Abstract
This paper presents analytical solutions for the stress and displacement field in elastic layered geo-materials induced by an arbitrary point load in the Cartesian coordinate system. The point load solutions can be obtained by referring to the integral transform and the transfer matrix technique. However, former solutions usually exist in the cylindrical coordinate system subjected to axisymmetric loading. Based on the proposed solutions in the Cartesian coordinate, it is very easy to solve asymmetric problems and consider the condition with internal loads in multi-layered geo-materials. Moreover, point load solutions can be used to construct solutions for analytical examination of elastic problems and incorporated into numerical schemes such as boundary element methods. The results discussed in this paper indicate that there is no problem in the evaluation of the point load solutions with high accuracy and efficiency, and that the material non-homogeneity has a significant effect on the elastic field due to adjacent loading.
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References
Boussinesq, J., Application des potentiels à l’ étude de l’ équilibre et du Mouvement des Solides élastiques. Paris: Gauthier-Villars, 1885(in French).
Cerruti, V., A Treatise on the Mathematical Theory of Elasticity, Fourth edition. New York: Dover Publications, 1882.
Mindlin, R.D., Force at a point in the interior of a semi-infinite solid. Journal of Applied Physics, 1936, 7(5): 195–202.
Burmister, D.M., The general theory of stresses and displacements in layered soil systems (I,II,III). Journal of Applied Physics, 1945, 6(2): 89–96; 6(3): 126–127; 6(5): 296–302.
Muki, R., Asymmetric problems of the theory of elasticity for a semi-infinite solid and a thick plate. Progress in Solid Mechanics. Amsterdam: North-Holland, 1960.
Michelow, J., Analysis of Stresses and Displacements in an N-layered Elastic System under a Load Uniformly Distributed on a Circular Area. Richmond, California: Chevron Research Corporation, 1963.
Schiffman, R.L., General analysis of stresses and displacements in layered elastic systems. In: Proceeding of First International Conference on the Structural Design and Asphalt Pavements. Michigan: University of Michigan, 1962.
Bufler, H., Theory of elasticity of a multilayered medium. Journal of Elasticity, 1971, 1(2): 125–143.
Bahar, L.Y., Transfer matrix approach to layered system. Journal of the Engineering Mechanics Division, 1972, 98(5): 1159–1172.
Yue, Z.Q. and Wang, R., Static solutions for transversely isotropic elastic N-layered systems. Acta Scientiarum Naturalium Universitatis Pekinensis, 1988, 24(2): 202–211(in Chinese).
Pan, E., Static Green’s functions in multilayered half-spaces. Applied Mathematical Modelling, 1997, 21(8): 509–521.
Wang, J. and Fang, S., The state vector methods of axisymmetric problems for multilayered anisotropic elastic system. Mechanics Research Communications, 1999, 26(6): 673–678.
Wang, W. and Ishikawa, H., A method for linear elastic-static analysis of multi-layered axisymmetrical bodies using Hankel’s transform. Computational Mechanics, 2001, 27(6): 474–483.
Zeng, S. and Liang, R., A fundamental solution of a multilayered half-space due to an impulsive ring source. Soil Dynamics and Earthquake Engineering, 2002, 22(7): 541–550.
Lu, J.F. and Hanyga, A., Fundamental solution for a layered porous half space subject to a vertical point force or a point fluid source. Computational Mechanics, 2005, 35(5): 376–391.
Han, F., Development of novel green’s functions and their applications to multiphase and multilayered structures. PhD Thesis, University of Akron, 2006.
Pan, E., Bevis, M., Han, F., Zhou, H. and Zhu, R., Surface deformation due to loading of a layered elastic halfspace: a rapid numerical kernel based on a circular loading element. Geophysical Journal International, 2007, 171(1): 11–24.
Alkasawneh, W., Pan, E., Han, F., Zhu, R.H. and Green, R., Effect of temperature variation on pavement responses using 3D multilayered elastic analysis. International Journal of Pavement Engineering, 2007, 8(3): 203–212.
Gerrard, C.M. and Harrison, W.J., The analysis of a loaded half space comprised anisotropic layers. SIRO, Division of Applied Geomechanics technical paper, 1971, 10: 25–132.
Davies, T.G. and Banerjee, P.K., The displacement field due to a point load at the interface of a two layer elastic half-space. Geotechnique, 1978, 28(1): 43–56.
Ai, Z.Y., Yue, Z.Q., Tham, L.G. and Yang, M., Extended Sneddon and Muki solutions for multilayered elastic materials. International Journal of Engineering Science, 2002, 40(13): 1453–1483.
Fukahata, Y. and Matsu’ur, M., General expressions for internal deformation fields due to a dislocation source in a multilayered elastic half-space. Geophysical Journal International, 2005, 161(2): 507–521.
Pastel, E.C. and Leckie, F.A., Matrix Methods in Elasto-Mechanics. New York: McGraw-Hill, 1963.
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Project supported by the National Natural Science Foundation of China (No. 51008188) and the China Postdoctoral Science Foundation (No. 20100470677).
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Zhang, Z., Li, Z. Analytical solutions for the layered geo-materials subjected to an arbitrary point load in the Cartesian coordinate. Acta Mech. Solida Sin. 24, 262–272 (2011). https://doi.org/10.1016/S0894-9166(11)60027-X
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DOI: https://doi.org/10.1016/S0894-9166(11)60027-X