Mechanics of Transformation Toughening and Related Topics

MECHANICS OF TRANSFORMATIONTOUGHENING AND RELATED TOPICS NORTH-HOLLAND SERIES IN APPLIED MATHEMATICS AND MECHANICS E...

Author: J.H. Andreasen | B.L. Karihaloo

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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

Data

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

01996 ELSEVIER SCIENCE B.V. All rights reserved No part of thispublication may be reproduced, stored in a retrieval system, or transmitted, in anyform or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of thepublishec Elsevier Science B. K Copyright& Permissions Department, PO. Box521, 1000AM Amsterdam, The Netherlands. Special regulationsf o r readers in the U.S.A. - This publication has been registeredwith the Copyright Clearance Center Inc. (CCC),222 Rosewood Drive Danvers, MA 01923. Information can be obtainedfrom the CCC about conditions under which photocopies ofparts of this publication may be made in the U.S.A. All other copyright questions, includingphotocopying outside of the U.S.A., should be referred to the publisher: No responsibility is assumed by the publisherfor any injury anuYordamage topersons orproperry as a matter ofproducts liability, negligence or otherwise, orfrom any use or operation of any methods, products, instructions or ideas contained in the material herein.

This book is printed on acid-free paper PRINTED IN THE NETHERLANDS

To Alla and Wivj

This Page Intentionally Left Blank

vii

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.


ix

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

xi

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

xii

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


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


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


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


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


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


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.


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


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


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, 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 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


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

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


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


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 )


3.3.2

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


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)


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


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 can be justified thermodynamically. If the transformation takes place with a martensitic combination (~,~,~,~) satisfying XijAijkiEk~' 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 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 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)~)~


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


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


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


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


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~


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)}


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


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 - 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,(

(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)


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


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


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 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


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


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.


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


F i g u r e 10.1: Symmetric dilatational spots with/~ - 10


F i g u r e 10.2: Symmetric dilatational spots with/~ - 20

313

314


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


Figure 10.5: Symmetric dilatational spots with ~ - 35


317

F i g u r e 10.6: Symmetric dilatational spots with fl - 35 (a - 0 plot filled the box region)

318


F i g u r e 10.7: Nonsymmetric dilatational spots with j3 - 25


F i g u r e 10.8" Nonsymmetric dilatational spots with ~ - 30

319

320


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 !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.


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


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


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


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-


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


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.


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


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


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


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)


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 will not lead to self-propagating transformations.

Part I11

Related Topics


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.


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


/

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


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


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


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


~-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)


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


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


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~. /- 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
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 /~ = ( 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 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

_ 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


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

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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


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

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