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Stochastic homogenization on perforated domains II – Application to nonlinear elasticity models
2022, Heida, Martin
Based on a recent work that exposed the lack of uniformly bounded (Formula presented.) extension operators on randomly perforated domains, we study stochastic homogenization of nonlinear p-elasticity, (Formula presented.), on such structures using instead the extension operators constructed in former works. We thereby introduce two-scale convergence methods on such random domains under the intrinsic loss of regularity and prove some generally useful calculus theorems on the probability space, for example, abstract Gauss theorems.
Singular limiting induced from continuum solutions and the problem of dynamic cavitation
2013, Giesselmann, Jan, Tzavaras, Athanasios E.
In the works of K.A. Pericak-Spector and S. Spector [9, 10] a class of self-similar solutions are constructed for the equations of radial isotropic lastodynamics that describe cavitating solutions. Cavitating solutions decrease the total mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions (for polyconvex energies) due to point-singularities at the cavity. To resolve this paradox, we introduce the concept of singular limiting induced from continuum solution (or slic-solution), according to which a discontinuous motion is a slic-solution if its averages form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for creating the cavity, which is captured by the notion of slic-solution but neglected by the usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the cavitating solution is in fact larger than that of the homogeneously deformed state. We also apply the notion of slic-solutions to a one-dimensional example describing the onset of fracture, and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan.