Discussion Forum

Dear Dhwajal,
as you suggest, you can use the Mindlin Plate application mode. In this case you have to set initial torques which are "equivalent" to the initial stress you have to model.
If you give a look to Comsol manual in the part about Mindlin Plate, you will find how these initial torques are expressed as function of the initial stresses.
In any case, bear in mind that you can reduce the compational effort of your calculation by using simmetries.
If your load is symmetric and your geometry is symmetric too, you can cut your model and study just a portion of it, thus highly reducing the computational cost.
I hope this help.
Hi!

Alessandro

Hi Alessandro,

Thank you for reply.

I have a sqare silicon nitride membrane and have a residual stress of 1 GPa. As mentioned by you, if I fill in values for Mx, My and Mxy in subdomain settings--> Include intial forces and moments, I dont see any change in eigen freq. value!! Did I get your hint right? Awaiting your comment.

Thank you again.

Dhwajal

Hi Dhwajal,
this fact could happen because of two reasons:

1) initial stresses do not actually affect the stiffness of your plate (maybe because boundary conditions allow the plate is to relax its internal stress)

2) the stiffness matrix employed in Eigenfreq calculation does not contain the contribution of the initial stresses.

In 3D Solid Stress-Strain module, for instance, you can activate thoese contributions to the stiffness matrix by setting "on" in the field "Large Deformation" under "Physics/Properties".
I saw that this option is not present in the Mindlin Plate application mode.
I suggest you to the exploit symmetries and try to solve the problem with the Solid Stress Strain appl. mode.
If you want, you can send to me your model and I will try to help you in a more "effective" way.
Hi!

Alessandro

Hi Allessandro & Dhwajal

I beleive we have already a couple of discussions about shell elements and the limitations of large deformation, try the search.

I would too advise to start with a 3D case (this time) but to use the symmetry. Note that for eigenfrequency analysis you must repeat 2^N times with also the antisymmetric BC's to catch all modes, slightly cumbersome to do, but the only way when hitting RAM liitations.

try it out on a simple case where you can find a handbook reference

And continue to keep us informed, always usefull to know

Have fun Comsoling
Ivar

Dear Ivar,

Thank you very much for your kind comment.

I tried using 3D solid:stress strain module but still I am stuck at meshing. I think, I should mention more about my geometry. My 'square plate' is actually a square membrane with dimensions 1 cm X 1cm X 50 nm. I think this huge dimension ratio is crashing the system while meshing. If I mesh too coarse, to reduce the mesh element, I am hit by minimum mesh quality. Any comments on this.

Best,

Dhwajal

Hi

for such geoemtries, first use the "equal" scale on/off functionality of the graphics window (bottom botrder, double click, side by side with GRID etc) to better select items, then use a surface mesh and sweep through the thin direction, or at least use the "advanced" manual mesh scaling in your "thin" direction.

This requires some mesh exercices on simple cases (even better, for a cick start, go to the COMSOL course on advanced meshing) you can find differnent info on meshing distributed around in the doc, use an "indexed search" on the pdf doc of comsol to find your way

Nothing like "exercices makes masters" ;)
Good luck
Ivar

Hi Ivar,

Thank you so much for your comments.

Today I tried exactly what you suggested. I created geometry (1mm X 1mm X 50nm) in 3D solid:stress strain and used interactive meshing, to first mesh the surface and then use sweep mesh parameters with 5 elements in thickenss direction. I got resonance freq. of 9.349 kHz without stress.

Just to check if these values make any scence, I modeled the same geomery in mindlin plate and got the resoance freq. of 828 Hz. So to check which value is true, I modeled in Shell module to find the reso. freq of 828 Hz. Which to me also sounds reasonable.

As finding resonance freq. with stress is my goal, I can't rely on 3D model and I can't solve it in mindlin or shell :(. The buck stops here!!!

Best,

Dhwajal

Hi

I find that you are going a bit quick in your conclusions, me too when I proposed 3D because: it's important that one understand well what one are demanding, I had also forgotten to check a few things:

I do not know how many elements you are using, but in 3D solid, 1x1mm^2 and 50nm thick that is a ratio of 1:20'000 hence to catch an effect in the thickness one should use a minimum of 3 elements in the thickness, 5 would be much better (as a minimum), and for elements to give good results they should not be more distorted than 1:10 in average, and 1:100 for some (as minimum quality), that means, if one use rectangular elements, an individual size of lets say 10[nm] in height for (100[nm])^2 area, one would need 10'000x10'000x5=500'000'000 elements, for second order quad items one have about 38Dof per elements, so some 19GDof, if I'm not wrong, I would assume one would need a 4TB RAM PC and at 3GHz clocking of today probably a year or more to solve. That is why one has made shell elements, as the effect in thickness can be reduced by an equation and one have no more this shape limitation in the thickness direction and a few tousand elements, a laptop and a few second CPU time, would do.

well I tried with 34x34x9 elements (quads), I get a few shapes that look like higher order modes at 2 and 2.2 ... kHz (for default steel), while the midline plate give me consistently some 442Hz in less than 8 sec, and a nice bellow shape with a few tousands elements. For me this is most probaby the shape quality <1E-4 that is so bad in 3D that one get numerical underflows, with this thin membrane

I did not fully understand your model previously, as I was thinking of a "small" feature embedded in a large volume, but that allows to increase the mesh size radially outwards, in your case with a fixed 50nm thickness you are indeed stuck.

But anyhow the theory and equations used in structural says that for your case of thin membrane the stress distribution should be symmetric in the thickness, with a centered midline, apart just at the edges and cornes, where you have certainly some fillets (in true life) and where 99.99% of the stres will concentrate, so the simulation of the correct shape of the jointure of the thin memebrane to the bulk surface is also important.

But I still was woundering why the frequency would change with stress, as the frequency is 1/2/pi*sqrt(k/m), the mass hardly changes (density certainly not), stiffness is related to the young module and the inertia, if you use a Young modulus as function of the stress OK then it is non-linear, else not, and the thickness will not change that much, or ?, which physical effect would induce a frequency change by adding initial stress ?

Cannot help more
Ivar

Dear Ivar,

Thank you again for your kind reply. I did get your point.

The increase in resonance freq. due to initial stress comes from the fact that you model the intial stress (residual stress for my MEMS structure) as tensile stress (as its SiN). For this I followed 'Theory of Plates and Shells' by Timoshenko. I have solved this par. diff. equ. analitically and was able to derive 'stiffness factor' which is function of thichness, radius and tensile stress, but my model fails, for this current dimension ratio. You can try to model a circular membrane of Radius= 150e-6 m and thickness= 1e-6,which will give resonance at ~200kHz. Now when you apply intial stress ?x, ?y and ?xy of about 175 MPa (what we typically get for Silicon Nitride membrane) we get the resonance freq. of ~ 620 kHz.

I will look for more literature which deals with stresses in membranes with this huge dimension ratio.

Again thanks for your all effort.

Best,

Dhwajal

Hi

Well the subject interest me as I have on my todo list some studies on AlN mems resonators ;)

But I have a few other complex simulations to close first ;)

Ivar