Optimal laminates in single-slip elastoplasticity. (English) Zbl 1454.74021

Summary: Recent progress in the mathematical analysis of variational models for the plastic deformation of crystals in a geometrically nonlinear setting is discussed. The focus lies on the first time-step and on situations where only one slip system is active, in two spatial dimensions. The interplay of invariance under finite rotations and plastic deformation leads to the emergence of microstructures, which can be analyzed in the framework of relaxation theory using the theory of quasiconvexity. A class of elastoplastic energies with one active slip system that converge asymptotically to a model with rigid elasticity is presented and the interplay between relaxation and asymptotics is investigated.


74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity)
74B20 Nonlinear elasticity
49J45 Methods involving semicontinuity and convergence; relaxation
Full Text: DOI


[1] N. Albin; S. Conti; G. Dolzmann, Infinite-order laminates in a model in crystal plasticity, Proc. Roy. Soc. Edinburgh Sect. A, 139, 685-708 (2009) · Zbl 1167.74010
[2] J. M. Ball; B. Kirchheim; J. Kristensen, Regularity of quasiconvex envelopes, Calc. Var. Partial Differential Equations, 11, 333-359 (2000) · Zbl 0972.49024
[3] S. Bartels; C. Carstensen; K. Hackl; U. Hoppe, Effective relaxation for microstructure simulations: Algorithms and applications, Comput. Methods Appl. Mech. Engrg., 193, 5143-5175 (2004) · Zbl 1112.74501
[4] S. Bartels, Linear convergence in the approximation of rank-one convex envelopes, M2AN Math. Model. Numer. Anal., 38, 811-820 (2004) · Zbl 1083.65058
[5] S. Bartels, Reliable and efficient approximation of polyconvex envelopes, SIAM J. Numer. Anal., 43, 363-385 (2005) · Zbl 1089.65052
[6] S. Bartels; T. Roubíček, Linear-programming approach to nonconvex variational problems, Numer. Math., 99, 251-287 (2004) · Zbl 1063.65051
[7] C. Carstensen, Nonconvex energy minimization and relaxation in computational material science, in IUTAM Symposium on Computational Mechanics of Solid {M}aterials at Large {S}trains (Stuttgart, 2001), Solid Mech. Appl., 108, Kluwer Acad. Publ., Dordrecht, 2003, 3-20. · Zbl 1081.74019
[8] C. Carstensen; S. Conti; A. Orlando, Mixed analytical-numerical relaxation in finite single-slip crystal plasticity, Contin. Mech. Thermodyn., 20, 275-301 (2008) · Zbl 1160.74326
[9] C. Carstensen; P. Plecháč, Numerical analysis of compatible phase transitions in elastic solids, SIAM J. Numer. Anal., 37, 2061-2081 (2000) · Zbl 1049.74062
[10] C. Carstensen, Numerical analysis of microstructure, in Theory and Numerics of Differential Equations (Durham, 2000), Universitext, Springer, Berlin, 2001, 59-126. · Zbl 1070.74033
[11] C. Carstensen; K. Hackl; A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458, 299-317 (2002) · Zbl 1008.74016
[12] C. Carstensen; S. Müller, Local stress regularity in scalar nonconvex variational problems, SIAM J. Math. Anal., 34, 495-509 (2002) · Zbl 1012.49027
[13] C. Carstensen; P. Plecháč, Numerical solution of the scalar double-well problem allowing microstructure, Math. Comp., 66, 997-1026 (1997) · Zbl 0870.65055
[14] C. Carstensen; T. Roubíček, Numerical approximation of Young measures in non-convex variational problems, Numer. Math., 84, 395-415 (2000) · Zbl 0945.65070
[15] M. Chipot, Numerical analysis of oscillations in nonconvex problems, Numer. Math., 59, 747-767 (1991) · Zbl 0712.65063
[16] M. Chipot and S. Müller, Sharp energy estimates for finite element approximations of non-convex problems, in Variations of Domain and Free-Boundary Problems in Solid Mechanics (Paris, 1997), Solid Mech. Appl., 66, Kluwer Acad. Publ., Dordrecht, 1999,317-325.
[17] M. Cicalese and N. Fusco, A note on relaxation with constraints on the determinant, ESAIM: Control Optim. Calc. Var., 25 (2019), 15pp. · Zbl 1437.49026
[18] S. Conti, Relaxation of single-slip single-crystal plasticity with linear hardening, in Multiscale Materials Modeling, Fraunhofer IRB, Freiburg, 2006, 30-35.
[19] S. Conti; A. DeSimone; G. Dolzmann, Soft elastic response of stretched sheets of nematic elastomers: A numerical study, J. Mech. Phys. Solids, 50, 1431-1451 (2002) · Zbl 1030.76006
[20] S. Conti; G. Dolzmann; C. Klust, Relaxation of a class of variational models in crystal plasticity, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465, 1735-1742 (2009) · Zbl 1186.74023
[21] S. Conti; G. Dolzmann; C. Kreisbeck, Asymptotic behavior of crystal plasticity with one slip system in the limit of rigid elasticity, SIAM J. Math. Anal., 43, 2337-2353 (2011) · Zbl 1233.35187
[22] S. Conti; G. Dolzmann, Relaxation of a model energy for the cubic to tetragonal phase transformation in two dimensions, Math. Models. Methods Appl. Sci., 24, 2929-2942 (2014) · Zbl 1304.49027
[23] S. Conti and G. Dolzmann, Quasiconvex envelope for a model of finite elastoplasticity with one active slip system and linear hardening, Continuum Mech. Thermodyn. (2019).
[24] S. Conti; G. Dolzmann, On the theory of relaxation in nonlinear elasticity with constraints on the determinant, Arch. Ration. Mech. Anal., 217, 413-437 (2015) · Zbl 1323.49010
[25] S. Conti; G. Dolzmann, Relaxation in crystal plasticity with three active slip systems, Contin. Mech. Thermodyn., 28, 1477-1494 (2016) · Zbl 1355.74060
[26] S. Conti; G. Dolzmann, An adaptive relaxation algorithm for multiscale problems and application to nematic elastomers, J. Mech. Phys. Solids, 113, 126-143 (2018) · Zbl 1441.74145
[27] S. Conti; G. Dolzmann, Numerical study of microstructures in single-slip finite elastoplasticity, J. Optim. Theory Appl., 184, 43-60 (2020) · Zbl 1433.49061
[28] S. Conti; G. Dolzmann; C. Kreisbeck, Relaxation of a model in finite plasticity with two slip systems, Math. Models Methods Appl. Sci., 23, 2111-2128 (2013) · Zbl 1281.49006
[29] S. Conti; F. Theil, Single-slip elastoplastic microstructures, Arch. Ration. Mech. Anal., 178, 125-148 (2005) · Zbl 1076.74017
[30] B. Dacorogna, Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, 78, Springer-Verlag, Berlin, 1989. · Zbl 0703.49001
[31] G. Dal Maso, An Introduction to {\( \Gamma \)-Convergence}, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser, Boston, 1993. · Zbl 0816.49001
[32] E. Davoli; G. A. Francfort, A critical revisiting of finite elasto-plasticity, SIAM J. Math. Anal., 47, 526-565 (2015) · Zbl 1317.74022
[33] E. De Giorgi, Sulla convergenza di alcune successioni d’integrali del tipo dell’area, Rend. Mat. (6), 8, 277-294 (1975) · Zbl 0316.35036
[34] A. DeSimone; G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of \(\rm SO(3)\)-invariant energies, Arch. Ration. Mech. Anal., 161, 181-204 (2002) · Zbl 1017.74049
[35] E. D. Giorgi; T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 58, 842-850 (1975) · Zbl 0339.49005
[36] K. Hackl; S. Heinz; A. Mielke, A model for the evolution of laminates in finite-strain elastoplasticity, ZAMM Z. Angew. Math. Mech., 92, 888-909 (2012) · Zbl 1348.74060
[37] W. Han and B. D. Reddy, phPlasticity, in Mathematical Theory and Numerical Analysis, Interdisciplinary Applied Mathematics, 9, Springer, New York, 2013. · Zbl 1258.74002
[38] E. Kröner, Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen, Arch. Rational Mech. Anal., 4, 273-334 (1960) · Zbl 0090.17601
[39] M. Kružík; T. Roubíček, Optimization problems with concentration and oscillation effects: Relaxation theory and numerical approximation, Numer. Funct. Anal. Optim., 20, 511-530 (1999) · Zbl 0940.49002
[40] G. Lauteri and S. Luckhaus, An energy estimate for dislocation configurations and the emergence of Cosserat-type structures in metal plasticity, preprint, arXiv: 1608.06155.
[41] H. Le Dret; A. Raoult, The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function, Proc. Roy. Soc. Edinburgh Sect. A, 125, 1179-1192 (1995) · Zbl 0843.73016
[42] E. H. Lee, Elastic-plastic deformation at finite strains, J. Appl. Mech., 36, 1-6 (1969) · Zbl 0179.55603
[43] S. Luckhaus; L. Mugnai, On a mesoscopic many-body {H}amiltonian describing elastic shears and dislocations, Contin. Mech. Thermodyn., 22, 251-290 (2010) · Zbl 1234.74012
[44] S. Luckhaus and J. Wohlgemuth, Study of a model for reference-free plasticity, preprint, arXiv: 1408.1355.
[45] M. Luskin, On the computation of crystalline microstructure, in Acta Numerica, 1996 , Acta Numer., 5, Cambridge Univ. Press, Cambridge, 1996, 191-257. · Zbl 0867.65033
[46] A. Mainik; A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, J. Nonlinear Sci., 19, 221-248 (2009) · Zbl 1173.49013
[47] C. Miehe, On the representation of Prandtl-Reuss tensors within the framework of multiplicative elastoplasticity, Int. J. Plasticity, 10, 609-621 (1994) · Zbl 0810.73016
[48] C. Miehe; M. Lambrecht; E. Gürses, 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, 2725-2769 (2004) · Zbl 1115.74323
[49] C. Miehe; E. Stein, A canonical model of multiplicative elasto-plasticity: Formulation and aspects of the numerical implementation, Europ. J. Mech. A/Solids, 11, 25-43 (1992)
[50] C. Miehe; M. Lambrecht, Analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials: Large-strain theory for standard dissipative solids, Internat. J. Numer. Methods Engrg., 58, 1-41 (2003) · Zbl 1032.74526
[51] A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Contin. Mech. Thermodyn., 15, 351-382 (2003) · Zbl 1068.74522
[52] A. Mielke and S. Müller, Lower semicontinuity and existence of minimizers in incremental finite-strain elastoplasticity, ZAMM Z. Angew. Math. Mech., 86 (2006), 233-250, . · Zbl 1102.74006
[53] A. Mielke; R. Rossi; G. Savaré, Global existence results for viscoplasticity at finite strain, Arch. Ration. Mech. Anal., 227, 423-475 (2018) · Zbl 1387.35574
[54] A. Mielke and T. Roubíček, Rate-independent systems, in Theory and Application, Applied Mathematical Sciences, 193, Springer, New York, 2015. · Zbl 1339.35006
[55] A. Mielke; T. Roubíček, Rate-independent elastoplasticity at finite strains and its numerical approximation, Math. Models Methods Appl. Sci., 26, 2203-2236 (2016) · Zbl 1349.35371
[56] A. Mielke; U. Stefanelli, Linearized plasticity is the evolutionary \(\Gamma \)-limit of finite plasticity, J. Eur. Math. Soc. (JEMS), 15, 923-948 (2013) · Zbl 1334.74021
[57] A. Mielke; F. Theil; V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162, 137-177 (2002) · Zbl 1012.74054
[58] J. Moreau, Sur les lois de frottement, de plasticité et de viscosité, Comptes Rendus de l’Académie des Sciences, 271, 608-611 (1970)
[59] J. C. B. Morrey, Multiple Integrals in the Calculus of Variations, Die Grundlehren der mathematischen Wissenschaften, Band, 130, Springer-Verlag New York, Inc., New York, 1966.
[60] S. Müller, Variational models for microstructure and phase transitions, in Calculus of Variations and Geometric Evolution Problems, Lecture Notes in Math., 1713, Springer, Berlin, 1999, 85-210.
[61] S. Müller; V. Šverák, Convex integration with constraints and applications to phase transitions and partial differential equations, J. Eur. Math. Soc. (JEMS), 1, 393-442 (1999) · Zbl 0953.35042
[62] S. Müller, L. Scardia and C. I. Zeppieri, Gradient theory for geometrically nonlinear plasticity via the homogenization of dislocations, in Analysis and Computation of Microstructure in Finite Plasticity, Lect. Notes Appl. Comput. Mech., 78, Springer, Cham, 2015,175-204. · Zbl 1456.74018
[63] M. Ortiz; E. A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals, J. Mech. Phys. Solids, 47, 397-462 (1999) · Zbl 0964.74012
[64] C. Reina; S. Conti, Kinematic description of crystal plasticity in the finite kinematic framework: A micromechanical understanding of \({F} = {F}^e{F}^p\), J. Mech. Phys. Solids, 67, 40-61 (2014) · Zbl 1323.74018
[65] C. Reina; S. Conti, Incompressible inelasticity as an essential ingredient for the validity of the kinematic decomposition \(F = F^eF^i\), J. Mech. Phys. Solids, 107, 322-342 (2017) · Zbl 1442.74004
[66] T. Roubíček, Relaxation in Optimization Theory and Variational Calculus, De Gruyter Series in Nonlinear Analysis and Applications, 4, Walter de Gruyter & Co., Berlin, 1997. · Zbl 0880.49002
[67] T. Roubíček, Numerical techniques in relaxed optimization problems, in Robust Optimization-Directed Design, Nonconvex Optim. Appl., 81, Springer, New York, 2006,157-178. · Zbl 1120.49026
[68] J. C. Simo, A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. I. Continuum formulation, Comput. Methods Appl. Mech. Engrg., 66, 199-219 (1988) · Zbl 0611.73057
[69] J. C. Simo; M. Ortiz, A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations, Comput. Methods. Appl. Mech. Engrg., 49, 221-245 (1985) · Zbl 0566.73035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.