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to the applied tension, provided that this tension is not too great. In interpreting this result it is assumed that the tension is uniform over the cross-section of the bar, and that the extension of longitudinal filaments is uniform throughout the bar; and then the result takes the form of a law of proportionality connecting stress and strain: The tension is proportional to the extension. Similar results are found for the same materials when other methods of experimenting are adopted, for example, when a bar is supported at the ends and bent by an attached load and the deflexion is measured, or when a bar is twisted by an axial couple and the relative angular displacement of two sections is measured. We have thus very numerous experimental verifications of the famous law first enunciated by Robert Hooke in 1678 in the words “Ut Tensio sic vis”; that is, “the Power of any spring is in the same proportion as the Tension (—stretching) thereof.” The most general statement of Hooke’s Law in modern language would be:—Each of the six components of stress at any point of a body is a linear function of the six components of strain at the point. It is evident from what has been said above as to the nature of the measurement of stresses and strains that this law in all its generality does not admit of complete experimental verification, and that the evidence for it consists largely in the agreement of the results which are deduced from it in a theoretical fashion with the results of experiments. Of such results one of a general character may be noted here. If the law is assumed to be true, and the equations of motion of the body (§ 5) are transformed by means of it into differential equations for determining the components of displacement, these differential equations admit of solutions which represent periodic vibratory displacements (see § 85 below). The fact that solid bodies can be thrown into states of isochronous vibration has been emphasized by G. G. Stokes as a peremptory proof of the truth of Hooke’s Law.

20. According to the statement of the generalized Hooke’s Law the stress-components vanish when the strain-components vanish. The strain-components contemplated in experiments upon which the law is founded are measured from a zero of reckoning which corresponds to the state of the body subjected to experiment before the experiment is made, and the stress-components referred to in the statement of the law are those which are called into action by the forces applied to the body in the course of the experiment. No account is taken of the stress which must already exist in the body owing to the force of gravity and the forces by which the body is supported. When it is desired to take account of this stress it is usual to suppose that the strains which would be produced in the body if it could be freed from the action of gravity and from the pressures of supports are so small that the strains produced by the forces which are applied in the course of the experiment can be compounded with them by simple superposition. This supposition comes to the same thing as measuring the strain in the body, not from the state in which it was before the experiment, but from an ideal state (the “unstressed” state) in which it would be entirely free from internal stress, and allowing for the strain which would be produced by gravity and the supporting forces if these forces were applied to the body when free from stress. In most practical cases the initial strain to be allowed for is unimportant (see §§ 91-93 below).

21. Hooke’s law of proportionality of stress and strain leads to the introduction of important physical constants: the moduluses of elasticity of a body. Let a bar of uniform section (of area $\omega$) be stretched with tension ${\text{T}},$ which is distributed uniformly over the section, so that the stretching force is ${\text{T}}\omega ,$ and let the bar be unsupported at the sides. The bar will undergo a longitudinal extension of magnitude ${\text{T}}/{\text{E}},$ where ${\text{E}}$ is a constant quantity depending upon the material. This constant is called Young's modulus after Thomas Young, who introduced it into the science in 1807. The quantity ${\text{E}}$ is of the same nature as a traction, that is to say, it is measured as a force estimated per unit of area. For steel it is about 2.04 × 10¹² dynes per square centimetre, or about 13,000 tons per sq. in.

22. The longitudinal extension of the bar under tension is not the only strain in the bar. It is accompanied by a lateral contraction by which all the transverse filaments of the bar are shortened. The amount of this contraction is $\sigma {\text{T}}/{\text{E}},$ where $\sigma$ is a certain number called Poisson’s ratio, because its importance was at first noted by S. D. Poisson in 1828. Poisson arrived at the existence of this contraction, and the corresponding number $\sigma ,$ from theoretical considerations, and his theory led him to assign to $\sigma$ the value ${\tfrac {1}{4}}\cdot$ Many experiments have been made with the view of determining $\sigma ,$ with the result that it has been found to be different for different materials, although for very many it does not differ much from ${\tfrac {1}{4}}\cdot$ For steel the best value (Amagat’s) is 0.268. Poisson’s theory admits of being modified so as to agree with the results of experiment.

23. The behaviour of an elastic solid body, strained within the limits of its elasticity, is entirely determined by the constants ${\text{E}}$ and $\sigma$ if the body is isotropic, that is to say, if it has the same quality in all directions around any point. Nevertheless it is convenient to introduce other constants which are related to the action of particular sorts of forces. The most important of these are the “modulus of compression” (or “bulk modulus”) and the “rigidity” (or “modulus of shear”). To define the modulus of compression, we suppose that a solid body of any form is subjected to uniform hydrostatic pressure of amount p. The state of stress within it will be one of uniform pressure, the same at all points, and the same in all directions round any point. There will be compression, the same at all points, and proportional to the pressure; and the amount of the compression can be expressed as $p/k.$ The quantity $k$ is the modulus of compression. In this case the linear contraction in any direction is $p/3k$; but in general the linear extension (or contraction) is not one-third of the cubical dilatation (or compression).

24. To define the rigidity, we suppose that a solid body is subjected to forces in such a way that there is shearing stress within it. For example, a cubical block may be subjected to opposing tractions on opposite faces acting in directions which are parallel to an edge of the cube and to both the faces. Let ${\text{S}}$ be the amount of the traction, and let it be uniformly distributed over the faces. As we have seen (§ 7), equal tractions must act upon two other faces in suitable directions in order to maintain equilibrium (see fig. 2 of § 7). The two directions involved may be chosen as axes of $x,y$ as in that figure. Then the state of stress will be one in which the stress-component denoted by ${\text{X}}_{y}$ is equal to ${\text{S}},$ and the remaining stress-components vanish; and the strain produced in the body is shearing strain of the type denoted by $e_{xy}.$ The amount of the shearing strain is ${\text{S}}/\mu ,$ and the quantity $\mu$ is the “rigidity.”

25. The modulus of compression and the rigidity are quantities of the same kind as Young’s modulus. The modulus of compression of steel is about 1.43 × 10¹² dynes per square centimetre, the rigidity is about 8.19 × 10¹¹ dynes per square centimetre. It must be understood that the values for different specimens of nominally the same material may differ considerably.

The modulus of compression $k$ and the rigidity $\mu$ of an isotropic material are connected with the Young’s modulus ${\text{E}}$ and Poisson’s ratio $\sigma$ of the material by the equations

$k={\text{E}}/3(1-2\sigma ),$⁠$\mu ={\text{E}}/2(1+\sigma ).$

26. Whatever the forces acting upon an isotropic solid body may be, provided that the body is strained within its limits of elasticity, the strain-components are expressed in terms of the stress-components by the equations

$e_{xx}=({\text{X}}_{x}-\sigma {\text{Y}}_{y}-\sigma {\text{Z}}_{z})/{\text{E}},$	⁠ $e_{yz}={\text{Y}}_{z}/\mu ,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ . . (1)
$e_{yy}=({\text{Y}}_{y}-\sigma {\text{Z}}_{z}-\sigma {\text{X}}_{x})/{\text{E}},$	⁠ $e_{zx}={\text{Z}}_{x}/\mu ,$
$e_{zz}=({\text{Z}}_{z}-\sigma {\text{X}}_{x}-\sigma {\text{Y}}_{y})/{\text{E}},$	⁠ $e_{xy}={\text{X}}_{y}/\mu .$

If we introduce a quantity $\lambda ,$ of the same nature as ${\text{E}}$ or $\mu ,$ by the equation}}

$\lambda ={\text{E}}\sigma /(1+\sigma )(1-2\sigma ),$

(2)

we may express the stress-components in terms of the strain-components by the equations

${\text{X}}_{x}=\lambda (e_{xx}+e_{yy}+e_{zz})+2\mu e_{xx},$	⁠ ${\text{Y}}_{z}=\mu e_{yz},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ . (3)
${\text{Y}}_{y}=\lambda (e_{xx}+e_{yy}+e_{zz})+2\mu e_{yy},$	⁠ ${\text{Z}}_{x}=\mu e_{zx},$
${\text{Z}}_{z}=\lambda (e_{xx}+e_{yy}+e_{zz})+2\mu e_{zz},$	⁠ ${\text{X}}_{y}=\mu e_{xy};$

and then the behaviour of the body under the action of any forces