Monolithic FE2 approach for the thermomechanical modeling of beam structures

Abstract

In the present contribution, the FE2 scheme for beam elements is extended to thermomechanically coupled problems. Beam elements have the advantage of drastically reducing the number of degrees of freedom compared to solid elements. However, the major challenge in modeling structures with beam elements lies in developing sophisticated non‐linear beam material models. This drawback resides in the fact that these elements require effective cross‐sectional properties involving material and geometric properties. The FE2 method, combined with a homogenization scheme based on the Hill‐Mandel condition, solves this problem. Within this scheme, homogenization of a representative volume element (RVE) on the mesoscopic scale provides effective cross‐sectional properties for the macroscopic scale. This homogenization procedure allows the consideration of non‐linear material formulations and cross‐sectional deformation within the analysis of a beam structure. The applicability of such a FE2 scheme for purely mechanical problems was already shown. In the present contribution, an extension to thermomechanically coupled problems is provided. In the proposed setting, the macroscopic scale is represented by beam elements with displacement, rotation, and temperature degrees of freedom. Solid elements with displacements and temperature degrees of freedom describe the behavior of the RVE. Hence, the proposed extension solves both scales in a monolithic approach. The assumption of a steady state problem at both scales allows a focus on a consistent scale transition and a discussion about the choice of suitable boundary conditions under the assumption of beam kinematics.