
Continuum Mechanics
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Myron B. Allen, PhD, is Professor of Mathematics at the University of Wyoming, Consulting Editor for Wiley's Pure and Applied Mathematics book series, and past provost and vice president for Academic Affairs. His research interests include numerical analysis, mathematical modeling, and fluid mechanics in porous media.
Content
Preface v
1 Geometric Setting 1
1.1 Vectors and Euclidean Point Space 2
1.1.1 Vectors 2
1.1.2 Euclidean Point Space 6
1.1.3 Summary 8
1.2 Tensors 8
1.2.1 First-Order Tensors and Vectors 8
1.2.2 Second-Order Tensors 11
1.2.3 Cross Products, Triple Products, and Determinants 15
1.2.4 Orthogonal Tensors 20
1.2.5 Invariants of a Tensor 21
1.2.6 Derivatives of Tensor-Valued Functions 24
1.2.7 Summary 27
2 Kinematics I: The Calculus of Motion 29
2.1 Bodies, Motions, and Deformations 29
2.1.1 Deformation 32
2.1.2 Examples of Motions 33
2.1.3 Summary 36
2.2 Derivatives of Motion 36
2.2.1 Time Derivatives 37
2.2.2 Derivatives with Respect to Position 38
2.2.3 The Deformation Gradient 40
2.2.4 Summary 42
2.3 Pathlines, Streamlines, and Streaklines 43
2.3.1 Three Types of Arc 43
2.3.2 An Example 45
2.3.3 Summary 49
2.4 Integrals Under Motion 49
2.4.1 Arc, Surface, and Volume Integrals 49
2.4.2 Reynolds Transport Theorem 55
2.4.3 Summary 57
3 Kinematics II: Strain and its Rates 59
3.1 Strain 59
3.1.1 Symmetric Tensors 60
3.1.2 Polar Decomposition and the Deformation Gradient 64
3.1.3 Examples 66
3.1.4 Cauchy-Green and Strain Tensors 68
3.1.5 Strain Invariants 70
3.1.6 Summary 71
3.2 Infinitesimal Strain 72
3.2.1 The Infinitesimal Strain Tensor 72
3.2.2 Summary 75
3.3 Strain Rates 75
3.3.1 Stretching and Spin Tensors 76
3.3.2 Skew Tensors, Spin, and Vorticity 79
3.3.3 Summary 84
3.4 Vorticity and Circulation 84
3.4.1 Circulation 84
3.4.2 Summary 88
3.5 Observer Transformations 89
3.5.1 Changes in Frame of Reference 89
3.5.2 Summary 95
4 Balance Laws 97
4.1 Mass Balance 98
4.1.1 Local Forms of Mass Balance 99
4.1.2 Summary 102
4.2 Momentum Balance 102
4.2.1 Analysis of Stress 104
4.2.2 Inertial Frames of Reference 110
4.2.3 Momentum Balance in Referential Coordinates 113
4.2.4 Summary 114
4.3 Angular Momentum Balance 115
4.3.1 Symmetry of the Stress Tensor 117
4.3.2 Summary 118
4.4 Energy Balance 119
4.4.1 Thermal Energy Balance 122
4.4.2 Summary 124
4.5 Entropy Inequality 124
4.5.1 Motivation 125
4.5.2 Clausius-Duhem Inequality 126
4.5.3 Summary 127
4.6 Jump Conditions 127
4.6.1 Singular Surfaces 129
4.6.2 Localization 132
4.6.3 Summary 135
5 Constitutive Relations: Examples of Mathematical Models 137
5.1 Heat Transfer 138
5.1.1 Properties of the Heat Equation 140
5.1.2 Summary 142
5.2 Potential Theory 143
5.2.1 Motivation 143
5.2.2 Boundary Conditions 144
5.2.3 Uniqueness of Solutions to the Poisson Equation 146
5.2.4 Maximum Principle 147
5.2.5 Mean Value Property 150
5.2.6 Summary 151
5.3 Fluid Mechanics 152
5.3.1 Ideal Fluids 152
5.3.2 An Ideal Fluid in a Rotating Frame of Reference 154
5.3.3 Acoustics 155
5.3.4 Incompressible Newtonian Fluids 158
5.3.5 Stokes Flow 159
5.3.6 Summary 163
5.4 Solid Mechanics 164
5.4.1 Static Displacements 164
5.4.2 Elastic Waves 167
5.4.3 Summary 170
6 Constitutive Theory 173
6.1 Conceptual Setting 174
6.1.1 The Need to Close the System 174
6.1.2 Summary 176
6.2 Determinism and Equipresence 177
6.2.1 Determinism 177
6.2.2 Equipresence 177
6.2.3 Summary 178
6.3 Objectivity 179
6.3.1 Reducing Functional Dependencies 180
6.3.2 Summary 182
6.4 SYMMETRY 183
6.4.1 Changes in Reference Configuration 183
6.4.2 Symmetry Groups 186
6.4.3 Classification of Materials 189
6.4.4 Implications for Thermoviscous Fluids 193
6.4.5 Summary 193
6.5 Admissibility 194
6.5.1 Implications of the Entropy Inequality 195
6.5.2 Analysis of Equilibrium 197
6.5.3 Linear, Isotropic, Thermoelastic Solids 199
6.5.4 Summary 202
7 Multiconstituent Continua 203
7.1 Constituents 204
7.1.1 Configurations and Motions 204
7.1.2 Volume Fractions and Densities 206
7.1.3 Summary 208
7.2 Multiconstituent Balance Laws 209
7.2.1 Multiconstituent Mass Balance 210
7.2.2 Multiconstituent Momentum Balance 212
7.2.3 Multiconstituent Angular Momentum Balance 214
7.2.4 Multiconstituent Energy Balance 215
7.2.5 Multiconstituent Entropy Inequality 216
7.2.6 Isothermal, Nonreacting Multiphase Mixtures 217
7.2.7 Summary 219
7.3 Fluid Flow in a Porous Solid 220
7.3.1 Modeling Assumptions for Porous Media 221
7.3.2 Balance Laws for the Fluid and Solid Phases 223
7.3.3 Equilibrium Constraints 225
7.3.4 Linear Extensions From Equilibrium 226
7.3.5 Commentary 228
7.3.6 Potential Formulation of Darcy's Law 229
7.3.7 Summary 233
7.4 Diffusion in a Binary Fluid Mixture 234
7.4.1 Modeling Assumptions for Binary Diffusion 235
7.4.2 Balance Laws for the Two Species 235
7.4.3 Constitutive Relationships for Diffusion 236
7.4.4 Modeling Solute Transport 239
7.4.5 Summary 242
A Guide to Notation 243
A.1 General Conventions 243
A.2 Letters Reserved for Dedicated Uses 244
A.3 Special Symbols 245
B Vector Integral Theorems 247
B.1 Stokes's Theorem 248
B.2 The Divergence Theorem 249
B.3 The Change-of-variables Theorem 252
C Hints and Solutions to Exercises 253
References 265
Index 269
CHAPTER 1
GEOMETRIC SETTING
Physicists recognize two branches of mechanics: classical and quantum. For the past century, the quantum view, emphasizing the corpuscular nature of matter at atomic and finer scales, has played a dominant role in most universities' physics curricula. In those settings, classical mechanics enjoys a distinguished mathematical pedigree, being based on ideas developed by Galileo Galilei, Johannes Kepler, Isaac Newton, Leonhard Euler, Joseph Louis Lagrange, William Rowan Hamilton, and others. Nevertheless, many academic physics departments treat classical mechanics as a mathematical training ground for undergraduates preparing to study the principles of subatomic particles and quantum fields, subjects commonly regarded as more fundamental.
Applied mathematicians and engineers tend to view classical mechanics from a different perspective. While the most elementary formulations of Newton's laws and the Lagrangian and Hamiltonian formalisms focus on idealized particles with mass, many natural phenomena appear to macroscopic observers-those whose scales of observation are significantly larger than meters-as continuous in space and time. For these phenomena, fruitful mathematical descriptions typically arise from extensions of classical mechanics pioneered, most notably, by Leonhard Euler and Augustin-Louis Cauchy and refined during the last half of the twentieth century by a large community of scientists, some of whom are mentioned in the preface. In these extensions, matter appears to be continuous, in a sense to be made more precise in the next chapter.
Continuum mechanics embodies these extensions, furnishing useful mathematical models of fluids, elastic solids, and viscoelastic materials. These models describe phenomena that we see and feel in our everyday interactions with the world: rocks in the Earth's crust, water on and beneath its surface, weather, the structures that humans build, and the biological tissues that we occupy. The models typically take the form of partial differential equations describing rates of change with respect to spatial position and time. Advances in our ability to understand and solve these types of equations-especially using high-performance computers-have made continuum mechanics one of the most powerful tools in applied mathematics and engineering. For this reason, in developing the elements of the subject, this book frequently draws connections between its core concepts and the qualitative theory of partial differential equations.
1.1 VECTORS AND EUCLIDEAN POINT SPACE
From a mathematical perspective, continuum mechanics has roots in geometry. In the most natural geometric setting, basic principles do not depend on any observer's particular frame of reference or choice of coordinate systems. One aim of this book is to develop the rudiments of continuum mechanics in a manner that minimizes reliance on particular coordinates, recognizing that using these concepts in specific problems often requires the adoption of a well-chosen coordinate system.
The geometric setting here is relatively simple, relying on ideas familiar to anyone who has studied multivariable calculus and linear algebra. For a more sophisticated approach, refer to [38].
1.1.1 Vectors
Fundamental to continuum mechanics is the three-dimensional Euclidean vector space over the field of real numbers. This space, which we denote as , has features that do not depend on any system for assigning numbers to the vectors in it. In particular, has three attributes beyond those common to all vector spaces. Although the attributes are elementary, it is useful to review them in coordinate-free language and to show how coordinates arise.
- Inner product. possesses an inner product, that is, a binary operation that maps each pair of vectors onto a real number with the following properties:
Geometry in arises from the inner product. It allows us to associate with each vector in a length,
and, when , a direction . Two nonzero vectors x and y have angle given by
Two vectors x and y are orthogonal if . For two arbitrary vectors , with , the orthogonal projection of y onto x is
1.1.1See Figure 1.1.
- Orthonormal basis. There is an orthonormal basis for , that is, a basis such that for and . when .
Figure 1.1 The orthogonal projection of y onto x.
The basis , shown in Figure 1.2, establishes a Cartesian coordinate system on . Using this system, we represent any vector as a point in the vector space of ordered triples of real numbers: if , then
denotes its representation in with respect to the basis. A different choice of basis vectors for -even a different choice of orthonormal basis-yields a different representation in , but the vector in remains fixed in magnitude and direction. For this reason we distinguish from .
Figure 1.2 Standard orthonormal basis vectors defining a Cartesian coordinate system.
Exercise 1
For a given orthonormal basis , it is possible to determine the coefficients knowing the vector x. Show that .
For consistency with the conventions of matrix multiplication, discussed later, we write representations of vectors in as column arrays. Under this convention, when the basis for is understood, we sometimes abuse notation by writing as if the vector equals its representation:
For typesetting convenience we sometimes denote column vectors as formal transposes of row vectors, for example,
With respect to the orthonormal basis . the inner product of two vectors has the value
The following exercise gives a coordinate-free expression for the inner product in terms of lengths.
Exercise 2
Prove the polarization identity:
- Cross product. admits a second form of vector multiplication. If have angle then their cross product is the vector that has length and direction orthogonal to both x and y, with sense given by the right-hand rule, as illustrated in Figure 1.3.
Figure 1.3 The cross product , showing the right-hand rule.
As a binary operation on vectors, the cross product is endemic to three space dimensions, as discussed further in Section 3.3. In all that follows, we assume that the orthonormal basis for has positive orientation, meaning that whenever or (2, 3, 1) or (3, 1, 2), that is, whenever is an even permutation of (1, 2, 3). Under this convention, with respect to the basis , the cross product has the value
1.1.2Later sections explore additional algebraic and geometric interpretations of the cross product.
Representations with respect to the basis can be useful for calculations, but two caveats are in order. First, they are not the only numerical representations available for vectors Infinitely many orthonormal bases exist for and it is possible to construct infinitely many nonorthonormal, non-Cartesian bases. Second, the principles of continuum mechanics do not require any choice of basis or associated coordinate system. Nevertheless, this book frequently uses and the Cartesian coordinate system it defines to discuss examples, since this basis furnishes a computationally familiar setting. The remainder of this chapter reviews further aspects of the algebra and geometry of with an attempt to minimize unnecessary references to coordinates.
Subsequent chapters refer to vectors that have a variety of physical dimensions. For example, the dimension of position vectors is length, denoted by [L], while velocity vectors have dimension length/time, or . Fastidious readers may anticipate some apparent anomalies associated with algebraic operations involving pairs of vectors having different physical dimensions. Section 1.2 proposes a resolution.
1.1.2 Euclidean Point Space
As fundamental as the Euclidean vector space may be to the mathematics of continuum mechanics, the objects of interest-sets of material points defined in Chapter 2-do not reside there. Instead, consistent with experience, these objects occupy points P in a type of space that has no intrinsic algebraic structure. To apply the tools of algebra and calculus, we attach the vector space to choosing a point to serve as the origin and assigning to every point a vector in that translates into This approach allows us to refer to points in a way that facilitates the mathematical analysis available for vectors This subsection furnishes details of the association between and the space .
Recognizing the distinction between the set of points and the algebraic structure may seem pedantic. But without doing so, we cannot correctly account for disparate descriptions made by different observers, who assign vectors to points in different ways. Sections 3.5 and 6.3 examine the effects of such differences.
Definition
A set of points is a Euclidean point space over if there is a translation mapping d : having the following properties:
- for every
- for all
- For every fixed the mapping is one-to-one and onto. In other words, for every point and...
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