ON SCALE EFFECT IN DYNAMIC FRAGMENTATION

The dynamic large-scale fragmentation resulting in a large impact or explosion is discussed. It is known that, unlike the lab-scale tests and independently on the type of explosion, energy and so on, the fragment size distribution presents maximums explained by block structure of the rocks and that this number grows with increasing the fragment size. A quantum-continuous model for this distribution in a rock cell under the longlength primary wave action is proposed. Thus, the differential and cumulative distributions over the whole fragmentation volume are calculated, and the free-parameters in the model are fitted to the experimental data. INTRODUCTION Fragmentation of solids involves many natural and artificial processes, ranging from sky body collisions to comminution of coffee grains. Such phenomena play an important role in the formation of various structures (asteroid clouds, impact/explosion craters, etc.) and are distinguished by diversity, dependence on many factors and complexity of description. Fundamental investigations in this area began from lab-scale tests in statics [1-3] -see [4] and later in dynamics [5,6]. Their findings serve as the basis firstly for empirical laws and then for theoretical predictions. Assuming the fractal law for the size distribution of particles, Carpinteri and Pugno [7,8] have unified the three comminution laws proposed by von Rittinger [1], Kick [2] and Bond [3] for predicting the energy consumption in fragmentation. A theoretical model [7] was followed by some experiments on drilling and compression of heterogeneous materials [8]. They also confirm the fractal nature of fragmentation and lead to the determination of the model parameters. In dynamics, the stone ball fracture resulting in impact by a projectile at 0.1-3 km/s shows that the distribution is subject to the same law, but it is described inserting two distinct fractal dimensions. The last peculiarity was later explained by a geometrical feature of the primary wave propagation inside the ball [9], where, based on the continuum damage theory of dynamic fragmentation developed by Grady and Kipp [10], the thorough computer-analytic simulation appears to be in a good agreement with the lab-scale impact experiments. Note that Carpinteri and Pugno [8] found the same result in perforations: the bi-fractality was explained as due to two different fracture mechanisms under the drilling tool, namely, cutting and crushing. However, all these experiments were performed only at the micrometer fragment size range. Meantime, it was repeatedly noted that the scale effect is of great importance in fragmentation. As a matter of fact, the fragment size distribution observed after many large explosions (to kilotons energy) in rocks, does not follow the simple (mono) fractal law in the range 1-100 cm [11]. Unlike the small-scale tests, it has maximums explained by the block structure of rocks and moreover, the number of pieces surprisingly grows when the fragment size increases. These peculiarities only slightly depend on the type of explosion, its energy, and the depth of the charge laying. To describe all scales of dynamic fragmentations resulting in a large impact or explosion in this work, we suggest a quantum-continuous model of the local fragmentation distribution using the evidence on how the primary wave traveling through a cell of the rock spends its energy into fracture and heat, as recently emphasized by Simonov [12]. For the finer fragmentation (<5 mm), Melosh’s et al. description is adopted [9]. When changing to the block fracture with a hierarchy of characteristic sizes, the degenerate distribution function, that presents singularities, is inserted into the model. By integrating the local distributions, the global distribution over the whole destroyed volume is found out. Forcing the free model parameters to fit the empirical data [11] leads to remarkable agreement between theory and observations. Nevertheless, the restriction of the empirical data concerning quantity and accuracy, as well as the usual incorrectness of the similar reverse problems, leave behind them a great deal with concerning the possibility of refining the present simulation. PHYSICAL ASUMPTIONS Consider the fragmentation resulting in a large (sub)surface blast or collision of a massive projectile with a large-size brittle solid body simulated by an elastic half-space. Assume that the semi-spherical primary wave propagates starting at the distance , a conventional radius from the point of action, and with initial magnitude in mass velocity 0 R s km 69 . 3 0 = U , where the imaginary melting/solid boundary takes place according to reference [12]. Then, a description of the shock wave propagation for accounting the thermodynamics, in particular, its specific energy lost the energy spent to destroy a cell of ( ) r f 3 m 1 will be identical to as reported in [12]. So, the wave, emerged by a large action, propagates according to the well-known empirical law in dependence on dimensionless coordinate 0 R r R = with the decay coefficients, α (U , 1 U > 0 U > s m 30 1 ≈ U ) and β (U 1 U < ) at stages of strong and weak shock wave, respectively (typically α =1.87 and β =1.6 for rocks [13, 14]). The radius and wave width, 0 R H , can be evaluated from the experimental data [13] and thus it follows that and for the 5 kilotons TNT explosion in granite, so that we choose m 2 . 4 0 ≈ R m 30 H 20 − ≈ 5 0 = R H for the future calculations. In the domain of the fine fragmentation, we employ the differential distribution of particle sizes over a cell given in reference [9]: ( ) ( ) max 1 3 4 max 0 , 1 , l l L L L l A r l p m c = − = − − (1) where is the number of particles per 1 with sizes between l and , is the peak size, is the normalizing constant and m is a free parameter ( ( ) l l pc d 0 A 3 m l l d + max l 10 ≤ m ). This is valid for . If , the characteristic sizes of the block structure of various rocks ( mm , 5 2 5 < l 1 mm 5 ≥ l , 20 3 mm 1000 , 500 , 140 , 50 6 4 5 = = = l l l