The Prandtl-Reuss elastoplastic plates

We prove an existence of solutions for the Prandtl-Reuss model of elastoplastic plates with cracks. The proof is based on a special combination of a parabolic regularization and the penalty method. With the appropriate a priori estimates, uniform with respect to the regularization and penalty parameters, a passage to the limit along the parameters is fulfilled. Both the smooth and nonsmooth domains are considered in the present section. The results obtained provide a fulfilment of all original boundary conditions. 

Here i — i is the convex and continuous function describing a plasticity yield condition, the dot denotes a derivative with respect to t, n = (ni,ri2) is the unit normal vector to the boundary F. The function v describes a vertical velocity of the plate, rriij are bending moments, (5.175) is the equilibrium equation, and equations (5.176) give a decomposition of the curvature velocities —Vij as a sum of elastic and plastic parts aijkiirikiy Vijy respectively. Let aijki x) = ajiki x) = akuj x), i,j,k,l = 1,2, and there exist two positive constants ci,C2 such that for all m = rriij  

In the sequel the known Green formula is used, namely, for all smooth functions v, rriij, iyj = 1 2, we have 

The Green formula (5.181) can be specified for the case where the functions V, rriij are not smooth enough. To this end, introduce the Hilbert space 

H T) is the space dual of the space iF (F), and ( , ), ( , )s,r denote the scalar product in and the duality pairing between 

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