In the previous chapter, we derived kinematic fields to describe the possible deformed configurations of a continuous medium. These fields on their own cannot predict the configuration a body will adopt as a result of a given applied loading. To do so requires a generalization of the laws of mechanics (originally developed for collections of particles) to a continuous medium, together with an application of the laws of thermodynamics. The result is a set of universal conservation and balance laws that apply to all bodies:

1. conservation of mass;

2. balance of linear and angular momentum;

3. thermal equilibrium (zeroth law of thermodynamics);

4. conservation of energy (first law of thermodynamics);

5. second law of thermodynamics.

These equations introduce four new important quantities to continuum mechanics. The concept of stress makes its appearance in the derivation of the momentum balance equations. Temperature, internal energy and entropy star in the zeroth, first and second laws, respectively. In this chapter we focus on the mechanical conservation laws (mass and momentum) leaving the thermodynamic laws to the next chapter.

Conservation of mass

A basic principle of classical mechanics is that mass is a fixed quantity that cannot be formed or destroyed, but only deformed by applied loads. Thus, the total amount of mass in a closed system is conserved. For a system of particles this is a trivial statement that requires no further clarification.

In this final chapter of Part I, we discuss the formulation and specification of well-defined problems in continuum mechanics. For simplicity, we restrict our attention to the purely mechanical behavior of materials. This means that, unless otherwise explicitly stated, in this chapter we will ignore thermodynamics. The resulting theory provides a reasonable approximation of real material behavior in two extreme conditions. The first scenario is that of isentropic processes (see Section 6.2.5), where the motion and deformation occurs at such a high temporal rate that essentially no flow of heat occurs. In this scenario the strain energy density function should be associated with the internal energy density at constant entropy. The second scenario is that of isothermal processes (see Section 6.2.5), where the motion and deformation occurs at such a low temporal rate that the temperature is essentially uniform and constant. In this scenario the strain energy density function should be associated with the Helmholtz free energy density at constant temperature.

We start by discussing the specification of initial boundary-value problems in Section 7.1. Then, in Section 7.2 we develop the principle of stationary potential energy. Finally, in Section 7.3 we introduce the idea of stability and ultimately derive the principle of minimum potential energy.

Initial boundary-value problems

So far we have laid out an extensive set of concepts and derived the local balance laws to which continuous physical systems (which satisfy the various assumptions we have made along the way) must conform.

The rapid growth of computer power since the 1960s has been accompanied by a similarly rapid growth and development of computational methods, to the point where the stress analysis of complex components is a routine part of almost any engineering design. To demonstrate how continuum mechanics problems can be accurately and efficiently solved by an approximate numerical representation on a computer, we will focus on the solution of static problems in solid mechanics, and we will not consider the effects of temperature. While there is certainly no shortage of numerical techniques to solve fluid mechanics, heat transfer or other continuum problems, our focus on solids reflects the emphasis of this book in general. And while we will start out on a relatively general footing applicable to many of the computational techniques available for solid mechanics, our focus will be on the finite element method (FEM). This is because the FEM has clearly emerged as the most common and powerful approach for solid mechanics and materials science. Further, we view the FEM as a natural bridge between continuum mechanics and atomistic methods. In Part IV of the companion book to this one [TM11], we explicitly use it as a way to build multiscale models combining atomistic and continuum frameworks.

A perusal of Chapter 11 on Further Reading makes it clear that the FEM is a subject that can easily fill an entire book on its own.

In this chapter we study solutions to the equations of continuum mechanics instead of the equations themselves. In particular, our aim will be to obtain general equilibrium solutions to the field equations of continuum mechanics that are independent, in a specific sense, of the material from which a body is composed. Such solutions are of fundamental importance to the practical application of the theory of continuum mechanics. This is because they provide valuable guidance to the experimentalist who would like to design experiments for the determination of a particular material's constitutive relations. Generally, in an experiment it is only possible to control and measure (to a greater or lesser extent) the tractions and displacements associated with the boundary of the body being studied. From this information one would like to infer the stress and deformation fields within the body and ultimately extract the functional form of the material's constitutive relations and the values of any coefficients belonging to this functional form. However, if the interior stress and deformation fields explicitly depend on the functional form of the constitutive relations, then it is essentially impossible to infer this information from a practical experiment.

According to Saccomandi [Sac01], a deformation which satisfies the equilibrium equations with zero body forces and is supported by suitable surface tractions alone is called a controllable solution. A controllable solution that is the same for all materials in a given class is a universal solution.

This appendix presents a brief summary of the main continuum mechanics and thermodynamics equations derived in Part I to serve as a quick reference. Each entry includes the relevant equation number in the main text, the equation in both indicial and invariant form (where applicable) and a brief description. The reader is referred back the text for details of the derivation and variables appearing in the equations.

A solid material subjected to mechanical and thermal loading will change its shape and develop internal stress and temperature variations. What is the best way to describe this behavior? In principle, the response of a material (neglecting relativistic effects) is dictated by that of its atoms, which are governed by quantum mechanics. Therefore, if we could solve Schrödinger's equation for all of the atoms in the material (there are about 1022=10 000 000 000 000 000 000 000 atoms in a gram of copper) and evolve the dynamics of the electrons and nuclei over “macroscopic times” (i.e. seconds, hours and days), we would be able to predict the material behavior. Of course, when we say “material,” we are already referring to a very complex system. In order to predict the response of the material we would first have to construct the material structure in the computer, which would require us to use Schrödinger's equation to simulate the process by which the material was manufactured. Conceptually, it may be useful to think of materials in this way, but we can quickly see the futility of the approach: the state of the art of quantum calculations involves just hundreds of atoms over a time of nanoseconds.

Fortunately, in many cases it is not necessary to keep track of all the atoms in a material to describe its behavior. Rather, the overall response of such a collection of atoms is often much more readily amenable to an elegant, mathematical description.

Continuum mechanics seeks to provide a fundamental model for material response. It is sensible to require that the predictions of such a theory should not depend on the irrelevant details of a particular coordinate system. The key is to write the theory in terms of variables that are unaffected by such changes; tensors (or tensor fields) are the measures that have this property. Tensors come in different flavors depending on the number of spatial directions that they couple. The simplest tensor has no directional dependence and is called a scalar invariant to distinguish it from a simple scalar. A vector has one direction. For two directions and higher the general term tensor is used.

Tensors are tricky things to define. Many books define tensors in a technical manner in terms of the rules that tensor components must satisfy under coordinate system transformations. While certainly correct, we find such definitions unilluminating when trying to answer the basic question of “what is a tensor?”. In this chapter, we provide an introduction to tensors from the perspective of linear algebra. This approach may appear rather mathematical at first, but in the end it provides a far deeper insight into the nature of tensors.

Before we can begin the discussion of the definition of tensors, we must start by defining “space” and “time” and the related concept of a “frame of reference,” which underlie the description of all physical objects.

The suggestions for further reading given below are divided according to the two parts of the book: theory and solutions.

Books related to Part I on theory

There exists an impressive assortment of books addressing the topics contained in the first part of this book. Here we list either those books that have become standard references in the field, or titles that focus on specific aspects of the theory and therefore provide a deeper presentation than the relatively few pages of this book will permit.

• Readers interested in the connection between continuum mechanics and more fundamental microscopic theories of material behavior are referred to the companion book to this one, written by two of the authors, called Modeling Materials: Continuum, Atomistic and Multiscale Techniques and also published by Cambridge University Press [TM11]. That book includes a concise summary of the continuum theory presented in this book (which serves as a good abbreviated reference to the subject), followed by a discussion of atomistics (quantum mechanics, atomistic models of materials and molecular statics), atomistic foundations of continuum concepts (statistical mechanics, microscopic expressions for continuum fields and molecular dynamics) and multiscale methods (atomistic constitutive relations and computational techniques for coupling continuum and atomistics). [TM11] is consistent in spirit and notation with this book and is likewise targeted at a broad readership including chemists, engineers, materials scientists and physicists.

• Although published in 1969, Malvern's book [Mal69] continues to be considered the classic text in the field. It is not the best organized of books, but it is thorough and correct. It will be found on most continuum mechanicians' book shelves.

• […]

This book on Continuum Mechanics and Thermodynamics (CMT) (together with the companion book, by Tadmor and Miller, on Modeling Materials (MM) [TM11]) is a comprehensive framework for understanding modern attempts at modeling materials phenomena from first principles. This is a challenging problem because material behavior is dictated by many different processes, occurring on vastly different length and time scales, that interact in complex ways to give the overall material response. Further, these processes have traditionally been studied by different researchers, from different fields, using different theories and tools. For example, the bonding between individual atoms making up a material is studied by physicists using quantum mechanics, while the macroscopic deformation of materials falls within the domain of engineers who use continuum mechanics. In the end a multiscale modeling approach – capable of predicting the behavior of materials at the macroscopic scale but built on the quantum foundations of atomic bonding – requires a deep understanding of topics from a broad range of disciplines and the connections between them. These include quantum mechanics, statistical mechanics and materials science, as well as continuum mechanics and thermodynamics, which are the focus of this book.

Together, continuum mechanics and thermodynamics form the fundamental theory lying at the heart of many disciplines in science and engineering. This is a nonlinear theory dealing with the macroscopic response of material bodies to mechanical and thermal loading. There are many books on continuum mechanics, but we believe that several factors set our book apart. First, is our emphasis on fundamental concepts.

Continuum mechanics deals with the change of shape (deformation) of bodies subjected to external mechanical and thermal loads. However, before we can discuss the physical laws governing deformation, we must develop measures that characterize and quantify it. This is the subject described by the kinematics of deformation. Kinematics does not deal with predicting the deformation resulting from a given loading, but rather with the machinery for describing all possible deformations a body can undergo.

The continuum particle

A material body B bounded by a surface ∂B is represented by a continuous distribution of an infinite number of continuum particles. On the macroscopic scale, each particle is a point of zero extent much like a point in a geometrical space. It should therefore not be thought of as a small piece of material. At the same time, it has to be realized that a continuum particle derives its properties from a finite-size region ℓ on the micro scale (see Fig. 3.1). One can think of the properties of the particle as an average over the atomic behavior within this domain. As one moves from one particle to its neighbor the microscopic domain moves over, largely overlapping the previous domain. In this way the smooth field-like behavior we expect in a continuum is obtained. A fundamental assumption of continuum mechanics is that it is possible to define a length ℓ that is large relative to atomic length scales and at the same time much smaller than the length scale associated with variations in the continuum fields.

Thermodynamics is typically defined as a theory dealing with the flow of heat and energy between material systems. This definition is certainly applicable here, however, Callen provides (in his excellent book on the subject [Cal85]) an alternative definition that highlights another role that thermodynamics plays in continuum mechanics: “Thermodynamics is the study of the restrictions on the possible properties of matter that follow from the symmetry properties of the fundamental laws of physics.” In this chapter (and the next), we address both of these aspects of thermodynamic theory in the context of continuum mechanics.

The theory of thermodynamics boils down to three fundamental laws, deduced from empirical observation, that all physical systems are assumed to obey. The zeroth law of thermodynamics is related to the concept of thermal equilibrium. The first law of thermodynamics is a statement of the conservation of energy. The second law of thermodynamics deals with the directionality of thermodynamic processes. We will discuss each of these laws in detail, but first we describe the basic concepts in which thermodynamics is phrased.

For the purposes of thermodynamic analysis, the universe is divided into two parts: the system whose behavior is of particular interest, and the system's surroundings (everything else). The behavior of the surroundings is of interest only insofar as is necessary to characterize its interactions with the system.