Applied Computational Intelligence and Soft Computing

Volume 2017, Article ID 8971353, 7 pages

https://doi.org/10.1155/2017/8971353

## On the Horizontal Deviation of a Spinning Projectile Penetrating into Granular Systems

Department of Mathematics and Statistics, University of Hail, Hail, Saudi Arabia

Correspondence should be addressed to Waseem Ghazi Alshanti; moc.oohay@itnahslameesaw

Received 6 March 2017; Accepted 17 May 2017; Published 6 June 2017

Academic Editor: José-Ignacio Hidalgo

Copyright © 2017 Waseem Ghazi Alshanti. 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

The absence of a general theory that describes the dynamical behavior of the particulate materials makes the numerical simulations the most current powerful tool that can grasp many mechanical problems relevant to the granular materials. In this paper, based on a two-dimensional soft particle discrete element method (DEM), a numerical approach is developed to investigate the consequence of the orthogonal impact into various granular beds of projectile rotating in both clockwise (CW) and counterclockwise (CCW) directions. Our results reveal that, depending on the rotation direction, there is a significant deviation of the -coordinate of the final stopping point of a spinning projectile from that of its original impact point. For CW rotations, a deviation to the right occurs while a left deviation has been recorded for CCW rotation case.

#### 1. Introduction

Any collection of many macroscopic discrete solids, whose typical size ranges from micrometers to centimeters, such that most of the particles are in contact with at least some of their neighboring particles, is called granular material. Sand, coal, sugar, corn, rice, and beads are classical examples for such materials. Granular materials exist almost everywhere in nature or any industry process. They are the base materials for most of the products, including food, agricultural, powder, pharmaceutical, mineral, and chemical products, and even the technology of missiles and penetrator bombs that burrow into ground to strike critical targets is affected by the properties of the grains within the ground. So far, there are still no reasonable theoretical justifications for many phenomena relevant to the particulate systems due to the rigorous mechanical behaviors of these systems. The dynamics of a granular bed impacted by a projectile penetrating at normal angle are still a challenging problem.

Even though the dynamics of penetration problem have long been of interest and remain an active research area attracting the attention of mathematicians and physicists, no consensus has emerged regarding the equation that describes the velocity of projectile during penetrating a granular bed [1–3]. Only a few studies have been conducted over the last three decades to better characterize the dynamics of penetration problem. Those studies, in general, focused on experimental investigations. In 1957, an equation was proposed which roughly expresses the negative acceleration of projectile penetrating randomly packed sand [4]. The equation has the formwhere , , and are positive constants and denotes the projectile penetration velocity at time . Also, it was suggested that the centrifuge modeling is an appropriate and powerful tool for investigating the penetration of projectile into granular soil [5]. For different soils, power relationships between projectile penetration depth and projectile mass-to-area ratio were presented. Moreover, the depth of penetration was proposed as a function of initial velocity and material properties of the granular bed [6]. A review of scaling laws for impact process and a brief for the current approaches that are used to study the processes of impacts was carried out [7, 8]. Walsh et al. [9] performed experiments on impact craters formed by dropping a steel ball vertically into a container of small glass beads. By explicit variation of ball density, diameter, and drop height, the crater diameter was confirmed to scale as the fourth power of the energy of the ball at impact. More and more scaling laws that govern the penetration process of a projectile into granular beds were presented by a number of researchers [10–13]. Amato and Williams investigated the dependence of the crater diameter on the kinetic energy of falling ball into a sand filled container [14]. Bruyn and Walsh [15] reported that the penetration depth of a steel sphere dropped vertically into a container of loosely packed small beads increases linearly with the incident momentum of the projectile. The effect of several impact conditions on the rebound velocity of steel projectile was discussed in detail via three-dimensional (DEM) simulations [16]. It was shown that the impact velocity does not greatly affect the general scattering behavior of particulate aggregation but affects the rebound velocity of the steel projectile from the particulate aggregation. Several attempts have been made by using two- and three-dimensional (DEM) simulations to investigate deep penetration mechanism, impact crater, and force distribution of a projectile impacting into a granular medium [17–24]. The effect of closed lateral walls on penetration depth of a projectile has experimentally been studied [25, 26]. A proposed model for the propagation of energy due to the impact of a projectile on a dense granular medium was established by Crassous et al. [27]. Further results can be found in [28, 29].

In the present work, based on this method, a mathematical model and numerical technique have been developed to study the dynamics of a rotating projectile impacting orthogonally into various granular beds. We investigate the effect of the rotation direction on the final position of the projectile after penetration.

#### 2. The Mathematical Model

During simulation, granular particles bear two types of forces: contact forces and gravitational body force. Any contact force between two particles is decomposed into normal and tangential components. The normal contact force is modeled by a damped linear spring, while the tangential contact force is modeled by a linear spring in series with a frictional sliding element. The formula that determines the contact force of particle and particle iswhere and are unit vectors in the normal and shear directions of the contact plane and and are, respectively, the magnitudes of the normal contact force and shear contact force; namely, where and are, respectively, the particle-particle normal and tangential spring coefficients, and are, respectively, the elastic contribution of the contact force between the particles and in the normal direction ( direction) and the friction coefficient of the granular particles, and are, respectively, the normal compression and the tangential displacement between the particles and over the time step , and and are the radii of the particles and . Under the contact forces and the gravitational body force, each particle has the following motion dynamic equations: where , , , and are, respectively, the mass, rotational moment of inertia, position, and rotational vectors of the centre of particle , , , , and are, respectively, contact force and moment acting on particle due to particle and external forces and moment acting on particle , and is the number of particles within the granular bed. Hence, , we have a system of first-order ordinary differential equations as follows:Therefore, by numerical integration of Newton’s equation of motion, the updated velocities and positions of all particles can be determined.

#### 3. Numerical Simulation

In 1979, Cundall and Strack [30] proposed the soft particle discrete method which is considered as one of the powerful tools for handling particulate systems. In the present work, for a typical simulation, a code was constructed by utilizing the C programming language. This code handles a two-dimensional discrete element computer simulation of circular particles subject to gravitational and contact forces. The code is able to model particles and walls with various properties, for example, contact properties, number of particles, size of particle, particle’s density, and number of walls. A java movie code is also included so that the results of the simulations can be viewed. Consequently, a granular bed of dimensions is generated by the random packing method where the particles are recognized as two-dimensional discs. To study the influence of the particle size on the ultimate position of the rotating projectile after penetration, monosized particle and multisized particle beds with equal dimensions are constructed. Each granular bed is subjected to normal impact of projectile with diameter , density , and fixed initial impact velocity .

The simulation environmental conditions include physical properties of the granular particles under consideration, initial conditions, and boundary conditions, all being normalized. All parameters are normalized using the density of particle, gravitational acceleration, and particle diameter. A list of used normalized environmental and mechanical simulation parameters is given in Tables 1 and 2.