Equilibrium, Universal Solutions, and Inflation

Abstract

Let us begin by recalling three important observations from Chapters 1 and 2. First, equilibrium requires that ΣF = 0 and ΣM = 0. Second, if a body is in equilibrium, then each of its parts are likewise in equilibrium. Third, there may exist at each point p in a body (cf. Fig. 2.4) nine components of stress, six of which are independent, which we denote as σ_{(face)(direction)} relative to the coordinate system of choice. Because stress may vary from point to point within a body, the components at a nearby point q may have different values. (Note: It is usually convenient to refer components at different points to the same coordinate system.) Now, if we consider a small cube of material, centered about point p which is located at (x, y, z) and has stresses σ _xx , σ _xy , ..., σ _zz , then the stresses on the faces of the cube may differ from those at the center; that is, if the xx component at the center of the cube is σ _xx , then on the positive and negative faces of the cube, at distances ±Δx/2 from the center, we may have σ _xx + Δσ _xx and σ _xx − Δσ _xx , respectively (i.e., values slightly greater than or less than that at point p).