plz refer this book

Hardcover. Transport Processes: Momentum, Heat and Mass (Allyn and Bacon series in engineering). $2.36. Hardcover. Books by Christie J. Geankoplis

Thank you : D

You will find a wealth of important answers with a search on vibrational modes of clamped membranes. The elastic modulus of the system mediates the amount of deformation in response to a force. It also drives the frequency and mode of the vibration in oscillations. Think of a spring in 1-D that follows Hook's law. Force and oscillation frequency both contain the spring constant. What you find in your search is the extension of this idea to 2-D and 3-D with membranes. Just because you see a non-linear deformation in response to the force may not mean that your membrane is in a non-linear (non-Hook's law) region. The limit of course is that you do not overdrive your clamped membrane system to a non-linear region (outside of Hook's law behavior). Your question is not clear whether you are doing this or not.

Dear Prof. Weimer,

Thank you very much.

I want to make clear the elastic property of 2 nm-thick silicon membranes.

Si floating membranes which we fabriacted is rectangular with all edges clamped. When the AFM tip loads on such floating membranes, the deformation was large and its variation is nonlinear. We want to obtain the Young's modulus from the force-deformation curve.

Best wishes,

Xiaodong

I recommend that you find a mechanical engineering textbook or equivalent source book with equations for the mechanical deformations in clamped membranes. The deformation versus force equation will use the geometry of the membrane component as well as the Young's modulus of the material. For a bi-axially clamped membrane, it will also use the Poisson ratio of the material. Since this is a point load, the shape of the deformation versus force curve will depend on the position where you load the membrane. The best experiment will be one where you load the membrane reproducibly at exactly its center position. To some extent, I can also imagine the use of a square or circular membrane will make it easier for you to apply the mechanical conversion equation.