A fully coupled particle method for dynamic analysis of saturated soil

Abstract

Among other numerical issues, it is well known that the finite element method (FEM) lacks objectivity in reproducing high deformation rates due to extreme external actions. In geotechnical applications, the coupling of large solid deformations with the pore fluid flow is a critical subject, being one of the multiple scenarios where FEM could have restricted applications. In order to overcome the aforementioned numerical drawbacks, the generic theoretical approach presented in this work is implemented in the context of an explicit numerical method known as the material point method (MPM). Since the MPM can be viewed as a special Lagrangian FEM with particle quadrature and continuous mesh updating, the improved formulation and numerical implementation presented here are well suited for the study of coupled water pore pressure and soil deformation models. One important aspect of the presented coupled formulation is the assumption of two independent sets of Lagrangian material points for each phase. This characteristic leads to a numerical tool oriented to large deformations simulations in saturated porous media, with a fully coupled thermodynamically consistent formulation. To illustrate its robustness and accuracy, the approach is applied to two different real engineering applications: progressive failure modeling of a granular slope and river levees. The obtained results show that the physics of fluid flow through porous media is adequately represented in each analyzed case. It is also proved that it accurately represents the kinematics of soil skeleton and water phase for fully saturated cases, ensuring mass conservation of all constituents.

Appendix: Matrix expressions of Eqs. (29)–(30)

In the following, the matrix expressions of discrete momentum balance equation of Eqs. (25) and (26) are presented:

$$\begin{aligned} \mathbf {Fi}^{w}_{I} =&-\sum _{wp=1}^{N_{w}} \sum ^{N_{s}}_{sp=1} N^{w}_{wp_{I}} nR^{w} \left( N^{w}_{wp_J}v^{w}_{wp_J} - N^{s}_{sp_J}v^{s}_{sp_J}\right) \Omega ^{w}_{wp}\nonumber \\&\quad + \sum _{p=1}^{N_{w}} \mathbf {G}^{w}_{I} n\bar{p}^{w}\Omega ^{w}_{p} \end{aligned}$$

(38)

$$\begin{aligned} \mathbf {Fe}^{w}_{I} =&\sum _{p=1}^{N_{w}} b^{w}_{p} m^{w}_{p} N^{w}_{p_I} - \int _{\partial \Omega _{w}} N^{w}_{p_{I}} t^{w}_{p}\text {d}\partial \Omega _{w} \end{aligned}$$

(39)

$$\begin{aligned} \mathbf {Fi}^{sw}_{I} =&- \sum _{p=1}^{N_{s}} \mathbf {G}^{s}_{I} \ \varvec{\sigma }'_{p} \ \Omega ^{s}_{p} + \sum _{p=1}^{N_{w}} \mathbf {G}^{w}_{I} \ \mathbf {I} \ \bar{p}^{w} \ \Omega ^{w}_{p} \end{aligned}$$

(40)

$$\begin{aligned} \mathbf {Fe}^{sw}_{I} =&\sum _{p=1}^{N_{s}} b^{s}_{p} m^{s}_{p} N^{s}_{p_I} + \sum _{p=1}^{N_{w}} b^{w}_{p} m^{w}_{p} N^{w}_{p_I}\nonumber \\&\quad +\int _{\partial \Omega _{s}} N^{s}_{p_{I}} t^{s}_{p}\text {d}\partial \Omega _{s}\nonumber \\&\quad + \int _{\partial \Omega _{w}} N^{w}_{p_{I}} t^{w}_{p}\text {d}\partial \Omega _{w} \end{aligned}$$

(41)