# In which cases should we consider non-zero traction term? **URL:** \ **Category:** variational formulation\ **Created:** [February 20, 2023, 9:45am UTC](https://fenicsproject.discourse.group/t/in-which-cases-should-we-consider-non-zero-traction-term/10474 "2023-02-20T09:45:14Z")\ **Posts on this page:** 5\ **Page:** 1
**Author:** ![anneker](https://avatars.discourse-cdn.com/v4/letter/a/7ba0ec/32.png) [@anneker](https://fenicsproject.discourse.group/u/anneker)\ **Post date:** [February 20, 2023, 9:45am UTC](https://fenicsproject.discourse.group/t/in-which-cases-should-we-consider-non-zero-traction-term/10474/1 "2023-02-20T09:45:14Z")
Hi, 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](https://jsdokken.com/dolfinx-tutorial/chapter2/linearelasticity_code.html), 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) Many thanks for your help, Anne ---
**Author:** ![Felipe\_Rocha](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/felipe_rocha/32/7071_2.png) [@Felipe\_Rocha](https://fenicsproject.discourse.group/u/Felipe_Rocha)\ **Post date:** [February 21, 2023, 3:12pm UTC](https://fenicsproject.discourse.group/t/in-which-cases-should-we-consider-non-zero-traction-term/10474/2 "2023-02-21T15:12:34Z")
Hi Anne, Some clarification: in your first equation you should write T \cdot v in the RHS, and as a natural result you will have \sigma \cdot n = T. 1. From the Cauchy theorem t(x, n) = \sigma(x) \cdot n , where x is the position. 2. If you forget traction terms in a primal formulation, it means zero-traction, i.e., free-boundary. 3. It does not work since your \sigma(u) and in the RHS you should only have v, i.e., no trials, only test functions. Hope that it helps. Felipe ---
**Author:** ![anneker](https://avatars.discourse-cdn.com/v4/letter/a/7ba0ec/32.png) [@anneker](https://fenicsproject.discourse.group/u/anneker)\ **Post date:** [February 24, 2023, 12:33pm UTC](https://fenicsproject.discourse.group/t/in-which-cases-should-we-consider-non-zero-traction-term/10474/3 "2023-02-24T12:33:26Z")
Many thanks for your answer @Felipe_Rocha, - 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](https://fenicsproject.org/pub/tutorial/html/._ftut1011.html)): - 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: Many thanks for your help, Anne ---
**Author:** ![Felipe\_Rocha](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/felipe_rocha/32/7071_2.png) [@Felipe\_Rocha](https://fenicsproject.discourse.group/u/Felipe_Rocha)\ **Post date:** [March 3, 2023, 6:30pm UTC](https://fenicsproject.discourse.group/t/in-which-cases-should-we-consider-non-zero-traction-term/10474/4 "2023-03-03T18:30:47Z")
Hi Anne, 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. Cheers, Felipe ---
**Author:** ![anneker](https://avatars.discourse-cdn.com/v4/letter/a/7ba0ec/32.png) [@anneker](https://fenicsproject.discourse.group/u/anneker)\ **Post date:** [March 6, 2023, 4:00pm UTC](https://fenicsproject.discourse.group/t/in-which-cases-should-we-consider-non-zero-traction-term/10474/5 "2023-03-06T16:00:01Z")
Hi Felipe, Many thanks for your feedback. You are rigtht about σ⋅n and the divergence! It’s a mistake. Kind regards, Anne