Multiscale Modeling of Heterogenous Materials E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 486

Veröffentlichungsjahr: 2013

Beschreibung

A material's various proprieties is based on its microscopic and nanoscale structures. This book provides an overview of recent advances in computational methods for linking phenomena in systems that span large ranges of time and spatial scales. Particular attention is given to predicting macroscopic properties based on subscale behaviors. Given the book's extensive coverage of multi-scale methods for modeling both metallic and geologic materials, it will be an invaluable reading for graduate students, scientists, and practitioners alike.

Leseprobe

Table of Contents

Foreword

Chapter 1. Accounting for Plastic Strain Heterogenities in Modeling Polycrystalline Plasticity: Microstructure-based Multi-laminate Approaches

1.1. Introduction

1.2. Polycrystal morphology in terms of grain and sub-grain boundaries

1.3. Sub-boundaries and multi-laminate structure for heterogenous plasticity

1.4. Application to polycrystal plasticity within the affine approximation

1.5. Conclusion

1.6. Bibliography

Chapter 2. Discrete Dislocation Dynamics: Principles and Recent Applications

2.1. Discrete Dislocation Dynamics as a link in multiscale modeling

2.2. Principle of Discrete Dislocation Dynamics

2.3. Example of scale transition: from DD to Continuum Mechanics

2.4. Example of DD analysis: simulations of crack initiation in fatigue

2.5. Conclusions

2.6. Bibliography

Chapter 3. Multiscale Modeling of Large Strain Phenomena in Polycrystalline Metals

3.1. Implementation of polycrystal plasticity in finite element analysis

3.2. Kinematics and constitutive framework

3.3. Forward Euler algorithm

3.4. Validation of the forward Euler algorithm

3.5. Time step issues in the forward Euler scheme

3.6. Comparisons of CPU times: the rate tangent versus the forward Euler methods

3.7. Conclusions

3.8. Acknowledgements

3.9. Bibliography

Chapter 4. Earth Mantle Rheology Inferred from Homogenization Theories

4.1. Introduction

4.2. Grain local behavior

4.3. Full-field reference solutions

4.4. Mean-field estimates

4.5. Concluding observations

4.6. Bibliography

Chapter 5. Modeling Plastic Anistropy and Strength Differential Effects in Metallic Materials

5.1. Introduction

5.2. Isotropic yield criteria

5.3. Anisotropic yield criteria with SD effects

5.4. Modeling anisotropic hardening due to texture evolution

5.5. Conclusions and future perspectives

5.6. Bibliography

Chapter 6. Shear Bands in Steel Plates under Impact Loading

6.1. Introduction

6.2. Viscoplasticity and constitutive modeling

6.3. Higher order gradient theory

6.4. Two-dimensional plate subjected to velocity boundary conditions

6.5. Shear band in steel plate punch

6.6. Conclusions

6.7. Bibliography

Chapter 7. Viscoplastic Modeling of Anisotropic Textured Metals

7.1. Introduction

7.2. Anisotropic elastoviscoplastic model

7.3. Application to zirconium

7.4. High strain-rate deformation of tantalum

7.5. Conclusions

7.6. Bibliography

Chapter 8. Non-linear Elastic Inhomogenous Materials: Uniform Strain Fields and Exact Relations

8.1. Introduction

8.2. Locally uniform strain fields

8.3. Exact relations for the effective elastic tangent moduli

8.4. Cubic polycrystals

8.5. Power-law fibrous composites

8.6. Conclusion

8.7. Bibliography

Chapter 9. 3D Continuous and Discrete Modeling of Bifurcations in Geomaterials

9.1. Introduction

9.2. 3D bifurcations exhibited by an incrementally non-linear constitutive relation

9.3. Discrete modeling of the failure mode related to second-order work criterion

9.4. Conclusions

9.5. Acknowledgements

9.6. Bibliography

Chapter 10. Non-linear Micro-cracked Geomaterials: Anisotropic Damage and Coupling with Plasticity

10.1. Introduction

10.2. Anisotropic elastic damage model with unilateral effects

10.3. A new model for ductile micro-cracked materials

10.4. Conclusions

10.5. Acknowledgement

10.6. Appendix

10.7. Bibliography

Chapter 11. Bifurcation in Granular Materials: A Multiscale Approach

11.1. Introduction

11.2. Microstructural origin of the vanishing of the second-order work

11.3. Some remarks on the basic micro-macro relation for the second-order work

11.4. Conclusion

11.5. Bibliography

Chapter 12. Direct Scale Transition Approach for Highly-filled Viscohyperelastic Particulate Composites: Computational Study

12.1. Morphological approach in the finite strain framework

12.2. Evaluation involving FEM/MA confrontations

12.3. Conclusions and prospects

12.4. Bibliography

Chapter 13. A Modified Incremental Homogenization Approach for Non-linear Behaviors of Heterogenous Cohesive Geomaterials

13.1. Introduction

13.2. Experimental observations on the Callovo-Oxfordian argillite behavior

13.3. Incremental formulation of the homogenized constitutive relation

13.4. Modifying of the local constituents’ behaviors

13.5. Implementation and numerical validation of the model

13.6. Calibration and experimental validations of the modified incremental micromechanical model

13.7. Conclusions

13.8. Acknowledgement

13.9. Bibliography

Chapter 14. Meso- to Macro-scale Probability Aspects for Size Effects and Heterogenous Materials Failure

14.1. Introduction

14.2. Meso-scale deterministic model

14.3. Probability aspects of inelastic localized failure for heterogenous materials

14.4. Results of the probabilistic characterization of the two phase material

14.5. Size effect modeling

14.6. Conclusion

14.7. Acknowledgments

14.8. Bibliography

Chapter 15. Damage and Permeability in Quasi-brittle Materials: from Diffuse to Localized Properties

15.1. Introduction

15.2. Mechanical problem — continuum damage modeling

15.3. Permeability matching law

15.4. Calculation of a crack opening in continuum damage calculations

15.5. Structural simulations

15.6. Conclusions

15.7. Acknowledgement

15.8. Bibliography

Chapter 16. A Multiscale Modeling of Granular Materials with Surface Energy Forces

16.1. Introduction

16.2. Stress-strain model

16.3. Results of numerical simulation without surface energy forces consideration

16.4. Granular material with surface energy forces: the example of lunar soil

16.5. Summary and conclusion

16.6. Bibliography

Chapter 17. Length Scales in Mechanics of Granular Solids

17.1. Introduction

17.2. Model description

17.3. Force chains

17.4. Fluctuating particle displacements

17.5. Friction mobilization

17.6. Conclusion

17.7. Acknowledgements

17.8. Bibliography

List of Authors

Index

First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Oana Cazacu to be identified as the author of this work have been asserted by her in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Multiscale modeling of heterogenous materials : from microstructure to macro-scale properties / edited by Oana Cazacu.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-047-9

1. Inhomogeneous materials--Mathematical models. I. Cazacu, Oana.

TA418.9.I53M85 2008

620.1’1015118--dc22

2008027555

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-047-9

Foreword

For several decades, the mechanical behavior of materials has been described by phenomenological constitutive relations in the general framework of the theory of elasto-visco-plasticity. These approaches, when combined with numerical methods (e.g., finite element methods), have led to very powerful tools — tools which simulate boundary value problems in a reasonably realistic way, making them relevant for real engineering problems.

However, this methodology has reached some limits, primarily because of the great number of constitutive parameters to be determined, the recurrent difficulty in simulating cyclic loading and limitations in predicting post-failure behavior. In recognition of these basic limitations, methods based on the micromechanics of heterogenous materials have been vigorously developed through so-called homogenization techniques. Molecular dynamics also appeared as a very useful tool to simulate the mechanical behavior of materials whose internal structure is easy to identify.

This book is timely and relevant, addressing recent advances which take into account the microstructure of materials. To that end, some chapters illustrate how it is possible to improve phenomenological constitutive relations by incorporating proper micromechanical ingredients, while others propose to build macroscopic relations by using localization-homogenization methods and a local microscopic relation, describing the interaction laws between the element grains or particles. Of course these interaction laws are usually rather simple — if a proper scale has been chosen — or at least well established. For these methods the most difficult step is probably to build a localization (or projection) operator, essentially because the solutions are not unique.

Other chapters relate the application of continuum-based multi-scale methods to metallic materials or geomaterials. New powerful methodologies for describing anisotropy both at single crystal and aggregate level are presented.

Here also the mechanical parameters are very few, the local laws (grain level or single crystal level) are quite simple and the derived macroscopic properties are surprisingly realistic. Thus, the macroscopic behavior, which appears in experiments as extremely complex, can be described with few mechanical ingredients. Hence, this macro-complexity may be due to the great number of elements in interaction or texture and not to an eventual micro-complexity. As in the case of molecular dynamics, such continuum-based multi-scale methods allow the real world to be rebuilt numerically.

Some essential difficulties appearing in a continuum mechanics framework (e.g., the description of an internal length) are solved in molecular dynamics in a very natural and elegant way. On the other hand, the likely remaining difficulty is to take into account a proper geometry for the element assembly.

With respect to practical problems, we now understand that it is essential to consider materials in their own environment. From this perspective, describing environmental coupling (as induced by chemo-thermo-hydro-mechanical interactions) will be more and more important in the future. Several chapters consider these “multi-physics” couplings.

From an engineering perspective, failure is always an essential question. An efficient analysis framework is provided by the bifurcation theory. The existence of bifurcated branches and the roles of imperfections and perturbations have been investigated with success. Moreover, for non-associated materials (and all materials whose behavior depends on the mean pressure seem to behave like this), the existence of a large stress domain of bifurcations or of material/geometric instabilities has now been established on firm theoretical, experimental and numerical bases. Various failure modes are associated with these bifurcations, and diffuse and localized failures are also discussed in several chapters.

Finally, another interesting facet of this book lies in the fact that a variety of solid materials are considered — which is quite a rare feature today. Readers will be interested in cross-linking the methods and tools developed to describe the macroscopic properties of metallic materials and geomaterials from their very different microscopic internal structures.

This book gathers together selected papers from the invited lectures presented during the 1st US-France Symposium held 28-30 March 2007 at the University of Florida in Shalimar, FL. It was locally organized by John “Row” Rogacki and Oana Cazacu under the auspices of the International Center for Applied Computational Mechanics (ICACM) with financial support provided by the Air Force Office of Research, Program Manager Dr. Fariba Fahroo (grant # FA9550-07-1-0234). This support is gratefully acknowledged.

Félix Darve

Grenoble Institute of Technology, France

Chapter 1

Accounting for Plastic Strain Heterogenities in Modeling Polycrystalline Plasticity: Microstructure-based Multi-laminate Approaches1

1.1. Introduction

Models of the mechanical or physical behavior of materials are most efficient when they are microstructure-based. However, reproducing reality is not achievable and energy savings also demand models that do not become highly consumptive of computer space and time. With the goal of bridging scales in models that start from elementary atomistic models to simulate an overall response, the search for compromises between the microstructural descriptions and the resulting simulation accuracy will remain a challenging area for a long time. A smart alternative to running huge “ab initio” computational calculations is to anticipate which microstructural features do really matter at the macro-scale concerned according to the considered situation.

In the specific field of modeling the plastic behavior of heterogenous non-linear metallic materials, effective properties are reasonably approached when simultaneously considering i) a good enough description of the evolving morphology, ii) an appropriate homogenization scheme according to the material morphology and behavior type, iii) a relevant microstructural modeling of intra-crystalline plasticity. In all these domains the last few decades have significantly enriched the available background. With regard to the third point, dislocation dynamic simulations have clarified many features that concern crystal hardening evolutions with strain [DEV 06], the anisotropy of which is one of the most complex questions not yet answered, even for the simplest cubic structures. With regard to the other two domains, enhanced morphological descriptions by integrating n-point statistics with increasing n [TAL 97], as well as improved first-and second-order developments to better describe the non-linearity of the phase plastic behavior laws [PON 98] have enabled remarkable gains in the accuracy of accessible behavior estimates or bounds. The possible further improvements in the global modeling of polycrystalline plasticity discussed here address points that concern in a combined manner the morphology description, the homogenization framework and the behavior approximation in terms of plastic flow criterion. For theoretical details and simulation examples relating to this book, see [FRA 07, FRA 08].

For such aggregates that deform by crystallographic shear mechanisms (i.e. slip, twinning or also transformation plasticity up to a volume change), we first question the granular description that is conventionally used, compared to an alternative description in terms of grain boundary and sub-boundary orientation distribution. Secondly, between the evolution of the sub-boundary spatial arrangement and the shear activity in the material, a link is made that is based on a multi-laminate approach of plastic heterogenities. Such a multi-laminate approach to describe the current morphology of the strained material in turn acts on the homogenization scheme that can be preferentially used. Comparing with the inclusion-based modeling, further advantages are pointed out, as the more natural reference to an equivalent homogenous super-crystal that justifies introducing a single plastic potential for the whole aggregate, or the possibility of accounting for a grain size effect. Section 1.2 illustrates some support to a sub-boundary based morphology description of polycrystal plasticity, section 1.3 introduces the considered multi-laminate representation and section 1.4 summarizes the proposed modeling framework that results.

1.2. Polycrystal morphology in terms of grain and sub-grain boundaries

1.2.1. Some evidence of piece-wise regularity for grain boundaries

When looking at micrographic or nanographic pictures of metallic aggregates, there is plenty of evidence that grains are polyhedral domains whose boundary facets result from the elaboration route. Figure 1.1 shows two examples that concern aggregates of micrometric (left) or nanometric (right) grains. On a topological ground, it is also obvious that if, ideally speaking, all the grains were convex and smooth — i.e. with rounded edges — some complementary matrix phase, of vanishing volume fraction, should be accounted for to ensure matter continuity and compactness. However, part of the grains can even be non-convex polyhedrons, some have sharp edges and the widely used assumption of ellipsoidal grain shape is definitely a rough one. Furthermore, a total space mapping is hardly obtained with a single grain shape, unless with the particular polyhedrons identified in stereology. The related rank-2 correlation functions between pairs of grain centroids (A-A, A-B, A-C, B-B, B-C, etc.) are in general different for shapes that are not congruent and they neither reduce to a single ellipsoidal symmetry nor to a set of different ellipsoidal symmetries, as discussed in [FRA 04a]. These are real limits to inclusion-based aggregate approaches.

Figure 1.1.TEM microstructures of (left) micrometric grains in steel (by courtesy of K. Zhu, LPMTM) and of (right) ultra-finely grained alpha iron (by courtesy of G. Dirras, LPMTM)

An alternative type of morphology description that has emerged is from considering the orientation distribution of the grain boundaries. Not entering into the details here, from the still open analysis domain of boundary networks [SHU 05] or coincident site lattice [COU 05], an essential feature of these aggregate structures is noteworthy: the crystallographic orientations of the boundary facets of such polyhedral grains are mainly oriented parallel to or close to dense planes.

1.2.2. Characteristics of plastic-strain due to sub-grain boundaries

Now, when a metallic aggregate with at least micrometric grains is plastically strained, a grain sub-structure is installed progressively that is intimately related to the shear mechanisms that operate in the considered crystallographic lattice. Figure 1.2 shows an example in the case of slip activity. According to the Kuhlsmann-Wilsdorf LEDS theory [KUL 02], dislocation cell sub-structures can be analyzed into “dislocation carpets”, parallel to the slip planes, and “dislocation walls”, more or less orthogonal to individual slip directions or at some mean orientation with regard to several different slip directions. In fact, as exemplified in Figure 1.2 for a sequence tension + shear with regard to the same direction, a dislocation substructure in an aggregate that has suffered a known strain path can be analyzed with regard to the orientation of the planes of maximum resolved shear stress (MRSS), closely related to the most active nearby slip planes. Most of the dislocation cell walls do appear either parallel or normal to the traces of the MRSS planes. A two-phase morphology (including dislocation-free cells in a matrix of dense walls) often proved to be an acceptable first order general description of these sub-structures. However, they are also better represented by typical polyhedral cells than by smooth inclusions. At this sub-grain scale, as well as at grain scale, even if here domain convexity is a more acceptable simplification, ellipsoidal approximations are still rough descriptions of the reality.

Figure 1.2.TEM microstructure of IF steel after 10% tension in the rolling direction followed by a 20% shear along the same direction SD (courtesy of V. Richard, LPMTM). The single (resp. double) lines mark the traces of the MRSS plane in tension (resp. shear)

Nowadays, other evidence of metal plasticity is that the intra-crystalline substructure (in large enough grains to allow it) rapidly overcomes the initial granular structure in terms of specific contribution to the yield surface or to the hardening evolution. In fact, efficient models for metal forming simulations must integrate that, after each strain path change, the previous dislocation sub-structure is erased to make room for a newly built sub-structure that is characteristic of the current path [HAD 06].

It is noteworthy that ultra finely grained aggregates do not easily deform by such intra-crystalline multiple slip processes. These nanometric structures are much harder than the micrometric structures and also much less ductile. At this grain scale where the grain boundary “matrix” carries a large part of the deformation, multiphase approaches of the inclusion/matrix type can be adapted, with little plasticity in the grain interior. However, in terms of a sub-boundary based approach as is the question here, this difference more directly connects with the lack of intra-granular plastic activity through the lack of related plastic-strain-due sub-boundaries [CUB 07].

1.3. Sub-boundaries and multi-laminate structure for heterogenous plasticity

Figure 1.3 illustrates the relative contributions of the grain boundary structure and the sub-grain structure that is due to plastic strain and evolves with it.

Figure 1.3.TEM microstructure that is similar on both sides of a grain boundary (arrows) in a rolled and annealed steel (courtesy of F. Gregori and A. Wauthier, LPMTM)

As observed, once the plastic straining is installed, the grain structure becomes second in terms of boundary density, compared with the cell sub-structure, and becomes negligible at large enough plastic strain. A consequence is that any modeling of aggregate behavior that is based on the grain shape evolution will miss the dominant contribution of the faster changes of the intra-crystalline cell structure. However, modeling the intra-granular heterogenities without excessive complexity is a difficult task. At least, the dominant feature to be accounted for certainly is their intimate connection to slip (or generally shear) activity.

One recently explored way to account for the cell sub-structure evolution prior to the grain evolution, among various approaches that have been developed with that goal, is to treat the aggregate as an evolving multi-laminate structure, with laminate orientations tightly connected with the plastic strain mechanisms of the constitutive crystals, here called shear mechanisms. Thorough analyses of the possible description of the plastic strain in metallic crystals in terms of evolving multi-laminate structures have been performed by Ortiz and co-workers while tentative multi-laminate models for polycrystalline plasticity have also been proposed by them [ORT 99] or by other groups [ELO 00]. If laminate structures are relevant for plastic straining due to such shear mechanisms, difficulties arise when attempting to follow successive branching that are expected to multiply when the plastic straining increases or when willing to ensure individual compatibility conditions between laminate orientations. Descriptions that bypass these difficulties, from statistical considerations for example, would be more easily manageable.

The approach proposed in [FRA 07, FRA 08] for single and polycrystals goes in this direction. The main additional point that is used there is that not only does the multi-laminate approach fit well with a plastic strain that results from shearing mechanisms, but it also matches with a structure description in terms of the sub-boundary orientation distribution and its evolution.

1.3.1. Effective moduli tensor and Green operator of multi-laminate structures

Multi- or rank-n laminate structures have been shown [MIL 88] to have simple connections with granular structures in the special case of two-phase composites made of a connected matrix (m) embedding a second included phase (i) that is fully characterized (domain shape and distribution) by an ellipsoidal volume V of “strain Green operator”1. The expression of the effective current tangent moduli in that latter case in terms of the phase moduli is:

[1.1]

The effective related moduli of a rank-n laminate structure with successive laminate layers of normal direction are obtained from repeatedly applying equation [1.1]. For alternated layers of the matrix phase and of the rank-(n-1) laminate, equation [1.1] is applied with, instead of , the operator at rank k that stands for the Green operator of the ωk -oriented laminate layers in the medium (m). The recurrent formula reads:

[1.2]

It yields for the moduli tensor of the rank-n laminate (ω1, ω2 , · · ·, ωn) the form:

[1.3]

[1.4]

In equation [1.4], , while the basic operators are the laminate operators normal to all the at directions in R3.

Expression [1.3] does not depend on the selected hierarchy of the successive laminations k. It only depends on their orientations ωk and on the weights ψk ≥ 0,

, that finally apply on each of them. Contrarily, the multi-laminate is hierarchical in cases of several included phases or of different embedding matrices. Let us generalize equation [1.1] specifically for congruent inclusions V of N different moduli Ci and concentrations fi embedded in a matrix of vanishing volume fraction . It yields the general form that iteratively provides the self-consistent (sc) estimate for the thus-defined aggregate structure, for example, at iteration p and with depending on the moduli at iteration (p-1):

[1.5]

If we extend equation [1.3] as equation [1.5] extends equation [1.1], we can obtain a self-consistent type of estimate for an aggregate with N phases treated as a rank-n laminate structure, where and fi(n) from [1.3] substitutes with and fi in [1.5] and that (sc) type of estimate does not depend on the lamination hierarchy. Thus, replacing an inclusion-based ellipsoidal description of some aggregate structure by a rank-n laminate description mainly amounts to replacing a continuously weighted distribution of laminate Green operators (over the unit sphere in R3) by a discrete weighted sum of n of them. In the inclusion-based description, the ellipsoidal anisotropy of the weight function of V obeys where is the breadth of the characteristic ellipsoid V in the ω direction (the isotropic weight function is . Similarly in the multi-laminate description taken to describe (congruent) polyhedral domains with n differently oriented facet types, the large weights in equation [1.3] would correspond to the directions normal to densely packed parallel boundaries. Any discrete sum of (k-oriented) laminate operators is part of the continuous distribution of the laminate operators over R3. This means that going from a granular morphology characterized by a continuous laminate operator distribution to a discrete multi-laminate structure (that would be induced by plastic straining in the present case) basically amounts to suppressing a part of the continuous distribution and to adapting the part that remains. The representative weight function and its evolution with the plastic strain will then result from the considered “boundary-based” morphology description for the aggregate concerned (nature of shear mechanisms, related orientation distribution of the boundaries and sub-boundaries, relation to the plastic strain, grain size influence, etc.).

[1.6]

In equation [1.6], q represents the dependency of function a with grain size effects and other morphological features to be accounted for, Ep (resp. Wp) is the — g(α(I)) plastic strain (resp. rotation) tensor and represents the distribution of all shear strains on all the shear mechanisms of the aggregate, to be defined in section 1.4. Function a is a priori an increasing function from zero to one with the plastic strain, the rate of which depends on the parameter(s) q. The dominant characteristic of equation [1.6] is the difference of plastic strain dependency between the two contributions: a mean grain shape V that is assigned to remain ellipsoidal and identical for all grains can only be stretched and rotated according to the material overall plastic deformation. This restricts the evolution of the weight function . In comparison, is the characteristic weight function of a rank-n laminate substructure where each individual term can be independently linked to one individual shear amplitude that is related to one shear (slip or twinning or transformation) plane (or direction) type in the aggregate, the most active of these planes (resp. directions) being expected to participate dominantly2. Function a progressively (rapidly) changes the reference operator from the grain (inclusion-based) operator defined according to equation [1.4] (and for an aggregate case, using the self-consistent approximation), to the “boundary-based” multi-laminate form defined using a weigh function such as Ψ(ω1,…ωn) in equation [1.3]. Such an evolution can be introduced in any homogenization framework that would be found relevant enough, as well as with any preferred plastic flow law for the material and any behavior laws for the phases. Regarding the flow criterion, the aggregate description in terms of a statistical distribution of boundary plus sub-boundary orientations allows us to see it as a homogenous equivalent super-crystal. This makes the use of a single plastic potential as flow criterion relevant. Regarding the homogenization framework, an important property of the multi-laminate structures can now be used to justify significant simplifications.

1.3.2. Multi-laminate structures and piece-wise homogenous plasticity

The stress and strain fields are uniform almost everywhere in the layers of any (hierarchical or not) multi-laminate structure. Here, “almost everywhere” means that heterogenities may remain close to the planar boundaries between layers but with infinitesimal volume fractions. This property also makes (at least statistically) the plastic strain field in the aggregate piecewise homogenous. Thanks to this property, we can make use of the advantageously simplifying TFA (Transformation Field Analysis) framework [DVO 92].

Thus, on the one hand, such (either hierarchical or not) multi-laminate descriptions of the aggregate morphology are justified in terms of grain and sub-grain boundary distributions evolving with the shear mechanisms and on the other hand they are claimed to validate the use of the TFA as a homogenization framework. This is now formalized when starting from the context of the affine homogenization approximation of which the TFA is a special case. As the TFA accommodates the incompatibilities purely elastically between differently strained domains, the affine comparison material for each non-linear phase is the elastic regime of the phase itself, with their individual plastic strains as homogenous eigenstrains. In the following, this TFA-HML modeling (where “HML” stands for “hierarchical multi-laminate”) is associated, according to [FRA 07, FRA 08], with the globally regularized Schmid law as a single plastic potential for the aggregate and with a local crystal hardening law that agrees with experimental data.

1.4. Application to polycrystal plasticity within the affine approximation

1.4.1. Constitutive relations

We denote by eα(I) the mean total strain over a sub-domain α(I) of phase I, and the corresponding mean stress. The affine approximation of non-linear laws results from the first order Taylor expansion:

where

The asterisks indicate that in each homogenous non-linear sub-domain α(I) we refer to a linear comparison material of moduli and to a homogenous eigenstrain . Disregarding specific features of finite rate-independent crystal plasticity [FRA 91], stress σα(I) in sub-domain α(I) can be written as:

[1.7]

The stress concentration and influence tensors depend on the current effective tangent elastic-plastic moduli of the material according to the self-consistent scheme3. So it is as well then for operators and that combine intra- and inter-granular heterogenity effects4. For a single plastic potential F, the plastic flow conditions will read F=0 and dF=0. We take the single plastic potential for the aggregate of the globally regularized Schmid form:

[1.8]

where in each domain α(I) a n-set of basic shear mechanisms g(oc(I)) is considered. We denote by the current applied resolved shear stress on the mechanism g(α(I)) defined by a shear direction and a shear plane of unit normal . The strain hardening law for the crystals is taken from the classical form for slip (that may be non-local):

[1.9]

in terms of the mean shear increments over the aggregate . Here we simplify to slip mechanisms. The plastic flow condition dF=0 in that case becomes:

[1.10]

The incremental macroscopic stress dΣ versus strain dE relation reads:

[1.11]

[1.12]

[1.13]

In equation [1.13], the effective hardening matrix terms can be stepwise written as:

[1.14]

[1.15]

By nature of the TFA, the operators are in general of the order of elasticity moduli, which yields H0(TFA) ≈ μ / 3 >> H0(h) for “standard” grain sizes where H0(h)≈ μ / 300 . This is not the case for ultra finely grained polycrystals where →. The influence tensors are weighted sums of laminate operators that combine a continuous and a discrete contribution for the inter-granular and the intra-granular heterogenity levels respectively. According to equation [1.7], they involve the influence operators that read:

[1.16]

[1.17]

For isotropic elasticity The related operators ti and t′i for the ellipsoid V in the above expressions can be replaced by operators t⟨G⟩ and t′⟨G⟩ of a weight function given by equation [1.6]. Similar relations can be derived for Fα(I)β(I) either considering sub-granular ellipsoidal partition or considering a multi-laminate type of structure in the grain, that yields an operator of the form of equation [1.3] instead of ti. For a set of basic operators related to the planes of active shear mechanism, precise methods of defining the respective weights from their shear activity are still unanswered questions. Simple forms can be incrementally derived in proportion to the slip increments , using the current coefficients .

1.4.2. Fundamental properties for multi-laminate modeling of plasticity

We now come to a key point for the proposed “sub-boundary-based” multi-laminate approach of the aggregate structure. Let us note first that when all laminates are oriented parallel to identical crystallographic planes, all laminate Green operators are identical up to a rotation and the apparent elasticity symmetry can be isotropic for in-plane isotropic laminates. More essentially, for the Green operators of all laminates in that are either parallel to the shear plane of a shear mechanism or normal to its shear direction, the product of the Green operator with the Schmid tensor of this mechanism has been shown to be zero [FRA 07]. We call such a shear mechanism an orthogonal mechanism. Consequently, for an evolution law of the operators (using a α-graded weight function such as equation [1.6]) that reduces or suppresses the non-vanishing contributions of the basic Green operators (i.e. those not associated with an orthogonal mechanism), the TFA accommodation can be reduced to a negligible level compared to the physical hardening term H0(h) . Since the lower this H0(h) term is the larger the constitutive grains are, the TFA accommodation needs be decreased more for large grains than for small ones to possibly become negligible. The physically allowed amount of reduction in terms of boundary and sub-boundary analyses will depend on the considered material.

Thus, when associating a multi-laminate approach with the TFA framework, it is possible for appropriately oriented, and hierarchically ordered if needed, laminates to reduce the well known overstiffness that results from the TFA. The necessary condition for this reduction that basic laminates are essentially “orthogonal” with the most active shear mechanisms is consistent with the description of the aggregate morphology in terms of plastic-strain-due sub-boundary orientation distributions rather than in terms of mean grain or domain shape. Sub-boundaries mainly oriented orthogonally with shear mechanisms in the sense of “orthogonal” defined here is a relevant assumption. Sub-boundaries have more influence when the grains are larger, which means a larger elastic accommodation part for small grain sizes. The superposition of several possible lamination structures, with possibly different occurrence probabilities, clearly gives a statistical nature to such an approach. Furthermore, thanks to the superposition of the several possible solutions or “modes” and to the coupling using the globally regularized Schmid flow law, this type of modeling is parent, for microstructure-based polycrystal plasticity, to the non-uniform TFA (NTFA) model proposed in [MIC 03].

1.5. Conclusion

From this discussion, it is possible to imagine various modeling sketches that may benefit from the discussed properties regarding multi-laminate structures both connected with crystallographic shear mechanisms and sub-boundary-based morphology description. Sketches will differ according to the shear mechanisms involved, the physical hardening law considered and the flow criterion. Cases have been explained and discussed: intra- and inter-granular hierarchical multi-laminate plasticity for slip [FRA 07], grain-size dependent overall HML polycrystal plasticity for slip and twinning [FRA 08]. Although some difficulties may remain hidden in the modeling details, these HML sub-boundary-based approaches of the evolving morphology of polycrystalline aggregates under plastic straining, here coupled with the TFA framework, seem an alternative of interest to the widely spread and used, but limited, inclusion-based modeling.

1.6. Bibliography

[COU 05] Couzinié J.P., Decamps B., Priester L. Interaction of dissociated lattice dislocation withaΣ3grainboundaryincopper. Int. J. Plasticity, vol. 21, 4, p. 759–775, 2005.

[CUB 07] Cubicza J., Dirras G., Szommer P., Bacroix B. Microstructure and yield strength of ultrafine grained aluminium processed by hot isostatic pressing. Mat. Sci. Engng. A, vol. 458, 1/2, p. 385–390, 2007.

[DEV 06] Devincre B., Kubin L., Hoct T. Physical analyses of crystal plasticity by DD simulations. Scripta Materialia, vol. 54, 5, p. 741–746, 2006.

[DVO 92] Dvorak, G.J. Transformation Field Analysis in inelastic composite materials. Proc. R. Soc. London A, vol. 437, p. 311–327, 1992.

[ELO 00] El Omri A., Fennan A., Sidoroff F., Hihi A. Elastic-plastic homogenization for layered composites. Eur. J. Mech. A/Solids, vol. 19, p. 585–601, 2000.

[FRA 91] Franciosi P., Zaoui A. Crystal Hardening and the Issue of Uniqueness. Int. J. Plasticity, vol. 7, p. 295–311, 1991.

[FRA 04a] Franciosi P., Lebail H. Anisotropy features of phase and particle spatial pair distributions, in various matrix/inclusions structures. Acta Materialia, vol. 52, 10, p. 3161–3172, 2004.

[FRA 04b] Franciosi P., Lormand G. Using the Radon transform to solve inclusion problems in elasticity. Int. J. Solids and Structures, vol. 41, 3/4, p. 585–606, 2004.

[FRA 07] Franciosi P., Berbenni S. Heterogeneous crystal and polycrystal plasticity modeling from transformation phase analysis within a regularized Schmid flow law. J. Mech. Phys. Solids, vol. 55, 11, p. 2265–2299, 2007.

[FRA 08] Franciosi P., Berbenni S. Multi-laminate plastic strain organization for non-uniform TFA modelling of polycrystal regularized plastic flow. Int. J. Plasticity, vol. 24, p. 1549-1580, 2008.

[HAD 06] Haddadi H., Bouvier S., Banu M., Maier C., Teodosiu C. Towards an accurate description of the anisotropic behaviour of sheet metals under large plastic deformations. Int. J. Plasticity, vol. 22, 12, p. 2226–2271, 2006.

[KUL 02] Kulhmann-Wilsdorf D. Why do dislocations assemble into interfaces in Epitaxy as well as in Crystal Plasticity? To minimize free energy. Metall. and Mat. Trans. A, vol. 33, 8/1, p. 2519–2539, 2002.

[MIC 03] Michel J.C., Suquet P. Non uniform Transformation Field Analysis. Int. J. Solids and Structures, vol. 40, p. 6937–6955, 2003.

[MIL 88] Milton G.W. Classical Hall effect in two-dimensional composites: a characterization of the set of realizable effective conductivity tensors. Phys. Rev. B, vol. 38, 16, p. 11296–11303, 1988.

[ORT 99] Ortiz M., Repetto E.A. Non-convex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids, vol. 47, p. 397–462, 1999.

[PON 98] Ponte-Castaneda P., Suquet P. Non-linear composites. Adv. Appl. Mechanics, vol. 34, p. 172–302, 1998.

[SCH 05] Schuh C.A., Kumar M., King W.E. Universal features of grain boundary networks in FCC materials. J. Materials Science, vol. 40, 4, p. 847–852, 2005.

[TAL 97] Talbot D.R.S., Willis J.R. Bounds of third order for the overall response of non linear composites. J. Mech. Phys. Solids, vol. 45, 1, p. 87–111, 1997.

1 Chapter written by Patrick FRANCIOSI.

1 For “the strain modified Green operator integral related to the infinite medium (m)”.

2 For the sake of simplicity, it is easier to only consider laminates parallel to shear planes, whose density or probability can be more easily linked to the in-plane shear activity.

3

4 In equation [1.7] we have set

Chapter 2

Discrete Dislocation Dynamics: Principles and Recent Applications1

2.1. Discrete Dislocation Dynamics as a link in multiscale modeling

In crystalline materials, a dislocation is a line defect which represents permanent deviations of atoms from their original crystallographic periodicity. Dislocation motion in the slip system of the crystal gives rise to macroscopic plastic deformation. A dislocation is thus a microscopic carrier of crystal plasticity.

Modeling the plasticity of crystalline materials involves understanding the nature of dislocations, which is defined at the atomistic scale and also evaluating the deformation behaviors at the macroscopic scale. Many models have been developed to understand the plasticity of metals. Since the features of plasticity vary a lot in size and time, the models also vary widely in length and time scales as depicted in Figure 2.1. Out of a range of models, most attention is given in this chapter to Molecular Dynamics (MD), Dislocation Dynamics (DD) and Continuum Mechanics (CM) with a special emphasis on DD simulations.

Atoms are the basic constituent elements of MD simulations. Atoms interact with each other through interatomic potentials. The temporal trajectory of a set of atoms under an external loading is simulated by minimizing the total potential energy of the system. The deviations of the position of the atoms from the lattice sites implicitly represent the dislocations. The atomistic scale topology of a dislocation line can thus be investigated by MD. MD simulations are employed mostly in studying physical properties of a single or a few dislocation lines due to the constraints of the simulation size (<(200 nm)3).

In DD methods, dislocation lines are represented explicitly. Each dislocation line is treated as an elastic inclusion embedded in an elastic medium. The collective evolution of a large number of interacting dislocations under an external loading is simulated using elastic properties of the crystal. Core properties of dislocations such as line mobility, junction strength etc., are input parameters of DD simulations that can be derived from MD simulations. DD simulations give access to the dislocation patterning but also to the mechanical response of the simulated volume (<(50µm)3).

Figure 2.1.Typical volume size and physical time covered by three models devoted to crystal plasticity

At a higher scale, CM treats the behavior of a continuum medium using a set of equations and boundary conditions. There is a wide range of numerical techniques which can solve the equations. Finite difference and finite element methods are two broad subsets of such techniques. In these methods, a continuum domain of interest is subdivided into discrete cells or elements, in which the values of certain physical quantities are determined by solving a system of equations. Ideally, CM should use a set of constitutive equations that accurately account for the physics at the origin of the plasticity which, in the case of crystalline materials is closely related to the dislocation dynamics. Thus, DD simulations can obviously feed CM models by calculating the constitutive equations. In CM, the maximum size of the simulated volume is not limited but instead imposed by the space resolution associated with the problem treated. However, each simulation cell must be big enough to be representative of a continuum medium in agreement with the constitutive equations.

As introduced briefly above, MD, DD and CM have their own characteristic length and time scale. Figure 2.1 shows such ranges of length and time scales for each method. As the performance of each numerical method is improved, the volume and the physical time which can be simulated increase (top and right domain limits of each method in Figure 2.1). Recently the length and time scales of the three methods begin to overlap. This gives a great impetus to exchange information between the different models in order to build up a unified description of crystal plasticity, which would ideally be able to predict the behavior of a material from the fundamental properties of the atoms.

2.2. Principle of Discrete Dislocation Dynamics

The concept of 3D discrete dislocation simulations was imagined by L. Kubin, Y. Bréchet and G. Canova in the early 1990s [KUB 92], [DEV 92]. The first code Micromégas was a simple model for which dislocation lines of a f.c.c. single crystal are sub-divided into sets of edge and screw dislocation segments embedded in a continuum medium as pictured in Figure 2.2a.

Figure 2.2.(a) Edge-screw discretization of dislocation lines; (b) internal stresses induced by edge and screw segments

Each dislocation segment generates a long range elastic stress field within the entire simulated sample. In the case of isotropic elasticity, analytical expressions for the internal stress generated by a finite segment have been established by J.C.M. Li [LI 64] and R. DeWit [DeWIT 67]. Taking into account the anisotropy of the elastic medium is more challenging in term of calculation time since there is no explicit relationship. We can however use the line integrals proposed by J. Willis [WIL 70].

Using linear elasticity properties, the effective stress applied on each dislocation is evaluated at the middle point of each segment as the superposition of the internal stress induced by all the segments in the simulated volume and the applied stress imposed by the loading. This induces a force given by the Peach-Koehler equation

[2.1]

where ξ is the unit vector of the line direction and b the Burgers vector of the dislocation segment. In DD simulations, most of the CPU time is devoted to evaluating equation [2.1] since this implies N2 computations of the internal stress tensors, and the number of segments, N, continuously increase with time. Thus, many efforts have been carried out in optimizing the calculation of the internal stresses. As an example, Bulatov et al. recently proposed a fast multipole decomposition leading to N operations [CAI 06]. These optimizations can still be improved by using parallel calculation which is now a common option in most DD codes [SCH 99] [SHI 06] [ARS 07].

Once the force is known on each dislocation segment, the segments are displaced according to mobility functions which depend on the material properties. In the case of FCC structure, the velocity is a linear function of the glide component of the force whereas in the case of materials with large Peierls valleys, like BCC for example, the velocity of screw segments is given by a thermally activated relationship using a Boltzman relation [TAN 98] [CHA 06].

Typical outputs of DD simulations are obviously the dislocation microstructure but also much statistical data such as the dislocation densities, the cumulated shear strain, the stored energy, the local stresses, etc. Recently, methods have been developed in order to calculate the actual shape of any part of the crystal deformed by the dislocations as illustrated in Figure 2.3.

Figure 2.3.Example of calculation of the actual shape of a crystal deformed by dislocation activity. Case of a copper single crystal loaded in uniaxial tension along (112) axis

From some points of view, DD simulations can be seen as an ideal tool to fill the gap between atomic simulations (MD) and continuum modeling. These properties explain the large dissemination of this modeling during the past few years. Today, there are more than a dozen discrete dislocation dynamics codes throughout the world [ZBI 98] [SCH 99] [GHO 00] [WEY 02] [BUL 06] dedicated to different crystallographic structures but all based on similar ingredients. The studies presented hereafter have all been performed using the code <inlinecode>Tridis developed in Grenoble, France [VER 98].

2.3. Example of scale transition: from DD to Continuum Mechanics

2.3.1. Introduction to a dislocation density model

The development of constitutive equations for crystal plasticity is still challenging today. The objective is always to derive a set of behavior laws including most of the physics involved during plastic deformation. In the case of crystalline materials, this involves accounting for dislocation properties at the continuum scale.

In this section, we present a crystal plasticity model for which the dislocation densities on the different slip systems are the internal variables. Three equations are needed to relate the stress to the plastic strain. We should note that each law is derived from physical considerations of dislocation motions.

2.3.1.1. Constitutive equations of a dislocation based model of crystal plasticity

Plastic behaviors of crystals have been studied since the early 1960s [TEO 75]. Let us consider the motion of a single dislocation gliding on slip system s and embedded in a heterogenous stress field. Assuming that the dislocation motion is governed by the obstacles (as usually admitted for FCC structures), these obstacles can be classified in two categories:

(i) obstacles inducing long-range stresses τμ like for example dislocations stored at grain boundaries or around precipitates;

(ii) obstacles inducing a short-range stress field, written as τ*, like in the case of forest dislocations or impurities.

τμ does not depend on temperature whereas τ* is thermally activated. Following this distinction, we can define a typical spatial evolution for the stress component as plotted in Figure 2.4.

Figure 2.4.Decomposition of the stress fields felt by a moving dislocation at a given temperature

When considering only the isotropic hardening, the spatial evolution of the athermal stress τμ is similar to a periodic function with a zero average and a large wavelength. When a dislocation meets an obstacle which acts within few atomic distances to the dislocation position, it needs an additional stress τ* to pass the obstacle. For each slip system s, the resolved shear stress needed for the dislocation motion is then

[2.2]

When the time for the dislocation flight is negligible in comparison to the waiting time in front of the obstacle, we can write the following expression for the dislocation velocity [TEO 76]:

[2.3]

where vD is the Debye frequency, b the Burgers vector magnitude, ΔG0 and ΔV* the activation energy and activation volume respectively.

Averaging the velocities out of the whole mobile dislocation density ρm of a given slip system is achieved using the Orowan equation:

[2.4]

When the effective resolved shear stress is moderate (less than 70% of the value at 0 K, we can neglect the inverse probability so that the sinh in equation [2.4] is replaced by the negative exponential part. Replacing , the first order approximation in terms of finally gives [RAU 93]:

[2.5]

Equation [2.5] is the flow law written under the classic form of a power relationship between strain rate and normalized stress.

Let us briefly recall the assumptions needed to establish this equation. Equation [2.5] is valid for moderate values of . Moreover, the first order approximation implies that . These conditions are fulfilled for most FCC metals within a range of temperatures lower than 0.3 Tf, (so-called cold regime of deformation). Parameters and n have been established from an average performed over all the mobile dislocations within a given slip system so that the three macroscopic variables , ΔG0 and ΔV* are now pseudo-phenomenological constants although the underlying physics is still there. In other words, this means that we need to identify the value of these parameters.

Strain hardening is defined by the increase of the athermal stress with the internal variables, i.e. the dislocation densities. Such a relationship is introduced through the Mecking and Kocks equation [MEC 81]:

[2.6]

where ρF is the forest dislocation density and α a coefficient close to 0.3. This equation can be split over the 12 slip systems of the FCC crystal using a matrix a which corresponds to the slip system interactions.

[2.7]

The closed form of the system of constitutive equations is obtained by the writing the evolution of the dislocation density with the deformation. The relationship can be explained from statistical considerations of dislocation production and annihilation [MEC 81] [ESS 79].

[2.8]

This equation involves a storage term with a mean free path of the form Λ=K√Vρ and an annihilation term based on the annihilation distance yc=βR. Equation [2.7] implies that dislocation densities will saturate when the production term equals the annihilation term leading to a saturation of the isotropic hardening.

Finally, the crystal plasticity model is defined by the set of three equations [2.5], [2.7] and [2.8]. They can be rewritten in a classical form by calculating the derivative of [2.7] and introducing given by [2.8]. This gives:

[2.9]

Note that these constitutive equations are only valid for monotonic loading since they do not account for any kinematic hardening.

2.3.1.2. Parameter identification

The DD model presented in section 2.2 is well adapted to identify the coefficients involved in this dislocation density-based model of crystal plasticity. As an example, any asu coefficient of the hardening law can be determined by simulating the interaction of slip systems (s) and (u) and measuring the shear stress applied on system (s) needed to force it to cross the population of dislocation on forest system (u) of density ρ(u) [FIV 97]. Following this idea, we can identify the 12x12 coefficients of matrix a (which is restricted to 5 independent values depending on the 3D geometry of the system interaction a1copla, a1ortho, a1coli, a2, a3). Recently, Devincre et al. [DEV 06] found the following values for copper single crystals: a1coli=0.625 ±0.044; a3=0.122 ±0.012; a2=0.137 ±0.014; a1copla ≈ a1ortho=0.0454 ±0.003.

Similarly, it is possible to measure the values of coefficients K and yc from DD curves giving the evolution of the dislocation densities on the different slip systems with the plastic deformation [FIV 98] [TAB 98].

For copper, we find a typical value K=32 for the mean free path and yc=3.b for the annihilation distance.

2.3.1.3. Application to copper simulations

The constitutive equations are easily introduced in a finite element code such as ABAQUS using the User MATerial routines. Using the parameters identified by DD simulations, we can now simulate the visco-plastic behavior of any single crystal under a given loading. Note that polycrystalline materials can also be simulated provided the mean free path involved in equation [2.8] is modified in order to account for the distance between the integration Gauss point and the grain boundary.

Figure 2.5 shows the simulation of a copper single crystal loaded in tension along the direction.

Figure 2.5.Simulation of a tensile test along

Initially, the Schmid factor is the highest on system . This corresponds to stage I where the hardening rate is weak since there is only a single slip system activated. If rotations are allowed in the grips as shown in the deformed mesh plotted in Figure 2.5, the Schmid factor changes with the plastic deformation cumulated in system B4. This activates a secondary slip system . Dislocation activity in system B4 is then modified by the forest dislocations of system C1 so that the hardening rate is more pronounced. This is the stage II regime. Finally, as soon as the two dislocation densities are closer to the saturation value, the stress tends to saturate so that the hardening rate decreases in the so-called stage III.

2.3.1.4. Taking into account kinematic hardening

Fatigue simulations performed in DD (see section 2.4) revealed the effect of dipole interactions on the mechanical response of a single crystal of copper submitted to a cyclic loading. This gave information on the intra-granular hardening and more precisely on the kinematic part of the hardening, i.e. the hardening stress which can be recovered when the loading is reversed. This section presents a model that can reproduce most of the experimental features of fatigue [DEP 08]. The model is based on dipole interactions and can nicely complete the set of constitutive equations presented in section 2.3.1. It can be shown from dislocation theory [HIR 82], [FRI 64] that a dipole of height h has a strength s written as

[2.10]

The distance h is easily estimated from DD simulations. Figure 2.6a shows a typical dislocation microstructure within a grain cyclically loaded in pure shear with a plastic strain amplitude Δγp=3.10−3. The distribution of the dipole heights is reported in Figure 2.6b.

Figure 2.6.Thin foil taken out from a DD fatigue simulation and distribution of dipole heights

Assuming that the height distribution follows a Gaussian function, experiments show that the mean value evolves as [CAT 05] so that the distance decreases when the dislocation density increases. The corresponding dipole strength can now be calculated using equation [2.10] as shown in Figure 2.7 for two values of the dislocation density in the DD simulation.

Figure 2.7.Distribution of the dipole height and corresponding dipole strength calculated for two values of the dislocation density

For the sake of simplicity, we will assume that the strength distribution also fits a Gaussian function f(s) defined by the average value and the standard deviation σs:

[2.11]

As shown in equation [2.11], the average value is supposed to be proportional to the dislocation density via a classical relationship proposed by Mughrabi [MUG 75]. Moreover, experiments show that the standard deviation is proportional to the average value using a constant coefficient λ [CAT 05]. This behavior was recently confirmed by DEPRES using DD simulations [DEP 04].

Locally, the stress may destabilize the weakest dipoles which can then contribute to the shear strain rate as:

[2.12]

where X is a long range internal stress induced by the strain gradients. The displacements induced by the dislocation on the grid located in Figure 2.6 point out these gradients. The corresponding deformed mesh is plotted in Figure 2.8. The three marks indicate that extra dislocations are located between marks 1 and 2 and also between marks 2 and 3.

Figure 2.8.Evidence of extra dislocations inducing strain gradients

These excess dislocations (often called geometrically necessary dislocations) are all pinned on dipoles. Thus, they induce a back stress on the source from which they have been emitted. This kinematic back stress can be roughly estimated as the stress induced by an equivalent dislocation loop of radius with a net Burgers vector Nb. The number of loops N is directly related to the difference between the local deformation at the dipole location and the macroscopic strain γ: N=(2R/b)(γ−γs). Thus, the back stress, X is given as where A is a geometric parameter depending on the actual geometry of the grain and the dislocation loops. Replacing N by its dependence over the shear strain, we finally obtain:

[2.13]

DD simulation shows that the geometric coefficient M is close to a constant M≈2. Equation [2.11] can now be introduced in equation [2.12] in order to define a new flaw law accounting for kinematic hardening.

The constitutive model based on equation [2.11], [2.12], [2.13] can be completed by dislocation evolution law [2.8] so that it can be used to derive the mechanical response of a grain submitted to a cyclic loading. A typical response is given in Figure 2.9 together with the experimental results of Mughrabi [MUG 78]. Numerical results were obtained with M=2 and λ=0.8.

Figure 2.9.(a) Predicted hysteresis loops for a given imposed plastic strain amplitudeΔγp=3.10−3 and loop shape obtained for different imposed strain amplitude; (b) correspondingexperiments published by Mughrabi [MUG 78]

The proposed model is in very good agreement with the experimental measurements of Mughrabi, both for the evolution of the hysteresis at a given imposed stain amplitude and also for the shape of the cycle reached at saturation for different strain amplitudes. To conclude this part on the kinematic hardening, let us recall that the proposed model only adds 2 constants M and λ to the original isotropic hardening model. Here, a simple scalar model is presented but it can easily be written in a tensor manner and implemented in finite elements to treat any complex path loading.

2.4. Example of DD analysis: simulations of crack initiation in fatigue

2.4.1. Case of single phase AISI 316L