The linear dynamics of axially functionally graded(AFG)cantilevered pipes conveying fluid is studied.Based on the differential equation of motion of uniform cantilevered pipes conveying fluid,the governing equation of an AFG cantilevered pipe is derived and further discretized by using the differential quadrature method(DQM).The elastic modulus and density of the pipe are assumed to be varied along the pipe length and the effects of elastic modulus gradient and density gradient on the critical flow velocity for flutter instability are studied.It is found that,compared with uniform pipes,the increase of elastic modulus from the fixed end to free end leads to a more stable system,but the opposite result may be obtained for the increase of density.In the curves of dimensionless critical flow velocity versus density gradient and of dimensionless critical velocity versus elastic modulus gradient,the anti-S-shape segments are observed when the mass ratio becomes large.Furthermore,mode exchange has also been detected with increasing density gradient or elastic modulus gradient within a certain range of mass ratio.