Structure–property–performance linkages for heterogenous energetic materials through multi-scale modeling

Abstract

In porous solid energetic materials, the mechanical processing technique (e.g. casting, pressing) creates defects such as voids, cracks, interfaces, and inclusions; these defects in the microstructure strongly influence the sensitivity of the material to imposed loading. Energy localization at defects causes hotspots; the ignition and growth of hotspots in the microstructure (i.e. the meso-scale) play a crucial role in the macroscale initiation of the material. Predictive models of shock response of energetic materials must connect the meso-scale heterogeneities (structure) and hotspot physics (properties) to macro-scale response (performance and safety). To achieve this structure–property–performance (S–P–P) linkage, SEM-imaged samples of neat pressed HMX are obtained, and morphometry is performed to quantify the microstructure. Since the microstructure is stochastic, the aleatory uncertainties in the morphological parameters are quantified. The link between the microstructure and the key meso-scale quantity of interest—the hotspot ignition and growth rates—is established using reactive meso-scale computations to construct meso-informed surrogate models for energy localization. The surrogate models are used to close homogenized macro-scale governing equations. The performance of the HE at the macro-scale, i.e. its sensitivity to shock loading, is measured via run-to-detonation distances (in Pop plots) and the critical energy required for initiation (in James plots). The predicted critical energy for the material is compared with experimental data. The methods established in this paper can be useful not only for establishing structure–property–performance (S–P–P) linkages for pressed energetic materials, but also for other heterogenous reactive composites such as propellants and plastic-bonded explosives.

Appendix

1.1 Governing equations at the meso- and macro-scales

The conservation laws for mass, momentum, and energy that apply at both meso- and the macro-scales in the MES-IG model are cast in Eulerian form, viz.:

$$\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho {u}_{i}\right)}{\partial {x}_{i}}=0,$$

(A1)

$$\frac{\partial \left(\rho {u}_{i}\right)}{\partial t}+\frac{\partial \left(\rho {u}_{i}{u}_{j}-{\sigma }_{ij}\right)}{\partial {x}_{j}}=0,$$

(A2)

and

$$\frac{\partial \left(\rho E\right)}{\partial t}+\frac{\partial \left(\rho E{u}_{j}+{\sigma }_{ij}{u}_{i}\right)}{\partial {x}_{j}}=\dot{\mathcal{E},}$$

(A3)

where ρ, and ${u}_{i}$ denote the density, and the velocity components, respectively, $E=e+\frac{1}{2}{u}_{i}{u}_{i}$ is the specific total energy, and e is the specific internal energy. The source term $\dot{\mathcal{E}}$ in Eq. (A3 is the rise in specific internal energy of the system due to heat released in the decomposition of solid HMX into gaseous reaction products. The Cauchy stress tensor, ${\sigma }_{ij}$ is of the form:

$${\sigma }_{ij}={S}_{ij}-p{\delta }_{ij},$$

(A4)

where ${S}_{ij}$ is the deviatoric stress tensor and p is the pressure.

The chemical species are evolved in time by solving the conservation equation

$$\frac{\partial \rho {Y}_{i}}{\partial t}+div\left(\rho \overrightarrow{V} {Y}_{i}\right)={\dot{Y}}_{i},$$

(A5)

where ${Y}_{i}$ is the mass fraction of the ${i}^{th}$ species and ${\dot{Y}}_{i}$ is the production rate source term for the ${i}^{th}$ species. The numerical stiffness in solving the reactive set of equations is circumvented using a Strang operator splitting approach (Strang 1968), where first the advection of species is performed using the flow time step to obtain predicted species values:

$$\frac{\partial \rho {{Y}_{i}}^{*}}{\partial t}+div\left(\rho {\overrightarrow{V}}^{n} {Y}_{i}^{*}\right)=0.$$

(A6)

In a second step, the evolution of the species mass fraction due to chemical reactions is calculated

$$\frac{d{Y}_{i}^{n+1}}{dt}={\dot{Y}}_{i}^{*n}$$

(A7)

The species evolution Eq. (A7) is advanced in time using a 5th order Runge–Kutta Fehlberg method (Fehlberg 1968), which uses an internal adaptive time-stepping scheme to deal with the stiffness of the chemical kinetic equations.

At the meso-scale, in the high-resolution reactive void collapse calculations performed in the setup shown in Fig. 7a, the HMX and void spaces are delineated using a sharp-interface Eulerian framework presented in previous work (Kapahi and Udaykumar 2015, 2013; Rai et al. , 2017; Rai and Udaykumar 2015, 2018). The collapse of voids due to shock loading and the formation of hot spots are modeled with the solid HMX modeled as an elasto-plastic material of constant yield strength, ${\sigma }_{y}$. Shock heating can lead to the melting of HMX; therefore, thermal softening of HMX is modeled using the Kraut–Kennedy relation (Menikoff and Sewell 2002). The pressure at the meso-scale is obtained from a Birch–Murnaghan equation of state (Menikoff and Sewell 2002; Sewell and Menikoff 2003),

$$p\left(\rho ,e\right)={p}_{k}\left(\rho \right)+\rho \Gamma \left[e-{e}_{k}\left(\rho \right)\right],$$

(A8)

where

$${p}_{k}\left(\rho \right)=\frac{3}{2}{K}_{T0}\left[{\left(\frac{\rho }{{\rho }_{0}}\right)}^{7/3}-{\left(\frac{\rho }{{\rho }_{0}}\right)}^{5/3}\right]\left[1+\frac{3}{4}\left({K}_{T0}^{^{\prime}}-4\right)\left[{\left(\frac{\rho }{{\rho }_{0}}\right)}^{2/3}\right]-1\right],$$

(A9)

$${e}_{k}\left(\rho \right)={e}_{0}- {\int }_{\frac{1}{{\rho }_{0}}}^{\frac{1}{\rho }}{p}_{k}\left(\rho \right)d\left(\frac{1}{\rho }\right),$$

(A10)

and

$$e={e}_{k}\left(\rho \right)+ {C}_{v}T,$$

(A11)

where $\Gamma$ is the Gruneisen co-efficient, and T is the temperature. The isochoric specific heat ${C}_{v}$ is computed from the isobaric specific heat ${C}_{p}$ using the following equation:

$${C}_{v}={C}_{p}-{max(\alpha }^{2}TV{K}_{T},200J/KgK),$$

(A12)

where α is the thermal expansion co-efficient, and ${K}_{T}$ is the bulk modulus.

The chemical decomposition of HMX is modeled using a three-step mechanism proposed by Tarver et al. (1996). A detailed description of the implementation of the three-step model in the current numerical framework is presented in previous work (Rai et al. 2017a). Here, a brief overview of the reaction model and its implementation is provided.

Chemical decomposition of HMX takes place in three steps involving four different species:

$$Reaction 1: HMX \left({C}_{4}{H}_{8}{N}_{8}{O}_{8}\right)\to fragments \left(C{H}_{2}NN{O}_{2}\right)$$

(A13)

$$Reaction 2: fragments \left(C{H}_{2}NN{O}_{2}\right)\to intermediate gases \left(C{H}_{2}O,{N}_{2}O,HCN,HN{O}_{2}\right)$$

(A14)

and

$$Reaction 3: 2 \times intermediate gases \left(C{H}_{2}O,{N}_{2}O,HCN,HN{O}_{2}\right)\to final gases \left({N}_{2},{H}_{2}O,C{O}_{2},CO\right)$$

(A15)

The solid HMX (species 1, mass fraction ${Y}_{1}$) under high temperature decomposes into fragments (species 2, ${Y}_{2}$). The fragments are further decomposed to intermediate gases (species 3, ${Y}_{3}$) which are later converted to the final gases (species 4, ${Y}_{4}$) through exothermic reactions leading to high temperatures in the hotspot.

The change in temperature due to chemical decomposition of HMX is calculated by solving the evolution equation,

$$\rho {C}_{p}\dot{T}={\dot{Q}}_{R}+k{\nabla }^{2}T$$

(A16)

where k is the thermal conductivity of HMX and ${\dot{Q}}_{R}$ is the total heat release rate because of the chemical reaction. The source term in Eq. (A3) is computed by setting $\dot{\mathcal{E}}={C}_{v}\dot{T}$.

At the macro-scale, the material is a homogenized mixture of solid HMX and gaseous reaction products. The mixture is assumed to behave hydro-dynamically (Kapila et al. 2007; Lee and Tarver 1980), i.e., ${S}_{ij}$ is neglected in comparison to $p{\delta }_{ij}$ in Eq. (A4). The chemical heat release due to the decomposition of HMX into gaseous products is accounted for by transitioning the mixture from a cold, unreacted solid Hugoniot to a product Hugoniot. The equations of state for the reactants and the products are given by a Cochran–Chan and a JWL equation of state, as described in Sen et al. (2018b).