A new approach to modeling the thermomechanical, orthotropic, elastic-inelastic response of soft materials
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ORIGINAL PAPER
A new approach to modeling the thermomechanical, orthotropic, elastic-inelastic response of soft materials M. B. Rubin1 Received: 18 September 2018 / Accepted: 8 November 2018 © Springer Nature Switzerland AG 2018
Abstract This paper generalizes six previously developed nonlinear distortional deformation invariants for general hyperelastic orthotropic materials to model the thermomechanical, orthotropic, elastic-inelastic response of soft materials. These new invariants depend on two independent functions of elastic dilatation and temperature and characterize elastic distortional deformations from Hydrostatic States of Stress (HSS). When the Helmholtz free energy depends on these invariants, elastic dilatation and temperature, the correct response in HSS is automatically satisfied so the determination of the functional form of the Helmholtz free energy is simplified and can focus on modeling the response causing deviatoric stress. In addition, the new invariants are based on an Eulerian formulation of evolution equations for microscructural vectors that describe elastic deformation and directions of anisotropy. In contrast with the standard Lagrangian formulation, the Eulerian formulation is unaffected by arbitrary choices of the reference configuration, an intermediate configuration, a total deformation measure, and an inelastic deformation measure. Keywords Elastic-inelastic response · Large deformation · Orthotropic · Soft material · Thermomechanical
1 Introduction Modeling the thermomechanical, elastic-inelastic response of anisotropic materials is an important challenging problem. One application, with small elastic deformations, is predicting the anisotropic response of metals due to forming processes. Another application, with large elastic deformations, is predicting the response of soft tissues with anisotropy due to fiber bundle orientations. Most models for elastically anisotropic response of metals are based on the Lagrangian formulation presented in [1–3]. Within the context of this approach the total deformation gradient F and the plastic deformation gradient Fp are determined by integrating evolution equations of the forms F˙ = LF ,
F˙ p = p Fp ,
(1)
where L is the velocity gradient and p is a second order tensor that controls the rate of plastic dissipation and needs to be specified by a constitutive equation. Also, the elastic deformation gradient Fe is defined by Fe = FF−1 p .
(2)
For soft tissues with growth, Fp , p are replaced by a growth tensor Fg and a rate of growth g (e.g. [4]). Eckart [5] seems to be the first to suggest that since stress is determined by elastic deformations, a theory of elasticinelastic response can be proposed which introduces an elastic deformation measure directly by an evolution equation. This model was proposed for elastically isotropic material response and the same model was proposed by Leonov [6] for polymeric liquids. Motivated by these ideas, an alternative to the Lagrangian formulation for elastically anisotropic response M. B. Rub
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