| |
| |
| 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 | |