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This is the equation of a surface in a three-dimensional diagram. The material is said to be linearly viscoelastic if, for any value of t, the line representing the variation of strain with stress is a straight one. For many substances this is approximately true, at least while the strains and stresses are small. Cheeses vary in this respect. For a linear viscoelastic material the two-dimensional graph of strain versus time will contain all the required rheological information.

The basic graph is the creep curve. If the experiment is prolonged indefinitely, some equilibrium will ultimately be established. If the material is essentially a solid, but one whose deformation is retarded, the strain will eventually become constant. There are obviously two parameters which can be used to denote the material behavior, one based on the ultimate deformation and the other on the rate at which it is (asymptotically) established. On the other hand, if the material is more akin to a liquid, the final equilibrium will be a dynamic one in which a constant rate of strain is established. As before, two parameters are needed to describe the behavior: one, the ultimate flow; the other, the rate of attaining it. The experiment may also be conducted rather differently. Instead of allowing it to proceed ad infinitum, the stress may be removed at a predetermined point. Should the material possess any elastic characteristics, some of the energy used in the deformation will be stored within it; on removal of the stress the strain will decrease again and once more eventually reach an equilibrium. This part of the graph is known as the recovery curve. Again there are two points of particular interest—the amount of recovery and the rate.

So far the description refers to what may be called an ideal viscoelastic material and is an oversimplification of real-life experience. If the material is not linearly viscoelastic, the creep and recovery curve can still be obtained, but a single curve will not include all the rheological information about the sample. A series of such curves is necessary, giving the full three-dimensional representation. It has also been assumed so far that the properties of the material are not altered in any way by the action of the strain itself. In the case of cheese it is very evident that at large strains the whole structure breaks down and cannot recover. Indeed, the stress and strain at the point of onset of this breakdown may be a useful quantity when describing the properties of any cheese, as will be seen later. Even at lower strains there may well be unobserved changes in the structure. These could occur during the early part of a cycle of measurement, in which case they may be revealed by the failure of the recovery curve to follow the expected pattern. However, they may occur during handling of the cheese at any stage between manufacture and sampling. A complete rheological description of a sample really needs not only the experimental curve or curves but a statement of the history of the straining of the sample up to the moment of measurement. This is most likely to be unavailable and it is necessary to presume that the sample is in good condition and to allow that variations in its history are included within the natural variation among samples.

It is often more convenient from the point of view of practical instrumentation to carry out the compression experiment in a different way. Instead of applying a fixed stress and observing the reaction, the sample is subjected to a known rate of strain and the build-up of the stress observed. Again, this may be prolonged indefinitely, if the operation of the instrument will allow it. If the straining is stopped at any point, it is not usually practicable to apply a reverse rate of strain. In this case, the strain is usually held constant and the relaxation of the stress observed. Again, a complete cycle of deformation and relaxation contains all the available information about the rheological properties of the sample. Quite sophisticated instruments have been developed which enable this type of test to be carried out with precision and then record the results, making this now the most widely used test. In principle, the operation of these instruments is simple. The sample is compressed between parallel plates at a known fixed rate and the force required is monitored.

However, before discussing the results of any measurements made on these instruments it is important to consider the exact mode of operation. As the measurement is usually carried out, the sample is in the form of a small cylinder, although other right prisms may be used. This is placed with its axis vertical on one plate and is then compressed along that axis. Since the compression is linear, the height of the sample at any instant after the compression has begun is given by h = h0( 1 — at)

where h() is the original height of the sample and a denotes the rate at which the two places approach each other, ie, a = —dh/dt

Using equation 1 for the strain leads to e = — ln(l - at) (11)

If the instrument's recorder treats time (or its equivalent linear distance traveled) as one axis, then this is not a true strain axis. As the compression increases from 0 to 100% the true strain increases from zero to infinity. In fact, whatever definition of strain is used (6), other than the linear compression, the time axis is a distortion of the strain axis. Second, as a sample is compressed its volume remains constant, so that the cross-sectional area increases as the height decreases. Assuming that the plates are perfectly smooth, so that there is no lateral friction between them and the end surfaces of the sample, a purely viscous, or equally a perfectly elastic solid, sample would deform uniformly and the cross-sectional area A at any height would vary precisely inversely with the height, ie,

A viscoelastic material does not deform so simply, but in such a way that the lateral movement is greatest near the ends, so that a concave shape results (32), as shown in Figure 6. In this case the stress is not uniform throughout the sample. When there is some friction between the plates and the sample, the lateral movement of the end layers is restricted and a rotational couple is set up within the sample, using the perimeter of the end surfaces as fulcrum. If the sample is homogeneous or at least cohesive, the effect of this is that the middle of the sample will spread outward, giving rise to barrel-shaped distortion (Fig. 6c). When the rotational forces which develop are sufficient to

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