3-DIMENSIONAL CONSTITUTIVE VISCOELASTIC LAWS WITH FRACTIONAL ORDER TIME DERIVATIVES

In this article the three-dimensional behavior of constitutive models containing fractional order time derivatives in their strain and stres s operators is investigated. Assuming isotropic viscoelastic behavior, it is shown that when the material is incompressible, then the one-di mensional constitutive law calibrated either from shear or elongation tests can be directly extended in three dimensions, and the order of f ractional differentiation is the same in all deformation patterns. Whe n 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 c ase where the material is elastically compressible, the constitutive l aws during elongation and shear are different; however the order of fr actional differentiation remains the same. It is shown that for an ela stically compressible material, the four-parameter fractional solid-th e rubbery, transition, and glassy model, which has been used extensive ly to approximate the elongation behavior of various polymers, can be constructed from the three-parameter fractional Kelvin-the rubbery tra nsition model in shear and the elastic bulk modulus of the material. S ome of the analytical results obtained herein with operational calculu s are in agreement with experimental observations reported in the lite rature. Results on the viscoelastic Poisson behavior of materials desc ribed with the fractional solid model are presented and it is shown th at at early times the Poisson function reaches negative values. (C) 19 97 The Society of Rheology.