DOI QR코드

DOI QR Code

A Solution for Green's Function of Orthotropic Plate

직교이방성 평판의 Green 함수에 대한 새로운 해

  • Published : 2007.03.01

Abstract

Revisited in this paper are Green's functions for unit concentrated forces in an infinite orthotropic Kirchhoff plate. Instead of obtaining Green's functions expressed in explicit forms in terms of Barnett-Lothe tensors and their associated tensors in cylindrical or dual coordinates systems, presented here are Green's functions expressed in two quasi-harmonic functions in a Cartesian coordinates system. These functions could be applied to thin plate problems regardless of whether the plate is homogeneous or inhomogeneous in the thickness direction. With a composite variable defined as $z=x_1+ipx_2$ which is adopted under the necessity of expressing the Green's functions in terms of two quasi-harmonic functions in a Cartesian coordinates system Stroh-like formalism for orthotropic Kirchhoffplates is evolved. Using some identities of logarithmic and arctangent functions given in this paper, the Green's functions are presented in terms of two quasi-harmonic functions. These forms of Green's functions are favorable to obtain the Newtonian potentials associated with defect problems. Thus, the defects in the orthotropic plate may be easily analyzed by way of the Green's function method.

Keywords

Green's Function;Stroh Formalism;Orthotropic Kirchhoff Plate

References

  1. Stroh, A. N., 1958, 'Dislocations and Cracks in Anisotropic Elasticity,' Phil. Mag., Vol. 3, pp. 625-646 https://doi.org/10.1080/14786435808565804
  2. Stroh, A. N., 1962, 'Steady State Problem in Anisotropic Elasticity,' J. Math. Phys., Vol. 41, pp. 77-103 https://doi.org/10.1002/sapm196241177
  3. Ting, T. C. T., 1996, Anisotropic elasticity: theory and applications, Oxford University Press, New York
  4. Cheng, Z. Q. and Reddy, J. N., 2002, 'Octet Formalism for Kirchhoff Anisotropic Plates,' Proc. R. Soc. Lond., Vol. A458, pp. 1499-1517 https://doi.org/10.1098/rspa.2001.0934
  5. Kelvin, Lord, 1882, 'Note on the Integration of the Equations of Equilibrium of an Elastic Solid,' Mathematical and Physical Papers 1, Cambridge Univ. Press, pp. 97-98
  6. Indenbom, V. L. and Orlov, S. S., 1968, 'Construction of Green's Function in Terms of Green's Function of Lower Dimension,' J. Appl. Math. Mech., Vol. 31, pp. 414-420 https://doi.org/10.1016/0021-8928(68)90059-2
  7. Cheng, Z. Q. and Reddy, J. N., 2003, 'Green's Functions for Infinite and Semi-infinite Anisotropic Thin Plates,' J. Appl. Mech., Vol. 70, pp. 260-267 https://doi.org/10.1115/1.1533806
  8. Cheng, Z. Q. and Reddy, J. N., 2004a, 'Green's Functions for an Anisotropic Thin Plate with a Crack or an Anticrack,' Int. J. Engng. Sci., Vol. 42, pp. 271-289 https://doi.org/10.1016/j.ijengsci.2003.06.001
  9. Cheng, Z. Q. and Reddy, J. N., 2004b, 'Laminated Anisotropic Thin Plate with an Elliptic Inhomogeneity,' Mech. Mater., Vol. 36, pp. 647-657 https://doi.org/10.1016/S0167-6636(03)00081-4
  10. Hsieh, M. C. and Hwu, C., 2002, 'Anisotropic Elastic Plates with Holes/Cracks/Inclusions Subjected to Out-of-Plane Bending Moments,' Int. J. Solids Struct., Vol. 39, pp. 4905-4925 https://doi.org/10.1016/S0020-7683(02)00335-9
  11. Mura, T., 1982, Micromechanics of defects in solids, Martinus Nijhoff Publishers, Hague, Netherlands
  12. Yang, K. J., 2005, 'Elastic Analysis of Defects in Orthotropic Kirchhoff Plate,' J. Appl. Mech., Accepted for Publication
  13. Jones, R. M., 1975, Mechanics of composite materials, McGraw-Hill, New York
  14. Eshebly, J. D., 1961, 'Elastic Inclusions and Inhomogeneities,' Progress in Solid Mechanics 2, I. N. Sneddon and R. Hill, eds., North-Holland, Amsterdam, pp. 89-140
  15. Eshelby, J. D., Read, W. T. and Shockley, W., 1953, 'Anisotropic Elasticity with Application to Dislocation Theory,' Acta Metallurgica, Vol. 1, pp. 251-259 https://doi.org/10.1016/0001-6160(53)90099-6
  16. Eshelby, J. D., Read, W. T. and Shockley, W., 1953, 'Anisotropic Elasticity with Applications to Dislocation Theory,' Acta Metall., Vol. 1, pp. 251-259 https://doi.org/10.1016/0001-6160(53)90099-6
  17. Hwu, C., 2004, 'Green's Function for the Composite Laminates with Bending Extension Coupling,' Compos. Struct., Vol. 63, pp. 283-292 https://doi.org/10.1016/S0263-8223(03)00175-2
  18. Yang, K. J., Beom, H. G. and Kang, K. J., 2005, 'Thermal Stress Analysis for an Inclusion with Nonuniform Temperature Distribution in an Infinite Kirchhoff Plate,' J. Thermal Stresses, Vol. 28, pp. 1123-1144 https://doi.org/10.1080/014957390967857