Three-dimensional constitutive viscoelastic laws with fractional order time derivatives

Abstract

In this article the three-dimensional behavior of constitutive models containing fractional order time derivatives in their strain and stress operators is investigated. Assuming isotropic viscoelastic behavior, it is shown that when the material is incompressible, then the one-dimensional constitutive law calibrated either from shear or elongation tests can be directly extended in three dimensions, and the order of fractional differentiation is the same in all deformation patterns. When the material is viscoelastically compressible, the constitutive law in elongation involves additional orders of fractional differentiation that do not appear in the constitutive law in shear. In the special case where the material is elastically compressible, the constitutive laws during elongation and shear are different; however the order of fractional differentiation remains the same. It is shown that for an elastically compressible material, the four-parameter fractional solid—the rubbery, transition, and glassy model, which has been used extensively to approximate the elongation behavior of various polymers, can be constructed from the three-parameter fractional Kelvin—the rubbery transition model in shear and the elastic bulk modulus of the material. Some of the analytical results obtained herein with operational calculus are in agreement with experimental observations reported in the literature. Results on the viscoelastic Poisson behavior of materials described with the fractional solid model are presented and it is shown that at early times the Poisson function reaches negative values.