E = 2000000
nu = 0.3
C = (E / ((1 + nu) * (1 - 2 * nu))) * np.array([
    [1 - nu, nu, nu, 0, 0, 0],
    [nu, 1 - nu, nu, 0, 0, 0],
    [nu, nu, 1 - nu, 0, 0, 0],
    [0, 0, 0, (1 - 2 * nu) / 2, 0, 0],
    [0, 0, 0, 0, (1 - 2 * nu) / 2, 0],
    [0, 0, 0, 0, 0, (1 - 2 * nu) / 2]

This C matrix I took from a textbook I have that says it should be good for three degrees of freedom for axial forces. How should such a C matrix be expanding if I also want to include three additional degrees of freedom for rotational forces (moment)?

  • $\begingroup$ There also is a B matrix which consists of 6 rows and 12 columns. So for the additional three degrees of freedom for moment B would have to also expand. I don't quite get what you mean by those are additive since it doesn't really make sense to me that B would have to expand to make room for those new dogs. Could it be explained what happens with B and c for the new dogs? Maybe you mean that C is added on a diagonal? $\endgroup$
    – prusso
    Commented May 23 at 3:18
  • 2
    $\begingroup$ That looks like a compliance tensor in Voigt notation. It relates stress and strain. It is not a stiffness matrix. $\endgroup$ Commented May 23 at 12:40
  • 1
    $\begingroup$ Either way, I think you have to more precisely say what the matrix represents, and in particular what exactly it is you want to do. I have no idea what your actual goal is. $\endgroup$ Commented May 23 at 12:40
  • $\begingroup$ I was hoping to expand the tetrahedron from 3 dofs to six dofs (for moment and axial forces). I'm finding there is little written how to do so and also it might not be an accepted practice. Plates and shells with six dofs are recommended. All I know so far is that B and C would have to expand if that were possible. If there is an answer, I hope for it to include how to expand B and C from three axial dofs to six dofs 3 for forces and 3 for moment. $\endgroup$
    – prusso
    Commented May 23 at 20:27

1 Answer 1


Four-node tetrahedral elements with rotational degrees of freedom (DOFS) have been tried in the past but have never found wide-spread use. See for example, this reference, Bucher, and the references contained within.

These rotational DOFS are often referred to as "drilling" freedoms in the finite element structural mechanics literature because they were first used to enhance the basis of three-node triangular membrane elements (the three additional DOFS are normal to the plane of the element at the nodes).

Introducing rotational DOFS in these types of elements is definitely an advanced topic and you should first consider whether you want to make the investment in time to pursue it. You did not mention WHY you need to do this. However, if you still want to pursue this research, I strongly recommend starting with the three-node triangle since it is much easier to understand.

As has already been pointed out in the comments, including rotational DOFS in the element is completely unrelated to making any changes to the basic stress-strain relations (the matrix you are denoting as C). These relations arise from the basic theory of elasticity for a linearly elastic material and are used in exactly the form you show for a wide variety of finite element formulations in solid mechanics.

  • $\begingroup$ The reason why would be to solve a multiple materials (multi-domain) problem where the two domains act as a beam element. $\endgroup$
    – prusso
    Commented May 26 at 0:35

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