Nonlinear constitutive formulation for a finite deformation beam model based on the mixing rule for composites

Abstract

Constitutive nonlinearity for beam models has been traditionally described by means of concentrated and distributed models, in the most cases, formulated assuming infinitesimal deformation. The concentrated models consider elastic elements equipped with plastic hinges at the ends. In the case of distributed plasticity models, inelasticity is evaluated at a fixed number of points on the cross sections and along the beam axis. These points corresponds to of fibers directed along the axis. Therefore, this approach is referred as fiber approach. Additionally, two versions of the models can be defined: the displacement based method, which is based on the interpolation of the strain field along the elements and force based method which obtains the sectional forces and moments interpolating the nodal values and satisfying the equilibrium equations even in the inelastic range. Both approaches are affected by the strain localization phenomenon when materials with softening behavior are employed and, therefore, the whole structural response becomes mesh dependent if no appropriate corrections are considered. On the another hand, one of the most invoked geometrically exact formulations for beams in finite deformation is that of Simo which generalize to the 3–D dynamic case the formulation developed by Reissner. Only a few works have developed fully geometrical and constitutive nonlinear formulations for beams, but they have been mainly focused on plasticity. Recently, Mata et.al. [14, 15] have extended the formulation due to Reissner-Simo for considering and arbitrary distribution of composite materials on the cross sections for the static and dynamic cases. The displacement based method is used for solving the resulting nonlinear problem. Thermodynamically consistent constitutive laws are used in describing the material behavior of simple materials and the parallel version of the mixing rule is used for composites. In this work a detailed presentation of the implementation of the mixing rule for the treatment of constitutive nonlinearity in the Reissner–Simo formulation for beams is presented. Finally, several numerical examples validating the proposed formulation are given.