Abstract. Experiments on elastomers have shown that sufficiently-large triaxial tensions induce the material to exhibit holes that were not previously evident. In this paper conditions are presented that allow one to use the direct method of the calculus of variations to deduce the existence of hole creating deformations that are global minimizers of a nonlinear, purely-elastic energy. The crucial physical assumption used is that there are a finite (possibly large) number of material points in the undeformed body that constitute the only points at which cavities can form. Each such point can be viewed as a preexisting flaw or an infinitesimal microvoid in the material.
We say that L ( x , U , Vu) is a null Lagrangian if and only if the corresponding integral functional g(u) = Jn L(x, U , Vu) dx has the property that g(uIn the homogeneous case, corresponding to L(x, U, Vu) = @(Vu), it is known that a necessary and sufficient condition for L to be a null Lagrangian is that @(Vu) is an affine combination of subdeterminants of Vu of all orders. In this paper we show that all inhomogeneous null Lagrangians may be constructed from these homogeneous ones by introducing appropriate potentials.
Abstract. Consider a nonlinearly elastic body which occupies the region Ω ⊂ R m (m = 2, 3) in its reference state and which is held in tension under prescribed boundary displacements on ∂Ω. Let x 0 ∈ Ω be any fixed point in the body. It is known from variational arguments that, for sufficiently large boundary displacements, there may exist discontinuous weak solutions of the equilibrium equations corresponding to a hole forming at x 0 in the deformed body (this is the phenomenon of cavitation). For each > 0, define the regularized domains Ω = Ω\B (x 0 ) which contain a preexisting hole of radius > 0 centered on x 0 . Now consider the corresponding mixed displacement/traction problem on Ω in which the boundary ∂Ω is subject to the same boundary displacements and the deformed cavity surface (i.e., the image of ∂B ) is required to be stress-free. It follows from variational arguments that there exists a weak solution u of this problem for each > 0. In this paper we prove convergence of these regularized minimizers u in the limit as → 0. In particular, we show that if n → 0, then, passing to a subsequence, u n → u, where u is a minimizer for the original pure displacement problem on Ω.Finally, we study the effect on cavitation of regularizing the variational problem by introducing a surface energy term which penalizes the formation and growth of cavities.
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