Elastoplastic Deformation of a Compound Disk under Impulsive Loading

Abstract

The small elastoplastic deformation theory and the finite-element method are used to analyze the behavior of a compound disk under axisymmetric impulsive loading. Numerical results are presented, which describe the development of plastic deformation, the effect of hardening and the duration of the loading impulse on the oscillatory elastoplastic deformation of the disk Keywords: solid of revolution, theory of plasticity, isotropic material, finite elements, impulsive loading, unloading A method for solving the axisymmetric dynamic problem for isotropic elastoplastic solids of revolution with arbitrary meridional section under impulsive thermomechanical loading was proposed in [11]. The method is based on small elastoplastic deformation theory [1, 2, 8, etc.] and involves the finite-element approximation of unknown displacements in spatial coordinates and finite-difference representation of time derivatives. The nonlinear problem is linearized by the method of variable parameters. In the present paper, this method is extended to compound solids of revolution of finite dimensions. By a compound solid is meant a discretely inhomogeneous solid of revolution whose constituents are solids of revolution. For the entire body and for its parts, there is a common axis of revolution coinciding with the z-axis of a cylindrical coordinate system z, r, ϕ. The constituents of the body are made of dissimilar isotropic materials characterized by different (real) limit σ-ε relationships and are in perfect mechanical contact with each other.1. Solution Method. Let the basic unknowns be the axial (U z (z, r, t)) and radial (U r (z, r, t)) displacements. We will use a triangular ring finite element with linear approximation of the displacement vector to partition the meridional section of the solid into triangles by a technique described in [4]. If the meridional section is approximated by N nodes and M triangles with node numbers i, j, and k, then we have the following recurrent formulas of the explicit scheme for computing the displacements at the time (t + ∆t) in terms of the displacement at the previous times t and (t -∆t):u t t u t u t t t M c u r i r i r i m i m M ii m z ( ) ( ) ( ) ( ) ( ) + = − − − = ∑ ∆ ∆ ∆ 2 2 1 1 [ i ii m r i m M t d u t ( ) ( ) ( ) + = ∑ 1 1

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Numerical solution to the initial-boundary-value problem of electroelasticity for a radially polarized hollow piezoceramic cylinder

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