Asymptotic beam theory for non-classical elastic materials

This paper is devoted to the study of the plane-stress deformation of a beam composed of non-classical elastic materials that are suitable for modeling certain inter-metallic alloys with a nonlinear constitutive relation between the linearized strain and stress. The aim is to derive a consistent asymptotic beam theory without the ad hoc assumptions usually made in the development of beam theories. The methodology involves expanding the displacement, the in-plane strain tensor and stress tensor in a Taylor series, leading to a system of nonlinear equations that is solved. An analytical iteration procedure is developed to solve the system of equations leading to an analytical solution. The beam theory and the approximate general analytical solutions are used to study four examples. For the purpose of validation of the approximate analytical solution, we use a spectral collocation method to carry out numerical simulations for the full 2D problem, which confirms the validity of the approximate analytical solution. The study also reveals that the Euler-Bernoulli type of hypotheses is not suitable for a certain class of problems.