Dynamic response of a functionally graded tube embedded in an elastic medium due to SH-Waves

Dynamic response of a cylindrical tube surrounded by an unbounded elastic medium due to plane harmonic SH-Waves is studied. A two-dimensional mathematical model is considered. Cylindrical coordinates are used for convenience. The surrounding medium is assumed to be homogeneous, isotropic and linear elastic. The tube is assumed to be made of linear elastic functionally graded materials (FGMs) such that shear modulus and shear wave velocity are assumed to change linearly from inner surface to outer surface. Material properties are constant along circumferential direction. It is assumed that the inner surface of the tube is traction-free and there is a welded contact between the tube and the surrounding medium. Governing equations are slightly different in the tube region and the unbounded region. Both of the governing equations are solved by applying Finite Fourier Transform in circumferential direction. The exact solution series are presented in terms of Fourier-Bessel series in the unbounded region and power series in the tube region. The presented numerical results show that when the incoming wave lengths decrease, shear stresses at the tube increase significantly. It was shown that for the shorter incoming wave lengths, tubes made of FGMs are subjected to smaller shear stresses compared to the tubes homogeneously made of outer surface material of the FG cases.