Continuum Mechanics and Thermodynamics From Fundamental Concepts to Governing Equations

ISBN-10: 1107008263
ISBN-13: 9781107008267
Edition: 2011
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Book details

List price: $125.00
Copyright year: 2011
Publisher: Cambridge University Press
Publication date: 12/15/2011
Binding: Hardcover
Pages: 372
Size: 7.50" wide x 9.75" long x 0.75" tall
Weight: 2.134
Language: English

Ellad B. Tadmor is Professor of Aerospace Engineering and Mechanics at the University of Minnesota. His research focuses on multiscale method development and the microscopic foundations of continuum mechanics.

Ryan S. Elliott is Assistant Professor of Aerospace Engineering and Mechanics at the University of Minnesota. An expert in stability of continuum and atomistic systems, he has received many awards for his work.

Preface
Acknowledgments
Notation
Introduction
Theory
Scalars, vectors and tensors
Frames of reference and Newton's laws
Tensor notation
Direct versus indicial notation
Summation and dummy indices
Free indices
Matrix notation
Kronecker delta
Permutation symbol
What is a tensor?
Vector spaces and the inner product and norm
Coordinate systems and their bases
Cross product
Change of basis
Vector component transformation
Generalization to higher-order tensors
Tensor component transformation
Tensor operations
Addition
Magnification
Transpose
Tensor products
Contraction
Tensor basis
Properties of tensors
Orthogonal tensors
Symmetric and antisymmetric tensors
Principal values and directions
Cayley-Hamilton theorem
The quadratic form of symmetric second-order tensors
Isotropic tensors
Tensor fields
Partial differentiation of a tensor field
Differential operators in Cartesian coordinates
Differential operators in curvilinear coordinates
Divergence theorem
Exercises
Kinematics of deformation
The continuum particle
The deformation mapping
Material and spatial field descriptions
Material and spatial tensor fields
Differentiation with respect to position
Description of local deformation
Deformation gradient
Volume changes
Area changes
Pull-back and push-forward operations
Polar decomposition theorem
Deformation measures and their physical significance
Spatial strain tensor
Linearized kinematics
Kinematic rates
Material time derivative
Rate of change of local deformation measures
Reynolds transport theorem
Exercises
Mechanical conservation and balance laws
Conservation of mass
Reynolds transport theorem for extensive properties
Balance of linear momentum
Newton's second law for a system of particles
Balance of linear momentum for a continuum system
Cauchy's stress principle
Cauchy stress tensor
An alternative ("tensorial") derivation of the stress tensor
Stress decomposition
Local form of the balance of linear momentum
Balance of angular momentum
Material form of the momentum balance equations
Material form of the balance of linear momentum
Material form of the balance of angular momentum
Second Piola�Kirchhoff stress
Exercises
Thermodynamics
Macroscopic observables, thermodynamic equilibrium and state variables
Macroscopically observable quantities
Thermodynamic equilibrium
State variables
Independent state variables and equations of state
Thermal equilibrium and the zeroth law of thermodynamics
Thermal equilibrium
Empirical temperature scales
Energy and the first law of thermodynamics
First law of thermodynamics
Internal energy of an ideal gas
Thermodynamic processes
General thermodynamic processes
Quasistatic processes
The second law of thermodynamics and the direction of time
Entropy
The second law of thermodynamics
Stability conditions associated with the second law
Thermal equilibrium from an entropy perspective
Internal energy and entropy as fundamental thermodynamic relations
Entropy form of the first law
Reversible and irreversible processes
Continuum thermodynamics
Local form of the first law (energy equation)
Local form of the second law (Clausius-Duhem inequality)
Exercises
Constitutive relations
Constraints on constitutive relations
Local action and the second law of thermodynamics
Specific internal energy constitutive relation
Coleman�Noll procedure
Onsager reciprocal relations
Constitutive relations for alternative stress variables
Thermodynamic potentials and connection with experiments
Material frame-indifference
Transformation between frames of reference
Objective tensors
Principle of material frame-indifference
Constraints on constitutive relations due to material frame-indifference
Reduced constitutive relations
Continuum field equations and material frame-indifference
Controversy regarding the principle of material frame-indifference
Material symmetry
Simple fluids
Isotropic solids
Linearized constitutive relations for anisotropic hyperelastic solids
Generalized Hooke's law and the elastic constants
Limitations of continuum constitutive relations
Exercises
Boundary-value problems, energy principles and stability
Initial boundary-value problems
Problems in the spatial description
Problems in the material description
Equilibrium and the principle of stationary potential energy (PSPE)
Stability of equilibrium configurations
Definition of a stable equilibrium configuration
Lyapunov's indirect method and the linearized equations of motion
Lyapunov's direct method and the principle of minimum potential energy (PMPE)
Exercises
Solutions
Universal equilibrium solutions
Universal equilibrium solutions for homogeneous simple elastic bodies
Universal solutions for isotropic and incompressible hyperelastic materials
Family 0: homogeneous deformations
Family 1: bending, stretching and shearing of a rectangular block
Family 2: straightening, stretching and shearing of a sector of a hollow cylinder
Family 3: inflation, bending, torsion, extension and shearing of an annular wedge
Family 4: inflation or eversion of a sector of a spherical shell
Family 5: inflation, bending, extension and azimuthal shearing of an annular wedge
Summary and the need for numerical solutions
Exercises
Numerical solutions: the finite element method
Discretization and interpolation
Energy minimization
Solving nonlinear problems: initial guesses
The generic nonlinear minimization algorithm
The steepest descent method
Line minimization
The Newton�Raphson (NR) method
Quasi-Newton methods
The finite element tangent stiffness matrix
Elements and shape functions
Element mapping and the isoparametric formulation
Gauss quadrature
Practical issues of implementation
Stiffness matrix assembly
Boundary conditions
The patch test
The linear elastic limit with small and finite strains
Exercises
Approximate solutions: reduction to the engineering theories
Mass transfer theory
Heat transfer theory
Fluid mechanics theory
Elasticity theory
Afterword
Further reading
Books related to Part I on theory
Books related to Part II on solutions
Heuristic microscopic derivation of the total energy
Summary of key continuum mechanics equations
References
Index

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