Compute stress tensor in finite element simulation

Question

I have implemented a finite element simulation using a tetrahedral mesh that minimizes a linear elasticity potential energy. Among the main quantities I compute for the simulation is the deformation gradient $F$ for each tetrahedron, from which I can then compute an elastic energy that I can minimize with respect to nodal positions.

I'm having trouble to understand how to compute "the stress tensor", the thing usually denoted as $\sigma$ which is apparently just a 3x3 matrix. This is one of those derived quantities that I have very little intuition of, since I don't even need to use in my finite element simulation to compute useful results, but I'm trying to understand it since many textbooks mention it. I understand the elastic energy and the deformation gradient of a tetrahedron. Is there a way to compute the stress tensor $\sigma$ for a tetrahedron using its deformation gradient $F$?

Tensors aren't really matrices (see why stress is a tensor here). — Kyle Kanos, Jun 18 '18 at 16:54
I know there is a difference. I just wanted to make my question very concrete and specific in terms of input - output. I want to compute 9 numbers for the stress tensor, I have access to the elastic energy (1 number) and deformation gradient (9 numbers) for a single tetrahedron. How do I do it? — yewang, Jun 18 '18 at 16:57
is apparently a 3x3 matrix strongly suggests you don't recognize or understand the difference, I'm pointing you to a resource to fill that gap. — Kyle Kanos, Jun 18 '18 at 16:59
Sure you can compute the stress tensor from the deformation gradient tensor. For small strains, this is just Hooke's Stress-Strain Law. — Chet Miller, Jun 19 '18 at 10:41
On behalf of @user3408085: http://www.cns.gatech.edu/PHYS-4421/lautrup/2.8/stress.pdf — Nat, Jun 19 '18 at 21:48

Compute stress tensor in finite element simulation

0 Answers0