The specific features of the pure bending of the elastic panel undergoing large strains

Abstract

Within the framework of the semi-inverse method of three-dimensional nonlinear elasticity we consider the problem of the equilibrium and stability of a rectangular panel undergoing pure bending. By using two different models of compressible nonlinear elastic media — semi-linear material and Blatz & Ko material — the boundary value problems of the panel equilibrium were formulated and their numerical analysis was performed. For both models it was found that the loading diagram — the dependence of the bending moment on the angle of the bend — has a maximum point followed by a falling part. Using the bifurcation approach the problem on the stability of bent panel was studied. For this purpose the linearization of the equilibrium equations in the neighborhood of the constructed solution was performed and the possibility of the existence of nontrivial solutions of the resulting linear problem was investigated. An unusual feature of the panel instability under bending, discovered in this paper, is the existence of bifurcation points on the increasing section of the loading diagram. Analytical transformations associated with the derivation of nonlinear boundary value problems and the generation of equations of neutral equilibrium were performed using the automation system for semi-inverse method of nonlinear elasticity developed by the authors in the environment of computer algebra system Maple.