Large bending strains of an inhomogeneous bar

This paper presents an analysis of the problem of pure bending of an inhomogeneous nonlinearly elastic bar with rectangular cross-section. The character of heterogeneity corresponds to the case of hard coating on the inner or outer bar surface. The linear and exponential types of heterogeneity are considered. The commonly used models of compressible nonlinearly elastic media are applied to describe the mechanical properties of bar’s material at large strains. The 2D problem is reduced to the boundary value problem for second order differential equations by means of a semi-inverse method. The nonlinear boundary value problems are solved using an automation package developed by the authors within the framework of the computer algebra system Maple. The stability analysis is performed in the context of a bifurcation approach when studying the linearized boundary value problem. The influence of coating on the load diagram and the stability of the bent bar is evaluated. The main purpose of this study is to find the relation between the maximum point on the loading diagram and the position of the stability loss point on this diagram. For the bar with a harder upper side, bifurcation points are found to be placed to the left of the maximum point. This means that such a bar will lose its stability on the rising part of the bending diagram.