Elasticity

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| Subject classification: this is an engineering resource.

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

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Welcome to the Introduction to Elasticity learning project. Here you will find notes, assignments, and other useful information that will introduce you to this exciting subject.

## Learning Project Summary

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* Project code:
* Suggested Prerequisites:
* A course in mechanics of materials.
* Partial differential equations.
* Time investment: 6 months
* Assessment suggestions:
* Portal:Engineering and technology
* School:Engineering
* Department:Mechanical engineering
* Stream:Applied mechanics
* Subject Area:Elasticity
* Level:First year graduate or higher

## Objectives

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The theory of elasticity deals with the deformations of elastic solids and has a well developed mathematical basis. This course will deal with applied engineering aspects of the theory and will include :

1. Definition of stresses, strains, equilibrium and compatibility.
2. Derivation of the governing equations.
3. Solution of problems in plane stress, plane strain, torsion, bending.
4. Introduction to three-dimensional problems.

Vectors and tensors will be discussed and used to enhance understanding of the theory where necessary. The course intends to provide the student with the tools and an understanding of the use of vectors and tensors in describing the deformation and motion of elastic solids, the formulation of the governing equations using physical laws, and the solution of simple linear elasticity problems using various analytical techniques.

## Syllabus

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1. Vectors, tensors and index notation
1. Vectors
2. Tensors
3. Coordinate transformations
4. Sample homework problems
* Example 1
2. Kinematics - deformation, strain measures, strain-displacement relations
1. Kinematics: Strains and Strain-displacement relations
2. Homogeneous and inhomogeneous displacements
3. Rigid body motions
4. Principal strains
5. Antiplane shear
6. Plane strain
7. Examples
* Example 1
* Example 2
* Example 3
* Example 4
3. Kinetics - stress measures and stress-traction relations
1. Kinetics: Stress-force relations
2. Principal stresses
3. VonMises stress
4. Plane stress
5. Examples
* Example 1
* Example 2
* Example 3
* Example 4
4. Constitutive equations - Hookes's law, linearized elasticity
1. Constitutive laws: Stress-strain relations
2. Examples
* Example 1
* Example 2
* Example 3
* Example 4
* Example 5
* Example 6
5. Equilibrium and compatibility, superposition, uniqueness
1. Equilibrium
* Example 1
* Example 2
* Example 3
2. Compatibility
* Example 1
6. Two-dimensional solutions - Plane stress and plane strain, St. Venant's principle, Airy stress function, Beam bending, Edge dislocation
1. Antiplane shear
* Example 1
2. Plane strain
* Example 1
3. Plane stress
4. St Venant's principle
5. Beam bending examples
* Example 1
6. Stress functions
7. Airy stress function (without body force)
* Example 1
8. Using the Airy stress function
1. Polynomial solutions
2. Fourier series solutions
3. Problems in polar coordinates
7. Two-dimensional problems - Michell solution, hole in a plate
1. Disk with a circular hole
* Example
* Plate with a circular hole in shear
2. General stress functions in polar coordinates (Michell's solution)
* Example
* Plate with a circular hole in tension
8. Two-dimensional problems with body forces - rotating beam, circular disk
1. Airy stress function with body force
2. Determining the body force potential
3. Examples
* Rotating rectangular beam
* Spinning disk
9. Two-dimensional problems - William's solution, Wedges
1. Wedge with boundary tractions
2. Williams' asymptotic solution
* Example 1: Axially loaded wedge
3. Flamant solution
* Example 1: Transversely loaded wedge
10. Axisymmetric problems
11. Torsion - Prandtl stress function, membrane analogy
1. Torsion of circular cylinders
2. Torsion of noncircular cylinders
3. Warping functions
* Warping of circular cylinder
* Warping of elliptical cylinder
* Warping of rectangular cylinder
4. Prandtl stress function
* Solution strategy for Prandtl stress function
* Example 1: Torsion of triangular cylinder
* Example 2: Torsion of thin walled open sections
* Example 3: Torsion of thin walled closed sections
* Example 4: Torsion of rectangular sections
12. Special problems - contact, dislocations, antiplane shear
1. Contact
* Concentrated force on half plane
* Distributed force on half plane
* Indentation due to a flat punch
* Hertz contact - indentation due to a cylinder
13. Thermoelasticity
14. Variational methods and energy principles
1. Energy methods in elasticity
2. Minimizing a functional
3. Kinematically admissible displacement field
4. Principle of minimum potential energy
5. Statically admissible stress field
6. Principle of minimum complementary energy
7. More variational principles
1. Hu-Washizu variational principle
2. Hellinger-Reissner variational principle
8. Approximate solutions
1. Rayleigh-Ritz method
9. Examples
* Example 1
* Example 2
* Example 3: Torsion of a cylinder
* Example 4: Bending of a cantilever beam

## Quizzes and Exams

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* Quizzes
1. Sample quiz 1
2. Sample quiz 2
3. Sample quiz 3
4. Sample quiz 4
5. Sample quiz 5

* Sample Midterm Exam
1. Sample midterm problem 1
2. Sample midterm problem 2
3. Sample midterm problem 3
4. Sample midterm problem 4
5. Sample midterm problem 5

* Sample Final Exam
1. Sample final problem 1
2. Sample final problem 2
3. Sample final problem 3
4. Sample final problem 4
5. Sample final problem 5
6. Sample final problem 6

## Assignments

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## Textbooks

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* The Linearized Theory of Elasticity by William S. Slaughter, Publisher: Birkhauser, Boston; ISBN 0-8176-4117-3
* Elasticity: Second Edition by J.R. Barber, Publisher: Kluwer Academic Publishers; ISBN 1-4020-0966-6; (2002)
* Elasticity in Engineering Mechanics: Second Edition by Arthur P. Boresi and Ken P. Chong

## Additional References

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* Elasticity
* Journal of Elasticity by Springer Science+Business Media
* Theory of Elasticity : Third Edition by S. P. Timoshenko and J. N. Goodier, McGraw-Hill, New York, 1970. This book is an excellent reference of the theory of elasticity.
* Theoretical Elasticity  by A. E. Green and W. Zerna, Dover Publications, New York, 1992. This is a paperback edition of the original. Students interested in a mathematical approach may wish to consult this book.
* A Treatise on the Mathematical Theory of Elasticity : Fourth Edition by A. E. H. Love, Dover Publications, New York, 1944. Another older but excellent reference on elasticity.
* Mechanics of Materials
* Mechanics of Materials : Fifth Edition by J. M. Gere and S. P. Timoshenko, Brooks/Cole, Pacific Grove, CA, 2001. An excellent introduction to mechanics of materials.
* Mechanics of Materials : Second Edition by F. P. Beer, E. R. Johnston, Jr., and J. T. DeWolf, McGraw-Hill,New York, 1992. Another widely used introduction to mechanics of materials.
* Vectors and Tensors
* A Brief on Tensor Analysis : Second Edition by James G. Simmonds, Springer-Verlag, New York, 1994. A brief introduction to tensors designed for undergraduates. A must read if you wish to introduce yourself to tensors.
* Vectors, Tensors and the Basic Equations of Fluid Mechanics by R. Aris, Dover Publications, New York, 1962. An excellent introduction to vectors and tensors.
* Dynamics of Polymeric Liquids : Volume 1 : Second Edition by R. B. Bird, R. C. Armstrong, O. Hassager, John Wiley & Sons, New York, 1987. The appendix contains a brief introduction to tensors and their transformations in various coordinate systems.
* Engineering Mathematics
* Foundations of Applied Mathematics by Michael D. Greenberg, Prentice-Hall, Englewood Cliffs, N.J., 1984. A wonderful text on applied mathematics - great for Fourier series, complex variables and ordinary and partial differential equations.
* Advanced Engineering Mathematics : Eighth Edition by Erwin Kreyszig, John Wiley & Sons, New York, 1999. An updated version of the classic text on engineering mathematics.

## Resources

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Below is a list of potentially useful links to mathematical, symbolic computing and other resources.

Maths Tutorials
Mathematics dictionary from Wolfram Research
Rebecca Brannon's tutorials on vectors, tensors and a lot more
Unix Tutorials
Unix tutorial from the Univeristy of Utah
Unix tutorial from University of North Carolina
Emacs Tutorials
Emacs tutorial from the University of Chicago
Emacs tutorial from Temple University
LaTeX Tutorials
Latex tutorial from UC Davis
Latex tutorial from UMIST
Maple Tutorials
Maple tutorial from Indiana
Maple tutorial from MapleSoft
Maple tutorial from the University of Utah
Maxima Tutorials
The Computer Algebra Program Maxima - a Tutorial
wxMaxima (a nice Maxima GUI) and Maxima tutorials list
Minimal Maxima - tutorial
Matlab Tutorials
Matlab tutorial from Michigan Tech
Matlab tutorial from the University of British Columbia
Matlab tutorial from the University of Waterloo
Matlab tutorial from the University of Utah

## Brief History of Experimental Linearized Elasticity

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The historical information presented here has been taken from The experimental foundations of solid mechanics by J.F. Bell in Handbuch der Physiks, Volume VIa/1.

> On Solid Mechanics:
> That this branch of physics remains a vital and provocative subject for fundamental study nearly three fourths of the way through the 20th century, is one of the lessons to be learned from perusing the 300 year history of the growth of the experimental foundations of solid mechanics since the inaugural measurements of Robert Hooke in the 17th century.
>  J.F. Bell, 1973.
>  On Linear Stress-Strain Response For Small Deformations:
> The dilemma of Leibniz in the 17th century over the apparently conflicting experiments of Hooke and James Bernoulli has been resolved in favor of the latter. The experiments of 280 years have demonstrated amply for every solid substance examined with sufficient care, that the strain resulting from small applied stress is not a linear function thereof.
>  J.F. Bell, 1973.

1678 : Robert Hooke
Discovers that force is a linear function of elongation based on expeiments on long, thin wires and springs. His anagram for this law was "ceiiinosssttuu" (published 1676) which was deciphered as "Ut tensio sic vis" in his 1678 paper.
1720 : Jordan Ricatti
Proposes that elastic properties of a body could be inferred from the frequency of vibration. The first experimental study of elastic E-moduli.
1729 : Pieter Van Musschenbroek
Publishes the first book showing testing machines for tension, compression, and flexure.
1766 : Leonhard Euler
Introduces the concept of "Young's modulus" eighty years before Thomas Young popularized Euler's concepts of the "height of the modulus" and the "weight of the modulus".
1780 : Charles Augustin Coulomb
First to measure the shear modulus in the modern sense.
1787 : Ernst Chladni
Calculates ratios of the velocity of sound in air to that in various solids. This work provided a major impetus for 19th century continuum mechanics.
1807 : Thomas Young
Publishes his "Lectures on Natural Philosophy". This work led to the popularization of the "height of the modulus". The units were in feet.
1809 : Jean Baptiste Biot
First direct measurement of the velocity of sound in a solid.
1813 : Alphonse Duleau
First quasi-static experiments for small deformation linear elasticity (by design). This work provided experimental evidence for numerous theoretical developments in elasticity, including St Venant's principle and the theoretical work of Cauchy, Poisson and Navier.
1841 : Guillaume Wertheim
Presents first definitive study of elastic properties of solids under various conditions to the French Academy. This study included results from Jean Victor Poncelet, Thomas Tregold, Antoin Masson, Felix Savart among others. Linear plots of stress versus strain begin to be widely used.
1848 : Guillaume Wertheim
First experiments showing that the Poisson's ratio of a solid does not have the constant value of 0.25.
1859 : Gustav Robert Kirchhoff
First measurement of Poisson's ratio indepenedent of the elastic modulus and specimen diameter.
1869 : Marie Alfred Cornu
First direct optical measurement of Poisson's ratio.
1882 : Woldemar Voigt
Performs experiments to prove the isotropy or otherwise of solids.
1904 : Arnulph Malloc
Devises a method to determine the quasi-static bulk modulus based on the theory of linear elasticity.
1908 : Eduard August Gruneisen
The Poisson's ratio is first determined experimentally as ratio of lateral and longitudinal strains. Uses Malloc's method to determine the compressibility of solids.

## Brief Early History of Theoretical Linearized Elasticity

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c 1630 : Isaac Beeckman
Realizes that strain (change in length/length) should enter an elastic law.
1687 : Isaac Newton
Publishes "Principia" which provide the laws of motion : inertia, conservation of momentum, and balance of forces, though inertia and momentum remained undefined.
1684 : Gottfried Wilhelm Leibniz
Finds the relation between bending moment and the moment of inertia of a linear elastic beam.
1691-1704 : James Bernoulli
Derives the general equations of equilibrium using different methods : balance of forces, balance of moments, and the principle of virtual work.
Finds that the stress (force/area) as a function of strain characterizes a material and thus proposes the first true stress-strain relation and a material property.
1713 : Antoine Parent
Determines the position of the neutral fiber and postulates the existence of shear stresses.
1736 : Leonhard Euler
Publishes "Mechanics" where he defines a mass-point and acceleration. Also introduces vectors. Most of the equations in mechanics in use today can be traced to the work of Euler.
1742 : John Bernoulli
First to refer all positions to a single, rectangular Cartesian co-ordinate system.
1743 : Jean le Rond d'Alembert
First to derive a partial differential equation as the statement of a law of motion.
1750-1758 : Leonhard Euler
Formulates the principles of conservation of linear momentum and moment of momentum. Distinguishes mass from inertia.
1773 : Charles Augustin de Coulomb
Proved that shear stresses exist in a bending beam.
1788 : Joseph Louis Lagrange
Publishes "Mechanique Analitique" which contains much of the mechanics known until that time.
1822 : Augustin Louis Cauchy
Discovers the stress principle - relating the total forces and total moment to internal and external tractions. Cartesian co-ordinate system. This is basically the first description of the stress tensor. Cauchy also presented the equations of equilibrium and showed that the stress tensor is symmetric.
1833 : Siméon Denis Poisson
Publishes statement and proof that a system of pairwise equilibriated and central forces exerts no torque. This is fundamental to the principle of conservation of moment of momentum.

More details can be found in the books by Timoshenko and Love.

## People

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Here are some people whose work you will encounter in engineering elasticity.

Abel (Abel integral equations)
Airy (Airy stress function)
Bernoulli, James (Beam bending)
Bernoulli, Daniel (Superposition)
Betti (Betti's theorem)
Boussinesq (Boussinesq solution)
Burger (Burger's vector)
Carter (Carter's problem)
Castigliano (Castigliano's theorem)
Cattaneo (Cattaneo's problem)
Cauchy (Cauchy-Green deformation tensor)
Christoffel (Christoffel symbols)
Collins (Collins' method)
Cosserat (Cosserat elasticity)
Coulomb (Coulomb friction)
D'Alembert (D'Alembert's principle)
Descartes (Cartesian coordinates)
Dirichlet (Dirichlet boundary conditions)
Duhamel (Rational mechanics)
Dundurs (Dundurs' theorem)
Euclid (Euclidean geometry)
Euler (Equations of equilibrium)
Flamant (Flamant solution)
Fourier (Fourier series)
Louis Fredholm (Fredholm integral equations)
Galileo (Bending of a beam)
Galerkin (Galerkin finite element method)
Gauss (Divergence theorem, potential theory)
Green, George (Green's function)
Hadamard (Elastodynamics)
Hankel (Hankel transform)
Helmholtz (Helmholtz potential)
Hertz (Hertzian contact)
Hooke (Hooke's law)

Jacobi (Jacobian)
Kirchhoff (Kirchhoff stress)
Kronecker (Kronecker delta)
Lagrange (Green-Lagrange strain tensor)
Lamb (Elasticity solutions)
Lame (Lame's constants)
Laplace (Laplacian)
Legendre (Legendre polynomials)
Love (Mathematical thoery of elasticity)
Maxwell (Maxwell's theorem)
Mellin (Mellin transform)
Michell (Michell's solution)
Mindlin (Mindlin's problem)
Mohr (Mohr's circle)
Navier (Navier's equation)
Neumann (Neumann boundary condition)
Newton (Laws of motion)
Noether (Invariants)
Ockham (Occam's razor)
Papkovich (Papkovich-Neuber solution)
Pascal (Pascal's triangle)
Poisson (Poisson's ratio)
Prandtl (Prandtl's stress function)
Rodrigues (Euler-Rodrigues formula for rotation)
Saint-Venant (Saint-Venant's principle)
Stokes (Stokes theorem)
von Mises (von Mises failure criterion)
Taylor (Taylor series expansion)
Taylor, Geoffrey (Dislocations in metals)
Williams (Williams' solution)
Young (Young's modulus)