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

Abstract

In this paper, the effect of material inhomogeneity on void formation and growth in incompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid composite sphere composed of two neo-Hookean materials perfectly bonded across a spherical interface. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the undeformed configuration. Such a configuration is the only stable solution for sufficiently large loads. In contrast to the situation for a homogeneous neo-Hookean sphere, bifurcation here may occur either locally to the right or to the left. In the latter case, the cavity has finite radius on first appearance. This discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon observed in certain structural mechanics problems.

Cavitation for incompressible anisotropic nonlinearly elastic spheres

“…Further studies for incompressible materials were carried out in [5,21] for elastostatics and in [6] for elastodynamics. The effects of material inhomogeneity on cavitation in incompressible materials were investigated by Horgan and Pence [17][18][19]. Void collapse for both incompressible and compressible materials has been examined in [2].…”

Cavitation, the incompressible limit, and material inhomogeneity

“…We show that the mathematical phenomenon of cavitation depends crucially on the nature of the material present at the origin of the ball. In particular, for a class of inhomogeneous incompressible materials, the critical load for cavitation is the same as the critical load for a homogeneous incompressible ball composed entirely of the material found at the origin of the inhomogeneous ball (this result appears in [11]: the related problem of a composite sphere made of two homogeneous incompressible materials was subsequently studied, independently, in Horgan and Pence [5,6] and analogous conclusions drawn). For the inhomogeneous compressible case we give sufficient conditions for cavitation to occur and demonstrate that any deformation which keeps the ball intact is unstable if the radial stretch at the origin exceeds , the critical boundary displacement for a ball composed entirely of the material present at the origin of the inhomogeneous ball.…”

“…In Theorem 3 we show that the critical displacements j.km at which cavitation To show convergence of these critical loads to the incompressible critical load poses a more difficult problem as it involves passing to the limit in (6) as Akrix -► 1 and k -> 0 simultaneously. We overcome this by an alternative characterisation of the critical load as the "stress at infinity" in an infinite body.…”

“…Thus, we may also assume that the corresponding symmetric function <f>mc(Vj, v2, v3) is also defined on all of R3 + . The equilibrium equations for incompressible materials are the Euler-Lagrange equations for the functional 6) where p(X), the pressure, is a Lagrange multiplier corresponding to the constraint of incompressibility (1.5).…”

On the stability of cavitating equilibria

“…The mathematical phenomenon of cavitation in the setting of finite elasticity was first demonstrated by Ball in [3]. This work was subsequently extended and generalised in [23,21,15] and many aspects of the problem are now well understood (see [1,9,10,11,12,15,17,22,18,8] and the references therein). It is known that these cavitating equilibria, which correspond to discontinuous radial deformations of a ball of hyperelastic material, are often global minimisers of the stored energy in classes of radial maps of the ball (see [3,21,22]).…”