Optimal laminates in single-slip elastoplasticity

Sergio Conti; Georg Dolzmann

doi:10.3934/dcdss.2020302

Article Contents

2021, Volume 14, Issue 1: 1-16. Doi: 10.3934/dcdss.2020302

This issue Previous Article Preface Next Article Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity

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
^* Corresponding author: Georg Dolzmann

Received: March 2019

Revised: August 2019

Early access: March 2020

Published: January 2021

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 74C15; Secondary: 74B20, 49J45.

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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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