Keywords: thermodynamic formulation, inelasticity, induced anisotropy, structure tensors.

In this work, a thermodynamic approach to the modeling and simulation of induced elastic and inelastic material behaviour in the phenomenological realm as based on the concept of evolving structure tensors is discussed. From the constitutive point of view, these quantities determine the material symmetry properties. In addition, the stress and other dependent constitutive fields are isotropic functions of these by definition. The evolution of these during loading then results in an evolution of the anisotropy of the material. From an algorithmic point of view, the current approach leads to constitutive models which are quite amenable to numerical implementation. To demonstrate the applicability of the resulting constitutive formulation, we apply it to the cases of (i), metal plasticity with combined hardening involving both deformation- and permanently-induced anisotropy relevant to the modeling of processes such as metal forming, and to (ii), deformation-induced anisotropy in an initially orthotropic pneumatic membrane consisting of a rubber matrix and nylon fibres.

The approach developed in this work offers a thermodynamically consistent framework to model materials which behave, in a broad sense, anisotropically. Within this framework, the approach taken here is based on two basic assumptions. The first is the concept of elastic isomorphism extended from the classical form of earlier works to now include the effects of deformation-induced anisotropy on the evolution of the vector- and tensor-valued internal variables. As shown in earlier work, in the context of this assumption, the local inelastic deformation in the material becomes associated with the plastic part of the standard multiplicative elastoplastic decomposition of the deformation gradient . The second assumption involves the modeling of induced anisotropy via evolving structure tensors, where in particular the free energy and inelastic potential are formulated as isotropic functions of their arguments, which include these tensors. At least in the context of anisotropy fixed with respect to the reference configuration, such a representation of the free energy density and inelastic potential should always be possible. In the present work, this has been exemplified by the example of the fibre-reinforced polymer membrane. And an example of the more general case of evolving anisotropy with respect to the reference configuration is given in the form of a combined hardening model. As shown in both of these case and more generally, the reduction of the free energy density and inelastic potential implied by the use of structure tensors leads in addition to a reduction of the dependence of the formulation on the nine-dimensional quantity to one on the related six-dimensional one in the context of the modeling of as an elastic material isomorphism. In this case, one obtains forms of model potentials which depend only on the symmetric part of . In this way, the present approach simplifies the modeling of both deformation-induced and persistent anisotropy (at least at the phenomenological level) significantly.

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