Elasticity High Deborah Number Measurements

Systematic measurements of stress and strain can be made and the results plotted as a rheogram. If our material behaves in a simple manner - and it is surprising how many materials do, especially if the strains (or stresses) are not too large - we find a linear dependence of stress on strain and we say our material obeys Hooke s law, i. e. our material is Hookean. This statement implies that the material is isotropic and that the pressure in the material is uniform. This latter point will not worry us if our material is incompressible but can be important if this is not the case. 

Here p is the pressure, which we are assuming is isotropic. We are also assuming that our material is incompressible, and the rigidity modulus, G, is what we would measure in a shear experiment. We can eliminate the pressure  

We are only pulling the ends of our block of elastomer and the sides are 

If we work at small strains so that we are in the linear (Hooke s law) region of the rheogram, then Equation (2.2) reduces to 

Therefore the extensional rigidity modulus or Young s modulus can be written in terms of the shear modulus as 

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The fluid s relaxation time A is the characteristic time of the fluid and, for oscillatory shearing, cu 1 can be taken as a measure of the characteristic time of the flow process, so De = A to. Thus, viscous behaviour occurs when the Deborah number is low, reflecting the fact that the fluid is able to relax. When the Deborah number is high, elastic behaviour is observed because the fluid is unable to relax sufficiently quickly. 

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