Determination of Young’s modulus of samples of arbitrary thickness from force distance curves: numerical investigations and simple approximate formulae

•Simple, piecewise polynomial approximations of the load - indentation relation for a linear - elastic layer of an arbitrary thickness indented with a rigid punch are presented.•Layers either free to slip or bonded to the rigid substrate, indented with a conical, paraboloidal or cylindrical punch, are considered.•Numerical solutions of the integral equation representations of the thin layer indentation problem were approximated with piecewise, high order Chebyshev polynomials, and then used as input for the Remez algorithm, producing low - degree polynomial approximations with a uniform relative error.•The proposed approximations have been implemented in an open software AtomicJ to facilitate their use for analysis of force - distance curves recorded with an Atomic Force Microscope.