Numerical Investigation of the Fragmentation Process in Marble Spheres Upon Dynamic Impact

Abstract

Three-dimensional discrete element simulations are performed to better understand the impact-induced complex fragmentation process of marble spheres. A 3D clumped particle method is used and a new calibration procedure to match both the quasi-static and dynamic mechanical behaviors of marble is proposed. The impact simulation results show that radial macrocracks occur, which break the intact marble spheres into large orange-slice-shaped fragments. Secondary macrocracks occur for impact velocities larger than 9 m/s, and the fragment size is further reduced. The fracture mechanisms due to local damage, radial, and secondary macrocracks significantly affect the impact process, energy dissipation, evolutions of the masses of the first and second largest fragments, and damage ratio. The numerical model is able to accurately capture all mechanisms, from local damage to disintegration, of the impact-induced fragmentation. Due to local damage and macrocracks, the obtained fragments consist of large and small fragments. The fragment size distributions based on mass and number can be fitted using a generalized extreme value law. The numerical predictions indicate that the translational velocities of some small fragments can be significantly higher than the impact velocity due to the instant high-tensile stress wave near the contact area. The results also suggest that there is no correlation between fragment mass and fragment kinetic energy.

Appendix: Energetic Formulation

In the DEM simulations, the energy components can be calculated for each time step. In the following formulation, it is assumed that the potential energy can be neglected, since the impact occurs at a notably small time scale. Hence, gravity is not considered and the total input energy of the numerical model is constant throughout the impact process. In addition, no numerical damping is considered in the impact simulations. The total kinetic energy $E_{\text{k}}$ of all particles (considering both translational and rotational motion) can be expressed as follows:

$$E_{\text{k}} = \sum\limits_{i = 1}^{{n_{\text{p}} }} {\frac{1}{2}\left( {m_{i} v_{i}^{2} + I_{i} \omega_{i}^{2} } \right)} ,$$

(4)

where $n_{\text{p}}$ is the number of particles; $m_{i}$ and $I_{i}$ are the mass and moment of inertia of particle $i$, respectively; and $v_{i}$ and $\omega_{i}$ are the translational and rotational velocities of particle $i$, respectively. When fragmentation occurs during impact, the translational kinetic energy of a fragment can be calculated using the mass and the velocity of the centroid of the fragment. The total kinetic energy of a fragment after impact can be calculated using Eq. (4). The rotational kinetic energy of a fragment can be obtained by subtracting the translational kinetic energy from the total kinetic energy of the fragment. The elastic deformation energy $E_{\text{el}}$ of the numerical model includes contact deformation energy $E_{\text{cdef}}$ and bond deformation energy $E_{\text{bdef}}$. Hence, $E_{\text{el}}$ can be defined as shown:

$$E_{\text{el}} = E_{\text{cdef}} + E_{\text{bdef}} .$$

(5)

According to the linear contact-stiffness model used in this study, the contact deformation energy $E_{\text{cdef}}$ can be expressed as follows:

$$E_{\text{cdef}} = \sum\limits_{j = 1}^{{n_{\text{c}} }} {\frac{1}{2}\left( {{{\left| {F_{j}^{\text{n}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {F_{j}^{\text{n}} } \right|^{2} } {k^{\text{n}} }}} \right. \kern-0pt} {k^{\text{n}} }} + {{\left| {F_{j}^{\text{s}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {F_{j}^{\text{s}} } \right|^{2} } {k^{\text{s}} }}} \right. \kern-0pt} {k^{\text{s}} }}} \right)} ,$$

(6)

where $n_{\text{c}}$ is the number of contacts; $\left| {F_{j}^{\text{n}} } \right|$ and $\left| {F_{j}^{\text{s}} } \right|$ are the magnitudes of the normal and shear components of the contact force, respectively; and $k^{\text{n}}$ and $k^{\text{s}}$ are the normal and shear contact stiffnesses, respectively. The bond deformation energy $E_{\text{bdef}}$ stored in the parallel bonds is expressed as follows:

$$\begin{aligned} E_{\text{bdef}} &= \sum\limits_{b = 1}^{{n_{{{\text{p}}b}} }} {\frac{1}{2}\left( {{{\left| {\bar{F}_{b}^{\text{n}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\bar{F}_{b}^{\text{n}} } \right|^{2} } {\left( {A\bar{k}^{\text{n}} } \right)}}} \right. \kern-0pt} {\left( {A\bar{k}^{\text{n}} } \right)}} + {{\left| {\bar{F}_{b}^{\text{s}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\bar{F}_{b}^{\text{s}} } \right|^{2} } {\left( {A\bar{k}^{\text{s}} } \right)}}} \right. \kern-0pt} {\left( {A\bar{k}^{\text{s}} } \right)}}}\right.}\\&\quad{\left.{ + {{\left| {\bar{M}_{b}^{\text{n}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\bar{M}_{b}^{\text{n}} } \right|^{2} } {\left( {J\bar{k}^{\text{s}} } \right)}}} \right. \kern-0pt} {\left( {J\bar{k}^{\text{s}} } \right)}} + {{\left| {\bar{M}_{b}^{\text{s}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\bar{M}_{b}^{\text{s}} } \right|^{2} } {\left( {I\bar{k}^{\text{n}} } \right)}}} \right. \kern-0pt} {\left( {I\bar{k}^{\text{n}} } \right)}}} \right)} , \end{aligned}$$

(7)

where $n_{{p{\text{b}}}}$ is the total number of parallel bonds; $\left| {\bar{F}_{b}^{\text{n}} } \right|$ and $\left| {\bar{F}_{b}^{\text{s}} } \right|$ are the normal and shear forces acting in the bond, respectively; $\left| {\bar{M}_{b}^{\text{n}} } \right|$ and $\left| {\bar{M}_{b}^{\text{s}} } \right|$ are the normal and shear moments, respectively; $\bar{k}^{\text{n}}$ and $\bar{k}^{\text{s}}$ are the normal and shear bond stiffnesses, respectively; $A$ is the cross-sectional area of the bond; $I$ is the moment of inertia; and $J$ is the polar moment of inertia. The shear ($E_{\text{bs}}$) and tensile ($E_{\text{bt}}$) deformation energies of a parallel bond can be expressed as below:

$$E_{\text{bs}} = \frac{1}{2}\left( {{{\left| {\bar{F}_{b}^{\text{s}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\bar{F}_{b}^{\text{s}} } \right|^{2} } {\left( {A\bar{k}^{s} } \right)}}} \right. \kern-0pt} {\left( {A\bar{k}^{s} } \right)}} + {{\left| {\bar{M}_{b}^{\text{s}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\bar{M}_{b}^{\text{s}} } \right|^{2} } {\left( {I\bar{k}^{n} } \right)}}} \right. \kern-0pt} {\left( {I\bar{k}^{n} } \right)}}} \right),$$

(8)

$$E_{\text{bt}} = \frac{1}{2}\left( {{{\left| {\bar{F}_{b}^{\text{n}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\bar{F}_{b}^{\text{n}} } \right|^{2} } {\left( {A\bar{k}^{\text{n}} } \right)}}} \right. \kern-0pt} {\left( {A\bar{k}^{\text{n}} } \right)}} + {{\left| {\bar{M}_{b}^{\text{n}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\bar{M}_{b}^{\text{n}} } \right|^{2} } {\left( {J\bar{k}^{\text{s}} } \right)}}} \right. \kern-0pt} {\left( {J\bar{k}^{\text{s}} } \right)}}} \right).$$

(9)

According to Eq. (7), the energy dissipated by bond breakage $W_{\text{b}}$ is the sum of the deformation energy of all broken bonds and can be calculated as shown:

$$\begin{aligned} W_{\text{b}} &= \sum\limits_{b = 1}^{{n_{\text{bpb}} }} {\frac{1}{2}\left( {{{\left| {\bar{F}_{b}^{\text{n}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\bar{F}_{b}^{\text{n}} } \right|^{2} } {\left( {A\bar{k}^{\text{n}} } \right)}}} \right. \kern-0pt} {\left( {A\bar{k}^{\text{n}} } \right)}} + {{\left| {\bar{F}_{b}^{\text{s}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\bar{F}_{b}^{\text{s}} } \right|^{2} } {\left( {A\bar{k}^{\text{s}} } \right)}}} \right. \kern-0pt} {\left( {A\bar{k}^{\text{s}} } \right)}}}\right.} \\&\quad {\left.{ + {{\left| {\bar{M}_{b}^{\text{n}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\bar{M}_{b}^{\text{n}} } \right|^{2} } {\left( {J\bar{k}^{\text{s}} } \right)}}} \right. \kern-0pt} {\left( {J\bar{k}^{\text{s}} } \right)}} + {{\left| {\bar{M}_{b}^{\text{s}} } \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\bar{M}_{b}^{\text{s}} } \right|^{2} } {\left( {I\bar{k}^{\text{n}} } \right)}}} \right. \kern-0pt} {\left( {I\bar{k}^{\text{n}} } \right)}}} \right)} , \end{aligned}$$

(10)

where $n_{\text{bpb}}$ is the number of broken parallel bonds. In addition to $W_{\text{b}}$, friction work $W_{\text{f}}$ is another energy dissipation mechanism. Since the loading paths influence the value of friction work, the friction work is given as an incremental formulation:

$$\left( {W_{\text{f}} } \right)^{t + \Delta t} = \left( {W_{\text{f}} } \right)^{t} + \sum\limits_{j = 1}^{{n_{\text{c}} }} {\left( {\left\langle {F_{j}^{\text{s}} } \right\rangle \left( {\Delta U_{j}^{\text{s}} } \right)^{\text{slip}} } \right),}$$

(11)

where $\left\langle {F_{j}^{\text{s}} } \right\rangle$ and $\left( {\Delta U_{j}^{\text{s}} } \right)^{\text{slip}}$ are the average shear force and the increment of slip displacement at the contact for the current time step, respectively. The increment of slip displacement is determined by decomposing the total displacement $\left( {\Delta U_{j}^{\text{s}} } \right)$ into a slip and an elastic portion $\left( {\Delta U_{j}^{\text{s}} } \right)^{\text{elas}}$, such that the following applies:

$$\left( {\Delta U_{j}^{\text{s}} } \right)^{\text{slip}} = \left( {\Delta U_{j}^{\text{s}} } \right) - \left( {\Delta U_{j}^{\text{s}} } \right)^{\text{elas}} = v_{j}^{\text{s}} \Delta t + \frac{{\left( {F_{j}^{\text{s}} } \right)^{{\left( {t + \Delta t} \right)}} - \left( {F_{j}^{\text{s}} } \right)^{\left( t \right)} }}{{k^{\text{s}} }},$$

(12)

where $v_{j}^{\text{s}}$ is the normal contact velocity with respect to the contact plane.