## Info

At the commencement of the compression the stress is zero and the initial slope is proportional to the elastic component. Initially the curve is convex upward, but as the compression proceeds a point of inflexion is reached, depending on the ratio of the viscous to the elastic components and the rate of compression. Finally, the curve becomes asymptotic to a line through the origin, with a slope given by the elastic constant.

If the stress-strain curve obtained for a particular sample can be recognized as being similar to one of these two patterns, the appropriate model can be identified and the material constants evaluated. It is seldom that such a simple fit occurs. A material as heterogeneous as cheese is unlikely to conform to the behavior of a two-element model. More sophisticated models may give a better representation, but adding further elements only makes the theory more complex and the analysis that much more difficult. One word of caution must be entered here. The Voigt body is characterized by the finite intercept on the stress axis. The converse is not necessarily true. A finite intercept also arises in the case of any material possessing a yield value, since this is the stress which must be overcome before any deformation takes place.

In the usual way in which compression tests are carried out, the compression is allowed to proceed to a point far beyond that at which any simple theory may be expected to apply. Often it reaches 80%, a true strain of 1.609. Long before this is reached, any structure which may have been present in the original sample of cheese can be seen to have broken down. If it is carried to this extent, only the early part of the test may be considered as measuring the rheo-logical properties of the original sample, while the latter part becomes a test of the mechanical strength of the structure. A typical compression curve is shown in Figure 7. It

20 40 60 Compression (%)

Figure 7. Typical force-compression curve.

20 40 60 Compression (%)

### Figure 7. Typical force-compression curve.

is often difficult to decide whether there is an intercept on the stress axis, ie, whether there is an instantaneous buildup of stress at the moment the compression begins or a rapid development of stress rising from zero but in a finite though brief time. Nevertheless, it is essential that the distinction be made if the correct model is to be assigned. The instrument itself necessarily has a finite, even if very rapid, acceleration from rest, and the response of the recording mechanism also takes a finite time, so that even if the stress were instantaneous, it would appear as a very steep rise (35).

However, much more serious than any limitations in the instrument is the problem of attaining perfection in the shape of the sample. Usually the dimensions of the sample and the rate of compression are chosen so that the strain rate at the commencement is of the order of 0.01 s_1. In a sample, say 20 mm in height, a lack of parallelism of the two end surfaces of only 0.2 mm means that for the whole of the first second of compression only part of the sample is being compressed. This will inevitably appear on the record as a flow, whatever the real properties of the cheese.

Assuming that any doubts about the shortcomings of either sample or instrumentation may be satisfactorily allayed, it is pertinent to consider the principal features of the compression curve. In the early stages, after the initial rise there is a smooth increase of stress with the strain. Should the compression be halted at this stage, the cheese would recover either completely or in part and a repeat of the curve could then be obtained, the new curve having the same general shape as the original, showing that the internal structure of the cheese had remained more or less intact, although there may have been some rearrangement. Eventually a point is reached, the point A in Figure 7, where the structure begins to break down and the slope becomes noticeably less. The conventional view is that this is the point at which cracks in the structure appear and the cracks then spread spontaneously (18,36). In the case of a hard cheese these cracks may be evident to the eye, but they may at first be so small and so localized that they do not become visible until the compression has proceeded somewhat further. If the cheese has a very homogeneous structure, it is to be expected that these cracks will develop throughout the structure at about the same time and the change of slope will be clearly defined. In a more heterogeneous cheese they may appear over a range of strains and the change of slope will be much more diffuse. Once this region has been passed, the cracks continue to develop at an increasing rate and become more evident, until at point B the rate at which the structure breaks down overtakes the rate of build-up of stress through further compression and a peak is reached. This value is obviously dependent on a balance between the spontaneous failure of the structure and the effect of increasing deformation applied by the instrument to any residual structure. It is a convenient parameter to determine and may be used as a measure of firmness or hardness of the cheese (18). Beyond this point the stress may continue to fall if the failure of the structure becomes catastrophic, until the fragments become compacted into a new arrangement which can take up the stress and this now rises again.

If the compression test is carried on beyond the point at which it is reasonable to attempt to interpret it in terms of an acceptable model, and hence to evaluate any of the customary rheological parameters, it becomes purely empirical. One of the points on the curve most frequently used is the peak (18,37) (point B). This is a kind of yield point and one can obtain from it two useful parameters, the stress required to cause catastrophic breakdown of the structure and the amount of deformation that the cheese will stand before this breakdown occurs. The two are not entirely independent. In a dynamic measurement such as this on an essentially viscoelastic material, the rate of strain as well as the strain itself influences the moment at which the breakdown occurs. For reliable comparisons, all measurements should be made at the same rate of straining. Fortunately, many workers have found it convenient to use similar sizes and rates of compression, so that at least approximate comparisons can be made between their results. The stress measured at point B is in fact sometimes called a yield value, but it should not be confused with the yield stress of a plastic material determined by other methods such as a cone penetrometer. The yield at point B is not a material constant defining the strength of the sample; breakdown has already been occurring at least since point A, and maybe earlier. The peak value at B only indicates the maximum to which the stress rises before the collapse of structure overtakes the build-up of stress in what remains of that structure.

The other point frequently used is the stress at 80% compression. The damage to the structure when the compression reaches this extent is generally so complete that it is unrealistic to regard the stress as applying to the original sample. This is particularly true in the case of hard cheeses such as the typical English and Grana varieties, which crumble long before this degree of compression is reached. Nevertheless, compressions of this order and much greater arise during mastication (38). The stress at 80% can therefore give some indication of the consumer's response to the firmness of the cheese, although it must be said that in mastication the rate of compression is also much greater.

Measurements have been made on four types of hard cheese, Cheddar, Cheshire, Double Gloucester (36), and Leicester (39), which confirm that the value of the stress at the peak point B is dependent on the rate at which the compression is carried out. It was shown that over at least a twentyfold range of the factor a in equation 11, the rate at which the plates approached each other, the peak stress was linearly related to the fourth root of a. This is shown in Figure 8, where straight lines have been drawn through the points for each cheese type. If one bears in mind that this is a destructive test, so that each measurement had to be made on a different sample, the fit of the lines to the experimental points is quite acceptable. This result is, however, purely empirical; as far as can be seen there is no theoretical justification for it and it is limited to those types of cheese which crumble on breakdown. But there is a practical benefit. Using this finding, it is reasonable to reduce measurements made at any practical rate of compression to a standard rate so that the results of workers in different laboratories may be compared.

(Compression rate)0-25, (mm/s)0-25

Figure 8. Effect of rate of compression on stress at yield point •, Double Gloucester; □, Cheshire; Leicester; and O, Cheddar.

(Compression rate)0-25, (mm/s)0-25

Figure 8. Effect of rate of compression on stress at yield point •, Double Gloucester; □, Cheshire; Leicester; and O, Cheddar.

The investigation was carried one step further on Double Gloucester cheese (36). Not only were the peak values found to obey this fourth-root law but so were the stresses at other degrees of compression; the results are shown in Figure 9. The fact that the relation between stress and rate of deformation is more or less constant before, at, and beyond the yield point may be considered to lend support to the hypothesis that the processes taking place within the cheese are similar throughout the compression. The stress at any point is the result of a balance between the rate of collapse of structure, ie, the spread of cracks, and the buildup of stress in that structure which remains. If it is assumed that the basic framework within the cheese has at least some structural strength, there will be some buildup of stress before any cracks appear, and the hypothesis predicts that the stress—strain curve will show an initial instantaneous rise due to viscoelastic deformation before

(Compression rate)0-25, (mm/s)0-25

Figure 9. Effect of rate of compression on stress at various compressions: •, 10%; O, 40%; and ■, 70%.

(Compression rate)0-25, (mm/s)0-25

Figure 9. Effect of rate of compression on stress at various compressions: •, 10%; O, 40%; and ■, 70%.

any cracks develop, followed by a rising curve, convex upward, as the discrete minute cracks increase, with a pronounced change of slope as they begin to coalesce.

There are alternative explanations which may be advanced for the shape of the curves. In the early stages the curve rises from the origin and is often convex upward. This is characteristic of a Maxwell body. The interpretation could be that cheese behaves as an elastic fluid with a very high viscosity term. Neither the elasticity or the viscosity are readily obtained from the curves, but it is possible to calculate an apparent elasticity from the slope. Sometimes the slope at the origin is taken; more usually, the slope over the middle portion of the rise.

On the other hand, it is arguable that the structure of cheese is basically a solid one, but that even under very small stresses minute cracks begin to appear in that structure (40,41), even though these are far too small to be observed by the naked eye and may be disguised to some extent by the fact that some liquid component from the fat could flow into some of the opening interstices. Experiments on cheese analogs (42) at very low strains have confirmed that the structure does indeed break down well before the compression reaches 1%. It is possible to analyze the curves further on the basis of this hypothesis. Suppose that the cheese has an initial rigid structure giving rise to an elasticity E0. Then at the instant at which the compression commences the slope of the stress—strain curve da/de is equal to E0. If the cheese breaks down continuously by the appearance of cracks, infinitessimal at first but becoming gradually more widespread and larger, the strength of the cheese is progressively reduced, so that at any subsequent instant the elasticity E = da/de is less than E0. If the breakdown of the structure is consequent upon the extent of the strain, the equation may be written

This is the equation of a curve through the origin with an initial slope E0 and subsequently convex upward, as is usually observed.

Pursuing this a little further, the function f(e) is a distribution function of the breaking strains of the interpar-ticulate junctions within the cheese. As a trial hypothesis, one may postulate that the distribution function is linear up to the point at which all the junctions are broken. Replacing f(e) by a constant c and integrating equation 17 one gets a = E0e - ce2(e < ecriticai) (18)

This is the equation of a parabola with its apex upward and a maximum stress of a = Eg/4c at a strain of e = E0/2c. Eventually the situation is reached, at ecritical, where all the structure is more or less completely destroyed and individual crumbs may move more or less independently as in the flow of a powder. If the cheese is spreadable, the stress required to maintain this flow may be expected to be constant. On the other hand, some further compaction may take place and the stress begin to rise again. In general, the distribution would not be expected to be a linear one. While the argument remains the same, leading to a convex upward curve with a peak and a subsequent trough, algebraic analysis is more involved. However, if the constant c is replaced by a general expression, such as a series expansion in e, the slope at the origin is still given by E0 but the parabolic form becomes distorted.

When studying individual curves it is not always possible to determine the slope at the origin with any confidence, particularly since this is the region where the response time of the recorder is most likely to introduce its own distortion. However, rewriting equation 18 as a/e = E0 — ce it is possible to construct fresh curves of a/e versus e and these may be easier to extrapolate to zero strain and thereby estimate the value of E0. The (negative) slope of this curve is then the distribution function of c. In a few cheeses c has been found to be more or less constant up to strains approaching unity (ie, about 60% compression) but with most cheeses the value of c decreases as the strain increases, indicating that the rate of breakdown of structure is actually greatest at the lowest strains. Figure 10 shows a few typical derived curves.

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