Cavitation for incompressible anisotropic nonlinearly elastic spheres

Abstract

In this paper, the effect of material anisotropy on void nucleation and growth in incompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid sphere composed of an incompressible homogeneous nonlinearly elastic material which is transversely isotropic about the radial direction. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the undeformed configuration at sufficiently large loads. Closed form analytic solutions are obtained for a specific material model, which may be viewed as a generalization of the classic neo-Hookean model to anisotropic materials. In contrast to the situation for a neo-Hookean sphere, bifurcation here may occur locally either to the right (supercritical) or to the left (subcritical), depending on the degree of anisotropy. In the latter case, the cavity has finite radius on first appearance. Such a discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon of structural mechanics. Such dramatic cavitational instabilities were previously encountered by Antman and Negr6n-Marrero [3] for anisotropic compressible solids and by Horgan and Pence 1-17] for composite incompressible spheres.

Emergence and disappearance of load induced fiber kinking surfaces in transversely isotropic hyperelastic materials

“…An alternating layer solution is a stack of (sub)layers, each with a constant shearing using one of the three different fixed values of κ determined by T app . For such alternating layer solutions, the separating planes have normal e 2 and hence traction continuity requires continuity of T 12 , T 22 and T 32 across these separating planes. Since T 32 vanishes identically, the last of these three conditions is automatic.…”

“…Anisotropic hyperelasticity theory provides a natural framework for the modeling of fiber reinforced materials subject to large deformation [22,23,25,26,28]. Such models are of great interest in a variety of fields, a notable example being that of soft tissue mechanical behavior [10].…”

The Numerical Computation of the Critical Boundary Displacement for Radial Cavitation

“…(Hence, in particular, the curve representing g(λ) in figure 1 intersects the λ axis with an infinite slope.) This problem of characterizing λ crit has been studied extensively in the past, see, for example, the works of [10], [5], [36], [11], and [23]. However, the emphasis in most of these papers is in deriving exact, closed form solutions for the cavitation solution for specific materials from which the critical boundary displacement can then be obtained (an exception is the work of [36] which gives interesting bounds on λ crit for stored energy functions with a special structure).…”

On the near-critical behavior of cavitation in elastic plane membranes

“…When the material is compressible, the two-point boundary value problem can be solved by a shooting procedure in the most general case (see, e.g., [8]), but many studies have focused on finding closed-form solutions for specific material models (see, e.g., [9][10][11][12]). There also exists a large body of literature concerned with the effects of anisotropy, material inhomogeneity, surface tension, and plastic behavior; see, e.g., [13][14][15][16][17][18][19]. We refer to [20] and [21] for a comprehensive review of the literature.…”