Three-dimensional time-harmonic elastodynamic Green’s functions for anisotropic solids

A method based on the Radon transform is presented to determine the displacement field in a general anisotropic solid due to the application of a time-harmonic point force. The Radon transform reduces the system of coupled partial differential equations for the displacement components to a system of coupled ordinary differential equations. This system is reduced to an uncoupled form by the use of properties of eigenvectors and eigenvalues. The resulting simplified system can be solved easily. A back transformation to the original coordinate system and a subsequent application of the inverse Radon transform yields the displacements as a summation of a regular elastodynamic term and a singular static term. Both terms are integrals over a unit sphere. For the regular dynamic term, the surface integration can be evaluated numerically without difficulty. For the singular static term, the surface integral has been reduced to a line integral over half a unit circle. Reductions to the cases of isotropy and transverse isotropy have been worked out in detail. Examples illustrate applications of the method.

[1]  R. Burridge,et al.  Fundamental elastodynamic solutions for anisotropic media with ellipsoidal slowness surfaces , 1993, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[2]  R. G. Payton,et al.  Elastic Wave Propagation in Transversely Isotropic Media , 1983 .

[3]  C. M. Fortunko,et al.  A computationally efficient representation for propagation of elastic waves in anisotropic solids , 1992 .

[4]  C. Y. Wang,et al.  A new look at 2-D time-domain elastodynamic Green's functions for general anisotropic solids , 1992 .

[5]  I. S. Sokolnikoff.  Mathematical theory of elasticity , 1946 .

[6]  F. John.  Plane Waves and Spherical Means: Applied To Partial Differential Equations , 1981 .

[7]  S. Deans.  The Radon Transform and Some of Its Applications , 1983 .

[8]  S. Helgason.  The Radon Transform , 1980 .

[9]  Hauser,et al.  Internal diffraction of ultrasound in crystals: Phonon focusing at long wavelengths. , 1992, Physical Review Letters.

[10]  J. Achenbach.  Wave propagation in elastic solids , 1962 .

[11]  W. C. Rheinboldt,et al.  The hypercircle in mathematical physics , 1958 .

[12]  N.Ya. Vilenkin,et al.  RADON TRANSFORM OF TEST FUNCTIONS AND GENERALIZED FUNCTIONS ON A REAL AFFINE SPACE , 1966 .

[13]  M. Lighthill,et al.  Studies on Magneto-Hydrodynamic Waves and other Anisotropic wave motions , 1960, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[14]  E. A. Kraut,et al.  Advances in the theory of anisotropic elastic wave propagation , 1963 .

[15]  J. Lothe,et al.  Elastic Strain Fields and Dislocation Mobility , 1992 .

[16]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[17]  Toshio Mura,et al.  Micromechanics of defects in solids , 1982 .

[18]  A new method to obtain 3-D Green's functions for anisotropic solids , 1993 .