On a model of nonlocal continuum mechanics, Part I: Existence and Regularity
by
R.L. Fosdick and D. Mason
in
J. Appl. Math., 58, 1278-1306, 1998.
Category: Journal Article
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Abstract:
. In this work we assume that the total stored energy functional for a body B depends not only on the local strain eld, but also on the spatial average of the strain eld over the body weighted with an in°uence kernel. We investigate the problem of minimizing the total stored energy subject to a given bulk displacement 0. After the general setup for this problem is reviewed, we give sucient conditions for an energy minimizing strain eld e() to satisfy an integro-di erential Euler{Lagrange equation. The result is general and applies to material energies that display a wide variety of singular behavior. Through analysis of this Euler{Lagrange equation for a special class of in°uence kernels, we arrive at a regularity theorem which ensures that energy minimizing strain elds must be periodic, piecewise smooth, and possess a nite number of simple discontinuities. We then combine this with a well-known existence result for relaxed minimization problems to arrive at a general existence theorem for the nonconvex problem
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