A direction sensitive detonation model for granular to continuum scale for shock initiation of pentaerythritol tetranitrate single crystal in multi-dimensions

Experiments have shown that the shock sensitivity of a single crystal pentaerythritol tetranitrate (PETN) has a strong dependence on the crystal orientation. The ignition and growth (I & G) model has been widely used in studies of the shock initiation of energetic materials while the model is independent of the direction of compression, and thus it is impossible to address anisotropic sensitivity of such material. In this paper, we base our new model in the recently proposed reactive flow concept that incorporates an anisotropic ignition mechanism that depends on both strain and strain rate which are given in the general tensor notation. A multi-dimensional simulation is performed in order to illustrate the strain dependence of the initiation of a PETN pellet. The model is applicable to any anisotropic energetic material subjected to a shock impact, not limited to single crystal PETN.


I. INTRODUCTION
2][3] For instance, the pressure (or mean stress) threshold for initiation of detonation for PETN crystals along the <100> direction is at least 4 times that of <110> direction. 4Plaksin 5 provided experimental evidence that the maximum shear stress along such preferred direction leads to an enhanced sensitivity to a detonation.Experimental findings [1][2][3]5 and molecular simulations 4,6 both suggest that special coupling between thermal, chemical, and mechanical effects is necessary to realistically address the mechanism of anisotropic sensitivity in a single crystal PETN.
Table I summarizes the run distance to detonation for various input stresses and crystal orientations.Both the <110> and <001> directions are shown to be sensitive to impact, even at a relatively low pressure of 8.6 GPa while <101> and <100> directions exhibit insensitivity to stresses up to a Chapman-Jouguet (C-J) pressure of 31 GPa.The microscopic mechanism behind the anomalous characteristics observed in experiments have been explained by the steric hindrance effect. 2The steric hindrance model consists of dislocation and preferred slip planes induced by the microstructure of PETN.The material properties comprised of detonation velocity (∼8.28 mm/µs), von Neumann spike (∼45 GPa), and C-J pressure (∼31.5 GPa) are independent of orientations when the steady detonation state is achieved.
The ignition and growth (I & G) continuum model, 8,9 based on empirical observations of the pressure-dependent initiation of detonation, has been widely used to date.The basis of the model is a hot spot theory, where inclusion of a compressibility (or density) factor to ignite the heterogeneous high explosive is addressed.Since the model is independent of the direction of compression, that is, the model is always isotropic; it is impossible to address the anisotropic sensitivity effects.There is a work that has focused on the anisotropic elastic and plastic response of PETN in the microscopic sense. 10More recently, our previous continuum model has incorporated what has been observed in both experiment and the atomistic calculation of PETN and proposed a crystal orientation dependent reactive flow model. 11Although the model is three-dimensional in theory, an explicit simulation of two-dimensional consideration was not provided.Here, the anisotropic reactive response of a two-dimensional PETN crystal subjected to an external shock impact is presented, which enhances the fidelity of the current state of direction sensitive detonation simulation from a granular to a continuum scale.

II. ANISOTROPIC CHEMICAL KINETICS
The anisotropic reactive flow model 11 is defined by: In this model, λ is the reaction progress variable where 0 corresponds to the unreacted while 1 corresponds to the reacted state.The model assumes three distinct phases in the reaction progress.The first phase corresponds to an ignition, which is triggered by a shock compression and creation of the hot spots.The second phase is the growth of the reaction front in low pressure regime in the solid phase, while the third phase describes completion of the reaction.
In developing an anisotropic ignition and growth model, we assume that dislocation and slip systems identified in earlier work [4][5][6] are effective mostly in the early stages of shock initiation.Therefore, ignition is anisotropic, while growth of reaction remains isotropic.The overall dislocation and slip are taken into consideration by calculating the strain field instead.Also, the shock impact is assumed sufficiently strong enough such that any elasto-plastic behavior of PETN is neglected.Finally the growth phase of reaction is assumedly dominantly pressure driven.The Cartesian tensor form of a direction-dependent detonation rate law is given by: where ε i j is the strain tensor, εi j is the strain rate tensor, J i j is a unit matrix of all ones, H( εi j ) is a strain rate threshold function and χ εexp is an assessment function whether a state of the explosive is in expansion or compression.The function f indicates the positive part of its argument: Quantities used in Eq. ( 2) are defined in Eqs. ( 4) and ( 5) such that The first term in Eq. ( 2) is the ignition term controlled by the strain relative to a critical strain of the crystal orientation, and by the threshold strain rate.The second and third terms of Eq. ( 2) describe the growth of reaction which is strongly dependent on pressure.The reference strain rates εi j,0 are defined in an intrinsic sense, and they are tabulated in Table II.For a typical detonation, a range of corresponding strain rates is between 10 7 ∼ 10 8 /s.The reference strain rate is used to prevent false initiation in response to a non-shock loading.
Nonetheless the strain tensor based ignition may be falsely triggered if a sufficiently large negative strain is present.In particular near the explosive edges where such negative strains typically occur, the physical anisotropy requires a correction for suitable description of the reactive flow.The negative-strain does not incur ignition in the outer region of an explosive since a regulator χ εexp is introduced as in Eq. ( 2).
An impact pressure high enough to reach a C-J value at 31 GPa along the insensitive direction can trigger an instantaneous detonation.One needs not worry about the direction sensitivity in such extreme compression case; it will detonate regardless.However, anisotropic feature embedded in the ignition term of Eq. ( 2) allows the initial pressure to build up along the sensitive direction, leading to a full detonation through the growth term.
An isotropic behavior similar to the classical I & G framework can be naturally recovered if one applies constant values for all reference strain and strain rates of the anisotropic coefficient tensor such as εi j,0 = ε0

III. GOVERNING EQUATIONS
The governing equations involving mass, momentum, energy conservation and species evolution are explicitly written for two-dimensional simulation in a cylindrical coordinate system: where ρ is density, u r , u z are velocity components in radial and axial directions, E is the total energy per unit mass with e being its specific internal energy, and p is the hydrostatic pressure.We assume that (i) dislocation and slip system in molecular scale are neglected; instead strain and strain rate dependent ignition is considered, (ii) anisotropic initiation of detonation is governed by the impact (pressure) direction, (iii) elastic-plastic behavior is negligible for strong shocks in early stage of the simulation, that is stress-strains are unchanged during that time, (iv) explosive runaway or growth is dominantly pressure driven and isotropic.The JWL equation of state (EOS) is used to model the reacted and unreacted HE, Tables III and IV summarize the chemical kinetics parameters and the EOS of PETN used for the numerical simulation.A detailed description on numerical implementation of the governing equations is similarly addressed in Refs.11 and 12.

A. One-dimensional problem
One-dimensional calculations of PETN test are compared for both isotropic and anisotropic models.The considered impact pressure varied from 8 to 19 GPa, and the uniform mesh size is set to 2.5 µm.Upon impacting one end of the PETN, the shock wave progresses into the crystal, and the run to detonation distance is extracted from the calculated result.[3]9 The calculation reproduces the measurements fairly well in <110> direction and provides the prediction in <001> direction.
The ignition experiment 13 using the Mylar flyer plate is reproduced by both models and again compared.The Mylar flyer plate with velocity close to 4.1 km/s imparts 19 GPa to a PETN during ∼0.06 µs.Calculated pressure histories for both models well resolve the shock initiation, overdriven super detonation, and pressure decay towards the C-J value in the sensitive <110> direction as shown in Fig. 2. The existence of a super detonation 1 for pre-compressed PETN that accelerates the reaction rate is captured by the model.The pressure results clearly mark the von Neumann spike of 45 GPa, and the C-J pressure of 31.5 GPa is also in a good agreement with the experimental value.Further details of the one-dimensional results may be found in Ref. 11.

B. Two-dimensional problem
The time-resolved temperature 3 measured by the photomultiplier tubes (PMTs) provides the temporal characteristics of the ignition, pressure build-up, and detonation.Temperature and detonation velocity at steady state are measured to be 4140 K and 8.2 mm/µs, respectively, independent of impact orientation.The explosion of a single crystal PETN involving the super-detonation makes it extremely difficult to predict the temperature using a constant specific heat capacity, because a fixed specific heat assumes the C-J state only.
In order to address multi-dimensional response, the unconfined rate stick test is performed.Computational domain is 10 mm in diameter with a thickness of 5 mm, and uniform mesh of 5 µm is selected for fully resolving the reaction length of PETN.Impact pressures of 15 and 31 GPa are head on to one end of an explosive along two orientations (<110> and <100>) as drawn in Fig. 3. Two representative orientations in a cylindrical coordinate are aligned along the axial direction.Calculated results in Figs. 4 and 5 show a noticeable difference depending on the direction of a shock impact on a crystal orientation.The chemical reaction is permitted when a relatively small reference strain rate is present along the shock travel direction of <110>.The reaction is also controlled by the strain and the pressure terms of the model.When a shock moves toward the charge oriented along the insensitive <100> direction, a detonation failure is observed.Current strain rate in <100> orientation is substantially below the reference strain rate corresponding to a C-J pressure.Hence, ignition is naturally prohibited in the entire domain.Figure 6 is the isotropic result showing the orientation independence in <110> and <100>.
In addition, Fig. 7 shows strongly anisotropic shock sensitivity in PETN in accordance to its orientation.The hot spot in the pre-compressed PETN of the <110> orientation gives rise to a sudden pressure build-up, and detonation is eventually configured by the anisotropic chemical model.The calculated results based on the anisotropic model that include time to detonation, von Neumann spike, detonation velocity (pressure), and super detonation velocity (pressure) are all in good agreement with the experimental results 3 for both sensitive and insensitive orientations.Whereas the isotropic model fails to describe the experiment 3 for both sensitive and insensitive orientations.
Then the impact of the C-J pressure is considered in both <110> and <100> orientations.Shock ignition begins almost instantly at 31.3 GPa, and the reaction rate increases drastically.The strain and pressure based-ignition terms are active in an extremely short duration, and such the growth term governs most of the detonative flow in all orientations.The results of the sensitive and insensitive orientations are shown identical in Figs. 8 and 9.

V. CONCLUSION
We have described a multi-dimensional anisotropic detonation rate model for impact ignition behavior of a single crystal PETN.Inclusion of the strain tensor allows for directional sensitivity to ignition and for reliable reproduction of the test data.The two-dimensional rate stick test showed realistic and anisotropic detonation responses of a PETN pellet when impacted along the sensitive orientation.We believe that the proposed multi-dimensional reaction model is suitable for describing the anisotropic sensitivity of energetic material not limited to a PETN crystal.The proposed model is expected to improve the current state of anisotropic detonation modeling from a granular to a continuum scale.

TABLE III .
Parameters for Anisotropic model.