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  January 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (1) : 1-16. doi: 10.3934/dcdss.2020302
References:
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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.

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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.

[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.

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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.

[5]

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

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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.

[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.

[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.

[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.

[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.

[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.

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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.

[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.

[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.

[15]

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

[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.

[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.

[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 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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.

[33]

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

[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.

[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. 

[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.

[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.

[38]

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

[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.

[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]

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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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