On pure bending in non-linear elasticity: A circular closed-form 2D solution for semi-linear orthotropic material

Abstract

Large-deformation pure bending of a straight beam of uniform rectangular cross-section in plane strain is re-visited and generalised to an orthotropic linear-elastic material and the case of plane stress. The beam is bent into a completely shearless state by means of normal surface traction of a specific profile. For a linear-elastic material expressed in terms of Biot strain and stress tensors (orthotropic generalisation of the so-called semi-linear material) the solution still exists in closed form and is expressed in terms of the applied cross-sectional traction. As such, it provides a convenient benchmark reference for numerical methods in non-linear elasticity.In agreement with the isotropic case, through-the-thickness stresses develop in the form a hyperbolic cosine, but now with the magnitude actually depending on a measure of orthotropy in the material, while the longitudinal stresses change over the thickness in the form of a hyperbolic sine. In general, the unstrained axis is displaced from the centroidal axis.For an ideal material with infinite yield strength, the theoretical maximum of the magnitude of the longitudinal traction is limited to the amount of the geometric mean of the effective Young's moduli in the longitudinal and the through-the-thickness directions while, for a brittle material, bending strength reduces by approximately 715 of the square of the ratio between the magnitude of the applied traction and this mean value.