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

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.

A coupled finite–hierarchic infinite element method for an inhomogeneous transversely isotropic soil layer under irregularly distributed strip load

“…In most cases in practice, the soil deposit is of limited thickness and underlain by a hard stratum. Wang et al [2,3] presented a detailed review of the works on inhomogeneous isotropic/transversely isotropic soils. The existing literature reveals that inhomogeneous transversely isotropic finite layers received far less attention from researchers than inhomogeneous transversely isotropic half-spaces.…”