Biaxial Multiaxial Fatigue and Fracture (European Structural Integrity Society)

.927 463

Fig. 19. Cut through the submodel, ay at bending

1.8SS 1.391

2.783 2 319

3.710 3.247

4.638 4.174

S.566 S.102

6.030

Fig. 20. Stresses CFy at bending and Ty^ at torsion along the weld root

Predicted lifetime is slightly longer for the old design. Now there are three regions with nearly identical damage sums. In addition to the failure-critical elements on the welding undercut on tube and forged arm (Fig. 21), there is another location with high damage at the weld root. This location is shown in the cut in Fig. 21. Table 3 contains calculated damage sums and blocks to failure for three multiaxial fatigue criteria, two load cases, interaction of load cases and load levels 1.0 and 1.4. Locations of failure have been determined to be the same for each criterion. Speculation that the weld root of the old geometry is failure-critical, too, is verified.

Evaluation of Fatigue of Fillet Welded Joints in Vehicle Components Under Multiaxial Service Loads 37

Intersection of weld root

complete submodel

Fig. 21. Calculated failure-critical locations, criterion maximum principal stress Table 3. Calculated fatigue lives with the local stress approach for the old design Multiaxial fatigue criterion

Load case

Interaction of load case 1 and 2 Maximum principal Load case 1 (bending) stress Load case 2 (torsion) Interaction of load case 1 and 2 Load case 1 (bending) Load case 2 (torsion) Interaction of load case 1 and 2 Critical plane Load case 1 (bending) normal stress (mode I) Load case 2 (torsion) Interaction of load case 1 and 2 Load case 1 (bending) Load case 2 (torsion) Interaction of load case 1 and 2 Critical plane - shear Load case 1 (bending) stress (modes n+ni) Load case 2 (torsion) Interaction of load case 1 and 2 Load case 1 (bending) Load case 2 (torsion)

Load level

Maximum damage sum

Calculated number of blocks to failure

1.0

0.00639 0.000414 3.75E-5

1565 2416 26667

1.4

0.00226 0.00146 0.000132

443 684 7551

1.0

0.000606 0.000414 3.75E-5

1650 2461 26667

1.4

0.00214 0.00146 0.000132

467 684 7551

1.0

0.000642 0.000352 9.89E-5

1558 2841 10111

1.4

0.00345 0.00189 0.000532

290 529 1880

38

G. SAVAIDIS ETAL

COMPARISON OF RESULTS CALCULATED WITH THE HOT SPOT STRESS APPROACH AND THE LOCAL STRESS APPROACH All fatigue results of the vehicle component as calculated and experimentally determined are plotted in Fig. 22. The Figure contains blocks to failure calculated with three different failure criteria (maximum principal normal stress, critical plane - normal stress (mode I), and critical plane - shear stress (modes n + HI)) in conjunction with the hot spot and the local stress approach. The experimental results are plotted as dots.

,40

8 5 76 4 3

^

1 \

^

•

experimental results |

\ \ \

0)

>

\ ,20

\

•

\

CO O

\ \ \ \

.;ee-

200

1

1

1

1

1

1

1

1

1

• -1

1

1

1000

r—r-|

1

10000

number of blocks to failure Dots: experiments, curves 1 to 8: calculations: 1: hot spot stress approach applied with critical plane shear stress criterion, 2: hot spot stress approach applied with critical plane normal stress criterion, 3: old geometry - local stress approach applied with critical plane normal stress criterion, 4: old geometry - local stress approach applied with max. principal normal stress criterion, 5: old geometry - local stress approach applied with critical plane shear stress criterion, 6: new geometry - local stress approach applied with critical plane normal stress criterion, 7: new geometry - local stress approach applied with mx. principal normal stress criterion, 8: new geometry - local stress approach applied with critical plane shear stress criterion. Fig. 22. Comparison between calculated and experimental fatigue lives.

The hot spot stress approach in conjunction with the criterion critical plane - normal stress shows a slight trend to conservative calculations at lower load levels. The use of the constant amplitude ^-A^ curve within the hot spot stress approach supported by the criterion critical plane shear stress seems to be unsuitable to calculate fatigue lives for this weld detail. Lifetimes predicted with the local stress approach are generally very conservative here. The predicted fatigue lives for the new geometry are slightly shorter than the ones calculated for the old geometry. However, the advantage of the new geometry compared to the old geometry is that the weld root is no longer failure-critical. Good agreement exists between the two criteria critical plane normal stress and maximum principal stress. But this was to be expected in a predominantly locally uniaxial situation.

Evaluation of Fatigue of Fillet Welded Joints in Vehicle Components Under Multiaxial Service Loads 39

Validation of calculated fatigue results Calculated fatigue lives or damage sums are influenced by several factors which are worth to be discussed in the following: As mentioned previously, the local stress approach is applied in practice in connection with the submodelling technique. Within the latter, the usual procedure is to take displacements (and rotations) from a coarse mesh as boundary conditions for the finely meshed submodel. Attention should be paid to the requirement that the stiffnesses of the models do not differ too much. For example, a very stiff submodel will yield much higher stresses under applied displacements than a compliant submodel would. In the current investigation we found differences in resulting bending and torsion moments in the tube in the order of 5%. The higher stresses occurred for the stiffer new design variant. This is the reason for the lower predicted lives for this design. Some inherent uncertainties are obviously linked with the submodelling technique itself. Calculated results also depend on the multiaxial fatigue criterion used. In general nonproportional loading cases, criteria based on conventional equivalent stresses (Tresca, von Mises) are inappropriate. In special cases with locally proportional stress situation, they might work to a limited extent as well. The hypothesis of the effective equivalent stress introduced by Sonsino and Klippers [21] showed good predictions for welded flange-tube joints from fine-grained steel FeE 460 under bending and torsion with constant and variable amplitudes. As mentioned above, these cases - dominating uniaxial stresses for example - have some practical relevance. In case of doubts on local stress states, critical plane criteria should be preferred: Critical plane - normal stress (mode I) or critical plane - shear stress (mode n+m). The first one should normally be used when normal stresses are dominant. The second criterion is appropriate in the case of dominating shear stresses. Scatter of fatigue lives is an unavoidable matter of fact. It should be taken into account when comparing calculated and experimentally determined lives. The number of tested components here is quite low; thus, even mean values of lives are subject to uncertainties. On the other hand, the baseline stress-life curves for prediction, Figs. 7 and 15, are only mean curves in a scatter band. Within the local stress approach, the ratio T for probabilities of survival of 10% to 90% (in stresses) is 1.5 for normal stresses and 1.39 for shear stresses, respectively [19, 20]. Thus, a factor of 2 to 3 in lives can easily arise from this fact and can be qualified as minor inaccuracies. At last, the real fatigue life mainly depends on the geometrical form and quality of a single weld. It is possible to model the real welded form or geometry from a drawing. In this report the geometry of already existing welds is modelled. However, the plane surface of the weld and the notch radius 1mm are idealisations. But geometry and stress distribution depend on each other. Therefore, the element with maximum damage sum is not the only one to be looked at. Other elements with similar damage sums should also be regarded as failure-critical as well. These failure-critical locations can be determined quite accurately using the local stress approach. Using the hot spot stress approach, only one critical location on the tube has been detected. This location is verified and one more location has been detected with the local stress approach. The corresponding finite element results for the older geometry verified the weld root as failure-critical. This is in good agreement to the experimental results. After modification of the tube to the new geometry the weld root is no longer failure-critical. Figure 23 shows a tested component with new geometry. A further effect is shown: The crack initiation does not start from the weld at all. The notch at the tapering of the tube far away from the weld undercut is failure-critical. Finally, this result prevented the presentation of experimental fatigue lives for the new design (for the weld undercut). On the one hand, it is possible to calculate lives for this location based on local stresses or strains and on cyclic

40

a SAVAIDIS ETAL

material data, but this is not the subject of the current paper. On the other hand the result reveals some aspects of the problems when dealing with life prediction in an industrial environment: As the tapered tube is known from experience to survive quite a number of truck lives further investigations on the component have minor priority. Nevertheless, gaining experience with life prediction techniques is the prerequisite to apply them in design process.

Tapering of tube Forged arm

Failure-critical location calculated with hot spot approach and local approach

Fig. 23. Experimental tested new geometry, crack initiation at tapering of tube

CONCLUSIONS The mechanical behaviour and fatigue life of a thin-walled tube joined to a forged component by fillet welding has been investigated. The component is loaded by nonproportional random sequences of bending and torsion as measured during operation. The stresses in the welded structure have been calculated using finite element analysis. The fatigue life has been determined theoretically, by means of the hot spot and local stresses in conjunction with the critical plane approach, and experimentally. Though the experimental data base is quite narrow, basic trends can be derived from the present investigation concerning the possible field of application and capability of the theoretical approaches calculating fatigue life of vehicle components. 1. A coarse finite element model of the component has been created in accordance with the n w guideline for application of the hot spot stress approach. Comparison of numerically and experimentally determined end-results (fatigue lifetimes) shows that the coarse model is suitable to determine lifetimes with the hot spot stress approach, if a reliable hot spot stress-life curve at constant amplitude loading is existing for the detail investigated. Because of the coarse mesh, the geometry of the weld is not modelled in detail. For instance, in this investigation only one failure-critical location has been detected. 2. If experimental results are missing, appropriate S-N curves from publications, e.g. the nW guideline or Eurocode can be used. In cases of multiaxial loading causing normal and shear stresses, attention must be paid to the slope of the S-N curve used, since various suggestions are reported in the literature. The major advantage of the hot spot stress approach is a relatively low expense to model and calculate.

Evaluation of Fatigue of Fillet Welded Joints in Vehicle Components Under Multiaxial Service Loads 41

3. The coarse finite element model is unsuitable for usage in conjunction with the local stress approach. Therefore, a submodel of the failure-critical detail has been created here. Due to the finer mesh, the number of elements is increasing. The results obtained here show that weld geometry optimisation is only possible with the local stress approach. 4. In general, higher numerical expense does only make sense, if failure-critical locations should be investigated more exactly. In the majority of engineering applications, a finer mesh or more detailed model is often an unrealisable option due to technical and commercial reasons. 5. To calculate fatigue lives in accordance with the local stress approach, no experimental input data are required, when using the universal local a-N curve.

REFERENCES 1. Hobbacher, A. (1996). Recommendations for Fatigue Design of Welded Joints and Components. Document Xm-1539-96/XV-845-96, International Institute of Welding, Paris. 2. Radaj, D. (1990). Design and Analysis of Fatigue Resistant Welded Structures. Abingdon Publishing Cambridge. 3. Radaj, D. and Sonsino, C M . (1998). Fatigue Assessment of Welded Joints by Local Approaches. Woodhead Publishing Limited, Cambrigde. 4. Maddox, S.J. (1991). Fatigue strength of welded structures, 2""^ edition, ISBN 1 85573 0138, Woodhead Publishing. 5. Bo vet-Griff on, M., Ehrstrom, J.C., Courbiere, M., Bignonnet, A., Thomas, J.J., Puchois, J.P., Rethery, S. and Liennard, C. (2001). Fatigue assessment of welded automotive aluminium components using the hot spot approach. Proc. 8^ INALCO, 28-30.03.2001, Munich. 6. Fayard, J.L, Bignonnet, A. and Dang Van, K. (1996): Fatigue design criterion for welded structures. Fat. Fract. Engng. Mater. Struct. 19, 723. 7. Savaidis, G. and Vormwald, M. (2000). Hot-spot stress evaluation of fatigue in welded structural connections supported by finite element analysis. Int. J. Fatigue 11, 85. 8. Niemi, E.J. (1995). Recommendations Concerning Stress Determination for Fatigue Analysis of Welded Components. Document Xm-1458-92/XV-797-92, International Institute of Welding, Abingdon Publishing, Cambridge. 9. Niemi, E.J. (2001). Structural Stress Approach to Fatigue Analysis of Welded Components. Document XHI-1819-00/XV-1090-01/Xm-WG3-06-99. International Institute of Welding, Abingdon Publishing, Cambridge. 10. Vormwald, M., Purkert, G. and Schliebner, R. (2000). Lebensdauerbewertung einer NFGFahrerhausschwinge mit dem Programm FALANCS. Technical report, Weimar. 11. Sonsino, C M . (1994). Festigkeitsverhalten von SchweiBverbindungen unter kombinierten phasengleichen und phasenverschobenen mehrachsigen Beanspruchungen. Materialwissenschaft und Werkstofftechnik 25, 353. 12. Sonsino, C M . (1995). Multiaxial Fatigue of Welded Joints Under In-Phase and Out-ofPhase Local Strains and Stresses. Int. J. Fatigue 17, 55. 13. N.N.: FALANCS. Version 2.9j, LMS Durability Technologies GmbH, Kaiserslautern. 14. Fatemi, A. and Socie D.F. (1988). A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fat. Fract. Engng. Mater. Struct. 11, 149. 15. Fatemi, A. and Kurath, P. (1988). Multiaxial fatigue life predictions under the influence of mean stresses. J. Engng Mat. Techn. 110, 380. 16. Maddox, S.J. and Ramzjoo, G.R. (2001). Interim fatigue design recommendations for fillet welded joints under complex loading. Fat. Fract. Engng. Mater. Struct. 24, 329.

42

a SAVAIDIS ETAL

17. Sonsino, C M . (1999). Overview of the State of the Art on Multiaxial Fatigue of Welds. ESIS, Publication 25, Elsevier. 18. Haibach, E. (1989). Betriebsfestigkeit. Verfahren und Daten zur Bauteilberechnung. VDIVerlag, Dtisseldorf. 19. Olivier, R. (2000). Experimentelle und numerische Untersuchungen zur Bemessung schwingbeanspruchter Schweifiverbindungen auf der Grundlage des ortlichen Konzeptes. Phd-thesis, TU-Darmstadt. 20. Olivier, R. and Amstutz, H. (2000). Fatigue strength of shear loaded welded joints according to the local concept. Materialwissenschaft und Werkstofftechnik 32. 287. 21. Sonsino, C M . and Kueppers M. (2001). Multiaxial fatigue of welded joints under constant and variable amplitude loadings. Fat. Fract. Engng. Mater. Struct. 24, 309.

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

43

MULTIAXIAL FATIGUE ASSESSMENT OF WELDED STRUCTURES BY LOCAL APPROACH

Florence LABESSE-JIED\ Bruno LEBRUN\ Eric PETITPAS^ and Jean-Louis ROBERT^ LERMES, Blaise Pascal University, lUT, B.P. 2235, 03101 Montlugon Cedex-France GIATIndustries, CRET/MOD, 7, Route de Guerry, 18023 Bourges-France

ABSTRACT The moving of vehicles on chaotic ground induces dynamic multiaxial loading on structures and mechanical components. As a consequence, early fatigue damage occurs especially in structural details such as notched areas and welded parts. A multiscale approach has been developed to design the structures against fatigue, starting from the dynamics of the vehicle and ending with the calculation of structural details using a local approach to assess the fatigue life. The methodology of the local approach developed is introduced. The evaluation of the prediction capability of this local approach is described. Finally, the application to the fatigue life assessment of welded elements is presented and compared to experimental results. The major parameters of the weld geometry that govern the material resistance against fatigue are pointed out. They concern geometrical features depending on the quality of the weld. Their influence on the weld durability is outlined and the way the proposed assessment method accounts for them in a quantitative manner is detailed. KEYWORDS Welding, fatigue life, local approach, multiaxial loading, damage cumulation

INTRODUCTION Welding is a technology commonly used nowadays in many applications of the mechanical industry because it allows the making of complex structures from simple components as bars or plates. Engineering designers and researchers have thus developed different evaluation techniques for ensuring fatigue resistance of such welded structures [1]. Structural details and welded joints are assessed from the fatigue point of view in design codes principally on the basis of the nominal stress range. Practically they are classified into many different classes depending on the geometry and the applied loading. A fatigue strength curve is attributed to each class, the denomination of which corresponds to the characteristic fatigue strength at 5.10^ cycles [2]. Most design codes allocated to fatigue assessment of welded structures refer to this principle. Figure 1 depicts the conventional Wohler S-N curves generally consulted as referenced standard quality fatigue properties. In the case where the nominal stress can not be easily defined within the fatigue-prone welded structure, the proof against fatigue uses either hot-spot or structural stresses [1,3]. The

44

E LABESSE-JIED ET AL

hot-spot stress is obtained by the hnear extrapolation of the stress distribution to the weld toe (Fig. 2). The location of this considered stress is due to the fact that it is generally the critical point of the weld, i.e. the crack initiation site. The structural stress is the measured or Finite Element calculated stress obtained at a given distance afar from the weld toe. Because of the stress concentration generated by the geometrical discontinuity at the weld toe and directly related to the transition radius, this structural stress is in general higher than the hot-spot one and leads to a more conservative dimensioning.

Constant amplitude fatigue limit

1e7

logN

Fig. 1. Wohler S-N curves for classified weld geometries (from [2])

a, : structural stress Chi • hot-spot s^ess

a^:

f:

maw mum notch stress CT, - f (weld geometry, loading mode)

,cr, structural stress •distribution (strain gauge or FE)

notch stress distnbuton (F£) Stress-ccncerrtration factor: K, =

/^' linear extrapotation

weld-toe 1

^ = f (tNckness, gauge length) • 1 to 2 mm

Fig. 2. Conventional design stresses in a weld joint (from [1,3])

Multiaxial Fatigue Assessment of Welded Structures by Local Approach

45

The advantage of these approaches is the relatively easy procedure for a design purpose. However two main drawbacks can be retained. The effective maximum stress existing at the weld toe is not the referred stress used for the assessment of the weld despite the fact indeed that this value is the origin of the fatigue damage. The second difficulty consists in the uniaxial stress states used for evaluating the weld durability. As a matter of fact the notch effect constraints existing at the weld toe and the possibly multiaxial loading of the structure generally induce multiaxial stress states. In the case of rotating principal stress directions, a multiaxial fatigue criterion is the only way to properly account for the effective influence of such stresses in fatigue. PRESENTATION OF THE PROPOSED LOCAL APPROACH This approach is developed for the design of welded structures of all-terrain vehicles. When such a vehicle is moving on chaotic ground at high speed, its structure and mechanical components are subjected to vibrations capable of inducing fatigue cracks especially in notched areas and welded parts. This is the reason why many all-terrain vehicles built all over the world suffer from progressive cracking. Controlling the fatigue dimensioning of structures under complex loading becomes thus an essential condition in the design process. Furthermore, controlling the fatigue strength of welded parts constitutes an important technico-economic advantage. The aim of this section is to present the approach developed and its application in order to ascertain fatigue strength under complex loading. Since the fatigue phenomenon is very local, whereas the phenomenon generating it is vehicle-wide, the design approach must necessarily be multiscale. The approach developed results in 3 steps. Starting from the terrain, the first step consists in simulating or testing a vehicle when rolling, in order to make it possible to determine the internal loads. Figure 3 shows for instance the forces induced by the suspension on the chassis due to the moving of a 6-wheeled vehicle on a training ground at high speed. As a matter of fact, the ratio between forces coming from the vehicle's suspension shows that the loadings are not proportional. The structure is actually subjected to random non-proportional multiaxial loading. These loads are used, during the second step of the multiscale approach, for the overall finite element calculation of the structure (dynamic or quasi-static), resulting in the identification of the potential fatigue damage areas [4]. The third step of the multiscale approach consists in assessing the lifetime of the critical areas by the local approach.

5.E+05

^

l.E+05

-l.E+05 O.E+00

2.E+01

4.E+01

6.E4-01

8.E+01

l.E+02

Time (s)

Fig. 3. Force versus time in one suspension of a 6-wheeled vehicle

E LABESSE-JIED ET AL

46

The approach adopted for the local calculation of fatigue damage under multiaxial random loading is shown on the flowchart of Fig. 4. The calculation is performed on each weak point of the surface of the structure. The inputs are the nodal plane stresses coming from a dynamic calculation (stationary random or transient) or from a quasi-static calculation.

rii

time

NiR

Fig. 4. Calculation of the fatigue damage by the local approach The operational service stresses obtained through the latter analysis have to be corrected for the effects of stress concentration resulting from geometric singularity overlooked during the calculation. This may concern a fillet, a hole, or even the micro-geometry of the weld toe or the weld root. The correction is performed with a two-dimensional stress concentration factor matrix [K], the components Ky of which are identified from a local calculation of the structural detail. The local stress states are thus obtained from the nominal stress states as: [ci]local = [ K ] . [ a ] n

(1)

It is well recognised now that residual manufacturing stresses, combined with service stresses, play an important role from the fatigue point of view. These stresses are taken into account in the proposed approach in the form of initial stresses to which the service stresses are added. [c?]total - [Cljresidual + [^] local

(2)

Multiaxial Fatigue Assessment of Welded Structures by Local Approach

47

Stress states histories may present peaks overshooting the material yield stress. The plasticity resulting from these stress peaks modifies the material stress-strain response because of stress hardening. Hence, plasticity modifies the damage kinetics. The plastic correction is made with the hypothesis that the total strain remains constant. The elastic-plastic model corresponding to the Chaboche non-linear isotropic and kinematic hardening model is used [5]. Eqs (3), (4) and (5) describe the yield surface and the kinematic and isotropic stress hardening rules of this model.

f ( s , X , R ) = J(s-X)-R(p) = [ | ( s - X ) : ( s - X ) ] 2 - R ( p ) = 0

(3)

R(p) = Ro+Q.[l-exp(-y.p)]

(4)

dX = -Ca.d8P-C.X.dp

(5)

where ? is the deviatoric stress tensor, X and R(p) are the kinematic and isotropic parts of the stress hardening respectively. R(p) is divided into the initial yield stress Ro and the isotropic hardening which maximum value is Q. Q < 0 corresponds to a softening of the material whereas Q > 0 corresponds to a material hardening. The multiaxial fatigue criterion used is based over the critical plane concept. Such a criterion defines a so-called damage indicator related to any material plane (or facet). This indicator is generally a fiinction of the shear and normal components acting onto this facet. The principle is to search the most damaged plane, i.e. the critical plane. The assumption is made that this material plane drives the fatigue behaviour of the material as it is the first plane to experience a fatigue crack initiation. Bannantine and Socie [6] showed that fatigue cracks initiate from free surface on only 2 sets of facets, 90° or 45° inclined from the normal to the free surface (Fig. 5).

Fig. 5. Expected crack initiation planes (from [6])

The fatigue criterion is thus applied to those 2 groups of facets in order to find out the one that is submitted to the highest amount of fatigue damage. On each facet, the shear stress history is calculated; the rainflow procedure applied to one projection of that shear stress makes it possible to identify and extract the cycles from the stress states histories. For each cycle, the criterion is constructed by combining the shear stress amplitude and the maximum hydrostatic stress encountered during the cycle, that is:

48

F LABESSE-JIED ET AL.

a | , = M A X [ T a ( t ) + |3.aH(t)]

(6)

cycle Ta(t) = T ( t ) - T

(7)

The damage associated with each cycle is calculated by using Miner's linear rule and the material fatigue strength curve expressed in the form of Basquin's law [7]. d^=S-^-S

!—r

i Nf

(8)

The lifetime is calculated by maximising the damage over all the examined planes. In other words, it means that the critical plane enforces its fatigue life to the material.

d = MAX[d"J

(9)

N - d

(10)

EVALUATION OF THE METHOD O N SPECIMENS The local approach is assessed on cylindrical and tubular shaft specimens from experiments reported in the round-robin program performed by the SAE in the 80's [8]. The shafts are submitted to proportional or non-proportional strain-controlled tension and torsion loading. The strain amplitudes are constant or variable. The random spectra come from measurements made on log skidder and agriculture tractor axles and recorded as Markov matrix. Fatigue tests are performed on shaft specimens with different deterministic or random amplitudes until complete fracture. Calculations are carried out according to the local approach flowchart (Fig. 4). Starting from the deterministic or random strain cycles, the stress states are calculated by using a cyclic constitutive law with non-linear kinematical and isotropic hardening. Then, the damage corresponding to the cycles extracted by the rainflow method is calculated and cumulated. The life corresponding to each strain sequence is then calculated by using the most damaged facet. The shaft is made up of C45 carbon steel. The mechanical properties of this steel are summarised in Table 1.

Table 1. Monotonic and cycUc properties of C45 carbon steel (fi-om [8]) Gy (MPa) 280

K (MPa) 1185

n 0.23

a'y (MPa) 180

K' (MPa) 1258

n' 0.208

aV (MPa) 948

I'f (MPa) 505

b -0.092

Figure 6 gives the comparison between the experiments and calculated lives for all kinds of loading: deterministic uniaxial tension, proportional tension-torsion, non proportional tension-

Multiaxial Fatigue Assessment of Welded Structures by Local Approach

49

torsion and random uniaxial tension for 3 different random spectrums. In the case of random loading, the life is expressed as the number of blocks up to crack initiation (Nf). The results are plotted with two straight lines indicating a deviation interval band between Nf/3 and 3 Nf. This comparison makes it possible for the prediction to be assessed within an interval between Nf/4 and 4 Nf. lE+06

lE+00 lE+00

lE+01

lE+02

lE+03

lE+04

lE+05

lE+06

Nf experimental Fig. 6. Comparison of predicted lives against experimental ones

The highest deviations between predictions and experiments are obtained for nonproportional loading and random amplitude, hi the case of non-proportional loading, the most important part of the error comes from the fatigue criterion used, hi fact, several authors [9,10] showed that, for non-proportional loading, the critical plane criterion taking account of the maximum of the shear stress amplitude encountered during one cycle give inaccurate results. The average of the shear stress on the facet over the cycle or the integration of shear stresses over all planes may give better results. This concept developed in some multiaxial fatigue criteria is the so-called integral approach. Considerations on how to integrate such a criterion for random loading are necessary, hi the case of random tension loading, the calculation is always optimistic, because the accelerating influence of previous small amplitude cycles onto the damage rate due to large amplitude cycles is not taken into account. To improve the prediction, a more physical damage parameter such as cumulative micro-plasticity on each facet or micro-cracks nucleation on each facet should be used.

APPLICATION OF THE LOCAL APPROACH TO WELDED COMPONENTS This section details all the steps of the local approach procedure which are successively run on in order to assess the fatigue lifetime of a welded mechanical assembly. Welded joints are particular structural details characterised by: - Stress concentrations induced by local shape of joint and manufacturing geometrical defects produced by the welding process (angular deviation and misalignment of the connected metal components).

50

E LABESSE-JIED ET AL

- Metallurgical transformation resulting in local changes of the mechanical static and fatigue properties, - Residual stresses induced by a non-homogeneous cooling of the welded components. The purpose of the procedure included in the local approach software is to take these peculiarities into account in the fatigue calculation. The approach is local as the fatigue life prediction is made from the local stress states in the most critical welded area. Studies concerning the influence of these three peculiarities showed that the stress concentration factor is the most important parameter to be considered. Figure 7 shows the stress concentration on the weld toe or on the weld root of a butt joint when it is subjected to axial loading. The stress concentration can easily exceed a factor equal to 2.

-EEEEiiEEEE}

(a) Fig. 7. Stress distributions within a butt-welded joint: (a) with angular distortion; (b) without angular distortion. Geometrical defects modify also the stress distributions and have consequently to be taken into account. Frequently geometrical defects are the angular distortion of a butt-welded joint and the misalignment of a cruciform welded joint as shown in Fig. 8. The importance of the angular distortion is illustrated on Fig. 7; as it shows that the location of the weakest point of the weld moves from the weld root (b) to the weld toe (a) when the angular distortion is taken into account for the stress calculation. The strong modifications regarding the stress distributions are explained by a significantly different change of the applied loading. For example, an angular distortion often generates a static bending moment when the welded sample is installed and clamped in the jaws of the fatigue testing machine. This bending moment is due to the fact that the sample is straightened because of jaws' alignment. This load is induced in fact by boundary conditions and is superimposed to the fatigue loading. The stress concentration is represented by the matrix [K] associated with the stress concentration located at the critical area. This critical zone may be the weld toe or the weld root. It depends in fact on the particular conditions as loading and local geometry of the case encountered. The partial penetration at the weld root can be interpreted as a geometrical defect too. The corresponding stress concentration factors are calculated from Finite Element Analysis of the real geometry of the weld subjected to the given particular loading. Figure 9 shows as an example the calculation results of the [K] stress concentration matrix.

Multiaxial Fatigue Assessment of Welded Structures by Local Approach

(a)

51

(b)

Fig. 8. Welding geometrical defects: (a) angular distortion of a butt-welded joint; (b) misalignment defect of a cruciform welded joint Global typical model

Local typical model

' 1 0.5 0' [K] defined as [ajiocai = [K].[a]nom with [K] = 0.5 1.9 0 0 0 1 Fig. 9. Calculation of the stress concentration matrix associated with a butt weld for a given loading The constitutive cyclic law and the fatigue properties correspond to those of the critical zone of the welded joint, i.e. the weld toe or the weld root. The material properties are identified from specimens simulating the metallurgical states of these overheated areas. Such properties may be available in the technical literature, as for example within reference [11]. Another possible way to identify these properties is to use inverse simulation method from some fatigue results obtained for welded joints of the same material [12]. The local residual stresses existing at the weld toe or at the weld root could be calculated using coupled simulation of the thermal, metallurgical and mechanical phenomena during welding. An alternative method is to measure the local residual stresses using X-ray diffraction

52

F. LABESSE-JIED ET AL

or the incremental hole method. These residual stresses are then combined with the local stresses in order to determine the total operational stress states from which the fatigue life procedure is performed: W,„,^, =Mres.dual+[KlWnom

dD

The plasticity correction is performed when it is necessary with the cyclic plasticity parameters of the critical area. The multiaxial fatigue criterion function and the Miner's damage amount are calculated on each material facet. By this way the fatigue life on the facet maximising the damage is assessed.

EVALUATION OF THE METHOD ON WELDED COMPONENTS The proposed fatigue analysis method has been performed on welded structures. Figure 10 shows experimental data on butt-welded joints. Two series of tests corresponding to loading ratios equal to -1 and 0.5 respectively are plotted in the figure. These results have been obtained for the 6 mm thick 16MnNiCrMo5 high strength steel plates welded in 2 layers with the MAG process. The real geometry and angular distortion of the joint has been measured and then modelled by the Finite Element method. Figure 7a shows the stress states involved when the joint is clamped in the jaws of the fatigue testing machine and subjected to nominal axial loading. The weakest point of the joint is focussed on the weld toe. The local stress states are calculated by the Finite Element method using the actual geometry of the weld. For an axial cyclic loading they are obtained by combining the cyclic nominal stresses modified by the stress concentration matrix and the initial stress states due to the distorted welded specimen clamped in the jaws of the testing machine. The local mechanical properties of the overheated thermally affected area corresponding to the weld toe are presented in Table 2. These properties come from cyclic loading tests performed on specimens on which a heat treatment simulating the overheat of the thermally affected area has been realised and validated using the inverse method [12]. Table 2. Mechanical properties of the thermally affected material 'y

(MPa) 565

K' (MPa) 1150

n' 0.111

aV (MPa) 2080

-0.13

As the welded joints are subjected to high-temperature stress relieving, the residual stresses are disregarded. The fatigue strength of the joint under tensile cyclic loading corresponding to the ratios R = -1 and R = 0.5 respectively are calculated. They are plotted with straight lines in Figure 10. The comparison against experimental fatigue tests results shows a rather good agreement of predicted fatigue behaviour despite the fact that the scatter of these results is rather wide.

53

Multiaxial Fatigue Assessment of Welded Structures by Local Approach

1000

•--

1

•

,,,^^

--| ! 1

•

CO

a.

•

V

•

X (0

\

E CO

—calculation R=0.5 •

calculation R=-1

^*i*^*

A

A A

1 A testR=-1 100 1E+03

1

A

test R=0.5

1E+04

1E+06

1E+05 cycles

j

1 1E+07

Fig. 10. Comparison between predicted and experimental lives of stress-relieved butt-welded steel

DEFINITION OF THE WELD QUALITY: THE GEOMETRICAL PARAMETERS The quality of welded joints is defined from a design point of view by geometrical parameters such as the width of the weld (denoted as Lpc and Lrc for the weld toe and the weld root respectively), the height of the joints (S), the transition radii, the axial misalignment and the angular distortion. Transition radii and geometrical defects (distortion and misalignment) constitute in fact the main parameters of the quality of welds. Figures 11 and 12 show the geometrical parameters of butt and fillet welded joints respectively.

Rl

Root

R2

Fig. 11. Geometrical and micro-geometrical parameters of butt welds

The mean values and standard deviations of the geometrical parameters measured on several butt welded joints in 7020 aluminium alloy of 20 mm thickness, welded using the multi-layer MIG process, are given in Table 3. The angular distortion and the axial misalignment measured are respectively 0.7 ± 0.9° and 0.5 mm.

54

F LABESSE-JIED ET AL

Table 3. Mean geometrical parameters measured on multi-pass MIG butt welds in 7020 aluminium alloy (thickness equal to 20 mm).

Mean value (mm) Standard deviation (mm)

Rl R2 12.6 4.3 9.1 4.5

Top bead Rmin Lpc 2.7 34 3.0 1.5

S 2.9 0.6

Rl 6.7 3.3

Root bead Rmin Lpc S 4.0 3.7 28.4 3.6 2.4 2.2 2.4 0.7 R2

R2

Fig. 12. Geometrical and micro-geometrical parameters of fillet welds The mean values and standard deviations of the geometrical parameters measured on several fillet welded joints in 7020 aluminium alloy of 30 mm or 20 mm thickness, welded using the muhi-layer MIG process, are given in Table 4. Table 4. Average geometrical parameters measured on multi-layer MIG fillet welds in 7020 aluminium alloy (thickness t and T equal to 20 mm and 30 mm respectively). Welds A & B Weld A Rl R2 Rl R2 2.9 2.8 2.6 2.9

Mean value (mm) 1.6 Standard deviation (mm)

1.8

1.2

1.2

Weld B Rl R2 1.7 1.3

Jl

J2

14.2

16.2

0.9

0.8

0.8

0.7

0.6

1.1

Multiaxial Fatigue Assessment of Welded Structures by Local Approach

55

The precise measure of the characteristic geometry of these welded components makes it possible to accurately determine the local stress states existing within them. This is of major importance as these stresses are responsible for the local fatigue behaviour of the welds.

ANALYSIS OF THE EFFECT OF THE WELD QUALITY ON THE FATIGUE STRENGTH The local approach method was applied to welded joints in aluminium alloy to analyse the effect of the welds quality upon the fatigue strength. The examples presented here concern butt-welded joints in 7020 aluminium alloy of 20 mm thickness. The welds were obtained by using multi-pass MIG process with a 5183 filler metal. Figure 13 shows a macrograph of the weld.

Fig. 13. Aluminium alloy butt welds with the fatigue crack occuring in the vicinity of the weld toe

Material properties and geometrical parameters

Mechanical properties. Fatigue cracks always start at the weld toe or at the weld root. The material parameters of these zones must therefore be determined and used for the local approach. In order to determine these parameters, the metallurgical state at the toe of the bead is reproduced by an overheating thermal treatment on test samples. The material parameters that were obtained by this way are given in Table 5.

Table 5. Mechanical properties of heat affected zones of aluminium samples Elasticity parameters E V (MPa) 7.3-10^ 03

Ro (MPa) 200

Cyclic plasticity parameters Ca C Q b (MPa) (MPa) iF 150 ToO 50

Fatigue parameters a'f b p (MPa) 670 -0.095 2.3-10"^

56

F LABESSE-JIED FT AL

Mean residual stresses. The mean residual stresses were measured on the surface of the toe of the bead with means of X-Rays diffraction. They reach 25 MPa and 37 MPa in the transversal and longitudinal directions respectively.

Mean geometrical parameters. Experimental comparisons are made on the basis of the average geometrical parameters measured on welded test specimens. Table 6 summarises these geometrical parameters.

Table 6. Geometrical parameters of the aluminium alloy butt welds

Rmin Mean value (mm) 2.67 Standard deviation (mm) 1.55

Top bead Lpc S 34.03 2.94 3.00 0.59

Rmin 3.69 2.20

Root bead Lrc S 28.40 3.64 2.43 0.68

Moreover, the average axial misalignment of 0.5 mm and angular distortion of 0.7° are taken into account for the fatigue life assessment. The following sections describe the sensitivity of the butt welds, regarding their predicted fatigue life under constant amplitude loading, to residual stresses, angular distortion and transition radius.

Influence of residual stresses Residual stresses are known for their determinant role on the fatigue strength of materials, including of course welding joints. This influence depends on the local sensitivity of the material to hydrostatic stress. Fatigue properties, determined from results of fatigue tests performed on the overheated metallurgical state corresponding to the weld toe, indicate an important sensitivity to hydrostatic stress. This sensitivity can also be quantified by the ratio of fatigue strengths obtained under alternate tensile test and alternate torsion test respectively, for a given number of cycles. The smaller this ratio is, the more the material is sensitive to the residual stresses. A material insensitive to the hydrostatic stress effect has the following strength ratio:

^

(12)

=S

The present overheated material located at the weld toe provides the following fatigue strength ratio, at 10^ cycles: I"

\

-1.4

(13)

weld toe, N=10 cycles

It is important to note that this ratio is about 1.6 for high-strength low alloy steels hardened/tempered at high temperature. This indicates a lower sensitivity of these steels to residual stresses.

57

Multiaxial Fatigue Assessment of Welded Structures by Local Approach

The analysis of the influence of the residual stresses level on the fatigue strength of butt welded joints subjected to alternate cyclic load has been carried out by the proposed local approach. These calculations were done with the mean measured geometry of the welds. Figure 14 shows the influence of the residual stresses on the fatigue strength of the butt welds. The different curves are examined from the left side to the right side, i.e. in the increasing order of the fatigue strength.

20

-nil SIG_res

experimental results

-minimum SIG_res measured

-maximised SIG_res

-SIG_res shot peening

maximum SIG_res measured 01E+4

' ' '^ ' i 1E+5

1E+6

1E+7

1E+8

Number of cycles

Fig. 14. Influence of residual stresses on predicted alternate tensile fatigue strength (R = -1) of butt welds

The lowest S-N curve has been obtained with the maximised residual stresses, it is to say CTiong = G^trans = 140 MPa. The sccond S-N curve was plotted by considering the maximum measured values of residual stresses which correspond to: aiong = 70 MPa and Qtrans = 60 MPa. These two assessed S-N curves are found to be quite conservative. The minimum measured residual stresses (aiong = 1 5 MPa and Gtrans = -20 MPa) lead approximately to the same simulation results as the calculation carried out accounting for no residual stresses. These assumptions seem to be a little optimistic regarding to the experimental S-N curve. The last SN curve was obtained assuming that residual stresses are compressive and reach the maximum absolute value, i.e. aiong = CTtrans = -100 MPa. This assumption brings quite non conservative predictions. It is similar in fact to the effect of pre-stressed shot peening. As a conclusion, there is clearly a very considerable effect of residual stresses on the fatigue strength of aluminium alloy welded joints. The influence is beneficial in the case of compressive residual stresses and detrimental in the other case. A pre-stress shot peening type finishing treatment makes strongly increase the fatigue strength of welded joints.

58

F. LABESSE-JIED ET AL

Influence of angular distortion defects The effect of geometrical defects is more difficult to analyse and must be looked at on a caseby-case basis. Generally speaking, an axial misalignment or an angular distortion creates a supplementary bending moment, which induces some compressive stresses on one side of the joint and some tensile stresses on the other side. The effect of these static stresses on the fatigue strength of the welded joint depends on the loading and on the micro-geometry of the bead: - generally, when the geometrical defect induces stresses opposite to those induced by the service loading, some improvement in fatigue strength can be expected, - when the micro-geometry is symmetrical with respect to the mean fibre, the geometrical defect causes a reduction in the fatigue strength of the welded joint, - when the transition radii are smaller on the side on which the geometrical defect induces compressive stresses, then the defect makes improve the fatigue strength of the welded joint. On the contrary, i.e. with tensile static stresses applied on the smallest transition radii side, the fatigue strength of the welded joint is reduced. To sum up, the effect of geometrical defects may be positive or negative. Well-controlled defects could in principle contribute to improve the fatigue strength of welded joints. The effect of angular distortion is quantified by calculation of the fatigue strength of buttwelded joints subjected to alternate tensile loading. The direction of the modelled distortion, in relation to the geometry of the joint, corresponds to the direction that was measured, namely a closing of the welded joint on the root side. The S-N curves of Fig. 15 show the importance of angular distortion upon the predicted fatigue strength of welds. All the calculations are realised with the average geometry and the mean residual stresses. The highest fatigue strength is obtained with a zero angular distortion. The general trend is that the fatigue strength decreases as the angular distortion increases. Most of the experimental test results are close to the predicted fatigue strengths obtained with considering the average measured angular distortion.

100

<5 40 £ E o z 20

•

experimental results angular distorsion = 3°, axial = 0.5 mm

~.»^.angular distorsion = 1.4°, axial = 0.5 mm angular distorsion = 0.7°, axial = 0.5 mm angular distorsion = 0°, axial = 0.5 mm

1E+3

1E+4

1E+5 Number of cycles

1E+6

1E+7

Fig. 15. Influence of angular distortion on the predicted alternate tensile S-N curve (R = -1) of butt welds

59

Multiaxial Fatigue Assessment of Welded Structures by Local Approach

Influence of the transition radius The preponderant parameter in welding quality is the transition radius at the weld toe and at the weld root. Indeed this parameter primarily determines the maximum stress concentration factor of the welded joint. The other micro-geometrical parameters, namely the width of the bead and the height of the weld dome, are secondary. Figure 16 shows the influence of the transition radius on the predicted alternate tensile fatigue strength of butt welds, assuming mean geometrical parameters and also intermediate level of residual stresses between the maximum and minimum measured values.

c

1o

z

20

0 1E+3

experimental results - r = 0.5 mm - r = 1.49 mm - r = 5.9 mm -r = 10 mm 1E+4

1E+5

1E+6

1E+7

Number of cycles

Fig. 16. Influence of the transition radius (uniform radius) on alternate tensile fatigue curves ( R = -1) of butt welds The transition radius is clearly shown to be a strongly influent parameter regarding to the fatigue strength of the butt weld. On the contrary, a treatment allowing to obtain large transition radii, as grinding for instance, makes increase the fatigue life by a factor of 3 approximately with respect to the average quality level. It can be noted that, within the assumptions stated in this study (mean level of residual stresses and average geometry), the tests results fairly fit with the predicted lives obtained with the mean transition radii minus the standard deviation.

CONCLUSIONS A multiscale approach has been developed to design against fatigue structures subjected to multiaxial constant or variable amplitude loading. The evaluation of the lifetime starts from the knowledge of the service conditions of the whole structure (i.e. operating loading spectrum) and requires to assess the stress states in its critical zones. Stresses are actually induced both by the external loading and the local geometry. They may be severely increased because of stress concentration generated by geometrical discontinuities and angular or axial misalignment. The main step of the proposed fatigue evaluation procedure is the determination of the local effective stresses as these govern the fatigue phenomenon, i.e. the crack initiation. An elastic-

60

E LABESSE-JIED ET AL

plastic stress response model is also applied for that purpose in the case where the stress levels exceed the material yield stress. A multiaxial critical plane criterion and the linear Miner's damage summation technique are employed for assessing lifetimes. The ratio between predicted results and largely scattered experimental ones are found to be less than 4. On the basis of experimental results and lifetimes calculated by the presented fatigue assessment procedure, the following items may be pointed out: - All the geometrical peculiarities of the welded assembly must be considered including angular and axial defects generated by the welding. The influence of these parameters is the most pronounced one with respect to the fatigue behaviour of the weld, - Fatigue predictions require to take account for S-N curves modifications involved by metallurgical thermally dependent changes, - The residual stresses remaining in the vicinity of the weld should be considered and superimposed to the actual operational stress states existing within the weld. The cyclic plasticity correction has to be run on when stress states exceed largely the initial material yield stress, - A multiaxial fatigue damage model as a fatigue stress-based criterion has to be used in order to account for multiaxial stress states generated within the weld. The fatigue assessment procedure proposed in this work is modular and opened so that it may include any multiaxial fatigue criterion, attended that the ductility level of materials reveals more or less suitability of multiaxial fatigue concepts [13].

REFERENCES 1.

2.

3.

4.

5. 6. 7. 8. 9. 10. 11. 12.

Sonsino, C M . (1997). "Overview of the state of the art on multiaxial fatigue of welds". Fifth International Conference on Biaxial/Multiaxial Fatigue and Fracture, Cracow, Poland. Macha E. and Mroz Z. Editors. Vol. I, pp. 395-419. Hobbacher A. and IIW Joint Working Group XIII-XV (1995). "Recommendations on fatigue of welded components". Edited by the International Institute of Welding. XIII1539-95/XV-845-95, 117 p. Sonsino, C M . and Maddox, S.J. (2001). "Multiaxial fatigue of welded structures Problems and present solutions". Sixth International Conference on Biaxial/Multiaxial Fatigue and Fracture, Lisbon, Portugal. De Freitas M. Editor. Vol. I, pp. 3-15. Campion, B., Petitpas, E. and Sauveterre, M. (1999). "Determination of the fatigue behaviour of a structure in a Gaussian random environment with the Abaqus software", 14th French Mechanics Congress, Toulouse, France. AUM Editor. N° 5029. Lemaitre, J. and Chaboche, J.L. (1988). Mecanique des materiaux solides. Dunod, Paris. 188 p. Bannantine, J.A. and Socie, D.F. (1992). "Advances in Fatigue Lifetime Predictive Techniques". ASTM STP 1122, ASTM, Philadelphia, pp. 249-275. Basquin C H . (1910). "The exponential law of endurance tests", ASTM. PV ac. 10, 625 p. Leese, G.E. and Socie, D.F. (1989). Multiaxial Fatigue Analysis and Experiments, SAE AE-14. Leese G.E. and Socie D.F. Editors. 162 p. Galtier, A. and Seguret, J. (1990). Revue Frangaise de Mecanique 4, pp. 291-299. Zenner H., Simburger A. and Liu J. (2000). Int. J. Fatigue 22, pp. 137-145. Boiler, Chr and Seeger, T. (1987). Materials Data for Cyclic Loading, Part E : Cast and Welded Metals. Elsevier. 201 p. Petitpas, E., Lebrun, B., De Meerleer, S., Foumier, P., Charrier, P., Pollet, F. (2000). "Use of the local approach for the calculation of the fatigue strength". Edited by the International Institute of Welding. XIII-1850-00, 13 p.

Multiaxial Fatigue Assessment of Welded Structures by Local Approach

61

13. Sonsino, CM. and Kueppers, M. (2001). "Fatigue behaviour of welded aluminium under multiaxial loading". Sixth International Conference on Biaxial/Multiaxial Fatigue and Fracture, Lisbon, Portugal. De Freitas M. Editor. Vol. I, pp. 57-64.

Appendix : NOMENCLATURE b

fatigue strength exponent; S-N curve slope

C

kinematic hardening coefficient

Ca

kinematic hardening modulus

d^,d,D

fatigue damage related to one material plane,, cycle, stress history

E

Young's modulus

J1,J2

joint size of fillet weld

K

strength coefficient

K'

cyclic strength coefficient

Lpc, Lrc

width of weld toe, weld root

n

strain hardening exponent

n

unit normal vector

n'

cyclic strain hardening exponent

N

fatigue life (number of blocks)

Nf

fatigue life (number of cycles)

P

cumulated plastic strain

Q,Y

isotropic hardening parameter

R

isotropic stress hardening

Ro

isotropic stress hardening coefficient

Rmin

mean value of transition radius

R1,R2

transition radii

S

height of the beads

t

time ; plate thickness

T

plate thickness

s'

deviatoric stress tensor

X

kinematic stress hardening

P

fatigue parameter

sP

plastic strain

V

Poisson's ratio

62

F LABESSE-JIED ET AL G

Stress state tensor

^fat

equivalent uniaxial stress amplitude according to one multiaxial criterion

a'f

fatigue strength coefficient

QHW

hydrostatic pressure

Gy

yield stress

a'y

cyclic yield stress coefficient

a.i

completely reversed (R=-l) tensile fatigue strength

T

mean shear stress over one cycle

T(t)

shear stress

I'f

fatigue strength coefficient

Ta(t)

ahemate shear stress

T-i

completely reversed (R=-l) torsion fatigue strength

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

63

MICRO-CRACK GROWTH BEHAVIOR IN WELDMENTS OF A NICKEL-BASE SUPERALLOY UNDER BIAXIAL LOW-CYCLE FATIGUE AT HIGH TEMPERATURE

Nobuhiro ISOBE* and Shigeo SAKURAI** ^Mechanical Engineering Research Laboratory, Hitachi Ltd., ** Thermal & Hydroelectric Systems Division, Power & Industrial Systems, Hitachi, Ltd., 3-1-1, Saiwai, Hitachi, Ibaraki, Japan

ABSTRACT Tensile-torsional-combined biaxial low-cycle fatigue tests on welded tubular specimens of the Ni-base superalloy Hastelloy-X were carried out and the micro-crack growth behavior in the weldments was investigated with the aim of improving life assessment methods for hightemperature components. Welded hollow cylindrical specimens of two types were prepared: one welded in the axial direction and the other welded in the circumferential direction. Fatigue lives of the welded specimens were about half that of the base metal. In both weld and base metal, the initiation of micro-cracks was observed in the early stage of life, but the initiated length of the micro-cracks in the weld metal was about 0.5 mm while the equivalent figure for base metal was about 0.1 mm. The crack growth life from 0.5 mm to failure in the base metal specimen almost coincided with the failure life in the welded specimen. The maximum principal strain was confirmed to be a good parameter for evaluating crack growth rates for both weldments and base metal. These results show that the reduction in fatigue strength is not due to the strain concentration at the weld. The fatigue life of weldments of Hastelloy-X is affected by the initiated lengths of micro-cracks affect. KEYWORDS Biaxial low-cycle fatigue, Micro-crack, Weldment, Crack growth rate. Principal strain

INTRODUCTION In recent years, the demands placed on fossil power plants, such as for high efficiency and frequent start-stop cycles, have harshened their service conditions. More advanced methods of life assessment and maintenance for reliability of such power plants have thus become more important. A suitable way of assessing reduction in the strength of weldments is necessary in the fatigue assessment of components, and a fatigue strength reduction factor (FSRF) is, in many cases, introduced to achieve this. For example, ASME Boiler and Pressure Vessel Code

64

N. ISOBE AND S. SAKURAI

Section IE, Division I, subsection NH [1] applies a factor of 2 as the FSRF for the fatigue life assessment of weldments. This FSRF derived from the discussion for carbon and stainless steels that show large inelastic deformation behavior at high temperature [2]. The suitable assessment method for the fatigue life of the weldment of Ni-base superalloys, which use higher temperature than stainless steels, has not been developed. Furthermore, such factors are usually derived from uniaxial fatigue testing, while many real components are used under complex multiaxial stress-strain conditions. It is, therefore, important to develop a life assessment method that takes multiaxial stress or strain into account suitably. The assessment of crack growth behavior is important in the life evaluation of components. This is because the growth of cracks of the same size as grains is observed in the early stages of life and their propagation dominates a component's life [3]. In this study, we investigated the initiation and growth of micro-cracks in weldments of a nickel-base superalloy, Hastelloy-X, under combined tensile and torsional loads at high temperatures. The evaluation of the crack growth rates in weldments is discussed in order to improve the life assessment of high-temperature components.

TEST PROCEDURE A Ni-base superalloy, Hastelloy-X, and its weldment were tested. The weld metal was also Hastelloy-X, but its content of trace elements, such as Si, Mn and W slightly differ from the base metal. Their chemical compositions are listed in Table 1 and mechanical properties in Table 2. Specimens were hollow cylinders 22 mm in diameter and 2 mm thick in the gauge portion (Fig.l). Welded specimens of two types were prepared: one was welded along the axial direction, the other was along the direction of the circumferential direction, as shown in Fig. 1(b) and (c). Specimens were machined from blocks of welded two plates as shown in Fig. 2. TIG welding was used and the weld was 7 mm wide. Figure 3 shows the microstructure of the weldment. The average grain diameters were about 50 |Lim in the weld metal and about 20 jxm in the base metal.

Table 1. Chemical compositions of materials (wt%). i) Base metal Mn 0.69

P S Ni 0.013 0.001 Bal.

Cr 21.7

Co 1.0

Mo 8.9

W 0.50

Fe 17.6

ii) Weld metal Mn C Si 0.10 0.006 0.57

P S Ni 0.009 0.006 Bal.

Cr 21.65

Co 0.93

Mo 8.9

W 0.73

Fe 18.6

C 0.06

Si 0.40

Table 2. Mechanical properties of the materials.

Base metal Weld metal

ao.2 (MPa) 382 415

GB

(MPa) 764 753

Elongation (%) 52.1 40.0

Cu 0.08

Micro-Crack Growth Behavior in Weldments of a Nickel-Base Superalloy Under .

<^7 A

r

L

- - ^ k - - r c\j cx)T

T

"^

85

85

60 230

(a) Base metal specimen(BM)

Weld metal

(b) Axially welded specimen(WAX) 25(gai^ge length) 12.5

^

Weld metal

(c) Circumferentially welded specimen(WCF) Fig. 1. Shape and dimensions of specimen and weld line.

Weld metal

Weld m 9tal /

/^/

\ /—^

/

/'"x A /

/

X

1 '
(b) Circumferentially welded specimen

Fig. 2. Sampling positions of welded specimens

65

66

A^. ISOBE AND S. SAKURAI

_.

"^iAr^ipf^J^ I*

Base metal < \ •Weld metal Fig. 3. Microstructure of welded specimen. A servo-hydraulic, axial-torsional machine with an axial load capacity of 245 kN and torque capacity of 2.8 kN*m, was used for the strain-controlled multiaxial fatigue tests. An axial extensometer with conical-point quartz extension rods that could independently control and measure both the axial and shear strain was used. Its gauge was 25 mm long. Strain-controlled multiaxial fatigue tests with combined axial and torsional loading were carried out in several von Mises* strain ranges (ACeq). Strain waves were in-phase with fully reversed triangular waves at a strain rate of 0.1 %/s. The principal strain ratios ()), which is the ratio of the minimum strain to the maximum principal strain {^-z-^Jz^) [4], employed were -1 (pure torsion), -0.64 (combined, E = Vsy), and -0.50 (purely axial). The strain measurement point in the axially welded specimen was at a position 90 degrees in the circumferential direction from the weld line. Specimens were heated by induction heating and tested at 7(X)°C. The failure of a specimen was defined as the number of cycles producing a 25% reduction in the stress range from the saturated value. Testing was occasionally interrupted to obtain cellulose-acetate film replicas and thus observe the growth of fatigue cracks. RESULTS AND DISCUSSION Results of Low-cycle Fatigue Testing Figure 4 shows the low-cycle fatigue life of base metal and welded specimens. Results for a round bar specimen with a diameter of 8 mm are also plotted in the figure. The uniaxial fatigue-test result was almost the same as for the round bar [5], so the failure life of the hollow cylindrical specimen was roughly equivalent to that of a round bar. The solid line in the figure indicates the fafigue curve for the uniaxial condition. Fafigue lives in the torsional fafigue test were 2 to 3 times longer than in the uniaxial test so that the Mises' equivalent strain was not an appropriate parameter for assessing the fafigue life of Hastelloy-X. Fafigue lives of welded

Micro-Crack Growth Behavior in Weldments of a Nickel-Base Superalloy Under .

67

specimens were about half those of base metal specimens regardless of the principal strain ratio and strain range. The broken line in the figure was determined by moving the solid line in the vertical direction to coincide with the uniaxial data of welded specimens. The exact fatigue strength reduction factor was about 1.4. Figure 5 shows the Micro-Vickers hardness distribution around the weld line. Both base and weld metal had hardened after tests, but the difference in hardness of base and weld metal was small. This suggests that no strain concentration is produced around the boundary between base and weld metal. Figure 6 shows the von Mises' equivalent stress-strain relationship of base metal and weldment. There was not a distinct difference between base metal and weldment so the stress-strain relation of both materials could be considered as the same.

5.0

11 l l l

\

1

1

1—1—1

1 1 1 111|

700°C

a(D

\ C CO i_

c CO

^—• CO

"co CD

if)

\

L t

• ^ Q swEI ^^^^

0.5

Circum. welded(WCF) @ 0=-O.5O [ 1 0 =-1.00 1

1 1 1 1 11

10^

-\

•

~^i

rWelded specimen [Axially welded(WAX) L o
1 r 1 1 1

Base metal(BM) • 0 =-0.50(barf ^ • 0 =-0.50 A (f) =-0.64 M
1

.

1 ^'"^-^

.

......1

10^

10^

Number of cycles to failure Nf Fig. 4. Relationship between Mises' strain range and failure cycle.

As received _^__

> X 300h

After the test WAX WCF

' i Aeeq=0.7% (t)^-0.50 (1)=-1.00

CO CO CD

c 250h CO CO CD

200h Weld metal

15a

-5

_L 0

5

Position (mm) Fig. 5 Micro Vickers hardness of welded specimen after the tests.

N. ISOBE AND S. SAKURAI

0.2

0.4

0.6

Mises' eq. strain amplitude

0.8

1.0

A£eq/2 (%)

Fig. 6. von Mises'equivalent stress-strain relationship between base metal and weldment.

0.4

SUS304 Welded joint. RT IJeWM

0.3 h JO. 0.2 Front surface side _l_

60

i 0.1 (I)

_L_

_1_

50 40 30 20 10 Distance from vy/eld bead (mm) Back surface side

i i fl

10

20

3P

0.2 0.3| 0.4'

Fig. 7. Longitudinal strain range distribution near the weld line at room temperature [7]. In the case of carbon and stainless steels, the reduction in fatigue life of weldment has been also observed. The fatigue strength reduction factors were determined through the discussion on the effect of joint configuration, welding process, cyclic plasticity, metallurgical factors, and so on [2]. For a metallugical aspect, the effect of a nongeometric metallurgical notch effect which is the hardness changes locally around the heat affected zones within the multi-pass weldment and causes larger strain within the softened regions, had been reported [6]. Figure 7 shows an example of strain distribution around the weld joint in a bending test of a large plate of 304 stainless steel [7]. This strain increasing has been considered as the reason of the life reduction in the weldment of stainless steels. On the other hand, such a distinct change in hardness was not observed in the weldment of Hastelloy-X, so the life reductions of weldment were not caused by the strain concentration.

Micro-Crack Growth Behavior in Weldments of a Nickel-Base Superalloy Under ...

69

Crack Growth Behavior Crack-growth behaviors observed in uniaxial and torsional fatigue testing with A8eq = 0.7 % are shown in Fig. 8. The measured crack lengths in these figures were the lengths projected onto the maximum principal strain plane. Crack lengths were almost proportional to the loading cycle on a semi-logarithm graph except for the early and fmal stage of life. Relatively large cracks were observed at the beginning of the lives of the welded specimen compared with the base metal specimens. Photographs of surface cracks observed at an early stage in the lives of two axially welded specimens are shown in Figs. 9 and 10. Cracking had been initiated at grain boundaries and the initiation direction of these micro-cracks did not depend on the principal strain plane. On the other hand, the propagation direction was the principal plane. Figure 11 shows the cracks observed in the torsional fatigue test of base metal with A£eq = 0.7 %. Small X-shaped cracks on the principal planes were observed.

2000

1000

Number of cycles N (a) Uniaxial fatigue test. 50r

• BM D WAX WCF 2000

4000

6000

Number of cycles N (b) Torsional fatigue test Fig. 8. Crack growth behaviors in low-cycle fatigue tests of Hastelloy-X at 700°C.

70

A^. ISOBE AND S. SAKURAI

>^r

(b) N=670 Fig. 9. Surface cracks observed in an axially welded specimen with (t)=-0.50 and A8eq=0.7%.

71

Micro-Crack Growth Behavior in Weldments of a Nickel-Base Superalloy Under .

§•

Weld metal

f Boundary

Base metal 0.5 m m I I

(a) N=45

-1 ' ^'/i

_

.

f

Boundary!

0.5 mm I

I

(b)N=1800 Fig. 10. Surface cracks observed in an axially welded specimen with (|)=-1.00 and A8eq=0.7%.

m f:s §^

I 0.5 mm I I

Fig.l 1 Surface cracks observed in a base metal specimen with (|)=-1.00 and A£eq=0.7%.

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A^. ISOBE AND S. SAKURAI

Figure 12 shows relationships between the growth rate and surface length of cracks. Crack growth rates in this figure were determined by the secant method [8]. In torsional fatigue tests of base metal and axially welded specimens, crack growth rates decreased when cracks reached about 0.5 mm long. In the uniaxial fatigue tests on all specimens and in the torsional fatigue test on circumferentially welded specimens, on the other hand, the cracks were relatively large and reached about 2.0 mm long before the growth rates decreased. X-shaped cracks were observed in the torsional fatigue tests of base metal [9] and axially welded specimen with ACeq = 0.7%, whereas cracks only propagated on one side of the principal plane in the torsional fatigue test on the circumferentially welded specimen. Figure 13 shows the crack morphologies observed in the uniaxial and torsional fatigue tests. In a torsional fatigue test, there are two equivalent principal planes. The minimum principal strain is the compressive strain parallel to the crack and its absolute value is the same as the maximum principal strain. Itoh [10] investigated the effect of the compressive strain parallel to the crack by finite element analysis. It has been reported that the crack opening displacement in the torsional condition, which corresponds to the case in Fig. 13(b), is smaller than that in a uniaxial test. This is because the maximum principal stress acting on the crack initiated plane is reduced when the minimum principal strain is small, i.e., its absolute value is large, due to the effect of the Poisson ratio. Furthermore, when small X-shaped cracks form, cracks on two equivalent principal planes affect each other and the stiffness around them also reduced so the crack growth rate decreases. This is because a crack shows complicated growth behavior in the early stage of life in the torsional test of base metal and axially welded specimen with Aeeq = 0.7%. On the other hand, a decrease in crack growth rate was observed when the crack length was about 1 mm in the torsional test of the circumferentially welded specimen. This phenomenon was also observed in the torsional fatigue test of the axially welded specimen with Ae^ = 0.5%, as shown in Fig. 12. In these tests, the crack was relatively long when it was initiated and no Xshaped cracks were observed. The crack propagated on one side of the principal plane with a direction close to its initiated direction.

0 10"

1 10-^ 1

— 1 — I — I — I

-•-BM,(t)=-(!).56

I I' 1 '1 "1 I

^^BM,(t)=-1.00 -O-WAX,(t)=-0.50 •-^^WAX,(t)=-1.00 -^-WCF,(t)=-0.50 -^WCF,(t)=-1.00

10-^1

CD

2 10"

I ioi I

2

^

^^10-1

.

I I • III

100

J I I 1111

1

i_

101

Crack length 2c (mm)

Fig. 12. Relationships between crack growth rate and crack length.

Micro-Crack Growth Behavior in Weldments of a Nickel-Base Superalloy Under .

^ (a) Uniaxial

(b) Torsion, single crack

73

^" (c) Torsion, X-shaped crack

Fig. 13. Morphology of cracks observed in uniaxial and torsional fatigue tests.

/^' ^M f>^

0.5 mm I I

Fig. 14. Surface cracks observed in the axially welded specimen with ())=-1.00 and Aeeq=0.5%.

The crack initiation point was about 1 mm away from the base-weld metal boundary in the torsional test of the axially welded specimen with A£eq = 0.7%, as shown in Fig. 10(a). In this test, cracks propagated mainly in weld metal and hardly propagated to the boundary. The strain concentration around the base-weld metal boundary may have been negligible according to Fig. 5, so this inbalance in crack growth direction would be the effect of micro-cracks initiated in the grain boundary. The number and size of these micro-cracks in weld metal were both larger than those in base metal. The main crack was formed by the coalescence of these micro-cracks so the crack propagated mainly in the weld metal.

Grain Boundary Oxidation and Life Reduction in Weldment Relatively large cracks were observed in the early stage of life in welded specimens, as shown in Fig. 8. Figure 15 shows a photograph of a replica taken from the same position as Fig. 10(a) before the test but after the specimen had been heated. The grain boundary where crack initiated was possible to distinguish from other ones. Figure 16 shows cross-sectional views through (a) a base metal and (b) a welded specimen after testing with Aeeq=0.7% in the pure torsional condition. Some grain boundary cracks were observed and surface cracks initiated at them. It was confirmed by an energy dispersive X-ray (EDX) analysis that these grain boundaries had been oxidized, so these cracks represented oxide spikes. Grain sizes affect the

74

N. ISOBE AND S. SAKURAI

0.2mm Fig. 15. Grain boundary where the crack in Fig. 10 initiated after the specimen had been heated.

(a) Base metal, (t)=-1.00, Aeeq=0.7%

(b) Axially welded specimen, (t)=-1.00, Aeeq=0.7% Fig. 16. Surface cracks and oxide spikes.

Micro-Crack Growth Behavior in Weldments of a Nickel-Base Superalloy Under ...

75

face) Oxidized grain boundary

(Cross section) (a) Grain boundary oxidation

(b)Crack growth in the early stage of life was mainly in the depth direction

Fig. 17. Schematic formation of grain boundary cracks and growth in the early stage of life. sizes of such oxide spikes and, in the early stage of life, crack lengths were greater in the weld metal than in the base metal. Thus, these relatively large surface oxide spikes in the weld metal caused the reduction in life of the welded specimens. Figure 12 shows that the crack growth rate was reduced after the rapid propagation in the early stages of life, especially in the torsional fatigue tests. The region where oxidation had a large influence was about one grain in deep as shown Fig. 16. These oxidation-induced microcracks are shallow when first initiated at the grain boundary even the crack covered several grains. Rapid crack growth was thus restricted to the surface regions and the growth rate fell as soon as the cracks extended further, as shown in Fig, 17. Figure 18 shows the relationship between crack surface length and life fraction. The crack lengths shown here are the length projected on the maximum principal plane, except for the result for the torsional fatigue test on the base metal with Aeeq=1.0% for which the maximum shear plane was used as the projection plane, because this crack propagated on the maximum shear plane [8]. The solid line in the figure was obtained by assuming that the initial crack length was 0.1 mm and final length was 5 mm. The broken lines indicate the factor of 2 scatter band. The initiated lengths of cracks in welded specimens were in the range 0.5-1.0 mm. Crack lengths in this range correspond to those in a base metal specimen at a life fraction of 0.5. The crack-growth life from 0.5-1.0 mm long to failure is equivalent to the fatigue life reduction ratio for a welded specimen that was about 0.5 as shown in Fig. 4. This result suggests that the reduction in fatigue life for a weldment is not due to the difference in crack growth rate. The grain boundary cracking induced by oxidation caused the fatigue life reduction in the weldment.

Evaluation of the Crack Growth Rate We have previously recommended the maximum principal strain as a good parameter for evaluating the crack growth rate of Hastelloy-X under biaxial fatigue [8]. Next, we discuss the crack growth rate evaluation based on a fracture mechanics approach by using the cyclic Jintegral range. The cyclic J-integral range is derived as [11] ;

76

N. ISOBE AND S. SAKURAI

100F

E E

WCF :-0.50, 0.7% -1.00,0.7%

-0.50, 0.7% (t)=-0.64, 0.7% -1.00,1.0% (t)=-1.00, 0.7% •1.00,0.5%

o (M

c CD

BM • (|)=-0.50, 0.7% • (t)=-0.64, 0.7% A (t)=-1.00, 1.0% J T (t)=-1,00, 0.7% 3

0.0

0.2

0.4

0.6

0.8

1.0

Life fraction N/Nf Fig. 18. Relationship between crack length and life fraction.

AJf =2;rY'

AcTAg^

f(n')AaA€^ IK

(1)

where Aa, A£e, and ASp are stress, elastic strain, and plastic strain ranges, respectively. In this discussion, we used von Mises' equivalent strain and the maximum principal strain to calculate the cyclic J-integral range. They are determined from von Mises' stress-strain and the principal stress-strain hysteresis loops of each test. Here, c is the crack length, and Y is the correlation factor of crack shape, and we used 0.714, which corresponds to a semi-circular surface crack in the uniaxial condition. The f(n') is a function of the cyclic hardening exponent n' in the cyclic stress-strain relationship and is given by /(A2') = 3 . 8 5 V ^ ( 1 — : ) + ;rAi'

(2)

Figure 19 shows the relationship between crack growth rate and the cyclic J-integral range. The solid line indicates the relationship determined from the uniaxial test with a round bar specimen of base metal. Uniaxial data for cylindrical specimens almost coincides with the solid line. The J-integral range obtained using the von Mises' equivalent stress-strain relationship overestimated crack growth rates in torsional tests. On the other hand, the accuracy of the estimation was improved by using the J-integral range based on the principal stress-strain relationship. Crack growth rates in Fig. 19 were determined by the secant method so these growth rates were influenced by the microstructure, localized oxidation, and so on. Figure 20 shows the relationship between and the von Mises' equivalent and the principal strain range. The nondimensional crack growth rate was obtained as the gradient of crack length and number of cycles on a semi-logarithm graph. This value can be considered as the overall crack growth rate through the life including the several effects as mentioned above. The principal strain range could evaluate the nondimensional crack growth rate under biaxial

Micro-Crack Growth Behavior in Weldments of a Nickel-Base Superalloy Under .

10

o ''

I ""I

•

BM.=-0.50

•

BM,<^=-0.64

Qoror^^^ir,,^o/RM^5)

I

BM;^=-1.00

Bar specimen (BM)

-11 O WAX,(t)=-0.50 ' O WAX,(t)=-0.64 A WAX,(t)=-1.00 @ WCF,(t)=-0.50 -21 ^ WCF,(t)=-1.00

'^

O

o 0

I 10-^1 p

10

I Mill

10

10 AJi (N/m)

10^

Cyclic J-integral range based on Mises' strain (a) J-integral range based on von Mises' strain

10'

o E E

10"

1 1 I I I lll| • •BM,(1)=-0.50

• A .O O A © .A

BM,(t)=-0.64 Bar specimen BM,(|)=-1.00 WAX.(t)=-0.50 WAX,(t)=-0.64 WAX,ct)=-1.00 WCF,(t)=-0.50 WCF,(t)=-1.00

o CD

10" o J^ 10' o

o 10 AJf(N/m) Cyclic J-integral range based on principal strain (b) J-integral range based on the principal strain Fig. 19. Relationship between crack growth rate and the cyclic J-integral range.

77

78

A^. ISOBE AND S. SAKURAI

0 ^

1

ia2|:

BM • (N-0.50 A (|)=-0.64 • (N-1.00

D)

W

/

-

c

0

E

/

' A'

// /

r

/

P

WAX O =-0.50 A (t)=-0.64 • (1)=-1.00

/'

4V

/ /' 1 / ' Factofbf 2| / '

•

o

10-4L

" 1 ^ ^

WCF @ (j)=-0.50 [Dl (t)=-1.00

C

Z

-

1

1

1

1 1

0.5 1 Mises' eq. strain range Aeeq (%) (a) von Mises'equivalent strain range 0

^-^ (0

o D)

10-2Fz t " " [

1

c E C

o

1

0-3

1

r-TT— 7 / i

/. /' / / W / /

t

/

L

/

J 1 /

/

h 1 U

/

/

/

/

/ f/' /

Ui 10-^'1

1

1

J

_

/ •

° ^/ ^' /Ii / /'

Factor oif 2 '

"1H

\

m^/ P

r

0

r

^

oZ CO"D O O

1

BM • (l)=-0.50 A (h-0.64 • (N-1.00

1

1

1

WAX O (N-0.50 A (t)=-0.64 D (1)=-1.00 WCF @ (j)=-0.50 0 (1)=-1.00

'i -

1

0.5 1 Maximum principal strain range Ae^ (%)

(b) Maximum principal strain range Fig. 20. Relationship between normalized crack growth rate and strain parameters stress within the factor of two scatter band, whereas the scatter of data was large when the von Mises equivalent strain range was employed. Therefore, the maximum principal strain is a better parameter for evaluating crack growth rates than von Mises' strain. There was little difference between cracking in base metal and in weldments. Therefore, the reduction in fatigue life seen in weldments was not because the base metal and weld metal have different degrees of resistance to cracks. The initiated length, which is affected by grain boundary oxidation, is the most important factor in the life reduction of weldments. The scatter band of the crack growth rate was large in the region where the J-integral range was less than 5 N/m in Fig. 19(b), especially for the torsional fatigue test. Figure 21 shows the relationship between the maximum principal strain range and failure life. Failure lives in torsional fatigue tests were larger than those in uniaxial tests although the principal stress and

Micro-Crack Growth Behavior in Weldments of a Nickel-Base Superalloy Under .

79

CD O)

c

5.0

1 1 1 11

1

03

1—1

1

. . . . .i

BM • # • •

^

l

^1.0|[

EN.

Q.

E 3

E X

5

tk^B^ D'^^'—-

[Welded specimen WCF \ WAX [ 1

O <|)=-0.50 A (j)=-0.64 D <|)=-1.00 1 . . . .1

0.1

10^

]

] ({)=-0.50(barf) " (i)=-0.50 J (N-0.64 (N-1.00 j

• (DJ

•i <0.5

1 1 1 1 ii|

.

@ =-0.50 0 =-1.00 1 1 1 III

1

10=^

Number of cycles to failure

.1

10^

Nf

Fig. 21. Relationship between maximum principal strain range and failure cycle. strain were good parameters for evaluating the crack growth rate, because of the reduction in stress intensity around the crack tip due to the effect of the X-like crack shape and the minimum principal strain parallel to the crack, as mentioned above. Fatigue strength reduction in a weldment is generally evaluated based on the failure life of standard specimens. In the case of a weldment of Hastelloy-X, fatigue life reduction can be explained by the grain boundary cracks induced by oxidation in the early stage of life. Therefore the fracture mechanics approach is suitable for rationalizing the evaluation of fatigue life reduction. The principal stress and strain are appropriate parameters for taking into account the stress multiaxiality because the crack growth dominates the fatigue life in both base metal and weldment.

SUMMARY We investigated crack growth behaviors in weldments of Hastelloy-X under biaxial low-cycle fatigue in order to improve the life assessment techniques for high-temperature components. From the results, the following conclusions were obtained. (1) The reduction in the fatigue life of a welded specimen is about 1/2 and the fatigue strength reduction factor is about 1.4. Failure lives in the torsional fatigue tests were about twice those in the uniaxial test. (2) In low-cycle fatigue tests of welded specimens, 0.5-1.0 mm long cracks were observed in the early stages of life. These cracks initiated at grain boundaries and the oxidation of the grain boundary affected the cracking. (3) In torsional fatigue tests, the crack growth rate in the early stage of life was smaller than in uniaxial tests. This is because of the reduction in stiffness around the crack due to the effect of the X-like crack shape and the minimum principal strain being parallel to the crack. (4) Crack-growth rates in weldments were almost the same as in base metal so fatigue life reduction is considered to be due to the initiated crack lengths. The cyclic J-integral range based on principal stress and strain is a suitable parameter for evaluating crack growth rates in biaxial stress states.

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N. ISOBE AND S. SAKURAI

REFERENCES l.ASME (1995): Boiler and Pressure Vessel Code, Section HI, Division 1 - Subsection NH. 2.Jaske, C. E. (2000): Fatigue-Strength-Reduction Factors for Welds in Pressure Vessels and Piping : Transactions of ASME, Journal Pressure Vessel Technology, 122, 297-304. B.Sakurai, S., Usami, S. and Miyata, H. (1987): Microcrack Initiation and Growth Behavior Under Creep-Fatigue in a Plain Specimen of Degraded CrMoV Cast Steel : JSME International Journal, 30, 1732-1740. 4.Sakane, M., Ohnami, M. and Sawada, M. (1987): Fracture Modes and Low Cycle Biaxial Fatigue at Elevated Temperature: Transactions of ASME, Journal of Pressure Vessel Technology, 109, 236-243. 5.Sakurai, S. and Umezawa, S. (1991): Fatigue Crack Growth Threshold for Distributed-SmallCracks in Hastelloy-X at High Temperature: Journal of the Society of Material Science Japan, 40, 1035-1041 (in Japanese). 6.Usami, S., Fukuda, Y. and Shida, S. (1986): Micro-Crack Initiation, Propagation and Threshold in Elevated Temperature Inelastic Fatigue: Transactions of ASME, Journal of Pressure Vessel Technology, 108, 214-225 7.Fukuda, Y. and Sato, Y. (1995): Micro-Crack Propagation and Fatigue Life of Large Welded Structures: Journal of the Society of Material Science Japan, 44, 65-70 (in Japanese). 8.ASTM (1996): Standard Test Method for Measurement of Fatigue Crack Growth Rates: ASTM E647, 587. 9.Isobe, N. and Sakurai, S. (2000): Micro-Crack Growth Modes and Their Propagation Rate under Multiaxial Low-Cycle Fatigue at High Temperature: Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP, 1387, 340-352. lO.Itoh, T., Sakane, M. and Ohnami, M. (1994): High Temperature Multiaxial Low Cycle Fatigue of Cruciform Specimen : Transactions of ASME, Journal of Engineering Material and Technology, 116, 90-95. ll.Dowling, N.E. (1976): Geometry Effects and the J-Integral Approach to Elastic-Plastic Fatigue Crack GRowth: Cracks and Fracture, ASTM STP, 601, 19-32

Appendix : NOMENCLATURE ^

Principal strain ratio ((t)=e3/8i)

£eq

von Mises equivalent strain

<7eq

von Mises equivalent stress

£1

Maximum principal strain

£3

Minimum principal strain

c

Half crack length

AJf

Cyclic J-integral range

2. HIGH CYCLE MULTIAXIAL FATIGUE

This Page Intentionally Left Blank

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

83

MULTIAXIAL FATIGUE LIFE ESTIMATIONS FOR 6082.T6 CYLINDRICAL SPECIMENS UNDER IN-PHASE AND OUT-OF-PHASE BIAXLVL LOADINGS

Luca SUSMEL^ and Nicola PETRONE^ ^Department of Engineering, University ofFerrara, Via Saragat 7, 44100 Ferrara, Italy Department of Mechanical Engineering, University ofPadova, Via Venezia 1, 35131 Padova, Italy

ABSTRACT Fully reversed bending/torsion fatigue tests were conducted on 6082-T6 solid cylindrical specimens under force control. Specimens were subjected to pure bending, pure torsion, in-phase and out-of-phase bending/torsion loadings and the investigated fatigue lives ranged between 10 and 2-10^ cycles to failure. The actual strains were measured by means of strain gauges positioned in correspondence of critical points. Experimental strain measurements highlighted that all the tests were conduced in pure elastic stress conditions. The material fatigue behaviour was studied by analysing the cracks pattern due to the considered biaxial loadings. All the tests showed that crack initiation was always MODE II dominated (that is, it occurred on the plain of maximum shear stress amplitude), whereas the crack propagation was MODE I governed. Just in the presence of pure torsional loadings cracks grew under MODE II loadings. A good correlation with measured fatigue lives was obtained by applying the Susmel and Lazzarin's criterion valid for homogeneous and isotropic materials, despite the slight degree of anisotropy showed by the material. KEYWORDS Biaxial fatigue loadings, multiaxial fatigue life prediction, cracking behaviour.

INTRODUCTION In the most general situations mechanical components are subjected to complex fatigue loadings that generate multiaxial stress states in correspondence of critical points. During the last 60 years the problem of multiaxial fatigue assessment has been extensively investigated by researchers in order to provide engineers with safe methods for the fatigue life prediction in the presence of complex stress states. The state of the art shows that the problem can be faced by using two different approaches [1, 2]: in the low cycle fatigue ambit (that is, when the damage contribution

84

L SUSMEL AND N. PETRONE

due to the plasticity cannot be disregarded), strain based methodologies are always suggested, whereas in the high cycle fatigue field, stress based techniques have to be employed to predict the multiaxial fatigue limit. Criteria valid for the fatigue lifetime calculation can be classified in three different categories: strain based methods, strain/stress based methods and energy based approaches. Brown and Miller [3] observed that the fatigue life prediction could be performed by considering the strain components normal and tangential to the crack initiation plane. Moreover, the multiaxial fatigue damage depends on the crack growth direction: different criteria are required if the crack grows on the component surface or inside the material. In the first case they proposed a relationship based on a combined use of a critical plane approach and a modified Manson-Coffin equation, where the critical plane is the one of maximum shear strain amplitude. Subsequently, Wang and Brown [4] introduced in the criterion formulation the mean stress normal to the critical plane in order to account for the mean stress influence. Socie [5, 6] observed that the Brown and Miller's idea could be successfully employed even by using the maximum stress normal to the critical plane, because the growth rate mainly depends on the stress component normal to the fatigue crack. Starting from this assumption, he proposed two different formulations according to the crack growth mechanism: when the crack propagation is mainly MODE I dominated, then the critical plane is the one that experiences the maximum normal stress amplitude and the fatigue lifetime can be calculated by means of the uniaxial Manson-Coffin curve [5]; on the other hand, when the growth is mainly MODE II governed, the critical plane is that of maximum shear stress amplitude and the fatigue life can be estimated by using the torsional Manson-Coffin curve [6]. Criteria based on energetic parameters calculation are founded on the assumption that the energy density is the unique parameter really significant of the fatigue damage. The employment of this quantity has a crucial advantage over the methodologies discussed above: theoretically, the amount of energy required for the fatigue failure is independent from the complexity of the stress state present at the critical point, therefore just a uniaxial fatigue curve is enough to predict fatigue lives even in the presence of complex loadings. Garud [7] suggested that the multiaxial fatigue assessment could be performed by using only the energy due to the plastic deformation. Subsequently, Ellyin [8, 9] stated that not only the plastic energy but also the positive elastic energy is crucial to estimate properly the fatigue damage. In particular, fatigue depends on the elastic energy due to the tensile stress components, because experimental investigations clearly showed that fatigue failures can occur in the high cycle fatigue field, where the plastic contribution is negligible. Moreover, it is well known that a positive mean stress component reduces the fatigue life more than a negative one. For these two reasons an energy based approach can give satisfactory results only when it can account for both the plastic and the positive elastic contribution. Recently, Lazzarin and Zambardi [10] employed successfully a linear-elastic energy method together with a critical distance approach for the static and fatigue assessment of sharply notched components under mixed mode loadings. In the ambit of high cycle fatigue most of the established theories are based on the use of stress components. The oldest high-cycle fatigue assessment method has been proposed by Gough [11] and it is based on an extensive experimental investigation conduced on smooth and notched specimens of different materials subjected to in-phase bending and torsion loadings. By reanalysing these results Gough proposed two empirical formulations valid for brittle and ductile materials.

Multiaxial Fatigue Life Estimations for 6082-T6 Cylindrical Specimens Under ...

Several criteria were subsequently proposed by other researchers adopting the critical plane approach. Among the most employed in the case of out-of-phase loadings, it is worth mentioning the criteria proposed by Findley [12], by Matake [13] and by McDiarmid [14]. The first method considers as critical plane the one where a linear combination of the shear and the normal stresses reaches its maximum value. The other two are based on the assumption that the critical plane is the one experiencing the maximum shear stress amplitude and the fatigue damage depends even on the stress normal to this plane. These criteria allow the evaluation in a component of "non-failure" conditions, but they are not suitable for the fatigue lifetime calculation. By starting from a mesoscopic approach. Dang Van [15] proposed an original criterion completely different if compared to the methodologies above discussed. In particular, the simplest version of this method is formalised as a linear combination of the shear and the hydrostatic stresses. This criterion postulates that the fatigue failure is avoided if, in each instant of the load history, the critical stress parameter is lower than a reference value depending on two fatigue limits experimentally determined in simple loading conditions (typically, the use of the uniaxial and the torsional fatigue limits is suggested). Papadopoulos [16] as well formulated a criterion founded on the mesoscopic approach fundamentals. By using rigorous mathematical procedures applied to the elastic shakedown state concept, he introduced two different formulations valid for brittle and ductile materials, which gave satisfactory results in the fatigue limit estimation of plane components subjected to in-phase and out-of-phase loadings [17]. Recently, Papadopoulos proposed a new development of this method for the fatigue life calculation in the medium-high cycle fatigue field [18]. Carpinteri and Spagnoli [19, 20] formulated a new method founded on an original reinterpretation of the classical critical plane approach. Their new criterion correlates the critical plane orientation with the weighted mean principal stress directions and the multiaxial fatigue assessment is performed by using a non-linear combination of the maximum normal stress and the shear stress amplitude acting on the critical plane. By using the theory of cyclic deformation in single crystal, Susmel and Lazzarin [21] presented a medium-high cycle multiaxial fatigue life prediction method developed under the hypothesis of homogeneous and isotropic material and based on the combination of a modified Wohler curve and a critical plane approach. This criterion correlated very well with the experimental results referred both to smooth and blunt notched specimens subjected to either in-phase or out-of-phase loads [21, 22]. All the approaches discussed above can be employed for the multiaxial fatigue assessment of mechanical components of homogeneous and isotropic materials, but they do not give satisfactory results in the presence of anisotropy, as it happens with composites. For this reason different theories have been formulated to predict the fatigue endurance of this kind of materials. Initially, it is important to highlight that the fatigue damage of composites under multiaxial loadings is mainly influenced by the biaxiality ratios [23]; in particular it depends on the shear stress components and it holds true both for plane and notched components. Another important role is played by the off-axis angle (lay-up), whereas the influence on the fatigue strength of the out-of-phase angle between load components seems to be negligible, at least for the glass/polyester laminates [23]. By reanalysing about 700 experimental data take from the literature Susmel and Quaresimin [24] showed that the best accuracy in the multiaxial fatigue prediction of composite materials can be obtained by using the criteria proposed by Tsai-Hill [25], by Fawaz & Ellyin [26] and by Smith & Pascoe [27]. In particular, the first one is founded on the application of the classical polynomial failure criterion due to Tsai-Hill to the multiaxial fatigue assessment. The second one is based on

85

86

L SVSMEL AND N. PETRONE

an extension of the Mandell semi-log linear equation and, finally, the criterion due to Smith & Pascoe is capable of modelling three different fatigue damage mechanisms: the rectilinear cracking, the shear deformation along the fibre plane and the combined rectilinear cracking and matrix shear deformation. These approaches are clearly developed for composites, but when the degree of anisotropy is not so high, different methods are again suggested. The criterion proposed by Lin et al. [28, 29] can be mentioned as a valid criterion to apply in this kind of situations. This method is based on an original employment of the strain vector. Aim of the present paper is the validation of the Susmel and Lazzarin's criterion on a widely used industrial aluminium alloy showing a slight degree of anisotropy to confirm if the employment of specific models for anisotropic materials can be avoided when subjected to inphase and out-of-phase multiaxial loadings. FUNDAMENTALS OF THE MODIFIED WOHLER CURVES METHOD The theoretical frame of the Susmel and Lazzarin's criterion is based on a combined use of a modified Wohler diagram and the initiation plane concept [21]. The theory of deformation in single crystal has been employed to give a physical interpretation of the fatigue damage: in a polycrystal this depends on the maximum shear stress amplitude, Xa, (determined by the minimum circumscribed circle concept [30]) and on the maximum stress an,max normal to the plane of maximum shear stress amplitude (initiation plane) [21].

1I

Pi
O^

-e-

^k"^ ^^"S^TT^

^

TA,Ref ( P l )

• ^ 't^A,Ref ( P i ) ^

increasing p

1^A,Ref ( P n ) 1 1 1

\ ^. NRef lOgNf Fig. 1. Frames of reference and definition of the polar co-ordinates ^ and 6.

Fig. 2. Modified Wohler diagram.

Consider a cylindrical specimen subjected to a multiaxial cyclic load (Figure 1). With reference to a maximum shear stress amplitude plane, located by cj)* and 0* angles, it is possible to define for this plane the stress ratio p as follows:

Multiaxial Fatigue Life Estimations for 6082-T6 Cylindrical Specimens Under ...

87

where in Eq. 1 the maximum value of normal stress is able to include the influence of mean stress on the fatigue strength, according to the Socle's fatigue damage model [6]. Consider now a log-log Wohler diagram where in the abscissa there are the number of cycles to failure Nf and in the ordinate the shear stress amplitude Ta((t)*, 6*) calculated on the initiation plane (Figure 2). It can be demonstrated [21] that different fatigue curves are generated in the modified Wohler diagram by changing the p values. Each single curve is identified by the inverse slope kx (p) and by the reference shear stress amplitude TA,Ref (p) corresponding to NRef cycles (usually 2-10^ cycles in several design codes). Moreover, experimental results showed [21, 22] that as the p ratio increases the fatigue curve moves downwards in the modified Wohler diagram (Figure 2). On the basis of this observation and by evaluating the functions TA,Ref (p) and kt (p) by a best fit procedure performed using experimental data, it is possible to predict the fatigue life for a multiaxial cyclic stress state by applying the following expression:

Nf

^A,Ref (P)

,(r,e*)

NRef

(2)

By reanalysing systematically experimental data taken from the literature [21, 22], it has been observed that a good correlation with experimental results can be obtained just by expressing 'CA,Ref (p) and kx (p) as linear functions and by using uniaxial and torsional fatigue data for their calibration. In particular, under this assumption TA,Ref (p) and kx(p) can be expressed as follows: ^A,Ref (Ph t A,Ref (P = l ) - '»^A,Ref (p == O)] P + -^A,Ref (p^^)

kx(p)-[kx(p = l ) - k , ( p = 0)}p + k,(p = 0)

(^)

(4)

The presented method can also be used to estimate the fatigue life of notched components by applying the correction based on the fatigue notch factor Kf to the fatigue curves used in the model calibration, and by performing the assessment in terms of nominal stresses [21, 22]. Given that the multiaxial fatigue behaviour can be described by a single modified Wohler curve as the p value changes, it can be highlighted that a multiaxial notch factor can be always defined as function of the p ratio. By applying this idea, it has been demonstrated that the multiaxial Kf factor is always a linear function of the stress ratio p [22]. Finally, it is interesting to mention the fact that this method can be reinterpreted even in terms of critical distance approach [31]. In particular, by using as critical distance a length depending on the El'Haddad short crack constant [32], it has been shown that this method is capable of predictions within an error of about 15%, when employed to estimate the fatigue limit of V-notched specimens of low carbon steel subjected to in-phase MODE I and MODE II loads [31].

L SUSMEL AND N. PETRONE

EXPERIMENTAL DETAILS Fatigue tests were carried out on solid cylindrical specimens subjected to both in-phase and out-of-phase bending/torsion loadings. The material used in this investigation was aluminium 6082 T6, supplied in 30-mm-diameter bars. The material chemical composition is reported in Table 1. The bars were produced by means of an extrusion process, which introduced a certain degree of anisotropy by orientating grains mainly along the bar axis. The specimen (Figure 3) was machined in a CNC lathe and its 50mm long gauge length surface was successively polished down to a 6-fAm diamond compound to obtain a mirror-like finish. The average values of the experimentally determined mechanical properties of the used material, along the longitudinal extrusion axis, were: tensile strength 343 MPa, yield stress 301 MPa and Young's modulus 69400 MPa. In Figure 4 it has been plotted the lowest axial stress-strain curve recorded from a test conducted to evaluate the material static properties. Strains reported on this diagram were directly measured by means of an axial/torsional extensometer. The fully reversed bending and torsion moments were generated by using two hydraulic actuators, which loaded the specimen by means of a friction clamped loading arm (Figure 5). The contemporary use of a Lab VIEW software and two MTS 407 digital controllers allowed to generate and control the applied forces during each test; axial and shear strains were monitored by means of strain gages (Figures 3 and 5) applied at known positions with respect to the nominal maximum bending stress point. Signals were gathered by a National Instruments SCXI-IOOODC data acquisition system. Length and orientation of the macro-cracks were measured by means of a Leika stereoscope. Finally, the fatigue failure was defined as 2% bending or torsion stiffness drop and the frequency of each test was equal to 4 Hz.

Fig. 3. Specimen geometry (a) and picture of a polished specimen with rosette (b). Table 1. Chemical composition of the tested 6082-T6 aluminium alloy [%].

Si

Mg

Mn

Fe

CT

Zn

Cu

Ti

0.9^L1

O.S^LO

0.5^0.9

0.5

0.25

0.20

0.1

0.10

Multiaxial Fatigue Life Estimations for 6082-T6 Cylindrical Specimens Under .

300 b cv=252 MPa 250-f--

_l

0.01

0.02

I

I

I

I

0.03

I

I

I L

-+-

0.04

e

0.05

Fig. 4. Lowest recorded static stress-strain curve (strain measured by exstensometer).

Fig. 5. Biaxial test machine.

89

90

L. SUSMEL AND N. PETRONE

EXPERIMENTAL RESULTS The multiaxial fatigue behaviour of the considered aluminium alloy was studied by testing 44 different cylindrical specimens. Experimental tests were subdivided into four different groups: pure bending tests, pure torsional tests, biaxial tests characterised by >-=Xxy,a/<^x,a less than 1 and biaxial tests with A, greater than 1. Moreover, each single biaxial experimental investigation was performed by applying in-phase loadings and out-of-phase loading having a nominal phase angle equal to 90° for X<\ and 126° for X>\. The amplitude and the mean value of stress components as well as the phase angle were calculated by using the average value of all the peaks of forces recorded during each single test. The experimental results have been summarised in Tables 2-5, where the listed values refer to: bending stress amplitude, ax,a; mean bending stress, Gx,m\ shear stress amplitude, Txy,a; mean shear stress, Txy,m; biaxial stress ratio, ^=Txy,a/cyx,a; phase angle between the applied stress components, 5; number of cycles to failure, Nf,2%, defined using the 2% bending or torsion stiffness decrease criterion. By observing Table 3 it can be seen that a small bending stress component was always present even under pure nominal torsional loadings. The value of measured ax,a ranged from 13 MPa up to 24 MPa and it was practically independent of the applied torsional stress amplitude. In any case, the influence of the bending has been always disregarded because its value was in general negligible if compared to the applied torque. The presence of this unwanted stress component under pure torsional loadings was due to intrinsic plays within the ball-socket joints used to connect the loading arm to the actuators. The presence of these plays generated further complications under out-of phase loadings. In particular, when A. was greater than 1 the measured out-of-phase angle ranged between 125° and 129°, that is, it was greater than the imposed nominal angle of 90°. As an example, in Figure 6 the Xxy vs. a^ diagrams for two different biaxial tests are reported. By observing these diagrams some signal perturbations generated by the joints plays can be noticed.

P32BT8

P23BT4

OH

150 100 50 0-50 -100 -150 -200 - 1 5 0 - 1 0 0 - 5 0

0

50

Gx [MPa]

100 150 200

200 150 ^ 100 50 0 - ^ -50 -100 -150 -200 -200 -150 -100 -50

>

0

50

100

150 200

CJx [MPa]

Fig. 6. Crossplots of tests named P23BT4 (ax,a=147 MPa, Txy,a=90 MPa, 5=-8°) and P32BT8 (ax,a=149 MPa, Txy,a=68 MPa, 5=93°)

91

Multiaxial Fatigue Life Estimations for 6082-T6 Cylindrical Specimens Under .

Table 2. Experimental results: fully-reversed bending tests. 6

-k

Nf^%

Specimen Code

ax,a

ax,in

'^xy,a

l-xyjin

[MPa]

[MPa]

[MPa]

[MPa]

[1

P5B5 PlBl P7B1 P2B2 P8B3 P3B3 P4B4 P6B4

224 190 188 180 162 165 145 145

-1 0 -1 -4 0 -2 -1 -1

4 5 4 4 3 4 4 4

0 7 0 -1 1 1 1 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

52990 159000 197275 244403 421560 437636 1060730 1235690

X.

Np% [Cycles]

9.9 7.7 7.4 6.9 7.6 4.1 5.7 5.8

14695 23052 67690 113455 196555 449997 497990 1100000

[Cycles]

Table 3. Experimental results: fully-reversed torsion tests. Specimen Code

Ox,a

CTx,in

T^xyja

^xy,in

6

[MPa]

[MPa]

[MPa]

[MPa]

[1

P14T2 P10T2 P11T3 P12T3 P13T1 P9T1 P16T4 P15T4

14 18 15 16 13 24 15 15

1 3 1 1 3 0 2 1

138 139 111 111 99 98 86 87

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0

CRACKING BEHAVIOUR The complex cracking behaviour showed by the material under complex fatigue loadings was a consequence of the morphological structure of the studied 6082-T6 generated by the extrusion manufacturing process. This orientated grains mainly along the extrusion axis and it favoured the initiation of small cracks evenly distributed on the specimen surface and oriented along the specimen axis. Under bending loading, shear crack nucleation was followed by growth on the plane of maximum principal stress (MODE I growth), independently of the bending stress amplitude levels. As expected, the crack initiation occurred in correspondence of the maximum bending stress points, so two different cracks were present on the surface of each specimen.

92

L SUSMEL AND N. PETRONE

In Figure 7^ the bi-dimensional development of gauge surface of specimens subjected to fullyreversed bending have been reported and, on each single surface, one of the two detected fatigue cracks has been drawn. The crack final patterns were obtained by interrupting tests when the bending stiffness drop was equal to 20%. By observing this figure it is possible to notice that the crack growth were mainly MODE I dominated and the cracking pattern was not influenced by the applied stress amplitude, that is, it was independent from the number of cycle to failure. Table 4. Experimental results: in-phase and out-of-phase bending/torsion tests (X=Txy,a/cTx,a>l)Specimen Code

[MPa]

[MPa]

[MPa]

P21BT3 P20BT3 P17BT1 P22BT1 P18BT2 P19BT2 P36BT11 P41BT14 P37BT12 P38BT13 P39BT13 P42BT16

70 71 59 61 53 52 79 79 69 68 68 60

-3 -1 -1 0 -1 -2 -1 -4 1 2 2 3

118 117 100 98 83 82 129 116 110 99 99 94

CTx,a

Ox,m

''^xy,a

6

[MPa]

n

X.

Nf^% [Cycles]

0 1 1 0 1 0 1 0 0 0 0 0

0 1 -7 -18 -2 2 129 125 126 128 125 126

1.69 1.65 1.69 1.61 1.57 1.58 1.63 1.47 1.59 1.46 1.46 1.57

71255 78730 230750 516985 1018775 1289550 20730 41490 188882 234725 368080 1016280

"^xyjin

In Figure 8 it has been reported the cracks patterns generated by torsional loadings. This figure shows that the growth always occurred in MODE II and fatigue cracks were always oriented along the specimen axis. Moreover, some small cracks were soon evident on the surface after few cycles (N<5000 cycles), independently of the applied shear stress level, but the stiffness decrease became evident only when a crack had grown along all the gauge length. The formation of these small cracks was favoured by the preferred grains orientation induced by the extrusion process; the number of cracks increased, and their length decreased, as the amplitude of the applied torque increased. Under biaxial loadings, and independently of the phase angle values, when X was greater than the unity, the failure was similar to those observed under pure torsion (Figures 9 and 10): a high density of cracks was evident on the gauge length surface and the cracks were always oriented along the specimen axis. On the contrary, when bending prevailed over torque (X<1), the cracks initiation was always MODE II dominated, but cracks grew on the plane of maximum normal stress for both in-phase and out-of-phase tests (Figures 11 and 12). All the cracks patterns reported in Figures 7-12 were obtained by interrupting tests when the bending or torsion stiffness drop was equal to 20%, but the fatigue data reanalysis has been

^ In Figures 7-12 x-axes always emanate from one of the two points of maximum bending amplitude. Moreover, the position on the gauge surfaces of each single x-axis changes, because the bi-dimensional develops have been plotted by trying to show the crack distributions as clear as possible.

Multiaxial Fatigue Life Estimations for 6082-16 Cylindrical Specimens Under ...

93

performed by extrapolating from the stiffness curve recorded during each single test the number of cycles to failure corresponding to a stiffness decrease of 2%. The rule used to define the failure had a strong influence on the width of Wohler fatigue curve scatter bands (Figure 13). In fact, under bending loadings just two fatigue cracks were always generated at the maximum stress points and the stiffness drop was a function of their total length; under torsional loadings the stiffness drop was related to the rate of appearance of some dominant cracks, oriented by the actual material morphology and usually generated by the coalescence and bridging of small adjacent cracks. On the contrary, the cracks density, which depended on the applied stress amplitude, seemed not to be correlated with the 20% stiffness drop. This situation became more critical in the presence of biaxial loadings and it caused the experimental points to fall in wider scatter bands when a torsional stress component was applied. Table 5. Experimental results: in-phase and out-of-phase bending/torsion tests (X=Txy,a/cTx,a
Specimen Code

Gfx,a

Gfx,m

'^xy,a

[MPa]

[MPa]

[MPa]

[MPa]

n

P25BT5 P26BT5 P24BT4 P23BT4 P27BT6 P28BT7 P43BT17 P46BT20 P35BT10 P44BT20 P45BT20 P34BT10 P32BT8 P30BT8 P33BT9 P31BT9

147 151 163 147 146 118 119 188 189 189 171 190 149 151 155 152

-2 -4 -2 1 -3 -3 1 0 -5 1 -4 -4 0 0 1 -1

106 104 81 90 76 82 72 106 106 106 99 105 68 67 72 47

1 0 0 -1 -1 1 -1 0 0 0 1 0 0 0 1 2

-4 -3 -5 -8 -6 -5 0 89 94 88 91 91 93 94 92 91

^xy,m

X

Nf;j%

[Cycles]

072 0.69 0.50 0.61 0.52 0.69 0.61 0.56 0.56 0.56 0.58 0.55 0.46 0.44 0.46 0.31

31000 64090 124460 132215 232370 315795 694062 5590 27420 34015 44750 47020 114845 273325 445560 456725

FATIGUE LIFETIME PREDICTION By using biaxial experimental results listed in Tables 2-5 it was possible to check the accuracy of Susmel and Lazzarin's criterion in the multiaxial fatigue life prediction, also in the presence of manufacturing anisotropy. The criterion was calibrated by means of the bending and torsional experimental fatigue results. The constants of each calibration curve are reported in Figure 13. The values of TA,Ref» kx and To (scatter ratio of shear stress amplitude calculated on initiation plane for 90% and 10% probability

L SUSMEL AND N. PETRONE

94

PlBl - Nf,2% = 159000 Cycles

P5B5-Nf,2%= 52990 Cycles

i f

E

X '

f X '

1 20jtmm

P7B1 - Nf,2%= 197275 Cycles

^. P2B2 - Nf,2% = 244403 Cycles

1 X '

P4B4 - Nf,2% = 1060730 Cycles

X

i

1

P6B4 - Nf,2% = 1235690 Cycles

1 X '

Fig. 7. Cracks pattern at 20% stiffness drop on the bidimensional development of the specimen gauge surface: fully-reversed bending tests (one critical point only).

Multiaxial Fatigue Life Estimations for 6082-T6 Cylindrical Specimens Under .

P14T2 - Nf,2% = 14695 Cycles

'

95

P10T2-Nf,2%= 23052 Cycles

' "l,

JM\ i \^'"'. 20jt mm

P11T3-Nf,2%= 67690 Cycles

P12T3 - Nf,2%= 113455 Cycles

P13T1 - Nf2%= 196555 Cycles

P16T4 - Nf 2% = 497990 Cycles

i

\

1

x\ ]

/

I'll

M\

n

i

\

\

/

U

\\\

\

'. 1

Fig. 8. Cracks pattern at 20% stiffness drop on the bidimensional development of the specimen gauge surface: fully-reversed torsional tests.

L SUSMEL AND N. PETRONE

96

P21BT3 - Nf,2% = 71255 Cycles

P20BT3 - Nf,2% = 78730 Cycles

P17BT1 - Nf,2% = 230750 Cycles

P22BT1 - Nf,2% = 516985 Cycles

P18BT2-Nf,2%= 1018775 Cycles

P19BT2 -Nf,2%= 1289550 Cycles

Fig. 9. Cracks pattern at 20% stiffness drop on the bidimensional development of the specimen gauge surface: in-phase bending/torsion tests (>^=Txy,a/cTx,a>l, 6sO°).

Multiaxial Fatigue Life Estimations for 6082-T6 Cylindrical Specimens Under ...

P36BT11 - Nf,2% = 20730 Cycles

97

P41BT14 - Nf,2% = 41490 Cycles

1—~— I

I

^•%\^>ll

) 1 f 1 \V

' il 1

U'

11i

'kl!

M'*' 1

20jr mm

P37BT12 - Nf,2% = 188882 Cycles

P38BT13 - Nf,2%= 234725 Cycles

P39BT13-Nf,2%= 368080 Cycles

P42BT16 - Nf 2% = 1016280 Cycles

Fig. 10. Cracks pattern at 20% stiffness drop on the bidimensional development of the specimen gauge surface: out-of-phase bending/torsion tests (X=Txy,a/ax,a>l, 6sl26°).

L SUSMEL AND N. PETRONE

98

P26BT5 - Nf,2% = 64090 Cycles

P25BT5-Nf2% == 31000 Cycles

1^ HsV

V

n

1

\

'J li

20JI

mm

i

\

V

\

1-^

;

f —^-

P24BT4 - Nf,2% = 124460 Cycles

P23BT4 - Nf,2%= 132215 Cycles

P27BT6-Nf,2% = 232370 Cycles

P28BT7 - Nf,2% = 315795 Cycles

f/ 1

x'f

Fig. 11. Cracks pattern at 20% stiffness drop on the bidimensional development of the specimen gauge surface: in-phase bending/torsion tests (X=Txy,a/ax,a
Multiaxial Fatigue Life Estimations for 6082-T6 Cylindrical Specimens Under .

P34BT10 - Nf,2% = 47020 Cycles

P35BT10 - Nf,2% = 45132 Cycles

1

!y\...

99

\J

A

E

xl \ 2031 mm

P32BT8-Nf,2%= 114845 Cycles

f

fx

^^ P30BT8 - N,,2%= 273325 Cycles

fx

P27BT6-Nf,2% = 445560 Cycles

Tx

P28BT7 - Nf,2% = 456725 Cycles

xf

Fig. 12. Cracks pattern at 20% stiffness drop on the bidimensional development of the specimen gauge surface: out-of-phase bending/torsion tests (X=Txy,a/ax,a
100

L. SUSMEL AND N. PETRONE

200

P

kc

'CA,Ref

To

0 1

7.99 6.98

76.8 66.5

1.285 1.073

^•^v

_ ^ —. D

..^

-

•^v^

Ta((t)*, e*) [MPa] -

^--

*"^^w ^ ^ ^ ^ - ^ s ^

^ ^

^*'^"*v*.^^%^^^*^^^^

^^

' ^

1

""^-5^^^^:^^:>.

100

"""-^5^^^Jj '*^

1

J

1

10^

\

1

1

1

\

I

1

1

10'

1

"Ox*

^""^^ ^** ^ > « * , ^

•"<», .^"^^

1

10'

:

210''

1

Nf;2% [Cycles] Fig. 13. Pure bending (solid line) and pure torsion (dashed line) Wohler fatigue curves with 10% and 90% probability of survival scatter bands (thinner lines).

6082-T6 lUUUUUUU -

Ps=10%

Nf,2%

— ^ ^ y /

[Cycles]

A •

*

1000000-

90% j^

—B— © ^ A •

100000-

•'' '

10000^ t./

10000

^"-y

D^Z^' 1 £.

• • ' • "1

100000

n

•

'

i;

Torsion X>1,6=(3° X<1,6=0° || >i>l,6=90° >i
• ••"•I

1000000

10000000

Nf,e [Cycles] Fig. 14. Experimental fatigue life Nf,2% vs. estimated fatigue life Nf,e.

Multiaxial Fatigue Life Estimations for 6082-T6 Cylindrical Specimens Under ...

101

of survival) were determined by using the p=l and p=0 series (Tables 2 and 3) and under the hypothesis of a log-normal distribution of the number of cycles for each stress level and for the 95% confidence level. By using the measured applied stresses and the actual out-of-phase angles, all the experimental data reported in Tables 2-5 were re-analysed in terms of shear stress amplitude and maximum stress normal to the initiation plane, defined as the plane that experienced the maximum shear stress amplitude. Assuming a linear expression for the TA,Ref (p) and kx(p) functions and using the bending (p=l) and torsional (p=0) fatigue curves for the model calibrations, Eqs. (3) and (4) can be rewritten as: ^A,Ref(p)--10.25-p^76.76

(5)

k^(p)=-1.01-p + 7.99

(6)

Figure 14 shows the correlation of experimental fatigue life Nf,2% vs. estimated fatigue life Nf,e calculated by Eq. 2. In this diagram both the multiaxial fatigue data and the calibration data have been plotted together; calibration data were the bending and torsional tests used to determine Eq. (5) and (6). In this figure the continuous straight lines were used to define the bending scatter band, while the dashed lines delimited the torsional scatter band. Even in this case the scatter bands were determined under the hypothesis of a log-normal distribution of the number of cycles to failure with a confidence level equal to 95%. By observing Figure 14 it is possible to notice that both in-phase and out-of-phase multiaxial data were located within the widest scatter band related either to axial or to torsional data. In particular, predictions concerning in-phase data were always conservative; out-of-phase data having X>1 were positioned in the non-conservative zone, whereas fatigue points having X<1 were located in the conservative zone.

CONCLUSIONS Specimens used in this investigation were manufactured by means of a process which did not allow to obtain a high degree of isotropy in the material. The main consequence of this situation was that 6082 T6 changed considerably its fatigue cracking behaviour as the value of applied ^=Xxy,a/Ox,a changed: when the bending prevailed over the torsion (X<1), cracks grew in MODE I and their initiation occurred at the maximum stress value points; on the contrary, when the applied torque prevailed over the bending {X>1), cracks were always generated over all the gauge surface and oriented mainly along the extrusion direction. Therefore, cracks grew either along the specimen axis or perpendicular to it and this trend was evident both for in-phase and out-of-phase tests. The fatigue life prediction was performed by using the method proposed by Susmel and Lazzarin without taking into account the material anisotropy. The model was calibrated by means of the 50% probability of survival bending and torsional fatigue curves. This method correlated reasonably well with the experimental results, despite the peculiar fatigue cracking behaviour showed by the investigated 6082 T6 aluminium alloy. Finally, two facts can be pointed out to confirm the soundness of the employed fatigue life prediction method: the first one is the ability of the wider scatter band coming from calibration data to contain all the multiaxial data independently of out-of-phase angles, biaxiality ratios and

102

L. SUSMEL AND N. PETRONE

fatigue lives; the second one is the capacity of predicting correct fatigue lives despite the evident degree of material anisotropy due to manufacturing process as commonly showed by real industrial components.

REFERENCES 1.

You, B. R. , Lee, S. B. (1996) A critical review on multiaxial fatigue assessments of metals. Int. J. Fatigue 18 4, 235-244. 2. Socie, D. F . , Marquis, G. B. (2000) Multiaxial Fatigue, SAE. 3. Brown, M. W., Miller, K. J. (1973). A theory for fatigue under multiaxial stress-strain conditions. In: Proc. Inst. Mech. Engrs, Vol. 187, 745-755. 4. Wang, C. H., Brown, M. W. (1993). A path-independent parameter for fatigue under proportional and non-proportional loading. Fatigue Fract. Engng. Mater. Struct. 16 12, 12851298. 5. Socie, D. F. (1987). Multiaxial Fatigue Damage Models. Trans. ASME, J. Eng. Mat. Techn. 109 4, 293-298. 6. Fatemi, A. , Socie, D. F. (1988). A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fatigue Fract. Engng. Mater. Struct. 11 3,149-166. 7. Garud, Y. S. (1981). A new approach to the evaluation of fatigue under multiaxial loadings. Trans. ASME, J. Eng. Mat. Techn. 103,119-125. 8. Ellyin, F. (1989). Cyclic Strain Energy Density as a Criterion for Multiaxial Fatigue Failure. In: Biaxial and Multiaxial Fatigue, EGF 3, Mechanical Engineering Publications, London, 571-583. 9. Ellyin, F., Golos K., Xia Z. (1991). In-Phase and Out-of-Phase Multiaxial Fatigue Trans. ASME, J. Eng. Mat. Techn. 113, 112-118. 10. Lazzarin, P., Zambardi, R. (2001). A finite-volume-energy based approach to predict the static and fatigue behaviour of components with V-shaped notches. Int. J. Fracture 112, 275-298. 11. Gough, H. J. (1949). Engineering Steels under Combined Cyclic and Static stresses. In: Proc. Inst. Mech. Engrs. 160, 417-440. 12. Findley, W. N. (1959). A theory for the effect of mean stress on fatigue under combined torsion and axial load or bending. Trans ASME Ser. B 81, 301-306. 13. Matake, T. (1977). An explanation on fatigue limit under combined stress. Bulletin ofJSME 20, 257-263. 14. McDiarmid, D. L. (1991). A general criterion for high-cycle multiaxial fatigue failure. Fatigue Fract. Engng. Mater. Struct. 14, 429-453. 15. Dang Van, K. (1993). Macro-Micro Approach in High-Cycle Multiaxial Fatigue. ASTM STP 1191, American Society for Testing Materials Philadelphia, 120-130. 16. Papadopoulos, I. V. (1987). Fatigue polycyclique des metaux: une novelle approche. These de Doctorat, Ecole Nationale des Fonts et Chaussees, Paris, France. 17. Papadopoulos, I. V. , Davoli P., Gorla C , Filippini M., Bemasconi A. (1997). A comparative study of multiaxial high-cycle fatigue criteria for metals. Int. J. Fatigue 19 3, 219-235. 18. Papadopoulos, I. V. (2001). Long life fatigue under multiaxial loading. Int. J. Fatigue 23 10, 839-849. 19. Carpinteri, A., Brighenti, R., Spagnoli, A. (2000). A fracture plane approach in multiaxial high-cycle fatigue of metals. Fatigue Fract. Engng. Mater. Struct. 23, 355-364.

Multiaxial Fatigue Life Estimations for 6082-T6 Cylindrical Specimens Under ...

20. Carpinteri, A. and Spagnoli, A. (2001). Multiaxial high-cycle fatigue criterion for hard metals. Int J Fatigue 23, 135-145. 21. Susmel, L., Lazzarin, P. (2002). A bi-parametric Wohler curve for high cycle multiaxial fatigue assessment. Fatigue Fract. Engng. Mater. Struct. 25, 63-78. 22. Lazzarin, P., Susmel, L. (2002). A Stress-Based Method to Predict Lifetime under Multiaxial Fatigue Loadings. Submitted for publication to Fatigue Fract. Engng. Mater. Struct. 23. Quaresimin, M., Susmel, L. (2002). Multiaxial fatigue behaviour of composite laminates. Key Engineering Materials Vols. 221-222, 71-80. 24. Susmel, L., Quaresimin, M. (2001). Multiaxial Fatigue Life Prediction Criteria for Composite Materials. In: Proc. of 6th International Conference on Biaxial/Multiaxial Fatigue & Fracture, Lisboa (Portugal), 379-386. 25. Azzi, V. D., Tsai, S. W. (1965). Anisotropic strength of composites. Experimental Mechanics 5 (9), 283-288. 26. Fawaz, Z. , Ellyin, F. (1994). Fatigue failure model for fibre-reinforced materials under general loading conditions. J. Comp. Mat. 20 15, 1432-1451. 27. Smith, E. W., Pascoe, K. J. (1989). Biaxial fatigue of Glass-Fibre reinforced composite. Part 2: failure criteria for fatigue and fracture. Biax./Multiax. Fatigue EGF 3, 367-396. 28. Lin, H., Nayeb-Hashemi, H., Pelloux, R. M. (1992). Consitutive relations and fatigue life prediction for anisotropic A1-6061-T6 rods under biaxial proportional loadings. Int. J. Fatigue 14 4, 249-259. 29. Lin, H., Nayeb-Hashemi, H., Pelloux, R. M. (1993). A multiaxial fatigue damage model for orthotropic materials under proportional loading. Fatigue Fract. Engng Mater. Struct. 7, 723742. 30. Papadopoulos, I.V. (1998). Critical plane approaches in high-cycle fatigue: on the definition of the amplitude and mean value of the shear stress acting on the critical plane. Fatigue Fract. Engng. Mater. Struct. 21, 269-285. 31. Susmel, L., Taylor, D. (2002). The critical distance method and the modified Wohler curves for the multiaxial fatigue assessment of sharply notched components. In Proc. of 8th International Fatigue Congress, FATIGUE 2002, Stockholm, Sweden, Vol. 3, 1889-1897. 32. El Haddad, M. H., Dowling, N. F., Topper, T. H., Smith, K. N. (1980). Int. J. Fract. 16, 1524.

Appendix : NOMENCLATURE kx

Inverse slope of the modified Wohler curve

Kf

Fatigue strength reduction factor

Nf

Number of cycles to failure

Nf e

Estimated number of cycles to failure

]\[j^^^

Number of cycles assumed as a reference value

Oxyz

Reference frame

Pg

Probability of survival

103

104

L SUSMEL AND N. PETRONE j^

'CA,Ref(Ps=90%)/TA,Ref(Ps=10%). Scatter ratio of reference shear stress amplitude for 90% and 10% probabilities of survival.

A^ 0

Angles which define the position of a generic plane A

^^ 0^ ^ P CTAOC

Angles which define the initiation plane Generic plane Stress ratio related to the initiation plane Fully reversed axial fatigue limit

/A* 0*\

Maximum stress normal to the initiation plane

^

Nominal axial stress amplitude

a x,m T ((b* 0*) ^

Nominal mean stress amplitude Shear stress amplitude on the plane of the maximum shear stress amplitude Fatigue strength corresponding to NRef cycles

T^Aoc

Fully reversed torsional fatigue limit

'^xy,.

Nominal shear stress amplitude

T xy,m

Nominal mean shear stress

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

105

LONG-LIFE MULTIAXIAL FATIGUE OF A NODULAR GRAPHITE CAST IRON

Gary B. MARQUIS and Paivi KARJALAINEN-ROIKONEN Lappeenranta University of Technology, Department of Mechanical Engineering, P.O. Box 20, FIN53851 Lappeenranta, Finland *yTTIndustrial Systems, P.O.Box 1705, FIN-02044 VTT, Finland

ABSTRACT Nodular graphite cast iron is one example of a material that fails in fatigue primarily by the initiation and growth of Mode I cracks. With this in mind, a tensile critical plane damage parameter would be expected to correlate life for different stress states. Long life fatigue tests have been performed on cast nodular graphite iron using uniaxial tension, torsion and equibiaxial loading. Results are compared to a Goodman-type fatigue limit criterion previously developed for this material that successfully correlates data for a variety of stress states: uniaxial tension with and without mean stresses, fully reversed torsion, cyclic torsion with mean shear stress, and cyclic torsion with static normal stresses. The criterion includes a multiaxial correction physically based on the growth of small cracks from notches and small defects, and is able to account for the detrimental effect of the negative second principal stress in torsion. The expected benefit of a positive second principal stress in biaxial tension was not observed in experiments. The equi-biaxial stress state greatly increased the observed tortuosity of fatigue cracking as compared to the uniaxial or torsion stress states. It is assumed that this produced a lower fatigue limit because the crack driving force was nearly equal in all directions thus allowing cracks to link up weaker regions of the complex cast iron microstructure. KEYWORDS Nodular graphite cast iron, fatigue limit, equi-biaxial stress, torsion fatigue, Goodman, critical plane

INTRODUCTION Early damage models for multiaxial fatigue were developed primarily empirically using numerical methods to fit data from two or more stress states. As more knowledge has been gained about the complex growth mechanisms of cracks in uniaxial and complex strain fields, newer damage models have been devised which attempt to capture the essential loading

106

G.B. MARQUIS AND P. KARJALAINEN-ROIKONEN

features which lead to fatigue damage. Small cracks grow in a complicated manner that is affected by crack closure, microstructure and crack size. Linear elastic fracture mechanics assumes that the stress and strain fields surrounding a crack are uniquely determined by the stress intensity factor. However, continuum mechanics assumptions are not valid for cracks of the size of the microstructure because, on this scale, a polycrystalline material does not resemble the ideal homogeneous solid. Closure conditions for small cracks are also different since the crack wake is small. These factors prevent direct modeling with linear elastic or elastic-plastic fracture mechanics. Critical plane models The so-called critical plane approaches to multiaxial fatigue have evolved from experimental observations of the nucleation and growth of cracks during cyclic loading. Models in this class attempt to compute fatigue damage on specific planes within a test specimen or component. A critical plane can, therefore, be defined as one or more planes within a solid subject to a limiting value of some damage parameter. For example, in some models the limiting damage parameter value is associated with the maximum alternating shear strain on a specific plane while other models define as critical the plane experiencing the maximum value of some combination of stress and strain components during a load history. The common element in all critical plane approaches is that fatigue life is considered to be controlled by the combination of stresses and strains acting on a specific critical plane or on a set of critical planes. Socie and Marquis have published a review on critical plane and other approaches to multiaxial fatigue [1]. It has been observed that fatigue life will usually be dominated either by crack growth along shear planes or along tensile planes. Crack growth mode will depend on material, stress state, environment and strain amplitude. A critical plane model will incorporate the dominant parameters governing either type of crack growth. Due to the different possible cracking modes, shear or tensile dominant, no single damage model should be expected to correlate test data for all materials in all life regimes. While critical plane models underscore the important role of crack nucleation and propagation on fatigue, they do not specifically model crack propagation. Normally they describe the complex crack nucleation and microcrack growth mechanisms in a general sense using load or strain values available in most engineering applications. These stress or strain terms are normally those values that are obtained directly from strain gage measurements or finite element analysis. Crack growth from an initial size to failure is assumed, but for the sake of simplicity most models avoid the integration of crack a propagation equation. Successful models are able to predict both the fatigue life and the dominant failure plane(s). Among the early critical plane fatigue damage criteria, the Findley model [2-4] is one of the most refined. Findley suggested that the normal stress, an, on a shear plane might have a linear influence on the allowable alternating shear stress, AT/2.

^ + tonl = / ^

(1)

^max

This model differs from earlier empirical methods [5-7] and methods based on extensions of static yield criteria in that it identifies the stress components acting on a specific plane within

Long-Life Multiaxial Fatigue of a Nodular Graphite Cast Iron

107

the material. Findley identified a critical plane for fatigue crack initiation and growth that is dependent on both alternating shear stress and maximum normal stress. The combined action of shear and normal stresses is responsible for fatigue damage and the maximum value of the quantity in parentheses is used rather than the maximum value of shear stress. Other successful long-life shear stress based models for multiaxial fatigue have the same general form as Eq. (1) and include both a shear stress and a normal stress terms to account for observed mean stress and combined loading effects. Several examples long-life shear stress based models and the shear and normal stress terms employed in these models are listed in Table 1. Of these examples, all except the Sines model fall into the critical plane category because Tresca shear stress is associated with a specific plane.

Model Findley [2-4] Sines [6-7] McDiarmid [8] Dang Van [9]

Table 1. Examples of long-life shear stress based models Shear stress term Normal stress term Tresca Normal stress, a^ Von Mises Hydrostatic stress Tresca Normal stress, a^ (on maximum shear plane) Tresca (microscopic value) Hydrostatic stress

Critical plane models for tension Numerous researchers [1,10,11] have emphasised the need for alternate fatigue damage models depending on whether a material fails predominately due to shear crack growth or due to tensile crack growth. Shear stress based criteria characterised by Eq. (1) have been developed based on observations of crack nucleation and growth in ductile materials. It is commonly found that crack nucleation and early growth occurs along planes of maximum shear stress and that tensile stresses along the plane of maximum shear stress accelerate fatigue damage while compressive stresses along this plane increases fatigue life. In contrast, cast iron and stainless steel under some loading conditions have been shown to be normal stress dominated and, therefore, require different parameters to correlate torsion and tension fatigue data [10, 12-15]. In these materials, micro-cracks may nucleate in shear, but fatigue life is dominated by early crack growth on planes perpendicular to the maximum principal stress or strain. Long-life fatigue of high strength steels may also be dominated by the propagation / nonpropagation of Mode I cracks from non-metallic inclusions of other defects. Smith et al. [16] proposed a suitable relationship that includes both the cyclic strain range and the maximum stress. This model, commonly referred to as the SWT model, was originally developed as a correction for mean stresses in uniaxial loading situations. The model is still widely used and is a common feature in most commercial fatigue analysis software. The SWT model can also be used in the analysis of both proportionally and non-proportionally loaded components fabricated from materials that fail primarily due to Mode I tensile cracking [17]. The SWT model for multiaxial loading is based on the principal strain range, Aei and maximum stress on the principal strain range plane, an,max,2 <^n,max ^

= ^ ( 2 N f )^^ + af 8f (2Nf )^+^

(2)

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G.B. MARQUIS AND P. KARJALAINEN-ROIKONEN

The Stress term in this model makes it suitable for describing mean stresses during multiaxial loading and it also takes into account the significant non-proportional hardening effects that occur in materials like stainless steel. The virtual strain energy (VSE) model of Liu [18] is a straightforward generalization of a uniaxial energy based fatigue life prediction model. However, it differs from most other energy type models in that work quantities are defined for specific planes within the material, and it can therefore be classified as a critical plane approach. The virtual strain energy quantity, AW, on a plane is divided into elastic and plastic work components. Elastic work, AW^, is the sum of the two darker shaded regions in Fig. 1, AaAe^, while the plastic work, AWP, is approximated as the product, AaAeP. Figure 1 shows that the product AaAeP is larger than the plastic hysteresis energy, i.e. the area inside the hysteresis loop. The difference is the nonshaded area outside the hysteresis loop and is commonly termed the complementary plastic work. The axial work per cycle AWj is defined as the sum: AWi = AWP + AWe

(3)

Similarly, the shear work per cycle AWn is defined by substituting AT, AyP and Ay^ for the corresponding normal stress / strain terms.

Fig. 1. Elastic and plastic strain energies for the VSE approach. For proportional multiaxial loading, the VSE model considers two possible failure modes, i.e., a mode for tensile failure and a mode for shear failure. In tension-torsion loading, AW is the sum of axial and shear work, AWj and AWn acting on a plane:

Long-Life Multiaxial Fatigue of a Nodular Graphite Cast Iron

AW = A W i + A W n

109

(4)

Failure is expected to occur on the material plane having the maximum VSE quantity. Depending on material, temperature, and loading, one of the VSE parameters will dominate. For materials that fail primarily along tensile planes, the quantity AW is computed by first identifying the plane on which AWj is maximized and then adding the shear work on that same plane. Both AWj and AWn are virtual quantities whose physical meaning is related but not equal to the elastic strain energy and plastic hysteresis energy. For this reason the model is not defined for out-of-phase or non-proportional loading. Murakami and associates have observed that the fatigue limit for many materials is not equal to the stress for crack initiation but the threshold stress for propagating a crack that emanates from a defect [19-21]. The defects may be small cracks, scratches, inclusions, or porosity. Many of these flaws are irregularly shaped, and it has been proposed to use the Varea as a measure of the flaw size. In torsion, the shear stress around a hole is zero, and cracks nucleate in tension at 45° to the applied shear stresses. Propagation of such cracks will be controlled by the Mode I stress intensity. Crack driving force near a notch is greater in the case of torsion as compared to uniaxial tension. For example, the stress concentration factor for a center hole is 3 in uniaxial loading and 4 in torsion suggesting that torsion fatigue strength would be 75% of the tensile fatigue strength. Figure 2 shows that the Mode I stress intensity factor is higher in torsion (?i = a2 / (7^ = -1) than in tension (K = 0) for a crack originating at the edge of a hole. The fatigue limit for tension must be decreased by the ratio of the stress intensity factors to obtain the estimate of the torsion curve. Using fracture mechanics arguments it has been estimated that the fatigue strength in torsion is 0.8-0.83 of the fatigue strength in tension for materials dominated by Mode I failure from small pores and defects [21-23]. The Murakami model defines fatigue limit values in tension and torsion as

^fl=

1.43(HV-hl20) r—

Tfi = O.SGfi =

1.15(HV+120) ^ r^

(^)

(6)

where HV is the Vickers hardness and varea is determined by projecting the defect onto the plane of maximum tensile stress. So far this parameter has been defined for the cases of tension only and torsion only, but similar facture mechanics arguments can be used for other proportional stress ratios as well. Nodular cast irons are an example of materials that normally fail in cyclic loading in a brittle fashion. Failure in this case is often initiated from shrinkage pores or other casting defects. Even when the applied loading is predominantly torsion, damage in nodular iron is observed to occur along maximum principal stress planes rather than shear planes as small cracks nucleate and propagate from naturally occurring inclusions and shrinkage pores [12-15].

no

G.B. MARQUIS AND P. KARJALAINEN-ROIKONEN

For thick-section nodular iron castings, Mode I fatigue cracks normally initiated from defects several hundreds microns diameter. A Goodman-type fatigue limit relation for nodular iron has been developed to correlate torsion and tension data as well as account for mean stress effects [12,13]. The following fatigue limit relation can be written:

^(5-

1 (l-0.25?i)

2ar

(7)

AGV

/?m+^ipO.2

where Aa is the fatigue limit stress range, am is the mean stress on the plane of maximum alternating normal stress and A a ^ is the R = -1 fatigue limit stress range. The term X is added to distinguish between the uniaxial and torsion load cases based on the difference in small crack driving force. The critical plane in this case is computed based on the combined effect of mean stress and alternating normal stress. In the case of torsion loading with mean torque, the static and mean shear stresses are resolved as alternating and mean tensile stresses.

1.50

K,=FoV7ca

1.00

1.50

2.00

2.50

3.00

a R Fig. 2. Stress intensity factors for cracks emanating from a hole. It is interesting to note that the critical plane models for materials that fail predominantly along tensile planes are generally simple extensions of models developed for uniaxial fatigue. There are no great difficulties in applying these damage parameters to proportional multiaxial loading cases other than uniaxial tension or pure torsion, but it is not immediately clear how they can be applied to more general non-proportional loading. The purpose of the current investigation is to consider the extension of Eq. (7) to proportional equi-biaxial (X, = 1) fatigue of thick-section nodular iron. As has already been seen in Fig. 2, the driving force of cracks emanating from defects in biaxial tension is significantly less than that for tension or torsion. With this in mind, it would be expected that the fatigue

Long-Life Multiaxial Fatigue of a Nodular Graphite Cast Iron

111

limit under equi-biaxial tension would be greater than for either uniaxial tesion or pure torsion. Unfortunately, equi-biaxial tension fatigue data is rarely published. EXPERIMENTS Materials Nodular cast iron is extensively used in the production of vehicles and heavy machinery components. When compared to grey iron, nodular cast irons have significantly higher fatigue strengths at long lives that can be used to great advantage in design. For complex parts, loading is often multiaxial and in some cases is also highly non-proportional. Material in this investigation was a nodular cast iron GRS 500 / ISO 1083 cast as ingots 100x100x300 mm in size. Castings were slow cooled but no post cast heat treatment was performed. Measured tensile properties were /?p0.2 = 340 MPa and /?ni = 620 MPa. Microstructure was predominantly pearlite with small regions of ferrite surrounding the graphite nodules. Figure 3 shows the typical microstructure for the material tested. Fractography of failed specimens cycled just above the fatigue limit has shown that failure is dominated by the nucleation and growth of individual cracks from shrinkage pores near the specimen surface [14,15]. Shrinkage pores several hundred microns in diameter are a common feature of thick-section castings such as the ingots in this investigation. Figure 4 shows the extreme value distribution of maximum defect size for a 3400-mm3 test volume. Defect sizes were approximated as the square root area of the smallest ellipse or semiellipse that just enclosed the defect. Ellipses were used for internal defects while semi-ellipses were used for surface breaking or near surface defects. The vertical axis in Fig. 4 is based on rank probability analysis. Measured defect sizes are ordered from smallest to greatest and assigned a rank number, /, that varies from 1 to n where n is the total number of defects measured. The value Prank is an estimate of the cumulative probability of finding a defect larger than that corresponding to rank number / and is approximated using the formula:

/z + 0.4

The value y = 0.37, therefore, corresponds to the median defect size while y = 6 corresponds to the defect size greater than 99.7% of all defects in the sample [24]. Also shown in Fig. 4 is a similar measured defect distribution for casting defects from a nominally identical material obtained from a different foundry. In this case the median defect size is clearly larger than for the median defect in the ingots and the reduced slope in the resulting regression line is an indication of a greater scatter in the sample. The statistical method described here is essentially identical to the inclusion rating method by statistics of extremes (IRMSE) proposed by Murakami [25] for evaluating the cleanliness of steels. The method has been shown to be especially powerful in its ability to discriminate between different ultra-clean bearing steels and to predict the size of defects larger that those found in a series of inspection domains. Such statistical methods are valuable for quality control monitoring and for specifying maximum allowable stresses for high reliability design.

112

G.B. MARQUIS AND P. KARJALAINEN-ROIKONEN

Fig. 3. Microstructure of the GRS 500 / ISO 1083 iron.

#

2

li

-2 0

500

1000

sqrt(area) in )im Fig. 4. Extreme value distributions of shrinkage pore defects.

Long-Life Multiaxial Fatigue of a Nodular Graphite Cast Iron

113

Specimens Both tubular and solid axial test specimens were used for this study. The cast ingots were first sawn into rectangular bars and finally turned to the dimensions shown in Fig. 5. Axial polishing using successively finer grades of emery paper and diamond paste was used. For torsion loading tubular specimens were chosen to minimize the effect of stress gradients. Either specimen geometry would be suitable for tension loading, but the solid specimens are significantly less expensive to manufacture. It can be noted that both the highly stressed volume and surface area of the tubular specimen are 221 Imm^ and 1206mm2, respectively. Comparable values for the solid specimen are 3392mm^ and 1 ISlmm^. These values are close enough so that size effects are not expected to significantly influence the test results.

136)f

37 •

23

62

-\

f-

20

35

24

012-

i

a)

T 30

y- 16

M22x1

b)

Fig. 5. a) Torsion/biaxial and b) axial test specimens (Dimensions mm). Testing Loading of the tubular specimens was stress controlled cyclic torsion or equi-biaxial tension, i.e., proportional axial load with internal pressure. The stress ratio for both torsion and biaxial testing was R = 0.1. Solid specimens were used for the uniaxial tests at a constant tensile mean stress of 182 MPa. Loading continued until a fatigue crack propagated through the 2.0 mm thick wall of the specimen or until 2 x 1 0 ^ fatigue cycles had been reached. Surface length of the cracks at failure was 3 to 10 mm. The test matrix is shown in Table 2. Table 2. Test matrix Stress state

\-(32l<3\

torsion

-1

biaxial tension uniaxial tension

^-^\m\n^^\

^ (MPa)

0.1

variable

0.98

0.1

variable

0

variable

182

114

G.B. MARQUIS AND P. KARJALAINEN-ROIKONEN

RESULTS AND DISCUSSION Fatigue life Fatigue test data are shown in Fig. 6. At comparable fatigue lives, the specimens subject to torsion loading failed at noticeably lower stress levels than did the uniaxial tension specimens. The fatigue limit was approximately 78% of the tensile fatigue limit. Under equi-biaxial loading the nodular iron had nearly the same fatigue limit as under uniaxial loading. It can be noted that the uniaxial test data in this figure was based on constant mean stress testing at a mean stress level of 182 MPa as compared to the constant R = 0.1 testing used for the other two stress states. For this reason the uniaxial curve required some adjustment. At the fatigue limit, however, the mean stress resulted in a stress ratio very similar to that used for the other stress states. The fatigue limit for uniaxial tension shown is based on Eq. (7) and the finite life curve is based of an extensive test program using this material [15]. Specific R values for each set of tension fatigue data are shown in Fig. 6.

250

200 CO QL

3

=

150

Q.

E

< CO

en

%

100 U

10

10

10

10

Number of Cycles to Failure

Fig. 6. Experimental results for tension, equi-biaxial, and torsion fatigue.

Long-Life Multiaxial Fatigue of a Nodular Graphite Cast Iron

115

Crack observations For all stress states tested, fatigue cracks in nodular iron specimens nucleated and propagated on maximum principal stress planes. Typical fatigue cracks under uniaxial tension, torsion and equi-biaxial tension are shown in Fig. 7. Cracks for torsion or tensile loading showed some tortuosity as the crack linked up numerous microstructural features, i.e., graphite nodules or shrinkage pores. However, it can be observed from Fig. 7 that the crack direction is fairly well defined. During equi-biaxial loading the cracks grew generally in the plane normal to the hoop stress, but in comparison to the torsion or tensile cracks there is significantly more branching and tortuosity. In these tests the stress state was nearly equi-biaxial. Both numerical calculations and strain gage measurements showed that the axial stress was only about 2% greater than the hoop stress. The Mode I crack driving force was thus nearly equal in all directions. For comparison, a torsion fatigue crack for normalised low carbon steel C45 tested in completely reversed torsion is shown in Fig. 8. Material was received as 40 mm diameter bar stock and later machined into the tubular specimens shown in Fig. 5a. Measured hardness was BHN=193. As the figure shows, small cracks in this C45 initiate on both 0° and 90° planes for torsion only loading. This is a well-documented phenomenon for ductile metals. Crack growth along the specimen surface is via Mode n crack growth and into the depth of the material by Mode in growth. After shear cracks reach a length of several hundred microns, crack branching along ±45° planes occurs. For near-fatigue limit testing, the crack branching occurs rather late in the total life of the material. Cracks produced by uniaxial tension fatigue are similar for both C45 and nodular cast iron, i.e. initiation and propagation is macroscopically always normal to the direction of maximum principal stress. For this reason it is very difficult to distinguish between shear and tensile dominated materials based only on axial tension fatigue loading. Effect of second principal stress on fatigue While the dominant failure mechanism of the nodular iron is due to Mode I crack growth, the fatigue limit stresses for torsion and equi-biaxial tension cannot be compared directly to the uniaxial fatigue limit without additional considerations of the stress state. Because failure in nodular iron is controlled by the nucleation and growth of small cracks from inclusions and pores, the second principal stress influences the fatigue strength. As previously discussed, crack driving force near a notch is greater in the case of ?i= -1 as compared to X^O. Endo and Murakami [22] and Beretta and Murakami [23] used fracture mechanics arguments to estimate the fatigue strength in torsion to be approximately 80-83% the fatigue limit in tension for materials dominated by Mode I failure from small defects. In materials where the fatigue limit strength is controlled by the propagation / nonpropagation of cracks nucleating at individual small holes or inclusions, it may be expected that the biaxial fatigue limit is higher than that for uniaxial fatigue, but the authors are not aware of experimental work on this subject. Once cracks have nucleated and growth can be modelled by linear elastic fracture mechanics, the role of the second principal stress is reduced. While only limited data is available, the trend is that transverse stress only marginally affects the Mode I crack propagation. Brown and Miller [26] reported the effect of biaxial stress on Mode I crack growth for AISI316 stainless steel. At low stresses the effect of the transverse stress

116

G.B. MARQUIS AND P. KARJALAINEN-ROIKONEN

a) Gi

D

02=0

b)

O'

02 =-01

C)

t

D

G2 = a i

Fig. 7. Fatigue cracks during a) axial tension, b) torsion, and c) equi-biaxial tension loading.

Long-Life Multiaxial Fatigue of a Nodular Graphite Cast Iron

117

0^

02 = -ai

Fig. 8. Fatigue cracks during torsion of C45 low carbon steel. is small with the biaxial tension (X = 1) having the lowest growth rate. This may be related to the slightly smaller plastic zone size for this loading case. The difference between the growth rates increases as the stress levels increase. Only as the nominal stress approached the yield strength did the above Authors find more significant differences in growth rates. Finite element analysis shows that under these high loading stresses, the plastic zones are much larger for X= -1 than those estimated from the elastic solution. In an extensive series of biaxial loading experiments on 2024 and 7075 aluminium alloys, Liu et al. [27] obtained similar results. Applied stresses were between 20 to 60 percent of the material tensile strength and the biaxial stress ratios ranged from -1.5 to 1.75. Mode I crack growth rate for this material was not affected by the transverse stress and all of the crack growth rate data fell within a single scatter band. The current study shows that the fatigue limit for this material is nearly the same for both uniaxial and equi-biaxial tension. By contrast the fatigue limit during torsion fatigue was significantly lower. Under equi-biaxial loading the crack driving force was nearly equal in all directions, so failure in nodular iron is not due to nucleation and crack propagation in a single plane but cracks are able to progress freely with no strong directional preference. All planes normal to the surface of the specimen are critical planes. It is well known that fatigue cracks always form and propagate in the weakest region of a material subject to high stresses. In his keynote paper on metal fatigue. Miller [28] has discussed the issue of non-propagating cracks and the fatigue limit both for notched and smooth specimens. Cracks may cease to grow: (1) as a result of a stress gradient and reduced crack tip driving force in the region of a stress concentration, or (2) as a result of the crack tip reaching some microstructural barrier. In an equi-biaxial stress field a fatigue crack will not have a strong preferential direction. It is therefore reasonable to assume that cracks can more easily grow around microstructural features that might otherwise halt or greatly retard crack advance. For this reason, the fatigue limit in biaxial tension for nodular iron is lower than what is expected based on simple elasticity considerations. Increased crack tortuosity for nearly equi-biaxial loading as compared to tension or torsion is clearly seen in a comparison of cracks in Fig. 7.

118

G.B. MARQUIS AND P. KARJALAINEN-ROIKONEN

Based on the current experiments, Eq. (7) should be modified for nodular iron so that stress ratio effect takes on a maximum value equal to zero regardless of the stress state. In other words, the fatigue limit is reduced for biaxial stress ratios less than zero but is not correspondingly increased for biaxial stress ratios greater than zero. The fatigue limit for this material cannot exceed that observed during uniaxial tension fatigue. Equation (7) is modified in the following way: 1

Aa:

[(1-0.25K)

K= ^ K= 0

2a m /?ni+V-2

I ^1

(9)

AGV

for^<0 forX>0

Figure 9 shows the current data together with previously collected uniaxial tension and torsion fatigue limit data for the GRS 500 / ISO 1083 nodular iron [12]. The line representing Eq. (9) allows the fatigue limit for this iron to be estimated based on static tensile properties and the measured endurance limit for one stress state, e.g., fully reversed axial fatigue. The equation includes modification factors for both mean stress and stress state. Based on Eq. (9), fatigue limit data for the three stress states reported here fell along a single line.

T [A(^/(Rp0.2 + Rm)]*0-0.25;i)

Current data O Torsion

AOw / (RpO.2 + Rm)

OEqui-biaxial

•:— R = 0.1

• Tension Previous data • Tension A Torsion

-e-+ -1

-0.5

0

0.5

1

[2aTi/(Rpo.2 + Rm)]*(1-0.25^)

Fig. 9. Predicted and measured mean fatigue limit values for nodular iron for different mean stresses and stress states.

Long-Life Multiaxial Fatigue of a Nodular Graphite Cast Iron

119

As previously discussed, the driving force for a crack emanating from a small defect in equibiaxial tension is significantly less than that for tension or torsion. It was expected that this would produce a correspondingly higher fatigue limit. For the nodular cast iron tested here, this was not observed. Cast irons represent materials with very complex microstructures. The relatively strong pearlite phase is interspersed with graphite nodules that are very weak in fatigue. With cracks free to propagate in any direction, it is assumed that cracks change directions as needed as they link up closely spaced weak regions of the microstructure. In the future, it would be of interest to perform this type of testing for a more homogeneous material with small and widely spaced defects. CONCLUSIONS Long-life fatigue tests of nodular graphite cast iron have been performed under uniaxial tension, torsion and nearly equi-biaxial tension (k = C52 I <3\ = 0.98). Test data for torsion loading in the long but finite life regime was significantly below the uniaxial fatigue data, and the fatigue limit was approximately 78% of the uniaxial fatigue limit. The nodular iron had nearly the same fatigue limit as under both biaxial and uniaxial loading. Fatigue cracks in biaxial fatigue were significantly more tortuous than in either torsion or uniaxial fatigue due to the nearly equal driving force in all directions. A fatigue limit relation previous developed for this material under tension and torsion loading with a variety of mean stress levels is slightly modified to include biaxial loading. In the future, tests involving plane strain would be of interest both because it is common in engineering design and because it is somewhere between uniaxial loading and equi-biaxial tension. Most critical plane damage parameters have been developed for ductile materials that fail by shear crack development. Several critical plane fatigue parameters suitable for tensile damage materials are discussed. These have primarily been developed for uniaxial load cases. Their application to proportional loading is possible, but further development is required before they can be applied to general non-proportional loading. ACKNOWLEDGEMENTS The authors are grateful to H. Laukkanen from VTT Industrial Systems for his careful assistance in executing the fatigue tests. Experimental work was partially supported through the project Fadel/Gjutdesign, which is funded by the Nordic Industrial Fund, Wartsila Technology, Metso Corp., Componenta Cast Components, Componenta Pistons and VTT Industrial Systems. REFERENCES 1. 2. 3.

Socie, D. F. and Marquis, G. B. (2000) Multiaxial Fatigue, SAE, Warrendale, PA. Findley, W. N. (1959), A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending, J. Engng Industry, pp. 301-306. Findley, W. N. and Mathur, P. N. (1956) Modified theories of fatigue under combined stress, Proc. Soc. Exp.. Stress Anal, 14, pp. 35-46.

120

4.

5. 6. 7. 8.

9.

10. 11. 12. 13.

14.

15. 16. 17.

18.

19. 20. 21.

G.B. MARQUIS AND R KARJALAINEN-ROIKONEN

Findley, W. N. (1953) Combined Fatigue Strength of 76S-T61 Aluminum Alloy with Superimposed Mean Stresses and Correction for Yielding, Tech. Note 2924, NACA Washington DC. Gough, H. J. and Pollard, H. V. (1935) The strength of metals under combined alternating stresses, Proc. Inst. Mech. Engrs., 131, pp. 3-103. Sines, G. (1959) Behavior of metals under complex static and alternating stresses. In: Metal Fatigue, G. Sines and J.L. Waisman (Eds.) McGraw Hill, pp. 145-169. Sines, G. (1955) Failure of materials under combined repeated stresses with superimposed static stresses. Tech. Note 3495, NACA, Washington DC. McDiarmid, D. L. (1994) A shear stress based critical-plane criterion of multiaxial fatigue failure for design and life prediction", Fatigue Fract. Engng. Mat. Struct. 17, pp. 1475-1485. Dang-Van, K. (1993) Macro-micro approach in high-cycle multiaxial fatigue. In: Advances in Multiaxial Fatigue, ASTM STP 1191, D.L. McDowell and R. Ellis (Eds.) ASTM, Philadelphia, pp. 120-130. Socie, D. F. (1987) Multiaxial fatigue damage models, J Eng. Mat. Tech. (ASME), 109, pp. 293-298. Carpinteri, A. and Spagnoli, A. (2001) Multiaxial high-cycle fatigue criterion for hard metals. Int. J. Fatigue, 23, pp. 135-145. Marquis, G. and Socie, D. (2000) Long-life torsion fatigue with normal mean stresses Fatigue Fract. Engng. Mater. Struct. 23, pp. 293-300. Marquis, G. (2000) Mean stress in long-life torsion fatigue. In: ECF 13 Fracture mechanics: applications and challenges, M. Fuentes, M. Elices, A. Martin-Meizoso and J. M. Martinez-Esnaola (Eds.), Elsevier Science, Amsterdam. Marquis, G., Rabb, R. and Siivonen, L. (1999) Endurance Limit Design of Spheroidal Graphite Cast Iron Components Based on Natural Defects, In: Fatigue Crack Growth Thresholds, Endurance Limits, and Design, ASTM STP 1372, J. C. Newman and R. Piascik (Eds.) ASTM, West Conshohocken, pp. 411-426. Marquis, G., and Solin J. (2001) Long-life Fatigue Design of GRP 500 Nodular Cast Iron Components, VTT research notes 2043, Espoo, Finland. Smith, R. N., Watson, P. and Topper, T. H. (1970) A stress-strain parameter for the fatigue of metals, J. Mater., 5, pp. 161-US. Socie, D. F. (1993) Critical plane approaches for multiaxial fatigue damage assessment. In: Advances in Multiaxial Fatigue, ASTM STP 1191, eds. D. L. McDowell and R. Ellis, ASTM, Philadelphia, pp. 7-36. Liu, K. C. (1993) A method based on virtual strain-energy parameters for multiaxial fatigue life prediction. In: Advances in Multiaxial Fatigue, ASTM STP 1191, eds. D. L. McDowell and R. Ellis, ASTM, Philadelphia, 1993, pp. 67-84. Murakami, Y. and Endo, M. (1994) Effects of defects, inclusions and inhomogeneities on fatigue strength. Int. J. Fatigue, 16, pp. 163-182. Murakami, Y. and Endo, T. (1980) Effects of small defects on the fatigue strength of metals. Int. J. Fatigue, 2, pp. 23-30. Abe, M. Ito, H. and Murakami, Y. (1990) Tension-compression and torsion low-cycle fatigue of maraging steel. In: Fatigue 90, H. Kitagawa and T. Tanaka (Eds.), MCE Publications, Birmingham, pp. 1661-1666.

Long-Life Multiaxial Fatigue of a Nodular Graphite Cast Iron

22. 23.

24.

25.

26.

27. 28.

121

Endo, M. and Murakami, Y. (1987) Effects of an artificial small defect on torsional fatigue strength of steels, / £ng. Mat. Tech. 109, pp. 124-129. Beretta, S. and Murakami, Y. (1998) The stress intensity factor for small cracks at micro-notches under torsion. In: ECF 12 Fracture from defects, M. W. Brown, E. R. de los Rios and K. J. Miller (Eds.), EMAS, pp. 55-60. Wallin, K. (1999) The probability of success using deterministic reliability. In: Fatigue Design and Reliability, ESIS Publication 23, G. Marquis and J. Solin (Eds.), Elsevier Science, Oxford, pp. 39-50. Murakami, Y. (1994) Inclusion rating by statistics of extreme values and its application to fatigue strength prediction and quality control of materials, J Res. National Inst Std Tec/i. 99, pp. 345-351. Brown M. W. and Miller, K. J. (1985) Mode I fatigue crack growth under biaxial stress at room and elevated temperature, In: Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown (Eds.), ASTM, Philadelphia, pp. 135-152. Liu, A.F., Allison, J.A., Dittmer, D.F., and Yamane, J. (1979) Effect of biaxial stresses on crack growth. In: Fracture Mechanics, ASTM STP 677, C. W. Smith (Ed.), pp. 5-22. Miller, K. J. (1991) 27^*" John Player lecture: Metal fatigue - past current and future, Proc. Inst. Mech. Engineers, London, 14 p.

NOMENCLATURE Varea

square root area of a defect projected onto maximum normal stress plane, |im

b BHN

Basquin exponent Brinell hardness

c E

Coffin-Manson exponent elastic moduus (MPa)

HV k Nf Prank

Vicker's hardness, kgf/mm^ normal stress sensitivity coefficient, Findley parameter cycles to failure cumulative defect size probability based on rank probability analysis

R /?po 2 R^ AW AWj AWjj AW^ AW^ ef' Ae^ AeP Ay X

stress ratio, a ^ i n / ^max 0.2% offset yield strength ultimate tensile strength total virtual stain energy on a plane (VSE model) virtual tensile stain energy on a plane (VSE model) virtual shear stain energy on a plane (VSE model) elastic work during a proportional cycle (VSE model) approximate plastic work during a proportional cycle (VSE model) fatigue ductility coefficient range of maximum principal normal strain range of normal plastic strain for proportional loading shear strain range biaxial stress ratio, (^2 / CT}

122

G.B. MARQUIS AND P. KARJALAINEN-ROIKONEN Gn

normal stress on a plane

(5{ Gfi

fatigue strength coefficient fatigue limit stress amplitude for axial loading, R = -1

^fl,R a^i

fatigue limit stress amplitude for axial loading, R ^ -1 mean stress on the plane of maximum alternating normal stress

^2' ^1 Aa Aa^ Tfi Ax

principal stresses fatigue limit stress range at mean stress (5^^ fatigue limit stress range a ^ = 0 fatigue limit stress amplitude for torsion loading shear stress range

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

123

THE INFLUENCE OF STATIC MEAN STRESSES APPLIED NORMAL TO THE MAXIMUM SHEAR PLANES IN MULTIAXIAL FATIGUE Rebecca Peace KAUFMAN and Tim TOPPER Department of Civil Engineering, University of Waterloo 200 University Ave W, Ontario, Canada

ABSTRACT Tubular specimens of two different hardnesses of SAE 1045 steel (BHN 456 and 203) were tested in axial tension-compression together with alternating internal/external pressure. For the SAE 1045 steel, BHN 456, static mean stresses ranging from ^ 0 0 MPa to 740 MPa were applied normal to the maximum shear stress amplitude planes together with alternating shear stress amplitudes from 150 MPa to 1000 MPa. For the SAE 1045 steel, BHN 203, static mean stresses ranging from - 6 0 MPa to 100 MPa were applied normal to the maximum shear stress amplitude planes together with alternating shear stress amplitudes from 125 MPa to 220 MPa. An approximately linear relationship was found between the applied normal static mean stress and the maximum cyclic shear stress on the critical shear planes for a given fatigue life. The fatigue life remained constant with increasing mean stress for tests with constant shear stress amplitude values and static tensile mean stresses larger than 500 MPa and 76 MPa for the hard and soft steel, respectively. It was assumed that, for static mean stresses larger than these values, the crack faces were fully separated thus allowing unhindered Mode II displacement, a condition defined as interference free crack growth by Bonnen and Topper [1]. A confocal scanning laser microscope was used to determine the validity of this assumption. With this apparatus, the crack depth profiles were measured as a function of the magnitude of the static mean stress applied normal to the shear plane. The 2-D line profiles of the fracture surfaces were obtained to investigate the impact of normal static mean stresses on asperity height and shape [2]. Current critical plane theories were investigated to determine their ability to predict fatigue life for alternating shear stresses and static mean stresses normal to the maximum shear planes. A modified Findley parameter gave the best fit to the experimentally obtained data. To include the interference free condition in the parameter, tensile static mean stresses larger than the interference free condition were replaced with the normal static mean stress that resulted in the interference free stress state in the modified Findley parameter. KEYWORDS Multiaxial fatigue, static mean stress, interference free crack growth.

124

R.P. KAUFMAN AND T.H. TOPPER

INTRODUCTION It has been observed that the fatigue resistance of machine parts can be increased by producing compressive residual stresses at the surface of a component [3-7]. The action in fatigue of the residual stresses in an engineering component should be the same as that of a mean stress in a test specimen if the stress state in the component is similar to that in the test specimen. Hence, the fatigue behavior of test specimens subjected to alternating shear stresses and static mean stresses normal to the maximum shear planes can be used to predict the effect of residual surface stresses in multiaxial fatigue. Residual stresses induced by surface heat treatments, such as induction hardening, result in static mean stresses normal to the maximum shear planes. Therefore, the stress state in the test specimens used during this program of study is similar to that of engineering components with these residual stresses. The purpose of this paper is to investigate the effects of static mean stresses normal to both shear planes and compare the effects with those predicted by current static mean stresses theories and a proposed static mean stress model. Current multiaxial fatigue life prediction techniques can be categorized into three cases: equivalent stress/strain, energy and critical plane approaches. The equivalent stress/strain approaches are easy to implement and allow the wealth of uniaxial data to be used in making multiaxial life predictions. Most energy-based methods utilize the plastic energy of stress/strain hysteresis loops as a correlating parameter and are insensitive to static mean stresses applied normal to the maximum shear planes. The critical plane approaches are based on the mode of crack initiation. For these approaches, a damage parameter based on a combination of shear stress/strain and/or normal stress/strain is computed on a number of planes. The plane with the highest value is taken to be the critical plane [8]. Some of the multiaxial critical plane static mean stress theories that the author has found to date are the following; Findley parameter Ppindley ==^max+/^«

(^)

where r ^ a x is the alternating shear stress, cr„ is the normal stress on the maximum shear plane and / is a constant for a given number of cycles to failure [4]. Several other parameters are very similar to Eq. 1 [3, 9-13]. Modified Kandil, Brown and Miller parameter PMKBN * = rmax + ^^ * ^n

(2)

where ;^max i^ ^^^ maximum shear strain and £*„ is the strain normal to the maximum shear plane. The value of S* is an empirical constant used to condense the experimental data into a parameter vs fatigue life curve [14]. Fatemi and Socie [15] Armax ^FS ~"

^

_. max 'n

(3)

The Influence of Static Mean Stresses Applied Normal to the Maximum Shear Planes in ...

125

where ^YmsiX ^^ ^^e largest shear strain range, (J„"^^^ is the maximum stress normal to the shear plane and dy is the yield stress of the material. Second Modified Kandil, Brown and Miller parameter [16] PMKBM

= A?^max / 2 + Af „ + C7„^ / E

(4)

where A^^ is the normal strain range and a^^ is the mean stress perpendicular to the maximum shear strain amplitude. A static tensile mean stress normal to one of the planes of maximum shear strain amplitude in a tension torsion machine has been shown by a number of investigators to reduce torsional fatigue strength [17, 18]. However, there is only a very small amount of data to document an increase of fatigue strength in shear due to static compressive stresses normal to the two planes of maximum shear stress range (if the compressive stress is applied to only one maximum shear plane, failures continue to occur on the other maximum shear plane at the same shear stress range). Sines and Ohgi [19] in an extensive 1981 review of fatigue under combined stresses show only three data points to support their conclusion that static compressive stresses normal to the planes of maximum shear are beneficial. Two of these points come from a reanalysis of tests on notched specimens by Seeger [20] in which the notch constrained cracks from growing on the second maximum shear plane. Other evidence of the beneficial effect of compressive normal stresses on shear fatigue that we have uncovered consists of crack growth experiments by Smith and Smith [21] who reported that a compressive normal stress decreased shear crack growth rates, and work by Bums and Parry [22] who showed that a superimposed hydrostatic pressure which introduced compressive stresses on the planes of maximum alternating shear increased fatigue strength. Although it was assumed by Sines [6] and subsequent authors [23, 24] that for a given fatigue life, there is a linear relationship between the maximum alternating shear stress and the static mean stress normal to the planes of maximum alternating shear for both tensile and compressive normal stresses, there is little data to support linearity in the compressive region. Recent work by Bonnen [25] and Varvani-Farahani [26] which relates the effect of tensile mean stress to crack face interference suggests that, once a tensile mean stress opens the crack so that there is no longer interference, higher tensile mean stresses will not cause a further reduction in fatigue strength. Both authors obtained interference free crack growth by combining the application of periodic Mode I overloads with an alternating shear stress. Bonnen verified that the crack faces were growing free of interference with each other by performing two tests with different overloads for a given shear stress amplitude. For the second test, the Mode I overload was increased to almost double the overstrain level of the first test. The equivalent fatigue life of the second test was the same as the first result and, consequently, crack face interaction was assumed to be at the lowest possible level. Varvani-Farahani measured biaxial crack opening stress with the confocal scanning laser microscope (CSLM). Biaxial specimens were internally pressurized to separate the crack faces, and increasing magnitudes of internal pressure were applied until the crack depth remained constant. His CSLM imaging revealed that, with the application of the Mode I overstrains (yield stress magnitude), the shear cracks were fully open at zero internal oil pressure. The research completed by Bonnen and Varvani-Farahani is summarized in Fig. 1, where

126

R.P KAUFMAN AND T.H. TOPPER

Alternating

Spring Force

Alternating Periodic Overload Fig. 1. Schematic of crack face interference free conditions. The magnitude of the periodic overload stress is significantly larger than the clamping force stress (spring force stress). the portion of the alternating shearing stress acting on the crack tips is dependent on the magnitude of the Mode 1 periodic overloads and the spring force of the material. Fig. 1 shows that the application of a large enough periodic Mode I overload causes an interference free condition to result. The magnitude of the periodic overload affects the amount of the total shearing force acting at the crack tips. Large periodic compressive overloads compress the asperities in the crack wake, thereby reducing crack closure/interference and increase the crack growth rate. Large periodic tensile overloads (yield point magnitude) leave shear cracks fully open. Once closure free crack growth is achieved, further increasing the magnitude of the tensile or compressive applied periodic overload (Mode I) does not decrease the fatigue life or increase the crack growth rate. Data from a test machine that uses internal and external pressure and axial loading on a tubular specimen to produce a wide range of static compressive and tensile mean stresses normal to the maximum alternating shear stress planes is presented in the following. It is expected that tensile mean stresses normal to one or both maximum shear planes will decrease the frictional force or surface interference on the shear plane which will result in a decrease in the fatigue life for a given applied alternating shear stress range. Once the normal static tensile mean stresses (5j„() are sufficiently large to fully separate the crack faces, the fatigue life will remain constant for larger tensile static mean stresses (astatic Mean)- The proposed model of the effect of static mean stresses together with alternating shear stress for a given fatigue life is illustrated in Fig. 2.

Material The material used during this program of study was SAE 1045 steel with two Brinell hardness numbers (BHN 456 and 203). The first hardness level, BHN 456, was chosen because its ratio of yield strength to fracture strength is high, and the onset of ratcheting in load control is delayed

The Influence of Static Mean Stresses Applied Normal to the Maximum Shear Planes in ...

127

Amplitude

int

int

Mean

Mean

Mean Fig. 2. Proposed static mean stress model for a given fatigue life [26].

compared to softer steels. This hardness level is similar to that of the case of induction hardened shafts tested during the next stage of this program of study. The second hardness, BHN 203, was chosen to match the material used in previous research by Bonnen [25] and Varvani-Farahani [26]. The material was in the form of 65 mm diameter round stock. The steel had the following composition (Wt%): 0.46 C, 0.17 Si, 0.81 Mn, 0.027 P, 0.0235 S and the remainder Fe. Longitudinally oriented (rolling direction) sulphide inclusions of approximately 0.1 to 0.3 mm were also noted [27].

Testing Apparatus The biaxial rig used for this program of study [28] includes two servo-controlled loading systems as is shown in Fig. 3. Specimens can be subjected to tension-compression loading in or out-ofphase with fluctuating internal-external pressure. The maximum load in the axial direction is 111.2 kN and the maximum pressure is 21 MPa. Each loading channel, axial and hoop, is independently controlled by two conventional servo-controlled, cyclic loading systems. Due to the independent control of the principal stresses and strains, both proportional straining at all stress ratios and non-proportional straining are possible.

128

R.P. KAUFMAN AND T.H. TOPPER

Fig. 3. Biaxial fatigue system [28].

Specimen Design Two different specimen designs were used in this study. Specimen design (A) shown in Fig. 4 was developed by Elkholy [29]. During this study, specimens with a 0.45 mm wall thickness and a 14 mm gauge length were used for the 456 BHN material. Stresses up to -800 MPa were applied without buckling. To achieve stresses larger than -800 MPa, a wall thickness thinner than 0.45 mm was required, since the specimen design (A) failed in buckling at this stress level. Consequently, a new specimen design was developed to achieve higher stresses without buckling.

R=36 14 mm , gauge length A

R = 25 2 nmi gauge length B

Fig. 4. Specimen designs A and B.

In the present test program, the stresses in the critical gauge length were restricted to the linear elastic region of metal behavior. Consequently, a linear elastic finite element program, IDEAS by SDRC (version 7m), was used to determine the stresses and strains in the 2 mm gauge length of specimen B. The finite element model was used to determine the relationship between the imposed end loads, the applied pressures, the stresses and the strains in the gauge length. These results were verified experimentally with internal and external 2 mm, 0°-45°-90° rosette strain gauges. A maximum difference of 4% was found between the experimentally obtained strains and the FEM analysis. These differences can be primarily attributed to neglecting the variations in wall thickness in the gauge length of the specimens. Following machining, the thickness of the wall of the tubular specimens varied by as much as 0.03 mm.

The Influence of Static Mean Stresses Applied Normal to the Maximum Shear Planes 129 in . The tubular specimens (designs A and B) were cut using a numerically controlled lathe. The exterior surfaces of the specimens were polished to a 6 |Lim finish. A surface finish of 12 to 16 |Lim was measured on the interior wall of the specimen after honing. Testing and Results The following loading sequences were applied to the biaxial specimens during the testing program. Pure Shear, 0 Static Mean Strain Tests. Parts (a) and (b) of Fig. 5 show an example of the loading history applied to a specimen to obtain a pure shear strain cycle on the maximum shear 0.005

-]

(a) 0.004

; '\ j \ ; \ / \ i \ j \ -

0.003

0.002

•

0 <

0.002

0.003

A

-I

^

V|

A

S

P

r—\ /—A ./ \

/—-\

V

^

A

/*

\

^

V

2

-j-

^

-4

Li. - S i r a " ' " • ^ " " ^ ^ ^axial

£hoop

^"°"

t

i-

/—\ V —

J-

3

/3

y

^

S.ra.ninax.al d ^ r ^ c ^

^axial

[[h

(b)

^ho(

7/2

0

£[

Max 450 Shear Plane

Fig. 5. Schematic of pure shear, zero mean stress loading.

130

R.P. KAUFMAN AND T.H. TOPPER

plane. A Mohr's circle representation in part (c) shows that the maximum shear plane is at 45° to the principal axes. The strain state on the maximum shear plane is illustrated in (d). A total of 15 pure shear tests were conducted on specimens of BHN 456. Table 1 lists the alternating shear stress amplitudes that were applied on the maximum shear planes and the resulting number of cycles until a crack depth of approximately 50 |Lim or final separation of the specimen occurred. Specimens that reached 1x10^ cycles without failing were removed and deemed to be run-outs.

Table 1. Alternating shear test results for SAE 1045 steel, 456 BHN. Test No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Shear Stress Amplitude on Maximum Shear

Mean Stress on the Maximum Shear Plane

Plane (AT/2) (MPa)

(amean) ( M P a )

335 344 352 387 387 438 433 465 464 437 515 519 497 609 642

0 0 -12 -36 23 -5 -13 -10 3 2 6 -5 0 5 0

Fatigue Life Cycles to Failure 1 000 000 (run-out) 1 000 000 (run-out) 1 000 000 (run-out) 1 000 000 (run-out) 100467 76839 72602 61620 45125 34925 20691 20299 15000 12191 8222

Zero mean stress fatigue tests for SAE 1045 steel, BHN 203, with varying levels of cyclic alternating shear stress were reported by Bonnen [25].

Alternating Shear Stress with Superimposed Static Mean Stresses. A total of 47 (No 16 to No 62 in Table 2) alternating shear stress tests with static tensile mean stresses and 15 (No. 63 to No. 77 in Table 2) tests with static compressive mean stresses normal to the maximum shear planes were performed for BHN 456. For BHN 203, a total of seven (No 1 to No. 7 in Table 3) alternating shear stress tests with static normal tensile mean stresses and five (No. 8 to No. 12 in Table 3) tests with static normal compressive mean stresses were performed. For the as received SAE 1045 steel (BHN 203), a run-out was listed if the specimen did not fail before 2x10^ cycles. Tables 2 and 3 list the results for the static mean stress fatigue tests with alternating shear stress for the hardened and as received SAE 1045 steels.

The Influence of Static Mean Stresses Applied Normal to the Maximum Shear Planes in .

131

Table 2. Fatigue tests with static mean stresses and fully alternating shear stresses for SAE 1045 steel, BHN 456. Test No.

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

Shear Stress

Mean Normal Stress

Amplitude (AT/2)

(amean) ( M P a )

(MPa) 152 175 210 160 148 167 197 176 175 265 199 241 194 251 305 192 323 238 271 268 358 483 310 251 237 308 371 446 266 288 345 287 301 295 261 420 456 254 327

491 390 360 500 500 502 392 500 497 350 665 360 635 405 360 584 369 684 405 547 374 106 411 450 449 711 270 212 500 390 414 428 646 741 495 350 403 456 566

Cycles to Failure

1 000 000 (run-out) 1 000 000 (run-out) 1 000 000 (run-out) 274363 233000 107036 62381 60000 51000 48624 46477 37000 35000 31500 31435 29606 29559 25640 24517 17913 17577 15956 15320 15277 13148 12818 12683 12292 12104 10000 9132 6578 6477 5730 4971 4798 4533 4507 4468

132

R.P. KAUFMAN AND T.H. TOPPER

Test No.

Shear Stress Amplitude (AT/2)

Mean Normal Stress (Gmean) (MPa)

Cycles to Failure

382 390 364 379 395 701 523 414

3742 3742 3317 2150

-382 -318 -150 -119 -292 -185 -123 -388 -138 -275

1 000 000 (run-out) 1 000 000 (run-out) 1 000 000 (run-out) 350000 264192 142000 104000 76907 55000 38789 30041 26000 22000 17000 12643

(MPa)

55 56 57 58 59 60 61 62 63 64 65

J

66 67 68 69 70 71 72 73 74 75 76 77

429 450 500 506 580 408 432 652 556 505 420 431 540 610 448 574 483 547 451 591 553 623 730

-97 -305 -141 -324 -316

646 458 340 255

Table 3 Fatigue tests with static mean stresses and fully reversed alternating shear stresses for the as received SAE 1045 steel. Test No.

1 2 3 4 5 6 7 8 9 10 11 12

Shear Stress Amplitude

Mean Normal Stress

(AT/2) (MPa)

(amean) (MPa)

125 140 147 130 160 137 150 186 187 190 200 220

100 40 40 100 40 100 100 -60 -60 -60 -60 -60

Cycles to Failure 2 000000 (run-out) 2 000 000 (run-out) 228285 166234 88845 58719 47935 2 000 000 (run-out) 941969 372048 99217 52041

The Influence of Static Mean Stresses Applied Normal to the Maximum Shear Planes in .

133

For the SAE 1045 steel, BHN 203, as the length of the fatigue crack increased in depth and surface length, the applied hoop pressure sine wave became pointed. The number of cycles until failure was taken as the number of cycles until this change in the applied hoop pressure sine wave was observed.

ANALYSIS OF RESULTS Fatigue life curves were obtained for various magnitudes of static mean stress applied normal to the maximum shear planes for the hard and soft SAE 1045 steel. Figures 6 and 7 show that the fatigue strength decreases with increasing static mean stress until the crack • surfaces are interference free for the hard (BHN 456) and soft (BHN 203) SAE 1045 steel, respectively. The fatigue strength remains constant for applied tensile static mean stresses larger than these mean stress values.

1000 X

(0

n ^A

CD

^

i_

>—^

55 ^ CD SZ

o ^/A

n n X -f nV*)K ° nc«b .• J

°<.

^^ V A O °

:= CL

'^

^^A

CO E

<

100 1.00E+02

1.00E+03

1.00E+04

1.00E+05

X

SA^

.

tF r 1.00E+06

1.00E+07

Life (Nf) A 500 MPa to 740 MPa Mean Stress o 350 MPa to 375 MPa Mean Stress D 6 MPa to -35 MPa Mean Stress • -120 MPa to -140 MPa Mean Stress X -290 MPa to -320 MPa Mean Stress Fig. 6. Fatigue life data for various static mean stresses levels applied normal to the planes of maximum alternating shear stress for SAE 1045 steel, BHN 456.

Figures 8 and 9 were obtained by constructing best fit curves to the test results in Figs 6 and 7 and taking the value of the alternating shear stress for given fatigue lives from these curves. In Figs 8 and 9, there is a linear relationship between the maximum alternating shear stress and the static mean stress on the critical planes for static mean stresses less then S[^i. When the tensile static mean stress exceeds the interference free stress, (5'int) , the alternating shear stress stops decreasing and remains constant.

R.R KAUFMAN AND T.H. TOPPER

134

1000

^^

°

AO

O

^Cr--

w a. S


rn (0 SI


•o

100

^Q.

U) f-

<

A O MPa Mean Stress [25] • 40 MPa Mean Stress • 100 MPa Mean Stress o-60 MPa Mean Stress + Fully Open (T-C Overload) [25] X Fully Open (C Overload) [26]

10 1.00E + 03

1.00E + 04

1.00E + 05

1.00E + 06

1.00E + 07

Fatigue Life

Fig. 7. Fatigue life data for various levels of static mean stress applied normal to the planes of maximum alternating shear stress for SAE 1045 steel, BHN 203. The"crack face interference free" data was reported by Bonnen [25] and Varvani-Farahani [26].

(0 QL

CO 0 JC CO

ID Q.

<

-400

-200

0

200

400

600

Mean Stress (MPa) o 10 000 Cycles n Fatigue Limit Fig. 8. Relationship between the alternating shear stress on the maximum shear planes and static mean stress for two fatigue lives for hard SAE 1045 steel, BHN 456.

The Influence of Static Mean Stresses Applied Normal to the Maximum Shear Planes in .

135

CO

-20

0

20

40

100

Mean Stress (MPa) | o 100000 Cycles D Fatigue Limit |

Fig. 9. Relationship between alternating shear stress on the maximum shear planes and static mean stress for a given fatigue life for as received SAE 1045 steel, BHN 203. A recently developed technique involving the confocal scanning laser microscope (CSLM) was used to determine the validity of the assumption that the crack faces were interference free at 500 MPa and 76 MPa for the hard and soft steel, respectively [2]. Upon the detection of a shear crack with a minimum depth of 50 )Lim, the specimen was removed from the biaxial fatigue rig shown in Fig. 3. The specimen was then internally pressurized and placed under the CSLM. For a given platform position, a raster scan was carried out on a focused spot across the tubular surface. The reflected light was collected through a pinhole and detected. The pinhole rejected light from above or below the plane of focus of the specimen thereby resulting in optical sectioning. To obtain the required number of optical sections to develop a 3-D image of the crack profile, images were collected starting from the point when the tubular surface was not visible during the raster scan because the platform was too far away from the objective microscope, until the tubular surface was once again not visible because the platform was too close to the objective microscope. Afterwards, computer software was used to merge the optical images based on the variation of intensity of the reflected light in each image. The combined images provided a 3-D view of the crack profile. With progressive increases in internal pressure, the increase in crack depth was found by "optically slicing" the crack profile until a stress was reached above which there was no further increase in depth. The static stress level at which the crack depth remained constant was taken to be the crack face interference free stress level as is shown in Fig. 10. The crack face interference free level for the hard steel was not determined using the same technique because of very short time between the detection of a shear crack and catastrophic failure of the specimen did not allow cracked specimens to be obtained. Figure 10 shows a plot of the normalized crack depth as a function of the static mean stress for SAE 1045 steel, BHN 203. The normalized crack depth represents the ratio between the crack depth at a given static mean stress level and the maximum crack depth at the interference free stress level. The data shows that the crack depth increased until a normal static mean stress of 78 MPa was applied. Above this static mean stress level, the crack faces were fully separated and further increases in static pressure did not increase the crack depth. EFFECT OF STATIC MEAN STESS ON ASPERITY HEIGHT AND PROHLE To determine the effect of the static mean stresses applied normal to the maximum shear stress

136

R.P. KAUFMAN AND T.H. TOPPER

11

•

0.9

1 0.8 J

i.0.7

• 4

J

•

•

4? I \

•

\

•

\

•

(D

Q ^0.6 1

•

\

CO

0 0.5 "D CD

E

1 0.3 0.2 0.1

0

20

40

60

80

100

Static Mean Stress (MPa)

Fig. 10. Normalized crack depth versus static mean stress on the maximum shear planes for as received SAE 1045 steel, BHN 203.

amplitude planes combined with alternating shearing stresses on asperities, their height and the fracture surface shape were examined with a stereographic microscope, a scanning electron microscope and a confocal scanning laser microscope. Changes in the amount of interference between the crack surfaces with varying magnitudes of the static mean stresses were deduced from asperity height and fracture surface appearance. The length of the fatigue crack before the onset of fast fracture was also determined. In the presence of a large tensile static mean stress, the fracture surface profiles revealed minimal crushing or deformation of the asperity tips. Consequently, all or almost all of the shearing force was transmitted directly to the crack tips, and the crack growth was assumed to be essentially interference free. As static mean stresses decreased from the interference free static mean stress level, there was an increase in the deformation of the asperities tips and a reduction in asperity height. These features suggest that there was a reduction in the shearing forces acting at the crack tips. This is consistent with the observed increase in fatigue life with decreasing mean stress. The fracture surfaces for the hard and as received steels listed in Table 4 were obtained from specimens removed from the fatigue rig when a through thickness crack was detected. For the tests conducted during this program, the number of cycles until the detection of a through thickness crack was recorded as the fatigue life. The crack location was ascertained using a stereographic microscope and a scanning electron microscope. The area encompassing the crack was separated from the specimen by plasma cutting around the initiation point of the fatigue crack and dipping the piece in liquid nitrogen. Pliers were then used to pull the fracture surfaces apart ensuring that the morphology of the fracture surfaces was not damaged. Table 4 lists the

The Influence of Static Mean Stresses Applied Normal to the Maximum Shear Planes in .

137

Table 4. Fracture surfaces investigated for the SAE 1045 steel. Specimen ID and Hardness Level

Alternating Shear Stress on Maximum Shear Plane (MPa)

A (BHN 456) B (BHN 456) C (BHN 456) D (BHN 203) E (BHN 203)

268 433 483 187 130

Static Mean Stress Normal to the Maximum Shear Plane (MPa) 547 -13 -138 -60 100

Fatigue Life (Cycles)

17913 72602 55000 941969 166234

test conditions of the specimens from which the fracture surfaces were taken for the hard and soft SAE 1045 steels. The fracture surfaces were examined using the CSLM technique described previously. The CSLM software enabled the authors to obtain 2-D line profiles of the fracture surfaces. The resolution of the confocal laser scanning microscope image processing technique was approximately 3 |Lim [30]. For specimens A, B and C, there was adequate resolution and a satisfactory signal to noise ratio to determine average asperity shape and height. For specimens D and E, there was too much noise to draw any conclusions from the 2-D line profiles. SURFACE ASPERITY APPEARANCE Fracture Surfaces A (BHN 456, a^^

= 547 MPa), B (BHN 456, a,,

= -\3MPsi) and

=-138 MPa j . The scanning electron microscope photographs of C (BHN 456, C7s, fracture surfaces A and B show small thumbnail cracks initiating on the outer surfaces of the tubes and propagating inwards. The fatigue cracks grew inwards approximately 50 |im and 60 |j,m in specimens A and B before fast fracture occurred. The presence of ductile dimples are clearly visible in Fig. 11 (a) and (b) indicating the region of fast fracture. The confocal scanning

Fig. 11. Fracture surface morphology for SAE 1045 steel, BHN 456. Images (a), (b), and (c) from specimens A, B and C, respectively.

138

R.R KAUFMAN AND T.H. TOPPER

laser microscope images clearly show the amount of deformation at the asperity tips. The scanning electron microscope image of the fracture surface from specimen C shown in Fig. 11 c) is relatively featureless. The flattened fracture surface is void of any significant surface features. It can be hypothesized that the compressive static mean stress normal to the maximum shear planes was large enough to prevent fast fracture until the fatigue crack propagated through the wall.

FRACTURE SURFACES D (BHN 203, CTst^ucMcsn = " ^ ^ ^ P a ) AND E (BHN 203, CT^^^^^^^^^^ = 1 0 0 M P a .;

Figure 12 shows two fracture surfaces that were examined with a scanning electron microscope and a confocal scanning laser microscope, respectively. Surface D is from a test with a 60 MPa compressive mean stress, whereas surface E is from a test with a 100 MPa tensile mean stress. The surface features in Fig. 12 are uniform. The fatigue failure regions cannot be distinguished from fast fracture regions on the fracture surfaces examined.

^

L^i^^S

Compressive Mean Stress (Specimen D)

Tensile Mean Stress (Specimen E)

Fig. 12. Fracture surface morphology for SAE 1045 steel, BHN 203, taken with a scanning electron microscope.

ASPERITY HEIGHT Average asperity height in the fatigue failure region of the fracture surfaces was measured with the confocal scanning laser microscope. Average asperity height decreased with decreasing static mean stress for the hard SAE 1045 steel. Similar results could not be ascertained from the fracture surfaces from specimens D and E since there was too much noise in the 2-D line profiles obtained. Figure 13 shows the CSLM asperity height profile of Specimen A with a tensile mean stress of 547 MPa, obtained with a confocal scanning laser microscope. Figure 13 was obtained by measuring the asperity heights through the wall thickness including the thumbnail crack shown in Figure 11 (a). It can be seen that the asperity peaks in both the fatigue crack region (1) and fast fracture region (2) are sharp. The abrupt vertical lines downwards represent noise in the CSLM system.

The Influence of Static Mean Stresses Applied Normal to the Maximum Shear Planes in .

139

2-D Line Profile

Wall Thickness

Fig. 13. A 2-D representation of the fracture surface morphology from specimen A taken with the CSLM. Sections 1 and 2 represent the fatigue crack growth and fast fracture regions, respectively.

DISCUSSION The present paper has shown that the fatigue life of specimens or components subjected to alternating shear stresses decreases as the magnitude of the applied static mean stress increases until it reaches the value when the crack faces are fully open. Confocal scanning laser microscope measurements have shown that, for the hard SAE 1045 steel (BHN 456), the average asperity height decreases with decreasing levels of tensile mean stress. For the soft steel, the interference free stress level deduced from fatigue data (^jnt)' ^^ MPa, was found to be very close to the static mean stress level, 78 MPa, determined using the CSLM technique. Current multiaxial static mean stress theories discussed in this paper predict a continuous linear relationship in which the alternating shear stress decreases as the static mean stress increases. However, the present results show that the decrease ends at the mean stress at which the crack faces fully separate. The following modifications to Findley's parameter are suggested. ^Findley - ^max + ^<^n *

(5)

for C7„ * = cr„ for a^ < ^int or C7„ * = ^jnt for a^ > ^int For SAE 1045 steel, BHN 456 and 203, a k value of 0.5 reduced the experimental data into a narrow band for the hard and soft steels. A k value of 0.5 was initially suggested by Findley. The interference free stress (Sint) is 500 MPa for the hard material and 76 MPa for the as received material. The modified Findley parameter is plotted versus fatigue life for the hard and soft steels in Figs 14 and 15, respectively. The use of the suggested modification to Findley's parameter results in the majority of the experimental parameter values falling within the 2x fatigue life boundaries drawn about the mean curve. The modified Findley parameter was found to be the best static mean stress parameter investigated during this study for condensing the experimental data. Imposing a cut off level, above which further increases in tensile mean stress no longer reduce fatigue life, accounted for the crack face interference free behavior found in this investigation.

R.P. KAUFMAN AND T.H. TOPPER

140

900 ^ 800 E

5 700 i

600 500 400 300 1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

Fatigue Life o 0 Mean Stress • Tensile Mean Stress A Compressive Mean Stress Regression — -ax ^ '2x - 5x - - '5x

Fig. 14. Modified Findley parameter versus fatigue life for BHN 456. 300

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08

Fatigue Life o 0 Mean Stress • Tensile Mean Stress A Compressive Mean Stress Regression — '2x -^ '2x

Fig. 15. Modified Findley parameter versus fatigue life for BHN 203.

The Influence of Static Mean Stresses Applied Normal to the Maximum Shear Planes in ...

CONCLUSIONS For a given fatigue life, data from this test program exhibit an inverse linear relationship between the alternating shear stress and the static mean stress normal to the maximum shear stress amplitude planes for static mean stresses smaller than 500 MPa and 76 MPa for the hard and soft steels, respectively. Increasing the tensile mean stress beyond these values does not result in a further decrease in the alternating shear stress for a given fatigue life. When static mean stresses normal to the plane of maximum alternating shear stress are high, fracture surface asperities are unmarked. Asperity heights increase with increasing tensile mean stress until the stress at which the crack faces no longer touch is achieved. The modified Findley parameter condensed the majority of the experimental results within a 2x band on a parameter versus fatigue life curve. For tensile static mean stresses larger then iSjnt' ^^^ crack faces were found to be fully separated, a condition previously identified as crack face interference free growth. This cracking mechanism is incorporated into the modified Findley parameter by placing a limiting value ^j^t above which the mean stress value used in the parameter is kept constant.

ACKNOWLEDGEMENTS The authors would like to thank NSERC for the funding to complete this program of study and the Canadian Armed Forces. The authors would also like to thank Dr. Bonnen, Dr. Conle, Mr. Barber, Mr. Boldt and Mr. Alberts for their contributions to this research.

REFERENCES 1. Bonnen, J., and Topper, T.H., (1999) Multiaxial Fatigue and Deformation: Testing and Prediction, STP 13872\3. 2. Varvani-Farahani, A. and Topper, T.H., (1997) "Short fatigue crack characterization and detection using confocal scanning laser microscopy (CSLM)," Nontraditional methods of sensing stress, strain and damage in Materials and Structures, ASTM STP 1318 43-55. 3. Stulen, F.B., and Cummings, H.N. (1954) "A failure criterion for multiaxial fatigue stresses," ASTM Proceedings 54, 822. 4. Findley, W.N., (1954) "Experiments in fatigue under ranges of stress in torsion and axial load from tension to extreme compression," ASTM Proceedings 54, 301. 5. Findley, W.N., (1959) "A theory for the effect of mean stress on fatigue of metals under combined torsion and axial loading or bending," Trans. ASME J. Eng. For Industry 81, 301306. 6. Sines, G. (1961) "The prediction of fatigue fracture under combined stresses at stress concentrations," Bulletin ofJSME 4, 443. 7. Mazelsky, B., Lin, T.H., Lin, S.R. and Yu, C.K., (1969) "Effect of axial compression on lowcycle fatigue of metals in torsion," 7 Ba^/c Eng.91, 780. 8. Chu, C.-C, (1994) "Critical plane analysis of variable amplitude tests for SAE 1045 steels,"5'A£ Technical Paper #940246.

141

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R.R KAUFMAN AND T.H. TOPPER

9. Guest, J.J., (1940) "On the strength of ductile materials under combined stresses," Proc. Inst of Automobile Engrs. 35, 33. 10. Gough, H.J., and Pollard, H.V., (1935) "The effect of specimen form on the resistance of metals to combined alternating stresses," Proc. Inst Mech Engrs. 131, 3. 11. Stanfield, G.,(1935) Discussion of the strength of metals under combined alternating stresses, by Gough, H. and Pollard, H., Proc. Inst Mech Engrs. 131 93. 12. McDiarmid, D.L., (1974) "Mean stress effects in biaxial fatigue where the stresses are outof-phase and at different frequencies," Aero J. 78 325. 13. Carpinteri, A., and Spagnoli, A. (2001) "Multiaxial high-cycle fatigue criterion for hard metals." Int J Fatigue 23, 135-145. 14. Kandil, F.A., Brown, M.W., and Miller, K.J., (1982) "Biaxial low-cycle fatigue fracture of 316 stainless steel at elevated temperatures," The Metals Society 280 203. 15. Fatemi, A., and Socie, D. (1988) "A critical plane approach to multiaxial fatigue damage including out-of-phase loading, "7 Fat and Fract Eng Mat and Struct.3 149. 16. Socie, D.F., Kurath, P. and Koch, J., "A multiaxial LCF parameter," (1985) Second International Symposium on Multiaxial Fatigue. 17. Fatemi, A., and Kurath, P., (1988) "Multiaxial fatigue life predictions under the influence of mean-stress," J Eng Mat and Tech 110 380. 18. Sines, G,. PhD Thesis, "Failure of materials under combined repeated stresses with superimposed static stresses," University of California, USA. 19. Sines, G., and Ohgi, G., (1981) "Fatigue criteria under combined stresses and strains," Trans ASME, 103 82. 20. Seeger, G., (1936) Zeitschiftd.V.I., 80 698. 21. Smith, M.C., and Smith, R.C., (1988) "Towards an understanding of Mode II fatigue crack growth," ASTM STP 924 260. 22. Burns, D.J., and Parry, J.S.C, (1964) "Effect of large hydrostatic pressures on the torsional fatigue strength of two steels," J Mech Eng Sci, 6 293. 23. Socie, D.F., and Waill, L.A., Dittmer, D.F., (1985) "Biaxial fatigue of inconel 718 including mean stress effects," ASTM STP 853. 24. Socie, D.F., and Shield, T.W., (1984) "Mean stress effects in biaxial fatigue of inconel 718," J Eng Mat Tech 106 221. 25. Bonnen, J., (1998) PhD Thesis, "Multiaxial fatigue response of normalized 1045 steel subjected to periodic overloads: experiments and analysis," University of Waterloo, Canada. 26. Varavani-Farahni, A., (1998) PhD Thesis, "Biaxial fatigue crack growth and crack closure under constant amplitude and periodic compressive overload histories in 1045 steel," University of Waterloo, Canada. 27. Kaufman, R., (2002) PhD Thesis, "Static mean stress effects in multiaxial fatigue and their implications on fatigue life predictions of induction hardened shafts," University of Waterloo, Canada. 28. Havard, D.G., (1970) PhD Thesis, "Fatigue and deformation of normalized mild steel subjected to cyclic loading," University of Waterloo, Canada. 29. Elkholy, A.H., (1975) Masters Thesis, University of Waterloo, Canada. 30. Snoddy, J., (2001) Work Term Report, "Design, test and use of a confocal laser microscope," University of Waterloo, Canada.

The Influence of Static Mean Stresses Applied Normal to the Maximum Shear Planes in ...

Appendix : NOMENCLATURE BHN

Brinell hardness number

E

Young's modulus

e

Local strain

f,k

Constant for Findley's parameter

k

Constant for Fatemi and Socie parameter

7

Engineering shear strain

P

Parameter

(J

Local stress

Gy

Yield stress

Rx

Shear stress ratio, """/

S

Stress

s*

Empirical constant in Modified Kandil, B

/

*' max

t

Local shear stress

X

Ratio between Xa and a a

SUBSCRIPTS, SUPERSCRIPTS AND OPERATORS 1, 2, 3

Orthogonal principal stress/strain coordinates

axial

Value of variable in axial direction

Hoop

Value of variable in hoop direction

int

Value of variable when crack faces are interference free

max

Maximum value of the variable during the load cycle

mean

Value of static variable normal to the maximum shear planes

n

Value of variable normal to the crack plane

no

Value of static variable perpendicular to the maximum shear strain amplitude

static mean ^

Value of static variable normal to the maximum shear planes Range of variable during the loading cycle

143

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3. NON-PROPORTIONAL AND VARIABLEAMPLITUDE LOADING

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Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

147

FATIGUE LIMIT OF DUCTILE METALS UNDER MULTIAXIAL LOADING

Jiping LKjl and Harald ZENNER^ 1 Volkswagen AG, 38436 Wolfsburg, Germany 2 TU Clausthal, 38678 Clausthal-Zellerfeld, Germany

ABSTRACT The further-developed Shear Stress Intensity Hypothesis (SIH) is presented for the calculation of the fatigue limit of ductile materials under multiaxial loading. The fatigue limit behaviour for different cases of multiaxial loading is analysed with SIH and experimental results, especially with the effect of mean stresses, phase difference, frequency difference, and wave form. In a statistical evaluation, the further-developed SIH provides a good agreement with the experimental results. KEYWORDS Fatigue limit, multiaxial loading, multiaxial criteria, weakest link theory

INTRODUCTION A multiaxial stress state which varies with time is generally present at the most severely stressed point in a structural component. The stress state is usually of complex nature. The individual stress components may vary in a mutually independent manner or at different frequencies, for instance, if the flexural and torsional stresses on a shaft are derived from two vibrational systems with different natural frequencies. For assessing this multiaxial stress state, the classical multiaxial criteria, such as the von Mises criterion or the maximum shear stress criterion, are not directly applicable. This is illustrated in Fig. 1 for two load cases. In the first case, an alternating normal stress occurs in combination with an alternating shear stress with a phase shift of 90°, Fig. la. The second case involves a normal pulsating tensile normal stress a^ and a compressively pulsating normal stress <x. Fig. lb. In both load cases, the principal stresses exhibit the same variation with time. In accordance with the classical multiaxial criteria, the same equivalent stresses are calculated in both cases. The endurance limits are very different, however, as is shown by experiments. This is explained by the fact that the principal direction can vary in the case of multiaxial stress. A variable principal direction is not taken into account by the classical multiaxial criteria.

148

J. LIU AND H. ZENNER

a) 4 0y

4 h. <^x = Oxa sin(a)t) "•^xy = txya sin(cot- 8xy) "^xya - 0,00x3

8^„ = 90°

Ox = Oxm+Oxa Sin(cot) Oy = aym+Oya sin (cot- 8y) •'ya ~ ''xa' ^ym = "^xm , = Oxa,5„ = 0° Ij^CTx

^1

r -1 90° 0° 27C(0t

27Ccot

-90°

a-iaD" Endurance limit for smooth specimens, steel 34Cr4:

158 MPa

240 MPa

Fig. 1. Coordinate stresses, principal stresses and direction of principal stresses for one cycle. Influence of variable principal stress direction on the endurance limit.

For calculating the fatigue limit, a number of multiaxial criteria have been developed in the past. The multiaxial criteria can be subdivided as follows: empirical approach, critical plane approach, and integral approach. The empirical theories were derived by extension of the classical criteria or were usually developed for specific load cases in correspondence with test results [1-8]. With the critical plane approach, the stress components in the critical plane with the maximal value of equivalent stress are considered as relevant for the damage [9-15]. In the case of the integral approach, the equivalent stress is calculated as an integral of the stresses over all cutting planes of a volume element, for instance, with the hypothesis of effective shear stress [16] or the shear stress intensity hypothesis [17]. The hypothesis of Papadopoulos [18,19] is based on the same principle and differs from the shear stress intensity hypothesis by the consideration of the mean stresses.

Fatigue Limit of Ductile Metals Under Multiaxial Loading

149

In the present paper, the further-developed shear stress intensity hypothesis (SIH) is described. The fatigue limit behaviour of ductile metallic materials is explained with special attention to the effects of the mean stresses, the phase difference, the frequency difference, and the wave form.

SHEAR STRESS INTENSITY HYPOTHESIS The development of the shear stress intensity hypothesis (SIH) can be retraced to the interpretation of the von Mises criterion in accordance with Novoshilov [20]. In the past, the von Mises criterion has been interpreted differently: - Distortion energy (Maxwell 1856, Huber 1904, Hencky 1924) - Octahedral shear stress (Nadaj 1939) - Root mean square of the principal shear stresses (Paul 1968) - Root mean square of the shear stresses for all intersection planes (Novoshilov 1952) Novoshilov proved that the mean square value of the shear stresses over all cutting planes is identical to the von Mises stress: 7t

2n

— j \[Ty^Ysinrdcpdr

={3i2)y^

(1)

Simbuerger [16] applied this new interpretation according to Novoshilov to cyclic multiaxial loading and developed the hypothesis of the effective shear stresses. Zenner [17,2123] has further developed this multiaxial criterion and designated the result as the shear stress intensity hypothesis (SIH). In [24] the classical multiaxial criteria, the maximum shear stress criterion and the von Mises criterion, have been derived as special cases of the weakest link theory. On the basis of this analysis, a general fatigue criterion has been formulated for multiaxial stresses. The existing multiaxial criteria of integral approach and of the critical plane approach can be derived as special cases from the general fatigue criterion. On the basis of this analysis of the weakest link theory in [24], the shear stress intensity hypothesis SIH is newly formulated and further developed. In the newly developed SIH, the equivalent shear stress amplitude and the equivalent normal stress amplitude are evaluated as an integral of the stresses over all cutting planes. Fig. 2: 1/ ^vn

T ~ ]

|— j

15 [

l^^y^sinyd^Pdy]^

(2)

In y^O (p=0

1/^2 J

150

J. LIU AND H. ZENNER

Fig. 2. Integration domain of the SIH and stress components in the intersection plane 7(p

The stress amplitudes r^^ and dj^ in each cutting plane are calculated from the time function of the stress components. For loading cases with sinusoidal time functions and same frequencies of stress components explicit equations for r^^ and cr^^ can be derived in dependence on the phase shift [24]. The exponents |H] and |i2 can be chosen between 2 and infinity. If very large values are selected for the exponents, the equivalent stress corresponds to the maximal stress over all the cutting planes according to the maximum norm of the algebra. In order to simplify the calculation, the exponents are selected as |JI=|J2=2. The equivalent stress amplitude is calculated by combination of the two equivalent stress amplitudes of the shear stress and normal stress (see Eqs (2) and (3)): 2

2

.f

(4)

The coefficients a and b are determined from the boundary conditions for pure altemadng tension-compression and pure alternating torsion: 1 a= — 5

(5)

b = - 6-2

(6)

From the conditions a>0 and b>0, the ranges of validity for the hypothesis SIH are defined by the fatigue limit ratio:

V3

Tw

(7)

Fatigue Limit of Ductile Metals Under Multiaxial Loading

151

For most ductile materials, the fatigue limit ratio is situated within these limits. For extending the range of validity for the hypothesis SIH, the exponents \i \ and \X2 can be selected to be greater than two; thus, the values between 1 and 2 can also be included. For the calculation of the equivalent mean stresses, the mean shear stresses are weighted over the shear stress amplitude, and the mean normal stresses over the normal stress amplitude in all cutting planes: n

In

\ l^m^'i^i'^'^y'^^^y ' vm,T

r=0(p=o n In

(8)

y=0(p=0

n

In

K

^2

(9)

In

Generally the sign of the mean shear stress is of no importance. A positive or negative mean shear stress has the same effect. Therefor, v ] is selected to be exactly equal to 2 for the equivalent mean shear stress. Thus, for a positive or negative mean shear stress, a positive equivalent value is always obtained; that is, the mean shear stress still exerts a reducing effect. For the evaluation of the normal mean stress, the exponent V2 is selected to be equal to unity; consequently, positive and negative mean normal stresses can be distinguished. For considering the effect of the mean stresses, the failure condition can be formulated in different ways. For example, an equivalent mean stress can be obtained by combining the equivalent mean shear stress and equivalent mean normal stress, (7^^^ -^<^vm,T +"^vm,(TThe equivalent stress amplitude, Eq. (4), and the equivalent mean stress can be compared with the help of a Haigh-diagram. In the following, the failure condition is formulated directly by a combination of the equivalent stresses from Eqs (4), (8), and (9): ^va + ^ (^vm,T + " ^vm,C7 = ^W

(10)

The coefficients m and n are determined from the fact that the failure condition is fulfilled in the case of both pulsating tension and pulsating torsion:

'W

2 (11) ^Sch

J. LIU AND H. ZENNER

152

<^Sch] Am 'OscH ~2\

^w-

^'

(12)

^Sch

In addition, the characteristic strength values for alternating axial loading o^v, pulsating axial loading 05^/1, alternating torsional loading %, and pulsating torsional loading T^ch ^^ necessary. For the pulsating torsional strength, the following value is assumed: 4x W

(13)

^Sch = •

1+ ^Sch

EFFECT OF MEAN STRESS If the time functions of the stress components of a plane stress state are synchronous, the following analytical equation can be deduced for the calculation of the equivalent stress amplitude from Eqs (2) to (6):

(14)

_J2 2 ^va,a ~]r.

1 2

^21

^xa "•" ^ ^ya "^ ^xa^ya

1/2

•*" ^^xya ( \2

{ola+o]fa

+

2-

l<^m^ya +

(15) l'/2 ^xya I

(16)

and for the equivalent mean stresses from Eqs (8) and (9): ^\2(^ym+^\i<^xm^

+ ^ 2 l ' ^ r y m '^ ^22^xm'^xym ^vm,a

= -[^?,\^xm

ym

(17)

+ ^23^>'m'^jcym J

+ ^32^ ym + A^S^xym J

(18)

The coefficients A/; depend only on the ratio of the stress amplitudes cr^-^, (Jy^, and T^^. The coefficients are indicated in Table 1. For the case of an alternating normal stress and an alternating shear stress (cyclic bending and torsion, for instance), the failure condition (Eq. (10)) yields the well-known elliptical equation for ductile materials: 2 ^xaP

( T

+

\

^xyaD K

^

=1 J

(19)

Fatigue Limit of Ductile Metals Under Multiaxial Loading

153

Table 1. Coefficients A/y for the calculation of equivalent mean stresses, x = cr^^, y = oy^ and z ^xya

J

i

1

2

x^ + y'^ -xy^'}>Z^

JC^+V^-X>' + 32^

A:^+v^-jry' + 3z^

7x^ + 7>-^-6xy + 36z^

10xz-6vz ,2^3,2__^,^3^2

-6xz + 10>'z

3 2x2 + 2y^-3x>' + 3z^

1 2 3

5x^ -^y^ ^Ixy + Az^ 3JC^ + 3V^ + 2X); + 4Z^

x^ + y ^ - x y + 3z^

x^ + 5y^ + 2.ry + 42^

8(x + y)z

3x^ + 3v^ + 2x>' + 4z^

3x^ + 3y^ + 2xy + 4z^

1

Vac/^W 0.8

1

>*V-^o

1 0.6 i

130 test results

0.4 J j

•

steel (bending&torsion)

o

steel (tension&torsion)

0.2 J

•

Al-alloy

j •

^ *

•^LJ

ellipse equaliuii

0

1 — ' — '

0.2

0.4

0.6

0.8

^xac/^W Fig. 3. Fatigue limit under alternating normal and shear stresses

If one combines the elliptical equation with collected test results [25] to yield a standardised diag ram, Fig, 3, Eq. (19) then agrees with the test results. For the case of an alternating normal stress with a superposed static shear stress, the fatigue limit is decreased by the superposed static shear stress. Fig. 4. Up to a static shear stress Tj^^ which is lower than the yield strength RpQ 2, the influence of the superposed shear stress is correctly described by the SIH. Beyond this value, however, the influence of the superposed shear stress is overestimated. With r^y,^ > /?p0.2' severe plastic deformations occur; consequently, this case is defined by a static strength design, and is of no importance for practical applications.

154

J. LIU AND H. ZENNER

^xac/^V ^xym

•t

t-^ ^4-^

0.2

0.4

0.6

0.8

Fig. 4. Effect of the mean shear stress on the fatigue hmit for cycHc normal stress

1.2

1 1

* ^

^

1 '^xyac/'^W 0.8 'xya

0.6

H

0.4 J

•t

Ck35 (SIH)

0.2 J

•

] 0

^V o

onr^i-M!r»^ ('Oil i\ J U U I M 0 4 ( o i l 1)

-1.5

'

<-•

30 CrMo4 [26]

o '

$-

Ck35 [26] '

'

1

-1

'

'

'

1

1

'

-0.5

'

1

'

0

0.5

1

1.5

^xrT/^pO.2

Fig. 5. Effect of the mean normal stress on the fatigue limit for cyclic shear stress

For the case of an alternating shear stress with a superposed static normal stress, the fatigue limit is decreased by the superposed static positive normal stress (tension). A superposed negative static normal stress (compression) increases the fatigue limit to a limiting value. Fig. 5. If the negative static stress exceeds a certain value, the fatigue limit decreases again with increasing compressive mean stress. The influence of the mean stress essentially comprises the effects of the equivalent mean shear stresses and the equivalent mean normal

Fatigue Limit of Ductile Metals Under Multiaxial Loading

155

Stresses. The equivalent mean shear stress is still positive and always decreases the fatigue limit. The equivalent mean normal stress can be positive or negative. A negative mean normal stress increases the fatigue strength, and a positive mean normal stress decreases the fatigue strength. In the case of compression, the two effects result in different behaviour of the mean stress; this depends on the ratio of the coefficients m to n (Eq. (10)). As is shown by the test results [27], the behaviour markedly differs within the compression range for different materials. The effect of a superposed static normal stress on the fatigue limit for cyclic normal stress depends on its direction with respect to the cyclic normal stress. As is shown by experiment and calculation in accordance with SIH, the effect of a superposed static normal stress is weaker in the direction perpendicular to the cyclic normal stress than in the direction of the cycHc normal stress. Fig. 6.

1 -1

-• "" ^ Jl ^

0.8 -

•

n

^md^yW 0.6 -

f ^y

0.4 • •

0.2

34 Cr4 [28] St60 [29]

i

0

n u

1

()

0.2

0.4 ^xm/RpO,2anc

'

0.6

1

'

0.8

^yrT/^pO,2

Fig. 6. Effect of the mean normal stress on the fatigue limit for cyclic normal stress in two different directions

EFFECT OF PHASE DIFFERENCE A phase shift between an alternating shear stress and an alternating normal stress results in a slight increase in fatigue strength, as is predicted by the shear stress intensity hypothesis. The maximal fatigue limit occurs at a phase shift 8^^^ of 90°, and is higher than the synchronous value by approximately 5 per cent. Fig. 7. The fatigue limit diagram is symmetrical with respect to a phase shift of 90°. Therefore, the dependence on the phase shift is plotted only in the range from 0° to 90°. As is shown by the test results, the situation is far from uniform. During a phase shift S^y, both an increase and a decrease in strength are observed experimentally. Fig. 7. It has hitherto not been possible to prove the existence of a unique material dependence for the relation ^jcaD(4y=90°)/a;,^£,( J^^=0°). The SIH is usually applicable.

156

J. LIU AND H. ZENNER

1.4 SIH

1.3 O

1.2 1.1

• •

34 Cr4 [28] 42 CrMo4 [30]

o •

25CrMo4[31] steels [32]

1 0.9 0.8 0.7 0.6 0°

30°

90°

60°

phase shift 6, xy Fig. 7. Effect of a phase shift between a cyclic normal stress and a cyclic shear stress

o

k

•

42CrMo4[11]

o

34Cr4 [28]

•

Q

G^xa=^ya

<^x,t a , ayRx=Ry=0.05

St35 [33]

"N /I / r\" \ A / \y \ \f / \

CO

;

^ • -

k^ >Vj/^"/

30°

60° 90° 120° phase shift 8y

150°

;^ w cot

180°

Fig. 8. Effect of a phase shift between two cyclic normal stresses

For two cyclic normal stresses, only test results with pulsating loads are available, because of experimental difficulties. The effect of a phase shift 5y between two pulsating normal stresses is small within the range from 0° to 90°, and results in a significant decrease in the fatigue limit within the range from 90° to 180°, Fig. 8. A slight increase in strength is observed in the vicinity of 60°. A phase shift ^ =180° between two normal stresses of equal magnitude

Fatigue Limit of Ductile Metals Under Multiaxial Loading

157

corresponds to the case of torsional loading. The minimum of the fatigue limit is situated here. The effect of the phase shift 8y is correctly described by the SIH.

EFFECT OF FREQUENCY DIFFERENCE In the case of uniaxial loading, the fatigue limit of metallic materials can usually be regarded as frequency-independent. In the case of multiaxial loading, however, the frequency difference between the stress components plays an important role. In contrast to the influence of the phase shift, considerably less attention has been paid to the experimental behaviour of the fatigue strength in the presence of differences in frequency of the stress components. The effect of the frequency ratio X^y between a shear stress r^y and a normal stress (J^ is illustrated in Fig. 9. The plotted curve is not continuous; that is, it applies only to discrete frequency ratios. The points calculated for discrete values of the frequency ratio have been connected with straight lines. As is shown by test results, the fatigue limit is decreased by a frequency difference between the normal and shear stresses. If T^yjo^^ is equal to 0.5, a frequency ratio X^y of 8 reduces the fatigue limit by about 30 per cent. This behaviour is described well by the SIH. For X^y > 1 as well as for A^y < 1, the behaviour of the fatigue limit is similar; that is, the behaviour of the fatigue limit is independent of whether the frequency of the normal stress is higher or lower than that of the shear stress. A frequency difference between two pulsating normal stresses also reduces the fatigue limit, Fig. 10. However, only two test results obtained at a frequency ratio ^, of 2 are available. At an initial frequency ratio A, of 2, the largest portion (by approximately 20 per cent) of decrease in fatigue strength is already achieved, for all practical purposes, as is predicted by the furtherdeveloped shear stress intensity hypothesis SIH.

1.4 -

1

L.

1 '^ X

Q

1

1 1 1 1 1 1

1

J

1 i__ 1 1 1 1

-SIH

1.2 -

•

34 Cr4 [28]

h

1.1 -

o

25CrMo4[31]

h

CO

I 1

Rx=Rxy=-1

_

L

0.9 0.8 -

o

o

0.7 0.6 0.1

1

r—1—1—1 1 1 11

1 —t

1

•

^'

1 O 1

1—i— 1 1 1 1 1

10

frequency ratio X.•xy Fig. 9. Effect of a frequency difference between a cyclic normal stress and a cyclic shear stress

J.LIU AND H. ZENNER

158

II

>>

X

Q (0

Rx=Ry=0.05

^^ Q to

3

4

5

6

frequency ratio l^

Fig. 10. Effect of a frequency difference between two pulsating normal stresses EFFECT OF WAVE FORM The wave form of the stress components does not generally affect the endurance limit, if loading is uniaxial, or if the stress components oscillate proportionally or synchronously to each other. This result is also predicted by the SIH. This conclusion has been experimentally confirmed [11,31]. However, the wave form does affect the fatigue limit if the stress components do not oscillate synchronously to one another. The influence of a phase shift ^y between a cyclic normal and a cyclic shear stress with different wave forms is shown in Fig. 11. In contrast to the sinusoidal wave form, the effect of a phase shift d^y on the endurance limit is more pronounced for the other two wave forms. A phase shift between an alternating normal and an alternating shear stress with a triangular wave form will increase the fatigue limit to a greater extent than with a sinusoidal wave form. If the influence of the phase shift between two normal stresses is compared for different wave forms, it is obvious that the effect of the wave form is not remarkable at a phase shift of 180°, Fig. 12. For different wave forms, significant differences exist in the phase shift range between 30° and 150°. In the case of a trapezoidal wave form, even a small phase shift of only 30° results in a significant decrease in fatigue limit. With a sinusoidal wave form, in contrast, a remarkable decrease in fatigue limit begins from a phase shift of 60°. In the case of cyclic normal stresses with a triangular wave form, the decrease in fatigue limit can be assumed to begin at an even larger value of the phase shift 8y.

Fatigue Limit of Ductile Metals Under Multiaxial Loading

159

1.4 1.3 O

I

1.2

> Q

triangle

'^x-'^xy—'*

1.1 1 0.9 A

5^ •'

I

Sinus

T

^^s^

Q CO

! / trapez

0.8 H

0.7

4

0.6 0°

30°

90°

60°

phase shift 5^,,

Fig. 11. Effect of wave forms and phase shift between an alternating normal stress and an alternating shear stress, test results from [28]

1.4 1.3

^xa=V R^=Ry=0.05

1.2 Q

1.1

CO

1 Q

0.9

triangle

CO

^ sinus

0.8

trapez -^V

0.7

•

0.6 0°

30°

60°

90°

120°

150°

180°

phase shift 5

Fig. 12. Effect of the wave form and phase shift between two pulsating normal stresses, test results from [11]

J. LIU AND H. ZENNER

160

STATISTICAL EVALUATION As a check on the accuracy, the ratio jc of the experimentally determined value to the calculated value of the fatigue limit is employed for such an evaluation: ^xaD,exp

(20)

^xaD,SIH

where CTxaD,exp denotes the estimated average value of the experimentally determined fatigue limit. For the calculation of (JxaD,cal i^ accordance with SIH, the estimated average values of errand a^ch ^^^ taken as a basis. At JC = 1, the result of the calculation is equal to the experimental resuh. At JC>1, the fatigue limit with the SIH is underestimated; at JC<1, the fatigue limit is overestimated. The results of the above statistical evaluation are listed in Table 2 for different load cases: the average value x and the standard deviation .s of the ratio JC are shown. In the case of a correct prediction, the average value of the ratio JC should be equal to about the unity and the standard deviation should be very small. 99.99 99.9

182 test series

/•

99

[%]

95 90 80 70 50

J

x= 0,984 s= 0,067 Xmin= 0.820

30 20 10 5 1

Xmax= 1.168

•

.1 .01 0.6

0.8

1

1.2

1.4

^"^xaD.exp'^xaD, cal.

Fig. 13. Comparison between the experimentally determined fatigue limit and the calculated one, as is predicted by the further-developed SIH The results of 182 test series with a maximal von Mises stress lower than \.\RpQ2 are considered. The 130 test results from Fig. 3 for the load case where a normal stress and a shear

Fatigue Limit of Ductile Metals Under Multiaxial Loading

161

Stress alternate without mean stresses are not considered in the statistical evaluation. As is shown in Fig. 3, the prediction according to SIH with the elliptic equation is very good for this simple load case. For each load case, the SIH provides a good prediction of the fatigue limits. The average values X are still near the unity and the standard deviations s are small. A greater scatter is estimated only for the load case where one shear stress alternates with (J^^, Oy^ or t ^ ^ , and for the load case where two normal stresses alternate with phase difference and mean stresses different from zero. The statistical distribution of the ratio x for all 182 test results is plotted in Fig. 13. For the cases considered, the further-developed shear stress intensity hypothesis SIH provides an accurate prediction of the fatigue limit. The ratio x of the experimental fatigue limit to the calculated fatigue limit has an average value equal to about 1.0, and ranges between 0.8 and 1.2. With these results, 90 per cent of the values are within the range between 0.85 and 1.1. The standard deviation s is equal to 0.067.

Table 2. Comparison between experimentally determined fatigue limit and calculated one according to the SIH, for different load cases. Ferritic and ferritic perlitic steel, ultimate steel ^m - 400 to 1600 MPa, n - number of test series (maximum von Mises stress < 1,1 Rpo^i)'^ ^ mean value, s - standard deviation Load case

n

X

s

•^max

•^mln

Gj^ alternating 26

1.003

0.063

1.088

0.864

14

0.944

0.077

1.059

0.820

72

0.988

0.050

1.162

0.849

12

0.905

0.055

1.022

0.842

29

0.988

0.092

1.168

0.860

12

1.031

0.028

1.089

0.970

with cr,^, Gy^ or/and T^y^

17

0.993

0.055

1.108

0.912

All results

182

0.984

0.067

1.168

0.820

with cj^^, Gy^ or/and r^,^ r ^ alternating with o,^^, Gy^ or z^y^ G^ and T^ alternating with o,^, Gy^ or r^,^ and with phase difference Gj^ and T^ alternating, sinusoidal with different frequencies and non-sinusoidal o^ and Gy alternating, with (7,^, Gy^

and with phase difference Gy^ and Gy alternating, with different frequencies G^, T^ and Gy alternating,

162

J. LIU AND H. ZENNER

CONCLUSIONS The further-developed shear stress intensity hypothesis (SIH) provides a useful prediction of the fatigue strength for multiaxial loading. It should be pointed out that deviations between experiment and calculation are not due only to the multiaxial criteria [34]. Possible reasons for deviations include anisotropic material behaviour (texture), non-statistical nature of the fatigue limit, and experimental error. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Marin, J. (1942). Interpretation of Experiments on Fatigue Strength of Metals Subjected to Combined Stresses. Weld. J. 24, 245. Sines, G. (1955). Fatigue of Materials under Combined Repeated Stresses with Superimposed Static Stresses. Technical Note 2495, NACA, Washington D. C. Kakuno, K. and Kawada, K. (1979). A New Criterion of Fatigue Strength of a Round Bar Subjected to Combined Static and Repeated Bending and Torsion. Fatigue Fract. Engng. Mater. Struct. 2, 229. Lee, S. B. (1985). A Criterion for Fully Reversed Out of Phase Torsion and Bending. In: Multiaxial Fatigue, pp. 553-568, Miller, K. J. and Brown, M. W. (Eds). ASTM STP 853, Philadephia Mertens, H. (1988). Kerbgrundund Nennspannungskonzepte zur Dauerfestigkeitsberechnung - Weiterentwicklung des Konzeptes der Richtlinie VDI 2226. VDI-Berichte Nr. 661, pp. 1-25 Mertens, H. and Hahn, M. (1993). Vergleichsspannungshypothese zur Schwingfestigkeit bei zweiachsiger Beanspruchung ohne und mit Phasenverschiebungen. Konstruktion 45, 196 Hahn, M. (1995). Festigkeitsberechnung und Lebensdauerabschatzung fiir metallische Bauteile unter mehrachsig schwingender Beanspruchung. Diss. TU Berlin Liipfert, H. P. (1994). Beurteilung der statischen Festigkeit und Dauerfestigkeit metallischer Werkstoffe bei mehrachsiger Beanspruchung. Habilitationsschrift TU Bergakademie Freiberg McDiarmid, D. L. (1973). A Criterion of Fatigue Failure under Multiaxial Stress. Research Mem. No. NL54, The City University London Nokleby, J. O. (1981). Fatigue under Multiaxial Stress Conditions. Report MD-81 001, Div. Masch. Elem., The Norw. Institute of Technology, Trondheim/Norwegen Bhongbhibhat, T. (1986). Festigkeitsverhalten von Stahlen unter mehrachsiger phasenverschobener Schwingbeanspruchung mit unterschiedlichen Schwingungsformen und Frequenzen. Diss. Uni. Stuttgart Troost, A., Akin, O. and Klubberg, F. (1987). Dauerfestigkeitsverhalten metallischer Werkstoffe bei zweiachsiger Beanspruchung durch drei phasenverschoben schwingende Lastspannungen. Konstruktion 39,479 Sonsino, C. M. and Grubisic, V. (1987). Multiaxial Fatigue Behaviour of Sintered Steels under Combined In and Out of Phase Bending and Torsion. Z. Werkstofftechn. 18, 148 Dang Van, K., Griveau, B., Message, O. (1989). On a new multiaxial fatigue criterion: Theory and application . In EGF 3, Biaxial and Multiaxial Fatigue, Brown, M.W. and Miller, K.J. (Eds), 459 Carpinteri, A. and Spagnoli, A. (2001). Multiaxial high-cycle fatigue criterion for hard metals. Int. J. Fatigue 23 (2), 135

Fatigue Limit of Ductile Metals Under Multiaxial Loading

16.

163

Simbiirger, A. (1975). Festigkeitsverhalten zaher Werkstoffe bei einer mehrachsigen, phasenverschobenen Schwingbeanspruchung mit korperfesten und veranderlichen Hauptspannungsrichtungen. Diss. TH Darmstadt 17. Zenner, H. and Richter, I. (1977). Eine Festigkeitshypothese fur die Dauerfestigkeit bei beliebigen Beanspruchungskombinationen. Konstruktion 32, 11 18. Papadopoulos, I. V. (1994). A New Criterion of Fatigue Strength for Out-of-Phase Bending and Torsion of Hard Metals. Int. J. Fatigue 16, 377 19. Papadopoulos I.V., Davoli, P., Gorla, C , Filippini, M. and Bemasconi, A. (1997). A comparative study of multiaxial high-cycle fatigue criteria for metals. Int J Fatigue 19, 219 20. Novoshilov, V. V. (1961). Theory of Elasticity (J. J. Sherrkon trans.). Jerusalem Israel Program for Scientific Translation 21. Zenner, H., Heidenreich, R. and Richter, I. (1980). Schubspannungsintensitatshypothese Erweiterung und experimentelle Absttitzung einer neuen Festigkeitshypothese flir schwingende Beanspruchung. Konstruktion 32, 143 22. Zenner, H. (1988). Dauerfestigkeit und Spannungszustand. VDI-Berichte Nr. 661, pp. 151-186 23. Zenner, H. and Liu, J. (1989). Berechnung der Dauerschwingfestigkeit bei mehrachsiger Beanspruchung. WGMK-Tagung, Clausthal 24. Liu, J. (1997). Weakest Link Theory and Multiaxial Criteria. In: Proc of the 5'^ Int Conf on Biaxial/Multiaxial Fatigue and Fracture, pp. 45-62, Cracow 25. Liu, J. (1991). Beitrag zur Verbesserung der Dauerfestigkeitsberechnung bei mehrachsiger Beanspruchung. Diss. TU Clausthal 26. Baier, F.-J. (1970). Zeit- und Dauerfestigkeit bei uberlagerter statischer und schwingender Zug-Druck- und Torsionsbeanspruchung. Diss. Uni. Stuttgart 27. Bergmann, J. W. (1988). Werkstoffdauerfestigkeit II. FVV- Forschungsberichte, Heft 410 28. Heidenreich, R., Zenner, H. and Richter, I. (1983). Dauerschwingfestigkeit bei mehrachsiger Beanspruchung. Forschungshefte FKM, Heft 105 29. El-Magd, E. and Mielke, S. (1977). Dauerfestigkeit bei uberlagerter zweiachsiger statischer Beanspruchung. Konstruktion 29, 253 30. Lempp, W. (1977). Festigkeitsverhalten von Stahlen bei mehrachsiger Dauerschwingbeanspruchung durch Normalspannungen mit liberlagerten phasengleichen und phasenverschobenen Schubspannungen. Diss. Uni Stuttgart 31. Mielke, S. (1980). Festigkeitsverhalten metallischer Werkstoffe unter zweiachsig schwingender Beanspruchung mit verschiedenen Spannungszeitverlaufen. Diss. RWTH Aachen 32. Nishihara, T. and Kawamoto, M. (1945). The Strength of Metal under Combined Alternating Bending and Torsion with Phase Difference. Mem. of the College of Engng., Kyoto Imperial University, 11, 85 33. Issler, L. (1973). Festigkeitsverhalten metallischer Werkstoffe bei mehrachsiger phasenverschobener Schwingbeanspruchung. Diss. Uni. Stuttgart 34. Zenner, H., Simburger, A. and Liu, J. (2000). On the fatigue limit of ductile metals under complex multiaxial loading. Int J Fatigue 22, 137

164

J. LIU AND H. ZENNER

Appendix: N O M E N C L A T U R E a, b, m, n

coefficients of SIH

Rp02

static yield strength of 0.2% plastic strain

Sj^, Sy

phase differences to a^^^

Xjfy, Ay

frequency ratio to CJ^

jU\^ Ml' ^2» ^2

exponents of SIH

^xm^ ^ym' ^xym

mean Stresses

(Jj-fl, (Jya, Txya

Stress amplitude

a^aD

fatigue limit for normal stress

^va^ ^va,(T> ^va,T

equivalent stress amplitude

^vm' ^vm,cT> ^vm,T

equivalent mean stresses

a^ch

fatigue limit (double amplitude) for pulsating tension

(%

fatigue limit for alternating tension-compression

a-yfpa

normal stress amplitude in the intersection plane ytp

a^fp^

mean normal stress in the intersection plane y(p

T^ch

fatigue limit for pulsating torsion (double amplitude)

Tj^aD

fatigue limit for shear stress

%

fatigue limit for alternating torsion

Tyfpa

shear stress amplitude in the intersection plane y(p

Tyfp^

mean shear stress in the intersection plane yig>

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) Published by Elsevier Science Ltd. and ESIS.

165

SEQUENCED AXIAL AND TORSIONAL CUMULATIVE FATIGUE: LOW AMPLITUDE FOLLOWED BY HIGH AMPLITUDE LOADING

Peter BONACUSE US Army Research Laboratory^ NASA Glenn Research Center, Brook Park, OH, USA and Sreeramesh KALLURI Ohio Aerospace Institute, NASA Glenn Research Center, Brook Park, OH, USA

ABSTRACT The experiments described herein were performed to determine whether damage imposed by axial loading interacts with damage imposed by torsional loading. This paper is a follow on to a study [1] that investigated effects of load-type sequencing on the cumulative fatigue behavior of a cobalt base superalloy, Haynes 188, at 538°C. Both the current and the previous study were used to test the applicability of cumulative fatigue damage models to conditions where damage is imposed by different loading modes. In the previous study, axial and torsional two load level cumulative fatigue experiments were conducted, in varied combinations, with the low-cycle fatigue (high amplitude loading) applied first. In present study, the low amplitude fatigue loading was applied initially. As in the previous study, four sequences (axial/axial, torsion/torsion, axial/torsion, and torsion/axial) of two load level cumulative fatigue experiments were performed. The amount of fatigue damage contributed by each of the imposed loads was estimated by both the Palmgren-Miner linear damage rule (LDR) and the non-linear, damage curve approach (DCA). Life predictions for the various cumulative loading combinations are compared with experimental results. Unlike the previous study where the DCA proved markedly superior, no clear advantage can be discerned for either of the cumulative fatigue damage models for the loading sequences performed. In addition, the cyclic deformation behavior under the various combinations of loading is presented. KEYWORDS Multiaxial, cumulative fatigue, axial loading, torsional loading, tubular specimens.

166

R BONACUSE AND S. KALLURI

INTRODUCTION Many multiaxial fatigue damage models are based on the premise that damage is a tensor, i.e., it has both magnitude and direction. These Sensorial' approaches imply that damage imposed by loading in one direction does not readily interact with subsequent damage imposed by loading in another direction. By subjecting thin-walled tubular specimens to fatigue in different loading directions, in sequence, this hypothesis can be tested. In this study, a block of lower amplitude loading, at a given fraction of the estimated life, preceded a second block of higher amplitude loading to failure. This work is a complement to a previous study [1] where higher amplitude cycles were imposed initially, followed by lower amplitude loading to failure. Various combinations of axial and torsional load-type and load-sequence interactions have been explored in both studies. The most common method of accounting for the fatigue damage accumulated in a material subject to variable amplitude loading is to estimate the fraction of the fatigue life expended for each cycle of a given amplitude and then sum up all the fractions for each of the load excursions. When this sum reaches unity, the material is said to have used up its available life. This model is commonly referred to as the linear damage or Palmgren-Miner rule (LDR) [2]. However, many studies have shown LDR to be inaccurate by as much as an order of magnitude for certain combinations of variable amplitude loading [3-6]. Halford [3] outlines the uniaxial loading combinations where the LDR is likely to break down and where alternative approaches may prove useful. In the present study, two cumulative fatigue damage models are assessed for their ability to predict the remaining cyclic life after prior loading at both different amplitudes and different loading directions. The first model is the LDR [2]. The second is the damage curve approach (DCA) of Manson and Halford [7]. In the previous study (high amplitude loading followed by low amplitude loading), the damage curve approach (DCA) was found to model the load interaction behavior remarkably well for all the cases investigated including the mixed load experiments (axial followed by torsional and torsional followed by axial). This seemed to indicate that the fatigue damage was most likely isotropic, at least when the loading sequence was high amplitude followed by low amplitude. A possible explanation for apparent isotropic nature of damage accumulation under fully reversed fatigue loading is that the cyclic hardening behavior of superalloys is, at least to some extent, isotropic in nature. This hardening should influence damage accumulation even when loading is imposed in another direction. The magnitude of plastic deformation is often used as a measure of the accumulated damage in a loading cycle. If the material is in a work hardened state from previous mechanical cycling it should be able to absorb a larger portion of ensuing deformations elastically, resulting in lower plastic strains. Thus, a smaller increment of damage would then be accrued in each subsequent cycle. This mechanism would be in competition with propagating cracks that may have initiated in the prior cycling; higher stresses due to work hardening would increase the crack propagation rate in the subsequent loading.

MATERL\L, SPECIMENS, AND TEST PARAMETERS Specimens used in this study were fabricated from hot rolled, solution annealed, Haynes 188 superalloy, 50.8 mm diameter bar stock (heat number: 1-1880-6-1714). This is the same heat of material used to perform the experiments described in Ref. [1]. The measured weight

Sequenced Axial and Torsional Cumulative Fatigue: ...

167

percent composition of the superalloy was: < 0.002 sulfur, 0.003 boron, < 0.005 phosphorus, 0.052 lanthanum, 0.09 carbon, 0.35 silicon, 0.8 manganese, 1.17 iron, 14.06 tungsten, 22.11 chromium, 22.66 nickel, with the balance made up of cobalt. All experiments were performed on thin walled tubes with nominal gage section dimensions of 26 mm outer diameter, 22 mm inner diameter, 41 mm straight section and 25 mm gage length. The interior surfaces of the tubes were honed in an attempt to preclude crack initiation on the inner surface of the specimen. Outer surfaces were polished with final polishing direction parallel to the specimen axis. Further details on the specimen geometry and machining specifications can be found in Ref. [8]. The baseline axial and torsional fatigue lives for this material, specimen geometry, and test temperature can be found in Ref. [1]. The specimens were heated to 538°C with an induction heating system. All specimens were subjected to sequential constant amplitude fatigue loading under strain control. A commercially available, water-cooled, biaxial, contacting extensometer with a 25 mm gage length, designed specifically for axial-torsion testing, was used. The loading actuator that was not being used for fatigue strain cycling (the torsional actuator during axial cycling or the axial actuator during torsional cycling) was maintained in load control at zero load. This procedure allowed strains to accumulate in the zero load direction. During the axial strain cycling segments, relatively small mean strains in the load controlled torsional axis were observed. However, the torsional strain cycling segments always showed increasing mean axial strains. When torsional strains were applied in the first segment, the magnitudes of these axial strains were recorded and then electronically set to zero prior to commencing the second loading segment. The specimen failure criterion programmed into the testing software was a 10% drop in the measured load in the strain controlled direction. Five experiments were terminated due to a controller interlock. Details on the testing system and test control procedures can be found in Ref. r 11.

TEST MATRIX The test matrix for this study is shown in Table 1. Seventeen different combinations of low amplitude followed by high amplitude, two load level experiments were performed. The loading sequences were axial followed by axial (axial/axial), torsion followed by torsion (torsion/torsion), axial followed by torsion (axial/torsion), and torsion followed by axial (torsion/axial), with at least four different, first load level life fractions imposed in each combination. A fifth life fraction was imposed in the torsion/torsion subset. One torsional experiment was repeated as a cursory check on the expected specimen-to-specimen variability in fatigue life. This summed to a total of 18 tests performed for this study. Table 1 also contains the stress range and mean stress at half-life for each load level, the number of cycles imposed, and the final crack orientation.

TABLE I: Test matrix and interaction fatigue data for Haynes 188 at 538°C

Specimen

AEI

First Load Level; v = 0.5 Hz A01 0ml A ~ I A Zml n~ (MPa) (MPa) (MPa) (MPa) (Cycles)

AxialIAxial HYII-103 0.0067 811 -9 ... ... HYII-116 0.0066 849 -6 ... ... HYII-119 0.0066 882 -10 ... ... ... HYII-114 0.0066 905 -9 ... Torsion/'Torsion ... 0.0120 515 HYII-117 ... ... ... 0.0120 536 HYLI-1 12 ... ... ... 0.01 19 554 HYII- 1 15 ... ... ... 0.0121 521 HYII- 109 ... ... ... 0.01 19 588 HYII-118 ... ... ... 0.0120 579 HYII- 104 ... ... Axial/Torsion HYII-I 10 0.0069 841 -10 ... ... ... HYII-111 0.0069 862 -8 ... ... HYII-108 0.0065 892 -8 ... HYII-105 0.0066 888 -9 ... ... Torsion/Axial ... 0.0121 498 HYII-120 ... ... ... HYII- 107 ... ... 0.0120 523 ... 0.01 19 540 HYII- 106 ... ... ... 0.0119 581 HYII-113 ... ... * Angle measured with respect to the specimen axis.

AE?

Second Load Level; v = 0.1 Hz A02 Gm2 Ay2 A Tm2 n2 Crack* (MPa) (MPa) (MPa) (MPa) (Cycles) Orientation

...

... ... ... ...

3926 7851 15702 23553

0.0203 0.0202 0.0203 0.0205

1254 1247 1244 1267

-14 -14 -1 1 -12

... ... ... ...

... ... ...

... ... ... ...

789 758 659 815

75" 80" 85" 90"

-1 0 1 0 0 -2

5857 11714 23427 23427 35141 40998

... ...

... ... ... ... ...

... ... ... ... ... ...

0.0345 0.0349 0.0346 0.0347 0.0347 0.0349

73 1 740 731 709 748 732

1 1 0 2 0 -2

1250 1100 816 1343 1467 1294

0" 0" 0" 0" 0" 0"

...

... ... ...

...

!m tu

0 0

5

sh

* 2b b

... ... ... ...

3926 7851 15702 23553

... ... ... ...

... ... ... ...

...

0 -1 0 -1

5857 11714 23427 35141

0.0201 0.0203 0.0200 0.0204

1400 1414 1432 1452

... ...

0.0348 0.0347 0.0344 0.0346

781 761 794 783

2 0

-15 -16 -20 -18

... ... ... ...

... ... ... ...

... ... ...

...

1

2

...

1189 1218 930 1253

0" 5" 0" 0"

560 494 459 427

90" 90" 75" 80"

C

?

Sequenced Axial and Torsional Cumulative Fatigue: ...

169

CUMULATIVE DAMAGE MODELS The results of the cumulative fatigue experiments performed for this study were compared with predictions of two load interaction models: the LDR (Eq. 1) [2], and the DCA (Eq. 2) [7].

n.

(2)

vNw The LDR assumes that damage accumulated during each load excursion can be simply added to the already accumulated damage in the material. Load sequence and changes in the properties of the material are not taken into account. The LDR has the distinct advantage of being straight forward to implement for virtually all loading histories, provided sufficient baseline fatigue data are available and an adequate cycle counting method is employed. The DCA attempts to model the observed non-linear interactions between two load-level experiments. The underlying assumption of the DCA is that high amplitude loading initiates cracking early in life whereas under lower amplitude loading measurable cracking does not occur until late in life. In the case of the experiments performed in this study, the implication is that the initially imposed lower amplitude loading might not initiate cracks. In the subsequent higher amplitude loading the material might then 'ignore' the previous cycling or even derive a benefit from it, allowing the sum of life fractions to be greater than unity.

RESULTS AND DISCUSSION The initially imposed lower amplitude cyclic loading (0.65% axial and L24% torsional nominal strain ranges) had sufficient plasticity to cause this solution-annealed material to isotropically harden. The magnitude of the hardening in the first load level was proportional to the number of imposed cycles. The cyclic hardening behavior for the lower amplitude, first load level, loading is presented in Fig. L The horizontal lines in each of these figures correspond to stress range for the last cycle of the lower amplitude loading. The vertical drop lines indicate the last cycle of the first load level for each experiment. There was some variation in the first cycle stress range for both the axial and torsional loading. In the first load levels, the average axial first cycle stress range (the left most data point in each of the curves in Fig.l (a) and (c)) was 646 MPa with a standard deviation of 15 MPa, while the torsional first cycle average stress range (the left most data point in each of the curves in Fig.l (b) and (d)) was 382 MPa with a standard deviation of 14 MPa. The most likely explanations for these variations include the natural variability in the material properties and discrepancies in the machining and/or measurement of the specimen gage section. A 0.1% error in the measurement of a gage section dimension (inner or outer radius), which is the approximate precision of the micrometers employed, would lead to a 0.2% error in the axial stress and a 0.4% error in the shear stress calculations. This dimensional measurement error would account for only about 10% of the variability observed. The specimen to specimen variation in the hardening rate, however, in all the axial and torsional experiments was small.

170

P. BONACUSE AND S. KALLURI

1800

1500 H

< CD D) C

= •O- "i = -y-- ^^ = = -^-^^ - • -

CO

Q.

^1

3926 7851 15702 23553

1200 H

CO

CL if) CO CD

900

1-

(D

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600

0 10°

10^

102

103

10^

10^

^Q4

10^

Number of Cycles, n^ (a) axial/axial 0 ...Q lUUU _^_ 'co' Q_^._ e 800 - _ » _•_ < innn

n^ = 5857 n^ = 11714 n^= 23427 n^= 23427 n^= 35141 n^ = 40998

1 600(/) (0 0

(f)

^

400^

CO

U

(/)

^ n-

0

\

10°

10^

102

^Q3

Number of Cycles, n^ (b) torsion/torsion

171

Sequenced Axial and Torsional Cumulative Fatigue:

1800

-#-

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o

—T--^•-

^1

1500

< 6

D)

^1 ^1

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^1

= 3926 = 7851 = 15702 = 23553

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c

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(/)

900 - b z : = = = = = = i :

CD i_

CO

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600

x <

10°

10^

102

-103

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10^

10^

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Number of Cycles, n^ (c) axial/torsion

1000

= 5857 O ^^ = 11714 -T-- ^1 = 23427 _^._ n^ = 35141 -

CT5

<

800

•

-

^1

U) C CO

DC

600

(/) (/)

E^^E^EE:^^^EEEEE^^EEEEEE^^^^^^

0 v_ •+->

CO ^ CO 0

400

CO 10°

10^

102

-103

Number of Cycles, n^ (d) torsion/axial Fig. 1.

First load level cyclic stress response for: (a) axial/axial, (b) torsion/torsion, (c) axial/torsion, and (d) torsion/axial cumulative fatigue experiments

172

P. BONACVSE AND S. KALLURI

The stress range vs. cycle number plots for the higher amplitude, second load level (Fig. 2) show some interesting behavior. The lower initial life fraction experiments continued to cyclically harden during the second load level. The higher initial life fraction experiments tended to cyclically soften (with exception of the axial-axial where all hardened to failure). The material in each type of load level combination tended to converge to a similar stress range as the cycles accumulated in the second load level, independent of the number of lower amplitude cycles previously imposed. The axial/axial and torsion/torsion experiments tended to stabilize to the same stress levels as the baseline (constant amplitude fatigue tests performed at the second load level) experiments (also plotted in Fig. 2), whereas the mixed loading experiments, axial/torsion and torsion/axial, stabilized at stress levels 6.5% and 12.0% above the baseline experiments, respectively. The extra hardening observed in the mixed loading experiments may be attributable to the same mechanism that causes additional hardening in mechanically out-of-phase multiaxial loading [9]. Experiments ending with torsional loading always failed on the maximum shear strain plane, i.e. the final failure cracks were parallel to the specimen axis. Experiments completed under axial loading all failed at or near the plane of maximum tensile stress; perpendicular to the specimen axis. In the tests completed with axial loading, the 10% load drop failure criteria corresponded to an average crack length of 20.5 mm. Somewhat longer surface cracks (27.4 mm average length) occurred in the tests completed with torsional loading. This result is not unexpected in that shear cracks (cracks that form and propagate on maximum shear stress/strain planes) tend to be longer at failure than cracks that form and propagate on maximum normal stress/strain planes [10]. Five of the eighteen tests were terminated due to controller interlocks. Controller interlocks are preprogrammed limits on: load, strain, and displacement. These limits were typically set to approximately 10% above or below the expected maximum values of the measured variables. An interlock can also occur when the difference between the command and feedback signal in the control loop reaches a preprogrammed threshold, which was 15% of the commanded strain for these experiments. A controller interlock shuts off hydraulic pressure and sends a signal to the control software to indicate that the test has been terminated. Larger final cracks with significant ductile tearing and specimen distortion were associated with the controller interlocks. The length of the cracks that propagated in fatigue, as identified on the fracture surfaces, were of the same order as those where the experiments ended at the 10% load drop. The difference in the number of cycles to failure associated with the interlock events, as compared to those terminated at a 10% load drop, is believed to be small. The results of the axial/axial, torsion/torsion, axial/torsion, and torsion/axial experiments are compared with the predictions of the LDR and the DC A in Fig. 3. It's clear that neither cumulative damage model appears to predict the observed behavior adequately. In most cases, the LDR seems to more closely approximate the observed behavior for lower initially applied life fractions (ni/Ni < 0.4), whereas the DC A model seems to do better with the higher (ni/Ni > 0.4) initially applied life fractions. Figure 4 displays this cumulctive fatigue data with the applied life fraction in the first load level on the horizontal axis and the observed sum of life fractions on the vertical axis. The horizontal line at 1.0 depicts where the data would fall if the LDR perfectly predicted the damage interaction. Again, the deviation from the LDR as the imposed life fraction increases is readily apparent. Figure 5 shows a comparison between the predicted and observed remaining cycle life, n2, for (a) the LDR and (b) the DCA. Both models, in general, predicted fatigue life within the expected experimental scatter-band (factors of two of the average life) for LCF. However, the

Sequenced Axial and Torsional Cumulative Fatigue: ...

Baseline 1 Baseline 2 - ^ ^ n^A = 3926 n^^ = 7851 - ^ - n , , = 15702

1800

10^

102

103

Number of Cycles, n^ (a) axial/axial Baseline 1 RiT^.plino P

0

CO

CL

e 800

n^^ = 5857

...O n^^=11714 _ ^ _ n^^ = 23427 _V._ n^, = 23427 _ » _ n^^ = 35141 _ Q _ n^^ = 40998

1000

<] (D O)

J

c CO

CC


0

173

600 A

1—

CO 1—

CO CD

sz

CO

10^

102 Number of Cycles, n^ (b) torsion/torsion

103

^o'^

174

P. BONACUSE

AND S.

KALLURI

Baseline 1 Baseline 2 n^A= 3926

1000

n^A= 7851 n. :15702

•o-

CO

CL

:23553

-V

800 S = - . ^ W W X ^ U) C CO

DC (/) CO 0 •*—•

CO

600

CO 0

CO 10^

102

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10^

Number of Cycles, n^ (c) axial/torsion

1800 CO

CL

§

1200 Baseline. 1 Baseline 2 ni-r= 5857

CO

0

900

= 11714

CO

- = 23427

x <

35141

-V10°

10^

102

103

10^

N u m b e r of Cycles, n^

(d) torsion/axial Fig. 2.

Second load level cyclic stress response with baseline data for: (a) axial/axial, (b) torsion/torsion, (c) axial/torsion, and (d) torsion/axial cumulative fatigue experiments

175

Sequenced Axial and Torsional Cumulative Fatigue: ...

1.0

0.8

0

-

^

\ ° \

\

°

\ \

c

\

q o

CO

\

0.6 h

\

CD

en

N,^ = 39 255

0.4

\

\

N , , = 825

c

\ \

\

-

\

•g

\

E CD

DC

0.2

O

.

0.0 0.0

1

\

Haynes 188 LDR DCA 1

1

\ 1

1

0.2

1

0.4

1

\

1

0.6

1

N

O.i

1.0

Applied Life Fraction, n/N^ (a) axial/axial

1.0

\

'

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n

°

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\

o

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CD

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\

N^T = 58 568

\ \

N^j = 1 751

E CD

DC

0.2 L

D

\ \ \

Haynes 188 LDR

\ \

DCA 0.0 0.0

•

I

0.2

I

I

1

0.4

0.6

.

, 0.8

Applied Life Fraction, n/N^ (b) torsion/torsion

\ . \ 1.0

P. BONACUSE AND S. KALLURI

176

1.0 .

,

1 —

-r-—1,^1

—r

'

I

'

j

\ \

0.8

\

cvi

c c o o

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c 'c

\ \ \

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0.4

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E 0.2

-

A

\

\ \

Haynes 188 LDR

\ \ \

DCA 1

0.0 0.0

-

\

CO

0 DC

1 1

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0.2

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Applied Life Fraction, n/N^

(c) axial/torsion

l.U

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1

I

r — - t - _

•

^ \ 7"

\

0.8

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q o

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\ ^

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c c

0.4 -

•

j

1

\\

1

\\ 1i

\

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\

1

\

\

^

\

N , , = 825

\ -

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-

\

DCA

\ \ nn

0.0

1

1

1

0.2

1

0.4

1

1

0.6

0.8

1.0

Applied Life Fraction, n/N^

(d) torsion/axial Fig. 3. Damage interaction plots for: (a) axial/axial, (b) torsion/torsion, (c) axial/torsion, and (d) torsion/axial cumulative fatigue experiments

Sequenced Axial and Torsional Cumulative Fatigue:

177

LDR under predicted most of the experimental results, whereas the DC A tended to over predict the life. This finding would seem to favor the LDR for loading conditions similar those imposed in this study. A minimum of 3 experiments for each of the load interaction conditions would be needed to separate the effect of experimental scatter from 'real' load interaction effects. Unfortunately, the resources allotted for this study did not permit an investigation with this level of detail.

+

c o o 03

E CO

0.2

0.4

0.6

0.8

1.0

Applied Life Fraction, n/N^ Fig. 4.

Applied life fraction in the first load level vs. the experimentally determined sum of life fractions.

The extent of isotropic hardening in the material during the first load level has a definite effect on the subsequent deformation. This should come as no surprise as Haynes 188 at 538°C cyclically hardens to failure at the constant amplitude strain ranges imposed in the first load segment (Fig. 1). Therefore, the magnitude of work hardening at the start of the second load level is directly correlated with the number of cycles imposed in the first load level. The work hardened state at the end of the first load level might reveal information about subsequent damage accumulation. As is shown in Fig. 6, the magnitude of the equivalent plastic strain in the first load level is inversely correlated with the sum of the life fractions. The axial (Eq. 3) and shear (Eq. 4) plastic strain ranges were calculated in the conventional way as follows: A

A

^ ^

A8p, = A 8 , , , - —

(3) (4)

178

P. BONACUSE AND S. KALLURI

10^

;^

1

1

1—1—1

1 1 1 1

Factors of 2 on 1 jfe

-1

1—r-71—1 1 1 u

/

_

^ .

/ : /

/

/

/ ' 103

/

i

4^4

\ /A

O O)

c c

\ /

(0

/ v

V/ /

£ DC

\ / i/.

_i

I

1

\ ^ / 0 D A V _i

1 1 1 1 1

.

/

/o

"^

102 L^L 102

/

/

": Axial/Axial Torsion/Torsion Axial/Torsion Torsion/Axial

1

1

1 1 I I I .

103

10^

Remaining Cyclic Life, {n2) OBSERVED

(a) LDR

lU

1

1—1—1

1 1 1 11

—1—1—r-71—1

111^

•

Q LU

\-

~^ <

•

0

Q LU

a.

(0

/

cc

^ /

^(\2

/

/

/

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0 D A V

/

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:

/

^^cxg

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c

/

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103 7

0

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/

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/ ^ ^

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Factors of 2 on Life '^~^—-

1 1 1 j__i_

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Axial/Axial Torsion/Torsion AxialyTorsion Torsion/Axial 1

1

1

^03

1 1 11

10^

Remaining Cyclic Life, (n2) OBSERVED

(b) DCA Fig. 5.

Observed vs. predicted lives for cumulative fatigue experiments using (a) the LDR and (b) the DCA

Sequenced Axial and Torsional Cumulative Fatigue: ...

179

The equivalent axial plastic strain for torsional loading is given in Eq. 5:

(5)

Ae,,.^-^

Figure 6 tends to support the assertion that the extent of work hardening can affect the subsequent rate of damage accumulation. The equivalent plastic strain range in the second load level is plotted against the sum of life fractions in Fig. 7. The plastic strain ranges in this plot were calculated at the approximately the mid point in the second load level. There seems to be no correlation between the plastic strain range at 'half-life' in the second load level and the sum of life fractions. For each combination of load types the 'stabilized' equivalent plastic strain range is approximately the same with the exception of the torsion/axial loading where it is slightly lower than the three other loading combinations.

j^

\J.KJKJO:J

CD -J

0 D A

"D 03

0

0.0030 -

^D

00

•_

V

va

LL

0.0025 -

0

Axial/Axial Torsion/Torsion Axial/Torsion Torsion/Axial

D

Q.

< A "5

0.0020 -

0

A

a 0

A

55 .0

-1—'

^

0.0015 -

-^

> iS"

0.0000 0.6

0.8

1.0

1.2

1.4

1.6

1.8

Sum of Life Fractions (n/N^ + n2/N2) Fig. 6.

Equivalent plastic strain range at the end of the first load level vs. the sum of life fractions.

In a literature review of cumulative fatigue modeling by Fatemi and Yang [11], many approaches to account for the observed deviations from the linear damage rule were enumerated. The conclusion of the review is that none of the reviewed models has proven to be universally applicable because each of the models only accounted for, at most, a few of the "phenomenological factors". Fatemi and Yang grouped the reviewed cumulative damage models into the following six categories: 1) linear damage accumulation, 2) non-linear damage accumulation, 3) life curve modifications for load interactions, 4) crack growth approaches, 5) continuum damage models, and 6) energy based methods.

180

P. BONACUSE AND S. KALLURI

For many of these model categories, the experimental results discussed in this paper present various difficulties, both in interpretation and implementation. Both the LDR and the DCA have been shown to be less than ideal predictors of the observed interaction behavior. As Haynes 188 has been shown not to have an endurance limit, many of the life curve modification models that incorporate this concept would tend to be inaccurate in many regimes. Crack growth approaches may not prove useful in describing this data set in light of the observed damage accumulation behavior in the mixed loading experiments. The final failure crack direction is only a function of the final loading direction (shear cracking for the axial/torsion experiments and maximum normal stress cracking for the torsion/axial experiments). The axial/torsion and torsion/axial experiments seem to closely follow the LDR (with the exception of the highest applied life fraction in the axial/torsion experiments). The mixed loading experiments performed in this study seem to indicate that for Haynes 188 at 538°C, there is no unusual interaction under the conditions imposed. Models that account for damage on the basis of accumulated plastic work implicitly incorporate the effect of strain hardening on the damage induced in the material (strain hardening materials will exhibit more plastic work at similar plastic strain ranges or less plastic work at similar total strain ranges). Because the accumulated plastic work at failure for most materials has been shown to be variable depending on the loading conditions [12], this class of models may not accurately predict the accumulated damage.

g

0.0160

CO

o

-J "D C

0.0150 H

0 CO

0.0140

o o

CO

d ^—•

0.0130

CO

o •fn

> LU

0.0120

0.0000 0.8

1.0

1.2

1.4

1.6

1.8

Sum of Life Fractions (n/N^ + n^H^) Fig. 7.

Equivalent plastic strain range in the second load level at the approximate mid-life point vs. the sum of life fractions

There is a very good chance that competing mechanisms, one linked to the cyclic deformation behavior and the other linked to crack growth, which may be modeled well individually but not in a combined sense, are the root of the discrepancies observed in the literature. Energy and plastic strain based methods do better in modeling crack 'initiation' and

Sequenced Axial and Torsional Cumulative Fatigue: ...

181

early growth and may therefore predict cumulative damage accumulation where observable cracking does not appear until late in the cyclic life. However, methods based on crack propagation arguments would seem to be more likely to produce good correlations where significant cracking develops early in life. Some method of transitioning between these regimes is required for reliable estimates of cumulative fatigue damage under complex loading conditions.

CONCLUSIONS The following conclusions are drawn from this study of the cumulative fatigue behavior of Haynes 188at538°C: 1) Neither the LDA nor the DC A have predicted the interactions in these experiments particularly well. Predictions by the LDA have been generally conservative (under predicted life), whereas the DC A predictions have been generally non-conservative (over predicted life). 2)

There appears to be a transition that occurs at or about an initially imposed life fraction of 0.4 in the set of experiments conducted. Below this level the material adheres to a linear damage rule. Above this level, the damage accumulation appears to be non-linear.

3)

The fatigue life data indicate a transition behavior that is linked to the extent of work hardening imposed by the initial load level: the greater the life fraction expended in lower amplitude loading, the greater the chance of having a sum of life fractions more than unity.

4)

The stabilized (half-life) stress response in the second load level does not show any correlation with damage accumulation for the loading conditions imposed.

REFERENCES 1.

2. 3.

4. 5.

6.

Kalluri, S. and Bonacuse, P. J. (2000), "Cumulative Axial and Torsional Fatigue: An Investigation of Load-Type Sequencing Effects," In: STP 1387 - Multiaxial Fatigue and Deformation: Testing and Prediction, pp. 281-301, S. Kalluri and P. J. Bonacuse (Eds), ASTM, West Conshohocken, PA. Miner, M. A. (1945), "Cumulative Damage in Fatigue," Journal of Applied Mechanics 12, No. 3, {Trans. ASME, Vol. 67), pp. A159-A164. Manson, S. S. and Halford, G. R. (1981), "Practical Implementation of the Double Linear Damage Rule and Damage Curve Approach for Treating Cumulative Fatigue Damage," Int. J. Fracture 17, pp. 169-192. Bui-Quoc, T., (1982), "Cumulative Damage with Interaction Effect due to Fatigue Under Torsion Loading," Experimental Mechanics, pp. 180-187. McGaw, M. A., et al., (1993), "The Cumulative Fatigue Damage Behavior of Mar-M 247 in Air and High Pressure Hydrogen," In: ASTM STP 1211, Advances in Fatigue Lifetime Predictive Techniques: Second Volume, pp. 117-131, M. R. Mitchell and R. W. Landgraf, (Eds.), ASTM. Halford, G. R., (1997), "Cumulative fatigue damage modeling - crack nucleation and early growth," Int. J. Fatigue 19, Supp. No.l, pp. S253-S260.

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P. BONACUSE AND S. KALLURI

I.

Manson, S. S., and Halford, G. R. (1986), "Re-examination of Cumulative Fatigue Damage Analysis ~ An Engineering Perspective", Engineering Fracture Mechanics 25, Nos. 5/6, pp. 539-571. 8. Bonacuse, P. J., and Kalluri, S. (1993), "Axial-Torsional Fatigue: A study of Tubular Specimen Thickness Effects," JTEVA 21, 160. 9. Bonacuse, P. J. and Kalluri, S. (1994), "Cyclic Axial-Torsional Deformation Behavior of a Cobalt-Base Superalloy," Cyclic Deformation, Fracture, and Nondestructive Evaluation of Advanced Materials: Second Volume, ASTM STP 1184, pp. 204-229, M. R. Mitchell and O. Buck (Eds.), ASTM, West Conshohocken, PA. 10. Bonacuse, P. J. and Kalluri, S. (1995), "Elevated Temperature Axial and Torsional Fatigue Behavior of Haynes 188," J. of Engineering Materials and Technology 117, pp. 191-199. II. Fatemi, A. and Yang, L. (1998), "Cumulative Fatigue Damage and Life Prediction Theories: a Survey of the State of the Art for Homogeneous Materials," Int. J. Fatigue 20, No. 1, pp. 9-34. 12. Halford, G. R. (1966), "The Energy Required for Fatigue," / of Materials 1, No.l, pp.3-18.

Appendix : NOMENCLATURE E

Elastic Modulus

G

Shear Modulus

Ni

Number of cycles to failure for first load level

N2

Number of cycles to failure for second load level

j^i

Number of cycles imposed during first load level

ni

Actual number of cycles to failure during second load level

AEI, A82

Imposed strain ranges for load levels 1 and 2

^£j^^

Total measured axial strain range

^£ J

Plastic axial strain range

^£ J ^ AYI, AY2

Equivalent plastic strain range Plastic shear strain ranges for load levels 1 and 2

^y J

Plastic shear strain range

^Ytot

Total measured shear strain range

^(j

Axial stress range

AGI , Aa2

Axial stress ranges for load levels 1 and 2

c^mi, C7m2

Axial mean stresses for load levels 1 and 2

^^

Shear stress range

All, AT2

Shear stress ranges for load levels 1 and 2

'Cmi, "Cmz

Mean shear stresses for load levels 1 and 2

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

183

ESTIMATION OF THE FATIGUE LIFE OF HIGH STRENGTH STEEL UNDER VARIABLE-AMPLITUDE TENSION WITH TORSION: USE OF THE ENERGY PARAMETER IN THE CRITICAL PLANE

Tadeusz L A G 0 D A \ Ewald M A C H A \ Adam NIESLONY^ and Franck MOREL^ ^ Technical University ofOpole, ulMikolajczyka 5, 45-271 Opole, Poland ENSMA, Futuroscope, France

ABSTRACT The paper concerns application of the energy parameter, being a sum of the elastic and plastic strain energy density in the critical plane, for describing experimental data obtained in fatigue tests of 35NCD16 steel, subjected to constant amplitude tension-compression, torsion and variable amplitude tension-compression, torsion and combined proportional tension with torsion. It has been shown that the normal strain energy density in the critical plane is a suitable parameter for correlation of fatigue lives of 35NCD16 steel under considered kinds of loading. The critical plane is the plane where the normal strain energy density reaches its maximum value. KEYWORDS Biaxial fatigue, proportional loading, variable amplitude, fracture plane, energy criterion, life time

INTRODUCTION There are three main models of multiaxial fatigue failure criteria applied for reduction of the complex loading state to the equivalent uniaxial state. They are stress, strain and energy - based models. The proposed energy criteria can be classified into three groups, depending on the strain energy density per cycle, assumed as a damage parameter under multiaxial fatigue [1 - 3]. They are: criteria based on the elastic strain energy for high-cycle fatigue, criteria based on the plastic strain energy for low-cycle fatigue, criteria based on the sum of the elastic and plastic strain energy for low- and high-cycle fatigue. At present, the criteria including the strain energy density in the critical plane or in the fracture plane become dominating in energy description of multiaxial fatigue. These criteria seem to be the most promising for future applications. The authors proposed the energy approach to fatigue life estimation under multiaxial random loading [ 4 - 9 ] . In the case of uniform stress

184

T. LAGODA ET AL

distribution (cruciform specimens and thin-walled cylinders), the criterion of the maximum normal strain energy in the critical plane, including the positive energy parameter under tension and the negative energy parameter under torsion was a suitable quantity for describing the test results of lOHNAP and SUS304 steels. All the analyses were based on a parameter of strain energy density which has been presented in details in [10, 11]. In [12 - 14] the mesoscopic approach to fatigue life estimation was proposed. The papers concern uniaxial tension-compression with torsion in cylindrical solid specimens made of 35NCD16 steel, and the proposed approach is based on the concepts of Orowan [15], Papadopulos [16] and Dang-Vang [17]. In this paper the authors will reanalyse the former test results of 35NCD16 steel and assess if the proposed energy parameter can correlate these experimental data. As it results from the previous papers [ 4 - 9 ] , this parameter seems to be very efficient for correlation of the experimental data obtained for steels lOHNAP, 18G2A, SUS304 and 12010.3 as well as cast irons GGG40 and GGG60. FATIGUE TESTS The tested 35NCD16 steel has the following chemical composition: Si - 0.37%, Mn - 0.39%, P - 0.01%, S - <0.003%, Ni - 3.81%, Cr - 1.7%, Mo - 0.28%, C - 0.364%, the rest Fe. Young's modulus is E = 205 GPa and Poisson's ratio is v = 0.3, the yield point is R0.2 = 930 MPa, tensile strength Rm = 1070 MPa, contraction Z = 20.7 %. Cylindrical specimens were tested under constant and variable-amplitude loading. In the case of constant amplitude loading, tests were performed under tension-compression and pure torsion. In the case of variable-amplitude loading, tests were performed under tension-compression, combined proportional tension with torsion for the ratio of shear and normal stress T(t)/a(t) = 0.5 and pure torsion [12-14]. All the fatigue tests were realized under controlled force and twisting moment. Basing on the tests under constant amplitude uniaxial tension-compression it was possible to determine the regression models of the fatigue characteristic according to [18] logNf = A - m l o g a ,

(1)

Parameters of the determined Wohler curve are as follows: A = 21.07, m = 5.86, the fatigue limit Gaf = 450 MPa, and the corresponding number of cycles No = 3.328 • 10^. The results of fatigue tests under torsion were approximated by a similar regression model: logNf =A^-m^-logT,j

(2)

where At = 44.51, mt= 15.08, the fatigue limit Taf = 330MPa, and the corresponding cycle number Not = 3.395- 10^ It is important to note that the numbers of cycles No and Not for the analysed material differ by one order. The results for variable-amplitude loading are shown in Table 1. In the tests the standard course CARLOS - lateral - Car Loading Sequence with 95180 cycles in a block was used. The cycles with low amplitudes were eliminated so as to obtain the block of f 1 type (with a high reduction of low amplitudes) and f2 type (with a low reduction of low amplitudes). The loading of CARLOS type is a typical loading used in tests of car elements realized in France and Germany (specially LBF Darmstadt). Its course is characterised by a high irregularity coefficient.

Estimation of the Fatigue Life of High Strength Steel Under .

185

Table 1. Fatigue test results under variable-amplitude loading Tension-compression

Tension-compression with

Torsion

torsion x(t) = 0.5a(t) a max

N exp

OCexp

^amax

N exp

'^exp

T ^amax

N

^exp

exp

[MPa]

[blocks]

[1

[MPa]

[blocks]

V]

[MPa]

[blocks]

[1

675*

126

0

595*

47

10

488*

174

45

675*

99

0

595*

64

12

488*

125

45

675*

82

0

595*

62

18

488*

102

45

675**

170

0

595**

71

15

488**

155

45

736**

37

0

595**

63

20

525**

47

45

736**

37

0

595**

64

20

525**

64

45

736**

46

0

525**

89

45

* - course CARLOS-fl (13568 extrema) ** - course CARLOS-f2 (46656 extrema) CARLOS - Car Loading Sequence

DEFINITION OF THE ENERGY PARAMETER In the case of random or variable-amplitude loading, the application of the energy models is difficult when calculation of the plastic strain energy density is based on the areas of the closed stress-strain hysteresis loops. The hysteresis loop area a - e can be easily determined under cyclic loading, but under random loading it is not strictly defined. A certain proposal of the area counting was presented by Kliman in [19]. However, in [3] it has been shown that in some cases the areas of the hysteresis loops can be enlarged and mistakes can occur in fatigue life calculating. Assuming the strain energy density in the following form iy(t) = la(t)e(t)

(3)

as a fatigue parameter is not a good solution, because we are not able to determine a loading type from history W(i). The product of stress and strain in Eq. (3) is positive under tension and compression. In [4 - 11] the authors defined histories of the parameter of strain energy density, taking into account the sign of strains sgn(e(t)) and stresses sgn(o(t)) in order to distinguish the energy under tension and the energy under compression, as follows )]+sgn[E(t)]

W(t) = la(t)e(t)sgn[a(t)] + la(t)e(t)sgn[8(t)] - l a ( t ) e ( t ) ^ ^ ^ ^

(4)

T. LAGODA ET AL

186

For distinguishing the positive and negative work in a fatigue cycle, we introduced functions sgn[8(t)] and sgn[a(t)] in Eq. (3). Then, during a tension half-period, the sign of work 0.25 8(t)a(t) sgn[8(t)] is determined by the strain sign and during a compression half-cycle the sign of work 0.25 e(t)a(t) sgn[a(t)] is influenced by the stress sign. Let us introduce the two-argument logical function sgn(x, y), sensitive to signs of variables X and y, defined as follows 1 when sgn(x) = sgn(y) = 1 0.5 when x = 0 and sgn(y) = l when sgn(x) = -sgn(y) sgn(x,y): 0 -0.5 when x = 0 and sgn(y) = - l -1 when sgn(x) = sgn(y) = - l

or y = 0 and sgn(x) = l (5a) or y = 0 and sgn(x) = - l

When sgn(x, y) = 0.5 or -0.5 according to Eq. (5a), then always W(t) = 0 according to Eq. (4). Thus, Eq. (5) can be written in a simplified way 1 sgn(x,y) = 0

when sgn(x) = sgn(y) = l when sgn(x) = -sgn(y)

or x=0, or y = 0

-1 when sgn(x) = sgn(y) = - l (5b) Then Eq. (4) can be written as follows

W(t) = ia(t)e(t)sgn[a(t),e(t)]

(6)

Let us notice that Eq. (6) expresses the positive and negative parameter of strain energy density in a fatigue cycle and it allows to distinguish energy (specific work) under tension and under compression. Expression (6) has another advantage: the parameter of strain energy history has the zero mean value, while cyclic stress and strain change symmetrically in relation to the zero levels. If we do not introduce the sign of stress and strain under cyclic loading, the frequency of the strain energy history W{i) would increase twice, the amplitudes would decrease twice and the mean value would be different from zero. It is not acceptable because it leads to counting of a greater number of fatigue cycles with smaller amplitudes [20]. Under cyclic loading, if stress and strain reach their maximum values Ga and 8a, the maximum parameter of strain energy density, Eq. (6) and its amplitude are equal to W =0.5a.8.

(7)

Figure 1 shows interpretations of amplitude of the strain energy density parameter. Assuming W(t) according to Eq. (6) as the damage parameter, we can replace the standard characteristics of cyclic fatigue (Ga - Nf) and (8a - Nf) by a new one, (Wa - Nf). In the case of the highcycle fatigue, when the characteristic (Ga - Nf) is used, the axis Ga should be replaced by Wa, where

Estimation of the Fatigue Life of High Strength Steel Under ...

w. =-

2E

187

(8)

Figure 2 shows a part of variable-amplitude histories of stress, strain and energy parameters calculated by Eqs (3) and (6). It can be seen that the energy parameter W(t) (Fig.2d) is a stochastic process with the zero expected value when the stress and strain are centred processes. Those properties are not observed in energy W(i) (Fig.2c).

aa G

Fig. 1. Interpretation of amplitude of strain energy density parameter W^ == O.Sa^e^

HOW SIMPLE LOADING INFLUENCES THE FATIGUE LIFE When the full section specimens are subjected to torsion, the shear stress gradient occurs and the fatigue strength of the material is greater than in the case of pure shearing with no stress gradient and damage accumulated in all the bar section. Let us take into account influence of

T. LAGODA ETAL

188

the Stress gradient under torsion and let us bring that kind of loading to pure shearing so as to obtain equivalence between the both loading.

•0.004-800-^

c) W(t) = la(t)e(t)sgn[a(t),e(t)] 1.000-

0.500-1 0.000

-0.500

-1.000

1.000-

Fig.2. Fragments of the variable amplitude histories of a) stress, b) strain and c) energy parameters W{i) and d) W(t) Transform the Eq. (2) of regression for pure torsion to the following form T„=TV,(2Nf)'''

(9)

Then, coefficient x'^ and exponent b^ of fatigue strength under torsion are equal to

m.

(10)

Estimation of the Fatigue Life of High Strength Steel Under ...

189

At+log2

T'ft = 10

"^^

.

(11)

respectively. Similarly to Eq.(9) we can write the equation for pure shearing T,=TV(2Nf)\

(12)

where T'^ and b are the coefficient and exponent of strength under shearing, respectively. Under uniaxial tension-compression the relationship o, = o ; ( 2 N f f

(13)

can be written with use of shear stresses in the plane of the maximum shearing as

r,=^{2^J=t,{2Nj.

(14)

a' TV-^. 2

(15)

Thus

From the ratio of shear stress amplitudes under shearing Eq. (12) and torsion Eq. (9) for a given number of cycles Nt-r we obtain ta _ x ' f ( 2 N t _ r f ^at

(16)

TVt(2N,_,f

and using Eq. (15) we have T,=^(2N,_,r^T,,

(17)

For a chosen number of cycles N^r from Eq. (17) we can calculate the shear stress amplitudes under shearing (in the section of uniform stress distribution), Xa equivalent to the maximum shear stress amplitude under torsion (non-uniform stress distribution in the section). Tat. Under random loading, application of Eq. (17) is time - consuming because it must be used many times, for each amplitude of cycle counted from the stress history (for example, with use of the rain flow algorithm). In order to reduce and simplify calculations we can rescale all the random history of shear stress with use of one constant coefficient. Applying the equation T(t) = k^T,(t)

(18)

190

T. LAGODA ET AL.

where ka is a coefficient, we can easily calculate a random history of stresses T(t), corresponding to history of the nominal shear stresses under torsion Tt(t) with the gradient of these stresses. Similarly as in the case of Eq. (18) we can transform a random history of shear strains. Applying the equation Y(t) = M , ( t )

(19)

where kg is a coefficient, we can also easily calculate the random history of strains y(t), corresponding to history of the nominal shear strains under torsion yt(t) with the gradient of these strains. In order to assume suitable coefficient k^ and kg please remember that under torsion and tension the exponents (inclinations) of the Wohler's curves are often different (see Fig.3). Moreover, from Fig.3 it appears that the knees of the considered lines occur at different numbers of cycles. Thus, for a number of cycles Nt-r less than the limit numbers of cycles under tension-compression No and torsion Not we can determine ratios of the permissible stresses and strains. For a high number of cycles we can assume the approximate formulas for calculations constant coefficient k^ and kg, like those in the paper by Lowisch [21] ,HCF,aV(2N,_J^

,HCF ^

^'f(2N.-,)''

= ^ ( 2 N , _ . r \

(20)

=^X_(2N._,)^-^.,

(21)

(l + v)T'ft(2Nt_r) '

0 + v)tft

where N,,<min{No,No,}

(22)

is a number of cycles for which we convert stresses and strains from torsion to tension. Hr^F

Let us note that in the range of HCF the coefficient k^ ff/^C

ample, in the case of the considered material k j

HCF

and kg

change a little. For ex/:

HCF

= 0.521 for Nf = 10 cycles and kg

0.663 forNf= 10^ cycles. FATIGUE LIFE CALCULATION BASED ON THE ENERGY PARAMETER The parameter of the normal strain energy density in the critical plane is verified in this paper. Fig. 4 shows the algorithm for fatigue life determination using the energy parameter. At first (stage 1) we register histories of normal stress axx(t) from tension and shear stress Txy(t) from torsion on the surfaces of tested specimens. Having histories of the stress state components, we are able to determine (stage 2) the four strain courses eo(t) = ^
(23)

191

Estimation of the Fatigue Life of High Strength Steel Under .

eyy(t) = - v - a , , ( t )

(24)

ezz(t) = - v - a , , ( t ) E

(25)

exy(t) = ^ Y x y ( t ) - ^ T , y ( t )

(26)

where v is the Poisson's ratio and G is the shear modulus.

400

[MPa]

C^a Nt-r

300

1 "^^\ ^ '^t V

CJaf

No

^"^\^

't:af

Not

Nf

200 1 10^

\

\ 1 rr"TTTl

10^

i

'n

^ T - n"^n

[cycles] ^

^ \ 1 1 1 1 11

10^

10^

Fig.3. Exemplary Wohler's curves for torsion and tension-compression Having the stress and strain tensor we can determine for push-pull with torsion the normal stress and strain in any direction of unit vector r[ described by the direction cosines l^,m^,n^ [2 - 6, 8, 9, 20, 22] with respect to the coordinate system of specimen Oxyz (stage 3)

T. lAGODA ETAL.

192

On(t) = l,CJ„(t) + 2 1 > J , , ( t ) ,

(27)

En (') = '.'^xx (t) + m^e„ (t) + n^e„ (t) + 21>^e,y (t).

(28)

1

Determination of the stress tensor ajj(t), (i,j = x, y, z)

2

Calculation of the strain tensor ejj(t), (i,j = x, y, z)

3

Determination of the normal strain and stress e^(t) and cj^(t)

4

Calculation of the equivalent energy history W^(t)

5

Determination of the critical plane (^, m^, fi ^ j

6

Counting the cycle and half-cycle amplitudes in the critical plane

7

Fatigue damage accumulation

8

Fatigue life determination

Fig.4. Algorithm for the fatigue life determination with use of energy parameter Since in our experiment shear stress and shear strain coming from torsion with the gradients of these quantities, we introduce the correction coefficients k "

, Eq. (20) and k^

, Eq. (21)

into Eqs (27) and (28). Then we obtain the following equations a , ( t ) = l,X,(t)4-2k^^^l,m,T,y(t) eri(t) = l K x ( t ) + m^eyy(t) + n^8,,(t) + 2 k f % m ^ e , y ( t )

(29) (30)

Estimation of the Fatigue Life of High Strength Steel Under ...

193

Thus, the parameter of the normal strain energy density in the critical plane with normal r\ according to Eq. (6) takes the following form (stage 4) W^(t)-0.5a^(t)e^(t)sgn[a^(t),e^(t)]

(31)

In plane stress state, the vector r\ normal to the fracture plane may be described with use of only one angle a in relation to the axis Ox. Thus, the direction cosines of the unit vector rf are l^=cosa,

m^^sina,

n^=0

(32)

Assuming that the equivalent time history of energy parameter is the parameter of normal strain energy density described in Eq. (31) Weq(t) = WTi(t), we introduce Eqs (23) - (26) into Eq. (30) and next we obtain the expression for the equivalent energy parameter of the normal strain in the critical plane Weq(t) in the elastic range. Weq (t) = 0.5(1^0,, (t) + 2k»CPi^m^T,y (t)]- [(i^ - vm^ - vn^ ) i a , , (t) + ke^^H^m^ ^ T , y (t) •sgn t^a^^(t) + 2k^^Pi^m^T^y(t)], l(i^^ (33) For torsion, Eq.(33) takes the form W,,(t) = k f ' = k f ^ ^ m ^ - ^ T ^ y ( t ) s g n [ T , , ( t ) ] .

(34)

(j

Equation (34) reaches the maximum for angle K/4, i.e. for

1,=A,=^.

(35)

and for random loading it can be written as Weq(t) = k^CPk^^P-^T^y(t)sgn[T,y(t)] . 4G

(36)

For cyclic loading the amplitude of the equivalent energy parameter is W.e,=krkr-^T:,,,

(37)

where the correction coefficients k^*^^, k^^^ for a high number of cycles were obtained according to Eqs (20) and (21).

194

T. LAGODA ET AL

At Stage 5 the critical plane is determined. In this paper the authors used the damage accumulation method, as in [4 - 7, 22]. According to this method, the critical plane is the plane where the calculated damage degree according to the assumed fatigue parameter is the maximum, and the fatigue life is the minimum. Thus, the fatigue lives were determined in many planes according to the successive stages. After determination of energy parameter histories in a given plane (stage 6), the rain flow algorithm was used for cycle and half-cycle counting and next damage was accumulated (stage 7) according to the Palmgren-Miner hypothesis [25, 26] for W, > a • W,, (38)

S(To) = 0

for W, < a - W . ,

where: S(To) - degree of the material damage at time To, j - number of the class intervals of the amplitude histogram, Waf - fatigue limit expressed in the strain energy density. No - number of cycles corresponding to the fatigue limit o^, tti- number of cycles with amplitude W^ , m - exponent of the fatigue characteristic (a^ - N^), m' = m/2 - exponent of the fatigue characteristic (w^ - N^), a - coefficient (< 1) allowing to consider amplitudes below the fatigue limit in the damage accumulation. Damage was also accumulated according the Serensen - Kogayev hypothesis [25]: for W, > a • W., S(To) =

(39)

b N, for W.,
where:

Jw,iti-a.W,f b = i=l W,, ^al - a - Waf U =-

Serensen - Kogayev coefficient.

• frequency of occurrence of particular levels Wai at time To, J

1=1

Wai - maximum amplitude of the energy parameter.

Estimation of the Fatigue Life of High Strength Steel Under ...

195

After determination of the damage degree at an observation time To according to Eqs (38) and (39), we calculated the fatigue life (stage 8)

or if we have time observation in blocks, we obtain calculated fatigue life in number of blocks Nb,cal

where S (Nb) is the damage degree accumulated in a single block of Nb cycles. Then the calculated number of cycles is equal to ^ c a l = ^ = Nb,,3pNb

(42)

COMPARISON OF THE CALCULATED AND ACTUAL FATIGUE LIVES In order to assess the algorithm for fatigue life calculation under proportional variableamplitude tension with torsion, presented in Fig.4, we must verify particular assumptions of that algorithm. At first we should verify the algorithm for constant amplitude loading. From the constants of the regression model for torsion, Eq. (2) and from Eqs (10) and (11) we obtain bt = -0.066 and x\ = 937 MPa. Table 2 shows the fatigue lives under constant amplitude torsion calculated according to Eq.(41). The calculated and experimental fatigue lives are also compared in Fig.5. From Table 2 and Fig. 5 it appears that the proposed model gives correct results of the fatigue life calculations for pure torsion. All the results are in the scatter band of the factor 3. Table 2. Calculated and experimental fatigue lives under constant amplitude torsion

-^at

Ncal

Nexp

[MPa]

[cycle]

[cycle]

450

27854

33000,35000

430

53326

60000

400

149842

130000,280000

370

456377

390000,600000, 1100000

350

1009463

700000,900000

196

T.LAGODAETAL

Fig.5. Comparison of the calculated fatigue life with experimental data for constant amplitude torsion A further verification was performed for variable-amplitude tension-compression. The results of fatigue life calculations Nb,cai under uniaxial variable-amplitude tension-compression according to Palmgren-Miner (PM) and Serensen-Kogayev (SK) hypothesis and for various coefficients a are shown in Table 3. The table also contains the relative mean error of the calculated fatigue lives according to the following equation

(43)

N b,exp Fatigue damage accumulation according to the Palmgren-Miner hypothesis leads to unrealistic estimation of the fatigue life. The mean relative error is 942%. Taking account of the amplitudes of the considered energy parameter below the fatigue limit (a = 0.25) gives the mean error 240%. The Serensen-Kogayev hypothesis gives a realistic estimation of the fatigue life.

Estimation of the Fatigue Life of High Strength Steel Under .

197

The calculations were made for coefficients a =1, 0.81, 0.64. The best agreement with the experimental results is obtained when a = 0.81. In such a case we obtain the mean error 18%. Moreover, assuming lower amplitudes leads to incorrect estimation of the fatigue life and a greater error in the life calculation. In Fig.6 the calculated and experimental fatigue lives for uniaxial tension-compression are compared. It can be seen that the calculation results for a = 0.81 are included in the scatter band with a factor 2, and for 0.64 ^ a ^ 1 in the scatter band with a factor 3.

1000 H

100 H

Fig.6. Comparison of the calculated fatigue life with experimental data for uniaxial variable amplitude tension-compression Basing on the previous calculations, we performed further calculations for all the tests realized under variable amplitude loading. The calculation results for damage accumulation according to the Serensen-Kogayev linear hypothesis, including the energy parameter amplitudes 81% of the fatigue limit expressed in energy are shown in Table 4. These calculated fatigue lives are compared with the experimental lives in Fig.7. In the case of variable amplitude tor-

198

T. LAGODA ETAL

sion the critical plane is inclined of n/A with respect to the specimen axis, like in cyclic tests, and the calculated fatigue lives are satisfactory. Under combined tension with torsion we obtained different critical planes and the fatigue life calculation results are very similar. All the calculated fatigue lives are satisfactory. Let us notice that for combined variable amplitude tension with torsion the mean angle of the fracture plane (Table 1) is oCexp = 15.8°, and the calculated angle of the critical plane, being also the fracture plane (Table 4), is Ocai = 18°. Thus, the error in the fracture plane determination is only 2.2°. J

\

I

U-J

L

Nb,cal

[blocks] 100

# Tension A Torsion <^ Tension with torsion

10 10

100

Fig.7. Comparison of the calculated fatigue life with experimental data for uniaxial variable amplitude tension-compression, torsion and proportional tension-compression with torsion according to Serensen-Kogayev hypothesis for a = 0.81

199

Estimation of the Fatigue Life of High Strength Steel Under .

Table 3. Calculated fatigue lives Nbxai [blocks] under uniaxial variable-amplitude tensioncompression according to the Palmgren-Miner (PM) and Serensen-Kogayev (SK) hypothesis for different coefficients a PM

PM

SK

SK

SK

[MPa]

a=l

a = 0.25

a=l

a = 0.81

a = 0.64

675*

1137

276

160

109

65

675**

1137

313

160

109

65

736**

421

180

59

41

32

ER

942%

240%

48%

18%

38%

* - course CARLOS-fl (13568 extrema), ** - course CARL0S-f2 (46656 extrema) Table 4. Results of fatigue life calculations under variable-amplitude loading according to the Serensen-Kogayev hypothesis with coefficient a = 0.81 Tension- compression

Tension with torsion

Torsion

^amax

^b,cal

^cal

^amax

^b,cal

^cal

'^amax

^b,cal

^cal

[MPa]

[blocks]

[°]

[MPa]

[blocks]

[°]

[MPa]

[block]

[°]

675*

109

0

595*

150

18

488*

208

45

675**

109

0

595**

150

18

488**

208

45

736**

41

0

525**

62

45

* - course CARLOS-fl (13568 extrema), ** - course CARL0S-f2 (46656 extrema)

CONCLUSIONS 1. The normal strain energy density parameter in the critical plane is an efficient quantity describing the fatigue life under variable amplitude tension-compression, torsion and proportional tension with torsion. 2. Under constant amplitude torsion where the stress gradient occurs, we can determine the equivalent stress amplitude and accumulate the damage with use of the Wohler curve for tension-compression. 3. Application of the linear hypothesis of damage accumulation, formulated by PalmgrenMiner, leads to too high fatigue life as compared with the experimental life. Application of the Serensen-Kogayev hypothesis, including amplitudes of the analysed energy parameter higher than 81%) of the fatigue limit (expressed in energy), leads to correct estimation of the fatigue life. 4. Application of the energy parameter distinguishing tension (positive value of the energy parameter) and compression (negative value of the energy parameter) for fatigue life es-

200

T. lAGODA ET AL

timation under variable amplitude proportional tension with torsion leads to satisfactory results of calculations, in agreement with experimental ones. 5. The critical planes determined with use of the damage accumulation method agree with the fracture planes determined in experiments. ACKNOWLEDGEMENTS The paper was realized within the research project 7 T07B 018 18 and Polonium 2002 partly financed by the Polish State Research Committee in 2000-2002 and NATO Advanced Fellowships Programme 1|J|2000.

REFERENCES 1.

Lagoda T. and Macha E., (1998) A review of high-cycle fatigue models under nonproportional loadings, in: Fracture from Defects, Proc. ECF-12, Sheffield, Eds. M.W.Brown, E.R. de los Rios and K.J.Miller, EMAS, Vol. I, pp. 73-78

2.

Macha E., (2001) A review of energy-based multiaxial fatigue failure criteria. The Archive of Mechanical Engineering, vol.XLVIII, No. 1, pp. 71-101

3.

Macha E. and Sonsino CM., (2000) Energy criteria of multiaxial fatigue failure. Fatigue Fract. Engng Mater. Struct., Vol 22, pp.1053-1070

4.

Lagoda T. and Macha E., (1999) Assessment of long-life time under uniaxial and biaxial random loading with energy parameter on the critical plane, Fatigue '99 Proc. 7th Int. Fatigue Congress, Eds. X.-R. Wu and Z.-G. Wang, EMAS, Vol.11, pp.965-970

5.

Lagoda T., Macha E. and B^dkowski W., (1999) A crifical plane approach based on energy concepts: application to biaxial random tension-compression high-cycle fatigue regime. Int. J.Fatigue, vol.21, pp.431-443

6.

Lagoda T. and Macha E., (2000) Generalization of energy-based mukiaxial fatigue criteria to random loading, Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1397, S.Kalluri and P.J.Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, pp. 173-190

7.

Lagoda T., Macha E. and Sakane M., (1998) Correlation of biaxial low-cycle fatigue lives of SUS304 stainless steel with energy parameter in critical plane at 923 K, 6th ISCCP Bialowieza, Eds. A.Jakowluk and Z.Mroz, Technical University of Bialystok, pp.343-356

8.

Lagoda T. and Macha E., (2001) Energy approach to fatigue life estimation under combined tension with torsion. Scientific Papers of the Technical University ofOpole, Z.67, No 269/2001, Opole, (Poland), pp.163-182

9.

Lagoda T., Macha E., Nieslony A. and Morel F., (2001) The energy approach to fatigue life of high strength steel under variable-amplitude tension with torsion, Proc. of the 6th ICBMFF, Lisbon, Ed. Manuel Moreira de Freitas, vol.1, pp.233-240

10. Lagoda T., (2001) Energy models for fatigue life estimation under random loading - Part I - The model elaboration, Int.J. Fatigue, vol.23. No 6, pp.467-480

Estimation of the Fatigue Life of High Strength Steel Under ...

201

11. Lagoda T., (2001) Energy models for fatigue life estimation under random loading - Part II - Verification of the model, InU. Fatigue, vol.23, No 6, pp.481-489 12. Morel F., (1998) A fatigue life prediction method based on a mesoscopic approach in constant amplitude multiaxial loading. Fatigue Fract. Engng. Mater. Struct., vol.21, pp.241-256 13. Morel F., (1998) A mesoscopic approach to describe the high cycle fatigue behaviour of metals submitted to multiaxial loading, Scientific Papers of the Institute of Materials Science and Applied Mechanics of the Wroclaw University of Technology No 60, Conferences No 8, Wroclaw, pp. 12-54 14. Morel F., (1996) Fatigue multiaxiale sous chargement d'amplitude variable, Docteur these, CNRS, ENSMA, Futuroscope, France, ps.288 15. Orowan E., (1939) Theory of the fatigue metals. Proceedings of the Royal Academy, London, A, 171 16. Papadopoulos Y.V., (1996) Exploring the high-cycle fatigue behaviour of metals from the mesoscopic scale, J. Mech. Behav. Mat., vol.6, pp.93-118 17. Dang-Van K., (1993) Macro-micro approach in high-cycle multiaxial fatigue, Advances in Multiaxial Fatigue, ASTM STP 1191, D.L.McDowell and R.Ellis, Eds., American Society for Testing and Materials, Philadelphia, pp. 120-130 18. ASTM E 739-91 (1998) Standard practice for: Statistical analysis of linear or linearized stress-life (S - N) and strain-life (e - N) fatigue data. Annual Book of ASTM Standards, vol. 03.01, Philadelphia, pp.614-620 19. Kliman V., (1985) Fatigue life estimation under random loading using the energy criteria, Int. J. Fatigue, vol.7. No 1, pp.39-44 20. Grzelak J., Lagoda T. and Macha E., (1991) Spectral analysis of the criteria for multiaxial random fatigue, Mat. -wiss. U. Werkstofftech, Nr 22, pp.85-98 21. Lowisch G., Bomas H. and Mayr P., (1994) Fatigue crack initiation and propagation in ductile steels under multiaxial loading. Fourth Int. Conf on Biaxial/Multiaxial Fatigue, St Germain en Laye, Vol. II, pp.27-42 22. Macha E., (1989) Simulation investigations of the position of fatigue fracture plane in materials with biaxial loads, Mat. -wiss. U. Werkstofftech., Heft 4/89, pp. 132-136; Heft 5/89,pp.l53-163 23. Miner M.A., (1945) Cumulative damage in fatigue, J. of Applied Mechanics, vol. 12, pp.159-164 24. Palmgren A., (1924) Die Lebensdauer von Kugellagern, VDI-Z, vol. 68, SS.339-341 25. Serensen S.V., Kogayev V.P. and Shneyderovich R.M., (1975) Nesushchaja sposobnost i raschet detaley mashin na prochnost, Izd. 3-e, Mashinostroenie, Moskva, ps.488 (in Russion)

202

T. LAGODA ETAL

Appendix: NOMENCLATURE a

-

E ER G

-

coefficient including influence of amplitudes less than the fatigue limit, Young's modulus, relative mean error. shear modulus. direction cosines. Wohler curve exponent, number of cycles. Palmgren-Miner yield point, tensile strength. damage degree. Serensen - Kogayev time. observation time. strain energy parameter. contraction, angle between the longitudinal axis of the specimen and the normal direction of critical plane, engineering shear stain. Poisson's ratio. normal stress. fatigue limit for tension. shear stress, fatigue limit for torsion.

In, ni^, r^n-

m N PM Ro.2

Rm

s

SK t To W Z a Y V

a CTaf

T

Taf 5: Subscripts:

a b cal eq exp max

— — .

amplitude. block. calculated. equivalent. experimental. maximum value.

"

Function: 1

for

X >0

sgn(x) = ' 0

for

X =0

-1

for

X <0

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

203

CRITICAL PLANE-ENERGY BASED APPROACH FOR ASSESSMENT OF BIAXIAL FATIGUE DAMAGE WHERE THE STRESS-TIME AXES ARE AT DIFFERENT FREQUENCIES

A. Varvani-Farahani Department of Mechanical, Aerospace and Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, Ontario, MSB 2K3, Canada

ABSTRACT A new fatigue parameter has been developed to assess the fatigue lives under in-phase and outof-phase biaxial constant amplitude fatigue stressing where the stresses are at different frequencies. In this parameter, the normal and shear stress and strain ranges have been calculated from the largest stress and strain Mohr's circles. The total damage accumulation in a block loading history has been calculated from the summation of the normal and shear energies on the basis of a cycle-by-cycle analysis. The normal and shear energies used in this parameter are divided by the tensile and shear fatigue properties, respectively, and the proposed parameter, unlike many other parameters, does not use an empirical fitting factor. The proposed fatigue parameter has successfully correlated biaxial fatigue lives of thin-walled EN24 steel tubular specimens tested under in-phase and out-of-phase biaxial fatigue stressing where stresses were at different frequencies and included mean values. KEYWORDS Fatigue damage parameter, loading frequency ratio, block loading history, critical plane, shear and normal energies

INTRODUCTION Many engineering components that undergo fatigue loading experience multiaxial stresses, in which two or three principal stresses fluctuate with time, i.e. the corresponding principal stresses are out-of-phase or the principal directions change during a cycle of loading. These components include car axles, helicopter propeller shafts, and airplane wings subjected to bending and torsion, which can be out-of-phase, and at different frequencies. To assess fatigue life of the components, many attempts have been made to derive theories based on equivalent stress/strain, elastic-plastic work/energy and critical plane approaches. General reviews of multiaxial fatigue life prediction methods are presented by Garud [1], Brown and Miller [2], and You and Lee [3]. As yet there is no universally accepted approach. It was shown that the von Mises criterion was limited to correlating multiaxial life data under proportional loading in the high cycle fatigue regime [4-5]. Energy approach has been applied in conjunction with

204

A. VARVANI-FARAHANI

incremental plasticity theory to predict fatigue life under complex non-proportional multiaxial loading conditions [6], however the approach did not successfully correlate the uniaxial and torsional fatigue data. In another attempt [7-8], a ratio of axial strain to maximum shear strain has been introduced as a correction factor to the energy approach. This ratio has been criticized because the hysteresis energy method used in Refs [7-8] does not hold any terms to reflect this strain ratio. Fatigue analysis using the concept of a critical plane of maximum shear strain is very effective because the critical plane concept is based on the fracture mode or the initiation mechanism of cracks. In the critical plane concept, after determining the maximum shear strain plane, many researchers have defined fatigue parameters as combinations of the maximum shear strain (or stress) and normal strain (or stress) on that plane to explain multiaxial fatigue behavior [9-13]. Some researchers [14-18] used the energy criterion in conjunction with the critical plane approach. Energy-critical plane parameters are defined on specific planes and account for states of stress through combinations of the normal and shear strain and stress ranges. These parameters depend upon the choice of the critical plane and the stress and strain ranges acting on that plane. For instance, Liu [14] calculated the virtual strain energy (VSE) in the critical plane by the use of crack initiation modes. The critical plane in Liu's parameter is associated with two different physical modes of failure and the parameter consists of Mode I and Mode n energy components. The Mode I energy is computed by first identifying the plane on which the axial energy is maximized and adding the shear energy on that plane. Similarly, the Mode 11 energy is calculated by first identifying the plane on which the shear energy is maximized and then adding the axial energy component. In the calculation of VSE, Liu induced both elastic energy and plastic energy while the elastic energy was not considered in Garud's model [6]. Chu et al. [15] formulated normal and shear energy components based on Smith-Watson-Topper parameter. They determined the critical plane and the largest damage parameter from the transformation of strains and stresses onto planes spaced at equal increments using a generalized Mroz model [19]. Glinka et al. [16] presented a multiaxial life parameter based on the summation of the products of shear and normal strains and stresses on the critical shear plane. In a recent study [17], a multiaxial fatigue parameter for various in-phase and out-of-phase loading paths has been proposed. The parameter is given by the integration of the normal energy range and the shear energy range calculated for the critical plane at which the stress and strain Mohr's circles are the largest during the reversals of a cycle. The maximum shear strains for in-phase and out-of-phase paths are numerically calculated at small increments through a fatigue cycle. The parameter has taken into account both elastic and plastic strain components in fatigue damage analysis. The normal and shear energies in this parameter have been weighted by the tensile and shear fatigue properties, respectively, and the parameter requires no empirical fitting factors. The parameter has taken into account the effect of mean stress applied normal to the critical plane and has also shown an increase when there is additional hardening, caused by out-of-phase loading, while strain-based parameters fail to take into account this effect. The present paper further extends the Varvani's damage parameter [17] for fatigue life assessment under biaxial loading condition by accumulating damage on a cycle-bycycle basis within an entire two-axis block loading history, where the stress-time axes are at different frequencies.

MATERL\L AND FATIGUE TESTS Table 1 tabulates the chemical composition and fatigue properties of EN 24 steel studied in the present paper.

Critical Plane-Energy

Based Approach for Assessment of Biaxial Fatigue Damage where .

205

Table 1. Chemical composition (Wt%) and fatigue properties of EN 24 steel. C: 0.35-0.45, Si: 0.10-0.55, Mg: 0.45-0.70, Ni: 1.30-1.80, Cr: 0.90-1.40, Mo:0.20-0.35, S and P: 0.05, and Fe: remaindeT

E=2Q7GPa

£f = 1.14

af = 1650MPa

G=80 GPa

Yf = 1.69

Tf':::115Q MPa

Due to the difficulty and expense of performing complex multiaxial fatigue tests particularly when two axes of loading are at different frequencies, a very few data of this type is available in the literature. Biaxial fatigue test data used in this study have been taken from References [20-22]. Biaxial fatigue tests have been conducted on thin-walled tubular EN24 steel specimens subjected to constant amplitude alternating longitudinal stress (aO and alternating differential pressure across the wall thickness (02) at different values of the frequency ratio a: Frequency of CJ, = 1,2, and 3. a= Frequency of (J2 Ten biaxial loading histories have been studied. In this paper, for the convenience of presentation, load paths and histories have been categorized into three kinds. Figure 1 presents in-phase and out-of-phase loading histories and load paths for: (a) a = l , (b) a=2, and (c) a=3. (a) Frequency ratio a = l : the ratio of loading frequencies in both axes is equal to unity and load paths are linear. In Fig. la, histories Al and A2 are in-phase biaxial loading histories. History Al contains no mean stress, while history A2 contains a transverse mean stress value different from zero. (b) Frequency ratio a=2: the ratio of loading frequencies of the longitudinal axis to the transverse axis is equal to two. Load history Bl contains no mean stress, while history B2 has a non-zero transverse mean stress value. Histories Bland B2 have butterfly-shaped load paths.

im

o2

(a)

>\ r

/ >"•

tin

>^

\y B3v.

C3J ^:J^

NW

Loi

al

Aa2

o2

C4\

LGI

/

(b) (c) Fig. 1. In-phase and out-of-phase biaxial loading histories where stresses are at different frequencies (a) a = l , (b) a=2 and (c) a=3.

206

A. VARVANI-FARAHANI

In load histories B3 and B4, two axes of loading start with a phase delay of (t)=90°: the former history contains no mean stress, while the latter history contains a transverse mean stress value. Histories B3 and B4 have non-Hnear C-shaped load paths (see Fig. 1(b)). (c) Frequency ratio a=3: two axes are loaded with a frequency ratio of 3:1. Histories CI and C3 correspond to biaxial tests including no mean value, however, histories C2 and C4 contain a non-zero transverse mean stress value. Histories CI and C2 have non-linear Z-shaped load paths, while histories C3 and C4 have non-linear S-shaped load paths (see Fig. 1(c)).

FATIGUE DAMAGE MODEL AND ANALYSIS

Cyclic stress and strain analyses The stress and strain tensors for a thin-walled tubular specimen are given by Eq (1) and Eq (2), respectively:

^.,=

=

0

^ 22

0

0

V

^ii

0

(1)

an J

0

0

0

£22

0

0

0

£•33

(2) J

Figure 2 illustrates a thin-walled tubular specimen subjected to biaxial fatigue loading. The two controlled applied stresses are the principal stresses in the plane shown on the tubular specimen. A Ell

Wrrrn (a)

(b)

(0

Fig. 2. (a) Thin-walled tubular specimen subjected to biaxial loading, (b) 3D presentation of principal stress state, and (c) of principal strain state.

Critical Plane-Energy

Based Approach for Assessment of Biaxial Fatigue Damage where ...

207

In equation (1) Gjj is the stress tensor (both i and j are equal to 1,2,3), and the stresses Gu, 022, and 033 are principal stresses. Since the thin-walled tubular specimens are in plane stress condition, a22=0. Principal stresses can be calculated from applied stress amplitudes: f^ll = ^ / n l + L ^ l S l n(^)]

(3a)

(722 = 0

(3b) (3c) a )]

where Gmi and (Jmi are the longitudinal and the transverse mean stresses, respectively. The angle 6 is the angle during a\ cycles of stressing in a block loading history at which the Mohr's circles are the largest and has the maximum value of shear strain. Angle 9 varies from 0 to 2an. Angle (j) corresponds to the phase delay between loads on the longitudinal and transverse axes. In Eq (2) 8ij is the strain tensor (both i and j are equal to 1,2,3) where the strains 8ii, 822, and 833 are principal strains calculated using the elastic-plastic strain constitutive relation (Eq(4)). In Eq (4), the first square bracket presents the elastic and the second square presents the plastic components of strains: cP 1 -eq_

\ + v.

- ^ . - ^ ^ ^ k^'j\

2 '^

(4)

where the deviatoric stress Sjj is defined as the difference between the tensorial stress Ci^ and hydrostatic stress (-cr^^t)"

S =a ij

where 5ij=l

if i=j

5.j=0

if i^j

--a,,S ;/

^

kk

(5)

ij

E is the elastic modulus, Ve=0.3 is the elastic Poisson's ratio for the material used in this study and Okk is the summation of principal stresses. The stress amplitude of the hysteresis loop at half-life cycle was associated with the stabilized cyclic stress-strain loop. The cyclic stressstrain curve can be described with the same mathematical expression as for the monotonic stress-strain curve. The relationship between the equivalent cyclic plastic strain e^^ and the equivalent cyclic stress crgq obtained from uniaxial stabilized cyclic stress-strain data is: ^1227(£-P )/2.78

(6)

The coefficient and the exponent in Eq (6) are the cyclic strength coefficient, K*=1227 MPa, and the cyclic strain hardening exponent, n*=0.36, respectively. Eq (4) presents elastic-plastic deformation and correlates the tensorial cyclic stress and strain components for 3D state of stress and strain. A simple form of this equation for the

208

A. VARVANI-FARAHANI

uniaxial

loading

condition

is

the

well-known

Ramberg-Osgood

relation

(^11 = ^ n / ^ + P i i / ^ j " )' which is commonly used to present cyclic uniaxial stress-strain curve. The range of maximum shear stress and the corresponding normal stress range are calculated from the largest stress Mohr's circle during the first reversal (at the angle 61) and the second reversal (at the angle 62) of a cycle as: _^ll-^33

AW=P^^^ >^u±^]

1

f (Til-0-33

-P^^^

(7a)

_£iLl£B|

(7b)

where a n , O22 and 033 are the principal stress values calculated using Eq (3). Similarly, the range of maximum shear strain and the corresponding normal strain range on the critical plane at which Mohr's circles are the largest during the first reversal (at the angle 9i) and the second reversal (at the angle 62) of a cycle are calculated as:

A|ijii^Vf£LLl£3ll _f£lLl£31| ^

^

l^lilfB

Je\ ^

^

_£iil£3i|

(8a)

Jei

(8b)

where en, £22, and 833 are the principal strain values (£n>£22> £33) which are calculated using Eq (4).

Proposed fatigue parameter In this paper for the convenience of presentation, first the proposed parameter and its capability to take into account the effects of out-of-phase strain hardening and mean stress are discussed for the load histories with frequency ratio of a=l [17,23,24] and then the damage model is extended for other ratios of a=2 and a=3. The range of maximum shear stress AXmax and shear strain AI-^-'^^ I obtained from the largest stress and strain Mohr's circles at angles 9i and 62 during a cycle and the corresponding normal stress range AGn and the normal strain range ASn on that plane are the components of the proposed parameter in the present paper. Multiaxial fatigue energy-based models have been long discussed in terms of normal and shear energy weights. In Garud's approach [8] he found that an empirical weighting factor equal to C=0.5 in the shear energy part of his model (Eq 10) gave a good correlation of multiaxial fatigue results for 1% Cr Mo V steel for both in-phase and out-of-phase loading conditions: Afilo- + CA}Ar = /(Nf)

(9)

Critical Plane-Energy Based Approach for Assessment of Biaxial Fatigue Damage where ... where Nf is the number of cycles to failure, hi Eq (9) Ae and Aa are the range of normal strain and stress, and Ay and Ax are the range of shear strain and stress, respectively. Tipton [25] found that a good multiaxial fatigue life correlation was obtained for 1045 steel with a scaling factor C of 0.90. Andrews [26] found that a C factor of 0.30 yielded the best correlation of multiaxial life data for AISI 316 stainless steel. Chu et al. [15] weighted the shear energy part of their formulation by a factor of C=2 to obtain a good correlation of fatigue results. Liu's [14] and Glinka et al. [16] formulations provided an equal weight of normal and shear energies. The empirical factor (C) suggested by each of the above authors gave a good fatigue life correlation for a specific material which suggests that the empirical weighting factor C is material dependent. In the present study, the proposed model correlates multiaxial fatigue lives by normalizing the normal and shear energies using the axial and shear material fatigue properties, respectively, and hence the parameter uses no empirical weighting factor. Both normal and shear strain energies are weighted by the axial and shear fatigue properties, respectively: pi-,(Aa„AfJ+

'

^AwA^i^jj=/(Nf)

(10a)

where dt and £f are the axial fatigue strength coefficient and axial fatigue ductility coefficient, respectively, and ri and /f are the shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively.

Out-of-phase strain hardening Under out-of-phase loading, the principal stress and strain axes rotate during fatigue loading (e.g. see [13]) often causing additional cyclic hardening of materials. A change of loading direction allows more grains to undergo their most favorable orientation for slip, and leads to more active slip systems in producing dislocation interactions and dislocation tangles to form dislocation cells. Literactions strongly affect the hardening behavior and as the degree of outof-phase increases, the number of active slip systems increases. Socie et al. [27] performed inphase and 90° out-of-phase fatigue tests with the same shear strain range on 304 stainless steel. Even though both loading histories had the same shear strain range, cyclic stabilized stressstrain hysteresis loops in the 90° out-of-phase tests had stress ranges twice as large as those of the in-phase tests. They concluded that the higher magnitude of strain and stress ranges in the out-of-phase tests was due to the effect of an additional strain hardening in the material [28]. During out-of-phase straining the magnitude of the normal strain and stress ranges is larger than that for in-phase straining with the same applied shear strain ranges per cycle. The proposed parameter via its stress range term increases with the additional hardening caused by out-of-phase tests whereas critical plane models that include only strain terms do not change when there is strain path dependent hardening. To calculate the additional hardening for out-ofphase fatigue tests, these approaches may be modified by a proportionality factor like the one proposed by Kanazawa et al. [29].

Mean stress correction Under multiaxial fatigue loading, mean tensile and compressive stresses have a substantial effect on fatigue life. Sines [30] showed compressive mean stresses are beneficial to the fatigue life while tensile mean stresses are detrimental. He also showed that a mean axial tensile stress

209

210

A. VARVANI-FARAHANI

superimposed on torsional loading has a significant effect on the fatigue life. In 1942 Smith [31] reported experimental results for twenty-seven different materials from which it was concluded that mean shear stresses have very little effect on fatigue life and endurance limit. Sines [30] reported his findings and Smith's results by plotting mean stress normalized by monotonic yield stress versus the amplitude of alternating stress normalized by fatigue limit (R=-l) values. The relation is linear as long as the maximum stress during a cycle does not exceed the yield stress of the material [28]. Concerning the effect of mean strain on fatigue life, Bergmann et al. [32] found almost no effect in the low-cycle fatigue region and very little effect in the high-cycle fatigue region. Mean stress effects are included into fatigue parameters in different ways [28]. One approach was applied earlier by Fatemi and Socie [33] to incorporate mean stress using the maximum value of normal stress during a cycle to modify the damage parameter. Considering the effect of mean axial stress, a mean stress correction factor 1 + ^

inserted into Eq (10a)

showed a good correlation of multiaxial fatigue data containing mean stress values for both inphase and out-of-phase straining conditions. This correction is based on the mean normal stress, &!!, applied to the critical plane. To take into account the effect of mean axial stress on the proposed parameter, Eq (10a) is rewritten as:

1+ ^ 1 ^^[^a,^e,)+vrty

^Ar,,, A Ym ^ ^ JJ= /(N,)

(10b)

where the normal mean stress cC acting on the critical plane is given by:

^n"=ik"+^n™")

(11)

where cf^"" and cC" are the maximum and minimum normal stresses, respectively, which are calculated from the stress Mohr's circles. The mean normal stress correction factor can be applied for both <j^ >0 and <j^ <0 at which a*^ >0, the tensile mean normal stress, increases the fatigue damage and cj^ <0, the compressive mean normal stress, decreases the fatigue damage. Equation (10b) takes into account the effects of compressive and tensile stress ranges on fatigue damage analysis. This coincides with a numerous fatigue tests and analyses performed earlier by Topper and his research group [34-38]. They have confirmed that the compressive portion of the stress cycle contributes significantly to the accumulation of fatigue damage, even when the maximum stress is below the fully reversed fatigue limit.

Proposed fatigue damage parameter for biaxial fatigue tests where the stresses are at different frequencies The proposed damage parameter has been further developed to assess the fatigue lives under in-phase and out-of-phase biaxial constant amplitude fatigue stressing where the stresses are at different frequencies. In this parameter, the normal and shear stress and strain ranges have been calculated from the largest stress and strain Mohr's circles corresponding to the most damaging planes in each cycle. The total damage accumulation in a block loading history has been calculated from the summation of the normal and shear energies on the basis of cycle-by-cycle

Critical Plane-Energy Based Approach for Assessment of Biaxial Fatigue Damage where .

211

analysis. Equation (12) has been developed to take into account the damage accumulation due to stress cycles within a block fatigue loading: P h = P | . t + P 9 n H + - . + PN

A w A | ^ | | > = /(Nf)

(12)

where N corresponds to the number of Gi cycles, i represents the i-th Gi cycle in a block loading history, subscript b represents the block loading, and subscript f represents the failure. Biaxial fatigue tests a-2\ Figure 3 presents an in-phase biaxial block fatigue loading at a frequency ratio a=2 and the largest stress and strain Mohr's circles required for damage analysis. This block contains two Q\ cycles and one (5i cycle. During the first Gi cycle, the reversals of Gi cycle loaded simultaneously with the first reversal of G2 cycle. At the first Gi cycle, the maximum shear strain and stress ranges were numerically calculated at 10° increments and the largest Mohr's circles have been obtained at 9i=100° and 02=250° angles. At the second Gi cycle, the largest Mohr's circles were numerically obtained at 63=470° and 64=620° angles (see Fig. 3). To distinct Mohr's circles obtained at 6t-angles (61, 62,..., 62N), Mohr's circles corresponding to 6-angles at which subscript t is an odd number have been illustrated by solid lines and those Mohr's circles corresponding to 6-angles at which subscript t is an even number have been illustrated by dashed lines. Stress Mohr's circles

Strain Mohr's circles ^

[<-(A£j^)eie2

i,Y/2

One block loading history

(Ao„),

Fig. 3. In-phase biaxial fatigue stressing (history Bl) where two axes are at different frequencies (a=2)

212

A. VARVANI-FARAHANI

Figure 4 presents an out-of-phase biaxial block fatigue loading at a frequency ratio oc=2 and the corresponding largest stress and strain Mohr's circles within a block loading history. A block loading history contains two d cycles and one G2 cycle having a phase delay of (t)=90°. The first reversal of d cycle loaded from zero level while the a i cycle is under tensile stress. At the first cycle, the maximum shear strain and stress ranges were numerically calculated at 10° increments and the largest Mohr's circles have been obtained at 0i=9O° and 02=270° angles. Similarly, during the second G\ cycle, the reversals of G\ cycle loaded simultaneously with the second reversal of 02 cycle. At the second a\ cycle, the largest Mohr's circles were numerically obtained at 93=450° and 64=630° angles (see Fig. 4). The proposed fatigue parameter for the first Gi cycle is expressed as:

1-^^ ^f

Plst - "

(13a)

Ar,

The proposed fatigue parameter for the second G\ cycle is expressed as:

1+ ^ ^ P2nd = / .

,\(Aq-nA£n )2nd ^^'

(13b)

Ar^n.A

The total damage in a block loading history is calculated by accumulating the parameter for the block loading history consisting of two Ci cycles (Pb=Pist+P2nd)Stress Mohr's circles

Strain Mohr's circles Y/2, U-l^^n)ei.92 k

i'h ^ V

MJ J

^' f

Y/2,_-(-..^

One block loading history ^(^£n)e3,e4

Fig. 4. Out-of-phase biaxial fatigue stressing (history B3) where two axes are at different frequencies (a=2)

Critical Plane-Energy Based Approach for Assessment of Biaxial Fatigue Damage where .

213

Biaxial fatigue tests a=3: Figure 5 presents an in-phase biaxial block fatigue loading at a frequency ratio of a=3 and the largest stress and strain Mohr's circles required for damage analysis. This block contains three a\ cycles and one <J2 cycle. During the l-^(5\ cycle, the first three reversals of ai cycle loaded simultaneously with the first reversal of 02 cycle. At the first Gi cycle, the maximum shear strain and stress ranges were numerically calculated at 10° increments and the largest Mohr's circles have been obtained at 61=80° and 02=270° angles. At the second G\ cycle, the largest Mohr's circles were numerically obtained at 93=460° and 84=620° angles, and finally at the third Gi cycle, the largest Mohr's circles were obtained at 85=810° and 86=1000° angles (see Fig. 5). Stress Mohr's circles

Strain Mohr's circles •n^ei,e2

(^<^n)ei.e2

T

(Aaj 05,96 <->

A Third cycle

^X ^ 66

\^ 05

V

1

/ • - -

*

\_j/ Ji <

One block loading history

Fig. 5. In-phase biaxial fatigue stressing (history CI) where two axes are at different frequencies (a=3) Figure 6 presents an out-of-phase biaxial block fatigue loading at a frequency ratio a=3 and the corresponding largest stress and strain Mohr's circles within a block loading history. A block loading history contains three G\ cycles and one 02 cycle having a phase delay (|)=180°. The first reversal of Gi cycle loaded from zero level while the G2 cycle is under tensile stress. At the first cycle, the maximum shear strain and stress ranges were numerically calculated at 10° increments and the largest Mohr's circles have been obtained at 81=110° and 82=268° angles. Similarly, during the second a\ cycle, the reversals of a\ cycle loaded simultaneously with the second reversal of a2 cycle. At the second (5\ cycle, the largest Mohr's circles were

214

A. VARVANI-FARAHANI

numerically obtained at 03=460° and 64=620° angles, and finally at the third G\ cycle, the largest Mohr's circles were obtained at 95=800° and 06=1000° angles (see Fig. 6). Stress Mohr's circles

Strain Mohr's circles

y/2

X,

gifc:(^£n)91,e2

One block loading history

Fig. 6. Out-of-phase biaxial fatigue stressing (history C3) where two axes are at different frequencies (a=3). The proposed fatigue parameter for the first (5\ cycle is expressed as:

1+ Put —

^

(AcTnAfn)lst

(14a)

(^f ^ f )

The proposed fatigue parameter for the second Gi cycle is expressed as:

(A(7nA£j2«^ {^f ^ f )

The proposed fatigue parameter for the third Gi cycle is expressed as:

(14b)

Critical Plane-Energy

215

Based Approach for Assessment of Biaxial Fatigue Damage where .

1+ ^f

;

(14c)

l^f rf J

3rd

The total damage in a block loading history is calculated by accumulating the parameter for the block loading history consisting of three Gi cycles (Pb=Pist+P2nd+ Psrd)Table 2 lists 9-angles, at which the largest Mohr's circles are obtained for various block loading histories shown in Fig 1. Table 2. The most damaging planes obtained for various block loading histories. Histories

a

A1-A2 B1-B2 B3-B4 C1-C2 C3-C4

1 2 2 3 3

6-angles (degrees) of most damaging planes of a\ cycles 65 66 64 03 92 61 ___ ___ ___ 270 90 ___ 620 250 470 100 ___ 630 270 450 90 810 1000 620 270 460 80 800 1000 620 260 460 110

FATIGUE LIFE RESULTS AND CORRELATIONS Stress amplitude-number of blocks to failure Figures 7a-7c present the number of blocks to failure versus the magnitude of biaxial stress amplitude for the load histories tabulated in Table 2. 400 F

400 OH

300

n ^

200 U 10'

n

Transverse mean stress (In-phase loading)

i(a)

No mean stress (In-phase loading)

o o o

^ o n

n

oo

10-^ Number of blocks to failure

300

•

•

•

200

;

• •

• •

:(b) 10"

rOut-oF-phase roadihg) o n

B3(a=:2) C3(a=3) Oo

• 0

• •

B4(a=2) C4(a=3) No mean stress

I Including transverse : mean stress\

200 10

•

10Number of blocks to failure

10-^

10

• ;\:

A2(a=l) B2(a=2) C2(a=3)

• i

;

o Al (a==1) o Bl (a^--2) n CI (a=3)

400

300

• • •

:(c) 10-^ Number of blocks to failure

10"

Fig. 7. Stress amplitude versus number of blocks to failure for various biaxial fatigue tests.

216

A. VARVANI-FARAHANI

In all biaxial tests, the amplitude of Ci and G2 was identical. Figure 7a compares three inphase biaxial fatigue tests (Al, Bl and CI) with frequency ratios a equal to 1, 2, and 3, respectively. Histories B3 (a=2) and C3 (a=3) in figure 7c present the out-of-phase test results when no mean stress is involved. For both in-phase and out-of-phase fatigue test series, the fatigue strength decreases as the frequency ratio a increases. The lower fatigue strength is accompanied with a greater fatigue damage due to a greater number of Oi cycles in a block loading history as the frequency ratio a increases. Figure 7c also presents the detrimental effect of mean stress on fatigue results when histories B3 and C3 (with no mean stress) are compared with histories B4 and C4 (with mean stress). The mean stress influence on fatigue damage can also be observed by comparison of Figs 7a and 7b.

Fatigue life correlation using proposed parameter Figures 8a-8d present biaxial fatigue life correlations based on the proposed parameter (equation (12)) for in-phase and out-of-phase tests, which are shown in Fig. 1. Li this parameter, the normal and shear stress and strain ranges have been calculated from the largest stress and strain Mohr's circles. The total damage accumulation in a block loading history has been calculated from the summation of the normal and shear energies on the basis of cycle-bycycle analysis. 0.1

,

' ' ' ' ' 1

No mean stress ; : (In-phase loading) i

,0.01 9a

o a. o

^ocP ;

a

^

'

•

•

'

•

•

•

•

,

0.1

, , 1

eo^

3 0.01

.

.

.

I

.

1

.

• A2(a=l)[ • •

B2(a=2) C2(a=3)

-

(a)

]

! Transverse mean stress (In-phase loading)

M

"°"-^ 0 0 p

10^ Number of blocks to failure

No mean stress : (Out-of-phase loading)

B

,

1

0.001 10'

:

.

0 Al (a=l)f ° Bl (a=2) 0 CI (a=3)

10°

2 0.001 lO'*

0.1

'! : 0 B3(a=2)E ° C3(a=3):

.S'o.oi

1(b)-

OH

10' Number of blocks to failure

:': Transverse mean stress ! ; (Out-of-phase loading)

'

• •

•

•

10"

•

'

'

!

-

B4(a=2)[ C4(a=3)[

T3O.OI l

0 0 0

O

(c) •

nnni

10-^

10' Number of blocks to failure

10"

o 0.00110^

i;d)10' Number of blocks to failure

10"

Fig. 8. Biaxial fatigue life correlation for various in-phase and out-phase histories and frequency ratios. Fig. 8 shows that the proposed fatigue parameter successfully correlates biaxial fatigue lives of thin-walled EN24 steel tubular specimens tested under in-phase and out-of-phase biaxial fatigue stressing where stresses were at different frequencies and included mean values. The proposed parameter successfully correlates biaxial fatigue lives in low-cycle-fatigue regime

Critical Plane-Energy Based Approach for Assessment of Biaxial Fatigue Damage where .

217

(Nf<10 ) within a factor that varies from 1.2 to 2.0. A much better correlation of fatigue lives is observed at high-cycle-fatigue regime (Nf>10''), within a factor froml.O to 1.2.

DISCUSSION Energy-critical plane parameters [15-17] including the parameter proposed in the present paper are defined on specific planes and account for states of stress through combinations of the normal and shear strain and stress ranges. These parameters depend upon the choice of the critical plane and the stress and strain ranges acting on that plane. For the proposed parameter, the critical plane is defined by the largest shear strain and stress Mohr's circles during the reversals of a cycle, and the parameter consists of tensorial stress and strain range components acting on this critical plane experiencing the maximum damage. The total damage accumulation in a block loading history has been computed from the summation of the normal and shear energies on the basis of a cycle-by-cycle analysis. Figure 9a presents the biaxial stress- fatigue life data tested by McDiarmid [20-22]. In this figure, the magnitude of the longitudinal stress (Oi) versus the number of C\ cycles to failure for ten different load histories has been presented. Figure 9b shows how closely biaxial fatigue data are correlated using the parameter proposed in this paper. In this figure, the proposed damage parameter has been plotted versus the number of blocks to failure. The proposed parameter successfully correlates biaxial fatigue lives within a factor that varies from 1.2 to 2.0 for 10^
400

300

o Al

n CI

o Bl

A

C3

V

X

C2 A2

+ B2

s B3

o B4 ' C4

J-Q.Ol o ex 00 200

0.001 Number of a cycles to failure

Number of blocks to failure

Fig. 9. In-phase and out-of-phase biaxial fatigue life correlation (a) Gi-Nai and (b) the proposed parameter vs. Number of blocks to failure. For example, in the block loading history Bl with a frequency ratio a=2 (as is shown in figure 3), the first G\ cycle at its first reversal contains the largest stress and strain Mohr's circles at Oi=100° and in its second reversal contains the largest Mohr's circles at ©2=250°, while (52 cycle experiences its first reversal. Similarly, at the second Gj cycle, two planes of 63=470° and 64=620° were obtained as the most damaging planes. The total damage in this block is calculated by accumulating of damage parameters from the first and the second a\ cycles in that block history. In this damage analysis, those points with the greatest strain and stress values (with the largest Mohr's circles) throughout a block loading history are

218

A. VARVANI-FARAHANI

responsible for the largest physical damage on the critical planes, where the debonding of materials occurs. The proposed damage parameter via its stress range term increases with the additional hardening caused by out-of-phase tests whereas critical plane parameters that include only strain terms do not change when there is strain path dependent hardening. The parameter also contains no empirical constants for weighting of normal and shear energy components, instead, the normal and shear energy components are normalized by the axial and shear fatigue properties. These weighting factors make the damage parameter dimensionless as the fatigue damage by definition has no unit. The proposed fatigue damage parameter seems to show some promises in correlating fatigue results conducted under random fatigue loading (in-progress). The random fatigue using the proposed parameter will be examined in a close future.

CONCLUSIONS A fatigue damage parameter has been developed to assess the fatigue lives under in-phase and out-of-phase biaxial constant amplitude fatigue stressing where the stresses are at different frequencies. The normal and shear stress and strain ranges, which are included in this parameter, have been calculated from the largest stress and strain Mohr's circles. The total damage accumulation has been calculated from the summation of the normal and shear energies in a block loading history on the basis of a cycle-by-cycle analysis. The parameter contains no empirical constants for weighting of the normal and shear energy components; instead, the normal and shear energy components are normalized by the axial and shear fatigue properties. The proposed fatigue parameter has successfully correlated biaxial fatigue lives of thin-walled EN24 steel tubular specimens tested under in-phase and out-of-phase biaxial fatigue stressing where stresses were at different frequencies and included mean values. Fatigue life correlation has varied from 1.2 to 2.0 for 10'^
ACKNOWLEDGEMENTS The financial support of Natural Science and Engineering Research Council (NSERC) of Canada (NO. 1-51-56137) is greatly appreciated.

REFERENCES 1. 2.

3. 4. 5.

Garud, Y.S. (1981), Multiaxial fatigue: a survey of the state of the art. J Test Eval 9 (3), 165-178. Brown, M.W. and Miller, K.J. (1982), Two decades of progress in the assessment of multiaxial low-cycle fatigue life. In.- Amzallag, C , Leis, B., and Rabbe, P. (Eds), Lowcycle fatigue and life prediction, ASTM STP 770. American Society for Testing and Materials, 482-499. You, B.R. and Lee, S.B. (1996) A critical review on multiaxial fatigue assessment of metals. Int. J. Fatigue, 18(4), 235-244. Jordan, E.H. (1982), Pressure vessel and piping design technology-A decade of progress. ASME. American Society of Mechanical Engineers, 507-518. Garud, Y.S. (1979), A new approach to the evaluation of fatigue under multiaxial loadings. Ostergren, W.J. and Whitehead, J.R., (Eds). Proceedings of the Symposium

Critical Plane-Energy Based Approach for Assessment of Biaxial Fatigue Damage where ...

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on Methods for Predicting Materials Life in Fatigue. New York: American Society of Mechanical Engineers, 247-263. Garud, Y.S. (1981), A new approach to the evaluation of fatigue under multiaxial loadings. J Engng Mater Tech. Transaction of the ASME, 103, 118-125. Ellyin, F. and Kujawski, F. (1993), In: Advances in Multiaxial fatigue. ASTM STP 1191. McDowell, D.L., and Ellis, R., (Eds), Philadelphia: American Society for Testing and Materials, 55-66. Ellyin, F., Golos, K., and Xia, Z. (1991), In-phase and Out-of-phase Multiaxial Fatigue. J Engng Mater Tech. 113,411-416. Brown, M.W. and Miller, K.J. (1973) A theory for fatigue under multiaxial stress-strain conditions. Proc Inst Mech Eng. 187, 745-755. Findley, W.N. (1959), A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending. J Engng hid. 301-306. Flavenot, J.F. and Skalli, N. (1989) A critical depth criterion for evaluation of long life fatigue strength under multiaxial loadings and stress gradient. In: Brown, M.W. and Miller, K.J. (Eds). Biaxial and Multiaxial Fatigue. London: ESIS Publication no. EGF3, 355-365. Stulen, F.B., and Cummings, H.N. (1954), Proc. ASTM, 54, 822-835. Carpinteri, A., and Spagnoli, A. (2001), Multiaxial high-cycle fatigue criterion for hard metals. Int. J. Fatigue, 23 (2), 135-145. Liu, K.C. (1993) A method based on virtual strain-energy parameters for multiaxial fatigue. In: McDowell, D.L.and Ellis, R., (Eds). ASTM STP 1191. Philadelphia: American Society for Testing and Materials, 67-84. Chu, C.C, Conle, F.A. and Bonnen, J.F. (1993) Multiaxial stress-strain modeling and fatigue life prediction of SAE axle shafts. In: McDowell, D.L. and Ellis, R, (Eds). ASTM STP 1191. Philadelphia: American Society for Testing and Materials, 37-54. Glinka, G., Shen, G., and Plumtree, A. (1995) A multiaxial fatigue strain energy density parameter related to the critical plane. Fatigue Fract Engng Mater Struct. 18, 37-46. Varvani-Farahani, A. (2000) A new energy-critical plane parameter for fatigue life assessment of various metallic materials subjected to in-phase and out-of-phase multiaxial fatigue loading conditions, Int. J. Fatigue, 22, 295-305. Macha, E. and Sonsino, C. M. (1999) Energy criterion of multiaxial fatigue failure, Fatigue Fract Engng Mater Struct, 22, 1053-1070. Mroz, Z. (1967) On the description of anisotropic workhardening, J.Mech. Phys. Solids, 15, 163-175. McDiarmid, D.L. (1985) Fatigue under out-of-phase biaxial stresses of different frequencies. In: Miller, K.J. and Brown, M.W., (Eds). Multiaxial Fatigue, ASTM STP 853 Philadelphia: American Society for Testing and Materials, 606-621. McDiarmid, D.L. (1989) The effect of mean stress on biaxial fatigue where the stresses are out-of-phase and at different frequencies. In: Brown, M.W. and Miller, K.J., (Eds). Biaxial and Multiaxial Fatigue, EGF3. London, 605-619. McDiarmid, D.L. (1991) Mean stress effects in biaxial fatigue where the stresses are out-of-phase and at different frequencies. ESIS 10, Kussmual, K., McDiarmid, D. and Socie, D. (Eds), London, 321-335. Varvani-Farahani, A. and Topper, T.H. (2000) A new multiaxial fatigue life and crack growth rate model for various in-phase and out-of-phase strain paths. In: Kalluri, S. and Bonacuse, P.J. (Eds). Multiaxial Fatigue Deformation: Testing and Prediction, ASTM STP 1387. Philadelphia: American Society for Testing and Materials, 305-322.

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24.

25.

26. 27. 28. 29. 30. 31. 32.

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Varvani-Farahani, A., and Topper, T. H. (2000) A new energy-based multiaxial fatigue parameter, Fatigue 2000: Fatigue and Durability Assessment of Materials, Components and Structures, 4^*^ International Conference of the engineering Integrity Society, University of Cambridge, UK, 313-332. Tipton, S.M. (1984) Fatigue behavior under multiaxial loading in the presence of a notch: methodologies for the prediction of life to crack initiation and life spent in crack propagation. Ph. D. thesis, Mechanical Engineering Department, Satnford University, Stanford, CA. Andrews, R.M. (1986) High temperature fatigue of AISI 316 stainless steel under complex biaxial loading, Ph.D. thesis, University of Sheffield, UK. Socie, D. (1987) Multiaxial fatigue damage models. J Eng Mater Technol, 109, 293298. Socie, D. and Marquis G, Multiaxial Fatigue, Society of Automotive Engineers SAE International, 2000. Kanazawa, K, Miller, K.J., and Brown, M.W. (1979) Cyclic deformation of 1% Cr-MoV steel under out-of-phase loads. Fatigue Eng Mater and Struct. 2, 217-228. Sines, G. (1961) The prediction of fatigue fracture under combined stresses at stress concentrations. Bull Jpn Soc Mech Eng. 4(15), 443-453. Smith, J.O. (1942) Effect of range of stress on fatigue strength of metals. University of Illinois, Engineering Experiment Station, Bui no. 334, 39(26). Bergmann J., Klee, S. and Seeger, T. (1977) Effect of mean strain and mean stress on the cyclic stress-strain and fracture behavior of steel StE70. Materialpruefung 19(1), 10-17. Fatemi, A. and Socie, D. (1988) A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fatigue Fract Engng Mater Struct, 11,149-165. Au, P., Topper, T.H., and El Haddad, M., (1982) The effect of compressive load on the threshold stress intensity for short cracks, AGARD Conference Proceedings No.328, In: Behavior of Short Cracks in Airframe Components, Toronto, Canada, 19-24. Yu, M.T., Topper, T.H., and Au, P. (1984) The effects of stress ratio, compressive load and underload on threshold behavior of a 2024-T351 aluminum alloy, proceedings, Fatigue 84 (C.J. Beevers Ed.), Vol.1, 179-190. Yu, M.T. and Topper, T.H. (1985) The effects of materials strength, stress ratio and compressive overload on the threshold behavior of a SAE 1045 steel, ASTM Journal of Engineering Materials and Technology, 107, 19-25. DuQuesnay, D.L., Pompetzki, M.A., Topper, T.H., and Yu, M.T. (1988) Effects of compression and compressive overloads on the fatigue behavior of a 2024-T351 aluminum alloy and a SAE 1045 steel, ASTM STP 942, (H.D. Solomon, G.R. Halford, L.R. Kaisand and B.N.Leis, Eds) American Society for Testing and Materials, Philadelphia, 173-183. Varvani-Farahani, A. and Topper, T.H. (1997) Crack growth and closure mechanisms of shear cracks under constant amplitude biaxial straining and periodic compressive overstraining in 1045 steel, Int. J. Fatigue, 9(7), 589-596.

Critical Plane-Energy Based Approach for Assessment of Biaxial Fatigue Damage where ...

221

Appendix: N O M E N C L A T U R E

b C E f G i K* n* N Nf Pb R Sij

Subscript b corresponds to a block history. A material constant Modulus of Elasticity Subscript f corresponds to fatigue failure. Shear Modulus The ith Gi cycle in a block loading history Cyclic strength coefficient Cyclic strain hardening exponent Number of a\ cycles in a block loading history Number of blocks to failure Damage accumulation due to stress cycles within a block loading Stress ratio (minimum stress/maximum stress) The deviatoric stress

£^eq

The equivalent plastic strain

CTJ"

T h e normal mean stress acting on the critical plane

(7 eq

The equivalent stress

^-^^^^

T h e range of m a x i m u m shear acting on critical plane

(7^^^ and a!^^^

T h e m a x i m u m and m i n i m u m normal stresses

A£n AGn AXmax a 811, 822, and 833 8f 8ij (j)

The The The The The The The The

Yf Ve 9

normal strain range acting on critical plane normal stress range acting on critical plane range of m a x i m u m shear stress acting on critical plane ratio of frequency of G]/ frequency of G2 principal strains axial fatigue ductility coefficient strain tensor (i and j=1,2,3) phase delay between two axes of loading

The shear fatigue ductility coefficient The elastic Poisson's ratio The angle during a cycle of stressing at which the M o h r ' s circle is the largest and has the maximum value of shear strain. Gi The alternating longitudinal stress CTii, G22, and G33 The principal stresses G2 The alternating differential pressure across the wall thickness Gf The axial fatigue strength coefficient Gij The stress tensor (i and j= 1,2,3) Gkk T h e summation of principal stresses Gmi and Gm2 The longitudinal and the transverse mean stresses, respectively If The shear fatigue strength coefficient

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Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

223

FATIGUE ANALYSIS OF MULTIAXIALLY LOADED COMPONENTS WITH THE FE-POSTPROCESSOR FEMFAT-MAX

Christian GAIER and Helmut DANNBAUER ECS - Engineering Center Steyr, Magna Steyr Steyrer Str. 32, A-4200 St. Valentin, Austria

ABSTRACT The fatigue behavior of machine components is influenced by a lot of parameters as geometry (notches), load (static - cyclic - stochastic, proportional - nonproportional), material (ductile semiductile - brittle), surface roughness, surface treatment, temperature, etc. At the ECS the software FEMFAT for fatigue analysis has been developed, which is based on elastic Finite Elements stress results. It can take those influences into account by generating local synthetic component S/N-curves. Dynamic material data like fatigue limit for tension-compression, bending and torsion are taken as input to consider notch effects and the different damaging effect of normal and shear stress. For nonproportional load an extended method of the wellknown critical plane approach is presented, which has been implemented into the multiaxiality module FEMFAT-MAX. It can be applied even for triaxial stress states and nonproportional stochastic loads. But for special multiaxial load situations (bending - torsion with phase shift) the extended critical plane approach delivers good results only for brittle materials. Results for ductile materials can be strongly on the unsafe side. Suitable corrections of the local S/Ncurves by introducing a new material parameter is presented to overcome these problems. KEYWORDS Fatigue analysis, synthetic S/N-curves, critical plane, multiaxial loads, nonproportional loads, rotating principal stresses.

INTRODUCTION With increasing capacities of computers the prediction of the lifetime of components by numerical methods becomes more and more important in the automotive industry, but also in other areas of mechanical engineering. The number of testing cycles can be reduced by finding the crack initiation point with suitable software tools and pre-optimization of components, which saves time and money. At the ECS the software tool FEMFAT for fatigue analysis has been developed, which is based on elastic Finite Elements stress results. It can be applied for a wide range of problems: the assessment of uniaxially and multiaxially loaded components as well as welding seams and spot joints is possible.

224

C GAIER AND H. DANNBAUER

The theory appHed in FEMFAT differs in some aspects from classical approaches like the nominal stress concept or the local one and can be characterized by the terms „influence parameter method", "concept of synthetic S/N-curves" or "local stress concept". As basic input data S/N-curves of unnotched specimens are used and modified locally at each node of the FEmesh to obtain computed component S/N-curve at critical spots. Several influence parameters can be considered as e.g. stress-gradient to take into account notch effects, mean-stress influence, surface roughness, surface treatments, temperature, technological size, etc. Also mean-stress rearrangements caused by plastic deformations are taken into account. The user can choose between several analysis methods for the quantification of the influence parameters, e.g. methods which are fixed by German guiding rules from the Research Committee for Mechanical Engineering FKM ("Forschungskuratorium Maschinenbau") [7] and such ones which have been developed by ECS ("Engineering Center Steyr") [1-6]. For proportional loads, which scale the magnitude of local stress states, but which do not change the orientation of the principal axes, classical damage hypotheses like the maximum principal stress hypothesis for brittle materials or the distortion energy (von Mises) criterion for ductile ones can be applied in the high cycle fatigue domain. These hypotheses deliver results with sufficient accuracy for the technical practice. The situation is different for complex load situations, when non-correlating stochastic external loads are applied to components with complicated shapes. The principal axes of local stress tensors will change their orientations with time. For very special load combinations these axes can even rotate, while the magnitude of the maximum principal stress remains almost constant. A widely accepted and applied method to deal with such general load situations is the critical plane approach (e.g. [8-13]). At the critical spot of the component it is assumed, that the plane is responsible ("critical") for fatigue failure, for which the accumulated damage exceeds a critical limit at first (theoretically 1). A still open question remains, how to combine the critical plane method with classical rainflow counting procedures in a general way, for the assessment of triaxial stress states and for different material behaviours (brittle/ductile). A method is developed using a similar nomenclature for basic stress quantities as in [12] and [13].

THEORETICAL BACKGROUND hi fatigue science closed hysteresis loops in the local stress-strain-path have been recognized to be responsible for local damages of the material. This has been realized for uniaxial stress or strain states with constant directions. A procedure for counting such closed hyteresis loops is the well known rainflow counting method (Matsuishi and Endo [14], see Fig. 1). hi reality the situation is much more complicated: Stresses and strains are symmetric tensors of second order in the 3D-space, where cycle counting methods cannot be applied directly. The stress tensor must be reduced by two steps to obtain scalar quantities, for which a rainflow counting of amplitudes and mean values can be applied (Fig. 2): The first step is the well known transformation into a plane, which is specified by its unit normal vector n: SnAO = ^xx(0 n, + o-,/0 n^ + a,,(0 n^

A time dependent stress vector S„(0 is obtained, consisting of three independent stress components SnM)^ ^n.yit) and Sn,z{t).

Fatigue Analysis of Multiaxially Loaded Components with the FE-Postprocessor FEMFAT-MAX

•

^

^

^

^

Rainflow I

<^^LI

time Fig. 1. Rainflow counting of closed hysteresis loops.

Tensor of 2ncl order (stress, strain)

Normal vector n of plane

Tensor of 1st order = Vector

Unit direction vector d

Tensor of 0th order = Scalar

Fig. 2. Tensor diminution.

Fig. 3. Stress Vector Projection.

225

226

C GAIER AND H. DANNBAUER

For the second step the stress vector is normal-projected onto a given Une, whose direction is specified by its unit direction vector d (Fig. 3). The resulting scalar quantity denotes the stress component SnAO ^^ direction d and can be obtained by simply applying the inner product on d and S„(0: S„,,{t) = d-S„{t) = d^ S„^^(0 + d^ 5„,^(0 + d^ S„^^(t)

(2)

There are two special cases: • With d=ii one obtains the normal component N„(t) of the stress vector Sn(t). • With d 11=0 one obtains the resolved shear stress Tn(t). For 3D-stress states, a unique direction d cannot be derived from the relation dn=0. This is true due the fact, that the shear stress can point in any direction inside the plane and for nonproportional loads it can change its direction or even rotate.

THE "CRITICAL PLANE - CRITICAL COMPONENT" - APPROACH For given unit vectors n and d, one obtains a scalar stress quantity SnAO ^V Eqs (1) and (2), for which rainflow counting and damage analysis can be performed following Miner's rule of linear damage accumulation. This procedure can be applied for each combination of n and d. This combination, where the damage reaches its maximum value, is denoted as "critical plane critical component" n^ and d^. Analogous to the simple critical plane criterion it is assumed, that the stress component in direction dc, acting on the plane DC, is responsible for local crack initiation.

DEPENDENCY OF FATIGUE PARAMETERS ON THE POLAR ANGLE For a given line specified by its direction d, the projected stress vector S„,XO = SnAO^ "^ (dS„(/))d can be resolved into its normal and shear component (Fig. 3): ^ M ( 0 = S„,,(0-n = (d-S„(OXd-n)=^„,,(Ocos^ T„,, (0 = |S „,, (0 - ^M (On| = S„^, (t) sin 0

(3) (4)

The type of loading can be characterized by the ratio of shear stress to normal stress, which depends only on the polar angle 6, but not on the azimuth one:

^M(OM,.(0 = tan^

(5)

Three basic load types can be simply specified by its polar angle: • (9 = 0° ...Tensile Stress • <9 = 90° ...Shear Stress • 0 = \ 80° ... Compressive Stress Other load types are combinations of these basic ones. Because all directions d with same polar angle 6 represent the same load type, the stress Sn_d(t) can be compared with the same fatigue limit for all possible directions d lying on a cone with opening angle 26. Therefore the fatigue limit is a function only of the polar angle 6, if the material is assumed to be isotropic. This

Fatigue Analysis of Multiaxially Loaded Components with the FE-Postprocessor FEMFAT-MAX

227

function can be obtained in principle by measurements on specimens applying different combinations of bending/torsional loads (Fig. 4).

^ 350 n ^y? a ^ 300 § 250

"^^^-^

(0 Z U U

1

0

I

1

1

!

1

1

i

1

10 20 30 40 50 60 70 80 90 Polar Angle [deg]

Fig. 4. Fatigue limit of a tempering steel (0° = tension/compression, 90° = shear).

Not only the fatigue limit can be specified as function of 6, but also other fatigue material parameters like S/N-curves for lifetime prediction or Haigh-diagrams to take into account the mean stress influence. FEMFAT is a Finite Element postprocessor for lifetime prediction, where the proposed criterion has been implemented and tested. According to the German FKM-guiding rules [7], in FEMFAT the mean stress influence is taken into account by Haigh-diagrams, which are constructed by polygonal lines. With such Haigh-diagrams the fatigue limit is decreased for tensile mean stress and increased for compressive one. Therefore the Haigh-diagram behaves to be unsymmetric according to tensile/compressive loading. The situation is different for torsional loading. For this case the Haigh-diagram must be symmetric. These different behaviours according to different load situations can be easily taken into account by specifying Haigh-diagrams in dependence on the polar angle 6. Fig. 5 shows such a Haigh-diagram for a ductile material (tempering steel), Fig. 6 for a brittle one (grey cast iron). At ^= 90° (equivalent to pure shear stress) the symmetry of the Haigh-diagram can be seen.

CONSTRUCTION OF HAIGH-DIAGRAMS IN FEMFAT-MAX Usually Haigh-diagrams in FEMFAT are composed of nine points (Fig. 7): The first point at the right hand side of the Haigh-diagram is fixed by the ultimate strength for tension. The ninth point at the left hand side is fixed by the ultimate strength for compression. The fifth point in the middle is fixed by the fatigue limit for alternating tension and compression. The fourth point with stress ratio R = 0 is defined by the pulsating fatigue limit for tension and the sixth point with stress ratio R = ± oo by that one for compression (if its value is known). The ultimate strength and the pulsating fatigue limit can be completely different for tension and compression loading, especially for grey cast iron as shown in Fig. 7.

C GAIER AND K DANNBAUER

228

5" a c

r 3 •o

Mean Stress [Mpa]

Fig. 5. Haigh-diagram of a tempering steel as function of the polar angle 0.

a c 3

-0

s r 3 3 •o

(0

90° Mean Stress [Mpa]

Fig. 6. Haigh-diagram of a grey cast iron as function of the polar angle 6. The situation is different for shear loading: it cannot be distinguished between tension and compression. Therefore the Haigh-diagram must be symmetric and the slope of the curve at the point with stress ratio R = -1 must be zero to be smooth. To obtain always technical meaningful Haigh-diagrams for all possible combinations of shear ultimate strength Tuts and shear fatigue limit 2/7, the nine points are set in the following way (see also Fig. 8):

Fatigue Analysis of Multiaxially Loaded Components with the FE-Postprocessor FEMFAT-MAX

Point 1:

^m

Point 2:

T^ m = T^ uts —

=

5

Point 3:

T

^ fl,m

^uts

= —T

Point 4:

^fl..

2

(6)

-1±

(7)

2

5

2

Point 5:

=0

2 3"^fl

^fl..

(8)

'^ fl,m ^^fl

(9) (10)

^fl^m =^fl

^ m = ^

229

The points 6, 7, 8 and 9 are symmetric to 4, 3, 2 and 1. Therefore the same relations (6)-(9) are valid with inverse sign for Zm. Point 3 has been placed on the centre of gravity of a triangle, spanned by the points 2, 4 and the point of intersection of the lines r^^ - TJJ and

-800

-700

-600

-500

-400

-300

-200

-100

0

100

Fig. 7. Haigh-diagram for grey cast iron for tension/compression loading.

Fig. 8. Haigh-diagram for grey cast iron for shear loading.

200

300

230

C GAIER AND H. DANNBAUER

VALIDATION A workshop was held in Germany by the Research Organization for Mechanical Engineering FKM to validate different numerical methods for fatigue analysis of multiaxially loaded components. Beside other examples two simple ones for numerical calibration have been chosen, which demonstrate very clearly the influence of nonproportional loads on the fatigue behaviour of a brittle and a ductile material, respectively. Figure 9 shows measured S/N-curves of a cast aluminium specimen and Fig. 10 such ones of a tempering steel. These tests have been performed at the LBF (Fraunhofer Institute Structural Durability) in Darmstadt/Germany [1517]. A constant amplitude bending load, a torsional one and combinations of these loads have been applied. For cast aluminium the S/N-curves appear in the following order: - bending torsion - combined bending and torsion with 90 degree phase shift - combined bending and torsion in phase For steel the order is completely different: - bending combined bending and torsion in phase - combined bending and torsion with 90 degree phase shift - torsion For FEMFAT analysis Finite Element stress results have been used as it can be seen in Fig. 11. To see clearly the suitability of methods for combined loading, the fatigue limits for tension/compression, bending, torsion and the slope of the material S/N-curve have been adapted to obtain the correct results for pure uniaxial bending and torsion. The important notch effect has been taken into account by calculating a local ratio of stress concentration factor Kt to fatigue notch factor A^in dependence on the relative stress gradient [1]. Figure 12 shows, that for cast aluminium quite good results are delivered even for the combined load case with phase shift by applying the "critical plane - critical component" method as described previously. But for steel the situation is quite different: For the combined load case with phase shift the calculated Hfetime is about 13 times higher than the test result. In Fig. 13 one can see that the computed S/N-curve is rather the same as the S/N-curve for bending. Test specimen (dimensions in mm)

^ Stress consentration factor: Bending Torsion

IC*=1.49 K„=1.24

Material: Cast aluminium G-AiSi 12 CuMgNi Ultimate tensile strength: 235 MPa Tensile yield strength: 175 MPa Test series: Symbol: O

Pure alternating bending

#

Pure alternating torsion

A

Combined altemating bending and torsion in phase (x^a„ = 1.0)

O

Combined alternating bending and torsion out of phase (phase difference 90")

Fig. 9. S/N-curves of cast aluminium (Ref CM. Sonsino (LBF))

Fatigue Analysis of Multiaxially Loaded Components with the FE-Postprocessor FEMFAT-MAX 231 800 P, in ( % 1 ~ 90 I

50 O-

/K5

J

10 ~

sL.feri

5 tests on each stress level ^ e a c h 5 run-outs

MateriQt: Ck i 5 { SAE 1045 ) | R „ = 850 MPa. Rp Q^ = 810 MPa 150 2

\

\

5

10*

\

\ 5

10^

2

5

lO''

1 2

1 5

Fatigue life to crack initiation N,^ (a = 0.25mml Ret.: A. Simburger

Fig. 10. S/N-curves of tempering steel (Ref. A. Simburger (LBF))

Fig. 11. Stress distribution for bending and torsion, on the surface and inside.

10^

232

C. GAIER AND H. DANNBAUER

The reason is the critical plane criterion: The stress amplitude in the critical plane is almost the same for combined load with 90 degree phase shift as for uniaxial bending load. The difference is, that for the combined load case the stress vector is rotating, whereas for the bending one it changes only its sign. This vector rotation is not considered by the critical plane criterion. The test results show, that for ductile materials there must be an interaction between planes, which could be taken into account e.g. by integral methods, where an average damage value is calculated over all planes. - Bending Test -Torsion Test -Connb.O°Test -Connb.90°Test

- -B - -^ — --A- -e -

Bendng Fenfat Torsion Fennfat Connb.0''Femfat Cart. 90° Forfat

- , 90

100000

1000000 hkjmber of cydes

Fig. 12. Computed S/N-curves for cast aluminium and comparison to tests

-m— Bending Test - • — T o r s i o n Test - ^ — C o m b . 0° Test - • — C o m b . 90° Test

-

-B -» -A -e -

Bending Femfat Torsion Femfat Comb. 0° Femfat Comb. 90° Femfat

100000

Number of cycles

Fig. 13. Computed S/N-curves for tempering steel and comparison to tests

Fatigue Analysis of Multiaxially Loaded Components with the FE-Postprocessor FEMFAT-MAX 233 A NEW METHOD TO TAKE INTO ACCOUNT PHASE SHIFTS Integral methods have a disadvantage: They would deliver different results also for uniaxial bending and torsional loading. But the numerical results are good and should not be affected. Therefore we propose to change the S/N-curve in dependence of a multiaxiality parameter. This parameter could be simply the phase shift. But in practice this is not a good choice, because a phase shift can be defined only for monofrequentic signals. It should be possible to define such a parameter which can be applied for stochastic signals too. A solution has been implemented in FEME AT, which was proposed by Chu et al [11]. The stress state is considered for each time step in the stress space (Fig. 14). The ratio of the axis lengths of the inertia ellipsoid in the stress space can be used as degree of multiaxiality (IM- It is a value in the range from 0 to 1. For the combined load case with 90 degree phase shift, as it can be seen in Fig. 10, we would obtain a value of (iM=0.58 or, if we scale the shear stress by a damage equivalent factor of (= ratio tension/shear fatigue limit), a value of JM=1, respectively. For all other load cases (uniaxial and combined without phase shift) we will get a value of dM=^- Now the local fatigue limit can be modified by the approximate formula

',/7,nn

(11)

(l-^M-^.

SM denotes the sensitivity of multiaxiality, which is a material parameter. Using a value of 5'M=0,26 for the considered tempering steel, a good result is obtained for the combined load case with phase shift for the fatigue limit (see Fig. 15). An additional minor correction of the cycle limit of endurance could lead to an even better result for the finite life domain.

k ^v

Almost uniaxial stress-distribution

Fig. 14: Local stress state history

Strongly multiaxial stress-distribution

C GAIER AND H. DANNBAUER

234

H i — Bending Test -•—Torsion Test -A—Comb. O^'Test -•—Comb. 90° Test

-

-B -» -A-e -

Bending Femfat Torsion Femfat Comb. 0° Femfat Comb. 90" Femfat

500

450

400 H

350

«» 300

250

200 1000

10000

100000

1000000

10000000

Number of cycles

Fig. 15: S/N-curves for tempering steel with additional corrections for out of phase loading

FURTHER INVESTIGATIONS ON COMBINED IN PHASE LOADING Nevertheless, FEMFAT delivers comparatively conservative results for in phase loading, as it can be seen in Fig. 15. A rather simple interpolation between fatigue limit for tension/compression and shear has been used: ^n.e-—-—

+ ^^

^^ cos(2(9)

(12)

It is investigated now, if the result quality can be improved by modifying Eq. (12). To obtain a better understanding of the selection of critical planes and critical stress components, Mohr's circle is considered for in phase loading (Fig. 16). For a stress state with z/cr= 0.58 the ratio of minimum and maximum principal stress oj/o/ is theoretically -0.21, if there are no notches. But for the considered specimen a much lower ratio of-0.085 is obtained by the Finite Element analysis, because the critical spot is located in the notch with 5 mm radius. The influence of the notch is considerable, because the local deformation will be partially suppressed. A biaxial stress state will be induced with modified principal stresses, leading to a more tensile stress state characterized by a lower ratio of cji/o/. Next, the safety factor SFe against fatigue limit is derived as a function of the polar angle 0: SF,

* fl,e

(13)

Fatigue Analysis of Multiaxially Loaded Components with the FE-Postprocessor FEMFAT-MAX

235

The polar angle 0, where 5F^ becomes a minimum, defines the critical stress component. 0/7,6'is the fatigue limit taken fi-om Eq. (12). By a suitable modification of Eq. (12), the result of Eq. (13) can be adapted to test results. Sn,d denotes the amplitude of the stress component Sn,Jj), with d-n=cos^. Nevertheless, Sn,d is not unique in Eq. (13), because it depends not only on the polar angle 0, but also on the plane specified by its unit normal vector n. Our main interest has to be focused on SFo for a given 6, which is a minimum over all planes. Therefore we have to look firstly for a minimum of SFe over all planes and secondly for an additional minimum ("minimum of minimum") over all inclinations ^ of direction d. As a result we will obtain both the critical plane Uc and the critical component in direction dc. For this purpose, Eq. (13) has to be redefined:

SFe=-^

(14)

*^«,^,max denotes the maximum stress over all planes for a given 0, with d n = cos^= constant. This stress can be found by constructing the tangent to Mohr's circle, which is perpendicular to a line through the origin of the coordinate system, with inclination ^ t o the cr-axis (Fig. 16). Using the following geometric relations, a formula can be derived for Sn,d.maxM^=^l±^ + ^lZ^cos0

(15)

T=?^^^smd 2

(16)

S„=P:+T:

(17)

fT ^ a = arctani —^ \

(18)

S„,,^,=S„cos{0-a)

(19)

a denotes the angle between the stress vector S^, and the plane's normal n, as it can be seen in Fig. 3. As an interesting result it can be pointed out, that the plane, where Sn,d.max is acting on, is inchned by an angle of 6/2 to the first principal axis. Figure 17 shows the normalized fafigue limit oj]_0 /a/j, the normalized stress amplitude Sn.d,maix/CTi and the resulting safety factor SFe for the ratio 03 /CJ/ = -0.085. The minimum ofSFe can be found at ^ = 50°. This value was also delivered by FEME AT. It means, that the angle between the critical stress component and the plane's normal is 50°. The result for in phase loading in Fig. 15 can be improved by increasing the fatigue limit at 6^ = 50°. For this case we have extended Eq. (12) by an additional term: ^^^ = ^ V : l £ + ^ A Z l £ c o s ( 2 5 ) + ^ £ 2 ^ ( l _ c o s ( 4 ^ ) )

(20)

236

C GAIER AND H. DANNBAUER

This curve can be seen in Fig. 4. The additional term has been chosen in this way by adjusting the second derivative of (20) to zero at ^ = 0°. The corresponding safety factors are shown in Fig. 18. The minimum has moved to ^= 0°, which means, that with Eq. (20) the normal stress will be critical for fatigue failure. The new results for the S/N-curves can be found in Fig. 19.

>-(T

Fig. 16: Mohr's circle Fatigue Limit - - - - -

-Stress Amplitude

Safety Factor

I 0.'

40

50

60

70

80

Polar Angle [deg]

Fig. 17: Safety factor against fatigue limit in dependence on the polar angle 6

Fatigue Analysis of Multiaxially Loaded Components with the FE-Postprocessor FEMFAT-MAX

Fatigue Limit

Stress Amplitude

Safety Factor

8 £

0,8

H

i

0,6

10

20

30

40

50

60

70

Polar Angle [deg]

Fig. 18: Safety factor against fatigue limit in dependence on the polar angle 6

- • — Bending Test - • — Torsion Test ~4—Comb. 0° Test "-•—Comb. 90° Test

1000

10000

- -s - -^ --L- -e -

100000

Bending Femfat Torsion Femfat Comb. 0° Femfat Comb. 90° Femfat

1000000

10000000

Number of cycles

Fig. 19: S/N-curves for tempering steel with additional corrections for in phase loading

237

238

C. GAIER AND H. DANNBAUER

The result quality for combined in phase loading has been moved in the right direction, but still it has not reached the test result. Additional improvements are not possible anymore, because a further increase of the fatigue limit at ^ = 0° would also change the result for pure bending. But the result for bending should not be affected, because it has been already adapted to the test result.

SUMMARY AND OUTLOOK The critical plane criterion is proved to deliver good results for brittle materials, but it is shown that it has some drawbacks when applying it to ductile ones in combination with out of phase loading. A method, where a new material parameter has been introduced, is proposed to overcome these problems. This method can be applied also for stochastic load combinations. Additionally a method has been introduced for the assessment of both normal and shear stress for stochastic nonproportional loadings. Advantages of the proposed "critical plane critical component" method are: - hidependency from coordinate system - Normal and shear stress and all possible combinations are taken into account. - rainflow-counting can be applied for lifetime evaluation without any problems. - Applicable for both brittle and ductile materials by specifying cyclic data in dependence on the polar angle 0 (S/N-curves, Haigh-diagrams, etc.) - The interpolation function between tension/compression and torsional load can be adapted to test resuhs. - Generally applicable for biaxial and triaxial stress (or strain) states - Physical interpretation possible as „critical component" of stress vector Disadvantages are: - High computational effort - Sometimes additional corrections are necessary for out-of-phase loading (e.g. for tempering steel). Alternatively, integral methods can be applied instead of a maximum search, to obtain an averaged value for the lifetime. This has been already proved for infinite lifetime evaluation (safety against fatigue limit) of hard metals, e.g. by Zenner et al. [9-10] and Papadopoulos et al. [12]. For lifetime prediction till crack initiation, further investigations are necessary. Future efforts take aim to extend the proposed method by integral solution algorithms. The methods, as presented here, are based merely on stresses and therefore applicable only to the high cycle fatigue domain (HCF) for lifetimes > -5000. Nevertheless, in principle these methods could be extended for low cycle fatigue (LCF) too by exchanging stresses for strains and introducing suitable damage parameters. Such strain based fatigue evaluations need to be combined with usually extensive non-linear elastic-plastic stress-strain analyses. Experiences in this direction are still outstanding.

Fatigue Analysis of Multiaxially Loaded Components with the FE-Postprocessor FEMFAT-MAX

239

REFERENCES 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11. 12.

13. 14. 15.

16.

17.

Eichlseder W. (1989), Rechnerische Lebensdaueranalyse von Nutzfahrzeugkomponenten mit der Finite Elemente Methode, Dissertation, University of Technology Graz, Austria. Eichlseder W. and linger B. (1994), Prediction of the Fatigue Life with the Finite Element Method, SAE Paper 940245. Unger B., Eichlseder W. and Raab G. (1996), Numerical Simulation of Fatigue Life - Is it more than a prelude to tests?, Proc. Fatigue'96, Berlin. Steinwender G., Gaier C. and Unger B. (1998), Improving the Life Time of Dynamically Loaded Components by Fatigue Simulation, SAE Paper 982220, pp. 465-470. Gaier C., Unger B. and Vogler J. (1999), Theory and Application of FEMFAT - a FEPostprocessing Tool for Fatigue Analysis, Proc. Fatigue'99, Beijing, pp. 821-826. Gaier C , Steinwender G. and Dannbauer H. (2000), FEMFAT-MAX: A FE-Postprocessor for Fatigue Analysis of Multiaxially Loaded Components, NAFEMS-Seminar Fatigue Analysis, Wiesbaden, Germany. German FKM Guiding Rule (1998), Rechnerischer Festigkeitsnachweis fuer Maschinenbauteile, VDMA Verlag Frankfurt/Main, 3"^^ edition. Nokleby J. O. (1981), Fatigue under Multiaxial Stress Conditions, Rep. MD-81001, Div. Mach. Elem., The Norway Inst. Technol., Trondheim,. Zenner H., Heidenreich R. and Richter I. (1985), Fatigue Strength under Nonsynchronous Multiaxial Stresses, Z. Werkstofftech. 16, Germany, pp. 101-112. Sanetra C. and Zenner H. (1991), Betriebsfestigkeit bei mehrachsiger Beanspruchung unter Biegung und Torsion, Konstruktion 43, Springer-Verlag, Germany, pp. 23-29. Chu C. C , Conle F. A. and Huebner A. (1996), An Integrated Uniaxial and Multiaxial Fatigue Life Prediction Method, VDI Berichte Nr. 1283, Germany, pp. 337-348. Papadopoulos I. V., DavoH P., Gorla C , Filippini M. and Bemasconi A. (1997), A comparative study of multiaxial high-cycle fatigue criteria for metals. Int. J. Fatigue, Vol. 19, No. 3, pp. 219-235. Socie D. F. and Marquis G. B. (2000), Multiaxial Fatigue, SAE, Warrendale, U.S.A. Matsuishi M., and Endo T. (1968), Fatigue of Metals Subjected to Varying Stress, Proc. Kyushi Branch JSME, pp. 37-^0. Simbuerger A. (1975), Festigkeitsverhalten zaeher Werkstoffe bei einer mehrachsigen phasenverschobenen Schwingbeanspruchung mit koerperfesten und veraenderlichen Hauptspannungsrichtungen, Fraunhofer Institute Structural Durability (LBF), Darmstadt, Germany, Report Nr. FB-121. Sonsino C. M. and Grubisic V. (1985), Mechanik von Schwingbruechen an gegossenen und gesinterten Konstruktionswerkstoffen unter mehrachsiger Beanspruchung, Konstruktion 37, Springer-Verlag, Germany, pp. 261-269. Sonsino C. M. (2001), Influence of load deformation-controlled multiaxial tests on fatigue life to crack initiation. Int. J. Fatigue 23, Elsevier, pp. 159-167.

Appendix : NOMENCLATURE d

Unit direction vector

dc

Unit direction vector of the critical stress component

dM

Degree of multiaxiality

240

C GAIER AND H. DANNBAUER

Kf

Fatigue notch factor

Kt

Stress concentration factor

n

Unit normal vector of a plane

He

Unit normal vector of the critical plane

A^;,

Normal stress component of S„

Nn,d

Normal stress component of S^,^

R

Stress ratio

SFe

Safety factor against fatigue limit for direction d inclined by 6 to the plane's normal n

SM

Sensitivity of multiaxiality

S„

Stress vector in the plane specified by its unit normal vector n

Sn,d

Stress component in direction d of stress vector acting on plane n

t

Time

Tn

Shear stress component of S„

Tn,d

Shear stress component of S„,^

6

Polar angle spanned by n and d

^

Normal stress

^

Maximum principal stress

^

Minimum principal stress

^

Alternating tension/compression fatigue limit

^

Tension/compression fatigue limit including mean stress influence

(J Q

Fatigue limit for direction d inclined by 6 to the plane's normal n

^

Mean tension/compression stress

^

Compession ultimate strength

^

Tension ultimate strength ^

Shear stress

^

Alternating shear fatigue limit

^

Shear fatigue limit including mean stress influence

Tr

Mean shear stress

'^uts

Shear ultimate strength

4. DEFECTS, NOTCHES, CRACK GROWTH

This Page Intentionally Left Blank

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

243

THE MULTIAXIAL FATIGUE STRENGTH OF SPECIMENS CONTAINING SMALL DEFECTS

Masahiro ENDO Department of Mechanical Engineering, Fukuoka University, Jonan-ku, Fukuoka 814-0180, Japan

ABSTRACT A criterion for multiaxial fatigue strength of a specimen containing a small defect is proposed. Based upon the criterion and the ^area parameter model, a unified method for the prediction of the fatigue limit of defect-containing specimens is presented. In making this prediction, no fatigue testing is necessary. To validate the prediction procedure, combined axial and torsional loading fatigue tests were carried out using smooth specimens as well as specimens containing holes of diameters ranging from 40 to 500 fim which acted as artificial defects. These tests were conducted under in-phase loading condition at i? = -1. The materials investigated were annealed 0.37 % carbon steel, quenched and tempered Cr-Mo steel, high strength brass and nodular cast irons. When the fatigue strength was influenced by a defect, the fatigue limit was determined by the threshold condition for propagation of a mode I crack emanating from the defect. The proposed method was used to analyze the behavior of the materials, and good agreement was found between predicted and experimental results. The relation between a smooth specimen and a specimen containing a defect is also discussed with respect to a critical size of defect below which the defect is not detrimental. KEYWORDS Multiaxial loading, fatigue thresholds, small defects, small cracks, -[area parameter model, steels, brass, cast irons.

INTRODUCTION Over a number of years a great deal of effort has been expended in the attempt to establish reliable predictive methods for the determination of the fatigue strength under both uniaxial and multiaxial loading conditions. However, prior to the 1970's, the methods proposed did not provide a useful mean for the analysis of materials which contained either non-metallic inclusions or small flaws that are usually encountered in engineering applications. This was in part because most of the proposed methods were applicable only to two-dimensional cracks or

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notches of simple shapes, whereas actual inclusions are often of a three-dimensional irregular shape. In addition, whereas the large crack problem has attracted attention in fatigue studies since the birth of fracture mechanics, the behavior of small cracks could not be analyzed in a similar way, for their behavior has been found to be anomalous with respect to large cracks, as pointed out by Kitagawa and Takahashi [1]. These authors, in the first quantitative characterization of the fatigue threshold behavior of small cracks, showed that the value of AK\\x decreased with decreasing crack size. This finding led the development of many subsequent studies on small or short cracks. Since their initial work a number of models and predictive methods for the determination of the fatigue strength of defect-containing components have been proposed, although most of these have dealt only with uniaxial fatigue. These models have been reviewed in detail by Murakami and Endo [2]. Research has shown [3,4] that the fatigue strength of metal specimens containing small defects above a critical size is essentially determined by the fatigue threshold for a small crack emanating from the defect. Based upon this consideration, Murakami and Endo [4] used linear elastic fracture mechanics (LEFM) to propose a geometrical parameter, 4area , which quantifies the effect of a small defect. Using this parameter they succeeded in deriving a simple equation [5] for predicting the fatigue strength of metals containing small defects. Subsequently, this model, referred to as the 4area parameter model, has been successfully employed in the analysis of a number of uniaxial fatigue problems which dealt with small defects and inhomogeneities [6,7]. However, in many applications, engineering components are often subjected to multiaxial cyclic loading involving combinations of bending and torsion. A number of studies have been concerned with this topic [8-13], but with the exception of pure torsional fatigue very few studies have been directed at the study of the behavior of small flaws under multiaxial fatigue loading conditions despite the importance of small flaws in design considerations. Nisitani and Kawano [14] performed rotating bending and reversed torsion fatigue tests on 0.36 % carbon steel specimens which contained defect-like holes of diameters ranging from 0.3 to 2 mm. They reported that the ratio of torsional fatigue limit to bending fatigue limit, ^ = r^ / cr^, was about 0.75 and attributed the result to the ratio of stress concentrations at the hole edge at fatigue limits; that is, 3 cr^ under bending and 4 r^ under torsion. (Here r^ and a^ are the fatigue strengths of specimens containing small flaws in reversed torsion and tension, respectively.) Mitchell [15] also predicted ^ = 0.75 for specimens having a hole in the similar way. Endo and Murakami [16] drilled superficial holes which simulated defects ranging from 40 to 500 |im in diameter in 0.46 % carbon steel specimens to investigate the effects of small defects on the fatigue strength in reversed torsion and rotating bending fatigue tests. Based upon the observation of cracking pattern at the holes, they correlated the fatigue strength under torsion with that under bending by comparing the stress intensity factors (SIFs) of a mode I crack emanating from a two-dimensional hole. They predicted ^ = -0.8 for specimens containing a surface hole. In that study, they also observed that there was a critical diameter of a hole below which the defect was not detrimental to the fatigue strength, and that the critical size under reversed torsion was much larger than under rotating bending. In recent papers [17-19], the further application of the -Jarea parameter to multiaxial fatigue problems has been made. Combined axial-torsional fatigue tests were carried out using annealed 0.37 % carbon steel specimens containing a small hole or a very shallow notch [17]. It was concluded that the fatigue strength was related to the threshold condition for propagation of a mode I crack emanating from a defect, and an empirical method for the prediction of the fatigue limit of a specimen containing a small defect was proposed [17]. Murakami and Takahashi [18] analyzed the fatigue threshold behavior of a small surface crack in a torsional

The Multiaxial Fatigue Strength of Specimens Containing Small Defects

IAS

shear stress state and extended the use of the ^area parameter to mixed-mode threshold problems. In addition, Nadot et al. [19] have discussed the extension of Dang Van's multiaxial fatigue criterion [20] to the defect problem by using the -Jarea parameter. Beretta and Murakami [21,22] used numerical analysis to calculate the stress intensity factor (SIF) for a three-dimensional mode I crack emanating from a drilled hole or a hemispherical pit under a biaxial stress state. By comparing with the previous experimental data [17], they concluded that the value of SIF at the tip of a crack emanating from a defect determined the fatigue strength of a specimen which contained a small defect above the critical size subjected to combined stresses. The present author [23] subsequently proposed a new criterion for fatigue failure which was also based upon the SIF. This criterion was expressed in the form of an equation which, by including within the criterion the -Jarea parameter model, provided a unified method for predicting the fatigue strength of a metal specimen containing a small defect. The applicability of the method was investigated with an annealed steel [23] and nodular cast irons [23,24]. The essence of this approach will be presented in the present paper. The principal objective of this study is to determine the generality of the author's predictive method [23] with additional experimental newly obtained data. In the present study the relation between the fatigue strengths of smooth specimens and specimens containing defects in multiaxial fatigue will also be discussed.

BACKGROUND FOR THE PREDICTION OF THE MULTIAXIAL FATIGUE STRENGTH

The ^area parameter model Murakami and Endo [4] have shown that the maximum value of the SIF, K^^^^^, at the crack front of a variety of geometrically different types of surface cracks can be determined within an accuracy of 10% as a function of ^area , where the area is the area of a defect or a crack projected onto the plane normal to the maximum tensile stress, see Fig. 1. The expression for ^imaxas a function of area (Poisson's ratio of 0.3) is:

area

Maximum tensile stress direction

Fig. 1. Definition of area.

246

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^imax =0.6500-0 V W ^

(1)

where CTQ is the remote applied stress. Thereafter Murakami and Endo [5] employed the Vickers hardness value as the representative material parameter and showed that the threshold level for small surface cracks or defects could be expressed by the following equation for uniaxial loading at the stress ratio, R, of-1: A^,, =l>3x\0-\HV

+ \2Q){4area)"'

(2)

In addition, they found that the fatigue limit could also be expressed as a function of 4area by: ^ 1 . 4 3 ( / / F + 120)

where AAr,^, the threshold SIF range, is in MPa Vm , CTW, the fatigue limit stress amplitude, is in MPa, HV, the Vickers hardness, is in kgf/mm^ and ^[area is in |im. Equations (2) and (3) were derived on the basis of LEFM considerations. More recently a justification for the exponents of 1/3 in Eq.(2) and -1/6 in Eq.(3) was provided by McEvily et al [25] in a modified LEFM analysis which also considered the role of crack closure in the wake of a newly formed crack. The prediction error involved in the use of Eqs (2) and (3) is generally less than 10 percent for values of V area less than 1000 jim, and for a wide range of HV [5,6]. Murakami and co-workers [26-28] further extended Eq.(3) to include the location of the defect, i.e., whether it was at the surface or sub-surface, and also to include the effect of mean stress. The generalized expression they developed for the fatigue strength is: _ C ( / / F +120) \-R (si area) 1/6

(4)

where the value of C depends on the location of the defect being 1.43 at the surface, 1.41 at a subsurface layer just below the free surface, and 1.56 for an interior defect. The value of the exponent a was related to the Vickers hardness by a= 0.226 + HV x 10"^. Equations (2)-(4) are useful for practical applications in that they require no fatigue testing in making predictions. The yfarea parameter model has been applied to deal with many uniaxial fatigue problems including the effects of small holes, small cracks, surface scratches, surface finish, non-metallic inclusions, corrosion pits, carbides in tool steels, second-phases in aluminum alloys, graphite nodules and casting defects in cast irons, inhomogeneities in super clean bearing steels, gigacycle fatigue, etc. They are summarized in detail in the literature [6,7].

Criterion for multiaxial fatigue failure of defect-containing specimens The author has previously shown [17,23,24] that, in in-phase combined axial and torsional fatigue loading tests, the fatigue limit for specimens containing small defects is determined by the threshold condition for propagation of a small crack emanating from a defect. The materials investigated were an annealed 0.37 % carbon steel [17,23] and nodular cast irons [23,24].

The Multiaxial Fatigue Strength of Specimens Containing Small Defects

247

Figure 2 shows typical examples of non-propagating cracks observed in those materials at the fatigue limit. The direction of a non-propagating crack is approximately normal to the principal stress, cTi, regardless of combined stress ratio, TIG. Under a stress slightly higher than the fatigue limit, a crack which propagated in a direction normal to a\ resulted in the failure of the specimen. Based upon such observations, the fatigue limit problem for specimens containing small surface defects when subjected to combined stress loading was considered to be equivalent to a fatigue threshold problem for a small mode I crack emanating from defects in the biaxial stress field of the maximum principal stress, ai, normal to the crack and the minimum principal stress, 02, parallel to the crack. Consider an axi-symmetric surface defect containing a mode I crack under the remote biaxial stresses, cjy and a^, as shown in Fig. 3. The mode I SIF, K\, at the crack tip is given by the

50 jim Axial direction

50jim , - ^ , i (a) 0.37% carbon steel; a ^ /? = 100 \xm, cTa 175 MPa, ra=87.5MPa[17].

;^3i.r Axial direction (b) FCD700 nodular cast iron; smooth specimen, cJa = ^a = 160 MPa [24].

Fig. 2. Small non-propagating cracks emanating from a defect observed at fatigue limit under combined loading.

+

(3=

Fig. 3. A three-dimensional defect leading to a crack subjected to biaxial stress.

248

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following superposition.

where FIA and FIB are the correction factors for the cases A and B in Fig. 3, respectively, and c is the representative crack length. It is hypothesized that the threshold SIF range under biaxial stress, A^th,bi, is equal to that under uniaxial stress, AArth,uni, or: A/:,,., = A/:„„„

(6)

This criterion has previously been used by Endo and Murakami [16] in the correlation of the pure torsional fatigue limit, w, the biaxial fatigue limit, with the rotating bending fatigue limit, ow; the uniaxial fatigue limit, for specimens having a small hole at the surface. Based upon this criterion, Beretta and Murakami [21,22] predicted that ^ , the ratio of the fatigue limit in torsion to that in tension, i.e., T^la^, for a mode I crack emanating from a three-dimensional surface defect under cyclic biaxial stressing should have a value between 0.83 and 0.87. They found that the predicted value of (j> agreed well with previously reported experimental results for various steel and cast iron specimens which contained small artificial defects. For fully reversed loading; /.e., i? = -1, AA\h,bi and A/Cth,uni were expressed using Eq.(5) as A^th,bi = ^lA (2cT,) V ^ + F,B (2cT2 ) V ^

(7)

A^.H,u„i=^.A(2cTjV^

(8)

where G\ and 02 are the maximum and minimum principal stress amplitudes resulting from the combined stress at fatigue limit, respectively, and ow is the threshold stress amplitude for a mode I crack under tension-compression cyclic loading; that is, the uniaxial fatigue limit of a specimen containing the same sized defect under /? = -1 loading. When the crack length, c, under uniaxial loading is equal to that under biaxial loading, Eq.(6) is reduced to G\ + kOi = CTw

(9)

where k = FIB/FIA, and represents the effect of stress biaxiality. If the torsional fatigue limit is designated by TW, since a\ = -02 = rw, then ^ = TJCTW = 1/(1 - k). Equation (9) as well as Eq.(6) provides a criterion for fatigue failure of specimens containing small defects when subjected to multi-axial loading. For round-bar specimens subjected to combined axial and torsional loading, Eq.(9) can be expressed as (\/(Pf(Tja^f

+ (\/(/>- l)(c7a/ow)' + (2 . l/^)(cTa/aw) = 1

(10)

where ok and Ta are the normal and shear stress amplitudes, respectively, at the fatigue limit under combined loading. Equation (10) is identical in form to Gough and Pollard's "ellipse arc" relationship [29], which has been used to fit the experimental data for brittle cast irons and specimens with a large notch [29,30]. The ellipse arc is empirical, and as such it requires fatigue tests for the determination of CTW and rw. In contrast, in the case of small defects, aw can

The Multiaxial Fatigue Strength of Specimens Containing Small Defects

249

be predicted using Eq.(3) without the need for a fatigue test. In addition, the value of (j) can be estimated by stress analysis, as verified by Beretta and Murakami [21,22]. If the average value ^calculated by Beretta and Murakami is used, i.e., 0.85, Eq.(lO) becomes: 1.38(ra/crw)^ + 0.176(cra/crw)^ + 0.824(cra/crw) = 1

(11)

This expression is considered to be applicable to specimens containing a round defect on or near the surface where plane stress condition is satisfied. The use of this expression will be demonstrated in the following.

MATERIALS AND EXPERIMENTAL PROCEDURE The materials investigated in this study are: steels, nodular cast irons and a high strength brass. Although experimental data for an annealed steel and the cast irons have previously reported elsewhere [23,24], they will be included below for purposes of discussion. The chemical compositions of the various metals are listed in Table 1. The 0.37 % carbon steel (JIS S35C) was annealed at 860°C for 1 hour. The Cr-Mo steel (JIS SCM435) was heat-treated by quenching from 860°C followed by tempering at 550°C . The two nodular cast irons (JIS FCD400 and FCD700) have different matrix structures; ferritic for FCD400 and almost pearlitic for FCD700. The cast irons and the brass were used in the as-received condition. Mechanical properties are given in Table 2, and microstructures are shown in Fig. 4.

Table 1. Chemical composition (wt.%)

S35C steel SCM435 steel FCD400 cast iron FCD700 cast iron High strength brass

C Si Mn 0.37 0.21 0.65 0.36 0.30 0.77 3.72 2.14 0.32 3.77 2.99 0.44 Cu Pb 59.1 0.0030

P 0.019 0.027 0.008 0.023

S 0.017 0.015 0.018 0.11 Fe 0.0032

Ni Cr Cu 0.13 0.06 0.14 0.02 0.02 1.06 0.04 0.47 Mn Zn 0.021 bal.

Mo 0.18 -

Mg

0.038 0.058 Al 0.0047

Table 2. Mechanical properties

Annealed S35C steel Quenched/tempered SCM435 steel FCD400 cast iron FCD700 cast iron High strength brass

Tensile strength MPa 586 1030 418 734 467

Elongation % (Gage length: 80 mm) 25 14 25 8.0 42

Vickers hardness HV (kgf/mm^) 160 380 190(ferrite) 330 (pearlite) 110

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lOO^im Transverse section Longitudinal section (a) Annealed S35C steel

Transverse section Longitudinal section (b) Quenched/tempered SCM435 steel

100 ^im

100 pm

Transverse section Longitudinal section (c) High strength brass

100 ^im

FCD400 FCD700 (d) Nodular cast irons

Fig. 4. Microstructures. Figure 5 shows the geometries of the smooth specimens. They have a uniform cross section which was either 8.5 or 10 mm in diameter, and 19 mm in length. After surface finish with an emery paper of grade #1000, about 30 jam of a thickness was removed by electro-polishing in order to remove the work-hardened layer. During electro-polishing the presence of inclusions or graphite nodules at the surface led to the development of undesired pits which were larger in size than the defect or nodule. These undesired pits were removed by polishing the surfaces of smooth specimens with alumina paste after electro-polishing. A small hole shown in Fig. 6 was then drilled into the surface of a number of specimens. Such specimens will be referred to as hole-containing specimens in this paper. The diameter d of the holes ranged from 40 to 500 jim. The depth h was equal to the diameter d\ so that the defects were geometrically similar. The hole-containing Cr-Mo steel specimens were electro-polished after drilling of a hole of 80 \mi in diameter in order to make a larger pit having a 4area of 83 |im. The other smooth and holecontaining specimens were slightly electro-polished to remove a surface layer of 1-2 |im in thickness before the fatigue tests. For steels, it was confirmed by an X-ray stress analyzer that the residual stresses on the specimen surface were almost zero. Uniaxial load tests were carried out using either a servo-hydraulic uniaxial fatigue testing machine with an operating speed of 50 Hz or a rotating bending testing machine with 57 Hz. Another servo-hydraulic axial/torsional fatigue testing machine which ran at 30-45 Hz was used for combined load and pure torsional fatigue tests. Pure torsional fatigue tests of FCD400 cast iron were conducted with a Shimadzu TBIO-B testing machine of the uniform moment type at 33 Hz with the specimen shown in Fig. 5c. All tests were performed under in-phase ftilly

251

The Multiaxial Fatigue Strength of Specimens Containing Small Defects

4 strain gauges attached

& ^

,_

\ ^ / in>^

1

20

20

50

(a) For combined axial/torsional load test or reversed torsion test 40

<

2d^, 34 117

1 31 20

(b) For tension-compression test and combined axial/torsional load test under r/a= 1/2

.,

0^

^1

1 •<—

- ^

'—p-Tcr—

20

96

"*

O '^ »_

.

50 *^

A

^

40 140

--^ 50

>

(c) For reversed torsion test

^/Hii J8!. 80

50 210

80

(d) For rotating bending test

Fig. 5. Shapes and dimensions of smooth specimens.

cy= /? = 40 ~ 5 0 0 / i m

'area=y/d{h-d/4V3)

Fig. 6. Hole geometries.

reversed (R = -1) loading and a sinusoidal waveform. The combined stress ratios of shear to normal stress amplitude, r/a, were chosen to be 0, 1/2, 1,2 and oo. For the tension-compression tests, in order to eliminate bending stresses each specimen was equipped with four strain gauges to facilitate proper alignment in the fixtures. The nominal stresses were defmed as (J = 4P /(TTD'^ ) for tension-compression

(12)

a = 32A/^ /(TTD^ ) for rotating bending

(13)

T = 16M, l{nD^) for reversed torsion

(14)

where a is the normal stress amplitude, r is the torsional shear stress amplitude, P is the axial load amplitude, Mb is the bending moment amplitude, M is the torsional moment amplitude and D is the specimen diameter. The fatigue limits under combined stress are defmed as the combination of the maximum nominal stresses, Ta and cTa, under which a specimen endured 10

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cycles for a fixed value of r/a. The minimum increment in stress level in determining the fatigue limit was 5 MPa for greater value of crand r, except for Cr-Mo steels where 10 MPa was used.

RESULTS AND DISCUSSION

Behavior of small cracks at the threshold level Figures 7 and 8 show the non-propagating cracks emanating from a hole at the fatigue limit of hole-containing Cr-Mo steel and brass specimens. As seen in these figures, the direction of non-

±o. -•—I

Axial direction

50 ^m (a) Tension-compression; ow = 340 MPa.

Axial direction

50 \xm (b) Combined loading; da = ra = 200 MPa.

±o. Axial direction

mg^^^mmM:-^'mmm^ - B I B B (c) Pure torsion; TW = 320 MPa Fig. 7. Small non-propagating cracks emanating from hole at fatigue limit of quenched/tempered SCM435 steel; 4area = 83 |im (c/= 90 |im).

The Multiaxial Fatigue Strength of Specimens Containing Small Defects

253

^^^^^^^^^^^MM

Axial direction

200 {im (a) Combined loading; cja = Ta = 70 MPa

p^lillil^ipiiiWIK^ iiiiiiplii»%.<;^^;::,i^;M

:pl

Axial direction

200 |im (b) Pure torsion; TW =115 MPa Fig. 8. Small non-propagating cracks emanating from hole at fatigue limit of high strength brass; ^ = 500 [im. propagating cracks is approximately normal to the principal stress, CTI, regardless of the combined stress ratio, r/cr. At a stress level just above the fatigue limit, the failure of the holecontaining specimens resulted from the continuous propagation of a crack emanating from the hole. These observations with Cr-Mo steel and brass further indicate that, as in the case of

254

M. ENDO

0.37 % carbon steel [17,23] and nodular cast irons [23,24] previously reported, the fatigue limit is determined by the threshold condition for a mode I crack emanating from a defect and propagating in the direction normal to a\ It has also been noted that holes 40 and 100 jim in diameter did not lower the fatigue strength of annealed 0.37 % carbon steels, under the special condition of a large combined stress ratio, r/cr. Those hole-containing specimens did not break as the result of a crack emanating from the hole, but rather from a crack which initiated from the smooth area at a stress level just above the fatigue limit. The critical size of a non-detrimental defect will be discussed below.

Comparison of experimental and predicted results for hole-containing specimens Figures 9-11 show the comparison of the experimental data with the predictions for holecontaining specimens for two steels and a high strength brass. The predicted curves were obtained by substituting y/area and //Finto Eq.(3) and by inserting the value of CTW into Eq.(l 1). The data points indicated by arrows in Fig. 9 are for the results of specimens containing a nondetrimental hole. Figure 12 presents the comparison for those three materials in a different manner. This figure has been normalized by dividing the values of (Ja and r^ on the axes by the uniaxial fatigue limit, ow, which was predicted by Eq.(3). In order to compare the predictions only with the experimental results controlled by a defect, in this figure, the results for the nondetrimental holes were not included. It is seen that the predictions for the fatigue limit of holecontaining specimens subjected to combined stress agree quite well with the experimental data. It is also emphasized that no experimental data were needed in making these predictions.

300

I I I I I I I I I I ^I I I I I I I I I I I I I I I I

CO Q.

2 )
200 F-

Experimental fatigue limit: • Smooth o d=40/zm J^: Hole was A d=100//m not detrimental D d=500/im

"5. E

Prediction: Smooth

CO

(0 CO 0)

d=40[Am

100 h

d =100 urn cf=500nm

CO

c o 52 .o 0

' ' ' ' • * ' ' • ' ' ' • ' '

0

I I • I

100 200 Axial stress amplitude, aa MPa

300

Fig. 9. Comparison of predicted and experimental results for annealed S35C steel specimens.

The Multiaxial Fatigue Strength of Specimens Containing Small Defects

400

I I I I I I I I I I I I M I 1 1 I I I I I I I I I I M I I I M I I I I l_

a. ^

Prediction by Eqs(3)and(11)

300

0

•D

3

E 200 OS

en

0) CD

75 100 c o

Experimental fatigue limit:

(0

o

o

90 ^m hole

•' • I •' •' I • • •' I • •' • I ' '

0

100

• • I • • •'

200

11

300

400

Axial stress amplitude, aa MPa Fig. 10. Comparison of predicted and experimental results for quenched and tempered SCM435 steel specimens containing a hole of 90 |im in diameter.

150 I—I—I—I—I—I—I—I—I—I—I—\—I—I—r 03

Prediction by Eqs(3)and(11) 0)

100

•o •*^ E CO

in (/> CD

S

50

CO

c g

Experimental fatigue limit: o

o .J

0 0

500 ^m hole I

i

'

'

50

I—ILOJ

•

100

L-

150

Axial stress amplitude, a^ MPa Fig. 11. Comparison of predicted and experimental results for high strength brass specimens containing a hole of 500 jam in diameter.

255

256

M. ENDO

h

1

I

T

1

T"• • • " r " " " " 1

1

I

1

^ Fatigue limit predicted byEqs(3)and(11) i 1

^^^^v/

V-"

0.5

Experimental r Material L o S35C 1 A S35C L D S35C O SCM435 L < Brass

data d >larea 40 37 100 93 500 463 90 83 500 463 (in ywm)

CA\

1 --1

H

1

Ar^

0.5 Oa/CTw

Fig. 12. Comparison of predicted and experimental results; era and Ta are normalized by CJW which is predicted by Eq.(3).

Fatigue strength prediction for nodular cast iron specimens The graphite nodules and microshrinkage cavities which are inherently present in nodular cast irons lead to a reduction in fatigue strength as well as increased scatter in the data. In order to create reliable designs, engineers must be able to determine the lower bound of the scatter band for the fatigue strength. Care must therefore be exercised when the fatigue strength of the materials containing a number of small defects in the structure is estimated from the results of fatigue tests using small number of small specimens, because the results may be nonconservative. A prediction method which does not require a fatigue test, such as the one proposed in this paper, is vital to the prediction of the lower bounds. It was shown [27], based upon LEFM considerations, that a defect can become most harmful when it is located just below the free surface, and the greater the defect is in size, the more detrimental it will be to the fatigue strength. Therefore, if in the extreme situation it is assumed that the maximum sized defect is present just below the surface, a lower bound of uniaxial fatigue limit,
The Multiaxial Fatigue Strength of Specimens Containing Small Defects

257

transverse section of a specimen. According to the instructions of [7], the mean value of the -Jarea m^^ measured was taken as appropriate value of the thickness, t, of the 3D inspection domain. Thus, the standard inspection volume, FQ, was defined by / ^o, and the expected value of ^area max for a given control volume, F, was obtained using the return period, T, defined by VIVQ. Table 3 lists the Vickers hardness values of matrices, the values of ^area^^^ which are expected from Fig. 13 for either 1, 5 or 10 specimens, and the predicted lower bounds of uniaxial fatigue limits, CTWI, at i? = -1 based upon Eq.(4) (C = 1.41). Figure 14 shows the comparison of experimental results with the lower bound predictions for nodular cast irons containing graphite nodules and cavities. The predicted curves were

Cumulative •frequency c

- F(%) .99.8r

_ ^ _ Graphite nodule ~^^- Microshrinkage cavity

2000 ^^ 1000

500 E 200 I 100 50 20

c B cc

150 max

(a) FCD400 cast iron Cumulative frequency Graphite nodule Microshrinkage cavitiy

2000 K 1000

500 O 200 0 100 r 50 0) 20 CC

150

max (b) FCD700 cast iron Fig. 13. Extreme-value statistics distributions of ^|area^ , for graphite nodules and microshrinkage cavities [24].

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Table 3. Lower bounds of uniaxial fatigue limits, CTWI, at /? = -1 predicted for smooth nodular cast iron specimens.

Material

Vickers microhardness of matrix

FCD400

190 (ferrite)

FCD700

330 (pearlite)

Graphite nodule Number of specimens 1 5 10 1 5 10

Microshrinkage cavity

CTwl

ylarea^^

jam

MPa

|im

MPa

115 125 129 104 112 116

198 196 195 293 289 287

227 261 276 288 335 355

177 173 171 247 241 238

obtained by inserting the value of CTWI for 5 specimens into ow in Eq.(l 1), since 3-6 specimens were used to determine each fatigue limit in the experiments. It is seen that the curve predicted for the cavities is the true lower bound, though the agreement with the curve for graphite nodule appears to be better. A small change in V^A^maxdoes not significantly affect the predicted results for CTWI, since a^^\ is proportional to the -1/6 power of yjarea max, see Eq.(4). Since, in FCD700, the ferrite envelopes surrounding the graphite nodules are thin and their effect is negligible, the value of y/area max only for graphite nodules was used for the prediction in this study. However, this approximation is not always valid, for example, for JIS FCD600 nodular cast iron [24] having a typical bull's eye structure in which the ferrite regions are large in comparison to the graphite nodules. A detailed discussion of how to apply the prediction method to such a microstructure is available in reference [24].

Relation of smooth and hole-containing specimens with respect to non-detrimental defects It is known that the pure torsional fatigue strength is insensitive to small defects or notches. For example, in the case of annealed 0.37 % carbon steel, a hole of 100 |im in diameter which caused about 20 % reduction in fatigue strength under uniaxial loading, under pure torsional loading was non-detrimental. Gough and Pollard [33] proposed the following empirical criterion, called ellipse quadrant, to fit the data for smooth specimens of ductile metallic materials

^aO

(15)

where CTWO is the smooth specimens uniaxial fatigue limit, TWO is the smooth specimens torsional fatigue limit, cJao is the axial or bending stress amplitude, and rao is the torsional shear stress amplitude at the fatigue limit under combined loading. The experimental fatigue limits of smooth specimens of annealed 0.37 % carbon steel were 145 MPa for TWO and 235 MPa for CTWO.

The Multiaxial Fatigue Strength of Specimens Containing Small Defects

300

11 1 1 1

\\^^^ \^^

1 1 { 1 1 1 1 1 1 1 1 1 1 1 1 r 1J

, Prediction by Eq. (17) for defect-free specimen

j j

Lower bound predicted by ' • • • - . Eqs (4) (C= 1.41) *and.(11): * Graphite nodule

J \ j 1 j

QT 200 13 +^

"5. E CO

£ 100

V //Microshrirrkage -\ V N / / cavity '\ J

in

15 c o

\ p Experimental data: \ t • Broken y o Not broken \ \

(0

'-H H

11 1 1 1 1 1 1 1 1 1 1 1

0

100 1 1 1 1 II 1 200 i n j r i l ^ 1 1 1 1 1 1 300 11 Axial stress amplitude, o MPa (a) FCD400 cast iron

400

I II I I I M 1 I I I I I { II I I I I M I [ I I II I I M [ I I I II.

(0 Q.

^ Prediction by Eq. (17) for defect-free specimen

V- 300 F0) •D D

Lower bound predicted by * q Eqs(4)(C=1.41)and(11): j Graphite nodule MIcroshrinkage :\ cavity

|-200 CO

CO (D

I 100 C

g f

Experimental data: • Broken o Not broken I I M I I I I I I I I M I II I I I I I

0

100 200 300 Axial stress amplitude, o MPa

400

(b) FCD700 cast iron Fig. 14. Comparison of predicted and experimental results for smooth nodular cast iron specimens.

259

260

M ENDO

The prediction for smooth specimens is given as the dashed curve in Fig. 9, based upon Eq.(15). It is now clear that when the defect is sufficiently small, the fatigue strength is determined by a competition between the fatigue strengths of the smooth region of the specimen and that part containing the hole. From the intersection of the predicted solid lines for the 40 |im and 100 jim cases with the dashed curve, it appears that the critical size of a non-detrimental defect increases with increase in the combined stress ratio, r/<x In the case of nodular cast irons, as already mentioned above, the fatigue strength of smooth specimens includes the contribution of the graphite nodules and the microshrinkage cavities present in the structure, so that the fatigue limit of defect-free specimens for these irons cannot be obtained in a fatigue test. In this case, the following empirical relation [7,26] is therefore employed: G^,=\.6HV

(16)

;R = -\

where CTWO is in MPa and HV, the Vickers hardness, is in kgf/mm^. It can be considered that Eq.(16) predicts the uniaxial fatigue limit of a defect-free specimen or of a specimen that contains non-detrimental defects. For ductile metals of relatively low strength, the ratio of TWO to (Two is about 0.6 [34,35], and Eq.(15) is reduced to

M\

\.6HV J

(17)

0.6(1.6//F)

The predictions based upon Eq.(17) are shown by dashed curves in Figs. 14a and 14b. Since the solid curves for graphite nodules and microshrinkage cavities are lower than dashed curves for defect-free specimens, it appears that nodules and cavities can act harmfully to lower the fatigue strength under all combinations of crand r. Figure 15 gives an example of the non-propagating cracks observed [36] in a smooth

50jUm

Axial direction

(a) Mode I crack

50^m

Axial direction

(b) Mode II crack

Fig. 15. Mode I and mode II non-propagating cracks observed at torsional fatigue limit in a smooth FCD400 cast iron specimen [36]

The Multiaxial Fatigue Strength of Specimens Containing Small Defects

261

FCD400 cast iron specimen just below the pure torsional fatigue limit. As seen in Fig. 15a, many of the cracks initiated at graphite nodules were in direction of about ±45° to specimen axis. In the same specimen, as shown in Fig. 15b, in the 0° or ± 90° shear planes mode II nonpropagating cracks were observed in the matrix remote from nodules. In smooth FCD700 specimens, no mode II non-propagating crack was observed at the torsional fatigue limit. In the case of FCD400, the fatigue limit of the matrix structure as predicted from Eq.(17) is 182 MPa, a value very close to the experimental value, 175 MPa. This stress appears to be sufficiently high to nucleate mode II cracks by slip repetitions in the matrix structure, and may be why the coexistence of both mode I and mode II non-propagating cracks was observed. In the case of FCD700, it appears that the reduction in torsional fatigue strength due to graphite nodules is so large that mode II non-propagating cracks cannot develop in the matrix, see Fig. 14b. In addition Fig. 14 indicates that these nodules or cavities can reduce the bending fatigue limit much more than the torsional fatigue limit.

CONCLUSIONS 1. A criterion for fatigue failure under multiaxial loading conditions for specimens containing defects has been presented. 2. With the aid of this criterion, a unified method for the prediction of the fatigue limit has been developed. A feature of the method is that predictions can be made without the need for fatigue testing. 3. Good agreement has been found between the predictions based upon this method and the experimental data obtained in fatigue tests which were carried out under either uniaxial, torsional or combined loading conditions for a variety of materials which included annealed medium carbon steel, quenched and tempered Cr-Mo steel, ferritic and pearlitic nodular cast irons and a high strength brass. 4. When the defect is quite small, the fatigue strength is determined by a competition between the fatigue strengths of the defect-containing part and a defect-free part. 5. There is a critical size below which defects are not deleterious. However this critical size is dependent upon the combined stress ratio.

ACKNOWLEDGEMENT The author wishes to thank Prof A. J. McEvily of University of Connecticut for fruitful discussions and for correcting the English manuscript of this paper.

REFERENCES 1. Kitagawa, H. and Takahashi, S. (1976). Applicability of Fracture Mechanics to Very Small Cracks or the Cracks in the Early Stage, In: Proc. 2nd Int. Conf. on Mechanical Behavior of Materials, pp. 627-631, Am. Soc. Metals, Metals Park, Ohio. 2. Murakami, Y. and Endo, M. (1994). Effects of Defects, Inclusions and Inhomogeneities on Fatigue Strength, Int. J. Fatigue 16, 163-182.

262

M. ENDO

3. Murakami, Y. and Endo, T. (1980). Effects of Small Defects on Fatigue Strength of Metals, Int. J. Fatigue 2, 23-30. 4. Murakami, Y. and Endo, M. (1983). Quantitative Evaluation of Fatigue Strength of Metals Containing Various Small Defects or Cracks, Engng. Fract. Meek 17, 1-15. 5. Murakami, Y. and Endo, M. (1986). Effects of Hardness and Crack Geometries on zlA^th of Small Cracks Emanating from Small Defects, In: The Behaviour of Short Fatigue Cracks; Miller, K. J. and de los Rios, E. R. (Eds), pp. 275-293, Mech. Engng. Publ., London. 6. Murakami, Y. and Endo, M. (1992). The ^farea Parameter Model for Small Defects and Nonmetallic Inclusions in Fatigue Strength: Experimental Evidences and Applications, In: Theoretical Concepts and Numerical Analysis of Fatigue; Blom, A. F. and Beevers, C. J. (Eds), pp. 51-71, EMAS Publ., London. 7. Murakami, Y. (2002). Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions. Elsevier, Oxford. 8. Miller, K. J. and Brown, M. W. (Eds) (1985). Multiaxial Fatigue. ASTM STP 853. Am. Soc. Test. Mater., Philadelphia. 9. Brown, M. W. and Miller, K. J. (Eds) (1989). Biaxial and Multiaxial Fatigue. Mech. Engng. Publ., London. 10. Kussmaul, K. F. and McDiarmid, D. L. (Eds) (1991). Fatigue under Biaxial and Multiaxial Loading. Mech. Engng. Publ., London. 11. Pineau, A., Cailletand, G. and Lindley, T. C. (Eds) (1996). Multiaxial Fatigue and Design. Mech. Engng. Publ., London. 12. Miller, K. J. and McDowell, D. L. (Eds) (1999). Mixed-Mode Crack Behavior. ASTM STP 1359, Am. Soc. Test. Mater., Philadelphia. 13. Socie, D. F. and Marquis, G. B. (2000). Multiaxial Fatigue. Soc. Automotive Engrs. Inc. Warrendale, PA. 14. Nisitani, H. and Kawano, K. (1971). Correlation between the Fatigue Limit of a Material with Defects and its Non-Propagating Crack - Some Considerations Based on the Bending or Torsional Fatigue of the Specimen with a Diametrical Hole, Trans. Japan. Soc. Mech. Engrs. 37, 1492-1496. 15. Mitchell, M. R. (1977). Review of the Mechanical Properties of Cast Steels with Emphasis on Fatigue Behavior and the Influence of Microdiscontinuities, Trans. ASME, J. Engng. Mater. Tech. 99, 329-343. 16. Endo, M. and Murakami, Y. (1987). Effects of an Artificial Small Defect on Torsional Fatigue Strength of Steels, Trans. ASME, J. Engng Mater. Tech. 109, 124-129. 17. Endo, M. (1998). Fatigue Strength of Annealed 0.37 % Carbon Steel Containing Small Defect under Combined Axial and Torsional Loading, In: Fracture from Defects, Proc. ECFI2 Vol. I, pp. 115-120, Brown, M. W., de los Rios, E. R. and Miller, K. J. (Eds), EMAS Publ., West Midlands, UK. 18. Murakami, Y. and Takahashi, K. (1998). Torsional Fatigue of a Medium Carbon Steel Containing an Initial Small Surface Crack Introduced by Tension-Compression Fatigue: Crack Branching, Non-propagation and Fatigue Limit, Fatigue Fract. Engng. Mater. Struct. 21, 1473-1484. 19. Nadot, Y., Bertheau, D., Denier, V. and Mendez, J. (2000) Integrity of Cast Components Containing Inhomogeneities, In: Fracture Mechanics: Applications and Challenges, Proc. ECFI3, on CD-ROM, Elsevier, Oxford.

The Multiaxial Fatigue Strength of Specimens Containing Small Defects

263

20. Dang Van, K. (1992). On Structural Integrity Assessment for Multiaxial Loading Paths, In: Theoretical Concepts and Numerical Analysis of Fatigue', Blom, A. F. and Beevers, C. J. (Eds), pp. 343-357, EMAS Publ., London. 21. Beretta, S. and Murakami, Y. (1998). The Stress Intensity Factor for Small Cracks at Micro-Notches under Torsion, In: Fracture from Defects, Proc. ECF12 I, pp. 55-60, Brown, M. W., de los Rios, E. R. and Miller, K. J. (Eds), EMAS Publ., West Midlands, UK. 22. Beretta, S. and Murakami, Y. (2000). SIF and Threshold for Small Cracks at Small Notches under Torsion, Fatigue Fract. Engng. Mater. Struct. 23, 97-104. 23. Endo, M. (1999). Effects of Small Defects on the Fatigue Strength of Steel and Ductile Iron under Combined Axial/Torsional Loading, In: Small Fatigue Cracks: Mechanics, Mechanisms and Applications, pp. 375-387, Ravichandran, K. S., Ritchie, R. O. and Murakami, Y. (Eds), Elsevier, Oxford. 24. Endo, M. (2000). Fatigue Strength Prediction of Ductile Irons Subjected to Combined Loading, In: Fracture Mechanics: Applications and Challenges, Proc. ECF13, on CDROM, Elsevier, Oxford. 25. McEvily, A. J., Endo, M. and Murakami, Y. (2002). On the -Jarea Relationship and the Short Fatigue Crack Threshold, to be published. 26. Murakami, Y., Kodama, S. and Konuma, S. (1989). Quantitative Evaluation of Effects of Non-metallic Inclusions on Fatigue Strength of High Strength Steels. I: Basic Fatigue Mechanism and Evaluation of Correlation between the Fatigue Fracture Stress and the Size and Location of Non-metallic Inclusions, Int. J. Fatigue 11, 291-298. 27. Murakami, Y. and Usuki, H. (1989). Quantitative Evaluation of Effects of Non-Metallic Inclusions on Fatigue Strength of High Strength Steels. II: Fatigue Limit Evaluation Based on Statistics for Extreme Values of Inclusion Size, Int. J. Fatigue 11, 299-307. 28. Murakami, Y., Uemura, Y., Natsume, Y. and Miyakawa, S. (1990). Effect of Mean Stress on the Fatigue Strength of High-Strength Steels Containing Small Defects or Nonmetallic Inclusions, Trans. Japan Soc. Mech. Engrs. 56, 1074. 29. Gough, H. J. and Pollard, H. V. (1937). Properties of Some Materials for Cast Crankshafts, with Special Reference to Combined Stresses, Proc. Instn. Automobile Engrs. 31, 821-893. 30. Gough, H. J. (1949). Engineering Steels under Combined Cyclic and Static Stresses, Proc. Instn. Mech. Engrs. 160, 417-440. 31. Endo, M. (1991). Fatigue Strength Prediction of Nodular Cast Irons Containing Small Defects, In: Impact of Improved Material Quality on Properties, Product Performance, and Design, MD-Vol. 28, pp. 125-137, Muralidharam, U. (Ed), Am. Soc. Mech. Engrs., New York. 32. Gumbel, E. J. (1957). Statistics of Extremes. Columbia Univ. Press, New York. 33. Gough, H. J. and Pollard, H. V. (1935). The Strength of Metals under Combined Alternating Stresses, Proc. Instn. Mech. Engrs. 131, 3-103. 34. Heywood, R. B. (1962). Designing against Fatigue. Chapman and Hall Ltd., London. 35. Forrest, P. G. (1966). Fatigue of Metals. Pergamon Press, Oxford. 36. Endo, M. (2002) Effects of Small Natural and Artificial Defects on Multiaxial Fatigue Strength of Nodular Cast Iron, Proc. Eighth Int. Fatigue Congress {Fatigue 2002), Stockholm, pp, 2791-2798, Blom, A. F. (Ed), EMAS Publ., West Midlands, UK.

264

M ENDO

Appendix: NOMENCLATURE ^a

C c D d

F\A, FIB h HV k •t^Imax

Mb Mt

P R

So t T V Vo A^th AArth,bi A/Lth,uni

a (J Ob
Gi CTa CTaO CTw CTwO

CTwl CTx, <Jy

r ^a

Geometrical parameter defined as the square root of the area of a defect or a crack projected onto the plane normal to the maximum tensile stress -Jarea of the largest defect Constant in Eq.(4), which is dependent on defect position but independent of material and defect size Representative crack length Specimen diameter Hole diameter Correction factors of stress intensity factor (SIF), see Fig. 3 Hole depth Vickers hardness Ratio of correction factors of SIF, F\^F\^ SIF in mode I Maximum value of A^i at the front of a three-dimensional crack Bending moment amplitude Torsional moment amplitude Axial load amplitude Stress ratio Standard inspection area Thickness of standard inspection domain Return period, VI VQ Control volume Standard inspection volume, t So Threshold SIF range AA^th under biaxial loading A^th under uniaxial loading Material constant as a function ofHV Ratio of torsional to uniaxial fatigue limit, rJcTyj, of defect-containing specimens Normal stress amplitude Remote applied stress Maximum principal stress amplitude at fatigue limit under combined loading Minimum principal stress amplitude at fatigue limit under combined loading crat fatigue limit of defect-containing specimens under combined loading crat fatigue limit of smooth specimens under combined loading crat uniaxial fatigue limit of defect-containing specimens crat uniaxial fatigue limit of smooth specimens Lower bound of scatter band for CTW Remote applied stresses in x-y coordinate system, see Fig. 3 Shear stress amplitude rat fatigue limit of defect-containing specimens under combined loading rat fatigue limit of smooth specimens under combined loading rat torsional fatigue limit of defect-containing specimens rat torsional fatigue limit of smooth specimens

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

265

AN ANALYSIS OF ELASTO-PLASTIC STRAINS AND STRESSES IN NOTCHED BODIES SUBJECTED TO CYCLIC NON-PROPORTIONAL LOADING PATHS Andrzej BUCZYNSKI^ and Grzegorz GLINKA^ ' Warsaw University of Technology, Institute of Heavy Machinery Engineering, ul. Narbutta 85, 02-524 Warsaw, Poland University of Waterloo,), Department of Mechanical Engineering, Waterloo, Ontario N2L 3 0 1 , Canada

ABSTRACT Fatigue and durability analyses of machine components and structures require calculation of elastic-plastic stresses and strains in notched bodies subjected to non-proportional loading sequences. Analytical methods are seldom feasible even in the case relatively simple geometrical configurations and therefore numerical methods are often employed. The numerical-analytical method discussed below is based on the incremental relationships relating the fictitious elastic and elastic-plastic Neuber type strain energy densities near the notch tip, and the material stressstrain behavior simulated according to the Mroz-Oarud cyclic plasticity model. The method consists of a set of algebraic incremental equations that can be easily solved for elastic-plastic stress and strain increments near a notch knowing the increments of the hypothetical elastic notch tip stress history and the material stress-strain curve. The validation of the proposed model against numerical data includes two non-proportional loading histories. In particular the basic equations involving the equivalence of the strain energy density are carefully examined. Finally, the numerical procedure for solving the basic set of equations is briefly described. The method is particularly suitable for fatigue life analyses of notched bodies subjected to cyclic multiaxial loading paths. KEYWORDS Elastic-plastic strains at notches, Multiaxial cyclic loading, Strain energy density INTRODUCTION Notches and other geometrical irregularities cause significant stress concentration. Such an increase of stresses results often in localized near the notch tip plastic deformation, leading to premature initiation of fatigue cracks. Therefore, the fatigue strength and durability estimations of notched components require detail knowledge of stresses and strains in such regions. The stress state in the notch tip region is in most cases multiaxial in nature. Axles and shafts may experience, for example, combined out of phase torsion and bending loads. Although modem Finite Element commercial software packages make possible to determine the notch tip stresses in elastic and elastic plastic bodies for short loading histories, such methods are still impractical in the case of long loading histories experienced by machines in service. A representative cyclic

266

A. BUCZYNSKJAND G. GLINKA

loading history may contain from a few thousands to a few millions of cycles. Therefore incremental elastic-plastic finite element analysis of such a history would required prohibitively long computing time. For this reason more efficient methods of elastic-plastic stress analysis are necessary in the case of fatigue life estimations of notched bodies subjected to lengthy cyclic stress histories. One of such methods, suitable for calculating multiaxial elastic-plastic stresses and strains in notched bodies subjected to proportional and non-proportional loading histories, is discussed in the paper.

LOADING HISTORIES The notch tip stresses and strains are dependent on the notch geometry, material properties and the loading history applied to the body. If all components of a stress tensor change proportionally, the loading is called proportional. When the applied load causes the direction and/or ratio of principal stresses to change in any load increment, the loading is termed nonproportional If plastic yielding takes place at the notch tip then the stress path in the notch tip region is almost always non-proportional regardless whether the remote loading is proportional or not. The non-proportional loading/stress paths are usually defined by successive increments of load/stress parameters and all calculations need to be carried out incrementally. In addition the material stress-strain response to non-proportional cyclic loading paths has to be appropriately simulated, including the stress-strain non-linearity and the material memory effects. THE STRESS STATE AT THE NOTCH TIP For the case of general multiaxial loading applied to a notched body (Fig. 1), the state of stress near the notch tip is tri-axial in nature. There are six unknown stress and strain components. Therefore there are twelve unknowns all together and a set of twelve independent equations is required for the determination of all stress and strain components at a point near the notch tip.

and

(1)

The material constitutive relationships provide six equations, leaving six additional equations to be established. 0

THE MATERIAL CONSTITUTIVE MODEL 0

In the case of proportional or nearly proportional notch tip stress paths the Hencky total deformation equations of plasticity can be used in the analysis. 1+v

E

.

V n ^ . 3 s\

S ^:--<,s,+-^s:

' E '' ' 2&

ij

(2)

267

An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic .

where:5';=cr;.--(j;^,^

^22

/

<^2i'

>^

o-,/ '12

^

k^^' / ^il"

^11

'13

Fig. 1. Stress state at a notch tip (notation). Non-proportional load histories require the application of the theory of incremental plasticity. The Hooke law and the Prandtl-Reuss flow rule are the most frequently used models. For an isotropic body, the two models together give the basic incremental stress-strain relationship.

"

E

•' E

" '

2 a'

"

(3)

The multiaxial incremental stress-strain relation (3) is obtained from the uniaxial stress-strain curve by relating the equivalent plastic strain increment to the equivalent stress increment such that:

dKcr" )

(4)

The function, ZQ^= f(aeq), is identical to the plastic strain - stress relationship obtained experimentally from an uni-axial cyclic tension-compression test.

268

A. BUCZYNSKIAND G. GLINKA

THE NOTCH TIP STRESS-STRAIN RELATIONS The load or any other parameter representing the load is usually given in the form of the nominal or reference stress being proportional to the remote applied load and the stress concentration factor Kt. However, the use of the stress concentration factor, Kt, is not sufficient in the case of multiaxial stress states near the notch tip because it supplies information about only one stress component. Therefore the fictitious "linear elastic" stresses which would exist near the notch tip in the absence of plasticity are used in the method discussed below. In the case of notched bodies in plane stress or plane strain state the relationship between the load and the elastic-plastic notch tip strains and stresses in the localized plastic zone is often approximated by the Neuber rule [1] or the Equivalent Strain Energy Density (ESED) equation [2]. It was shown later [3, 4] that both methods can be extended for multiaxial proportional and non-proportional modes of loading. Similar methods were also proposed by Hoffman and Seeger [ 5] and Barkey et al. [6 ]. All methods consist of two parts namely the constitutive equations and the relationships linking the fictitious linear elastic stress-strain state (aij^,8ij^) at the notch tip with the actual elastic-plastic stress-strain response (aij^8ij^), as shown in Fig. 2.

Fig. 2. Stress states in geometrically identical elastic and elastic-plastic bodies subjected to identical boundary conditions. The Neuber rule [2,3] for proportional loading, where the Hencky stress-strain relationships are applicable, can be written for the uni-axial and multiaxial stress state in the form of equation (5a) and (5b) respectively. <^22^22 = <^22^22 (uniaxial stress state)

(5a)

An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic .

ij

ij

ij

ij

269

(5b)

(multiaxial stress state)

The Neuber rule (5a) represents the equality of the total strain energy (the strain energy and the complimentary strain energy density) at the notch tip, represented by the rectangles A and B in Fig.3.

Fig. 3. Graphical interpretation of the Neuber rule. The total strain energy density equation written in engineering notation takes the form of expression (6). <1<1 + <^ll^n + <3<3 + <^l2^l2 + ^2>23 + ^lAz

(6) i^ii

+ (7-,->£-f

The overall strain energy density equivalence represented by eq.(6) relating the pseudo-elastic and the actual elastic-plastic notch tip strains and stresses at the notch tip has been generally accepted as a good approximation rule but the additional conditions, necessary for the complete formulation of a multiaxial stress state problem, are still the subject of discussion. Hoffman and Seeger [5] assumed that the ratio of the actual principal strains at the notch tip is to be equal to the ratio of the fictitious elastic principal strain components while Barkey et al. [6] suggested using the ratio of principal stresses. The data presented by Moftakhar [7] indicate that the accuracy of the stress or strain ratio based analysis depends on the degree of constraint at the notch tip. Therefore, Moftakhar et al. [7] have proposed to use the ratio of the strain energy

270

A. BUCZYNSKI AND G. GLINKA

density contributed by each pair of corresponding stress and strain components. It was confirmed later by Singh et al. [4] that the accuracy of the additional energy equations was also good when used in an incremental form. Because the ratios of strain energy density increments seem to be less dependent on the geometry and constraint conditions at the notch tip than the ratios of stresses or strains the analyst is not forced to make any arbitrary decisions about the constraint while using these equations. However, the additional strain energy density equations [4, 7] have a theoretical drawback indicated by Chu [8], namely the estimated elastic-plastic notch tip strains and stresses may depend slightly on the selected system of coordinates. Fortunately, the dependence is not very strong and with suitably chosen system of axis it could be sufficiently accurate for a variety of engineering applications. It was also found that the set of equations involving the strain, stress and the strain energy density increments can be singular at some specific ratios of stress components, which is due to the conflict between the plasticity model (normality rule) and strain energy density equations. Such a conflict can be avoided if the principal idea of Neuber involving only shear stresses is implemented in the incremental form. Namely, it should be noted that the original Neuber rule (5a) was derived for bodies in pure shear stress state. It means that the Neuber equation states the equivalence of only distortional strain energy density. Therefore, in order to formulate the set of necessary equations for a multiaxial non-proportional analysis of elastic-plastic stresses and strains at the notch tip, the equality of increments of the total distortional strain energy density was used. As a consequence all equations were written in terms of deviatoric stresses and strains.

DEVIATORIC STRESS-STRAIN RELATIONSHIPS The notch tip deviatoric stresses of the hypothetical linear-elastic input are determined as:

S;=cr;--crl,S^

(7)

The elastic deviatoric strains and strain increments can be calculated from the Hooke law. AS'

The actual near the notch tip deviatoric stress components in the notch tip can analogously be defined as:

ij

ij

^

kk

ij

^ ^

The incremental deviatoric elastic-plastic stress-strain relationships based on the associated Prandtl-Reuss flow rule can be subsequently written as: AS' Ae:=^ + S:dA. ' 2G ' In engineering notation the deviatoric stress-strain relations are written as:

(10a)

An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic ...

^^ ^^

271

(10b)

2G

^'

"•^ 2G

^^

AC«

where: 3A£/;

^^=;^^ kF=^<7^^; 2 (T, ^ eq

^

^^eq

The specific form of the stress-strain function, 8eq''= f(cTeq), must be obtained experimentally from an uniaxial cyclic test. EQUIVALENCE OF INCREMENTS OF THE TOTAL DISTORTIONAL STRAIN ENERGY DENSITY It is proposed, analogously to the original Neuber rule, to use the equivalence of increments of the total distortional strain energy density contributed by each pair of corresponding stress and strain components. ^r,A<,+<,A5f,=5r,A<,+<,A^ri 5f3 A<3 + et.AS:, = 5^3 A<3 + e^.AS^, S',,Ael, + 4 A^2^2 = 52^2^4 + 4 ^5^2 52^3 A ^ + e',,AS^,, = ^,^3 A 4 + 4 A^,^3

(11)

53^3 A 4 + 4 ^^3^3 = ^3^3 Ae3^3 + 4^5-3^3 The equalities of strain energy increments for each set of corresponding hypothetical elastic and actual elastic-plastic strain and stress increments at the notch tip can be shown graphically (Fig. 4) as the equality of areas of the two pairs of rectangles representing the increments of total strain energy density (including the complementary one). The area of dotted rectangles represents the total strain energy increment of the hypothetical (fictitious) elastic notch tip input

272

A. BUCZYNSKIAND G. GLINKA

while the area of the hatched rectangles represents the total strain energy density of the actual elastic-plastic material response at the notch tip.

Fig. 4. Graphical representation of the incremental Neuber rule.

^9

A

Fig. 5. Piecewise linearisation of the material cr-f curve and the corresponding work hardening surfaces.

An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic ...

273

Equations (lO)-(ll) form a set of twelve simultaneous equations from which all deviatoric strain and stress increments can be determined, based on the hypothetical linear elastic notch tip stress path data, i.e. increments Aay^ (obtained from the linear-elastic analysis) and the constitutive stress-strain curve (3).

"

2G

A<, =

AS,", 2G

2 <,

3 + - -^ ^ u 2 < "

A^" A<3 = 2G

A4 =

2 <

"

A5°3

^el, = A<3 =

"

2G

2G

+ -^5,-3 2 o-^ ' ' 3 Af" + -^5^5"

2G

^

a

a

2 % SlM^+ <, + <,AS,",

SIM: + <2 A5,", = Sn^^n + <2^n SIM, + <3AS,"3 =S:M. +<,As:, Sl,Ael,+el,AS;, ,_=S",M2+42^22

(12)

5*23 Ae23 + ^ 2 3 ^ 2 3 ~ "^rS^^B "*" ^ 2 3 ^ 2 3

S';^Ael^ + 4 ^ 3 3 = ^ ' 3 ^ 4 + <3^3'3

For each increment of the external load, represented by increments of pseudo-elastic (fictitious) deviatoric stresses, ASy^, the deviatoric elastic-plastic notch tip strain and stress increments, Aey^ and ASij^, are computed from the equation set (12). With the help of equation (9) the calculated deviatoric stress increments, ASij^ can subsequently be converted into the actual stress increments, Aay^. A5,"=A<,-i(A<,+AcT,>Aa3"3) ^"2 = A<3 -^(ACT,", + ACT,", + A<3)

A5°3 = Aa3" - ^ ( A < , + Aat, + Aa3"3) AS:,

= ACT:,

AS:,=Aa:,

ASl,=Aal,

(13)

274

A. BUCZYNSKIAND G. GLINKA

The deviatoric and the actual stress components Sy^ and ay^ at the end of each load increment are determined from eq.(14-15).

5;" = 5r + 2A5;* + A5r.

(14)

k=l

af = cyf + f^A(Tf+AaT

(15)

k=l

where: n denotes the number of the load increments. The actual strain increments, Asy^, canfinallybe determined from the constitutive equation (3). CYCLIC PLASTICITY MODEL In order to predict the notch tip stress-strain response of a notched component subjected to multiaxial cyclic loading, the incremental equations discussed above have to be linked with the cyclic plasticity model. Several plasticity models are available in the literature. The most popular is the model of Ref [9] proposed by Mroz. According to Mroz [9] the uniaxial stress-strain material curve can be represented in a multiaxial stress space by a set of work-hardening surfaces.

In the case of a two-dimensional stress state (G\\^ = ai2^ = au^ = 0), such as that one at a notch tip, the work-hardening surfaces can be represented by ellipses in the coordinate plane for which the axes are defined by the directions of principal stress components (Fig. 5). The equation of each work-hardening ellipse in the two-dimensional principal stress space is:

•

^

>0

(17)

The essential elements of the plasticity model can be presented in such a case graphically in a two-dimensional principal stress space. The load path dependency effects are modeled by prescribing a rule for the translation of ellipses in the a2^-a3^ plane. The translation of these ellipses is assumed to be caused by the sought stress increment, represented in the principal stress space as a vector. The ellipses can be translated with respect to each other over distances dependent on the magnitude of the stress/load increment. The ellipses move within the boundaries of each other, but they do not intersect. If an ellipse comes in contact with another, they move together as one rigid body. However, it has been found that the ellipses in the original Mroz model may sometimes intersect each other, which is not permitted. Therefore, Garud proposed [10] an improved translation rule that prevents any intersections of plasticity surfaces. The principle idea of the Garud translation rule is illustrated in Fig. 6. a) The line of action of the stress increment, Aa^, is extended to intersect the next larger nonactive surface, fj, at point B2.

An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic ...

275

b) Point B2 is connected to the center, O2, of the surface f2. c) A line is extended through the center of the smaller active surface , Oi, parallel to the line O2B2 to find point Bi on surface fi. d) The conjugate points Bi and B2 are connected by the line B1B2. e) Surface fi is translated from point Oi to point Oi' such that vector OiOi' is parallel to line B1B2. The translation is complete when the end of the vector defined by the stress increment, Aa, lies on the translated surface fi'.

Fig. 6. Geometrical illustration of the translation rule in the Garud incremental plasticity model. The mathematics reflecting these operations can be found in the original paper of Mroz [9] or Garud [10] or in any recent textbook on the theory of plasticity. The Mroz and Garud models are relatively simple but they are not very efficient numerically, especially in the case of long load histories with a large number of small increments. If the computation time is of some concern the model based on infinite number of plasticity surfaces proposed by Chu [11] can be used in lengthy fatigue analyses. The cyclic plasticity models enable the relationship, ASeq^'-Aagq^, to be established providing the actual plastic modulus for given stress/load increment, AGJ. In other words the plasticity model determines which piece of the stress-strain curve (Fig. 5) is to be utilized during given stress/load increment. Two or more tangent ellipses translate together as rigid bodies and the largest moving ellipse indicates which linear piece of the constitutive relationship should be used for a given

276

A. BUCZYNSKI AND G. GLINKA

Stress increment. The slope of the current linear segment of the stress-strain curve defines the plastic modulus, Aaeq/Aseq'', necessary for the determination of the parameter, dX, in the constitutive equation (10). The plasticity models are described in most publications, as algorithms for calculating strain increments that result from given series of stress increments or vice versa. This is called as the stress or strain controlled input. In the case of the notch analysis neither stresses nor strains are directly inputted into the plasticity model. The input is given in the form of the total deviatoric strain energy density increments and both the deviatoric strain and stress increments are to be found simultaneously by solving the equation set (12). Therefore, the plasticity model is needed only to indicate which work-hardening surface is to be active during current load increment, which subsequently determines the instantaneous value of the parameter dX. In order to find the actual elastic-plastic deviatoric stress and strain increment ASy^ and Ae/ from the equation set (12), the value of parameter dX is determined first according to the current configuration of plasticity surfaces. After calculating all stress increments, ASjj^, and subsequently, Aay^, the plasticity surfaces are translated as shown in Fig. 6. The process is repeated for each subsequent increment of the "elastic" input, Aay^. The Mroz [9] and Garud [10] models were chosen here as an illustration. Obviously, any other plasticity model can be associated with the incremental stress-strain notch analysis proposed above. The Garud plasticity model was employed in the analysis discussed below. COMPARISON OF CALCULATED ELASTIC-PLASTIC NOTCH TIP STRAINS AND STRESSES WITH ELASTIC-PLASTIC FINITE ELEMENT DATA The component used for the numerical validation was a cylindrical specimen (Fig. 7) with circumferential notch subjected to simultaneous tensile and torsion loading. The basic dimensions of the cylindrical component were p = 3 mm, R = 70 mm and t = 3 mm resulting in the torsional and tensile stress concentration factor K^ = 1.82 and Kta = 2.80 respectively. The ratio of the notch tip hoop to axial stress under tensile loading was ^^^^Icsii = 0.2179.

R=70 mm, t=3mm, p=3 mm

Kt =2.80, Kt =1.82, a,,/a„=0.2179

Fig. 7. Cylindrical specimen with circumferential notch subjected to tension-torsion loading The stress concentration factors for the axial and torsional loads were defined as:

An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic ...

K^

— K, =

and

32

277

(18)

The nominal stresses in the net cross section were determined as:

4R-tf

and

T„ =-

2T

4R-tf

(19)

The elastic-plastic finite element stress and strain stress results were obtained using the ABAQUS finite element package with built-in plasticity model. The torque T induced the 'linear elastic' shear stress G23^ at the notch tip and the axial load F induced the normal axial stress component a22^ and the normal hoop stress ass^. The fictitious elastic normal stress components maintained constant ratio throughout the entire loading history, i.e. a337a22^=const. The increments of the hypothetical "elastic" stress components Aa23^, Aa22^ and AGBS^ and associated strains were used as the input into the equation set (12). Two linear segments as shown in Fig. 8 were used to approximate the material stress-strain curve. The material properties were E = 200000 MPa, Ei= 4142.50 MPa, v = 0.3 and ay = 200 MPa.

E=200000 MPa Ei=4142.5MPa v=0.3

Fig. 8. Material elastic-plastic stress-strain curve The maximum applied load levels F and T were chosen to be higher than it would be required to induce yielding at the notch tip if each load was applied separately. The axial-shear notch tip elastic stress paths, a22^-a23^ applied to the notched component were those shown in Fig.9a and 9b. There were 10 full loading cycles applied in the case of the Stress Path #1 and 6 full cycles in the case of the Load Path #2. The FEM calculations for the Stress Path #1 required 72 hours CPU time on a Personal Computer with 800 MHZ processor. The finite element mesh of the notched component is shown in Fig. 10. The calculated from eq. (12)

A. BUCZYNSKI AND G. GLINKA

278

and the FEM determined axial and shear strain components, 822^ and 823^, and the axial and shear stress components, a22^ and a23^, respectively are shown in Figs. 11a - 12b. The resuhs corresponding to the Stress Path #1 (Fig. 9a) are shown in Figs. 11a and l i b . Note, that the calculated stresses and strains and the finite element data are identical in the elastic range. Load path #1

Axial elastic notch tip stress, MPa

Fig. 9a. Non-proportional load/stress path #1 Load path #2

25

50

75

100

125

150

175

200

-25 H -50 -75 J Axial elastic notch tip stress, MPa

Fig. 9b. Non-proportional load/stress path #2

225

250

275

300

An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic ...

279

Fig. 10. Finite element model of the notched component Load Path #1

0.002

Axial strain

Fig. 1 la. Evolution of the elastic-plastic axial strain and axial stress at the notch tip induced by the Stress Path #1

280

A. BUCZYNSKIAND G. GLINKA

Q.

-0.0012

^

0.0008

0.0012

-120 Shear strain

Fig. 1 lb. Evolution of the elastic-plastic shear strain and shear stress at the notch tip induced by the Stress Path #1 Load path #2

,>|?002 y /

0.0025

0.003

-FEM I -ENERGY!

Axial strain

Fig. 12a. Evolution of the elastic-plastic axial strain and axial stress at the notch tip induced by the Stress Path #2

An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic .

281

Load Path #2

0.0025

-120 Shear strain

Fig. 12b. Evolution of the elastic-plastic shear strain and shear stress at the notch tip induced by the Stress Path #1 Just above the onset of yielding at the notch tip, the strains predicted using the proposed model and the finite element data gradually deviate form the linear elastic behavior. Both sets of results reveal some ratcheting of the axial strain (Fig. 1 la); however it appears that the proposed model stabilizes quicker than the FEM data based on the ABAQUS software package plasticity model. The data obtained for the Stress Path #2 are shown in Figs. 12a and 12b. The proposed model overestimates the notch tip strains in comparison with the FEM data similarly to the classical uniaxial Neuber rule. However, this overestimation is not significant and it could be acceptable in many practical applications.

CONCLUSIONS A method for calculating elastic-plastic strains and stresses near notches induced by multiaxial loading paths has been proposed. The method has been formulated using the equivalence of the total distortional strain energy density. The generalized equations of the total equivalent strain energy density yielded a conservative solution for the notch tip strains and stresses in the case of cyclic non-proportional loading paths. The method has been verified by comparison with finite element data obtained for identical notched components subjected to non-proportional loading paths. The notch tip strains and stresses calculated for cyclic load paths can be used for the estimation of fatigue lives for multiaxial cyclic loading histories. The calculated from equation (12) and the FEM determined axial and shear strain components, Z2i and 823^, and the axial and shear stress components, a22^and a23^, respectively are shown in Figures 11 and 12.

282

A. BUCZYNSKI AND G. GLINKA

REFERENCES 1. Neuber, H., " Theory of Stress Concentration of Shear Strained Prismatic Bodies with Arbitrary Non Linear Stress-Strain Law", ASME Journal ofApplied Mechanics, vol. 28, 1961, pp. 544-550. 2. Molski, K. and Glinka, G., " A Method of ElasticPlastic Stress and Strain Calculation at a Notch Root", Material Science and Engineering, vol. 50, 1981, pp. 93-100. 3. Moftakhar, A., Buczynski, A. and Glinka, G., "Calculation of Elasto-Plastic Strains and Stresses in Notches under Multiaxial Loading", International Journal of Fracture, vol. 70, 1995, pp. 357-373. 4. Singh, M.N.K., "Notch Tip Stress-Strain Analysis in Bodies Subjected to NonProportional Cyclic Loads", Ph.D. Dissertation, Dept. Mech. Eng., University of Waterloo, Ontario, Canada, 1998. 5. Seeger, T. and Hoffman, M., "The Use of Hencky's Equations for the Estimation of Multiaxial Elastic-Plastic Notch Stresses and Strains", Report No. FB-3/1986, Technische Hochschule Darmstadt, Darmstadt, 1986. 6. Barkey, M.E., Socie, D.F. and Hsia, K.J., "A Yield Surface Approach to the Estimation of Notch Strains for Proportional and Non-proportional Cyclic Loading", ASME Journal of Engineering Materials and Technology, vol. 116, 1994, pp. 173. 7. Moftakhar, A. A., "Calculation of Time Independent and Time-Dependent Strains and Stresses in Notches", Ph. D. Dissertation, University of Waterloo, Department of Mechanical Engineering, Waterloo, Ontario, Canada, 1994. 8. Chu, C. -C, "Incremental Multiaxial Neuber Correction for Fatigue Analysis", International Congress and Exposition, Detroit, 1995, SAE Technical Paper No.950705, Warrendale, 1995. 9. Mroz, Z., "On the Description of Anisotropic Workhardening", Journal of Mechanics and Physics of Solids, vol 15, 1967, pp. 163-175. 10. Garud, Y. S., "A New Approach to the Evaluation of Fatigue under Multiaxial Loading, Journal of Engineering Materials and Technology, ASME, vol. 103, 1981, pp. 118-125. 11. Chu, C. -C, "A Three-Dimensional Model of Anisotropic Hardening in Metals and Its Application to the Analysis of Sheet Metal Forming", Journal of Mechanics and Physics of Solids, vol.32, 1984, pp. 197-212. 12. Barkey, M. E., "Calculation of Notch Strains under Multiaxial Nominal Loading", Ph. D. Dissertation, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, 1993.

Appendix: N O M E N C L A T U R E E

- modulus of elasticity

El

- plastic modulus

ey^

- actual elastic-plastic deviatoric strains at the notch tip

Cij^

- hypothetical elastic devaitoric strains at the notch tip

FEM

- finite element eethod

G

- shear modulus of elasticity

An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic ...

KtCT

- Stress concentration factor due to axial load

KtT

- stress concentration factor due to torsional load

k, n

- load increment number

5ij

- Kronecker delta, Sy = 1 for i = j and Sy = 0 for i T^ j

AsyP

- plastic strain increments

Asy^

- elastic strain increments

A8y^

- actual elastic-plastic strain increments

Aseq^^

- equivalent plastic strain increment

Aay^

- pseudo-elastic stress components

Aay^

- actual stress components

Aaeq^ - actual equivalent stress increment Sy^

- deviatoric stresses of the elastic input

Sy^

- actual deviatoric stresses

Seq''^

- actual equivalent plastic strain

8y^

- actual elasto-plastic notch-tip strains

sy^

- elastic notch tip strain components

Sn

- nominal strain

V

- Poisson's ratio

Qeq^

- actual equivalent stress at the notch tip

ay^

- actual stress tensor components in the notch tip

Gy^

- notch tip stress tensor components of the elastic input

GY

- parameter of the material stress-strain curve (yield strength)

P

- axial load

T

- torque

R

- radius of the cylindrical specimen

283

This Page Intentionally Left Blank

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

285

THE BACKGROUND OF FATIGUE LIMIT RATIO OF TORSIONAL FATIGUE TO ROTATING BENDING FATIGUE IN ISOTROPIC MATERL4LS AND MATEIUALS WITH CLEAR-BANDED STRUCTURE

Takayuki FUKUDA^ and Hironobu NTSITANI^ Department of Mechanical Engineering, Sasebo National College of Technology, Sasebo, 857-1193, Japan Department of Mechanical Engineering, Kyushu Sangyo University Fukuoka, 813-8503, Japan

ABSTRACT The fatigue limit ratios of torsional to rotating bending fatigue (zjoy^) have been determined in several steels. This ratio was found to be equal to about 0.55 in an annealed rolled carbon steel (C=0.2, 0.45%) and about 0.65 in an annealed cast carbon steel, in a diffusion annealed carbon steel (C=0.45%) and in a quenched & tempered carbon steel (C=0.45%). The difference between the two values is due to the microstructure. The annealed rolled carbon steels delivered as round bars have a clear-banded structure of ferrite and pearlite. The other three materials have no banded structure. Namely, the former materials are regarded as anisotropic, while the latter ones are considered as isotropic materials. The clear-banded structure in the axial direction greatly affects the torsional fatigue limit, but it does not affect the rotating bending fatigue limit. This is due to the fact that, in the carbon steel with a clear-banded structure, large local strain concentration occurs within the ferrite in torsional fatigue but not in rotating bending fatigue. Because of this fact, the torsional fatigue limit in a banded structure relatively decreases and, therefore, the value of fatigue ratio TW/CTW becomes small. KEYWORDS Fatigue limit ratio. Torsional fatigue. Rotating bending fatigue. Isotropic material. Banded structure, Carbon steel.

INTRODUCTION A large number of studies have been made on the fatigue behavior under combined stresses [1-8]. In particular, Nishihara [2], Gough [3] and Findley [4] have studied the fatigue strength under torsional and rotating bending combined stresses. They proposed methods according to which the fatigue limit under general combined stresses is calculated from individual fatigue limits Qwandiw, where aw is the rotating bending fatigue limit and TW is the torsional fatigue limit. Since the fatigue limit under general combined stresses is usually discussed according to the

286

T. FUKUDA AND H. NISITANI

individual fatigue limits, it is important to understand the physical meaning of the fatigue limit ratio, zj cjw, of torsional to rotating bending fatigue. Most of the results of the above studies have been discussed using macroscopic criteria for plastic yielding. For example, it is said that the fatigue limit ratios zj Oy^oi many ductile materials follow the maximum distortion energy criterion (Mises' criterion; TS/(TS=0.58, where Is, as are the yield strengths measured in shear and tensile loading, respectively)[9]. In many experimental results, however, the ratio of fatigue limits ( zjo^) does not always follow this macroscopic criterion of yielding. In carbon steels, which are ductile materials, the ratio varies from 0.55 to 0.7 depending on the microstructure. According to the tensile and torsional static tests of two kinds of carbon steels in which one is a fully annealed rolled carbon steel having a clear-banded structure and another is an annealed cast carbon steel having no banded structure, the clear-banded structure greatly affects the local strain concentration in torsion, but it hardly affects that in tension [10]. Taking into consideration these facts, rotating bending and torsional fatigue tests have been carried out on plain specimens of several kinds of carbon steels which have clear-banded structure or not, and the fatigue limit ratio ( rjaw) values are herein discussed based on whether the specimen has a clear-banded structure or not. FATIGUE LIMIT RATIO OF BENDING AND TORSIONAL FATIGUE [DATA MAINLY OBTAINED IN THE PAST] Classification of fatigue limit Table 1 shows experimental data, mainly obtained in the past, of fatigue limit ratio of torsional to bending fatigue for various kinds of materials. The value of fatigue limit ratio zj aw varies between 0.5 and 1 depending on the material. As is well known, fatigue process depends on crack initiation and crack propagation. Crack initiation is controlled by the maximum shear stress ( Tmax) while crack propagation is mainly related to the maximum tensile stress ( a max). They are related to the microstructure or properties of material in a complex way. Because of these facts, there are three different cases concerning the dependence of fatigue limit. (a) The fatigue limit is controlled by the maximum tensile stress, amax(b) The fatigue limit is controlled by the maximum shear stress, imax(c) The fatigue limit is controlled by both amax and Tmax In the case where the fatigue limit is controlled by the maximum tensile stress only (defective material) In a defective material, defects, for example inclusions or small holes, are regarded as a kind of crack. Thus, in this case, crack initiation can be ignored and only crack propagation can be considered. Gray cast iron or nodular cast iron can be treated in this way, since flake or spheroidal graphite is regarded as a defect. In this case, the main factor deciding the fatigue limit is the maximum tensile stress (amax). In torsion, the principal stress ( ai) is equal to the principal shear stress( r i), as is shown in Fig. 1. Therefore, if the fatigue limit is controlled by the maximum tensile stress (amax) only.

The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in ...

287

Table 1. Fatigue limit ratio between bending and torsional fatigue Material

Mechanical properties

Heat treatment gs

Op

' s

^

Fatigue limit HB,HV

a«

r «•

0^

239

138

0.58

265

147

0.56

Carbon steel

Annealed

L292

521

-

45.1

Carbon steel n

Annealed

L228

376

72.9

72.9

Normalized

L353

705

32.7

32.7

194

294

196

0.67

NiCr steel

Nothing

U482

704

59.2

59.2

237

392

255

0.65

Thermal refined

U879

962

44.9

44.9

295

530

304

0.57

"

HB103

Cast iron

Nothing

-

205

127

103

0.81

70/30 brass

Nothing

U337

409

70.4

70.4

122

131

54

0.41

Carbon steel

Normalized

U252

424

178

70.0

HB127

264

149

0.56

356

638

255

58.2

195

327

204

0.62

Spheroidized

283

470

178

67.0

144

271

154

0.57

Normalized

834

248

347

237

0.68

518

-

18.0

850t:oil que., 7001:air cooled

-

72.5

165

337

202

0.6

"

SSO'Coil que., 610'Cair cooled

578

711

422

67.5

237

438

263

0.6

CrVa steel

850t:oil que., 7001: air cooled

670

740

468

66.0

229

423

254

0.6

NiCr steel

830'Coil que., 620'C water cooled

748

882

574

65.0

282

532

347

0.65

"

" " " Ni steel

"

216

SSO'Coil que., two step cooled

743

883

-

60.5

278

502

319

0.64

NiCrMo steel

SSO'Coil que., 600'Cair cooled

1224

898

60.0

394

651

337

0.52

NiCr steel

8201: and 2001: air cooled

1642

479

798

445

0.56

Nothing

-

52.0

Silal cast iron

-

0

322

237

216

0.91

216

1.3

182

249

208

0.84

68.0

-

501

302

0.6

581

327

0.56

658

384

0.58 0.55

1 Nicrosolal "

Nothing

226

CrMo steel

900t:oil que., 580t:oil que.

844

939

NiCrMo steel

830'Coil que., 575X^011 que.

949

1087

CrMoVa steel

900T:oil que., 600t:oil que.

-

1376

NiCr steel

830t:oil que., 200'Cair cooled

970

NiCrMo steel

845'Coil que., 677'Ccooled

L876

70/30 brass

Annealed

113

WT80C

Thermal refined

761

809

Nothing

240

477

1 Nodular cast iron

54.0

1368

-

64.0

-

657

364

776

452

-

HRC25

462

286

0.62

320

-

73.1

83.4

73.5

0.88

461

275

0.6

10.6

-

196

186

0.95

CrMo steel

850'Cque., 600'Ctempering

U923

1041

" "

850X3que., 450X3tempering

1243

1373

850T3que., 300X3 tempering

1500

1756

850t3oil que., 630X3tempering

883

974

I NiCrMo steel Carbon steel n

845X1 water que., 6(K)X^tempering

U564

748

825X^ water que., 6U0X)tempering

700

860

Cr Steel

855X^oil que., 550X1 water cooled

935

1051

855X;oiI que., 6(X)X^water cooled

838

969

855X;oil que., 550X^water cooled

1046

1118

1081

1139

"

CrMo steel

66.4

61.3

Hv331

589

368

0.62

51.1

434

699

397

0.57

41.8

535

791

456

0.58

63.9

314

539

351

0.65

Nishihara,T(2)

Gough,H.J.(3)

Firth,P.H.(5)(6)

Findley,W.N.(7)

Matake,T(8)

Nishijima,S. (11)(12)

68.0

Hv245

415

284

0.69

283

458

318

0.69

61.0

328

582

401

0.69

NRIM fatigue

62.0

299

522

358

0.69

data sheet

59.0

362

586

405

0.69

(12)

62.0

372

610

416

0.68

60.0

339

85513oil que., 60()X1 water cooled

975

983

549

382

0.7

Carbon steel

Annealed

324

570

1170

51.3

235

140

0.59

Nodular cast iron

Nothing

343

596

-

3.7

206

186

0.9

Age-hardened

250

309

479

48.0

131

65

0.5

I Al-alloy

Nishihara,T(l)

58.0

-

" "

"

-

61.0

Reference

-

(13) Nisitani,H.(14)

as, Is- Yield stress(L:Lower, U:Upper)(MPa), o^\ Ultimate tensile strength (MPa), ^ : Reduction of area (%), HB,HV,HR : Hardness, Gw, T w: Fatigue limit (MPa)

(15)

288

T. FUKUDA AND H. NISITANI

f Fig.l. Principal stress and principal shear stress in torsion

the relation between both fatigue limits of bending and torsion becomes (Jw='^w in torsion. Then, the fatigue limit ratio of torsional fatigue to bending fatigue is obtained by the following equation: ^w/C^w=l

(1)

This equation corresponds to the maximum tensile stress criterion, which is related to the macroscopic criterion of yielding ( rs/as = 1). A defective material, which contains sharp defects like gray cast iron, mostly follows the Eq.(l). However, in the artificial defects of small circular holes or nodular cast irons, it is necessary to consider the stress concentration of a circular hole. Since the maximum stress due to the stress concentration of a circular hole is equal to approximately 3 a in bending and 4 r in torsion ( o max= "^ max), the ratio of fatigue limits is reduced to the following equation: L:^/a^ = 0.75

(2)

Nisitani presented some experimental results for an annealed rolled carbon steel bar with a small hole [13]. In the case of a single hole specimen, r J a ^ w a s equal to approximately 0.75, whereas it was equal to about 0.9 in the case of the connected hole specimen. It was also shown that Zj o^ was equal to approximately 0.9 (not 0.75) for a nodular cast iron. This means that spheroidal graphite is not always independently distributed and is regarded as a kind of connected hole. As is mentioned above, the fatigue limit of a defective material including cast iron is mainly controlled by the maximum tensile stress, and the value of fatigue limit ratio ij a^ follows Eq.(3), depending on the type or distribution of defects, etc. z^,/0^ = 0.75 ^ 1

(3)

In the case where the fatigue limit is controlled by the maximum shear stress only (non-defective material) If the cracks once initiated do not stop propagating, the fatigue limit is determined by whether a crack initiates or not. According to the successive observations by Nisitani, age-hardened Al-alloy corresponds to this case [15]. In this material a crack initiates in a fairly small region

The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in ...

( ' ^ l / x m ) like a point, and gradually grows as the number of cycles N increases. Then the crack does not stop propagating due to the work softening effect [16], which finally causes the specimen to break. This phenomenon is quite different from the case of an annealed carbon steel, where fatigue process can clearly be divided into crack initiation and crack propagation [16], Figure 2 shows schematic illustrations of the crack initiation process of the age-hardened Al-alloy and annealed metals [17]. In Fig.2(a), the starting region of fatigue cracking of an age-hardened Al-alloy is much smaller than a grain size and extends gradually toward the grain boundary. In this material, it is difficult to distinguish the small slip band from a microcrack, or the initiation process from the propagation process. On the other hand, in Fig.2(b), the crack initiation process of the an annealed low carbon steel is entirely different from the crack propagation process. Until the initiation of a crack, fatigue damage is accumulated gradually in the same region (shaded area in Fig.2(b)), dimensions of which are closely related to the grain size, and then the region (a grain boundary or a slip band) turns into a crack as a whole. Since the factor controlling crack initiation is the maximum shear stress, the fatigue limit in this case is determined by only the maximum shear stress. The value of imax for the fatigue

Surface

r_[ }-•'N/Ni=0

<1

=1

>l Fracture surface

(a)

Age-hardened Al-alloy , Surface

N/Ni=0


=1

>1 Fracture surface

(b)

Annealed low carbon steel, a -brass, Al-alloy

Fig.2. Schematic illustrations of fatigue crack initiation process [17]

289

290

T. FUKUDA AND H. NISITANI

limit in bending is equal to that in torsion. Thus considering o^.J 'rniax=2 in bending and o^.^J rniax=l in torsion, the fatigue limit ratio of torsional fatigue to bending fatigue is obtained by the following equation: r , , / a ^ = 05

(4)

This equation corresponds to the maximum shear stress criterion, which is related to the macroscopic criterion of yielding ( zJo^={)5).

In the case where the fatigue limit is controlled by both Of^ax cL^d tmax In annealed metals, the fatigue process can be divided into two different processes, i.e., an initiation stage and a propagation stage, as is shown in Fig.2 (b) [18,19]. (a) Process ( I ): By slip repetitions, the slip band or the grain boundary (or the part near the grain boundary), which is going to become a crack, is disrupted as a whole and is gradually turned into a free surface. Then a crack of the size of a crystal grain initiates. (b) Process (11): The crack initiated in process ( I ) increases in length and depth, and finally causes the specimen to break. As is mentioned above, since process ( I ) is controlled by the maximum shear stress and process ( A ) is controlled by the maximum tensile stress, the factor deciding the fatigue limit is not simple to be determined. That is, the fatigue limit in both bending and torsion is determined by the limting stress for propagation of a non-propagating microcrack. Considering a max/ '^max=2 in bending and a^ax/ '^max=l in torsion, the crack once initiated in bending is easier to propagate than that in torsion. Consequently, since the fatigue limit in bending becomes small compared to that in torsion, the value of fatigue limit ratio is more than 0.5, and the following equation holds on the basis of our present data: r^/a^ = 055^0.7

(5)

The fatigue limit ratios have frequently been discussed based on the maximum distortion energy criterion (Mises' criterion, rs/as = 038), which is related to the macroscopic criterion of yielding [9,20]. In this case, since the fatigue limit depends on the microstructure and the material properties, and is controlled by both crack initiation and crack propagation, the factor deciding the fatigue limit is complex to be determined. Therefore, it is difficult to apply the macroscopic criterion of yielding for deciding the fatigue limit. In the present study, through the successive observations of fatigue processes and the grid line method, the physical background of the value of r J a^ in carbon steel will be made clear.

MATERIALS AND EXPERIMENTAL METHODS The materials used are rolled carbon steel (S20C, S45C) and cast carbon steel (SC450). The chemical composition is shown in Table 2. Three kinds of specimens were prepared from a rolled round carbon steel bar (S45C) by heat treatment: the first kind is annealed (S45C), the second kind is diffusion annealed (S45C-DA) and the third one is quenched and tempered (S45C-H). Conditions of heat treatment and mechanical properties are shown in Table 3. The microstructures are shown in Table 4 in the following. In the longitudinal section, the banded

The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in . 291

Structures of ferrite and pearlite areas are recognized in annealed rolled carbon steels (S20C, S45C), and not in cast carbon steel (SC450) nor in S45C-H or S45C-DA material. Namely, the former two are regarded as anisotropic materials, and the latter three are regarded as isotropic materials (see Table 4). In both tensile and torsional static tests, two types of specimens (S45C, SC450) were used. The diameter of the specimen was 8 mm, and all specimens were electropolished to the depth of about 20/xm to remove the surface layer. For the observation of the specimen surface, plastic replicas were taken from the surface after applying definite macroscopic strains. The surface states were taken from the replicas to negative films with the optical microscope, then digitized by using a film scanner and stored in the computer's memory. The local strain was measured using the image-processing method [21]. The macroscopic overall strains for the tensile tests are £g= 3, 6, 9, 12 %, and the macroscopic shear strains for the torsional test are Tg = 4.5, 9, 13.5, 18 %. The local strain for tensile test corresponds to the normal logarithmic strain which is calculated from the elongation of an axial grid line, and the local strain for torsional test corresponds to the shear strain which is calculated from the change of an intersectional angle of two grid lines. The fine grid lines were also drawn on the surface of the specimen with a diamond point needle [10]. The width of the lines was made less than 1 /i m and the depth of the lines less than 0.5 U m. The mesh size of the grid is 20 X 20 Mm. In the rotating bending and torsional fatigue tests, we used all the specimens shown in Table 3. The diameter of each specimen was 8 mm for SC450, S45C-DA and S45C-H, and 5 mm for

Table 2. Chemical composition

(%)

Material

C

Si

Mn

P

S

Cu

Ni

Cr

S20C

0.21

0.21

0.47

0.014

0.017

0.21

0.06

0.09

S45C

0.44

0.20

0.69

0.009

0.009

0.007

0.04

0.005

SC450

0.21

0.37

0.70

0.008

0.005

Table 3. Heat treatment and mechanical properties

Type Anisotropy

Isotropy

Material

Heat treatment

Osl

^B

S20C

890°C Ihr -^ F.C.

276

469

864

58.7

S45C

845°C Ihr -^ F.C.

335

569

1061

57.2

SC450

920°C Ihr ^ F.C. 1200°C 6hr -^ F.C. 845°C 0.5hr -> A.C. 845°C Ihr -> F.C. 845°C 0.5hr ^ A.C. 845°C Ihr -^ W.C. 600°C Ihr ^ W.C.

271

462

793

56.2

328

560

1050

55.5

528

751

1571

52.2

S45C-DA S45C-H

Osl '• Lower yield stress MPa o^:Ultimate tensile strength MPa Oj'.Tvuc fracture strength MPa ^ :Reduction of area %

Oj

^

292

T. FUKUDA AND H. NISITANI

S20C and S45C. After turning, the specimens were annealed in vacuum at 650°C for one hour, and were then electropolished to the depth of about 20 /i m to remove the surface layer. EXPERIMENTAL RESULTS AND DISCUSSION Local normal strain in tensile static test Figures 3 and 4 show the changes of surface state in the tensile tests of S45C and SC450 materials. The axial direction is the loading direction. It can be seen that square grids are deformed into nearly rectangular grids due to the axial elongation by tension. The degrees of their deformations are almost the same, since circumferential grid lines are nearly straight. Namely, there is not a large difference between the deformations of ferrite and pearlite bands. Consequently, the strain concentration in tension is small.

(a)

fg = 0 %

(b)

£g=12%

Fig.3. The changes of surface state in the tensile test of S45C

Axial direction

(a)

fg = 0 %

(b)

£g=12%

Fig.4. The changes of surface state in the tensile test of SC450

60 /i m

The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in ...

50

t Axial direction I I I I I I I I I S45C(Tension)

I II

X CO

I I I M I I I I I 1 2345678910 Axial line number n (a) S45C

t Axial direction I I I I I II SC450(Tension) •

O ' ' 0 0 0 / ^ I I I I I I I I ? I 1 2345678910 Axial line number n (b) SC450

Fig.5. Local strains in tensile test ( £ g = 12%)

Figure 5 shows local strains at £g=12%, which are calculated from the elongation of one hundred axial grid lines (initial length of around 20/im). The ordinate (Tmax=l-5 £ ) is the maximum local shear strain in tension. The abscissa is the location number for grid lines. That is, the abscissa is the axial line number n shown in Fig.3 and 4. The local strains of ten axial grid lines are plotted in Fig.5. Each horizontal line indicates the average of one hundred values for maximum local shear strains, which approximately coincides with the applied macroscopic strain (Tgmax=l-5X 12=18%). The local strains are not uniform. That is, there exist various local strains, which are from several percent to some thirty percent, although the applied macroscopic shear strain is 18%. As can be seen in Fig.5, local strains in the tensile test are not uniform but the strain concentration due to the banded structure is small.

Local shear strain in torsional static test Figures 6 and 7 show the changes of surface state in the torsional test of both materials. It is recognized that square grids are deformed into nearly lozenge-shaped grids due to the distortion caused by torsion. Deformation of each grid is not uniform and circumferential lines are not straight. Especially in S45C steel, there is a large difference between the deformations of ferrite band and those of pearlite band. That is, as in the case of deformation of an elastic body sandwiched between two rigid bodies, the strain concentrates within the ferrite band sandwiched between two pearlite bands. Figure 8 shows the electron micrographs of the surface state of the specimens having the fine grid lines drawn with a diamond needle. In SG450 steel, circumferential lines are almost straight. In S45C steel, however, those are not straight and deformation of each grid is not uniform. There is a large difference between the deformation in ferrite band (gray portion) and pearlite band (white portion). Figure 9 shows one hundred values of the local shear strain measured at an overall shear strain of Tg=18%, which are calculated by the same method as that used in the tensile static test. The local strains in torsion are not uniform in both materials, and there exist various values of local strains, which are from several percent to some forty percent, although the

293

T. FUKUDA AND H. NISITANI

294

(a)

7g=0%

(b)

yg=18%

Fig.6. The changes of surface state in the torsional test of S45C

(a)

7g = 0 %

(b)

7g=18%

Fig.7. The changes of surface state in the torsional test of SC450

(a)

S45C

(b)

SC450

Fig.8. The local deformation in the static torsion test (SEM, y g=18%)

The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in ...

T Axial direction

t Axial direction

1

2345678910 Axial line number n (a) S45C

295

1

2345678910 Axial line number n (b) SC450

Fig.9. Local strains in torsional test ( T g = 18%)

applied macroscopic strain is 18%. The variation of local strains in torsion is larger than that in tension. In S45C steel, local strains are unevenly distributed. That is, most of them on each axial grid line have a tendency to be distributed only above and/or below the average line. When the axial grid line exists on the pearlite band, which is harder than ferrite, most of local strains on the line are distributed below the average line, and when the axial grid line exists on the ferrite band, most of local strains on the line are distributed above the average line. This means that the strain concentrates within the ferrite in S45C steel which has a clear banded structure.

Local deformation and crack initiation in torsional fatigue of carbon steel with clear-handed structure Cracks initiated near grain boundaries. Figure 10 shows the changes of surface state in the torsional fatigue test of S45C steel. This is an example of the crack initiated near a grain boundary. Figure 10(a) shows optical micrographs (X 200). In the early stage of stress cycling ('^O.OlNf ,Nf: fatigue life), the long and narrow shadow, which will later become a crack, already appears near the grain boundary. Its darkness increases with increasing number of cycles N without increase in its size, and it develops later into a long crack. As is stated above, this fatigue process can be divided into two stages, crack initiation and crack propagation. Figure 10(b) shows the same region observed by scanning electron microscopy. At first, the region of the ferrite crystal grain which later becomes a crack (the long and narrow region that is white in color) slips. Then the width of the slip band and the extent of disruption increase with increasing N. Due to the large deformation on one side of the line, shown by the arrow in the figure, the deformation by slip is quite large in spite of high-cycle fatigue. Judging from the deformation, it seems that the region becomes more active due to work softening after work hardening by slip. On the other hand, the region near the pearlite side is scarcely deformed. Therefore, because of the compensation of deformation, large strains are concentrated near the grain boundary between pearlite and ferrite. Consequently, the cracks apparently appear near the grain boundary.

296

T. FUKUDA AND H. NISITANI

From the above observation, it can be said that this type of crack initiation in the region near the grain boundary is due to the nonconformity of deformation between peariite and ferrite. Cracks initiated in ferrite crystal grains. Figure 11 shows an example of a crack initiated in the ferrite crystal grain of the same specimen. In the optical micrographs [Fig. 11(a)], the entire region of stratiform ferrite areas sandwiched between peariite colonies with a width of about 10 M m is greatly damaged and dark. According to the observation of electron micrographs [Fig. 11(b)], only the ferrite zones are greatly damaged, and the first crack appears in the most damaged crystal grain. This is also confirmed from the fact that the central line in the ferrite is greatly and continuously deformed on one side. Judging from the inclination of the line, it seems that the shear strain of the region amounts to several hundred %. However, the macroscopic shear strain corresponding to nominal stress is about 0.2% (y ). As in the case of deformation of an elastic body sandwiched between rigid bodies, the strain is concentrated in the layer of ferrite bands sandwiched

\ Axial direction (a) N=0 (N/Nf)

N=0.5 X 10^ (0.011)

Optical micrograph N=2X10^ (0.044)

N=3X10^ (0.066)

N=9X10^ (0.198)

(b) Electron micrograph Fig. 10. Crack near a grain boundary in torsional fatigue (S45C)

1 1 20/im N=22X10' (0.484)

10/i m

The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in .

297

between pearlite colonies. As can be deduced from the above observations, this type of crack initiation is due to the fact that the entire region of ferrite suffers large amounts of concentrated cyclic strain caused by the nonconformity in the deformation between the nondeformed pearlite and deformed ferrite areas. It seems that the slipped portion in the ferrite is softened after some stress repetitions, and then the slip concentrated there initiates cracks. The cracks are not necessarily initiated near the grain boundary. In the area where the cracks are initiated, several hundred % local shear strain is confirmed, which accumulated on one side. The circumstance is similar to the case as is stated above. For the reasons mentioned above, almost all the cracks were initiated and propagated first in the axial direction in this material. The reason for this is that this material has a clear-banded structure of ferrite and pearlite, that is, the ferrite and pearlite grains of the specimens are distributed in layers parallel to the axial direction, as is observed in Table 4. The ferrite zones sandwiched between pearlite colonies become active as a whole. Consequently, the crack is

(a) Optical micrograph N=0 (N/Nf)

N=0.5 X lO'* (0.011)

N=2X10 (0.044)

N=3 X lO'* (0.066)

20 Aim

N=9X10^ (0.198)

N=22X10^ (0.484)

(b) Electron micrograph Fig. 11. Crack in the ferrite crystal grain in torsional fatigue (S45C)

\Qjim

298

T. FUKUDA AND H. NISITANI

easily initiated in the ferrite grain in torsional fatigue of carbon steel which has a clear-banded structure.

Fatigue test The S-N curves of all specimens are shown in Fig. 12. Table 4 shows the fatigue limits a^and Tw and the fatigue limit ratios of torsional fatigue to rotating bending fatigue (^Jo^) of all materials. In anisotropic materials which have a clear-banded structure (S20C, S45C), the ratios are 0.55^0.57, and in isotropic materials which have no banded-structure (SC450, S45C-DA, S45C-H), the ratios are 0.65^0.68. As is mentioned above, in the torsional fatigue of carbon steel, which has a clear-banded structure, the large local strain concentration occurs in the ferrite. On the other hand, in the rotating bending fatigue of the same steel, the local strain concentration hardly occurs in the ferrite band. This produces a relative reduction of the torsional fatigue limit. Namely, the fatigue limit ratio ('Ojo,^) depends on whether the specimen has a clear-banded structure or not, and does not follow the macroscopic criterion for plastic yielding (Mises' criterion; TJO^=0.5S). The fatigue behavior of a specimen is related to the local microstructural details of the specimen. On the other hand, the macroscopic yielding is related to the average properties of the specimen. This is the reason why the criterion for macroscopic yielding is not applicable to fatigue loading under combined stresses in general. In discussing the results of fatigue tests under combined stresses (bending and torsion), it is important to consider whether the material used has a banded structure or not.

Classification of fatigue limit ratio Tj o^ The fatigue limit ratio zj a^ varies between 0.5 and 1 depending on the material as is shown in Table 1. The fatigue process consists of crack initiation and crack propagation. Therefore, by considering crack initiation, crack propagation, microstructure and type of defect, the value of fatigue limit ratio can be classified as is shown in Table 5.

CONCLUSIONS (1) The local deformations are not uniformly distributed in both static tension and static torsion, and therefore most of local strains are not the same as the applied macroscopic strain. (2) A clear-banded structure greatly affects the local strain in static torsion but it hardly affects that in static tension. That is, large shear strain concentration in torsion occurs in the ferrite band sandwiched between pearlite bands but the mean normal strain in tension in a ferrite band is nearly equal to that of a pearlite band. (3) Because of the above conclusion, the fatigue cracks of carbon steel with a clear-banded structure are initiated more easily in torsional fatigue than in bending fatigue. That is the reason why the fatigue limit ratio of the steel is close to 0.58 (Mises' criterion). (4) The fatigue limit ratio ( x j a^) depends on whether the specimen has a clear-banded structure or not, and does not follow the macroscopic criterion of yielding. In discussing the results of fatigue tests under combined stresses (bending and torsion), it is important to consider whether the material used has a banded structure or not.

The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in . 299 500

ouu

S20C

(0 0.

§. 4 0 0 | -

S45C

400

^^"•QBending

Bending

300

300 h

MPa a^=235

0

(&

0 •o

1 200 h

"5. E

Q.

E CO

MPa =245

200 • U Torsion MPa 1 -Cw=135

CO CO

0)

CO

0)

100 10'

1

inn 10^

10^

10^

10

lO''

10°

1 10'

10"

N u m b e r of cycles N

N u m b e r of cycles N

(a) S20C

(b) S45C 500

500 ^

1

10^

S45C-DA

SC450

400 h

^

400 Bending

300

300 h

MPa 0^=240

Bending 0

0

1 200 h

1 200

Q.

Q.

E

MPa| rTv,=i55

E

CO

CO

Torsion

U) CO 0

0) 100"— 10^

CO

10"

10"

10^

lO''

100 10'

Number of cycles N

J-

_L

10"^

10°

(d) S45C-DA

(c) SC450 500 ^^Vi

Bending

r-

MPa

S. 400 h 300 h

"

MPa T^=255

^ Torsion

i

10'

Number of cycles N

•fc

200

Q.

E CO

S45C-H

CO 100 10^

L_ lO''

_L

J_

lO''

10'

N u m b e r of cycles N

(e) S45C-H Fig. 12. S-N curves

10°

10'

300

T. FUKUDA AND K NISITANI

Table 4. Fatigue limits

REFERENCES 1. Nishihara, T. and Kawamoto, M. (1940), 'The Fatigue Test of Steel under Combined Bending and Torsion", Trans. Jpn. Soc. Mech. Eng., Vol.6, No.24, 1. 2. Nishihara, T. and Kawamoto, M. (1941), "The Strength of Metals under Combined Alternating Bending and Torsion", Trans. Jpn. Soc. Mech. Eng., Vol.7, No.29, 85-95. 3. Gough, H.J. (1949), "Engineering Steel under Combined Cyclic and Static Stresses", Proc. Inst. Mech. Eng., Vol. 160-4, 417.

The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in .

301

Table 5. Classification of fatigue limit ratios i J o, Characteristics of material Fatigue Crack limit does initiates not depend from on defects Fatigue Shape limit of depends on defect defects

4.

Point region

Work softening

'^ w/ ^ w

0.5

Example material Age-hardened Al-alloy

Banded structure

0.55--0.6

Rolled carbon steel

Isotropic

0.65-0.7

Cast carbon steel

Single hole

0.75

Steel with a hole

Connected hole

0.9

Nodular cast iron

Finite region

Crack-like defect

1

Gray cast iron

Findley, W.N. (1957), "Fatigue of Metals Under Combination of Stresses", Trans. ASME, Vol.79-6,1337. 5. Firth, P.H. (1956), "Fatigue of Wrought High-Tensile Alloy Steels", Int. Conf Fatigue, Inst. Mech. Eng., 462-499. 6. McDiarmid, D.L (1991), "A General Criterion for High Cycle Multiaxial Fatigue Failure", Fatigue Fract. Eng. Mater. Struct., Vol. 14-4, 429-453. 7. Findley, W.N. (1956), "Theory for Combined Bending and Torsion Fatigue with Data for SAE4340 Steel",/«r. Conf Fatigue Metal, Inst. Mech. Eng., 150-157. 8. Matake, T. (1976), "A Consideration for Fatigue Limit under Combined Stresses", Trans. Jpn. Soc. Mech. Eng., Vol.42, No.359, 1947-1953. 9. Peterson, R.E. (1956), "Torsion and Tension Relations for Slip and Fatigue", Colloquium on Fatigue, International Union of Theoretical and Applied Mechanics, 186-195. 10. Nisitani, H. and Fukuda, T. (1994), "Non-Uniformity of Local Strain Concentration in Static Deformation of Plain Specimens of Rolled Round Carbon Steel Bars", Proc.4^^ ISOPE, 222-227. 11. JSME, (19S2)JSMEData Book, Fatigue of Material 1, 16-73. 12. Nisijima. S. (1977), National Research Institute for Metals Fatigue Data Sheets, Vol.19, 119, 227. 13. Nisitani, H. and Kawano, K. (1972), "Correlation between the Fatigue Limit of a Material with Defects and Its Non-Propagating Crack", Bull. JSME, Vol.15, No.82, 433-438. 14. Nisitani, H. and Murakami, Y. (1973), "Part of Spheroidal Graphite of Nodular Iron Casting under Bending or Torsional Fatigue", Research of Machine, Vol.25, No.4, 543-546. 15. Nisitani, H. and Goto, T. (1976), "Notch Sensitivity in Fatigue of an Al-Alloy", Trans. Jpn. Soc. Mech. Eng., Vol.42, No.361, 2666-2672. 16. Forsyth, P.J.E. (1963), "Fatigue Damage and Crack Growth in Aluminium Alloys", Acta Metallurggica, Vol.11, 703-715. 17. Nisitani, H. (1985), "Behavior of Small Cracks in Fatigue and Relating Phenomena", Current Research on Fatigue Cracks, Jpn. Soc. Mat. Sci., 1-22. 18. Nisitani, H. and Takao, K. (1974), "Successive Observations of Fatigue Process in Carbon Steel, 7:3-Brass and Al-Alloy by Electron Microscope", Trans. Jpn. Soc. Mech. Eng., Vol.40, No.340, 3254-3266. 19. Nisitani, H. (1968), "Fatigue under Two-Step Loading in Electropolished Specimen of

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T. FUKUDA AND H. NISITANI

S25C Steel", Trans. Jpn, Soc. Mech. Eng., Vol.34, No.258, 220-223. 20. Gough, H.J. and Pollard, H.V., (1935), "The Strength of Metals under Combined Alternating Stresses", Proc. Inst. Mech. Eng., Vol.131, 3-103. 21. Fukuda, T., Kawasue, K. and Nisitani, H. (1999), "Quantitative Measurement of Local Deformation using an Image Correlation Technique", Proc. Eighth International Conference on the Mechanical Behaviour of Materials, Vol.2, 471-476.

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

303

INFLUENCE OF DEFECTS ON FATIGUE LIFE OF ALUMINIUM PRESSURE DIECASTINGS

Fernando Jorge LINO\ Rui Jorge NETO^ Alfredo OLIVEIRA^ and Fernando Manuel Femandes de OLIVEIRA* Faculdade de Engenhaha, Universidade do Porto, Departamento de Engenharia Mecdnica e Gestdo Industrial Rua Dr. Roberto Frias, 4200-465 Porto, Portugal ^INEGI, Instituto de Engenharia Mecdnica e Gestdo Industrial, Rua do Barroco, 174-214, 4465-591 Lega do Balio, Porto, Portugal

ABSTRACT Fatigue life of aluminium pressure diecastings is strongly dependent on the microporosity level of the parts. Even in a very controlled production process, it is almost impossible to obtain aluminium parts w^ithout micropores, which means that a considerable amount of parts are rejected, in accordance to internal companies criteria. Although these criteria are based on standards, they change from company to company, and depend on the type of the parts and the amount, size and location of the micropores. Many times, parts that could have a good fatigue life are rejected based on these criteria. The aim of the present work is to study the influence of the microporosity level and size on the fatigue life of aluminium pressure diecastings. Two different lots of samples, removed from aluminium components (considered unacceptable and acceptable) were tested using the staircase fatigue test. All the fractured parts were analysed macro and microscopically and the images obtained were digitalized in order to classify the size and amount of the micropores. The results obtained were compared with the fatigue life curves, in order to evaluate the influence of the microporosity on the fatigue life of aluminium components. KEYWORDS Fatigue life, manufacturing defects, aluminium, staircase fatigue test, diecasting.

INTRODUCTION One of today's greatest challenges in the foundry industry is the production of complex and structurally sound parts. At the forefront of this challenge is the porosity and inclusions size and levels, and how and where they develop. The presence, even in small levels, of these types of defects in pressure diecastings can lead to a significant reduction in the tensile strength, ductility, pressure tightness and fatigue life, affecting the life and the integrity of the cast parts [1].

304

FJ.LINOETAL

Three factors can lead to the presence of porosity: shrinkage, coupled with a lack of interdendritic feeding during mushy zone solidification, evolution of hydrogen gas bubbles due to a sudden decrease in hydrogen solubility during solidification, and collapsed air [1, 2]. Aluminium diecasting alloys present very interesting properties, namely: good machinability, low weight, low transformation cost with the possibility of obtaining complex shapes, and, moreover, they are recyclable. However, they are very prone to present casting defects. Although the recent developments in pressure diecasting industry (use of low injection velocities, special feeding [2, 3], vacuum and "true isostatic pressure" [1]) and the use of simulation processes contributed to the improvement of the aluminium cast parts quality (possibility of structural parts production), it is almost impossible to avoid the presence of defects in the parts that are supplied to the customers [4-6]. Considering this, foundry companies follow international standard criteria (for example, ASTM Standard E 505 [7]) and also develop internal standards to classify the parts as acceptable or unacceptable. Frequently, a location or size of one defect is not critical in one part, but is unacceptable in other type of parts. This is especially important in structural parts, where the loading type during service can conduct to fatigue initiation. Pressure aluminium diecasting alloys have extremely low ductility (1-3%) [4, 8], which means that once initiated, cracks easily propagate until the failure of the component [9]. In the light of the above, it is obvious the interest, emphasized by current studies [1, 5, 10], in evaluating the effect of the presence of defects on fatigue life reduction of pressure diecastings. For example, some authors have been trying to determine the relationship between casting conditions and the amount of porosity in a casting. The majority of the models capable of providing a qualitative description of the level of microporosity fail to give accurate values because the prediction of microporosity requires a detailed understanding of pore nucleation and growth in the melt [1]. In the aluminium pressure diecasting industry, the main type of defects susceptible of being observed are oxidized surfaces, foreign material inclusions (oxide), gas cavities (hydrogen and gas porosity), macro shrinkage (cavities and sponginess), and microshrinkage (feathery, sponge, intercrystalline or interdendritic) [6, 11]. Gas pores, shrinkage pores and gas-shrinkage pores represent the main types of porosity that are detected in diecastings. In gas pores, liquid aluminium reacts with water vapour in the atmosphere to produce aluminium oxide and hydrogen gas. Gas porosity arises during the solidification, due to the difference in solubility of hydrogen gas in liquid and solid aluminium. If a casting is poorly fed during solidification, shrinkage will cause a hydrostatic stress in the liquid metal. This stress increases until a pore forms with the aid of a nucleus. Gas-shrinkage pores results from the fact that gas evolution and shrinkage occur in the same volume of liquid metal at the same time [1]. In this paper, the effect of the presence of porosity on fatigue life reduction of aluminium pressure diecastings is quantified. Such an effect will then be used in a software, under development, which will be able to predict fatigue life of high strength castings [12]. One of the advantageous of this study is the fact that all the fatigue tests are conducted in samples removed from real components or in real components and not in samples pressure diecasted separately in the metallic moulds. These last samples frequently have a section thickness that does not represent the usual parts thickness obtained in pressure diecasting.

Influence of Defects on Fatigue Life of Aluminium Pressure Diecastings

305

EXPERIMENTAL ANALYSIS OF DEFECTS To study the influence of the presence of defects on fatigue life of aluminium pressure diecastings, two lots of 50 parts each, considered acceptable and unacceptable, were supplied by a foundry company (SONAFI, Porto, Portugal). The selection was made in accordance to the foundry internal criteria for the part selected, and using adequate quality control techniques (visual control, X-Rays, etc.). Figure 1 presents the component selected, with the fatigue test sample placed in the position from where it was removed.

Fig. 1. Brake pedal in an aluminium diecasting alloy AS9U3 (NF A57-703) [8] with the fatigue test sample. Figure 2 shows an image of the X-Rays control performed in one pedal. As one can see, different defects sizes can be detected in this control.

Fig. 2. X-Rays control in a brake pedal. The central part of the pedal shows pores with different sizes and geometries. The fatigue test sample was removed from region A.

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FJ. LINOETAL

All the parts were pressure diecasted in the AS9U3 (NF A57-703) [8] aluminium alloy. The typical microstructure of this alloy consists essentially in a aluminium dendrites, aluminiumsilicon eutectic cells, and intermetallics AbCu, Al7Cu2Fe and AlFeSiMg [6, 11] (see Fig. 3). Table 1 presents the alloy chemical composition specified by the Standard NF A57-703 and the medium values measured in the components using a spark spectrometer. This table shows that there are no significant differences between the two compositions.

Fig. 3. Typical microstructure of the pressure diecasting AS9U3 alloy (etched with HF 0.5%). This microstructure is composed by a aluminium dendrites, aluminium-silicon eutectic cells, and intermetallics A^Cu, AlyCuiFe and AlFeSiMg.

Table 1. Chemical composition (%) of the aluminium alloy AS9U3 (NF A57-703) [8].

NF A 57-703

Fe

Si

Cu

Zn

Mg

Mn

Ni

Pb

Sn

Ti

1.3

7.5-10

2.5-4.0

A-1.2

0.3

0.5

0.5

0.2

0.2

0.2

0.2

0.23

0.05

0.15

0.04

0.02

B-1.3 Measured in

0.78

9.24

2.99

0.81

the components

Figure 4 presents the cross section of an optical micrograph showing interdendritic (region A) and gas cavities (region B) regions, and also a macrograph of a transversal section of an unacceptable pedal, where these defects can also be observed. As one can see, pressure diecastings present a significant amount of micropores. This fact is responsible for considerable research in the foundry area, in order to produce aluminium diecastings with better characteristics [1,5, 13-15].

Influence of Defects on Fatigue Life of Aluminium Pressure Diecastings

307

Fig. 4. Porosity in pressure diecastings: a) optical micrograph indicating a region A of interdendritic cavities and a region B of gas cavities, and b) macrograph of a polished transversal section of an unacceptable pedal showing small and large pores.

Table 2 presents the main properties of the aluminium alloy AS9U3 (NF A57-702/703) in the non heat-treated condition [8]. Table 2. Properties of the AS9U3 (NF A57-703) pressure diecasting alloy [8]. Alloy Properties Gr (MPa)

Elongation (%) Density (g/cc)

Value 200 0.5-1.5 2.8

Image analysis was performed with software called PAQUI (developed by the Centre of Materials of University of Porto - CEMUP, Porto, Portugal) in an Olympus optical microscope. The analyses were done in 10 brake pedals (5 acceptable and 5 unacceptable), in the transversal section of the fatigue test samples region. A routine to perform a specific analysis was defined in order to count and measure the defects (considered spherical) in the samples section. 50 to 80fieldswere characterised in each sample, using an optical microscope with a 5x magnification. Very small defects, with diameter <50 jam, were counted but not measured in order to reduce the amount of data to be treated. The mean reletive defects area was obtained dividing the total defects area by the total fields area. Figure 5 presents one of the fields analysed in an unacceptable sample. Figure 6 presents the mean number of defects detected in each field, classified in five diameter (jim) classes. As expected, unacceptable parts have higher amount of defects in all classes.

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FJ.LINOETAL

Fig. 5. Optical micrograph of one field analysed in an unacceptable sample showing a large amount of defects and a large pore with a diameter around 5 jim.

110-50 • 50-100 D100-150 [3150-200 • >200

Acceptable

Unacceptable

Fig. 6. Mean number of defects detected in each field analysed. The defects are divided in five area (|Lim) classes for acceptable and unacceptable samples. The mean reletive defects area obtained in acceptable parts was 1.9%, while for unacceptable ones was 2.9%. This is a very small difference between the two lots, which can be explained by the fact that frequently a single defect is the reason for a part rejection.

FATIGUE TESTS One fatigue sample was removed from each pedal by EDM and machined for the following dimensions, 90 mm length, 20 mm width and 3 mm thickness. The geometry of the sample was defined according to the ASTM Standard E 466 [16] and is represented in Fig. 7.

Influence of Defects on Fatigue Life of Aluminium Pressure Diecastings

309

Fig. 7. Fatigue test sample based on the ASTM Standard E 466 [16]. Two tensile tests were performed in both sets of samples in order to determine the main mechanical properties of the AS9U3 aluminium alloy. The geometry of the sample for the tensile test was modified in accordance with Standard E 8M-89b [17]. A constant cross section of 10 mm and 3 mm thickness was adopted. The mechanical properties obtained were very similar for acceptable and unacceptable samples. Considering this, the values indicated on Tab. 3 represent the medium value obtained from the 4 tensile tests performed in both acceptable and unacceptable samples. These values are similar with the ones specified by the Standard (NF A57-703) [8] (see Tab. 2), which can be the result of the natural defects presence in pressure diecastings. Density values, obtained from three samples of each set, are also lower due to the presence of micropores in the components and due to the slight difference in the alloy chemical composition. This table also includes the hardness values, which are equal for both sets of samples.

Table 3. Measured properties of the aluminium alloy AS9U3 (NF A57-703). Main Properties Gr (MPa) ao,2 (MPa) E (GPa) Elongation (%) Density (g/cc) Acceptable/Unacceptable Hardness (HB)

Value 204 148 64 1 2.70/2.72 93

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Fatigue tests were performed in a 250 kN Servo-Hydraulic MTS machine (see Fig. 8), and using the staircase method (DIN EN ISO 9964 [14]). The loading ratio aMin/c^Max selected for the tests was 0.1, and tests started with 50 % of the aluminium alloy tensile strength. Figure 9 presents one pedal and some broken samples.

Fig. 8. Servo-hydraulic MTS machine for the fatigue tests.

Fig. 9. Brake pedal and broken fatigue aluminium samples. To start the staircase fatigue test [16, 18], it was estimated that the fatigue limit was 50% of the tensile strength. The number of levels used for each set of 20 samples was 7. Tables 4 and 5 indicate the values employed in the test and the resuhs obtained, respectively. Surprisingly, the unacceptable samples only have a small decrease in the fatigue limit;

311

Influence of Defects on Fatigue Life of Aluminium Pressure Diecastings

however, they present a slightly higher standard deviation. Figures 10 and 11 present the Woehler curves obtained (in accordance with the ASTM Standard E 468 [19]) using these fatigue limits and the best fit of fatigue tests with three different levels of alternating stress, using 5 samples for each level. Table 4. Staircase fatigue test parameters. Staircase Method Value Estimated fatigue limit 102 50 % Tensile Strength (MPa) First applied range of stress (MPa) 103 Frequency of the test (Hz) 20 Number of samples 20 7 Number of levels 0.1 CTMin/c^Max Table 5. Fatigue test results. Data Fatigue limit (MPa) Standard deviation

Acceptable | Unacceptable 87 91 5 7

Curve S-N

a>

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Fig. 10. Woehler curve for acceptable samples.

1000000

10000000

312

F.J. LINO ET AL

10000000

Fig. 11. Woehler curve for unacceptable samples. As one can see, unacceptable samples have a much broader number of cycles-to-failure distribution (correlation coefficient R^ equal to 0.25) for the high alternating stress regime, which means that the defects (essentially micropores) have a considerable effect on fatigue life reduction for high alternating stresses. All the fatigue fracture surfaces were analysed in an Olympus stereograph microscope, connected to a PC, and defects bigger than 50 |am classified by manual detection with a DPSOFT software program (version 3.0 from Olympus). Figures 12 and 13 show the fracture surface of two unacceptable samples. During the defects detection, for counting and area measurement, it was necessary to frequently compare the image in the computer monitor and in the microscope, since the image in the computer was not three dimensional, which caused difficulties in the defects characterisation.

Fig. 12. Macrograph of a fracture surface of an unacceptable sample presenting a large pore.

Influence of Defects on Fatigue Life of Aluminium Pressure Diecastings

313

Fig. 13. Stereographic picture of the fracture surface of an unacceptable sample. Some of the fracture surfaces were analysed in a SEM microscope. Figures 14 and 15 present some of the images obtained characterised by a brittle facture. In fact, Fig. 14 shows very clearly this type of fracture. However, small regions with some deformation were also detected. Figure 14 b) shows a pore interior with a dendrite and small deformation. Figure 15 shows other fracture surfaces presenting some regions of small deformation and large fragile intermetallic plates.

Fig. 14. Fracture surface, a) the fracture surface presents regions of small deformation, b) image inside a pore with a dendrite and small deformation. Figure 16 presents, for both the acceptable and unacceptable sets, the frequency of defects classified in seven classes of mean pore size diameters. This figure shows the presence of defects with diameter higher than 250 jim and higher amounts of large size defects in the fracture surfaces of unacceptable samples, which means that they play an important role in the fatigue life of these parts.

F.J. UNO ET AL

314

Fig. 15. Fracture surface, a) the fracture surface presents regions of small deformation, b) some deformation with large fragile plates between the aluminium.

Figures 17 and 18 present the area percentage of pores detected in the fracture surface divided by the total planar fracture surface area (sample reference names are reported along the X axis). As one can see, the pore surface area in the fracture surfaces of both sets of samples is very similar, around 2.7% for the unacceptable samples and approximately 2.2% for acceptable ones. These values are similar to the ones obtained in the cross sections of the pedals, which were presented previously: 2.9% for unacceptable samples and 1.9% for acceptable samples.

Classes (^m)

a) Fig. 16. Frequency of defects divided in seven classes of mean pore sizes diameters for; a) acceptable samples, and b) unacceptable samples.

Influence of Defects on Fatigue Life of Aluminium Pressure Diecastings

^ (0

£

(0

a>

i "
3

315

5,0 4,b 4,0 3,5 3,0

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fe.

5s^;

2,5

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so

2.0

1 £ ! £ o 0.

1,0

tn

1.5

1 m

0,5 0,0

lU 1 <<<<<<<<<<<<<<< Unacceptable samples

1i

Figure 17. Percentage of pore area with respect to planar fracture surface area, and mean value for unacceptable samples.

Acceptable samples

Figure 18. Percentage of pore area with respect to planar fracture surface area, and mean value for acceptable samples.

316

F.J. LINO ETAL

Figure 19 presents the mean, maximum and minimum defect diameters for the two sets of aluminium samples. This figure confirms that even though the mean value is very similar for both sets (around 150-170 |Lim), unacceptable samples have a much broader distribution.

600

• MaximumI value • Mean value A Minimum value

Acceptable

Unacceptable

Fig. 19. Mean, maximum and minimum defect diameters for the two sets (acceptable and unacceptable) of broken fatigue samples. The data obtained with the pore area measurements and the fatigue test results can be joined in a single graph. The result obtained is indicated in Fig. 20, which presents the effect of the stress amplitude (upper curve) on the number of fatigue cycles, A^, for all tested samples. The graph also indicates the defects mean diameter (with the standard deviation) of each sample, as a function of the number of cycles to failure. As expected, large pore diameters and large standard deviations are associated with shorter fatigue lives. Using the data of this figure and a multiple linear regression analysis, it is possible to establish an equation (with a correlation coefficient of 0.43) that relates the number of fatigue cycles with the applied stress and the mean defect diameter of the components: A^ = 6217357-446440--3109^,

(1)

where, A^ is the number of cycles to failure, G is the stress amplitude (MPa), and cj)^ the defect mean diameter (|am). It should be emphasized that although the correlation coefficient obtained for this equation is low, it can be considered an acceptable value for a multiple linear regression. A more suitable equation (with a correlation coefficient of 0.45) can be obtained if the mean diameter is replaced by the maximum defect size (^^^^) detected: N = 6232263 - 45314(j - 825^,

(2)

Influence of Defects on Fatigue Life of Aluminium Pressure Diecastings

317

This equation represents a good tool for fatigue life prediction of diecastings. In fact, a diecasting company, using a non-destructive test, like the X-Rays control, can detect the maximum defect of one component. Considering the size of the detected defect and the estimated service stress level of the component, it is possible to estimate the number of the cycles that the component can withstand until breaking.

1000

150

05 CL

t

^100 CD

"Q.

E < S 50 C/)

i I nmf IT

100

1000

10

10

10

10'

Number of Cycles Fig. 20. Effect of the stress amplitude (upper curve) and mean pore diameter (lower curve) on the number of cycles of all samples. The mean diameter of each sample is indicated with the standard deviation. The results achieved represent the first part of a more complete study about the influence of defects on the fatigue life of diecasting components. The second part of the work will include fatigue tests in the aluminium rocking arm of the Fig. 21. Using the Solid Works software, the displacements and the equivalent stress levels will be calculated by the finite element method (FEM). This allows the localization of the regions, which will preferentially suffer fatigue fracture. Using the X-Rays analysis, all large size defects present in the parts will be identified and the parts separated in different lots, characterized by the maximum defect size and location. The results of the fatigue tests will then be compared with the ones just presented. It is expected that the conclusions of the study can supply interesting data for the software under development.

318

F.J. LINO ET AL

CONCLUSIONS Fatigue life of aluminium pressure diecastings is influenced by the size and amount of defects (essentially pores). The mean level of pores detected in the aluminium brake pedals, considered acceptable and unacceptable, was 1.9% and 2.9%, respectively. The comparison of acceptable and unacceptable lots of parts, have shown similar fatigue limits (91 and 87 MPa), however with a slightly larger standard deviation for the unacceptable samples. The amount of pores in the fracture surface, 2.2% for acceptable samples and 2.1% for unacceptable ones, was very similar to the one obtained in the cross section of the brake pedals, 1.9% for acceptable and 2.9% for unacceptable ones. The Woehler curves obtained have shown that for high alternating stresses, defects size have a significant effect on fatigue life reduction. Combining the results obtained with the image analysis and the fatigue tests it was possible to develop an equation that relates simultaneously the stress amplitude applied to an aluminium component and the maximum defect size detected in the non-destructive control. The equation developed in this study represent one first rough tool for fatigue life prediction of pressure diecastings. It is expected that more fatigue tests, to be performed soon in real components, will improve the reliability of the model presented here.

Fig. 21. Aluminium rocking arm.

REFERENCES 1. 2. 3.

Spada, A. T. (1999). Aluminium Casters Discuss Porosity, Melt Quality, Modem Casting. 5th International AFS Conference on Molten Aluminium Processing, 58-60. Nguyen, T., and Carrig J. (1986). Water Analogue Studies of Gravity Tilt Casting Copper Alloy Components, AFS Transactions 94, MEM 92, 519-528. Nyamekye, K., An, Y. K., Bain, R., Cunningham, M., Askeland, D., Ramsay, C. (1994). Expert System for Designing Gating System for Permanent Mould Tilt-Pour Casting Process. AFS Transactions 150, 127-131.

Influence of Defects on Fatigue Life of Aluminium Pressure Diecastings

4. 5.

6. 7. 8. 9. 10.

11. 12.

13. 14.

15. 16. 17. 18. 19.

319

ASM INTERNATIONAL (1992). Casting, ASM Handbook, Metals Handbook, Formerly Ninth Edition, Volume 15, ASM International, Materials Park, Ohio, U.S.A. Tsumagari, N., Brevick, J. R., Mobley, C. E. (1998). Control of Microstructures in Aluminium Diecastings, AFS Transactions, 15-20, Milwaukee, Wisconsin, Ohio State University, Columbus. Backerud, L., Chai, G. and Tamminen, J. (1990). AFS/SKANALUMINIUM: Solidification Characteristics of Aluminium Alloys, Volume 2, Foundry Alloys. ASTM (1996). E 505, Reference Radiographs for Inspection of Aluminium and Magnesium Die Castings, Philadelphia, U.S.A. NF (1970). A57- 703, Produits de Fonderie, Pieces Moulees Sous Pression en Aluminium ou enAlliages D'alluminium, 487-505. Hertzberg, R. W. (1989). Deformation and Fracture Mechanics of Engineering Metals, John Wiley & Sons, Inc., USA. Viswanathan, S., Duncan, A. J., Sabau, A. S., Han, Q., Porter, W. D. and Riemer, B. W. (1998). Modeling of Solidification and Porosity in Aluminum Alloy Castings, AFS Transactions, 411-417, ORNL, Oak Ridge Tennessee. Russo, E. (1993). EDIMET: The Atlas of Microstructures of Aluminium Casting Alloys, Italy. Lino, F. J. (2000). Final Report of the CRAFT Project Darcast (BRST-CT98-5328, Project N° BES2-5637, Coordinated by CIMNE, Barcelona, Spain), INEGI (Institute of Mechanical Engineering and Industrial Management), Porto, Portugal. Kim, S. B., Yeom, K. D., and Hong C. P. (1997). Cast Metals 10. Dighe, M. D., Jiang, X. G., Tewari, A., Rahardjo, A. S. B., Gokhale, A. M. (1998). Quantitative Microstructural Analysis of Microporosity in Cast A356 Aluminium Alloy, AFS Transactions 13, 181-190. Klansky, J. (1999). Image Analysis of Aluminum Alloys, Advanced Materials & Processes 7, 23-26. ASTM (1982). E 466, Standard Practice for Conducting Constant Amplitude Axial Fatigue Tests of Metallic Materials, 465-469, Philadelphia, USA. ASTM (1989). E 8M-89b, Standard Test Methods for Tension Testing of Metallic Materials [Metric], 146-160, Philadelphia, USA. ADHESIVES (1995). DIN EN ISO 9664, The Methods for Fatigue Properties of Structural Adhesives in Tensile Shear. ASTM (1990). E 468, Standard Practice for Presentation of Constant Amplitude Fatigue Test Results for Metallic Materials, 475-480, Philadelphia, USA.

Appendix: NOMENCLATURE

E

Young's modulus

N

Number of cycles

R'

Correlation coefficient

a

Stress amplitude

CTr

Tensile strength

^0,2

Yield strength

320

F.J. LINO ET AL. GMax

M a x i m u m stress

QMin

M i n i m u m stress

^„, (b .

Mean defect diameter Maximum defect diameter

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

321

VARIABILITY IN FATIGUE LIVES: AN EFFECT OF THE ELASTIC ANISOTROPY OF GRAINS?

Sylvie POMMIER MSSMat, Ecole Centrale Paris, Grande Vote des Vignes, 92295 Chatenay Malabry Cedex

ABSTRACT Probabilistic approaches are developed to describe scale effects and scatter in fatigue. When fatigue cracks are nucleated on defects, four main sources of scatter are usually acknowledged: the probability to find a defect per unit volume, the variability of the defects sizes, of their distance to the surfaces and of their mutual distance. When fatigue cracks are not nucleated on defects, the mechanisms at the origin of the scatter remain unclear. In a "defect free" metal under bulk elastic conditions, crack nucleation is attributed to cyclic slip in "w^eak" grains. Therefore, the variability of grain's size and orientation is expected to be firstly responsible for the scatter in fatigue lives. The main material dimension, to take into account for predicting a scale effect, is therefore the grain size. However according to the elastic anisotropy of grains, a material dimension larger than the grain size may prevail. As a matter of fact, using 3D elastic finite element analyses and experiments, it is shown that the spatial distribution of stresses and strains is self-organized in a polycrystal, at a scale larger than the grain size. This scale is approaching for example 15 grains in copper. This effect is analogous to the "arching" effect, widely studied in granular material.

KEYWORDS Fatigue, scale effect, scatter, elasticity, anisotropy, percolation, grains.

INTRODUCTION Life prediction methods for industrial components are usually based on crack nucleation criteria. A vital point for the design of industrial component is to evaluate both the fatigue life of the component and the confidence associated with this prediction. For this purpose, the key parameters driving crack nucleation and the sources of variability have to be clearly identified. On the one hand, most fatigue criteria rely on the assumption that in both LCF and HCF, fatigue damage occurs in grains as a consequence of cyclic slip in "weak" grains or aggregates of grains. The models enabling the transfer of data collected on samples to industrial components are therefore evaluating the slip ability in "weak" grains. Under proportional

322

S.POMMIER

loading, an equivalent shear stress invariant [1] is sufficient. Under non-proportional loading conditions, a critical plane has usually to be determined and the damage is calculated in this plane. The Dang Van criterion [2] belongs to this category. The damage is calculated using a combination of the shear stress on the critical plane and of the hydrostatic pressure. On the other hand, probabilistic approaches that are developed to predict scale effect and scatter in fatigue rely on the assumption that cracks are nucleated on defects. Murakami and coauthors [3] clearly showed that the fatigue limit is decreasing when the defect size is increased and that under a critical dimension defects become non-damaging. Four main sources of scatter are usually acknowledged. The first one is the probability to find a defect within a given volume of material. The others are the variability of the defect's size, of their distance to the surfaces and of their mutual distance [4]. The Dang Van's criterion can be successfully employed even when cracks are initiated on defects [5], since cracks are nucleated by cyclic slip within grains around the defects. The scatter can be introduced in the model through a statistic distribution of the fatigue limit, which is evaluated, for example, from the distribution of the defects sizes, through the widely used yfarea model by Murakami&Endo [6]. When the material is "defect free" or at least when defects are non-damaging, crack nucleation occurs in "weak" grains. This is the case in particular in titanium alloys, in aluminium alloys and in superalloys used by the aeronautic industry, possibly because, on the one hand, the defects sizes are small in these alloys, and because, on the other hand, fatigue crack nucleation in grains is assisted by environmental effects at the surfaces. In any case, when crack nucleation occurs in "weak" grains, the scatter in fatigue lives does not completely disappear and should be evaluated from life prediction methods, which is not the case by now. For example, Susmel&Petrone [7] have found in a 6082 T6 aluminium alloys fatigued under various testing condition a scatter factor equal to +/-2 on the experimental fatigue lives. In their experiments, the surface of material in the gauge length is close to 15700 mml In this case, the authors mention clearly that crack nucleation does not occur on defects. It would be useful to be able to predict the scatter in such cases, and to identify its origins. Concerning crack nucleation in "weak" grains, if an analogy with defects is made the scatter should be controlled by: the volume fraction of "weak" grains, the variability of the size of "weak" grains, of their distance to the surfaces and of their mutual distance. The variability of the grain sizes is easy to determine from micrographs. The effect of the surfaces is clear, since in a "defect-free" material, cracks are almost always nucleated on the surfaces. In order to estimate the volume fraction of "weak" grains and the distance between "weak" grains, we should at first identify clearly which are the "weak" grains. A "weak" grain is usually a grain within which cyclic slip occurs during fatigue cycling. Therefore, under bulk elastic conditions, this is a grain within which the maximum resolved shear stress over all the slip systems is higher than elsewhere. The resolved shear stress is high when two conditions are satisfied: 1- the crystalline orientation of the grain towards the principal stress directions is favourable for slip and 2- the stress applied on the "weak" grain is high. 1- Concerning the crystalline orientation of the grain, if the texture of the material is known, the statistic distribution of the Schmid factor can be calculated (Fig. 1. (a)). This distribution allows determining the probability to find a grain with a high value of the Schmid factor within a given volume of material. The mutual distance between grains with a high Schmid factor could also be determined, for example, using crystal orientation maps as measured from electron back scattering patterns. This information is important to evaluate the probability of micro-cracks coalescence during the first stages of crack growth.

Variability in Fatigue Lives: An Effect of the Elastic Anisotropy of Grains?

323

0.5 Max Schmid Factor S

o- lower than x -•- greater than x

0.25

0

0.5

(a)

(b)

Fig. 1. (a) Statistic distribution of the maximum Schmid factor for a uniaxial stress state and a random orientation of FCC grains. For this calculations the 12 principal FCC slip systems (111)<110> have been taken into account, (b) Illustration of the load percolation effect in sand. 2- Concerning the stress state within the grain, the problem is complex. As a matter of fact, since the mechanical behaviour of grains is usually anisotropic within their elastic domain, the stress distribution is heterogeneous within the polycrystal. First of all, the object of this paper is to identify the scale associated with the heterogeneity of the stress distribution, in order to predict scale effects in fatigue. Since we aim at determining, both the volume fraction of overstressed grains and their mutual distance, the spatial distribution of the overstressed areas within a polycrystal is studied. This problem is a well-known problem in granular materials. As a matter of fact, heterogeneous stress distributions are commonly observed in granular media, such as sand, stone or com heap [8,9]. Savage published recently a state of art in that area [9]. Early experimental observations of the stresses in a 2D media constituted of photoelastic cylinders [10] showed that the loads are transmitted through contacts between grains. A load percolation network, coincident with a percolating network of grains in mutual contact, is formed through the granular media. The links of that network follow the isostatic lines in the equivalent elastic continuum material [10]. The load percolation network is carrying a force larger than the mean one [11, 12]. This effect, so-called "arching effect", is a widely studied problem since it controls the constitutive behaviour of granular materials, such a soils for example. The problem of the percolation of a quantity is usually observed when, on the one hand, the material is heterogeneous and when, on the second hand, the capability of the material to transfer that quantity varies very significantly from one point to another. If we consider a sand for example (Fig. 1. (b)), solid grains are able to transfer the deviatoric part of the stress, while water can only sustain hydrostatic pressure. Therefore, the load percolation network carries all the deviatoric part of the stress [11, 12]. Consequently (Fig. 1. (b)), type I grains located within the load percolation network are subjected to shear, while type II solid grains located out of the load percolation network are only subjected to hydrostatic pressure. Polycrystals are constituted of grains but they differ essentially from granular media since all grains are able to transfer the applied load. However, their rigidity, and as a consequence, the elastic energy stored at a given load level, is varying according to their crystalline

324

S. POMMIER

orientation towards the load direction. In a polycrystal, the crystalline orientation varies significantly and suddenly at grain boundaries. This effect could lead to the formation, in the polycrystal, of a load percolation network analogous to that observed in a granular material. In such a case, the intrinsic scale of the material would be larger than the grain size, since it would be related to the scale of this load percolation network. Experiments and FEM analyses have been conducted in order to check if this scale exists in elastic polycrystals and what would be its importance in fatigue.

EXPERIMENTS A few experiments have been conducted using the photostress technique. When a photoelastically coated sample is subjected to loads, the resulting stresses cause strains to exist over its surface. Because the photoelastic coating is bonded to the surface of the sample, the strains in the sample are transmitted to the coating. The strains in the coating produce proportional optical effects, which appear as isochromatic fringes when viewed with a reflection polariscope.

10 mm

800 MPa

850 MPa

Fig. 2. Observations of the strain at the surface during a tensile test using the photostress technique on a TA6V titanium alloy.

The experiments were conducted on a duplex TA6V, for which sets of a nodules were shown to display the same crystallographic orientation over areas, so-called "macrozones" [13,14], whose diameter is close to one millimetre. The conventional yield stress of this alloy is 850 MPa. In Fig. 2 are displayed pictures taken during a tensile test at various stress levels. At 356 MPa the sample is fully elastic, however it is observed that inclined bands are crossing the specimen. It was checked using micro-strain gauges that these bands do not correspond to plastic strain localisation bands. Moreover, in the points indicated as black dots in Fig. 2., the principal strain directions were determined by photostress analysis. It was found that the local principal strain direction in the inclined bands is aligned with the load axis, showing that the observed inclined bands are not shear bands. At 800 MPa the bands are obvious and located at

Variability in Fatigue Lives: An Effect of the Elastic Anisotropy of Grains?

325

the same place of those at 356 MPa. If the specimen is loaded up to the yield stress (i.e. 850 MPa), the contrast is increased and particularly evident close to the surfaces. Though plastic strain has occurred at 850 MPa, the shape of the patterns observed on the surface is not significantly modified as compared with that observed at 356 MPa. (Note that the greyscale levels does not reflect a modification of the strain distribution. The isochromatic fringes belong all to the first order at 356 MPa, while they belong to two consecutive fringe orders at 850 MPa, which explains the reversion of the greyscale levels though the strain distribution is similar). It can be concluded from these experiments that the scale associated with the strain heterogeneity in the TA6V titanium alloy is larger than the grain size, approaching 10 grains. The following experiments have also been performed, to reveal a possible scale associated with the heterogeneous distribution of strain in a different material. Thin sheets of OFHC polycrystalline copper have been subjected at room temperature both to a monotonic tensile test and to a cyclic creep test. The sample is tested with Gmin =0 and with Cmax increasing slowly by 2.5 MPa every 500 cycles up to failure. The grain size is close to 20 |J.m (Fig. 3. (a)). After a monotonic tensile test, the surface of the sample is rough due to plastic strain. However, it is not possible to distinguish any regular pattern with a scale larger than the grain size on the surface. On the contrary after a cyclic creep test, fine inclined lines forming a regular pattem are observed at the surface of the sample (Fig. 3. (b)), revealing a scale for the heterogeneity of strain larger than one millimetre.

Fig. 3. OFHC polycrystalline copper, cyclically creep tested, with Gmin =0 and with a^ax increasing by 2.5 MPa every 500 cycles up to failure. The observations are performed out of the striction zone, (a) slip lines at the surface of the sample revealing a grain size close to 20 jxm. (b) patterns generated by cyclic creep at the surface of the sample. FINITE ELEMENT ANALYSES Finite elements calculations were performed, in order to understand the above-mentioned effects observed in the experiments and to discuss their importance for fatigue crack nucleation. A polycrystalline thin sheet was modelled by 3D FEM analysis. The grains were modelled as 3D regular hexagons, as shown in Fig. 4. (a). The element's edge lengths are 0.2 x 0.2 x 0.25 mm. The hexagons have radii of 1.2 mm while their thickness is 0.5 mm. The size of the model is 10 mm in width, 20 mm in length and 0.5 mm in thickness. In each hexagon, the crystal

326

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orientation is set to be constant. The boundary conditions consist in imposed displacements. For example, in uniaxial extension, a displacement of the upper side of the model is applied so that the mean strain in the model is <€yy>=0.1%, the bottom side is fixed and the other sides are free. As the problem is fully linear elastic this choice is arbitrary. The elastic constants of the studied crystals are displayed in Table 1. The local orientations (1,2,3) of the hexagons are randomly selected in order to create an isotropic texture. The number of grains per "sample" is of 228 (Fig. 4. (b)).

(3, y)

10 grains

Fig. 4. (a) Mesh of an individual grain by linear tetrahedrons, (b) fmite element model of a thin sheet. The intensity map corresponds to the angle between the direction i, of the local coordinate system attached to the crystal in each grain (1,2,3), with the direction y of the coordinate system of the model (x,y,z) Table 1. Elastic constants (GPa) of the studied crystals. The elastic behaviour of the hexagonal crystals (Zr, Ti and Zn) is modelled by an isotropic transverse elasticity (5 independent elastic constants) and that of the cubic crystals (Al, Fe and Cu) is modelled by a cubic elasticity (3 independent elastic constants). 2C66=Cii-Ci2. Crystal Zirconium Titanium Zinc Aluminium Iron Copper

Cll 144. 162. 165. 107. 231.4 168.4

C12 72.8 92. 31. 60.8 134.6 121.4

C13 65.3 69. 50. 60.8 134.6 121.4

C33 165. 180.7 62. 107. 231.4 168.4

C66 35.6 35.2 67. 28.3 116.4 75.5

C44 32.1 46.7 39.6 28.3 116.4 75.5

Spatial distribution of stress and strain in a poly crystal. In Fig. 5 are displayed iso-contours of the maximum principal stress component in copper for the same model, with the same set of orientations in uniaxial extension, shear and biaxial extension. The local stress is very heterogeneous, which was expected from the high elastic

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anisotropy of the copper crystal (Table 1). The most interesting result is that regular patterns are found in the calculations. Though the distribution of the local orientations was defined to be random, the local stress distribution is definitely not random. In uniaxial extension, it was checked that the direction of the maximum principal stress is close to the vertical axis everywhere in the sample. However, vertical links appear in the model that sustain higher stresses. With the same model, if the mean principal stress directions are rotated, the directions of the links follow the principal stress directions (e.g. see Fig. 5 (b)). This observation is very similar to previous results obtained with granular media [10].

Fig. 5. hitensity maps (MPa) of the maximum principal stress component in the case of copper, (a) uniaxial extension Eyy = 0.1 %, (b) shear strain Yxy = 0.1 %, (c) biaxial extension EXX = Cyy = 0.1 %. The displacements are magnified by a factor 100.

Fig. 6. Intensity maps of the maximum principal strain component (in %) in the case of copper, (a) uniaxial extension Eyy = 0.1 %, (b) shear strain Yxy = 0.1 %, (c) biaxial extension Exx = Cyy = 0.1 %. The displacements are magnified by a factor 100.

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S. POMMIER

In Fig. 6 are displayed iso-contours of the maximum principal strain component in copper for the same model, with the same set of orientations in uniaxial extension, shear and biaxial extension. As in Fig. 5, though the crystalline orientations of grains were defined to be random, the spatial distribution of strain in the model is not random. However, the patterns observed for the strain are different from those observed for the stress. There is roughly a rotation of 45° between the links of the maximum principal stress network and that of the maximum principal strain network. In uniaxial extension for example, the intensity of maximum principal strain is higher within links inclined of about 45° with the y axis, though the maximum principal strain direction is roughly aligned with the y axis. This result is consistent with the experimental observations on TA6V (Fig. 2). Aluminium

Iron

Copper

Zirconium

Tiianiiim

Zinc

Fig. 7. Intensity maps of the maximum principal stress component (in MPa) in uniaxial extension £yy = 0.1 %, (a) aluminium, (b) iron, (c) copper, (d) zirconium, (e) titanium, (f) zinc. The same calculations have been performed for aluminium, iron, copper, zirconium, titanium and zinc. Similar results are obtained. If the same set of orientations is employed, the spatial distribution of the maximum principal stress in the three cubic crystals (aluminium, iron and copper) is very similar (Fig. 7. a,b,c). The only difference is in the intensity of these links, which is higher if the elastic anisotropy of the crystal is increased. For example, while the

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329

relative difference between the maximum principal stress in overstressed and understressed links is lower than 15% in aluminium, it exceeds 70% in copper, h is worth noticing that the length of an overstressed links is approaching 15 grains for example in copper, but there is no proof that it could not be larger if the number of grains in the model was increased. The distribution of the maximum principal stress in zirconium, titanium and zinc is also selforganized (Fig. 7 d,e,f). As for the cubic crystals, the distributions are similar in the three cases expect for the intensity of the load links, which is maximum in the case of zinc. The mechanism leading to the formation of a load percolation network in a polycrystal can be explained as follows. Let consider a particle with a high rigidity embedded within a homogeneous material (Fig. 8.). The material is loaded homogeneously far from the particle. Away from the particle the isostatic lines are aligned with the maximum principal stress direction Z?, but they turn in the vicinity of the particle, which is a stress concentrator. The curvature of the isostatic lines corresponds to a load transfer by shear within the homogeneous material. If, in the planes that contain the principal stress direction /?, the shear modulus of the material is low, the load transfer by shear is inefficient. Consequently, the stress state is disturbed by the particle over a distance /, which is large, hi a polycrystal, the formation of load links is due, on the one hand, to the existence of grains (or defects) with a higher or a lower rigidity related to the principal stress direction than the mean one, and on the other hand, to the inefficiency of the material to homogenize the stress state around that grain. This inefficiency is directly connected to the shear modulus of the material in the planes that contain the maximum principal stress direction. If the texture of the material is random, the length of the links is dependent on the probability to find consecutive grains with a low shear modulus related to the principal stress direction [15], i.e. to the width of the distribution of the shear modulus.

\ \ i \b ' . '

\\V //

» \\\

1 \i 111

! 'I

\ Material with a high shear modulus and a particle with a high rigidity

11

Material with a low shear modulus and a particle with a high rigidity

Fig. 8. Illustration of the mechanism leading to the formation of a load percolation network in a polycrystal. It can be concluded from these first results that the anisotropy of the grains in their elastic domain lead to the formation of a load percolation network, analogous to that observed in a granular material. This network possesses an intrinsic scale larger than the grain size.

330

S. POMMIER

Quantitative evaluation of the importance of the load percolation network. In order to quantify the importance of the formation of a load percolation network for the fatigue crack nucleation process, the following calculations have been performed. A single hexagon located at the centre of the thin sheet was set to have a fixed crystal orientation, while the crystal orientations of the other hexagons in the model were randomly selected before each calculation. Seventy computations have been performed. For each computation the maximum principal stress component was determined at the centre of the grain for which the crystal orientation is fixed. The distribution of this value was calculated for the above seventy random configurations. This distribution is found to be a Gaussian. Therefore the distribution width is calculated here, as three times the standard deviation divided by the mean value of the distribution. The results of these computations are gathered in Table 2. The maximum principal stress in aluminium is found to vary of+/- 7 % around the mean value. This variability is rather low as compared with copper and zinc for which the variation is of +/- 35 %. The variability of the maximum principal stress found for iron, is also very high, i.e. +/- 25 %. It was mentioned in the introduction that two conditions should be fulfilled to promote fatigue crack nucleation. On the one hand, a "weak" grain should be heavily loaded and on the other hand, its crystalline orientation should be favourable for slip. It is interesting, at first, to check if one effect is of prime importance as compared with the other. Table 2. Maximum principal stress at the centre of the grain, with a crystalline orientation as follows: (9i, \)/, (P2) =(0,0,0), located at the centre of the model. The results have been calculatedfi*om70 random configurations of its neighbours and uniaxial mean extension <eyy>=0.1%. Comparison with the distribution of the Schmid factor (SF).

Minimum value (MPa) Maximum value (MPa) Mean Value (MPa) Distribution's width (%) *

Al 64.0 72.6 68.1 ±7

Fe 137.9 210.7 179.9 ± 24

Cu 67.9 128.3 101.8 ±35

Zr 92.6 104.6 97.6 ±8.5

Ti 105.8 122.5 114.1 ±9.25

Zn 79.2 126.9 103.6 ±35

HCP Symmetry HCP FCC HCP BCC FCC 0.462 Mean value of SF 0.462 0.462 SF distribution's width (%) + 7.6 + 7.6 + 7.6 -23.5 -23.5 -23.5 ** * for a Gaussian distribution 99.865 % of the results are lower than the mean value of the distribution plus three times the standard deviation of the distribution ** only 4% of the results are over the upper bound and under the lower bound. For this purpose, the distribution of the Schmid factor (SF) was calculated according to the crystalline orientation of the grain. The Euler space was mapped by 512000 orientations. The maximum Schmid factor was calculated over the 12 slip systems of the FCC and of the BCC systems for each orientation. The distributions of the Schmid factor were determined from these calculations (Fig. 1. (a)). They are very similar for the FCC and BCC systems, and they

Variability in Fatigue Lives: An Effect of the Elastic Anisotropy of Grains?

331

are not centred on their mean value. Low values of the maximum Schmid factor are not excluded, but their probability is very small. The mean value of the Schmid factor is found to be high (<SF>=0.462). Therefore, in grains for which the Schmid factor is at its maximum (SF=0.5), the Schmid factor is only 8% higher than the mean Schmid factor in the polycrystal. In copper, for a given crystalline orientation of the grain, the maximum principal stress can be higher by 35 % than the mean value in the polycrystal, according to the configuration of its neighbours. In comparison, for a given stress state in the grain, the resolved shear stress can be higher by only 8 % than the mean value in the polycrystal, according to the crystalline orientation of the grain. Therefore, in iron and copper, the load percolation network should be dominant, in the crack nucleation process, as compared with the effect of the crystalline orientation of the grains. In aluminium, the two effects appear to be of the same magnitude (Table 2). In order to check this assumption the following calculations have been performed. In each point of the model (Fig. 9), the maximum resolved shear stress over the twelve slip systems of the FCC crystal was calculated as follows, using the stress tensor obtained from the FEM: (l\

^-^ = 7 6 - -

{Tioy

(l\

,(ioT)c 1

Maximum resolved shear stress (MPs) Aluminium

, etc..

(1)

vly

Maximum resolved shear stress (MPa) Copper

1 lalf I rcsca equivalent stress (MPa) Copper

Fig. 9. (a) Schmid factor intensity maps for copper and aluminium; (b,c) Maximum resolved shear stress intensity maps (MPa), over the twelve (111)<110> slip systems of the FCC crystal (b) in aluminium, (c) in copper; (d) Half Tresca equivalent stress in copper. The thin sheet is subjected to a uniaxial extension Eyy^O.l %. The sample is subjected to a uniaxial extension. If the material is isotropic in its elastic domain, Xmax is equal to the Schmid factor. When the anisotropy of the crystal increases, the stress heterogeneity increases within the polycrystal and modifies the value of Xmax- In order to visualize which effect is dominant on the distribution of Xmax in the polycrystal, the maps for Tmax are plotted out and compared with the intensity maps of the Schmid factor and with the Tresca equivalent stress (divided by a factor of 2), which measures the maximum shear stress.

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S. POMMIER

This map is plotted for copper only, since the spatial distribution of the Tresca equivalent stress is very similar in aluminium, except that the intensity of the links is much lower. It is obvious from Fig. 9, that in aluminium Xmax is mostly dominated by the crystalline orientation of the grain, while in copper, Xmax is mostly determined by the location of the grain with respect to the load percolation network.

Aluminium

Iron

Copper

Fig. 10. Maximum resolved shear stress intensity maps (MPa), on the (111)<110> slip systems: (a) in aluminium, and (c) in copper; on the (110)<111> slip systems: (b) in iron; on the (001) slip systems: (d) in zirconium, (e) in titanium, (f) in zinc. The same calculations have been performed for iron, zirconium, titanium and zinc (Fig. 10.). It appears that in iron also, the intensity of the maximum resolved shear stress on the BCC slip systems is mostly defined by the location of the grain in the load percolation network (Fig. 10 (b)). In the three hexagonal materials, the effect of the crystalline orientation of the grain appears to be dominant. However, in these calculations only three slip systems were taken into account, which increases the weight of the Schmid factor as compared with FCC or BCC crystals. As a matter of fact, the distribution of the maximum Schmid factor over 1 plane and 3 directions is obviously larger than the distribution of the maximum Schmid factor over 6 planes and 2 directions. To be realistic, a higher number of slip systems should be taken into account, but the primary slip systems being different in these three materials, the comparison would

Variability in Fatigue Lives: An Effect of the Elastic Anisotropy of Grains?

333

become difficult. One family of slip systems (the basal one) was therefore chosen arbitrarily for this comparison. Nevertheless, it appears that, even if only three slip systems are considered, the effect of the load percolation network is not negligible in zinc. It can be concluded from these results, that the importance for fatigue crack nucleation of the formation of a load percolation network through the polycrystal depends on the elastic anisotropy of the material on the one hand, and on the number of primary slip systems, on the other hand. If the number of primary slip systems and the elastic anisotropy of grains are high, the nucleation process should be dominated by the self-organisation of the stress and strain heterogeneity within the polycrystal. On the contrary, if the number of primary slip system and the elastic anisotropy of grains are low, the nucleation process should be dominated by the crystalline orientation of grains.

Multiaxial fatigue. If multiaxial loading conditions are applied, up to three load percolation networks, aligned with the three principal stress directions, can develop simultaneously. This is shown for example in Fig. 11, where the spatial distributions of the maximum (Fig. 11. (a)) and of the minimum (Fig. 11. (b)) principal stress components were plotted for a pure shear strain. In this case the biaxiality ratio is equal to - 1 , and the principal stress directions form an angle of 45*^ with the axes of the sample. The maximum principal stress component is higher in links inclined at +45°, while the minimum principal stress component is higher in links inclined at -45° with the axes of the samples. Consequently, the Tresca equivalent stress is maximum at the intersections between the two networks (Fig. 11. (c)). Though the pattern is somehow different for the maximum resolved shear stress, Xmax is also higher around the intersections between the two networks.

Fig. 11. Intensity maps (MPa), in the case of copper, for a pure shear deformation equal to Yxy = 0.1 % (a) maximum principal stress component, (b) minimum principal stress component, (c) Tresca equivalent stress, (d) maximum resolved shear stress on the (110)<111> slip systems. The displacements are magnified by a factor 100. This effect is liable to modify the statistic distribution of the shear stress in an individual grain. As a matter of fact, there is a dispersion of both the maximum and the minimum

334

S. POMMIER

principal stress component around their mean values. This dispersion reaches +/- 35% in copper under uniaxial loading conditions. If a biaxial stress state is applied on a sample, with a mean value of the Tresca equivalent stress equal to that applied under uniaxial conditions (Eq. (2)), does it mean that the dispersion around this mean value (Eq. (3)) is also the same? There is no reason to consider that the load percolation networks associated with each direction are uncorrected, hi such a case, the dispersion of the difference between the maximum and the minimum principal stress components is not the sum of the dispersions for each component. The same problem occurs for determining the dispersion on the maximum resolved shear stress on slip systems. /

\shear

/ ^ p max

y,

/

\ shear

i

^ p min /

\ shear pmin/

^^[^eJ=^'^Kmax-^pmin] ^ shear ^ ^ -> shear *- ' ^ _l

/.

V

y'^/jmaxy

^ ^ K max ] " ^^'« K m i n ] = ^^^ [ ^ , max ] v/,^,^;. L / J ^.f,ear *~- ' -* extension *- ' -•

(2)

(3)

This question is important, since the fatigue life of the material is limited by the "weakest" grains, and consequently by the maximum bound of the distribution of Tmax and not by its mean value - It is therefore important to check if, with similar , but different loading conditions, the maximum bounds for Xmax are similar. To answer this question, the crystalline orientation of one grain at the centre of the model was fixed, while the orientations of the other grains in the model were selected randomly before each computation. The calculations were performed for copper. One hundred and fifty computations have been performed in uniaxial extension and in shear. The data are "measured" at the centre of the grain with a fixed crystalline orientation.

100«b

t

509o

0% 50«(

lOOOo

150«i

a e q Tiesca / < Oeq Tiesca >

Fig. 12. Probability that the Tresca equivalent stress, in a grain with a given orientation at the centre of the thin sheet, is greater than a given value. Distribution calculated using the FEM, for a given orientation and 150 configurations of the neighbours. Solid symbols: uniaxial extension, empty symbols: shear.

Variability in Fatigue Hues: An Effect of the Elastic Anisotropy of Grains?

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The statistic distributions of the Tresca equivalent stress calculated in shear and in tension are plotted in Fig. 12. The scatter is larger for shear than for a uniaxial extension. About 7% of the values obtained in shear are over the upper bounds obtained in uniaxial extension. Up to 150 calculations have been performed to build each distribution, in order to check that this effect was not an artefact, and this is not, since it corresponds to 11 points. This effect is interpreted as the effect of the intersections between the links of the load percolation networks. As a matter of fact, in the intensity maps of the Tresca equivalent stress (Fig. 11. (d)), the overstressed grains are always found at the intersections between the links of the load percolation networks associated with each principal direction. Therefore, a few grains at the intersections between the two links are heavily loaded, the grains located within the links are moderately loaded, while those located out of the two loads percolation networks sustain very low levels of the Tresca equivalent stress. In comparison, in uniaxial extension, the grains are heavily loaded if they belong to the load percolation network and sustain low values of the Tresca equivalent stress if they are out of the links. Consequently, there is an increase of the scatter on the Tresca equivalent stress in shear as compared with uniaxial loading conditions.

100*0 9-

50«o

50%

10090

1500o

t max / < T max ^

Fig. 13. Probability that the maximum resolved shear stress on FCC slip systems, in a grain with a given orientation at the centre of the thin sheet, is greater than a given value. Distribution calculated using the FEM, for a given orientation and 150 configurations of the neighbours. Solid symbols: uniaxial extension, empty symbols: shear. In Fig. 13, are plotted the distributions of the maximum resolved shear stress on the FCC slip systems. As for the Tresca equivalent stress, a significant increase of the scatter is found when a shear stress state is applied as compared with a uniaxial stress state. The parameters characterizing the distributions of the Tresca equivalent stress and of the maximum resolved shear stress are gathered in Table 3. The difference between the standard deviation of the distributions obtained in shear and in uniaxial extension is lower for the maximum resolved

336

S. POMMIER

shear stress than for the Tresca equivalent stress. Nevertheless this increase is not negligible, since in tension, the standard deviation for Xmax is close to 12 % of < Tmax > while it approaches 15% of < Xmax > in shear. Although the scatter is higher in shear than in uniaxial extension, the maximum bound of the distribution of the Tresca stress is found to be very similar in these two cases (Table 3). As a matter of fact, the boundary conditions, applied on the model of the thin sheet, were not adjusted to obtain similar mean values of the mean Tresca equivalent stress or of <Xmax> in the two cases, but the same equivalent strain. Consequently, with such boundary conditions, the higher level of the scatter in shear, as compared with a uniaxial extension, results in a lower value of the mean Tresca equivalent stress, and not in an increase of the probabilities for high values of the Tresca equivalent stress. Table 3. Values calculated at the centre of the grain located at the centre of the model with a crystalline orientation as follows: ((pi, \\f, (^) =(0,0,0). The standard deviations are calculated with 150 random configurations of its neighbours. The model is tested either in uniaxial extension <£yy>=0.1% or in shear =0.1%. The standard deviations (6) are given as a percentage of the mean value of the distribution.

150



8(aeq)

< a e q > ( l + 3 8 ( Geq))

<Xn,ax>

8( X^ax)

< Xmax>(l+38( X^ax))

analyses Uniaxial extension Shear

MPa 101.7

(%) 11.2

(MPa) 135.8

(MPa) 44.2

(%) 12.0

(MPa) 60.2

87.6

17.2

132.6

33.5

14.8

48.3

It can be concluded from these calculations, that with a similar value of < Xmax^ in the polycrystal, two different loading conditions are not equivalent in terms of the nucleation of micro-cracks. As a matter of fact, these calculations show that, with the same value of <Xmax>, the maximum resolved shear stress is significantly higher, within a few grains, in torsion as compared with the case of tension. However, the heavily loaded grains are sparse since they are located at the intersections between the heavily loaded links associated with each principal direction. Therefore, on the one hand, this effect should lead to a reduction of the threshold for crack nucleation in torsion as compared with tension, if the threshold is given as a critical value for <Xmax>- However, on the other hand, it may also modify the conditions for crack coalescence during subsequent crack growth. Therefore, it is not easy at this stage, to provide definite conclusions on the effect of the increase of the scatter in an individual grain, on the fatigue life of the entire sample

Rotating loads The last point that is worth to be discussed is the possible effect of the load percolation network in the case of rotating loads. In the first place, the previous results show that for a similar value of <Xmax>, there is a higher probability for high values of the maximum resolved shear stress under multiaxial than under uniaxial loading conditions (Fig. 12). Consequently, even if the mean values of Xmax are

Variability in Fatigue Lives: An Effect of the Elastic Anisotropy of Grains?

337

comparable, the probabilities for high values of Xmax should be higher under in-phase as compared with out-of-phase multiaxial loading conditions. In the second place, the spatial distribution of damage at the surface of the material should be different, if turning loads are applied as compared with cyclic loads with a constant direction. As a matter of fact, if damage appears in grains with a high level of Xmax, under uniaxial loading conditions micro-cracks should appear within the links of the load percolation network. It was shown above, that the links of the load percolation network are aligned with the principal stress directions. Therefore if the maximum principal stress direction is turning, the number of damaged grains per unit surface should increase. This effect is illustrated in Fig. 14, the model was subjected either to an extension along the y axis (Fig. 14 (a)), or to an extension along the x axis (Fig. 14 (b)) or to a cycle with an extension along y at first and then along x (Fig. 14 (c)). An element is considered as damaged, and plotted in black, if once during the fatigue cycle, the maximum resolved shear stress exceeds 65 MPa. The fraction of damaged grains is much higher in Fig 14 (c) than in (a) or (b). This effect should not modify the critical shear stress for crack nucleation in the polycrystal. However, if the number of damaged grains per unit surface is increased, the crack growth rate in the short crack regime is expected to increase due to micro-crack coalescence.

max

(MPa) 30

I

^ .

r *' \ '

.\*

^'iV'"'

; '>

u

*f~€

m ^ >1

1'

*v- 'P

65

< - # •

\ \

'•'» -I ^ .

^"

;' --, ^

(a) 1 ^ *.

>

1

/

'1

^

!

r^

(b)

(c)

'^ max > 65 MPa Fig. 14. Intensity maps of the maximum resolved shear stress in copper: (a) uniaxial extension 8yy=0.1 %, (b) uniaxial extension exx=0.1 %, (c) non-proportional uniaxial extension at first 8yy=0.1 % and then 8xx=0.1 %. An element is considered as damaged (in black) if once during the fatigue cycle Tmax exceeds 65 MPa.

Discussion The problem of the distribution of stress and strain in the polycrystal was modelled here using a thin sheet. The first reason for this choice was to limit the number of elements, in order to perform statistic analyses within realistic times and with a number of grains in the model

338

S. POMMIER

sufficient to let a load percolation network appear. The second reason is that micro-crack nucleation occurs mostly at the surfaces of the samples. However, in a massive sample, grains located under the surface, may contribute to homogenize the stress state within the polycrystal. It was shown before [15], with a rough model of a massive sample, that the scatter is lower in the bulk than at the surfaces. Nevertheless, the scatter at the surfaces is very similar in the case of a thin sheet and in the case of a grain located at the surface of a massive sample. This point would need further investigation, since the difference between the stress and strain heterogeneity at the surfaces and in the bulk of a sample could contribute, with environmental effects, to explain the preferential nucleation of micro-cracks at the surfaces.

CONCLUSIONS Probabilistic approaches are developed to describe scale effects and scatter in fatigue. When fatigue cracks are nucleated on defects, the origin of the scatter is clear. When defects are nondamaging, micro-cracks are usually nucleated by cyclic slip in "weak" grains. In this case, the origins of the scatter in fatigue lives remain unclear. Under bulk elastic conditions, a "weak" grain is a grain within which the maximum resolved shear stress on the slip systems is the highest, which happens when two conditions are satisfied: the Schmid factor of the grain is high and the stress applied on the "weak" grain is high. The object of the paper was to discuss the second condition. Since the elastic behaviour of the grains is anisotropic, the stress and strain distribution is heterogeneous in a polycrystal. The spatial distribution of this heterogeneity was studied using experiments and finite element analyses. The spatial distribution of strain at the surface of a sample of TA6V titanium alloy was observed using the photostress method. Though the material is fully elastic, fine inclined lines appeared at the surface of the sample, where the strain is higher than the mean one in the sample. However the direction of the principal strain remains mostly coincident with the load axis. This experiment showed that there is a scale associated with the strain heterogeneity in the TA6V titanium alloy, which is larger than the grain size, approaching 10 grains. In order to reveal a scale for the spatial distribution of strain in a different material, thin sheet of OFHC polycrystalline copper have been subjected to a cyclic creep test. After failure, fine inclined lines forming a regular pattern are observed at the surface of the sample, revealing a scale for the heterogeneity of strain larger than one millimetre. Finite elements calculations were performed, in order to understand the above-mentioned effects. A polycrystalline thin sheet was modelled by FEM analysis. These computations showed that a load percolation network, analogous to that observed in a granular material, is formed through the polycrystal. The load is transferred through heavily loaded links whose direction is coincident with the principal stress directions of the equivalent homogeneous problem. This network possesses an intrinsic scale larger than the grain size. The probability of a given value of the maximum principal stress within a grain was calculated using the FEM. One grain located at the centre of the thin sheet was set to have a fixed orientation, while the crystalline orientations of the other grains in the model were selected randomly before each calculation. The variability of the maximum principal stress under uniaxial loading conditions depends on the elastic anisotropy of the grain. For a given crystal orientation, the maximum principal stress vary up to +/- 35 % for zinc and copper and

Variability in Fatigue Lives: An Effect of the Elastic Anisotropy of Grains?

339

of +/- 24 % for iron. This variability is very high as compared with the width of the distribution of the Schmid factor in FCC crystal. The spatial distribution of the maximum resolved shear stress on slips systems was calculated by the FEM, and compared with the distribution of the Tresca equivalent stress on the one hand and with the distribution of the Schmid factor in the model on the other hand. It can be concluded from these calculations that the importance for fatigue crack nucleation of this load percolation network depends on the elastic anisotropy of the material on the one hand, and on the number of primary slip systems on the other hand. If the number of primary slip systems and the elastic anisotropy of grains are high, the nucleation process should be dominated by the self-organization of the stress and strain heterogeneity within the polycrystal. On the contrary, if the number of primary slip system and the elastic anisotropy of grains are low, the nucleation process should be dominated by the crystalline orientation of grains. When the distribution of Xmax is dominated by the load percolation effect, some effects of that network may arise in multiaxial fatigue. It was shown, that with a similar mean value of the maximum resolved shear stress on slip systems in a polycrystal, two different loading conditions are not equivalent in terms of the nucleation of micro-cracks, since the maximum bounds for Xmax can be different. For example, these calculations show that with a similar mean value , the maximum bound for Xmax is higher in torsion as compared with tension. The grains with the highest value of imax are located around the intersections between the heavily loaded links associated with each principal direction. These grains are sparse but overstressed. The role in fatigue of the self-organized spatial distribution of stress and strain in the polycrystal, which is described in this paper, should also be important for crack coalescence during subsequent crack growth, since it controls the number of damaged grains per unit surface and their mutual distance.

REFERENCES 1.

Sines, G. and Ohgi, G. (1981). Fatigue criteria under combined stresses or strains. Journal of Engineering Materials and Technology 103, 82-90. 2. Dang Van, K. (1993). Macro-micro approach in high-cycle multiaxial fatigue. In: Advances in Multiaxial Fatigue, ASTM STP 1191, pp. 120-130, Mc Dowell, D.L. and Ellis, R. (Eds.), ASTM, Philadelphia. 3. Murakami, Y., Toriyama and T.,Coudert, E.M. (1994). Instructions for a New Method of Inclusion Rating and Correlations with the Fatigue Limit. Journal of Testing & Evaluation 22,318-326. 4. Hild, F., Billardon, R. and Beranger, A.S. (1996). Fatigue failure maps of heterogeneous materials. Mechanics of Materials 22, 11-21. 5. Beretta, S. (2001). Analysis of multiaxial fatigue criteria for materials containing defects. In: ICB/MF&F, pp. 755-762, de Freitas, M., (Eds.), ESIS, Lisboa. 6. Murakami, Y. and Endo, M. (1994). Effects of defects, inclusions and inhomogeneities on fatigue strength. Int. J. Fatigue 16, 163-182. 7. Susmel, L. and Petrone, N. (2001). Fatigue life prediction for 6082-T6 cylindrical specimens subjected to in-phase and out of phase bending/torsion loadings. In: ICB/MF&F, pp. 125-142, de Freitas, M., (Eds.), ESIS, Lisboa. 8. Guyon, E. and Troadec, J.P., (1994). Du sac de billes au tas de sable, Odile Jacob (Eds), Paris.

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9. 10. 11.

12.

13.

14. 15.

S. POMMIER

Savage, S.B. (1997). Problems in the static and dynamics of granular materials. In: Powder and grains, pp. 185-194, Behringer, R.P. and Jenkins, J.T. (Eds.), Balkema, Rotterdam. Dantu, P. (1968). Etude statistique des forces intergranulaires dans un milieu pulverulent. Geotechnique. 18. 50-55 Radjai, P., Wolf, D.E., Roux, S., Jean, M. and Moreau, J.J. (1997). Force networks in dense granular media. In: Powder and grains, pp. 211-214, Behringer, R.P. and Jenkins, J.T. (Eds.), Rotterdam. Roux, J.N. (1997). Contact disorder and nonlinear elasticity of granular packings: A simple model. In: Powder and grains, pp. 215-218, Behringer, R.P. and Jenkins, J.T. (Eds.), Balkema, Rotterdam. Le Biavant, K., Pommier, S. and Prioul, C. (1999), Ghost structure effect on fatigue crack initiation and growth in a Ti-6A1-4V alloy. In : Titane 99:Science and technology, pp 481487, Goryin, I.V. and Ushkov, S.S. (Eds), Saint Petersburg, Russia. Le Biavant, K., Pommier, S. and Prioul, C. (2002). Local texture and fatigue crack initiation in a Ti-6A1-4V Titanium alloy. Fat. Fract. Engng. Mater. Struct 25, 527-545. Pommier, S. (2002). "Arching" effect in elastic polycrystals. Fat. Fract. Engng. Mater. 5rrMcr. 25, 331-348.

Appendix : NOMENCLATURE OFHC FEM (1,2,3) and (x,y,z)

Oxygen Free High Conductivity Copper Finite element method Coordinate systems attached to the grain and to the model

(j^^

Stress tensor as calculated using the FEM in (1,2,3)

SF

Schmid factor

Tmax

Maximum resolved shear stress on the slip systems of a crystal

cr^

Tresca equivalent stress

G^

Principal stress

<^/;max ' <^/; min

Maximum, miulmum principal stress components



Mean value of/

5(/)

Standard deviation of/

FCC, BCC, HCP

Face centered cubic, body centered cubic, hexagonal close-packed

Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.

341

THREE-DIMENSIONAL CRACK GROWTH: NUMERICAL EVALUATIONS AND EXPERIMENTAL TESTS

Calogero CALI, Roberto CITARELLA, Michele PERRELLA Department of Mechanical Engineering, University of Salerno via Ponte don Melillo I, 84084 Fisciano (SA), Italy

ABSTRACT Experimental observations of three- and two-dimensional fatigue crack growth are compared to numerical predictions from the computer code BEASY. The two dimensional propagation occur in a Multiple Site Damage (MSD) scenario created on a pre-notched specimen, undergoing a traction fatigue load as defined by a general load spectrum. Experimental analyses on a fatigue machine were carried out in order to validate the numerical simulation and to provide the necessary material fatigue data for the aluminium plates. The numerical code adopted (BEASY) is based on Dual Boundary Element Method (DBEM). General modelling capabilities are allowed by this approach, with the allowance for general crack front shape and a fully automatic propagation process. By means of a non-linear regression analysis, applied on in house obtained experimental data, the material parameters for the NASGRO 2.0 crack propagation law were defined, capable to effectively keep into account the threshold effect and the unstable final propagation (the crack closure option was switched off). A satisfactory agreement between numerical and experimental crack growth rates was obtained, even starting from a complex MSD scenario, created by the presence of three holes in the plate. Moreover the load introduction to the specimen was monitored by strain gauge equipment. The numerical simulation include also the through the thickness propagation, corresponding to 3D part-through cracks; in this case some specimen were pre-notched by a comer crack on one of the holes and the 3D experimental crack propagation monitored. KEYWORDS MSD, Part-through crack. Load Spectrum, Dual BEM, NASGRO 2.0, BEASY

INTRODUCTION Damage Tolerance is used in the design of many types of structures, such as bridges, military ships, commercial aircraft, space vehicle and merchant ships. Damage tolerant design requires accurate prediction of fatigue crack growth under service conditions and typically this is accomplished with the aid of a numerical code. Many aspects of fracture mechanics are more complicated in practice than in two-dimensional laboratory tests, textbook examples, or overly

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C CALI, R. CITARELLA AND M. PERRELLA

simplified computer programs. Load spectrum, threshold effects, enviromnental conditions, microstructural effects, small crack effects, Multiple Site Damage (MSD) conditions, material parameters scatter, mixed loading conditions (material flaws or pre-cracks, which may have been introduced unintentionally during the manufacturing process, can have an arbitrary orientation with respect to the loading applied to the component) and complex three dimensional geometry, all complicate the process of predicting fatigue crack growth in real word applications. This paper focuses on some of these complications (see also [1]): load spectrum influence, complex three dimensional geometry, fatigue material parameters assessment, mixed loading conditions and MSD conditions. A series of laboratory tests have been designed and implemented to assess fatigue material parameters and to evaluate three dimensional fatigue life prediction capabilities for a numerical life prediction code (BEASY), based on Dual Boundary Element Method (DBEM). With such code the geometry of the test specimen and the shapes of evolving crack fronts, are not restricted to the simplified configurations found in the libraries of many commercial codes. Many complications were purposely minimised: initial crack sizes and loading were such that small crack and threshold effects had little consequence on the propagation phenomena; environmental and microstructural effects were considered part of the experimental scatter. EXPERIMENTS Experimental fatigue tests were performed on complex geometry notched plates undergoing cyclic axial load. The crack initiation process and the crack propagation were monitored and a general traction load spectrum was applied on the specimen. Experimental crack growth rate and crack path were compared with those obtained with a numerical procedure based on DBEM and the correct simulation of the load introduction to the specimen was checked by strain gauge measurements. Simple geometry cracked plates and two-dimensional crack propagation The fatigue data necessary for the numerical analysis were previously obtained by experimental crack growth tests on simple geometry specimens [2]. All of the test aluminium alloy specimens described in this paper were manufactured from a single lot. The first part of the fatigue experimental tests was carried out on 3 simple notched (hole/slut) specimens (Fig.l), in such a way to work out, with statistical significance (for R=0.1), the material fatigue parameters for crack growth simulation. The initial through the thickness notch was cut by a thin saw. The plate geometry as well as the whole testing procedure was consistent with the ASTM E 647-91 specifications. A constant amplitude fatigue traction load {Pmax=14 kN, R=Pmir/Pmax="O.I) was applied by a servo-hydraulic machine (Instron 8502), with a frequency f=5 Hz, at ambient temperature. During the experimental fatigue test on simple specimens, crack length data, measured by optical systems, were recorded and used to work out, by analytical formula, the corresponding SIF's (Stress Intensity Factors) and crack growth rates. The overall specimen size was chosen sufficiently large in order to get a mainly elastic behaviour, with plasticity effects confined to the final part of crack propagation, whose monitoring can start only after a pre-cracking phase, as prescribed by the ASTM E 647 standard. The observed rectilinear crack propagation path turned out to be consistent with symmetric

Three-Dimensional Crack Growth: Numerical Evaluations and Experimental Tests

343

boundary conditions but, whenever out of tolerance deviations from the ideal rectilinear path came out, the corresponding specimens were discarded from the analysis. In the fmal part of crack propagation, due to high plasticity effects, a 45° deflection of the crack surface preceded the ductile failure [3]. 370 ^00"

I \ 1 ^,. 0.3

185 1

^ \R20

f CO

85

Fig.l. Simple specimen (N. 5-7) geometry adopted for material fatigue parameters assessment. For each valid specimen (N° 5-7), according to the mentioned standard, a chart with crack length against the number of cycles was plotted, in order to assess crack growth rates da/dN, calculated by the secant method and SIF's, calculated by analytical formulas. Crack growth rate values were then plotted against AK in a bilogarithmic chart and a linear regression was performed to estimate the Paris law material constants C and n (Fig. 2). ! 1

1

J

* /

i F—1

II 1 £ 1 Z 5

•

1.00Er04

l\

j

!i ^• ! _

^T3

,•

i

i

N.5-6.7 specimens data

I — Paris

C=2.21E-13 m=3.J5^ l.OOB-05 1 )0

1000 1/2-,

AK [MP amm J

Fig. 2. Regression curve using Paris law.

Fig. 3. Notched complex specimens (N.l2): initial MSD scenario (dimensions mm).

344

C CALI, R. CITARELLA AND M. PERRELLA

Complex geometry cracked plates: two- and three-dimensional crack propagation Two dimensional. For that concern the two-dimensional long crack propagation, a complex MSD initial scenario was artificially created on a rectangular plate (Fig. 3), by means of notched holes (a triangular notch was cut in correspondence of each hole) and, after the precracking process, four different cracks simultaneously propagating were obtained. The propagating crack lengths were monitored on both sides of the specimen in order to check the correctness of the load introduction on the specimen: any misalignment between the machine grips could create a bending condition for the specimen and consequently an elliptical propagating crack front [4]. This check, together with the strain gauge measurements, allowed assessing the validity of the 2D hypothesis for such long crack propagation analysis. Three dimensional. To study the through the thickness crack propagation a triangular notch emanating from hole 2 (corresponding to crack 2 in Fig. 3) was cut. After the crack initiation and pre-cracking period, a quarter elliptical comer crack started to propagate. Such crack was monitored by an optical measuring system based on a moving camera (Fig. 4), in order to alternatively follow the two break points of the elliptical crack front. The images obtained by the (manually) moving camera (Fig. 5) were elaborated by an image analysis software in order to obtain the two ellipse semi-axes measures. For long crack initiation period it is possible, with an in house made software, to automatically make periodic photo that can be recorded and postprocessed, avoiding the need for a continuous surveillance of the specimen by the operator. In the latter case two camera are needed for measuring the two semi-axes and only a short propagation length can be monitored before the crack tip get out of focus (but this drawback can be overcome by using special lenses or with an automatically moving camera).

Fig. 4. Equipment adopted for monitoring the through the thickness crack propagation.

Three-Dimensional Crack Growth: Numerical Evaluations and Experimental Tests

345

Fig. 5. Images of the propagating crack: size A (left) and size C (right). In the latter only the crack tip at the internal hole is visible, few cycle before getting a through the thickness crack. NUMERICAL ANALYSIS Two and three-dimensional analysis are respectively needed to simulate the through the plate and through the thickness crack propagation. The material fatigue parameters, obtained by the experimental analysis previously described, are useful to perform a crack growth simulation on a complex geometry specimen made of the same material. The results of such numerical analysis were compared with those from the experimental tests, in order to validate and improve the numerical procedure, based on DBEM [5-9]. Two dimensional simulation on MSB plates Two loading conditions were considered on different complex specimens: 1. Cyclic load with constant amplitude (Pmax-Pmin=12.6 KN) and stress ratio {R=0.1), on specimen N.l. The Paris law was adopted for numerical crack growth assessment (the same values had been used for the simple notched specimens). Crack paths (Figs. 6-7) and propagation times (Figs. 8-11), obtained by BEASY code [10], were compared with the experimental ones, getting a satisfactory agreement; 2. Cyclic load with variable amplitude and stress ratio on specimen N.2. With the experimental data coming from simple notched specimens and from the previous complex specimen (N.l), a non-linear regression was attempted, in order to model the threshold phenomena and the fracture toughness for the final unstable crack propagation. To this aim the NASGRO 2.0 law was chosen for numerical crack growth assessment (for such case the Paris law did not give satisfactory results). The crack paths are the same as for the previous case (Figs. 6-7) and the propagation times (Figs. 8-11), obtained by BEASY code, were compared with the experimental ones, getting a satisfactory agreement especially in the first part of the propagation. The differences, however limited, in the final part suggest the need of an improved correlation; this can be done by increasing the experimental data coming from simple specimens (it is necessary to test the simple specimens with different R values). The parameter values for the NASGRO 2.0 Eq.(l), with no crack closure effect, are

346

C. CALl R. CITARELLA AND M. PERRELLA

indicated in Table 1. Such parameters were extracted from the NASGRO database, in correspondence of Al 2219-T87 which was reputed the most similar to the aluminium used, in order to get the unknowns C and n from the non-linear regression (made with MATHEMATICA 4) which is based on in house obtained experimental data:

da ^ dN~

V (

^K t^

^ C = 4.26£-ll,« = 2.69

(l-^)-^c

(1)

In table 2 the load spectrum adopted for the specimen N° 2 is illustrated. From Fig. 8 which is related to the first appearing crack 1, it is possible to note the relatively little difference between initiation times for the two complex geometry specimens. Table 1. Material parameters adopted for crack growth simulation Y o u n g s M odulus [M Pa] Poissons ratio Yield stress (Ys) [ M P a ] Ultimate tensile strength ( U T S ) [ M P a ] Plane strain fracture t o u g h n e s s [MPamm^^^] Empirical constant [Exponent Threshold SIF at R = 0 [MPamm^^^] |Cut off stress ratio [intrinsic crack length [mm] [._

E V ^YS CTUTS

^Ic

Ak, Bk

P. q AKo Rci

lo

"7^0E + 0 4 | 0.3 283 309 900 1 1 120 0.7 0.102 1

The NASGRO 2.0 form of the equation for AKth is given by:

A/:.

• arctan(l - /?)

A^n

• arctan(l - R^i)

if

R
if

R>R.,

A^. J V a^Qc,

The fracture toughness Kc for 2D plane stress and 3D crack models that have been identified as "all-through", is given by:

K.^

l + B,e

-if

K,

Three-Dimensional Crack Growth: Numerical Evaluations and Experimental Tests

_ DISPLACED SHAPE

Scale f actor'30.000

- DISPLACtD SHAPE

Scale factor 150.00

Fig. 7. Numerical crack propagation paths at 774000 (upper) and 1070000 cycles (lower) for specimen N°2.

Fig. 6. Experimental crack propagation paths (specimen N°2).

Table 2. Load spectrum applied to specimen N. 2. Number of cycles 581000 774000 866000 968000 1046000 1076000

347

Pmax (KN)

14 16.3 21.5 19.2 17.2 19.2

R 0.10 0.22 0.41 0.58 0.77 0.58

348

C. CALI, R. CITARELLA AND M. PERRELLA

Crack 1

570000 620000 670000 720000 770000 820000 870000

N

920000 970000 1020000 1070000 1120000

[cycles]

Fig. 8. Crack length versus number of cycles for crack 1, as obtained by simulation with Paris law (applied to specimen 1) and NASGRO 2.0 law (applied to specimen 2), together with related experimental data.

Crack 2 18 17 16 15 14 13 12 ^11 B 10 E 9

• •

Numeric n°l Experimental n°l Experimental n°2 Numeric n°2

6 5 4 3 2 1 570000

630000

690000

750000

810000

870000

930000

990000

1050000

1110000

N [cycles]

Fig. 9. Crack length versus number of cycles for crack 2, as obtained by simulation with Paris law (applied to specimen 1) and NASGRO 2.0 law (applied to specimen 2), together with related experimental data.

Three-Dimensional Crack Growth: Numerical Evaluations and Experimental Tests

349

Crack 3 7 -r

]

•

6

Experimental n°l •Numeric n°l

J^^ ^

t

- / .

5 1 •^—

•Numeric n°2

^^00^^'^k

^ ^""^

•

C8 3

1

i

^^

^^jt^

:

^J^^^^-^""' i

^

0 570000

620000

670000

^ 720000

^ 770000

\

.

870000

920000

^ 820000

:

^

970000 1020000 1070000 1120000

N [cycles]

Fig. 10. Crack length versus number of cycles for crack 3, as obtained by simulation with Paris law (applied to specimen 1) and NASGRO 2.0 law (applied to specimen 2), together with related experimental data. Crack 4 24 22 20 18 16

570000

A Experimental n°2i i •^Numeric n°2

620000

670000

720000

770000

820000

870000

920000

970000

1020000

1070000

N [cycles] Fig. 11. Crack length versus number of cycles for crack 4, as obtained by simulation with Paris law (applied to specimen 2) and NASGRO 2.0 law (applied to specimen 2), together with related experimental data.

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C. CALL R. CITARELLA AND M. PERRELLA

Three dimensional simulation: theoretical aspects and results. Surface crack solutions are widely used in applications of fracture mechanics to fatigue and monotonic loading. A semi-elliptical surface crack lying perpendicular to the surface and subject to applied stresses with no shear component parallel to the crack, experience mode I conditions around its edge, as in the problem presented in the following. Generally this orientation is the critical one, also in mixed mode conditions if the fatigue crack threshold is governed by (AG I)TH, where G is the energy release rate. The dual boundary element method. BEASY uses dual elements for 3D crack growth analysis. The dual boundary element method (DBEM) incorporates two independent boundary integral equations: the displacement equation applied at the collocation point on one of the crack surfaces and the traction equation on the other surface. Application of the DBEM to threedimensional crack growth analysis has been presented in [11-13]. In the absence of body force, the displacement boundary equation can be written as: C^^{X^\UXMT,J{X\X^

(2) where ij denote Cartesian components, Ty {x » and Uij{x » represent the Kelvin traction and displacement fundamental solutions at a boundary point x, respectively. The symbol J stands for the Cauchy principal value integral. Assuming continuity of both strains and traction at JC ' on a smooth boundary, the stress components ay are given by: \cr,^{x')+ ]T,„{x',x).u,{x').dr{x)= ]u„^{x',x)-t,{x')-dr{x) 2 r r

(3)

where Tky {x » and Ukij{x » contain derivatives of Tjj (jc» and Uij{x ',x), respectively. The symbol J stands for the Hadamard principal value integral. The traction components tj are given by: \t^{x')^n^{x^\]T,,^{x\x\u,{x^^^ ^

r

(4)

where «/ (JC ') denotes the component of outward unit normal to the boundary at x'. Equations (2) and (4) constitute the basis of DBEM. A more complete description is given in [11].

Three-Dimensional Crack Growth: Numerical Evaluations and Experimental Tests

351

Stress Intensity Factors. Since Irwin [14] demonstrated the importance of the stress intensity factor (SIF) in determining crack-tip stress fields in two dimensional problems, many different methods have been devised for obtaining SIF's [15]. The stress intensity factors (SIF's) are computed using crack opening displacement method. When one point formulae are employed, the Mode I, II and III SIF's are evaluated as:

/,;=^L^ |Z.(„;| _„;| ) 4(l-v ) V2r ^ '^^'^

^^=-"'

(5) (6)

4(1-v^j \2r

^ ^^-"

^^=-"^

(7)

where the displacement u^ is evaluated at point P as in Fig. 12; w/, u/ and Ut^ are projections of u^ on the coordinate directions of the local crack front coordinate system and 0=7i and 6=-7C denote upper and lower crack surfaces respectively. Kj^, J^// and Ki/ are approximations of SIF's corresponding to the point P' along the crack front and on a normal line to the front as in Fig. 12.

Fig. 12. Calculation of SIF using crack opening displacement. During the crack growth, instabilities may cause peaks in the SIF values along the crack front. It is advisable to enable the BEASY option that allows these peaks to be smoothed by either smoothing the stress intensity factors or by smoothing the crack growth increment or by performing both smoothing operations. This should also be used if the values computed at the crack breakout points are not very accurate and give excessively high stress intensity factor values, as it happens in the problem under study. As a matter of fact the asymptotic stress field at the crack tip is usually obtained by performing analytical two-dimensional plane stress or plain strain calculations. This yields the well known f^^ singularity which prevails along the crack front within the body. However, at the intersection of the crack front with a free boundary, the stress state is a genuine three-dimensional one for which two dimensional approximations are not applicable: the mode I stress singularity decreases at the free boundary and takes the value r'^"^^ ^, whereas the mode II and mode III singularity increase in strength

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C CAU

R. CITARELLA AND M. PERRELLA

and attain a value of f^^^^ [16]. At the end, high resolution accuracy at the comer may not be worth pursuing, in any case, because the geometry where the crack intersects the surface is expected to adjust under loading so as to ameliorate the stronger singularity [17]. The incremental direction. The crack growth direction and SIF equivalent are computed by the minimum strain energy density criterion. The criterion for three dimensional problems can be found in [18]. The explicit expression of strain energy density around the crack front can be written as: ^ = ^^.0(1) dV rcos(/>

/g\

where S(Q) is given by S{0) = a,,-Kj^2a,,-Kj

K, +a,,-Kl

^a,,-Kli

(9)

and a^^ = 16/r/i ^12 =

( 3 - 4 v - c o s ^ ) ( l + cos^) sinO' {cosO-1 + 4v)

%7VJU

a^2 =^—•[4(l-v).(l-cos<9)+(3cos(9-l)(l + cos6>)] 16;r// 1 4;r// in which |i stands for the shear modulus of elasticity and v is the Poisson ratio. S/cos(/> represents the amplitude of intensity of the strain energy density field and it varies with the angle ^ and 0. It is apparent that the minimum of S/cos