Application of the Theory to Real Materials

Now a circle could be drawn in the diagram to represent any other state of stress for which the limit of proportionality had been determined : all that is needed is a knowledge of the greatest and least of the principal stresses, to fix the ends of its horizontal diameter. According to Mohr's hypothesis, circles drawn in this way for all conceivable tests would be touched by an enveloping curve, such as the dotted curve CEA in fig. 12 ; and this curve will represent the relation of tangential to normal stress which is the real criterion of elastic failure. For practical purposes the form of CEA may be determined by drawing only a few circles: for example, in addition to the tensile and compressive tests which have been considered already, we might carry out a test of the material in torsion (i.e., in simple shear), and so obtain a circle centred at the origin, as illustrated by FEG in fig. 12. If the stress-difference criterion were correct, we should find that all three circles had the same diameter, so that the form of CEA would became a straight line parallel to DOB.

Mohr's theory is a logical development of assumptions which are certainly reasonable. Moreover, it offers a way of escape from the difficulties, mentioned above, in regard to the permissible limits for "hydrostatic" tension and compression; for there is no reason why the enveloping curve should not meet the hori zontal axis at some point F which (regarded as a circle of zero diameter) represents a limiting hydrostatic tension of finite inten sity, whereas on the compression side it may very well cut the axis at infinity. These questions are left to be settled by experiment.

Strain-energy Theories.-45.

A criterion of quite another type was suggested by Beltrami in 1885 (Roma. Acc. Lincei Rend. 1885; Math. Annalen, 1903), and more recently by B. P. Haigh. (Brit. Ass. Reports, 1919 and 1921) : it is based on the assump tion that any definite volume of material has only a limited capacity for absorbing and storing energy in the form of elastic strain. In isotropic material, the energy stored per unit volume in a state of stress defined by pi, is and according to Beltrami's theory, failure will occur if where C is a physical constant of the material.

On this basis, the limits for "hydrostatic" tension and com pression will evidently be identical, and this fact constitutes an objection which has been urged already, in relation to other theories.

46. Hiiber, followed by Hencky (Zeitschr. f. Ang. Math. u. Mech., 1924), has proposed a modified energy criterion, in which that part of W which corresponds to change of volume (as op posed to change of form) is neglected. This procedure is equiva lent to the assumption of an infinite bulk-modulus,—that is (§ 28) to the assumption that has the value 4: inserting this value in (34), we have the modified criterion, that elastic failure will occur if where C has the same value as before.

No limit is now imposed on the resistance of the material either to hydrostatic tension or compression, and hence one difficulty (in theory) still remains. The criterion is found to accord closely

with the results of experiment, within the ranges of stress which have been imposed.

47. The second branch of our subject—that known as "testing of materials"—is concerned with those properties of real materials which determine their value to the engineer. His most important requirements are rigidity, which demands a knowledge of the elastic constants, and strength, which requires that the criterion of elastic failure shall be expressed in exact numerical form ; but he desires in addition that his materials shall possess qualities, such as hardness and tenacity (or "toughness"), which are more difficult to define with precision, because they relate to be haviour in that range of stress and strain wherein Hooke's Law, and hence the mathematical theory of elasticity, do not apply.

Stress-strain Diagrams.-48.

The customary procedure for determining elastic constants is to plot diagrams of the type exemplified by fig. 11, showing the relation of stress to strain. Young's modulus (E) is generally obtained by subjecting a long rod to tension, and measuring the consequent increase in the distance between two definite marks. The stress is calculated, in accordance with § 29, as the ratio (total load)/(area of cross section), and the extension as the ratio (increase of length be tween marks)/(original length between marks). To find the modulus of rigidity (µ), it is usual to subject a circular rod to torsion, and to interpret the results in accordance with the mathe matical solution of this problem (§ 31), whereby the shear stress (q) at the outer surface can be calculated in terms of the twisting couple, and the shear strain (0) in terms of the rotation, relative to one another, of two cross-sections separated by a known length. Knowing E and we can, theoretically, deduce any other of the elastic constants by the formulae of § 28. But special methods are of ten employed to determine par ticular constants directly : thus, Poisson's Ratio 6, or the "bulk modulus" k, can be found by tests which depend upon the mathematical solution for uni form bending, or for a closed tube subjected to internal pressure (§ In all cases, when the diagram has been constructed, the relevent elastic constant can be deduced as the ratio of stress to strain within the range for which these quantities are proportional. The extent of this range is also deter mined by the test, and it pro vides a part of the information required to fix the criterion of elastic failure (see §§ 41-46). But, for reasons which we have indicated, the whole diagram (up to the point of ultimate fracture) is important : thus it is essential that we should be able, in these tests, to apply a measured load through a considerable distance.