Variational approach to non-linear boundary value problems for elasto-plastic incompressible bending plate

Abstract

Non-linear boundary value problems for inelastic isotropic homogeneous incompressible bending plate, within the range of J 2 -deformation theory, are considered. An existence of the weak solution of the non-linear problem with clamped boundary condition is obtained in H 2 ( Ω ) by using monotone operator theory and Browder–Minty theorem. For linearization of the non-linear problem a monotone iteration scheme is constructed. It is shown that the sequence of potentials obtained from the sequence of approximate solutions (i.e. iterations), is a monotone decreasing one. Convergence of the iteration process in H 2 -norm is proved by using the convexity argument. Numerical solutions, based on finite-difference scheme, are given for linear bending problems with rigid clamped as well as simply supported boundary conditions. Further numerical examples are presented to illustrate the convergence of approximate solutions and monotonicity of the potentials as applied to the non-linear problems.