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Research Papers

# Plane-Strain Shear Dislocation on a Leaky Plane in a Poroelastic Solid

[+] Author and Article Information
Yongjia Song

Department of Astronautics and Mechanics,
Harbin Institute of Technology,
Postbox 344,
92 West Dazhi Street,
Harbin 150001, China;
Department of Civil and Environmental Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: songyongjia061220110@126.com

John W. Rudnicki

Fellow ASME
Department of Civil
and Environmental Engineering,
Northwestern University;
Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: jwrudn@northwestern.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 14, 2016; final manuscript received November 3, 2016; published online November 21, 2016. Editor: Yonggang Huang.

J. Appl. Mech 84(2), 021008 (Nov 21, 2016) (8 pages) Paper No: JAM-16-1451; doi: 10.1115/1.4035179 History: Received September 14, 2016; Revised November 03, 2016

## Abstract

Solutions for the stress and pore pressure $p$ are derived due to sudden introduction of a plane strain shear dislocation on a leaky plane in a linear poroelastic, fluid-infiltrated solid. For a leaky plane, $y=0$, the fluid mass flux is proportional to the difference in pore pressure across the plane requiring that $Δp=R∂p/∂y$, where $R$ is a constant resistance. For $R=0$ and $R→∞$, the expressions for the stress and pore pressure reduce to previous solutions for the limiting cases of a permeable or impermeable plane, respectively. Solutions for the pore pressure and shear stress on and near $y=0$ depend significantly on the ratio of $x$ and $R$. For the leaky plane, the shear stress at $y=0$ initially increases from the undrained value, as it does from the impermeable plane, but the peak becomes less prominent as the distance $x$ from the dislocation increases. The slope ($∂σxy/∂t$) at $t=0$ for the leaky plane is always equal to that of the impermeable plane for any large but finite $x$. In contrast, the slope $∂σxy/∂t$ for the permeable fault is negative at $t=0$. The pore pressure on $y=0$ initially increases as it does for the impermeable plane and then decays to zero, but as for the shear stress, the increase becomes less with increasing distance $x$ from the dislocation. The rate of increase at $t=0$ is equal to that for the impermeable fault.

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## Figures

Fig. 1

A shear dislocation at the origin corresponds to cutting the negative x axis, displacing the top to the right and the bottom to the left by the same amount, then bonding the cut elastic plane back together

Fig. 2

Schematic representation of a thin porous layer

Fig. 4

Time-dependence of the nondimensional pore pressure induced on y=0+ ahead (x>0) of a shear dislocation at the origin for different values of x/R. (Results for the impermeable plane here and in Fig. 5 have been calculated from Ref. [6], but on the scale of the plot they are indistinguishable from those for small x/R. The pore pressure induced on the permeable plane is zero. Values of the pore pressure on y=0− are equal in magnitude and opposite in sign to those on y=0+. The pore pressure induced on the permeable slip plane is zero.

Fig. 3

Time-dependence of the dimensionless shear stress ((σxy(x,0+,t)−σxy(x,0+,∞))/(σxy(x,0+,0+)−σxy(x,0+,∞))) on y=0 for a shear dislocation at the origin for various values of x/R. Values for the permeable and impermeable planes were calculated using the expressions from Refs. [5,6], but calculations for sufficiently large (impermeable) and small (permeable) R were indistinguishable from these on the scale of the plot.

Fig. 5

Nondimensional pore pressure induced on plane y=0 by a shear dislocation at the origin as a function of position x at a fixed time t≠0. The plot is for various values of ct/R2. The pore pressures for the shear dislocation are shown for y=0+; values for y=0− are the negative of those shown. The pore pressure on the permeable plane is zero.

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