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Computer Methods in Applied Mechanics and Engineering manuscript No.

(will be inserted by the editor)

Domain partitioning material point method for simulating shock in

polycrystalline energetic materials

Ran Ma ·WaiChing Sun ·Catalin R. Picu ·Tommy Sewell

Received: November 19, 2022/ Accepted: date

Abstract

Heterogeneous energetic materials (EMs) subjected to mechanical shock loading exhibit complex

thermo-mechanical processes which are driven by the high temperature, pressure, and strain rate behind

the shock. These lead to spatial energy localization in the microstructure, colloquially known as “hotspots”,

where chemistry may commence possibly culminating in detonation. Shock-induced pore collapse is one of

the dominant mechanisms by which localization occurs. In order to physically predict the shock sensitivity

of energetic materials under these extreme conditions, we formulate a multiplicative crystal plasticity

model with key features inferred from molecular dynamics (MD) simulations. Within the framework of

thermodynamics, we incorporate the pressure dependence of both monoclinic elasticity and critical resolved

shear stress into the crystal plasticity formulation. Other fundamental mechanisms, such as strain hardening

and pressure-dependent melting curves, are all inferred from atomic-scale computations performed across

relevant intervals of pressure and temperature. To handle the extremely large deformation and the evolving

geometry of the self-contact due to pore collapse, we leverage the capabilities of the Material Point Method

(MPM) to track the interface via the Lagrangian motion of material points and the Eulerian residual update

to avoid the mesh distortion issue. This combination of features enables us to simulate the shock-induced

pore collapse and associated hotspot evolution with a more comprehensive physical underpinning, which

we apply to the monoclinic crystal

β

-HMX. Treating MD predictions of the pore collapse as ground truth,

head-to-head validation comparisons between MD and MPM predictions are made for samples with

identical sample geometry and similar boundary conditions, for reverse-ballistic impact speeds ranging

from

0.5 km s−1

to

2.0 km s−1

. Comparative studies are performed to reveal the importance of incorporating

a frictional contact algorithm, pressure-dependent elastic stiffness, and non-Schmid type critical resolved

shear stress in the mesoscale model.

Keywords HMX, Energetic material, Shock, Pore collapse, Material Point Method

1 Introduction

Polymer-bonded explosives (PBXs) are highly ﬁlled composites comprising energetic crystallite ﬁller

ensconced in a continuous polymer matrix. They are widely used in many civil and military applications.

Due to the stored chemical energy of the energetic constituent, these materials may initiate detonation

under unexpected external impact and cause vast damage if not stored or processed properly. Numerical

Ran Ma, WaiChing Sun (corresponding author)

Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, New York

Catalin R. Picu

Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, New York

Tommy Sewell

Department of Chemistry, University of Missouri, Columbia, Missouri

2 Ran Ma et al.

simulation assisted-design of microstructure and shock sensitivity evaluation is important in avoiding such

disasters.

This paper focuses on material point modeling of shock-induced pore collapse in both single-crystal and

polycrystalline energetic materials undergoing extremely large deformation and evolving contacts. In all

numerical examples, the spatial domain is composed of an energetic substance commonly referred to as

β-HMX, a monoclinic polymorph of octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine.

Numerous mesoscale models have been proposed to correlate the shock resistance of energetic materials

under a variety of loading conditions. For example, the statistical crack mechanical (SCRAM) model

[Dienes,1978] was extended with viscoelasticity for the polymer phase to predict the non-shock ignition

and mechanical response of PBX [Bennett et al.,1998]. This model is further extended with frictional heating

of microcracks, melting, ignition, and fast burn for PBX containing microcracks to model the ignition as

well as the following late-stage fast burn [Dienes et al.,2006]. Although most energetic materials are brittle

under atmospheric pressure, considerable ductility is observed under conﬁning pressure [Wiegand et al.,

2011], which is the typical stress state under conﬁned shock loading. Therefore, the SCRAM model is further

extended with viscoplasticity [Yang et al.,2018]. In another independent study, a viscoplasticity model for

tungsten alloy [Zhou et al.,1994] is extended as the constitutive relation for

β

-HMX which, in combination

with the cohesive zone model for fracture and friction [Barua and Zhou,2011], is used to study the ignition

probability of PBX under shock loading [Kim et al.,2018]. Besides these mesoscale models, the Johnson-Cook

viscoplasticity model is also used for energetic materials, for example, PMMA [Rai et al.,2020] and TATB

[Zhao et al.,2020]. With the pressure-dependent melting temperature and viscosity evaluated by atomistic

models [Kroonblawd and Austin,2021], different extensions of the Johnson-Cook model are compared with

the atomistic shock simulation as the benchmark. These material models often assume isotropic constitutive

response and material frame indifference is enforced by an objective stress rate. However, the manufacturing

defect on the order of

10 µm

or smaller is also an important factor that triggers the shock ignition [Barton

et al.,2009]. These defects cannot be observed easily in experiments and therefore corresponding studies rely

mainly on theoretical studies. At these characteristic length scales, the material anisotropy of the embedded

energetic material may have a strong inﬂuence on the predicted shock resistance. Therefore, anisotropic

crystal plasticity models are necessary to precisely capture the microstructural evolution and the resultant

evolving anisotropy at the crystal level.

Previous work, such as [Barton et al.,2009], develops a rate-dependent crystal plasticity model for

energetic material suitable for deformation within a large spectrum of strain rates. In this work, two

dislocation motion resistances, including the thermal activation barrier and the phonon drag limit, are

combined in the constitutive relation representing major strain hardening mechanisms at different strain

rates. This model is further extended with a chemical reaction model [Austin et al.,2015], and shock

simulations with different shock pressure, pore size, thermal conductivity, crystal ﬂow strength, and liquid

viscosity are performed to study their effects on the reactive mass, the shear band morphology, and the peak

temperature. A parametric study of this model shows that the plastic dissipation and the temperature rise

are less sensitive to the anisotropic elastic coefﬁcients than to the plasticity model [Zhang and Oskay,2019],

which is consistent with an independent parametric study using the Johnson-Cook model [Das et al.,2021].

In this work, the interpenetration of crystals can be prevented via the level set function that represents the

interface. However, since there is only one velocity ﬁeld deﬁned across the interface, the contacts of the

collapsed region of the pores are essentially glued together without any tangential slip.

The mesoscale crystal plasticity models could be a feasible alternative that allows direct validation

inferred from underlying physical observations (e.g. the plastic slip of a slip system), and may be easily

extended by incorporating other deformation mechanisms such as fracture and deformation twinning

[Clayton and Knap,2011,Ma and Sun,2021]. For instance, combining a pressure-dependent thermoelasticity

free energy of RDX [Austin and McDowell,2011] and a dislocation-density based crystal plasticity model

suitable for high strain rates [Austin and McDowell,2011], [Luscher et al.,2017] introduced a mesoscale

material model for crystalline RDX with further validation against an impact experiment. The crystal

plasticity ﬁnite element model is further compared with the Knoop indentation test to determine the

active slip systems in

β

-HMX [Zecevic et al.,2021]. Recently, phase-ﬁeld based fracture models [Grilli

and Koslowski,2019], twinning models with explicit representation [Ma and Sun,2021] and implicit

representation [Zhang and Oskay,2019,Zecevic et al.,2020] are also combined with mesoscale crystal

plasticity models for more physical insights into the shock responses observed in energetic materials.

Domain partitioning MPM for simulating shock in energetic materials 3

Nevertheless, introducing high-ﬁdelity simulations may unavoidably require a substantial amount of

material parameters. While a subset of these parameters could be physically associated with dislocation

theories, the rest of these parameters may lack physical underpinning. Calibrating these parameters is

usually achieved through solving inverse problems to reproduce the experimentally measured shock

velocity [Dick et al.,2004]. The lack of justiﬁcations from underlying physics might lead to overﬁtting and

hence increase the difﬁculty of generating reliable simulations.

In Ma et al. [2021], an attempt is made to incorporate atomistic simulation results to calibrate a small-

deformation non-Schmid crystal plasticity model. The result suggests that incorporating the pressure

sensitivity into both the elasticity and the yield function for the slip system may lead to a mesoscale model

more compatible with the atomistic counterpart. However, the geometrical nonlinearity, the evolving contact

kinematics, the constitutive responses of the crystal interfaces, and the pressure dependence of the melting

curves that govern the phase transition from solid to liquid have not yet been captured.

The objective of this paper is to leverage the salient features of the material point method to capture

evolving geometry such that a more realistic multiplicative mesoscale plasticity model for the bulk and

interfaces of energetic materials can be used to replicate the contact mechanics of the pore collapses and

ultimately predict the hotspot formation, a crucial mechanism that triggers an explosion. In most of the

hydrocodes for explosion simulations (e.g., Pierazzo and Melosh [2000], Sambasivan et al. [2013], Mudalige

et al. [2014]), a single velocity ﬁeld is used for contacting bodies where the surface friction associated with

the pore collapse process is ignored. This simpliﬁcation may lead to under-evaluated hotspot temperature

due to the negligence of frictional heating, especially for irregularly shaped embedding defects. In particular,

the ﬁeld-gradient domain partition treatment in the material point method (MPM) enables us to capture the

evolving contact geometry among multiple bodies as well as self-contacts. As demonstrated in our numerical

examples on both single- and poly-crystal simulations and the comparisons with the MD simulations, this

improvement is crucial for us to accurately simulate the secondary shock due to the pore collapse.

This paper will proceed as follows. Section 2describes in detail the atomistic-model informed crystal

plasticity model. The stress update algorithm and its implementation in the material point method are also

described in detail. Section 3presents the parameter determination procedure using the atomistic results.

Section 4presents the validation against the atomistic-scale shock simulations. In Section 5, a comparative

study is performed on the importance of the frictional contact algorithm, the pressure-dependent hyperelas-

ticity, and the non-Schmid crystal plasticity within the mesoscale model. Section 6summarizes the major

results and concluding remarks.

2 Constitutive model

We start with the multiplicative decomposition theory, where the deformation gradient

F

is decomposed

into the elastic part Feand the plastic part Fpas

F=FeFp. (1)

Taking

v

as the spatial velocity vector, the spatial velocity gradient

L=grad v

, which is power conjugate to

the Kirchhoff stress τ, is decomposed into the elastic part and the plastic part as:

L=˙

FF−1=Le+FeLpFe−1=˙

FeFe−1+Fe˙

FpFp−1Fe−1, (2)

where Leis the elastic velocity gradient, and Lpis the plastic velocity gradient.

We further deﬁne the elastic right Cauchy-Green deformation tensor

Ce

and the elastic Green-Lagrangian

deformation tensor Eeto measure the elastic deformation:

Ce=FeT Fe,Ee=1

2(Ce−I). (3)

Then, the stress Seis deﬁned as the contravariant pull-back of the Kirchhoff stress τ[Anand,2004],

Se=Fe−1τFe−T, (4)

which is power-conjugate to the elastic deformation tensor Ee, that is, Se:˙

Ee=τ: sym[Le].

4 Ran Ma et al.

A general form of the total free energy function per unit mass takes the form

ψ=ψ(Fe,γ,T), (5)

where

γ

represents a collection of internal variables, and

T

is the absolute temperature. The total free energy

consists of three parts: the elastic strain energy due to the reversible elastic deformation, the stored energy

due to the accumulation of crystal defects, and the thermal energy. In order to further simplify the model

development, we assume that the elastic free energy is temperature independent, which means that the

elastic stiffness is independent of the temperature and the thermal expansion is neglected. This assumption

is justiﬁed by atomistic results indicating that the temperature dependence of the elastic stiffness is much

weaker than the pressure dependence [Pereverzev and Sewell,2020]. We neglect the strain energy associated

with dislocation storage, as customary in constitutive modeling of plasticity. Therefore, a simpliﬁed form of

the free energy function is assumed as:

ψ=ψe(Ee) + ψT(T), (6)

where the speciﬁc form of the free energy function is discussed in detail in the following Sections.

2.1 Atomistically-informed hyperelasticity

Traditionally, when modeling the thermo-mechanical response of energetic material under strong shock,

the pressure-dependence of the elasticity is modeled through the equation-of-state (EOS), which relates

the pressure to the volumetric strain, while keeping the anisotropy constant [Barton et al.,2009]. The

Mie-Gr

¨

uneisen EOS is typically used for

β

-HMX, which is calibrated by the atomistic-scale simulations at

various pressure and temperature [Menikoff and Sewell,2002]. However, due to the monoclinic symmetry

of the

β

-HMX single crystal, the volumetric part and the deviatoric part of the elastic free energy cannot

be fully decoupled, so the pressure depends on the volumetric strain as well as the deviatoric strain. Also,

the shear components of the elastic stiffness are pressure-dependent as well [Pereverzev and Sewell,2020],

which is usually neglected in most mesoscale models.

Taking into account the aforementioned discussions, we develop a nonlinear hyperelasticity model

to reproduce the pressure-dependent elastic stiffness observed in atomistic evaluations [Pereverzev and

Sewell,2020]. Departing from the inﬁnitesimal strain counterpart [Ma et al.,2021], we generalize this model

to the ﬁnite strain formulation. The geometric nonlinearity of the ﬁnite strain formulation is also considered

in the calibration procedures, which will be introduced in Section 3.

The

β

-HMX unit cell is deﬁned in the

P

2

1/n

space group throughout this paper, as shown in Figure

1. The covariant basis vectors

M1

,

M2

, and

M3

are deﬁned to represent the three lattice vectors of the

monoclinic unit cell. Taking

Ee

as the elastic strain measure, we deﬁne three isotropic and four anisotropic

strain invariants:

I1=tr Ee,I2=1

2tr[Ee]2−tr[Ee2],I3=det Ee,

I4=M1·Ee·M1,I5=M2·Ee·M2,I6=M3·Ee·M3,I7=M1·Ee·M3.

(7)

Note that these strain invariants form a subset of the general set of eight strain invariants for materials with

monoclinic symmetry [Vergori et al.,2013]. The elastic stiffness at

0 GPa

, as well as these strain invariants,

are utilized to construct the elastic free energy.

Based on the atomistic-scale evaluations where the elastic stiffness coefﬁcients are estimated within a

set of pressures between

10−4GPa

and

30 GPa

and temperatures between

300 K

and

1100 K

[Pereverzev

and Sewell,2020], the elastic stiffness increases substantially with pressure while decreases mildly with

increasing temperature. This justiﬁes the approximation made here that the elastic free energy is temperature

independent. We propose the following form of elastic free energy as

ρ0ψe=f(I1,I2,I3,I4,I5,I6,I7)Ee:Ce

0:Ee, (8)

where

Ce

0

is the elastic stiffness at atmospheric pressure, function

f

is an arbitrary function that fulﬁlls typical

stability requirements, and ρ0is the initial density of the energetic material under ambient conditions.

Domain partitioning MPM for simulating shock in energetic materials 5

α

γ

β

M1 = [100]

M2= [010]

M3M3= [001]

a

b

c

c*

e1e2

e3

Fig. 1: Monoclinic unit cell of the

β

-HMX crystal in

P

2

1/n

space group. The lattice constants are

a=6.53 ˚

A

,

b=11.03 ˚

A

,

c=7.35 ˚

A

,

α=γ=90◦

, and

β=102.689◦

(at

295 K

) [Eiland and Pepinsky,1954]. The vectors

e1

,

e2

, and

e3

denote the basis vectors of the global Cartesian coordinate system. The vectors

M1

,

M2

, and

M3indicate the coordinate system of the monoclinic crystal.

One advantage of the speciﬁc form of the elastic free energy in Equation

(8)

is that the elastic stiffness at

atmospheric pressure is exactly reproduced. This property signiﬁcantly simpliﬁes the determination of the

arbitrary function

f

and the calibration process, as discussed in Section 3. Also, to improve the ﬁdelity of

the predicted thermo-mechanical response, the temperature dependence of the elastic free energy should be

considered.

2.2 Atomistic-model informed crystal plasticity

Due to the large strain rates involved in the shock simulations, crystal plasticity models applicable to a wide

range of strain rates are generally used where the thermal activation barrier and the phonon drag limit are

typically considered in the model [Barton et al.,2009,Luscher et al.,2017]. These models are calibrated and

validated against shock experiments with different single-crystal orientations, specimen thicknesses, and

ﬂyer velocities for HMX [Dick et al.,2004], RDX [Hooks et al.,2006], and other energetic materials.

These dislocation-based crystal plasticity models, though based on homogenized dislocation theories,

are validated only against macroscale velocity measurements from shock experiments and lack further

validations from the underlying physics at the atomistic scale. Also, some key properties observed in the

atomistic scale models, which may have strong inﬂuences on the predicted hotspot evolution, are not fully

accounted for. For example, it is observed in a recent atomistic scale study that the Peierls-Nabarro stress of

β

-HMX increases

5

to

50

times depending on the slip system when the conﬁning pressure increases from

10−4GPa to 27 GPa at 300K [Pal and Picu,2019].

In this section, we develop a crystal plasticity model where the pressure dependence of the critical

resolved shear stress (CRSS) is accounted for. However, since limited atomistic-scale information regarding

the strain rate sensitivity of this crystal is available at this time, we restrict our crystal plasticity model to be

rate-independent. We acknowledge that, to improve the ﬁdelity of our model, the strain-rate dependence

should be accounted for; this would also simplify the implementation as well as improve the numerical

robustness. However, sufﬁcient data from MD simulations to calibrate the rate sensitivity was not available

to us within the scope of this research and hence this will be considered in the future.

This multiplicative crystal plasticity is formulated via the large-deformation crystal plasticity framework

ﬁrst introduced in Anand and Kothari [1996]. In the original framework [Anand and Kothari,1996], the

pseudo-inverse of the Jacobian of the active yield criterion functions is approximated through singular-value

decomposition (SVD), and slip systems that violate the consistency condition are eliminated from the

potential active set. Miehe and Schr

¨

oder (2001) further improved this algorithm and compared different

pseudo-inverse approaches to approximate the pseudo-inverse of the Jacobian [Miehe and Schr

¨

oder,2001].

Considering the strongly anisotropic nature of energetic materials with both pressure sensitivity and

temperature sensitivity, we generalize the SVD-based slip system selection algorithm such that it is applicable

to any hyperelasticity model coupled with the non-Schmid slip system activation rule.

Let

Nslip

represents the total number of slip systems in the crystal. In the intermediate conﬁguration,

we deﬁne

s(α)

0

and

m(α)

0(α=

1,

. . .

,

Nslip)

as the slip direction and the slip plane normal of slip system

(α)

.

6 Ran Ma et al.

Following the standard ﬂow rule of crystal plasticity, the plastic velocity gradient

Lp

is decomposed into the

summation of plastic slip rates on each slip system:

Lp=

2Nslip

∑

α=1

˙

γ(α)s(α)

0⊗m(α)

0, (9)

where

˙

γ(α)

is the plastic shear strain rate of slip system

(α)

. Note that the summation is performed over

2

Nslip

slip systems where each slip system is counted twice with opposite slip directions. This is important

for the rate-independent crystal plasticity model, where the plastic multiplier

˙

γ(α)

is non-negative to ensure

that the Karush-Kuhn-Tucker (KKT) conditions in Equation (13) are fulﬁlled.

We further introduce the Mandel stress Ξas

Ξ=CeSe=FeT τFe−T. (10)

Then, the resolved shear stress τ(α)is further derived using the Schmid law,

τ(α)=Ξ:s(α)

0⊗m(α)

0,α=1, . . . , 2Nslip . (11)

Note that the Mandel stress

Ξ

is power-conjugate to the plastic velocity gradient

Lp

, and the resolved shear

stress

τ(α)

is power-conjugate to the plastic slip rate

γ(α)

, so that the plastic work per unit volume

Wp

reads:

Wp=τ:FeLpFe−1=Ξ:Lp=

2Nslip

∑

α=1

τ(α)˙

γ(α)(12)

The slip system constitutive relation follows the KKT conditions, that is, the plastic shear strain is

non-negative, the yielding function is non-positive, and the consistency condition applies:

˙

γ(α)≥0, φ(α)=hτ(α)−g(α)i≤0, ˙

γ(α)φ(α)=0, α=1, . . . , 2 Nslip , (13)

where φ(α)and g(α)are the yielding function and the slip system resistance of slip system (α).

The slip system resistance (CRSS) has two components: the Peierls-Nabarro stress representing the lattice

resistance and the strain hardening associated with the interaction of dislocations. The atomistic scale results

suggest that the Peierls-Nabarro stress is strongly pressure-dependent [Pal and Picu,2019] and weakly

temperature-dependent [Khan et al.,2018]. We assume that the temperature effect and the pressure effect

are separable. The strain hardening component is a function of the dislocation density and was studied in

ambient conditions using atomistic models [Khan and Picu,2021]. We represent the slip resistance g(α)as:

gα=gα

h(γ)

1− T−Tref

Tm(p)−Tr e f !Mh

+gα

p(p)

1− T−Tref

Tm(p)−Tr e f !Mp

,α=1, . . . , 2 Nslip , (14)

where

T

is the absolute temperature,

Tm(p)

is the pressure-dependent melting temperature,

Tref =298 K

is

the reference temperature,

gα

p(p)

represents the Peierls-Nabarro stress at the reference temperature,

gα

h(γ)

represents the strain hardening component at the reference temperature, and

Mh

and

Mp

are material

constants. The temperature should have different softening effects on the Peierls-Nabarro stress

g(α)

p

and the

hardening component

g(α)

h

, since the physical basis of the temperature dependence of these two terms are

different. Therefore, two thermal softening coefﬁcients Mpand Mhare introduced separately.

The pressure-dependent lattice friction (Peierls-Nabarro stress), which produces the non-Schmid type

yielding criterion, could be deﬁned in multiple different approaches. One possibility is to deﬁne the pressure

as the surface compression applied on the slip plane, that is

p(α)=m(α)

0·Se·m(α)

0

. However, we adopt a

different approach where the pressure is deﬁned as the hydrostatic stress

p=−tr[σ]/

3. This approach is

consistent with the corresponding atomistic models where the pressure-dependent Peierls-Nabarro stress

Domain partitioning MPM for simulating shock in energetic materials 7

is evaluated [Pal and Picu,2019]. A polynomial model is used to approximate the pressure-dependent

Peierls-Nabarro stress g(α)

p(p)as

g(α)

p(p) = (c(α)

1p2+c(α)

2p+c(α)

3,p>0

c(α)

3,p≤0α=1, 2, . . . , 2 Nslip , (15)

where c(α)

1to c(α)

3are material constants calibrated from the atomistic models.

The strain hardening contribution to the ﬂow stress was evaluated in [Khan and Picu,2021] using a

combination of atomistic and mesoscale models, and the corresponding ﬂow stress at dislocation densities

ranging from

1011 m−2

to

1015 m−2

was evaluated. Since the dislocation density

ρ=1015 m−2

is large

compared with the initial dislocation density and is hardly achievable in shock simulations, we take the

slip system resistance at

1015 m−2

as the saturation stress. Therefore, the strain hardening contribution

gα

h

is

approximated as

g(α)

h(γ) = g(α)

h∞−c(α)

3tanh H0γ

g(α)

h∞−c(α)

3!,γ=

2Nslip

∑

α=1Zt

0|˙

γ(α)|dt, (16)

where

H0

is the initial hardening rate, and

g(α)

h∞

is the saturation stress of the slip system

(α)

evaluated at a

dislocation density of 1015 m−2and in ambient conditions.

The melting temperature

Tm

also inﬂuences the Peierls-Nabarro stress and strain hardening according

to the atomistic model, as shown in Equation

(14)

. It is also a function of the conﬁning pressure

p

based on

the atomistic models, and is approximated by the Simon-Glatzel relation [Kroonblawd and Austin,2021],

which takes the following form:

Tm(p) = Tm01+p−pre f

a01/c0

, (17)

where Tm0is the melting point at pre f =0 GPa, and a0and c0are material constants.

The crystal plasticity model is replaced by a Newtonian ﬂow model when the temperature

T

reaches

the melting point

Tm

. Although temperature and pressure-dependent viscosity is suggested by atomistic

models [Kroonblawd and Austin,2021], a constant viscosity is used in this paper:

dev[σ] = 2ηdev[D], (18)

where

η

is the ﬂuid viscosity,

D=sym[L]

is the symmetric part of the velocity gradient, and

σ=τ/J

is the

Cauchy stress. Also note that we do not model the solidiﬁcation process and the chemical reaction in this

paper, since our primary focus is on the triggering mechanisms of the detonation but not the physics in the

post-detonation regime,

2.3 Thermodynamic consistency

To accurately capture the thermo-mechanical behavior of energetic materials under shock loading, our next

goal is to ensure that the proposed mesoscale crystal plasticity model fulﬁlls the thermodynamic consistency.

We restrict our focus to the shock wave propagation within orders of

ps

. At this temporal regime, it is

reasonable to assume that the material is under adiabatic conditions, and hence the heat ﬂux and heat source

can be ignored.

The local form of the ﬁrst law of thermodynamics (energy balance) is

ρ0˙

e=τ:D=Se:˙

Ee+

Nslip

∑

α=1

τ(α)˙

γα, (19)

8 Ran Ma et al.

where

e

is the internal energy per unit mass. The second law of thermodynamics (Clausius-Duhem inequal-

ity) requires that the changing rate of entropy is non-negative under adiabatic conditions. Taking advantage

of the Legendre transformation ψ=e−TS, the Clausius-Duhem inequality is equivalently derived as:

Dint =ρ0T˙

S=Se−ρ0

∂ψ

∂Ee:˙

Ee+

Nslip

∑

α=1

τ(α)˙

γα−ρ0S+∂ψ

∂T˙

T≥0. (20)

where Sis the entropy per unit mass. Then, the Coleman-Noll conditions read,

Se=ρ0

∂ψ

∂Ee,S=−∂ψ

∂T, (21)

which combines with enforcing the KKT conditions Equation

(13)

, guarantee that the dissipation inequality

is always fulﬁlled.

Due to the lack of the thermal free energy

ψT(T)

justiﬁed by the atomistic scale models, we instead

employ a phenomenological model to characterize the temperature evolution based on the corresponding

atomistic evaluations [Menikoff and Sewell,2002]:

T=Tref exp "Ta1

J−12

sgn 1

J−1#+∑2Nslip

α=1τ(α)˙

γ(α)

ρcv, (22)

where

J=det[F]

is the volumetric strain,

cv

is the isochoric speciﬁc heat, and

Ta=

4.5 is a manually

adjusted material parameter to reproduce the Hugoniot relations. A temperature-dependent speciﬁc heat is

used [Sewell and Menikoff,2004],

cv=˜

T3

cv0+cv1˜

T3+cv2˜

T3+cv3˜

T3(23)

where

cv0

to

cv3

are material parameters. The absolute temperature

T

is normalized by the Debye temperature

θ(J)as

˜

T=T

θ(J),θ(J) = θ0J−Γaexp Γb1

J−1, (24)

where θ0,Γa, and Γbare material parameters.

2.4 Stress update algorithm

We developed an active slip system selection algorithm for the aforementioned rate-independent crystal

plasticity model based on the stress update procedure initially proposed by [Anand and Kothari,1996] and

later improved by [Miehe and Schr

¨

oder,2001]. Our key contribution is the improvement of this algorithm

to accommodate a general hyperelastic model and a non-Schmid slip system activation rule with pressure

sensitivity and temperature sensitivity.

The time integration is semi-implicit in the sense that the temperature of the last converged step

Tn

is

used to calculate the thermal-softening coefﬁcient. Since the time step required for the shock simulation

is very small, the critical time step to maintain stability for this semi-implicit treatment is likely to be

signiﬁcantly larger than the actual time step used. Hence, it is reasonable to assume that the semi-implicit

time integration algorithm may remain stable.

The internal variable

g(α)

, which is deﬁned in Equation

(14)

, is a function of the accumulated shear

strain, pressure, and temperature. In order to simplify the return mapping algorithm, the trial pressure

˜

p

is

used to compute the pressure-dependent yield stress

g(α)

p

and the melting temperature

Tm

. This pressure

is corrected at the end of the stress update algorithm in Step 9, followed by a repeating time integration

with the corrected pressure, internal variables, and the melting temperature. Only one additional iteration

is normally required to reach a converged pressure because of the small time steps usually used in shock

simulations.

Domain partitioning MPM for simulating shock in energetic materials 9

When determining the resolved shear stress

τ(α)

, one may assume that the elastic stretch is small

compared with the plastic strain such that

τ(α)

approximately equals to

Se:s(α)

0⊗m(α)

0

[Anand and

Kothari,1996]. However, in the shock simulation, the volume change

J=det F

is not negligible, while

the volume-preserving part of the elastic deformation gradient

¯

Fe=J−1/3 F

is comparably less signiﬁcant.

Therefore, we introduce the following approximation to the resolved shear stress τ(α):

τ(α)=Ξ:s(α)

0⊗m(α)

0=[CeSe]:s(α)

0⊗m(α)

0≈J2/3Se:s(α)

0⊗m(α)

0,α=1, . . . , Nslip .

Note that the volume change Jis constant within one time step.

The Newton-Raphson method is used to solve this system of nonlinear equations. When the active set

A

contains redundant slip systems, the slip system increments cannot be uniquely determined, and the

linearization of the residual produces a

n×n

singular matrix. Upon the detection of a singular lineariza-

tion matrix, the singular value decomposition (SVD) strategy is used to approximate the inverse of the

linearization matrix [Miehe and Schr¨

oder,2001].

In the following stress update algorithm, the subscript

n

denotes variables evaluated at time step

tn

which is assumed to be known a prior.

Given: Fn+1,Fn,Fp

n,γn,An,Tn,

Find: σn+1,Fp

n+1,γn+1,An+1,Tn+1.

Here, the active set

A

is a subset of the 2

Nslip

slip systems. The slip systems within

A

is active, that is,

the shear strain rate

˙

γ(α)>

0 and the yield criterion

φ(α)=hτ(α)−g(α)i=

0 for slip system

α∈ A

, as

demonstrated in the KKT condition in Equation (13).

Step 1. Compute the trial elastic states

Trial elastic strain: Fe

tr =Fn+1Fp−1

n,Ce

tr =Fe T

tr Fe

tr,Ee

tr = (Ce

tr −I)/2, J=det[Fn+1],

Trial stress: Se

tr =ρ0(∂ψ/∂Ee

tr),σtr = (1/J)Fe

trSe

tr Fe T

tr ,˜

p=−tr[σtr ]/3,

Trial internal variable: g(α)

tr =g(γn,Tn,˜

p).

Step 2. Elastic predictor

Assemble trial active set Atr ={α|φ(α)=τ(α)

tr −g(α)

tr >0}

If Atr =∅, then

set σn+1=σtr,Fp

n+1=Fp

n,γn+1=γn,An+1=∅

go to Step 9

Otherwise, An+1=An,∆γ(α)=0.

Step 3. Stress and stiffness update

Fp

n+1=hI+∑An+1∆γ(α)s(α)

0⊗m(α)

0iFp

n,Fe=Fn+1Fp−1

n+1,Ee= (FeFeT −I)/2,

Se=ρ0(∂ψ/∂Ee),CSE

IJK L =ρ0[∂2ψe/(∂Ee

I J ∂Ee

KL )],γn+1=γn+∑An+1∆γ(α).

Step 4. Compute residual and Jacobian for active slip systems

R(α)=τ(α)−g(α),α∈ An+1,

D(αβ)=S(α)

0:CSE :Ce

trS(β)

0+H0

1− Tn−Tr e f

Tm−Tr e f !Mh

"1−tanh2 H0γn+1

g(α)

h∞−c(α)

3!#,

with τ(α)≈J2/3Se:s(α)

0⊗m(α)

0and g(α)=g(α)(γn+1,Tn,˜

p)is deﬁned in Equation (14).

Step 5. Update incremental plastic slip and check convergence

If Dis singular, then compute its pseudo-inverse ¯

D−1=V¯

Σ−1UT

Otherwise, compute its regular inverse ¯

D−1=D−1

∆γ(α)←∆γ(α)+∑β∈An+1(¯

D−1)(αβ)R(α)

If q∑α∈An+1[R(α)]2>tol go to Step 3.

10 Ran Ma et al.

Step 6. Active set update I: Drop inactive slip systems

α=arg min hφ(α)i,α∈ An+1

If ∆γ(α)<0: Update active set An+1← An+1/{α}and go to Step 3.

Step 7. Active set update II: Add potential active slip systems

α=arg max hφ(α)i,α/∈ An+1

If φ(α)>0: Update active set An+1← An+1∪ {α}and go to Step 3.

Step 8. Plastic deformation gradient and stress

Fp

n+1=hI+∑An+1∆γ(α)s(α)

0⊗m(α)

0iFp

n,Fe=Fn+1Fp−1

n+1,Ee= (FeFeT −I)/2,

Se=ρ0(∂ψ/∂Ee),σn+1= (1/ J)FeSeFe T ,γn+1=γn+∑An+1∆γ(α).

Step 9. Update pressure and temperature

Pressure p=−tr[σ]/3, temperature Tn+1=T(J, dissipation)

If |p−˜

p|>tol: Set ˜

p=pand go to Step 2

Otherwise, exit

2.5 Material point method for shock simulations

The shock wave propagation problem is solved by the material point method (MPM) based on the following

considerations. First, the material point method is suitable to treat the challenging numerical issues involved

in the shock wave propagation problems, including the local large deformation associated with the inter-

action between the shock wave and the pore, and the frictional contact associated with the pore collapse

process. Furthermore, as the Lagrangian material particles store the deformation history and the internal

variables, this historical information can be easily tracked through the trajectory of the material points,

which is a property also shared by other meshfree methods, such as the reproducing kernel particle method

(RKPM) [Wang et al.,2014] and the smoothed particle hydrodynamics (SPH) [Liu and Liu,2010]. This treat-

ment greatly simpliﬁes the implementation of the path-dependent models formulated with multiplicative

kinematics (such as the crystal plasticity used in this paper) with profound geometrical nonlinearity [Liu

and Sun,2020a,b,Ma and Sun,2022]. The MPM-based shock wave simulation, which is regularized by the

artiﬁcial viscosity, is veriﬁed against the analytical solutions of one-dimensional Riemann problems [Ma

et al.,2009], and was further applied to model the blast and fragmentation of concrete walls [Hu and Chen,

2006]. The MPM could also be extended with a phase-ﬁeld fracture model to study the dynamic fracture

behavior under large distortion [Kakouris and Triantafyllou,2017].

In the following numerical examples, our focus is on predicting the temperature and pressure ﬁeld

evolution as a result of the primary shock wave induced by impact and of the secondary shock wave due

to the pore collapse. So the duration of the simulations is not tremendously larger than the CFL stability

condition, and as a result, we use the MPM formulation with semi-implicit time integration. This time

integration algorithm is conditionally stable, so a sufﬁciently small time step is picked to fulﬁll the CFL

stability condition. The B-spline function is used as the shape function of the background mesh to avoid

cell-crossing instability [Steffen et al.,2008]. The MPM implementation uses TaiChi [Hu et al.,2019], an

open-source programming language designed for high-performance computing in the Python environment.

2.5.1 Classical material point method

Consider a deformable body occupying region

Ω

in the Euclidean space. The boundary

∂Ω

is divided into

the Dirichlet boundary

∂Ωu

and Neumann boundary

∂Ωt

. Taking

b

as the body force per unit mass, the

strong form and weak form of the balance law of linear momentum is expressed as:

ρ˙v=div σ+ρb,

ZΩρw·˙vdV=ZΩ(ρw·b−grad w:σ)dV+Z∂Ωt

w·tdS,

where w∈H1

Ωis the test function and t=σ·nis the surface traction.

Domain partitioning MPM for simulating shock in energetic materials 11

In the classical material point method, the domain

Ω

is discretized into particles, each representing a

simply connected subdomain of

Ω

. Each particle

p

carries the mass

mp

, velocity

vp

, deformation gradient

Fp

,

and other internal variables of the subdomain it represents. The algorithm for solving the initial-boundary

value problem is brieﬂy summarized as follows, which serves as the starting point of our modiﬁcation to

capture pore collapse induced secondary shock with frictional contact.

Step 1. Particle-to-grid projection

. Mass and linear momentum are transferred from particles to grids as

mn

i=∑pwip mp

and

(mv)n

i=∑pwip mpvn

p

, where

mn

i

and

mp

represent the mass of grid

i

and

particle

p

,

(mv)n

i

and

mpvn

p

represent the velocity of grid

i

and particle

p

, and

wip

represents the

value of the ith b-spline shape function evaluated at the position of particle p.

Step 2. Solve grid velocity

. The internal force is assembled as

fi=∑pgrad wip ·σpJpVp

, where

σp

is

the Cauchy stress of particle

p

,

Jp

is the Jacobian of the deformation gradient, and

Vp

is the initial

volume of particle p. The grid momentum is updated explicitly as (mv)n+1

i=(mv)n

i−∆tfi.

Step 3. Grid-to-particle projection

. Particle velocity and its spatial gradient is then projected from the

background mesh as: particle velocity

vn+1

p=∑iwip vn+1

i

and spatial velocity gradient

Ln+1

p=

∑ivn+1

i⊗grad wip .

Step 4. Convection

. The current particle position

xn+1

p

and its deformation gradient

Fn+1

p

are then updated

as: xn+1

p=xn

p+∆tvn+1

pand Fn+1

p= (I+∆tLn+1

p)Fn

p.

2.5.2 Self contact with frictional sliding

When evaluating the shock sensitivity of energetic materials using mesoscale simulations, it is inevitable to

model frictional contact associated with the free surfaces embedded in the material, including micro-crack

surfaces and pore surfaces. Under the shock wave pressure, the local impact and friction of these free

surfaces are also key contributions to the hotspot formation. In the classical hydrocode or material point

method where a single velocity ﬁeld is used, two pieces of materials are welded together upon contact and

frictional sliding is ignored. Therefore, the hot spot temperature is underestimated as a result of ignoring the

frictional heating, especially for irregularly shaped pores. In order to model the self contact with frictional

sliding, we utilize the damage-ﬁeld gradient (DFG) algorithm [Homel and Herbold,2017], with speciﬁc

adjustments summarized below.

When discretizing the initial conﬁguration of the specimen into material particles, one layer of surface

particles is extracted and assigned the surface indicator, with the layer thickness equal to the element size of

the background mesh. The gradient of the surface indicator is used to separate two groups of contacting

particles associated with one grid of the background mesh, such that the multi-body contact algorithm

[Huang et al.,2011,Xiao et al.,2021] can be applied. The Coulomb friction model is used throughout this

paper, and the friction coefﬁcient

µ

takes a constant value

0.25

based on the experimental measurements of

multiple energetic materials (including HMX, RDX, and PETN) at high pressure [Wu and Huang,2010].

Unlike the DFG algorithm where the damage ﬁeld is used as an additional surface separation indicator, we

limit our mesoscale model within the continuum mechanics range, and the fracture and associated surface

friction will be pursued in future work. We also assume that the plastic dissipation associated with pore

collapse is much larger than the pore surface friction, such that the temperature increase due to the surface

friction is negligible.

Preparation:

Deﬁne surface indicator

ζp

for each particle, such that

ζp=

0 for interior particles and

ζp=

1

for particles within the surface region. Note that the surface indicator is only used to separate contacting

bodies instead of computing the surface normal direction.

Step 1. Reset surface-indicator gradient of particles

. Assemble the surface indicator from particles to grid

as

ζi= (∑pwipmpζp)/(∑pwi p mp)

. The surface-indicator gradient at particle

p

is then computed as

∇ζp=∑iζigrad wip

.Note that instead of using the spherically symmetric cubic kernel functions to

approximate the continuous damage ﬁeld [Homel and Herbold,2017], we use the B-spline shape

functions and the classical MPM approach which also avoids the edge effect and produces C1

continuous damage ﬁeld.

12 Ran Ma et al.

Step 2. Reset surface-indicator gradient of grids

. Search within the compact set of the kernel function

of grid

i

for the particle with maximum surface-indicator gradient, and assign it to grid

i

:

∇ζi=

arg max(wip >0)k∇ζpk.

Step 3. Partitioning

. For each grid point

i

, if the particle set

{p|wip ∇ζp· ∇ζi<

0

}

is not empty, then

grid point

i

is within the contact region, and particle quantities are assembled to one of the two

background meshes denoted by ς∈ {0, 1}, where ς=0 if ∇ζp· ∇ζi≥0 and ς=1 otherwise.

Step 4. Particle-to-grid projection

. Based on the partitioning indicator

ς

, mass and linear momentum are

assembled from particles to grids as

mn

ςi=∑pwip mp

and

(mv)n

ςi=∑pwip mpvn

p

. The internal force

is also assembled as

fςi=∑pgrad wip ·σpJpVp

. The surface indicators at grid

i

are deﬁned as

ζςi= (∑pwipmpζp)/(∑pwi p mp).

Step 5. Check separability condition

. The separability condition is deﬁned as

ζςi>ζmin =

0.5,

∀ς∈ {

0, 1

}

.

If grid

i

is not separable, the contact force is determined to enforce the continuity of the velocity

ﬁeld:

fc

0i=−fc

1i=[m0i(mv)1i−m1i(mv)0i]/(m0i+m1i)

. Otherwise, the frictional contact force is

determined through the Coulomb model.

Step 6. Contact force

. The normal directions of the two contacting bodies are

ˆnςi=∑pmpgrad wip /

k∑pmpgrad wip k

. Consider that the two normal directions may not be parallel, a corrected

normal direction is deﬁned as

n0i=−n1i=(ˆn0i−ˆn1i)/kˆn0i−ˆn1ik

. Deﬁne

ˆ

f0i=−ˆ

f1i=

[m0i(mv)1i−m1i(mv)0i]/(m0i+m1i)

, where the normal component

ˆ

fn

ςi=ˆ

fςi·nςinςi

and the

shear component

ˆ

ft

ςi=ˆ

f0i−ˆ

fn

ςi

. If grid

i

is separable and the two contacting bodies are penetrating

each other, that is,

[m1i(mv)0i−m0i(mv)1i]·n0i>

0, the actual contacting force, taking into account

the friction, is then fc

ςi=ˆ

fn

ςi+min µkˆ

fn

ςik,kˆ

ft

ςikˆ

ft

ςi/kˆ

ft

ςik.

Step 7. Solve grid velocity. The grid velocity is updated explicitly as (mv)n+1

ςi=(mv)n

ςi−∆tfςi+fc

ςi.

Step 8. Grid-to-particle projection

. Based on the partitioning indicator

ς

, grid velocity and its spatial

gradient are then projected to the particles as: vn+1

p=∑iwip vn+1

ςiand Ln+1

p=∑ivn+1

ςi⊗grad wip .

Step 9. Convection. The convection part is the same as the classical material point methods.

2.5.3 Artiﬁcial viscosity

Artiﬁcial viscosity is commonly used to stabilize the shock wave propagation and suppress the spurious

oscillation behind the wavefront [Benson,1991,Mattsson and Rider,2015]:

Sv=c0ρ0

˙

J|˙

J|

J2h2I+c1ρ0cs

˙

J

JhI, (25)

where

c0

and

c1

are material constants,

cs

is the bulk sound speed, and

h

is the characteristic length scale

of the MPM particles. The second-order viscosity term smears the stress discontinuity but introduces

high-frequency oscillation, while the ﬁrst-order viscosity term is helpful in suppressing this oscillation. By

introducing the artiﬁcial viscosity, the discontinuous stress wave is smeared into a continuous function, and

the width of the transition zone is restrained within several ﬁnite elements. Note that the artiﬁcial viscosity

has a negligible inﬂuence on the results outside the transition zone, and the Hugoniot jump condition is still

fulﬁlled.

3 Effective material properties inferred from molecular dynamics simulations

In this section, results from multiple types of molecular dynamics simulations are interpreted such that

they can be systematically incorporated into the crystal plasticity model in Section 2in the calibration and

validation process.

Speciﬁcally, the hyperelasticity model is calibrated against the elastic moduli evaluated by atomistic

models at various pressures and temperatures [Pereverzev and Sewell,2020]. Then, the pressure dependence

of the critical resolved shear stress in Equation

(15)

is calibrated based on results from Pal and Picu [2019]

Domain partitioning MPM for simulating shock in energetic materials 13

while the temperature dependence is based on results from [Khan et al.,2018]. Furthermore, the hardening

of each slip system in Equation

(16)

is calibrated using data from [Khan and Picu,2021]. Meanwhile, the

melting temperature, the speciﬁc heat, and the melt viscosity are directly inferred from atomistic simulations.

To avoid excessive curve-ﬁtting of the model, this subset of material parameters is not ﬁne-tuned via an

optimization algorithm.

3.1 Pressure-dependent hyperelasticity

The elastic stiffness tensor at various pressures ranging from

10−4GPa

to

30 GPa

and various temperatures

ranging from

300 K

to

1000 K

was evaluated using atomistic models in [Pereverzev and Sewell,2020]; this

provides the relation between the material time derivative of the Cauchy stress

˙

σ

and the symmetric velocity

gradient

D

. It is observed that, within the parameter range, the pressure has a much larger inﬂuence on the

elastic stiffness than the temperature, and hence the effect of pressure is accounted for in the constitutive

model.

Our hyperelasticity model in Equation

(8)

is developed to reproduce this nonlinear material behavior.

The speciﬁc form of the unknown function

f

in Equation

(8)

needs to be determined to reproduce the

nonlinear material behavior while maintaining the material symmetry. We use the following polynomial

function to achieve this goal:

f=b1I2

4+b2(I5+I6)2+b3(I4+I5)2+b4(I4+I6)2

+b5(I4−I7)2+b6(I5+I7)2+b7(I6+I7)2+1, (26)

where

I1

to

I7

are strain invariants deﬁned in Equation

(7)

, and

b1

to

b7

are material constants. The convexity

of the free energy function

ψe

is not guaranteed for every elastic strain in the strain space, but the tangent

stiffness matrix is always positive deﬁnite when the stress is within a small neighborhood of a pressure state

between 0 GPa and 30 GPa, as imposed by the data used for calibration.

The elastic stiffness

Ce

0

in ambient conditions in Equation

(8)

corresponds to the stiffness evaluated

with atomistic models at

10−4GPa

and

300 K

. The material parameters

b1

to

b7

are calibrated based on the

atomistic results at various pressures. The resulting parameters are listed in Table 1.

Table 1: Parameters in Equations (8) and (26) which deﬁnes the hyperelastic free energy.

b1b2b3b4b5b6b7

13.0 4.4 6.4 7.4 8.3 5.2 4.2

In our previous work [Ma et al.,2021], the inﬁnitesimal strain theory is assumed and thus the free-energy

based stiffness is compared directly with the atomistic scale evaluation. In the current ﬁnite-deformation

hyperelasticity model, in order to achieve a fair comparison between the free-energy based stiffness and the

atomistic evaluation, the geometric nonlinearity caused by the hydrostatic compression before the stiffness

evaluation needs to be considered. The atomistic-scale stiffness

CMD

correlates the material time derivative

of the Cauchy stress ˙

σand the symmetric part of the velocity gradient (the rate of deformation):

˙

σ=CMD :D,D=sym L.

Although this elastic constitutive relation is not objective, the elastic evaluation is still reliable because

the strain perturbation during the evaluation process is small and the spin

W=skw L

vanishes. The

relationship between the atomistic evaluated stiffness CMD and the hyperelastic stiffness CSE is:

˙

σij =CMD

ijkl :Dkl ≈1

J˙

τij =1

JFi J FjJ FkK FlLCSE

IJK L −pIi jkl Dkl,CS E

IJK L =ρ0

∂2ψe

∂Ee

I J ∂Ee

KL

, (27)

where the stress state is assumed to be volumetric, and the material time derivative of the volume change is

assumed to be negligible. The deformation gradient

F

in Equation

(27)

needs to be determined through the

elastic stretch caused by the pressure as well as the crystal orientation during the atomistic evaluation.

14 Ran Ma et al.

The deformation gradient in Equation

(27)

is determined to reach the same crystal orientation as the

atomistic model [Pereverzev and Sewell,2020]. The hydrostatic pressure is ﬁrst applied which leads to an

elastic stretch

U

. Note that the crystal symmetry remains monoclinic under hydrostatic pressure. Then, a

rigid rotation

R

is applied to the sample, such that the

[

100

]

axis of the deformed conﬁguration is aligned

with the

x

-axis of the global Cartesian coordinate system and the

[

010

]

axis is aligned with the

y

axis. Then,

the deformation gradient in Equation (27) is determined as F=RU.

The comparison between the atomistic stiffness and the hyperelastic stiffness is shown in Figure 2, where

three representative stiffness coefﬁcients are compared at various pressures. In general, the hyperelastic

stiffness is able to reproduce the pressure dependence observed in the atomistic evaluations. The atomistic

model reveals that every elastic stiffness coefﬁcient of

β

-HMX has different pressure sensitivity, which poses

challenges to the hand-crafted hyperelasticity model. For example, in Figure 2, the pressure sensitivity of

elastic stiffness

C11

is well reproduced, but

C22

and

C33

are less accurate. Therefore, in our recent study, we

also attempted to construct the hyperelastic model of

β

-HMX through machine learning [Vlassis et al.,2022],

but we limit ourselves to the hand-crafted hyperelasticity model in this paper.

0 5 10 15 20 25 30

pressure (GPa)

0

50

100

150

200

250

300

stiffness (GPa)

Fig. 2: Comparison of the hyperelastic stiffness and the elastic stiffness evaluated by atomistic models.

3.2 Non-Schmid crystal plasticity

The CRSS of single-crystal

β

-HMX was evaluated by atomistic models at

300 K

and under various pressures

ranging from

10−4GPa

to

30 GPa

[Pal and Picu,2019]. Similar evaluations were also performed at ambient

pressure but with two different temperatures at

300 K

and

400 K

[Khan et al.,2018]. According to these

atomistic evaluations, the pressure dependence is much stronger than the temperature dependence within

these parameter ranges. Therefore, precisely replicating the pressure dependence of each slip system in the

constitutive crystal plasticity model is important.

The strain hardening with various dislocation densities, including

1011 m−2

,

1013 m−2

, and

1015 m−2

, is

also evaluated in [Khan and Picu,2021].

In the mesoscale model, six slip systems with the smallest CRSS are chosen as potentially active slip

systems. The material parameters in Equation

(15)

are calibrated for these slip systems to reproduce the

pressure-dependent CRSS. The calibrated results are listed in Table 2, while detailed procedures of the

calibration process are discussed in our previous work [Ma et al.,2021].

The contribution to the ﬂow stress of strain hardening is much weaker than that of lattice friction [Khan

and Picu,2021]. When the dislocation density is

1015 m−2

, the ﬂow stresses of

(

011

)[

01

¯

1]

and

(

011

)[

11

¯

1]

slip

systems increase by

55 %

over the lattice resistance, while the ﬂow stresses of other slip systems increase

much less. In the shock simulation, the dislocation density can hardly reach

1015 m−2

, which is considered

here as the upper limit of the dislocation density and the corresponding contribution to the ﬂow stress

Domain partitioning MPM for simulating shock in energetic materials 15

Table 2: Calibrated material parameters for Peierls-Nabarro stress and strain hardening of single crystal

β-HMX. Only attractive forest dislocations are considered for strain hardening.

(011)[01 ¯

1] (011)[100] (101)[10¯

1] (101)[010] (01 ¯

1)[011] (011)[11¯

1]

c(α)

1(GPa−1) 2.01 ×10−30.0 0.0 2.96 ×10−32.01 ×10−30.0

c(α)

23.75 ×10−27.38 ×10−23.76 ×10−2−2.20 ×10−23.75 ×10−24.30 ×10−2

c(α)

3(GPa) 3.75 ×10−23.59 ×10−19.85 ×10−21.22 ×10−13.75 ×10−21.24 ×10−1

g(α)

h∞−c(α)

3(MPa) 45.4 19.0 59.1 46.2 45.4 67.1

is taken as the saturation stress

(g(α)

h∞−c(α)

3)

in Equation

(16)

as listed in Table 2. The hardening rate is

controlled by parameter H0, which is set here at 10 MPa.

3.3 Other material properties of β-HMX

The remaining material parameters of the mesoscale model are listed in Table 3, including the speciﬁc

heat in Equation

(23)

, the melting temperature in Equation

(17)

, and the viscosity in Equation

(18)

. These

models are also justiﬁed by the atomistic models [Sewell and Menikoff,2004,Kroonblawd and Austin,2021],

therefore we directly incorporate them into our mesoscale model.

Table 3: Other material parameters related to the mesoscale model of single crystal β-HMX.

Variable Unit Value Reference

Γa- 1.1 [Sewell and Menikoff,2004]

Tm0K 551 [Kroonblawd and Austin,2021]

Γb-−0.2 [Sewell and Menikoff,2004]

ηcP 5.5 [Barton et al.,2009]

Tref K 298 -

cv0K kg J−15.265 ×10−7[Sewell and Menikoff,2004]

θ0K 1 [Sewell and Menikoff,2004]

cv1K kg J−13.073 ×10−4[Sewell and Menikoff,2004]

a0GPa 0.305 [Kroonblawd and Austin,2021]

cv2K kg J−11.831 ×10−1[Sewell and Menikoff,2004]

c0- 3.27 [Kroonblawd and Austin,2021]

cv3K kg J−14.194 ×10−4[Sewell and Menikoff,2004]

Mh- 3.0 [Springer et al.,2018]

Mp- 3.0 [Springer et al.,2018]

µ- 0.25 [Wu and Huang,2010]

4 Model validation

In this example, the mesoscale crystal plasticity model is compared directly with the atomistic scale model

through three shock simulations performed on single-crystal

β

-HMX under various shock velocities. Key

emphasis is placed on the Hugoniot relations and the hotspot evolution adjacent to the pore inclusions.

The model and the boundary conditions are shown in Figure 3. The shock loading is applied along the

[

010

]

crystal direction, while plane strain constraint is applied on the

(

100

)

crystal plane. Three different

initial velocities are applied to the single crystal

β

-HMX as the shock velocity, including the low-velocity

shock 0.5 km s−1, the medium-velocity shock 1.0 km s−1, and the high-velocity shock 2.0 km s−1.

A two-dimensional circular vacuum void is placed in the center of the specimen which serves as an

initial defect embedded in the single crystal

β

-HMX. Upon shock loading, the embedded pore may collapse

16 Ran Ma et al.

6

x, [001]

y, [010]

150 nm

150 nm

50 nm

v0

(a) Atomistic model

x, [001]

y, [010]

150 nm

150 nm

50 nm

v0

(b) Continuum model

Fig. 3: The model and boundary conditions for the shock simulation of a single crystal

β

-HMX containing a

circular vacuum pore.

depending on the shock velocity, and the local temperature increases due to the local plastic dissipation and

the pore surface impacting, which serves as the hotspots that may eventually trigger the detonation.

4.1 Setup of the molecular dynamics model

The MD pore-collapse results shown in this paper were obtained using data from MD trajectories originally

reported by Das et al. [2021]. Similar MD simulations are also reported by Duarte et al. [2021] with exactly

the same crystal orientation and piston velocity, but the specimen size is increased by a factor of

1.6

. The

simulations were performed using the LAMMPS code [Thompson et al.,2022] in conjunction with a variant

of the all-atom fully ﬂexible non-reactive force ﬁeld due to Smith and Bharadwaj (S-B) [Smith and Bharadwaj,

1999,Bedrov et al.,2000]. The S-B force ﬁeld is well validated and has been used in a variety of MD studies

for HMX [Mathew and Sewell,2018]. Potentials for covalent bonds, three-center angles, and improper

dihedral angles are modeled using harmonic functions, and for dihedrals using truncated cosine series.

Non-bonded pair interactions between atoms belonging to different molecules, or separated by three or

more covalent bonds within a molecule, are modeled using the Buckingham+Coulomb (exp-6-1) potential.

Long-range forces were evaluated using the particle-particle particle-mesh (PPPM) solver [Hockney and

Eastwood,1988]. The speciﬁc S-B version used here is described in our previous work [Zhao et al.,2020]. The

differences relative to the original ﬂexible-bond S-B model [Bedrov et al.,2000] are adjustments to the CH

and NO covalent bond-stretching force constants to yield a vibrational density of states more consistent with

the experiment and the addition of a very-short-range repulsive non-bonded pair potential that prevents the

”Buckingham catastrophe” (in which the forces between non-bonded atoms diverge for sufﬁciently short

interatomic distances due to the divergence of the potential energy to negative inﬁnity for distances below

that corresponding to the global maximum in the exp-6-1 potential).

Shocks initially propagating parallel to the

[

010

]

crystal direction in the

P

2

1/n

space-group setting were

simulated using a reverse-ballistic conﬁguration wherein a ﬂexible sample of HMX impacts with normal

incidence onto a rigid, stationary piston composed of the same material. The initially 1D shock wave that

results propagates in the direction opposite to the impact vector until it scatters at the pore wall. The starting

HMX slab is quasi-2D and monoclinic shaped, with initial edge lengths of

∼5 nm ×150 nm ×150 nm

. The

3D-periodic monoclinic-shaped computational domain (i.e., primary simulation cell) is also quasi-2D but

with cell edge lengths of

∼5 nm ×160 nm ×150 nm

. The extra

10 nm

added along

ˆ

y

(

||[

010

]

= initial shock

direction) is a vacuum region that is introduced at the “top” of the sample and which serves to minimize

Domain partitioning MPM for simulating shock in energetic materials 17

long-range force interactions between the free surface of the sample and the piston across the

[

010

]

periodic

boundary.

A right-cylindrical pore with an initial diameter

50 nm

is located at the center of the sample, with the

pore axis parallel to the thin direction. The resulting 3D-periodic primary cell containing the slab with pore

and the vacuum region at the top is equilibrated in the isochoric-isothermal (NVT) ensemble. Using the ﬁnal

phase space point from the equilibration, the ﬁrst three unit cells at the bottom of the HMX slab, comprising

∼3 nm

of material along the shock direction, are assigned to the piston. Velocities and forces for atoms in the

piston are set to and maintained at zero for the remainder of the simulation. Initial conditions selection for

the shock is completed by adding the Cartesian impact-velocity vector

up= (

0,

−up

, 0

)

to the instantaneous

thermal velocities of the atoms in the sample. The shocks were simulated in the isochoric-isoenergetic (NVE)

ensemble until the sample lengths rebounded by

10 %

relative to the values at maximum compression.

Atomic positions, velocities, and per-atoms stress tensors were stored for subsequent analysis.

Instantaneous 2D spatial maps of local temperature and stress in the samples were obtained using

the methods described by those authors. Brieﬂy, a 3D Cartesian Eulerian grid was superposed on the

computational domain. The square-grid edge spacing in the plane of the sample was

152.6 nm ×150.4 nm

.

The spacing in the thin direction,

5.3 nm

, spans the sample. Instantaneous atomic positions were mapped

into the grid. Atoms belonging to a given cell were used to calculate the instantaneous local temperature

and stress for that cell, assuming local equilibrium and taking proper account of the periodic boundary

conditions. These quantities, or ones computed from them, were assembled into spatial maps.

4.2 Setup of the continuum scale model

The material point method (MPM) is used to numerically approximate the solution of the continuum-scale

boundary value problem. The spatial domain is ﬁrst discretized into Lagrangian particles that move in

an Eulerian background mesh. The background mesh is a structured partition of quadrilateral elements

with the average element size equal to

0.33 nm

. The initial positions of the particles coincide with the Gauss

quadrature points of the background mesh, resulting in a total number of 745,980 particles. A convergence

study has been performed to ensure that the spatial discretization is sufﬁciently reﬁned to approximate the

solution but, for brevity, is not included in the paper. The time step is ﬁxed as

0.01 ps

, which satisﬁes the

CFL condition and guarantees the convergence of the return mapping algorithm.

The symmetry boundary condition is applied on the three side walls as shown in Figure 3(b), such

that no ﬂow across the walls is permitted but the tangential ﬂow is allowed. This is different from the

atomistic model as shown in Figure 3(a), where transmissive periodic boundary conditions are applied on

the lateral sidewalls and a

3 nm

thick piston is placed on the bottom. However, because of the particular

crystal orientation considered, the two should be equivalent, that is, the lateral velocity (or displacement) at

the boundary should be zero in the atomistic model due to symmetry considerations.

4.3 Comparisons between continuum and molecular dynamics simulations

The shock-induced pore collapse simulated with the mesoscale model is validated against atomistic scale

models, with an emphasis on the Hugoniot relations and the pressure/temperature contour. The comparison

between the mesoscale model and the atomistic model is qualitative based on the following considerations.

First, dislocations must nucleate in the atomistic model, so the atomistic model goes through a transient

process in which stress overshoots to create dislocations, after which plasticity may happen at smaller

stresses. But the mesoscale model assumes dislocations already exist in the material, so the critical resolved

shear stress is much smaller than the atomistic model. Second, the periodic boundary condition is applied

along the thickness direction in the atomistic model, while the plane strain condition is prescribed in the

mesoscale model. Such differences may lead to different slip system activation even though the orientation

of the single crystal is selected to minimize 3D effects.

Figure 4compares the Hugoniot relations simulated by the atomistic and mesoscale models, which

are both evaluated before the shock wave reaches the embedded pore. It is observed that the mesoscale

continuum model is capable of replicating the trend of the relationships among the pressure, shock velocity,

18 Ran Ma et al.

and shock wave velocity, as shown in Figures 4(b) and (c) whereas discrepancy in the temperature vs. shock

velocity is also observed. This inconsistency could be attributed to the temperature evolution model used in

Equation

(22)

where the Gr

¨

uneisen coefﬁcient is obtained from a ﬁrst-order Taylor expansion. Presumably, a

better curve-ﬁtting result can be achieved after more MD simulation data or quantum chemistry calculation

to re-calibrate the equation of the state (EOS) model at the high shock velocity range [Menikoff and Sewell,

2002]. However, a more comprehensive analysis would require a substantial amount of additional atomistic

simulations and is therefore outside the scope of this study.

0.5 1 1.5 2

Shock velocity (km/s)

300

400

500

600

700

Temperature (K)

atomistic

continuum

(a) Temperature

0.5 1 1.5 2

Shock velocity (km/s)

0

5

10

15

20

25

Pressure (GPa)

atomistic

continuum

(b) Pressure

0.5 1 1.5 2

Shock velocity (km/s)

4.5

5

5.5

6

6.5

7

7.5

Shock wave velocity (km/s)

atomistic

continuum

(c) Shock wave speed

Fig. 4: Comparison of the Hugoniot relations simulated by the atomistic model and the mesoscale model.

Case 1: Shock velocity 0.5 km s−1

Figure 5compares the temperature distribution and the pore geometry at

t=32 ps

and

42 ps

evaluated

by the atomistic model and the mesoscale model. Lateral jetting from the equator of the pore, as well as

the extension of the shear bands from the pore surface to the bulk material, is observed in the atomistic

model. This shear band formation is also replicated in the mesoscale continuum model. The temperature

increase within the shear band, which is mainly due to the plastic dissipation rather than the volumetric

compression, is also comparable between the results obtained from these two models.

The shear band appears to be more smeared in the mesoscale model than in the atomistic model. One

possible reason is that the shear band formation in the atomistic model goes through a transient process in

which stress overshoots to create dislocations, after which plasticity happens at smaller stress. The strain-

softening in the mesoscale model is less severe to trigger the sharp shear bands observed in the atomistic

model. It is also observed that the pore collapses faster in the mesoscale model than in the atomistic model.

One possible explanation is that the rate dependence is not considered in the crystal plasticity model, so the

critical resolved shear stress is much lower in the mesoscale model.

Case 2: Shock velocity 1.0 km s−1

Figure 6and Figure 7compare the atomistic model and the mesoscale model regarding the temperature and

pressure distributions predicted under the

1.0 km s−1

shock. The lateral jetting and the associated shear band

are still observed in the atomistic model similar to the

0.5 km s−1

shock case but with higher temperatures

both within the shear band and within the bulk material. But in the mesoscale model, the material jetting

forms from the upstream of the pore which is closer to the pore collapse pattern under

2.0 km s−1

shock

loading as shown in Figure 8and Figure 9. At time

t=23 ps

when the pore starts to shrink but before

collapsing, the hotspots are located within the shear band region. Compared with the atomistic model, the

mesoscale model underestimates the peak temperature within the shear band as shown in Figures 6(a)

and (c), but the pressure ﬁeld is well reproduced as shown in Figures 7(a) and (c). Again, the temperature

rise with the shear band is more related to the plastic dissipation than the volumetric compression. At time

Domain partitioning MPM for simulating shock in energetic materials 19

(a) Atomistic 32 ps (b) Atomistic 42 ps

(c) Continuum 32 ps (d) Continuum 42 ps

Fig. 5: Temperature contour of low velocity shock (0.5 km s−1) before and after pore collapse.

t=35 ps

when the pore is fully collapsed, the hotspots are located at the pore surface impacting region,

and the peak temperature reaches as high as

2000 K

as shown in Figures 6(b) and (d). The secondary shock

wave due to the pore surface impact is also reproduced in the mesoscale model, but the mesoscale model

overestimates the pressure of the secondary shock wave as we compare Figures 7(b) and (d). The major

reason is that the hydrodynamic-force collapse mechanism observed in the mesoscale model produces

larger secondary shock pressure compared with the shear band mechanism observed in the atomistic model,

which is further demonstrated in detail as follows.

Two possible pore collapse modes may manifest in the energetic materials under shock loading, that is,

the shear band mechanism and the hydrodynamic force mechanism [Rai et al.,2020]. These two mechanisms

are triggered at different shock velocities. When the shock velocity is not high enough to liquefy the

bulk material, the shear band emanating from the lateral surface of the pore is the major mechanism that

accompanies the pore collapse. As the shock velocity further increases and the temperature of the bulk

material approaches the melting point, the pore collapse is dominated by hydrodynamic force in the form

of material jetting from the shock direction, whereas the shear band is less likely to form.

In this example, we observe the transition from the shear band mechanism to the hydrodynamic force

mechanism in both the mesoscale and the atomistic simulations, which indicates that the essence of the

transition is captured at the continuum scale. However, we also observe that the transition from the shear

band mode to the hydrodynamic force dominated mode occurs at a lower shock velocity in the mesoscale

model than in the atomistic model. One possible explanation is that the rate dependence of the critical

resolved shear stress is not incorporated in our current plasticity model. As such, the yield stress of the

20 Ran Ma et al.

mesoscale model is not large enough to withhold the upstream material jetting compared with the atomistic

model.

An implementation of the crystal plasticity model to incorporate rate dependence for each plastic

slip system is technically feasible. However, this will require a signiﬁcant amount of molecular dynamic

simulations to rigorously identify the corresponding material parameters under different strain rates,

pressure, and temperature. As such, we will consider this extension in the future study but is out of this

study. The simulation approach presented in this paper is also different from Duarte et al. [2021] in which a

power-law crystal plasticity model is used without taking into account the underlying physics of individual

slip systems.

(a) Atomistic 23 ps (b) Atomistic 35 ps

(c) Continuum 23 ps (d) Continuum 35 ps

Fig. 6: Temperature contour of medium velocity shock (1.0 km s−1) before and after pore collapse.

Case 3: Shock velocity 2.0 km s−1

The atomistic and mesoscale simulations are further compared in the case where a shock velocity of

2.0 km s−1

is applied. At the time

t=16 ps

, the material jetting from the upstream surface of the pore

suggests that the pore collapse process is dominated by the hydrodynamic force mechanism, which is

observed both in the atomistic model (Figure 8(a)) and in the mesoscale model (Figure 8(c)). The shape

of the embedded pore, as well as the peak temperature of the hotspot region, is also reproduced in the

mesoscale model. At the time

t=22 ps

, the embedded pore is fully collapsed and the hotspot temperature

reaches as high as

3000 K

, as shown in Figures 8(b) and (d). By employing the gradient-partition technique

Domain partitioning MPM for simulating shock in energetic materials 21

(a) Atomistic 23 ps (b) Atomistic 35 ps

(c) Continuum 23 ps (d) Continuum 35 ps

Fig. 7: Pressure contour of medium velocity shock (1.0 km s−1) before and after pore collapse.

[Homel and Herbold,2017], we are able to continue the simulation after the pore collapse. This is important

to capture the growth of the secondary shock wave that occurs after the pore collapse. As shown in Figure

9(d), the secondary shock wave emanating from the embedded pore with about

25 GPa

shock pressure

exhibited in the atomistic model (Figure 9(b)) is replicated in the mesoscale model.

5 Parametric study

In this section, three numerical examples are performed to demonstrate the capability of the non-Schmid

crystal plasticity model and the frictional contact algorithm for predicting the shock response of energetic

materials. In Parametric Study 1, a polycrystal shock simulation is designed to demonstrate the interaction

between grain boundary sliding/cohesion and pore surface contact. In Parametric Study 2 and 3, the results

of the

1 km s−1

shock simulation are used as the control case to study the effects of pressure sensitivity in

the mesoscale model.

5.1 Parametric Study 1: effects of pressure-dependent hyperelasticity

In this numerical experiment, we explore the effect of pressure-dependent hyperelasticity by replacing

the elasticity model of the control case with the EOS-based elasticity while keeping the other components

of the material model and the setup of the boundary value problem identical. The Mie-Gr

¨

uneisen EOS is

22 Ran Ma et al.

(a) Atomistic 16 ps (b) Atomistic 22 ps

(c) Continuum 16 ps (d) Continuum 22 ps

Fig. 8: Temperature contour of high velocity shock (2.0 km s−1) before and after pore collapse.

implemented [Menikoff and Sewell,2002], and the pressure

p

is a function of the volumetric strain

J

and the

internal energy e:

p=pc(V) + Γ

V[e−ec(V)],Γ(J) = Γa+ΓbJ

where

Γ

is the Gr

¨

uneisen coefﬁcient,

V=

1

/ρ=JV0

is the speciﬁc volume,

V0=

1

/ρ0

is the initial speciﬁc

volume at ambient condition, and

Γa

and

Γb

are material constants deﬁned in Equation

(24)

. The cold

pressure pcis:

pc=3

2K0"V

V0−7

3−V

V0−5

3#×"1+3

4K0

0−4(V

V0−7

3−1)#,

where

K=16.5 GPa

and

K0=

8.7 are material constants. Meanwhile,

ec

is the internal energy corresponding

to the isothermal state:

ec(V) = e0−ZV

V0