Stress intensity factors' behaviour is studied for long plane cracks interacting with a region of functionally graded elastic material. The region is assumed embedded into a large body treated as a homogeneous elastic continuum. The analysis is limited to small deviations of the graded region's elastic modulus from that of the surrounding body (Poisson's ratio is kept constant) and analytical solutions are sought using a perturbation technique. Emphasis is laid on the case of an infinite strip, which admits a closed form solution. A cosine change of the modulus of elasticity is treated, furnishing the solution for arbitrary variation in the form of a Fourier's expansion. Finite element analysis is subsequently performed for investigating the scope of validity of the analytical solution. The results for a set of finite changes of the elastic modulus are compared with the analytical predictions, and a remarkably wide range of validity is demonstrated. New functions, suitable for non-homogeneous material description, are introduced to approach the case of non-constant Poisson's ratio. The properties and possible applications of these functions are examined.