Simulating programmable morphing of shape memory polymer beam systems with complex geometry and topology

TL;DR

The paper develops a comprehensive framework to simulate programmable morphing in shape memory polymer beam systems with complex geometry using a temperature-dependent generalized Maxwell viscoelastic model coupled to thermo-mechanical beam theory. It combines time-temperature superposition with viscoelastic state evolution and a strong-form isogeometric collocation discretization to achieve high accuracy on curved, multi-patch beam geometries while maintaining a minimal, rotation-friendly set of unknowns. A key contribution is the SO(3)–consistent linearization and time stepping that enable stable, large-deformation simulations of shape programming and recovery without extra variables. The work demonstrates the capability to reproduce full shape memory cycles in challenging topologies and curved configurations, with implications for design of 4D-printed morphing devices and patient-tailored SMP structures in biomedical and aerospace contexts.

Abstract

We propose a novel approach to the analysis of programmable geometrically exact shear deformable beam systems made of shape memory polymers. The proposed method combines the viscoelastic Generalized Maxwell model with the Williams, Landel and Ferry relaxation principle, enabling the reproduction of the shape memory effect of structural systems featuring complex geometry and topology. Very high efficiency is pursued by discretizing the differential problem in space through the isogeometric collocation (IGA-C) method. The method, in addition to the desirable attributes of isogeometric analysis (IGA), such as exactness of the geometric reconstruction of complex shapes and high-order accuracy, circumvents the need for numerical integration since it discretizes the problem in the strong form. Other distinguishing features of the proposed formulation are: i) ${\rm SO}(3)$-consistency for the linearization of the problem and for the time stepping; ii) minimal (finite) rotation parametrization, that means only three rotational unknowns are used; iii) no additional unknowns are needed to account for the rate-dependent material compared to the purely elastic case. Through different numerical applications involving challenging initial geometries, we show that the proposed formulation possesses all the sought attributes in terms of programmability of complex systems, geometric flexibility, and high order accuracy.

Figures (2)

• Figure 1: The shape memory cycle in SMP.
• Figure 2: Shift factor $a_{\textnormal{T}}$ as a function of temperature vanManen_2021sLin_2020.