The Relation of Stress to Strain 21

Isotropy is not a property of wood, which is well known to have its greatest strength "along the grain"; nor is it found, in experiment, to be a property of crystals : to represent such ma terials, we must assume this ideal material to be aeoiotropic that is, to have elastic constants which vary with direction. Wrought metals, on the other hand, behave as isotropic sub stances, in the sense that specimens cut from the same material, but in different directions, behave similarly under tests. Thus the assumption of isotropy, which greatly simplifies our calculations, is legitimate for most practical applications of the theory. We shall not discuss aeolotropic materials further in this article.

Stress-strain Relations in Isotropic Material.-24.

For isotropic materials we may show that the number of independent constants cannot exceed two. For we have seen, in the analysis of stress (§ 6), that a small rectangular block can be found, at any point in the material, whose faces are subjected to purely nor mal stresses; and in the analysis of strain (§ 2o), that a small rectangular block can be found whose faces remain rectangular after strain. The stresses on the faces, in the first case, are termed "principal stresses," and the extensions of the edges, in the second case, are termed "principal extensions, or strains." Now it is clear that, in material which has no directional property, the directions of the principal stresses and of the principal strains must coincide ; for there is no reason why a symmetrical system of purely normal stresses should produce asymmetrical distortion, as would be the case if the block ceased to be rectangular. There fore, in the most general statement of Hooke's law for isotropic materials, we have to relate three principal stresses with three principal strains ; and our formulae will thus be three of the type The quantityµ is termed the "modulus of rigidity," or "shear modulus" of the material. From the physical standpoint, k and are to be regarded as the fundamental elastic constants: k measures the resistance to change of volume unaccompanied by change of form, whilst ,u measures the resistance to change of form unaccompanied by change of volume.

Young's Modulus and Poisson's Ratio.-27. The conditions in a simple tensile test are such that, very approximately, P2 = P3 = 0, so that el, e2, will all be proportional to pi, the longitudinal tension.

Considerations of symmetry require that

and shall be equal, but we may not assert that they are zero: in tests on actual materials it is found that and are finite and opposite in sign to If then we write These results apply, strictly, to a beam which is bent by forces applied in a particular way (i.e., so as to produce the foregoing

distribution of stress on the terminal sections) ; but theory indi cates reasons why the results may, with an accuracy sufficient for practical purposes, be extended to beams bent in any manner, and even to beams of varying cross-section. The results (22)–(24) thus form the basis of a very general theory of bending.

Much use is made in mechanical engineering of circular shafts which transmit couples from one end to the other : for example, the propeller shaft of a steamship transmits a couple from the engine or turbine to the propeller, where this couple is opposed by the reaction of the water. Such couples tend to twist the shaft, and in this problem again the state of stress is, for tunately, simple. Let AB (fig. 9) be a uniform circular shaft held at the end A and twisted by a couple applied in the plane B. In consequence of the strain due to twisting (this is very much mag nified in the diagram) a radius CD in the plane B remains straight but turns round to CD', and a line AD, originally parallel to the axis of the shaft and at a distance r from it, distorts into a helix AD'—that is, into a curve which makes a constant angle 0 with lines parallel to the axis. Cross-sections of the shaft remain plane when the shaft is twisted and hence 0 is the angle of shear : so, at a distance r from the axis, we have a shearing stress, of intensity acting on each cross-section in a direction perpendicular to the radius.

Now let i denote the angle of twist DCD' , and let 1 be the length of the shaft. Then i and 0 (both being assumed very small) are related by the equation Now it is an observed property of real materials (and an evident condition of their persistence) that the elastic constants E, k, shall all be positive. It follows from (r7) that cannot be nega tive and numerically greater than 1; for otherwise the ratio Eh.1, would be negative. And since (I is thus shown to be positive, it follows from ('9) that o- cannot be positive and greater thanl; for otherwise the ratio WA would be negative. Thus we have as the condition for a stable material,