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Computational Materials Science

journal homepage: www.elsevier.com/locate/commatsci

Meso-scale failure simulation of polymer bonded explosive with initialdefects by the numerical manifold method

Ge Kanga, Youjun Ningb,⁎, Pengwan Chena,⁎

a State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, Chinab School of Mechatronic Engineering, Southwest Petroleum University, Chengdu 610500, Sichuan, China

A R T I C L E I N F O

Keywords:Polymer bonded explosives (PBXs)Initial defectsConstitutive modelsCohesive contactNumerical manifold method (NMM)

A B S T R A C T

Polymer bonded explosive (PBX) is a composite consisting of the polymer binder and embedded explosiveparticles, along with a large number of the particle/binder interfaces as the third constituent. The particlevolume fraction is often higher than 90%. In the present work, a visco-elastic constitutive model, an elastic visco-plastic constitutive model and a bilinear cohesive contact relationship model are implemented into the nu-merical manifold method (NMM) program, an open source code programmed with C language, to describe thedeformations of the polymer binder, the explosive particles and the particle/binder interfaces, respectively. Thefracturing of the polymer binder and explosive particles is described based on the maximum tensile stress and theMohr-Coulomb criteria. Three categories of initial defects, including the initial interfacial debonding, initialvoids in the polymer binder, and initial micro-cracks in the explosive particles, are considered in the PBX meso-structures under both uniaxial tensile and compressive conditions. The tension-compression asymmetry, theinfluence of the initial defects on the meso-structure failure modes/patterns and the macroscopic effectivetensile/compressive strength of PBXs are investigated. The factors that cause the differences between the NMMresults and other numerical or experimental results are analyzed and discussed. This work enables and provesthe NMM to be an robust numerical tool for further simulation studies of the mechanical performances of PBXs,as well as other particle-filled composites, at the meso-scale.

1. Introduction

Polymer bonded explosive (PBX) is a kind of energetic materialmainly consisting of the polymer binder and filled explosive particlesalong with the particle/binder interfaces. Taking the HMX-basedPBX9501 as an example, the volume fraction of HMX particles is about92–93%, while the content of Estane (a kind of polymer binder) is onlyabout 7–8% [1]. Due to the extremely lower elastic modulus of thebinder [2], it allows PBXs to deform largely and absorb most of thedeformation energy. Meanwhile, due to the high rigidity and fracturestrength of the particle, failures/damages in PBXs are easier to evolvealong the weaker particle/binder interfaces. Therefore, besides of thepolymer binder and explosive particles, the particle/binder interfacesalso have significant effects on the mechanical response of the PBXs [3].

It is a big challenge for the stability and safety of PBX because of itsdiverse and complex situated environments. In the past years, manyresearchers have been focusing on experimental characterization ofPBXs to investigate the deformation, fracture behavior and fracturemechanism of the PBX at the meso-scale [4–8]. Some researches

devoted to the mechanical properties of the particle/binder interfacesand developed different micro-scale cohesive interfacial laws [3,9,10].With the development of SEM and MCT Technologies, the formationreason, distribution pattern and evolution process of some main in-trinsic defect forms (voids and micro-cracks) at the meso-scale couldalso be observed [11,12], especially the particle/binder delaminationand flow of the particles [13]. In addition, in recent years, a lot ofnumerical simulation works were also conducted to investigate themeso-scale mechanical properties of composites. In the respect of PBX,different constitutive models were proposed for the polymer binder andexplosive particles [14,15,16], respectively. The constitutive models ofdamaged interface, which is considered as a mixed binder-void inter-phase layer [17,18], were also proposed. Besides, some new numericalframeworks were proposed specially to investigate the deformation anddamage sensing capabilities of nano-composite bonded explosives(NCBX) [19] and PBX [20]. Moreover, many researchers also had donea lot of numerical works to study effective modulus of PBX [21–26]. Inthe respect of other composites, a PERMIX software framework [27]and a methodology for stochastic modelling of the fracture [28] were

https://doi.org/10.1016/j.commatsci.2019.109425Received 23 July 2019; Received in revised form 8 November 2019; Accepted 18 November 2019

⁎ Corresponding authors.E-mail addresses: cnningyj@foxmail.com (Y. Ning), pwchen@bit.edu.cn (P. Chen).

Computational Materials Science 173 (2020) 109425

Available online 02 December 20190927-0256/ © 2019 Elsevier B.V. All rights reserved.

T

proposed for polymer/particle nanocomposites.At the meso-scale, the SEM results of PBX show that it always

contains a large proportion of initial defects, such as interfacial de-bonding, voild in the binder and micro-cracks in the particles, and theywill influence the mechanical performances of PBXs to a large extent[11,12]. The large number of displacement jumps and stress dis-continuities across the debonded interfaces and micro-cracks in the PBXmeso-structure bring challenges to continuum-based numericalmethods, such as the FEM. In the present work, the NMM, a continuous-discontinuous numerical method, is further developed on the basis ofour previous work [29] to simulate the deformation and failure of PBXsat the meso-scale, by mainly considering the effects of initial defects. Avisco-elastic constitutive model and an elastic visco-plastic constitutivemodel are implemented for the polymer binder and explosive particles,respectively, and a bilinear cohesive contact relationship (BCCR) modelis implemented for the particle/binder interfaces. The fracturing of thepolymer binder and explosive particles bases on the maximum tensilestress and the Mohr-Coulomb criterion [30,31]. The failure process ofthe PBX meso-structures is simulated under both uniaxial tensile andcompressive conditions. Besides, the volume fraction of explosive par-ticles in this present work is 90% more or less, much higher than that ofother work [2].

2. Basic concepts of the NMM

2.1. Concepts of MC and PC

The numerical manifold method (NMM) is a unified continuous-discontinuous numerical method firstly proposed by Shi [32]. Thismethod employs an implicit solving scheme and can be adopted to si-mulate the deformation and failure behaviors of solids in both static/dynamic conditions. Especially, two cover systems, i.e., the mathema-tical cover (MC) and the physical cover (PC), are two important con-cepts in the NMM. For the example, in Fig. 1, the MC system is com-posed of regular triangles which completely cover the whole physicalregion (Ω). For each node of the MC system, the hexagonal piece, whichis composed of six triangles, such as Mi, Mj, Mk and Ml at nodes “i”, “j”,“k”, and “l”, is called a mathematical cover (MC). The overlapping partof each MC and the physical region is called a physical cover (PC), suchas Pi, Pj, and Pk (case (i), (ii), (iii) in Fig. 1). Besides, if a MC is com-pletely divided into two or more totally separated parts by cracks, eachpart will form a PC, such as Pl1 and Pl2 derived from Ml (case (iv) inFig. 1).

Theoretically, in the NMM framework, arbitrary shapes of MCs areapplicable [30]. However, for the convenience of the cover systemgeneration, definition of the weight function, and integration of the

weak form, at present, the triangular mesh topology, as shown in Fig. 1,is most adopted. Based on this kind of topology, the common area ofthree, and must only three PCs is defined as a manifold element (ME),which is also an important concept in the NMM.

2.2. Concept of ME

The concept of ME in the NMM is illustrated in Fig. 2. For the tri-angle e1, its three nodes are i, j and k, respectively. Due to the existenceof the crack, Mj and Mk are completely divided into Pj1 and Pj2, Pk1 andPk2, respectively, while Mi is not completely divided, as shown inFig. 2(b). Therefore, for ME e11, its three PCs are Pi, Pj1 and Pk1; whilefor ME e12, its three PCs are Pi, Pj2 and Pk2. Similarly, in Fig. 2(c), Ml,Mm and Mn are all completely divided by the crack into two separateparts. Thus, three PCs of the ME e51 are Pl1, Pm1 and Pn1, and for ME e52,its three PCs are Pl2, Pm2 and Pn2. In addition, in Fig. 2(d), Mr, Ms and Mt

are all not completely divided by the crack. Thus, the triangle e8 formsonly one ME with three PCs of Pr, Ps and Pt. It can be observed that anFig. 1. Illustration of MC and PC in the NMM.

Fig. 2. Illustration of ME in the NMM (“□” denotes an MC completely split by acrack, while “○” denotes an MC isn't or not completely split by a crack).

G. Kang, et al. Computational Materials Science 173 (2020) 109425

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ME may be a triangle or an irregular polygon, however, always de-termined by three and only three PCs. The three PCs for an ME are alsodefined as the nodes of this manifold element.

Furthermore, taking ME e8 for instance, its displacement field can beexpressed by the displacements of its three nodes as follows

∑= ∈ =xw e h r s tu x x u x( ) ( ) ( ), ( , , , )h h 8 (1)

where uh(x) is the displacements of the node h (h = r, s, t); wh(x) istermed as the weight function, which satisfies

≤ ≤ ∀ ∈= ∀ ∉ =

∑ =

w Mw M h r s t

w

x xx x

x

0 ( ) 1,( ) 0, , ( , , )

( ) 1

h h

h h

h (2)

However, for MEs e51 and e52 across the crack surface, the dis-placement jump across can be expressed as

∑ ∑= − =w w h l m nu x u u( ) , ( , , )ch h h h1 1 2 2

(3)

where uc(x) represents the displacement jump across the crack; uh1, uh

2

and wh1, wh

2 (h = l, m, n) represent the local independent unknownsand weight functions, respectively.

2.3. Concept of contact

The contact also is an important concept in NMM. As shown inFig. 3(a), assume two MEs locate along two sides of a crack or inter-facial surface, three categories of contact exist, in which category I andIII will be transferred into category II in contact treatment in NMM.Category I can be transferred into two “vertex-edge” contacts (p1 to p2p3and p4 to p2p3), while category III only can be transferred into one (p1 top2p3 or p2 to p1p3), as marked by red dashed circles in the figure. For theparticle/binder interface and micro-crack simulations in the presentstudy, most contacts are category I and they are transferred into cate-gory II in contact treatment. Fig. 3(b) shows how the invasion of avertex-edge contact is judged and treated. At time t, the sequence ofvertexes p1, p2, and p3 is anti-clockwise (as marked by black arrows inFig. 3(b)), which satisfies

= >x yx yx y

Δ111

01 1

2 2

3 3 (4)

where (xi, yi) is the coordinates of vertex pi (i = 1, 2, 3). At momentt + Δt, assume p1 moves to p1′, then the sequences of p1′, p2, p3 turns

clockwise, which satisfies

=+ ++ ++ +

<x u y vx u y vx u y v

Δ111

01 1 1 1

2 2 2 2

3 3 3 3 (5)

where (ui, vi) is the displacement increment of vertex pi (i = 1, 2, 3) inΔt. Therefore, Δ < 0 is a sign of invasion. If invasion happens, anormal penalty spring is added to constrain the normal movement ofvertex p1. Meanwhile, if it does not slide (according to the Mohr-Cou-lomb criterion) in shear direction, a shear penalty is added to constrainits shear movement simultaneously; otherwise, no shear spring will beadopted. The normal and shear invasion displacements can be calcu-lated by projecting vector p0p1′ into the local n-o-s coordinate system,in which p0 is the projection point of p1 on segment p2p3. If the contactis open (Δ > 0), a normal spring can also be added to represent thetensile strength on crack or interfacial surfaces.

Besides the penalty method described above, the Lagrange multi-plier method, or the augmented Lagrange multiplier method, also canbe employed to prevent the invasion between the two sides of a dis-continuity. Details can be found in relevant references [33]. The cor-responding sub-matrices of contact are added into the NMM systemequilibrium equation as follows

=K u fΔ Δ (6)

where K is the global stiffness matrix, Δu is unknowns of displacementincrement, and Δf is global loading matrix. Then, by solving the systemequilibrium equation, the displacement field of a problem can be ob-tained. Different from that in FEM, here, Δu can include the displace-ment jumps across discontinuity surfaces due to the two cover systemsin NMM.

3. Contact and material models for PBX simulations

In practice, the categories of energetic particles in PBXs mainly in-clude HMX, RDX, TATB, and PETN, et al. Among these, HMX is widelyused and has drawn intensive interest for several decades because of itshigh energy density. For HMX-based PBX9501, it mainly contains 92%HMX particles and 8% Estane [1] in volume. Its particle/binder inter-face can be treated as the third constituent with cohesive mechanicalproperties [3]. Studies [2,14] also indicated that the HMX particle is ahydrostatic strain rate dependent elastic visco-plastic material, and thepolymer binder, such as Estane in the PBX9501, Nitrocellulose in thePBX9404, is a visco-elastic material which is extremely sensitive tostrain rates and ambient temperatures.

3.1. Bilinear cohesive contact relationship model for particle/binderinterfaces

For the particle/binder interfaces, the bilinear cohesive contact re-lationship (BCCR) model is applied both on the normal and shear di-rection. In Fig. 4, d is the open or sliding contact displacement; F is thecorresponding contact force; k0 and k1 are the spring stiffness of theascending and descending stages, respectively; d0 is the contact dis-placement corresponding to the interface tensile or shear strength; dc isthe critical open or sliding displacement. A damage variable D is in-troduced to describe the damage degree of the contact when d > d0,and the damaged spring stiffness k′ can be calculated as k′ = (1-D)k0.When d > d0, if unloading occurs, k′ will be used in the unloadingprocess and further used in the subsequent loading stage until D gets anew value. The area of the triangle below the two solid lines is definedas the fracture energy GF.

Tan et al. [3] used the extended Mori-Tanaka method and theequivalence of cohesive energy on the macro-scale and micro-scale,respectively, to link the macro-scale compact tension experiment withthe micro-scale cohesive law for particle/matrix interfaces. TheFig. 3. Illustration of contact technique in NMM.

G. Kang, et al. Computational Materials Science 173 (2020) 109425

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parameters for particle/binder interfaces they measured are listed inTable 1. For the BCCR model, its implementation in NMM and theverification can be found in reference [29].

3.2. Visco-elastic constitutive model for the polymer binder (Estane)

A visco-elastic constitutive model, which contains N generalizedMaxwell elements, is introduced into the NMM. As described in re-ference [34], the shear modulus is expressed as a Prony series form

∑= +=

−G t G G e( ) ei

N

it τ

1

/ i

(7)

where τi is the relaxation time of the ith Maxwell element, Gi is ith shearrelaxation modulus, and Ge is the long term modulus when the binder isfully relaxed. Here, N = 22 and Ge = 0. The frequency-modulus rela-tions for different temperatures are shifted and superposed by using theWilliams-Landell-Ferry (WLF) shift function

= − −+ −

α T TT T

log( ) 6.5 ( )120T

0

0 (8)

in which T0 = 19 °C is called the reduced temperature [34], and αT isthe time-temperature shift factor. Then, the relaxation time τi can becalculated by αT, as follows

= = …−τ α i1.5· ( 1, 2, ,22)i Ti(7 ) (9)

For the case of uniaxial stress, the relevant Young's modulus is givenby

= +E t G t v( ) 2· ( )·(1 ) (10)

in which ν = 0.495 is the Poisson’s ratio of the polymer binder.Therefore, the Young's modulus can be approximately expressed as:

= + ≈E t G t v G t( ) 2· ( )·(1 ) 3· ( ) (11)

Suppose there exists a series of Young’s relaxation modulus Ei(i = 1, 2, … , 22), according to Eq. (11), the Young’s relaxation mod-ulus is given by

≈E G3·i i (12)

According to the WLF shift function, the shear relaxation modulus Gi

(i = 1, 2, …, 22) is fitted with a series of measured storage modulus inreference [34]. Then, the Young's relaxation modulus Ei and the re-laxation time τi at the room temperature (25 °C) can be calculated,respectively, as shown in Table 2.

According to generalized Maxwell elements, the stress–strain be-havior of the binder is determined as [34]

∫ ∫∑= − ==

− −σ E t τ ε τ dτ E e ε τ dτ( ) ( ) ( )

t

i

N

it t τ

τ0

10

i(13)

Extend equation (10) to the two dimensional (2D) state:

∫∑= ∂∂=

− −σ A εE

τe dτ[ ]

ii

t t ττ

1

22

0i

(14)

in which σ is the Kirchhoff stress tensor, ε is the Green strain tensor.Here, for the plane stress or plane strain, [A] can be expressed as below,respectively:

=−

⎛

⎝⎜

−

⎞

⎠⎟

= −+ −

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−

−−

−

ν

νν

ν

νν ν

A

A

[ ] 11

1 01 0

0 0 (1 )/2,

[ ] 1(1 )(1 2 )

1 0

1 0

0 0

νν

νν

νν

2

1

11 2

2(1 ) (15)

where ν is the Poisson’s ratio of the polymer binder.The implementation of the above visco-elastic constitutive model in

NMM was repoeted in detail in reference [29]. In the present work, theelastic term of the visco-elastic constitutive model is neglected in thedescription of Estane in PBX9501.

Here, to verify the implemented visco-elastic constitutive model inNMM for the polymer binder of Estane, a single element mode l asshown in Fig. 5 is simulated, The length and width of the element modelare 1 m and 0.1 m, respectively. The left boundary is fixed. On the rightboundary, a displacement loading for different strain rates(0.004933 s−1, 0.04933 s−1, 0.4995 s−1, 1.568 s−1, 14.95 s−1) of theelement is applied. The strain rates are identical to that in reference[34]. The larger length/width ratio (10:1) of the element model guar-antees its one-dimensional deformation to some extent. The compar-isons of NMM simulation results and theoretical/experimental results[34] are shown in Fig. 6. It can be found that quite satisfactoryagreements are achieved.

3.3. Elastic visco-plastic constitutive model for the explosive particles(HMX)

In one-dimensional case, the elastic visco-plastic constitutive modelcan be expressed as

= −σ E ε ε ( )p (16)

Here, E is the elastic modulus, and ε is the total strain rate, which canbe decomposed into an elastic term and a visco-plastic term as

= +ε ε ε e p (17)

According to the flow rule of the Mises materials [35], under thecomplicated stress state, the visco-plastic strain rateεpsatisfies

= =ε Sεσ

σ S S3¯

2 ¯, with ¯ 3

2:p

p

(18)

Fig. 4. Illustration of the BCCR model.

Table 1Parameters for particle/binder interface [3].

Normal spring stiffness (GPa) Shear spring stiffness (GPa) Tensile strength (MPa) Shear strength (MPa) Fracture energy (J/m)

300 120 1.66 1.66 81

G. Kang, et al. Computational Materials Science 173 (2020) 109425

4

in which εp and σ are the equivalent visco-plastic strain rate and theequivalent stress, respectively, which both are scalar quantity, and S isthe Kirchhoff stress tensor. Thus, Eq. (16) can be further extended totwo-dimensional as

= − = −σ M ε ε M ε M Sεσ

: ( ) : :3¯

2 ¯pp

(19)

where M is the matrix of elastic modulus. At plane stress or strain state,it can be expressed as followings, respectively

=−

⎛

⎝⎜

−

⎞

⎠⎟

= −+ −

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−

−−

−

M

M

Eν

νν

ν

E νν ν

1

1 01 0

0 0 (1 )/2,

(1 )(1 )(1 2 )

1 0

1 0

0 0

νν

νν

νν

2

1

11 2

2(1 ) (20)

in which E and ν are the elastic modulus and the Poisson's ratio of

materials, respectively.Actually, the most critical term in Eq. (19) is the equivalent visco-

plastic strain rate εp. It mainly reflects the plastic flow of the material. Inthe present work, according to references [36,37], εp is expressed as

=

= ⎡⎣

⎤⎦

= −

= + − ⎡⎣

− ⎤⎦

+

{ }( ) ( )

ε

ε ε

ε ε a ε T σ

ε T σ β

¯

¯ ¯

¯ ¯ exp[ g(¯ , )/ ¯]

g(¯ , ) 1 1 1

pε ε

ε ε

σε T

m

m p

pεε

N TT

κ

¯ ·¯ ¯ ¯

1 0¯

g(¯ , )

2

0¯

p

p

1 21 2

0 0 (21)

where εp is the equivalent plastic strain, ε0 and εm are reference strainrate for a low regime of strain rate and a high regime of strain rate,respectively, and m and a are relevant strain rate sensitivity parameters.σ0 is the quasi-static yield stress, ε0 is a reference strain, N is the strainhardening exponent, T0 is a reference temperature, and β and κ arethermal softening parameters. The function ε Tg(¯ , )p represents thequasi-static stress-strain response at ambient temperature. Obviously,Eq. (18) has considered the strain hardening and strain-rate dependenceof plasticity.

When the calculation proceeds from step n to step n + 1, supposethe time increment Δt is very small, then, Eq. (19) can be further ex-pressed as

Table 2The shear relaxation modulus Gi, the Young's relaxation modulus Ei, and the relaxation time τi [14].

Shear relaxation Prony terms (/MPa) G∞ G1 G2 G3 G4 G5

0 0.00417 0.00741 0.00159 0.0380 0.0676G6 G7 G8 G9 G10 G11

0.0891 0.0056 0.1622 0.2218 0.4753 2.104G12 G13 G14 G15 G16 G17

2.618 12.882 52.481 223.872 436.516 457.088G18 G19 G20 G21 G22

346.737 251.188 177.83 117.489 75.8

Normal relaxation Prony terms (/MPa) E1 E2 E3 E4 E5 E60.0125 0.0222 0.0475 0.114 0.203 0.2673E7 E8 E9 E10 E11 E120.0168 0.4866 0.6654 1.426 6.312 7.854E13 E14 E15 E16 E17 E1838.647 157.44 671.616 1309.54 1371.26 1040.21E19 E20 E21 E22753.56 533.48 352.46 227.573

Relaxation time Prony terms (/s) τ1 τ2 τ3 τ4 τ5 τ67.355e5 7.355e4 7.355e3 7.355e2 7.355e1 7.355τ7 τ8 τ9 τ10 τ11 τ127.355e−1 7.355e−2 7.355e−3 7.355e−4 7.355e−5 7.355e−6

τ13 τ14 τ15 τ16 τ17 τ187.355e−7 7.355e−8 7.355e−9 7.355e−10 7.355e−11 7.355e−12

τ19 τ20 τ21 τ227.355e−13 7.355e−14 7.355e−15 7.355e−16

Fig. 5. Single element model.

Fig. 6. Comparisons of NMM simulation results with theoretical/experimental results under different strain rates at T = 25°.

G. Kang, et al. Computational Materials Science 173 (2020) 109425

5

= −σ M ε M Sεσ

tΔ : Δ :3¯

2 ¯Δ :p

(22)

in which Δσ is the increment of the stress tensor, while S is the accu-mulated total stress tensor. It is written in the matrix form as

⎛

⎝⎜⎜

⎞

⎠⎟⎟

=⎛

⎝⎜⎜

⎞

⎠⎟⎟

− ⎛

⎝⎜

⎞

⎠⎟M M

σσσ

εεε

εσ

tσσσ

ΔΔΔ

ΔΔΔ

· 32

¯

¯Δ ·p

11

22

12

11

22

12

112212 (23)

Assuming matrix Q as

= − ⎛

⎝⎜

⎞

⎠⎟Q M

εσ

tσσσ

· 32

¯

¯Δ ·p 11

2212 (24)

Then Eq. (23) can be written as

⎛

⎝⎜⎜

⎞

⎠⎟⎟

=⎛

⎝⎜⎜

⎞

⎠⎟⎟

+M Qσσσ

εεε

ΔΔΔ

ΔΔΔ

11

22

12

11

22

12 (25)

Now, the strain energy of a manifold element can be deduced as:

∐ ∬

∬

∬ ∬

=⎛

⎝⎜⎜

⎞

⎠⎟⎟

=⎧

⎨⎩

⎧⎨⎩

⎫⎬⎭

+⎫

⎬⎭

=⎧⎨⎩

⎫⎬⎭

+

= +

M Q

M Q

M Q

ε ε εσσσ

dxdy

ε ε εεεε

dxdy

ε ε εεεε

dxdy ε ε ε dxdy

D A B B D D A B

(Δ Δ Δ )ΔΔΔ

(Δ Δ Δ )ΔΔΔ

(Δ Δ Δ )ΔΔΔ

(Δ Δ Δ )

[ ] [ ] [ ][ ] [ ] [ ]

e A

A

A A

eT e

eT

e e eT e

eT

12 11 22 12

12

12

12

12 11 22 12

11

22

12

12 11 22 12

11

22

12

12 11 22 12

12

12

(26)

in which Ae is the area of the manifold element, [De]T=(De(1) De(2) De(3))is the displacement unknowns of three PCs, [Be]=(Be(1) Be(2) Be(3)) is adeformation matrix, and e(i) is the ith PC, i = 1, 2, 3.

Therefore, the following stiffness matrix for a manifold element isadded to the global stiffness matrix K:

=⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

M MA B B A

B

B

B

B B B[ ] [ ] ( )ee

Te

eeT

eT

eT

e e e

(1)

(2)

(3)

(1) (2) (3)

(27)

That is

→ =M KA B B r s[ ] [ ] [ ], , 1, 2, 3ee

Te e r e s(r) (s) ( ) ( ) (28)

and the following loading matrix for a manifold element is added to theglobal loading matrix Δf:

− = −⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Q QA B A

B

B

B

12

[ ] 12

ee

T eeT

eT

eT

(1)

(2)

(3) (29)

That is

− → =QA B rF12

[ ] { }, 1, 2, 3ee r

Te r( ) ( ) (30)

The single element model in Fig. 5 is used again to verify the elasticvisco-plastic constitutive model for the HMX particles. The parametersfor the constitutive model are listed in Table 3, in which the elastic

parameters (E and ν) are deduced through 13 elastic constants for the β-HMX monoclinic single crystals at room temperature in [14], while theplastic parameters are derived by inversion of constitutive equation in[36,37].

Fig. 7 shows the comparisons of NMM simulation results with the-oretical results under different strain rates and different temperatures.It can be found that, each stress-strain curve will undergo three dif-ferent stages in the loading process. In Stage I, due to the small value ofthe visco-plastic strain rate, the deformation is dominated by the elas-ticity. The curve is close to a straight line with a slope of the elasticmodulus E. In Stage II, as the increase of the visco-plastic strain, theslope of the stress-strain curve decreases gradually towards horizontal.In Stage III, the stress-strain curve finally approaches to horizontal andthe final stress level is close to the yield stress of the material(260 MPa). Moreover, in the second and third stage, a higher loadingstrain rate, or a lower temperature, means a higher stress level of thematerial response. It can be found that good agreement between thetheoretical and NMM simulation results is achieved, even under largestrain (> 20%). The implemented elastic visco-plastic constitutivemodel is verified and it represents the mechanical character of non-linear elasticity, deformation hysteresis, strain rate and temperaturedependence of visco-plastic materials well.

3.4. Fracturing criterion

The maximum tensile stress criterion and Mohr-Coulomb criterionhave been extensively used in rock mechanics due to the simplicity andthe capability to predict peak strength and failure directions effectively.Assuming there is an inclined plane with an angle β to the direction ofthe minor principal stress σ3, according to the Mohr-Coulomb criterion,the shear strength on this inclined plane is

= +s c σ φtann (31)

where c is the cohesion, φ is the internal friction angle, and σn is thenormal stress. On this inclined plane, the normal stress is

= + + −σ σ σ σ σ β12

( ) 12

( ) cos 2n 1 3 1 3 (32)

and the shear stress is

= −τ σ σ β12

( ) sin 21 3 (33)

Substituting Eq. (32) to Eq. (31), and letting s = τ, the limit stresscondition on the inclined plane is calculated as

=+ + −

− +σ

c σ β φ ββ φ β

2 (sin 2 tan (1 cos 2 ))sin 2 tan (1 cos 2 )1

1

(34)

Fig. 8 shows the Mohr-Coulomb criterion combined with the max-imum tensile stress criterion. There must have a critical plane, on whichthe available shear strength will be firstly reached when σ1 increases.Fig. 8(a) also gives the orientation of this critical plane as

= +β π φ4 2 (35)

which can also be obtained by solving d(s-τ)/dβ = 0. On the criticalplane, sin2β = cosφ and cos2β = −sinφ, then Eq. (34) reduces to

=+ ++

σc φ σ φ

φ2 cos (1 sin )

1 sin13

(36)

The linear relationship between σ3 and σ1 is plotted in Fig. 8(b).

Table 3Parameters for the elastic visco-plastic constitutive model [14,36,37].

E (GPa) ν ε0 (s−1) m εm (s−1) a (1/MPa) σ0 (MPa) ε0 (s−1) T0 (K) N β κ

13.75 0.32 1e-4 100 8e12 22.5 260 5.88e-4 293 0.05 2.4 0.15

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If the Mohr-Coulomb envelop is extrapolated to σ1 = 0, it will in-tersect the σ3 axis (T0′ in Fig. 8(b)) at an apparent value of the uniaxialtension strength. However, the experimentally measured uniaxial ten-sile strength is generally lower. Therefore, a tension cutoff is applied ata selected value of the uniaxial tensile stress, which derives the max-imum tensile stress criterion as

=σ T- 3 0 (37)

4. Simulations and analysis

In reference [38], the construction process of meso-structures forPBX 9501 was described in great detail. In that work [38], 5 groups ofmeso-structures with different sizes (0.6 mm, 0.9 mm, 1.2 mm, 1.5 mm,1.8 mm) had been simulated to investigate the influence of the meso-structure size on the effective modulus. It was found that the influencebecomes negligible when size of the meso-structure reaches 1.2 mm,thus the model characteristic length was decided to be 1.2 mm, around5 times of the size of the maximum particle. Therefore, in the presentwork, the meso-structure model size of 1.2 mm is used directly. InFig. 9, a meso-structure model is displayed. The corresponding particlesize distribution is shown in Fig. 9(c) and the particle volume fraction isabout 90.06%.

The boundary condition of the meso-structure model under uniaxialtension and compression are shown in Fig. 9(a) and (b), respectively.For the left boundary, the x-displacement is constrained and the y-displacement is free. For the bottom boundary, the y-displacement isconstrained and the x-displacement is free. For the right boundary, thekinetic-couple condition is applied to guarantee that the meso-structurealways keeps a square shape during its deformation [24] as illustratedby the dashed lines. For the top boundary, a displacement loading withspeed 0.01 mm/s is applied, thus the strain rate of the whole model inthe y direction is about 0.008 s−1. The mechanical parameters of thePBX constituents in Table 1, Table 2, and Table 3 are used in the NMM

simulation.

4.1. The influence of initial interface debonding

Fig. 10 illustrates the initial interface debonding in PBX meso-structure, in which 4 randomly distributed debonded interfaces areindicated by arrows. The initial debonded interfaces are set with zerotensile and shear strengths. To describe the content of initial interfacedebonding quantitatively, an interface debonding coefficient α is de-fined as follows:

=∑

×=αl

L100%i

ni

p

1

(38)

In which, li is the length of the ith debonded interface, n is the totalnumber of the debonded interfaces, and Lp is the total length of all theparticle-binder interfaces in the whole meso-structure.

As shown in Fig. 11, a group of PBX meso-structures with α varyingfrom 0% to 50% is constructed. These models are simulated with theloading strain rate and environment temperature of 0.08 s−1 and 25 °C,respectively. The same loading strain rate and environment tempera-ture are considered in all the following simulations in this paper.

Fig. 12 shows the failure patterns of the PBX meso-structures withdifferent α values under uniaxial tension. It can be found that, themicro-failure mainly occur on the particle/binder interfaces because ofthe obviously lower strength of the interface as compared with that ofthe other two bulk constituents. The fracture of the binder will takeplace thereafter due to the large deformation under tension. The in-terface debonding is a usual phenomenon observed in experiments[39]. The debonded interfaces gradually coalesce to form main cracksalmost perpendicular to the loading direction and lead to the ultimatemacroscopic tensile failure of the whole meso-structures. The locationof the formed main cracks is quite different in the six models withdifferent α values. This is because the different distributions of the in-itial debonded interfaces provide different opportunities for the

Fig. 7. Comparisons of NMM simulation results with theoretical results under different strain rates and different temperatures.

Fig. 8. Mohr-Coulomb criterion with a tension cutoff.

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debonding to spread. However, it also can be observed that the newinterface debonding generally likes to take place around bigger ex-plosive particles.

Fig. 13 shows the failure patterns of PBX meso-structures with dif-ferent α values under uniaxial compression. It can be found that, theinterfacial debonding is also the main failure mode along with largedeformation of the binder, similar to that in tensile condition. Gen-erally, the main cracks, or crack bands, formed by coalesced debondedinterfaces eventually lead to the failure of the whole meso-structures ina compression-shear pattern. When α ≤ 30%, two main shear crack

bands are formed clearly, indicating an obvious shear failure in themacroscopic. However, when α ≥ 40%, the main shear crack bandsturn vague with more scattered micro-cracks in the whole meso-struc-ture. Moreover, in all the six cases, the new interface debonding alsolikes to take place around bigger explosive particles, along with aphenomenon of the flow of small particles together with the deforma-tion of the polymer binder.

Fig. 14 shows the macroscopic effective stress-strain curves of themeso-structures with different α values under uniaxial tension andcompression. It can be found that, the effective tensile and compressivestrength of the meso-structures (peak value of curves) has an obviousasymmetric character. This is because that the particle/binder inter-facial mechanical properties play a dominant role under the tensileloading condition, while under the compressive loading condition, themechanical properties of all the three constituents of PBXs play im-portant roles. This asymmetric phenomenon is also widely observed inrelevant experiments [39]. It is also found that, the increase of the in-itial debonding content decreases the macroscopic effective tensile andcompressive strength dramatically. From Fig. 13, it can be found thatthe global large strain under tension and compression is mainly at-tributed to the large deformation of the polymer binder and failure ofthe particle/binder interfaces.

Table 4 shows NMM simulated effective tensile and compressivestrength along with some ABAQUS simulation and experimental results.For the NMM simulation results, when α = 0%, the effective tensilestrength is close to 1.66 MPa (given interfacial tensile strength). Itfurther verifies the dominant role of the interfacial mechanical propertyunder tensile condition. When α = 0%, the ABAQUS and experimental

Fig. 9. PBX meso-structure model under uniaxial tension/compression.

Fig. 10. Illustration of the initial interface debonding of PBX meso-structure(Only particle/binder interfaces are displayed, in which black lines representdebonded interfaces).

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tensile strength is lower than the NMM simulation result, but very closeto that when α = 10%. The differences between NMM results and otherresults may be caused by the application of a complex PBX meso-structure in the NMM simulation, in which a large number of contactsexist on the particle/binder interfaces.

For the effective compressive strength, the NMM results are ob-viously lower than other results. This can probably be attributed by thefollowing factors: (1) The shear spring stiffness value on the particle/binder interfaces may be unreasonable in the present BCCR; (2) Thefriction force on the particle/binder interfaces are neglected to achieve

Fig. 11. A group of PBX meso-structures with different α values (Red lines represent bonded interfaces; Black lines represent debonded interfaces).

Fig. 12. Simulated failure patterns of PBX meso-structures with different α values under uniaxial tension.

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better convergence speed of the computation; (3) Even thickness of thebinder layer is assumed. Other factors, such as the size of the meso-structure, the particle size distribution and particle gradation, etc., mayalso have certain degree of influence on the results because of thecomplexity under the compressive condition. The specific influences ofthese factors could be investigated by simulations in the future.

As compared with the case α = 0%, when α = 50%, the effectivetensile and compressive strength decreases 74.26% and 60.53%, re-spectively. The initial interfacial debonding defect reduces the tensilestrength more seriously.

4.2. The influence of initial voids

Fig. 15 illustrates the initial void defect in the polymer binder of

Fig. 13. Simulated failure patterns of PBX meso-structures with different α values under uniaxial compression.

Fig. 14. Effective stress-strain curves of the meso-structures with different αvalues under uniaxial tension and compression.

Table 4Comparison of NMM simulated effective tensile and compressive strength with other numerical or experimental results under different initial interface debondingcoefficient.

NMM results under different α values (MPa) Other results (MPa)

0% 10% 20% 30% 40% 50% ABAQUS results Experimental results

Tension 1.589 1.117 0.935 0.737 0.555 0.409 1.18 [14] 1.07 [40]Compression 6.686 5.764 4.782 4.291 3.381 2.639 9.58 [14] 11.11 [41]

Fig. 15. Illustration of the initial void in the binder of PBX meso-structure (Onlythe polymer binder layer is displayed).

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Fig. 16. A group of PBX meso-structures with different β values (Blank parts represent voids).

Fig. 17. Simulated failure patterns of PBX meso-structures with different β values under uniaxial tension.

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PBXs, in which 4 randomly distributed voids are indicated by arrows.The voids are generated by randomly deleting some manifold elementsin the polymer binder domain. Similarly, to describe the content ofvoids in the meso-structure quantitatively, a void volume fraction β isdefined as follows:

=∑

×=βA

A100%i

ni

p

1

(39)

In which Ai is the area of the ith void, n is the total number of thevoids, and Ap is the total area of the whole meso-structure. The produceof voids will lead to the decrease of interface content obviously whenthe void content is high, which may influence the simulation results.However, the maximum β value considered below is only 2.0%.

The study [42] pointed out that the β value for PBX9501 with thePVF of 92% is around 1%~3%. As shown in Fig. 16, a group of PBX

Fig. 18. Simulated failure patterns of PBX meso-structures with different β values under uniaxial compression.

Fig. 19. Effective stress-strain curves of the meso-structures with different βvalues under uniaxial tension and compression.

Table 5CComparison of NMM simulated effective tensile and compressive strength with other numerical or experimental results under different initial void volume fraction.

NMM results under different void volume fraction β Other results

0% 0.4% 0.8% 1.2% 1.6% 2.0% ABAQUS results Experimental results

Tension (/MPa) 1.589 1.354 1.173 0.953 0.899 0.851 1.18 [14] 1.07 [40]Compression (/MPa) 6.686 5.961 5.344 5.288 3.946 3.692 9.58 [14] 11.11 [41]

Fig. 20. Illustration of the initial cracks in the explosive particles (Only ex-plosive particles are displayed).

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Fig. 21. A group of PBX meso-structures with different d values (Black segments represent micro-cracks in the explosive particles).

Fig. 22. Simulated failure patterns of PBX meso-structures with different d values under uniaxial tension.

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meso-structures with β value varying from 0% to 2.0% is constructed.Fig. 17 shows the failure patterns of PBX meso-structures with dif-

ferent β values under uniaxial tension. It can be found that, interfacialdebonding is still the main failure mode, and different content of theinitial voids in the binder brings different locations of the approxi-mately horizontal main cracks that lead to the final tensile failure of thewhole meso-structures. Meanwhile, with the increase of the β value,

there are more scattered micro-cracks generated simultaneously in themeso-structure.

Fig. 18 shows the failure patterns of PBX meso-structures with dif-ferent β values under uniaxial compression. It can be found that, in-terfacial debonding also is the main failure mode. When β ≤ 1.2%,obvious main shear cracks, or crack bands, are formed, which indicatescompression-shear failure of the whole meso-structures. However,when β ≥ 1.6%, the main shear cracks become vague and there aremore scattered micro-cracks generated in the whole model.

Fig. 19 shows the macroscopic effective stress-strain curves of themeso-structures with different β values under uniaxial tension andcompression. The tensile and compressive strength of the meso-struc-tures also shows an obvious asymmetric character, and the increase ofthe void volume fraction β reduces the effect tensile and compressivestrength of the meso-structures obviously.

As shown in Table 5, when β = 0.8% or 1.2%, the NMM simulatedeffective tensile strength is close to the ABAQUS and experimental re-sults, and the NMM simulated effective compressive strength is alsoobviously smaller than the ABAQUS and experimental results. Ascompared with the case with zero initial defects, when β = 2.0%, theeffective tensile and compressive strength reduces 46.44% and 44.78%,respectively, a quite close proportion.

4.3. The influence of initial micro-cracks

Based on the fact that random distribution of micro-cracks exists inreal explosive particles, some manifold element nodes in the particles

Fig. 23. Simulated failure patterns of PBX meso-structures with different d values under uniaxial compression.

Fig. 24. Effective stress-strain curves of the meso-structures with different dvalues under uniaxial tension and compression.

Table 6Comparison of NMM simulated effective tensile and compressive strength with other numerical or experimental results under different initial micro-crack density.

NMM results under different crack density d Other results

0 1.0 2.0 3.0 4.0 5.0 ABAQUS results Experimental results

Tension (/MPa) 1.589 1.432 1.406 1.392 1.275 1.054 1.18 [14] 1.07 [40]Compression (/MPa) 6.686 6.183 5.945 5.879 5.755 5.748 9.58 [14] 11.11 [41]

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domain are separated randomly in the initial PBX meso-structure modeland zero tensile and shear strengths are set to their surfaces to simulateinitial cracks in PBX. As shown in Figs. 20, 4 randomly distributed cracksegments are indicated by arrows. To describe the density of crack inthe meso-structure quantitatively, a parameter of crack density d (inmm/mm2) is defined as follows:

=∑ =d

dA

in

i

p

1

(40)

In which di is the length of the ith crack segment, n is the totalnumber of the cracks, and Ap is the total area of the whole meso-structure. As shown in Fig. 21, a group of PBX meso-structures with dvarying from 0 to 5.0 is constructed.

Fig. 22 shows the failure patterns of PBX meso-structures with dif-ferent d values under uniaxial tension. It can be found that, along withthe interfacial debonding, the fracture of explosive particles also be-comes an obvious micro-failure mode, as signed with arrows. The wholemeso-structures fail eventually due to the generation of tensile maincracks that are almost perpendicular to the loading direction. The maincracks like to evolve around bigger particles, and different d valuebrings different locations of the failure routes as well.

Fig. 23 shows the failure patterns of PBX meso-structures with dif-ferent d values under uniaxial compression. It can be found that, similarto that in the tensile condition, the interfacial debonding along with thefracture of explosive particles produce the main cracks, or crack bands,to lead to the macroscopic failure of the meso-structure models.Meanwhile, the macroscopic shear failure routes seem do not changeapparently until d increases to 5.0. This is because the initial defect ofmicro-cracks is located in the explosive particles. Although particlefracture is involved under the compressive loading, interfacial de-bonding and binder large deformation are still the main failure factors.Moreover, similar to all the previous examples in the present study, thefailures in the meso-structures are likely to take place around biggerparticles and the small particles mainly perform as flow matter with thedeformation of the polymer binder.

Fig. 24 shows the macroscopic effective stress-strain curves of themeso-structures with different d values under uniaxial tension andcompression. The tensile and compressive strength of the meso-struc-tures also shows an obvious asymmetric character. The increases of thed value leads to the decease of the tensile and compressive strengthsimultaneously.

As shown in Table 6, the NMM simulation result of the effectivetensile strength with a d value of 5.0 is close to the ABAQUS and ex-perimental results. However, similar to the previous examples in thepresent study, the NMM simulated effective compressive strength isobviously lower than the ABAQUS and experimental results. Whend = 5.0%, the effective tensile and compressive strength of the meso-structure decreases 25.74% and 14.03%, respectively, as comparedwith case without initial defects. The influence of the initial micro-cracks in the explosive particles on the effective tensile strength is muchmore obvious than that on the effective compressive strength of themeso-structures. It indicates that the influences of the existence ofclosed micro-cracks on tensile failure are greater than that on com-pressive failure.

5. Conclusions

In the present paper, a bilinear cohesive contact relationship model,a visco-elastic constitutive model, and an elastic visco-plastic con-stitutive model are implemented into NMM to simulate the three con-stituents of PBX, respectively. Based on the maximum tensile stress andthe Mohr-Coulomb criteria, the microscopic fracturing of the particlesand polymer binder is simulated. The deformation and failure behaviorsof PBX meso-structures with three categories of initial defects includingthe initial interfacial debonding, the initial voids in the polymer binder,

and the initial micro-cracks in the explosive particles, are simulatedwith the extended NMM. Results indicate that:

(1) Under the tension condition, the particle/binder interface propertyplays the dominant role in the determination of the macroscopiceffective tension strength of the PBX meso-structure. Under thecompression condition, along with that of the particle/binder in-terface, the properties of the explosive particles and polymer binderalso play important roles to determine the effective compressionstrength. Therefore, the PBX performs an obvious tension-com-pression strength asymmetry.

(2) The initial defects will lead to the decrease of the tension andcompression strength of the PBX in different degree. For initial in-terfacial debonding and micro-cracks, the tension strength is morelikely to be reduced as compared with the compression strength.For initial voids, its influence on the tensile and compressivestrength is almost identical. In addition, the initial defects will leadto different microscopic failure modes and macroscopic failurepatterns of the PBX meso-structures as well. Under both the tensionand compression conditions, the interfacial debonding, along withlarge deformation and possible subsequent fracture of the binder, isa major microscopic failure mode. The initial micro-cracks in theexplosive particles also bring an obvious failure mode of particlefracture. In the macroscopic, different types and content of the in-itial defects will lead to different locations of tensile main cracksunder the tension condition, and different remarkableness of theshear main cracks (or crack bands) under the compression condi-tion. However, an unchanged phenomenon is that failures are liketo take place around bigger particles.

(3) The microscopic deformation and failure behavior of PBX meso-structures under compression is of much complexity and bringschallenges to precise reproduction of it through numerical simula-tion, which indicates a direction of our future work. Moreover, inthe present work, to obtain high volume fraction of the explosiveparticles, an ideal PBX meso-structure with identical thickness ofthe polymer binder layer is used. This ideal model is different fromthe real PBX meso-structure and could influence the simulationresults to some extent. How to obtain more realistic PBX meso-structures with high particle volume fraction is also a future workdirection. Generally, the developed NMM in the present workshould be a promising tool for the simulation of particle-filledcomposites at the meso-scale.

CRediT authorship contribution statement

Ge Kang: Software, Formal analysis, Validation, Writing - originaldraft. Youjun Ning: . : Software, Methodology, Supervision, Writing -review & editing. Pengwan Chen: Conceptualization, Resources,Funding acquisition, Project administration.

Declaration of Competing Interest

The authors declare that they have no known competing financialinterests or personal relationships that could have appeared to influ-ence the work reported in this paper.

Acknowledgements

This work was supported by the Science and Technology ProgramProject of Sichuan Province under Grant number 2017JY0128, and theNSAF project under Grant number U1330202.

Data availability

The data used to support the findings of this study are availablefrom the corresponding author upon request.

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