The atomic force microscope (AFM) is increasingly becoming a first line tool for measuring material properties at microscopic scales. When it comes to soft biological materials such as cells, however, issues with experimental techniques and with data analysis remain. We have developed methodologies for thin, soft samples based on the assumption of material isotropy and homogeneity. Most biological systems, however, depending on the scale of interest may be neither. We are developing experimental techniques further to establish the types of measurements needed to acquire data that would allow detailed material characterization in the case of material anisotropies. In parallel, we are developing theoretical models for the corresponding contact problems which are being used to analyze experimental data. Our motivation is to measure the material parameters of the tectorial membrane (TM) in-vitro in order to establish its role in the mechanical filtering and amplification that occur in the mammalian cochlea. The TM is a collagenous membrane which has been shown by electron microscopy to have strongly anisotropic fiber organization. This anisotropic patterning of the fibers are suspected to play an important role in determining the mechanical response of the TM, both locally, at the stereociliary attachments and globally.