- Uses direct notation for a clear and straightforward presentation of the mathematics, leading to a better understanding of the underlying physics
- Covers high-interest research areas such as small- and large-deformation continuum electrodynamics, with application to smart materials used in intelligent systems and structures
- Offers a unique approach to modeling incompressibility and thermal expansion, based on the authors' own research
Stephen Bechtel is a professor emeritus in the Department of Mechanical & Aerospace Engineering at The Ohio State University. He obtained his Ph.D. in Mechanical Engineering from the University of California, Berkeley. He is a Fellow of the American Society of Mechanical Engineers (ASME) and a two-time winner of the Ohio State University College of Engineering Lumley Research Award. His research interests include advanced materials, including polymer/nanoparticle composites, magnetorheological fluids, ferroic solids, and piezoelectric crystals; industrial polymer processing and fiber manufacturing; and shear and extensional characterization of polymer melts and solutions.
Fundamentals of Continuum Mechanics provides a clear and rigorous presentation of continuum mechanics for engineers, physicists, applied mathematicians, and materials scientists. This book emphasizes the role of thermodynamics in constitutive modeling, with detailed application to nonlinear elastic solids, viscous fluids, and modern smart materials. While emphasizing advanced material modeling, special attention is also devoted to developing novel theories for incompressible and thermally expanding materials. A wealth of carefully chosen examples and exercises illuminate the subject matter and facilitate self-study. Uses direct notation for a clear and straightforward presentation of the mathematics, leading to a better understanding of the underlying physics Covers high-interest research areas such as small- and large-deformation continuum electrodynamics, with application to smart materials used in intelligent systems and structures Offers a unique approach to modeling incompressibility and thermal expansion, based on the authors' own research
Front Cover 1
Fundamentals of Continuum Mechanics: With Applications to Mechanical, Thermomechanical, and Smart Materials 4
Copyright 5
Dedication 6
Contents 8
Preface 14
Continuum mechanics: the new pedagogy 14
Acknowledgments 15
Part I: The Beginning 18
Chapter 1: What Is a Continuum? 20
Chapter 2: Our Mathematical Playground 22
2.1 Real numbers and Euclidean space 22
2.1.1 Properties of real numbers 22
2.1.2 Properties of Euclidean space 24
2.2 Tensor algebra 29
2.2.1 Second-order tensors, zero tensor, identity tensor 29
2.2.2 Product, transpose, symmetry 33
2.2.3 Dyadic product 39
2.2.4 Cartesian Components, Indicial Notation, Summation Convention 41
2.2.5 Trace, scalar product, determinant 52
2.2.6 Inverse, orthogonality, positive definiteness 56
2.2.7 Vector product, scalar triple product 59
2.3 Eigenvalues, eigenvectors, polar decomposition, invariants 61
2.4 Tensors of order three and four 64
2.5 Tensor calculus 65
2.5.1 Partial derivatives 65
2.5.2 Chain rule, gradient, divergence, curl, divergence theorem 69
2.5.3 Tensor calculus in Cartesian component form 73
2.6 Curvilinear coordinates 76
2.6.1 Covariant and contravariant basis vectors 77
2.6.2 Physical components 80
2.6.3 Spatial derivatives: Covariant differentiation 82
2.6.3.1 Gradient and divergence of a vector 83
2.6.3.2 Divergence of a tensor 87
Part II: Kinematics, Kinetics, and the Fundamental Laws of Mechanics and Thermodynamics 90
Chapter 3: Kinematics: Motion and Deformation 92
3.1 Body, configuration, motion, displacement 92
3.2 Material derivative, velocity, acceleration 97
3.3 Deformation and strain 102
3.3.1 Deformation gradient 102
3.3.2 Stretch, rotation, Green's deformation tensor, Cauchy deformation tensor 105
3.3.3 Polar decomposition, stretch tensors, rotation tensor 108
3.3.4 Principal stretches and principal directions 111
3.3.4.1 Directions of pure stretch in the map U 111
3.3.4.2 Directions of pure stretch in the map V 113
3.3.5 Other measures of deformation and strain 113
3.4 Velocity gradient, rate of deformation tensor, vorticity tensor 121
3.5 Material point, material line, material surface, material volume 126
3.6 Volume elements and surface elements in volume and surface integrations 127
Chapter 4: The Fundamental Laws of Thermomechanics 132
4.1 Mass 132
4.2 Forces and moments, linear and angular momentum 133
4.3 Equations of motion (mechanical conservation laws) 134
4.4 The first law of thermodynamics (conservation of energy) 135
4.5 The transport and localization theorems 137
4.5.1 The transport theorem 137
4.5.2 The localization theorem 139
4.6 Cauchy stress tensor, heat flux vector 141
4.7 The energy theorem and stress power 147
4.8 Local forms of the conservation laws 148
4.9 Lagrangian forms of the integral conservation laws 154
4.9.1 Mass, forces, moments, linear and angular momentum 156
4.9.2 Conservation of mass, linear momentum, and angular momentum 157
4.9.3 First law of thermodynamics 158
4.9.4 Summary 158
4.10 Piola-Kirchhoff stress tensors, referential heat flux vector 159
4.10.1 Relations between spatial and referential quantities 159
4.11 The Lagrangian form of the energy theorem 160
4.12 Local conservation laws in Lagrangian form 161
4.13 The second law of thermodynamics 165
Part III: Constitutive Modeling 172
Chapter 5: Constitutive Modeling in Mechanics and Thermomechanics 174
Part I: Mechanics 174
5.1 Fundamental laws, constitutive equations, a well-posed initial-value boundary-value problem 174
5.2 Restrictions on the constitutive equations 176
5.2.1 Invariance under superposed rigid body motions 177
5.2.1.1 Superposed rigid body motions 177
5.2.1.2 Relationships between geometric and kinematic quantitiesunder a SRBM 181
5.2.1.3 Relationships between kinetic quantities under a SRBM 186
5.2.1.4 Invariance requirements 188
5.2.2 Material symmetry 188
Part II: Thermomechanics 190
5.3 Fundamental laws, constitutive equations, thermomechanical processes 192
5.4 Restrictions on the constitutive equations 195
5.4.1 Invariance under superposed rigid body motions 195
5.4.1.1 Relationships between thermal quantities under a SRBM 195
5.4.1.2 Invariance requirements 196
Chapter 6: Nonlinear Elasticity 198
6.1 Mechanical theory 198
6.2 Thermomechanical theory 200
6.2.1 Restrictions imposed by the second law of thermodynamics 200
6.2.2 Restrictions imposed by invariance under superposed rigid body motions and conservation of angular momentum 204
6.2.3 Restrictions imposed by material symmetry: Isotropy 208
6.3 Strain energy models 211
Chapter 7: Fluid Mechanics 214
7.1 Mechanical theory 214
7.1.1 Viscous fluids 214
7.1.1.1 Restrictions imposed by invariance under superposed rigid body motions 215
7.1.1.2 Linear viscous (Newtonian) fluids 217
7.1.1.3 The Navier-Stokes equations 219
7.1.2 Inviscid fluids 222
7.2 Thermomechanical theory 223
7.2.1 Viscous fluids 223
7.2.1.1 Restrictions imposed by invariance under SRBMs andthe second law of thermodynamics 224
7.2.1.2 Linear thermoviscous (Newtonian) fluids 226
7.2.2 Inviscid fluids 229
Chapter 8: Incompressibility and Thermal Expansion 232
8.1 Introduction 232
8.1.1 Motion-temperature constraints 233
8.1.2 Motion-entropy constraints 234
8.2 Newtonian fluids 235
8.2.1 The compressible theory: a brief review 235
8.2.2 Incompressibility 237
8.2.3 Incompressibility as a constitutive limit: an alternative perspective 244
8.2.4 Thermal expansion 246
8.2.4.1 A density-temperature constraint (isothermal incompressibility) 246
8.2.4.2 A density-entropy constraint (isentropic incompressibility) 249
8.2.5 Thermal expansion as a constitutive limit: an alternative perspective 251
8.2.5.1 Isothermal incompressibility 251
8.2.5.2 Isentropic incompressibility 252
8.3 Nonlinear elastic solids 253
8.3.1 The compressible theory: a brief review 253
8.3.2 Incompressibility 254
8.3.3 Incompressible strain energy models 258
Part IV: Beyond Mechanics and Thermomechanics 264
Chapter: 9 Modeling of Thermo-Electro-Magneto-Mechanical Behavior, with Application to Smart Materials 266
9.1 The fundamental laws of continuum electrodynamics: Integral forms 267
9.1.1 Notation and nomenclature 267
9.1.2 Conservation of mass 268
9.1.3 Balance of linear momentum 269
9.1.4 Balance of angular momentum 273
9.1.5 First law of thermodynamics 274
9.1.6 Second law of thermodynamics 276
9.1.7 Conservation of electric charge 277
9.1.8 Faraday's law 279
9.1.9 Gauss's law for magnetism 279
9.1.10 Gauss's law for electricity 280
9.1.11 Ampère-Maxwell law 281
9.1.12 Transformations between spatial and referential TEMM quantities 282
9.2 The fundamental laws of continuum electrodynamics: Pointwise forms 286
9.2.1 Eulerian fundamental laws 286
9.2.2 Lagrangian fundamental laws 292
9.3 Modeling of the effective electromagnetic fields 294
9.3.1 Minkowski model 295
9.3.2 Lorentz model 295
9.3.3 Statistical model 295
9.3.4 Chu model 296
9.3.5 A comparison of the four models 296
9.4 Modeling of the electromagnetically induced coupling terms 297
9.4.1 An alternative approach 298
9.5 Thermo-electro-magneto-mechanical process 300
9.6 Constitutive model development for thermo-electro-magneto-elastic materials: Large-deformation theory 301
9.6.1 The reduced Clausius-Duhem inequality, work conjugates 301
9.6.2 The all-extensive formulation 302
9.6.2.1 Polarization and magnetization as independent variables 304
9.6.3 Other formulations 305
9.6.3.1 The deformation-temperature-electric field-magnetic field formulation 307
9.6.4 Restrictions imposed by invariance under superposed rigid body motions and conservation of angular momentum 310
9.7 Constitutive model development for TEME materials 311
9.7.1 Small-deformation kinematics, kinetics, electromagnetic fields, and fundamental laws 311
9.7.2 Linear constitutive equations 313
9.7.3 Material symmetry 315
9.8 Linear, reversible, thermo-electro-magneto-mechanical processes 316
9.9 Specialization of the small-deformation TEME framework 319
Appendix A: Different Notions of Invariance 322
Appendix
324
Appendix
326
Appendix
328
Appendix
330
E.1 Governing equations 330
E.1.1 Density-entropy formulation 331
E.1.2 Density-temperature formulation 332
E.1.3 Pressure-entropy formulation 332
E.1.4 Pressure-temperature formulation 333
E.2 Stability conditions 334
Appendix
336
F.1 Deformation-temperature-electric displacement-magnetic induction formulation 336
Bibliography 338
Index 342
Our Mathematical Playground
Abstract
The purpose of this chapter is to enable the reader to become fluent in the language of this textbook: tensor algebra and tensor calculus. In particular, special attention is devoted to rigorously developing the mathematical foundations underlying tensor algebra and tensor calculus from the ground up, starting with the fundamental concepts of a vector space and an inner product space. Throughout, we favor a direct (or coordinate-free) presentation of the mathematics. Although direct notation requires some effort to master, it ultimately lends itself to a more transparent presentation of the physical concepts. Care is taken to provide the reader with sufficient background to specialize the coordinate-free results to Cartesian or curvilinear coordinate systems. Almost all examples are worked by the authors to facilitate self-study and to ensure the reader has a firm mathematical foundation.
Keywords
direct notation
vector space
inner product space
tensor algebra
tensor calculus
cartesian coordinates
curvilinear coordinates
This textbook is primarily a course in physics. The physical notions, however, must be expressed through the language of mathematics. When this mathematical language becomes cumbersome, there is a danger that the mathematics will obscure the physics, and the subject will appear to be mere symbol manipulation. It is therefore desirable to present the physics in the simplest possible mathematics, which is direct notation. This direct presentation of the mathematics exists independently of any coordinate system. Once the theory has been developed and presented in direct form, it may be referred to any coordinate system when applied to a particular problem. This chapter will acquaint the student with, or serve as a review of, direct notation.
In this chapter, as well as the remainder of the book, we employ for brevity the following logical notation (beyond the customary operational and ordering symbols +, −, =, <, >, ≤, ≥): ∀ abbreviates “for all” or “for any,” ∈ abbreviates “an element of,” ∃ abbreviates “there exists,” ∋ abbreviates “such that,” ⊂ or ⊆ abbreviates “a subset of,” abbreviates “the set of real numbers,” ∪ abbreviates “the union of,” ⇒ abbreviates “implies,” ⇔ abbreviates “if and only if,” and ≡ abbreviates “is defined as.”
2.1 Real numbers and euclidean space
The interplay of mathematics and physics in the development of continuum mechanics was as follows1: In their observations of the world around them, physically minded scientists encountered two types of quantities. Some quantities, such as temperature, mass, and pressure, were ordered sets (see Figure 2.1). From this concept were constructed the real numbers. Other quantities, such as velocity, acceleration, and force, had both a magnitude and a direction, and combined as shown in Figure 2.2. From these observations came a vector space endowed with an inner product called a Euclidean space.
2.1.1 Properties of real numbers
Physical quantities such as temperature, pressure, and mass are described by real numbers. For all scalars α, β, γ that are elements of the set of real numbers (or, in simplified notation, α,β,γ∈R), the following properties hold:
α+β∈R;commutativity of addition,α+β=β+α;associativity of addition,α+(β+γ)=(α+β)+γ;existence of an additive identity,∃0∋α+0=α;existence of an additive inverse,∃(−α)∋α+(−α)=0;closure of multiplication,αβ∈R;commutativity of multiplication,αβ=βα;associativity of multiplication,(αβ)γ=α(βγ);existence of a multiplicative identity,1α=α;existence of a multiplicative inverse(or reciprocal),∃1α∋α1α=1,α≠0;zero product,0α=0;distributivity of multiplication over addition,(α+β)γ=αγ+βγ.
(2.1)
Real numbers are an ordered set, so any pair of scalars α and β that are elements of the set of real numbers satisfy one and only one of
<β,α=β,α>β.
(2.2)
2.1.2 Properties of euclidean space
In this section, we arrive at Euclidean space by progressing from vector spaces, to metric spaces, to normed spaces, and finally to inner product spaces. The vector space (whose elements are called vectors) postulates the algebraic concepts of vector addition, scalar multiplication, and the zero element (or origin) of the space. The metric space (whose elements are called points) postulates topological concepts such as the distance between two points. The normed space (a vector space endowed with a norm) postulates the concept of the length of a vector. Finally, the inner product space (a vector space endowed with an inner product) postulates the concept of an angle between two vectors. Ultimately, we illustrate that every inner product space is also a vector space, a metric space, and a normed space, and is hence endowed with all of their separate properties (refer to Figure 2.3). An n-dimensional inner product space, where n is a positive integer, is known as a Euclidean space n.
Vector space X. The elements of vector space X are called vectors. For all vectors u, v, w in vector space X, and for all scalars α, β that are elements of the set of real numbers (or, in simplified notation, ∀ u, v, w ∈ X and α,β∈R), the following properties hold:
u+v∈X;commutativity of vector addition,u+v=v+u;associativity of vector addition,u+(v+w)=(u+v)+w;existence of an additive identity,∃0∋u+0=u;existence of an additive inverse,∃(−u)∋u+(−u)=0;closure of scalar multiplication,αu∈X;associativity of scalar multiplication,α(βu)=(αβ)u;existence of a multiplicative identity,1u=u;distributivity of scalar multiplication over scalar addition,(α+β)u=αu+βu;distributivity of scalar multiplication over vector addition,α(u+v)=αu+αv.
(2.3)
For a vector space we have the algebraic concepts of linear combination, independence, dependence, span, linear manifold, basis, and dimension.
Metric space X. The elements of metric space X are called points. The real-valued function d(u, v) is called the metric of X; it accepts points u and v as inputs, and provides the real-valued distance between points u and v as output. The metric d(u, v) is defined such that the following properties hold ∀ u, v, w ∈ X:
≠v⇒d(u,v)>0,d(u,u)=0,d(u,v)=d(v,u),d(u,w)≤d(u,v)+d(v,w).
(2.4)
For a metric space we have the topological concepts of open sets, closed sets, continuity, convergence, completeness, compactness, connectedness, and boundedness. Note that we can have a vector space without the notion of a metric, and a metric space without the notions of scalar multiplication or a zero element (i.e., an origin).
Normed space X. The normed space X is a vector space in which there exists a real-valued function |u| known as the norm of the vector u; the norm accepts vector u as input, and...
| Erscheint lt. Verlag | 2.12.2014 |
|---|---|
| Sprache | englisch |
| Themenwelt | Naturwissenschaften ► Physik / Astronomie ► Mechanik |
| Naturwissenschaften ► Physik / Astronomie ► Strömungsmechanik | |
| Technik ► Bauwesen | |
| ISBN-10 | 0-12-394834-7 / 0123948347 |
| ISBN-13 | 978-0-12-394834-2 / 9780123948342 |
| Haben Sie eine Frage zum Produkt? |
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