American Institute of Mathematical Sciences

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January  2021, 14(1): 1-16. doi: 10.3934/dcdss.2020302

Optimal laminates in single-slip elastoplasticity

Sergio Conti 1, and  Georg Dolzmann 2,,

1. 

Universität Bonn, Bonn, D-53115, Germany
2. 

Universität Regensburg, Regensburg, D-93053, Germany

* Corresponding author: Georg Dolzmann

Received  March 2019 Revised  August 2019 Published  March 2020

Fund Project: The first author is supported by DFG SFB 1060 "The mathematics of emergent effects", project 211504053/A05

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.

Keywords: Elastoplasticity, single slip, quasiconvexity, relaxation, optimal laminates.
Mathematics Subject Classification: Primary: 74C15; Secondary: 74B20, 49J45.
Citation: Sergio Conti, Georg Dolzmann. Optimal laminates in single-slip elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 1-16. doi: 10.3934/dcdss.2020302
References:
[1]

N. Albin, S. Conti and G. Dolzmann, Infinite-order laminates in a model in crystal plasticity, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 685-708.  doi: 10.1017/S0308210508000127.  Google Scholar

[2]

J. M. Ball, B. Kirchheim and J. Kristensen, Regularity of quasiconvex envelopes, Calc. Var. Partial Differential Equations, 11 (2000), 333-359.  doi: 10.1007/s005260000041.  Google Scholar

[3]

S. Bartels, C. Carstensen, K. Hackl and U. Hoppe, Effective relaxation for microstructure simulations: Algorithms and applications, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5143-5175.  doi: 10.1016/j.cma.2003.12.065.  Google Scholar

[4]

S. Bartels, Linear convergence in the approximation of rank-one convex envelopes, M2AN Math. Model. Numer. Anal., 38 (2004), 811-820.  doi: 10.1051/m2an:2004040.  Google Scholar

[5]

S. Bartels, Reliable and efficient approximation of polyconvex envelopes, SIAM J. Numer. Anal., 43 (2005), 363-385.  doi: 10.1137/S0036142903428840.  Google Scholar

[6]

S. Bartels and T. Roubíček, Linear-programming approach to nonconvex variational problems, Numer. Math., 99 (2004), 251-287.  doi: 10.1007/s00211-004-0549-2.  Google Scholar

[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. doi: 10.1007/978-94-017-0297-3_1.  Google Scholar

[8]

C. Carstensen, S. Conti and A. Orlando, Mixed analytical-numerical relaxation in finite single-slip crystal plasticity, Contin. Mech. Thermodyn., 20 (2008), 275-301.  doi: 10.1007/s00161-008-0082-0.  Google Scholar

[9]

C. Carstensen and P. Plecháč, Numerical analysis of compatible phase transitions in elastic solids, SIAM J. Numer. Anal., 37 (2000), 2061-2081.  doi: 10.1137/S0036142998337697.  Google Scholar

[10]

C. Carstensen, Numerical analysis of microstructure, in Theory and Numerics of Differential Equations (Durham, 2000), Universitext, Springer, Berlin, 2001, 59–126. doi: 10.1007/978-3-662-04354-7_2.  Google Scholar

[11]

C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458 (2002), 299-317.  doi: 10.1098/rspa.2001.0864.  Google Scholar

[12]

C. Carstensen and S. Müller, Local stress regularity in scalar nonconvex variational problems, SIAM J. Math. Anal., 34 (2002), 495-509.  doi: 10.1137/S0036141001396436.  Google Scholar

[13]

C. Carstensen and P. Plecháč, Numerical solution of the scalar double-well problem allowing microstructure, Math. Comp., 66 (1997), 997-1026.  doi: 10.1090/S0025-5718-97-00849-1.  Google Scholar

[14]

C. Carstensen and T. Roubíček, Numerical approximation of Young measures in non-convex variational problems, Numer. Math., 84 (2000), 395-415.  doi: 10.1007/s002110050003.  Google Scholar

[15]

M. Chipot, Numerical analysis of oscillations in nonconvex problems, Numer. Math., 59 (1991), 747-767.  doi: 10.1007/BF01385808.  Google Scholar

[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. doi: 10.1007/978-94-011-4738-5_38.  Google Scholar

[17]

M. Cicalese and N. Fusco, A note on relaxation with constraints on the determinant, ESAIM: Control Optim. Calc. Var., 25 (2019), 15pp. doi: 10.1051/cocv/2018030.  Google Scholar

[18]

S. Conti, Relaxation of single-slip single-crystal plasticity with linear hardening, in Multiscale Materials Modeling, Fraunhofer IRB, Freiburg, 2006, 30–35. Google Scholar

[19]

S. Conti, A. DeSimone and G. Dolzmann, Soft elastic response of stretched sheets of nematic elastomers: A numerical study, J. Mech. Phys. Solids, 50 (2002), 1431-1451.  doi: 10.1016/S0022-5096(01)00120-X.  Google Scholar

[20]

S. Conti, G. Dolzmann and C. Klust, Relaxation of a class of variational models in crystal plasticity, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 1735-1742.  doi: 10.1098/rspa.2008.0390.  Google Scholar

[21]

S. Conti, G. Dolzmann and C. Kreisbeck, Asymptotic behavior of crystal plasticity with one slip system in the limit of rigid elasticity, SIAM J. Math. Anal., 43 (2011), 2337-2353.  doi: 10.1137/100810320.  Google Scholar

[22]

S. Conti and G. Dolzmann, Relaxation of a model energy for the cubic to tetragonal phase transformation in two dimensions, Math. Models. Methods Appl. Sci., 24 (2014), 2929-2942.  doi: 10.1142/S0218202514500419.  Google Scholar

[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). doi: 10.1007/s00161-019-00825-8.  Google Scholar

[24]

S. Conti and G. Dolzmann, On the theory of relaxation in nonlinear elasticity with constraints on the determinant, Arch. Ration. Mech. Anal., 217 (2015), 413-437.  doi: 10.1007/s00205-014-0835-9.  Google Scholar

[25]

S. Conti and G. Dolzmann, Relaxation in crystal plasticity with three active slip systems, Contin. Mech. Thermodyn., 28 (2016), 1477-1494.  doi: 10.1007/s00161-015-0490-x.  Google Scholar

[26]

S. Conti and G. Dolzmann, An adaptive relaxation algorithm for multiscale problems and application to nematic elastomers, J. Mech. Phys. Solids, 113 (2018), 126-143.  doi: 10.1016/j.jmps.2018.02.001.  Google Scholar

[27]

S. Conti and G. Dolzmann, Numerical study of microstructures in single-slip finite elastoplasticity, J. Optim. Theory Appl., 184 (2020), 43-60.  doi: 10.1007/s10957-018-01460-0.  Google Scholar

[28]

S. Conti, G. Dolzmann and C. Kreisbeck, Relaxation of a model in finite plasticity with two slip systems, Math. Models Methods Appl. Sci., 23 (2013), 2111-2128.  doi: 10.1142/S0218202513500279.  Google Scholar

[29]

S. Conti and F. Theil, Single-slip elastoplastic microstructures, Arch. Ration. Mech. Anal., 178 (2005), 125-148.  doi: 10.1007/s00205-005-0371-8.  Google Scholar

[30]

B. Dacorogna, Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, 78, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-51440-1.  Google Scholar

[31]

G. Dal Maso, An Introduction to {$\Gamma$-Convergence}, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[32]

E. Davoli and G. A. Francfort, A critical revisiting of finite elasto-plasticity, SIAM J. Math. Anal., 47 (2015), 526-565.  doi: 10.1137/140965090.  Google Scholar

[33]

E. De Giorgi, Sulla convergenza di alcune successioni d'integrali del tipo dell'area, Rend. Mat. (6), 8 (1975), 277-294.   Google Scholar

[34]

A. DeSimone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of $\rm SO(3)$-invariant energies, Arch. Ration. Mech. Anal., 161 (2002), 181-204.  doi: 10.1007/s002050100174.  Google Scholar

[35]

E. D. Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 58 (1975), 842-850.   Google Scholar

[36]

K. Hackl, S. Heinz and A. Mielke, A model for the evolution of laminates in finite-strain elastoplasticity, ZAMM Z. Angew. Math. Mech., 92 (2012), 888-909.  doi: 10.1002/zamm.201100155.  Google Scholar

[37]

W. Han and B. D. Reddy, phPlasticity, in Mathematical Theory and Numerical Analysis, Interdisciplinary Applied Mathematics, 9, Springer, New York, 2013. doi: 10.1007/978-1-4614-5940-8.  Google Scholar

[38]

E. Kröner, Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen, Arch. Rational Mech. Anal., 4 (1960), 273-334.  doi: 10.1007/BF00281393.  Google Scholar

[39]

M. Kružík and T. Roubíček, Optimization problems with concentration and oscillation effects: Relaxation theory and numerical approximation, Numer. Funct. Anal. Optim., 20 (1999), 511-530.  doi: 10.1080/01630569908816908.  Google Scholar

[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. Google Scholar

[41]

H. Le Dret and A. Raoult, The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 1179-1192.  doi: 10.1017/S0308210500030456.  Google Scholar

[42]

E. H. Lee, Elastic-plastic deformation at finite strains, J. Appl. Mech., 36 (1969), 1-6.  doi: 10.21236/AD0678483.  Google Scholar

[43]

S. Luckhaus and L. Mugnai, On a mesoscopic many-body {H}amiltonian describing elastic shears and dislocations, Contin. Mech. Thermodyn., 22 (2010), 251-290.  doi: 10.1007/s00161-010-0142-0.  Google Scholar

[44]

S. Luckhaus and J. Wohlgemuth, Study of a model for reference-free plasticity, preprint, arXiv: 1408.1355. Google Scholar

[45]

M. Luskin, On the computation of crystalline microstructure, in Acta Numerica, 1996 , Acta Numer., 5, Cambridge Univ. Press, Cambridge, 1996, 191-257. doi: 10.1017/S0962492900002658.  Google Scholar

[46]

A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, J. Nonlinear Sci., 19 (2009), 221-248.  doi: 10.1007/s00332-008-9033-y.  Google Scholar

[47]

C. Miehe, On the representation of Prandtl-Reuss tensors within the framework of multiplicative elastoplasticity, Int. J. Plasticity, 10 (1994), 609-621.  doi: 10.1016/0749-6419(94)90025-6.  Google Scholar

[48]

C. Miehe, M. Lambrecht and 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 (2004), 2725-2769.  doi: 10.1016/j.jmps.2004.05.011.  Google Scholar

[49]

C. Miehe and E. Stein, A canonical model of multiplicative elasto-plasticity: Formulation and aspects of the numerical implementation, Europ. J. Mech. A/Solids, 11 (1992), 25-43.   Google Scholar

[50]

C. Miehe and 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 (2003), 1-41.  doi: 10.1002/nme.726.  Google Scholar

[51]

A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Contin. Mech. Thermodyn., 15 (2003), 351-382.  doi: 10.1007/s00161-003-0120-x.  Google Scholar

[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, . doi: 10.1002/zamm.200510245.  Google Scholar

[53]

A. Mielke, R. Rossi and G. Savaré, Global existence results for viscoplasticity at finite strain, Arch. Ration. Mech. Anal., 227 (2018), 423-475.  doi: 10.1007/s00205-017-1164-6.  Google Scholar

[54]

A. Mielke and T. Roubíček, Rate-independent systems, in Theory and Application, Applied Mathematical Sciences, 193, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[55]

A. Mielke and T. Roubíček, Rate-independent elastoplasticity at finite strains and its numerical approximation, Math. Models Methods Appl. Sci., 26 (2016), 2203-2236.  doi: 10.1142/S0218202516500512.  Google Scholar

[56]

A. Mielke and U. Stefanelli, Linearized plasticity is the evolutionary $\Gamma$-limit of finite plasticity, J. Eur. Math. Soc. (JEMS), 15 (2013), 923-948.  doi: 10.4171/JEMS/381.  Google Scholar

[57]

A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177.  doi: 10.1007/s002050200194.  Google Scholar

[58]

J. Moreau, Sur les lois de frottement, de plasticité et de viscosité, Comptes Rendus de l'Académie des Sciences, 271 (1970), 608-611.   Google Scholar

[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. doi: 10.1007/978-3-540-69952-1.  Google Scholar

[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. doi: 10.1007/BFb0092670.  Google Scholar

[61]

S. Müller and V. Šverák, Convex integration with constraints and applications to phase transitions and partial differential equations, J. Eur. Math. Soc. (JEMS), 1 (1999), 393-442.  doi: 10.1007/s100970050012.  Google Scholar

[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. doi: 10.1007/978-3-319-18242-1_7.  Google Scholar

[63]

M. Ortiz and E. A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals, J. Mech. Phys. Solids, 47 (1999), 397-462.  doi: 10.1016/S0022-5096(97)00096-3.  Google Scholar

[64]

C. Reina and 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 (2014), 40-61.  doi: 10.1016/j.jmps.2014.01.014.  Google Scholar

[65]

C. Reina and S. Conti, Incompressible inelasticity as an essential ingredient for the validity of the kinematic decomposition $F = F^eF^i$, J. Mech. Phys. Solids, 107 (2017), 322-342.  doi: 10.1016/j.jmps.2017.07.004.  Google Scholar

[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. doi: 10.1515/9783110811919.  Google Scholar

[67]

T. Roubíček, Numerical techniques in relaxed optimization problems, in Robust Optimization-Directed Design, Nonconvex Optim. Appl., 81, Springer, New York, 2006,157–178. doi: 10.1007/0-387-28654-3_8.  Google Scholar

[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 (1988), 199-219.  doi: 10.1016/0045-7825(88)90076-X.  Google Scholar

[69]

J. C. Simo and M. Ortiz, A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations, Comput. Methods. Appl. Mech. Engrg., 49 (1985), 221-245.  doi: 10.1016/0045-7825(85)90061-1.  Google Scholar

show all references

References:
[1]

N. Albin, S. Conti and G. Dolzmann, Infinite-order laminates in a model in crystal plasticity, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 685-708.  doi: 10.1017/S0308210508000127.  Google Scholar

[2]

J. M. Ball, B. Kirchheim and J. Kristensen, Regularity of quasiconvex envelopes, Calc. Var. Partial Differential Equations, 11 (2000), 333-359.  doi: 10.1007/s005260000041.  Google Scholar

[3]

S. Bartels, C. Carstensen, K. Hackl and U. Hoppe, Effective relaxation for microstructure simulations: Algorithms and applications, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5143-5175.  doi: 10.1016/j.cma.2003.12.065.  Google Scholar

[4]

S. Bartels, Linear convergence in the approximation of rank-one convex envelopes, M2AN Math. Model. Numer. Anal., 38 (2004), 811-820.  doi: 10.1051/m2an:2004040.  Google Scholar

[5]

S. Bartels, Reliable and efficient approximation of polyconvex envelopes, SIAM J. Numer. Anal., 43 (2005), 363-385.  doi: 10.1137/S0036142903428840.  Google Scholar

[6]

S. Bartels and T. Roubíček, Linear-programming approach to nonconvex variational problems, Numer. Math., 99 (2004), 251-287.  doi: 10.1007/s00211-004-0549-2.  Google Scholar

[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. doi: 10.1007/978-94-017-0297-3_1.  Google Scholar

[8]

C. Carstensen, S. Conti and A. Orlando, Mixed analytical-numerical relaxation in finite single-slip crystal plasticity, Contin. Mech. Thermodyn., 20 (2008), 275-301.  doi: 10.1007/s00161-008-0082-0.  Google Scholar

[9]

C. Carstensen and P. Plecháč, Numerical analysis of compatible phase transitions in elastic solids, SIAM J. Numer. Anal., 37 (2000), 2061-2081.  doi: 10.1137/S0036142998337697.  Google Scholar

[10]

C. Carstensen, Numerical analysis of microstructure, in Theory and Numerics of Differential Equations (Durham, 2000), Universitext, Springer, Berlin, 2001, 59–126. doi: 10.1007/978-3-662-04354-7_2.  Google Scholar

[11]

C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458 (2002), 299-317.  doi: 10.1098/rspa.2001.0864.  Google Scholar

[12]

C. Carstensen and S. Müller, Local stress regularity in scalar nonconvex variational problems, SIAM J. Math. Anal., 34 (2002), 495-509.  doi: 10.1137/S0036141001396436.  Google Scholar

[13]

C. Carstensen and P. Plecháč, Numerical solution of the scalar double-well problem allowing microstructure, Math. Comp., 66 (1997), 997-1026.  doi: 10.1090/S0025-5718-97-00849-1.  Google Scholar

[14]

C. Carstensen and T. Roubíček, Numerical approximation of Young measures in non-convex variational problems, Numer. Math., 84 (2000), 395-415.  doi: 10.1007/s002110050003.  Google Scholar

[15]

M. Chipot, Numerical analysis of oscillations in nonconvex problems, Numer. Math., 59 (1991), 747-767.  doi: 10.1007/BF01385808.  Google Scholar

[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. doi: 10.1007/978-94-011-4738-5_38.  Google Scholar

[17]

M. Cicalese and N. Fusco, A note on relaxation with constraints on the determinant, ESAIM: Control Optim. Calc. Var., 25 (2019), 15pp. doi: 10.1051/cocv/2018030.  Google Scholar

[18]

S. Conti, Relaxation of single-slip single-crystal plasticity with linear hardening, in Multiscale Materials Modeling, Fraunhofer IRB, Freiburg, 2006, 30–35. Google Scholar

[19]

S. Conti, A. DeSimone and G. Dolzmann, Soft elastic response of stretched sheets of nematic elastomers: A numerical study, J. Mech. Phys. Solids, 50 (2002), 1431-1451.  doi: 10.1016/S0022-5096(01)00120-X.  Google Scholar

[20]

S. Conti, G. Dolzmann and C. Klust, Relaxation of a class of variational models in crystal plasticity, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 1735-1742.  doi: 10.1098/rspa.2008.0390.  Google Scholar

[21]

S. Conti, G. Dolzmann and C. Kreisbeck, Asymptotic behavior of crystal plasticity with one slip system in the limit of rigid elasticity, SIAM J. Math. Anal., 43 (2011), 2337-2353.  doi: 10.1137/100810320.  Google Scholar

[22]

S. Conti and G. Dolzmann, Relaxation of a model energy for the cubic to tetragonal phase transformation in two dimensions, Math. Models. Methods Appl. Sci., 24 (2014), 2929-2942.  doi: 10.1142/S0218202514500419.  Google Scholar

[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). doi: 10.1007/s00161-019-00825-8.  Google Scholar

[24]

S. Conti and G. Dolzmann, On the theory of relaxation in nonlinear elasticity with constraints on the determinant, Arch. Ration. Mech. Anal., 217 (2015), 413-437.  doi: 10.1007/s00205-014-0835-9.  Google Scholar

[25]

S. Conti and G. Dolzmann, Relaxation in crystal plasticity with three active slip systems, Contin. Mech. Thermodyn., 28 (2016), 1477-1494.  doi: 10.1007/s00161-015-0490-x.  Google Scholar

[26]

S. Conti and G. Dolzmann, An adaptive relaxation algorithm for multiscale problems and application to nematic elastomers, J. Mech. Phys. Solids, 113 (2018), 126-143.  doi: 10.1016/j.jmps.2018.02.001.  Google Scholar

[27]

S. Conti and G. Dolzmann, Numerical study of microstructures in single-slip finite elastoplasticity, J. Optim. Theory Appl., 184 (2020), 43-60.  doi: 10.1007/s10957-018-01460-0.  Google Scholar

[28]

S. Conti, G. Dolzmann and C. Kreisbeck, Relaxation of a model in finite plasticity with two slip systems, Math. Models Methods Appl. Sci., 23 (2013), 2111-2128.  doi: 10.1142/S0218202513500279.  Google Scholar

[29]

S. Conti and F. Theil, Single-slip elastoplastic microstructures, Arch. Ration. Mech. Anal., 178 (2005), 125-148.  doi: 10.1007/s00205-005-0371-8.  Google Scholar

[30]

B. Dacorogna, Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, 78, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-51440-1.  Google Scholar

[31]

G. Dal Maso, An Introduction to {$\Gamma$-Convergence}, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[32]

E. Davoli and G. A. Francfort, A critical revisiting of finite elasto-plasticity, SIAM J. Math. Anal., 47 (2015), 526-565.  doi: 10.1137/140965090.  Google Scholar

[33]

E. De Giorgi, Sulla convergenza di alcune successioni d'integrali del tipo dell'area, Rend. Mat. (6), 8 (1975), 277-294.   Google Scholar

[34]

A. DeSimone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of $\rm SO(3)$-invariant energies, Arch. Ration. Mech. Anal., 161 (2002), 181-204.  doi: 10.1007/s002050100174.  Google Scholar

[35]

E. D. Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 58 (1975), 842-850.   Google Scholar

[36]

K. Hackl, S. Heinz and A. Mielke, A model for the evolution of laminates in finite-strain elastoplasticity, ZAMM Z. Angew. Math. Mech., 92 (2012), 888-909.  doi: 10.1002/zamm.201100155.  Google Scholar

[37]

W. Han and B. D. Reddy, phPlasticity, in Mathematical Theory and Numerical Analysis, Interdisciplinary Applied Mathematics, 9, Springer, New York, 2013. doi: 10.1007/978-1-4614-5940-8.  Google Scholar

[38]

E. Kröner, Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen, Arch. Rational Mech. Anal., 4 (1960), 273-334.  doi: 10.1007/BF00281393.  Google Scholar

[39]

M. Kružík and T. Roubíček, Optimization problems with concentration and oscillation effects: Relaxation theory and numerical approximation, Numer. Funct. Anal. Optim., 20 (1999), 511-530.  doi: 10.1080/01630569908816908.  Google Scholar

[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. Google Scholar

[41]

H. Le Dret and A. Raoult, The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 1179-1192.  doi: 10.1017/S0308210500030456.  Google Scholar

[42]

E. H. Lee, Elastic-plastic deformation at finite strains, J. Appl. Mech., 36 (1969), 1-6.  doi: 10.21236/AD0678483.  Google Scholar

[43]

S. Luckhaus and L. Mugnai, On a mesoscopic many-body {H}amiltonian describing elastic shears and dislocations, Contin. Mech. Thermodyn., 22 (2010), 251-290.  doi: 10.1007/s00161-010-0142-0.  Google Scholar

[44]

S. Luckhaus and J. Wohlgemuth, Study of a model for reference-free plasticity, preprint, arXiv: 1408.1355. Google Scholar

[45]

M. Luskin, On the computation of crystalline microstructure, in Acta Numerica, 1996 , Acta Numer., 5, Cambridge Univ. Press, Cambridge, 1996, 191-257. doi: 10.1017/S0962492900002658.  Google Scholar

[46]

A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, J. Nonlinear Sci., 19 (2009), 221-248.  doi: 10.1007/s00332-008-9033-y.  Google Scholar

[47]

C. Miehe, On the representation of Prandtl-Reuss tensors within the framework of multiplicative elastoplasticity, Int. J. Plasticity, 10 (1994), 609-621.  doi: 10.1016/0749-6419(94)90025-6.  Google Scholar

[48]

C. Miehe, M. Lambrecht and 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 (2004), 2725-2769.  doi: 10.1016/j.jmps.2004.05.011.  Google Scholar

[49]

C. Miehe and E. Stein, A canonical model of multiplicative elasto-plasticity: Formulation and aspects of the numerical implementation, Europ. J. Mech. A/Solids, 11 (1992), 25-43.   Google Scholar

[50]

C. Miehe and 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 (2003), 1-41.  doi: 10.1002/nme.726.  Google Scholar

[51]

A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Contin. Mech. Thermodyn., 15 (2003), 351-382.  doi: 10.1007/s00161-003-0120-x.  Google Scholar

[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, . doi: 10.1002/zamm.200510245.  Google Scholar

[53]

A. Mielke, R. Rossi and G. Savaré, Global existence results for viscoplasticity at finite strain, Arch. Ration. Mech. Anal., 227 (2018), 423-475.  doi: 10.1007/s00205-017-1164-6.  Google Scholar

[54]

A. Mielke and T. Roubíček, Rate-independent systems, in Theory and Application, Applied Mathematical Sciences, 193, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[55]

A. Mielke and T. Roubíček, Rate-independent elastoplasticity at finite strains and its numerical approximation, Math. Models Methods Appl. Sci., 26 (2016), 2203-2236.  doi: 10.1142/S0218202516500512.  Google Scholar

[56]

A. Mielke and U. Stefanelli, Linearized plasticity is the evolutionary $\Gamma$-limit of finite plasticity, J. Eur. Math. Soc. (JEMS), 15 (2013), 923-948.  doi: 10.4171/JEMS/381.  Google Scholar

[57]

A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177.  doi: 10.1007/s002050200194.  Google Scholar

[58]

J. Moreau, Sur les lois de frottement, de plasticité et de viscosité, Comptes Rendus de l'Académie des Sciences, 271 (1970), 608-611.   Google Scholar

[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. doi: 10.1007/978-3-540-69952-1.  Google Scholar

[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. doi: 10.1007/BFb0092670.  Google Scholar

[61]

S. Müller and V. Šverák, Convex integration with constraints and applications to phase transitions and partial differential equations, J. Eur. Math. Soc. (JEMS), 1 (1999), 393-442.  doi: 10.1007/s100970050012.  Google Scholar

[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. doi: 10.1007/978-3-319-18242-1_7.  Google Scholar

[63]

M. Ortiz and E. A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals, J. Mech. Phys. Solids, 47 (1999), 397-462.  doi: 10.1016/S0022-5096(97)00096-3.  Google Scholar

[64]

C. Reina and 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 (2014), 40-61.  doi: 10.1016/j.jmps.2014.01.014.  Google Scholar

[65]

C. Reina and S. Conti, Incompressible inelasticity as an essential ingredient for the validity of the kinematic decomposition $F = F^eF^i$, J. Mech. Phys. Solids, 107 (2017), 322-342.  doi: 10.1016/j.jmps.2017.07.004.  Google Scholar

[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. doi: 10.1515/9783110811919.  Google Scholar

[67]

T. Roubíček, Numerical techniques in relaxed optimization problems, in Robust Optimization-Directed Design, Nonconvex Optim. Appl., 81, Springer, New York, 2006,157–178. doi: 10.1007/0-387-28654-3_8.  Google Scholar

[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 (1988), 199-219.  doi: 10.1016/0045-7825(88)90076-X.  Google Scholar

[69]

J. C. Simo and M. Ortiz, A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations, Comput. Methods. Appl. Mech. Engrg., 49 (1985), 221-245.  doi: 10.1016/0045-7825(85)90061-1.  Google Scholar

11,Figure 1] for the plot with $ h = 0 $.">Figure 1.  $ W_{\mathrm{CHM}} $ with $ \mu = 2 $ along the rank-one line (3.6) with $ h = 0.1 $ and $ \tau = 1 $ (left panel) and $ h = 1 $ and $ \tau = 0 $ (right panel), see also [11,Figure 1] for the plot with $ h = 0 $.
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Figure 2.  Left panel: relation between $ W_j $, $ W_{\mathrm{rigid,0}} $, $ W_j^{\mathrm{qc}} $, and $ W_{\mathrm{rigid,0}}^{\mathrm{qc}} $. Right panel: known relations between the corresponding functionals $ E_j $, $ E_{\mathrm{rigid,0}} $, $ E_j^* $, and $ E_{\mathrm{rigid,0}}^* $. See Section 5 for the definitions and details
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Table 1.  Relaxation results in the literature with {rigid elasticity and} plastic energy density proportional to $ |\gamma|^\alpha $, $ \alpha = 1,2 $ and $ h\tau = 0 $. In the three cases with $ W^{\mathrm{rc}} = W^{\mathrm{qc}} = W^{\mathrm{pc}} $ an explicit formula is given in the mentioned papers, in the others there are only partial results
slip systems $ h=0 $ $ \tau=0 $
$ N=1 $ $ W^{\mathrm{rc}} = W^{\mathrm{qc}} = W^{\mathrm{pc}} $ [29] $ W^{\mathrm{rc}} = W^{\mathrm{qc}} = W^{\mathrm{pc}} $ [18]
$ N=2 $ at $ 90^\circ $ $ W^{\mathrm{rc}} \neq W^{\mathrm{pc}} $ [1] $ W^{\mathrm{rc}} = W^{\mathrm{qc}} = W^{\mathrm{pc}} $ [28]
$ N=3 $ at $ 120^\circ $ no results partial results [25]
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slip systems $ h=0 $ $ \tau=0 $
$ N=1 $ $ W^{\mathrm{rc}} = W^{\mathrm{qc}} = W^{\mathrm{pc}} $ [29] $ W^{\mathrm{rc}} = W^{\mathrm{qc}} = W^{\mathrm{pc}} $ [18]
$ N=2 $ at $ 90^\circ $ $ W^{\mathrm{rc}} \neq W^{\mathrm{pc}} $ [1] $ W^{\mathrm{rc}} = W^{\mathrm{qc}} = W^{\mathrm{pc}} $ [28]
$ N=3 $ at $ 120^\circ $ no results partial results [25]
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