Thermodynamics of stressed systems

As every school child knows, the difference between a solid and a liquid is that a liquid takes the shape of the container in which it is placed while the shape of a solid is independent of the shape of the container (providing the container is big enough). In other words, we must apply a force in order to change the shape of a solid. However, the thermodynamic functions described heretofore have no terms that depend on shape. In this chapter, we extend the thermodynamics discussed above to include such effects and therefore make it applicable to solids. However, since this is a thermodynamics, rather than a mechanics text, we focus more on the relationship between stress and thermodynamics rather than on a general description of the mechanical properties of solids. We start out discussion of mechanical deformation by describing the change of shape of a solid. We define the displacement vector at any point in the solid u(x, y, z) as the change in location of the material point (x, y, z) upon deformation: that is, ux(x, y, z) = x' - x, where the prime indicates the coordinates of the material that was at the unprimed position prior to the deformation. In linear elasticity, we explicitly assume that the displacement vector varies slowly from point to point within the solid where i and j denote the directions along the three axes, x, y, and z. Consider the small parallel-piped section of a solid with perpendicular edges shown in Fig. 7.1. We label the first corner as O, located at position (xO, yO, zO) and subsequent corners as A, B, . . . located at positions (xA, yA, zA), (xB, yB, zB), . . . The edge lengths are Δx, Δy, and Δz such that, for example, xA = xO + Δx. As a result of the deformation, the material originally at point O is displaced to point O' with coordinates (x'O, y'O, z'O).