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A calibration method for estimating critical cavitation loads from below in 3D nonlinear elasticity

Bevan, Jonathan and Deane, Jonathan (2018) A calibration method for estimating critical cavitation loads from below in 3D nonlinear elasticity SIAM Journal on Mathematical Analysis.

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Abstract

In this paper we give an explicit sufficient condition for the affine map uλ(x):=λx to be the global energy minimizer of a general class of elastic stored-energy functionals I(u)=∫ΩW(∇u)dx in three space dimensions, where W is a polyconvex function of 3×3 matrices. The function space setting is such that cavitating (i.e., discontinuous) deformations are admissible. In the language of the calculus of variations, the condition ensures the quasiconvexity of I(⋅) at λ1, where 1 is the 3×3 identity matrix. Our approach relies on arguments involving null Lagrangians (in this case, affine combinations of the minors of 3×3 matrices), on the previous work Bevan &amp; Zeppieri, 2015, and on a careful numerical treatment to make the calculation of certain constants tractable. We also derive a new condition, which seems to depend heavily on the smallest singular value λ1(∇u) of a competing deformation u, that is necessary for the inequality I(u)<I(uλ), and which, in particular, does not exclude the possibility of cavitation.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
NameEmailORCID
Bevan, JonathanJ.Bevan@surrey.ac.ukUNSPECIFIED
Deane, JonathanJ.Deane@surrey.ac.ukUNSPECIFIED
Date : 2018
Copyright Disclaimer : © 2018, Society for Industrial and Applied Mathematics
Related URLs :
Depositing User : Clive Harris
Date Deposited : 19 Jan 2018 14:54
Last Modified : 19 Jan 2018 14:54
URI: http://epubs.surrey.ac.uk/id/eprint/845650

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