The radial volume derivative and the critical boundary displacement for cavitation

We study the displacement boundary value problem of minimizing the total energy E(u) stored in a nonlinearly elastic body occupying a spherical domain B in its reference configuration over (possibly discontinuous) radial deformations u of the body subject to affine boundary data u(x) = λx for x ∈ ∂B. For a given value of λ, we define what we call the radial volume derivative at λ, denoted by G(λ), which measures the stability or instability of the underlying homogeneous deformation u h λ(x) ≡ λx to the formation of holes. We give conditions under which the critical boundary displacement λ crit for radial cavitation is the unique solution of G(λ) = 0. Moreover, we prove that the radial volume derivative G(λ) can be approximated by the corresponding volume derivative for a punctured ball B ∈, containing a pre-existing cavity of radius ∈ > 0 in its reference state, in the limit ∈ → 0 and we use this to propose a numerical scheme to determine λ crit. We illustrate these general results with analytical and numerical examples.