A Characterisation of the Boundary Displacements Which Induce Cavitation in an Elastic Body

The Radial Volume Derivative and the Critical Boundary Displacement for Cavitation

“…We note that the volume derivative studied in the current paper is purely re stricted to radial deformations and hence may not coincide with the general volume derivative as defined in [14] (the derivative in [14] is defined using the class of all, possibly nonsymmetric, deformations, and it is currently not known whether these energy minimizers coincide with radial minimizers in the compressible case 4 ).…”

“…In this paper we give an alternative characterization of the unstable set of bound ary strains U which we then use as the basis for a numerical method to evaluate ∂U for a number of energy functions. This characterization is based on the approach given in [14] and is motivated in part by calculations of Varvaruca [23]. We first define the set of deformations (10) A λ,c = {r ∈ A λ | r(0) = c}, which produce a hole of a fixed radius c > 0 and volume V = ω n n c n .…”

“…Thus G(λ) = 0 if and only if h (λ n ) = 0, which is exactly the same condition obtained in [4] for the criteria for the initiation of cavitation. This example can be generalized to the case of nonradial deformations (see [14]).…”

“…Furthermore, works such as Hou and Abeyaratne [6], Lopez-Pamies, Idiart, and Nakamura [9], Lopez-Pamies, Nakamura, and Idiart [10], Huang, Hutchinson, and Tvegaard [7], and Tvergaard, Huang, and Hutchinson [22] have proposed methods to approximate the onset of cavitation in nonsymmetric situations. A new approach for determining the onset of cavitation for nonsymmetric boundary conditions, which is based on the method presented in the current paper, is given in [14].…”

“…A central problem in the general nonsymmetric case is to identify the set of affine displacement boundary conditions for which a corresponding energy minimizer produces holes. In [14], the authors introduce a notion of a derivative of the energy functional with respect to hole formation and conjecture that the zero set of this derivative corresponds to the boundary of the set of matrices which, when used to define affine displacement boundary data, result in a discontinuous energy minimizer. The current paper considers a radial version of this conjecture, and we prove that the conjecture is true in this case.…”