The case of two-dimensional cracks

In this section we shall prove the existence of a solution of the elastoplastic boundary value problem for the particular case of a nonsmooth boundary which arises if we remove a two-dimensional surface from the interior of the body. 

Let fl C he an open, bounded and connected set with a smooth boundary T, and Tc C R be a smooth orientable two-dimensional surface. We assume that this surface can be extended up to the outer boundary T in such a way that fl is divided into two subdomains Ri, fl2 with Lipschitz boundaries. We assume that this inner surface Tc is described parametrically by the equations 

We use the same notation as in the previous subsection. The boundary of flc consists of three components r,r+,Tj, where T correspond to the positive and negative directions of the normal n, respectively. We introduce the space 

Notice that boundary values on T+ and Tj of any element u G (which we may think of as one-sided limits) are different, in general. Accordingly, for all functions on flc to be discussed below, their traces, if they exist, will in general differ on T+ and T . 

As before, the Neumann boundary conditions (5.37) and (5.38) enforce a function space decomposition based on the conditions 

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