Relationship between pressure and thickness 3

[Lucifer17]

I have an elastic material between two plates. One of the plates is stationary. Does anyone know the mathematical formula for calculating the change of distance between the two plates due to perpendicular pressure?

 

[bvi]

If the surface contact between the plates is frictionless and the material behavior elastic, then it is d=st/E where d is displacement, s is contact stress, t is thickness of plate being compressed, and E is elastic modulus. If you have friction, the change in distance will be less. You will develop stresses in the plane of the plate being compressed.

 

[desertfox]

Hi Lucifer17

The formula given by cb4 is correct, however this only applies if the elastic material follows Hookes law ie:- I
mean the Modulus of Elasticity of the material is a straight
line which is derived from plotting stress versus strain.
You quote an elastic material between plates an elastic material to a Mechanical Engineer would include Steel,Brass,
Aluminium etc, but I think you mean a Rubber or some other
non-metalic material in which case the above formula would not necessarily apply.
Assuming that all the materials are in accordance with Hookes law then the true change in overall height of your
composite would be:-

d*s*t/E for the elastic material
and 2*(d*s*t/E)for the plate material

If you can be more specific about your materials ie:- what are the materials involved and what are the dimensions of the plates and elastic material, also how are the plates supported and by what means is the pressure applied we may be able to help you further.


regards desertfox

 

[march1971]

desertfox is right, there is need for you to give mor details so that useful solutions can be profered.

 

[Lucifer17]

First, I want to thank you all for your replies.
Here are more details about my problem:
The plates are very small and metallic(copper) and have a circular surface: Radius: 17u Thickness: 1u
The elastic material between the plates is non-metalic as Desertfox assumed and very soft (this is actually a second problem that I have. The material must be non-conductive with high permittivity(3-4), any suggestions?) and is 4u thick (distance between the plates).
I like to know with some force(200-800 g) how far I can press down the moving plate.

 

[desertfox]

Hi Lucifer

Thankyou for your response, what is 17u? don't understand the units do you mean 17 micrometres. Also we need the mechanical properties of the non metallic ie:- Modulus of elasticity and maximum compressive strength, well I don't know about permittivity figures but how about using a silicon rubber?
You need to decide on your materials before we can help you with the mechanics.


regards desertfox

 

[Keegan1]

Hi everyone,

I'm helping Lucifer17 with this. Let me try to describe the problem in more detail: Imagine two cookies stacked on a drinking glass. The top cookie is an elastomer. The bottom cookie is thinner, but made of a slightly more rigid material. Assume the glass is rigid, and supports the cookies only at the outer rim. What is desired to know is the deflection of the center of the bottom cookie with an applied force downward on the top one.

Now for the scale of the problem: The top cookie is 4 microns thick. Not sure yet what this material is, but lets assume it's a membrane. The bottom cookie is 1 micron thick; material is copper. The glass O.D. and cookie diameters are 34 microns. The wall thickness of the glass is 3 microns.

I think with a point-force downward at the center of the top piece (elastomer) it will translate to pressure (force over area) evenly distributed across the top of the bottom piece. [I think this because of the 4/1 ratio of thicknesses of the top layer to the bottom; maybe I'm off-base] So, imagine the problem as a very thin plate (the bottom one) with pressure applied to one side. The plate is circular, supported simply at the outer boundary.

I've poked around looking for equations to describe the deflection of a round plate, but haven't found anything useful.

Hope my description helps to clarify the problem. Thanks in advance for your help.

 

[desertfox]

Hi Lucifer & Keegan1

My approach to this problem would be :-

1/ Material properties of both supporting plate and
elastomer ie strengths of both materials and Modulus
of elasticity of both materials.

2/ Workout the equivalent thickness of the elastomer
as if it was made of copper, you can do this using the
formula below:-

t'= (Ec/Ee)*tc

where t'= equivalent thickness in copper of the elastomer
Ec= modulus of elasticity of copper
Ee= modulus of elasticity of elastomer
tc= thickness of copper disc.

Now use The book "formula's for stress and strain" by Roark
& Young look for flat circular plates in the book.Your case, or the nearest would be "case 16" table 24 page 366 in my book which is the 5th edition. If you're loading the disc via a central point load the stresses will be very high, at the begining of the flat plates chapter they give an equivalent radius based on the thickness of your disc, at which you can assume the load will act over for your situation.
Again here I don't know how accurate you need to be, but be careful choosing your elastomer and try to find out whether
the modulus of elasticity is linear over the range of your loading as I mentioned in an earlier post.
Now the problem you have described here is one of bending so the formula given in earlier posts are not applicable as they are for tension and compression loads.

regards desertfox

 

[jte]

Lucifer17 and Keegan1-

To add to desertfox's commentary:

According to Timoshenko, Poisson established the solution to the bending of circular plates in 1829. Timoshenko covers this issue with Woinowsky-Krieger in "Theory of Plates and Shells" published by McGraw.

Here's the trick part of desertfox's commentary: There was a subtle change between the 5th edition, "Formulas for Stress and Strain" by Roark and Young, and the 6th (as far as I'm aware, the latest) edition, "Roarks Formulas for Stress and Strain" by Young (also published by McGraw). Gotta know your author if you're searching for the book. I'd recommend getting this book over Timoshenko simply because "Roarks" covers the same stuff - in the 6th ed. chapter 10, Table 24 Case 16 (page 432) seems to apply - and then goes beyond Timoshenko.

The danger with using the formulas in Timoshenko or "Roark's" Table 24 is that they are limited to a deflection of 1/2 the plate thickness (page 391, the intro. to chapter 10). These formulas are based on the plate resisting the load through bending. This is all well and good until the plate deforms enough to start to carry the load through membrane stress which resists the load much more efficiently. So... read section 10.11 on pages 457 and 477 and use the iterative approach for your solution.

Although it's more involved, you'll find that your solution is FAR more accurate if you use the formulas which account for large deflection. Actual deflection in the center of the plate tracks test data ('course, I was looking at carbon steel at more "normal" sizes) much better. The basic formulas can overestimate deflection by a factor of 2 when the actual deflection is on the order of one plate thickness and a factor of 3 at 2t.

As a final option, you could use FEA, but you must deal with a few nonlinearities - geometric and probably material - so that could get tedious. Fortunately the problem is axisymmetric so at least the modeling should be easy.

jt