The exponential and power material functions are often applied to functionally gradient materials (FGM). Obviously, it is of fundamental significance to study FGM with arbitrary material function. Because an arbitrary function can be treated as finite linear segments approximately, it is essential to research FGM with a linear material function. Crack-tip higher order stress and displacement fields for an anti-plane crack perpendicular to the direction of property variation in a FGM with a linear shear modulus along the gradient direction are obtained through the asymptotic analysis. The asymptotic expansions of crack tip stress fields bring out explicitly the influence of non-homogeneity on the structure of the stress field. The analysis reveals that only the higher order terms in the expansion are influenced by the material non-homogeneity. Moreover, it can be seen from expressions of higher order stress fields that at least three terms must be considered in the case of FGM in order to explicitly and theoretically account for non-homogeneity effects on crack tip stress fields.