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 ω) be stretched with tension T, which is distributed uniformly over the section, so that the stretching force is Twω, and let the bar be unsupported at the sides. The bar will undergo a longitudinal extension of magnitude T/E, where 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 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 σT/E, where σ 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 σ, from theoretical considerations, and his theory led him to assign to σ the value ¼. Many experiments have been made with the view of determining σ, with the result that it has been found to be different for different materials, although for very many it does not differ much from ¼. 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 E and σ 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 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 Xy is equal to S, and the remaining stress-components vanish; and the strain produced in the body is shearing strain of the type denoted by exy. The amount of the shearing strain is S/μ, and the quantity μ 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 μ of an isotropic material are connected with the Young’s modulus E and Poisson’s ratio σ of the material by the equations
k = E / 3(1 − 2σ), μ = E / 2(1 + σ).
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
exx = (Xx − σYy − σZz) / E, eyz = Yz / μ,
eyy = (Yy − σZz − σXx) / E, ezx = Zx / μ,