Matrix, vector, and tensor algebras are often used in the theory of continuum mechanics in order to have a simpler and more tractable presentation of the subject. In this chapter, the mathematical preliminaries required to understand the matrix, vector, and tensor operations used repeatedly in this book are presented. Principles of mechanics and approximation methods that represent the basis for the formulation of the kinematic and dynamic equations developed in this book are also reviewed in this chapter. In the first two sections of this chapter, matrix and vector notations are introduced and some of their important identities are presented. Some of the vector and matrix results are presented without proofs because it is assumed that the reader has some familiarity with matrix and vector notations. In Section 3, the summation convention, which is widely used in continuum mechanics texts, is introduced. This introduction is made despite the fact that the summation convention is rarely used in this book. Tensor notations, on the other hand, are frequently used in this book and, for this reason, tensors are discussed in Section 4. In Section 5, the polar decomposition theorem, which is fundamental in continuum mechanics, is presented. This theorem states that any nonsingular square matrix can be decomposed as the product of an orthogonal matrix and a symmetric matrix. Other matrix decompositions that are used in computational mechanics are also discussed. In Section 6, D'Alembert's principle is introduced, while Section 7 discusses the virtual work principle.

In this chapter, the general kinematic equations for the continuum are developed. The kinematic analysis presented in this chapter is purely geometric and does not involve any force analysis. The continuum is assumed to undergo an arbitrary displacement and no simplifying assumptions are made except when special cases are discussed. Recall that in the special case of an unconstrained three-dimensional rigid-body motion, six independent coordinates are required in order to describe arbitrary rigid-body translation and rotation displacements. The general displacement of an infinitesimal material volume on a deformable body, on the other hand, can be described in terms of twelve independent variables; three translation parameters, three rigid-body rotation parameters, and six deformation parameters. One can visualize these modes of displacements by considering a cube that may undergo an arbitrary displacement. The cube can be translated in three independent orthogonal directions (translation degrees of freedom), it can be rotated as a rigid body about three orthogonal axes, and it can experience six independent modes of deformation. These deformation modes are elongations or contractions in three different directions and three shear deformation modes. It is shown in this chapter that the rotations and the deformations can be completely described using the matrix of the position vector gradients, which in general has nine independent elements. This fact can be mathematically proven using the polar decomposition theorem discussed in the preceding chapter. The deformation at the material points on the body can be described in terms of six independent strain components.

The analysis of plastic deformation is important in many engineering applications including crashworthiness, impact analysis, manufacturing problems, among many others. When materials undergo plastic deformations, permanent strains are developed when the load is removed. Many materials exhibit elastic–plastic behaviors, that is, the material exhibits elastic behavior up to a certain stress limit called the yield strength after which plastic deformation occurs. If the stress of elastic-plastic materials depends on the strain rate, one has a rate-dependent material, otherwise the material is called rate independent. In the classical plasticity analysis of solids, a nonunique stress–strain relationship that is independent of the rate of loading but does depend on the loading sequence is used (Zienkiewicz and Taylor, 2000). In rate-dependent plasticity, on the other hand, the stress–strain relationship depends on the rate of the loading.

The yield strength of elastic–plastic materials can increase after the initial yield. This phenomenon is known as strain hardening. In the theory of plasticity, there are two types of strain hardening, isotropic and kinematic hardening. In the case of isotropic hardening, the yield strength changes as the result of the plastic deformation. In the case of kinematic hardening, on the other hand, the center of the yield surface experiences a motion in the direction of the plastic flow. The kinematic hardening behavior is closely related to a phenomenon known as the Bauschinger effect, which is the result of a reduction in the compressive yield strength following an initial tensile yield.

In the preceding chapters, the general nonlinear continuum mechanics theory was presented. In order to make use of this theory in many practical applications, a finite dimensional model must be developed. In this model, the partial differential equations of equilibrium are written using approximation methods as a finite set of ordinary differential equations. One of the most popular approximation methods that can be used to achieve this goal is the finite element method. In this method, the spatial domain of the body is divided into small regions called elements. Each element has a set of nodes, called nodal points that are used to connect this element with other elements used in the discretization of the body. The displacement of the material points of an element is approximated using a set of shape functions and the displacements of the nodes and possibly their derivatives with respect to the spatial coordinates. In this case, the dimension of the problem depends on the number of nodes and number and type of the nodal coordinates used.

In the literature, there are many finite element formulations that are developed for the deformation analysis of mechanical, aerospace, structural, and biological systems. Some of these formulations are developed for small-deformation and small-rotation linear problems, some for large-deformation and large-rotation nonlinear analysis, and the others for large-rotation and small-deformation nonlinear problems. Several numerical solution procedures and computational algorithms are also proposed for solving the resulting system of finite element differential equations.

Nonlinear continuum mechanics is one of the fundamental subjects that form the foundation of modern computational mechanics. The study of the motion and behavior of materials under different loading conditions requires understanding of basic, general, and nonlinear, kinematic and dynamic relationships that are covered in continuum mechanics courses. The finite element method, on the other hand, has emerged as a powerful tool for solving many problems in engineering and physics. The finite element method became a popular and widely used computational approach because of its versatility and generality in solving large-scale and complex physics and engineering problems. Nonetheless, the success of using the continuum-mechanics-based finite element method in the analysis of the motion of bodies that experience general displacements, including arbitrary large rotations, has been limited. The solution to this problem requires resorting to some of the basic concepts in continuum mechanics and putting the emphasis on developing sound formulations that satisfy the principles of mechanics. Some researchers, however, have tried to solve fundamental formulation problems using numerical techniques that lead to approximations. Although numerical methods are an integral part of modern computational algorithms and can be effectively used in some applications to obtain efficient and accurate solutions, it is the opinion of many researchers that numerical methods should only be used as a last resort to fix formulation problems. Sound formulations must be first developed and tested to make sure that these formulations satisfy the basic principles of mechanics.

In the theory of continuum mechanics, stresses are used as measures of the forces and pressures. As in the case of strains, different definitions can be used for the stresses; some of these definitions are associated with the reference configuration, whereas the others are associated with the current deformed configuration. The effect of the forces on the body dynamics can only be taken into consideration by using both stresses and strains. These stress and strain components must be defined in the same coordinate system in order to have a consistent formulation. In this chapter, several stress measures are introduced and the relationship between them is discussed. The Cauchy stress formula is presented and used to develop the partial differential equations of equilibrium of the continuous body. The equations of equilibrium are used to develop an expression for the virtual work of the stress forces expressed in terms of the stress and strain components. The objectivity of the stress rate and the energy balance equations are also among the topics that will be discussed in this chapter.

EQUILIBRIUM OF FORCES

In this section, the equilibrium of forces acting in the interior of a continuous body is considered. Let P be a point on the surface of the body, n be a unit vector directed along the outward normal to the surface at P, and be the area of an element of the surface that contains P in the current configuration.

This book is designed for a one- or two-quarter course in continuum mechanics for first-year graduate students and advanced undergraduates in the mathematical and engineering sciences. It was developed, and continually improved, by over four years of teaching of a graduate engineering course (ME 238) at Stanford University, USA, followed by over four years of teaching of an advanced undergraduate mathematics course (MA3G2) at the University of Warwick, UK. The resulting text, we believe, is suitable for use by both applied mathematicians and engineers. Prerequisites include an introductory undergraduate knowledge of linear algebra, multivariable calculus, differential equations and physics.

This book is intended both for use in a classroom and for self-study. Each chapter contains a wealth of exercises, with fully worked solutions to odd-numbered questions. A complete solutions manual is available to instructors upon request. A short bibliography appears at the end of each chapter, pointing to material which underpins, or expands upon, the material discussed here. Throughout the book we have aimed to strike a balance between two classic notational presentations of the subject: coordinate-free notation and index notation. We believe both types of notation are helpful in developing a clear understanding of the subject, and have attempted to use both in the statement, derivation and interpretation of major results. Moreover, we have made a conscious effort to include both types of notation in the exercises.

Chapters 1 and 2 provide necessary background material on tensor algebra and calculus in three-dimensional Euclidean space.

In this chapter we state various axioms which form the basis for a thermo-mechanical theory of continuum bodies. These axioms provide a set of balance laws which describe how the mass, momentum, energy and entropy of a body change in time under prescribed external influences. We first state these laws in global or integral form, then derive various corresponding local statements, primarily in the form of partial differential equations. The balance laws stated here apply to all bodies regardless of their constitution. In Chapters 6–9 these laws are specialized to various classes of bodies with specific material properties, via constitutive models.

The important ideas in this chapter are: (i) the balance laws of mass, momentum, energy and entropy for continuum bodies; (ii) the difference between the integral form of a law and its local Eulerian and Lagrangian forms; (iii) the axiom of material frame-indifference and its role in constitutive modeling; (iv) the idea of a material constraint and its implications for the stress field in a body; (v) the balance laws relevant to the isothermal modeling of continuum bodies.

Motivation

In order to motivate the contents of this chapter it is useful to recall some basic ideas from the mechanics of particle systems. To this end, we consider a system of N particles with masses mi and positions xi as illustrated in Figure 5.1. It will be helpful to think of these particles as the atoms making up a continuum body.