von Mises: cylinder, circle, or ellipse?

The answer is…all of the above!   The von Mises cylinder is centered about the {1,1,1} direction in principal stress space.  The ellipse that we learn about as undergrads applies only to plane stress (where the third principal stress is zero), and this is just the intersection of the cylinder with the plane.  The circle applies to the octahedral plane, which is the view of the cylinder down its {1,1,1} axis.  This animation should clarify what is going on:

MisesCylinderAndMisesEllipse

 

PowerPoint slides for Mohr’s circle

The link below provides a collection of slides used to explain Mohr’s circle in an undergraduate mechanics course at the University of Utah.  If you use a Mac, it is unlikely that these will render properly (so go sit at a PC in your university computer lab to look at them).  Make sure to use slideshow mode, as these have many animations!

MohrCircleFiles  (zip file contains two PPT lectures and one Mathematica file)

F-tables for prescribed deformation

Motion without superimposed rotation

Motion without superimposed rotation

Same deformation with superimposed rotation

Same deformation with superimposed rotation

When developing constitutive models, it is crucial to run the model under a variety of standard (and some nonstandard) homogeneous deformations. To do this, you must first describe the motion mathematically. As indicated in http://csm.mech.utah.edu/content/wp-content/uploads/2011/03/GoBagDeformation.pdf, a good way to do that is to give the deformation gradient tensor, F. The component matrix [F] contains the deformed edge vectors of an initially unit cube, making this a very easy to way to prescribe deformations.

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Hyphenation in technical (or other) writing.

HyphenKeyCircled

Should you say “finite element” or “finite-element?” Which is better: a “beautifully-written” manuscript, or “beautifully written” one? Are your equations non-linear or nonlinear? Our one-page list of hyphenation rules summarizes information found in a variety of authoritative sources (Princeton Review, Strunk & White, etc.). Happy technical writing!

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