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

Abstract

While dynamic comminution is of interest to many processes and situations, this work is focused on the projectile impact onto concrete walls, in which predictions have been hampered by the problem of the so-called 'dynamic overstress'. Recently, in analogy with turbulence, Bažant and Caner modeled the overstress as an additional viscous stress generated by apparent viscosity that accounts for the energy dissipation due to kinetic comminution of concrete into small particles at very high shear strain rate. Their viscosity estimation, however, was approximate since it did not satisfy the energy balance exactly. Here their model is extended and refined by ensuring that the drop of local kinetic energy of high shear strain rate of form- ing particles must be exactly equal to the energy dissipated by interface fracture of these particles. The basic hypothesis is that the interface fracture occurs instantly, as soon as the energy balance is satisfied. Like in the preceding work, this additional apparent viscosity is a power function of the rate of the deviatoric strain invariant. But here the power exponent is different, equal to −7/3, and the apparent viscosity is found to be proportional also to the time derivative of the rate of that invariant, i.e., to the second derivative of the shear strain. It is assumed that the interface fracture that comminutes the material into small particles occurs instantly, as soon as the local kinetic energy of shear strain rate in the forming particles becomes equal to the energy required to form interface fractures. The post-comminution behavior, including subsequent further comminution and clustering into bigger particle groups to release the kinetic energy that is being dissipated by inter-group friction, is discussed and modeled. The present formulation makes it possible to eliminate the artificial damping of all types, which is normally embedded in commercial finite element codes but is not predictive since it is not justified physically. With the aforementioned improvements, and after implementation into the new microplane model M7 for fracturing damage in concrete (which includes the quasi-static rate effects), the finite element predictions give superior agreement with the measured exit velocities of steel projectiles penetrating concrete walls of different thicknesses and with the measured depths of penetration into concrete blocks by projectiles of different veloci-123 78 Y. Su et al. ties. Finally it is pointed out that the theory presented may also be used to predict proximate fragmentation and permeability enhancement of gas shale by powerful electric pulsed-arc explosions in the borehole.

Strain Rate Dependent Microplane Constitutive Model for Comminution of Concrete under Projectile Impact

“…Note that the right-hand side of this expression is consistent with [25]. To obtain the scalar value ∆f , we now calculate the effective values of both sides, by taking square root of the inner product with itself.…”

“…(4) adds, or integrates, the energy dissipations for different particle sizes and is calculated under the assumption that for each size there is a regular array of equal-size particles. Then, considering the dynamic fracture criterion [23,25] to be −∆K = SΓ, an overall energy balance is imposed (where Γ = interface fracture energy of the comminuting particles). Substitution of eqs 3 and 4 gives the minimum particle size as…”

“…So it is useful to obtain the drop in kinetic energy in one time increment of the comminution process. Accordingly, as proposed in [25] the increment of drop in kinetic energy is given by…”

“…The dynamic comminution (i.e., fragmentation, pulverization and crushing) of materials is of interest for many practical purposes, such as explosion effects on concrete structures, impact of metals, composites and ceramics, rock blasting and fracturing of gas or oil shale by chemical explosions or electro-hydraulic pulsed arc in a horizontal borehole [1,2,2-5,5-17, [17][18][19][19][20][21][22][23][24][25][26][27][28][29][30][31]. This article deals with the projectile impact onto concrete walls.…”

Fragmentation as an aggregation process: the role of defects

“…The amount of kinetic energy initially communicated to the ring is available to create a new interface. Grady's approach extends from the classical von Rittinger's law [22], which simply matches the available kinetic energy and the surface energy, by recognizing that the amount of kinetic energy available for fragmentation is the material dilatational energy (see also [23]). This balance leads to a fragment size…”