Fragmentation of Shells

Abstract

We present a theoretical and experimental study of the fragmentation of closed thin shells made of a disordered brittle material. Experiments were performed on brown and white hen egg shells under two different loading conditions: impact with a hard wall and explosion by a combustible mixture. Both give rise to power law fragment size distributions. A three-dimensional discrete element model of shells is worked out. Based on simulations of the model, we give evidence that power law fragment mass distributions arise due to an underlying phase transition which proved to be abrupt for explosion and continuous for impact. We demonstrate that the fragmentation of closed shells defines a new universality class of fragmentation phenomena.

New Universality Class for the Fragmentation of Plastic Materials

“…Several possible mechanisms have been put forward to understand the emergence of the universal power law behavior. For rapid break-up of heterogeneous bulk solids with a high degree of brittleness, the self-similar branching-merging scenario of propagating unstable cracks governed by tensile stresses can explain the main features of the fragment mass distribution [5,[11][12][13], while for shell systems an additional sequential binary breakup mechanism has to be taken into account [7,8]. It is an important question of broad scientific and technological interest how plasticity, and the emergence of complicated stress states like shear affect the fragmentation process.…”

“…In the data analysis, 99 − 99.5% of the total mass of the samples was recovered. In order to evaluate the mass distribution of the fragments, we scanned the pieces with an open scanner obtaining digital images where fragments appear as white spots on the black background [7,8]. This way the identification of fragments is reduced to cluster searching of white pixels.…”

“…Fragmentation phenomena are ubiquitous in nature and play a crucial role in numerous industrial processes related to mining and ore processing [1]. A large variety of measurements starting from the breakup of heavy nuclei through the usage of explosives in mining or fragmenting asteroids revealed the existence of a striking universality in fragmentation phenomena [1][2][3][4][5][6][7][8][9][10]: fragment mass distributions exhibit a power law decay, independent on the type of energy input (impact, explosion, ...), the relevant length scales or the dominating microscopic interactions involved. Detailed laboratory experiments on the breakup of disordered solids have revealed that mainly the effective dimensionality of the system determines the value of the exponent, according to which universality classes of fragmentation phenomena can be distinguished.…”

Emergence of energy dependence in the fragmentation of heterogeneous materials

“…During the past decades research on fragmentation mainly focused on the statistics of fragment masses (sizes) obtained by the breakup of heterogeneous materials [1,9,10]. A large number of experimental [1,[5][6][7][8][9][10][11][12][13][14][15][16][17] and theoretical studies [13,[18][19][20][21][22][23][24] have confirmed that the mass distribution of fragments is described by a power law functional form. The exponent of the distribution was found to show a high degree of robustness, i.e., investigations revealed that the value of the exponent does not depend on the type of materials, amount of input energy, and on the way the energy is imparted to the system until materials of a high degree of heterogeneity are fragmented [1,9,10].…”

“…The value of the exponent is mainly determined by the dimensionality of the system [13,15,18,19,21,23,[25][26][27] and by the brittle or ductile mechanical response of the material [28]. The universality of fragmenting has been shown to be the fingerprint of an underlying phase transition from the damaged to the fragmented phase of the breakup process [5,6,14,18].…”