broken, rather than being sheared off - see Table 1. The fracture toughnesses estimated from the breaking forces are constant within the range of error. Thus, despite the unusual stress distributions in the two angle ply laminates, fracture toughnesses have probably been measured. In the case of the quasi isotropic laminate, for the 10mm and 12.5mm notches the fracture toughness is in agreement, within experimental error, of the value given for a
M.R. PIGGOTTAND W. ZHANG
452
[90/-45/0/+45]4s carbon-epoxy laminate in the ASTM El922 standard. Our value was 50±10MPaVm; the round robin value given in the standard was 57±3MPaVm. This confirms that our test specimen was efficacious.
^
t l±45J,
30
ID Q.
o c
20 A
JZ
3
P ^
10
— 1 —
10
I '"
15
Notch Depth (mm)
Fig. 9. Apparent fracture toughnesses of glass fibre laminates, with laminate angles indicated on the curves. At higher ply angles than 30°, this is probably not the case. Thus for 45°, Fig 7, the results show a monotonic increase with increasing notch depth, and no fibres were broken. So we conclude that the notch could not be made deep enough to avoid edge softening, and still permit an accurate analysis. For the higher angles, the results seem to be compatible with plateau values having have been reached. However, since no fibres were broken, the apparent plateau is associated with the fibres shearing out of the matrix, with a process zone involving the whole surface of both notches. Thus the process zone was unduly large, so that lineiT elastic fracture conditions were not satisfied, and therefore fracture toughness was not measured in these cases either. The glass fibre laminate results are less well characterized than the carbon. In the case of the lower angle results, Fig. 8, there seems to be a peak value between about 8 and 12mm notch depth. While the 15° angle ply laminates suffered much fibre breakage, the 30° laminates suffered relatively little, except at 15mm notch depth. Table 1. However, this was associated with a lower fracture toughness, Fig. 8. (The aberrant result at 3mm for (p = 30° is probably due to statistical variation.) Thus fibre shearing was contributing significantly to the fracture toughness at the smaller notch depths, in both cases. (This was apparently not true for the lower angle carbons.) This again suggests large process zones, casting doubt on whether fracture toughness was being measured.
453
Fracture Toughness of Angle Ply Laminates
At the higher angles, fig. 9, while plateau regions are indicated by the results, it again seems unlikely that true fracture toughnesses were being measured as there was no significant fibre breakage, so that the process zones again involved the whole notch surface. When we plot the apparent toughnesses taken from the "most likely" plateau values in Figs. 6-9 we get the result shown in Fig. 10. The quasi isotropic value has been inserted at 40°, as suggested by the fit shown in Fig. 4. The results in Fig. 10 show that the carbon composites have higher apparent toughnesses than the glass fibre laminates at all angles. The quasi isotropic result fits very well at 40°, indicating that a 40° angle ply laminate might serve as well for strength and toughness, yet not have the premature splitting problem associated with the incorporation of 90° plies in the quasi isotropic. The laminate is much weaker, and less tough than the [±30]s, suggesting that a balanced angle ply laminate with cp somewhere between 31 ° and 39° would be better and well worth testing for use in critical applications, such as aerospace. 150
Q.
S,
100 H
M tf) 0)
c
JZ CJ> 3
o H
10,+45,90r451,
0)
Ply Angle ( Degrees)
Fig. 10. Apparent fracture toughnesses of carbon and glass fibre angle ply laminates. We can estimate the works of fracture, Gx, from the fracture toughnesses, using Gx - Kx /EKX
(2)
where EKX is a function of the laminate compliances, *§/,, see equations (7.4) and (7.5) in [7] (these equations are simplified forms applicable to balanced laminates). The calculations give roughly 1.2MJm'^ for both the carbon and the glass [±15]s laminates, about 0.4MJm'^ for the carbon and 0.5MJm"^ for the glass [±30]s laminates, and 0.054MJm"^ for the quasi isotropic laminate.
454
M.R. PIGGOTTAND W. ZHANG
5.
Conclusions
The fracture toughness of angle ply laminates can only be measured with any confidence under very specific conditions. These include wide and short gauge len^'ths and very deep notches. Even then clear-cut results are only obtained at small ply angl<js, with carbon fibre laminates, and not so clear with glass, since some of the fibres sheared out rather than breaking. The test works well with quasi isotropic carbon-epoxy laminates, yielding a result which is consistent with one measured using the ASTM standard. Angle ply laminates appear to be very tough, with fracture toughnesses of up to BOMPaVm and corresponding works of fracture of 1.2MJm'^ (although it must not be forgotten that, like all laminati^s, their resistance to delamination is weak). They are also very strong, and for (p < 35° ajjpear to perform very much better than quasi isotropic laminates. It is recommended that the aerospace industry investigate their usefulness for critical applications.
Acknovv^ledgements The authors are grateful to NSERC (Canada) for financial support which made this research possible. They also thank Boeing (Winnipeg) and Hexcel (Dublin, CA) for gifts of prepregs.
References 1. 2. 3. 4. 5. 6. 7.
Piggott, M. R. (2000) Polymer Composites 21, 506-13. Piggott, M.R., Liu, K and Wang, J (2000) ASTM STP 1383, 324-33. Tsai, S. W. (1965) NASA CR 225. Piggott, M. R. and Quian, M (2002) Proc 47'^ SAMPE Symp. 145-53. Piggott, M. R. and Zhang, W (2001) Proc 46'^ SAMPE Symp. 2371-7. Williams, J. G. (1973) Stress Analysis of Polymers, pp 265-70, Longmans, London. Piggott, M.R. (2002) Load Bearing Fibre Composites, 2^ Edition, pp 225-6, Kluwer Academic Publishers, Boston.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
455
STRAIN ENERGY RELEASE RATE FOR CRACK TIP DELAMINATIONS IN ANGLE-PLY CONTINUOUS FIBRE REINFORCED COMPOSITE LAMINATES C. Soutis and M. Kashtalyan Department of Aeronautics Imperial College London Prince Consort Road London SW7 2BY, UK ABSTRACT The expressions for the strain energy release delaminations growing from the tips of angle ply laminates loaded in tension are presented. Strain laminate residual stiffness properties are predicted density and delamination length.
rate associated with local matrix cracks in orthotropic energy release rate and the as functions of matrix crack
C SOUTISANDM. KASHTALYAN
456
INTRODUCTION Fracture process in multidirectional composite laminates subjected to in-plane static or fatigue tensile loading involves sequential accumulation of damage in the form of matrix cracks that appear parallel to the fibres in the off-axis plies, edge delamination and local delamination long before catastrophic failure. These resin dominated failure modes significantly reduce the laminate stiffness and are detrimental to its strength. Comprehensive observations of damage accumulation in [02/02^~^2^s carbon/epoxy laminates under quasi-static and fatigue tensile loading have been reported in [1, 2]. Stresses responsible for the matrix cracking and associated local delamination were calculated by means of the finite element analysis. Expressions for strain energy release rate were also suggested. The present paper is concerned with theoretical modelling of cracked orthotropic composite laminates loaded in tension. The approach employs the Equivalent Constraint Model (ECM) of the damaged laminate, earlier applied to [±6^/90^]^ laminates damaged by transverse cracking in the 90° plies and local delaminations growing uniformly from the matrix crack tips at the (-6/90) interface [3] and to cross-ply [0^ /90„]^ laminates with delaminations growing along transverse cracks and splits [4]. In this work, the approach is extended to local delaminations associated with angle ply matrix cracks.
X, X'vv . . . / <
^
^ X
^
matrix cracks delaminations
(P
- i-e^„)
2K
^(M)
^
Fig. 1. Front and edge views of a [0^ 10^ /~^J.y laminate subjected to in-plane tensile loading and damaged by matrix cracking and local delaminations in the inner ply
Strain Energy Release Rate for Crack Tip Delaminations
457
Figure 1 shows a schematic of [0^/^„/-^„]^. laminate subjected to in-plane tensile loading and damaged by matrix cracking and delaminations. The laminate referred to the global xyz and local x\^^x[^^xf'' co-ordinate systems, with the xl^"* axis directed along the fibres in the damaged {-6^) layer. Henceforth all quantities referred to the local co-ordinate system will be denoted with the index //. Matrix cracks in the damaged layer are assumed to span the whole width of the laminate and be spaced uniformly at a distance 2s^. Local delaminations growing from the tips of matrix cracks at the {61- 6) interface are assumed to be strip-shaped, with a delamination length (strip width) 21 ^ .
FRACTURE ANALYSIS The total strain energy release rate G'"' associated with local delaminations growing from the tips of matrix cracks is equal to the first partial derivative of the total strain energy U stored in the damaged laminate with respect to the total delamination area A''^ provided the applied strains {£} are fixed and the matrix crack density C = (25'^)"^ remains unchanged dU
(1) {e},C
The strain energy release rate can be effectively calculated if instead of the damaged laminate one considers the 'equivalent' laminate, in which the damaged layer is replaced with an 'equivalent' homogeneous one with degraded stiffness properties. The residual stiffness matrix [Q ] of the 'equivalent' layer is a function of the relative delamination area D-i^ls^ and the relative crack density D"^^ =h^ I s^ (i.e. normalised by the thickness h^ of the damaged {-6^) layer). Its determination is discussed in the next section. If hygrothermal effects are neglected, the total strain energy stored in the laminate element with a finite gauge length L and width w is U=^wL{8Y{A]{e} where [A] = ^[Q].h^
(2)
is the residual extension stiffness matrix of the 'equivalent' laminate.
Noting that the area of a single crack tip delamination is a'"^ = 2^^w/sin^, Fig. 1, the total delamination area is equal to A^"^ = 2^'"^CL = 2LwD^''/sin^. Then the strain energy release rate, calculated from Eqs. (1) and (2), is G'\e,D;\D'')
'^ '^"^
Under uniaxial strain, Eq. (3) simplifies to
= -^{eY^{e}sine
2 * ' dD'ffi
(3)
C SOUTISANDM. KASHTALYAN
458
•yid,
sin^
(4)
The residual in-plane stiffness matrix [Q ] of the 'equivalent' layer in the global co-ordinates can be obtained from the residual in-plane stiffness matrix [Q ^^^ ] in the local co-ordinates by the well-known transformation formulae [3]. The residual in-plane stiffness matrix [Q ^^^ ] of the 'equivalent' layer in the local co-ordinates is related to the in-plane stiffness matrix [2^^^] of the undamaged material via the introduced in [5] In-situ Damage Effective Functions (IDEFs) Af
= A^f(D^\
D'^\ j = 2,6 as 0
[g(.)]^[2(.)]_
(5) 0
0
e^fA^,-,'
Calculation of the residual in-plane axial stiffness g ^ using Eq. (5) and transformation formulae [3] yields the strain energy release rate associated with local delamination in terms of IDEFs and Qlf^ as
G^^^,z)7,z);^)=-^^^
^ i ^ c o s ' ^ + 2G,f s i n ' ^ c o s ' ^ +
aA^,^, '^^4Q^^fsm'9cos'e^'' -neiTsin"^
3^r
M'' dD-
(6) smO
RESIDUAL STIFFNESS Substituting the residual stiffness matrix [Q^^^], Eq. (5), into the constitutive equations for the 'equivalent' layer {^^'^^} = [Q^^^]{£^'^^} gives the IDEFs A^^^,K^^l in terms of the lamina macrostresses [G'"'^^] and macrostrains [s^'^^] as Tr^d)
T^id) KiM)
A*,';'-i>^12 *^11 ~ i^22
^2.
\(M)y(d) . /I
(7)
To determine the IDEFs as functions of the damage parameters DJ"", D^ , micromechanical analysis of the damaged laminate has to be performed. Since cracks and delaminations are spaced uniformly, a representative segment of the laminate, containing one matrix crack and two crack tip delaminations, may be considen^d. The
Strain Energy Release Rate for Crack Tip Delaminations
459
representative segment can be segregated into the laminated and delaminated portions. Due to the symmetry, the analysis can be confined to its quarter, Fig. 2.
delamination
(o„/^„)
i-e„)
M
matrix crack
Fig. 2. A quarter of the representative segment of the damaged laminate
Let alp denote the in-plane microstresses in the damaged layer (i.e. stresses averaged across the layer thickness). In the delaminated portion, we have (72^ = (7^2^ = 0. In the laminated portion, the in-plane microstresses may be determined by means of a 2-D shear lag analysis. The equilibrium equations in terms of microstresses take the form r(d) dot' T. (8) ' —h„ ^ = 0, 7 = 1,2 dxf By averaging the out-of-plane constitutive equations, the interface shear stresses TJ in Eqs. (8)
are expressed in terms of the in-plane displacements ulj^^ and ull'\ averaged across the thickness of respectively the damaged (-0J layer and the outer sublaminate (0^ /OJ,so that
:K^M''-^l''')^Kj,(uf-0
(9)
The shear lag parameters K^^,K22 and K^2(= ^21) ^^ determined on the assumption that the out-of-plane shear stresses in the damaged layer and outer sublaminate vary linearly with x^^^. Substitution of Eqs. (9) into Eqs. (8) and subsequent differentiation with respect to Xj^^ lead to the equilibrium equations in terms of microstresses and microstrains (i.e. strains averaged across the layer thickness). To exclude the latter, constitutive equations for the damaged layer and the outer sublaminate, equations of the global equilibrium of the laminate as well as generalised plane strain conditions are employed. Finally, a system of coupled second order non-homogeneous ordinary differential equations is obtained
rfv
-N,jal',> -N,ja',t> - A,a, =0, 7 = 1,2
(10)
460
C SOUTIS AND M. KASHTALYAN
Here a^ is the applied stress, while N^j,k = \,2 and A^. are constants depending on the inplane stiffness properties of the intact material [Q^^^], shear lag parameters K^^,K22 and K^^ and angle 6. Equations (10) can be uncoupled at the expense of increasing the order of differentiation, resulting in a fourth order non-homogeneous ordinary differential ecjuation, with boundary conditions prescribed at ^2^^ = ^^ • The in-plane microstresses in the laminated portion of the damaged layer are then found as yid) ^
cosh \ (x^^^ i-^^. ...K . : _ . v ^ ^ ^' ^=1 coshA^(^^-^^)
^•=i'2
(11)
V
Here X^ are the roots of the characteristic equation, resulting from the forth order differential equation, and A^^. and Cj are constants depending on Nj^^ and A^.. The lamina macrostresses {G^"^^}, involved in Eq. (6), are obtained by averaging the microstresses, Eqs. (11), across the length of the representative segment as explicit functions of the relative crack density D "'^ and relative delamination area D !f
YA,—^tanh^-^^
^ + C.{\-&')
(12)
The macrostrains in the 'equivalent' layer {e^'^^} are assumed to be equal to those in the outer sublaminate {e^""^]. They are calculated from the constitutive equations for the outer sublaminate and equations of the global equilibrium of the laminate as {£'''} = {e'^'} = [S'^']{a'"'] = [S'''Wo\{l + K){n}a
(13)
where {n} = {cos^^, sin^^, cos^sin ^ } ^ . Thus, the lamina macrostresses, Eq. (12), and macrostrains, Eq. (13), are determined as explicit functions of the damage parameters Z)J^, D^^. Consequently, first partial derivatives of IDEFs, Eq. (7), involved into the expressions for the strain energy release rate, Eq. (6), can be calculated analytically. RESULTS AND DISCUSSION Predictions of strain energy release rate G'"^ associated with local delamination are made for the AS4/3506-1 graphite/epoxy material system which was examined in [1, 2]. Its lamina properties are as follows: £^ii=135GPa, £"22=1 IGPa, Gi2=5.8GPa, Vi2=0.301, single ply thickness r=0.124mm. Residual engineering properties of the laminate, damaged b> matrix cracks and local delaminations are also predicted.
Strain Energy Release Rate for Crack Tip Delaminations
461
O'Brien [1] suggested a simple closed-form expression for the strain energy release rate associated with local delamination in a [02/^2/~^2L laminate. In the nomenclature of this paper it is given by &'
e'
3E'h^ 1 4^,.
1 6E^
(14)
where h is the laminate thickness and E^ and E^^ are respectively the laminate modulus and the modulus of the locally delaminated sublaminate as calculated from the laminated plate theory. In Eq. (14), the strain energy release rate is independent from the delamination length. Also, the effect of matrix cracking is not taken into account when calculating the laminate modulus. Figure 3 shows the normalised strain energy release rate G^"^ /e^^, calculated from Eq. (6) as a function of £/t, i.e. the delamination length normalised by the single ply thickness. The laminate lay-up is [Oj /252 / - ^Sj],, and crack half-spacings are s = 40t and s = 20t. This is equivalent to the crack densities of C = l.OOScm"^ and C = 2.016cm"^ respectively. It may be seen that the present approach gives the strain energy release rate for local delamination that depends both on the crack density and delamination length. The result of Eq. (14) for the same lay-up is found equal to 21.45 MJ/m^ and can be reduced to 12.7 MJ/m^ if shear-extension coupling and bending-extension coupling are taken into account [1, 2]. Still, it is much higher than our predictions, since it does not account for matrix cracking and is independent of delamination length. Figure 4 shows the variation of the laminate axial modulus E^, transverse modulus E^,, shear modulus G^,, major Poisson's ratio v^, (normalised by their undamaged value) as a function of the relative delamination area D^"^ for the [O2 /3O2 /-302]^ laminate Matrix crack density was taken equal to C = 3cm'^. Values at ^ = 0 indicate residual engineering properties of the laminate at this crack density without delaminations. Local delaminations further decrease the laminate moduli and, for this lay-up, increase the Poisson's ratio. Matrix cracking in angle-ply laminates also introduces the coupling between extension and shear. The axial/transverse shear-extension coupling coefficients [5] that characterise shearing in the xy plane caused by respectively axial/transverse stress are plotted in Fig. 5 as a function of the relative delamination area D'"^. There is no experimental data to compare our analytical predictions and this is the topic of current work. Results will be presented at a future event.
C SOUTISANDM. KASHTALYAN
462
•2 "5
u.oo ~
—•—s=40t
0)
w (0 CD
2
0.36 n ^ ^ ^ ^ - • ^ ^ ^
-o—s=20t
>.
D) ^^
o eg
H
0.34 -
•i(0 ^^ CO •o
0 .<2
0.32 -
"cd
E o
z
^^
0.3-
0
0.8
1.6
2.4
3.2
4
Normalised delamination width
Fig. 3. Normalised strain energy release rate G''^/£^ associated with local delamination in a cracked [O2 /252 / - I S j ] , AS4/3506-1 laminate as a function of normalised delamination length l/t
30 60 Relative delamination area, %
Fig. 4. Normalised engineering properties of a [Oj fSO^ /-3O2], AS4/3506-1 laminate as a function of relative delamination area D''' , %. Crack density 3cm'^
Strain Energy Release Rate for Crack Tip Delaminations
463
0.07
^
0.05
0.03
3
0.01
-0.01
30 60 Relative delamination area, % Fig. 5. Shear-extension coupling coefficients of a [Oj /3O2 /-SOj]^ AS4/3506-1 laminate as a function of relative delamination area D''' ,%. Crack density 3cm"^
REFERENCES 1.
O'Brien, T.K., and S.J. Hooper. 1991. "Local delamination in laminates with angle ply matrix cracks: Part I Tension tests and stress analysis", NASA Technical Memorandum 104055. 2. Salpekar, S.A., O'Brien, T.K., Shivakumar, K.N. 1996. "Analysis of local delamination caused by angle ply matrix cracks", /. Composite Materials, 30(4). 3. Zhang, J., J. Fan and C. Soutis. 1994. "Strain energy release rate associated with local delamination in cracked composite laminates". Composites, 25(9): 851-862. 4. Kashtalyan, M. and C. Soutis. 2000. "The effect of delaminations induced by transverse cracking and splitting on stiffness properties of composite laminates". Composites Part A 31(2): 107-119. 5. Jones, R.M. 1999. Mechanics of composite materials: 2"^^ edition. Philadelphia, PA: Taylor & Francis.
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
465
THE EFFECT OF RESIDUAL STRESS ON TRANSVERSE CRACKING IN CROSSPLY CARBON-POLYETHERIMIDE LAMINATES UNDER BENDING
L.L. WARNET, R. AKKERMAN and P.E. REED Composites Group, Department ofMechanical Engineering, University of Twente, 7500AE Enschede, The Netherlands.
ABSTRACT Transverse cracking in cross-ply laminated composite beams is investigated as a function of the level of thermal residual stresses. A range of residual stresses was obtained by varying the lay-up of the beam considered. Carbon -and glass- reinforced polyetherimide was chosen for their capacity to produce high levels of thermal stress. The experimental work focussing on the occurrence of the first transverse crack in 5 types of laminated beam is reported. The data obtained is analysed using both continuum mechanics andfi-acturemechanics principle. The results show that for this particular case, the first transverse crack can be described with a maximum stress criterion. The authors also suggest that the data obtained can be used to define a critical energy release rate for the occurrence of a transverse crack in mode I. The experimental data also show a relation between the critical energy release rate and the strength transverse to the fibres KEYWORDS Laminated Composite, Residual stress, transverse cracking,fi-acturemechanics INTRODUCTION Thermal residual stresses are inherent to fibre reinforced composites due to the heterogeneity of the thermo-mechanical properties of their two constituents. Such stresses build up when composite structures are cooled down from the processing temperature to the test temperature. Residual stresses will be present on both a fibre-matrix scale (micro-scale), and on a ply-toply scale (macro-scale) in laminates built up from layers with different orientations. It is recognised that these stresses should be taken into account in any stress analysis. This paper studies the effects of residual stresses on the transverse cracking of cross-ply [90/0]s laminated beams under 3 point bend loading. Transverse cracking (transverse to the fibres as shown in Fig. 1) are recognised as being the first damage mechanism occurring under bending/impact condition. Experiments were performed on carbon-polyetherimide laminates having different levels of thermal stresses. Different levels of thermal stresses were obtained by producing the laminates of different lay-ups and by alternatively using carbonpolyetherimide or glass-polyetherimide plies for the central 0° layer. A carbon-polyetherimide system was chosen for its capacity of producing high levels of thermal stresses, mainly arising from the large difference between the temperature at which the stresses build up (215°C) and the test temperature (23 °C). The number of outer carbon-polyetherimide 90° layers in the beams was kept constant for all the lay-ups used.
466
L.L. WARNET, R. AKKERMANAND P.E. REED
The results of the experiments were analysed according to continuum mechanics as well as fracture mechanics principles. The evaluation of the stress at failure as well as the onergy released are used to evaluate the validity of, respectively, a maximum stress criterion iind an energy approach as a failure criterion. First an short analysis of the level of thermal residual stresses is presented. Then some background is given for calculating stress at the occurrence of the first transverse crack as well as the energy released at the formation of the transverse crack. The experimental program and the results are then presented.
Igg IL
(a) ^ (b) Fig. 1. Three point bend set-up with occurrence of a transverse crack in the bottom 90° layer. BEAM BEND TEST ANALYSIS The quantification of the thermal residual stresses will be shortly highlighted. Then, the occurrence of the first transverse crack will be analysed using the level of stress reached as well as fracture mechanics principles. For both these techniques, the composite beams are considered from a macroscopic point of view, where the layers are assumed to behave in a transversal isotropic way. Thermal residual stresses Quantification of residual stresses after manufacture. The build up of thermal stresses starts during fabrication of the laminate when it is cooled from the stress free temperature to room temperature. The stress free temperature in the case of an amorphous thermoplastic as used in this study is taken as the glass transition temperature [1] {Tg of the Polyetherimide used is 215°C). On a fibre-matrix scale, the contraction of the matrix {am = 57 x 10"^ /°C) is constrained by the presence of the fibre {Of = -1 x 10"^ /°C for the carbon in the fibre direction). This results in residual stresses on a fibre-matrix scale (microscale). On a macroscopic scale, the properties of a unidirectional layer can be considered trans\ ersally isotropic. This means, in turn, that a multidirectional composite will not only contain stresses on a microscale, but also on a ply-to-ply (macroscopic) scale. The analytical quantification of the residual stresses on the macroscopic scale is generally based on a simple ID model [2] or the 2D plane-stress classical lamination theory [3] (CLT).
The Effect of Residual Stress on Transverse Cracking
467
An expression for the residual in-plane stress in the 90° layer of a cross-ply [0/90]s laminate according to a ID model is: {90) _ r,l*
E2toEj(a2-aj)AT
(1)
tgEj + tgoE2
where Ei and a, are the in-plane elasticity moduli and the coefficients of thermal expansion of the layer in the fibre direction (/=1) and transverse to the fibre direction (/=2), to and tgo are the total thickness of the 0° and 90° layer respectively and AT is the temperature difference between Tg and room temperature {AT =TR-Tg). The different levels of thermal stresses in this study were achieved by producing cross-ply plates having different lay-up. Since this study focuses on the failure behaviour of the 90° layer, the amount of 90° layers was kept constant for all the lay-up used in order to prevent geometrical effects. It was shown by Parvizi, Garrett and Bailey [4] that the stress at which transverse cracking occurs is inversely proportional to the thickness of the 90° layer. The effect is referred to as the constraining effect of the 0° layer. The variation of lay-up was therefore limited to the 0° layer, where not only its thickness was changed, but also the material used. Still based on polyetherimide, a glass-reinforced layer provide different thermo-mechanical properties as shown in Table 1, leading to lower levels of thermal residual stress. This table gives the thermo-mechanical properties of both carbon- and glass reinforced polyetherimide layer. Table 1. Thermo-mechanical properties of the two types of layers used. Ei(GPa) E2(GPa) Gu (GPa) vn
V23
ai (/°C) as (/"C)
Carbon-PEI
117.2
8.2
3.5
0.32
0.45
1x10"'
Glass-PEI
43.1
14.3
5.5
0.27
0.45
7.5x10"^ 26x10"^
32x10"'
Based on these properties, the level of residual stress in the 90° layer for the different crossply lay up used in this study as calculated with the classical lamination theory (CLT) range from 0 to 43.5 MPa as shown in Table 2. Table 2. Macroscopic residual stress in the 90° layer after fabrication, according to the CLT [906c]s U90)
(MPa)
[904c/0ig]s
[904c/03g]s
[904c/0lc]s
[904c/04c]s
19.7
27.3
36.9
43.5
Stress relaxation between manufacture and testing. Between manufacturing and testing, relaxation of the residual stress will occur as a result of the viscoelastic properties of the matrix, which controls the behaviour of the 90° layer. Measurements to determine the relaxation of the ply-to-ply residual stress of the carbon-polyetherimide lay-up, with the highest \CYG\ of residual stresses of the laminates tested ([904c/04c]s), have been presented previously [5]. It shows that the level of residual stress follows a power law with time. For the purpose of this study, it was chosen to perform the bending experiments 240 hours after fabrication, as the level of residual stress remains more or less constant. The level of residual stress as calculated in (1) is then altered by a reduction factor,/v. The relaxed residual stress.
468
LL WARNET, R. AKKERMANAND P.E. REED
in the 90° layer is then defined as, c^S/* (^) = /v {240hours) cr^f/;
(2)
The factor/, for the lay-up considered are reproduced in Table 3. Table 3. Viscoelasticity induced correction factor /^, 240hours after fabrication.
/v
[906c]s
[904c/0lg]s
[904c/03g]s
[904c/0lc]s
[904c/04c]s
1
0.955
0.925
0.905
0.865
Bending stress A relation for the bending stress in the 90° layer as a function of its coordinate in the thickness direction z is obtained from simple beam theory: (Qo^,
F(L/ 2-\d\)zEy
,
J
2(EjIo + E2l9o) where d is the longitudinal distance between the crack plane and the vertical axis through the loading nose as shown in Fig. 1. This distance is necessary to calculate the actual bending stress where the crack forms, since the bending moment is not constant over the length of the beam. The bending stress has its maximum at the beam free surface, (i.e. for z = h/2 with h the total beam thickness) and this is the location where the transverse crack is assumed to initiate. Energy released at crackformation The energy released at the formation of the first transverse crack will be calculated from the experimental results. The values obtained will be used to evaluate the applicability of an energy failure criterion. The transverse crack in the 90° layer of a cross-ply [904c/0x]s beam studied here is assumed to grow in the thickness direction according to a mode I failure mode (opening mode according to the fracture mechanics theory [6]). The energy release rate associated with the growth of a crack is defined as [6]:
dA where U is the strain energy, W the external energy and dA an increment in crack area. In the present study, it was observed that the first transverse crack mostly occurred in an unstable way, with a sudden force drop and therefore constant displacement. In the case of constant displacement (^and assuming linear elasticity, equation (4) reduces to:
dA S=const
2C
"^
The Effect of Residual Stress on Transverse Cracking
469
with dcldA the rate of change of compliance with crack growth. This last well known result [6] is not correct when residual stresses are present. However, it presents a convenient way to estimate an experimental 'partial' energy release rate, since the beam deflection 5 and the change in compliance dc are easily measurable. Assuming that the crack grows in the thickness direction, equation (5) becomes:
^exp
2Bdac
(6)
\^2-Ci)
where B is the specimen width, ci and C2 are respectively the compliance at a crack length a and the compliance at crack length a+da. It is worth noting that equation (6) is only applicable for small compliance changes. This means that this procedure cannot be used for the unidirectional [906c]s beams, since the occurrence of the first transverse crack also means the total fracture of the specimen (i.e. C2=oo). It was observed during the experiments that most cracks grow in an unstable way and therefore this equation will be applied for a crack length change from an initial crack length Ui to the total transverse crack length tgoll. In equation (6), the value of the initial crack length at still has to be worked out. The length of the initial crack is obtained by performing an analysis with a finite element model. A 2D model with plane strain, four noded quadrilateral PLANE 42 elements was used for this purpose. In this model, the increase of compliance in a cross-ply beam due to the growth of a crack from an initial crack length at to frill length tgoll is calculated. It is then compared to the measured compliance increase due to the formation of the first transverse crack. As shown in the meshed geometry representing half the beam in Fig. 2, this was achieved by releasing the nodes corresponding to the crack length at the boundary condition for symmetry. Further variables necessary for the model, like the deflection J, the beam width B and the beam thickness h, were taken for each lay-up as the averaged values from the tested specimens. The model is loaded with the deflection 5 as well as with a thermal loading corresponding to the relaxed thermal residual stresses. Two models for each lay-up need to be solved, the first having a starter crack length at, the second the total 90° layer thickness tgoll.
1^
d[
90^
1
A
1Y
j
Fig. 2. Mesh used for the calculation of the compliance of the cracked cross-ply beam In order to obtain the 'total' energy release rate, which can be compared to the critical energy release rate Gc, the same finite element model is used, where the energy release rate is calculated according to [7]:
B da
5=const
2B
\o.{£-aM)dV \v
lG.{£-aAT)dV Ja=t,J2
\v
(7) ^a=a,-
with a, 6 and a the total stress, total strain and thermal expansion tensors respectively.
470
LL WARNET, R. AKKERMANAND P.E. REED
EXPERIMENTAL PROGRAM Specimen preparation 250x250mm^ laminates made from carbon and glass polyetherimide prepreg material (Ten Gate Advanced Gomposites bv) were consolidated in a closed form mould at 325°G, at a pressure of 0.7MPa for 20min. Gooling to room temperature was done at the same pressure over a period of 40min. The plates were subsequently inspected by G-scan. The matrix content of the resulting Garbon-PEI laminates was measured using a Soxhlet system with chloroform as the matrix solvent. An average of 41.4% mass matrix fraction was obtained (relative standard deviation 3.3%), based on coupons extracted from 6 different plates. The average ply thickness was 0.162mm with a relative standard deviation of 3%). Around 10 specimens per lay-up (see list in Table 2) were cut using a water-cooled diamond sav/ from different locations in the plates for statistical purposes. For the unidirectional specimen [906c]s and the cross-ply beams having the highest level of residual stress [904c/04c]s, 20 and respectively 40 specimens cut from different laminates were tested. It is worth noting that a few specimens coming from the plates having the highest level of internal stresses ([904c/0ic]s & [904c/04c]s) did have obvious transverse cracks in the 90° layer and were not tested. The cut specimens were conditioned at 23°G and 50% relative humidity until testing. An arbitrary time of 240 hours after fabrication was chosen for the bending testing. Experimental set-up All experiments were performed on a universal screw-driven testing machine fitted with a three point bending set-up, at a constant cross-head velocity of 0.5mm/min. The span-length L used was 30mm, with 5mm diameter cylindrical fixed supports, and a 9.5mm diameter cylindrical loading nose. The beams were around 50mm long and nominally 12mm wide (b), the precise width being measured for each specimen. The deflection of the beam was monitored using an inductive displacement gauge (Mahr Pupitron) with a sensitivity of 0.06mm/V. The transducer measured the deflection at the bottom side of the beam, directly under the loading nose main axis. No extra correction for indentation was necessary. The detection of the transverse cracking took advantage of the 90° layer being outer layer. It was therefore possible to follow the transverse cracking growth process in its width using simple optical recording techniques. Furthermore, the formation of a crack in the outer layer of the cross-ply laminates mostly lead to a significant increase in bending compliance which can induce a force drop if the crack growth is unstable. The force at which the crack occurs is not sufficient for determining the stress at fracture but also requires the position of the crack along the beam. For this purpose, cracking was visualised and recorded by a GGD camera connected to a time-coded video recorder. As sketched in Fig. 1, the recording occurred through an optical mirror placed under the beam and fixed to the loading nose in order to keep the required picture sharpness during the test. The recorder produced 24 images per second. Relating the recorder time scale to the forcedisplacement time scale made it possible to relate every cracking event to the forcedisplacement data. Within the scope of this paper, only the occurrence of the first transverse crack will be considered.
The Effect of Residual Stress on Transverse Cracking
471
RESULTS AND ANALYSIS Stress at the onset of the first transverse crack The crack growth was unstable in most cases, which resulted in a sudden force drop and an increase in beam compliance, as shown in Fig. 3. The transverse cracks typically started at one sawn edge of the beam. The crack growth was stable on only some [904c/04c]s specimens. Crack dimm) 1
-3.81
2
-0.99
3
4.09
4
1.51
5
-8.3
Deflection S (mm)
Fig. 3. Three point bend set-up with occurrence of a transverse crack in the bottom 90° layer. The axial stress at the initiation of the first visible transverse crack is now considered and results are given in Table 4. These average values include for each lay-up the relaxed residual stress, the bending stress and the total stress at the occurrence of the first transverse crack, as well as the relative standard deviation. The detailed results of the total stress at first ply failure are shown in Fig. 4, where the stress for each test specimen as well as the average for the considered group of specimens is set against the relaxed level of residual stresses. Table 4. Residual and bending stresses at the initiation of the first transverse crack. [906c]s
[904c/0ig]s
[904c/03g]s
[904c/0ic]s
[904c/04c]s
0
18.8
25.2
33.4
37.6
70.4
51
41.3
31.3
32
Total stress <]*(MPa)
70.4
69.8
66.5
64.7
69.6
rsd (%)
9.1
111
9.5
12.9
6.1
Residual stress Bending stress
LL WARNET, R. AKKERMAN AND RE. REED
472 y\j -
cd
&
s
X
80 i \
•
X
+
X
5 B
o 0
t
0
-1-
•= 6 0 j
^
O
tS w
C/3
S
00
B
0 0
*
8
504030 -
1
\
1
1
10
15
20
25
—1
r
30
35
40
Residual stress cr^^i (MPa) O POJsXPVOigls + [904Ag]s O P04c/0ic]s D [904e/04Js - Average Fig. 4. Axial stress at first transverse crack as a function of the relaxed residual stresses The total stress at the occurrence of the first transverse crack is the principal stress, and it is therefore possible to use a Maximum Stress Criterion as a failure criterion. It is simply performed by comparing the stress at the occurrence of the first transverse crack to the strength of the layer. In other words, failure occurs when: ^ftc,I* -
^c,90
(8)
Where ac,9o is the stress at the fracture of the [906c]s specimens presented in Table 4. In Fig. 4, this criterion is represented by the horizontal line at 70.4 MPa. Although the scatter is high it shows that whatever the level of thermal stress, the simple stress criterion describes well the first transverse cracking for this material in this loading situation. It should be added that no conclusions should be drawn on the applicability of this criterion for any types of layup, as it is recognised that a continuum mechanics based criterion is geometry dependent [4]. Energy release rate at the crackformation It was already mentioned that for the beam tested, transverse cracking was mostly associated with unstable crack growth. It is worth adding that a micrographic study of the transverse cracks showed that the release of energy induced by transverse cracking was not enough to lead to the initiation of a crack for delamination between the 90° and the 0° layer. Such an observation was made on carbon-epoxy systems [8]. In order to calculate the experimental energy release rate Gexp using relation (6), it is necessary to evaluate the length of the initial crack at. It is proposed to compare the measured change in compliance dcexp due to the formation of the first transverse crack with a modelled change of compliance calculated for different length of starter crack. A 2D plane strain ANSYS finite element model is used for this purpose, where the theoretical increase in compliance dcfem due to the formation of the transverse crack from at to tgoll is computed for different values of at. For each lay-up considered, the model is loaded thermally to account for the relaxed residual stresses, as well as the average measured deflection, ^for each lay-up.
The Effect of Residual Stress on Transverse Cracking
473
An example of the type of graph obtained is shown in Fig. 5 for the [904c/04c]s beams, where the modelled increase in compliance dcfem is plotted as a function of the initial crack length at. For the type of laminate considered, this example shows that the measured increase in compUance of 1.3.10'^ mm/N corresponds to an initial crack length at of about 0.23mm. Results for the other lay-up are given in Table 5.
Fig. 5. With finite element modelled change of compliance due to the occurrence of the first transverse crack as a function of the initial crack length for the [904c/04c]s lay-up. Table 5: Assumed length of the starter crack.
Evaluated at (mm)
[904c/0ig]s
[904c/03g]s
0.5
0.5
[904c/0ic]s
[904c/04c]s
2.2
Two distinctive behaviours can be observedfi-omthe results in Table 5. Li the case of the layups having the highest level of residual stress ([904c/04c]s and [904c/0ic]s), the measured increase of compliance due to the formation of the first crack corresponds to a starter crack in the FEM model of respectively 2.2 x 10"^ m and 3 xlO""* m. For the lay-ups having the lowest level of residual stresses ([904c/0ig]s and [904c/03g]s), the measured increase of compUance is always higher than the calculated increase in compliance, whatever the assumed length of the starter crack, hi other words, more energy was released than could be theoretically predicted in these two cases. Since a negative initial crack length is not physically acceptable, an arbitrary initial crack length of 0.5 x lO"'* m was chosen. The difference in behaviour can be indicative of the presence of defects or micro-cracks induced by the presence of residual stress. At a high level of residual stress, the presence of micro-cracks leads to a comparatively lower level of energy being required for the starter crack initiation. At low levels of residual stresses, less or no micro-cracking means that extra energy is required for the starter crack initiation.
LL WARNET, R. AKKERMAN AND RE. REED
474
Based on the estimated initial crack length at, the energy release rate can be calculated using the stress-strain situation provided by the finite element model and equation (7), v^ith as loading the average experimental deflection S and a thermal loading. Results are given in Table 6, where it can be observed that the average energy release rate is reasonably constant as a function of the level of thermal stress. This gives confidence in the validity of the results. Table 6: Average energy release rate
G (J/m^)
[904c/0ig]s
[904c/03g]s
[904c/0ic]s
[904c/04c]s
625
591
716
677
A representation of the scatter obtained if the energy release rate had been calculated for each specimen instead of the average on the sample is obtained by plotting the partial energy release rate as a function of the level of thermal stress. Although the scatter is large, it can be observed that the experimental energy release rate decreases with increasing level of residual stress. uvu -
800 -
B
^
600-
I
u
400 -
o
D D
o
+
i
X
I §
+
200 -
n
n
B 8
0-
1
10
20
30
40
1
\
1
50 Residual stress O^^^ji (MPa)
1
1
60
1
70
1
'
80
X P V 0 i g ] s + [904c/03gL oP04c/0ic]s n P^4c/04c]s « Average
Fig. 6. Partial energy release rate as a function of the relaxed residual stress. The question rise whether the behaviour described can be evaluated on afi-acturemechanics basis. The critical energy release rate associated with the formation of a transverse crack is technically difficult to measure. As a first approximation, it is possible to use the critical energy release rate obtained from a double cantilever beam fracture mechanics test (DCB). This test concerns the growth of a delamination between two layers (mostly oriented al 0°) in opening mode I. Tests performed on the same carbon-polyetherimide at 0°/0° interface as in this study were reported recently [9] and gave a value of 1200 j W . An other approximation is to use the energy release rate evaluated on basis of the experimental as the critical energy release rate, since it is hardly dependant on the level of residual stress (see Table 6 as well as Fig. 7). This would give a critical energy release rate of about 650 J/m^. Worth adding is that this value is only valid in mode I, and not in mixed
The Effect of Residual Stress on Transverse Cracking
475
mode. Such a situation often occurs for transverse crack within a laminate, where it grows under the influence of normal and shear stresses. An extra observation can be made when plotting a linear fit on the data describing the dependence of the partial energy release rate with the level of thermal stress. Fig. 7 shows that the linear fit crosses the G-axis at a value of the energy release rate of around 650 j W . At this point where no residual stress is present, the partial energy release rate should be equal to the total energy release rate, which is now the case. Moreover, the linear fit crosses the residual stress axis at a value of about 70MPa, which is the strength of the 90° layer as measured in this study. This is the value where theoretically only residual stress would be enough to let transverse cracking occur. 1000
20
30
40
50
60
Residual stress ^"-r (MPa) D ^ total
- G.
Fig. 7. Average energy release rate as a function of the relaxed residual stress. CONCLUSION hi this paper the influence of the level of thermal residual stresses on the development of transverse cracking in cross-ply carbon-polyetherimide laminated beams subjected to bending has been considered. Different levels of ply-to-ply residual stresses were obtained by producing cross-ply ([90/0]s) laminates having different lay-ups in the 0° layer. Two methods were used to analyse the bend tests. The first considered the stress at the formation of the first transverse crack, the second the energy released at the formation of this crack. A maximum stress criterion (using the strength of a unidirectional beam as a reference) applies for the structure considered when taking into account the residual stress relaxation. The results based on a fracture mechanics analysis show that the experiments were able to give an (expensive) approximation of the critical energy release rate for transverse cracking in carbon-polyetherimide under mode I. Limitation is that the choice of an initial crack length is critical. This should be of less importance when considering multiple transverse cracking.
476
L.L WARNER R- AKKERMANAND P.E. REED
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
Nairn, J.A., Zoller, P. (1980) J. ofMaterials Science 20, 355. Collings, T.A., Stone, D.E.W. (1985) Composites 16, 307. Eijpe, M.P.I.M., Powell, P.C. (1996)/. of Thermoplastic Composite Materials 10, 145. Parvizi, A., Garrett, K.W., Bailey, J.E. (1978) J, ofMaterials Science 13, 195. Wamet, L. (2000). Ph D Thesis, University of Twente, The Netherlands. Williams, J.G. (1984) Fracture mechanics of polymers, Wiley. Nairn, J. A. (2000) Int. J. of Fracture 105, 243. Lammerant, L., Verpoest, I. Composites Science & Technology 51, 505. Akkerman, R., Reed, P.E., Huang, K.Y., Wamet, L. 2"^ ESIS TC4 conf -Fracture of Polymers, Composites and Adhesives, Elsevier.
3.3 Z-Pinned Laminates and Bridging Analysis
This Page Intentionally Left Blank
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
479
DEDUCING BRIDGING STRESSES AND DAMAGE FROM G/c TESTS ON FIBRE COMPOSITES A.J. BRUNNER^ B.R.K. BLACKMAN* & J.G. WILLIAMS* ^EMPA, Polymers/Composites Laboratory Ueberlandstrasse 129 CH-8600 Duebendorf Switzerland *Imperial College London Department of Mechanical Engineering Exhibition Road London SW7 2BX. UK. ABSTRACT An analysis is given for determining a damage factor for the transverse and shear moduli of a composite and also for the local stress prior to delamination. It is determined from the correction factor A which arises in the standard scheme for measuring Gio It is noted that large variations occur in both A and the predicted modulus and this is explained in terms of microcracking which renders the crack length measurements inaccurate. When suitable corrections are made to A the resulting damage factors suggest reductions in stiffness by factors of up to 20 and of local stress by factors of 5. Microcracking is suggested as the primary mechanism causing rising R curves. KEYWORDS Composite, delamination, bridging stress, damage, R-curve. INTRODUCTION There has been considerable effort expended in recent years [1-3] on developing a methodology for determining the critical energy release rate, G^^' ^^^ ^^e delamination of fibre reinforced, polymer matrix, composites culminating in an international standard [3]. The test determines a resistance, or R, curve for the delamination process as Gj^^ versus the crack growth and does this by measuring the load, P, the displacement, J, and the crack length, a simultaneously during stable delamination growth in a double cantilever beam (DCB) specimen. Thus the compliance, C of the specimen (where C=S/P) is determined as a function of a and Gj^ is found from the derivative of this relationship via,
G . = ^ - ^ ^ (1) '"^ 2b da ^ ^ where b is the specimen width. The derivative can be determined in two ways. A simple power law of the form.
480
A J. BRUNNER, B.R.K. BLACKMAN AND J.G. WILLIAMS C = Aa"
(A and n are constants)
(2)
may be used or corrected beam theory can be employed,
where E^ is the axial modulus, h the thickness of one arm of the DCB, A is a length correction applied to account for the deformation beyond the crack tip in the beam and A^ is a finite displacement correction factor [1] to give average values of A and Ej. Equation (3) is usually preferred to (2) because when it is fitted to the experimental data, E^ provides a cross check on the method if it is measured independently e.g. by three point bending (3PB) and A can be estimated from an elastic analysis [4,5]. The constant « » 3 but there are no independent checks on the values obtained. The calibration procedure requires that yyirj
is plotted as a function of a and E] is derived
from the slope and A from the intercept. Generally this process gives answers to within 1015% of the 3PB value [e.g. 1,2] and for fibre-polymer laminates, A / = ;{f«2.1.
It was
observed in some of the original work on carbon-epoxy and APC 2 materials [1] however, that while El was usually close to the expected value, A was often higher than that calculated and this was attributed to damage reducing the lateral stiffness. There have subsequently been numerous round-robin exercises on a wide range of materials and several general observations may be made on the^*! and A values obtained. If there is a limited increase in Gjc with a, i.e. a flat R curve, this is usually associated with E^ and A being about the expected values. Such data usually show very Httle evidence of damage or of fibre bridging where unbroken fibres straddle the opened crack surfaces. When there is evidence of such bridging and damage there is usually a rising R curve with significant variations in both E^ and A. There is generally a relationship between E^ and A in that they increase together and A is always larger, often greatly so, than the elastic value. Such effects occur in materials in which the fibre-matrix bonding is weak, as in many glass fibre materials, and in cross-ply laminates. Such observations are consistent with the notion of high A values being associated with damage and perhaps a difficulty in defining a. This paper will analyse these variations in E^ and A with a view to rationalising the observations and then examine the A values to determine if they may be used to characterise the damage process via a damage factor and a bridging stress. Such parameters are additional items of information which may be deduced from the usual Gj^ test which may prove useful in understanding damage in composites caused by interlaminar crack growth. ANALYSIS A recent analysis [5] has developed that used previously [4] and has included the lateral stress distribution in the beam beyond the crack tip. For anisotropic composites^ with a transverse modulus E2, a shear modulus ju and Poisson's ratio v we have,
t here Gu=n: E\\^E\ and Eji^Ei (with directions as defined in Fig. 1)
Deducing Bridging Stresses and Damage from G,c Tests on Fibre Composites
^
+ 0.24 1 ^
10
481
(4)
The first term is the contribution from the shear deformation and the second that from the transverse deformation. The transverse stress at the crack tip is given by; .2.4/E,Gj^
(5)
We now assume that the damage incurred in advance of the crack tip can be characterised by a factor 0 < ^ < 1 such that both /u and E2 are reduced by this factor. A ^ of unity implies no damage and a ^ of zero implies total loss of transverse and shear stiffness. If x is measured in a test, (/> may be determined from,
;^2^_Lf^_2v 1 + 0.24,
(6)
For most laminates the shear effect is dominant and thus. (7)
The damage, or bridging, stress is now given by,
h
Z i
Mh
X may be found from equation (3) from the standard procedure in which \r/\jj
(8)
is plotted
against a. It may also be evaluated on a point by point basis for each crack length if an independently measured value ofE], is used and in this case,
^4>'-'/'
(9)
The G/c values may also be calculated on a point-by-point basis from
^ic
=
'3Pc^^ bh
and the R curve found.
1 X
E,b
(10)
A.J. BRUNNER, B.R.K. BLACKMANAND J.G. WILLIAMS
482
A possible model of microcracking to describe the damage is shown in Fig. 1. m mic roc racks of length 2c occur in each half of the beam in the local zone A (3 are shown). Assuming no interaction the lateral modulus, ^E^, is given by, 1
1
l + TT
hA
1 , n (mc - = 1+— —
(11)
The elevation in Gic from these microcracks can be computed from simple area addition giving, Gjc _. 4 f mc (12) R= Gjc{init) x\^ The initiation value, Gjc(init), is assumed to be that of the matrix and G/c is the maximum value, seen in the R curve. Thus elevations in toughness can be computed from ^ thougli only mc the parameter I -^^^ I is determined. In practice h is usually 1-3 mm and cwO.l mm so that ^
can be expected to be about 0.05 to give an estimate of m.
Fig. 1. Microcracking model EXPERIMENTAL Tables 1-5 give values of Gic at initiation, ^'jand A determined from the ISO protocol. In Table 1 a glass fibre PMMA material was tested by six laboratories and showed considerable fibre bridging though it was not quantified. The Gjc values are for initiation and, apart from one result, are reasonably consistent. There was a pronounced R curve with G/c rising to about 2,500 J/m^ over the crack growth of 40 mm. The A and Ei values derived from the compHance measurements show a large variation with a mean modulus of 57 GPa. Three point bending tests on the GF-PMMA gave modulus values of between 32 and 34 GPa. In the data analysis presented in Figs. 4-5, a value ofEj of 33 GPa has been used. The average value ofh was 1.6 mm so the minimum A would be about 4 mm so the values obtained between 6 and 26 mm are all above this. This set is given because it is extreme and highlights the huge variation in E\ and A though no such large variations are seen in Gio Table 2 shows a much less extreme example and is for 6 specimens of a uniaxial carbon fibre epoxy material tested in one laboratory. There is some variation in Ei but much less tian in the previous example and the mean value is close to the true value of 145 GPa. The minimum
483
Deducing Bridging Stresses and Damage from Gjc Tests on Fibre Composites
A is again 4 mm so the measured values are all higher. An attempt was made to quantify the bridging here and A does increase with the amount observed. There was a modest R curve with Gic rising to about 200 j W over 60 mm from the initiation value of 130 J/m^. The data in Tables 3, 4 and 5 are for another CF-Epoxy system and here a uniaxial and two cross-ply laminates, a symmetrical interface design [0/90]6s and an unsymmetrical one [0/90] 12, were used. The uniaxial material gave small variations in both Ei and A with good agreement of the mean E\ to the 3PB value. Very httle bridging was observed and the R curve rose from 270 J/m^ to about 350 J/m^. The cross-ply materials showed considerable bridging and R curves and there were large variations in both A and Ei. Indeed one A value was negative but this occurred in a specimen where the crack deviated from the central interface and the value is not included in the analysis. The minimum value of A is again about 4 mm so the fitted values are close to and actually lower than this for the unsymmetric material. The large amount of bridging and the R curves would suggest much higher values and this will be addressed later. 40
'
, , , , , , _
1
: —•—GF/PMMA - -•— CF/epoxy : -B - CF/epoxy [0]24 : --A--CF/epoxy [0/90]6S 30 : • CF/epoxy [0/90] 12
-_
25 S
20
• / :
A
/^
-
V•
A'
• •
'
1
1
1
1
1
1
1
1
1
100
> /
,
/
m
-
n
: 1
150
,
,
,
, 1 200
E, [GPal Fig. 2. Variations of A with £/ from standard procedure. The A values are plotted versus the E\ values for all five sets in Fig. 2 and they form similar trends with more pronounced scatter in the GF-PMMA data. The values are obtained from equation (3) which can be written as.
Ehlf
(a + A)
(13)
The form of the plot is shown in Fig. 3. The fact that the "back-calculated" modulus Ei for the GF-PMMA is 57 GPa, nearly twice the experimental value of 33 GPa is worth noting. The cause(s) for this are not clear. Evidence pointing to possible explanations include large thickness variations along the beam (observed between 2.9 and 3.4 mm, i.e., up to 0.5 mm, more than the 0.1 mm allowed by the test protocol [3]), deviation of the delamination from the
484
A J. BRUNNER, B.R.K. BIACKMAN AND J.G. WILLIAMS
mid-plane (experimentally observed in some cases, and again, not permissible [3]), and, probably to a lesser extent, the large bending deformation in the relatively thin beams (5 mm is recommended for glass-fibre reinforced laminates [3]). Two sources of error are likely. Providing that a stiff machine is used there is unlikely to be any significant error in C which leaves the dimensions b and h and the crack length a. The width b does not vary significantly but h can be a source of error because of uneven thickness in the sheet or non-central cracking, particularly with bridging. The former source is excluded in the protocol since specimens with significant thickness variations are excluded (the protocol [3] sets a maximum allowed variation in measured thickness, 2h of 0.1mm, implying an maximum permissible thickness variation of 3.3% in 3mm thick carbon fibre specimens or 2% in 5mm thick glassfibre specimens). The latter is possible but can usually be recognised as very low A values. Additionally, if a was correct but h in error we would expect A to be consistent and £, to vary. a is measured at the side of the specimen usually by the cracking of a paint layer, and this becomes increasingly difficult as bridging increases since it is preceded by microcracking around the crack tip. In addition, there can be inaccuracies from crack front curvature.
Fig. 3. Variation in A with errors in a. If one supposes an error of Sa in measuring a, as shown in Fig. 3, then the fitted slope can rotate about some mean crack length a giving a change in A, the true value, of 5A (this argument holds also if Sa is assumed to increase with increasing delamination length). From the diagram Sa =
'-^\SA a + AJ
(14)
For the GF-PMMA, for example, a = 60 mm, a^ = 40 mm and A = 12 mm and variations in A are about +14 mm to -6 mm indicating errors in Sa of+2 mm i.e. ±3%. For the CF-Epoxy materials a =90 mm, a^=60 mm and variations in A for the uniaxial materials are about + 5 mm for A = 9 mm, i.e. Sa^ ± 2 mm, i.e. +2%. For the cross plies the variations are much larger suggesting crack length errors of about ± 5 mm, i.e. ±5%. Such a picture is consistent with the notion of an increasing difficulty of defining a when microcracking and bridi ung is occurring. The variations in E^ and A can be deduced from equation (13) since
485
Deducing Bridging Stresses and Damage from Gjc Tests on Fibre Composites
^ ( ^ + A ) - ^ ( a + A)
(15)
where E. and A are the true values. Thus,
3
£-1
+A
(16)
for A « ( 3 . The lines drawn in Fig. 2 are best fits and, on comparing the slopes predicted from equation (16), the uniaxial materials give quite good agreement with the mean a values reflecting expected variations, i.e. 2%. For the other materials, which showed bridging, this was not so and the measured slopes were much less than those expected, suggesting lower a values. It is possible that this arises from a systematic error in a because of the bridging in that the true, or effective, value of a is given by, ^true =ka + a{\-k)
(17)
This reflects the stiffening effect of the bridging and is an idea given in [9] where an effective crack length is used to model bridging. The k value for each test can be found from k = EI/E^ and there is assumed to be no correction at a=a. The corrected A value is, from equation (15) 'A = kA-a{\-k)
(18)
Average A values can be found for each data set and these are given as j = A/h in Table 6, together with values of G\c, k, crmax, ^, f^c/h and R. The detailed point by point analysis is explored in Figs. 4-7 where examples are given of specimens of the GF-PMMA and the [0/90]6s epoxy materials. Both are analysed using the k correction to crack length chosen to make Ei the 3PB value. For the GF-PMMA, k=0.S6 and in Figs. 4 and 5 there is a steadily rising R curve from 400 j W to about 4,500 J/m^ with o^^^ rising from about 100 MPa to 300 MPa and % is constant at about 5 giving a (p value of 0.04 and a rising from 25 MPa to 70 MPa. This suggests a strong initial damage mechanism which decreases as the crack grows, possibly by some form'of crack tip shielding. Figs. 6 and 7 are the results for the CF-Epoxy cross-ply material [0/90]6s which show similar effects.
CONCLUSIONS The general conclusions of this analysis may be determined by an examination of table 6. Ei has now been fixed at the 3PB value and a corrected by k to give the effective value. All Gic values are changed since a becomes ka and the values alter by a factor of k'\ For the GFPMMA these changes are around 30% which, since£"1 oc k Ei, translate into errors of a factor of 3 in Ei. For the other materials k is closer to unity giving errors in a of about 10%. The a values are in the range 25-80 MPa going from high to low damage and those figures are consistent with measured
A J. BRUNNER, B.R.K. BLACKMAN AND J.G. WILLIAMS
486
5000
4000 0M
3000
2000
1000
50
55
60
65
Effective crack length [mml
Fig. 4. R curve, CTmax and % for the uniaxial GF/PMMA material.
0.15
0.1
^
0.05
40
45
50
55
60
65
70
Effective crack length [mm]
Fig. 5. Bridging stress cr and damage factor ^ for the uniaxial GF/PMMA material.
Deducing Bridging Stresses and Damage from Gjc Tests on Fibre Composites
50
60
70
80
90
100
487
110
Effective crack length [mm] Fig. 6. R c u r v e , CJmax and X for the cross-ply CF/epoxy [0/90]6s material.
200
50
70
80
90
Effective cracli lengtli [mm]
Fig. 7. Bridging stress cr and damage factor (|) for the cross-ply CF/epoxy [0/90]6s material.
488
A.J. BRUNNER, B.R.K. BLACKMANAND J.G. WILLIAMS
transverse strengths [8]. The values of c, derived from (m//z)=0.05, are in the range 50-10 and these give rise to R values in the range 4.5-1.7. The latter values are for the uniaxial CF-i. poxy systems while the observations suggest that /?~1.5. For the GF-PMMA the observed values are about 13 compared to 4.5 predicted while for the CF-epoxy cross plies 5 is observed and 2.3 - 2.9 is predicted. Thus the notion of the observed R-curve being due to microcracks and being a genuine effect is confirmed by the observations. It is also of interest to note that in Figs. 5 and 7, (/) remains constant during crack growth while a increases suggesting that stiffness loss rather than a limiting stress is the controlhng factor. These observations are somewhat at odds with those given in [9] for a GF-epoxy system. It was observed that an effective crack length of about 0.75 times the measured crack growth value removed the R-curve effect using uncorrected beam theory. In an elegant exper ment the bridging fibres were removed by soaking in acid after which initiation occurred iit the same value. This implies that the R-curve calculated with uncorrected beam theory is due to bridging and that when this is corrected for, or removed, constant Gic is achieved No discussion of modulus values or of correction factors is given and it is possible that such an analysis, whilst giving constant Gic, may result in a varying E\. A more detailed study v/ill be necessary to resolve this issue. It is clear, however, that the crucial issue in these tests is the determination of crack length. When microcracking and bridging is small then there is no problem since a may be defined visually. For larger effects such a definition becomes problematical and the definition of an effective value determines the apparent behaviour. REFERENCES 1 Hashemi, S., Kinloch, A. J., Wilhams, J.G., (1990), Proc. R. Soc. London. A247, pp. 173-199. 2 Brunner, A.J., Tanner, S., Davies, P., Wittich, H. Proceedings ECCM-CTS2, Eds. P.J. Hogg, K. Schulte, H. Wittich, Woodhead PubHshing, pp. 523-532. 3 ISO 15024. (2001) Standard test method for mode I interlaminar fracture toughness, Gic, of unidirectional fibre-reinforced polymer matrix composites. 4 Wilhams, J.G. (1989) Comp. Set & Tech., 35, pp. 367-376. 5 Williams, J.G. and Hadavinia, H. (September 2002) "Elastic and elastic-plastic correction factors for DCB specimens." Presented SitECF14, Cracow, Poland. 6 Wilhams, J.G. (1985) "Fracture mechanics of polymers." Elhs Horwood, Chichester. 7 Unpublished values determined at Imperial College (GF-PMMA) and EMPA (CF-Epoxy). 8 Daniel, I.M. and Ishai, O. (1994) "Engineering mechanics of composite materials." Oxford University Press. 9 Huang, X.N. and Hull D. (1989), Comp. ScL & Tech. 35, pp. 283-299.
Deducing Bridging Stresses and Damage from Gjc Tests on Fibre Composites
489
Table 1. Round robin results for fracture toughness testing of a modified GF-PMMA Laboratory/Specimen Gic init[J/m^] El [GPaJ ^""^ Amount of fibre A [mm] bridging 6.9 Medium/high 1/1 355 33 17.4 348 1/2 70 8.9 2/1 320" 52 High 6.3 41 2/2 295 High 14.5 606 68 3/1 25.7 4/1 328 115 10.3 4/2 146 65 6.0 259" 47 5/1 9.0 41 5/2 315" 10.8 386 47 6/1 11.3 45 6/2 401" 366*** 11.6 57 Average 92.1(25.4%) 5.8 (50.0%) 22.8(40.1%) SD (CoV) Notes: (*) Analysis using the 5% offset value and the corrected beam theory method. (**) Visual initiation rather than 5% offset value. (***) If specimen 3/1 is considered an outlier and eUminated, the average is 342+49 J/m^. C) From compliance measurement. Table 2. Test data for a CF -epoxy uniaxial composite Laboratory/Specimen GIC* init [J/m'] A [mm]
El [GPa] ^^^
Amount of fibre bridging Medium Low Low Low/medium Medium High
132 10.6 1/1 148 134 5.3 1/2 126 8.2 127 133 1/3 4.3 1/4 147 115 8.8 135 143 1/5 16.6 1/6 120 165 9.0 133 138 Average 8.8(6.7%) 4.4 (49.0%) 18(12.7%) SD (CoV) Notes: (*) Analysis using the 5% offset value and the corrected beam theory method. ( ) From compliance measurement. Table 3. Test data for a CF-Epoxy uniaxial composite Laboratory/Specimen
GIC* init [J/m^]
A [mm]
El [GPa] ^^^
Amount of fibre bridging
8.3 162 2/1 258 2/2 7.1 156 6.7 2/3 287 155 3.5 2/4 248 138 5.0 304 2/5 137 274 6.1 Average 150 26 (9.4%) 1.9(31.2%) SD (CoV) 11.4(7.6%) Notes: (*) Analysis using the visual initiation values from the insert and the corrected beam theory. From 3PB, Ei=145GPa. C) From compliance measurement.
490
A J. BRUNNER, B.R.K. BLACKMAN AND J.G. WILLIAMS
Table 4 Test data for a CF-Epoxy cross-ply [0/90]6s composite Laboratory/Specimen
El [GPa] ^""^ Amount of fibre bridginj4 2/1 4.6 68 330 2/2 95 323 9.7 10.2 93 2/3 2/4 15.0 376 117 299 2/5 -11.8^^ 46 ^'^ Average 332 9.9 93 SD (CoV) 32 (9.7%) 4.3 (42.9%) 20(21.5%) Notes: (*)Analysis using the visual initiation values from the insert and the corrected beam theory. (**) Crack deviated from the mid-plane and the value was excluded from the statistics, from 3PB, Ei=73GPa. C^) From compliance measurement. Gic* init [J/m'l
A [mm]
Table 5 Test data for a CF-Epoxy cross-ply [0/90] 12 composite Laboratory/Specimen
El [GPa] ^^^ Amount of fibre bridging 2/1 287 3.7 67 2/2 208 6.1 77 474 2/3 2.3 62 395 4.4 2/4 69 2/5 238 2.1 68 Average 320 3.7 69 111(34.7%) 1 SD (CoV) 1.6(44.3%) 5.4(7.8%) Notes: (*)Analysis using the visual initiation values from the insert and the corrected beam theory. From 3PB, Ei=62GPa. C) From compliance measurement. Gic* init [J/m^]
A [mm]
Table 6. Average values Material
h (mm)
El
(GPa)
Z
(J/m^)
k
Cmax
*
mc
R
(MPa) (MPa) 342 GF-PMMA 33 3.1 0.83 111 0.12 39 1.6 n.i 4.5 6.2 1.02 145 133 69 CF-Epoxy 1.6 0.11 23 4.0 3.6 274 2.3 85 CF-Epoxy 2.2 145 0.99 0.83 78 0.4 1.7 332 2.5 0.92 98 73 0.36 59 1.2 CF-Epoxy 6S 2.0 2.9 370 62 1.8 0.96 96 0.60 75 CF-Epoxy 12 0.6 2.0 2.3 (*) Initiation value. Assumed values: E2 = lOGPa, // = 4.2GPa and v = 0.3. These values are largely dependent on the matrix properties and volume fraction, which are approximately the same for all materials.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and' ESIS. All rights reserved.
491
Z-PIN BRIDGING FORCE IN COMPOSITE DELAMINATION HONG-YUAN LIU\ WENYIYAN^ AND YIU-WING MAI^'^ ^ Centre for Advanced Materials Technology (CAMT), School ofAerospace, Mechanical and Mechatronic Engineering J07 The University of Sydney, Sydney, NSW 2006, Australia ^ MEEM, City University ofHong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
ABSTRACT This paper presents a theoretical model of z-pin reinforced mode I delamination. Study is focused on the role of the z-pin bridging law on crack growth. Three typical z-pin pull-out curves: pull-out with friction only; pull-out with an initial bonding and friction; and pull-out with an initial bonding, stable debonding and friction, are used as the bridging laws in the delamination crack growth analyses. Solutions are given for a Double-Cantilever-Beam (DCB) with different bridging laws. Comparison of the R-curves due to different bridging laws shows that a simple bi-linear function can be used as a reliable bridging law for the z-pin reinforcement. Computer simulation of mode I delamination of DCB with z-pins shows good agreement between the present model and available experimental results. KEYWORDS Through-thickness reinforcement, z-pinning, bridging law, mode I delamination, DoubleCantilever-Beam (DCB), pull-out. INTRODUCTION Through-thickness reinforcements, such as stitching and z-pinning, are now widely considered as a successfiil method to improve the delamination fracture toughness of laminated composites. To study the mechanisms of this type of reinforcement, Jain and Mai developed the first micro-mechanics models for interlaminar mode I [1] and mode 11 [2] crack growth in DCB specimens reinforced by through-thickness stitching, hi their models, the inter -face between the stitching and laminate was assumed fully frictional. The bridging force of the stitching was calculated by assuming a constant frictional shear stress between the stitch and laminate. Numerical results from their models show that through-thickness stitching can greatly improve the fracture toughness in both mode I and mode II delamination. Later, Cox presented his model of mode II delamination with through-thickness fibre tow [3]. Here, the bridging tow was assumed to deform internally in shear as a rigid/perfectly plastic material.
492
H.-Y. LIU, W. YANAND Y.-W. MAI
The axial sliding of the tow relative to the laminate is opposed by friction, which is assumed as uniform shear tractions, hi his numerical example, both the shear and closure tractions of the tow were given by assumed values based on observations from experiments. Recently, Liu and Mai presented a theoretical model of mode I delamination of DCB with z-pinning [4]. hi their model, the bridging stress of z-pin is calculated by a single fibre pullout model [5], which includes the whole process of the z-pin pull-out: elastic deformation before z-pin debonding, elastic deformation and fiictional sliding during debonding growth and, finally, frictional sliding. Computer simulations are given for mode I delamination fracture with z-pin reinforcement. Effects of areal density, radius, Young's modulus of z-pin, and especially, the interfacial friction between the pin and laminate are studied in depth. Results of all these theoretical models show that the efficiency of the through-thickness reinforcement greatly depends on its bridging mechanism. Therefore, the relationship between the bridging force of the reinforcement and its displacement, that is, the bridging law, becomes the most important issue in theoretical and numerical analyses of z-pin reinforcement. However, a z-pin pull-out is a complicated process which is affected by many parameters, such as material properties, geometry and, especially, the interfacial properties between the pin and laminate. To-date, there are no available experimental details of the bridging law to support all those previous analyses. All those results and conclusions depend on the assumed bridging law or single pin pull-out model with assumed interface properties. So, what is the inaccuracy in these results caused by simplification of the bridging law? Which parameters in the bridging law can provide the dominant influence on the delamination growth? These questions need to be answered before any further refinement of these past theoretical models can be advanced. This paper presents a detailed study on the effects of the bridging law relationships on the delamination crack growth of z-pinned laminate. Solutions are given for a Double-Cantilever -Beam (DCB) with three typical bridging laws: pull-out with friction only; pull-out with initial bonding and friction; and pull-out with initial bonding, stable debonding and friction. By changing the bridging law parameters, their effect on crack growth is examined. Based on this parametric study, a simplified bridging law is suggested for further analyses.
THEORETICAL MODEL OF DCB MODE I DELAMINATION
Figure. 1 Double-Cantilever-Beam with z-pins
Z-Pin Bridging Force in Composite Delamination
493
Figure 2 Mechanics model of z-pins bridging delamination, in which Lc is the delamination crack length; P is the load corresponding to the applied displacement, 5. Figure 1 shows a DCB specimen with z-pins. During delamination growth, a reinforcing z-pin provides a closure force to the opening crack. Li the meantime, the z-pin experiences elastic deformation, debonding from the laminates and, finally, frictional pullout. Due to symmetry only one single beam is considered in the mechanics model of a delamination bridged by zpins as shown in Figure 2. The bridging force caused by the f^ z-pin, at its location x/, is added to the beam as an external force, P/. From a generalised beam theory, the differential equation of the deflection curve is given by Eh" = M(x)
(1)
where, EI is the flexural rigidity of the laminated beam and Mis the bending moment. Before the delamination tip reaches the first pin, 0 < Z^ < x,, there is no bridging force on the beam, the total bending moment applied to the beam is M(x) = Px
(0<x
(2)
The solution of Eq. (1) at this initial stage can be obtained as follows: S z=^(^
-3xi:+2Z:)
(0<x
(3)
where, ^is the applied displacement at the loading end, S=z(0). The fracture energy method is used as the delamination criterion [6]. The strain energy release rate is calculated by: Gj =
1 dU
(4)
in which, w is the width of the laminated beam; U is the total strain energy in the bent beam and is:
494
H.-Y. LIU, W. YAN AND Y.-W. MAI
1 '^'
U = —JM\x)dx EI
(5)
0
From Eqs. (2), (4) and (5), we can obtain
If the energy release rate of the bent beam is greater than a critical intrinsic toughness of the unpinned DCB, Gic, the delamination will propagate. After the crack reaches the first pin, L^ > Xj, the first pin starts to provide a bridging fcrce to the opening crack. The differential equation of the deflection curve in each interval is given by Px
{0<x<x,)
Eh'
(7) Px-P^{z){x-x^)
{x^ ^x
in which, P\ is the bridging force from the first z-pin at the location x=xi. The boundary conditions in each interval are z(x,r = z(x,r,
z'(x,r = z'(x,y,
ZXLJ = Z{L^) = O,
(S)
hi Eq (7), the bridging force. Pi, is a function of the flexural displacement z(x\) which is given by the bridging law. Thus, it is mathematically difficult to obtain a closed-form solution of Eq. (7). histead, an iteration method is used to obtain a numerical solution, hi the iterative calculation, we add the displacement, 5, and the crack length, Lc, step by step by a tiny increment, hi the first step, we give a tiny increase in the crack length, L,=x,+AL^
(9)
Since the increment ALc is very small, the displacement of the first pin is very small. Thus, the bridging force of the z-pin can be neglected in Eq. (7). According to Eq. (3), the displacement of the first pin can be approximately calculated by z{x,) = ^(x',-3Llx,+2Ll)
(10)
Substituting the obtained displacement, z(x\), into the pin bridging law, the bridging force, Pi, can be obtained, hicreasing the applied displacement, S=3^AS, and applying the bridging force. Pi from the previous step to Eqs. (7) and (8), the deflection of the beam under i resent applied displacement, S=3^AS, can be approximately calculated:
Z-Pin Bridging Force in Composite Delamination
1^„3
495
(0<x<;c,)
Px^ +CX + EI5
Eh
(11)
-Px' --PAx-x,f+Cx 6 6 ' '
+ EIS
in which P ,, -2,-,. P = ^(L,-x,n2L,+x,)+
C, =^(4-x,)^
- 3£7<5 ^3
(12) (13)
2
The new displacement of the first pin z{x\) can be obtained from Eq. (11). Substituting z(x\) into the bridging law, a new bridging force at this step can be obtained. This bridging force is subsequently applied to calculate the deflection of the beam in the next step. If the number of the bridging z-pins is A^ when the delamination crack length is Lc, the differential equation of the deflection curve in each interval becomes: (0<x<x^)
Px
EIz" =
Px-Y,Pjiz)(x-Xj)
(x. <x<X.,,,
i = l,2...N-1)
(14)
7=1
Px-Y^Pj{z\x-Xj)
(x^<x
Following the above iteration method, the solution at this step can be solved as
1
EIz
(0<x<x^)
Px'-hC x + EIS
1 ' 1 -Px' -Y-P.ix-x.y 6 73'6 ' 1 ^1 -Px' -y-P.(x-x.y
+C x + EIS
(x. < x < x . , ,
+C x + EIS
/ = 1,2,...A^-1)
(x^<x
(15)
)
in which. (16)
496
H.-Y. LIU, W. YANAND Y.-W. MAI
P = ^[Y,Pj{K-Xj)\2L^^x.) ^L'c
(17)
+ 6Em
y=i
Then, the displacement of the /th pin is: Px^ M p. EIz{x.) = —^-T-^{x.-Xjf+Cx.-¥EId 6 y.i 6
/=1,2,
N
(18)
Adding the appHed displacement step by step and using the solved displacement to calculate the current bridging force, Pi in Eq. (17), a new set of displacement z(jc/) can be obtained. Combining Eqs. (1), (4), (5) and (14), the strain energy release rate at the current applied displacement can also be obtained. The above process is repeated until the energy release rate is large enough to cause the delamination crack to grow. BRIDGING LAW Three typical pull-out curves are shown in Fig.3, in which h is half-length of a z-pin and is equal to the thickness of one cantilever beam. Here, the pull-out curves are simply given by bi-linear or tri-linear functions which can be defined by the parameters: maximum debonding force Pd, maximum frictional pull-out force Pf and their corresponding displacements 5y, 82 and 6J. Figure 3(a) represents a pull-out process with initial bonding, stable debonding and friction (S-I). In the first stage, Q<5
5, P,=
{Q
(S-I)
(19)
Z-Pin Bridging Force in Composite Delamination
497
{0<S<S,) P.= Pr--
h-S.
•(S-S,)
(S-II)
(20)
(s-m)
(21)
(S^<S< h)
'-LS
(0 <
P; =
iS,<S< h)
h-S,
U
S-I
'^ Pr
.9 S
82
83
Zrpin displacement 5 (a)
..
Pa
0
2 Pr
S-II
&D
a
&i}
g
v^ D
z-pin displacement 5 (b)
z-pin displacement 5 (c)
Figure 3. Three typical forms of z-pin bridging law.
H.-Y. LIU, W. YANAND Y.-W. MAI
498
NUMERICAL EXAMPLES Numerical examples are given for a DCB specimen with z-pins. The input data for calculation were taken from experiments conducted by Cartie and Partridge [7]. The Young's modulus of the beam was El^^^=\3% GPa. The beam was 150 mm long and 20 mm wide (w); the half-pin length (h) was equal to a single beam thickness =1.5 mm; the flexural rigidity of the beam was calculated by: EI = E^^^^
; and the initial crack length was 50 mm. Li the following
examples, the first column of z-pins was located at 5 mm away from the initial delamination tip (see Figure 1) and the length of pinned region was 25 mm. There are 5 rows x8 columns of z-pins in the z-pinned area. The pin diameter was 0.28 mm. The fracture energy for delamination was assumed as: Gjc=250 J/m^. hi our calculations, the z-pins are assumed uniform in the pinned area and vertical to the laminate. The inaccuracies due to non-uniform z-pin spacing and orientation were not considered in this work. They are being studied in our laboratory and the results will be reported elsewhere. Effect of stable debonding Figure 4 shows the R-curves (represented by the applied loads at the loaded-ends of the DCB as a function of delamination length) for mode I delamination with different bridging laws, SI in Figure 3(a) and S-II in Figure 3(b). The bridging forces for these two cases are given by Eqs. (19) and (20), respectively. The interval, 82 <S <S^, designates the stable debonding process along the z-pin. hi the calculation, the parameters in the bridging laws are, Pff=20 N, P/=15 N and Sz/h^O.Ol, (%//i=0.01, 0.04 and 0.1, respectively. The special case of S^ =^S^ corresponds to the bridging law of S-II, where stable debonding is negligible. Compared to the unpinned samples (dashed line), it is clear that z-pinning can greatly increase the fracture load P applied at the ends of the DCB. With increasing stable debonding region (S^-S2),P also increases but this is not very significant.
o
50
60
70
80
90
crack length L c (mm) Figure 4. Effect of stable debonding.
100
Z-Pin Bridging Force in Composite Delamination
499
Effect of the maximum pull-out force Pf The effect of the maximum pull-out force P/is shown in Figure 5, where the bridging law has the form of S-III in Figure 3(c). hi this case, the bonding strength between the pin and laminate is relatively weak compared to its friction resistance. When the pull-out force is large enough to overcome the fiictional resistance, the z-pin is slowly pulled-out from the laminate, hi Fig. 5, the maximum pull-out forces are assumed to be: ION, 15 N and 20 N. It is clearly shown that the pull-out force plays an important role in the z-pin reinforcement, hicreasing the friction between the pins and the laminate can greatly enhance the fracture toughness of the laminated composite. \J\J
80 -
N/""^
ION / 0 < ^
60 ^3
P/=20 N 15 N
40 20 0-
50
unpinned
]
5i//^=0.01 1
1
60
70
-H
80
\
90
100
crack length L c (mm) Figure 5. Effect of maximum pull-out force.
Effect of maximum debonding force Pd Effect of bonding strength is shown in Figure 6, in which the R-curves are calculated by the bridging laws: S-II and S-IIL The maximum frictional pull-out force P/is ION and maximum debonding forces Pd are assumed as: 30 N, 20 N and ION. Maximum pull-out displacement before debonding, d2/h=0.0l. It is clear that the differences between the three curves are very small. Therefore, enhancement of interfacial bonding can only result in slight improvement in z-pin reinforcement. Figure 6 shows a saw-tooth pattern in the applied load - crack length curve consistent with the presence of a debonding force. This behaviour is caused by the large drop in the pull-out force during debonding as shown in Figure 3(b).
H.-Y. LIU, W. YANAND Y.-W. MAI
500
ft.
50
60
70
80
90
100
crack length L c (mm) Figure 6. Effect of debonding force.
Effect ofz-pin deformation Figure 7 shows the effect of the deformation of z-pin on the bridged delamination. As shown in Figure 3(c), 5\ is the maximum displacement of the z-pin before shding and is completely caused by the z-pin deformation. Because of the high stiffness of the z-pin, the value of d\/h should be very small. Li Figure 7, the values of di/h are chosen as 0.01, 0.03 and 0.1. However, the differences among these three curves are negligible. 100 n 80 /*-s
^
60 -
ft. •o
40 -
di/h=om 0.03 0.1
\^^^^^
mpinned
o 20 0 -
50
1
\
—1
1
60
70
80
90
crack length L c (mm) Figure 7. Effect ofz-pin deformation.
100
501
Z-Pin Bridging Force in Composite Delamination
Simulation on mode I z-pinned delamination
250 low density
200 £ ft. •2
1
Pf
=20 N, 17N, 15N,13N
150
inpinned
ft.
100 50 0\ 50
\— 60
h
70
H
80
\
90
100
60
crack length L c (mm)
(a)
70
80
90
100
crack length Z^ (mm)
(b)
Figure 8. Computer simulation of mode I delamination of z-pinned laminated composites. Solid lines represent the simulation results and dots represent the experimental data [7]. Comparison of the above results shows that the maximum frictional pull-out force is the most sensitive parameter in the z-pin bridging law. However, the fracture load is not sensitive to the form of the bridging law. (It should be noted that in the numerical examples provided in the previous sections, except in Figure 7, the areas for the z-pin bridging laws, that is the pullout work for the z-pin, as the chosen parameters change, have not been kept constant). Thus, it is reasonable and practical to use the simple bi-linear bridging law, S-III shown in Figure 3(c), to simulate the z-pin effect on mode I delamination in the DCB specimens. Figure 8 gives the computer simulated results for the mode I delamination crack growth using the present model and the bi-linear bridging law of Eq. (21). The test data were taken from Cartie and Partridge [7]. Three cases are considered, in which the laminates are reinforced by no pins, 5 rowsxS columns pins (low density. Figure 8a) and 11 rowsxl6 columns pins (high density. Figure 8b), respectively. The maximum displacement of z-pin before sliding is assumed to be, 5\lh=0.0\. The exact pull-out force P/should be measured by separate pull-out tests. However, since such data for z-pin pull-out do not exist, we have used four values of P/, 13 N, 15 N, 17 N and 20 N, to fit the data. It is noted that the predicted curves for P/=13N and 20N appear to provide the lower and upper bounds to the test results. Also, the low z-pin density data in Figure 8(a) appear to show a saw-tooth pattern. This behaviour is caused by the reduction of the bridging force during debonding, generally referred to as "stick-slip". In our simulation, however, we did not consider initial debonding in the bi-linear bridging law (That is, Pd = Pj). So this sawtooth pattern could not be predicted. CONCLUSIONS Theoretical model of mode I delamination of z-pinned DCB has been developed to study the bridging mechanism of z-pin reinforcement. The numerical results show that the fracture load for delamination in the z-pinned DCB is greatly dependant on the frictional pull-out force Pf. However, since the deformation of the z-pin is relatively small compared to sliding distance, the effects of debonding force and debonding length on the fracture resistance of the DCB are
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insignificant. It can be concluded that the delamination behavior of the z-pinned DCB is insensitive to the form of the bridging law. This conclusion is consistent with the study of cohesive zone models by WilHams and Hadavinia [8]. Based on this conclusion, a simple form of bi-linear bridging law (S-III) is suggested for future studies on delamination of z-pinned laminated composites. Using this bridging law, numerical results are obtained using the analytical model developed in this paper for the applied forces at the ends of the DCB to sustain delamination growth. These predictions agree well with experimental data obtained by Cartie and Partridge [7].
ACKNOWLEDGEMENTS The authors wish to thank the Australian Research Council for supporting this project and for the awards of an Australian Research Fellowship to H.-Y. Liu and an Australian Federation Fellowship to Y.-W. Mai. Continuing financial support of this work by the HKSAR Research Grants Council (CERG Project # 9040613) is also appreciated. Numerous discussions with our international collaborators, Professor Ivana Partridge, Dr Denis Cartie of Cranfield University and Professor Gordon Williams of hnperial College, London are gratefully acknowledged.
REFERENCES 1. 2. 3. 4. 5. 6. 7 8.
Jain L. K. and Mai Y.-W, (1994), Composites Science and Technology, 51, 331-345. Jain L. K. and Mai Y.-W. (1994), Int J Fracture, 68, 219-244. Cox B. N, (1999), Advanced Composites Letters, 8, 249-256. Liu H.-Y. and Mai Y.-W., (2001) hi Proceedings of ICCM13, 25^^-29'^ of June 2001, Beijing, China, editor: Y. Zhang Liu H.-Y., Zhang X, Mai Y.-W. and Diao X.-X., (1999), Composites Science and Technology, 59, 2191-2199. WilHams J. G., (1989), "Fracture Mechanics of Anisotropic Materials", Application of Fracture Mechanics to Composite Materials (ed. K. Friedrich). Elsevier Science Pubhshers. Cartie D. D. R. and Partridge I. K., (1999), the 2"^ ESIS TC4 Conference, 13^^-15'^ of September 1999, Les Diablerets, Switzerland, ESIS Publication 27, 27-36, Editors; J. G. Williams and A. Pavan. Wilhams J. G. and Hadavinia, H., (2002), Journal of the Mechanics and Physics of Solids, 50, 809-825.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackm'an, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
503
EFFECTS OF MESOSTRUCTURE ON CRACK GROWTH CONTROL CHARACTERISTICS IN Z-PINNED LAMINATES Denis D.R. Cartie^*, Andreas J. Brunner^ and Ivana K. Partridge^ ^ Advanced Materials Dept, SIMS, Cranfield University, MK43 OAL, UK ^ Polymers/Composites, EMPA, CH-8600 Dubendorf, Switzerland
ABSTRACT Standard Mode I Double Cantilever Beam specimens for delamination testing of a unidirectional (UD) IM7/977-2 composite were Z-pinned with two separate blocks of ZFiber® reinforcement. The reinforced beam configuration was such as to provoke an unstable delamination, propagating between the two Z-pin blocks. Crack resistance curves for these specific geometry specimens of IM7/977-2 indicate that the unstable delamination cracks are arrested by the second Z-pin block, with the crack propagation resistance being dictated primarily by the Z-pinning density within a block. Acoustic emission analysis is used to interpret visual observations and other test data. KEYWORDS: Z-Fiber®, Acoustic Emission (AE), delamination, unstable crack propagation, crack arrest, Z-pin blocks, IM7/977-2
BACKGROUND Z-Fiber® pinning is a recently developed alternative to stitching when aiming to enhance the through-the-thickness properties of polymer matrix composite laminates. The reinforcement is achieved by inserting rods through-the-thickness of a composite part, prior to its cure. The manufacturing process has been developed by Aztex Inc. (Foster Miller) in Boston and a detailed description of the manufacture of Z-Fiber® (termed 'Z-pin'), of the preforms and of other related products can be found in a recent review [1]. The insertion of carbonfibre/bismaleimide resin (T600/BMI) Z-pins into an uncured laminate is shown schematically in Fig. 1. The presence of Z-pins as the through-the-thickness reinforcement has been shown to result in dramatic increases in the apparent resistance to crack propagation under Mode I and Mode II loading conditions, in laboratory tests on standard unidirectional (UD) beam samples [2]. The Z-pin pull-out has been identified as the dominant energy micro-mechanism responsible for the resistance to delamination under Mode I conditions. The behaviour of individual Z-pins in pull-out from a UD-laminate has been characterised and modelled and the single Z-pin pull-out curves used as input into a 2D Finite Element (FE) model of delamination under Mode I [3, 4].
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D.D.R. CARTIE, A J. BRUNNERANDI. PARTRIDGE
Preliminary trials in complex aerospace type structures suggest that selectively placed blocks of Z-pins, of dimensions similar to those used in the laboratory specimens, have a significant capacity to resist or even stop the propagation of an inter-laminar crack in the structure. The success of even relatively simple FE models in predicting the behaviour of any specific block of Z-pins in the laboratory specimens would seem to indicate that the same approach will be successful in more sophisticated local-global models of structures containing such reinforcement.
Preform
Fig 1: Schematic of Z-pin insertion The work reported here addresses specific questions on (1) how the density of Z-pins in both areas affects the delamination behaviour, (2) whether Z-pinning will be able to stop unstable delamination cracks, growing at relatively high speed, and (3) when in time or, alternatively, at which load levels damage mechanisms become active in the Z-pinned areas or other parts of the test specimens. Analysis of the acoustic emission recorded during the test was used to interpret visual observations and other test data. EXPERIMENTAL Experimental design In this study, we have induced stable and unstable crack propagation behaviour, by achieving higher crack speeds without using specialised high rate loading apparatus. The FE-method, referred to above, has been used in order to design specific test samples in which the Mode I delamination resistance was artificially enhanced by the use of localised pinning. Because of the presence of the Z-pins, the loads required to propagate the crack were greater than in the unpinned case, artificially increasing the total elastic energy stored in the beam. After the crack had passed through the first pinned area (see Fig. 2) the elastic energy was released leading to fast crack propagation (average crack speed of 0.25 m/s) into the second block of Zpins.
Effects of Mesostructure on Crack Growth Control Characteristics
505
Fig 2: Specimen configuration (used for all Z-pinning densities) Specimens All specimens were made from IM7/977-2 prepreg supplied by Cytec. One single panel, 300 mm X 750 mm was laid up by hand on a warm table because the prepreg was not particularly tacky. Hot de-bulks (vacuum only, 60°C) were carried out after every fourth layer. A 10 ^im PTFE film was inserted in the mid-plane of the laminate in order to simulate an artificial crack. A final pressurised de-bulk (120 C, 3 bar pressure, 30 minutes) was carried out in order to reach a laminate thickness close to neat cured thickness, prior to Z-pinning with 0.28 mm diameter pins (T600/BMI), at densities of 0.5%, 2% and 4% in the selected locations. Table 1 summarises the range of specimens made. Table 1: Sample designat ion and Z -pin blocks densities Sample designation Density i n first pinned area 0.5% SO.5/0 SO.5/0.5 0.5% SO.5/2 0.5% SO.5/4 0.5% S2/2 2% S4/0 4%
Density in second pinned area 0% 0.5% 2% 4% 2% 0%
The presence of the Z-pins affected at least one series of specimens, resulting in a local thickness reduction of up to 0.25 mm. This was taken into account when the data from these specimens were interpreted. Test methods The specimens were tested following a procedure based on the ISO standard 15024 for unidirectional fibre reinforced laminates under Mode I loading conditions, using a Zwick ZNIO computer controlled test apparatus (lOkN load frame, 5kN class 1 load cell, maintaining 1% accuracy at nominal loads of lOON). Precracking from the insert has not been performed as the beam length was already limited and the focus of the work is very much on crack propagation. Corrected beam theory analysis is used to calculate the delamination toughness Gic- Since the
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D.D.R. CARTIE, A.J. BRUNNER AND I. PARTRIDGE
specimens do not fulfil the requirements of ISO 15024, Gic values shall be regarded as 'apparent' and not valid material data. Most specimens were loaded with a constant cross-head speed of 2 mm/min, and some at 5 mm/min for comparison. The event marker facility was used to correlate the load displacement curve with the visually observed crack length. Selected tests were monitored with acoustic emission (AE) using an equipment type AMS-3 (from Vallen Systeme) and two resonant sensors (type SE 150-M, resonance frequency around 150 kHz). With a preamplifier gain of 34 dB, the detection threshold was set at 40 dB (60 dB in some preliminary tests). Sensor No. 1 was mounted near the end of the specimen using a spring clamp, No. 2 on top of the load-block using duct tape. A silicon-free vacuum-grease was used as couplant. For each AE signal, a set of signal parameters was recorded, including arrival time at sensor, maximum amplitude, signal duration, number of counts (threshold crossing within the signal duration) and signal energy [5]. The coupling was verified with simulated acoustic emission (pencil lead breaks). The linear location of the AE sources along the specimen was also verified with pencil lead breaks at both ends, at the position of the insert film tip and of the first row of pins of each pinned area. RESULTS AND DISCUSSION Crack propagation through the first Z-pin block Figure 3 shows the apparent Delamination resistance versus Crack length curves for a representative set of specimens, covering the range of Z-pinning block combinations investigated. Crack propagation through the 15 mm of unpinned material immediately behind the insert film allowed an accurate measurement of the delamination toughness of IM7/977-2 to be made. Considering first the region before the first Z-pin block: initiation from the insert was unstable, leading to an initiation value from the maximum load (MAX-point) Gicmax= 320 J/m^. Fibre bridging was visible in the crack wake, increasing the toughness to a plateau value of Gip = 450J/ml As can be seen from Fig. 4 and 5, the AE activity (measured by the number of hits per 8 seconds) is relatively high and continuous whilst the crack propagates in the unpinned area. The activity from both AE sensors is comparable, with a slightly higher activity from sensor No. 2. The AE activity from both sensors drops when the delamination reaches the first row of pins. Crack propagation in the first Z-pinned block: all specimens except S4/0 show a stick-slip type of behaviour. Some discrepancies in toughness level can be recognised within the set of S0.5specimens. SO.5/4 consistently showed higher apparent Gic than both, SO.5/2 and SO.5/0.5. This is due to a significant inclination of the Z-pins in the first block in the SO.5/4 samples, resulting from a specific localised manufacturing fault. Sensitivity of the delamination resistance to the manufacturing quality of low pinning density laminates has been observed before. The AE activity, while persisting at a certain level lower than during delamination propagation in the unpinned region, mirrors the stick and slip nature of the behaviour of the SO.5 and S2/2 specimens. When the crack is stopped by a row of pins, the AE activity drops and increases again when the cracks propagates to the next row of pins. This is particularly noticeable in the case of the 0.5% pinning density blocks where the pin to pin spacing is sufficient to allow a clear distinction between the propagation and arrest of the delamination crack to be made. The number of rows of pins can be counted by the number of peaks in the
507
Effects of Mesostructure on Crack Growth Control Characteristics
AE activity plots. Following the unstable delamination growth that is observed when the delamination reaches the end of the first pinned area, AE activity from both sensors ceases completely. Generally, the levels of AE activity seem to be reproduced fairly well for comparable test conditions. This is evident from the initial AE activity levels in the unpinned area beyond the tip of the starter film. The pinning density seems to affect the speed of the delamination crack through the pinned area more than it affects the average AE activity observed (compare for example the first pinned area of Figs. 4 and 5). A change in loading rate, however, seems to affect the AE activity but the number of tests is too small for firm conclusions. In all the S4/0 specimens the delamination crack stops after only 5 mm of propagation in the pinned block. The failure mode then changes from delamination crack propagation to a bending failure of the arms. The toughness at the time of failure is 7000 J/m . The calculated bending stress in the beam is ab=1100MPa. The local curvature leads to a compressive failure strain in the outside layers of 1.3%. Bending failure >*^.S4/0 6000
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Crack Length (mm) Fig 3: Apparent delamination resistance curves of representative specimens, the position of the pinned areas is shown by the dashed lines.
508
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Crack propagation between the Z-pin blocks Several extra long SO.5/0 specimens were used to find out how far an unstable delamination would propagate from the back end of the first Z-pin block in a 4 mm thick UD beam of
Effects of Mesostructure on Crack Growth Control Characteristics
509
IM7/977-2. The experiments showed that the increased elastic energy stored in the beam by the presence of the first 0.5% pinning density region generates unstable delamination growth at an average speed of 0.25 m/s for more than 100mm. Crack propagation through the second Z-pin block The fast cracks thus generated and propagated across the 40 mm separating the end of the first Z-pin block and the start of the second, in all cases were stopped within the second block of Zpins. The 'stopping distances' measured from the first row of pins were 6 mm for the SO.5/0.5 specimens, 4 mm for the S2/2 and SO.5/2 specimens and 3 mm for the SO.5/4 specimens, i.e., are clearly seen to decrease with increasing pinning density of the second block. This indicates that the minimum pinning length required to stop an unstable delamination is between 5 and 10 mm for the lowest pinning density (0.5%) used in these tests. Whether the minimum length also depends on the distance the unstable crack grows, can not be answered at present. Once the arrested crack is re-loaded, the subsequent growth of the delamination within the second Z-pin block essentially exhibits the same R-curve as that noted in the first Z-pin blocks for a given pinning density (see Fig. 3). The specimen with the highest pinning density, SO.5/4, again failed by a bending failure at Qb = 1030 MPa. After a reloading period with no detectable AE activity up to several minutes, AE activity from both sensors appears again. As in the first pinned area, there are distinct peaks of the AE activity while the delamination propagates through the second pinned area. However, these activity peaks are no longer evenly spaced and the correlation with the number of pinning rows observed during delamination growth in the first block seems lost. The AE activity during delamination propagation in the second pinned area is distinctly lower than that observed both in the unpinned area at the beginning and in the first pinned area (at least for all specimens tested at 2 mm/min). With increasing delamination length, the AE activity from sensor No. 1 positioned at the end of the specimen is becoming higher than that from sensor No. 2 on the load block. Sensor No. 2 is expected to show a lower sensitivity because of signal attenuation at the additional interfaces between load-block and beam and sensor, respectively. This also explains why even at the beginning of the test sensor No. 2 yields AE activity roughly comparable to that of No. 1 in spite of the larger distance between AE source and sensor. In Fig. 4, the AE activity, resulting from the progressive bending failure of the specimen, is increasing in both sensors towards the end of the test. This is not the case for the specimen shown in Fig. 5 and can probably be attributed to reduced coupling of the sensors caused by rapid movements of the specimen due to unstable delamination growth beyond the second pinned area. Sensors frequently were "shaken off as the specimen broke in two half-beams at this stage. Loading rate effects Exploratory tests with a higher loading rate of 5mm/min (maximum allowed by ISO 15024) were conducted on two SO.5/0.5 and one S2/2 specimen in order to check for indications of loading rate effects. In Fig. 6, the dashed lines show the apparent delamination resistance curves of specimens tested at a loading speed of 5mm/min, the full lines the resistance curves of the specimens tested at 2mm/min. The S2/2 specimens are represented by filled circles.
D.D.R. CARTIE, A J. BRUNNERANDL
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SO.5/0.5 samples are represented by open triangles. It is evident from Fig. 6 that the general behaviour of the SO.5/0.5 is not changed by the increased loading rate. -•— •^~ -i^ -^
40
80
120
Crack Length (mm)
2%-2%-2mm/min 2%-2%-5mm/min 0.5%-0.5%-2mm/min 0.5%-0.5%-5mm/min
160
Fig 6: Comparison of the apparent delamination resistance with the different loading rates The S2/2 specimens tested at a loading rate of 2 mm/min show a rising R-curve from 4500 J/m^ to 6500 J/m^ through the pinned areas. One S2/2 specimen was loaded at 5 mm/min and found to exhibit an apparent R-curve dropping from 4500 J/m^ to 3500 J/m^ within the first Zpin block. At present, it is not clear whether this behaviour has to be expected for this type of specimen, and if confirmed by additional tests, why it occurs. Visual inspection of all S2/2 specimens after the test showed that during tests carried out at 5 mm/min, the Z-pins located in the first block failed in tension, whereas they pulled-out from the specimens tested at a loading speed of 2 mm/min. Fig. 7 shows the effects of loading rate on the AE activity from selected individual specimens. Specimen S2/2-1, tested at 2mm/min, is shown on the upper plot, while the lower plot represents the AE activity of specimen S2/2-5 tested at 5mm/min. In the case of the specimens tested at a loading rate of 2mm/min, the AE activity is consistently higher during the crack propagation in the unpinned area than within the Z-pinned blocks. In all specimens tested at 5 mm/min, this trend is inverted, the AE activity (measured as the number of hits per 8 s) is higher in the pinned areas. The attention of the reader is pointed to the fact that AE activity values are only comparable if the same integration time (here 8s) for the acoustic emission activities is used. The total test duration is inversely proportional to the loading rate, as expected.
Effects of Mesostructure on Crack Growth Control Characteristics
511
Fig 7: Acoustic emission activity for specimen S2/2-1 tested at 2 mm/min (top) and specimen S2/2-5 tested at 5 mm/min (bottom), please note the difference in the time scales Location ofAE signal sources An example of the location of acoustic emission sources as a function of time is shown in Fig. 8. Each symbol is plotted at the location of its source along the specimen beam at the arrival time at the sensor. Linear location using arrival time differences between two sensors allows location along one axis only, plotted on the abscissa. The co-ordinate system is such that the origin is at the end of the beam. Sensor 1 is at 15 mm and sensor 2 at 225 mm, i.e., the sensors are a distance of 210 mm apart. It is well-known that linear location of AE sources is more accurate within the area enclosed by the sensor array and that the uncertainty is rapidly increasing outside this area. The tip of the insert film is at 164 mm, the first pinning area extends between 125 and 149 mm, and the second between 59 and 84 mm as indicated in Fig. Fig. 8 shows all located events. The signal attenuation in CFRP-materials is usually quite high, typically 50 - 60 dB within 200 - 300 mm. Attenuation correction for located events has not been attempted. The sensor distance of 210 mm therefore implies that events located at the centre between both sensors (minimum propagation distance) show a lower detection threshold (amplitude) than those located near either sensor. A similar plot but applying an amplitude filter criterion of > 60 dB published by Bohse et al. [6] clearly showed the delamination propagation as a function of time and also allowed a quantitative estimate of the size of the damage zone produced by the growing delamination.
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D.D.R. CARTIE, A J. BRUNNER AND I. PARTRIDGE
Even though in Fig. 8, there are AE source locations in the whole range comprised by the insert starter film right from the start of loading and continuing until the fast, unstable delamination growth beyond the first pinned area occurs, there is a distinct, dense band of AE source locations whose edge initially coincides with the position of the insert starter film. With time, this edge progresses further towards the end of the beam (towards the origin of the co-ordinate system). The "slope" of this edge (equivalent to the time-derivative of the position, i.e., the speed) shows the speed of the tip of the delamination. The solid Hne shown in Fig. 8 is derived from the independent, visual delamination length readings marked on the load-displacement plot. The agreement between the position of the delamination tip determined from AE and from visual observation is remarkable. The changes in slope agree well with the visual observations during the test. The only significant difference between both measurements occurs when the delamination approaches the end of the first pinned block. There, AE source location shows relatively high activity throughout the remainder of the pinned area and considerable activity between the first and second block of Z-pins. This can, quite likely, be attributed to the large stresses exerted on the few remaining Z-pins. Except for the time when the delamination is approaching the last rows of Z-pins, the width of the dense band of AE source locations remains roughly constant with time. However, beyond the edge of this dense band of AE source locations, there are sporadic locations that increase in numbers with time. As early as 500 s into the test (in the case shown in Fig. 8), there are AE source locations that coincide with the position of the second pinned area. Even though the micro-mechanisms responsible for these signals can not be identified from the location plot, the locations can be taken as a sign of "stress release" in the second pinned area at this time. Once the delamination has reached the 11^^ row of pins in the first pinned area, the number of locations in the second pinned area is increasing significantly and the locations themselves are spread over the whole range of the second pinned area. Sporadic signals now occur even between the end of the second pinned area and the end of the specimen. The attenuation effects discussed above indicate that AE signals that yield source locations have AE signal amplitudes above a certain threshold level. Empirically and in a statistical sense (the exact source mechanism of an individual AE signal can only be identified in a few, rather artificial model experiments [e.g. Ref 7]), AE signals with amplitudes in the range around and above 60 dB are usually interpreted as coming from crack or delamination growth [6]. Of course, initial de-bonding between laminate plies and pins could be such a source. However, a complete discussion of AE source identification is beyond the scope of the present paper. On the other hand, the conclusion that effects of the second pinned area start quite early and reach a significant level even while the delamination is still progressing through fie first pinned area is almost inevitable. The delamination propagation in the second pinned area is not seen as clearly as that m the unpinned and in the first pinned area. A rather broad band of AE source locations seems to follow the area where the half-beams of the specimen are subject to increasing b(inding. Excessive bending may induce micro-mechanisms leading to slow but steady damage accumulation in the half-beams. Fibre failure of so-called fibre "bridges" may also contribute to the signals located there. For the interpretation of the data, it is important to note that the coupling of the sensors during the test could not be periodically checked. Reduced couplmg ol the sensors due, e.g., to mechanical shock during unstable delamination growth can not be excluded.
Effects of Mesostructure on Crack Growth Control Characteristics
513
With respect to the specific questions posed in the introduction, it can be concluded that AE monitoring yields a means to (a) determine the instantaneous delamination speed, to (b) note changes in this speed, to (c) identify the instants when significant activity sets in at particular positions along the beam (e.g., within the second pinned area), and to (d) compare instantaneous activity in different areas of the specimen. It seems, therefore, feasible, to compare different pinning arrangements (e.g., with differing density or different length/size) in detail and thus to experimentally arrive at "optimal" pinning arrangements. If, in the future, detailed identification of AE source mechanisms becomes feasible (e.g., based on pattern recognition analysis of recorded AE signal waveforms), effectiveness of Z-pinning in laminates or composite parts could be further improved. Sensor 2" 200 H
2500 Sensor 1'
Fig 8: Plot of AE source location for specimen S2/2, the solid Une is derived from the independent, visual delamination length readings, sensor arrangement and pin blocks are schematically shown on the right CONCLUSION In this paper, the data analysis was carried out according to the Mode I ISO standard for unidirectional laminates (ISO 15024). It is not yet clear whether application of this analysis scheme yields valid materials data, but in any case, initiation of delamination growth in the Zpinned materials requires loading to comparatively higher loads, i.e. a higher energy input. Delamination propagation within the pinned areas yields relatively high apparent delamination resistance curves and produces unstable delamination growth behind the Z-pin block. In all specimens tested, the second block of Z-pins stopped the unstable delamination growth within a few mm. The work presented here has confirmed the ability of Z-pin blocks to arrest a propagating delamination crack, for a range of crack speeds. From the perspective of modelling and predictions of behaviour of more complex Z-pinned structures, it is encouraging to see that the
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D.D.R. CARTIE, A J. BRUNNER AND I. PARTRIDGE
'apparent fracture toughness' within a Z-pin block is strongly dependent on the pinning density. However, the influence of the local strain rate upon the micromechanics of failure of the Z-pins remains to be clarified. Acoustic emission monitoring provides a tool for independent crack speed measurements and thus verification of visual observation, as well as for identifying the onset and development of stress-induced activity in the pinned areas. ACKNOWLEDGEMENTS The work reported here was part funded by EPSRC (UK) (GR/ R 90369). The supply of prepreg by Cytec for this work and for TC4 round robin activities is gratefully acknowledged. The technical assistance of Mr M. Heusser (EMPA) for part of the tests is gratefully acknowledged. REFERENCES 1. Partridge I.K. Cartie D.D.R and Bonnington, T. (2003), 'Manufacture and performance of Z-pinned composites', Ch 3 in Advanced Polymeric Materials: Structure-property relationships, Eds S.Advani and G. Shonaike, CRC Press (April 2003). 2. Cartie D.D.R and Partridge I.K., (2000) 'Delamination behaviour of Z-pinned laminates', Proceedings ESIS Conference on Fracture of Polymers, Composites and Adhesives, ESIS Publication 27, pp. 27-36, Elsevier, Oxford. Proceedings of 2^^ ESIS TC4 conference, (Les Diablerets, Switzerland, 13-15 September 1999), eds Williams J. G. and Pavan A., Elsevier 2000, ISBN 0-08-043710 9. 3. Cartie D.D.R and Partridge I.K., (2001) 'A finite element tool for parametric studies of delamination in Z-pinned laminates'. Proceedings of DFC6, Manchester Conference Centre, Manchester, UK, pp. 49-55, loM communications. 4. Liu H.-Y. and Mai Y.W. (2000), 'Effects of Z-pin reinforcement on mode I delamination'. Proceedings oflCCM 13, Beijing, China. 5. Miller, R.K., Mclntire, P., (1987) Nondestructive Testing Handboook, Vol. 5, Acoustic Emission Testing, American Society For Nondestructive Testing, Columbus OH (USA). 6. Bohse, J., Krietsch, T., Chen, J., Brunner, A.J., (2000) 'Acoustic Emission Analysis and Micromechanical Interpretation of Mode I Fracture Toughness Tests on Composite Materials', Proceedings ESIS Conference on Fracture of Polymers, Composites and Adhesives, ESIS Publication 27, pp. 15-26, Elsevier, Oxford. 7. Brunner A.J., Nordstrom, R.A., Flueler, P., (1997), 'Fracture Phenomena Characterization in FRP-Composites by Acoustic Emission', Proceedings European Conference on Macromolecular Physics: Surfaces and Interfaces in Polymers and Composites, EPS Vol. 21B, pp. 83-84, European Physical Society.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
515
FRACTURE TOUGHNESS AND BRIDGEVG LAW OF 3D WOVEN COMPOSITES V.TAMUZS and S.TARASOVS Institute of Polymer Mechanics, University of Latvia Riga, Latvia
ABSTRACT In the paper, the methodology of measuring Gic by using DCB specimens in the presence of large-scale bridging is adapted for investigating the delamination in 3D woven composites. The main novelties introduced in the DCB testing concern a) the loading device, b) the specimen geometry, c) the measurement scheme, and d) the determination of Gic and the bridging law from experimental data. The geometry-independent material characteristic Gic as a function of the initial crack opening displacement (ICOD) is introduced. Typical values of the delamination fracture toughness of 3D glass and carbon composites are presented. KEYWORDS Fracture toughness, delamination, 3D composite, bridging law, crack opening displacement. INTRODUCTION The delamination properties of laminated composites are very important material characteristics. The fracture toughness of regular laminated composites has rather low values (the critical delamination energy release rate Gic in mode I usually does not exceed 0.2-0.4 kJ/m^). One way of improving the delamination toughness consists in introducing a small amount of fibers in the thickness direction of plates. The through-the-thickness reinforcement acts as a crack bridging, which requires new concepts and methods to be used for the fracture resistance calculation. The delamination of UD composites in the presence of large-scale bridging has been modelled and studied experimentally rather well, and the list of some references can be found in [1, 2].
516
V. TAMUZS AND S. TARASOVS
It was shown that, in the presence of large-scale bridging (i.e., when the size of the bridging region is comparable to the crack size), the R-CUTVQ is not a material characteristic md its shape depends on the specimen geometry. In the case of a double-cantilever beam, tht shape of the i?-curve depends on the stiffness of specimen arms. In [3], it was noticed that DCB specimens with a higher bending stiffness required a longer crack extension before the steady-state crack growth resistance was attained. In [4], the concept of a bridging law was introduced to characterise the R-curvQ for DCB specimens In the present paper, the methodology of testing and modelling the delamination of UD composites in the presence of large-scale bridging is adapted for the delamination stud> in 3D woven materials. The following four key alterations used in this study are essential: 1. an original loading device is utilized to apply high mechanical loads; 2. a special form of tabbed DCB specimens is selected to increase the strength and stiffness of specimen arms; 3. as in [2], a simple analytical formula for calculating Gu is used; 4. the fracture toughness G\c as a function of the crack opening displacement is introduced as a material characteristic instead of the traditional R-CUTWQ Gic(Aa), because the latter strongly depends on the specimen geometry (on the sample thickness). THE MATERIAL AND SPECIMENS INVESTIGATED The orthogonal 3D woven laminate used in this work was elaborated and produced by the 3TEX Inc (USA). These laminates consist of unidirectional in-plane layers orientated in the 0° and 90° directions. The transverse reinforcing components are yarns orientated perpendicularly to the midplane. These yarns bind the material and hold the in-plane fibers together. This is achieved without interweaving the in-plane fibers, and hence the reduction in the in-plane stiffness caused by the fiber waviness is avoided. The amount of transverse yarns is about 3 percent of total fiber volume. The longitudinal modulus E of beams were 17.9 GPa and 56.3 GPa for glass and carbon fiber specimens respectively (measured in simple tension) The mechanical properties of the materials are described in detail in [5]. Four types of composites were studied: glass- and carbon- fiber composites with and without fibers in the thickness direction were tested and the increase in the fracture toughness caused by this reinforcement was estimated. The specimens had the classical DCB shape of thickness 10 mm for the carbon-fibijr and 6.1 mm for the glass-fiber specimens and width equal to 15 mm and contained an initial precrack 25-mm long. The initial notches of width equal to 0.3 mm for the 3D glass-fiber and 0.5 mm for the 3D carbon-fiber specimens were introduced by a thin saw. The crack tip was sharpened by a thin blade. In order to monitor the position of the crack front, the specimen edges were painted white with a brittle paint (typewriter correction fluid) and fine lines were drawn at 2-mm intervals on the white surface. The operator measured the crack length visually with the help of a magnifying glass.
Fracture Toughness and Bridging Law of 3D Woven Composites
517
The materials with through-the-thickness fibers had a much higher fracture toughness than ordinary layered composites, therefore the specimen arms were broken before the onset of crack propagation. In order to prevent this, metal tabs were glued to the bottom and upper sides of the specimens. Steel tabs 30 XTCA (so called "spring steel") of thickness 2.0, 2.2, and 2.5 mm and width 17 mm were used. Different adhesives were checked and "Bison" has been found the strongest. Nevertheless in testing the tabbed 3D carbon-fiber specimens, the separation of the tabs from the specimen surface before the crack initiation was observed in all initial experiments. To prevent this, side-grooved specimens were used. The 1 mm wide side grooves were machined by milling cutter. The depth of grooves equal to 4 mm was chosen to cut the side lines of transverse yarns leaving intact only 2 transverse yarn lines. PROCEDURE OF EXPERIMENTAL LOADING AND MEASUREMENTS An MTS testing rig (model 309) with a 20-kN load cell (loading scale in range 0-1000 N) was used. Because of the rather high load level, the force was applied not through glued metal blocks (as recommended in [6]), but by a specially designed test device. By means of the "comb teeth" device (teeth thickness equals 1 mm), the load was applied directly to the faces of the initial precrack (see Figs 1 and 2). The loading was performed at a constant displacement rate of 2 mm/min.
Fig. 1. The loading device in a separated position.
Fig. 2. A specimen and the loading device before testing.
518
V. TAMUZSANDS. TARASOVS
The geometry of a DCB specimen is shown in Fig. 3, where 2h is the total thickness of the specimen, a^ is the length of the initial notch, a is the length of a propagated crack, d is the crack opening under an applied wedge force P, and 5 is the crack opening at the tip of the initial notch. Two extensometers, one attached to the test jig of the loading device for measuring the crack opening along the load line and another attached to the top and bottom sides of the specimen at the end of the initial crack, were used. The displacement of the actuator of the testing machine was also recorded.
strain clip gage
Fig. 3. Specimen geometry and the parameters measured. CALCULATION OF THE ENERGY RELEASE RATE The determination of Gic by DCB specimens is prescribed by [6]. Let us discuss the features of different approaches following [2]. The energy release rate G in a DCB specimen is defined in the usual way: (1)
2b da where c = — is the compliance of the system. Formula (1) is well known and widely used.
Neglecting the bridging effect, the deflection of an ideal cantilever beam of length a, width b, Ebh a'P and bending stiffness EI = under a load P is equal to . The full opening of the 12 ^ 3EI DCB equals the doubled deflection a=
, 3EI
(2)
and the compliance is 2a' 3EI
(3)
Fracture Toughness and Bridging Law of 3D Woven Composites
519
Using Eqs. (1) and (3), the most popular formula for the DCB specimen is obtained, G(P,a)^
(4)
Elb
Combining Eqs. (4) and (2), we get three other, modified formulas for G: G(P,a,d) =
3Pd 2ba '
(5) 2/3
^ G{a,d)
(6)
EIb{ 2P 9EI£_ Aba*
(7)
Applying Eqs. (4)-(7) to an ideal isotropic cantilever beam, equal results will be obtained. But, strictly speaking, they are all invalid for the DCB specimens, since the boundary conditions at the end of the cracked part of the specimens are not the same as at the clamped end of a cantilever beam. As result, the deflection of an actual specimen under a given load is always greater then that predicted by the beam theory (Eq. 2) neglecting the bridging. The difference is still greater for unidirectional composites, since Eq. (2) neglects the interlaminar shear. The error is very high for a short crack, but it diminishes when the crack propagates. Therefore, formulas (4)-(7) will give different results and it is expedient to compare the predictions obtained by Eqs. (4)-(7) and by general formula (1). To this end, a finite-element model of a DCB specimen with and without bridging was used [2]. In Fig. 4, the value of G calculated directly from the finite-element analysis is compared with those obtained by formulas (4)-(7). •—•—G(Ra)/G{FEM) 1 -Hil-G(dP,a)/G(FEM)| —A—G(d,P)/Q(FEM) r
2-
-•—^.dWGFEMi
S 1,5-
1
Ll_
CD
;
O
0
2
^X
i5
8
10
—I
12
14
a/h
Fig. 4. A comparison of the energy release rates calculated from Eq. (4)-(7) with G obtained directly from the finite-element model for different crack length to thickness ratios [2]. It was found that formula (6) leads to better results even for very short cracks. In this comparative theoretical analysis, the deflection d in formulas (5)-(7) is taken from finiteelement calculations.
V. TAMUZSANDS. TARASOVS
520
In applying these formulas to the experimental results, the deflection is taken from experimental graphs and therefore it reflects the true compliance of specimens, including the influence of the bridging effect as well. The ASTM standard [6] recommends one to use formula (5), which overestimates the value of G, as is seen from Fig. 4. To correct the results, it is advised to select some fictitious crack size a+Aa, where Aa is determined by generating a least-square plot of the cube-root compliance C^^^ as a function of the crack length a. The method is called "Modified Beam Theory (MBT)". It is clear that the introduction of an artificially increased crack size will improve the results of formulas (5) and (4) as well. By this method, an accuracy similar to that of Eq. 6 can be achieved, where the crack length is excluded and replaced by the experimentally measured compliance. However, the MBT method has two disadvantages. First, the reduction techniques assume that the energy release rate of a material has a constant value during the crack propagation, and, therefore, it cannot be used for materials with extensive bridging. Second, the fitting procedure requires many experimental points in order to increase the accuracy. Applying this method in situations where only few experimental points are available can lead to large errors. Finally, the energy consumed in crack propagation can be measured directly by the area method, with unloading of the specimen after some crack advance, namely (8)
bAa where S is the area of one loading-unloading cycle, b is the specimen width, and Aa is the crack length increment. The area method was used for comparison with the calculated Gu values, where it was possible. RESULTS AND ANALYSIS Fracture toughness of the 3D glass-fiber composite A load-deflection curve for a specimen is presented in Fig. 5.
Fig. 5. Load-deflection curve of a 3D glass-fiber specimen. The corresponding R-CUTVQS are presented in Figs. 6 and 7. It is seen that the results obtained by the recommended analytical formula and the area method are rather close. The initial i^alue of the fracture toughness is 0.3 kJ/m^, and the fracture toughness during the steady-state 3rack
Fracture Toughness and Bridging Law of SD Woven Composites
521
propagation is about 8 kjW. The crack growth was initiated by cracking of the matrix at the values of G typical of regular composites, and then G increased. The i^-curves show (Fig. 6) that G increases significantly after the crack propagation by 10 mm. But it is not a material characteristic, because the slope of the i^-curve depends on the thickness of the composite and glued tabs. In Fig. 5 it is seen that unloading lines reveal some residual "plastic" deflection (the lines do not go back through zero). Apparently the reason is the uncompleted pull-out and frictional push back of transverse yarns in their sockets which can be observed by the close examination of the failure zone.
0
I
\
\
50
60
70
\
iB^L 10
20
30
40
80
90
Aa,mm
Fig. 6. R-curves of six 3D glass-fiber specimens, calculated by Eq.6.
10 8 6 42 0 10
20
30
40
50
60
70
80
Aa,mm
Fig.7. R-curves of six 3D glass-fiber specimens, calculated by the area method. In [1, 2], it was shown that the bridging law for a UD composite in the presence of large-scale bridging can be obtained by the formula o(5) =
^
522
V. TAMUZSANDS. TARASOVS
where a is the traction between the crack faces as a function of crack opening 5, Gu is the experimentally measured or calculated energy release rate, and 5* is the crack opening outside the bridging area, i.e., at the tip of the initial precrack (ICOD). It was shown in [2] that the resulting bridging law allows one to predict the traditional /^-curves for different specimen thicknesses, therefore it can be used as a material characteristic. This means that the function Gic(5*) or Gic (ICOD) is a material characteristic, contrary to the traditional R-curve. In Fig. 8, the fracture toughness as a function of the ICOD is shown. l^
-
p 5
10 -
8
-./V ^
^
E
4
—
» •
=i^
^
^ ^ ^—
_
ol I
10
12
5*, mm
Fig. 8. Fracture toughness versus the crack opening at the initial precrack tip. It is seen that, at the initial crack opening equal to 0.2 mm, the fracture resistance increases tenfold (to 3 kjW) and reaches 7 kJ/m^ at an ICOD equal to 1 mm. Fracture toughness of the 3D carbon-fiber composite The load-displacement curve for a specimen is presented in Fig. 9. The corresponding incurves, the fracture toughness vs. the crack opening, and the bridging law are presented in Figs. 10-12, respectively. The values of the critical fracture toughness in the plots are calculated by using formula (6). The initial fracture toughness is 0.6 k j W and the fracture toughness of the steady-state propagation is about 20 kJ/m^. It is seen from Fig. 11 that., at an ICOD equal to 0.25 mm, the value of G increases to 6-10 kjW and reaches a steady-state propagation value exceeding 20 k j W at an ICOD equal to 0.5-1 mm.
»u -
ft"^ 40 -
/ /^
J/
0
/ / ^
^ / / / /
• V \ / '\ -n . ^ .-^
Y Ay /'^
y ^-n^ S\
V
c^
4
6
8
10
12
(/,mm
Fig. 9. Load-displacement curve of a 3D carbon-fiber specimen.
Fracture Toughness and Bridging Law of 3D Woven Composites
523
Fig. 10. R-curves of 3D carbon-fiber specimens.
30 25 A
h-A
20 15 10
0
1
0.5
2
1.5
2.5
5*, mm
Fig. 11. Fracture toughness versus the crack opening at the initial precrack position.
454035 w 30 2 256 20 15 105 0-
\
1
1
0.2
0.4
0.6
\
\
0.8
1
^
1
1.2
•
1.4
8, mm
Fig. 12. Bridging law (traction at the crack faces versus the crack opening).
524
V. TAMUZSANDS. TARASOVS
CONCLUSION In order to perform delamination experiments on 3D composite materials, modified DCB specimens were designed. The ordinary composite plates are rather thin and the arms of DCB specimens are broken before the onset of crack propagation. The specimen arms should be stiffened by additional tabs glued on both sides of the specimens. A comb-like loading device was used and can be recommended for applying high tensile loads directly to crack faces. The analytical formula Gic(P, 5) is found to give the best results, which agree well with the Gic measurements by the area method. The additional parameter — the crack opening displacement at the initial precrack (ICOD) — was measured to obtain the bridging law, which is a material characteristic independent of the specimen geometry. It is recommended to use the graphs Gic (ICOD) instead of the traditional 7^-curves Gic(Aa) for characterizing the delamination fracture resistance of composites with extensive bridging. The microcracking at the tip of a crack in 3D woven materials starts at the same values of Gic as the matrix cracking: 0.3-0.6 kjW. At a crack opening equal to 0.2 mm, the crack propagation resistance increases 10 to 20 times and, for the steady-state propagation, at a crack opening of 0.5 mm, reaches very high values (up to 20 kJ/m^). ACKNOWLEDGMENT The authors express their gratitude to the 3TEX Inc. and personally Dr. A.Bogdanovich for supplying the composite plates investigated. REFERENCES 1. Sorensen, B.F., Jacobsen, T.K. (1998) Composites, Part A, 29A, 1443-145 L 2. Tamuzs, V., Tarasovs, S. and Vilks, U. (2001) Engineering Fracture Mechanics, 68, 513525. 3. Spearing, S.M., Evans, A.G. (1992) ActaMetallMater, 40, 2191. 4. Suo, Z., Bao, G., Fan, B. (1992) JMech Phys Solids, 40, 1. 5. Mohamed, M.H., Bogdanovich, A.E., Dickinson, L.C., Singletary, J.N., Lienhart. R.B. Sampe Journal, 37, No. 3,8-17 (2001, May/June). 6. ASTM D 5528-94a. Standard test method for mode I interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites. Annual Book of ASTM Standards, American Society for Testing and Materials, Philadelphia.
3.4 Modelling and Lifetime Prediction
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
527
3D MODELLING OF IMPACT FAILURE IN SANDWICH STRUCTURES
C. YU*, M. ORTIZ and A J. ROSAKIS *E. T. S. de Ingenieros, Canales y Puertos, Universidad de Castilla-La Mancha 13071 Ciudad Real, Spain Graduate Aeronautical Laboratories, California Institute of Technology Pasadena, CA 91125, USA ABSTRACT Cohesive theories of fracture are applied to simulate the complex failure modes in sandwich structures subjected to low-speed impact. The particular configuration contemplated in this study refers to the experiments performed by Xu and Rosakis [1], where the model specimens involving a compliant polymer core sandwiched between two metal layers, were adopted to simulate failure evolution mechanisms in real sandwich structures. Fracture has been modeled by recourse to an irreversible cohesive law embedded into three-dimensional cohesive elements. These cohesive elements govern all aspects of the separation of the incipient cracks. The cohesive behavior of the material is assumed to be rate independent and, consequently, all rate effects predicted by the calculations are due to inertia. The fidelity of the model has been validated by several previous simulations [2,3]. The numerical simulations have proved highly predictive of a number of observed features, including: the complex sequences of the failure mode, shear-dominated inter-sonic (a speed that is greater than shear wave speed but less than the longitudinal wave speed of the material) inter-layer cracks, the transition from inter-layer crack growth to intra-layer crack formation and the core branching later on. KEYWORDS Impact failure, sandwich structures, intra-layer crack, inter-layer crack, finite elements and cohesive elements. INTRODUCTION Sandwich structures have relative advantages over other structural materials in terms of improved stability, weight savings, crash worthiness and corrosion resistance. Therefore layered materials and sandwich structures have diverse and technologically interesting applications in many areas of engineering. These includes the increased use of composite laminates in aerospace and automotive engineering; the introduction of layered concrete pavements in civil engineering; the use of thin films and layered structures in micro-electric components, and very recently, the introduction of naval engineering applications like nonferrous ship hulls [1].
528
C. YU, M. ORTIZ AND AS ROSAKIS
However, one of the shortcomings of sandwich structures is the still incomplete understanding of damage, as it can develop as a consequence of low-velocity impacts (such as tool drops) or high-energy events (such as ballistic penetration) or an unusual level of the design loading. The damage characterization of sandwich structures is certainly more complicated than the one of conventional laminates. A first reason, related to their special structure, is that the damage often does not develop uniformly across the thickness. A second reason is that besides typical failure modes (such as penetration and de-lamination), also crushing and facet sheets debonding must be addressed. The shear load transfer through the core is often combined with unsymmetrical damage, which requires a detailed study in order to understand the damage progression and evaluate the residual strength. Motivated by those concerns, Xu and Rosakis [1] conducted a series of experiments investigating the generation and subsequent evolution of dynamic failure modes in layered materials subjected to impact. Model experiments in plane stress configuration were chosen to simplify the 3D problem of the out-of-plane impact of the real sandwich structures. High-speed photography and dynamic photo-elasticity were utilized to study the nature and sequence of the failure modes. A series of complex failure modes were documented. They found that in all cases, the dominant dynamic failure mode is the inter-layer failure (de-lamination between panel and core), which is shear-driven and proceeds at inter-sonic speed even under moderate impact speeds. The shear inter-layer cracks kinked into the core layer, propagated as openingdominated intra-layer cracks and eventually branched as they attained high enough growth speeds causing more fragmentation [1]. The effects of impact speed and interfacial bond strength on the dynamic failure of model sandwich structures were also reported. Cohesive theories have been proved to be one of the most appealing and reliable approaches to the numerical simulation of complex fracture processes. Cohesive models fiimish a complete theory of fracture which is not limited by any consideration of material behavior, finite kinematics, non-proportional loading, dynamics, or geometry of the specimen. In addition, cohesive theories fit naturally within the conventional framework of finite-element analysis, and have proved effective in the simulation of complex fracture processes [2,3]. Cohesive theories are characterized by an intrinsic length scale and an intrinsic time scale. The accurate description of fracture processes by means of cohesive elements requires the resolution of the characteristic cohesive length of the material [4,5]. In some materials, such as ceramics and glass, this length may be exceedingly small. Thus calculations based on cohesive elements inevitably have a multi-scale character, in as much as the numerical model must resolve two disparate length scales commensurate with the macroscopic dimensions of the solid and the cohesive length of the material. The intrinsic time scale, in conjunction with inertia, introduces a rate dependency able to modify the response of the model for slow and fast processes. Another interesting feature of the fmite-element model adopted is the explicit treatment of fracture and fragmentation [6,7]. It tracks individual cracks as they nucleate, propagate, branch, and possibly link up to form fragments. It is incumbent upon the mesh to provide a rich enough set of possible fracture paths, since the model allows de-cohesion to occur along element boundaries only. However, no mesh dependency is expected as long as the cohesive elements adequately resolve the fracture process zone of the material [5]. It is also interesting to notice that the micro inertia attendant to the material in the dynamic fragmentation process contributes to the correct simulation of the rate effects [2,3].
3D Modelling of Impact Failure in Sandwich Structures
529
The simulations in this paper give failure modes sequences very similar to the actual ones observed in the experiments. The model predicts the formation of shear-dominated inter-layer (or interfacial) cracks that initiate first and that such cracks grow very dynamically, their speeds and shear nature being enhanced by the large wave mismatch between the core and the face sheet. The triggering of the complex mechanism of the intra-layer failure of the core structure is also well reproduced. The organization of the paper is the following. The material characteristics and the dynamic experimental setup are described in the next section. Finite element implementations are presented afterwards and the simulation results and the comparisons with the experiments are discussed at the end. EXPERIMENTAL SETUP Material description The sandwich specimen used in the experiments of Xu and Rosakis [1] is a thin plate obtained by bonding along their thickness two 4340-steel (external) plates to an Homalite-100 (internal) plate. The geometry for a typical specimen used in the experiments is shown in Fig. 1. Both steel and Homalite are characterized by elastic material properties, reported in Table 1 [1]. The fracture parameters for the two bulk materials and the interface bond Weldon-10 adopted in the experiments are reported in Table 2 [1], where Oc is the maximum cohesive stress or static tensile strength of the material; TC is the maximum shear stress; 6c is critical opening displacement of the cohesive law; Kic is the fracture toughness; Gc is the fracture energy; R is the characteristic length of the material.
Steel Homalite Steel
mi
Area =57.15
mm^
Fig. 1 Geometry for a typical specimen (three layers with equal thickness) used in the experiments of Xu and Rosakis Table 1. Elastic material properties for Homalite-100 and 4340 Steel Material Young's modulus Poisson's ratio Mass density Longitudinal speed Shear wave speed Rayleigh wave speed
E {GPa) V
Pikg/m') Cj(m/s)
c^{mls) c,im/s)
Homalite-100 5.3
4340 Steel 208
0.35 1230 2200 1255 1185
0.3 7830 5500 3320 2950
530
C. YU, M. ORTIZ AND A J. ROSAKIS
Table 2. Fracture and cohesive parameters for Homalite-100,4340 Steel and Weldon-10 Homalite-100
Material Tensile strength
cr, {MPa)
Shear strength Fracture energy Opening displacement Characteristic length
r^ G^ d^ R
35
40 88.1 6.6 2.8
(MPa) (N/m) (M^) (mm)
4340 Steel 1490
Weldon-10 14
1054.0 10620 14.3 14
22 45.0 176.6 18
Experimental technique Photo-elasticity is an experimental technique for stress and strain analysis that is particularly useful for members having complicated geometry, complicated loading conditions, or both. The photo-elastic method is based upon a unique property of some transparent materials, in particular, some plastics. When the model is stressed and a ray of light enters along one of the directions of principal stress, the light is divided into two component waves, each with its plane of vibration (plane of polarization) parallel to one of the remaining two principal planes (on which shear stress is zero). Furthermore, the light travels along these two paths with different velocities, which depend on the magnitudes of the remaining two principal stresses in the material, so the waves emerge with relative retardation, which is the number of wave cycles experienced by the two rays traveling inside the body. The two waves are brought together in the photo-elastic polari-scope, and permitted to come into optical interference. A photo-elastic pattern, iso-chromatic pattern of dark and light bands called/rmge^ is formed, and it is related to the stress system by the stress-optic law in plane stress configuration [1]. Laser (100 nun Beam Diameter)
Polarizer 1
l^mm liil • P Specimen/'^
^ ; ^
Polarizer!
f
^„.-^ I { ^]
^^
^ . .,. Rotating Mirror Type High Speed Camera (Cordin330A) J^tt^
Fig. 2 Schematic of the dynamic photo elasticity setup (Xu and Rosakis) A schematic setup of the photo-elasticity technique is given in Fig. 2. Two sheets of a circular polarizer were placed on both sides of the specimen. An Innova Sabre argon-ion pulsed laser
3D Modelling of Impact Failure in Sandwich Structures
531
served as the light source. The coherent, monochromatic, plane-polarized light output was collimated to a beam of 100 mm diameter. The laser beam was transmitted through the specimen, and the fringe patterns were recorded by the high-speed camera. During the impact test, the projectile was fired by a gas gun and hit the specimen on the side to trigger the recording system. Under dynamic deformation, the generation of iso-chromatic fringe patterns is governed by the stress optic law. For the case of monochromatic light, the condition for the formation of fringes can be expressed as (1) h where a^ - a^ is the principle stress difference of the stress tensor averaged through the thickness; f^ is the material fringe constant associated with the incident light wave length (/^ = l^.lkN Im for Homalite-100); h is the thickness of the specimen; N is the number of fringes. The iso-chromatic fringe patterns observed are proportional to the contour levels of the ^{(J^-G^)!!. maximum shear stress, i.e., r -(J,
=•
FINITE ELEMENT MODEL The experiments of Xu and Rosakis were simulated by recourse to a finite element discretization of the continuum. In order to simulate crack initiation and growth, the cohesive model proposed by Camacho and Ortiz [5], and subsequently extended to three dimensions by Ortiz and Pandolfi [6], were emloyed. In adopting a cohesive description of fracture, the formation of a crack is regarded as a gradual process of separation, either by opening or by shearing, leading to the formation of new free surfaces. The cohesive law furnishes the traction vector / across the cohesive surface as a function of the opening displacement ^' = |wj. Following Camacho and Ortiz [5], this cohesive behavior is formulated in terms of the effective opening displacement
Gc= ob&/2
^inax/ ^
Fig. 3 Decomposition of the opening displacement.
Fig. 4 Linearly decreasing cohesive law.
d = 4p'8l+5l
(2) ^n -^'U is the normal component of the opening displacement and S^ ~\ ^~^n21 is the shear component magnitude of the tangential opening displacement, see Fig. 3. The cohesive behavior under monotonic loading is then assumed to be governed by a cohesive potential. The resulting tractions are of the form:
532
C. YU, M. ORTIZ AND A.J. ROSAKIS
t=^
= ^iP'S,+S„n)
(3)
do o —* where t = ^JP ^t^ -^t^ is an effective cohesive traction. Here t„=l-n is the normal component of the cohesive traction and t^ = \l-t^n\ is the magnitude of the tangential opening traction. It follows from Eqn. 3 that the parameter /? measures the ratio of shear and normal cohesive strength of the material. It also roughly defines the ratio of Kuc to Kic of the material [2]. The particular monotonic envelope adopted in calculations is shown in Fig. 4. Thus, potential cohesive surfaces are assumed to be rigid up to the attainment of the cohesive strength GC, and the effective cohesive traction to subsequently decrease linearly and vanish upon the attainment of a critical effective opening displacement 5c. The resulting fracture energy of the material is Gc = CTC5C/2, [6]. The cohesive element is rendered irreversible by unloading to the origin from the monotonic envelop just described, Fig. 4. It bears emphasis that upon closure, the cohesive surfaces are subjected to the contact unilateral constraint, including friction. Contact and friction are regarded as independent phenomena to be modeled outside the cohesive law. NUMERICAL RESULTS Mesh description In order to reduce the computational effort, without losing correspondence with the experimental counterparts, a slice of the specimen, 1.6 mm thick (1/4 of the real specimen thickness), is modelled. The out of plane displacements along the surfaces of the plate are constrained to avoid buckling. The computational mesh comprises 65365 nodes and 32482 10node quadratic tetrahedrons. Fig. 5. The core is modeled using a uniform minimum mesh size of 0.8 mm; the mesh coarsens in the steel layer, with one tetrahedral element across the thickness. The cohesive elements are adaptively inserted along previously coherent interfaces when the local effective stress attains a critical value [6].
Fig. 5 Computational mesh comprising 65365 nodes and 32482 tetrahedrons.
3D Modelling of Impact Failure in Sandwich Structures
533
Boundary and loading conditions The effect of the bullet impact is approximated by prescribing a velocity profile to the nodes lying on the contact area. The load history is presented in Fig. 6. The impulse duration tp is 18.4 jis, estimated from the bullet length; both the rise time tr and the step down time td are 2 jis. The impact speed assumed in the calculation is 32.4 m/s. The analysis has been conducted up to 300 |Lis after impact, with a stable time step of 0.01 jis.
Fig. 6 Impact velocity profile, with rising time tr = 2 |is, stepping down time td impulse time tp =18.3 jis.
2 |is; the
Typical failure patterns The typical failure sequence observed in the experiments of Xu and Rosakis [1] is sketched in Fig. 7. One of the major conclusions of Xu and Rosakis is that shear-dominated inter-layer (or interfacial) cracks are the ones that initiate first and that such cracks grow very dynamically, their speeds and shear nature being enhanced by the large wave mismatch between the core and the face-sheets (the ratio of shear wave speeds of steel to Homalite is 2.6, Table 2). It is the kinking of these cracks into the sandwich core that triggers the complex mechanisms of intralayer failure. •
,
.
.
'
- •
'.
•
^
'
'
"
'
•
•
^
.
'
-
•
. .
"
'
'
' •
* • '
Q (b)
(a)
T-y^''
• " " " V ^ " ' " " " " ''""
: ^
^__ '"'*'{'••"['••
''','>*"'.'''''•
Q (c)
"''''"' "i,1
(d)
Fig. 7 Typical failure sequence observed in the experiments of Xu and Rosakis, where the arrows indicate the direction of the crack propagation, while the central line in (c) and (d) depicts the Rayleigh wave propagation after the interface separation.
534
C YU, M. ORTIZ AND A J. ROSAKIS
Fig. 8 Typical crack patterns of the core material Homalite-100, observed in the experiments. At the end of the experiments, the specimens were completely broken, the pieces were collected and glued together to give a crack pattern, and one of those specimens is de])icted in Fig.8. The damage sequence was caught by the rotating mirror shown in Fig. 2. It is observed that the damage patterns are not necessarily symmetric because of small variations of loading conditions. In order to facilitate the view of numerical crack patterns, the highlighted contour plots of a damage variable, which is defined as the ratio between the consumed cohesive energy and the total fracture energy per unit surface (critical energy release rate Gc), were applied. The comparison between the experimental and numerical crack patterns in the same observation window is shown in Fig. 9. Through the photo-elastic facility, the early stages of the damage pattern were recorded. Fig. 9 shows the experimental fringe pattern and the numerical contour plots of maximum shear stress at about 42 |as after impact. Note the clear shock wave pattern in both cases. This implies that the crack propagates at a velocity above the shear wave speed of the core material Homalite, and is therefore inter-sonic.
c (a)
(b)
(c)
Fig.9 (a) Observation window; (b) experimental fringe patterns; (c) numerical contour plots of maximum shear stress at 42 |as after impact. Dynamic crack arrest and re-initiation In order to have a closer look of the crack speed history, the crack initiated from the left edge of the lower interface was chosen to compare with the experimental results in Fig. 10. One can
535
3D Modelling of Impact Failure in Sandwich Structures
notice that the crack speed along the lower interface reaches the Rayleigh wave speed soon 100
80
140
100
Time (jos)
(a)
£.
T
—- • 1.5
E
^
2 a 0)
1
^\ 1
f\i\
^ / /
Cs -€3-
-/L.J y / 1
'"~°N
'h
.3
\
^ H
\
t
PI
//' //
\\
1—
I
60
\
\\ !
•
/
•' ^
1
f
w J
/
\ \ •i w^
1/
\\
0.5
i^
^ fy"
\\
O
n
tal
-
/ /
o
- Numerical
__l
L _ _
80
Time (|is)
L 100
-1
L 120
LJ
(b) Fig. 10 Crack-tip position (a) and crack speed (b) history of the lower interface, comparison between experimental (filled triangles) and numerical (empty squares) results.
536
C YU, M. ORTIZ AND A J. ROSAKIS
initiation and drops to a very low value around 90 |is after impact (the plateau part in Fig. 10, both numerical and experimental); next, it increases again dramatically up to the shear wave speed of Homalite-100. In previous research on interfacial cracks, Lambros and Rosakis [8] showed that as soon as an interfacial crack accelerates to the Rayleigh wave speed, it keeps a stable speed as long as constant energy is provided to the crack tip. If the energy sipply is suddenly increased, the crack accelerates unstably to another discreet constant level within the inter-sonic regime, otherwise if an unloading wave reaches the crack tip, the crack quickly arrests. Here the complex wave interaction and structural vibration response would result in temporary loss of driving force which accounts for the observed crack arrest and re-initi ition. The effects of the interfacial strength and impact speeds The effects of different interfacial strengths and impact speeds were also investigated. The results show that even small variations in impact speed and bond strength could substantially influence the initiation behavior of de-lamination (location and nucleation time) and lead to substantially different inter-layer crack speed histories and therefore influence the timing sequence and final extent of subsequent intra-layer damage within the sandwich structures, Fig. 11-12, which agrees very well with the observations by Xu and Rosakis [1].
(a)
(b)
Fig. 11 Numerical crack patterns at time 300 |LIS after impact, with the same shear strength x = 22 MPa, for impact speed of (a) 33 m/s and (b) 46 m/s.
Y'-xr-— (a)
-jj-
TTT (b)
Fig. 12 Numerical crack patterns at time 300 jis after impact, with impact speed 33 ni/s, for inter-facial shear strength of (a) x = 22 MPa and (b) x = 7.74 MPa. SUMMARY AND CONCLUSIONS The model experiments of Xu and Rosakis on low-speed impact over sandwich structures were simulated applying cohesive models. The simulation captures qualitatively the main experimental observations. The most relevant correspondence is in the development of the first crack at the interface between the layers, the presence of shear stresses along the interface, which renders the crack shear driven and often inter-sonic, and the transition between interlayer crack growth and intra-layer crack branching. The effects of impact speed and bond shear strength are also investigated and highly satisfactory predictions are obtained.
3D Modelling of Impact Failure in Sandwich Structures
537
ACKNOWLEDGEMENTS CY thanks Ministerio de Educacion, Cultura y Deporte, Spain, for the fellowship SB20000191, which makes possible her stay at ETSIde Caminos, C.,y P., Universidad de Castilla-La Mancha (UCLM). She also acknowledges the financial support from the Vicerrectorado de Investigacion of UCLM. CY and MO are grateful for DOE support provided through Caltech's ASCI/ASAP Center for the Simulation of the Dynamic Response of Solids. AJR acknowledge the support of the Office of Naval Research through grant N00014-9 to Caltech and the support of the National Science Foundation grant CMS9813100.
REFERENCES 1. Xu, L.R. and Rosakis, A.J. (2002), Int. J. Solids Structures, 39 (16). 2. Ruiz, G., Pandolfi, A. and Ortiz, M. (2000) Int. J. Numer. Methods Engrg. 48(7). 3. Yu, C, Pandolfi, A., Ortiz, M, Coker, D. and Rosakis, A. J. (2002) Int. J. Solids Structures, 39(25). 4. Mi, Y., Crisfield, M.A., Davies, Gao and Hellweg, H.B. (1998) J. Compos. Mater. 32(14). 5. Camacho, G.T. and Ortiz, M. (1996) Int. J. Solids Structures, 33 (20-22). 6. Pandolfi, A. and Ortiz, M. (1998) Engrg. Comp. 14 (4). 7. Ortiz, M. and Pandolfi, A. (1999) Int. J. Numer. Methods Engrg. 44. 8. Lambros, J. and Rosakis, A.J. (1995) Proc. Roy. Soc. London A451.
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Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003 Elsevier Ltd. and ESIS. All rights reserved.
539
INTERFACIAL STRESS CONCENTRATIONS N E A R FREE EDGES A N D CRACKS B Y THE BOUNDARY FINITE ELEMENT METHOD Jan Lindemann and Wilfried Becker Universitat Siegen, Institut fiir Mechanik und Regelungstechnik, D-57068 Siegen, Germany [email protected], [email protected]
ABSTRACT In the present paper the Boundary Finite Element Method is presented as a boundary discretization method for the numerical investigation of interfacial stress concentrations in composite laminates. In contrast to the classical boundary element method, the element formulation is finite element based, which avoids the necessity of a fundamental solution. Comparative results from finite element calculations show good agreement both for the laminate free-edge effect and for the example of the stress concentrations near cracks in composite laminates. Keywords: Boundary Finite Element Method, composite laminates, stress localization, laminate free-edge effect, free-edge stresses, crack problems, transverse matrix crack, numerical methods, boundary discretization INTRODUCTION In practical applications of composite laminates stress concentrations may have a detrimental effect on the strength of lightweight structures. Such stress concentrations (or localizations) occur where the elastic or geometric properties change discontinuously. In the framework of linear elasticity, the stress concentrations may occur with finite or with infinite stresses. Typical examples for such stress localizations are the concentration of stresses near free edges of composite laminates or near cracks in a laminate. These stress localizations are not taken into account by classical laminate theory [1] where basically a layerwise plane stress state is assumed. In general, the stress state near free edges and cracks is fully three-dimensional with high stress gradients, especially of the out-of-plane stresses. The interfacial stresses, which occur, may lead to delamination failure of the laminate before the strength limit from the inplane stresses is reached. Because of this importance of stress localizations many investigations have been dedicated both to the laminate free-edge effect, starting with the finite difference analyses of Pipes and Pagano 1970 [2] and the matrix crack problem. For both cases, closed-form analytical solutions are of a more or less approximate character. On the other hand, numerical methods as the finite elements (FEM) require a high discretizational effort because of
540
J. LINDEMANNAND W. BECKER
the strong localization of the stress concentration together with the high stress gradients. An overview of the different approaches for the laminate free-edge effect has been given e.g., by Herakovich [3], Stiftinger [4] and Kant and Swaminathan [5]. Since the presence of transverse matrix cracks in laminates means a loss of strength and stiffness, besides experimental studies, e.g., [6], several theoretical investigations have been dedicated to this subject. Because of the local stress concentration near the crack tips, the presence of a crack may induce delamination in a layered structure, which has been investigated in more detail e.g., in [7]. One example for the question of the stiffness degradation due to transverse cracking is the work of Kashtalyan and Soutis [8]. In the current paper, results of finite element analyses presented by McCartney et al. [9] are used for comparison. For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for example the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM). Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name 'Consistent Finite Element Cell Method' for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. B O U N D A R Y FINITE ELEMENT FORMULATION FOR C O N T I N U A WITH EQUIVALENT CROSS-SECTIONAL PROPERTIES In the following the governing equations are presented for the case that there exists no similarity center, but that equivalent properties for cross-sections parallel to the boundary are given. The starting point for the Boundary Finite Element Method is the boundary discretization of the continuum (Fig. 1) which is here depicted for the free-edge effect situation of a laminate with a symmetric layup, where only one strip of lengtl A/ of the continuum needs to be considered. This is always the case, when the field qu intities are independent of the ^-direction, as it is for the example in the case for a laminate under uniaxial tension in the y-direction. The Boundary Finite Element Method 13ads to a local solution for the field quantities near the boundary which will vanish witliin the continuum, so that the total stress field is achieved in the framework of linear elasticity by superposition of this local solution near the boundary and an appropriate solution for the stress field within the continuum, which is given for laminates by classical laminate theory. One layer of isoparametric volume elements of infinitesimal width Ax is introduced. The local coordinates of the isoparametric parent element are denoted by ^, 77, C with the values varying from —1 to + 1 . The ^-axis of the parent element corresponds with the x-axis shown in Fig. 1. On the
541
Interfacial Stress Concentrations Near Free Edges and Cracks
L
0°
:9o°| X
Ax
Fig. 1: Considered part of the continuum with boundary finite element discretization element surfaces at x = 0 and x = Ax there are eight nodes each, so that the shape functions on the element surfaces are given by NiivX),
1 = 1,...,8
or
N =
{Ni,...,Nsy
(1)
Regarding the ^-direction the shape functions become
N, = ^(l + ejON, j = 0,l.
(2)
Herein, the subscripts j denote the element surfaces at the boundary (j = 0) and in the interior at x = Ax {j = 1) with the corresponding coordinate values ^ =^ ~1 and ^ = + 1 respectively so that the shape function for the node j , k results in:
Njk = l{l + ^jONk.
(3)
With this the shape functions of the volume element become N = (No, N i ) • In the case of equivalent cross-sections in x-direction the presumed geometric similarity is given by: xi = xo + /^ c, yi = Yo, zi = zo (4) with the length scale K — Ax and constant vectors XQ = (XQ, . . . , XQY ^^^ c = ( c , . . . , cY. The isoparametric mapping rule for the link between the ^,77, C-coordinates of the parent element and the element coordinates x, y, z becomes: X =
N^x
-
N^z
=
Nj^xo + N f x i
= xo + § ( H - O c , (5)
z
taking into account that the sum of the shape functions over all nodes in one element equals one. With c—\ the Jacobian matrix and its inverse result in: ^,4
y.i
^,^
^,77
2/,ry
^^T]
L ^,c y,c ^A
0
0
0 Njjyo Njzo 0
N^^yo
N^zo
J-i =
!JU
0
0
0
J22
i23
0
J32
i33
(6)
The determinant of the Jacobian reads det J = - det J
with
det J = N ^ yo N ^ ZQ - N ^ ZQ N ^ yo-
(7)
J. LINDEMANNAND W. BECKER
542
The derivatives of the shape functions result in: ( N,,,
/ N^k.x \ N.jk,y
-
1^*
\
J-
=
J
V Njk,z I
(8)
A;,77
v
1+M A^/ fc,C
/
so that the standard strain-displacement operator of the finite element formulation becomes
B 3k
jk,x
0
0 0 0
•^^jk,y
0 0
jk,z
0
^^ jk,x
K jk,x
0
0
Njk,z
Njk,z
K jk,y
y_T5i • B [ _+L I I I L I B ^
^^jk,y
(9)
with the components of the strain tensor written as a vector e = (e^, e^, e^, jyz, Jxz^ IxyY with the matrices B^ and B^ given by: ill
Bi
0 0 0 0 0
0 0 0 0 0 ill
0 " " 0 0 0 0 0 J22 0 0 0 Nk\ Bl = 0 0 32.2 J32 0 ill 0 . i22 0
" 0 i32 J22 0 0 .
Nk,rj +
' 0 0 0 i23 0 0 0 i33 i33 0 . i23 0
0 0 i33 i23
Nk,i-
(10)
0 0
The total strain-displacement operators B'^ and B^ of all nodes are assembled by B^ and B | . Subsequently the complete strain-displacement operator B = [BQ B I ] is partitioned corresponding to the interfaces j = 0 and j = 1 into the matrices BQ and B i : B, = ^ B i + — ^ ^ ^ B ^ 2^ J = 0 , 1 .
(11)
Application of the same decomposition to the static stiffness matrix of the finite element formulation K-j W TiBdV (12) results in
K,/
=
/
=
t l t l i^l
B^'DBidV B J D B, detJ de dr^ < , 3, / = 0,1
(13)
with the determinant of the Jacobian matrix defined by (7) and the elasticity matrix D. The integral for the ^-direction is evaluated in a closed-form analytical manner under consideration of the strain-displacement operators (11), which leads to the following form of the static stiflfness submatrices: (14) with the coefficient matrices given by
K
: ^j e, E°,
E° = j
r^
B ^ ^ D B^ det J drj dC,
(15)
543
Interfacial Stress Concentrations Near Free Edges and Cracks
K]. = | E ^ " and
2
1 M
4
12
(16)
J-i J-i + 1 /•+!
E^
/ : / :
B^^DB^detJdjydC.
(17)
In order to obtain the boundary finite element formulation, on the one hand the forcedisplacement relation of the discretized element layer between nodal forces P and nodal displacements u is considered, which can be written in the decomposed form: Koo Koi Kio K i i
Po Pi
Uo
(18)
On the other hand the relationship between the loads R and the displacements u at the discrete degrees of freedom at the nodal locations of the finite element discretization can be given by means of an elastic stiffness matrix K°°, which is independent from the considered cross-section: R,-K°°u„ i-0,1. (19) In accordance with equilibrium the external loads R j and the nodal loads P j are interrelated as Po — Ro, (20) Pi = - R i With the relations (18) and (19) this leads to: Koo Koi Kio K n
Uo Ul
0 0
Uo Ul
(21)
From this system of linear equations the displacements Ui may be eliminated. Since the resulting equation must be valid for arbitrary displacements UQ, the coeflacent matrix must vanish, which eventually leads to: (22)
( K u + K ~ ) K o / ( K ~ - Koo) + Kio = 0.
Herein, the submatrices Kj/ are given by (14) with the coefl&cients defined by (15), (16) and (17). The inverse K ^ / is approximated by the following polynomial of order three K^/ = -/^ E°"^ + /i:^ A + /^^ B
(23)
with the unknowns A and B resulting from KQI K ^ / = I. Eventually this leads to K O - 1 - - / . E O - ' - / . 2 E O " ' K J I E ° " ' - / . 3 E ° " ' (Kg^-fKjiE°"'Kji)
E^"\
(24)
Inserting the matrices (14) and the expression (24) for the inverse K ^ / in equation (22) and performing the limit /^ —> 0 results in the following Matrix-Riccati equation for the unknown elastic stiflFness matrix K°°: j^oo J.0 1 K°° + E^ E°"^ K^ -h K°° E°"^ E^^ + E^ E°"^ E^^
E^ - 0.
(25)
J. LINDEMANNAND W. BECKER
544
Equivalent to this equation is the eigenvalue problem:
1 T?.0-1 T E^E°~ E? ! ^
.
E2
*11
*12
A
*21
*22 .
0 -A
-E'0 - 1
*11
*12
E 1^ ET?o-i '
^21
*22
0
,
A=
Ai
0
0
A2
0
•••
(26)
where the boundary stiffness matrix results in: (27) In order to determine the field quantities such as displacements, strains and stresses inside the laminate, away from the discretized boundary, the first part of equation (21) is considered: KooUo + K o i U i = K ^ U o . (28) The displacement vector Ui at the cross section j = 1 can be expanded as Ui = Uo +
9u dx
Ax
(29)
with the infinitesimal length Ax which equals the introduced width scale parameter /^ so that equation (28) becomes (Koo + Koi) Uo + Ko
dn dx
/^ = K ^ uo.
(30)
Taking into account the decomposition (14) for K^i and the coefficient matrices (15) to (17) and performing the limit K —>^ 0 it follows: du dx
+ E°"^ (EI'^ + K ~ ) U = 0
(31)
where KQJ = —E° has been used. This differential equation can be rewritten with help of the first set of the eigenvalue problem of equation (26) as: dn
K A * - ) U + ^ = 0.
(32)
This differential equation for the behaviour of the displacements in x-direction can be solved easily in a closed-form analytical manner: u = *iidiag(e^^^)*i/uo
(33)
with the displacements UQ resulting from the boundary conditions, i.e., from the forcedisplacement relation R = K°° UQ at the boundary. The strains (arranged as a vector) follow with the introduced strain-displacement operator (11) from: e = B u = [BoBi]
Uo 111
(34)
Interfacial Stress Concentrations Near Free Edges and Cracks
545
Using the expansion (29) and performing the Umit /^ —)• 0 finally yields: (35)
OX
Eventually the components of the stress tensor within the continuum can be obtained as: ^ — [Pxi ^yi
^zi
^yzi '^xz-) ^^xy)
Tie.
(36)
The equation (33) and the subsequent equations show the main benefit of the Boundary Finite Element Method: The field quantities inside the continuum can be calculated easily in closed-form analytical manner. RESULTS In the following results are presented for the application of the Boundary Finite Element Method both for the case of the laminate free-edge eflFect and for the case of a single transverse matrix crack in the framework of linear elasticity theory. a^/MPa, T^^/MPa
0.5
1 1.5 2 x/mm (distance from free edge)
2.5
3
Fig. 2: Interfacial stresses at the 0°/90°-interface near the free edges Free-Edge Effect As a first example, the interfacial stress concentration at the 0°/90°-interface near the free edges of a symmetric [0°/90°]5-cross-ply laminate of 0.5 mm single ply thickness is considered. The BFEM discretization and the coordinate system used in the analyses for this case can be seen in Fig. 1. The laminate is subjected to thermal loading of AT = -100 K. The transversely isotropic single ply properties are given in the form of the engineering constants as Ei = 135.0 GPa, E2 = Es = 10.0 GPa, Gu = Gu = 5.0 GPa, G23 = 5.0 GPa, ^12 = ^13 = ^23 =' 0.27, Qi = -0.6 • 10-VK and a2 = a^ = 30.0 • lO-V^. For the laminate a plane strain state with respect to the x-z-plane is assumed, i.e., the strain component £y in the longitudinal direction vanishes. Whereas in the finite element analyses, performed with the programme MSC/NASTRAN, 24 layers of 48 isoparametric volume
546
J. LINDEMANNAND W. BECKER
elements, of which each consists of 20 nodes, have been used, the analysis by Bcundary Finite Element method shows already qualitatively and quantitatively reasonable results with a discretization by eight 8-noded isoparamatetric surface elements per ply only. In Fig. 2 the BFEM results are depicted as lines, the FEM results are denoted by ruarkers. The concentration of the interlaminar stress components GZ and r^z in the vicinit^ of the free edge of the laminate can clearly be seen. These interlaminar stresses vanish wii hin thi3 laminate after a length in the order of the total laminate thickness. The high gradieit of a. in the immediate neighbourhood of the free edge is an indicator for a singular behaviour of this stress component. From the theory it is expected that the interlaminar shear stress T^Z is zero directly at the free edge because of the boundary conditon of a traction fiee surface. This behaviour is better fulfilled by the BFEM results than by the FEM results. A short distance away from the free edge, in the interior of the continuum, the absolute value of this shear stress becomes a maximum and then the stress tends towards zeri > again. considered strip of the continuum
matrix crack BFEM discretization in crack plane
Fig. 3: Application of the BFEM analysis for transverse matrix cracking Transverse Matrix Crack In this case the stress field near a transverse matrix crack in a symmetric cross-ply laminate is investigated, which consists of five transversely isotropic single plies with 0.2mm thickness and which is subjected to a thermal loading of A T = — lOOK and a mechanical loading by a bending moment per unit length of M = l.OkN (see left illustration of Fig. 3). The transversely isotropic material properties are given by E*! = 45.6 GPa, E2 = E^, — 16.2 GPa, G12 = Gi3 = 5.83 GPa, G23 = 5.79 GPa, v^^ = i^u = 0.278, z/23 = 0.4, ai = 8.6 • 1 0 - V K and a2 = 0^3 = 26.4 • 10~^/K. In order to perform the BFEM analyses, the discretization is introduced in the ^-z-plane (see Fig. 3), which is identical with the crack plane. Since plane strain conditions with respect to the x-z-plsine are assumed and the field quantities are independent of the ^/-direction, it is sufficent to consider only one strip of the continuum of unit width. For the given laminate, first the stresses for the uncracked laminate according to the classical laminate theory have been calculated. In order to satisfy the condition that the crack surface is free of tractions, the resulting layer stress from the classical laminate theory has been applied as external load in the Boundary Finite E;lement analysis. The complete solution is then given by the superposition of these load cases (see Fig. 4). The results of the Boundary Finite Element Method are compared with finite element results obtained in [9], where a two-dimensional finite element model using th(; ANSYS element type TLANE82' has been applied. In the BFEM analyses 64 layers of the eight-node surface elements per ply have been used. The results for the axial stress Ox along the crack plane show an excellent agreement with the results from literature (Fig. 5). The
Interfacial Stress Concentrations Near Free Edges and Cracks
—^T
/ \
/ \
'Z
0" 90'*( QO 1
(;
)
' ) - \
90* 0"
thermal loading of AT= -lOOK and mechanical bending moment per unit width of M=l .OkN
547
'Z
0«' 90" 0" 90* 0"
^N •M)
V BFEM-modelling with 64 layers of 8-node-surface elements per ply
homogeneous case with CLT-solution
Fig. 4: Boundary Finite Element analysis and load application 15000
j
.2
t
10000
m\ • w
1 ••
BFEM
V
0°
• • ~\\
0° — ^T —
FEM (McCartney + etal. 2000)
7 0^
5000
^ -t
-5000
2000
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 z/mm aJMPa z
1500 crack ):
1000 500 i
J
0790°-interface (z=0.1 mm)
\
-500
0° 9q« C— y 90° 0"
^^W 1 1
^..Y^^fcdfciS^^^'
' ''' '' '
9070°-interface (z=0.3nr m)
0.05
0.1 0.15 0.2 0.25 x/mm (distance from crack plane)
0.3
Fig. 5: Results for the axial stress along the crack plane and the interlaminar normal stress in axial direction starting from the crack tips
548
J. LINDEMANNAND W. BECKER
high gradients of the crack plane axial stress at the crack tips at the respective interfaces are indicating a singular behaviour of this stress component at the crack tip. F irthermore the concentration of the interlaminar normal stress at the interface near the crack tips can clearly be seen. Here the markers do not denote FEM results, but are di^picted in order to show where the field quantities in the BFEM analysis have been calculated within the continuum. These stresses vanish away from the crack after a length in the order of one single ply thickness. Again an almost singular behaviour, which is of course not exactly included in the boundary finite element solution, is revealed near the crack tips. CONCLUSIONS The Boundary Finite Element Method is presented as a numerical method which coiiibines characteristics of the finite element method and the boundary element method. For certain geometric situations, this method allows an easy investigation of stress localization problems with less discretizational effort in comparison with the finite element method. For both the example of a transverse matrix crack and the example of the laminate fr(^e-edge eff'ect, the results are shown to be in excellent agreement with comparative finite element results. ACKNOWLEDGEMENT This work has been performed under the financial support of 'Deutsche Forschungsgemeinschaft' under BE 1090/8-1 which is gratefully acknowledged. REFERENCES [1] Jones, R.M. (1975). Mechanics of Composite Materials. McGraw-Hill, New York. [2] Pipes, R.B., Pagano, N.J. (1970) J. Comp. Mat. 4, 538. [3] Herakovich, C.T. (1989). In: Handbook of Composites, Vol. 2 — Structure and Design. pp. 187-230, Herakovich, C.T., Tarnopol'skii, Y.M. (Eds). North-Holland, Amsterdam, [4] Stiftinger, M.A. Diisseldorf.
{1996).Fortschritt-Berichte
VDI,
Reihe
18
203
VDI-Verlag.
[5] Kant, T., Swaminathan, K. (2000) Compos. Struct. 49, 69. [6] Adolfsson, E., Gudmundson, P. (1999) Int. J. Sol. Struct. 36, 3131. [7] Berthelot, J.-M., Le Corre, J.-F. (2000) Comp. Scien. Tech. 60, 1055. [8] Kashtalyan, M., Soutis, C. (2000) Composites A 3 1 , 335. [9] McCartney, L.N., Schoeppner, G.A., Becker, W. (2000) Comp. Scien. Tech. 60, 2347, [10] Wolf, J.P., Song, C. (1996). Finite-Element Wiley and Sons, Chichester.
Modelling of Unbounded Medic. John
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
549
STABILITY OF J-CONTROLLED CRACKS IN PIPES J.LELLEP Tartu University, Institute of Applied Mathematics, 46 Vanemuise str., 51014 Tartu, Estonia E-mail: lellep(a)-ut.ee ABSTRACT A stability analysis is presented for circumferential cracks of constant depth in cantilevered pipings. The analysis is based on the tearing modulus concept and the tearing stability criterion. Assuming that the cracked cross-section is subjected to limit moment the crack growth is studied in the case of pipings subjected to impact loading. KEY WORDS: J-integral, tearing modulus, crack propagation, plastic fracture NOTATION A c E h / k KQ / M MQ M^ Mp NQ A^o P R t
- crack area - crack size (length) - Young modulus -thickness of the tube - J- integral - material parameter (ratio of yield stresses in the circumferential and longitudinal directions, respectively) - initial kinetic energy - length of the tube - applied bending moment - yield moment - axial moment - plastic limit moment - circumferential force - yield force - axial force -radius of the cylinder - time
550
J. LELLEP
/, T^
- final moment - applied tearing modulus
7].
- surface tractions
T^^^
- material modulus
u^
- displacements
VQ - initial velosity W ' radial deflection W - energy density ;c - longitudinal coordinate a - angle corresponding to the neutral axis Ag, A^ - elastic and plastic crack opening displacements (p ^
- rotation angle - angle acceleration - final angle of rotation
(p^ Y r K^,SQ
- non-dimensional crack length - a curve surrounding the crack tip - strain components
ju
- mass per unit length
(TQ
- yield stress
co,v,T 20
- auxiliary parameters - crack length in the circumferential direction
INTRODUCTION It is well known that the methods of elastic-plastic firacture mechanics provide more realistic models of cracked structures with high toughness compared with the methods of the linear elastic fracture mechanics. Ductile materials are used in structural elements not only in piping systems of power plants but in chemical industry, in aircraft propulsion systems and elsewhere [1-8]. Evidently, cracked elements in chemical or power plants pose a serious threat to operation of these structures. Therefore, it is extremely important that the crack will not spread unstably through the pipe thickness. A useful tool in the investigation of elastic-plastic crack growth is the concept of the tearing modulus [9-16]. A theory of stability analysis of J-controlled crack growth was developed by Hutchinson and Paris [4]. The concept of the tearing modulus was accommodated to circumferential cracks in reactor piping systems by Tada, Paris, Gamble [14] and to pipes with internal flaws by Rajab and Zahoor [9,10]. Zahoor and Kanninen [16] studied a fourpoint loading system. In the present paper the behaviour of cracks in tubes made of a fiber-reinforced composite is studied. It is assumed that the fibers are oriented circumferentially and that the material behaviour follows the concept of an ideal plastic body.
551
Stability of J-Controlled Cracks in Pipes
THE J-INTEGRAL AND TEARING MODULUS The J-integral suggested by Rice [11] and crack opening displacement [13] are the parameters most often used to characterize non-linear fracture behaviour of cracked pipes [13,14]. In the present study the tearing modulus stability criterion is used.
The path independent integral was presented for a plane problem as [11]
•'•I
Wdx^-T^^ds
where T is the curve surrounding the crack tip, W - energy density, 7^. - surface tractions, w. - displacements. The tearing modulus is defined as [7] o-Q
dc
Note that sometimes the modulus T^ is called applied tearing modulus. In (1) E is the Young's modulus, a^ is the yield stress or some weighted-average value ofthe yield and ultimate stresses and c is the relevant flaw size. Crack stability is assured when the applied tearing modulus is less than its material counterpart at the prescribed J value. Thus the condition of stability ofthe crack growth is given by [4,7,13] Ta
(2)
If (2) is not satisfied, e.g. if T^ > T^^^, then the crack growth is unstable. In the latter relation F dJ T
—
/ON
mat
a,
dc ^ T
The material tearing modulus is derived from the materials J-resistance curve, e.g. —^^ dc is the slope ofthe J-resistance curve. In general the material tearing modulus varies with J , decreasing with increasing values of ^ [7] . The J-integral was originally defined by J. Rice [11] for elastic materials. In the case of an ideal plastic element with a plastic hinge the J-integral can be calculated as [9,13,14]
552
J. lELLEP
where A is the crack area, M - appHed bending moment and cp - rotation angle of the cracked body. NET-SECTION ANALYSIS As an example of application of (l)-(4) let us consider a cantilevered piping run with cracked root section. Two kinds of cracks presented in Fig. 1 and 2, respectively, used for modelling of stress corrosion cracks in boiling water nuclear reactor steam supply systems will be considered in present study. Consider first the case of bending of the tube as a beam. It is assumed that the bending moment applied to the root section of the tube is the plastic limit moment. Let the neutral axis be defined by the angle a=
-(9 + Acj^R
(5)
for the case presented in Fig.l. Here P is the axial force which can be induced by the internal pressure and G^ stands for the yield stress in the longitudinal direction. It is expected herein that the crack does not extend below the neutral axis and the complications due to crack closure can be neglected.
Fig. 1. a) Part-through crack in a circular cylindrical tube
b) Stress distribution
Stability of J-Controlled Cracks in Pipes
553
Fig. 2. Internal part through-wall crack of constant depth The limit moment corresponding to Fig. 1 and to the beam theory is M = 4croMncosa
sin 6>+Pi? sin a .
(6)
Note that the relations (5), (6) correspond to thin-walled pipings when h is much smaller than R. The crack surfaces can be presented as A = 2Rc9.
(7)
In the subsequent analysis we consider an axisymmetric deformation of the tube with a crack in the root section. YIELD CONDITION FOR A TUBE OF FIBER REINFORCED COMPOSITE MATERIAL Let us consider a cantilever piping having an internal crack of depth c at the root section. The root section corresponds to Fig. 2. It is assumed that the tube is made of a composite material which is composed of stiff ductile fibers arranged in a parallel uniform array in a ductile matrix. The composites of metal-metal type are considered, first of all. However, the combination of a ductile matrix with brittle fibers can be easily accounted for, as well. As an example of the latter case serves the aluminium alloy reinforced with boron fibers. Departing from the maximum shear stress theory of plastic flow R. H. Lance and D. N. Robinson [6] developed yield conditions for fiber reinforced composite materials. The authors of [6] assumed that the material could flow plastically if (i) the shear stress on planes parallel to fibers, and in a direction perpendicular to them, reaches a critical value or
554
J. LELLEP
(ii) the shear stress on the same planes, but in a direction, parallel to fibers, reaches another critical value or (iii) the maximum shear stress on planes oriented at 45V to the fiber direction reaches a critical value not equal to either of the previous two values. The latter case accounts for the ductility of fibers themselves. Non-linear yield surfaces for plastically anisotropic materials which are applicable in the continuum theory of inelastic composites are suggested by R. Hill [3], W. Prager [8], S. W. Tsai and E. Wu [15], F. Barlat et al [1], M. Gotoh [2].
^Mx/Mo
Fig. 3. Yield condition for a cylindrical shell In the present paper we are seeking for simple theoretical predictions of the relationship between crack length and tearing modulus. Since it is reasonable to assume that the material obeys the piece wise linear yield condition developed by R. H. Lance and D. N. Robinson [6]. Although the yield surface strongly depends on the orientation of fibers we confine our attention to the case of circumferential arrangement of fibers in the matrix material, provided kcj^ is the yield stress in the circumferential direction. The yield condition for circumferentially reinforced cylindrical tubes is presented in Fig. 3. Here ^0 = ^0^'
Mo
CT.h'
. Note that the theory to be used is a continuum theory, in
principle. Such effects as fiber splitting, fiber buckling and debonding at the fiber-matrix interface as well as fiber bridging are not considered herein. The contribution of elastic deformations is neglected also. It has been emphasized by R. H. Lance and D. N. Robinson [6] that although the current theory is a continuum theory it can prescribe the behaviour of a fiber reinforced composite. The presence of fibers is acknowledged in terms of their effect on the yield of the matrix itself and their contribution to the strength of the composite in the fiber direction.
Stability of J-Controlled Cracks in Pipes
555
DETERMINATION OF THE ANGLE OF ROTATION Assume that the cantilever piping is subjected to the internal impulsive loading. The intensity of the pressure is assumed to be high enough in order to cause plastic deformations. Evidently, the deformation of the tube subjected to the uniform internal pressure can be considered as an axisymmetric deformation.
T 2R
Fig. 4. Cylindrical shell subjected to the initial impulsive loading The equilibrium of an axisymmetric cylindrical element is prescribed by the equation [5] a'M, dx'
Nn
- = -//
d^w dt'
(8)
where M^ and A^^ stand for the bending moment in the longitudinal direction and membrane force in the circumferential direction, respectively. Here W is the transverse deflection and ju is the mass per unit length of the tube. The strain rates consistent with (8) have the form [5] K,
= -
d^W . _W_ dx' ' "'' R
(9)
where dots denote the differentiation with respect to time t. It is reasonable to assume that the stress-strain state of the tube corresponds to the regime BC (Fig. 3). Thus (10) and according to the associated flow law k,=(i,
Sg>Q
(11)
It easily follows from (10) and (11) that W = (p{t)-x,
(12)
556
J. LELLEP
where the boundary conditions WiO.t) = 0, W(l,t) = ^ • / have been taken into account. Relations (9), (11) and (12) show that the associated flow law is satisfied, if ^o - ^ ^^ each moment of time. Substituting (10) and (12) in (8) leads to the equation (13)
-^(f>{t)'X .
R
Integrating (13) twice with respect to the coordinate x gives dM^
kNo
- =
and
-X-
dx , .
R X
..x^
^
^^ 2
^
(14)
u(p— + Ci
KISIQ
.,X
^
^
(15)
where C\ and C2 are arbitrary functions of time. The right hand end of the tube is assumed to be free. Thus the boundary conditions at x=l are
M(/,0 = 0, f^(/,0 = 0. ox
(16)
At the root section a plastic hinge circle is to be located so that each generator of the middle surface of the cylinder rotates around the hinge according to (12). Thus the longitudinal bending moment at x=0 is to be equal to the plastic limit moment, e.g. M(0,0 = M^. (17) Boundary conditions (16), (17) with (14), (15) give the constants of integration C, = C2 =
2
\ R 2R
3
/
(18) I'
and the acceleration
^
2R
(19)
It can be seen from (19) that the angular acceleration of a generator of the cylinder is a constant.
Stability of J-Controlled Cracks in Pipes
557
Thus (20)
(p = (P't + -
and ^
2
(21)
/
In (20) and (21) the initial conditions ^(0) = 0 and ^(0) = — are taken into account. The quantity VQ can be treated as a given constant if the initial velocity is given. However, VQ can be considered as an unknown constant if the initial velocity distribution is not fixed but initial kinetic energy is given. The motion ceases at the moment MVQI
U =-
3
^
(22)
2R
The maximal angle of rotation can be defined according to (19), (21), (22) as MVQI ^1 =
"
(23)
2R
Let us consider now the case when the initial kinetic energy ^o is given but the velocity VQ is unknown. In this case according to (12) one has 1
/
(24)
Since VQ = /^(O) one easily obtains from (24) 6^ Ml' TEARING MODULUS AND STABILITY OF THE CRACK GROWTH As a rule, the J-integral in (l)-(4) is split into elastic and plastic components, e.g.
whereas the center-crack-opening-displacement A = A +A„ . e
p
(25)
J. LELLEP
558
However, in the case of gross yielding at the cracked section of a pipe the contribution of J^ is small compared to J^ and the elastic part can be neglected in the foregoing analysis. Thus J = Jp in (4). The exact calculation of the J-integral and the tearing modulus (1) is quite complicated. It was shown by several authors that a good approximation for the J-integral in the case of rotation around a plastic hinge can be presented as [13,14] dM
(26) (D . dA ^ In (26) M is the moment applied to the section which is fully plastic and cp is the corresponding angle of rotation. J =
In the case of axisymmetric deformation of the tube with the internal flaw of constant depth one can take M = M
- ITTR where
M,=^{h-cy.
(27)
Thus according to (26), (27) and (19), (21) .2
j =
^(h-c)\^ 2 fir
(^ojh-cf
^j.crohl^ 2R
'T
(28)
r-th
(29)
and J. ^ EVQ \ 3o-o/^^ laj
h)
SJUVQP
Rh
Here T^ stands for the applied tearing modulus in the case of radial crack growth.
DISCUSSION The variation of Ta in time is presented in Fig. 5 for different values of the nondimensional crack length y = —. It is somewhat surprising that the tearing modulus Ta is h relatively weakly sensitive with respect to the crack length. In Fig. 5 and 6
r = ^ r ; .= '''' Ev^
hR
(30)
Stability of J-Controlled Cracks in Pipes
559
v=30, (D=20 6x0,2 0,4 0,6 0,8
1 1,2 1,4 1,6 1,8
2 2,2 2,4 2,6 2,8 3
-0,2
-0,4
-0,6
-0,8
Fig. 5. Variation of the tearing modulus in time In Fig. 5 €0 = 20. The curves labeled with 1, 2, 3, ...,9 in Fig. 5 correspond to values of / = 0.1 , ;^ = 0.2 ,..., 7 = 0.9 , respectively. It can be seen from Fig. 5 that for sufficiently long time period the maximum of the tearing modulus is achieved at the final moment of the period. Thus it might be expected that the stability criterion (2) is to be checked at the final moment of motion, first of all. Calculating the value of the tearing modulus at the moment T=TI making use of (22) and (29), (30) one obtains
TA)-^-^.
j_£|
2kV hR
1-^
2kl^ hR
(31)
560
J. LELLEP
The variation of TI=TC(TI} with respect to co is presented in Fig. 6 for several values of the crack length ^ . Here v = 40.
V=40
Fig. 6. Applied tearing modulus for a pipe It is interesting to remark that (19)-(23) and (28), (29) remain valid, if M^ = 0, e.g. c=h. Moreover, the relation T^ = T^ (co) depends relatively weakly on c. Calculations carried out showed that in a wide range of values of parameters co and v the tearing modulus has negative values (Fig. 5 and 6). However, it is not out of keeping with the results of H. Tada et al [13,14] which predict negative values of the tearing modulus for short pipe runs and large crack angles 6 . In the case of a negative value of the tearing modulus the function decreasing function with respect to c.
J = J{c,0)
is a
It can be seen from Fig. 6 that the tearing modulus increases when the parameter o) increases. Thus the cracks appear to have the tendency to be stable. Calculations showed that in the range of very small values of the parameter o) the tearing modulus has nonnegative values. However, in this case the tube is either very short or has a great diameter. Thus for reasonable values of geometrical parameters the crack propagation is stable, provided the material modulus is positive. It was shown in [13,14] , that for the stainless steel T^^f is about 200.
Stability of J-Controlled Cracks in Pipes
561
CONCLUDING REMARKS The problem of the stability of a crack in a piping run subject to an internal impact has been studied. Assuming that the crack has spread circumferentially over the tube with constant length in the radial direction the deformation is considered to be axisymmetric. The motion of a generator of the cylinder has been defined making use of the methods of plastic analysis of cylindrical shells. This motion in turn has been employed to calculate the Jintegral and tearing modulus for a cylindrical cantilever tube with a crack in the root section. Calculations carried out showed that the cracks are stable in the early stage of motion. However, in the final stage of motion cracks can be unstable in tubes with very small values of the parameter
/HL •
ACKNOWLEDGEMENT The research work was partly supported by the Estonian Science Foundation. The support through the Grant Nr 4377 is gratefully acknowledged. REFERENCES 1. Barlat, F., Lege, D.J. and Brem, J.C, (1991), A six-component yield function for anisotropic materials. Int. J. Plasticity, 7, 693. 2. Gotoh, M., (1977), A theory of plastic anisotropy based on a yield function of fourth order. Int. J. Mech. Sci., 19, 505. 3. Hill, R., (1993), A user-friendly theory of orthotropic plasticity in sheet metals. Int. J. Mech. Sci., 35, 19. 4. Hutchinson, J.W. and Paris, P.C., (1979), Stability analysis of J-controlled crack growth. In: J.D. Landes, J.A. Begley and G.A. Clarke (Eds), Elastic-Plastic Fracture, ASTM STP 668. American Society for Testing Materials, Philadelphia, 37. 5. Jones, N., (1989), Structural Impact, CUP, Cambridge. 6. Lance, R.H. and Robinson, D.N., (1972), A maximum shear stress theory of plastic failure of fiber-reinforced materials. J. Mech. Phys. SoHds, 19, 49. 7. Paris, P.C, Tada, H., Zahoor, A. and Ernst, H., (1979), The theory of instability of the tearing mode of elastic-plastic crack growth. In: J.D. Landes, J.A. Begley and G.A. Clarke (Eds), Elastic-Plastic Fracture, ASTM STP 668. American Society for Testing Materials, Philadelphia, 5. 8. Prager, W., (1969), Plastic failure of fiber-reinforced materials. Trans ASME, 1969, E34, 542. 9. Rajab, M.D. and Zahoor, A., (1991), Tearing modulus analysis for pipes containing constant depth internal flaw. J. Pressure Vessel Technology, 113, 1, 156. 10. Rajab, M.D., Zahoor, A., (1990), Fracture analysis for pipes containing full circumference internal part-throughwall flaw. Int. J. Pressure Vessels Piping, 41, 11.
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11. Rice, J.R., (1968), A path-independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech., 35, 379. 12. Stronge, W.J. and Yu, T.X., (1993), Dynamic Models for Structural Plasticity. Springer, London. 13. Tada, H., Paris, P.C. and Irwin, G.R., (2000), The Stress Analysis of Cracks Handbook. ASME Press, New York. 14. Tada, H., Paris, P.C. and Gamble, R.M., (1979), Stability analysis of circumferential cracks in reactor piping systems. U.S. Nuclear Regulatory Commission Report, NUREG/CR-0838. 15. Tsai, S.W. and Wu, E.M., (1971), A general theory of strength for anisotropic materials. J. Compos. Mater., 5, 58. 16. Zahoor, A. and Kanninen, M.F., (1981), A plastic fracture mechanics prediction of fracture instability in a circumferentially cracked pipe in bending - Part I: J-integral analysis. J. Pressure Vessel Technology, 103, 352.
Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.
^^3
ALTERNATIVE FATIGUE FORMULATIONS FOR VARIABLE AMPLITUDE LOADING OF FIBRE COMPOSITES FOR WIND TURBINE ROTOR BLADES
R.P.L. NUSSEN and D.R.V. VAN DELFT Delft University of Technology, Wind turbine Materials and Constructions-Group, Delft, the Netherlands ABSTRACT This paper looks at modem wind turbine rotor blades from the point of view of material fatigue. Characteristics of the rotor blades and loads are discussed and a simple commonly used lifetime prediction method is reviewed. Also, possible modifications to the various components of the fatigue calculations are discussed. KEYWORDS Fatigue, variable amplitude loading, composites, lifetime prediction. INTRODUCTION Composite materials are extensively used in modem wind turbine rotor blades. These blades have exhibited a significant increase in length over the past few years (blade lengths exceed 30 m). In addition, wind turbines are increasingly being installed collectively in wind farms, to serve as large electricity plants rather than stand-alone power supply. The associated high investments call for accurate and reliable lifetime prediction methods for the whole turbine, but especially for the blades, which are difficult to repair and to reinstall. However, there are indications that the commonly used lifetime prediction methods can lead to inaccurate results for the composite blade's lifetime under the strongly varying loads to which they are subjected. The present paper presents the preliminaries to an extensive study into the fatigue behaviour of wind turbine rotor blade materials under variable amplitude loading. This paper will discuss some of the problems and drawbacks of the currently prescribed lifetime prediction methods, and describes possible ways of tackling these. First, the most important loads on wind turbine rotor blades are described and a brief review of fatigue research in composites is presented to place the following approach in a broader perspective. The approach described in this paper concems modification of the existing formulations. Possible modifications are illustrated with predictions for an existing data set, which was generated by subjecting composite specimens to different variable amplitude load sequences.
R.P.L. NIJSSEN AND D.R.V. VAN DELFT
564
WIND TURBINE ROTOR BLADES Loads In terms of loads, the rotating wings of a wind turbine may best be compared to a helicopter rotor, which has been tilted 90°, but with the upper side of the airfoil shaped blade crosssection located downwind instead of upwind. However, where a helicopter blade is connected flexibly to the rotor axis, a wind turbine rotor blade is fixed rigidly to the hub. This results in significant bending moment loads in the blade. The loads can be separated into wind loads (constant wind speed + gusts), and mass loads (dynamic loads, gravity loads, centrifugal loads). The wind load causes high forces perpendicular to the rotor plane (comparable to rotor thrust in a helicopter) and a torque on the hub (facilitating energy production). An alternating load component is found in edgewise (or in-plane) direction due to the mass loading of the rotor blades. Also, the blade is loaded by centrifugal forces, (see Fig. 1). Large variations in loads occur due to the variable nature of the wind. The estimated amount of load cycles lies in the order of magnitude of 10^ cycles for a modem (large) wind turbine, see Mandell et.al. [1]. Construction The high lift loads on the rotor blades are carried by the main spar construction similar to the one found in most aircraft wing constructions. The spar as well as the low-load-carrying sections of the rotor blade is manufactured of composite materials. Several layouts and associated manufacturing techniques are applied in the rotor blade industry. Some possible cross-section layouts are shown in Fig. 2. In most cases, the spar web is made of a composite with a high degree of multi-axial lay-up (to withstand shear forces), whereas the spar flanges contain a large amount of unidirectional composites. In order to withstand the large bending moments (distributed load increases towards the tip), special airfoils with high thickness-to-chord ratios have been developed, e.g. the DU-airfoil series designed and tested by Timmer and van Rooij [e.g. 2]. Near the rotor hub, the blade geometry usually converges to a cylindrical construction.
Mass loads (gravity, centrifugal)
W
Fig. 1. Loads on a wind turbine rotor blade
Fig. 2. Characteristic cross-sections
Materials & Manufacturing All contemporary wind turbine rotor blades used on megawatt-sized turbines for electricity generation are almost completely made of fibre reinforced composite materials (the hub construction and attachment of the blades is usually manufactured in steels). The more production-friendly epoxy resins are increasingly replacing polyester resins. The most commonly used fibre material is E-glass. Because of the increased self-induced mass loading of the blades due to the increasing size, the implementation of lighter and stiffer
Alternative Fatigue Formulations for Variable Amplitude Loading
565
fibre materials is investigated. The first carbon-fibre blades are in their development stage today. Automated production technologies are slowly emerging. Where possible, some substructures are manufactured by automated prepreg-winding rather than labour intensive hand-layup methods. Numerous material coupon test results have been collected over the years by researchers in the field of composites. To the best knowledge of the authors, there are two public databases containing test results specific to wind turbine composites. These are the TACT'-database [3] and the database composed by the U.S. Department of Energy in collaboration with Montana State University (D0E/MSU)[4]. This paper illustrates results of lifetime prediction methods using an sample data set of unnotched fatigue specimens and is not specifically concerned with the complete structure or structural details. Although research into fatigue behaviour of virgin composite material seems remote from the ability to predict fatigue behaviour of full-scale rotor blade constructions, recent work by van Leeuwen et. al. [5] suggests that probability distributions of both rotor blade- and material failure in fatigue are similar. Also, Mandell et al. [6] compared the fatigue behaviour of a composite I-beam to coupons of the materials used in the I-beam and concluded, that no major difference was evident in the results. Nevertheless, the dominant failure mechanism in a construction may be different from that in laboratory material specimens, e.g. due to failure of connections or production errors, so one or more rotor blade(s) or substructure(s) should always be subjected to a full scale test. FATIGUE OF FIBRE REINFORCED COMPOSITES Over several decades, researchers have attempted to predict life and strength in composites subjected to fatigue. No reliable prediction models for fatigue life and residual strength in composite structures have been developed yet. Some of the most important difficulties are illustrated in a paper by Owen and Howe [7], who state that fatigue life prediction model classification can be based on assumed relationships between damage and cycle ratio {r\/N later in this paper), and distinguish linear- vs. non-linear damage accumulation, stressamplitude-dependency (no mention is made of the influence of mean stress in the cited reference), and stress interaction (damage development influenced by transition between stress amplitude levels). Numerous experimental efforts concentrating on detailing any of these three relationships are often frustrated by the variability in material properties, as well as lack of and scatter in test results. There are many approaches to the modelling of cumulative fatigue damage, as Fatemi and Yang note in their recent extensive survey of models for homogeneous materials [8]. Some of these models were later adopted for composites. As noted by Curtis [9], the development of models is classically divided into two camps: phenomenological modelling, i.e. (semi-) empirical modelling, and mechanistic models [9]. Due to the scarcity of consistent experimental investigations of the fatigue damage mechanisms, most existing damage accumulation theories are (semi-) empirical. Over the years, many studies have focused on collecting the appropriate information to feed and develop these semi-empirical models. Extensive reviews and useful overviews are provided by Yang and Du [10], and Hwang and Han [11]. Conventional fracture mechanics methods are unpractical for application to composite materials, or invalid because the damage area is large with respect to crack length and other specimen dimensions. Moreover, in related experiments, the interaction of various damage modes complicates the characterisation of each singular damage mode by parameters such as strain energy release rate [9]. Usually, several failure mechanisms are distributed among many (unknown) locations in the composite, and failure mechanisms may interact, creating a complicated damage accumulation or strength reduction in the material. Predicting the
566
R.P.L. NIJSSEN AND D.KV. VAN DELFT
onset and growth of damage from a seemingly undamaged initial state using fracture mechanics has therefore not yet reached a universal solution, see e.g. Lindhagen et al. [12]. However, implementation of accurate models of the failure mechanisms in combination with some statistical description of the failure locations may be facilitated in the near fui ure. Pending the development of appropriate mechanistic theoretical modeling, semi-empiiical models derived from dedicated constant amplitude and variable amplitude tests are the itiost likely to result in engineering design rules. LIFETIME PREDICTION METHODS As has been indicated in the previous chapter, quite sophisticated lifetime prediction algorithms are continuously being developed. Unfortunately, no appropriate validation has been performed for any universally applicable theory. Therefore, current guidelines for lifetime prediction methods for variable amplitude loading of composites lag the intricate modeling efforts described in the literature, and resort to the classical and well-known method. This is referred to as the baseline method. Baseline Method This method, which is outlined in Fig. 3, together with results for a sample data set to be discussed later, is relatively simple to perform, and is based on the principle of the Palmgren-Miner summation of a Rainflow counted signal. Classically, a degree of conservatism is attributed to calculations using the Palmgren-Miner summation, but there are sources that indicate that this is not necessarily the case. In the following experimental work is quoted which demonstrates that for wind turbine composites non-conservative calculations can be made, but this depends also on the other components of the calculation. The interrelation of the blocks constituting the baseline method will be discussed in more detail and some alternatives will be discussed. Rainflow count
Linear Goodman diagram
1
Miner summation •
Lifetime prediction
0.6
^
0.4 0.2 I—prediction WISPER X WISPER test 0
10
t •
--prediction WISPERX • WISPERX test
100
1000
no. of sequences
10000
Fig. 3. Baseline method for fatigue calculations, and results for WISPER(X), compared to fatigue test results for an sample data set of glass-polyester specimens [14]
ALTERNATIVE LIFETIME PREDICTION FORMULATIONS There are several ways to improve lifetime prediction formulations. In order to obtain a realistic lifetime prediction, many researchers concur that it is of paramount importance to base the models on the fatigue mechanisms present in the (damaged) material [13]. However, the theoretical modeling efforts have not reached a satisfactory calculation method yet. The method applied in this paper is to maintain the baseline formulations with minor adaptations in such a way that the predictions improve. Although there are no additional
Alternative Fatigue Formulations for Variable Amplitude Loading
567
mechanistical fundaments to this approach, the simplicity of the method is preserved and within a small range of situations, the effectivity is acceptable. Each of the blocks in Fig. 3 can be modified in order to change the lifetime prediction. In the following, examples of some possible modifications are discussed in order to show the sensitivity of the prediction method to the modifications. Experimental data from a sample data set (generated by van Delft et al. [14]) of coupons subjected to the WISPER and WISPERX standardized load spectra are compared to predictions using the baseline method and the modified methods. WISPER and WISPERX are standardized test loading sequences for wind turbine rotor blades, more or less representative for a two-month flap-loading signal in the vicinity of the blade root. They were derived from normal strain measurements on the upwind side of the blade root of different turbines. Note, that the standardized sequences are by no means intended for use in design calculations. WISPERX is a version of WISPER where the small amplitude cycles have been extracted to save testing time. In Fig. 4, the size of the bubble indicates the frequency of occurrence of each cycle type. 'X'es were added to show the locations of bubbles which are too small to be visible. The x-es above the dotted line refer to both WISPER- and WISPERX-cycles, whereas the x-es below the dotted line only show the location of WISPER-cycles as a consequence of the elimination of small amplitude cycles. The sequences contain some 130,000 and 13,000 cycles, respectively, but were intended to cause the same fatigue damage if applied to the same specimens. The majority of cycles in WISPER(X) is in the region of 0.1
O
WISPER
%
WISPERX
X
location of WISPER(X) cycles
R-average for WISPER
mean stress [levels]
Fig. 4. WISPER and WISPERX For a detailed description of the spectra, the reader is referred to ten Have et al. [15,16,17]. In the following, the WISPER and WISPERX-spectra are used to compare different modifications of the baseline method. Fig. 3 gives the prediction using the baseline prediction method vs. coupon test data. The prediction is highly non-conservative and fails to predict the observed difference between WISPER and WISPERX coupon test data. Note that this difference was unintended by the developers and unexpected. The large extra damage caused by WISPER in comparison to WISPERX may be caused by the small amplitude cycles in WISPER. Schutz and Gerharz already mentioned this possible detrimental effect of small amplitude cycles in the presence of larger ones [18]. Modification of the Damage Accumulation Rule The Palmgren-Miner rule is the most commonly used method of accounting for damage accumulation:
568
R.P.L. NIJSSEN AND D.R.V. VAN DELFT
D=y^
(1)
^N,
where / indicates cycle type (combination of mean stress and amplitude), n, is the number of cycles of type / in one sequence and A^, is the number of cycles of type / that can be expected to lead to failure in a constant-amplitude load. A^, is taken from the SN-curve (described later). In some references, n/N is called 'cycle ratio' [7]. D is the damage number. A construction can be expected to fail when the total damage reaches a value of 1. The guidelines prescribe the Palmgren-Miner rule for wind turbine fatigue calculations [19]. The most important drawbacks of the baseline method are, that it does not allow for poss: ble non-linearity of damage accumulation, or for any sequence effects. Coupon tests show, ihat Palmgren-Miner's sum at failure may vary strongly. Various sources confirm that PalmgrenMiner calculations may lead to either conservative or non-conservative predictions (see e.g. Yang and Du [10]). Others [20] note a predominant trend towards non-conservatism. For block loading tests, PSlmgren-Miner's sum at failure may be assumed to be as small as 0.1. (Mandell et al. [21]). Fig. 5 shows a prediction using this value for Palmgren-Miner's sum.
--prediction WISPERX • WISPERX test
10
100
1000
no. of sequences
Fig.
10000
5. Lifetime prediction using Palmgren-Miner sum = 0.1
A similar way of modifying Palmgren-Miner's rule is by adding empirical constants, e.g. rewriting Palmgren-Miner's rule as was proposed by Owen and Howe [7]:
/ \^^ D=i: A n-,+ B- BL Ni
[w,)
\
(2)
Bond [22] used this expression and an iterative procedure to find the best values of A, B, and C. An example of a similar prediction for the current dataset is shown in Fig. 6. Although this procedure can result in an accurate match with experimental results for a certain combination of material and test conditions, and the method is based on the same Rainflow counted signal as was used for equation 1, there are some deficiencies. The prediction result is not very sensitive to the values of some of the constants. This means that for different situations, the values of the constants may be very different, as was indeed seen in the iteratively determined values in Bond's work. Originally, equation 2 was intended by Owen and Howe to describe damage in terms of normalized resin cracking. When the abovementioned curve fit procedure is used, all physical meaning to the parameters is lost. In equation 2 for different cycle types the parameters are assumed constant, but they may very well depend on e.g. R-value.
569
Alternative Fatigue Formulations for Variable Amplitude Loading
Equation 2 is not to be confused with residual strength approaches to damage accumulation models. In the above equation, for each cycle type, the damage is assumed to start growing from zero damage. This neglects the fact that, when transitioning between cycle types, the damage incurred by the previous cycle type should be the starting point for the damage evolution caused by the next cycle type.
10
100
1000
no. of sequences
10000
Fig. 6. Results of prediction using alternative Miner summation for WISPER (D=Z(3.1-ni/Ni + 0.05-(ni/Nif ^^)). Finally, the mode of application of different forms of Palmgren-Miner's rule modifications or residual strength approaches is linked also to the counting algorithm involved, see also the next chapter. Modification of the counting algorithm The range-pair-range and Rainflow counting algorithms, as described by Matsuishi and Endo (1968) and de Jonge (1969) [23], were meant as methods to quantify the stress-strain hysteretic loops in metallic materials under fatigue. It is not certain that counting hysteretic loops in composite materials is fundamentally relevant for the fatigue damage quantification, but generally, these counting methods are recognised as being superior to other methods such as level crossing, peak-counting methods etc. The loading signal is described as a collection of load cycles (stress/strain loops-loops). Hereby, a cycle can be characterized in many ways. In the following, mean stress, stress amplitude and 'R-value' are used to distinguish different cycle types. The R-value is the ratio between minimum and maximum stress: (3) The 'range-pair-range' method yields the same results as the Rainflow count, but is easier to implement in a computer program. Several algorithms have been developed over the years, a survey of which has been given by Bishop and Sherrat [24]. According to them, all algorithms yield the same result as long as the signal is rearranged to start with the highest peak or the lowest trough. During the execution of all algorithms, a residual of uncounted load cycles accumulates. This residual usually has a shape similar to the fictitious residual depicted in Fig. 7. The Fig. 7. Typical shape for the residual of a cycles in this residual are counted once all Rainflow counted signal other cycles have passed through the algorithm. There are different algorithms for counting the residual, e.g. see also [25, 26, 27].
570
R.P.L NIJSSEN AND D.R.V. VAN DELFT
Counting methods and damage accumulation rules are interrelated. The baseline method as described above assumes that counting of the peaks and troughs in the signal precedes the Palmgren-Miner summation. This means, that the damage calculation can only be performed for a signal once the signal has been completely counted, i.e. both the signal itself and the residual. Contrarily, in some damage accumulation rules or residual strength degradation methods, the signal needs to be followed extreme-by-extreme, which means that cycles which in a normal Rainflow count first are allocated to the residual need to be counted ad-hoc [e.g. 20]. This calls for a different cycle counting method, yielding different results. Therefore, tare must be taken in comparing lifetime predictions from the baseline method to those using e.g. a Palmgren-Miner summation or a Residual Strength Degradation method. Not only the damage accumulation theory is different in this case, but often the counting method as wt 11. Fig. 8 shows the prediction using an alternative counting method, where only the maxima above 0 have been counted and assumed to have minima of zero, so as to get cycles v/ith R=0 only. An improvement in the accuracy of the prediction can be seen. For each type of cycle, the contribution to the total damage {n/N in equation 1) must be evaluated. After determining n,- from a cycle count, Nt must be found. For this, an SN-curve and a constant-life diagram (CLD) are needed. In the following, some possible modifications to the SN-formulation and the constant-life diagram are discussed. Modification of the SN-curve formulation The SN-curve is a line in the Qamp-N-plane (N is a measure of number of cycles to failure, S is a general symbol referring to stress, strain, load, displacement etc. In this case, 'S' denotes aamp). When constant-amplitude fatigue data are available for the material, an SNcurve can be found using regression analysis of the test data. Usually, it is appropriate to
--prediction WISPERX • WISPERX test
10
100
1000
no. of sequences
Fig.
8.
10000
Lifetime prediction using alternative cycle count
interpolate data using a straight line in an SN-diagram with logarithmical axes. This 'loglog-* or *power law-' SN-curve, is of the form: log{N) = a'log(7^^p + b
(4)
The constants a and b depend on the material and R-value. Usually, the SN-curve for R=-l can be assumed to cross N=l at Ultimate Tensile Strength (UTS). Then, the above formulation can be simplified to: N--
1
a
(5)
Alternative fatigue Formulations for Variable Amplitude Loading
571
For the rotor blade material under consideration (glass-fibre reinforced polyester), a representative value for a is -10. This log-log expression is prescribed in the guidelines if no test data for the material are available [19]. It has been shown by Rink et al. [28], that up to a certain number of cycles, interpolation of the experimental data from the example with a power law expression or an exponential expression is equally valid. In a previous study, several alternatives were given for the SNcurve [29]. An example of modified lifetime prediction results are depicted in Fig. 9. Here, formulations were used using an SN-curve which is steeper than the SN-curve found from tests at R=-l. Such alternative SN-curves may prove useful if they can be expressed as parametric SN-curves, where e.g. the slope parameter is a function of the material and/or spectral properties. 1
<2
•
s
0.8
H o . a - ^^"^--^^Sg^^.
Ht.«
1 0.4 0.2
—prediction WISPER
0 • 1
9.
-^^
N
^
S^C:J;;»~._ - -prediction WISPER)?*
X WISPER test
Fig.
[:^
10
• WISPERX test 100
1000
no. of sequences Lifetime prediction alternative SN-curve
10000
using
Strictly speaking, a lifetime prediction should be done with a specified failure probability and confidence level, see for instance Ronold and Echtermeyer [30], or Sutherland and Veers [31], but for the sake of simplicity in comparing the mean values of different models to the experimentally determined mean lifetime, these statistics and the associated myriad of possibilities for the choice of distribution types and statistical methods have been omitted. Moreover, for the comparison of models, comparing the mean of the data is sufficient. Modification of the Constant-life Diagram (CLD) The relation between the SN-curve and the constant-life diagram is depicted in Fig. 10. The experimentally determined SN-curve for a certain R-value can be used to estimate the fatigue life at another R-value. This is done by defining a constant-life interpolation. This means that from the experimental SN-curves lines of constant life are drawn in a constantlife diagram (mean stress or strain vs. stress- or strain amplitude, see Fig. 10). A constantlife diagram (CLD) consists of a set of lines that connect points in the amean-cJamp-plane with the same expected number of cycles to failure. In the baseline method, the CLD is approximated by a linear Goodman diagram, relating the expected lifetime for any possible load cycle type to experiments for R=-l. In the Goodman diagram, points on the Gamp-axis (R=-l axis, amean=0) are connected to UTS on the right-hand-side and Ultimate Compressive Strength (UCS) on the left-hand-side, respectively. The constant-life diagram A ^amp
Increasing N
Fig. 10.
Baseline Goodman diagram and relation between CLD and SN-curve
572
KP.L NIJSSEN AND D.R.V. VAN DELFT
is bounded by the N=l line and all constant-life lines emanate from UTS on the amean-'ixis (for negative mean stresses, the lines originate from UCS). The points on the Gamp-axis are found using the SN-curve for R=-l. Note that in this case, modification of the SN-curve interpolation only concerns modification of the SN-curves, which were obtained by experiments. However, as depicted in Fig. 10, the SN-curve and CLD are closely interrelated, so a change in SN-curve also significates a change in constant-life diagram. In the case where equation 5 is valid, the number of cycles to failure for a cycle type characterized by Gmean and Qamp can then be calculated by:
(6)
A/ =
(^
1,
UTS
'UTS' in the amean-term is replaced with '-UCS' for negative mean stresses (UTS and UCS positive). The other UTS-term remains, even for negative QmeanAs was indicated, modification of the CLD is related to modification of the SN-curves. With a modification of the CLD, a modification is meant which does not involve any changes to the experimental SN-curves, which were used to construct the CLD. On the other hand, any SN-curves that are derived from the interpolation can be affected by the CLD-modification. In order to allocate more damage to the small amplitude cycles, some alternative CLDs have been proposed previously [32]. For example, in one alternative CLD the N=l line is the same as in the linear Goodman diagram, but all other lines are drawn parallel to the N=l line. In this case, the permissible number of cycles at a certain stress level and mean stress can be found for a lin-log SN-curve using: log{N) = c-a^^ -ofM
(6)
UTS
Similar to equation 6, replace 'UTS' with '-UCS' to evaluate cycles with a negative mean stress (UCS = positive number). A similar formulation may be used for a log-log SN-curve. The expression for this reads: log(N) = a • log\
UTS
+d
(7)
For the values of the constants a, h, c and d for the rotor blade material of the specimens for which the calculations were performed, the reader is referred to [32]. Fig. 11 shows a prediction according to equation 7. 1 -1 —prediction WISPER 0.8 -|
--prediction WISPERX
X WISPER test
• WISPERX test
X>«<MH
1
Fig. 11.
10
•
100
1000
no.of sequences
10000
Prediction using parallel lines in CLD.
Alternative Fatigue Formulations for Variable Amplitude Loading
573
The prediction using parallel lines is far too conservative, but a possible feature of this type of modification emerges from the figure: employing parallel lines may enable the fatigue formulations to sufficiently take into account the smaller cycles in the sequences, thus showing a clear distinction between predictions for WISPER and WISPERX (which was not seen in the baseline prediction). Extra knowledge on the material behaviour should facilitate better predictions, so the lifetime prediction should improve with the inclusion of R=0.1 data. Two alternative formulations that include the R=0.1 data were investigated. The two formulations are the same in the region between R=-l and R=0.1 (straight lines between corresponding points on the projected SN-curves). Beyond R=0.1, two versions of an interpolation were used, the first reminiscent of the Goodman diagram, the second similar to the parallel lines-alternative. These formulations and the corresponding prediction results are displayed for log-log SN-curves in Fig. 12. 1 1
Cfamp
0.8
Poe
SKX
• " * * * < ^ j ^
5
X )0<
XX
J 0.4 —prediction WISPER X WISPER test
--prediclion WISPERX
0.2
• WISPERX test
CJmean
• • • • • * * * ^ , ^ ^ ^ ^ * * * * - « » . ^
—prediction WISPER
--prediction WISPERX
X WISPER test
• WISPERX test
0 10
100
no. of sequences
1000
10
^
100
no. of sequences
1000
Fig. 12: Prediction with parallel lines in CLD and R=0.1 data (left) and prediction including R=0.1 data, with goodman type for R>0.1. Both for log-log SN-curve definition Not surprisingly, the lifetime predictions using the parallel lines yield more conservative predictions than those using the Goodman type interpolation. From Fig. 12 it seems, that the conservatism and ability to discriminate between WISPER and WISPERX needed relative to the 'classical' (i.e. employing Goodman-lines) fatigue formulations may be found in a constant-life formulation employing parallel lines, whereas the accuracy of the prediction is enhanced by including R=0.1 data. CONCLUDING REMARKS Since no satisfactory theory has been developed to describe the fatigue behaviour of composite materials as a function of (micro) mechanical material properties, and as the engineering practice calls for less complex and more easily applied design rules, means of modifying lifetime prediction formulations to improve the predictions have been investigated. It is shown by an example that there are several ways to modify the most commonly used lifetime prediction method, in order to improve its predicting capabilities. These modifications provide an easy and efficient way to perform lifetime predictions. The drawback is, that the modifications have little or no validated physical background. Any correlation between model parameters and macroscopical material properties or damage development characteristics should be established by extensive test programs. This constrains the applicability of these methods to limited situations. The ultimate success on the road to accurate, reliable and realistic lifetime prediction methods for composite materials may be found in damage mechanics. However, more experimental efforts are necessary to develop and validate the micro mechanical theory.
574
KP.L NIJSSENAND D.R. V. VAN DELFT
Pending the development of appropriate mechanistic theoretical modeling, semi-empirical models derived from dedicated constant amplitude and variable amplitude tests are the most likely to lead to engineering design rules. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.
Mandell, J.F., Reed, R.M., Samborsky, D.D. (1992), Sandia National Laboratories, report nr. 92-7005 Timmer, W.A., van Rooij, R.PJ.O.M. (1996), IvW rep. nr. IW-96102R, Delft UT de Smet, B.J., Bach, P.W. (August 1994), ECN-C—94-045 Mandell, J. F., Samborsky, D. D. (1997), rep. no. SAND97-3002, Sandia National Laboratories van Leeuwen, J.L., van Delft, D.R.V., et al. (2002), proc. ASME/AL\A Wind Energy Symp., Reno, NV Mandell, J.F., Samborsky, D. D., Sutheriand, H. J. (1999), proc. EWEC Owen, M.J., Howe, R.J. (1972), J. Phys. D: Appl. Phys., Vol. 5, pp.1637-1649 Fatemi, A., Yang, L. (1998), Int. J. Fatigue, Vol 20 (1), pp. 9-34 Curtis, P.T., Davies, A.J. (June 2000), pi. lect. ECCM9, Brighton, UK Yang, J.N., Du, S. (1983), J. Comp. Mat., 17, pp. 511-526 Hwang, W., Han, K.S. (March 1986), J. Comp. Mat., Vol. 20, pp 125-153 Lindhagen, J.E., Gamstedt, E.K. Berglund, L.A. (2000), Comp. Sc. Tech., 60, pp. 2883-2894 Gamstedt, E. K., 1997, Ph.D.Thesis, Lulea University of Technology van Delft, D.R.V., de Winkel, G.D., Joosse, P.A. (1997), proc. ASME/AIAA Wind Energy Symp., Reno, NV ten Have, A.A. (1992), NLR TP 91476 U, NLR, the Netheriands ten Have, A.A. (1993), NLR TP 92410 U, NLR, the Netheriands, proc. ASME/AIAA Wind Energy Symp., Houson, TX ten Have, A.A. (1988), proc. EWEC, pp. 448-452 Schutz, D., Gerharz, J.J. (1977), Composites 8,4, pp. 245-250 Germanischer Lloyd (1993) Rules and Regulations, IV - Non-Marine Technology, Part 1 Wind Energy, Hamburg, Germany Wahl, N.K. (2001), Ph.D. thesis, Montana State University Kensche, C.W. (ed.) (1996), Eur. Comm., ISBN 92-827-4361-6, Brussels Bond, LP. (1999), Comp. A, 30, pp. 961-970 de Jonge, J.B (November 1983), subch. 3.4 of AGARDograph no. 292 (ed. F. Liard), ISBN 92-83-0341-4 Bishop, N.W.M. and Sherrat, F. (March 1989), Environment. Eng., 2 (1), pp. 1M4 Downing, S.D., Socie, D.F. (January 1982), Int. J. Fatigue, 4, pp. 31-40 Amzallag, C, Gerey, J.P., Robert, J.L., Bahuaud, J. (June 1994), Fatigue, Vol. 16, pp. 287-293 Glinka, G., Kam, J.C.P. (1987), Int. J. Fatigue, 9, No. 3, pp. 223-228 Rink, H.D., van Delft, D.R.V., de Winkel, G.D. (September 1995), Stevin report 694-30, Delft UT Nijssen, R.P.L., van Delft, D.R.V. (2001), proc. SAMPE Conf., Seattle, WA, ISBN 0-938994-91-3 Ronold, K.O., Echtermeyer, A.T. (1996), Comp. part A, 27A, pp. 485-491 Sutheriand, H. J., Veers, Paul S. (2000), proc. ASME/AIAA Wind Energy Symp., Reno, NV Nijssen, R.P.L., van Delft, D.R.V. (Jan. 2002), proc. ASME/AIAA Wind Energy Symp., Reno, NV
575
AUTHOR INDEX
Akkerman, R. 465 Amis, E.J. 365 Amundsen, K. 421
Ivankovic, A.
317
Janssen, M. Bakker, A. 115 Basu, S. 155 Becker, W. 539 Beguelin, Ph. 129 Billon, N. 65 Blackman, B.R.K. 293, 433,
479
Bomal, Y. 39 Brule, B. 15 Brunner, A.J. 433, 479, 503 Burgel, A. 175 Carianni, G. 103 Cartie, D.D.R. 503 Chateauminois, A. 51 Chevalier, J. 27 Chiang, M.Y.M. 365 Chilese, C.G. 221 Davies, R 279 Dijkstra, K. 115 Dubourg, M.C. 51 Eligabe, G. 265 Estevez, R. 155, 27 Fara, S. 387 Fiedler, B. 421 Frassine, R. 103 Frontini, R 265 Georgiou, I. 317 Ghanem, A. 3 Goldberg, A. 3 Gomina, M. 399 Gordillo, A. 77 Grein, C. 129 Hagglund,R. 355 Halary, J.L. 15 Hazra, S. 167 He, J. 365 Hojo, M. 421 Horikawa, N. 329 Horsfall, I. 221
143, 199, 253,
115
Kao-Walter, S. 355 Karac, A. 253 Karim, A. 365 Kashtalyan, M. 455 Kausch, H.-H. 129 Kinloch, A.J. 293,317 Kobayashi, T. 175 Kuipers, N.B. 115 Kusaka, T. 329 Lang, R.W. 187 Lange, R.F.M. 115 Lee, J.J. 373 Lee, W-S. 231 Leevers, R 167 Lellep, J. 549 Lin, H-L. 231 Lindemann, J. 539 Liu, H.-Y. 491 Mai, Y.-M. 491 Major, Z. 187 Mariani, R 103 Marissen, R. 115 Martinez, A.B. 77 Maspoch, M.Ll. 77 Matsuda, S. 421 Meyer, J.-R 65 Michel, B. 241 Monnerie, L. 15 Moore, D.R. 341 Moreau, R. 399 Nakache, E. 399 Namiki, H. 329 Nijssen, R.RL. 563 Ochiai, S. 421 Olagnon, C. 27 Orange, G. 39 Ortiz, M. 527 Paraschi, M. 293 Partridge, I.K. 503 Pavan, A. 387
Pegoretti, A. 89 Pettarin, V 265 Piggott, M.R. 445 Pinot, L. 399 Plummer, C.J.G. 3 Rager, A. 199 Reed, RE. 465 Ricco, T. 89 Riemslag, A.C. 115 Rink, M. 103 Rosakis, A.J. 527 Saad, N. 27 Sanchez, J.J. 77 Santana, 0 . 0 . 77 Sargent, J. 279 Shin, K.C. 373 Shockey, D.A. 175 Simon, F 305 Song, R. 365 Soutis, C. 455 Sprafke, R 241 Stable, R 355 Tamuzs, V 515 Tanaka, M. 421 Tarasovs, S. 515 Ting, S.K.M. 143 Tropsa, V 317 Valentin, G. 305 van Delft, D.R.V 563 Van der Giessen, E. 155 Wang, L. 167 Wamet, L.L. 465 Watanabe, M. 207 Watson, C.H. 221 Williams, J.G. 143, 199,
479
Wittier, 0. 241 Wu, W.L. 365 Yagi, H. 329 Yan, W. 491 Yu, C. 527 Zhang, W. 445
This Page Intentionally Left Blank
577
KEYWORD INDEX
ABS 231 Acoustic Emission (AE) 399, 503 Activation volume 231 Adhesion 341,365 Adhesive 317 Adhesive fracture toughness 341 Adhesive interface 329 Adhesive joints 293 Adiabatic 167 Alumina fiber 421 Aluminium foil 355 Analogical model 265 Angle ply laminates 445, 455 Bending and tensile type specimens 187 Bending force 265 Boundary discretization 539 Boundary finite element method 539 Bridging law 491, 515 Bridging stress 479 Brittle fracture 207 Calcium carbonate 39 Carbon fiber 421 Carbon fiber sheet 329 Cavitation 3, 39 Charpy 199 Co-cured lap joints 373 Co-poly(ester ester) 115 Cohesive elements 527 Cohesive failure 293, 305 Cohesive stress 167 Cohesive zone model 317 Combinatorial approach 365 Composite laminates 539 Composites 433, 455, 479, 563 Concrete structure 329 Constraint 143 Contact fatigue 51 Core-shell 65 Crack 355 Crack arrest 175, 503 Crack closure 199 Crack closure integral 241 Crack growth initiation 241 Cracking 231 Crack opening displacement 515 Crack problems 539 Crack propagation 549 Crack resistance 39 Crack wave interaction 207 Craze 143, 167 Crazing 27, 155 Creep 103, 115
Critical fibre angle 387 Cross-ply laminates 433 Damage 479 Debonding 39, 279 Decohesion 167 Delamination 479, 503, 515 Detergent 103 Double-Cantilever-Beam (DCB) 491 Drop impact 253 Drop tower 221 Ductile-brittle transition 129 Dynamic crack branching 207 Dynamic fracture 231 Dynamic fracture toughness 175 Dynamic modulus 221 Dynamic properties 221 Dynamic testing 221 Edge delamination 365 Elastic modulus 39 Energy release rate 199 Engineering polymers 187 Entanglements 15 Environmental Stress Cracking (ESC) 103, 115 Epoxy 51 Epoxy matrix 421 Essential Work of Fracture (EWF) 65, 77, 89, 253 Fatigue 3, 563 Fatigue characteristics 373 Fibre debonding 387 Fibre orientation 387 Fibre pullout 387 Filler particle 39 Finite elements 365, 527 Finite element simulation 241 Finite Volume Method (FVM) 199, 253, 317 Fixed arm peel 341 Fluid-structure interaction 253 Force-based analysis and dynamic data reduction 187 Fractographic observations 399 Fracture 15, 143 Fracture criterion 329 Fracture energy 221 Fracture envelope 279 Fracture mechanics 305, 341, 365, 465 Fracture mechanisms 387 Fracture of polymers 241 Fracture toughness 39, 89, 187, 221, 329, 355, 387, 399,445,455,515 Free-edge stresses 539 Fretting 51
578 Glass fibre 399 High strain rate 221 Hydrolysis 115 IM7/977-2 503 Image analysis 279, 399 Impact 167 Impact failure 527 Impact tests 199, 221 Impact wedge peel test 317 Injection mould 77 Instrumented impact tests 265 Inter-layer crack 527 Interfacial debonding 365 Interlaminar fracture toughness 421 Intra-layer crack 527 Intrinsic parameters 129 Inverse problem 265 iPP/EPR blends 129 J-integral 199, 399, 549 Laminated composite 465 Laminate free-edge effect 539 Laminates 341, 355 LEFM 129 Lifetime prediction 563 Liquid nitrogen 421 Manufacturing pressure 373 Mechanical tests 399 Metal substrates 341 Microcracking 293 Microdeformation 3 Mixed mode 329 Mode I delamination 491 Mode I delamination fracture 433 Mode II 293 Modifier 65 Molecular mass 103 Molecular mobility 15 New toughness test method 455 Nucleation 51 Numerical methods 539 Orientation 77 PET 77 PET/PC blends 77 Plane stress 89 Plastic containers 253 Plastic fracture 549 Plasticity 27, 155 Plastic zone 15 Plastic zone correction 129 Poly(butylene adipate) 115 Poly(butylene terephthalate) 115 Polyethylene 3, 103, 143, 167 Polyethylene pipes 175 Poly(ethylene terephthalate) 65, 89 Polymer fracture 27, 155 Polymers 341, 355 Polypropylene 39
Pressure vessels 175 Process zone 39 Propagation 51 Pull-out 491 R-curve 293,433,479 Rapid crack propagation 175 Rate 143 Rate dependence 187, 241 Relaxations 15 Residual stress 465 Retrofitting 329 Rubber-toughened thermoplastic 399 Sandwich structures 527 Semi-aromatic polymides 15 Shear loading 305 Short fibre composites 387 Silica 39 Slow crack growth 3 Stacking sequence 373 Stiffener 279 Strain-rate 231 Stress localization 539 Surface roughness 373 T-peel 341 TDCB 317 Tearing modulus 549 TEM 3 Tensile load bearing capacity 373 Ternary blends 399 Thermal effects 155 Thermomechanical reliability 241 Thermoplastic Elastomer (TPE) 115 Thin films 89, 365 3D composite 515 Three-point bending 265 Through-thickness reinforcement 491 Time-to-failure 115 Top-hat 279 Toughness 27, 51, 65 Toughness evaluation 129 Traction-separation 143, 253 Transverse cracking 465 Transverse matrix crack 539 Unstable crack propagation 503 Variable amplitude loading 563 Viscoelasticity 89, 241 Wood adhesive joints 305 Z-Fiber® 503 Z-pin blocks 503 Z-pinning 491
