Universal relations in linear thermoelastic theories of thermally-responsive shape memory polymers

Abstract

In this paper we formulate a one-dimensional linear thermo-elastic (LTE) model in integral form for describing the behavior of thermally-responsive shape memory polymers (SMPs), which unifies and slightly generalizes numerous theories proposed in the literature starting with the seminal approach proposed by Liu et al. (2006). The presented model in its most general form requires the calibration of three response functions. Detailed analysis of four types of shape memory cycles (SMCs) used to quantify the shape memory effect in thermally-responsive SMPs and the corresponding forms of constitutive relations derived within LTE model display a number of critical properties. In particular, two of three response functions may be determined in many different ways from strain and stress storage/recovery profiles measured in SMCs (the third response function may be determined from an independent test). As implication of this fact, we show that the LTE model predicts a number of inter-relations between the measured strain and stress storage/recovery profiles. All these relations are universal in the sense that for any shape memory polymer, which may be correctly described by LTE model, the strain/stress storage/recovery profiles measured in SMCs must satisfy (at least approximately) these relations. Their role within the LTE model is analogous to the role of universal relations in the theory of finite deformation elasticity. In particular, these universal relationships provide a theoretical basis for the validation of any model within the LTE class. The basic theoretical results derived in this paper are illustrated using data obtained by Liu et al. (2006).