Crack in a solid. Nonpenetration conditions

Let a solid occupy a bounded domain fl c with the smooth boundary r (see Fig.1.1). Let fl contain a smooth unclosed surface Fc, probably intersecting F. We assume that Fc is an oriented surface such that there exists a mapping 

Let us denote by n = (711,712,713) a unit outer normal to F and choose the direction 7/ = (i/i, 7/2,1 3) of a unit normal vector to Fc. Then v defines the positive side F+ of the surface Fc with the outer normal —v and the negative side Fj of Fc with the outer normal v. Thus we get the domain flc = Fc disposed between the outer boundary F and the inner boundary 

Let V be some known function defined in the domain fic- If v and the boundary dflc = F U F+ U Fj are sufficiently smooth, then we can define values of v at the boundary (the exact smoothness conditions are studied in Section 1.4). In particular, having the values u p+ and u p-, we introduce the jump of v at Fc by the formula 

As before, let u = (771,772,773) denote the displacement field in the domain fic- If the boundary is clamped then (1.44) provides 

The stress free boundary condition (1.45) for crack surfaces implies 

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