# Correctly setting boundary condition for periodic linear elasticity problem

From an old, wise engineering book Peterson's Stress Concentration Factors (http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470048247.html page 324) I've got the following problem:

There is 2D infinite periodic structure with round holes of radius r with tension applied to it. The relation of maximum stress in the body to the applied one should follow nice 1/x like curve, see the picture. While I'm trying to reproduce this example by numerical simulations, I'm getting that max stress is almost independent of hole size.

I use displacement based weak formulation for elasticity with periodic boundary conditions remaping top to bottom and left to right, with exception of corners (0,0), (0,1), (1,0), (1,1). For problem to be well-posed I've fixed displacement vector in one point (0,0) to be equal (0,0). Am I right? Is fixing one point enough? I still can rotate the unit square though. Do I need to fix an additional point?

The tension is applied via Neumann boundary conditions with tension applied in the following way. First, I define vector $g = (1.0, 0)$ and then I create right hand side for the weak form of equation: $$\int C\varepsilon(u):\varepsilon(v)dx = \int_{left border} (-g,v)ds + \int_{right border} (g,v)ds,$$ where the differential operator $\varepsilon$ is the symmetric part of the gradient and $v$ is a test function obeying periodic boundary conditions. This is the place I'm most doubted about applying tension correctly. Is it correct? Is my problem well-posed?

p.s. I'm solving it with Fenics and code can be found here https://fenicsproject.org/qa/13464/verifying-elasticity-benchmark-structure-boundary-conditions

• What is the reference for the original elasticity solution to this problem? It is unclear to me that periodic boundary conditions are appropriate for this problem. – Bill Greene May 14 '17 at 21:40
• @BillGreene yes, you are right! The displacement itself is not a periodic, it was mine mistake. But, the displacement could be represented as $u = A\tilde{u} + u^*$ where $u^*$ is a periodic part. So, if I can somehow remove linear part $A\tilde{u}$ I would get a periodic problem. Not sure though, how do I remove the linear part first. Would like to get any help. – Moonwalker May 15 '17 at 11:13
• Sorry, I do not know how to apply boundary conditions like that for a FE model. If I were solving this problem, I would create several quarter models with symmetry BC applied at $x=0$ and $y=0$. The first model would have two holes, the second three, etc. As you increase the number of holes, the stresses at the center hole will converge relatively rapidly due to St Venant's principle. Not an elegant approach, but a straightforward one. – Bill Greene May 15 '17 at 11:54
• I would insist in the reference for the problem, as @BillGreene asked. – nicoguaro May 17 '17 at 19:45
• @BillGreene it is from Peterson's Stress Concentration Factors eu.wiley.com/WileyCDA/WileyTitle/productCd-0470048247.html page 324 – Moonwalker May 18 '17 at 16:03

Here is a description of a small FE model that might approximate the case of an infinite number of holes in an infinite plate.

Create a model of a single repeating element with $1/4$ of a hole centered at $x=0, y=0$. Along the boundaries at $x=0$ and $y=0$ apply classical symmetry boundary conditions.

On the two remaining edges, apply constraints on the displacements simply to keep the edges straight and parallel to either the x-axis or y-axis, as appropriate. In other words, all nodes along the boundary $x=l/2$ would have the same $u$ displacement and all nodes along the boundary $y=l/2$ would have the same $v$ displacement. This type of boundary condition is often referred to as an MPC (multi-point constraint) (e.g. NASTRAN, ABAQUS).

Either or both of these edges would also have an applied stress loading normal to the edge of $\sigma_1$ or $\sigma_2$.

The idea is that the straight-edge constraints would enforce compatibility with the adjoining cells but not restrict the motion unnecessarily.

• The stresses on $x=a$ and $y=b$ would not be uniform in this case, and their distribution along these boundaries would be unknown a priori. – DanielRch May 21 '17 at 18:14
• Yes, that's the basic idea. The total applied normal force at (say) $x=l/2$ would be $\sigma_1 l/2$ and the equilibrating internal stress distribution would be calculated. – Bill Greene May 21 '17 at 18:48

As a first approach, you could do as @BillGreene suggests, that is increase the size of your model, increasing the number of cells each time. I guess that around 10 cells in each direction you should have a value that won't change much for further increasing.

Another option is to constrain the displacements at opposite sides of the cell, for example, in the horizontal direction, you would have

$$u(\text{left}, y) = u(\text{right}, y) + g(y)\, ,$$

a common choice is to have $g(y) = u_0$, constant.

If you combine both approaches, you would have better results, I guess.

• @BillGreene it is from Peterson's Stress Concentration Factors eu.wiley.com/WileyCDA/WileyTitle/productCd-0470048247.html page 324 and it seems, that my mistake is solving for periodic displacement component. To obtain picture from the book I need add constant component. – Moonwalker May 18 '17 at 16:04