In the present work the problem of thin surface coatings of an elastic layer subject to a residual thermal stress is analytically investigated. The system adopts a curvature as a result of the misfit strain due to the different thermal expansivities of the layer and coatings. The problem is governed by a singular integral equation which can be reduced to a linear algebraic system by approximating the unknown interfacial shear stress via series expansion of Chebyshev polynomials. The present analysis is developed in order to assess the mechanical behaviour of a crystalline undulator designed for obtaining high emission radiations through channelling phenomenon.