Introduction

In contrast to the study of solid mechanics, in which we were preoccupied with deformed geometries, our main concern in fluid mechanics is to view fluid particles in motion as continua and to determine the velocities at any given point in space (Navier, 1821; Lamb, 1879; Lamb, 1959; Milne-Thompson, 1960; Robertson, 1965). For this purpose we invoke the Eulerian coordinates whose properties were detailed in Subsection 2.2.2. Most fluids, whether gases or liquids, are called Newtonian fluids, in which the stress tensor is linearly proportional to the velocity gradients. On the other hand, certain chemical fluids, polymers, rheologically complex fluids, and suspensions, among others, are referred to as non-Newtonian fluids, in which the stress tensor is nonlinearly proportional to the velocity gradients. Non-Newtonian fluid mechanics is discussed in Section 7.2.

The subject of Newtonian fluids is generally referred to as fluid mechanics, which encompasses diverse topics, such as the motion of airplanes and missiles through the atmosphere, satellites through the outer atmosphere, submarines and ships through water, the flow of liquids and gases through ducts, the transfer of heat and mass by fluid motion, propagation of sound through gases and liquids, the study of ocean waves and tides, the study of air masses in the atmosphere, astrophysical, geophysical, and meteorological problems, and reacting fluids.

Fluids, like all matter, are made up of molecules. Thus the properties of fluid motion, such as those observed, may be studied on the basis of the mechanics of the molecules that compose the fluid.

General

The notion of stress originates from the need to quantify internal or external forces distributed, respectively, in a body or along its boundary in equilibrium. Body forces such as gravity act inside the body whereas surface forces act on its bounding surface. Stresses are those forces distributed over an infinitesimal unit area cut out of a body in certain directions or over an infinitesimal unit area on the bounding surface. Stresses may also arise from hydrodynamic pressure and/or velocity gradients in a fluid. Furthermore, changes in temperature in solids or fluids give rise to stresses.

Stresses may be related to strains in solids or rates of deformation in fluids through constitutive laws. Stresses are described in many different ways, depending, for example, on the coordinate systems used, the magnitudes of the strains (in solids) or the velocity gradients (in fluids), or the types of substances involved. The basic groundwork for stresses is developed in this chapter, but related subjects are discussed throughout the remainder of this book. The study of stress-related problems, in general, is referred to as kinetics for the body in equilibrium.

Discussions in this chapter include definitions of forces and stresses, conservation laws of linear and angular momentum, coordinate transformations of stresses, and deviatoric stresses. These topics, together with the concept of kinematics discussed in Chap. 2, constitute the basis for the theories of elasticity in Chap. 4 and fluid mechanics in Chap. 5.

Continuum mechanics is one of the most important interdisciplinary studies in our pursuit toward physical phenomena and mathematical aspects of universal laws. In the past, however, continuum mechanics was regarded as a discipline mainly associated with various branches of engineering dictated by Newton's theory of mechanics. As such, one focuses on solid mechanics and fluid mechanics governed by the speed of sound. However, the aim of this book is to include in continuum mechanics the physical phenomena governed by the speed of light as well. Doing so involves Maxwell's electromagnetic continuum and Einstein's relativistic spacetime continuum.

My previous books Continuum Mechanics (Prentice-Hall, 1988) and Applied Continuum Mechanics (Cambridge University Press, 1996) were written primarily for engineers. The present book is intended for a broader audience, including engineers, physicists, and applied mathematicians. Although the material presented is diversified, my desire is to address the importance of unifying and placing in perspective the concept of mass, momentum, and energy associated with all physical phenomena on earth and beyond, thus assisting the reader to be interdisciplinary.

This book is divided into two parts: Part I, Basic Topics, and Part II, Special Topics. Part I covers mathematical preliminaries, kinematics, kinetics, linear elasticity, and Newtonian fluid mechanics. Only Cartesian tensors are used in Part I. Part II includes curvilinear continuum, nonlinear continuum, electromagnetic continuum, and differential geometry continuum. Curvilinear tensors are used extensively in Part II.

The attempt was made in this book to show the analogy between the physical behavior of motions of solids and fluids on earth and that of astrophysical objects in the universe. We saw the conservation laws arising from the thermodynamics laws for solids in Chap. 4 and for fluids in Chap. 5 in a very similar fashion. The curvilinear continuum of Chap. 6 was extended to nonlinear mechanics for large deformations in Chap. 7. We witnessed the electromagnetic continuum of Chap. 8 combined with the spacetime geometries to arrive at the relativistic gravitohydromagnetics in Chap. 9. Differential geometries with curvilinear continuum introduced in Chap. 6 were applied in Chap. 9 to shell geometries for engineering and to spacetime geometries of Einstein's theory of relativity. Thus we have seen actions of nature on earth and throughout the universe under one roof, as called in this book, General Continuum Mechanics. Thus the energy and momentum contained in a solid bar and in fluids under external disturbances are controlled by the same principles: the first and second laws of thermodynamics. Similarly, deformations of a solid shell in engineering on earth and spacetimes of Einstein's theory of relativity in the universe are controlled by the same principles: the first and second fundamental tensors. They all belong to the mechanics of continuum.

Equations of strains and stresses as well as all governing equations of solids and fluids may be written in curvilinear coordinates for curved geometries such as in cylinders, spheres, and large deformation problems. Mathematics involving curvilinear coordinates originated with Riemann in the middle of the 19th century and subsequently was used by Einstein in the early 20th century in describing curved spacetime geometries of the universe for the theory of relativity. The curvilinear coordinates were then utilized in engineering to describe the large deformations of solids (nonlinear elasticity), viscous fluids (non-Newtonian fluid dynamics), and differential geometries (shell structures).

The basic formulations and geometries of curvilinear continuum are employed in nonlinear continuum presented in Chap. 7 and differential geometry continuum discussed in Chap. 9, including shell theories and relativity theories. Tensor analysis in curvilinear continuum, nonlinear continuum, and differential geometry continuum presented in Part II is more rigorous than in Part I. References for tensor analysis include Erickson (1960), Eringen (1962, 1967), and Gurtin (1981), among others.

Tensor Properties of Curvilinear Continuum

General

Recall that tangent vectors and metric tensors associated with deformed states were discussed in Section 2.2. In the analysis of cylinders or spheres, the undeformed state itself must be non-Cartesian. Figure 6.1.1 shows the use of a general Lagrangian non-Cartesian or curvilinear coordinate system for the undeformed state and its convected coordinates for the deformed state.

Tangent Vectors

The tangent vectors (base vectors) g i are drawn tangent to the initial undeformed curvilinear coordinates ξi.

On learning that a new text on quantum field theory has appeared, one is surely tempted to respond with Isidor Rabi's famous comment about the muon: “Who ordered that?” After all, many excellent textbooks on quantum field theory are already available. I, for example, would not want to be without my well-worn copies of Quantum Field Theory by Lowell S. Brown (Cambridge 1994), Aspects of Symmetry by Sidney Coleman (Cambridge 1985), Introduction to Quantum Field Theory by Michael E. Peskin and Daniel V. Schroeder (Westview 1995), Field Theory: A Modern Primer by Pierre Ramond (Addison-Wesley 1990), Fields by Warren Siegel (arXiv.org 2005), The Quantum Theory of Fields, Volumes I, II, and III, by Steven Weinberg (Cambridge 1995), and Quantum Field Theory in a Nutshell by my colleague Tony Zee (Princeton 2003), to name just a few of the more recent texts. Nevertheless, despite the excellence of these and other books, I have never followed any of them very closely in my twenty years of on-and-off teaching of a year-long course in relativistic quantum field theory.

As discussed in the Preface for Students, this book is based on the notion that quantum field theory is most readily learned by starting with the simplest examples and working through their details in a logical fashion. To this end, I have tried to set things up at the very beginning to anticipate the eventual need for renormalization, and not be cavalier about how the fields are normalized and the parameters defined.