Theoretical Study Of Dynamics And Scalings Of Projectile Impacting Granular Materials

Granular material under the action of external force is a fundamental problem for the dynamics of granular media. The high discreteness makes granular media exhibit rich dynamical behaviors under the effect of external collision. After impact, yield will occur on granular materials, and the localized strain field will result in the mixture of solid and liquid phases and the solid-liquid state transition. In addition, contact interactions between particles, including friction and inelastic collisions are highly nonlinear and dissipative mechanisms. Therefore study of the fundamental dynamic processes and the mechanical behavior of a projectile impacting a granular media is helpful to further understand the complexity of some natural phenomena related to the impact with a granular material, and on the other hand from a desire to understand the drag force on a projectile moving through a granular media, and to reveal the complex rheological process in granular materials.This dissertation systematically investigated the dynamical process and the scaling of two-dimensional granular bed subjected to oblique impact of a spherical projectile numerically using the discrete element model. Then, the influence of projectile rotation on dynamic response of granular bed obliquely impacted by a rotational projectile was investigated, especially for the scaling of penetration depth. At last, taking into account the probability density distribution of stress fluctuation amplitude in granular materials and the coupling effect between shear rate and volume fraction in iterative calculation, this paper proposed a modified non-local rheological model in order to describe dense granular flows more accurately. This modified model was then applied to predict the rheological characteristics of flows down a rough inclined plane, aiming at explanation the avalanche behavior of gravitationally-forced granular layers in the collapse of impact craters. The main works are concluded as follows:1. Established a discrete dynamic model capable of describing the basic mechanical behaviors of granular materials, and proporsed corresponding statistical methods. The dynamic response of two-dimensional granular bed subjected to the oblique impact of a spherical projectile was numerically examined based on the discrete element method. The influences of projectile’s initial velocities and impacting angles on the trajectory of projectile, as well as the dependence of penetration depth and stopping time on impact velocity were quantitatively analyzed. Based on the microscopic contact force of particles, a phenomenological model for the impacting angle larger than the critical angle is eventually proposed, which accurately describe the nature of resistance force exerted on the projectile by the granular medium during the impacting process. Finally, the validity of this model has been checked.2. The dynamical response of granular bed obliquely impacted by a rotational projectile has been investigated, especially for the influence of particle rotation on the penetration depth. Simulations reveal that, the influence of particle rotation on its penetration depth exhibits an obvious criticality, and the critical angular velocity is mainly affected by the impact velocity and impact angle. Furthermore, the variation of critical angular velocity with the impact velocity and impact angle was presented. Last, the influence law of the initial particle rotation on the linear scaling law of the penetration depth and the initial impact velocity was investigated.3. The influence of different system parameters (density ratio and diameter ratio between projectiles and bed granular) and impact velocities on the post-impact motion of projectiles were considered. A criterion of the three motion patterns was presented according to the trajectory of the impact projectile at low velocities. Then, this paper examined the post-impact motion of the projectile in different density ratios (bulk density/projectile density) and diameter ratios (granular diameter/projectile diameter) at the initial space of impact velocity and angle. In the end, the dependence of the critical rebounding and penetrating angles of the projectile on impact velocity and system parameters (density ratio and diameter ratio) were discussed.4. In dense granular flows, a non-local rheology theory was proposed by Pouliquen et al.(2009) based the idea of a self-activated process, in which a rearrangement at one position was trigged by stress fluctuation due to rearrangements elsewhere in the material. Taking into account the probability density distribution of stress fluctuation amplitude in granular materials and the coupling effect between shear rate and volume fraction in iterative calculation, we proposed a modified non-local Theological model in order to describe dense granular flows more accurately. Due to dense granular flows down inclines preserve this complexity but remain simple enough for detailed analysis, this modified model was applied to predict the rheological characteristics of the flow down a rough inclined plane. Compared to the previous non-local rheological model, the predicted results of the modified model, such as critical thickness, depth-averaged velocity and shear rate profile, are all better consistent with the existing experimental and simulating results.Through numerically investigated dynamic response of two-dimensional granular bed subjected to the oblique impact of a spherical projectile, systematically revealed the scaling laws, as well as commendably modified the rheological model for predicting the dynamical characteristics of the flow down a rough inclined plane in the collapse of impact craters, we established a theoretical models and statistical methods capable of describing the mechanical behaviors of granular materials. These investigations not only reveal and analyze the complex dynamic process of granular materials under the effect of external collision, but also provide important theoretical foundation and quantitative analysis methods fundamental problem for the dynamics of granular media. The dissertation presented foreground of mechanical analysis methods in exploring discrete complex systems, and established a foundation for intensive study mechanical problems related to granular materials.