Theory question about linear viscoelasticity in Numerical Tours

1

Hello!

I am working with the linear elasticity demo https://comet-fenics.readthedocs.io/en/latest/demo/viscoelasticity/linear_viscoelasticity.html

What exactly is going on in these lines of code:

incremental_potential = strain_energy(u, epsv)*dx \
                        + dt*dissipation_potential((epsv-epsv_old)/dt)*dx \
                        - Traction*u[1]*ds(1)
F = derivative(incremental_potential, w, w_)
form = replace(F, {w: dw})

I am trying to map this onto Equations 5, 6, and 7 of Lahellec and Suquet (2008), https://hal.archives-ouvertes.fr/hal-00214209/document, which I think is where this formulation was proposed (?).

Is it correct that the second and third lines of code, above, implement the infimum (i.e., LS08 Equation 6)?

Thanks!

Brad

2

Hi,
more or less yes. Solving F==0 corresponds to the first-order optimality condition of the variational principle consisting in minimizing the incremental potential. However, contrary to what s written in Lahellec and Suquet, here minimization is performed both on u and epsv : minimization wrt epsv yields indeed (6) but minimization wrt to u solves the balance equation with the stress/strain relation (7).

form is then used to extract the bilinear and linear parts of the corresponding linear variational problem

Hope this helps

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3

This helps a lot. I have a followup question, too. @bleyerj, looking at your publication list it seems like this might be an area that you would be able to comment on.

Background. I asked about the incremental potential function because I’d like to modify the viscoelasticity demo to describe a viscoelastic Euler beam. Before starting out, I didn’t realize that FEniCS doesn’t have (I think?) any type of H^2 element fully implemented. Hence why the FEniCS biharmonic demo uses a C/DG approach.

So my question is, is there a straightforward way to implement the DG penalty terms from the biharmonic demo into the incremental potential approach?

4

Indeed, FEniCS does not have such elements available.
A possible remedy is to use a Timoshenko model (see for instance these demos https://comet-fenics.readthedocs.io/en/latest/demo/beam_buckling/beam_buckling.html and https://comet-fenics.readthedocs.io/en/latest/demo/beams_3D/beams_3D.html) and maybe assuming viscoelasticity only on the normal stress part of the behaviour, the shear behaviour remaining elastic if this is OK for you.
Or indeed as you mention use a DG approach, but then I am not so sure about what would be the corresponding formulation in a viscoelastic setting.
Basically, the DG approach amounts to add rotational springs with some large stiffness scaling with 1/mesh_size. Do you need to have such springs behave as viscoelastic, that I cannot tell you precisely. I think you should have less trouble with the Timoshenko model approach…

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