Some dynamic problems for elastic materials with functional inhomogeneities: anti-plane deformations

The paper considers two dynamical problems for an isotropic elastic media with spatially varying functional inhomogeneity, the propagation of surface anti-plane shear SH waves, and the stress deformation state of an anti-plane vibrating medium with a semi-infinite crack. These problems are considered for five different types of inhomogeneity. It is shown that the propagation of surface anti-plane shear waves is possible in all these cases. The existence conditions and the speed of propagation of surface waves have been found. In the section devoted to the investigation of the stress deformation state of a vibrating medium with a semi-infinite crack, Fourier transforms along with the Wiener Hopf technique are employed to solve the equations of motion. The asymptotic expression for the stress near the crack tip is analyzed, which leads to a closed form solution of the dynamic stress intensity factor (DSIF). Here also the problem is considered for five different functional inhomogeneities. From the formulae for DSIF thus obtained one can see that the inhomogeneity can have both a quantitative and qualitative impact on the character of the stress distribution near the crack.