Recent advances in nanotechnology have revealed numerous new methods of manufacturing functionally graded coatings and materials, but progress in this field is limited by the lack of knowledge about the mechanical behavior of such structures. Existing models of the mechanics of layered structures are not generally adequate for this purpose, since functionally graded structures can exhibit both qualitative and quantitative behavioral differences in comparison with homogeneous or layered structures, particularly if there is a significant gradient of elastic properties in the coating. In applications, interest is focused mainly on the deformation fields and stresses inside the inhomogeneous material caused by the contact tractions. Stresses at the interface between the functionally graded coating and the elastic half-space are of particular interest because of their influence on the propagation of cracks and other defects on this interface. Shear stresses at this interface associated with rapid variation in elastic properties with depth are particularly dangerous because of potential delaminations. In their work the authors: • develop a precise mathematical model and of the computational methods which makes it possible to achieve stable numerical results while analyzing the mechanical properties of functionally graded coatings; • study the variation effect in elastic properties on the maximum stresses in the surface layers of materials with functionally graded coatings caused by indentation.