Three Dimensional Viscoelasticity in Finite Strain: Formulation of a Rate-Type Constitutive Law Consistent with Dissipation

Abstract

Viscoelastic constitutive models can predict transient phenomena like creep and stress relaxation. Among the materials that can exhibit such effects in finite strain are biological soft tissues which are commonly modeled using a multiphasic continuum theory. Under infinitesimal strain, the classical 1-D Standard Linear Model (1-D SLM) is a simple law containing a stress rate and exhibiting the desired transient and equilibrium behavior observable in many soft tissues. The derivation of a rate-type constitutive law appropriate for modeling the non-linear viscoelasticity of soft tissues is the focus of this study. Well-posed laws should be objective and consistent with thermodynamic considerations of dissipation and energy. Infinitesimal models are not objective, while many non-linear analogies to the 1-D SLM fail to address dissipation. In the current study, internal variables are introduced, and employed in the derivation of a 3-D non-linear rate-type viscoelastic constitutive law. Evolution of the internal variables is assumed to involve first order rates. Properties of the 1-D SLM as well as existing non-linear models of soft tissues are used to motivate the constitutive assumptions and additional requirements. These requirements include symmetry of the stress, isotropy, reduction to hyperelasticity (via material parameters) and the existence of a hyperelastic equilibrium state. A class of objective rate-type constitutive laws satisfying dissipation and the additional requirements is derived. As an illustration, a compressible finite linear model is formulated. In infinitesimal strain, this model provides a 3-D analogy to the 1-D SLM with a set of constraints on the material parameters. The finite linear model is analyzed under simple time-dependent compression, extension and shear and shown to be consistent with expected behavior.