The Mechanical Coupling of Fluid-Filled Granular Material Under Shear

Abstract

The coupled mechanics of fluid-filled granular media controls the physics of many Earth systems, for example saturated soils, fault gouge, and landslide shear zones. It is well established that when the pore fluid pressure rises, the shear resistance of fluid-filled granular systems decreases, and, as a result, catastrophic events such as soil liquefaction, earthquakes, and accelerating landslides may be triggered. Alternatively, when the pore pressure drops, the shear resistance of these geosystems increases. Despite the great importance of the coupled mechanics of grain–fluid systems, the basic physics that controls this coupling is far from understood. Fundamental questions that must be addressed include: what are the processes that control pore fluid pressurization and depressurization in response to deformation of the granular skeleton? and how do variations of pore pressure affect the mechanical strength of the grains skeleton? To answer these questions, a formulation for the pore fluid pressure and flow has been developed from mass and momentum conservation, and is coupled with a granular dynamics algorithm that solves the grain dynamics, to form a fully coupled model. The pore fluid formulation reveals that the evolution of pore pressure obeys viscoelastic rheology in response to pore space variations. Under undrained conditions elastic-like behavior dominates and leads to a linear relationship between pore pressure and overall volumetric strain. Viscous-like behavior dominates under well-drained conditions and leads to a linear relationship between pore pressure and volumetric strain rate. Numerical simulations reveal the possibility of liquefaction under drained and initially over-compacted conditions, which were often believed to be resistant to liquefaction. Under such conditions liquefaction occurs during short compactive phases that punctuate the overall dilative trend. In addition, the previously recognized generation of elevated pore pressure under undrained compactive conditions is observed. Simulations also show that during liquefaction events stress chains are detached, the external load becomes completely supported by the pressurized pore fluid, and shear resistance vanishes.

Appendix 1: Pore Fluid Pressure Evolution for De ≪ 1

In this section, the evolution of pore pressure is studied for drained boundaries with De ≪ 1. Under such conditions, the time-dependent term in Eqs. 15 and 18 becomes negligible compared with the diffusion term because De ≪ 1 in the non-dimensional Eq. 30. Equation 15 then becomes:

$$ \nabla \cdot\left[k({\mathbf{x}},t)\nabla P({\mathbf{x}},t)\right] = \eta\nabla\cdot{\mathbf{u_s}}({\mathbf{x}},t). $$

(44)

Formulation similar to Eq. 44 was developed by Iverson (1993) for drained conditions. For the 1D case, after integration, Eq. 44 becomes:

$$ \frac{\partial P(z,t)}{\partial z} = \frac{\eta}{k_0}u_{sz}(z,t) + C(t), $$

(45)

where C(t) is an integration factor, k(z, t) is approximated as the permeability scale factor, k ₀, and u _sz is the horizontally averaged z component of the solid velocity. In order to express the pressure as a function of the temporal derivative of the porosity, $\partial \Upphi/\partial t$ as in Eq. 18, we use the 1D form of Eq. 17:

$$ \frac{\partial u_{sz}}{\partial z} = \frac{1}{1-\Upphi}\frac{\partial \Upphi}{\partial t}. $$

(46)

Integrating Eq. 46 between the center of the layer at z = 0 and some distance z from the center (Fig. 7) results in:

$$ \begin{aligned} \int_0^z \frac{\partial u_{sz}(z^{\prime},t)}{\partial z^{\prime}}\hbox{d}z^{\prime} =& \int_0^z \frac{1}{1-\Upphi(z^{\prime},t)}\frac{\partial \Upphi(z^{\prime},t)} {\partial t}\hbox{d}z^{\prime}\\ =& \int_0^z -\frac{\partial [\ln(1-\Upphi(z^{\prime},t))]}{\partial t}\hbox{d}z^{\prime}\\ =& -\frac{\partial}{\partial t}\int_0^z \ln(1-\Upphi(z^{\prime},t))\hbox{d}z^{\prime}\\ \approx& -\frac{\partial}{\partial t}\int_0^z \left( -\Upphi(z^{\prime},t) - \frac{\Upphi(z^{\prime},t)^2}{2}\right)\hbox{d}z^{\prime}\\ \approx & -\frac{\partial}{\partial t}\int_0^z -\Upphi(z^{\prime},t) \hbox{d}z^{\prime} \\ =& \frac{\partial\langle\Upphi(z,t)\rangle}{\partial t}z, \\ \end{aligned} $$

(47)

where $\langle\Upphi(z,t)\rangle$ is the average porosity between the system’s center and distance z from the center. Equation 47 then leads to the relationship:

$$ u_{sz}(z,t) = u_{sz}(0,t) + \frac{\partial\langle\Upphi(z,t)\rangle}{\partial t}z. $$

(48)

Substituting Eq. 48 in Eq. 45 results in:

$$ \frac{\partial P(z,t)}{\partial z} = \frac{\eta}{k_0}\frac{\partial\langle\Upphi(z,t)\rangle}{\partial t}z + C_1(t). $$

(49)

Integrating Eq. 49 between the layer’s center and distance z leads to:

$$ P(z,t) = P(0,t) + \frac{\eta}{k_0}\frac{\hbox{d}\langle\Upphi(z,t)\rangle}{\hbox{d} t}\frac{z^2}{2} + C_1(t) z, $$

(50)

where the rate of change of the average porosity, $\hbox{d}\langle\Upphi(z,t)\rangle/\hbox{d} t,$ is approximated as uniform in space. Requiring complete drainage across the boundaries, i.e. $P(\zeta,t) = P(-\zeta,t) = 0,$ Eq. 50 leads to:

$$ P(z,t) = -\frac{\eta}{2k_0}\frac{\hbox{d}\langle\Upphi(\zeta,t)\rangle}{\hbox{d} t}\left(\zeta^2 - z^2\right). $$

(51)