Comminution of solids caused by kinetic energy of high shear strain rate, with implications for impact, shock, and shale fracturing

Abstract

Although there exists a vast literature on the dynamic comminution or fragmentation of rocks, concrete, metals, and ceramics, none of the known models suffices for macroscopic dynamic finite element analysis. This paper outlines the basic idea of the macroscopic model. Unlike static fracture, in which the driving force is the release of strain energy, here the essential idea is that the driving force of comminution under high-rate compression is the release of the local kinetic energy of shear strain rate. The density of this energy at strain rates >1,000/s is found to exceed the maximum possible strain energy density by orders of magnitude, making the strain energy irrelevant. It is shown that particle size is proportional to the −2/3 power of the shear strain rate and the 2/3 power of the interface fracture energy or interface shear stress, and that the comminution process is macroscopically equivalent to an apparent shear viscosity that is proportional (at constant interface stress) to the −1/3 power of this rate. A dimensionless indicator of the comminution intensity is formulated. The theory was inspired by noting that the local kinetic energy of shear strain rate plays a role analogous to the local kinetic energy of eddies in turbulent flow.fracture mechanics | dynamic fracture | shale gas | dimensional analysis T he previous studies of high-rate dynamic fracture of rocks, concretes, ceramics, composites, and metals have dealt mainly with the nucleation, propagation, and branching of dynamically propagating cracks, their interference with elastic or shock waves, and the mechanism of development of the zones of densely distributed fractures, called the Mescall zones (1-7). However, a comminution model in the form of a macroscopic constitutive equation that could be used in large dynamic finite element programs for global response of structures is apparently still unavailable.Seeking such a constitutive model, we begin with the analysis of a simple idealized process in which the solid is comminuted to identical particles (Fig. 1). Among simple space-filling regular subdivisions in the plane of maximum shear, regular hexagons are the most likely because they give the smallest surface-to-volume ratio (Fig. 1A) and thus require the minimum energy to form. In the direction normal to the hexagons, we assume the particles to be prismatic.Consider that, at a certain moment, the strain rate (shown in Fig. 1B as a displacement regarded as infinitesimal) becomes high enough for the kinetic energy of shear strain rate to suffice for creating the fractures and interface slips that separate the particles of as yet unknown size. As that happens, the particles release their local kinetic energy, slip against each other, and regain their original undeformed shape, while the particle centers conform to the same macroscopic velocity field (Fig. 1C).For the sake of simplicity and clarity, we will first outline a 2D analysis of comminution in the plane of maximum shear strain rate denoted as _ e D , and leave the 3D generaliza...

Viscous energy dissipation of kinetic energy of particles comminuted by high-rate shearing in projectile penetration, with potential ramification to gas shale

“…the 2nd time derivative of strain. The comminution occurs only if the effective deviatoric strain rate changes, and no additional comminution occurs during the time intervals in which the rate is constant, which is a significant difference from the previous formulation (Bažant and Caner 2013;Bažant and Caner 2014).…”

“…In Bažant and Caner (2013) and Bažant and Caner (2014), it was proposed that the type of behavior is decided by the dimensionless number…”

“…For concrete, the transition occurs at˙ D ≈ 1/s. In Bažant and Caner (2013) and Bažant and Caner (2014), Eq. (11) could be transformed to the form:…”

“…A sound way to model the dynamic overstress must be a physically justified refinement, rather than ad hoc modification, of the damage constitutive model. Such refinement has recently been attempted with considerable success in the work of Bažant and Caner (2013), Bažant and Caner (2014) and Caner and Bažant (2014). The underlying constitutive model was the microplane model M7 for concrete, which includes only the quasi-static rate effects.…”

“…The oversimplification in the previous work (Bažant and Caner 2013;Bažant and Caner 2014;Caner and Bažant 2014) was due to calculating the additional shear viscosity so as to yield a dissipation rate proportional to the deviatoric stress invariant of the shear stress required to drive interface fracture. What prevented this simplification from being unrealistic was that this invariant had the correct physical dimension, the energy per unit volume.…”