Continuum MechanicsIn this book the basic principles of continuum mechanics and thermodynam ics are treated in the tradition of the rational framework established in the 1960s, typically in the fundamental memoir "The Non-Linear Field Theories of Mechanics" by Truesdell and Noll. The theoretical aspect of constitutive theories for materials in general has been carefully developed in mathemati cal clarity - from general kinematics, balance equations, material objectivity, and isotropic representations to the framework of rational thermodynamics based on the entropy principle. However, I make no claim that the subjects are covered completely, nor does this book cover solutions and examples that can usually be found in textbooks of fluid mechanics and linear elasticity. However, some of the interesting examples of finite deformations in elastic materials, such as biaxial stretching of an elastic membrane and inflation of a rubber balloon, are discussed. In the last two chapters of the book, some recent developments in ther modynamic theories are considered. Specifically, they emphasize the use of Lagrange multipliers, which enables the exploitation of the entropy principle in a systematic manner for constitutive equations, and introduce some basic notions of extended thermodynamics. Although extended thermodynamics is closely related to the kinetic theory of ideal gases, very limited knowledge of kinetic theory is needed. |
Contents
1 Kinematics | 1 |
111 Change of Reference Configuration | 4 |
13 Linear Strain Tensors | 8 |
14 Motion | 13 |
141 Material and Spatial Descriptions | 14 |
15 Relative Deformation | 17 |
16 Rate of Deformation | 20 |
17 Change of Frame and Objective Tensors | 22 |
6 Isotropic Elastic Solids | 153 |
62 Boundary Value Problems in Elasticity | 155 |
63 Homogeneous Stretch | 157 |
631 Uniaxial Stretch | 158 |
632 Biaxial Stretch | 159 |
64 Symmetric Loading of a Square Sheet | 160 |
641 Stability of a Square Sheet | 162 |
65 Simple Shear | 166 |
171 Transformation Property of Motion | 25 |
172 Property of Some Kinematic Quantities | 26 |
2 Balance Laws | 31 |
211 Field Equation and Jump Condition | 35 |
212 Balance Equations in Material Coordinates | 36 |
22 Conservation of Mass | 38 |
23 Laws of Dynamics | 41 |
231 Forces and Moments | 42 |
232 Stress Tensor | 43 |
233 Conservation of Linear and Angular Momenta | 50 |
24 Conservation of Energy | 51 |
25 Summary of Basic Equations | 54 |
251 Basic Equations in Material Coordinates | 56 |
252 Boundary Conditions of a Material Body | 57 |
26 Field Equations in Arbitrary Frames | 58 |
3 Basic Principles of Constitutive Theories | 63 |
32 Principle of Material Objectivity | 65 |
321 In Referential Description | 68 |
a Particular Class of Materials | 70 |
33 Simple Material Bodies | 72 |
34 Reduced Constitutive Relations | 75 |
35 Material Symmetry | 77 |
351 Constitutive Equation for a Simple Solid Body | 81 |
352 Constitutive Equation for a Simple Fluid | 82 |
353 Fluid Crystal with an Intrinsic Direction | 84 |
36 Isotropic Materials | 86 |
361 Constitutive Equation of an Isotropic Material | 88 |
37 Fading Memory | 89 |
371 Linear Viscoelasticity | 90 |
372 BoltzmannVolterra Theory of Viscoelasticity | 92 |
373 Linear Viscoelasticity of Rate Type | 93 |
374 Remark on Objectivity of Linear Elasticity | 94 |
4 Representation of Constitutive Functions | 97 |
42 Isotropic Functions | 98 |
421 Isotropic Elastic Materials and Linear Elasticity | 107 |
422 ReinerRivlin Fluids and NavierStokes Fluids | 109 |
423 Elastic Fluids | 111 |
43 Representation of Isotropic Functions | 112 |
431 Isotropic Thermoelastic Solids and Viscous HeatConducting Fluids | 118 |
44 Hemitropic Invariants | 119 |
45 Anisotropic Invariants | 122 |
451 Transverse Isotropy and Orthotropy | 124 |
452 On Irreducibility of Invariant Sets | 126 |
5 Entropy Principle | 129 |
52 Entropy Principle | 131 |
53 Thermodynamics of Elastic Materials | 132 |
531 Linear Thermoelasticity | 135 |
54 Elastic Materials with Internal Constraints | 139 |
55 Stability of Equilibrium | 144 |
551 Thermodynamic Stability Criteria | 148 |
66 Pure Shear of a Square Block | 169 |
67 Finite Deformation of Spherical Shells | 173 |
671 Eversion of a Spherical Shell | 175 |
672 Inflation of a Spherical Shell | 176 |
68 Stability of Spherical Shells | 179 |
681 Stability under Constant Pressures | 180 |
682 Stability for an Enclosed Spherical Shell | 181 |
7 Thermodynamics with Lagrange Multipliers | 183 |
72 Viscous HeatConducting Fluid | 184 |
721 General Results | 186 |
722 NavierStokesFourier Fluids | 188 |
73 Method of Lagrange Multipliers | 189 |
731 An Algebraic Problem | 190 |
732 Local Solvability | 191 |
74 Relation Between Entropy Flux and Heat Flux | 194 |
8 Rational Extended Thermodynamics | 199 |
82 Formal Structure of System of Balance Equations | 200 |
821 Symmetric Hyperbolic System | 201 |
822 Galilean Invariance | 204 |
83 System of Moment Equations | 207 |
84 Closure Problem | 213 |
841 Entropy Principle | 214 |
842 Formal Procedures | 216 |
85 ThirteenMoment Theory of Viscous HeatConducting Fluid | 217 |
851 Field Equations | 223 |
852 Entropy and Entropy Flux | 225 |
86 Monatomic Ideal Gases | 226 |
861 ThirteenMoment Theory | 227 |
862 Constitutive Equations | 228 |
871 Fouriers Law and Heat Conduction | 229 |
873 Remark on Boundary Value Problems | 232 |
A Elementary Tensor Analysis | 233 |
A11 Inner Product | 234 |
A12 Dual Bases | 235 |
A13 Tensor Product | 238 |
A14 Transformation Rules for Components | 243 |
A15 Determinant and Trace | 245 |
A16 Exterior Product and Vector Product | 251 |
A17 SecondOrder Tensors | 254 |
A18 Some Theorems of Linear Algebra | 256 |
A2 Tensor Calculus | 262 |
A22 Differentiation | 263 |
A23 Coordinate System | 272 |
A24 Covariant Derivatives | 275 |
A25 Other Differential Operators | 277 |
A26 Physical Components | 281 |
A27 Orthogonal Coordinate Systems | 282 |
| 289 | |
| 293 | |
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Common terms and phrases
A₁ arbitrary balance equations basis boundary called Cartesian coordinate system change of frame consider constant constitutive equations constitutive functions constitutive relation coordinate system covariant defined definition deformation gradient denoted density derivative entropy entropy inequality entropy principle equilibrium Euclidean Euclidean transformation Example Exercise field equations fluid given grad heat conduction heat flux hence implies incompressible inner product internal energy invariants isotropic functions jump condition Lagrange multipliers linear transformation material body material objectivity matrix metric tensor motion notation obtain principle of material reference configuration relative representation rotation satisfy scalar second-order tensor simple shear skew-symmetric skew-symmetric tensor solution space spin tensor strain tensor stress tensor Sym(V symmetric tensor temperature tensor field theorem theory thermoelastic vector field velocity θε Λε др მი მს