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A remark on a stability criterion for the radial cavitating map in nonlinear elasticity

Bevan, JJ (2015) A remark on a stability criterion for the radial cavitating map in nonlinear elasticity Journal of Mathematical Analysis and Applications, 425 (2). pp. 954-982.

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Abstract

We study the integral functional I(w) := ∫|adj∇w(w|w|)|qdx on suitable maps w:B⊂R → R and where 2q∈(2, 3). The inequality I(w)≥I(i), which we establish on a subclass of the admissible maps, was first proposed in [13] as one of two possible necessary conditions for the stability, i.e. local minimality, of the radial cavitating map in nonlinear elasticity. Here, i is the identity map. Admissible maps w either do not vanish (and in this case possess a single discontinuity x in B which produces a cavity about the origin), or vanish at exactly one point x in B, in which case w is a diffeomorphism in a neighbourhood of x. We show that I({dot operator}) behaves like a polyconvex functional and associate with it another functional, K({dot operator}), satisfying I(w)≥I(i)+q(K(w)-K(i)). We give conditions under which K(w)=K(i), and from these infer I(w)≥I(i). It is also shown that (i) K is strictly decreasing along paths of admissible functions that move x away from the origin and (ii) K(w) exhibits some quite pathological behaviour when w is sufficiently close to i.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
AuthorsEmailORCID
Bevan, JJUNSPECIFIEDUNSPECIFIED
Date : 15 May 2015
Identification Number : 10.1016/j.jmaa.2015.01.007
Additional Information : © 2015, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/
Depositing User : Symplectic Elements
Date Deposited : 12 Jun 2015 17:02
Last Modified : 15 May 2016 01:08
URI: http://epubs.surrey.ac.uk/id/eprint/807916

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