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Finite Deformation Continuum Mechanics

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with emphasis on incompressible viscoelastic materials

Description

mapping

This website presents the principles of finite deformation continuum mechanics with many example applications to metals and incompressible viscoelastic materials (rubber). Examples also cover both rectangular and cylindrical coordinates.

Special Schedule

No extra class on May 10, after all. We're on schedule and don't need it. Still no classes the following week, May 13-17.

Homework

Homework #1
Homework #2
Homework #3
Homework #4
Homework #5
Homework #6
Homework #7
Homework #8
Homework #9
Homework #10
Homework #11
Homework #12 - Due 8 May 2013

Homework Solutions

Homework #1 Solutions
Homework #2 Solutions
Homework #3 Solutions
Homework #4 Solutions
Homework #5 Solutions
Homework #6 Solutions
Homework #7 Solutions
Homework #8 Solutions
Homework #9 Solutions
Homework #10 Solutions
Homework #11 Solutions

Kick-Off Survey

Here is the MS-Word file. And here is the text.

Schedule

Feb 4, 2013 - June 5, 2013
Mondays and Wednesdays
10:00am - 11:30am EDT
Author   Bob McGinty, PhD, PE
Email    bmcginty@gmail.com



Table of Contents

  1. INTRODUCTION

  2. BASIC MATHEMATICS

    1. Vectors
    2. Matrices & Tensors
    3. Vector Calculus
    4. Tensor Notation (Basic)
    5. Tensor Notation (Advanced)
    6. Divergence Theorem
    7. Coordinate Transformations
    8. Transformation Matrices
    9. Cylindrical Coordinates

  3. INTRODUCTORY MECHANICS

    1. Stress
    2. Strain
    3. Hooke's Law

  4. DEFORMATIONS AND STRAIN

    1. Deformation Gradients
    2. Polar Decompositions
    3. Rotation Matrices
    4. Finite Element Mapping
    5. Small Scale Strains
    6. Green & Almansi Strains
    7. Principal Strains & Invariants
    8. Hydrostatic & Deviatoric Strains
    9. Velocity Gradients
    10. True Strain
    11. Material Derivatives
    12. Special Strain Topics

  5. STRESS

    1. Stress Introduction
    2. Traction Vectors
    3. Energetic Conjugates
    4. Stress Transformations
    5. Principal Stresses & Invariants
    6. Hydrostatic & Deviatoric Stresses
    7. Von Mises Stress
    8. Corotational Derivatives
    9. Equilibrium

  6. MATERIAL BEHAVIOR

    1. Continuity Equation
    2. Navier Stokes Equation
    3. Thermodynamics
    4. Hooke's Law
    5. Mooney-Rivlin Models
    6. Dynamic Material Properties
    7. Fourier Transforms
    8. Materials and Tire Behavior
    9. Metal Plasticity


Textbook

Introduction to the Mechanics of a Continuous Medium, Lawrence E. Malvern, 1969.

Also available electronically here (24MB):
http://www.scribd.com/doc/5987971/MALVERN-LE-Introduction-to-the-Mechanics-of-a-Continuous-Medium


Additional References

A Note About The Web Technologies Used Here

Two relatively new web technologies are used on these pages. The first technology is Scalable Vector Graphics, or SVG. Pages on this site will display SVG files on compatible browsers, and PNG files on incompatible ones. The advantage of SVG over PNG is that SVG graphics can be scaled to any size without the onset of pixelization. SVG files used here were created using Inkscape, an excellent graphics program available free on the internet here.

Update: SVG is hopelessly broke on Firefox at the moment, so I've had to turn much of it off. Sigh...

The second new technology being used here is MathJax, a Javascript based display engine for mathematical equations programmed in the LaTeX language. MathJax eliminates the need to display equations as GIF or PNG graphics files (or SVG for that matter). MathJax requires only the following line of code in the <HEAD> segment of a webpage.

<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=default"></script>

It is then possible to program any math expression in the HTML source using the LaTeX language. For example, typing \(\sigma_{ij}\) produces \( \sigma_{ij} \).