Using 10-node tetrahedron, is strain continuous between neighbouing tetrahedons?

Question

I'm trying to implementing a Finite Element Analysis algorithm. I solve K u = f to get the displacement u, and then calculate strain with u, then calculate the stress. Finally, I use the stress to calculate the Von Mises Stress, and visualize this. From the result I find the strain is not continuous between tetrahedrons.

I use 10 nodes tetrahedron as the element, so the displacement is a second-order polynomial in every element. The displacement should be enforced to be continuous between tetrahedrons. And the strain, which is the first order derivatives of the displacements should be continuous inside every tetrahedron. But I'm not sure: is this true across the interface between tetrahedrons?

Answer 1

You should not compute the stress and strain at the nodes but inside the elements. You can choose for example 4 Gauss points and compute the values there. You then have to think about a scheme on how to get the values computed at the Gauss points onto the tet nodes.

There is a Mathematica application example which illustrates this. Unfortunately the web page is no longer available, but the notebooks are here. You'll find the example in the application example section under Finite Element Method, Structural Mechanics 3D (in the old HelpBrowser). If you have difficulties I could convert it to PDF and send it you.

Answer 2

Only the components of strain tangent to the adjoining face are guaranteed continuous. This follows from the displacement continuity, when you take derivatives in the direction of the interface they are the same.

Commercial FEM programs typically do some post process averaging to make the other components look continuous. Note the strain components normal to an element boundary are only expected to be continuous if the underlying constitutive model is continuous, so such averaging is not always appropriate.