Continuum mechanics

Welcome to this learning project about continuum mechanics!

Introduction

Topic of continuum mechanics deals with the basics equations of motion that governs the mechanics of both solid and fluid objects in nature.

Project metadata

Educational level: this is a tertiary (university) resource.

Subject classification: this is an engineering resource.

• Suggested Prerequisites:
  • Linear algebra
  • Partial differential equations
• Time investment: 6 months
• Portal: Engineering and Technology
• School: Engineering
• Department: Mechanical engineering, Civil engineering, Aeronautical engineering, Applied mechanics
• Level: Senior year undergraduate and graduate students

Content summary

This is an introductory course on the continuum mechanics of materials. Small deformation theory is generalized for finite deformation scenarios and applied for both solid and fluids.

Goals

This learning project aims to.

• provide the mathematical foundations of continuum mechanics
• expose students to extension of small deformation theory into the finite deformation regime in solids and fluids

Contents

Syllabus and Learning Materials Syllabus and Learning Materials

Assignments, tests and quizzes Assignments

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.

Tests and Quizzes

• Quiz 1
  • Solutions
• Kinematics Test

Textbooks and References

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Textbooks

• Continuum Mechanics by A.J.M Spencer, Dover Publications, 2004
• Introduction to the Mechanics of a Continuous Medium by L.E. Malvern, Prentice-Hall, 1969

References

• Mathematics:
  • R. M. Brannon (2004), Elementary Vector and Tensor Analysis for Engineers.
  • R. M. Brannon (2004), Curvilinear Coordinates.
  • A.P.S. Selvadurai (2000), Partial Differential Equations in Mechanics 1,2. Springer

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