The Hencky model. Problem formulation

Let C be a bounded domain with a smooth boundary L, and Lc be a smooth curve without selfintersections, Lc C fl. We assume that Lc contains its end points. Denote by flc the mid-surface of the plate, flc = fl Lc- We choose a unit normal vector n = 1, 2) to the curve Lc- The curve Tc corresponds to the crack in the plate. The crack shape as a surface in can be described as a G Tc, —h z h, where x = xi,X2) G fl, 2h is the thickness of the plate, is a distance to fl. The domain flc contains, therefore, three components of the boundary T, T+, Tj. Here T fit to the positive and negative directions of the normal n, respectively. Let n = (ni,n2) be the external unit normal vector to L. 

Denote by the Sobolev space of functions having the first 

Hereafter the known Green formula will be used, namely, for all smooth functions w, rriij, i,j = 1,2, we have 

The same formula is valid for the domain flc- In this case the additional integrals over r+,Tj will appear. By we denote the space 

Formulation of the elastoplastic problem for the plate having the crack is as follows. In the domain flc we want to find functions w, m = rriij, ijy bi = 2, satisfying the following equations and inequalities  

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