John Craighead's blog

I'm trying to better understand the 3-noded triangular element with drilling rotations.

I think that the 3-noded triangle with drilling rotations is derived from the 6-noded linear strain triangle by, for mid-side nodes, constraining out-of-plane dof & converting in-plane dof to nodal rotation.

In particular, I'm wondering about the following:

Like others, I'm very sorry to learn of Jerry Marsden's passing. He kindly helped all who asked (including me) despite a very busy life and recent illness. It seems fitting to mention this before citing him below. 

According to Marsden/Hughes (Mathematical Foundations of Elasticity), stretch & shear (from homogeneous elastic deformation) are determined soley by U in the polar decomposition of the deformation gradient F=RU where R = rot'n & U = right stretch.

Hello !

I have been reading quite a bit about decomposing deformation gradients into F=RU = vR where R = rigid rot'n, U = right stretch & v = left stretch. Since the principle stretch axes & basis vectors don't usually coincide, such stretches produce both stretch & shear as shown in animation at link below.

A singular value decomposition of U (or v) can be used to isolate the pure stretches. In my case, U is postive semi-definite so SVD given by eigenvalue decompostion as follows:

I'm working on a MSc thesis involving both rigid body (zero energy) & deformational (elastic) modes and their orthogonality wrt mass matrix. I want to prove that elastic modes are mutually orthogonal (done !), that elastic & rigid body modes are so (sort of done !) & that rigid body modes also are so (unsure how ?). Can anyone offer any advice ?

Thanks, John.