Elastons: Basics 2 | Geometry saves the Earth!

Stress

After having defined the strain, i.e. the amount of deformation at a point in the object, the authors proceed to define stress, the measure of internal forces acting in the object as a result of strain.

The formula looks benign enough:

$\sigma = C : \epsilon$

but keep in mind that what we see here is actually a 4-tensor $C$ , which is basically a 3x3x3x3 construct with 81 entries. Looking at this wikipedia page, we see that only 21 of those entries are independent. $\epsilon$ is the strain we saw earlier, which looks like a 3×3 matrix with 9 entries.

The funky thing here is the $:$ operator, which is defined as tensor contraction. To be honest, I only have a slight understanding of what tensors are in the first place, and this is the first time I actually saw a tensor contraction.

This paper has some of the same definitions as the Elaston paper and also explains the tensor contraction better. In a way we are defining a linear relationship between stress and strain. The tensor $C$ determines how the individual entries of the stress and strain tensors are connected.

Implementation issues

I have made some good progress with the implementation issue concerning the CHOLMOD library some days ago. I managed to set up a Visual Studio Solution for compiling it as well as all the dependencies. So far, it seems to work well, being able to compile the demo programs and running them (haven’t checked for correctness yet). The biggest problem was actually compiling the linear algebra library LAPACK, since it requires installing the Intel Fortran compiler and compiling the whole thing. Afterwards, I had to change all names of BLAS (a library associated with LAPACK) functions in CHOLMOD to upper case. No idea why they are not like that in the first place, since in the Fortran files, the names seem to be in upper case. Anyway, it doesn’t look like this part will be a large problem. The larger problem is how to construct the input to the CHOLMOD package…

Concerning this, I think I have worked out the two most important things coming up, how to construct the basis functions for the MLS approach and the mass and stiffness matrices.