Figure 10. Derived curves of apparent modulus versus strain.
where a0 is the stress at the instant of halting. The ratio v/G has the dimensions of time and is known as the relaxation time. This is readily determined from the experimental curve, either by redrawing it in logarithmic form or as the time for the stress to fall to 1/e of its starting value. The relaxation curve by itself does not allow the elastic and viscous components to be obtained separately. When the simple Maxwell model is evidently insufficient to describe the relaxation behavior, an analysis of the curve may become more difficult. The sample may be described by a more complex model created by combining several simple binary models. There are computational procedures which enable the various constants of the model to be evaluated in such a case, but the precision falls off rapidly as the number of constants increases. The rheologist may sometimes be interested in determining several relaxation times and moduli for a particular sample, especially if these can be assigned to recognizable structural elements in the material, but more often the proliferation of constants does little to clarify the behavior of cheese.
There is an empirical treatment which is sometimes useful in analyzing relaxation data (43,44), particularly in cases where the precision of the data may be in some doubt. If one takes Yt as the fraction of the stress which has decayed in time t, ie,
where <r0 is the stress at the commencement of the relaxation and at the stress after time t, it has been found that many complex viscoelastic materials relax in such a way that the relation
holds to a fair degree of approximation; k1 and k2 may easily be found. Then Hk2 is the extent to which the stress eventually will decay, while the ratio k2/ki is a measure of the rate of decay. For a perfectly elastic body or a Voigt body Hk2 is zero; there is no relaxation. For a liquid or a Maxwell body it is unity. For more complex models it lies somewhere in between. This is a useful device for deciding whether a simple model will suffice. For a Cheddar cheese Hk2 was found to be in the region of 0.8, so a simple model is not adequate for describing the relaxation behavior of this cheese (43). An alternative treatment has been suggested (45), which, it is claimed, sometimes fits experimental data more closely. This may be derived by considering a binary model of the Maxwell type, but one in which the viscosity associated with the dashpot is not constant but follows a power-law variation with stress. The resulting expression is
where n is the exponent of the power law. This is an interesting suggestion: One of the consequences of the power-law variation of viscosity is that it becomes infinite at zero stress, and a Maxwell body with an infinitely viscous dash-
pot is in fact a solid. Thus this is a model for a material which is solid while at rest but becomes progressively more fluid as the stress increases. This is not too different from the description of cheese given above: a solid material whose structure breaks down progressively when subject to strain.
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