Linear Constitutive Relations in Isotropic Finite Viscoelasticity

Batra [1] used a linear relationship between the second Piola-Kirchhoff stress tensor S and the Green-St. Venant strain tensor E to study finite simple shearing and finite simple extension deformations of an elastic body. In each case he found that the tangent modulus (i.e. the slope of the shear stress vs. the shear strain curve or the slope of the axial stress vs. the axial stretch curve) is a monotonically nondecreasing function of the pertinent measure of strain. This contradicts the response observed for most materials. However, a linear relation between the Cauchy stress tensor σ and the left Cauchy–Green tensor B was found to give a response similar to that observed in experiments [2] for most materials. Here we study the corresponding problem for an incompressible linear viscoelastic material. We use two constitutive relations: in one the relationship between S and the history of E is linear and in the other σ is linearly related to the history of the relative Green-St. Venant strain tensor Et . It is shown that the instantaneous elastic response given by the former constitutive relation is unrealistic in the sense described above but that obtained with the latter one agrees with the expected one. For an incompressible linear viscoelastic material, the aforestated two constitutive relations are (e.g. see Christensen [3] for Equation (1a) and Fosdick and Yu [4] for Equation (1b))