Page:Elementary Text-book of Physics (Anthony, 1897).djvu/131

§ 104] shearing stresses, and all strains to dilatations and shearing strains, the knowledge of the voluminal elasticity and of the elasticity exhibited during a shear, or the rigidity, is sufficient to describe the elasticity of the body under any form of stress.

103. Voluminal Elasticity.—Let a body of volume $$V$$ be subjected to a uniform hydrostatic stress $$P$$, by which it undergoes a change of volume, given by $$v$$. From Hooke's law we know that, at least within certain limits of stress and consequent deformation, $$V$$ and $$P$$ are proportional. The dilatation or the change of the unit of volume is $$\frac{v}{V}$$. The modulus of elasticity in this case, or the voluminal elasticity of the body, is therefore $$\frac{PV}{v}$$. The voluminal elasticity is denoted by $$k$$.

104. Rigidity.—Let $$S$$ be one of the tangential stresses which constitute a simple shearing stress, that is, a shearing stress of which the elements act in one plane; then the deformation produced is a simple shear. The modulus of rigidity is measured by the ratio of the shearing stress to the amount of the shear (§ 98); it is denoted by $$n$$.

The amount of the shear may be defined in a more convenient form as follows: Let us suppose that the rhombus $$ACDB$$ (Fig. 38) has been strained by a simple shear into the rhombus $$AC'D'B$$, and that this deformation is infinitesimal. The elongation of the diagonal $$AD$$ is then $$FD'$$. The triangle $$DFD'$$ is then an isosceles triangle, since the angle $$DFD'$$ is a right angle, and the angle $$DD'F$$ differs from half a right angle only by an infinitesimal. Therefore $$FD' \sqrt{2} = DD'$$. Now $$AD$$, being the diagonal of a rhombus that is only infinitesimally different from a square, is equal to $$BE \sqrt{2}$$; and therefore the amount of the shear, or $$\frac{DD'}{BE}$$, equals $$\frac{2FD'}{AD}$$, that is, equals twice the elongation along the axis of the shear.