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

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.