A beam theory for a new class of constitutive relation for elastic bodies

A theory for two-dimensional (2D) beams is presented for the case of considering new constitutive equations, wherein the linearized (infinitesimal) strain tensor is assumed to be a nonlinear function of the Cauchy stress tensor. An incremental formulation is developed to solve numerically the resultant equations. Three constitutive equations for elastic bodies are considered: a model wherein we have strain-limiting behavior, a nonlinear model for rock, and a bimodular constitutive equation for rock (which can also be considered as a nonlinear model for rock). Two boundary value problems are studied: the deformation of a cantilever beam with a point load, and a three-point beam with a point load applied on the middle. The numerical results are compared with the predictions of the classical theory for beams for the case of linearized elastic solids. The results for the stresses can be notoriously different, when comparing the predictions of the nonlinear constitutive equations with the results obtained using the linearized theory of elasticity.