Supposing I have this variational form of the linear elasticity problem to solve: \int\limits_\Omega (\sigma:∇v) dx = \int\limits_{\delta\Omega} (\sigma.n.v) ds + \int\limits_\Omega f.v dx, where T = \sigma.n

In this tutorial, the traction T is set to 0. But I guess, this is not always the case for elastic equation, is not it?

Here are my questions:

What is the physical sense of the traction term? Is it an external constraint imposed on the boundary (like external forces) or rather an internal-behaviour term due to the equation itself?

In which cases should I consider and leave the traction term in the variational form? For instance, I read non-zero traction was “artificially” used to avoid simulation divergence when potential fluid backflow, but it was in the case of Navier-Stokes equation → So what about elastic equation?

In the examples I read on non-zero traction boundaries, it was proposed an imposed value for the traction (e.g. Constant((1.0, 1.0)). But is it possible/what happens if I leave the theoretical expression of traction as it is: \int\limits_{\delta\Omega} (\sigma.n.v) ds ? (I tried to do so but it seems to crash and I had to put it at T=0)

I made some other researches. From what I understood, that are the different steps required to formulate the variational form which lead to then “inject” traction T into the equation (as in https://fenicsproject.org/pub/tutorial/html/._ftut1011.html):

Taking v=0 on \Gamma_D and injecting the traction boundary condition: \int_{\Omega}(\sigma:\nabla v) dx = \int_{\Omega}(f. v) dx + \int_{\Gamma_N}(T. v) ds , \mbox{∀v∈V}

As a matter of fact, T can be “choosen” as a external surface loading force, which is physicially equivalent to internal stresses at boundary (\sigma(u)). Is that correct?

So to conclude, here are my refined questions:

I can choose the traction term expression:

either T = non-zero fenics.Constant (homogeneous traction loading)

either T = f(x,y,z) (heterogenous traction loading)

either T = 0 traction-free) Is that correct?

I don’t need to write the term \int_{\delta\Omega}(\sigma(u).n. v) ds in the fenics variational form (LHS in that case), but I replace it by \int_{\Gamma_N}(T. v) ds with the imposed traction T in RHS, isn’t it?

I can consider non-zero traction term if I want to consider external surface loading in my model. Isn’t it?

For your middle question: Traction is equivalent to \sigma \cdot n.
Yes for your 3 refined questions. In your initial problem I suppose you wanted to write the divergence instead of the Laplacian.