Diffraction of a plane shear elastic wave by a circular rigid disk and a penny-shaped crack

Abstract

Introduction.Diffraction of P-waves by disk-shaped obstacles and cracks embedded in a homogeneous and isotropic elastic medium has recently been given considerable attention. A bibliography of this subject is given in [1]. The corresponding problems of diffraction of S-waves are also of considerable interest in seismology and geophysics. Knowledge of the modification of S-waves in the vicinity of a crack is helpful in predicting the nature of flaws in the material. The dynamic stress intensity factors in the vicinity of a crack also help in predicting the fracture and failure of the material. Furthermore, the observation of diffraction pattern of S-waves generated by earthquakes or explosions reveal the presence of inhomogeneities and structural discontinuities in the medium. However, this problem has not received much attention. The main difficulty is that diffraction of S-waves leads to an asymmetric boundary value problem which is rather difficult to solve.We present here the solution of the problem of diffraction of a plane S-wave for two cases. In the first case we discuss diffraction of S-waves by a rigid circular disk which is embedded in an infinite homogeneous and isotropic elastic medium. In the second case we analyze the corresponding problem of diffraction of a plane S-wave by a penny-shaped crack. It is assumed that the two faces of the crack are separated by a small distance so that they do not come into contact during vibration. The incident S-wave is assumed to be time-harmonic and is polarized in planes perpendicular to those of the disk and the crack. Furthermore, it is assumed to propagate along their axes of

Diffraction of elastic waves by a penny-shaped crack: analytical and numerical results

“…They can also be used to obtain low-frequency asymptotic solutions of the integral equations, for we can expand as an ascending power series in £; see Appendix B for these expansions. In Appendix C , we derive low-frequency approximations for the diffraction of a normally-incident SV-wave, and compare our results with those obtained by Mai (1968c) and Jain & Kanwal (1972); we find complete agreement with those of Jain & Kanwal.…”

“…Taking k = K sin 0, letting 0 -> 0 and putting = 1, we find th a t Jain & Kanwal (1972) have also obtained low-frequency asymptotic expansions for Ic and Is. We find complete agreement with their results (apart from a typo graphical error in their equation (100): the second term inside the first pair of square brackets should be -( l + r 4) / ( l -r 2)).…”

“…Using his previous representations for (j), \J r and y, integral equations, from which he obtained low-frequency approximations for the dynamic stress-intensity factors. Jain & Kanwal (1972) have studied the same asymmetric problem as Mai. They used different integral representations for the scalar potentials, leading to a pair of Fredholm integral equations of the first kind, which they formally converted into Fredholm equations of the second kind.…”