Introduction

Material or matter is composed of discrete molecules, which in turn are made up of atoms. An atom consists of negatively charged electrons, positively charged protons, and neutrons. Electrons form chemical bonds. The study of matter at molecular or atomistic levels is very useful for understanding a variety of phenomena, but studies at these scales are not useful to solve common engineering problems. Continuum mechanics is concerned with a study of various forms of matter at macroscopic level. Central to this study is the assumption that the discrete nature of matter can be overlooked, provided the length scales of interest are large compared with the length scales of discrete molecular structure. Thus, matter at sufficiently large length scales can be treated as a continuum in which all physical quantities of interest, including density, are continuously differentiable.

Engineers and scientists undertake the study of continuous systems to understand their behavior under “working conditions,” so that the systems can be designed to function properly and produced economically. For example, if we were to repair or replace a damaged artery in human body, we must understand the function of the original artery and the conditions that lead to its damage.

Background and Overview

In the mathematical description of equations governing a continuous medium, we derive relations between various quantities that characterize the stress and deformation of the continuum by means of the laws of nature (such as Newton's laws, conservation of energy, and so on). As a means of expressing a natural law, a coordinate system in a chosen frame of reference is often introduced. The mathematical form of the law thus depends on the chosen coordinate system and may appear different in another type of coordinate system. The laws of nature, however, should be independent of the choice of a coordinate system, and we may seek to represent the law in a manner independent of a particular coordinate system. A way of doing this is provided by vector and tensor analysis. When vector notation is used, a particular coordinate system need not be introduced. Consequently, the use of vector notation in formulating natural laws leaves them invariant to coordinate transformations. A study of physical phenomena by means of vector equations often leads to a deeper understanding of the problem in addition to bringing simplicity and versatility into the analysis.

Introduction

Virtually every phenomenon in nature, whether mechanical, biological, chemical, geological, or geophysical, can be described in terms of mathematical relations among various quantities of interest. Most mathematical models of physical phenomena are based on fundamental scientific laws of physics that are extracted from centuries of research on the behavior of mechanical systems subjected to the action of natural forces. What is most exciting is that the laws of physics, which are also termed principles of mechanics, govern biological systems as well (because of mass and energy transports). However, biological systems may require additional laws, yet to be discovered, from biology and chemistry to complete their description.

This chapter is devoted to the study of fundamental laws of physics as applied to mechanical systems. The laws of physics are expressed in analytical form with the aid of the concepts and quantities introduced in previous chapters. The laws or principles of physics that we study here are (1) the principle of conservation of mass, (2) the principle of conservation of linear momentum, (3) the principle of conservation of angular momentum, and (4) the principle of conservation of energy.

Introduction

The kinematic relations developed in Chapter 3, and the principles of conservation of mass and momenta and thermodynamic principles discussed in Chapter 5, are applicable to any continuum irrespective of its physical constitution. The kinematic variables such as the strains and temperature gradient, and kinetic variables such as the stresses and heat flux were introduced independently of each other. Constitutive equations are those relations that connect the primary field variables (e.g., ρ, T, x, and u or v) to the secondary field variables (e.g., e, q, and σ). Constitutive equations are not derived from any physical principles, although they are subject to obeying certain rules and the entropy inequality. In essence, constitutive equations are mathematical models of the behavior of materials that are validated against experimental results. The differences between theoretical predictions and experimental findings are often attributed to inaccurate representation of the constitutive behavior.

First, we review certain terminologies that were already introduced in beginning courses on mechanics of materials. A material body is said to be homogeneous if the material properties are the same throughout the body (i.e., independent of position). In a heterogeneous body, the material properties are a function of position.

Introduction

This chapter is dedicated to the study of deformation and stress in solid bodies under a prescribed set of forces and kinematic constraints. We assume that stresses and strains are small so that linear strain–displacement relations and Hooke's law are valid, and we use appropriate governing equations, called field equations, derived in the previous chapters. Mathematically, we seek solutions to coupled partial differential equations over an elastic domain occupied by the reference (or undeformed) configuration of the body, subject to specified boundary conditions on displacements and forces. Such problems are called boundary value problems of elasticity.

Most practical problems of even linearized elasticity involve geometries that are complicated and analytical solutions to such problems cannot be obtained. Therefore, the objective here is to familiarize the reader with the certain solution methods as applied to simple boundary value problems. Problems discussed in most elasticity books are about the same and they illustrate the methodologies used in the analytical solution of problems of elasticity. Since this is a book on continuum mechanics, the coverage is some what limited. Most problems discussed here can be found in elasticity books, for example, by Timoshenko and Goodier (1970) and Slaughter (2002).

Introduction

In the beginning of Chapter 3, we have briefly discussed the need to study deformation and stresses in material systems that we may design for engineering applications. All materials have certain threshold to withstand forces, beyond which they “fail” to perform their intended function. The force per unit area, called stress, is a measure of the capacity of the material to carry loads, and all designs are based on the criterion that the materials used have the capacity to carry the working loads of the system. Thus, it is necessary to determine the state of stress in a material.

In the present chapter, we study the concept of stress and its various measures. For instance, stress can be measured per unit deformed area or undeformed area. As we shall see shortly, stress at a point in a three-dimensional continuum can be measured in terms of nine quantities, three per plane, on three mutually perpendicular planes at the point. These nine quantities may be viewed as the components of a second-order tensor, called stress tensor. Coordinate transformations and principal values associated with the stress tensor and stress equilibrium equations will also be discussed.

Cauchy Stress Tensor and Cauchy's Formula

First we introduce the true stress, that is, stress in the deformed configuration κ that is measured per unit area of the deformed configuration κ.

Many of the mathematical models of natural phenomena are based on fundamental scientific laws of physics or otherwise are extracted from centuries of research on the behavior of physical systems under the action of natural forces. Today this subject is referred to simply as mechanics – a phrase that encompasses broad fields of science concerned with the behavior of fluids, solids, and complex materials. Mechanics is vitally important to virtually every area of technology and remains an intellectually rich subject taught in all major universities. It is also the focus of research in departments of aerospace, chemical, civil, and mechanical engineering, in engineering science and mechanics, and in applied mathematics and physics. The past several decades have witnessed a great deal of research in continuum mechanics and its application to a variety of problems. As most modern technologies are no longer discipline-specific but involve multidisciplinary approaches, scientists and engineers should be trained to think and work in such environments. Therefore, it is necessary to introduce the subject of mechanics to senior undergraduate and beginning graduate students so that they have a strong background in the basic principles common to all major engineering fields. A first course on continuum mechanics or elasticity is the one that provides the basic principles of mechanics and prepares engineers and scientists for advanced courses in traditional as well as emerging fields such as biomechanics and nanomechanics.