Derivation of the nonpenetration condition

Let the mid-surface of the Kirchhoff-Love plate occupy a domain flc = fl Tc, where C is a bounded domain with the smooth boundary T, and Tc is the smooth curve without self-intersections recumbent in fl (see Fig.3.4). The mid-surface of the plate is in the plane z = 0. Coordinate system (xi,X2,z) is assumed to be Descartes and orthogonal, x = xi,X2)- 

Projection Hip of the surface can be represented as the union of two sets in accordance with the direction of the axis namely. Hip = H U H. We denote by H the part of the projection of provided that this part is obtained by moving along the positive direction of the axis Respectively, we find H. In particular, the curve Fc belongs both to H and H (see Fig.3.6). We assume the direction of the normal n = ( 1,1 2) to the curve 

We recall that in the Kirchhoff-Love plate theory the horizontal displacements depend linearly on the coordinate i.e. 

The nonpenetration condition of the crack faces at the point (a , z) G, X G has the following form  

the mutual nonpenetration condition between the crack faces is described by the inequalities (3.173), (3.176). The inequalities have a nonlocal character in particular, they contain values of the functions both at the point X and the point y = Px moreover the last values (i.e. at the point y = Px) are taken at the opposite crack faces. 

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