A New Thermoviscoplastic Beam Theory Model for Structural and Stress Relaxation in Shape Memory Polymers

This paper presents a finite element formulation to solve the beam problem for a visco-plastic shape memory polymer constitutive model under bending load. The method employs the von Karman beam theory, which addresses geometric nonlinearity. An approach based on Euler–Bernoulli beam theory, which does not account for geometric nonlinearity, is also developed to highlight its effects. The presented method can address moderate deflections and rotations, with loading conditions that include bending shear forces and bending moments. The formulation is developed by linearizing the virtual work of the beam, allowing for the derivation of a tangent stiffness matrix. To ensure convergence, an iterative numerical process known as the Newton–Raphson technique is used to satisfy the equilibrium laws. Although this method is initially applied to a specific material, it can be adapted for use with any material. Using the ABAQUS explicit solver, a VUMAT subroutine has been created to implement the visco-plastic thermomechanical constitutive model in a 3D finite element framework. The results from the 3D ABAQUS/Explicit simulation are used to validate the results of the proposed method. Notably, the findings from the developed model and the 3D finite element model (VUMAT) show a maximum error of only 9.4%. Key results are reported, including shape recovery, force recovery, force–deflection over time, cold programming, ABAQUS/Explicit mesh dependency and stress and strain contours for both the von Karman beam theory element and the 3D ABAQUS/Explicit model. The shape and force recovery responses from the Euler–Bernoulli beam theory element are also documented. The analysis includes an investigation into the mesh dependency of convergence and the overall results of the method. This formulation proves suitable for analyzing and optimizing smart structures comprising shape memory polymers or alloys, offering low computational costs.