Analytical solution for the problem of pure bending of orthotropic micropolar plate

Abstract

When analysing the problem of pure bending of a plate, it can be noticed that, for cylindrical bending, a spatial problem collapses into a plane-strain problem. For such a boundary-value problem of the Cosserats' continuum, three engineering parameters are required: Young’s modulus, Poisson’s ratio and the characteristic length for bending. Here we consider an orthotropic form of such a problem, whereby two Young’s moduli, four Poison’s ratios and one characteristic length for bending are found to be sufficient to propose a mathematical model of this problem. General equations of the isotropic micropolar continuum are introduced, and the analytical solution for the pure bending of an isotropic micropolar plate is generalized to the case of orthotropic microstructure. By defining the ratio of a distributed force and a distributed moment boundary conditions required for the pure-bending state, a closed-form solution to this problem is obtained in terms of displacement, strains and stress functions. It is shown that the derived results reduce to the isotropic ones if a material isotropy is assumed.