Finite element schemes for the integral constitutive models

2 Finite element schemes for the integral constitutive models 

As mentioned in Chapter 1, in general, the solution of the integral viscoelastic models should be based on Lagrangian frameworks. In certain types of flow 

If the position of the fluid particle at current time t is given by x t) then its motion is defined using the following position vector 

The right Cauchy-Green strain tensor corresponding to this deformation gradient is thus expressed as 

The inverse of the Cauchy-Green tensor, Cf, is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress Tv for a fluid particle as a time integral of the defonnation history, i.e. 

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