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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 |
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Vol. 5 (2010), No. 2, 241–262
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Abstract |
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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
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Milestones
Received: 13 January 2009
Revised: 8 August 2009
Accepted: 13 August 2009
Published: 30 August 2010
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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 |
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| 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 |
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| 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 |
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