Single-integral constitutive equations for viscoelastic fluids

3 Single-integral constitutive equations for viscoelastic fluids 

Here r(t) is the stress at a fluid particle given by an integral of deformation history along the fluid particle trajectory between a deformed configuration at time f and the current reference time t. 

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4 i and 4 2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as 

Some of the integral or differential constitutive equations presented in this and the previous section have an exact equivalent in the other group. There are, however, equations in both groups that have no equivalent in the other category. 

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