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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