For large membrane deflections, see

Osamu Tabata et al., "Mechanical Property Measurements of Thin Films Using
Load-Deflection of Composite Rectangular Membranes,"
Sensors and Actuators A, vol. 20, 1989, pp. 135-141.

and

Mark G. Allen et al., "In Situ Measurement of Residual Stress, Young's
Modulus,
and Ultimate Strain of Thin Films,"
Applied Physics Letters, vol. 51, no. 4, July 1987, pp. 241-243.

        --Kirt