The Numerical Computation of the Critical Boundary Displacement for Radial Cavitation

Abstract

We study radial solutions of the equations of isotropic elasticity in two dimensions (for a disc) and three dimensions (for a sphere). We describe a numerical scheme for computing the critical boundary displacement for cavitation based on the solution of a sequence of initial value problems for punctured domains. We give examples for specific materials and compare our numerical computations with some previous analytical results. A key observation in the formulation of the method is that the strong-ellipticity condition implies that the specification of the normal component of the Cauchy stress on an inner pre-existing but small cavity, leads to a relation for the radial strain as a function of the circumferential strain. To establish the convergence of the numerical scheme we prove a monotonicity property for the inner deformed radius for punctured balls.

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

“…the third term in (30) is not even a power function). We note, however, that in each case when δ = 0 the amplitude r(0) is proportional to √ λ − λ cr , which is consistent with the general result given by Negrón-Marrero and Sivaloganathan [40].…”

“…We note, however, that these expansions would break down in the limit λ → λ cr , which is the parameter regime to be examined in the current paper. Some asymptotic results have been derived by Negrón-Marrero and Sivaloganathan [40] to aid their numerical calculations. In particular, they showed that even for a general strain-energy function the deformed cavity radius is given by r(0) = C(λ−λ cr ) 1/n to leading order as λ approaches λ cr , where C is a positive constant and n is the dimension of the cavitation problem.…”

The Radial Volume Derivative and the Critical Boundary Displacement for Cavitation

“…Also, as previously noted, our expression (41) for the vanishing of the radial volume deriva tive coincides with the condition for the critical boundary displacement for radial cavitation obtained, using a shooting argument, by Stuart in the interesting paper [21]. However, the methods in [13] and that in [21] do not generalize to nonsymmet ric problems. On the contrary, the approach of the current paper, identifying the critical boundary displacements as the zero set of the volume derivative, provides a novel, and potentially very effective, criterion for the computation of the boundary of the set of linear displacement boundary conditions for which cavitation occurs in multidimensional, nonsymmetric problems (see [14]).…”

“…In section 6 we regularize the problem of minimizing (5) over A λ,c in the volume derivative (12) or (13), by replacing the solid ball B by a punctured ball B with a pre-existing hole of radius > 0 in its reference configuration. Specifically, we prove that if we correspondingly write…”

“…For the radial problem, a more efficient numerical method to approximate λ crit than the one used in section 7 is given in [13]. In that paper, the approach in section 6, using regularized problems on punctured balls, is used as the basis for a very efficient numerical method for computing the critical boundary displacement λ crit .…”