Cavitation In Nonlinear Elasticity Research Articles

Radial Symmetry of p-Harmonic Minimizers

"It is still not known if the radial cavitating minimizers obtained by Ball [J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. R. Soc. Lond. A 306 (1982) 557--611] (and subsequently by many others) are global minimizers of any physically reasonable nonlinearly elastic energy". The quotation is from [J. Sivaloganathan and S. J. Spector, Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity, Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008), no. 1, 201--213] and seems to be still accurate. The model case of the $p$-harmonic energy is considered here. We prove that the planar radial minimizers are indeed the global minimizers provided we prescribe the admissible deformations on the boundary. In the traction free setting, however, even the identity map need not be a global minimizer.

Mathematical theory and numerical computation of cavitation in nonlinear elasticity

Many soft nonlinear elastic materials, such as polymers, organic tissues, etc., can produce special singular deformation called cavitation under certain conditions. The study of cavitation is an important part of the effort to understand, characterize, analyze and apply the materials. In this paper, we will try to show the readers a rough picture of the computation and simulation of the cavitation phenomenon in nonlinear elasticity by a brief review on some of the representative mathematical models and theories, numerical methods and corresponding analytical and numerical results.

Γ-convergence Approximation of Fracture and Cavitation in Nonlinear Elasticity

Our starting point is a variational model in nonlinear elasticity that allows for cavitation and fracture that was introduced by Henao and Mora-Corral (Arch Rational Mech Anal 197:619–655, 2010). The total energy to minimize is the sum of the elastic energy plus the energy produced by crack and surface formation. It is a free discontinuity problem, since the crack set and the set of new surface are unknowns of the problem. The expression of the functional involves a volume integral and two surface integrals, and this fact makes the problem numerically intractable. In this paper we propose an approximation (in the sense of Γ-convergence) by functionals involving only volume integrals, which makes a numerical approximation by finite elements feasible. This approximation has some similarities to the Modica–Mortola approximation of the perimeter and the Ambrosio–Tortorelli approximation of the Mumford–Shah functional, but with the added difficulties typical of nonlinear elasticity, in which the deformation is assumed to be one-to-one and orientation-preserving.

The problem of dynamic cavitation in nonlinear elasticity

The notion of singular limiting induced from continuum solutions (slic-solutions) is applied to the problem of cavitation in nonlinear elasticity, in order to re-assess an example of non-uniqueness of entropic weak solutions (with polyconvex energy) due to a forming cavity.

AN EFFICIENT NUMERICAL METHOD FOR CAVITATION IN NONLINEAR ELASTICITY

This paper is concerned with the numerical computation of cavitation in nonlinear elasticity. The Crouzeix–Raviart nonconforming finite element method is shown to prevent the degeneration of the mesh provoked by the conventional finite element approximation of this problem. Upon the addition of a suitable stabilizing term to the elastic energy, the method is used to solve cavitation problems in both radially symmetric and non-radially symmetric settings. While the radially symmetric examples serve to illustrate the efficiency of the method, and for validation purposes, the experiments with non-centered and multiple cavities (carried out for the first time) yield novel observations of situations potentially leading to void coalescence.

A dual-parametric finite element method for cavitation in nonlinear elasticity

A dual-parametric finite element method is introduced in this paper for the computation of singular minimizers in the 2D cavitation problem in nonlinear elasticity. The method overcomes the difficulties, such as the mesh entanglement and material interpenetration, generally encountered in the finite element approximation of problems with extremely large expansionary deformation. Numerical experiments show that the method is highly efficient in the computation of cavitation problems. Numerical experiments are also conducted on discrete problems without the radial symmetry to show the validity of the method to more general settings and the potential of its application to the study of mechanism of cavity nucleation in nonlinear elastic materials.

Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity

It is still not known if the radial cavitating minimizers obtained by Ball [J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. R. Soc. Lond. A 306 (1982) 557–611] (and subsequently by many others) are global minimizers of any physically reasonable nonlinearly elastic energy. We therefore consider in this paper the related problem of obtaining necessary conditions for these radial solutions to be minimizers with respect to nonradial perturbations. A standard blowup argument applied to either an inner or an outer variation yields an apparently new inequality that, for most constitutive relations, has yet to be verified. However, in the special case of a compressible neo-Hookean material, W(F)=μ2|F|2+h(detF), we show that the inequality produced by an outer variation clearly holds whilst that produced by an inner variation is a well-known inequality (first proven by Brezis, Coron, and Lieb [H. Brezis, J.-M. Coron, E.H. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986) 649–705] ) which arises in the theory of nematic liquid crystals:∫B|∇n(x)|2dx⩾∫B|∇(x|x|)|2dx=8π for all n∈W1,2(B,∂B) (so that |n|=1 a.e.) that satisfy n(x)=x/|x| on ∂B, where B is the unit ball in R3.

On cavitation and degenerate cavitation under internal hydrostatic pressure

In this paper we present some interesting variants on the mathematical phenomenon of cavitation in nonlinear elasticity. The paper is motivated by the experimental work of Gent and Tompkins on pressurized elastomers, the fundamental mathematical work of Ball and an example of degenerate cavitation due to Sivaloganathan.

Discontinuous equilibrium solutions and cavitation in nonlinear elasticity

A study is made of a class of singular solutions to the equations of nonlinear elastostatics in which a spherical cavity forms at the centre of a ball of isotropic material placed in tension by means of given surface tractions or displacements. The existence of such solutions depends on the growth properties of the stored-energy functionWfor large strains and is consistent with strong ellipticity ofW. Under appropriate hypotheses it is shown that a singular solution bifurcates from a trivial (homogeneous) solution at a critical value of the surface traction or displacement, at which the trivial solution becomes unstable. For incompressible materials both the singular solution and the critical surface traction are given explicitly, and the stability of all solutions with respect to radial motion is determined. For compressible materials the existence of singular solutions is proved for a class of strongly elliptic materials by means of the direct method of the calculus of variations, an important step in the analysis being to show that the only radial equilibrium solutions without cavities are homogeneous. Work of Gent & Lindley (1958) shows that the critical surface tractions obtained agree with those observed in the internal rupture of rubber.