Elastoplasticity

In elastoplastic models, it is assumed that there exist plastic deformations denoted by ij. The Hencky law implies that the following relations hold (Annin, Cherepanov, 1983 Duvaut, Lions, 1972) ... 

The function is assumed to be convex and continuous. Thus, we have the static elastoplastic model... 

Here ij denotes a plastic deformation velocity. Adding the relations (1.10), (1.11), we obtain the quasi-static elastoplastic model... 

The considered constitutive laws for elastoplastic models generalize ones used in elasticity. The main peculiarity of elastoplastic models consists in an existence of inequality type restrictions imposed upon the stresses. Omitting the mentioned restrictions, elastoplastic models turn into elastic ones. 

Now we formulate the models for perfectly elastoplastic plates considered in Chapter 5. By the Hencky law (1.9), the vertical component w of the plate displacements satisfies the equations (Erkhov, 1978)... 

The flow model of Prandtl-Reuss for elastoplastic plate is as follows ... 

Let us emphasize that not model can be presented as a minimization problem like (1.55) or (1.57). Thus, elastoplastic problems considered in Chapter 5 can be formulated as variational inequalities, but we do not consider any minimization problems in plasticity. In all cases, we have to study variational problems or variational inequalities. It is a principal topic of the following two sections. As for general variational principles in mechanics and physics we refer the reader to (Washizu, 1968 Chernous ko, Banichuk, 1973 Ekeland, Temam, 1976 Telega, 1987 Panagiotopoulos, 1985 Morel, Solimini, 1995). 

We prove the existence of solutions for the three-dimensional elastoplastic problem with Hencky s law and Neumann boundary conditions by elliptic regularization and the penalty method, both for the case of a smooth boundary and of an interior two-dimensional crack (see Brokate, Khludnev, 1998). It is shown in particular that the variational solution satisfies all boundary conditions. 

In this section we shall prove the existence of a solution of the elastoplastic boundary value problem for the particular case of a nonsmooth boundary which arises if we remove a two-dimensional surface from the interior of the body. 

In this section the existence of a solution to the three-dimensional elastoplastic problem with the Prandtl-Reuss constitutive law and the Neumann boundary conditions is obtained. The proof is based on a suitable combination of the parabolic regularization of equations and the penalty method for the elastoplastic yield condition. The method is applied in the case of the domain with a smooth boundary as well as in the case of an interior two-dimensional crack. It is shown that the weak solutions to the elastoplastic problem satisfying the variational inequality meet all boundary conditions. The results of this section can be found in (Khludnev, Sokolowski, 1998a). 

In this subsection we prove an existence theorem for the elastoplastic problem in the case where the domain has a nonsmooth boundary. 

We prove an existence theorem for elastoplastic plates having cracks. The presence of the cracks entails the domain to have a nonsmooth boundary. The proof of the theorem combines an elliptic regularization and the penalty method. We show that the solution satisfies all boundary conditions imposed at the external boundary and at the crack faces. The results of this section follow the paper (Khludnev, 1998). 

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 ... 

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. 

In this subsection we prove the solvability of the elastoplastic problem for a plate having a nonsmooth boundary. A solution of the problem will satisfy all boundary conditions both at the exterior boundary and at the crack faces. 

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