The mathematical foundations of anelasticity: existence of smooth global intermediate configurations

Abstract

A central tool of nonlinear anelasticity is the multiplicative decomposition of the deformation tensor that assumes that the deformation gradient can be decomposed as a product of an elastic and an anelastic tensor. It is usually justified by the existence of an intermediate configuration. Yet, this configuration cannot exist in Euclidean space, in general, and the mathematical basis for this assumption is on unsatisfactory ground. Here, we derive a sufficient condition for the existence of global intermediate configurations, starting from a multiplicative decomposition of the deformation gradient. We show that these global configurations are unique up to isometry. We examine the result of isometrically embedding these configurations in higher-dimensional Euclidean space, and construct multiplicative decompositions of the deformation gradient reflecting these embeddings. As an example, for a family of radially symmetric deformations, we construct isometric embeddings of the resulting intermediate configurations, and compute the residual stress fields explicitly.

Instabilities in liquid crystal elastomers

“…Within this theoretical framework, the material deformation due to the interaction between external stimuli and mechanical loads can be expressed as a composite deformation from a reference configuration to the current configuration, via an elastic deformation followed by a natural (stress free) shape change. The multiplicative decomposition of the associated gradient tensor is similar in some respects to those found in the constitutive theories of thermoelasticity, elastoplasticity, and growth [47,63] (see also [46,92]), but is different on one major aspect, namely: the stress-free geometrical change is superposed on the elastic deformation, which is applied directly to the reference state. This difference is important since, although the elastic configuration obtained by this deformation may not be observed in practice, it may still be possible for the nematic body to assume such a configuration under suitable external stimuli.…”

A Rod Theory for Liquid Crystalline Elastomers

“…Here, we assume the existence of an isotropic phase that can be reached from a suitably imprinted defect field in the original fabrication process. The exact conditions under which such an intermediary configuration can be mapped to a stressfree reference configuration that is isotropic has been discussed in the general framework of anelasticity in [18]. In (3), a > 0 represents a temperature-dependent stretch parameter, ⊗ denotes the tensor product of two vectors, and I = diag(1, 1, 1) is the second-order identity tensor.…”

“…We exploit theoretically the multiplicative decomposition of the deformation gradient from the reference configuration to the current configuration into an elastic distortion followed by a natural shape change [20,[41][42][43][44]. This multiplicative decomposition is similar to those found in the constitutive theories of thermoelasticity, elastoplasticity, and morphoelasticity [19,37] (see [18,53] as well), but it is also different in the sense that the stress-free geometric change is superposed on the elastic deformation, which is directly applied to the reference state.…”

Universality in Anisotropic Linear Anelasticity

“…In nonlinear anelasticity, in the notion first defined in [7], strain has an elastic and an anelastic part. In terms of deformation gradient it is written as F = F e F a , where F e and F a are the elastic and anelastic deformation tensors, respectively [17,37,39]. The hybrid German-English portmanteau term eigenstrain has its origin in the pioneering paper of Hans Reissner [31] (Eigenspannung means proper or self stress) and was further popularized by Mura [25,30].…”