ABSTRACT
Real solids are always three dimensional (3-D) in nature; however, most textbooks on elasticity have been restricted to two-dimensional (2-D) situations (either plane strain or plane stress). The main reason is that the equations of motion or equilibrium (the so-called Navier equation in displacement formulation) or the compatibility equations (or the BeltramiMichell equation in stress formulation) for 3-D solids are difficult to solve. This chapter introduces the method of solution for 3-D elasticity. More specifically, three coupled differential equations have to be solved for three unknown variables in the displacement formulation. Two major approaches have been adopted to uncouple these equations by either introducing displacement potentials or stress functions. Even though the resulting governing equations for displacement potentials or stress functions are uncoupled, the method of solutions for them is by no means straightforward even for simple practical problems. As discussed in Chapter 2, plane stress or plane strain condition is normally assumed to idealize real situations. The problem of 2-D elasticity is much simpler than 3-D elasticity. Even when numerical methods (such as the finite element method) are used to solve real problems, 2-D idealization is usually adopted. In geomechanics, 3-D solutions are, however, essential in engineering applications. Examples include the Kelvin problem (point force in a full space), Boussinesq’s problem (vertical surface point force applied on an elastic half-space), Cerruti’s problem (horizontal surface point force applied on an elastic half-space), and Mindlin’s problem (point force in an elastic half-space). These solutions have been used extensively to generate solutions for other practical problems. For example, the widely used Newmark influence charts and Fadum charts in soil mechanics were both obtained by superimposing (or integrating) the solution of Boussinesq’s problem. These solutions also provide the fundamental solutions to the Green’s method, the body force method, the boundary integral equation method, and the boundary element method. Because of its mathematical complexity, the solution technique in solving these fundamental solutions is not covered in most textbooks in geomechanics or in elasticity. For example, the classical textbook on elasticity by Timoshenko and Goodier did not include a complete treatment on the method of solutions for 3-D elastic solid. Only some ad hoc 3-D elastic problems were considered. We believe, however, that the method of solutions for 3-D elasticity is of utmost importance and must be covered in a textbook on geomechanics or elasticity. This is the purpose of this chapter.
