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

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[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

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[16]

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[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

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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

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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

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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

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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.  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]

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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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