M4-5n numerical solution using the mixed FEM, validation against the finite difference method

Abstract

The final aim of this work is to build a tool dedicated to the calculation of the mechanical fields in pavements incorporating possible vertical cracks in some layers or partial debonding at the interface between layers. The development of this tool is based on a specific layer-wise modeling of the structure so-called M4-5n. In this model the stress fields are approached through polynomial approximations in the vertical direction for each layer. Its construction is based on the Hellinger-Reissner (H-R) variational principle of continuum mechanics. One advantage of the M4-5n is to reduce by one the dimension of the problem. Moreover this model leads to finite values of the generalized interface stresses at the crack lips of the structures studied. This approach is thus particularly adapted to parametric studies and might be considered for analyzing crack growth in layered structures such as pavements. The contribution of the present paper to this model is focused on the computation of its numerical solution by means of the mixed Finite Element Method (FEM). The developed method is based on the maximum of the complementary energy theorem using Lagrangian multipliers to ensure the equilibrium equations. The resulting formulation is equivalent to the H-R variational principle applied to the generalized displacement and stress fields. This approach is applied to a beam structure composed of four elastic homogenous layers resting on Winkler’s springs. Vertical cracks across some layers are introduced. The results obtained are compared with those from an earlier approach using the Finite Difference Method (FDM).