Informal Write-ups in Mechanics

• TUTORIALS: straightforward primers on particular topics.
• MYTH BUSTERS: Misconceptions in mechanics
• DUSTY CORNERS: little-known or interesting aspects of mechanics issues.
• BACK DOORS: "Better ways" to do common tasks.

iMechanica.org

Elementary write-ups*

• Functional and Structured Tensor Analysis for Engineers: A step-by-step introduction to tensor analysis that assumes you know nothing but basic calculus. Considerable emphasis is placed on a notation style that works well for applications in materials modeling, but other notation styles are also reviewed to help you better decipher the literature. Topics include: matrix and vector analysis, properties of tensors (such as "orthogonal", "diagonalizable", etc.), dyads and outer products, axial vectors, axial tensors, scalar invariants and spectral analysis (eigenvalues/eigenvectors), geometry (e.g., the equations for planes, ellipsoids, etc.), material symmetry such as transverse isotropy, polar decomposition, and vector/tensor calculus theorems such as the divergence theorem and Stokes theorem. (A draft of this document was last released publically on Aug. 3, 2003. The non-public version is significantly expanded in anticipation of formal publication.)
  
  
• Large deformation kinematics: Summarizes the meaning of the deformation gradient tensor, stretches, rotations, etc. Also shows how material line segments, volumes, and area vectors change in response to deformation.
  
  
• Rotation: A REALLY BIG (long download time) tutorial on how to describe rotation. Topics include coordinate transformations, tensor transformations, converting an axis and angle of rotation into a rotation tensor, Euler angles, quaternions, and generating a uniformly random rotation tensor. This document also discusses the common numerical problem of "mixing" rotations in such a way that the mixed rotation is physically reasonable. The pages in the document that deal with random rotations contain some complicated figures, so don't worry if your pdf reader pauses for a while on those pages. As a matter of fact, watching the pdf viewer render the figures is like an informative movie because it draws the random dots in the same order as I computed them. By watching the rendering, you can see the nonuniform clustering quite clearly.] (Last posted here 020509, but a formal publication is anticipated)
  
  
• Mohr's Circle: A self-study refresher with interesting tidbits such as Pole Point and how to do Mohr's circle for nonsymmetric matrices -- very useful for quickly doing a polar decomposition! (Last posted 2003, but considerable work has been performed recently to incorporate Mohr's circle as part of the opensource VTK for visualization in finite element simulations)
  
  
• DEFINE YOUR STRAIN! This single-page document emphasizes the need for experimentalists and theorists alike to ALWAYS define their strain measure. For every percent increase in strain, the most popular measures of strain will disagree by as much as 1.5%. This might not sound like much, but try running a simple shear Von Mises strain cycle using log strain and engineering strain. You will find that the engineering strain calculation produces anomalous PRESSURES because volumetric strain does NOT equal the trace of strain EXCEPT for logarithmic strain. (Last updated 010531)
  
  
• FORTRAN compared with C++ : A table that has FORTRAN coding on one side and the equivalent C++ coding on the other. This is very useful if you know one language well and wish to do a similar task in the other (weak) language.
  
  
• Laws of motion . Summary of the basic equations of mechanics, along with abbreviated derivations.
  
  
• Thermostatics derivative simplification tables. When studying thermodynamics, do you feel lost in an alphabet soup of too many derivatives? Do you take a "random walk" through various identities hoping to stumble upon the right answer? If so, then this document will help! It describes a systematic way to express any thermostatics derivative in terms of fundamental thermodynamic state variables (pressure, temperature, entropy, and specific volume) and basic material properties such as specific heat, bulk modulus, thermal expansion coefficient, Gruneisen's parameter, etc. (Last updated 051223)
  
  
• The thermoelastic square. A very kewl mnemonic device for recalling thermodynamic identities (the Gibbsian relations, the Maxwell relations, the contact or Legendre transformations, etc.) I am working on a new version of this document that will clarify why property definitions for solids do NOT, in general, reduce to those for fluids when the tensors are isotropic. Stay tuned...
  
  
• The "cnv" program. FORTRAN source code for a program that does simple pre-processing tasks, many of which are especially useful for converting fortran source code (e.g., changing all in-line comments to ANSI standard F77 style).
  
  
• The "elas" program. FORTRAN source code for a program that computes all elastic constants and wave speeds given any two independent ones. If you want to look at an example to help you decide if you want this program, click here.
  
  
• Lectures from Dan Segalman's advanced vibrations course NOTE: when viewing these pdf documents, you may want to select "rotate pages" from the Document pull-down menu in Adobe acrobat reader. (Last updated 010615)
  
  
• Classical constitutive material models for engineering materials, an overview book chapter by Kaspar Willam.
  
  
• Computational Methods in Lagrangian and Eulerian Hydrocodes, an overview book chapter by Dave Benson.
  
  
• Miscellaneous documents from Rebecca Brannon's Introduction to Continuum Mechanics course
  • Homework
  • A bonus homework question
  • The final (makeup)
  • Nomenclature
  • Single page drawing to help illustrate polar decomposition

Intermediate write-ups*

• Curvilinear coordinates: A self-study introduction to the general theory applied to 3-D Euclidean space. (It helps to read the "Elementary Vector and Tensor Analysis" document before attempting this one). (last update 020509)
  
  
• Radial Return (or viewgraphs): A tutorial on the underlying theory of projecting a stress state back to the plastic yield surface. (Last updated 010531)
  
  
• Caveats concerning conjugate stress and strain measures and related viewgraphs entitled The distinction between large distortion and large deformation: This document contains an analytical solution for the stress in a fiber reinforced composite in the limit as the matrix material goes to zero stiffness. Because the solution is exact for arbitrarily large deformations, it can be used to illustrate subtle concepts in large deformation continuum mechanics.

Please direct comments and suggestions about this page to Rebecca Brannon at .