×

zbMATH — the first resource for mathematics

Three-dimensional time-harmonic elastodynamic Green’s functions for anisotropic solids. (English) Zbl 0852.73011
A three-dimensional displacement elastodynamic Green’s function (second order tensor) for a homogeneous anisotropic unbounded solid subject to a time-harmonic point force is obtained in the form of a sum of two integrals over a unit sphere, by using the Radon transform technique. The first term of the sum represents a static singular part of the Green’s function, while the second term describes a dynamic regular part of this function. The integrands of both integrals are expressed in terms of the eigenvectors and eigenvalues of a scaled acoustic tensor [cf. M. E. Gurtin, The linear theory of elasticity, in Encyclopedia of Physics, Springer-Verlag, Berlin (1972), vol. 6a/2, p. 244. The general anisotropic Green’s function is shown to be reducible to a well-known closed-form Green’s function for a homogeneous isotropic unbounded elastic body, and to a particular integral form for a homogeneous transversely isotropic unbounded elastic body. Numerical results are also presented for a transversely isotropic body subject to a point force parallel to the axis of isotropy.
The reviewer believes that the formula for \(c_3\) in eqs. (7.19) is misprinted and should read \(c_3= \sqrt {\kappa_4 (1- n^2_3)+ n^2_3}\).

MSC:
74E10 Anisotropy in solid mechanics
74B05 Classical linear elasticity
44A12 Radon transform
PDF BibTeX XML Cite
Full Text: DOI