APPLICATION OF THE THEORY TO REAL MATERIALS Scope of the Theory.-37. The theory of elasticity, and hence the solutions which it provides, are based upon three main assump tions, namely: (a) that the material, and therefore the displacement, are "con tinuous " ; (b) that "Hooke's law" is satisfied; (c) that the strains and displacements are everywhere small.
Only when these assumptions are justified will the foregoing solu tions be strictly valid.
Now real materials, as we have seen, are not continuous at all: timber, for example, has a "grain" which can be seen with the naked eye, and metals, examined under the microscope, are found to be conglomerates of small crystals, in general arranged at random. But their structure is invariably fine in relation to the sizes of members used in engineering construction, or of the pieces in which they are tested for strength and rigidity; hence its exact nature is a question of minor importance to the practical designer, who (with certain exceptions which will be stated later) is con cerned to know the general nature, rather than the finer details, of the stress-distributions which occur. We may legitimately employ "statistical" rather than exact methods, carrying over to real problems the concepts and notation which our theory has developed in relation to ideal continuous materials.
This is the standpoint adopted in the second branch of our subject, the science of testing of materials. When we stretch a rod of steel in a testing machine, and measure the resulting extension, we interpret our results as giving relations between stress and 35. A similar argument may be employed to find the stress im posed by uniform pressure on a thin spherical shell. Using the same notation, we see that a total force due to pressure, of amount 36. If the ends of a circular tube are closed, the pressure on the ends will impose a stress, on planes which are perpendicular to the axis, of intensity given by (31). The stress on axial planes is given by (3o), and the radial stress (which varies from —P at the inner face to zero at the outer face) is relatively negligible. So the tube will undergo circumferential and axial extensions which, by the formulae (i8) of § 27, are strain. We calculate the stress, in accordance with our theoretical
definition (§ 4), as the quotient of the total load divided by the cross-sectional area of the rod; we measure the extension of some definite length of the rod, and calculate the strain, again in accord ance with theory (§ 14), as the corresponding fractional exten sion; and we assume, on grounds which are afforded by the theory (§ 29), that the stress and strain are uniform. That is to say, we treat the material as continuous, disregarding all effects which depend upon its detailed structure. Further, in dealing with steel or other wrought metal, we treat the material as isotropic, although we know that the crystals composing it have properties which depend on direction. This is legitimate from the practical standpoint, because the random orientation of the crystals renders the material "statistically isotropic" (cf. R. V. Southwell and H. J. Gough, Phil. Mag., Jan. 1926, § 1) : it would evidently not be permissible in relation to wood or other materials which exhibit a definite "grain." 38. Fig. I I gives some results, due to Kirkaldy (Experiments on the Mechanical Properties of Steel by a Committee of Civil Engineers, London, 1868 and 1870), which have been interpreted in this manner; they relate to tests made on long rods in tension. It will be seen that the stress and strain are very closely propor tional—i.e., that Hooke's law is satisfied—up to a definite limit for each material which represents a considerable stress but a very small extension (less than 1%). This limit is termed the "limit of proportionality" for stress of the type considered. After it has been passed, the strain increases at a much greater rate in relation to the stress; moreover, as we shall see later, the strain does not disappear when the load is removed, so that the elasticity of the material, as well as the proportionality of stress to strain, has broken down. Therefore, in real materials, the second assump tion (b) of the mathematical theory is satisfied only within the limits of proportionality : if these are exceeded, we may no longer assume that the stresses are distributed in accordance with our calculations.