Continuum mechanics

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Welcome to this learning project about Continuum mechanics!

Contents 1 Learning Project Summary 2 Content summary 3 Goals 4 Contents 4.1 Syllabus and Learning Materials 4.2 Assignments 4.3 Tests and Quizzes 4.4 Textbooks and References 4.5 Reading List 5 Learning materials

[edit] Learning Project Summary

• Project code:
• Suggested Prerequisites:
  • Linear algebra
  • 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

[edit] Content summary

This is an introductory course on the continuum mechanics of solids.

[edit] Goals

This learning project aims to.

• provide the mathematical foundations of continuum mechanics.
• expose students to some of the numerous constitutive models of solids.

[edit] Contents

[edit] Syllabus and Learning Materials

1. Mathematical Preliminaries
   1. Set notation
   2. Functions
   3. Vectors
   4. Matrices
   5. Tensors
      1. Useful tensor algebra identities
      2. Useful relations between tensors and vectors
      3. Curl of the gradient of a vector
      4. Relations between surface and volume integrals
      5. Leibniz formula in one dimension
      6. Leibniz formula in three dimensions
   6. Partial differential equations
   7. Variational calculus
2. Kinematics
   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
3. Stress measures and stress rates
   1. Stress measures
   2. Deviatoric and volumetric stress
   3. Objective stress rates
4. Balance laws
   1. Governing equations and thermodynamics
   2. Balance of mass
   3. Balance of linear momentum
   4. Balance of angular momentum
   5. Balance of energy
   6. Clausius-Duhem inequality
5. Constitutive relations
   1. Thermoelasticity
      1. Relation between Cauchy stress and Green strain
      2. Maxwell relations for thermoelasticity
      3. Balance of energy
      4. Clausius-Duhem inequality
      5. Specific heat relations
   2. Nonlinear Elasticity
   3. Plasticity
   4. Viscoplasticity
   5. Viscoelasticity.

[edit] Assignments

• 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

[edit] Tests and Quizzes

• Quiz 1
  • Solutions

[edit] Textbooks and References

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Textbooks

• 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.
• Nonlinear Finite Elements for Continua and Structures by T. Belytschko, W. K. Liu, and B. Moran, John Wiley and Sons, 2000.
• Computational Inelasticity by J. C. Simo and T. J. R. Hughes, Springer, 1998.

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.

[edit] Reading List

• ...

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