An exact solution to layered transversely isotropic poroelastic media under vertical rectangular moving loads

Vertical and rocking motions of a flexible strip foundation placed on a layered transversely isotropic poroelastic half‐space

“…The fundamental solutions for layered poroelastic media are derived herein to determine the vertical displacement due to a certain stress distribution. For the layered poroelastic media, the extended precise integration method 41,42 is introduced in this section to solve the ordinary differential matrix equations, which are derived from the basic equations under the Biot theory 32,33 …”

“…The ordinary differential matrix form for the extended precise integration method 41,42 can be formulated by means of differential equations from Equation (8) to Equation (: $$\begin{equation}\frac{{\mathop{\rm d}\nolimits} }{{{\rm{d}}z}}\left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{{\bf V}}(\xi ,\;z)}\\[6pt] {{{\bf U}}(\xi ,\;z)} \end{array} } \right] = \left[ { \def\eqcellsep{&}\begin{array}{ll} {{\bf A}}&{{\bf D}}\\[6pt] {{\bf B}}&{{\bf C}} \end{array} } \right] \cdot \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{{\bf V}}(\xi ,\;z)}\\[6pt] {{{\bf U}}(\xi ,\;z)} \end{array} } \right]\end{equation}$$where the general stress vector $$\mathbf{V}(\xi ,z)=[ \def\eqcellsep{&}\begin{array}{ccc}\mathrm{i}{\bar{\tau}}_{\textit{xz}}& {\bar{\sigma}}_{z}& \bar{p}\end{array} ]^{\mathrm{T}}$$ and general displacement vector $${{\bf U}}(\xi ,\;z) = {[ { \def\eqcellsep{&}\begin{array}{*{20}{c}} {{\rm{i}}{{\bar{u}}}_x}&{{{\bar{u}}}_z}&{{{\bar{Q}}}_f} \end{array} } ]}^{\rm{T}}$$ are associated with block matrices A , B , C and D , in which …”

“…For layered poroleastic media, the general stress vector $${{\bf V}}(\xi ,\;z)$$ and general displacement vector $${{\bf U}}(\xi ,\;z)$$ at the arbitrary depth z can be calculated by solving Equation under the standard procedure of the extended precise integration method 41,42 . Then, the Fourier inversion transform (7) is applied to the state vectors $${{\bf V}}(\xi ,\;z)$$ and $${{\bf U}}(\xi ,\;z)$$, which yields the corresponding variables in the physical domain.…”

“…For dynamic SSI problem, the mixed boundary conditions from Equation to Equation describe the vibration source acting on the layered transversely isotropic poroelastic media. The corresponding response can be calculated by the extended precise integration method 41,42 with the surface flexibility coefficient $${f}_{zz}(\xi )$$, which determines the relationship between the surface displacement $${\bar{u}}_z(\xi ,0)$$ and stress $${\bar{\sigma }}_z(\xi ,0)$$. The surface flexibility coefficient $${f}_{zz}(\xi )$$ has an equal value of the surface displacement $${\bar{u}}_z(\xi ,0)$$ induced by a unit stress wavenumber in the transformed domain.…”

“…In this paper, vertical and rocking motions of a flexible strip foundation on a layered transversely isotropic poroelastic half‐space are investigated. For the fundamental solution, the scheme of the precise integration method 36 is introduced to obtain dynamic response of the underlying media, in consideration of its successful applications on the wave propagation problem, 37,38 consolidation of poroelastic layered media, 39,40 and dynamic analysis of a moving load 41,42 . Considering the flexible strip foundation, the analytical solutions for deflection are obtained by applying the Fourier transform technique to the dynamic equilibrium of the thin plate.…”