Discontinuous energy minimizers in nonlinear elastostatics: an example of J. Ball revisited

Abstract

Simple direct methods of the Calculus of Variations, together with mild a priori restrictions of a constitutive nature, are exploited to show that a uniform radial displacement at the boundary fails to induce a homogeneous radial expansion of a compressible elastic ball when the boundary displacement reaches a critical value (of which an explicit lower bound is offered): rather, above the critical value, energy minimizers are radial deformations with a hole at the center, and are accompanied by a stress field with radial stress vanishing, and circumferential stress unbounded, at the surface of the hole.

Void collapse in an elastic solid

“…In Bali's study he carried out a detailed analysis of this problem for the case of an isotropic, incompressible material: in particular, he determined the precise sub-class of such materials which exhibit this pheneomenon, and for such materials, derived an explicit expression for the load level at which the cavity radius becomes infinite. (Ball (1982), Stuart (1985), , Podio-Guidugli et al (1986) and Sivaloganathan (1986aSivaloganathan ( , 1986b have considered various aspects of the corresponding problem for a compressible medium. See also Horgan and Pence (1989) and James and Spector (1989).…”

Void nucleation in tensile dead-loading of a composite incompressible nonlinearly elastic sphere

“…In a number of recent papers [1][2][3][4][5][6][7] the phenomenon of void formation and growth has been examined analytically within the framework of nonlinear elasticity and plasticity. Such phenomena have long been of concern to metallurgists (see e.g., [8]).…”

“…3.23) has been used to obtain the last equality in (3.27). Since B(Q) < 0, Q > 0, we see that (3.27) yields the desired result (3.26).…”

Cavitation, the incompressible limit, and material inhomogeneity

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Motivated by work of Gent and Lindley [3], a rigorous treatment of cavitation in finite elasticity was first given by Ball [2] in a fundamental paper, and subsequently by a number of authors [4,10,12,16] (see also [ 1 ] for aelotropic materials and [9] for the dynamic problem). The setting for these works is a ball of initially perfect material which is held in a state of tension under prescribed radial loads or displacements on the boundary.

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“…Thus, under the corresponding deformation u (given by (1.7)) a cavity of radius r(0) forms at the centre of the ball and (iii) is the natural boundary condition that the cavity surface is stress free. The existence of cavitating solutions is studied in [2,4,10,12,16], For incompressible materials the constraint (1. parametrised by k e (0, kQ), where <I>inc represents the stored energy function of an incompressible material and h is a compressibility term. We will denote by Ik the corresponding energy functional (1.11).…”