## Table of Contents

1. Objectives and Applications of Solid Mechanics1.1 Defining a Problem in Solid Mechanics1.1.1 Deciding what to calculate 1.1.2 Defining the geometry of the solid 1.1.3 Defining loading 1.1.4 Deciding what physics to include in the model 1.1.5 Defining material behavior 1.1.6 A representative initial value problem in solid mechanics 1.1.7 Choosing a method of analysis

2. Governing Equations2.1 Mathematical Description of Shape Changes in Solids2.1.1 The displacement and velocity fields 2.1.2 The displacement gradient and deformation gradient tensors 2.1.3 Deformation gradient resulting from two successive deformations 2.1.4 The Jacobian of the deformation gradient 2.1.5 The Lagrange strain tensor 2.1.6 The Eulerian strain tensor 2.1.7 The infinitesimal strain tensor 2.1.8 Engineering shear strains 2.1.9 Decomposition of infinitesimal strain into volumetric and deviatoric parts 2.1.10 The infinitesimal rotation tensor 2.1.11 Principal values and directions of the infinitesimal strain tensor 2.1.12 Cauchy-Green deformation tensors 2.1.13 Rotation tensor, Left and Right stretch tensors 2.1.14 Principal stretches 2.1.15 Generalized strain measures 2.1.16 The velocity gradient 2.1.17 Stretch rate and spin tensors 2.1.18 Infinitesimal strain rate and rotation rate 2.1.19 Other deformation rate measures 2.1.20 Strain equations of compatibility for infinitesimal strains2.2 Mathematical Description of Internal Forces in Solids2.2.1 Surface traction and body force 2.2.2 Traction acting on planes within a solid 2.2.3 The Cauchy (true) stress tensor 2.2.4 Other stress measures - Kirchhoff, Nominal and Material stress 2.2.5 Stress measures for infinitesimal deformations 2.2.6 Principal stresses and directions 2.2.7 Hydrostatic stress, Deviatoric stress, and Von Mises effective stress 2.2.8 Stresses near an external surface or edge – boundary conditions2.3 Equations of motion and equilibrium for deformable solids2.3.1 Linear momentum balance in terms of Cauchy stress 2.3.2 Angular momentum balance in terms of Cauchy stress 2.3.3 Equations of motion in terms of other stress measures2.4 Work done by stresses; Principle of Virtual Work2.4.1 Work done by Cauchy stresses 2.4.2 Rate of mechanical work in terms of other stress measures 2.4.3 Rate of mechanical work for infinitesimal deformations 2.4.4 The principle of Virtual Work 2.4.5 The Virtual Work equation in terms of other stress measures 2.4.6 The Virtual Work equation for infinitesimal deformations3. Constitutive Equations: Relations between Stress and Strain3.1 General Requirements for Constitutive Equations3.2 Linear Elastic Material Behavior3.2.1 Isotropic, linear elastic material behavior 3.2.2 Stress—strain relations for isotropic, linear elastic materials 3.2.3 Reduced stress-strain equations for plane deformation of isotropic solids 3.2.4 Representative values for density, and elastic constants of isotropic solids 3.2.5 Other elastic constants – bulk, shear and Lame modulus 3.2.6 Physical Interpretation of the elastic constants for isotropic materials 3.2.7 Strain energy density for isotropic solids 3.2.8 Stress-strain relation for a general anisotropic linear elastic material 3.2.9 Physical Interpretation of the anisotropic elastic constants. 3.2.10 Strain energy density for anisotropic, linear elastic solids 3.2.11 Basis change formulas for the anisotropic elastic constants 3.2.12 The effect of material symmetry on anisotropic stress-strain relations 3.2.13 Stress-strain relations for linear elastic orthotropic materials 3.2.14 Stress-strain relations for linear elastic transversely isotropic material 3.2.15 Values for elastic constants of transversely isotropic crystals 3.2.16 Linear elastic stress-strain relations for cubic materials 3.2.17 Representative values for elastic properties of cubic crystals3.3 Hypoelasticity - elasticity with nonlinear stress-strain behavior3.4 Generalized Hooke’s law – elastic materials subjected to small stretches but large rotations3.5 Hyperelasticity - time independent behavior of rubbers and foams subjected to large strains3.5.1 Deformation measures used in finite elasticity 3.5.2 Stress measures used in finite elasticity 3.5.3 Calculating stress-strain relations from the strain energy density 3.5.4 A note on perfectly incompressible materials 3.5.5 Specific forms of the strain energy density 3.5.6 Calibrating nonlinear elasticity models 3.5.7 Representative values of material properties for rubbers and foams3.6 Viscoelasticity - time dependent behavior of polymers at small strains3.6.1 Features of the small-strain rate dependent response of polymers 3.6.2 General constitutive equations for linear viscoelastic solids 3.6.3 Spring-damper approximations to the relaxation modulus 3.6.4 Prony series representation for the relaxation modulus 3.6.5 Calibrating the constitutive laws for linear viscoelastic solids 3.6.6 Representative values for viscoelastic properties of polymers3.7 Small strain, rate independent plasticity - metals loaded beyond yield3.7.1 Features of the inelastic response of metals 3.7.2 Decomposition of strain into elastic and plastic parts 3.7.3 Yield criteria 3.7.4 Graphical representation of the yield surface 3.7.5 Hardening laws 3.7.6 The plastic flow law 3.7.7 The elastic unloading condition 3.7.8 Complete incremental stress-strain relations for an elastic-plastic solid 3.7.9 Typical values for yield stress of polycrystalline metals 3.7.10 Perspectives on plastic constitutive equations - The Principle of Maximum Plastic Resistance 3.7.11 Perspectives on plastic constitutive equations - Drucker’s Postulate 3.7.12 Microscopic perspectives on plastic flow in metals3.8 Small strain viscoplasticity - creep and high rate deformation of metals3.8.1 Features of creep behavior 3.8.2 Features of high-strain rate behavior 3.8.3 Small-strain, viscoplastic constitutive equations 3.8.4 Representative values of parameters for creeping solids 3.8.5 Representative values of parameters for high rate deformation3.9 Large strain, rate dependent plasticity3.9.1 Kinematics of finite strain plasticity 3.9.2 Stress measures for finite deformation plasticity 3.9.3 Elastic stress-strain relation for finite strain plasticity 3.9.4 Plastic constitutive law for finite strain viscoplasticity3.10 Large strain viscoelasticity3.10.1 Kinematics for finite strain viscoelasticity 3.10.2 Stress measures for finite strain viscoelasticity 3.10.3 Relation between stress, strain and strain energy density 3.10.4 Strain relaxation 3.10.5 Representative values for material parameters3.11 Critical State Models for Soils3.11.1 Features of soil behavior 3.11.2 Constitutive equations for Cam-Clay 3.11.3 Application of constitutive equations to simple 2D loading 3.11.4 Typical values of material properties for soils3.12 Constitutive models for metal single crystals3.12.1 Review of some important concepts from crystallography 3.12.2 Features of plastic flow in single crystals 3.12.3 Kinematic descriptions used in constitutive models of single crystals 3.12.4 Stress measures used crystal plasticity 3.12.5 Elastic stress-strain relation used in crystal plasticity 3.12.6 Plastic stress-strain relation used in crystal plasticity 3.12.7 Representative values of plastic properties of single crystals3.13 Constitutive models for contacting surfaces and interfaces in solids3.13.1 Cohesive zone models of interfaces 3.13.2 Models of contact and friction between surfaces4. Solutions to simple boundary and initial value problems4.1 Axially & Spherically Symmetric Solutions for Linear Elastic Solids Under Quasi-Static Loading4.1.1 Governing equations of linear elasticity in Cartesian components 4.1.2 Simplified equations for spherically symmetric linear elasticity problems 4.1.3 General solution to the spherically symmetric linear elasticity problem 4.1.4 Pressurized hollow sphere 4.1.5 Gravitating sphere 4.1.6 Sphere with steady state heat flow 4.1.7 Simplified equations for axially symmetric linear elasticity problems 4.1.8 General solution to the axisymmetric boundary value problem 4.1.9 Long cylinder subjected to internal and external pressure 4.1.10 Spinning circular plate 4.1.11 Stresses induced by an interference fit between two cylinders4.2 Axially & Spherically Symmetric Solutions for Elastic-Plastic Solids Under Quasi-Static Loading4.2.1 Summary of governing equations 4.2.2 Simplified equations for spherically symmetric problems 4.2.3 Hollow sphere subjected to monotonically increasing internal pressure 4.2.4 Hollow sphere subjected to cyclic internal pressure 4.2.5 Simplified equations for plane strain axially symmetric solids 4.2.6 Long (plane strain) cylinder subjected to internal pressure.4.3 Spherically Symmetric Solution for Large Strain Elasticity Problems4.3.1 Summary of governing equations in Cartesian coordinates 4.3.2 Simplified equations for incompressible spherically symmetric solids 4.3.3 Pressurized hollow sphere made from an incompressible rubber4.4 Simple Dynamic Solutions for Linear Elastic Solids4.4.1 Surface subjected to time varying normal pressure 4.4.2 Surface subjected to time varying shear traction 4.4.3 1-D bar subjected to end loading 4.4.4 Plane waves in an infinite solid 4.4.5 Summary of wave speeds in isotropic elastic solids. 4.4.6 Reflection of waves traveling normal to a free surface 4.4.7 Reflection and transmission of waves normal to an interface 4.4.8 Simple example involving plane waves - the plate impact experiment5. Analytical Techniques and Solutions for Linear Elastic Solids5.1 General Principles5.1.1 Summary of the governing equations of linear elasticity 5.1.2 Alternative form of the governing equations – the Navier equation 5.1.3 Superposition and linearity of solutions 5.1.4 Uniqueness and existence of solutions to the linear elasticity equations 5.1.5 Saint-Venant's Principle5.2 Airy Function Solutions to Plane Stress and Plane Strain Problems for Linear Elastic Solids5.2.1 The Airy solution in rectangular coordinates 5.2.2 Demonstration that the Airy solution satisfies the governing equations 5.2.3 The Airy solution in cylindrical-polar coordinates 5.2.4 Airy function solution to the end loaded cantilever 5.2.5 2D Line load acting perpendicular to the surface of an infinite solid 5.2.6 2D Line load acting parallel to the surface of an infinite solid 5.2.7 Arbitrary pressure acting on a flat surface 5.2.8 Uniform normal pressure acting on a strip 5.2.9 Stresses near the tip of a crack5.3 Complex Variable Solution to Plane Strain Static Elasticity Problems5.3.1 Complex variable solutions to elasticity problems 5.3.2 Demonstration that the complex variable solution satisfies the governing equations 5.3.3 Complex variable solution for a line force in an infinite solid 5.3.4 Complex variable solution for an edge dislocation in an infinite solid 5.3.5 Cylindrical hole in an infinite solid under remote loading 5.3.6 Crack in an infinite elastic solid under remote loading 5.3.7 Fields near the tip of a crack on a bimaterial interface 5.3.8 Frictionless rigid flat indenter in contact with a half-space 5.3.9 Frictionless parabolic (cylindrical) indenter in contact with a half-space 5.3.10 Line contact between two non-conformal frictionless elastic solids 5.3.11 Sliding contact between two rough elastic cylinders 5.3.12 Dislocation near the surface of a half-space5.4 Solutions to 3D Static Elasticity Problems5.4.1 Papkovich-Neuber Potential representations for 3D solutions 5.4.2 Demonstration that the Papkovich-Neuber solution satisfies the governing equations 5.4.3 Point force in an infinite solid 5.4.4 Point force normal to the surface of an infinite half-space 5.4.5 Point force tangent to the surface of an infinite half-space 5.4.6 The Eshelby Inclusion Problem 5.4.7 Elastically mismatched ellipsoidal inclusion in an infinite solid 5.4.8 Spherical cavity in an infinite solid subjected to remote stress 5.4.9 Flat ended cylindrical indenter in contact with an elastic half-space 5.4.10 Frictionless contact between two elastic spheres 5.4.11 Contact area, pressure, stiffness and elastic limit for general non-conformal contacts 5.4.12 Load-displacement-contact area relations for general axisymmetric contacts5.5 Solutions to plane problems for anisotropic elastic solids5.5.1 Governing equations of elasticity for anisotropic solids 5.5.2 Stroh representation for fields in anisotropic elastic solids 5.5.3 Demonstration that the Stroh representation satisfies the governing equations 5.5.4 Stroh eigenvalues and anisotropy matrices for cubic materials 5.5.5 Degenerate materials 5.5.6 Fundamental elasticity matrix 5.5.7 Orthogonal properties of Stroh matrices A and B 5.5.8 Barnett-Lothe tensors and the Impedance Tensor. 5.5.9 Useful properties of matrices in anisotropic elasticity 5.5.10 Basis change formulas for matrices used in anisotropic elasticity 5.5.11 Barnett-Lothe integrals 5.5.12 Stroh representation for a state of uniform stress 5.5.13 Line load and dislocation in an infinite anisotropic solid 5.5.14 Line load and dislocation below the surface of an elastic half-space5.6 Solutions to dynamic problems for isotropic elastic solids5.6.1 Love potentials for dynamic solutions for isotropic solids 5.6.2 Pressure suddenly applied to the surface of a spherical cavity in an infinite solid 5.6.3 Rayleigh waves 5.6.4 Love waves 5.6.5 Elastic waves in waveguides5.7 Energy Methods for Solving Static Linear Elasticity Problems5.7.1 Definition of the potential energy of a linear elastic solid 5.7.2 The principle of stationary and minimum potential energy 5.7.3 Uniaxial compression of a cylinder solved by energy methods 5.7.4 Variational derivation of the beam equations 5.7.5 Energy methods for calculating stiffness5.8 The Reciprocal Theorem and its Applications5.8.1 Statement and proof of the reciprocal theorem 5.8.2 Simple example using the reciprocal theorem 5.8.3 Formulas relating internal and boundary values of field quantities 5.8.4 Solutions for 3D dislocation loops in an infinite solid5.9 Energetics of Dislocations in Elastic Solids5.9.1 Classical solution for potential energy of an isolated dislocation loop in an infinite solid 5.9.2 Non-singular dislocation theory 5.9.3 Energy of a dislocation loop in a stressed, finite elastic solid 5.9.4 Energy of two interacting dislocation loops 5.9.5 Driving force for dislocation motion – The Peach-Koehler formula5.10 Rayleigh-Ritz Method for Estimating Natural Frequencies For Vibrating Elastic Solids5.10.1 Mode shapes and natural frequencies, orthogonality of mode shapes, and Rayleigh's principle 5.10.2 Estimate of natural frequency of vibration for a beam6. Analytical Techniques and Solutions for Plastic Solids6.1 Slip Line Field Theory6.1.1 Interpreting a slip-line field 6.1.2 Derivation of the slip-line field method. 6.1.3 Examples of slip-line field solutions to boundary value problems6.2 Bounding theorems in plasticity and their applications6.2.1 Definition of the plastic dissipation 6.2.2 The Principle of Minimum Plastic Dissipation 6.2.3 The Upper Bound Plastic Collapse Theorem 6.2.4 Examples of applications of the upper bound theorem 6.2.5 The lower bound plastic collapse theorem 6.2.6 Examples using the lower bound plastic collapse theorem 6.2.7 The lower bound shakedown theorem 6.2.8 Examples using the lower bound shakedown theorem 6.2.9 The upper bound shakedown theorem 6.2.10 Examples using the upper bound shakedown theorem7. Introduction to Finite Element Analysis in Solid Mechanics7.1 A Guide to Using Finite Element Software7.1.1 The Finite Element Mesh for a 2D or 3D component 7.1.2 Nodes and Elements in a Mesh 7.1.3 Special Elements – Beams, Plates, Shells and Truss elements 7.1.4 Material Behavior 7.1.5 Boundary conditions 7.1.6 Constraints 7.1.7 Contacting Surfaces and Interfaces 7.1.8 Initial Conditions and external fields 7.1.9 Solution procedures and time increments 7.1.10 Output 7.1.11 Units in finite element computations 7.1.12 Using Dimensional Analysis to simplify FEA analysis 7.1.13 Simplifying FEA analysis by scaling the governing equations 7.1.14 Dimensional analysis – closing remarks7.2 A Simple Finite Element Program7.2.1 The finite element mesh and element connectivity 7.2.2 The global displacement vector 7.2.3 Element interpolation functions 7.2.4 Element strains, stresses and strain energy density 7.2.5 The element stiffness matrix 7.2.6 The global stiffness matrix 7.2.7 Boundary loading 7.2.8 Global residual force vector 7.2.9 Minimizing the Potential Energy 7.2.10 Eliminating prescribed displacements 7.2.11 Solution 7.2.12 Post processing 7.2.13 Example FEA code8. Theory and Implementation of the Finite Element Method8.1 Generalized FEA for static linear elasticity8.1.1 Review of the principle of virtual work 8.1.2 Integral (weak) form of the governing equations of linear elasticity 8.1.3 Interpolating the displacement field and the virtual velocity field 8.1.4 Finite element equations 8.1.5 Simple 1D Implementation of the finite element method 8.1.6 Summary of the 1D finite element procedure 8.1.7 Example FEM Code and solution 8.1.8 Extending the 1D finite element method to 2 and 3 dimensions 8.1.9 Interpolation functions for 2D elements 8.1.10 Interpolation Functions for 3D elements 8.1.11 Volume integrals for stiffness and force 8.1.12 Numerical integration schemes for 2D and 3D elements 8.1.13 Summary of formulas for element stiffness and force matrices 8.1.14 Sample 2D/3D linear elastostatic FEM code8.2 The Finite Element Method for Dynamic Linear Elasticity8.2.1 Review of the governing equations of dynamic linear elasticity 8.2.2 Expressing the governing equations using the principle of virtual work 8.2.3 The finite element equations of motion for linear elastic solids 8.2.4 Newmark time integration for elastodynamics 8.2.5 1-D implementation of a Newmark scheme 8.2.6 Example 1D dynamic FEM code and solution 8.2.7 Lumped mass matrices 8.2.8 Example 2D and 3D dynamic linear elastic code and solution 8.2.9 Modal method of time integration 8.2.10 Natural frequencies and mode shapes 8.2.11 Example 1D code with modal dynamics 8.2.12 Example 2D/3D FEM code to compute natural frequencies8.3 The Finite Element Method for Nonlinear (Hypoelastic) Materials8.3.1 Summary of governing equations 8.3.2 Governing equations in terms of the Virtual Work Principle 8.3.3 Finite element equations 8.3.4 Solving the finite element equations using Newton Raphson Iteration 8.3.5 Tangent moduli for the hypoelastic solid 8.3.6 Summary of the Newton-Raphson procedure for hypoelastic solids 8.3.7 What to do if the Newton-Raphson iterations don’t converge 8.3.8 Variations on Newton-Raphson iteration 8.3.9 Example hypoelastic FEM code8.4 The Finite Element Method for Large Deformations: Hyperelasticity8.4.1 Summary of governing equations 8.4.2 Governing equations in terms of the principle of virtual work 8.4.3 Finite element equations 8.4.4 Solution using Consistent Newton Raphson Iteration 8.4.5 Tangent stiffness for the neo-Hookean material 8.4.6 Evaluating the boundary traction integrals 8.4.7 Example hyperelastic finite element code8.5 The Finite Element Method for Viscoplasticity8.5.1 Summary of governing equations 8.5.2 Governing equations in terms of the Virtual Work Principle 8.5.3 Finite element equations 8.5.4 Integrating the plastic stress-strain law 8.5.5 Material Tangent 8.5.6 Solution using Consistent Newton Raphson Iteration 8.5.7 Example small-strain plastic FEM code8.6 Advanced Element Formulations: Incompatible Modes; Reduced Integration and Hybrid Elements8.6.1 Shear locking and incompatible mode elements 8.6.2 Volumetric locking and reduced integration elements 8.6.3 Hybrid elements for modeling near-incompressible materials8.7 List of FEA Codes and Input Files9. Modeling Material Failure9.1 Summary of Mechanisms of Fracture and Fatigue under Static and Cyclic Loading9.1.1 Failure under monotonic loading 9.1.2 Failure under cyclic loading9.2 Stress and Strain Based Failure Criteria9.2.1 Stress based failure criteria for brittle solids and composites 9.2.2 Probabilistic Design Methods for Brittle Fracture (Weibull Statistics) 9.2.3 Static Fatigue Criterion for Brittle Materials 9.2.4 Constitutive laws for crushing failure of brittle materials 9.2.5 Ductile Fracture Criteria 9.2.6 Ductile failure by strain localization 9.2.7 Criteria for failure by high cycle fatigue under constant amplitude loading 9.2.8 Criteria for failure by low cycle fatigue 9.2.9 Criteria for failure under variable amplitude cyclic loading9.3 Modeling Failure by Crack Growth:Linear Elastic Fracture Mechanics9.3.1 Crack tip fields in an isotropic, linear elastic solid. 9.3.2 The assumptions and application of linear elastic fracture mechanics 9.3.3 Calculating stress intensity factors 9.3.4 Calculating stress intensity factors using finite element analysis 9.3.5 Measuring fracture toughness 9.3.6 Typical values for fracture toughness 9.3.7 Stable Tearing – Kr curves and Crack Stability 9.3.8 Mixed Mode fracture criteria 9.3.9 Static fatigue crack growth 9.3.10 Cyclic fatigue crack growth 9.3.11 Finding cracks in structures9.4 Energy Methods in Fracture Mechanics9.4.1 Definition of crack tip energy release rate for cracks in linear elastic solids 9.4.2 Energy release rate as a fracture criterion 9.4.3 Relation between energy release rate and stress intensity factor 9.4.4 Relation between energy release rate and compliance 9.4.5 Calculating stress intensity factors using compliance 9.4.6 Integral expressions for energy flux to a crack tip 9.4.7 Rice’s J integral 9.4.8 Calculating energy release rates using the J integral9.5 Plastic Fracture Mechanics9.5.1 Dugdale-Barenblatt cohesive zone model of plastic zone at a crack tip 9.5.2 Hutchinson-Rice-Rosengren crack tip fields for a power-law hardening solid 9.5.3 Plastic Fracture Mechanics based on J9.6 Linear Elastic Fracture Mechanics for Interfaces9.6.1 Crack tip fields for a crack on an interface 9.6.2 Phenomenological Theory of Interface Fracture 9.6.3 Stress Intensity Factors for some Interface Cracks 9.6.4 Crack Path Selection10. Approximate Theories for Solids with Special Shapes: Rods, Beams, Plates and Shells10.1 Preliminaries: Dyadic Notation for Vectors and Tensors10.2 Motion and Deformation of Slender Rods10.2.1 Variables characterizing the geometry of the rod’s cross-section 10.2.2 Coordinate systems and variables characterizing deformation of a rod 10.2.3 Additional deformation measures and useful kinematic relations 10.2.4 Approximating the displacement, velocity and acceleration in the rod 10.2.5 Approximating the deformation gradient 10.2.6 Other strain measures 10.2.7 Kinematics of rods that are bent and twisted in the unstressed state 10.2.8 Representation of forces and moments in slender rods 10.2.9 Equations of motion and boundary conditions 10.2.10 Constitutive equations relating forces to deformation measures in elastic rods 10.2.11 Strain energy of an elastic rod10.3 Simplified Versions of the General Theory of Deformable Rods10.3.1 Stretched flexible string with small transverse deflections. 10.3.2 Straight elastic beam with small deflections and no axial force 10.3.3 Straight elastic beam with small transverse deflections and significant axial force10.4 Exact Solutions to Simple Problems Involving Elastic Rods10.4.1 Free vibration of a straight beam without axial force 10.4.2 Buckling of a column subjected to gravitational loading 10.4.3 Post-buckled shape of an initially straight rod subjected to end thrust 10.4.4 Rod bent and twisted into a helix 10.4.5 Helical spring10.5 Motion and Deformation of Thin Shells: General Theory10.5.1 Coordinate systems and variables characterizing deformation of shells 10.5.2 Vectors and tensors in non-orthogonal bases: Covariant and Contravariant components 10.5.3 Additional deformation measures and kinematic relations 10.5.4 Approximating the displacement and velocity field 10.5.5 Approximating the deformation gradient 10.5.6 Other deformation measures. 10.5.7 Representation of forces and moments in shells 10.5.8 Equations of motion and boundary conditions 10.5.9 Constitutive equations relating forces to deformation measures in elastic shells 10.5.10 Strain energy and kinetic energy of an elastic shell10.6 Simplified Versions of the General Shell Theory10.6.1 Flat plates with small out-of-plane deflections and negligible in-plane loading 10.6.2 Flat plates with small out-of-plane deflections and significant in-plane loading 10.6.3 Flat plates with small in-plane and large transverse deflections 10.6.4 Stretched, flat membrane with small out-of-plane deflections 10.6.5 Membrane equations in cylindrical-polar coordinates10.7 Solutions to Problems Involving Membranes, Plates and Shells10.7.1 Thin circular plate bent by pressure applied to one face 10.7.2 Vibration modes and natural frequencies for a circular membrane 10.7.3 Estimate for the fundamental frequency of vibration of a rectangular flat plate 10.7.4 Bending induced by inelastic strain in a thin film on a substrate 10.7.5 Bending of a circular plate due to a through-thickness temperature gradient 10.7.6 Buckling of a cylindrical shell subjected to axial loading 10.7.7 Torsion of an open-walled circular cylinder 10.7.8 Membrane shell theory analysis of a spherical dome under gravitational loadingAppendix A: Review of Vectors and MatricesAppendix B: A Brief Introduction to Tensors and their PropertiesAppendix C: Index Notation for Vector and Tensor OperationsAppendix D: Vector and Tensor Operations in Polar CoordinatesAppendix E: Miscellaneous Derivations