Advances in Materials Science and Engineering

Volume 2018 (2018), Article ID 1368713, 6 pages

https://doi.org/10.1155/2018/1368713

## Numerical Simulations for Large Deformation of Geomaterials Using Molecular Dynamics

Correspondence should be addressed to Jun Zhang

Received 22 September 2017; Accepted 15 November 2017; Published 28 January 2018

Academic Editor: Francesco Ruffino

Copyright © 2018 Ziyang Zhao and Jun Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

From the microperspective, this paper presents a model based on a new type of noncontinuous theoretical mechanical method, molecular dynamics (MD), to simulate the typical soil granular flow. The Hertzian friction formula and viscous damping force are introduced in the MD governing equations to model the granular flow. To show the validity of the proposed approach, a benchmark problem of 2D viscous material flow is simulated. The calculated final flow runout distance of the viscous material agrees well with the result of constrained interpolated profile (CIP) method as reported in the literature. Numerical modeling of the propagation of the collapse of three-dimensional axisymmetric sand columns is performed by the application of MD models. Comparison of the MD computational runout distance and the obtained distance by experiment shows a high degree of similarity. This indicates that the proposed MD model can accurately represent the evolution of the granular flow. The model developed may thus find applications in various problems involving dense granular flow and large deformations, such as landslides and debris flow. It provides a means for predicting fluidization characteristics of soil large deformation flow disasters and for identification and design of appropriate protective measures.

#### 1. Introduction

As a kind of typical granular material, soil causes the common flow forms mainly for earthwork excavation, flow of soil slope by filling, and sudden slip on a weak foundation soil. The large deformation disasters caused by the flow of soil particles have received considerable attention and researches both at home and abroad.

Molecular dynamics originated in the 50s and began to be widely concerned in the mid-70s. In 1957, the state equations of gas and liquid were studied, firstly using molecular dynamics in the hard sphere model, thus setting a precedent for studying the macroscopic properties in the molecular dynamics method [1]. Molecular dynamics started to develop after 70s, gradually perfected its system in the late 80s and got rapid development along with the flourish of computer hardware in the 90s. Molecular dynamics is a connecting link between macroscopic characteristics and microstructure, which can make certain microscopic interpretation for theoretical analysis and experimental researchdifficult to understand. Especially in chemistry, biology, physics, and material science or related fields, it gets better application [2–4]. In recent years, molecular dynamics is also applied to the research on granular flow of geomaterials. The inducing factors of shallow landslide were studied using the molecular dynamics method [5]. Factors affecting the occurrence of microsliding in the sand pile were discussed with the molecular dynamics method [6]. Molecular dynamics simulations are particularly well suited to investigate the effects of materials properties of liquid-solid interfaces on flow boundary conditions [7].

Molecular dynamics method, which is a combination of physics, mathematics, and chemistry, mainly relies on Newtonian mechanics to simulate the movement of molecular system and can simulate the movement of a single particle in a large number of particle collection system (solid, gas, or liquid). Its key is the concept of movement, that is, to compute the evolution of the position, velocity, and orientation of particles over time. The particle in molecular dynamics can be atom, molecule, or larger set of particles [8]. The basic principle of molecular dynamics is the application of Newton’s classical mechanics to firstly calculate the trajectories of all particles in the physical system and then by using relevant statistical methods to calculate mechanics, thermodynamics, and dynamics characteristics of the system. That is to say, firstly, the system of particles is abstracted to many interacting particles, and the interaction force between particles is obtained by using the potential energy function derivation method. And then, the acceleration and velocity of each particle are calculated according to the Newtonian mechanics, and the particles have the corresponding trajectories after a certain integral steps, which must be preserved by setting a certain time interval. Finally, calculation results of the interested quantities are extracted based on statistical principle [9–11].

At present, the molecular dynamics simulation researches mostly focus on physics, biology, and chemistry, but the application in the field of geotechnical engineering is rare. The molecular dynamics approach is suitable to model granular material and to observe the trajectory of a single particle, so as to possibly identify its dynamical properties. Thus, the molecular dynamics method is tried to be applied to the granular flow characteristics research, and a 3D MD model that simulates granular flow is presented in this work, which is dedicated to provide a strong scientific basis for solving geotechnical engineering problem.

#### 2. Numerical Approach

The accuracy of the model depends on the accuracy of description of external forces acting on the particle and the treatment of collisions. The particles can be either rigid and/or soft. A soft particle model [12] is used in analyzing particle-particle and particle-wall interactions in this paper. The particles are considered to be spherical and identical. The collisions are assumed to be central and involve both linear elastic and damping forces. In the study, 3D ordered granular packings made up of spherical particles are considered. The motion equation for the sphere is written in a general form aswhere is the displacement, is the force, is the mass, and is the total number of particles. is the composition of contact force between spheres *j* and *i*, where the interaction force between any contacting spheres under compression is given by the Hertzian law [13]:where is the overlap distance of two particles and the and stand for the radius of the particles *i* and *j*, respectively. The expression of is defined as [13]where and are the elastic constants for normal contact and tangential contact, respectively; and are the viscoelastic damping constants for normal contact and tangential contact, respectively; is the effective mass of two particles of mass and ; is the tangential displacement vector between two spherical particles which is truncated to satisfy a frictional yield criterion; is the unit vector along the line connecting the centers of the two particles; and and are the normal and tangential components of the relative velocity of the two particles, respectively.

Viscous damping force calculation formula of the particle is based on the Stokes drag law of fluid mechanics [14], which can be expressed aswhere is the dynamic viscosity of the frictional fluid, is the diameter of particle, and is the velocity.

At present, in the field of molecular dynamics, the integral of the motion equation mainly adopts Verlet algorithm [15]. The integral algorithm putted forward by Verlet is the most widely used in the molecular dynamics. The numerical simulations are performed using the molecular dynamics package LAMMPS [10] in the work, by considering spheres as point masses connected by nonlinear springs.

The calculation formulas of Verlet algorithm are as follows:

#### 3. Validation of the MD Model

The performance of the new model will be verified in this section. It is essential to create an input script containing the desired commands before running LAMMPS. The Verlet algorithm is applied to the integral of motion equation in the input scripts. When numerical calculation starts, LAMMPS reads the input script, and the displacement, velocity, and acceleration of every granular particle are calculated and updated, thus obtaining the evolution trajectory of the whole calculation system over time. A two-dimensional simulation of viscous material flow was conducted to verify the reliability of the MD method.

Viscous materials flow on the plane under its own gravity, and a literature [16] has detailed calculation and analysis on this flow process using constrained interpolated profile (CIP). CIP method was developed for prediction of large deformations associated with a geomaterial flow, so it can be used as a contrast with the MD method in this work. The model [17] is shown in Figure 1.