On non-linear stability in unconstrained non-linear elasticity

Haughton, D.M. (2004) On non-linear stability in unconstrained non-linear elasticity. International Journal of Non-Linear Mechanics, 39(7), pp. 1181-1192. (doi: 10.1016/j.ijnonlinmec.2003.07.002)

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Publisher's URL: http://dx.doi.org/10.1016/j.ijnonlinmec.2003.07.002


A method of obtaining a full three-dimensional non-linear Hadamard stability analysis of inhomogeneous deformations of arbitrary, unconstrained, hyperelastic materials is presented. The analysis is an extension of that given by Chen and Haughton (Proc. Roy. Soc. London A 459 (2003) 137) for two-dimensional incompressible problems. The process that we present replaces the second variation condition expressed as an integral involving a quadratic in three arbitrary perturbations, with an equivalent sixth-order system of ordinary differential equations. The positive definiteness condition is thereby reduced to the simple numerical evaluation of zeros of a well-behaved function. The general theory is illustrated by applying it to the problem of the inflation of a thick-walled spherical shell. The present analysis provides a simpler alternative approach to bifurcation problems approached by using the incremental equations of non-linear elasticity.

Item Type:Articles
Keywords:Second variation, Non-linear, Hadamard stability, Elasticity
Glasgow Author(s) Enlighten ID:Haughton, Dr David
Authors: Haughton, D.M.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:International Journal of Non-Linear Mechanics
Journal Abbr.:Int. J, Non-Linear Mech.
Published Online:28 August 2003

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