Vol. 5, No. 2, 2010

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ISSN: 1559-3959
Fundamental solutions for an inhomogeneous cross-anisotropic material due to horizontal and vertical plane strain line loads

Cheng-Der Wang, Jia-Yan Hou and Wei-Jer Wang

Vol. 5 (2010), No. 2, 241–262
Abstract

This work derives the fundamental solutions for displacements and stresses due to horizontal and vertical line loads acting in a continuously inhomogeneous plane strain cross-anisotropic full space with Young’s and shear moduli varying exponentially with depth. The governing equations can be obtained by combining the generalized Hooke’s law, the strain-displacement relationships, and the equilibrium equations. Then, utilizing Fourier transforms, the governing equations are transformed into ordinary differential equations. Additionally, by using the variation of parameters, the solutions of the displacements in the Fourier domain are found. However, the stress solutions in the same domain can also be found by employing the stress-strain-displacement relationships. Eventually, performing inverse Fourier transforms by means of the numerical integration program QDAGI, the displacements and stresses induced by horizontal and vertical plane strain line loads can be calculated. The solutions indicate that the displacements and stresses are profoundly influenced by the nondimensional inhomogeneity parameter, the type and degree of material anisotropy, the types of loading, and the nondimensional horizontal distance. The proposed solutions are identical to those of Wang and Liao after suitable integration, as derived in an appendix, when the full space is a homogeneous cross-anisotropic material. A series of parametric studies are conducted to demonstrate the present solutions, and to elucidate the effects of aforementioned factors on the vertical normal stress. The results reveal that estimates of displacement and stress should take the inhomogeneity into account when studying cross-anisotropic materials under applied line loads.

Keywords
inhomogeneity, cross-anisotropic full space, horizontal and vertical plane strain line loads, Fourier transforms, fundamental solutions of displacements and stresses, numerical integrations
Milestones
Received: 13 January 2009
Revised: 8 August 2009
Accepted: 13 August 2009
Published: 30 August 2010
Authors
Cheng-Der Wang
Department of Civil and Disaster Prevention Engineering
National United University
No. 1, Lien Da, Kung-Ching Li
Miao-Li, Taiwan 360
Taiwan
Jia-Yan Hou
Department of Civil and Disaster Prevention Engineering
National United University
No. 1, Lien Da, Kung-Ching Li
Miao-Li, Taiwan 360
Taiwan
Wei-Jer Wang
Department of Civil and Disaster Prevention Engineering
National United University
No. 1, Lien Da, Kung-Ching Li
Miao-Li, Taiwan 360
Taiwan