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is in every case caused by the gradual growth of minute flaws from the beginning of the series of tests, or whether the elastic quality of the material suffers deterioration apart from such flaws. It appears, however, to be an ascertained result that, so long as the limit of linear elasticity is not exceeded, repeated loads and rapidly alternating loads do not produce failure of the material.

36. The question of the conditions of safety, or of the conditions in which rupture is produced, is one upon which there has been much speculation, but no completely satisfactory result has been obtained. It has been variously held that rupture occurs when the numerically greatest principal stress exceeds a certain limit, or when this stress is tension and exceeds a certain limit, or when the greatest difference of two principal stresses (called the “stress-difference”) exceeds a certain limit, or when the greatest extension or the greatest shearing strain or the greatest strain of any type exceeds a certain limit. Some of these hypotheses appear to have been disproved. It was held by G. F. Fitzgerald (Nature, Nov. 5, 1896) that rupture is not produced by pressure symmetrically applied all round a body, and this opinion has been confirmed by the recent experiments of A. Föppl. This result disposes of the greatest stress hypothesis and also of the greatest strain hypothesis. The fact that short pillars can be crushed by longitudinal pressure disposes of the greatest tension hypothesis, for there is no tension in the pillar. The greatest extension hypothesis failed to satisfy some tests imposed by H. Wehage, who experimented with blocks of wrought iron subjected to equal pressures in two directions at right angles to each other. The greatest stress difference hypothesis and the greatest shearing strain hypothesis would lead to practically identical results, and these results have been held by J. J. Guest to accord well with his experiments on metal tubes subjected to various systems of combined stress; but these experiments and Guest’s conclusion have been criticized adversely by O. Mohr, and the question cannot be regarded as settled. The fact seems to be that the conditions of rupture depend largely upon the nature of the test (tensional, torsional, flexural, or whatever it may be) that is applied to a specimen, and that no general formula holds for all kinds of tests. The best modern technical writings emphasize the importance of the limits of linear elasticity and of tests of dynamical resistance (§ 87 below) as well as of statical resistance.

37. The question of the conditions of rupture belongs rather to the science of the strength of materials than to the science of elasticity (§ 1); but it has been necessary to refer to it briefly here, because there is no method except the methods of the theory of elasticity for determining the state of stress or strain in a body subjected to forces. Whatever view may ultimately be adopted as to the relation between the conditions of safety of a structure and the state of stress or strain in it, the calculation of this state by means of the theory or by experimental means (as in § 18) cannot be dispensed with.

38. Methods of determining the Stress in a Body subjected to given Forces.—To determine the state of stress, or the state of strain, in an isotropic solid body strained within its limits of elasticity by given forces, we have to use (i.) the equations of equilibrium, (ii.) the conditions which hold at the bounding surface, (iii.) the relations between stress-components and strain-components, (iv.) the relations between strain-components and displacement. The equations of equilibrium are (with notation already used) three partial differential equations of the type

${\frac {\partial {\text{X}}_{x}}{\partial x}}+{\frac {\partial {\text{Y}}_{y}}{\partial y}}+{\frac {\partial {\text{Z}}_{z}}{\partial z}}+\rho {\text{X}}=0$

(1)

The conditions which hold at the bounding surface are three equations of the type

${\text{X}}_{x}\cos(x,\nu )+{\text{Y}}_{y}\cos(y,\nu )+{\text{Z}}_{z}\cos(z,\nu )+={\overline {\text{X}}}_{\nu },$

(2)

where $\nu$ denotes the direction of the outward-drawn normal to the bounding surface, and ${\overline {\text{X}}}_{\nu }$ denotes the $x$-component of the applied surface traction. The relations between stress-components and strain-components are expressed by either of the sets of equations (1) or (3) of § 26. The relations between, strain-components and displacement are the equations (1) of § 11, or the equivalent conditions of compatibility expressed in equations (1) and (2) of § 16.

39. We may proceed by either of two methods. In one method we eliminate the stress-components and the strain-components and retain only the components of displacement. This method leads (with notation already used) to three partial differential equations of the type

$(\lambda +\mu ){\frac {\partial }{\partial x}}\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}\right)+\mu \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}\right)+\rho {\text{X}}=0,$

(3)

and three boundary conditions of the type

$\lambda \cos(x,\nu )\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}\right)+\mu \left\{2\cos(x,\nu ){\frac {\partial u}{\partial x}}+\cos(y,\nu )\left({\frac {\partial v}{\partial x}}+{\frac {\partial u}{\partial y}}\right)+\cos(z,\nu )\left({\frac {\partial u}{\partial z}}+{\frac {\partial w}{\partial x}}\right)\right\}\quad ={\overline {\text{X}}}_{\nu }.$

(4)

In the alternative method we eliminate the strain-components and the displacements. This method leads to a system of partial differential equations to be satisfied by the stress-components. In this system there are three equations of the type

${\frac {\partial {\text{X}}_{x}}{\partial x}}+{\frac {\partial {\text{Y}}_{y}}{\partial y}}+{\frac {\partial {\text{Z}}_{z}}{\partial z}}+\rho {\text{X}}=0,$

(1 bis)

three of the type

${\frac {\partial ^{2}{\text{X}}_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}{\text{Y}}_{y}}{\partial y^{2}}}+{\frac {\partial ^{2}{\text{Z}}_{z}}{\partial z^{2}}}+{\frac {1}{1+\sigma }}{\frac {\partial ^{2}}{\partial x^{2}}}\left({\text{X}}_{x}+{\text{Y}}_{y}+{\text{Z}}_{z}\right)=-{\frac {\sigma }{1-\sigma }}\rho \left({\frac {\partial {\text{X}}}{\partial x}}+{\frac {\partial {\text{Y}}}{\partial y}}+{\frac {\partial {\text{Z}}}{\partial z}}\right)-2\sigma {\frac {\partial {\text{X}}}{\partial x}},$

(5)

and three of the type

${\frac {\partial ^{2}{\text{Y}}_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}{\text{Y}}_{x}}{\partial y^{2}}}+{\frac {\partial ^{2}{\text{Y}}_{x}}{\partial z^{2}}}+{\frac {1}{1+\sigma }}{\frac {\partial ^{2}}{\partial y\partial z}}({\text{X}}_{x}+{\text{Y}}_{y}+{\text{Z}}_{z})=-\rho \left({\frac {\partial {\text{Z}}}{\partial y}}+{\frac {\partial {\text{Y}}}{\partial z}}\right),$

(6)

the equations of the two latter types being necessitated by the conditions of compatibility of strain-components. The solutions of these equations have to be adjusted so that the boundary conditions of the type (2) may be satisfied.

40. It is evident that whichever method is adopted the mathematical problem is in general very complicated. It is also evident that, if we attempt to proceed by help of some intuition as to the nature of the stress or strain, our intuition ought to satisfy the tests provided by the above systems of equations. Neglect of this precaution has led to many errors. Another source of frequent error lies in the neglect of the conditions in which the above systems of equations are correct. They are obtained by help of the supposition that the relative displacements of the parts of the strained body are small. The solutions of them must therefore satisfy the test of smallness of the relative displacements.

41. Torsion.—As a first example of the application of the theory we take the problem of the torsion of prisms. This problem, considered first by C. A. Coulomb in 1784, was finally solved by B. de Saint-Venant in 1855. The problem is this:—A cylindrical or prismatic bar is held twisted by terminal couples; it is required to determine the state of stress and strain in the interior. When the bar is a circular cylinder the problem is easy. Any section is displaced by rotation about the central-line through a small angle, which is proportional to the distance $z$ of the section from a fixed plane at right angles to this line. This plane is a terminal section if one of the two terminal sections is not displaced. The angle through which the section $z$ rotates is $\tau z,$ where $\tau$ is a constant, called the amount of the twist; and this constant $\tau$ is equal to ${\text{G}}/\mu {\text{I}},$ where ${\text{G}}$ is the twisting couple, and ${\text{I}}$ is the moment of inertia of the cross-section about the central-line. This result is often called “Coulomb’s law.” The stress within the bar is shearing stress, consisting, as it must, of two sets of equal tangential tractions on two sets of planes which are at right angles to each other. These planes are the cross-sections and the axial planes of the bar. The tangential traction at any point of the cross-section is directed at right angles to the axial plane through the point, and the tangential traction on the axial plane is directed parallel to the length of the bar. The amount of either at a distance r from the axis is $\,i\tau r$ or ${\text{G}}r/{\text{I}}.$ The result that ${\text{G}}=\mu \tau {\text{I}}$ can be used to determine $\mu$ experimentally, for $\tau$ may be measured and ${\text{G}}$ and ${\text{I}}$ are known.

42. When the cross-section of the bar is not circular it is clear that this solution fails; for the existence of tangential traction, near the prismatic bounding surface, on any plane which does not cut this surface at right angles, implies the existence of traction applied to this surface. We may attempt to modify the theory by retaining the supposition that the stress consists of shearing stress, involving tangential traction distributed in some way over the cross-sections. Such traction is obviously a necessary constituent of any stress-system which could be produced by terminal couples around the axis.