Semi-Analytical Large-Strain Bending of Thermoviscoplastic SMP Beams with Plastic Softening

The accurate prediction of the large-strain thermomechanical response of shape memory polymer (SMP) beams is challenging because geometric nonlinearities must be coupled with temperature-dependent viscoelastic and viscoplastic mechanisms, including plastic softening. This study develops a semi-analytical framework for the large-strain pure bending of thermoviscoplastic SMP beams. The formulation combines finite-deformation bending kinematics with the thermoviscoplastic constitutive model of Zeng et al., while a mapping-based fitting strategy is introduced to approximate the through-thickness distribution of the viscous mechanical stretches. The resulting stress field is obtained by enforcing radial equilibrium in polar coordinates and satisfying the traction-free inner and outer surfaces. The formulation is implemented in MATLAB R2026a and independently verified against a three-dimensional ABAQUS/Explicit model incorporating the same constitutive equations through a user-defined VUMAT. Shape recovery, bending-force recovery, stress distributions, deformation fields, and mesh sensitivity are investigated for two SMP material parameter sets. The semi-analytical and finite element predictions show close agreement throughout the thermomechanical cycles. For example, at the end of the relaxation stage, the predicted inner radii are 27.27 mm and 27.26 mm for the semi-analytical and finite element approaches, respectively, corresponding to a difference of approximately 0.04%. The results demonstrate that the proposed formulation provides an efficient alternative for analyzing large-strain SMP bending while retaining the essential path-dependent thermoviscoplastic behavior.