Applications of the Constitutive Equation

We shall illustrate the applications of the constitutive Eq. (6.59) in two cases stress relaxation following a step shear deformation and steady-state shear flow. 

For a step shear deformation applied at t = 0, the displacement functions are 

In maintaining a steady-state shear flow, the displacement functions are given by 

The viscosity result (Eq. (6.70b)) is the same as what will be obtained from substituting the relaxation modulus (Eq. (6.64b)) into the linear viscoelastic relation (Eq. (4.30)). Similar to Eq. (6.65), Eq. (6.71) is a nonlinear result. 

Equation (6.70) indicates that the viscosity is independent of the shear rate Aq. However, it is well known that the polymeric liquid exhibits non-Newtonian behavior, namely, that the viscosity value decreases with increasing shear rate after the rate reaches a certain value. This discrepancy is a weak point of the elastic dumbbell model and arises from an inherent weakness in the Gaussian distribution assumed for the connector vector. We can see the cause of this deficiency from the following analysis of how the dumbbell configuration changes with shear rate. 

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In general, the numerical simulation approach has several advantages in predicting and simulating local impact effects, due to its capabihty to capture the details of the impact process, time and cost-effectiveness, and reliability. However, it should be noted that the stabihty and accuracy of algorithms for the application of the constitutive equations must be assured, because the sensitivity of some material parameters is often unknown to the user, requiring a detailed and comprehensive parameter analysis. The successftd modelling of an impact event is based on vahd boundary descriptions and material models with parameters obtained from rehable material tests. 

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