Simplified nonpenetration condition

Consider a vertical plane 11 passed over a fixed point y G Tc in the direction n(y) (i.e. n(y) lies in 11 ). We assume that the intersection Cy = Ily n is a straight line for every y G Tc, and denote by a(y) the angle between Cy and z = 0. The normal to in 11 will not depend on because of 

supposing a to be small enough (leading to small flip), we assume that the displacements in flip do not vary along the normal p y), namely, 

This assumption combined with Kirchhoff-Love s formula 

Dividing this inequality by cosa(y), we finally deduce the required relation 

The obtained nonpenetration condition (3.185) is local as compared to (3.173), (3.176) since this condition is considered only at the curve Tc. Let us recall that we have assumed that the angle between the crack surface and the axis is small. By this assumption, the small deflection x — Px has been neglected in (3.173), (3.176). It is of importance to deduce (3.177) from (3.185). Indeed, if is transformed into a vertical crack, then Cy is a straight line, a y) = 0, and from (3.185) we obtain the nonpenetration condition 

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Thus, (1.53) is a complete nonpenetration condition of the crack surfaces for the Kirchhoff-Love plates and shallow shells. By putting the thickness 2h to be zero, one reduces (1.53) to the simplified nonpenetration condition (1.50). 

In this section we deal with the simplified nonpenetration condition of the crack faces considered in the previous section. We formulate the model of a plate with a crack accounting for only horizontal displacements and construct approximate equations using penalty and iterative methods. The convergence of these solutions is proved and its application to the onedimensional problem is discussed. Analytical solutions for the model of a bar with a cut are obtained. The results of this section can be found in (Kovtunenko, 1996c, 1996d). 

Let us now obtain a complete system of boundary conditions fulfilled at Lc provided that the simplified nonpenetration condition (3.185) holds. We assume the solution x G iX is smooth enough and use Green s formulas for smooth functions (see Section 1.4),... 

We consider a problem similar to the one considered in Section 2.8. The nonpenetration condition between crack faces is taken in simplified form. Our aim is to obtain some qualitative properties of solutions for a contact problem for a plate having a crack. 

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