Application of the Theory to Real Materials

39. A restriction is thus imposed upon the application of mathe matical theory. But its practical utility is not seriously affected, for the reason that stresses which would produce elastic failure should be avoided in design: they would evidently involve perma nent distortion of the member concerned, and in most cases they would be liable to cause actual rupture of the material. It is a general rule of design that stresses must not be allowed to exceed the "limits of proportionality" for the material concerned; so we may say that assumption (b) will be satisfied by actual materials, within the permissible range of stress.

It follows that assumption (c) will also be satisfied, since, as we have just seen, the permissible range of stress corresponds to a very small permissible range of strain. To sum up:—Unless we are concerned to know the distribution of stress in such detail that the actual structure of our material is a contributory factor, we may rely on the mathematical theory to predict either (I) the distortion or (2) the stresses which will be produced by specified loads, up to that point at which the strains first cease to be purely elastic. After this point is reached, we are less concerned to know the precise distribution of stress, because we know in advance that the stresses will be unsafe.

But a problem of fundamental importance still remains to be examined, in that the mathematical theory, of itself, affords no answer to the general question, whether specified forces will in volve stresses that are safe. Curves of the type of fig. II, derived from tensile tests, will of course give the required information in relation to members which have to withstand pure tension ; but they will not supply adequate data for the design, e.g., of a crank subjected to combined twist and bending, since this, as we have seen (§ 33), involves a state of compound stress.

The Question of Elastic Failure.-4o.

The circumstances which determine failure of elasticity in real materials, when the state of stress predicted by theory is of the most general type (involving three unequal principal stresses) have been a subject of much controversy. At the outset we observe that this is one of the exceptional cases, mentioned earlier, in which mathematical theory, with its fundamental assumption of continuity, ceases to give an adequate account of the real state of affairs. For whatever may be the factors affecting strength, they must depend upon the structure of the material, which is not in fact uniform or contin uous. Failure will commence at some point of greatest weakness,

and will be determined, not by the general distribution of stress, considered statistically, but by the actual state of stress at that point.

Thus our mathematical concept of stress, and the calculations which are based upon it, break down as applied to the "problem of ' elastic failure," which strictly falls within the province of the physicist rather than the engineer. On the other hand, we have no alternative method of calculation, and some basis for design must be laid down, even though it may not be strictly logical. What is attempted is to find rules whereby, when the stress-distribution has been calculated by the mathematical theory, it may be exam ined to determine whether the stresses fall within the elastic limit. We suppose that our material has been tested under simple conditions (e.g., in tension, where the state of stress is as illus trated in diagram A of fig. r), and that its limits of proportion ality have been ascertained. We are confronted, in the general problem, with the state of stress which is illustrated by diagram D of the same figure; and we seek a criterion which shall deter mine limits of proportionality for this state. The evidence for the criterion will be experimental, but it must be interpreted in accordance with "statistical" ideas, which have no strict validity; so it is not surprising that general agreement on a definite criterion has not in fact been attained.

Alternative Theories.-4r.

Lame, followed by Rankine, adopted a Maximum Stress Theory, according to which the crite rion_of elastic failure is pi, the (numerically) greatest of the principal stresses. Let f be the limit of proportionality as meas ured in a simple tensile test : then, according to the theory, failure occurs if and and have no effect. In particular, f is the limiting value of the hydrostatic pressure which can be sustained without per manent distortion of the material.

Experiment contradicts this conclusion, for it is found that much greater pressures can be sustained': moreover, the theory is at variance with results obtained in torsion tests, where the mate rial is subjected to simple shear (§ 3r). The criterion is no longer accepted, except as a working rule for design in brittle materials, such as cast iron.