From nonlinear to linearized elasticity via Γ-convergence: The case of multiwell energies satisfying weak coercivity conditions

Abstract

From nonlinear to linearized elasticity via -convergence: the Γ case of multiwell energies satisfying weak coercivity conditions Copyright and reuse:Sussex Research Online is a digital repository of the research output of the University.Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable, the material made available in SRO has been checked for eligibility before being made available.Copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. Abstract. Linearized elasticity models are derived, via Γ-convergence, from suitably rescaled nonlinear energies when the corresponding energy densities have a multiwell structure and satisfy a weak coercivity condition, in the sense that the typical quadratic bound from below is replaced by a weaker p bound, 1 < p < 2, away from the wells. This study is motivated by, and our results are applied to, energies arising in the modeling of nematic elastomers.

Derivation of a Linearised Elasticity Model from Singularly Perturbed Multiwell Energy Functionals

“…In this case, the convergence of minimisers holds in W 1,q (Ω; R d ). Further generalisations of these results have been obtained in [17,1] in the case of multi-well energies where the distance between the wells is of order ε. It is worth mentioning that, in the context of Γ-convergence, linear elastic energies have been also obtained as macroscopic limit of nonlinear atomistic energies in [5,18].…”

“…The above formal derivation of linear elasticity can be made rigorous in terms of convergence of absolute minimisers of boundary value problems associated with the functionals (0.1) and (0.2). This has been done in the framework of Γ-convergence in [9,17,2,1]. In the first paper [9], convergence of minimisers was proven in H 1 (Ω; R d ) under the condition (0.4)…”

The Gap Between Linear Elasticity and the Variational Limit of Finite Elasticity in Pure Traction Problems

“…This result clarified the mathematical consistency of the linear boundary value problems, under displacements and forces Date: March 14, 2019. 1 prescribed on the boundary of a three-dimensional material body, via a rigorous deduction from the nonlinear elasticity theory. We mention several papers facing issues in elasticity which are connected with the context of our paper: [1], [2], [3], [4], [5] [6], [7], [8], [9], [20] [22], [23], [24], [26], [27], [28], [29].…”

Dimension reduction via $$\Gamma $$ Γ -convergence for soft active materials

“…In particular, we focus our attention on nematic elastomers [1,2,5,14,15,28]. These are polymeric materials that, thanks to the coupling between elasticity and orientational nematic order, undergo spontaneous deformations as a consequence of a temperature-driven phase transformation.…”