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INSTITUTE OF PHYSICS PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING

Modelling Simul. Mater. Sci. Eng. 14 (2006) 537–561 doi:10.1088/0965-0393/14/4/001

Numerical simulation of shock wave propagation inspatially-resolved particle systems

Ryan A Austin1, David L McDowell1,3 and David J Benson2,3

1 G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, 801 FerstDr. N.W., Atlanta, GA 30332-0405, USA2 Department of Mechanical and Aerospace Engineering, University of California, San Diego,9500 Gilman Drive, La Jolla, CA 92093-0411, USA

E-mail: [email protected], [email protected] and [email protected]

Received 6 June 2005, in final form 28 December 2005Published 11 April 2006

Online at stacks.iop.org/MSMSE/14/537

Abstract

The shock compression of spatially-resolved particle systems is studied at the

mesoscale through a series of finite element simulations. The simulations in-

volve propagating shock waves through aluminium–iron oxidethermite systems

(Al+Fe2O3) composed of micron-size particles suspended in a polymer binder.

Shock-induced chemical reactions are not considered in this work; the particle

systems are modelled as inert mixtures. Eulerian formulations are used to ac-

commodate the highly dynamic nature of particulate shock compression. The

stress–strain responses of the constituent phases are modelled explicitly at high

strain rates and elevated temperatures. Dynamic behaviour of the model sys-tem is computed for a set of mixtures (20% and 50% epoxy content by weight)

subjected to a range of loading conditions (particle velocities that span 0.300–

1.700kms−1). Spatial profiles of pressure and temperature obtained from the

numerical simulations provide insight into thermomechanical responses at the

particle level; such resolution is not available in experiments. Finally, Hugoniot

data are calculated for the particle mixtures. Stationary pressure calculations

are in excellent agreement with experiments, while shock velocity calculations

exhibit larger deviations due to the 2D approximation of the microstructure.

Nomenclature

µD mean of the inclusion size distributionσ D standard deviation of the inclusion size distribution

Af area fraction

ρ mass density

t time

3 Author to whom any correspondence should be addressed.

0965-0393/06/040537+25$30.00 © 2006 IOP Publishing Ltd Printed in the UK 537

http://dx.doi.org/10.1088/0965-0393/14/4/001

mailto:%[email protected]




http://stacks.iop.org/ms/14/537

http://stacks.iop.org/ms/14/537




http://dx.doi.org/10.1088/0965-0393/14/4/001



538 R A Austin et al

x position

u velocity

u acceleration

σ Cauchy stress

f specific body forcee specific internal energy

T temperature (absolute)

E internal energy per unit volume

EC cold compression energy per unit volume

P pressure

U P particle velocity

U S shock wave velocity

ε strain rate

φ generalized solution variable

Φ source term

Fext

external force Fint internal force

B discrete gradient operator

N shape function

τ surface traction

M lumped diagonal mass matrix

V volume

η relative density

CP specific heat at constant pressure

CV specific heat at constant volume

K bulk modulus

µ shear modulus

κ thermal conductivity

EH latent heat of melting

C acoustic wave speed

ε uniaxial strain

τ effective shear stress

τ flow stress in shear

b Burgers vector

ρ dislocation density

k Boltzmann’s constant

σ y current flow stress

σ y,0 initial flow stress

KH Knoop hardnessH linear strain hardening coefficient

1. Introduction

Reactive particle metal mixtures (RPMMs) are highly heterogeneous materials that provide

energy release due to exothermic chemical reactions initiated under shock-loading conditions.



Spatially-resolved particle systems 539

RPMMs are typically composed of micron- or nano-scale particles. Such mixtures are inert

under ambient conditions, but reactions are initiated due to the energy supplied by the passage

of sufficiently strong shock waves. Design of energetic materials that also serve structural load

bearing purposes has been addressed in recent work [1], with the goal of replacing part of the

non-energetic structure by energetic material for purposes of enhancing overall specific energyrelease characteristics. A detailed understanding of a candidate material’s mechanical and

thermal behaviour at multiple length and time scales is required to exploit its multifunctional

character. A grasp of this behaviour will (1) enable informed decision making in designing

particle mixtures to meet structural and energetic requirements, (2) promote rapid insertion of

energetic materials due to reduced experimental testing and (3) provide capability to tailor the

amount of energy released upon detonation.

An extensive review of the shock compression of particle systems has been published by

Thadhani [2]. Shock waves induce high rates of energy deposition at particle surfaces, which

are responsible for large plastic strains, the formation of strong temperature gradients, melting

at particle surfaces, mass mixing and the generation of high defect densities. The unique

environments created during shock compression promote chemical reactivity. Reactions initi-

ated by shock waves may be classified as shock-induced or shock-assisted [3]. By definition,shock-induced chemical reactions (SICRs) are initiated within the time scales of mechanical

(pressure) equilibration; shock-assisted chemical reactions (SACRs) are initiated on the longer

time scales of thermal equilibration, after release waves have allowed mechanical relaxation.

A brief description of experimental methods utilized in shock compression is presented

next. Comprehensive experimental details are beyond the scope of this manuscript; the

literature may be referenced [2, 4] for complete descriptions. Shock waves are typically

generated in experiments by the impact of a flyer plate accelerated by compressed gas or

explosives. Post-shock recoverytechniques allow microscopy analyses to be performedon final

microstructure morphologies. Time-resolved measurement techniques [5,6] provide temporal

histories of macroscopic shock data. Time-resolved measurements are useful in many cases

because they yield information pertaining to reaction initiation and kinetics; however, such

experimental techniques are challenging due to the short time scales of shock wave propagation

and reaction initiation (SICRs can be initiated in less than 100 ns [3]). Shock-compressionexperiments have been performed in a number of silicide systems, including Ni–Si [7], Nb–

Si [8, 9], Mo–Si [6, 10, 11] and Ti–Si [5]. Experimental work has also been performed in both

loose-particle [12,13] and polymer-bonded[14] Al+Fe2O3 systems. The experimental work of

Thadhani etal [3,5] and Vandersall and Thadani [6] shows that predictions of reaction initiation

must take into account particle-level behaviour. However, time-resolved experimental methods

cannot achieve full spatial resolution at the particle level during shock wave propagation;

therefore, numerical methods must be introduced.

Williamson [15] is credited with the earliest spatially-resolved numerical simulation

of particulate shock compression. Williamson studied thermal and mechanical responses

of a unit cell containing a small set of closely-packed steel particles. Simulations were

performed in a 2D Eulerian finite difference code because the high levels of pressure, severe

plastic strains, material jetting and melting rendered traditional Lagrangian finite elementmethods inadequate. Highly-structured materials, which display substantial periodicity, can be

represented by repeated sets of unit cells; however, the heterogeneous nature of many particle

systems requires the use of statistical volume elements (SVEs), which reflect the random

spacing and size variations found in actual particle distributions. Benson [16, 17] advanced

the study of Williamson by considering larger domains of randomly-distributed, micron-size

Cu particles. Benson developed a 2D multi-material Eulerian hydrocode [18] to handle the

dynamic nature of particulate shock compression. Methods for calculating the shock wave




velocity–particle velocity relation [17, 19] were used to characterize the Hugoniot of the Cu

system and compare simulations with experimental results. In another set of finite element

simulations, particle-level responses of shock-loaded titanium–silicon carbide systems (Ti–

SiC) were modelled by Benson et al [20] to study initiation and growth of interfacial reaction

layers. Benson and Conley [21] later studied thermomechanical responses of shock-loadedHMX by performing numerical simulations on highly resolved, digitally imported micrographs

of mixture microstructures. Thisset of simulations tookinto account the explicit morphologyof

the microstructure, in contrast to the majority of prior modelling efforts that assumed simplified

microstructure geometries.

This body of work seeks to model particle-level thermomechanical responses of discrete

microscale particle systems during shock compression. The specific particle system of interest

is a thermite system composed of aluminium and iron oxide particles (Al+Fe2O3) embedded in

an epoxy matrix. Shock compression will be studied for a range of mixtures and loading con-

ditions using finite element simulation. The Al + Fe2O3 powder mixtures are, in fact, reactive

systems; however, chemical reactions are not modelled in this work, which aims to serve as a

precursor to studies of initiation of chemical reactions. Methods involved in the analysis will

be presented, including techniques used in microstructure reconstruction and the Eulerian finiteelement method. The initial-value boundary value problem will be defined and the constitutive

models will be discussed for each phase. Shock wave velocities and stationary pressures will

be calculated and compared with experimental data to determine the accuracy of the numeri-

cal simulations. The results will provide insight into (1) spatially-resolved thermomechanical

responses at the particle level and (2) Hugoniot data for the polymer-bonded thermite system.

2. Shock compression of the aluminium–iron oxide thermite system

The analysis of the thermite system is conducted by propagating shock waves through the

particle mixtures. Two mixture classes were selected for this study. The first class corresponds

to particle mixtures that have been tested experimentally, which contain 50% epoxy by weight.

The second class corresponds to particle mixtures that will be fabricated in future experiments

with 20% epoxy by weight. The mixture classes will be refered to as 20 wt% and 50 wt%

mixtures in the remainder of this paper.

2.1. Microstructure reconstruction

An objective of this research is to observe and predict the thermomechanical behaviour of

inert particle systems based on mixture parameters that can be controlled (e.g. particle size and

phase content). Therefore, it is necessary to implement algorithms that synthetically generate

microstructures based on prescribed mixture parameters. An ability to generate physically

realistic microstructures will allow material designers to explore performance ranges beyond

those considered experimentally, based on parameters that characterize the morphological

attributes of the mixture.

A micrograph of a non-planar fracture surface from a 20 wt% mixture is shown in figure 1.Constituent phases include aluminium particles (Al-1100), agglomerates of iron oxide particles

(Fe2O3) and the epoxy matrix (Epon 828). The smooth, large entities are aluminium particles.

The smaller, granular entities are iron oxide agglomerates. The dark regions are epoxy. Voids

are present but are difficult to differentiate from the epoxy phase. It should be noted that the

iron oxide agglomerates comprise 3–5 subparticles.

Thepowder microstructure is approximatedby extracting a randomplanar section from the

3D microstructure. Thus, the complexity of the actual microstructure is simplified assuming




Figure 1. Experimental micrograph of a 20 wt% epoxy mixture. Markers indicate (a) epoxy, (b)aluminium particles and (c) iron oxide agglomerates.

Table 1. Mixture parameters for simulated microstructures.

wt% epoxy Material µD (µm) σ D (µm) Af

Al 1.5 0.30 0.21

20 Fe2O3 0.4 0.10 0.31

Voids 0.4 0.10 0.02

Al 1.5 0.30 0.09

50 Fe2O3 0.3 0.06 0.13

Voids 0.3 0.06 0.02

a 2D distribution of circular particle/void sections that intersect the extraction plane. The 2D

size distributions of the inclusion phases are lognormal. Here, the lognormal size distributions

are defined by the mean inclusion diameter (µD) and the standard deviation of the inclusion

diameter (σ D) on the 2D section. The area fractions (Af ) of the particulate phasesare calculated

assuming the reactants are in stoichiometric proportions (volume fractions of constituents in

3D are equivalent to area fractions in 2D [22]). Estimates of porosity for fabricated mixtures

were provided by Ferranti and Thadhani [14]. The mixture parameters used to reconstruct

microstructures for simulation are given in table 1. Even though each weight class is simulated

with thesamemixture parameters, everysimulation will be uniquedue to therandom techniques

used to generate the microstructures. In other words, our analyses address the stochasticity of

the microstructure.

The process of reconstructing microstructures is efficient. Discrete sets of inclusions

(aluminium particles, iron oxide agglomerates and voids) are randomly generated withdiameters taken from the specified lognormal size distributions. The number of inclusions

generated for each phase is controlled by the prescribed area fraction.

The inclusions are added to the domain sequentially using a constrained Poisson process

(this technique is also known as a random sequential addition process [23]). During

this process, it is permissible for inclusions to come into contact. The positions of the

aluminium particles are located first. A two-point spatial correlation, i.e. the nearest-neighbour

distribution, is used to quantify the spatial positioning of the aluminium particles. Initial




Figure 2. First-nearest-neighbour (NN1) distributions calculated in models and estimated fromexperiments for the aluminium phase (a) before simulated annealing and (b) after simulatedannealing.

Figure 3. A reconstructed 20 wt% epoxy powder microstructure. Markers indicate (a) iron oxideagglomerates and (b) aluminium particles; the darkest circular entities are voids.

calculations of the nearest-neighbour distribution in the models do not agree with normally

distributed estimates from experimental micrographs [14], asshown infigure 2(a). The shaded

bars in figure 2 represent the probability that a given aluminium particle’s closest neighbour

(also an aluminium particle) is within the designated class. Spatial locations of the aluminium

particles are randomly perturbed, using an iterative simulated annealing technique [24], until

nearest-neighbour distributions match (within tolerance) those estimated from experimental

measurements. The nearest-neighbour distributions of the aluminium phase after simulated

annealing is depicted in figure 2(b). Once the positions of the aluminium particles are set, the

iron oxide agglomeratesand voids are placedin the gaps between the aluminium particles using

the aforementioned constrained Poisson process. Experimental measures of nearest-neighbour

distributions forthe iron oxide andvoid phasesare notavailabledue to thedifficultiesassociated

with resolving these phases using optical microscopy. The epoxy matrix fills in the space notoccupied by inclusions. The components of thefinal SVEfor a model thermite system(20 wt%)

are shown in figure 3.

2.2. The Eulerian finite element method

The finite element simulations are performed in a 2D multi-material Eulerian hydrocode [18]

developed by Benson. Eulerian formulations are used in these calculations because element




distortions are too severe to handle with traditional Lagrangian formulations. A broad

overview of the numerical techniques implemented in this work will be presented. An in-

depth description of the computational methods used in Eulerian hydrocodes is beyond the

scope of this paper, the interested reader may consult [25].

2.2.1. Governing equations. The conservation of mass, momentum and energy serve as the

governing equations in the Eulerian finite element method, i.e.

∂ρ

∂t + ∇ · (ρu) = 0, (1)

∂ρu

∂t + ∇ · (ρu⊗ u) = ∇ · σ + ρ f , (2)

∂ρe

∂t + ∇ · (ρeu) = σ : ε. (3)

In the above equations, ρ is the mass density, t is time, u is the velocity vector, σ is the Cauchystress tensor, f is the specific body force vector, e is the specific internal energy and ε is the

strain rate tensor (rate of deformation tensor).

2.2.2. Operator splitting. Eulerian formulations allow the material to flow through the

elements that define the Eulerian mesh. Here, the spatial domain is discretized by two

computational meshes (Eulerian and Lagrangian) that are coincident at the beginning of a

time step. This separation allows the Eulerian conservation equations to be solved by operator

splitting [18]. Operator splitting is a method of decoupling the material transport through

the mesh, so that the governing differential equations are replaced by sets of equations that

can be solved sequentially. This solution technique involves two steps. The Lagrangian step,

which is performed first, advances the solution in time. During this step, the Lagrangian

mesh deforms with the material. The Eulerian step, which is performed second, accountsfor material transport between the elements of the spatially-fixed Eulerian mesh while time

remains constant.

A general transport equation is expressed as

∂

∂t φ + u · ∇φ = Φ, (4)

where φ represents a generalized solution variable and Φ is a source term. Operator splitting

is used to decompose the transport equation into the following terms:

∂

∂t φ = Φ, (5)

∂

∂t φ + u · ∇φ = 0, (6)

where equation (5) defines the Lagrangian step and equation (6) defines the Eulerian step. If

the generalized solution variable φ is replaced by ρu, and the source term is replaced by

∇ · σ + ρ f , then equation (5) may be solved using the principle of virtual work [18].

The Lagrangian step uses the central difference method to advance the solution in

time. Nodal accelerations, velocities and positions are calculated from an equilibrium force




Figure 4. TheLagrangian meshremainsattached to thematerialas it deformsduringthe Lagrangianstep. The solution is re-mapped to the Eulerian mesh during the Eulerian step.

balance, i.e.

Fext − Fint =

ρ f n N dx +

τ

n N d −

BT : σn dx, (7)

un

= M −1[ Fext

− Fint], (8)

un+1/2 = un−1/2 + unt, (9)

xn+1 = xn−1 + un+1/2t. (10)

Here, n refers to the current time step, Fext is the vector of externally applied loads, Fint is

the vector of internal forces due to the stress state, B is the discrete gradient operator, N is a

vector of shape functions, τ is the surface traction vector, u is the acceleration vector, M is

the lumped diagonal mass matrix and x is the vector of current nodal positions. As shown in

equations (9) and (10), the nodal acceleration at the start of the time step is used to calculate

the time-centred nodal velocity during the update, which is subsequently used to update the

position at the end of the time step. Special techniques are needed during the Lagrangian step

to handle (1) the discontinuous nature of shock loading and (2) the allowance of more than

one material in an element. The former is addressed through the introduction of an artificialshock viscosity; the latter is handled by defining a mixture theory. These techniques will be

discussed later.

The Eulerian step, which accounts for a greater computational expense [17], is responsible

for re-mapping the deformed Lagrangian mesh back to the spatially-fixed Eulerian mesh. This

is achieved by projecting the material and solution variables described by the Lagrangian mesh

onto the Eulerian mesh with advection techniques. It is easiest to visualize this process by

considering a domain that is discretized by two computational meshes, as in figure 4. Here,

the thin lines represent the Eulerian mesh and a portion of the Lagrangian mesh is depicted

in bold. The Lagrangian mesh remains attached to the material as the solution is advanced

in time. At the end of a time step, the current material state is described by the Lagrangian

mesh, which is projected onto the Eulerian mesh while time remains constant. The meshes are

effectively reconciled. The Lagrangian mesh at the beginning of the next time step is definedby the Eulerian mesh at the end of the current time step, thus avoiding the formation of severe

element distortions. The arbitrary Lagrangian–Eulerian (ALE) formulation is an extension of

the aforementioned technique, in which the motion of the Eulerian mesh is predefined.

2.2.3. Transport algorithms. Element-centred solution quantities (e.g. stress, internal energy

and density) are advected during the Eulerian step using the MUSCL (monotone upwind

scheme for conservation laws) transport algorithm developed by Van Leer [26]. This technique




may be described by considering a 1D mesh, which may be extended to 2D. A general solution

variable φ is updated according to

φ+j +1/2

=

V −j +1/2φ−j +1/2 + V j φj − V j +1φj +1

V −j +1/2 + V j − V j +1

. (11)

Here, nodes j and j +1 define the endpoints of the element j +1/2 and the − and + superscripts

indicate quantitiesbefore and afterthe transport, respectively. The volume in the element before

transport is denoted by V −j +1/2. The volume transported between elements, V j , is a purely

geometric calculation based on the relative motion of the meshes. The φj and φj +1 terms are

calculated by linear interpolation of the solution variable within an element [18].

It is quite common for elements to contain more than one material in multi-material

Eulerian calculations; such elements are known as mixed elements. Consequently, interfaces

must be tracked between dissimilar materials. A nine-point finite difference stencil is used to

track the orientations of material interfaces in an algorithm developed by Youngs [27]. The

algorithm represents material interfaces with a straight line. Material identification numbers

are used to differentiate multiple materials within a single element. This is an important

consideration in the construction of discrete particle models. If two separate entities of thesame material identification number come into contact, they will coalesce (similar to a fluid).

Thus, it is necessary to define an adequate number of materials (with the same constitutive

models and material parameters) to keep track of distinct contacting entities.

2.2.4. Shock viscosity. Special computational methods are necessary to resolve the jump

discontinuities associated with shock waves. This is addressed by introducing a viscous term,

known as the artificial shock viscosity, which smears the shock front over a transition region

that is sharp but continuous. In solids, the shock viscosity introduces the proper amount of

irreversibility (damping) to prevent oscillations in the solution variables in the vicinity of the

shock front. It should be noted that the shock viscosity must be included as a hydrostatic

component in the Cauchy stress tensor. Viscous effects have been studied in the context of

discrete particle simulations by Benson and Conley [21]; as expected, the inviscid case leadsto flows with higher levels of turbulence.

2.2.5. The mixture theory. The evaluation of stress in an element containing more than

one material is performed using a mixture theory [28]. The mixture theory defines how the

total strain rate in a mixed element is distributed among the materials. Strain increments and

rotations are used to calculate the stress increment in each material. The mean stress of an

element, which is used in the nodal force calculations, is taken as the volume-fraction-weighted

sum of the constant stresses in each material.

The mixture theory must be selected carefully, as it can have a profound effect on the

modelling of local temperature spikes (hot spots). Hot spots in porous systems typically arise

from the collapse of voids, which is simply the thermomechanical interaction between a void

and the surrounding material. An understanding of how the different mixture theories affectthermal behaviour is essential for assessing the validity of predicted hot spots. The two mixture

theories of interest are (1) the mean strain rate mixture theory and (2) the single-iteration

pressure-equilibration mixture theory [28].

The mean strain rate mixture theory assigns the mean strain rate and spin (vorticity) to

each material within an element. This theory is inadequate in mixed elements for which

there is a substantial difference between the compressibility of the constituent materials. For

example, if a mixed element contained a metal and a gas, both materials would be subjected




to the same strain rates; this is obviously contrary to the physics of the situation, as the stiffer

material should carry a larger portion of the load. This is an important consideration because

the microstructure of the thermite system contains pores.

The single-iteration pressure-equilibration mixture theory calculates the strain rates in

each material by allowing the pressure in each material to approach an equilibrium state.Full mechanical equilibrium is not imposed because a shock wave would traverse an element

multiple times before the pressure equilibrated completely. The equations that define the

equilibrium process are non-linear; they are solved using a single iteration in the Newton–

Raphson method, so that the pressures are relaxed towards equilibrium. This theory yields

better results in the cases of the substantially different compressibility of constituent materials.

Therefore, the single-iteration pressure-equilibration mixture theory is used for calculations in

the thermite system. It should be noted that the material interfaces are perfectly bonded. The

effects of friction are not implemented because techniques for its inclusion are currently not

robust enough to handle strong shock loading.

2.2.6. Temperature calculations. Temperature is calculated from the thermal energy content

of a material according to the following relation,

T = E − EC (η)

ρCP

. (12)

Here, E is the internal energy (per unit volume), EC is the cold compression energy (per unit

volume), η is the relative density and CP is the specific heat at constant pressure. The latent

heat of melting is taken into account for materials that reach their melting temperatures.

The conduction of heat in all phases is modelled by Fourier’s law. This may not be a

precise description of heat transfer modes for phasesthat experience melting or decomposition;

however, such an approximation is on the order of other approximations introduced in these

analyses.

3. The initial value-boundary value problem

Shock compression is idealized by the passage of a single 1D shock wave. The initial value-

boundary value problem, which aims to replicate idealized shock loading, is depicted in

figure 5.

The Eulerian finite element method necessitates the application of boundary conditions

to both the Lagrangian and Eulerian meshes. The sets of boundary conditions applied in these

calculations aresimilar to those used in theprior modelling work of Benson etal [16,17,21,29].

A compressive shockwave is propagatedthrough the mixture by applying a Lagrangian velocity

boundary condition (i.e. the particle velocity U P) to the left surface of the SVE. The velocity

boundary condition is rampedup duringthe first stage of thesimulation according to a quadratic

function of time to avoid spurious oscillations in the solution associated with instantaneous

loading. ‘Roller’ boundary conditions, which constrain displacements in the normal directionto be equal to zero, are applied to the remaining surfaces of the SVE. The boundary conditions

applied to the top and bottom surfaces of the SVE serve as reflecting planes (i.e. planes of

symmetry). The simulation is terminated once the shock wave front has traversed 90–95% of

the SVE, so that wave reflections are avoided. The Eulerian boundary conditions are applied

in such a manner that the Eulerian mesh contracts in the direction of wave propagation, so

that it is always filled by the computational domain. Therefore, the calculations are classified

as ALE.




Figure 5. Boundary conditions applied to the SVE for idealized 1D shock wave propagation.

Table 2. Loading conditions.

Mixture class Particle velocity, U P (kms−1)

20wt% epoxy 0.500–1.00050 wt% epoxy 0.364–0.607, 1.403–1.683

The particle velocity ranges considered for each mixture class are provided in table 2.

Particle velocities selected for the 50 wt% mixture correspond to shock compression

experiments performed by Ferranti and Thadhani [14] and Jordan et al [30]; experimental

data have not been obtained for the 20 wt% mixtures.

The explicit time step is controlled by the Courant stability criterion, which limits

the time step to the minimum time necessary for an elastic wave to propagate across the

shortest edge of an element. The time step is further reduced by a factor of two to be

conservative.

The plane strain condition is imposed to reduce the true 3D nature of shock loading

to a more tractable 2D case. Thus, particles in the 2D cross section are considered to be

cylinders in 3D, i.e. extended into the plane. The computational domain is discretized by 2D

solid quadrilateral elements with a single integration point. An hourglass viscosity is used to

control hourglass modes that are known to develop in calculations utilizing reduced integration

due to an excess of unconstrained degrees-of-freedom.

Mesh-density and domain-size studies were performed in order to balance the speed

and accuracy of statistically-representative calculations. A domain size of 14 µm × 7µm

is sufficient for distributions containing 20% epoxy by weight; the domain size is increased

to 22 µm × 11µm for distributions containing 50% epoxy by weight in order to include an

adequate number of aluminium particles. Three mesh densities were evaluated: 150 × 80,

250 × 135 and 300 × 165 elements. It was found that further refinements to a mesh with

250 × 135 elements yielded insignificant changes to the results in the 20 wt% calculations; themesh density is scaledappropriately for thelarger domain. In theprevious work of Benson[17],

it was recommended that the smallest particles be spanned by 2–3 elements, at a minimum.

The mesh density that has been selected for these calculations spans the smallest particles

(iron oxide agglomerates) with 3–4 elements. It is noted that shock propagation is not highly

resolved in the smallest of particles, but there are usually only a few of these particles (5–10)

in a typical particle distribution. The majority of iron oxide particles are spanned by 6–8

elements.




Table 3. Material properties.

Properties Units Al-1100 Fe2O3 Epon 828

Initial mass density, ρ0 g cm−3 2.71 5.27 1.20

Shear modulus, µ GPa 26.8a 225 1.393

Specific heat, CP kJkg−1 K−1 0.904 0.607 2.1

Thermal conductivity, κ W m−1 K−1 222 5b 0.2

Melting temperature, T m K 926 1780 533

Latent heat of melting, EH kJkg−1 390 600b —

a Reported at T = 300 K [38].b Estimated from a survey of oxides [34].

4. Material properties

Material properties used in the finite element calculations are given in table 3. Material

properties are well defined for the aluminium phase; the properties listed may be found in most

metal handbooks, e.g. [31]. This cannot be said for the remaining phases; therefore, certain

approximations are made in the absence of data. The initial density and specific heat of Fe 2O3

are provided in [32]; here, the specific heat at constant volume CV is substituted for the specific

heat at constant pressure CP , which is a reasonable assumption for this pressure–temperature

range. The shear modulus of Fe2O3 is approximated from its relation to the elastic wave speed

C [33] and the bulk modulus K [32], i.e. ρ20 C = √

(4/3)µ + K. The thermal conductivity and

latent heat of melting for Fe2O3 are estimated from a survey of oxides (e.g. FeO, Fe3O4, MnO,

TiO2, ZrO2, Al2O3, NiO) in [34]; the melting temperature of Fe2O3 is provided in [34]. The

initial density and shear modulus of Epon 828 were reported from experiments performed by

Ferranti and Thadhani [14]. The specific heat and thermal conductivity of the epoxy resin is

provided in [35]. Thermoset polymers, such as Epon 828, do not melt at a distinct temperature;

instead, polymeric chains dissociate at the decomposition temperature. The decomposition

temperature of Epon 828 is provided in [36]. Any additional energy pertaining to a phase

change in the epoxy phase is neglected since Epon 828 is noncrystalline [37].

5. Constitutive models

Hydrostatic and deviatoric components of the stress–strain response in each material are

distinguished and calculated explicitly from the respective constitutive models for each phase.

An equation of state (EOS) is used in pressure calculations. A strength model is used in

deviatoric stress calculations. The constitutive models used for each phase are discussed next.

5.1. Equations of state

The Mie–Gruneisen EOS is used to model the hydrostatic response of the aluminium and

epoxy phases. The pressure, expressed in terms of density and internal energy, is given as

P = ρ0C2ξ [1 + (1 − /2)ξ − aξ 2/2]

[1 − (S 1 − 1)ξ − S 2ξ 2/(ξ + 1) − S 3ξ 3/(ξ + 1)2]2+ ( + aξ)E, (13)

in compression and as

P = ρ0C2ξ + ( + aξ)E, (14)

in tension, where ξ = ρ/ρ0 − 1, C is the acoustic wave speed in the material at the reference

temperature and density, is the Gruneisen gamma, a is a first-order correction term,




Table 4. EOS parameters.

Parameters Units Al-1100 Epon 828 Fe2O3

Mie-Gruneisen

C km s−1 5.38 2.60

S 1 1.34 1.59S 2 0 0

S 3 0 0

1.68 2.18

a 0 0

Murnaghan

K0 GPa 20.27

CV kJkg−1 K−1 0.607

a 4.35

1.99

S i (i

=1, 2, 3) are coefficients of a cubic fit of the shock velocity–particle velocity relation, and

E is the internal energy per reference volume. Model parameters for Al-1100 [33] and Epon828 [33] are given in table 4. For simplicity, the Gruneisen parameter is assumed constant; it

is estimated using the common approximation = 2S 1 − 1, as in [39]. The correction term,

a, is assumed to be zero in the absence of data.

The Murnaghan EOS [40] is used as the hydrostatic model for the iron oxide phase. The

pressure, given in equation (15), is expressed in terms of density and temperature:

P = K0

a

ρ0

ρ

−a

− 1

+ ρ0CV (T − T 0) . (15)

Here, K0 is the bulk modulus at 0 K, CV is the specific heat at constant volume, a and are

material parameters and T 0 is the reference temperature. Model parameters for iron oxide [32]

are given in table 4.

5.2. Deviatoric strength models

Physically based constitutive models for deviatoric stress–strain behaviour have been used

when possible. Models for aluminium and epoxy are well defined for a range of strain rates.

Unfortunately, such models are not currently available for the iron oxide phase, so the strength

behaviour is approximated. A brief description of the models will be provided here; the

literature should be referenced for complete details and material parameters.

5.2.1. Aluminium. A constitutive model for the stress–strain response of high-purity

aluminium has been proposed by Klepaczko et al [38]. Here, thermally activated mechanisms

are used to model the kinetics of dislocation glide and the rate-sensitive generation of

dislocations is taken into account. Thus, the model is sensitive to variations in temperatureand strain rate. The current state of the microstructure is tracked through the dislocation

density, which serves as a single, physically based internal state variable. The evolution of

the dislocation density is based on the competing processes of dislocation generation and

annihilation.

It is assumed that the flow stress in shear may be decomposed as τ = τ u + τ ∗, where

τ u is the internal (athermal) stress and τ ∗ is the effective (thermally activated) stress. The

decomposition of the flow stress assumes the existence of different barriers that impede the




motion of dislocations. The internal stress is associated with long-range obstacles that are

athermal in character (e.g. grain walls, cell walls and cell dislocations). The internal stress is

expressed as

τ u = αµ(T )b

ρ, (16)where α is the dislocation/obstacle interaction coefficient, µ(T ) is the temperature-dependent

shear modulus, b is the Burgers vector and ρ is the dislocation density. The effective stress

is associated with weaker obstacles (e.g. forest dislocations and Peierls barriers) that may be

overcome by thermal activation. The effective stress is cast in a generalized Arrhenius form,

τ ∗ = τ ∗0 (ρ)

1 −

kT

G0

ln

ν0(ρ)

˙γ p

1/q1/p

, (17)

where ˙γ p

is the effective plastic shear strain rate, τ ∗0 (ρ) is the thermally activated part of the

threshold stress (i.e. the stress barrier associated with short range obstacles at 0 K), ν0(ρ) is

the attempt frequency factor at 0 K, k is Boltzmann’s constant, G0 is the activation energy at

0 K, and p and q are constants that describe the shape of the energy barrier. The dislocationdensity is updated to account for the evolution of the microstructure during a general load

history. The rate of change in the dislocation density with respect to the effective plastic shear

strain is expressed as

∂ρ

∂ ˙γ p = M I I ( ˙γ

p) − ka ( ˙γ

p, T )[ρ − ρ0]. (18)

Here, M I I ( ˙γ p

) is the rate-dependent dislocation multiplication term, ka( ˙γ p

, T ) is the

dislocation annihilation factor and ρ0 is the initial dislocation density.

5.2.2. Epoxy. Constitutive models for glassy polymers at high strain rates are limited in the

open literature. Chen and Zhou [41] proposed an empirical model for Epon 828, but it only

agreed with experimental results at low strain levels (ε < 0.10). Lu et al [42] later fit materialparameters for Epon 828 to a physically based viscoelastic–viscoplastic model proposed by

Hasan and Boyce [43], which is accurate in the dynamic regime for uniaxial strains up to 0.35.

The Hasan–Boyce model is used to model deviatoric stress–strain responses in the epoxy

phase.

According to the Hasan–Boyce model, the deformation of amorphous glassy polymers

is controlled by the energetically distributed nature of the microstructure and its thermally

activated evolution with time. The distributed nature of the microstructure is characterized

by activation energy barriers; plastic deformation occurs when local energy barriers are

overcome. Here, plastic deformation is achieved by local shear transformations (i.e.

molecular re-arrangements). Local rearrangements in glassy polymers are thermoreversible,

so mechanisms for both forward transformations and reverse transformations (strain recovery)

must be included. Drawing on the mechanisms of thermal activation, the effective plastic strainrate may be represented as

˙γ p = (γ

p1 (a∗) + γ

p2 (α−1))

exp

τυ(T)

kT

− exp

−τυ(T) + S

kT

. (19)

Here, γ p

1 (a∗) and γ p

2 (α−1) are functions related to the distribution of activation energy in

the microstructure, τ is the effective shear stress, υ(T ) is the temperature-dependent shear

activation volume, a∗ and α−1 are internal state variables that describe the probability of




inelastic shear transformation events and S is an internal state variable that represents the

locally stored strain energy. The internal state variables evolve with respect to time as follows:

a∗ = −(a∗ − a∗eq)f (γ p)ω, (20)

α−1 = −(α−1 − α−1eq )ω, (21)

S = β(γ p)τ ˙γ p − Sω , (22)

where a∗eq and α−1

eq are the steady-state values of the respective internal state variables, ω is the

effective frequency of local shear transformations, f (γ p) is a function chosen to reflect the

fact that the creation of sites with low activation energies (soft sites) occurs at a negligible rate

until the effective plastic strain is sufficiently large and β(γ p) is a function that accounts for a

decrease in the storage rate of strain energy with increasing plastic strain.

5.2.3. Iron oxide. Experimental data for the deviatoric stress–strain response of iron oxide

are extremely sparse. Consequently, a physically based constitutive model for iron oxide was

not available for this study. A simple elastic–plastic model has been adopted, consisting of an

initial linear elastic response followed by linear istotropic strain-hardening, with flow stressgiven by

σ y = σ y,0 + H εp. (23)

In the above equation, σ y is the current flow stress, σ y,0 is the initial flow stress, H is the linear

strain-hardening coefficient and εp is the effective plastic strain. One may recall that the iron

oxide phase is composed of agglomerates of smaller iron oxide subparticles. It is proposed

that once the initial yield strength of the agglomerate is reached, the collection of particles will

flow, with a small degree of hardening. This seems to be a reasonable assumption, as the iron

oxide phase is much stronger and stiffer than the aluminium and epoxy phases.

The initial flow stress of iron oxide is estimated from the Knoop hardness (KH ) of a bulk

sample as σ y,0 ≈ KH /3; the Knoop hardness of iron oxide is provided in [44]. The initial

flow stress is scaled back by a factor of two to account for the strength decreases associatedwith agglomerates of subparticles. The strain-hardening coefficient is estimated as 0.1% of

the elastic shear modulus. Therefore, the parameters in the approximate model are estimated

as σ y,0 = 0.800 GPa, H = 0.200 GPa.

It is recognized that the assumed constitutive relation for the iron oxide phase is insensitive

to variations in strain rate and temperature, which is certainly a shortcoming of the model.

Future work should include an experimentally validated phenomenological model for the iron

oxide phase.

5.2.4. Numerical implementation of the constitutive models. The updated internal energy

and pressure are solved simulateneously because they are related implicitly, i.e.

P n+1 = P (ρn+1, En+1), (24)

En+1V n+1 = EnV n+1 − 12

[P n + P n+1]V . (25)

Here, P (ρn+1, En+1) refers to the pressure evaluated from the EOS at t n+1, V n+1 is the updated

volume, En+1 is the updated internal energy per unit volume and V is the change in material

volume over the time step.

Deviatoric stress–strain models adhere to J2-flow theory. Stress increments from the

constitutive models are calculated explicitly using forward-Euler integration. The time-rate

of change in the deviatoric stress tenor is defined simply as σ = 2µ(ε − ε

p), where εp is




Figure 6. Post-shock microstructure morphologies for (a) U P = 0.500kms−1, and (b) U P =1.000kms−1.

the plastic part of the total deviatoric strain rate tensor, ε. A radial return algorithm [45] is

used to return stress states to the yield surface when necessary (advection techniques can cause

the stress state to fall outside the current yield surface). Deviatoric stresses are set equal to

zero when local material temperatures reach the melting temperature (or the decomposition

temperature in the case of epoxy) to reflect a loss of strength. Hydrostatic stress components

remain; therefore, melted materials flow easily, but still carry hydrostatic loads.

6. Thermomechanical responses

Thermal and mechanical responses of the thermite system were studied through (1) relative

amounts of material deformation, (2) spatial distributions of temperature and pressure and (3)

spatial distributions of melted materials. The simulation results are discussed next.

6.1. Microstructure morphologies in the deformed state

Relativeamounts of material deformation forthe 20 wt%caseare depicted in figure 6. Thesame

powder microstructure was subjected to different loading conditions to highlight differences. It

is clear that the high-velocity case (U P = 1.000kms−1) exhibits substantially greater amounts

of plastic deformation as compared with the low-velocity case (U P = 0.500kms−1). Plasticdeformation in the aluminium phase is particularly severe. The iron oxide phase exhibits a

considerably stiffer response, as agglomerates remain fairly circular in the low-velocity case;

iron oxide inclusions are deformed into a bowed shape in the high-velocity case. Higher levels

of ‘turbulent mixing’ are observed for the high-velocity case. The aluminium particles appear

to flow in between and around the strong iron oxide obstacles. This could be an important

consideration in the design of energetic powder mixtures. For example, if it is desirable to

delay the onset of reaction, the spatial positioning and content of the aluminium particles could




be manipulated to control relative amounts of intimate contact between reactants. It should

be noted that damage criteria for material failure (fracture) have not been implemented for

the aluminium phase. Consequently, the aluminium particles probably exhibit higher levels

of ductility than would be considered normal; however, the temperatures of these particles

approach their melting temperature, so increases in ductility might be expected. Unfortunately,post-shock micrographs of the microstructure are not available from experiments, as impact

tests destroy the samples.

Spurious artefacts tend to arise within 1 µm of the velocity boundary condition (left

surface), as shown in figure 6(b). Material deformations in this region are more severe than

those found away from theleft surface of themodel. Temperature anomalieswere also observed

near the surface where the velocity boundary condition was imposed for the high-velocity

case; this ‘wall heating’ phenomenon is caused by the addition of shock viscosity [ 20]. Future

sampling of results will omit consideration of this thin strip of the domain. Significant artefacts

were not found near the top and bottom surfaces of the SVE.

The simulations of this study employ a 2D approximation of the microstructure. The

degree of constraint is certainly higher for 2D calculations as compared with fully 3D

calculations due to the plane strain condition. In 2D calculations, the aligned cylindricalinclusions are constrained against out-of-plane displacements. Had the calculations been

performed in 3D, one would expect spherical particles to experience limited out-of-plane

displacements during wave propagation. This would probably result in reduced amounts of

deformation in the particulate phases due to the higher compressibility of the matrix. Although

the deformation results are of 2D character, they are useful for comparative purposes and for

observing general trends in deformation responses to support design of such mixtures.

6.2. Spatial profiles of pressure and temperature

Spatial profiles of pressure and temperature quantify the thermomechanical responses of the

SVE. Such measures resolve particle-level interactions and highlight the heterogeneous nature

of the microstructure during shock compression. Furthermore, the spatial profiles of pressure

allow us to compare simulation results with experimental data.Shock pressures are measured experimentally in a macroscopic sense. Stress gauges on

the free surfaces of a specimen record the impact stress and the propagated stress during,

for example, a Taylor impact test. However, the spatial distributions of pressure calculated

in the finite element simulations are highly non-uniform due to the heterogeneity of powder

microstructures. Therefore, averaging techniques must be used to convert the heterogeneous

pressure distributions to measures that can be compared with experimental data.

In a method similar to that used by Benson and Conley [21], a slice of the microstructure

has been taken at the horizontal centreline of the SVE. The values of pressure for the nodes

that fall on the horizontal centreline are recorded. This construction yields a 1D distribution

of pressure along the axial direction (x) of the microstructure, which shall be known as the

centreline pressure P c (averaging has not been performed here). If one imagines taking a

horizontal slice of the microstructure at every nodal position in the transverse direction (z)

and averaging the values, transversely averaged 1D profiles of pressure are obtained along the

axial direction of the SVE as

P j = 1

N z

N zk=1

P j,k . (26)

Here, P j is the transversely averaged pressure on the centreline at node j , N z is the number of

nodes in the transverse direction and P j,k is the pressure value at the node (j,k). The 1D




(a)(b)

(c)

C e n t r e l i n e

C e n t r e l i n e

Figure 7. Spatial profiles of pressure and temperature in distinct 20 wt% mixtures plotted againstthe axial coordinate (x) of the powder microstructure at a time when the shock wave has passedsufficiently far into the SVE; (a) centreline pressure profiles at selected particle velocities, (b)transverselyaveraged pressureprofilesat selectedparticlevelocities and(c) a centreline temperatureprofile for a particle velocity of 1.000km s−1.

transversely averaged pressure profiles are averaged along the axial position of the SVE

(behind the shock front) to obtain the stationary pressure P st, which may be compared with

the propagated stress in experiments.

The method outlined above is also used to construct centreline temperature ( T c) profiles.

However, it is not desirable to construct transversely averaged profiles of temperature.

Averaging schemes would smear out the temperature gradients in the microstructure, which is

undesirable because local temperature fields are of primary interest. Therefore, our interests

shall reside in the centreline temperature profiles; if necessary, a representative temperature of

the SVE may be inferred from the centreline profiles.

Centreline pressure profiles and transversely averaged pressure profiles for three distinct

20 wt% mixtures subjected to a range of particle velocities are provided in figures 7(a) and

(b), respectively. A centreline temperature profile for a 20 wt% mixture subjected to a particlevelocity of 1.000 km s−1 is provided in figure 7(c). The sudden drop in pressure depicted

in figure 7(a) represents the shock wave front (i.e. the boundary between virgin material

and shocked material). The width of the shock front is dependent on the magnitude of the

particle velocity; higher particle velocities generate shock waves with sharp, well-defined

fronts. Here, the shock front is spread over 8–10 elements (∼0.4 µm) for the particle

velocities applied to the 20 wt% mixtures. Centreline pressure profiles and transversely

averaged pressure profiles for three distinct 50 wt% mixtures subjected to a range of particle




(a) (b)

(c)

C e n t r e l i n e

C e n t r e l i n e

Figure 8. Spatial profiles of pressure and temperature in distinct 50 wt% mixtures plotted againstthe axial coordinate (x) of the powder microstructure at a time when the shock wave has passedsufficiently far into the SVE; (a) centreline pressure profiles at selected particle velocities, (b)transverselyaveraged pressureprofilesat selectedparticlevelocities and(c) a centreline temperatureprofile for a particle velocity of 1.600km s−1.

velocities are provided in figures 8(a) and (b), respectively. A centreline temperature

profile for a 50 wt% mixture subjected to a particle velocity of 1.600km s−1 is provided in

figure 8(c).

Higher levels of variability are contained in the centreline pressure profiles, as compared

with the transversely averaged pressure profiles, due to the heterogeneity of the mixture

(e.g. moderate pressure spikes occur at particle interfaces). Transversely averaged pressure

profiles are relatively flat due to the averaging scheme described above. Centreline profiles

of temperature are extremely erratic due to temperature spikes found at contacting particle

boundaries and in void-collapsed regions and temperature minima at the aluminium particle

interiors.

6.3. Melting behaviour

The melting of constituent phases is an important consideration. In particular, the melting

of the Al phase will be a concern in future studies of SICRs, as reactions in experiments (at

ambient pressure) initiate near the melting temperature of Al (926 K). Melt fractions for a

20 wt% mixture subjected to the high-velocity load case (U P = 1.000kms−1) are depicted

in figure 9 (a melt fraction equal to 1.0 indicates complete melting or decomposition). It

can be seen that the decomposition of the epoxy phase closely follows the shock front as

temperatures typically reach 600 K directly behind the shock front. A few Fe2O3 particles




Figure 9. A spatial distribution of melt fractions (or polymer decomposition) in a 20 wt% mixturesubjected to a particle velocity U

P =1.000kms

−1.

Figure 10. Spatial distributionof meltfractions in an enlargedregionof a 50 wt% mixturesubjectedto a particle velocity U P = 1.600kms−1(decomposition of the epoxy phase is omitted).

display very limited amounts of melting at particle surfaces; theAl particlesdisplay no melting.

Insignificant amounts of melting should be expected for the iron oxide phase due to its high

melting temperature, but it is interesting that the surfaces of the Al particles do not melt despiteintense plastic deformation and elevated surrounding temperatures. This indicates that the time

scales involved in these shock passage events are too short for the effects of heat conduction

to be realized. Melt fractions in an enlarged region of a 50 wt% mixture subjected to a particle

velocity U P = 1.600kms−1 are depicted in figure 10. The complete decomposition of the

epoxy phase is omitted here for clarity. Bold circles have been placed around particles that

experience melting. As shown in figure 10, small fragments of both Al and Fe2O3 melt

sporadically throughout the SVE for the 1.600km s−1 load case.




6.4. Particle-level observations for the case of minimum epoxy content

The observations in this section pertain to the 20 wt% mixture. Peak temperatures at reactant

interfaces will be important in future work that addresses the conditions that lead to microscale

reaction initiation. In the low-velocity case (U P = 0.500kms−1

), temperatures ranging from400 to 500 K and from 500 to 550 K were found at the surfaces of aluminium particles and iron

oxide agglomerates, respectively. Interior temperatures of the particles remained at 360–400 K

for both phases. Typical temperatures in the epoxy phase were 600 K. In the high-velocity case

(U P = 1.000kms−1), temperatures ranging from 500 to 600 K and from 900 to 1300 K were

found at the surfaces of aluminium particles and iron oxide agglomerates, respectively. Interior

temperatures of the aluminium particles remained near 380 K, while the centres of the iron

oxide agglomerates reached 700 K. Typical temperatures in the epoxy phase were 1000 K. The

depth of elevated temperatures from the surface of the particles was approximately 0.125 µm

in both particulate phases for both low- and high-velocity cases. This depth corresponds to

approximately 10% and 30% of the respective diameters of the aluminium particles and iron

oxide agglomerates.

Void collapse events were a potent source of heat generation in the mixture; such events

are known to play a key role in the initiation of chemical reactions [46]. Local temperaturespikes of approximately 1200 and 1800K were found in the vicinity of collapsed voids (near

the shock front) for the low- and high-velocity cases, respectively. Hot spots remain intact

after the shock front has moved past the void-collapsed region.

7. Hugoniot characterization

The collection of Hugoniot data by experimental methods is time consuming and expensive.

Homogeneous materialsrequire, at a minimum, 10–20 data pointsto constructa setof Hugoniot

curves; this establishes a lower bound on the number of experiments (e.g. Taylor impact tests)

needed to characterize the high-pressure response of the material. Obtaining Hugoniot data

for a range of particle mixtures (different phase contents and particle morphologies) would be

prohibitively expensive. To address this problem, a method for predicting the Hugoniot for

different particle mixtures has been implemented in the finite element calculations.

The Hugoniot can be expressed in terms of five quantities (pressure, density, internal

energy, particle velocity and shock wave velocity). The Rankine–Hugoniot relations, which

are derived from the conservation of mass, momentum and energy, are used in combination

with any two of the aforementioned quantities to uniquely define the Hugoniot. In this study,

the shock wave velocity (U S) and stationary pressure (P st) are calculated. This yields two

distinct definitions of the Hugoniot for each mixture, as the particle velocity (U P) is known.

The particle velocity is simply the velocity boundary condition that is applied to the SVE.

The shock wave velocity is calculated by tracking the position of the shock front over

time. An averaging technique must be used to define the location of the shock front because

shock waves do not remain precisely planar when propagating through heterogeneous media.

Materials with dissimilar acoustic wave speeds cause bowing of the wave front and mixturesthat aredensely packed with particlesintroducepercolation effects . [39]. The stochastic nature

of shock wave propagation in random heterogeneous media has been studied extensively by

Ostoja-Starzewski [47, 48].

The position of the shock front is determined by averaging the number of elements in each

row of the Eulerian mesh that are above a small pressure threshold. It is important to note

that the size of the elements must also be tracked because the Eulerian mesh contracts as the

simulation proceeds. The shock wave velocity is obtained through a least-squares fit of the




Figure 11. Hugoniot data obtained from simulations and experiments; (a) U S–U P relations for the50 wt% mixtures, (b) P –U P relations for the 50wt% mixtures, (c) U S–U P relation for the 20 wt%mixtures and (d ) P –U P relation for the 20 wt% mixtures.

shock front position with respect to time. As mentioned earlier, the plateau-like behaviour of

transversely averaged pressure profiles facilitates the calculation of a stationary pressure by

averaging along the axial coordinate.

The U S–U P and P –U P relations for the 20 wt% and 50 wt% mixtures are plotted in

figure 11. Three particle mixtures were simulated for each particle velocity. The scatter

in the numerical results is small for mixtures simulated in the low-velocity regime (U P <

1.000kms−1); the scatter is somewhat larger for mixtures simulated in the high-velocity

regime (U P 1.000kms−1). The simulation results are plotted alongside experimental data

for the 50 wt% mixtures obtained by Ferranti and Thadhani [14] at the Georgia Institute of

Technology and Jordan and co-workers [30] at Eglin Air Force Base. As shown in figure 11(a),

shockwave velocity calculations agreewell with experimental data in the high-velocity regime,

as differences are less than 8%; differences are somewhat larger (capped at 14%) in the low-

velocity regime. This trend is in agreement with prior observations by Benson [17] for pure Cupowders, where larger deviations in the shock wave speed calculations were observed at lower

particle velocities. As shown in figure 11(b), stationary pressure calculations show excellent

agreement with experimental data in the low-velocity regime, as differences are less than

7%; differences are somewhat larger (capped at 12%) in the high-velocity regime. However,

all stationary pressure calculations in the numerical models for the high-velocity case fall

within the uncertainty bars of the experimental data points (measures of uncertainty were not

quantified for experiments performed in the low-velocity case). The simulation results from




the 20 wt% mixtures (figures 11(c) and (d )) follow the same general trends as those from

the 50 wt% mixtures. Unfortunately, experimental data are not available to enable a critical

evaluation of the results for the 20 wt% mixtures; however, a goal of this research is to predict

the Hugoniot of an arbitrary particle mixture with limited data from experiments.

Despite discrepancies in the U S–U P relation, it is encouraging to observe a lineardependence between the shock wave velocity and particle velocity, as such a relation holds for

many materials. The deviations in calculations of U S and P st are attributed primarily to the 2D

approximation of the microstructure. It is conceivable that a correction could be applied for

this effect of inclusion rendering but may not be necessary depending on objectives.

8. Summary and conclusions

The shock compression of spatially resolved microscale particle systems was simulated for

a set of discrete particle mixtures and imposed particle velocities. Local thermomechanical

responses were observed and Hugoniot data were collected. Particular observations of interest

may be enumerated:

1. Transversely averaged pressure profiles from numerical simulations provide mesoscale

data that agree with macroscale experimental measurements.

2. Centreline temperature profiles from numerical simulations quantify particle-level

temperature fields in the microstructure; such results are not available from experiments.

3. The particulate phases experience insignificant melting; a few small, isolated fragments

are the only volumes that melt in certain load cases. The surfaces of the aluminium

particles fail to melt because the time scales of the simulation are too short to realize

substantial heat conduction.

4. The Hugoniot of the 50 wt% particle mixtures is well characterized by the P –U P and

U S–U P relations calculated in the finite element models; numerical results are in excellent

agreement with experimental data.

A long-term goal of this research is to gain predictive capabilities pertaining to the onset

of local reaction initiation at the mesoscale and the ensuing propagation of reactions at the

macroscale. The local initiation of reactions at reactant interfaces is sensitive to mechanical

conditions (i.e. local states of pressure and shear) and the thermal environment. The methods

established in this work have resolved the thermomechanical responses at the particle level

that will be used in future work to evaluate suitable reaction initiation criteria. Such studies

are beyond the scope of this paper, which has focused primarily on the details of the solution

methodology, constitutive equations, numerical implementation, thermomechanical responses

and Hugoniot calculations.

The results of this work suggest that: (1) costly impact experiments may be partially

replaced by numerical simulations for the efficient characterization of particle mixture

Hugoniots and (2) highly resolved thermomechanical responses can be predicted at the

mesoscale for a range of particle mixtures in exploration of the design space. The predictivecapabilities established here provide opportunities for obvious extension: (1) frameworks may

be established in future work to assess the likelihood of local microscale reaction initiation and

(2) the effects of porosity and particle morphology on predictions of reaction initiation could

be evaluated through parametric sets of simulations. The combination of these numerical

simulations with future experimental work will certainly advance our understanding of the

thermal, mechanical and chemical behaviour of such reactive particle systems, with an aim of

supporting the design of energetic materials.




Acknowledgments

This research was funded by the AFOSR MURI 1606U81 on Multifunctional Energetic

Structural Materials at the Georgia Institute of Technology. RAA gratefully acknowledges

J E Shepherd for his aid in scripting numerical simulations and L Ferranti, N N Thadhani andL J Jordan for private communications of experimental results.

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ryan a austin, david l mcdowell and david j benson- numerical simulation of shock wave propagation...

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