Page:Elementary Text-book of Physics (Anthony, 1897).djvu/127

§ 100] To examine the relations which must hold among the components of pressure in different directions at any point within a body subjected to stress, we consider a small cube described in a body, and examine the relations among the pressures on its faces necessary to maintain it in equilibrium. We assume that no external forces act directly on the matter contained in the cube. In general, each of the faces of the cube will be subjected to a stress. This stress may be resolved into a normal component and two tangential components taken parallel with the sides of the face to which the stress is applied. Calling the normal components acting on two opposite faces $$P$$ and $$P'$$, those acting on another pair of opposite faces $$Q$$ and $$Q'$$, and those acting on the third pair $$R$$ and $$R'$$, we may express the conditions that the centre of mass of the cube will not be displaced by the equations $$P = P', Q = Q', R = R'$$.

Since the forces which act upon the cube are in equilibrium, and since their normal components maintain the equilibrium of the centre of mass, their tangential components give rise to couples, and these couples are also in equilibrium. These couples are arranged as shown in Fig. 35, for those lying in the plane of one pair of faces. Since equilibrium exists, the two couples formed by the forces $$S$$ and the forces $$S'$$ are equal, and therefore $$S = S'$$, where $$S$$ and $$S'$$ may be used to denote the tangential pressures on the surfaces of the cube. Similar couples in equilibrium will act on the cube in two other planes at right angles with this one so that the whole set of pressures acting on the cube are the three normal pressures $$P, Q, R$$, and the three tangential pressures $$S, T, U$$. It may be shown, by an analytical method that need not be given, that if a small sphere be described about a point in the body and the pressures applied to its surfaces examined, there will be three radii at right angles to each other, at the extremities of which the pressures are normal to the surface of the sphere. These three directions are called the principal axes of stress.