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This book develops Initial Boundary Value equations for solving physical problems. Covers Tensors, Continuum Kinematics, Stress, Fundamental Equations, Constitutive Equations, Linear Elasticity, Hyperelasticity, Plasticity, Thermoelasticity and more.

This publication is aimed at students, teachers, and researchers of Continuum Mechanics and focused extensively on stating and developing Initial Boundary Value equations used to solve physical problems. With respect to notation, the tensorial, indicial and Voigt notations have been used indiscriminately. The book is divided into twelve chapters with the following topics: Tensors, Continuum Kinematics, Stress, The Objectivity of Tensors, The Fundamental Equations of Continuum Mechanics, An Introduction to Constitutive Equations, Linear Elasticity, Hyperelasticity, Plasticity (small and large deformations), Thermoelasticity (small and large deformations), Damage Mechanics (small and large deformations), and An Introduction to Fluids. Moreover, the text is supplemented with over 280 figures, over 100 solved problems, and 130 references.

Arvustused

From the reviews:

The book is meant as a textbook for master and doctoral students and researchers. It is based on lecture notes of civil engineering courses of the author given at the University of Castillia-La Mancha (Spain). So the reader can expect a careful and detailed introduction to the subject without too much novelty. The book is perhaps helpful for those readers who have already a strong background in continuum mechanics and want to find additional information on topics . (Albrecht Bertram, zbMATH, Vol. 1277, 2014)

Preface xvii
Abbreviations xix
Operators and Symbols xx
SI-Units xxi
Introduction 1(8)
1 Mechanics
1(1)
2 What is Continuum Mechanics?
1(2)
2.1 Hypothesis of Continuum Mechanics
1(1)
2.2 The Continuum
2(1)
3 Scales of material Studies
3(3)
3.1 Scale Study of Continuum Mechanics
3(3)
4 The Initial Boundary Value Problem (IBVP)
6(3)
4.1 Solving the IBVP
6(1)
4.2 Simplifying the IBVP
7(2)
1 Tensors
9(116)
1.1 Introduction
9(1)
1.2 Algebraic Operations with Vectors
10(6)
1.3 Coordinate Systems
16(4)
1.3.1 Cartesian Coordinate System
16(1)
1.3.2 Vector Representation in the Cartesian Coordinate System
17(3)
1.3.3 Einstein Summation Convention (Einstein Notation)
20(1)
1.4 Indicial Notation
20(8)
1.4.1 Some Operators
22(1)
1.4.1.1 Kronecker Delta
22(1)
1.4.1.2 Permutation Symbol
23(5)
1.5 Algebraic Operations with Tensors
28(63)
1.5.1 Dyadic
28(4)
1.5.1.1 Component Representation of a Second-Order Tensor in the Cartesian Basis
32(2)
1.5.2 Properties of Tensors
34(1)
1.5.2.1 Tensor Transpose
34(2)
1.5.2.2 Symmetry and Antisymmetry
36(6)
1.5.2.3 Cofactor Tensor. Adjugate of a Tensor
42(1)
1.5.2.4 Tensor Trace
42(2)
1.5.2.5 Particular Tensors
44(1)
1.5.2.6 Determinant of a Tensor
45(3)
1.5.2.7 Inverse of a Tensor
48(3)
1.5.2.8 Orthogonal Tensors
51(1)
1.5.2.9 Positive Definite Tensor, Negative Definite Tensor and Semi-Definite Tensors
52(1)
1.5.2.10 Additive Decomposition of Tensors
53(1)
1.5.3 Transformation Law of the Tensor Components
54(7)
1.5.3.1 Component Transformation Law in Two Dimensions (2D)
61(4)
1.5.4 Eigenvalue and Eigenvector Problem
65(2)
1.5.4.1 The Orthogonality of the Eigenvectors
67(2)
1.5.4.2 Solution of the Cubic Equation
69(3)
1.5.5 Spectral Representation of Tensors
72(4)
1.5.6 Cayley-Hamilton Theorem
76(2)
1.5.7 Norms of Tensors
78(1)
1.5.8 Isotropic and Anisotropic Tensor
79(1)
1.5.9 Coaxial Tensors
80(1)
1.5.10 Polar Decomposition
81(2)
1.5.11 Partial Derivative with Tensors
83(2)
1.5.11.1 Partial Derivative of Invariants
85(1)
1.5.11.2 Time Derivative of Tensors
86(1)
1.5.12 Spherical and Deviatoric Tensors
86(1)
1.5.12.1 First Invariant of the Deviatoric Tensor
87(1)
1.5.12.2 Second Invariant of the Deviatoric Tensor
87(2)
1.5.12.3 Third Invariant of Deviatoric Tensor
89(2)
1.6 The Tensor-Valued Tensor Function
91(5)
1.6.1 The Tensor Series
91(1)
1.6.2 The Tensor-Valued Isotropic Tensor Function
92(2)
1.6.3 The Derivative of the Tensor-Valued Tensor Function
94(2)
1.7 The Voigt Notation
96(9)
1.7.1 The Unit Tensors in Voigt Notation
97(1)
1.7.2 The Scalar Product in Voigt Notation
98(1)
1.7.3 The Component Transformation Law in Voigt Notation
99(1)
1.7.4 Spectral Representation in Voigt Notation
100(1)
1.7.5 Deviatoric Tensor Components in Voigt Notation
101(4)
1.8 Tensor Fields
105(12)
1.8.1 Scalar Fields
106(1)
1.8.2 Gradient
106(5)
1.8.3 Divergence
111(2)
1.8.4 The Curl
113(2)
1.8.5 The Conservative Field
115(2)
1.9 Theorems Involving Integrals
117(8)
1.9.1 Integration by Parts
117(1)
1.9.2 The Divergence Theorem
117(3)
1.9.3 Independence of Path
120(1)
1.9.4 The Kelvin-Stokes' Theorem
121(1)
1.9.5 Green's Identities
122(3)
Appendix A A Graphical Representation Of A Second-Order Tensor
125(534)
A.1 Projecting a Second-order Tensor onto a Particular Direction
125(5)
A.1.1 Normal and Tangential Components
125(2)
A.1.2 The Maximum and Minimum Normal Components
127(1)
A.1.3 The Maximum and Minimum Tangential Component
128(2)
A.2 Graphical Representation of an Arbitrary Second-order Tensor
130(8)
A.2.1 Graphical Representation of a Symmetric Second-Order Tensor (Mohr's Circle)
134(4)
A.3 The Tensor Ellipsoid
138(1)
A.4 Graphical Representation of the Spherical and Deviatoric Parts
139(6)
A.4.1 The Octahedral Vector
139(6)
2 Continuum Kinematics
145(100)
2.1 Introduction
145(1)
2.2 The Continuous medium
146(5)
2.2.1 Kinds of Motion
147(1)
2.2.1.1 Rigid Body Motion
147(2)
2.2.2 Types of Configurations
149(1)
2.2.2.1 Mass Density
150(1)
2.3 Description of Motion
151(5)
2.3.1 Material and Spatial Coordinates
151(1)
2.3.2 The Displacement Vector
152(1)
2.3.3 The Velocity Vector
152(1)
2.3.4 The Acceleration Vector
152(1)
2.3.5 Lagrangian and Eulerian Descriptions
152(1)
2.3.5.1 Lagrangian Description of Motion
152(1)
2.3.5.2 Eulerian Description of Motion
153(1)
2.3.5.3 Lagrangian and Eulerian Variables
153(3)
2.4 The Material Time Derivative
156(7)
2.4.1 Velocity and Acceleration in Eulerian Description
158(1)
2.4.2 Stationary Fields
159(2)
2.4.3 Streamlines
161(2)
2.5 The Deformation Gradient
163(13)
2.5.1 Introduction
163(1)
2.5.2 Stretch and Unit Extension
163(2)
2.5.3 The Material and Spatial Deformation Gradient
165(3)
2.5.4 Displacement Gradient Tensors (Material and Spatial)
168(3)
2.5.5 Material Time Derivative of the Deformation Gradient. Material Time Derivative of the Jacobian Determinant
171(1)
2.5.5.1 Material Time Derivative of F . The Spatial Velocity Gradient
171(1)
2.5.5.2 Rate-of-Deformation and Spin Tensors
172(2)
2.5.5.3 The Material Time Derivative of F-1
174(1)
2.5.5.4 The Material Time Derivative of the Jacobian Determinant
174(2)
2.6 Finite Strain Tensors
176(15)
2.6.1 The Material Finite Strain Tensor
177(4)
2.6.2 The Spatial Finite Strain Tensor (The Almansi Strain Tensor)
181(2)
2.6.3 The Material Time Derivative of Strain Tensors
183(1)
2.6.3.1 The Material Time Derivative of the Right Cauchy-Green Deformation Tensor
183(1)
2.6.3.2 The Material Time Derivative of the Green-Lagrange Strain Tensor
183(1)
2.6.3.3 The Material Time Derivative of C-1
184(1)
2.6.3.4 Material Time Derivative of the Left Cauchy-Green Deformation Tensor
184(1)
2.6.3.5 The Material Time Derivative of the Almansi Strain Tensor
185(1)
2.6.4 Interpreting Deformation/Strain Tensors
186(1)
2.6.4.1 The Relationship between the Strain and Stretch Tensors
187(1)
2.6.4.2 Change of Angle
188(1)
2.6.4.3 The Physical Interpretation of the Deformation/Strain Tensor Components. The Right Stretch Tensor
189(2)
2.7 Particular Cases of Motion
191(4)
2.7.1 Homogeneous Deformation
191(1)
2.7.2 Rigid Body Motion
192(3)
2.8 Polar Decomposition of F
195(20)
2.8.1 Spectral Representation of Kinematic Tensors
197(6)
2.8.2 Evolution of the Polar Decomposition
203(5)
2.8.2.1 The Alternative Way to Express the Rate of Kinematic Tensors
208(7)
2.9 Area and Volume Elements Deformation
215(5)
2.9.1 Area Element Deformation
215(2)
2.9.1.1 The Material Time Derivative of the Area Element
217(1)
2.9.2 The Volume Element Deformation
218(1)
2.9.2.1 The Material Time Derivative of the Volume Element
219(1)
2.9.2.2 Dilatation
220(1)
2.9.2.3 Isochoric Motion. Incompressibility
220(1)
2.10 Material and Control Domains
220(2)
2.10.1 The Material Domain
220(1)
2.10.2 The Control Domain
221(1)
2.11 Transport Equations
222(2)
2.12 Circulation and Vorticity
224(1)
2.13 Motion Decomposition: Volumetric and Isochoric Motions
225(3)
2.13.1 The Principal Invariants
227(1)
2.14 The Small Deformation Regime
228(11)
2.14.1 Introduction
228(1)
2.14.2 Infinitesimal Strain and Spin Tensors
229(2)
2.14.3 Stretch and Unit Extension
231(1)
2.14.4 Change of Angle
232(1)
2.14.5 The Physical Interpretation of the Infinitesimal Strain Tensor
232(1)
2.14.5.1 Engineering Strain
233(2)
2.14.6 The Volume Ratio (Dilatation)
235(1)
2.14.7 The Plane Strain
236(3)
2.15 Other Ways to Define Strain
239(6)
2.15.1 Motivation
239(2)
2.15.2 The Logarithmic Strain Tensor
241(1)
2.15.3 The Biot Strain Tensor
242(1)
2.15.4 Unifying the Strain Tensors
242(1)
2.15.5 One Dimensional Measurements of Strain (1D)
243(1)
2.15.5.1 Cauchy's strain or Engineering strain or the Linear strain
243(1)
2.15.5.2 The Logarithmic or True strain
243(1)
2.15.5.3 The Green-Lagrange strain
243(1)
2.15.5.4 The Almansi strain
243(1)
2.15.5.5 The Swaiger strain
244(1)
2.15.5.6 The Kuhn strain
244(1)
3 Stress
245(24)
3.1 Introduction
245(1)
3.2 Forces
245(2)
3.2.1 Surface Forces (Traction)
245(1)
3.2.2 Gravitational Force (Body Force)
246(1)
3.3 Stress Tensors
247(22)
3.3.1 The Cauchy Stress Tensor
248(1)
3.3.1.1 The Traction Vector
248(1)
3.3.1.2 Cauchy's Fundamental Postulate
248(4)
3.3.2 The Relationship between the Traction and the Cauchy Stress Tensor
252(8)
3.3.3 Other Measures of Stress
260(1)
3.3.3.1 The First Piola-Kirchhoff Stress Tensor
260(2)
3.3.3.2 The Kirchhoff Stress Tensor
262(1)
3.3.3.3 The Second Piola-Kirchhoff Stress Tensor
262(2)
3.3.3.4 The Biot Stress Tensor
264(1)
3.3.3.5 The Mandel Stress Tensor
264(1)
3.3.4 Spectral Representation of the Stress Tensors
265(4)
4 Objectivity Of Tensors
269(16)
4.1 Introduction
269(1)
4.2 The Objectivity Of Tensors
270(7)
4.2.1 The Deformation Gradient
272(1)
4.2.2 Kinematic Tensors
273(2)
4.2.3 Stress Tensors
275(2)
4.3 Tensor Rates
277(8)
4.3.1 Objective Rates
278(1)
4.3.1.1 The Convective Rate
279(1)
4.3.1.2 The Oldroyd Rate
279(1)
4.3.1.3 The Cotter-Rivlin Rate
280(1)
4.3.1.4 The Jaumann-Zaremba Rate
280(2)
4.3.1.5 The Green-Naghdi Rate (Polar Rate)
282(1)
4.3.2 The Objective Rate of Stress Tensors
282(3)
5 The Fundamental Equations Of Continuum Mechanics
285(56)
5.1 Introduction
285(1)
5.2 Density
285(1)
5.2.1 Mass Density
286(1)
5.3 Flux
286(1)
5.4 The: Reynolds Transport Theorem
287(4)
5.4.1 Reynolds' Transport Theorem for Volumes with Discontinuities
288(3)
5.5 Conservation Law
291(1)
5.6 The Principle of Conservation of Mass. The Mass Continuity Equation
291(6)
5.6.1 The Mass Continuity Equation in Lagrangian Description
293(2)
5.6.2 Incompressibility
295(1)
5.6.3 The Mass Continuity Equation for Volume with Discontinuities
295(2)
5.7 The Principle of Conservation of Linear Momentum. The Equations of Motion
297(5)
5.7.1 Linear Momentum
297(1)
5.7.2 The Principle of Conservation of Linear Momentum
297(1)
5.7.2.1 The Equilibrium Equations
298(3)
5.7.3 The Equations of Motion with Discontinuities
301(1)
5.8 The Principle of Conservation of Angular Momentum. Symmetry of the Cauchy Stress Tensor
302(5)
5.8.1 Angular Momentum
302(1)
5.8.2 The Principle of Conservation of Angular Momentum
303(4)
5.9 The Principle of Conservation of Energy. The Energy Equation
307(11)
5.9.1 Kinetic Energy
307(1)
5.9.2 External and Internal Mechanical Power
307(3)
5.9.3 The Balance of Mechanical Energy
310(2)
5.9.4 The Internal Energy
312(1)
5.9.5 Thermal Power
313(1)
5.9.6 The First Law of Thermodynamics. The Energy Equation
314(1)
5.9.6.1 The Energy Equation in Lagrangian Description
315(1)
5.9.7 The Energy Equation with Discontinuity
316(2)
5.10 The Principle of Irreversibility. Entropy Inequality
318(8)
5.10.1 The Second Law of Thermodynamics
318(2)
5.10.2 The Clausius-Duhem Inequality
320(1)
5.10.3 The Clausius-Planck Inequality
321(1)
5.10.4 The Alternative Form to Express the Clausius-Duhem Inequality
321(2)
5.10.5 The Alternative Form of the Clausius-Planck Inequality
323(1)
5.10.6 Reversible Process
323(1)
5.10.7 Entropy Inequality for a Domain with Discontinuity
324(2)
5.11 Fundamental Equations of Continuum Mechanics
326(2)
5.11.1 Particular Cases
327(1)
5.11.1.1 Rigid Body Motion
327(1)
5.11.1.2 Flux Problems
327(1)
5.12 Flux Problems
328(6)
5.12.1 Heat Transfer
328(1)
5.12.1.1 Thermal Conduction
328(2)
5.12.1.2 Thermal Convection Transfer
330(1)
5.12.1.3 Thermal Radiation
330(1)
5.12.1.4 The Heat Flux Equation
330(4)
5.13 Fluid Flow in Porous Media (Filtration)
334(1)
5.14 The Convection-Diffusion Equation
335(3)
5.14.1 The Generalization of the Flux Problem
338(1)
5.15 Initial Boundary Value Problem (IBVP) and Computational Mechanics
338(3)
6 Introduction To Constitutive Equations
341(34)
6.1 Introduction
341(2)
6.2 The Constitutive Principles
343(2)
6.2.1 The Principle of Determinism
344(1)
6.2.2 The Principle of Local Action
344(1)
6.2.3 The Principle of Equipresence
344(1)
6.2.4 The Principle of Objectivity
344(1)
6.2.5 The Principle of Dissipation
344(1)
6.3 Characterization of Constitutive Equations for Simple Thermoelastic Materials
345(6)
6.4 Characterization of the Constitutive Equations for a Thermoviscoelastic Material
351(9)
6.4.1 Constitutive Equations with Internal Variables
355(5)
6.5 Some Experimental Evidence
360(15)
6.5.1 Behavior of Solids
360(2)
6.5.1.1 Temperature Effect
362(1)
6.5.1.2 Some Mechanical Properties of Solids
362(7)
6.5.2 Behavior of Fluids
369(1)
6.5.2.1 Viscosity
370(1)
6.5.3 Behavior of Viscoelastic Materials
371(1)
6.5.4 Rheological Models
372(3)
7 Linear Elasticity
375(48)
7.1 Introduction
375(1)
7.2 Initial Boundary Value Problem of Linear Elasticity
376(1)
7.2.1 Governing Equations
376(1)
7.2.2 Initial and Boundary Conditions
377(1)
7.3 Generalized Hooke's Law
377(4)
7.3.1 The Generalized Hooke's Law in Voigt Notation
378(1)
7.3.2 The Component Transformation Law for the Generalized Hooke's Law
379(1)
7.3.2.1 The Matrix Transformation for Stress and Strain Components
380(1)
7.3.2.2 The Transformation Matrix of the Elasticity Tensor Components
381(1)
7.4 The Elasticity Tensor
381(11)
7.4.1 Anisotropy and Isotropy
381(1)
7.4.2 Types of Elasticity Tensor Symmetry
382(1)
7.4.2.1 Triclinic Materials
382(1)
7.4.2.2 Monoclinic Symmetry (One Plane of Symmetry)
383(1)
7.4.2.3 Orthotropic Symmetry (Two Planes of Symmetry)
384(1)
7.4.2.4 Tetragonal Symmetry
384(2)
7.4.2.5 Transversely Isotropic Symmetry (Hexagonal Symmetry)
386(2)
7.4.2.6 Cubic Symmetry
388(2)
7.4.2.7 Symmetry in All Directions (Isotropy)
390(2)
7.5 Isotropic Materials
392(7)
7.5.1 Constitutive Equations
392(1)
7.5.2 Experimental Determination of Elastic Constants
393(1)
7.5.2.1 Young's Modulus and Poisson's Ratio
393(1)
7.5.2.2 The Shear and Bulk Moduli
394(4)
7.5.3 Restrictions on Elastic Mechanical Properties
398(1)
7.6 Strain Energy Density
399(5)
7.6.1 Decoupling Strain Energy Density
402(2)
7.7 The Constitutive Law for Orthotropic Material
404(1)
7.8 Transversely Isotropic Materials
405(1)
7.9 The saint-Venant's and Superposition Principles
406(2)
7.10 Initial Stress/Strain
408(2)
7.10.1 Thermal Deformation
408(2)
7.11 The Navier-Lame Equations
410(1)
7.12 Two-Dimensional Elasticity
410(8)
7.12.1 The State of Plane Stress
411(1)
7.12.1.1 The Initial Strain
412(1)
7.12.2 The State of Plane Strain
413(2)
7.12.2.1 Thermal Strain
415(2)
7.12.3 Axisymmetric Solids
417(1)
7.13 The Unidimensional Approach
418(5)
7.13.1 Beam Structural Elements
418(2)
7.13.1.1 The Internal Normal Force and the Bending Moments
420(1)
7.13.1.2 The Shear Forces and the Torsional Moment
421(1)
7.13.1.3 The Strain Energy
422(1)
8 Hyperelasticity
423(42)
8.1 Introduction
423(1)
8.2 Constitutive Equations
424(8)
8.2.1 Elastic Tangent Stiffness Tensors
427(1)
8.2.1.1 The Material Elastic Tangent Stiffness Tensor
427(1)
8.2.1.2 The Spatial Elastic Tangent Stiffness Tensor
428(2)
8.2.1.3 The Instantaneous Elastic Tangent Stiffness Tensor
430(1)
8.2.1.4 The Elastic Tangent Stiffness Pseudo-Tensor
431(1)
8.3 Isotropic Hyperelastic Materials
432(8)
8.3.1 The Constitutive Equation in terms of Invariants
434(1)
8.3.1.1 The Constitutive Equation in terms
434(2)
8.3.1.2 The Constitutive Equation in terms of
436(1)
8.3.2 Series Expansion of the Energy Function
436(1)
8.3.3 Constitutive Equations in terms of the Principal Stretches
437(3)
8.4 Compressible Materials
440(7)
8.4.1 The Stress Tensors
442(3)
8.4.2 Compressible Isotropic Materials
445(1)
8.4.2.1 Compressible Isotropic Material in terms of the Invariants
446(1)
8.5 Incompressible Materials
447(4)
8.5.1 Geometrical Interpretation
449(1)
8.5.2 Isotropic Incompressible Hyperelastic Materials
450(1)
8.5.2.1 Series Expansion of the Energy Function for an Isotropic Incompressible Hyperelastic Materials
451(1)
8.6 Examples of Hyperelastic Models
451(11)
8.6.1 The Neo-Hookean Material Model
452(1)
8.6.2 The Ogden Material Model
452(1)
8.6.2.1 The Incompressible Ogden Material Model
452(1)
8.6.2.2 The Hadamard Material Model
453(1)
8.6.3 The Mooney-Rivlin Material Model
453(1)
8.6.3.1 Strain Energy Density
453(1)
8.6.3.2 The Stress Tensor
454(1)
8.6.4 The Yeoh Material Model
454(1)
8.6.4.1 Strain Energy Density
454(1)
8.6.4.2 The Stress Tensor
454(1)
8.6.5 The Arruda-Boyce Material Model
454(1)
8.6.6 The Blatz-Ko Hvperelastic Model
455(1)
8.6.7 The Saint Venant-Kirchhoff Model
455(1)
8.6.7.1 Strain Energy Density
455(1)
8.6.7.2 The Stress Tensor
456(1)
8.6.7.3 The Elastic Tangent Stiffness Tensor
456(1)
8.6.8 The Compressible Neo-Hookean Material Model
457(1)
8.6.8.1 Strain Energy Density
457(1)
8.6.8.2 The Stress Tensor
457(1)
8.6.8.3 The Elastic Tangent Stiffness Tensor
458(2)
8.6.9 The Gent Model
460(1)
8.6.10 The Statistical Model
460(1)
8.6.11 The Eight-Parameter Model
461(1)
8.7 Anisotropic Hyperelasticity
462(3)
8.7.1 Transversely Isotropic Material
463(2)
9 Plasticity
465(82)
9.1 Introduction
465(2)
9.2 The Yield Criterion
467(24)
9.2.1 The Yield Surface for Anisotropic Materials
468(1)
9.2.1.1 The Yield Surface Gradient
468(1)
9.2.2 The Yield Surface for Isotropic Materials
468(3)
9.2.3 The Yield Surface for Materials Independent of Pressure
471(1)
9.2.3.1 The von Mises Yield Criterion
471(4)
9.2.3.2 The Tresca Yield Criterion
475(2)
9.2.4 The Yield Criteria for Pressure-Dependent Materials
477(1)
9.2.4.1 The Mohr-Coulomb Criterion
478(4)
9.2.4.2 The Drucker-Prager Yield Criterion
482(4)
9.2.4.3 The Rankine Yield Criterion
486(3)
9.2.5 Evolution of the Yield Surface
489(2)
9.3 Plasticity Models in Small Deformation Regime (Uniaxial Cases)
491(12)
9.3.1 Rate-Independent Plasticity Models (Uniaxial Case)
491(1)
9.3.1.1 Perfect Elastoplastic Behavior
491(4)
9.3.1.2 Isotropic Hardening Elastoplastic Behavior
495(5)
9.3.1.3 Kinematic Hardening Elastoplastic Behavior
500(2)
9.3.1.4 Isotropic-Kinematic Elastoplastic Behavior
502(1)
9.4 Plasticity in Small Deformation Regime (The Classical Plasticity Theory)
503(12)
9.4.1 The Infinitesimal Strain Tensor and Constitutive Equation
504(1)
9.4.2 Helmholtz Free Energy
505(1)
9.4.3 Internal Energy Dissipation and the Evolution of the Internal Variables
505(2)
9.4.4 The Elastoplastic Tangent Stiffness Tensor
507(5)
9.4.5 The Classical Flow Theory
512(1)
9.4.5.1 Perfect Plasticity
512(1)
9.4.5.2 Isotropic-Kinematic Hardening Plasticity
513(2)
9.5 Plastic Potential Theory
515(3)
9.6 Plasticity in Large Deformation Regime
518(1)
9.7 Large-Deformation Plasticity Based on the Multiplicative Decomposition of the Deformation Gradient
518(29)
9.7.1 Kinematic Tensors
518(2)
9.7.1.1 Deformation and Strain Tensors
520(4)
9.7.1.2 Area and Volume Elements Deformation
524(1)
9.7.1.3 The Spatial Velocity Gradient
525(3)
9.7.1.4 The Oldrovd Rate
528(1)
9.7.1.5 The Cotter-Rivlin Rate
529(2)
9.7.2 The Stress Tensors
531(1)
9.7.2.1 Stress Tensor Rates
532(1)
9.7.3 The Helmholtz Free Energy
533(1)
9.7.3.1 Decoupling the Helmholtz Free Energy
533(1)
9.7.3.2 The Objectivity Principle for the Helmholtz Free Energy
533(1)
9.7.3.3 The Isotropic Helmholtz Free Energy
534(1)
9.7.3.4 The Rate of Change of the Isotropic Helmholtz Free Energy
534(2)
9.7.4 The Plastic Potential and the Yield Criterion
536(1)
9.7.5 The Dissipation and the Constitutive Equation
537(1)
9.7.6 Evolution of the Internal Variables
538(1)
9.7.7 The Elastoplastic Tangent Stiffness Tensors
539(1)
9.7.7.1 The Elastoplastic Tangent Stiffness Tensor
540(2)
9.7.8 The Hyperelastoplastic Model with von Mises Yield Criterion
542(1)
9.7.8.1 The Helmholtz Free Energy
542(1)
9.7.8.2 The Stress Tensor
543(1)
9.7.8.3 Formulation Considering the Transformation as an Isochoric Transformation
544(1)
9.7.8.4 The Rate of Change of the Helmholtz Free Energy
545(1)
9.7.8.5 Yield Criterion and Evolution of the Internal Variables
546(1)
10 Thermoelasticity
547(40)
10.1 Thermodynamic Potentials
547(5)
10.1.1 The Specific Internal Energy
548(1)
10.1.2 The Specific Helmholtz Free Energy
548(1)
10.1.3 The Specific Gibbs Free Energy
549(1)
10.1.4 The Specific Enthalpy
550(2)
10.2 Thermomechanical Parameters
552(4)
10.2.1 Isothermal and Isentropic Processes
552(1)
10.2.2 Specific Heats and Latent Heat Tensors
553(3)
10.3 Linear Thermoelasticity
556(9)
10.3.1 Linearization of the Constitutive Equations
556(1)
10.3.1.1 The Linearized Piola-Kirchhoff Stress Tensor
557(1)
10.3.1.2 The Linearized Heat Flux Vector
558(2)
10.3.1.3 Linearized Entropy
560(1)
10.3.1.4 The Helmholtz Free Energy Approach
560(1)
10.3.1.5 Linearization of the Constitutive Equations
561(1)
10.3.1.6 Linear Thermoelasticity in a Small Deformation Regime
561(1)
10.3.1.7 Linear Thermoelasticity in a Small Deformation Regime
562(3)
10.4 The Decoupled Thermo-Mechanical Problem in a Small Deformation Regime
565(4)
10.4.1 The Purely Thermal Problem
567(1)
10.4.2 The Purely Mechanical Problem
568(1)
10.5 The Classical theory of Thermoelasticity in Finite Strain (Large Deformation Regime)
569(4)
10.5.1 The Coupled Heat Flux Equation
570(2)
10.5.2 The Specific Helmholtz Free Energy
572(1)
10.6 Thermoelasticity based on the Multiplicative Decomposition of the Deformation Gradient
573(10)
10.6.1 Kinematic Tensors
574(2)
10.6.2 The Stress Tensor
576(1)
10.6.3 Area and Volume Elements
576(2)
10.6.4 Isotropic Materials
578(1)
10.6.5 The Constitutive Equations
579(1)
10.6.5.1 The Constitutive Equation for Energy
579(1)
10.6.5.2 The Constitutive Equations for Stress
580(2)
10.6.5.3 The Constitutive Equation for Entropy
582(1)
10.7 Thermoplasticity in a Small Deformation Regime
583(4)
10.7.1 The Specific Helmholtz Free Energy
583(1)
10.7.2 Internal Energy Dissipation
584(3)
11 Damage Mechanics
587(48)
11.1 Introduction
587(1)
11.2 The Isotropic Damage Model in a Small Deformation Regime
588(17)
11.2.1 Description of the Isotropic Damage Model in Uniaxial Cases
588(1)
11.2.1.1 The Constitutive Equation
589(1)
11.2.2 The Three-Dimensional Isotropic Damage Model
590(1)
11.2.2.1 Helmholtz Free Energy
590(1)
11.2.2.2 Internal Energy Dissipation and the Constitutive Equations
591(2)
11.2.2.3 "Ingredients" of the Damage Model
593(6)
11.2.2.4 The Hardening/Softening Law
599(2)
11.2.3 The Elastic-Damage Tangent Stiffness Tensor
601(1)
11.2.4 The Energy Norms
602(1)
11.2.4.1 The Symmetrical Damage Model (Tension-Compression) -- Model I
602(1)
11.2.4.2 The Tension-Only Damage Model -- Model II
603(1)
11.2.4.3 The Non-Symmetrical Damage Model -- Model III
604(1)
11.3 The Generalized Isotropic Damage Model
605(4)
11.3.1 The Strain Energy Function
606(1)
11.3.2 Spherical and Deviatoric Effective Stress
607(1)
11.3.3 Thermodynamic Considerations
607(1)
11.3.4 The Elastic-Damage Tangent Stiffness Tensor
608(1)
11.4 The Elastoplastic-Damage Model in a Small Deformation Regime
609(6)
11.4.1 The Elasto-Plastic Damage Model by Simo & Ju (1987) in a Small Deformation Regime
610(1)
11.4.1.1 Helmholtz Free Energy
610(1)
11.4.1.2 Internal Energy Dissipation. Constitutive Equations. Thermodynamic Considerations
611(1)
11.4.1.3 Damage Characterization
612(1)
11.4.1.4 The Elastic-Damage Tangent Stiffness Tensor
612(1)
11.4.1.5 Characterization of the Plastic Response. The Elastoplastic-Damage Tangent Stiffness Tensor
613(2)
11.5 The Tensile-Compressive Plastic-Damage Model
615(6)
11.5.1 Helmholtz Free Energy
616(1)
11.5.2 Damage Characterization
617(1)
11.5.3 Evolution of the Damage Parameters
618(1)
11.5.4 Evolution of the Plastic Strain Tensor
619(1)
11.5.5 Internal Energy Dissipation
619(2)
11.6 Damage in a Large Deformation Regime
621(14)
11.6.1 Gurtin & Francis' One-Dimensional Model
622(1)
11.6.2 The Rate Independent 3D Elastic-Damage Model
622(1)
11.6.3 The Damage Variable. Damage Evolution
623(1)
11.6.4 The Plastic-Damage Model by Simo & Ju (1989)
624(1)
11.6.4.1 Specific Helmholtz Free Energy
624(1)
11.6.4.2 Internal Energy Dissipation. Constitutive Equations. Thermodynamic Considerations
624(2)
11.6.4.3 Damage Characterization
626(1)
11.6.4.4 The Hyperelastic-Damage Tangent Stiffness Tensor
626(1)
11.6.4.5 Characterization of the Plastic Response. The Effective Elastoplastic - Damage Tangent Stiffness Tensor
627(1)
11.6.4.6 The Elastoplastic-Damage Tangent Stiffness Tensor
628(1)
11.6.5 The Plastic-Damage Model by Ju(1989)
628(1)
11.6.5.1 Helmholtz Free Energy
629(1)
11.6.5.2 Internal Energy Dissipation. Constitutive Equation. Thermodynamic Considerations
629(1)
11.6.5.3 Characterization of Damage. The Tangent Damage Hyperelasticity Tensor
630(1)
11.6.5.4 The Elastic-Damage Tangent Stiffness Tensor
630(1)
11.6.5.5 Characterization of Plastic Response. The elastoplastic Tangent Stiffness Tensor
631(1)
11.6.5.6 The Elastoplastic-Damage Tangent Stiffness Tensor
632(3)
12 Introduction To Fluids
635(24)
12.1 Introduction
635(1)
12.2 Fluids at Rest and in Motion
636(1)
12.2.1 Fluids at Rest
636(1)
12.2.2 Fluids in Motion
637(1)
12.3 Viscous and Non-Viscous Fluids
637(2)
12.3.1 Non-Viscous Fluids (Perfect Fluids)
638(1)
12.3.2 Viscous Fluids
638(1)
12.4 Laminar Turbulent Flow
639(1)
12.5 Particular Cases
640(2)
12.5.1 Incompressible Fluids
640(1)
12.5.2 Irrotational Flow
641(1)
12.5.3 Steady Flow
641(1)
12.6 Newtonian Fluids
642(3)
12.6.1 The Stokes' Condition
645(1)
12.7 Stress, Dissipated and Recoverable Powers
645(2)
12.8 The Fundamental Equations for Newtonian Fluids
647(12)
12.8.1 The Navier-Stokes-Duhem Equations of Motion
648(1)
12.8.1.1 Alternative Form of the Fundamental Equations for Newtonian Fluids
648(1)
12.8.1.2 The Fundamental Equations for Incompressible Newtonian Fluid
649(1)
12.8.2 The Navier-Stokes Equations of Motion
650(1)
12.8.3 The Euler Equations of Motion
650(1)
12.8.3.1 Non Viscous and Incompressible Fluids
651(1)
12.8.3.2 Bernoulli's Equation
652(1)
12.8.4 The Equation of Vorticity
653(6)
Bibliography 659(8)
Index 667
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