Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity

Samuel Amstutz; Nicolas Van Goethem

doi:10.3934/dcdsb.2020240

Article Contents

2020, Volume 25, Issue 10: 3769-3805. Doi: 10.3934/dcdsb.2020240

This issue Previous Article Preface Next Article Higher-order time-stepping schemes for fluid-structure interaction problems

Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity

Samuel Amstutz^1, and
Nicolas Van Goethem^2,

1.
Ecole polytechnique, CMAP, Route de Saclay, 91128 Palaiseau, France
2.
Universidade de Lisboa, Faculdade de Ciências, Departamento de Matemática, CMAFcIO, Portugal

Received: August 2019

Revised: June 2020

Early access: July 2020

Published: October 2020

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Abstract

A general model of incompatible small-strain elasticity is presented and analyzed, based on the linearized strain and its associated incompatibility tensor field. Strain incompatibility accounts for the presence of dislocations, whose motion is ultimately responsible for the plastic behaviour of solids. The specific functional setting is built up, on which existence results are proved. Our solution strategy is essentially based on the projection of the governing equations on appropriate subspaces in the spirit of the Leray decomposition of solenoidal square-integrable velocity fields in hydrodynamics. It is also strongly related with the Beltrami decomposition of symmetric tensor fields in the wake of previous works by the authors. Moreover a novel model parameter is introduced, the incompatibility modulus, that measures the resistance of the elastic material to incompatible deformations. An important result of our study is that classical linearized elasticity is recovered as the limit case when the incompatibility modulus goes to infinity. Several examples are provided to illustrate this property and the physical meaning of the incompatibility modulus in connection with the dissipative nature of the processes under consideration.

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Mathematics Subject Classification: 35J48, 35J58, 49S05, 49K20, 74C05, 74G99, 74A05, 74A15, 80A17.

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Figure 1. In-plane strain ($ \varphi(z) $, top left) and vertical strain ($ \psi(z) $, top right) with $ z $ on the horizontal axis, for $ \ell = -10 $ (blue), $ \ell = -100 $ (red), $ \ell = -1000 $ (yellow). Value of $ u(h) $ as a function of $ \ell $ (bottom right)

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Figure 2. Strain components in cylindrical coordinates, as functions of $ r $, for $ \ell = -1000 $ (blue), $ \ell = -100 $ (red), $ \ell = -20 $ (yellow), classical plane strain elastic solution (dashed)

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Figure 3. Functions $ \varphi $, $ \psi $ and $ u $ for $ \ell = -1000 $ (blue), $ \ell = -100 $ (red), $ \ell = -10 $ (yellow), elastic solution (dashed)

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Table 1. External work

$ \ell $ $ -1000 $ $ -100 $ $ -20 $

$ W $ $ 0.3006 $ $ 0.4616 $ $ 1.0620 $

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