Abstract
This paper gives a brief review of the Green's function method for solution of the general three-dimensional Boussinesq problem for advanced materials that are highly anisotropic. The Boussinesq problem refers to calculation of stress and/or strain fields in semi-infinite solids, subject to surface loading by solving the equations of elastostatic equilibrium. Analytical and semi-analytical expressions are derived for the elastostatic Green's functions based upon the delta-function representation developed earlier. The Green's function provides a computationally efficient method for solving the anisotropic Boussinesq problem. The Green's function should be useful for modeling physical systems of topical interest such as nanostructures in semiconductors, interpretation of nanoindentation measurements, and application to the boundary-element method of stress analysis of advanced materials. Numerical results for displacement and stress fields are presented for carbon-fiber composites having general orthotropic, tetrahedral, and hexagonal symmetries, and single-crystal silicon having cubic symmetry.
Similar content being viewed by others
References
D. Bimberg, M. Grundmann and N.N. Ledentsov, Quantum Dot Heterostructures. New York: John Wiley (1999) 328pp.
M. Hebbache and M. Zemzemi, Nanoindentation of silicon and structural transformation: Three-dimensional contact theory. Phys. Rev. B 67 (2003) article number 233302.
T.A. Cruse, Boundary-element Analysis in Computational Fracture Mechanics. Dordrecht: Kluwer Academic Publishers (1988) 162pp.
A.G. Every and K.Y. Kim, Determination of elastic constants of anisotropic solids from elastodynamic Green's Functions. Ultrasonics 34 (1996) 471–472.
T. Dutta, T.K. Ballabh and T.R. Middya, Green-Function calculation of effective elastic constants of polycrystalline materials. J. Phys. D-Appl. Phys. 26 (1993) 667–675.
R. Arias, Elastic fields of stationary and moving dislocations in finite samples. Phil. Mag. B 78 (1998) 109–113.
T.C.T. Ting and D.M. Barnett, Image force on line dislocations in anisotropic elastic half-spaces with a fixed boundary. Int. J. Solids Struct. 30 (1993) 313–323.
. J.P.Hirth and J. Lothe, Theory of Dislocations. Malabar, Florida: Kreiger Publishing Company (1992) 857pp.
V.K. Tewary, Multiscale Green's-function method for modeling point defects and extended defects in anisotropic solids: application to a vacancy and free surface in copper. Phys. Rev. B (2004) article number 094109.
T.C.T. Ting, Anisotropic Elasticity: Theory and Applications. New York: Oxford University Press (1996) 570pp.
T. Mura, Microechanics of Defects in Solids. The Hague: Martinus Nijhoff Publishers (1982) 494pp.
T.C.T. Ting and V.G. Lee, The three-dimensional elastostatic Green's function for general anisotropic linear elastic solids. Q. J. Mech. Appl. Math. 50 (1997) 407–426.
D.M. Barnett, The precise evaluation of derivatives of the anisotropic elastic Green's functions. Physica Status Solidi (b) 49 (1972) 741–748.
J.W. Deutz and H.R. Schober, Boundary value problems using elastic Green's functions. Comp. Phys. Comm. (Netherlands) 30 (1983) 87–91.
S.A. Gunderson and J. Lothe, A new method for numerical calculations in anisotropic elasticity problems. Physica Status Solidi (b) 143 (1987) 73–85.
Y. Hisada, Efficient method for computing green's functions for a layered half-space with sources and receivers at close depths. Bull. Seismolog. Soc. Am. 85 (1995) 1525–1526.
E. Pan, Static green's functions in multilayered half spaces. Appl. Math. Modell. 21 (1997) 509–521.
T.C.T. Ting, Green's functions for an anisotropic elliptic inclusion under generalized plane strain deformations. Q. J. Mech. Appl. Math. 49 (1996) 1–18.
E. Pan and F.G. Yuan, Three-dimensional Green's functions in anisotropic bimaterials. Int. J. Solids Struct. 37 (2000) 5329–5351.
C.Y. Wang, 2-Dimensional elastostatic Green's Functions for general anisotropic solids and generalization of Stroh's Formalism. Int. J. Solids and Struct. 31 (1994) 2591–2597.
C.Y.Wang and J.D. Achenbach, A new method to obtain 3-D Green's functions for anisotropic solids. Wave Motion 18 (1993) 273–289.
C.Y. Wang, Elastic fields produced by a point source in solids of general anisotropy. J. Engng. Math. 32 (1997) 41–52.
J.R. Willis, Self-similar problems in elastodynamics. Phil. Trans. R. Soc. A274 (1973) 435–491.
I.N. Sneddon, Fourier Transforms. New York: McGraw Hill (1951) 542 pp.
E. Pan and B. Yang, Elastostatic fields in an anisotropic substrate due to a buried quantum dot. J. Appl. Phys. 90 (2001) 6190–6196.
E. Pan, Three-dimensional Green's functions in an anisotropic half space with general boundary conditions. J. Appl. Mech.-Trans. ASME 70 (2003) 101–110.
K.P.Walker, Fourier integral representation of the Green function for an anisotropic elastic half-space. Proc. R. Soc., London A 443 (1993) 367–389.
B. Yang and V.K. Tewary, Formation of a surface quantum dot near laterally and vertically neighboring dots. Phys. Rev. B 68 (2003) article number 035301.
H. Hasegawa, V.G. Lee and T. Mura, Green-functions for axisymmetrical problems of dissimilar elastic solids. J. Appl. Mech.-Trans. ASME 59 (1992) 312–320.
R. Rajapakse and Y. Wang, Greens-functions for transversely isotropic elastic half-space. J. Engng. Mech.-ASCE 119 (1993) 1724–1746.
I.N. Sneddon, Fourier-transform solution of a boussinesq problem for a hexagonally aeolotropic elastic halfspace. Q. J. Mech. Appl. Math. 45 (1992) 607–616.
H.Y. Yu, S.C. Sanday and C.I. Chang, Elastic inclusions and inhomogeneities in transversely isotropic solids. Proc. R. Soc. London A 444 (1994) 239–252.
Z.Q. Yue, Closed-form Green's functions for transversely isotropic bi-solids with a slipping interface. Struct. Engng. Mech. 4 (1996) 469–484.
R.Y.S. Pak and F. Ji, Axisymmetrical stress-transfers from an embedded elastic cylindrical-shell to a halfspace. Proc. R. Soc. London A 441 (1993) 237–259.
M.A. Sales and L.J. Gray, Evaluation of the anisotropic Green's function and its derivative. Comp. Struct. 69 (1998) 247–254.
Y.C. Pan and T.W. Chou, Point force Solution for an infinite transversely isotropic solid. J. Appl. Mech.-Trans. ASME 43 (1976) 608–612.
Y.C. Pan and T.W. Chou, Green's function solutions for semi-infinite transversely isotropic materials. Int. J. Engng. Sci. 17 (1979) 545–551.
Y.C. Pan and T.W. Chou, Green's functions for two-phase transversely isotropic materials. J. Appl. Mech.-Trans. ASME 46 (1979) 551–556.
C.D. Wang, C.S. Tzeng, E. Pan, J.J. Liao, Displacements and stresses due to a vertical point load in an inhomogeneous transversely isotropic half-space. Int. J. Rock Mech. Mining Sci. 40 (2003) 667–685.
D.M. Barnett and J. Lothe, Line force loadings on anisotropic half-spaces and wedges. Physica Norvegica 8 (1975) 13–22.
J.R. Barber, Some polynomial solutions for the non-axisymmetric boussinesq problem. J. Elasticity 14 (1984) 217–221.
E. Pan and F.G. Yuan, Boundary element analysis of three-dimensional cracks in anisotropic solids. Int. J. Num. Meth. Engng. 48 (2000) 211–237.
F. Tonon, E. Pan and B. Amadei, Green's functions and boundary element method formulation for 3D anisotropic media. Comp. Struct. 79 (2001) 469–482.
M. Gellert, Discrete numerical-solution of the generalized Boussinesq problem. Int. J. Num. Meth. Engng. 21 (1985) 2131–2144.
J.R. Willis, Hertzian contact of anisotropic bodies. J. Mech. Phys. Solids 14 (1966) 163–176.
A.G. Every, Displacement field of a point force acting on the surface of an elastically anisotropic half-space. J. Physics A 27 (1994) 7905–7914.
V.K. Tewary, Computationally efficient representation for elastodynamic and elastostatic Green's functions for anisotropic solids. Phys. Rev. B51 (1995) 15695–15702.
P.M. Morse and H. Feshbach, Methods of Mathematical Physics: Part I. New York: McGraw-Hill (1953) 1037pp.
V.K. Tewary, R.H. Wagoner and J.P. Hirth, Elastic Green's function for a composite solid with a planar interface. J. Materials Res. 4 (1989) 113–123.
V.K. Tewary, Elastic Greens-function for a bimaterial composite solid, containing a free-surface normal to the interface. J. Materials Res. 6 (1991) 2592–2608.
P.A. Martin, On Green's function for a biomaterial elastic half-plane. Int. J. Solids Struct. 40 (2003) 2101–2119.
L. Pan, Boundary-Element Strategies and Discretzied Green's Functions. Ph.D. Thesis, Iowa State University (1997) 156pp.
P.A. Martin and F.J. Rizzo, Partitioning, boundary integral equations, and exact Green's functions. Int. J. Num. Meth. Engng. 38 (1995) 3483–3495.
V.K. Tewary, M. Mahapatra and C.M. Fortunko, Green's function for anisotropic half-space solids in frequency space and calculation of mechanical impedance. J. Acoust. Soc. Am. 100 (1996) 2960–2963.
K.Y. Kim, T. Ohtani, A.R. Baker and W. Sachse, Determination of all elastic constants of orthotropic plate specimens from group velocity data. Res. Nondestruct. Eval. 7 (1995) 13–29.
V.K. Tewary, Mechanics of Fiber Composites. New York: John Wiley (1979) 288pp.
I.M. Gel'fand and G.E. Shilov, Generalized functions-Vol 1. New York: Academic Press (1964) 423 pp.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tewary, V. Elastostatic Green's function for advanced materials subject to surface loading. Journal of Engineering Mathematics 49, 289–304 (2004). https://doi.org/10.1023/B:ENGI.0000031191.64358.a8
Issue Date:
DOI: https://doi.org/10.1023/B:ENGI.0000031191.64358.a8