Universal form of the equations governing membrane deformation under hydrostatic pressure for simpler design of sensors and tunable optical devices

Flexible membranes have applications in liquid filled lenses and pressure sensors. They deform under hydrostatic pressure, thus changing the asphericity of the lens and its focal length. This behavior enables tuning of the lens by changing the pressure of the fluid inside. A universal form of the nonlinear differential equations describing the deformation of a flexible membrane is presented here, showing that their solution is valid for membranes having the same thickness to radius ratio and made of materials having the same flexural rigidity and Poisson ratios. Hence by solving the equations once, a simple scaling allows obtaining a set of solutions that matches these ratios. This should simplify the design of tunable lenses and pressure sensors based on flexible membranes. In addition, approximate analytic solutions are presented in a normalized form.