ON A MODEL OF NONLOCAL CONTINUUM-MECHANICS PART-I - EXISTENCE AND REGULARITY

In this work we assume that the total stored energy functional for a b ody depends not only on the local strain field, but also on the spatia l average of the strain field over B the body weighted with an influen ce kernel. We investigate the problem of minimizing the total stored e nergy subject to a given bulk displacement Delta greater than or equal to 0. After the general setup for this problem is reviewed, we give s ufficient conditions for an energy minimizing strain field e(.) to sat isfy an integro-differential Euler-Lagrange equation. The result is ge neral and applies to material energies that display a wide variety of singular behavior. Through analysis of this Euler-Lagrange equation fo r a special class of influence kernels, we arrive at a regularity theo rem which ensures that energy minimizing strain fields must be periodi c, piecewise smooth, and possess a finite number of simple discontinui ties. We then combine this with a well-known existence result for rela xed minimization problems to arrive at a general existence theorem for the nonconvex problem.