Hardness (kN/m2)

Figure 5. Penetrometer readings compared with compression modulus.

Another test of a similar nature is the use of a cone to penetrate the sample (22). This test was originally developed for use with high-viscosity lubricants. A loaded cone with its apex pointed vertically downward and initially just making contact with the upper surface of the sample is allowed to penetrate the sample until equilibrium is reached. If a suitable combination of the apical angle of the cone and the load is used, the equilibrium is usually reached within a few seconds and the penetration is deep enough to be measured comfortably. The test assumes that the sample behaves as a plastic or Bingham body; that is, when acted upon by small stresses it behaves as a solid, but once a critical stress, the yield stress, is exceeded, it flows as a viscous liquid.

In fact, the test works in reverse. At the commencement the stress is infinite, since the area of application, which is the tip of the cone, is zero. As the cone penetrates, the material is caused to flow laterally and the rate of penetration is controlled by this lateral flow. As the penetration proceeds, the cross-sectional area increases proportionally with the square of the depth of penetration and the stress decreases correspondingly. When the stress no longer exceeds the critical or yield point, motion ceases and an equilibrium is established. It is a matter of contention among rheologists whether a yield value really exists. Some maintain that there is always some flow, albeit minimal, however low the stress. For practical purposes, however, the cone soon appears to be stationary and the depth of penetration can be recorded. This is a simple, quick test and, while it is not entirely nondestructive, it does only a little damage near the surface of the sample. The vertical stress Y may be calculated from the penetration and the angle of the cone:

where a is the apical angle of the cone, h the penetration depth, M the applied load, and g the gravitational constant. This is greater than the yield stress since it does not take into account the stresses involved in causing the sample to flow laterally. An estimate of the yield stress may be obtained by multiplying the equilibrium vertical stress by the factor 1/2 sin a (23). It is only an estimate, though, because it takes no account of any friction between the surface of the cone and the cheese.

Another form of penetrometer-type test is that sometimes referred to as the puncture test (24). In place of a needle or cone, a rod is driven into the sample and the resistance to its motion measured. The mechanism of the deformation of the sample is more complicated in this case. At least four principal factors are involved. In the first place, the sample ahead of the rod is compressed. Second, the rod must cut through the sample at its leading edge: hence the name puncture, and this requires a force. Third, there is the frictional resistance between the surface of the rod and the cheese, and finally, there is the force required to set up the lateral flow within the sample. If the sample is large enough compared with the dimensions of the rod so that any compression in directions perpendicular to the motion may be ignored, once a dynamic equilibrium has been set up the first two factors will be constant and the

third will increase linearly with the penetration. By using rods of different cross-sectional areas and keeping the perimeter the same it is possible to separate these effects (25,26).

Instead of performing this test on an extensive sample, such as a whole cheese, it is more often carried out on a small sample contained within a rigid box (27-29). When done in this manner, additional forces are called into play. The lateral forces on the rod as the sample is compressed between the rod and the walls of the container now become important. In addition, the compressive force on the leading face of the rod is no longer constant as the distance between this face and the bottom of the container decreases. If the sample is shallow, it is arguable (27,28,30) that this may indeed be the largest single contributory force, and, by ignoring the others in comparison with it, one can calculate a modulus from the ratio of the measured stress to the compression. It is obvious that this will be an overestimate of any true modulus which could have been derived in a simple unrestrained test and that the excess will be arbitrary and unknown. In spite of this the test is useful because it gives an indication of the magnitude of the rigidity of the sample and allows comparisons to be made between similar types of cheese.

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