Hello fellow Stressers!!!

Just for curiosity.. We know that Force (F)= Stiffness (K) x Displacement (D).

And for a simple bar, the stiffness (K) is given by AE/L.

From the equation as L approaches 0 the stiffness K is going to infinity.

And as the stiffness K is going to infinity the Force (F) is going to infinity.

But I think, the displacement (D) will be very small also, so maybe ultimately the Force (F) will be not going to infinity.

Anyway, is this one case of singularity in FEA? Where at sharp corners, as you increase the mesh density, the mesh length is approaching zero and thereby the stress is not converging and keep on increasing in value as you increase the mesh density.

Just correct me if any wrong mathematical statement I've made.

Cheers!!!

In my opinion, for your case you would recall the meaning of that formula which is Hooke's law:
F/A=E*(l-l0)/l0
When F->0 you get l->l0 which is simply "No force applied to, no elongation as effect". As math only, l->l0 means also F->0 but this is not cause-effect.
I cannot see any singularity in this physics.

If you try to investigate l0=0 you are losing the physics behind. Because you try to find what happened with a cylindrical body with zero initial length when apply an axial force; physically the force cannot elongate such cylinder because the cylinder does not exist there. As the cylinder has any length- even one incredible small for practice- physics works correctly. And in your query, when say l0 tends to zero (as limit) does not means l0=0 but l0 with a value near zero and physics works with, as above written.

Thanks Mariog for that great explanation. Anyway, the direct stiffness approach is a closed form of solution unlike Finite Element Approach which is approximate. So the Physics for the axial stiffness of a bar works.

I am just curious what part of the finite element approach is causing the singularity in a sharp corner as the mesh is increased. What part of the math (particular number) is causing this stress singularity?

Just correct any wrong mathematical statement I've made.

Cheers!!!

An explanation is in place, of course, but it is too detailed to be exposed here. I'll try to say something with the risk to be ridiculous in front of both mathematicians and engineers.

The math roots of FEA are linked to what a mathematician calls Energetic Space, in fact a math Hilbert space having an "inner product" fundamented by "them" on some physical facts.
Ultimately the goal is to minimize the energy associated with a system- as a functional in the energetic space- and there is an equivalence in this approach with solving some differential or partial equations systems, discovered long time ago by Euler.
In this process of minimization, a "uh" approximation is involved (something similar with Galerkin method in vibrations, if one would be more familiar with- where you may use quite ridiculous approximation of modal response with good final result for first period). Let's say "uh" is a function in a finite-dimensional subspace of the solution space usually replaced by a segmented function.

In this process- that God and software made it hidden for an engineer- such approximation would fail in cases as you've observed. Simply said, you try to squeeze too much from approximation of math in such cases because you try to enter too much in details by meshing. And as you can see all the construction/ math behind gives the troubles. But software can be written to solve such problems, in case this is the goal- I mean there isn't any math wall that cannot be removed. Ultimately an accessible (for us) FEA is a commercial software intended to satisfy the majority of users, rather than be the top tool for scientists.

So long as you treat k and E as static numeric values, then Hooke's law, etc. are only applicable over a narrow range of conditions, most notably temperature and whether or not the material has previously exceeded its elastic limit.

So long as the stress/strain curve remains linear, you can generally safely assume that the resultant k value remains constant so long as E remains constant.

You must also remain cognizant that another form of linearization is assuming the cross section remains the same, which it doesn't when you noticeably deform it.

With regards to divergence errors within FEA through further discretization:

If you were to calculate FEA by hand, you would find that you can transmit forces through your sharp corner. And you could then take your loads, divide by area, and then come up with a guess on the stress.

Then if you were to double that mesh density, you would find that your forces aren't converging to 0 as fast as your areas are, thus your calculated stresses go up.

Continue doubling the density, but otherwise keeping the same topology creates this divergence error.

Ideally, you would end up with a "magic mesh" where increased discretization results in a linear relationship between "F" and "A" between increases in discretization, and stresses converge to an answer.

Taking your cube for example, you know that F at end one will transmit to the other end. Thus, splitting it into smaller chunks will result in each chunk being 1/nth of the original, and summing them all back up comes to F.

But if your cube instead has a hole in the middle, if you end up with vertices at 12:00, 3:00, 6:00, and 9:00, you're going to have divergence errors, because mathematically, you can't turn a vertical load into a horizontal one.

Borzki,

In fact to skip the math trick in your original question you must start with F (action and cause of the deformation).
By math a finite F=k*x for case k-->infinity (because l-->0) requires x-->0 to generate in the right member of equation an indeterminate case of limits (as to be solved giving a finite result).
I know it's sounds like a joke, but the interpretation- linked to continuous medium hypothesis- is : in linear domain, a finite force will generate almost no deformation to a bar than tends to disappear! And, as I've written previously, when the bar physically disappeared, no bar and no deformation are there, that's why there isn't any singularity.

Thanks Mariog & Michael for the enlightenment. As engineers, our concern with sharp corners is the stress concentration & ultimately if fatigue is a concern, then we have a way around this, by designing a smooth transition in the corners to have a well defined stress that is converging to a reasonable value. I remember an article regarding the old design of airplane windows which are rectangular & now becomes rounded in shape.

This is why many FEA consultants have been performing physical testing to match the theory & reality as close as practically possible.

Any other opinion is highly appreciated!!



Cheers!!

I have tried this one 2D element following the energy principle to derive the stiffness matrix "BTCBxJacobianxwixwj". As I make the coordinates closer to each other the Jacobian gets smaller & the stiffness matrix K also gets smaller & therefore the Displacement D=K^-1*F gets higher since it's inverse of stiffness & the stress gets higher S=CBD. There is really a point away from the corner that we need to stop in order get a stress that will converge on the same point as the mesh is increased.

Just correct me if I have made any wrong mathematical statement.

Cheers!!!

Which is your physical interpretation of the fact the Jacobian dictates the behavior of your FEA model?

Hi Mariog,

My interpretation of the Jacobian, it is some sort of scaling factor for an element used to facilitate the numerical integration from one coordinate system (e.g. 0 to L) to another (-1 to +1) or a left over number to compensate for the conversion from one domain to another. Maybe the distance b/w the coordinates in the element, I've assigned is too small, less than the gaussian integration point of +/- 0.57735 for a 2x2 integration, so the number just blows up.

So I think, there is really some point which we need to stop the refining of mesh. There are some consultant that extract the forces at the toe of the weld from FEA model to calculate the stress from the equilibrium of forces which make it mesh insensitive.

Any other opinion is highly appreciated.

Just correct any wrong statement I've made.

Cheers!!

I agree with your last conclusions. The Jacobian is a map method form "real" to "natural" coordinates (and back) and shouldn't inflate the results.