I am planning to run numerical simulations in metal elastoplasticity (von-Mises yield condition with and without isotropic hardening). However, I am completely new to this subject and I am unsure about the choice of appropriate finite element spaces.
I read that a problem called “volumetric locking” may occur, which is related to the fact that standard finite element functions are not flexible enough to deal with the incompressible plastic deformations and that mixed formulations may be necessary.
Jeremy Bleyer points out in his standard example on fenics plasticity
that in the 2D case it is sufficient to choose quadratic triangles for the displacement field in a non-mixed formulation.
However, I am mainly interested in AXISYMMETRIC situations (also in the general 3D case) and I wasn’t able to find precise information on this topic in the literature.
Hi,
I am not an expert on this topic but I think P^2 interpolation on triangle should be OK in axisymmetric conditions. However, in 3D I think that even P^2 on tetrahedra will exhibit volumetric locking.
Classical remedy for that is to resort to mixed methods where displacement and plastic strains are independently discretized, pretty much like velocity-pressure discretization for Stokes problem in which incompressibility condition poses the same issue.
I haven’t implemented that in FEniCS though.
which says, if I’m not mistaken, that in 2D even linear triangles are sufficient while in the axisymmetric case it is quite difficult to find suitable finite element spaces without resorting to mixed methods or similar.
Hi,
OK good to know, I wasn’t aware that axisymmetric conditions were more problematic than plane strain.
However, the above table seems wrong to me especially when looking at the 3 node triangle which is clearly not suitable for incompressible materials. It is very easy to construct a structured mesh with left orientation with both left and bottom boundaries fixed which completely lock the whole solution to being zero.
For appropriate mixed finite element spaces for elasticity you could use this
Mixed finite elements for elasticity Numer. Math. (2002) 92: 401–419 Arnold + Winther
and
Section 7 of Finite Element Exterior Calculus (2010) Arnold + Falk + Winther
@bleyerj: Hmm, ok, that’s good to know. This table is in fact taken from a PhD thesis which I do not have access to. So, I am not sure if I am misinterpreting it somehow (possibly related to the notion of “full integration”?).
@Thomas2: Thank you for your suggestion. I will take a closer look. However, at first glance, the first paper seems to deal with the plane situation only, not with an axisymmetric setting?
