Wave propagation in a nonhomogeneous elastic rod



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Texas Tech University


The longitudinal displacement and stress in a semi-infinite nonhomogeneous elastic rod subject to an arbitrary pressure at its end are calculated. Material inhomogeneity is introduced through the assumption of an exponentially varying Young's modulus. The Laplace transform is utilized to solve the equation of motion for the displacement in the case that the modulus varies throughout the rod. The method is then extended to the problem of a bimaterial rod consisting of a finite inhomogeneous section bonded to a semi-infinite homogeneous part. The Yotmg's modulus is continuous at the interface. For both problems integral representations of the stress and displacement are derived. As a limiting case of the analysis, the corresponding results for a homogeneous rod axe obtained. For the semi-infinite problem a short time asymptotic analysis is carried out. The effect of material inhomogeneity upon wave propagation is then deduced. Special results for the bimaterial rod axe obtained in the case that the variation in the modulus is small.



Shear waves, Elasticity, Wave mechanics