IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials: Proceedings of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials: Proceedings of the IUTAM Symposium on Variational Concepts

IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials IUTAM BOOKSERIES Volume 21 Se...

Author: Klaus Hackl

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IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials

IUTAM BOOKSERIES Volume 21 Series Editors G.M.L. Gladwell, University of Waterloo, Waterloo, Ontario, Canada R. Moreau, INPG, Grenoble, France Editorial Board J. Engelbrecht, Institute of Cybernetics, Tallinn, Estonia L.B. Freund, Brown University, Providence, USA A. Kluwick, Technische Universität, Vienna, Austria H.K. Moffatt, University of Cambridge, Cambridge, UK N. Olhoff Aalborg University, Aalborg, Denmark K. Tsutomu, IIDS, Tokyo, Japan D. van Campen, Technical University Eindhoven, Eindhoven, The Netherlands Z. Zheng, Chinese Academy of Sciences, Beijing, China

Aims and Scope of the Series The IUTAM Bookseries publishes the proceedings of IUTAM symposia under the auspices of the IUTAM Board.

For other titles published in this series, go to www.springer.com/series/7695

Klaus Hackl Editor

IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials Proceedings of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, Bochum, Germany, September 22–26, 2008

Editor Klaus Hackl Ruhr University Bochum Institute of Mechanics Bochum, Germany [email protected]

e-ISSN 1875-3493 ISSN 1875-3507 ISBN 978-90-481-9194-9 e-ISBN 978-90-481-9195-6 DOI 10.1007/978-90-481-9195-6 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2010927406 © Springer Science+Business Media B.V. 2010 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Table of Contents

Preface

vii

List of Committee Members and Sponsors

ix

List of Participants

xi

Stability of Quasi-Static Crack Evolution through Dimensional Reduction J.-F. Babadjian

1

FE2 -Simulation of Microheterogeneous Steels Based on Statistically Similar RVEs D. Balzani, J. Schröder and D. Brands Advancements in the Computational Calculus of Variations C. Carstensen and C. Manigrasso

15

29

A Phase Field Approach to Wetting and Contact Angle Hysteresis Phenomena A. DeSimone, L. Fedeli and A. Turco

51

Application of Relaxation Methods in Materials Science: From the Macroscopic Response of Elastomers to Crystal Plasticity G. Dolzmann

65

Variational Concepts with Applications to Microstructural Evolution F.D. Fischer, J. Svoboda and K. Hackl A Micromechanical Model for Polycrystalline Shape Memory Alloys – Formulation and Numerical Validation R. Heinen and K. Hackl

79

91

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Table of Contents

Solution-Precipitation Creep – Modeling and Extended FE Implementation S. Ilic and K. Hackl

105

Time-Continuous Evolution of Microstructures in Finite Plasticity D.M. Kochmann and K. Hackl

117

Models for Dynamic Fracture Based on Griffith’s Criterion C.J. Larsen

131

An Energetic Approach to Deformation Twinning K.C. Le and D.M. Kochmann

141

Computational Homogenization of Confined Frictional Granular Matter H.A. Meier, P. Steinmann and E. Kuhl

157

Existence Theory for Finite-Strain Crystal Plasticity with Gradient Regularization A. Mielke

171

Error Bounds for Space-Time Discretizations of a 3D Model for Shape-Memory Materials A. Mielke, L. Paoli, A. Petrov and U. Stefanelli

185

On the Implementation of Variational Constitutive Updates at Finite Strains J. Mosler and O.T. Bruhns

199

Phase-Field Modeling of Nonlinear Material Behavior Y.-P. Pellegrini, C. Denoual and L. Truskinovsky

209

Polyconvex Energies for Trigonal, Tetragonal and Cubic Symmetry Groups J. Schröder, P. Neff and V. Ebbing

221

Phase Transitions with Interfacial Energy: Interface Null Lagrangians, Polyconvexity, and Existence M. Šilhavý

233

A Unified Variational Setting and Algorithmic Framework for Monoand Polycrystalline Martensitic Phase Transformations E. Stein and G. Sagar

245

Dissipative Systems in Contact with a Heat Bath: Application to Andrade Creep F. Theil, T. Sullivan, M. Koslovski and M. Ortiz

261

Preface

Variational calculus has been the basis of a variety of powerful methods in the field of mechanics of materials for a long time. Examples range from numerical schemes like the finite element method to the determination of effective material properties via homogenization and multiscale approaches. In recent years, however, a broad range of novel applications of variational concepts has been developed. This comprises the modeling of the evolution of internal variables in inelastic materials as well as the initiation and development of material patterns and microstructures. The IUTAM Symposium on “Variational Concepts with Applications to the Mechanics of Materials” took place at the Ruhr-University of Bochum, Germany, on September 22–26, 2008. The symposium was attended by 55 delegates from 10 countries. Altogether 31 lectures were presented. The objective of the symposium was to give an overview of the new developments sketched above, to bring together leading experts in these fields, and to provide a forum for discussing recent advances and identifying open problems to work on in the future. The symposium focused on the development of new material models as well as the advancement of the corresponding computational techniques. Specific emphasis is put on the treatment of materials possessing an inherent microstructure and thus exhibiting a behavior which fundamentally involves multiple scales. Among the topics addressed at the symposium were: 1. Energy-based modeling of material microstructures via envelopes of nonquasiconvex potentials and applications to plastic behavior and phasetransformations. 2. Modeling of the evolution of material microstructures in time and the associated thermodynamics using suitable variational principles. 3. Micromechanical modeling of shape-memory alloys and the evolution of martensitic microstructures. 4. Variational multiscale methods and associated numerical procedures. 5. Micromechanics of multifield and multiphysics problems.

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Preface

A variety of novel results and methods were presented, especially in the realm of applying energy-based concepts to complex materials and structures. It was shown that the topic of the symposium is undergoing a rapid development boosted by the advancement of mathematical techniques in conjunction with new experimental findings concerning material microstructures. The contributions in this volume have undergone a full peer-review. I would like to thank the reviewers involved for their valuable comments which certainly led to improvements in the papers contained in this book. Finally, thanks are due to Jolanda Karada of Karada Publishing Services for her professional preparation of the volume prior to its submission to Springer. It is my hope that it will give the reader an insight into the exciting new developments of the field of variational methods and contribute to its popularity within the mechanics and physics communities. Klaus Hackl Bochum, November 2009

List of Committee Members and Sponsors

Scientific Committee K. Hackl, Bochum (Chair) J. Ball, Oxford K. Bhattacharya, Pasadena P. Ponte Castaneda, Palaiseau Cedex S. Conti, Duisburg G. Francfort, Paris M. Ortiz, Pasadena O. Sigmund, Lyngby M.P. Bendsøe (IUTAM representative) Sponsors IUTAM (International Union of Theoretical and Applied Mechanics) Deutsche Forschungsgemeinschaft (DFG) through research group FOR 797 “Analysis and Computation of Microstructures in Finite Plasticity” Springer Verlag Ruhr-Universität Bochum

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Babadjian, Jean-François CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France [email protected] Balzani, Daniel Universität Duisburg-Essen, Fakultät für Ingenieurwissenschaften – Abteilung Bauwissenschaften, Institut für Mechanik, Universitätsstr. 15, 45117 Essen, Germany [email protected] Bartel, Thorsten Universität Dortmund, Fakultät Maschinenbau, Leonard-Euler-Str. 5, 44227 Dortmund, Germany [email protected] Bourdin, Blaise Department of Mathematics, Lousiana State University, 344, Locket Hall, Baton Rouge, LA 70803-4918, USA [email protected] Carstensen, Carsten Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, 10099 Berlin, Germany [email protected] Cherkaev, Andrej University of Utah, Department of Mathematics, Salt Lake City, UT 84112, USA [email protected]

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Conti, Sergio Universität Bonn, Institut für Angewandte Mathematik, Abteilung Angewandte Analysis, Beringstr. 6, D-53115 Bonn, Germany [email protected] Dacorogna, Bernard Ecole Polytechnique Fédérale de Lausanne, Abteilung für Mathematik, Station 8, 1015 Lausanne, Switzerland [email protected] DeSimone, Antonio SISSA – International School for Advanced, Studies, via Beirut 2–4, 34014 Trieste, Italy [email protected] deBotton, Gal Ben-Gurion University, Department of Mechanical Engineering, P.O. Box 653, Beer-Sheva 84105, Israel [email protected] Dmitrieva, Olga Max-Planck-Institute for Iron Research, Max-Planck-Str. 1, 40237 Düsseldorf, Germany [email protected] Dimitrijevic, Bojan Ruhr University of Bochum, Institute of Mechanics, 44780 Bochum, Germany [email protected] Dolzmann, Georg Universität Regensburg, NwFI-Mathematik, 93040 Regensburg, Germany [email protected] Ebbing, Vera Universität Duisburg-Essen, Fakultät für Ingenieurwissenschaften – Abteilung Bauwissenschaften, Institut für Mechanik, Universitätsstr. 15, 45117 Essen, Germany [email protected] Fischer, Franz Dieter Montanuniversität Leoben, Institut für Mechanik, Franz-Josef-Str. 18, 8700 Leoben, Austria [email protected]


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Francfort, Gilles L.P.M.T.M., Université Paris-Nord, Avenue J.-B. Clément, 93430 Villetaneuse, France [email protected] Garroni, Adriana Università di Roma ‘La Sapienza’, Dipartimento di Matematica ‘G. Castelnuovo’, Piazzale A. Moro, 2, 00185 Roma, Italy [email protected] Hackl, Klaus Ruhr University of Bochum, Institute of Mechanics, 44780 Bochum, Germany [email protected] Heinen, Rainer Werkstoffkompetenzzentrum, Division Auto, ThyssenKrupp Steel AG, Kaiser-Wilhelm-Str. 100, D-47166 Duisburg, Germany [email protected] Heinz, Sebastian Weierstrass-Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, 10117 Berlin, Germany [email protected] Homayonifar, Malek GKSS Research Centre, Institute of Materials Research, Mechanics of Materials (WMS), Max-Planck-Str.1, D-21502 Geesthacht, Germany [email protected] Hoppe, Ulrich Ruhr University of Bochum, Institute of Mechanics, 44780 Bochum, Germany [email protected] Ilic, Sandra Ruhr University of Bochum, Institute of Mechanics, 44780 Bochum, Germany [email protected] Junker, Philipp Ruhr University of Bochum, Institute of Mechanics, 44780 Bochum, Germany [email protected] Kintzel, Olaf GKSS Research Centre, Institute of Materials Research, Mechanics of Materials (WMS), Max-Planck-Str.1, D-21502 Geesthacht, Germany [email protected]

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Kochmann, Dennis M. Ruhr University of Bochum, Institute of Mechanics, 44780 Bochum, Germany [email protected] Krieger, Stefan Ruhr University of Bochum, Institute of Mechanics, 44780 Bochum, Germany [email protected] Langhoff, Tom-Alexander Institut für Technische Mechanik, Kontinuumsmechanik im Maschinenbau, Universität Karlsruhe (TH), Postfach 6980, 76128 Karlsruhe, Germany [email protected] Larsen, Christopher J. Worcester Polytechnic Institute, Department of Mathematical Sciences, 100 Institute Road, Worcester, A 01609, USA [email protected] Le, Khanh Chau Ruhr University of Bochum, Institute of Mechanics, 44780 Bochum, Germany [email protected] Löbach, Dominique Universität Bonn, Institut für Angewandte Mathematik, Abteilung Angewandte Analysis, Beringstr. 6, D-53115 Bonn, Germany [email protected] Makowska, Rasa RWTH Aachen University, Department of Continuum Mechanics, Eilfschornsteinstr. 18, 52062 Aachen, Germany [email protected] Makowski, Jerzy Ruhr University of Bochum, Institute of Mechanics, 44780 Bochum, Germany [email protected] Meier, Holger Technische Universität Kaiserslautern, Lehrstuhl für Technische Mechanik, Postfach 3049, D-67653 Kaiserslautern, Germany [email protected] Miehe, Christian Universität Stuttgart, Institut für Mechanik im Bauwesen, Lehrstuhl I, Pfaffenwaldring 7, 70550 Stuttgart, Germany [email protected]


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Mielke, Alexander Weierstrass-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany [email protected] Milton, Graeme W. University of Utah, Department of Mathematics, Salt Lake City, UT 84112, USA [email protected] Mosler, Jörn GKSS Forschungszentrum, Max-Planck-Straße 1, 21502 Geesthacht, Germany [email protected] Neff, Patrizio Nichtlineare Analysis und Modellierung, Fachbereich Mathematik, Universität Duisburg-Essen, Universitätsstr. 2, 45141 Essen, Germany [email protected] Nguyen, Quoc Son Laboratoire de Mecanique des Solides, École Polytechnique, 91128 Palaiseau, France [email protected] Ortiz, Michael California Institute of Technology, Graduate Aeronautical Laboratories, 1200 E. California Blvd, Mail Stop: 105-50, Pasadena, CA 91125, USA [email protected] Pellegrini, Yves-Patrick Département de Physique Théorique et Appliquée, CEA-DAM Ile-de-France, F-91927 Arpajon Cedex, France [email protected] Petrov, Adrien Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin, Germany [email protected] Racherla, Vikranth Laboratoire de Mécanique des Solides, École Polytechnique, 91128 Palaiseau, France [email protected]

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Schröder, Jörg Universität Duisburg-Essen, Fakultät für Ingenieurwissenschaften – Abteilung Bauwissenschaften, Institut für Mechanik, Universitätsstr. 15, 45117 Essen, Germany [email protected] Schütte, Henning Ruhr University of Bochum, Institute of Mechanics, 44780 Bochum, Germany [email protected] Sembiring, Pramio Ruhr University of Bochum, Institute of Mechanics, 44780 Bochum, Germany [email protected] Sengul, Yasemin University of Oxford, Mathematical Institute, 24-29 St.Giles’, OX1 3LB Oxford, United Kingdom [email protected] Silhavy, Miroslav Academy of Sciences of the Czech Republic, Institute of Mathematics, Zitna 25, CZ-115 67 Praha 1, Czech Republic [email protected] Stein, Erwin Leibniz Universität Hannover, Institut für Baumechanik und Numerische Mechanik, Appelstr. 9a, 30167 Hannover, Germany [email protected] Steinmann, Paul Universität Erlangen-Nürnberg, Lehrstuhl für Technische Mechanik, Egerlandstraße 5, 91058 Erlangen, Germany [email protected] Theil, Florian University of Warwick, Mathematic Institute, Zeeman Building, Coventry CV 7AL, United Kingdom [email protected] Trinh, Tuyet Ruhr University of Bochum, Institute of Mechanics, 44780 Bochum, Germany [email protected]


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Turteltaub, Sergio R. Delft University of Technology, Department of Aerospace Engineering, Kluyverweg 6, 2629 HT Delft, The Netherlands [email protected] Weinberg, Kerstin Universität Siegen, Institut für Mechanik und Regelungstechnik, Lehrstuhl für Festkörpermechanik, Paul-Bomatz-Str. 9-11, 57076 Siegen, Germany [email protected]

Stability of Quasi-Static Crack Evolution through Dimensional Reduction Jean-François Babadjian

Abstract This paper deals with quasi-static crack growth in thin films. We show that, when the thickness of the film tends to zero, any three-dimensional quasi-static crack evolution converges to a two-dimensional one, in a sense related to the convergence of the associated total energy. We extend the prior analysis of [2] by adding conservative body and surface forces which allow us to remove the boundedness assumption on the deformation field.

1 Introduction In this paper, we study the evolution of cracks in thin structures in a quasi-static setting. Our approach of fracture mechanics is based on a variational model proposed in [10] (see also the monograph [4]) where the (quasi-static) evolution results from the competition – at each time – between a bulk and a surface energy, under a growth constraint on the crack. Many existence results have been obtained (see e.g. [8, 9, 11] and references therein). Sometimes a small parameter is involved in the model, and it is an interesting question to study the asymptotic behavior of the model when the parameter tends to zero (see e.g. [14] for the homogenization and [2] for the dimension reduction of quasi-static crack evolution). When dealing static problems, the notion of -convergence (see [7]) has proven to be a powerful tool to capture the asymptotic behavior of minimizers, or even minimizing sequences. It turns out that even in the quasi-static case, one can define a notion of convergence related to the convergence of the associated total energy (see [16] for an abstract theory in the more general framework of rate independent processes).

Jean-François Babadjian Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France; E-mail: [email protected]

K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_1, © Springer Science+Business Media B.V. 2010

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J.-F. Babadjian

We present here an extension of the result in [2] on the convergence of a quasistatic crack evolution in thin films, as the thickness tends to zero. In [2], an empirical L∞ bound was done on the deformation field (as in [9]) in order to gain compactness in the space SBV p of special functions of bounded variation. It is sometimes possible to justify this assumption as in the antiplanar case (see [11] when N = 2) where it follows from a consequence of the maximum principle. Unfortunately, in the full three-dimensional elasticity, the maximum principle does not hold anymore. We propose here to remove this hypothesis adding suitable conservative bulk and surface forces as in [8]. The price to pay is that the deformation field in not anymore p compact in SBV p but in a larger subspace GSBVq of generalized special functions of bounded variation. The arguments we use in the present paper are very close to those of [2], and for this reason we will only mention the main differences without giving the precise proofs of the results. The paper is organized as follows: in Section 2, we will describe the model in the physical configuration and state the existence result of [8]. Then, in Section 3, we will reformulate the problem on a rescaled configuration in order to work on a fixed domain. In Section 4, we will perform an asymptotic analysis of the total energy of the system in a static setting, thanks to a -convergence method. Finally, we will address the asymptotic of the quasi-static problem in Section 5, proving that it converges to a quasi-static evolution associated to the -limit model.

2 Description of the Model We consider a homogeneous thin film occupying in its reference configuration the cylinder ε := ω × (−ε, ε), where ε > 0 and ω is a bounded open subset of R2 with Lipschitz boundary. The Dirichlet part of the boundary where the deformation is prescribed is the lateral boundary ∂D ε := ∂ω × (−ε, ε), while the Neumann part ∂N ε = ω × {−ε, ε} is made of the lower and upper sections. On the lateral boundary ∂D ε , we impose a time dependent boundary deformation φ ε (t) on a finite time interval [0, T ], where t → φ ε (t) ∈ W 1,1 ([0, T ]; W 1,p (ε ; R3 ) ∩ Lq (ε ; R3 )), for some p > 1 and q ≥ 1. On the remaining part of the boundary ∂N ε , we impose a time dependent surface conservative force which will be described in Section 2.3.2.

2.1 Admissible Cracks We fix an open subset εB of ε of the form εB := ω × (−ε + εη, ε − εη) for some η ∈ (0, 1), to that the set εB represents the brittle part of the body. The set of all

Stability of Quasi-Static Crack Evolution through Dimensional Reduction

3

admissible cracks is given by εB and H 2 (K) < +∞}. R(εB ) := {K : K is rectifiable, K ⊂ Note that any admissible crack must lie far enough from the upper and lower sections. The safety region ε \εB can be interpreted as a layer of unbreakable material (see [8, remark 3.8]). We denote by H N−1 the (N − 1)-dimensional Hausdorff measure in RN (we (resp. ∼ shall only consider the cases N = 2 or 3), and by ⊂ =) inclusion (resp. N−1 equality) up to a set of zero H -measure. We assume that the energy spent to produce a crack K is of Griffith type, i.e., K(ε)(K) := H 2 (K).

(1)

2.2 Admissible Deformations We refer to [1] for the usual definitions and results on geometric measure theory, BV , SBV and GSBV spaces. Precise definitions of the jump set Su and of the approximate gradient ∇u of a function u ∈ GSBV (U ; Rd ), where U is an open subset of RN , can be found in that reference. Following Dal Maso et al. [8], we further define for p > 1 GSBV p (U ; Rd ) : = u ∈ GSBV (U ; Rd ) : ∇u ∈ Lp (U ; Rd×N ) and H N−1 (Su ) < +∞ , p

and if q ≥ 1, GSBVq (U ; Rd ) := GSBV p (U ; Rd ) ∩ Lq (U ; Rd ). Moreover, we p say that a sequence un u in GSBVq (U ; Rd ) if un → u a.e. in U , un u in Lq (U ; Rd ), ∇un ∇u in Lp (U ; Rd×N ) and H N−1 (Sun ) is uniformly bounded. For a given admissible crack K ∈ R(εB ) and a boundary deformation φ ∈ 1,p W (ε ; R3 ) ∩ Lq (ε ; R3 ), we define the set of admissible deformations with finite energy relative to (K, φ) by p K, u = φ H 2 -a.e. on ∂D ε \ K}. AD ε (φ, K) := {u ∈ GSBVq (ε ; R3 ) : Su ⊂

The associate bulk energy is defined by W (ε)(∇u) :=

W (∇u(x)) dx,

(2)

ε

where W : R3×3 → [0, +∞), the stored energy density, is a quasiconvex function of class C 1 satisfying standard p-growth and p-coercivity conditions (p > 1): there exist 0 < β < β < +∞ such that β |ξ |p ≤ W (ξ ) ≤ β(1 + |ξ |p )

for every ξ ∈ R3×3 .

(3)

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J.-F. Babadjian

In particular, the functional W (ε) : Lp (ε ; R3×3 ) → [0, +∞) defined by W (ε)() := W ((x)) dx ε

is differentiable on Lp (ε ; R3×3 ), and its differential DW (ε) : Lp (ε ; R3×3 ) →

Lp (ε ; R3×3 ), with p = p/(p − 1), is given by DW (ε)(), = DW ((x)) : (x) dx for every , ∈ Lp (ε ; R3×3 ). ε

On the left-hand side of the previous equality, we have denoted by ·, · the duality

pairing between Lp (ε ; R3×3 ) and Lp (ε ; R3×3 ).

2.3 The Forces We assume that the body is subjected to the action of conservative body and surface forces with potentials F and Gε = εG respectively. Note that the order of magnitude of the applied forces are exactly those inducing a limiting membrane model (see [12, 13]).

2.3.1 The Body Forces Let q ≥ 1, the density of the applied body forces per unit volume at time t ∈ [0, T ] is given by Dz F (t, u(x)), where F : [0, T ] × R3 → R and the map z → F (t, z) belongs to C 1 (R3 ) for every t ∈ [0, T ]. We suppose that for every t ∈ [0, T ], the functional F (ε)(t)(u) := F (t, u(x)) dx (4) ε

is of class C 1 on the space Lq (ε ; R3 ), and its differential DF (ε)(t) : Lq (ε ; R3 )

→ Lq (ε ; R3 ), with q := q/(q − 1), is given by DF (ε)(t)(u), v = Dz F (t, u(x)) · v(x) dx for every u, v ∈ Lq (ε ; R3 ). ε

We have denoted by ·, · the duality pairing between Lq (ε ; R3 ) and Lq (ε ; R3 ). Concerning the regularity in time, we assume that there exist an exponent q˙ < q and, for a.e. t ∈ [0, T ], a functional F˙ (ε)(t) : Lq˙ (ε ; R3 ) → R of class C 1 ,

with differential D F˙ (ε)(t) : Lq˙ (ε ; R3 ) → Lq˙ (ε ; R3 ), where q˙ = q/( ˙ q˙ − 1), such that for every u, v ∈ Lq (ε ; R3 ), the functions t → F˙ (ε)(t)(u) and t → D F˙ (ε)(t)(u), v are integrable on [0, T ], and


t

F (ε)(t)(u) = F (ε)(0)(u) +

5

F˙ (ε)(s)(u) ds,

(5)

0

t

DF (ε)(t)(u), v = DF (ε)(0)(u), v +

D F˙ (ε)(s)(u), v ds

(6)

0

for every t ∈ [0, T ]. We further assume that F (ε)(t) is upper semicontinuous in Lq (ε ; R3 ) with respect to the pointwise almost everywhere convergence. Finally, we suppose that F (ε)(t), DF (ε)(t), F˙ (ε)(t) and D F˙ (ε)(t) satisfy suitable q-growth conditions: there exist constants a0 > 0, a1 > 0, a2 > 0, b0 ≥ 0, b1 ≥ 0, b2 ≥ 0, and nonnegative integrable functions on [0, T ], a3 , a4 , b3 and b4 (uniform in ε) such that ⎧ q q a0 uLq (ε ;R3 ) − b0 ≤ −F (ε)(t)(u) ≤ a1 uLq (ε ;R3 ) + b1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ | DF (ε)(t)(u), v | ≤ (a2 uq−1 + b2 )vLq (ε ;R3 ) , Lq (ε ;R3 ) (7) q˙ ⎪ |F˙ (ε)(t)(u)| ≤ a3 (t)uLq˙ (ε ;R3 ) + b3 (t), ⎪ ⎪ ⎪ ⎪ ˙ ⎩ | D F˙ (ε)(t)(u), v | ≤ (a (t)uq−1 + b4 (t))vLq˙ (ε ;R3 ) . 4 Lq˙ (ε ;R3 ) 2.3.2 The Surface Forces The density of the surface forces on ∂N ε at time t ∈ [0, T ], under the deformation u is given by εDz G(t, u(x)), where G : [0, T ] × R3 → R is such that z → G(t, z) is of class C 1 (R3 ) for every t ∈ [0, T ]. We fix an exponent r, related to the trace theorem in Sobolev spaces, such that r ∈ [p, p/(3 − p)] if p < 3, while r ≥ p if p ≥ 3. We assume that for every t ∈ [0, T ], the functional G(t, u(x)) dH 2(x) (8) G(ε)(t)(u) := ε ∂N ε

is of class C 1 on Lr (∂N ε ; R3 ), with differential DG(ε)(t) : Lr (∂N ε ; R3 ) →

Lr (∂N ε ; R3 ), where r = r/(r − 1), given by G(ε)(t)(u), v = ε Dz G(t, u(x)) · v(x) dH 2 (x) ∂N ε

for all u, v ∈ Lr (∂N ε ; R3 ),

where ·, · denotes the duality pairing between Lr (∂N ε ; R3 ) and Lr (∂N ε ; R3 ). As for the regularity in time, we suppose that for a.e. t ∈ [0, T ], there exists a ˙ ˙ functional G(ε)(t) : Lr (∂N ε ; R3 ) → R of class C 1 , with differential D G(ε)(t) :

r ε 3 r ε 3 r ε L (∂N ; R ) → L (∂N ; R ), such that for every u, v ∈ L (∂N ; R3 ), the ˙ ˙ mappings t → G(ε)(t)(u) and t → D G(ε)(t)(u), v are integrable on [0, T ], and

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J.-F. Babadjian

t

G(ε)(t)(u) = G(ε)(0)(u) +

˙ G(ε)(s)(u) ds,

(9)

0

t

DG(ε)(t)(u), v = DG(ε)(0)(u), v +

˙ D G(ε)(s)(u), v ds

(10)

0

for every t ∈ [0, T ]. ˙ ˙ Finally, we suppose that G(ε)(t), DG(ε)(t), G(ε)(t) and D G(ε)(t) satisfy suitable r-growth conditions: there exist nonnegative constants α0 , α1 , α2 , β0 , β1 , β2 , and nonnegative integrable functions on [0, T ], α3 , α4 , β3 and β4 such that ⎧ q −α0 εurLr (∂ ε ;R3 ) − β0 ε ≤ −G(ε)(t)(u) ≤ α1 εuLr (∂ ε ;R3 ) + β1 ε, ⎪ N N ⎪ ⎪ ⎪ ⎨ | DG(ε)(t)(u), v | ≤ (α2 εur−1 + β ε)v r 2 L (∂N ε ;R3 ) , Lr (∂N ε ;R3 ) (11) r ˙ ⎪ |G(ε)(t)(u)| ≤ α3 (t)εuLr (∂ ε ;R3 ) + β3 (t)ε, ⎪ N ⎪ ⎪ ⎩ ˙ | D G(ε)(t)(u), v | ≤ (α4 (t)εur−1 + β4 (t)ε)vLr (∂N ε ;R3 ) . Lr (∂ ε ;R3 ) N

The reason why all the previous coercivity and growth constants/functions are of order ε is due to the fact that the surface force density Gε = εG of scales like ε.

2.4 Quasi-Static Evolution For a given admissible crack K ∈ R(εB ) and a given boundary deformation φ ∈ W 1,p (ε ; R3 ) ∩ Lq (ε ; R3 ), the total energy of the configuration (K, u), with u ∈ AD ε (φ, K), at time t ∈ [0, T ] is given by E (ε)(t)(u, K) := W (ε)(∇u) − F (ε)(t)(u) − G(ε)(t)(u) + K(ε)(K). We define a quasi-static evolution with boundary condition t → φ ε (t) as a map p t → (v ε (t), K ε (t)) from [0, T ] to GSBVq (ε ; R3 ) × R(εB ) with the following properties: (i)

Global stability: for all t ∈ [0, T ], we have v ε (t) ∈ AD ε (φ ε (t), K ε (t)) and K

E (ε)(t)(vε (t), K ε (t)) = min{E (ε)(v , K ) : K ∈ R(εB ), K ε (t) ⊂ and v ∈ AD ε (φ ε (t), K )}.

K ε (t) whenever s ≤ t. (ii) Irreversibility: K ε (s) ⊂ (iii) Energy balance: the mapping t → E(ε)(t) := E (ε)(t)(v ε (t), K ε (t)) is absolutely continuous on [0, T ] and ˙ E(ε)(t) = DW (ε)(∇v ε (t)), ∇ φ˙ ε (t) − DF (ε)(t)(v ε (t)), φ˙ ε (t) − F˙ (ε)(t)(v ε (t))


7

ε ˙ − DG(ε)(t)(v ε (t)), φ˙ ε (t) − G(ε)(t)(v (t)).

The following existence result has been proven in [8]. Theorem 1. Let K0ε ∈ R(εB ) and v0ε ∈ AD ε (φ ε (0), K0ε ) such that E (ε)(0)(v0ε , K0ε ) ≤ E (ε)(0)(v , K ) K , and every v ∈ AD ε (φ ε (0), K ). Then there for every K ∈ R(εB ) with K0ε ⊂ exists a quasi-static evolution t → (v ε (t), K ε (t)) with boundary deformation φ ε (t) such that (v ε (0), K ε (0)) = (v0ε , K0ε ).

3 The Rescaled Configuration Our goal is to perform an asymptotic analysis of the quasi-static evolution as the thickness of the film ε → 0. As usual in dimension reduction problems (see e.g. [6, 15]) we rescale the problem into an equivalent one with the advantage of being stated over a fixed domain. Before doing this we shall make some assumptions on the initial crack. We assume that it is compatible with the geometry of the problem, i.e., that K0ε = γ0 × (−ε + εη, ε − εη), for some countably H 1 -rectifiable set γ0 ⊂ ω. We now define := 1 , B := 1B , ∂D := ∂D 1 and ∂N := ∂N 1 . For x ∈ , we denote by xα := (x1 , x2 ) ∈ ω the in-plane variable. We set ψ ε (t, xα , x3 ) := φ ε (t, xα , εx3 ), uε0 (xα , x3 ) := v0ε (xα , εx3 ), uε (t)(xα , x3 ) := v ε (t)(xα , εx3 ), ε (t) := {(xα , x3 ) ∈ B : (xα , εx3 ) ∈ K ε (t)}. Changing variables in (1), (2), (4) and (8) leads to

1

ε ε ε W (ε)(∇v (t)) = ε W ∇α u (t) ∇3 u (t) dx =: εW ε (∇uε (t)), ε

1

ε 2 ε ε

ε ε K(ε)(K (t)) = ε

ν (t ) α ε ν (t ) 3 dH =: εK ( (t)), ε (t ) ε F (t, uε (t)) dx =: εF (t)(uε (t)), F (ε)(t)(v (t)) = ε

G(t, uε (t)) dH 2 (x) =: εG(t)(uε (t)),

G(ε)(t)(v ε (t)) = ε ∂N

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J.-F. Babadjian

where ∇α (resp. ∇3 ) denotes the approximate gradient with respect to xα (resp. x3 ), and ν ε (t ) = ((ν ε (t ))α , (ν ε (t ) )3 ) is the normal to ε (t). ˙ so that there hold the As in Section 2.3, we define the functionals F˙ (t) and G(t) ˙ ˙ analogue of (5), (6), (9) and (10) with F , F , G and G instead of F (ε), F˙ (ε), G(ε) ˙ ˙ DG, D G, ˙ F , F˙ , DF and D F˙ satisfy analogue growth and G(ε). Note that G, G, and coercivity conditions as (7)–(11) with the same exponents, and with coercivity and growth constants/functions independent of ε. We write E ε (t) for the rescaled total energy at time t. Then, we deduce that the p mapping t → (uε (t), ε (t)) from [0, T ] to GSBVq (; R3 ) × R(B ) satisfies the following properties: (i)

Global stability: for all t ∈ [0, T ], we have uε (t) ∈ AD 1 (ψ ε (t), ε (t)) and

E ε (t)(uε (t), ε (t)) = min{E ε (u , ) : ∈ R(B ), ε (t) ⊂ and u ∈ AD 1 (ψ ε (t), )}.

ε (t) whenever s ≤ t. (ii) Irreversibility: ε (s) ⊂ (iii) Energy balance: the mapping t → E ε (t) := E ε (t)(uε (t), ε (t)) is absolutely continuous on [0, T ] and E˙ ε (t) = DW ε (∇uε (t)), ∇ ψ˙ ε (t) − DF (t)(uε (t)), ψ˙ ε (t) − F˙ (t)(uε (t)) ε ˙ − DG(t)(uε (t)), ψ˙ ε (t) − G(t)(u (t)).

4 Analysis of Static Problem by -Convergence Before going to the study of the quasi-static problem, we will discuss the asymptotic behavior of the total energy as ε → 0 thanks to a -convergence method. Since the work of external forces F + G corresponds to a continuous perturbation of the sum of the bulk and surface energies, we will not take it into account. We will actually study a weak formulation of the problem replacing the crack by the jump set of the deformation field. Indeed, Let us define Iε : L1 (; R3 ) → [0, +∞] by

1

1

2

Iε (u) := W ∇α u ∇3 u dx +

(νu )α ε (νu )3 dH ε Su if u ∈ GSBV p (; R3 ), and +∞ if u ∈ L1 (; R3) \ GSBV p (; R3 ). Then, the following -convergence result holds: Theorem 2. Let ω be a bounded open subset of R2 and W : R3×3 → R be a continuous function satisfying (3). Then the functional Iε -converges for the strong L1 (; R3 )-topology to I : L1 (; R3) → [0, +∞] defined by


9

⎧ ⎨ 2 QW0 (∇α u) dxα + 2H 1 (Su ) if u ∈ GSBV p (ω; R3 ), I(u) := ⎩ ω +∞ otherwise, where W0 (ξ ) := inf{W (ξ |z) : z ∈ R3 } for every ξ ∈ R3×2 , and QW0 is the quasiconvexification of W0 . This result has been proven in [2] (see also [3]) in a SBV p framework, and one can notice that much easier arguments lead to the analogue in GSBV p as stated in Theorem 2. Indeed, in GSBV p , there is no lack of compactness and it is not necessary to appeal to a truncation argument as in [2, lemma 3.3]. It follows immediately from the GSBV -compactness theorem [1, theorem 4.36] that any minimizing sequence (uε ) ⊂ GSBV p (; R3 ) with uniformly bounded energy is relatively compact in GSBV p (; R3 ), and that any accumulation point is independent of x3 (we identify those functions to GSBV p (ω; R3 )). The proof of the lower bound is exactly the same than [2, lemma 3.9] using the lower semicontinuity result in GSBV p (see e.g. [8, theorem 2.8]). The construction of a recovery (u¯ ε ) can be performed as in [15]: it suffices to take u¯ ε (xα , x3 ) := u(xα ) + εx3 bε (xα ) for some suitable function bε ∈ Cc∞ (ω; R3 ), and then we appeal to a classical relaxation result in GSBV p . We also refer to [5] for an alternative proof using a singular perturbation argument.

5 Analysis of the Quasi-Static Problem In view of Theorem 2, one can guess that the 3D quasi-static evolution – whose existence is ensured by Theorem 1 – will converge in a certain sense to a 2D quasistatic evolution associated to the -limit model. We assume that ψ ε → ψ in W 1,1 ([0, T ]; W 1,p (; R3 ) ∩ Lq (; R3 )) and that the sequence ((1/ε)∇3 ψ ε ) is strongly converging in W 1,1 ([0, T ]; Lp (; R3)). In particular, the limit function ψ ∈ W 1,1 ([0, T ]; W 1,p (ω; R3 ) ∩ Lq (ω; R3 )) is independent of x3 . We first derive some compactness of (uε (t), ε (t)). Indeed taking (ψ ε (t), ε (t)) as competitor in the minimality, and using the growth an coercivity properties satisfied by the functionals Wε , F and G implies that the sequence of approximate scaled gradients (∇α uε (t)|(1/ε)∇3 uε (t)) is bounded in Lp (; R3×3 ), and the sequence (uε (t)) is bounded in Lq˙ (; R3) ∩ Lq (; R3 ). Moreover, since uε (t) ∈ W 1,p ( \ B ; R3 ), the trace theorem and the choice of the exponent r ensures that (uε (t)) is compact in Lr (∂N ; R3 ). Then we use the energy balance together with ˙ to the growth and coercivity conditions satisfied by DF (t), DG(t), F˙ (t) and G(t) ensure that

1

ε ε (ν (t ))α (ν (t ))3

dH 2 < +∞. sup

ε ε>0 ε (t ) At this step, we are in position to use a mean convergence for rectifiable sets introduced in [2], very close to the σ p -convergence in [8, 9]:

10

J.-F. Babadjian

Definition 1. Let εn 0+ , n ⊂ be a sequence of countably H 2 -rectifiable sets, and γ ⊂ ω be a countably H 1 -rectifiable set. We say that n converges to γ in if

(ν )α

1 (ν )3 dH 2 ≤ C, n n

εn n and the following properties hold:

nk and ∇α uk |(1/εnk )∇3 uk p dx ≤ C, (a) if uk u in SBV p (), Suk ⊂ γ; for some subsequence (εnk ) ⊂ (εn ), then u ∈ SBV p (ω) and Su ⊂ p () such a sequence (u ) ⊂ SBV (b) there exist a function u ∈ SBV p (ω) and n

p p n , ∇α un |(1/εn )∇3 un dx ≤ C, and that un u in SBV (), Sun ⊂ Su = γ. Then, using [2, proposition 4.3], one can extract a subsequence (εn ) (independently of t) and find a countably H 1 -rectifiable set γ (t) increasing with respect to t such that εn (t) converges to γ (t) in sense of Definition 1. Note that it is possible to prove that γ (0) = γ0 . Moreover the estimates we have on the sequence (uεn (t)) allow us to apply the GSBV -Compactness Theorem (see [1, theorem 4.36]) which ensures, for each t ∈ [0, T ], the existence of a t-dependent subsequence (εnt ) ⊂ p (εn ) such that uεnt (t) u(t) in GSBVq (; R3). Moreover the limit deformation p 3 u ∈ GSBVq (ω; R ), i.e., it is independent of x3 . We claim that the pair (u(t), γ (t)) is a quasi-static evolution associated to the limit model. We already proved the irreversibility condition. To show the minimality property we use the following jump transfer theorem whose proof can be obtained exactly as in [2, theorem 4.4], using [8, theorem 5.3] instead of [11, theorem 2.1]. Let ω ⊂ R2 a bounded open set containing ω, and define := ω × (−1, 1). Theorem 3. Let n ∈ R(B ) be a sequence of countably H 2 -rectifiable sets conp verging to γ in the sense of Definition 1. Then, for every v ∈ GSBVq (ω ; R3 ), there p exists a sequence (vn ) ⊂ GSBVq ( ; R3 ) such that vn = v a.e. on \ , • • •

vn → v in Lq ( ; R3 ),

1

∇α vn ∇3 vn → (∇α v|0) in Lp ( ; R3×3 ),

εn

(νvn )α

1 (νvn )3 dH 2 ≤ 2H 1 (Sv \ γ ). lim sup

εn n→+∞ Svn \n

Arguing exactly as in the proof of [2, lemma 5.5] and using the upper semicontinuity property of F (t) together with the continuity of G(t) (which comes from the trace theorem in W 1,p and the choice of the exponent r), one can show that for every t ∈ [0, T ], u(t) minimizes v → 2 QW0 (∇α v) dxα + 2H 1 (Sv \ γ (t)) − 2F (t)(v) − 2G(t)(v), ω p

among {v ∈ GSBVq (ω; R3 ) : v = ψ(t) H 1 -a.e. on ∂ω}. Moreover one has convergence of the bulk energy (for the sequence εn )


11

1

W ∇α uεn (t) ∇3 uεn (t) dx → 2 QW0 (∇α u(t)) dxα , εn ω

as well as weak convergence of the stress (for the subsequence εnt )

1

εnt εnt DW ∇α u (t) ∇3 u (t) D(QW0 )(∇α u(t)|0) in Lp (; R3×3 ) εnt (12) at every time. Remark that by [2, proposition 4.7], the function QW0 is of class p C 1 . For every v ∈ GSBVq (ω; R3 ) such that v = ψ(t) H 1 -a.e. on ∂ω, and every 1 countably H -rectifiable set γ ⊂ ω, we define E (t)(v, γ ) = 2 QW0 (∇α v) dxα + 2H 1 (γ ) − 2F (t)(v) − 2G(t)(v). ω

The minimality property proven above exactly says that E (t)(u(t), γ (t)) = min{E (t)(v, γ ) : γ countably H 1 −rectifiable set in ω, p

v ∈ GSBVq (ω; R3 ) such that v = ψ(t) H 1 −a.e. on ∂ω} (see e.g. [2, remark 5.4]). Moreover, since we have reduced dimension, the functional G(t) becomes a bulk force as well as F (t). It remains to prove the energy balance. Arguing word for word as in [8], approximating Bochner integrals by suitable Riemann sums, one can show that t ˙ E (t)(u(t), γ (t)) ≥ E (0)(u(0), γ (0)) + 2 DW0 (∇α u(s)), ∇α ψ(s)) 0

− DF (t)(u(s)), ψ˙ (s) − F˙ (s)(u(s)) ˙ − DG(s)(u(s)), ψ˙ (s) − G(s)(u(s)) ds, where W0 : Lp (ω; R3×2 ) → [0, +∞) is defined by W0 () := 1 ω QW0 ((x)) dxα . By [2, Proposition 4.7], we deduce that W0 is of class C with

p 3×2 p 3×2 differential DW0 : L (ω; R ) → L (ω; R ) given by DW0 (), = D(QW0 )((x)) : (x) dxα . ω

We prove the other inequality exactly as in [2, lemma 5.8], using the upper semicontinuity property of the functional F (t) and the weak convergence of the stresses (12) already mentioned above. We also deduce the convergence of the surface energy (for the sequence εn ):

(ν εn (t ) )α

1 (ν εn (t ) )3 dH 2 → 2H 1 (γ (t)).

εn εn (t )

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J.-F. Babadjian

In conclusion, we have proven the following result which states that any 3D quasistatic crack evolution converges to a 2D quasi-static evolution associated to the limit model: Theorem 4. There exist a two-dimensional quasi-static evolution t → (u(t), γ (t)) relative to the boundary data ψ(t) for the -limit model, and a sequence εn 0+ such that for any t ∈ [0, T ], • • •

εn (t) converges to γ (t) in sense of Definition 1; p uεnt (t) u(t) in GSBVq (; R3 ) for some t-dependent subsequence (εnt ) ⊂ (εn ); the total energy E εn (t) converges to E (t), and more precisely

1

εn εn W ∇α u (t) ∇3 u (t) dx → 2 QW0 (∇α u(t)) dxα , εn ω

(ν εn (t ) )α

1 (ν εn (t ) )3 dH 2 → 2H 1 (γ (t)).

ε ε n n (t )

Acknowledgement The research of the author has been supported by the Chair “Mathematical Modelling and Numerical Simulation, F-EADS Ecole Polytechnique INRIA F-X”.

References 1. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000. 2. J.-F. Babadjian, Quasistatic evolution of a brittle thin film, Calc. Var. and PDEs 26(1), 2006, 69–118. 3. J.-F. Babadjian: Lower semicontinuity of quasiconvex bulk energies in SBV and integral representation in dimension reduction, SIAM J. Math. Anal. 39(6), 2008, 1921–1950. 4. B. Bourdin, G.A. Francfort and J.-J. Marigo: The Variational Approach to Fracture, Springer, Amsterdam, 2008. 5. A. Braides and I. Fonseca: Brittle thin films, Appl. Math. Optim. 44, 2001, 299–323. 6. A. Braides, I. Fonseca and G.A. Francfort: 3D-2D Asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J. 49, 2000, 1367–1404. 7. G. Dal Maso: An Introduction to -Convergence, Birkhäuser, Boston, 1993. 8. G. Dal Maso, G.A. Francfort and R. Toader: Quasi-static crack growth in nonlinear elasticity, Arch. Rational Mech. Anal. 176, 2005, 165–225. 9. G. Dal Maso, G.A. Francfort and R. Toader: Quasi-static evolution in brittle fracture: The case of bounded solutions, Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi. Quaderni di Matematica 14, 2005, 247–265. 10. G.A. Francfort and J.-J. Marigo: Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46, 1998, 1319–1342.


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11. G.A. Francfort and G.J. Larsen: Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math. 56, 2003, 1465–1500. 12. G. Friesecke, R. James and S. Müller: A hierarchy of plate models derived from nonlinear elasticity by -convergence, Arch. Rational Mech. Anal. 180(2), 2006, 183–236. 13. D. Fox, A. Raoult and J.C. Simo: A justification of nonlinear properly invariant plate theories, Arch. Rational. Mech. Anal. 25, 1992, 157–199. 14. A. Giacomini and M. Ponsiglione: A -convergence approach to stability of unilateral minimality properties in fracture mechanics and applications, Arch. Rational Mech. Anal. 180, 2006, 399–447. 15. H. Le Dret and A. Raoult: The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J.Math. Pures Appl. 74, 1995, 549–578. 16. A. Mielke, T. Roubicek and U. Stefanelli: -limits and relaxations for rate-independent evolutionary problems, Calc. Var. and PDEs 31, 2008, 387–416.

FE2-Simulation of Microheterogeneous Steels Based on Statistically Similar RVEs D. Balzani, J. Schröder and D. Brands

Abstract A main problem of direct homogenization methods is the high computational cost, when we have to deal with large random microstructures. This leads to a large number of history variables which needs a large amount of memory, and moreover a high computation time. We focus on random microstructures consisting of a continuous matrix phase with a high number of embedded inclusions. In this contribution a method is presented for the construction of statistically similar representative volume elements (SSRVEs) which are characterized by a much less complexity than usual random RVEs in order to obtain an efficient simulation tool. The basic idea of the underlying procedure is to find a simplified SSRVE, whose selected statistical measures under consideration are as close as possible to the ones of the original microstructure.

1 Introduction Nowadays, the main task in designing high-tech-steels for automotive applications is to optimize stiffness and ductility while minimizing dead weight. To achieve the demands for high strength and good formability modern steels make use of (random) multi-phase structures. For an accurate phenomenological description of the macroscopic material behavior of these complex micro-heterogeneous materials the interactions of the individual constituents on the micro-scale have to be taken into account. A suitable numerical tool for the direct incorporation of micromechanical information is a direct/numerical two-scale homogenization scheme, also known as the direct micro-macro-transition procedure or the FE2 -method, see e.g. [4, 8, 10] and D. Balzani · J. Schröder · D. Brands Institute of Mechanics, Faculty of Engineering Sciences, University of Duisburg-Essen, Universitätsstr. 15, 45117 Essen, Germany; E-mail: {daniel.balzani, j.schroeder}@uni-due.de


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references therein. In this scheme an additional microscopic boundary value problem, associated to each macroscopic point, has to be solved. For the multiphase steels under consideration the associated microstructures are characterized by representative volume elements (RVEs) of random nature, in this context, see [9] where this direct transition approach is applied with respect to DP-steels. The considered RVEs contain a very large set of inclusions and must satisfy the conditions of statistical homogeneity and ergodicity. The associated spatial finite element discretization leads to a large number of history variables which requires a large amount of memory, and moreover a high computation time. In order to overcome these high computational costs statistically similar representative volume elements (SSRVEs) have to be constructed which are characterized by a much less complexity than usual random RVEs. The proposed procedure is based on the assumption that some statistical measures describing the morphology of the real large microstructure are similar to the ones of the SSRVE. Possible statistical descriptors are the volume fraction, the specific internal surface, n-point probability functions or herefrom deduced quantities. In [6] it has been shown that the power spectral density of a microstructure can be interpreted as a probability density function in the frequency domain. Motivated by this work, a method for the determination of substructures that are statistically similar to more complex two-phase microstructures has been proposed in [7]. There, the author optimizes a periodic statistically similar substructure by minimizing a least square function considering the power spectral densities of the real microstructure and the substructure, where a fixed inclusion geometry is considered. Thus, the basic idea of the underlying procedure is to find simplified SSRVEs, whose statistical measures under consideration are as close as possible to the ones of the original microstructure. Therefore, we formulate a suitable least-square function accounting for the above mentioned demand, wherein splines are used for the parameterization of the inclusion phase, in this context we refer to [1]. The arising problem of minimizing the least-square function is treated by applying a moving frame algorithm which is combined with a line-search algorithm. Finally, we provide a numerical example showing the performance of the proposed method.

2 Discrete Multiscale Approach The classical continuum mechanics focusses on the mechanical properties being described in a material point. Herewith, the principle of local agency is postulated, which assumes a homogeneous stress- and strain distribution in an infinitesimal neighborhood of the considered material point. Due to the heterogeneity of the material this assumption does not hold at the microscale. We are interested in obtaining continuum mechanical quantities associated to an infinitesimal vicinity of a material point at the macroscale X ∈ B based on informations at the microscale. For the definition of the macroscopic quantities we consider a representative volume element (RVE), parametrized in X ∈ B, where the mi-

FE2 -Simulation of Microheterogeneous Steels

17

Fig. 1 Visualization of the concept of micro-macro modeling approach.

croscopic field quantities are determined, see Figure 1. In general the macroscopic deformation gradient F and the macroscopic first Piola–Kirchhoff stresses P are defined by suitable surface integrals, see e.g. [8]. Applying some technically useful assumptions allows for the reformulation of the macroscopic quantities in terms of the volume averages over their microscopic counterparts, i.e. 1 1 F= F dV and P = P dV , (1) vol(RVE ) B vol(RVE ) B respectively. In the sequel, macroscopic quantities are generally denoted by overlines, i.e. (•). In a variety of applications we are interested in the incremental overall response of the material (2) P = A : F , wherein A := ∂F P denotes the macroscopic nominal moduli. Macroscopic Boundary Value Problem. Neglecting acceleration terms the balance of linear momentum at the macroscale reads Div[P] + f = 0 in B wherein f denotes the macroscopic volumetric force vector, respectively. The boundary conditions are given by u = u˜ on ∂B u and t = P · n = ˜t on ∂B P , where u and t denote the macroscopic displacement and traction vectors, respectively. Microscopic Boundary Value Problem. At the microscale the boundary value problem is given by Div[P] = 0 in B, where we have neglected acceleration terms and volumetric forces. The boundary conditions are derived from the macrohomogeneity condition, also referred to as Hill-condition, see [3]. It postulates that the macroscopic power is equal to the volumetric average of the microscopic powers, i.e. 1 ˙ P·F= P · F˙ dV . (3) vol(RVE ) B

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After some algebraic manipulations we obtain the three possible types of boundary conditions for the microscopic boundary value problem: (i) the stress boundary condition t = PN on ∂B; (ii) the linear boundary displacements x = FX on ∂B; and (iii) the periodic boundary conditions ˜ x = FX + w,

˜+ = w ˜ −, w

t+ = −t−

on ∂B .

(4)

˜ denotes fluctuations of the displacement field and that (•)+ , (•)− means Note that w quantities at periodically associated points of the RVE-boundary, for further details we refer to [4, 8]. Discrete Formulation of the coupled Micro-Macro-Transition. The basic idea of the FE2 -method is that a standard macroscopic boundary value problem is considered, where at each Gauss point a microscopic boundary value problem is driven by some of the above mentioned boundary conditions. Focusing on periodic boundary conditions (iii) we solve the microscopic BVP and return the average of resulting microscopic stresses P according to (1)2 to the macroscale. At the microscale we consider the weak form and its linear increment for a typical finite element ˜ dV , Ge = δ F˜ · P dV and Ge = δ F˜ · (A : F) (5) Be

Be

with the microscopic nominal moduli A := ∂F P. The fluctuation parts of the actual, virtual and incremental deformation gradient can be approximated by using standard ansatz functions for the fluctuation displacements interpolating between ˜ virtual displacements δ d, ˜ and incrementhe fluctuation parts of displacements d, ˜ Then we obtain the discrete representation of the linearized tal displacements d. problem nele ˜ δ d) ˜ + Ge (d, ˜ δ d, ˜ d) ˜ = 0, Ge (d, (6) e=1

where nele denotes the number of finite elements. After assembling all finite elements this leads to the global microscopic problem ˜ T {KD ˜ + R} = 0 , δD

(7)

˜ and residual with the global vectors of incremental fluctuation displacements D forces R, and with the global microscopic stiffness matrix K. In each iteration the actual fluctuations of displacements are computed from (7) and updated, i.e. ˜ ⇐D ˜ + D, ˜ until |R| < tol. D At the macroscale a standard FE-discretization is considered where the macroscopic moduli entering the macroscopic stiffness matrix are computed by 1 LT K−1 L A = A − vol(RVE )

with

L=

nele

A e=1

Be

BT A dV ,

(8)


19

Fig. 2 Schematical illustration of the basic concept: (a) usual RVE with arbitrary inclusion morphology and (b) periodic microstructure with SSRVE.

1 wherein denotes a suitable assembling operator and A = vol(RVE) B A dV the classical Voigt-bound. For details on deriving the consistent macroscopic moduli we refer to [4, 8].

A

3 Generation of Statistically Similar RVEs In the context of Finite-Element simulations based on coupled micro-macro transition approaches a usual RVE is determined by taking the smallest possible subdomain of the whole microstructure which is still able to represent the macroscopic stress-strain response. Allthough these volume elements are the smallest possible sub-structures, they are typically still too complex for efficient discrete micro-macro calculations. Therefore, in this contribution we construct statistically similar microstructures that are characterized by a lower complexity, cf. [1]. The basic idea in this context is to replace a RVE with an arbitrary complex inclusion morphology by a periodic one composed of periodically arranged unit cells, see Figure 2. Then the main effort is that in FE2 calculations only the unitcell needs to be considered as a RVE if periodic boundary conditions are applied. A method for the construction of such periodic structures is proposed for the special case of randomly distributed circular inclusions with constant equal diameters in [7]. The underlying idea there is to find the position of the circular inclusions with given diameter such that a least-square functional taking into account the side condition that the spectral density of the periodic RVE should be similar to the one of the non-periodic microstructure, is minimized. Motivated by this approach we propose to consider the generalized minimization problem

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L(γ ) → min with

L(γ ) =

nsm

2 ωi Pireal − PiSSRV E (γ ) ,

(9)

i=1

where P real and P SSRV E denote appropriate statistical measures describing the inclusion morphology computed for the real microstructure and for the SSRVE, respectively. The number of considered statistical measures is represented by nsm, whereas the weighting factor ω levels the influence of the individual measures. For the description of a general inclusion phase morphology in the SSRVE we assume a suitable two-dimensional parameterization controlled by the vector γ .

3.1 Statistical Measures for the Description of the Inclusion Morphology A variety of statistical measures exist to describe the morphology of a single phase in a multi-phase composite, see e.g. [11]. It is mentioned that in the sequel we are only interested in the morphology of the inclusion phase in a two-phase material. There exist four basic parameters for its description, see e.g. [5]. The first two are defined by VI SI PV := and PS := , (10) V V with VI , SI denoting the volume and internal surface of the inclusion phase and V being the total volume of the considered material segment. These two parameters basically represent the volume fraction and the specific internal surface. The third and fourth parameter, the specific integral of mean curvature and the specific integral of total curvature are defined by 1 {min[κ] + max[κ]} ds 2V 1 min[κ] max[κ] ds , := V

PM := PK

and (11)

respectively, wherein κ denotes the curvature computed at an infinitesimal surface element ds. A possibility to cover more statistical information may be the (discrete) spectral density computed for the inclusion phase by the multiplication of the (discrete) Fourier transform with its conjugate complex. The discrete spectral density is defined by 1 PSD (m, k) := |F (m, k)|2 (12) 2π Nx Ny with the discrete Fourier transform given by


F (m, k) =

Ny Nx p=1 q=1

21

2iπ kq 2iπ mp exp χ(p, q) . exp Nx Ny

(13)

The maximal numbers of pixels in the considered binary image are given by Nx and Ny ; the indicator function is defined by χ :=

1, if (p, q) is in inclusion phase 0, else .

(14)

Due to the fact that the spectral density covers information concerning the periodicity of a given microstructure, this measure is of major importance for us since we are interested in finding a simplified periodic microstructure which represents the mechanical response of the real microstructure. Another possibility to obtain even more statistical information is to compute the n-point probability function PNP (x1 , . . . , xn ) = χr1 (x1 , α) · · · χrn (xn , α) ,

(15)

which is defined to be the ensemble average over samples α of the multiplication of indicator functions (14) evaluated at n points. For the definition of ensemble averages we refer to [2]. The n-point probability function represents the probability that a number of n points are located in the inclusion phase. For practical applications the discrete one-point and two-point probability functions are of high importance and computed by Pr =

y −1 N x −1 N 1 χr (p, q) , Nx Ny

p=0 q=0

Prs (m, k) =

(pM

p M −1 q K −1 1 χr (p, q)χs (p + m, q + k) , − pm )(qK − qk ) p=p q=q m

(16)

k

with the summation limits given by pm = max[0, −m] , pM = min[Nx , Nx − m] qk = max[0, −k] ,

qK = min[Ny , Ny − k] .

Interestingly, the one-point probability function is equivalent to the first basic parameter, the volume fraction, since it represents the probability that one point is situated in the inclusion phase. In adition, the two-point probability function is strongly correlated with the spectral density since it can be computed in terms of the Fourier transform. This further motivates the spectral density to be a suitable statistical measure for the generation of SSRVEs. In order to include further statistical information one could take into account higher-order probability functions.

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D. Balzani et al.

(a)

(b)

(c)

Fig. 3 Example: normalized (a) original and (b) rebinned spectral density of a typical microstructure; (c) relevant area of spectral density. Herein, black and white colors mean the values of 0 and 1, respectively.

3.2 Proposed Method The volume fraction covers a very important overall information with view to the influence of the statistical morphology description on the mechanical properties, thus, we propose to incorporate this first-order probability function as a first measure in (9). The spectral density seems to be a suitable measure for the characterization of periodic information in a microstructure and covers also directional information. Therefore, we also take into account the side condition that the spectral density of a real microstructure should be as similar as possible to the spectral density of the SSRVE. Then we set nsm = 2 and end up in the least-square functional L(γ ) := ω1 LSD + ω2 LV with LSD (γ ) =

1 Nx Ny

LV (γ ) = 1 −

Ny Nx

2 real SSRV E PSD (m, k) − PSD (m, k, γ ) , m=1 k=1 PVSSRV E (γ ) PVreal

(17)

2 ,

where we introduced a reasonable normalization and set ω1 = ω2 = 1. For the parameterization of the inclusion morphology of the SSRVE splines are used, thus, the coordinates of the sampling points arranged in γ represent the degrees of freedom in the optimization problem. In order to get reasonable results it may be necessary not to consider the spectral density at a very fine resolution level. Therefore, first the spectral density is computed at a high resolution and the trivial entry of the SD is discarded. Second, the spectral density is rebinned such that a lower resolution is obtained and then the SD is normalized. In Figures 3a and 3b the original and rebinned spectral density of a typical microstructure is shown, where the x- and yaxis are the frequency axis and greyscale colors show the normalized values of the SD. Please note that the origin of the frequency domain is located in the center of the illustrations. In order to improve the efficiency of the method a threshold value


23

A (a)

(b)

Fig. 4 Simple test example: (a) generated target structure and (b) initial configuration in the optimization process for finding the SSRVE.

ς is defined as a lower bound of values in the spectral density that characterize a relevant SD for the optimization problem, see the blue frame in Figure 3b and its amplification in Figure 3c. Finally only this relevant area of the SD is considered for the computation of the associated term in the objective function LSD . The volume fractions required for the computation of LV are directly computed from the input image of the given real microstructure and the image of the SSRVE obtained by the spline parameterization.

3.3 Numerical Optimization Method For an illustration of the main characteristics of the above mentioned optimization problem let us first consider a simple test example, where we consider an assumed real two-dimensional microstructure with one inclusion generated by randomly distributing four sampling points, see Figure 4a. In the sequel we denote this microstructure as the target structure. This target structure should be recovered by minimizing the objective function L. In this example we start from the microstructure depicted in Figure 4b, where we fix three sampling points and let only one point free (point A), so we obtain two degrees of freedom. The objective function is plotted over the degrees of freedom, see Figure 5, and we notice by zooming into the minimum that the objective function is far away from being smooth.

Fig. 5 Visualization of the objective function for a simple test example with two magnifications of the minimum area.

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D. Balzani et al. a

frame k + 2 = frame k + 1

frame k + 1 a if L(M1,k )

M3 d

M1

L(M0,k+1 ) M0

M3 M0

for j = 1...4:

M4

new random Mj M2

Frame k

(a)

(b)

M4 M2

(c)

Fig. 6 Schematic illustration of moving frame algorithm.

Apparently, the computation of the discrete spectral density and the volume fraction takes into account a specific discrete image resolution. Hence, this leads to a non-smooth function and precludes the application of standard gradient-based optimization procedures. Furthermore, we have to deal with many local minima when increasing the number of degrees of freedom, so optimization becomes even more difficult. To overcome the difficulties arising from the particular minimization problem a moving frame algorithm is applied. For this purpose we first randomly generate a starting point M0,k and then generate further nmov random points in a frame of the size 2a × 2a, see Figure 6a. Then the objective function L is evaluated at these points, the frame center is moved to the point M0,k+1 defined by the lowest value of L and the iteration counter is initialized lit ermax = 0. If the frame center remains unaltered, i.e. no new minimum of L is found in this iteration step (M0,k+1 = M0,k+2 ), we set lit er = lit er + 1. If lit er = lit ermax the stopping criterion is reached and the actual minimal value of L is interpreted as local minimum associated to the starting value. In addition, this procedure is repeated a predefined number of cycles with different random starting values. If a high fraction of minimizers of the individual optimization cycles leads to a similar SSRVE, then we choose this result as an appropriate solution. In order to improve the method the frame size a can be modified depending on the difference |d| and lit er . Furthermore, a combination with a line-search algorithm is implemented, where L is also evaluated at a number of nline points interconnecting the frame center point M0 with the random points M1 , M2 , . . . , Mnmov .

4 Numerical Example For the analysis of several “real” microstructures that are characterized by a completely different inclusion morphology it is reasonable to consider randomly generated microstructures that are treated as target structures. One possibility for the generation of such target structures is provided by the Boolean method, where ellipsoids built from the matrix material are inserted at random points in a pure inclusion material until a predefined volume fraction is reached. As an example, Figure 7a shows


(b)

(a) rx /ry = 5, rx ∈ [1, 3]

25

(c) PV = 0.1859

(d) nele = 1888

Fig. 7 Steps for the generation of a target structure: (a) result of the Boolean method, (b) smoothed target structure, (c) relevant area of the spectral density, and (d) discretization of the target structure.

the result of applying the Boolean method for an aspect ratio of the semi-principal axis of rx /ry = 5 and randomly generated rx ∈ [1, 3]. The stopping criterion for the Boolean method is given by a volume fraction of the inclusion phase of 0.2 ± 1%. In the next step we smoothen the boundaries of the inclusions in order to avoid singularities, see Figure 7b. The resulting volume fraction is PV = 0.1859 and the relevant spectral density is shown in Figure 7c. For Finite-Element simulations the smoothed target structure is discretized by 1888 triangular elements with quadratic shape functions, see Figure 7d. For an illustration of the performance of the proposed method four different types of inclusion morphologies are taken into account: one inclusion with three sampling points (type I) leading to convex inclusions, one inclusion with four sampling points (type II), and two inclusions with three and four sampling points each (type III and type IV), respectively. The target structure shown in Figure 7b is considered and the optimization process is performed for the four different types of SSRVEs. The results are shown in Figure 8, where the discretizations and the relevant spectral densities are depicted. Type I

Type II

L = 0.022 nele = 174

L = 0.0157 nele = 224

Type III

Type IV

L = 0.0125 nele = 174

L = 0.0055 nele = 496

Fig. 8 Discretizations of optimized SSRVEs and the associated relevant areas of spectral densities; nele denotes the number of finite elements.

D. Balzani et al.

26 σx [MPa]

Horizontal Tensiontension x1

600

550

Tension x1 σx [MPa] Horizontal tension

500

2500

450

2000

400

1500

target I II III IV

350

300

target pure matrix pure inclusion

1000

500

250

0

0

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

lx / lx,0

lx / lx,0

σy [MPa]

Vertical tension Tension x2

600 550

σy [MPa]

500

2500

450

2000

400

1500

target I II III IV

350 300

Tension x2 Vertical tension


1000 500

250

0

0

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

ly / ly,0

ly / ly,0

σxy [MPa]

Shear Shear

240

σxy [MPa]

220

Shear Shear

1400

200

1200

180

1000

160 140

120 100

0


800

target I II III IV

600 400 200

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

ux / ly,0

0

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

ux / ly,0

Fig. 9 Stress-strain response of the target structure and the SSRVE type I to IV (left) and of the pure matrix and inclusion material (right) for the three virtual experiments horizontal tension, vertical tension and shear.


27

As expected, the values of the objective function decrease with an increasing number of sampling points, and therefore with increasing complexity of the SSRVE. In order to study the SSRVE’s capability to reflect the mechanical response of the target structure we compare the stress-strain response of the SSRVEs with the response of the target structure in three virtual experiments: tension in (i) horizontal and (ii) vertical direction, and (iii) shear. The resulting stress-strain curves of the tests are shown in Figure 9. For the vertical tension test all SSRVEs fit the curve accurately. But for the shear test and especially for the horizontal tension test there is a significant discrepancy between the results of types I, II, III and IV. In addition, we observe a decreasing error for increasing degrees of freedom in the SSRVE generation for the horizontal tension test. Summarizing, type IV leads to the best mechanical results while having the most complex inclusion morphology, although the connections between the inclusions seem unphysical. For type III, where we consider separated convex inclusions, we also observe a very good aggreement with the target structure response. Therefore, we conclude that more than one inclusion may be necessary to describe the mechanical response. It is remarked that although the computation time of one virtual experiment using the SSRVE is much less than using the target microstructure, the main effort of the proposed method becomes significant when FE2 -simulations are performed for relevant non-homogeneous macroscopic boundary value problems where more than only a few macroscopic finite elements are necessary. Then, the computation time for minimizing the least-square functional is of minor importance since this process needs only to be performed once for each type of microstructure.

5 Conclusion In this contribution a method for the generation of statistically similar representative volume elements (SSRVEs) is proposed using statistical measures describing the inclusion morphology. In this context an objective function is considered, which takes into account the difference of the spectral density and the volume fraction of a real microstructure and the SSRVE. For the minimization of the objective function a moving frame algorithm is applied. A numerical example comparing the mechanical response of a real microstructure and the generated SSRVEs shows the performance of the proposed method.

Acknowledgement The financial support of the “Deutsche Forschungsgemeinschaft” (DFG), project no. SCHR 570-8/1, is gratefully acknowledged.

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References 1. D. Balzani and J. Schröder, Some basic ideas for the reconstruction of statistically similar microstructures for multiscale simulations, Proceedings of Applied Mathematics and Mechanics 8, 2008, 10533–10534. 2. M. Beran, Statistical Continuum Theories, Wiley, 1968. 3. R. Hill, Elastic properties of reinforced solids: some theoretical principles, Journal of the Mechanics and Physics of Solids 11, 1963, 357–372. 4. C. Miehe, J. Schröder and J. Schotte, Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials, Computer Methods in Applied Mechanics and Engineering 171, 1999, 387–418. 5. J. Ohser and F. Mücklich, Statistical Analysis of Microstructures in Materials Science, J. Wiley & Sons, 2000. 6. E. Parzen, Stochastic Processes. Holden-Day, San Francisco, CA, 1992. 7. G.L. Povirk. Incorporation of microstructural information into models of two-phase materials, Acta Metallurgica et Materialia 43(8), 1995, 3199–3206. 8. J. Schröder, Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Stabilitätsproblemen, Habilitationsschrift, Institut für Mechanik (Bauwesen), Lehrstuhl I, Universität Stuttgart, 2000. 9. J. Schröder, D. Balzani, H. Richter, H.P. Schmitz and L. Kessler, Simulation of microheterogeneous steels based on a discrete multiscale approach, in Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, P. Hora (Ed.), 2008, pp. 379–383. 10. R.J.M. Smit, W.A.M. Brekelmans and H.E.H. Meijer, Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling, Computer Methods in Applied Mechanics and Engineering 155, 1998, 181–192. 11. S. Torquato, Random Heterogeneous Materials. Microstructure and Macroscopic Properties, Springer, 2002.

Advancements in the Computational Calculus of Variations Carsten Carstensen and Cataldo Manigrasso

Abstract The aim of this paper is a survey on some state-of-the-art techniques in the numerical analysis of minimisation problems in nonlinear elasticity and on the concepts of quasi and polyconvexity in the calculus of variations. The issue of the approximation of singular minimisers is addressed via a penalty finite element method (PFEM) and macroscopic deformations are computed via the relaxation finite element method (RFEM). New numerical results are presented for the scalar doublewell problem as a benchmark in computational microstructures with adaptive meshrefining, in order to recover optimal convergence in the presence of singularities at interfaces.

1 Introduction This paper is devoted to nonlinear phenomena modelled as minimisation problems and to their numerical simulation. Particular attention is given to applications in material science and in nonlinear elasticity, such as phase transitions in solids and fluids, but the possible fields of application span from mechanics and optics to geometry [14]. The direct method of calculus of variations allows a proof of existence of minimisers for the energy functional W (x, Dv) dx − φ(x, v) dx. (1) E(v) :=

Carsten Carstensen Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany and Department of Computational Science and Engineering, Yonsei University, 120-749 Seoul, Korea Cataldo Manigrasso Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany; E-mail: [email protected] K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_3, © Springer Science+Business Media B.V. 2010

29

30

Advancements in the Computational Calculus of Variations

Here v : → Rm belongs to the set of admissible deformations A on the bounded open Lipschitz domain ⊂ Rn , W is the stored-energy function and φ represents the body force and the low order contributions. If W is convex with respect to Dv and satisfies certain growth and smoothness conditions (see for example [14]), Tonelli’s theorem shows that E is weakly lower semicontinuous and as a consequence that a minimiser can be found. While in the scalar case, i.e. m=1 or n=1, convexity is a necessary and sufficient condition for the existence of minimisers, in higher dimensions it is only a sufficient one. Moreover, in this setting the assumption of convexity is not physically reasonable [1]. Morrey [19] proposed the more suitable class of quasiconvex functions, where f quasiconvex ⇔ E weakly lower semicontinuous also for dimensions higher than one (for m=1 or n=1 convexity = quasiconvexity). We recall that some function W : × Rm×n → R is quasiconvex if W (x0 , F0 ) dx ≤ W (x0 , F0 + Dξ(x)) dx, (2) ω

ω

holds for any x0 ∈ , any constant F0 ∈ Rm×n and any ξ ∈ C0∞ (ω, Rm ), with ω ⊂ a bounded open set with boundary ∂ω of zero volume [1, 19]. If m = n = 3 for example, this means that for an homogeneous body made from the material found in x0 and of the form given by ω, the homogeneous strain F0 must be a minimiser of the energy functional. Although it is shown under this and other suitable smoothness and growth conditions that some minimisers exist, quasiconvexity is a difficult concept to handle, since it is a global condition and it is tricky to verify if a given function is quasiconvex [18, 22]. On top of that, the computation of the quasiconvex envelope of a function, which plays an important role in the relaxation theory, is still an open problem. A weaker but easier to handle condition is rank-one convexity, i.e. W (A + ta ⊗ b) is convex in the real parameter t for all A and for all rank-one matrices a ⊗ b [14]. It has already been excluded that for n ≥ 2 and m ≥ 3 quasiconvexity and rank-one convexity are equivalent [22], but this question is still unanswered when m = 2 [20]. We finally recall the relations between the generalised notions of convexity introduced up to now. Given f : Rm×n → R, then f convex ⇒ f quasiconvex ⇒ f rank-one convex and if m = 1 or n = 1 all these notions are equivalent. The computational calculus of variations addresses the numerical evaluation of corresponding semiconvex hulls [15] and the analysis of resulting minimisation problems. The remaining parts of this paper are structured as follows. Section 2 introduces the notion of polyconvexity, presents some existence and uniqueness results obtained in a two dimensional problem with small loads and reports on an a priori estimate of the finite element discretisation error. Section 3 deals with a class of problems whose minimisers are not regular enough to be detected by the standard

C. Carstensen and C. Manigrasso

31

finite element method. Some reformulation of the problem via a penalty scheme decouples the deformation and the strain, recovers the convergence of the discrete energies to the exact value and allows for the computation of singular minimisers in the presence of the Lavrentiev phenomenon. Section 4 analyses non-convex minimisation problems, which in fact do not allow for a minimiser, even if they possess weakly limited infimising sequences. The attention of the numerical analysis is then drawn to relaxation schemes or to reformulations of the problem in the Young measure relaxation framework that aim at the calculation of such a limit.

2 Polyconvex Minimisation Ball introduced a stronger condition than quasiconvexity in [1] and proposed a new class of functions of the form W (x, F ) = g(x, F, det F ) W (x, F ) = g(x, F, adjF, det F )

for n = 2 for n = 3

with convex functions g(x, ·, ·) or respectively g(x, ·, ·, ·) for each x, and called them polyconvex functions W (x, ·). He showed then the existence of minimisers under more physically suitable conditions than those proposed by Morrey. In contrast to the aforementioned quasiconvexity condition, this one enables an easy treatment of the constraint det F > 0 and so includes many physically interesting materials like those of Mooney–Rivlin or Ogden. Polyconvexity turns out to be a sufficient condition for quasiconvexity. In particular, given f : Rm×n → R ∪ {+∞}, the following implications hold f convex ⇒ f polyconvex ⇒ f quasiconvex ⇒ f rank-one convex. Moreover, the implications are true also in the other direction if m = 1 or n = 1. Carstensen and Dolzmann [9] studied a special class of polyconvex energy functionals in the case m = n = 2 and obtained an optimal a priori estimate for the related finite element minimisers following Zhang [24]. Given an energy functional (1) with E(v) := (W (Dv) − f · v) dx, (3)

the key assumptions (H1)–(H5) of [24] read: (H1) ⊂ R2 bounded C 3 -Domain; (H2) G ∈ C 3 (M2×2 × (0, ∞); R) is convex and G(F, δ) ≥ α + β|F |p

∀F ∈ M2×2 , ∀δ > 0,

with p > 2, β > 0, α ∈ R; 2×2 : det A > 0} (H3) for γ > 0 and F ∈ M2×2 + := {A ∈ M

32


W (F ) = γ |F |p + G(F, det F ) ; (H4) g ∈ C 3 (; R2 ), f ∈ Lr (; R2 ) with r > 2, and A, defined by A := u ∈ W 1,p (; R2 ) : u = id + g on ∂, det Du > 0 a.e. in , is not empty; (H5) for all sequences (Fj )j ∈ in M2×2 and (δj )j ∈ in (0, ∞), with limj →∞ Fj = F and limj →∞ δj = 0, we have lim G(Fj , δj ) = +∞.

j →∞

The main result in the above mentioned paper by Ball guarantees the existence of a minimiser under these hypotheses, while a result by Zhang [24] implies its uniqueness and higher regularity provided the boundary conditions are a small perturbation of the identity. Theorem 2.1 (Zhang). Suppose (H1)–(H5). Then there exist ε > 0 and a function η : [0, ε] → R+ , which is continuous at zero with η(0) = 0, such that for all g ∈ C 3 (; R2 ), f ∈ Lr (; R2) with g W 3,∞ () + f + div

∂W (id + Dg) W r () < ε ∂F

(4)

there exists a unique weak solution u with u − id W 2,r () < η(ε) of div

∂W (Du) + f = 0 in ∂F

and u =id + g on ∂.

A stability estimate for the global minimiser u ∈ A and for an approximation w ∈ A ∩ W 1,∞ (; R2 ) was then derived, always in the case that both are not far from the identity. Under the same hypotheses of the preceding theorem, with |u − id|1,∞ and |w − id|1,∞ small enough, it was proved that for any v ∈ A such that E(u) ≤ E(v) ≤ E(w) the estimate p

|u − v|W 1,p () + |u − v|2W 1,2 () |u − w|2W 1,2 () .

(5)

holds, where the symbol is an abbreviation of ≤C for a certain constant C. Such a result can be applied directly to the discrete case where the minimiser is sought in the space of piecewise affine functions A (its existence is shown using a result of Ball similar to the one mentioned above). A modified interpolation operator : ˆ 1 (u − id − g), so that it is exact A → A is introduced, with u = id + g − on the boundary and it is the usual nodal interpolation operator elsewhere. Finally, the optimal a priori estimate is obtained as an application of Equation (5), since the discrete minimiser satisfies the inequality E(u) ≤ E(u ) ≤ E( u). Theorem 2.2. For all g ∈ C 3 (; R2 ), f ∈ Lr (; R2) with (H1)–(H5) and (4) and ε depending on the approximation properties of the triangulation, the unique


33

minimiser u of E and any minimiser u ∈ A satisfy p

|u − u |W 1,p () + |u − u |2W 1,2 () |u − u|2W 1,2 () h 2L∞ () (ε + η(ε)). Here h is a piecewise constant function such that h (x) is the diameter of the element of the triangulation that contains x. Notice also that η(ε) is the same as in Theorem 2.1.

3 Penalty Finite Element Method If the minimiser u is singular, i.e. it is discontinuous or its gradient is unbounded, its computation can be very challenging but it is nonetheless of interest since u possibly describes cavitation or fracture of materials. The difficulty to overcome in these cases is that any conforming finite element space is not capable of approximating sufficiently well the singular minimising function so to detect the minimum of the energy, a phenomenon named after Lavrentiev (1928). In order to have a more precise formulation of this phenomenon, let us define the space of admissible deformations as 1,p

Ap := W0 (; Rm ) = {v ∈ W 1,p ()m : v|∂ = 0}, and the problem as find inf E(A1 ) := inf E(v), v∈A1

with an energy E : A1 → R ∪ {+∞} as in (2). The inclusion A∞ ⊂ A1 implies that, if v ∈ A∞ such that E(v) < +∞ exists, then −∞ < inf E(A1 ) ≤ inf E(A∞ ) < +∞. The Lavrentiev phenomenon consists in a gap between the two values above, so that in fact inf E(A1 ) = inf E(A∞ ). (6) This precludes the possibility of approximating with an arbitrary accuracy inf E(A1 ) with a conforming discrete method, since any conforming finite element space is contained in A∞ . Moreover, what typically happens is that, if a minimiser u exists, for any sequence (uj ) ⊂ A∞ such that uj → u a.e., a sort of repulsion effect occurs, so that E(uj ) → +∞ and no minimising sequence can converge to u (see for example [5] for a one-dimensional example). If one knows in advance where the singularities occur and which order they have, it is possible to choose appropriate basis functions in a non-standard enhanced finite element method so to reproduce this behaviour. In general, however, this information is missing and one has to proceed differently trying to reformulate the problem so to decouple the deformation

34


and its gradient. In this way one manages to move around this sort of repulsion barrier and to enter the basin of attraction of the minimiser (see Ball and Knowles in [4] for the one-dimensional case and Negrón–Marrero in [21] for higher dimensions). We will here describe in more detail the procedure adopted by Carstensen and Ortner, which stems out of the previous ones but seems to be of more practical use. In [12] they proved the equivalence of finite element failure and Lavrentiev phenomenon and then proposed a penalty scheme to detect singular minimisers. Given a sequence of finite element spaces (not necessarily nested) V0 , V1 , V2 , · · · ⊂ ∪∞ =0 V ⊆ A∞ , we say that the method is convergent if inf E(A1 ) = lim inf E(V ),

(7)

→∞

or that it fails otherwise. Now, it is clear that if the method is convergent, then the Lavrentiev phenomenon (6) cannot occur, but the converse is not so straightforward. We just sketch here how to prove it and refer always to [12] for the details. Given a vector v ∈ A∞ and its interpolation v := I v through a nodal interpolation operator I : A∞ → V , it can be shown that lim →∞ E(v ) = E(v) thank to a density argument, and then that lim →∞ inf E(V ) → inf E(A∞ ) = inf E(A1 ), namely the method is convergent. ¯ × Theorem 3.1 (Finite Element Failure ⇔ Lavrentiev Phenomenon). If W : m m×n → R is continuous, then lim →∞ inf E(V ) = inf E(A∞ ), and in R ×R particular, lim inf E(V ) = inf E(A1 )

→∞

⇐⇒

inf E(A1 ) = inf E(A∞ ).

Regarding the reformulation of the problem via a decoupling of the deformation and of its gradient, some techniques can be shown in the framework of polyconvex, convex and quasiconvex energy densities. As far as the polyconvex case is concerned, let us consider an energy functional like in Equation (3), with f ∈ Lq ()n for some q > 1, and an energy density of the form W (x, v, F ) = φ(x, F, det F ), (8) with φ : × Rn×n × R → [0, +∞]. Assume moreover that φ satisfies |F |n + (η) φ(x, F, η) 1 + |F |n + (η), and φ(x, ·, ·) is convex and l.s.c. in Rn×n × R

for a.a. x ∈ ,

(9)

where : R → [0, +∞] is convex and has superlinear growth. We take as space of admissible functions V = uD + W01,n ()n ,


35

with uD ∈ W 1,n ()n and E(uD ) < +∞. Under these hypotheses the minimisation problem has at least one solution u (see [14]). We then define a penalty functional : L1 ()2 → [0, +∞] of the form ψ(|η − ζ |)dx, (10) (η, ζ ) =

where ψ has 1-growth at infinity. Discretizing V and L1 () by V = uD, + P10 (T )n

and Y = P0 (T ),

with uD, → uD strongly in W01,n ()n , we finally define the penalised energy functional using a sequence ε 0 E (v , η ) = (v , η ) + ε −1 (det Dv , η ) φ(x, Dv , η ) − f · v dx + ε −1 (det Dv , η ). =

Notice that with this formulation the terms decoupled are the gradient and its determinant. The discrete problem takes then the form (u , η ) ∈ argmin E (V , Y ). The following theorem shows then that this discrete method is capable of detecting the singular minimiser, provided that the sequence (ε ) is aptly tuned. Theorem 3.2. Assume that (8), (9), and (10) hold. Then there exists a sequence ε 0 such that, for any sequence of (u , ξ ) ∈ argmin E (V , Y ), we have (u , ξ ) → inf E(V )

and ε −1 (det Du , ξ ) → 0.

Moreover, the family (u ) is precompact in the weak topology of W 1,n ()n in the sense that there exist accumulation points, and each accumulation point u is a minimiser of E in V . In particular, there exists a subsequence k ∞ such that u k u ξ k det Du

weakly in W 1,n ()n , weakly in L1 (),

where u is the solution of the exact minimisation problem. We would like to remark that no hint is given how to construct the sequence (ε ), which is anyway a crucial ingredient for the convergence. In the implementation of the method this sequence has to be designed carefully: choosing ε too small at the beginning may result in giving too much weight to the penalty functional and in coming too near to the original Galerkin discretisation. Is is therefore advisable to choose ε relatively large at the beginning so to obtain a minimiser far from the Galerkin solution, and then to gradually reduce it as the mesh is refined. More details on the so called continuation algorithm can be found always in [12].

36


Rather than showing analogous convergence results for the convex and quasiconvex cases, that anyway include relevant problems such as those of Manià type [5] and those proposed by Foss et al. [17] and Ball [2], we would like to state more precisely the relation between this new discrete formulation and the original continuous problem. To this end, let us define the functionals ⎧ if v ∈ V , η = det Dv, ε = 0, ⎨ E(v) F (v, η, ε) = (v, η) + ε−1 (det Dv, η) if v ∈ V , 0 < ε < ∞, ⎩ +∞ otherwise; (v, η) + ε−1 (det Dv, η) if v ∈ V , η ∈ Y , 0 < ε < ∞, F (v, η, ε) = +∞ otherwise. Slightly modifying some arguments used in the proof of theorem 3.2, one can show that both a liminf and limsup condition hold for the sequence of functionals F as compared with the limit F in the entire space W 1,n ()n × L1 () × [0, ∞). This is equivalent to saying that F is the -limit of the sequence (F ), written −lim F = F. →∞

This is the weakest convergence notion and states essentially that the arguments in the direct method of the calculus of variations are applicable [7].

4 Non-Convex Minimisation 4.1 Benchmark in Computational Microstructure If the energy density W is non-convex, then in general the energy functional E is not weakly lower semicontinuous and the existence of minimisers is not ensured. We address here the scalar double-well problem, that is discussed for example in [10] (or in a more general framework in [13]) and whose energy functional is defined by E(v) := W (Dv) dx + |v − f |2 dx,

with W : R2 → R, F → |F − F1 |2 |F − F2 |2 , √ where F1 := (3, 2)/ 13, F2 := −F1 and is the open rectangle (0, 1) × (0, 3/2). The problem (P ) consists in finding a minimiser of E in the set of all admissible functions A := uD +W01,4 (), where uD specifies the Dirichlet conditions. We refer the reader always to [10] for further details on the setting of the problem and for the expressions of the know exact solution u and of its stress σ , which will be later used to evaluate the convergence of the numerical method. What happens in this case is


37

that, although an infimising sequence developing finer and finer oscillations exists in A, the minimum can not be attained. At this point one has two choices, that in fact lead to equivalent results: one can substitute the energy density W with its lower convex envelope W ∗∗ to create artificially the properties sufficient for the attainment of the minimum (RP ), or, otherwise, the definition of solution can be generalised in the sense of the Young measure (GP ). The price that has to be paid in both cases is the loss of information on the microscopical level, that is the microstructure is averaged out of the solution (RP ) or is described only in a statistical sense (GP ). Indeed, we are just interested on some macroscale quantities, like the macroscopical deformation u, that is understood as the weak limit of infimising sequences (uj ), or the macroscopical stress σ = DW (Du). For the sake of clarity we state here below the problem (RP ) u = arg min E ∗∗ (v) v∈A

where E ∗∗ (v) :=

and set σRP := DW ∗∗ (Du),

W ∗∗ (Dv) dx +

|v − f |2 dx,

2

W ∗∗ (F ) := max{0, (|F |2 − 1)} + 4 |F |2 − ((3, 2) · F )2 /13 . On the other hand, we would like to remind the reader some technicalities before stating (GP ). Using standard arguments, one can argue that an infimising sequence of (P ) converges weakly uj u in W 1,4 (), and, as a consequence (see for example [14]), that this limit is the solution of the relaxed problem (RP ). The above sequence is in fact also bounded in W 1,4 () and this enables one to apply the theorem below, that can be found in [3], and to formulate the problem in the Young measure framework. The reader is reminded that a family of probability measures ν := {νx }x∈ is said a Young measure if the function ν, g : → R, x → νx , g := g(F ) dνx (F ) Rn

is measurable for all g ∈ C0 (Rn ). Moreover, a sequence (Fk ) in Lp (; Rn ) is said to generate a Young measure ν if ∗

g(Fk ) ν, g

in L∞ (),

for all g ∈ C0 (Rn ).

To have a more pictorial view, one can think of a ball centred in x, Bδ (x), and define (k) νx,δ as the probability distribution of the values of Fk (y) when y is picked uniformly at random from Bδ (x). Then (k)

νx = lim lim νx,δ , δ→0 k→∞

38


and the limits are understood in the sense of the weak* convergence [3]. A result reported in [13] reads Theorem 4.1. Assume that the sequence (uj ) is bounded in W 1,4 (), 1 ≤ p ≤ ∞. Then there exists a subsequence (ujk ) such that (Dujk ) generates a Young measure ν. Moreover, if g ∈ C(Rn ) and g(Dujk ) is sequentially weakly relatively compact in L1 (K), with K a measurable subset of , then g(F ) dν(F ) in L1 (K). g(Dujk ) Rn

Thanks to the above mentioned boundedness of the infimising sequence (uj ), we can apply this result and finally say that there exists a solution for the problem (GP ) (u, ν) ¯ = arg

min

v∈A ν∈GY M(;Rn)

E(v, ν)

and set σGP := where

Rn

D W (F ) d ν(F ¯ ),

E(v, ν) :=

|v − f |2 dx

νx , W (Dv) dx +

and GY M(; Rn ) is a suitable space for the Gradient Young Measures. For the explicit expression of ν¯ see [10]. The equivalence of (RP ) and (GP ) consists in the fact that the stress of the relaxed problem σRP and that of the generalised one σGP coincide, that is σRP = σGP . Moreover, they equal also the macroscopical stress σ .

4.2 Numerical Treatment The numerical discretisation of the original non-convex problem, (P ), leads to severe difficulties. The existence of a global discrete minimiser is ensured by the finite dimensionality of the problem, but the global minimisers are difficult to compute because they are surrounded by a cluster of local minimisers. On top of that, with this formulation the deformation u is also prone to develop oscillations dependent on the mesh size and orientation, and which therefore cannot describe the real microstructure configuration. On the other hand, (RP ) does not show the same disadvantages, since, thanks to the convexity of W ∗∗ , every local minimiser is also a global minimiser. The Galerkin discretisation is based on a regular triangulation T of into triangles and the space A is approximated through


39

A = uD, + P01 (T ; R)

(11)

¯ is the space of piecewise affine functions vanishing on where P01 (T ; R) ⊂ C() the border and uD, ∈ P 1 (T ; R) is the interpolation of uD on the boundary nodes. In [10] an a priori estimate of the error in (RP ) can be found. Theorem 4.2. (a) There exists a unique solution u of (RP ) and the functional E ∗∗ is uniformly convex on A . (b) Let u denote the generalised solution of (P), σ = DW ∗∗ (Du) and σ = DW ∗∗ (Du ). Then σ − σ L4/3 () + u − u L2 () ≤ C|u − I u|W 1,4 () with C independent from the mesh size. Here the phrase “generalised solution of (P )” indicates the weak limit of an infimising sequence (uj ) of E. Unfortunately, the known regularity of u is not enough to deduce from this theorem an optimal convergence rate. In fact, u ∈ W 3/2−ε,4 for any ε > 0 and so we can estimate the interpolation term on the right side of the in1/4 equality at best with a term h , where h is the maximal diameter of the triangles of T . Altough the outcome of the numerical experiments with uniform refinement is more encouraging, since both the error on the stress and that on the deformation 3/4 seem to converge with speed h , this is still suboptimal and is a clear hint that an adaptive algorithm should be used. The adaptive mesh-refining strategy that has been adopted works as follows. Given an initial triangulation T0 , the adaptive finite element algorithm (AFEM) generates a sequence of triangulations T1 , T2 , . . . and associated discrete subspaces A1 , A2 , . . . , and solves the discrete problems (RP1 ), (RP2 ), . . . , which yield the deformations u1 , u2 , . . . and the stresses σ1 , σ2 , . . . , the whole process being driven by a refinement indicator ηT . The loop that occurs at each level consists of the steps SOLVE → ESTIMATE → MARK → REFINE as explained more in detail below. INPUT – As it is observable in the exact solution [10], singularities are expected along the diagonal of the rectangular domain with (0, 3/2) and (1, 0) as extremes. In order to test the capability of the adaptive algorithm to refine where the solution is irregular, the starting triangulation T0 is chosen so that it cannot resolve the diagonal along which the singular behaviour appears, that is no side of a triangle lies on the diagonal. SOLVE – The discrete spaces A described in Equation (11) are used to approximate the solution u. The minimisation problem u = arg min E ∗∗ (v ) v ∈A

and set σ := DW ∗∗ (Du ),

40


is solved finding the zero of the first derivative of the convex functional E ∗∗ DE ∗∗ = 0 through a Newton–Raphson method that solves D 2 E ∗∗ (u ,j )r ,j = −DE ∗∗ (u ,j ), and then sets u ,j +1 = u ,j + αr ,j , α being a parameter. The starting term of the iteration u ,0 is a nodal interpolation of u −1 on A and the iteration on j stops when r ,j is less than a certain tolerance threshold. Once u is obtained, the calculation of the discrete stress σ is straightforward. ESTIMATE – Three different estimators have been proposed and justified in [10], namely •

the residual-based error estimator η R

:=

3/4

ηTR

,

T ∈T

with 4/3

ηTR = hT

•

|2 (f − u )|4/3 dx + T

hE |J (σ · nE )|4/3 ds,

(12)

(∂T )∩

where nE the external normal of a side E of a triangle T and J (σ · nE ) the jump of the normal component of σ on that side; the averaging error estimator η A

:=

T ∈T

with

4/3 σ − A(σ ) L4/3 (T )

3/4 ,

1 A(σ ) := σ (y) dy φz , |ωz | ωz z∈N

•

where ωz is the patch made out of the triangles that share the node z and φz the nodal basis function of A centred in z; the D2-error estimator

ηTD2 , η D2 := T ∈T

with


ηTD2

3/4

7/3 4/3 := 2h2T (f − u )) L4/3 (T ) + hE J (σ · nE ) L4/3 (E) ×

41

E∈E(T )

4 h−1 E |[∇u ] · nE |

1/4

,

E∈E(T )

where the last multiplicative term is a heuristical approximation of D 2 u L4 (E) [10]. The three error estimators are shown to be reliable. Theorem 4.3. Let u and u solve (RP) and (RP ) with σ := DW ∗∗ (Du) and σ := DW ∗∗ (Du ), respectively. Then there hold the following a posteriori error estimates. (a) σ − σ 2L4/3 () + u − u 2L2 () ≤ c1 η R + h.o.t. (b) σ − σ 2L4/3 () + u − u 2L2 () ≤ c2 η A + h.o.t. (c) If u ∈ W 2,4 () then σ − σ 2L4/3 () + u − u 2L2 () ≤ c3 η D2 + h.o.t. The constants c1 , c2 , c3 depend on E ∗∗ (u) + E ∗∗ (u ) and the shape of the elements in T . For η R the theory predicts an efficiency-reliability gap, that is η R − h.o.t. σ − σ L4/3 () (η R )1/2 + h.o.t., so we would expect that the stress error behaves like (η R )β , for some β such that 1/2 ≤ β ≤ 1. Anyway, what we obtain in the numerical experiments clearly shows that η R is a good estimator σ − σh L4/3 () ≈ η R . The same can be repeated for η A , that anyway gives a significantly sharper approximation of the stress error in comparison to η R , as can be seen from the figures. The estimation procedure consists in the computation of ηT for each T in T and of the upper error bound

α ηT , η = T ∈T

where α = 3/4 for ηTR , ηTA , and α = 1 for ηTD2 . MARK – The marking was done selecting a minimal subset M of T with the bulk criterion

1/α ηT η ≤ T ∈M

for some global parameter 0 < < 1.

42


REFINE – The triangles in the set M are refined via the red-refinement procedure. Further refinement then occurs in order to avoid hanging nodes. This computes new triangulation with red-green-blue refinement. Explanations on the red-green-blue refinement procedure here applied can be found in [23]. We would like to emphasise that thanks to the adaptive method described above with any of the three indicators the numerical method recovers its optimal convergence rate in the sense that σ − σ L4/3 () ∝ N −1/2 ≈ h .

4.3 Improvements on the Estimates – Oscillations The first term in the residual-based error estimator (12), namely the volume term, can be substituted by oscillations of the known term f , that are defined as follows in a patchwise way:

osc(f, K )4/3 := osc(f, ωz )4/3 , z∈K

where K is the set of internal nodes and osc(f, ω) := diam(ω) f − fω L4/3 (ω) , with fω the average of f over the patch ω. The following reliability estimate, which is known from [10] and [13] and which is presented in a more general setting in [8], shows in fact that σ − σ 2L4/3 () η + osc(f, K ), where η :=

3/4 ηT

and ηT =

hE |J (σ · nE )|4/3 ds. (∂T )∩

T ∈T

Since we are now dealing with both element-based and patch-based error indicators we need a suitable marking criterion that substitutes the one proposed in the preceding subsection. We used the one suggested by [11]. MARK – Seek the subset M ⊂ T ∪ K of minimal complexity with:

4/3 ηT + osc(f, ωz )4/3 . (η + osc(f, K )4/3 ) ≤ T ∈M ∩E

T ∈M ∩K

As pointed out in a remark in the same paper, if the edge residuals dominate the oscillations, this criterion becomes a much simpler marking strategy based only on


43

the edge residual indicators ηT , of the type described previously. This is in fact our case, since the known term f is a polynomial whose oscillations are not significant if compared to the surface terms.

4.4 Improvements on the Estimates – A New Estimator In the following we recall briefly the last part of the proof of the reliability of the error estimator η D2 presented in [10], in order to obtain a new estimator. Let us consider a function w ∈ W 1,4 () that takes into account the error approximation on the boundary, that is w = uD − uD, on ∂. A remarkable result presented in [10] and used in the proof of theorem 4.3 is the following estimate of the stress and displacement error c −1 σ − σ 2L4/3 () + 2 u − u 2L2 () ≤ c 3 |w|4W 1,4 () + 4 w 2L2 () + 4 Res(u − u − w),

(13)

where the functional on the right hand side indicates the residual σ · Dv dx. Res(v) := 2 (f − u ) v dx −

We choose w so to minimise the first two terms on the right hand side of (13). Namely we set wT = 0 if T ∩ ∂ = ∅ and otherwise we damp linearly to zero the value of w on the border. Be z the centre of the largest inner circle contained in T and y ∈ T a point on ∂. For any point on the segment that joins them, x = λy + (1 − λ)z, we set w(x) = λw(y). In this way we can control both w and its gradient |∇w|W 1,∞ (T ) w/ hT L∞ (∂T ) + ∂E w/∂s L∞ (∂T ) hT ∂E2 uD /∂s 2 L∞ (D ) , w L∞ (T ) w L∞ (T ) h2T ∂E2 uD /∂s 2 L∞ (D ) . Besides, since w = 0 only on a boundary layer of elements, we conclude that |w|4W 1,4 () + w 2L2 () = o(h) = h.o.t. Always in [10] it is shown that the constant c can be controlled from above. At this point the only term in Equation (13) still to be estimated is Res(u − u − w). We know from the discrete Euler–Lagrange equations that P01 (T ) ⊂ Ker(Res) and so, after setting v := u − u − w, we take v ∈ P01 (T ) and use the linearity of Res to write Res(v) = Res(v − v ). Choosing v = I u − u , where I u is the nodal interpolation of the exact solution in P01 (T ), we obtain after an elementwise integration by parts

44


Res(v − v ) =

(div T σ + 2(f − u ))(v − v ) dx +

J (σ · nE ) (v − v ) dx, E

(14) with Cauchy inequalities, this leads to

Res(v − v ) ≤ div σ + 2(f − u ) L4/3 (T ) u − I u − w L4 (T ) T ∈T

+

J (σ · nE ) L4/3 (E) u − I u − w L4 (E) .

(15)

E∈E

Since piecewise linear functions are used to approximate the deformation u, the vanishing term div σ can be dropped. Notice also that the sum on the edges actually runs only over the inner edges, because the jump of the stress is zero on the border of . Moreover, thanks to the above choice, w = 0 on the interior edges and so we can drop it in the second line. The weight of the surface term on the right hand side of the equation can be reshaped using a trace inequality [6] 4 3 4 u − I u 4L4 (E) h−1 T u − I u L4 (T ) + hT D(u − I u) L4 (T ) .

(16)

This yields Res(v − v )

2(f − u ) L4/3 (T ) u − I u − w L4 (T )

T ∈T

+

J (σ · nE ) L4/3 (E)

E∈E

1/4 4 3 4 × h−1 . T u − I u L4 (T ) + hT D(u − I u) L4 (T ) The main difficulty encountered in the implementation of an estimator based on the right-hand side of the above inequality is finding a suitable substitute for the unknown difference u − I u. What we did, following a widely used averaging technique, was to use the discrete solution u instead of the first order interpolation of the exact solution I u, and a second-order interpolation of u instead of u u − I u − w L4 (T ) ≈ J2 u − u − w L4 (T ) . Here the second-order interpolation operator is defined as J2 : A → A ,2 := P02 (T ; R) + uD, ,2 , where uD, ,2 ∈ P 2 (T ; R) is an interpolation of uD . The operator J2 was first presented in [25] and is based on a patch-oriented least-square fitting of the piecewise linear function to be approximated. More in detail, given a piecewise affine function u , in order to determine the coefficient of the nodal function φz of A ,2 , we look for a polynomial in P 2 (ωz ) that best fits the values of u in the midpoints of the sides of the triangles of the patch ωz . The value of this polynomial in z becomes


45

the coefficient of φz . The coefficient of a degree of freedom lying on the midpoint of a side is then determined via a linear interpolation of the coefficients of the two vertices that were just found. This procedure is of course not adopted on the boundary, where we always use a nodal interpolation uD, ,2 of the known Dirichlet data. Summing up, the proposed error estimator reads

η P 2 := 2(f − u ) L4/3 (T ) J2 u − u − w L4 (T ) T ∈T

+

J (σ · nE ) L4/3 (E)

E∈E

1/4 4 3 4 × h−1 . T J2 u − u L4 (T ) + hT D(J2 u − u ) L4 (T ) The adaptive scheme used for the mesh refinement with this error estimator is analogous to that used for η D2 . The figures clearly show that it exhibits the same good behaviour as the aforementioned ones. Figures 1 to 4 show the behaviour of the new error estimator when refinement is performed with the already known error indicators. Figure 5 shows that the convergence of all the error estimators presented is preserved also with the new error indicator η P 2 . Moreover, Figure 6 confirms that the new indicator induces refinement along the diagonal interface and in the lower half of the mesh, where oscillations are expected to occur as explained in [10]. Finally Figure 7 displays the discrete solution obtained with the adaptive method explained above.

5 Conclusions After a review of the most recent techniques PFEM and RFEM of the computational calculus of variation, focus was on the numerical analysis of the relaxed formulation of a non-convex minimisation benchmark problem in computational microstructures. The result of the numerical experiments performed in [10] have been fully confirmed and the analysis of [8] has suggested a new treatment of the oscillations. A new estimator η P 2 for the error of the macroscopical deformation u and the stress σ has been proposed, whose reliability was proved developing arguments presented in [10]. The implementation of this estimator needed however some heuristical approximation of u−I u 4L (T ), the norm of the difference between the exact solution and its interpolation on the discrete space. Following Rannacher’s paradigm [6], u was substituted with a second-order interpolation of the discrete solution u , and I u with u itself. Such empirical arguments had already proven successful in the implementation of the error estimator η D2 in [10], and in fact the behaviour of the new estimator confirms that of those [10]. In practical experiments, the new estimator shows also efficiency, reproducing up to a multiplicative constant the behaviour of the stress error of the solution, which converges optimally as the mesh size h tends to zero. It is also to be remarked that Figures 1 to 5 show an initial transient in which

46


Fig. 1 Convergence history of estimators η R , η A , η D2 , η P 2 as functions of the number of unknowns N = 1/ h2 for meshes computed by AFEM with refinement indicated by η R .

Fig. 2 Convergence history of estimators η (with oscillations), η A , η D2 , η P 2 as functions of the number of unknowns N = 1/ h2 for meshes computed by AFEM with refinement indicated by η (with oscillations).


47

Fig. 3 Convergence history of estimators η R , η A , η D2 , η P 2 as functions of the number of unknowns N = 1/ h2 for meshes computed by AFEM with refinement indicated by η Z .

Fig. 4 Convergence history of estimators η R , η A , η D2 , η P 2 as functions of the number of unknowns N = 1/ h2 for meshes computed by AFEM with refinement indicated by η D2 .

48


Fig. 5 Convergence history of estimators η R , η A , η D2 , η P 2 as functions of the number of unknowns N = 1/ h2 for meshes computed by AFEM with refinement indicated by η P 2 .

Fig. 6 Triangulation obtained refining with η P 2 (1025 DoF).


49

Fig. 7 Discrete solution obtained refining with η P 2 (1025 DoF).

η P 2 seems to converge slowlier in adaptive schemes than in uniform ones. This is however just a pre-asymptotic phenomenon, which is expected when such higherorder interpolation occurs, while the asymptotical behaviour is indeed optimal.

Acknowledgments The authors used some MATLAB realisation of SOLVE due to Robert Huth, whose contribution is thankfully appreciated. The work of the first author was partly supported by the WCU program through KOSEF (R31-2008-000-10049-0). The work of the second author was supported by the Research Training Group Graduiertenkolleg 1128 of the DFG Research Center MATHEON.

50


References 1. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63, 1977, 337–403. 2. J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A 306(1496), 1982, 557–611. 3. J.M. Ball, A version of the fundamental theorem for Young measures, in ıPartial Differential Equations and Continuum Models of Phase Transitions, Proceedings, M. Rascle, D. Serre, and M. Slemrod (Eds.), Springer Lecture Notes in Physics, Vol. 359, Springer, 1989, pp. 207–215. 4. J.M. Ball and G. Knowles, A numerical method for detecting singular minimisers, Numer. Math. 51, 1987, 181–197. 5. J.M. Ball and V.J. Mizel, Singular minimisers for regular one-dimensional problems in the calculus of variations, Bull. Am. Math. Soc., New Ser. 11, 1984, 143–146. 6. R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numerica, Cambridge University Press, 2001, pp. 1102. 7. A. Braides, Gamma-Convergence for Beginners, Oxford University Press, USA, 2002. 8. C. Carstensen, Convergence of adaptive FEM for a class of degenerate convex minimisation problems, IMA J. Num. Math., 2008. 9. C. Carstensen and G. Dolzmann, An a priori error estimate for finite element discretisations in nonlinear elasticity for polyconvex materials under small loads, Numer. Math. 97, 2004, 67–80. 10. C. Carstensen and K. Jochimsen, Adaptive finite element error control for non-convex minimization problems: Numerical two-well model example allowing microstructures, Computing 71, 2003, 175–204. 11. C. Carstensen and Y. Kondratyuk, Adaptive finite element methods for uniformly convex minimisation problems of optimal complexity, in preparation, 2008. 12. C. Carstensen and C. Ortner, Computation of the Lavrentiev phenomenon, in preparation, 2008. 13. C. Carstensen and P. Plecháˇc, Numerical solution of the scalar double-well problem allowing microstructure, Math. Comp. 66, 1997, 997–1026. 14. B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, 2007. 15. G. Dolzmann, Variational Methods for Crystalline Microstructure – Analysis and Computation, Lecture Notes in Mathematics, Vol. 1803, Springer, 2003. 16. L.C. Evans, Partial Differential Equations, American Mathematical Society, 2002. 17. M. Foss, W.J. Hrusa and V.J. Mizel, The Lavrentiev gap phenomenon in nonlinear elasticity, Arch. Ration. Mech. Anal. 167(4), 2003, 337–365. 18. J. Kristensen, On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré Anal. Non Linéaire 16(1), 1999, 1–13. 19. C.B. Morrey, Quasiconvexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2, 1952, 25–53. 20. S. Müller, private communication. 21. P.V. Negrón-Marrero, A numerical method for detecting singular minimisers of multidimensional problems in nonlinear elasticity, Numer. Math. 58(2), 1990, 135–144. ˘ 22. V. Sverák, Rank-one convexity does not imply quasiconvexity, Proc. Royal Soc. Edinburgh, Sec. A. Math. 120(1–2), 1992, 185–189. 23. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, Wiley-Teubner, 1996. 24. K. Zhang, Energy minimisers in nonlinear elastostatics and the implicit function theorem, Arch. Rat. Mech. Anal. 114, 1991, 95–117. 25. O.C. Zienkiewicz and J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Int. J. Numer. Meth. Engrg. 33, 1992, 1331–1364.

A Phase Field Approach to Wetting and Contact Angle Hysteresis Phenomena Antonio DeSimone, Livio Fedeli and Alessandro Turco

Abstract We discuss a phase field model for the numerical simulation of contact angle hysteresis phenomena in wetting. The performance of the model is assessed by comparing its predictions with experimental data on the critical size of drops that can stick on a vertical glass plate.

1 Introduction It is well known that liquid drops can adhere to vertical or inclined plates by exploiting surface tension and frictional forces that can pin the contact line. These forces give rise to a phenomenon, called contact angle hysteresis, that allows the drop to adapt its contact angles and resist to gravity [1]. If gravity does not exceed the pinning forces, the drop can stick and stand in an equilibrium state; otherwise it begins to roll down. Since hysteresis is affected only by the type and the microscopic geometry of the solid, once the inclination of the plate is fixed there is a critical volume beyond which the gravity component along the slope of the surface cannot be balanced, and motion starts. This scheme is quite simple in a two dimensional geometry (in such a case we deal with a critical area Acrit ), see Figure 1. A 2-D drop is in equilibrium if ρgA + σLV cos θA − σLV cos θR ≤ 0,

(1)

where σLV stands for the surface tension (which is a force per unit length), ρ for the liquid density, g for the usual gravity acceleration, and θA and θR for the advancing contact angle and the receding contact angle, respectively. The maximum value Acrit of A compatible with (1) is Antonio DeSimone · Livio Fedeli · Alessandro Turco SISSA – International School for Advanced Studies, Via Beirut 2-4, 34014 Trieste, Italy; E-mail: {desimone, fedeli, turco}@sissa.it K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_4, © Springer Science+Business Media B.V. 2010

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σLV cos θR

σLV

θR

W σLV cos θA θA

Fig. 1 Physical scheme of the forces involved in the problem. The ability of the drop to change its contact angle allows it to stick.

Acrit =

σLV (cos θR − cos θA ) . ρg

σLV

(2)

Formula (2) prompts two remarks. The first is that necessary condition for adhesion is that cos θR = cos θA (this implies that Young’s law is violated, i.e. the drop in Figure 1 is not a stationary point for the energy of capillary surfaces). The second one is that the critical area is proportional to the difference (cos θR − cos θA ). The three dimensional case, which is the one of interest for applications, is more complex. Dussan describes in [2] the critical configuration of a drop on a tilted plane, with small hysteresis (i.e. cos θR − cos θA small), by studying its dynamics in the limit of vanishing gravitational force. In this way, he estimates that the largest drop that can stick to the surface of the solid inclined at a given angle γ is 3 3 1 96 2 σLV (cos θR − cos θA ) 2 (1 + cos θA ) 4 1 − 32 cos θA + 12 cos3 θA Vcrit ∼ . 1 3 π ρg sin γ (cos θA + 2) 2 (1 − cos θA ) 4 (3) The 3/2 power in the second factor on the right hand side of (3) is obvious from dimensional analysis. However, the critical volume depends on cos θR and cos θA not only through their difference, but also through a non-dimensional correction factor depending on cos θA alone. In this paper we investigate the problem of finding the shape and the critical size of a water drop surrounded by air and in contact with a vertically tilted glass. The mathematical scheme we use to find a solution is based on a phase field model, which we adapt to take into account the presence of hysteresis. While the use

A Phase Field Approach to Wetting and Contact Angle Hysteresis Phenomena

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of phase field models to study wetting problems is by now classical (see, e.g., [3,4]), their application to the study of contact angle hysteresis is new. In particular, we replace the problem of minimizing capillary energy with a family of discrete incremental problems (a one-parameter family parametrized by the evolving loading conditions) in which the sum of energy change and dissipation due to the motion of the contact line is minimized. The geometry of the drop is described by a phase function φ that takes the value 1 in the liquid phase, the value 0 in the surrounding vapor, and spans the whole [0, 1] interval in a liquid-vapor transition region. The equilibrium shape of the drop at time t + δt is obtained from the one at time t by setting up a steepest descent dynamics for φ, driven by capillary energy and dissipation. The dynamics leads φ towards a critical point of an -regularized version of the capillary energy (augmented by the pseudo-potential of the pinning forces at the contact line). In the limit as tends to 0, one recovers the solution of the capillary problem, with sharp interfaces between the phases. The presence of the solid is modeled by imposing suitable (Dirichlet or) Neumann boundary condition, that are tuned in order to reproduce the advancing and receding angles.

2 Phase Field Approach to Wetting The energy of a liquid drop ω in contact with a solid S and surrounded by a fluid (usually air, so it can be referred to as vapor) is given by [1, 5]: E (ω) = σSL (x)dAx + σLV (x)dAx + σSV (x)dAx ∂S ω ∂V ω ∂S\∂S ω + ρL (x)G(x)dVx , (4) ω

where ∂S ω is the interface between liquid and solid, also denoted by SL , ∂V ω = ∂ω \ ∂S ω is the liquid-vapor interface LV and ∂S \ ∂ω is the solid-vapor one, SV . Since we consider a homogeneous fluid, ρL is set equal to 1, while G(x) stands for a generic potential related to an external force field (gravity in our case). As already mentioned, the terms σAB (x) are the surface tensions at a point x on the AB interface. When S represents a homogeneous solid, σSL and σSV are constant and (4) simplifies to E (ω) = σSL |SL | + σLV |LV | + σSV |SV | + G(x)dVx = (σSL − σSV )|SL | + σLV |LV | +

ω

G(x)dVx + k,

(5)

ω

where |A| denotes the measure of the set A and k is a constant that does not affect the search for the minima of the functional, and so it will be omitted. Given a fixed volume V > 0, the (geometric) problem of capillarity is to find

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ω∗ = argmin{E(ω)}. |ω|=V

(6)

It is well known (see, e.g., [5]) that solutions of this problem and, in fact, all stationary points of energy (5) have constant mean curvature (Laplace’s law) and meet the solid surface with a contact angle θ = θY depending only on the chemical properties of the three phases L, S, V and given by Young’s law cos θY = −

σSL − σSV . σLV

(7)

We move now to the phase field formulation. We consider a bounded region ⊂ R3 (the container) whose boundary ∂ describes a homogeneous solid surface. Let W (s) = a 2 s 2 (1 − s)2 be a non-negative potential (with a > 0 to be specified later) and σ : [0, +∞) → R a continuous function. A phase field formulation for the capillary problem is obtained by considering an energy of the type 1 E (φ) = |∇φ|2 + W (φ) + φG(x) dx + σ( φ)dHn−1 (x), (8) ∂ where φ is the phase function and φ stands for the trace of φ on ∂ (clearly we have to choose an appropriate function space). The potential W , that vanishes only for the values 0 and 1 of φ representing the vapor and the liquid phases, has to be tuned in order to produce the correct surface tension values from the corresponding interphase transition layers in the limit as → 0 (this is done at the end of this section). We extend E , given by (8), to φ ∈ L1 by setting E (φ) if φ ∈ H 1 ( , R), (9) E (φ) = +∞ otherwise in L1

and define E0 (φ) =

E(φ) if φ ∈ BV ( , {0, 1}), +∞

otherwise in L1 .

(10)

Notice that if φ ∈ BV ( , {0, 1}), then φ = χL , the characteristic function of the region of space occupied by the liquid phase. Now, defining s

W (y)dy , σˆ (t) := inf σ (s) + 2 s≥0

t

c0 :=

1

0

it can be proved that (9) -converges to

W (y)dy,


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⎧ 2c0 |LV | + σˆ (1)|SL | + σˆ (0)|SV |+ ⎪ ⎪ ⎨ if φ ∈ BV ( , {0, 1}), (11) 0 (φ) = + φ(x)G(x)dx E ⎪ ⎪ ⎩ +∞ otherwise in L1 and that if φ∗ is a family of minimizers of E , and if φ ∗ is its limit in L1 , then φ ∗ is 0 . This shows that φ ∗ converges to the characteristic function of a a minimizer for E minimizer of a geometric capillary problem. In order to obtain convergence to solutions of (6) we simply need to tune the values of c0 , σˆ (0), and σˆ (1) appropriately. In fact, choosing as in [6] σ (t) := Nt, where N is a constant, the Euler–Lagrange equations for (8) yield the boundary condition −2

∂φ = N. ∂n

Here n is the outward unit normal vector to ∂ . So, in order to obtain that (8) converges to (5), we just have to choose appropriate values for a and N. This is an easy calculation. Indeed, setting 2c0 =

a = σLV , 3

σˆ (0) = 0 , 3 s2 1 s − + = σSL − σSV , σˆ (1) = inf Ns + 2a s≥0 3 2 6

(12)

we see by comparing (11) with (5) that (12) tells us that the contact angle θ we are imposing is precisely the one given by Young’s law (7). Thus, the study of the NY minimum problem (12) selects the value of NY of N such that, if ∂φ ∂n = − 2 , then the contact angle is θY .

3 The Hysteresis Model Topographical and chemical imperfection on the solid surface at the sub-micron scale generate frictional forces that are able to pin the contact line. This is the origin of the phenomenon of contact angle hysteresis, namely, the possibility that equilibrium drops adopt contact angles different from the one given by Young’s law (7). We focus on quasi-static evolutions, namely, evolutions through equilibria driven either by time-dependent constraints (as in the case of an evaporating droplet), or by timedependent external forces (as in the case of a drop subject to gravity and resting on a surface which is initially horizontal and then tilted), or both. In these cases, solutions at time t depend not only on the instantaneous values of the data, but also on their entire time history. We approach such problems by defining a one-parameter

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family of incremental minimization problems in which, based on the knowledge of the unknowns at time t, we determine their values at time t + δt, with δt small. Following [7,8], we begin by considering the following discrete incremental formulation for the problem of quasistatic evolution of a drop whose volume changes according to the prescribed law |ω| = V(t). For simplicity, we neglect gravity in this first example, i.e., we set G = 0 in (5), and we illustrate the behavior of a small droplet on a horizontal solid surface. Given the configuration ω∗ (t) of a drop at time t, the one at time t + δt is given by ω∗ (t + δt) = argmin {E(ω, t + δt) + D(ω, ω∗ (t))} |ω|=V(t +δt )

(13)

where the dissipation D(ω1 , ω2 ) is given by D(ω1 , ω2 ) = µ|∂S ω1 ∂S ω2 |.

(14)

Here A B = (A \ B) ∪ (B \ A) denotes the symmetric difference of the sets A and B; µ > 0 is a parameter giving the dissipated energy per unit variation of the wetted area. For ω∗ (t) a spherical cap, energy and dissipation can be written simply as E = (σSL − σSV )πa 2 + σLV A,

(15)

D(ω1 , ω2 ) = D(a1 , a2 ) = µπ|a12 − a22 |,

(16)

where A = 2πRh is the area of the spherical cap of radius R and height h, while a is the radius of the wetted area, namely, the interface between the solid and the liquid. Studying the Euler–Lagrange equations associated with (13), we obtain that the drop contact angle θ is deduced from ⎧ if a < a(t), ⎪ ⎨ {cos θR } cos θ ∈ [cos θR , cos θA ] if a = a(t), (17) ⎪ ⎩ if a > a(t), {cos θA } where the advancing contact θA and the receding contact angle θR are given by cos θA = cos θY −

µ , σLV

cos θR = cos θY +

µ . σLV

The meaning of condition (17) is the following. Any value of the contact angle θ such that cos θ ∈ [cos θR , cos θA ] can be seen in an equilibrium configuration. However, at a point of the contact line which is advancing (resp., receding), the contact angle must be θA (resp., θR ). Using these angle conditions, and the fact that a drop which is initially a spherical cap always remains a spherical cap, the solutions to (13) can be computed analytically for any given volume history t → V(t).


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Considering now the general case (13), in which time-dependent external forces may be present, we look for solutions ω = ω(t + δt) minimizing the incremental problem: F (ω, t +δt) = (σSL −σSV )|∂S ω|+σLV |∂V ω|+µ|∂S ω ∂S ω(t)|+ G(x, t)dVx ω

(18) by the phase field method discussed in the previous section. In practice, this means to solve φ = V(t + δt) (19) φ∗ (t + δt) = arg min E (φ, t + δt), subject to

with

∂φ ∗ −2 = ∂n

NA on ∂ A , NR on ∂ R ,

(20)

where NA and NR are the Neumann boundary conditions associated with the advancing and receding angle respectively, computed by the use of (12). The regions ∂ R and ∂ A are -approximations of the parts of the solid which are wet and dry at time t, defined as appropriate super- and sub-level sets of φ∗ (t). We remark that (19)–(20) can be regarded as the phase field formulation of a standard capillarity problem on a heterogeneous surface whose properties are timedependent.

4 Numerical Implementation 4.1 The Basic Numerical Scheme We first discuss the numerical scheme to solve the case without hysteresis. We set the problem in the computational domain ∗ ⊂ , a bounded subset of R3 with piecewise C 2 and Lipschitz boundary. We decompose it into two parts: ∂ ∗ = ∂S ∗ ∪ ∂V ∗ ; the set ∂S ∗ is the part of ∂ ∗ which is common with the solid surface, denoted by ∂S in (4) and by ∂ in (8), and it is a proper subset of ∂ ; ∂V ∗ is contained in the interior of and we can suppose it is occupied by the vapor phase. Notice that when the computational domain is much smaller than the mathematical one, the effects of the boundary parts of far from the liquid phase are negligible. In this setting the Euler–Lagrange equations for the phase-field model are (here we assume G = 0 and a = 1 for simplicity):

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⎧ 1 ⎪ ⎨ − φ + φ(1 − φ)(1 − 2φ) + λ = 0 ∂φ N = − 2 ⎪ ⎩ ∂n φ=0

in ∗ on ∂S ∗ on ∂V ∗

(21)

where λ is a Lagrangemultiplier whose value has to be calculated in order to meet the volume constraint ∗ φ = V. To solve the equilibrium equation, we transform the problem into a parabolic PDE generated by a gradient flow [9], and we follow an artificial relaxation dynamics until the system reaches the equilibrium configuration. The gradient flow is introduced by setting φ = φ(τ, x), where τ is a fictitious time, and solving 1 φτ = φ − φ(1 − φ)(1 − 2φ) − λ.

(22)

Here the Laplacian is with respect to the space derivatives and the subscript τ denotes a time derivative. The solution of our original equation (21) will be lim φ(τ, ·). In fact, along the flow (22), the energy is decreasing in time

τ →+∞

d E = −2 dτ

∗

|φτ2 |dx 0.

For the time integration of the equation, we have used a forward Euler scheme, that unfortunately is stable only for small values of dτ . In order to find at each iteration the correct value for the Lagrange multiplier λ, we have implemented a splitting method: given an initial guess φ0 that satisfies the appropriate boundary condition on the derivatives, we follow the scheme:

1

φ N+ 2 = dτ

1 φ N − φ N (1 − φ N )(1 − 2φ N ) + φ N λ = N

V−

N+ 1 φ 2 1 1

φ N+1 = φ N+ 2 + λN where V = ∗ φ 0 . By construction we have that ∗ φ N = V stays constant during the iterations; the space derivatives have been calculated by means of a seven-point finite differences stencil which guarantees a second order accuracy.


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4.2 Accounting for Hysteresis Moving now to the case with hysteresis, we apply the numerical scheme presented in the previous subsection to the incremental problem discussed in Section 3. Within one time increment δt, the external loads and the constraints are frozen at their value at time t. We can obtain that the drop advance (resp. recede) with contact angle θA (resp. θA ) by imposing through Equation (20) that in the regions that are dry (resp. wet) at time t the contact angle be the advancing (resp. the receding) one. Thus, the algorithm of the previous subsection is easily adapted to accommodate the presence of frictional forces at the contact line: we solve a one-parameter family of standard capillarity problems obtained by replacing the Neumann boundary condition (212 ) by (20), in which ∂ A and ∂ A are -approximations of the parts of the solid which are dry and wet at time t. Studying the volume-driven evolution of a spherical cap in the presence of hysteresis, a case for which an exact solution is available (see Section 3), we have noticed that condition (12), which has been derived for the limit configurations as goes to zero, is not strong enough to guarantee the correct evolution of the contact line when = 0. In fact, at this stage, the interface is not yet sharp, but has a width which depends on . This has the consequence that our computed drops tend to advance or recede slightly too early when compared with the behavior of the exact solution. In order to stabilize the triple junction among the phases we add a third zone on the solid surface which is in contact with the liquid-vapor interface. On this third zone we put a boundary condition that correspond to a bigger advancing contact angle near the dry area and a condition for a smaller receding angle near the wet zone.

4.3 The Multigrid Architecture To deal with the large number of degrees of freedom necessary to resolve a 3-D geometry, we need to use adaptive mesh refinement. We use static refinement in the region close to the solid, and dynamic refinement close to the liquid-vapor interface LV . The criterion for dynamic refinement is the following: we regrid using up to two levels of refinement in the regions where 0.05 ≤ φ ≤ 0.95. Accuracy needs to be preserved across regions where the computational mesh changes from coarse to fine. This is done using interpolation techniques which preserve the second order accuracy of the Laplacian across a level boundary. This can be handled using ghost cells: a layer of fictitious nodes that contribute to the seven-point stencil at the boundary of real cells. The solution is updated thanks to a V cycle: an iterative scheme that prescribes at each step to update first the solution on the finest level, then to pass the new information down to the coarsest, so to update φ and finally to come back up. Therefore there are two main problems to handle: the passage of the information from a fine to

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Fig. 2 A phase field drop: a slice parallel and close to the solid surface (left) and one perpendicular to it (right). Note the -layer representing the liquid-vapor interface.

a coarse level and the opposite one, that are the descending and the ascending part of the “V ”. Details of the implementation are described in [10, 11]. The complex structure of the grid hierarchy, with its ghost and real cells, the interpolations and the indexing of such a large number of degrees of freedom, was made possible by the use of an existing ad hoc library: SAMRAI, a C++ library specifically developed for adaptive mesh refinement. Parallelization of the code on hg1 and mercurio partitions of the high performance computing grid available at Sissa was performed, in collaboration with C. De Vittoria.

5 Examples and Results This section contains some of the results obtained by the use of the algorithm described in the previous section. In particular, inspired by experiments by Carre and Shanahan [12], we have determined intervals to which critical volumes of water drops on differently treated vertical glasses must be confined (confidence intervals). We compare our results with the data in [12] and those deduced by the application of Dussan’s formula (3). As it is common in the study of hysteretic evolutions, solutions depend not only on the instantaneous value of the data, but also on their entire time history and, in particular, on the initial conditions. Our protocol to initialize the simulations is the following. We first set θA = θY = θR , where the value of θY is taken from the data in [12], and let the system relax to an equilibrium configuration in the absence of gravity. In these conditions, the equilibrium configuration of the drop is a spherical cap meeting the solid surface at an angle given by Young’s law. Then we “turn on” gravity and hysteresis gradually and simultaneously, and record whether the


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Table 1 Contact angle data (θY , θA , θR ) for untreated glass plates (control) and those treated with a commercial anti-rain composition (CAC) and with the Corning surface treatment (CST), before and after ageing in boiling water and before and after ageing in running water. Glass treatment

No ageing

After 1 hr in 100◦ C–H2 O

After 30 hr in running water

Control

θY = 42.5 θA = 51 θR = 32.5

θY = 51.5 θA = 61 θR = 41

θY = 44 θA = 54 θR = 32

CAC

θY = 108 θA = 113 θR = 103

θY = 72 θA = 91 θR = 51

θY = 96.3 θA = 111 θR = 82

CST

θY = 105 θA = 111 θR = 99.5

θY = 99.5 θA = 105 θR = 94

θY = 103 θA = 113 θR = 94

Table 2 Critical volume data in µl; VcD is the one deduced from Dussan’s formula (3), VcE is the experimental value and VcC is our computational result. After 1 hr in 100◦ C–H2 O

Glass treatment

No ageing

Control

VcD VcE VcC

CAC

VcD = 1.83 VcE = 1.5 VcC ∈ (1.12, 2.51)

VcD = 19.74 VcE = 17.7 VcC ∈ (16, 20.12)

VcD = 9.89 VcE = 8.6 VcC ∈ (7.52, 10.648)

CST

VcD = 2.56 VcE = 1.8 VcC ∈ (1.4, 2.98)

VcD = 2.55 VcE = 2.0 VcC ∈ (1.72, 2.98)

VcD = 4.95 VcE = 4.0 VcC ∈ (3.7, 5.4)

= 6.34 = 4.9 ∈ (4.44, 7.2)

VcD VcE VcC

= 7.68 = 6.3 ∈ (5.72, 8.47)

After 30 hr in running water VcD = 7.92 VcE = 6.5 VcC ∈ (6.07, 8.47)

drop remains in equilibrium with the full value of the gravitational force, or it starts sliding. The meaning of the confidence interval we have used to present our results is the following: the minimum value of the specified range is the volume of the largest drop that we have observed to be stable; the maximum is the size of the smallest drop we saw rolling down. The critical volume belongs to this interval. The level of accuracy reached in this evaluation depends mostly on the computational time spent to reduce the error that affects the solution. We are working towards narrowing the width of our confidence interval, in order to obtain a more precise estimate of the critical volume. In Table 1 we report the contact angles measured for water drops on the different materials, subjected to several treatments. Table 2 shows the critical volume obtained with the different approaches, which are also plotted in Figure 3 as a function of (cos θR − cos θA ).

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Fig. 3 Comparison of the results of simulations based on our model with the experimental data in [12] and with Dussan’s formula (3).

Looking at Figure 3, the agreement of the experimental data with both Dussan’s formula and with our numerical simulations is satisfactory. In particular, with reference to (3), it is interesting to observe that, in a 3-D framework, the threshold on the volume beyond which motion can start does not depend only on the difference (cos θR − cos θA ) like in the 2-D scheme, but also on a second term, a correction factor which depends on θA . Using a Taylor expansion centered in π2 and defined for convenience on the interval [50◦, 120◦ ] (that is the range where θA varies in our examples), (3) can be rewritten as: 3 3 45 Vcrit ∼ C (cos θR − cos θA )3/2 1 − t + t 2 − t 3 (23) 4 16 64 3 2 3 is far from 1, namely, where t = θA − π2 . When the term 1 − 34 t + 16 t − 45 64 t when θA is far from π2 , then Vcrit deviates from the monotone curve C(cos θR − cos θA )3/2 , hence explaining the non-monotone behavior of the data in Figure 3.


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6 Summary and Discussion In this paper we have presented a diffuse interface model for the solution of the capillarity problem in the presence of contact angle hysteresis which is capable of approximating the solution of the geometrical problem. We have checked its validity by comparing some experimental results with our numerical simulations, obtaining a satisfactory agreement. A natural continuation of this research is to implement simulations with solid substrates which are heterogeneous at microscopic scales (micropillars) checking whether the model is able to capture both Wenzel and Cassie-Baxter equilibrium states, see, e.g., [13]. The latter display a complex free-boundary at the composite interface between the rough solid and the liquid, where pockets of vapor may be trapped at the bottom of asperities. From the numerical point of view, we plan to replace the explicit scheme in the calculation of the solution with a more performant semi-implicit one, in order to beat the limitations on the stable time step.

References 1. P.-G. De Gennes, F. Brochard-Wyart and D. Quéré, Capillarity and Wetting Phenomena, Springer, 2004. 2. E.B. Dussan, On the ability of drops or bubbles to stick to non-horizontal surfaces of solids, J. Fluid Mech. 151, 1985, 1–20. 3. P. Seppecher, Moving contact lines in the Cahn-Hilliard theory, Int. J. Engrg. Sci. 34, 1996, 977–992. 4. H. Garcke, B. Nestler and B. Stoth, A multi-phase-field concept: Numerical simulations of moving phase boundaries and multiple junctions, SIAM J. Appl. Math. 60, 1999, 295–315. 5. R. Finn, Equilibrium Capillary Surfaces, Springer, 1986. 6. L. Modica, Gradient theory of phase transitions with boundary contact energy, Ann. Inst. H. Poincaré Anal. Non Linéaire 5, 1987, 497. 7. G. Alberti and A. DeSimone, Quasistatic evolution of sessile drops and contact angle hysteresis, forthcoming, 2009. 8. A. DeSimone, N. Grunewald and F. Otto, A new model for contact angle hysteresis, Networks and Heterogeneous Media 2, 2007, 211–225. 9. W. Bao and Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput. 25, 2004, 1674. 10. A. Turco, F. Alouges and A. DeSimone, Wetting on rough surfaces and contact angle hysteresis: Numerical experiment based on a phase field model, ESAIM: M2AN, 2009. 11. A. Turco, Variational techniques in the numerical simulation of molecular dynamics trajectories and of wetting on rough surfaces, PhD Thesis, 2008, http://digitallibrary.sissa.it/handle/1963/3405. 12. A. Carre and M.E.R. Shanahan, Drop motion on an inclined plane and evaluation of hydrophobic treatments to glass, J. Adhesion 49, 1995, 177–185. 13. G. Alberti and A. DeSimone, Wetting of rough surfaces: A homogenization approach, Proc. R. Soc. A 461, 2005, 79–97.

Application of Relaxation Methods in Materials Science: From the Macroscopic Response of Elastomers to Crystal Plasticity Georg Dolzmann

Abstract In this contribution we review some aspects of the mathematical analysis of fine structures in materials in three distinct systems: small deformations within the range of pseudoelasticity in shape memory materials, soft elasticity for nematic and smectic elastomers, and relaxation via formation of microstructures in single crystal plasticity.

1 Mathematical Framework It has been known for centuries that fine structures in materials across a wide range of scales, up to the atomistic one, have a tremendous impact on the mechanical response of materials to applied forces. In this contribution we review some aspects of the mathematical analysis of these phenomena in three distinct systems: small deformations within the range of pseudoelasticity in shape memory materials, soft elasticity for nematic and smectic elastomers, and relaxation via formation of microstructures in single crystal plasticity. The observed phenomena are the result of the combined effects of a large number of quite different mechanism. While it is certainly possible to include all of these effects in an abstract mathematical model, the analytical techniques available today are not yet powerful enough to draw meaningful conclusions from such a general formulation. For this reason we single out a very particular facet, namely the question of what predictions can be made based on the principle of minimization of energy.

Georg Dolzmann NWF-I Mathematik, Universität Regensburg, D-93040 Regensburg, Germany K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_5, © Springer Science+Business Media B.V. 2010

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1.1 Quasiconvexity, Effective Energies, and Relaxation The unifying thread behind the phenomena discussed in this contribution is a lack of convexity of the free energy densities for these systems proposed in the literature. All of the models considered here are models in nonlinear elasticity and therefore the fundamental variable is the deformation u mapping a reference configuration ⊂ Rn onto the observed or deformed configuration u() ⊂ Rn . Typically n = 2 or n = 3 are the cases of physical relevance. More generally, one can consider a variational integral I [u] = W (∇u)dx

Rn

where ⊂ and u : → is a vector-valued map. Under suitable growth and coercivity conditions, existence of a minimizer follows from the direct method in the calculus of variations [19] if and only if the energy density W : Rm×n → R is quasiconvex in the sense of Morrey, that is, if for all F ∈ Rm×n and all Lipschitz functions φ ∈ W01,∞ (; Rm ) the inequality W F + ∇φ dx ≥ W F dx (1) Rm

(0,1)n

(0,1)n

holds. This can be interpreted as saying that for given affine boundary data x → F x on ∂ the homogeneous affine map x → u(x) = F x minimizes the variational functional. In particular, the material does not have a spontaneous tendency to form microstructure. In the applications discussed in the subsequent sections the energy densities fail to be quasiconvex and existence of minimizers cannot be deduced by the direct method in the calculus of variations. Various powerful methods have been developed to understand the behavior of variational functionals of this type, including the analysis of oscillations in minimizing sequences via gradient Young measures and their relatives [4, 32, 33], the definition of the relaxed functional [19], and the construction of minimizers based on Gromov’s idea of convex integration in a Lipschitz setting [20, 25, 29]. We focus in this contribution mostly on the aspect of relaxation, but occasionally we take advantage of some powerful results in convex integration. If the density W fails to be quasiconvex, then one defines the effective or macroscopic energy as the energy (per unit volume) needed by the system to deform an infinitesimal volume element subject to affine boundary conditions x → F x on ∂. For our purposes we may use the representation W qc (F ) = inf W F + ∇φ dx . φ∈W01,∞ (;Rm ) (0,1)n

It turns out that the function W qc is indeed quasiconvex if W satisfies standard growth and coercivity conditions. In this case, the relaxed functional defined by

Relaxation Methods in Materials Science

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I qc [u] =

W qc (∇u)dx

satisfies standard growth and coercivity conditions as well and it is a remarkable fact that the two functionals are closely related in the sense that minimizing sequence for I converge to minimizers of I qc and vice versa.

1.2 Verification of Quasiconvexity and Convex Integration Two related notions of convexity are fundamental in verifying that a given function W is quasiconvex or not [19,28]. A function W is said to be polyconvex, if it can be written as a convex function of all subdeterminants of F , or equivalently, if for all F there exists a supporting polyaffine function pF . In the specific dimensions relevant here this can be stated as follows: A function W : Rn×n → R is polyconvex, if for n = 2 with P2 = {p : p(X) = A : X + c det X, A ∈ R2×2 , c ∈ R} W (F ) = sup p(F ) : p ∈ P2 , p ≤ W and for n = 3 with P3 = {p : p(X) = A : X + B : cof X + c det X, A, B ∈ R3×3 , c ∈ R} W (F ) = sup p(F ) : p ∈ P3 , p ≤ W . Here cof F denotes the cofactor matrix of F , i.e., the matrix of all 2 × 2 subdeterminants of F that satisfies F (cof F )T = (det F )I . A function W : Rm×n is rank-one convex, if it is convex on rank-one lines in matrix space, that is, if t → W (F + tR)

is convex in t for all F, R ∈ Rm×n , rank(R) = 1 .

It is not hard to see that (for W finite-valued) polyconvexity is a sufficient condition for quasiconvexity while rank-one convexity is a necessary one. In analogy to the convex envelope of a function W : Rm×n → R as the largest convex function less than or equal to W we define the rank-one convex envelope by W rc (F ) = sup V (F ) : V ≤ W, V rank-one convex . Similarly one defines W qc and W pc and in fact this definition for W qc is equivalent to the one given before. With these notions of convexity at hand it follows for W finite-valued that W ≥ W pc ≥ W qc ≥ W rc and a natural strategy to characterize W qc is to check whether W pc and W rc agree. A very useful approach to the computation of W rc relates to an algorithmic version suggested in [26]. The key idea is the following: if W is not rank-one convex,

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then there exist matrices F , F1 , F2 with rank(F1 − F2 ) = 1 and λ ∈ (0, 1) such that F = λF1 + (1 − λ)F2 and W (F ) > λW (F1 ) + (1 − λ)W (F2 ). One can therefore define an approximation to W rc by seeking for each F a pair of good matrices F1 , F2 with rank(F1 − F2 ) = 1 and λ ∈ (0, 1) such that F = λF1 + (1 − λ)F2 and setting (F ) = λW (F1 ) + (1 − λ)W (F2 ) . W instead of W ) and then one tries to This process can be iterated if needed (using W is polyconvex. If this is the case, then necessarily W pc = W = W rc . verify that W All examples discussed in the subsequent sections rely on this strategy. If W is . finite-valued, then also W qc = W A very challenging obstacle to this approach is the fact that most of the foregoing facts concerning W qc are valid only for finite-valued energies. However, in the modelling of incompressible materials one often imposes this incompressibility via the requirement that the energy be infinite if the argument F , the deformation gradient, does not correspond to a volume preserving deformation, that is, if det F = 1. As observed in [7], in this case quasiconvexity does not imply rank-one convexity and the equality of W pc and W rc does not necessarily provide a formula for W qc . However, under very general assumptions the missing link can be obtained by construction of suitable test functions for (1) based on convex integration, see [13] for a discussion of the relation of quasiconvexity and rank-one convexity for energies related to incompressible materials.

2 Shape Memory Materials The papers [5, 6, 11] sparked a renewed interest in the mathematical description of microstructures arising in connection with diffusionless solid-solid phase transformations, see, e.g., [8] for a review of the background in materials science. In the case of an austenite – martensite transformation, the material can accommodate certain deformations in the martensitic phase by the formation of suitable mixtures of the different martensitic phases on a small length scale. Upon heating of the sample to the austenitic phase, the martensitic phases lose their stability, the specimen returns to the austenitic phase and recovers its original shape (shape memory effect). A natural question in this context is to ask for the set A of all affine deformations that can be recovered upon heating, at least within a simple model capturing the main features of the process. In the case of a cubic to tetragonal phase transformation, such a model can be based on the minimization of a free energy density W that satisfies the following hypotheses: 1. Characterization of ground states: W ≥ 0 and W (F ) = 0 if and only if F ∈ K with


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K = SO(3)U1 ∪ SO(3)U2 ∪ SO(3)U3 , 1 1 − λ2 ei ⊗ ei , λ > 0 ; Ui = I − λ λ 2. Frame indifference: W (RF ) = W (F ) for all R ∈ SO(3). A possible choice is W (F ) = dist2 F, K where dist(·, K) denotes the Euclidean distance of the matrix F to the set K and where SO(3) is the group of all proper rotations in R3 . It turns out that the set A is equal to K qc , the zero set of the relaxation of W , that is, K qc = (W qc )−1 (0), where K = W −1 (0) for W ≥ 0, see, e.g., [28]. Despite this precise characterization on an abstract level, one of the fundamental open problems related to this specific model problem is the question of how to characterize K qc more explicitly. Despite this lack of information which is typically needed in arguments based on convex integration, it was shown in [22] that this set has relatively open interior in the manifold of all matrices with determinant equal to one. Theorem 1 ([22]). Suppose that λ = 1 and that = {X : det X = 1}. Then there exists an > 0 such that B(I, ) ∩ ⊂ K qc . Moreover, for all F ∈ B(I, ) ∩ there exists a ground state u ∈ W 1,∞ (; R3) with Du ∈ K a.e. and u(x) = F x on ∂. One interpretation of this result is as follows: any deformation of a given sample in a reference configuration (in the austenitic phase) can be realized in the martensitic phase with given boundary conditions x → F x on ∂ with F ∈ B(I, ) ∩ at no energy cost since ∇u lies in the zero set of W a.e. However, these minimizers have an intrinsically complicated geometry and the microstructures arise at all (in particular arbitrarily small) length scales. The existence of these special solutions would be ruled out if one included for example a term proportional to the length of the interface between the different phases SO(3)Ui in the variational integral [23, 24].

3 Nematic and Smectic Elastomers A fascinating model system in which the program of understanding surprising material behavior based on relaxation can be carried out completely are special classes of elastomers, rubber-like materials with a particular molecular structure, see [35] and the references therein for the physics of the systems and [14, 15, 21, 31] for the relaxation results. In these materials, nematic mesogens, rigid, rod-like molecules, are attached via spacers to the long backbone chains forming the underlying network. For high temperatures these molecules have a random orientation, but below

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a critical temperature Tcr the mesogens tend to align locally in one direction which can be described by a director, a vector field n : → S2 of unit length. The free energy density proposed by Bladon, Terentjev, and Warner [9] depends on the deformation gradient F and the director n and is given by ⎧ ⎨ µ r 1/3 |F |2 − r − 1 |F T n|2 − 3 if det F = 1, WBTW (F, n) = 2 r ⎩ +∞ else where µ > 0 and r ≥ 1 are temperature dependent material constants. In the isotropic phase for T > Tcr the parameter r is equal to one and the energy reduces to the Neo-Hookean energy of a rubber, W (F ) = (µ/2)(|F |2 − 3) on det F = 1. Upon the transition to the low temperature or nematic phase, the parameter r becomes larger than one and minimization in n leads to W (F ) = min WBTW (F, n) = n∈S2

µ 1/3 2 r λ1 + r 1/3 λ22 + r −2/3 λ23 − 3 2

(on det F = 1) where 0 ≤ λ1 ≤ λ2 ≤ λ3 denote the singular values of F , i.e., the eigenvalues of the symmetric matrix (F T F )1/2. We rewrite W as ⎧ ⎨ λ1 p + λ2 p + λ3 p − 3 if det F = 1, γ1 γ2 γ3 W (F ) = ⎩ +∞ else, with 0 < γ1 ≤ γ2 ≤ γ3 and γ1 γ2 γ3 = 1. The Bladon, Terentjev, Warner energy is a special case of this general family of energies that corresponds to the choices p = 2,

γ1 = γ2 = r −1/6 ,

γ3 = r 1/3 .

The following remarkable relaxation result was obtained in [21]. Theorem 2 ([21]). The macroscopic energy of the system is given by ⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ λ1 p + 2 γ1 p/2 − 3 ⎪ ⎪ ⎪ ⎪ λ1 ⎨ γ1 qc W (F ) W (F ) = ⎪ ⎪ ⎪ ⎪ ⎪ λ3 p + 2 γ3 p/2 − 3 ⎪ ⎪ γ3 λ3 ⎪ ⎪ ⎩ +∞

if F ∈ L, if F ∈ I1 , if F ∈ S, if F ∈ I3 , else.

The phases L, I1,3 and S are sketched in Figure 1. Since the energy is frame invariant and finite only on the manifold {det X = 1}, two parameters suffice to describe a state of the system. Note also that λ3 (F ) = λmax (F ), λ1 (F ) = λmin (F ) = λmax (cof F ) if det F = 1. Numerical simulations based on this relaxation formula were presented in [15].


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Fig. 1 The different phases in the relaxation of the macroscopic energy of a nematic elastomer. The material is ideally soft in the phase I and shows the typical response of a rubber in one direction in the intermediate phases I1,3 . Note that in this phase, the relaxed energy depends only on one of the three singular values λi . No relaxation occurs in the phase S.

As in the case of models for shape memory materials it should be noted that the relaxation mechanism is again based on the formation of fine structures at all scales. Stretching experiments [27] are usually done on thin sheets of monodomain samples in which the director is initially uniformly aligned in direction e2 and reoriented through an applied stretch in direction e1 . Experimental observations show that this reorientation is not uniform but leads to stripe patterns. The Bladon, Terentjev, Warner model, based on ideal assumptions on the sample, predicts in this case a stress–strain curve which is flat as long as the reorientation process happens. In experiments one observes initially an increase of stresses in response to applied strains before the flat part in the stress-strain diagram is reached. This has been explained in [34] by particularities of the synthetisation process which leads to the presence of a favored director n0 . The corresponding correction to the energy is given by W (F, n, n0 ) = |F |2 − α|F T n|2 − β|F n0 |2 + c for suitable constants α, β, and c (up to a global multiplicative factor). This energy is not isotropic and the techniques applied for the characterization of the relaxation of the Bladon, Terentjev, and Warner energy cannot be used. It is an open question to find a closed form for the relaxation in this situation. A heuristic approach via a reduction to a two-dimensional problem was discussed in [14]. A different variant of these elastomers is obtained by confining the backbone chains to thin layers between layers of mesogens. In these systems, the lowtemperature phase displays a smectic order in which the mesogens align parallel to the layer normal n0 . The Adams–Warner model [2]

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WAW (F ) =

⎧ ⎨ |F |2 + kd 2 ⎩

0

d0 | cof F n0 | − 1

2

∞

if det F = 1 , else

with k 1 and d0 constants coincides up to leading order with ⎧ ⎨ |F |2 + k | cof F n0 | − d0 2 if det F = 1 , Wk (F ) = ⎩∞ else. Note that Wk is transversally isotropic, i.e., W (QF R) = W (F ) for all Q, R ∈ SO(3) such that Rn0 = n0 . This additional invariance allows one to find an explicit formula for the relaxation. Theorem 3 ([1]). Let P = I − n0 ⊗ n0 . Then |F n0 |2 + f λ2max (F P ), | cof F n0 | if det F = 1 , qc Wk (F ) = ∞ else . Here the function f is given by f (x) =

⎧ ⎨ ⎩

x2 + 2d0

d02 x2

1/2

if x > d0

,

else.

4 Single Crystal Plasticity Models in finite plasticity are based on a multiplicative decomposition of the deformation gradient, F = ∇u = Fel Fpl where Fel and Fpl are the elastic and the plastic part of the deformation, respectively. In the special situation of a single crystal and a monotone loading path modeled via incremental problems in a suitable time discretization, the corresponding deformations in the first time step can be obtained via a nonconvex variational principle [10, 30] in which the free energy density is given as the sum of three terms, W (F ) = Wel (Fel ) + Wpl (Fpl ) + Diss(Fpl ) . Here Wel (·) is the elastic potential, Wpl (·) the plastic potential, and Diss(·) the energy dissipated by the plastic deformation. This approach is now based on the fundamental assumption that the system tries to minimize its energy among all possible multiplicative decompositions, that is, that the macroscopic or condensed energy is given by


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Wcond (F ) = inf Wel (Fel ) + Wpl (Fpl ) + Diss(Fpl ) : F = Fel Fpl . As pointed out in [10, 30], Wcond fails to be quasiconvex and the system tends to form microstructures in order to reduce its energy. The following theorem describes the relaxed energies for two models in two space dimensions, one without hardening and one with linear hardening, under the special assumptions of one active slip system and rigid elasticity. Theorem 4 ([12, 18]). Suppose that 0 , if Fel ∈ SO(2), Wel (Fel ) = ∞ , else . 1. No hardening: Suppose that WCT (F ) =

|γ | , if F ∈ M (2) , ∞ , else ,

with M (2) = {F ∈ M2×2 : F = R(I + γ s ⊗ m), R ∈ SO(2), γ ∈ R}. Then qc |F |2 − 2 det F , if F ∈ N (2) , WCT (F ) = ∞, else . rc Here N (2) = {F ∈ M2×2 | det F = 1, |F s| ≤ 1} = M (2) . 2. Linear hardening: Suppose that 2 |γ | , if F ∈ M (2) , WC (F ) = ∞ , else . Then qc

WC (F ) =

|F m|2 − 1 , if F ∈ N (2) , ∞, else .

It is natural to ask what happens if we relax the very restrictive assumptions that only one slip system is active and that the only allowed elastic deformations are rigid body motions. The situation in which two slip systems are active is already much more complicated since in this case the rank-one convex and the polyconvex envelopes do not coincide [3]. Since it is unknown whether in two dimensions rankone convexity is equivalent to quasiconvexity it is an open problem to characterize the relaxation. We now turn towards exploring the impact of elastic energy by replacing the assumption of rigid elasticity by a penalization of elastic deformations, i.e., we define W,el (Fel ) =

1 distq (Fel , SO(2)) ,

> 0, q ≥ 1.

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Theorem 5 ([16]). Suppose that we consider a model with one active slip system and elastic approximation with > 0 and q ≥ 1. 1. No hardening: Let WNH, be given by 1 distq (F (I − γ s ⊗ m) , SO(2)) + |γ | . WNH, (F ) = min γ ∈R Then (WNH, )qc (F ) = 0 for all F ∈ N (2) . 2. Linear hardening: Let WLH, be given by 1 q 2 dist (F (I − γ s ⊗ m) , SO(2)) + |γ | . WLH, (F ) = min γ ∈R Then (WLH, )qc (F ) > 0 for all F ∈ SO(2). Note that WNH, approximates the corresponding density with rigid elasticity pointwise as → 0. Thus there is a qualitative difference in the case without hardening between the pointwise limit of the densities and their relaxation. This assertion remains valid under more general assumptions, in particular the elastic energy can be infinite for matrices Fel with det Fel ≤ 0. The key observation in the proof is that the multiplicative decomposition F = Fel Fpl leads to non-standard growth conditions of Wcond , in particular sublinear growth along certain directions. Consider for example the matrices α α−1 t t t 0 1t Ft = = Fe Fp . = 0t 0t 01 For t 1 we have |Fel | ∼ |Ft |1/α and |Fpl | ∼ |Ft |(α−1)/α , therefore W (Ft ) ≤ C |Fel |q + |t α−1 | ∼ C |Fel |q + |Fpl | ∼ C |Ft |q/α + |Ft |(α−1)/α . If we choose α = q + 1 then we find that W (Ft ) ≤ C|Ft |q/(q+1). In the case with linear hardening the situation is different and it is in fact possible to establish a rigorous link between between the models with rigid elasticity and those with penalized elasticity, at least for q ≥ 4. Theorem 6 ([17]). Suppose that = (0, 1)2 ⊂ R2 , that s = e1 , m = e2 and that 1,4/3 2 X= u∈W (; R ) : u dx = 0 .

For > 0 and q ≥ 4 we define 1,4/3 2 X = u ∈ W (; R ) : WLH, (∇u )dx < ∞ ,

where we endow both X and X with the weak W 1,4/3 -topology. Let


J [u] = and J [u] =

⎧ ⎨ ⎩

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WLH, (∇u)dx if u ∈ X ,

⎧ ⎨ ⎩

+∞

else

qc

WC (∇u)dx if u ∈ X0 , +∞

else

where u dx = 0, ∇u ∈ N (2) a.e. in . X0 = u ∈ W 1,2 (; R2×2) :

Then

J → J in X as → 0 in the sense of -convergence with respect to weak convergence in W 1,4/3 .

5 Concluding Discussion The examples presented in this contribution demonstrate the remarkable progress that has been achieved in the quest to understand suprising material behavior based on relaxation of free energy densities. At the same time they also illustrate the limitations of the methods that have been employed so far. One of the key aspects that need to be addressed in the future concerns the development of techniques for the characterization of (an approximation of) the relaxation of energy densities in threedimensional models and of densities with less invariance properties. The ultimate goal of this approach is to contribute to the new branch of predictive sciences in which experimental techniques are guided by numerical simulations of models in the framework of continuum mechanics and nonlinear elasticity. The reliable and efficient implementation and algorithmic realization seems to require relaxed models in which the convexity of the macroscopic energy densities allows one to obtain stable numerical solutions without artificial oscillations at the length scale of the underlying discretization. If the relaxation is based on a rigorous mathematical derivation, then the information gathered in the analytical relaxation process can be used to resolve all necessary microstructures a posteriori.

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Acknowledgement The research reported in the paper was partially supported by the NSF through grants DMS0104118 and DMS0405853 and by the German Research Foundation (DFG) through Forschergruppe FOR 797 “Analysis and computation of microstructure in finite plasticity” project DO 633/2-1.

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20. Dacorogna, B. and Marcellini, P., Implicit Partial Differential Equations, Progress in Nonlinear Differential Equations and Their Applications, Vol. 37, Birkhäuser, Boston, MA, 1999. 21. DeSimone, A. and Dolzmann, G. Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies, Arch. Rational Mech. Anal. 161, 2002, 181–204. 22. Dolzmann, G. and Kirchheim, B., Liquid-like behavior of shape memory alloys, C. R. Math. Acad. Sci. Paris 336(5), 2003, 441–446. 23. Dolzmann, G. and Müller, S., Microstructures with finite surface energy: The two-well problem, Arch. Ration. Mech. Anal. 132(2), 1995, 101–141. 24. Kirchheim, B., Lipschitz minimizers of the 3-well problem having gradients of bounded variation, Preprints, Max Planck Institute for Mathematics in the Sciences, Leipzig, 12, 1998. 25. Kirchheim, B., Rigidity and Geometry of Microstructures, Lecture Notes 16, Max Planck Institute for Mathematics in the Sciences, Leipzig, 2003. 26. Kohn, R.V. and Strang, D., Optimal design and relaxation of variational problems I, II, III. Comm. Pure Appl. Math. 39, 1986, 113–137, 139–182, 353–377. 27. Kundler, I. and Finkelmann, H., Strain-induced director reorientation in nematic liquid single crystal elastomers, Macromol. Rapid Commun. 16, 1995, 679–686. 28. Müller, S., Variational models for microstructure and phase transitions, in Calculus of variations and geometric evolution problems (Cetraro, 1996), Lecture Notes in Math., Vol. 1713, Springer, Berlin, 1999, pp. 85–210. 29. Müller, S. and Šverák, V., Convex integration with constraints and applications to phase transitions and partial differential equations, J. Eur. Math. Soc. (JEMS) 1(4), 1999, 393–422. 30. Ortiz, M. and Repetto, E.A., Nonconvex energy minimization and dislocation structures in ductile single crystals, J. Mech. Phys. Solids 47(2), 1999, 397–462. 31. Šilhavý, M., Ideally soft nematic elastomers, Netw. Heterog. Media 2, 2007, 279–311. 32. Tartar, L., The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Equations, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., Vol. 111, Reidel, 1983, pp. 263–285. 33. Tartar, L., H -measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Royal Soc. Edinburgh Sect. A 115, 1990, 193–230. 34. Verwey, G.C., Warner, M. and Terentjev, E.M., Elastic instability and stripe domains in liquid crystalline elastomers, J. Phys. II France 6, 1996, 1273–1290. 35. Warner, M. and Terentjev, E.M., Liquid Crystal Elastomers, International Series of Monographs on Physics, Vol. 120, Oxford University Press, 2003.

Variational Concepts with Applications to Microstructural Evolution F.D. Fischer, J. Svoboda and K. Hackl

Abstract In systems at elevated temperature the development of the microstructure of a material is controlled by diffusional and interface migration processes. As first step the description of the microstructure is reduced to a finite number of time-dependent characteristic parameters (CPs). Then the Thermodynamic Extremal Principle (TEP) is engaged to develop the evolution equations for these characteristic parameters. This treatment is demonstrated on a bamboo-structured material system predicting the spatial and time distribution of chemical composition as well as the deformation state.

1 Introduction Let us consider a material as a rather complicated system evolving with time. We concentrate here on structural materials like steels subjected to a time history with respect to its thermal and mechanical treatment. As an example we think on precipitates which may influence the mechanical properties of steel to an extreme amount. First, precipitates nucleate and start to grow, the change in the chemical composition of the supersaturated matrix being still negligible. Then, during the growth stage of the precipitates, the supersaturation of the matrix decreases substantially. Finally, in the coarsening stage, large precipitates grow at the expense of smaller ones and the matrix supersaturation is very low and gradually decreases towards equilibrium. F.D. Fischer Institute of Mechanics, Montanuniversität Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria; E-mail: [email protected] J. Svoboda Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 22, CZ-616 62 Brno, Czech Republic K. Hackl Lehrstuhl für Allgemeine Mechanik, Ruhr-Universität, Bochum, 44780 Bochum, Germany K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_6, © Springer Science+Business Media B.V. 2010

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The picture is complicated by the fact that the evolution of the microstructure is influenced by misfit interactions between precipitates and that, for multi-component alloys, the chemical composition of the precipitates may be changing (and far from their equilibrium values) at all stages of phase separation. As already mentioned the whole process is an irreversible, thermodynamically non-equilibrium process. In the further context we concentrate on linear nonequilibrium thermodynamics. We will, therefore, describe the system by a set of characteristic parameters. The system treated in material science are usually rather complex. In most cases, however, only a limited number of characteristic parameters (CPs), denominated as qi (i = 1, . . . , N), can be used for the description of the state and evolution of the system, and only a limited number of CPs are of interest implying a certain degree of idealization of the system. The number of CPs depends on the required accuracy of the description and on the complexity of the system. Moreover, constraints of local or global character amongst the CPs must often be taken into account. The goal is the determination of the evolution equations for the CPs, resulting in the time evolution of the rates q˙i (i = 1, . . . , N). The classical way of determining the evolution of a system is represented by the solution of phenomenological equations complemented by conservation laws and proper boundary and contact as well as initial conditions. By setting some assumptions on the system geometry a certain degree of idealization of the system can be achieved, so that the phenomenological equations can be solved providing the time evolution of the CPs. However, in many practical cases the amount of necessary idealization may be completely lacking of the physical substance of the problem. Obviously, the solution of the standard phenomenological equations need not to be the most efficient way. This problem can be overcome by the application of the Thermodynamic Extremal Principle (TEP), which represents a tool for deriving evolution equations for the CPs in a direct way but is very rarely exploited. The present paper is aimed to show a concept of how the TEP can be applied. Before doing this we give an example for the qi , which can be considered as internal variables within the context of continuum thermodynamics. In the case of precipitates the q˙k can be the rates of the radii ρk (k = 1, . . . , m) of the precipitates k, and the rates c˙ki of the mean concentration cki of a component i (i = 1, . . . , n) in the precipitate k. These rates can be related directly via the mass conservation equation to the fluxes jki ; for details see. e.g. [23].

2 The Thermodynamical Extremum Principle (TEP) Remark: If practicable we use the Einstein summation convention.

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2.1 General Formulation of the TEP The story of TEP starts with the variational formulation of a thermodynamic problem by Onsager in 1931 [19], who showed that the equations for heat conduction in an anisotropic system could be derived from the requirement of the maximum of a functional having a close relation to the total entropy production in the system. By the way, Onsager [20] applied the TEP also for a variational description of diffusion. However, this paper has been “lost” in the open literature. We refer the reader here to an overview and rather basic paper by Svoboda et al. [23] with a lot of according references. Let us explain the TEP shortly with an exercise. We distinguish between the entropy production P of a process and the dissipation Q. In the notation of continuum thermodynamics we have thermodynamic forces fi , which can be found as derivatives −∂φ/∂qi of the free energy φ(ext; qi , . . . , qN ). The symbol “ext” stands for the set of external thermodynamic variables. Then P follows as P = fi q˙i = −

∂φ q˙i . ∂qi

(1)

The entropy production P reflects the thermal, mechanical and chemical process to which a system is subjected. On the other hand we have a dissipation Q which reflects how the internal variables qi develop due to the microstructure of a system. Q is a positive function of the qi and q˙i , Q = Q(qi , q˙i ). (2) and can be a positive definite quadratic form of q˙i or, in a more general view, the first non-zero Taylor series term for a development of Q with respect to q˙i at q˙i = 0. The TEP says now that Q obtains a maximum for the side condition that P ≡ Q, max{Q(qi , q˙i )|q˙i ; Q = fi q˙i }.

(3)

The necessary conditions are now ∂ (Q + λ(Q − fi q˙i )) = 0, ∂ q˙j

j = 1, . . . , N.

(4)

The quantity λ represents a Lagrange multiplier. Of course, some further constraints can be added with further Lagrange multipliers, see later. The solution of (4) yields after some analysis fi =

Q ∂Q . ∂ q˙i

∂Q ∂ q˙j q˙ j

(5)

If one can invert Eq. (5), one finds evolution equations of the type q˙i = q˙i (fi , . . .).

(6)

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Hackl and Fischer [11] showed that the TEP is equivalent to Minimum Principle for the Dissipation Potential, if the dissipation potential and Q are functions of homogeneous functions of the order ≥ 1. It is interesting to note that for an order = 1 one can deal also with time-independent plasticity.

2.2 Application of the TEP to Problems in Mechanics and Thermodynamics of Materials First applications of the TEP were published by Svoboda and coauthors for creep and sintering [22]. A series of papers has been devoted to understand diffusion taking into account the role of vacancies with sources and sinks [24]. These studies have opened the door to a wide field of specific problems connected with diffusive processes in materials. Phase transformations, meeting both the role of a thick interface and the so-called Kirkendall effect, were dealt with, see e.g. [24, 25]. Also a quick melting process was investigated [9]. The TEP helped also to understand the solute drag in real (thick) interfaces [10] and allows to understand also the thermodynamic stability of interfaces [8]. The phenomenon of grain growth and coarsening can also be described in a unique way by the TEP, see [26]. From the practical point of view a specific emphasis has been laid to predict the evolving microstructure in multi-component alloys as steels, see e.g. [23]. The generation and annihilation of several types of carbides in multi-component steels can now be predicted in an accurate way [21]. It was also possible to include into the TEP shape parameter terms allowing for predicting the varying shape of different kinds of precipitates, from needles to spheres, see [27]. An independent approach, deriving an equivalent formulation to the TEP, was followed by Cocks and coworkers [4]. Most recently Moroz [17] used the TEP for nonlinear chemical thermodynamics. Finally it should be mentioned that the TEP has been overtaken also in the book by Hillert [13, section 17].

3 Multi-Component Diffusion and the TEP It should be mentioned that the literature on vacancy-mediated diffusion is rather rare, specifically with respect to non-ideal sources and sinks for vacancies. If vacancies are considered, usually tacitly an equilibrium concentration of vacancies is assumed. This concept was firstly followed by Darken [7] solving the so-called “Kirkendall” problem for a binary alloy (see [5, 18] for a mathematical treatment). Later this concept was extended to multi-component diffusion, see e.g. the contributions of the Swedish group [1, 3, 12] and the recent overview by Kattner and Campbell [14]. In the following context non-ideal sources and sinks for vacancies are dealt with, which enforces also constitutive laws for the generation or annihilation of vacancies.


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3.1 Mass Conservation We investigate a system consisting of n substitutional components (k = 1, . . . , n) and m interstitial components (k = n + 1, . . . , n + m). Instead of molar fractions we use site fractions yk for the component k, since we are dealing also with vacancies with a site fraction y0 . In the case of ideal sources and sinks of vacancies y0 obtains eq its equilibrium value y0 , which is the so-called “Darken” case, see above. To each component k ≥ 1 a scalar partial molar volume k is addressed in the isotropic ¯ 0, case. To the vacancies we address the partial molar volume ¯0 =

n yk k . 1 − y0

(7)

k=1

This leads to the molar volume of one mole of lattice sites as ¯ = ¯0+

n+m

yk k ;

(8)

k=n+1

for details see [24]. To each component (also for the vacancies) we address a flux jk , described in the actual configuration. Then the mass conservation enforces the following relations ¯ div jk − αyk , k ≥ 1, (9) y˙k = − ¯ div j0 + α(1 − y0 ), y˙0 = − n

jk = −j0 ;

(10) (11)

k=1

for details see [24]. The quantity α represents the rate at which vacancies are generated (α > 0) or annihilated (α < 0). One has to treat α, or in general the tensor α with the trace α = δ : α and δ being the unity tensor, as additional thermodynamic eq fluxes. Only in the case of y0 = y0 we can calculate α from (10) keeping in mind eq eq that y˙0 ≡ 0 and y0 1, yielding with (11) n ¯ div ¯ div j0 = − (12) jk . α= k=1

One can immediately see from (12) that all the fluxes of the substitutional components (k ≥ 1) determine α. Furthermore, since the fluxes are generally functions of the yi , i = 1, . . . , n + m, the quantity α is a rather complicated and nonlinear expression in the yi (acting as the CPs, or expressed in variables, as the qi ). We can now define an eigenstrain tensor ε gc (in the actual configuration), whose rate ε˙ gc can be decomposed additively into two parts as ε˙ gc = 1 ε˙ gc + 2 ε˙ gc ,

(13)

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with 1

2

ε˙ gc

ε˙ gc =

¯0 α, ¯

⎞ ⎛ n n+m ¯ 0) δ (k − =− ⎝ divjk + k divjk ⎠ . 3 1 − y0 k=1

1ε ˙

(14)

(15)

k=n+1

refers due to generation and annihilation of vacancies, 2 ε˙ gc as the second term is rather standard and takes into account the different size of the individual molar volumes (for details see e.g. [24]). Note that spherical symmetry (isotropic material) is assumed for sake of simplicity. Otherwise, one has to move to tensorial molar volumes. gc

3.2 Entropy Production and Dissipation First, we consider a constant load stress state σ and constant temperature T . Furthermore, we neglect the local elastic strain energy term. Second, we work with a representative volume element with average quantities, so we can now move from total quantities P and Q to specific quantities p and q (please do not mix it with the qi ). Without details, which can be found in [24], we can express the specific entropy production p as n+m µ∗0 : α ∗ p=− (16) grad µk · jk , + ¯ k=1 and the specific dissipation q as

q = Fg : α +

n+m k=1

(jk )2 Ak

2

n =1

+

j

A0

.

(17)

The following comments and definitions are necessary: • As thermodynamic fluxes we have α and jk , k = 1, . . . , n + m. • The chemical potential for the vacancies is described as following: eq

µ0 = RT n(y0 /y0 ), R is the gas constant, T the temperature. Furthermore, we introduce the tensor potential of the vacancies as ¯ 0. µ∗0 = δµ0 − σ

(18)

Several papers have been published, mainly based on Darken’s concept, where ¯ 0 is ignored in (18), see e.g. [6]. the term −σ • The generalized chemical potentials of the individual components follow as


¯ H /(1 − y0 ), µ∗k = µk − µ0 − (k − )σ µ∗k = µk − k σH ,

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i = 1, . . . , n,

i = n + 1, . . . , n + m.

As σH we denote the hydrostatic stress defined as σ : δ/3. The quantity µk is the sum of the chemical potential (energy) of the individual components, which may depend on the site fractions of all the other components, and the so-called “entropy of mixture” term RT nyk . • The quantities Ak , k = 1, . . . , n + m are so-called atomic mobility terms written as eq yk Dk y0 · Ak = , k = 1, . . . , n, eq ¯ RT y 0

eq

Ak =

yk Dk , ¯ RT

k = n + 1, . . . , n + m .

eq

The Dk are the tracer diffusion coefficients of the individual components in eq the case of y0 = y0 . The mobility term for the vacancies follows according to Manning’s derivations [15] A0 = −

n 1 Ak , 1−f

(19)

k=1

with f being the geometric correlation factor for random walks (f = 0.7815 for fcc- and 0.7272 for bcc-alloys), for an excellent overview on this topic, see [2]. It should be mentioned that also an improved concept exists, elaborated by Moleko [16], with individual correlation factors for each component. However, we recommend to follow the simpler Manning concept. • Fg is a yet unknown tensor of thermodynamic forces due to the generation and annihilation of vacancies. • The specific dissipation q is obviously a linear form in α and a quadratic form in jk , k = 1, . . . , n + m. Inspecting p shows that also two separate terms exist with respect to α and jk . Therefore, one can conclude that two independent processes are still coexistent, the first one occurs due to the generation and annihilation of vacancies, the second one is a diffusion process.

3.3 Application of TEP to q with q = p Since the processes with respect to α and jk (k = 1, . . . , n + m), are decoupled, also the application of the TEP can be decoupled yielding immediately for α Fg : α = −

µ∗0 : α, ¯

(20)

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and as a consequence Fg = −

µ∗0 . ¯

(21)

To ensure q ≥ 0 the simplest evolution law for α is a linear one, α = −R :

µ∗0 ,

(22)

with R being a positive definite fourth order tensor, details see below. With respect to jk the specific dissipation q contains a quadratic form yielding immediately λ = −2 in (4) and after some analysis jk = −

n+m

Lik grad µ∗i ,

i, k = 1, . . . , n + m,

(23)

i=1

Lik = Ak δik +

(1 − f ) Ai Ak n k , f j =1 Aj

i, k, 1, . . . , n + m.

(24)

with no summation and the indicator k = 1 for k ≤ n, otherwise 0. Following comments seem to be valuable: • The Lik are often denominated the “Onsager”-terms, mainly due to the symmetry Lik = Lki . • The relations (23), (24) are derived in the actual configuration; of course the gradand div-operations have also to be performed in this configuration. • The Lik are a direct outcome of the TEP and do not stem from transformations between, often obscure, reference configurations. The reader may convince him/herself inspecting chapters of diverse books on diffusion. • The off-diagonal terms disappear, if f is set to 1. This has been done very often in the open literature – we refer here to the already mentioned Darken solution for the Kirkendall effect, see e.g. [24]. However (1 − f )/f is in the order of 1/3 and can usually not be neglected a-priori. • Of course, one can transfer the set of equations to a specific initial or reference configuration, e.g. the deformed configuration only due to the initial stress state σ 0 . Here we refer to the usual manipulations between configurations, working with the deformation gradient F.

4 Examples for Evolving Microstructures Let us investigate a so-called bamboo-structure as it may be present in very thin metallic wires consisting of a sequence of grains in the longitudinal (1-) direction with the cross-section of only one grain. The grains act in their interior (bulk) as non-ideal sources and sinks for vacancies, mainly due to the existence of jogs at dislocation lines. The grains are separated by so-called grain-boundary regions, which


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are rather prominent ideal sources and sinks for vacancies. For sake of simplicity we assume an unstressed situation and concentrate on the proper description of α. For the bulk (being a grain) an isotropic tensor α from Section 3.1 is given by equations (54) and (62) in [24] as α=−

¯ 2πµ0 aH A0 δ, 3

(25)

where H is the jog density and a is the lattice spacing. For the grain boundary region normal to the 1-direction the tensor is given by equation (67) in [24] as ⎛ ⎞ n+m ¯ div j0 δ ⎝ ¯ δ˜ − α= yk k ⎠ , (26) ¯ 0 (1 − y eq ) 3 0 k=n+1 where δ˜ is a tensor with the only non-zero component δ˜11 = 1. Let us assume a Fe-Mn-C bamboo structured system with 10 grain boundaries. The thermodynamics of the Fe-Mn-C system has been calculated using a twosublattice regular solution model and diffusion coefficients taken from the literature, for details see [28]. As the initial state a wire of the length 10 µm is chosen with the chemical composition yFe = 0.900, yMn = 0.099, yC = 0.010 in the left half of the wire and yFe = 0.960, yMn = 0.039, yC = 0.030 in the right half of the wire. In the whole wire the initial and equilibrium site fraction of vacancies eq y0 (t = 0) = y0 = 0.001 is supposed. The jog density H is varied to simulate the influence of sources and sinks for vacancies in the bulk on the system evolution. The kinetics of the system is described by the time evolution of the site fraction profiles (inclusive y0 ) and of the strain parallel (εx ) and normal (εr ) to the wire axis. The strain compents εx and εr are the time integrals of the only two components of the strain tensor ε˙ gc in (13–15). In Figure 1 the results of the simulation of the wire with grain boundaries for a value a · H = 1 · 1014 m−2 are presented. The evolution of the site fraction profiles is presented in Figures 1a–c. The site fraction of vacancies is kept at its equilibrium value in those elements which include a grain boundary (see Figure 1c). The strain components due to the generation and annihilation of vacancies at the grain boundaries dominate (compare Figures 1d and 1e). The vacancy flux shows significant jumps at grain boundaries and keeps practically constant inside the grains (Figure 1f). In Figure 2 the results of simulation for the wire with grain boundaries and the value a · H = 1 · 1018m−2 are presented. The change in profiles of the components in comparison to Figures 1a, b is insignificant and, thus, corresponding plots are not presented. However, in comparison to Figure 1c the deviation from the equilibrium site fraction of vacancies has significantly decreased (Figure 2a), and also the axial strain due to generation and annihilation of vacancies at grain boundaries decreased markedly (compare Figures 1d and 2b). On the contrary the profile of the radial strain has changed also qualitatively (compare Figures 1e and 2c). The explanation

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Fig. 1 Time evolution of profiles for a · H = 1 · 1014 m−2 . (a) Site fraction of Mn, (b) site fraction of C, (c) site fraction of vacancies, (d) axial strain (e) radial strain, (f) vacancy flux.

of this effect can be found in [24]. Since the sources and sinks for vacancies are sufficiently active in the whole system, the profile of the vacancy flux is nearly smooth (see Figure 2d).


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Fig. 2 Time evolution of profiles for a · H = 1 · 1018 m−2 . (a) Site fraction of vacancies, (b) axial strain (c) radial strain, (d) vacancy flux.

5 Conclusion Extremum principles for non-equilibrium states, as worked out by the authors, are in the position to treat rather difficult microstructure evolution problems. Onsager’s principle is shown to deliver the evolution equations for combined diffusional and interface migration processes. The mostly not well understood role of vacancies can be treated within this concept in a rational way. Finally all necessary evolution equations are provided. A rather difficult bamboo-structured material system is investigated with respect to the spatial and time distribution of chemical composition and deformation state. The concept is open also for much more difficult microstructures with interacting families of precipitates and vacancy-induced pores.

References 1. Andersson, J.O. and Agren, J.: Models for numerical treatment of multicomponent diffusion in simple phases, J. Appl. Phys. 72, 1992, 1350–1355. 2. Belova, I.V. and Murch, G.E.: Analysis of interdiffusion data in multicomponent alloys to extract fundamental diffusion information, J. Phase Equil. Diff. 27, 2006, 629–637.

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3. Chen, Q., Jeppsson, J. and Agren, J.: Analytical treatment of diffusion during precipitate growth in multicomponent systems, Acta Mater. 56, 2008, 1890–1896. 4. Cocks, A.C.F., Gill, S.P.A. and Pan, J.-Z.: Modeling microstructure evolution in engineering materials, in E. van der Giessen and Th.Y. Wu (Eds.), Advances in Applied Mechanics, Vol. 36, Academic Press, San Diego, 1999, pp. 81–162. 5. Cornet, J.-F. and Calais, D.: Etude de l’effet Kirkendall d’apres les equations de Darken. J. Phys. Chem. Solids 33, 1972, 1675–1684. 6. Danielewski, M., Wierzba, B., Bachorczyk-Nagy, R. and Pietrzyk M.: Three-dimensional interdiffusion under stress field in Fe-Ni-Cu alloys, J. Phase Equilibria Diff. 27, 2006, 691–698. 7. Darken, L.S.: Diffusion, mobility and their interrelation through free energy in binary metallic systems, Trans. Amer. Inst. Min. Metall. Engrg. 175, 1948, 184–201. 8. Fratzl, P., Fischer, F.D. and Svoboda, J.: Energy dissipation and stability of propagating interfaces. Phys. Rev. Lett. 95, 2005, 195702-1–195702-4. 9. Gamsjäger, E., Svoboda, J., Fischer, F.D. and Rettenmayr, M.: Kinetics of solute driven melting and solidification, Acta Mater. 55, 2007, 2599–2607. 10. Gamsjäger, E., Svoboda, J. and Fischer, F.D.: Solute drag or diffusion processes in a migrating thick interface, Philos. Mag. Lett. 88, 2008, 415–420. 11. Hackl, K. and Fischer, F.D.: On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials, Proc. Royal Soc. A 464, 2008, 117–132. 12. Helander, T. and Agren, J.: Diffusion in the B2-B.C.C. Phase of the Al-Fe-Ni system – Application of a phenomenological model, Acta Mater. 47, 1999, 3291–3300. 13. Hillert, M.: Phase Equilibria, Phase Diagrams and Phase Transformations, 2nd edn., Cambridge University Press, Cambridge, 2008. 14. Kattner, U.R. and Campbell, C.E.: Modelling of thermodynamics and diffusion in multicomponent systems, Mat. Sci. Technol. 25, 2009, 443–459. 15. Manning, J.R.: Diffusion Kinetics for Atoms in Crystals, Van Nostrand, Princeton, NJ, 1968. 16. Moleko, L.K. and Allnatt, A.R.: Exact linear relations between the phenomenological coefficients for matter transport in a random alloy, Philos. Mag. A 58, 1988, 677–681. 17. Moroz, A.: A variational framework for nonlinear chemical thermodynamics employing the maximum energy dissipation principle. J. Phys. Chem. B 113, 2009, 8086–8090. 18. Nakajima, H.: The discovery and acceptance of the Kirkendall effect: The result of a short research career. JOM 49, 1997, 15–19. 19. Onsager, L.: Reciprocal relations in irreversible processes I. Phys. Rev. 37, 1931, 405–426. 20. Onsager, L.: Theories and problems of liquid diffusion. Ann. N.Y. Acad. Sci. 46, 1945, 241– 265. 21. Sonderegger, B., Kozeschnik, E., Leitner, H., Clemens, H., Svoboda, J. and Fischer, FD.: Computational analysis of the precipitation kinetics in a complex tool steel. Int. J. Mat. Res. 99, 2008, 410–415. 22. Svoboda, J. and Riedel, H.: Quasi-equilibrium sintering for coupled grain-boundary and surface diffusion, Acta Mater. 43, 1995, 499–506. 23. Svoboda, J., Fischer, F.D., Fratzl, P. and Kozeschnik, E.: Modelling of kinetics in multicomponent multi-phase systems with spherical precipitates I. – Theory, Mat. Sci. Engrg. A 385, 2004, 166–174. 24. Svoboda, J., Fischer, F.D. and Fratzl, P.: Diffusion and creep in multi-component alloys with non-ideal sources and sinks for vacancies, Acta Mater. 54, 2006, 3043–3053. 25. Svoboda, J., Vala, J., Gamsjäger, E. and Fischer, F.D.: A thick-interface model for diffusive and massive phase transformation in substitutional alloys, Acta Mater. 54, 2006, 3953–3960. 26. Svoboda, J. and Fischer, F.D.: A new approach to modelling of non-steady grain growth, Acta Mater. 55, 2007, 4467–4474. 27. Svoboda, J., Fischer, F.D. and Mayrhofer, P.H.: A model for evolution of shape changing precipitates in multicomponent systems, Acta Mater. 56, 2008, 4896–4904. 28. Svoboda, J., Fischer, F.D. and Gamsjäger, E.: Simulation of chemically driven inelastic strain in multi-component systems with non-ideal sources and sinks for vacancies, Acta Mater. 56, 2008, 351–357.

A Micromechanical Model for Polycrystalline Shape Memory Alloys – Formulation and Numerical Validation Rainer Heinen and Klaus Hackl

Abstract The specific material properties of shape memory alloys are due to the formation of martensitic microstructures. In this contribution, we develop a strategy to model the material behavior based on energy considerations: we first present narrow bounds to the elastic energy obtained by lamination of the multi-well problem in the monocrystalline case. These considerations are then extended to polycrystals and compared to a convexification bound. Due to the acceptably low difference between convexification lower and lamination upper bound, we use the convexification bound to establish a micromechanical model which, on the basis of physically well motivated parameters such as elastic constants and transformation strains, is able to represent a variety of aspects of the material behavior such as pseudoelasticity, pseudoplasticity and martensite reorientation.

1 Introduction Since shortly after the discovery of the special aspects in the material behavior of shape memory alloys, there have been many efforts to model its important features. The main focuses are, on the one hand, to find appropriate phenomenological descriptions for macroscopic applications and to explain microscopic properties by physical considerations, on the other hand. The literature on these topics is too extensive to be summarized here; well-known examples are, however, Bouvet et al. [4] and Helm and Haupt [18] for phenomenological, as well as Ball and James [1], Rainer Heinen Werkstoffkompetenzzentrum, Division Auto, ThyssenKrupp Steel AG, Duisburg, Germany, and Lehrstuhl für Allgemeine Mechanik, Ruhr-Universität Bochum, Germany; E-mail: [email protected] Klaus Hackl Lehrstuhl für Allgemeine Mechanik, Ruhr-Universität Bochum, Germany; E-mail: [email protected] K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_7, © Springer Science+Business Media B.V. 2010

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Stupkiewicz and Petryk [28], Kohn [19], and Truskinovsky [30] for microscopic modeling. In this paper, we connect earlier works on the energy density computation for monocrystals [8, 9, 16] with others on the modeling of polycrystals [13]. The connection is established based on an approach by Bruno et al. [5] and Smyshlyaev and Willis [26] as described in [14]. The material behavior of shape memory alloys is characterized by transformation strains η, chosen to be η0 = 0 for the austenite, and elastic constants collected in the fourth-order elasticity tensor C. Furthermore, the chemical energy αi is introduced to distinguish between the height of the energy wells of different crystallographic phases: αA = α0 for the austenite and αM = αi , 1 ≤ i ≤ n, for the different martensitic variants. Assuming linear elastic material behavior, the energy density for pure variants in the monocrystalline case is W (ε, ii ) =

1 (ε − ηi ) : C : (ε − ηi ) + αi , 2

where ε is the linearized strain and ii are the unit-vectors in n + 1-dimensional variant space. Since deformations of several percent are observed in the material behavior of shape memory alloys, it may, at first glance, seem surprising that a linear elastic model is used here. However, most of this deformation is realized by phase transformation processes while the additional elastic straining of the different phases remains clearly limited. For this reason, a linear elastic model is suitable for most technical applications which are, in general, designed to bear loads in the regime of the transformation stress where the influence of dislocation-mediated plasticity is still negligible. Phenomenological modeling for structures experiencing large deformations and rotations has been performed in [6, 22, 24], for instance. The energy at every microscopic material point is determined as the one minimizing the elastic energy for a given strain ε W (ε) = min [W (ε, ii )] .

(1)

i

Due to the non-convexity of this formulation, further energy reduction can be reached by microstructure formation on the mesoscopic level. Mathematically, this corresponds to the quasiconvexification of Eq. (1) in the sense of [7] QW mono (ε) = min QW mono (ε, c) with c ⎧ ⎨ mono s W ε + ∇ φ (y) , χ (y) dy QW (ε, c) = inf χ,φ ⎩

1,2 n+1 φ ∈ Wper , () , χ (y) ∈ Ppure

(2) ⎫ ⎬ χ (y) dy = c . ⎭

Micromechanics of Polycrystalline SMA

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Expanding Eq. (2) gives rise to the definition of the so-called energy of mixing ⎧

n ⎨ 1 s ∇ φ− ∇sφ : C : χi ηi dy wmix (c) = inf χ,φ ⎩ 2 i=0 ⎫ ⎬ 1,2 n+1 φ ∈ Wper , χ (y) dy = c () , χ (y) ∈ Ppure ⎭

and the alternative formulation of the quasiconvex energy density QW mono (ε, c) =

n

[ci W (ε, ii )] + wmix (c) ,

(3)

i=0

where the first addend is a Taylor-type upper bound corresponding to the assumption of a constant strain throughout the representative volume element. This formulation has also been employed in several earlier works on the energy-based modeling of shape memory alloys [9, 10, 16]. Due to the complexity of its constraints, no direct expression is known for the energy of mixing unless a maximum of two [19] or three variants are considered [27].

2 Lamination Upper Bound As an estimate to the energy of mixing, we introduce a lamination upper bound based on the microstructural pattern shown in Figure 1, which consists of austenite and twinned martensite. The choice of the twinned martensitic variants is constrained by the linearized lamination mixture formula ηi − ηj =

1 (n ⊗ a + a ⊗ n) 2

as developed in [1] and [3]. This formula determines wether variants i and j are able to form a stress free interface with each other. Only those pairs of variants that fulfill this equation are combined to twins which are then numbered pairwise (variant 1 twinned with variant 2, 3 with 4, and so on) while the austenite remains to be variant 0. The suggested microstructure is employed to derive the second-order lamination upper bound to the energy of mixing wmix (c) ≤ wlam (c) =

M K=1

dK θK (1 − θK ) φ (i2K−1 − i2K )

(4)

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Austenite Martensite 1 Martensite 2 Martensite 3 Martensite 4 Martensite 5 Martensite 6

Fig. 1 Assumed microstructural pattern for a seven-variant material, yielding three martensitic twins in addition to the austenite.

M (d0 + d1 + · · · + dK−1 ) dK (K−1) + φ d − θ (K) d0 + d1 + · · · + dK K=1

with φ (κ) = inf −G (ω) : (κ ⊗ κ)| ω ∈ Sd−1 G (ω) =

1 (ω · C : ηk ) · T (ω)−1 · (ω · C : ηl ) ik il 2

u · T(ω) · u = (ω ⊗s u) : C : (ω ⊗s u) ∈ Rd×d sym , c

2J −1 where d is the spacial dimension of the problem considered, θJ = c2J −1 +c2J is the volume fraction of the first variant within the J th twin (where J = 1, . . . , m runs over the different twins), θ J = θJ e2J −1 + (1 − θJ ) e2J is the normalized phase fraction vector for the J th twin, dJ = c2J −1 + c2J is the volume fraction of the J th twin within the microstructure, and dJ = JK=0 dK θ K / JK=0 dK is a vector which contains the normalized sum of the austenite and all martensitic twin phase fractions up to a certain twin J . In Eq. (4), each permutation of the ordering of the martensitic variants into twins as well as of the order of these twins gives a new value for the upper bound. A detailed derivation and proof of this bound may be found in [8].

3 Extension to Polycrystals Now, let us consider a polycrystalline domain consisting of a large, but finite number of crystallites N. Each crystallite j has a certain orientation given by the corresponding rotational tensor Rj ; the volume fraction corresponding to this crystal


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orientation is denoted by ξ j . The transformation strains have to be rotated to the T j local crystallographic orientation and result in ηi = Rj · ηi · Rj . Consequently, the elastic energy of variant i in grain j yields 1 j j W¯ ε, ii , Rj = ε − ηi : C : ε − ηi + αi 2 and the mesoscopic energy of the polycrystal is given by ⎫ ⎧ ⎬ ⎨ 1,p QW mono R (y) ε + ∇ s φ˜ (y) RT (y) dy φ˜ ∈ Wper () . QW¯ (ε) = inf ⎭ φ˜ ⎩

Here, the monocrystalline quasiconvexification, which is not known in explicit form, would have to be computed for the crystallographic orientation present in every point of the representative volume element . In a sophisticated paper by Smyshlyaev and Willis [26], these authors present a strategy to employ monocrystalline upper estimates for bounding the energy of polycrystals from above: QW¯ (ε) ≤ QW¯ lam (ε) = inf QW¯ lam (ε, c) c

with QW¯ lam (ε, c) =

1 1 (5) (ε − η) : C : (ε − η) − η : Q : η + α 2 2 N N mono j j 1 j j j mono + η ξ η : Q : η + ξ j QWlam ,c 2 j =1

j =1

and η =

j j

ξ j ci ηi ,

i,j

ηj =

j j

ci ηi ,

ηmono

i

j

=

i

j

ci ηi ,

α =

j

ξ j ci αi .

i,j

Here, we have plugged in the second order lamination bound presented above as an estimate of the relaxation with fixed volume fractions in the monocrystalline case. The fourth order texture tensor Q used in Eq. (5) is defined as follows: 1 ˜ ∞ (ζ ) dζ , Q = d−1 S Sd−1

where ˜∞ klop (ζ ) =

2λ¯ µ¯ Ukl (ζ ) Uop (ζ ) + µ¯ Uko (ζ ) Ulp (ζ ) + Ukp (ζ ) Ulo (ζ ) , ¯λ + 2µ ¯

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Ukl (ζ ) = δkl − ζk ζl , d−1 and S is the surface of the unit sphere. A detailed discussion of this bound including the influence of the polycrystalline texture may be found in [14].

4 Convexification A simple and intuitional way to obtain an estimate for the energy density of polycrystals is to assume a constant stress within each variant of each crystallite and neglecting the compatibility constraints. Since, in this case, minimization is performed over a larger set of microstructures than physically available, this so-called convexification estimate is a lower bound: ⎧ ⎫ ⎨ ⎬ j j j j ξ j ci W εi , ii , Rj ε = ξ j ci ε i . QW¯ conv (ε, c) = inf (6) j ⎭ εi ⎩ i,j i,j The remaining constraint in Eq. (6) makes sure that the overall strain is preserved. In a finite strain formulation, further refinement of the bound could be reached by additionally incorporating minors and the determinant of the overall strain which would lead to the polyconvexification of the energy density. The formulation given in Eq. (6) has the advantage that the minimization can be performed analytically. The solution QW¯ conv (ε, c) =

1 (ε − η) : C : (ε − η) + α 2

(7)

is discussed in [13] in detail for the more general case of anisotropic material properties for austenite and martensite. In the monocrystalline case, the convexification lower bound may be used to derive an alternative estimate for the energy of mixing: 1 1 ci ηi : C : ηi + ci ck ηi : C : ηk . (8) 2 2 n

wmix (c) ≥ wconv (c) = −

i=0

n

n

i=0 k=0

5 Comparison of Upper and Lower Bounds For evaluating the energy estimates obtained by upper and lower bounds, we first present a comparison of the monocrystalline energy of mixing computed from lamination and convexification, respectively, as described in Eqs. (3), (4), and (8). As an example, Figure 2 displays this strain-independent part of the energy for a linear


c

97

(1, 1, 5, 5, 5, 2, 2)/21 + (2, 4, 5, 0, 3, 2, 1)/17

Fig. 2 Monocrystal: Comparison of lamination upper and convexification lower bound, phase fraction variation from one random configuration to another, N = 1, αA = αM = 0.

Fig. 3 Polycrystal: Comparison of upper and lower bounds, phase fraction variation from one random configuration to another, isotropic texture, N = 50, αA = αM = 0.

path between two randomly chosen phase fraction vectors. The maximum difference between both bounds observed in this example is two per mill which means that the actual elastic energy is determined with a small remaining uncertainty. The convexification, lamination and Taylor bound in the case of isotropic polycrystals are compared in Figure 3. Here, the Taylor estimate has been obtained by setting wmix to zero in Eq. (3) and plugging this into Eq. (5) as a rough estimate of the relaxation for given volume fractions. The vicinity of lamination upper and con-

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vexification lower is clearly visible: the difference between these is only ten percent while the Taylor bound yields several times higher values. The isotropic elastic constants λ¯ = 89.7 GPa and µ¯ = 60.8 GPa and the transformation strains in these example have been calculated from the data determined in [25] by elaborate experimental investigations of cubic-orthorhombic transforming CuAlNi. The ability of the model to simulate the material behavior of cubic-monoclinic transforming NiTi has been presented in [17] based on earlier first-principles investigations to determine the elastic constants of B19’ NiTimartensite [29].

6 Micromechanical Model Given the good agreement between upper and lower bounds, we choose the more straightforward convexification estimate to establish an energy-based micromechanical model for polycrystalline shape memory alloys. In order to close the formulation, we introduce a dissipation function which is homogeneous of first order in the volume fraction change rates n N j 2 (˙c) = r c˙i . ξj j =1

i=0

This kind of dissipation function has been shown to be suitable for describing rateindependent materials in [20, 21], while the rate-independence of shape memory alloys has been demonstrated experimentally in [11]. The general procedure of describing the dissipative aspects of the behavior of rate-independent materials by dissipation functions is discussed in [12] and [23]. A more sophisticated approach to model dissipation in phase transforming materials may be found in [2]. The material behavior is then derived by minimizing the total power L (c, c˙ ) =

d QW¯ rel (ε, c) + (˙c) = −q · c˙ + (˙c) dt

(9)

at constant strain ε, where we have introduced the thermodynamically conjugated driving force to c˙ , q = −∂QW¯ rel /∂c, as an abbreviation. Furthermore, we define j j j the active sets Aj = {i | ci > 0 ∨ (ci = 0 ∧ c˙i > 0)} as well as the active deviator j

j

(devAj q)i = qi −

1

nAj

k∈Aj

j

qk ,

nAj being the number of elements in Aj , to finally obtain the evolution equation ρ j j c˙i = j devAj qi j A ξ


99

along with the Kuhn–Tucker conditions ρ≥0 1 j 2 = devAj qi − r 2 ≤ 0 j ξ i,j

ρ=0 and the consistency condition j

devAj qi < 0,

∀i ∈ / Aj

(10)

which serves as a “switch” and determines whether a certain variant that has been inactive before has to become active in a current time step. A detailed discussion of this model may be found in [13].

7 Numerical Examples As an example for the capability of the model to reproduce the material behavior of polycrystalline shape memory alloys, the upper left part of Figure 4 shows a stressstrain curve computed for axial loading. For this example, the material parameters have been chosen as follows: αA = −25 Nmm/mm3 , αM = 0 Nmm/mm3 , and r = 6 Nmm/mm3 . Due to the austenite chemical energy being lower than the martensite one, the material behaves pseudoelastically here. As can be seen from the plot, both the hysteresis and the well-known tensioncompression assymetry follow without any further data fitting or adjustment of any additional parameters. The right hand side of Figure 4 shows the development of one of the martensitic variants for certain loadsteps marked in the stress-strain diagram. Here, ϕ and θ are the polar coordinates corresponding to the (1, 0, 0)-plane normals of each of the 20 000 crystal orientations considered. Gray level and size of the dots are scaled with the volume fraction of variant 5 in each crystallite, thick black points meaning 100%. The reorientation pattern observed for this variant can be explained by considering the component of the transformation strain oriented parallely to the load direction, see lower left part of Figure 4. A similar illustration for the case of pure shear loading may be found in Figure 5. It can be seen from the volume fraction evolution of the same martensitic variant as in Figure 4, the texture observed in this example is very different from the case of axial loading. Again, the specific form of the texture is explained by favorable orientations of the transformation strain considered with respect to the loading direction.

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Fig. 4 Stress-strain curve, evolution of martensite variant 5 and xx-component of transformation strain. N = 20 000, axial loading.

Similar numerical examples show that pseudoplastic material behavior can equally be reproduced by appropriatly adjusting the chemical energies without changing the structure of the model, see [13] for further details.

8 Conclusion After bounding the elastic energy density narrowly from above and below by second order lamination and convexification, respectively, we have developed a micromechanical model capable of predicting the most important aspects of the material behavior of polycrystalline shape memory alloys such as pseudoelasticity and pseudoplasticity. The model relies on physically relevant parameters such as transformation strains and elastic constants. The martensite orientation distribution predicted by the model has been validated successfully by synchrotron diffraction experiments in [15]. Apparently, the very rough assumption of a constant stress throughout the whole polycrystalline domain is sufficient for estimating the material behavior of shape memory alloys.


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Fig. 5 Stress-strain curve, evolution of martensite variant 5 and yz-component of transformation strain. N = 20 000, shear loading.

Future work will include the simulation of cubic to monoclinic transforming, and thus thirteen-variant, materials and intermediate crystallographic phases such as the well-known R-Phase.

Acknowledgements This work has been funded by the Deutsche Forschungsgesellschaft under project A9 of the Collaborative Research Center 459 “Shape Memory Technology” at the Ruhr-Universität Bochum.

References 1. J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal. 100, 1987, 13–52. 2. T. Bartel and K. Hackl, A novel approach to the modelling of single-crystalline materials undergoing martensitic phase-transformations, Mat. Sci. Eng. A 481, 2008 371–375.

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3. K. Bhattacharya, Microstructure of Martensite – Why It Forms and How It Gives Rise to the Shape-Memory Effect, Oxford University Press, New York, 2003. 4. C. Bouvet, S. Calloch and C. Lexcellent, A phenomenological model for pseudoelasticity of shape memory alloys under multiaxial proportional and nonproportional loadings, Eur. J. Mech. A-Solids 23, 2004, 37–61. 5. O.P. Bruno, F. Reitich and P.H. Leo, The overall elastic energy of polycrystalline martensitic solids, J. Mech. Phys. Solids 44, 1996, 1051–1101. 6. D. Christ and S. Reese, Finite-element modelling of shape memory alloys – A comparison between small-strain and large-strain formulations, Mat. Sci. Eng. A 481, 2008, 343–346. 7. B. Dacorogna, Quasiconvexification and relaxation of nonconvex problems in the calculus of variations, J. Funct. Anal. 46, 1982, 102–118. 8. S. Govindjee, K. Hackl and R. Heinen, An upper bound to the free energy of mixing by twincompatible lamination for n-variant martensitic phase transformations, Continuum Mech. Thermodyn. 18, 2007, 443–453. 9. S. Govindjee, A. Mielke and G.J. Hall, The free energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis, J. Mech. Phys. Solids 51, 2003, 763. 10. S. Govindjee and C. Miehe, A mult-variant martensitic phase transformation model: Formulation and numerical implementation, Comput. Meth. Appl. Mech. Engrg. 191, 2001, 215–238. 11. S. Grabe and O.T. Bruhns, On the viscous and strain rate dependent behavior of polycrys˝ talline NiTi, Int. J. Solids Struct. 45, 2008, 1876-U1895. 12. K. Hackl and F.D. Fischer, On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials, Proc. R. Soc. A 464, 2008, 117–132 13. K. Hackl and R. Heinen, A micromechanical model for pre-textured polycrystalline shape memory alloys including elastic anisotropy, Continuum Mech. Thermodyn. 19, 2008, 499– 510. 14. K. Hackl and R. Heinen, A lamination upper bound to the free energy of n-variant polycrystalline shape memory alloys, J. Mech. Phys. Solids 56, 2008, 2832–2843. 15. K. Hackl, R. Heinen, W.W. Schmahl and M. Hasan, Experimental verification of a micromechanical model for polycrystalline shape memory alloys in dependence of martensite orientation distributions, Mat. Sci. Eng. A 481, 2008, 347–350. 16. R. Heinen and K. Hackl, On the calculation of energy-minimizing phase fractions in shape memory alloys, Comput. Meth. Appl. Mech. Engrg. 196, 2007, 2401–2412. 17. R. Heinen, K. Hackl, W. Windl and M. Wagner, Microstructural evolution during multiaxial deformation of pseudoelastic NiTi studied by first-principles-based micromechanical modeling, Acta Materialia 57, 2009, 3856–3867. 18. D. Helm and P. Haupt, Shape memory behaviour: Modelling within continuum thermodynamics, Int. J. Solids Struct. 40, 2003, 827–849. 19. R. Kohn, The relaxation of a double-well energy, Continuum Mech. Thermodyn. 3, 1991, 193–236. 20. A. Mielke, F. Theil and V.I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Rat. Mech. Anal. 162, 2002, 137–177. 21. A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Continuum Mech. Thermodyn. 15, 2003, 351–382. 22. C. Müller and O.T. Bruhns, A thermodynamic finite-strain model for pseudoelastic shape memory alloys, Int. J. Plasticity 22, 2006, 1658–1682. 23. M. Ortiz and L. Stainier, The variational formulation of viscoplastic constitutive updates, Comput. Meth. Appl. Mech. Engrg. 171, 1999, 419–444. 24. S. Reese and C. Christ, Finite deformation pseudo-elasticity of shape memory alloys – Constitutive modelling and finite element implementation, Int. J. Plasticity 24, 2008, 455–482. 25. P. Sedlák, H. Seiner, M. Landa, V. Novák, P. Sittner and Ll. Mañosa, Elastic constants of bcc austenite and 2H orthorhombic martensite in CuAlNi shape memory alloy. Acta Materialia 53, 2005, 3643–3661. 26. V. Smyshlyaev and J. Willis, A ‘non-local’ variational approach to the elastic energy minimization of martensitic polycrystals, Proc. R. Soc. London A 454, 1998, 1573–1613.


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27. V. Smyshlyaev and J. Willis, On the relaxation of a three-well energy, Proc. R. Soc. London A 455, 1998, 779–814. 28. S. Stupkiewicz and H. Petryk, Modelling of laminated microstructures in stressinduced martensitic transformations, J. Mech. Phys. Sol. 50, 2002, 2303–2331. 29. M.F.-X. Wagner and W. Windl, Lattice stability, elastic constants and macroscopic moduli of NiTi martensites from first principles, Acta Materialia 56, 2008, 6232–6245. 30. L. Truskinovsky, About the normal growth apporximation in the dynamical theory of phasetransitions, Continuum Mech. Thermodyn. 6, 1994, 185–208.

Solution-Precipitation Creep – Modeling and Extended FE Implementation Sandra Ilic and Klaus Hackl

Abstract The topic of this contribution is the mechanical modeling of solutionprecipitation creep, a process occurring in polycrystalline and granular structures under specific temperature and pressure conditions. The model presented has a variational structure and is based on a novel proposal for the dissipation while the elastic energy is kept in the standard form. The assumed dissipation term depends on two kinds of velocities characteristic for the process: velocity of material transfer and velocity of inelastic deformations, both manifesting themselves on the boundaries of the grains. For the numerical implementation, the standard finite element program FEAP together with the pre- and postprocessing software package GID are used. The simulations are illustrated by two examples, a polycrystal with regular hexagonal microstructure and a polycrystal with random microstructure.

1 Introduction Solution-precipitation creep is a deformation process typical for high temperatures and moderate stresses whereby 0.4 of the melting temperature and 10−7 of the shear modulus are threshold values for its initiation [21]. The process has a diffusional character and is based on the particle migration through the intercrystalline space of the polycrystal. The precipitation phase leads to the extension of existing crystals or to the formation of completely new ones, mostly with a fibrous habit and interleaved with inclusions and micropores. For itself, solution-precipitation creep is a “self-exhausting” process: on observing an extended period of time, the grains always become more elongated, which causes a significant decrease in strain rate and driving forces. Consequently, this deformation type is accompanied by rotation of the individual grains reestablishing new sources of driving forces. The solution Sandra Ilic · Klaus Hackl Institute of Mechanics, Ruhr University Bochum, D-44780 Bochum, Germany; E-mail: {sandra.ilic, klaus.hackl}@rub.de K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_8, © Springer Science+Business Media B.V. 2010

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precipitation creep is often mistaken for the process of grain boundary migration, although it has a quite distinct nature. The investigation of solution-precipitation creep has a long history. It started in the nineteenth century with Sorby [23,24], but still remains the subject of many new publications [3–7, 19, 20]. In the scope of a short overview, at first, a model known as Corren’s equation will be mentioned: c νσn = µ − µ0 = R T ln . (1) c0 Here σn denotes the normal stress, ν Poisson’s ratio, µ and c are the chemical potential and concentration, and superscript "0" denotes the reference state when σn = 0. R and T have the usual meaning in thermodynamics: they represent the universal gas constant and the absolute temperature respectively. In contrast to (1) which is mostly used in the geology, the following expression is customary in metallurgy: e˙ =

KD σ. d 2 kT

(2)

Here e˙ represents the strain rate, K is a proportionality coefficient between the normal stress gradient on the local scale and the macroscopic stress σ acting on the polycrystalline sample, D is the diffusion coefficient, the atomic volume, k Boltzmann’s constant and d is the diameter of a representative grain. Although at first view different from (1), expression (2) can be derived from it by using Fick’s law. Expression (2) can be also compared with Nabarro–Herring’s equation [11, 17] describing the diffusion process through the bulk crystal lattice e˙n =

KDn σ d 2r 2k2T

(3)

or with Coble’s equation [2] corresponding to the diffusion process which takes place along the crystal boundary e˙c =

kπDc h σ. d 3 r 3/2 kT

(4)

In the last two equations, e˙n and e˙c are strain rates, Dn and Dc diffusion coefficients, r is the ratio of the longest to the shortest dimension of the grain, and h is the width of the grain boundaries. In contrast equations above which are based on the modeling of so called governing (slowest) part of the entire process, our intention in this paper is to present a model considering solution-precipitation creep as a coupled process. To this end, Section 2 explains the definitions of elastic power and dissipation term combined to a Lagrangian. Furthermore, minimization yields Euler’s equations which describe inelastic processes on the grain boundaries. The elastic potential is assumed in the three-field formulation proposed by Simo, Taylor and Pister (Section 3). The Lagrangian presented in Section 2 together with the specification of the elastic part

Solution-Precipitation Creep – Modeling and Extended FE Implementation

107

(Section 3) are used as a basis for the derivation of the weak form necessary for an implementation of the finite element method (Section 4). Finally, Section 5 considers illustrative numerical examples and the paper ends with conclusions and outlook.

2 Continuum-Mechanical Model In order to postulate the continuum mechanical model for solution-precipitation creep, the behavior of a single crystal i with the boundary ∂i within a polycrystal is considered. Furthermore, the inelastic deformations are introduced as internal variables in such a way, that decomposition of the deformations φ into elastic and inelastic part φ Ei and φ Ii leads to the multiplicative decomposition of the deformation gradient F: (5) φ = φ Ei ◦ φ Ii ⇒ F = FEi · FIi . It is well known, that in plasticity only the decomposition of the deformation gradient is meaningful and in general cannot be integrated up to the level of deformations. Here, however, the situation is different. The mapping φ Ii describe the change of shape of an individual grain and this is well defined within each grain. Incompatibility is only encountered in between different grains. A further step in modeling is to define elastic energy and dissipation term, and to this end a short description of the physics lying behind the whole process will be recalled. During the solution-precipitation creep the material particles of a crystal dissolve from a boundary part, move through the intercrystalline space and precipitate on some other boundary part of the same or other crystals. The presence of the fluid phase especially intensifies the process as it accelerates the material transport. Naturally, the material transport causes the inelastic deformations of the crystal φ Ii and the boundary motion whose normal component vin is related to the inelastic deformations in the following way: vin =

d I φ · n¯ i . dt i

(6)

Note that hereafter the overbar symbol will be used in correlation with the inelastically deformed intermediate configuration. Accordingly, n¯ i represents the normal to the inelastically deformed surface of crystal i . The material motion and the boundary motion are related to each other, too, and this coupling condition is the consequence of the continuity equation vin = ∇ · Qi .

(7)

Here the symbol Qi is introduced to denote the velocity of the material transport. Finally, the dissipation term is assumed to be dependent on squares of described velocities γ κ ¯ = Q2i + (vin )2 dS. (8) 2 ∂i 2 i

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Here κ and γ are material constants so that κ/γ 0 be a hardening modulus and µ > 0 denote a Lamé parameter. Multiplicative split of the deformation gradient into an elastic part Fe and an irreversible, plastic part Fp yields the standard multiplicative decomposition F = Fe Fp . Then, we employ the energy density for an incompressible neo-Hookean material in the form (Fe , p) = 12 µ(tr, FTe Fe − 3) + κ

n

pi4 ,

det F = 1.

(17)

i

Each of the n active slip-systems i is characterized by its slip direction si and slip plane normal mi (|si | = |mi | = 1, si · mi = 0). pi denotes the hardening variable corresponding to slip system i. Consider the following flow rules with slip rates γ˙i [3], n (18) γ˙i si ⊗ mi , p˙ i = |γ˙i | F˙ p F−1 p = i

with initial conditions γi (0) = 0 and pi (0) = 0. Condition (18)2 assumes no crossslip, or infinite latent hardening. Time-integration of (18)1 for generally oriented slip systems may become quite complex [11]. Therefore, we limit our considerations here to slip systems with a common slip plane normal m. In this case, we can infer via time-integration that F−1 p =1−

n

γi si ⊗ m.

(19)

i

For the dissipation functional we simply assume the form (γ˙ ) = r

n i

with some material constant r ≥ 0, see [3].

|γ˙i |,

(20)

122

D.M. Kochmann and K. Hackl Y

cond

j

3

Y

j = 150° k = 0.000 k = 0.003 k = 0.010 k = 0.020

g

3.5

g

3

2.5

2.5

2

2

1.5

1.5

k= j= j= j=

1 0.5 -2

cond

-1

1

2

3

0.01 128.6° 138.5° 150.0°

0.5

g

5

4

1

-2

-1

1

(a)

2

3

4

5

g

(b)

Fig. 2 Condensed energy of a sample material subject to plane-strain simple shear deformation, undergoing a homogeneous deformation (no microstructure formation) for various hardening parameters and slip system orientations (only one active slip system).

Figure 2 illustrates the condensed energy path for a simple shear test of a homogeneous sample material which is not allowed to form microstructures. The nonconvex energy density of the above formulation clearly becomes apparent for γ > 0. Also, the influence of the slip system orientation (angle ϕ) and the hardening parameter κ can be observed. Note that for γ < 0, the energy density is convex and hence no microstructures arise; therefore, we only consider the non-convex regimes in the following. Due to the non-convex condensed energy, microstructures arise as energy minimizers. Let us assume a first-order laminate microstructure with N phases having interfaces with unit normal b. We define the deformation gradient in phase i according to Eq. (12). To ensure incompressibility of each laminate phase, we must enforce that det Fi = 1 or ai · b = 0. Taking this constraint into account, the approximated relaxed energy takes the form rel (F, λ, γij , pij , b) =κ

N

λi

n

i

j

⎡ µ 1 pij4 + ⎣ N 2 i +

N

λi

i

+

N

⎛ λi tr ⎝1 −

i

where

λi bi ·b

⎛ ⎞ N N λj λk bj · Cbk 1 ⎝ ⎠ − −1 bj · b bk · b b · C b j k

bi · b b · C−1 b

n

−

bi · Cbi bi · b ⎞ ⎛

γij m ⊗ sj ⎠ C ⎝1 −

j

⎛ bi = ⎝1 −

n j

−3 n

⎞⎤ γij sj ⊗ m⎠⎦ ,

(21)

j

⎞⎛ γij s ⊗ mj ⎠ ⎝1 −

n j

⎞ γij mj ⊗ s⎠ · b.

(22)

Time-Continuous Evolution of Microstructures in Finite Plasticity

123

C = FT F is the right Cauchy–Green tensor, pij is the hardening history and γij the plastic slip in phase i on slip system j . The solution for single-slip plasticity was shown in [7]. For simplicity, let us reduce the present model to a two-phase laminate (N = 2) with two active slip systems (n = 2). We define the volume fraction of phase 2 as λ such that, taking into account (20), the dissipation potential for only one slip system can be given in the form [7] ˙ γ˙i ) = r (1 − λ) |γ˙1 | + λ |γ˙2 | + ˙λ(γ1 − γ2 ) . ∗ (λ, γi , λ, (23) For multiple slip systems, the definition of ∗ is no longer unique. We assume a dissipation potential of the following type: ˙ γ˙ij ) = r [(1 − λ) (|γ˙11 | + |γ˙12 |) + λ (|γ˙21| + |γ˙22|) ∗ (λ, γij , λ, ˙ 11 − γ21 )| + |λ(γ ˙ 12 − γ22 )| . + |λ(γ

(24)

Note that this dissipation potential not only accounts for dissipation due to changes of the plastic variables in both phases [2, 12] but also due to changes of the volume fractions, where the latter contribution depends on the existing microstructure at the onset of the time step (viz. of values γij ). The Lagrange functional corresponding to (6) now takes the form ˙ γ˙ij , b) = L(F, λ, γij , pij , λ,

d rel ˙ γ˙ij ). (25) (F, λ, γij , pij , b) + (λ, γij , λ, dt

Via the principle given in (7) we now arrive at five evolution equations for λ and γij from the above Lagrange functional (for j = 1, 2) ∂ rel , −r |γ11 − γ12 | + |γ21 − γ22 | sign λ˙ ∈ −q = ∂λ ∂ rel ∂ rel −r (1 − λ) sign γ˙1j ∈ + sign γ˙1j , ∂γ1j ∂p1j −r λ sign γ˙2j ∈

∂ rel ∂ rel + sign γ˙2j . ∂γ2j ∂p2j

(26) (27) (28)

To obtain a numerical scheme for computing the evolution of plastic microstructures, we develope an incremental formulation to be solved numerically for finite deformation increments Fn , Fn+1 . Note that a change of λ results in mixing the formerly pure phases in a small part of . We proposed to obtain the updated p values by taking the energetic average [7]. Here, we adopt this in principle but for each slip system independently (no slip system interaction), i.e. for λn+1 = λn +λ and e. g. λ > 0 we have 4 4 4 (λn + λ)p21,n+1 = λp21,n + λp11,n ,

p11,n+1 = p11,n ,

(29)

4 (λn + λ)p22,n+1

p21,n+1 = p21,n .

(30)

=

4 λp22,n

4 + λp12,n ,

124

D.M. Kochmann and K. Hackl

To find the onset of microstructure formation, we investigate upon each increment if there exists a combination (b, γ21 , γ22 ) such that for some value 1 we have sup q (Fn+1 , λ = , γij , p11 , p12 , p21 = |γ21 |, p22 = |γ22 |, b − r (|γ11 − γ21 | + |γ12 − γ22 |) b, γ21 , γ22 ; |b| = 1 ≥ 0, (31) and a laminate forms with this vector b and plastic slips γ21 , γ22. (Numerical experiments indicate that the above form converges towards the sought vector b as → 0.) Otherwise no microstructure originates. Once a laminate has formed, we check with each further increment whether or not a rotation of the laminate is energetically admissible by means of (16), i.e. by finding bn+1 from rel (bn+1 )− rel (bn )+2rλn (1−λn ) |γ12,n − γ22,n | + |γ21,n − γ11,n | ≤ 0, (32) where the right-hand side represents the dissipation given by (15). Our numerical scheme is outlined in Algorithm 1. This algorithm computes the microstructure evolution (i.e. plastic slips γij and volume fraction λ) by incrementally minimizing functional (25). As we use the (partially) relaxed energy and dissipation functional, this constitutes a well-posed problem and we can resort to solving the stationarity conditions (26), (27), and (28). Each step starts with the current state as the initial condition and solves the stationarity conditions to update the internal variables. For an initially homogeneous material (i.e. no laminate present) the interface normal b as well as the slips γ2j for the originating second laminate phase are determined via maximization of the driving force according to condition (31). Once a laminate has formed, the evolution of variables λ and γij is computed by a staggered algorithm: In a first step a time-discretized version of (26) is solved for the increment λ for fixed γij . Afterwards pij are updated via (29) and (30). Then, in a second step, (27) and (28) (for j = 1, 2) are solved for the increments γij for fixed λ. We also do this in a staggered form (γ1j are updated first, then follow γ2j ) to reduce complexity of the minimization problem. Numerical experiments indicate a rather neglible influence of the order of the staggered algorithm for a suffiently small time step. Finally the updated values of λ and γij are transfered to the next time-step. Algorithm 1. Incremental formulation for double-slip plasticity: (a) incremental load update: Fn+1 = Fn + F (b) find λn+1 (assume γij = const. = γij,n ): –

for the initially uniform single-crystal (λ = 0): find b and γ2j from (31) find λn+1 from

⇒ γ21,n+1 , γ22,n+1 , bn+1 ,


125

q(Fn+1 , λ, γij,n , pij,n ) sign λ = r |γ11,n − γ21,n | + |γ12,n − γ22,n | ⇒ λn+1 = λ –

for an existing laminate microstructure (λ > 0) solve: q(Fn+1 , λn + λ, γij,n , pij,n ) sign λ = r |γ11,n − γ21,n | + |γ12,n − γ22,n |

(33)

⇒ λn+1 = λn + λ –

check for a laminate rotation by finding bn+1 from (32).

–

update pij,n according to (29) and (30) or the analogous form if λ < 0

(c) find γij,n+1 (assume λ = const. = λn+1 ) by solving in a staggered form: ∂ rel ∂ rel ⎢ ∂γ + ∂p sign γ11 γ1j,n +γ1j = −r (1 − λn+1 ) sign γ11 11 11 ⎢ p1j,n +|γ1j | ⎢ ⎢ rel rel ⎢ ∂ ∂ ⎣ + sign γ12 γ +γ = −r (1 − λn+1 ) sign γ12 1j,n 1j ∂γ12 ∂p12 ⎡

p1j,n +|γ1j |

⇒

γ1j,n+1 = γ1j,n + γ1j ,

p1j,n+1 = p1j,n + |γ1j |

⎡

∂ rel ∂ rel + sign γ = −r λn+1 sign γ21 21 γ ⎢ ∂γ 2j,n +γ2j ∂p21 21 ⎢ p2j,n +|γ2j | ⎢ ⎢ ⎢ ∂ rel ∂ rel ⎣ + sign γ22 γ +γ = −r λn+1 sign γ22 2j,n 2j ∂γ22 ∂p22 p2j,n +|γ2j |

⇒

γ2j,n+1 = γ2j,n + γ2j ,

p2j,n+1 = p2j,n + |γ2j |

3 Numerical Results The numerical scheme outlined above can be applied to arbitrary deformations, so long as the deformation remains volume-preserving to ensure material incompressibility. The first example treats the microstructure evolution during a plane-strain tension-compression test parametrized by the macroscopic deformation gradient ⎛ ⎞ 1+δ 0 0 F = ⎝ 0 1/(1 + δ) 0 ⎠ . (34) 0 0 1

126


Y

Y

0.4

0.7

k = 0.001 j = 70°

j

0.6 0.5 0.4 0.3

0.3

Yunrel

0.2

0.2 0.1

0.1

Y

rel

0.5

1

1.5

(a)

2

2.5

3 d

k = 0.01 j = 70°

j

Y

unrel

Y

rel

0.5

1

1.5

2

2.5

3 d

(b)

Fig. 3 Plane-strain tension-compression test with one active slip system: Comparison of condensed and relaxed energy density.

Computations were performed with µ = 2, r = 0.001 and with constant increments δ = 0.0004 up to a maximum load of δmax = 3. To compare the influence of hardening, Figures 3 and 4 show results obtained for κ = 0.001 and for κ = 0.01. The slip system was oriented under an angle of ϕ = 70◦ (see Figure 3 for the definition of ϕ). Because of the non-aligned slip system the material stability of the homogeneous deformation is lost and microstructures arise for δ > 0. Due to the (quasi-)convexity of cond for δ < 0, no microstructures form with negative strain δ. Figure 3 compares the condensed and relaxed energy for the two hardening parameters, where we term the energy obtain from Algorithm 1 relaxed. It becomes apparent that the present approach considerably decreases the energy of the laminate (predominantly due to the updating procedure of the hardening variables) even beyond the recovery of convexity of the unrelaxed solution. Figure 4 summarizes the corresponding evolution of the laminate microstructure by illustrating the plastic slips in both phases, γ1 and γ2 , respectively, the volume fraction of phase 2, λ, and the Cauchy stress components upon straining. In this example, the body behaves elastically first until a second phase with finite, non-zero slip γ2 originates from the uniform ground state. During the course of deformation both phases exhibit plastic flow until finally only one homogeneous phase remains. Once the laminate has formed with a distinct orientation vector b, we do not observe laminate rotations due to the large amount of dissipation necessary for a rotation. Rotations commonly only occur when the body is in an almost uniform state (i.e. λ ≈ 0 or λ ≈ 1). A second example investigates the microstructure evolution during a plane-strain simple shear experiment on a sample with two active slip-systems. The macroscopic deformation gradient takes the form ⎛ ⎞ 1γ 0 F = ⎝0 1 0⎠, (35) 001


(a)

gi 2

0.5 -2

1

1.5

2

2.5

g

3

0.5

g2

1.5

2

2.5

3

g

g2

-2

l

1

1

0.8

0.8 s

0.6

m

0.4

b

0.6

s

0.4

m

b

0.2

0.2

0.5

1

1.5

2

2.5

3

g

0.5

(c)

s11

s11

0.8

0.8

0.6

s11,unrel. 0.5

0.4 0.2

s11,rel

1

s22

1.5

2

2.5

3

g

0.5

1

1.5

2

s11,unrel.

2.5

3

g

0.5

-0.2

-0.2

-0.3

-0.3

-0.4

-0.4

3g

2.5

1

1.5

2

2.5

3

g

2

2.5

3

g

s22,unrel. 0.5

-0.1

2

s11,rel

0.1

s22,rel

0.1

1.5

j

s22

s22,unrel.

0.2

1

0.6

j

(d)

-0.1

1

-1

l

0.2

g1

1

(b)

0.4

k = 0.01 j = 70°

2

g1

1

-1

gi

k = 0.001 j = 70°

3

127

1

1.5

s22,rel

Fig. 4 Plane-strain tension-compression test with one active slip system: Comparison of the plastic slips, volume fraction of phase 2, and Cauchy stress components.

and we assume two active slip systems as depicted in Figure 5, with ϕ = 150◦, α1 = 5.71◦, α2 = 16.70◦, so that m1 = m2 = m = (− sin ϕ, cos ϕ, 0)T . Results


128 Y

g

0.8

j

l

j = 150° k= 0

g

0.15

g

s2

0.125

0.6

0.1

Ycond

0.4

j

g

0.05 0.025

Yrel 1

s1 m

0.075

0.2 0.5

a1 a2

1.5

2

2.5

3 g

gij

0.5

1.5

1

2

2.5

3 g

10 s12

8

0.8 0.6 0.2 -0.2

6

s12,cond

0.4

s12,rel 0.5

1

g21

4 1.5

2

2.5

3 g

-0.4

2

g22 g11

0.5

1

1.5

2 g12 2.5

3 g

Fig. 5 Plane-strain simple shear test with two active slip systems: Comparison of condensed and relaxed energy, volume fraction of phase 2, Cauchy shear stress, and plastic slips γij .

for the relaxed energy, the volume fractions, plastic slips and the Cauchy shear stress are summarized in Figure 5. We see that the non-convex energy density gives rise to the formation of a laminate microstructure of the same type as before: The material first deforms elastically, now along both slip directions on the common slip plane. Then a second laminate phase arises with finite plastic slips γ21 and γ22 along the two slip directions, respectively. Due to the non-symmetric alignment of both slip directions the plastic slips in the two directions are not the same. The originating laminate exhibits an orientation as sketched in Figure 5. Of course, relaxation of the energy via lamination also affects the stress-strain behavior, as can clearly be seen from the plotted Cauchy shear stress.

4 Conclusions and Discussion We have shown how the time-continuous evolution of microstructures can be described efficiently by employing a relaxation of the nonconvex potentials. We specified the general procedure for first-order laminates in elasto-plastic single crystals, and we derived explicit evolution equations for the example of an incompressible neo-Hookean material with multiple active slip systems. Note that we employed a partially-relaxed energy only, i.e. the laminate energy is minimized only with respect to the elastically changing variables while the evolution of all remaining unknowns


129

is determined from dissipative evolution equations. Numerical results illustrate the applicability of the employed formulation and demonstrate the evolution of the arising laminate microstructures. In contrast to earlier approaches in the literature we persued an incremental strategy that captures the actual microstructural changes during each time step for multislip plasticity. Many reported examples [1, 2, 12] were for consiceness restricted to single-slip plasticity (and often to two dimensions only). The present approach accounts for, in principle, arbitrarily many active slip systems in three-dimensional problems while the numerical procedure outlined in this work involved two active slip systems within the same glide plane. Furthermore, the present approach incorporates the actual changes of the microstructure during each time step, which was precluded in those analyses based on condensed energy functionals [1]. These microstructural changes (with an already existing microstructure at the beginning of the time step) cannot be accounted for by a pure minimization method [1, 2] but call for an incremental strategy as specified above; see also [12]. In particular, those methods based on the minimization of a condensed energy functional are appropriate for monotonic loading only – the present approach in turn can readily be applied to cyclic deformation histories [7]. Also, any incremental strategy should correctly reflect all microstructural rearrangements during a time step. In contrast to many earlier models we accounted for the actual amount of dissipation required to change not only the plastic slips but also the volume fractions during each time step (this amount of dissipation depends on the existing internal variables at the beginning of the time step) and we proposed an updating procedure for the hardening variables due to changes of the plastic slips and of the volume fractions during a time increment. As most of the aforementioned literature approaches were for single-slip only, a comparison of the results with the present method is possible when the outlined strategy is applied to a single active slip system only. In this case it was shown that the present incremental strategy considerably reduces the stored energy during a given straining path even beyond the recovery of convexity of the unrelaxed solution [9]. This behavior was mainly due to the particular form chosen for the updates of the internal variables upon changes of the volume fractions. Without this updating procedure (or for only negligible hardening) the solution would exhibit a high agreement with earlier approaches. Generalizations of the oultined method in several directions are underway.

References 1. Bartels, S., Carstensen, C., Hackl, K. and Hoppe, U., Effective relaxation for microstructure simulation: Algorithms and applications. Comp. Meth. Appl. Meth. Eng. 193, 2004, 5143– 5175. 2. Carstensen, C., Conti, S. and Orlando, A., Mixed analytical-numerical relaxation in finite single-slip crystal plasticity. Cont. Mech. Thermodyn. 20, 2008, 275–301.

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3. Carstensen, C., Hackl, K. and Mielke, A., Non-convex potentials and microstructures in finitestrain plasticity. Proc. R. Soc. London A 458, 2002, 299–317. 4. Conti, S. and Ortiz, M., Minimum principles for the trajectories of systems governed by rate problems. J. Mech. Phys. Solids 56, 2008, 1885–1904. 5. Conti, S. and Theil, F., Single-slip elastoplastic microstructures. Arch. Rat. Mech. Anal. 178, 2005, 125–148. 6. Hackl, K. and Fischer, F.D., On the relation between the principle of maximum dissipation ˝ and inelastic evolution given by dissipation. Proc. Royal Soc. London A 464, 2008, 117U-132. 7. Hackl, K. and Kochmann, D.M., Relaxed potentials and evolution equations for inelastic microstructures. In: Daya Reddy, B. (Ed.), IUTAM Symposium on Theoretical, Computational and Modelling Aspects of Inelastic Media. Springer, Dordrecht, 2008, pp. 27–39. 8. Hackl, K., Schmidt-Baldassari, M. and Zhang, W., A micromechanical model for polycrystalline shape-memory alloys. Mat. Sci. Eng. A 378, 2003, 503–506. 9. Kochmann, D.M. and Hackl, K., An incremental strategy for modeling laminate microstructures in finite plasticity – Energy reduction, laminate orientation and cyclic behavior. Lecture Notes Appl. Comp. Mech., Springer, 2009, accepted for publication. 10. Lambrecht, M., Miehe, C. and Dettmar, J., Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic-plastic bar. Int. J. Solids Struct. 40, 2003, 1369– 1391. 11. Miehe, C., Schotte, J. and Lambrecht, M., Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals. J. Mech. Phys. Solids 50 2002, 2123–2167. 12. Miehe, C., Lambrecht, M. and Gürses, E., Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity. J. Mech. Phys. Solids 52, 2004, 2725–2769. 13. Mielke, A., Finite elastoplasticity, Lie groups and geodesics on SL(d). In: Newton, P., Weinstein, A., Holmes, P. (Eds.), Geometry, Dynamics, and Mechanics, Springer, Berlin, 2002. 14. Mielke, A., Deriving new evolution equations for microstructures via relaxation of variational incremental problems. Comp. Meth. Appl. Meth. Eng. 193, 2004, 5095–5127. 15. Mielke, A. and Ortiz, M., A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems. ESAIM Control Optim. Calc. Var. 14, 2007, 494– 516. 16. Ortiz, M. and Repetto, E.A., Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids. 47, 1999, 397–462.

Models for Dynamic Fracture Based on Griffith’s Criterion Christopher J. Larsen

Abstract There has been much recent progress in extending Griffith’s criterion for crack growth into mathematical models for quasi-static crack evolution that are well-posed, in the sense that there exist solutions that can be numerically approximated. However, mathematical progress toward dynamic fracture (crack growth consistent with Griffith’s criterion, together with elastodynamics) has been more meager. We describe some recent results on a phase-field model of dynamic fracture, and introduce models for “sharp interface” dynamic fracture.

1 Introduction Models for crack evolution begin with Griffith’s criterion, stated for the quasi-static case. The idea is that as boundary conditions or loads slowly vary, the material can be assumed to be in elastic equilibrium at all times, subject also to the varying crack set, where the displacements are allowed to be discontinuous. Because the displacements correspond to equilibria, as a crack grows, there is a corresponding instantaneous change in the stored elastic energy. Griffith’s criterion states that a crack grows so that the decrease in stored elastic energy (compared with the stored elastic energy corresponding to a stationary crack) balances the increase in surface energy, postulated to be proportional to the surface area of the crack (see [9]). This principle was turned into a precise definition of quasi-static fracture evolution in [7] together with [6]. For a given varying Dirichlet condition g on a time interval [0, Tf ], such an evolution is a family (u(t), K(t)) with u(t) ∈ SBDg(t ) (), K(t) =

S(u(τ ))

τ ∈Q

τ ≤t

Christopher J. Larsen Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA; E-mail: [email protected] K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_10, © Springer Science+Business Media B.V. 2010

131

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where SBDg is the space of Special functions of Bounded Deformation with Dirichlet condition g, and S(u(τ )) is the discontinuity set of u(τ ) (considered to include the part of ∂ on which u does not have the correct Dirichlet data). It further must satisfy, for all T ∈ [0, Tf ], 1 1 Ae(u(T )) : e(u(T ))dx ≤ 2 Ae(w) : e(w)dx + H N−1 (S(w) \ K(T )) (1) 2

1 2

∀w ∈ SBDg(T ) ()

Ae(u(T )) : e(u(T ))dx + H N−1 (K(T )) =

T

+ H N−1 (K(0)) + 0

1 2

Ae(u(0)) : e(u(0))dx (2)

Ae(u(t)) : e(g(t))dxdt. ˙

Equation (1) represents global (unilateral) minimality (with given elasticity tensor A), and (2) represents energy balance. Francfort and Marigo [7] proposed a discrete minimization scheme for proving existence, which has been carried out under some assumptions in [5, 8]. Note that in this model, loads cannot be considered in a reasonable way, due to the global minimality (1), but in most of the models below, they can.

2 Phase-Field Models Computing solutions to problem (1)–(2) is quite difficult, but fortunately Ambrosio and Tortorelli [1] introduced more regular functionals that converge to the above energy. We define, for ε > 0, the elastic energy E : H10 × H1 → R ∪ {+∞}, and the (phase-field) surface energy H : H1 → R, respectively, as 1 2 2 (4ε)−1 (1−v)2 +ε|∇v|2 dx, E (u, v) := 2 (v +ηε )|∇u| dx and H(v) :=

where ηε → 0 as ε → 0. Ambrosio and Tortorelli [1] showed that E + H -converges to E(u) :=

1 2

|∇u|2 + H N−1 (S(u))

defined for u ∈ SBV (). E and E can be easily altered for linear elasticity, with Ae(u) : e(u) replacing |∇u|2 , and it is this elastic energy that we will consider (see [4] for an analysis of linear elasticity in this context). We will also consider below the kinetic energy K : H1 → R, 1 K(u) ˙ := 2 |u| ˙ 2 dx,

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which is unaffected by the presence of a phase field in the model. The external loads at time t are collected into a functional (t) ∈ H−1 , f (t) · ϕ dx + h(t) · ϕ ds ∀ϕ ∈ H1 , (t), ϕ =

N

where f (t) ∈ L2 (; R3 ) and h(t) ∈ L2 (N ; R3 ). We assume throughout that ∈ C1 ([0, Tf ]; H−1 ). Finally, the total energy is given by F (t; u, u, ˙ v) := K(u) ˙ + E (u, v) − (t), u + H(v) in the case of dynamics; we also write F (t; u, v) for the quasi-static energy, dropping the kinetic term. For simplicity below, we will take h = 0.

2.1 Quasi-Statics Following the global minimization approach of Francfort and Marigo [7], a pair (u, v) is a quasi-static evolution if it satisfies global minimality and energy balance at every time: for every T , (u(T ), v(T )) minimizes (u , v ) → F (T ; u , v ) subject to v ≤ v(T ) T ˙ u dt , F (T ; u(T ), v(T )) = F (0; u(0), v(0)) − + 0

T

(3) (4)

0

Ae(u(t)) : e(g(t))dxdt. ˙

This problem was studied in [10], where it was shown that solutions converge to solutions of (1)–(2) (assuming = 0 and the antiplane case). However, such global minimizers cannot in general be computed, which would seem to negate the advantage of the phase field approach. Yet, the natural way of trying to compute minimizers of (3)–(4) is to alternately minimize in u and v (see [2]), which produces pairs that are in some ways superior to solutions of (3)–(4) and (1)–(2). First, these pairs are only stable, and are not generally global minimizers. Second, this alternate minimization can handle loads ( = 0), since this minimization will not constantly “see” that breaking off a piece of the material and sending it far enough away (in the direction of f ) will always reduce the total energy (of course, this is related to producing stable states, not global minimizers). Finally, this alternate minimization approach suggests an approach to dynamics.

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2.2 Dynamics A natural extension to dynamics, proposed in [3, 11], is the following: Definition 1. (u, v) is a (phase field) dynamic fracture evolution if u¨ − div((v2 + ηε ) Ae(u)) = f F (T ; u(T ), u(T ˙ ), v(T )) = F (0; u(0), u(0), ˙ v(0)) −

in ,

(5)

˙ u dt ,

(6)

v(T ) minimizes v → E (u(T ), v ) + H(v ) subject to v ≤ v(T )

(7)

T 0

where (6) and (7) hold for every T (and where for simplicity we take h = 0 and zero Dirichlet data) with initial conditions u(0) = u0 and u(0) ˙ = u1 and as initial condition for v we prescribe an arbitrary v0 ∈ H1 which satisfies the unilateral minimality condition (7). Naturally, we look for u(T ) ∈ H01 , v(T ) ∈ H 1 . The idea, originating in [3], is that the principle for dynamic crack growth, (7), should be identical to that for the quasi-static setting, while u should obey a wave equation corresponding to elastic stiffness (v 2 + ηε )A. The numerical method to solve the above system, proposed in [3], is to iterate between updates in u using time steps in (5), and updates in v using (7). That such solutions converge, as the time step goes to zero, to a solution also obeying (5) and (7), while also satisfying (6), was proved in [11] (assuming a certain dissipation in the dynamics). It is natural to then consider sharp interface models, i.e., models with crack sets rather than phase fields. The advantage of the above phase field approach is the fact that v 2 multiplies Ae(u) : e(u), so that, even though there is no instantaneous update in u due to a crack increment, there is still a decrease in stored elastic energy due to the decrease in v, which models a crack increment. The quasi-static setting can in this way be mimicked using a phase field. But, a new principle is needed for the sharp interface model. This can be seen from the fact that a limiting version of condition (7) is meaningless in the sharp interface case – crack increments have no effect on the stored elastic energy, so there can be no minimality condition. We need to find other (precise) conditions for crack growth, without relying on minimality. Below we propose two, both of which extend to the quasi-static setting; we will show that there, the first principle is stronger than minimality, while the second is equivalent to it.

3 Sharp Interface Models In observing solutions to phase field dynamic fracture [3], a striking feature is crack branching, which seems to only occur at high stress rates (and not surprisingly, is not seen in quasi-statics). This can only happen because u develops large stresses at the crack “tip”, in the directions of the branch, so that the minimality drives v to


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decrease there, which “softens” the material, and results in still larger stresses (just like in quasi-statics, even though u is following a wave equation and not minimality). For quasi-statics, u is always in equilibrium, and so the largest stresses are straight ahead of the tip (at least in the anti-plane setting). In seeking an equivalent (in principle) condition for crack growth, to replace minimality, it is natural to consider that a (5)–(7) solution with growing crack, and a solution for which the crack is prevented from growing, both balance energy. Furthermore, if the stresses at the crack tip are not big enough for minimality to drive the crack to grow, then a growing crack could not balance energy – the cost of the crack would exceed the reduction in elastic energy. These considerations lead to the first definition below, based only on the principle that if a crack can grow while balancing energy, then it must grow. We first redefine the total energy F in the natural way, removing the phase field v and the parameter ηε (i.e., v 2 + ηε ≡ 1), and replacing H(v) with H N−1 (K). We then have Definition 2. (u, K) is a Maximal Dissipation (MD) dynamic fracture evolution if: 1. u is a solution of the wave equation on \ K: u¨ − div(Ae(u)) = f

in \ K,

with traction-free boundary condition on K (and imposed initial conditions) 2. Energy balance: for all T , F (T ; u(T ), u(T ˙ ), K(T )) = F (0; u(0), u(0), ˙ K(0)) −

T

˙ u dt ,

0

3. Maximality: for all T , if a pair (w, L) satisfies 1 and 2, with K(t) ⊂ L(t) ∀t ∈ [0, T ], then K(t) = L(t) for all t ∈ [0, T ] Note that 3 is just maximal dissipation with respect to set inclusion (hence the label (MD)), where subset and equality relations are taken to hold up to sets of H N−1 measure zero, and we consider solutions u(t) ∈ SBD, or SBV in the anti-plane case. While the meaning of 1 is clear enough if K is closed, and we conjecture that solutions will have (essentially) closed crack sets, it is worth stating a weak version (stated for zero loads): ∞ u, φt t + Ae(u), e(φ) dt = 0 0

∀φ ∈ C02 ((0, ∞); SBD) with S(φ(t)) ⊂ K(t) ∀t. The second model is motivated by the fact that the (MD) principle is very strong, and in fact strictly stronger, in the quasi-static setting, than minimality (the fact that it is stronger is explained below, after its definition, but we can note here that the “strictly” stronger follows from the fact that in the case of an energetic tie between a solution with a crack and a solution without, (MD) will select the crack solution).

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Showing existence is expected to be quite difficult, which suggests attempting to formulate a weaker model. The idea here is that, if the stresses at a crack tip have just become large enough for the crack to run while balancing energy, then if the crack does not run, a moment later, the stresses will be larger than necessary for the crack to run while balancing energy; in fact, it will now be possible for the crack to run and have the future energy below that for energy balance. To see this, notice that if the stresses are larger than necessary, then they are large enough for energy balance corresponding to a larger fracture toughness, i.e., a factor in front of H N−1 (K) larger than 1. But that means an energy decrease since the factor is just 1. Definition 3. (u, K) is a (no) Decreasing Energy Extension (DEE) dynamic fracture evolution if: 1. u is a solution of the wave equation on \ K, just as in Definition 2 2. (u, K) balances energy just as in Definition 2 3. Maximally: for all T , if a pair (w, L) satisfies 1 with (w(t), L(t)) = (u(t), K(t)) for all t ∈ [0, T ], then d d F (t; w(t), w(t), ˙ L(t))|t =T + ≥ F (t; u(t), u(t), ˙ K(t))t =T + dt dt Note that condition 3 should hold for all possible futures of and g, so that in particular it should hold if they are all continued, from time T , as constants. In that case, 3 becomes: for all T , if a pair (w, L) satisfies 1 with (w(t), L(t)) = (u(t), K(t)) for all d t ∈ [0, T ], then dt F (t; w(t), w(t), ˙ L(t))|t =T + ≥ 0. This means that there should be no extension of (u, K) (corresponding to fixed future loads and boundary conditions) with decreasing energy, hence the label (no) DEE. This is the form of condition 3 that we will consider below.

4 General Formulations: Revisiting Quasi-Statics A natural question is to find the relationship between these maximal dissipation principles (so far stated only in a dynamic setting) and the minimality principles in quasi-statics and in phase-field dynamic fracture. We therefore give general definitions of (MD) and (DEE) evolutions that encompass both dynamics and quasi-statics (and possibly others), both of which are examples of evolution problems (EP). Below, satisfying (EP) in condition 1 means, e.g., that at every time t, u(t) is a global minimizer of v → E (v) subject to H N−1 (S(v) \ K(t)) = 0 (in the quasi-static setting), or that u satisfies condition 1 in Definition 2 (in the dynamic setting). Definition 4 (Maximal Dissipation (MD)). (u, K) is a (MD) solution to an evolution problem (EP) if:


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1. (u, K) satisfies (EP) 2. (u, K) balances total energy 3. ∀T , if (w, L) satisfies 1 and 2, and K(t) ⊂ L(t) ∀t ∈ [0, T ], then K(t) = L(t) ∀t ∈ [0, T ]. Definition 5 (Decreasing Energy Extension (DEE)). (u, K) is a (DEE) solution to an evolution problem (EP) if: 1. (u, K) satisfies (EP) 2. (u, K) balances total energy 3. ∀T , if a pair (w, L) satisfies 1 with (w(t), L(t)) = (u(t), K(t)) for all t ∈ [0, T ], d then dt F (t, w(t), L(t))|t =T + ≥ 0 where in condition 3, we assume loads and boundary conditions are continued as constants from time T . We now investigate the relationship between these solutions in the context of quasistatics, and globally minimizing quasi-static evolutions (GM). For simplicity, we will assume the anti-plane case, and A ≡ I . We will show that (MD) ⇒ (DEE) ⇔ (GM) Proposition 1. If (u, K) is a (MD) quasi-static evolution (globally minimizing), then it is a globally minimizing quasi-static evolution in the sense of (1)–(2). Proof. Suppose that (u, K) is a (MD) solution but not a globally minimizing quasistatic evolution. Then there exists a time t and v ∈ SBV such that 2 N−1 |∇v| dx + H (S(v) \ K(t)) < |∇u(t)|2 dx,

which is equivalent to |∇v|2 dx + H N−1 (S(v) ∪ K(t)) < |∇u(t)|2 dx + H N−1 (K(t)).

Since |∇u(t)|2 dx+H N−1 (K(t)) = |∇u(t −δ)|2dx+H N−1 (K(t −δ))−

t

t −δ

˙ u , ,

by continuity of the last term we have that for δ > 0 small enough, 2 N−1 |∇v| dx + H (S(v) ∪ K(t − δ)) < |∇u(t − δ)|2 dx + H N−1 (K(t − δ))

and hence

|∇v|2 dx + H N−1 (S(v) \ K(t − δ)) <

|∇u(t − δ)|2 dx.

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We then define a competitor (w, L) as follows. We first consider a smooth curve with two endpoints C ⊂ (in 2-D; 3-D is similarly handled by choosing a smooth surface) with length greater than the total energy of (u, K) for all times, and such that H N−1 (C ∩ K(t)) = 0 for all t. For λ ∈ [0, H N−1 (C)], set C(λ) to be the curve in C that, beginning at one designated endpoint of C, has length λ. We then choose λ such that, with vλ the elastic minimizer for time t − δ with S(vλ ) ⊂ K(t − δ) ∪ S(v) ∪ C(λ) =: Kλ , we have |∇vλ |2 dx + H N−1 (Kλ \ K(t − δ)) = |∇u(t − δ)|2 dx

(we can do this since, for λ = 0, the left hand side above is less than the right, and for λ = H N−1 (C), we have the reverse inequality, while as λ increases, the elastic energy is decreasing, and the surface energy is increasing and continuous). We then set w(t − δ) := vλ and L(t − δ) := Kλ . The pair (w, L) is continued for τ > t − δ as a globally minimizing quasi-static evolution, as in [8], so that energy balance is maintained. For τ < t − δ, we set (w, L) := (u, K). (w, L) is then a competitor for (u, K) in Definition 4, but condition 3 in that definition is violated since H N−1 (L(τ ) \ K(τ )) > 0 for τ ∈ [t − δ, t], contradicting (u, K) being a (MD) solution. This completes the proof. Proposition 2. If (u, K) is a (DEE) quasi-static evolution (globally minimizing), then it is a globally minimizing quasi-static evolution in the sense of (1)–(2). Proof. If at some time T (u, K) is not a global minimizer, it is immediate that we can have an extension with a jump decrease in energy, so that we have d F (t; w(t), L(t))|t =T + < 0, dt contradicting (w, L) being a (DEE) solution. Proposition 3. If (u, K) is a a globally minimizing quasi-static evolution in the sense of (1)-(2), then it is a (DEE) quasi-static evolution (globally minimizing). Proof. We give a proof of a stronger version, without the assumption that the boundary conditions are continued as constants. Suppose that (u, K) is a globally minimizing quasi-static evolution, and let t and an extension (w, L) starting at t be given. Note that (u(t), K(t)) is a unilateral global minimizer, and we consider a discrete globally minimizing extension: define u(t ˜ + t) to be the minimizer of the total energy corresponding to time t + t, subject to the irreversibility constraint, so that the surface energy in the total energy is H N−1 (S(u(t ˜ + t)) \ K(t)). By usual energy estimates (see section 3.1 in [8]) we have that


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F (t + t; u(t ˜ + t), K(t) ∪ S(u(t ˜ + t))) = F (t; u(t), K(t)) t + t + ∇ g˙ · ∇udxdt + o( t), t

which also holds for u(t + t) replacing u(t ˜ + t). By the minimality of u, ˜ we have d d F (τ ; w(τ ), L(τ ))|τ =t + ≥ F (τ ; u(τ ), K(τ ))|τ =t + , dτ dτ completing the proof. Remark 1. We conclude by posing the following questions: 1. Are solutions of these dynamic models consistent with Griffith’s criterion, in the sense that if u0 is in equilibrium such that the initial crack set has energy release below that necessary for crack growth, and u1 = 0, then the dynamic crack does not run? 2. Is there much stronger regularity of dynamic solutions than (is provable) for quasi-static evolutions? 3. Are any of these dynamic models the limit of the phase-field models? (Perhaps in principle and some situations, but probably not always true.) 4. What is the quasi-static limit of the phase-field dynamic model? (Probably it is not a phase-field quasi-static global minimizer, except when H is continuous in time.) 5. What is the quasi-static limit of the sharp-interface dynamic model? (Probably it is not a quasi-static global minimizer, as in 4.)

Acknowledgement This material is based on work supported by the National Science Foundation under Grant No. DMS-0505660 and No. DMS-0807825.

References 1. Ambrosio, L. and Tortorelli, V.M.: On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. 6-B, 1992, 105–123. 2. Bourdin, B.: Numerical implementation of the variational formulation of brittle fracture, Interfaces Free Bound. 9, 2007, 411–430. 3. Bourdin, B., Larsen, C.J. and Richardson, C.L.: A time-discrete model for dynamic fracture based on crack regularization, submitted. 4. Chambolle, A.: A density result in two-dimensional linearized elasticity, and applications, Arch. Ration. Mech. Anal. 167, 2003, 211–233. 5. Dal Maso, G., Francfort, G.A. and Toader, R.: Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal. 176, 2005, 165–225.

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6. Dal Maso, G. and Toader, R.: A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Ration. Mech. Anal. 162, 2002, 101–135. 7. Francfort, F. and Marigo, J.-J.: Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids. 46(8), 1998, 1319–1342. 8. Francfort, G.A. and Larsen, C.J.: Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math. 56, 2003, 1465–1500. 9. Griffith, A.: The phenomena of rupture and flow in solids, Phil. Trans. Roy. Soc. London CCXXI-A, 1920, 163–198. 10. Giacomini, A.: Ambrosio–Tortorelli approximation of quasi-static evolution of brittle cracks, Calc. Variat. PDE 22, 2005, 129–172. 11. Larsen, C.J., Ortner, C. and Süli, E.: Existence of solutions to a regularized model of dynamic fracture, Mathematical Models and Methods in Applied Sciences, to appear.

An Energetic Approach to Deformation Twinning Khanh C. Le and Dennis M. Kochmann

Abstract Within continuum dislocation theory the plastic deformation of a single crystal with one active slip system under plane-strain constrained shear is investigated. By introducing a twinning shear into the energy of the crystal, we show that in a certain range of straining the formation of deformation twins becomes energetically preferable. Energetic thresholds for the onset of twinning and of plastic flow are determined and investigated. A rough analysis qualitatively describes not only the evolving volume fractions of twins but also their number during straining. Finally, we analyze the evolution of deformation twins and of the dislocation network at non-zero dissipation. We present the corresponding stress-strain hysteresis, the evolution of the plastic distortion and the twin volume fractions.

1 Introduction Slip and twinning are the major deformation modes which accommodate a change of shape under the action of applied tractions or displacements. Experimental evidence for deformation twinning was found long time ago (see e.g. [9] for a comprehensive overview), and dislocation-based description were developed [8, 10]. Deformation twins have been reported to occur especially in b.c.c., h.c.p. and lower symmetry metals and alloys but also in many f.c.c. metals and alloys with low stacking-fault energy, or other intermetallic compounds as well as in geological materials such as calcite or quartz. Deformation twinning basically divides the originally uniform single crystal into two volumetric parts – a parent phase (with unaltered crystal lattice) and a twin phase (with a different crystal lattice orientation, commonly symmetric with respect to the twin interface). Both phases normally occur in the form of lamellar structures, where a bicrystal consisting of neighboring parent and twin Khanh C. Le · Dennis M. Kochmann Lehrstuhl für Allgemeine Mechanik, Ruhr-Universität Bochum, D-44780 Bochum, Germany; E-mail: {chau.le, dennis.kochmann}@rub.de, [email protected] K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_11, © Springer Science+Business Media B.V. 2010

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phase is commonly referred to as a twin. This topic is closely related to martensitic phase transformations and the observed effect of transformation induced plasticity (TRIP). In this paper, however, we will limit our analysis to the effects of twinning induced plasticity (TWIP). The formation of deformation twins has a significant impact on the macroscopic stress-strain response. The evolution of twins provides TWIP alloys with excellent hardening behavior [1], allowing for higher stresses and larger strains than in common f.c.c. or b.c.c. metals. As a special characteristic the onset of twinning, i.e. the rapid nucleation of deformation twins, often gives rise to a large load drop in the stress-strain behavior [9]. The increase of strength and work hardening during microstructure refinement by twinning in manganese steels or other TWIP-alloys remains until now not quite well understood. Perhaps the dislocation pile-up near the twin boundaries (raising the boundary energy) and the related size-effects play an important role here. In this paper we present a micro-mechanical model to describe the initiation and evolution of deformation twins in metals and alloys by employing a continuum dislocation approach. This approach is dictated by the high dislocation densities accompanying plastic deformations, which may be treated using continuum dislocation theory [2,6,16,17,20,21,23]. Here, we adopt the logarithmic energy formulation for the dependence of the defect energy on the scalar dislocation density obtained from Nye’s dislocation density [23], as recently proposed [4]. Based on this energetic approach, the analytical solution of an anti-plane constrained shear problem was reported for single crystals [5]. Interesting features of the solution are the energetic and dissipative yielding thresholds, the Bauschinger translational work hardening and a size effect. The dislocation nucleation admits a clear characterization by the variational principle for the final plastic states [4]. Further works investigated the plane-strain deformation of single and double-slip plasticity in single and bicrystals [13, 14, 18]. Comparison with the results of discrete dislocation simulations [22, 25] shows good agreement between the discrete and the present continuum approach [18]. In this paper, we modify the energetic basis of the aforementioned models to allow for a characterization of the initiation and evolution of deformation twins [15].

2 Plane-Strain Constrained Shear of Twins Let us first consider a single crystal under plane-strain constrained shear. We assume a homogeneous crystal in form of a thin strip with width a and height h, 0 ≤ x ≤ a, 0 ≤ y ≤ h as shown in Figure 1. The crystal is subjected to a plane-strain constrained shear deformation rigidly enforcing displacements on its upper and lower sides as u(0) = 0, v(0) = 0, u(h) = γ h, v(h) = 0, (1)

An Energetic Approach to Deformation Twinning

143

rotation

y

y

bt shear

g TT

m

T

TT

TT

s a

ju jl TT

TT

T

TT

h

y

g

x

x

hs h(1-s) x

Fig. 1 Schematic of the model crystal with one active slip system (s, m) and the formation of deformation twin by a twinning shear through movement of dislocations to the twin boundary followed by a rigid rotation.

where u(y) and v(y) are the longitudinal and transverse displacements, respectively, with γ being the overall shear strain. First, we assume that the thickness of the strip in the z-direction L is large and the width a is much greater than the height h (L a h) to neglect end effects and to have the stresses and strains depending only on one variable y in the central part of the strip. For the plane-strain state the in-plane components of the strain tensor read εxx = 0,

εxy = εyx =

1 u,y , 2

εyy = v,y ,

(2)

where the comma in indices denotes differentiation with respect to the corresponding coordinates. If the overall shear strain γ is sufficiently small, then the crystal deforms elastically and u = γ y, v = 0 everywhere in the strip. If γ exceeds some critical threshold, then edge dislocations may appear to reduce the crystal’s energy. We assume that the crystal is initially uniform with only one active slip system, with the slip direction (or the direction of the Burgers vector) s = (cos ϕ, sin ϕ, 0)T perpendicular to the z-axis and inclined at an angle ϕ with the x-axis, and the dislocation lines parallel to the z-axis. The normal vector to the slip plane is denoted by m = (− sin ϕ, cos ϕ, 0)T . Therefore, we have for the plastic distortion βij = βsi mj , and we may assume translational invariance, i.e. β = β(y). For an investigation of the influence of multiple slip systems on this approach see [19]. When loading the initially uniform single crystal, edge dislocations are nucleated along the given slip lines and accumulate upon further straining. As γ increases further, it might by energetically more preferable to form the twin phase by means of a twinning shear (comparable to Bain’s strain in the theory of phase transformations), followed by a rigid rotation as depicted in Figure 1. This decomposition was originally proposed by Bullough [7]. Let the volume fraction of the twin phase be s and that of the parent phase hence 1 − s. For simplicity we first limit our analysis to only

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one twin with the parent phase in the lower portion and the twin phase in the upper portion of the crystal (the order of phases does not qualitatively change results). The twinning shear is realized by the motion of already existing edge dislocations to the twin boundary along slip lines, leaving the crystal again dislocation- and stress-free. The subsequent rigid rotation of the twin phase does not change the energy of the crystal but changes the existing slip system of the parent phase (sl , ml ) with angle ϕl into the new slip system (su , mu ) of the twin phase with angle ϕu . After the onset of deformation twinning, the plastic distortion can be additively decomposed as βij (y) =

l l βsi mj βsiu muj

for 0 < y < h(1 − s), +

βt sil mlj

+ ωij

for h(1 − s) < y < h,

with βt denoting the constant twinning shear and ωij being the constant skewsymmetric rotation tensor. This additive split of the plastic distortion is a good approximation for small twinning shear and small rotation. As we intend to investigate deformation twinning, we assume the rotation of the upper part such that the active slip systems after rotation are symmetric with respect to the interface (ϕu = −ϕl = ϕ). A rather simple geometric analysis shows that in this case the twinning shear is given by βt = −2 cot ϕ. Because of the prescribed displacements (1) dislocations cannot penetrate the boundaries at y = 0 and y = h, and we assume that they cannot penetrate the twin boundary either because the neighboring crystal does not admit the same slip system, therefore β(0) = β(h) = 0,

β (h(1 − s)) = 0.

(3)

Introducing the following piecewise defined quantities in the upper and lower part of the crystal for h(1 − s) < y < h, βu (y), ϕu , βt β(y), ϕ, βT = (4) for 0 < y < h(1 − s), βl (y), ϕl , 0 p

we obtain the in-plane components of the elastic strain tensor εije = εij − εij as 1 1 e (β sin 2ϕ + βT sin 2ϕl ), εxy = (u,y − β cos 2ϕ − βT cos 2ϕl ), 2 2 1 = v,y − (β sin 2ϕ + βT sin 2ϕl ). (5) 2

e εxx = e εyy

As β depends on y only, there are two non-zero components of Nye’s dislocation density tensor αij = j kl βil,k [23] (with j kl the permutation symbol), namely αxz = β,y sin ϕ cos ϕ and αyz = β,y sin2 ϕ. Thus, the resultant Burgers vector of all dislocations whose lines cut the area perpendicular to the z-axis is parallel to the


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slip direction s. The scalar dislocation density equals ρ=

1 2 1 2 = |β sin ϕ|, αxz + αyz ,y b b

where b is the magnitude of the Burgers vector. Note that we do not include a contribution to the dislocation density from the twinning shear βt as well as from the rigid rotation [15]. Assuming that the crystal is elastically isotropic with equal elastic moduli, we write for the energy per unit volume [3, 4] U (εije , αij ) =

1 e 2 1 λ εii + µεije εije + µk ln , |β,y sin ϕ| 2 1−

(6)

bρs

where µ and λ are the Lamé elastic moduli, ρs is the saturated dislocation density, and k a material constant. The first and second term of (6) describe the elastic energy, the last term represents the energy of the dislocation network. The logarithmic nature of the energy of the dislocation network was extensively discussed in [13,14], and it basically provides an energetic barrier against over-saturation of the crystal with dislocations. With (4), (5) and (6) the total energy functional becomes h 1 1 2 λv,y + µ(u,y − β cos 2ϕ − βT cos 2ϕl )2 E(u, v, β, s) = aL 2 2 0

2 1 1 1 + µ(β sin 2ϕ + βT sin 2ϕl )2 + µ v,y − β sin 2ϕ − βT sin 2ϕl 4 2 2 1 dy. (7) + µk ln |β,y sin ϕ| 1 − bρs TWIP-alloys have rather low stacking fault energies, so the contribution of surface energy to this functional may be neglected [15]. Functional (7) can be reduced to a functional depending on β(y) and s only via standard variational calculus: h

1 1 (1−κ) (β sin 2ϕ + βT sin 2ϕl )2 + κ (β sin 2ϕ + sβt sin 2ϕl )2 2 2 0 2 sin2 ϕ |β,y sin ϕ| 1 β,y 1 2 dy, (8) + (γ − β cos 2ϕ − sβt cos 2ϕl ) + k + 2 bρs 2 (bρs )2

E(β, s) = aLµ

h µ , and · = h1 0 ·dy. Note that we approximated the logarithmic where κ = λ+2µ term in (7) by keeping only the two leading terms of a Taylor expansion. If dissipation is negligible, then the plastic distortion β and the volume fraction s minimize (8) under the constraints (3) as well as 0 ≤ s ≤ 1. If the resistance to

146


dislocation and twin boundary motion cannot be neglected, then the energy minimization must be replaced by the variational equation [24] δE + aL 0

h

∂Dp ∂Dt δs = 0, δβdy + aL ∂ s˙ ∂ β˙

(9)

where, in case of rate-independent theory, ˙ Dp = K|β|,

Dt = µεt |˙s |.

These functions are the dissipation potentials due to plastic slip and motion of the twin boundary, respectively, with K being a constant called critical resolved shear stress, εt a material constant, and the dot above a function denoting its time derivative. Thus, from (9) we derive the evolution equations ∂Dp δγ U , =− ˙ δβ ∂β

δγ U ∂Dt =− . ∂ s˙ δs

(10)

The right hand side is the variational derivative of the energy with respect to β and s, respectively. For β˙ = 0, the flow rule (10)1 is replaced by the equation β˙ = 0 (analogously, for (10)2 in case s˙ = 0).

3 Energy Minimizers and the Initiation of Slip and Twinning 3.1 Energetic Thresholds for Slip and Twinning First, we investigate the plastic deformation of the model crystal at zero dissipation and thereby estimate the onset of plastic flow and of twinning. The crystal is initially uniform and behaves elastically. From experimental evidence and also from the problem formulation above, we conclude two competing mechanisms of the microstructure to reduce the crystal’s energy upon further straining. On the one hand, macroscopic plastic slip may occur throughout the entire crystal along the principal slip system (s, m) and we should find a distinct threshold for the first occurrence of β = 0 (onset of plastic flow). On the other hand, we can expect a twinning threshold at which deformation twins begin to grow into the material, characterized by s > 0. Both thresholds will be determined from energy minimization. If the resistance to dislocation and twin boundary motion can be neglected (and hence the energy dissipation is zero) the determination of β(y) and s reduces to the minimization of the total energy (8). The results obtained in our works [5, 13, 14, 18, 19] as well as from discrete dislocation simulations [22, 25] suggest to seek the minimizer in the form of thin boundary layers of concentrated dislocations, leaving the central part of each crystal dislocation-free. Thus, we write


β(y) =

⎧ βl (y), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ βlm , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨βl (h(1 − s) − y),

147

for y ∈ (0, ll ), for y ∈ (ll , h(1 − s) − ll ), for y ∈ (h(1 − s) − ll , h(1 − s)),

(11)

⎪ ⎪ for y ∈ (h(1 − s), h(1 − s) + lu ), βu (y), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ for y ∈ (h(1 − s) + lu , h − lu ), β , ⎪ ⎪ um ⎪ ⎩ βu (h(2 − s) − y), for y ∈ (h − lu , h),

where βlm and βum are constants, ll and lu are unknown constant boundary lengths with 0 ≤ ll ≤ 12 h(1 − s) and 0 ≤ lu ≤ 12 hs, and βl (ll ) = βlm and accordingly βu (h(1 − s) + lu ) = βum . Varying the energy functional (1) with respect to all unknows together with the boundary conditions (8), introducing the dimensionless variable y¯ = bρs y and dropping the overbar in the sequel for short, we obtain the analytical solution (12) βl (y) = βlp 1 − cosh ηy + tanh ηll sinh ηy , βu (y) = βup 1 − cosh η(y − h(1 − s)) + tanh ηlu sinh η(y − h(1 − s)) , (13) with η = 2 βlp = βup =

1−κ k

|cos ϕ| and

cos 2ϕ (γ − β cos 2ϕ) + κ sin 2ϕβ sin 2ϕ − sβt (cos2 2ϕ + κ sin2 2ϕ) (1 − κ) sin2 2ϕ

,

cos 2ϕ (γ − β cos 2ϕ) − κ sin 2ϕβ sin 2ϕ − sβt (cos2 2ϕ − κ sin2 2ϕ) (1 − κ) sin2 2ϕ

+ βt

(14) along with βim = βip (1 − 1/ cosh ηli ) (i = l, u). The average quantities can be obtained as sin 2ϕ −2βlp (ll − tanh(ηll )/η) − βlm (h(1 − s) − 2ll ) h [4pt] +2βup (lu − tanh(ηlu )/η) + βum (hs − 2ll ) ,

β sin 2ϕ =

β cos 2ϕ =

cos 2ϕ 2βlp (ll − tanh(ηll )/η) + βlm (h(1 − s) − 2ll ) h +2βup (lu − tanh(ηlu )/η) + βum (hs − 2ll ) .

s is obtained form the following condition:

(15)

148


1 βt cos 2ϕ (β cos 2ϕ − γ ) − κ sin 2ϕβ sin 2ϕ + (1 − κ)βt2 sin2 2ϕ (16) 2 1 + sβt2 (cos2 2ϕ + κ sin2 2ϕ) − k sin2 ϕ βu2 (h(1 − s)) − βl2 (h(1 − s)) = 0, 2 where the latter term stems from the Weierstrass–Erdmann corner condition [11] and characterizes the jump of the dislocation density at the moving interface. Applying solution (12) together with (14) and (15) to the energy functional (8), we can write the total energy of the crystal as (not written out here for brevity) E = E(βlp , βup , ll , lu , s).

(17)

To find the solution, we need to find the global minimum of the total energy with respect to all five arguments with the additional constraints 0 ≤ s ≤ 1,

0 ≤ ll ≤

1 h(1 − s), 2

0 ≤ lu ≤

1 hs. 2

(18)

Figure 2 shows the evolution of all model parameters as functions of shear strain γ , obtained from numerical minimization of (17). Results show that plastic slip occurs first at rather small strain (in this example γen ≈ 6.83·10−5) and dislocations are nucleated and accumulate in the uniform crystal. Since this threshold occurs before the onset of twinning, the threshold value is the same as for a single crystal, as presented in [18], 2k sin ϕ . (19) γen = hbρs |cos 2ϕ| Upon further straining, twinning becomes energetically favorable and s jumps to some finite value, here at γt w ≈ 0.030. From this point on, existing dislocations appear to be consumed by the newly created twin boundaries, the parent part of the crystal remains elastic whereas the twin crystal starts to nucleate and accumulate dislocations. Thanks to the finite jump of s, free space (or mean free path) is provided for the immediate dislocation pile-up in the twin phase. Further straining results in a steady increase of s, until finally s = 1, i.e. the parent phase vanishes, and the whole crystal exhibits dislocation pile-ups along the twin slip system.

3.2 Influence of the Slip System Orientation In our recent contribution [15] we always assumed a fixed slip system orientation (or twin angle) ϕ and never analyzed the influence of varying slip system orientation. It turns out, however, that the orientation of the active slip system has an important influence on the mechanical behavior. Figure 3 illustrates the twinning threshold γt w as a function of the orientation angle ϕ and for different crystal heights h. The twinning threshold is smallest if the slip system is oriented close to ϕ ≈ 90◦ because, in this case, no dislocations are required to form the twin boundary. On the one hand,


E

149

s 1

0.0003 0.00025

0.8

0.0002

0.6

0.00015

0.4

0.0001

0.2

0.00005

blp

0.05

0.1

0.15

0.2

0.25

0.3

0.05

0.1

0.15

0.2

0.25

0.3

-0.005

g g

bup

0.05

0.1

0.15

0.2

0.25

0.3

0.05

0.1

0.15

0.2

0.25

0.3

0.05

0.1

0.15

0.2

0.25

0.3

g g

-0.05

-0.01 -0.015

-0.1

-0.02

-0.15

-0.025 -0.03

-0.2

lu

ll

0.5

1

0.4

0.8 0.6

0.3

0.4

0.2

0.2

0.1 0.05

0.1

0.15

0.2

0.25

0.3

g

g

Fig. 2 Results from minimizing the total energy functional: (a) total energy, (b) volume fraction s, (c) βlp , (d) βup , (e) ll and (f) lu , as functions of the overall shear strain γ . Computed with ϕ = 87◦ , k = 1.56 · 10−4 , ν = 0.33, ρs = 1.834 · 1015 m−2 , b = 2.5 · 10−10 m, h = 100 µm.

it becomes apparent that the twinning threshold gradually increases with increasing deviation from this 90◦-orientation since the twinning shear strain βt rises and more dislocations are required to constitute the twin interfaces. Note that the same behavior can be stated for the onset of plastic flow: following (19), the yield threshold is lowest if ϕ = 90◦ and increases with increasing deviation of the angle from ϕ ◦ . On the other hand, the classical size effect already observed for single and bicrystals [13, 14, 18, 19] becomes obvious: with decreasing crystal height h the twinning threshold rises considerably, i.e. the smaller the crystal the higher the material’s resistance to initiate twin structures. The influence of the slip system orientation can also be observed in Figure 4, where the minimizing paths of the total energy for a plane-constrained shear test are displayed for various active slip system orientations. It becomes obvious that with increasing deviation of ϕ from 90◦ the twinning threshold increases and the broad

150


gtw

h = 50 mm h = 20 mm h = 10 mm h = 5 mm

0.06 0.05 0.04 0.03 0.02 0.01 75

80

85

90

j [°]

Fig. 3 Influence of the slip system orientation angle ϕ and the crystal height h on the onset of twinning.

E 0.0004 0.0003

j j j j

0.0002

= = = =

88° 87° 86° 85°

0.0001

0.1

0.2

0.3

0.4

g

Fig. 4 Paths of the minimizing energy path for a plane-constrained shear test with varying active slip system orientation ϕ.

plateau, during which the crystal transforms from the parent phase gradually into the twin phase, expands. For ϕ = 88◦ e.g. the crystal has completely adopted the twin phase already at about 0.13% of strain, where as for ϕ = 85◦ finally consists of purely twin phase not before about 0.34% of strain. Hence, the orientation of the active slip system will considerably influence the stress-strain behavior of the crystal. Note that the present approach is based on the geometrically linear theory and thus restricted angles ϕ close to 90◦ such that βt is small.

3.3 Surface Energy and Number of Twins To estimate the actual number of twins, we extend the total energy of the crystal by also taking into account the energy stored at the side surfaces of the crystal (i.e. at x = 0 and x = a) at which boundary conditions u(0, y) = u(a, y) = γ y,


y

151

n

g

100 80 60 40

x d

d

20

0.025 0.05 0.075 0.01 0.125 0.15 0.175 0.2

g

Fig. 5 Crystal with boundary layers of thickness δ and evolution of the number of twins n with increasing overall shear strain γ . For computations we used ϕ = 87◦ , k = 1.56 · 10−4 , ν = 0.33, ρs = 1.834 · 1015 m−2 , b = 2.5 · 10−10 m, h = 100 µm, a = h, = 0.044 J/m2 , δ as the triangle thickness of the side boundaries.

v(0, y) = v(a, y) = 0 are assumed (see Figure 5). Expanding the above energy funcational by an elastic boundary energy contribution, we have two competing contributions to the total energy of the crystal, which consists of inner energy stored in the central part of the crystal as described above and the total elastic energy of the side boundary layers. Numerical results for evolving number of twins n as function of γ via the outlined method are shown in Figure 5 with = 0.044 J/m2 which is the surface energy density of twin boundaries in cooper [12]. For details about this rather qualitative estimate see [15].

4 Plastic Deformation at Non-Zero Dissipation Having shown the existence of energetic thresholds for both plastic flow and deformation twinning, we now analyze the deformation of the model crystal at non-zero dissipation which gives a physically more realistic picture. If the resistance to dislocation motion (and hence the dissipation of energy) cannot be neglected, the plastic distortion and the volume fraction may evolve only if the corresponding yield condition is satisfied, see (10). Computing the variational derivative of (8), we derive from (10) the yield conditions for β in the parent and twin part of the crystal, from which we obtain the solution with dimensionless y upon loading:

1 βl (y) = βlp 1 − cosh ηy + tanh ηh(1 − s) sinh ηy , 2

1 ηhs sinh η(y − h(1 − s)) βu (y) = βup 1 − cosh η(y − h(1 − s)) + tanh 2 (20)

152


with η = 2 βlp = βup =

1−κ k

|cos ϕ| and

cos 2ϕ (γr − β cos 2ϕ) + κ sin 2ϕβ sin 2ϕ − sβt (cos 22ϕ + κ sin 22ϕ) (1 − κ) sin2 2ϕ

,

cos 2ϕ (γr − β cos 2ϕ) − κ sin 2ϕβ sin 2ϕ − sβt (cos2 2ϕ − κ sin2 2ϕ) (1 − κ) sin2 2ϕ

+ βt .

(21) The average quantities read

2 sin 2ϕ 1 1 −βlp h(1 − s) − tanh ηh(1 − s) η h 2 2

1 1 +βup hs − tanh ηhs η , (22) 2 2

2 cos 2ϕ 1 1 βlp h(1 − s) − tanh ηh(1 − s) η β cos 2ϕ = h 2 2

1 1 hs − tanh ηhs η . (23) +βup 2 2 β sin 2ϕ =

Solving the above evolution equations is numerically cumbersome. Instead, we transform the problem into a minimization problem, which can easily be solved using standard energy minimization techniques. Therefore, we modify the total energy functional by removing the linear term with β and adding an artificial term to obtain ∗

E(β) =

h

1 1 (1 −κ) (β sin 2ϕ + βT sin 2ϕl )2 + κ (β sin 2ϕ + sβt sin 2ϕl )2 2 2 0 1 1 2 2 2 + (γr − β cos 2ϕ − sβt cos 2ϕl ) + kβ sin ϕ + st dy. (24) 2 2

Applying solution (20) together with (21) and (22) to the energy functional (24), we may write the total energy of the crystal as (again not written out here for brevity) E = E(βlp , βup , s).

(25)

To find the solution, we minimize the modified energy (25) with respect to these quantities with the additional constraint 0 ≤ s ≤ 1. A similar solution can be found for the case of unloading [15]. It is interesting to calculate the shear stress τ which is a measurable quantity. During positive loading beyond the critical shear strain, we have for the normalized shear stress (or the elastic shear strain) γe =

τ = γcr + γr (1 − (β + sβt ) cos 2ϕ) , µ

(26)

An Energetic Approach to Deformation Twinning t/m

D

0.001

E

0.0005

B

O -0.1 G -0.0005

H

s

D

0.6

0.1

0.3 g

0.2 F

0.4

0.2 H

-0.001

G

B

-0.1

blp

0.1

0.2

0.3

g

0.2

0.3

g

bup

0.1

0.1

0.05

F

H D

G

0.1

A

-0.025 -0.05

C

0.8

0.075

-0.1

F

1

C

A

I H

153

0.2

0.3

G

-0.1

g

0.1 -0.1

F

C

C B

B

-0.2

D

Fig. 6 Results from minimizing the total energy functional: (a) stress-strain curve, (b) volume fraction s, (c) βlp , (d) βup as functions of the overall shear strain γ . Computed with ϕ = 87◦ , k = 1.56 · 10−4 , ν = 0.33, ρs = 1.834 · 1015 m−2 , b = 2.5 · 10−10 m, h = 100 µm, εt = 5 · 10−5 .

where γr = γ −γcr . The second term of (26) causes hardening due to the dislocation pile-up and a load drop at the initiation of deformation twinning. Note that this stress-strain behavior also highly depends on the number of twins and the choice of εt . During inverse loading, the stress-strain curve follows analogously [15]. Figure 6 illustrates the evolution of all variables along with the normalized shear stress (or elastic shear strain) versus shear strain curve for the following straining path (with γcr = γen from energy minimization): γ is first increased from zero to some arbitrary value γ ∗ = 0.3, then decreased to −0.1, and finally increased to zero (note that the rate of change of γ( t) does not affect the results due to the rate independence of dissipation). The straight line OA corresponds to purely elastic loading with γ increasing from zero to γcr . Section AB corresponds to plastic flow without deformation twinning with κ = K. Yielding begins at point A with the yield stress σy = K, and we can observe a work hardening section due to dislocation pile-up. The initiation of twinning at point B is indicated by the characteristic sharp load drop [9]. During further straining the volume fractions of parent and twin phases evolve and the load remains almost constant (see the plateau in section BC). Finally, at point C we again observe a uniform crystal with s = 1, i.e. deformation twinning has ended and the common hardening by dislocation pile-up occurs in the single crystal which has now completely adopted the twin lattice. During unloading, as γ decreases from γ ∗ to γ ∗ − 2γcr (line DE), the plastic distortion β and volume fraction s are frozen. As γ

154


decreases further to γ∗ , plastic yielding occurs (section EF) followed by detwinning. The onset of twinning during inverse loading is clearly indicated in the cycle by the beginning plateau at point F, which ends at point G with a homogeneous parent phase, followed by hardening of the single crystal.

5 Conclusions We presented an energy-based micro-mechanical model within continuum dislocation theory to describe the initiation and evolution of deformation twins which arise from a single crystal with one active slip system under constrained shear deformation. As interesting characteristics, we showed distinct thresholds for the onset of plastic flow and the initiation of deformation twinning, the stress-strain response exhibiting a sharp load drop (followed by a stress plateau) upon the onset of twinning, and the influence of dissipation resulting in the typical stress-strain hysteresis.

References 1. Allain, K.E., Chateau, J.P. and Bouaziz, O., A physical model of the twinning-induced plasticity effect in a high manganese austenitic steel. Mater. Sci. Eng. A 387, 2004, 143–147. 2. Bilby, B.A., Bullough, R. and Smith, E., Continuous distributions of dislocations: A new application of the methods of non-Riemannian geometry. Proc. Royal Soc. A 231, 1955, 263–273. 3. Berdichevsky, V.L., On thermodynamics of crystal plasticity. Scripta Mater. 54, 2006, 711– 716. 4. Berdichevsky, V.L., Continuum theory of dislocations revisited. Cont. Mech. Thermodyn. 18, 2006, 195–222. 5. Berdichevsky, V.L. and Le, K.C., Dislocation nucleation and work hardening in anti-planed constrained shear. Cont. Mech. Thermodyn. 18, 2007, 455–467. 6. Berdichevsky, V.L. and Sedov, L.I., Dynamic theory of continuously distributed dislocations. Its relation to plasticity theory. J. Appl. Math. Mech. (PMM) 31, 1967, 989–1006. 7. Bullough, R., Deformation twinning in the diamond structure. Proc. Royal Soc. London A 241, 1957, 568–577. 8. Cahn, R.W., Twinned crystals. Adv. Phys. 3, 1954, 363–445. 9. Christian, J.W. and Mahajan, S., Deformation twinning. Progr. in Mater. Sci. 39, 1995, 1–157. 10. Frenkel, J. and Kontorova, T., On the theory of plastic deformation and twinning. Acad. Sci. USSR J. Phys. 1, 1939, 137–149. 11. Gelfand, I.M. and Fomin, S.V., Calculus of Variations. Dover Publications, New York, 2000. 12. Hirth, J.P. and Lothe, J., Theory of Dislocations, 2nd edition. Krieger Publishing Company, Malabar, Florida, 1982. 13. Kochmann, D.M. and Le, K.C., Plastic deformation of bicrystals within continuum dislocation theory. J. Math. Mech. Solids, doi: 10.1177/1081286507087322, 2008. 14. Kochmann, D.M. and Le, K.C., Dislocaton pile-ups in bicrystals within continuum dislocation theory. Int. J. Plasticity 24, 2008, 2125–2147. 15. Kochmann, D.M. and Le, K.C., A continuum model for initiation and evolution of deformation twinning. J. Mech. Phys. Solids 57, 2009, 987–1002. 16. Kondo, K., On the geometrical and physical foundations of the theory of yielding. In Proc. 2nd Japan Congr. Appl. Mech., 1952, pp. 41–47.


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17. Kröner, E., Kontinuumstheorie der Versetzungen und Eigenspannungen. Springer, Berlin, 1958. 18. Le, K.C. and Sembiring, P., Plane constrained shear of single crystals within continuum dislocation theory. Arch. Appl. Mech. 78, 2008, 587–597. 19. Le, K.C. and Sembiring, P., Plane-constrained shear of a single crystal strip with two active slip-systems. J. Mech. Phys. Solids 56, 2008, 2541–2554. 20. Le, K.C. and Stumpf, H., A model of elastoplastic bodies with continuously distributed dislocations. Int. J. Plasticity 12, 1996, :611–627. 21. Le, K.C. and Stumpf, H., On the determination of the crystal reference in nonlinear continuum theory of dislocations. Proc. Royal Soc. London A 452, 1996, 359–371. 22. Needleman, A. and Van der Giessen, E., Discrete dislocation and continuum descriptions of plastic flow. Mater. Sci. Eng. A 309–310, 2001, 1–13. 23. Nye, J.F., Some geometrical relations in dislocated crystals. Acta Metall. 1, 1953, 153–162. 24. Sedov, L.I., Variational methods of constructing models of continuous media. In: H. Parkus and L.I. Sedov (Eds.), Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids. Springer Verlag, Wien/New York, 1968. 25. Shu, J.Y., Fleck, N.A., Van der Giessen, E. and Needleman, A., Boundary layers in constrained plastic flow: Comparison of nonlocal and discrete dislocation plasticity. J. Mech. Phys. Solids 49, 2001, 1361–1395.

Computational Homogenization of Confined Frictional Granular Matter H.A. Meier, P. Steinmann and E. Kuhl

Abstract Multiscale modeling and computation of confined granular media opens a novel way of simulating and understanding the complicated behavior of granular structures. Phenomenological continuum approaches are often not capable of reproducing distinguishing features of granular media, like, e.g., the breaking and forming of particle contacts. Alternatively, a multiscale homogenization procedure, based on a discrete element method, allows to capture such distinguishing features. The present manuscript deals with the variationally based computational homogenization and simulation of granular media, whereby the macroscopic impact of inter-particle friction defines the overall focal point. To bridge the gap between both scales, we apply the concept of a representative volume element, essentially linking both scales through variational considerations. As the beneficial outcome of the variational approach, the Piola stress is derived from the overall macroscopic energy density.

1 Introduction The behavior of confined granular media is of great interest to the industry as well as research. Both disciplines are mainly concerned with the so-called homogenized response of a granular assembly. Thereby, individual particle interactions are of minor H.A. Meier Department of Mechanical Engineering, University of Kaiserslautern, D-67653 Kaiserslautern, Germany; E-mail: [email protected] P. Steinmann Department of Mechanical Engineering, University of Erlangen-Nuremberg, D-91058 Erlangen, Germany; E-mail: [email protected] E. Kuhl Department of Mechanical Engineering, Stanford University, Stanford, CA 95305-4040, USA; E-mail [email protected] K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_12, © Springer Science+Business Media B.V. 2010

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interest. Yet, it is well established that the apparent properties of granular assemblies strongly influence the properties of the incorporated grains. One particular choice to model and simulate confined granular matter consists of utilizing a homogenization strategy. In the context of computational mechanics, homogenization methods require the existence of two scales. One scale (microscale) is used to describe the details of the underlying microstructure, while the other scale (macroscale) focuses on the effective material response, see, e.g., [12] or [19]. Both scales are linked by an averaging procedure, typically relying on the existence of a representative volume element (rve). For the present case, the microscale level is defined by unbonded disk-like discrete elements, see, e.g., the discrete element method (dem) proposed by Cundall and Strack [2,3]. Macroscopically, the resultant energy density as well as the resultant stress are of great importance. This publication focuses on the macroscopic impact of inter-particle friction of confined particulate matter. Inter-particle friction describes one form of possible energy dissipation on the grain-scale level. Next to the presence of dilatancy, inter-particle friction is accounted as a principle of granular assemblies, compare [4]. Phenomenological based frictional contact models are for example proposed by Vu-Quoc and Zhang [15], Vu-Quoc, Zhang and Walton [16], Walton [17] and Walton and Braun [18]. We attempt to derive a frictional tangential inter-particle contact model by applying a variational approach. In the context of continuum mechanics, variational approaches are used by numerous investigators to describe dissipative processes, see, e.g., Comi and Perego [1] or Ortiz and Stainier [11]. Recently, Ortiz and Pandolfi employed a variational approach to define a formulation of the Cam-clay theory of plasticity, see [10]. Miehe and colleagues extended the variational concept to allow the variational formulation of homogenization of inelastic microstructures, compare [7] and [8]. We deploy a variational model to define inter-particle tangential friction forces. The adopted procedure relies on the minimization of a general work function with respect to the internal variables, whereby the generalized work function includes an elastic and an inelastic contribution. Internal variables are updated subsequently during the minimization process. The result of the minimization process defines an incremental potential. The incremental potential is used to define the elastic tangential contact force as well as the related couples. A discrepancy between the obtained results does not allow for the conservation of the global angular momentum. We provide the reader with a suitable algorithmic update procedure to circumvent this problem. Section 2 introduces the reader to the behavior of frictional granular matter. Macroscopic responses are defined in Section 3. Examples providing information about the macroscopic impact of inter-particle friction are granted in Section 4. Uni-axial compression, simple shear as well as cyclic loading tests are performed.

Computational Homogenization of Confined Frictional Granular Matter

159

2 Microscale 2.1 Kinematics The assumption of Taylor [14] defines the spatial positions of the individual grains on the microscale level by mapping their material positions Xi via the macroscopic deformation gradient F, xi = F · Xi .

(1)

Individual microscopic fluctuations are a priori neglected, leading to a homogeneous deformation field over the locally attached microstructure. The so-called spatial branch vector constructs by subtracting the spatial positions of two grains, lij = F · Xj − Xi . Its length measures the distance between two grains, while its direction nij = lij /lij defines the direction of contact. Penalty type related contact mechanics requires knowledge of the overlap between neighboring objects. The disk-like shape of the grains allows for a trivial specification of the spatial interparticle overlap εij , εij = ri + rj − lij ,

(2)

whereby ri and rj present the radii of the particles. For two grains being in contact, εij delivers a value greater than or equal to zero. The value itself measures the overlap. Next to the inter-particle overlap, the tangential contribution of the slip between the contacting grains represents a relevant quantity. Its derivation follows from its rate, computed by subtracting the current tangential velocities of the contact points, lij − ˙lij · tij , γ˙ij = θ˙i ri + θ˙j rj ri + rj

(3)

with the vector tij = ω · nij defining the tangential contact direction, θ˙i the angular velocity of particle i and ˙lij the velocity of the branch vector. The second order tensor ω specifies a standard two dimensional rotation matrix, evaluated for an angle of 90◦ . Observe that, in contrast to the classical formulation, the scaling factor lij /[ri + rj ] is included. The scaling procedure originates from the allowance of inter-particle overlap and is thought of as a compensation. Related to the assumed rigidity of the grains, the scaling factor defines as a constant. By applying a standard backward Euler integration, the discrete expression of the tangential inter-particle slip yields: n+1 lij − lnij tnij · nn+1 γijn+1 = γijn + [θin+1 − θin ]ri + [θjn+1 − θjn ]rj ij . ri + rj

(4)

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The derived expression includes the incremental update of the tangential interparticle slip. This update is composed by two shares. The first share originates from pure particle rotation, while the second part is related to pure particle translation. We additively split (4) in an elastic and a plastic part, γij = γelij + γplij . The elastic part of the tangential inter-particle slip relates to the adhesion between the contacting grains. The plastic part is interpreted as inter-particle sliding.

2.2 Contact Potential For the contact normal direction, a one-sided harmonic inter-particle contact potential is postulated, solely depending on the inter-particle overlap. nij =

1 H(εij )Enij [εij ]2 . 2

(5)

Letting the Heaviside function H act on εij allows to restrict to repulsive contacts between the individual objects. With respect to the tangential contact direction, we introduce an energy storage function. tij =

1 H(εij ) Etij [γelij ]2 + hij [αij ]2 . 2

(6)

This function is based on the elastic part of the tangential slip, as well as on a contact related internal hardening variable αij . Permitting the Heaviside function to act on the inter-particle overlap, (6) solely outputs a value different from zero if the particles are in contact. The tangential contact stiffness is remarked by Etij , while the hardening modulus is denoted by hij . Considering the results of Mindlin [9], we relate the tangential contact stiffness to the normal contact stiffness by Etij = ϑEnij with ϑ ∈ [2/3, 1]. For a single contact, the discrete Clausius–Planck inequality reads, ˙ tij ≥ 0, Dij = ftij γ˙ij −

(7)

including the discrete internal dissipation power, Dij , the tangential stress power, ftij γ˙ij and the rate of (6). Inserting the rate of the energy storage function in (7), reorganization of the result and the application of the Coleman–Noll argument result in a definition of the magnitude of the discrete tangential contact force, as well as in a reduced discrete dissipation power statement, ∂tij ∂tij ∂tij ftij = (8) , Dij = − γ˙plij + α˙ ij ≥ 0. ∂γij ∂γplij ∂αij Included are the thermodynamical forces


Q1ij = −

161

∂tij ∂tij = Etij H(εij )γelij and Q2ij = − = −hij H(εij )αij , (9) ∂γplij ∂αij

which are are conjugated to the rates of the internal variables γ˙plij and α˙ ij , respectively. We introduce a yield function of Coulomb type which includes the thermodynamical forces, Yij = |Q1ij | + Q2ij ≤ cij .

(10)

The constant cij is set to cij = fnij tan (φ), with φ being the friction angle. Solving the classical optimization problem, i.e., satisfying the principle of maximum dissipation, compare for example [13], yields the evolution equations for the internal variables, γ˙plij = λij Etij sgn(γelij )

and α˙ ij = λij ,

(11)

with λij being the contact specific plastic multiplier. With this, the discrete interparticle dissipation power yields: Dij = λij fnij tan(φ) ≥ 0,

(12)

depending on the contact specific plastic multiplier as well as on the magnitude of the current normal contact force. To obtain the incremental potential Wtn+1 ij , we insert the discrete inter-particle dissipation power in the Clausius–Planck inequality and integrate the result with respect to time. Finding the reduced incremental potential relates to a minimization problem with respect to the incremental plastic multiplier: n+1 = inf W n+1 (γ n+1 , λij ) , with W n+1 = n+1 − n + λij cn+1 . W tij tij tij tij tij ij ij

λij

(13) The solution of the minimization problem usually involves a Newton-type iteration scheme, however, based on the nature of the here assumed energy storage function, the incremental plastic multiplier is calculated in closed form.

λij =

n+1 fnn+1 ij tan (φ) − Yij

Etij + hij

.

(14)

The update procedures for the plastic share of the inter-particle slip, as well as for the internal hardening variable result to: n+1 n γpln+1 ij = γplij + λij sgn(ftij )

and αijn+1 = αijn + λij .

In the event of contact breaking, γijn+1 and γpln+1 ij are set to zero.

(15)

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2.3 Contact Forces The contact normal force, resulting from a single contact between two grains, equals the negative derivative of the normal contact potential, see (5), with respect to the spatial particle position. fnij = −

∂nij = −Eij H(εij )εij nij ∂xi

(16)

Obviously, based on the selected one-sided harmonic pair potential, the resulting contact force is linear dependent on the inter-particle overlap. The magnitude of the contact normal force is defined by fnij = Enij H(εij )εij . Its direction is given by the negative contact normal nj i . The frictional tangential contact forces generate by taking the negative derivative of the reduced incremental potential, given in (13), with respect to the current particle positions, =− ftn+1 ij

n+1 ∂W tij ∂xn+1 i

n+1 lnij n n+1 n+1 εij = −Etij γeln+1 H tij · tij tij . ij ln+1 ij

(17)

In contrast to the contact normal force, compare (16), the derivative of the kinematic variable with respect to the current particle position introduces a type of scaling facn+1 n tor, sijn+1 = lnij /ln+1 ij [tij · tij ], whereby the factor itself relates to the possible change of the contact direction. In particular, the included fraction of length between the branch vectors may be interpreted as an inverse incremental stretch of the distance between the particle centers, whereas the product of the tangent vectors defines an incremental change of the contact direction. Note the reciprocal character of the corresponding contact forces, which itself is related to Newton’s third axiom. Based on the orientation of the tangential contact force as well as on its origin, couples act on the individual particles. These couples derive by taking the negative derivative of the incremental potential function; at this point with respect to the individual current particle rotation, mn+1 ij

=−

n+1 ∂W tij ∂θin+1

n+1 = −Etij γeln+1 ij H εij

ri ln+1 ri + rj ij

(18)

Carefully observe that the derived couples do not fit the obtained tangential forces, n+1 n+1 i.e., mn+1 ij e3 = ri /[ri +rj ]lij ×ftij . Hence, the use of the derived quantities leads to an unbalanced angular momentum of the granular assembly. Remedy is found by an algorithmic modification of the particle couples. Therefore, the individual couples are scaled by sijn+1 . n+1 n+1 mn+1 algoij = sij mij

(19)

This modification ensures the balance of the local as well as global angular mon+1 mentum, i.e., mn+1 × ftn+1 algoij e3 = ri /[ri + rj ]lij ij . The particle forces and resultant


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couples compute by a summation over all particles, denoted by P, in the discrete system, n+1 fin+1 = mn+1 (20) fnn+1 and mn+1 ij + ftij algoi = algoij . i∈P

i∈P

j=i

j=i

Related to the assumption of Taylor, the positions of the individual grains are prescribed, solely leaving the rotational degrees of freedom unconstrained. Thus, for the individual grains to be in static equilibrium, we require the resultant particle couples to vanish: !

∀ i ∈ P.

mn+1 algoi = 0,

(21)

3 Macroscale Starting with the normal contact potential as well as reduced incremental tangential potential, see (5) and (13), we focus on relating these particular results to the macroscale. We find an energy density with respect to the material configuration to be an appropriate quantity to relate between the two scales. Smearing out the microscopic energy over the volume of the undeformed rve, yields a macroscopic energy density,

n+1

=

1 n+1 n+1 nij + Wtij , 2Vrve

(22)

i∈P j ∈P j=i

whereby Vrve refers to the undeformed volume of the particulate rve. As known from classical continuum mechanics, the Piola stress associates to the derivative of a potential function with respect to the deformation gradient tensor. Applying this relation to link the macroscopic Piola stress to the macroscopic energy density, we take the derivative of the macroscopic energy density, see (22), with respect to the macroscopic deformation gradient. P

n+1

=

∂

n+1

n+1

∂F

=−

1 n+1 fi ⊗ Xi Vrve

(23)

i∈P

We observe that the macroscopic Piola stress, concerning the assumption of Taylor, consists of a volume weighted summation of the tensor product between the particle forces in the spatial and the particle positions in the material configuration over all particles in the rve.

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grain ∅ [mm]

0.315

0.250

0.160

0.125

0.100

particles [/]

0

1

40

74

35

Fig. 1 Exemplary shown microscopic rve, subjected to uni-axial compression. Left: Initial undeformed setup with λ = 0. Center: Horizontally compressed rve at λ = 0.3. Right: Vertically compressed rve at λ = 0.3.

4 Examples The proposed computational homogenization is illustrated by selected numerical examples. Employing the generation algorithm of Meier et al. [5], we generate 15 rves, containing 150 primary particles. Related input parameters for the generation algorithm are listed in Table 1. The obtained rves incorporate a mean volume density of approximately 0.84. We subject the set of rves to three different deformation scenarios. At first, we perform uni-axial compression as well as simple shear tests. These test are described in Sections 4.1 and 4.2 and are performed in horizontal as well as vertical direction. Having studied the macroscopic output under basic deformation, we tend to the application of cyclic loading, see Section 4.3. For all examples we select the normal as well as the tangential contact stiffness to 1E+03, i.e., En = Et = E = 1E+03, while the relative convergence threshold for the particle couples is selected to 1E-08.

4.1 Uni-Axial Compression As a first example, we consider a uni-axial compression test in horizontal as well as vertical direction. The compression is applied by the macroscopic deformation gradient F = δij ei ⊗ ej − λek ⊗ ek . For the horizontal compression, k is set to 1. Vertical compression requires k = 2. The loading is chosen to λ ∈ [0 → 0.3], applied in 300 equidistant load steps. An exemplary selected rve in its initial and deformed configuration is shown in Figure 1. Observe that the relative high number of load steps relates to the large deformation of the microstructure. We focus on

Computational Homogenization of Confined Frictional Granular Matter ·1E-01 0◦ 10◦ -3.0 20◦ 30◦ -2.0

-4.0

σ /E

-1.0

·1E-02 0◦ 10◦ 0.5 20◦ 30◦ 0.0

σ /E

-1.0

0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 1 − F 11 1.0

·1E-01 0◦ 10◦ -3.0 20◦ 30◦ -2.0

-4.0

165

σ 12 /E

0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 1 − F 22 ·1E-02 0◦ 10◦ 0.5 20◦ 30◦ 0.0

1.0

σ 12 /E

-0.5

-0.5

-1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30

-1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 1 − F 22

1 − F 11

Fig. 2 Macroscopic influence of the microscopic friction angle φ ∈ {0◦ , 10◦ , 20◦ , 30◦ } under uniaxial compression. Errorbars show the standard deviation. All curves and errorbars are scaled by the normal contact stiffness. Left column: Horizontal compression. Right column: Vertical compression. Top row: Macroscopic Cauchy stress. Solid lines denote the stress components in horizontal direction, while dashed lines refer to the vertical stress components. Bottom row: Shear component of the macroscopic Cauchy stress.

the macroscopic influence of different microscopic friction angles. Therefore, we perform 4 calculations with different microscopic friction angles between 0◦ and 30◦ , setting the hardening modulus to hij = h = 0. The obtained stress results are presented in Figure 2. The left side of Figure 2 refers to the horizontal compression, while the right side shows results in regards to the vertical compression. The top row of Figure 2 depicts the horizontal as well as vertical components of the macroscopic Cauchy stress. We identify the horizontal components of the macroscopic Cauchy stress by solid lines. Vertical components of the macroscopic Cauchy stress are denoted by dashed lines. We observe the switched directions of loading to yield similar macroscopic results. Overall, we find the depicted results satisfactory, considering the size of the errorbars. For a friction angle equal to zero, the obtained results equal frictionless observations, compare Meier et al. [6]. The increase of the microscopic friction angle yields a stiffness increase in the direction of loading. Orthogonal to the direction of loading, the increase of the microscopic friction angle enforces a decrease in stress. This behavior, i.e., the widening between the two curves, explained by the micro mechanics of frictional contacts. In detail, the direction as well

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Fig. 3 Exemplary shown microscopic rve, subjected to simple shear. Left: Initial undeformed setup with λ = 0. Center: Horizontally sheared rve at λ = 1. Right: Vertically sheared rve at λ = 1.

as the magnitude of the normal contact force stay equivalent for all variations of the microscopic friction angle. Hence, the tangential contact direction does not change either. However, the magnitude of the tangential contact force differs depending on the angle of inter-particle friction. This phenomena allows for the rise of loading directed forces, while orthogonal to the direction of loading a decrease in force is imposed. The apparent linear behavior between the individual results relates to the formulation of the inter-particle friction and is strongly connected to the behavior of the tangent function. The related shear components of the macroscopic Cauchy stress are located in the bottom row of Figure 2.

4.2 Simple Shear For closer observations, we perform a simple shear test in horizontal as well as vertical direction. The related macroscopic deformation gradient F = δij ei ⊗ej +λek ⊗el is used to apply the deformation. Simple shear in horizontal direction requires k = 1 and l = 2, while switched values of k and l define simple shear in vertical direction. The loading parameter is chosen to λ ∈ [0 → 1]. The total deformation applies in 1000 equally sized load steps. Observe that we do not apply compression prior to the simple shear deformation. Hence, the volume of the rve is kept constant. As in the previous example, we observe the deformation in the horizontal as well as in the vertical direction. Initial and deformed configurations of an exemplary selected rve are shown in Figure 3. Our investigations focus on the macroscopic impact of the inter-particle friction angle, keeping the hardening modulus to zero. For this purpose, identical to Section 4.1, we plot the components of the macroscopic Cauchy

Computational Homogenization of Confined Frictional Granular Matter ·1E-01 0◦ 10◦ -1.2 20◦ 30◦ -0.8

-1.6

·1E-01 0◦ 10◦ -1.2 20◦ 30◦ -0.8

σ /E

-1.6

-0.4 0.0 0.00

σ /E

-0.4 0.20

·1E-02 0◦ 10◦ 6.0 20◦ 30◦ 4.0 8.0

0.40 0.60 F 12

0.80

1.00

0.0 0.00

0.20

·1E-02 0◦ 10◦ 6.0 20◦ 30◦ 4.0

σ 12 /E

8.0

2.0 0.0 0.00

167

0.40 0.60 F 21

0.80

1.00

0.80

1.00

σ 12 /E

2.0 0.20

0.40 0.60 F 12

0.80

1.00

0.0 0.00

0.20

0.40 0.60 F 21

Fig. 4 Macroscopic influence of the microscopic friction angle φ ∈ {0◦ , 10◦ , 20◦ , 30◦ } under simple shear. Errorbars show the standard deviation. All curves and errorbars are scaled by the normal contact stiffness. Left column: Horizontal compression. Right column: Vertical compression. Top row: Macroscopic Cauchy stress. Solid lines denote the stress components in horizontal direction, while dashed lines refer to the vertical stress components. Bottom row: Shear component of the macroscopic Cauchy stress.

stress. Overall, we find identical behaviors for the switched directions of loading. The horizontal as well as vertical components of the macroscopic Cauchy stress shows a remarkable equivalence. No influences in regards to the variation of the inter-particle friction angle are noticed. In particular, we find all plots to deliver almost identical values which are congruent with the frictionless results of Meier et al. [6]. The coarse scale behavior of the shear components of the macroscopic Cauchy stress show a nearly identical output.

4.3 Cyclic Loading Next, we investigate on the macroscopic impact of the inter-particle hardening modulus. Therefore, we select a cyclic loading deformation scenario. We compute the response for hij = h = 0 as well as hij = h = E, keeping the inter-particle friction angle to 30◦. Based on the discussed results, we restrict this study to horizontal deformation. At first, the rves are compressed in a bi-axial manner, i.e., we

168 -1.0

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σ 11 /E

-1.0

-0.8

-0.8

-0.6

-0.6

-0.4

-0.4

-0.2

-0.2

0 E 0.0 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 F 12

0.8 0.4

·1E-02 0 E

σ 12 /E

0.0 -0.4 -0.8 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 F 12

·1E-01

σ 22 /E

0 E 0.0 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 F 12

6.0

·1E-01

F 12

4.0 2.0 0.0 -2.0 -4.0 -6.0 0

500

1000 load step

1500

2000

Fig. 5 Macroscopic influence of the microscopic hardening modulus hij ∈ {0, E} under cyclic loading. Errorbars show the standard deviation. All stress related curves and errorbars are scaled by the normal contact stiffness. Top left: Horizontal component of the macroscopic Cauchy stress. Top right: Vertical component of the macroscopic Cauchy stress. Bottom left: Shear component of the macroscopic Cauchy stress. Bottom right: Prescribed behavior of the shear component of the macroscopic deformation gradient.

apply the macroscopic deformation gradient F = δij [1 − λc ]ei ⊗ ej . The loading parameter, governing the compression, is set to λc ∈ [0 → 0.01]. The compression applies in 10 uniform load steps. Reaching the compressed state, the shearing deformation is initiated. Shearing applies by F = δij [1 − λc ]ei ⊗ ej + λs e1 ⊗ e2 . The loading parameter λc is held constant during the shearing, i.e., λc = 0.01 = const. The loading parameter λs , which defines the shear deformation, applies in three stages [0 → 0.5], [0.5 → −0.5] and [−0.5 → 0]. The complete cycle uses 2000 uniform load steps. We provide the reader with a plot of the significant change of the shear component of the macroscopic deformation gradient, compare Figure 5 (bottom right). The results of the macroscopic Cauchy stress are depicted in the remaining graphs of Figure 5. The top row, presenting the horizontal (left) as well as vertical (right) component of the macroscopic Cauchy stress, show almost identical behaviors for both selections of the microscopic hardening variable. A zoom in on the fine scale of the plotted graph allows for further observations. For the horizontal as well as vertical stress component a history dependent behavior is noticed. The horizontal component shows a decreasing tendency, while the vertical component depicts an increasing behavior. The shear components of the macroscopic


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Cauchy stress allow to identify significant differences between the different hardening moduli. We observe a distinct hysteresis for h = E, while for h = 0 a rather flat hysteresis curve is noticed. In contrast to the cyclic loading of frictionless particles, the overall results show a macroscopic history dependence which solely relates to microscopic dissipation processes of the proposed inter-particle friction.

References 1. Comi, C. and Perego, U., A unified approach for variationally consistent finite elements in elastoplasticity. Comput. Meth. Appl. Mech. Eng. 121, 1995, 323–344. 2. Cundall, P.A. and Strack, O.D.L., The distinct element method as a tool for research in granular media. Tech. Rep., Report to the National Science Foundation Concerning NSF Grant ENG76-20711, Parts I, II, 1978. 3. Cundall, P.A. and Strack, O.D.L., A discrete numerical model for granular assemblies. Géotechnique 29(1), 1979, 47–65. 4. Duran, J., Sands, Powders, and Grains. Springer, 1999. 5. Meier, H.A., Kuhl, E. and Steinmann, P., A note on the generation of periodic granular microstructures based on grain size distributions. Int. J. Numer. Anal. Meth. Geomech. 32(5), 2008, 509–522. 6. Meier, H.A., Steinmann, P. and Kuhl, E., Towards multiscale computation of confined granular media – Contact forces, stresses and tangent operators. Tech. Mech. 28, 2008, 32–42. 7. Miehe, C., Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. Int. J. Numer. Meth. Eng. 55(11), 2002, 1285–1322. 8. Miehe, C., Schotte, J. and Lambrecht, M., Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals. J. Mech. Phys. Solid 50(10), 2002, 2123–2167. 9. Mindlin, D., Compliance of elastic bodies in contact. ASME J. Appl. Mech. 16(3), 1949, 259– 268. 10. Ortiz, M. and Pandolfi, A., A variational Cam-clay theory of plasticity. Comput. Meth. Appl. Mech. Eng. 193, 2004, 2645–2666. 11. Ortiz, M. and Stainier, L., The variational formulation of viscoplastic constitutive updates. Comput. Meth. Appl. Mech. Eng. 171, 1999, 419–444. 12. Schröder, J., Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Stabilitätsproblemen. Habilitation, Bericht Nr. I-7 des Instituts für Mechanik (Bauwesen) Lehrstuhl I, Universität Stuttgart, 2000. 13. Simo, J.C. and Hughes, T.J.R., Computational Inelasticity. Springer-Verlag, 2000. 14. Taylor, G.I., Plastic strain in metals. J. Inst. Met. 62, 1938, 307–324. 15. Vu-Quoc, L. and Zhang, X., An accurate and efficient tangential force-displacement model for elastic frictional contact in particle-flow simulations. Mech. Mater. 31, 1999, 235–269. 16. Vu-Quoc, L., Zhang, X. and Walton, O.R., A 3-d discrete-element method for dry granular flows of ellipsoidal particles. Mech. Mater. 31, 2000, 483–528. 17. Walton, O.R., Numerical simulation of inclined chute flows of monodisperse inelastic, frictional spheres. Mech. Mater. 16, 1993, 239–247. 18. Walton, O.R. and Braun, R.L., Viscosity granular-temperature and stress calculations for shearing assemblies of inelastic frictional disks. J. Rheol. 30, 1986, 949–980. 19. Zohdi, T.I., Homogenization methods and multiscale modeling: Linear problems. In Encyclopedia of Computational Mechanics. E. Stein, R. de Borst and T. Hughes (Eds.), Wiley, 2004.

Existence Theory for Finite-Strain Crystal Plasticity with Gradient Regularization Alexander Mielke

Abstract We provide a global existence result for the time-continuous elastoplasticity problem using the energetic formulation. The deformation gradient is decomposed multiplicatively into an elastic part and the plastic tensor P, which is driven by the plastic slip strain rates p˙ j . We allow for self-hardening as well as crosshardening. The strain gradients ∇pj and ∇P are used to regularize the problem, thus introducing a length scale and preventing the formation of microstructure.

1 Introduction Elastoplasticity at finite strain is usually based on the multiplicative decomposition ∇ϕ = F = Fel Fpl , introduced in [17]. This decomposition reflects the Lie group def structure of GL+ (d) = { F ∈ Rd×d | det F > 0 }, where the elastic part Fel contributes to the energy storage whereas the plastic tensor P = Fpl evolves according to a plastic flow rule. The plastic tensor maps the material frame (crystallographic def lattice) onto itself and is usually assumed to lie in SL(d) = { P ∈ Rd×d | det P = 1 }. In this paper we combine the formal ideas for single-crystal plasticity from [20, 26] with the recent analytical developments in [18] proving a global-in-time existence result for solution in finite-strain elastoplasticity. The difficulty is to find a formulation that allows us to use functional analytical tools that are compatible with the strong nonlinearities generated by the Lie group structures resulting from GL+ (d) and SL(d). We use here the recently developed geometric theory of energetic solutions for on abstract topological spaces (cf. [11, 21]) was strongly motivated by the present application and, thus, provides the first mathematical foundation to treat the existence theory for rate-independent finite-strain elastoplasticity. Alexander Mielke Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, D-10117 Berlin, Germany; E-mail: [email protected] K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_13, © Springer Science+Business Media B.V. 2010

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To be more specific we introduce some notations. Let ϕ : → Rd denote the deformation, P : → SL(d) the plastic tensor, and p : → [0, ∞[m is the vector of slip strains. Then, we assume that the stored-energy functional takes the form E(t, ϕ, P, p) = W (x, ∇ϕP−1 , p, ∇P, ∇p) dx − (t), ϕ.

Here Felast = ∇ϕP−1 represents the multiplicative decomposition. The gradients (∇P, ∇p) introduce a length scale and are essential to provide compactness, thus preventing the formation of microstructure, cf. [3, 5]. Such regularizing terms are also common in engineering models, cf. [1,9,10,13,14,23]. In the quasistatic setting we assume further that ϕ(t) minimizes the energy E(t, ·, P(t), p(t)) subject to ϕ(t, x) = gDir (t, x) for x ∈ Dir ,

(1)

which provides the usual elastic equilibrium equation div σ = fvol in and σ · ν = ftract on the Neumann part of the boundary ∂, where σ = ∂F W is the first Piola– Kirchhoff stress tensor. The evolution P and p is governed by the plastic flow rule which will be assumed ˙ p) to be formulated by a dissipation potential R(x, P, p, P, ˙ such that W (· · · ) − div ∂ W (· · · ) ∂ P ∇P ˙ p) R(x, P, p, P, ˙ + . (2) 0 ∈ ∂(sub ˙ p) P, ˙ ∂p W (· · · ) − div ∂∇p W (· · · ) It is possible to supplement E by a surface integral involving the plastic variables: (x, P(x), p(x), ν(x)) dx with ν being the normal vector, ∂

where : ∂×SL(d)×Rm ×Sd−1 → R is a nonnegative Caratheodory function. This term could be used to account for surface effects due to plasticity (i.e., accumulation of dislocation). The boundary conditions associated with (2) are ∂∇P W (· · · )ν + ∂P = 0,

∂∇p W (· · · )ν + ∂p = 0.

where ν is the outer normal vector. To simplify the presentation we omit this term. In (2) R(x, P, p, ·, ·) is convex on the tangent space and ∂(sub R denotes the ˙ p) P, ˙ corresponding subdifferential. This flow rule is rate independent if R(x, P, p, ·, ·) is ˙ p)) ˙ p). positively homogeneous of degree 1, i.e., R(x, P, p, λ(P, ˙ = λR(x, P, p, P, ˙ By the proper choice of R we will guarantee that this flow rule contains the essential kinematic relation between the plastic tensor and the slip strains, namely m P˙ = where Sα = mα ⊗ nα , α = 1, . . . , m, p˙α Sα P, (3) α=1

Existence for Finite-Strain Crystal Plasticity

173

are the the slip systems. Here nα ∈ Rd is the (unit) normal vector of the slip system Sα and mα ∈ Rm is the slip direction satisfying mα · nα = 0. A crucial step for deriving an existence theory is replacing the dissipation potential R by the associated dissipation distance D, see (5). The dissipation functional D(x, P0 (x), p0 (x), P1 (x), p1 (x)) dx D(P0 , p0 , P1 , p1 ) =

measures the minimal amount of energy dissipated when going from the state (P0 , p0 ) to (P1 , p1 ). An important fact is that D satisfies the (unsymmetric) triangle inequality. A major difficulty arises from the fact that D has only logarithmic growth because of plastic invariance, see (6). As a consequence D cannot be coercive on linear function spaces. The energetic approach for rate-independent systems (RIS) (Q, E, D) is exactly suited for this situation. However, we still will have extra work to establish coercivity of the energy, see Section 3.1. The energetic formulation provides a weak form of the system (1) and (2). For this we choose a state space Q for q = (ϕ, P, p) by identifying suitable weakly closed subsets of Sobolev spaces over . A mapping q = (ϕ, P, p) : [0, T ] → Q is called energetic solution for the RIS (Q, E, D), if for all t ∈ [0, T ] the stability condition (S) and the energy balance (E) hold: (S) E(t, q(t)) ≤ E(t, q) + D(q(t), q) for all q ∈ Q, t (E) E(t, q(t)) + DissD(q; [0, t]) = E(0, q(0)) + ∂s E(s, q(s)) ds.

(4)

0

Here DissD(q; [r, s]) = sup N 1 D(P(τj −1 ), p(τj −1 ), P(τj ), p(τj )), where the supremum is taken over all partitions of [r, s]. In the case of external loadings and ˙ time-independent boundary conditions we have ∂t E(t, q) = −(t), ϕ. However, if gDir depends on time the power of the displacement loadings is more difficult to express in a mathematically correct way, since the stresses on the boundary are not well defined. Following [11, 18] we write the unknown displacement as a composition ϕ(t, x) = gDir (t, y(t, x)), where y : → Rd is the new unknown satisfying y(t, x) = x for x ∈ Dir . With q = (y, z) we write E(t, q) = E(t, gDir (t)◦y, z) and find that ∂t E(t, q) can be expressed in terms of the Kirchhoff stress tensor and a convected derivative. In Section 2 we follow [20] for discussing the mechanical modeling of elastoplasticity and for explaining why the concept of energetic solutions can be seen as a weak version of the classical plasticity formulation. The major advantage of (S) and (E) is that it avoids derivatives and is based solely on the functionals E and D, which need not be smooth or even continuous. In Section 2.2 we formulate precise assumptions on W, D, and gDir that allow us to construct solutions in suitable Sobolev spaces. The main result is Theorem 1 which states the global existence of energetic solutions for single-crystal plasticity. For the cases of kinematical hardening and isotropic hardening we refer to [18].

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Rigorous mathematical work in finite-strain plasticity is still rare. Relaxation methods for one incremental step are studied in some detail, because the strong methods from the calculus of variations apply, see e.g. [4, 7, 26]. Local existence results for plastic evolutions is contained in [24], where the multiplicative split was used together with a small strain assumption and a grain boundary relaxation, which replaces our regularization. In [8] a relaxed singe-slip system was investigated. For a comparison of numerical approaches to finite-strain plasticity we refer to [6,12,25].

2 Modeling Assumptions and Results We first provide an exact description of the mechanical model in terms of the constitutive functions, namely the stored-energy density W and the dissipation potential R, which follows the concept of generalized standard materials, see e.g. [15]. Here we discuss the main symmetries and the basic kinematic relations. Next we discuss the assumptions that are necessary to develop a mathematical existence theory. Finally, this section closes by stating the main existence result and the underlying abstract theory developed in [18].

2.1 Mechanical Modeling We recall the multiplicative decomposition ∇ϕ = F = Fel Fpl , where the plastic tensor P = Fpl ∈ SL(d) maps the material space crystallographic lattice onto itself. The slip strains pj are combined into a vector p ∈ [0, ∞[m . To simplify notations we let z = (P, p) ∈ Z = SL(d)× [0, ∞[m and use A as a place holder for ∇z = (∇P, ∇p). (x, F, P, p, A) and the dissipation potential The stored-energy density W = W ˙ R = R(x, P, p, P, p) ˙ have to satisfy the following symmetry properties: (Sy1) Objectivity (frame indifference) (x, QF, P, p, A) = W (x, F, P, p, A) for all Q ∈ SO(d); W (Sy2) Plastic indifference (x, F (x, F, P, p, A) W P, P P, p, A) = W ˙ p) P P, p, P, R(x, P, p, P˙ P, p) ˙ = R(x, ˙

for all P ∈ SL(d);

(Sy3) Material symmetry (x, F, PS, πS p, S A) = W (x, F, P, p, A) W ˙ πS p) ˙ p) PS, πS p, PS, P, p, P, R(x, ˙ = R(x, ˙

for all S ∈ S ⊂ O(d).


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In (Sy3) the group S is the material-symmetry group which acts on the plastic strain by a permutation πQ : p → (pπQ (1) , . . . , pπQ (m) ), see [20, Sect. 3.4.4], and

S (∇P, p) = ∇(PS, πS p). In the sequel we will drop the explicit dependence on x and R depend for notational simplicity. However, the whole theory is still valid if W on x ∈ , which would be the case for polycrystals. and R can be written in a reduced form via A consequence of (Sy2) is that W ˙ p) ˙ −1 , p), p, P, (F, P, p, A) = W (F P−1 , p, A) and R(P, ˙ = R(p, PP ˙ W ˙ −1 ∈ sl(d) = T1 SL(d) = { ξ ∈ Rd×d | tr ξ = 0 }. We now define the where ξ = PP dissipation distance D(·, ·) on Z×Z via

1 1 R(z(s), z˙ (s)) ds z ∈ C ([0, 1], Z), z(0) = z0 , z(1) = z1 . D(z0 , z1 ) = inf 0

(5) Thus, D has the dimension of an energy density and measures the amount of energy per volume that has to be spent to transform a material point from the internal state z0 into z1 . The plastic indifference (Sy2) implies that the dissipation distance D is right-invariant, namely D(P1 , p1 , P2 , p2 ) = D(1, 0, P2 P−1 1 , p2 −p1 ) for all P1 , P2 , p1 , p2 .

(6)

We specify R further in such a way that the slip kinematics (3) is enforced automatically by R(p, ξ, ν) < ∞, namely

m

m for ξ = m α=1 κα να α=1 να Sα and v ∈ [0, ∞[ , R(p, ξ, ν) = (7) ∞ otherwise, where the threshold parameters κα (cf. [13, 26]) are assumed to be bounded positive constants. Note that the slip strain behave monotonically and are not allowed to decrease. However, often Sα+m/2 = −Sα for α ≤ m/2, then P˙ may take any value. Since v = p˙ and P˙ = ξ P, the flow rule (2) implies the slip kinematics (3), because the subdifferential ∂(P, ˙ p) ˙ R is nonempty if and only if R is finite. We assume that the set of slip systems S = { Sα | α = 1, . . . , m } is large enough to generate the whole group SL(d). More precisely, a slip system Sα has to be considered as an element of sl(d) = T1 SL(d), such that Pα (τ ) = eτ Sα = 1+τ Sα is a simple shear. We say that SL(d) is generated by S, if each P ∈ SL(d) can be written in the form Pα1 (τ1 ) · · · PαN (τN ), where N ∈ N, αk ∈ {1, . . . , m}, and τk ∈ R. By the standard theory of Lie groups and their Lie algebras this is equivalent to saying that sl(d) is the smallest Lie algebra containing S (with respect to the standard Lie bracket [ξ1 , ξ2 ] = ξ1 ξ2 − ξ2 ξ1 ). Obviously, S generates SL(d), if the linear hull of S equals sl(d), and this is the case in many cases of crystal plasticity, see [7]. However, this is by far not necessary, for an example consider S = {e1 ⊗e2 , e2 ⊗e1 }, which generates SL(2), see [16]. def

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Subsequently we will not write down this condition on S, since it is not essential. If it is not satisfied, we just have to replace SL(d) by the smaller Lie group G ⊂ SL(d) that is generated by S. The whole theory will still hold for any such subgroup.

2.2 Precise Mathematical Assumptions For notational simplicity we restrict to the case of displacement boundary conditions that are independent of time and use volume and surface forces to drive the system. We refer to [11, 18] to the case of time-dependent boundary conditions. The domain ⊂ Rd is bounded and has a Lipschitz boundary. The Dirichlet part Dir of the boundary is assumed to have positive surface measure. For gDir we assume that it can be extended to all of Rd as follows: gDir ∈ C1 ([0, T ]×Rd ; Rd ),

∇gDir ∈ BC1 ([0, T ]×Rd , Lin(Rd ; Rd ))

and |∇gDir (t, x)−1 | ≤ C for all (t, x) ∈ [0, T ]×Rd ,

(8)

where “BC1 ” means C1 and boundedness. Thus, for each t ∈ [0, T ] the mapping gDir (t, ·) : Rd → Rd is a global diffeomorphism. We seek for ϕ(t, ·) in the form ϕ(t, x) = gDir (t, y(x)) with y ∈ Y = { y ∈ Y|y|Dir = id } and Y = W1,qY (; Rd ). def

We set q = (y, z) and E(t, y, P, p) = E(t, gDir (t, ·)◦y, P, p). Since no confudef sion can arise, we denote E again by E. The internal variable z = (P, p) ∈ Z = m SL(d)× [0, ∞[ lies in the space Z of internal states, which is chosen as Z = { (P, p) ∈ Z | (P(x), p(x)) ∈ Z and D(1, 0, P(x), p(x)) < ∞ a.e. in }, def

where Z = W1,r (; Rd×d ×Rm ) with r > d. The stored-energy functional E and the dissipation distance D take the forms def E(t, y, z) = W (∇gDir (t, y(x))∇y(x)P(x)−1 , z(x), ∇z(x)) dx, def D(z0 , z1 ) = D(z0 (x), z1 (x)) dx,

where D is defined in (5) via R in (7). The conditions on W are much more involved. In particular, they include coercivity and convexity assumptions to obtain def lower semicontinuity. To shorten notation we let L(d,m) = Rd×d×d ×Rm×d and use A for ∇z = (∇P, ∇p) ∈ L(d,m) . The function M : Rd×d → Rµd with 2

µd = ds=1 ds = 2d d −1 maps a matrix to all its minors (subdeterminants). The Kirchhoff stress tensor is defined via K(F, p, A) = ∂F W (F, p, A)F T . We impose the following:


177

there exists W : Rµd ×Rm ×L(d,m) → R∞ : (i)

W is lower semicontinuous,

(ii) W (F, p, A) = W(M(F ), p, A), (iii) W(x, ·, p, ·): Rµd ×L(d,m) → R∞ is convex; there exist c > 0, qY , r > d, qp > 1 such that W (F, p, A) ≥ c |F |qY + |p|qp + |A|r − 1/c.

(9a)

(9b)

there exist c0W , c1W , and α ∈ (0, 1] such that for |N| ≤ 1/(2d) |K(F, p, A)| ≤ c1W (W (F, p, A)+c0w ) |K((1+N)F, p, A)−K(F, p, A)| ≤

(9c)

c1W (W (F, p, A)+c0w )|N|α .

Thus, (9a) implies that the mapping F → W (x, F, z, A) is polyconvex, cf. [2]. Condition (9b) implies the necessary coercivity, which includes (self or cross) hardening via the lower bound c|p|qp . Note that we do not assume a coercivity in P. Condition (9c) will be used to control the power of the time-dependent Dirichlet boundary data.

2.3 Statement of the Result We now formulate our existence result, which will be proved in Section 3. Theorem 1. Let the spaces Q = Y×Z ⊂ Y×Z = Q and the functionals E and D be defined as above such that the conditions (8), (9), and (11) hold. Let q0 = (y0 , z0 ) ∈ Q ∩ (Y×Z) be a stable initial condition, i.e., E(0, q0 ) < ∞ and E(0, q0 ) ≤ E(0, q) + D(q0 , q) for all q ∈ Q. Then, for the elastoplasticity problem defined via the RIS (Q, E, D) there exists an energetic solution q : [0, T ] → Q with q(0) = q0 and q ∈ L∞ ([0, T ]; Y×Z). For similar results involving kinematic or isotropic hardening models in finite-strain plasticity, we refer to [18]. All these existence result are based on the abstract theory of energetic solutions for RIS on topological spaces developed in [11, 21]. We consider two reflexive and separable Banach spaces Y and Z and weakly closed subsets Y and Z, respectively. def The state space for the full system is then given by Q = Y×Z ⊂ Q = Y×Z, and the states are denoted by q = (y, z). The evolution is described in terms of the stored-energy functional E : [0, T ]×Q → R∞ and the dissipation distance D : Z×Z → [0, ∞]. The set in which E takes finite values is denoted by dom E = { (t, q) ∈ [0, T ]×Q | E(t, q) < ∞ }. def

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For the stored-energy functional E and the dissipation distance D : Z×Z → [0, ∞] we impose the abstract conditions (E1), (E2) and (D1), (D2), respectively. In Section 3 we then show that these conditions are consequences of (8), (9), and (11). Compactness of energy sublevels: for all t ∈ [0, T ] and E > 0 the sublevels { q ∈ Q | E(t, q) ≤ E } are bounded and weakly closed in Q.

(E1)

Uniform control of the power ∂t E: there exist c0E , c1E > 0 such that for all (t∗ , q) ∈ dom E : E(·, q) ∈ C1 ([0, T ]) and |∂t E(t, q)| ≤ c1E (c0E +E(t, q)) for all t.

(E2)

Extended quasi-distance: (i) ∀ z1 , z2 ∈ Z : D(z1 , z2 ) = 0 ⇐⇒ z1 = z2 , (ii) ∀ z1 , z2 , z3 ∈ Z : D(z1 , z3 ) ≤ D(z1 , z2 ) + D(z2 , z3 ).

(D1)

Weak lower semi-continuity: zk z, zk z =⇒ D(z, z) ≤ lim infk→∞ D(zk , zk ).

(D2)

To formulate the abstract existence result we need to impose additional conditions which provide a suitable compatibility between the two functionals E and D. For this we define the set of stable states at time t via S(t) = { q ∈ Q | E(t, q) < ∞, E(t, q) ≤ E(t, q) + D(q, q) for all q }. def

Moreover, we define the notion of a stable sequence (tk , qk )k∈N via supk∈N E(tk , qk ) < ∞ and qk ∈ S(tk ) for all k ∈ N. A function q : [0, T ] → Q is called an energetic solution of (Q, E, D), if t → ∂t E(t, q(t)) is integrable and if for all t ∈ [0, T ] we have global stability (S) and energy balance (E) in (4). Theorem 2. Let the RIS (Q, E, D) satisfy (E) and (D). Moreover, let the following compatibility condition hold: For all stable seq. (tj , qj )j ∈N with (tj , qj ) (t∗ , q∗ ): ∂t E(t∗ , qj ) → ∂t E(t∗ , q∗ ),

(C1)

q∗ ∈ S(t∗ ).

(C2)

Then, for each q0 ∈ S(0) there exists an energetic solution q : [0, T ] → Q for (Q, E, D) with q(0) = q0 and q : [0, T ] → Q measurable. The existence result for plasticity in Theorem 1 will be obtained as a special case of this abstract result.


179

3 Coercivity and Lower Semicontinuity In this section we show that the assumptions in Section 2.2 for the elastoplastic problem are sufficient to establish the abstract assumption (E) for the stored-energy functional E, (D) for the dissipation distance D, and the compatibility conditions (C). Having done this, the Existence Theorem 1 for the elastoplastic problem is a direct consequence of the abstract existence result in Theorem 2.

3.1 Stored Energy Potential To establish the coercivity of E we note that we always use the matrix norm |F | = (F :F )1/2 . In particular, we have |AB| ≤ |A| |B|, which implies def

|∇gDir ∇y P−1 | ≥ |∇y|/(|∇gDir| |P|) ≥ c|∇y|/|P|, where here and in the sequel c and C denote small and large positive constants that may vary from occurrence to occurrence. These constants only depend on the data and are independent of the states q. Integrating the last estimate we obtain, for all q ∈ Q, the estimate ∇gDir ∇y P−1 LYqY ≥ c∇yLYqY /PLY∞ ≥ c∇yLYqY e−CpL∞ ≥ c log ∇yLqY − CpL∞ − C, q

q

q

q

where we used (8) and (11c), which can be applied since (1, 0, P(x), p(x)) ∈ D by the definition of Z. The last estimate follows from the rough lower estimate eβ ≥ β. It is the missing coercivity in P that forces us to use such weak logarithmic estimates. Using the coercivity (9b) of W and the embedding W1,r () ⊂ C() we obtain q E(t, q) ≥ c log ∇yLqY − CpL∞ + cpLpqp + c(∇P, ∇p)rLr − C q ≥ c log ∇yLqY + c log PL∞ + cpLp∞ + c(∇P, ∇p)rLr − C, where we used (11c) once again. This proves coercivity, since qk Q → ∞ implies E(t, qk ) → ∞. The weak sequential lower semi-continuity of E(t, ·) follows similarly as in [18, Thm. 5.2]. In fact, the proof is even simpler, since the weak convergence qk q implies the uniform convergence of (Pk , pk ) → (P, p) in C0 (; Rd×d ×Rm ). Thus, the convexity semi-conditions (9a) for W (via W) allow us to use the standard techniques developed in [2]. Thus, we have established the following result, which implies that the abstract assumption (E1) holds. Lemma 1. Assume (9) and (11c) hold. Then the functional E(t, ·) restricted to Y×Z is weakly lower semicontinuous and coercive.

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Finally, we investigate the differentiability of E(t, q) with respect to time. For this we recall the definition of the Kirchhoff stress tensor from (9), namely K(F, p, A) = ∂F W (F, p, A)F T ∈ sl(d) = T1 SL(d). For q = (y, P, p) ∈ Q with E(0, q) < ∞ we introduce the abbreviation Kq (x, F ) = ∂F W (x, F P(x)−1 , P(x), p(x), ∇P(x), ∇p(x))(F P(x)−1 )T . def

The following result was established in [18] (by combining Propositions 4.3 and 4.4 with Theorem 5.3 there) and using the property (12) established below. Lemma 2 (Power of the boundary conditions). If assumption (8) and (9) hold, then E satisfies (E2) and (C1). In particular, there exist constants c0E ∈ R and c1E > 0 and a modulus of continuity ω such that the following holds: For (t, q) ∈ dom E we have E(·, q) ∈ C1 ([0, T ]) with Kq (x, ∇gDir (t, y(x))∇y(x)):V (t, y(x)) dx, (10a) ∂t E(t, q) =

−1 ∂ ∇gDir (t, y), where V (t, y) = ∇gDir (t, y) ∂t |∂t E(t, q)| ≤ c1E E(t, q)+c0E , and |∂t E(t1 , q)−∂t E(t2 , q)| ≤ ω(|t2 −t1 |) E(t1 , q)+c0E .

(10b) (10c)

The importance of formula (10a) is that Kq is in L1 () for all q ∈ dom E(t, ·), whereas V lies in C() because of the smoothness of the given boundary data gDir .

3.2 Dissipation Potential D The first result provides some elementary properties for the dissipation distance D def defined via (7) and (5). We let D = { (P0 , p0 , P1 , p1 ) | D(P0 , p0 , P1 , p1 ) < ∞ }, which is a closed subset of (SL(d)×Rm )2 , and η = (1, 1, . . . , 1) ∈ Rm . Lemma 3. Assume that R has the form (7). Then D defined in (5) satisfies D: Z×Z → [0, ∞] is lower semicontinuous and D: D → [0, ∞[ is continuous;

(11a)

D(z1 , z2 ) = 0 ⇐⇒ z1 = z2 ;

(11b)

D(z1 , z3 ) ≤ D(z1 , z2 ) + D(z2 , z3 );

there exist constants c1 , c2 > 0 such that (P0 , p0 , P1 , p1 ) ∈ D =⇒ |P1 −P0 | ≤ c1 ec2 |p1 −p0 | −1 ;

(11c)

for each ε > 0 there exists Pε ∈ SL(d) and ρε > 0 such that (1, 0, P, εη) ∈ D for all P ∈ SL(d) with |P−Pε | ≤ ρε .

(11d)


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While the proof of the properties (11a)–(11c) is standard, see [18, 19], the property (11d) is not so obvious. To show this, we recall the implicit assumption that {S1 , . . . , Sm } generates SL(d). Moreover, we let Aε = { ν ∈ C1 ([0, 1]; Rm | ν˙ α ≥ 0, ν(1)−ν(0) = εη }, −1 ∈ A }, ˙ Pε = { P(1) | P ∈ C1 ([0, 1]; SL(d)), P(0) = 1, P(·)P(·) ε

m

m = α=1 Sα , κ∗ = α=1 κα , and Nε = exp(ε), and obtain Nε ∈ Pε and D(1, 0, P, εη) ≤ εκ∗ < ∞ for all P ∈ Pε . Now the control theory on non-commutative Lie groups shows that Nε is in fact an interior point of the reachable set Pε . Thus we may set Pε = Nε and have found ρε > 0, such that (11d) holds. Condition (11a) implies that D is well defined and the positivity (D1)(i) follows from (11b). Integrating the pointwise triangle inequality (11b) we see that (D1)(ii) holds. Using again that zk z in Z implies zk → z in C0 () and that D is nonnegative and lower semicontinuous in both z-variables, the classical lower semicontinuity theory implies the lower semicontinuity of D, namely (D2).

3.3 Compatibility Conditions (C2) To apply Theorem 2 it remains to establish the compatibility condition (C2), which states that weak limits of stable sequences are stable again. We establish this by constructing so-called joint recovery sequences, cf. [22]. Assume that a stable sequence (tj , qj )j ∈N with tj → t∗ and qj q∗ is given. We have to show q∗ ∈ S(t∗ ). For any given test state q we have to show E(t∗ , q∗ ) ≤ E(t∗ , q) + D(z∗ , z). If D(z∗ , z) = ∞ the estimate holds and nothing needs to be shown. z) < ∞ we establish this condition by construction a joint For the case D(z∗ , recovery sequence ( qj )j ∈N that satisfies (a) E(tj , qj ) → E(t∗ , q),

(b) D(zj , zj ) → D(z∗ , z).

(12)

From these conditions the stability of q∗ follows using the stability of qj , i.e., E(tj , qj ) ≤ E(tj , qj ) + D(zj , zj ). Letting j → ∞ the left-hand side can be estimated by weak lower semi-continuity and the right-hand side converges to the desired limit. The problem in deriving (12b) is the lack of continuity of the integrand D of Pj , p j ) carefully. For this we use property D. Hence, we have to choose zj = ( (11d) where we additionally observe that (Pε , ρε ) must satisfy Pε → 1 and ρε → 0 because of (11c). We let δj = zj −zL∞ and choose (εj )j such that δj < ρεj → 0. We set

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j = p+εj η, Pj = Nεj +εj η, Pj = Nεj P, p P, p j = p

(13)

and find by the triangle inequality D(Pj , pj , Pj , p j ) ≤ D(Pj , pj , Pj , p j ) + D( Pj , p j , Pj , p j ) −1 = D(1, 0, Nεj PP , p−pj +εj η) dx + D(P, p, P, p ),

j

where we have used plastic invariance for the second term. By construction the integrand of the first term can be estimated by κ∗ εj and we obtain lim supj →∞ D(zj , zj ) ≤ D(z, z). Since the opposite estimate follows by lower semi-continuity we have established (12b). The convergence (12a) follows easily y and applying Lebesgue’s dominated convergence theorem. by setting yj =

Acknowledgements The author is grateful to Khanh Chau Le for stimulating discussions. The research was partially supported by the DFG through FOR 797 Analysis and Computation of Microstructures in Finite Plasticity under Mie 459/5-1.

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Error Bounds for Space-Time Discretizations of a 3D Model for Shape-Memory Materials Alexander Mielke, Laetitia Paoli, Adrien Petrov and Ulisse Stefanelli

Abstract This paper deals with error estimates for space-time discretizations of a three-dimensional model for isothermal stress-induced transformations in shapememory materials. After recalling existence and uniqueness results, a fully-discrete approximation is presented and an explicit space-time convergence rate of order hα/2 + τ 1/2 for some α ∈ (0, 1] is derived.

1 Introduction This note is concerned with error control for fully-discrete approximations in the context of solids undergoing martensitic transformations. More specifically, we address the description of the isothermal 3D quasistatic evolution of shape-memory alloys (SMAs). The latter are metallic alloys showing some surprising thermomechanical behavior, namely, strongly deformed specimens regain their original shape after a thermal cycle (shape-memory effect). Moreover, within some specific (suitably high) temperature range, SMAs are superelastic, meaning that they fully recover comparably large deformations. These features are not present in most materials traditionally used in Engineering and, thus, are at the basis of innovative and commercially valuable applications. Nowadays, SMAs are successfully used

Alexander Mielke · Adrien Petrov Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany; E-mail: {mielke, petrov}@wias-berlin.de Laetitia Paoli LaMUSE (Laboratoire de Mathématiques de l’Université de Saint-Etienne), 23 rue Paul Michelon, 42023 Saint-Etienne Cedex 02, France; E-mail: [email protected] Ulisse Stefanelli Istituto di Matematica Applicata e Tecnologie Informatiche – CNR, via Ferrata 1, 27100 Pavia, Italy; E-mail: [email protected] K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_14, © Springer Science+Business Media B.V. 2010

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in many applications among which biomedical devices (vascular stents, archwires, endo-guidewires) and MEMS (actuators, valves, mini-grippers and positioners). We will focus on a phenomenological, small-deformation model for polycrystalline materials describing both the shape memory and the superelastic effect. In the present isothermal reduction shape-memory effect is actually not reproduced, and we refer to [17,22] for models driven by temperature. The model has been originally advanced by Souza et al. [25] and then combined with finite elements by Auricchio and coworkers [7, 8]. The state of the material is determined by its displacement u : → Rd with respect to the reference configuration ⊂ Rd (d = 2, 3) and by a tensorial internal variable z : → Rd×d dev (deviatoric d-tensors) which represents the inelastic part of the deformation ε, namely z = ε − C−1 σ where C is the elasticity tensor and σ is the stress. In fact, z corresponds the oriented effective strain of detwinned martensites (product phase) with respect to twinned martensites and austenite (parent phase). Our interest in this model is motivated by its ability to describe (at least to a qualitative extent) the thermomechanical behavior of SMAs by means of a small number of easily fitted material parameters (7 material constants in 3D). Another interesting feature of the Souza–Auricchio model is that it turns out to be quite naturally posed in the frame of the variational theory of rate-independent systems [15]. This feature was indeed exploited in [4], where wellposedness issues for continuous problems (constitutive relation and quasistatic evolution) as well as the convergence of discretizations and regularizations has been discussed. In particular some fullydiscrete approximations (uτ,h , zτ,h ) obtained by implicit Euler discretization in time (τ is the fineness of the time-partition) and piecewise linear finite elements in space (h is the mesh size) are proved in [4, theorem 7.1] to converge to the unique solution of the time-continuous quasistatic evolution problem. The focus of this note is to provide explicit convergence rates in space and time for these fully-discrete approximations. In particular, we check that ∃α ∈ (0, 1] : u−uτ,h H1 (;Rd ) +z−zτ,h H1 (;Rd×d ) ≤ O(hα/2 +τ 1/2 ). In the special case of a convex polyhedron and homogeneous Dirichlet conditions for the displacement the parameter α can be chosen to be α = 1. A more elaborate and general theory will be developed in [21]. The above quantitative control is, to our knowledge, the first result in this direction in the context of the mechanics of solid-solid phase transformations. Note that our error estimate is derived under natural regularity requirements. Namely, it depends solely on data and no extra-smoothness of the solution (u, z) is assumed. This specific feature sets this result apart from the existing literature on error control for time- or space-time discretizations of variational evolution problems (inequalities) arising in elasto-plasticity (see [1, 12]). Related numerical approaches to rate-independent models for SMA are given in [14, 19, 22]. However, there the method of -convergence is employed, which guarantees the convergence of subsequences only and provides no quantitative error estimates.

Space-Time Discretizations for Shape-Memory Materials

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2 The Mechanical Model We briefly review the model, the interested reader being referred to the original papers [5–7, 25] for additional details. Let the reference configuration be a nonempty, bounded, and connected polyhedron in Rd (d = 2, 3). We assume the boundary ∂ to be partitioned in two disjoint open sets Neu and Dir with ∂Neu = ∂Dir (in ∂) such that Dir has positive surface measure. Adopting the framework of Generalized Standard Materials (see e.g., [16] and within the small-strain regime), we additively decompose the linearized deformation ε = ε(u) = 12 (∇u+∇uT ), where u is the displacement, into the elastic part εel ∈ d×d Rd×d sym and the inelastic (or transformation) part z ∈ Rdev . The free energy density of the material depends on ε only via εel = ε−z: W (ε, z) =

ν 1 C(ε−z):(ε−z) + H (z) + |∇z|2 . 2 2

(1)

Here, C is a positive definite elasticity tensor (for isotropic materials, for simplicity), ν > 0 is expected to measure some nonlocal interaction effect for the internal variable z, and ∇z stands for the usual gradient with respect to spatial variables. Indeed, gradients of inelastic strains have already been considered in the frame of shapememory materials by Frémond [9] and the reader is also referred to [2, 10, 18] for examples and discussions on nonlocal energy contributions. Finally, the hardening function H : Rd×d dev → R is given by (|z|−c3 )4+ c2 H (z) = c1 ρ 2 +|z|2 + |z|2 + 2 ρ(1+|z|2 )

(2)

where the user-defined parameter ρ > 0 is small and c1 , c2 , and c3 are given and represent a superelastic-transformation stress-activation level, a hardening modulus with respect to the internal variable z, and the maximum modulus of transformation strain that can be obtained by alignment (detwinning) of the martensitic variants, respectively. One has to mention that this specific form of W can be much generalized and is here fixed for definiteness only. In particular, W is a ρ-approximation of the original choice of [25] which in turn corresponds to the limit (ρ, ν) → (0, 0) (see [4]). Note that the current choice of H is just one of the many possible and it is motivated as a smoothing of the hardening function HSo (z) = c1 |z| + c22 |z|2 + χ(z) originally introduced by Souza et al. [25]. There χ : Rd×d dev → [0, +∞] denotes the indicator function of the ball {z ∈ Rd×d : |z| ≤ c }. 3 The proposed model in this dev note is a macroscopic phenomenological model for shape-memory polycrystalline materials undergoing phase transformations driven by stress. The motivation as well as detailed explanations for the form of HSo can be found in [6]. The constitutive relations are given in the form

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σ = ∂W /∂ε = C(ε−z),

(3a)

ξ = −∂W /∂z = C(ε−z) − Dz H (z) + ν z,

(3b)

where ξ denotes the thermodynamic force associated with z. The evolution of the material will be described by the following classical relations: ξ ∈ R∂|˙z|,

(3c)

div σ + f = 0 in , σ n = T in Neu , u = 0 in Dir .

(3d)

The latter equation gives the equilibrium equations, where f and T are a given body force and a surface tension, respectively. The flow rule (3c) corresponds to the classical generalized normality assumption (R > 0 is the fixed transformation radius), and the symbol ∂ stands for the subdifferential in the sense of convex analysis, viz., ξ ∈ R∂|˙z|

if and only if

− ξ :(w−˙z) + R|w| − R|˙z| ≥ 0 for all w ∈ Rd×d dev .

3 The Variational Formulation For the admissible displacements u and the internal states z we choose the natural function spaces def def def U = u ∈ H1 (; Rd ) u = 0 on Dir , Z = H1 (; Rd×d dev ), Q = U × Z. Later we will also need the larger space X = L2 (, Rd × Rd×d dev ). The symbol ·, · denotes the duality pairing between Q and Q. For the loadings f and T in (3d) we require that defined via def

(t), q = f (t) · u dx + T (t) · u dx, def

Neu

satisfies ∈ C1 ([0, T ]; X ). Furthermore, we choose an initial datum q0 = (u0 , z0 ) ∈ S(0) where the set S(t) of stable states at time t ∈ [0, T ] is defined as the set of all q = (u, z) ∈ Q satisfying the condition W (u, z) dx − (t), q ≤ W ( u, z) dx − (t), q + R| z − z| dx (4)

for all q = ( u, z) ∈ Q. The variational formulation of (3) consists in finding q : [0, T ] → Q such that q(0) = q0 , C(ε(u)−z):ε(v) dx = , v

(5a) for all v ∈ U,

(5b)


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(C(z−ε(u)) + Dz H (z)):(w−˙z) + ν∇z:∇(w−˙z) dx

+

R|w| dx −

R|˙z| dx ≥ 0

for all w ∈ Z,

(5c)

almost everywhere in time. The following wellposedness theorem is proved in [4], see also [16, sec. 5.3]. Theorem 1 (Wellposedness). For each q0 ∈ S(0) problem (5) admits a unique solution q : [0, T ] → Q, which even lies in CLip ([0, T ]; Q). Using the general theory of rate-independent systems (cf. [15]) it is still possible to show the existence of energetic solutions for cases where the elasticity tensor C is z-dependent, as along as C depends continuously on z and lies between suitable bounds. However, the uniqueness of the solutions and the Lipschitz continuity in time strongly depend on the fact that C is independent of z.

4 Space-Time Discretization: Main Result Let us now introduce our space-time discretization of (5). To this aim, we choose a sequence ( τ )τ >0 of partitions { 0 = tτ0 < tτ1 < · · · < tτkτ = T } of the time interval [0, T ] with max{ tτk − tτk−1 : k = 1, . . . , kτ } ≤ τ and a sequence (Qh )h>0 of finitedimensional spaces exhausting Q. In particular, assume to be given a regular triangulation {Tk } of [24] and choose Uh and Zh to be the subspaces of continuous, def piecewise polynomials of fixed degree m ≥ 1 on {Tk }. Finally, let Qh = Uh × Zh . As for the initial value, we shall ask for q0,h ∈ Sh (0) where the set of approximate stable states is defined as in (4) by replacing Q by Qh . i i Our space-time discretization of (5) consists in finding qτ,h = (uiτ,h , zτ,h ) ∈ Qh for i = 0, 1, . . . , kτ such that 0 qτ,h = q0,h , i C(ε(uiτ,h )−zτ,h ):ε(v h ) dx = (tτi ), v h

(6a) for all v h ∈ Uh ,

(6b)

i i i i i −ε(uiτ,h ))+Dz H (zτ,h ):(wh −δzτ,h ) + ν∇zτ,h :∇(wh −δzτ,h ) dx C(zτ,h

+

R|wh | dx −

i R|δzτ,h | dx ≥ 0

for all wh ∈ Zh

for i = 1, . . . , kτ . Here we used the short-hand notation i = δzτ,h def

1 tτi −tτi−1

i−1 i (zτ,h −zτ,h ),

(6c)

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i later on. Because of convexity the conditions (6b)– which will also be used for qτ,h (6c) are equivalent to solving incremental minimization problems, see [15]. We shall denote by qτ,h = (uτ,h , zτ,h ) : [0, T ] → Qh ⊂ Q the piecewiseconstant-in-time interpolants of the above fully-discrete solutions. In particular, we have k−1 qτ,h (t) = qτ,h def

for tτk−1 ≤ t < tτk , k = 1, . . . , kτ

kτ and qτ,h (T ) = qτ,h . def

The above scheme has been proved to be wellposed and convergent in [4] (but see also [20] and the detailed analysis in [21, appendix]). Theorem 2 (Wellposedness, stability, and convergence). For all q0,h ∈ Sh (0), i solving (6). Moreover, there exists C there exists a unique qτ,h stab > 0 such that i i qτ,h Q + δqτ,h ≤ Cstab Q

for all i = 1, . . . , kτ and all h > 0.

If additionally q0,h → q0 in Q, then maxt ∈[0,T ] qτ,h (t)−q(t)Q converges to 0 as τ + h → 0, where q : [0, T ] → Q is the unique solution of Theorem 1. The purpose of this work is to establish a quantitative convergence result giving explicit convergence rates with respect to the mesh size h of the spatial discretization and the timestep τ . Our main result reads as follows: Theorem 3 (Space-time convergence rates). There exist α ∈ (0, 1] and Cerr > 0 (all independent from τ and h) such that the following holds. For each q0 ∈ S(0) there exists a sequence q0,h ∈ Sh (0) of approximating initial data, such that max q(t)−qτ,h (t)Q ≤ Cerr (hα/2 +τ 1/2 )

t ∈[0,T ]

where q and qτ,h are the unique solutions of (5) and (6), respectively. In case is convex and Neu = ∅, one can choose α = 1. A proof of this error estimate has been obtained in [21] in the more general setting of an abstract evolutionary inequality. In the present concrete situation of SMAs the error-control argument is somehow simpler. Hence we are able to provide a full proof below.

5 Proof of the Error Estimate Define the functionals E : [0, T ] × Q → R, H : Q → R, and : Q → [0, ∞] via def W (u, z) dx − (t), q , E (t, q) =

c2 def H(q) = H (z)− |z|2 dx, 2

(q) = def

R|z| dx.


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In particular, note that there exists C > 0 such that, for all q ∈ Q, we have that def (q) ≤ C qX . Let A ∈ Lin(Q, Q ) be defined by Aq = Dq E (t, q)−Dq H(q)+ (t), that is, for all (v, w) ∈ Q,

def

A(u, z), (v, w) = C(ε(u)−z):(ε(v)−w) + c2 z:w + ν∇z:∇w dx.

Note that A is symmetric and coercive, namely there exists κ > 0 such that

Aq, q ≥ κq2Q for all q ∈ Q. Finally, let Ph : Q → Qh be the Galerkin projector via A, which is defined such that Ph q is the unique solution of

APh q, ph = Aq, ph

for all ph ∈ Qh .

(7)

The Galerkin projectors Ph are uniformly bounded with respect to h and commute with A, i.e., P∗h A = APh . The next lemma provides a useful approximation property of the Galerkin projectors. Note that this lemma crucially relies on the H1+s -regularity of the associated linearized stationary problem for (5). Let I be the identity on Q. Lemma 1 (Approximation property). There exist α ∈ (0, 1] and CP > 0 such that (Ph −I)qX ≤ CP hα qQ

for all h > 0 and all q ∈ Q.

(8)

If is convex and Neu = ∅ then α can be chosen as α = 1. Proof. Within this proof, the symbol C stands for a generic positive constant, possibly depending on data only. Let us start by recalling that, in the present setting, given f ∈ L2 (; Rd ), the unique solution u ∈ U of the boundary value problem of linearized elastostatics Cε(v):ε(u) dx = f ·v dx for all v ∈ U

belongs to H1+s (; Rd ) for some s ∈ (0, 1] and s = 1 for convex and Neu = ∅ [11, Section 4.6, p. 148]. At the same time, given g ∈ L2 (; Rd×d dev ), the unique solution z ∈ Z of the elliptic system

Cw:z + ν∇w:∇z + c2 w:z dx = g:w dx for all w ∈ Z

is such that z ∈ H1+r (; Rd×d dev ) for some r ∈ (0, 1] with r = 1 if is convex [11, corollary 2.6.7, p. 79]. Let now η = (f , g) ∈ X be given and ϕ = (u, z) ∈ Q be the unique solution of A∗ ϕ = η. By the very definition of A we get that

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Cε(v):ε(u) dx =

(f −div(Cz))·v dx

(Cw:z+ν∇w:∇z + c2 w:z) dx =

for all v ∈ U,

(g+Cε(u)):w dx

for all w ∈ Z.

Owing to the regularity theory we have that α = min{s, r} ∈ (0, 1] is such that def

ϕH1+α (;Rd ×Rd×d ) ≤ C(f −div(Cz), g+Cε(u))X ≤ C(ηX +ϕQ ) dev

where α = 1 for convex and Neu = ∅. In particular, as clearly ϕQ ≤ CηX , we have proved the regularity ϕH1+α (;Rd ×Rd×d ) ≤ CηX .

(9)

dev

Next, we exploit the classical duality technique by Aubin and Nitsche [3, 23]. Assume to be given a (linear) projector h : Q → Qh fulfilling for all σ ∈ (0, 1] ∃C > 0 for all ϕ∈Q : ϕ −h ϕ Q ≤ Chσ ϕ H1+σ (;Rd ×Rd×d ) .

(10)

dev

The latter can be realized, for instance, by taking L2 -projections and the interpolation error control of (10) follows from [13, lemma 5.6]. Let q ∈ Q be fixed and define ϕ ∈ Q as the unique solution of A∗ ϕ = (Ph −I)q ∈ X . We have that, for all ϕh ∈ Qh ,

(7) (Ph −I)q2X = A∗ ϕ, (Ph −I)q = A(Ph −I)q, ϕ = A(Ph −I)q, ϕ−ϕh ≤ ALin(Q,Q ) (Ph −I)qQ ϕ−ϕh Q ≤ CqQ ϕ−ϕh Q . By choosing ϕh = h ϕ we get that def

(Ph −I)q2X ≤ CqQ ϕ−h ϕQ (10)

(9)

≤ CqQ hα ϕH1+α (;Rd ×Rd×d ) ≤ CqQ hα (Ph −I)qX dev

and the assertion follows.

The core of the proof of Theorem 3 is contained in the following proposition. Proposition 1 (Key estimate). There exist α ∈ (0, 1] and Ckey > 0 independent of τ and h such that max q(t)−qτ,h (t)Q ≤ Ckey q0 −q0,h Q +hα/2 +τ 1/2 . t ∈[0,T ]

Moreover, if is convex and Neu = ∅ then α can be chosen as α = 1.


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Proof. We clearly have that, for all t ∈ [0, T ], qτ,h (t)−q(t)Q ≤ qτ,h (t)−qh (t)Q + qh (t)−q(t)Q ,

(11)

where qh : [0, T ] → Qh is a limit of a subsequence of qτ,h as τ tends to 0. The first term in the above right-hand side can be estimated by using the same ideas of [20, prop. 7.2, theorem 7.3]. Namely, there exists C1 > 0 such that qτ,h (t)−qh (t)Q ≤ C1 τ 1/2 .

(12)

We estimate now the second term on the right hand side of (11). Since q and qh solve (Q, E , , q0 ) and (Qh , E , , q0,h ), respectively, we have

Dq E (t, qh (t)), vh −q˙h (t) + (vh ) − (q˙h (t)) ≥ 0

Dq E (t, q(t)), v−q(t) ˙ + (v) − (q(t)) ˙ ≥0

for all vh ∈ Qh ,

(13)

for all v ∈ Q,

(14)

which hold a.e. in [0, T ]. Choosing v = q˙h (t) in (14) and adding it to (13) we obtain

Dq E (t, qh (t)), vh −q˙h (t) + Dq E (t, q(t)), q˙h (t)−q(t) ˙ + (vh ) − (q(t)) ˙ ≥ 0, for all vh ∈ Qh . Using the triangle inequality this implies

Dq E (t, qh (t))−Dq E (t, q(t)), q˙ h (t)−q(t) ˙ ≤ Dq E (t, qh (t)), vh −q(t) ˙ + (vh −q(t)) ˙

for all vh ∈ Qh .

(15)

Let us now evaluate the right-hand side of (15) by computing, for all vh ∈ Qh ,

Dq E (t, qh (t)), vh −q(t) ˙ + (vh −q(t)) ˙ ≤ Aqh (t), vh −q(t) ˙ + Dq H(qh (t))X + (t)X + C vh −q(t) ˙ X. ˙ we find C2 > 0 such that Using Dq H ∈ CLip(Q, X ) and letting vh = Ph q(t),

Aqh (t)+Dq H(qh (t))−(t), vh −q(t) ˙ + (vh −q(t)) ˙ ≤ Aqh (t), (Ph −I)q(t) ˙ + C2 1+qh (t)Q +q(t)Q (Ph −I)q(t) ˙ X. Theorems 1 and 2 give qh (t)Q ≤ Cstab , q(t)Q ≤ Cstab and q(t) ˙ Q ≤ CLip . def Hence, using (7)–(8) and setting C3 = 0 + C2 CP (1+2Cstab) CLip we infer from (15) that ˙ ≤ C3 hα . (16)

Dq E (t, qh (t))−Dq E (t, q(t)), q˙h (t)−q(t) Define γ (t) = Dq E (t, qh (t))−Dq E (t, q(t)), qh (t)−q(t) ≥ κqh (t)−q(t)2Q where the lower bound stems from the coercivity of A and the convexity of H. We have def

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γ˙ (t) = 2 Dq E (t, qh )−Dq E (t, q), q˙h −q ˙ + ∂t Dq E (t, qh )−∂t Dq E (t, q), qh−q + Dq E (t, q)−Dq E (t, qh )+D2q E (t, qh )[qh −q], q˙h ˙ + Dq E (t, qh )−Dq E (t, q)+D2q E (t, q)[q−qh ], q. Exploiting Dq H ∈ C1,Lip(Q, Q ) and estimate (16) provides C4 > 0 such that ˙ Q +q˙h Q qh −q2Q . γ˙ ≤ 2C3 hα + C4 0+q

(17)

Setting C5 = max(2C3 , C4(0+2CLip )), we deduce from the definition of γ and (17) that we have γ˙ (t) ≤ C5 hα +γ (t)/κ . Hence Gronwall’s lemma yields γ (t) ≤ C t /κ α e 5 −1 κh + eC5 t /κ γ (0). As we readily infer that γ (0) ≤ C6 q0,h −q0 2Q with def def C6 = ALin(Q,Q ) + CH for CH = Dq HLip , we have obtained that def

qh (t)−q(t)2Q ≤ eC5 t /κ −1 hα + C6 eC5 t /κ q0,h −q0 2Q /κ.

(18)

Carrying (12) and (18) into (11) we obtain the desired result.

Once Proposition 1 is established, the proof of Theorem 3 is concluded by the following approximation result for initial data. Lemma 2 (Approximation of initial data). There exists C0 > 0 and a choice of approximated initial conditions q0,h ∈ Sh (0) such that, for h small, q0 −q0,h Q ≤ C0 hα/2 where α ∈ (0, 1] is the same as in (8). Proof. The approximations q0,h may be obtained by solving the following problem qh)+( qh −Ph q0 )}. By the triangle inequality, we find q0,h = Argmin qh ∈Qh {E (0, qh ) + ( qh −Ph q0 ) − (q0,h −Ph q0 ) E (0, q0,h ) ≤ E (0, ≤ E (0, qh ) + ( qh −q0,h ),

(19)

for all qh ∈ Qh . Namely, we have proved that q0,h ∈ Sh (0). Since q0 ∈ S(0), we q −q0 2Q ≤ E (0, q ) + ( q −q0 ) for all q ∈ Q. Letting q = q0,h have E (0, q0) + κ2 and using the triangle inequality and the minimality of q0,h we obtain κ q0,h −q0 2Q ≤ E (0, q0,h) − E (0, q0 ) + (q0,h −Ph q0 ) + ((Ph −I)q0 ) 2 (19)

≤ E (0, qh ) − E (0, q0) + ( qh −Ph q0 ) + ((Ph −I)q0 )

for all qh ∈ Qh . When choosing qh = Ph q0 we find κ q0,h −q0 2Q ≤ E (0, Ph q0 ) − E (0, q0) + ((Ph −I)q0 ). 2 Next, we evaluate the right hand side of (20) as follows

(20)


195

κ q0,h −q0 2Q ≤ E (0, Ph q0 ) − E (0, q0) + ((Ph −I)q0 ) 2 1 = APh q0 , (Ph −I)q0 − A(Ph −I)q0 , (Ph −I)q0 2 + H(Ph q0 ) − H(q0 ) − (0), (Ph −I)q0 + ((Ph −I)q0 ) 1 = − A(Ph −I)q0 , (Ph −I)q0 − (0), (Ph −I)q0 + ((Ph −I)q0 ) 2 1

Dq H(q0 +s(Ph −I)q0 ), (Ph −I)q0 ds. +

(7)

(21)

0

The integral term in the above right-hand side can be estimated as follows

1

Dq H(q0 +s(Ph −I)q0 ), (Ph −I)q0 ds

0

1

= 0

Dq H(q0 +s(Ph −I)q0 )−Dq H(q0 ), (Ph −I)q0 ds

1

+

Dq H(q0 ), (Ph −I)q0 ds

0

1

≤

sCH (Ph −I)q0 Q (Ph −I)q0 X ds + Dq H(q0 )X (Ph −I)q0 X .

0

Hence, using (8) and (21), we find κ κ q0,h −q0 2Q + (Ph −I)q0 2Q 2 2

(21) CH (Ph −I)q0 Q +Dq H(q0 )X . ≤ CP hα q0 Q (0)X +C + 2 The assertion follows by taking h small.

Our proof shows some flexibility in the choice of the hardening function H , which may just be asked to be convex and smooth and to have reasonable growth. Yet, we surely need the regularizing gradient term for obtaining the convergence rate, since it makes the nonlinearity Dq H and the nonsmoothness through lower order. Moreover, the quadratic form of the elastic part plays a crucial role. Thus, it is not clear how this proof can be adapted in the case of C is z-dependent; in particular, since we are not able to show uniqueness of solutions in this case.

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Acknowledgements AP is supported by the Deutsche Forschungsgemeinschaft through the project C18 “Analysis and numerics of multidimensional models for elastic phase transformation in a shape-memory alloys” of the Research Center M ATHEON. US is partially supported by FP7-IDEAS-ERC-StG Grant #200497 (BioSMA). Finally, LP and US gratefully acknowledge the kind hospitality of WIAS.

References 1. J. Alberty, C. Carstensen, and D. Zarrabi, Adaptive numerical analysis in primal elastoplasticity with hardening, Comput. Methods Appl. Mech. Engrg. 171(3–4), 1999, 175–204. 2. M. Arndt, M. Griebel, and T. Roubícek, Modelling and numerical simulation of martensitic transformation in shape memory alloys, Cont. Mech. Thermodyn. 15, 2003, 463–485. 3. J.-P. Aubin, Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods, Ann. Scuola Norm. Sup. Pisa (3) 21, 1967, 599–637. 4. F. Auricchio, A. Mielke and U. Stefanelli, A rate-independent model for the isothermal quasistatic evolution of shape-memory materials, M3AS Math. Models Meth. Appl. Sci. 18(1), 2008, 125–164. 5. F. Auricchio, A. Reali and U. Stefanelli, A three-dimensional model describing stress-induces solid phase transformation with residual plasticity, Int. J. Plasticity 23(2), 2007, 207–226. 6. F. Auricchio and L. Petrini, Improvements and algorithmical considerations on a recent threedimensional model describing stress-induced solid phase transformations, Int. J. Numer. Meth. Engrg. 55, 2002, 1255–1284. 7. F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations. Part II: Thermomechanical coupling and hybrid composite applications, Int. J. Numer. Meth. Engrg. 61, 2004, 716–737. 8. F. Auricchio and E. Sacco, Thermo-mechanical modelling of a superelastic shape-memory wire under cyclic stretching-bending loadings, Int. J. Solids Struct. 38, 2001, 6123–6145. 9. M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. 10. E. Fried and M. Gurtin, Dynamic solid-solid transitions with phase characterized by an order parameter, Physica D 72(4), 1994, 287–308. 11. P. Grisvard, Singularities in Boundary Value Problems, Recherches en Mathématiques Appliquées (Research in Applied Mathematics), Vol. 22, Masson, Paris, 1992. 12. W. Han and B.D. Reddy, Plasticity (Mathematical Theory and Numerical Analysis), Interdisciplinary Applied Mathematics, Vol. 9. Springer-Verlag, New York, 1999. 13. P. Houston, I. Perugia, A. Schneebeli and D. Schötzau, Mixed discontinuous Galerkin approximation of the Maxwell operator: The indefinite case, M2AN Math. Model. Numer. Anal. 39(4), 2005, 727–753. 14. M. Kružík, A. Mielke and T. Roubícek, Modelling of microstructure and its evolution in shapememory-alloy single-crystals, in particular in CuAlNi, Meccanica 40, 2005, 389–418. 15. A. Mielke, Evolution in rate-independent systems, in C. Dafermos and E. Feireisl (Eds.), Handbook of Differential Equations, Evolutionary Equations, Vol. 2, Elsevier, Amsterdam, 2005, pp. 461–559. 16. A. Mielke, A mathematical framework for generalized standard materials in the rateindependent case, in R. Helmig, A. Mielke and B.I. Wohlmuth (Eds.), Multifield Problems in Solid and Fluid Mechanics, Lecture Notes in Applied and Computational Mechanics, Vol. 28, Springer-Verlag, Berlin, 2006, pp. 351–379.


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17. A. Mielke, A model for temperature-induced phase transformations in finite-strain elasticity, IMA J. Appl. Math. 72(5), 2007, 644–658. 18. A. Mielke and T. Roubícek, A rate-independent model for inelastic behavior of shape-memory alloys, Multiscale Model. Simul. 1, 2003, 571–597. 19. A. Mielke and T. Roubícek, Numerical approaches to rate-independent processes and applications in inelasticity, M2AN Math. Model. Numer. Anal. 43, 2009, 399–428. 20. A. Mielke and F. Theil, On rate-independent hysteresis models, Nonl. Diff. Eqns. Appl. (NoDEA) 11, 2004, 151–189 (accepted July 2001). 21. A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error estimates for space-time discretizations of a rate-independent variational inequality, SIAM J. Numer. Anal., 2009, submitted. 22. A. Mielke, L. Paoli and A. Petrov, On the existence and approximation for a 3D model of thermally induced phase transformations in shape-memory alloys, SIAM J. Math. Anal. 41, 2009, 1388–1414. 23. J. Nitsche, Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens, Numer. Math. 11, 1968, 346–348. 24. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, Vol. 23, Springer-Verlag, Berlin, 1994. 25. A. Souza, E. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stressinduced phase transformations, Eur. J. Mech. A/Solids 17, 1998, 789–806.

On the Implementation of Variational Constitutive Updates at Finite Strains J. Mosler and O.T. Bruhns

Abstract In this paper an efficient, variationally consistent, algorithmic formulation for rate-independent dissipative solids at finite strain is presented. Focusing on finite strain plasticity theory and adopting the formalism of standard dissipative solids, the considered class of constitutive models can be defined by means of only two potentials being the Helmholtz energy and the yield function (or equivalently, a dissipation functional). More importantly, by assuming associative evolution equations, these potentials allow to recast finite strain plasticity into an equivalent, variationally consistent minimization problem, cf. [1–4]. Based on this physically sound theoretical approach, a novel numerical implementation is discussed. Analogously to the theoretical part, it is variationally consistent, i.e., all unknown variables follow naturally from minimizing the energy of the respective system. Extending previously published works on such methods, the advocated numerical scheme does not rely on any material symmetry regarding the elastic and the plastic response and covers isotropic, kinematic and combined hardening, cf. [5, 6].

1 Introduction In line with classical computational plasticity theory, the goal of variational constitutive updates is the calculation of all state and history variables X at time tn+1 based on the previous solution, i.e., Xn → Xn+1 . However, in contrast to conventional schemes such as the by now standard return-mapping algorithm, variational constitutive updates are based on a variational principle. More precisely, all unknown Jörn Mosler GKSS Research Centre, Institute for Materials Research, Materials Mechanics, D-21502 Geesthacht, Germany; E-mail: [email protected] Otto T. Bruhns Institute of Mechanics, Ruhr University Bochum, D-44780 Bochum, Germany; E-mail: [email protected] K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_15, © Springer Science+Business Media B.V. 2010

199

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J. Mosler and O.T. Bruhns

history variables, together with the deformation mapping, follow from minimizing a suitable (pseudo) potential. The advantages of variational constitutive updates compared to classical computational approaches are manifold. For instance, from a theoretical point of view, the existence of solutions can be analyzed by using the same tools originally designed for hyperelastic material models, cf. [2, 7, 8]. Furthermore, a minimum principle induces some sort of canonical metric which can be taken for error estimation, cf. [4, 9–11]. From a practical point of view, variational constitutive updates are advantageous as well. For example, a minimization principle opens up the possibility to apply state of the art optimization algorithms. Particularly for multisurface plasticity models such as single-crystal plasticity this represents an interesting feature, cf. [1]. Comi and co-workers were the first who advocated a constitutive update based on a minimization principle, cf. [12, 13]. In the respective numerical implementation, the constitutive model was enforced in a weak sense, i.e., a Hu-Washizu-type formulation. Probably inspired by the works [12, 13], Ortiz and co-workers proposed a constitutive update based on a minimization principle as well, cf. [1, 8, 9]. However, in contrast to the previous works, the presented algorithmic formulation coincides with the structure of standard finite element codes, i.e., the update is performed pointwise at the integration points. Since the works by Ortiz and coworkers [1,8,9], variational constitutive updates have been significantly further elaborated, cf. [2, 14–19]. Although variational constitutive updates are already relatively well developed from a theoretical point of view, the proposed implementations are mostly restricted to very simple constitutive prototype models, cf. [2, 14–19]. For this reason, a variationally consistent algorithmic formulation suitable for a broad range of different plasticity models including those characterized by a pronounced anisotropic mechanical response is presented.

2 Finite Strain Plasticity – A Short Summary In this section, a short summary associated with finite strain plasticity based on a multiplicative decomposition of the deformation gradient F := GRAD ϕ of the type F = F e · F p is given, cf. [20]. For a comprehensive overview and critical comments on different plasticity formulations at finite strains, refer to [21–23]. Since this class of constitutive models is well known nowadays, the respective constituents are simply summarized without any additional comment: • Multiplicative decomposition of the deformation gradient F into an elastic part F e and a plastic part F p F = Fe · Fp (1) • Additive decomposition of the Helmholtz energy (α is a collection of suitable strain-like internal variables)

Implementation of Variational Constitutive Updates at Finite Strains

¯ e (F e ) + p (α) =

201

(2)

• Elastic response (S and C are the second Piola–Kirchhoff stress tensor and the right Cauchy–Green strain tensor) S=2

∂ ∂ −1 −T = 2F p · · Fp e ∂C ∂C

(3)

• Reduced dissipation inequality (, Lp and Q are the Mandel stresses, Lp = F˙ p · −1 F p the plastic velocity gradient and Q = −∂α denotes the stress-like internal variables, respectively (all objects belong to the intermediate configuration) ) D = : Lp + Q · α˙ ≥ 0 • Space of admissible stresses (φ represents the yield function) Eσ = (, Q) ∈ R9+n | φ(, Q) ≤ 0

(4)

(5)

• Evolution equations and flow rule (λ is the plastic multiplier) Lp = F˙ p · F p

−1

= λ∂ φ

α˙ = λ∂Q φ

(6)

• Karush–Kuhn–Tucker conditions λ≥0

φλ ≥ 0

φ≤0

(7)

3 Finite Strain Plasticity – Standard Dissipative Solids This section is concerned with standard dissipative solids in the sense of Halphen and Nguyen [24], i.e., the plasticity framework discussed before is recast into an equivalent minimization problem. This section follows to a large extent [1, 2] (see also the more recent works [5, 6]). Conceptually, the underlying idea of the aforementioned variational principle is to minimize the stress power P˜ P = P : F˙ .

(8)

Here, P denotes the first Piola–Kirchhoff stress tensor. Clearly, Eq. (8) makes only sense, if the stresses are physically admissible. In line with [1,2], this functional can be extended to all states (including those which are physically not admissible), i.e., ˜ ϕ, ˙ ϕ, E( ˙ F˙ p , α, ˙ , Q) = ( ˙ F˙ p , α) ˙ + D(F˙ p , α, ˙ , Q) + J (, Q) with

(9)

202


J (, Q) :=

0

∀(, Q) ∈ Eσ

∞

otherwise

(10)

being the characteristic function of Eσ . As a result, for admissible stress states, the identity P = E˜ holds true. The interesting properties of functional (9) become apparent, if the stationarity conditions are computed. According to [1, 2] for a fixed deformation mapping ϕ, they result in the flow rule (6)1 , the constitutive relation for the internal stress-like variables Q and the constitutive relation for the Mandel stresses . The remaining variation of E˜ with respect to ϕ˙ will be discussed later. Following [1, 2] and applying the postulate of maximum dissipation (normality rules) ˙¯ = sup : L¯ p + Q · α˙¯ | (, Q) ∈ Eσ (11) J ∗ (L¯ p , α) the stress power (more precisely E˜ ) can be re-written as ˙ ϕ, E (ϕ, ˙ F˙ p , α) ˙ = ( ˙ F˙ p , α) ˙ + J ∗ (L˙ p , α). ˙

(12)

It bears emphasis that mathematically J ∗ represents the Legendre-transformation of J. ˙ Even As evident from Eq. (12), the only remaining variables are ϕ, ˙ F˙ p and α. p more importantly, the strain-like internal variables F and α follow jointly from the minimization principle ◦

˙ := inf E (ϕ, ˙ F˙ p , α) ˙ red (ϕ)

(13)

F˙ p ,α˙

◦

which, itself, gives rise to the introduction of the reduced functional red depending only on the deformation mapping. It can be shown in a straightforward manner that ◦

red acts like a hyperelastic (pseudo) potential. More precisely, the stresses derive ◦ from red , i.e., ◦ (14) P = ∂F˙ red .

4 Finite Strain Plasticity – Variational Constitutive Updates In this section, the key contribution of the present paper is addressed, i.e., a variational constitutive update which can be applied to a broad family of different plasticity models. This family is spanned by yield functions fulfilling the following properties: • The internal stress-like variables Q contain two parts. Qk is a back-stress and thus, it is associated with kinematic hardening, while Qi corresponds to the isotropic counterpart. Hence, the yield function reads φ(, Qk , Qi ) = φ( − Qk , Qi ).

(15)


203

• The yield function is of the type equi

φ(, Qk , Qi ) = equi (, Qk , Qi ) − Q0 ,

(16)

with equi (c(, Qk , Qi )) = cn equi (, Qk , Qi ),

c ∈ R+

(17)

being an equivalent stress which is a positively homogeneous function of degree n. equi defines the shape of the admissible stress space. According to Eq. (16), equi its initial diamater is denoted as Q0 . It is noteworthy that properties (15)–(17) are not very restrictive and they are fulfilled for almost all frequently applied yield functions. According to Remark 1 and without loss of generality, n can be assumed as n = 1. In this case, the reduced dissipation simplifies significantly. More precisely, equi

D = λQ0

≥0

(18)

Hence, for admissible states, the stress power is given by ˙ + λQequi . E = 0

(19)

In line with numerical implementations for standard finite strain plasticity models (not variationally consistent) such as [25, 26], Eq. (19) is approximated by applying a consistent time discretization to the evolution equations (see [1, 2]). For the sake of concreteness, a first-order fully implicit scheme is adopted. More precisely, with the notation tn+1 λ := λ dt ≥ 0 (20) tn

the following approximations are used: p p Fn+1 = exp λ ∂ φ|n+1 · Fn , αi |n+1 = αi |n + λ∂Qi φ, αk |n+1 = αk |n + λ ∂Qk

φ

(21) n+1

.

Inserting Eqs. (20) and (21) into Eq. (20), the time integration of E yields p Iinc (Fn+1 , αk |n+1 , αi |n+1 , λ)

tn+1 equi = E dt = n+1 − n + λQ0 .

(22)

tn

So far, variational constitutive updates are relatively simple and hence, the respective implementation seems to be straightforward. Unfortunately, this is not the case. The reasons for that are manifold. For instance, a direct minimization of Iinc p with respect to Fn+1 is not admissible, since F p has to comply with physical con-

204


straints resulting from the flow rule (and of course, det F p > 0). Furthermore, if plastic loading is considered, the additional restriction φ = 0 has to be enforced relating the stresses (and thus the strains) to the internal variables α k and αi . Clearly, within the return-mapping algorithm, the aforementioned problems do not occur. Within this robust scheme, the stresses, the internal stress-like variables and the plastic multiplier are usually chosen as unknowns, cf. [25, 26]. Evidently, the stresses do not have to fulfill a constraint such as det F p > 0. Therefore, it seems to be promising to replace the unknowns in Eq. (22). Here, the following re-parameterization is considered: ˜ := ∂ φ| ˜ , M = M()

(23)

Hk = ∂Qk φ = −M, Hi = Hi (Q˜ i ) := ∂Qi φ Q˜ ,

(24) (25)

λ = λ(a) := a 2 .

(26)

i

˜ and Q˜ i denoting pseudo stresses, and a pseudo stressIt depends on the unknowns like hardening variable. It has to be emphasized that these pseudo variables are not necessarily identical to their physical counterparts in general, i.e., ˜ = ,

Q˜ i = Qi .

(27)

˜ and Q˜ i define only the flow and hardening direcMore precisely, the variables , tions. Thus, they are related to their physical counterparts by ˜ = M(), M()

Hi (Q˜ i ) = Hi (Qi ).

(28)

Finally, by inserting the new parameterizations (23)–(26) into the approximations (21) and subsequently, into Eq. (22), the novel variational constitutive update results in inf Iinc ,

Iinc = Iinc (X),

with

˜ Qi , a] X = [,

⇒

dim[X] = 11.

(29)

This unconstrained optimization problem can be solved in a standard manner, e.g, by employing gradient type methods, cf. [27, 28]. Further details are omitted. They can be found in [5, 6]. Remark 1. The presented algorithm is based on a positively homogeneous yield function of degree one. However, any yield surface φ = 0 associated with a yield function being positively homogeneous of degree n can always be re-written as (see Eq. (16))

n n n n equi equi equi φ = 0 ⇔ equi = Q0 ⇔ equi = Q0 ⇔ φ˜ := equi − Q0 = 0. (30) √ n Clearly, equi is positively homogeneous of degree one. Hence, the proposed constitutive update can indeed be applied to any positively homogeneous yield function


205

of degree n. It bears emphasis that almost all frequently employed plasticity models are thus covered.

5 Numerical Example: A Modified Barlat-Type Model The applicability, the performance as well as the robustness of the proposed variational constitutive update are demonstrated here by means of a numerical example, cf. [5]. Following Cazacu and Barlat [29], an orthotropic yield function of the type φ() = (J2◦ )3/2 − J3◦ − (τ0 )3 , eq

(31)

with J2◦ =

a1 a2 a3 (11 − 22 )2 + (22 − 33 )2 + (33 − 11 )2 6 6 6 2 + a 2 + a 2 + a4 12 5 13 6 23

(32)

and J3◦ =

1 1 1 3 3 3 [2 (b1 + b4 ) − (b2 + b3 )] 33 + + (b1 + b2 ) 11 (b3 + b4 ) 22 27 27 27 1 1 2 2 − (b1 22 + b2 33 ) 11 − (b3 33 + b4 11 ) 22 9 9 1 2 2 − [(b1 − b2 + b4 )11 +(b1 − b3 + b4 )22 ]33 + (b1 + b4 )11 22 33 9 9 1 2 + 2b1112 13 23 − 13 [2b922 − b8 33 − (2b9 − b8 ) 11 ] 3 1 2 [2b10 33 − b5 22 − (2b10 − b5 ) 11 ] − 12 3 1 2 [(b6 + b7 ) 11 − b6 22 − b7 33 ] − 23 3

(33)

is considered. Here, ai and bi are material parameters defining the anisotropy of the equi respective material and τ0 denotes the shear strength (in the original work [29], there is a typing error). Clearly, the yield surface (φ = 0) associated with Eq. (31) being positively homogeneous of degree three can be re-written into the positively homogeneous of degree one counterpart 1/3 eq φ(, αi ) = (J2◦ )3/2 − (J3◦ ) − τ0 − Qi (αi )

(34)

where Qi := −∂ p /∂αi corresponds to isotropic hardening. It is assumed that this internal variable saturates and converges to Q∞ i , i.e.,

206


Table 1 Barlat-type model: material parameters chosen according to [31] (only the non-zero parameters are shown). a1 3.01742

a2 4.63451

a3 0.979594

E [MPa]

ν [–]

45000

0.33

equi

Qi

b1 –3.08191

equi

τ0

[MPa]

48.5757

b2 2.77759

b3 –2.30817

Q∞ i [MPa]

αu [–]

80.0

0.1

b4 1.89439

(αi ) = Q∞ i (1 − exp[αi /αu ]) ,

(35)

with αu representing an additional material parameter. The model is completed by the elastic response. In line with [30] (see also [6]) a functional based on a deviatoric/volumetric split is adopted, i.e., e = with

1 1 e µ(tr[Cdev ] − 3) + K(J e − 1)2 2 4

e Fdev := (J e )−1/3 F e ,

e e T e Cdev := (Fdev ) · Fdev = (J e )−2/3 C e .

(36)

(37)

The material parameters used for the numerical analysis correspond to sheets of Mg and are summarized in Table 1. In line with the experiments reported in [31], the material response is investigated by a uniaxial tensile test. Since Mg shows a pronounced tension/compression asymmetry, both loading cases are analyzed. The predicted mechanical responses are shown in Figure 1. Although the model is highly anisotropic, the variational constitutive update works very robustly and efficiently. Convergence problems did not occur. The computed mechanical response is in good agreement with the experiments reported in [31].

6 Conclusions In this paper, a constitutive update for so-called standard dissipative solids has been presented. In line with the previous works [1, 2], the unknown history variables, together with the deformation mapping, follow jointly from minimizing an incrementally defined (energy) potential. However, in contrast to already existing prototype models [2, 14–19], the proposed method covers a broad range of different constitutive models including anisotropic elasticity, anisotropic yield functions and isotropic as well as kinematic hardening. The key idea of the advocated variational constitutive update is to conveniently parameterize the evolution equations and the


207

Fig. 1 Barlat-type model: stress-strain-diagram predicted by the variational constitutive update for simple tension and simple compression.

flow rule. By doing so, all physical restrictions such as det F p > 0 are naturally included within the resulting unconstrained minimization principle.

References 1. M. Ortiz and L. Stainier, The variational formulation of viscoplastic constitutive updates, Computer Methods in Applied Mechanics and Engineering 171, 1999, 419–444. 2. C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finitestrain plasticity, Proc. R. Soc. London A 458, 2002, 299–317. 3. C. Miehe, Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation, International Journal for Numerical Methods in Engineering 55, 2002, 1285–1322. 4. J. Mosler, On the numerical modeling of localized material failure at finite strains by means of variational mesh adaption and cohesive elements, Habilitation, Ruhr University Bochum, Germany, 2007. 5. J. Mosler and O.T. Bruhns, On the implementation of rate-independent standard dissipative solids at finite strain – Variational constitutive updates, Computer Methods in Applied Mechanics and Engineering, 2009, in press. 6. J. Mosler and O.T. Bruhns, Towards variational constitutive updates for non-associative plasticity models at finite strain: Models based on a volumetric-deviatoric split, International Journal for Solids and Structures 46(7–8), 2009, 1676–1684. 7. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63, 1978, 337–403.

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Phase-Field Modeling of Nonlinear Material Behavior Y.-P. Pellegrini, C. Denoual and L. Truskinovsky

Abstract Materials that undergo internal transformations are usually described in solid mechanics by multi-well energy functions that account for both elastic and transformational behavior. In order to separate the two effects, physicists use instead phase-field-type theories where conventional linear elastic strain is quadratically coupled to an additional field that describes the evolution of the reference state and solely accounts for nonlinearity. In this paper we propose a systematic method allowing one to split the nonconvex energy into harmonic and nonharmonic parts and to convert a nonconvex mechanical problem into a partially linearized phasefield problem. The main ideas are illustrated using the simplest framework of the Peierls–Nabarro dislocation model.

1 Introduction Nonconvex energy potentials are used in solid mechanics for the modeling of martensitic transformations [9], plasticity [1] and fracture [25]. Parts of the resulting energy landscapes correspond to sufficiently smooth deformations preserving the locally affine structure of the lattice environment of each atom. Other parts represent highly distorted atomic arrangements associated with either loss or reacquisition of nearest neighbors. While deformations of the first type can (often) be described by the conventional strain tensor of (linear) elasticity theory, a representation of the deformations of the second type requires introducing additional internal variables accounting for deviations from the local affinity of the stressed atomic configuraY.-P. Pellegrini · C. Denoual CEA, DAM, DIF, F-91297 Arpajon, France; e-mail: [email protected], [email protected] L. Truskinovsky Laboratoire de Mécanique des Solides, CNRS UMR-7649, École Polytechnique, Route de Saclay, F-91128 Palaiseau Cedex, France; E-mail: [email protected] K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_16, © Springer Science+Business Media B.V. 2010

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tions. In particular, these supplementary variables describe the evolution of the local reference state (LRS) from which the elastic deformations are measured [3, 11, 26]. The main difference between the elastic strains and these supplementary internal variables is that the dynamics of the former is typically inertial, while that of the latter is usually overdamped. Sometimes the nonelastic variables can be minimized out as in the case of deformational plasticity (e.g., [2]). In this paper we deal instead with situations where the internal variables have to revealed rather than hidden. We assume that the coarse-grained nonconvex energy density f (ε) is known either from extrapolations of experimental measurements or from ab-initio calculations involving atomic homogeneity constraints. We suppose that the argument ε of this function, that represents a coarse-grained strain, is small and can be additively split into the linear elastic part e, and a phase-field part η that accounts for the nonelastic evolution of the LRS. Our next assumption is that f can be represented as a sum of two terms: the elastic energy fe , which depends on e = ε − η and the phase-field energy g, which depends on η. We interpret f (ε) as the outcome of adiabatic elimination of the variable η and consider the inverse problem of recovering the phase-field energy g(η) from the function f (ε) under the assumption that the function fe (e) is quadratic. The problem of the identification of g(η) reduces to a problem of optimization and the relation between the ‘optimally’ related functions f (ε) and g(η) is studied in some prototypical cases. If, in contrast, the function g(η) is chosen independently, the corresponding function f (ε) is typically non-smooth and non single-valued, e.g. [6]. To motivate the need for the phase-field variables we consider in full detail a specific physical example. It deals with the mixed, discrete-continuum representation of a dislocation core [12, 16]. More specifically, we develop a modified version of the classical Peierls–Nabarro (PN) model that accounts for a finite thickness of the slip region. In this problem the coarse-grained description of the slip zone is provided by the so-called γ -potential [5,27]. The phase field represents an ‘atomically sharp’ slip and the part of the interaction potential related to g gives rise to the slip-related pull-back force [7, 16, 23]. Our general method of recovering the expression for this force represents an extension of Rice’s transform, which was first introduced in the context of a dislocation nucleation problem [19]. In this paper only the simplest scalar problem in a one-dimensional setting is considered. The slightly more general question of extracting from the coarse-grained energy a convex (instead of quadratic) component will be examined elsewhere [17].

2 Surface Problem We begin with the special case when the phase field is localized on a surface. In problems involving fracture or slip it often proves convenient to represent the energy of a body as the sum of a bulk term depending on strain gradients and a surface term penalizing displacement discontinuities. The bulk term is usually modeled by linear elasticity. The modeling of the surface energy is less straightforward [6, 25].

Phase-Field Modeling of Nonlinear Material Behavior

211

For instance, the models will be different depending on whether the location of the discontinuities is known a priori or not. In a 1D setting with a known fracture set the equilibrium problem reduces to minimizing the following energy functional

1

W [u] =

dx fe (ux ) +

0

fa (δ(x)).

(1)

a

Here fe (e) = (E/2)e2, E > 0 is the elastic modulus and a is a coarse-graining length scale that typically exceeds several atomic sizes. The set a in (1) represents discontinuity points resolved at scale a and δ(x) = [u]a (x) is the corresponding displacement discontinuity. The surface energy fa (δ) is then an effective interaction over the distance a; in particular, the shear-related component of fa (δ) coincides the γ -potential mentioned in the Introduction. In the case when the fracture set is unknown the surface energy has to be chosen differently. The reason is that in this model the displacement discontinuity at scale a does not represent the microscopic slip between neighboring atomic planes, and therefore the difference between elastic deformation and inelastic slip has to be yet resolved at this scale [19]. More precisely, linear elasticity, which has nothing to do with slip and which is already accounted for in the bulk term, has not been excluded from fa (δ). The identification of the surface energy in (1) with fa (δ), which is quadratic at the origin, leads in a free discontinuity problem to a degenerate solution with infinitely many infinitely small discontinuities [6]. To remove linear elasticity from the surface term, one should replace the coarsegrained discontinuity [u]a by the atomically sharp slip η(x) = [u](x) that does not depend on a. The energy (1) is then rewritten as

1

W [u] = 0

dx fe (ux ) +

g(η),

(2)

where now is the set of discontinuity points corresponding to a = 0. The problem is to find the relation between the function fa (δ), representing an empirical input, and the unknown function g(η). To define g(η) we divide the total slip δ into an elastic part, a e, where e is an equivalent elastic strain, and an inelastic part η. The function g(η) is defined by the condition that fa (δ) is a relaxation of the energy afe (e) + g(η) under the condition that ae + η = δ, namely: E δ−η 2 fa (δ) = inf a + g(η) . (3) η 2 a If the energy fa (δ) is a single-well function and the infimum is unique, the function g is completely defined. If fa (δ) is periodic as in the case of dislocations, in order to have a uniquely-defined g(η), we need to replace in definition (3) the global

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Fig. 1 Parametric transforms (6), (7) applied to a Lennard–Jones potential f (δ) = δ −12 − δ −6 . (a) f (δ) (dashed), and the two branches of g(η) (solid), where ‘p’ (‘u’) labels the physical (nonphysical) branch; (b) f (δ) and the ‘p’ branch of g (η); (c) η(δ).

minimization by a properly-defined local minimization denoted hereafter by ‘infloc ’ (minimization over η starting from the minimum of fa closest to δ). In what follows, our task will be to reverse definition (3) and to recover the nonequilibrium energy g(η) from fa (δ). What allows us to proceed is the specific (harmonic) structure of the elastic part of the energy. We observe that the function g must satisfy the following necessary condition (E/a)(δ − η) = g (η).

(4)

Moreover, differentiation of (3) with regard to δ gives fa (δ) = (E/a)(δ − η).

(5)

These two equations allow one to represent g (η) in the following parametric form [7, 8, 19] (6) (η, g (η)) = δ − (a/E)fa (δ), fa (δ) . The parametric representation for g(η) then reads (η, g(η)) = δ − (a/E) fa (δ), fa (δ) − [a/(2E)][fa (δ)]2 .

(7)

Since for nonconvex fa (δ) this representation may lead to a multivalued function g(η) formula (7) must be supplemented by an additional branch selection procedure. To illustrate the mapping f (δ) → g(η) given by (7) and the selection of a physical branch we consider a Lennard–Jones potential fa , with a = 1 and assume that E = f (δ0 ), where δ0 is the only minimum of f (see Figure 1). Notice that the resulting function g (η) has an infinite slope at η = δ0 and that for η δ0 we must have g(η) ∝ (η − δ0 )3/2 . The removal of the linear elastic part of the energy becomes important in PNtype modeling of dislocations. Consider, for instance, a straight screw dislocation in an isotropic linear-elastic body and assume that the sharp discontinuity plane,


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y = 0, lies between the two effective gliding surfaces located at y = ±a/2. To account for the finite thickness of the core region a we need to modify the classical PN model [12]. According to our interpretation the linear elastic stress outside the slip region (−a/2, a/2) must be balanced by the coarse-grained pull-back stress that is resolved at the spatial scale a. We therefore interpret the pull-back stress at this scale as fa (δ(x)), where fa is the γ -potential, a periodic function with period b and with fa (0) = 0. The expression for the linear stress outside the slip region is derived in the Appendix. With these considerations in mind we obtain for the unknown function η(x) representing a mathematical slip at y = 0 the following system of equations +∞ µ a + σ a (x) = fa (δ(x)), − dx η (x ) arctan (8) πa −∞ 2(x − x ) δ(x) = (a/µ)fa (δ(x)) + η(x),

(9)

where σ a is the resolved applied stress at scale a. If we match the linear elastic behavior at η = 0 with that in the bulk regions we obtain that µ = afa (0). Using in this relation the physical shear modulus and the value of fa (0) from the γ -potential provides a rough estimate for a, the effective interaction range. We notice that parameter a enters both equations (8, 9), which makes this system different from the one studied in [16,19]. The ideas behind our nonlocal extension of the PN model are also different from that of Miller et al. [13] where a nonlocal kernel was introduced empirically as part of the pull-back stress, and the usual 1/(x − x ) kernel was used for the bulk stress. To bring the system (8, 9) into the framework of phase-field models, we identify the effective pull-back force fa (δ(η)) with g (η) and rewrite Eq. (8) as µ 2

+∞ −∞

dx Ka (x − x ) η (x ) + σ a (x) = g (η).

(10)

where Ka (x) = −(2/πa) arctan(a/2x). It is now easy to see that g (η) enjoys the parametric representation a (11) (η, g (η)) = δ − fa (δ), fa (δ) , µ where we recognize the mapping (6) (see also [16, 19, 23]). To make the link with the classical PN model one needs to consider the limit a → 0. By computing η in terms of δ and expanding (10) in powers of a, we obtain to order O(a) the following ‘gradient’ extension of the PN model +∞ µ δ (x ) − dx + λδ (x) + σa (x) = f (δ(x)), (12) 2π −∞ x − x where λ = aµ/4. For different weakly or strongly nonlocal generalizations of the PN model see [13, 21]. Equation (12) features an effective applied stress that differs

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from σa (x, 0) defined in the Appendix by an O(a) correction, namely σa (x) ≡ σa (x, 0) + (a/2)[(1/2)∂y σa (x, 0) − K0 ∂x σa (x, 0)]. The classical PN model is retrieved by letting a = 0.

3 Bulk Problem Now let us place the problem in a more general framework. The task is to approximate locally the empirical potential f (ε) by a quadratic function with an optimally chosen reference state η, and to associate with this state a reference energy g(η). Behind such construction is the assumption that all the nonlinearity of the problem is related to the evolution of the reference state. The simplest setting to pose formally the problem is the one-dimensional geometrically linearized theory of nonlinear elastic bars. According to our interpretation the empirical energy is represented as

E 2 f (ε) = inf (ε − η) + g(η) (13) η,loc 2 and the problem is to find the intrinsic phase-field function g(η). Following the previous section we write the parametric representation for g (η) in the form f (ε) η, g (η) = ε − , f (ε) . (14) E The function g(η) is then given by the mapping f (ε)2 f (ε) , f (ε) − . η, g(η) = ε − E 2E

(15)

The consistency of this procedure requires the parameter E and the function f (ε) to be related. If we expand the parametric definition of g(η) near a reference state ε0 where f (ε0 ) = 0, we obtain g(ε0 ) = f (ε0 ), g (ε0 ) = 0 and g (ε0 ) = f (ε0 )/[1 − f (ε0 )/E]. The natural choice E = f (ε0 ) makes g (ε0 ) infinite. The behavior of the higher derivatives of g(η) near η = ε0 is found by assuming (without loss of generality) that derivatives f (k) (ε0 ) vanish for k = 3, . . . , n − 1. The order of the asymptotics depends on n > 2, which is the first integer such that f (n) (ε0 ) = 0:

f (n) (ε0 ) (n−1) (n − 1)f (n) (ε0 ) n (16) δε δε . , f (ε0 ) − (η, g(η)) ε0 − (n − 1)!E n! Hence g behaves near its minimum as: |g(η)−g(ε0)| ∼ |η −ε0 |n/(n−1) . The generic case is n = 3; the case n = 4 corresponds to periodic potential relevant for dislocations; for f locally harmonic, n = ∞.


215

Fig. 2 Geometrical illustration of the construction defined by Eqs. (14–17).

Observe now that the function g computed from (14, 15), can also be viewed as a solution of the following optimization problem:

E (17) g(η) = sup f (ε) − (ε − η)2 , 2 ε,loc which is a natural inverse of (13) (see also [18, 20]). Since the equation η = ε − f (ε)/E may have several solutions ε(η), the representation (17) removes the ambiguity by always selecting the upper branch. The working of Eqs. (14-17) with E = f (ε0 ) is illustrated in Figure 2. In the domain at the left of ε0 , where f grows faster than harmonic the desired tangency point does not exist. In this case the difference f (ε) − E2 (ε − η)2 is maximized at ε = −∞. This situation takes place in the Lennard–Jones example of Section 2 where we have to use g(δ) = +∞ for δ < δ0 (hatched area of Figure 1a). To handle general multi-well energies, we first introduce the Stillinger–Weber mapping ε0 (ε) that links to any state ε the local minimum ε0 of f (ε) that would be attained from this state by steepest-descent [22]. Next, we modify equations (13) and (17) as:

1 f (ε0 (ε))(ε − η)2 + g(η) , f (ε) = inf (18) η,loc 2

1 (19) g(η) = sup f (ε) − f ε0 (η) (ε − η)2 . 2 ε,loc Whereas (19) defines g, equation (18) states that knowing f is equivalent to knowing g plus the linear-elastic behavior of f near its local minima. The precise meaning of the ‘loc’ in Eqs. (18, 19) is as follows. Operationally, the minimization in the definition of f is carried out over η, starting from ε0 (ε), the local minimum nearest to ε determined by the SW mapping; the corresponding elastic modulus is also determined by the starting point. The maximization in the definition of g(η) proceeds along similar lines except that now the relevant elastic

216


Fig. 3 Phase-field representation of a double-well potential f (ε) = (ε − 1)2 (2ε + 1)2 + 0.2ε.

Fig. 4 Transform (19) applied to the piecewise-harmonic periodic potential f (δ) = (δ − [δ])2 /2. Notation [·] stands for the integer part. The function η(δ) is determined as the argmin in (18).

modulus is determined by the local minimum closest to η, and is fixed during the maximization. The optimization is carried out starting from ε = η. Figure 3 illustrates the case of a double-well potential with unequal curvatures of the wells. Notice that in contrast to what we saw in Figure 1 the function η(ε) is now bounded. Another interesting case is the periodic potential that is used in the description of reconstructive phase transitions (e.g., [4]). Consider, for instance, the piecewiseharmonic periodic case shown in Figure 4 that is often used in analytical studies [13, 21]. The parametric representation (14) of g is here useless and the definition (19) must be used instead. In this extreme case, all elasticity has been removed from g and the resulting g(η) is cone-shaped at its minima (Figure 4a) as predicted by Eq. (16) for n → ∞. The force g(η) is discontinuous (Figure 4b) and its extreme values provide thresholds for the evolution of η, whose stepwise character is an artifact due to the absence of smooth spinodal regions in f . It is also instructive to consider for comparison the case of an unbounded harmonic potential f (ε) = (Ef /2)(ε − ε0 )2 . From (17) with E = Ef , one deduces that g(η) = +∞ if η = ε0 and g(η = ε0 ) = 0. This trivial example indicates that in a purely linear-elastic model, the reference state does not have to evolve.


217

4 Concluding Remarks The goal of this paper was to reveal in the simplest setting the variational nature of the generalized Rice transform. The problem consists in splitting a coarse-grained lattice potential f , describing the overall deformation of a sufficiently large number of atoms, into a (quasi) convex elastic potential and an inelastic potential g dealing with structural rearrangements. Here the potential f is assumed to be measurable by molecular statics along a prescribed deformation path relevant to the material transformation in question. For simplicity, the elastic potential is assumed in this paper to be a standard quadratic function of the macroscopic strain. The inelastic potential g must be a function of the phase-field variable η, whose identification represents an important part of the problem. While our precise construction solving the above problem is presented in the static setting (see Eq. (19)), the motivation for the splitting concerns, first of all, dynamical applications (e.g., [7]). Thus we assume that material displacement u associated to the strain ε = ∂x u evolves inertially almost without damping (standard elastodynamics), while the dynamics of the phase-field variable is overdamped. More precisely we assume that the relaxation of the variable η follows the time-dependent Ginzburg–Landau (TDGL) equation. By means of an empirical ‘viscosity’ parameter ν we can write the evolution equation in the form

1 ∂ 1 2 η˙ = − f ε0 (ε) (ε − η) + g(η) , ν ∂η 2 where we have omitted for simplicity the conventional gradient-penalizing terms (e.g., [7, 24]). In the static setting the above equation reduces to our basic Eq. (18). The definitive advantage of separating the wave motion from an overdamped TDGL relaxation is the possibility to attribute effective damping only to large atomic displacements. Our preliminary investigations [17] indicate that extending the variational set-up presented in this paper to higher dimensions and generalizing it in the direction of extracting (quasi) convex, rather than merely quadratic elastic components, is feasible. These issues will be addressed systematically in a separate publication.

Appendix The following computations are largely based on the Eshelby’s arguments presented in [10]. Consider a Volterra screw dislocation with zero-width core and with Burgers vector b. The displacement uz (x, y) has the form uz (x, y) =

b y b b Arg(x + iy) = arctan + sign(y)θ (−x), 2π 2π x 2

(20)

where θ is the Heaviside function, and where the indeterminacy in the discontinuity of uz is resolved by specifying the glide plane (y = 0). The distributional part in the

218


r.h.s. of Eq. (20), usually omitted in the literature (e.g. [12]), is crucial to the present derivation because it represents the irreversible atomic displacements on the plane y = 0. We introduce the eigendistortion, βij∗ , as the part of the dislocation-induced distortion βij = ui,j , that is not linear-elastic. For our dislocation, its only non∗ (x, y) = b θ (−x)δ (y), where δ is the Dirac distribution zero component is βyz D D [14]. The linear-elastic distortion, βije , is defined through the additive decomposition of the total distortion βij , namely βije ≡ βij − βij∗ [14]. The elastic strains are eij = sym βije . By using the identity [arctan(1/x)] = πδD (x) − 1/(1 + x 2 ), we obtain that the distributional parts in βij and βij∗ mutually cancel out giving the standard result [12] exz (x, y) = −

b y , 4π x 2 + y 2

eyz (x, y) =

b x . 4π x 2 + y 2

(21)

The stress induced by the eigenstrain is then σiz∗ (x, y) = 2µ eiz (x, y), where i = x, y. In the presence of an applied shear stress [15] σa ≡ σa yz , Eq. (20) becomes y b 1 y b arctan + sign(y)θ (−x), (22) dy σa (x, y ) + uz (x, y) = µ 0 2π x 2 The total stress is then σ = σ ∗ + σa and e = σ/2µ. Now, the key step consists in averaging the stress over the layer of width a containing the glide plane. Introduce: +a/2 σ ij (x) ≡ a1 −a/2 dy σij (x, y). From (22), the x component of the total relative atomic lattice displacement between the atomic planes at y = ±a/2 reads:

a a µb arctan + b θ (−x). σ a (x) + δ(x) ≡ uz (x, +a/2) − uz (x, −a/2) = µ πa 2x (23) Furthermore on account of (21) the average shear stress σ yz (x) in the layer is: σ yz (x) = σ a (x) +

µb 1 2π a

a/2

−a/2

dy

a x µb arctan . = σ (x) + a πa 2x x2 + y2

(24)

Comparison of (24) and(23) shows that: δ(x) = (a/µ)σ yz (x) + b θ (−x).

(25)

Consider next an Eshelby screw dislocation with an extended core described by a continuous function η(x). The distortion β ∗ becomes: +∞ ∗ (x, y) = δD (y)η(x) = −δD (y) dx η (x); βyz x

the Volterra dislocation corresponds to the limiting case η(x) = bθ (−x). Dis∗ (x, y) ≡ placements, strains and stresses are obtained by convolution using dβyz −δD (y)η (x)dx [10] as elementary distortions. The analogs of Eqs. (24, 25) are:


σ yz (x) = σ a (x) −

µ πa

+∞ −∞

dx arctan

δ(x) = (a/µ)σ yz (x) + η(x).

219

a 2(x − x )

η (x ),

(26) (27)

Equation (27), which we use in the paper, shows the relation between the coarsegrained displacement δ, and the discontinuity η.

References 1. Carpio, A. and Bonilla, L.L.: Discrete models for dislocations and their motion in cubic crystals. Phys. Rev. B 12, 2005, 1087–1097. 2. Carstensen, C., Hackl, K. and Mielke, A.: Nonconvex potentials and microstructures in finitestrain plasticity. Proc. R. Soc. London A 458, 2002, 299–317. 3. Choksi, R., Del Piero, G., Fonseca, I. and Owen, D.R.: Structural deformations as energy minimizers in models of fracture and hysteresis. Math. Mech. Solids 4, 1999, 321–356. 4. Conti, S. and Zanzotto, G.: A variational model for reconstructive phase transformations, and their relation to dislocations and plasticity. Arch. Rational Mech. Anal. 173, 2004, 69–88. 5. Christian, J.W. and Vitek, V.: Dislocations and stacking faults. Rep. Prog. Phys. 33, 1970, 307–411. 6. Del Piero, G. and Truskinovsky L.: Macro and micro-cracking in 1D elasticity. Int. J. Solids Struct. 38, 2001, 1135–1148. 7. Denoual, C.: Dynamic dislocation modeling by combining Peierls–Nabarro and Galerkin methods. Phys. Rev. B 70, 2004, 024106. 8. Denoual, C.: Modeling dislocations by coupling Peierls–Nabarro and element-free Galerkin methods. Comput. Meth. Appl. Mech. Engrg. 196, 2007, 1915–1923. 9. Ericksen, J.: Equilibrium of bars. J. Elast. 5, 1975, 191–202. 10. Eshelby, J.D.: Uniformly moving dislocations. Proc. Phys. Soc. London A 62, 1949, 307–314. 11. Hakim, V. and Karma, A.: Crack path prediction in anisotropic brittle materials. Phys. Rev. Lett. 95, 2005, 235501. 12. Hirth, J.P. and Lothe J.: Theory of Dislocations, 2nd edn. Wiley & Sons, New York, 1982. 13. Miller, R., Phillips, R., Beltz, G. and Ortiz, M.: A non-local formulation of the Peierls dislocation model. J. Mech. Phys. Solids 46, 1998, 1845–1867. 14. Mura, T.: Micromechanics of Defects in Solids, 2nd edn. Martinus Nijhof, Dordrecht, 1987. 15. Nabarro, F.R.N.: Dislocations in a simple cubic lattice. Proc. Phys. Soc. 59, 1947, 256–272. 16. Ortiz, M. and Phillips, R.: Nanomechanics of defects in solids. Adv. Appl. Mech. 36, 1999, 1–79. 17. Pellegrini, Y.-P., Denoual C. and Truskinovsky, L., in preparation. 18. Ponte Castañeda, P. and Suquet, P.: Nonlinear composites. Adv. Appl. Mech. 34, 2002, 171– 302. 19. Rice, J.R.: Dislocation nucleation from a crack tip: An analysis based on the Peierls concept. J. Mech. Phys. Solids 40, 1992, 239–271. 20. Rockafellar, T.: Convex Analysis. Princeton University Press, Princeton, 1997. 21. Rosakis, P.: Supersonic dislocation from an augmented Peierls model. Phys. Rev. Lett. 86, 2001, 95–98. 22. Stillinger, F.H. and Weber, T.A.: Packing structures and transitions in liquids and solids. Science 225, 1984, 983–989. 23. Sun, Y., Beltz, G.E. and Rice, J.R.: Estimates from atomic models of tension-shear coupling in dislocation nucleation from a crack tip. Mat. Sci. Eng. A 170, 1993, 69–85. 24. Truskinovsky, L.: Kinks versus shocks. In: Fosdick, R., Dunn, E. and Slemrod, M. (Eds.), Shock Induced Transitions and Phase Structures in General Media, IMA, Vol. 52, SpringerVerlag, 1993, pp. 185–229.

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25. Truskinovsky, L.: Fracture as a phase transformation. In: Batra, R. and Beatty, M. (Eds.), Contemporary Research in Mechanics and Mathematics of Materials, CIMNE, Barcelona, 1996, pp. 322–332. 26. Wang, Y. and Khachaturyan, A.G.: Three-dimensional field model and computer modeling of martensitic transformations. Acta Mater. 45, 1997, 759–773. 27. Woodward, C.: First-principles simulations of dislocation cores, Mat. Sci. Engrg. A 400–401, 2005, 59–67.

Polyconvex Energies for Trigonal, Tetragonal and Cubic Symmetry Groups Jörg Schröder, Patrizio Neff and Vera Ebbing

Abstract In large strain elasticity the existence of minimizers is guaranteed if the variational functional to be minimized is sequentially weakly lower semicontinuous (s.w.l.s.) and coercive. Therefore, for the description of hyperelastic materials polyconvex functions which are always s.w.l.s. should be preferably used. A variety of isotropic and anisotropic polyconvex energies, in particular for the triclinic, monoclinic, rhombic and transversely isotropic symmetry groups, already exist. In this contribution we propose a new approach for the description of trigonal, tetragonal and cubic hyperelastic materials in the framework of polyconvexity. The anisotropy of the material is described by invariants in terms of the right Cauchy–Green tensor and a specific fourth-order structural tensor. In order to show the adaptability of the introduced polyconvex energies for the approximation of real anisotropic material behavior we focus on the fitting of a trigonal fourth-order tangent moduli near the reference state to experimental data.

1 Introduction If we are interested in the construction of anisotropic energies Neumann’s Principle [16], which states that the symmetry group of a considered material must be included in the symmetry group of any tensor function of the constitutive laws of the material, plays an important role. This principle leads to several restrictions of the specific form of anisotropic energies and therefore has a great importance for Jörg Schröder · Vera Ebbing Institut für Mechanik, Abteilung Bauwissenschaften, Fakultät für Ingenieurwissenschaften, Universität Duisburg-Essen, Universitätsstraße 15, 45117 Essen, Germany; E-mail: [email protected] Patrizio Neff Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Universitätsstraße 2, 45117 Essen, Germany; E-mail: [email protected] K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_17, © Springer Science+Business Media B.V. 2010

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the representations of tensor functions, Wang [25–29], Smith [22,23], Spencer [24]. Based on the principle of material symmetry, the isotropicization theorem [32] and Neumann’s Principle we arrive at the Principle of Isotropy of Space introduced by Boehler in the 1980s. This principle states, that tensor functions of anisotropic constitutive laws are expressible as isotropic tensor functions with structural tensors as additional agencies. The structural tensors, characterizing the symmetry group of the anisotropy class of concern, are the bases for an irreducible and coordinatefree representation of the anisotropic constitutive equations, in this context we refer to [33] and [32]. For the description of the transversely isotropic, triclinic, monoclinic and rhombic symmetry groups first and second-order structural tensors are sufficient, see [7, 8, 15, 33]. In this case the tensor-representation theory yields a complete polynomial basis of principal and mixed invariants, which are suitable for the construction of specific energy functions. For the representation of the remaining eight mechanically important anisotropy types structural tensors of order up to six are required: for the trigonal, tetragonal and cubic systems fourth-order and for the hexagonal systems sixth-order structural tensors are necessary, see [30–33]. A direct generalization of [7,8] with respect to the usage of structural tensors of order higher than two as variables within the isotropic tensor function is not exhaustively discussed in literature yet, except for some particular cases: Betten [2–4], Boehler [9] and Betten and Helisch [5, 6]. An analysis of the inherent symmetry of elastic materials based on the notion of a symmetry transformation is given in [13]. In the framework of polyconvexity, in the sense of Ball [1], coordinate-free representations of transversely isotropic and orthotropic energies were first proposed by Schröder and Neff [17, 18]. A direct extension of [18], by introducing a single fourth-order structural tensor, has been used by Kambouchev et al. [14] for the set up of a polyconvex cubic free-energy. Polyconvex triclinic, monoclinic and rhombic anisotropic functions, based on crystallographically motivated structural (metric) tensors, are discussed in [10–12, 19]. In this contribution a framework for developing polyconvex energy functions for the tetragonal, trigonal and cubic symmetry classes is proposed. Therefore, specific fourth-order single structural tensors, which can be decomposed in a sum of dyadic products, are introduced. In order to show the applicability of this procedure, we present the fitting of an trigonal fourth-order elasticity tensor near the reference state with polyconvex anisotropic invariants and compare the results with experimental data taken from the literature.

2 Mathematical and Mechanical Preliminaries In finite elasticity the mathematical treatment of boundary value problems is based on the direct methods of variations, i.e., finding a deformation ϕ which minimizes the elastic free-energy function W (F = Gradϕ) subject to specific boundary conditions. To guarantee the existence of minimizers of certain variational principles in finite elasticity the variational functional must be sequentially weakly lower semi-

Polyconvex Energies for Trigonal, Tetragonal and Cubic Symmetry Groups

223

continuous (s.w.l.s.) and coercive. Polyconvex functions are always s.w.l.s., quasiconvex and rank-one convex, see [1]. By considering smooth energy functions, the latter condition ensures the ellipticity of the corresponding acoustic tensor, i.e., material stability is then guaranteed. In anisotropic elasticity the principle of objectivity and the principle of material symmetry are important for the construction of the constitutive equations. The principle of objecticity is fulfilled since reduced constitutive equations in terms of the right Cauchy–Green tensor are used, i.e., ψ(C) = W (F)

with C = FT F .

(1)

The principle of material symmetry enforces the invariance of the constitutive equations with respect to the transformations Q ∈ G ⊂ O(3) of the material symmetry group G ⊂ O(3). Thus, the invariance condition ψ(C) = ψ(Q C QT )

∀Q∈G

(2)

must hold. For the description of the anisotropy of the material structural tensors, see e.g. [8], can be inserted as additional arguments into the free-energy function (2). Second-order structural tensors G and fourth-order structural tensors G characterize the underlying material symmetry group G ⊂ O(3) if G = Q G QT ∀ Q ∈ G ⊂ O(3) (3) G = Q Q : G : QT QT is valid. The index representation of G = Q Q : G : QT QT is given by ¯¯ ¯ ¯ GABCD = QAA¯ QB B¯ QC C¯ QD D¯ GAB C D . Inserting second-order and fourth-order structural tensors as further tensorial arguments into the free-energy function (2) leads to the representation ψ(C, G) = ψ(Q C QT , Q G QT , Q Q : G : QT QT )

∀ Q ∈ O(3). (4)

A comparison of (2) with (4) shows that the introduction of structural tensors as additional tensor agencies in the free-energy function extends the G-invariant functions (2) to isotropic tensor functions (4). For the construction of energy functions which satisfy this property we use the principal and mixed invariants in terms of the right Cauchy–Green tensor and the structural tensors as a polynomial basis.

3 Fourth-Order Single Structural Tensors In [33] it is shown that each mechanically important anisotropic symmetry group (C1 − C13 ) can be characterized by one single structural tensor (of order up to six). For the representation of the tetragonal, trigonal and cubic anisotropy fourth-order structural tensors are needed.

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In this work we propose fourth-order crystallographically motivated single structural tensors for tetragonal, trigonal and cubic symmetry classes of the form G=

3

ai ⊗ ai ⊗ ai ⊗ ai ,

(5)

i=1

where the vectors ai , i = 1, 2, 3 are the base vectors of the underlying crystallographically motivated base system. In the sequel the cartesian base vectors are denoted by e1 = (1, 0, 0)T , e2 = (0, 1, 0)T , e3 = (0, 0, 1)T . The base system of the tetragonal symmetry classes C4 and C5 is characterized by two orthogonal base vectors a1 and a2 perpendicular to the four-fold axis a3 . Whereas for C4 a1 and a2 can be taken as any two orthogonal vectors perpendicular to a3 , for C5 they must be aligned in directions of two mutually orthogonal two-fold axes of C5 , e.g., a 1 = a e1 , a 2 = a e2 , a 3 = c e 3 , (6) 2 | see Figure 1b. The anisotropic elasticity tensors at reference state C = 4 ψCC C=1 for the crystal type 4 and 5 in Voigt notation are given by the expressions ⎤ ⎡ C1111 C1122 C1133 C1112 0 0 ⎢ C1111 C1133 −C1112 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ C3333 0 0 0 ⎥ ⎥, ⎢ (7) C4 : C = ⎢ C1212 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ sym. C2323 0 ⎦ ⎣ C2323

⎡

C5 :

C1111 C1122 C1133 0 0 ⎢ C1111 C1133 0 0 ⎢ ⎢ ⎢ C3333 0 0 C=⎢ ⎢ C1212 0 ⎢ ⎢ sym. C2323 ⎣

0 0 0 0 0 C2323

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦

(8)

In the trigonal case (C8 and C9 ) we consider the basis of the associated rhombohedral cell denoted by a1 , a2 , a3 of equal lengths and including the same angle with one another. The three-fold axis is given along the (a1 + a2 + a3 )-direction. Whereas for C8 the base vectors ai can be taken as any of the described base vectors, for C9 the base vectors ai must be perpendicular to the three two-fold axes of C9 , see Figure 1a. Taking the vectors fi , i = 1, 2, 3 in directions of the three two-fold axes into account √ √ 1 1 3 3 e2 , f 3 = − e1 − e2 , (9) f 1 = e1 , f 2 = − e1 + 2 2 2 2 the base vectors are then given by a1 ⊥ f1 , a2 ⊥ f2 , a3 ⊥ f3 . In detail we obtain for the symmetry group C9


225

Fig. 1 (a) Trigonal (rhombohedral) cell (C9 ), (b) tetragonal cell (C5 ), (c) cubic cell (C7 ).

√

1 √ 1 3 3 − 3 b e2 + c e 3 , a 2 = b e1 + b e2 + c e 3 , a1 = 3 3 2 2 √ 3 1 3 a3 = − b e1 + b e2 + c e 3 . 3 2 2 The trigonal elasticity tensors of type 8 and type 9 appear in the forms ⎤ ⎡ C1111 C1122 C1133 0 C1123 C1113 ⎢ C1111 C1133 0 −C1123 −C1113 ⎥ ⎥ ⎢ ⎥ ⎢ C3333 0 0 0 ⎥ ⎢ ⎥, ⎢ C8 : C = ⎢ 1 ⎥ (C − C ) −C C 1111 1122 1113 1123 ⎥ ⎢ 2 ⎥ ⎢ sym. C 0 2323 ⎦ ⎣ C2323 ⎤ C1111 C1122 C1133 0 C1123 0 ⎢ C1111 C1133 0 −C1123 0 ⎥ ⎥ ⎢ ⎥ ⎢ C3333 0 0 0 ⎥ ⎢ ⎥. ⎢ C=⎢ 1 0 C1123 ⎥ ⎥ ⎢ 2 (C1111 − C1122 ) ⎢ sym. C2323 0 ⎥ ⎦ ⎣ C2323

(10)

(11)

⎡

C9 :

(12)

In the cubic case (C7 ) the three mutually orthogonal base vectors ai coincide with the three four-fold symmetry axes of C7 , i.e., a i = a ei ,

i = 1, 2, 3,

(13)

which are depicted in Figure 1c. The coefficient scheme of the elasticity tensor is

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J. Schröder et al. Table 1 Fourth-order structural tensors. Tetragonal system, type 4: ⎡

a1 = (a, b, 0)T , a2 = (−b, a, 0)T , a3 = (0, 0, c)T a 4 + b4

⎢ ⎢ ⎢ ⎢ ⎢ G=⎢ ⎢ ⎢ ⎢ ⎣

2a 2 b2

0

a 3 b − b3 a

0

0

a 4 + b4

0

b3 a − a 3 b

0

0

0

0

0

2a 2 b2

0

0

0

0

c4 sym.

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0 Tetragonal system, type 5: a1 = (a, 0, 0)T , a2 = (0, a, 0)T , a3 = (0, 0, c)T G = diag[a 4 , a 4 , c4 , 0, 0, 0] ⎛ ⎜ a1 = ⎝

√a 3 − √b 3 c 3 ⎡ 1 8

⎢ ⎢ ⎢ ⎢ ⎢ G=⎢ ⎢ ⎢ ⎢ ⎣

Trigonal system, type 8: ⎞ ⎛ ⎛ a a + b2 − b2 − √ − √ 2 3 2 3 ⎟ ⎜ ⎜ ⎟ a ⎟ a b ⎜ b ⎠, a2 = ⎜ ⎝ 2√3 + 2 ⎠, a3 = ⎝ 2√3 − 2 ⎞

C2

1 2 24 C 1 2 8 C

c 3 1 2 18 C c 1 2 18 C c 1 4 27 c

sym.

c 3

0

A

B

0

−A

−B

0 1 24

C2

0

0

−B

A

1 18

C c2

0 1 18

with

A=−

1 √ a 2 bc 4 3

+

1√ 3 b c, 12 3

B=−

1 √ ab2 c 4 3

+

⎞ ⎟ ⎟ ⎠

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

C c2

1√ 3 a c, 12 3

C = a 2 + b2

Trigonal system, type 9: √ √ √ a1 = (0, −b/ 3, c/3)T , a2 = (b/2, b/(2 3), c/3)T , a3 = (−b/2, b/(2 3), c/3)T ⎤ ⎡ 1 4 1 2 2 1 4 1√ 3 0 b c 0 8 b 24 b 18 b c 12 3 ⎥ ⎢ 1 2 2 1 4 ⎥ ⎢ 0 − 1√ b3 c 0 ⎥ ⎢ 8b 18 b c 12 3 ⎥ ⎢ 1 4 ⎥ ⎢ c 0 0 0 27 ⎥ G=⎢ ⎥ ⎢ 1 4 1√ 3 sym. b 0 b c ⎥ ⎢ 24 12 3 ⎥ ⎢ ⎥ ⎢ 1 2 2 0 ⎦ ⎣ 18 b c 1 2 2 18 b c Cubic system, type 7: a1 = (a, 0, 0)T , a2 = (0, a, 0)T , a3 = (0, 0, a)T G = diag[a 4 , a 4 , a 4 , 0, 0, 0]


⎡

C7 :

⎢ ⎢ ⎢ C=⎢ ⎢ ⎢ ⎣

C1111 C1122 C1122 C1111 C1122 C1111

0 0 0 C1212

sym.

0 0 0 0 C1212

227

⎤

0 0 0 0 0

⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(14)

C1212

The parameters a, b, c which appear in (6), (10) and (13) can be interpreted as additional material parameters. The resulting fourth-order structural tensors obtained by inserting e.g. (6), (10) and (13), respectively, into (5) are listed in Table 1. The used invariants in our formulation are the principal invariants I1 := tr C, I2 := tr[Cof C], I3 := detC, and mixed invariants J7 = C : G : C,

J8 = C : G : C2 ,

J9 = C2 : G : C2 .

(15)

Based on our experience for the construction of polyconvex anisotropic functions it is helpful to generate invariants, which are given in terms of the cofactor of the right Cauchy–Green tensor. This has two reasons: (i) this structure is directly correlated to the arguments appearing in the definition of polyconvexity itself and (ii) to the non-uniqueness of polyconvex functions. Therefore we introduce the additional invariants J10 = Cof C : G : C,

J11 = Cof C : G : Cof C,

J12 = Cof C : G : C2 . (16)

Further useful invariants are J4 = C : G : 1,

J5 = Cof C : G : 1,

J6 = C2 : G : 1 .

(17)

The invariants appearing in (16) can be expressed in terms of the invariants (15), (17) and the principal invariants, by exploiting the Cayley–Hamilton theorem. The functions J4 , J5 , J7 and J11 are polyconvex. Outline of the proof : Each function Ji , i = 4, 5, 7, 11 can be additively decomposed into the polconvex functions (tr[C(ai ⊗ ai )])k , k = 1, 2, i = 1, 2, 3 and (tr[Cof [C](ai ⊗ ai )])k , k = 1, 2, i = 1, 2, 3, respectively. In detail, we obtain the expressions J4 = J7 =

3 i=1 3 i=1

3

tr(ai ⊗ ai )tr[C(ai ⊗ ai )], J5 = tr[C(ai ⊗ ai

)]2 ,

tr(ai ⊗ ai )tr[Cof [C](ai ⊗ ai )],

i=1 3

J11 =

(18) tr[Cof [C](ai ⊗ ai

)]2 .

i=1

The proof of polyconvexity of the individual terms is documented in [17, 18], using the fact that the sum of polyconvex functions is polyconvex.

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4 Polyconvex Anisotropic Energy Function In the following we use a simple polyconvex energy function of the form ψ=

m

fj (J4j ) + fj (J5j ) + f j (J7j ) + gj (I3 ) ,

(19)

j =1

with the j -th invariants J4j = 1 : Gj : C,

J5j = 1 : Gj : Cof C,

J7j = C : Gj : C.

In detail, the polyconvex coercive anisotropic energy is given by ⎡ m n 3 1 (αrj +1) (β +1) ⎣ + J rj ψ= αrj J4j βrj 5j (αrj + 1)mj (βrj + 1)mj r=1 j =1 3 mj −γrj 1 (ηrj +1) + + I η J7j γrj 3 (ηrj + 1)mj rj

(20)

(21)

with the abbreviation mj := 1 : Gj : 1 and the restrictions αrj , βrj , ηrj ≥ 0, γrj ≥ −1/2. It should be remarked that the function 1 J (α + 1)mαj 4

(α+1)

+

1

(β

J β 5 + 1)mj

(β+1)

+

mj −γ I γ 3

(22)

satisfies the coercivity condition, see the proof in [19]. Additive combination of functions of the form (22) are also coercive, therefore, (21) is coercive. The second Piola–Kirchhoff stress tensor appears in the form m n −γ S= 2 (−3 mj I3 rj + r=1 j =1

3 1 β +1 α J5jrj )C−1 + J4jrj ( Gj : 1) α β rj rj (mj ) (mj ) (23) 3 2 β ηrj − J5jrj I3 C−1 ( Gj : 1)C−1 + ηrj J7j ( Gj : C) . β rj (m ) (mj ) j

Furthermore, the stress-free reference configuration condition S(C = 1) = 0 is automatically satisfied, i.e., 3 βrj +1 1+ m (mj )βrj j 2 ηrj ( Gj : 1) = 0 . + m (mj )ηrj j

n m S(C = 1) = 2 −3 mj +

r=1 j =1

3 1 α β mj rj − mj rj α β rj rj (mj ) (mj )

(24)


229

5 Approximation of Trigonal Elasticity Tensors In order to approximate the phenomenological response of real trigonal materials with the above-mentioned anisotropic polyconvex function (21) we fit the linearized fourth-order elasticity tensor near the reference state C0 , with C0 := 4∂CC ψ|C=1 ,

(25)

to experimental measurements. The tangent moduli C0 at the reference state is identical to the classical representation of the elasticity tensor in the small strain regime. This fact results from the linearization of the stress response functions at a natural state, i.e., ∂S Lin[S] = C0 : Lin[E] with C0 := 2 and S|C=1 = 0 (26) ∂C C=1 with the Green–Lagrange strain tensor E :=

1 (C − 1), 2

(27)

which reduces to the linear relation σ = C0 : , because σ = Lin[S] and the linear strain tensor is given by = Lin[E]. For the approximations we use experimental data on elasticities. In detail, the fitting of moduli is done by the minimization of the error function C(V )comp − C(V )exp , (28) e= C(V )exp where C(V )comp ∈ R6×6 denotes the computed tangent moduli C0 in Voigt notation. Furthermore, C(V )exp ∈ R6×6 designates the associated coefficient scheme of experimental values. The used norm of the matrix schemes is defined by 6 6 (V ) (V ) C = (Cij )2 . (29) i=1 j =1

The material parameter adjustments have been performed by the evolution strategy proposed by Schwefel [20]. In order to get a better imagination of the underlying anisotropic material behavior we plot the characteristic surfaces of Young’s moduli of the adjusted material; from this we get an impression of the anisotropy ratios. As a representative example we show the results of the fitting of the elasticities of the trigonal material Lithium niobate to the fourth-order elasticity tensor obtained from the proposed model, see (21). Here we set n = 1, m = 2 in (21) and take the trigonal structural tensor of crystal type 9 presented in Table 1 into account. The experimentally determined elasticities of Lithium niobate are taken from Simmons and Wang [21] and are depicted together with the computed elasticity

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⎡ C

(V )exp

203

⎢ ⎢ ⎢ =⎢ ⎢ ⎣

53 203

75 75 245

sym.

0 9 0 −9 0 0 75 0 60

(a) ⎡

203.0

53.0 203.0

⎢ ⎢ ⎢ C(V )comp = ⎢ ⎢ ⎣

75.0 75.0 245.0

sym.

0 0 0 9 0 60

z

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0 8.9 0 −8.9 0 0 75.0 0 60.1

(b)

0 0 0 8.9 0 60.1

⎤ ⎥ ⎥ ⎥x ⎥ ⎥ ⎦

y

(c)

Fig. 2 Trigonal material: (a) Experimental elasticities C(V )exp and (b) computed elasticities C(V )comp , rounded up/down to the given accuracy of the experimental elasticities, in [GPa]. (c) Characteristic surface of Young’s moduli. Table 2 Material parameter set for Lithium niobate. i

j

bj

cj

αij

βij

γij

ηij

1 1

1 2

2.27 0.00

1.97 1.61

0.84 2.28

0.79 2.75

–0.47 –0.17

0.67 13.97

moduli in Figure 2. Furthermore, the characteristic surface of Young’s moduli is also presented in Figure 2. The optimization yields a set of material parameters given in Table 2. The optimized structural tensor G1 exhibits the explicit expression ⎡ ⎤ ⎢ ⎢ G1 = ⎢ ⎢ ⎣

3.30 1.10 1.11 3.30 1.11 0.56 sym.

0 1.11 0 0 −1.11 0 0 0 0 1.10 0 1.11 1.11 0 1.11

⎥ ⎥ ⎥ ⎥ ⎦

(30)

and G2 appears in diagonal form as G2 = diag(0.0 , 0.0 , 0.25 , 0.0 , 0.0 , 0.0).

(31)

A comparison of the experimental data, see Figure 2 a, with the fitted numerical values for the elasticities, see Figure 2b, shows that the relative error e, defined in (28), is less than 0.1% and therefore negligible from the engineering point of view.


231

6 Conclusion In this contribution we have presented a framework for the construction of polyconvex energy functions suitable for the approximative description of trigonal, tetragonal and cubic hyperelastic materials. For this we have used polyconvex polynomial invariants in terms of the right Cauchy–Green tensor and fourth-order structural tensors. For the construction of the structural tensor the base system of the underlying crystal class has been taken into account. A representative example has been discussed in order to show the capability of the proposed energy function for the fitting of real anisotropic material behavior.

Acknowledgement Financial support from DFG (research grant SCHR 570/6, Ne 902/2) is gratefully acknowledged.

References 1. Ball, J.M.: Convexity conditions and existence theorems in non-linear elasticity. Archive for Rational Mechanics and Analysis 63, 1977, 337–403. 2. Betten, J.: Integrity basis for a second-order and a fourth-order tensor. International Journal of Mathematics and Mathematical Sciences 5(1), 1982, 87–96. 3. Betten, J.: Recent advances in applications of tensor functions in solid mechanics. Advances in Mechanics 14(1), 1991, 79–109. 4. Betten, J.: Anwendungen von Tensorfunktionen in der Kontinuumsmechanik anisotroper Materialien. Zeitschrift für Angewandte Mathematik und Mechanik 78(8), 1998, 507–521. 5. Betten, J. and Helisch, W.: Irreduzible Invarianten eines Tensors vierter Stufe. Zeitschrift für Angewandte Mathematik und Mechanik 72(1), 1992, 45–57. 6. Betten, J. and Helisch, W.: Tensorgeneratoren bei Systemen von Tensoren zweiter und vierter Stufe. Zeitschrift für Angewandte Mathematik und Mechanik 76(2), 1996, 87–92. 7. Boehler, J.P.: Lois de comportement anisotrope des milieux continus. Journal de Mécanique 17(2), 1978, 153–190. 8. Boehler, J.P.: A simple derivation of representations for non-polynomial constitutive equations in some cases of anisotropy. Zeitschrift für Angewandte Mathematik und Mechanik 59, 1979, 157–167. 9. Boehler, J.P.: Introduction to the invariant formulation of anisotropic constitutive equations. In: J.P. Boehler (Ed.), Applications of Tensor Functions in Solid Mechanics, CISM Courses and Lectures, Vol. 292, Springer, 1987, pp. 13–30. 10. Ebbing, V., Schröder, J. and Neff, P.: On the construction of anisotropic polyconex energy densities. In: Proceedings in Applied Mathematics and Mechanics, Vol. 7, 2007, pp. 4060,009– 4060,010. 11. Ebbing, V., Schröder, J. and Neff, P.: Polyconvex models for arbitrary anisotropic materials. In: Proceedings in Applied Mathematics and Mechanics, Vol. 8, 2008, pp. 10,415–10,416. 12. Ebbing, V., Schröder, J. and Neff, P.: Approximation of anisotropic elasticity tensors at the reference state with polyconvex energies. Archive of Applied Mechanics 79, 2009, 651–657.

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13. Jarić, J.P., Kuzmanović, D. and Golubović, Z.: On tensors of elasticity. Theoretical and Applied Mechanics 35(1–3), 2008, 119–136. 14. Kambouchev, N., Fernandez, J. and Radovitzky, R.: A polyconvex model for materials with cubic symmetry. Modelling and Simulation in Material Science and Engineering 15, 2007, 451–467. 15. Liu, I.S.: On representations of anisotropic invariants. International Journal of Engineering Science 20, 1982, 1099–1109. 16. Neumann, F.E.: Vorlesungen über die Theorie der Elastizität der festen Körper und des Lichtäthers, Teubner, 1885. 17. Schröder, J. and Neff, P.: On the construction of polyconvex anisotropic free energy functions. In: C. Miehe (Ed.), Proceedings of the IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains, Kluwer Academic Publishers, 2001, pp. 171–180. 18. Schröder, J. and Neff, P.: Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. International Journal of Solids and Structures 40, 2003, 401–445. 19. Schröder, J., Neff, P. and Ebbing, V.: Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors. Journal of the Mechanics and Physics of Solids 56(12), 2008, 3486–3506. 20. Schwefel, H.P.: Evolution and Optimum Seeking, Wiley, 1996. 21. Simmons, G. and Wang, H.: Single Crystal Elastic Constants and Calculated Aggregate Properties, The MIT Press, Massachusetts Institute of Technology, 1971. 22. Smith, G.F.: On a fundamental error in two papers of C.-C. Wang “On representations for isotropic functions, Parts I and II”. Archive for Rational Mechanics and Analysis 36, 1970, 161–165. 23. Smith, G.F.: On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. International Journal of Engineering Science 9, 1971, 899–916. 24. Spencer, A.J.M.: Theory of invariants. In: A. Eringen (Ed.), Continuum Physics, Vol. 1, Academic Press, 1971, pp. 239–353. 25. Wang, C.C.: On representations for isotropic functions. Part I. Isotropic functions of symmetric tensors and vectors. Archive for Rational Mechanics and Analysis 33, 1969, 249–267. 26. Wang, C.C.: On representations for isotropic functions. Part II. Isotropic functions of skewsymmetric tensors, symmetric tensors, and vectors. Archive for Rational Mechanics and Analysis 33, 1969, 268–287. 27. Wang, C.C.: A new representation theorem for isotropic functions: An answer to Professor G.F. Smith’s criticism of my papers on representations for isotropic functions. Part 1. Scalarvalued isotropic functions. Archive for Rational Mechanics and Analysis 36, 1970, 166–197. 28. Wang, C.C.: A new representation theorem for isotropic functions: An answer to professor G.F. Smith’s criticism of my papers on representations for isotropic functions. Part 2. Vectorvalued isotropic functions, symmetric tensor-valued isotropic functions, and skew-symmetric tensor-valued isotropic functions. Archive for Rational Mechanics and Analysis 36, 1970, 198–223. 29. Wang, C.C.: Corrigendum to my recent papers on “Representations for isotropic functions”. Archive for Rational Mechanics and Analysis 43, 1971, 392–395. 30. Xiao, H.: On isotropic extension of anisotropic tensor functions. Zeitschrift für Angewandte Mathematik und Mechanik 76(4), 1996, 205–214. 31. Zhang, J. and Rychlewski, J.: Structural tensors for anisotropic solids. Archives of Mechanics 42, 1990, 267–277. 32. Zheng, Q.S.: Theory of representations for tensor functions – A unified invariant approach to constitutive equations. Applied Mechanics Reviews 47, 1994, 545–587. 33. Zheng, Q.S. and Spencer, A.J.M.: Tensors which characterize anisotropies. International Journal of Engineering Science 31(5), 1993, 679–693.

Phase Transitions with Interfacial Energy: Interface Null Lagrangians, Polyconvexity, and Existence M. Šilhavý

Abstract For interfacial interactions of “separable type” the existence is proved of stable multiphase equilibrium states minimizing the total energy. The total energy includes a deformation dependent contribution along sharp interfaces separating the phases. The second gradients of deformation do not occur; the theory is based on interfacial null Lagrangians as determined in [11, 12]. The interfacial interaction is always of separable type if the number of phases does not exceed 3; for the number of phases ≥ 4, the separable nature of the interface interaction is an assumption.

1 Introduction We consider a body that can exist in states consisting of r inhomogeneous solid phases indexed by a = 1, . . . , r. We identify the body with the reference configuration represented by a bounded open set ⊂ R3 with Lipschitzian boundary. The states are pairs (y, P ) where y : → R3 is a deformation function and P = (E1 , . . . , Er ) is a partition of into subsets Ea of where Ea is the region occupied by phase a. The fact that one or several of the sets Ea is empty is not excluded. The total energy E(y, P ) of the state (y, P ) is given by E(y, P ) = Eb (y, P ) + Eif (y, P )

(1)

where Eb (y, P ) and Eif (y, P ) are the bulk and interfacial energies defined as follows. The bulk energy is Eb (y, P ) =

r

fâ (∇y) dL3

(2)

a=1 Ea

M. Šilhavý ˇ Žitná 25, 115 67 Prague 1, Czech Republic; Institute of Mathematics of the AV CR, E-mail: [email protected] K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_18, © Springer Science+Business Media B.V. 2010

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where fâ : Lin+ → R is the bulk free energy density of phase a expressed as a function of the deformation gradient F = ∇y. Throughout, Lin denotes the set of all second order tensors in R3 , interpreted as linear transformations from R3 to R3 , Lin+ is the set of all second order tensors with positive determinant, and L3 denotes the Lebesgue measure in R3 . The interfacial energy is given by Eif (y, P ) = (3) fˆ a,b (V y, na,b ) dH 2 . 1≤a 0 for L3 a.e.point of .

(12)

We allow fâ to take the value ∞ not only to incorporate Condition (iv), which leads to the orientation preserving property (12), but also to allow the effective domains eff dom fâ = {F ∈ Lin : fâ (F) < ∞}

Sharp Phase Interfaces

241

be strictly smaller than Lin+ . Thus one may assume that the effective domains are disjoint, and/or exclude states with deformation gradient in the spinodal region. Proof. Let M(, V ) denote the space of measures on with values in a finite dimensional vectorspace V and let M(µ) denote the mass of the measure µ ∈ M(, V ), i.e., M(µ) = |µ|() where |µ| denotes the total variation of µ. Let (yi , P i ) ∈ A(z0 ) be a minimizing sequence where we write P i = (Eai , . . . , Eri ). By the coercivity assumptions on fâ and a the sequences |∇yi |Lp , | cof ∇yi |Lq , H 2 (bd∗ Eai ), M(H a (yi , P i )) and M(pa (yi , P i )) are bounded. Combining the boundedness of |∇yi |Lp with the Dirichlet boundary data, one obtains the boundedness of |yi |W 1,p . Standard compactness theorems for Sobolev space and for the spaces of measures give that for some subsequence of (yi , P i ), denoted again (yi , P i ), we have (13) yi y in W 1,p (, R3 ),

nia H 2

cof ∇yi C in Lq (, Lin) bd∗ Eai , H a (yi , P i ), pa (yi , P i ) ∗ ∆a

in M(, X)

(14)

a = 1, . . . , r, for some y ∈ W 1,p (, R3 ), C ∈ Lq (, Lin), and ∆ ∈ M(, X) where nia is the measure theoretic normal to Eai . The boundedness of nia H 2 Eai says that the sequence of the derivatives of the characteristic functions 1Eai of Eai in is bounded in M(, R3 ). The imbedding theorem from BV functions (e.g., [1, corollary 3.49, chapter 3]) implies 1Eai → 1Ea

in L1 ().

(15)

for some set Ea ⊂ of finite perimeter, i.e., L3 ((Eai , Ea )) → 0,

(16)

where (Eai , Ea ) is the symmetric difference of Eai and Ea . Moreover, the limit in r r1 a Eai = 1 on gives a 1Ea = 1 and thus P := (E1 , . . . , Er ) is a partition of into sets of finite perimeter. Furthermore, if we write ∆a = (∆1a , ∆2a , ∆3a )

(17)

for the components of the X valued measure ∆a in the product X := R3 × Lin × R3 , then nia H 2 bd∗ Eai ∗ ∆1a in M(, R3 ) (18) and ∆1a = na H 2

bd∗ Ea

where na is the measure theoretic normal to Ea . The condition E(yi , P i ) < ∞ for each i and Hypothesis (iv) imply that det ∇yi > 0 for every i and L3 a.e. point of . From (13) by [9, lemma 4.1] then cof ∇yi cof ∇y in Lq (, Lin),

(19)

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det ∇yi det ∇y in L1 (K, R)

(20)

for each compact subset K of ; recall also that [see (13)] ∇yi ∇y in Lp (, Lin).

(21)

The equiintegrability of the sequences ∇yi and cof ∇yi and (21) and (16) yield 1Eai ∇yi 1Ea ∇y,

1Eai cof ∇yi 1Ea cof ∇y in L1 (, Lin)

and in particular, d H (yi , P i )v =

Eai

∇yi curl v dL3 →

d p(y , P )v = i

i

∇y curl v dL3 , Ea

3

Eai

∇y · ∇v dL3

cof ∇y · ∇v dL → i

Ea

for each v ∈ C0∞ (, R3 ). Hence (14) yields ∇y curl v dL3 = d∆2a v, cof ∇y · ∇v dL3 = d∆3a v Ea

Ea

p,q Gr ()

where we use the notation (17). Thus (y, P ) ∈ and H a (y, P ) = ∆2a and 3 pa (y, P ) = ∆a . Equations (14) and (18) give i 2 na H bd∗ Eai , H a (yi , P i ), pa (yi , P i ) ∗ (22) in M(, X). na H 2 bd∗ Ea , H a (y, P ), pa (y, P ) We now recall that a is nonnegative and convex and apply the Ioffe lowersemicontinuity theorem [1, theorem 5.8, chapter 5]. One then deduces from the weak convergences (21), (19) and (20) and the strong convergence (15) that for any compact subset K of we have a (∇yi , cof ∇yi , det ∇yi ) dL3 fâ (∇yi ) dL3 ≥ lim inf lim inf i i→∞ Eai i→∞ Ea ∩K ≥ a (∇y, cof ∇y, det ∇y) dL3 Ea ∩K = fâ (∇y) dL3 . Ea ∩K

The arbitrariness of K then gives i 3 ˆ lim inf fa (∇y ) dL ≥ i→∞

which implies

Eai

Ea

fâ (∇y) dL3 ,

243

Sharp Phase Interfaces

lim inf Eb (yi , P i ) ≥ Eb (y, P ). i→∞

(23)

Using (22) and the Reshetnyak lower-semicontinuity theorem (e.g., [1, theorem 2.38, chapter 2]), one obtains lim inf Eif (yi , P i ) ≥ Eif (y, P ). i→∞

(24)

Thus (23) and (24) provide lim inf E(yi , P i ) ≥ E(y, P ). i→∞

Clearly, (y, P ) ∈ A(z0 ).

4 Conclusion In the above we outlined a theory of equilibrium of states of coexistent phases in solids separated by interfaces bearing deformation dependent energy. Such interfaces generally support nonzero standard stresses [11, 12], which is not the case of an interface energy depending merely on the interface normal. The phase equilibrium is governed by the balance of the configurational stress [7, 8]. For phase interfaces separating more than 4 phases the theory is restricted by the separability assumption defined above.

References 1. Ambrosio, L., Fusco, N. and Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000. 2. Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63, 1977, 337–403. 3. Ball, J.M. and James, R.D., Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal. 100, 1987, 13–52. 4. Ball, J.M. and James, R.D., Proposed experimental tests of a theory of fine microstructure and the two-well problem, Phil. Trans. Royal Soc. Lond. 338, 1992, 389–450. 5. Ball, J.M. and Mora-Corral, C., A variational model allowing both smooth and sharp phase boundaries in solids, Comm. Pure Appl. Anal. 8, 2009, 55–81. 6. Fonseca, I., Interfacial energy and the Maxwell rule, Arch. Rational Mech. Anal. 106, 1989, 63–95. 7. Gurtin, M.E., The nature of configurational forces, Arch. Rational Mech. Anal. 131, 1995, 67–100. 8. Gurtin, M.E. and Struthers, A., Multiphase thermomechanics with interfacial structure, 3. Evolving phase boundaries in the presence of bulk deformation, Arch. Rational Mech. Anal. 112, 1990, 97–160. 9. Müller, S., Tang, Q. and Yan, B.S., On a new class of elastic deformations not allowing for cavitation, Ann. Inst. H. Poincaré, Analyse non linéaire 11 , 1994, 217–243. 10. Parry, G.P., On shear bands in unloaded crystals, J. Mech. Phys. Solids 35, 1987, 367–382.

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11. Šilhavý, M., Phase transitions with interfacial energy: A variational approach, Preprint, Institute of Mathematics, AS CR, Prague. 2008-10-22, 2008. 12. Šilhavý, M., Phase transitions with interfacial energy: Convexity conditions and the existence of minimizers, Preprint, Institute of Mathematics, AS CR, Prague. 2009-2-16, 2009.

A Unified Variational Setting and Algorithmic Framework for Mono- and Polycrystalline Martensitic Phase Transformations Erwin Stein and Gautam Sagar

Abstract The unified setting presented here is based on phase transformations (PTs) of monocrystalline shape memory alloys (SMAs) and includes polycrystalline SMAs whose microstructure is modeled using lattice variants of RVEs consisting of equal convex isotropically elastic grains. A pre-averaging scheme for randomly distributed polycrystalline variants of PT-strains is used transforming them into those of a fictitious monocrystal. A major point is the overall finite element based integration algorithm in time and space for both mono- and polycrystalline PTs with differences only on the material level, whereas the parametric time integration with quasi-convexification and the solution algorithm in space remains the same. Examples for full PT cycles and comparisons with experiment are presented.

1 Introduction Due to growing applications of shape memory alloys (SMAs), there is a great need to develop fairly accurate and efficient models to describe this complicated material response. There are several lines of development for engineering modeling at different length scales of crystal properties and related phase transformations (PTs) in mono- and polycrystalline SMAs. The literature on it is very rich, hence references are restricted to the specific topics of this paper. The micromechanical material properties are modeled at a macroscopic C 1 continuum theory using Bain’s principle [1] and the Cauchy–Born hypothesis of lattice-continuum link [2]. Based on these kinematic assumptions theories and computations of monocrystalline PTs have been published in, e.g., [3–7], and polycrystalline research results can be found in, e.g., [8–13]. Mono- and polycrystalline PTs are treated separately in these cited papers. Erwin Stein · Gautam Sagar Institute of Mechanics and Computational Mechanics (IBNM), Leibniz Universität Hannover, Appelstr. 9A, D-30167, Germany; E-mail: {stein, sagar}@ibnm.uni-hannover.de K. Hackl (ed.), IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries 21, DOI 10.1007/978-90-481-9195-6_19, © Springer Science+Business Media B.V. 2010

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Fig. 1 SME with quasiplasticity (a) and superelasticity (b), Ms : martensitic start temperature, Mf : martensitic finish temperature, As : austenite start temperature, Af : austenite finish temperature, θ: ambient temperature and Md : a critical temperature beyond which SE is not observed.

The intention and goal of this paper is a combined methodology in conjunction with a unified computational algorithm to describe martensitic PTs in mono- and polycrystalline SMAs for engineering applications.

2 Martensitic Phase Transformation of Metallic SMAs Martensite PT is usually considered as a diffusionless first-order transformation between ‘high’ temperature austenitic and ‘low’ temperature martensitic phases [1]. The specific feature of those shape memory alloys is the ability to ‘remember’ their initial state. In case of shape memory effect (SME) due to quasiplastic (QP) behavior after elastic deformation and subsequent PT (due to a critical driving force at a subcritical temperatures) a SMA will only recover its old shape after unloading if a second (higher) critical temperature is reached by heating. On the other hand, in case of superelasticity (SE) a SMA returns immediately to its initial shape during elastic unloading if the temperature of the material has at least the second critical value from the beginning of the process (Figure 1). SMAs are widely used, e.g. as peripheral stents which are designed for supporting the blood vessels or orthodontic wires to correct irregularities in the position of the teeth, etc., making use of superelasticity. PTs are also employed in actuating devices for many engineering applications e.g. in robotic muscle by applying quasiplasticity.

3 PT-models for Mono- and Polycrystals Both models are finally represented within the classical phenomenological thermomechanical theory for a macro C 1 continuum describing the transition from austenitic (with higher symmetry) to less stable martensitic (with lower symmetry) crystal configuration. Both models are based on Bain’s principle which states that the martensitic crystal structure is built along the smallest lattice strain [1]. Austenitic and martensitic crystallographic lattices and their deformations are described by Bravais lattices, using adequate linearly independent lattice vectors.

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247

According to the Cauchy–Born hypothesis [2], the deformation of the Bravais lattice vectors can be presented by the deformation gradient F of the C 1 point continuum. The coefficient matrices of the deformation gradients for phases i = 1, . . . , n of one austenitic to several martensitic phases are called the Bain-matrices or transformation matrices and usually marked by U ti . At linearized kinematics, the transformation strain tensor εti for phase i reads εti =

1 t2 (U − I) ≈ U ti − I , ∀x ∈ ⊂ R3 , 2 i

(1)

with I as the rank-2 identity tensor. Additional Constitutive Assumptions for Polycrystals: With the aim to develop a unified computational PT concept the following reasonable assumptions are introduced, yielding a sort of first order consistent approximation of the whole problem which is fairly approved by experiments. 1. All grains of a polycrystalline representative volume element (RVE) have the same topology and they are convex without empty volume. A RVE consists of a sufficient finite number of grains [8, 9]. 2. All grains of a RVE have same size (volume) [8, 10]. 3. All grains are kinematically C 1 compatible. 4. All grains of an alloy have the same number of phase variants. 5. The volumetric size of a particular variant is the same in every grain. 6. All grain orientations are deterministic and represented by Eulerian angles [8,10]. Remark: This is a further approximation because generally, there is a stochastic distribution of grain orientations which can be described by Young’s measure, provided the required data are available, see [9]. Available calculations show that the influence of this stochastic behavior can be neglected approximately for computed examples [9]. Deeper knowledge need further research. 7. A constant transformation strain is assumed for each phase variant of a grain [8–10]. Remark: This is motivated from the micromechanical theory of an inclusion in an elastic body for which a constant strain is assumed (instead of a phase variant in this paper). 8. The elastic anisotropy of martensitic and austenitic phases is neglected [3–8, 10– 15]. Remark: In general, elastic anisotropy in the material can arise especially due to rolling process which is considered in [9]. Here, it is assumed that this anisotropy can be neglected in the macro model. 9. The Reuß assumption (yielding lower bound) is used for homogenization, stating that all grains of a RVE are subjected to a uniform stress state which is equal to the applied macroscopic stress state [10].

248


4 PT-Modeling of Monocrystals Using C1 Macro Continuum According to Ball and James [16], the effective macroscopic free strain energy is given as the minimum of the free energies of all possible active phases of a crystal, = min ψi , i=1,...,n

with

ψi = ψiel + ψich ,

(2)

where ψiel and ψich are the elastic and the so-called chemical energy of the i-th phase variant. ψich was given by Abeyratane et al. [14] for 3 phases and used for n phase material in [4–7] as θ θ ch + ρli , ψi = ρci θ 1 − log θ0 θ0 where θ = θ − θ0 is the temperature difference from the current to the reference temperature θ0 , and li = l is the same latent heat for all phases. ρ is the mass density, and ci is the specific heat capacity of the i-th phase with the assumption ci = c. The heat energy related to volumetric expansion is neglected. For the defined type of problem is not quasiconvex, which implies nonexistence of deformations minimizing for prescribed boundary data and hence indicates the formation of microstructure [15]. To overcome the problem, quasiconvex relaxation is used connected with homogenization of the microstructure at minimum energy. The free energy, depending on the phase fractions can be decoupled as [5, 15], (•, ξ ) = ξ · ψ(•) + M (ξ ), • := a strain measure, (3) with the phase fraction vector ξ = ni=1 ξi e i , ei · ej = δij , −ξi ≤ 0, and the mass conservation condition e∗ · ξ − 1 = 0; e∗ = ni=1 ei∗ , |ei∗ | = 1, is the normal vector n−1 ⊂ Rn . ψ is the vector of phase energies, with of the convex n PT-polytope P ψ(•) = i=1 ψi (•i ) ei , where ei ∈ Rn , ψ ∈ Rn , and M (ξ ) is the free energy of mixing which has to be convex and non-positive. These conditions restrict the admissible range of ξ ∈ Pn−1 . el The elastic energy for phase i at linearized strains reads ψiel = 12 εel i : Ci : ε i , t el θ with the elastic strains εi = (ε − εi − ε i ), where ε is the total (kinematic) strain, and εθi εti (hence neglected) is thermal strain. Then ψiel follows as ψiel =

1 (ε − εti ) : Ci : (ε − εti ) , 2

(4)

˜ εt R ˜ T , where Ci is the rank-4 elasticity tensor of the addressed phase with, ε ti = R i ˜ is the crystal orientation matrix w.r.t to the global system (e.g. specimen). i, and R It is assumed that each phase has the same elasticity tensor, Ci = Cn .


249

At linearized strains the mixing energy for the general n-variant problem M ( was investigated by Govindjee et al. [15] using Reuß bound as LS εt , ξ ) = 1 n 1 n n t t t t −2 εi : C : εi + 2 εi : C : εj . i=1 ξi i=1 j =1 ξi ξj The mathematical model at finite strain is based on the multiplicative decomposition of the total deformation gradient F into elastic F e and transformation part F t , and using Neo-Hookean hyperelastic isotropic material [7]. The elastic free energy of phase i is split into the volumetric and deviatoric terms as ψiel = Wi (Jie ) + Wi (bei ), with the deviatoric part of elastic left Cauchy–Green −1 tensor bei := Jie −2/3 F ei F ei T ≡ Jie −2/3 bei , bei = F U ti 2 F T , Wi is a convex function of Jie := det F ei . Herein F ei is elastic deformation gradient of i-th phase and F ti is phase transformation gradient of i-th phase for n-phase system. The following explicit forms of a Neo-Hookean hyperelastic material are considered 1 1 1 e2 e e Wi (Ji ) := κ (Ji − 1) − ln Ji , Wi (bei ) := µ ( tr[bei ] − 3) , (5) 2 2 2 where µ and κ are the shear modulus and bulk modulus for linearized strains, respectively. The extension of the decoupled form of energy, Eq. (3), to finite strain was given in [7] which has the same form as for small strain. So far, a generalized analytical expression for mixing energy at finite strain obviously could not be found. An incremental extension of the free energy of mixing from small to finite strains for Neo-Hookean material with convex stress-strain function was given in [7]. The resulting free energy of mixing for a 2-phase system (1 means austenite, 2 means t t t martensite) is generally given as FMS ( b , ξ ) ≈ −ξ2 ψ2t ( b2 , J2t ) + ξ2 ξ2 ψ2t ( b2 , J2t ), T t ˜ bt R ˜ , which was used for the study of full PT cycle in CuAlNi [7] with b =R where only one martensitic phase was active besides the austenitic parent phase.

5 Approximated Modeling of Polycrystals Referring to the set of assumptions in section 3, an averaging scheme is used for polycrystalline PT strains associated to each variant of the crystals which are assumed to be randomly distributed, in contrast to the assumption of equally distributed textured crystals (e.g. achieved by rolling and/or pre-training) as applied by Jung et al. [12]. In [12] a simple habit plane-based multi-variant model has been proposed as extension of earlier model by Siredey et al. [13], and the macroscopic Lagrangian transformation strains were computed as in monocrystal. In order to simulate polycrystalline material, each finite element was assumed there to correspond to a crystal grain with proper texture. The material model was formulated in a large deformation material setting restricted to St. Venant elastic material i.e. finite rotations only. In this contribution, the averaged PT strains for each variant, ε ti , are treated similar to the transformed monocrystalline PT strains εti . By pre-homogenization,

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E. Stein and G. Sagar 1 1 1 V 2

Macro FE grid with GPs

V

V

V

3

1

V

2

2 1

2

V

2

3

V

1

N

g=1

g

n

N

å V

2

V

1 3

V V= å V = å

3

V

3

V

g

V

3

2

3

i=1 g=1 i

RVE

Fig. 2 Homogenization of the phases within the RVE yielding macro-data at Gauss points of finite elements.

the material properties of randomly oriented grains of a RVE are represented approximately at Gauss points (GPs) of finite elements (Figure 2). Under the Reuß assumption number 9, Section 3, the average PT-strain of variant i in an equivalent monocrystal of a polycrystalline RVE with N-grains follows as εti

=

N g V i

Vi

g=1

R g εti R gT ,

(6)

where g = 1, . . . , N is the number of grains in a RVE, i = 1..n isgthe number of g variants Vi is the volume of i-th variant in grain g, Vi = N g=1 Vi is the sum of i-th variant over all N-grains of a RVE, and R g is the orientation matrix of grain g. The orientation of g-th grain is described by the three random Euler angles (α g , θ g , & ηg ), where α g is a rotation with respect to axis x3 , ηg is a rotation w.r.t. axis x2 , and ηg is a rotation w.r.t. axis x3 again. The rotation matrices for these three angles are [8] ⎡ ⎡ ⎤ ⎤ cos α g − sin α g 0 cos θ g 0 sin θ g 0 1 0 ⎦ R 3 (α g ) = ⎣ sin α g cos α g 0 ⎦ , R 2 (θ g ) = ⎣ g 0 0 1 − sin θ 0 cos θ g ⎤ cos ηg − sin ηg 0 and R 3 (ηg ) = ⎣ sin ηg cos ηg 0 ⎦ . 0 0 1 ⎡

The combined rotation matrix – the orientation matrix – is R g = R 3 (α g ) R 2 (θ g )R 3 (ηg ). It is assumed that Vi1 = Vi2 = · · · = ViN , which yields ε ti =

N 1 g t gT R εi R . N g=1

(7)


251

This simple pre-averaging technique of PT-strains in randomly oriented grains is appropriate for pronounced texture as achieved by rolling of specimen and by pretraining. The average (effective) PT-strains for polycrystals using Young’s measure, derived in [9], can be simplified to the presented one, Eq. (7), when all the grains are assumed to have the same size. The pre-averaging technique transforms the polycrystalline material into an equivalent monocrystalline one. Hence the assumption of Ball and James [16], Eq. (8), can be used to get the following effective macroscopic strain energy of a RVE = min ψ¯i (•), with ψ¯i = ψ¯ iel + ψich , for RV E , (8) i=1..n

where the elastic energy at linearized strains, neglecting thermal strain, reads 1 ψ¯ iel = (ε − ε ti ) : Ci : (ε − ε ti ). 2

(9)

As in Section 4, the decoupled free energy for polycrystalline material is now given as ¯ ¯ ¯ M (ξ ), (•, ξ ) = ξ · ψ(•) + (10) n ¯ ψ¯ is the vector of the phase energies with ψ(•) = i=1 ψ¯i (•i ) ei , ei ∈ Rn , ψ¯ ∈ n M ¯ R , and (ξ ) is the free energy of mixing which is convex and non-positive. Here the phase fraction vector ξ holds for a representative monocrystal of the given polycrystalline material. The mixing energy for general n-variant problem given for monocrystal by Govindjee et al. [15], is then presented for polycrystal as M ¯ LS (ξ ) = −

n n n 1 1 ξi ε ti : C : ε ti + ξi ξj ε ti : C : εtj . 2 2

(11)

i=1 j =1

i=1

Remark: In the case of an incremental finite strain model the averaging technique using R g yields the averaged left Cauchy–Green PT tensor bti

N 1 g t gT = R bi R . N g=1

This can be used to extend the presented polycrystalline PT-model at linearized strain to finite strain.

6 Global Lagrangian Formulation and Time Integration The presence of kinematic constraints of the material suggests the enhancement of the global free energy function by a Lagrangian functional as it was presented by Siredy et al. [13] and Huang and Brinson [3], also used in [5,6] for small strains and in [7] for finite strains, now presented as

252


L(•, ξ , γ , δ) = ♦(•, ξ ) − γ · ξ + δ(e∗ · ξ − 1), for

(12)

¯ which depends on the problem, with the vector ♦ represents the free energy or γ (partial phase energy vector) and the scalar δ as Lagrangian parameter for mass conservation for n phases, fulfilling the Kuhn–Tucker conditions of the saddle point problem γi ≥ 0 , −ξi ≤ 0; i = 1 to n, and γ · ξ = 0.

(13)

The stress response of the material is the Gâteaux derivative σ ∗ (•, ξ ) = ∂• L =

n

ξi σ i (•), ∀x ∈ ,

(14)

i=1

σ i (•) is the stress contribution from phase i. The driving force (thermodynamical conjugate force), f = −∂ξ L ⇒ f +(•)−γ +∂ξ (ξ )+δe∗ = 0, and the specific dissipation condition reads D = f · ξ˙ ≥ 0. represents the vector of elastic energy ¯ represents the free energy of mixing M , M or ¯M. ψ or ψ. LS FS LS The phase fraction vector ξ is determined using the local maximum dissipation principle f · ξ˙ → Max . The Lagrangian functional for the convex PT function φ reads (f , λ) = −D + λφ −→ stat,

(15)

with the Kuhn–Tucker conditions λ ≥ 0, φ ≤ 0, and λφ = 0, where the Lagrangian parameter λ is the loading factor. The stationary condition (partial derivative of has to be 0) yields the evolution equation of the phase fraction vector, ξ˙ = λ∂f φ

(16)

with the normality rule, i.e. ξ is normal to the PT-surface φ. With analogies to the theory of stable inelastic deformations in elastoplasticity it can be deduced that the ‘phase transformation function’ φ has to be convex (similar to a convex yield function in plasticity) as φ = f − fc ≤ 0 ,

(17)

with the critical driving force fc as an energy barrier for initiating PT. This completes the constitutive equations of the PT-model. PT can only take place if an adequate norm of the driving force is equal to the critical valuefc . The L2 -norm f 2 is used to determine an incremental elastic step (implicit time step) approached by an ‘elastic trail’ method in the space of thermodynamical conjugate forces [5]. Backward Euler method is used for time integration. To integrate the resulting equations first in time, an implicit finite difference method is applied to the evolution equation (16) with incremental load factor λ = tλn+1 during the process time increment t, yielding


ξ n+1 − ξ n − λ∂f φ = 0

253

(18)

for every Gauss point. The Kuhn–Tucker conditions for the Lagrange multiplier λ result in λφ = 0 and λ ≥ 0.The subscript n + 1 indicating time is omitted in the sequel. The further local time-invariant constraints for the phase fractions are: e∗ ·ξ −1 = 0 (mass conservation condition), −ξi ≤ 0 (positiveness of the phase fractions), −γ · ξ = 0 (first Kuhn–Tucker condition) and γi ≥ 0 (second Kuhn-Tucker condition). The fundamental difficulty to know which variant will have active constraints associated with positiveness of phase fraction is overcome by using an active set strategy, as given in [5]. In order to detect whether a deformation increment is still elastic or already in the state of PT, the following operator split (at a frozen deformation state) is introduced with a trial driving force f t r at the beginning of a (process) time increment t as < 0 ⇒ λ for elastic step tr φ(f ) (19) ≥ 0 ⇒ λ for PT step. Due to the presence of γ and δ, f t r is an implicit vector functional. δ can be eliminated by using dot product of f t r with e∗ , but determination of γ and δ results in a local nonlinear optimization problem for f t r [5]. A unified variational algorithm for mono- and polycrystalline martensitic PTs is shown in Figure 3. In total, a local system of non-linear equations for the unknowns f n+1 , γ n+1 , λn+1 , δn+1 , ξ n+1 , results for every GP of the FE-mesh as f + (•) − γ + ∂ξ (ξ n + λ∂f φ) + δe∗ = 0

(20)

φ(f ) = f − fc = 0

(21)

ξnα

+ λ∂f α φ = 0 for all α ∈ active set

e∗ · (ξ n + λ∂f φ) − 1 = 0.

(22) (23)

They are solved iteratively by Newton’s method. In case of evolution of the phase fractions, n + m + 2 unknowns are to be determined by n + m + 2, Eqs. (20 to 23); n is the total number of phases and m is the number of constraints for an active set. All the unknowns are stored as the vector X := {f , γ , λ, δ} and the residuals in the vector R := {(20), (21), (22), (23)}. The iteration tangent reads (see also [5]) K=

∂R , ∂X

(24)

from which the material tangent at small strain follows as CCO = C −

n n i=1 j =1

Dij σ i (ε) ⊗ σ j (ε) ,

(25)


254

Fig. 3 Unified variational algorithm for mono- and polycrystalline martensitic PTs.

and full algorithmic tangent at finite strain is given as c

algo

=

n i=1

ξi ci −

n n

cPT ij ,

(26)

i=1 i=1

e e where cPT ij = D ij [σ i (bi ) ⊗ σ j (bj )] is the spatial PT-tangent for interacting phases i and j , (27) with D = ∂f φ ⊗ v + λ∂f ∂f φ A = Dij ei ⊗ ej .

Herein A is the upper left n x n block of K −1 , and v is the vector of the first n entries of the (n + m + 1)-th row of K −1 . The time integration of presented model is performed within the FE-program Abaqus via UMAT interface; in case of nonlinear deformation processes this interface requires the Jaumann rate of Kirchoff stress tensor as the tangent in the current configuration in order to get quadratic convergence [17].

7 Examples First, the strain-controlled SE tension experiment with polycrystalline NiTi alloy is computed, and comparisons of experimental data [18] with numerical results are presented. Next the comparison of computational results of a strain-controlled SE tension test for both, mono- and polycrystalline CuAlNi alloys are presented. The computations, using Abaqus, are strain-controlled and performed in two steps with a prescribed total axial displacement in the first step and the related load


(a) Geometry and boundary conditions of specimen.

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(b) Comparison of computed and experimental stress-strain data in the loading direction.

Fig. 4 Superelastic polycrystalline NiTi. Table 1 Material data of polycrystalline NiTi SE specimen. Young’s modulus E Poisson ratio ν Mass density ρ Specific heat c

68,200 N/mm2 0.3 6.45 kg/m3 460 J/(kg·K)

Latent heat l Energy barrier fc Equilibrium temperature θ0 Ambient temperature θ

167,000 J/kg 7.5 N/mm2 293 K 296 K

reduction to zero in the second step. The temperature is kept constant during the whole cyclic PT- deformation processes.

7.1 Superelastic Polycrystalline NiTi Numerical results obtained for a strain controlled uniaxial tensile test are compared with experimental results carried out by Lammering and Vishnevsky [18]. The wire material chosen in [18] is a nearly equiatomic NiTi polycrystalline alloy which shows superelastic behavior at room temperature. The experimental setup details are given in [18]. The spatial dimensions of the wire and the boundary conditions for uniaxial tension test are shown in Figure 4(a). For finite element computation the discretization was carried out with 432 linear B-bar hexahedral element (of the type C3D8 in Abaqus) which is sufficiently accurate due to convergence studies. The material parameters used in computation are: Young modulus, specific heat, mass density, critical force, Poisson’s ratio [12] and latent heat [19]. The remaining relevant material parameters are chosen to be θ0 = 293 K and θ = 296 K. All parameters are listed in Table 1. The Bain-matrices (transformation stretch matrices) of the cubic to monoclinic-I transformation of NiTi crystals for α = 1.0243, γ = 0.9563, δ = 0.058 and = −0.0427 are obtained from [1], where

256


α, γ = 0.9563, δ = 0.058, are the transformation stretches determined from the lattice parameters of the parent phase (austenite) and product phase (martensite). For numerical calculations of the tensile behavior of NiTi polycrystals, the RVE is consists of 27 grains (which is sufficient according to test with 73 and 103 grains), are randomly oriented with respect to the loading axis. The orientations of grains are described by Eulerian angles which are obtained from all possible combinations of assumed angles 10◦ , 20◦ and 30◦ . The averaged PT-stains of an equivalent monocrystal of N-grain polycrystalline RVE are obtained from Eq. (7). The numerical results are compared with the strain-controlled SE tension experiment for polycrystalline NiTi, see Figure 4(b). The computed stress and strain data are obtained from averaging the values over all the Gauss points in the cross section in the middle of the prismatic specimen. Three numerical tests for 3, 4 and 5% axial tensile strain are presented. Comparison of the experimentally measured stress-strain function with the numerically obtained data at linearized strains shows good agreement. The small reduction in the stiffness during elastic loading is evident in the experimental result due to the presence of a R-phase. NiTi transforms from the cubic to trigonal or rhombohedral ‘R-phase’ before transforming to martensite. Thus two transformation matrices describe the total transformation. The first, describes the deformation from austenite to the R-phase and the second from austenite to martensite phase. Since the presence of R-phase is not included in the presented PT-model, the numerically computed elastic loading curve has deviation from the experimental one. Furthermore, the slight deviation of the computed results with same elastic modulus from the experimental one during elastic unloading is due to the different elastic moduli of austenite and martensitic phases with EA > EM .

7.2 Superelastic Mono- and Polycrystalline CuAlNi A finite element computation shows the effect of pre-homogenization, Eq. (7), on PT- strains. The dimensions of the specimen are taken from the experiment carried out on superelastic monocrystalline CuAlNi material by Xiangyang et al. [20]. The shape of the specimen with flat rectangular cross-section and the 3D finite element meshing with tri-linear tetrahedral elements are depicted in Figure 5(a). The discretization is carried out with 1,350 linear tetrahedral elements (of the type C3D4 in Abaqus). Material parameters used in computation, given in Table 2 are taken from [6]. The latent heat l is updated to 4,550 J/kg. The transformation matrices of the cubic to orthorhombic transformation of CuAlNi crystals are taken from [1] with α = 1.0619, β = 0.9178 and γ = 1.023. of monocrystalline specimen [20] reads The spatial orientation matrix R ⎞ ⎛ 0.2019 −0.777 −0.596 ⎟ =⎜ R ⎝ −0.0756 0.5934 −0.8017 ⎠ . 0.9767 0.2062 0.0597


(a) Geometry, boundary conditions and 3D finite element discretization of specimen.

257

(b) Computed stress-strain curves of mono- and polycrystalline SE of CuAlNi.

Fig. 5 Superelastic mono- and polycrystalline CuAlNi. Table 2 Material data of mono- and polycrystalline CuAlNi SE specimen. Young’s modulus E Poisson ratio ν Mass density ρ Specific heat c

25,960 N/mm2 0.35 8.0 kg/m3 400 J/(kg·K)

Latent heat l Energy barrier fc Equilibrium temperature θ0 Ambient temperature θ

4,550 J/kg 0.001 N/mm2 236 K 296 K

tR T . The computations are done for a It transforms the PT-strain as εti =: Rε i monocrystalline and a fictitious polycrystalline SE CuAlNi specimen. The RVE of polycrystalline CuAlNi material consists of 27 grains which are described in similar fashion (as for NiTi in previous subsection) by Eulerian angles. The averaged PT-strains of an equivalent monocrystal for N-grain polycrystalline RVE is obtained from Eq. (7). Figure 5(b) shows the computed stress-strain functions of mono- and polycrystalline SE CuAlNi for the loading direction. One can see that the transformation strain in monocrystal is nearly 6% whereas in polycrystal the value is about 2.5% which correctly captures the experimental observation [1]. Hence the presented preaveraging technique is a reasonably good engineering approach to model polycrystalline SMAs under the condition that the used assumptions are fairly fulfilled.

8 Concluding Remarks The unified variational concept for monocrystalline PTs (with QP-SME and SE) is extended to polycrystalline SMAs with pre-homogenized PT-strains using a number of randomly oriented single crystal grains characterized by Eulerian angles. For

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both mono- and polycrystalline PT-models, a unified algorithmic structure is obtained. The new polycrystalline PT-model successfully captures basic features such as stress-strain behavior and gradual transformation which is approved by comparisons with experimental data.

References 1. K. Bhattacharya, Microstructure of Martensite: Why It Forms and How It Gives Rise to the Shape-Memory Effect, Oxford University Press, Oxford, 2003. 2. J.L. Ericksen (Ed.), The Cauchy and Born Hypothesis for Crystals, Academic Press, 1984. 3. M. Huang and L.C. Brinson, A multivariant model for single crystal shape memory alloys behavior, Journal of the Mechanics and Physics of Solids 46, 1998, 1379–1409. 4. G.J. Hall and S. Govindjee, Application of the relaxed free energy of mixing to problems in shape memory alloy simulation, Journal of Intelligent Material Systems Structures 13, 2002, 773–782. 5. S. Govindjee and C. Miehe, A multi-variant martensitic phase transformation model: Formulation and numerical implementation, Comput. Methods Appl. Mech. Engrg. 191, 2001, 215–238. 6. E. Stein and O. Zwickert, Theory and finite element computations of a unified cyclic phase transformation model for monocrystalline materials at small strains, Comput. Mech. 40, 2007, 429–445. 7. E. Stein and G. Sagar, Theory and finite element computation of cyclic martensitic phase transformation at finite strain, Int. J. Numer. Meth. Engng. 74(1), 2008, 1–31. 8. M. Huang, X. Gao and L.C. Brinson, A multivariant micromechanical model for SMAS. Part 2. Polycrystal model, Int. J. Plasticity 16, 2000, 1371–1390. 9. K. Hackl and R. Heinen, A micromechanical model for pretextured polycrystalline shapememory alloys including elastic anisotropy, Continuum Mech. Thermodyn. 19, 2008, 499– 510. 10. F.A. Nae, Y. Matsuzaki and T. Ikeda, Micromechanical modeling of polycrystalline shapememory alloys including thermo-mechanical coupling, Smart Mater. Struct. 12, 2003, 6–17. 11. L.C. Brinson, R. Lammering and I. Schmidt, Stress induced transformation behaviour of a polycrystalline NiTi shape memory alloy: Micro and macromechanical investigations via in situ optical microscopy, J. Mech. Phys. Solids 52, 2004, 1549–1572. 12. Y. Jung, P. Papadopoulos and R.O. Ritchie, Constitutive modelling and numerical simulation of multivariant phase transformation in superelastic shape-memory alloys, Int. J. Num. Methods Engrg. 60, 2004, 429–460. 13. N. Siredey, E. Patoor, M. Berveiller and A. Eberhardt, Constitutive equations for polycrystalline thermoelastic shape memory alloys. Part I: Intergranular interactions and behavior of the grain, Int. J. Solids Struct. 36, 1999, 4289–4315. 14. R. Abeyaratne, S. Kim and J. Knowels, A one-dimensional continum model for shape memory alloys, Int. J. Solids Structures 31, 1994, 2229–2249. 15. S. Govindjee, A. Mielke and G.J. Hall, The free-energy of mixing for n-variant martensitic phase transformations using qausi-convex analysis, J. Mech. Phys. Solids 52, 2003, I–XXVI. 16. J.M. Ball and R.D. James, Fine phase mixtures and minimizers of energy, Arch. Rat. Mech. Anal. 100, 1987, 13–52. 17. E. Stein and G. Sagar, Convergence behavior of 3-D finite elements for Neo-Hookean material models using Abaqus-Umat, Int. J. Computer-Aided Engrg. Software 25(3), 2008, 220–232. 18. R. Lammering and A. Vishnevsky, Investigation of local strain and temperature behaviour of superelastic NiTi wires, in Proceedings of the First Seminar on the Mechanics of Multifunctional Materials, Rep. No. 5, 2007, pp. 77–82.


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19. K.B. Gilleo, Photon-conducting media alignment using a thermokinetic material, US Patent 6863447, http://www.patentstorm.us/patents/6863447/fulltext.html, 2005. 20. Z. Xiangyang, S. Qingping and Y. Shouwen, A non-invariant plane model for the interface in cualni single crystal shapememory alloys, J. Mech. Phy. Solids 48, 2000, 2163–2182.

Dissipative Systems in Contact with a Heat Bath: Application to Andrade Creep Florian Theil, Tim Sullivan, Marisol Koslovski and Michael Ortiz

Abstract We develop a theory of statistical mechanics for dissipative systems governed by equations of evolution that assigns probabilities to individual trajectories of the system. The theory is made mathematically rigorous and leads to precise predictions regarding the behavior of dissipative systems at finite temperature. Such predictions include the effect of temperature on yield phenomena and rheological time exponents. The particular case of an ensemble of dislocations moving in a slip plane through a random array of obstacles is studied numerically in detail. The numerical results bear out the analytical predictions regarding the mean response of the system, which exhibits Andrade creep.

1 Introduction The fundamental question addressed in this work is: How does a dissipative system behave when it is placed in contact with a heat bath? Specifically, it is desired to derive a variational characterization of the trajectories of the system, i.e., a functional whose minimizers are the system trajectories. For definiteness, consider systems whose state is described by an N-dimensional array x ∈ RN of generalized coordinates. The energetics of the system are described by an energy function E(t, x). The explicit time dependence of E may arise, for example, as a result of the application Florian Theil · Tim Sullivan Mathematics Institute, Warwick University, Coventry, CV47AL, United Kingdom; E-mail: {f.theil, t.j.sullivanl}@warwick.ac.uk Marisol Koslowski School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907-2088, U.S.A.; E-mail: [email protected] Michael Ortiz Engineering and Applied Sciences Division, California Institute of Technology, Pasadena, CA 91125, U.S.A.; E-mail: [email protected]


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to the system of a time-dependent external field. In addition, the system is assumed to be dissipative, and, therefore, the equilibrium equations are of the form: 0 ∈ ∂x E(t, x) + ∂(x), ˙

(1)

where is a dissipation potential. These equilibrium equations define a set of ordinary differential equations which, given appropriate conditions at, say, t = 0, can be solved uniquely for the trajectory x(t), t ≥ 0. If (z) = 12 |z|2 one obtains the classic gradient flow of the potential E. If is homogeneous of degree one, then the system becomes rate-independent: i.e., if x(t) solves (1) for E and ϕ is a reparame˜ ·) = E(ϕ(t), ·). Another trization of time, then x(t) ˜ = x(ϕ(t)) solves (1) for E(t, direct consequence is the stability requirement ∗ (−∂x E(t, x)) < ∞,

(2)

where ∗ is the Legendre-transform of . Rigorous analysis and a review of the literature on equation (1) in the rate-independent case can be found in [12, 13]. To include the effect of thermal noise we leave the purely deterministic framework and consider stochastic systems where the the trajectories are no longer unique but random and define a probability measure on path space. More precisely, as a first step we replace the time-continuum {t ≥ 0} by a discrete set 0 = t0 < t1 < . . . of times and consider the Markov chain with transition probability densities proportional to xn+1 − xn exp − E(tn+1 , xn+1 ) + |tn+1 − tn | εn . tn+1 − tn The parameter ε measures the strength of the thermal fluctuations. As a second step we determine the limiting process which is obtained when εn = θ (tn+1 − tn ) and the fineness of the time discretization h = supn (tn+1 − tn ) tends to 0. The constant θ is proportional to the temperature, the proportionality constant is not known and has to be estimated by comparing the model predictions with measurements. In the classical case of linear kinetics, in which (z) = 12 |z|2 , the limiting pathmeasure is generated by well-known stochastic differential equations. In this paper we will mostly focus on the rate-independent case where is 1-homogeneous. Here the limiting measure is generated by ordinary differential equations, i.e. the evolution is deterministic. However, the differential equation depends in a nontrivial way on E and . As a representative example of application, the theory is applied to the case of Andrade creep in metals; the theory shows that Andrade creep represents the meanfield behavior of an ensemble of linear elastic dislocations moving on a slip plane through a random array of obstacles under the action of an applied resolved shear stress at finite temperature. In order to establish this connection, the energetics and dynamics of the dislocation ensemble are described by means of the Koslowski– Cuitiño–Ortiz phase-field model [10]. The resulting governing equations are sim-

Disspative Systems in Contact with a Heat Bath

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plified by means of a mean-field approximation. The analysis then shows that the slip strain grows as a 1/3 power of time, i.e., the system exhibits Andrade creep.

2 Heuristic Derivation We proceed to introduce the basic scheme that accords probabilities to the trajectories of a dissipative system. The first step in the derivation is to discretize the equations of evolution in time and to reinterpret the time-discretized equations as defining the optimal transport of probability measures. An interior-point regularization of the Kantorovich form of the transport problem then effectively places the system in contact with a thermal bath. Conveniently, the regularized incremental problem can be solved explicitly in closed form for the transition probability disbribution. The small-time-step limit is considered in Section 3.

2.1 Heuristic Derivation If the system is conservative, i.e. (x) ˙ ≡ 0, then the instantaneous state x(t) of the system at time t follows directly from energy minimization, i.e., from the problem: E(t, x(t)) = inf E(t, y). y∈RN

(3)

This variational framework can be extended to dissipative systems by time discretization [12,13,15–17]. Specifically, consider a discrete time incremental process consisting of a sequence of states xn ∈ RN at times tn = nh, h > 0 fixed, and introduce the incremental work function W : xn+1 − xn . (4) Wn (xn , xn+1 ) = E(tn+1 , xn+1 ) − E(tn , xn ) + |tn+1 − tn | tn+1 − tn It should be carefully noted that W (xn , xn+1 ) is determined by both the energetics and the kinetics of the system and depends on both the initial and the final configuration of the system over the time step. For a given state xn the state xn+1 is a minimizer of Wn . The incremental problem above can be re-interpreted as an optimal transport problem in which the incremental work function W as defined by (4) is regarded as a cost function, i.e., as the cost of rearranging the system from one configuration, xn , to another one, xn+1 . Consider the probability density functions ρn : RN → R describing the state of the system at each time step tn = nh. The corresponding optimal transport problem for the (n + 1)-th time step is:

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inf Fh [u], u(xn , xn+1 ) dxn+1 = ρn (xn ), subject to: RN u(xn , xn+1 ) dxn dxn+1 = 1,

(5a) (5b) (5c)

RN

where u(xn , xn+1 ) = ρn+1 (xn+1 |xn )ρn (xn ) is the joint probability density of two states xn and xn+1 at successive time steps and W (xn , xn+1 )u(xn , xn+1 ) dxn dxn+1 . (6) Fh [u] := RN

In addition, it follows that ρn+1 (xn+1 ) =

RN

u(xn , xn+1 ) dxn .

(7)

Problem (5) is a linear programming problem. Introducing Lagrange multipliers λ : RN → R and µ ∈ R, define the Lagrangian Lh [u] := Fh [u] + u(xn , xn+1 )λ(xn ) dxn dxn+1 + µ u(xn , xn+1 ) dxn dxn+1 . RN

RN

(8) In order to model the effect of a heat bath of “temperature” ε > 0, additionally introduce the interior-point regularization: ε Lh [u] := Lh [u] + ε u(xn , xn+1 ) log u(xn , xn+1 ) dxn dxn+1 . (9) RN

The second term on the right-hand side is the negative of the well-known Gibbs– Boltzmann entropy functional for the probability density u, and has the effect of heavily penalizing deterministic, coherent evolutions. The unique minimizer of the extended Lagrangian subject to the constraints (5b) and (5c) is given by W (xn , xn+1 ) ρn (xn ) u(xn , xn+1 ) = exp − , (10) Z(xn ) ε where the partition function Z is given by W (xn , xn+1 ) Z(xn ) := dxn+1 . exp − ε RN Finally, (7) yields the explicit representation W (xn , xn+1 ) ρn (xn ) ρn+1 (xn+1 ) = exp − dxn . ε RN Z(xn )

(11)

(12)


265

Note that, in order for (12) to make sense, the work function W (xn , xn+1 ) must grow rapidly enough as |xn+1 | → +∞ to ensure that Z(xn ) is finite for each xn ∈ RN . Physically, setting physical temperature = ε/kB in terms of Bolzmann’s constant kB , ε can be interpreted as the temperature of heat bath with which the system is coupled. From now on ρnh,ε we be interpreted as the transition probability of a Markov chain, i.e. h,ε (· | Xn ). (13) law(Xn+1 | X1 , . . . , Xn ) = ρn+1 In [18] we give a more extensive discussion together with illustrative examples.

3 Mathematical Results We consider the limiting behavior of the Markov chain (13) as h tends to 0 and ε = θ h, where θ is a temperature-like constant. If the friction potential is 1homogeneous, and the potentials E and are convex and coercive we can characterize the asymptotic behavior completely. It turns out that the limiting process is deterministic and can be obtained as a solution of an ordinary differential equation. Theorem 1. Let be 1-homogeneous, coercive and convex and define ∗ ∗ F (v ) = log dv exp −v , v − (v) . Let Xnh,ε be a realization of the Markov chain (13). If ε = h and E has the property that F (DE(·, t)) be convex, then as h ↓ 0, (X[th,θh / h] ) converges in probability in ∞ N L ([0, T ], R ) to the differentiable, deterministic process y = y θ satisfying the ordinary differential equation y˙ = −θ DF [Dy E(t, y)], (14)

where F (v ∗ ) = log dv exp (−v ∗ , v − (v)) . More precisely, for any T > 0, λ > 0,

h,θh

√ P sup Xt − ytθ 2 ≥ λ ∈ O h as h ↓ 0. (15) 0≤t ≤T

The proof can be found in [18]. Note that F (v ∗ ) < ∞ if and only if ∗ (−v ∗ ) < ∞, i.e. we recover the stability criterion (2) although the evolution (14) is no longer rate-independent. In many applications, the dissipation potential may not be coercive, so that F is infinite everywhere. In some cases the analysis can be carried over, an important application of this generalization of the KCO-model which is discussed in the next section.

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Corollary 1 (Degenerate friction potential ). (a) Let V be a finite-dimensional vector space and : V → [0, ∞) be given by (z) = σi |zi | i

where σi > 0 for all i ∈ {1 . . . n}. Then log σi2 − (vi∗ )2 . F (v ∗ ) = − i

(b) Let X be another vector space such that V is imbedded into X, E(t, u) be timedependent, coercive quadratic form on X and the reduced quadratic form Ered : [0, T ] × V → R be defined by Ered (t, ξ ) = min{E(t, u) | π(u) = ξ }, where π is the orthogonal projection induced by the imbedding. Then for each θ > 0, T > 0, as h ↓ 0 the projection π(X[th,θh / h] ) converges in probability in ∞ n L [0, T ]; R to the differentiable, deterministic process y = y θ satisfying y˙t = −θ DF (Dξ Ered (t, ξ )).

(16)

The proof can be found in [18].

4 Application to Crystallographic Slip and Creep In this section, the preceding formalism and analysis are applied to Andrade creep, also known as β-creep. In 1910, Andrade [1, 2] reported that as a function of time, t, the creep deformation, y, of soft metals at constant temperature and applied stress can be described by a power law y(t) ∼ t 1/3 . Similar behavior has been observed in many classes of materials, including non-crystalline materials. Recently, it has been suggested that Andrade creep is an example of critical behavior [5–7]. In these theories, creep is considered as an instance of critical avalanche formation analogous to sandhill models introduced to explain self-organized criticality. However, creep differs from sandhill behavior in two notable respects: the avalanches are nucleated by thermal activation; and the system exhibits strain hardening. Several numerical simulations are able to predict the power law behavior of the plastic deformation at constant temperature and applied stress in agreement with experimental observation [3, 7, 14]. Those models also predict a transition to steady state creep at later times, which results from recovery. The analysis will proceed in four steps: the first is a presentation of an infinitedimensional phase-field model for dislocation dynamics on a single slip plane (thought of as the 2-torus, T2 ). This model has been proposed by Koslowski, Cuitiño


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and Ortiz in [10] and a mathematically rigorous analysis of the line-tension limit can be found in [8, 9]. Secondly, this infinite-dimensional evolution will be reduced to a finite-dimensional one that takes account of the values of the phase field at certain “obstacle sites”. Thirdly, a mean-field approximation will be derived to reduce this large finite-dimensional evolution to a one-dimensional model of the energeticsand-dissipation type analyzed earlier. It will be shown that Andrade’s t 1/3 creep law follows from the reduced model when linear hardening is assumed. Finally, numerical simulations of phase-field model for dislocation dynamics are carried out. These simulations show excellent agreement with the reduced model.

4.1 Derivation of the Mean-Field Model The simplest setting is obtained if the reference configuration of the material is given by the unit torus T2 = [0, 1]2 and the potential energy of a phase-field u, which represents the amount of slip in units of the Burgers vector, is given by E(s, u) :=

k∈Z2 \{0}

µb2 1 2 |u(k)| ˆ −b·s 4 1/K + d/2

T2

u(x) dx,

(17)

where k is the wavenumber, K :=

ν k22 1 |k|2 − , 1−ν 1 − ν |k|2

u(k) ˆ := T2 e2πix·k u(x) dx is the k-th Fourier component of u, b is the Burgers vector, s is the applied shear stress, µ is the shear modulus, ν is Poisson’s ratio and d is the interplanar distance. The first term in the energy E is invariant under translations, i.e. E(s, u + u ) = E(s, u) + s · b u (18) if u is a constant field. The obstacles are assumed to be disk-like subsets Br (pi ) ⊂ T2 where p1 , . . . , pn ∈ T2 are the centers of the obstacles and Br (pi ) is the disk with radius 0 < r 1 and center at pi . Typically it will be assumed that the positions pi are random and that the phase field u is constant within each obstacle, i.e. u(x) = ξi if x ∈ Br (pi ). The dissipation functional depends on the obstacles and is defined as follows: (u) ˙ = κ |u(x)| ˙ dx. (19) p∈O

Br (p)

Note that in this formulation the friction functional vanishes on an infinite dimensional set, hence the assumptions of Theorem 1 are not satisfied. However, since is a weighted L1 -norm, Corollary 1 implies formally that the projection of the infi-

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nite dimensional Markov chain onto the values of the elastic field on the obstacles converges to the solution of a reduced differential equation. This differential equation is determined by constructing the appropriate Schur complement of the elastic energy, i.e. by minimizing out the the degrees of freedom that do not experience friction. For each ξ ∈ RO , define the reduced elastic energy by

Ered (s, ξ ) := inf E(s, u) u ∈ H 1/2 (T2 ) and u(x) = ξi if x ∈ Br (pi ) and the reduced dissipative potential: red (ξ˙ ) := κ| Br |

|ξ˙p |.

p∈O

Note that Ered is a quadratic form, i.e., there exists a matrix G ∈ RO×O , a vector τ ∈ RO and a scalar h ∈ R such that Ered (s, ξ ) = 12 ξ ·Gξ −(s ·b)τ ·ξ + h2 (s ·b)2 . The coefficients of G and τ and the value of h are random and depend on the realization of the obstacle positions. The resulting free energy is log κ 2 |Br |2 − (v ∗ · ep )2 , F (v ∗ ) = − p∈O

where ep ∈ RO denotes the unit dislocation over the obstacle site p ∈ O. Since the number of obstacles is typically large the differential equation (16) is difficult to analyze. The situation simplifies when one considers the asymptotic behavior of the solutions in the limit where the number of obstacles |O| tends to infinity and r tends to zero such that κ|O| |Br | = σ . We conjecture that if ξ(t = 0) is constant over the obstacles, then in the limit ξ(t) remains constant, i.e. there exists a fixed function ξ¯ (t) ∈ R such that ξp (t) → ξ¯ (t) for each p ∈ O, as r tends to 0. Moreover, there exists τ¯ ∈ R such that (Gξ −(s·b)τ )·ep converges to (s·b)τ¯ )·ep and thus the evolution of ξ¯ is characterized by the the following mean-field free energy: FMF (v¯ ∗ ) := − log(σ 2 − (v¯ ∗ )2 ).

4.2 Derivation of the Andrade Creep Law These reduction steps yield a scalar differential equation 2θ τ¯ dξ¯

= θ FMF ((s · b)τ¯ )2 = 2 dt σ − τ¯ 2 with appropriate initial conditions; in order to simplify the notation it is assumed that s · b = 1. Assume a constant applied load τ¯ and that the resistance is σ , with


269

0 ≤ |τ¯ | < σ – once again, this is the stability criterion (2). At least formally, σ may vary provided that the inequality σ > |τ¯ | ≥ 0 still holds, and under the additional assumption of linear strain hardening, i.e. σ = σ0 ξ¯ , the previous equation becomes 2θ τ¯ dξ¯ = 2 . dt σ0 ξ¯ (t)2 − τ¯ 2

(20)

Clearly, the behaviour of ξ¯ (t) for small t will depend upon the magnitude of τ¯ as compared to σ0 . However, under the assumption that the effective applied stress τˆ is small in comparison to the frictional resistance σ = σ0 ξ¯ , 2θ τ¯ dξ¯ = 2 . dt σ0 ξ¯ (t)2 This yields

1/3 ¯ξ (t) ≈ C1 + 6θ τ¯ t ∼ t 1/3 as t → +∞, σ02

where C1 is a constant of integration, by the general solution 1/1+2β kθ kθ 1−2α t dy = =⇒ y(t) = (1 + 2β) C1 + . dt (σ0 θ α y(t)β )2 σ02

(21)

Thus, as illustrated by the plots of numerical solutions to (20) in Figure 1, Andrade’s t 1/3 creep law follows as a straightforward consequence of the limiting dynamics. Furthermore, the theory also also predicts more rapid creep for larger values of the temperature-like parameter θ , as intuition would suggest.

4.3 Numerical Simulations This subsection describes numerical simulations of the phase-field dislocation theory proposed by Koslowski, Cuitiño and Ortiz in [10]. Proceeding to the numerical solution of the phase-field dislocation model, the slip plane is discretized in a square of 50 by 50 grid points with periodic boundary conditions and a random array of obstacles. The potential energy is computed in Fourier space following (17). The irreversible dislocation obstacle interactions are built into the framework developed in Section 2 by the introduction of the incremental work function (4). The simulations are performed with a Metropolis Monte Carlo algorithm using the transition probability (4) from state un to un+1 as derived in Section 2.1 and the frictional potential (19). Given an initial state u0 (x), the phase field is set to be constant within each obstacle, i.e. u0 (x) = ξi0 if x ∈ Br (pi ). The system is allowed to relax for a given applied stress τ¯ . After the system reaches a relaxed state the values of the phase

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F. Theil et al. log ξ¯ (t) 2.5

2

1.5

1

0.5

1

2

3

4

5

log t

Fig. 1 A log-log plot of ξ¯ (t) (colour) against t for (20), illustrating Andrade’s t 1/3 Andrade creep law behaviour by comparison with t → t 1/3 (straight). Parameters: initial condition ξ¯ (0) = 1; θ = 1; σ0 = 1; τ¯ = 0.1 (lower), 0.5 (middle), 0.9 (upper).

field at the obstacles points are updated with the new values. These values are used to compute the friction at the obstacles for the next time step. In the present model hardening is represented by an increase in the frictional resistance, σ following ¯ β σ = σ0 u(t)

(22)

where σ0 is a hardening coefficient and u¯ is the average slip. Figure 2 shows the numerical simulations of the phase field dislocation model for different values of hardening coefficient at an applied stress τ¯ = 0.5 and θ = 1. The red line corresponds to β = 0 and shows a power law dependence on t of the form given in equation (21) with 1/1 + 2β = 0.94 ± 0.02. Figure 2 also shows the simulations for linear hardening, i.e. β = 1. The simulated average slip follows a power law relation with respect to time with exponents 1/1 + 2β = 0.27 ± 0.01 and 1/1 + 2β = 0.28 ± 0.01 for hardening coefficients σ0 = 1 and σ0 = 2 respectively, in agreement with Andrade creep.

5 Summary and Conclusion We have developed a theory of statistical mechanics for dissipative systems governed by equations of evolution that assigns probabilities to individual trajectories of the system. The theory can be made mathematically rigorous and leads to precise predictions regarding the behavior of dissipative systems at finite temperature.


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Fig. 2 A log-log plot of u(t)/u ¯ 0 (third from top) against t. Numerical simulations of the phasefield dislocation model with no hardening (bottom), and linear hardening with σ0 = 1 (second from top) and σ0 = 2 (top).

Such predictions include the effect of temperature on yield phenomena and rheological time exponents. The particular case of an ensemble of dislocations moving in a slip plane through a random array of obstacles has been studied numerically in detail. The numerical results bear out the analytical predictions regarding the mean response of the system, which exhibits Andrade creep. This finite-temperature “phase-space exploration” is expected to be most revealing when the energy landscape of the system is rough and the system tends to develop microstructure (cf., e.g., [4, 11]). Of particular interest are continuum systems whose states are described by fields in infinite-dimensional functional spaces and governed by energy functionals lacking lower semi-continuity. In such cases, the preferred or most likely trajectories may be viewed as defining the effective or macroscopic energetics and kinetics of the system.

References 1. Edward N. da C. Andrade, Proceedings of the Royal Society of London A84, 1910, 1. 2. Edward N. da C. Andrade, Proceedings of the Royal Society of London A90, 1914, 329.

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3. Daryl C. Chrzan and M.J. Mills, Criticality in the plastic deformation of Ni3 Al. Physical Review Letters 69(19), 1992, 2795–2798. 4. S. Conti and M. Ortiz, Minimum principles for the trajectories of systems governed by rate problems. Journal of the Mechanics and Physics of Solids 56(5), 2008, 1885–1904. 5. A.H. Cottrell, Criticality in andrade creep. Philosophical Magazine A 74(4), 1996, 1041–1046. 6. A.H. Cottrell, Strain hardening in andrade creep. Philosophical Magazine Letters 74(5), 1996, 375–379. 7. G.S. Daehn, Primary creep transients due to non-uniform obstacle sizes. Materials Science and Engineering A 319–321, 2001, 765–769. 8. A. Garroni and S. Müller, -limit of a phase-field model of dislocations. SIAM Journal on Mathematical Analysis 36(6), 2005, 1943–1964 (electronic). 9. A. Garroni and S. Müller. A variational model for dislocations in the line tension limit. Archive for Rationional Mechanics and Analysis 181(3), 2006, 535–578. 10. M. Koslowski, A.M. Cuitiño and M. Ortiz, A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals. Journal of the Mechanics and Physics in Solids 50(12), 2002, 2597–2635. 11. A. Mielke and M. Ortiz, A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems. ESAIM: Control, Optimisation and Calculus of Variations, 14, 2008, 494–516. 12. A. Mielke and F. Theil, On rate-independent hysteresis models. NoDEA Nonlinear Differential Equations Appl. 11(2), 2004, 151–189. 13. A. Mielke, F. Theil and V.I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle. Archive for Rationional Mechanics and Analysis 162(2), 2002, 137–177. 14. M.C. Miguel, A. Vespignani, M. Zaiser and S. Zapperi, Dislocation jamming and Andrade creep. Physical Review Letters 89(16), 2002, 165501–1–4. 15. M. Ortiz and E.A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals. Journal of the Mechanics and Physics of Solids 47(2), 1999, 397–462. 16. M. Ortiz and L. Stainier, The variational formulation of viscoplastic constitutive updates. Computer Methods in Applied Mechanics and Engineering 171(3–4), 1999, 419–444. 17. R.A. Radovitzky and M. Ortiz, Error estimation and adaptive meshing in strongly nonlinear dynamic problems. Computer Methods in Applied Mechanics and Engineering 172(1–4), 1999, 203–240. 18. T. Sullivan, M. Koslowski, F. Theil and M. Ortiz, Dissipative systems in contact with a heat bath: Application to andrade creep. Journal of the Mechanics and Physics of Solids, 2009, to appear.

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