Analysis of boundary-value problems describing the non-isothermal processes of elastoplastic deformation taking into account the loading history

We discuss the theory and approximate methods for solving boundary-value problems of thermoplasticity in a quasi-static formulation when the process of non-isothermal elastoplastic deformation of a body is a sequence of equilibrium states. In this case, the stress-strain state depends on the loading history, and the process of inelastic deformation is to be observed over the whole time interval being studied. The boundary-value problem is stated as a non-linear operator equation in the Hilbertian space. The conditions that provide the existence, uniqueness and continuous dependence of the generalized solution on the applied loads and initial strains are defined. A convergence of the methods of elastic solutions and variable elastic parameters is studied to solve the boundary-value problems describing the non-isothermal processes of active loading taking into account the initial strains dependent on the deformation history and heating. __________ Translated from Problemy Prochnosti, No. 1, pp. 69–99, January–February, 2006.     

Iteration Algorithms for Solving Boundary-Value Problems of the Theory of Small Elastic-Plastic Strains on the Basis of the Mixed Finite Element Method

A mixed projection-mesh scheme has been formulated for the solution of nonlinear boundary-value problems of the theory of small elastic-plastic strains in displacements, strains, and stresses. Iteration algorithms of the solution of nonlinear equations of the mixed method have been considered. Numerical results of the solution of two model problems are presented. The data obtained on the basis of the classical and mixed finite element methods are compared.__________Translated from Problemy Prochnosti, No. 3, pp. 111 – 127, May – June, 2005.     

Solution for the 3D Temperature Problem of the Thermoelasticity Theory in Displacements

A new method of solution for the 3D problem of the thermoelasticity theory in displacements is proposed. The resultant general solution can be used in solving thermoelastic problems both for half-space and an elastic layer. The second boundary-value problem for half-space is detailed and numerical calculation is given.__________Translated from Problemy Prochnosti, No. 3, pp. 86 – 95, May – June, 2005.     

Construction of Constitutive Relationships of the Deformation Theory for Simple in Noll's Sense Hardening Materials with Elastoplastic Behavior

Based on the theory of simple hardening materials with elastoplastic behavior, the general constitutive relationships of the deformation theory of plasticity are mathematically strictly constructed for arbitrary continuous, piece-wise continuously differentiable deformation trajectories, any strains and symmetry types of the material properties. Two conditions under which this is possible are considered. The approaches to a strict specialization of general constitutive relationships of the deformation theory of plasticity have been developed by imposing restrictions on the material strains, deformation processes and properties. In this case, the restrictions on the properties of materials formalize the data obtained in the experimental investigations. A series of both new and known constitutive relationships have been constructed that are arranged into a hierarchy according to the level of complexity of the response to deformation. The area of applicability of the derived physical equations has been defined. Special attention has been given to the modeling of finite and infinitesimal strains of isotropic materials. __________ Translated from Problemy Prochnosti, No. 6, pp. 35 – 49, November – December, 2005.