Different Forms of Constitutive Relation

In this case, one has only two invariants of the internal tensor, which makes the general relations for the tensor functions simpler. However, it does not mean that the final relations will be simpler. We can see later (see Section 9.3.5) that there is a relation between the recoverable strain and the deformation of macromolecular coil (see formula (9.75)), so a transfer from one formalism to the other can be performed and the results of the two approaches can be compared. 

All the constitutive relation that we have discussed in this chapter include some relaxation equations for the internal tensor variables which ought to be considered to be independent variables in the system of equations for the dynamics of a viscoelastic liquid. 

However, in the earlier times, the constitutive relation for a viscoelastic liquid were formulated when the equations for relaxation processes could not be written down in an explicit form. In these cases the constitutive relation was formulated as relation between the stress tensor and the kinetic characteristics of the deformation of the medium (Astarita and Marrucci 1974). 

In this section, we shall show that the constitutive relation with internal variables is followed by two types of constitutive relations which do not include internal variables. For the sake of simplicity, we shall consider the simplest set of equations 

Indeed, we can obtain a relation between the stress tensor and the velocity gradient tensor if we exclude tensor from the set of equations (8.32)-(8.33). This can be done in two different ways. 

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