Classical Continuum Mechanics (Applied and Computational Mechanics)

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
PREFACE
ABOUT THE AUTHOR
LIST OF ABBREVIATIONS
1. INTRODUCTION
2. CONCEPTS AND MATHEMATICAL PRELIMINARIES
	2.1. Introduction
	2.2. Notations
		2.2.1. Einstein notations
		2.2.2. Index notations and Kronecker delta
		2.2.3. Vector and matrix notations
		2.2.4. Remarks
	2.3. Permutation tensor
	2.4. -identity
	2.5. Operations using vector, matrix and Einstein notations
		2.5.1. Multiplication
		2.5.2. Substitution
		2.5.3. Factoring
		2.5.4. Contraction (or trace)
	2.6. Reference frame and reference frame transformation
	2.7. Coordinate frames and transformations
		2.7.1. Cartesian frame and orthogonal coordinate
		2.7.2. Curvilinear coordinates (or frame)
	2.8. Curvilinear frames, covariant and contravariant
		2.8.1. Covariant basis
		2.8.2. Contravariant basis
		2.8.3. Alternate way to visualize covariant and contravariant bases ~gi, ~gi
	2.9. Scalars, vectors and tensors
	2.10. Coordinate transformations, definitions and operations
		2.10.1. Coordinate transformation T
		2.10.2. Induced transformations
		2.10.3. Isomorphism between coordinate transformations and induced transformations
		2.10.4. Transformations by covariance and contravariance
		2.10.5. The tensor concept: Covariant and contravariant tensors
	2.11. Tensors in three-dimensional x-frame, tensor operations, orthogonal coordinate transformations and invariance
		2.11.1. Tensors in Cartesian x-frame
		2.11.2. Tensor operations
		2.11.3. Transformations of tensors dened in orthogonal frames due to orthogonal coordinate transformation
		2.11.4. Invariants of tensors
		2.11.5. Hamilton-Cayley theorem
		2.11.6. Differential calculus of tensors
	2.12. Some useful relations
	2.13. Summary
3. KINEMATICS OF MOTION, DEFORMATION AND THEIR MEASURES
	3.1. Description of motion
	3.2. Material particle displacements
	3.3. Lagrangian, Eulerian descriptions and descriptions in  fuid mechanics
		3.3.1. Lagrangian or referential description of motion
		3.3.2. Eulerian or spatial description of motion
		3.3.3. Descriptions in fuid mechanics
		3.3.4. Notations
	3.4. Material derivative
		3.4.1. Material derivative in Lagrangian or referential description
		3.4.2. Material derivative in Eulerian or spatial description
	3.5. Acceleration of a material particle
		3.5.1. Lagrangian or referential description
		3.5.2. Eulerian or spatial description
	3.6. Deformation Gradient Tensor
	3.7. Continuous deformation of matter, restrictions on the description of
motion
	3.8. Change of description, co- and contra-variant measures
	3.9. Notations for covariant and contravariant measures
	3.10. Deformation, measures of length and change
		3.10.1. Covariant measures of length and change in length
		3.10.2. Contravariant measures of length and change in length
	3.11. Covariant and contravariant measures of nite strain in Lagrangian and Eulerian descriptions
		3.11.1. Covariant measures of finite strains
		3.11.2. Contravariant measures of finite strains
	3.12. Changes in strain measures due to rigid rotation of frames
		3.12.1. Change in covariant Lagrangian descriptions of strain when xi are changed to x0 i due to rigid rotation
		3.12.2. Changes in contravariant Eulerian measures of strains when xi are changed to x0 i due to rigid rotation
		3.12.3. Change in covariant Lagrangian measures of strain when xi are changed to x0 i by rigid rotation [ Q]
		3.12.4. Changes in Eulerian measures of strain when xi are changed to x0 i by rigid rotation [Q]
	3.13. Invariants of strain tensors
	3.14. Expanded form of strain tensors
		3.14.1. Green's strain measure: ["[0]]
		3.14.2.Almansi strain measure : ["[0]]
	3.15. Physical meaning of strains
		3.15.1. Extensions and stretches parallel to ox1; ox2; ox3 axes in the x-frame: covariant measure of strain in Lagrangian description using "[0], Green's strain in Lagrangian description
		3.15.2. Extensions and stretches parallel to x-frame axes
using ["[0]]
		3.15.3. Angles between the bers or material lines
	3.16. Small deformation, small strain deformation physics
		3.16.1. Green's strain: Lagrangian description
		3.16.2. Almansi strain tensor: Eulerian description
	3.17. Additive and multiplicative decompositions of deformation gradient tensor [J]
		3.17.1. Additive decomposition of [J]
		3.17.2. Multiplicative decomposition of [J]: Polar decomposition
into stretch and rotation tensor
		3.17.3. Strain measures in terms of [Sr], [Sl] and [R]
	3.18. Invariants of [C[0]], [B[0]], [Sr] and [Sl] in terms of principal stretches of [Sr] and [Sl] .
		3.18.1. Principal stretches of [Sr] and [Sl]
		3.18.2. Principal invariants of [C[0]] in terms of ri
		3.18.3. Principal Invariants of [Sr]
		3.18.4. Principal invariants of [B[0]] in terms of li
		3.18.5. Principal Invariants of [Sl]
	3.19. Deformation of areas and volumes
		3.19.1. Areas
		3.19.2. Volumes
		3.19.3.Integral form of fdAg over @V
	3.20. Summary
4. DEFINITIONS AND MEASURES OF STRESSES
	4.1. Concept of stress
	4.2. Cut Principle of Cauchy
	4.3. Deffinition of stress on area
	4.4. Cauchy stress tensor
		4.4.1. Force balance
		4.4.2. Moment of Forces
		4.4.3. Cauchy principle
	4.5. Stress measures: finite deformation, finite strain
		4.5.1. Contravariant Cauchy stress tensor ˙(0) and ˙(0) in Lagrangian and Eulerian descriptions
		4.5.2. Covariant Cauchy stress tensor in ˙(0) and ˙(0) in Eulerian and Lagrangian descriptions
		4.5.3. Mixed stress tensors: Jaumann stress tensor
	4.6. Contravariant Second Piola-Kirchhoff stress tensor ˙[0] or ˙[0]
	4.7. Contravariant First Piola-Kirchhoff or Lagrange stress tensor ˙
	4.8. Covariant Second Piola-Kirchhoff stress tensor ˙[0] or ˙[0]
	4.9. General Remarks
	4.10. Summary of stress measures
		4.10.1. Cauchy stress tensors
		4.10.2. Jaumann stress tensors
		4.10.3. Second Piola-Kirchhoff stress tensors
		4.10.4. First Piola-Kirchhoff stress tensor
	4.11. Conjugate strain measures
	4.12. Relations between different stress measures and some other useful relations
	4.13. Summary
5. RATES, CONVECTED TIME DERIVATIVES AND OBJECTIVITY
	5.1. Rate of deformation
		5.1.1. Lagrangian description
		5.1.2. Eulerian description
	5.2. Additive decomposition of velocity gradient tensor
	5.3. Interpretation of the components of [D]
		5.3.1. Diagonal components of [D]
		5.3.2. Off diagonal components of [D]: physical interpretation
	5.4. Rate of change or material derivative of strain tensors [C[0]] and ["[0]]
	5.5. Physical meaning of spin tensor [ W ]
	5.6. Vorticity vector and vorticity
	5.7. Rate of change of [J], i.e., material derivative of [J]
	5.8. Rate of change of [ J], i.e., material derivative of [ J]
	5.9. Rate of change of det[J], i.e material derivative of jJj
	5.10. Rate of change of det[  J], i.e., material derivative of det[  J]
	5.11. Rate of change of volume, i.e., material derivative of volume
	5.12. Rate of change of area: material derivative of area
	5.13. Convected time derivatives of stress and strain tensors
		5.13.1. Stress and strain measures for convected time derivatives
		5.13.2. Convected time derivatives of the Cauchy stress tensor: compressible matter
		5.13.3. Convected time derivatives of the Cauchy stress tensor: incompressible matter
		5.13.4. Remarks
		5.13.5. Convected time derivatives of the strain tensors
	5.14. Conjugate pairs of convected time derivatives of stress and strain tensors
	5.15. Objective tensors and objective rates
		5.15.1. Galilean transformation: tensors
		5.15.2. Euclidean or Non-Galilean transformations: tensors
		5.15.3. Objective rates
		5.15.4. Remarks
	5.16. Summary
6. CONSERVATION AND BALANCE LAWS IN EULERIAN DESCRIPTION
	6.1. Introduction
	6.2. Localization theorem
	6.3. Mass density
	6.4. Conservation of mass: CM
	6.5. Transport theorem
		6.5.1. Approach I
		6.5.2. Approach II
		6.5.3. Continued development of transport theorem
	6.6. Conservation of mass: CM
		6.6.1. Integral form of CM
		6.6.2. Differential form of CM
	6.7. Kinematics of continuous media: BLM
		6.7.1. Preliminary considerations
		6.7.2. Derivation of equations of BLM
		6.7.3. Approach I: Integral and differential forms of BLM
		6.7.4. Approach II
	6.8. Kinematics of continuous media: BAM
		6.8.1. Integral form of BAM
		6.8.2. Differential form of BAM
	6.9. First law of thermodynamics: FLT
		6.9.1. Integral form of FLT
		6.9.2. Differential form of FLT
	6.10. Second law of thermodynamics: SLT
		6.10.1. Integral form of SLT
		6.10.2. Differential form of SLT
	6.11. Summary of mathematical model from CBL
		6.11.1. Differential form of the CBL
		6.11.2. Integral form of the CBL
	6.12. Summary
7. CONSERVATION AND BALANCE LAWS IN LAGRANGIAN DESCRIPTION
	7.1. Introduction
	7.2. Mathematical model for deforming continua in Lagrangian description
	7.3. Conservation of mass: (CM)
		7.3.1. Integral form of CM
		7.3.2. Differential form of CM
	7.4. Balance of linear momenta: (BLM)
		7.4.1. Differential form of BLM
	7.5. Balance of angular momenta: (BAM)
		7.5.1. Differential form of BAM
	7.6. First law of thermodynamics: (FLT)
		7.6.1. Differential form of the FLT
		7.6.2. Rate of mechanical work conjugate pairs in the energy equation (differential form)
		7.6.3. Energy equation in equivalent rate of work conjugate measures
	7.7. Second law of thermodynamics: (SLT)
		7.7.1. Differential form of SLT
	7.8. Second law of thermodynamics using Gibbs potential (differential form)
		7.8.1. Using ˙[0] and "[0] as conjugate pair
		7.8.2. Using ˙[0] and C[0] as conjugate pair
	7.9. Summary of differential form of CBL
	7.10. Summary
8. CONSTITUTIVE THEORIES
	8.1. Introduction
	8.2. Axioms of constitutive theory
	8.3. Approaches of deriving constitutive theories
		8.3.1. Thermodynamic approach
		8.3.2. Other approaches (not strictly thermodynamic)
	8.4. Considerations in the constitutive theories
		8.4.1. Common deformation physics
	8.5. Thermodynamic approach of deriving constitutive theories
		8.5.1. Entropy inequality in Eulerian description
		8.5.2. Additive decomposition of Cauchy stress tensor (0) 
		8.5.3. Constitutive tensors, their argument tensors and SLT
		8.5.4. Constitutive theory for equilibrium Cauchy stress tensor (0) e ˙ (volumetric deformation physics): Eulerian description, Helmholtz free energy density
		8.5.5. Constitutive theories for equilibrium stress (0) e˙ (volumetric deformation): Lagrangian description
		8.5.6. Constitutive theories for equilibrium contravariant second Piola-Kirchho stress tensor e˙[0]
	8.6. Representation Theorem
		8.6.1. Def: Representation theorem
		8.6.2. Constitutive theory for a symmetric tensor of rank two
		8.6.3. Constitutive variable is a tensor of rank one
	8.7. Other approaches of deriving constitutive theories
	8.8. Summary
9. CONSTITUTIVE THEORIES FOR THERMOELASTIC SOLIDS
	9.1. Introduction
	9.2. Thermodynamic approach
		9.2.1. Finite deformation, nite strain, compressible, non-isothermal
		9.2.2. Finite deformation, nite strain, compressible, isothermal: reversible deformation
		9.2.3. Finite deformation, nite strain, incompressible, non-isothermal
		9.2.4. Finite deformation, nite strain, incompressible, isothermal: reversible deformation
		9.2.5. Small deformation, small strain
	9.3. Other approaches of deriving constitutive theories (not necessarily thermodynamic)
		9.3.1. Constitutive theory for ˙[0] using : reversible
		9.3.2. Constitutive theory for ˙[0] using strain energy density ˇ
		9.3.3. Constitutive theory for ˙[0] using ˇ = ˇ("[0]) and Taylor series expansion: non-isotropic, non-homogeneous solid continua
		9.3.4. Constitutive theory for ˙ using ˇ("): small deformation, small strain
	9.4. Constitutive theories for the heat vector
		9.4.1. Constitutive theory for q using entropy inequality
		9.4.2. Constitutive theories for q using representation theorem
	9.5. General remarks
	9.6. Summary
10. CONSTITUTIVE THEORIES FOR THERMOVISCOELASTIC SOLIDS WITHOUT MEMORY
	10.1. Introduction
	10.2. Finite deformation, nite strain
		10.2.1. Equilibrium stress tensor e˙[0]
		10.2.2. Deviatoric stress d˙[0]
	10.3. Small strain, small deformation
		10.3.1.Equilibrium stress tensor e
		10.3.2. Deviatoric stress d˙
	10.4. Kelvin-Voigt Model
	10.5. 1D wave propagation in viscoelastic solid continua
		10.5.1. Alternate phenomenological model for dissipation
		10.5.2. Model Problem: Numerical Studies
	10.6. Constitutive theories for heat vector q
	10.7. General remarks
	10.8. Summary
11. CONSTITUTIVE THEORIES FOR THERMOVISCOELASTIC SOLIDS WITH MEMORY
	11.1. Introduction
	11.2. Finite deformation, nite strain
		11.2.1. Constitutive theory for d˙[0]
	11.3. Small deformation, small strain
		11.3.1. Constitutive theory for d˙
	11.4. Memory modulus or relaxation modulus
	11.5. Zener constitutive model
	11.6. Model problem: numerical studies
	11.7. Constitutive theories for heat vector q
	11.8. General remarks
	11.9. Summary
12. CONSTITUTIVE THEORIES FOR THERMOVISCOUS FLUIDS
	12.1. Introduction
	12.2. Preliminary considerations
	12.3. Constitutive theory for equilibrium Cauchy stress
	12.4. Constitutive theory for deviatoric Cauchy stress
		12.4.1. A constitutive theory of order one (n = 1) for (0) d ˙: Newtonian and generalized Newtonian fluids
		12.4.2. Linear constitutive theory of order n for (0) d 
		12.4.3. Linear constitutive theory of order one (n = 1): Newtonian and generalized Newtonian fluids
		12.4.4. Generalized Newtonian 
uids: variable transport properties
	12.5. Constitutive theory for heat vector, Eulerian description
		12.5.1. Constitutive theory for q using entropy inequality
		12.5.2. Constitutive theory for q using representation theorem
	12.6. General remarks
	12.7. Summary
13. CONSTITUTIVE THEORIES FOR THERMOVISCOELASTIC FLUIDS
	13.1. Introduction
	13.2. Preliminary considerations
	13.3. Considerations in the constitutive theories
	13.4. Constitutive theory for equilibrium stress (0)e
	13.5. Constitutive theory for deviatoric stress (0)d ˙
		13.5.1. Simplified constitutive theories for (m)d ˙: m = 1, n = 1
		13.5.2. Maxwell constitutive model
		13.5.3. Giesekus Constitutive Model
		13.5.4. Discussion on the Giesekus constitutive model derived here and the constitutive model used currently
		13.5.5. Simplified constitutive theory for deviatoric Cauchy stress tensor d˙(0): m = 1, n = 2
		13.5.6. Oldroyd-B constitutive model (m = 1, n = 2)
		13.5.7. A single constitutive theory for dilute and dense polymeric fluids (m = 1, n = 2)
	13.6. Constitutive theory for heat vector
	13.7. Numerical studies using Giesekus constitutive model
		13.7.1. Model Problem 1: fully developed ow between parallel plates
		13.7.2. Model Problem 2: fully developed ow between parallel plates using 2D formulation
	13.8. General remarks
	13.9. Summary
14. CONSTITUTIVE THEORIES FOR THERMO HYPO-ELASTIC SOLIDS
	14.1. Introduction
	14.2. Preliminary considerations
	14.3. Constitutive theory for equilibrium Cauchy stress (0)e ˙
	14.4. Constitutive theory for deviatoric Cauchy stress
		14.4.1. Linear constitutive theory of order n for (1) d ˙
		14.4.2. Linear constitutive theory of order one (n = 1)
	14.5. Constitutive theory for heat vector q
	14.6. General remarks
	14.7. Summary
15. THERMODYNAMIC RELATIONS AND COMPLETE MATHEMATICAL MODELS
	15.1. Introduction
	15.2. Thermodynamic pressure: equation of state
		15.2.1. Perfect or ideal gas law
		15.2.2. Real gas models
		15.2.3. Compressible solids
	15.3. Internal energy
		15.3.1. Compressible matter
		15.3.2. Incompressible matter
	15.4. Differential form of complete mathematical models in Lagrangian description
		15.4.1. CBL: Finite deformation, finite strain
		15.4.2. Constitutive theory for e˙[0]: finite deformation, finite strain
		15.4.3. Constitutive theory for d˙[0]: finite deformation, finite strain
		15.4.4. CBL: Small deformation, small strain
		15.4.5. Constitutive theory for e˙: small deformation, small strain
		15.4.6. Constitutive theory for d˙: small deformation, small strain
		15.4.7. Constitutive theory for heat vector q
	15.5. Differential form of complete mathematical model in Eulerian description
		15.5.1. CBL: Conservation and balance laws
		15.5.2. Constitutive theory for (0)
e ˙
		15.5.3. Constitutive theory for (0)
d ˙
		15.5.4. Constitutive theory for heat vector 
	15.6. Summary
16. ENERGY METHODS, PRINCIPLE OF VIRTUAL WORK, CALCULUS OF VARIATIONS
	16.1. Introduction
	16.2. Boundary value problems (BVPs)
		16.2.1. Mathematical classi cation of differential operators
		16.2.2. Calculus of variations
		16.2.3. Principle of Virtual work: BVPs
		16.2.4. Final remarks (BVPs)
	16.3. IVPs, energy methods and principle of virtual work
		16.3.1. Energy functional from the differential form of IVP
		16.3.2. Differential form of IVP from energy functional
		16.3.3. Principle of virtual work in IVPs
	16.4. Summary
Appendix A: Combined generators and invariants
Appendix B: Transformations and operations in Cartesian, cylindrical and spherical coordinate systems
	B.1. Cartesian frame x1; x2; x3 and cylindrical frame r; ; z
		B.1.1. Relationship between coordinates of a point in x1; x2; x3 and r; ; z-frames
		B.1.2. Converting derivatives of a scalar with respect to x1; x2; x3 into its derivatives with respect to r; ; z
		B.1.3. Relationship between bases in x1; x2; x3- and r; ; z-frames
	B.2. Cartesian frame x1; x2; x3 and spherical frame r; ; ˚
		B.2.1. Relationship between coordinates of a point in x1; x2; x3 and r; ; ˚-frames
		B.2.2. Converting derivatives of a scalar with respect to x1; x2; x3 into derivatives with respect to r; ; ˚
		B.2.3. Relationship between bases, i.e., unit vectors in x1; x2; x3- and r; ; ˚-frames
	B.3. Differential operations in r; ; z- and r; ; ˚-frames
		B.3.1. r; ; z-frame
		B.3.2. r; ; ˚-frame
	B.4. Some examples: r; ; z-frame
		B.4.1. Symmetric part of the velocity gradient tensor D
		B.4.2. Skew-symmetric part of the velocity gradient tensor W
	B.5. Summary
BIBLIOGRAPHY
INDEX