Abstract
We derive a scaling-relation, for the infimum of the energy J(epsilon,delta)(u, gamma) = integral(Omega) 1/epsilon dist(q) (del u(1 - gamma(e) over arrow(1) circle times (e) over arrow(2)) , SO(2)) + vertical bar gamma vertical bar(p) d lambda(2) (x, y) + delta V-y (chi({gamma = 0}), Omega) , for small epsilon, delta > 0, where p, q >= 1, u : Omega -> R-2 is a deformation with suitable affine ...
Abstract
We derive a scaling-relation, for the infimum of the energy J(epsilon,delta)(u, gamma) = integral(Omega) 1/epsilon dist(q) (del u(1 - gamma(e) over arrow(1) circle times (e) over arrow(2)) , SO(2)) + vertical bar gamma vertical bar(p) d lambda(2) (x, y) + delta V-y (chi({gamma = 0}), Omega) , for small epsilon, delta > 0, where p, q >= 1, u : Omega -> R-2 is a deformation with suitable affine boundary conditions and gamma : Omega -> R is a suitable slip variable. This model is motivated by a two-dimensional single-slip model in finite crystal plasticity. We show that the infimum of the energy J(epsilon,delta) scales as delta(q/q+1)/epsilon(1/q+1). This scaling-relation is attained by an asymptotically self-similar branching construction. (C) 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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