Non-shearing Fields and Extension

In order to proceed with the evaluation of the time-dependent Poisson ratio v(0, both sets of relaxation behaviour are required. Now from Chapter 2 we know the Poisson ratio is the ratio of the contractile to the tensile strain and that for an incompressible fluid the Poisson ratio v = 0.5. Suppose we were able to apply a step deformation as we did for a shear stress relaxation experiment. The derivation then follows the same course as that to Equation (4.69)  

This represents the relaxation of that stress in terms of the Young s relaxation function. Now we can express this as a viscosity by multiplying both sides by time t and integrating to get 

The first term on the left is the viscosity in extension, an extensional viscosity rje  

The constant Tr is called the Trouton ratio10 and has a value of 3 in this experiment with an incompressible fluid in the linear viscoelastic limit. The elongational behaviour of fluids is probably the most significant of the non-shear parameters, because many complex fluids in practical applications are forced to extend and deform. Studying this parameter is an area of great interest for theoreticians and experimentalists. 

Viscoelastic Properties of Polymers, 3rd edn, Wiley, New York, 1980. 

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