Advanced elasticity

Welcome to this learning project about Advanced elasticity!

• Project code:
• Suggested Prerequisites:
  • Introduction to Elasticity
  • Continuum mechanics
  • Partial differential equations
• Time investment: 6 months
• Assessment suggestions:
• Portal:Engineering and Technology
• School:Engineering
• Department:Mechanical Engineering
• Stream:Applied Mechanics
• Level: First year graduate

This is an advanced course on the elastic deformations of solids.

This learning project aims to.

• provide the mathematical foundations of advanced elasticity.

1. Nonlinear Elasticity
   1. Deformation measures
      1. Motion, displacement, velocity, acceleration
      2. Strains and deformations
      3. Polar decomposition
      4. Spectral decompositions of kinematic quantities
      5. Volume change and area change
      6. Time derivatives and rate quantities
      7. Objectivity of kinematic quantities
   2. Stress measures
      1. Stress measures
      2. Deviatoric and volumetric stress
      3. Objective stress rates
   3. Balance laws and entropy inequality
   4. Constitutive relations
      1. Cauchy elastic materials
      2. Thermoelastic materials
      3. Relation between Cauchy stress and Green strain
      4. Maxwell relations for thermoelasticity
      5. Balance of energy
      6. Clausius-Duhem inequality
      7. Specific heat relations
      8. Nonlinear elastic materials
         1. Incompressible hyperelastic material
         2. Hyperelastic material models
            1. /Neo-Hookean material
            2. /Mooney-Rivlin material
            3. /Ogden material
            4. /Yeoh material
   5. Examples
2. Buckling
   1. Refresh your basics
      1. Curvature
   2. /Linear buckling
   3. /Nonlinear buckling
   4. Examples
3. Three-dimensional elastcity
   1. /Elastic potentials
   2. Examples
4. Plates and Shells
   1. /Plates
   2. /Shells
   3. Examples
5. Dynamics
   1. /Vibrations
   2. Examples
6. Fracture
   1. Examples

• Homework 1 Problem set
  • Solutions
• Homework 2 Problem set
  • Solutions
• Homework 3 Problem set
  • Solutions
• Homework 4 Problem set
  • Solutions
• Homework 5 Problem set
  • Solutions
• Homework 6 Problem set
  • Solutions
• Homework 7 Problem set
  • Solutions
• Homework 8 Problem set
  • Solutions
• Homework 9 Problem set
  • Solutions
• Homework 10 Problem set
  • Solutions
• Homework 11 Problem set
  • Solutions

• Quiz 1
  • Solutions

Textbooks

• R.W. Ogden, (1984), Non-linear Elastic Deformations, Dover Publications.
• A.J.M Spencer (2004), Continuum Mechanics, Dover Publications. An excellent introduction to continuum mechanics. You should own a copy for your personal library.
• L.E. Malvern (1969), Introduction to the Mechanics of a Continuous Medium, Prentice-Hall. A good reference for continuum mechanics.

References

• Continuum Mechanics:
  • R. M. Brannon (2004), Large Deformation Kinematics. An excellent introduction to kinematics.
  • R. M. Brannon (2004), Rotation. Almost everything you ever wanted to know about rotations.

• Mathematics:
  • R. M. Brannon (2004), Elementary Vector and Tensor Analysis for Engineers. This free online book is the best introduction I have seen for vector and tensor analysis for nonlinear mechanics.
  • R. M. Brannon (2004), Curvilinear Coordinates. Very useful if you wish to follow the literature on the nonlinear deformation of shells.
  • A.P.S. Selvadurai (2000), Partial Differential Equations in Mechanics 1,2. Springer. Excellent introductory text on partial differential equations with engineers in mind.

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