Definitions

Before the rheological measurements which have been made on cheese can be discussed it is necessary to define a few of the terms used by rheologists. In the preceding paragraphs the words effort and force have been used in their everyday connotation. The term used by the rheologist is stress, denoted by the Greek letter a. This is formally defined as the force measured in newtons (N) divided by the area in square meters (m2) over which it is applied, and is measured in pascals (Pa). A convenient aide-mémoire for the size of these units is that a newton is approximately the weight, under the action of gravity, of a medium-sized apple. The stress may be applied normal to the surface, as in Figure la, or tangentially, as in Figure lb. In either case it is equal to the force divided by the area over which it is applied.

In the case of Figure la, which depicts a weight resting on the upper surface of a sample, the deformation which

Force I Force

Figure 1. Application of stress to a sample: (a) compressive strain; and (b) shear.

occurs is a change (Ah) in the height (h) of the sample. The fractional change in height, Ah/h, defines the strain, which is denoted by the Greek letter e. There is a practical difficulty here: the height is continuously changing during the deformation, so there is some uncertainty over what value should be used for h (6). For small deformations this is relatively unimportant, but when large deformations occur, such as is usual when examining cheese, it is necessary to be clear on this point. For the sake of clarity, the ratio of the change in height to the original height will be referred to as the fractional compression and expressed as a percentage. When the change is related to the varying height, the formula for the strain becomes e = ln(MM (1)

This is sometimes called the true strain and will be given in decimal form.

If the stress is applied tangentially, as in Figure lb, the resulting strain is described as a shear. It is defined as the distance through which one plane has traveled relative to another divided by the distance between them and is given the symbol y. Both the strain and the shear are the quotients of two lengths and are therefore dimensionless numbers.

With these definitions of the physical quantities stress, strain, and shear, it is now possible to define the properties of some ideal materials. The rigidity of a solid, hitherto used loosely, can now be defined formally as the ratio of the tangential stress to the shear produced and is given the symbol n:

The dimensions of n are evidently the same as the dimensions of the stress, but it is usual to express the modulus of rigidity in newtons per square meter and to reserve the use of pascals for stresses only, thereby preserving an easily recognizable distinction between them.

The second property of an elastic solid which may be defined can be seen by referring again to Figure la. This is the modulus of elasticity and is the ratio of the stress to the strain. If the material is ideal, it will be isotropic; that is, the properties will be the same in all directions. During compression the volume will remain constant, so that while the height of the sample is changing in the direction of the applied stress, the dimensions in the two directions at right angles will change simultaneously. At any point within the sample only one-third of the stress will be used in causing the change in height. Another third will be used in each of the two other directions normal to this, causing the sample to expand laterally. Hence, for any ideal elastic material the modulus of elasticity will be three times the modulus of rigidity. However, if the material is compressible, some of the stress will be used up in compressing it and the factor 3 relating the modulus of elasticity to that of rigidity falls toward 2. The importance of this is that some authors express their results in terms of elasticity and some in terms of rigidity. An accurate comparison is possible only when some information is available concerning the compressibility of the material. In practice the ideal incompressible material is as unlikely to exist as any other ideal material. Almost all cheese is somewhat compressible, but in general not so compressible that any serious error is introduced by using the factor 3 to convert from one unit to the other.

In the case of liquids there is only one material constant that is of interest. The rate of flow or deformation is now given by the quotient of the strain and the time taken to reach it if the motion is constant, or in the unsteady state by the first derivative with respect to time of the strain, è. The ratio of the applied stress to the rate of flow is called the viscosity and this is denoted by the Greek letter tj, so that n = a/e (3)

The unit of viscosity is the pascal-second. A rough guide to its magnitude is that the viscosity of water at room temperature is very nearly 1/1000 Pa-s.

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