Emergence of energy dependence in the fragmentation of heterogeneous materials

The most important characteristics of the fragmentation of heterogeneous solids is that the mass (size) distribution of pieces is described by a power law functional form. The exponent of the distribution displays a high degree of universality depending mainly on the dimensionality and on the brittle-ductile mechanical response of the system. Recently, experiments and computer simulations have reported an energy dependence of the exponent increasing with the imparted energy. These novel findings question the phase transition picture of fragmentation phenomena, and have also practical importance for industrial applications. Based on large scale computer simulations here we uncover a robust mechanism which leads to the emergence of energy dependence in fragmentation processes resolving controversial issues on the problem: studying the impact induced breakup of platelike objects with varying thickness in three dimensions we show that energy dependence occurs when a lower dimensional fragmenting object is embedded into a higher dimensional space. The reason is an underlying transition between two distinct fragmentation mechanisms controlled by the impact velocity at low plate thicknesses, while it is hindered for three-dimensional bulk systems. The mass distributions of the subsets of fragments dominated by the two cracking mechanisms proved to have an astonishing robustness at all plate thicknesses, which implies that the nonuniversality of the complete mass distribution is the consequence of blending the contributions of universal partial processes.

Figure 1

Geometrical setup of the simulations: rectangular samples of a square shaped basis were considered in such a way that the side length $L/\langle d\rangle =30$ of the square was fixed and the height $H/\langle d\rangle$ of the sample was varied from 3 to 15. Here a sample is presented for $H/\langle d\rangle =11$. A particle in the middle of the front side of the sample was selected; together with its neighbors it got an initial velocity pointing into sample. The cylinders connecting the particles represent beams and the white arrow indicates the direction of the impact velocity. The inset shows a closer view on a small segment of the sample.

Figure 2

Time evolution of a fragmenting plate of thickness $H/\langle d\rangle =3$ embedded in the three-dimensional space. The impact velocity $v_0/c=0.2$ is slightly above the critical point of fragmentation. Compressed beams are green while the stretched ones are red so that the propagation of a shockwave can be observed. Particles are colored according to the fragment they belong to while the color (grey) of fragments is randomly selected from a palette. Panel (d) presents the final state of the system where the sample is reassembled by placing the particles back to their original position.

Figure 3

Inset: Average mass of fragments $\langle M_2/M_1\rangle$ as a function of the impact velocity $v_0$ for all plate thicknesses $H$ considered. The main panel presents the same data rescaled with appropriate powers of the plate thickness $H$ to obtain collapse of the curves.

Figure 4

Mass distribution of fragments at different impact velocities for four different plate thicknesses $H/\langle d\rangle$: (a) 3, (b) 5, (c) 11, (d) 15. In the damage phase the distributions are composed of two distinct parts, i.e., the large residues form a peak at $m/M_{\mathrm{tot}}\approx 1$, while the small fragments have a rapidly decreasing distribution. The two regimes are separated by a gap which gradually disappears as the critical velocity $v_c$ is approached from below. The dashed straight lines represent power laws of exponents (a) 1.7 and 2.4, (b) 1.7, (c) 1.9, and (d) 1.9.

Figure 5

Mass distribution of fragments $p\left(m\right)$ obtained at the critical point of fragmentation $v_c$ for the smallest thickness $H/\langle d\rangle =3$. Fragments giving dominating contribution to different ranges of $p\left(m\right)$ originate from well defined spatial regions of the sample. These regions are highlighted in a single sample in the inset using the same colors (characters) as for the corresponding ranges of the mass distribution. For comparison we also present the mass distribution at the highest impact velocity $v_0/c=0.5$ where the power law regime is significantly steeper. The slopes of the two straight lines are $\tau =1.7$ and $\tau =2.4$.

Figure 6

Identification of fragment subsets based on the bounding box of fragments and of the complete sample. Two plates are shown with different thicknesses and impact velocities (a) $H/\langle d\rangle =5$ and $v_0/c=0.23$, (b) $H/\langle d\rangle =11$ and $v_0/c=0.3$. For each subset the largest fragment is highlighted with different colors: light blue (light grey), green (medium grey), and red (dark grey) stand for the spanning, surface, and bulk fragments, respectively. The bounding boxes are indicated by the wire frames.

Figure 7

Mass distribution of subsets of surface, spanning, and bulk fragments together with the complete distribution of all fragments for two plate thicknesses $H/\langle d\rangle =5$ and 15 in the upper and lower rows, respectively. In both cases the results are presented for two values of the impact velocity $v_0$ slightly above the corresponding $v_c$ (left column) and for the limit of high speed impact (right column).

Figure 8

Data collapse analysis of the mass distribution of subsets of surface and bulk fragments for two plate thicknesses $H/\langle d\rangle =$ 5 and 15 in the upper and lower rows, respectively. Rescaling the distributions with appropriate powers of the impact velocity above the critical point good quality data collapse is achieved. The bold lines represent fits of the master curves with Eq. (7).