Foundations of Solid Mechanics

I Mathematical Foundations.- 1.1 Tensors and continuum mechanics.- 1.2 Scalars and vectors.- 1.3 Indicial notation.- 1.4 Algebra of Cartesian tensors.- 1.5 Matrices and determinants.- 1.6 Linear equations and Eigenvalue problem.- 1.7 Theorems on tensor fields.- 1.8 Differential geometry.- 1.9 Dirac-delta and Heaviside step functions.- 1.10 Bessel functions.- 1.11 Laplace transforms.- 1.12 Inverse Laplace transforms.- 1.13 One-to-one mappings.- 1.14 Curvilinear coordinates.- 1.15 Derivatives with respect to curvilinear coordinates.- 1.16 Exercise problems.- II Stress and Strain Tensors.- 2.1 Introduction.- 2.2 Force distribution and stresses.- 2.3 Stress vector and equations of mation.- 2.4 Euler's laws of motion.- 2.5 Stress tensor.- 2.6 Stationary shear stresses.- 2.7 Octahedral shear stress and stress deviator.- 2.8 Strain tensor.- 2.9 Compatibility conditions.- 2.10 Cylindrical and spherical coordinates.- 2.11 Problems.- 2.11.1 Stress or strain computation from three different normal components.- 2.11.2 Cylindrical and spherical rotation components.- 2.11.3 Rigid-body rotation and translation components.- 2.12 Exercise problems.- III Linear Elasticity.- 3.1 Strain energy function.- 3.2 Orthotopic and isotropic elastic solids.- 3.3 Young's moduli and Poisson's ratios for orthotropic elastic solids.- 3.4 Solution schemes.- 3.5 Field equations in tenns of displacements.- 3.6 Problems.- 3.6.1 Beltrami-Michell compatibility conditions.- 3.6.2 Conservative forces and potentials.- 3.6.3 Elastostatic displacement potentials.- 3.6.4 Elastodynamic displacement potentials.- 3.6.5 Positive definiteness of the strain energy function.- 3.6.6 Stress and strain computation from measured results.- 3.6.7 Saturated porous elastic media.- IV Elastostatic Plane Problems.- 4.1 Plane problems of orthotropic elastic materials.- 4.2 Airy function for isotropic plane problems.- 4.3 Isotropic elastic plane problems in cylindrical coordinates.- 4.4 Displacement for a given biharmonic function.- 4.5 Examples of infinite plane problems.- 4.6 Particular solutions for concentrated forces.- 4.7 Exercise problems.- Table 4.1 Complementary and particular solutions for elastostatics of isotropic planes.- V Bending of Elastic Thin Plates.- 5.1 Basic assumptions.- 5.2 Equilibrium, boundary conditions and stress resultants.- 5.3 Physical meaning of stress resultants.- 5.4 Governing conditions for isotropic plates.- 5.5 Solutions for rectangular plates.- 5.6 Closed form solutions for circular plates.- 5.7 Series solutions for circular plates.- 5.8 Polygonal plates supported at corners.- 5.9 Plates on elastic foundation.- 5.10 Exercise problems.- Table 5.1 Complementary and particular solutions for elastostatic bending of thin isotropic plates.- VI Elastostatics with Displacements as Unknowns.- 6.1 Field equations for plane problems.- 6.2 Solution scheme for large planes.- 6.3 Solution scheme for large spaces.- 6.4 Homogeneous half planes and half spaces.- 6.5 Concentrated force inside a half space.- 6.6 Load transfer problems.- 6.7 Infinite elements for multilayered half spaces.- 6.8 Saturated large spaces.- 6.9 Exercise problems.- VII Linear Viscoelasticity.- 7.1 Linear elasticity and Newtonian viscosity.- 7.2 Creep and relaxation.- 7.3 Compliance and modulus of mechanical models.- 7.4 Differential equations for stress-strain relationship.- 7.5 Steady state harmonic oscillation.- 7.6 Thermorheologically simple solids.- 7.7 Three-dimensional theory.- 7.8 Quasi-static solution by separation of variables.- 7.9 Steady state harmonic solution scheme.- 7.10 Integral transform methods and their limitations.- 7.11 Three-dimensional thermoviscoelasticity.- 7.12 Problems.- 7.12.1 Reciprocal theorem for harmonic oscillation.- 7.12.2 Vibration of a bar with a viscoelastic support.- 7.12.3 Indentation on a viscoelastic half space.- 7.12.4 Torsional oscillation of a hollow cylinder.- 7.12.5 Quasi-static torsional oscillation of a hollow cylinder.- 7.12.6 Dynamic response of an incompressible cylinder.- 7.12.7 Isothermal harmonic vibration.- 7.12.8 Isothermal effects on stretched string.- 7.12.9 Varying temperature effects on stretched string.- 7.12.10 Heating of an infinite slab.- VIII Wave Propagation.- 8.1 Terminology in wave propagation.- 8.2 Wavefront and jumps.- 8.3 Velocity jumps in isotropic elastic domains.- 8.4 Reflection and tansmission at interfaces and boundaries.- 8.5 Waves in isotropic viscoelastic media.- 8.6 In-plane harmonic surface waves.- 8.7 Antiplane harmonic surface waves.- 8.8 Vibration of multilayered elastic half spaces.- 8.9 Asymmetric vibration of a homogeneous half space.- 8.10 Axisymmetric torsion of a layered half space.- 8.11 Total solution to vibration of half planes.- 8.12 Vibration of viscoelastic half spaces.- 8.13 Infinite elements for a homogeneous half space.- 8.14 Exercise problems.- IX Plasticity.- 9.1 Facts from simple tests.- 9.2 Basic assumptions and common characteristics of various theories.- 9.3 Various yield functions.- 9.4 Hardening and flow rules.- 9.5 Incremental formulation for isotropic hardening.- 9.6 Viscoplasticity.- X Finite Deformation.- 10.1 Different descriptions of changing configuration.- 10.2 Material derivative and conservation of mass.- 10.3 Stress tensors in different descriptions.- 10.4 Equations of motion in different descriptions.- 10.5 Finite strain tensors.- 10.6 Reformed Lagrangian description.- 10.7 Strain tensors in curvilinear coordinates.- 10.8 Equilibrium equations and stress tensors in curvilinear coordinates.- 10.9 Physical components of vectors and tensors.- 10.10 Boundary conditions and constitutive relationship in curvilinear coordinates.- 10.11 Compatibility conditions.- 10.12 Problems.- 10.12.1 Constitutive law in Eulerian description.- 10.12.2 Maxwell-Betti reciprocal theorem for finite deformation.- References.- Author Index.