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  • 1. Objectives and Applications >
    • 1.1 Defining a Problem >
      • 1.1.1 Deciding what to calculate
      • 1.1.2 Defining geometry
      • 1.1.3 Defining loading
      • 1.1.4 Choosing physics
      • 1.1.5 Defining material behavior
      • 1.1.6 A representative problem
      • 1.1.7 Choosing a method of analysis
  • 2. Governing Equations >
  • 2.1 Deformation measures >
  • 2.1.1 Displacement and Velocity
  • 2.1.2 Deformation gradient
  • 2.1.3 Deformation gradient from two deformations
  • 2.1.4 Jacobian of deformation gradient
  • 2.1.5 Lagrange strain
  • 2.1.6 Eulerian strain
  • 2.1.7 Infinitesimal Strain
  • 2.1.8 Engineering Shear Strain
  • 2.1.9 Volumetric and Deviatoric strain
  • 2.1.10 Infinitesimal rotation
  • 2.1.11 Principal strains
  • 2.1.12 Cauchy-Green deformation tensors
  • 2.1.13 Rotation tensor, Stretch tensors
  • 2.1.14 Principal stretches
  • 2.1.15 Generalized strain measures
  • 2.1.16 Velocity gradient
  • 2.1.17 Stretch rate and spin
  • 2.1.18 Infinitesimal strain/rotation rate
  • 2.1.19 Other deformation rates
  • 2.1.20 Strain equations of compatibility
  • 2.2 Internal forces >
  • 2.2.1 Surface traction/body force
  • 2.2.2 Internal tractions
  • 2.2.3 Cauchy stress
  • 2.2.4 Kirchhoff, Nominal, Material stress
  • 2.2.5 Stress for infinitesimal motions
  • 2.2.6 Principal stresses
  • 2.2.7 Hydrostatic, Deviatoric, Von Mises stress
  • 2.2.8 Stresses at a boundary
  • 2.3 Equations of motion >
  • 2.3.1 Linear momentum balance
  • 2.3.2 Angular momentum balance
  • 2.3.3 Equations using other stresses
  • 2.4 Work and Virtual Work >
  • 2.4.1 Work done by Cauchy stress
  • 2.4.2 Work done by other stresses
  • 2.4.3 Work for infinitesimal motions
  • 2.4.4 Principle of virtual work
  • 2.4.5 Virtual work with other stresses
  • 2.4.6 Virtual work for small strains
  • 3. Constitutive Equations >
  • 3.1 General requirements
  • 3.2 Linear elasticity >
  • 3.2.1 Isotropic elastic behavior
  • 3.2.2 Isotropic stress-strain laws
  • 3.2.3 Plane stress & strain
  • 3.2.4 Isotropic material data
  • 3.2.5 Lame, Shear, & Bulk modulus
  • 3.2.6 Interpreting elastic constants
  • 3.2.7 Strain energy density (isotropic)
  • 3.2.8 Anisotropic stress-strain laws
  • 3.2.9 Interpreting anisotropic constants
  • 3.2.10 Anisotropic strain energy density
  • 3.2.11 Basis change formulas
  • 3.2.12 Effect of material symmetry
  • 3.2.13 Orthotropic materials
  • 3.2.14 Transversely isotropic materials
  • 3.2.15 Transversely isotropic data
  • 3.2.16 Cubic materials
  • 3.2.17 Cubic material data
  • 3.3 Hypoelasticity
  • 3.4 Elasticity w/ large rotations
  • 3.5 Hyperelasticity >
  • 3.5.1 Deformation measures
  • 3.5.2 Stress measures
  • 3.5.3 Strain energy density
  • 3.5.4 Incompressible materials
  • 3.5.5 Energy density functions
  • 3.5.6 Calibrating material models
  • 3.5.7 Representative properties
  • 3.6 Viscoelasticity >
  • 3.6.1 Polymer behavior
  • 3.6.2 General constitutive equations
  • 3.6.3 Spring-damper approximations
  • 3.6.4 Prony series
  • 3.6.5 Calibrating constitutive laws
  • 3.6.6 Calibrating material models
  • 3.6.7 Representative properties
  • 3.7 Rate independent plasticity >
  • 3.7.1 Plastic metal behavior
  • 3.7.2 Elastic/plastic strain decomposition
  • 3.7.3 Yield criteria
  • 3.7.4 Graphical yield surfaces
  • 3.7.5 Hardening laws
  • 3.7.6 Plastic flow law
  • 3.7.7 Unloading condition
  • 3.7.8 Summary of stress-strain relations
  • 3.7.9 Representative properties
  • 3.7.10 Principle of max. plastic resistance
  • 3.7.11 Drucker's postulate
  • 3.7.12 Microscopic perspectives
  • 3.8 Viscoplasticity >
  • 3.8.1 Creep behavior
  • 3.8.2 High strain rate behavior
  • 3.8.3 Constitutive equations
  • 3.8.4 Representative creep properties
  • 3.8.5 Representative high rate properties
  • 3.9 Large strain plasticity >
  • 3.9.1 Deformation measures
  • 3.9.2 Stress measures
  • 3.9.3 Elastic stress-strain relations
  • 3.5.4 Plastic stress-strain relations
  • 3.10 Large strain viscoelasticity >
  • 3.10.1 Deformation measures
  • 3.10.2 Stress measures
  • 3.10.3 Stress-strain energy relations
  • 3.10.4 Strain relaxation
  • 3.10.5 Representative properties
  • 3.11 Critical state soils >
  • 3.11.1 Soil behavior
  • 3.11.2 Constitutive laws (Cam-clay)
  • 3.11.3 Response to 2D loading
  • 3.11.4 Representative properties
  • 3.12 Crystal plasticity >
  • 3.12.1 Basic crystallography
  • 3.12.2 Features of crystal plasticity
  • 3.12.3 Deformation measures
  • 3.12.4 Stress measures
  • 3.12.5 Elastic stress-strain relations
  • 3.12.6 Plastic stress-strain relations
  • 3.12.7 Representative properties
  • 3.13 Surfaces and interfaces >
  • 3.13.1 Cohesive interface models
  • 3.13.2 Contact and friction
  • 4. Solutions to simple problems >
  • 4.1 Axial/Spherical linear elasticity >
  • 4.1.1 Elastic governing equations
  • 4.1.2 Spherically symmetric equations
  • 4.1.3 General spherical solution
  • 4.1.4 Pressurized sphere
  • 4.1.5 Gravitating sphere
  • 4.1.6 Heated spherical shell
  • 4.1.7 Axially symmetric equations
  • 4.1.8 General axisymmetric solution
  • 4.1.9 Pressurized cylinder
  • 4.1.10 Spinning circular disk
  • 4.1.11 Interference fit
  • 4.2 Axial/Spherical elastoplasticity >
  • 4.2.1 Plastic governing equations
  • 4.2.2 Spherically symmetric equations
  • 4.2.3 Pressurized sphere
  • 4.2.4 Cyclically pressurized sphere
  • 4.2.5 Axisymmetric equations
  • 4.2.6 Pressurized cylinder
  • 4.3 Spherical hyperelasticity >
  • 4.3.1 Governing equations
  • 4.3.2 Spherically symmetric equations
  • 4.3.3 Pressurized sphere
  • 4.4 1D elastodynamics >
  • 4.4.1 Surface subjected to pressure
  • 4.4.2 Surface under tangential loading
  • 4.4.3 1-D bar
  • 4.4.4 Plane waves
  • 4.4.5 Wave speeds in isotropic solid
  • 4.4.6 Reflection at a surface
  • 4.4.7 Reflection at an interface
  • 4.4.8 Plate impact experiment
  • 5. Solutions for elastic solids >
  • 5.1 General Principles >
  • 5.1.1 Governing equations
  • 5.1.2 Navier equation
  • 5.1.3 Superposition & linearity
  • 5.1.4 Uniqueness & existence
  • 5.1.5 Saint-Venants principle
  • 5.2 2D Airy function solutions >
  • 5.2.1 Airy solution in rectangular coords
  • 5.2.2 Demonstration of Airy solution
  • 5.2.3 Airy solution in polar coords
  • 5.2.4 End loaded cantilever
  • 5.2.5 Line load perpendicular to surface
  • 5.2.6 Line load parallel to surface
  • 4.4.7 Pressure on a surface
  • 4.4.8 Uniform pressure on a strip
  • 4.4.8 Stress near a crack tip
  • 5.3 2D Complex variable solutions >
  • 5.3.1 Complex variable solution
  • 5.3.2 Demonstration of CV solution
  • 5.3.3 Line force
  • 5.3.4 Edge dislocation
  • 5.3.5 Circular hole in infinite solid
  • 5.3.6 Slit crack
  • 5.3.7 Bimaterial interface crack
  • 5.3.8 Rigid flat punch on a surface
  • 5.3.9 Parabolic punch on a surface
  • 5.3.10 General line contact
  • 4.3.11 Frictional sliding contact
  • 4.3.12 Dislocation near a surface
  • 5.4 3D static problems >
  • 5.4.1 Papkovich-Neuber potentials
  • 5.4.2 Demonstration of PN potentials
  • 5.4.3 Point force in infinite solid
  • 5.4.4 Point force normal to surface
  • 5.4.5 Point force tangent to surface
  • 5.4.6 Eshelby inclusion problem
  • 5.4.7 Inclusion in an elastic solid
  • 5.4.8 Spherical cavity in infinite solid
  • 5.4.9 Flat cylindrical punch on surface
  • 5.4.10 Contact between spheres
  • 4.4.11 Relations for general contacts
  • 4.4.12 P-d relations for axisymmetric contact
  • 5.5 2D Anisotropic elasticity >
  • 5.5.1 Governing equations
  • 5.5.2 Stroh solution
  • 5.5.3 Demonstration of Stroh solution
  • 5.5.4 Stroh matrices for cubic materials
  • 5.5.5 Degenerate materials
  • 5.5.6 Fundamental elasticity matrix
  • 5.5.7 Orthogonality of Stroh matrices
  • 5.5.8 Barnett/Lothe & Impedance tensors
  • 5.5.9 Properties of matrices
  • 5.5.10 Basis change formulas
  • 5.5.11 Barnett-Lothe integrals
  • 5.5.12 Uniform stress state
  • 5.5.13 Line load/dislocation in infinite solid
  • 5.5.14 Line load/dislocation near a surface
  • 5.6 Dynamic problems >
  • 5.6.1 Love potentials
  • 5.6.2 Pressurized spherical cavity
  • 5.6.3 Rayleigh waves
  • 5.6.4 Love waves
  • 5.6.5 Elastic waves in waveguides
  • 5.7 Energy methods >
  • 5.7.1 Definition of potential energy
  • 5.7.2 Minimum energy theorem
  • 5.7.3 Simple example of energy minimization
  • 5.7.4 Variational approach to beam theory
  • 5.7.5 Estimating stiffness
  • 5.8 Reciprocal theorem >
  • 5.8.1 Statement and proof of theorem
  • 5.8.2 Simple example
  • 5.8.3 Boundary-internal value relations
  • 5.8.4 3D dislocation loops
  • 5.9 Energetics of dislocations >
  • 5.9.1 Potential energy of isolated loop
  • 5.9.2 Nonsingular dislocation theory
  • 5.9.3 Dislocation in bounded solid
  • 5.9.4 Energy of interacting loops
  • 5.9.5 Peach-Koehler formula
  • 5.10 Rayleigh Ritz method >
  • 5.10.1 Mode shapes, nat. frequencies, Rayleigh's principle
  • 5.10.2 Natural frequency of a beam
  • 6. Solutions for plastic solids >
  • 6.1 Slip-line fields >
  • 6.1.1 Interpreting slip-line fields
  • 6.1.2 Derivation of slip-line fields
  • 6.1.3 Examples of solutions
  • 6.2 Bounding theorems >
  • 6.2.1 Definition of plastic dissipation
  • 6.2.2 Principle of min plastic dissipation
  • 6.2.3 Upper bound collapse theorem
  • 6.2.4 Lower bound collapse theorem
  • 6.2.5 Examples of bounding theorems
  • 6.2.6 Lower bound shakedown theorem
  • 6.2.7 Examples of lower bound shakedown theorem
  • 6.2.8 Upper bound shakedown theorem
  • 6.2.9 Examples of upper bound shakedown theorem
  • 7. Introduction to FEA >
  • 7.1 Guide to FEA >
  • 7.1.1 FE mesh
  • 7.1.2 Nodes and elements
  • 7.1.3 Special elements
  • 7.1.4 Material behavior
  • 7.1.5 Boundary conditions
  • 7.1.6 Constraints
  • 7.1.7 Contacting surface/interfaces
  • 7.1.8 Initial conditions/external fields
  • 7.1.9 Soln procedures / time increments
  • 7.1.10 Output
  • 7.1.11 Units in FEA calculations
  • 7.1.12 Using dimensional analysis
  • 7.1.13 Scaling governing equations
  • 7.1.14 Remarks on dimensional analysis
  • 7.2 Simple FEA program >
  • 7.2.1 FE mesh and connectivity
  • 7.2.2 Global displacement vector
  • 7.2.3 Interpolation functions
  • 7.2.4 Element strains & energy density
  • 7.2.5 Element stiffness matrix
  • 7.2.6 Global stiffness matrix
  • 7.2.7 Boundary loading
  • 7.2.8 Global force vector
  • 7.2.9 Minimizing potential energy
  • 7.2.10 Eliminating prescribed displacements
  • 7.2.11 Solution
  • 7.2.12 Post processing
  • 7.2.13 Example code
  • 8. Theory & Implementation of FEA >
  • 8.1 Static linear elasticity >
  • 8.1.1 Review of virtual work
  • 8.1.2 Weak form of governing equns
  • 8.1.3 Interpolating displacements
  • 8.1.4 Finite element equations
  • 8.1.5 Simple 1D implementation
  • 8.1.6 Summary of 1D procedure
  • 8.1.7 Example 1D code
  • 8.1.8 Extension to 2D/3D
  • 8.1.9 2D interpolation functions
  • 8.1.10 3D interpolation functions
  • 8.1.11 Volume integrals
  • 8.1.12 2D/3D integration schemes
  • 8.1.13 Summary of element matrices
  • 8.1.14 Sample 2D/3D code
  • 8.2 Dynamic elasticity >
  • 8.2.1 Governing equations
  • 8.2.2 Weak form of governing eqns
  • 8.2.3 Finite element equations
  • 8.2.4 Newmark time integration
  • 8.2.5 Simple 1D implementation
  • 8.2.6 Example 1D code
  • 8.2.7 Lumped mass matrices
  • 8.2.8 Example 2D/3D code
  • 8.2.9 Modal time integration
  • 8.2.10 Natural frequencies/mode shapes
  • 8.2.11 Example 1D modal dynamic code
  • 8.2.12 Example 2D/3D modal dynamic code
  • 8.3 Hypoelasticity >
  • 8.3.1 Governing equations
  • 8.3.2 Weak form of governing eqns
  • 8.3.3 Finite element equations
  • 8.3.4 Newton-Raphson iteration
  • 8.3.5 Tangent moduli for hypoelastic solid
  • 8.3.6 Summary of Newton-Raphson method
  • 8.3.7 Convergence problems
  • 8.3.8 Variations on Newton-Raphson
  • 8.3.9 Example code
  • 8.4 Hyperelasticity >
  • 8.4.1 Governing equations
  • 8.4.2 Weak form of governing eqns
  • 8.4.3 Finite element equations
  • 8.4.4 Newton-Raphson iteration
  • 8.4.5 Neo-Hookean tangent moduli
  • 8.4.6 Evaluating boundary integrals
  • 8.4.7 Convergence problems
  • 8.4.8 Example code
  • 8.5 Viscoplasticity >
  • 8.5.1 Governing equations
  • 8.5.2 Weak form of governing eqns
  • 8.5.3 Finite element equations
  • 8.5.4 Integrating the stress-strain law
  • 8.5.5 Material tangent
  • 8.5.6 Newton-Raphson solution
  • 8.5.7 Example code
  • 8.6 Advanced elements >
  • 8.6.1 Shear locking/incompatible modes
  • 8.6.2 Volumetric locking/Reduced integration
  • 8.6.3 Incompressible materials/Hybrid elements
  • 9. Modeling Material Failure >
  • 9.1 Mechanisms of failure >
  • 9.1.1 Monotonic loading
  • 9.1.2 Cyclic loading
  • 9.2 Stress/strain based criteria >
  • 9.2.1 Stress based criteria
  • 9.2.2 Probabilistic methods
  • 9.2.3 Static fatigue criterion
  • 9.2.4 Models of crushing failure
  • 9.2.5 Ductile failure criteria
  • 9.2.6 Strain localization
  • 9.2.7 High cycle fatigue
  • 9.2.8 Low cycle fatigue
  • 9.2.9 Variable amplitude loading
  • 9.3 Elastic fracture mechanics >
  • 9.3.1 Crack tip fields
  • 9.3.2 Linear elastic fracture mechanics
  • 9.3.3 Calculating stress intensities
  • 9.3.4 Using FEA
  • 9.3.5 Measuring toughness
  • 9.3.6 Values of fracture toughness
  • 9.3.7 Stable tearing
  • 9.3.8 Mixed mode fracture
  • 9.3.9 Static fatigue
  • 9.3.10 Cyclic fatigue
  • 9.3.11 Finding cracks
  • 9.4 Energy methods in fracture >
  • 9.4.1 Definition of energy release rate
  • 9.4.2 Energy based fracture criterion
  • 9.4.3 G-K relations
  • 9.4.4 G-compliance relation
  • 9.4.5 Calculating K with compliance
  • 9.4.6 Integral expression for G
  • 9.4.7 The J integral
  • 9.4.8 Calculating K using J
  • 9.5 Plastic fracture mechanics >
  • 9.5.1 Dugdale-Barenblatt model
  • 9.5.2 HRR crack tip fields
  • 9.5.3 J based fracture mechanics
  • 9.6 Interface fracture mechanics >
  • 9.6.1 Interface crack tip fields
  • 9.6.2 Interface fracture mechanics
  • 9.6.3 Stress intensity factors
  • 9.6.4 Crack path selection
  • 10. Rods, Beams, Plates & Shells >
  • 10.1 Dyadic notation
  • 10.2 Deformable rods - general >
  • 10.2.1 Characterizing the x-section
  • 10.2.2 Coordinate systems
  • 10.2.3 Kinematic relations
  • 10.2.4 Displacement, velocity and acceleration
  • 10.2.5 Deformation gradient
  • 10.2.6 Strain measures
  • 10.2.7 Kinematics of bent rods
  • 10.2.8 Internal forces and moments
  • 10.2.9 Equations of motion
  • 10.2.10 Constitutive equations
  • 10.2.11 Strain energy density
  • 10.3 String / beam theory >
  • 10.3.1 Stretched string
  • 10.3.2 Straight beam (small deflections)
  • 10.3.3 Axially loaded beam
  • 10.4 Solutions for rods >
  • 10.4.1 Vibration of a straight beam
  • 10.4.2 Buckling under gravitational loading
  • 10.4.3 Post buckled shape of a rod
  • 10.4.4 Rod bent into a helix
  • 10.4.5 Helical spring
  • 10.5 Shells - general >
  • 10.5.1 Coordinate systems
  • 10.5.2 Using non-orthogonal bases
  • 10.5.3 Deformation measures
  • 10.5.4 Displacement and velocity
  • 10.5.5 Deformation gradient
  • 10.5.6 Other strain measures
  • 10.5.7 Internal forces and moments
  • 10.5.8 Equations of motion
  • 10.5.9 Constitutive relations
  • 10.5.10 Strain energy
  • 10.6 Plates and membranes >
  • 10.6.1 Flat plates (small strain)
  • 10.6.2 Flat plates with in-plane loading
  • 10.6.3 Plates with large displacements
  • 10.6.4 Membranes
  • 10.6.5 Membranes in polar coordinates
  • 10.7 Solutions for shells >
  • 10.7.1 Circular plate bent by pressure
  • 10.7.2 Vibrating circular membrane
  • 10.7.3 Natural frequency of rectangular plate
  • 10.7.4 Thin film on a substrate (Stoney eqs)
  • 10.7.5 Buckling of heated plate
  • 10.7.6 Cylindrical shell under axial load
  • 10.7.7 Twisted open walled cylinder
  • 10.7.8 Gravity loaded spherical shell
  • A: Vectors & Matrices
  • B: Intro to tensors
  • C: Index Notation
  • D: Using polar coordinates
  • E: Misc derivations
• Problems
  • 1. Objectives and Applications >
    • 1.1 Defining a Problem
  • 2. Governing Equations >
  • 2.1 Deformation measures
  • 2.2 Internal forces
  • 2.3 Equations of motion
  • 2.4 Work and Virtual Work
  • 3. Constitutive Equations >
  • 3.1 General requirements
  • 3.2 Linear elasticity
  • 3.3 Hypoelasticity
  • 3.4 Elasticity w/ large rotations
  • 3.5 Hyperelasticity
  • 3.6 Viscoelasticity
  • 3.7 Rate independent plasticity
  • 3.8 Viscoplasticity
  • 3.9 Large strain plasticity
  • 3.10 Large strain viscoelasticity
  • 3.11 Critical state soils
  • 3.12 Crystal plasticity
  • 3.13 Surfaces and interfaces
  • 4. Solutions to simple problems >
  • 4.1 Axial/Spherical linear elasticity
  • 4.2 Axial/Spherical elastoplasticity
  • 4.3 Spherical hyperelasticity
  • 4.4 1D elastodynamics
  • 5. Solutions for elastic solids >
  • 5.1 General Principles
  • 5.2 2D Airy function solutions
  • 5.3 2D Complex variable solutions
  • 5.4 3D static problems
  • 5.5 2D Anisotropic elasticity
  • 5.6 Dynamic problems
  • 5.7 Energy methods
  • 5.8 Reciprocal theorem
  • 5.9 Energetics of dislocations
  • 5.10 Rayleigh Ritz method
  • 6. Solutions for plastic solids >
  • 6.1 Slip-line fields
  • 6.2 Bounding theorems
  • 7. Introduction to FEA >
  • 7.1 Guide to FEA
  • 7.2 Simple FEA program
  • 8. Theory & Implementation of FEA >
  • 8.1 Static linear elasticity
  • 8.2 Dynamic elasticity
  • 8.3 Hypoelasticity
  • 8.4 Hyperelasticity
  • 8.5 Viscoplasticity
  • 8.6 Advanced elements
  • 9. Modeling Material Failure >
  • 9.1 Mechanisms of failure
  • 9.2 Stress/strain based criteria
  • 9.3 Elastic fracture mechanics
  • 9.4 Energy methods in fracture
  • 9.5 Plastic fracture mechanics
  • 9.6 Interface fracture mechanics
  • 10. Rods, Beams, Plates & Shells >
  • 10.1 Dyadic notation
  • 10.2 Deformable rods - general
  • 10.3 String / beam theory
  • 10.4 Solutions for rods
  • 10.5 Shells - general
  • 10.6 Plates and membranes
  • 10.7 Solutions for shells
  • A: Vectors & Matrices
  • B: Intro to tensors
  • C: Index Notation
  • D: Using polar coordinates
  • E: Misc derivations
• FEA codes
  • Maple
  • Matlab
• Report an error

Problems for Chapter 3

Constitutive Models: Relations between Stress and Strain

3.1.  General Restrictions on Constitutive Equations

The following problems illustrate the consequences of general restrictions on constitutive equations:

Linear elastic materials: 3.2.6, 3.2.7

Hyperelastic materials: 3.3.2

Generalized Hooke’s law: 3.4.3

Hyperelastic materials: 3.5.8, 3.5.9.

(c) A.F. Bower, 2008
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