Chapter 4 - Nonlinear Constitutive Models with Finite Strains

Abstract

This chapter discusses constitutive relations for viscoelastic media with finite strains. A brief survey of differential models in finite viscoelasticity is presented. Fractional derivative of an objective tensor is introduced, and fractional analogs are constructed for differential models with finite strains. Fractional differential models provide fair agreement between numerical prediction and experimental data for viscoelastic solids and fluids. Integral models for nonlinear viscoelastic media with large deformations, a model of adaptive links, and constitutive equations are derived based on the Lagrange variational principle. Optimal choice of a strain energy density for adaptive links is also discussed. An analog of this model in finite viscoelasticity is described by some differential equations but the total number of these equations essentially exceeds the number of constitutive relations in the linear theory because rheological models that are equivalent to each other at infinitesimal strains differ significantly at finite strains. The Maxwell model consists of an elastic and a viscous element connected in series. Unlike the Kelvin-Voigt model, two approaches are distinguished in the design of the Maxwell models with finite strains. The standard viscoelastic solid is treated as a system consisting of two springs and a dashpot.