Determining the layers' Young's moduli and thickness from the indentation of a bilayer structure

The inverse problem of determining three parameters: the film thickness, the film and substrate Young's moduli of a film/substrate bilayer by indentation, is formulated and solved. The physical mechanism for the solvability of the inverse problem is that these three parameters have different impacts at different indentation depth. Their impacts are systematically studied, which also provides a different approach of finding the three parameters or refining their range. Compared with various atomic force microscopy based techniques of detecting subsurface structures, which have to deal with an extremely difficult or even an insurmountable inverse problem with the integral equation of dynamics, the inverse problem here formulated by statics is much more straightforward and simpler. Formulating and solving such an inverse problem can be of some help to the applications such as characterizing subsurface structures, the out-of-plane properties of two-dimensional (2D) materials and various bilayer structures.