Three-Dimensional Interfacial Green’s Function for Exponentially Graded Transversely Isotropic Bi-Materials

Document Type: Research Papers

Authors

1 School of Civil Engineering, College of Engineering, University of Tehran

2 School of Engineering Science, College of Engineering, University of Tehran

Abstract

By virtue of a complete set of two displacement potentials, an analytical derivation of the elastostatic Green’s functions of an exponentially graded transversely isotropic bi-material full-space was presented. Three-dimensional point-load Green’s functions for stresses and displacements were given in line-integral representations. The formulation included a complete set of transformed stress-potential and displacement-potential relations, with the utilization of Fourier series and Hankel transform. As illustrations, the present Green’s functions were analytically degenerated into special cases, such as exponentially graded half-space and homogeneous full-space bi-material Green’s functions. Owing to the complicated integrand functions, the integrals were evaluated numerically, and in computing the integrals numerically, a robust and effective methodology was laid out which provided the necessary account of the presence of singularities of integration. Some typical numerical examples were also illustrated to demonstrate the general features of the exponentially graded bi-material Green’s functions which will be recognized by the effect of degree of variation of material properties.

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Main Subjects


Apsel, R.J., and Luco, J.E. (1983). “On the Green’s functions for a layered half space. Part II”, Bulletin of the Seismological Society of America, 73(4), 931-951.
Ardeshir-Behrestaghi, A., and Eskandari-Ghadi, M. (2009). “Transversely isotropic two-layered half-space under the tangent load to the surface in the frequency domain”, Journal of Faculty of Engineering, University of Tehran, 43(4), 335-348 (In Persian).
Birman, V., and Byrd, L.W. (2007). “Modeling and Analysis of Functionally Graded Materials and Structures”, Applied Mechanics Reviews, ASME, 60(5), 195-216.
Chan, Y.S., Gray, L.J., Kaplan, T., and Paulino, G.H. (2004). “Green’s function for a two-dimensional exponentially graded elastic medium”, Proceedings of the Royal Society of London, Series A, 460 (2046), 1689-1706.
Eskandari, M., and Shodja, H.M. (2010). “Green’s functions of an exponentially graded transversely isotropic half-space”, International Journal of Solids and Structures, 47 (11-12), 1537-1545.
Eskandari-Ghadi, M. (2005). “A complete solution of the wave equations in the transversely isotropic media”, Journal of Elasticity, 81 (1), 1-19.
Eskandari-Ghadi, M. (2007). “Potential function method for transversely isotropic with axially symmetric”, Journal of Faculty of Engineering, University of Tehran, 41(6), 675-681 (In Persian).
Eskandari-Ghadi, M., Pak, R.Y.S., and Ardeshir-Behrestaghi, A. (2009a). “Elastostatic Green's functions for an arbitrary internal load in a transversely isotropic bi-material full-space”, International Journal of Enginering Science, 47(4), 631-641.
Eskandari-Ghadi, M., Sture, S., Pak, R.Y.S., and Ardeshir-Behrestaghi, A. (2009b). “A tri-material elastodynamic solution for a transversely isotropic full-space”, International Journal of Solids & Structures, 46(5), 1121–1133.
Eskandari-Ghadi, M., Pak, R.Y.S., and Ardeshir-Behrestaghi, A. (2008). “Transversely isotropic elastodynamic solution of a finite layer on an, infinite subgrade under surface loads”, Soil Dynamics and Earthquake Engineering, 28(12), 986-1003.
Eskandari-Ghadi, M., and Amiri-Hezaveh, A. (2014). “Wave propagations in exponentially graded transversely isotropic half-space with potential function method”, Mechanics of Materials, 68, 275-292. 
Kalantari, M., Khojasteh, A., Mohammadnezhad, H., Rahimian, M., and Pak, R.Y.S. (2015). “An inextensible membrane at the interface of a transversely isotropic bi-material full-space”, International Journal of Engineering Science, 91, 34-48. 
Kashtalyan, M., and Rushchitsky, J.J. (2009). “Revisiting displacement functions in threedimensional elasticity of inhomogeneous media”, International Journal of Solids and Structures, 46 (18-19), 3463-3470.
Khojasteh, A., Rahimian, M., and Eskandari-Ghadi, M. (2006). “Three dimensional analysis of a transversely isotropic half-space under the tangent load to the surface in the frequency domain”, Journal of Faculty of Engineering, University of Tehran, 40(5), 611-624 (In Persian).
Khojasteh, A., Rahimian, M., and Pak, R.Y.S. (2008a). “Three-dimensional dynamic Green’s functions in transversely isotropic bi-materials”, International Journal of Solids and Structures, 45 (18-19), 4952-4972. 
Khojasteh, A., Rahimian, M., Eskandari, M. and Pak, R.Y.S. (2008b). “Asymmetric wave propagation in a transversely isotropic half-space in displacement potentials”, International Journal of Engineering science, 46 (7), 690-710. 
Khojasteh, A., Rahimian, M., Pak, R.Y.S., and Eskandari, M. (2008c). “Asymmetric dynamic Green’s functions in a two-layered transversely isotropic half-space”, International Journal of Engineering Science, ASCE, 134(9), 777-787.
Khojasteh, A., Rahimian, M., Eskandari, M., and Pak, R.Y.S. (2011). “Three-dimensional dynamic Green’s functions for a multilayered transversely isotropic half-space”, International Journal of Solids and Structures, 48 (9), 1349-1361.
Khojasteh, A., Rahimian, M. and Eskandari, M. (2013). “Three-dimensional dynamic Green’s functions in transversely isotropic tri-materials”, Applied Mathematical Modelling, 37 (5), 3164-3180. 
Lambros, J., and Rosakis, A.J. (1995). “Dynamic decohesion of biomaterials: experimental observations and failure criteria”, International Journal of Solids and Structures, 32(17-18), 2677-2702.
Lekhnitskii, S.G. (1963). Theory of Elasticity of an Anisotropic Elastic Body, Holden Day, San Francisco.
Li, X.F., Tang, G.J., Shen, Z.B., and Lee, K.Y. (2015). “Axisymmetric problems of a penny-shaped crack at the interface of a bi-material under shear and compression”, International Journal of Solids and Structures, 69-70, 403-414. 
Martin, P.A., Richardson, J.D., Gray, L.J., and Berger, J.R. (2002). “On Green’s function for a three-dimensional exponentially graded elastic solid”, Proceeding of the Royal Society of London, Series A, 458 (2024), 1931-1947. 
Noijen, S.P.M., Van der Sluis, O., Timmermans, P.H.M., and Zhang, G.Q. (2012). “A semi-analytic method for crack kinking analysis at isotropic bi-material interfaces”, Engineering Fracture Mechanics, 83, 8-25.
Pak, R.Y.S. and Guzina, B.B. (2002). “Three-dimensional Green’s functions for a multi-layered half-space displacement potentials”, Journal of Engineering Mechanics, ASCE, 128(4), 449-461.
Pan, E., and Yang, B. (2003). “Three-dimensional interfacial Green’s functions in anisotropic bimaterials”, Applied Mathematical Modeling, 27 (4), 307-326.
Rahimian, M., Eskandari-Ghadi, M., Pak, R.Y.S. and Khojasteh, A. (2007). “Elastodynamic potential method for transversely isotropic solid”, Journal of Engineering Mechanics, ASCE, 133(10), 1134-1145.
Rajapakse, R.K.N.D. and Wang, Y. (1993). “Green’s functions for transversely isotropic elastic half-space”, Journal of Engineering Mechanics, ASCE, 119(9), 1724-1746.
Sallah, O.M., Gray, L.J., Amer, M.A. and Matbuly, M.S. (2010). “Green’s function expansion for exponentially graded elasticity”, International Journal for Numerical Method in Engineering, 82 (6), 756-772.
Selvadurai, A.P.S. and Katebi, A. (2013). “Mindlin’s problem for an incompressible elastic half-space with an exponential variation in the linear elastic shear modulus”, International Journal of Engineering Science, 65, 9-21. 
Sneddon, I.N. (1951). Fourier transforms, McGraw-Hill, New York.
Sneddon, I.N. (1972). The use of integral transforms, McGraw-Hill, New York.
Wang, C.D., Tzeng, C.S., Pan, E. and Liao, J.J. (2003). “Displacements and stresses due to a vertical point load in an inhomogeneous transversely isotropic half-space”, International Journal of Rock Mechanic and Mining Sciences, 40(5), 667-685.
Wang, C.D., Pan, E., Tzeng, C.S., Han, F. and Liao, J.J. (2006). “Displacements and stresses due to a uniform vertical circular load in an inhomogeneous cross-anisotropic half-space”, International Journal of Geomechanic, 6(1), 1-10. 
Wang, C.D. and Tzeng, C.S. (2009). “Displacements and stresses due to non-uniform circular loadings in an inhomogeneous cross-anisotropic material”, Mechanics Research Communications, 36(8), 921-932.  
Zhao, Y.F., Zhao, M.H., Pan, E. and Fan, C.Y. (2015). “Green’s functions and extended displacement discontinuity method for interfacial cracks in three-dimensional transversely isotropic magneto-electro-elastic bi-materials”, International Journal of Solids and Structures, 52, 56-71.