A Time-Dependent Finite Element Formulation for Thick Shape Memory Polymer Beams Considering Shear Effects

Abstract

In this paper, employing a thermomechanical small strain constitutive model for shape memory polymers (SMP), a beam element made of SMPs is presented based on the kinematic assumptions of Timoshenko beam theory. Considering the low stiffness of SMPs, the necessity for developing a Timoshenko beam element becomes more prominent. This is due to the fact that relatively thicker beams are required in the design procedure of smart structures. Furthermore, in the design and optimization process of these structures which involves a large number of simulations, we cannot rely only on the time consuming 3D finite element analyses. In order to properly validate the developed formulations, the numeric results of the present work are compared with those of 3D finite element results of the authors, previously available in the literature. The parametric study on the material parameters, e.g., hard segment volume fracture, viscosity coefficient of different phases, and the external force applied on the structure (during the recovery stage) are conducted on the thermomechanical response of a short I-shape SMP beam. For instance, the maximum beam deflection error in one of the studied examples for the Euler–Bernoulli beam theory is 7.3%, while for the Timoshenko beam theory, is 1.5% with respect to the 3D FE solution. It is noted that for thicker or shorter beams, the error of the Euler–Bernoulli beam theory even more increases. The proposed beam element in this work could be a fast and reliable alternative tool for modeling 3D computationally expensive simulations.