Characterization of material parameters from experimental data remains challenging, especially on biological structures. One of such techniques allowing for the inverse determination of material parameters from measurement data is the virtual fields method (VFM). However, application of the VFM on general structures of complicated shape has not yet been extensively investigated. In this paper, we extend the framework of the VFM method to thin curved solids in three-dimensional, commonly denoted shells. Our method is then used to estimate the Young's modulus and hysteretic damping of the human eardrum. By utilizing Kirchhoff plate theory, we assume that the behavior of the shell varies linearly through the thickness. The total strain of the shell can then be separated in a bending and membrane strain. This in turn allowed for an application of the VFM based only on data of the outer surface of the shell. We validated our method on simulated and experimental data of a human eardrum made to vibrate at certain frequencies. It was shown that the identified material properties were accurately determined based only on data from the outer surface and are in agreement with literature. Additionally, we observed that neither the bending nor the membrane strain in an human eardrum can be neglected and both contribute significantly to the total strain found experimentally.