Effective finite-difference method for elastoplastic boundary value problems

On the basis of deformation theory of plasticity and the strain space plasticity theories the boundary value problems are formulated. Discrete analogies of the boundary-value problems are constructed by the finite-difference method. By resolving the finite-difference equations with respect to central and boundary nodal displacements, recurrence relations, separately for internal and boundary points are obtained. This relations allow to find the desired quantities using the iterative method under zero initial conditions. Note that the boundary value problem based on strain space plasticity is formulated with respect to the increments of the displacement. In this case, the external load is applied gradually with small increments, and the total solution is found, in accordance with these as a sum of displacement increments corresponding to every increments. As an example, the problem of compressing a plastic rod with constant force has been solved; a comparison of the numerical results of the plastic boundary value problems formulated by the deformation theory and the strain space theories obtained by the iterative and the elimination methods show that they are very close thereby ensuring the validity of the proposed iterative method based on finite-difference equations.

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