New Universality Class for the Fragmentation of Plastic Materials

We present an experimental and theoretical study of the fragmentation of polymeric materials by impacting polypropylene particles of spherical shape against a hard wall. Experiments reveal a power law mass distribution of fragments with an exponent close to 1.2, which is significantly different from the known exponents of three-dimensional bulk materials. A 3D discrete element model is introduced which reproduces both the large permanent deformation of the polymer during impact and the novel value of the mass distribution exponent. We demonstrate that the dominance of shear in the crack formation and the plastic response of the material are the key features which give rise to the emergence of the novel universality class of fragmentation phenomena.

Figure 1

Final states of impact at low impact velocities in the experiment (a) and in the simulation (b). In the contact area with the hard wall, large permanent deformations occur due to compression, while above it vertical cracks are formed due to tensile stresses. The simulations are in very good agreement with the measurements.

Figure 2

Mass distribution of fragments obtained from experiments at three different impact velocities. $m_0$ denotes the average mass of PP spheres. At the highest impact velocity $v_0=75\mathrm{m}/\mathrm{s}$, $F(m)$ shows a power law behavior over 3 orders of magnitude followed by an exponential cutoff for the very large pieces. Simulation results obtained with the parameters ${\Theta }_{\mathrm{th}}=1$ and $t_h=0$ (bending&healing) are in a very good agreement with the experimental findings. Inset: the shape parameter $\sqrt{I_1/I_2}$ of fragments as a function of mass for the experiments of the main figure.

Figure 3

(a) The average mass of the largest $⟨M_{\max }⟩$ and second largest $⟨M_{\max }^{2\mathrm{nd}}⟩$ fragments as a function of the impact velocity $v_0$ normalized by the critical velocity $v_c$ of fragmentation for two parameter sets. (b) Fragment mass distributions obtained for the limiting cases of tension, bending, and bending&healing dominated breakups giving the corresponding exponents ${\tau }_{\mathrm{br}}$, ${\tau }_{\mathrm{sh}}$, and ${\tau }_{\mathrm{sh}\mathrm{\text{−}}\mathrm{pl}}$, respectively.

Figure 4

Probability distribution of the angle $\Phi$ between the velocity vector of fragments and the direction of impact for tension ${\Theta }_{\mathrm{th}}=200$ and shear ${\Theta }_{\mathrm{th}}=1.0$ dominated breakup. Insets: final states of fragmentation for tension (a) and shear (b) dominance. The arrows indicate the direction of the velocity of a few fragments.