Mechanics of Transformation Toughening and Related Topics

MECHANICS OF TRANSFORMATIONTOUGHENING AND RELATED TOPICS

NORTH-HOLLAND SERIES IN

APPLIED MATHEMATICS AND MECHANICS EDITORS:

J.D. ACHENBACH Northwestern University

B. BUDIANSKY Harvard University

H.A. LAUWERIER University ofAmsterdam

P.G. SAFFMAN California Institute of Technology

L. VAN WIJNGAARDEN Twente University of Technology

J.R. WILLIS University of Bath

VOLUME 40

ELSEVIER AMSTERDAM*LAUSANNE*NEWYORK*OXFORD*SHANNON*TOKYO

MECHANICS OF TRANSFORMATION TOUGHENING AND RELATED TOPICS B.L. KARIHALOO School of Civil andMining Engineering The University of Sydney Australia

J.H. ANDREASEN Institute of Mechanical Engineering Aalborg University Denmark

1996

ELSEVIER AMSTERDAM LAUSANNE *NEW YORK*OXFORD*SHANNON TOKYO

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. B o x 211, 1000 AE Amsterdam, The Netherlands

L i b r a r y o f C o n g r e s s Cataloging-in-Publication

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K a r i h a l o o . B . L. M e c h a n i c s o f t r a n s f o r m a t on t o u g h e n i n g and r e l a t e d t o p l c s / B . L . K a r i h a l o o . J.H. A n d r e a s e n . p. cm. -- ( N o r t h - H o land s e r i e s in a p p l i e d m a t h e m a t i c s and m e c h a n l c s ; v . 40) I n c l u d e s b i b l i o g r a p h i c a l r e f e r e n c e s and indexes. ISBN 0-444-81930-4 1. Ceramic materials--Thernomechanical properties--Mathematical m o d e l s . 2. F r a c t u r e mechanics--Mathematical m o d e l s . 3. M a r t e n s i t i c transformations--Mathematical models. I. A n d r e a s e n , J. H. 11. T i t l e . 111. S e r i e s . T A 4 5 5 . C 4 3 K 3 7 1996 620.1'40426--dc20 96-1174

CIP

ISBN: 0-444-81930-4

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To Alla and Wivj

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Preface Since the benefit of stress-induced tetragonal to monoclinic phase transformation of confined tetragonal zirconia particles was first recognized in 1975, the phenomenon has been widely studied and exploited in the development of a new class of materials known as transformation toughened ceramics (TTC). In all materials belonging to this class, the microstructure is so controlled that the tetragonal to monoclinic transformation is induced as a result of a high applied stress field (e.g. a t a crack tip), rather than as a result of cooling the material below the martensitic start temperature. T h e significance of microstructure to the enhancement of thermomechanical properties of TTC is now well understood, as are the mechanisms that contribute beneficially to their fracture toughness. The micromechanics of these mechanisms has been extensively studied and is now ripe for introduction to a wide audience in a cogent manner. The description of the toughening mechanisms responsible for the high fracture toughness of TTC requires concepts of fracture mechanics, dislocation formalism for the modelling of cracks and of Eshelby’s technique. This has presented us with the opportunity to review these concepts briefly for the benefit of the reader who is unfamiliar with them. The advanced readers have our sympathy, if they find this revision superfluous to their needs. The monograph has its origin in the sets of notes that the first author wrote on two separate occasions for lectures read to participants from research and industrial organizations. The preparation of the monograph has meant that the lecture notes had to be brought up to date and substantially enlarged to include several topics which have only recently been fully investigated. We are indebted to the whole community of researchers who have contributed to our present understanding of the mechanics of transformation toughening in TTC. Nothing would have given us greater pleasure than to thank all of them individually, but we were bound to miss some names and to give offence unintentionally. We therefore offer them a collective thank you and hope reference to their contributions in the monograph at least partly compensates for this omission on our part.

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Contents I

Introduction and Theory

1

1 Introduction

3

2 Transformation Toughening Materials 2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Modern Zirconia-Based Ceramics . . . . . . . . . . . . . . 2.3 Martensitic Transformation . . . . . . . . . . . . . . . . . 2.3.1 Retention of the t-phase . . . . . . . . . . . . . . . 2.4 Fabrication and Microstructure of PSZ . . . . . . . . . . . 2.5 Microstructural Development . . . . . . . . . . . . . . . . . 2.5.1 Ca-PSZ . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Mg-PSZ . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Y-PSZ . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Fabrication and Microstructure of TZP . . . . . . . . . . 2.6.1 Y-TZP . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Ce-TZP . . . . . . . . . . . . . . . . . . . . . . . .

9

3 Constitutive Modelling 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Constitutive Model for Dilatant Transformation Behaviour 3.3 Constitutive Model for Shear and Dilatant Transformation Behaviour . . . . . . . . . . . . . . . . . . 3.3.1 Stress-Strain Relations during Transformation . . . 3.3.2 Transformation Criterion and Transformed Fraction of Material . . . . . . . . . . . . . . . . . . . . 3.3.3 Comparison between the Two Constitutive Models 3.3.4 Comparison with Experiment . . . . . . . . . . . .

9 10 11 15 17 18 18 21 27 28 29 30 35 35 36 43 44 47 54 56

X

Con2e nt s 3.4 Constitutive Model for ZTC . . . . . . . . . . . . . . . . . 3.4.1 Equivalent Inclusion Method for Inhomogeneity Problem . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Transformation Yielding and Twinning of a Single Zirconia Particle . . . . . . . . . . . . . . . . . . . 3.4.3 Overall Properties and Local Fields . . . . . . . . 3.4.4 Transformation Yielding and Twinning of T T C . . 3.4.5 Transformation Yielding under Uniaxial Loading .

58 59 62 69 73 76

4 Elastic Solutions for Isolated Transformable Spots 81 4.1 Centres of Transformation . . . . . . . . . . . . . . . . . . 81 4.1.1 Centre of Dilatation . . . . . . . . . . . . . . . . . 84 4.1.2 Centre of Shear . . . . . . . . . . . . . . . . . . . . 85 4.1.3 Planar Transformation Strains . . . . . . . . . . . 87 4.1.4 Complex Representation . . . . . . . . . . . . . . . 89 4.2 Transformation Spots . . . . . . . . . . . . . . . . . . . . 93 4.2.1 Transformation Spots in an Infinite Plane . . . . . 93 4.2.2 Transformation Spots in a Half-Plane . . . . . . . 96 4.3 Homogeneous Dilatant Inclusions . . . . . . . . . . . . . . 97 4.3.1 Weight Functions for a Single Inclusion in an Infinite Plane . . . . . . . . . . . . . . . . . . . . . 101 4.3.2 Weight Functions for a Subsurface Inclusion . . . . 102 4.3.3 Weight Functions for a Row of Inclusions . . . . . 103 4.3.4 Weight Functions for a Stack of Inclusions . . . . . 105 4.3.5 Weight Functions for a Row of Subsurface Inclusions . . . . . . . . . . . . . . . . . . . . . . . 106 5

Interaction between Cracks and Isolated Transformable Particles 109 5.1 Interaction of a Spot with a Crack . . . . . . . . . . . . . 109 5.1.1 Finite crack . . . . . . . . . . . . . . . . . . . . . . 109 5.1.2 Semi-Infinite Crack . . . . . . . . . . . . . . . . . . 114 5.2 Stress Intensity Factors . . . . . . . . . . . . . . . . . . . 117 5.3 Mode-I Spot Distributions . . . . . . . . . . . . . . . . . . 118

6 Modelling of Cracks by Dislocations 129 6.1 Dislocation Formalism . . . . . . . . . . . . . . . . . . . . 129 6.1.1 Complex Representation of Dislocations . . . . . . 131 6.2 Representation of Cracks by Dislocations . . . . . . . . . 132 6.2.1 Weight Functions for an Edge Dislocations in an Infinite Plane . . . . . . . . . . . . . . . . . . . . . 136

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Contents

6.2.2 6.2.3 6.2.4 6.2.5

I1

Weight Functions Weight Functions Weight Functions Weight Functions Dislocations . . .

for a Subsurface Dislocation . . 137 for a Row of Dislocations . . . . 140 for a Stack of Dislocations . . . 143 for a Row of Subsurface . . . . . . . . . . . . . . . . . . . 146

Transformation Toughening

151

7 Steady-State Toughening due to Dilatation 153 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.2 Toughness Increment for a Semi-Infinite Stationary Crack 155 7.3 Toughening due to Steady-State Crack Growth . . . . . . 157 7.3.1 Steady-State Crack Growth in Super-critically Transforming Materials . . . . . . . . . . . . . . . 160 7.3.2 Steady-State Crack Growth in Sub-critically Transforming Materials . . . . . . . . . . . . . . . 166 7.3.3 Influence of Shear on Super-critical Steady-State Toughening . . . . . . . . . . . . . . . . . . . . . . 182 8 R-Curve Analysis 187 8.1 Semi-Infinite Cracks . . . . . . . . . . . . . . . . . . . . . 187 8.1.1 Stationary and Growing Semi-Infinite Crack . . . . 191 8.2 Single Internal Cracks . . . . . . . . . . . . . . . . . . . . 196 8.2.1 Stationary and Growing Internal Crack . . . . . . 199 8.2.2 Relation Between Toughening and Strengthening . 205 8.2.3 Biaxially Loaded Internal Crack . . . . . . . . . . 208 8.3 Array of Internal Cracks . . . . . . . . . . . . . . . . . . . 214 8.3.1 Mathematical Formulation . . . . . . . . . . . . . 214 8.3.2 Onset. of crack growth . . . . . . . . . . . . . . . . 217 8.3.3 Growing cracks . . . . . . . . . . . . . . . . . . . . 226 8.4 Surface Cracks . . . . . . . . . . . . . . . . . . . . . . . . 231 8.4.1 Model Description and Theory . . . . . . . . . . . 232 8.4.2 Single Surface Cracks . . . . . . . . . . . . . . . . 236 8.5 Array of Surface Cracks . . . . . . . . . . . . . . . . . . . 242 8.6 Steady-State Analysis of an Array of Semi-Infinite Cracks 247 8.6.1 Results and Discussion . . . . . . . . . . . . . . . . 253 8.7 Solution Strategies for Interacting Cracks and Inclusions . 261

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Contents

9 Three-Dimensional Transformation Toughening

9.1 Introduction . . . . . . . . . . . . . . . . . 9.2 Three-Dimensional Weight Functions . . 9.3 Dilatational Transformation Strains . . 9.4 Shear Transformation Strains . . . . . . 9.4.1 Simple Transformation Domains

271 . . . . . . . . . 271 . . . . . . . . . . 272 . . . . . . . . . . 285 . . . . . . . . . . 286 . . . . . . . . . . 289

10 Transformation Zones from Discrete Particles 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Semi-Infinite Stationary Crack . . . . . . . . . . . . . . 10.3 Semi-Infinite Quasi-Statically Growing Crack . . . . . . 10.4 Self-propagating Transformation (Autocatalysis) . . . . 10.4.1 A Strip of Transformable Material . . . . . . . . 10.4.2 A Row of Transformable Particles . . . . . . . .

I11 Related Topics

303 303 . 305 . 324 . 331 . 331 . 337

341

11 Toughening in DZC 343 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 343 11.2 Contribution of Phase Transformation to the Toughening of DZC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 11.2.1 Experimental Evidence . . . . . . . . . . . . . . . 345 11.2.2 Dilatational Contribution to the Toughening of ZTA . . . . . . . . . . . . . . . . . . . . . . . . 347 11.3 Contribution of Microcracking to the Toughening of DZC 351 11.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 351 11.3.2 Reduction in Moduli and Release of Residual Stress . . . . . . . . . . . . . . . . . . . . 353 11.3.3 I < t i P / I < Q P P ' for Arbitrarily Shaped Regions Containing a Dilute Distribution of Randomly Oriented Microcracks . . . . . . . . . . . . . . . . . . 358 11.3.4 I i ' i P / I < a P P ' for two Nucleation Criteria for Stlationary and Steadily-Growing Cracks . . . . . . . . 359 11.4 Contribution of Small Moduli Differences to the Toughening of T T C . . . . . . . . . . . . . . . . . . . . . 367 11.4.1 Introduction - . . . . . . . . . . . . . . . . . . . . . 367 11.4.2 Mathematical Formulation . . . . . . . . . . . . . 368 11.4.3 Calculation of Displacement Field . . . . . . . . . 375 11.4.4 Evaluation of Some Integrals . . . . . . . . . . . . 378 11.4.5 Correction for Moduli Differences . . . . . . . . . . 384

Contents

...

Xlll

11.4.6 Results and Discussion . . . . . . . . . . . . . . . . 389 11.5 Effective Transformation Strain in Binary Composites . . 390 11.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . 390 11.5.2 Effective Transformation Strains . . . . . . . . . . 391 11.5.3 General Bounds and Dilute Estimates . . . . . . . 393

395 12 Toughening in DZC by Crack Trapping 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 395 12.2 Small-Scale Crack Bridging . . . . . . . . . . . . . . . . . 396 12.3 Crack Trapping by Second-Phase Dispersion . . . . . . . . 402 12.3.1 Two-Dimensional Crack Trapping Model . . . . . . 402 12.3.2 Three-Dimensional Small-Scale Crack Trapping . . 408 12.4 Crack Trapping by Transformable Second-Phase Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 13 Toughening in DZC by Crack Deflection 13.1 Stress Intensity Factors at a Kinked Crack Tip . 13.2 Interaction Between Crack Deflection and Phase Transformation Mechanisms . . . . . . . . . . . . 13.3 Crack Deflection in a Zone of Non-homogeneous Transformable Particles . . . . . . . . . . . . . . 13.3.1 Computational Procedure . . . . . . . . .

425 . . . . . 426 . . . . . 429

. . . . . 434 . . . . . 437

14 Fatigue Crack Growth in Transformation Toughening Ceramics 443 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 443 14.2 Fatigue Crack Growth From Small Surface Cracks in Transformation Toughening Ceramics . . . . . . . . . . . 444 14.2.1 Examples of Fatigue Crack Growth . . . . . . . . . 446 14.3 Arrest of Fatigue Cracks in Transformation Toughened Ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 14.4 Improved Endurance Limit of Zirconia Ceramics by Overloading . . . . . . . . . . . . . . . . . . . . . . . . . . 459 15 Wear in ZTC 465 15.1 Experiment'al Observations of Wear in Zirconia Ceramics 465 15.1.1 Partially Stabilized Zirconia, PSZ . . . . . . . . . . 465 15.1.2 Tetragonal Zirconia Polycrystal, TZP . . . . . . . 468 15.1.3 Zirconia Toughened Alumina, ZTA . . . . . . . . . 469 15.2 Subsurface and Surface Cracks under Contact Loading in Transformation Toughened Ceramics . . . . . . . . . . . . 470

Cont e n2s

xiv

15.3 15.4 15.5 15.6

Mathematical Formulation . . . . . . . . . Subsurface Crack under Contact Loading Surface Crack under Contact Loading . . Concluding Remarks . . . . . . . . . . . . .

. . . . . . . . . 472 . . . . . . . . . 478 . . . . . . . . . 485 . . . . . . . . 497

Bibliography

501

Author Index

517

Subject Index

521

Part I

Introduction and Theory

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3

Chapter 1

Introduction All ceramics as a rule have very low resistance to crack propagation, i.e. very low fracture toughness. Cubic zirconia ceramics are no exception to this rule. They suffer two phase transformations between the melting point of zirconia a t about 277OOC and room temperature. These transformations which result in profuse microcracking can be eliminated by stabilizing the high temperature cubic phase with calcia (CaO), magnesia (MgO), yttria (Y203) or ceria (CeOz). However, the fully stabilized cubic zirconia still has low toughness and hardness and is not especially strong for engineering application. A development in 1975 exploited the phase transformation by using insufficient amount of stabilizer in order to inhibit the tetragonal ( t ) to monoclinic ( m ) transformation that would normally occur at about 1100°C on cooling from the cubic phase. This leaves the t-ZrOz phase in a metastable state. Substantial toughening is achieved when the retained metastable t-ZrOz is induced to transform to the monoclinic phase under high applied stresses, such as those a t a crack tip. T h e t --+ m phase transformation is martensitic in nature and is accompanied by a dilatation of about 4% and deviatoric shear strains of about IS%, if the t-Zr02 precipitates are unconstrained. The range of ceramic materials which exploit the controlled t m transformation has grown extensively. They are collectively called the zirconia-toughened ceramics (ZTC). Depending upon the matrix in which the metastable t-ZrOz precipitates are embedded, the ZTC are further subdivided into three main groups. These are designated partially stabilized zirconia (PSZ), tetragonal zirconia polycrystals (TZP) and dispersed zirconia ceramics (DZC).

-

4

Introduction

The metastable t-phase in PSZ exists as precipitates dispersed within a cubic stabilized zirconia matrix, the common stabilizing addition being CaO, MgO and Y2O3. In T Z P the entire polycrystalline body generally consists of t-phase. This is achieved by alloying with oxides which have a relatively high solubility in ZrOa at low temperatures, e.g. Y ~ 0 3and CeO2. In DZC materials, the metastable t-phase is dispersed in a nonZrO2 matrix which may be either an oxide or a non-oxide, e.g. A1203, Sic,Si3N4, TiB2, T I N . The athermal t -+ m transformation induced as a result of an applied stress field, e.g. a t a crack tip results in a net dilatation of 4%, but because of shear accommodation processes, e.g. twinning, the net deviatoric shear strain is much less than that of an unconstrained particle. It is not surprising therefore that. many works dealing with the toughening induced by the t rn transformation ignore the shear component and consider only the dilatation. However, as we shall see in this monograph the shear transformation strains not only contribute to the toughening but, more importantly, induce some new effects which are not known to exist under dilatational strains alone. Among these is the phenomenon of autocatalysis, whereby the stresses induced by the transformation of a few tetragonal particles are sufficient to induce further transformation which thus becomes a self-propagating process. The exact mechanism that triggers stress-induced (athermal) 1 + m transformation is still on open question. Some investigators believe that the transformation occurs spontaneously when the critical mean stress a t the location of a tetragonal precipitate attains a critical value. Others seem to favour a transformation criterion that includes both hydrostatic and deviatoric stress components. All known transformation triggering criteria have been discussed in this monograph to a certain extent, although the greater part of the discussion relies on the critical mean stress criterion. The contents of the monograph have been arranged in three parts. Part I (Chapters 2-6) gives a description of materials, and their constitutive equations and introduces the mathematical tools necessary for studying the interaction between isolated transformable particles and cracks. On the basis of these tools, the toughening induced by t + m in T T C is described in Part I1 (Chapters 7-10). The interaction of transformation toughening mechanism with other toughening mechanisms is investigated in Part 111 (Chapters 11-15) of the monograph, as are the fatigue and wear characteristics of TTC. A brief outline of the contents of each chapter follows. Chapter 2 gives a description of the various transformation-toughened ceramics -+

Introduction

5

(TTC), emphasizing the fabrication processes necessary for the ret,ention of tetragonal zirconia in a metastable state. Chapter 3 is devoted to the constitutive description of the whole class of T T C in the spirit of the classical theory of plasticity. It begins with a description of dilatational transformation plasticity and introduces the concepts of sub- and super-critical transformation. The influence of shear transformation strains upon the stress-strain relations of T T C is descrihed next, followed by that of DZC in which the elastic constants of the matrix and transformable phases are different. The complex stress potentials for small circular spots of arbitrary transformation strain are derived in Chapter 4 using Muskhelishvili’s method and Eshelby’s formalism. A second method based on the concepts of force doublets and dipoles (strain centres) is also used to develop Green’s functions for infinitesimal transformable spots. It is shown that in the limit of vanishing transformable spot size, the two methods yield identical complex stress potentials. T h e complex stress potentials for an isolated transformable spot of arbitrary shape are used in Chapter 5 to derive image potentials for semiinfinite and finite cracks interacting with such a spot. These potentials are then used to calculate the stress intensity factors a t the tips of the cracks as a function of the location of spot and the transformation strains in it. For many two-dimensional crack problems, especially those involving single or multiple surface cracks, the Muskhelishvili complex stress potentials cannot be constructed in a closed form. In such cases it is expedient to resort to the equivalence between appropriate line dislocations and cracks. The essential features of this equivalence (i.e. the dislocation formalism) are briefly outlined in Chapter 6. In particular, the weight functions necessary for the representation of single and multiple cracks in a plane or half-plane are derived for use in subsequent chapters. Chapter 7 is devoted to a continuum or macroscopic description of the composite zirconia systems in which the discrete transformable spots are smeared out into a transformation zone. The change in the stress intensity factor induced by the presence of the transformation zone near the tip of a semi-infinite crack is calculated when the crack is stationary or when it is growing under steady-state conditions. The emphasis is on sub- and super-critical dilatational transformation, although the role of shear transformation strain is also briefly explored. The complete analysis of the quasi-static growth of a crack, taking into account the progressive development of the transformation zone around the crack, is given in Chapter 8. As in the preceeding Chapter it

6

Introduction

is assumed that super-critical transformation occurs a t a critical mean stress and induces only dilatational strains. The analysis begins with that of a semi-infinite crack i n an infinite medium, followed by that of a single finite crack and an array of collinear finite cracks, and ends with that of a single and a periodic array of edge cracks. The computational difficulties arising from the penetration of a crack into a transformable inclusion are identified and strategies developed to overcome these. The plane strain continuum description given in Chapters 7 and 8 to what is essentially a discrete, three-dimensional problem is quite adequate if the number of transformed particles is large. When the transformation zone spans only a few particles and when the remote loading contains mode I1 and mode I11 components, besides that of mode I, a three-dimensional approach is called for. Chapter 9 is devoted to the derivation of analytical expressions for stress intensity factors induced along a half-plane crack front by unconstrained dilatational and shear transformation strains using three-dimensional weight functions. The discrete transformable domain is assumed to be in the shape of a sphere or a spheroid, and the influence of the orientation of an oblate spheroid relative to the half-plane crack front upon the transformation toughening is highlighted. The role of shear stresses, besides that of the hydrostatic stress, in triggering the t -+ m transformation is studied in Chapter 10, together with the contribution of transformation-induced shear strains to the toughening. This study is conducted not in the continuum plane strain approximation but by assuming that the tip of a semi-infinite crack is surrounded by a distribution of small transformable spots. It transpires that shear stresses created by super-critical transformation of a few spots may he sufficient by themselves to trigger the transformation of neighbouring spots, thereby creating a self-propagating autocatalytic reaction. Chapter 11 is devoted to the study of toughening mechanisms in dispersed zirconia ceramics (DZC), such as ZTA. T h e toughening in these ceramics can arise from two complementary mechanisms depending on the volume fraction of tetragonal zirconia. At low volume fractions, there is practically no stress-induced phase transformation, and the increase in toughness is primarily due to microcrack-induced dilatation around thermally formed monoclinic zirconia precipitates. At high volume fractions, on the other hand, the stress-induced transformation toughening mechanism seems to dominate over the microcrack mechanism. Both mechanisms are studied with reference to two ZTA compositions. Although the contribution of microcracking mechanism to the toughening

In trod uc t ion

7

of PSZ or TZP is believed to be minimal, even in these materials the slight mismatch in the elastic constants of tetragonal and monoclinic polyinorphs can give a significant effect upon the toughening process. This question is also addressed in Chapter 11. When the differences in the elastic moduli of matrix and transformable phases are large, as in all DZC, the concept of effective transformation strain is introduced. T h e toughening of DZC by the shielding of a macrocrack front by a zone of transformation or microcracks is sensitive to temperature. In these materials, toughening can also result from the inhibition of propagating cracks by second phase particles, i.e. by crack bridging. This toughening mechanism is not sensitive to temperature and often acts in conjunction with the transformation toughening mechanism. The mechanics of toughening by crack bridging is studied in Chapter 12, together with its interaction with the transformation toughening mechanism. Another potential mechanism of toughening in DZC is by crack deflection in combination with phase transformation. Cracks deviate from their planes when they encounter second phase particles, the deviation being all the more noticeable when these particles are non-homogeneously distributed in the matrix, as is always the case in DZC. The interaction between crack deflection and phase transformation toughening mechanisms is investigated in Chapter 13. Transformation-toughened ceramics have been found to be susceptible to mechanical degradation under cyclic loading. As in metals, the rate of growth of long cracks shows a power-law dependence on the applied stress intensity range. However, small cracks - the size of naturally occurring surface flaws - are found to grow a t stress intensity levels below the long-crack fatigue threshold, at which fatigue cracks are presumed dormant in damage-tolerant designs. Chapter 14 is devoted to the development of fatigue crack growth models, which predict the known longand short-crack fatigue behaviour of TTC. A detailed study is also made of the microstructural parameters that ensure crack arrest a t a given applied stress amplitude. It is demonstrated that occasional overloading can improve the endurance limit of T T C . T T C are known to exhibit poor wear performance under rolling/sliding conditions. The role of surface and subsurface cracks under sliding contact load in this poor performance is investigated in Chapter 15. The tetragonal precipitates are modelled as discrete circular spots. AS in Chapter 10, the influence of shear tractions under the contact load in the triggering of transformation is also examined.

9

Chapter 2

Transformation Toughening Materials 2.1

General

A ceramic is a combination of one or more metals or semi-metals (such as Si), with a non-metallic element. Depending on the non-metallic element present in the composition, a ceramic is classified as being an oxide (if the non-metallic element is 0 2 ) or a non-oxide. The comparatively large non-metallic ions serve as a matrix with the small metallic ions tucked into the spaces in between. The basic elements are linked by either ionic or covalent bonds (or both). The ceramics can be either amorphous or crystalline. Ceramics cover a vast field. One of the earliest known materials was a ceramic, viz. stone. Glass and pottery (and even concrete) are all ceramics, but the ceramics of most engineering interest today are the new high-performance ceramics that find application for cutting tools, dies, internal combustion engine parts, and wear-resistant parts. An excellent summary may be found in the paper by Morrell (1984). Diamond is the ultimate engineering ceramic and has been used for many years for cutting tools, dies, rock drills, and as an abrasive. But it is expensive. The strength of a ceramic is largely determined by its grain size, distribution of microcracks and processing technique. A new class of fully dense, high strength ceramics is emerging that is competitive on a price basis with metals for cutting tools, dies, human implants and engine parts. Ceramics are potentially cheap materials.

10

Transformation Toughening Materials

2.2

Modern Zirconia-Based Ceramics

Pure zirconia (ZrO2) suffers two transformations between its melting point (T,) a t about 277OOC and room temperature. These transformations which result in profuse microcracking can be eliminated by stabilizing the high temperature cubic form with calcia (CaO), magnesia (MgO), yttria (Y2O3) or ceria (CeOz). Cubic stabilized zirconia has a low toughness and hardness and is not especially strong. However an Australian development by Garvie et al. (1975) exploited the phase transformations by using insufficient amount of stabilizer in order to inhibit the tetragonal ( t ) to monoclinic ( m ) transformationthat would normally occur a t about 1100°C on cooling from the cubic phase. Toughening is achieved when this retained metastable t-phase is induced to transform t o the monoclinic phase under high stresses, such as those a t a crack tip. The t m transformation results in dilatation and shear strains that impede the progress of a growing crack. This class of ceramics is now known as transformation-toughened ceramics (TTC). The t + m transformation of confined t-ZrO2 particles under stress leads to the enhancement of other mechanical properties such as strength, thermal shock resistance, as well as of fracture toughness. An excellent account of the dependence of thermomechanical properties of T T C on microstructure may be found in the article by Hannink (1988) who is one of the three original developers of T T C . We provide a brief outline of material from that paper. The range of ceramics exploiting the athermal (i.e. high stress induced) t --t m martensitic transformation of ZrOz is large and is called the zirconia-toughened ceramics (ZTC). ZTC may be further classified into three groups:

-

1. Partially stabilized zirconia (PSZ) in which submicron-size t-ZrO2-precipitates are uniformly dispersed in an essentially c-Zr02-matrix. The amount of t-phase can be found by Xray diffraction techniques;

2. Tetragonal zirconia polycrystals (TZP) in which the principal constituent is the very fine-grained t-ZrOZ. Usually prepared with Y 2 0 3 as the stabilizer (1.75-3.5 inol%), though lately CeO2 is also being used; 3. Dispersed zirconia ceramics (DZC) in which t-ZrO2 precipitates are dispersed in a non-ZrOz matrix, the most common of which are Alz03, S i c , Si3N4, TiB2, TIN. When A1203 is

2.3. Martensi tic Transformation

11

the matrix, the corresponding toughened ceramic is called ZTA . T h e shape of the dispersed t-ZrO2 is determined by the constraint provided by the matrix. Thus, t-ZrO2 precipitates appear as spheres in ZTA and as thin oblate spheroids in PSZ (aspect ratio 5:l). In all ZTC, optimum mechanical properties are intimately connected with the size of t-ZrO2 precipitates and the metastability of t-phase. TO ensure that the t -+ m transformation does not occur a t the martensitic start temperature M , requires extreme care and control in sintering, hold (ageing) and cooling treatments. Thcsc will be briefly discussed later in this introduction. But first a few words on the t + m martensitic transformation.

2.3

Martensitic Transformation

The t -+m phase transformation in Zr02 is martensitic in nature. An important feature of this transformation is the existence of lattice correspondence (LC) between the unit cells of the parent (t-ZrOz) and product (m-ZrO2) phases. Adoption of a correspondence implies that the change in polymorphic structure can be approximated by a homogeneous lattice deformation i n which the principal axes of the parent lattice remain orthogonal except for a rigid body rotation. This is tantamount to minimization of the strain energy. The lattice strains (Bain strains) are given by eqn (2.1). Table 2.1 and Fig. 2.1 give the lattice parameters of the three polymorphic phases of ZrO2, from which it is possible to calculate the Bain strains due to transformation, e: ( i , j = 1 , 2 , 3 refer to the co-ordinate axes of c-ZrO2). In view of the close similarity in lattice parameters in all three polymorphs, the t ---* m transformation in ZrO2 has resulted in three nearly identical correspondences. These are distinguished on the basis of which monoclinic axis a m , b , or cnz is parallel to the tetragonal ct axis and are designated LC A, B and C, respectively. am eyl = - cosp - 1 at

12

Transformation Toughening Materials T 1 x e I 3 = e3', = - t a n ( p - -1 2 2

It is clear t h a t the unconstrained t + m transformation results in both dilatational (e: = eTl + e;? + eT3) and deviatoric (e?. ; i # j) strains '3. with about 4-6% dilatation and 14-16% deviatoric strains. However, since the transformation usually takes place in a constrained matrix ( a c-Zr02 in the case of PSZ and a non-Zr02-matrix in the case of DZC) a large proportion of deviatoric strains is cancelled as a result of twinning. T h e net deviatoric strains in a constrained matrix are likely to be of the same order as dilatation. In all Z T C , the aim is to produce materials whose martensitic start temperature A l , for t m is less than or equal to room temperature in order to prevent spontaneous transformation taking place during cooling. M S depends on the particle size of t-Zr02. T h e thermal treatment therefore aims to bring t-Zr02 to a critical size. T h e critical size of metastably retained t-Zr02 may be roughly calculated from thermodynamic considerations. T h e transformation of a metastably retained t - Z r 0 2 to a lower energy m - Z r 0 2 state requires that an activation (nucleation) barrier AF" (Fig. 2.2) be overcome. This means a critical (embryonic) nucleus must be activated before the martensitic t m transformation can occur a t a temperature designated M,. T h e actual state of metastability of the I-phase will depend upon a number of physical and structural factors. This state is depicted schematically with the use of a free energy diagram in Fig. 2.2 for the

-

-+

Lattice Parameter (nm) Material

Monoclrnic am

bm

Ca-PSZ

8.4

0.5132

0.5094 0.5180

0.5171 0.5182 0.5296 98.67"

Mg-PSZ

9.4

0.5080

0.5080 0.5183

0.5117 0.5177 0.5303 98.91"

Y-PSZ

7.4

0.5130

0.5116 0.5157

Ce-TZP

12

-

0.5132 0.5228

-

0.5193 0.5204 0.5362 98.80"

Table 2.1: Lattice parameters

2.3. Mar tensi tic Transforma tion

13

f

Figure 2.1: Crystal polymorphs. (a) monoclinic, ( b ) tetragonal, and (c) cubic

14

Transform ation Toughening Materials

'1

L

1

b

Reaction coordinate

Figure 2.2: Schematic illustration of the free energy forms associated with the t + m transformation relative to the free energy of the initial constrained and/or doped t-phase, as a function of the initial particle size - small (S), crit,ical (C) and large ( L ) . Also shown is the driving force for the reactions, AFchern and A F , for unconstrained and constrained t particle respectively, and the activation energy A F a for the metastable S and C particles

t-phase with respect t o the resultant. m-phases. T h e free energy F T ~ ~ of the t-phase constrained or chemically doped, is shown on the left of the diagram. T h e free energies for various states of the resultant mphase F M relative ~ ~ to ~the normalized t-phase condition are shown on the right hand side of the diagram. A t the top right is the free energy F ~ ~ ~of ~small i s orl highly doped rn particles. At room temperature in this condition, the M, of the particle is well below room temperature so t h a t the particle is stable within a constraining matrix and the free energy of the resulting m-phase is higher t h a t of the t-phase; thus there is no net driving force to complete the transformation. Also the activation energy for the nucleation stage AF; is virtually insurmountable. A particle whose M, is well above room temperature, e.g. F M ~ ~ ~ ( L ) will transform on cooling, as AF' will be lower and easily surmounted by thermal energy k T (where k is Boltzniann's constant) a t tempera-

15

2.3. Martensi tic Transforma tion

tures sufficiently below M,, i.e. the particle is unstable with sufficient undercooling and the driving force A F is large. In the intermediate case, e.g. F M ~ ~where ~ ( M, ~ )is just below room temperature AFC is small and can be surmounted as a result of an applied stress, i.e. the particle is metast,ahle. Therefore there exists a critical 1 size or state, attainable through thermal treatment and/or doping, below which the particle can be induced to transform with the aid of an applied stress, and above which it will not. Therefore, it is the aim of the thermal treatment process and chemical doping to bring the t particles to this critical condition.

2.3.1

Retention of the t-phase

As described above, retention of the t-phase is the most important factor for utilizing the transformation toughening phenomenon. As depicted in Fig. 2.2 the stability (or metastability) of the t-phase is most readily described in terms of the thermodynamics of the transformation. General models entail the total free energy change associated with the transformation of a spherical particle by considering variables such as bulk, surface and twinning free energy terms. We shall briefly look at the practical situation using the ”end-point’’ approach in which only the final energies are considered without touching the nucleation question. Simplistically, for an unconstrained crystal the free energy F is given by

4 F = , 7 r r 3 F c ~+ 4 a r 2 S c ~ i)

where r is the radius of the crystal, FCH and SCH are the free energy per unit volume and surface energy of the crystal respectively. The difference in free energy between the t and rn polymorphs is given by

4

AFo = ;“r3(Ft i)

-

F,n)

+ 4wr2(St - S m )

The t-form can exist at a critical value rc when AFo is zero, at a particular temperature below the normal tra.nsformation temperature. Thus we can write

Realistically, however, when modelling particles constrained within a confining matrix, as in PSZ and ZTC,.it is not sufficient to consider only

16

Transform a tion Tough ening Materials

the bulk chemical and surface energy terms to account for the stability of t h e t-phase. Study of the coarsening behaviour of precipitates in Ca-PSZ showed t h a t the transformation temperature could be predicted if the total free energy change also included such terms as dilatational strain plus changes in the particle-matrix int.erfacia1 energy. Subsequently, this approach has been taken even further to include the energy contribution of twinning and microcracking. Expanding ( 2 . 3 ) to include the additional terms, the free energy description for the transformation of a spherical constrained particle of radius r , in the absence of a n applied stress, becomes

where A F o is the total free energy change, A F the free energy change per unit volume, and subscripts C H , D and SH refer to chemical, dilat,ational and shear energy contributions. A S denotes the free energy change per unit area. and subscripts TW and P refer to the contributions from twin boundaries and precipitate (particle/matrix) interface, respectively. By summing the various terms (2.5) can be written as

C

where V is the volume of the particle, AFST is the sum of strain energy and C A S the s u m of interfacial energy contributions. Expressing A FCH in terms of experimental variables and applying conditions for equilibrium, (2.6) becomes

where rc again denotes the critical particle size, q and Tb are respectively the enthalpy of the transformation and the transformation temperature of a crystal of "infinite" radius, and T is the transformation temperature ( M s )of a zirconia particle of radius r c . Coinparison with experimental d a t a gives very good agreement for this model of particle stability. Nucleation of the transformation is another consideration, and this is now often discussed in terms of a soft mode mechanism. T h e actual nucleation mechanism shall not concern us here, except to say t h a t for t h e systems under discussion nucleation has not been a problem when

2.4. Fabrication and Microstructure of PSZ

17

the materials have been suitably processed. In the following sections we shall briefly describe the most essential steps required in the fabrication of PSZ and TZP. Those interested in learning more about DZC should consult the paper by Hannink (1988).

2.4

Fabrication and Microstructure of PSZ

The manufacture of PSZ requires high purity ZrOa powder (submicron grain size) with stabilizers which are soluble in ZrOz. This means primarily CaO and MgO, although in the early development phase of PSZ, Y2O3 was also used but it is only soluble in Z r 0 2 a t rather high temperature (around 1800°C). We shall here explain the fabrication process of PSZ only with respect to MgO. The optimum performance of this composition, denoted MgPSZ, results from a stabilizer content in the range 8-10 mol% (see the zirconia-rich end of the phase diagram in Fig. 2.3). If the stabilizer content is less than 8 mol% there is less control over the precipitation of t-ZrO, because sintering has to take place a t a higher temperat.ure. On the other hand, if it is more than lo%, then it has been found that the t-ZrOa precipitate content is not enough for attaining the best possible fracture toughness. Even with the use of high purity powders, silica can be a significant contaminant which preferentially reacts with the MgO stabilizer. The contaminant can be removed by the addition of SrO during sintering. The compact of Mg-ZrOz is heated to about 1700°C: (i.e. to just above the c-ZrO2 phase boundary, Fig. 2 . 3 ) . The high solution temperature results in a somewhat coarse grain size (30-60 p m ) . The heating is followed by a rapid cooling (cooling rate x 50O0C/hour) to a temperature just above or below the eutectoid boundary (Z 14OO0C, Fig. 2.3). The cooling process results in the production and retention of uniform tZrOa precipitates (needle shaped wit.h c-axis = 30-60nm) in c-ZrOa grains and in the prevention of pro-eutectoid ZrO2 at grain boundaries. T h e uniform distribution of precipitates and prevention of pro-eutectoid ZrOz formation is readily achieved by rapid undercooling to a temperature below the eutectoid boundary. In this treatment, classical homogeneous nucleation and precipitation operates. If cooling is too slow, large areas of pro-eutectoid ZrOa form at grain boundaries due to heterogeneous nucleation. If allowed to develop, the grain boundary phase will control the thermomechanical properties. For commercial expediency, the initial rapid cool or quench is often slowed to allow some coarsen-

Transform ation Toughening Materials

18

O b k Solid Solution

Monoclinic + MgO

0

5

10

#I

15

Mol% MgO

Figure 2.3: Phase diagram of Mg-PSZ ing of the homogeneously precipitated phase, so that a large fraction of precipitates will be metastable at room temperature. The ultimate precipitate morphology in a fabricated body will depend upon the type of stabilizer used. The microstructures resulting from the various cooling procedures and subsequent heat treatments for the various alloy systems will be described below.

2.5 2.5.1

Microstructural Development Ca-PSZ

Commercially viable Ca-PSZ compositions occur in the range 3-4.5% CaO-ZrOz. Sintering and solution treatment is thus determined by the phase boundary position. For an 8.4 mol% (3.8 wt%) Ca-PSZ alloy, firing is carried out at 18OOOC. Figure 2.4 shows the t-precipitate coarsening sequence of an 8.4 mol% Ca-PSZ alloy. For the three precipitate sizes shown in Fig. 2.4a-c; (a) is the "as-fired" (UA) size obtained from the sintering solution treatment

2 5 . Microstruct ural Development

19

Figure 2.4: Microstructural development of the precipitate phase in Ca-PSZ (a) as-fired, (b) peak-aged, and (c) over-aged material. Dark field transmission electron microscopy images

Transformation Toughening Materials

20

B

800

c

3.3% CaO

200

I

20

I

I

60

I

I

,

I

I

I

100 140 ,180 Ageing time at 1300 C, h

Figure 2.5: Flexural strength attainable for various Ca-PSZ compositions (wt% CaO) as a function of ageing time a t 1300°C:

followed by a rapid cooling sequence; Figures 2.4b and c are the "peakaged" (PA) and "over-aged'' (OA) sizes, respectively. These precipitate sizes can be compared with the schematic free energy situations of the m-phase shown in Fig. 2.2, where UA, PA, and OA equal S, C, and L , respectively. The S material cannot be induced to transform by stress ( A F * too large and free energy difference positive), and the L material transforms during cooling to room temperature ( A F * virtually zero and free energy difference large, MS=38O0C).The C material has an M , a t approximately 20°C and an M j well below room temperature so that the precipitates can be considered critically sized. A number of workers have studied the growth rate (ageing kinetics) of the Ca-PSZ system using transmission electron microscopy and Xray measurements. They showed that while the 2 precipitates remained predominantly coherent with the cubic matrix up to xlOOnm, precipitate coarsening obeyed a t ' J 3 relationship. Growth in excess of z l 0 0 n m resulted in a loss of coherency a t the ageing temperature, impingement of the growing precipitates and a break down of the t ' l 3 relat,ionship. Also with precipitate sizes >100nm, M , is above room temperature so that the precipitates transform to m during cooling and the material is said to be over-aged. The precipitate size of x lOOnm also coincides with the ageing time at which the optimum mechanical properties are

2.5. Microstruc t ural Development

21

A more recent study however found that initially a t'/' (quadratic) relationship is obeyed for the growth of precipitates up to 30nm. When precipitate growth exceeds this value a t ' / 3 is followed. These observations indicate that interfacial growth rather than lattice diffusion is the growth-rate controlling mechanism. T h e t precipitate size and metastability is most strongly reflected in the mechanical properties. Figure 2.5 shows the flexural strength as a function of ageing time a t 13OOOC for a series of Ca-PSZ alloys. The trend in flexural strength as a function of ageing time illustrates very well the effect of stabilizer content on thermal processing treatment. The precipitate growth for 4wt% CaO is such that approximately 6Oh ageing is required to achieve the PA condition. Figure 2.5 also shows the rapid drop-off in strength as the sample is overaged (OA). While Ca-PSZ was the first transformation toughening system to be developed it has found little commercial application owing to the greater versatility and better mechanical properties achievable in the Mg-PSZ system.

2.5.2

Mg-PSZ

Rapid and controlled cooling treatment

Due to the variety of possible treatments and the resultant microstruct u r d features, the Mg-PSZ system is possibly the most complicated. Typical commercial compositions are shown by the hatched region of Fig. 2.3. After firing, the ceramic is generally rapidly or controlled cooled as described above. A typical rapid cooling rate is x 500°C/h which results in a t precipitate size with the long axis in the range 30-60nm. The precipitates in this system occur as lenticular ellipsoids with {loo}, habit planes and the ct-axis always parallel to the axis of rotation of the lens, i.e. the short axis. Figure 2.6 shows a dark field electron microscope image of typical as-fired rapidly cooled precipitates. The M, of these precipitates is well below room temperature so that further ageing is necessary. With a starting precipitate size this small, and because of eutectoid decomposition, only ageing treatments above the eutectoid temperature (14OOOC; see Fig. 2.3) should be used. A heat treatment in the sub-eutectoid region can be beneficially employed under certain circumstances, as will be described later. Figures 2.6b and c illustrate the coarsening sequence of the small precipitates using an ageing treatment of 14OO0C. Optimally aged (PA) precipitates have mean diameters of ~180nm and a thickness of ~ 4 0 n mi.e. , an aspect ratio of x 4 . 5 : l .

22

Transform at ion Toughening Materials

Figure 2.6: Microstructural development of the precipitate phase in Mg-PSZ. (a) as-fired (UA or S), (b) aged for 2h (PA or C) , and (c) fired for 4h (OA or L) a t 142OOC. Transmission electron micrographs (a) dark field, (b) and (c) bright images. Bar length = 0.5pm

2.5. Microst ruc t ural Development

23

x l 8 0 n m and a thickness of x40nm, i.e. an aspect ratio of x 4.5:l For commercial reasons, controlled cooling is more economical and, when used to greatest effect can produce materials of maximum strength from a single firing. Optimum sized precipitates from a controlled cool are just below the PA size, so that the material can be sub-eutectoid aged to achieve maximum toughness or a variety of properties in between depending upon the industrial application. Isothermal hold treatments

Recently, detailed studies have been performed to examine the precipitation process in a 9.7 mol% Mg-PSZ material using isothermal arrests during a 500°C/h cooling curve which started a t 1700OC. From the cooling cycle and different hold times employed, three different precipitate forms were identified: (1) primary; (2) large random, and (3) secondary. Three further forms could be identified after prolonged isothermal hold treatments for subsequent sub-eutectoid ageing. These were (4) intermediate precipitates, resulting from the growth of primary precipitates when a sample is isothermally held above the eutectoid temperature; (5) tertiary precipitates, which grow in the matrix and in regions surrounding large random precipitates, and which result from supersaturation of the cubic stabilized zirconia matrix phase when material containing pro-eutectoid precipitates is aged below 120OoCc;and (6) a eutectoid decomposition product which nucleates and grows from the grain boundaries during ageing a t 1100OC. We shall concern ourselves only with the first three precipitate forms. Primary precipitates are similar to those described for the rapid cooling sequence (Fig. 2.6a), and are a consequence of homogeneous nucleation and groth from a supersaturated solid solution when the material is cooled below the c / ( c t ) phase boundary (Fig. 2.3). Large random precipitates occur in materials continuously cooled to room temperature a t 500 OC/h and nucleate a t matrix inhomogeneities such as pores and inclusions (Fig. 2.7a). They are more numerous as a result of autocatalytic nucleation and growth, in samples isothermally held a t temperatures above 14OO0C (Fig. 2.7b). They do not contribute to the transformation toughening behaviour as they are all monoclinic a t room temperature. These precipitates do, nevertheless contribute to the toughness through their crack deflection and bridging contributions. Secondary precipitate growth (SPG), once nucleated, occurs rapidly in a temperature zone of about 1250-14OO0C, just below the eutectoid temperature and approximately 350°C below the c/(c + t ) phase boundary.

+

24

Transforma tion Toughen ing Materials

Figure 2.7: SEM images of Mg-PSZ (a) cooled a t 500°C/h to room temperature showing large random precipitates (arrowed) nucleating on a second phase particle in a matrix of primary precipitates, (b) large random precipitate distribution after isothermal hold of 250 min a t 155OOC showing precipitates nucleating on pores Polished and HF-etched surfaces, examined by scanning electron microscopy, show that nucleation of SPG originates within the grain but near grain boundaries. Once nucleated, growth of these precipitates sweeps rapidly around the grain periphery and proceeds inwards t o the grain interior. When the growth process has not gone to completion the surface exhibits a spotty or mottled appearance at low magnifications. At high magnifications the spot region reveals the presence of two precipitate sizes. These sizes are primary within the spot and secondary a t the periphery. Often some large random precipitates are also present within the spot. Serial sectioning shows that once spots are observed on a surface, all grains within the sample have incomplete SPG. Apparent completion of SPG in some grains is a reflection of the sectioning process. The SPG process results in precipitates which are very uniform in size and once formed are remarkably stable t o further coarsening at the hold temperature. The actual precipitate size is a function of the isothermal hold temperature. An example of this stability is illustrated in Fig. 2.8a and b , which shows that precipitate and spot development after an isothermal hold of 140 min and 1290 min at 1375"C, respectively. In Fig. 2.8b it is evident that the spot area has been almost entirely

2.5. Microstruct ural Development

25

Figure 2.8: SEM image illustrating secondary precipitate stability against further coarsening. Mg-PSZ isothermally held at 1375OC for (a) 140 niin, and (b) 1290 min. Note no observable change in SPG regions; white spots have been almost completely consumed by large random precipitates consumed by large random precipitates while the precipitates in the outer SPG region have not grown noticeably when compared to Fig. 2.8a.

Sub-eutectoid ageing One of the toughest sintered ceramics known may be produced from suitably prefiring Mg-PSZ and optimally ageing a t llOO°C, i.e. by subeutectoid ageing. Suitably prefired materials are ones which contain precipitates near PS condition for transformation toughening, and are most conveniently produced by either controlled or isothermal hold cooling sequences. The microstructural influence of the sub-eutectoid ageing treatment on optimally prefired materials (and hence the resultant mechanical properties), may be summarized into four main features. First, the anticipated decomposition of the c-ZrO2 matrix phase, (see Fig. 2.3) occurs at the grain boundaries and pores. This decomposition reaction is not significant in terms of mechanical property degradation for the ageing times generally used. The decomposition reaction may be considerably slowed by use of suitable sintering aids. The other three processes occur

Transformation Toughening Materials

26

.

N

E 600

r

Ageing p m p 1100 c

CI

N r

2 a

c

0

2 200

iL

~~

1

5

50

10

Time (h)

Figure 2.9: Comparison of fracture surface energy for 9.4mol% MgPSZ when optimally aged above and below the eutectoid temperature (1400°C) within the grains, and are

1. formation of an ordered anion vacancy phase phase) at the precipitate-matrix interface;

Mg2Zr5012

(5-

2. precipitation of very small t particles within the cubic matrix of the precipitate-laden grains; and

3. the transformation on cooling of some of the original t precipitates. As described above strength may be optimized by a number of heat treatment regimes but optimum toughness coupled t o strength is more readily and reliably attained by sub-eutectoid ageing. Figure 2.9 compares the fracture surface energy after ageing treatments above and below the eutectoid temperature. The significant increase in fracture surface energy is also a reflection of the increase in fracture toughness. Another benefit of the llOO°C ageing treatment is the display of transformation plasticity (see below) and crack growth stability which is indicative of R-curve behaviour.

2.5. Microstruc t ural Development

27

2.5.3 Y-PSZ As shown at the top of Fig. 2.10, Y-PSZ materials occupy a wide range of compositions. Y-PSZ in the 3-6mol% composition range is of little commercial interest as an engineering ceramic because the high solid solution temperatures and sluggish nature of the precipitate growth process make the materials commercially unviable. Solution treatment and firing is generally carried out for this system at 1700-2000°C and results in large 50-70pm grains. Firing is followed by a rapid cool and subsequent ageing at 1300-14OO0C. Controlled and isothermal hold cooling sequences have not been reported for these materials. Figure 2.11 shows the microstructural development from (a) the ”asfired” state to (b) the coarsening stage after ageing at 1300°C. Upon coarsening the precipitates agglomerate into rectangular plates and colonies composed of twin related variants. Microstructures in this system are complicated by the presence of two tetragonal forms, t and t’. The t phase is low in solute and when suitably sized may be stress induced to transform. The t’ phase is high in solute and must be decomposed into the fully stabilized cubic form and

Figure 2.10: Phase diagram of Y-PSZ and Y-TZP

28

Transformation Toughening Materials

Figure 2.11: Microstructural development of (a) 9.4mol% Y-PSZ as fired "tweed" structure and ( b ) same sample aged for lOOh a t 13OO0C. Note rafting and rectangular shaped plates of impinging 2 precipitates. Transmission electron micrographs (a) bright field, (b) dark field image metastable t before transformation toughening benefits can be obtained. The optimal mechanical properties of Y-PSZ materials can approach those of the other PSZ systems, but are of little interest because better properties may be obtained from Y-TZP materials.

2.6

Fabrication and Microstructure of TZP

The manufacture of T Z P requires high purity ZrO2 powders (usually with size distribution in the range 10-200nm) with Y203 or CeO2 as a stabilizer. We shall describe briefly the development of microstructure in both Y-TZP and Ce-TZP, because of their very distinct features.

2.6. Fabrication and Microstructure of T Z P

2.6.1

29

Y-TZP

The compact of Y203-Zr02 is fired a t a temperature of between 1300 and 1500OC. Optimum performance is ensured by a stabilizer content of between 1.75 and 3.5mol% (3.5-8.7wt%), as seen from the zirconia-rich end of the phase diagram in Fig. 2.10

2 ma1 %

2.5 mol 96

Figure 2.12: SEM images of Y-TZP revealing equiaxed grains The grains in the fired product are equiaxed (see SEM image on Fig. 2.12) and are around 0.5-2pm in size depending upon firing time, temperature and solute content. The t-Zr02 content can vary between 100-60% (the remaining being mostly c-ZrOz). As seen from Fig. 2.13, Y-TZP materials can be made exceedingly strong, however they suffer from instability and severe strength degradation after exposure to moist air or hot water in an autocla.ve a t M 230OC. The loss in strength has been attributed to a number of factors. One of the most plausible is that gross destabilization of the t phase and subsequent transformation of the surface layers to m results in the introduction of incipient flaws. Several approaches have been used to overcome this instability. These range from reducing the initial grain size to increasing the grain boundary silica with a thin layer of stable c-ZrOz. All the approaches have generally resulted in a reduction of the desired mechanical properties. Thus, while no overall satisfactory solution has been found a decrease in the metastability of the t-phase appears the most favoured approach. More recently strengths of 2.0-2.4GPa and fracture toughness of 3.56.0 M P a 6 have been achieved by the addition of 5-30wt% A1203 to 2.5mol% Y-TZP. Aside from improved processing procedures, the in-

30

Tkansforma tion Toughening Materials

z

5

2 0

-5 E a x

E

2wor HIP 14OO0C 140MPa Normal sintering

I.Iol

1500

14oooc

1000

u

i

2

3

4

Y,O, mol%

Figure 2.13: Flexural strength dependence on Y 2 0 3 content of sintered and isostatically hot pressed Y-TZP. HIP sample was sintered at 1350°C for 2h prior to HIP treatment crease in strength is not fully understood in terms of transformation toughening.

2.6.2

Ce-TZP

Tetragonal phase stabilization in the CeOz-ZrO2 system can occur over a wide range of compositions, 12-20 mol% CeOz. The preferred composition is 12 mol% CeOz-ZrOz . Fabrication is normally carried out by firing at ~ 1 5 0 0 ° Cfor l h . Further consolidation by HIPing is not used due to the ready reducibility of CeO2 to CezOs, in conditions such as those encountered in HIP units with graphite dies and heaters. The grain size after fabrication is in the range 2-3,um with microstructures of equiaxed grains similar to those for Y-TZP, as shown in Fig. 2.12. Ce-TZP materials display considerably greater stability over Y-TZP under similar environmental conditions. The mechanical properties of Ce-TZP as function of Ce02 composition are shown in Fig. 2.14. From this figure it is evident that, while flexural strength is not as high as that of Y-TZP (Fig. 2.13) the toughness can be considerably greater

2.6. Fabrication and Microstructure of T Z P

31

h

k

u, >

r

h

r -

E

30-

6

E

20-

0

hi-

10 I

1

14

16

C e 0 2 mol%

Figure 2.14: Mechanical properties of Ce-TZP materials as a function of composition and grain size, (top) Vickers hardness, (middle) flexural strength, and (bottom) critical stress intensity factor (maximum K I for ~ Y-TZP M 1 0 M P a m . The response of Ce-TZP to various stress situations has attracted considerable attention due to its ability t o undergo considerable plastic strain. They have particularly interesting mechanical properties at sub-zero temperatures. For instance, the fracture toughness at -5OOC is almost twice the value at 2OoC making this ceramic composition of particular interest for low temperature application. Moreover, increased toughness is accompanied by an increasingly pronounced deviation of the macro-

Transforma tion Toughening Materials

32

I

m

Figure 2.15: Optical observations of the transformed zone about cracks in Ce-TZP at various temperatures. (a) 100°C, (b) 2OoC, (c) -lO°C, and (d) -4OOC

2.6. Fabrication and Microstructure of T Z P

33

scopic stress-strain curve from the linear relationship. Values of the overall inelastic strain a t failure up to 0.3% in tension or flexure have been reported. In that sense, phase transformation induces a sort of plasticity into the Ce-TZP material, as is evident from the optical images in Fig. 2.15. It is clear that the transformed zone at sub-zero temperatures resembles closely the crazing a t crack tips in polymers with very distinct shear bands.

This Page Intentionally Left Blank

35

Chapter 3

Constitutive Modelling 3.1

Introduction

In this Chapter we will discuss three constitutive models of TTC materials. In contrast to conventional ceramic materials which exhibit linear elastic behaviour up t o failure these materials exhibit a nonlinear stressstrain behaviour once a certain stress level is reached. It is believed that the occurrence of stress induced t -, m transformation results in the formation of inelastic strains, which allow the material to redistribute stress and are an important feature in the crack growth resistance of T T C. To reduce the complexity of the transformation we assume that we can identify a material sample which is small compared to all macro-

Figure 3.1: Schematic representation of the microstructure of PSZ ceramics, with transformed penny-shaped particles

36

Constitutive Modelling

scopic dimensions, but which is large enough that statistical averaging over all transformable particles is meaningful. For example, a continuum element of PSZ ceramics may look like the schematic drawing in the left of Fig. 3.1. Such a material sample can then be treated as a continuum element for which all (macroscopic) quantities are averages over the sample. Phenomena on smaller scale are discarded. This means, for instance, that local stress and strain fields around individual particles are not considered, but only the macroscopic average of these fields over all particles in the sample is relevant. The differences in the three models to be discussed in $83.2, 3.3, and 3.4 arise from the assumption of the influence of the shear component of the transformation. The first model is based on the assumption that the shear component may be neglected completely and includes only the dilatant part. The second model includes both the dilatant and the shear transformation strains but assumes that the strain deviator is parallel to the stress deviator. The third model does not make this assumption. In the first two models, the elastic constants E and v of matrix and inclusions are assumed to be identical. The third model relaxes this restrictive assumption. As the martensitic transformation proceeds with the speed of sound, we neglect any dynamical effects and assume that the transformation occurs instantaneously and is time independent.

3.2

Constitutive Model for Dilatant Transformation Behaviour

As described in Chapter 2 the transformation from tetragonal to monoclinic structure involves both dilatant and shear components. However, the first constitutive model developed by McMeeking and Evans (1982), and Budiansky et al. (1983) neglects the shear component, arguing that when the tetragonal particle is embedded in the surrounding matrix it transforms into a number of bands with the sense of the shear alternating from one band to the next. In this viay the average shear component is much less than 1696, and may even play no role a t all, while the dilatation remains about 4.5%. The effect of twinning on the residual shear is schematically demonstrated in Fig. 3 . 2 . Of course, transformations which produce a single twin variant per particle will also interact with the shear strains, depending on the orientation of the particle. However, this is not considered in the modelling of dilatant transformation behaviour, for which we will follow closely the

3 . 2 . Dilatant Transformation Behaviour

37

Figure 3.2: Schematic representation of the influence of twinning on the transformation shear, demonstrating that twinning reduces the influence of the macroscopic shear component exposition of Budiansky et al. (1983). Consider a special two component material comprising a linear elastic matrix material with embedded particles which undergo an irreversible inelastic dilatation. The assumed constitutive behaviour of the individual particles is not, in general, the same as that of a particle undergoing a true dilatant phase transformation. Nevertheless, the results for this idealized composite do lend insight to what can be expected for a more complicated system. Each component of the composite is assumed isotropic. The particle and matrix have identical linear shear behaviour so that in each

where sij is the stress deviator, eij is the strain deviator, and p is the shear modulus. The matrix material responds linearly under hydrostatic tension and compression with bulk modulus B according to 6,

1 = 3flPP = B E P P

(3.2)

where uij and ~ i i jare the stress and strain tensors, u, is the mean stress, and E~~ is the total dilatation. The dilatant response of the particles is depicted in Fig. 3.3. Under monotonically increasing E ~ the ~ particles , satisfy ( 3 . 2 ) with the same bulk modulus B as the matrix material as long as

where 6; is the critical mean stress associated with the start of transformation. On the intermediate segment of the curve the incremental response is governed by B’ according to

Constitutive Modelling

38

%r

%P

Matrix

Particles

Composite

Figure 3.3: Dilatant stress-strain behaviour of particles and matrix material making up a two-phase composite. The shear behaviour is linear with the same shear modulus in both phases. The macroscopic behaviour of the composite is also shown

U,

= B’ipp

(3.4)

T h e inelastic, or transformed, part of the dilatation in the particle is denoted by 6, and it is defined as the difference between the total dilatation and the linear elastic dilatation, i.e.

0, =

E PP - U m / B

T h e maximum transformed dilatation in each particle is 6; (3.4) holds in the interval

(3.5) and thus

For larger values of E~~ the incremental response is again governed by the elastic modulus according to urn = BiPP

(3.7)

T h e volume fraction of the particles is denoted by c and no assumptions on their shapes need be made. The stress-strain behaviour to be shown for the composite is exact with the usual interpretation of average,

3.2. Dilatant Transformation Behaviour

39

or overall, stress and strain for a multi-phase material. The shear modulus of the composite is p at all strains and the bulk modulus B governs for cpv _< o ' C / B and again, incrementally, at sufficiently large Spy as shown in Fig. 3.3. Once cpv exceeds o'~n/B the response is incrementally linear with the macroscopic relation --

(3.s)

where B satisfies 1

+ 4p/3

=

C

B' + 4p/3

+

1--C

B + 4p/3

(3.9)

B is derived using Hill's (1963) method for the determination of the overall moduli of two-phase isotropic elastic composites both of whose phases have the same shear modulus. The macroscopic stress and strain of the composite are denoted by upper case letters. The dilatation is uniform in each particle and is the same in all particles. The overall transformed dilatation of the composite is defined by 0 -

Epp - E r n / B

(3.10)

On the intermediate segment of the curve 0 -

(1- B/B)(Epv - E~/B)

(3.11)

with the overall and local transformed dilatations related by (9 - c0v. The maximum, or complete, transformed dilatation of the composite is 0T

--

(3.12)

cOT

The strain range of the intermediate branch is ,~ < Ev v < 2 ~ B --B+

Or

[

1-

(3.13)

For Epv above the upper limit in (3.13) no further transformation occurs and (3.7) applies. If at any strain level beyond (r~/B the dilatation starts to decrease, the transformed dilatation in the particles is frozen and the incremental unloading response is governed by B. Equation (3.9) for B remains valid even for negative B', although this result is not necessarily unique when B' < - 4 # / 3 . If B / > - 4 # / 3 , the

Constitutive Modelling

40

equations governing the incremental behaviour of the particles are elliptic so that the stress and strain fields in the particles are necessarily smooth and unique. If B I < - 4 p / 3 the incremental equations are hyperbolic and certain discontinuities in the stress and strain fields in the particles become possible. Furthermore, if B I < - 4 # / 3 a particle in an infinite elastic matrix with moduli B and p can transform completely to Op T as soon as the critical mean stress is reached. From (3.9) it is seen that the transition from elliptic to hyperbolic behaviour in the particles at B ~ - - 4 # / 3 gives B - - 4 p / 3 , so that the corresponding transition for the composite coincides with that of the particles in this special system. In what follows we adopt the dilatant stress-strain behaviour for the composite shown in Fig. 3.3 and again in more detail in Fig. 3.4. It must be emphasized that we are not assuming that the particle response in an actual system is that specified above. Experimental observations suggest that partially transformed states of individual particles do not exist. A given particle is either untransformed or fully transformed. On the other hand, an actual composite system will usually have a distribution of particle sizes with an associated distribution of critical stresses. Thus, even though each particle transforms completely when its own critical stress is attained, a distribution of critical stresses may result in a composite whose incremental bulk modulus never drops below - 4 # / 3 . Indeed, if the distribution is sufficiently wide it is conceivable that the incremental bulk modulus of the composite might not even become negative. For the stress-strain behaviour in Fig. 3.4 with B > - 4 p / 3 , it can be shown that the spatial distribution of the macroscopic transformed di-

~m

~m Loading//B c

C

Zm

B/ Unloading /0 ~ II

II II s

B=O

,

Zm

/ t) i

/; ~Unloading /

d B<0

F i g u r e 3.4: Dilatant stress-strain behaviour of the composite

3.2. Dilatant Transformation Behaviour

41

latation 0 must be continuous. Thus, in general, there will exist regions in any nonhomogeneously deformed composite in which 0 < 0T, corresponding to partially transformed macroscopic states. If B < -4#/3, discontinuities in 0 can occur. The condition B < - 4 # / 3 has additional significance for the incremental behaviour of the composite. With # > 0, this is the condition for a real longitudinal wave speed. It is also the condition which excludes internal stretching necking modes that can, in principle, develop when ellipticity is lost. Not only is the physical response of the composite strongly dependent on whether B is greater or less than -4#/3, the mathematical techniques used to generate solutions to the crack problems posed in this Monograph also differ depending on whether B is above or below this transition. For this reason we introduce the following to distinguish the two ranges of behaviour B > -4#/3

<=> sub-critically transforming composite

B--4#/3

r

critically transforming composite

B <-4#/3

r

super-critically transforming composite

m

m

(3.14)

The incremental form of the equations for the composite are analogous to those for an elastic-plastic solid. For any stress state not on the transforming branch

Em-

BEpp

and

0-0

(3.15)

On the transforming branch there are two possible incremental responses which will be called loading and unloading, borrowing plasticity terminology. As shown in Fig. 3.4, loading occurs if Epp > 0 and

Em

-

-BEpp

with O - ( 1 - B/B)/~pp

(3.16)

Unloading occurs if/~pp < 0 and then (3.15) applies. Upon unloading to zero mean stress the transformed, or permanent, dilatation is 0. If 0 is less than 0 T the material is said to be partially transformed, in the sense described above, while it is called fully transformed when 0 attains 0T. The full transformation 0 T c a n be identified with cOT. Given a transformed dilatation 0 the integrated stress-strain relations in three dimensions are

Constitutive Modelling

42

Eij = 2---~Ei~j+

E~Sij +-~05iy

(3.17)

and

r~ - 2,E~j + B(E~p - 0 ) ~

(3.18)

where superscript s denotes the deviatoric component. In plane strain (E33 = E13 = E23 = 0) the above reduce to

(

1

E ~ - 2# E ~ - -~Euu6~Z

) + B(Euu - 0)6~

(3.19)

2

E33 - - ~/.tEuu + B(Euu - o )

(3.20)

Em = -l -+-v- ~ E u . - ~E0

(3.21)

and E~

1

-

~..(2~ - uE..6~) + aft

l+u 06~ 3

(3.22)

where Greek subscripts range over 1 and 2 with a repeated Greek subscript indicating a sum over just 1 and 2. Here, u is Poisson's ratio of the elastic branch so that

# -

E 2(1+ u)'

B =

E 3(1- 2u)

(3.23)

where E is Young's modulus. In the sub-critical case, the transformation strain rate b0T can be calculated from ce~

_ -} < E ~ < r ~

R

(1-~ eel

B)Epp when E~ +

--B--

-

0

1

-

-

-

~

+ c m 0~~ 1-~-

(3.24/

elsewhere

whereas in the critical and super-critical cases, the transformation strain rate is undefined, but the transformation strains are simply given by

3.2. Shear and

Dilatant

cOT

coT

-

43

Transformation

when

0

-- CrnOT

Epp <

when

Epp>

B

(3.25)

B

In the constitutive model under discussion, six material parameters determine the material behaviour" Poisson's ratio u, Young's modulus E, the bulk modulus on the intermediate segment B, a critical mean c the maximum volume fraction c m of transformable material stress ~rrn, and the unconstrained particle dilatation 0p. T Dimensional analysis and close examination of the governing equations reveal that the constitutive behaviour can be captured by three dimensionless variables" Poisson's ration u, the ratio B / # governing the slope of the intermediate stressstrain curve and the transformation strength parameter w, as defined by Amazigo & Budiansky (1988): i

-

3.3

cr~

[1+:] 1-

(3.26)

C o n s t i t u t i v e M o d e l for Shear and Dilatant Transformation Behaviour

Since the pioneering work of Budiansky et al. (1983) described above, much effort has gone into understanding the role of transformation shear strains and how to incorporate these in a constitutive description. Lambropoulos (1986) was the first to consider the twinning effect in a constitutive description in an approximate manner by ignoring interactions between transformable precipitates. As in the model described above, the martensitic phase transformation was assumed to be super-critical in the sense that it took place when a function of the macroscopic stress state attained a critical value. The resulting constitutive law was similar in structure to the incremental theories of metal plasticity in that it was characterized by a yield function, a loading criterion and a set of flow equations. It cannot however describe the material behaviour after initiation of transformation nor can it predict the real volume fraction of transformed material. Chen & Reyes-Morel (1986) also proposed a phenomenological transformation yield criterion which was pressure

44

Constitutive Modelling

sensitive to reflect the experimental data on transformation plasticity in compression. In the following we shall describe briefly the continuum model of Sun et al. (1991) which uses terminology and ideas from conventional plasticity theory. The exposition follows closely the work of Stam (1994).

3.3.1

Stress-Strain Relations during Transformation

The transformation plasticity model due to Sun et al. (1991) assumes the representative continuous element to consist of a large number of transformable inclusions (index I) coherently embedded in an elastic matrix (index M), as shown schematically in Fig. 3.1. Microscopic quantities (in the representative element) are denoted by lower case characters. The macroscopic quantities are found by taking the volume average ( ) of the microscopic quantities over the element. Thus the microscopic stress and strain tensors are denoted by O'ij and cij, respectively, for a given volume fraction of transformable metastable tetragonal inclusions c. The relation between microscopic and macroscopic stresses is

E0 -

(o'ij)v - -V

(rij dV - c(criJ)v, + (1 - c)(c~iJ>v.

(3.27)

where the volumes of the element, matrix and inclusions are given by V, VM, and VI respectively and c is the actual fraction of transformed material which is obviously less than or equal to cTM. The macroscopic strains are assumed to be small, and under isothermal conditions can be decomposed into an elastic part E~j and a "plastic" part E~j induced by the t --* m transformation in the inclusions

Eij -

Ei~ + EPj - MO.klrkl + c(cPj )VI

Here M ~ (Li~ -1, with Li~ inclusions and matrix

Lij kl -- 1 + u

(3.28)

being the elastic moduli of both

k~jl -'~ 6jk6il) + 1 - lJ 2u

~ij6kl]

(3.29)

The inelastic strain due to t ---, m transformation can in turn be written as a sum of dilatant and deviatoric parts distinguished by with superscripts d and s, respectively

3.3. Shear and Dilatant Transformation

-

El/+

45

E,? -

+

(3.30)

The rate of inelastic strain (designated by a superposed dot) during progressive transformation, 5 > 0, can be obtained by differentiating (3.30) or by averaging the transformation strain civj over the freshly transformed inclusions (per unit time) occupying the volume dVi, i.e. 9

"ps

.

pd

9

pd

ps

= c(ciJ )dV, § c(CiJ )dV,

(3.31)

pd within each inclusion can be written in The dilatant part of strain Qj terms of the constant stress-free lattice dilatation gvd (_0V /3) which typically takes a value of 1.5% at room temperature, i.e. _

pd

1 Pd~ij

gpd(~ij

T

(3.32)

p$

The deviatoric part (gij)Vz is significantly less than the stress-free lattice shear strain of 16% because of twinning. Based on the earlier work of Reyes-Morel & Chen (1988), and Reyes-Morel et al. (1988), the rate of change b(Ci~s)dU~ is assumed to depend on the average deviatoric stress siM in the matrix according to

Here, A is a material function, which may be regarded as a measure of the constraint provided by the elastic matrix, and ~M is the von Mises stress in the matrix, which will be specified later. When ~M _ 0, A should be put equal to zero because there is no stress bias. However, experimental data of Chen & Reyes-Morel (1986, 1987), and Reyes-Morel Chen (1988) show that under proportional loading the value of A is almost constant during the transformation process. The macroscopic constitutive relationship (3.33) is assumed to apply to the ensemble of transformable particles in the continuum element. The deviatoric transformation strain over individual transformed particles will not depend on the local matrix stresses in such a simple manner because of twinning along well-defined directions on specific crystallographic planes and also

Constitutive Modelling

46

because the amount of twinning within a particle is dependent on its size (Evans & Cannon, 1986). Although much research has been devoted to nucleation and twinning in a single particle, these phenomena are still not well understood and need further attention. Some light will be shed on them when we describe the third constitutive model (w For the present model (3.33) is an acceptable approximation in the average sense over many grains with different orientations within d~/). From (3.31)-(3.33), the inelastic strain rate is

(3.34)

EPj - c(gPd~ij -]-(CiPS.)dVi )

The total macroscopic strain rate is obtained by adding (3.34) to the elastic strain rate E[j - M~

E~. _ E~j + Ef~

o

'

-- Mij]r162

~ ~- C(~Pd~ij ~- (~ij )dV I )

(3.35)

In an inverted form, we have

~ij -- 21-t(Eij -- Ern(~ij ) @ 3BErn~ij - c(3BgPd~ij + ~#@iPs}dV ) (3.36) where Em - Epp/3. For future reference, we note that under plane strain conditions E33 = E13 = E23 = 0, so that (3.28) and (3.36) reduce to 1

(3.37) and

1Et..5~Z) + BE.t.6~Z _ d(3BcPd6~z 2.k..

+ BE.. -~(aBc ~ + 2.(4;)..

Zm - B ( / ~ . . - 3de pd)

l+vE~ -

E

) E.

+

2~(g~)dVi )

3.3. Shear and Dilatant Transformation

3.3.2

47

T r a n s f o r m a t i o n Criterion and T r a n s f o r m e d Fraction of Material

The constitutive description is complete when the transformation condition and the evolution equation for ~ have been prescribed. This requires derivation of (forward and reverse) transformation yield conditions, for which we need to calculate the free energy of the continuum element by summing its elastic strain energy, the chemical free energy and surface energy (w We present without detail (Sun et al., 1991) the various energy components. The elastic strain energy W per unit volume of the continuum element is given by

W-

-1~ i j MOklrkl - ~ 1

1

= -~r~,:~ M ~

+-ilc2

- c

I v -o'T gijPdV 1 B~ A2 + ~B~(~d) 3 ~

[~l@iPS)vl(giPS)v 1 4- 3B2(gPd) 2]

(3.39)

where

(3.40)

-~T _ Cri7 + (-ffiJ )v M __ ~ i7 -c(cr~7)v,

is the transformation induced internal stress or eigenstress in the inclusion as defined by Mura (1987), and

0.i7 __ Li jOkl(Aklmn -- Iklmn )C~,~

(3.41)

is the Eshelby stress in an inclusion (Eshelby, 1957; 1961). The elements of the Eshelby tensor Akzm, for a spherical inclusion are Al111 -

A2222 -

A3333 =

7-5u 1 5 ( 1 - ~)

Al122-

A2233-

A3311 -

Al133-

A1212- A2323- A3131 =

A2211 -

4-5~ 15(1 - u)

A3322 =

5~- 1 15(1 - u) (3.42)

Although the shape of the transforming particle influences the stress

48

Constitutive Modelling

field, the spherical shape has been assumed here for simplicity. M The deviatoric and mean stresses in the matrix siM and ~rm can be found using the averaging method of Mori & Tanaka (1973) for a body containing many transforming spherical particles

(3.43)

aiM _- .:-,,~v" - cB2c pd

where Sij = E i j -- Ern~ij and E,~ = Epp/3 are the deviatoric part and the mean stress of the macroscopic stress tensor Eij, and B1 -

5u- 7 2# 15(1 - u)' -

B2 -

2u- 1 2B ~1 - u

(3.44)

These two parameters which resemble bulk moduli result from Mura's (1987) approach. The change in chemical free energy per unit volume is obtained by subtracting the chemical free energy in the martensitic phase from the chemical free energy in the tetragonal phase. This temperature (T) dependent contribution of the free energy is given by AGch~m(T)

= cAGt_m(T)

(3.45)

For equisized spherical particles, the total change in surface energy per unit volume is 67pc A~,r - c A o (3.46) a0

where a0 is the diameter of a particle, and 7p is the surface energy change per unit area during the t ---+rn transformation. The Helmoltz (or free) energy per unit volume, (I), can now be written by adding (3.45)and (3.46)to (3.39) O ( E i j , T, c, <eiPS>v ) -

W + As~,,. + AGch~,~

The complementary free energy is given by

(3.47)

3.3. Shear and Dilatant Transformation

49

1

1 A2 3 B +c 5B~ + ~ ~(c~) ~]

--lc22[B1@iPS)vi(EiSS)vx -Jr-3B2(cPd) 2] -cAo - cAGt_m

(3.48)

It is clear from (3.47) that the thermodynamic state of the representative continuum element is completely described by the variables Eij, T, c, and (ci~S)y~, in which c and (ci~S)u~ are the internal variables describing the microstructural changes in the material during transformation. Denoting the thermodynamic force conjugate to internal variables c and (ci~Sly~, by F c and Fief respectively it follows from the second law of thermodynamics that

,~v = Oq,

OqJ

b~

(3.49) where (vide (3.48))

-

1 A2 - -~B2(cPd) 3 Ao + AGt__.m- -~B1 2]

--C[/~1(ciPS)v@iPS)v-t-3B2(cpd)2 1

(3.50)

and (3.51)

- c

Let us denote the total energy dissipation per unit volume by

W d = Do ccu

(3.52)

50

Constitutive Modelling

where Do is a material constant which can be determined by direct measurement or microstructural calculations, and the cumulative fraction of the transformed material during the whole deformation history is -

Ccu

f

(3.53)

Idcl

The energy dissipation rate W d is thus Wd -

Docc~, -

I

Dob

( -D0b

(forward transformation,

~ >_ 0)

(reverse transformation,

b <_ 0)

(3.54)

Since ~P must be equal to W d, (3.49) and (3.54) give for forward transformation Dob

(3.55)

Feb + Fi~ s (@iS)v, - -Doi:

(a.56)

FC~. + Fi]" (eiP])v, -

while for reverse transformation

Substitution of (3.50) and (3.51) into (3.55) and (3.56) and elimination of d by using (3.31) and (3.34) gives the yield condition in stress space for forward F+ and reverse F_ transformation F + ( ~ i j , C, T , (ciP.S}v1)

M c pd - Co(T, c) - 0 (3.57) Ao M + 3o m

-

F _ ( E i j , c, T, (ei~)v ) -

2-AaM + 3erM e pd - Co(T, c) - 0 (3.58) 3

where Co(T,c)-

Do + Ao + A G t - m ( T )

- ~3 B2(cpe)2 +

1

- -~BIA 2

c~B0(cPe)2c

(3.59)

and Co(T, c) -

Co(T, c) - 2D0

(3.60)

As the constitutive model is derived at the macroscopic level, the hard-

3.3. Shear and Dilatant Transformation

51

ening effect does not follow from the derivation itself, and Sun et al. (1991) introduced the last term in (3.59) on purely phenomenological grounds. The hardening response may be expected on a microstructural scale because of the following: 1. particle size distribution; it takes a higher stress level to transform smaller particles; 2. crystallographic orientation of particles; particles oriented along favourable planes transform first; and 3. the mutual interference of transformed particles. Introducing the parameters B0 and h0 Bo = 4 # ( l + v ) + P h ~ ( 2 8 - 2 0 v ) " 1_ v 5(1 _ v)

'

A h o - 3gPd

(361) "

the yield function in forward transformation (3.57) can be written as 2AcrM + 3cPda M + B o ( 1 - a)c(r

F+- 5

2- Co(T)

(3.62)

with an explicit linear hardening term, dependent on c. The yield condition (3.57) describes a "transformation surface" in stress space (Eij) within which transformation is excluded, similar to a yield surface in the theory of plasticity. When the transformation proceeds, the transformation surface expands (or contracts) as well as translates in stress space. Borrowing further from plasticity theory, the present material with transformation plasticity may be deemed to exhibit mixed hardening; isotropic expansion (or contraction) as a function of c is governed by Co(T, c), while kinematic hardening originates from the internal stresses appearing in the relations (3.43) between the matrix and macroscopic stresses. The incremental stress-strain relations for forward transformation may be obtained by the usual routine of internal variable constitutive theory (Rice, 1971) 9

.

OF

+ c

~

or;;

Constitutive Modelling

52

= M~

A,7)

+ ~ ePdsij --[- O"M

(3.63)

is determined by the consistency condition

oF+

oF+

b.

O{eiS~)v' (ciJ )v, - 0

(3.64)

The solution of the above equation gives

OF+ O~-,ij

2BIA2 + (3B2 + o~Bo)(r 3

2

(3.65)

- 2B1A2 + (3B2 + c~Bo)(ePd)2 3

Expression (3.65) holds when the transformation is in progress, i.e. the current stress state satisfies the transformation condition (3.57) and there is still some transformable material left c < cm, and as long as no unloading takes place. We will call this the loading or transformation branch. If the response is elastic, because the criterion (3.57) is not satisfied, F+ < 0, or because elastic unloading occurs from a plastic state, F+ - 0 and ~ < 0 according to (3.65), ~ must be set equal to 0. To summarize,

(r

Sij "4- 3gPd~m

-~B1A 2 + (3B2 + aBo)(r 0 when

2

when

(F+-Oand~_>O)

(3.66)

(F+ < 0 ) o r ( F + - 0 a n d ~ < 0 )

Following along similar lines for reverse transformation, we have

2B1A2 + (3B2 + aBo)(cpd)2 0 when

when

(F_-0

and~<_O)

(F_ < 0 ) or (F_ - 0 and ~ >_ 0)

(3.67)

3.3. Shear and Dilatant Transformation

53

From eqns (3.63) and (3.65)it follows that

9 (

si,)

i.e. the inelastic strain rate is normal to the yield surface (3.57) in stress space. The normality rule is thus identically satisfied and there is no need to assume it a priori, as is sometimes necessary in conventional phenomenological treatments. The constitutive equations may be rearranged into a form convenient for numerical analysis. For this we introduce the designations

~

=

~]

OF+

OEij -- epdSij + A aM

g - -~B1 A2 + 3B2(cPd) 2 + e~Bo(gPd) 2 2

_

1 OF+ Ely

(3.69)

g ~ij

and rewrite (3.63) as

E~ - M~

+ 1T~jT~,~, g

Next, we multiply (3.70) by (M~zij)- 1 1

_

(3.70)

Lk,i j o to give

0

LklijEij - Ekl + -LklijTijTmn~mn g

(3.71)

and again by Tkz to obtain

Tk,L~

- (1 + 1TabL~ g

(3.72)

or

1 o T o _[~ij TmnEmn - (1 +-TabLabcd cd) -1 TmnLmnij g

Eliminating TktEkt from (3.71)and (3.73), we get

(3.73)

Constitutive Modelling

54

1

_ L O k l p q T p q T m n L mo n i j ~kl

--

LOklij --

1

(3.74)

o

1 +-TabLabcdTcd g which may be rewritten in the following compact form ~ij

(3.75)

-- L i j k l E k l

The instantaneous moduli Lijkt in the above rate equation are --1L~ m n

Tm,~Tpq L p~q k l

g

when F+ - 0 and ~ > 0

LOj kl -Lijkl

--

0

1 + 1Tab L~b~dTr g LiOkl

3.3.3

(3.76)

when F+ -r 0 or 5 _~ 0

Comparison Models

between

the

Two

Constitutive

The continuum model of Sun et al. (1991) reduces to that of Budiansky et ps al 9(1983) when the shear transformation strains are ignored, i.e. Qj - 0, for which we set h0 - 0. Noting that the dilatation 0pT in the first model corresponds to 3c pd in the second, it follows from (3.63) that

1L~ + 1 ~ , ~ + ~~,~ k,j = 2, 5~

(3.77)

which is identical to the rate expression of the integrated stress-strain equation (3.17), with the dilatational strain rate 9

E~ -

1.

~r~

+ 3e~ ~

(3.78)

The corresponding b is given by (3.66)

-

B2r d 0

when

F+ - 0

and (~ > 0

when

F+ # 0 or b _< 0

(3.79)

3.3. Shear a n d D i l a t a n t T r a n s f o r m a t i o n

55

On the transformation branch, the volumetric inelastic strain rate is given by (vide (3.44)and (3.79)) 3F, pp B2

3ccPd --

=

1+

(3.80)

Epp

(y+31 This result also follows from the model of Budiansky et al. (1983), when B - 4 p / 3 is substituted into (3.24). a > 0 corresponds to subcritical formulation and the relationship between a and B is now a-

9BB-

3BB2 + 3B2B

--

or

--

BBo - BoB

B-

B(B2 + 89

(3.81)

B2 + gc~Bo + 3B 1

The similarity between the two models extends to the onset of transformation when c - 0. In the model of Budiansky et al. (1983) this happens at the critical mean stress

r ~ - r~m

(3.82)

In the continuum model of Sun et al. (1991) the initial yield condition is obtained by putting c - 0 and (cPS)y~ - 0 in (3.57) F+ -

2hocPdEe 4- 3cPdEm -- C o ( T , O) - 0

(3.83)

where 2e

C o ( T , O) -

--

~ij ~ij,

~ij -- ~ i j -- ~ m ,

2 m -- -~ Epp

Do + Ao + A G t . . . . ~ ( T ) - 3Bl(hovpd) 2

(3.84)

3 )~ ~B~(c ~d (3.85)

If we denote by E C the critical stress

rc

_ Co(T, 0)

--

3r

(3.86)

then (3.83) reduces to 2-hoEe + E m 3

-

Ec

(3.87)

This initial yield condition reduces to (3.82) when shear strains are ne-

Constitutive Modelling

56

glected, i.e. h0 = 0, and E c is appropriately reinterpreted.

3.3.4

Comparison with Experiment

To illustrate the capabilities of the continuum model of Sun et al. (1991), the predicted stress-strain relations are compared in Fig. 3.5 with experimental stress-strain curves for TZP and PSZ. The experimental data of TZP and PSZ were obtained under triaxial compression by Reyes-Morel & Chen (1988), and Chen & Reyes-Morel (1986), respectively. The broken lines show the theoretical response in the axial (xl) and radial (x2 and xa) directions. The strain in the axial direction is negative and the strains in the radial directions are positive. Obviously, inelastic strains of that nature cannot be predicted by the purely dilatant model. For TZP, the experimental data point towards a nearly perfect inelastic regime after the initiation of transformation followed by linear hardening. This bilinear transformation behaviour in TZP cannot be fully explained by the model of Sun et al. (1991). For PSZ, the transition from linear elasticity to transformation behaviour occurs more gradually, whereas in the theoretical model the transition is sharp. The experimental data may be used to estimate the various material parameters appearing in the constitutive model (v, w, h0 and a). For TZP tested by Reyes-Morel & Chen (1988) in hydrostatic compression the elastic properties are given as E=190 GPa and v=0.3. For the fully tetragonal material cm=l.0, and the constant lattice dilatation is always assumed to be cvd=0.015. The experimental stress-strain curve for TZP in Fig. 3.5(a) is used to derive the shear transformation strains in the Xl and x2 directions as a function of the transformed fraction. These strains are shown in Fig. 3.6. As the loading was almost proportional in the test, the shear transformation strains are given by

EiJ -- 3ch~

(3ss)

Ee

Using this relation the value of h0 can be estimated from the experimental curves of Fig. 3.6. The broken lines show the theoretical response for h0=l.4 in the axial (Xl) and radial (x2 and x3) directions. The critical transformation stress E c can be determined using the condition at the onset of transformation (3.87) 2 F+(Eij, (ci~)u) - -~hoEe +

Em -

EC

- 0

(3.89)

3.3. Shear and Dilatant Transformation

57

xl

x2 ~

/

T

x3 -Ell .............

•ll

s

[MPa]

4000 ,'•

3000

,:; ,;;,

2500 2000 1500

El c! [MPa] 2500

/,

3500

i, ,,v'

,,

2000

,,~f~~,'"

,,'s

E22

,

1500

.... ," " / /

"""

.,

,

"

p

-

"

J

,,'"',,"

1000 r/

1000

500

500 0

i

O r

0.00 0.01 0.02 0.03 0.04 0.05 Eli

a)

TZP

I

I

i

J

0.0 0.005 0.01 0.015 0.02 El,

b)

PSZ

F i g u r e 3.5" Stress-strain curves for T Z P and PSZ obtained by triaxial compression at room temperature. The broken lines correspond to the predicted response

For a specimen loaded in compression E l l in addition to hydrostatic pressure P, Fig. 3.5(a), it follows from (3.89) that Ec _ 2h0- 1 - -----~--Ell

- P

(3.90)

From Fig. 3.5(a) the stress level at which the transformation is initiated is E~1=-1300 MPa, so that with P = 1 2 5 MPa, we find E c = 6 6 0 MPa. The value of the hardening parameter c~ is found by comparising the slope of predicted inelastic branch of the stress-strain curves of Fig. 3.5 with that of the experimental curves. In this manner the hardening parameter c~ is

Constitutive Modelling

58

Eij ps

0.05 [ p$

-E11 0.04

0.03

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

gg

--

~II SIllS

0.02 s l

s

l

0.01

s -

s

i I I s

0.00 ~ 0.0

, ..~

0.2

I

I

1

0.4

0.6

0.8

I

1.0

c

F i g u r e 3.6: Contribution of shear transformation to the transformed fraction for TZP. Broken lines show the predictions of the constitutive model estimated to be 1.16. As mentioned earlier, the experimentally observed bilinear regime of the transformation cannot be modelled, as the present model only takes into account linear hardening. For PSZ E=190 GPa, ~=0.3, cPd=0.015 and cm=0.35. The remaining parameters can be calculated in a similar way, as discussed above and are estimated to be: c~=1.2, h0=l.3, and E C ,,~490 MPa (based on E/11 - - 1 1 5 0 MPa and P = 1 2 0 MPa from Fig. 3.5(b)).

3.4

C o n s t i t u t i v e M o d e l for Z T C

We now present an incremental plasticity formulation due to Lam et al. (1995) which overcomes the two basic limitations of the model described in w namely that the transformable particles and the matrix have the same elastic properties and that the strain deviator is parallel to the

3.4. Constitutive Model for ZTC

59

F i g u r e 3.7: Configuration of the matrix-particle system stress deviator during transformation. This will allow the constitutive model to be used for all zirconia toughened ceramics, including dispersed zirconia ceramics. The incremental plasticity formulation for the transformation plasticity behaviour of zirconia-toughened ceramics is based on the micromechanics of heterogeneous media, which requires the solution of the inhomogenous inclusion problem. This will be accomplished via Eshelby's equivalent inclusion method. A single-particle model for the transformation yielding and twinning of an ellipsoidal zirconia particle, based on the end-point thermodynamics of martensitic phase transformation will be developed. The average eigenstrain inside the transformed particle is determined by the volume fractions of the four possible martensitic variants which minimize the total free energy of the matrix-particle system after transformation. The model can consider thermal expansion mismatch besides the elastic mismatch between the matrix and the particle, as well as thermoelastic anisotropy for both the matrix and the particle. First the overall fields of the particulate-toughened composites will be related to their local counterparts. An incremental plasticity model is then established by incorporating the single-particle model with the Mori-Tanaka (1973) estimate for the overall inelastic strain. 3.4.1

Equivalent Inclusion Method Inhomogeneity Problem

for

An ellipsoidal tetragonal zirconia particle (Fig. 3.7) is embedded in an

Constitutive Modelling

60

infinite elastic medium subjected to a load which produces a homoge0 in the matrix in the absence of the particle. The neous stress f i e l d ~-'ij homogeneous eigenstrains are EiT M and EiTP in the matrix and particle, respectively. Denote the elasticity tensors of the matrix and of the particle by cijMkt and ciPkz, respectively. They are not required to be the same and can be isotropic or anisotropic. The solution for the matrix-particle system is the superposition of the stress-free homogeneous eigenstrain E[~ M and the solution with a uniform eigenstrain (E T M - E TP) inside the particle. The latter is obtained by Eshelby's equivalent inclusion method for an inhomogeneity (Eshelby, 1957; Mura, 1987) which we shall briefly review below. Let E g be the stress perturbation caused by the inhomogeneity with eigenstrain E I - ( E T M - E TP) and let EiD be the corresponding perturbation of strains. The constitutive relations for the particle and the matrix are as follows Eij + E D

- ciPk,(E~, + E D - EI,)

Eij0

- cijM,(E ~ + E D)

0

+ E D

in 9

outside

f~

(3.91)

where E O

_

M (Cijk,)

- 1

Z~

(3.92)

is the homogeneous strain due to the external loading E~ in the absence of the ellipsoidal particle. According to the equivalent inclusion method, the inhomogeneity can be modelled by an inclusion (with the M elastic properties of the matrix Cijkz) in the homogeneous matrix with eigenstrain E/. plus an equivalent eigenstrain Ei) M Eij0 + ED _ Cijk,(E~ ' + E D _ E I,

Eijo +

-

__

(E ~ + E D)

* Eij)

in [2

outside

f~

(3.93)

Equations (3.91) and (3.93) are equivalent when the fictitious eigenstrain E'~j satisfies the following equation in the region f~ C iM j

+

-

El,-

*

-

P Cijk,(

+

ED

-

El,)

(3.94)

For an ellipsoidal particle, the solution of (3.94) has been given by Eshelby (1957). The strain disturbance E D is uniform inside the particle and is given by

3.4. Constitutive Model for ZTC

E,~

61

A , ~ , ( E I, + E;,)

-

(3.95)

where Aijkt is the so-called Eshelby tensor that depends on the elasticity cijMkl and the geometry of the particle (The components of Aijkz for a spherical inclusion are given in (3.42)). Substituting (3.95) into (3.94) gives tensor

E i *j -

Pi;~, [(CMmn - C P m n ) E mon + C kPl m n E mIn ]

(3.96)

- EI

where Pijkl

-

-

P (Cijmn

-

-

(7.M M "-',;ran., ) A m n k l + C i j k l

(3.97)

Therefore, the total uniform strain and stress inside the inhomogeneity due to the external load Eij0 and the eigenstrains EiT M and EiTP are D

TM

_ Eo + E , ? , -1 +AijklPklmn

M q [(Cmnp

- E-' ~ ~m. ~ , , ~~, 0 +

P

TP

TM

o - CijMklEkl0 + CijMkz(Akzmn- i k l m n ) p - 1mnpq

•

Cpqrs - c ~P, ~ , ) < , o + c ~P, ~ , ( < , T P -

<,T M

)]

(3.9s)

where Iijkl is the fourth-order identity tensor. Another expression for the average total strain E P can be obtained by substituting the second eqn (3.98) into (3.94)

= EiT M + Oijk,E~, M -1 Cpq~, P (E~TP - E~, TM ) +Q~ik,Ak,m,,(Cmnvq)

where

Q~jkz -

(I~ik~A~jm,~ ( c 2M~ ) - 1 M( c ~ - c ~ , )

p)-I

(3.99)

(3.100)

Constitutive Modelling

62 3.4.2

Transformation Single Zirconia

Yielding Particle

and

Twinning

of a

When the applied load ~ij0 increases to a critical level, the embedded zirconia particle experiences phase transformation from tetragonal to monoclinic form which is accompanied by dilatational and shear transformation straining. When only first-order effects are accounted for, the transformation strain can be assumed to be homogeneous inside the whole particle f~ and is denoted here by E n. According to Lam and Zhang (1992), for the transformation of zirconia crystal from tetragonal to monoclinic symmetry, there are only four variants for the right stretch tensor Uij, denoted by U1, U~j, U3 , and U4 , although there are many more variants of deformation gradients. The transformation of the monoclinic particle can be approximately considered as a uniform eigenstrain that is the volume average of the strains of the four martensitic variants

Ea -

4

E

1

4

fin EiN _ -2 E / 3 N ( u N u ~ - Iij)

N=I

(3.101)

N=I

where Iij is the unit tensor of the second order and /3N is the volume fraction of the martensitic variant UiN. The right-stretch tensors of all the four martensitic variants for the transformation of ZrO2 particles in Mg-PSZ (see Table 2.1 for lattice parameters) at room temperature are 1.0102 0 0.0826

0 1.0256 0

0.0824 0 1.0260

1.0102 0 -0.0826

0 1.0256 0

-0.0824 0 1.0260

1.0256 0 0

0 1.0102 0.0824

0 0.0824 1.0260

1.0256 0 0

0 0 1.0102 -0.0824 -0.0824 1.0260

(3.102)

3.4. Constitutive Model for ZTC

63

The average strain (3.101) is calculated in the following manner. Let F i j be the transformation gradient defined with respect to an orthonormal basis whose directions are consistent with three lattice vectors. Assume that (Fi 1, F~j ..... Fi~ ) are the deformation gradients of all the variants which may be twin-related to Fij. That is to say, for each deformation gradient F~ e {Fi~, F 2, ..., Fi')), there exist two orthogonal I N[) satisfying symmetry tensors RijI E O, H I E ~ and two vectors (ai, and jump conditions associated with twinning. Now, we will investigate which deformation gradients of the variants are twin-related to F / . In fact, if we replace (RiI , H I , ai, I N[) by (Rij' * H , / * , a i' * N , ~ * ) and Fij, by F / , we will easily find that the new equations of symmetry I and discontinuity have exactly n solutions (RilF]kHl,,...,RijF~kglt). This implies that there may be a variant whose deformation gradient is I J J I RijRjkFkIHlmHmn with respect to the given coordinate system. Similarly, we can obtain the variant with deformation gradient FIJ,...,K ij

:

I K K J I RikR~t...RzmFm,~Hnp...Hpqgqj

(3.103)

where I, J, K can be arbitrary integers among [1, 2, ..., hi. We now have a clear and complete picture of the twinned crystal. It IJ...K is built up of the homogeneous deformation regions Fij' ' which interface with each other only on a finite number of well-defined twin planes Fig. 3.8. As we are mainly interested in the macroscopic transformation strains of the twinned particle there are two approaches to averaging the deformations of the variants involved. One is based on the assumption that the averaged deformation gradient F~. of the particle is the volume average of the deformation gradient variants

B

C D A

t

F i g u r e 3.8: A schematic description of the twin structure of the transformed zirconia particle. A, B, C, D denote four possible martensitic variants

Constitutive Modelling

64

FP"-

~"

E

VIj .... ,KF/~' J ' ' K

(3.104)

I,J,...,K

where VI,j,...,K is the volume of the region with transformation deformation gradient F~ ' J ' ' K as defined by eqn (3.103), and V is the total volume of the particle considered. The strain tensor E~j of the particle is determined by 1

E,) - ~(r~F~% -

1(1

~) (3.105)

I J . . . K 0t3...7

This averaging approach cannot be used directly for the calculation of transformation shear in the case of t ---, m phase transformation since there are too many gradient variants given by eqn (3.103). We shall employ an alternative approach which has been widely used in composite mechanics in which the macroscopic strains of a multi-phase material are considered as the volume average of the strains of the phases involved. For a given deformation gradient variant FiI J n the Green strain tensor E~ J K is

EI/...K__ -~(F~ 1 : ~ c F ; IJ...K s

-

I~)

1

....K _ I~j ) = - 2 ( u [ / ..K ~sl/k:

(3.106)

The right stretch tensor U / J n must take on one of the values Uff of (3.102). The average transformation strains are therefore expressed by z

-

1

v , , . . . ,< z

IJ...K

. . '< -

z IJ

(3 . o7)

N

When there are only four possible martensitic variants we obtain (3.101). The change in total potential energy of the matrix-particle system due to phase transformation is essential in determining the transformation yielding and twinning. To calculate this change we need to know the perturbation caused to the total potential energy by the inhomogeneity and the eigenstrain E~ relative to the homogenous stress field Eij0 and

3.4. Constitutive Model for ZTC

65

strain field E~ It can be expressed as follows (Mura, 1987) W D

-_

-

1/~(~i~

)dV f~

-~

_

~

r, u

(3.108)

Substituting (3.97) and (3.98) into (3.108) and taking into account the fact that the stress and strain fields inside the inhomogeneity are uniform, we have WD_

- - ~1V p { ~ i jo

+

[Pi;~, ((CMmn

--

P 0 + CijklEkl)] P I Cklmn)Emn

jk~(Aktmn Ikzmn)P mnpq -1 -

• ((CpqM~- cpP,.,)E~ + cpP~,E[~)] E/~ + E~ where Vp is the volume of the particle. The change in the total potential energy of the matrix-particle system during the phase transformation is the difference between the potential energy of the system after and before the transformation. Taking into account the fact that the eigenstrain inside the particle changes from E~TP to EiTP + Ei~ during the phase transformation it follows from (3.109) 1

AGM -- --~Vp[Ei~

+ E[jBijktZ~ + Ei~DijktZ~]

(3.110)

where -1 p Aij kl - Pijmn Cmnkl .jr.(Cijmn) M -1 (Cm~v~ M P -T ~ , + Iijkl - Cmnpq)Ppar,(hr, ,~ -- I,.~a, ) C ,M P -T M Bij kl -- Cijmn Pmnpq (Apqrs - Ipqrs )Crskl M

+Cijrnn (Amnpq - Imnpq )Pp-'qlrsCrPskl M Dijk, - Cijmn(Am,~vq - Im,~vq)Pp--ql,.,cPkt

(3.111)

According to the so-called end-point thermodynamics of phase transformation (Lange, 1982), the transformation can proceed only if the change in the free energy AG between the initial untransformed tetragonal state and its transformed monoclinic state is non-positive

66

Constitutive Modelling

AG = AGM + AGcH + AGs = AGcn < 0

(3.112)

where A G c n is a critical value that AG has to attain for the phase transformation to be thermodynamically favourable, AGM is the net change in the potential energy of the whole matrix-particle given by (3.110), A G s is the surface energy change both in the matrix-particle interface and in the interfaces between the twinned variants and A G c H is the change in chemical energy. When martensite is the low-temperature phase, A G c H is given approximately by AGcH Vp

_ g(1 -- t~ ~-)

(3.113)

with g being the enthalpy of the transformation of an infinitely large single crystal and ~, the stress-free equilibrium transformation temperature (Garvie & Swain, 1985). The expression for the interfacial energy change is quite complicated. Although its complete description requires full knowledge of the twin structure of the transformed particle, it can be reasonably approximated (Garvie & Swain, 1985) as follows AG______~s= ~--~(1 + ~2~(-)) vp r](n) ~

( n - 1)(n + 1) ll

,

~;1, ~2 > 0

(3.114)

where ~1 and ~2 are material constants, and n is the number of twins in the particle. Among the three components of the free energy change (3.112), only the change in potential energy AGM depends on the twinning combination (ill, ~2, ~3, and ~4), once we have accepted the approximation (3.114). Thermodynamically, the twinning combination (ill, fl~, 133, and f14) should be such that the whole matrix-particle system assumes a minimum free energy state after the transformation. If we know the critical stress E C at which the phase transformation takes place, the twinning combination is determined by minimizing the objective function AGM

ve subject to

=--

1

2

f

y'~

[E~)Aijkz

E kt a

+

E[jBij

kz E a

+ EaDij

kz E a]

(3 9115)

3.4. Constitutive Model for ZTC

0 _< ~i _~ 1,

67

fll + f12 + f13 + f14 - 1

(3.116)

The minimum value of AGM/Vp should be (3.112)

AGM 1 = AGc - ~(AGcRVe vp

A G s - AGcH)

(3.117)

Conversely, if we know AGc, we can determine the yield stress E C and the twinning combination. Define the applied stress as

E~ - X ~ + ~rXij

(3.118)

where cr is a scalar loading parameter characterizing the loading level and is always taken as positive, X ~ is the initial stress tensor and Xij is a second-order symmetric tensor characterizing the loading direction and complexity. From (3.112) the loading parameter cr can be determined as follows 0"(~1, "", ~ 4 ) --

- 2 A G c - ( E [ j B i j k , + E~Dijk, +X~ Xij Aij kzEka

(3.119)

The critical loading parameter ~rcR is given by

acn - min{a(fll,fl2,~3,~4)}

(3.120)

subject to the constraint (3.116). It can be shown that subject to the condition (3.116) and XijAijktEt:~ > 0, the minimum of cr(fll, ~2, f13,/?4) also minimizes the potential energy change (3.115). Let the minimum of (3.120) be attained at (ill, /?2, /33, f14). For any other combination (fl~, fl~, ~ , ~ ) of the martensitic variants, satisfying Xij AijkiEk~' > 0, we always have ~cR

- ~(fl~, &, &, ~4) <

- 2 A G e - ( E I B i j k z + Eij Dijkt + X ~ Aijk,)Ek, XijAijkzE~t

(3.121)

where Eij is the average particle strain corresponding to the martensitic combination (fl~, fl~, fl~, fl~). From (3.121) it follows that the martensitic combination (ill, f12, f13, f14) minimizes the free energy change

Constitutive Modelling

68

Vp[

AGcR<_-~

~,

~,

a,

a,]

(X~ +o'cRXij)AijklEkl +EIjBijklEij +Eij DijklEkl

+AGcH +AGs

(3.122)

in which AGcH and AGs are independent of the martensitic combination (13~,fl~, 13~,13~). The equality holds when (fl~, fl~, fl~, fl~) is equal to (~1, ~2, ~3, ~4)" The requirement X i j A i j k I E k ~ ~ > 0 can be justified thermodynamically. If the transformation takes place with a martensitic combination (~,~,~,~) satisfying XijAijkiEk~' <_ O, then it follows from (3.115) that the particle-matrix system will assume a lower or equal free energy state if the particle transforms with the same martensitic combination ( ~ , ~ , ~ , ~ ) before external stress is imposed on it. Therefore, the martensitic combination ( ~ , ~ , ~ , ~ ) could not be a twinning mode for a particle which has not transformed under the action of the original stress X ~ . The dependence of yield stress crcR on the particle size can be examined using (3.119). Let Vp = n~03, n > 0, so that

1

t~1

ACc -- vp~(/XacR- AGcH) - ~--(1 + ~271(n))

(3.123)

The first term at the right-hand side of (3.123) is found experimentally to be independent of particle size. I<1, g2 and g all are positive, and AGe decreases with decreasing particle size ~o. Therefore, smaller particles will transform at a higher critical loading parameter ~rcR, in agreement with test data. For given particle shape and loading complexity tensor X~ the twinning combination (~1, ~2,133, ~4) and the transformation strain E~ can be determined numerically by solving the constrained optimization problem (3.120). For the special case where the matrix and the particle share the same elastic stiffness tensors, the objective function (3.119) becomes

5r(~1,/~2, ~3, ~4) ----

1

XijE~

{ - 2 A G c - E~ __

[(Aiyk,- Iijk,) Cklmn M

M C~jkl(hkzmn Ikzm,~)]

--EiajcijMk,(Ak,mn -- Ik,m~ ) E g -

--

X~

E~rt

n

3.4. Constitutive Model for ZTC

69

Extensive numerical analysis of the yielding behaviour of a single particle according to (3.124) has been given by Zhang & Lam (1994).

3.4.3

Overall Properties and Local Fields

To establish a proper constitutive relation between the overall macroscopic strain and stress fields based on the known thermostatic properties of the constituents we consider aligned ellipsoidal tetragonal particles embedded in an infinite matrix. The shape of the ellipsoids and the orientations of their tetragonal lattice relative to the principal axes of the respective ellipsoid are assumed to be the same for all particles. The particles are aligned along a given direction, but otherwise randomly distributed. Let us consider a representative volume V with surface S, the volume average of the mechanical response of the matrix and the included particles represents the macroscopic response of the ZTC. The matrix phase occupies a volume VM, whereas the particles occupy a total volume Vp so that

VM + V p - - V

)~M + )~P- 1,

,~M -- VM/V,

)~p - V p / V

(3.125)

Suppose that the constitutive relations of the matrix and the particle are known and are formally

~ ( ~ ) _ c,~, M (E~,(~) - El?' (x__)) 2~(~)

-

P

X_~ VM

x__E Vp

(3.126)

where NO(x_) and Eij(x_) are stress and strain tensors and EiTM(x) and EiTP(x) are eigenstrains in the matrix and particle, respectively. They need not be uniform. Define the overall stresses and strains as averages of the local quantities over the respective volumes

~,~ - ~ r~'7 + ~ r~ --Eij - ~ M EiM + ~ P E P

(3.127)

Constitutive Modelling

70 where M

P

E i M - - VM 1 / V ~ Eij(x__)dV, E P

1 l = ~pp

Eij(x__)dV

(3.128)

Substituting (3.126) into the first of (3.127) we get

(3.129) where

EiTM -- VM1 fY EijTM(x)dV_ M

EiTP - ~1/vpETP(x)d V_

(3.130)

are the average eigenstrains within the matrix and the particle, respectively. Now the problem is the computation of the average strains EiM and E P in the respective phases caused by uniform overall strain -Eij and by uniform eigenstrains both within the matrix and within the particles. The solution of this problem can be generally expressed in the form (Dvorak, 1990) w

I-)MPI2TP

z g - Ag

, + v ,'; z

+ v ,';f , z r2

(3.131)

where AijMkt and APkt are the mechanical concentration factor tensors MM MP PM PP and Dijkt , Dijkz , Dijkt , and Dijkz are the so-called eigenstrain concentration factor tensors. Once these concentration factor tensors are known, the overall elastic stiffness and eigenstrain tensors can be obtained by substituting (3.131) into (3.129) to give

Eij -- Cijkl(Ekl- Ekl)

.

where

.

.

.

I

(3.132)

3.4. Constitutive Model for Z T C

Cijkl-

M

71

M

P

P

,~MCijmn Amnkl + )tPCijmnAmn kl _-- cijMk l -[- /~p( Cijm, P ~

M ) Am,,kt P Cijm,~

--

(3.133)

--I M MM [~P r)PM C i j k l Z k , -- (,~MCijMk, __ ~MCijmn Dmnkl -- ,~p ,..,ijrnn~_,,mnkl ) E l y l--

M

MP

P

mmnkl

PP mnDmnk )ETI P (3.134)

The concentration factor tensors are derived using the Mori-Tanaka (1973) method. Assume that the average strain E P in the interacting inhomogeneities in which there exists a uniform eigenstrain EiTP can be approximated by that of a single inhomogeneity with uniform eigenstrain EiT M embedded in an infinite matrix subjected to the uniform average matrix strain EiM. The solution of this problem can be easily derived from (3.98)

E P - Ei~ M "4- EiT M "4- Aij klPklmn-1 X

M [(Cmnpq - CmP

SM

q)E q

+ C

P

.

q(E

TP q -

TM

)]

(3.135)

where EiT M is the average strain caused by stresses in the matrix which are to be determined. Substituting (3.135) into the second equation (3.127) the average strain Ei~ M is given by -1 P )ErsTM Ei~ M - I'(ijkzEkz - Kijk,(Iktrs - ,~pAkzmnPmnpqCpqrs -

)tKijkzAkzmn Pmnpq -1 Cpq,.sE~TP

(3.136)

where Kijkl -- {Iijkl + ~ p A i j m n Pmnpq(-1 Cpqk _ Cpqkl)P } -1

(3. 137)

Substituting (3.136)into (3.135) and comparing the latter with (3.131), we have AijMkl -- Kij kl APkz - Kijkl + Aijmn P~npq -1 (CpqrsM - Cpq,., P )Kr, kl

72

Constitutive MM

Diikt

-

Iijkz - f f i j m n ( l m n k l

Modelling

- ) ~ p A m n r s P~s~u ~Ct uPk l / ~

MP

Dij kl -- - ~ P K i j m n Amnpq Pp-qlr, c P k , PM

MM

-1

M

P

Dij kz - Dij kz -- Aij m n Pmnpq {( Cpqrs - Cpqrs ) I~[rstu --1

P

x ( It~,k, - ,kpAt~,abPab~dC~dkt) + c P k , } PP MP - 1 M P D i j k l - D i j k l - A i j m n P~nnpq { A P ( C p q r s - Cpq,. s ) -1

P

P

X Kvstu AtuabPabcaVcakl -- Cpqkl

}(3.1 38)

It can be verified that the concentration factor tensors satisfy the following relations (Dvorak, 1992) / ~ M A iM j k l + ) ~ P A iPj k l - - I i j kl MM

MP

M

Dij kl + Dij kz -- Iij kl -- Aij kz DijP Mkz + DijP Pkt - Iij kl -- A p kt

(3 139)

In the present formulation it is possible to include the eigenstrain due to thermal expansion mismatch between the particle and matrix besides the eigenstrain due to phase transformation. It is however assumed that the matrix is purely elastic. The eigenstrains in the particle E T P and matrix EiT M are

(3.140) where m p and m iM are the thermal expansion coefficients of the particle and the matrix, respectively, OF is the fabrication temperature and E P is the average transformation strain in the zirconia particles. Substituting (3.140) into (3.133) gives the constitutive relation for ZTC .

.

.

.

p

Eij -- C i j k t ( E k l - -mkl(v ~ -- ~ F ) -- E k l ) ~I

~P

E i j - m--~j(t9 -- I~F) + E i j

where

(3.141)

3.4. Constitutive Model for Z T C

-m- i j -- -Ci~ 1k l { ( ~ M C M p q

_

) i M C M m n D mMnM p q _ lPCPmn

--P Eij

73

PM )m M Dmnpq

Dmnpq - I P C k l m n D m n p q ) m p q

1 (A,, P Cklp q - ) i u C M m n Dm.~p M P q -- ApCPmnD...~pq)Epq PP P kl

(3 9142)

--p

m--~j is

the overall thermal expansion coefficient tensor and Eij the overall transformation inelastic strain. As pointed out by Hill (1967), the -=_P

inelastic strain Eij is usually not equal to the volume average A p E P. Only for the special case where the matrix and particles share the same elastic stiffness t e n s o r s ( C i M j k l -- C iPj k l ) , we have AijMk --

-- Iij

MM D i j k l -- ) i F A i j kl MP D i j kl -- --.'~p A i j kl PM D i j k I -- --,'~M A i j kl PP D i j kl -- ,'~M A i j kl -mij - ) i M m i M + ~ P m P ..=p

(3.143)

Eij - )iRE P

3.4.4

Transformation

Yielding

and Twinning

of TTC

To complete the description of the transformation plasticity of TTC we need a yield criterion e,

-

0

(3.144)

and a flow rule --P dEij

_ O P (Eij, v~, Q)d0

(3.145)

The former determines the critical overall stress level at which the zirconia particles with characteristic radius 0 transform from tetragonal to

Constitutive Modelling

74

monoclinic phase, and the latter the increment to the average inelastic --P

strains Eij due to the phase transformation of particles with radii between ~ - d~0 and ~. We already considered this problem in w where we introduced both the transformation yield condition and the flow rule. If we assume that under the action of overall external load Eij, all zirconia particles with radii larger than ~ have transformed to monoclinic --p phase resulting in an overall inelastic transformation strain Eij , then the overall strain is . . . . I - ---:P E i j - M i j k l ~ k l + Eij -- M i j k l ~ k l + m i j ( v q -- ~ F ) + E i j

(3.146)

Mijkl--Cijkl --p

with m~j and Eij defined by (3.142). Substituting (3.146) into (3.136) we obtain the average matrix stress ~.,.S.M ~j ~ S M _ y~M K,SM ij ~ i j kl-~kl

M{ Kklm.--Mmnpq~pq

---- Cijkl

+ [Kklmnmm.

-

M Kkzm,~mm.

-1 CPtu(mtu M - mPu)] (v~- OF) +ApKklmnAmnpqPpqr, -J"Kklmn'-E --mnP

- ApKkzmnAmnpq pparsCPtu-1

Etu

.147)

Next we impose the Mori-Tanaka (1973) assumption again. We assume that under an increment AEij - do'-Xij(O) to the overall stress Eij, an untransformed tetragonal zirconia particle of radius between Q and 0 - d o will transform to monoclinic phase with the same martensitic combination (/~l,/32,/~3, ~4) as when it is embedded alone in an infinite matrix with uniform external loading E~ M + AE~ M -

EiTM + dcrCijM,Iik,,~.-M~,~pq-Kpq(~)

(3.148)

In accordance with the minimization problem (3.115)-(3.116) the transformation plastic strain of particles transformed under increasing external loading is determined by the minimization of the objective function AGM

Ye

1

3.4. Constitutive Model for ZTC

75

+ E I Bijk,E~ + EiUjDijk,E~]

(3.149)

subject to the constraint (3.116) with E~ depending on the twinning combination (/31, t32,/33,/34), as in (3.101). The increment in the average particle deformation is 1 --p

dEij _ Mij kl( )~pCPpq _ ,,~MCMmn Dmnp --

PC m. Dmnpq PP )dE

(3.150)

where f(co)dco is the total volume of the zirconia particles with radii between co- dco and co, satisfying

fo ~ f(co)dco - Vp

(3.151)

The relation between dco and dcr is determined by the function AGe. It is related to the minimum of the objective function as follows

min(AGM/Vp)

> AGc(co);

unloading

- AGc(CO);

neutral loading

< AGc(e);

loading

(3.152)

In the first two cases, no particles transform to monoclinic phase, i.e. dco = 0, whereas in the third case

CO- dco - AGcl(min(AGM/Vp))

(3.153)

where A G c 1 is the inverse of the monotonically decreasing function

AGe. Equivalently, the relation between do and d(r can be determined by minimizing the objective function d ~ r - min

1 M

,"

- -

- -

[Ci i ktIi~klm,~MmnpqXpq( O)]Aij~, Ea.

Constitutive Modelling

76

x [-2AGc(~o - dQ) -(E[jBiik, + E~Dijkt + EijSM Aijk,) 3.4.5

Transformation Loading

Yielding

under

Ekl%(3./]}

154)

Uniaxial

Among the geometric and physical parameters required in the preceding constitutive description, the temperatures t~, OF, ~),, the thermal expansion coefficients miM and m P, the elastic stiffness tensors cijMt and P and the particle size distribution f(~o) are easily measured and have Cijkt been reported separately for different zirconia toughened ceramics. The energy parameter AGc that is essential to the calculation of yield stress and plastic flow rate, involves the energy barrier AGcR that depends on the size and volume fraction of transformable precipitates, the chemical energy change AGcH and the interfacial energy change AGs. The last two terms can be approximately estimated in terms of the temperature t9 and particle radius ~0using (3.113) and (3.114), while the first term is assumed to be zero in most of the previous calculations (Lange, 1982; Garvie & Swain, 1985). To discuss how the parameter AGc can be established by experiment on ceramics toughened with aligned zirconia particles, we investigate the transformation yielding of ceramics with aligned zirconia particles of equal size under uniaxial loading. Theoretically, all these particles will transform simultaneously, so that ~ij

--

E~(Q)

o'~UA, -

O,

f(o) - Vp6(O- Oc) --P

Z~(o)

(3.155)

- 0

where 5(L0- Oc) is the Dirac delta function,

I~21nl

nlFt2 Y/lYt31

(3.156)

[n3nl n3n2 n3n3_] and (nl, n2, n 3 ) T z (cos/~ Cos ct, COS/3 sin a, sin/3) T is the loading direction relative to the principal axes of the ellipsoidal particle. Substituting (3.155) into (3.149) gives

2AGc

min

UA

[(E~ M~ + o-cRCij~tKktmn'-Mmr~pqEpq )Aijmr~

12

3.4.

Constitutive Model for ZTC

77

(3.157) where ~SM0

M M -- Cij kl { I'(klrnn m m n -- Iiklrnn m m n

+API~klmnAmnpq

-1

P

M

is the average thermal residual stress in the matrix. Once the yield stress crcR is known from uniaxial tension tests, the critical parameter AGc can be calculated. The accuracy of the above constitutive model for the transformation yield stress and flow rate is best demonstrated by comparing the predicted values with experimental data. For this let us consider the pressure sensitivity of the uniaxial tensile yield stress of ceramics toughened with zirconia spheres at the initiation of transformation under uniaxial tensile loading q ~ u a superposed on given confining pressure -plij, -

-;Ii

+

(3.159)

Both the matrix and the particles are assumed to be elastically isotropic with elastic stiffness tensor

C i j k l -- #

~ik~jl + 6il6jk "t-

1 "2lz2u 6ij6kl)

(3.160)

where # is the shear modulus, v is the Poisson ratio and 6ij is the Kronecker delta, p = 8 3 . 2 G P a and v=0.30 will be used for both the matrix and the particle (Garvie & Swain, 1985). Figure 3.9 shows plots of the uniaxial tensile yield stress Yc against the hydrostatic pressure p for prolate spheroidal particles. The aspect ratio is r = c/a = c/b = 0.2, with a = b > c being the three principal radii of the ellipsoid. It is assumed that there is no thermal expansion mismatch between the matrix and the particles. The volume fraction of the tetragonal zirconia particles is taken as 0.35. For the five loading directions considered, the uniaxial tensile yield stress Yc is approximately linearly dependent on the hydrostatic pressure p. The pressure sensitivity coefficients r are between 1.118 and 1.318 except for the case a = 0,/3 = 7r/2. The tensile yield stress at zero hydrostatic pressure is found to be strongly dependent on the critical energy parameter AGc. In the numerical calculations, A G c = --0.22AGcH/Vp,

Constitutive Modelling

78 1700 Experimental // [ - Mg-PSZ ( M S ) / ( [ ,, Mg-PSZ (TS) / 1500

1300 r cD r~ tD

~9 1100

900 ~ / 7

o o

0

,~6

rd6

~/4

~,,//"

zx 0 rd2 o ~6 rd6 v ~3 ~/3 700 , I , l 0 100 200 300 400 500 Confining Pressure, p (MPa)

F i g u r e 3.9" Uniaxial tensile yield stress versus hydrostatic pressure" theoretical and experimental results with AGcu/~/~ = - 0 . 2 8 5 ( 1 - 10/1448)GPa (Garvie & Swain, 1985). If it is assumed that AGcn = 0 and Garvie & Swain's (1985) estimate for the total interfacial energy change AGs/Vp = 5.41~o-ljm -2 is used, the particle size corresponding to the critical energy parameter AGe is found to be Q=24.51nm. It can also be seen that for ceramics toughened with aligned zirconia particles the transformation yield surface is anisotropic. The tensile yield stresses are higher along loading directions in the tetragonal lattice planes (/3 = 0 or /3 = 7r/2) than along other loading directions. The anisotropy is quite similar to that for a single zirconia particle, embedded in an infinite matrix (Zhang & Lam, 1994). Experimental results on the pressure sensitivity coefficients for two grades of Mg-PSZ, MS (maximum stress) and TS (thermal shock resistance), have been reported by Reyes-Morel (1986), and C h e n & ReyesMorel (1986). The theoretical values of the tensile transformation yield stress for such ceramics should be understood as orientation averages of the results for ceramics with aligned particles because the lenticular

3.4. Constitutive Model for ZTC

79

tetragonal zirconia precipitates inside the tested ceramics are randomly orientated. The experimental points are also shown in Fig. 3.9. The pressure sensitivity coefficients according to these experimental data are 1.26 for Mg-PSZ MS and 1.21 for Mg-PSZ TS (Reyes-Morel, 1986) - in good agreement with the theoretical predictions.

This Page Intentionally Left Blank

81

Chapter 4

Elastic Solutions for Isolated Transformable Spots 4.1

Centres of Transformation

The theory of strain centres in isotropic elastic continua has been extensively used in the modelling of transformation toughening. Most efforts have involved two dimensional approximations in conjunction with fracture toughness studies. Several different methods for deriving the potentials of transformable spots have been reported in the literature. Hutchinson (1974) has applied the Muskhelishvili (1954) method and the Eshelby (1957) formalism to obtain complex stress potentials for small circular spots of arbitrary transformation strain. A second method, proposed by Rose (1987a), uses the concepts of force doublets and dipoles to develop Green's functions for infinitesimal strain centres. In the limit of vanishing spot size, both methods yield equivalent results. It has been argued, however, that the complex stress potential method is the superior of the two for two reasons. First, the governing expressions take on a particularly compact form in terms of complex variables. Secondly, this method provides directly the potentials for finite size spots, that are necessary when the shear transformation strain contribution is involved. The Green function method can also be used in this case but it requires a good deal more work. For these reasons, the complex representation is emphasized in

82

Isolated Transformable Spots

this Monograph, but the Green function method is considered first to obtain the fundamental solutions for strain centres to demonstrate the equivalence of the two methods. 2-D Green's functions Two-dimensional Green's functions under plane strain conditions are

G~(x_,x_') -

8 7 r p ( 1 - u ) [ _r2 . - ( 3 - 4 u ) 5 ~ z l o g

(4.1)

x ~ - x~' and _r 2 - x.rx.r, a , ~ , 7 x , y , and repeated inwhere - ~ dex denotes summation, p is the shear modulus and u Poisson's ratio. Plane stress Green functions can be obtained by replacing u in (4.1) with -

u/(l+u).

Green's function G~z(x__,x_') is the component of displacement at x in the direction c~ due to a unit point force at x__'applied in the direction fl, such that the displacement u~(x__) due to an applied load P~(x_') can be written as u~(x__) -

G~z(x_, x_')Pz(x_')

(4.2)

As an example, the displacements at the point (x, y) due to a single force P, acting in the x-direction at the origin are

u~=87r#(1_u) P~

~-(3-4u)l~ xy

uy = 87r#(1 - u) -'-~ r where r 2 - x 2 + y 2 Force Doublets An example of a force doublet is a force P, at x_' - (h, 0) and an equal and opposite force - P ~ at the origin xZ - (0,0), see Fig. 4.1. The displacement in the x-direction from these forces can be written as -

(h, 0)) -

which may be formally rewritten as

(0, 0)]

4.1.

83

Centres of Transformation

Y

T Y

T P~

P~ H

h

F i g u r e 4.1" Force doublets

( G ~ ( x , (h, 0)) - ( G ~ (x__,(0, 0)) ux =

h

(hP~)

Now, if we let h ---, 0 such that (hP~) ---, B~, where B, is a constant, then the limiting process is effectively a differentiation of the Green function u~ -

0

B~:--~Tx~a ~ ( x , x__') -

-B~G~,~:(x__, x_')

(4.3)

The displacement uy in the y-direction may be written by inspection 0

~ - B. b~,G~.(~, ~') - - B . C ~ . , . ( ~ , ~')

(4.4)

For a similar force doublet B~ with a force Py applied at (0, h) in the y-direction and an equal and opposite force - P y at the origin (see Fig. 4.1) the displacements are given by

~. - B~ 0--~a.~(~, ~') - - B ~ a . ~ , ~ ( a , ~')

~

- B~ ~0 G ~(~,

~'1

-

-B~a~

,y (~, ~')

(4.5)

The derivatives of the Green functions appearing in the expressions (4.3)-(4.5) are (vide (4.1))

84

Isolated Transformable

c,~,,~,(~,

1 [6~Z 8rrU(1-v) [ r 2

~__') -

+

613.~-2o,

2-s a x z x. r

r 2

r 4

-(3-4v)6~Z ~_~]

Spots

(4.6)

where 6~/~ is the Kronecker delta. 4.1.1

Centre

of Dilatation

I

D

F i g u r e 4.2: Centre of dilatation A centre of dilatation is obtained by superposition of the force doublets Bx and By with B D -- B x - B y as shown in Fig. 4.2. The displacements are given by (4.3)-(4.6) B D ( 1 -- 2v) x u~ = 47r#(1 -- u) -~ r BD(1-

2u) y

u~ = 4 ~ , ( 1 - ~) 7

(4.7)

where r 2 = x 2 + y2. Stresses corresponding to (4.7) are obtained from Hooke's law for an isotropic material o',~

-

2tt

e~

+ 1 - 2 v 6~e'Y'~

with the strain e~Z given in terms of displacements by

(4.8)

4.1. Centres of Transformation

-

r

85

1 ~ (u.,~ + uZ,.)

(4.9)

From (4.8) and (4.9) the stresses a~Z in terms of the displacements are ( r ~ - ju (5~'r~6 + (Sf~'r/i~6 + 1 -2v2v 5~5.r

u~,~

(4.10)

Substituting the derivatives of the displacements (4.7) into (4.10), we get the stresses due to a centre of dilatation O'xx

%uO'xy - -

4.1.2

Centre

_

BD(1--2v)(1 27r(1 - u) r2

2x 2)

B n ( 1 - - 2v) ( 1 2~r(1 - ~) r2

2y 2

r4

BD (1 -- 2v) xy 7r(1- v) -r~

(4.11)

of Shear y

T

F i g u r e 4.3: Centre of shear at origin A centre of shear at the origin shown in Fig. 4.3 is obtained in a similar manner to that of a centre of dilatation through the superposition of the force doublets (4.3)-(4.5) with Bs - B~ - - B u. The displacements due to a centre of shear are

86

Isolated Transformable S p o t s

u,~ -

[G.~,.(x, x') - G.~,,~(x_, x_')]

-Bs

Bs

2r(1 - u)

~ - -Bs

(

x

xy 2 )

( 1 - u) ~--~+ - r T

[G~,~(~, d ) - G~,~(~, ~')]

Bs

(

2~(1-~)

(1-~),~

y

~y) ~4

(4.12)

The stresses are obtained from (4.12) using (4.10)

O'xx :

O'yy =

O'xy =

Bs

x 2 ( x ~ - 3y 2)

~r(1 - u)

r6

Bs

y2(3x2 _ y2)

7r(1 - ~)

r6

Bs

x y ( x 2 _ y2)

~(1 - u)

r6

(4.13)

T Y

X

F i g u r e 4.4: Centre of shear at (x', y') and rotated by an angle c~ with respect to the x-axis The displacements for a centre of shear located at the point (x', y') and rotated by the angle c~ with respect to the x-axis (Fig. 4.4) are

Bs u~ = 2rr#(1 - u)

{( (1 - u) .-~' -r2

( ~ - ~ ' ) ( y - v')~) E4 cos 2c~

Centres of Transformation

4.1.

+

( 1 -22 p y - yr_'2

B s { ( Uy= 2 w p ( 1 - u ) -

( x - x ' ) 2r4( y - Y ' ) ) sin 2 a }

(l-u)

1-2#x-x' 2 r_2

+

87

( ~ - ~ ' ) ~ ( ~ - ~')) r__4 cos 2~

y-y' r-2

(x-x')(~-~')~) -r4

sin 2a

}

(4.14)

from which the stresses can be written as O'XX

----

__

r(1u )- B s

{ (x - x') 2 ( x - x') 2 -_r63(y - y,)2 cos 2a

+(~ _ ~,)(v _ y,)3(~ - ~')~ - (v - v')~ sin 2a } r6 O'yy

~

-

-

Bs ~

~(1 - u)

{

(y -

~, ~3(x- ~')~ - ( ~ - y')~ COS2~ )

(x-x')2-3(y-y')2_r 6 sin 2a }

-(x-x')(y-y') O'xy

z

--

Bs

27r(1 - u)

{

(y -

_r6

y,

)(x - x')

(~ - ~')~ - ( y - ~')~ -r6

cos 2a

(X -- X/) 4 -~- ( y - y/)4 __ 6 ( X - X/)2(y- y/)2 } r6 sin 2a (4.15) where r 2 - ( x - x') 2 - t - ( y - y,)2

4.1.3

Planar Transformation Strains

Following Rose (1987a), the planar transformation strains c TaZ can be expressed as T

1

(4.16)

where D - c~v T is the planar dilatation and S is the maximum shear strain with the principal strain axis inclined at an angle a to the reference coordinate system. The rotation tensor ;~Z(c~) is given by

Isolated Transformable Spots

88

x~(~)

=

-x,,~,(,~) = cos 2,~

Xxu(c~) = Xyx(a) = sin 2a

(4.17)

To relate the transformation strains (4.16) to the strengths of the transformation centres introduced in w167 and 4.1.2, let us consider a small inclusion situated at the origin. Following the Eshelby procedure (Eshelby, 1957, 1961), the displacement field outside the inclusion is obtained by applying tractions along the boundary of the inclusion which would induce strains in the inclusion in the absence of the surrounding material equal to the transformation strains ~T# . The appropriate tractions are FaT

_

T a~#n#

(4.18)

with

o'~# - #

1 - 2u +2S~:~

where n# is the outward normal to the boundary of the inclusion, and T cry# is obtained by introducing the planar transformation strain tensor (4.16) into the isotropic Hooke's law (4.8). In terms of Green's functions (4.1) and the Eshelby tractions (4.18), the displacements outside the inclusion can be written as

-/s G~z(x_,x')~.yn.yds where S represents the boundary of the inclusion which should not be confused with the maximum shear strain introduced at (4.16). Applying the divergence theorem to us gives

az'~

tto~

= - f

JA

o']ff~ d A

G~#,;(z_, z')~.ydA

4.1. Centres of Transformation

89

where A is the area (volume) of the inclusion. Substituting (4.18) and letting x > > x' this becomes us - -AG~z,~ (x_,x_')a~ ( D6~ 1 - 2v + 2 s x ~ ( ~ ) )

= -AG~,.~(~,~')p

Thus for the unrotated centre of transformation (c~ - 0) the displacements are

tt x

--

ADp

[G~x,x(x, x') + Gxu,u(x__, x')]

1 - 2 u

-

-

- 2 " s [G~,~(~_, ~_') - G~,~(~_, ~_')] ADp 1 --2u

[G~,~(~_, ~') + c ~ ~(~_,~_')] -

'

- 2 ~ s [c~,~(~, ~') - G~,~ (~, ~')]

(4.19)

Comparison with the results of w167 and 4.1.2 gives the following expressions for the strengths of transformation centres BD and B s in terms of the characteristic transformation strain parameters D and S ADp 1-2u

BD--

(4.20)

Bs - 2ASp

where A is the area of the inclusion. 4.1.4

Complex

Representation

Throughout this Monograph we will make extensive use of the complex potential method detailed in Muskhelishvili (1954). In cartesian coordinates (x, y) the stresses at z - x + iy, (i - v/-Z-f) may be expressed in terms of the complex potential functions O(z) and ~(z) by (

~

- ~

+ 2i~

"1

- 2 { ~ ' ( z ) + ~(z)}

(4.21)

90

Isolated Transformable Spots

where a prime denotes differentiation with respect to z, and an overbar complex conjugation, as in - 5 - x - iy. An alternative formulation, which is more convenient for crack problems, uses the Westergaard stress function

e'(z) - r

zr

r

(4.22)

2 { ( ~ - z)fp'(z) + ~ ' ( z ) - (I)(z)}

(4.23)

Substitution of (4.22) into (4.21) yields ayy - ax, + 2iaxy -

In terms of complex potentials, displacements are given by 2#(ux + iuu) -

xr

zr

r

(4.24)

where r = (I)(z) and r = ~(z). In terms of Poisson's ration u, x is given by x = 3 - 4 u for plane strain conditions and by x = ( 3 - u ) / ( 1 + u) for plane stress conditions. The stress potentials under a translation of the origin to the point z0 are given by (Muskhelishvili, 1954) (I)(z) -- (I)I(Z -- Z0)

9 (z) - ~ l ( z - z0) - ~0(I)~(z - z0)

(4.25)

where (I)l(Z) and ~ l ( z ) are the original (untranslated) potentials. Similarly a rotation of the cartesian axes through an angle a with respect to the x-axis results in the following stress potentials (~(Z)--(~l(Ze

i~

~ff ( z ) -- ~ff l ( z e i ~ ) e - 2 i cr

(4.26)

The complex potentials for point forces P. in the x-direction and Py in the y-direction applied at z0 are P. + iPu 1 (I)(z)- - 27r(1 + ~r z - z0

@ ( z ) - ~(P* - iPy) 1 2rr(1 + x) z - zo

P. + iPy ~o 2~r(1+ ~ ) ( z - zo) 2

(4.27)

4.1. Centres of Transformation

91

Together with (4.24) these potentials give the Green functions (4.1), when P, = Py = 1. The centres of transformation given in w167 and 4.1.2 can now be reformulated in terms of complex potentials (see Rose, 1987a). It is easily verified that the complex potentials for a centre of dilatation of strength D at the origin (Fig. 4.2) are

r

=0

Dlt 9 ( z ) - ( 1 - p)27rz 2

(4.28)

where we have used the fact that ~ra~(- ~rxx+tryy) vanishes corresponding to the field (4.11). Likewise, the complex potentials for a centre of shear of strength S at the origin (Fig. 4.3) are Spe 2ia

(I)(z) - - ( 1 - u)2~rz 2 q*(z) = 0

(4.29)

The complex potentials for a centre of shear at z0 at an angle to the x-axis are obtained by applying the translational and rotational transformation formulae (4.25)and (4.26)to (4.29) ~ t e 2ier

(I)(z) - - ( 1 - v)27r(z - z0) ~-

qI(z) -- -Sl'te2ie' -50 7r(1 - u) (z - z0)3

a)

(4.30)

T 0

T ~

x

~

-~ x

F i g u r e 4.5: (a) Half-line of dilatation; (b) half-line of strain centres

Isolated Transformable Spots

92

It will prove useful later (Chapters 5 and 7) when we consider steadystate transformation toughening to define a half-line of dilatation of unit strength (D = 1), with the end-point at the origin (Fig. 4.5a)

D(x, y) = H ( - x ) a ( y ) ;

S(x, y) = 0

(4.31)

where H(x) denotes the Heaviside step function. The infinite-body potentials corresponding to (4.31) are readily derived from (4.28) to be 9 (z) = 0 r

- -

(1 -

u)2rrz

(4.32)

In a like manner, from (4.29) one obtains the infinite-body potentials for a half-line of shear of unit strength (S = 1) defined by D(x,y) = 0;

S(x,y) = H(-x)~5(y),

X(x,y)

=

~

(4.33)

The potentials are

~(z) -- --

pe2i~ (1 -- v)2rrz

9 (z) = --(I)(z)

(4.34)

The non-zero ~(z) results from the translational property of the potential (4.25). The superposition of a half-line of dilatation (4.31) and a half-line of shear (4.33)is called a half-line of strain centres (Fig. 4.5b)

YT F i g u r e 4.6" Notation for integration theorems For future reference, we record here two theorems on the contour

4.2. Transformation Spots

93

integration of complex functions. The Cauchy integral theorem states (Fig. 4.6)

Os

zER-

0; it; 2it;

d__r _ r- z

(4.35)

zES z E R+

and Stokes theorem in two forms for complex functions (Milne-Thomson, 1968)

/f(z,-s

2i :/I~+ ff-ff-~f ( z , -s) d x d y

f(z,-5)d-5 - -2i

4.2

+ ~zf(Z ,-s

(4.36)

Transformation Spots

The problem of determining the elastic fields for inclusions with uniform transformation strains was completely solved by Eshelby (1957, 1961). Using Green's functions for single forces in three dimensions, Eshelby obtained the solutions in terms of certain integrals of these fundamental solutions. The method has become known as Eshelby's technique. In the following, the solutions to the simpler two-dimensional problem of a circular homogeneous inclusion is given in terms of Muskhelishvili's complex potentials. 4.2.1

Transformation

Spots

in an Infinite

Plane

A circular spot of transformation embedded in an infinite plane is shown in Fig. 4.7. The spot is centred in the planar coordinate system (x, y), with the third coordinate pointing out of the page. For a thick plate the spot is a long cylinder, while for a very thin plate the spot is a disk. The Eshelby (1957) procedure is used to obtain the potentials. Prior to transformation, the spot and the matrix form a homogeneous plane. The spot is cut out and allowed to undergo a stress-free transformation with strains r z. The transformation strain tensor is decomposed into dilatational and shear contributions as in (4.16) and (4.17). Next, the spot is squeezed back to its original radius using appropriate tractions, is reinserted into the hole, and the tractions are relaxed. The potentials are derived by integrating the plane strain or plane stress

Isolated Transformable Spots

94

F i g u r e 4.7" Transformation spot at the origin of an uncracked plane

point force potentials (4.27) around the layer on the circumference of the spot. The potentials for points z outside the plane strain spot are

O~ 9~ ( z ) -

i.tSe2ia Ao - - 27r(1 _ u)z2 pDAo 31tSA2 e 2ic' 27r(1 - u)z 2 - 27r2(1 - u)z 4

(4.37)

where Ao = 7ra2 is the spot area. The corresponding plane stress potentials are (I) ( z ) - _itS(1 + u)e2i~Ao c~ 27rz2

(z) - pD(1 + u)mo 3pS(1 + u)A2e 2i~ o~ 2:n.z2 -2~.2z4

(4.38)

The subscript c~ indicates a spot in an infinite uncracked plane. These forms are generalizations of Hutchinson's (1974) plane strain expressions. Notice the presence of the A02 terms in the ~oo potentials. This term will vanish when the Green's function limit of A0 ---* 0 is taken leaving only the leading order terms, as in (4.28)-(4.29). However, it is recovered upon integrating infinitesimal strain centres over a finite size spot. In two-dimensional problems subject only to traction boundary conditions, there is no difference between the plane stress and plane strain situations. The differences between the strain centre potentials (4.37) and (4.38) arise from two contributions. First, the planar tractions that must be applied to annul the transformation strains are different and

4.2.

Transformation Spots

95

secondly, the expressions for the point forces differ. Throughout most of this Monograph, we will be primarily concerned with plane strain conditions. In a sense, the plane stress potentials are quite contrived, since nothing has been done to suppress the out-of-plane strain of the spot. Thus, if the plane were viewed on its edge, a discontinuous bulge would occur over the transformed spot. The potentials for a spot shifted to the point z0 are given by the translation formulae (4.25). Substitution of (4.37)into (4.25) gives

lzSe2ia

Ao

#D

Ao

(I)oo( z ) - 27r(1 - , ) ( z 9 oo(z)-

zo) 2

2 ~ ( 1 - ~ , ) ( z - zo) 2

#Se2'~ {

27r(1 - r,)

3A2o 2~oAo } 7r(z - zo) 4 -I- (z - zo) 3

(4.39)

The Westergaard function associated with (4.39)is (vide (4.22))

~(z)-

#D Ao 27r(1 - ~ ) ( z - zo) 2

#Se 2i~Ao f 3Ao 27r(1 - ~,2) ], 7 r ( z - zo) 4

2(z - ~o) +

(z-zo)~

1 }

(z-zo),

(4.40)

The stresses inside the transformed spot can be calculated by integrating the layer of surface tractions with the observation point z taken inside the spot boundary and then adding the stresses that result from the reversal of the transformation strains. The plane strain potentials inside the spot are #D (I)oo(z)--4(1_~)

~(z)

#Se2i'~

-- --2(1 _ v)

while those for plane stress are (~oo (z) = -

#D(1 + v)

(4.41)

Isolated Transformable Spots

96 9~ ( z ) -

_ # S ( 1 + v)e 2i~ 2

(4.42)

Consistent with Eshelby's results, the stresses inside the spot corresponding to (4.41) and (4.42) are constant. 4.2.2

Transformation

Spots

in a Half-Plane

In the following the free surface problem in combination with a subsurface transformation spot is solved analytically by means of Muskhelishvili's theory of plane elasticity.

F i g u r e 4.8: Subsurface transformation spot The infinite plane potentials in the absence of a free surface for a dilatant inclusion are given by (4.37) or (4.38) with S = 0. From these potentials the stresses are obtained using (4.21). In order to meet the stress free condition (tryy = ~xy = 0) on the surface y = 0, it is necessary to add an elastic field which cancels the stresses appearing from the above infinite plane potentials on this surface. A potential r for the half-plane with prescribed stresses Cryy and tr~y on the surface y = 0 can be determined from a standard method devised by Muskhelishvili

r

-

- 1 / _ ~ tryy - i~rxydt 2~ri ~ t-z

(4.43)

4.3. Homogeneous Dilatant Inclusions

97

Plane problems generally need two potentials for the complete determination of stresses. Half-plane problems can be solved using only one potential determined by (4.43). This potential is defined in the whole plane by analytical continuation, which is inherent in the formulation given by (4.43). The definitions of infinite plane potentials and halfplane potentials are different and the complete half-plane problem is solved by superposition of stresses. In evaluating the integral in (4.43) care should be taken not to confuse the results for the upper and lower half planes. Stresses from the half-plane potential are determined from

+

_ 2 {Co(z)+

o -~rxx o + 2icr~ Cryy

o ) - "~o( z) } { ( 2 - z)O~o(Z) - ( I )(z

(4.44)

where the definition (I)0(z) - (I)0(2) is employed. For a circular inclusion centred at z0 in the half-plane and with an area A ( - zr a2), the complex potential outside of the inclusion is

pA

[

(I)(z) - 2 7r(1 - u)

D

- ( z - ~o) 2 - (z - zo) 2

Se -2i~ + (z - ~o)' -

4.3

S e 2i~

Homogeneous

2Se-2i~

- zo)

(z - ~o) 3

Dilatant

3 S e -2i~ a 2] +

(z - ~o) 4

(4.45)

Inclusions

Imagine that the inclusion before the transformation is cut put from the matrix and is then allowed to transform with uniform dilatation. The inclusion is now reverted to its original shape by applying a hydrostatic traction to its surface, and is then put back into the matrix, such that no stresses act across the interface. Applying an equal and opposite traction to the interface now gives the required stress field. Neglecting the constant stresses in the inclusion arising from the traction applied only to it, we obtain the elastic problem depicted in Fig. 4.9. A surface S in a homogeneous elastic medium is subjected to certain tractions p,, py. This line of argument was first followed by Eshelby (1957) for the more general homogeneous inclusion with constant arbitrary transformation strains. The notations used in the sequel are shown in Fig. 4.9. The shape of

Isolated Transformable Spots

98

the inclusion R is arbitrary, but its boundary S is assumed to be smooth.

F i g u r e 4.9: A dilatant inclusion in a plane The potentials determining the strain fields are obtained by integrating the fundamental solutions (4.27) over the boundary S of the inclusion R, and equating the Eshelby tractions determined by the stress-free strains induced by the transformation in the inclusion to the applied tractions p~ and pv, as given below. The complex potentials for the plane strain inclusion obtained by integrating (4.27) are

1

(I)(z) - - 87r(1 - v) ~(z)-8~(1

1 -

v)

~s Px + ipy ds z - zo

(3-4v) px-ipv z-zo

_ - 5 o ~

(z_zo)

ds(4.46)

The Eshelby tractions are determined by the stresses giving the same uniform strains Q*j in an unrestrained and untransformed inclusion as prevails in an unrestrained inclusion after the stress-free transformation has occurred, i.e.

, E v ( ,ciJ + 1 - v 2v 5ij ekk , )n j Pi -- o'ijnj -- 1 +

(4.47)

i , j = 1,2,3 For uniform dilatation with r 9 __ r 9 __ ~339 __ gT , the Eshelby tractions under plane strain conditions simplify to

Eg T P~

( 1 - 2t,)n~'

a

z,y

(4.48)

4.3.

Homogeneous Dilatant Inclusions

99

Introducing (4.48)in (4.46), the O-potential becomes

-Ec T

~ n~ + in v ds

-E~ T

~

O ( z ) - 8r(1 - v ) ( 1 - 2v)

_

z - z0

87r(1- v ) ( 1 - 2v)

idzo z - z0

Applying Cauchy's theorem of residues (4.35) to the contour integral along S, the integral is seen to vanish outside S, whereas for z inside S the integral is constant, and so giving Eg T

O(z) -

4(1 - v)(1 - 2v)

(4.49)

The #-potential is derived along the same lines, as the O-potential. Introducing (4.48)in (4.46) gives (3 - 4v) n~z --

~(z) - 8r(1 - v)(1 - 2v)

=

Eg T 8n'(1 - v)(1 - 2v)

nx + iny ) zoinV -2O(z _ zo) 2 ds

~s ((3 _ 4u) id-2o + -2oidzo ) z - zo

( z - zo)

Applying the Stokes theorem (4.36) for z inside the inclusion excluding a possible singularity around z in S' (Fig. 4.9), the q-potential becomes

E~ T _

3 - 4v Z --

-

ZO

+ 2 -(:9~0 - (z - z0)2

+ fs ( ( 3 - 4 u ) i d - 5 o ,

)

dxodyo

+ -#oidzo ) }

z-zo

(z-zo)

The line integral over the circle S' vanishes, and the area integral reduces to a simple expression, so that 9 (z) -

2~'(1

- v)

(z - zo) 2 dxodyo

Applying the Stokes theorem (4.36) a second time and again excluding

Isolated Transformable Spots

100

the singular point at z with a circle S ~ the O-potential finally reduces to

EgT

@(z) -

27r(1-v)

( fs id-s

z-z0

fs

idz~ )

,z-z0

For later use, the above expression can be conveniently rewritten in one of the following two forms

O ( z ) - 2 ~ ( 1 - u)

-

EcT

2r(1 - u)

z - z0 (~s

1 dY~ z - z0

(4.50)

For z outside the inclusion the integral over the circular path S ~ does not enter the derivation, so that the constant part of the O-potential is absent. The potentials obtained above completely determine the strain fields. In order to determine the stress fields the potentials must be altered inside the inclusion to compensate for the stress-free transformation strains. For pure dilatation, the O-potential does not change, but the O-potential has to be changed by a constant given by

-Eg T . (I)c(z) -

2 ( 1 - 2u)' 0;

z GR

(4.51)

z~R

Finally, the complex potentials for a homogeneous inclusion undergoing a uniform dilatational transformation under plane strain conditions are obtained by adding (4.51) to (4.49) and from (4.50)

-Eg T . O(z)--

4(1--u)' 0;

Eg T ~(z) - 2~r(1 - u)

zER

z~R ~s

~s

i dxo - {~'; 0; z - zo

zER z qL R

{~"'

zER z~R

1

z-zo

dyo +

0;

where two alternative formulations for the O-potential are given.

(4.52)

4.3. Homogeneous Dilatant Inclusions

4.3.1

Weight Infinite

Functions Plane

101

for a Single

Inclusion

in an

F i g u r e 4.10: Internal inclusion In the second part of this Monograph, we shall require stress components T ~ O'yy T and the mean stress t rT~ - tr~ + o'yy T T in setting up the govO'xx erning equations for a number of problems involving various cracks and transformation zones. These stresses are most conveniently represented through weight functions gxx(z, T T zo), gyy(z, zo), and g T~ (z, z0), as follows T

O'x x

__

EcT

27r(1 - ~,)

~s T )dxo g~x(z'z~

T __ E'cT ~S T (Z zo)dyo O'yy - 27r(1 - v) gYY ' T

n

O'ot ot - -

E~T

~S T x g ~ (z, zo)dxo

EcT

~S T y

27r(1 - v)

27r(1 - v)

g ~ (z, z0

)dyo

(4.53)

The weight functions are real-valued functions which for an arbitrarily placed single inclusion are obtained from the infinite plane potentials (4.52) and the complex stress formulae (4.21)

T gx~(Z, Zo) -- _ T (Z, Zo)-gyy

Y-- YO

(~- ~o)~ + ( y - ~o)~ Y-- YO

( y - yo)~ + ( y - yo)~

T (z, z0) - 0 ga~~(z, z0 ) _ g~;

(4.54)

In this formulation the expressions for gx~ T (z, z0) and gyy T ( z , zo ) are not

102

Isolated

Transformable

Spots

altered whether z is inside or outside the inclusion. The weight functions Tx Ty g~g (z, z0) and g~g (z z0) pertain only to z outside the inclusion. The two alternative forms of g T (z, z0) distinguished by superscripts x and y will be useful later when the weight functions are used in crack problems. For the crack along the x-axis, the weight function g T~ y (z, zo) will be used, while for a crack along the y-axis we will use gTh~(z, zo). 4.3.2

Weight

Functions

for a Subsurface

Inclusion

F i g u r e 4.11: Single subsurface inclusion The half-plane potential (I)T(z) for an inclusion, which by superposition of stresses from the infinite plane potentials gives a stress-free surface, y - 0, is obtained from (4.43) with o'uu - io'~, u - - ( o ' u ~ - i o ' ~ ) , o'u~ i~r~ are the stresses from the infinite plane potentials for an inclusion obtained from (4.52) and (4.21), and so giving E~ 9 ~o(Z) -

/~

2~(1-~,)

i

~dxo;

z

0;

-~o

y <0

(4.55) y>0

The superposition of stresses obtained from (4.52) and (4.21) with those obtained from (4.55) and (4.44) gives the solution for an inclusion in the half-plane. The stresses are given by (4.53) for an arbitrarily located subsurface inclusion, with the weight functions T

gxx ( z , zo ) -

(y 4- y0) 2 - (x - x0) 2 3(y 4- Yo) (y 4- y0) 2 4- ( x - x0) 2 - 2u ((~ + yo) ~ + (~ _ ~o)~)~ Y - Y0 (x - x0) 2 4- (y - y0) 2

4.3. Homogeneous Dilatant Inclusions

103

(y + y o ) ( x - xo) - ( . - .0) T (z. z o ) +4y gyy (y + y0) 2 + ( x - x0) 2 ((y + vo) ~ + (~ - ~o)~) ~

+ Tx g.'~ (z. zo) -

xo) 2 + (y - yo) 2

4(y + Y0) (y + yo) 2 + (x - xo) 2

g T~y( z , z o ) - -

4.3.3

X -- X0

(x -

Weight

4(x - xo) (y + yo) 2 + ( x - xo) 2 Functions

for a Row

(4.56) of Inclusions

F i g u r e 4.12: A row of inclusions in a plane For a self-similar array of inclusions, as depicted in Fig. 4.12 the weight functions can be obtained by adding stresses from all the inclusions in the array onto the central inclusion indicated by the thick contour. These weight functions are designated by capital letters OO

CL(z, zo)- ~

g.. ~ (z, zo + kd)

k=-oo O0

Gyy ~(z.

zo)-

~

g!'ly ( z ' Zo + kd)

k=-oo O0

G T,x ~(z.zo)-

E

g".'.T~ x (z . zo + kd)

k=-oo OO

Ty

c ~ (z. zo) -

Z k=-oo

g ~ ~(z.zo + k~)

(4.57)

Isolated Transformable Spots

104

weight functions g T~ ( z , zo), g T~ ( z , zo), g T~ x (z, z0), and g T~ y (z, z0) are given by (4.54). Stresses are obtained from (4.53) after replacing the weight functions (4.54) by (4.57)

where

the

T

O'x x

r

T

i

27r(1 - t~)

EcT

Jfs T ( Z zo )dxo Cxx '

EcT

~S T ( Z zo )dyo

2~'(1 -- ~)

EcT

~8 a ~T x (z, zo)dxo

EgT

~S G~; T y (z, zo )dyo

27r(1- v)

T

O'aa --

2~'(1 - v)

Gyy

,

(4.58)

The weight functions for an array of inclusions can be further simplified by the use of the following standard summation formulae C~

E

k=-cx~

r/

s,n-~,-rT)

r]2 + (b + dk) 2 = 2 cosh(27rd~ ) - c o s ( 2 ~ ) b+dk

k=-c~

7r

~"

~ + (b + dk)~ = - ~ r

sin(2~'-~)

- ~osh(2~)

(4.59)

The weight functions for a row of internal inclusions with a period d can be rewritten with the help of the above summations and (4.54) to read ~-

sinh(-~-(y- yo))

G.T(z, Zo) -- --~ cosh(-~-(y - Yo)) - cos(-~(x - xo)) ~s i n ( - ~ ( x - xo)) GuT(z, zo) - --~ cos( -Z2~ (x - xo)) - cosh( -Z-(Y2~ yo)) a . T,x ~ (z. zo) - a . T,y . (z. zo) - o

(4.60)

4.3. Homogeneous Dilatant Inclusions 4.3.4

Weight

Functions

105

for a Stack of Inclusions

F i g u r e 4.13: A stack of inclusions in a plane For a self-similar stack of inclusions as depicted in Fig. 4.13, the weight functions can be obtained by adding stresses from all the inclusions in the stack onto the central inclusion indicated by the thick contour. Denoting, O0

a~5(z, zo) -

~

g ~ ( z , zo +

~kd))

k=-oo O0

T

T (z, (Zo + ikd)) k=-c~ (X)

a~,:(z, zo) -

~

~ ~ (z, zo + ~kd)) gag

k=-c~ O0

T,y

gag (z, (z0 + ]r

(4.61)

~ OO

and using the weight functions for a single internal inclusion (4.54)

Isolated Transformable Spots

106

and the summation formulae (4.59), the weight functions (4.61) can be rewritten to read 7r

s i n ( - ~ ( y - Yo))

G~T(z, Zo) -- -~ c o s ( ~ ( y - Yo)) - cosh(-~(x - xo)) s i n h ( - ~ ( x - xo))

~-

GyT(z, Zo) -- -~ cosh(-~(x - xo)) - cos(-~(y - Yo))

c..T.~(z . zo ) 4.3.5

c.~ ~ (z.

-o

(4.62)

W e i g h t Functions for a R o w of Subsurface Inclusions

F i g u r e 4.14: A row of subsurface inclusions For a self-similar array of subsurface inclusions as depicted in Fig. 4.14, the weight functions can be obtained by adding the stresses from all the inclusions in the array onto the central inclusion indicated by the thick contour. The weight functions are obtained through the summation indicated in (4.57) of the weight functions for a subsurface inclusion (4.56). To simplify this derivation, it is expedient to rewrite the weight functions gT ( z , z0) and gyy T (z, Zo) of (4.56) as follows

g ~ ( z , z0 + k d ) -

[

3 + 2y

(~ + ~01~ + ( ~ - ~0 + kd)~

Y - Y0 ( y - y0) 2 + (x0 + kd) ~

4.3. Homogeneous Dilatant Inclusions ~__~]

T gyy (z, zo + kd) - -

1 + 2y

107 z - x o + kd (y + yo) 2 + (x - xo + kd) 2

Y - Yo ( y - yo) 2 + ( x - xo + kd) 2

(4.63)

Using the summation formulae (4.59) and performing the indicated differentiations, gives the weight functions for an array of subsurface inclusions separated by the distance d n" f 3 sinh( -~(y + yo)) G.T (z. zo) -- d ~. cosh( -~-(y + Yo)) - cos( --d--xo) 2~ 4~-

1 - cosh(-~(y + yo))cos(-~-xo)

+--J-~ ( c o s h ( ~ ( ~ + ~o)) - r

:

sinh(2~ } -z-(y- ~o)) ~osh ( ~ ( ~ - ~o)) - ~os( ~ ~o) ~- f sin(-~-(x - xo)) ayT(z. Zo) -- -d ~ cos(~-~-(x - x o ) ) - cosh( 2.~(yd+ Yo)) sin( ~d( x - x0)) sinh( ~d( y + Y0)) d y (cos(-~(x - xo)) - cosh(-~(y + yo))) 2

471"

sin(-~(x - xo)) } cos(-2-~-(x - x o ) ) - c o s h ( - ~ ( y - yo)) T,~ 47r sinh(~(y + Y0)) Gas (z, z0) - ~ cosh(-~-(y + y 0 ) ) - cos(?-~-(x - x0)) 4~"

G~; zo Ty(z, ) --dcos(

s i n ( ? - ~ ( x - x0))

- x o ) ) - cosh( T(v + u0))

(4.64)

As before, the stresses are calculated from (4.58) after the weight functions have been replaced by (4.64).

This Page Intentionally Left Blank

109

Chapter 5

Interaction between Cracks and I s o l a t e d Transformable Particles 5.1

Interaction of a Spot with a Crack

The stress potentials for an arbitrary transformable planar spot derived in Chapter 4 are now used to obtain the image functions for semi-infinite and finite cracks interacting with such a spot. Some results for stress intensity factors as a function of spot location are also presented for the finite and semi-infinite crack problems for which closed form solutions exist. The discussion then turns to the type of stress criteria for triggering transformation and is followed by results of specific studies. In this Section we derive the image potentials ~i(z) and ~i(z) which produce traction-free crack faces when added to the potentials for a transformation spot in an infinite uncracked plane. We first consider a finite crack and then specialize the approach to a semi-infinite crack. Image functions for transformation spots interacting with cracks in a half-plane are also developed. 5.1.1

Finite

crack

A transformation spot centred at z0 near a finite crack is shown in Fig. 5.1. The crack occupies the interval Ix I _< c and is not touched by the spot. The total stress potentials are

Cracks and Isolated Particles

110

F i g u r e 5.1: Transformation spot near a finite crack

(5.1)

(~(Z) -- ~ c ~ ( Z ) --~- (~Pi(C)

where the uncracked plane potentials which are analytic everywhere outside the spot (I)oo(z), f~oo(z) are given by (4.39). The image potentials are assumed to be analytic everywhere in the plane except for branch points at the crack tips. As z approaches the crack faces from the top and bottom (i.e. z x + ' - ; Ixl _< c, where the superscripts + , - denote respectively the upper and lower crack faces), the stresses (4.23) must vanish, i.e. ~ry~ + i~r~y -

0;

z---. x +'-"

Ixl _< c

(5.2)

(Note that as z --~ x + , - , g ---. x-,+). Substitution of (5.1) into (4.23) and the application of (5.2) gives the two equations - ~ i ( x - ) + ali(x +) -

f(x)

Ix] <_ c

(5.3)

-~i(x +) + a ~ ( x - )

f(x)

Ix I ~_ c

(5.4)

-

where the uncracked plane potentials have been combined to yield f(x)

-

S - D

( X - Z0) 2

+

S

( X - Z'0)2

-

2S(x - -5o)

( X - Z0)3

+

3SAo

71"(X-Z0)4

(5.5)

The continuity of tractions on the uncracked portion of the x-axis stipulates that (4.23) be the same as z ~ x from either the upper or lower half-planes. This results in

5.1. Interaction of a Spot with a Crack

9 ~(x-) + ~(~+)

-

~(~+)

+ ~(x-)

111

-

f(~)

-c_<x_
(5.6)

Subtracting (5.4) from (5.3) and combining the result with (5.6) gives -~i(x- ) - f ~ ( x - )

-- -~i(x +) - ~ ( x +) -- f ( x )

- oo <_ x <_ oo (5.7)

Based on arguments from the theory of complex functions, (5.7) represents the analytic continuation of the entire function E ( z ) - O i ( x ) fl~(x) between the upper and lower half-planes. By Liouville's theorem for entire functions, it can be argued that E ( z ) is a constant. Since the image potentials must decrease at least as rapidly as the uncracked plane potentials for large z (i.e. ' ~ i ( z ) , ~ ( x ) < 1/z 2, Izl---+ oo), we conclude E ( z ) - O, so that r

(5.8)

- n~(z)

where (Pi(z) - (Pi(-#). The elimination of (~i(z) from (5.3) and (5.4) gives ~ ( x - ) -4- K2~(x+) - f(x)

Ixl ~ c

(5.9)

which constitutes a Hilbert problem on the interval ]x I < c. Following Rice (1968), the solution is ~2'(z) -

X(z) /_+~ f(t)dt 2~-i ~ x ( t + ) ( t - z) + X ( z ) P ( z )

(5.10)

where, for the finite crack, Ix I < c

X(z) -

1 ,/z~ - c~

(5.11)

and P ( z ) is the two-term polynomial P(z) -

ao + alz

(5.12)

The branch cut of X(z) lies along the crack faces, and the branch )t(z) ..~ 1/z as Izl ~ oc is chosen. It can be shown that P ( z ) i s constant by insisting that the displacements, which are proportional to fl'(z), vanish for large z; this requires al = 0. If the path encircling the crack is

112

C r a c k s a n d I s o l a t e d Particles

traced, the displacement field must be single-valued, which leads directly to a0 = 0. The combination of (5.5), (5.10) and (5.11) now provides the formula

~'(z)-27rx/z

1 /_+c x/c 2 - t 2 2-c 2 ~ t-z u

•

(t-z0) 2 + (t-~0) 2-

( t - z 0 ) 3 + 7 r ( t - z 0 ) 4 dt

(5.13)

In order to evaluate the integral, we introduce the functions

~I(z, zo) -

/ /_

+~ v/c 2 _ t 2 ~ ( t - z l ( t - zo) dt ~-~

~H(z, zo) -

(5.14)

t x/c 2 - t 2

(5.15)

~ ( t - z l ( t - zo) ~ dt

It follows from (5.14) that

/_

~

x/'c 2 - t 2 1 ON /_+~ v/c 2 - t 2 (t - z ) ( t - zo) '~+1 dt - n! Oz~ ~ (t - z ) ( t - zo) dt 7r

On

--~I(z, n! Oz~

zo) -

71"

-~.I~(z, zo)

(5.16)

where the subscript n indicates the number of derivatives with respect to z0. (Note that the form of (5.16) remains unchanged if z0 is replaced with the conjugate ~0.) An analogous process for (5.15) produces the result

/

+~

tv/c 2 _ t 2

( t - z ) ( t - zo)~+~ d t -

1 0,~ /_+c tv/c2 _ t 2 (n + l)! Oz'~ ~ (t - z ) ( t - zo) 2 dt

7r 0~ ~H(z, (n + 1)! 8 z 8 ~ H ~ ( z ,

(n + 1)!

zo) zo)

(5.17)

The functions defined by (5.14) and (5.15) can be evaluated by contour integration, whence

5.1. Interaction of a Spot with a Crack

I(z, zo)-

z + zo ~ / z 2 - c~ + V/Zo~ - c~

H ( z , zo) -

z , / z ~ - c' - z o v / z [ ( z - zo)~

113

1

c' +

(5.18)

2Z~o - c' - 1 (zo - z ) V z ~ - c2

(5.19)

The use of (5.16) and (5.17) to simplify (5.13) results in

n'(z)

-

2Vz

2 -

c2

I

(,_g-- D)II(z, zo) + SlI(Z, z0)

SAo } +S-ZoI2(z, zo) - SHl (z, zo) + -~-~--I3(z, zo) (5.20) where the auxiliary functions I1, 12,/3 and HI are found by differentiating (5.18) and (5.19). Taking the conjugate of the terms independent of z in (5.20), in accordance with (5.8), provides the other image function

r

1

-

2~/z2,. -

c2

{ (S - D)I1 (z -50) + r

(z, zo)

+SzoI2(z,-Zo)- SHl(z,-fo) + SAo I3(z, T0)} -

-

(5.21)

For completeness, this Section is closed with the total finite-crack potentials. The addition of (4.40) and (5.20) gives f~'(z)-D

I(z, z0) } 2~[~ -C,

1 (z-z0) 2 -1

+s

(z-

+ 2~/z

1

2 -

2(z - T0)

z0) ~- -

c2

[

(z-

3A0

zo) ~ - ~ ( z -

z0) ~

,

+ SIl (z, -5o) 2x/z

2 -

Ao,lzzo,]}

II(z -Zo)+-~oI2(z zo)-Hl(z, zo)+-~-~

c2

and the combination of (4.37)and (5.21)provides

(5.22)

Cracks and Isolated Particles

114

O(z) _ _ DI1 (z,-s

+S {

2 ~ / z 2 - c2

S 2x/z

2 -

-1 ( z - zo) 2

+ I1(z,.5o) } 2ffz ~ - c~

f

c2

I1 (z, .50) - zoI2(z, .50)

Ao

- H l ( z , .5o) + ~--~I3(z,.50)

5.1.2

Semi-Infinite

}

(5.23)

Crack

F i g u r e 5.2: Transformation spot near a semi-infinite crack A transformation spot centred at z0 near a semi-infinite crack is shown in Fig. 5.2. The crack lies along the negative x-axis, and the spot is located so that it does not touch the crack faces. The solution process for the image potentials is essentially the same as for the finite crack. The assertion of traction-free crack faces and the continuity of tractions along the uncracked portion of the x-axis leads to (5.7). Analytic continuation arguments again establish (5.8) and result in the Hilbert problem ~ ' ( ~ - ) + ~'(~+) - f(~)

9< 0

(5.24)

where f(x) is given by (5.5). Here, the form of the solution (Rice, 1968) is

~l(z)

_ X(z) f 2~i

f(t)dt

]_oo x ( t +

)(t - z)

+ x(z)P(z)

(5.25)

5.1. Interaction of a Spot with a Crack

115

where 1

X(z)-

/-Vz

(5.26)

and P(z) - ao. The branch cut for (5.26) lies along the half-line x < 0, and the branch v ~ - 1 is taken. By imposing the condition that the remote displacements vanish, it can be shown a0 - 0. The introduction of (5.5) and (5.26) into (5.25) produces the result

a'(z)-

1/o

2~v~

S-D

oo t - z

(t - zo)2

S 2S(t - -fo) 3SAo ] + (t - ~o)2 - (t - zo)3 -~- 7r(t - - Z O ) 4 dt

(5.27)

Following the approach of the last Section, the integrals are evaluated by introducing the expressions

7rM(z, zo) -

ff oo ( t - z,/~ )(t-

7rP(z, zo) -

zo) dt

(5.28)

oo (t - z)(t - zo) 2 dt

(5.29)

from which we derive the formulae

ff

,/~

c~ ( t - z ) ( t - ZO)n+l d t -

10~ ~ x/%~ dt n! Oz~ oo ( t - z)(t - zo) 7r

= ~ M ~ ( z , zo)

oo ( t - z ) ( t - zo)~+ ~ d t -

(5.30)

1 on /o___ txfX-t dt (n + 1)! Oz~ o~ (t - z)(t - zo) ~ ~P~(z, (n + 1)!

zo)

(5.31)

where the subscript n again denotes the number of partial derivatives with respect to z0. Contour integration is used to establish

116

Cracks and Isolated Particles

1

M(z, zo) - v'ff + x / ~

(5.32)

3/2 3x/~ P(z z o ) - z3/2 - z~ ' ( z - zo)= + 2 ( 2 - zo)

(5.33)

The use of (5.30) and (5.31)in (5.27) leads to the formula

1{

~'(z)- -~

( S - D)Ml(z, zo) + SMI(Z,T0)

SAo +SToM2(z, zo) -- coP1(z, zo) + ~

M3(z, zo)} (5 34)

Conjugation of the non-z terms in (5.34) yields

4p(z) -- ~

1{

(S - D)MI (z, To) + SMI (Z, zo)

+SzoM2(z, To) - SPI(z, T0)+

SAo

N

M3(z, To) t (5.35)

For future reference, this Section is closed with the total potentials for the semi-infinite crack. The addition of (4.40) and (5.34) provides 1 _ I ( z , zo) ( z - zo)~

~'(z)- D +s + ~

+

(z

1[

}

- 1 2(z - To) 3Ao zo) ~ - ( z - zo) ~ - ~ ( z - zo) ~ Ml(Z, go)+ -s

SIl(z, T0) 2,/7

(5.36)

and the combination of (4.37)and (5.35) gives

,~(z) _ DMl(z,-~o) -

2e7

(zzo,]}

z o ) - Pl(z, zo)+ -~-~Ma ,

+s

{

-1

(z_zo) ~ +

Ml(z,T0)} ~

5.2.

Stress Intensity Factors

117

s{

-k--~-~ Ml (z,-s

A0

(Z,-s

)

~-'~M3(z,-zo) (5.37)

It should be noted that for Mode-I spot distributions, Rose (1987a) obtained the semi-infinite image potentials from the corresponding finite crack expressions through a coordinate shifting process. As we will subsequently show, the Mode-I symmetric expressions contain only I and M functions, for which the substitutions s = z - c and so = z0 - c can be used to derive M from I in the limit c ---. oo. However, such a formula reduction is not as easy for a single spot of transformation since it is much more difficult to obtain P from H via a coordinate shifting process.

5.2

Stress I n t e n s i t y Factors

The Mode-I and Mode-II stress intensity factors, KI and right crack tip of Fig. 5.1 are defined by

KI + iKII

KII, at the

lim V/2~'(x - c){o'uu(x ) + io'.u(x)}

-

X ..., C-I-

(5.3s)

where the stresses along the x-axis are given by (4.23) and (4.21). Since the stresses due to the image potentials are singular .at the crack tip, the combination of (4.23) and (4.21), and (5.20), (5.21), and (5.38) provides

Is "4ciKII -

(S - D)I1 (c, zo) + XI1 (C,-Zo) SAo

+S-~oI2(c, zo)

-

SHI(c, zo) + ~

} In(c, zo) (5.39)

where S~(c, z 0 ) = -

I2(c, zo) = I~(c, zo) = and

( zo - c) ~/Z~o - c~ c

(zo

-

c)~ v/~o

-

c~

-3c (zo - c ) ~ v / 4

- c~

+ +

CZo

(zo

-

c)(z~o

-

c~)~/~

-3cz~ (zo - c)(z~ - c~)~l~

(5.40)

Cracks and Isolated Particles

118

C2

Hl(C, Zo) -

C3

(zo_c)2v/z~_c2

+ (zo_c)(z~_c2)3/2

(5.41)

The stress intensity factors for the semi-infinite crack of Fig. 5.2 are given by

KI -t- iKII -

lim x/27rx {ay~ + icr~y}

z-.0+

(5.42)

Inserting the singular image terms (5.34) and (5.35) into (4.23) and (4.21), and employing (5.42)leads to

I~2i + i K i i - V f ~ {(S - D)M1 (0, zo) + SM1 (0,-#o) SAo 543(0, z0)} (5.43) +SgoM2(O, zo) - SP1 (0, z0) + ~ where M I ( 0 , zo) -- 2 P 1 ( 0 , zo) --

3/2 2z o

M2(0, zo) - 4z~o/2 15 7/2

Ma(O, z o ) -

(5.44)

8z 0

5.3

Mode-I

Spot

Distributions

F i n i t e crack

Two transformation spots in a Mode-I symmetric distribution near a finite crack are shown in 5.3. The x-axis is a mirror plane with spots located at z0 and go, and their principal shear-strain axes inclined at opposite angles -t-a. The potentials for the spot at z0 are given by (5.22) and (5.23), while those for the spot at g0 are given by the same expressions with the non-z terms conjugated. Thus, the overall potentials can be written as

-

5.3. Mode-I Spot Distributions

119

F i g u r e 5.3: Two transformation spots symmetrically positioned about the x-axis near a finite crack ai(z)

-

a'(z)+

(5.45)

(z)

where the subscript I denotes a Mode-I spot symmetric distribution. The expressions are simplified by noting

2Ii(z, z o ) - H l ( z , z o ) -

2zo [c

x/'c2 - t2 dt

J_~ ( t - z)(t- zo)~

= -zoI2(z, zo)

(5.46)

which results from a manipulation of (5.14) and (5.15). The form of (5.46) also holds when g0 replaces z0. The resulting potentials are

a'(z) - D {Q(z, zo) + Q(z, go)} 2(z - ~o) +s

3Ao

}

i z - - z o i ~ - ~(z - zo) ~ + R(z, z0)

+ ~ { 2 ( z - zo) 3Ao } z - g0) 3 - 7r(z - go) 4 + R(z, go)

(5.47)

and ~(z)

-

-

D[I1 (z, zo) + I1 (Z, Z'0)] 2~/z 2 -

where

c2

+ SR(z, zo) + SR(z, ~0) (5.48)

120

Cracks and Isolated Particles

Q ( z , zo) -

1

Ii(z, zo)

(z - zo) ~ -

2~/z~ - c~

(5.49)

and

R ( z , z0) -

-1

(z - z0)~ +

(-io - zo)I2(z, zo) + AoI3(z, z0)/(27r) 2 , / z ~ - c~

(5.50)

The expressions for symmetrically oriented point/strain nuclei obtained by Rose (1987a) are recovered, if terms of order A02 are dropped from (5.47), (5.48) and (5.50). a)

T

Y

C",,

b)

T

X

Y

.<1 ~0

r, /

'",

Figure 5.4" Two strain centres (point nuclei) symmetrically positioned about the x-axis near a finite crack (a), a semi-infinite crack (b) For completeness, let us briefly describe the procedure adopted by Rose (1987a). Consider the mode I configuration of strain centres in the vicinity of a finite crack, shown in Fig. 5.4 for which the appropriate

5.3. M o d e - I S p o t D i s t r i b u t i o n s

121

source density is

D(x, y) = DS(x - xo) {5(y - yo) + 5(Y + Yo)}

S(x, y) = S~(x - ~o) {5(U - ~o) + ~(~ + Yo)} ~(~0, ~0) = - ~ ( ~ 0 , - y 0 )

(5.51)

=

where D, S, and a are arbitrary constants. As before, the perturbation potentials due to the strain centres can be decomposed into

~(z)=~(z)+~(z) 9 (z) = ~ ( z ) +

~(z)

(5.52)

where (I)~(z)and ~ ( z ) denote the infinite-body (uncracked) potentials obtained from (4.28) and (4.29) by translation to zo = xo + iyo and T0, using the formulae (4.25), whereas (I)i(z)and ~i(z) describe the sourcefree image field which is required to satisfy the traction-free condition on crack faces. For mode I symmetry, the image stress field can be specified by a single potential, say (I), because of the following relation between the image potentials (cf. 4.21)

9 ~(z) -

-~(z)

(5.53)

We now recall that a centre of strain is the superposition of a centre of dilatation and a centre of shear, and consider first a single centre of dilatation of strength D located at z0. From (4.28) we obtain via the translational formula (4.25), the following infinite-body potential

9~ ( z ) = 0 9~ ( z ) -

D# 1 2~(1 - u)(z - z0) 2

(5.54)

Then from (4.21) we find that the stresses on the crack line Ixl _~ c, y - 0 due to (I)~(z) and ~ ( z ) are

y:0 -

(1 - u) 0T0

27r(z- T0)

(5.55)

122

Cracks and Isolated Particles

To cancel the stresses (5.55)on the crack, we use Muskhelishvili procedure and find x(t) (o'uy -io'~y)] dpi(Z)

__

1 L 2rrix(z)

t-z

~=~

(5.56)

where X(z) = x/z 2 - c 2 for internal cracks and C spans the interval x E [-c;e], whereas X(z) = ~ for semi-infinite cracks and C spans x E [-oo; 0]. Substituting (5.55)into (5.56) and performing the indicated integration, we find 9 i(z)

-

D--------~-~A 1(z, g0) (l-u)

(5.57)

where 0A A1 (z, zo) - Ozo

A(z, z0) -

x(zo) - x(z) 4rr(z- zo))((z)

(5.58)

For a symmetrically placed centre of dilatation of strength D, but located at g0, we find the image potential from (5.57) by replacing z0 with ~0. By superposition, the image potential for a pair of symmetrically located (with respect to the x-axis) centres of dilatation is Oi(z) -

D# (1 - u) [A1 (z, z0) + A1 (z, g0)]

(5.59)

Consider next a shear centre of strength S located at z0. The infinitebody potentials for this centre are given by (4.30) which we rewrite in a more compact form r

Y;1 ( z , z o ) e 2i ~

9~(z)--(l_ ~)

~162 ~) g0 E2 (z, zo)e 2i~ 9oo(z)--(l_ where

(5.60)

5.3. M o d e - I S p o t Distributions

X~(z, z0) -

123

0~ { 1 } Oz~ 2,~(z - zo)

The stresses on the crack line Ixl _< c, y from (4.21) (%y - ia~y)[y=0 = - ( 1 -s ' u)

(5.61)

0 due to (5.60) are obtained

[ ~ r l ( ~ ' z0) + ~-~'~r~l(~ ' ~0) +(zo - -io)e-2i~E2(x, 20)] (5.62)

The image potential that annuls the above crack-line stresses is now found from (5.56) to be Oi(z) -

- ( 1 s~ - u)

[~i C~A1(z ,Z0 ) -- ~- 2iaA 1(Z,Z0) +(Z -- -5o)e-2i~A2(z, g0)]

(5.63)

Superimposing on (5.63) the image potential of a shear centre of strength S at g0, we find the image potential of a pair of symmetrically located shear centres to be r

-

( 1S# -v)(z~176

[e 2/~ A2 ( z , )z0 - e -2i~A2 (z,~0)] (5.64)

Finally, superposition of (5.59) and (5.64) gives the image potential for the pair of strain centres shown in Fig. 5.4

r

D~ [ = (1 - u) Al(z, z0)+ Al(z,z0) +~(zo - ~o) [~A~(z, zo) - ~-~A~(z, ~o)]

(5.65)

S e m i - i n f i n i t e crack Two transformation spots in Mode-I symmetry near a semi-infinite crack are shown in Fig. 5.5. The spots are located at z0 and T0, and have principal strain axes inclined at opposite angles. The approach of the previous Section continues to hold, if we replace the potentials on the

124

Cracks and Isolated Particles

F i g u r e 5.5- Two transformation spots symmetrically positioned about the x-axis near a semi-infinite crack right side of (5.45) with (5.36) and (5.37). The overall expressions are simplified by noting the relation 2z0/_:

2Ml(z, z o ) - Pl(z, z o ) -

-x/ZTdt

(t- z)(t- zo)

= - z o M 2 ( z , zo)

(5.66)

which follows from (5.30) and (5.31). Rather than write out the full expressions, we point out that the forms can be quickly obtained from (5.47)-(5.50) by replacing x/z 2 - c2 with xF. The resulting expressions are generalizations of those obtained by Hutchinson (1974). For illustration, we give here the potential for the mode I configuration but for a semi-infinite crack. For this we substitute Iz0 - c I << c into (5.58) and get A((, (o) -

-1 47rx~(v~ + x/~)

(5.67)

where - {+iyo-re

ie,

~ - x-c

(5.68)

The image potential is again given by (5.65), with A((,~0) defined by (5.67).

5.3. Mode-I Spot Distributions

I

I

125

I

I

I

1

I

L 1

i 2

i 3

K<0

y/c 0 -1

--

-2

--

K<0 -3

-

-4 _z

-3

-2

-1

1 0

4

x/c F i g u r e 5.6" K-sign diagram; dilatation only (i.e. S / D = O)

Stress

Intensity

Factors

for Mode-I

Spot

Distributions

As an example, we consider the Mode-I stress intensity factor (hereafter abbreviated as just K) induced at the right tip of a finite crack by a set of symmetrically distributed spots, see Fig. 5.3. Further, the spots are treated as points of transformation so that all second order terms in area are discarded. The combination of (5.38) and the conjugate expression provides K -

2Re ~ { - D I 1 (c, z) + S(-Z - z)I2(c, z)} C

(5.69)

where (5.46) has been used, and the subscript on the spot central coordinate has been dropped. For plane strain, the coefficients are

D -

EOT(dA) 6 (1 - . )

s

=

E s ( d A ) e 2i~ 4 (1 -

(5.7o)

with the infinitesimal spot area denoted by dA. The introduction of (5.70) and the normalized coordinates z/c allows (5.69)to be put in the nondimensional form

126

Cracks and Isolated Particles

_, y/c 0

-2 -3 -4

-4

1

I/

I

I

I

1

-3

-2

-1

0 x/c

1

2

\1

3

F i g u r e 5.7: K-sign diagram; ( S / D - 4, r

3Kc3/2(1 - u)v/~ EOT(dA)

Re

1

(z. . 1). . z~ C

4

O)

20T(1 + u)

c

z)

c

Z •

1

2i

2

-

+

c

(~-1) (~)-1 (i- 1) (~) - 1 z

z

Z

2

3/2

(5.71)

c

which, for fixed S / D , a and u, can be used to map out the regions where spots induce positive (opening) and negative (closing) K. Figures 5.6-5.8 show the sign of K as a function of spot location for several different parameter combinations. In all figures the crack lies on the interval - 1 _~ x/c _~ 1 , y - 0, and the value u - 0.25 has been assumed. Figure 5.6 shows the results for dilatation only (i.e. S / D - 0). The two boundaries emanating from the right tip are asymptotic to radial segments inclined at the angles +7r/3 (Hutchinson, 1974). Figure

5.3. Mode-I Spot Distributions

4t

I

I

127

1

I

I

I

3

2

K>0

K<0

K<0

y/c 0

g<0 /

K>0

-2

-4 t -4

I -3

I -2

I -1

I 1 I 0 1 x/c

1

I

I

2

3

4

F i g u r e 5.8: /f-sign diagram; (S/D - 4, r - 7r/4) 5.7 shows a similar plot for S / D = 4 and r = 0. The presence of the shear strain component causes considerable complications in the diagram with seven regions converging at the right tip. Figure 5.8 is a companion plot that shows the effect of changing the inclination angle to r = 7r/4. The crack line has been omitted from this figure so that the very small region of shielding which surrounds the crack faces is clearly visible. The potentials derived in this Chapter can be used to simulate the effect of the transformed particle region surrounding a crack tip in zirconiareinforced ceramics by replacing the terms D and S with the appropriate microstructural strains. As described in Chapter 3, D is given by the lattice transformation value of 0.04, but, as explained there, S can range from 0 to 0.16. For a random aggregate of transformed particles, the value of the inclination angle r can be assumed to take on a uniform distribution in the range -7r/2 < r < 7r/2. Simulations are conducted by applying a set of remote stresses and specifying a spot transformation criterion. As the magnitude of the applied stresses is increased, spots around the crack tip will begin to transform and influence both the stresses at the untransformed spot centres and the K at the tip. As the example shows the presence of shear strains is expected to have an important influence on the process. The effect of various transformation

128

Cracks and Isolated Particles

strain and critical stress criteria on the evolution of the transformed region will be described in Part II of this Monograph. Finally, let us give an expression for the mode I stress intensity factor at the tip of a semi-infinite crack (Fig. 5.4b). We again omit the subscript I and simply denote the stress intensity factor K K -

2 lim ~ O i ( ( , ~ ' o ) ~---,0+

(5.72)

Substitution of (5.65) and (5.67) into the above equation gives

(i - u)~/27rr3 cos ~ + 3~ sin 0 sin(2c~- --)

(5.73)

129

Chapter 6

Modelling of Cracks by Dislocations 6.1

Dislocation Formalism

For many two-dimensional problems which we shall consider in this Monograph it is not easy to construct appropriate Muskhelishvili complex potentials. This is especially so in the case of single or multiple edge crack problems. In these cases we shall resort to the dislocation formalism that has been expounded in many treatises. The discussion of this formalism by Bilby & Eshelby (1968) is particularly enlightening and succinct. For the sake of completeness though, we shall reproduce certain essential features of this formalism. The replacement of slitlike cracks by dislocations is based on the fact that both these entities are essentially displacement discontinuities. A dislocation is a line bounding a surface across which the displacement field suffers a discontinuity, measured by Burgers vector of magnitude b. The simplest kind of dislocation is a straight line whose associated Burgers vector is constant. Dislocations are discrete entities but if they are packed closely it is reasonably accurate to smear them out as a continuous distribution of infinitesimal dislocations. Two types of dislocations (line singularities) will be made use of in this Monograph- the so-called screw and edge dislocations, respectively. In a cartesian coordinate system Oxyz with x the horizontal axis, y the vertical axis and the z-axis pointed normal to the plane of paper, screw and edge dislocation are straight line singularities parallel to the z-axis

130

Modelling of Cracks by Dislocations

with constant Burgers vector parallel to z-axis and y-axis (or z-axis), respectively. If an edge dislocation has its Burgers vector parallel to the x-axis it is said to be in glide, while if the Burgers vector is parallel to the y-axis it is said to be in climb. The screw and edge line singularities, for the case that they pass through the origin O, induce a state of anti-plane strain and plane strain, respectively, with the following stress fields (see e.g. Weertman & Weertman, 1964). Anti-plane strain state due to a screw dislocation: ~yz - - A

Y x2 +y2 X

axz = A x 2~ + y 2 ax~ = auy = trzz = erxy = 0

(6.1)

Plane strain state due to an edge dislocation in climb: x ( x 2 - y2) a~x - A (x 2 + y2)2 x ( x 2 + 3y 2) ayy - A (x2 + y2)2 a~y - A y ( x 2 - y2)

(x2 + y2)2

a z~ = u(cr~ + ~ryy)

ax~ = ay~ = 0

(6.2)

Plane strain state due to an edge dislocation in glide: y(3x 2

o'~ - - A

+

y2)

(x 2 + y2)2

auu - A y ( x 2 - y2)

(x 2 + y2)2 o'xu - A x ( x 2 - y2)

(x2 + y2)2

azz = u ( a ~ + ,Tyy)

6.1. Dislocation Formalism

131

a~z - avz - 0

(6.3)

where A - p b / 2 r g , # is the shear modulus, b the magnitude of Burgers vector, and g takes the values 1 and ( 1 - v) in the anti-plane strain and plane strain states, respectively, v being Poisson's ratio. 6.1.1

Complex

Representation

of Dislocations

The above stress fields for single dislocations can also be constructed from Muskhelishvili's theory of plane elasticity. For instance, the complex potentials for an edge dislocation in glide (i.e. Burgers vector is directed along the negative x-axis) located at zo = xo + iyo are iA r

-

-2(z-

,D (z)

--

-~-

zo)

iA[ 1

z - zo

] z0

( z - zo) 2

(6.4)

Likewise the potentials for a dislocation in climb (i.e. Burgers vector along the negative y-axis) are

r

A (z)

-

2 ( z - zo)

~D (z) _

A [ 2

1

~0

z-zo

(z-zo)

]

(6.5)

2

The stresses (6.2) and (6.3) can be obtained from (6.4) and (6.5) with the aid of the stress formulae (4.21). A general edge dislocation situated at z0 can be represented by CO(z) - c~ln(z - z0) n

CD(z)

-

~ln(z

- z0) - ~ ~

Z0

Z ~

(6.6) Z0

where a = # b / r i ( n + 1) with b = b, + iby being the complex Burgers + 1), where uc = [at] + i[uo], vector. Alternatively, a = t t u c e i ~ J a r ] - u + -u~-, [ u 0 ] - u + - u ~ , x - 3 - 4 v (plane strain conditions). [u~] and [u0] are the jumps in tangential and normal displacements across the dislocation line respectively, as indicated in Fig. 6.1.

Modelling of Cracks by Dislocations

132

uo

X

F i g u r e 6.1" An arbitrary dislocation in an infinite plane

6.2

Representation of Cracks by Dislocations

Now, let us consider an isolated central crack of length 2a, occupying the position Ixl < a, y = 0, i.e. the midpoint of the crack is assumed to coincide with the origin. If the solid is subjected to a uniform far-field stress ~ryu - cr, the crack may be represented by a distribution of infinitesimal edge dislocations in "climb" which allow for a displacement discontinuity across the crack in the y-direction (i.e. opening Mode I deformation). It is evident from (6.2) that the only non-vanishing components of the stress tensor are cryy and cr**. Furthermore, as the faces of the crack are traction-free it is clear that the resultant stress on any dislocation in the distribution must vanish. In other words, the resultant of the far-field stress ~ and the stress ~ryy due to the interaction of the dislocations representing the crack must vanish. Now, from (6.2) it follows that the stress ~uu at z due to a dislocation at x ~ is

~ruu(z ) -

A f ( x ' ) dz'z '

(6.7)

X--

where f ( x ~ ) d z ~ is the magnitude of the Burgers vector of all infinitesimal dislocations between x ~ and x ~ + dx ~, f ( x ~) being their distribution function. Thus, the traction-free requirement of the crack faces leads to the following singular integral equation f ( x ' ) dx'-+-

JfD X

-

-

Xt

Ao"

-

0

(6.8)

6.2. R e p r e s e n t a t i o n

133

of Cracks by Dislocations

where D is the region of the z-axis, Ix[ < c, over which the dislocations are distributed. The solution of singular integral equations of the type (6.8) - so-called Cauchy singular integral equations- is discussed in detail by Muskhelishvili (1954). The singular integral equation (6.8) also results if the solid is subjected to a uniform far-field stress ~r~ - a, when the crack is represented by a continuous distribution of edge dislocations in "glide" which allow for a displacement discontinuity of the crack faces in the z-direction. After taking into account symmetry, the only non-zero component of the stress tensor (6.3) in this case is azy. A crack subjected to in-plane shear stress is said to undergo sliding Mode II deformation. Similarly, the singular integral equation (6.8) describes a solid subjected to a uniform far-field stress ay~ - a, when the crack is represented by a continuous distribution of screw dislocations which allow for a displacement discontinuity of the crack faces in the z-direction. The only non-zero component of the stress tensor (6.1) is %z. A crack subjected to uniform longitudinal shear stress is said to be in Mode III (i.e. tearing mode) or in anti-plane strain deformation. It should be mentioned that the distribution function f ( z ' ) is an odd function for z' both for edge and screw dislocations. This follows from symmetry considerations and the inverted pile-up of dislocations used to represent the cracks, whereby positive and negative dislocations lie in the intervals 0 < z < c and - c < z < 0, respectively. Moreover, in the anti-plane strain deformation mode represented by screw dislocations there is no traction across the face z - 0. Thus the solution for x > 0 in an infinite plane can be used to represent an edge crack in a semi-infinite plane with free boundary along z - 0. The solution of the singular integral equation (6.8) gives the distribution function for a central crack lying along y - 0, [z[ < c

f (x ) -

1

7r2 A x/~2 _ x ~

/_~ ~x/c2 c

z '2

x - x'

D

(6 9)

d x ' + 7rx / c 2 - x2

where D is an arbitrary constant. 1 From the distribution function we can calculate the number of dislocations, N ( x ) , between 0 and x and, hence, the relative displacement, A(z), of the crack faces. N ( z ) is obtained by 1The solution of the singular integral equation (6.8), as given by Muskhelishvili (1954), involves the theory of functions of a complex variable, wherein certain singular terms are brought into the solution almost blatantly. An alternative solution of (6.8), involving a change of variables, is given by Bilby & Eshelby (1968) which is very instructive to understanding the solution of this simple singular integral equation.

Modelling of Cracks by Dislocations

134

integrating f ( x ) between the limits 0 and x

N(x) -

f(x')dx'

(6.ao)

and the relative displacement A(x) is given by

A(x)-

bN(x)

(6.11)

where b is the magnitude of the Burgers vector of an individual dislocation. However, to ensure A(4-c) = 0, the net sum of dislocations representing the crack must vanish, and hence the right hand side of (6.10) must vanish, whence it follows that the arbitrary constant D is equal to zero. Besides being able to find the crack opening displacement, A(x), it is possible to find the stress intensity factor K, (Bilby & Eshelby, 1968) ii2 -

27r3A2 limc(c- x)f2(x)

(6.12)

So far in our discussion of the dislocation formalism we restricted our attention to a single crack for the clarity of presentation. We shall now extend this formalism to arrays of cracks. We shall find, in particular, that the procedure is similar, except that we have to consider interaction between dislocations representing each crack together with those representing other cracks in the array. Of course, the solution of the resulting singular integral equation is unlikely to have a closed form. We shall demonstrate the procedure on the array of cracks shown in Fig. 6.2. An infinite elastic solid contains a sequence of slitlike cracks with a constant distance of vertical separation, hi (a stack of cracks). Note that the solid is deforming under one of the three far-field stresses shown. The inverted array of dislocations representing the cracks is illustrated in Fig. 6.2, where edge dislocations are shown for simplicity. The traction-free cracks occupy the positions - c <_ x < c; y - nhl (n - 0, 4-1, 4-2, ...). The governing singular integral equation can be derived in exactly the same manner as in the case of a single (i.e. isolated) crack in an infinite elastic solid. However, due consideration has to be given to the interaction of dislocations in the array. Clearly, each dislocation in the freely-slipping cracks is subjected to interaction stresses from all the other dislocations in the distributions. However, because of symmetry, each crack can be represented by the same continuous distribution of infinitesimal dislocations. Again because of symmetry, the interactions of all but one component of the stress tensor (ayy in Mode I,

6.2.

Representation of Cracks by Dislocations

TY |

|

|

135

Ix}

~ |

|

_.L

_.L_L_.L_.L

_J_

_.L_L_J__J_

_L

_.L_J__.L_.L

13).),-13

T

|

:-o

"IP"IP'T''T''IP

"f'"lff"lff"lP'

"~

~'"~"IP'Iff'T'

T T

u

2c _J_ "r'"~'IP'"IP

T

'~"~"IP''T

T

..L._L..L_.L

x

..L_I_..L_L

|

@

@

@

@

"-V "-V "-V "-V "-V

T

13yz = 13

F i g u r e 6.2: An infinite elastic solid containing a sequence of cracks with constant distance of vertical separation and subjected to a far-field stress cri~ - cr, and the equivalent distribution of long straight dislocations replacing the traction-free cracks

r in Mode II, and ~ryz in Mode III) will cancel each other, and we will be left with one singular integral equation in each mode which assures a traction-free state of crack faces. An expression for the interactive stresses is readily obtained by considering the influence of a vertical array of relevant dislocations situated along a plane x = x' upon a point x along one of the planes y = + n h l . Thus, for example, the shear stress ~rxy due to such an array of edge dislocations in glide situated along the plane x = x' is

~,~ -

A(x-

x') E

_ .

hl

[ ( x - x ' ) 2 + n2h~] 2

(6.13)

n~--l:X)

at a point x along one of the planes y - +nhl in the case of a stack of cracks in Mode II deformation. It is possible to evaluate the infinite sum over n - the number of rows in the stack - in terms of hyperbolic

Modelling of Cracks by Dislocations

136 functions, whence 71"2

a~y - A--~(x- x')cosech2[~r(x- xt)/hl]

(6.14)

For illustration purposes we presented above the total interactive stress only in Mode II deformation. The expressions for the respective non-zero stress tensor components can be derived for Modes I and III in a similar manner. We have yet to satisfy the traction-free requirement of the crack faces in respect of the non-zero stress tensor component due to crack interaction. This requirement, when translated into the language of dislocation kinematics, demands that the resultant stress on any dislocation in the distribution representing any of the cracks vanish when the system is in equilibrium, whence it follows that

f a

f(x')a(x', x)dx'+ F ( x ) - 0 X

--

(6.15)

X t

where G(x ~, x)- the term due to interaction of dislocations- is listed in the following sections for various crack configurations and deformation modes, and F(x), as before, takes the value a in the interval Ixl _< c. By a we understand the respective far-field stress, 9 (~uy ~ - ( ~ i n M o d e I and SO o n .

6.2.1

W e i g h t F u n c t i o n s for an E d g e D i s l o c a t i o n s an I n f i n i t e P l a n e

u,

-~ U0

Ur

~

in

-1

U0

F i g u r e 6.3: Dislocations in an infinite plane The in-plane stresses for edge dislocations (6.2) and (6.3) can in general D,3' be written in terms of certain weight functions h~z (z, z0) such that the stresses at z = x + iy due to a dislocation with a Burgers vector in the negative 7-direction situated at zo = xo + iyo are given by

6.2. Representation of Cracks by Dislocations

cruz -

137

Eb

47r(1 - ~)hDSY(z' Zo)

(6.16)

The appropriate weight functions are obtained through a translation of (6.2) and (6.3) as

h~(z,

( x - xo) 2 - ( y - yo) 2 z o ) - (~ - ~o) ((~ _ xo)~ + ( y _ ~o)~)~

h D,y ~ (z, zo) - (~ - ~o)

h~(z,

zo) - (y - ~o)

h~(z,

zo) -

( x - x0) 2 + 3 ( y - y0) 2

((~ - ~o)~ + (~ - ~o)~)~ (x - x0) 2 - ( y - y0) 2

((~ - ~o)~ + ( ~ - yo)~)~

(x - x0) 2 + ( y - y0) 2

3(~ - ~o)~ + (u - uo)~ h ~ (z, zo) - - ( y - ~ o ) ( ~ - - x ~ u i~ :: ~o~i~ D~x

h~

(z, z0) - (y - v0)

(x - x0) 2 - ( y - y0) 2

((~ - ~o)~ + ( y - ~o)~)~

( x - xo) 2 - ( y - yo) 2 h~'~(z, zo) - (~ - ~o) ((~ _ ~o)~ + (~ _ ~o)~)~ h~(z,

-2(~-

yo)

zo) - (~ _ xo)~ + ( ~ _ ~o)~

(6.17)

Superscript D refers to dislocations, superscript x to a Burgers vector in the negative x-direction and superscript y to a Burgers vector in the negative y-direction, as indicated in Fig. 6.3. For future reference, the mean stress weight functions h~D&~(z,zo) -- hxD~(z, zo) + hyD"Y(z, Zo) are included. 6.2.2

Weight

Functions

for a Subsurface

Dislocation

For a subsurface dislocation, the tractions on the free surface, y = 0, have to be annulled. For this the complex potentials (6.6) for an edge dislocation in an infinite plane have to be modified following the procedure suggested by Muskhelishvili. The half-plane problems can however be

Modelling of Cracks by Dislocations

138

T

Y

_t. z0 F i g u r e 6.4: A subsurface dislocation solved using just one potential which tions ~ryy and axy on the free surface The prescribed tractions are obtained and substituted into (4.43) to give the by subscript H A

is constructed for prescribed tracy = 0, as described in Chapter 4. from (6.6) with the signs reversed half-plane potential, distinguished

z - z0

i~- (z - ~o) 2

y< 0

'I'~-

(6.18) A 1 -i2 z-zo

y>0

The stresses corresponding to the half-plane potential (6.18) are determined from (4.44). The weight functions for an edge dislocation in a half-plane with the free surface at y = 0 are given by the superposition of the potentials for dislocations (6.6) and the stress formulae (4.21) and the half-plane potential (6.18) and the half-plane stress formulae (4.44), and are

h.-;~ (z, zo) -

-(~ - ~o) (y + ~o) ~ + ( ~ - ~o)~

+ 2(y - 3yo)(y + yo)(, - ~o) ((~ + ~o) ~ + ( ~ - ~o)2)~

3(~ + yo)~ - (~ - ~o) ~ + 4 y y o ( x - xo) ((y + yo) 2 + (x - xo)2) 3

(x - xo) 2 - ( y - yo) 2

+(x - ~o) ((~ _ xo)~ + ( y _ yo)~)~ ",~(z z o ) -

hy~

,

-2(~-

~o)

(y + yo) 2 - (x - xo) 2

(y + so)2 + (~ _ ~o)2 - (~ - ~o)((y + yo)~ + (~ _ ;oi~-)~

6.2.

Representation of Cracks by Dislocations

139

3(u + uo) ~ - (~ - ~0) ~ -4yyo(x - xo)((y + yo) 2 + (x - xo)2) 3 +(~ - ~o) (~ - ~~

+ 3 ( u - uo) ~ ~o) ~ + ( u - uo)~) ~

((~-

h~(z,

z o ) - ( u - yo)

(y + yo) 2 - (x - xo) 2 ((v + uo) ~ + (~ - ~o)~):

+ 4 YYo (Y + Yo )

(y + yo) 2 - 3(x - xo) 2

((u + vo) ~ + (~ - ~o)~) ~

(x-

+(u-

h ' # ( z , zo) -

y0) 2

uo)((x - ~o) ~ + (u - uo)~) ~ -3(x-

xo)

(y + yo) ~ + (x - Xo) 2

+(x+

h~';~(z, zo) -

x0) 2 -- ( y -

(x-

~o)

(Y + Y o ) ( y - 7yo) + (x - xo) 2 ((v + vo) ~ + ( x - xo)2) 2

x0) 2 + ( y -

y0) 2

2(~ + ~o) (y + yo) 2 + (x - xo) 2

§

+ Yo )

- ( y - ~o)

h ~D~x (z, zo) - (v - vo)

- ( y + 3yo)

(y + yo) 2 - (x - xo) 2

((~ + yo) ~ + ( ~ - ~o)~)~

(y + yo) 2 - 3(x - xo) 2

((v + ~o) ~ + (~ - ~o)~)~

3(x - Xo) 2 + ( y -

yo) 2

((~ - ~o)~ + (~ - ~o)~)~

(~ + ~o) ~ - (~ - ~o) ~ ((y + ~o) ~ + ( ~ - xo)~)~

(~ + ~o) 2 - 3(x - xo) 2 - 4 y y o ( y + Yo)((y + yo) 2 + (x - xo)2) 3 (x-

+(y-

Xo) 2 - ( y -

yo) 2

~o)((~ _ xo)~ + (~ _ yo)~)~

Modelling of Cracks by Dislocations

140

h~(z,

(y + y0) 2 - (x - x0) 2 z o ) - (~ - ~o)((~ + ~o)~ + (~ _ ~o)~)~

3(y + yo)2 - ( x - xo)2 -4yyo(x - xo) ((y + yo) 2 + (x - xo)2) 3

(~ - ~o)~ _ ( ~ - ~o) ~ + ( x - xo)((~ _ ~o)~ + ( y _ ~o)~)~ ~ h ~ (z, z0) -

(y + yo) 2 - ( ~ ~o) 2 2(y + ~o) - 4y0 (y + y0) ~ + (~ - ~0)~ ((y + ~o) ~ + (~ - ~o)~)~ 2(y - yo) (x - x0) 2 + ( y - y0) 2

6.2.3

Weight

Functions

I .1.

_L.

_L.

(6.19)

for a Row

of Dislocations

Y z0

_L.

.L

.1.

F i g u r e 6.5- Array of internal dislocations The weight functions for an internal array of dislocations (i.e. an array in an infinite plane) can be obtained by superposition of stresses from each dislocation in the array. The superposition gives the weight functions for the array in terms of the weight functions for a single dislocation (6.17) through the following summations CK)

h~j (z, zo + kd)

H ~ Y ( z , zo) k=-oo CK)

D,y Hyy (z, z0) -

~ k=-~

D,y ( Z , ZO -+" k d ) h~y

6.2.

Representation of Cracks by Dislocations

141

O0

HxD'y (z, z0) k~--(::K) (:x:)

HaD~y (z, Zo) -k=-cx:)

h"Y(zo.

, zo + kd)

(:X)

Hxn;x ( z , Zo ) -k=-c~ (:X3

D~x

Hvv (z, zo) -

E D,x (z, zo + kd) hv~ k=-c~ O0

HxD'x ( z , Zo ) --

zo + kd) k=-c~ OO

H~D~(z, Zo) --

E

h~D~(z' Zo + kd)

(6.20)

k=-oo

where capital letters are introduced to indicate weight functions referring to arrays of dislocations. In order to simplify the expressions of the weight functions for the horizontal array of dislocations in Fig. 6.5, it is expedient to write the formulae (6.17) as

h~';~(z, zo) -

[1- (v- v0)0-~0] (x - ~0)~x-~0 + (v - v0)~

D'v(z Zo)-- [l + (y-- y o ) ~ ]

h~

,

(x-

h~:~Y(z, z o ) - 2

h~,;~(z, zo)_

(x-

x - xo ~o) ~ + ( v -

X-- Xo

(~ _ ~o)~ + (y _ v~ ~ xo) ~ + ( v -

vo)~

vo) ~

[_2 + (v_ v0)~v0] x -

v-v0

xo 2 + (y - yo) 2

Modelling of Cracks by Dislocations

142

o,x

h~ (z, z o ) - -

[

,-yo

( ~ - ~o)~y~ x - ~o ~ + ( y - ~o) ~

zo)- [1_

X --

XO

_xo + ( y - vo)~

x-

(6.21)

x02 + ( y - y0) 2

To obtain the weight functions for an array of dislocations the summ a t i o n formulae (4.59) are needed. Introducing the auxiliary functions Pk (z, z0)

Fo(z, zo) =

-sin(~-~(x-

x0))

cos(~-~(x - xo)) - c o s h ( - ~ ( y - yo))

2n

sin(-~(x- xo))sinh(-~(y- yo))

Pl(z, zo) - ~ (cos(-~(x - xo)) - cosh(-~-(y - yo))) 2

(6.22)

and Qk(z, zo)

Qo(z, zo) :

Ql(z, z o ) -

sinh(-~(y-

r

yo))

yo))- r

xo))

_ Xo)) cosh(?-~(y - Yo)) - 1 2w cos( 2~(x d d ( c o s h ( ~ ( y - y0)) - cos(2~( x d - x~ 2

(6.23)

where Pk(z, zo) and Qk(z, zo) are related by Ok

P~(z zo) - ~..k Po(z, zo) '

coy~

(Ok Qk(z, zo) - ~yko QO(Z, zo)

(6.24)

the weight functions for an array of dislocations separated by a distance d can now be determined from

{Po(z, zo) - (~ - ~o)p~(z, zo)} H~D'~(z, z o ) - a~ {Po(z, zo) + ( y - ~o)P~(z, zo)}

6.2. Representation of Cracks by Dislocations

143

H~D'u(Z, Zo) - -~ {--(Y- yo)Ql(Z, Zo)} 7r

H~DDY(z, zo) - -~ {2P0(z, z0)} 71"

H~D~;~(Z,Z o ) - ~ {-2Qo(z, zo) + ( y - yo)Ql(z, zo)} D ,x 7r Hyy (z, z o ) - -~ {-(y- yo)Qi(z, zo)} 7r

HxD'x(z, Zo) -- -~ {P0(z, zo) - (y - yo)Pl(z, z0)} H~D~(z, Zo) - -~ {-2Qo(z, z0)}

(6.25)

with the aid of (6.22) and (6.23). 6.2.4

Weight

Functions

I

for a Stack of Dislocations Y

zo "l

F i g u r e 6.6" Stack of internal dislocations The weight functions for a vertical array, or a stack, of dislocations can be obtained following the same superposition procedure as for the

Modelling of Cracks by Dislocations

144

internal row of dislocations. The weight functions for the stack in terms of the weight functions for a single dislocation (6.17) are O3

Hff; y (z, zo)

-

\

-

k=-c~

D,y (Z, Zo) Hvv

O3

E hvv D,y (z, zo + ikd) k=-O3 O3

HrD'v ( z , zo ) k=-oo O3

h~D;~y (z, zo + ikd)

HaD~y (Z, Zo) -k=-O3 O0

Hxh;~ ( z , Zo) --

E

h,D2:~(z, zo + ikd

k=-o3 O3

v,'(z, zo) Hy9

E

HD, x(Z, Zo) -

E

H~D?~(z, Zo) --

E

hyvD'~(z'zo+ikd"

O3

h ~ x(z' zo + ikd)

CK)

h'~D;~(z'Zo + ikd)

(6.26)

k=-o3 In order to simplify the specific expressions of the weight functions for the vertical array of dislocations in Fig. 6.6, it is expedient to write the formulae of (6.17) as

h ~ ( z z o ) - [ ( x - ~o)O ] '

L a X o J

~-~o

(x - xo) 2 + ( y - yo) 2

~,~(z , zo) - [2- (x- xo) ~-~o] (~_ ~o)2 ~ - + ~o hyy (v- v~ 2 h~Y(z, z o ) -

0 ] Y-Yo - l + ( x - xol-~x ~ ( x - xo) 2 + ( y - y o ) 2

6.2. Representation of Cracks by Dislocations

h.D:?(z,

Zo) -- 2

145

9 -- ~ o

(x - xo) 2 + (y - yo) 2 x-

xo 2 + ( y - yo) 2

D,~(Z, ZO) _ [_I _t_(X _ Xo) o~O] x - xo 2Y-hyy + (Yo y - yo) 2

h.~(z, zo) - ( ~ - ~ o ) ~ h,';'(z,

zo) -

-2

0

X

--

X, 0

x - xo 2 + ( y - yo) 2

y - ~o

x - xo 2 + (y - yo) 2

(6.27)

Using the summation formulae (4.59) and introducing the auxiliary functions Pk (z, zo) s i n h ( ? - ~ ( x - xo))

Po(z, z o ) - cosh(~_~_(x _ xo)) - cos( 72'~( y -

yo))

271" COS(--3-(Y2~" Yo))cosh(?-~(x- x o ) ) - 1 x o ) ) - c o s ( - ~ ( y - yo))) 2

Pl(z, z o ) - d ( c o s h ( ~ ( x -

(6.28)

and Qk(z, zo)

Qo(z, zo) =

- sin ( ~ (y - Yo ))

r

vo))- r

sin( ~ ( y - yo)) sinh( -2~r ( ~ - ~o)) d ( c o s ( ~ ( y - y o ) ) - c o s h ( ~ ( x - xo))) 2

2u

Ql(z, z o ) -

- ~o)) (6.29)

where P~(z, zo) and Qk(z, zo) are related by

~k

P~Iz, zot - ~o~ PoIz, zo) 0k

Q~(z, zo) - ~o~ Qo(z, zo)

(6.30)

the weight functions for a stack of dislocations with a distance of separation d can be written as

Modelling of Cracks by Dislocations

146

71"

HxD~;u(Z, Zo) - ~ {(x - xo)Pl(Z, Zo)} D ,y

7r

Hyy (z, zo) - -~ {2P0(z, z o ) - (x - xo)Pl(z, z0)} 71"

HxD'Y(Z, Z o ) - -~ {-Qo(z, zo) + ( y - yo)Ql(Z, Zo)} g ~ D g y ( z , Zo) - -

7r {2P0(z, z0)} 7r

H,D~;*(Z, Zo) -- -~ {--Oo(z, zo) -- (x -- xo)Ql(z, zo)} D ,x

7r

Huu (z, zo) - -~ {-Qo(z, zo) + ( y - yo)Ql(z, zo)} 7r

H~D'~(Z, Zo) - -~ { ( x - xo)Pl(z, zo)} 7r

g~Dg~'(z, z0) -- 2 {-2Q0(z, z0)}

6.2.5

(6.31)

W e i g h t F u n c t i o n s for a R o w of S u b s u r f a c e Dislocations Y

I-

t-

Z0

I-

I-

F-

F i g u r e 6.7: Array of subsurface dislocations As for the internal array of dislocations w the weight functions for a subsurface array of dislocations can be written by a summation of the weight functions for a single subsurface dislocation (6.19) as oo

HxD~y ( z , Zo) -k=-c~

6.2. Representation of Cracks by Dislocations

Hyy o,~(z, zo)

147

O0

-

~

D'Y (z, ZO + k d) hvv

(:X3

HxD'y ( z, zo ) ]g-~--CX:) O0

HD~Y (z, Zo) -k=-oo (:X)

H;x' (z, zo) D

x

hx~ ( z , zo + k=-oo O0

D,X(Z, Zo) -Hyy

hvv (z, zo+ k=-e~ O0

HxD'x ( z , Zo ) --

h ~ (z, zo + kd) k=-oo (X)

HaD~x(z, Zo) -

E hD':~(zc, a , Zo + kd) k=-oo

(6.32)

where capital letters are again introduced to indicate weight functions for arrays of dislocations. Rewriting the weight functions for a single subsurface dislocation (6.19) as h~D;v(Z, Zo) -

0

02 ]

x-xo --1-(y-3yo)~--~yo + 2yyo--~y2o (y + yo) 2 + ( x - Xo) 2

+ [1 _ ( y _ yO)~yo]

x-xo

( x - xo) ~ + ( v - vo) ~

D,y (z, z o ) - [ - 1 + (y + yo)--~y (~~ hvv

- - 2vvo

+ [1 + (y_ yo)o~o]

( v + vo) ~ + (x - ~o) ~

x-xo

( x - x0) ~ + ( v - v0) ~

148

Modelling of Cracks by Dislocations c9

02 ]

Y + Yo

h ~ ( z , zo)- - ( ~ - yo)-~ ~ + 2yyo-~ ~ (~ + ~o)~ + (~ _ ~o)~ - [(~-~o)s

h~D~Y(Z, Zo) -

[--2 + 4yo +2

h ~ J ( z , zo) -

~-yo (~ - ~o)~ + ( ~ - yo)~

x_xo

(y+yo)2+(x_xo) X --

2

XO

(x - xo) 2 + ( y - yo) 2

[2 + (y + 3~o)-~y~ + 2y~o

(y + yo) ~ + (~ _ ~o)~

Y-Yo x - xo 2 4- ( y - yo) 2

+ [_2 + (y _ yo)o_~o ]

yo ~,~(z, zo)- [ - ( y - yo)-~o ~ - 2~yo o~~o] (~ + ~o)y+ h~ ~ + (~_ ~o) ~ - [(y-yo)s

hx~(z, zo)-

y-~o

x - xo 2 + ( y - yo) 2

[

- 1 - ( y + yo)-~y ~ - 2yyo

+ [ 1 _ ( y _ yO)~yo]

h~D~*(z, Z o ) -

2 + 4yo -2

(y + yo) 2 + (x - xo) 2

X--Xo x - xo 2 + ( y - yo) 2

(y + yo) 2 + (x -- Xo) 2

Y - Yo x - xo 2 + ( y - yo) 2

and introducing the auxiliary functions P~:(z, zo)

P~(z, zo) -

- s i n ( -z(~2~

~o))

cos(?-3~(x - x o ) ) - cosh(?-3~(y + yo))

(6.33)

6.2. Representation of Cracks by Dislocations U ( z , zo) =

P ; ( z , zo) =

sin( -T(x 2~ - xo))sinh(-~(y + yo))

2r (r

4~r2 { d~

149

- ~o)) - r

+ vo))) ~

sin( -2~T ( x - xol)cosh(~(y + Yol)

- ~o)) - ~ o s h ( ~ ( v + vo))) ~

(~os(~(~

2sin(-~-(x - xo))sinh2(~-(y + yo)) }(6 34) + (r

- ~o11- r

+ vo))) ~

"

and Q*k(z, zo) Q ; ( z , zo) -

Q~(z, zo) -

sinh( 72~( v + vo)) cosh(?-~-(Y + Yo)) - cos( -2~r z ( ~ - ~o)) 27r cos(-~(x- Xo))cosh(-~(y + Yo))- 1 d (cosh( 2,r(y d, + YO))_ cos(-~(x _ Xo)))2

9 47r2 f c~ z - Xo)) sinh( 2'~-T(y+ yo)) Q2(z, z o ) - ~ [ (cosh( -z(v 2'~ + v o ) ) - r - ~o))) ~ s i n 2 ( ~ ( z - x o ) ) s i n h ( ~ ( y + yo)) ( ~ o s h ( ~ ( v + vo)) - cos( -a-(~ ~ - ~o))) ~

]

f (6.35)

where P~ (z, zo) and Q;(z, zo) are related to P~ (z, zo) and Q; (z, zo) through differentiation with respect to yo, as in (6.24), the weight functions for the array of subsurface dislocations (Fig 6.7) are 7r

HxD~Y(z, zo) - -~ {-P~(z, zo) - (y - 3yo)P{(z, zo) + 2yyoP~(z, zo) + Po(z, z o ) - ( y - yo)Pl(z, zo)} D,y Hyy (z, zo) - ~ {-P~) (z, zo) + (y + yo)P~(z, zo) - 2yyoP~(z, zo)

+ Po(z, zo)+ ( y - yo)Pl(z, zo)} 7F

,

,

H~D'v(z, z o ) - ~ {--(Y- yo)Ql(Z, Zo) + 2yyoQ2(z, zo) - ( v - v o ) Q l ( z , zo)} 7I"

H,~DgY(z,Zo) - -~ {-2P~(z, zo) + 4yoP[(z, zo) + 2Po(z, zo)}

Modelling of Cracks by Dislocations

150

H D;*(Z, Zo)- -~ {2Qo(z, zo)+ (y + 3yo)Ql(z, zo) + 2yyoQ2(z, zo) -2Qo(z, zo) + (y - yo)Ql(z, zo)} D ,x

7r

,

,

Hyy (z, zo) - -~ { - ( y - yo)Q1 (z, zo) - 2yyoQ2(z, zo)

- ( y - yo)Ql(z, zo)} 7r

gxD'~(Z, Zo) - -~ {-P~) (z, zo) - (y + yo)P:(z, zo) - 2yyoP;(z, zo) +Po(z, zo) - (y - Yo)P1 (z, zo)} 7["

,

,

H,~Dg~(Z,Zo)- -~ {2Qo(z, zo)+ 4yoQl(z, z o ) - 2Qo(z, zo)}

(6.36)

Part I1

Transformation Toughening

This Page Intentionally Left Blank

153

Chapter 7

Steady-State Toughening due to Dilatation 7.1

Introduction

In Chapter 5 we studied the interaction between cracks and isolated transformable particles. We shall study more about this interaction in three-dimensions in Chapter 10. In the present Chapter we shall assume that the zirconia systems (e.g. PSZ, DZC) contain sufficiently many small transformable precipitates such that the transformation zone at the crack tip spans many particles. Under these assumptions, the transformable particles can be represented by a continuous distribution of strain centres or spots (Chapter 5) so that a continuum or macroscopic description of the composite zirconia system is quite adequate. We shall consider both super-critically and sub-critically transforming tetragonal precipitates (Chapter 3). In both instances, the transforming phase will be assumed to have the same (isotropic) elastic properties as those of the matrix, both before and after transformation. This is a reasonable assumption to make for PSZ and TZP compositions, as we shall confirm later in Chapter 11. For DZC on the other hand, the elastic properties of the transforming phase can differ substantially from those of the matrix, so that appropriate effective elastic moduli of the composite before and after transformation need to be considered. We shall return to this topic in Chapter 11. For the present two-dimensional continuum description the planar (w can be related to the three-dimentransformation strains c~Z T

154

Steady-State Toughening

sional lattice transformation strains eT, introduced in Chapter 2. The plane strain dilatational component D of c~zT in (4.16) can be related to the dilatation associated with Bain strains eT of a particle (2.1) via D(x_; a) -

2 ~(1 + ~)c(x_; a)e T ,)

(7.1)

where c(x_;a) denotes the local value of the volume fraction of transformed precipitates. In the continuum two-dimensional description the volume fraction c(x; a) is assumed to be independent of the position vector x_. Moreover, under steady-state conditions both c(x_;a) and D(x_; a) are independent of the crack length a. In other words, the plane strain dilatation D due to total lattice dilatation cOT is D -

~2 ( 1 + ~,)c0T

(7.2)

where Op T is the lattice dilatation of a single tetragonal precipitate. As in Chapter 3, we shall denote for simplicity cOT - 0T. From the continuum constitutive description given in w it will be recalled that (7.2) is applicable to the so-called super-critically transforming particles. For partially or sub-critically transforming particles, in the sense described in w the permanent dilatation ~ will be less than 0T. In contrast to D, the effective two-dimensional shear S(x_)in (4.16) does not bear a simple relation to the shear components of the lattice transformation strain eT because of the general occurrence of twinning and other shear-accommodation processes. Partly for that reason, and partly because of mathematical difficulties, the shear components have largely been ignored in most theoretical analyses. However, as we have seen in w there is increasing evidence that the macroscopic transformation shear has an important bearing on transformation toughening and other phenomena. We shall return to this topic in Chapter 9. In this Chapter though, we shall mostly be interested in toughening induced by dilatation, although we shall also consider very briefly the role of shear. We consider next the change in the stress intensity factor induced by the presence of transformation zone near the crack tip. In particular, we will be interested in knowing by how much the crack tip stress intensity factor K tip is reduced from the applied stress intensity factor K appz for the actual geometry of the cracked body at a given load. In other words, by how much is the body toughened by dilatant transformation, i.e.

155

7.1. Toughness Increment

what is the toughness increment. For illustration we consider only the semi-infinite crack configuration here. Finite crack configurations will be considered in the next Chapter.

7.2

Toughness Increment for a Semi-Infinite Stationary Crack

In the continuum approximation being considered here, it is assumed that the height and frontal size of the transformation zone are small in comparison with the length of the crack and other dimensions of the cracked specimen. The stress field remote from the tip of a semi-infinite crack (Fig. 7.1)can then be expressed via Kapp

trij = ~ where the universal applied loading and fracture mechanics. is fully transformed, form as (7.3) except

l

fij(r

(r ---, co)

(7.3)

Williams' functions fij(r that are independent of specimen geometry may be found in any text on As the tip of the crack is approached the material so that asymptotically the stress field has the same that the amplitude now equals K tip Ii'tip

o'ij = ~

fij(r

(r ---, O)

(7.4)

This also follows from the fact that in the absence of transformation I i ' t i p - - I~[ a p p I "

Somewhat surprisingly the relation K t i p = K appl is found to remain true even in the presence of transformation when the crack is stationary and K appz is increased monotonically. The proof of this assertion based on an application of the J-integral was given by Budiansky et al. (1983). For completeness, we reproduce the simple proof here. Assuming that no unloading occurs anywhere in the transformation zone under a monotonic increase in K appl, i.e. cpp increases monotonically at every point in the zone with increasing K appz, the material can be regarded as a nonlinearly elastic, small strain solid with the three-segment dilatation curve shown in Fig. 3.3. The formalism of J-integral therefore is applicable. On a contour F (Fig. 7.1) chosen remote from the crack tip, g is evaluated on the basis of (7.3) giving

Steady-State Toughening

156

T

Y

O= 0T

l

Fully transformed zone

0<0

T Partially transformed zone

c

(Im=(~m

n

F i g u r e 7.1: T r a n s f o r m a t i o n zone ahead of a s t a t i o n a r y crack and contours for the evaluation of J-integral

J -

(1 - v2)(K~PPt)2/E

(7.5)

An identical calculation on a contour shrunk down around the tip gives on the basis of (7.4) J -

(1 - v2)(KtiP)2/E

(7.6)

By p a t h - i n d e p e n d e n c e of J-integral it follows t h a t K tip - K appz regardless of the location of the b o u n d a r y of t r a n s f o r m a t i o n zone or of the distribution of r in it, as long as the latter increases monotonically with increasing KaPPz.

7.3. Toughening due to S t e a d y - S t a t e Crack Growth

7.3

157

Toughening due to Steady-State Crack Growth

In Chapter 5 (w we calculated the stress intensity factor (5.73) induced at the tip of a semi-infinite crack by two centres of strain located symmetrically with respect to the crack in mode I configuration. The stress intensity factor induced by two centres of dilatation of strength D which we shall designate A K tip may be written by setting S = 0 in (5.71) or (5.73) A Kti p -_

2-I~D Ro

[(o ~

- ~] + -Co

(77)

or

AKti p =

#D (1 - v)v/27rr 3

cos

3r 2

--

(7.8)

The centres of dilatation are located at re •162 with respect to the tip of the crack, and the plane strain dilatation D is given by D-

2 + u)O ~(1

(7.9)

Equation (7.9) is a generalisation of eqn (7.2) to include the case of partially transformed centres of dilatation for which 0 < 0 T -- COT . For super-critical transformation (7.9)reverts to (7.2) From eqn (7.8) it follows that any transformable precipitates which fall within 1200 fan ahead of the tip (i.e. -7r/3 < r < ~/3) give a positive A K tip, while those behind this fan give a negative A K tip. Thus, only transformed precipitates left behind in the wake of an advancing tip will lower the crack tip stress intensity factor K tip. For this it is necessary to study the change in K tip with crack growth. The basic solution (7.8) permits the calculation of K tip if the transformation zone and the distribution of 6(r, r within it are known. Denoting by A the region of transformation above y = 0 (Fig. 7.1), it follows by superposition that E - u) Ktiv - Kavvl + 12x/2-~(1 = K~vv t +

E

6 x / ~ ( 1 - u)

fs

O(z o

3 +-5 0 3)dxodyo

//AO(r,r162

A

(7.10)

158

Steady-State Toughening

where we have used (7.9) and replaced p with E / [ 2 ( l + u)]. zo = xo + iyo refers to a point within A. In general, neither A nor O(r, r is known in advance. Moreover, A changes with crack growth depending on the transformation criterion. Before proceeding with the calculation of K tip, it is useful to report briefly a calculation carried out by McMeeking &: Evans (1982) for a stationary semi-infinite crack in super-critically transforming material with t9 = 0 T everywhere in the zone. They approximated the boundary of the transformation zone by using the unperturbed elastic solution (7.3) to calculate the plane strain mean stress (l+u)

=

Kappl(1 q-lI)(Trr)- 89 3

-~-

cos

(~)

(7.11)

Equating (7.11) to the critical mean stress for transformation a~, they found R(r

-

~--~r(1+ u) 2

cos 2

(7.12)

as the distance to the boundary of the transformation zone A (Fig 7.1). The double integral in (7.10) is readily evaluated for the zone shape (7.12) and found to vanish, whence it can be concluded that K t i p - K appl t o the order (oT) 2, since (7.12) is only valid as OT ---, O. However, the previous proof using the J-integral formalism was free of this restriction. To evaluate the double integral (7.10)for the closed contour (7.12) (it represents a cardioid), we can use Green's theorem to reduce the area integral to a contour integral along the boundary of the transformation zone. Alternatively, we can write

Ktip - Kappl +

6X/c~(1 -- u)

r

cos

r dr de (7.13)

--

Formal integration with respect to r gives KtiP -

K"PP' + 6 x / ~E~T (1-

u)

/0

R(r189

--

de

(7.14)

7.3. Toughening due to Steady-State Crack Growth

159

The integral (7.14) can be further simplified by using the identity

0

R 89(r sin ~ sin r - -2~-~

~

R 89(r cos ~ sin r

}

r

+2R 89(r cos ~ cos r

r dR +2R- 89(r cos ~ sin r d---r

(7.15)

Substitution of (7.15)into (7.14) gives Ktip _

Kappl

2 E OT j~o'r r d(Rsin r de (7.16) R- 89162 cos : de 6x/~(1 - v)

It is now readily verified that the integral vanishes for R(r

(7.:2).

given by

We now consider the situation when the crack grows quasi-statically with K tip maintained at a critical value Kc equal to the intrinsic fracture toughness of the matrix material. This would require that the applied stress intensity factor K appz increase with crack advance. Otherwise, as we have seen above, the wake of the transformed material left behind the advancing tip would reduce K tip below Kc. By simulating this growth process in the super-critically transforming material, McMeeking & Evans (1982) have shown that K appz must increase with crack advance Aa to produce a resistance curve such as that shown in Fig. 7.2. We shall consider the resistance curves in detail in Chapter 8. Here we present a detailed study of the limiting steady-state problem which is depicted in Fig. 7.3. This problem supplies the minimum possible value of K t i p / K appz and correspondingly, the maximum possible toughness enhancement. The crack is imagined to have advanced at constant K appz leaving behind a semi-infinite wake of half-height H. To an observer moving with the tip, the stress and strain fields remain unchanged. It is implicit in the situation being considered that no reverse transformation is possible when O"m drops below ~r~n in the wake. The boundary of the loading zone in the region ahead of the tip occurs where crm - crm. All stress components are continuous across the boundary when B > - 4 # / 3 (Chapter 3), but ~m experiences a discontinuous drop in a super-critically transforming material element (B < -4~t/3) as it passes into the zone. For this material the condition cr~ - ~ c m u s t be attained as the boundary is approached from ahead of the zone. For

160

Steady-State Toughening

F i g u r e 7.2: Transient history of K with crack advance for growth at K tip = Kc

the sub-critically transforming materials the zone has a region which is only partially transformed (O < 8 T) as well as a completely transformed portion (0 = 8 T) as shown in Fig. 7.3. Loading occurs in the forward part of the transformation zone where 0epp/0a > 0, and the rear boundary of the loading zone occurs where Oepp/Oa = O. Behind this rear boundary, in the region called the wake, the material unloads and no further transformation takes place. Therefore, the distribution of the transformation in the wake is independent of z so that 0 = O(y) in the wake. Of course, for the super-critically transforming material 0 - o T everywhere in the loading zone and wake.

7.3.1

Steady-State Crack Growth in Super-Critically Transforming Materials

In this Section, results for the super-critically transforming material are presented. We start by deriving an asymptotic result valid for sufficiently small 0T . For sufficiently small OT, the size and shape of the transformation

7.3. Toughening due to Steady-State Crack Growth

161

T

Y

t

-H . . . . . . . .

Fully transformed wake P~tial]y ir~L'nsformed-w-a~ke

x

Loading zone

F i g u r e 7.3: Steady-state growth under constant K appl zone can be approximately found from the elastic solution. The condition am - ~r~ for the front edge of the zone again gives (7.12) when the elastic solution (7.11)is used. The wake boundary (y = H ) m e r g e s with the leading zone boundary at the point on the zone (7.12) corresponding to its maximum height above the x - a x i s where its tangent is horizontal. This occurs at r = 7r/3. The resulting estimate for the boundary of the entire half of the zone can be obtained as follows. From geometrical considerations R(r = H~ sine0 = 2H/x~'3. On the other hand, from (7.12) it follows that R(r = (1 + v)2(K"PPz)2/[67r(~r~)2]. Equating these two values, we get the half-height of the wake

H = x/~(l+v)2 (KappZ) ... 2 127r

a~n

(7.17)

Substitution of (7.17)into (7.12) now gives R(r

8_~ H cos2 ~; r

- 3V~

= H/sin r

r

0 _< r _< r < r < r

(7.18) (7.19)

With 0 = 0T, the integrations required for evaluation of K tip in (7.10) can be carried out analytically using (7.16). We observe that y0 = R ( r 1 6 2 on the boundary remains constant and equal to H, beyond r = r (= r / 3 ) , so that (7.16) can be written as an integral along the front of the transformation zone boundary

Steady-State Toughening

162

Ktip

~

r

2EOT

Kapp I _

jr0r176 R- 89162 cos ~dy0(r

(7.20)

Substituting (7.18) into (7.20), performing the indicated integration, and equating K tip to the intrinsic toughness Kc gives

EOT(H) -

K

-

2(1

1/2

-

(7.21)

in agreement with the result of McMeeking & Evans (1982), who used numerical integration and a somewhat different approach to obtain their result. By eliminating H using (7.17) one can also write the toughening as

Kr v/3 K~PP' = 1 - 1--~w

(7.22)

where w is the nondimensional transformation parameter which was introduced in (3.26). For arbitrary values of w the zone is no longer given by (7.17)-(r.19). It is now necessary to use an iterative procedure for determining the zone boundary. Once it is found, K 'ip is evaluated using (7.10). In order to determine the perturbation in the zone boundary due to transformation, we need to know the mean stress induced by transforming particles which T we designate a,~. The plane strain mean stress am = (a~, + ~ryv + O'zz)/3 taking the out-of-plane stress into account is given by

~m=

1 + u((rx. +ay ) - 4(l+V)Rer

--g-

Y

3

zo)

(7.23)

T developed by two centres of dilatation located The mean stress ~,~ symmetrically about the crack is determined from (7.23) where (I)(z, z0) is given by (5.72) with S = 0. Equation (7.23)follows from (4.21) and the fact that infinite-body potential (I)oo(z) (4.28) does not contribute to the mean stress. Substitution of ~i(z, zo) into (7.23) gives

erm cr~n _

1 18----TRe Y ~ ( x / ~ + w

1 x/~)2 + i ~ - 0 ( v ~ +

~0)2

(7.24)

The mean stress due to the applied field is given by (7.11) when

163

7.3. Toughening due to Steady-State Crack Growth

ylL ~3.0

i

/:

2.5 2.0 /

1.5 1.0

//

/

//' 0

15

5

10

0.5 I

........... t__.--0.0

0.0 0.5

1.0

1.5 2.0 2.5 3.0 3.5 4.0 x/L

F i g u r e 7.4: Steady-state transformation zone shapes. The dashed curve separates the transformation zone front and wake the boundary of the leading edge of transformation zone is approached from outside. The effect of the entire transformation zone on the mean stress can be calculated by integrating (7.24) over the upper-half of the transformation zone. The leading edge of the boundary is governed by

1 er~ _- K~PPZ( l + u) m 3

7r

r

+f /A F(Z;zo)dxodyo,

cos2 0 _< r < r

(7.25)

where F(z; zo) stands for the right hand side of (7.24) after replacing (without ambiguity)( by z and (0 by z0. It is possible to reduce (7.25) by carrying out the integration over x0 (with y0 fixed) from to the leading edge of the zone boundary. In performing the integration, the fact that ~i(z, z0) in (5.72) is proportional to the sum of the differentials of A(z; z0) with respect to z0 and :0 (el. 5.58) can be exploited. That step gives the integral equation for R(r 1 -

Kr

R

) cos

+ ~

M(r162162 0_~r162

(7.26)

164

Steady-State

H/L 5-

Toughening

~Y B

32-

'f

0

0

I

I

I

I

I

I

5

10

15

20

25

30

F i g u r e 7.5: Half-height of transformation zone for super-critically transforming materials where 1 - Re v/_~(~r + x/~)

M(r r with z -

R(r

ir

, zo - R ( r

ir

+

1

]

(7.27)

. w is given by (3.26) and

L---9~.2 [ K r 2 o ' ~

(7.28)

Note that L is the frontal intercept at r = 0 of the leading edge when w ---, 0, i.e. when the transformation strains are small. It is equal to R(0) obtained from (7.17) and (7.18)in this limit. The unknown angle r is obtained from the condition that the leading edge of the boundary must join the wake smoothly, i.e. R(r must satisfy the condition dy0(r

dr

_ d(nsin r

-

dr

= 0

at

r - r

This condition together with (7.26) determines both r

(0,r

(7.29) and R(r

in

An approximate solution for R(r and r was obtained by Budiansky

7.3. Toughening due to Steady-State Crack Growth

165

g c]K appl

1.0 R,x ~ , 0.8

0.20.40"6

0.0

0

Numerical Linear

,

"'-.::: ::~:~i'.~:............... ................,................

5

10

15

20

25

30

Figure 7.6: Steady-state toughening et al. (1983) by expanding R(r

a series

N k=l where the coefficients Ak (and r are unknown. The series representation (7.30) satisfies identically the condition (7.29). For any given set of coefficients Ak and r the left hand side of (7.26) is a well defined function of r Denoting it by Q(r the solution of integral equation (7.26) can be obtained by minimizing V(A1, A2,..., AN, r

-- ~0 r176[ Q ( r

1]2 dr

(7.31)

with respect to Ak (k = 1, ..., N) and r excluding the trivial minimum at r = 0. The minimization was carried out numerically using Newton-Raphson method. Budiansky et al. (1983) generated solutions with N = I , 2 or 3. They found that the quantities H and Ktip/K '~pvz were obtained to within 1% with just N = 1. They also found that to a very good approximation, the shape remains described by (7.18) and (7.19) but the half-height H of the zone is less than the prediction (7.17)

Steady-State Toughening

166

when w is not small. Examples of steady-state transformation zone shape are shown in Fig. 7.4. The half-height is plotted as a function of w in Fig. 7.5. The transformation zone size diverges for w .~ 30.0. This phenomenon is called lock-up, and was discovered by Rose (1986). The angle r separating the transformation zone front and wake increases from 60 ~ for w - 0 to 62.80 for w - 10 through to 68.0 ~ for w - 20 to 97 o at lock-up. The reciprocal toughening ratio K c / K appz is shown in Fig. 7.6 with the linear estimate given in (7.22) shown as the dashed line. The results were given by Amazigo & Budiansky (1988). 7.3.2

Steady-State Transforming

Crack Growth Materials

in Sub-Critically

For this exposition we again follow Budiansky et al. (1983). An energy balance relation for the plane-strain, steady-state problem is (1 - u 2) (K~ppz)2 = (1 - u 2) (Ktip) 2 + 2 E E

j[0H U(y)dy

(7.32)

where U(y) is the residual energy density (i.e. residual stress work per unit area per unit thickness in the z-direction) left behind in the wake as x ---, - c ~ . Equation (7.32) states that of the energy released from the remote field per unit crack extension part is deposited in the wake and the remainder is released by the advancing crack tip through the term involving (Ktip) 2. It is derived from the field equations governing the steady-state problem. Let U be the stress work density at any material point defined by U(c, history) -

jr0~

crijdcij

(7.33)

where it is assumed that there is no initial spatial variation in material properties. U is a history-dependent function in that the linear unloading response of the material is not suppressed. For plane deformations under steady-state growth conditions, the following line integral will be shown to be path-independent

f I - ]r(Unx - ~rijnjui,x)ds

(7.34)

where the crack is taken to be aligned with the x - a x i s and F is a contour such as that in Fig. 7.1. As emphasized by Palmer & Rice (1973), this

7.3.

Toughening due to Steady-State Crack Growth

167

reduces to the well known J-integral for small strain nonlinear elastic (deformation theory) materials; for the irreversible behaviour of interest here path-independence holds only in the steady-state. To prove path independence, let C be any closed contour in the xy-plane, let the region A within C contain no singularities, and let be the outward unit normal to C. By the divergence theorem

Un~ds -

U,~ dA - -

-~-~adA

c

(7.35)

c

The last equality in (7.35) follows from the relation between partial derivatives

O( )/Oa = - 0 ( )/0x

(7.36)

which holds for steady-state growth in the z-direction. It is to be understood that 0( )/Oa denotes the derivative with respect to crack advance at a fixed material point and 0( )/Ox denotes differentiation with the crack length a held fixed. Continuing with (7.35), we can write

-/a~ --~-~adA--/a OU

0~#

~r,j--ff~adA- /a ~ o'ijgij,xdA

= /A (~rijui,x)dA - /c ~rijnjui,~ds

(7.37)

c

The first step in the above continued equality is permissible because the derivative of U is taken at a fixed material point. The second step again uses (7.36), while the remaining steps use equilibrium and the divergence theorem. Thus, by (7.35)and (7.37) C U rtx -- o'ij nj u i , x ) d s

- 0

Since the integrand of I in (7.34) vanishes on the crack faces, it follows that I has the same value for all contours r encircling the tip. Since (7.4) holds at the crack tip, by shrinking I' down to the tip one finds (for plane strain) I - ( 1 - u :~)

(i,f,ip)2 E

(7.38)

Next take F as a circular contour of radius R which is very large compared to the zone height. On this contour there are two contributions to

168

Steady-State Toughening

I; one from the remote field (7.3) outside the wake and the other from the wake itself. Letting R -~ oc, one finds I -

( 1 - v 2)

(KOppz) 2 E

U(y)dy

(7.39)

where U(y) - limx-~_o~ U(x, y). The term vanishes in the remote wake. Taken together, (7.38) and (7.39) give the energy balance (7.32). The above derivation of the path-independence of I requires continuity of the stress and strain fields and thus is limited to the sub-critically transforming materials, although it does apply in the limit B - - 4 # / 3 for the critically transforming material. For the super-critically transforming materials (B < - 4 # / 3 ) , the transformation no longer occurs quasistatically. We have not attempted to write a general energy balance relation for the super-critically transforming materials. However, since the energy balance (7.32) does apply to the case B - - 4 # / 3 and since the solution for all the super-critically transforming materials is the same as that for B - - 4 p / 3 , it follows that (7.32) holds for the super-critical material when U is evaluated using B - - 4 p / 3 . We shall exploit this fact below and show how the steady-state crack growth in super-critically transforming materials discussed above in w can also be studied by finite element approximation. But first we shall discuss the finite element approximation for sub-critically transforming materials. The residual energy density has two components: a part associated with the transformation and a contribution due to residual stresses, 0.11(y) and 0.33(y), left behind in the wake as x ---, - o c . The transformational contribution (refer to Fig. 3.3) is the net work associated with the hysteresis loop leaving a residual dilatation 0 at 0"m - 0. This is vide (3.24) fo e

-B (1

-

B/B)

0d0 + 0"m0 ~ -

0+ a~m

B02 _ 2(1

-

B/B)

(7.40)

Far downstream in the wake, 0"22 = 0 while the constraint of the elastic half-spaces above and below the wake requires r = 0. From the plane strain relations (3.21), one finds that the non-zero stresses in the wake as x ---+-r are given by fill -- 0"33-- - E 0 / [ 3 ( 1 -

v)]

(7.41)

which results in a residual elastic strain energy density of f : 0.rod0

-

7.3. Toughening due to Steady-State Crack Growth

169

EO2/[9(1-u)]. In terms of the distribution of O(y)in the wake, the residual energy deposited into the wake per unit crack extension is therefore

2

/0

U(y)dy - 2

02)

o~0 + 2(1 - B / B ) + 9(1 - u)

dy (7.42)

The energy balance (7.32) with (7.42) provides an alternative expression to (7.10) for calculating K tip if the distribution of 0 is known. Both expressions were used as a consistency check on the accuracy of the numerical solution. The expression (7.42) for the residual energy in the wake is limited to B > - 4 # / 3 . As discussed above, for supercritically transforming materials it remains valid if (7.42) is evaluated using B - - 4 p / 3 . Then 2

v(y)ey-

r

(7.43)

since the full transformation takes place across the wake and the two quadratic terms in 0 cancel each other. Introducing (7.43) into (7.32) and eliminating H using the toughness estimate (7.22) gives the asymptotic result obtained by Amazigo & Budiansky (1988) Kr K app,

,/g

=

1 - 2--~rw

1+

v/~

(7.44)

This toughening estimate for super-critically transforming materials was included in Fig. 7.6 as the dotted curve. Budiansky et al. (1983) used for the sub-critically transforming materials the numerical method developed for the analogous steady-state elastic-plastic crack growth problem by Dean & Hutchinson (1980). The method is an iterative one in which estimates of the zone shape and distribution of 0 in any iterative step are used to generate improved estimates in the subsequent step. The steady-state character of the problem enters in that the solution at any point in the zone involves the history of that material point over its path in the negative x-direction from its intersection with the leading edge of the zone. In the steady-state problem the dilatation-rate is related to the spatial gradient of the dilatation by Oevp _ Oevp Oa Ox

170

Steady-State Toughening

or

A ~pp = - A a(9epp ~cOx where Aa is the part of the zone part of the zone by cOcpp/Ox = 0

(7.45)

crack growth rate. Thus, loading occurs in the forward in which cOcpp/Ox < 0 and unloading occurs in the rear (i.e. the wake) whose forward boundary is characterized (Fig. 7.2).

I I

|1 |1 |1 iii |1

II II Crack tip F i g u r e 7.7: Coarse layout of finite element grid The numerical method of Dean & Hutchinson (1980) is based on displacement finite element discretization of the field equations. A coarse representation of the finite element grid is shown in Fig. 7.7. The actual grid contains a total of 2592 quadrilateral elements, each of which is comprised of four constant strain triangular elements. Tractions determined from the remote field (7.3) are imposed on the outer boundary, whose closest distance to the crack tip is chosen to be about 150 times the half-height H of the transformation zone. A highly refined grid is laid out within a region around the tip. The dimension of the smallest elements is about H/27. A total of 17 quadrilateral elements spanned the half-wake in the y-direction and 22 elements fell between the crack tip and the leading edge of the zone. The finite element equations have the general form Mu -

P-

NO

(7.46)

where M is the elastic stiffness matrix, u is the nodal displacement vector, P is the load vector associated with the boundary tractions, and NO is the contribution of the transformed dilatation which enters in the

7.3.

Toughening due to Steady-State Crack Growth

171

form of a distributed body force. Given an estimate of/9 in any iterative step, (7.46) is solved for u and then the next estimate of the total strain in each element is obtained. From this new estimate of the total strain distribution the loading and wake regions of the transformation zone and of/9 are estimated. In the loading portion of the zone/9 can be expressed directly in terms of ~pp according to Cpp <Ec PP

0-0;

- (1 - B / B ) ( c p p - Cpp);

c < Cpp < Co Cpp pp

-- oT.

o Cpp

- -

C

<

(7.47)

Cpp

c -- c r ~ / B and Cpp o -- Cpp c A - o T ( 1 - B / B ) -1 In the wake 0 is where Cpp independent of x, as already discussed, and O(y) is assigned the value attained at the rear boundary of the loading zone where (gCpp/(gz- O. The elastic solution with 0 - 0 is used to initiate the process. Extremely rapid convergence was obtained, with 7 or 8 iterations sufficient to obtain convergence of computed quantities to three or four significant figures. In the small scale transformation problem use of the dimensionless coordinates x(o'~/Kappl) 2 and y(o'C/Kappl) 2 incorporates all of the dependence on K appz. The three remaining dimensionless parameters which fully specify the problem are w of (3.26), B / # , and Poisson's ratio u which is taken to be 0.3 in all the calculations to be reported below. (For the super-critically transforming materials the dependence on v is incorporated in w but u must be specified independently in the calculations for the sub-critically transforming materials. As already seen, one solution applies to all the super-critically transforming materials, as it is independent of B / # ) . The computed ratio K t i p / K appz for super-critical transforming materials as a function of w is the lowermost of the solid line curves in Fig. 7.8. There it can be seen that the asymptotic formula (7.22), which is plotted as a dashed line overestimates K t i p / K appl. A plot of K t i p / K appl as a function of the dimensionless parameter EOT v/-ff/[(1 - t,)Ii ~ppt] is shown in Fig. 7.9, where the lowermost solid line curve again applies to the super-critically transforming material. For this material the simple asymptotic solution (7.22) provides an excellent approximation over the full range computed, as was conjectured by McMeeking & Evans (1982). The other solid line curves in Figs. 7.8 and 7.9 apply to the sub-critically transforming materials.

Steady-State Toughening

172

gc/K appl 1.0

I

0.8

. .......

B/B = 0

0.6 0.4 0.2 0.0

""" """"'-,,. """"""" ..

i

ml

"

_

~ "...

1- 1--~m

0

4/3

I

t

t

t

5

10

15

20

m

F i g u r e 7.8: Ratio of tip to applied stress intensity factor. The dashed line is the asymptotic result for small w. The curves for the sub-critically transforming materials ( B / # > - 4 / 3 ) were calculated with L , - 0.3 Plots of the transformed dilatation 0 ahead of the crack tip (x > 0, y - 0) are shown in Fig. 7.10 for various values of w for the case - 0 . Two normalizations of 9 have been used in these plots: 0/0 T in Fig 7.10a and in Fig 7.10b we have used 0/r where Cpp ~rm ~/Bis the elastic dilatation at the onset of the transformation. Similar plots of the distributions O(y) across the wake are shown in Fig. 7.11. From Figs. 7.10a and 7.11a it is seen that only a relatively small portion of the zone is fully transformed (i.e. 0 - 0T) when ~o is not small. However in the m a t h e m a t i c a l limit ~o -+ 0, the entire zone becomes fully transformed and the zone boundary is described by (7.18)-(7.19). The normalization used in Figs 7.10b and 7.11b brings out the fact that the 0 - d i s t r i b u t i o n is essentially independent of 0T in the partially transformed zone where 0 < 0T until full transformation is achieved. Similar distributions are found for B / # - - 1 / 2 and B / # - - 1 . However, for a fixed value of w the portion of the zone which is partially transformed shrinks as B / # becomes more negative and must vanish as B --~ - 4 p / 3 . Plots of ~rm/~r~n and cr22/cr~ ahead of the crack tip are displayed in Fig. 7.12 again for various w with B - 0. Curves of K tip/Is appz as a function of w and EO T v/-ff/[(1 - u ) K ~ppz] have been included in Figs. 7.8 and 7.9, respectively, for B / # - O, - 1 / 2 9

.

~

~

-

7.3. Toughening due to Steady-State Crack Growth

KclK

173

appl

1.O O.8

B/U=

0

0.6 0.4-

~

. -4/3

0.2

1- 0.2143 E 0r~-H

........

( 1-v)K appl 0.0

0

I

1

t

2

I

3

E 0 T'd'~,,t ( 1-v)K appl

F i g u r e 7.9: Ratio of tip to applied stress intensity factor. The dashed line is the asymptotic result for small 0T. The curves for the subcritically transforming materials B / # > - 4 / 3 were calculated with u-0.3 and - 1 . The results shown were computed by means of the area integral (7.10). The energy balance relation (7.32) was also used to compute Ktip/KappZ. The difference between the two computed values of (Ktip/K'~PPt- 1) was less than 3% when a - 2w/9~ was less than 0.5 and was less than 7% when a = 2w/9r - 1. As expected, the subcritically transforming materials give rise to smaller reductions in the tip intensity factor than the super-critically transforming materials primarily because much of the zone is only partially transformed. As foreshadowed above, the finite element method used for subcritically transforming materials can also be used for super-critically transforming materials. Now, 0 = 0 T everywhere, but the body forces are not the same throughout the transformation zone. The growing crack will have an unloading effect on the material adjoining its faces leaving behind a wake with residual compressive stresses. On the other hand, the material ahead of the advancing crack tip is still under increasing load from the external sources and there are no residual stresses. The body forces in the front zone are equal to the stresses corresponding to the transformation dilatational strain 0T of a free particle.

S t e a d y - S t a t e Toughening

174

o/o r 1.o

i i

0.8-

1 ot--)oo i

0.6

i i i

0.2 0.4

i

0.5 0.2

a=l.O

i i

a)

0.0

0.00

0.02

0.04

i

0.06

0.08

0.10

o12 x,(<

0.06

0.08

0.10

0.12 x I (tim

KappI

B0 C

O' m

6.4 6.2

ot=l.O

3.0-

2.0 .5

1.0-

o~=0.2 o~=0.1

b)

o.o

0.00

I

0.02

0.04

(2

)2

Kappl "

F i g u r e 7 . 1 0 : D i s t r i b u t i o n of t r a n s f o r m e d d i l a t a t i o n a h e a d of crack tip for various a - 2w/97r w i t h B - 0

7.3.

0/0 T 1.0

---

1 ~---)oo II I

0.8 0.1

I I

o.6

i i i

o.2 o.4

i

,I

0.5 O.2

a)

175

T o u g h e n i n g due to S t e a d y - S t a t e Crack G r o w t h

l I

0.0 0.00

i

i

0.02

0.04

J

0.06

0.08 y ( ~jc )2 Kappl

B0 C (Ym 6.4

3.0

0.5

2.0

a=0.2 1.0 a=0.1

b)

0.0 0.00

i

i

0.02

0.04

i

0.06

0.08

J c 0.10 y ( (Ym ]2 K applz

F i g u r e 7 . 1 1 : D i s t r i b u t i o n of t r a n s f o r m e d d i l a t a t i o n across t h e wake for various c~ - 2w/97r with B - 0

S t e a d y - S t a t e Toughening

176

(Ym/(Ym 3.0 astic solution)

-

2.0

1.0 o~-0

:lO

1

0.5 a)

0.0

0.oo

(y~2/o"c 3.0

I

I

~

I

0.02

0.04

0.06

0.08

].OI

I

O.lO 0.12

I

0.14

~(~:Kapp I )~

"••=0.0

(Elastic solution)

2.0 a---0 5

l~I b)

0.0

0.00

,

0.02

,

0.04

,

0.06

,

0.08

,

O. 10

,

O.12

,

O.14

,~m

c

2

x ~, K appl

)

F i g u r e 7.12: Mean-stress and normal stress ahead of crack tip for various a - 2~o/97r with B - 0

The residual plane strain mean stress in the wake is obtained from (7.41)

R_ crm

lo.ini _ 2EOT 3 - - 9 ( 1 - - u)

(7.48)

whereas the hydrostatic pressure in the front zone applied as body forces are

7.3. Toughening due to Steady-State Crack Growth

F am -

-

2(1 -- t~)EOT 9(1 - 2tJ)

177

(7.49)

The boundary between the loading zone in front of the crack tip and the unloading zone at the crack surfaces is again determined from eqn (7.45) (Fig. 7.2). We follow closely the exposition by Thomsen et al. (1992) who used the finite element program ADINA with appropriate modifications for transformation-induced body forces. The mesh of four-noded isoparametric elements was generated for a four-point bend specimen which is usually used in the experimental determination of the fracture toughness of PSZ. A coarse layout of the mesh for one-half of the specimen is shown in Fig. 7.13. The mesh in Fig. 7.13 is coarse throughout most of the specimen. Only at the crack tip is a very fine grid (mesh) generated to achieve the increased resolution necessary for accurate monitoring of the transformation zone. The fine mesh at the crack tip is shown in Fig. 7.14. The size of the smallest elements is 0.3125 pm which is anticipated to give the desired resolution. The height of the region using the fine mesh is 10 pm to cater for the expected height of the fully developed transformation zone. The solution of the steadily growing crack in a transformation zone is obtained iteratively as follows:

- -

_ _ . . . _

f F i g u r e 7.13: Finite element mesh of one-half of a four-point bend specimen, showing the boundary conditions along the plane of symmetry

178

Steady-State Toughening

I:11111111111111111111111|11111111

::HH!!NH]HH||HN]]]]]]!!!

In" .............. "" ........... 11

,-!imimNii!ii m iiiNii = I=111=I..I.1111..I inmummimimnmmmmim imnmmminmmnmmmml IE111=11111 .....

,=||||||||||l|||||||||||||||li|||i ::::::::::::::::::::::::::::::::::

10 gm

::111|I|I:|||||I||11||11|:

........ :::::::::::::::::::::::::::::::::: |||||||||||:::|:||||||||||||||||'|I ........... |l,|||llllllllll|llll||l

iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii L

C r a c ~

r"

30 gm

"~

F i g u r e 7.14: Finite element mesh at crack tip. There are a total of 1560 elements 1. The external load is incremented and the matrix eqn (7.46) solved with 0 = 0, until K tip = Kc. 2. The crack is incremented by Aa and the boundary between loading and unloading parts of the transformation zone is determined using eqn (7.45). 3. The body forces due to residual stresses in the unloading part (the wake) are calculated from eqn (7.48) and those due to hydrostatic stresses in the front zone from eqn (7.49). 4. The matrix equation Mu = P-

NOT

(7.50)

where the matrix N is calculated using eqns (7.48) and (7.49), is solved and a fresh K tip calculated from eqns (7.32) and (7.43). 5. The external load is incremented and steps (3) and (4) repeated until K tip = Kc. 6. The crack is further incremented and steps (2)-(5) repeated until no further increase in K appz or H is observed, i.e. until the transformation zone is fully developed.

7.3.

Toughening due to Steady-State Crack Growth

E (GPa) v O~ c c (MPa) /7 m

(MP v

179

205.00 0.23 0.04 0.30 700.00

4.00

T a b l e 7.1: Material parameters used in numerical calculations; these correspond to Mg-PSZ Numerical calculations were carried out for the set of parameters corresponding to Mg-PSZ given in Table 7.1. Calculations showed that the height of the transformation zone stabilized after eight to nine iterations. The first iteration corresponding to purely elastic calculations resulted in the transformation zone in good agreement with the theoretically predicted zone shape (7.12). During the iterative process the stress intensity factor at the crack tip K tip was maintained as close as possible to the intrinsic matrix value Kc = 4MPax/~. The nominal stress intensity factor calculated from the applied load and specimen geometry was found to increase from the untransformed (elastic) value of 4.0 MPav/'~ to about 5.4 M P a v f ~ when the steady-state conditions were reached. In other words, a toughening of about 35% was achieved as a result of t ---. m transformation. The stresses within the transformation zone are compared with the stress field in a purely elastic material in front of a crack tip in Fig. 7.15. The results shown in Fig. 7.15 correspond to iteration 4. It is clear that t ---+ m transformation decreases the stresses ahead of the crack tip. The reduction in the normal stress ~22 in front of the crack leads to corresponding reduction in K tip, i.e. crack shielding due to t ~ m transformation. At the boundary of the zone the stresses abruptly increase to their purely elastic values, as is to be expected. The abrupt transition in stresses is due to the super-critical nature of transformation. The stress distribution throughout the transformation zone at the eighth iteration is shown in Fig. 7.16. The stress distributions are similar to the one in Fig. 7.15 except at the frontal boundary of the transformation zone. The stresses at this boundary oscillate because of the untransformed particles (elements) within the loading zone. This does not happen in the sub-critical material model, considered above.

180

Steady-State Toughening

5

l

a)

1 -

,

I

,

J

,

I

t

,

(o:

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 x K appl~"

z22/z. 10

E lution .....

b)

0

t

t

t..,J"

t

t

,

'

(0 m .)2 C

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 x ,,K appl"

F i g u r e 7.15: Mean and normal stresses in front of crack tip, with and without transformation zone The numerically calculated toughening ratio at steady-state Kapp~Kc = 1.35 compares favourably with the analytical result reported by Budiansky et al. (1983), viz KaPPZ/Kc - 1.30, for the material parameters listed in Table 7.1. It should, however, be noted that the actual toughening ratio may be much larger, because the intrinsic fracture toughness Kc is likely to be closer to 2 MPav/-m, rather than the value of 4 M P a x / ~ used in these calculations.

Toughening due to Steady-State Crack Growth

7.3.

181

o,do; 54

3

Linear elastic solution

2 1

a)

0 0.00

'

C

)2

0.02 0.04 0.06

0.08

0.10 0.12

0.14

0.16 x ( (~m

~ ~ ~ 0.02 0.04 0.06

I , ~ ~ 0.08 0.10 0.12

I 0.14

~ ( o ~~pp )2K 0.16 x l

Kapp I

o22/o~ 10

b)

-2 0.00

F i g u r e 7.16: Mean and normal stresses in front of crack tip at the eighth iteration In connection with the calculation of K tip it should be mentioned that the expression used for its numerical determination [eqns (7.32) and (7.43)] assumes a fully developed wake. In actual fact, this assumption is not strictly valid during intermediate iterations. It is difficult to estimate the error, although Budiansky et al. (1983) state that the error involved is less than 3% since K tip is essentially determined by half-height of transformation zone H which does not change significantly in successive

Steady-State Toughening

182

iterations. However, some additional error is inevitable in view of the diffusivity of the zone boundary making an accurate determination of H almost impossible. The sharpness of zone boundary can be improved by using a finer mesh. There are several factors which further influence the accuracy of numerical modelling. For example, it is not quite known when and by how much the crack should be incremented at each iteration. Likewise, it is not clear whether the transformation stresses (body forces) should be applied before, during or after the crack has been incremented. Here it is worth recalling that McMeeking & Evans (1982) claim that the steadystate conditions are attained when the crack has advanced by 2 - 3H. The numerical calculations showed that these conditions were attained when the crack had barely advanced by 0.6H. This may indicate that the crack increment Aa used in the numerical calculations was too small. On the other hand, the assumed crack increment should not be excessive (> H/3), for otherwise it would be difficult to monitor the growth of the transformation zone. Finally, in evaluating the above numerical results the two basic assumptions made should be borne in mind, and these are first that the composite is a continuum and secondly that the elastic constants of the composite remain unchanged by transformation. 7.3.3

Influence of Shear State Toughening

on

Super-Critical

Steady-

A comparison between theory and experiment for two PSZ materials reveals that the approximate solution (7.21) (Fig. 7.17) consistently underestimates the steady-steady toughness even for small w i.e. small H. In Fig. 7.17, AKd refers to the second term on the right hand side of (7.21). It has been suggested that the disparity arises because shear effects have not been incorporated. To get an idea of these effects, let us consider again the simple situation when the transformation zone is small. In order to include the shear transformation strains in (4.16) it is expedient to introduce an energy criterion for transformation (Rose, 1987a), such that the energy density ( ~ r ~ c ~ ) attains a critical value at the transformation zone boundary. Under steady-state conditions, Rose (1987a) has shown that the energy density can be replaced by ~r~zcTz, where cruz is the plane-strain elastic field and Cc~ ~T is given by (4.16). At the boundary of transforma-

7.3. Toughening due to Steady-State Crack Growth

183

tion zone T ~r,ze~ -

We

(7.51)

where We is the critical strain energy density corresponding to (r~ in the critical mean stress criterion. Substituting the elastic plane-strain stress field and (4.16) into (7.51) gives the following equation for the transformation zone boundary (Rose, 1987a)

R( r ) - ( K aPPZWr where F

S )

r

F r r

(7.52/

S

- cos~+~sinCsin(2a-~r

3

(7.53)

The frontal intercept R(0) - L is again given by (7.28)in which Wc/OT is identified with cry. (Note for constant c(x;a)e T in (7.1), Wc/D 3Wc/[2(1 + v)c0T]). With this identification R(0) - L remains the same whether or not one considers transformation shear strains. In order to calculate the height of the transformed zone, we use the relation I

[~ 10 ~

8

I

I

PSZ

A --'

r

TEM

Interferometer

MgO

9

-o-

n

0

CaO

o

-o-

[]

9

02r

0.0

0.5

I

I

1.0

1.5

c4-ff[ ~ ]

F i g u r e 7.17: Steady-state applied stress intensity factor - experimental values compared with eqn (7.21)

184

Steady-State Toughening

S/D = 1, a =rd2

iI /

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

7 - ~ iI

iI

i

,

II

i

iI

i

/

i

iI

SID = O 1

0

~xx

-

~x x

xx

xx

x

,

\

xx x

I IS

F i g u r e 7.18: Steady-state transformation zone shapes with and without shear strains d[R(r sin r162 - 0, with R(r given by (7.52). The shape of the transformation zone boundary given by (7.52) is shown in Fig. 7.18 for two special cases. The angle r at which the tangent to R(r is parallel to the crack increases from ~r/3 when S / D = 0 (i.e. for pure dilatation) to 69.910 when S / D = 1, and tends to 72 o as D / S ~ O. As seen above, if Wc/O T is identified with the previous critical mean stress (r~ the shear component S does not affect L. However, for fixed L, the zone halfheight H depends on S. With a = ~/2 (corresponding to shear bands at 45 ~ to the crack plane as in Mg-PSZ), this dependence has been found by Rose (1987a) to satisfy the following relation (error < 1%) H _ 3x/'-3 (1 + 1 07 L 8

9

a - 7r/2

(7 54)

The toughness increment is again calculated using (7.10). As before, the integral can be reduced to that along the front of the transformation zone, and so giving (cf. (7.22)) Ktip Kappl

= 1 - 1--~-~rw 1 + 1 . 0 7

(7.55)

7.3. Toughening due to Steady-State Crack Growth

185

For purely dilatational transformation, (7.55) reduces to the (7.22). It appears that the dilatational result (7.22) is in good agreement with the experimental value for Y-PSZ. On the other hand, for Mg-PSZ, the value given by (7.55) with S / D = 1 would seem to be in better agreement with experiment. Shear bands have been observed in the transformation zone around cracks in Mg-PSZ. However, the necessary zone shapes (see, e.g. Fig. 7.18 with S / D = 1) do not seem to be consistent with experiment. This is obviously connected with the previously mentioned shear accommodation processes, such as twinning.

This Page Intentionally Left Blank

187

Chapter 8

R-Curve Analysis In Chapter 7 we studied the toughness increment induced by phase transformation under steady-state conditions. In this Chapter, we will present a complete R-curve analysis of the quasi-static growth of a crack, taking into account the development of transformation zone. We begin with the analysis of a semi-infinite crack in an infinite medium, followed by that of a finite internal crack and an array of collinear internal cracks. We conclude the Chapter with the analysis of a single and of a periodic array of edge cracks. We shall assume throughout that the phase transformation only induces dilatation and that super-critical transformation takes place according to the critical mean stress criterion (Chapter 3).

8.1

Semi-Infinite

Cracks y

T H

ro F i g u r e 8.1: Semi-infinite crack

R-Curve Analysis

188

The model considered in the present Section consists of a semi-infinite crack C in an infinite plane body subjected to a load which induces an applied stress intensity factor K appl, as depicted in Fig. 8.1. The exposition follows that of Stump & Budiansky (1989a). Muskhelishvili's theory of plane elasticity is applied (Muskhelishvili, 1954), and the notations are in accordance those introduced in Chapters 4 and 5. In order to solve the problem depicted in Fig. 8.1 the traction induced on the crack-line by the transformation zone has to be cancelled. Due T (X), which is to symmetry the traction is given by crack-line stress cry~ obtained from (4.21) and (4.52)

T

%y(x)

-

EeTJs

4~(1-u)

x-z0

- ~ - ~1 ) x-~0

dyo

(8.1)

This equation holds irrespective of whether the crack tip is inside or outside the transformation zone. Thus the constant terms in the stress potentials of (4.51) do not enter the expression for the crack-line stress. There is however a jump in this stress across the zone boundary, which is embedded in the integral (8.1). The potentials for a crack with crack-line stresses balancing out the stresses from transformation are determined by the standard method devised by Muskhelishvili (1954). The image potential Oi(z) arising from the cancellation of the crack-line stress is determined from (5.56) by inserting the stress (8.1). The image potential O~(z) becomes

r

8 (1-

+

)VZ

+ (vz+

1

) dy0

(8.2)

The potential (~avvZ(z)from the applied loads is obtained from standard results (Muskhelishvili, 1954) Kapp l

2 /Y z

(8.3)

The potentials Oi(z) and ~aPPt(z) associated with the O-potentials of (8.2) and (8.3) which are necessary for the complete determination of the elastic fields are obtainable from the respective q~-potentials, but for the present analysis these potentials are not needed. The complete O-potentials are obtained by superposition of the image potential due to transformation (8.2) and the potential due to the applied load (8.3). In the following these results are used to obtain

8.1. Semi-Intinite Cracks

189

equations determining the transformation zone shape and crack growth behaviour. The mean stress cr~pt from the applied load is obtained from (8.3) and (7.23)

a~n-

g-----~Re

(8.4)

where the characteristic length L (7.28) has been used. The mean stress due to the image potential (hi(z) is obtained from (8.2) and (7.23) o'm_

1

w

1

( v/'~ + V/-~ ) v/-~ + ( V/-~ + ~ o ) v/-~ d y o (8.5)

~r~ - - lS-'-~Re

where the transformation parameter w (3.26) has been used. From (4.52) and (4.21)it is seen, that the mean stress due to the transformation zone itself vanishes outside the zone and takes on a constant value inside it. Thus the mean stress from transformation itself does not enter the equations needed here. The applied stress intensity factor K appt in (8.4) has to be tied to the intrinsic toughness Kc through a crack growth criterion in order to obtain a complete system of equations for the present problem. The stress intensity at the crack tip K tip can be viewed as consisting of the applied stress intensity factor K ~ppt determined by the far field load and an additional contribution from transformation A K tip

K tip = KappI+ A K tip

(8.6)

A K tip is the stress intensity factor from the image stress and is given by

AKtip_

Re{2O/(z) 2x/~-xrz}~__.0+

(8.7)

Introducing (8.2)into (8.7)gives

Kr

= 18r

+

dyo/L

(8.8)

To determine the transformation zone boundary prior to crack growth in accordance with the critical mean stress criterion for transformation,

190

R-Curve Analysis

the mean stress must approach the critical mean stress as the zone is approached from the outside. At the onset of crack growth the stress intensity factor at the crack tip K tip equals the intrinsic toughness Kc of the material. Adding the image mean stress from (8.5) to the applied mean stress from (8.4) and equating the sum to the critical mean stress results in the following system of equations for determining the transformation zone boundary at the onset of crack growth

K~

1

1 + (x/T + e~-5) e 7

- 1S----~Re

) zES

(s.9)

Kc = K tip = K appt + A K tip

The unknowns in the two equations (8.9) are the zone shape S and the applied stress intensity factor K appz. At the onset of crack growth the stress intensity factor at the crack tip K tip must equal the intrinsic toughness of the material Kc. This acts as a side condition to the nonlinear integral equation (8.9) determining the zone shape S. For a growing crack it is assumed that a wake of transformed material will develop behind the crack tip due to nonreversible transformation. To model the growing crack, the crack length is incremented. The crack tip therefore moves into the transformed zone. This will result in an increase in the mean stress ahead of the crack tip, while the mean stress behind it will decline. If the zone shape behind the crack tip is fixed and the stress intensity factor at the crack tip K tip is maintained at Kc, a new zone front and the applied stress intensity factor K appz necessary for quasi-static crack growth can be determined. For a small increment of the crack Aa the equations determining the transformation zone boundary can be written as ~ R e Kr

lSr

Re

(

1

(V/_~. + v/~)x/,_~

K~ = K tip = K "ppz + A K tip

+

(e7 +

1

)e7 )

z f: Sfron t

8.1. Semi-Infinite Cracks

191 (8.10)

Swake(a -4- Aa) = Swake(a)

where the wake Swake(a) is obtained by a reverse translation of the transformation zone by the distance Aa. The unknowns in eqn (8.10) are the zone front Sf,-ont and the applied stress intensity factor K appz. These are determined incrementally using the solution of (8.9) as the initial zone shape. As the crack is incremented, the integral equation (8.10) determines the new transformation zone front. The corresponding applied stress intensity factor t~ appz equalling the apparent toughness of the material is obtained from the side condition of (8.10) with the aid of (8.8). For sufficiently small crack increments Aa the transformation zone wake S,~ak~ joins smoothly with the front SI,.ont. These increments are however too small for an efficient computational scheme. Larger increments can be handled by imposing a smoothness condition on the zone shape. By joining the wake and the front by common tangents (Fig. 8.2) much larger increments of crack lengths can be handled without causing numerical difficulties.

F i g u r e 8.2: Transformation zone of a growing crack joined by common tangent to the wake

8.1.1

Stationary

and

Growing

Semi-Infinite

Crack

The transformation zone shapes for various values of w ranging from 0 to 30 are shown in Fig. 8.3. The transformation zone size is seen to increase with the transformation parameter w. (The transformation zone for w = 0 is the cardioid, described in w The transformation zone dissociates from the crack tip for non-zero values of w and approaches

R-Curve Analysis

192

1.2 I--

~

0.8

o~=30 25 20 15 10

0.6

5

1.0

0.4 0.2 0.0 -0.2

0.0 0.2 0.4 0.6 0.8

J

1.0 1.2 1.4 1.6 1.8x/L

F i g u r e 8.3: Initial transformation zone shapes for a semi-infinite crack

the crack face at right angles, in contrast to the situation for w = 0 where it terminates smoothly at the crack tip. The toughness increment for a stationary crack is negligible, less than 0.5% for 0 < w < 30. There is no increment when w = 0, (w Once the initial zone shape for a stationary crack has been found, it is possible to determine the growth around an advancing crack tip from (8.10). As a growing crack tip moves into the zone, material in its vicinity attains the critical mean stress and transforms, while due to the irreversibility of the transformation a wake of transformed material is left behind. Along a frontal portion of the transformed zone boundary the mean stress criterion is satisfied, while on the wake portion of the c . As we have seen boundary the mean stress will have dropped below ~rm (w the transformed region behind the radial lines running through the tip at +7r/3 reduces the stress intensity at the tip. To continue driving the crack forward, the applied stress intensity must be adjusted. Consequently to solve the growing crack problem, both the stress intensity and the zone shape must be determined as functions of the crack extension. Zone shapes for growing cracks for w = 5 and 10 are shown in Fig. 8.4. The development of toughening ratio Kappt/Kc or the R-curves corresponding to these growing cracks are shown in Fig. 8.5. The zone half-height H/L and the toughening ratio Kappt/Kc both overshoot the steady-state levels (broken lines; w for finite amounts of crack growth before approaching them asymptotically from above. The half-height of transformation zone at initiation of crack growth is

8.1. Semi-Infinite Cracks

193

y/L 1.O 0.8 0.6

i :

i i

,, i

'

i

i

1

0.4 0.2 a)

0.0

0

8

4 ~

~

4

~

I

I

8

12 I

I

12

Aa= 20

16 I

I

I

16

20

x/L

y/L 2.0 1.5 ,

1.O

,

,

0.5 b)

0.0

J 10

10 J

J 20

20 J

J 30

30 I

, 40

40 ,

Aa = 50 I 50

x/L

F i g u r e 8.4: Transformation zone shapes for a growing semi-infinite crack, ( a ) w - 5, (b) w - 10

shown in Fig. 8.6, together with the half-height at peak toughening and under steady-state conditions. The value at peak toughening diverges as the transformation parameter w attains the value of approximately 20.2. This is accompanied by unlimited increase in toughness, i.e. by lock-up. Under steady-state conditions the lock-up value of w is approximately 30.0, as described in Chapter 7. Reciprocal peak toughening is shown in Fig. 8.7 and compared with the reciprocal peak toughening under steady-state conditions, described

194

R-Curve

K

appl

Analysis

IK c

1.30 1.25 1.20

1.15 1.10 1.05

1.O0 a)

0.95

K

i 5

0

t 10

t 15

J 20

AalL

I 50

Aa/L

appl _

IK c

1.8 1.6 1.4 1.2 1.0 b)

0.8

0

J 10

I 20

I 30

I 40

P i g u r e 8.5: Development of a p p a r e n t toughness for a growing semiinfinite crack, (a) w - 5, (b) w - 10 in C h a p t e r 7. Surprisingly, the linear e s t i m a t e of s t e a d y - s t a t e toughening (see Fig. 7.6) agrees very well with the peak toughness values up to w~18.

195

8.1. Semi-Intinite Cracks H/L 5

_ ~~../Steady-state Initial 0

0

!

I

I

I

I

I

5

10

15

20

25

30

F i g u r e 8.6: Half-height of transformation zone at initiation of crack growth, at peak-toughening, and under steady-state conditions

gc/K appl 10

L!near 0.8 0.6 0.4 0.2 0.0

0

5

10

15

20

25

30 o)

F i g u r e 8.7: Reciprocal toughening at peak transformation and under steady-state conditions

R-Curve Analysis

196

8.2

Single Internal Cracks

In this Section we shall analyse the R-curve behaviour of T T C containing short internal cracks whose size can be commensurate with flaws that inevitably form in such materials. We shall consider first a single short internal crack and then an array of internal cracks.

F i g u r e 8.8" Internal crack with transformation zones The micromechanical model consists of a central crack C of length 2a in an infinite plane body subjected to a remote transverse stress ~r~ as shown in Fig. 8.8. The transformation zones of dilated material are bounded by S + at the right hand tip of the crack and by S - at the left tip. (For later use, a T-stress parallel to the crack is also included in the model). To solve the problem depicted in Fig. 8.8 the normal stress induced on the crack-line by the two transformation zones is to be cancelled. This stress is given by (4.21) and (4.52) T (X)

%v

--

47r(1-v)

+

x-zo

+ -

1 47r(1 - u)

+

x -

X

- -

dyo

x-:o

Z0

+

gg

- -

ZO

1

-} zo

+

x -

1 :o

x +

zo

dyo

1 )dyo

x+:o

(8.

11)

8.2. Single Internal Cracks

197

Here, E is Young's modulus, u Poisson's ratio, e T the plane dilatational strain, and z the complex coordinate z = x + iy. The boundary S + (and S - ) is determined by the critical mean stress criterion. In arriving at (8.11) the double symmetry of the problem has been exploited. The image potential (I)i(z) arising from the cancellation of the crackline stress (8.11) is determined from eqn (5.56)

ePi(z)- 2 (EeT 1 - u) fS + (A(z, z0) +A(z,T0) - A ( z , - z 0 ) - A(z,-To)) dyo

(8.12)

where A(z, A) -

x/A2 - a2 - x/z2 - a2 4~'(z - A)x/z 2 - a 2

(8.13)

The potential (~avVl(z) corresponding to the applied loads is r

/

) _

( z \ 2~/z 2 - a 2

l2

T )\ /

(814)

The mean stress (r,~pt from the applied loads is obtained from (8.14) and (7.23)

cr,~v' o'~ -

KaVV' i2La K---~

Re

{

z

l-T}

x / z 2 _ a 2 -- ~

(815)

The applied stress intensity factor is given by K appt = cr~ V/-~. The mean stress due to the image potential epi(z) is obtained from (8.12) and (7.23) O'm - C

O'rn

Re +

{A(z, zo) + A(z,-Zo) - A ( z , - z 0 ) - A(z,-T0)} dyo

(8.16)

The toughening increment A K tip for the internal crack is given by (8.17) whence, vide (8.12)

R-Curve Analysis

198

A K tip

-w

K~

+ a .... z0 - a

zo

367r

+

-i

+

a -+ zo

-5o + a - a

l a - - z+~ ~o )

(8.18)

At the onset of crack growth the stress intensity factor at the crack tip K tip must equal the intrinsic toughness K~ of the material. Adding the image mean stress from (8.16) to the applied mean stress from (8.15) and equating the sum to the critical mean stress results in the following system of equations for determining the transformation zone boundary at the onset of crack growth K~Pi~ K~ +--~--Re

{ Re

z ~/z 2 - a 2

+ (A(z, z0) +

h(z, ~0)

l-T} 2

- A ( z , - z 0 ) - h ( z , - ~ 0 ) ) d~0

K~ = K tip -- Kappl -t- A K tip

zES+ (8.19)

As in w for a small increment of the crack Aa the equations determining the transformation zone boundary can be written as

_

_

Kc

+ --~-Re

a + Aa

X//z2 - (a + Aa) 2

+ (h(z, z0) + h(z, ~0) - A(z,

2

-zo) - h(z, -~o)) dUO[z

ES;ron t

Kc = K tip

= Kappl w

~wake(a zt" Aa) --

A K tip

~wake(a)

(8.20)

The procedure for the solution of (8.20) is the same as for the system (8.10). We shall therefore omit the details and present only the results.

8.2. Single Internal Cracks

8.2.1

199

Stationary and Growing Internal Crack

Some results obtained by solving eqns (8.19) and (8.20) for imminent crack growth and for growing cracks, respectively are given in the following. The shape of the transformed zone at the onset of crack growth depends on the crack length and the transformation parameter w. The shapes of transformation zone at the onset of crack growth obtained from (8.19) are shown in Fig. 8.9 for ao/L = 5 and 10 and several values of w. The intermediate values of transformation parameter w and of initial crack length are of principal interest. This is because when the cracks are very short the transformation zone will diverge before crack growth appears, as the mean stress induced by applied load will exceed the

y/t, 0.7 0.6 0.5

~ / ~

0.4 0.3 0.2

a)

0)=30 /25 20 15 10 5 0

0.1 0.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

J

1.2 x/L

y/L 0.7 0.6 0.5

/

0)=30 25 2O 15 10 5

0.4 0.3

b)

0.2 0.1 0.0 -0.2

i 0.0

0.2

0.4

I

0.6

0.8

1.0

1.2 x/L

F i g u r e 8.9: Initial transformation zone shapes under uniaxial load (T = 0), ( a ) a o / L - 5, ( b ) a o / L - 10

200

R - C u r v e Analysis

rdL 5.0 4.0 3.0

aolL=500 50 10

2.0 1.0

0.0

0

5

10

15

:

_-

i

-

20

25

30

35

.

0.5

40 o3

F i g u r e 8.10" Frontal zone intercept r0 for uniaxial load. The transformation zone size diverges for ao/L = 0.5, as w ---* 0

critical mean stress before the stress intensity factor at their tips attains the critical value Kc. The critical crack length at which this occurs for uniaxially loaded cracks is easily obtained from eqn (8.19) by letting w = 0 and K appz = Kc; the critical crack length is ao/L = 0.5. For very large values of transformation parameter w self-cracking will occur (Stump & Budiansky, 1989b), but these values of w are well beyond those of practical interest. Two measures of transformation zone size are of interest. In analyzing steady-state toughening, the more practical size measure from an experimental point of view is the transformation zone height. For growing cracks, the height is not easily measured, but the distance between the crack tip and the transformation zone boundary intercept with the crack-line ahead of it is uniquely defined. Under steady-state conditions the zone height and the frontal intercept are proportional to each other and thus easily interchanged (see (7.17) and (7.28)). The frontal zone intercept r0 for several initial crack lengths is shown in Fig. 8.10. For very small cracks the transformation zone size diverges as the transformation parameter w tends to zero. The critical crack length for this to occur is ao/L = 0.5, as previously mentioned. Due to a toughness decrement at the onset of crack growth, finite values of the transformation parameter w give nondiverging transformation zone sizes for this critical crack length.

8.2. Single Internal Cracks

201

appl

K

1.0

/Kc

adL=~ 50

0.9

10

0.8

3

0.7

1

0.5 0.5

0.5 . . . .

0

I

I

I

1

I

I

I

I

5

10

15

20

25

30

35

40

F i g u r e 8.11: Stress intensity factor at the onset of crack growth. The broken lines indicate first order linear estimates The apparent toughness at the onset of crack growth is shown in Fig. 8.11 for various values of crack length and transformation parameter w. Prior to crack growth, transformation reduces the apparent toughness, and the more so, the smaller the crack. For very long cracks however a small increment in toughness of approximately 0.5% at w = 40 appears. One of the few linear approximations for the present theory as w ---. 0 can be readily obtained by first solving (8.19) with w = 0 and K appz = Kc for the transformation zone shape and then using this result in (8.18) to obtain the toughness decrement at the onset of crack growth. The linear approximations are shown in Fig. 8.11 by broken lines. For growing cracks, R-curves for several initial crack lengths obtained by solving eqns (8.20) are shown in Fig. 8.12 for two values of the transformation parameter w. The apparent toughness goes through a peak before reaching the steady-state value from below. The peak gets shallower as the initial crack length gets smaller. For short cracks, the peak value is less than the steady-state value, whereas for long cracks the peak overshoots it. There is experimental evidence (Swain & Hannink, 1984) in support of this. The applied stress cr~176 necessary for quasi-static crack growth corresponding to the R-curve results of Fig. 8.12 can be calculated from the following relation between toughness, stress, and crack length cr~176 cro

_ Kappt(Aa)/ ao Kc V a o + Aa

(8.21)

202

R-Curve

g

Analysis

appl

/K c

1.30 1.25 .,

1.20

1.15 1.10 1.05

1.00 a)

0.95

K

l 5

0

t 15

10

i 20

Aa/L

appl

IK c

1.6 1.4 1.2 1.0 I b)

0.8

0

I 10

l 20

I 30

,1 40

I 50 AalL

F i g u r e 8.12: R-curves under uniaxial load, (a) w = 5, (b) w : 10

The normalizing stress or0 is the stress necessary for initiating the growth of a crack of length a0 in the absence of transformation toughening, i.e. the inherent strength of the material. The stress needed for quasi-static crack growth is shown in Fig. 8.13. Due to the R-curve behaviour the applied stress has to be raised in order to maintain crack growth. After a finite amount of crack growth the stress curves go through a peak as the effect of the increased crack length becomes the dominating factor. The ultimate strength of the transforming ceramic

8.2. Single Internal Cracks

203

o~176 1.4

F

.a_~L~_-_=

1.2

5--~-"-~

1.0 0.8-

5

0.6 0.4 0.2 a)

0.0

i

0

i

5

J

10

i

15

20

Aa/L

o**/co 1.8

_

1.6 2~

a0{__L~_~

-

I4tL.

---

1.2 1.0 0.8

b)

0"6 0.4 f 0.2 0.0 0

5 l 10

t 20

~ I 30

J 40

J 50 Aa/L

F i g u r e 8.13: Applied uniaxial stress necessary for crack growth, (a) -5,(b)~-10 is given by the peak value ap of the applied stress cr~176 The peaks in the curves shown in Fig. 8.13 indicate that a certain amount of stable crack growth can be sustained before a transformation toughening ceramic will fail catastrophically. Assuming that a certain population of small cracks is present in the ceramic, the stable initial growth of the most critical crack among this population can cause other

R-Curve Analysis

204

y/L

ao/L=5 10 50

4

o,o

0 -2 -4 -5

tl

-30

I

I

I

I

I

I

-25

-20

-15

-10

-5

0

x/L

F i g u r e 8.14: Transformation zone boundaries under uniaxial load, w 10

K c/K appl 1.0 0.8 0.6 0.4 0.2 0.0

0

5

10

I

15

20

~

25

"'t

30

0)

F i g u r e 8.15: Reciprocal peak toughening under uniaxial load

smaller or less critical cracks to develop transformation zones around them and even to start growing before the critical crack itself eventually becomes unstable. If the population of inherent flaws or cracks has a sufficiently narrow size distribution a large number of such cracks may become active before catastrophic failure. This would give rise to a certain deviation from linearity in the stress-strain behaviour for these ceramics apart from the nonlinearity induced by the transformation itself.

205

8.2. Single Internal Cracks ~o/t~ p 1.0 0.8 0.6 0.4 0.2 0.0

0

5

10

15

20

F i g u r e 8.16" Reciprocal peak strengthening ratio under uniaxial load The peaks in the R and applied stress curves are reflected in the transformation zone shapes (Fig. 8.14) through a certain zone widening before steady-state conditions are reached. Reciprocal peak toughening ratio is depicted in Fig. 8.15. For comparison the steady-state toughness estimate consistent with the present theory is shown by the broken curve (Amazigo & Budiansky, 1988). The reciprocal peak strengthening ratio is shown in Fig. 8.16; the strengthening ratio decreases with decreasing internal crack size. Thus initially strong materials are less susceptible to strengthening by transformation toughening than are the initially weak materials. This effect can reduce the scatter in strengths, and thereby increase the Weibull modulus of these materials (Shetty & Wang, 1989). 8.2.2

Relation Between Strengthening

Toughening

and

The reciprocal peak strengthening ratio (Fig. 8.16) can be correlated with the steady-state toughening ratio (broken curve in Fig. 8.15). The result is shown in Fig. 8.17. Three microstructural parameters enter this correlation, namely the initial crack length a0, the characteristic length L (7.28), and the transformation strength parameter w (3.26). The last two parameters are defined with the critical mean stress ~r~ in the denominator. Besides,

206

R-Curve Analysis

the transformation parameter w is proportional the transformation density cOT . In correlating the peak strength and steady-state toughness data, as in Fig. 8.17 it is expedient to associate each curve with a specific microstructure. For variable transformation density cOT , the curves in Fig. 8.17 can be related to a microstructure with a specific initial c For this microstructure, crack length a0 and critical mean stress (rm. the strength increases monotonically with toughness, so that even without changing the critical mean stress, additional transformable particles improve the strength, as well as the toughness. oP/%

3.5 3.0 2.5 2.0 1.5-

~'

I

1.0

1.0

1.2

1.4

l

1.6

1.8

2.0

2.2

2.4

$$

K IK c

F i g u r e 8.17: Peak strengthening versus steady-state toughening for increasing transformation density cOT An alternative way of increasing the amount of transformation accompanying the crack growth is by lowering the critical mean stress a~n" c is decreased, both the characteristic length L and the transformaIf (rm tion parameter w are increased. From Fig. 8.16 it can be seen that this produces two opposite effects in peak strengthening. An increase in L reduces the reciprocal peak strengthening ratio, whereas an increase in w increases it. Both effects can be captured through the parameter

/3 -

w2ao/L - ~

K~(1-u)

(8.22)

8.2.

Single

Internal

207

Cracks

The peak strength and steady-state toughness relation is shown in Fig. 8.18 for fixed values of transformation density cO T and initial crack length a0, i.e. ~ is fixed and corresponding values of w and L are obtained through (8.22). The peak strengthening ratio reaches a maximum at finite amounts of toughening. In an ageing process, where coarsening dominates over precipitation, it can be expected that the density of transformable precipitates is constant, whereas due to the coarsening the critical mean stress for transformation decreases. In that case the result in Fig. 8.18 can be interpreted as leading to peak strengthening before peak toughening is reached. Eventually, toughening ceases as lowering the critical mean stress leads to spontaneous transformation during cooling. The results shown in Fig. 8.18 are in qualitative agreement with the results reported by Swain (1986), and Swain & Rose (1986). The present theory offers the possibility of predicting the effect of alloying, precipitation, ageing and other treatments of transformation toughening ceramics on the strength-toughness relationship, if.the influence of the c and c~T specific treatment upon the microstructural parameters a0, a m, is known. Alternatively, a theoretical estimate for the optimal critical mean stress necessary for maximum strengthening can be obtained for a given microstructure with minimized initial flaw sizes and maximized density of transformable particles.

oP/oo 1.4

-

1.3

-

1.2 1.1 1.0 9

0.9

1.0

I

1.2

l

1.4

l

1.6

C

Decreasing ~m -~ !

1.8

l

2.0

I

2.2

I

2.4

ss K /K c

Figure 8.18: Strengthening versus steady-state toughening for various values of/3

208

8.2.3

R - C u r v e Analysis

Biaxially Loaded Internal Crack

Some results obtained by solving eqns (8.19) and (8.20) for imminent crack growth and for growing cracks under equal biaxial tension (T = 1) are given in the following. The shape of the transformed zone at the onset of crack growth depends on the applied load, the crack length and the transformation parameter w. Some examples of transformation zone at the onset of crack growth obtained from eqn (8.19) are shown in Fig. 8.19 for two initial crack lengths ao/L. The corresponding frontal zone intercept r0 for sev-

y/t, 1.0

{o=30

25 20 15 10 5 0

0.8 0.6 0.4 0.2

a)

0.0

0.0

0.4

i

0.8

1.2

1.6

x/L

y/L 1.0

=30 25 20 15 10

0.8 0.6 0.4 0.2

b)

0.0

0.0

0.4

0.8

1.2

1.6

x/L

F i g u r e 8.19: Initial transformation zone shapes under equal biaxial tension (T = 1), ( a ) a o / L = 5, (b) ao/L = 10

8.2.

Single Internal Cracks

209

rdL 5.0 4.0

~

3.0 2.0

ao/L=500

~ ~ ~

--~~"

1.O 0.0 0

i

i

i

i

i

i

i

i

5

10

15

20

25

30

35

40

5 3 2

co

F i g u r e 8 . 2 0 : Frontal zone intercept r0 under equal biaxial tension. T h e t r a n s f o r m a t i o n zone size diverges for ao/L - 2, as w ---+ 0

r

appl

IKc ao/L=oo

1.0

50 0.9 0.8

10

0.7

5

0.6

3 2

0.5

0

i

i

i

I

J

I

!

J

5

10

15

20

25

30

35

40

to

F i g u r e 8.21" T o u g h e n i n g ratio at the onset of crack growth u n d e r equal biaxial tension. T h e broken lines indicate first order linear e s t i m a t e s

210

R-Curve Analysis appl /K c

K

1.3 1.2 1.1 1.0 0.9 a)

0.8

K

t

0

i

5

1

10

J

15

20

Aa/L

appl /K c

2.0 1.8 1.6 1.4 1.2 1.O

b)

0.8

Figure

8.22:

0

I.

l

10

20

.d

30

..

l

I

40

50

R-curves under equal biaxial tension, ( a ) w

Aa/L

-

5, (b)

w-lO eral initial crack lengths is shown in Fig. 8.20. For very small cracks the transformation zone diverges as the transformation parameter r tends to zero. The critical crack length at which this occurs is a o / L = 2. However, due to a toughness decrement at the onset of crack growth, finite values of the transformation parameter w give nondiverging transformation zones for this critical crack length. The apparent toughness at the onset of crack growth is shown in

8.2. Single Internal Cracks

211

0~1760

15t

1.4

aolL=**

1.2 1.0 0~

-

0.6

0.4 0.2 0.0

a)

0

"

I

t

I

10

5

I

15

20

Aa/L

0~1760

2.0 1.8 1.6 1.4 1.2

adL=** 5O

1.0 0.8 0.6 0.4 0.2 b)

0.0

0

1

i

i

i

10

20

30

40

i

50 Aa/L

F i g u r e 8.23: Applied equal biaxial tension necessary for crack growth,

(~) ~

-

5, (b)

~

-

~0

Fig. 8.21 for various values of initial crack length and transformation parameter w. For growing cracks the R-curves for several initial crack lengths obtained by solving eqn (8.20) are shown in Fig. 8.22 for two values of the transformation parameter w. The apparent toughness goes through a

212

R - C u r v e Analysis

y/L 5 4

5

2 0 -2 -4 -5

) -30

-25

- 0

-15

-10

-

0

x/L

F i g u r e 8.24: Transformation zone boundaries for equal biaxial tension, w-10

i~

,,, a p p l c/l{

1.0

Steady-state

0.8 0.6 0.4

I

0.2 \ \

0.0

0

5

10

15

20

I

i,

25

30

F i g u r e 8.25" Reciprocal peak toughening ratio under equal biaxial tension peak before reaching the steady-state value from above. The peak values are the larger the shorter the initial crack length, which is just the opposite of that observed under uniaxial tension (Fig. 8.12). Moreover, the peak value is always above the steady-state value (Andreasen, 1990; Andreasen & Karihaloo 1993a). Another noticeable dissimilarity in R-curves between the equal bi-

8.2. Single Internal Cracks

213

axial load and the uniaxial load is in the crack advance needed before the steady-state conditions are reached. Under equal biaxial tension, the crack advance necessary for attaining the peak toughness increases with diminishing initial crack length, while under uniaxial tension it reduces. The applied equal biaxial tension necessary for quasi-static crack growth corresponding to the R-curves of Fig. 8.22 can be calculated from (8.21). The results are shown in Fig. 8.23. As under uniaxial tension, so also under equal biaxial tension, the peaks in the R and stress curves are reflected in the transformation zone shapes through a zone widening before the steady-state conditions are reached. Under equal biaxial tension, as opposed to uniaxial tension, the zone widening is more pronounced for shorter initial cracks (cf. Figs. 8.14 and 8.24). The reciprocal peak toughening ratio under equal biaxial tension is shown in Fig. 8.25. For comparison the steady-state toughness estimate consistent with the present theory is shown by the broken curve (Amazigo & Budiansky, 1988). The appearance of peaks lead to diverging toughening or "lock-up" for values of transformation strength w lower than that expected from steady-state analysis, as reported by Rose (1987a). For finite crack lengths, the lock-up values of w are above the lock-up value w = 20.2 for semi-infinite cracks (Stump & Budiansky, 1989a) and less than the lock-up value of w = 30.0 for steady-state

1.O 0.8 0.6

-

10

50

0.4 0.2 0.0

0

5

10

15

20

F i g u r e 8.26: Reciprocal peak strengthening ratio under equal biaxial tension

214

R-Curve Analysis

conditions (Amazigo & Budiansky, 1988). The reciprocal peak strengthening ratio is shown in Fig. 8.26. It diminishes with diminishing initial crack length, in much the same manner as under uniaxial tension (Fig. 8.16).

8.3

Array of Internal Cracks

We will extend the discussion of Section 8.2 to a collinear array of internal cracks. The method used is based on dislocation formalism and complex potentials and it is similar to the method used in the previous Section. A collinear array of equally spaced cracks is illustrated in Fig. 8.27. The spacing is denoted d, and c is half the length of each crack designated C. For simplicity, one of the cracks in the array is assumed to be situated with its centre at the origin. This crack is referred to as the central crack. The plane is loaded at infinity by an external stress a ~ , normal to the cracks resulting in pure opening mode I. At the tip of each crack a transformation zone with the boundary, S develops when the plane is loaded and the criterion for transformation is satisfied, i.e. c It is also assumed that the the mean stress reaches a critical value, am. transformation is accompanied by a purely dilatational strain inside the transformation zone. 8.3.1

Mathematical

Formulation

Two governing equations are derived from the following conditions 1. Traction-free crack faces, ayy(z) = 0, z E C. C 2. Critical mean stress on S, am - am, z E S.

In a detailed form these conditions are

T ( z ) + ayy ~(z) - O,zzC ayy + ayy cr ~m + ~ T( z )

+ ~ D( z )

-

~ c ,z ~ s

(8.23)

where superscripts oc, T and D denote stress contributions from the remote applied stress, transformation and dislocations, respectively. The crack-line stress from the applied load a r can be written as

[~[appl ay~ = B 0 x f ~

(8.24)

8.3. Array of InternM Cracks

215

F i g u r e 8.27: Collinear array of internal plane cracks in a transforming ceramic where B0 a geometry factor given by B0 -

I

d 7rc ~cc tan(--~-)

(8.25)

The crack-line stress due to the transformation can be written as r

T (z) -

EcT [ r 27r(1- u) Js [GuT(x, Zo)- Guu(x,-~0)] dyo

x6.C

(8.26)

where GuT(z, zo) are given by (4.60), and symmetry has been exploited to reduce the integration along the zone boundaries to that along the right hand zone for the central crack. The crack-line stress due to the dislocations is given by (YyDy (z)

--

/0cD" (t) [HuD'u(x, t) -- HyyD'u(x , --t)] dt I

(8.27)

xEC

D,y where Hyy (z, zl) are given by (6.25). Again, symmetry has been exploited to reduce the integration to only the right hand half of the crack C. The mean stress along S from the dislocations can be written as

~(z) -

fc

D* (t) [H~D~u (z, t) - H~Dg~y (z, --t)] dt z6_S

(8.28)

216

R-Curve Analysis

where HaDg~y (z, zo) are given by (6.25). Introducing the dislocation density function D*(t)

D*(t) -

E/a~n

127r(1 - u)

D(t)

(8.29)

the condition of imminent crack growth (K tip = Kc) can be written as / lim 27r~/C- XD,(x) 9~ c v L

1

(8.30)

The system of equations determining the transformation zone shape at the onset of crack growth can now be written as O~

K appl

w J~s D* (t) [Hy u

Kappl + 1 - BoK~

iC

, z0) -

- Huy

dy0

at xEC

D* (t) [ HD'Y(Z D'Y(Z, --t)] dt ~ , t) - H,~

z6.S

(8.31)

The effect of the interaction of cracks on the strength and toughness of a transformation toughened ceramic can be studied in two ways. First, the initial crack length, co is kept constant and the distance between the cracks varied. This procedure is useful for understanding the effect of crack separation. It is also the physically most comprehensible way, especially for comparison with the results of a single internal crack. Secondly, the initial length of the unbroken material between the cracks, = ( d - 2 c 0 ) is kept constant by varying the distance between the cracks and the initial crack length. This procedure is relevant to a ceramic with a non-transformable matrix toughened by transformable particles, as shown in Fig. 8.28. Following this procedure the interaction is studied for different particle sizes and area fractions, Aj = ( d - 2co)2/d 2 of transformable particles. If the cracks in the array are very close to one another either initially or after growth, neighbouring transformations zones would merge together. This situation however will not be treated in the present analysis.

8.3. Array of Internal Cracks

217

F i g u r e 8.28: Cross-section of a particulate transformation toughened ceramic The shape and size of the transformation zone forming at the tip of each crack depend on the load, the crack length, the distance between the cracks and the transformation strength parameter, w. Due to the symmetry of the transformation zones only the upper right half of a zone will be shown in the following. The lower limit on the length of a single internal crack liable to grow was shown to be co/L = 0.5 (w For shorter cracks the applied far field mean stress necessary to initiate crack growth would exceed the critical mean stress for transformation leading to spontaneous transformation of the whole material. This limit will also be used here, even though the limit for an array of cracks would be lower than that for a single crack due to the interaction of the cracks in the array and the resulting reduction in the applied load necessary for ~=rack initiation. 8.3.2

Onset

of crack

growth

The transformation zones at the onset of crack growth are shown in Figs. 8.29-8.32 for ~o = 10 and four initial crack lengths, co/L = 0.5, 1, 5, 50. In each figure the distance, d / L is varied to show the effect of interaction between the cracks and their transformation zones. Generally, the transformation zones increase in height and length, as the cracks are brought closer. The length of the zone is characterized by the distance from the crack tip to the frontal intercept of the zone boundary with the z-axis. The intercept behind the crack tip, xc is virtually unchanged. For the longer cracks, co/L = 5 and co/L = 50, an increase in both transformation zone height and frontal intercept is observed. The same behaviour is observed in Fig. 8.33 for w = 30 and co/L = 5, but, as expected with larger zones than the corresponding zones for w = 10. For the very short cracks (e.g. co/L = 0.5) both the frontal intercept

R-Curve Analysis

218

y/L 0.5 0.4

.... dlL=10 i --

f

. . . . .

. . . . . . . . .

3.8 .....

0.3

.

.

.

.

.

0.2 O.1 0.0 0.4

0.6

I

I

0.8

1.O

I

1.2

1.4 x/L

F i g u r e 8.29: Transformation zones at the onset of crack growth for co/L - 0.5, w - 10 and d/L - oc, 10, 5, 4, 3.8, 3.5

y/t,

~f 0.4

=

0.3

.

5

0.2 0.1 0.0

1.0

J

J

i

1.2

1.4

1.6

1.8

x/L

F i g u r e 8.30" Transformation zones at the onset of crack growth for co/L - 1, w - 10 and d/L - oc, 10, 5, 4.5, 4.25 and the zone height decrease for d/L = 10 and d/L = 5, compared to the single crack. As the cracks are brought even closer the height keeps decreasing, while the frontal intercept increases. For coiL = 1 the long crack behaviour described above prevails, except when the cracks are very close, e.g. d/L = 4.25, where a reduction in height is observed. Figures 8.34 and 8.35 show the frontal intercept, r0 and the zone height, H respectively, as a function of log d-2c~ for different initial crack lengths. L

8.3.

Array of Internal Cracks

219

y~

~f

0.5 0.4

13.5

0.3 0.2 0.1 0.0

5.0

5.2

5.4

5.6

5.8

6.0

x/L

F i g u r e 8.31: Transformation zones at the onset of crack growth for coiL = 5, w = 10 and d/L = co, 50, 20, 15, 14, 13.5

y/L

~ F 0.7 0.6 0.5 0.4 0.3

0.0

.7

t

50.0

I

t

I

-'i ''=

I

50.2 50.4 50.6 50.8 51.0 51.2 51.4 x/L

F i g u r e 8.32: Transformation zones at the onset of crack growth for 50, w = 10 and d/L = co, 200,150,125,110,107,105,104.7

coiL =

From Fig. 8.34 it can be concluded that the frontal intercept progressively increases as the distance between the cracks decreases, with the exception of the very short cracks when a weak decrease is observed at relatively large values of d/L. This behaviour could possibly be also observed for the longer cracks, at similar long separation distances. Consistency of the general behaviour might also have been observed for the height, i f a larger range of (d-2co)/L was included in the analysis.

R - C u r v e Analysis

220

y/t, 0.6 0.5 0.4 0.3 0.2 0.1 0.0

4.8

5.0

5.2

5.4

5.6

5.8

6.0

x/L

F i g u r e 8 . 3 3 : T r a n s f o r m a t i o n zones at the onset of crack g r o w t h for co/L - 5, w - 30 a n d d / L - oc, 50, 20, 18, 17

rdt, 1.4 1.3 -

o)=10

/

1.1 1.0 0.9 0.8 0.7 2.0

! 1.8

i 1.6

I 1.4

i ----r1.2 1.0

I 0.8

i 0.6

I 0.4

i d_2c ~ 0.2 log L

F i g u r e 8 . 3 4 : F r o n t a l t r a n s f o r m a t i o n zone i n t e r c e p t as a f u n c t i o n of l o g ( d - ~ c ~ at the onset of crack g r o w t h for w - 10, co/L - 0.5, 1,5, 50 a n d for w - 30, co/L - 5

8.3. Array of Internal Cracks

221

H/L

0.75

-

0.70

r

~

0.65

co/L=50

0.60 0.55 0.50 0.45 0.40 0.35

2.0

l 1.8

i 1.6

1 1.4

i 1.2

.---------1 ---~-- ~ 0.5 i L J I J d_2c 0 1.0 0.8 0.6 0.4 0.2 log L

F i g u r e 8.35: Transformation zone height as a function of log(d-2c~ L ) at the onset of crack growth for w - 10, c o / L - 0.5, 1, 5, 50 and for w - 30,

coiL - 5

For the height, the general trend is an increase up to a peak value, as the cracks are brought closer, and a subsequent decrease when the cracks are very close. Figures 8.36-8.38 show the zone shapes for constant length of unbroken material between the cracks, )~ = ( d - 2 c 0 ) . From the different values of )~/L, different ratios of broken to unbroken material can be calculated in order to study the effect of the area fraction of transformable material on the mechanical properties of a particulate transformation toughened ceramic. In these figures each value of c o / L corresponds to a certain area fraction. It is seen that the height and length decrease as c o / L decreases (i.e. area fraction increases) except for very short cracks ( c o / L = 0.5) where the opposite happens. This pattern was also observed in the single crack analysis when c o / L is varied with the other parameters kept constant. The differences in zone shapes in Figs. 8.36-8.38 must be seen in the light of the change in the applied load as d / L is varied in order to m a i n t a i n K tip = Kr The toughening ratio KapVz/Kc at the onset of crack growth is plotted in Fig. 8.39 and 8.40 for different values of c o / L and A / L as a function of log d-2co and A/ respectively. The curves L also represent the normalized strength, cr~176 where a0 - Kc/~v/'ff-~ is the strength in the absence of transformation, i.e. KavVz/Kc = ~r~176

R-Curve Analysis

222

y/L 0.7 0.6

~

co/L=10

0.5 0.4 0.3 0.2 0.1 0.0 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

i

1.2 (x-q,)/L

F i g u r e 8.36: Transformation zones at the onset of crack growth for $/L = 4, w = 10 and co = 0.5, 1 , 2 , 3 , 4 , 5 , 10

y/t, 0.8 0.7 0.6

-f

cdL=50

0.50.4 0.3 0.2

J

0.1 0.0 -0.2

I

0.0

0.2

0.4

0.6

0.8

1.0

1.2 (x-q~)lL

F i g u r e 8.37: Transformation zones at the onset of crack growth for )~/L = 10, w = 10 and co = 0.5, 1,2, 3, 4, 5, 7, 10, 15, 50

at the onset of crack growth. When the initial crack length is kept constant and d/L is varied, it is seen from Fig. 8.39 that the curves decay monotonically, as the length of unbroken material is reduced. The initially long cracks start at a high level of applied stress intensity at long distances but decrease

8.3.

Array of Internal Cracks

223

y/t, 0.8 0.7 0.6

,~,

0.5

~

0.4

\

co/L=200 AF--'0.040

~,

0.3 0.2

0.1

A 0.98

0.0

~,xl

-0.2

0.0

I

I

I

0.2

0.4

0.6

II

I

0.8

1.0

1.2

(x-q))/L

F i g u r e 8 . 3 8 : T r a n s f o r m a t i o n zones at the onset of crack g r o w t h for ~ / L - 100, w - 10 and co - 0.5, 1, 5, 10, 20, 50, 70,100, 150,200

Kappt/Kc =_ 0"~/00

1.00 ~

5

0

oo=lO

0.95

.......

0.90 0.85

f

0.70 / 0.65 2.0

i

i

I

1.8

1.6

1.4

I 1.2

I 1.0

"

I 0.8

J 0.6

I

0.4

J d-2c o 0.2 log

F i g u r e 8 . 3 9 : N o r m a l i z e d applied stress intensity factor or applied stress as a f u n c t i o n of log d-2cQ at the onset of crack g r o w t h for w - 10, L c o / L = 0.5, 1, 5, 50 a n d for w = 30, c o / L = 5

R-Curve Analysis

224

K

appl

/Kc = 0"*/~o

1.O0 ML=IO0 0.95 0.90 0.85 0.80 0.75 t 0.0 0.1

t I I t I t l I J 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 A t

F i g u r e 8.40: Normalized applied stress intensity factor or applied stress as a function of area fraction at the onset of crack growth for w - 10 and A/L = 4, 10,100

more rapidly to a lower level than the shorter cracks. To conclude, the normalized strength and toughness are seen to decrease, the Closer the cracks are for a constant initial crack length. In other words, the detrimental effect of the transformation at the onset of crack growth is amplified by the interaction of the crack tips and transformation zones, compared to a single crack. By comparing the curves for w = 10 and w - 30 in Fig. 8.39 it is again seen that a higher value of w has a detrimental effect on the normalized toughness and strength (KaPPt/Kc and ~r~176 are always below unity). This means that the transformation weakens the material at the onset of crack growth. This result is in agreement with previously obtained results for a single crack. In Fig. 8.40 the amount of unbroken material between the cracks is kept constant by an appropriate variation of co/L and d/L. For moderate )~/L(= 4, 10) it is seen that the toughness at the onset of crack growth goes through a peak whose location depends on the value of )~/L. For a particulate toughened material it is concluded that at the onset of crack growth an o p t i m u m area fraction of particles exists, and that large particles are favourable with respect to the toughness and the strength, relative to a non-toughened material with a similar crack configuration.

8.3.

A r r a y of I n t e r n a l C r a c k s

225

o~.,E Kc co/t~o.5

0.7

{o=10 ~ {0=30 ........

0.6 0.5 0.4 0.3 5

0.2 O.1

--------I I 0.0 2.0 1.8 1.6

I

I

I

1.4

1.2

1.0

50

7---

I

0.8

0.6

i 0.4

i d_2c ~ 0.2 log L

F i g u r e 8 . 4 1 : Actual normalized applied far field stress as a function of log d-2c0 at the onset of crack growth for w - 10 , c o / L - 0.5 , 1 , 5 , 50 L and for ~ - 30, c o / L 5

0.7 0.6 0.5

-

l0

/

0.4 0.3 0.2 0.1 0.0 0.0

I

0.1

I

0.2

I

l

0.3 0.4

1

I

I

i

I

0.5

0.6

0.7

0.8

0.9

I

1.0 At

F i g u r e 8 . 4 2 : Actual normalized applied far field stress as a function of area fraction at the onset of crack growth for w - 10 and A/L - 4, 10,100

R-Curve Analysis

226

Figures 8.41 and 8.42 show the actual normalized applied far field stress corresponding to Figs. 8.39 and 8.40. Figure 8.42 shows that the actual normalized strength increases monotonically with increasing fraction of transformable particles.

8.3.3

Growing cracks

Each individual crack in the array is grown quasi-statically by the same amount by adjusting the applied load, so that K tip = Kc, as described in Section 8.2. The toughening ratio is plotted in Figs. 8.43 and 8.44 as a function of the crack advance for co/L = 5 and co/L = 50, respectively and different values of normalized distance, d/L. The curves correspond to the R-curves for an array of cracks. For large separation distances, e.g. d/L=lO0 in Fig. 8.43, the curve follows the single crack R-curve relatively closely until the cracks have grown sufficiently to interact, whereafter the curve decreases rapidly. For the smaller values of d/L the curves peak earlier as the neighboring crack tips and the transformation zones interact at an earlier stage of crack advance. Two factors are responsible for this behaviour. First, the initial de-

Kapp I

Kc 1.6

dlL= .~

1.5

I00

1.4 1.3 1.2

~20

1.1 l.O 0.9

0

I

I

I

I

I

I

I

.J

5

10

15

20

25

30

35

40

C-C O

L

F i g u r e 8.43: Normalized applied stress intensity factor as a function of crack advance for w - 10, c o / L - 5 and d/L - oo, 100, 50, 30, 20

8.3.

A r r a y of Internal Cracks

227

Kapp I

Kc 1.7 1.6

d/L

-

oo

1.5 1.4 1.3 1.2 1.1

112

1.0 0.9 0

i 5

i 10

i 15

i 20

I

I

I

I

25

30

35

40

c-c o

L

F i g u r e 8 . 4 4 : N o r m a l i z e d a p p l i e d stress i n t e n s i t y factor as a f u n c t i o n of crack a d v a n c e for w - 10, c o / L - 50 a n d d / L - oo, 1 2 5 , 1 1 2

~'/~o 1.3 F 1.2 1.1 1.0 0.9 0.8 -

30 30"

~~-...._

0.7 _ 0.6

Figure for w -

0

5 I 2

I 4

I 6

t 8

0 t 10

d/L = ,,,, ~ I 12

100 I 14

C.Co L

8 . 4 5 : N o r m a l i z e d a p p l i e d stress as a f u n c t i o n of crack a d v a n c e 10, co/L - 5 a n d d/L - oo, 100, 50, 30, 20

228

R-Curve

Analysis

~~176 1.6 1

.5

1.4

5

~ - d / L

~

= oo

1.3 1.2 1.1

~112

1.0 0.9 0

I 2

I 4

I 6

J 8

I 10

I 12

t 14

c-c o

L

F i g u r e 8 . 4 6 : N o r m a l i z e d a p p l i e d stress as a f u n c t i o n of crack a d v a n c e for w - 10, c o / L - 50 a n d d / L - ~ , 1 2 5 , 1 1 2

~/(~0 0.90

co/L = 2

0.88

3

1 ------------4

0.86 0.84 0.82 0.80

Figure for w -

0

i 0.02

i

i

i

0.04

0.06

0.08

c-c O

L

8 . 4 7 : N o r m a l i z e d a p p l i e d stress as a f u n c t i o n of crack a d v a n c e 10, A / L - 4 a n d c o / L - 1, 2, 3, 4, 10

8.3.

Array o f Internal Cracks

229

o*"/o 0 1.15 1.10

15

-

_-

4

co~L= 10

5

1.05

1.O0 0.95 1

0.90 0.85

F---0.5

0.80 0

Figure for w -

I 0.2

I 0.4

I 0.6

i 0.8

I 1.0

I c-~ 1.2 L

8 . 4 8 : N o r m a l i z e d a p p l i e d s t r e s s as a f u n c t i o n of c r a c k a d v a n c e 10, A / L - 10 a n d c o / L - 0.5, 1, 2, 3, 4, 5, 10, 15, 50

~**/ao 1.7 [1.6 1.5 ].4

co~L= 200

t

2-0

1.3 1.2 1.1 1.0 0.9 I 1 0.8 0

Figure for w -

I

I

1

2

2 3

I 4

I 5

I 6

I 7

I

8

c-c o

L

8 . 4 9 : N o r m a l i z e d a p p l i e d s t r e s s as a f u n c t i o n of c r a c k a d v a n c e 10, A / L - 100 a n d c o / L - 1,5, 10, 20, 5 0 , 1 0 0 , 2 0 0

230

R-Curve Analysis

velopment of transformation zone wakes behind the crack tips reduces K tip, and consequently K appz must be increased to maintain K tip - Kc. Secondly, the interaction of the crack tips and transformation zones in the array of cracks increases K tip as the cracks come closer, so that K appz must be reduced to maintain K tip = Kc. The interaction of the crack tips is determined by the geometrical factor B0 used to calculate K "ppt (8.25). For a single crack, these two factors will result in the R-curve going through a peak before reaching a steady-state level. For both c o / L = 5 and c o / L = 50 it is evident that the interaction of the crack tips and transformation zones drastically reduces the peak toughness of the material. Figures 8.45 and 8.46 show the normalized applied stress against the crack advance. It is seen that the stress curves peak earlier than the corresponding R-curves, but still allow for some subcritical crack growth, if loaded by a monotonically increasing far field stress. The applied stress, normalized with respect to ~0, is shown as a function of crack advance in Figs. 8 . 4 7 - 8.49 for the three values of )~/L = 4, 10,100, respectively. The peak stresses from these figures are then plotted in Fig. 8.50. For the small particle sizes ~ / L = 4, 10, the strength peaks at a finite area fraction but for )~/L = 4 it is below the

cP/c0 1.7 1.5 1.5 1.4 1.3 1.2 1.1 1.0 0.9 J 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

A]'

F i g u r e 8.50: Normalized applied peak stress as a function of area fraction, w - 10 and )~/L - 4, 10,100

8.4. S u r f a c e C r a c k s

231

o"4Z Kc 0.7 0.6 0.5

k/L = 4

0.4

10

0.3 0.2 0.1 0.0

I

I

I

I

I

I

I

1

i

I

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Af

F i g u r e 8.51: Actual normalized applied peak stress as a function of area fraction, w = 10 and )~/L = 4, 10,100

strength of a non-toughened material. From this it can be concluded that the transformation toughening is only efficient for large particle sizes for which an optimum particle fraction for the relative strength exists, but as shown in Fig. 8.51 the actual normalized strength is highest for small particles in high concentration.

8.4

Surface Cracks

In this Section a model for a single surface crack in transformation toughening ceramics is described with a view to examining the development of transformation zone, toughening, and strengthening along the lines of analysis presented for an internal crack in Section 8.2 (Andreasen & Karihaloo, 1994). Surface damage is a fundamental issue in the analysis of transformation toughened ceramics. The model described below is expected to form a good basis for analyzing thermal shock, fatigue, wear and other phenomena in which the presence of surface cracks plays a fundamental role. Some of these phenomena will be further examined in Part III of this Monograph.

R-Curve Analysis

232 8.4.1

Model

Description

and

Theory

The problem of surface cracks in the absence of transformation has been solved by a number of investigators, see e.g. Nemat-Nasser et al. (1978), Keer et al. (1979), and Nemat-Nasser et al. (1980). In these references the stability of thermally induced surface crack growth was studied.

F i g u r e 8.52: Model configuration for a surface crack The model for a surface crack in T T C is shown in Fig. 8.52. A surface crack C of length c is situated in the half plane and loaded at infinity by a constant transverse stress (r~ . At the tip of the crack a zone of transformed material bounded by the contour S develops as the load is applied. The transformation strains are assumed to be purely dilatational and constant inside the zone. The transformation is assumed to be induced by a critical mean stress crc In the analysis to follow the free-surface problem is solved analytically for a dislocation and for a homogeneous inclusion of arbitrary shape by means of Muskhelishvili's theory of plane elasticity (Chapter 4). The crack is modelled by a pile-up of dislocations (Chapter 6). The density of dislocations in the pile-up is adjusted to meet the traction-free crack condition. The transformation zone boundary is determined by the critical mean stress criterion. The traction-free condition and the critical mean stress criterion lead to two coupled singular integral equations which are solved numerically. The pile-up of dislocations can be described through a dislocation density function D(yo), such that D(yo)dyo is the Burgers vector of the dislocations between y0 and Yo + dyo. The stress due to this pile-up on the crack-line (r~xc gives an integral equation from which the dislocation density function can be determined such that the stress across the crack

8.4. Surface Cracks

233

C vanishes as described in Chapter 6. Formally, the crack-line stress (Fig. 8.52)can be written as ~(z)

-

0 -

~oo + ~ (~~ )

+

~(~) 1 ~

(8.32)

In terms of the applied stress intensity factor K appz the load r ~176 can be written as ~oo =

Kappl

(8.33)

B0x/~

For a single crack B0 is approximately 1.1215, (see e.g. Tada, 1985 or Murakami et al., 1987). The crack-line stress r from the transformation is calculated from

T gxx

o'xx

(ir, zo)dxo]

(8.34) rEC" zoE S

where the weight function gT~(z, zo) is given by (4.56), with z - 0-4- iv. The crack-line stress from the dislocations is calculated from

E

crD~(z)- 47r(1- t~2) / c D(yo )hxD~x (it, is)ds

(8.35) r,sEC

where the weight function h~DJ(r, s)is given by (6.19) (with z - 0 + iv and z0 = 0 + is). The transformation zone boundary is determined from the critical mean stress criterion, which can be written as crr m 1 + t, (~r~ + a~,(z)) eo + ~r,,(z) T D I m --

3

zES

(8.36)

The mean stress from the applied load is ~r~a~176 _ ~oo and it is given by (8.33) in terms of the applied stress intensity factor. The mean stress from transformation can be written as

l3q~ l,, ~T( z )

T - ~w / s g~(z, zold~o I~,~o~S

(8.37)

where g~a T ( z , )z0 is given by (4.56). Finally, the mean stress from the dislocations can be written as

R-Curve Analysis

234

l+u

D _

3(r~ aa~

E(l+u) 127r(1- v2)a~

/cD(yo)hD~(z, is)dsl

zes. ,ec

(8.38)

where the weight function h~D3~(z,s) is given by (6.19). The two equations for zero traction across the crack and for the transformation zone boundary can be rewritten in similar forms. For this we introduce the dislocation density function D*(s) via D*(s) -

E/cr~

127r(1 - v)

D(s)

(8.39)

and substitute it into (8.35) and (8.38). From (8.32)-(8.35), and (8.36)(8.38), we finally get O~

rEC Kappi~~c Tx (z, zo)dxo + /C D* (s)ha~; Dx (z, is)ds [(8.40) BoKc + ~w ~s ga'd

IzES

The dislocation density function D*(s) and the transformation zone boundary S can be obtained from (8.40) for a given load K appt and transformation parameter w. In order to analyse initial toughening and R-curve behaviour, the stress intensity factor at the crack tip K tip is required so as to impose a steady-state crack growth condition. The dislocation density function D* (s) has a square root singularity as the crack tip is approached and the stress intensity factor K tip is given through the following limit

Ktip Kc

=

lim 27rD*(s) i ~--~+

c+s L

(8.41)

At the onset of crack growth, as well as for quasi-static crack growth, the stress intensity factor at the crack tip K tip must equal the intrinsic toughness Kc, i.e. J

1-

lim

8--~-- C+

r

27rD*(s),/c~ s v

L

(8.42)

The steady-state growth condition supplements the two eqns (8.40), thereby allowing K appl - or the R-curve - to be determined.

8.4.

Surface Cracks

235

The two integral equations (8.40) contain a number of singularities. As already mentioned, the dislocation density function D* (y) has a square root singularity at the crack tip. The crack-line stress imposed by the transformation zone contains a discontinuity as the transformation zone boundary is crossed. This leads to a logarithmic singularity in the dislocation density function. The weight functions contain singularities of the ordinary Cauchy type, as well as weak singularities at the surface and at the transformation zone boundary intersection by the crack. In order to obtain accurate numerical solutions, it is imperative to have good control over these singularities. Thus, the singularities are isolated and treated analytically as far as possible, in order to ensure that only regular functions are numerically integrated. Solutions to (8.40) are obtained by improving a guess for the transformation zone shape through a number of perturbations. Inverting the first integral equation (8.40) for each perturbed shape, an improved zone shape is obtained by Newton-Raphson's method. For a growing crack the two coupled integral equations (8.40) can be restated in an incremental form as follows:

O

g appl

/

BoKr

V2(c + Ac)

L

gTx( ir' zo)dxo + Iv D*(s)hxDf~x(ir'is)ds K appl /

BoKr +~

rEC

L

V2(c + Ac)

gag (z,

lim 27rD* (s) s~-(c+ac)+ S(c + ac)wok, =

+

D*

D,~

is)ds zeS/,'ont

/Cv + Ac + s L

(8.43)

The assumption of no reverse transformation is imposed by the last side condition in (8.43). The transformation zone shape is only changed at the front Sf,-o,t of the crack tip, where the mean stress is rising, while a wake S,oake of transformed material is allowed to develop behind the

R-Curve Analysis

236

tip of the growing crack, where the mean stress is declining. The procedure for solving (8.43) is based on a guess for the transformation zone front SIront and the solution of the first integral equation (8.43) using this guess. The resulting dislocation density function D*(yo) is substituted into the second integral equation (8.43), whose solution gives an improved estimate of Si,,o,~t. This procedure is applied repeatedly until convergence criteria are met. The two side conditions in (8.43) are met by adjusting K appz iteratively and by joining the transformation zone wake S~ake and front S],-ont by common tangents until sufficient accuracy is attained (see w

8.4.2

Single Surface Cracks

In the following initial transformation zone shapes, initial toughening, and R-curve behaviour are discussed. Some general results on peak toughness and strengthening are also presented. The discussion follows the same lines as for internal cracks (w167 and where appropriate comparisons between surface cracks and internal cracks are made to emphasize relevant differences or similarities. Transformation zone shapes at the onset of crack growth obtained by solving (8.40) are shown in Fig. 8.53. The detachment of the trans-

y//.,

/{o=30

1.0

//is -/10

0.8 0.6 0.4 0.2 0.0

I

0.0

0.4

0.8

1.2

1.4

x/L

F i g u r e 8.53: Initial transformation zone shapes for single surface crack,

co/L- 10

8.4.

Surface Cracks

237

gappl/g c 1.0

co/L=500 50

0.9

lO

0.8

_

5

0.7 0.6

0

I 5

t 10

t 15

l 20

I 25

J 30

F i g u r e 8.54: Toughening ratio at the onset of crack growth for single surface crack formation zone wake from the crack tip is characteristic of the model being studied. The transformation zone size increases monotonically with the transformation parameter ~o when the initial crack is not too small. The critical initial crack length co/L at which the transformation zone diverges for vanishing r because the mean stress from the applied load cr~176 exceeds the critical value a ~ before crack growth is initiated is approximately co/L -- 0.3975 for a single surface crack, as opposed to co/L = 0.5 for a single internal crack (w However, as with internal cracks, the toughness decreases for w :/: 0, so that the transformation zone is bounded, even for an initial crack length equal to the critical value. The apparent toughness (i.e. the toughening ratio) at the onset of crack growth for several initial crack lengths is shown in Fig. 8.54. In general, the ratio decreases before crack growth, except for very long cracks when a slight increase is observed. The toughening ratio for a semi-infinite crack is within 0.5% of the toughening ratio for the surface crack of length c0/L = 500. The R-curves determined from the solution of (8.43) are shown in Fig. 8.55 for several initial crack lengths and for two values of the transformation parameter w. The limiting case of an infinite crack, shown with a broken line was first solved by Stump & Budiansky (1989a). The toughening ratio peaks before the reaching steady-state level. The pres-

238

R - C u r v e Analysis

K

appl /K c

1.3

m

1.2 1.1 oo

1.O 0.9 a)

0.8

K

0

I

I

I

5

l0

15

I

20 Ac/L

appl /K c

1.8 1.6 1.4 1.2

b)

,~ I 0.8

0

9

t

10

t

20

l

30

I

40

J

50

Ac/L

F i g u r e 8.55" R-curves for single surface crack, (a) w - 5, (b) w - 10

ence of a free surface causes the peak value to drop for short initial cracks; the peak value may even drop below the steady-state level. In Fig. 8.55b this is evident for the crack of length co/L = 5. Monotonically rising R-curves are also observed (e.g. for co/L = 5 in Fig. 8.55a), but this behaviour is the exception rather than the rule. The appearance of peaks in the toughening ratio prior to the attainment of the steady-state level seems to be an inherent feature of models based on the critical mean stress transformation criterion. They ap-

8.4. Surface Cracks

239

~**/% 1.4

co/L=500

1.2

50

1.0 0.8 0.6 0.4 0.2 a)

0.0

'

0

'

5

'

10

'

15

20

Ac/L

t~**/% 1.8 1.6

co~L=500

1.4

50

1.2 1.0 0.8

10

0.6

5

0.4 0.2 b)

0.0

0

I

I

I

I

10

20

30

40

I

50 Ac/L

F i g u r e 8.56: Strengthening ratio for single surface crack, (a) w - 5, (b) w - 10 pear in R-curves for semi-infinite cracks (w for internal cracks under uniaxial tension (w for internal cracks under equal biaxial tension (w and now for surface cracks. The peaks get shallower as the initial crack gets shorter. This trend is the same as we observed for internal cracks under uniaxial tension (w but it is contrary to that seen for internal cracks under equal biaxial tension (w The toughening ratio for a semi-infinite crack shown in Fig. 8.55 has converged to within a

R-Curve Analysis

240

fraction of a percent of the steady-state toughening value. In comparison with semi-infinite cracks, the convergence to steady-state values of the R-curves for finite initial crack lengths is seen to be quite slow. The applied stress ~r~176 at infinity necessary for maintaining quasistatic crack growth is shown in Fig. 8.56 for several initial crack lengths and for two values of the transformation parameter w. The strengthening ratio in Fig. 8.56 corresponds to the toughening ratio of Fig. 8.55. The strengthening ratio is obtained from the following relation between stress and toughness

~176176 _- K avptKr i co+c~Ac

(8.44)

where rr0 is the stress necessary to induce crack growth in the absence of transformation. Comparison of Figs. 8.55 and 8.56 shows that the peak strengthening ratio that determines the ultimate strength is attained at a shorter crack advance than is necessary to obtain peak toughness. Thus the peak toughness is not fully available for strengthening of the material, except for very long initial cracks. The peaks in the R-curve behaviour have their origin in the widening of the transformation zone (Fig. 8.57). The reciprocal peak toughening is shown in Fig. 8.58 for values of

y/t, 1.6 1.2 0.8 0.4 0.0

-30

-25

-20

-15

-10

I

I

-5

0

x/L

F i g u r e 8.57: Transformation zone shapes for single surface cracks, w 10

8.4. Surface Cracks

241

Kc/Kpeak 1.o 0.8

cd~5

lO 50 500

0.6 0.4 0.2 0.0 0

i

I

t

I

I

i

i

J

2

4

6

8

10

12

14

16

F i g u r e 8.58: Reciprocal peak toughening for single surface cracks

1.0 0.8 J

0.6 50 0.4 0.2 0.0

0

I

I

I

5

10

15

20

F i g u r e 8.59: Reciprocal peak strengthening

R-Curve Analysis

242

the transformation parameter w up to 16. Lock-up values of the transformation parameter w at which the transformation zone diverges and the peak toughness tends to infinity are not known for the geometry of Fig. 8.52. For initial cracks of length co/L = 5 and 10, the lockup values are less than the lock-up value w = 20.2 for a semi-infinite crack (Stump & Budiansky 1989a), whereas the lock-up values for initial cracks of length co/L = 50 and 500 are expected to lie between this value for a semi-infinite crack and the value under steady-state conditions (w = 30.0). The peak toughening ratio reduces with decreasing initial crack length for moderate values of w. For w larger than about 12, this trend is reversed (Fig. 8.58). The reciprocal peak strengthening is shown in Fig. 8.59 for values of the transformation parameter up to 20. These results are consistent with the results for internal cracks in that initially weak materials with long inherent cracks are more susceptible to strengthening than initially strong materials. The qualitative similarity of the strengthening results for single surface cracks with those for single internal cracks presented in w suggests similarity in their peak strengthening correlation with toughness (see Figs. 8.17 and 8.18).

8.5

Array of Surface Cracks

We shall now extend the discussion of the previous Section to an array of surface cracks. Surface damage is a fundamental issue in the application of ceramic materials. As a first step towards modelling this damage in transformation toughening ceramics, an array of surface cracks is introduced and analysed in a manner similar to that for a single surface crack. The configuration for an array of surface cracks is shown in Fig. 8.60. An infinite array of equally spaced (spacing d) surface cracks C of length c is situated in the half plane and loaded at infinity by a constant transverse stress aoo. At the tip of each crack a zone of transformed material bounded by the contour S develops as the load is applied. Effects of elastic mismatch between matrix and transforming particles are neglected, and reverse transformation is assumed not to take place. The applied stress can be expressed in terms of the applied stress intensity factor K avvt via a~ =

Kappl

Bo~/~

(8.45)

8.5. Array

of Surface Cracks

243

F i g u r e 8.60: An array of surface cracks where B0 varies with the crack spacing (see e.g. Tada, 1985 or Murakami et al., 1987). The governing equations determining the transformation zone shape and the dislocation density function at the onset of crack growth are obtained from (8.40) by replacing the weight functions for a single surface crack with those for an array of surface cracks

O m

EC

BoKc

+~

G~

, )dxo +

D*

is)ds zES

1-

lim 2 r D * ( s ) ~/ / c + s u---- c+ V L

(8.46)

zo) T The weight functions due to transformation GTx(z, zo) and a~.(z, are given by (4.64), and the weight functions due to dislocations D x D,x g,~, (z, z0)and H,~, (z, zo) by (6.36). For a growing crack (8.46) can be restated in an incremental form

/ L BoKc V2(c + Ac)

K appl O

__

+ ~

fs G T~ (ir' z~176 + /c D* (s)H~Da;x(ir' is)ds[

rEC

R-Curve Analysis

244

K appl/ L BoK~ V2(c + Ac)

__

+ ~W ~ s C~a T,x (z, zo)dxo+/cD. (s)H~ D,x(z, is)ds 1 -

lim

~-.-(c+Ac)+

S(c + Ac)wake

=

27rD*(s) ,/c

zESfront

+ Ac + s

V

L

S(c)wake

(8.47)

where the assumption of no reverse transformations is imposed by the last condition. Examples of initial transformation zone shape for an array of surface cracks obtained by solving (8.46) are shown in Fig. 8.61 for several crack spacings. The initial toughening accompanying the zones varies from a decrement of approximately 3% for infinite crack spacing to an increment of approximately 8% for crack spacing equal to the crack length (cold = 1; d/L = 10). The increment in apparent toughness at the onset of crack growth for crack spacing d/L less than approximately 40 is in contrast to the results for a single crack on initial toughening where only a slight increment in toughness appears for very long cracks and relatively high values of the

x/L

1.6f 1.4 1.2

1.0 0.8 0.6 0.4 0.2 0.0 -0.5

30 40

0.0

I

I!

0.5

1.0

II!

I

I

I

1.5

2.0

I

I

2.5

y/L

F i g u r e 8.61" Initial transformation zone shapes for arrays of surface cracks, w = 10 and c0/L = 10

8.5. Array o f Surface Cracks

g

245

appl

/K c

2.0 1.8 1.6 1.4 1.2 oo

1.0 0.8

Figure

co/L-

8.62:

0

J

'

10

20

'

'

30

40

'

50 Ac/L

R-curves for an array of surface cracks, w -

10 and

10

transformation parameter w. R-curves for various initial crack lengths obtained by solving the equations of (8.47) are shown in Fig. 8.62. The peaks in the apparent toughness induced by the transformation are the steeper, the smaller the crack spacing. The strengthening ratio ~r~176 corresponding to the toughening ratio of Fig 8.62 is shown in Fig. 8.63. This ratio is obtained from the relation 0"~

_

Cro -

K appl B o [

Kr

co

B Vco+Ac

(8.48)

Note that the geometry factor B depends on the crack length and therefore on the crack growth increment Ac. The peak value of the strengthening ratio determines the ultimate strength of the ceramic. As the cracks grow, the effect of the free surface diminishes and the stress necessary for continued crack growth depends more on the crack spacing, rather than on the crack length. In the limiting case where the steady-state conditions prevail, the stress needed to give quasi-static crack growth becomes constant and vanishes for a single surface crack (d/L = cxz). In contrast to the single surface crack model previously presented, the strengthening effect of transformation is enhanced by the presence

246

R-Curve Analysis

of multiple surface cracks, so that initially strong materials with closely spaced cracks are more amenable to strengthening that initially weak materials with widely spaced cracks. The above model for interacting surface cracks is expected to be a good first approximation for the analysis of surface damage. However, it is important to consider stability of growing cracks. The R-curve behaviour induced by transformation ensures a certain degree of stability in the crack growth. In the absence of transformation, small variations in crack length in ideally brittle materials will cause only the longest crack to grow. In the presence of transformation however, the R-curve behaviour counteracts this tendency and a large number of surface cracks can be expected to grow together before failure eventually is caused by the growth of the longest crack. Another important factor to be considered is the crack path stability. Small variations in crack length or crack spacing will cause the crack paths to depart from the initial crack plane implied in the present model. This can cause cracks to coalesce or provide additional shielding of the smaller cracks, whereby the plasticity effects are reduced by lowering the number of growing cracks.

o

/o o

2.0 1.6 1.2 5O 1~

0.8 0.4 0.0

0

a

J

I

t

10

2O

3O

4O

I

5O Ac/L

F i g u r e 8.63: Strengthening ratio for arrays of surface cracks, w - 10 and co/L = 10

8.6. Steady-State Analysis of an Array of Semi-Infinite Cracks

8.6

247

Steady-State Analysis of an Array of Semi-Infinite Cracks

In this Section an analysis of an array of semi-infinite edge cracks in transformation toughening ceramics under steady-state conditions is presented. It transpires that the transformation zones between the cracks cannot coalesce, but that for transformation densities above a critical value two transformation zone solutions are possible. One solution pertains to quasi-static crack growth and the other to pretransformed materials. The latter can cause excessive transformation to appear during loading before crack growth is initiated. The multiplicity of solutions is a consequence of the semi-infinite crack length. As was shown above, an array of finite surface cracks very effectively shields the crack tips in comparison with single surface cracks. For crack spacing less than about 5 times the crack length the stress intensity factor for an array of finite surface cracks is within 2% of the stress intensity factor for a similar array of semi-infinite surface cracks (Tada, 1985). The multiplicity of solutions that emerges from the study of steady-

F i g u r e 8.64: Model configuration for an array of semi-infinite cracks

248

R-Curve Analysis

state growth of semi-infinite edge cracks was not found in the similar study of finite surface cracks (w Surface grinding of transformation toughening ceramics can induce a certain strengthening of the component if the grinding gives rise to transformation. The grinding-induced transformation can be the result of at least two mechanisms. First, as the contact stresses between the grinding agent and the ceramic are locally very large, and possibly singular if the grinding agent consists of irregular particles, transformation is likely to take place in the vicinity of contact. A second, less direct mechanism is that the grinding just induces small cracks in the surface, but the transformation is brought about by subsequent loading of the ceramic during either the grinding process or service. The latter mechanism can be expected to give rise to transformation in a thicker surface layer in comparison with the former mechanism. Limited crack growth can be sustained by an array of cracks, where R-curve behaviour induced by transformation prevents the instability of this configuration that would otherwise occur. In the following, this mechanism where the transformation is a result of crack growth will be considered. The model configuration of an array of semi-infinite edge cracks is depicted in Fig. 8.64. An infinite array of equally spaced parallel cracks C (spacing d) is loaded at infinity by a constant normal stress cr~ . Each crack is bounded by a zone S of transformation formed during loading and crack growth. The zones are assumed to continue along the crack faces to infinity along the negative x-axis, such that steady-state conditions prevail. The transformation strains are assumed to be constant and purely dilatational in the zones in accordance with the super-critical transformation assumption. The transformation zone boundary ahead of a crack is determined by the critical mean stress criterion. A similar model for a single semi-infinite crack was described in Chapter 7. The transformation toughening behaviour at the onset of crack growth, and during stable crack growth, for an array of finite surface cracks was presented in the previous Section. The analysis in this Section gives the results appropriate for the limiting case of semi-infinite edge cracks in steady-state conditions. Multiple solutions, which did not emerge in the growth of finite cracks, now seem possible. These solutions suggest the possibility of having crack systems in transformation toughening brittle materials, whose growth is preceded by excessive transformation, thus leading to inelastic behaviour before failure. In the theoretical analysis to follow the cracks are modelled by a pile-up of appropriate dislocations. The density of the dislocations in the pile-up is adjusted to meet the traction free crack condition. The

8.6. Steady-State Analysis of an Array of Semi-Infinite Cracks

249

transformation zone is modelled as a homogeneous inclusion, and the transformation zone boundary ahead of the crack is determined by the critical mean stress criterion. The two conditions mentioned are expressed through two coupled integral equations which are solved numerically. The first condition, that of zero traction will be considered in some detail. Each crack is modelled by a pile-up of dislocations described through a dislocation density function D(t) such that Burgers vector b between t and t + dt is b = D(t)dt. Summing the stresses from each dislocation in the array of cracks and integrating the pile-up along the central crack, the following crack-line stress due to these dislocations is obtained

r (X)

_

E

47r(1 - ~2) j_oo D(t)H D'u (x, t)dt

(8.49)

where HuD,Y(z, zo ) is given by (6.31). x and t are along the central crack C. Taking advantage of the fact that y - y0 - 0 in uyD'Y(z, zo)in (6.31) the weight function Hy~,y(x, t) reduces to Hyy (x,t) -

~

2coth(Tr

d

)- ~(x-

t)cosech(~r

d

) (8.50)

The crack-line stress from the array of transformation zones is similarly obtained by summation as

r

T (.)

_

27r(1 - v)

r(* , z o ) d y o

ayy

(8.51)

where Gyu(x, T zo)is given by (4.62). The integration along the transformation zone boundary S in the crack-line stress in eqn (8.51) can be reduced to a line integral over the transformation zone front by exploiting the fact that the integrand of this equation reduces to a constant at infinity along the negative direction of the x-axis, and so giving

r

-

271"(1 -- P)

H

, z0) -

dy0

(8.52)

The bounds of the integral + H in eqn (8.52) are the half-height of the transformation zone (see Fig. 8.64). It is determined from the transformation zone front as the point where dyo/dxo vanishes along the zone boundary. It will be shown below that H is limited to a quarter

R-Curve Analysis

250

of the crack spacing, so that coalescence of neighbouring transformation zones cannot occur. The transformation zone wakes along the crack faces are a result of the assumption of no reverse transformation. The term -Tr/d is due to the closed part of the transformation boundary S at infinity along the negative z-axis. If the boundary S is assumed not to close at - c ~ , a non-zero far-field stress appears in the x-direction. Elimination of this stress at infinity yields the same governing equation, as (8.52) above. The condition of no traction across the cracks can now be obtained by adding the crack-line stresses from (8.49) and (8.52) and a constant stress dryy 0 needed to ensure the stress conditions at infinity

0 -

"

Z(

0 + 2~'(1 - u) o'yy

g

E

)

r Guy

-~ dyo

/~__o~ D(t)HyyD,y (z, t)dt

+ 4 r ( 1 - u2)

(8.53)

Due to symmetry the crack-line shear stress automatically vanishes. The constant stress ~ryy 0 introduced in eqn (8.53) is determined by considering the stresses at infinity in the y-direction. The stress from the dislocations is obtained by letting x tend to infinity in eqn (8.49). The stress from the transformation zones given by eqn (8.52) vanishes, so the resulting stress ~r~ at infinity is given by

cr~

-

o +

~r~y

2d(1

-

u 2)

oo

D(t)dt

(8.54)

The stress at infinity in the z-direction automatically vanishes. Eliminating the constant stress ~ryy 0 from eqn (8.53), by introducing the more practical stress at infinity ~r~176 corresponding to the far field loading (see Fig. 8.64) from eqn (8.54), finally gives the following condition of vanishing crack-line stress

0

-

or~17627r(1 - u) + 4 r ( 1 - u2 )

H

~ D(t)

zo)

-

-j

HyD'y(x,t) - 2-~ dt

(8.55)

The transformation zone boundary is determined from the critical

8.6. Steady-State Analysis of an Array of Semi-Infinite Cracks

251

mean stress criterion. Following the same method as for the crack-line stresses, the mean stresses are obtained through the summation for all cracks in the array and integration along the central crack. The resulting c the mean stress gives through the critical mean stress criterion am _ am following equation for the transformation zone boundary 1_

l+v(aco 3r

E f ( 7r) ) co D(t) u D ~ ( z , t) -- 2-~ dt (8.56) + 27r(1 - v2 )

where H,D&~ (z, Zo) is given by (6.31). The transformation itself does not contribute directly to the mean stress outside the transformation zone (see eqn (4.62)), so the additional mean stress from the transformation appears only indirectly as a result of the change imposed on the dislocation density function D(t) from eqn (8.55). The two integral equations for zero traction across each crack in the array (8.55) and for the transformation zone boundary (8.56) can be rewritten in similar forms. For this we introduce the transformation parameter w given in (3.26) and the length measure L given in (7.28). The stress at infinity is conveniently normalized by the critical applied stress or0 which would induce transformation in an uncracked specimen, namely 3 r = ~a~ (8.57) l+u Introducing a new dislocation density function D0(t) through

E/ L

Do(t)-

(8.58)

1 2 ( 1 _ v) D(t)

finally gives the integral equations for the traction-free crack (8.55) and the transformation zone boundary (8.56) 0 - -- + if0 ~ +-

1/

7f"

1 - --

frO

H

Do(t)

Guy (x, zo) -

dyo

S yD'y (x, t) - 2

dt

(

co

+ -

7r

co

Do(t)

H~D~(z, t) -- 2

dt

R-Curve Analysis

252

From (8.59) the dislocation density function Do(t) and the transformation zone boundary S can be obtained for a given load ac~ and a value of the transformation parameter w. In order to obtain the specific solution for quasi-static crack growth at steady-state conditions the stress intensity factor at the crack tip K tip is needed to impose a crack growth criterion. The dislocation density function Do(t) has a square root singularity as the crack tip is approached and the stress intensity factor K tip is given through the following limit

Ktiv Is

lim 2D0(x) ~--,o-

(8.60)

L

For quasi-static crack growth the stress intensity factor at the crack tip K tip equals the intrinsic toughness Kc

K tip -- Kc

(8.61)

Combining eqns (8.60) and (8.61) and the condition for determining the transformation zone height H gives the following supplementary conditions for the solution of the system of equations (8.59) 1 - z-.0-1im2 D o ( X ) I L x

0 - odU~

y0)

S

(8.62)

Solutions to eqns (8.59) and (8.62) for specific values of the transformation parameter w are obtained by initially guessing a transformation zone boundary and then iteratively solving the first eqn (8.59) to obtain the dislocation density function Do(t). This is used to obtain an improved guess for the transformation zone boundary S from the second eqn (8.59) and the second side condition (8.62). The side condition of imminent crack growth expressed through the first eqn (8.62) is met by adjusting the applied stress ~r~176 at each iteration. The roles of the transformation parameter w and the applied stress ~r~176 can be interchanged such that the applied stress is fixed and the transformation parameter acts as the unknown to be obtained from eqns (8.59) and (8.62). The first eqn (8.59) is Cauchy singular, as can be seen fi'om eqn

8.6. Steady-State Analysis of an Array of Semi-Infinite Cracks

253

(8.50), and the integral is to be evaluated in its principal value sense. The dislocation density function Do(t) has a square root singularity appropriate for the singular stress field at the crack tip but is otherwise well behaved, that is it is continuous and differentiable except at the crack tip. Consequently the inversion of this equation can be performed by standard Gauss-quadrature techniques (Erdogan et al., 1972) without difficulty and with good accuracy. 8.6.1

Results

and Discussion

On the basis of the model described above for an array of parallel semiinfinite edge cracks some results relating to the strengthening of ceramics with damaged surfaces and transformation induced by crack growth are presented in the following. Before presenting general results, it is interesting to consider the limiting case of diverging transformation zones, to delineate the conditions under which such solutions exist. An upper limit on the transformation zone height H can be obtained by considering the second eqn (8.59) for the critical mean stress criterion. As the applied stress goo is less than the critical applied stress ~r0, the D due to the dislocations given by the integral in the second mean stress grn eqn (8.59) and by (6.31) must give a positive contribution to the mean stress, so that the following inequality must hold

0 <--

//

oo Do(t)

(

sinh(27r~~d ) c o s h ( 2 7 r ~ ) - cos(27r~) - 1 dt

(8.63)

Numerical studies show that the dislocation density function Do(t) is always positive. For y0 > d/4 the integrand increases monotonically for fixed t, and tends to zero from below as x0 is allowed to increase. Therefore the integral is negative for y0 > d/4 and the inequality (8.63) is violateit. The limiting value of the transformation zone height is therefore H = d/4, and coalescence of neighbouring transformation zones cannot take place. At this limit for H, the zone front diverges, i.e. x0 ---* exp. When x0 ---* cxz and g ---. d/4 eqns (8.59) reduce to (with x ~ -cx~)

0-

0 "c~

o'o (Too

1 --

0"0

r 4/~ dyo Do(t)dt 9d J-d/4 "d oo (8.64)

254

R-Curve

Analysis

The second of the two eqns (8.64) gives the value of the remote load ~r~176 at which divergent transformation zones are possible. The integral of the dislocation density function D o ( t ) is related to the crack opening displacement, which in turn is related to the stress intensity factor. Considering the fundamental solution to the crack problem in the absence of transformation, the stress intensity factor is given by K - c r ~ 1 7 6 (Tada 1985), and by elementary analysis the opening of the crack is equal to the displacement of a plane strain strip of height d in the direction of the applied stress aoo, i.e. v+ - v- = aoo d(1 - u 2 ) / E . Imposing the side condition of imminent crack growth (8.61) and expressing the crack opening in terms of the dislocation density function D ( t ) gives v + - v-

-

o

D(t)dt -

X/~(1 -- ~2)

oo

Rewriting this in terms of D o ( t ) using (8.58) gives -~

Do(t)dt

-

(8.65)

oo

Substituting (8.65) into (8.64) then gives the critical value for the transformation parameter wc for which a diverged transformation zone is the solution

F i g u r e 8.65: Transformation zone for a single semi-infinite crack used for analyzing possible divergence

8.6. Steady-State Analysis of an Array of Semi-Infinite Cracks

w~-

18 ( 1 -

~/-~)

255

(8.66)

For the limiting case when the crack spacing d/L tends to infinity, i.e. for a single semi-infinite crack, the critical value for the transformation parameter is wc = 18. This result was obtained by initially assuming that the transformation zone has diverged. If a finite transformation zone at a single semi-infinite crack were initially assumed and the limit of diverging transformation zones were obtained by gradually increasing the transformation parameter w in eqns (8.59) and (8.62), a lock-up value for the transformation parameter of approximately 30.0 would be obtained (Amazigo & Budiansky, 1988). The result (8.66) indicates that a diverged zone can be a solution for a single semi-infinite crack for w = 18. It is worthwhile considering this point in a little more detail. The closing stresses on a single semi-infinite crack can be obtained from (4.52) and (4.21) as T(Z) _ R e ( Ec T ~ 1 dyo} ~r~y 27r(1 - v) x - z0

x<0

The rectangular zone depicted in Fig. 8.65 is the limiting convex shape giving the maximum closing stress. Integrating along this zone front gives (ryy

- v--------~ 2(1 c~

I

x < 0

H

ira -

arctan

Y0 x

-- r 0

_< 1

(8.67)

-H

The equality sign holds when the zone height H is first allowed to diverge, thereby rendering the size of the frontal zone intercept r0 irrelevant. If on the other hand, r0 tends to infinity with H but the ratio r 0 / H remains fixed, the inequality sign holds, and a will be less than one. Rewriting the inequality in terms of the transformation parameter gives l+v T > E~ T 1 + v = _ _ _w 3~r#~ (ryy _ 6a~n 1 - v 18 In order to annul the crack-line stress, it is necessary to apply a load cr~ equal and opposite to ~u~ at infinity. The applied stress must be

R-Curve Analysis

256

sufficiently large to yield a mean stress at infinity equal to the critical mean stress in order that the diverged zone is a solution. Thus ~ ( 1 + ~ ) / 3 - ~r~, and the inequality finally becomes

>

(8.68)

In the above line of reasoning the limiting behaviour was obtained directly by considering a single semi-infinite crack, and the limit is in agreement with that obtained in (8.66) by considering the limiting behaviour of an array of parallel semi-infinite cracks with increasing spacing. The limiting value actually obtained depends on how the limiting process is performed, i.e. the actual value of a in (8.67). As already mentioned, the value obtained by increasing the transformation parameter w for a single semi-infinite crack is approximately 30.0 (Amazigo Budiansky 1988). On the other hand, the limiting value obtained by increasing w and the crack spacing d successively is approximately 36.6. For sufficiently close spacing of the cracks in the array the applied load for crack growth in the absence of transformation crc~ - Kc/v/-d-/2 (Tada, 1985) is sufficient to induce transformation by exceeding the critical mean stress ~rm. Thus the transformation zones will diverge and cover the entire plane ahead of the cracks. From (8.66) this critical 7r. crack spacing is obtained as dr

(3"1q30

1.0 0.8

6

)

0.6

8

12 2O

0.4 0"2 f 0.0

0

l

5

~

10

i

15

i

20

i

25

i

30

F i g u r e 8.66" Strengthening ratio for array of semi-infinite cracks

8.6. Steady-State Analysis of an Array of Semi-Infinite Cracks

257

CO

4030 20 (Omax

l0 0

0.0

0.2

0.4

0.6

0.8

~.0 x--E

F i g u r e 8.67: Critical and maximum values for the transformation parameter For nondiverging transformation zones, solutions to eqns (8.59) and (8.62) are obtained numerically. The strengthening ratio for various crack spacings d/L is depicted in Fig. 8.66. It is seen that solutions to eqns (8.59) and (8.62) can be obtained for the transformation parameter equal to the critical transformation parameter wc of eqn (8.66) but with lower strengthening than the critical strengthening given by eqn (8.63). For these solutions, the transformation zones do not diverge, and for the transformation parameter w greater than the critical value wc but less than a certain maximum Wmaz, two finite transformation zone solutions to eqn (8.59) are possible. These limits are shown in Fig. 8.67. The result for the crack spacing d/L = 50 shown in Fig. 8.66 is redrawn in a slightly more explicit form in Fig. 8.68a. The stable region is now above the curve, the unstable region below it, and the curve itself pertains to quasi-static crack growth. For w = 22, the line A-D is indicated in the diagram. This line is followed from A to D as the applied load cr~176 is increased. The part from A to B is in the stable region, and as the load is increased from the point A no crack growth appears. When point B is reached quasi-static crack growth is possible. A further increase in the load will lead to unstable crack growth, as indicated by the broken line between B and C. The derivative of the crack tip stress intensity factor K tip with respect to the applied load a ~176 is positive at the point B, as indicated in Fig. 8.68b. Therefore it is not possible to go

R-Curve Analysis

258

20 15

lI a)

0 ~ 0

dg*/K,

I 0.2

I 0.4

0.6

i 0.8

I 0.2

I 0.4

I 0.6

J 0.8

0.2

0.4

0.6

0.8

?---X

1-~

IJo

~

oo

o Io o

-5 -lO b)

-15 0

{y ~0

xclL, YclL 25 20 15

C

lO

c)

0

0

F i g u r e 8.68: C h a r a c t e r i s t i c r e s u l t s for

d/L-

1-~oo 50

8.6. Steady-State Analysis of an Array of Semi-Intinite Cracks

259

from B to C just by increasing the load on the specimen. If however the situation pertaining to point C is brought about by some other means, quasi-static crack growth is possible at a higher load at C compared to the load at B. Increasing the load from point C towards point D leads to a decrease in the crack tip stress intensity factor, as indicated by the negative derivative in Fig. 8.68b at point C. Therefore a new stable region is reached and the point C is a "superstable" point at which an increase in the load stops crack growth by enhancing the transformation, i.e. the toughening effect of the transformation grows more rapidly than the increase in applied stress intensity factor. Under these circumstances failure will initiate first by divergence of the transformation zones as the applied load ~ approaches the critical load ~0 (see eqn (8.64)) and thereafter by crack growth as the surrounding matrix material loses its ability to enclose the transformation zone. Due to the assumption of no reverse transformation the configuration of larger transformation zones pertaining to the left branch cannot revert to the right branch simply by lowering the applied load, as the derivative of the crack tip stress intensity factor K tip with respect to the applied load ~r~ is positive for fixed transformation zone shapes, as indicated by the dotted line in Fig. 8.68b. The transformation zone boundary intercept with the crack line extension zc and the height of the transformation zone y~ - H associated with the quasi-static solutions of Fig. 8.68a are shown in Fig. 8.68c, with the points B and C indicating the load cases just described. Transformation zone shapes for crack spacing d/L = 50 and various loading ratios ~r are depicted in Fig. 8.69. The toughening ratio Kappz/Ke corresponding to the strengthening ratio of Fig. 8.69 is depicted in Fig. 8.70. In terms of the strength values shown in Fig. 8.66 the toughening ratio is

Kappl K~

-

-

O.c~ I

~/

(to v ~

,.

d

(8

69)

The broken curve is the limiting result for a single semi-infinite crack obtained in w 7.3 and the dotted line pertains to the critical value of the transformation parameter wc given by (8.65). From the above analysis it is evident that diverging transformation zones can exist for transformation strengths less than those expected from conventional lock-up analyses, and crack configurations in transforming ceramics can exist which induce excessive transformation for quite low transformation strengths before crack growth is initiated.

R-Curve Analysis

260

y/L 45 I

o'0/o0= 0.9

|

o.8

3 0.7 2

0"5"6

i

0 0

.4 1

2

3

4

5

6

7

8

x/L

F i g u r e 8.69: Transformation zone shapes for various strengthening ratios, and one crack spacing d/L = 50

Kc/Kappl 1.0 0.8 0.6 0.4 0.2 0.0

0

5

10

15

20

25

30

F i g u r e 8.70: Toughening ratio for array of semi-infinite cracks

8.7. Solution Strategies for Interacting Cracks and Inclusions

261

These circumstances cannot however be brought about simply by loading a precracked transformation toughening ceramic without an initial transformation zone. The latter must be induced by some other means, such as surface grinding or thermal chock.

8.7

Solution Strategies for Interacting Cracks and Inclusions

A numerical method for the integration of the singular integral equation resulting from the interaction of a surface crack with a subsurface inclusion is presented (Andreasen & Karihaloo, 1993b). This examplities the solution method applied in w167 The dislocation density function is partitioned into three parts: A singular term due to the load discontinuity imposed by the inclusion, a square-root singular term from the crack tip, and a bounded and continuous residual term. By integrating the singular terms explicitly the well-behaved residual dislocation density function only has to be determined numerically, together with the intensity of the square-root singular term. The method is applied to the determination of the stress intensity factor for a surface crack growing towards, and through, a circular inclusion. The objective is to provide an accurate numerical solution method for this problem in order to develop solution strategies applicable to the determination of transformation zones of arbitrary shape. In the latter problem, it is imperative to have good control over the singularities contained in the mathematical formulation in order to be able accurately to determine the boundary of the transformed region. The integral equation for determining the dislocation density function contains a number of noticeable features. At the crack tip the solution is square-root singular; at the intersection of the crack-line by the inclusion boundary the solution has a logarithmic singularity, and at the free surface the otherwise Cauchy singular kernel must vanish. All of these features have to be taken into account, if accurate numerical solutions are to be obtained. A widely used and very effective numerical solution method for integral equations with Cauchy singular kernels was given by Erdogan et al. (1972). By means of certain Gauss-quadrature formulas which explicitly take possible singular endpoints into account the integral equations are transformed into a set of linear algebraic equations. The quadrature formulas can be applied directly to the singular integral provided that the collocation points are appropriately chosen. Due to these features

R-Curve Analysis

262

the method is simple to implement and has gained widespread acceptance. A disadvantage of the method is that little freedom is left for choosing collocation and integration points. In the problem at hand the residual dislocation density function may vary rapidly in the vicinity of the point of intersection of the crack-line by the transformation zone boundary, thus control over the position of collocation and integration points is important in order to obtain sufficient numerical accuracy. Another drawback in relation to the present problem is that the common quadrature formulas are not readily applicable to surface crack problems. This can be overcome by symmetric continuation of the singular integral across the free surface (Gupta & Erdogan, 1974), but in general the dislocation density function cannot be continued in a smooth manner, and the numerical accuracy suffers. In the solution method to be described below the accuracy of the solution is of prime concern. Accordingly, the singularities of the problem are isolated and handled analytically in order to avoid any numerical difficulties. As we have seen above, the problem of interaction between a transformation zone and a surface crack reduces to the solution of two coupled singular integral equations, one ensuring zero crack-line stress and the other determining the transformation zone boundary by a critical mean stress criterion. The interaction of crackline by the transformation boundary introduces a discontinuity in crack-line stress. To simplify the discussion, the transformation zone boundary is fixed a priori, and the coupling between the equations is thereby avoided. The crack-line stress due to an arbitrary inclusion can be written as T

O'xx

_

~rT/s ( 3(y + y0) +

+2Y x~176

+

Y - Yo ) - x g + ( y - y0) 2

+

dxo

+

(8.70)

where S is the boundary of the inclusion, see Fig. 8.71. The singular term induces a discontinuity in the crack-line stress imposed by the inclusion. This discontinuity is fixed at 0 " T irrespective of the shape of the inclusion. Therefore without limiting the generality of the analysis to follow, the shape of the inclusion is fixed to be circular, so that the above crack-line stress can be analytically integrated. For a circular region, (8.70) becomes (Mura, 1987)

8.7. Solution Strategies for Interacting Cracks and Inclusions

263

F i g u r e 8.71: Model configuration

T _ ~rT ( ~==

T

r2 4r2y + (~~+ h)~ z r (u-h)~

3r2

(u--h) ~

;)

(s.71)

-lzE

where r is the radius of the circular inclusion and h is the distance from the surface to its centre, h - a + r, and R is the region occupied by the inclusion. The uniform dilatational transformation strain in the inclusion is described through the parameter a T which is given by ~T =

E• T

(8.72)

3(1 - u)

where 0T is the dilatation, E is Young's modulus and u Poisson's ratio. The parameter a T was introduced by Rose (1987a). a T (8.72) equals the crack-line stress discontinuity appearing from (8.71), when it is crossed by the boundary of the inclusion. The crack-line stress from a dislocation can be written as D

Eb

~r~ -

D,x

47r(1 - u 2)

h~x (y, y0)

(8.73)

where the weight function h=nj(y, yo) is given by (6.19). Taking advantage of the central position of the crack (x - x0 - 0), it reduces to

h=D~=(Y, Yo) -

1 +u0

+

6y (u+u0)~

-

4y 2 (~-u0)~

1 u-u0

(s.74)

R-Curve Analysis

264

It should be noted that (8.74) is Cauchy singular and vanishes at the free surface (y = 0). A dislocation density function D(yo) can be introduced such that D(yo)dyo is proportional to the Burgers vector b between y0 and yo +dyo

D(yo)dyo =

Eb 47r(1

-

v 2)

(8.75)

The integral equation determining the dislocation density function

D(yo) for a surface crack which annuls crack-line stress due to the inclusion can now be written from (8.71)-(8.75)

0 - ~r** T+ l

D(yo)g(y, yo)dyo

(8.76)

C

where c is the crack length, and g(y, yo) is given by (8.74). Before proceeding with the numerical inversion of the integral equation (8.76), the singular nature of the dislocation density function D(yo) is discussed in some detail. The displacement jump across the crack faces v(s) near the crack tip can be expanded as v(s) = A181/2 +0(8 3/2) (Barenblatt, 1962), s being the positive distance ahead of the crack tip. The dislocation density function can be obtained by differentiation of the crack face displacement to within a multiplying constant; thus the expansion of the dislocation density function near the crack tip can be written as D(s) - A2s -1/2 + 0(sl/2). A1 and A2 are proportional to the stress intensity factor KI. It should be noted that apart from the inverse square-root singularity, the near-tip expansion implies that the dislocation density function vanishes at the crack tip. At the crack load discontinuity induced by the inclusion, the crack face displacement contains a term proportional to s in Isl (Bilby et al., 1963), which leads to a logarithmic singularity in the dislocation density function with the expansion D(s) = A3 In Isl+O(s~ with s now being the distance from the crack load discontinuity. A3 is proportional to the crack load discontinuity a T . Bearing in mind the singular behaviour, the dislocation density function D(yo) is conveniently written as a sum of three parts, as follows

D(yo ) -

KtiP / -yo 7r 2x/"2~ c + Yo

aTi:+YOln

7r2

+ Yc

Yc - Y0 + Do(yo) c + Y0

(8.77)

The first part gives the square-root singularity pertinent to the stress

8.7. Solution Strategies for Interacting Cracks and Inclusions

265

intensity factor at the crack tip K tip. That this term indeed gives the singularity consistent with the stress intensity factor K tip is seen by expanding the stress O'xxD(8.73)-(8.75) ahead of the crack through the following limit K tip - limr--.0 crD~2X/~-~, where r is the distance on a straight extension of the crack-line. The second part in (8.77) gives a logarithmic singularity at y~, which leads to a crack-line stress discontinuity equal to O"T at yr without violating the near-tip expansion for the dislocation density function, as discussed above. The logarithmic term has a very simple integral formulation which will be exploited later. The last term Do(Yo) is a nonsingular and continuous function. In order not to violate the near-tip expansion the condition D o ( - c ) = 0 is imposed. This condition also ensures that no part of the crack tip singularity in the dislocation density function D(yo) is captured in the residual dislocation density function Do(yo). A more common way of representing a dislocation density function in terms of singular and regular functions is by products rather than sums (Erdogan et al., 1972). The representation chosen here offers some advantages over a product representation in the analytical integrations performed below, and simplifies the transformation of the integral equation (8.76) into an ordinary integral equation with a continuous crack-line load. Introducing the dislocation density function (8.77) into the integral equation (8.76) gives

0 -- O'xxC+ / ~ J ( ~" 2~-~~c-Y~

Yo + D0(Y0))g(y, yo)dyo

(8.78)

with c axx -

T axx

~2

c

+ Y0 log Yc -- Yo g(y, yo)dyo + Yc c + Yo

(8.79)

The unknowns of the singular integral equation (8.78) are the stress intensity factor K tip and the residual dislocation density function Do(yo). The modified crack-line stress r c is bounded and continuous, as it will be demonstrated below. The integral term in (8.79) is discontinuous due to the singularities of the integrand. As is demonstrated below, integration of the logarithmic term together with the Cauchy singularity of (8.79) creates a discontinuity which cancels out the discontinuity induced by the inclusion, thus rendering a~xc continuous over the entire crack. For a better understanding of the subsequent calculations, the crackline stress induced by the dislocations is written as the following limit

R-Curve Analysis

266 along any line z not coinciding with the crack-line C

~r~,:c(y) - ~,--.olimJc D(Y~ (g'~ (Y' Y~ + Re z --iiyo } ) dyo (8.80) where the Cauchy singular term of the weight function g(y, yo) (8.74) has been separated out, such that gn,(y, yo) is nonsingular, i.e. g(y, yo)-

gn,(y, Yo) = 1/(y - Yo).

The logarithmic part of the dislocation density function (8.77) is conveniently rewritten in an integral form as

f (Yo ) log Yc - Yo -- f(Yo) c+yo

Yp _ y--------~dyp

O"T ~/C + Y0

f(Yo)- ---~

(8.81)

c + Yc

where the function f(yo) is finite and differentiable. Introducing (8.81) into (8.80) and for the moment disregarding the nonsingular part of the weight function gns(y, y0), the crack-line stress can be written as the limit of a double integral lim

-i

x ~ o

as

c

c

Yp

-

Yo

z

-

i yo

Changing the order of integration, the integral (8.82) can be rewritten lim Re x~o

/v__.~jo__ f(yo) 1 ~dyodyp c Y p - yo iz + yo

= lim Re { J j ~

~-o +

1

~ iz + yp

vc 1 c iz+yp

f(yp)

c

yp - yo

cYp-Yo

+ f(y)

f(Y_O)-zz+ Yof(Y)) dyodyp} ciz+yo

dyp

(8.83)

f(y) is the value at y for z tending to zero along any path z. Provided that f(y) is bounded and continuous the first double integral is real and nonsingular. To see this, consider the integrands of the inner integral. These integrands are continuous and differentiable by virtue of the properties of f(y). Formally integrating the inner integral shows by the same reasoning that the integrand of the outer integral is continuous as

8.7. Solution Strategies for Interacting Cracks and Inclusions

267

well. From the fundamental theory of Cauchy integrals (Gakhov, 1966) it follows that in this case the limiting process and the integrations can be interchanged, provided that only the real part is needed, as in the present case. Adding and subtracting the Cauchy principal value integral for x = 0 of the last double integral in the above equality, gives

v__~j: f(Yo) ~ d 1y o d y p c c Yp - Yo iz + Yo

lim Re

x~o f~

-1

+jffr r

(f(Yo)_-f(Yp)f(Yo)_-f(Y))

f -1

J;

f

+f(y) lim Re f ~ ~o

9__ dyo ) c Y - Yo dyp

dyo

vo - I ( Y )

yp -

l

~ i z + yp

f ( 1 ~

i z + yo

+

Y-

1

Yo

) dypdyo (8.84)

By similar reasoning as applied above, the second integral is seen to be nonsingular. The discontinuity induced by the logarithmic part of the dislocation density function can now be obtained by carrying out the integrations of the singular double integral and taking the limit as follows fc 1 j_~ 1 1 ) f(y) lim Re + dyodyp

~o

~ iz + yp

= f(y) lim Re f ~ 9~ o

= f(y) lim Re x---.o

1 ~ iz+yp

~ iz + yo

( log ~ iz

-iz+c

Y - Yo

+ i~r - log - y ) y+c

dyp

f c l ( iz -y) log - log dyp c iz + yp -iz + c y+c

+ f (Y) ~-,olimRe { ilr l~ iz +- yrc

=f(v)

-~r 2

-c<_y
- 2/2 0

v-v yc
(8.85)

268

R - C u r v e Analysis

which through (8.81) gives a crack-line stress discontinuity opposite to that induced by the inclusion. The discontinuity can be obtained simply by integrating (8.82) not in the principal value sense but by retaining first the imaginary terms and then taking the real part only. This gives the correct discontinuity but the value at y = yr is not obtained. This value can be important if the transformation zone boundary is to be determined as a part of the problem. The full expression for the crack-line stress (8.79) induced by the logarithmic singularity can be written as c

~

T _

-- ~

y)

~r2x/c + yc

+[1 + 6y

r

+ 4y2~y2]A(c, yp, - y )

whereA= 1 fory
dyp - mv/c + y (8.86)

1/2 f o r y = y r

1 ( x/c + 7/log

A(c, 7/, ~) - 7/- ~

(ssT)

-V/c + ~ log

The integral has been transformed from one in y0 from - c to 0 with the logarithmic singular integrand (8.79) to a regular integral in yp from - c to yc (8.86). In order to invert the integral equation (8.78) the residual dislocation density function Do(yo) is linearly interpolated so that it is transformed into a system of linear algebraic equations i~ti p

0 __ O.xxC(Yi) +

~~-~F(yi)r

n + E Do(Yk)Vk(Yi)

k=l

(8.88)

The functions F(y) and Vk(y) are obtained by analytical integrations as given below. The system (8.88) consists of n + 1 linear equations with unknowns Do(yi), i = 1...n and K tip. The inversion of the integral equation (8.78) can now be performed simply by Gaussian elimination of the system of linear equations (8.88). The function r(y) in (8.88) is obtained by analytical integration of

8.7.

269

S o l u t i o n S t r a t e g i e s for I n t e r a c t i n g Cracks a n d Inclusions

square-root singular part of the dislocation density function (8.78) and the weight function (8.74) as F(y)

1 f~ -

./-Yo

dyo-

u)

y(2y2-6yc+7c

-~ j _ ~ g ( v , vo ) V c + vo

2(c - v) ~ v / - v ( c

- v)

(S.89)

T h e functions Vk(y) in (8.88) are obtained by linear interpolation of the residual dislocation density function and analytical integration of the weight function (8.74) as follows vk(v) -

yk vo - v k - ~ vk-~ Yk - Yk-1

g(Y, yo )dyo

+

Y - Yk-1 log Yk - Y Yk - Yk- 1 Yk-1 - y 7y+yk_llog Yk -- Yk-1

jfyyk+l Yo Y k + l g(y, yo )dyo k Yk - Yk+l

Y - Yk+l log Yk - Y Yk - Yk+l Yk+l -- Y

Yk+Y _ 7y+yk+llog Yk-1 + y Yk -- Yk+l

Yk+Y Yk+l + Y

yk+l - yk-1

(s.90)

__,.,y2O (Yk-1 + Y)(Yk + Y)(Yk+I -~- Y)

T h e integration and collocation points are n u m b e r e d from the crack tip towards the free surface. Defining yk = - c for k = 0 and yk = 0 for k = n + 1 formula (8.88) holds for k = 1 and the condition D o ( - c ) = 0 is a u t o m a t i c a l l y satisfied, but for k = n - 1, n, i.e. at the surface, additional t e r m s appear if the residual dislocation density function is linearly e x t r a p o l a t e d from the last integration point to the surface. T h e additional terms A Vn-1 and A Vn are to be added to the corresponding functions from (8.90)

A Yrt-1 --

A V,~ -

-1 Yn -- Y n -

1

1 Y,~ - Y,~- 1

(Yo - Y,~ )g(y, yo )dyo - (y,~ - y ) l o g

,., 0

(vo - v,~ )g(v, vo )dvo

Yn Y -- Yn Y

(Y0 - Y,~)g(Y, yo)dyo

270

R-Curve Analysis - ( 7 y + yn)log y + y" + 6y,~ y

2y~ (8.91) Y+ Y.

This concludes the problem of inverting the integral equation (8.78) for a surface crack with a continuous crack load. If collocation points are placed at the midpoints of the integration points used in (8.90), the numerical implementation is somewhat simplified because the integration points do not coincide with the collocation points, the special points at the surface, the point at which the transformation zone crosses the crack-line or the crack tip where otherwise limiting processes may be needed for determining function values.

271

Chapter 9

Three-Dimensional Transformation Toughening 9.1

Introduction

In the preceding two Chapters, the problem of transformation toughening was considered in two dimensions. This two-dimensional plane strain approximation to what is essentially a discrete, three-dimensional problem is adequate if the number of transformed particles is large but unlikely to be valid if the transformed zone spans only a few particles. The three-dimensional approach is also required if the remote loading contains mode II and mode III components, besides that of mode I. Fortunately, some progress has already been made in this direction, thanks to the recent availability of three-dimensional "weight functions" (Bueckner, 1987). These weight functions play the same role in a threedimensional body containing a half-plane crack, say, as do the Green functions (Chapter 4) in an uncracked body. It should be mentioned that Rice (1985a) had given exact analytical expressions for the stress intensity factor in mode I due to threedimensional elastic interactions between a half-plane crack and a source of internal stress such as transformation strains. Rice's treatment was generalized to modes I, II and III by Gao (1989) using the three-dimensional weight functions of Bueckner (1987). However, the pioneering work reported by Rice (1985a) and Gao (1989) is restricted to dilata-

272

Three-Dimensional Transformation Toughening

tional transformation strains for which the shape of transforming particles is irrelevant. As we have seen, transformation shear strains and shape and orientation of the transforming particles also have a significant effect upon transformation toughening. These have been considered by Karihaloo & Huang (1989), from which we borrow much material for this Chapter. We will give analytical expressions for stress intensity factors induced along a half-plane crack front by unconstrained dilatational and shear components of transformation strain for several simple shapes of the transforming particles using Bueckner's three-dimensional weight functions. When only shear strains are involved analytical treatment is practicable only for a transformation domain centred at the origin. In all other cases the analytical expressions are too cumbersome to be of much practical value. In such instances, numerical results are presented for transformation domain in the shape of a sphere and a spheroid. The influence of the orientation of an oblate spheroid relative to the half-plane crack upon the transformation toughening will be highlighted.

9.2

Three-Dimensional Weight Functions

For an infinite elastic body with a half-plane crack in the plane y = 0 (its front is parallel to z - a x i s along x = 0, so that the region x < 0 is cracked, Fig. 9.1), Rice (1985a) shows that the stress intensity factors K~(z ~) at location z ~ along the crack front (a = I, II, I I I refer to loading modes) due to an arbitrary distribution of body forces f - fij(x, y, z) per unit volume are given by

K,~(z') - Iv ha(x, y, z - z ' ) f dV,

(9.1)

where dV denotes an element of volume and the integral extends over all loaded elements. The functions ha = haj(x, y, z - z I) are called weight functions and they are universal for a given geometry of cracked body in the sense of having no dependence on the particular distribution of fij. They give the mode a stress intensity factor induced at the location z I along the crack front due to a unit point force in the j-th direction at the position x, y, z. We now consider the cracked body with transformation strains c T throughout a region V, where the latin indices (excepting x, y and z, if they appear explicitly)stand for x, y, or z. The transformation strains are understood in the sense that

9.2.

Three-Dimensional Weight Functions

273

Y

/

X

C-

V

F i g u r e 9.1" A transformable volume near a half-plane crack in an infinite body

,~

-

c~jkz (uk,~ - c ~ ) ,

(9.2)

where

Cijkz (,~, #) = ~ 6ij 3~l + # (6i~ 6jl + 3iz 6jk)

(9.3)

is the three-dimensional elasticity tensor (with v = ~/[2(~+#)]) which is assumed to remain unaltered by transformation. By Eshelby's equivalent inclusion technique (a succinct description may be found in a paper by Karihaloo & Viswanathan, 1988) it follows that the displacement field u produced by c T is the same as that produced in an identical body with c T - 0 subject to certain effective body forces over V, i.e. in V

~ , ~ + f~ -

(c~kz uk,z),~ + f~ - ( c ~ k t

c~),~ - 0.

(9.4)

On traction-free elements of the boundary having outer normal n T~ -

nj ~

-

n~ C ~ k ~ u k , ~

-

njC~j~z~T~

--

O.

(9.5)

Thus the displacement field produced by c T is the same as induced by an effective body force distribution

Three-Dimensional Transformation Toughening

274

f f ly - -(Cijkt cTz),j

(9.6)

throughout V plus a Dirac singular layer of effective tractions along the crack surfaces

Ti ~ly - nj Ci.ikt cT .

(9.7)

Inserting (9.6)into (9.1)and applying the divergence theorem (which results in the cancellation of the surface integral along the traction-free crack surfaces), gives

K~(z')

I v h ~ j , i ( x , y , z - z')CijklcTtdxdydz,

(9.8)

where the diffrentiation in haj,i is with respect to i = x, y, or z. For isotropic materials, (9.8) may be further simplified to read

K~(z') - 2p Iv U,~n (x, y, z - z') Vm,~T(x, y, z) dx dy dz,

(9.9)

where the fundamental stress field in mode a is defined by

2p U,~,~ = Cmnij h,~i,j,

(9.10)

with U ~ . - ~ l(h,~m,. + h,~. m) + ~ S m n (1 - 2u)

h~j,j

(9.11)

Note that if the crack front is located at x = a, y = yl so that an observation point along the crack front would have coordinates a, yl, z' then U~,~ (x, y, z - z') must be replaced with U~n (x - a, y - y', z - z') because of the translational invariance of the weight functions. Expressions for Um~n are given below (Gao, 1989) and will be further developed in the following. An explicit solution for hlj was given by Rice (1985a). Recently, Bueckner (1987) derived h~j for all modes in terms of a Papkovitch-Neuber potential function G(x, y, z)

G(x,y,z-

z') - ~Co ln( qq -+ ~F)

(9.12)

where

- v/[x + i(z - z')],

q - Re ~/[2(x + iy)] - ~ / ~ cos

(9.13)

9.2. T h r e e - D i m e n s i o n a l W e i g h t Functions

275

and -

~/~(x2 + y2),

tanr - y/x,

Co - - 1 / [ 4 ( 1 -

v)lr a/2] (9.14)

and r represent polar co-ordinates in the (x, y) plane such that the crack faces C + and C - can be distinguished by r = ~r,-~r, respectively (as in Fig. 9.1). In terms of G ( x , y, z - z'), the x, y, z components of h~j are hl~ = - ( 1

- 2 u ) G , : ~ - yG,~ u

hlu = 2(1 - u)G,y - yG,u u hlz = - ( 1 - 2u)G,z - yG,zu

(9.15)

h~. = - 2 ( 1 - v)g~ + y ~ , . hey = - ( 1 - 2 ~ ) ~ + Y~t~,y

fl = / / , / I /

h~z = - 2 ( 1 - t~)h~ + yq~,z

(9.16)

where ( 2 - t~)gli = - G , y + 2(1 - v)H,r + 2iH, z (2 - t~)hii = - i ( G y (2- v)~ii =-G,~

+ 2 H , ~ ) + 2(1 - v)S,~ - iG ~ - 2(1 - t,)U,y

(9.17)

(2 - tJ)glii = i(1 - v)(G,y + 2H,~) - 2H,z (2-

tP)hIII -- - ( 1 - v)G,y + 2H,~ + 2i(1 - t~)H,z

(2 - t~)r

= i(1 - t,)(G,~ + iG,z - 2 U y )

(9.18)

and H = yG,~: - xG,y

Define p(x, y) -

Im { ~/2(x + iy) } - X / ~ sin(e/2)

Using the relations q,z = q/28,

q,u = p / 2 8 ,

P,:c = - p / 2 8

(9.19)

276

T h r e e - D i m e n s i o n a l Transformation T o u g h e n i n g

(,. = 1/2t~,

p,y = q/20,

(,z = i/2~

and bearing in mind that

a = a[q(~,

~), ~(~, z)]

we define two functions which will be used in the sequel P-

G,y - G q(p/20) - C0x/~ Im[(x + iY)-l/2] ' O-i(zz')

Q-

G . + iG z - G q(q/20) - -Cox/"2 Re[(x + iy)-1/2] ' ' ' ~-i(z-z')

(9 20) "

In terms of the above functions, U,~ n can be represented as follows: u~ --a~

- 2~G~ - uP,~

U[z - - G , z z

- 2uG,:~.- yPzz

U~y - - Y P , xu U~'z - - v P ~ U[:~ - - ( 1

U~x - - 2 ( 1

- 2u)G,~z - yP,.~

(9.21)

- u ) g z , x - 2u~Z,y + y ~ , x x

C y - Y~ Z,yy U~zz = - 2 ( 1 - ~,)h~,z - 2 ~ , ~ , ~ + ~ , ~

U~z - - ( 1 - u)h~,~ + ~ , ~ , z + y ~ , y z

u L = - ( 1 - ~)(g,,~ + h , , ~ ) + ~ , , ~ z

(9.22)

U~,~ (~ = I I , I I I , m r n) can be written in terms of functions P, Q and

and their partial derivatives:

9.2.

Three-Dimensional Weight Functions

277

Uz1~(2 - u) - - y [(5 - 4u)Q,z~ - 4(1 - u)2Py~ - 2 ( 1 - u ) P ~ z r + 2(1 - u ) i P x y ] - ( 1 - u)x [ - 4 ( 1 - u)P,~z + 2iPx~ - 2iPz~] + ( 1 - u)[(5 - 4 u ) P z - 3iP,~] + 2 ( 1 - u)(1 - 2 u ) y G , . . . --(1 -- u){[--3(1 -- u ) i P z - ( 5 - u ) P . ]

U.zHZ(2 -- p ) -

+ y [ - 2 i P ~ z r - (4 - 4u)iP, yz - 2P:~y + 3iQ,~z]

+~[-4(1 - -)p~z + 2P~ - 2P~] - 2 i y ( 1 - 2.)C,z~z }

U/~(2 - u ) -

(2 - u)B2~y - y[2(1 - u ) P ~

u~"~(2 - ~,)- (2- .)B~ y II

Uuz (2 - u ) -

(9.23)

+ uQ,~y]

- ~2(1 - . ) p ~

(2 - u)B2yz - y[2(1 - u)P,z~ + uQ,uz ]

IIZ(9 - u)Z \--

(2-

u)B3u Z - y2(1 - u ) P z, z

(9.24)

T h e derivatives of h a r m o n i c functions P, Q, as well as B2~u, B2yz, B3~u and B3yz can be expressed a n a l y t i c a l l y in t e r m s of ~, r R, as follows:

B2xy

Coq

=-

2~R2(2-

u)

{

(2 - 3~) + ~

}

[ - 8 + 8~ + (8 - 4 ~ ) c o s r

-2u Coq(zB2~z = 2 - u

B3xy

--

z')~, 2 ~o2 R 2 R 2

- 2 u ( 1 - u) Coq(z - z')o 2

B3y z -- _

2- u

- ( 1 - .)C0q

OR2R 2

(2 + . ) +

[ - s + (s - 4 . ) cos r

(9.25)

T h e first order and m i x e d second order derivatives of P and Q are

Three-DimensionM Transformation Toughening

278

_

[1 + 4~--~ cosr ]

Cop P . - 20R 2

+ i Cop( z - z') [ 202 R 2

Coq 20R 2

Q~X ---

1- 4

"-~

_i Coq(z - z') [ 202 R2

Coq

By --

1-4

20R 2

02

1 + 2 cos r + 4 ~

Cop[

Q,y = 20R2

02 ] 1 - 2 cos r - 4 ~ cos r (1 - cosr

'~

, =

02

1 - 4~--7(1 + c o s r

2Cop(z - z') R4

Q,z 2Coq(z-z') =

R4

Cop

P,yr = 40R2

1 +4

Cop[

+i~-ff

Q,zr = -

]

d

]

] + 4~--~(1 + cosr

02 ]

1 - 2~--ff

c0q[1 - 2 ~---~ 02]

+i~-~

~-ff(

+i Cop(z - z') [ 402R 2

o2

1 - 2 cos r + 4 ~---ff(1 - cos r

+iCop(zz') [1 + 2cosr 202R 2 p~

]

cos r

+i Coq(z - z') [ 202 R2

cos r

1 + 3cosr

02

3 + 6cosr + 4~-7(1 + 3 cos r

Cop(z- z ~) Cop [ 02 ] R4 + i 20R2 -1 + 2--~

Second order derivatives of P and Q are as follows"

]

(9.26)

9.2.

Three-Dimensional Weight Functions

Q,yz _ Cop(z - z') [1 --

oR4

-i, Cop 202R2 ~4

- 8~

( 1 + cos

1+2cosr

--~

r

279

]

(5+4cosr

- 16~--~-(1 + cos r

p~

_ C0q(z-z') [1 - 8

--

0R 4

(1-r162

]

202R 2 - l + 2 c o s r 1 6 2 1 6 2 04

+16~(1 - cone) Cop

(1+2~osr

Q,.y = 4 e 2 R 2

-32~--~i- cos r

_iCop(z- z') [ 403 R2

(1+2~osr + cos r

- 3(1 - 2 cos r - 4 cos 2 r

~o2 -4~---~-(1 - 2 cos r

p ~y =

Coq [

4~o2R2

4cos 2 r

+ cosr

]

+ 4~--~(1 - 2 c o s r

+ 3 2~~ r162 - ~osr 4~o3R2

~o4 + 32~-~- cos r

~2

(1 - 2 c o s r

+iCoq(z- z') [

]

]

- 3(1 + 2 cos r - 4 cos 2 r

~o2 - 4 ~ (1 + 2 cos r - 4 cos 2 r

~o4 - 32 ~-~ cos r

- cos r

]

Three-Dimensional Transformation Toughening

280

Q ,zlr, ~"

1 - 2~osr

+i202R 2

2~(1

+4~o~r

16~cosr

, __ C o poR( z -4 z') [1 + 8 --~~ cos r ]

P z~

-

-

cop[ 2Coq [

~4 ] 02 ] Coq(z- z') [ 4_~7 0~]

+i202R 2 -1-2cosr162162

Q~ZZ ~

PZZ ~

P~XX ~"

/i~4

-3+4~--~

+ 2i

~2

coR4

-1 +

[ ~] ~o~,z_z,,[ ~]

2Cop - 3 + 4 R4 ~ Cop

402R 2

+2i

[1 + 2 cos r -

-1+4

oR 4

02

e4

8~-5(1 - cos r + 32~-~ cos 2 r

]

+iCop,z4Q 3R2( - z') ['[3(1 - 2cosr - 4cos 2 r

~02

+ 8~

Pyy =

04

(1 - ~os r - 2 ~os 2 r - 32 ~

1 + 2cosr + 8~-ff(2 + cosr

~os 2 r

]

- 32~--~ sin 2 r

- i Cop(z - z') [3(1 - 2 cos r - 4 cos 2 r 4~o3R2

[

~02

-

04

]

(l+3cosr

]

8~-7(cos r + 2 cos 2 r + 32~-~ sin 2 r

(9.27)

Third order mixed derivatives are"

Cop ( z - z ' )

Pyzr - -

2eR4

-i

Cop 4~o2R2

[

[ ~ 1+8

~e

3(1 + 2 cos r + 6--~(1 + 4 cos r

]

9.2. Three-Dimensional Weight Functions

281

~4

-16~--~(1 + 3 cos r

Pxyr

-

-

Cop 8~02R2

[3(1 + 2 cos r

-32~(2

+i

12~-~(1 + 2 cos r

- ~os r - 5 cos 2

Cop(zz') [15(1 ~5~2

r

]

2cosr - 4cos ~ r

Q2 +20 ~--5(1 - 2 cos r - 4 cos 2 r +32~(2

P ZX(/) ~

--

- cos r - 5 cos 2 r

C o q ( z - z') 1 - 8 2eR 4

Coq

4e2R 2

3(i - 2 cos

]

( 2 - 3cosr r + 6 ~-~(3 - 4 cos r

+ 16 ~--~-(-2 + 3 cos r

(9.28)

The real and imaginary parts of G,zzz which are contained only in U~z ,Ufx and UzZz (~ - II, III) as well as the real part of G,zz contained in UIxz,UIx and UIzz cannot be fully expressed analytically. However, many simplifications are possible as shown below. For an arbitrary domain V, U~z contains the term Re G,~z which may be written in a non-dimensional form using (9.20) ReG~z Z') , = - R e Q ,z + I m G ,zz _ 2 c o q ( Z -R4

+ ImG,zz (9.29)

Likewise, U/zI and UI1zI contain the terms Re G,~:cz and ImG,~xz, respectively, which may be rewritten as Re G,xxz - - R e Pyz - Re G,zzz

Three-Dimensional Transformation Toughening

282

=_coq(Z-

Z') {

R4

~o2

1 -- 8~-5(1 -- cos r

}

-- Re G,~.zz (9.30)

Im G,xxz = - I m P, yz - Im G,zzz = - C o 2~2R q 2 { (-1 + 2cosr

( 1 0 - 8cosr

~~

- ImG,zzz

+ 16(1 - cosr

(9.31)

To complete the discussion, it only remains to express explicitly G,zz and G,zzz. We begin with G,~z. First, note the following identities" 1

(q2

~ + i(z - z')~

~2)~4

_

-(z

- z') ~ - 2 i ( z - z ' ) ~

R2 1

R2d 2 +

{ ~(~

d4 _ R ~ + ~)+

2~(R ~ -

~)\

d2

f

~

~2 _ (z - z') 2 + 2i~(z - z') x - i(z - z') R4 d2

(q2 _ ~2)2~2

~(~2 _ R 2 + ~2) + 2 ~ ( R 2 _ e2)

d 2R 4

+i(z - z ' ) [ 2 ~ x - ~o2 + R ~ - Q2] d 2R4

(9.32)

Now, C,~ - C,~,~,~z + C,~,~,~ ----

Coq Coq d2R2

+

3Coq

3G

2~4(q2 _~2)

4~4

~o2 (20- x)+ 2~(x-

3 x2 01 - ~ [ ( 2 x - 01 + 2 ~ ( - x

+ ~1]

}

Three-DimensionalWeight Functions

9.2.

_iCoq(z~2 d2R2 z') { 1 - 2~-ff(1 3 02 + ~ [( 1 - 2 ~-~ cos r

283

- cosr r - 1)]

}

3G 4~4

(9.33)

where

=

~ / ~ + (z - z')~

(9.34)

and

Re-~4- Co < Rew Re -~s - Imw Im ~ } Im~4 - Co { Rew Im ~ + Imw Im ~xs}

(9.35)

In (9.35) w

-

q-~

such that ~,

c(~,

and

z -

(9.36)

In q + ~

z')

-

Co ~-~

(9.37)

1 q2 +d+qv/2(d+ x,) q2 + d- qv/2(d + x,)

Rew - ~ In

Im w = sin- 1

1 Re ~5

2q Im v/(q 2 - d) 2 + (2q Imp) 2

i { ( d~dX)l/2[x2 - (z -- d4d1/2

(9.38)

z,)2]

+2(d2d )i/2x(z- z')} -

1

Im ~-g

r

1 {_2(d+ x l12z

-

d4d1/2

2d )

(z-

z,

+(d~ ~)1/~[~- (: - :')~]}

)] (9.39)

284

Three-Dimensional Transformation Toughening

where the negative sign is chosen when ( z - z~) < 0. Finally, G,zz~ is given by

2Coqz(~0{7 4x) + ( ~1 2+ ~-2)(202)(x - 0) }

a,~zz = d2R4

2Coqz{7 1 2 } + d4R2 (~0- 4x) + (~--ff+ ~--ff)(-3x2)(0- x) -~ - 2~-7(1 - c o s r

-id2R2

2Coqz2{5 02 +i d2R4 ~ - 2 ( 1 - c o s r 2Coqz 2 {

+i d4R2

5

~ - 3o ~ o s r 1 6 2

3~cosr162 1

1)

2)}

2}

1 + -~)

- 1)(~

3Coq { x2 2] z, 2]} -i 4d4R 2 4~--ff[0x+(z-z') - [ 3 0 x + ( z )

3Coq(z-z'){(0 - 3x) + 4~-(x x2 } 85G - 0) + i--~-g

+~

d4R2

(9.40)

where

and

R e ~ - Co RewRe

-ImwIm

I m ~ - Co RewIm

-ImwRe

(9.41)

1 1 { d -.]-x )1/2[X 3 _ 3x(z - z') 2] Re ~ - d6dl/2 ( 2d +(

1

2d

1 {

)1/2(z

-

-

z')[3x2-(z - z') 2]

}

d+x

Im ,~--7- d6d~/2 -( 2d )~/2(z - z')[3x2 - ( z - z') ~] +( d2d-x )1/2[x3 _ 3x(z- z')2] }

(9.42)

9.3. Dilatational Transformation Strains

285

Gao (1989) has confirmed the accuracy of three-dimensional weight functions by deducing the two-dimensional weight functions via an integration with respect to z from -oo to oc and comparing the results with existing solutions in the literature (Bueckner, 1970; Tada et al., 1985; Hutchinson, 1974). 9.3

Dilatational

Transformation

Strains

For pure dilatational transformation strains T gmn

where

~mn

(x, Y, z) - 0T ( x , y, z) ran 3

(9.43)

is the Kronecker delta, eqn (9.9) is considerably simplified

Ky _ ~2tt Iv UiJ~ OT (x, y, z)dx dy dz,

(9.44)

where superscript D identifies dilatation, and Uj~ are given by

2(1 + v )

Uj/j

A0

Uj~-

ao2

c~

[1-8~-'~2 sin2 (2r

2(1 + v)

A----~ g~Z,u (13 - II, III)

-

(9.45)

where

~ll,u ~lll,y

-

--

sin(C)[2_3v 4~, (z - z')

R2

+ 8

#2 (

c o s(~

)r }

]1

sin ~ ) .

-

v

(9.46)

In (9.45)-(9.46), R 2 - x 2 +y2 +(z_z,)2 and A0 - 2(l-v)(2~r)~ ~89R 2. If we now assume that 0T is independent of z and note that /)~

1

r

- ~ d z - -0;

/))

1

7r

--~ dz - 203

and integrate (9.45) from -(x~ to +cx) we find

(9.47)

Three-Dimensional Transformation Toughening

286

gi

--

~-~ cos ( r3~

3(1# (- l +uu) )~ / ~ /A

oT(x,y)dxdy

xy

KII

~

-

+ u) 3(1p(1 - u) x/~

/A

0-~ sin

( r3

OT(x, y) dx dy

xy

KHI

=0,

(9.48)

where Axu is the cross-section of the transformed area. If the transformed area is symmetric about y = 0, then KII will also vanish, but KI given by (9.48) can be doubled with the area of transformation region halved to that above y = 0. This last result is in complete agreement with (7.10) for the increment in mode I stress intensity factor induced by transformation.

9.4

Shear Transformation Strains

We will now obtain the solution for unconstrained shear transformation _ CnmT(n :fim), for which eqn (9.9) reduces to strains with r

KS~(z') - 4P /v Ur~n CmnT dxdydz

(m~-n)

(9.49)

where we have used the superscript S to identify the pure shear contribution to the stress intensity factors. If K appz denotes the mode a stress intensity factor induced at the crack front by external loading, then under any arbitrary (internal) transformation strain field and external loading the stress intensity factors may be obtained by superposition

K~(z') - K~PV'(z') + KD(z') + KS(z')

(9.50)

The calculation of stress intensity factors due to unconstrained shear transformation strains presents several difficulties because the derivatives of G(x, y, z- z') with respect to x and z such as G,~z, G,~:xz and therefore the functions Ui~(i,j r y)involve real and/or imaginary parts of log-like complex functions. General expressions for G,xz and G,xxz were given above (9.29)-(9.31). The expressions (9.49) may be formally written in the following nondimensional form

9.4. Shear

Transformation Strains

KT(z') - (1 + .) +4_~2

287

I

% 4oR~ (1 - ~osr - 2~os ~ r 04

( 1 - ~os r - 2 cos 2 r + 32 ~

r Cov(z- z') [

-cy~

R4

R4

(1 + v)

-

8~--ff(1 - cos 2 r

]

o2

dV

(9.51)

7v)cosr - 2(2 - 3v)cos 2 r

+32~--i(v cos r + 2(1 - v)cos 2 r - (2 - v)cos 3 r

T Coq(z- z') [

-C~z (2 -

~,)o2R~ . ( - 2

+ 5~, + (2 - 3~,) cos r

- 8 ~ - i ( v + 2(1 - v)cos r - (2 - v)cos 2 r

+~L 4Cov(zz') [ o~ ~ - ~,~2R2 4~--ff(4 -

T 2(1 -- v)(1 -- 2v) 2- v y Re

(1 + ~,)

%

]

e2

]

13v + 8v 2 + v(3 - 4 v ) c o s r

+32~--i-(v cos r + v(5 - 4v) cos 2 r + vcos 3 r

-

]

I

Q2

KSii(z ')

]

c~,~ 40R2(2 _ v) (2 - 3 v ) ( - 1 + cosr - 2cos 2 r

+ 4 ~ 5 - ( - v + (4 -

-r

2r

(1 - cos r + 8 ~--2-(cos r - cos ~ r

--CzT(1 -- 2V)ReG,~z}

KSi(z ')

02

(1 + cos r

+~r Coq(z- ~') [

cos r 1 - r

]

}

G,:e~z dV

( 2 - ~,)o~R~

( 2 - 2~,- 2r162

(9.52)

Three-Dimensional Transformation Toughening

288

+ 1 6 ~ 4 (cos r ~ ~T

cos 2 r

C0q(1 11) - u)OR 2

Y~ 2 ( 2

(1

_

[(2 +.)+ (-32 + (32-

cos 0)]

7" C 0 p ( 1 - v ) [ -ez~ 2(2 - u)oR 2 . - 5u - (2 - 4u) cosr - (4 - 8u) cos 2 r +~-~ff2( - 4 + 12u - (28 + 4 u ) c o s r - (16 - 32u) cos 2 r

:

+32~-i(cos r + (1 - 2u)cos 2 r

]

+ezT 2(1 - v)(1 - 2u) } 2- u yIm(G,x~)

dV

(9.53)

To arrive at the non-dimensional form of stress intensity factors (9.51) - (9.53), we first introduced the non-dimensional variables identified by an asterisk

x*--x/a, K;S (z *')

-

y*--y/a,

z*--z/a,

KS (z')/(Ex/~)

dV* = dV/a 3, a3

G,*r. ~. - ~ G , ~ . ,

a4

G,*~.r.~. -

~G,~.~,

(9.54)

and then for brevity omitted the asterisk. Calculation of K s for any arbitrary V involves complicated in-

tegrands such as ICe(G,~), I m ( G . . . ) a n d

Im(G~)

((9.40)-(9.42))

which require numerical integration. However, the last term in the expression for KS(z ') will vanish if s T is a function of y alone. Likewise, the last term in the expressions for KS1(z ') and K/S11(z ') will vanish provided either ez~T is a function of y alone or e T~ is an even function of y and the region V is symmetric with respect to y. The expressions are further simplified if the unconstrained shear

9.4. Shear Transformation Strains

289

transformation strains are functions of y alone and the region V is symmetric with respect to y. In this case, besides the last term involving the derivatives of G(x, y, z), the terms containing the variable p = v/~sin(r also vanish.

9.4.1

Simple

Transformation

Domains

l ay/b

r rl(~.O )

v x c

F i g u r e 9.2: a in cartesian and ellipsoidal coordinate systems As mentioned above, for an arbitrary transformation domain V the integrals (9.51)-(9.53)must be evaluated numerically. In several instances however it is possible to express the integrals as infinite series, provided there is no contribution from the terms involving the derivatives of G(x, y, z). In this Section, we will first demonstrate this "analytical" procedure on the spheroidal region (Fig. 9.2)

+ u2 +

a2

z0) < 1

(9.55)

and then obtain complete solutions for the spheroidal region centred at the origin (x0 - y0 - z0 - 0). For the spheroidal region (9.55), and indeed for an arbitrary ellipsoidal region to be considered numerically in the next Section, we introduce ellipsoidal coordinates r, 9, r (Figs. 9.3, 9.4)

Three-Dimensional Transformation Toughening

290

m

x "

7

m

Figure yo - 0

9.3:

R a n g e of variation of r in the plane z -

zo for zo -

Xo

azo/C

=x0 +

F i g u r e 9.4: R a n g e of v a r i a t i o n of/9 in the plane y -

y0 - 0

0,

9.4. Shear Transformation Strains

x -

291

r c o s 0 cos r

b y - - r cos 0 sin r a

C

z - -rsin 9

(9.56)

a

such that in general the coordinates of the centroid of region V, x0, y0, z0 transform into r0,90, r We have used an overbar to distinguish the ellipsoidal coordinate r from the cylindrical coordinate r appearing in integrals (9.51)-(9.53) and elsewhere. r is related to r through b2

cosr -- c o s r 1 6 2 2 4 7

~-~sin2r 1/~

(9.57)

For a spheroidal region, r - r The lower (l) and upper (u) limits of integration for an arbitrary ellipsoidal region are calculated as follows. For the radius r -

(

a2

a2 ) 1/2

x2 + ~_y2 + ~ z 2

(9.58)

we have rl _< r _< ru

(9.59)

where ru,z -

/

_

r 0 c o s a : t : ~ / [ ( r 0 c o s a ) 2 - ( r ~ - 1)]

(9.60)

and c~ is the angle between radius vectors ro and r such that (Fig. 9.2): cos c~ -

cos Ocos 00 cos(r - r

+ sin Osin 80

(9.61)

For O0 _ 1 (where O0 - [(x02)+ (ayo/b)2]l/2; not to be confused with cylindrical coordinate ~ - (x 2 + y2)1/2 appearing in integrals (9.51)(9.53) and elsewhere), m

F0-~-

m

r162162

-

F0+~

(9.62)

where _ ~sin-l(1/00);

( 0;

00 _> 1 00 < 1

(9.63)

Three-Dimensional Transformation Toughening

292

ayo

xo>O

t a n - 1 (b-~0); Fo --

ayo

t a n - l ( b - ~ a ) -4- 7r;

xo < O, Yo > 0

ayo

(9.64)

xo < O , y o < O

t a n - 1 (b-~o) - 7r;

For 80 < 1, we h a v e - T r < r < 7r. Finally, the limits on 0 are calculated from the solution to the inequality, resulting from (9.61) 1 - %2sin s a >_ 0

(9.65)

We note first t h a t

Oo

O;

zo -

7r/2;

oo - O, zo > 0

-7r/2;

Oo - O , zo < 0

t a n - 1( z__~o); ~ooc

0

(9.66)

otherwise

The procedure for solving (9.65) is as follows. C o m b i n i n g (9.65) with (9.61) yields Cos ~ o cos ~ Oo ~os ~ ( ~ -

r

+ s i n s 0 sin s Oo + 2 cos 0 sin 0 cos Oo sin Oo c o s ( r - r

- c

where c - 1 - 1/r~. The above equation m a y be further reduced to an equation in tan 0, by dividing both sides by cos s 0. During this operation, the solutions 0 - 7r,-Tr are excluded. As a result, we get tan s O(sin s Oo - c) + 2 tan 0 sin Oo cos Oo cos(r - r

+ c~

cos 2 (r - r

- c - 0

We now consider two cases: If sin 2 0 o - c :/: 0 then there are two distinct solutions given by c~1~ - tan -1 [~V/(ro2 - 1)[1 - r~)cos 20o s i n S ( r

r

9.4. S h e a r T r a n s f o r m a t i o n S t r a i n s

293

OU

ro

-r/2

<1

-

0o=0

1

Oo = 7r/2 Oo = - r / 2 0 < 0o < r/2 -r/2

>1

< Oo < 0

~o0 < 1, zo _ 0 zo<0 ~ o - 1 , zo > 0

0

0 _ tan_l [cos(r

0o

_tan-1 [cos(V-~o)]tan 0o

-r/2 +

-r/2 O~2

zo
~2

~o>0

T a b l e 9.1: Lower and upper limits on 19

x

- r o2 cos60 sin 0o c o s ( r 1 - ro 2 cos 20o

r

J

(9.67)

If on the other hand, sin 2 0 o - c - 0, then there exists one unique solution: c~2- tan-l[ r2-1-c~176 _ r02 sin 20o cos(r - r

(9.68)

Note that the condition sin 2 190- c = 0 is equivalent to ~oo = 1 which has been exploited in obtaining (9.68). The lower and upper limits on 19 obtained from the solution of (9.65) are given in Tablem 9.1. For the special region described by (9.55), a - b, r - 0 and r - r such that LOo = z0, F0 = 0, thereby considerably simplifying the limits of integration. It should however be borne in mind that when Izo[ < 0 and [yo[ < b the integrals have a meaning only in the mathematical

Three-Dimensional Transformation Toughening

294

sense. For Ix0l _< 0 the integrals can be interpreted physically only when lY0] >_ b. For the region (9.55) and constant transformation strains the integrals (9.51)-(9.53) simplify to read

c -~ez~: ~ K [ ( o ) - (1 + u-------~C~ 2~ ( 2 - v)(1 + v)

/s

8~

-

-

Fz', (o , r162

a~:~y coCTfo~f~ !

( 2 - u)(1 + v) K'SII(O)

!

F~IIy(o, r162

Co~L c~ fo~fo t~ F~"z(O,~)~Od~

8 V / 2 ( 1 - v) Co--a~E~T ( 2 - v)(1 + v)

z - xy ,-, r162

4V/2(1 - u) c T foil fo 0" p l l I ( o r162 - yz ,-, (2 v ) ( l + v ) C ~ a z

(9.69)

where (m # n, a - I, II, I I I )

F.% - ( C C - V77)bk -4"[Vg.](eosOI sin 0)r162

'~ki

Jcos 0

r

r (9.70)

and "/" between c/s and between cos 0/sin 0 means that either cos 0 or sin 0 term appears in (9.70) together with the corresponding coefficients a~i[U~mn] or a~i[U,~n]. In (9.70), k = 0, 1,2, i = 0, 1,2,3, and bk =

cos 2k 0 (cos 2 0 + ~c 2 sin 2 O)k+l

(9.71)

The only non-zero coefficients a~i[U~n ] and a~i[U,.~n ] are

a~i[Ulz] 9 alo " - 1 , all " --1 a~k~[U~]

9 a~o - - ( 2 -

3v),

,

" -8 , a21

a~l - - ( 2 -

" --8 a22

3v),

a~2 -- - 2 ( 2 -

a~0 - -4v,

a~l - 4 ( 4 - 7u),

a ~ 2 - 8 ( 2 - 3v)

a~l-32v,

A~2 - 6 4 ( 1 - v ) ,

a~3--32(2-v)

3v)

9.4. Shear Transformation Strains

a~i[uil] 9a'lo -- 5 v - 2, a2o' - ' aSkit[UIIZl" v x y J hi0 a~it v[lfllI]j

-8v,

aSll

a s21 - -

a2o - 32,

--

2-

-16(1

3t/

-

a~l - - 2 ,

--v+2,

a~o

a~o -- 2 + v,

295

a 21 r -

-

v),

' a22

8(2-

~)

a~2--16

-32

a~l - 3 2 - 4v (9.72)

-32

The next step in the evaluation of integrals (9.69) is to expand ru, rz and bk appearing in (9.70) in powers of sin 0 and cos 0. This allows us to express the integrals as multiple sums involving powers of sin 0 and cos 0, but we find that the resulting expressions for K~s (0) are too cumbersome to be of any practical use, thereby negating any advantage this procedure may have over the direct numerical integration of (9.51)-(9.53) that will be presented below. However, for a spheroidal region (a = b) centred at the origin, the integrals (9.69) can in fact be evaluated as single infinite sums using the procedure involving expansion in powers of sin0 and COSO. In this special case, it can be shown that the only non-zero components of the fundamental fields U~mn are U / / a n d fflll..vz. This follows from the s y m m e t r y with respect to z and y, (see Table 9.2) resulting from the fact that when x0 = 0 and y0 = 0, p is an odd function and q an even function of y, Re G,~z and Re G,x~z are odd functions and I m G ~ z an even function of z. Therefore, for constant transformation strains, we have K s -0

KT,(o)

K[.(0)

-

2

T

(1 + v) c~y

~v

U~udV II

T Iv UIIIdv - (1 +2 ~)C~z ~ ~ --

(9.73)

where a2z2 V " x2 + y2 +-~7 <1

(9.74)

Next, we consider several special cases of the region (9.74). For an oblate spheroid, a > c, eqns (9.73) reduce to

296

Three-Dimensional

Transformation

u ~I I (-y) - u ~I I (y) -

(-z)

U~'z( - y ) - - u ~ ( ~ )

-

(z)

II II U~z (-~1 - U~z (y)

Uxlil

( - - Z ) - - U ~ yIII (z)

u ~III ( - ~ ) -

UIII,

~ i i i ,~z U~z )

uy~ (-z) - -u~,[ (z)

Uylll

Uz~(-~) - Uz~(y)

u S (-~) - -Uz'J (y)

UIII,

-

-U~z,~(z)

U~z'~(-z)

-US(z)

-

__ f]'IlI(y) "~ xy

N II(--Y) "-"Ixy

u~(-z) - -u~(z)

U[~(-z)

Toughening

~z ~y~

z (-z)-

zx

(-Y)

~

__

UIII,

-

U z I I I ( - Z) -

zx

T T I I I ,[ z ) Uz~

T a b l e 9.2: Behaviour of U m~t ~'

K/S(0)- 0

Ksi(O) -

_

8j~

Coac~y C

(2- ~,)(1 + v)

T

oo c2 { 4 3 1 x E(1-~-~lm g(2-3v)B(~,m+~) rn--0

16 7 1 +]-~(1 + ~,)(m + llB(~, m + ~1 64 m 11 1 } 105 (11 - 9v)(---~)_ B(-~-, m + ~) K[.(0)

-

4~/2(1 - ~) Co ( 2 - v)(1 + ~) • E(1m--O

c_ T aCyZ

{

3

(Y)

1

ac~) 2 m -(2+v)B(~ ,m+~)

9.4. Shear Transformation Strains

297

4 7 1 +~(16 + L,)(m + 1)B(~, m + ~) 64 m 11 3 (---3)-B(-T

1}

(9.75)

rn, ~1

+

For a prolate spheroid, a < c I '7 (o) - o

Ks'(0) -

--

8V~ a T (2 - ~,)(1 + ~,)Co 7~xy

oo

a2)m {4

• E (1 - ~-~

3

1

g(2 - 3vlB(~ + m, ~1

m--0

16 a2 7 1 +~-~(1 + t,l(m + 11~7B(~ + m, ~)

1} 64 m a 4 11 105(11 - 9t,)(_--~)_ ~-~B(~- + m, ~) 4V/2(1- ~')

KsII(O)- ( 2 - ~,)(1 + ~,)C~

a T

x ~ ( 1 - a2'm {-- (2 + C2 /

m=O

3

1

u)B(~ + m, ~)

4 a2 7 1 +~(16 + u)(m + 1)~TB(~ + m, ~) 64 m a 4 11 1 } 3 (---31-7~B(-4 - + m, ~)

(9.76)

For a sphere a = c KT(o)- 0

2x/2

KSI(O)- ( 2 - u)(1 + u)C~

8X/2(1-u)

KflI(O)- 5 ( 2 - u)(1 + v)C~

T 32 16v)B(3 1 9

T ~ (

5

4' 2)

3 1

- u)B(~, [)

(9.77)

Three-Dimensional Transformation Toughening

298 In (9.75)-(9.77)

B(p,q) -

fo

(1 - x)P-lxq-ldx

(9.78)

The notation (-~3)_ in the above expressions stands for (1 -

x) p -

Z. P n--0

The analytical expressions (9.77) are also useful for checking the accuracy of the general numerical integration procedure which is described next. For an arbitrary ellipsoidal domain V centred at x0, y0, z0 a2

( x - xo) 2 + ~ ( y -

a2

yo) 2 + ~-~(z - zo) 2 _< 1

(9.79)

and constant shear transformation strains granT, the integrals (9.51)(9.53) are evaluated numerically using the Gaussian method. To enhance the rate of convergence of the singular integrals as R - . 0, ellipsoidal coordinates (9.56) are expedient because they allow formal integration of (9.51)-(9.53) which respect to the coordinate r such that the integrands take the form

(x/~u - x/"~)f( O, r

(9.80)

where the upper and lower limits of integration on r are given by (9.60). The ranges of integration with respect to 0 and r given by expressions (9.62)-(9.66) are each divided into 23 - 8 intervals, with each interval being represented by six Gauss points. The accuracy of the numerical integration scheme is checked in two ways. First, by comparing the numerical results for a spherical region with the analytical results (9.77). It is found that, for v - 0.3, the numerical results differ from the analytical ones by less than +0.2%. Secondly, by increasing the number of intervals to 24 - 16. The results hardly differ from those obtained with only 23 - 8 intervals. The numerical integration scheme can also be used for an exploration of the problem at hand for a general ellipsoidal domain V. In particular, the spherical domain (a - b - c) and the oblate spheroidal domain with a - b and ale - 5 are of interest. The choice of these two shapes is dictated by the fact that zirconia particles are spherical when found in an alumina matrix or as thin oblate spheroids when found in partially

9.4. Shear T r a n s f o r m a t i o n

299

Strains

stabilized zirconia (Chapter 2). In an extensive numerical exploration of complex and singular integrals it is essential to have at least some independent checks on the qualitative, if not the quantitative, accuracy of the results. This is done by a close examination of the properties of U~n appearing in integrals (9.51)-(9.53). For instance U ~ , possess several symmetry properties with respect to y and z which influence the final result (i.e. KS). The behaviour of Um~n with respect to y and z for various m, n and c~ is evident from Table 9.2. Moreover, it is found that

K,~(-V) -K~ (v);

Zo -- 0

Kf(-z) -Kf (z);

YO -

0

Kf,(-v) -KS(v);

z0 -

0

-

-

-

I

zo=O, yo = 1, a/c=-1

I

(9.81)

I

3 2 1 Ks

o

-1 -2 -3 -4

-1.O

Zo 0 ~Yo

-0.5

1 a/c=- l j

0.0 x0

J

0.5

1.0

F i g u r e 9.5: Variation of K / (0) with z0 for two typical values of y0 and of a / c (a - b)

300

Three-Dimensional Transformation Toughening 10

I

I

I

Zo=- 112, yo= 1, a/c= 1 zo=- 1, yo= 1, a/c= l -

B

KHs

0 -2

m

-4 -6

-8 -10 -1.0

z 0 = - 1, y 0 = - 1,

alc=

1 -

zo=-112, y0=-l, alc=l I

I

I

-0.5

0.0 Xo

0.5

1.0

F i g u r e 9.6: Variation of KSl(O) with x0 for two typical values of y0 and z0 for a spherical region (a - b - c)

Numerical results show (Huang, 1992) that the dominant contributions to K s, I f [ i and K S l l come from U ~ , U ~ and U I N , respectively. The remaining components of Um~n make but little contribution to K s. Therefore, It' S, It'Si, It'Sii exhibit essentially the respective symmetry behaviour of U~/ , UI[ and U / u (Table 9.2.). This observation is useful not only for performing a qualitative check on the numerical results, but also in simplifying the graphical presentati6n of the latter, in that the stress intensity factors can be plotted to within a factor of the strain T corresponding to the dominant U,~,~ Thus in the figures to follow, the scale for It"s should actually be read as It'S/exT . In dimensioned quantities this scale represents ( E K S v / ' d ) / c T u. Similarly, the scale for K S in dimensioned quantities should read ( E K S l x / ~ ) / e T and for It'Sii should read (EIt'~llV/-a)/exTz . It is now possible to comment on the typical behaviour of K s as demonstrated by the numerical integration procedure (for u = 0.3). This behaviour was found to be qualitatively totally consistent with the above

9.4. Shear Transformation Strains

I

301

I

5

I

Zo--O,yo=1, a/c=1

4

3 2 1

KISI 0 -1 -2 -3 -4

-5 -6

-1.0

I

I

I

-0.5

0.0 x0

0.5

1.0

F i g u r e 9.7: Variation of KSllI(O) with x0 for two typical values of y0 and of a/c (a - b) observations. Typically, IKffl for a/c- 1 was found to be consistently larger than that for a / c - 5 (Fig. 9.5). However, the variation of [/its[ with [z0[ > 0 was similar for both shapes; [/its[ decreased with increasing Iz01 _> 0, initially rather slowly but then more rapidly. The shape of region V has a far more pronounced influence upon IKSi[ than upon IKSl. [KSl[ for h i e - 5 is not only much smaller than for a/c = 1 but it diminishes rapidly with increasing [z01 > 0. For hie = 1, [K]I[ seems to achieve (Fig. 9.6) the maximum value at or near ]z0] = 1/2 and then decreases with increasing [z0[ > 1/2. As far as IKSliiI is concerned, the shape and location of V seem to have an effect on it similar to that on [KtSl. Thus, [IiS/i[ for a l e - 1 is consistently larger than for a/c = 5 (Fig. 9.7). However, for both shapes it decreases with increasing Iz0[ > 0, although the rate of decrease is different. For a/c = 5, it decreases rapidly with increasing [z0[ > 0, but for a/c = 1 it decreases rather slowly in the beginning. Up to now, we were only interested in the shape of the transformation

302

Three-Dimensional Transformation Toughening

domain. We found in particular that spherical particles give a larger ]KS l than do oblate spheroids with long axis normal to the crack front (i.e. parallel to x-direction). We have observed in Chapter 3 that the orientation of the long axis of oblate spheroids relative to the crack plane has a significant influence on the extent of change in K D due to dilatational transformation strains. It is therefore of some interest to examine the influence of orientation of oblate spheroids on the stress intensity factor K s due to shear transformation strains. We consider an additional orientation, namely when the long axis is parallel to the crack front (i.e. parallel to z-direction). An example of this orientation of oblate spheroids which corresponds in size to the previously considered orientation (long axis parallel to x-direction) is simulated by choosing a = b, a/c = 0.2. When the long axis is parallel to the crack front (i.e. parallel to z-direction) numerical computations for a = b, a/c = 0.2 show, as expected, that the variation of K s(0) is more pronounced with [z0[ than was the case for a = b, a/c = 5. Thus the m a x i m u m value of [KI[ increases from around 4 at z0 = 0 to about 14 at [z0[ = 0.5 and further to about 75 at [z0[ - 1.0. The increase is even greater in [KII[. Its m a x i m u m value increases from about 0.6 at z0 = 0 through about 22 at z0 = 0.5 to about 850 at [z0[ = 1.0. The largest increase is noticed in [KIII[. Its m a x i m u m value increases from about 8 at z0 = 0 through about 14 at [z0[ = 0.5 to a colossal 1850 at [z0] = 1.0. This behaviour is quite consistent with that expected from analytical considerations. Similar changes to K s ( 0 ) can be expected when the long axis is normal to the crack plane (i.e. parallel to y-direction).

303

Chapter 10

T r a n s f o r m a t i o n Zones from D i s c r e t e P a r t i c l e s 10.1

Introduction

In Chapter 7 we discussed steady-state transformation in which the transformed particles were assumed to be continuously distributed. The profile of the transformed region ahead of the (semi-infinite) crack tip was found to approximate a partial cardioid when the super-critical dilatational transformation was triggered at a critical level of the mean stress. We also noticed (w 7.3.3) that the shape of this region was significantly altered by the presence of transformation shear strains, in addition to dilatation. The boundary of the leading edge of transformed region in the latter case was assumed to be governed by a critical value of strain energy density (7.51). We argued earlier that although there is still no consensus on the real triggering mechanism, the phenomenological stress criterion (3.87) formulated by Chen & Reyes-Morel (1986) at least appears to be validated by available experimental data. According to this criterion, transformation is expected to occur when O'm

--

C

(7 m

~

+ (1 - c~) v~g

--

1

(10.1)

Tc

where 0 < c~ < 1 is an empirical constant, rc is the critical value of effective shear r ~ g - (~1 sij s i j ) t/2 , where sij is the deviatoric stress tensor, i.e. sij - ~rij - ( T i n ~ij. The value ~ - 1 corresponds to the

304

T r a n s f o r m a t i o n Zones from Discrete Particles

critical mean stress that we have used in Chapters 7 and 8, while c~ = c and 0 gives a maximum shear stress criterion. The values of a, trm re are usually obtained by fitting experimental data for a particular transformable material composition. It is interesting to understand the role of critical stress criterion, i.e. of a, in the development of transformation zone. We shall examine this question in the present Chapter, together with the role of transformation shear strains. We already know from w167 7.3.3 and 9.3 the importance of these strains. We shall examine both these questions not in the continuum approximation but by assuming that the tip of a semi-infinite crack is surrounded by a distribution of small circular transformable spots. We shall increment the applied stress to simulate the spontaneous (supercritical) transformation of each spot according to the chosen triggering criterion, e.g. (10.1). We shall find that the shear stresses in the stress criterion for transformation and the shear strains induced by it have a dramatic influence upon the size and shape of the region containing the transformed spots. The shapes can be radically different from those predicted by the continuum model of dilatational strain triggered by a r i.e. a - 1 in (101). They extend far critical level of mean stress trm ahead of the crack tip and may reach millimetre proportions. The shear stress induced by the interaction between spots may even trigger an autocatalytic reaction, whereby the stresses created by the transformation are sufficient by themselves to trigger the transformation of neighbouring spots. We shall study the phenomenon of autocatalysis. For the exposition to follow we shall draw heavily from the works of Stump (1991, 1993, 1994). It will parallel the exposition in Chapter 7. We will first consider a semi-infinite stationary crack (w 7.2) and follow it by a quasi-statically growing semi-infinite crack (w 7.3). In both instances, we shall consider only super-critical transformation. In other words, when the stresses at the location of a transformable spot satisfy the prescribed triggering criterion, e.g. (10.1) it spontaneously transforms inducing plane strain dilatation D and shear strain S (4.16). In order not to introduce an unmanageable number of parameters, we shall adopt a simplified version of (10.1), namely O'm 8c

+ (1 -

Trn a x

= 1

(10.2/

8c

where Crm is given by (7.11) and the maximum shear stress Vmax = 1 [~(~yy - ~r~) + i crxy [ is the modulus of the complex stress, sc is usually identified with the stress at which the uniaxial tensile stress-strain curve

10.2. S e m i - I n f i n i t e S t a t i o n a r y Crack

305

of the bulk material sample deviates from linearity. Most of the mathematical expressions necessary in this Chapter were developed in Chapter 5 when we discussed elastic solutions for isolated transformable spots. We shall make repeated reference to those expressions.

10.2

Semi-Infinite Stationary Crack

Throughout most of this Section, attention is focused on spots in mode-I symmetric distributions, such as those in Fig. 5.4. The total potentials (i.e. the sum of the infinite plane and image contributions) are obtainable from (5.34) and (5.35) by adding to them the potentials for the spot at ~0. The latter are obtained from (5.34) and (5.35) when the non-z terms are conjugated. Likewise, for symmetric spots at z0 and 70 the mode I stress intensity factor at crack tip which we shall designate A K tips (s for symmetric spots) can be obtained from (5.43) after replacing v/z 2 - c 2 by x/7 to give D Re

A K tips -- V / ~

5A0]

i o;zo + 47rz7o/2

+ - - ~ S Re ! - 5/~ k Zo

(10.3)

where D and S are given by (5.70) with d A - Ao. For a single spot (Fig. 5.2), it is readily verified that A K tip ,~ - ~1 A K tips where superscript n distinguishes nonsymmetric spot distribution from a symmetric one. As the critical stress criterion for transformation (10.2) now includes the maximum shear stress vm~ besides the mean stress rrm, it is expedient to introduce also the stress potentials corresponding to the applied mode-I stress field _

I~( app l

(~appl(Z) -- a'appl(Z ) --

~(~Trz)l/2

(10.4)

The stress field given by the above potentials is still (7.3). The near-tip stress field will contain the contribution from mode II, if an asymmetric spot distribution is being considered (cf. (7.4))

///ip =

.tip

KII I j(o) + 2x/~_~ gij (0);

(r -+0)

(10.5)

306

Transformation Zones from Discrete Particles

where gij(O) a r e the mode II universal angular functions. For symmetric is identically zero spot distributions , ~~.tip II As K tip is increased, spots transform if the critical stress criterion is met by the stresses at spot centres. At the beginning of simulations, transformable spots are distributed over an area near the tip of the semiinfinite crack in such a way that they do not overlap or intersect the crack fine. Simulations are continued until the tip is on the verge of growth, i.e 9 until ~~.tip attains the intrinsic toughness value of the matrix Kc 'I Itip

The influence of non-zero K II for asymmetric spot distribution upon the onset of crack growth is ignored. In the absence of transformation, the remote (7.3) and near-tip (10.5) stress fields are identical so that at the instant of crack growth I ~ tip = Ktt ip - K~. The boundary R(O) of the region in which the critical stress criterion (10.2) is met in the absence of transformation is given by

R(o) Lo

-

-

{4(1 + v ) ( ~ ) ~ C~COS

}2 + (1 - a)[ sin O[

(10.6)

where the characteristic length (proportional to the frontal intercept of the boundary at 0 - 0) is

Lo-

1

(10.7)

In order to obtain (10.6) we used the stress potential (10.4) in the formulae (4.21)-(4.23) to calculate the plane strain stresses (r~, ~ryy and (r~v necessary for determining ~rm and rmaz appearing in (10.2). The boundary of the region described by (10.6) enables one to assess the effect of interactions when the spots transform in front of a stationary crack. The shape of the boundary (10.6) reduces to the cardioid (7.12) for c~ = 1, but changes to a figure of eight as c~ approaches zero. During simulation of transformation under increasing A = K t i p / K c , the combined effect of K tip and any previously transformed spots are continuously monitored at the centres of untransformed spots. This process continues until A ---. 1. For this it is necessary to calculate the stresses ~rm/sc and rma~/s~ outside the spots. Substitution of (5.34)(5.35) into (4.21)-(4.23)gives N

s~-

3

~

~=I

10.2. Semi-Infinite Stationary Crack

307 (10.8)

+-~F~(z, zn,An,r T m ax 8c

,~(1- ~/z)~-~ 2zl/2

~_~

--[-

[(Z'-- Z) F~(z, Zn, An)

n--1

+Gd(z, Zn, An) - Fd(z, zn, An)]

co

+ -~-~ [(-i - z ) F~ ( z , z,~ , An , r ) +Gs(z, z n , A n , r

Fs(z, z n , A n , r

(10.9)

where a prime denotes differentiation with respect to z, and parameters 3 and co are related to plane strain dilatation of a single spot 0T and shear strain S (4.16) as follows 3 -

E T 0v . so(1 - u)'

CO-

ES s ~ ( 1 - u 2)

(10.10)

The spot contributions to (10.8)-(10.9) are calculated from all transformed spots N. For symmetric distributions, the summation is carried over only transformed spots in the upper half-plane, whereas for arbitrary asymmetric distributions the summation extends over all transformed spots. For both distributions, expressions (10.8)-(10.9) retain the form but of course the various functions appearing in them are different. These functions are obtainable from (5.34)-(5.35), but for completeness we list them here for symmetric and asymmetric spot distributions. For a spot of area A0 whose centre is located at zo = roe ir176with respect to the crack tip (Fig. 5.2), the various functions appearing in (10.8)(10.9) are nothing but a regrouping of the functions (5.34)-(5.35) with an appropriate modification to the multipliers to account for the new parameters (10.10). Thus

Fd(Z, zo,Ao)--Ao Gd(z, Zo, Ao) - Ao F~(z, zo, Ao, r

/1 (z, ~0)

(10.11)

2zl/2 1

(z - zo)2

{1

-

/1 (z, z0)] 2zi.]~

- A0 e 2'r176 - ( z - z0) 2

I~ (z, zo) } + 2-~

(10.12)

308

Transformation Zones from Discrete Particles

1 [s~ (z,-~o) + zo S~(z,~o)

+Ao e- 2ir 2z~l~

Ao

- H l ( z , z o ) 4- ~ I3(z, zo) 1 - Ao e 2ir176 - (z - zo)2 +

G8 (z, zo, Ao, r

]

(10.13)

3Ao }

2(z-~o) (z- zo)~

~(z - zo )4

1 [I1 (z, zo) + -2o I2(z, zo)

+Ao e 2ir1762zl/2

Ao ] - H l ( z , zo) + ~ I3(z, ~o) + Ao e- 2/r I1 (z, ~o) 2zl/2

(10.14)

In (10.11)-(10.14)the auxiliary functions are 1

S~(z, zo)I2(z

'

-

(10.15)

1/2 1/ 2~o (z'l~+zo~) ~

1

Zo)

3/2 1/2 1/2)2 4z 0 (z + z0

S~(z, zo) - -

1 +

3 5/2 zl/2 Zo ( + Zo1/2 )2

1/

2zo (21/2 + z 0 2) 3

3/2

zo

3

1/2, 3

(zl/2 + Zo )

(10.17)

3/2 1/2)4 z o (zl/2 + Zo H l ( z , zo)

-

3/2 2(z3/2 - z~ ) (z - zo) 3

-

3 4Zlo/2(z - zo)

(10.16)

-

3z~/2 (z - zo) 2

(10.18)

For two spots of equal area Ao whose centres are located at zo and To with respect to the crack tip (Fig. 5.4), the various functions appearing in (10.8)-(10.9)are obtained from (10.11)-(10.14) by adding to these the terms corresponding to To - roe -i~~ and simplifying, and so giving Fd(Z, zo,Ao) -

Ao [11(z, zo) + Ii(z, ~o)] 2zl/2

(10 19)

10.2.

Semi-Int~nite Stationary Crack

Gd(z, Zo, Ao)

1 (z - zo) 2 +

- Ao

309 1

(z

~o)~

-

-- II (Z' ZO)2zl/2~"/I (Z, Z'0)] Fs(z, zo, Ao, r

(10.20)

1

- Ao e 2ir176 - (z - zo) 2

1/

+ 221/~ (~o + Ao e- 2ir [_

-

zo) i~(z, zo) + ~ i~(z, zo)

1

[ (z-~o) ~

1{ (zo

+2zl/~

Gs(z, zo, Ao, r

1

-

~o1 i~(z,-~o1 + ~ i~(z,-~o1

- Ao e 2ir176 - (z - zo) 2 +

+

(-5o

)]

2(z - To)

3Ao

( z - zo) ~

~ ( z - zo) ~

}]

zo)I2(z, zo) Ao I3(z, zo)] 2zl/2 -~- 47r ~ / 5

-

1 2(z +Ao e -2ir176 - (z - 2o) 2 +

zo)

(z - ~o)~

3Ao

~(z - ~o)~

~_ (Zo---Zo)I2(z,-zo) Ao I3(zi~o)] 2zl/2 + 4-~ z 1

(10.21)

The crack-tip stress intensity factor Iil ip is obtained by superposing the contribution of each transformed spot (10.3) to the applied stress intensity factor K tip. For symmetric distribution of transformed spots, the normalized toughness ratio is

/.---2=a+ ~

~ R~ ~ n=l

3~ ~

+ 3--~ .=1

Zn

An Re [e2ir ( z--n- z~/2 z.

4~z~/~

)]

(lO.22)

where the spot area has been normalized by Lo2. It is easily verified that

310

Transformation Zones from Discrete Particles

for an arbitrary (asymmetric) spot distribution, the contribution from the transformed spots is exactly one half of that in expression (10.22). Apart from this difference, KtliP/Kc for an asymmetric distribution is also determined by (10.22). We shall concentrate on the influence of the triggering mechanism, i.e. of the parameters a and sr in (10.2) upon the number and locations of transformed spots. To facilitate comparison with experimental studies the following material properties are used in all simulations described below: E - 200 GPa, ~, - 0.25, Kr - 3 MPax/~, 6T - 0.04. All spots 1 are assumed to be of constant diameter equal to ~#m, but two values of S are studied, S = 0 and 0.05 corresponding to 8 = 0 and/3, respectively in (10.10). Table 10.1 gives the values of L0, # and normalized spot area (Ao/L2o) for several values of sc in the range of 300 MPa < sc < 1.1 GPa. The upper limit of sr corresponds to L0 (10.7) of about the spot size. Thus, large values of sr require that spots of small sizes transform. It is known that very small transformable particles are prone to spontaneous thermally-induced t to m transformation.

sc (MPa)

#

L0 (pm)

Ao/L 2

1067 711 533 427 356 305

10 15 2O 25 30 35

0.315 0.709 1.260 1.980 2.840 3.860

0.878 0.173 5.49x 10 2 2.26x 10 2 1.08x 10 2 5.85x10 -3

T a b l e 10.1" Material and spot properties The simulation process is initiated by sequentially depositing untransformed spots at random locations within a rectangular box surrounding the crack tip. The box is typically about 2L0 in height on either face of the crack and extends to about 6L0 in front of the crack tip and to about 3L0 behind it. For non-zero S (i.e. non-zero 6) the inclination of the principal axes (angle c~ in (4.17)) is randomly chosen in the interval from -7r/2 to 7r/2. New spots which overlap existing ones or touch the crack faces are disregarded. The distribution process is terminated when the total area occupied by the spots reaches a quarter of the box area. As mentioned above, for symmetric distributions, the spots are constrained to lie in the upper half of the box.

10.2. Semi-Infinite Stationary Crack

311

The simulation is carried out iteratively by increasing the ratio ~ in increments of 0.1. At each iteration, the stresses at the centres of all untransformed spots are calculated using (10.8) and (10.9) and substituted into the transformation criterion (10.2). If the left hand side of the latter equals or exceeds unity for a particular spot, that spot is tagged, and the check resumed. After the current status of all hitherto untransformed spots has been checked, the group of tagged (transformed) spots is added to the transformed spots from the previous iterations. The ratio A is further incremented until it reaches the value unity. Increments which take ~ beyond this value require that it be adjusted in such a manner as to ensure that KI ip remains equal to Kc. The simulation is terminated when no additional spots transform during an iteration while ts ip - Kc. The results of simulations are presented for dilatational (e = 0) and mixed strain (e = 3) categories. Emphasis is placed on the dilatational transformations for both symmetric and nonsymmetric spot distributions; the mixed strain simulations consider only symmetric distributions. Each simulation consists of the 'family' of microstructures for the three values of c~ = 0, 0.5, and 1 (referred to respectively as the 'shear', 'mixed' and 'mean' stress criteria). These values are chosen to sample a cross-section of possible microstructures. The results do not necessarily preclude other possibilities. Multiple simulations are conducted for each set of parameters, however, only select results are presented below. Figures 10.1-10.6 show a series of symmetric simulation 'families' for various values of the dilatational strain parameter 3. (Two sets of microstructures are shown for 3 = 35). The parameter values and A at the instant of crack-growth initiation are shown in inset. The spot diameter is always S1 pm, and the size of the scale mark L0 can be found in Table 10.1. The shaded areas show the critical stress boundaries (10.6) in the absence of transformation. The regions governed by the 'mean' stress criterion are well approximated by a cardioid shape for all /3 with some variability in size with respect to the dashed boundaries (this is discussed further below). On the other hand, the pictures for the 'mixed' and 'shear' stress criteria show that the shear stresses have a profound effect on the shape of the transformed region. As/~ increases, claw-like regions of particles extend ahead, and to the side, of the tip. Autocatalytic features also develop with streams of neighbouring spots transforming, see Figs. 10.5 and 10.6. These streams are not necessarily parallel to the major thrust of the zone with respect to the tip. Exclusion zones where no spots transform are also visible around the horizontal axis of some 'mixed' and 'shear' simulations.

312

Transformation Zones from Discrete Particles

F i g u r e 10.1: Symmetric dilatational spots with/~ - 10

10.2. Semi-Infinite Stationary Crack

F i g u r e 10.2: Symmetric dilatational spots with/~ - 20

313

314

Transformation Zones from Discrete Particles

Figure 10.3: Symmetric dilatational spots with ~ - 25

10.2. Semi-Int~nite Stationary Crack

Figure 10.4: Symmetric dilatational spots with/~-- 30

315

316

Transformation Zones from Discrete Particles

Figure 10.5: Symmetric dilatational spots with ~ - 35

10.2. Semi-Infinite Stationary Crack

317

F i g u r e 10.6: Symmetric dilatational spots with fl - 35 (a - 0 plot filled the box region)

318

Transformation Zones from Discrete Particles

F i g u r e 10.7: Nonsymmetric dilatational spots with j3 - 25

10.2. Semi-Infinite Stationary Crack

F i g u r e 10.8" Nonsymmetric dilatational spots with ~ - 30

319

320

Transformation Zones from Discrete Particles

Figure 10.9: Symmetric 'mixed strain' simulations for ~ - 8 - 15

10.2. Semi-Intinite Stationary Crack

321

Figure 10.10: Symmetric 'mixed strain' simulations for f l - 8 - 20

322

T r a n s f o r m a t i o n Zones f r o m Discrete Particles

The reason for the pronounced difference in behaviour with and without shear stresses can be ascertained from an examination of the governing expressions (10.8)-(10.9). In general, the stresses due to dilatational spots have the form crm

H(0)

(10.23)

sc ~" 11-''-'-5r

~-,.o~ ~(o)

N

1

s~ ~ r - ; ~ +/~ n ~ l reiO -- Zn

(10.24)

where H and G are well-behaved functions. The mean-stress field (10.23) decreases monotonically with distance from the crack tip; there is no direct interaction between spots. As a result, the mean stress at untransformed spots situated beyond a certain maximum distance never reaches the critical value. The shear stress (10.24), however, contains both a radially dependent term and a term which depends on the distance between spots. For small/3 (i.e. ~ _< 20), the radial term dominates the behaviour so that the zone shapes are perturbations on those due to the applied stresses. However, for larger/3 the direct interaction term in (10.24) provides a substantial contribution, particularly at spot centres far from the tip. Thus at large ~, an autocatalytic process can occur as the transformation of a series of spots can induce local stresses sufficient to trigger the transformation of nearby spots. The process is conjectured to give rise to the streams of transformed particles which appear under 'mixed' and 'shear' stress criteria. The toughening ratio A depends strongly on the arrangement of transformed spots in the vicinity (i.e. several spot diameters) of the tip. The formula (10.22) for the toughness ratio shows that the spot contribution is a function of both r and 0. For dilatational transformations, spots 1 9 tip in the sector -~Tr <_ 0 <_ 51 r mcrease K I , whereas spots in the ranges 1 [0[ > ~r decrease it. The magnitude of the spot contribution drops off as r - 2 / 3 and the simulation results indicate that A is determined primarily by the spots within 3 diameters of the tip. Careful examination of the figures shows that in simulations with near-tip spots lying mainly in the 'shielding ' regions [0[ > !3r , A is consistently higher than in simulations with spots lying in the 'anti-shielding' r e g i o n , - 8 9 < 0 _< 89 The high A tends to produce large transformed regions (cf. Figs. 10.5 and 10.6), due to the high stresses farther from the tip. The sensitivity of A to the near-tip distribution of transformed spots may explain some of the zone variability, particularly in the 'mean' stress criterion zones.

10.2. Semi-Infinite Stationary Crack

323

Figures 10.1-10.6 clearly indicate that the development of unusual transformation zone shapes under the 'mixed' and 'shear' stress criteria is an essential consequence of transformation criteria which include shear stresses. It is conjectured that at large ~3, microstructures for c~ close to unity will be substantially different from 'mean' stress criterion results owing to the powerful effect of shear stresses. The effect of nonsymmetric spot distributions is illustrated in Figs. 10.7-10.8 for/3 - 25 and 30. For the 'mean' stress criterion the effect of asymmetry is relatively minor; the zone shapes are still approximated by cardioids. The microstructures for the 'mixed' and 'shear' stress criteria show more deviation when compared with symmetric results. It is conjectured that the direct interaction contribution of the shear stress (10.24) is more sensitive to spot distribution than the radial term and accounts for the lopsidedness of the nonsymmetric microstructures. However, a lack of symmetry does not appear to alter the basic trend of the results. Some 'mixed strain' simulations are shown in Figs. 10.9 and 10.10 for ~ = ~ = 15 and 20. The 'mixed strain' zone shapes show less identifiable structure than the dilatational shapes. Altering the distribution for a fixed set of parameters can produce dramatically different zone sizes. For example, the values ~ = ~0 = 20 often produced transformed spots which filled the distribution box. (In fact, for larger values of/3 and ~, it proves impossible to contain the transformed region, even with simulations of thousands of spots in a very large box). All the 'mixed strain' microstructures appear similar to the 'mixed' and 'shear' stress criteria results under dilatation alone. Streams of transformed particles again emerge, but with more variability due to the randomness of the principal-axes orientations. This is not unexpected since for these simulations both the mean and shear stresses depend on direct interactions as well as terms which decrease monotonically with the radial distance. In general, the 'mixed strain' simulations showed more nonuniqueness than the dilatational microstructures. The above simulations show that both shear transformation strains and critical shear stresses have a profound effect on the development of the transformed particle region in zirconia-reinforced ceramics. The commonly used continuum model of a dilatational transformation triggered by a critical mean stress (Chapters 7 and 8) appears to be a special case that yields remarkably simple and stable regions. The explosive growth in the size of the dilatant spot regions under 'mixed' and 'shear' stress criteria as the critical stress sc decreases is in qualitative agreement with experimental observations in PSZ (Inghels et al., 1990, and Marshall et

Transformation Zones from Discrete Particles

324

al., 1990) where critical stress levels of about 250 MPa produced zones in the order of 1 m m in size. These trends may also explain the large zones observed by Lutz et al. (1991) for duplex ceramic composites.

10.3

Semi-Infinite Quasi-Statically Growing Crack

Quasi-static growth of a semi-infinite crack is studied in two stages. The first stage is identical to that described above for the stationary crack from which we obtain the transformation zone shape and size when the crack is on the verge of growth. In the second stage, the crack is allowed to grow in small increments Aa in the range 0.5 < Aa/Lo < 1.0. At each increment, the instantaneous value of A is adjusted to 90% of its final value at the preceding crack tip position, and stage one calculations are repeated. Note that the expressions (10.8), (10.9) and (10.22) are valid for a growing crack provided all coordinates are measured from the current tip position. After the first check for transformed spots, the value of A is increased by either 5% of the starting value or the amount necessary to satisfy the transformation criterion (10.2), whichever is the smaller. The process of identifying transformed spots is repeated after each increment of A. Once K~ ip reaches Kc, )~ is adjusted to maintain this condition as long as the tip is at its current position. The incrementation of A is continued until no further spots transform at the current tip position, whereafter the latter can again be incremented. In order to identify the essential features of the transformation zone during quasi-static crack growth, we restrict ourselves to symmetric distributions of spots of identical area A0 that undergo only dilatational ..tip transformation (~ - S - 0). Thus, K H will be identically zero in rptip (10.5), and the expressions for stresses (10.8) and (10.9) and for ix, (10.22) are grossly simplified to read

~rm 4( l + V) Re [ )~ ~A~ N ] sr = 3 --~ + -~r E Fd(z, zn)

(10.25)

rt--1

Trn a x 8c

(1 -

2vz

3Ao

N

1

1

+ 67r E (z - -2)FJ(z, zn) - (z - z,~)2 - (z - -2,~)2 n--1

(o.26)

10.3. Semi-Infinite Quasi-Statically Growing Crack

325

a)

.

7__q~e

.~?

9

n ~

.

9

.~,l

b)

F i g u r e 10.11: Transformed spots after crack growth by (a) Aa/Lo - 5, (b) A a / L o - 15. f l - 2 0 , c~- 1.0

K~ ip fl Ao N 1 K~ = )~ 4-~127rL"'~-"-'- n~= l Re Zn3/2

(10.27)

where 1

Fd(z, Zo) -- 4(ZZo),/2 (v/.~ + x/~) 2

1

+ 4(z20)x/2 (v/_~ + x/~) 2 (10.28)

For clarity of presentation only two values of/3 (= 20 and 30) will 1 be studied under the mean stress (c~ - 1), mixed stress (c~ - ~), and shear stress (a = 0) transformation criteria. All material parameters remain unchanged from the previous Section. Figures 10.11-10.13 show the transformed particles for ~ = 20 under the three transformation

Transformation Zones from Discrete Particles

326

9 e~rO

I I

I

a)

0 0 ~

9

w - "

9

~

00-@

x:..":. :_'....-._'. - "go_

~

~"

_oo

~_ ".,,"% ".,T ~ _

Lo

o~u .Oo-~,-

__

9"~o 9-,,o~,e *o ~ " o ' "

_

o ,r~_ " o,>osJ@,

. ~ . . s..-,,~r~,r_~-,~:.~t.~,~. .,v'~

b)

L0

F i g u r e 10.12: Transformed spots after crack growth by (a) Aa/Lo = 5, (b) Aa/Lo = 15. /3 = 20, a = 0.5

criteria, respectively when the crack has advanced by Aa/Lo = 5 and 15 in increments of 0.5. The corresponding zone shapes at the instant of crack growth (Aa/Lo = 0) are given in Fig. 10.2. The frontal profile of the 'mean' stress zone has extended along the crack compared to that 1 simulation shows a at crack-growth initiation (Fig. 10.2a). For a - 3, decrease in the proportional extent of the finger-like spot streams ahead of the tip. For a = 0 the zone has developed a claw-shaped region ahead of the tip and, in contrast to the other two zones, appears to broaden normal to the crack line. As crack advance continues, the zones for a - 1 and ~1 remain almost constant in shape. In particular, the zone height, that is, the extent of the zone in the direction normal to the crack plane, stays almost constant. In contrast, for c~ = 0 the zone exhibits a changing profile throughout crack advance, with an enlargement of

10.3. Semi-Infinite Quasi-Statically Growing Crack

a)

327

~ o0ooe~

b) F i g u r e 10.13: Transformed spots after crack growth by (a) Aa/Lo = 5, (b) Aa/Lo = 15. fl = 20, c~ = 0.0

the claw structure ahead of the tip. Although simulations for larger tip advances are not shown it is noted that the claw region for c~ - 0 continues to extend farther in front of the tip. The zone height, however, remains at about the same level as shown in Fig. 10.13b. A comparison of multiple sets of simulations (not included here) reveals an important distinction between the results for a - 1 and those for a ~1 and 0 The zones for c~ - 1 have almost exactly the same length and width at all crack tip increments. For a 89and O, on the other hand, the zones show a much more pronounced sensitivity to the spot distribution. In particular, the location and orientation of streams that emerge from the densely packed region of transformed spots surrounding the tip and bordering the crack faces depend strongly on the details of the distribution. It seems that the development of such local-

Transformation Zones from Discrete Particles

328

L0

a)

b) F i g u r e 10.14: Transformed spots after crack growth by (a) Aa/Lo = 4, (b) Aa/Lo = 10. fl = 30, a = 1.0

ized features is dependent on a critical density and orientation of spots, but it is difficult to assess what these features imply for a continuum analysis. Figures 10.14-10.16 show the transformed particles for ~ - 30 under the three transformation criteria, respectively when the crack has advanced by Aa/Lo = 4 and 10 (Figs. 10.14-10.15)or by Aa/Lo = 2 and 4 (Fig. 10.16). The corresponding zone shapes at the instant of crack growth (Aa/Lo = 0) are shown in Fig. 10.4. For c~ = 1 (Fig. 10.14) the frontal zone profile remains practically unchanged and similar to a cardioid during crack growth. However, consistent with the continuum model (Chapter 8) the height of the zone goes through a peak. The zone shape for a - 1 exhibits a different kind of behaviour. The wings extending ahead of the tip at initiation of crack growth (Fig. 10.4b) develop into two sets of finger-like projections

10.3. Semi-Infinite Quasi-Statically Growing Crack

329

Lo

a)

b) F i g u r e 10.15: Transformed spots after crack growth by (a) Aa/Lo - 4, (b) A a / L o - 10. f l - 30, a - 0.5

ahead of a dense transformation zone. Interestingly, regions devoid of transformed spots also appear. The general frontal profile remains almost the same as the crack advances from Aa/Lo = 4 to 10. However, the zone height continues to grow without any sign of levelling off. For a = 0 (pure shear transformation criterion) the zone is very small at the instant of growth (Fig. 10.4c) but grows rapidly when the crack advances by Aa/Lo = 2. When the crack tip has reached Aa/Lo = 4, the zone has grown substantially and two claw-like projections have developed ahead of the tip. Additional crack increments (not reported here) show that the zone continues to broaden but the frontal profile retains the same basic features. The reasons for the differing zone shapes under mean stress (a = 1) and pure shear stress (c~ = 0) transformation criteria lie in the long-range nature of spot to spot interaction, as discussed in w 10.2.

Transformation Zones from Discrete Particles

330

%

L0 a)

F i g u r e 10.16: Transformed spots after crack growth by (a) Aa/Lo = 2, (b) Aa/Lo = 4. /3 = 30, a = 0.0 We have not attempted to plot R-curves (A against Aa/Lo) for the growing cracks, as we did in Chapter 8 under continuum approximation, because of considerable scatter in the simulation results. Nevertheless, )~ tends to increase as more and more transformed spots are left in the wake of the tip. Thus, the discrete model used in this Chapter on the one hand confirms qualitatively the continuum model results of Chapter 8, but on the other reveals certain interesting features not captured by the continuum model. Among the most notable features are the long streams of transformed spots extending far out of the dense zone of transformed spots surrounding the tip. We shall explore this feature further in the next Section.

10.4.

Self-Propagating Transformation (Autocatalysis)

10.4

331

Self-Propagating Transformation (Autocatalysis)

Long streams of transformed spots near crack tips have been noticed in experiments on supertough zirconia ceramics and have been attributed to self-propagating transformation (autocatalysis) by Heuer et al. (1988), and Dickerson et al. (1987). The notion of autocatalysis implies that the presence of transformed material alone is sufficient to trigger further transformations and has been put forward as an explanation for the room temperature recovery of transformed zones in some high toughness zirconia ceramics by Shaw et al. (1992). Autocatalytic processes have implications for potential increase in toughening levels by extending the size of the transformed zone and for also providing a lower limit on the critical stress below which spontaneous transformations occur. To give a more exact meaning to the term autocatalysis, two specific transformed regions in an infinite sheet are considered here: a continuum strip and a row of transforming particles. These examples are chosen for their ease of analysis. 10.4.1

A Strip

of Transformable

Material

F i g u r e 10.17: Continuum strip of transformation In order to study how a transformed region can assist or suppress additional transformations, consider a simplified model of a rectangular box of transformation spanning the interval - d < x _< d and - h _< y _< h and embedded in an infinite plane, see Fig. 10.17. The geometry of this region is purely contrived, but it may prove useful in approximating what happens to a grain subjected to a stress field which varies slowly over the size scale of the zone. Because of the lack of any stress concentrations that could conceivably cause such a region to develop, we may imagine that the box region has been cut from an initially untransformed

Transformation Zones from Discrete Particles

332

plane, the unconstrained transformation has been allowed to occur, and then the box has been reinserted into the plane. Attention is focused on the stress combinations that cause the critical transformation criterion (10.2) to be exceeded along the midpoints of the two vertical sides. Once the transformation criterion has been reached at these points, the box can then be imagined to extend laterally while the vertical sides remain straight and perpendicular to the horizontal sides. For autocatalysis to occur, the transformation must provide an increasing contribution to the critical stress combination with increasing box length and must be capable of sustaining the critical stress at the midpoints for boxes longer than a critical size. Basing the length of the box on just the stresses at the midpoints may seem questionable. However, as the stresses at points along the vertical sides close to the corners are dominated by log-type singularities, the predictions based upon just the midpoint locations, where the stresses are lower, represent a lower bound on what might actually happen if the boundary were free to adjust its shape to maintain a particular critical stress combination. For the rectangular strip of transformable material shown in Fig. 10.17, the potentials are obtained from (4.28)-(4.30) by integration over the strip

E c Sie 2i~ {

(I)~(z) - -47r (1 - u 2)

E c Si e 2 i a { ~~

(z + d + ih)

(z - d - ih) } (10.29)

log -~ + d - ih) + log (z - d + ih)

d - ih

(1- u2)z+d+ih

d + ih z+d-ih E ci OT

+ 47r (1 - u)

- d + ih + z-d-ih

- d - ih ] z-d+ih

{log (Z + d + ih) (z - d - ih) t -(z + d - ih) + log (z - d + ih)

(10.30)

where, in the spirit of the continuum model, the particle volume fraction c has been used to smear out the region of matrix and particles into an effective continuum. Using the potentials (10.29) and (10.30), the formulae (4.21) and setting z - x, the normalized mean stress and maximum in-plane shear stress directly ahead of the strip are readily calculated

~m_ (I+u) 8c

3

I ~+~UYOO 8c

10.4. Self-Propagating Transformation (A utocatalysis)

(

+ Tm ax

arctan ~

o'y~ - ~

8c

+

(x - d)h

~, (z - d) 2 + h 2

cZ(arctan h

+ ~

- arctan

x-

(10.31)

+ 2 i ~r~u 2so

cQe 2i~ ~ ~r

x+d

333

x-d

(x +d)h } (x + d) 2 + h 2

arctan

h)

x+d

(10.32)

where/3 and ~0 are given by (10.10) Note that the dilatational component (i.e. the ~-term) makes no contribution to the mean stress and provides a monotonically increasing contribution to the shear stress as x ---. d from x > d. The contributions of the ~-term are more complicated and depend on the angle a (not to be confused with c~ in (10.2)). Hereafter, the analysis is restricted to a - 7r/4 and ~r~ >__0. The choice a - 7r/4 aligns the pure shear axes of the transformation with the planar axis system, while positive values of (r~ ensure that the applied shear stresses do positive work through the transformation shear strain in the band. Under these stipulations and allowing x ---, d from the right gives the stresses __

Trn a x 8c

/10

~ry~ - ~r~ + 2i(r~% 2ic~(d/h) c~ 2d 2sc - 7r[4(d/h) 2 + 1] + ~ arctan -h-

0.34) (

at the edge of the boundary. Interestingly, the mean stress is completely unaffected by the transformation. The/3 contribution to the maximum in-plane shear stress increases monotonically with strip length while the contribution reaches a peak reduction at d/h = 1/2. This suggests that autocatalysis may be linked to the dilatational transformation triggered by shear stresses. As a case study, the pure shear criterion (i.e. c~ = 0) is considered for the three cases of: dilatation alone, shear strain alone, and combination of shear and dilatational strains, all subjected to just a remote shear stress. Setting ~ = 0 in (10.34) and equating 7",na~/Sc to unity provides the formula

Transformation Zones from Discrete Particles

334

cr~-

sc -

11__ ( c ~

2d) 2

-~r arctan ~

(10.35)

(The negative square root has been neglected, since the analysis is restricted to positive shear stresses). This formula determines the applied shear stress necessary to cause the transformation strip to grow in the horizontal direction as a function of starting strip length d/h and the dilatational parameter c~. Since the contribution of the dilatational transformation to rmax/s~ increases monotonically with strip length, the application of the computed value of (r~/sr would cause the strip to propagate unstably. (In interpreting this formula it is essential to consider h as a fixed, non-zero number due to the presence of the log-type singularities at the box corners. Letting h ~ 0 corresponds to allowing the sides of the box to collapse onto the x-axis, for which the analysis breaks down).

Oxy/Sc

c~ =0

1.00 0.75

I

___3__

0.50

6

7

0.25 0.00

I

0

l

2

3

4

9

5

d/h

F i g u r e 10.18" Applied stress ~r~/Sc vs. strip length d/h for various cfl Figure 10.18 shows some plots of cr~/sc against d/h for various c3. For c~ - 0, no stress intensification occurs and the critical stress sc must be applied remotely. As d --+ ~ , (arctan 2d/h) ---+ w/2, so that c~ _< 6 provide the asymptotic stress a~/Sc - V / 1 - (c3)2/36. For c3 > 6, the critical stress drops to zero at a finite strip length de, the

10.4. Self-Propagating Transformation (.4utocatalysis)

335

(Y•v]Sc 3.5

_

3.0 2.5 2.0 1.5 1.O

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0.5 0.0

0

l

,

1

2

a

3

,

4

i

5

d/h

F i g u r e 10.19" Applied stress cr~/sc vs. strip length dlh for various c~

value for autocatalysis. Setting cr~ - 0 and d the relationship de 1 3~" m = _ tan ~ h 2 c/3

de in (10.35) provides (10.36)

As c/3 ---. cx~, d~ vanishes since only a small amount of the powerful transformation is needed to trigger autocatalysis. For a volume fraction c = 0.25 and a m a x i m u m value /3 ~ 40, the peak values of c/3 -~ 10 are consistent with the appearance of extensive transformation regions in materials with sr ,,~ 250 MPa, as reported by Heuer et al. (1988). Now consider shear strains alone. Setting/3 = 0 in (10.34) and solving for the critical stress with a = 0 provides the formula ~r,y = 1 +

sc

2co(d/h) 7r[4(d/h) 2 + 1]"

(10.37)

Some plots of cr~/sc against d/h for various cQ are shown in Fig. 10.19. The peak in reinforcement at d/h = 1/2 means that "short" strips have a significant resistance to further extension, while for "long" strips, the effect of the shear strain vanishes. The peak in the ~ contribution implies that the strip would advance stably up to d/h = 1/2 and unstably thereafter. This result hints at significant interactions between shear and

Transformation Zones from Discrete Particles

336

dilatational strains for short strip lengths and parameter values when the contributions of the ~ and ~ terms are of comparable magnitude. Note that for values of shear angle c~ other than ~r/4, the Q contribution to the maximum in-plane shear stress still reaches a peak value at d/h = 1/2 so that, in general, the conclusions remain the same, although for some orientation angles the shear strain provides a degradation rather than reinforcement. If mean stress is included in the transformation criterion (i.e. c~ > 0), then the shear strains can provide a reinforcing contribution to (10.31) that increases monotonically with strip length. Thus, for "mixed" stress criteria involving both mean stress and maximum in-plane shear stress the shear strains can also drive autocatalysis. When both dilatational and shear strains are present we set the magnitude of (10.34) equal to unity and solve for ~ / s c to obtain the relationship sr

-

~[4(d/h)2+ 1] +

1-

~

arctan

(10.38)

The total stress must overcome the reinforcement provided by the shear strain while being driven by the dilatation. Interestingly, autocatalysis is determined by the dilatation alone and occurs when the argument of the square root function becomes zero so that the formula (10.36) continues to apply. Note that in general, autocatalysis now occurs at non-zero applied stresses. It is also possible to solve (10.38) and find a relationship between ~, Q and a critical strip length dc/h for autocatalysis in the absence of any applied stresses. There are several different solution regimes, depending on the relative magnitudes of the fl and ~0 terms, but the values given by considering just the dilatational term represent conservative estimates. The important result from the continuum strip model is that autocatalysis can be associated with dilatational transformation triggered by shear stresses. The values of c/3 .-- 10 corresponding to the lowest critical stress sc in real materials are consistent with autocatalysis. Further, the sensitivity of the critical strip length to values of cfl > 6 (Fig. 10.18) shows that slight changes in ~ can significantly increase the possibility of realizing autocatalysis. While these results were obtained by considering only the applied shear stress cry, the same general conclusions apply CX3 when non-zero cr~ and ~yy are considered. While we have concentrated on dilatational strains triggered by shear stresses, it should be emphasized that autocatalysis can also be expected

10.4.

Self-Propagating Transformation (A utocatalysis)

337

to occur for shear strains triggered by mean stress. That these particular stress and strain combinations should play a role in autocatalysis is not completely surprising since, in the terminology of plasticity theory, these are non-associative constitutive relations. If the transformations occurred sub-critically rather than super-critically, the above results for the strip length would show that the lack of normality between the normal to the yield surface and the incremental strain vector in stress-strain space would ultimately result in sudden localization when the two vectors became orthogonal.

10.4.2

A R o w of T r a n s f o r m a b l e P a r t i c l e s

l.g a

3--]

F i g u r e 10.20: Infinite row of spots with uniform spacing a In order to study how the strip results apply to a material with transforming particles, this section considers an infinite row of equally spaced spots lying along the z - a x i s of a plane subjected to a remote shear stress ( ~ , see Fig. 10.20. The spot radius is chosen as unity and the nondimensional separation between spot centres is a. Since the transforming particles in actual composites often have randomly oriented shear strain axes, we neglect the effect of shear strains and concentrate exclusively on dilatant particles triggered by m a x i m u m in-plane shear stress. In analogy with the continuum analysis, we will be interested in calculating the applied shear stress necessary to continue the growth of a row of transformed spots. For convenience, the planar coordinate system is placed at the centre of a row of 2N + 1 transformed spots. The transformed spot centres have the coordinates x0 = { - g a , - ( g - 1)a, .... ,0, .... , ( N - 1)a, Na}. From (10.26) we obtain with z0 - ~0 - x0, A0 - ~" Trn a x 8c

9~ ~ ~ax----Y ' -Y+ P sc

-6

g E

'

f't-'--g

(x-

1 na) 2

for locations z - x outside of any transformed spots.

(10.39)

Transformation Zones from Discrete Particles

338

Ox~,/sc 1.25 1.00

6a 2 ~-..__

-0

0.3

0.75 0.50 0.25 " 0.00

Figure

0.6

0.63 \ v.t~z 20

10.21:

t 40

I 60

~ 80

J 100 N

Applied stress a ~ / s c vs. number of particles N for

various ~/6a 2

In order to examine the growth of the spots, we set x - (N + 1)a, enforce rmax/Sc equals unity, and rearrange the expression to obtain the formula

8c -

1-

(

/3 ~

1 (N + f _ n)2

(10.40)

which gives the stress to cause the next spot to transform as a function of total number of spots and the parameter /3. Interestingly, the combination of p a r a m e t e r s / 3 / 6 a 2 enters the formula, so that the spot separation distance scales out of the problem. Figure 10.21 shows some plots of cr~/sc against N for several values o f / 3 / 6 a 2. For ease of visualization, lines have been plotted through the discrete values. In the limit N ---. oc, the series is a form of the Riemann Zeta function and converges to the value 7r/6 ~ 1.645. Thus, the curves for values fl/6a 2 < 1/1.645 ,~ 0.608 approach non-zero asymptotic limits for N --+ oc. For values of ~/6a 2 > 0.608, the argument within the squareroot radical becomes negative corresponding to autocatalysis, at finite oo N. The last real values of (r~u/sc, occurring at the critical value N - No, are indicated by the abrupt termination of the curves in Fig. 10.21. The

10.4. Self-Propagating Transformation (A utocatalysis)

339

6a 2 ~c

1.7

#

1.6 1.5

f

1.4 1.3 1.2 1.1 1.0

0

I

I

I

I

I

20

40

60

80

100

Nc

F i g u r e 10.22: Critical autocatalysis ratio 6a2//3~ vs. critical number of particles Nc acute sensitivity of Arc to values /3/6a 2 just greater than 0.61 shows how powerfully the shear stress criterion is driven by dilatational transformation. (A similar trend for cfl just greater than 6 is seen for the continuum strip results in Fig. 10.18). Finally, setting N - 0 in (10.40) shows that values of ~/6a 2 > 1 cause such severe stress increases that the transformation of a single spot is sufficient to trigger the transformation of the infinite row. This behaviour is referred to as "spontaneous transformation". Some general comments on Fig. 10.21 are in order. First, unlike the strip model, in which the strip could always become vanishingly small, there is an upper limit to the practical value of/3 in particle-reinforced materials, since it is virtually impossible that none of the particles will have spontaneously transformed due to internal weaknesses or flaws. Secondly, as a must be equal to or greater than 2 to prevent spots overlapping, the analysis predicts that/3 >_ 15 for autocatalysis to occur and /3 _> 24 for spontaneous transformation. For particle volume fractions of 1/4, it is reasonable to expect that some particles will be distributed close enough that a value of a ~ 3 might apply over a particular region. One would then expect to see autocatalysis appear a t / 3 -~ 33. Again,

340

T r a n s f o r m a t i o n Zones from Discrete Particles

the sensitivity of autocatalysis to values just above j3/6a 2 ~ 0.61 shows that a very abrupt transition in behaviour is to be expected. To examine autocatalysis, we solve for the values of 13/6a 2 ..~ 0.61 as function of N = Nc that cause the argument in (10.40) to vanish. The resulting formula is 6a 2 Z~ -

gc

~

n--- Nc

1

(No+l-n)

2

(10.41)

where /3 = /3c is plotted against Arc in Fig. 10.22. A curve has been plotted through the discrete values to aid visualization. In the limit Nc ~ oo, 6a2//3c ---, 1.645, as discussed above. The curve of Fig. 10.22 has a very similar appearance to that obtained by plotting the tangent function in (10.36). Here, however, the plot terminates at N --- 1 rather than running through the origin, since there must always be at least one transformed particle for interactions to occur. In summary, values of 6a2/fl~ _< 1 lead to spontaneous transformations, the parameter range 1 < 6a2/fl~ <_ 1.645 exhibits autocatalysis, while 6a2/~3~ > 1.645 will not lead to self-propagating transformations.

Part I11

Related Topics

This Page Intentionally Left Blank

343

Chapter 11

Toughening in DZC 11.1

Introduction

In this Chapter we shall consider ceramics which contain dispersed zirconia precipitates in various proportions. An example of such dispersed zirconia ceramics (DZC) is the commonly used zirconia toughened alumina (ZTA). The toughening in DZC can arise from two complementary mechanisms depending on the content of t-ZrO2. Recent in situ transmission electron microscopic studies by Riihle et al. (1986) on various ZTA compositions containing a fixed total content of ZrO2 (15 vol%) but a variable proportion of t-ZrO2 (between 0.23 and 0.86) have demonstrated the complementary nature of phase transformation and microcrack mechanisms in the toughening of these ZTA. They found that at low volume fractions of t-ZrO2 there was no stress-induced phase transformation, so that the toughness increment was primarily due to microcrack-induced dilatation around thermally formed m-ZrO2 precipitates (see the low t-ZrO2 end of Fig. 11.1). With an increase in the volume fraction of t-ZrO2, the proportion of t ---, m transformation due to the high stress at a sharp crack tip was seen to increase. In fact, at the largest volume fraction of t-ZrO2 studied (86% of the total ZrO2 content) the stress induced t ---. m transformation toughening mechanism would seem completely to dominate over the microcrack mechanism (see the high t-ZrO2 end of Fig 11.1). This is because the stress reduction generated by the t ---, m dilatation would not permit stress-induced microcrack initiation from m-ZrO2 created by the phase transformation. In this Chapter we shall first (w consider the extreme situation when the ZTA composition contains mostly t-ZrO2 precipitates, so that

344

Toughening in DZC

the toughening is a result of phase transformation alone. We shall then (w consider the other extreme situation when the ZTA composition contains mostly m-ZrO2 precipitates, so that the toughening is primarily induced by microcracking. We note en passant that the contribution of microcracking to the toughening of PSZ or TZP is believed to be only minimal. But even in these materials slight mismatch in the elastic constants of t-ZrO2 and m-ZrO2 can have a significant effect upon the toughening process. We shall study the effect of small moduli differences upon the toughening of TTC in Section 11.4. When the differences in the elastic moduli are large, as in all DZC, the perturbation approach taken in w11.4 is no longer applicable. In these cases we shall introduce in w an approach based on the concept of effective transformation strain.

11.2

Contribution of P h a s e Transformation to the Toughening of D Z C

We shall calculate the toughness increment resulting from the stressinduced dilatational component of the t ---. m transformation in a DZC on the example of a zirconia-toughened-alumina (ZTA) composition containing a high proportion of tetragonal zirconia precipitates and show that it agrees very well with the experimental value. The good agreement is made possible by allowing for the mismatch in the elastic constants between the zirconia particles and the alumina matrix, and for the observed variation in the size of the transformable tetragonal particles with the height of the transformation zone. The actual variation is estimated from experimental data (Riihle et al., 1986) which indicate that large particles (_>0.18#m) are more prone to stress-induced transformation than are the small ones. As far as the mismatch in the elastic constants is concerned, it is taken into account by calculating the two-dimensional dilatation appropriate to the composite of ZrO2 and A1203 (McMeeking, 1986; Rose, 1987a; see w No attempt will be made to estimate the influence of the shear component of the phase transformation or of a stress-induced transformation criterion other than the critical mean stress criterion. The exposition will follow closely the paper by Karihaloo (1991).

345

11.2. Phase Transformation and Toughening of DZC

1300 7

1200

t

5 4 3 exo

=

2

1100 ~ 1000 ~

900

~

800

r~

7OO

1 0

I

0

20

I

I

I

600

100 40 60 80 Tetragonal ZrO2 [%]

F i g u r e 11.1" Bend strength and fracture toughness of ZTA (total ZrO2 content - 15 vol%) as function of t-ZrO2

11.2.1

Experimental

Evidence

For future use and completeness of presentation, it is convenient to summarize briefly the experimental evidence on mechanical properties, transformation characteristics, and microcrack density (Riihle et al., 1986; Evans, 1989). Figure 11.1 shows the variation of four-point-bend strength and ISB (indentation strength in bending) fracture toughness with increasing tZrO2 content. All ZTA compositions containing a fixed total content (15 vol%) of ZrO2 have much higher toughness than pure A1203 (~3.5 to 4 MPax/~). The ZrO2 size distributions in the compositions containing the highest (86% t-ZrO2) and lowest (23% t-ZrO2) fractions of t-ZrO2 have been studied stereologically in a TEM. These studies have shown that the mean particle size of ZTA with 86% t-ZrO2 is 0.4pm and that a critical particle size of 0.6#m exists for spontaneous t ---. m transformation on cooling. From in situ straining experiments in TEM it was found that ZTA with 86% t-ZrO2 had a well-defined transformation zone and that larger particles (>0.18#m) were more prone to stress-induced transformation

346

Toughening in DZC

Nm Win+N,

5

-10

-5

~_4_ 4 4

~4

5_5 .

.5_

L

676__6

6--

3_ -.37 7

7__

s ssr 0 5 10 Distance from crack plane [gm]

F i g u r e 11.2: Variation of ZrO2 particle size in the transformation zone of ZTA containing 86% t-ZrO2. Numbers refer to size range groups of Table 11.1. Nm and Nt are the fractions of m- and t-ZrO2

Size group

Size range log scale (gm)

4 5 6 7 8

0.18-0.24 0.24-0.33 0.33-0.44 0.44-0.59 0.59-0.79

9 N f ( o ) - N2~N, Fig.ll.2 0.140 0.225 0.349 0.186 0.070

VI 0.100 0.200 0.325 0.250 0.125

Weighted NF(O) = Nf(O)Vf 0.014 0.051 0.106 0.046 0.009

T a b l e 11.1: Calculation of f(0)

than were the smaller particles. The size distribution of particles was found to vary along the height of the transformation zone, as can be seen from Fig. 11.2. The various size groups noted on this figure are defined in Table 11.1. In ZTA with only 23% t-ZrO2, on the other hand, no transformation zone was observed. However, from thin foils of known thickness TEM studies showed radial matrix microcracks. All such radial microcracks occurred along grain boundaries in A1203. Moreover, the interface between the A1203 and ZrO2 was usually debonded at the origin of the microcracks. We shall study the microcracking mechanism below in w11.3.

11.2. Phase Transformation and Toughening of DZC

347

E

x

a) A1203 = 85.00 % t-ZrO 2 = 3.45 %

0.i5 " "Microcrack density variation

Crack \

m-ZrO 2 = 11.55 %

/Transformation process zone / /Microcrack process zone

E

9y

X

0.15 "

b) A1203 = 85.00 %

Crack \

t-ZrO 2 = 12.9 % m-ZrO 2 = 2.1%

F i g u r e 11.3: Steady-state t ~ m transformation (hatched) and microcrack (cross-hatched) process zone around a macrocrack in two ZTA compositions

11.2.2

D i l a t a t i o n a l C o n t r i b u t i o n to the T o u g h e n i n g of ZTA

The aforementioned experimental evidence is graphically illustrated in Fig. 11.3, which shows the steady-state t ~ m transformation (hatched) and microcrack (cross-hatched) process zones around a macrocrack in the two ZTA compositions. Also shown are the distribution of microcrack density parameter and, where applicable, the size distribution of transformed particles, f(y) along the height of the process zone. As mentioned above, the toughness increment in the ZTA composition containing only 23% t-ZrO2 is due mainly to microcracking around the thermally formed m-ZrO2 particles (see Fig. l l.3a). This increment will be calculated in w

Toughening in DZC

348

Here we calculate the toughness increment in the ZTA composition containing 86% t-ZrO2 (i.e. 12.9 vol% of t-ZrO2 out of the total 15 vol% of ZrO2). It is evident from Fig. l l.3b that the toughness increment for this composition must result from both the microcracking around the thermally formed m-ZrO2 particles and the t ~ m transformation of the particles. Note, however, that since t --~ m dilatation results in a reduction in the hydrostatic stress in the transformation zone, no microcracking can be expected from the m-ZrO2 particles formed by stress-induced t ~ m transformation. From the measured heights of process zones due to t ---, m transformation and microcracking, it is clear that the latter zone is completely enveloped by the former. It is therefore reasonable to ignore the minor contribution of microcrack mechanism to the toughening of this compositi.on and to assume that the toughness increment is due almost exclusively to the dilatation resulting from the t ---. m transformation. A rough estimate of the contribution from microcracking and from the deflection of the microcracks may be obtained from the following relation (Li & Huang, 1990) F =

V/1 § 0.S7V! ~/1

-

:'~ Y/(1-

(11.1) ~,~)

where F is the ratio of the effective fracture toughness of the composite in the presence of microcracks and crack deflection to the fracture toughness of the matrix. Vl is the volume fraction of ZrO2 (=0.15) and um (=0.2) is Poisson's ratio of A1203. For the composition under study, the toughening ratio is just 1.07. Neglecting the small contribution from microcracking, we can formally write an expression for the toughness increment from the dilatational component of the phase transformation under steady-state plane strain conditions (w AI~"'iP

--

-ilia

(1 - P)

D(y)

o 2V~

cos

(3r162

(112)

where Ac refers to the area of transformation zone above the crack plane, and P are the effective shear modulus and Poisson's ratio of the ZTA composite, and D(y) is the two-dimensional plane strain dilatation corresponding to the lattice dilatation eT (~0.04) due to t ---. m transformation of a t-ZrO2 particle (Fig. 11.4; see (7.1)). Since the elastic constants of the transforming ZrO2 inclusion (#i 78GPa, ui ~ 0.31) differ much from those of the nontransforming A1203

11.2. Phase Transformation and Toughening of DZC

/

Transformationprocess zone

~Y

349

r

L 0) J r-

v

~)

w!

F i g u r e 11.4: Steady-state t --, m transformation zone in ZTA containing 86% t-ZrO2, showing the coordinate system and the approximate size distribution of transformable t-ZrO2 particles matrix (#.~ ,,~ 169GPa, ~m ~ 0.2), the effective elastic moduli are appropriate for relating D(y) to eT. The effective moduli of the two-phase composite ~ and ~ can be estimated by using Hill's (1963) self-consistent approach which requires the solution of the following two nonlinear simultaneous equations Vm

t~

N-B----~. + ~ - B m Vm - #~

[

t~ -]2

--

-- ~m

3

- 3B+4~ 6 (B + 2~) 5~ (3B + 4~) __

(11.3)

where B, Bi and Bm refer to the effective bulk modulus, the bulk modulus of ZrO2 (~ 180GPa), and the bulk modulus of A1203 (,~ 226GPa), respectively, and Vm (=0.85) and ~ (-0.15) are the volume fractions of the matrix and the transforming phase. The solution of eqns (11.3) with the indicated values of Bi, t~, Brn, Vm is ~=151GPa, B=218GPa. The relation between the two-dimensional plane strain dilatation D(y) and the lattice dilatation, eT can also be estimated in the spirit of self-consistent theory by considering the deformation of a single, quasispherical transforming ZrO2 particle within a homogeneous matrix that has the elastic constants of the composite (Rose, 1987a)

D(y)-

2(1---ff)f(y)eT 1 + 4-fi/3Bi

(11.4)

where f(y) is the local value of the volume fraction of transformed ma-

Toughening in DZC

350

terial. Experiments show (Rfihle et al., 1986) (see Fig. 11.2) that follows a bell shaped curve along the height of the transformation (H ~ :t:10ttm). However, to simplify calculations, we assume that diminishes linearly along the height of the transformation zone, that in the upper half of the zone (y >_ 0)

s(.) - s(0)(,-

Y

f(y) zone

f(y)

such

(11.5)

where the volume fraction of transformed material adjacent to the crack faces, f(0), is estimated from the experimental data as follows. The fraction of various size groups that have transformed within a height of lpm on either side of the crack plane are averaged (Fig. 11.2) and weighted by the corresponding volume fraction of the size group. This is explained in Table 11.1. The sum of these weighted fractions gives f(0) = ENf (0)Vf = 0.226. Substituting eqn (11.5) into eqn (11.4) with this value of f(0), and referring to Fig. 11.4, the steady-state toughness increment (11.1) from t --* m transformation may be rewritten as

AK tip-A

/~'/3J0Br176162 L"

+A where H

r-l/2

.so

L hlsin(r

r -1/2

1 (

1

/3

H rsinr

cos ( 3- r-

cos

H

drdr

(3~) - - drdr (11 6) "

10pm,B - 8H/(ax/3) (see (7.18)), and 'fif (O)eT A-

(1 + 4~/3Bi)

= 0.00341

(11.7)

The first of the two integrals in eqn (11.6) that gives the contribution from the zone in front of the crack tip bounded within the fan between r - 0 and r - 7r/3 can be evaluated analytically and is equal to 1.344Av/-ff. The second integral that gives the contribution from the wake of transformation zone (7r/3 < r < 7r) has been evaluated numerically and is equal to -2.149Ax/~. Substituting these two contributions together with the constant A from eqn (11.7) into eqn (11.6) gives AKtip=-I.31MPax/~, the negative sign indicating the shielding effect of transformation on the crack tip. The fracture toughness of ZTA containing 85 vol% A1203, 12.9 vol% t-ZrO2, and 2.1 vol% m-ZrO2 is therefore approximately equal to K~41umina- AKtip=4.81 to 5.31 MPavfm. This

11.3. Contribution of Microcracking to the Toughening of DZC

351

value is in close agreement with the measured ISB fracture toughness (5.25 5= 0.35 MPav/m-) for this composition (Fig. 11.1). It is interesting to note that were f(y) assumed to remain constant and equal to f(0) over 0 _< y < H as is customarily done (see w the resulting fracture toughness of the composition would work out to be 6.12 to 6.62 MPax/~, which would overestimate the measured value.

11.3 11.3.1

Contribution of Microcracking to the Toughening of DZC Introduction

As mentioned above, there is growing evidence (Riihle et al. 1987) that microcracking in regions of high stress concentration or at the tip of a macroscopic crack may postpone the onset of unstable macroscopic crack propagation in brittle solids such as DZC. For this mechanism to operate it is essential that the microcracks arrest at grain boundaries or particle interfaces and be highly stable in the arrested configuration. Ultimately the macroscopic crack advances by interaction and coalescence of the microcracks. But the microcrack zone can also have a shielding effect on the macroscopic crack tip, redistributing and reducing the average near-tip stresses. There are two sources of the redistribution of stresses in the near-tip stress field of the macroscopic crack. One is due to the reduction in the effective elastic moduli resulting from microcracking. The other is the strain arising from the release of residual stresses when microcracks are formed. The residual stresses in question develop in the fabrication of polycrystalline or multi-phase materials due to thermal mismatches between phases or thermal anisotropies of the single crystals. The spatial variation of these stresses is set by the grain size or by the scale of second phase particles. These residual stresses play an important role in determining the onset and extent of microcracking. Moreover, the microcracks partly relieve the residual stresses producing strains which are manifest on the macroscopic scale as inelastic strains. A continuum approach developed by Hutchinson (1987) will be described below in which it is assumed that a typical material element contains a cloud of microcracks. The stress-strain behaviour of the element is obtained as an average over many microcracks. A characteristic tensile stress-strain curve is shown in Fig. 11.5. The Young modulus E of the uncracked material governs for stresses below ~rc where microcracking

Toughening in DZC

352

~s

~c E ,,,,l

L..

,,

E

T

F i g u r e 11.5: Characteristic tensile stress-strain curve

first sets in. It will be assumed that microcracking ceases, or saturates, above some stress ~r~. The assumption of the existence of a saturated state of microcracking is fairly essential to the analysis described below, as will become evident later. It does seem reasonable to expect that the sites for nucleation of microcracks will tend to become exhausted above some applied stress level when local residual stresses are playing a central role in the microcracking process. Thus, it is tacitly assumed that there exists a zone of nominally constant reduced moduli surrounding an even smaller fracture process zone within which the microcracks ultimately link up. A reduced modulus E8 governs incremental behaviour for stresses above cry. The offset of this branch of the stress-strain curve with the strain axis, ~T, is the contribution from microcracking due to release of the local residual stresses. It can be thought of as a transformation strain. Two of the most important assumptions involved in the formulation of the constitutive law deal with the distribution of the orientations of the microcracks, whether the reduced moduli are isotropic or anisotropic for example, and the stress conditions for the nucleation of the microcracks. Recent microscopic observations of a zirconia toughened alumina (Rfihle et al. (1986) suggest that the microcracks which form in this material have a more-or-less random orientation with no preferred orientation relative to the applied stress. This would be consistent with the random nature of the residual stresses expected for this system. Nevertheless, there is not yet nearly enough observational information or theoretical

11.3. Contribution of Microcracking to the Toughening of DZC

353

insight to justify any one constitutive assumption. The approach taken below is to consider a number of reasonable options, so that the results discussed here will serve to bracket actual behaviour and give some indication of which uncertainties are most crucial to further development.

Y

E,v

F i g u r e 11.6: Geometry of the microcracked zone surrounding the tip of a semi-infinite crack From the point of view of mechanics, we consider the problem shown in Fig. 11.6. A microcracked region, Ac, surrounding the crack tip has reduced moduli which are uniform and isotropic. In analogy with phase transformation, a uniform dilatation ~T is also present associated with the release of residual stress. The crack is semi-infinite with a remote stress field specified by the applied stress intensity factor K appt, modelling a finite length crack under small scale microcracking conditions. The near-tip fields have the same classical form but their stress intensity factor, K tip, is different. It is the toughening ratio KaPPZ/Ktip which is sought as a function of the moduli differences, ~T and the shape of the zone. The knowledge of this ratio is not sufficient to predict the toughening increment due to shielding, because microcracking reduces the intrinsic toughness Kc of the matrix. The knowledge of KaPPZ/Ktip for different situations, such as stationary or growing cracks, can be used to make comparative assessments of macrocracking behaviour and to gain insight into phenomena such as stable crack growth.

11.3.2

Reduction Stress

in Moduli

and Release

of Residual

The following two examples are chosen to illustrate the way microcracking can reduce the moduli of a brittle material and give rise to inelastic strain by release of residual stress.

354

T o u g h e n i n g in D Z C

P e n n y - s h a p e d m i c r o c r a c k s in a p r e s t r e s s e d s p h e r i c a l p a r t i c l e

E,v

a) ,,

2b E,v

b) r

-I

F i g u r e 11.7: Two prototypical microcrack geometries" (a) pennyshaped microcrack in a spherical particle, (b) annular microcrack outside a spherical particle Consider the configuration of Fig. l l . 7 a which shows an isolated spherical particle or grain of radius b embedded in an infinite matrix. Both particle and matrix are assumed isotropic and with common Young's modulus E and Poisson's ratio u. Suppose the particle sustains a uniform residual stress prior to cracking. Let ~rR denote the normal component (assumed positive) acting across the plane where the microcrack will form. There is zero tangential traction on this plane. Now suppose a penny-shaped microcrack is nucleated which arrests at the interface of the particle and the matrix as shown in Fig. 11.7a. The volume of the opened microcrack is 16b3 aR AV - T ( 1 - u 2) E (11.8) The release of the residual stress creates an inelastic strain contribution. If the microcrack forms within a material element of volume V and if interaction with other microcracks is ignored, the inelastic strain contribution is Aeij

= A V V ninj -

16 b3 O'R 3 V (1 - u 2 ) - - f f n i n j

(11.9)

where ni is the unit normal to the plane of the microcrack. This is a

11.3. Contribution of Microcracking to the Toughening of DZC

355

uniaxial strain contribution with dilatational component Ackk =

16b a an (1 - u2)__~ 15 3 V

(11.10)

The formulation of the microcrack also increases the compliance of the material element (Budiansky & O'Connell, 1976). If crij is the macroscopic stress experienced by the material element, the increase in strain due to a component of stress acting normal to the plane of the microcrack (i.e. crnn = O ' i j n i n j ) is b3 finn (11 11) A~nn = -3- ( 1 - u 2) V E Any component of stress acting tangential to the plane of the microcrack (i.e. crnt - ~rijnitj, where ti is parallel to the crack face) gives rise to an increase in the corresponding strain component which is 16 (1 - g2)b 3 ~ t Ac~, = y ( 2 _ u ) V E

(11.12)

These contributions to the strain are also based on the assumption that interaction between the microcrack and its neighbours can be ignored. If the microcracks have random orientation with no preferred alignment, the microcracked material will be elastically isotropic on the macroscale and the strain due to the release of the residual stresses will be a pure dilatation. Suppose there are N microcracks per unit volume and let 0 be the measure of the microcrack density, where g is the average of Nb 3. With E and P denoting Young's modulus and Poisson's ratio of the microcracked material, the total strain following microcracking is obtained by averaging the contributions (11.9)-(11.12) over all orientations with the result (cf. (3.17))

+ v O'ij -- -~O'kk~ij ~~ gij -- 1 __ Jr- -310T~i j

(11 13)

0T _ ~ ( 1 - - U 2 ) 0 E

(11.14)

E

where

The notation here is deliberately chosen to be the same as that for a dilatational phase transformation since at the macroscopic level the dilatation due to release of the residual stress is indistinguishable from that due to phase transformation. The modulus E and Poisson's ratio

Toughening in DZC

356

of the microcracked material can be obtained from 3 2 ( 1 - v ) ( 5 - v) P = 1 + -~0 45 (2-v)

(11.15)

and B

--

B

-

1 + -

16 (1 - v 2) 9 (1 - 2u)

~

(11.16)

where # and B are the shear and bulk moduli of the uncracked material and ~ and B are the corresponding moduli for the microcracked material. These estimates of the moduli, which ignore microcrack interaction, agree with the dilute limit of estimates which approximate interaction (Budiansky &: O'Connell, 1976). They are reasonably accurate for values of Q less that about 0.2 and 0.3, and it is expected that the residual stress contribution in (11.13) will be accurate within this range as well. Annular

microcrack

around

a prestressed

spherical particle

Now, consider a spherical particle which has a residual compressive stress due, for example, to transformation or developed during processing as a result of thermal mismatch between particle and matrix. Referring to Fig. 11.7b, we suppose that the particle nucleates an axisymmetric microcrack at its equator with the outer edge of the crack arrested by some feature of the microstructure. Usually such a microcrack runs along a grain boundary and arrests at a boundary junction. We model the situation by taking the particle to be under a residual uniform hydrostatic compression O'ij "-- --O'R6ij prior to cracking. If, for example, this residual stress arises as a result of a dilatational transformation strain in the particle of r l~T~ij, then crn =

2EOT 9 ( 1 - v)

(11.17)

The moduli of the matrix and particle are again taken to be the same. The residual normal traction in the matrix acting across the plane of the potential crack is a-

a0

(11.18)

where ~0 - crn/2 is the tensile circumferential stress in the matrix just outside the particle, and r is the distance from the centre of the particle.

11.3.

Contribution of Microcracking to the Toughening of DZC

357

The volume of the annular crack due to the partial release of the residual stress (11.18) is given approximately by AV -- 7r2(1-

v2)ab2 ( 1 _ ab ) 2 (r0 E

(11.19)

Once the microcrack is nucleated it gives rise to an additional strain contribution (in a material element of volume V) in the direction normal to the crack plane

bc2-A~,~n - ~r2(1-v2)-~

( 1 + ~2c) -ann ~ F(c/b)

(11.20)

where ann is again the macroscopic stress component normal to the plane of the crack, and c = a - b. The function E(c/b)is 1 when c / b - 0 and monotonically decreases to 0.81 when c/b --+ co; it is very close to 1 for c/b _< 1. (The formula (11.20) can be derived from results given in the handbook by Tada et al. (1985). The counterpart to (11.20) for the shear strain contribution Ac,~t is not available). With N noninteracting annular, randomly oriented microcracks per unit volume, the strain is still given in terms of the macroscopic stress by (11.13) where now from (11.19)

0T -

NTr2(1-

~,Z)ab2 1 -

-~-

(11.21 /

The result (11.20) is not sufficient to determine estimates for E and since A~nt is also needed. However, if one assumes that the ratio of ent/trnt to ~nn/trnn is the same, or at least approximately the same, for the annular crack as for the penny-shaped crack, then E and F can still be obtained from (11.15) and (11.16). Now, however, by comparing (11.11) and (11.20), one sees that the crack density parameter must be taken as

37r2Nbc2 1 + ~-

16

F(c/b)

(11.22)

This formulation provides the density of annular microcracks measured in an equivalent density of penny-shaped cracks for the purpose of determining the reduction in moduli. The parameter proposed for arbitrarily shaped microcracks, Q = 2NA2/(TrP) where A and P are the area and perimeter (inner plus outer) of the crack, provides an excellent simple

358

T o u g h e n i n g in D Z C

approximation to (11.22). Riihle et al. (1987) found in ZTA containing a low volume fraction of t-ZrO2 that each ZrO2 particle is circumvented by a radial microcrack, consistent with the symmetry of the residual strain field around each particle. They also found that the microcrack density diminished with distance from the crack plane; the maximum density ~0c adjacent to the crack faces suggests a saturation value determined by m-ZrO2 content (Fig. 11.3a).

11.3.3

Ktip/K ~ppt for Arbitrarily Shaped Regions Containing a Dilute Distribution of Randomly Oriented Microcracks

Uniformly distributed microcracks Some general results for the plane deformation problem depicted in Fig 11.6 will now be presented. A semi-infinite crack lies on the negative x-axis. Within the microcracked region r _< R(0) , the material is governed by (11.13) where 0T can be thought of as a stress-free dilatational transformation strain. Within Ac, E, P and 0T are taken to be uniform. Outside this region the material is governed by s

--

l+v

E

v

O'ij -- --~O'kk~ij

(11.23)

The region Ac is restricted to be symmetric with respect to the x-axis. In analogy with phase transformation, when E - E and P - v, K tip is given by (7.14). When E and V differ from E and v, numerical work is generally required to obtain the relation K tip and K ~ppz. However, this relation can be obtained in closed form to lowest order in the differences between the moduli governing behaviour within and without A~. Moreover, to lowest order in these differences the contributions to K tip from 0 T and from the reduction in moduli within Ar can be superimposed. We shall see later in w that the superposition assumption is only partially valid. We proceed by considering the case 0 T - O, when one can conclude from dimensional analysis alone that l~[ t i p

Kapp z -

--~ F(---~, u,-if)

(11.24)

where F also depends on the shape of Ac, but not on its size. However, it is known that this relationship can be reduced to dependence on just two special combinations of the moduli (the so-called Dunders' parameters).

11.3.

Contribution of Microcracking to the Toughening of DZC

359

For present purposes the most convenient choice of moduli parameters is 1

61--i; v

1

62 - 1 ; / /

u~ - u

]

(11 26)

which both vanish in the absence of any discontinuity across the boundary of Ac. These parameters emerge naturally in the analysis which we shall here omit. Interested readers may consult the paper by Hutchinson (1987). With this choice Ktip I~appl --

f(61,~2,shape of

(11.27)

Ac)

The following result is exact to lowest order in 61 and 62 K tip i~appl

where kl -

k2 =

~

lf0~

3 ---- 1 + (kl - ~5)61 + (k2 + )52

(11 cos 0 + 8 cos 20 - 3 cos 30)

(11.28)

ln[R(O)]dO

27rl~0?i"(cosO+cos20)ln[R(O)]dO

(11.29) (11.30)

The integrals defining kl and k2 also appear in a different context, as we shall see in the next section (w11.4). Since the collection of terms in each integrand multiplying ln[R(0)] integrates to zero, kl and k2 are unchanged when R(O) is replaced by AR(O) and are thus dependent on the shape, but not on the size, of Ac. If Ac is a circular region centred at the tip, kl = k2 = 0. 11.3.4

KtiP/K~PPt f o r tionary

and

two Nucleation Steadily-Growing

Criteria

for Sta-

Cracks

The results of the previous Section are now specialized to specific zone shapes dictated by two possible microcrack nucleation criteria. The first is based on the mean stress; the second is based on the maximum normal stress. In each case, it will be assumed that there is no preferred orientation of microcracks so that the reduced moduli are isotropic. Results

Toughening in DZC

360

for both stationary cracks and cracks which have achieved steady-state growth conditions will be given so as to assess the potential for crack growth resistance following initiation. In every example, the zone shape and size are determined using the unperturbed elastic stress field (7.3) since this is consistent with our limited aim of obtaining just lowest order contribution to K tip. The perturbation of the size of the zone is likely to be relatively unimportant for the effect of the reduced moduli even for non-dilute crack distribution since the lowest order results for Ktip/K appl are independent of zone size, as discussed in the previous Section.

S t a t i o n a r y Crack w i t h N u c l e a t i o n at a Critical M e a n S t r e s s Zm

Saturated state

$

Zm

Saturated state

C

Zm

Zm C

Zm

Simplified criterion

,

a)

N

b)

N

F i g u r e 11.8: Variation of microcrack density N with mean stress With Em = ~rkk/3 as the mean stress, suppose microcracks begin nucleating at E ~ and the nucleation is complete at E~ with a variation in microcrack density N as indicated in Fig. 11.8a. To lowest order the elastic stress distribution (7.3) can be used to determine the zone shape and the distribution of the microcrack density within the zone. The distribution of the density and the relation of the inner region of uniform muduli to the full microcrack region fits precisely into the situation discussed in the preceding Section. Thus, the change in K tip due to the moduli reduction is the same, to lowest order, as when the microcracks are uniformly distributed throughout the zone. We will therefore restrict attention to the simplified nucleation criterion indicated in Fig. l l.8b and take

11.3. Contribution of Microcracking to the Toughening of DZC

~-0

361

for (Y]m)max < ~Crn (11.31)

= N for ( ~ m ) m a x ~ ~ c

The 0T-contribution to K tip does depend on the distribution of the microcrack density, but this can be evaluated fairly simply using (7.10) if desired. Here only the results for the simplified nucleation criterion (11.31) will be given. There will be a transition region just within the boundary to Ac in which the microcrack density varies from zero to the saturated value, but in the limit corresponding to the lowest order problem the transition region shrinks to zero. Imposing E m - E~ on the elastic field (7.3) one finds R(O) -

2

~--~(1 + u) 2

(Kappt)20

cos 2 2

E~

(11.32)

which is identical in form to the transformation zone boundary (7.12), except that the critical mean stresses for transformation and microcrack nucleation can be significantly different. The boundary of the microcracked zone is shown in Fig. 11.9a. Then, evaluating kl and k2 in (11.29) and (11.30), one obtains ]r -- 3/16 and k2 = - 1 / 4 . The 0T-contribution is found to be identically zero (as for a stationary crack under phase transformation (w so that the combined effect is given by just (11.28)

K tip

K.pp t = 1 -

_~

1

61 + ~62

(11.33)

To specialize the result even further we will use the results (11.15) and (11.16) for the reduced moduli ~ and B in terms of the crack density parameter ~0 which in turn is given by the average of Nb 3, or by (11.22), or by any other appropriate choice depending on the nature of microcracking. To lowest order in ~ one can show that

and

32(5-

(~1 -- ~

u-

163 (3 - v)(1 - v 2) 1--5( 2 - u) 0

_ U) ~0 -- 1.0990

1

(11.34)

Toughening in DZC

362

2 appl

1.0 Boundary of wake for steady-state problem

(l+v) (K

/

0.5

a)

Boundary for stationary roblem

0

0.0

c 2

/ Era)

I

'

I

0.5

-

1.0

x

c 2 (1 +v) 2(g appl/ ]~m)

(KPPl[ ]E1c 2)

0.4

H

b) 0.0

0.4

0.2

(Fppl

c

2

/Z I )

F i g u r e 11.9: Zones of microcracked material for stationary and steadily growing cracks for two nucleation criteria. (a) Critical mean stress criterion, (b) critical m a x i m u m principal stress criterion

62 = 16u(1 - 8u + 3u 2) 4 5 ( 2 - u) ~and thus

-0.095~o

(u - 1 ~)

(11.35)

Ktip

= 1 - 2 ( 3 5 - l l u + 32u 2 - 12u 3)~, Kappl 4 5 ( 2 - u) - 1-0.9196

(u-

1

~)

(11.36)

11.3. Contribution of Microcracking to the Toughening of DZC

363

S t e a d i l y - G r o w i n g crack w i t h N u c l e a t i o n at a C r i t i c a l M e a n Stress A crack which has extended at constant K appl has a wake of microcracks as indicated in Fig. 11.9a. With the nucleation criterion (11.31) in effect, the leading edge of the microcracked zone is given by (11.32) for [01 < 600 , and the half-height of the zone is given by H -

x / ~ ( l + u ) 2 (KaVVt) 127r \ ~

(11.37)

The values of kl and k2, which have been computed by numerical integration, are kl - 0 . 0 1 6 6

and k2 = - 0 . 0 4 3 3

(11.38)

The combined effect of moduli reduction and microcrack dilatation is given by (11.28) and (7.14) K tip

is

-

-

1 - 0.60861 4- 0.70762

1

(1 4- u) EO T

47rx/~ (1 - u) E~

= 1 - 0.60861 4- 0.70762 - 2(1 - u)Kavv z

(11.39)

where the 0T-contribution is the same as that for the corresponding transformation problem (7.21). Equations (11.35) for 51 and 52 still pertain and for tt = 1/3 Ktip EOT v/-H Kapp I -- 1 - 1.2780-0.3215 i~app-"-'-------------[-

(11.40)

By comparing (11.36) and (11.40), one notes that the shielding contribution due to moduli reduction is about 40% larger for the growing crack than for the stationary crack. This will add to crack growth resistance but the major source of resistance is likely to come from the release of residual stress (i.e. from 0w). Even without growth, however, moduli reduction provides some shielding according to (11.36) although how much extra toughness this generates cannot be predicted without knowledge of the toughness of the microcracked material within Ac, as already emphasized.

Toughening in DZC

364

Stationary C r a c k w i t h N u c l e a t i o n at a C r i t i c a l M a x i m u m N o r -

mal Stress

Now suppose that the microcracks are still nucleated with no preferred orientation so that within Ac the stress-strain relation is still (11.13), but suppose that nucleation occurs when a maximum principal stress O"I reaches a critical value E~, i.e. ~0- 0 for (~I)max < ~ (11.41)

= N for (~I)max _> ~

where as before, ( )max signifies the maximum value attained over the history. The boundary Ac as determined by (7.3) is now (of. (10.6)) 1 (

R(O) - ~

0 1 )2(I.~vPz)2 cos ~ + ~sin [0[ El

(11.42)

and this is shown in Fig. 11.9b. The value of kl and k2 have been obtained by numerical integration of (11.29) and (11.30) with the result kl

--

0.0779,

k2 - -0.0756

(11.43)

The combined effect of moduli reduction and microcrack dilatation is given by (11.28) and (7.14)

Ktip EoT Kapp I - 1 - 0.54761 + 0.67462- 6 r ( 1 - v)E~ = 1 - 0.54761 + 0.67462 - O.1060EOT (1 - l])]~ appl (11.44) where the half-height of Ar from (11.42) is obtained at 0 is

Hfor v -

1/3 and with

61

Ktip

i~appl

74.840 and

0.2504(KappZ) 2 E~

(11.45)

and 62 given (11.35), (11.44)reduces to ---- 1 -

EOT v ~

1.1530-0.159 K,~pp-----------7

(11.46)

11.3. Contribution of Microcracking to the Toughening of D Z C

365

S t e a d i l y - g r o w i n g c r a c k with n u c l e a t i o n at a c r i t i c a l m a x i m u m normal stress Now the zone Ac is specified by (11.42) for IOl < 74.840 and by H for 101 > 74.840 where H is given by (11.45). Evaluating the integrals in (11.29), (11.30) and (7.14) numerically, one finds

KtiV K apvt

= 1 - 0.6736a + 0.82262 - 0.1329

= 1 - 0.67361 + 0.82262 - 0.2656 which for u -

EO T ( 1 - v) E~I

EOr~/g (1 - u)Kapp i

(11.47)

1/3 and 61 and 62 given by (11.36) becomes

Ktip EOT v/-~ Kapp I = 1 - 1 . 4 1 7 0 - 0 . 3 9 8 Kapp-----------~

(11.48)

The predictions for this case are not very different from those based on a critical mean stress. The shielding due to reduction in moduli is larger in each case by about the same amount for the steadily-growing crack compared to the stationary crack. For nucleation at a critical maximum principal stress there is some shielding even for the stationary problems due to 0T. This is not the case for nucleation at a critical mean stress. Effect of z o n e s h a p e o n shielding For cases, such as those discussed above, in which the moduli of the microcracked material are isotropic and the release of residual stress gives a pure dilatation 0T, the general result can be used to gain qualitative insight into the effect of zone shape on shielding. For the 0T-contribution, it follows immediately that, because R1/2(0) is modulated by cos(30/2)in (7.14), decreases in R in the range 101 < 600 and increases in the range 101 > 600 will increase shielding. The trend is similar for shielding due to the reduction in moduli. Note from (11.36) that 61 is generally much larger in magnitude than 62 and will be the dominant of the two parameters in determining K tip. Therefore, the influence of shape on K tip comes about mainly through kl. By (11.29), the integral for kl involves ln[R(0)] modulated by

f(O) -

1 327r (11 cosO + 8cos20 - 3cos30)

(11.49)

Toughening in DZC

366 0.2 -

f ( 0 ) = ( l l c o s 0 + 8cos 2 0 - 3cos 30)/(32x)

0.1 -

~

i

Increasing R(0) increases shielding

0.0 ..................... '------~---

-0.1 Decreasing R(0) increases shielding -0.2 0~

~ 90 ~

l 180 ~

F i g u r e 11.10: Plot of the function f(O) appearing in the expression for kl and its implication for change in shielding stemming from changes in shape of microcracked zone

This function is plotted in Fig. 11.10, and is seen to be positive for 101 < 70.50 and negative for 101 > 70.5 ~ Thus, with a circular shape of reference (kl = k2 = 0), shape changes involving decreases in R for 101 < 70.50 and increases for 101 > 70.50 will increase shielding. However, the influence of shape change is not nearly as strong as in the case of the 0T-contribution. The examples worked out above suggest that k l and k2 are generally quite small so that shielding will not be markedly different than that afforded by a circular zone centred at the tip. Even the addition of the wake in the steady-state problems only increases the shielding by 30-40% over the circular zone. To summarize, the increase in shielding of the growing crack over the stationary crack due to the reduction in moduli (the Q--contribution) is between 30 and 40%. Values of ~ of about 0.3 near the crack tip have been observed by Riihle et al. (1987), corresponding to about a 40% reduction in K tip due to this effect. The shielding contribution due to release in residual stress (the OT-contribution) is exactly the same as in the corresponding transformation problem, and the shielding is significantly greater for the steadily growing crack than for the stationary crack. It would appear that strong resistance curve behaviour would stem mainly from the release of the residual stresses.

11.3. Small Moduli Differences and Toughening of TTC

11.4

Contribution of Small M o d u l i Differences to the T o u g h e n i n g of TTC

11.4.1

Introduction

367

In all chapters dealing with toughening induced by phase transformation it was assumed that the elastic constants of the transformed particles are identical to those of the untransformed matrix material and in the case of toughened zirconia this is very nearly true. However even in this material there is a small difference between the elastic constants of the tetragonal and monoclinic phases (Green et al., 1989). In this Section a perturbation expansion, that exploits the smallness of this difference in elastic constants, will be used to make an approximate estimation of the effect of this difference on the fracture toughness of the material. The exposition will follow closely the paper by Huang et al. (1993). As expected the influence of lowest order moduli differences is negligible, but the perturbation technique reveals two rather unexpected features of the solution. First, it shows that, even to the lowest order, the fracture toughness cannot be assumed to be a simple superposition of contributions from dilatation and moduli mismatch considered in isolation from each other. Secondly, it shows that the joint effect of the two is qualitatively different from the prediction based on the concept of effective dilatational strain (see w below). In view of the well-known similarity between the crack tip shielding by transformation and microcrack induced dilatation, the above features are likely to carry over to the microcracking problem that we considered in w We will again consider the plane strain model for steady-state crack growth that we investigated in w It will be assumed that the elasticity tensors Ca~76 of the transformed material (t-ZrO2) and Ca~7~ of the composite material (t-ZrO2 + m-ZrO2) consisting of particles that have undergone a mean stress-induced dilatant transformation embedded in a matrix of untransformed material in the neighbourhood of the crack are isotropic. In common with the study in w the shear strains induced by the phase transformation will not be included in the analysis. In order to make any progress it will be further assumed that Cap76 and Ca~-y6 are proportional so that (11.50)

368

T o u g h e n i n g in D Z C

As will be seen below this last assumption entails that Poisson's ratios v, ~ of the untransformed precipitates and the composite material be equal. The plane strain elasticity tensors are

C ~ . y 6 - 2~

-

-

C ~ ~.y 6 - 2-fi

{ {-

1

1 - 2v 6~6"y6 + ~(6,~.y6z6 + 6~.y6~6) v 6~.y6 1 - 20

+

}

1(6~.y6~6 + ~.y~,~ ) )

-2

6

(11.51)

with ~-(1

+e)p;

V- v

and the Greek subscripts range over x and y. The effective shear modulus ~ for the composite material in the neighbourhood of the crack can be calculated from the moduli p and pt of the untransformed and transformed materials respectively by Hill's selfconsistent method. For this we need to solve the two equations (11.3). In the case of toughened zirconia with volume fractions of the matrix Vm and transforming phase Vt equal to 0.7 and 0.3, shear moduli # 78.93 G P a and #t _ 96.29 GPa, bulk moduli B - 143.06 G P a and B t - 174.54 GPa and Poisson's ratio v ~ v t - 0.267 (Green et al., 1989) the moduli of the composite are ~ - 84.08 GPa and B - 151.73 GPa. This leads to the value : - 0.065 for the small parameter. 11.4.2

Mathematical

Formulation

Let the x, y plane D contain a semi-infinite crack coincident with the negative x-axis y - 0, x _< 0 subject to a remote mode I loading that would induce a stress intensity factor K appl at the crack tip in the absence of transformation. Let ~ be the region of steady-state transformed material surrounding the crack consisting of a parallel sided wake region of width 2H behind a small region of transformation ahead of the crack tip bounded by a smooth curve C. The rest of the D-plane exterior to ~ will be designated D - f~ (Fig. 11.11). It will be assumed that the concentration c of transformed particles is constant throughout ~. The plane strain c~Z T due to transformation is related to the stress free dilatation 0T that would occur in an unconstrained particle by equation

:.~ - g ( l +

(11.52)

11.4. Small Moduli Differences and Toughening of T T C

369

where cOT is the volumetric transformation strain. The concept of effective transformation strain introduced by McMeeking (1986) (see w below) for transforming composites in which the elastic properties of the transforming particles differ from those of the matrix would require that c in (11.52) be replaced by an effective coefficient ~. For purely dilatant transformation strains, E = { B t ( B B)}/{-B(B t - B)}. For the zirconia composition under consideration, the effective dilatational strain would be ~0T, with ~ - 0.3168. In analogy with the three-dimensional Eshelby formalism used in Chapter 9 (w if c~z is the total strain then the stress is given by -

/C~z~6

t

in D - f~ (11.53)

--

T)inf ]

The equilibrium equations are

T )] , Z _ 0 in

(11.54)

and continuity of surface traction across the boundary 0f2 of the transformed region gives C

OUT IN T ~.y~%~ n~ - -C~-y6(e~6 - c.y6 )n~

(11.55)

where nz is the outward normal to the boundary of the transformed region, assumed positive if pointing from the inside (IN) to the outside (OUT) of this region. The perturbation scheme is straight forward; all the dependent variables are expanded in power series in the small parameter c 0 1 (ra~ -- ~ra~ + cera~ + ...

(11.56)

and so on. When these expansions are substituted into the above equations and the coefficients of like powers of c on both sides of the resulting equations are equated the following hierarchy of perturbation equations is obtained: The O(1) equations are

370

Toughening in D Z C

appl

g, v

(D-~)

Transformation zone

~Y

Crac~k':,,,,,,,,,,,~Cx

I Kapp I

Figure 11.11" Steady-state transformation zone surrounding a semiinfinite crack, showing the coordinate system

o

Co~~.r6c.r 6,~

~0

-

T ( Co~76E76,~

-

in D-f~

(11.57)

in f~

with OUT

C~Z~6 [r176

T

nz -- -C~z.r6c.r6n ~

(11.58)

on the boundary OQ. The O(c) equations are

1 _{0 C~z'r6 c~6'Z

in D - f ~ o

(11.59)

_OIN )nil nZ -- -Cafl'r6(c'r6T - 6-'r6

(11.60)

with Cafl76 [516]~IN

on the boundary 0f~. From (11.57) it can be seen that the O(1) strain r

is due to a

11.4. S m a l l M o d u l i Differences and Toughening o f T T C

371

T inf2(cf. (9.6)), and surdistribution of body force F ~ - -Ca~6~76,Z face traction T ~ - Co, Z.y6c.~Tnz on the surface 0f~ (cf. (9.7)) plus the strain due to the external load. In Chapter 9 we showed how the weight function ha can be used to calculate the change in stress intensity factor due to the transformation. The same is applicable to two-dimensional problems, so that we may write

K~176176

We have retained the symbol ha for the two-dimensional weight function. It will be defined later. When the expressions for F ~ and T~ are substituted into this expression and the divergence theorem is used to reduce the second integral above the final result is

K~ f f

(11.61)

Equation (11.61) is the two-dimensional mode I counterpart of the three-dimensional eqn (9.9). The O(~) change in the stress intensity factor can be calculated by the same method from (11.59). The result is K1 - f i n

C,~ z'y ~( ayT - ~-y 6 ,~ d A o )ha

= K~- /

(11.62)

fr~ Caz.y6r o ha ,~ d A

and the stress intensity factor at the crack tip is K tip = K avpl + K ~ + c K 1 "b O(c 2)

(11.63)

In order to calculate the above integrals we introduce the independent complex variable z = x + iy (-5 = x - iy), the complex displacement w~ -

u~

(11.64)

+ iu ~

{-X

and the complex plane strain weight function 1

1

4~--x/~

(11.65)

where X = 3 - 4~. The two-dimensional mode I weight function was

372

T o u g h e n i n g in D Z C

derived by Bueckner (1972) and Rice (1972) before the corresponding three-dimensional weight functions (w 1987). It is however instructive to derive it from the three-dimensional mode I counterpart (9.15). This requires an integration of eqns (9.20) from -cx~ to oc with respect to the coordinate z (which we here donote s to avoid confusion with the complex variable z), and so giving F ~ P ( ~ , y, ~)d~ - - 2(1 _ 1~ , ) , ~ I m

P -

f -

~ Q ( x , y, s ) d s -

f

1/

= -~

1 2(1 - u ) v ~ Re

{ 1~} -

- a , ~ (11.66)

{1}

- G,, (11.67)

~

f,zaZ+f

1/ (d,. -d,~ )d~

( d , . + d , ~ )dz + -~

1

- 4(1 - u)v/~

(/1

-~dz +

_ 1 x/~) - 2(1- u)v/~ (V~ +

(11.68)

In common with the three-dimensional weight functions which were all expressed in terms of the derivatives of G, the two-dimensional mode I weight function can be expressed in terms of the derivatives of (~ (cf. (9.15)) hlx=-(1 hly=-(2-

- 2 u ) C Rez -1/2 2 u ) C I m z -1/2 --

CYlmz-3/2 2

CYRez-3/2 2

(11.69)

where C = [2(1 - u ) ~ ] -1 . We are now in a position to construct the complex weight function

h - nix "t- ihly

(11.70)

As we shall only consider mode I loading in this Section, hr, and hIu will be simply denoted as hi and h2 (11.65). The two-dimensional weight

373

11.4. Small Moduli Differences and Toughening of TTC function may finally be written as h=-(1- 2v)-~-

ilm~

2

1 ( -~z~z 1 2x/~(1 - u) + x/~

1)

Vff

z+:) 4:a/2

(11.71)

Substituting (11.51) and (11.52)into (11.61) gives g 0

2/~(1 + u)cOT 3(1- 2u)

_-

Next,.we show that h~,~ - 2Re

f/,h.,odA

(11.72)

{oh}

(11.73)

The following identities are true by definition h,x - h,z + h ~ - h:,~ + ih2,~ -ih,u - h,z - h y - -ih:,u + h2,u.

whence we obtain

(

1

( X

1)} 4~/2

ho~O~- 2Re v/~( ~_ u) 4~/2 X-1 { 1 1 } 4Vc~(1_ u)z--3~ + z~2 (1-2u) { 1 1} 4x/~(1 - u) ~ + ~

(11.74)

Substitution of (11.74)into (11.72) gives (cf. (7.10)) K~ = 6v~-~(1- u)

~

+~

1}

dg

(11.75)

Integration with respect to x further reduces this result to a contour integral around the curved boundary C of the transformed region ahead of the crack tip (Fig. 11.11)

374

Toughening in D Z C

/fa

{ 1

1

c

1

z-~ + ~-nT~} dA - /_oo d~ / dY { z-~ 1 + ~--~}

1 } - - 2 / C { z -1~ + -vv dv + lim ~0 0 ~0 2~"2 cos(O/2) cos(O)dOv/Tdr

0---*0

(11.76)

The last term results from the exclusion of a small (singular) circular region from the area integral. It vanishes as ~ tends to zero, so that KO -

P(I+u)cO T /c{

- 3x/2"~(1 - u)

1

1 }

- ~ + -~z dy

(11.77)

If the effect of the transformation on the location of the boundary C is neglected, i.e. only the far-field stresses due to K ~ppz are taken into account, such that the mean stress is (7.11) then the shape of C is given by (7.12)

( l + u)Kavpt { 1

3,/~~

~ +

1 }

7r < a r g ( z ) < 7r 3 _ ~ (11.78)

--1,

where ~m c is the critical mean stress that induces the tetragonal to monoclinic phase transformation. As shown on several occasions, on this boundary the integral on the right of eqn (11.77) can be calculated exactly

T c #cOv ~m [~" - - (1 7-~-~Tvv, jy dy

KO _

l

where from

(11.79)

( 11.78) y~, - - y , - rl~ sin(Tr/2) - 3-~23 ((1 + u)I'(app') 3v/~a~

so that ooK app l K~= - ~

4,/~

(11.80)

where r (3.26) is the parameter introduced by Amazigo & Budiansky (1988) which is a measure of the strength of transformation in region f~.

11.4. S m a l l M o d u l i D i f f e r e n c e s a n d T o u g h e n i n g o f T T C

375

In the method proposed by McMeeking (1986) for binary transforming composites which we shall describe in w below, the expression for K 0 retains the form of (11.80), but in the definition ofw (3.26) the matrix elastic constants and c must be replaced with the composite elastic constants and ~, respectively, The corresponding strength of transformation will be designated ~-. When expression (11.51) is inserted into the integral in (11.62) it becomes I -

C~6e~6

- 2p

j/o{.

0

1 - 2v ~`~`~h~'z + ~ o' ~ h ' ~ ' z

}

dA

(11.81)

In order to calculate this integral it is necessary to calculate c~ from the complex displacement w ~ (11.64). Either the weight function method of Rice (1985a) or the method of Rose (1987a) can be used to do this. 11.4.3

Calculation

of Displacement

Field

The complex displacement field w ~ consists of three parts: the displacement field w T due to transformation in the uncracked body; w L due to the external load K appl and w tip due to transformation in the cracked body. First we derive w T. The Muskhelishvili complex potentials (I)(z) and ~(z) for a centre of dilatation of unit strength, lying at any point z0 in an uncracked infinite body are given in (4.28) and (4.25). The corresponding displacements in plane strain are then obtained from (4.24), with w T = u , + iu u. For constant dilatational strain inside the transformation zone, D in (4.28)is given by (7.9). Thus, for a single centre of dilatation at z0 wT =

(1 + ~,)cOT 6~r(1 -- ~,)(z- z0)

(11.82)

Next, the plane strain displacement due to external load K appt can be easily found out UL

K~VVti r

Toughening in DZC

376 vL - K appl i ' r

- 4#

2--~r[(2X +

1) sin(C/2) - sin(3r

The complex displacement field is W L -- UL + iv L - ~

e-

(

e-

z+:)

(11.83)

Finally, we adopt the method of Rice (1985b) to derive the complex displacement field due to transformation strain in a cracked body ~. From the invariance property of weight functions, if the crack tip is moved from x = l to the neighbouring position x = l + 51, the body is subjected to O(1) change in the stress intensity factor (i.e. the change in stress intensity factor due to transformation strain without moduli difference) denoted K ~ given by

Ow tip Ol =

2(1 -

E

v 2)

K~

l)

(11.84)

where w tip = u tip + iv tip is the part of the displacement field due to the crack growing from l = -cxD to l = 0 into the region Ft

2(1--v2)#(1+V)c0T //~{

cgwtiP 0l

-

E 1

2 ~ / ~ ( 1 - ,,)

6 x / ~ ( 1 - v) -X

.. ~

2,/2 - t

1

1

(zo - 0 3/2 § (:o - l ) 3/2

1 ~/:-t

z+-2-21 4(:- l)~/:-

} l

dAo (11.85)

Integration with respect to l gives

wtip

( 1 + v)cO~ = 247r(1- v ) / / a +

v ~ ( ~ / 5 + v~) z+:

X { - ,/~(~%- + 48)

X

2

+

+ v~) 2

}

+

1

]

377

11.4. S m a l l M o d u l i Differences a n d T o u g h e n i n g o f T T C

1

1

(v/_~_ 4- v/~) 2 - ( v f ~ 4- x/~) 2

}

dAo

(11.86)

The total displacement is the sum of(11.82), (11.83) and (11.86) W 0

{

---

4-

(i4-U)cSpT//~ { 4

247r(1 - v)

.5- .50

4- r

~, zo) + r

~, To) } d A o ( 1 1 . 8 7 )

where

r

zo) -

v~(~/~ + v~)

~ ( ~ / ~ + vq)

1

z4-.5

( v ~ + v~) ~ In terms of w ~ '-2#//~

2~ - ~ ( v ~

+ ~)~

and h(z,.5) the integral (11.81)is ~ R2 e { ( 1 _ 2u)

{ Oh

4 - 2 R e { O - h O w05~

(11.88)

where the identities resulting from the definition h - hi + ih2 introduced above have been used, as well as similar identities for w ~ - w~ + iw ~ w o,X - w o,z + w ~,Z ~ wO,x 4- iw~ _iw~ , - w o, z _ w ~, z

_ i w o , y 4- w 2,y ~

All the derivatives with respect to x, y may now be transferred to z,.5"

hl,y - -Im{h,z - h,~-}, h2,x - Im{h,z + h,~-}

w ~ - Re{w~ 4- w ~

- Re{w?z - w,~

w~ - -Im{w~ - w,~-}, w~ - Im{ w~ 4- w~

(11.89)

Toughening in DZC

378 As _

_

(11.90)

+

or

~~2 - C~ - Imw~

(11.91)

it follows that e~

- 2[Reh ~Rew ~ + Reh ~-Rew ~ + Imh ~-Imw ~ = 2[Re{h~-w~

+

RehzRew~ ]

(11.92)

Some of the integrals in (11.88) can be calculated analytically, while the rest have to be evaluated numerically. In analytical integration care has to be exercised to isolate any non-integrable singularity at the crack tip. This is done by surrounding the latter with a circular core with the matrix moduli as suggested by Hutchinson (1987). (This procedure was used by Hutchinson in arriving at eqns (11.28)-(11.30)). It is of course now essential to realize that the corresponding value of the integral is not a contribution to the desired K tip, but to the singular fields within this inner circular core. To obtain the correct contribution to K tip, the procedure proposed by Hutchinson (1987) for the corresponding microcrack shielding problem is adopted here. Of course, this procedure is only approximate for our purposes as it ignores the interaction effects, but to the lowest order differences in moduli the error is expected to be negligible.

11.4.4

Evaluation

of Some

Integrals

The calculation of the integrals in (11.88) will be given in some detail. For the first term, the first step is to simplify the derivative of the displacement: 2

Re{0W ~

Kavpl { 1 +

1 }

(I+v)cOT f /a {g(z, zo) + 247r(1 -- u)

zo)

+g(z,-2o) + g(-s T0)} dAo

(11.93)

11.4.

Small Moduli Differences and Toughening of TTC

where

g(z, zo) =

379

1 zv~(v ~ + ~-)~

Integration with respect to x0 reduces the integral in the above equation to a contour integral around the boundary C 2

Is

Re(0W ~

( 1

(1- 2u------) -~-z } - 2#v/~ ~ + 2(1 + u)cOT / c 24~(1 - u)

1} {G(z, zo) + G(-5,zo)

+C(z, ~0) + G(~, ~0)} duo

(11.94)

where

C(z, zo) = ~(~/~-~ + This term is multiplied by

Re{0h

(1-2u)

z} -

8V~-~(1-v)

4#

/f,~

( 1

1 1

z - - a ~ + z~-]-~

The first term (1 - 2u) ~ , ,

Oh

Ow~

Re{ ~zz } R e { - ~ z

}dA

(11.95)

in (11.88) can now be calculated (after isolating any non-integrable singularities). Some of the integrals can be evaluated exactly analytically for the approximate transformation zone boundary (11.78), the rest can be reduced to contour integrals over C which then have to be evaluated numerically. Examples are given below. The leading term, after isolation of the non-integrable singularity at z - 0, is

//o(1

+ ~

1)(1 1) ~

+ ~

dA - - 3 ~ +

2_~ (11.96)

where the underlined contribution is from Izl = ~ which is independent of ~. The area integral (11.96) can be reduced by integrating with respect to x to a contour integral over the front of the transformation zone C and a circular region C e isolating z = 0

Toughening in DZC

380

+ __/c(1

+ 1 )

dA (

1 )

cos 8dO

=e

= - 3 v / 3 + 27r

(11.97)

There are four terms in (11.94) of the type

H(z, z o ) - / L

z-~ 1 Iv G(z, zo)dyodxdy

- / c d Y / c dy~

Zo-~ In

x/~+~/~

21}

z0v/~

zV ~

(11.98)

and when these four terms are combined the non-logarithmic part of the integral can be evaluated exactly (again after isolating the point z = 0 with Izl = 8), and the remaining part evaluated numerically. The result is

H(z, zo) + H(-5,zo) + H(z,-fo) + H(-5,-5o)= 3

- (5V/37r + g

)

(11.99)

where B = {2(1 + v)KaPP'}/{3V'~a'i}. The numerical factor AI(= -1.1188) is the contribution from the logarithmic terms in the integrand which have been evaluated numerically, whereas the underlined term is again the contribution from ]z I = ~owhich is independent of 8. The four terms remaining in the integral (11.95) have the form H1 (z, z0)=

--

~

G( z, zo)dyodxdy

/cd v /c{v " g (2( z V~- ~)(z0 z - +~) z + ~)

2 (zo - z + ~)3/2

11.4.

Small Moduli Differences and Toughening of TTC

381

When these four terms are combined the apparent singularity on the x-axis when (z - 2) - 0 is removed. This is easy to see when the sum of the first terms in the four integrals H (z, z0), H (2, zo), H (z, 2o), H(2, 20) is evaluated giving

~/~

(zo + 2iy)(iy)[z[

+

4~ z

(-go - 2iy)(-iy)[z[

v~

v~ z

+ (20 + 2iy)(iy)lz I + (zo - 2iy)(-iy)lz[ X/2~ox(-4y0)

[zlir.].r 2_ +

(-v~o

+ (-4~o

ff~-'ozox (-4yo)

Izlir~r 2_

- v ~ z o ) - 2 ( 4 ~ - ~ + v~Tz)

- 4~7zo) - 2 ( v ~ izlr 2

+ v~)

(11.1Ol)

where ~= - ~o~ + (so 9 2u) ~

The non-vanishing parts of the integrals are now integrated numerically to give

Hl(z, zo) + Hl(2, zo) + Hx(z,2o) + H1(2,2o) _- 2(1 +

v)h'~'VP'A2

(11.102)

where the numerical factor A2 - - 1 . 6 9 1 5 . The second term in (11.88)

2u / /a 2Re ( O-fiOw~ Oz 02 ) d A will now be evaluated. As above, the derivative bw~ calculated from (11.87) can be simplified by integration with respect to xo. The result is

Toughening in DZC

382

~w ~

,,.~

O-e

8~~

(z_:) -e3/

[ 4

12~r(1 - u)

-5 - -50

§

z--5 /

~-~(4~ + 4~) ~

2

1

~~o

1

~ ~~o

1

z4~(4~ + 4~)~

}] dy0(11 103)

This term is to be multiplied by

.~Oz m

--_

, 16x/2~(1- u)

( z 5/2 ) Z---5

Some of the integrals can again be evaluated analytically for the zone boundary (11.78) while the rest can be reduced to contour integrals over C which can then be evaluated numerically. After an integration with respect to x, the leading term reduces to G1 - 2Re

//~

(z - -5)2 dA

~/2z~/2

:_,e/,(z_'~)~~,/~-~(-)"'z 1 -5

} dy

(11.104) The next term must vanish

z-~)

-5o -~/ 2

d yo - 0

(11.105)

because a non-zero value would imply a contribution to the stress intensity factor from phase transformation in the absence of the crack which is clearly absurd. It is easily verified that this integral does indeed vanish. The contour integral re(-5--5o)-1 dyo can be easily calculated. The indefinite integral is

11.4. Small Moduli Differences and Toughening of T T C

dyo yo Y = - arctan -f - -20 xo - x

383

i In Iz - z012 2

(11.106)

Substitution of (11.106) into the left hand side of (11.105) gives -16Re +i

~ ~ cos ~9 In Iz IL"{(-' '

ycos2

Yo - Y) zol' + ysin 5t~ 2 arctan

5 In Iz - zo arctan Yoo -- Y x _t_y ~ sin ~e

[2)t

X 0 --

X

(11.107)

If Zo E Ce, zo ---, 0, the above integral may be rewritten as -16Re

~

- ~ ~os ~O In Izl ~ + ~sin ~e a,~tan -

X

The integrand is an odd function of 9, whereas gt is symmetric with respect to 9, so that the above integral must vanish for z0 E Ce. For z0 E C, it is found that the integrand F(9, 9o) in (11.107) satisfies

F (0, 00) + F (0, -00 ) + F (-0, 00) + F (-0, -00 ) - 0 so that the integral over the contour C which is also symmetric with respect to 0 again vanishes, thereby confirming the validity of (11.105). The next two integrals together, may be written as

/cG1(~o-2

+-~ol )dyo - i

3

o-~

~

(11108).

where G1 is given by (11.104) The penultimate term

"~

1

( z - -2)2

d yo

2

z~/~v~(v~+ v ~ ) ~

(11.109)

is to be evaluated numerically and is equal to 2(1 + v)K appt

3v57~,

A3

where the numerical factor A3 = 8.7426. The last term

(11.110)

Toughening in DZC

384

( z - -5)2 dyo 1 v~ 2 z~/~(,/7+

,e/l

V~) ~

(11.111)

must also be evaluated numerically, but is equal to the previous integral (11.110). This completes the evaluation of all integrals appearing in (11.88). 11.4.5

Correction

for Moduli

Differences

To summarise, 2# /

2 ~Re{~} Jn ( 1 - 2v ) (1 -

16~r(1

Re{ -N-z 0w~}dA-

2v) -

v) Kapp' (3v/~- 2~)

2887r2(1 - v) K"PPz -1.1188 + 6v/'3

7r3

4 ) - 1.6915] -(5V~r + 57r2 2#s

ljo {o o o} 2Re

Oz 0-5

8~r(1 - u)

+

02

1927r2(I - v)

(1!.112)

dA-

- 6

KaPP z

2v/3

- 6

- 8.7426 (11.113)

Finally from (11.80), (11.112)-(11.113)the O(c) change in stress intensity factor (for v = 0.267) can be evaluated: /~-1 = K 0 _ 0.0070K,Ppz _ 0.0022wKappz

(11.114)

where K ~ is given by (11.80). The total stress intensity factor at the "crack-tip" is (e = 0.065)

11.4. S m a l l M o d u l i Differences and T o u g h e n i n g o f T T C

385

~Y

Primary problem A

I Li -

~,V

", ,,

3

. X

~1, V

Auxiliary problem B

y IH

].t,V

]a,V

//'

.~ x

~, V

F i g u r e 11.12: Primary problem A (Fig. 11.11) and the definition of auxiliary problem B

~.tiv _ KappZ + e(_O.O481wKavvt _ O.O070Kappt) _ O.0459wKavpZ ~,tip _ K,ppl _ O.O031~zKappt _ O.O005Kappt _ O.0459~Kappt (11.115)

As previously noted, the above f(tip is not the desired value (hence the use of distinguishing tilde). It has to be corrected (albeit approximately) by the procedure adopted by Hutchinson (1987), as follows. It is worth recalling that our primary task is the determination of K t i p / K "pvz for the geometry of Fig. 11.11, which is designated as primary problem A in Fig. 11.12. We have so far actually solved the auxiliary problem B in which the composite material with elasticity tensor C~Z76 covers only the region ~ - Fte where 12e is a disk centred at ori-

386

Toughening in D Z C

gin with radius ~ and boundary Ce. Thus, the integrations in integral I (11.88) were carried over ~ - ~e only. When ~ ---+ 0, a logarithmic singularity will appear in I, so that we have to limit 8 to be positive, albeit infinitesimally small. A tilde over [~[tip and /~-i indicates that they correspond to Problem B. In dimensionless form, the solution to Problem B looks like ktip

Ko

k I

K aPPz = I + K aPvz + ~ K aPv---7 '

whereas that of Problem A would look like Ktip Ko K1 K aPvz = 1 + K avvz + e K aPv---------i

The contribution from transformation alone to Problem A is exactly the same as to Problem B, and is equal to K ~ avvz. We need only consider the correction to the remaining part (1 + K 1 / K avvz) in which K 1 is due to the moduli mismatch within f~ and its interaction with transformation strain 0~. /-<1, which we have already calculated is due to the reduced moduli within f~ - f~e and its interaction with transformation strain opT in f~. To adopt the technique used by Hutchinson (1987), we will neglect the contribution to K 1 and K 1 from the interaction of transformation over f~ and moduli mismatch. This is equivalent to the assumption that the ratio K appl + K 1 Kapp I + K 1

is invariant with respect to a change in opT. The technique of Hutchinson (1987), on the other hand, is equivalent to the assumption that this ratio is invariant with respect to a change in the geometry of fl, such that if a solution to a special geometry of f~ to primary problem A and auxiliary problem B were obtained then the solution to the problem of any geometry of fl can be factored through the following identity KaPVt + K 1

K'~r'Pz+ K 1

K appt

K"Pvl + ~"1

K"r'vz

K appl .q_ ~"1

Applying this identity to our problem, and using the already known solution of Problem B, the solution of problem A may be obtained through

11.4. S m a l l M o d u l i Differences and T o u g h e n i n g o f T T C

K "VPl + K 1 K"vvz

387

K "r'Pz + K 1 K "VVz + f f 1 -

K"vvt + r~"1

K"vvz

The problems A and B have been solved by Hutchinson (1987) for the special geometry of an infinite strip to obtain K appl + ~.1

5

3

K appl -t- K 1 = 1 + ~ 1 - ~ 2

(11.116)

which is not limited to the lowest order moduli differences. Here ~1 = [ ( # / ~ ) - 1 ] / ( 1 - u)and 52 - U~l. Inserting (11.80)into (11.114) we have ~.1 _ _O.O070KaVpz _ O.04814wKavpZ

where the last term is the contribution from the interaction between transformation and moduli mismatch over D - D e. This, plus the contribution from the external load gives K"vPt + K i Kapp I

K"PPt+ ~. 1 K x + K appZ --

K"vvz

/~- I + K"Pvl

1 - 0.0070- 0.04814w l + g 561 - ~ 3~ 2

(11.117)

Comparison of (11.115) and (11.117) shows that the net result is that the first three terms of (11.115) get divided by (11.116). The last term is the (correct) contribution from dilatation alone (11.80). For the zirconia composition under study, 61 = -0.0835, ~2 = -0.0223, so that for the lowest order differences in moduli we have from (11.115) and (11.117) K tip -

1 . 0 3 6 2 K appt - 0 . 0 0 3 2 ~ K appl'- 0 . 0 4 5 9 ~ K appt (11.118)

The first term in the right hand side of (11.118) is the contrubution of moduli changes alone and this contribution would appear to be deleterious to the overall toughening of the material under study. The second term is the contribution from the joint effect of the phase transformation and the lowest order moduli changes induced by this transformation. It is seen that this effect is synergistic. The last term is the contribution of dilatation alone. It is instructive to compare (11.118) with the corresponding expression for microcrack induced shielding obtained by adding (11.80) to (11.28) (w

388

Toughening in DZC

Ktip

= l + ( k l + g l 651 + ( k 2 +

K ,,ppl

3)52 - 0.0459w

(11.119)

where w is still defined by (3.26) but with cOT reinterpreted as the dilatation due to the formation of microcracks, and kl and k2 are given by (11.29) and (11.30), respectively, k 1 and k2 can be shown to correspond to the leading terms in (11.112) and (11.113), i.e. the terms independent of w. To within constant multipliers involving # and u the integrals giving these terms are (11.96) and (11.104). From (11.96), we have

//n(

1

-

fo

-

4

1

os(3O/2)cos(0/2) 4

/o"

In R ( 0 / ( ~ o s 2O + cos 0td0

From (11.104), we have Gl(z)=Re

f/(~

-~)~

~ d A _53/2z5/2

- - 1 6 fo '~ In R(0) sin 2 0 cos 0dO When w

-

0, we have therefore from (11.62), (11.112) and (11.113) K 1

Kapp I

:--C

(i

-

47r(1 8a'(1

-

29) u)

In R(0)(cos 20 + cos O)dO

3 u) f0~In R(O) sin 2 0 cos OdO -

(1 - ~,)

g ( 1 1 ~os 0 + 8 ~os 2o - a cos a0)

-2u(cos 0 + cos 20) } In R(O)dO = ki61 + k262

(ll.12o)

11.4. Small Moduli Differences and Toughening of TTC

11.4.6

389

R e s u l t s and D i s c u s s i o n

The contribution from first order moduli differences considered in isolation from phase transformation is given by (11.120). For the steadilygrowing crack (Fig. 11.11), we calculated kl = -0.0166, k2 = -0.0433 in w (see (11.38)). If we now assume, as we did in w that the contributions from moduli differences and phase transformation can be superimposed we will get from (11.120) and (11.80) Ktip I~ app i

= 1.0350 - 0.0459w

(11.121)

Comparison of (11.121) and (11.118) shows that, even to the lowest order differences in moduli, the fracture toughness is not a simple superposition of individual contributions from the dilatation and the moduli mismatch, but that there is a coupling between the two. Equation (11.118) also shows that, the net shielding effect in a binary transforming composite cannot be calculated by a simple replacement of w with ~ in the definition of K ~ (11.80). To see this, let us calculate ~from w (3.26) after replaceing p, u and c with/~, v and ~, respectively. For the zirconia composition under study this gives ~- = 1.1246w, so that the toughness ratio according to the effective dilatational strain approach is given by (7.22) Ktip

Kapp z = 1 - 0 . 0 4 5 9 ~ - 1 - 0.0516w

(11.122)

whereas the perturbation technique gives (11.118) Ktip I~ app I

= 1.0362 - 0.0491w

(11.123)

The effective transformation strain technique only calculates the coupling effect of dilatation and moduli mismatch. It does not take into account the effect of moduli mismatch alone. A comparison of (11.122) and (11.123) shows that, already for lowest order moduli mismatch, this can make not only a small quantitative, but also an important qualitative difference to the predicted shielding effect. Thus, for example if w = 5, (11.122) and (11.123) give KtiV/K appt = 0.742 and 0.7907, respectively. Comparison with the shielding effect of dilatation alone (Ktip/K appz - 1 - 0.0459w - 0.7705)shows that whereas the present perturbation technique predicts a reduction of about 3%, the effective transformation strain technique predicts an increase of nearly 4%. This

Toughening in DZC

390

needs to be borne in mind when studying the crack tip sheilding in composites, such as ZTA which have large differences in the elastic properties of the transforming and matrix phases.

11.5

Effective Transformation Strain in Binary Composites

11.5.1

Introduction

Notwithstanding the cautionary remark at the end of Section 11.4, we shall introduce the effective transformation strain concept for binary composites containing one transformable phase. This is the only way of treating binary composites whose phases have very different elastic moduli, although the treatment is approximate and based on analogy with thermoelastic properties of binary isotropic composites (Budiansky, 1970; Laws, 1973). It will be recalled that for composites with homogeneous elastic properties, an effective volumetric transformation strain ~T was defined by (3.12) in terms of the volume fraction c of transformed particles and of the volumetric transformation strain of a particle 0T unconstrained by the matrix. The quantity 0 T served as the effective transformation strain in regions of the continuum in which many particles were undergoing a stress-induced martensitic phase transformation. In this Section, the relation (3.12) will be generalized to transforming composites in which the elastic properties of the transforming phase differ significantly from those of the matrix phase. The ZTA composite that we studied in w is a good example. There we replaced relation (3.12) with the effective dilatation (11.4), dependent not only on the particle transformation strain but also on the composite elastic properties. We shall exploit the notion that thermal expansion and transformation strains in binary elastic composites are equivalent and restate the results found in the literature on thermal expansion strains in a convenient form for the transformation strains. This will in particular allow us to state bounds and estimates on effective transformation strains when the composite contains dilute concentrations of transformable phase. We shall find that the effective dilatant transformation is less than cOT, if the nontransforming matrix is stiffer than the transforming particle, as in ZTA. The exposition follows closely the paper by McMeeking (1986).

11.5. Effective Transformation 11.5.2

Effective

Strain in Binary C o m p o s i t e s

Transformation

391

Strains

Laws (1973) investigated the effective thermal expansion coefficient of a binary anisotropic composite. The interpretation of Laws' result for dilatant transformation is as follows. The elasticity law for the nontransforming phase is (rij -

(11.124)

Ci'~klCkz

where o'ij is the stress, cij the strain and Ci'~k z is the tensor of elastic moduli. The elasticity law for the transforming phase is o'ij -

(11.125)

C~ktckz

before transformation and ~ij

--

C~kz(CkZ -

Tp

(11.126)

CkZ )

after the phase change has taken place, where cTp is the homogeneous particle transformation strain. The superscripts m and p denote matrix and particles, respectively, but the two phases can also be intermingled in any way that retains the binary feature. The behaviour of the composite in a macroscopic sense is determined by O'ij -- CijklCki

(11.127)

-

(11.128)

before transformation and ~i

C~ikt(ckz- c~z)

after transformation. In (11.127) and (11.128) the stresses and strains are macroscopic averages in the sense of Hill (1963) and thus c T. is the effective transformation strain of the composite. The exact result obtained by Rosen & Hashin (1970) and Laws (1973) can now be restated as gT

-

~ Ci~l(Cklmn -- Cklmn)(

c mP. ~ - c ~ . ~ )

-~

T~ (11 . 129) C;qrsCrs

Hence, when each phase and the mixture are isotropic but the deviatoric part of the transformation strain in every particle is uniformly aligned 1 oT + eT eT _ ~8,i

(11.130/

Toughening in DZC

392

oT _ B p ( B - Bin) T -- B(Bp

-

B m ) Op -

-- p ( i t p _ l z m ) e i j

T

-ddOp

-- c s e i j

(11.131) (11.132)

Tp where tt and B are the shear and bulk moduli, respectively, and eij is the deviatoric part of the particle transformation strain. The terms Cd and cs have been introduced to represent the effective coefficients. The alignment of the shear part of the transformation among all the particles Tp obviously precludes a locally random orientation of eij in which case eT would be zero. For purely dilatant transformations, eqn (3.12) should be replaced by (11.131) and inserted into (7 21) to give the toughening effect. In (7.21), Young's modulus and Poisson's ratio for the composite should be used, as we did in arriving at (11.122). When the shear moduli are the same for both phases, the limit of (11.132) can be found by first bounding through the use of the Reuss and Voigt estimates (Hill, 1963) of it, and then allowing ttp to approach ttm. Both bounds approach c and so

Bp ( B - Bm ) 15 T ceTP sT = B ( B p - B m ) - 3 'jOp +

(11.133)

which can be simplified by use of the result (Hill, 1963) B - Bp +

1

1--C

Bm - B,

+

c

(11.134)

B, +

which leads to cT. _

!cS'J

1+

( 1 - c)(Bm - Bp)(34-p)

(11.135)

Bp (Bin + g

The effective dilatant transformation strain is thus increased above c8T if Bp > Bin. In addition, it can be seen clearly that when the two phases have identical elastic properties ~T _ cQTP confirming generally the expression (3.12).

(11.136)

11.5. Effective Transformation Strain in Binary Composites 11.5.3

General

Bounds

and

Dilute

393

Estimates

In the case of an arbitrary isotropic mixture, there are no general results for B and #; hence, measurements, bounds, or estimates must be used. Hill (1963) gives bounds on the bulk modulus for an isotropic mixture of two phases with the configuration otherwise arbitrary. The bounds are

c 1"~-

< B - Bm <

(Bp-Bm)(1-c)

- Bp-Bm

(Bm..~_ 4#m)

-

C

(Bp

Bin)(1

-

1-Jr-

-

c) (11.137)

4

when the signs of Bp - Bm and #v - P m are the same (the inequalities are reversed when the signs are different). As an example, consider two phases with the same Poisson's ratio, but with B,~/Bp - 2, pertinent to ZTA. The bounds in (11.137) become 9c

< 5d _<

13 -- 4c --

7c 9 -- 2c

(11.138)

Notice the satisfactory results when c - 0 and c - 1. For the shear modulus, the Hashin and Shtrikman (1963) bounds can be used. These are PV +

(ttm

--

# p ) ( 1

--

c) ~ _< P _< #m +

1

)

(#v #,~)c l + J 3 m ( 1 - c ) ( 2z~ -1).,,, -

(11.139)

where #m - # p and B m - Bp are both positive (the inequalities are reversed when both are negative) and ~ = [ 2 ( 4 - 5 u ) ] / [ 1 5 ( 1 - u ) ] . Taking the case Izm/l~p = 2, as before, and u = 0.25 45c 67c < 5, _< 6 8 - 23c 9 0 - 23c

(11.140)

Alternatively, estimates for B and/~ (and subsequently for 5d and 5s) can be obtained using the self-consistent method of Hill (1965), and Budiansky (1965). However, the results must be calculated numerically for each composition and particle shape. Instead, for simplicity, estimates for 5d and 58 which are valid when the concentration of the transforming phase is dilute, are based on the average of the Voigt Bv and Reuss BR estimates

Bv - (1 - c)Bm + cBp

(11.141)

394

Toughening in D Z C

1

1-c

Bn =

Bm

c

(11.142)

~ Bv

The resulting average leads to an estimate of ~d in a series expansion in c truncated at 2 terms _ Cd-

(Bp + B.~) c 2Bin

1-

c(B 2BpBm

(11 143) "

Similarly for cs ~-

c(PV#+P'~) ~ ) [] 1 -2 c(#~ t 2ttp#.~ t -m

(11.144)

Note that, when B m / B p = l*m/PV = 2, the estimates for Cd and Cs lie near the upper limits in (11.138) and (11.139). As a result, the expressions (11.143) and (11.144)should be taken with caution.

395

Chapter 12

Toughening in DZC by Crack Trapping 12.1

Introduction

In the preceding Chapters we considered toughening mechanisms in which the toughening resulted from the shielding of a macrocrack front (tip) by a zone of transformation or microcracks. These two mechanisms are sometimes grouped under a single category called the process zone mechanism. Another category of toughening mechanisms relies on the inhibition of propagating cracks by the presence of a second phase which is here understood in a very broad sense. It could for instance refer to the second phase particles in T T C or the external reinforcement by small particles (as in metal-ceramic composites) or by continuous fibres (as in whisker reinforced ceramics or ordinary resin-based composites). We shall only describe this category with respect to second phase particles which are intrinsic to the T T C material, as for example t-ZrO2 in a c-ZrO2 matrix (PSZ) or t-ZrO2 in an A1203 matrix (DZC). This toughening mechanism has the added advantage over the process zone mechanism in that it is temperature insensitive. The restraining effect of the second-phase inclusions on an advancing crack front which is the basis for the reduction of crack-front stress intensity factor (i.e. for toughness increment) may be visualized in two ways. First, it can be presumed that parts of an advancing crack in the ceramic matrix are pinned together by unbroken second-phase inclusions over a certain distance behind the crack tip.

396

Toughening in DZC by Crack Trapping

The length of the bridged crack portions and the consequent toughness increment depend on the breaking strength of the inclusions. Figure 12.1 shows how parts of an advancing crack are pinned in Mg-PSZ, resulting in bridging over the uncracked ligaments (the crack is advancing from left to right in this optical micrograph). Secondly, it may be viewed as the trapping of a crack front against forward advance by contact with an array of obstacles. The toughness increment in this viewpoint is determined by the ease (or difficulty) of cutting through or circumventing the obstacles. We shall however find that these two viewpoints overlap. We shall present below models based on both viewpoints. The latter viewpoint has the added advantage in that it permits us also to describe (by analogy with the two-dimensional model) the three-dimensional model to the first order. For clarity of presentation, we shall first consider the models in isolation from phase transformation, that is we shall assume that just this mechanism of toughening is operative. We shall then present the interaction effects between the crack-bridging and transformation toughening mechanisms, by assuming that the obstacles are transformable precipitates which not only transform to monoclinic form but offer resistance to crack propagation by trapping it.

12.2

Small-Scale Crack Bridging

Several equivalent approaches are available for studying the mechanism of toughening due to the bridging of some portions of an advancing crack by second phase particles. In all approaches the bridged portions are modelled by a continuous distribution of springs between crack faces. These springs may be linear or non-linear, although the non-linear spring model does not lend itself to easy mathematical analysis, with just one exception considered by Nemat-Nasser & Hori (1987). Three equivalent approaches have been proposed by Nemat-Nasser & Hori (1987), by Rose (19875), and by Budiansky, Amazigo & Evans (1988). The first two approaches stress the mathematical viewpoint, whereas the third the physical viewpoint at the expense of some mathematical rigour. We shall take this last approach and draw heavily from the above paper. In all three approaches, the implementation of the crack bridging model involves two distinct steps. First, a stress analysis must be performed for an assumed constitutive equation for the reinforcing spring (i.e. assumed force-displacement law). Secondly, an appropriate forcedisplacement law for the springs must be established, either theoretically

12.2. Small-Scale Crack Bridging

397

F i g u r e 12.1: An optical micrograph of a bridged crack or experimentally. Significant simplifications in the first (analysis) step result from the assumption that the bridged length is small relative to crack length and specimen dimensions and that the reinforcing springs have a linear force-displacement characteristic. This small-scale bridging assumption is akin to the small-scale yielding assumption made in the Dugdale model for metals with limited plasticity (e.g. high-strength steels). We shall bring out more clearly this similarity between smallscale bridging and yielding later in this Section. Following Budiansky et al. (1988), and Rose (1987b) we assume in the analysis step that the spring stress tr(x) is linearly proportional to the crack-face displacement in the bridged zone (Fig. 12.2) a(x) = kE'v(x)

(12.1)

Here, E' - E (plane stress) or E' - E / ( 1 - t, 2) (plane strain), k is the spring stiffness which we shall calculate later (in the second step), and 2v(x) is the crack-face displacement. We now use an argument based on simple energy considerations (also follows from the J-integral formalism adopted in w to relate the energy input to the energy consumed by the snapping of last spring and by the crack growth. The energy input is provided by the applied field K appz whereas the energy consumed by the crack growth is related to the critical stress intensity factor Kc for crack growth in the matrix (Kc has the same meaning as in transformation toughening mechanism), and the energy loss caused by the fracture of

Toughening in DZC by Crack Trapping

398

F i g u r e 12.2: Crack bridging by second phase particles and equivalent reinforcing spring model the last spring (which is required for crack growth to occur) is related to the spring breaking strength a(L) - a~. The above energy balance condition at the instant of imminent crack growth may be written as (Kappt) 2

EI

=

Kc 2

EI

,

o.y 2

(12.2)

EEI

In other words, crack propagation will occur when the toughening ratio A - KaPPZ/Kr attains the value A -

~/1-~- kK-----~r~ ~

(12.3)

The analysis problem therefore reduces to determining the relationship between A and the bridge length L, for which the spring stress distribution a(x) is required. This is because the stress intensity factor at x - L may be written as K appz less the contribution from the spring stress over 0 < x < L. This last contribution is known from handbooks (see, e.g. Tada et al., 1985), and so Kc may be written as Kc

-

K appl -

foL

~

dx

(12.4)

Instead of calculating a(x) from (12.4), one can first calculate v(x) from the same argument,

12.2. Small-Scale Crack Bridging

v(x)-

4Ii~PPZv/7 E'vf~

399

2 fo L

7rE'

dx'

cr(x') log

(12.5)

and then calculate a(x) from (12.1). Whichever route one takes, the solution of a singular integral equation is unavoidable. We shall not reproduce the solution steps here, but give just the final result. The interested reader will find the details in the paper by Budiansky et al. (1988). It transpires that A -

1

(12.6)

V / 1 - 2g2(c~)

where g(s) - ~r(s)/[KaPPtx/~], s - 4kx/Tr, and a - 4kL/Tr. The results for the non-dimensional spring stress g(a; s) are shown in Fig. 12.3 as a function of the non-dimensional distance (a - s) from the last spring, for a = 2, 5 and 20. The limiting case c~ = c~ (Rose, 1987b) is shown for comparison. g(s;a)

~ I 0.6 0.4 0.2 0~=oo

0.0

0

5

I

I

10

15

~20 2O

O~-s

F i g u r e 12.3: Variation of spring stress along the bridging zone as a function of spring stiffness We now proceed to the next step, that of estimating the effective spring constant k. It depends on the shape, size, distribution, and content of the second-phase particles in the matrix. For spherical particles,

Toughening in DZC by Crack Trapping

400

Budiansky et al. (19881 estimate k to be 2c)

E"

(12.7)

where E-~ is the effective elastic constant (it should actually be two constants, E-~ and P, in plane strain situation) of the composite material consisting of the ceramic matrix, and a random distribution of spherical particles (radius a) and volume fraction c. The effective elastic constant may be calculated on the basis of Hill's self-consistent method (see w The parameter ~ which also depends on c can be approximated by fl ,~ (1 - c)(1 - x/~). Because the above estimate is obtained on the assumption that the spherical particles are "smeared" out in the thickness direction, it is appropriate to replace ~ry by cS (where now S is the strength of one particle at fracture) and to account for the reduction in the length of advancing crack by (1 - c) in (12.2), so that it now reads

(Koppt)2

K~

~ S 2ac(1 - c)(1 - v ~)

Elm (1 - c ) + ~

E~

(12.8)

Then, for the case of particles that break elastically, the modified toughening ratio (cf. (12.311

-

A

~/'7(1 - c)

_ ~

r S2ac(1 - x/~)

1+ ~

K~

(12.9)

where 7 - - ~ / Z ~ . It is clear from (12.9) that the modified toughening ratio depends on the strength, size and concentration of the second phase particles (see Fig. 12.4). To conclude this section, we assume that the springs are perfectly rigid plastic in the spirit of ideal plasticity inherent to Dugdale-type models for small-scale yielding in metals. In other words, we assume that the energy loss caused by the fracture of the last spring is simply equal to 2cSv/ (the last term in (12.8)), where 2vl is the relative displacement of crack faces at fracture. In this idealized case, (12.81 gives the modified toughening ratio as

A

-

-

K appl K~V/7(1 - c)

_ ~

~C 1 + (1

2E~Sv/ c)

K2

(12"10/

401

12.2. Small-Scale Crack Bridging

lOh

K

appl

/K c

- - .

~/~'(l-c) 2 E( 1-v m ) 2

Em(1-v ) c L P = (1-c)(1-~c) Ta 0

0

1

I

I

I

10

20

30

40

I

50 P

F i g u r e 12.4: Toughening ratio for bridging by elastic spherical particles

In Dugdale-type models one often defines the fracture toughness K:, i.e. the critical stress intensity factor corresponding to the critical crackface displacement 2vl at fracture. K/ is related to 2v: as follows

K:

-

~JE-4S(2v/)

(12.11)

Note that both K: and v: refer to the composite. In metallurgical terminology K] is the fracture toughness and 2v! the critical crack opening displacement of an ideally-plastic material (yield strength S) exhibiting only small-scale yielding. In analogy with Dugdale-model, the modified toughening ratio (12.10)simplifies to give Is appI

KC

-

i(

1-c)+c

gr

(12.12)

Equation (12.12) is nothing but the law of mixtures on the work of fracture. For small amounts of second phase particles (c < < 1) it reduces to the law of mixtures on the toughnesses K ~ppz K: = (1 - c ) + c Kr K~

(12.13)

Toughening in DZC by Crack Trapping

402

We can continue the analogy with Dugdale model and express A =

KaVPZ/Kc in terms of the bridge length Lc at fracture. This is readily given by (12.4) after substituting or(x) = S and integrating from 0 to

Lc.

12.3

Crack Trapping by Second-Phase Dispersion

We now adopt the second viewpoint in which the toughness increment is thought to result from the trapping of a crack front against forward advance by contact with an array of obstacles. We shall in particular identify the conditions under which this viewpoint coincides with the crack-bridging viewpoint, as well as with the three-dimensional crack trapping model that will be described subsequently in this Section 12.3.1 Two-Dimensional Crack Trapping Model

F i g u r e 12.5: An idealized representation of crack front trapping by second-phase inclusions We follow the analysis by Rose (1987c, 1987d) and consider a planar crack (in the plane y = 0) with an initially straight front coinciding with the z-axis and growing in the direction of x-axis. Figure 12.5 shows an idealized representation of the crack front which is held up by second-phase inclusions, indicated by shaded circles. The idealization is intended to suggest that the major portion of the crack front can be regarded as having progressed to a distance 2A ahead of the line of inclusions but is trapped by unbroken ligaments stemming from

12.3. Crack Trapping by Second-Phase Dispersion

403

-..-.----__.._~~

a) J

Lr

7

Z

C

"

~

~,

~

~.

~

~,

F i g u r e 12.6: (a) Crack bridging represented by springs over the segment ]z] < A, (b) a section along x - 0, and (c) the unbroken ligaments between cracks replaced by springs the inclusions. We assume for simplicity that these deep crack front perturbations (A > > A) are periodic in nature and approximate this configuration by regarding the crack to extend to x = A, but with a uniform distribution of springs acting between the crack faces over the segment Iz[ < A, as in Fig. 12.6a. These springs (which we shall assume, as before to be linear) represent the restraining action of the unbroken ligaments. In other words, we use the linear spring stress-displacement relationship (12.1), but for clarity affix appropriate suffices to the stress and displacement o'uy = E'kuy (12.14)

Toughening in DZC by Crack Trapping

404

To obtain the appropriate spring constant k, we consider a thin slice taken along x - 0, as shown in Fig. 12.6b. The section through this slice is nothing but a periodic array of through cracks consisting of cracks of length 2s separated by unbroken ligaments of length 2r along the zaxis. Its response to a mode I stress field prescribed by K appl c a n be determined analytically (see, e.g. Tada et al., 1985), with the net y-axis displacement between y = 0 + and y = 0- given by 2uu(z )

- 4cruu~ log cos ~ + v/cos 2 ~- - cos 2 ~ 7rE'

The average crack opening (2Uy(Z)) over the gap Izl < , uy(z) - 0 over z covering the particles)

(2uu(z))-~

(12.15)

cos

s 2uy(z)dz-

7rE' log sec -~-

is (note

(12.16)

so that the average spring constant k is given by (12.14) and (12.16) to be

k~ - ~'/{21og (ser

}

(12.17)

)~ = (2r + 2s) is the spacing between the inclusions. It may now be shown using the previously described distributed spring model (12.4) that the ratio of the actual stress intensity factor K tip at the crack tip x = A to the nominal factor K appz depends only on the non-dimensional ratio kl, provided 2A(= l) is less than the crack and specimen dimensions,

Ktip Kappl

= Fl(kl)

(12.18)

where

Fl(x) -

i

1 + 0.355x 1 + 2.90x + 2.23x 2

(12 19)

The maximum spring stretch is given by (vide (12.5))

6m,~ - 2uv =

4K appl E'

Vl(kl)

(12.20)

12.3. Crack Trapping by Second-Phase Dispersion

405

where

VI(x) -

i

1 + 0.656x

i + 2.68x + 1.67x 2

(12.21)

FI(x) and VI(x) are interpolating functions constructed so as to reproduce the correct asymptotic behaviour for soft springs (kl < < 1) and for hard springs (kl > > 1). These functions were obtained by Rose (1987b). We mention en passant that there is a misprint in the corresponding equation in the paper by Rose (1987c) from which we have drawn much of the material in this Section. We now need to specify the dynamical conditions for quasi-static (steady-state) crack growth. We assume that failure occurs at a value of the ap.plied stress field K appl when K tip achieves the critical value for the material If/. As before, we shall assume the corresponding value of the maximum spring stretch 8ma~: to be 2vy (see (12.10)). It is convenient to specify the failure criterion at the failure of the spring (at x = - A ) in terms of a limiting stress intensity factor KI, rather than a limiting spring stretch 2yr. The connection between these two specifications (i.e. K/ at x = A and 2v/ at x = - A ) is established via the collinear crack array model mentioned above. For such an array (see, e.g. Tada et al., 1985) K/

-

crvv A tan -~ - E'kv!

tan -~-

(12.22)

From (12.20) and (12.22), we obtain the following equation for describing the onset of fracture

If/ _ 2 V l ( k l ) i k / t a n ~ ) K~PPz log(sec - -

(12.23)

Ky may be regarded as the critical stress intensity for the lateral growth of the crack front, that is for increasing s with A fixed, because the growth in the x-direction (increasing A for fixed s) occurs only when the spring at x = - A is stretched to its maximum value 2vy. It should be obvious that this K/ (at x = A), as well as v! (at x = - A ) are different in physical interpretation and magnitude from the Ky and vy used in the previous model (12.11). It is also convenient to express the critical value of K tip for the composite (i.e. KF) in terms of the intrinsic matrix toughness Kr and the toughness K / f o r lateral crack growth. K r

Toughening in DZC by Crack Trapping

406

which should now be regarded as the critical stress intensity factor of lateral and forward growth of the crack front is chosen to be related to K l and I~'~ by a law of mixtures on the work of fracture (cf. (12.12)) in proportion to the contents of matrix and inclusions -

/2s

2 + 2r K}

(12.24)

Then from (12.18) and (12.24) we get the following toughening ratio

K ~ppt

V/(1 -

c) +

c(Kf/K~) 2

(12.25)

where, in analogy with (12.12), we have denoted by c = 2r/,~ the area (volume) fraction of inclusions, such that 2s/,~ = ( 1 - c) is the area (volume) fraction of matrix. It is worth stressing again that the K] in (12.25) has a different physical interpretation from the Kf appearing in (12.12). Eliminating K appl between (12.23) and (12.25), one obtains an equation for determining kl in terms of the material parameters Kf/K~, and c

4- v (kt)

(l-c)+c

log

7r(12"-c )

(12.26)

Solving (12.26) for kl and substituting into (12.25), one obtains the toughening ratio K appz/Kc. The corresponding equation in the paper by Rose (1987c) is unfortunately incorrect casting doubts on his results and conclusions. The correct solution of (12.26) for several values of Kf/Kc and 0 <_ c _ 1 is shown in Fig. 12.7. The corresponding values of the toughening ratio (12.25) are shown in Fig. 12.8. For comparison, we have also shown (by broken lines) the toughening ratio Kappz/K~ predicted by a simple law of mixtures (i.e. by the numerator of eqn (12.25)). It is evident that by allowing the broken segments of the crack front to grow laterally (in the z-direction) before they advance in the x-direction, the toughening effect is amplified. We may therefore regard 1/Fl(kl) > 1 as the amplification factor in (12.25). The lateral growth of broken segments of the crack front without their forward progress is akin to an increase in the volume (area) fraction of the matrix traversed by the segments and a corresponding decrease in the effective area fraction of the inclusions. We may therefore introduce

12.3.

Crack Trapping by Second-Phase Dispersion

407

kl 0.8

12

0.6

0.4

0.2

t

I

I

1

l

0.0

0.2

0.4

0.6

0.8

0.0

Figure

-~._1

1.0 c

12.7: Spring stiffness for various values of

K//Kc

appl

K

/Kc Kf/K =12

12 10 ..--""""

9

8

6 4

4

2

3 2

0 0.0

Figure

l 0.2

i 0.4

t 0.6

i 0.8

1 2 . 8 : T o u g h e n i n g ratio for various values of

i 1.0 c

KI/Kc

Toughening in DZC by Crack Trapping

408

the effective area fraction of inclusions through f (which will now depend on current position in the z-direction), so that the area fraction of the matrix is (1 - f), and rewrite (12.25) as follows

-

where f - { 1 - ( 1 - c ) / F l ( k ' ) }

(1 - / )

+ /

(12.27)

and K ; -

K! {1 + ( F ? i k l ) - 1 ) / f } l , 2 .

We can now identify K~ with the K l appearing in (12.12), so that the latter equation is identical in form to (12.27). The above argument may be viewed as an attempt by the broken segments of crack front to join up (because it is easier to grow in the matrix even under the influence of inclusions) before making forward progress. This is quite likely to happen when A > > A, but not so when A < < A because of the strong influence of the inclusions. In the latter case, the law of mixtures (numerator in (12.25))is more likely than (12.27) to be closest to the toughening ratio, as we shall see in the next Section where we consider this case. 12.3.2

Three-Dimensional ping

Small-Scale

Crack

Trap-

The discussion in the preceding Section was based on a two-dimensional plane strain configuration, although there are elements of three-dimensionality implied in the lateral growth of the broken crack segments. Here we present a three-dimensional analysis of limited validity (A < < A) based on the first-order perturbation of the stress intensity factor distribution along the front of a half-plane crack, when the location of that front differs moderately from a straight line. We shall use this perturbation solution to the configuration of the front of a planar crack that is trapped against forward advance by contact with a periodic array of closely spaced obstacles. For this description we borrow heavily from a paper by Rice (1988). Consider a half-plane crack in the plane y = 0, growing in the direction z and having a straight front along x = a0 (Fig. 12.9). For fixed loading the stress intensity factor along the straight crack front parallel to the z-axis, K~ a0], may be obtained from (9.1) with c~ = I and x replaced by (z - a0). If the crack front is not straight but lies along the arc x = a(z)

12.3. Crack Trapping by Second-Phase Dispersion

409

F i g u r e 12.9: (a) A half-plane crack with a straight front at x = a0, (b) a half-plane crack with a moderately curved front at z = a(z), and (c) crack front trapped by impenetrable obstacles

(Fig. 12.9b) in the plane y = 0, then Rice (1988) has shown that, provided a ( z ) i s small such that in an average sense (on a large scale) the crack front is still straight and the variation of K~ a0] with a0, i.e. OK~ is much smaller that K ~ itself, the stress intensity factor K[a(z), 0, z] (denoted simply by K(z)) along the moderately curved crack front is

K(z) - K~ K~

a0]

a0]

1 / = ~ da(z')/dz'

= 2-~

co (z' - z) dz'

(12.28)

to the first-order deviation of a(z) from a0, i.e. to the first order in da(z)/dz. The singular integral in (12.28)is to be understood in the sense of Cauchy principal value. Next, consider the situation depicted in Fig. 12.9c. The crack front is trapped by impenetrable obstacles of some given distribution. Then a(z) is known along the contact zones Ltrap but K(z) is unknown there. Conversely, K(z) = Kc is known along the penetration zones Lpe,~ (i.e. the matrix part) but the depth of penetration a(z) is unknown there. These two conditions together reduce (12.28) to the singular integral equation

K~

ao] - Kc K~ ao]

1 [ da(z')/dZ'dz , 27r JLt,.~p (z - z')

Toughening in DZC by Crack Trapping

410

= 1/L

da(z')/dZ'dz'

(z-z')

(12.29)

for all z included in Lven. Once (12.29) is solved for a(z) along Lv,=, K(z) along Lt,-ap can be found from (12.28). In fact, the solution of (12.29) can be lifted by analogy from known solutions in two-dimensional crack theory. To explore the analogy consider a two-dimensional medium in (y, z) plane containing a single crack or an array of cracks. The medium is loaded remotely in mode I such that the stress avy = cravPZ corresponding to K appl and the opening gap (i.e. net y-direction displacement) between y = 0 + and y = 0- is 2uy(z) = 5(z). Then cryy(z) along z-axis is obtained as

E' f ? 5(z')/dz' aY~ - ~raPP'+ -~r oo ( z ' - z) dz'

(12.30)

For a single (finite or semi-infinite)crack (12.30) can be readily derived from (4.21), (4.24), and (8.14) or (8.3). Equation (12.30) may be rearranged to coincide exactly with (12.28) if one makes the identifications 2[cryy(z) - ~,pvt] .-..+ [K(z) - K ~

E'

K~

5(z) ~ a ( z )

(12.31)

In this analogy, along Lt,-av the opening displacement is 6(z) = a(z), whereas along Lven the crack faces sustain the stress ayy = crappz + E ' ( K c - K~ ~ Note that cryu will be less than crappz, since K ~ > Kc. Thus, the crack faces will open with the opening displacement 5(z) cor-

F i g u r e 12.10: Periodic array of impenetrable obstacles in the path of a half-plane crack

12.3. Crack Trapping by Second-Phase Dispersion

411

responding to the crack front penetration a(z) in the three-dimensional trapping problem. Rice (1988) extends the analogy to obstacles which are not completely impenetrable, but we shall here limit ourselves to impenetrable obstacles and consider the periodic array shown in Fig. 12.10. The impenetrable obstacles with centre-to-centre spacing 2L (= A used in the preceding Section) have a gap 2H(= 2s of the preceding Section) between them, into which the crack front can penetrate. We have already used the solution for this periodic configuration in the preceding Section (see (12.15), (12.16)). So with substitutions for avv and $(z) identified above, (12.15)and (12.16) read (for - H < z < H)

a(z) = 2Uy(Z)

_( 4L 7r

1-

(ap~,> -

K~

log

{

COS~_L" + r

4L2(

(2uv(z)> - -~ff

2 ~'z

,rH

~-T - c~ cos xH 2--X-

I<~

~-

(rH)

1 - KO) log sec ~

}

(12 32)

(12.33)

Substituting (12.32)into (12.29) one obtains along Lt,-~p (H < ]z I < 2 L - H) the following stress intensity factor K(z) =

+ (K ~ -

sin

~z

~//COS 2 ~ ~'H - - c~

~'z 2---E

(12.34)

which is singular, as expected, at the borders of the trap zone. The average value of K(z), denoted (K(z)) over the trap zone, (H < [z I < 2 L - H) is

(K(z)) -

LK ~ - HKc L-H

(12.35)

Denoting by f - ( L - H)/L the fraction of contact, so that (1 - f) is the fraction of penetration, eqn (12.35) can be written as

K~ (K(z)> /~ = ( l - f ) + f Kc

(12 36)

The limiting value of K ~ denoted as before K appl corresponds to the instant at which the crack front just breaks through the obstacles.

Toughening in DZC by Crack Trapping

412

This value can be calculated exactly. For the small perturbation approximation considered here (i.e. amax < < 2L, equivalent to A < < ~ of the preceding Section), it is necessary that

(K~ 2 - {(K(z))} 2

(12.37)

This is obvious from the observation made earlier that the assumption of small-scale deviation of crack front from straightness is akin to the assumption that the function a(z) fluctuates in z about a mean value so that, in an average sense, the crack front is still straight. K ~ is indeed the stress-intensity factor for the straight crack front a(z) = ao. Since K(z) is known everywhere at breakthrough (it is equal to Kc o n - H < z < H and Kp o n H < z < 2 L - H , where Kp > K ~ is the stress-intensity factor for circumventing the particles), eqn (12.37) gives the exact value of K~ K appz) at breakthrough

IiaPPZI(c

1 - f) + f

~

(12.38)

When (Kp - Kc)/Kc < < 1, (K(z)) ~ Kp, so that at the instant of breakthrough (12.36) reduces to

=(1-y)+y

(12.a91

which also follows from (12.38), as it should. It should be stressed that the linear perturbation theory is in fact applicable only to the case when ( K p - gc)/Kc < < 1. A comparison of (12.38) with (12.13), and with (12.27) quickly establishes the connection, at least for small crack front penetrations, between the three-dimensional and two-dimensional models of crack trapping on the one hand, and between the crack trapping and crack bridging models, on the other. All that is required for establishing this connections is an appropriate interpretation of K.t , K~, and lip, as discussed above.

12.4

Crack Trapping by Transformable Second-Phase Dispersion

We now consider the situation when the crack is trapped by second-phase dispersed precipitates which can also transform to monoclinic phase. The

12.4. Crack Trapping by Transformable Second-Phase Dispersion 413 ceramic matrix in such ZTC is toughened not only by the phase transformation of the tetragonal precipitates but also because the precipitates impede the progress of the macrocrack and trap it due to the mismatch in elastic properties. The length of the trapped zones is determined by the size, volume fraction and phase transformation characteristics of zirconia precipitates which we shall assume, for simplicity, to be periodically distributed. These parameters will therefore also determine the spring stiffness which will vary along the bridging zone. In this Section, we shall assume in the spirit of Dugdale (1960) and Bilby, Cottrell & Swinden (1963) model that the transformation zone has no thickness and is coplanar with the discontinuous macrocrack fragments, as shown in Fig. 12.11. We shall follow closely the paper by Jcrgensen (1990) to determine the spring stiffness in (12.1). The tetragonal precipitates that have impeded the progress of the macrocrack and have fragmented it will transform into monoclinic phase because of the very high stresses at the tips of the fragments, thus reducing the stress intensity factor at these tips. The discontinuous crack front cannot therefore grow laterally until the external loading is increased to overcome the shielding effect due to phase transformation. It will be assumed that the transformation is accompanied by dilatation alone so that the transformation zone at each crack tip in the periodic array (Fig. 12.11) can be regarded as planar to which the BCS model with the modification by Rose & Swain (1988) is applicable. In this modified BCS formulation, the stress distribution in the transformation zone at each crack tip in the array is obtained by superposition of two fields. The first stress field corresponds to a stress intensity factor equal to the fracture toughness of the matrix, Kc. The second field is due to a stress intensity factor (K appt- Kc), where K appt corresponds to the applied stress crappz, together with a constant stress (aappz- or*) acting across the transformation zone length I. The cohesive stress a* due to transformation-induced dilatation has to be determined from the dynamic condition for transformation, according to which the total stress O'zz at the end of transformation zone must equal the characteristic value a0 for a tetragonal precipitate, or0 is in turn related to the critical mean stress crm for tetragonal to monoclinic transformation via 3 fro = 2(1 + v)tr~ (12.40) Now following Bilby et al. (1964), and Rose and Swain (1988) it can be shown that or* and l are related to the loading and transformation

414

Toughening in DZC by Crack Trapping

F"

Vl

Figure 12.11: Crack trapping by periodic array of transformable precipitates

12.4. Crack Trapping by Transformable Second-Phase Dispersion 415 characteristics as follows 2 /Trs K~ o'* - o'o

1+

l -

(1

~--(Is s

-1

~"

'~ KaPPz K~ ~/ rs K.ppz ) tan -~-

(12.41)

- Is

(12.42)

tan 7r___ss

8(~*) 2

Next we need to calculate the opening of each of the cracks in the array. For this we again follow Bilby et al. (1964), and calculate the average opening over each crack (with its transformation zones)

2 [(s+l)

fy

I

'~ H(y')dy'dy'

(12.43)

I

where H(y') is given by

g(r

- ~

~osh -~

~'(r - a')

(r162

(12.44)

c' - sin ( ~ - ~ )

(12.45)

_ ~osh-~ ((a')~ +

and y' - sin (~~-~) , a'-- sin

Now assuming that the crack face displacement at section x of the macrocrack (Fig. 12.11) is given by the average opening of the lateral crack array with the transformation zones at this section, eqn (12.1) may be rewritten as k(x)

-

6r a p p l Et(uz(x)),

"

o "appl

--

I~[ appl

~)i

7r8

tan-~-

(12.46)

Figure 12.12 shows an example of the variation of k(x) with K.ppz/Kc

416

Toughening in D Z C by Crack Trapping

"~ 1.O o t~

,I--4

E

0.8

0

z

0.6 0.4 0.2 0.0

0

I

I

i

I

I

I

I

1

2

3

4

5

6

7

appl

K

/K c

F i g u r e 12.12: Normalized spring stiffness at a given instant (i.e. given x). k(x) is normalized by k corresponding to a linear spring model in the absence of phase transformation (12.17) which is equivalent in the present formulation to the condition K appl <_ Kc. It is clear that in the presence of a phase transformation the springs are softer. Having calculated the spring stiffness, we determine the macrocrack opening displacement using (12.5). v(x) - (uz(x)) in this equation is approximated by a second-order spline, and the resulting non-linear equation is solved iteratively to calculate ~r(x) for various values of Aa. The stress intensity factor at the tip of the macrocrack is then calculated from (12.4). The length of the bridging zone Aa is finally established from the dynamic conditions for growth of the macrocrack. The latter will grow when K tip (left hand side of (12.4)) reaches the intrinsic fracture toughness of the matrix Kc, while simultaneously the spring at x - Aa stretches to its limit and snaps. The limiting value of the spring extension at x - Aa is in turn determined by the dynamic condition for unstable crack growth in the lateral direction (z-direction). The unstable lateral growth occurs when the unbroken ligaments between the cracks join to regain the continuous front that pertained prior to its trapping. This is equivalent to saying that the phase transformation capacity of tetragonal precipitates has been exhausted and there is no further crack shielding available from them to prevent unstable lateral crack growth. At this instant, the applied stress

12.4. Crack Trapping by Transformable Second-Phase Dispersion

K

417

appl

/Kc 1.5-

1.4 ~ 1.3

~

1.2

~

~,-2s = 2001xm 1501xm lO0~m 50lam

1.1 1.0

1

J 5

I 9

i

I

13

17

I

21 L/2s

F i g u r e 12.13: Toughening ratio as a function of the size and volume fraction of transformable precipitates intensity factor K appt will equal the fracture toughness of ZTC in which the only toughening mechanism is the phase transformation. Having satisfied the two dynamic conditions for the growth of the macrocrack (in the x-direction), one can estimate the toughening resulting from the joint action of the phase transformation of dispersed tetragonal precipitates and crack trapping, using eqn (12.4). The resulting toughening ratio Kappt/K c is shown in Fig. 12.13 as a function of the volume fraction of precipitates (A/2s) for several values of the precipitates size ( A - 2s). It is interesting to note that in non-transformable ceramics this ratio is independent of precipitate size (cf. Fig. 12.8). This is explained by the fact that the toughening ratio is controlled by two distinct mechanisms. The contribution from the phase transformation to the toughening effect is most pronounced for small (A/2s), but diminishes with increasing (A/2s) when the bridging mechanism progressively takes over. It would appear that the transition in dominant mechanisms occurs at A/2s ~ 3. In applying the above results to ZTC it is worth remembering that the model assumes a small transformation zone compared to crack size and ignores any transformation of tetragonal precipitates in the x-direction. It is for this last reason that the model is likely to be more accurate for small rather than large values of (A/2s) because of the assumption that

Toughening in DZC by Crack Trapping

418

the opening displacement of the macrocrack at any x equals the average opening of the crack array in the y-direction for this section, i.e. the fracture toughness of precipitates themselves is not very different from that of the matrix. We now remove the restriction that the transformation zones are coplanar with the macrocrack fragments and allow them to grow out in the z-direction as far as required by the critical mean stress criterion (Fig. 12.14). For this we utilize the R-curve analysis for a collinear array of internal cracks in a TTC from w The exposition below follows closely the paper by M011er & Karihaloo (1995).

F i g u r e 12.14: Collinear array of plane cracks in a transforming ceramic As before, the effect of crack front trapping by transformable particles is modelled in two dimensions by a shielding stress, ~r' (x) acting over the shielded length Aa , as depicted in Fig. 12.15. The magnitudes of ~r8(x) and Aa are related to the volume fraction and size of transformable particles and to the intrinsic toughness of the matrix material, as shown below. The relative displacement of the crack faces for the above problem is again given by (12.5). Consider the point of instability of the macrocrack under a monotonically increasing load. At this point, the displacement of the crack faces at x = Aa is assumed to have reached a critical value, u(Aa)= (uc) so that a further increase in load would result in breakdown of the shielding mechanism. This dynamic condition for macrocrack growth gives the first governing equation

8K avpzv / ~

E'x/

4 f a,

Jo

o" (t) log

dr-(u

) = 0 (12.47)

12.4. Crack Trapping by Transformable Second-Phase Dispersion

419

Aa F i g u r e 12.15: Tip of macrocrack shielded due to crack front trapping

The crack will not however grow in a catastrophic manner unless the effective stress intensity factor at its tip simultaneously reaches the intrinsic toughness of the matrix material. The effective stress intensity factor for the trapped crack K tip is given by (12.4) with L replaced by Aa. At catastrophic growth K tip = Kc, which forms the second governing equation written in normalized form as

KavvZ

KC

r~ /o t' a

o's(x)

dx - 1 -

0

(12.48)

The macrocrack resumes its continuous front when the two dynamic conditions (12.47) and (12.48) are simultaneously satisfied. To relate cr~(z) and ur to the volume fraction and size of transformable particles, let us consider a cut t - t as indicated in Fig. 12.11. It is assumed that the relative displacement of the crack faces of the macrocrack at instability is equal to the average opening (ur of the array of cracks at peak applied stress ~rp. This stress can also be thought of as the apparent strength of the material. By introducing the length parameter L (7.28) and the critical average crack face displacement (re)

E/~, (vc) -

( u c ) 1 2 r ( 1 - v)

eqn (12.47) can be written in normalized form as

(12.49)

Toughening in DZC by Crack Trapping

420

O m

~/Aa ~Kc L

K appl

1

+ 7rIi~ 2vrff~"

fo

crs(t) log

v/S-d_ e7

dt

(vc) L

(12.50)

Equations (12.48) and (12.50) constitute the governing equations for the crack trapping problem in the two unknowns Aa and K appl. r and (re/ are determined from the analysis of the array of cracks in the transverse y-direction as discussed in Chapter 8 (w The critical average crack face displacement at x = Aa is obtained by integrating the dislocation density function -

dx

(12.51)

D* ( x 0 ) d z 0

~r'(t) along Aa is approximated by a linear or quadratic function and the two governing equations are solved for Aa and K appl using the standard iterative procedure, described in w

( appl YY

0.24

p~-

0.23 0.22 0.21 0.20 0.0

0.5

i 1.0

I

1.5

J

2.0 (C-Co)/L

F i g u r e 12.16: Normalized peak applied stress during quasi-static crack growth, w - 10, AlL - 20. The normalized initial crack length is

coil - 5

12.4. Crack Trapping by Transformable Second-Phase Dispersion

a)

x

[-F"

-

-

421

~

Aa

t3

P

__________=

b)

[-r

Aa

"

F i g u r e 12.17: Crack tip zone shielded by (a) linearly decreasing stress, (b) quadratically decreasing stress

The applied stress at which the transformation capacity of the particles is exhausted is shown in Fig. 12.16 for a given initial crack length. As foreshadowed above, the shielding stress will be approximated over Aa by a linear or quadratic function, as shown in Fig. 12.17. The applied stress intensity factor at the tip of the trapped crack K appt and the shielded length Aa are determined after the values of (uc) and a p have been calculated for different particle sizes s and area (volume) fractions A / f r o m the analysis of the array of cracks mI -

(12.52)

The logarithm of the normalized shielding length is shown in Figs.

Toughening in DZC by Crack Trapping

422

log Aa L 2 1

0 -1 -2 -3 -4

-5 0.0

I 0.2

t 0.4

I 0.6

I 0.8

I

1.0

Af

F i g u r e 1 2 . 1 8 : Shielding l e n g t h as a f u n c t i o n of a r e a f r a c t i o n for linear shielding stress; s/L - 2, 5, 50

log Aa L

0 -1 -2 -3 -4

-5 0.0

i 0.2

i 0.4

i 0.6

t 0.8

I

1.0

Af

F i g u r e 1 2 . 1 9 : Shielding l e n g t h as a f u n c t i o n of a r e a f r a c t i o n for q u a d r a t i c shielding stress; s/L - 2, 5, 20

12.4.

Crack Trapping by Transformable Second-Phase Dispersion

423

appl

K

IKc_

1.16 1.12

0

5

1.08

1.04 1.00 0.0

I

I

I

l,

0.2

0.4

0.6

0.8

I

1.0 JA,

Figure 1 2 . 2 0 : Applied stress intensity factor as a function of area fraction for linear shielding stress; s / L - 2, 5, 50

appl

K

IKc

1.20 -

1.16 1.12

5

1.08

1.04 1.00 L_ _ _

0.0

0.2

1

0.4

_ _ L

_ _

0.6

I

0.8

__1

1.0 ,4, ./

F i g u r e 1 2 . 2 1 : Applied stress intensity factor as a function of area fraction for q u a d r a t i c shielding stress; s/l - 2, 5, 50

424

Toughening in DZC by Crack Trapping

12.18-12.19 as a function of area fraction of transformable particles for three normalized particle sizes under the linear and quadratic approximations to crS(z), respectively. The corresponding applied stress intensity factors are shown in Figs. 12.20-12.21. For a given transformable particle size s the shielded length decreases with an increase in the volume (area) fraction. This is a consequence of the behaviour of the critical crack face displacement (vc) obtained from the analysis of the array of cracks. Low volume (area) fractions of a given particle size, s correspond to large values of ~ (12.52), i.e. long cracks between the particles. This leads to a large average displacement before instability sets in, even though the applied load is relatively low. The large critical displacement of the macrocrack at x = Aa results in a large shielded length for the low volume (area) fractions, and vice versa. This is in agreement with the results of Budiansky et al. (1988) for tough non-transforming particles in a brittle matrix. As expected the quadratically varying shielding stress distribution results in a higher level of toughening and longer shielding lengths than does the linear distribution. The behaviour described above is reflected in the behaviour of K appl in Figs. 12.20-12.21 since the applied stress intensity factor is calculated at the tip of the shielded crack. All curves in Figs. 12.20-12.21 peak at moderate volume (area) fraction, but the magnitude of K appt increases with increasing particle size. The same trend is forecast by the theoretical analysis of Rose (1987c) who also found agreement of his results with experimental observations for an epoxy matrix containing a dispersion of alumina trihydrate inclusions (Lange & Radford, 1971). There is also experimental evidence (Lange, 1982) to support that peak toughness of an alumina ceramic toughened by yttria stabilized tetragonal zirconia particles is attained at a moderate fraction of dispersed zirconia particles. For a given set of material parameters the above analysis permits determination of the toughness of a brittle matrix ceramic toughened by dispersed transformable particles. As the peak toughness is attained at moderate volume (area) fractions of transformable particles, the model is useful for tailoring the microstructure of these materials to suit specific applications.

425

Chapter 13

Toughening in DZC by Crack Deflection The toughening of DZC by crack deflection is the least studied of all toughening mechanisms. Yet, it could turn out to make the most significant contribution to the overall fracture toughness of ZTC materials from internal sources, especially at low volume fractions of zirconia precipitates. In all materials, crack deflection out of its plane may result from

F i g u r e 13.1: Crack deflection in Mg-PSZ

Toughening in DZC by Crack Deflection

426

numerous internal or external (loading) sources. Among the internal sources are first, the second phase particles which, because of their generally high toughness, force the crack to deviate out of its plane rather than trap it, and secondly, the grain boundaries. The fracture toughness of the latter is generally well below that of the single grains or of the polycrystalline aggregate. The application of a non-symmetric remote stress field naturally leads to crack deflection because of the mixed-mode character of the local stress in the loading device. It is not surprising therefore to notice out-of-plane crack growth in specimens which are nominally under mode-I loading. In fact, according to one of the most commonly used criteria for determining the deflected path - the so-called criterion of local symmetry - a straight crack front will follow a path along which the local stress-field is one of pure mode I. A comprehensive review of planar crack deflection under almost mode-I loading may be found in the paper by Karihaloo (1982).

13.1

Stress I n t e n s i t y Factors at a K i n k e d Crack Tip

In TTC materials the major source of crack deflection is likely to be int e r n a l - second phase particles, and grain boundaries. Figure 13.1 shows an optical micrograph of Mg-PSZ with substantial crack deflection. The macrocrack is growing from left to right in this micrograph. To get an idea of how much toughening can be expected as a result of crack deflection under arbitrary mixed-mode loading, we reproduce here a result from the paper by Karihaloo (1982) ]r ppl --

611K~ppl

-~- C12 A ' ~ pl

kI~ pl --

C21KIppl

r..appl -q- C221~II

(13.1)

where K~ppz, K ~ pt are the nominal (applied) stress-intensity factors for a straight crack and k~ ppl , "~II bappl are the nominal stress-intensity factors for a vanishingly small kink growing at the tip of a straight crack. C~#(c~, fl = 1, 2) are functions of the kink angle r (Hayashi ~: NematNasser, 1981). These are shown in Fig. 13.2, together with the IrwinWilliams functions (broken lines). The latter denoted C ~ represent the angular dependence of the near-tip stress field of a straight crack (origin of polar co-ordinates r, r at crack tip) (Williams, 1957)

13.1. Stress Intensity Factors at a Kinked Crack Tip

427

1.O 0.5 0.0 -0.5 -1.O -1.5 0.0

J 0.1

I 0.2

J 0.3

I 0.4

I 0.5

I 0.6

J 0.7

t 0.8 (I)/~:

F i g u r e 13.2: Functions C~Z (solid lines) compared with Irwin-Williams asymptotic near-tip angular functions C~Z

.

Vii

1(

-

-

~

C12 -- -~

.

C21

.

--

0

3 cos ~ + cos ~r sin ~r + cos

~

1(

sin ~r + cos

1(

3cos~r

C22-~

0) 0)

(13.2)

The good agreement between Can and C ~ at least for small kink angles, r shows that the stress-intensity factors at the tip of a vanishingly small kink from a straight crack may be calculated from (13.1) after replacing C~Z with the Irwin-Williams functions C~Z. In other words, k~ppz and appt

// may be calculated from the stress-field that existed at the tip of the straight crack prior to its kinking. The good agreement was predicted to the first order in kink angle by Cotterell & Rice (1980) and to the second order by Karihaloo et al. (1981). Using the small angle approximation of trigonometric functions in (13.2) and retaining terms of the order r inclusive, (13.1) reduce to

Toughening in DZC by Crack Deflection

428

3r

)KI

-

3 ~qr,.appl

-~ wl~ II

k~f p' = -~rK] pp' + (1 - 87r

(13.3)

From the second of the two equations (13.3), one can calculate the critical kink angle r from the criterion of local symmetry k ~ pz - 0 and substitute the critical kink angle r into the first equation to obtain (to within the terms of order r inclusive) an expression for k/ppz. We do not reproduce this expression here in full. For our purposes it is sufficient to assume further that the applied stress field is predominantly mode I in character, such that its mode II component is an order of magnitude smaller than mode I. With this assumption, r - - 2 K I f vt/KI vv~ (which confirms the earlier observation about crack deflection under a nominally ,r,.appl mode I loading). r may actually be written as r ~ 21J~Zl [/KI ppt, because under predominantly mode I applied stress field the shear stress component giving K..appl II will always be of the opposite sign to the kink angle, for the deflection drives the crack extension towards pure mode I. From (13.3) we find therefore

k~vvz - K]VVz-3

k,.appl L~II

iiivvt

)2

(13.4)

It is evident from (13.4) that crack deflection leads to a reduction in the stress intensity factor at the deflected tip, i.e. to toughening. The above discussion was of course restricted to vanishingly small kinks, and small angles of deflections. In fact, a similar toughening effect has been observed when the deflections are extensive both in amplitude and wavelength. Figure 13.3 demonstrates this point (Rubinstein, 1990). The results are for sinusoidal large scale deflections of a straight crack. It is interesting to point out that the results validate the criterion of local symmetry" the ratio t'aPpt/K~ ppz is almost zero even for such large '"II and extensive crack deflections. A word on the toughness G shown in Fig. 13.3. The Irwin relationship between G and K extended to mixed mode situations, is also applicable to non-straight crack configurations, (see the paper by Karihaloo, 1982) provided the stress intensity factors at the kink tip are used '

g~Vpz

=

(kip;Z)2 + ~zI

EI

EI

)9.

13.1. Interaction Between Crack Deflection and Transformation

g'1kfppl' .2- kl?ppl

A$I ~

1.0

429

[

~L'~L"-L'~'L"-I~

0.8-

I/KlPPl

0.6 0.4 0.2 0.0 -0.2

_ ~

0

appl,..,appl g /19"

i klalppl/g/ppl -

i

i

,

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

I 1

t 2

I 3

t 4

J 5

A/L

F i g u r e 13.3: Stress intensity factors and energy release rate at the tip of a sinusoidal extension to a straight crack We have used the lower case letter g to distinguish it from the applied toughness G for a straight crack.

13.2

I n t e r a c t i o n B e t w e e n Crack D e f l e c t i o n and P h a s e T r a n s f o r m a t i o n Mechanisms

Let us now give an indication of the interaction between the crack deflection and phase transformation mechanisms following the preliminary study by Andreasen (1992) who considered a pre-existing straight crack and a transformable particle in the vicinity of its tip under plane strain conditions (Fig. 13.4). It is assumed that the particle transforms under an applied load level that is below the critical load level required for crack extension. We shall denote the applied stress intensity factor at particle transformation by K PT (a - I, I I ) to distinguish it from the applied stress intensity factor at the instant of crack growth that we have throughout denoted by K appz. We now use the subscript a ( = I, I I ) not so much because of the possible mixed-mode nature of the applied loading, but primarily because of the crack deflection.

Toughening in DZC by Crack Deflection

430

z0-

l

y

0

Figure 13.4: A transformable particle (strain centre) near a kinked crack For a strain centre (strength of centre of dilatation D, the strength of shear centre, S) located at zo(= roe i~176in relation to the crack tip, we use the analysis described in w and let A0 ---* 0 in (4.39) to give #~

O~(z)--

e 2ia

27r(1 - u ) ( z - zo) 2

#D 9~ ( z ) -

27r(1 - , ) ( z -

#S

-5oe2i~

zo) 2 - 2~(1 - , ) ( z - zo) 3

(13.6)

We note in passing that (13.6) is nothing but the sum of corresponding potentials for a centre of dilatation (5.54) and a centre of shear (5.60). The stresses on the crack line (y = 0) induced by the potentials (13.6) are given by the sum of stresses (5.55) and (5.62). The corresponding image potentials required to annul these stresses are given by the sum of potentials (5.57) and (5.63) which we rewrite here for clarity of presentation

ED

(I)i = 2 ( 1 - ~2)

{ A I ( Z , ~ o ) - ~S

[e2i~Al( z ,zo)

~-~'~A (z, ~o) + (zo ~o)~-=~A~(z, ~)] } 9 ~-

~(-~)

-

~(z)

-

(13.7)

z~'~(z)

where a prime denotes differentiation with respect to z, and from (5.58) and (5.67)

Am =

az~

,

4~(v~ + x/~)x/7

}

(13.8)

13.2. Interaction Between Crack Deflection and Transformation

431

The stress intensity factors K T and KTI at the tip of the semi-infinite are given crack due to the single strain centre (Fig. 13.4) at zo - roe i~176 by K T + iKTI -- lim ~ ~x/27rx [ayy(x)+ icrxy(x)]} x-*O+

=

lim

x-.O+

{X/~Trx(~,(x)}

(13.9)

Substitution of (13.7)into (13.9) gives ED

KT = 2(1 - u 2) V/27rroa

cos - ~ + 3 ~ sin 00 cos

2a

~--

ED { 300 KTI = 2 ( 1 - u2)V/27rr 3 - s i n 2

s [2sin(2o 3 o) 3sin00cos

5Oo

,1310,

If we also considered a strain centre at T0 and superimposed its contribution on (13.10), we would retrieve the result (5.73). From the above limited analysis it is difficult to draw any firm conclusions on the combined effect of transformation and kinking on the toughening in TTC but it is instructive in revealing some interesting features. Consider first the situation when a straight crack is approaching a transformable strain centre. Let us assume that the centre transforms when the applied mode I stress induces a stress intensity factor K PT at the crack tip. After transformation the total stress intensity factors at the crack tip are

I 'F- I,'f + I 'T (13 11)

Ix'tip II -- KTI .tip

where K/T and KTI are given by (13.10). As I~.ii ~ O, the crack must deviate from its plane at a certain angle. As we have seen previously (cf. t.tip at the kink tip will drop below (13.4)), the stress intensity factor ~I tip K I , so that it will no longer be equal to Kc and the crack will be arrested. (We do not consider the unstable situation in which the crack growth takes place without inducing the strain centre to transform). To restore quasi-static crack growth conditions along the kinked path, the

Toughening in DZC by Crack Deflection

432

Oc/~ laD

0.15 -

0.20

=

(1-V)Kc

24~nr~

s =o

0.10 0.05 0.00

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-0.05 -0.10 --0.15

0.0

0.4 0.3 0.2

I

I

I

1

0.2

0.4

0.6

0.8

-

laD

0.20 =

]

0.1 -

Y / ~ X ~

0.0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

I

1.0 00/~

S/D=I, .

.

.

.

.

.

a=~12

.

-0.1 -0.2 -0.3

0.0

I

I

I

I

0.2

0.4

0.6

0.8

I

1.0 00/~

F i g u r e 13.5- Critical crack kinking angle as a function of the location of strain centre (r0, 00)

applied mode I stress has to be increased until the dynamical condition - Kc) is met. This may induce further kinking for crack growth ( t"tiP 'I which we will assume to be at shallow angles such that the following two conditions are simultaneously met

13.2. Interaction Between Crack Deflection and Transformation

433

appl

Ki /Kc 1.4 lID

0.20 =

1.2

4(1-V)Kc2"~~r~

1.O

0.1

0.05

0.8

S=O 0.6 0.0

I 0.2

j 0.4

I 0.6

j 0.8

j 1.0 O0/n

KIPPllKc 2.0

0.20 =

~tD

1.5

].o

0.]5 7 / D.IO ' / 0.05 '

0.5

S/D = I, a = n/2 0.0 0.0

i

i

i

0.2

0.4

0.6

J

0.8

i

1.o Oo/n

F i g u r e 13.6: Toughening due to combined influence of crack kinking and transformation of a strain centre at (r0, 00)

(13.12) The two equations (13.12) are solved simultaneously to determine 0c and K~ ppl. Figures 13.5 and 13.6 show the critical kink angle 0c and toughening ratio KTPPt/K c induced by a single strain centre at (r0, 00) for various positions of the centre and several values of its dilatational

Toughening in DZC by Crack Deflection

434

strength D. Two values of S/D are chosen; for equal shear and dilatation (S/D = 1), the shear bands are assumed to be at 450 to x-axis (a = The influence of several asymmetrically placed strain centres or transformable spots upon the crack growth direction and toughening ratio is studied next. This will also allow us to investigate the influence of the non-homogeneous distribution of transformable phase on steady-state homogeneous distribution, described in Chapter 7.

13.3

C r a c k D e f l e c t i o n in a Z o n e o f N o n homogeneous Transformable Particles

In Chapter 11 we presented experimental evidence in support of nonhomogeneous distribution of transformation around a crack plane (Fig 11.2, Table 11.1) in a zirconia toughened alumina alloy (ZTA). The influence of this non-homogeneity upon the crack growth path was ignored. We shall rectify this omission in the present Section and assume that the t-ZrO2 precipitates are distributed in a periodic pattern in the transformation zone around a steadily growing crack. Moreover, we shall assume that the distribution is not symmetric about the crack plane y = 0. A schematic representation of such a distribution is shown in Fig. 13.7. The distribution function D(x, y) is assumed to be of the form

D(x,y) - l+acos

~

cos

Lc

,

aE[0,1](13.13)

where a is the amplitude of variation in cm and the non-dimensionalising parameter Lc governs the distance between the maxima in the distribution function in either direction (Fig. 13.7). The distribution function is shifted in the y-direction by AL (~ C [0; 1]) to simulate asymmetry about y = 0. The characteristic length L is the frontal intercept along the extension of pre-existing straight crack when both dilatational and deviatoric transformation strains are present. It can be calculated using

(3.87)

L-

2 [Kr

9---~

Ec

2

(13.14)

It reduces to (7.28) when deviatoric transformation strains are ignored,

13.3. Crack Deflection in a Non-homogeneous Material

._

t - - ' J J ~ ~

u

w

I.'::

O

9

9

Lc

~

I,

...~:-: o. 9

99

i,;o

o

,,

: '

l ~ o l' , ~

oO

9

U

f:t,-

o 9

oo -

O

9

9

...~ .......:, .......

0 ~ 0 o| " q ~

..........90 ....... ~ o . ~

D 9 ~D ,,,o D.o__o 9 l?,~OoO_. oo~ ! I~,~ r ......................0S.o . . . . . :.'..

lO t

o 9

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

. ..... 9.........

......_8........ '

~ 1

0 ~ o|

, , _ .....90 ....... ~ o ~

12

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

oO o_ ,._o-,7.

Lc ~.,..:L:7.. i',.:L:7. r ~ .

distance from the crack surface in I.tm i,

...~:-:

0

9

]-2a

i

..........90 ..........

.............. 1 ...... 6. .......

o.I

...... 5 ....... .......

_4 .....

I......

O..m q,, 9 O,.am al I 9 T~ o _ T~ I 9 " 9 9 / 9 9. ~ ...... 9 ............... 0II;.o ..... 0l ................................ ;--_ --:.t------:-'

, ~-',- ,'f, ~-',. ,'~ ~-'.- ,'~

1......

P

]

1 ~tm

--O" 41

m

o4/ ~

---It

m

O"

oljJO m

~"~

9

435

Ov oil m,

m"l

/

:

.

0 100 % transformed material

F i g u r e 13.7: Schematic representation of the microstructure of a ZTA material, where zirconia particles appear dark. The distribution of transformed material perpendicular to the crack surface is shown on the right (cf. Fig. 11.2 ) i.e. h0 - 0 and Ec _ Crm c . In a physical sense the amplitude a in (13.13) gives the degree of non-homogeneity, whereas Lc represents its characteristic period. In an average sense, the amount of transformable phase remains constant, so that the strength of transformation, designated by w (3.26) is unaffected. Once the parameters a, Lc and )~ are chosen, the degree of non-homogeneity, and asymmetry of the particle distribution are completely defined. The effect of crack deflection on the toughness development during crack growth will now be investigated using the computational procedure developed by Stam (1994). This procedure will be briefly described later in this Chapter. Here we shall introduce a criterion for determining the direction of crack growth under mixed mode loading. It will be recalled that we already used the criterion of local symmetry in w for this purpose. There are two other criteria available in the literature (Broek, 1982). First, there is the maximum principal tensile stress criterion according to which a crack under mixed mode

436

Toughening in DZC by Crack Deflection

loading will grow in the direction normal to the maximum principal tensile stress. Then there is the minimum strain energy density criterion, according to which the growth will take place along the direction of minimum strain energy density. It is known (Karihaloo, 1982) that these criteria, as indeed the criterion of local symmetry, predict directions of crack growth under mixed mode loading which are very close to one another. The choice among the three criteria is therefore dictated by the mathematical and/or computational effort required for its application. Generally speaking, the criterion of local symmetry is the easiest to use, if the stress intensity factors at the tip of a growing (kinked) crack are known in a closed form, as was the case above in w When however the stresses ahead of the crack tip, necessary for computing the stress intensity factors, are to be computed numerically, the maximum principal tensile stress criterion would appear to be the most convenient. For this reason, this criterion will be used in the following. The Irwin-Williams asymptotic stress fields ahead of a crack tip in modes I and II can be resolved along the circumferential and tangential directions to give (see, e.g. Broek, 1982) crr162 =

1 [.C l l ( r

o',.r = ~

1[

.

,.a,,]

-.[- 6 1 2 ( r

C~1 ( r

pp' + C~2(r

p'

]

(13.15)

where C~z(c~,fl - 1,2) are given by (13.2), and r and r as before, are measured from the crack tip (see, e.g. Fig. 7.1). According to the criterion of maximum principal tensile stress, the crack will grow in the direction r normal to crr162 along which a,r vanishes, i.e. ,

C21(r

,

ppt + C22(r

,.appl

-

0

(13.16)

The similarity of this criterion to the criterion of local symmetry is obvious, at least for small angles. Comparison of the second eqn (13.15) with bappl the second eqn (13.1) shows, that crrr 2x/~-~-r can be identified with ~zI at the tip of an infinitesimal kink along 0, so that for small kink angles the vanishing of cr,r is tantamount to the vanishing of "bappl I I " In fact for small kink angles (13.16) reduces exactly to the second eqn (13.3), with K,.appt , and C22 , from (13.2) into H - 0. In general, substitution of C21 (13.16) gives

13.3. Crack Deflection in a Non-homogeneous Material

r 20tan 2 -~-C - tan -~-. ~ -

0

437

(13.17)

provided r162~: 0 or lr, and K I ppz :/: O, from which the critical kink angle can be calculated for given mixed mode loading

-

appl II

(13.18)

K~pp z

When the loading is predominantly of mode I type, 0 will be very small (g < < 1), so that the straight crack will form a very shallow kink at an angle r - - 2 0 - - "9~f4"appl ' H / K ] ppz, exactly as predicted by the criterion of local symmetry. In general, for any mixed mode loading, (13.17) can be solved to give tan r1622 13.3.1

Computational

1 :t= V/1 + 802 4Q

(13.19)

Procedure

A displacement-based finite element method was used by Stam (1994) to study the influence of non-homogeneous distribution of transformable particles (13.13) upon the development of toughness. Quadrilateral elements, built up from four constant strain triangular elements, were used. The potential region of transformation around a crack was chosen to be a large circle which was in turn divided into four zones, as shown in Fig. 13.8. The rectangular zone 1 closest to the crack tip was covered with 92• 12 fine quadrilateral elements, whereas the numbers of elements over the height of zones 2, 3, and 4 were 9, 5, and 30 respectively. A close-up of the finite element mesh is shown in Fig. 13.9, from which it is seen that at the start of the crack growth simulation the crack tip is three elements into zone 1. The crack tip is positioned at the centre of the potential transformation region. On the boundary of this region the displacement boundary conditions (u, v) corresponding to the applied stress intensity factors I s ppl and K]~ pl are prescribed. The computations under mixed mode loading are initiated by calculating the stress intensity factors at the tip of an incipient kink .~ktip -- k appl + _A.k. t~i p ,

~-

I,

II

(13.20)

Toughening in DZC by Crack Deflection

438

V

)

U

Crack [r(r0,0]

~

t~

[r(0),0]

Figure 13.8: Subdivision of potential transformation region around a crack into four zones where k appl is calculated at each step using (13.1) with the approximate functions C~Z from (13.2). The contribution of transformable particles is calculated by summing the contributions of all individual particles that have transformed in the domain ~ (Fig. 13.8)

Aktip.-a -- //~2 ---adktip

(13.21)

The contribution of each transformation particle (area A0) dkt~p is given by (5.43) in which now S and D depend on the location of the particle (see, (13.13)). In the procedure reported by Stam (1994) that we are following in this exposition, the contribution of each transformed particle was calculated using the two-dimensional weight functions

dktff~p - 2#/A -di~(x' y)E~j(x, y)dxdy

(13.22)

o

where the two-dimensional plane strain transformation strains E/Pj(x, y) are given by (3.30) with the fraction of transformed material c(x, y) = croD(x, y)

(13.23)

Note that the calculations will require Poisson's ratio u, the strength of transformation w (3.26), the transformation shear strain parameter

13.3.

Crack Deflection in a Non-homogeneous Material

439

F i g u r e 13.9: Finite element meshes in various transformation zones h0 (3.61), the hardening parameter a (3.62) and the parameter M =

C(T, O)/Co(T, 0) (3.60). However, as reverse transformation is assumed not to occur M - -cx~. In (13.22) Ui~ are the two-dimensionM counterparts of the threedimensional functions Ui~ that we studied in Chapter 9 (see (9.11)). They are obtained by integrating (9.11) with respect to z from - c ~ to +c~ (Gao, 1989). The non-zero components in modes I and II are Ull - ~-r-~

cos

+ 3cos--

Toughening in DZC by Crack Deflection

440

( 7cos y

--/ Cr-} U 2 2 - ~--I

- 3costa

30

a

U 3 3 - uCr-~ c o s y

_,U i = --1

8 r --t

-sin-~-+sinm --I

Ull -+- U22 -~- U33 - (1 + u)Cr

Ull - - ~ - r - ~ ~II

U22 - -

~II

_}

cos

30

5sin-~- + 3sin m

~I UI2

~I

U12- Ull

--H

U33

mll

_

-uCr-~ sin mll

RII

Ull -{'-U22 -~- U33

_

30 2 0 -(1 - u)Cr-} sin 3m 2

(13.24)

where C - 1/[2(1- u)x/~]. Note that (13.22), together with U~k from (13.24) reduces to (9.48) for dilatational transformation. The dynamical condition for crack growth is assumed to be satisfied tip when the reference stress intensity factor at the kink tip b"~ref given by ,-#

(~z

+ t~,I

(13.25)

attains the intrinsic toughness Kc of the matrix material. When the dynamical condition for crack growth is met, the direction of crack path is iv" In calculated by solving (13.17) in which now Q is replaced by this way, the crack is incremented locally and its direction of growth calculated with respect to its direction at the previous step. It is implied in this incremental procedure that the crack changes its direction gradually in small straight increments. The crack growth is simulated by reducing the stiffness of the particular element linearly to zero in several steps. The non-homogeneity in the distribution of transformable precipitates is simulated by the shift parameter A (13.13) relative to the starting crack

"~IlbtiP/k~

13.3.

Crack Deflection in a Non-homogeneous Material

Lr -

-

-

-

ml" n

..! ....... : ": . ..... " :

~.....max ...../~!....~' iIi - - -;: ] I .............::....mm ~...[...,0 ~ ....1~ " !'.',', \ / ,"~ i! ~i',",,~/,'ii: ..

:!i';

..... k = 0.375 ',, "'- . . . . . . Zc

-..

0

;

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

~k=0i125/

' . " ........ ".

I

X

/

I ',

' :' ~ '

"--" ' / ' ' :".. ""..i

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

..... ~ = 0.250

- -- ma

441

......

,9

mln "

4

-.

:

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

". i :',,"f'--/ :.. . . .:. . . . ...

:' : ,

t,

~,

~ . ma

X

I

F i g u r e 1 3 . 1 0 : Crack paths for three values of )~ over Ac -- 2L with m a t e r i a l p a r a m e t e r s u - 0.3, c~ - 1.15, w - 5, h0 - 1.25, M - -cx~, a - 1. T h e contours of m a x i m a and m i n i m a of the distribution function are also shown tip. T h e crack growth paths for three values of ,~ are shown in Fig. 13.10, t o g e t h e r with the contours of m i n i m a and m a x i m a in the distribution function (13.13). T h e deviation from the initial straight p a t h is the strongest for ,~ = 0.25. It will be seen from Fig 13.10 t h a t the crack is first a t t r a c t e d towards the area with small content of t r a n s f o r m a b l e phase before deflecting towards the area with large content of this phase. This is reflected in the oscillatory n a t u r e of the R - c u r v e , shown in Fig. 13.11. Note t h a t for ~ = 0 and 0.5 the t r a n s f o r m a b l e phase is d i s t r i b u t e d s y m m e t r i c a l l y with respect to x-axis. C o m p u t a t i o n s showed the crack did not deflect from x - a x i s for these two values of A It is seen from Fig. 13.11 t h a t the toughening ratio (K~PfZ/Ktip) is significantly affected by the value of ~. Interestingly, the oscillations in the t o u g h e n i n g ratio are bounded from above by ~ = 0 and from below by )~ = 0.5. Note t h a t for A = 0.35 the oscillations are very shallow so t h a t the corresponding R-curve coincides with t h a t for a h o m o g e n e o u s distribution of t r a n s f o r m a b l e phase (a = 0). T h e p r e l i m i n a r y study reported in this and the previous Sections d e m o n s t r a t e s t h a t n o n - h o m o g e n e o u s distribution of t r a n s f o r m a b l e phase in DZC can cause oscillations in the R-curve whose a m p l i t u d e depends on the degree of non-homogeneity. This m i g h t explain the scatter in the R-curves d e t e r m i n e d from tests on DZC. -

- j

Toughening in DZC by Crack Deflection

442

l( appl [K c

ref

t

2.0 ~ 1.5

....a.~[ 0.0

k=O.O ~ //

.

.

0

,

/

2

5 \~

.............a.._--..9.

I

I

I

1

i

0.5

1.0

1.5

2.0

2.5

J

3.0 AclL

F i g u r e 13.11: Development of toughness with crack growth for the materials parameters of Fig. 13.10. Also shown is the R-curve for a homogeneous distribution (a - 0)

443

Chapter 14

Fatigue Crack Growth in Transformation Toughening Ceramics 14.1

Introduction

Evidence of the susceptibility of zirconia toughened ceramics to mechanical degradation under cyclic tensile loading has been provided by a number of studies (see e.g. Dauskardt et al., 1987; Swain ~: Zelizko, 1988). Similar to metals the crack growth rate d c / d N for long cracks follows the well-known Paris power-law dependence on the applied stress intensity range, A K appz Ka~P~ - - K:..,~ppt mi n . Using the steady-state shielding value, K ss (w to calculate the effective stress intensity range at the crack tip, A K tip - KamPaP~- K s`, Dauskardt et al. (1990) showed that long crack data for materials of different toughnesses fell close to a universal curve of the form dc (14.1) = A (AKtip) n , -

-

dN

where A and n are material parameters. A more recent study by Steffen et al. (1991) showed a different behaviour for small cracks. It revealed a negative dependency of the crack growth rate on the applied stress intensity range for small (250 pm) naturally occurring surface cracks. Furthermore, small cracks were observed to grow at stress intensity levels below the long crack fatigue threshold, at which fatigue cracks are presumed dormant in damage tolerant de-

Fatigue of Ceramics

444

sign. The small crack behaviour is therefore of serious concern to fatigue life prediction procedures, where the use of long crack data could lead to a nonconservative design. It was also shown schematically in that paper that the crack growth rates for long cracks fell within the region of small crack behaviour, when plotted against the maximum effective stress intensity factor, K tip. Again the steady-state shielding was used to calculate K tip. The presentation in this Chapter suggests that a relation similar to eqn (14.1) can be used for the fatigue behaviour of small, as well as long cracks by computing the shielding effect as the crack grows rather than using the steady-state value to calculate K tip. The exposition follows closely the paper by Andreasen et al. (1995).

14.2

Fatigue Crack Growth From Small Surface Cracks in Transformation Toughening Ceramics

In this Section we will employ the model for a small surface crack presented in w and modify it to accommodate crack growth under fatigue conditions. The results of the theoretical analysis will be used to explain some of the above-mentioned experimental findings, together with other results obtained by in situ observations in a scanning electron microscope. These experiments confirmed the decrease in crack growth rate in the initial stage of loading.

F i g u r e 14.1: A Surface crack model for fatigue crack growth

14.2. Fatigue From Small Surface Cracks

445

In Fig. 14.1 the model of a single surface crack in a transforming ceramic is illustrated. The crack, C, is subjected to an oscillating far-field load, with a peak value of a ~ , normal to its faces resulting in a pure mode I opening of the crack. At the tip of the crack a transformation zone with the boundary, S develops when the load is applied. Full transformation is assumed to occur when the mean stress reaches a critical C value, ~rm . It is also assumed that the transformation is accompanied by a purely dilatational strain inside the transformation zone, i.e. the deviatoric part of the tetragonal to monoclinic transformation is not taken into account, and the transformation is assumed to be irreversible leaving a transformation zone wake behind as the crack grows. The crack is modelled by a pile-up of dislocations. Two singular integral equations for determining the dislocation density function and transformation zone result from the following two conditions: 1. The crack faces are traction-free, ax=(z) - 0, z E C, 2. The mean stress on S is equal the critical mean stress for C super-critical transformation, a,~ - am, z E S. The applied stress is written in terms of a dimensionless stress, a0 as ~0 -

l+v

~~o~

(14.2)

30" c m

Introducing this in the derivations given in w the traction-free condition for the crack and the critical mean stress condition for the transformation zone boundary can be written as the following singular integral equations 0-

a0 + ~

g== r6.C

1 - ~o + ~

D,= is)ds fs g,g (z, zo )do + /c D(s) h..(z,

(14.3) zES

From (14.3) the unknown dislocation density function D(s) and transformation zone boundary S can be calculated for a given applied load ~r0. The dislocation density function D(s) has a square root singularity as the crack tip is approached, where the stress intensity factor K tip is given through the following limit

Fatigue of Ceramics

446

Ktip Kr = ,--.-~+lim2rD(s)

s +L y

(14.4)

The applied stress intensity factor K appt can be written as

Kappl

KC

= Y~ro ~/2c/L

(14.5)

where Y = 1.1215 is a geometrical factor. As mentioned above, in the simulation of the fatigue crack growth it is assumed that the transformation is irreversible, so a wake of transformed material is left behind as the crack grows. In the numerical procedure the dislocation density function is computed from the first eqn (14.3) using the zone shape prior to the crack increment. A new zone is then determined from the second eqn (14.3) and joined with the old according to the assumption of irreversibility of transformation. An improved dislocation density function can then be predicted from this joined zone. This procedure is repeated until convergence, whereupon the crack can again be incremented. At each crack increment, K tip and K appt are calculated from eqns (14.4) and (14.5). Obviously the model does not take the cyclic nature of the loading into consideration when the transformation zone is determined, but it is assumed that the zone shape develops during peak loading. Raman spectroscopy and surface uplift measurements made under monotonic and cyclic loading by Dauskardt et al. (1990) support the contention that the transformation zone shape and size are not significantly affected by the cyclic character of the loading. 14.2.1

Examples

of Fatigue

Crack

Growth

In analogy with eqn (14.1) the crack growth law is formulated as dc

dN = B

(Ktip) n K~

(14.6)

where K tip -- K a p p l - Ks denotes the maximum stress intensity factor at the crack tip in a loading cycle, K, representing the shielding effect due to transformation. Taking the logarithm of eqn (14.6) gives log K tip _- 1 Kc n

( log ~de - log B )

(14.7)

14.2. Fatigue From Small Surface Cracks

447

where the equivalent crack growth rate log(dc/dN) is seen to be proportional to log(KtiP/Kc). When log(KtiP/Kc) is plotted against the applied stress intensity factor curves shown in Figs. 14.2(a)-(d) are obtained for various values of the transformation strength parameter w, applied load cr0, and initial crack length co/L. In Fig. 14.2(a) the equivalent crack growth rates for an initial crack length co/L = 5 and applied load o'0 = 0.10 are depicted for three values of the transformation parameter w = 10, 15, and 20. For w = 10 and 15 the initial crack growth rates quickly decline to a minimum but increase thereafter and follow the normal power law for long cracks. The initial drop in K tip can be explained by the development of a transformation zone wake increasing the shielding as the crack grows. This corresponds to a rise in K appz in the initial stage of an R-curve analysis. At the minimum the shielding effect due to the formation of the transformation zone wake is overshadowed by an increase in K appz due to the increased crack length. The minimum is related to the local maximum in the R-curve, as described in Chapter 8. For ~o = 20, the initial decline is sufficient for the crack-tip stress intensity factor to drop to zero, which for this example happens at a crack growth increment of A c / L = 0.126. The shift in the power-law region between the curves for w = 10 and 15 in Figs. 14.2(a)-(d)is due to different amounts of transformation toughening. The numerical results presented in Figs. 14.2(a)-(d) show that the crack growth rates decrease initially with increasing applied stress intensity until the crack has grown a sufficient amount. Thereafter the rates approach the normal long-crack behaviour. The initial decrease for small cracks is confirmed by experimental evidence gathered by Steffen et al. (1991). The arrest of small cracks observed experimentally is also in agreement with the numerical results. The arrest can occur when K tip has declined to some threshold value. The numerical simulation terminates when K tip reaches zero. In this connection it should be mentioned that the decrease in K tip could in some cases lead to compression at the crack tip in the unloading part of the fatigue loading cycle. Compression as part of the loading cycle has been shown to have a substantial influence on the fatigue crack growth (Steffen et al., 1991), but no definite results are available at present. In Fig. 14.3 experimental fatigue crack growth results obtained by in situ observations in a Scanning Electron Microscope (Mouritsen & Karihaloo, 1993; Mr & Karihaloo, 1990) are shown. Since the experiment was conducted in the vacuum chamber of the SEM, the effect of environmental assisted crack growth was excluded. The results originate

Fatigue of Ceramics

448

log(K tiPlKc) 0.0

--

-0.2 -0.4 -0.6 -0.8 -1.0

{0=10 {0=15 {0=20

-1.2 -1.4 a)

_1._60.50

I

i

I

-0.45

-0.40

-0.35

I

-0.30

I

I

-0.25

-0.20

log(K appl]Kc)

log(K tiP/Kc)

~176 f

-0.2

-0.4 -0.6

-0.8 -1.0

{0=10 {o=15 {0=20

-1.2 -1.4 b)

-1.6 -0.30

, -0.25

, -0.20

J -0.15

I -0.10

I

I

-0.05

0.00

log(K appl/Kc)

F i g u r e 1 4 . 2 : F a t i g u e crack g r o w t h r a t e s for (a) a n d (b) c o / L - 5 a n d ~ o - 0.15

co/L

- 5 a n d ~0 - 0.10,

14.2.

Fatigue From Small Surface Cracks

log(K

l/p~

449

,.

/Kc)

0.0-

C)

-0.2

-

-0.4

-

-0.6

-

-0.8

-

-1.0

-

-1.2

-

-1.4

-

-1.6'

-0.35

(0=10 (0=15 (0=20 I

I

1

I

1

I

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

log( K appl[Kc)

log(KtiP/Kc) 0.0-0.2 -0.4 --0.6 -0.8 -1.0

(0=10 (0=15 (0=20

-1.2 -1.4 d)

-1.6 -0.15

I -0.10

I -0.05

t 0.00

t 0.05

I

I

0.10

0.15 log(K appl/g c)

F i g u r e 14.2" (Continued) Fatigue crack growth rates for (c) co/L - 10 and ~ro - 0 . 1 0 and (d) c o / L - 10 and ~ r o - 0.15

450

Fatigue of Ceramics log(dc/dN) [log(mm/cycle)] -3.0 -4.0 -5.0 -6.0 -7.0 -8.0 0.70

I

I

I

I

I

I

0.75

0.80

0.85

0.90

0.95

1.00

log(Kappl ) [log(MPa~/'m)] F i g u r e 14.3: Experimental fatigue crack growth rates in mm/cycle as a function of K appz from a single Mg-PSZ specimen containing a long sharp surface crack of 2.6mm initiated in four-point bending from a notch. Prior to tensiontension in situ loading the specimen was annealed at l l 0 0 ~ to remove residual stresses due to cutting and crack initiation, and to reverse any transformation. A curve has been drawn through the measured points to illustrate similarity with the numerical results of Figs. 14.2(a)-(d). The large scatter in the growth rate seen in Fig. 14.3 can be attributed to the relatively high level of magnification at which the crack growth was observed. Despite this scatter, the crack growth rate clearly shows a decreasing trend in the initial stage. After a crack growth of approximately 0.9mm, the normal power-law region for long cracks was reached. This indicates that small crack growth behaviour is primarily due to the fact that the crack is initially free from transformation and the tip is only shielded when a transformation zone wake develops. The initial physical size of the crack is therefore only of secondary importance. The amount of transformation toughening is measured as the ratio of the applied stress intensity factor K appz to the crack-tip stress intensity

14.2.

451

Fatigue From Small Surface Cracks appl tip /K

K

2.0 [1.8 1.6

~_-=

1.4 lff 1.2 ~

~\

1.0 ~ /

a)

\ c0/L=5 O0/Om-O.15

0.8 0.0

K

appl

/K

c~ O0/om=O'lO ~ \ co/L=5 O0/Om=O.lO

i

1.0

i 2.0

i 3.0

t 4.0

i 5.0

J 6.0 Ac/L

tip

5.0 4.0 3.0 2.0 1.0

b)

0.0 0.0

\

O0/Om=O.15

\ ' , co/r=lO O0/Om-O.lO

~ \ Cd/.~50'dO'm=O. 10 , \co/L'~lO O0/?m=O.15

1.0

2.0

3.0

l

I

4.0

5.0

i 6.0 Ac/L

F i g u r e 14.4: Transformation toughening for (a) w - 10 and (b) ~v - 15

factor K tip. This is shown as a function of crack advance in Figs. 14.4(a)(b) for w = 10 and 15, respectively. The steady-state toughening ratio K a p p I / K tip for w = 10 varies between 1.56 and 1.58, and that for w = 15 between 2.08 and 2.09. These values are within 5% of the steadystate toughening values of 1.63 and 2.16 for w = 10 and 15, respectively obtained for quasi-static crack growth (7.22). An example of the resulting transformation zone shape is given in

452

Fatigue of Ceramics

y/L

1.2 0.8 0.4 0.0 -40

-35

-30

-25

-20

-15

-10

-5

0

I

5

x/L

F i g u r e 14.5: Transformation zone shape for w - 10, cr0 - 0.15, and c o / L - 10 Fig. 14.5 for w - 10, cr0 - 0.10 and co/L - 10, with the steady-state transformation zone shape obtained for quasi-static crack growth (w shown by the dashed curve for comparison. Though the difference in scaling between the x- and y-axes tends to exaggerate the difference between the two transformation zone shapes, it is clear they are distinctly dissimilar. In view of the large differences in zone shapes, one should not be lulled into concluding that the value of toughening obtained via quasi-static crack growth will always be close to the steady-state value obtained from the present analysis. The differences in toughening values mentioned above may very well reflect the different shapes of the associated transformation zones. The results for crack growth rates from Figs. 14.2(a)-(d) can be unified by normalizing the applied stress intensity factor by the steady-state toughening from quasi-static crack growth K 8s, rather than by the intrinsic toughness Kc. The normalized curves are shown in Fig. 14.6. The quasi-static steady-state toughening used to normalize the curves for a~ - 20 is 3.05. The crack growth rate curves in Fig. 14.6 that reach the power-law region are simulated until unstable crack growth appears with K tip = Kc. These curves fall within a very narrow band, thereby showing that replacement of the steady-state toughening from fatigue crack growth with values from quasi-static crack growth is a good approximation. From the results presented above, it would appear that the power-law

dc : C

dN

( i~(tip ) n K'"

(14.8)

14.2. Arrest of Fatigue Cracks

log(K 0.0,

d a

453

d

d

C

-0.5 a

-1.O

-1.5 -2.0

I

.0

-0.8

~,

-0.6

I

-0.4

I

-0.2

I

0.0 log(Kappl/Kss)

F i g u r e 14.6: Normalized fatigue crack growth rates (Letters refer to the three values of w in Figs. 14.2 is valid for both the initially small and the subsequently long crack stages of fatigue crack growth in transformation toughening ceramics.

14.3

Arrest of Fatigue Cracks in Transformation Toughened Ceramics

In the previous Section it was demonstrated that certain combinations of transformation strength and load cause the effective stress intensity factor to drop to zero, indicating crack arrest. A more detailed parametric study of this phenomenon to be discussed in this Section reveals that the applied load and minimum transformation strength parameter necessary to cause crack arrest are linearly related, independently of the initial crack length. This suggests that a threshold stress similar to the endurance limit in the conventional S/N approach can be used in the design of TTC against fatigue, instead of the threshold stress intensity factor. In this Section a detailed study is made of the parameters responsible for the behaviour of K tip during fatigue crack growth. The parameters considered are the initial crack length co, the transformation strength and the applied load tr ~~ The exposition follows closely the paper by Mr & Karihaloo (1995)

Fatigue of Ceramics

454

tip K /Kc

co~L=30

1.0

15

10

5

1 0.25 ............

0.8 0.6 0.4 0.2 0.0

0

5

10

15

20

25

30

35

40

45

J

50

(C-Co)/L

F i g u r e 14.7: Normalized effective stress intensity factor as a function of crack advance for a single surface crack with w = 10, or0 = 0.15 and co/L = 0.25, 1, 5, 10, 15, 30

The normalized effective stress intensity factor is plotted in Fig. 14.7 as a function of crack advance for different initial crack lengths, co and fixed values of w = 10 and cr0 = 0.15. All the curves generally follow the same pattern, irrespective of the initial crack length. Initially, there is a drop in K tip, as the shielding increases due to the formation of a transformation zone in the wake of the crack tip. At a finite crack advance K tip reaches a m i n i m u m and a monotonic increase in K appz overrides the increasing shielding. After the minimum, a monotonically increasing K tip eventually leads to failure, as Ktip/Kc reaches unity at a finite crack advance. For the given transformation strength and load, cracks initially longer than about co/L = 20, would experience Ktip/Kc > 1 using the procedure described above. In this case an R-curve analysis is needed to simulate the first cycle by letting the crack grow quasi-statically under monotonically increasing load and maintaining Ktip/Kc = 1. For co/L = 50, the peak applied stress is reached at cr0 = 0.14, thereby causing the crack to grow spontaneously in the first cycle before the desired load level is reached. Increasing the transformation strength parameter, w to 15 does not change the essential behaviour of the effective stress intensity factor, apart from a reduction in its level. However, the initial crack length at which the subcritical

14.3. Arrest of Fatigue Cracks

K

455

tw /K c

1.0

\

0.8

........................................................ ~cdL=50 40

0.6 0.4 0.2 0.0

0

0.5

~ 1.0

1.5

2.0

2.5

3.0

3.5

i 4.0 (C-Co)/L

Figure 14.8: Normalized effective stress intensity factor as a function of crack advance for a single surface crack with w = 20, (r0 = 0.15 and co/L = 0.25, 5, 10, 15, 20, 30, 40, 50. On the scale of the plot, the curve for co/L = 0.25 is barely visible crack growth is possible, and the amount of crack advance before instability sets in, are both longer compared to their values for ~ = 10. This illustrates the importance of the transformation strength for the fatigue crack growth. In a practical application there would be a large difference in the lifetime of a component for w = 10 and w = 15, due to the different effective stress intensity factor that the crack tip would experience. The lifetime is further extended for w - 15 by the longer critical crack length before instability sets in. For w = 20 the scenario changes completely as illustrated in Fig. 14.8. Now the curves all drop to zero prior to the minimum being reached irrespective of the initial crack length. In Fig. 14.9 the crack advance until arrest is presented as a function of the normalized initial crack length. It shows a linear dependency on the initial crack length with a slight bend at about co/L - 15. For the longer cracks a certain amount of R-curve crack growth is experienced in the first cycle in contrast to the shorter cracks for which K tip/Kc does not reach unity. For c0/L > 15, - 20 and (r0 = 0.15, the linear relationship is approximated by Cart -- CO = C~lC0 = 0.086C0

(14.9)

456

Fatigue

of Ceramics

(Carr'Co)[Z 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

0

5

10

15

20

I

I

I

1

I

I

25

30

35

40

45

50

co/L

F i g u r e 14.9: Normalized crack advance at arrest as a function of normalized initial crack length for w - 20 and ~r0 - 0.15

where the gradient, c~1 depends on the applied load and the transformation strength parameter. The above results for variable w and constant ~r0 indicate t h a t the combination of the transformation strength parameter and load determines conclusively whether crack arrest will occur or subcritical crack growth will eventually lead to failure. To investigate this further, let us fix the initial crack length at co/L = 5, and the transformation strength p a r a m e t e r at w = 20, and calculate the effective stress intensity factor for different load levels ~r0. The results are shown in Fig. 14.10, from which it is seen that crack arrest occurs at load levels lower than a value between a0 = 0.29 and ~r0 = 0.30. The initial crack length is therefore varied at load levels just above or just below this narrow range. The cracks either arrested, as at load level ~r0 = 0.29, or eventually grew unstably, as at ~r0 = 0.30. This confirms that the combination of transformation strength and load determines whether or not crack arrest will occur, irrespective of the initial crack length as long as it does not exceed the critical value at which spontaneous fracture will occur in the first loading cycle. The behaviour of the effective stress intensity factor for a fixed initial crack length, co/L = 5 has been investigated during crack growth for six different loads and various values of w; one typical example is shown in Fig. 14.11 for cr0 = 0.10. From this investigation a relation between

14.3.

Arrest of Fatigue Cracks

457

t/p._

r~

1.O 0.8 0.30 \0.29

0.6

_

0.4 0.2 0.0

t

0

\

1

I

I

2

3

\

i

I

I

4

5

6

(C-Co)/L

F i g u r e 1 4 . 1 0 : N o r m a l i z e d effective stress intensity factor as a function of crack advance for a single surface crack with w - 20, co/L - 5 and cr0 - 0.10, 0.15, 0.20, 0.25, 0.29, 0.30

t/p_

K /Kc 0.40 0.35 0.30 0.25 0.20 -

t.o=lO ~

15

0.15 0.10 0.05 0.00 0.0

0.2

0.4

I

0.6

I

0.8

I

1.O (C'Co)/L

F i g u r e 14.11" N o r m a l i z e d effective stress intensity factor as a function of crack a d v a n c e for a single surface crack with ~0 - 0.10, co/L - 5 and - 10, 15, 16.3, 16.4, 16.45, 16.5, 17, 20

Fatigue of Ceramics

458 ap Kmin/gc

1.21.0 ~

.

3

0

o.8

o..

o.6

0.15

0.4 0.2 0.0

i

0

5

i

10

15

20

F i g u r e 14.12: Normalized m i n i m u m effective stress intensity factor as a function transformation strength parameter for co/L - 5 and ~r0 = 0.025, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30 the applied load and transformation strength necessary to cause crack arrest can be developed. For the particular load shown in Fig. 14.11 a transformation strength, w = 16.5 will just cause crack arrest. The mini m u m effective stress intensity factor experienced by the cracks during .tip their growth history, K m i n is shown in Fig. 14.12 as a function of the transformation strength. The intercept with the z-axis determines the lower limiting value w~r~ where arrest will occur at a given load. An upper limit ~.a~ at which crack arrest is possible can also be delineated for large values of or0. Thus crack arrest is only possible for values of between the lower and upper limits. These limits are shown in Fig. 14.13 as a function of the normalized applied load. A linear relationship with or0 is observed for both limits wff r - a2~o + j32 - 8.62ao + 15.66

(14.10)

w~rr - a3~ro +/33 - - l 1 4 a o + 53.55

(14.11)

The existence of the upper limit can be explained from the shape and size of the transformation zones. For transformation strengths above ~arr , a large portion of the zone is formed ahead of the crack tip which is known to have a detrimental effect on the toughness of the material.

14.3. Improved Endurance Limit by Overloading

459

arr

{o

24-

..

22-

", ., .,,

20181614 0.00

~

J

I

0.05

I

I

I

0 . 1 0 0 . 1 5 0.20

I

I

0.25

0.30

I

0.35 %

F i g u r e 14.13: Minimum (solid line) and m a x i m u m (broken line) transformation strength parameter necessary for crack arrest as a function of the normalized applied stress for c o / L - 5 This phenomenon is akin to the so called autocatalysis (Stump, 1994), whereby stresses due to the transformation itself induce additional transformation. In the design against fatigue the above equations would be very useful to predict the m a x i m u m load at which crack arrest can be expected in a T T C with a known transformation strength parameter, irrespective of the natural flaw size. This load could be regarded as a threshold stress which can be used in the design against fatigue instead of the conventional threshold stress intensity factor, at least for loading conditions for which the effective stress intensity factor is below the intrinsic toughness of the material. This is equivalent to the endurance limit in the conventional S/N design approach.

14.4

Improved Endurance Limit of Zirconia Ceramics by Overloading

In Fig. 14.14 the effective stress intensity factor at the crack tip is plotted as a function of crack advance for different normalized applied stresses and fixed values of w - 15 and normalized initial crack length, co/L - 5. All the curves generally follow the same pattern at different levels of K tip

Fatigue of Ceramics

460

t~, K /Kc 1.0

08t

:: %= 0.25 After overload

~ \

-

04 I\ 0.2 ~ 0.0

0

~ 0 . 1 0

I

2

I

4

6

l

8

,J

10 (C-Co)/L

F i g u r e 14.14: Effective stress intensity factor as a function of crack advance for a single surface crack with a~ - 15, co/L - 5 and cr0 = 0.10, 0.15,0.20,0.25, and ~r0 - 0.25 after an initial overload at ~r~ = 0.3455 depending on the applied load. Initially, there is a drop in 1s tip, a S the shielding increases due to the formation of a transformation zone in the wake of the crack tip. At a finite crack advance, K tip reaches a m i n i m u m and a monotonic increase in K appl overrides the increasing shielding. For a sufficiently high transformation strength w, Is tip will drop to zero before the m i n i m u m is reached and thereby cause crack arrest. For w - 15 however, after the minimum, a monotonically increasing K tip eventually leads to failure, as K tip/Kc reaches unity at a finite crack advance. This is qualitatively consistent with the known R-curve behaviour. The above curves are compared with the curve for an identical crack which is initially overloaded to cr8 - 0.3455, which is equal to the ultim a t e strength of this composition, before being loaded at the constant value tr0 - 0.25. The crack is grown as in an R-curve analysis under an overload (i.e. it is allowed to grow quasi-statically under monotonically increasing load while K t i p / K c is maintainined at unity). Subsequently the load is lowered to tr0 - 0.25 to simulate fatigue crack growth. The overload has a significant effect on the behaviour of K tip after the load has been reduced leading to a quick arrest of the crack; without the overload Is tip would only experience a small drop at the load level cr0 - 0.25 and the crack would eventually lead to failure.

14.4. Improved Endurance Limit by Overloading

461

coiL 60 50403020100 0.15

J

0.20

0.25

0.30

0.35

0.40

0.45 Go'

F i g u r e 14.15: Maximum tolerable initial crack length as a function of normalized applied overload for w - 15 The beneficial effect of an initial overload is well-known from metals, e.g. in the proof testing of high pressure vessels. In transformation toughened ceramics the effect can be theoretically explained by the development of the transformation zone. The initial overload causes crack advance and formation of a large transformation zone wake. During the subsequent low level fatigue loading the crack merely grows into the initial zone without any additional transformation zone being formed in front of it. In this way, there is an overall increase in the amount of transformed material behind the crack tip. Since it is the transformation zone in the wake that provides the bulk of the shielding (i.e. a decrease in Ktip), whereas the transformation zone in front of the crack tip has the opposite effect, the overload generally improves the fatigue behaviour of a transformation toughened material. For a given overload level ~r~, only T T C components containing cracks smaller than a certain length, c~ would survive the overload. This corresponds to a conventional proof test for screening out components not capable of sustaining the proof load. Figure 14.15 shows the critical crack length that components made from T T C with w = 15 can tolerate without failure at different overload levels. At the overload level g~ - 0.3455, the critical crack length is c~/L 5. In Fig. 14.16 the effective stress intensity factor is plotted as a function of crack advance for different values of stress applied after the initial

Fatigue o f Ceramics

462 t/t, K /K c 1.0 0.80.60.40.2 0.0

33~~ 0 O. 15 i

~

0

I

2

J

4

6

8

.2985

J

I

l

10

12

14

(C-Co)/L

F i g u r e 1 4 . 1 6 : Effective stress intensity factor as a function of crack advance for a single surface crack with ~o = 15, co/L = 5 and cr0 = 0.15, 0.20, 0.25, 0.28, 0.29, 0.2975, 0.2985, 0.30, 0.31,0.32 after an initial overload at ~r8 - 0.3455

e (7O

0.300 0.295 0.290 0.285 0.280 0.275 0.270 0.265 0.260

0

1

I

I

I

I

I

1

2

3

4

5

6

co/L

F i g u r e 1 4 . 1 7 : E n d u r a n c e limit as a function initial crack length with w - 15 and ~r~ - 0 . 3 4 5 5

14.4.

Improved Endurance Limit by Overloading

463

e Go

0.32 0.30 0.28 0.26 0.240.22 0.20 0.18 0.16 0.15

I

0.20

0.25

0.30

0.35

0.40

0.45 %'

F i g u r e 14.18: Minimum and m a x i m u m endurance limits as a function of overload level with w - 15 overloading. The stress level at which crack arrest is possible is o'0 = 0.298. This value is regarded as the fatigue endurance limit ~r~ for the given material, overload level and crack configuration. An analysis of shorter cracks with the same transformation strength and subjected to the same overload surprisingly showed (Fig. 14.17) a decreasing endurance limit with decreasing initial crack length. The longest critical crack in a proof test is therefore not the worst case crack length in the post-overload fatigue crack growth. However, the endurance limit approaches a m i n i m u m value asymptotically as the crack length decreases. The m i n i m u m value is reached (to within the numerical accuracy) for an initial crack length co/L ~, 2 which did not grow during overloading. It should also be noted that crack arrest for w - 15 is not possible without overload, irrespective of the initial crack length and applied load (Mr & Karihaloo, 1990). The proof test could therefore play an i m p o r t a n t role in the improvement of the fatigue performance of T T C , especially of ceramics with low transformation strength. Figure 14.18 shows the normalized m i n i m u m and m a x i m u m endurance limits as a function of the applied overload level for w - 15. The m i n i m u m limit is of course the most relevant from a practical point of view, if the crack size distribution is not known. Both limits increase monotonically with the applied overload level. The m a x i m u m achievable endurance limit is bounded by the maxi-

Fatigue of Ceramics

464

e

cIo

0.35 {o=20

0.30

17

0.25

1510

0.20 0.15 0.10 0.05 0.00 0.00

I

0.10

0.20

0.30

0.40

I

0.50 %'

F i g u r e 14.19: M i n i m u m endurance limit as a function of overload, for - 5, 10, 15, 17, 20 m u m value of cr~ that can be allowed without causing global transform a t i o n in the component (i.e. the applied mean stress equals the critic cal mean stress for transformation, g~ - gin)" For instance, the critical mean stress is reached globally when g8 - 0.433 for a T T C with u - 0.3. The m i n i m u m endurance limit is plotted in Fig. 14.19 as a function of the applied overload for different values of w. The curves for w = 5, 10 follow the same pattern as for w = 15 in Fig. 14.18. In the range w = 1 5 . 5 - 19.5, there exists a lower bound on the overload level below which crack arrest occurs without any overload, as for r - 17 in the figure. For higher values of w no significant change in the endurance limit due to initial overloads is detected. The above observations are all consistent with the results from the previous Section. If the transformation strength is known, a proof test can be designed from the above results to achieve a prescribed endurance limit. The statistical crack size distribution however dictates how m a n y or how few components will survive the proof test. It has been shown above how initial overloads can be exploited to improve the endurance limit of transformation toughened ceramics. Most importantly, it was demonstrated t h a t an overload will facilitate crack arrest in a low transformation strength material which would otherwise be prone to cyclic fatigue degradation and eventual failure.

465

Chapter 15

W e a r in Z T C 15.1

E x p e r i m e n t a l Observations of Wear in Zirconia Ceramics

An abundant amount of wear test data is available in the literature for various types of ceramics (Macmillan, 1989). Among the ceramics most likely to be used in engineering applications are A1203, silicon carbide SiC, silicon nitride Si3N4, sialons SiA1ON, and zirconia ZrO2 based ceramics. The review below will be confined to experimental observations of wear of zirconia ceramics. Overviews of a number of ceramics for wear applications have been given by nreznak et al. (1985), Cranmer (1987), and by Garvie (1983). 15.1.1

Partially

Stabilized

Zirconia,

PSZ

Information on friction and wear properties of zirconia ceramics is quite sparse compared to other ceramics. The major contribution on wear of partially stabilized zirconia and especially of Mg-PSZ, is from Hannink and co-workers (Hannink et al., 1983, 1984; and Scott, 1985). Aronov (1986; 1987), Braza et al. (1987), Gane & Beardsley (1987), Rainforth et al. (1989), Zhu & Cheng (1991), and Wang et al. (1993) have also made experimental investigations of wear in PSZ. All of these investigations have primarily focused on the sliding and rolling wear of magnesia partially stabilized zirconia. Abrasive wear testing of Mg-PSZ has been performed by Thomsen (1991) and Naghash et al. (1994). The coefficient of friction #s is of great importance for a material with

466

Wear in Z T C

a potential for use in a wear application. For Mg-PSZ the coefficient of friction reportedly varies quite dramatically. In dry testing of Mg-PSZ sliding against itself, #8 was found to lie in the range from about 0.2 to about 1.0 (Hannink et al., 1984; Gane &: Beardsley, 1987; Aronov, 1986, 1987), with strong dependence on the test conditions, i.e. load, temperature, sliding speed, etc. The influence of temperature (Hannink et al., 1984; Aronov, 1986, 1987) indicates a maximum coefficient of friction for Mg-PSZ at ~ 400~ At low loads, Mg-PSZ seems to have quite unique low friction properties (Gane &: Beardsley, 1987). When Mg-PSZ is tested against other materials such as steel, copper, tungsten carbide so that the conditions are different, the experiments show that the coefficient of friction generally stays within the aforementioned range (Scott, 1985; Gane & Beardsley, 1987). However, Wang et al. (1993) have shown that a thin copper solid lubricant film on Mg-PSZ can reduce the coeflic.ient of friction to values below 0.1. Scott (1985) has observed an increase in coefficient of friction from 0.25-0.30 to about 0.5 when water was used as a lubricant. On the other hand, in stearic acid p8 reduced to 0.09. From the sliding wear results presented by Scott (1985), Gane &: Beardsley (1987), Braza et al. (1987), and Aronov, (1986; 1987) zirconia shows promising wear properties in sliding, especially at low sliding speeds. The dry sliding wear rates are comparable with those of tool grade alumina and better than those of Y-SiA1ON (Scott, 1985) and Si3N4 (Braza et al., 1987). However, the physical conditions during the tests (load, temperature, sliding speeds, environment, etc.) play a major role in determining the wear rates. Temperature has a significant effect on the wear rate. Both Scott (1985), and Aronov (1986, 1987) obtained a minimum wear rate in the 200-300~ range. However, Scott (1985) reported that in a narrow temperature range around 200~ very rapid wear took place which is believed to be linked to the tetragonal to monoclinic phase transformation. The presence of water increased the wear rate by an order of magnitude (Scott, 1985) in line with the increase in coefficient of friction, but for stearic acid an increase in wear rate was also seen, although the coefficient of friction decreased. Obviously, a tribo-chemical effect may be an important factor in the wear of zirconia; this makes the choice of lubricant crucial. In combined "cyclic" rolling and sliding wear tests, Braza et al. (1987) have shown that Mg-PSZ performed poorly compared to a pure "static" sliding test. During the static test the sliding load remains at a fixed position on the test specimen leading to a static or monotonic loading

15.1. Experimental Observations of Wear in Zirconia Ceramics

467

of the material, while in the cyclic test the applied load moves along the circumference of the circular test specimen leading to a cyclic loading of the material. The alternating compression and tension (compression in the front of the contact load and tension at the trailing edge caused by shear tractions due to friction) of the material below the wear surface in the cyclic test seems to change the characteristics of the material. In the static sliding test, the wear rate of PSZ was lower than that of Si3N4 used for comparison, whereas in the cyclic test the wear rate of PSZ was several orders of magnitude higher than that of Si3N4. This is believed to be a result of the phase transformaton in the near-surface region caused by the tensile stresses from the contact load. Braza et al. (1987) suggested that the transformation toughening effect in the cyclic test may be negated by the stress reversal. They based their assumption on cyclic fatigue data obtained by Dauskardt et al. (1987) who found that the fatigue cracks in PSZ propagated at much lower values of AK than those predicted from the fracture toughness values. However, Dauskardt et al. (1990) have later shown that there is no significant difference in the transformation strains under cyclic and monotonic loading. The reason for the difference in wear performance may rather be due to a change in the rolling/sliding conditions on the wear surface (i.e. coefficient of friction and pressure distribution in the contact area), caused by the subsurface transformation in the cyclic test. The transformed particles may introduce a surface uplift due to the dilatational and shear strains accompanying transformation leading to an alteration of the surface topography. The near-surface transformation may not only introduce uplift, but also compressive stresses in the near-surface layer which induces a (desirable) shielding effect on surface cracks. However, the compressive stress field, in combination with the change in surface topography and subsurface flaws like cracks, may also lead to material spalling off. This is confirmed by observations of the wear surfaces in Mg-PSZ by Rainforth et al. (1989). They observed in a pin on disc test that many grains of Mg-PSZ in the wear track contained parallel grooves and isolated pits which did not cross grain boundaries and were almost perpendicular to the sliding direction. The extent of grooving increased with load and was dominant in the Mg-PSZ disc (which experiences the cyclic sliding load) worn against stainless steel, but comparatively rare in the Mg-PSZ pins (which experiences more of a static loading condition) worn against a bearing steel disc. Rainforth et al. (1989) further showed that the groove direction coincided with the orientation of the tetragonal precipitates and the grooving

468

Wear in ZTC

F i g u r e 15.1: Optical Micrographs (Normanski contrast) of wear track and worn slider after 380min at 4.9N and an average speed of 6mm/s in acetic acid buffer solution (pH 4). From Hannink et al. (1984) areas coincided with an increase in the monoclinic content near the surface. A similar grooving effect has been reported for Mg-PSZ by Hannink et al. (1985). Scott (1985) has observed a pronounced grain relief on the surface of worn Mg-PSZ. This may be caused by a higher wear rate in certain grains or an uplift in the remaining grains. The grain relief is shown in Fig. 15.1. The texturing (surface uplift) of the worn surfaces is believed to be dependent on the crystallographic orientation of each grain. An initially textured tetragonal parent phase may lead to a preferred orientation of the transformed monoclinic product phase (Bowman &Chen, 1993). Other reasons for the relief may be non-uniform distribution of transformable particles throughout the grains or between individual grains. Tribo-chemical effects on the surface could also lower the surface energy leading to enhanced crack propagation at selected areas (Scott, 1985). 15.1.2

Tetragonal

Zirconia

Polycrystal,

TZP

Wear testing of TZP has been performed by Breznak et al. (1985), Birkby et al. (1988; 1989), Rainforth et al. (1989), van den Berg et al. (1993), Fischer et al. (1988), Braza et al. (1992), and Liang et al. (1993). Very little information on the measurement of coefficient of friction exists for this material. Based on reciprocating unlubricated sliding tests at room temperature, Breznak et al. (1985) have reported coefficients of friction ranging from about 0.2 to about 0.4 for TZP sliding against

15.1. Experimental Observations of Wear in Zirconia Ceramics

469

itself. For similar condition but in a pin on disc sliding test configuration Fischer et al. (1988) found the coefficient of friction to be 0.5-0.6. From another reciprocating test with a TZP pin sliding against various sialons, but otherwise the same condition as above, van den Berg et al. (1993) obtained coefficients of friction varying over a wide range (~0.3-0.9). The values for the coefficient of friction for TZP indicated above seem to be comparable with the values for PSZ. It is suggested that the wear rate of TZP is closely related to the toughness of the material, i.e. the amount of transformable tetragonal phase. Fischer et al. (1988) compared the wear resistance of fully tetragonal zirconia, mixed tetragonal and cubic zirconia, and the fully cubic zirconia. The wear resistance from a pin on disc test increases by a factor of 1200 from the brittle pure cubic to the toughest pure tetragonal material. This increase is proportional to the fourth power of fracture toughness. The wear tracks in the tetragonal material were characterized by an absence of fracture compared to the cubic material. This is believed to be related to the transformation of tetragonal to monoclinic phase. However, Rainforth et al. (1989) reported from comparable tests a high degree of intergranular fracture leading to grain pop-out. They suggested the grain loss was initiated by the transformation and, in absence of a matrix constraint, the dilatation and microcracking resulted in the grain pop-out. Thus, the phase transformation may enhance the fracture toughness and wear resistance, but may also be detrimental to the wear of surface layer, as in the PSZ materials. 15.1.3

Zirconia

Toughened

Alumina,

ZTA

ZTA has been tested for possible application in the automobile industry as a replacement for the conventional materials used in roller bearings. Tests have been carried out by Zhu & Cheng (1991), Braza (1994), and Braza et al. (1992). Abrasive wear testing of ZTA has been conducted by Naghash et al. (1994). Some rolling friction data have been obtained by Zhu L: Cheng (1991), Braza (1994), and Braza et al. (1992) for lubricated rolling of ZTA against cast iron. The coefficient of friction in rolling (~ 0.5) is low compared to the results previously presented for the sliding of PSZ and TZP. The wear of ZTA is reported to be primarily by grain pull-out due to a weak glassy grain boundary phase. For the lubricated rolling test this is not necessarily deleterious as the cavities left by the pulled out grains

470

Wear in Z T C

act as lubricant reservoirs (Braza, 1994). In the lubricated rolling wear tests, the performance of ZTA was comparable with the more expensive transformation toughened zirconia (TTC) and silicon nitride Si3N4 (Braza et al., 1992).

15.2

Subsurface and Surface Cracks under Contact Loading in Transformation Toughened Ceramics

The rolling and sliding wear process has attracted the interest of many researchers over the past few decades. A general review of wear is given by Macmillan (1989) and a more specific overview of delamination wear arising from rolling and sliding contact is given by Suh (1977). The bulk of that research was focused on the experimental aspects of wear, as discussed in w Only a few attempts have been made to develop analytical models for predicting the wear process. Fleming & Suh (1977), and Rosenfield (1980) have presented a fracture mechanical approach to delamination wear. They analysed a subsurface crack parallel to the surface of a half-plane subjected to a Hertzian surface load. The stress intensity factors at the crack tips were calculated from an approximate stress field induced by the contact load in a crack-free half-plane. Keer et al. (1982) presented an improved analysis for delamination and surface cracks. They calculated the stress field in the half-plane under surface Hertzian contact stresses using dislocation theory to model the cracks. More recently, several attempts have been made to explain the wear mechanisms in brittle materials (see, e.g. Braza et al., 1989; Keer & Worden, 1990; Bower & Fleck, 1994) and to predict the fatigue crack growth due to contact loading (see, e.g. Keer & Bryant, 1983; Miller et al., 1985). Extensive work has also been done by Dundurs, Comninou and others on the interface conditions, i.e. slip, stick and separation, especially of subsurface cracks (see, e.g. Chang et al., 1984; Comninou et al., 1983; Schmueser et al., 1981; and Sheppard et al., 1987). In this Chapter the wearing surface in rolling and sliding wear is approximated by a half-plane subjected to frictional contact loading. Under such loading inherent flaws, such as microcracks, voids and inhomogeneities eventually develop into macrocracks with a resultant deterioration in wear performance. Two macrocrack configurations are studied. A straight subsurface crack parallel to the surface and a straight edge crack perpendicular to the surface. The cracks are modelled us-

15.2. Cracks under Contact Loading

471

ing the dislocation formalism, and the contact load is assumed to be Hertzian. In the vicinity of the crack tips, where a high tensile stress field is anticipated, the material is allowed to undergo an irreversible phase transformation in accordance with an appropriate transformation criterion. The transformation zone around the crack tips is modelled as a collection of minute circular regions as in Chapter 10. Such a model is a good representation of the transformable precipitates, for example in partially stabilized zirconia. The elastic field from the transformed circular regions is obtained by applying the Eshelby formalism. The phase transformation exerts a strong influence on the behaviour of the cracked material compared to the purely elastic material through a significant closing effect on the crack faces. This results in the mode I stress intensity factor being almost zero. However, as the transformation zone is highly asymmetric relative to the crack faces, because of the nonuniform stress field due to the applied contact load, the mode II loading of the crack tips can be severely enhanced. This results in high residual near tip deformations after unloading, i.e. a high residual mode II stress intensity factor. The exposition follows that of Thomsen (1994). The plane strain model considered below is a good first approximation to many rolling/sliding problems, e.g. the roller-race problem. The half-plane contains a subsurface or a surface crack. A contact load on the free surface gives rise to normal and shear tractions on the surface, and a stress field in the half-plane. The presence of a crack creates a disturbance to this stress field which has to be calculated. If moreover phase transformation of material takes place, additional stresses are generated and these must be included in the analysis. The applied load stress field is determined by integration of the Flamant solutions for concentrated vertical and horizontal forces on the surface of a half-plane; the stress field from the presence of a crack is found by using the dislocation formalism. Finally, the stresses due to phase transformation are calculated by applying the Eshelby formalism for inclusions undergoing inelastic transformation strains. For the latter two analyses extensive use is made of Muskhelishvili's theory of plane elasticity, see {}4.1.4. Summing the various stress fields and observing the correct conditions on the crack faces gives a set of singular integral equations which have to be solved for the unknown dislocation distribution functions representing the sliding and opening mode deformations of the crack, see Chapter 6. Once these are known, a complete solution can be constructed for the stress and displacement fields throughout the half-plane.

Wear in Z T C

472

15.3

Mathematical Formulation

If a straight crack is positioned transversely to a uniaxial tensile or compressive stress field, it will always be either completely open or completely closed and there will be no sliding of the crack faces relative to each other. However, when the stress field from the applied load is highly non-uniform and asymmetric, as due to a contact load, a subsurface or surface crack in the half-plane may either be open or closed, or parts of it may be open depending on its position relative to the contact load. The crack faces may slide relative to each other or stick to each other due to friction between the faces if the compressive stresses are sufficiently high. Four basic configurations are therefore possible: separation, forward slip, backward slip, and stick, as shown in Fig. 15.2. For each of the four configurations, a set of equations and inequalities must be obeyed by the stress and crack face displacement fields (see, e.g. Chang et al., 1984, and Sheppard et al., 1987). For a subsurface crack, parallel to the free surface, as illustrated in Fig. 15.2, these are S e p a r a t i o n ( o p e n crack):

N C(x c) = O;

S C(x c) = O;

v c (x c) > 0

(15.1)

F o r w a r d Slip:

S c (x c) - - p c N C (xC); du c

( dt

~xcJ> 0 ;

N C (x C) < O; vc(x c)-O

(15.2)

B a c k w a r d Slip: Sc

(x c) - Pc N C (xC); duC(x C) dt

<0;

NC(xc)

< 0;

vC(xC)=o

(15.3)

Stick: IS C (xC)l < - # c N C(xc);

N c(xc)

< O;

15.3. M a t h e m a t i c a l Formulation

473

a)

b) ~

U

c)

d) I

II

/

q 4 p s ~ 4 ~ p ~ / / / / / / / / /

/

i ~ / /

F i g u r e 15.2: The four basic crack face conditions. (a) Separation (open crack), (b) Forward slip, (c) Backward slip, (d) Stick

duC(xc) dt

= O;

v c (x c ) - 0

(15.4)

Here, N C and S C are the normal and shear tractions on the crack faces, x C indicates a position on the crack face,/~c is the coefficient of friction between the crack faces, u C and v C are the relative crack face displacements in the horizontal and vertical directions, respectively and dt is an increment of the position of the applied load. In situations where the crack is partly open and partly closed, or where other combinations of basic configurations exist along the crack, a set of basic conditions from (15.1)-(15.4) is applied to the respective part of the crack. Consider an elastic half-plane containing an straight subsurface crack parallel to the free surface. The surface of the half-plane is loaded with a moving contact load of the Hertzian type, see Fig. 15.3. The contact load gives rise to normal and shear tractions on the surface N(w)

-

d v/d2 - w2;

S(w)-#~N(w)

(15.5)

where p0 is the maximum Hertzian pressure at w - 0, w is a variable describing the position within the contact zone - d < w < d, and p, is the coefficient of friction between the roller and half-plane. The length of the crack is c. For the straight subsurface crack (Fig. 15.3) the tractions N C ( z C) and S c (x c ) are

Wear in ZTC

474

N(w)

(E,v)

J

X

h

F- 1

Sl l

C

S2

c

rl

F i g u r e 15.3: Half-plane containing a straight subsurface crack parallel to the free surface. The half-plane is subjected to a moving Hertzian contact load. Possible high stress fields at crack tips are indicated by the zones S/ (i = 1, 2)

NC(x C) - a~y(xC)ny;

s c

_

(15.6)

where ny is the outward normal to the crack face, and for a straight surface crack perpendicular to the surface the tractions on the crack faces are

NC(x C) - crxx(yC)n~;

SC(y C) - (rxy(xC)nx

(15.7)

Each of the stress components is a sum of stress fields from different sources as follows D T L cruz - a~g + a,Z + a,~

(15.8)

where superscripts D, T and L refer to stresses from the presence of the crack, the transformation zone and the applied load, respectively. The stress field due to the crack is found by using the dislocation formalism (Chapter 6) and that from the transformation is found by using the Eshelby formalism for an inclusion (Chapter 4), together with Muskhelishvili's theory of plane elasticity. The stress field due the contact load is determined by integration of the Flamant solutions for concentrated vertical and horizontal forces on a free surface.

15.3. M a t f i e m a t i c a I F o r m u l a t i o n

475

After introducing the dislocation density functions for the crack and appropiate weight functions for dislocations in the half-plane and for transformation, together with the integrated Flamant solutions, the stresses in (15.8) become cruz(z) - 27r(1 - v)

~,(z)-

5r~

(z; t) -

27r(1-v)

f

(15.9)

D ' ( z-~ ) ~ ~ ( z , z_o c ) d x Co Lee

(15.10)

LeG

,=,

[~%(z, ~. , t) + ~ , ( z ,

(15.11)

~. , t)] d~

d

LEC

where p = E~ [2(1 + v)], E is the elastic modulus, v Poisson's ratio, Di(z_oc) are the unknown dislocation distribution functions, h ~ are weight functions for dislocations (Keer & Chantaramungkorn, 1975), m is the number of transformed particles, and a~/3,g(z, w, t) and a ~ , s ( z , w, t) are stresses arising from single normal and shear forces, respectively, z, z0c and z0T indicate functional dependence on the real variables x, y, x0c, y0c, x0T and y0T. The weight functions hT~(z, z__0 T) for the transformation are obtained using the well-known Eshelby formalism. Using the results from eqns (4.44) and (4.45)these can be written as

~'~(z,~)

F hxT(z, z_0 T) -- 2 ReO(z_) - Re [(z_-- z) O'(z_) - r

- ~(z_)]

F hyvT (z, z_0 T) - 2 ReO(z)_ + Re [(z_-- z) O'(z) - O(z) - ~(z_)]

r h~(z, d ) - I m [(~_- z) r

- r

- ~(z)]

(15.12)

where F - It/[2 7r (1 - v)]. The stress field at _zdue to the contact loading cr~t ~L (z; t) is obtained by integration over the contact zone of Flamant's solutions for concentrated vertical and horizontal forces on the surface of a half-plane (Timoshenko & Goodier, 1982)

N (z, ~) O'xx

2po

(~ - ~)~ v

j(~

_ ~)

476

Wear in Z T C

~

N (Z, W) - -

2/9 0

y3

~ d ~ / ( ~ _ w)~ + ~ v/(d~ - ~ 1

N (z, ~) - 2 po (. - w) ~ ~ d v / ( , _ ~)~ + y~ v/(d~ - ~ ) ~*~ s (z, ~ ) -

6r x x

6ry y (z, ~ ) -

~

2 ~ po (~ - ~)~ x/(d~ - w~) ~ , d v / ( ~ - w)2 + y2 2 #8 Po

;id

(x - w) y2

v / ( ~ - ~1~ + ~ v/(d~ - w

s (z_, w) - 2 p8 po (x - w) 2 y v/(d 2 - w 2) ~ d v / ( ~ _ w)~ + V~

2)

(15 13)

Here, z _ - x + i y ( i - x/~T). The stress intensity factors Ki (i - 1, 2) at the tip of an open crack are calculated from I(.i -- 2(1 g-- v) ~

lim ~ / l ~ - -el Di(z_~)), ~--.~

(15.14)

c when c - xtic p + i Ytip is the location of the crack tip. By introducing the stresses given in (15.8)-(15.11)into the expressions for the tractions on the crack faces (15.6)-(15.7) for the two crack configurations and observing the crack face conditions set out in (15.1)(15.4) over respective parts of the crack a set of singular integral equations is obtained. The two singular integral equations have to be solved numerically. A transformable circular region is assumed to transform if the critical mean stress criterion is satisfied (w

l+v 3 6rcm

(6r~ + 6ruy) >_ 1

(15.15)

Alternatively, the stress invariant criterion is used (w 6rm

6re

c~6r--T + (1 - ~ 1 - - > 1 rn

In general three-dimensional analysis

6rec --

(15.16)

15.3.

Mathematical Formulation

477

1

O.m -- -~ O'i i

a e -- r

sij/2,

Sij -- (rij -- O'mSij

(15.17)

c and (rec are characteristic stress levels for the transformation process. ~rm lies in the interval [0; 1] and for ~ - 1 the stress invariant criterion reduces to the mean stress criterion. A random distribution of transformable circular particles is assumed to be present throughout the material. A particle is assumed to transform according to the mean stress criterion (15.15), if the following holds

#

2 ~'(1 - u)

/c Di ( z_oC) h ,~D~i( z_,z_oC) dx C I

zeSj

m

2~(1-~,)

~._ hT (z,ff) ~ LE $i

+

;

[ , ~ ( z , ~ ; t) + a ~~ (z, ~; t)] d~ d

I

>- 13,~,~ + ~,

(15.18)

zE Sj

where Sj indicate the individual circular regions of transformable particles. Whenever the mean stress within the region of a particle exceeds the critical value for transformation, it is included in the collection of transformed particles and the effect from its stress field is taken into account in subsequent calculations. For the subsurface crack there is a stress singularity at both crack tips if they are open and transformation can take place at either one. This has to be taken into account when the distribution of transformable particles is modelled. The Cauchy-type singular integral equations for the stress field (15.8) are numerically solved for the unknown dislocation distribution functions. The condition (15.18) is imposed for determining the location and number of transformed particles. The set of singular integral equations is solved by appropriate Gaussian integration formulae. The complete numerical procedure for solving the problem of Fig. 15.3 with a subsurface crack and a moving contact load is outlined in the following. The first step is the solution of the singular integral equations for

478

Wear in ZTC

an assumed set of crack face boundary conditions, say (15.1), with the load at the initial position (t = 0). The second step is to check the high stress areas around the crack tips for possible phase transformation at any of the randomly distributed transformable particles using (15.18). It is doubtful whether the quasi-static approach to the transformation process is appropriate, i.e. the load is kept at the same position while the transformation zone is allowed to stabilize. However, it seems to be the most suitable initial approach. If any particles have met the transformation criterion the integral equations are solved again with the effect of transformation taken into account. In each successive iteration, only one newly transformed particle is included in the analysis to ensure that all interaction effects are accounted for. This procedure is repeated until no more particles transform. Thereafter the crack surface conditions are checked and if these are violated, a new set of crack face boundary conditions and the transition points are determined and the calculations repeated. When none of the crack face conditions are violated in successive iterations, the values of K/, KII, COD and CSD etc. are calculated. This procedure is repeated at various positions of the contact load (t is incremented) until the final load position is reached, i.e. one load pass is completed. The singular integral equations for the surface crack are solved, using the same numerical procedure as for the subsurface crack, with an appropriate modification to the Gaussian quadrature.

15.4

Subsurface Crack under Contact Loading

A selection of results for subsurface crack in the half-plane subjected to a moving contact load is presented in this Section. The following elastic properties for a typical PSZ material are used in the calculations: E - 205000 MPa and ~, = 0.3. All linear dimensions are normalized with respect to the basic crack length c. The actual crack size in a material like Mg-PSZ is likely to be in the order of the grain size, which is 3060 #m. The half-width of contact d is set to the same order of magnitude as the grain size. However, in actual applications the contact distance depends strongly on the material properties of both the roller and the race, as well as the force that is applied to the roller. If the maximum pressure P0 is set to, say, 100 MPa, it will correspond to a total force of about 8 N / m m for d = 50 pm. However, the magnitude of the pressure does not affect the nature of the results presented in the following.

479

15.4. S u b s u r f a c e Crack u n d e r C o n t a c t L o a d i n g

The transformation is accompanied by permanent inelastic dilatational transformation strain, ~ T __ 0.04. For brevity, results for shear transformation strains are not presented. A random distribution of transformable circular inclusions in the vicinity of the crack tip is considered. The radius of each inclusion is chosen to be a / c = 0.002. If this is related to a real PSZ material with a grain size of ~ 50 pm, the radius of the transformable inclusion will be ~ 0.1 #m, which is not unusual for tetragonal precipitates in Mg-PSZ (see, e.g. Hannink 1988). The transformable phase makes up 30% of the composite, i.e. a volume fraction of V! = 0.3, which may be low compared to some peak-aged Mg-PSZ, but it is not unusual for less optimal material systems. The exact volume fraction is not important in the present analysis in which emphasis is placed on the general behaviour of the material system containing transformable particles, rather than on a specific material system. The relative crack face displacements for four load positions t and crmc/po - 1.0 are shown in Fig.. 15.4. The displacements have been normalized by their respective maxima for clarity of presentation. It is however important to note that CSD is much larger than COD and that the crack faces are not displaced by an equal amount, as would appear from the figure. At load position t l, no transformation has taken place,

I

I

I

I

1

_

u

t3 t4

COD/CSD

A

B

I

I

I

I

I

F i g u r e 15.4" Crack face displacements C O D / C S D for four contact load positions, tl = 2.1, t2 - 4.0, t3 = 5.5, t4 = 8.0. Crack tip A is positioned at t A - x / c - 3.0 and B at tBc -- x / c - - 4.0. h / c - 0.5, #s - 0.5, c / P 0 - - 1 . 0 , ~ T _ 0 . 0 4 , ,,fT _ 0 . 0 d / c _ 0.5, ~,~

480

Wear i n Z T C 0.14 0.12 0.10

• 0

0.08 0.06 0.04 0.02

a)

0.00 0.25

J

1

1

w

I

I

t 3.2

t 3.4

t 3.6

,

tl l~22

0.20 0.15 0.10 • r~

0.05 0.00 -0.05 -0.10 -0.15

b)

-0.20 3.0

i 3.8

4.0

t,x/c

F i g u r e 15.5" Crack face o p e n i n g C O D (a) and sliding C S D (b) disp l a c e m e n t s for four c o n t a c t load positions, tl - 2.1, t2 - 4.0, i~3 - - 5 . 5 , t4 8.0. Crack tip A is positioned at t A - x / c - 3.0 a n d B at t B - x/c4.0. h / c 0.5, p~ - 0.5, d / c 0.5, c r ~ / V o - 1.0, 0 T - 0.04, 7 T -- 0.0

leading to a fairly small o p e n i n g of the crack tip B (Fig. 15.5a). At this load p o s i t i o n the crack is in forward sliding over the entire length. At load position t = 2.55, t r a n s f o r m a t i o n occurs at crack tip A, and at load p o s i t i o n t = 2.9 at crack tip B. T h e t r a n s f o r m a t i o n induces a wedging effect at b o t h crack tips for t2 = 4.0 leading to a small open zone near each crack tip which is itself still closed. At load position t2, the crack is in

15.4. Subsurface Crack under Contact Loading Crack tip A

a)

b)

r

481

Crack tip B O O

%

0

0

c) O0 000

d)

~ o

O 9

9

9 9

O

F i g u r e 15.6- Transformed circular particles for four values of critical mean stress tr~ after a completed load pass. (a) cr~/po = 1.1, (b) ~r~/po = 1.0, (c) ~ / p o = 0.9, (d) cr~/po = 0.8. Crack tip A is positioned at t A - x / c - 3.0 and B at t B - x / c - 4.0. h / c - 0.5, #~ - 0.5, d/c = 0.5, 0 T - - 0.04, ~fT __ 0.0 forward sliding within the zones of transformed particles near the crack tips, but in backward sliding mode over the remaining part of the crack (see Fig. 15.5b). The crack face displacements are quite complicated due to the large sliding deformations induced by the transformation near the crack tips. At load positions t3 and t4, the crack is open over most of its length, with only the tips being kept closed by the transformed particles. The amount of transformed material is considerably larger at crack tip B than at A. This is reflected in the much larger opening due to wedging at crack tip B. At both of these load positions the entire crack is in forward sliding (see Fig. 15.5b). Figure 15.5 shows the variations of unnormalized COD and CSD. The COD curves resemble the curves in Fig. 15.4a, whereas the CSD

Wear in ZTC

482

0.8

0.7

I

I

I

_

I

KII, A

_

KII, B ........

0.6 0.5

9 ............

~ . . . . . . . . . .. .......... . . . . . .

~

0.4

po~

!

:

i

0.3 0.2

'

:,,

,i '._/

0.1

a)

0.0 0.003

I

1

i

1

1

i

K/,A

,,""\ ,'~

I

~B

i

o,' o

0.002

~ /' ~ ~

po~

,/

/"

0.001

b)

0.000 0.0

tI 1

2.0

12

t3

I

4.0

I

6.0

t4 1

8.0

10.0

t, x/c

F i g u r e 15.7- Stress intensity factors for four contact load positions, tl - 2.1, t 2 - 4.0, t 3 - 5.5, t 4 - 8.0. (a) KII and (b) K I . Crack tip A is positioned at t A - x / c - 3.0 and B at t g - x / c - 4.0. h / c - 0.5, ~ s -- 0 . 5 ,

die-

0 . 5 , o'Cm/P0 -- 1.0,

0T

--

0.04, ~T _ 0.0

curves give a clearer view of the sliding deformation of the crack faces. It is worth noting that at load position t2, CSD is actually negative over most of the crack face except close to the crack tips. This is consistent with the applied stress field. The small j u m p in CSD near crack tip B at load position t2 is due to a small gap in the transformation zone (see Fig. 15.6b). This j u m p is also the reason for the very short, but wide open zone near crack tip B at load position t2 (see COD curve in

15.4.

Subsurface Crack under Contact Loading 0.16 [ 0.14 0.12

r~!~.

0.10

v

i

I

I

483

I

t4

0.08 0.06 0.04 0.02

a)

I

0.00 0.25

I

I

I

tl

t2

i

tl

0.20 0.15 0.10

x

0.05 0.00 -0.05 -0.10

b)

-0.15 3.0

3.2

3.4

3.6

3.8

4.0

t, x/c

F i g u r e 15.8: Crack face opening C O D (a) and sliding displacements CSD (b) for four contact load positions, tl - 2.1, t2 - 4.0, t3 - 5.5, t 4 - 8.0. Crack tip A is positioned at t ~ - x / c - 3.0 and B a t t B = x / c - 4.0. h / c - 0.5, #s - 0.5, d / c - 0.5, a ~ / p o - 0.8, 0T - 0.04, 7 T - 0.0

Fig. 15.5). T h e m o d e II stress intensity factors (Fig. 15.7a) at both crack tips are severely affected by the t r a n s f o r m a t i o n which is reflected in a high offset of the KII values, when the contact load has moved well past the crack. At crack tip A, t r a n s f o r m a t i o n occurs only on the approach of the contact load (t = 2.55), whereas at crack tip B m a t e r i a l t r a n s f o r m s

484

Wear in Z T C

0.8 0.7 I II

0.6 0.5

po4i

~~ !

,,

0.4

I

Ktt.A

~8 / ,

0.3 0.2 O.1

a)

0.0

I

0.004

I

t

-

/ /

-

/ -

0.001

b)

~A ~B

i

--

po,fd 0.002

I

I

I

0.003

I

,,J

i

J

t

t

IIi ,'I ~ i

i

/

/

i

0.000 0.0

F i g u r e 15.9" tl - 2.1, t 2 A is positioned #s - 0.5, d / c -

I

~

i

i

2.0

4.0

6.0

8.0

Stress intensity factors 4.0, t 3 - 5.5, t 4 - 8.0. at t A - x / c - 3.0 and 0.5, cr~/po - 0.8, 0 T - -

10.0

t, x/c

for four contact load positions, (a) K I I and (b) K~. Crack tip B at t B - x / c - 4.0. h / c - 0.5, 0.04, ,),T _ 0 . 0

on the approach (t ,~ 2.9), as well as after the load has passed the crack and moved away from the tip (t ~ 4.7). After transformation has taken place both crack tips remain closed (KI,A -- K I , B = 0; Fig. 15.7b). Figures 15.8-15.9 show the crack face displacements ( C O D / C S D ) and the stress intensity factors for a lower value of the critical mean stress (cr~/po = 0.8). It is obvious that the lower critical mean stress for transformation leads to more transformed material and therefore to a

15.5. Surface Crack under Contact Loading

485

higher degree of local deformation of the area near the crack tips. Also, the transformation occurs at an earlier load position (for crack tip A: t = 2.35, and for crack tip B: t = 2.7). K I I , on the other hand does not show any significant difference from the higher value of o'~/po. This leads to the conclusion that the extended transformation zone does not seem to alter significantly the local deformations of the crack tips. The COD and CSD curves are very similar to the curves for the higher transformation stress, although larger absolute values of the crack face displacements are observed. Small kinks in the CSD curve at load position t2 are seen at both crack tips. These may again be attributed to gaps in the transformation zones near the crack faces (see Fig. 15.6d), which also cause the short open zone at tip B. This effect is not noticeable on the considerably longer open zone at crack tip A. The zones of transformed particles corresponding to the above analyses are shown in Fig . 15.6. In addition to the zones for crm ~ /Po - 1.0 and o'~/po - 0.8, the zones for O'm/PO - 1.1 and tr~/po - 0.9 are also presented. A decrease in the critical mean stress leads to additional particles transforming, giving larger zones which extend farther away from the crack tips. The location and shape of the zones are similar to the constant mean stress contours in the analysis of the linear elastic material. However, for crack tip B the transformation zone shows signs of extending to both sides of the crack. This effect arises as a combination of the stress field from the initial transformation zone below the crackline and the stress field from the passing contact load.

15.5

Surface Crack under Contact Loading

Figure 15.10 shows the half-plane containing a straight surface crack. The surface of the half-plane is subjected to a moving contact load. For c /po - 1.0 the relative crack face displacements a critical mean stress cr,,~ and the stress intensity factors are shown in Fig. 15.11. The area near the crack tip is now severely deformed due to the transformed material. At load position tl = 3.5, i.e. when the contact load is behind the crack and the crack faces are subjected to compressive stresses, there is a clear wedging effect close to the tip which opens a small, but significant zone near the crack tip. Due to a very large mode II deformation near the crack tip and the difference in scales between the two types of deformation, the graphical representation of the displacements does not give a correct picture of the real situation. It is also again i m p o r t a n t to note that the two faces are not displaced by an equal amount, as would appear

Wear in ZTC

486

N(x)

2d iT

J

(E,v)

X

C

"3

F i g u r e 15.10" Half-plane containing a straight surface crack perpendicular to the free surface. The half-plane is subjected to a moving Hertzian contact load. Possible high stress field at the crack tip is indicated by the zone S from Fig. 15.11a. As the load moves past the crack mouth (t = 4.0) and causes more particles to transform, the crack opening due to the transformation strains become more significant. At load positions t3 and t4 the deformation of the area near the tip becomes very complicated since transformation now occurs on both sides of the tip, but not symmetrically. It appears that at load positions t 3 and t4 a folding of at least one of the crack faces is taking place. It is quite possible, for instance for the right face of the crack to fold, because the transformation makes the material close to it expand along this face, whereas the untransformed material a little farther away from it is preventing this expansion. The compressive stress in the vertical direction due to the contact load aids this process. The stress intensity factors of Fig. 15.11b clearly show the onset of transformation at t ~ 3.2 when the mode II stress intensity factor reaches a critical level for transformation. The figure also shows that compressive forces from transformation keep the crack tip closed (KI = 0.0) during the entire load pass. It is interesting to note that for lower critical mean stress values (rr~C, transformation also occurs when KII is negative, for instance when the load is anywhere in the range t = 4 . 3 - 6.0. This is caused by a combination of the lower critical mean stress and the permanent deformation of the crack tip area from the transformation when the load was in the range t = 3 . 2 - 3.8.

487

15.5. Surface Crack under Contact Loading I

I

I

I

I

I

I

I

I

1

Free surface

COD/CSD

Crack tip

a)

I

0.4

1

I

1

1

I

i

0.3 0.2 0.1

Ki

po~

b)

0.0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-0.1 -0.2 -0.3

-

-0.4

-

-0.5

-

-0.6 0.0

l

I

I

I

2.0

4.0

6.0

8.0

10.0

t, x/c

F i g u r e 15.11" Crack face displacements C O D / C S D and stress intensity factors for four contact load positions, t z - 3.5, t2 - 4.5, ta - 5.5, t4 - 7.5. (a) C O D / C S D , (b) KI and Is The crack is located at tc - x / c - 4.0. /L, - 0.5, d / c - 0.5, ~r~/po - 1.0, 0 T _ 0 . 0 4 , '7' T - - 0 . 0

As the contact load moves well away from the crack (t > 10), the tip is left with a large permanent deformation from the transformation strains. This is reflected in a residual mode II stress intensity factor of more than half of its m a x i m u m value during the entire load pass. Moreover the positive m a x i m u m value of K I I is significantly reduced in the presence of transformation (to about 65%). However, the m a x i m u m negative K I I is approximately the same as in the absence of transformation.

488

Wear in Z T C 0.75

I

I

I

1

-

t3

/

0.50 x

t2

-

o

0.25

a)

0.00

.

I'

I

1

1.5

,

I

II

_

t3

_

1.2

~1~.

0.8

•

t..)

0.4 0.0

b)

-

-0.4 ~

-1.0

tz

~

l

-0.8

-0.6

l

-0.4

-

L

-0.2

0.0 y/c

F i g u r e 15.12: Crack face opening COD (a) and sliding displacements CSD (b) for four contact load positions, tl = 3.5, t2 = 4.5, t3 = 5.5, t4 = 7.5. The crack located at is t c = x / c = 4.0. #s = 0.5, d / c = 0.5, O'Cm/P0 - - 1 . 0 , /9T - - 0 . 0 4 ,

,,fT _ 0 . 0

Figure 15.12 presents the actual variation of the two crack face displacements, COD and CSD. At load positions t 3 and t4, there is a sharp decrease in COD near the crack tip. This can be explained by looking at the corresponding transformation zone (Fig. 15.13c). The zone is unevenly distributed around the crack tip. The crack is closed from the tip past the dense collection of transformed particles to the right of the crack. Beyond this point it opens up very rapidly owing to the wedging

15.5. Surface Crack under Contact Loading

a) Edge crack

b)

489

c)

-.L.

Crack tip

F i g u r e 15.13: Zones of transformed particles for three critical mean stresses a,~ after a completed load pass. (a) a ~ / p 0 = 1.5, (b) cry/p0 = 1.25, (c) tr~/po = 1.0. tc = z/c = 4.0, p, = 0.5, 0T = 0.04, .)IT _. 0 . 0 effect from this dense collection of transformed particles. At load positions t l and t2, only the bottom collection of particles has transformed, thus closing the crack tip but wedging open the crack faces away from it. At load positions t3 and t4 the upper collections of transformed particles have developed, giving rise to a closing effect on the crack farther away from the tip causing the sharp decrease in COD. The effect from the complex transformation zone is also felt on the CSD (Fig. 15.12b). At load positions tl and t2 only the dense collection of particles has transformed. This causes a negative CSD close to the crack tip (y/c = - 1 ) , while above it CSD is positive. The maximum CSD is seen at the upper boundary of this dense zone for all load positions. CSD at load positions t3 and t4 shows several local extrema near the crack tip owing to the the complex shape of the transformation zone. Figure 15.14 shows the stress intensity factors for four values of the c governing the transformation From the K I I critical mean stress ~m c decreases transformation takes place at lower curves it is seen that as a m values of KII and t. In addition, the maximum value of KII is reduced due to increased number of transformed particles at the crack tip. However, the absolute value of the negative minimum of KII increases, so that the permanent deformation at the crack tip due to transformation strains increases significantly with a decrease in a~. This permanent deformation is reflected in an offset in Kli, when the load is well removed from the crack (t > 10).

Wear in Z T C

490 0.6 I

T

0.4 0.2

po~

~

I

~

1.25

::ii:.

-

1.5 1.75

.

0.0

1;,-

i, 1',: I;-

-0.2

I

Z,,:,;:,,,:: .yi.

--

........

............. -

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

"". ................. . .*" .. ." .." ,' -"

I; , i I ,; i I

-0.4 a)

-0.6

1

I

I

I

0.10

~3~Po=l.O 1.25

0.08

1.5 ........ 1.75

KI

.

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

1

1

I

I

.......................... .... "..... .: 9

. ' . . .9. . . . . . . . .

_

9

..........-

-:

"

0.06

0.04

_

-

0.02 -

b)

0.00 0.0

I

,

i

i

2.0

4.0

6.0

8.0

10.0

t, x/c

F i g u r e 1 5 . 1 4 : Stress intensity factors for four values of critical m e a n stress a ~ as a function of contact load position t. (a) KII, (b) KI. T h e crack is located at tc - x / c - 4.0. ps - 0.5, 6T - 0.04, ~ T _ 0.0 T h e very high negative value of KII at low values of critical transf o r m a t i o n m e a n stress m a y at first sight seem to be d e t r i m e n t a l to the material. However, in a real wear situation where the load passes the crack repeatedly, this is not quite so since at low negative values of K H t r a n s f o r m a t i o n is expected to take place to the left of the crackline (for K I I > 0 t r a n s f o r m a t i o n occurs to the right) which therefore contributes to a b u i l d - u p of t r a n s f o r m e d m a t e r i a l on both sides of the crack (see Fig. 15.13). Similarly to the earlier results for the pure dilatational

15.5. Surface Crack under Contact Loading

491

transformation, the effect on the mode I stress intensity is very significant (Fig. 15.14b). KI is zero throughout the load pass, i.e. the tip of the crack never opens due to the closure forces from transformation. Figure 15.13 shows the transformation zones corresponding to three critical transformation mean stresses. For the highest critical mean stress (Crm/P0 = 1.5) only one particle transforms in the high mean stress field to the right of the crack tip when the load is at t = 3.45. Decreasing ~r~n/p0 to 1.25 leads to additional particles transforming giving a larger zone to the right of, and primarily behind, the crack tip when the loading moves from t = 3.3 to t = 3.55 (Fig. 15.13b). The transformation zone expands when cry/p0 is further lowered to 1.0 (Fig. 15.13c). The dense collection of particles nearest to the crack tip develops on the approach of the load from t = 3.15 to t = 3.8. However, due to large deformations arising from this collection of transformed particles and a change in the applied mode II loading, the material to the left, and above the closed part of the crack, is now free to transform. The particles to the left of the crack transform when the load moves between t = 4 . 2 - 5.3. The top particles of the zone to the right of the crackline, as well as the two uppermost particles to its left transform at t = 5 . 3 - 6.0. The deformation of the crack due to this alternating pattern of transformed particles may lead to interlocking of the crack faces and thereby prevent their sliding past each other. Crack faces may also be prevented from sliding by the very high compressive stresses due to the transformation across the crack faces in the closed part of the crack in the presence on friction between the crack them. Together, the interlocking and frictional resistance to sliding will reduce the mode II deformation in the crack tip area. The influence of the combination of dilatational and shear transformation strains on the stress intensity factors may be judged from Fig. 15.15. The analysis considers only a modest amount of transformation shear strain (7 T = D/4 = 0.0087), corresponding to just about 10% of the shear strain of an unconstrained transforming tetragonal zirconia particle. This relatively low value has been adopted due to the high degree of twinning and other shear accommodating mechanisms that operate when the transforming particles are embedded in a matrix. The shear direction c~ is measured anti-clockwise from the horizontal position. From the curves in Fig. 15.15a it is evident that even a small amount of transformation shear strain has a significant effect on stress intensity factor KII. It is also clear that the shear direction plays an important role. For both c~ = 0 ~ and c~ = 1350 the effect on KII is quite significant

Wear in Z T C

492

0.8

l

:. r 0~=0 ~

45 ~ 90 ~ ........ 135 ~ ............. _

0.6 0.4

po.~c

0.2 0.0 -0.2

a)

~..t

-0.4 0.10

I

ct=O ~

45 ~ 90 ~ ........ 135 ~ ............. -

0.08

r~ po~

0.06

0.04

-

-..<

0.02 w

b)

0.00

0.0

I

2.0

i~

4.0

6.0

8.0

10.0

t, x/c

F i g u r e 1 5 . 1 5 " Stress i n t e n s i t y factors for four shear angles a as a funct i o n of c o n t a c t load p o s i t i o n t. (a) KII, (b) KI. T h e crack is l o c a t e d at t C -- X/C4 . 0 . ]1 s - - 0 . 5 , t Y m / P o - - 1 . 5 , 0 T - - 0 . 0 4 , . f T _ D / 4 - 0.0087

w h e r e a s for a = 450 a n d a = 900 it is less so. T h e differences are clearly e v i d e n t in the p e r m a n e n t d e f o r m a t i o n w h e n the e x t e r n a l c o n t a c t load is well r e m o v e d (t ~ 10). T h e b e h a v i o u r of the m o d e I stress intensity factor (Fig. 15.15b) u n d e r t h e c o m b i n a t i o n of d i l a t a t i o n a l a n d shear s t r a i n s is different f r o m its b e h a v i o u r u n d e r p u r e d i l a t a t i o n (Fig. 15.14b). K I is non-zero for two different intervals d u r i n g the load pass w h e n a = 0 ~ a n d d u r i n g a very s h o r t interval w h e n a = 135 ~ T h e crack tip opens ( K / > 0) because

15.5.

493

Surface Crack under Contact Loading

transformation takes place in front of the crack tip. For c~ = 135 ~ the crack tip is open only for a very short load interval as additional transformation occurs behind the crack tip, whereas for c~ = 0 ~ the effect is more permanent, leaving the crack tip open even when the contact load has moved well away from the crack. It is not clear though, whether the crack tip will remain permanently open when the contact load is completely removed. The transformation zones for four shear angles are shown in Fig. 15.16. The location and number of transformed particles in the zones are consistent with the behaviour of the stress intensity factors of Fig. 15.15. For c~ - 0 ~ and c~ = 135 ~ the zones consist of a larger number of particles than for c~ = 45 o and a = 900 which explains the differences in K I I , mentioned above. The presence of transformed particles in front of the crack tip for c~ = 0 ~ and c~ = 1350 explains why the crack tip opens ( K I > 0) for parts of the load pass. The shape of the zone is closely related to the shear direction c~. For instance, for c~ = 0 ~ the shear direction is horizontal, so that the shear strains produce an elongation of the transformed particles in the horizontal direction and a contraction in the vertical direction. The contraction in the vertical direction is particularly i m p o r t a n t here, since it reduces the compressive stresses on the neighbouring particles above and below the transforming particle. This reduction in combination with the high positive mean stress from the contact load results in an

Edge crack

~ g

Crack tip a)

b)

c)

d)

F i g u r e 15.16: Zones of transformed particles for four shear angles a after a completed load pass. (a) a - 0 ~ (b) a - 450 , (c) c~ - 900 , (d) a - 1350 . t c - z / c 4.0, ps - 0.5, a ~ / p o - 1.5, 0T - 0.04, ~ T __ D / 4 = 0.0087

494

Wear in ZTC

auto-catalytic effect perpendicular to the shear direction whereby the transformation zone extends vertically along the right side of the crack for c~ - 0 ~. For c~ - 450 and c~ = 90 ~ the auto-catalytic process is not favoured since the direction in which the compressive stress is reduced by the shear strains is perpendicular to the maximum principal stress arising from the contact load. For c~ - 135 ~ these two directions are again approximately coincident, and the auto-catalytic effect is again experienced perpendicular to the shear direction. From the above results it is clear that the magnitude and direction of shear transformation strains have a significant effect on the development of the size and shape of the transformation zone. However, it is difficult to assess the effect from shear strains in a real material because of the existence of a range of shear directions and the uncertainty associated with the magnitude of macroscopic shear strain. Finally, the variation of the stress intensity factors for four values of critical invariant stress (a~ + cr~)/2p0 (~ = 0.5) are presented in Fig. 15.17. Only dilatational strains are included. There is no transformation when (cr~ + cr~)/2p0 = 2.3 and it therefore corresponds to r C the elastic solution. From the KII curves it is seen that as (~rm + ~re) decreases transformation takes place at lower values of KII and t. The transformation leads to an increase in KII which is contrary to what was seen from the results for the critical mean stress criterion (see Fig. 15.14). This is due to the fact that the major part of the transformed particles are positioned in front of the crack tip (see Fig. 15.18) and not behind it, as for the mean stress criterion (see Fig. 15.13). When the leading edge of the contact load is about to pass the crack mouth (t = 3.5), 1s begins to decrease and reaches a local minimum when the trailing edge has almost passed the crack. However, when the load has fully passed (t = 4.5), K i t again increases. At this stage a large number of particles has already transformed (Fig. 15.18). For load positions beyond t ~ 4.5, the behaviour of [(II becomes very erratic. The calculations were stopped prematurely due to a lack of convergence for a large number of load positions. This is believed to be due to the development of very large transformation zones, primarily by autocatalysis. The KI curves in Fig. 15.17b also reflect the development of the transformation zone in front of the crack tip leading to its opening and therefore to non-zero IQ values. However, this only appears when the load has passed the crack (t > 5). The transformation zones corresponding to three critical stress invari-

[

15.5.

Surface Crack under Contact Loading 1.2 1.0

I

-

I

K//

poxl-c

0.5

1

=2~

2.1

........ 2.2 0.8 -. ............2.3

495

A "l: ~

~

,

'~

_

:...../'

0.4 0.2 0.0

a)

Crack

-0.2

i

0.25

~ 0.20 0.15

-

r

I

C

i'

:-.

I

!

(am+~e)/2Po=2.0

..... 2.1 ........ 2.2 ............2.3

!. i".., i i "",...,i

i i

0.10 0.05

. .... , ................

..

...........

9 "i

b)

0.00 0.0

Figure 15.17: invariant (irOn + t. (a) K I I , (b) 0 T = 0.04, AfT :

I

2.0

j

4.0

i

J 6.0

j

8.0

10.0

t, x/c

Stress intensity factors for four values of critical stress try)/2 p0 ( ~ = 0.5) as a function of contact load position K I . T h e crack is located at t c = x / c = 4.0. #s = 0.5, 0.0

ant values for t r a n s f o r m a t i o n , are shown in Fig. 15.18. Note the zones correspond to load position t - 4.5 and not after the load pass has been completed. Each of the zones includes a large n u m b e r of t r a n s f o r m e d particles c o m p a r e d to the zones presented earlier for the m e a n stress criterion (Fig. 15.13). T h e zone shapes corresponding to the two transf o r m a t i o n criteria are distinctly different. T h e t r a n s f o r m a t i o n zones for the stress invariant criterion extend over a larger area behind, as well as

496

W e a r in Z T C

,~~

Edge crack tip

e9 ,~p,m. 9 ee ,~~176 o~ 9 9 -IoS.oqP8

-s "; .~too..oo.~~d,..~,~. "~-. a)

-..ok..

.

b)

).re c)

..~

9. . ...~ " " " ~"' ~ ! I , ' :..-~ ......'.~,~'.~,~I,~. "~..

F i g u r e 15.18: Zones of transformed particles for three critical stress invariants (cr~ + cr~)/2 (~ = 0.5) at t = 4.5. (a) (cr~ + ~ ) / 2 p o = 2.0, (b) (cr~ + cr~)/2p0 = 2.1, (c) (cr~ + cr~)/2p0 = 2.2. t c = z / c = 4.0, Us = 0.5, oT ._ 0.04, .)IT = 0.0

"

15.6. Concluding Remarks

497

in front of the crack tip. The transformation zone for the mean stress criterion almost exclusively extends behind and close to the crack tip. This especially applies to the pure dilatational strains. The development of transformation zone for the mean stress criterion is predominantly governed by the contact load stress field. In contrast, the zone shape for the stress invariant criterion is predominantly governed by an auto-catalytic process (i.e. an interaction effect between the individually transformed particles) and to a lesser extent by the applied contact load. This also explains the discrete extension of the zones in selected directions. The tendency of the zones to develop as long "arms" in selected directions has also been observed by Stump (1994) for cracks in the infinite plane under a uniform tensile stress field. However, the applied load does play a role in positioning the zones, in that they predominantly develop from, and around, the highly stressed area to the right of the crack tip, when the contact load is approaching the crack (at load positions t = 3.5-4.0). The decrease in the critical value of (~r~ +a~)/2 P0 for transformation does not significantly affect the number of transformed particles (only about 10% more particles transform at the next lower value of the critical stress invariant). The large number of transformed particles in the transformation zone and its extension far away from the crack tip, makes the computational process very time consuming. As the load position is incremented beyond t = 4.5, the number of transformed particles keeps increasing leading to even larger zones which eventually extend beyond the assumed area for transformation. The zone in Fig. 15.18c is already beginning to touch the assumed lower boundary of the area for transformation. This effect eventually leads to the premature termination of the calculations. It is clear from the results that the inclusion of both stress invariants cr,~ and ae in the transformation criterion gives significantly different results compared to the pure mean stress criterion. The large discrepancy between the two criteria is caused by the values of the characteristic quantities crm, c cre, c and ~. A more reliable transformation criterion may be obtained, if these values were determined accurately from experiments.

15.6

Concluding Remarks

The results obtained in the elastic analysis of the subsurface and surface cracks in the half-plane subjected to contact loading (but not reported here) are in good agreement with results presented by other researchers,

498

Wear in ZTC

under identical assumptions (see, e.g. Keer et al., 1982; Hearle & Johnson, 1985; Yu & Keer, 1989; and Dubourg et al., 1988). However, it is important to note that results have not been obtained when frictional contact between crack faces is present. It is recognized that frictional contact may have a significant influence especially upon the sliding mode (mode II) deformation of the closed crack. In the analysis for a phase-transforming material, the frictional contact between the crack faces may even be of greater importance, since the transformation zones at the crack tip(s) showed a marked closing effect on the tips, thus exerting significant compressive stresses there. These compressive stresses would in turn induce frictional stresses opposing the sliding deformation of the crack tips. The frictional resistance component needs to be considered in future studies. The dilatation of the transforming material leads to a permanent deformation near the transformation zone. After the external load is removed, this permanent deformation may keep parts of the crack or even the entire crack open. It may also lead to heaving of the wearing surface, i.e. surface uplift, which may be further exacerbated by any residual opening of subsurface crack. The residual opening forces the material above the crack to buckle towards the free surface. Both these effects change the surface topography and thereby the rolling/sliding conditions of the surface to the detriment of the tribological performance of the material. The shapes of the transformation zones at the tips of the cracks turned out to be highly asymmetric about the crackline, with a consequent effect on the mode II crack tip deformation. This effect has not previously been investigated, and earlier research of transformation toughening always assumed the build-up of transformed material along the crack surface to be beneficial. This study shows that that is not necessarily so. From the results presented here, this effect would appear not to be beneficial, since it enhances the mode II deformation at the crack tip. However, as the transformation imposes closure forces on the crack tip, it is effectively closed during most of the load pass. The shear transformation strains have a significant effect on the development and extension of the transformation zone. The influence of the shear direction is not to be underestimated. The stress invariant criterion for the transformation of tetragonal phase has a significant effect on the development of the transformation zone. After the transformation has been initiated by the applied contact load, additional transformation of particles is largely governed by an auto-catalytic process and to a lesser extent by the contact load.

15.6. Concluding Remarks

499

The random distribution of transformed particles may also influence the local deformation at the crack tip as well as the development of the transformation zone. The onset of transformation is often accompanied by an abrupt jump in KII for a fixed load position. This may be due to the quasi-static transformation regime adopted in the analysis, whereby the transformation process was allowed to complete and stabilize (no more particles transformed) before the load position was incremented. The computational time in the presence of transformation is much longer than in its absence. This is partly due to the iterative procedure used for the determination of the transformation zone and partly due to the increase in the complexity of the crack face conditions. Thus, if in the absence of transformation, the crack would be either wholly open or wholly closed, or partly open and partly closed, so it was necessary only to determine one (or in rare cases two) transition points along the crack length, in the presence of transformation, the number of transition points could be up to five (five for the subsurface crack and four for the surface crack) during a load pass. The increased number was caused by the wedging effect induced by the transformation at the crack tip. The number of possible combinations of the basic crack face configurations delineated in (15.1)-(15.4) reached a maximum of twelve during a load pass in the presence of transformation.

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517

Author Index Amazigo, J. C., 43, 166, 169,205, 213,214, 255,256,374,396, 397, 399,400,424 Anderson, M. P., 468,469 Andreasen, J. H., 212,231,261, 429,444 Aronov, V., 465,466 Barber, J. R., 470,472 Barenblatt, G. I., 264 Beardsley, R., 465,466 Bilby, B. A., 129, 133, 134,264, 413,415 Birkby, I., 468 Bower, A. F., 470 Bowman, K. J., 468 Braza, J. F., 465,466,467,468, 469,470 Breval, E., 465,468 Breznak, J., 465,468 Broek, D., 435,436 Bryant, M. D., 470 Budiansky, B., 36, 37, 43, 54, 55, 155, 165, 166, 169, 180, 181, 188, 200,205,213,214,237, 242, 255,256, 355, 356,374, 390,393,396,397, 399,400,424 Bueckner, H. F., 271,274, 285, 372 Cannon, R. M., 46 Cao, G. Z., 468,469

Carry, C., 468 Chang, F. K., 470,472 Chantaramungkorn, K., 475 Charalambides, P. G., 351,358, 366 Chen, I. W., 43, 45, 56, 78,303, 468 Cheng, H. S., 465,466,467, 469, 470 Cheng, J. S., 45 Claussen, N., 324, 343,344, 345, 350, 352 Comninou, M., 470,472 Cook, T. S., 253,261,265 Cotterell, B., 427 Cottrell, A. H., 264,413,415 Cranmer, D. C., 465 Dauskardt, R. H., 324,443,446, 447, 467 Dean, R. H., 169, 170 de With, G., 468,469 Dickerson, R. M., 331 Dortmans, L., 468,469 Dubourg, M.C., 498 Dugdale, D. S., 413 Dundurs, J., 470 Dvorak, G. J., 70, 72 Erdogan, F., 253,261,262,265 Eshelby, J. D., 47, 60, 81, 88, 93, 97, 129, 133, 134

518

Author Index

Evans, A. G., 36, 46, 158, 159, 162, 171,182, 345, 351,358, 366,396,397, 399,400,424

Jahanmir, S., 468,469 Johnson, K. L., 498 Jcrgensen, M. H., 413

Fine, M. E., 465,466,467, 470 Fischer, T. E., 468,469 Fleck, N. A., 470 Fleming, J. R., 470 Fraser, W. B., 367

Karihaloo, B. L., 177, 212, 231, 261,272, 273,344, 367,418,426, 427, 428,436,444, 447, 453,463 Keer, L. M., 232,427, 470,475, 498 Kokmeijer, E., 468,469

Gakhov, F. D., 267 Gane, N., 465,466 Gangopadhyay, A. K., 470 Gao, H., 271,274, 285, 439 Garvie, R. C., 10, 66, 76, 77, 78, 465 Goodier, J. N., 475 Green, D. J., 367, 368 Gupta, G. D., 253,261,262,265 Hannink, R. H. J., 10, 17, 201, 367,368,465,466,468,479 Haritos, G. K., 470 Harrison, P., 468 Hashin, Z., 391,393 Hayashi, K., 426 Hearle, A. D., 498 Heuer, A. H., 323,324, 331,335, 343, 344, 345, 350,352 Hill, R., 39, 73, 349, 391,392, 393 Hori, M., 396 Huang, J., 348 Huang, X., 272,300, 367 Hutchinson, J. W., 36, 37, 43, 54, 55, 81, 94, 124, 126, 155, 165, 166, 169, 170, 180, 181,285,351, 358,359,366,378,385,386, 387 Hwang, K. C., 44, 47, 51, 54, 55, 56 Inghels, E., 323, 331

Lam, K. Y., 58, 62, 69, 78 Lambropoulos, J. C., 36, 37, 43, 54, 55, 155, 165, 166, 169, 180, 181 Lange, F. F., 65, 76,424 Laws, N., 390,391 La Violette, R. A., 304 Lefkow A. R., 465,466 Li, V. C., 348 Liang, H., 468 Licht, R. H., 468,469,470 Lilley, E., 468,469,470 Lutz, E. H., 324 Macmillan, N. H., 465,468,470 Marmach, M., 465 Marshall, D. B., 324,331,443, 446,467 McMeeking, R. M., 36, 158, 159, 162, 171, 182, 344, 351,358, 366, 369, 375,390 Miller, G. R., 470 Milne-Thomson, L. M., 93 Mr C. V., 418,444, 447,453, 463 Mori, T., 48, 59, 71, 74 Morrell, R., 9 Mouritsen, O. O., 447 Mouwakeh, M., 498 Mura, T., 47, 48, 60, 65, 262

519

Author Index

Murakami, Y., 233, 243 Murray, M. J., 465,466,468 Muskhelishvili, N. I., 81, 89, 90, 133, 188 Naghash, A. R., 465,469 Nauer, M., 468 Nemat-Nasser, S., 232,396,426, 427 Nielsen, B. N., 177 Nutting, J., 465,467,468,469 O'Connell, R. J., 355,356 Ong, P. P., 58 Oranratnachai, A., 232,427 Palmer, A. C., 166 Parihar K. S., 232 Pascoe, R. T., 10 Prokopovich, S. A., 465,469 Radford, K. C., 424 Rainforth, W. M., 465,467,468, 469 Ready, M. J., 324, 331,335 Reyes-Morel, P. E., 43, 45, 56, 78, 79,303 Rice, J. R., 51, 111, 166,271,272, 274, 372,375, 376,408,411,427 Ritchie. R. O., 324, 443,446,447, 467 Rose, L. R. F., 81, 87, 91, 117, 120, 166, 182, 183, 184,207, 213,263,344, 349, 375,396, 397, 399,402,405,406,413,424 Rosen, B. W., 391 Rosenfield, A. R., 470 Rubinstein, A. A., 428 Rfihle, M., 343,344, 345,350,351, 352, 358, 366

Schmueser, D., 470 Scott, H. G., 465,466,468 Shaw, M. C., 324, 331 Sheppard, S., 470,472 Shetty, D. K., 205 Shtrikman, S., 393 Smith, E., 413,415 Sorrell, C. C., 465,469 Stam, G., 44,435,437,438 Steffen, A. A., 443,447 Steinbrech, R. W., 323,331,335 Stevens, R., 465,467,468,469 Stump, D. M., 188,200,213,237, 242,304, 459,497 Suh, N. P., 470 Sumi, Y., 232 Sun, Q. P., 44, 47, 51, 54, 55, 56 Swain, M. V., 66, 76, 77, 78,201, 207, 324, 331,367, 368,413,443 Swinden, K. H., 264, 413,415 Tada, H., 233, 243,247, 254, 256, 285,357, 398,404,405 Tanaka, K., 48, 59, 71, 74 Thomsen, N. B., 177,465,471 Timoshenko, S. P., 475 van den Berg, P. H. J., 468,469 Villechaise, B., 498 Viswanathan, K., 273 Wang, J. S., 205 Wang, Y., 465,466 Weertman, J., 130 Weertman, J. R., 130 Williams, M. L., 426 Worden, R. E., 470 Worzala, F. J., 465,466 Yu, M. M. H., 498 Yu, S. U., 44, 47, 51, 54, 55, 56

520 Yu, W., 443,467 Zelizko, V., 443 Zhang, J. M., 58, 62, 69, 78 Zhu, D., 465,469

Author Index

521

Subject Index Ageing, 11, 18, 162 Annular microcrack, 354, 356 Array of dislocations, 134, 140, 143 inclusions, 103, 105 internal cracks, 134,214, 404 obstacles, 402,408 semi-infinite cracks, 247 spots, 337 subsurface dislocations, 146 inclusions, 106 surface cracks, 242 transformable precipitates, 414 Arrest of fatigue cracks, 447, 453 microcracks, 351,354,356 Autocatalysis, 304, 311,331,459 Average deviatoric stress, 45 opening displacement, 415 shear, 36 strain, 44, 61, 69 stress, 44, 69 Bain strain, 11 Biaxial tension, 208 Bounds on bulk modulus, 393 shear modulus, 393 transformation strain, 392

Bridging, 23, 396 Calcia, 10 -PSZ, 18 Cauchy integral theorem, 93 Centre of dilatation, 84 shear, 85 transformation, 81 Ceria, 10 -TZP, 30 Chemical free energy, 47, 48, 66 Circular transformable spots, 93, 96,119, 124, 303,312, 325,481,493,493 Cohesive stress, 413 Complementary free energy, 48 Complex representation of displacement, 90,371 stress, 89 stress intensity factors, 117, 120 potentials, 89 centre of dilatation, 91 shear, 91 dislocations, 131 half-plane, 102 inclusion, 100 point forces, 90 Compression, 56,356,447,467

522 Contact loading, 470,478,485 Crack array of internal, 134, 214,404 semi-infinite, 247 surface, 242 bridging, 23,396 deflection, 425 dislocation modelling, 129 fatigue, 443 finite, 109, 196, 214 half-plane, 271 kinking, 426 opening displacement, 134 average, 415 critical, 401 semi-infinite, 114, 155, 187, 242, 305, 344,396,426 subsurface, 470,478 surface, 231,242,444, 470,485 Criterion of crack growth, 189 local symmetry, 426,428 maximum principal tensile stress, 435 microcracking, 359 minimum strain energy, 436 spring failure, 405 transformation, 37, 47, 73,303 Critical crack-face displacement, 401, 419 density of spots, 328 energy parameter, 77 kink angle, 428 mean stress microcrack nucleation, 361 transformation, 37 particle size, 12, 15, 345 strain energy density, 183 stress intensity factor, 401

Subject Index transformation, 41 parameter, 254 Cubic zirconia, 13 Deflection, 425 Delamination, 470 Deviatoric strain, 12 stress, 45,303 Dilatant inclusions, 97 stress-strain behaviour, 38, 40 transformation, 36, 43,285 Dilatation, 12, 38 centre of, 84 effective, 389 microcrack, 343 planar, 154 Dilute concentration of microcracks, 356, 358 transformable phase, 390,393 Dislocation array of edge, 134, 140, 143 subsurface, 146 crack modelling, 129 density function, 132,215,233, 243,249,445,475 edge, 130, 131, 136 pile-up, 133, 232,248,445 screw, 130 stress fields, 130 subsurface, 138 DZC, 10, 153, 343,351,395,425 Edge dislocation, 130, 131, 136 Effective dilatational strain, 389 fracture toughness, 348 moduli, 349 spring constant, 399

9

Subject

523

Index

transformation strain, 390 Elastic mismatch, 344 Endurance limit, 459 Equivalent inclusion, 59, 273 Fatigue, 443 Force doublets, 82 Fracture toughness ceria-TZP, 31 magnesia-PSZ, 179 yttria-TZP, 29 ZTA, 345 Free energy chemical, 47, 48, 66 complementary, 48 Helmholz, 48 of transformation, 47 Green's functions, 81 Half-plane crack, 271 Helmholz free energy, 48 Hertzian contact load, 474 Hydrostatic compression, 56 I-integral, 166 Imminent crack growth, 199, 208, 218, 236, 244,305,360, 364, 398 Inclusion array of internal, 103, 105 subsurface, 106 dilatant, 97 equivalent, 59,273 internal, 101 subsurface, 102 Inherent flaws, 204, 470 Integral theorems, 93 Intrinsic toughness, 159, 162, 179, 180, 189,306,353,405,

416,418,440,452,459 J-integral, 155 Kink angles, 427 critical, 428 Kinking, 426 Lattice parameters, 11 strain, 11 Lock-up, 166, 193, 213,255 Magnesia, 10 -PSZ, 17, 21, 78, 179, 185,425, 450 material parameters, 179 phase diagram, 18 Martensitic start temperature, 12 transformation, 11 variants, 63 Mean stress criterion microcracking, 361 transformation, 37, 40, 55, 311, 336 Microcracking, 343, 351 Microcrack annular 354, 356 coalescence, 351 density, 355 dilatation, 343 dilute distribution, 358 nucleation criteria, 359 parameter, 355, 357, 360 penny-shaped, 354 shielding, 351 toughening, 343, 351 zone, 347, 353, 362 Mismatch elastic, 344

524 thermal, 351,356 Mixed stress criterion, 311,336 Moduli bounds on, 393 effective, 349 mismatch, 386,389 parameters, 359 Monoclinic zirconia, 13 Non-homogeneous distribution of particles, 434 Oblate spheroid, 295 Overaged, 18 Overloading, 459 Peak aged, 18 strengthening, 205,213,241 stress, 230,420 toughening, 195,204, 212, 241 Penny-shaped microcrack, 354 Perturbation of crack front, 408 Perturbation technique, 367 Phase diagram magnesia PSZ, 18 yttria TZP, PSZ, 27 Plasticity, 44, 73 Polymorphs, 13 PSZ, 10, 15, 17, 27, 36, 56, 153, 179, 185,323,343,395, 450,465,478 Quasi-static crack growth, 159, 187, 190,201,213,226, 240,245,247,252,257, 324,405,420,431,452 R-curve array of surface cracks, 245 behaviour, 187

b'ubject index internal cracks, 196,202,210 semi-infinite cracks, 187, 194 surface cracks, 238 Reinforcing Spring, 396 Residual stress, 353 Rolling wear, 465,470 Row of cracks, 214 dislocations, 140, 146,368, 393 inclusions, 103, 106 spots, 337 Screw dislocation, 130 Self-consistent method, 349,368, 393,400 Self-propagating transformation, 331 Semi-infinite cracks, 114, 155, 187, 242,305, 344,396,426 Shear angle, 86, centre of, 85 stress criterion, 311 transformation, 45, 58,286,491 Size distribution of transformed particles, 345 Sliding wear, 465,470 Slip, 472 Spherical particle, 354 Spontaneous transformation, 12, 339,340 Stack of dislocations, 143 inclusions, 105 Stationary cracks, 199,208,218, 236,244, 305,360,364 Steady-state crack growth, 157, 348,405 transformation toughening, 153, 165, 172, 182, 260,347

525

Subject Index

microcrack toughening, 347, 350, 363 Stick, 472 Stokes theorem, 93 Strain average, 44, 61, 69 Bain, 11 deviatoric, 12 lattice, 11 Strength-toughnnes relations, 205 Strengthening array of semi-infinite cracks, 256 surface cracks, 246 internal crack, 205, 213 surface crack, 241 Stress average, 44, 45, 69 cohesive, 413 deviatoric, 45,303 intensity factor, 117, 286,299, 482 critical, 401 peak, 230,420 residual, 353 Stress-strain behaviour dilatation, 38, 40 microcracking, 352 TZP, PSZ, 57 Strip of transformation, 331 Subcritical crack growth, 454 transformation, 41, 166 Subsurface crack, 470,478 dislocations, 138, 146 inclusions, 102, 106 Super-critical transformation, 41, 157 Surface energy, 47

uplift, 467,498 Tetragonal zirconia, 13 Thermal anisotropies, 351 mismatch, 351,356,406 Toughening bridging, 396 deflection, 425 kinking, 426 microcracking, 343,351 peak, see peak toughening ratio, see toughness increment second phase particles, 395 steady-state, see steady-stale toughening

transformation see transformation toughening

trapping, 395 Toughness increment, 155,344, 348 Transformation autocatalytic, 304, 311,331, 459 centre of, 81 criterion, 37, 47, 73,303 critical, 41 dilatant, 36, 43,285 free energy of, 15 martensitic, 11 parameter, 43 plasticity, 44, 73 self-propagating, 331 spontaneous, 12, 339, 340 spot, 81, 93, 109, 114, 118 strain bounds on, 392 effective, 390 planar, 87 shear, 45, 58,286,304 three dimensional, 285,286

~)26

b'ubject i n d e x

strip of, 331 sub-critical, 41, 166 super-critical, 41,157 t--~m

, lO

three-dimensional, 271 dilatational, 285 shear, 286 toughening initial, 237 steady-state see steady-sLate peak see peak toughening yielding, 62 zone shapes discrete particles, 303,312, 325,481,489,493 growing cracks, 193, 204,212, 240 initial, 191,199,208,217, 236, 244 steady-state, 260 Trapping, 395 transformable particles, 412 Triaxial compression, 56 TTC, 10, 35, 73, 196, 232, 344, 367, 395,418,426,453, 461 Twinning, 37, 43, 62 TZP, 10, 27, 28, 56, 153,344, 468 Wake of microcracks, 363 transformation, 157, 160, 164, Wear, 465 Weight function edge dislocation, 136 row of

dislocations, 140 inclusions, 103 subsurface dislocations, 146 inclusions, 106

single inclusion, 101 subsurface dislocation, 137 inclusion, 102 stack of dislocations, 143 inclusions, 105 three dimensional, 272 Westergaard stress function, 90, 95 Yielding, 62 Yttria, 10 -PSZ, 27, 185 phase diagram, 27 -TZP, 29 phase diagram, 27 Zirconia, 10 cubic, 13 lattice parameters, 12 monoclinic, 13 tetragonal, 13 Zone widening, 193, 205,213,240 ZTA, 11,343, 344, 347, 390,434, 469 ZTC, 10, 15, 58, 69, 72,413,417, 425,465