Bechtel / Lowe Fundamentals of Continuum Mechanics

With Applications to Mechanical, Thermomechanical, and Smart Materials
1. Auflage 2014
ISBN: 978-0-12-394834-2
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 340 Seiten

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

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.

Autoren/Hrsg.

Bechtel, Stephen

Lowe, Robert

Inhaltsverzeichnis

1;Front Cover;1
2;Fundamentals of Continuum Mechanics: With Applications to Mechanical, Thermomechanical, and Smart Materials;4
3;Copyright;5
4;Dedication;6
5;Contents;8
6;Preface;14
6.1; Continuum mechanics: the new pedagogy;14
6.2; Acknowledgments;15
7;Part I: The Beginning;18
7.1;Chapter 1: What Is a Continuum?;20
7.2;Chapter 2: Our Mathematical Playground;22
7.2.1;2.1 Real numbers and Euclidean space;22
7.2.1.1;2.1.1 Properties of real numbers;22
7.2.1.2;2.1.2 Properties of Euclidean space;24
7.2.2;2.2 Tensor algebra;29
7.2.2.1;2.2.1 Second-order tensors, zero tensor, identity tensor;29
7.2.2.2;2.2.2 Product, transpose, symmetry;33
7.2.2.3;2.2.3 Dyadic product;39
7.2.2.4;2.2.4 Cartesian Components, Indicial Notation, Summation Convention;41
7.2.2.5;2.2.5 Trace, scalar product, determinant;52
7.2.2.6;2.2.6 Inverse, orthogonality, positive definiteness;56
7.2.2.7;2.2.7 Vector product, scalar triple product;59
7.2.3;2.3 Eigenvalues, eigenvectors, polar decomposition, invariants;61
7.2.4;2.4 Tensors of order three and four;64
7.2.5;2.5 Tensor calculus;65
7.2.5.1;2.5.1 Partial derivatives;65
7.2.5.2;2.5.2 Chain rule, gradient, divergence, curl, divergence theorem;69
7.2.5.3;2.5.3 Tensor calculus in Cartesian component form;73
7.2.6;2.6 Curvilinear coordinates;76
7.2.6.1;2.6.1 Covariant and contravariant basis vectors;77
7.2.6.2;2.6.2 Physical components;80
7.2.6.3;2.6.3 Spatial derivatives: Covariant differentiation;82
7.2.6.3.1;2.6.3.1 Gradient and divergence of a vector;83
7.2.6.3.2;2.6.3.2 Divergence of a tensor;87
8;Part II: Kinematics, Kinetics, and the Fundamental Laws of Mechanics and Thermodynamics;90
8.1;Chapter 3: Kinematics: Motion and Deformation;92
8.1.1;3.1 Body, configuration, motion, displacement;92
8.1.2;3.2 Material derivative, velocity, acceleration;97
8.1.3;3.3 Deformation and strain;102
8.1.3.1;3.3.1 Deformation gradient;102
8.1.3.2;3.3.2 Stretch, rotation, Green's deformation tensor, Cauchy deformation tensor;105
8.1.3.3;3.3.3 Polar decomposition, stretch tensors, rotation tensor;108
8.1.3.4;3.3.4 Principal stretches and principal directions;111
8.1.3.4.1;3.3.4.1 Directions of pure stretch in the map U;111
8.1.3.4.2;3.3.4.2 Directions of pure stretch in the map V;113
8.1.3.5;3.3.5 Other measures of deformation and strain;113
8.1.4;3.4 Velocity gradient, rate of deformation tensor, vorticity tensor;121
8.1.5;3.5 Material point, material line, material surface, material volume;126
8.1.6;3.6 Volume elements and surface elements in volume and surface integrations;127
8.2;Chapter 4: The Fundamental Laws of Thermomechanics;132
8.2.1;4.1 Mass;132
8.2.2;4.2 Forces and moments, linear and angular momentum;133
8.2.3;4.3 Equations of motion (mechanical conservation laws);134
8.2.4;4.4 The first law of thermodynamics (conservation of energy);135
8.2.5;4.5 The transport and localization theorems;137
8.2.5.1;4.5.1 The transport theorem;137
8.2.5.2;4.5.2 The localization theorem;139
8.2.6;4.6 Cauchy stress tensor, heat flux vector;141
8.2.7;4.7 The energy theorem and stress power;147
8.2.8;4.8 Local forms of the conservation laws;148
8.2.9;4.9 Lagrangian forms of the integral conservation laws;154
8.2.9.1;4.9.1 Mass, forces, moments, linear and angular momentum;156
8.2.9.2;4.9.2 Conservation of mass, linear momentum, and angular momentum;157
8.2.9.3;4.9.3 First law of thermodynamics;158
8.2.9.4;4.9.4 Summary;158
8.2.10;4.10 Piola-Kirchhoff stress tensors, referential heat flux vector;159
8.2.10.1;4.10.1 Relations between spatial and referential quantities;159
8.2.11;4.11 The Lagrangian form of the energy theorem;160
8.2.12;4.12 Local conservation laws in Lagrangian form;161
8.2.13;4.13 The second law of thermodynamics;165
9;Part III: Constitutive Modeling;172
9.1;Chapter 5: Constitutive Modeling in Mechanics and Thermomechanics;174
9.1.1;Part I: Mechanics;174
9.1.2;5.1 Fundamental laws, constitutive equations, a well-posed initial-value boundary-value problem;174
9.1.3;5.2 Restrictions on the constitutive equations;176
9.1.3.1;5.2.1 Invariance under superposed rigid body motions;177
9.1.3.1.1;5.2.1.1 Superposed rigid body motions;177
9.1.3.1.2;5.2.1.2 Relationships between geometric and kinematic quantitiesunder a SRBM;181
9.1.3.1.3;5.2.1.3 Relationships between kinetic quantities under a SRBM;186
9.1.3.1.4;5.2.1.4 Invariance requirements;188
9.1.3.2;5.2.2 Material symmetry;188
9.1.4;Part II: Thermomechanics;190
9.1.5;5.3 Fundamental laws, constitutive equations, thermomechanical processes;192
9.1.6;5.4 Restrictions on the constitutive equations;195
9.1.6.1;5.4.1 Invariance under superposed rigid body motions;195
9.1.6.1.1;5.4.1.1 Relationships between thermal quantities under a SRBM;195
9.1.6.1.2;5.4.1.2 Invariance requirements;196
9.2;Chapter 6: Nonlinear Elasticity;198
9.2.1;6.1 Mechanical theory;198
9.2.2;6.2 Thermomechanical theory;200
9.2.2.1;6.2.1 Restrictions imposed by the second law of thermodynamics;200
9.2.2.2;6.2.2 Restrictions imposed by invariance under superposed rigid body motions and conservation of angular momentum;204
9.2.2.3;6.2.3 Restrictions imposed by material symmetry: Isotropy;208
9.2.3;6.3 Strain energy models;211
9.3;Chapter 7: Fluid Mechanics;214
9.3.1;7.1 Mechanical theory;214
9.3.1.1;7.1.1 Viscous fluids;214
9.3.1.1.1;7.1.1.1 Restrictions imposed by invariance under superposed rigid body motions;215
9.3.1.1.2;7.1.1.2 Linear viscous (Newtonian) fluids;217
9.3.1.1.3;7.1.1.3 The Navier-Stokes equations;219
9.3.1.2;7.1.2 Inviscid fluids;222
9.3.2;7.2 Thermomechanical theory;223
9.3.2.1;7.2.1 Viscous fluids;223
9.3.2.1.1;7.2.1.1 Restrictions imposed by invariance under SRBMs andthe second law of thermodynamics;224
9.3.2.1.2;7.2.1.2 Linear thermoviscous (Newtonian) fluids;226
9.3.2.2;7.2.2 Inviscid fluids;229
9.4;Chapter 8: Incompressibility and Thermal Expansion;232
9.4.1;8.1 Introduction;232
9.4.1.1;8.1.1 Motion-temperature constraints;233
9.4.1.2;8.1.2 Motion-entropy constraints;234
9.4.2;8.2 Newtonian fluids;235
9.4.2.1;8.2.1 The compressible theory: a brief review;235
9.4.2.2;8.2.2 Incompressibility;237
9.4.2.3;8.2.3 Incompressibility as a constitutive limit: an alternative perspective;244
9.4.2.4;8.2.4 Thermal expansion;246
9.4.2.4.1;8.2.4.1 A density-temperature constraint (isothermal incompressibility);246
9.4.2.4.2;8.2.4.2 A density-entropy constraint (isentropic incompressibility);249
9.4.2.5;8.2.5 Thermal expansion as a constitutive limit: an alternative perspective;251
9.4.2.5.1;8.2.5.1 Isothermal incompressibility;251
9.4.2.5.2;8.2.5.2 Isentropic incompressibility;252
9.4.3;8.3 Nonlinear elastic solids;253
9.4.3.1;8.3.1 The compressible theory: a brief review;253
9.4.3.2;8.3.2 Incompressibility;254
9.4.3.3;8.3.3 Incompressible strain energy models;258
10;Part IV: Beyond Mechanics and Thermomechanics;264
10.1;Chapter: 9 Modeling of Thermo-Electro-Magneto-Mechanical Behavior, with Application to Smart Materials;266
10.1.1;9.1 The fundamental laws of continuum electrodynamics: Integral forms;267
10.1.1.1;9.1.1 Notation and nomenclature;267
10.1.1.2;9.1.2 Conservation of mass;268
10.1.1.3;9.1.3 Balance of linear momentum;269
10.1.1.4;9.1.4 Balance of angular momentum;273
10.1.1.5;9.1.5 First law of thermodynamics;274
10.1.1.6;9.1.6 Second law of thermodynamics;276
10.1.1.7;9.1.7 Conservation of electric charge;277
10.1.1.8;9.1.8 Faraday's law;279
10.1.1.9;9.1.9 Gauss's law for magnetism;279
10.1.1.10;9.1.10 Gauss's law for electricity;280
10.1.1.11;9.1.11 Ampère-Maxwell law;281
10.1.1.12;9.1.12 Transformations between spatial and referential TEMM quantities;282
10.1.2;9.2 The fundamental laws of continuum electrodynamics: Pointwise forms;286
10.1.2.1;9.2.1 Eulerian fundamental laws;286
10.1.2.2;9.2.2 Lagrangian fundamental laws;292
10.1.3;9.3 Modeling of the effective electromagnetic fields;294
10.1.3.1;9.3.1 Minkowski model;295
10.1.3.2;9.3.2 Lorentz model;295
10.1.3.3;9.3.3 Statistical model;295
10.1.3.4;9.3.4 Chu model;296
10.1.3.5;9.3.5 A comparison of the four models;296
10.1.4;9.4 Modeling of the electromagnetically induced coupling terms;297
10.1.4.1;9.4.1 An alternative approach;298
10.1.5;9.5 Thermo-electro-magneto-mechanical process;300
10.1.6;9.6 Constitutive model development for thermo-electro-magneto-elastic materials: Large-deformation theory;301
10.1.6.1;9.6.1 The reduced Clausius-Duhem inequality, work conjugates;301
10.1.6.2;9.6.2 The all-extensive formulation;302
10.1.6.2.1;9.6.2.1 Polarization and magnetization as independent variables;304
10.1.6.3;9.6.3 Other formulations;305
10.1.6.3.1;9.6.3.1 The deformation-temperature-electric field-magnetic field formulation;307
10.1.6.4;9.6.4 Restrictions imposed by invariance under superposed rigid body motions and conservation of angular momentum;310
10.1.7;9.7 Constitutive model development for TEME materials;311
10.1.7.1;9.7.1 Small-deformation kinematics, kinetics, electromagnetic fields, and fundamental laws;311
10.1.7.2;9.7.2 Linear constitutive equations;313
10.1.7.3;9.7.3 Material symmetry;315
10.1.8;9.8 Linear, reversible, thermo-electro-magneto-mechanical processes;316
10.1.9;9.9 Specialization of the small-deformation TEME framework;319
11;Appendix A: Different Notions of Invariance;322
12;Appendix
B: The Physical Basis of Constitutive Assumptions;324
13;Appendix
C: Isotropic Tensors;326
14;Appendix
D: A Family of Thermomechanical Processes;328
15;Appendix
E: Energy Formulations and Stability Conditions for Newtonian Fluids;330
15.1;E.1 Governing equations;330
15.1.1;E.1.1 Density-entropy formulation;331
15.1.2;E.1.2 Density-temperature formulation;332
15.1.3;E.1.3 Pressure-entropy formulation;332
15.1.4;E.1.4 Pressure-temperature formulation;333
15.2;E.2 Stability conditions;334
16;Appendix
F: Additional Energy Formulations for Thermo-Electro-Magneto- Mechanical Materials;336
16.1;F.1 Deformation-temperature-electric displacement-magnetic induction formulation;336
17;Bibliography;338
18;Index;342

Leseproben

Chapter 2

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.

Figure 2.1 The real number line, illustrating an ordered set of masses 0 < m1 < m2.

Figure 2.2 Physical observations of the interaction between kinematic and kinetic quantities in mechanics. (a) The combination of two forces f1 and f2, yielding the resultant force f1+2 (concept of vector addition). (b) The product of mass m with acceleration a, yielding force f (concept of scalar multiplication of a vector).

2.1.1 Properties of real numbers

Physical quantities such as temperature, pressure, and mass are described by real numbers. For all scalars a, ß, ? that are elements of the set of real numbers (or, in simplified notation, a,ß,??R), the following properties hold:

a+ß?R;commutativity of addition,a+ß=ß+a;associativity of addition,a+(ß+?)=(a+ß)+?;existence of an additive identity,?0?a+0=a;existence of an additive inverse,?(-a)?a+(-a)=0;closure of multiplication,aß?R;commutativity of multiplication,aß=ßa;associativity of multiplication,(aß)?=a(ß?);existence of a multiplicative identity,1a=a;existence of a multiplicative inverse(or reciprocal),?1a?a1a=1,a?0;zero product,0a=0;distributivity of multiplication over addition,(a+ß)?=a?+ß?.

  (2.1)

Real numbers are an ordered set, so any pair of scalars a and ß that are elements of the set of real numbers satisfy one and only one of

  (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.

Figure 2.3 A schematic illustrating the interplay between the properties of vector, metric, normed, and inner product spaces.

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 a, ß that are elements of the set of real numbers (or, in simplified notation, ? u, v, w ? X and a,ß?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,au?X;associativity of scalar multiplication,a(ßu)=(aß)u;existence of a multiplicative identity,1u=u;distributivity of scalar multiplication over scalar addition,(a+ß)u=au+ßu;distributivity of scalar multiplication over vector addition,a(u+v)=au+av.

  (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...