Figure 10. Derived curves of apparent modulus versus strain.

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 - ceV < 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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