ABSTRACT
Elasticity is perhaps the most successful theory ever developed to model the mechanical response of solids. Many important fields in solid mechanics, such as fracture mechanics and the theory of dislocation, are developed from the firm basis of the mathematical theory of elasticity. Thus, the theory of elasticity itself is a pre-requisite for and provides a fundamental background to any graduate student who wants to study more advanced topics in geomechanics or in engineering in general. The mathematical theory of elasticity had occupied the minds of great scientists since the time of Galileo in the seventeenth century. Despite its development in the last 360 years, research on elasticity remains active today; it is fair to say that many theoretical and practical problems in elasticity remain to be solved. Elasticity, in general, can be regarded as a branch of science that deals with the mechanical deformations of solids that deform under applied loads, such as applied traction, displacements, and temperature gradients, and then are able to recover their original shape upon unloading. The scope of elasticity can be extremely wide, depending on the type of elastic solids, the types of loading, and the form of deformation that we are interested in. There are about 100 textbooks directly devoted to or closely related to the theory of elasticity. The solids can be modeled as one, two or three dimensional although most analytic solutions exist only for one-or two-dimensional problems. The loading can be time dependent such that inertia effect may play a key role; for example, wave propagation is a typical phenomenon due to dynamic loads. Wave propagations and soil dynamics will be discussed in Chapter 9. Compared to the dimensions of the original solid, the deformations and strains in the body can be either infinitesimally small or finite. The deformation of solids may be either proportional to the applied loads (linear elasticity) or nonproportional to the loads (nonlinear elasticity). When nonlinear constitutive behavior sets in or when large deformation and strain are allowed to occur, the uniqueness theorem fails. The solution for elasticity problems becomes the result of solving nonlinear differential equations, which, in general, cannot be solved analytically. Approximate techniques, such as perturbation, have been developed to solve such problems. Numerical methods for solving these problems may yield unreliable solutions if no special care is taken. At certain critical loading conditions a trivial solution may yield to a nontrivial solution (this is normally called bifurcation in mathematical terms). Bifurcation problems in geomechanics have also been considered quite extensively (e.g., Rudnicki and Rice, 1975; Chau and Rudnicki, 1990, Chau, 1992, 1993, 1994a, 1995ac, 1998a, 1999b; Chau and Choi, 1998; Muhlhaus et al., 1996;
Vardoulakis, 1979, 1983; Sulem and Vardoulakis, 1990; Vardoulakis and Sulem, 1996; Bigoni, 2012). In addition, the constitutive response of solids may be loading direction dependent; that is, the solid is not isotropic or it is anisotropic in response. For example, specimens taken horizontally from anisotropic solids will deform differently from those taken vertically. To date, most of the analytical solutions exist only for isotropic solids. Due to recent development of composite materials for the aerospace industry and other research, much effort has focused on the mechanical behavior of anisotropic elastic solids (e.g., Chau, 1994b, 1998b). In terms of the existence of strain energy, elasticity can further be divided into two groups: hyperelasticity, for solids having an elastic potential or strain energy function; and hypoelasticity, for solids attaining a linear relationship between strain rate and stress rate. Strictly speaking, hyperelasticity is more than elasticity, which simply requires the recovery of strain and deformation upon unloading; on the other hand, hypoelasticity is less than elastic since it does not even require proportionality between stress and strain. This chapter covers only problems with small deformations (i.e., linear elasticity). Both isotropic and anisotropic solids will be discussed, but the focus will mainly be on isotropic solids. Some practical examples will be used to illustrate the power of elasticity. The application of elasticity to model dislocation will also be introduced.
