Dear Arti,

>Dear All, I have a membrane 24 microns thick, 3mm*3mm square. I used
>profilometer to get the deflection on it by applying 40mN force. I got a
>deflection of 40,000microns. I guess its huge and hence maybe its wrong.

Might the profilometer have slipped off the end, or torn the membrane
(depending on how you'd attached things), and kept going for 4 cm?  You can't
push a 3mm long membrane more than a couple of mm before you slip off the end.

>I want to calculate therotically the deflcetion of the membrane. I tried to
>get formula but unfortunately all are for pressure. Do anyone know a formula
>to calculate deflection when force is applied? If yes please let me know.
>Thanks
>Arti

There is (I hope) an easy-to-use formula for transverse cantilever
displacement Y (a force applied to the tip to bend it about the fulcrum)
in terms of longitudinal displacement X under the same force (grasping the
tip and stretching it in a straight line directly away from the fulcrum,
picturing the cantilever as being held parallel to the X-axis).  Just multiply
the latter by the square of twice the length over the thickness, i.e.
Y = X(2L/T)^2.

One way of remembering this formula is to start out with a mental picture of
a horizontal cantilever twice as thick as it is long (so pretty short!) and
store away the "fact" that the transverse and longitudinal displacements
of the tip are equal, *according to the formula* ("fact" because I imagine
this is actually somewhat inaccurate for L << 3T).  The remaining thing
to keep track of is that this displacement ratio varies quadratically with
increasing length and decreasing thickness (width doesn't enter, obviously).

This assumes a rectangular cross-section uniform along the length of
the cantilever, and is good for small displacements and large L/T.
I wouldn't swear to its accuracy for L << 3T, the idea of transverse
and longitudinal displacements exactly agreeing at 2L=T and not at 3L=T
sounds a bit suspicious but I don't know anything about the asymptotics
there--hopefully I calculated that factor of 2 correctly, I did this math
just now and could have screwed up.

Your membrane being 125 times as long as it is thick, the 2L/T factor is 250.
So your displacement will be 250^2 = 62500 times what you'd get from tugging
on the membrane with the same force.

Your membrane has cross-sectional area A = 72 * 10^-9.  If it has elastic
modulus E = 2.5 gigaPascals say, then a force of E*A = 2.5*72 = 180 newtons
would double its length if it didn't tear first, a 3 mm displacement.
40 millinewtons is 1/4500 of this and so would stretch the membrane 2/3 of a
micron.  The above factor of 62500 turns this into a 41,700 micron transverse
displacement, in excellent agreement with your profilometer's measurement!

With such close agreement of theory with experiment, it seems a shame
to have to mention that the theory is inapplicable for displacements even
approaching L, let alone exceeding it by a factor of 13 as here, quite apart
from whether 2.5 gPa is anywhere near the Young's modulus of your membrane,
and that the equipment was probably off looking at some other object far
away from the membrane.

Vaughan Pratt