Tensor analysis and continuum mechanics /
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| Author / Creator: | Talpaert, Yves. |
|---|---|
| Imprint: | Dordrecht ; Boston : Kluwer Academic Publishers, c2002. |
| Description: | xvi, 591 p. : ill. ; 25 cm. |
| Language: | English |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/4861131 |
Table of Contents:
- Preface
- Chapter 1. Tensors
- 1.. First Steps with Tensors
- 1.1. Multilinear forms
- Linear mapping
- Multilinear form
- 1.2. Dual space, vectors and covectors
- Dual space
- Expression of a covector
- Einstein summation convention
- Change of basis and cobasis
- 1.3. Tensors and tensor product
- Tensor product of multilinear forms
- Tensor of type (0/1)
- Tensor of type (1/0)
- Tensor of type (0/2)
- Tensor of type (2/0)
- Tensor of type (1/1)
- Tensor of type (q/p)
- Symmetric and antisymmetric tensors
- 2.. Operations on Tensors
- 2.1. Tensor algebra
- Addition of tensors
- Multiplication of a tensor by a scalar
- Tensor multiplication
- 2.2. Contraction and tensor criteria
- Contraction
- Tensor criteria
- 3.. Euclidean Vector Space
- 3.1. Pre-Euclidean vector space
- Scalar multiplication and pre-Euclidean space
- Fundamental tensor
- 3.2. Canonical isomorphism and conjugate tensor
- Canonical isomorphism
- Conjugate tensor and reciprocal basis
- Covariant and contravariant representations of vectors
- Representation of tensors of order 2 and contracted products
- 3.3. Euclidean vector spaces
- 4.. Exterior Algebra
- 4.1. p-forms
- Definition of a p-form
- Exterior product of 1-forms
- Expression of a p-form
- Exterior product of p-forms
- Exterior algebra
- 4.2. q-vectors
- 5.. Point Spaces
- 5.1. Point space and natural frame
- Point space
- Coordinate system and frame of reference
- Natural frame
- 5.2. Tensor fields and metric element
- Transformations of curvilinear coordinates
- Tensor fields
- Metric element
- 5.3. Christoffel symbols
- Definition of Christoffel symbols
- Ricci identities and Christoffel formulae
- 5.4. Absolute differential, Covariant derivative, Geodesic
- Absolute differential of a vector, covariant derivatives
- Absolute differential of a tensor, covariant derivatives
- Geodesic and Euler's equations
- Absolute derivative of a vector (along a curve)
- 5.5. Volume form and adjoint
- Volume form
- Adjoint
- 5.6. Differential operators
- Gradient
- Divergence
- Curl
- Laplacian
- Exercises
- Chapter 2. Lagrangian and Eulerian Descriptions
- 1.. Lagrangian Description
- 1.1. Configuration
- 1.2. Deformation and Lagrangian Description
- 1.3. Flow and hypotheses of continuity
- 1.4. Trajectories
- 1.5. Streakline
- 1.6. Velocity and acceleration of a particle
- 1.7. Abstract configuration
- 2.. Eulerian Description
- 2.1. Definition; Comparison between L- and E-descriptions
- 2.2. Trajectory and velocity
- 2.3. Streamline
- 2.4. Steady motion
- Exercises
- Chapter 3. Deformations
- 1.. Homogeneous Transformation
- 1.1. Definition of homogeneous transformations
- 1.2. Convective transport
- Convective transport of a vector
- Convective transport of a volume
- Simple shear
- 1.3. Cauchy-Green deformation tensor and stretch
- (Right) Cauchy-Green deformation tensor
- Stretch
- Shear angle
- Principal stretches
- 1.4. Finite strain tensor
- 1.5. Polar decomposition
- Pure stretch and rotation
- Euler-Almansi strain tensor
- 1.6. Rigid body transformation
- 2.. Tangential Homogeneous Transformation
- 2.1. Deformation gradient
- 2.2. Homogeneous transformations of elements
- Transport of vectors, volume deformation, and area deformation
- Stretches
- Strain
- 2.3. Displacement and gradient
- Material displacement gradient
- Spatial displacement gradient
- Curvilinear coordinate system
- 3.. Infinitesimal Transformation
- 3.1. Tensor notions relating to infinitesimal transformations
- 3.2. Compatibility conditions
- 3.3. Rigid body transformation
- Exercises
- Chapter 4. Kinematics of Continua
- 1.. Lagrangian Kinematics
- 1.1. Homogeneous transformation motion
- 1.2. General motion and gradient
- 2.. Eulerian Kinematics
- 2.1. Homogeneous transformation motion
- Velocity field
- Material derivative of a vector
- Material derivative of a volume
- Eulerian rates
- 2.2. General motion and velocity gradient
- Velocity gradient tensor and Eulerian rates
- Lagrangian and Eulerian strain tensors
- Rate of rotation
- Decomposition of motion
- 2.3. Rigid body motion
- 3.. Material Derivatives of Circulation, Flux, and Volume
- 3.1. About the particle derivative
- Physical quantity
- Vector field
- Tensor field
- 3.2. Material derivative of circulation
- 3.3. Material derivative of flux
- 3.4. Material derivative of volume integral
- Lagrangian and Eulerian approaches
- Proper motion case
- Exercises
- Chapter 5. Fundamental Laws; Principle of Virtual Work
- 1.. Conservation of Mass and Continuity Equation
- 1.1. Axiom of mass conservation
- 1.2. Continuity equation
- Continuity equation in the Lagrangian description
- Continuity equation in the Eulerian description
- Mass flow rate
- 1.3. Material derivative of integral of mass density
- 1.4. Isochoric motion, steady and irrotational flows
- Isochoric motion
- Steady flow
- Steady isochoric flow
- Irrotational flow
- Isochoric irrotational flow
- 2.. Fundamental Laws of Dynamics
- 2.1. Body forces and surface forces
- 2.2. Principles of linear momentum and moment of momentum
- 2.3. Cauchy's stress tensor
- 2.4. Cauchy's stress tensor and principles of dynamics
- Linear momentum principle and equilibrium equations
- Moment of momentum principle
- The generalized Cauchy's theorem
- Poisson's theorem
- 3.. Theorem of Kinetic Energy
- 3.1. Theorem of kinetic energy in the Eulerian description
- 3.2. Theorem of kinetic energy in the Lagrangian description
- 4.. Study of Stresses
- 4.1. Reciprocity of stresses
- 4.2. Principal stresses
- 4.3. Stress invariants; deviator
- 4.4. Stress quadric of Cauchy and Lame stress ellipsoid
- 4.5. Geometrical constructions and Mohr's circles
- (Mohr's) stress plane
- Stress vector and plane of Mohr
- Description of Mohr's circles
- Particular stresses
- 5.. Principle of Virtual Work
- 5.1. Preliminary recalls
- 5.2. Rigid body motion
- 5.3. Expressions of virtual power (and virtual work)
- 5.4. Principle of virtual work
- 6.. Thermomechanics and Balance Equations
- 6.1. Balance equation
- Proper motion
- Material domain
- Fixed domain
- 6.2. First principle of thermodynamics
- Principle
- Balance equations and local forms
- Potential energy of body forces
- Internal energy and balance equation
- 6.3. Second principle of thermodynamics
- Principle
- Clausius-Duhem inequality
- Dissipation and reversibility
- 6.4. Conclusion and constitutive equations
- Exercises
- Chapter 6. Linear Elasticity
- 1.. Elasticity and Tests
- 2.. Generalized Hooke's Law in Linear Elasticity
- 2.1. Generalized Hooke's law
- 2.2. Quadratic forms and strain energy function
- 2.3. Isotropic material and Lame coefficients
- Constitutive equations
- Young's modulus and Poisson's ratio
- Bulk modulus
- Shear modulus
- Hooke's law's expression in a general coordinate system
- Navier's equations of motion
- 3.. Equations and Principles in Elastostatics
- 3.1. Navier's equation; the Beltrami equations of compatibility
- 3.2. Principle of superposition
- 3.3. Saint-Venant's principle
- 4.. Classical Problems
- 4.1. Plane problems
- Plane stress problems
- Plane strain problems
- 4.2. Classical problems in elastostatics
- Uniaxial stresses
- Torsion of a circular cylinder body
- Torsion of cylindrical shafts
- Exercises
- Summary of Formulae
- Bibliography
- Glossary of Symbols
- Index
