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DQEM for analyzing dynamic characteristics of layered fluid-saturated porous elastic media

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Abstract

Based on the Porous Media Theory presented by de Boer, the governing differential equations for a layered space-axisymmetrical fluid-saturated porous elastic body are firstly established, in which the suitable interface conditions between layers are presented. Then, a differential quadrature element method (DQEM) is developed, and the DQEM and the second-order backward difference scheme are applied to discretize the governing differential equations of the problem in the spatial and temporal domain, respectively. In order to show the validity of the present analysis, the dynamic response of a fluid-saturated porous medium is analyzed, and the obtained numerical results are directly compared with the existing analytical results. The effects of the numbers of the elements and grid points on the convergence of the numerical results are considered. Finally, the dynamic characteristics of a layered fluid-saturated elastic soil cylinder subjected to a water pressure or a dynamic loading are studied, and the effects of material parameters are considered in detail. From the above numerical results, it can be found that the DQEM has advantages, such as little amount in computation, good stability and convergence as well as high accuracy, so it is a very efficient method for solving the problems in soil mechanics, especially such problems with discontinuities.

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Abbreviations

PMT:

Porous Media Theory

DQEM:

Differential quadrature element method

FEM:

Finite element method

BEM:

Boundary element method

EFGM:

Element-free Galerkin method

DQM:

Differential quadrature method

\({r_\infty}\) :

Radius of a layered fluid-saturated porous elastic cylinder

H :

Height of the cylinder

H i (i = 1,2, . . . , n):

Height of the i th layer medium

n :

Number of layers

\({{\it {\bf u}}^{Si} = ({u_r^i ,u_\theta ^i ,u_z^i})}\) :

Displacement vector of the solid skeleton of the i th layer medium in the cylinder coordinate system

\({{\it {\bf w}}^{Fi} = ({w_r^i ,w_\theta ^i ,w_z^i})}\) :

Displacement vector of the fluid phase of the i th layer medium in the cylinder coordinate system

\({\lambda ^{Si}, \mu ^{Si}}\) :

Lame coefficients of the ith layer medium

p i :

Effective pore pressure of the fluid phase

\({S_v^i = \frac{(n^{Fi})^{2} \gamma ^{FRi}}{\kappa ^{Fi}}}\) :

Coupled interaction coefficient between the solid skeleton and fluid phase

\({\gamma ^{FRi}}\) :

Effective specific weight of the fluid

\({\kappa ^{Fi}}\) :

Darcy permeability coefficient

\({\rho ^{\alpha i}(\alpha = S,F)}\) :

Partial densities of the solid skeleton and fluid phase

\({n^{\alpha i}(\alpha = S,F)}\) :

Volume fractions of the solid skeleton and fluid phase

\({n_{^{0S}}^{Si}}\) :

Volume fraction of the solid skeleton at the initial state

\({\tilde {w}_r^i = \dot {w}_r^i -\dot {u}_r^i}\) :

Relative velocity of the fluid phase corresponding to the solid skeleton along the r-direction

\({\tilde {w}_z^i = \dot {w}_z^i -\dot {u}_z^i}\) :

Relative velocity of the fluid phase corresponding to the solid skeleton along the z-direction

\({\varepsilon _r^{Si} ,\varepsilon _\theta ^{Si} ,\varepsilon _z^{Si} ,\gamma _{rz}^{Si}}\) :

Strain components of the solid skeleton for the ith layer medium

\({\sigma _r^{SEi} ,\sigma _\theta ^{SEi} ,\sigma _z^{SEi} ,\tau_{rz}^{SEi}}\) :

Effective stress components of the solid skeleton for the ith layer medium

\({\sigma _r^{Si} ,\sigma _\theta ^{Si} ,\sigma _z^{Si} ,\tau_{rz}^{Si}}\) :

Total stress components of the solid skeleton for the ith layer medium

\({N_r^i \times N_z^i}\) :

Number of Grid points collocated along the r- and z-directions in the ith element

\({\psi _{\zeta \eta}^i = \psi ^{i}(r_\zeta ,z_\eta)}\) :

Value of function \({\psi ^{i}({r,z})}\) at the grid point \({({r,z}) = (r_\zeta ,z_\eta )}\)

\({A_{\zeta k}^{(j,i)}}\) :

Weighting coefficient of the jth order derivative of \({\psi ^{i}({r,z})}\) with respect to r in the ith element

\({B_{\eta l}^{(s,i)}}\) :

Weighting coefficient of the sth order derivative of \({\psi ^{i}({r,z})}\) with respect to z in the ith element

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Authors and Affiliations

  1. Shanghai Normal University, Shanghai, 200234, People’s Republic of China

    Yuan-Yuan Zhu

  2. University of Shanghai for Science and Technology, Shanghai, 200093, People’s Republic of China

    Yu-Jia Hu

  3. Shanghai University, Shanghai, 200072, People’s Republic of China

    Chang-Jun Cheng

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  1. Yuan-Yuan Zhu
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  2. Yu-Jia Hu
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Correspondence to Yuan-Yuan Zhu.

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Zhu, YY., Hu, YJ. & Cheng, CJ. DQEM for analyzing dynamic characteristics of layered fluid-saturated porous elastic media. Acta Mech 224, 1977–1998 (2013). https://doi.org/10.1007/s00707-013-0851-0

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