ABSTRACT
Continuum mechanics has been very successful in modeling physical phenomena in geomaterials, such as rock, soil, and concrete. Elasticity, plasticity, fracture mechanics, damage mechanics, viscoelasticity, and poroelasticity all can be considered branches of continuum mechanics that have found applications in geomechanics. Continuum mechanics, in fact, also includes fluid mechanics, but we will not deal with this aspect in this book. In order to understand and apply continuum mechanics more efficiently, tensor notation and analysis have been developed as the basic mathematical language for communication. This chapter deals only with elementary tensor analysis. The physical laws, if they really describe the physical world, should be independent of the position and orientation of observers, that is, independent of the coordinates used in describing these phenomena. For this reason, physical laws are ideally written in tensor equations because tensor equations are invariant under coordinate transformation. If a tensor equation holds in one coordinate system, it also holds in any other coordinate system in the same reference frame. As we will see in later sections, many physical laws (such as the equation of equilibrium) in terms of a special coordinate system (such as a cylindrical or a spherical coordinate system) can be obtained by simply specializing the tensor equation to its component form. This coordinate-invariant property makes tensor analysis a very attractive technique for analysis of geomechanics problems. The physical quantities involved in the formulation of continuum mechanics, such as displacement, stress, strain, and modulus of elasticity, are more conveniently referred to as tensors. Mathematically, such tensors can either be expressed in polyadic or indicial forms. Tensor has its existence independent of any coordinate system, yet when it is specified in a particular coordinate system, it contains certain sets of quantities called components, identified by free index (or indices). Nowadays, technical papers in geomechanics or continuum mechanics are very often written in terms of tensors, taking advantage of their conciseness property. Tensor analysis, therefore, becomes a pre-requisite for any graduate student who wants access to the state-ofthe-art information available in journal publications and advanced textbooks. This chapter will give a concise treatment of elementary tensor analysis for an orthogonal coordinate system, with particular reference to applications in geomechanics.
