Engineering Analysis with Boundary Elements 96 (2018) 123–137

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Engineering Analysis with Boundary Elements

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

Simulations of meso-scale deformation and damage of polymer bonded

explosives by the numerical manifold method

Ge Kang

a , Pengwan Chen

a , Xuan Guo

b , Guowei Ma

c , Youjun Ning

b , d , ∗

a State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China b School of Manufacturing Science and Engineering, Southwest University of Science and Technology, Mianyang 621010, China c School of Civil and Transportation Engineering, Hebei University of Technology, Tianjin 300401, China d Shock and Vibration of Engineering Materials and Structures Key Laboratory of Sichuan Province, Southwest University of Science and Technology, Mianyang 621000,

China

a r t i c l e i n f o

Keywords:

Polymer bonded explosive (PBX)

Meso-scale simulation

Bilinear cohesive contact relationship

Visco-elastic constitutive model

Numerical manifold method (NMM)

a b s t r a c t

Polymer bonded explosive (PBX) is a particle-matix composite consisting of explosive particles, polymer ma-

trix/binder, and the interface between the particles and the binder, and the particle volume fraction (PVF) in the

PBX is extremely high. To simulate the meso-scale deformation and damage behaviors of PBX with the numerical

manifold method (NMM), a bilinear cohesive contact relationship (BCCR) model with three parameters is incor-

porated in the NMM to describe the particle-binder interface, a visco-elastic constitutive model of prony series

with 22 parameters is incorporated in the NMM to describe the polymer binder, and a fracturing algorithm based

on the maximum tensile stress criterion and the Mohr–Coulomb criterion is employed to describe the fracturing

failures of the particles as well as the binder. The PBX meso-scale deformation and damage process of microcrack

initiation, crack propagation and formation of crack bands under tensile or compressive conditions are stud-

ied through NMM simulations, and the influenes of the PVF and the explosive particle geometrical distribution

(PGD) on PBX mechanical performaces are specially investigated. This work enables and proves the NMM to be

an promissing roubust numerical tool for further simulation studies of the meso-scale mechanical performances

of PBX, as well as other particle-fillled polymer composites.

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. Introduction

Polymer bonded explosive (PBX) is a kind of important composite ex-

losives consisting of explosive particles, a soft polymer matrix/binder

nd the interface between the particles and the binder. The explosive

article volume fraction (PVF) in the PBX is extremely high and can

each around 90–95% [1,2] . The meso-scale, as well as macro-scale me-

hanical performances of the PBX, which is of great importance, strongly

epends on the binder, because the elastic modulus of the binder is ap-

roximately three to four orders of magnitude lower than that of the

xplosive particles [3] . The low modulus of the binder allows it to de-

orm and absorb most of the mechanical work imparted to the PBX com-

osite under different loading conditions. Hence, to some extent, the

inder provides structural toughness and reduces the impact sensitivity

f the PBX. In addition, the properties of the particle/binder interface

lso have significantly influences on the macroscopic mechanical re-

ponse of the PBX. For example, the bulk modulus of PBX 9501 is more

han 40% lower than that of the same material with perfect interfaces

ithout any debonding [4] . Meanwhile, because the particle is much

∗ Corresponding author at: School of Manufacturing Science and Engineering, Sout

E-mail address: cnningyj@foxmail.com (Y. Ning).

ttps://doi.org/10.1016/j.enganabound.2018.08.011

eceived 30 June 2018; Received in revised form 21 August 2018; Accepted 21 Augu

955-7997/© 2018 Elsevier Ltd. All rights reserved.

tronger than the matrix, cracks are more likely to evolve along the

nterface between the particles and the binder [5 , 6] , as well as in the

inder matrix [7] , especially under tensile conditions [8] .

In recent years, numerical simulation became an important ap-

roach for PBX meso-scale deformation and damage studies due to

ig progresses of computer capacity and numerical simulation tech-

iques. The numerical methods used for the meso-scale mechanical anal-

sis of the PBX mainly include the molecular dynamics (MD) method

nd continuum-based numerical methods, such as the finite element

ethod (FEM). MD simulations are generally conducted to investigate

he pressure-volume-temperature behaviors [9 , 10] , the binding energies

11–14] , or the relationships between sensitivity and energy properties

15] . While for the FEM approach, Barua and co-workers [16 , 17] stud-

ed the thermal-mechanical response of the HMX/PBX over the strain

ate 10 4 –10 5 s − 1 by a two-dimensional cohesive finite element method

CFEM); Wang et al. [18] performed a series of three-pseudo dimen-

ional meso-scale calculations also with the concept of cohesive FEM

y considering a certain depth of the PBX model as a plate; Banerjee

t al. [19] investigated the effective elastic modulus of monodisperse

hwest University of Science and Technology, Mianyang 621010, China.

st 2018

G. Kang et al. Engineering Analysis with Boundary Elements 96 (2018) 123–137

Fig. 1. Illustration of cover systems of the NMM.

Fig. 2. Two cases of co-edges.

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Fig. 3. Geometric model and two meshing methods in the NMM.

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lass-estane mock PBX through two- and three-dimensional FEM simu-

ations, respectively, and the results indicated no significant difference

etween the two- and three-dimensional simulations; Arora et al . [20] .

imulated crystal debonding of the PBX two-dimensional meso-structure

hrough the FEM, where different particle formations are considered,

nd the results showed that the geometry of the particles plays a crucial

ole in determining the onset of the debonding failure and the severity

f fracturing in relation to whether it is a purely local or global failure.

In the present paper, another numerical method, namely, the nu-

erical manifold method (NMM), is adopted to be developed for the

eso-mechanical behavior simulations of the PBX. The NMM is a uni-

ed continuous-discontinuous numerical method first put forward by

hi [21] . It takes an implicit solving scheme no matter static or dy-

amic problems are analyzed at present. In the NMM, two cover sys-

ems, i.e., the mathematical cover (MC) and the physical cover (PC) are

mployed. As shown in Fig. 1 (a), the triangular meshes which cover the

hole physical region generate the MC system. These triangular meshes

re similar to the finite element meshes in the FEM, therefore, are called

he finite MC system. For each node of the MC system, the hexagonal

athematical piece, such as MC i , MC j , MC k and MC l at nodes i, j, k and

, respectively, in Fig. 1 (a), as a combination of six triangular elements,

124

s called a mathematical cover (MC). In addition, the overlapping of the

Cs with the physical region further generate the physical covers (PCs),

uch as PC i , PC j , PC k and PC l signed in Fig. 1 (a). In fact, when a MC is

ompletely divided by the internal cracks into two or more separated

G. Kang et al. Engineering Analysis with Boundary Elements 96 (2018) 123–137

Fig. 4. Force analysis of a block on a base under tensile and shear conditions.

Fig. 5. Comparison of contact force-loading displacement relationships be-

tween two meshing methods along with the corresponding theoretical results

when the LCR model is used in the NMM.

Fig. 6. The bilinear cohesive contact relationship (BCCR) model.

Fig. 7. Comparison of contact force-loading displacement relationships be-

tween two meshing methods along with the corresponding theoretical results

when the BCCR model is used in the NMM.

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arts inside the physical region, two or more PCs will be generated,

uch as PC l 1 and PC l

2 ; otherwise, only one PC will be formed, such as

C i , PC j and PC k .

Now assuming there does not exist an internal crack in the physical

egion, as shown in Fig. 1 (b), as a special case, when the triangles in

he mesh turn to much bigger, the physical region can be completely

ontained in one triangular element i –j –k as signed in Fig. 1 (b). Accord-

ng to the definitions of MCs and PCs illustrated above, it is found that

G. Kang et al. Engineering Analysis with Boundary Elements 96 (2018) 123–137

Fig. 8. A single manifold element model used to verify the Prony series constitutive model.

Fig. 9. Comparison of theoretical and simulation results of the one-element

model in Fig. 8 under different strain rates.

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Fig. 10. Schematic of the PBX meso-structure model.

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his physical region itself forms three totally overlapping PCs, namely,

C i , PC j and PC k corresponding to the three MCs, namely, MC i , MC j and

C k , respectively. In the NMM, the global displacement function is ob-

ained by weighting the displacement functions defined on the PCs, so

he physical region in Fig. 1 (b) will have its own displacement variables

ndependent of other physical regions, thus this kind of MC system is

alled the independent MC system.

Theoretically, the NMM integrates the FEM and the discontinuous

eformation analysis (DDA) [21] as its two special cases, making it able

o simulate continuum, transition from continuum to discontinuum, and

iscontinuum problems in a single framework [22–24] . So far, there

re very few works reported on the application of the NMM for the

126

imulations of particle-filled composites, such as the PBX [25] or the

oncrete [26,27] . It is demonstrated that the mechanical response of the

article/binder interface of the PBX comply with the bilinear cohesive

riterion [3] , and the polymer binder is a visco-elastic material which is

ensitive to both the strain rate and the temperature. In addition, a rea-

onable and applicable fracturing modeling algorithm is also required in

he NMM for the fracturing damage simulation of the explosive particles

s well as the polymer binder in the PBX.

In respect of the simulation of composite material interfaces by the

MM, An et al. [28,29] described the weak discontinuity across the ma-

erial interface through two special types of PCs with customized cover

unctions. In respect of material constitutive models, Wu et al. [30] in-

orporated a modified three-element visco-elastic constitutive model in

he NMM, and analyzed the effects of the loading rates on the crack be-

avior of a sedimentary rock; He et al. [31] incorporated an incremental

isco-elastic constitutive model based on the generalized Kelvin-Voigt

odel, and they also enriched the local displacements around the crack

ips with the visco-elastic crack-tip asymptotic displacement field. In

espect of the fracturing simulation, Ning et al. [22,32] implemented

fracturing algorithm based on the Mohr–Coulomb criterion with a

ensile cutoff, and simulated the opening and sliding along pre-existing

racks, fracturing through the intact rock, as well as the kinematics in

ock slope failures; An et al. [33] adopted the same fracturing simula-

ion algorithm to investigate the fracturing and failure process of rock

lopes with non-preexist joints; Wu et al. [34] also applied the Mohr–

oulomb criterion to investigate the effects of the friction and cohesion

n the crack growth from a closed crack under compressive condition,

nd the results showed that the NMM based on the Mohr–Coulomb crite-

ion predict not only the pure tensile or shear crack, but also the mixed

ensile-shear crack growth satisfactorily.

In the present paper, to simulate the meso-scale deformation and

amage behaviors of the PBX, a bilinear cohesive contact relationship

BCCR) model with three parameters [3] and a visco-elastic constitu-

ive model of prony series with 22 parameters [35,36] are incorporated

n the NMM to represent the particle-binder interface and the poly-

er binder in the PBX, respectively, and the fracturing of the explosive

G. Kang et al. Engineering Analysis with Boundary Elements 96 (2018) 123–137

Fig. 11. Meso-scale deformation and fracturing process of the PBX with different PVFs under uniaxial tensile loading.

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articles and the binder in the PBX is captured by the fracturing algo-

ithm [22,33] based on the maximum tensile stress criterion and the

ohr–Coulomb criterion. The remain parts of the paper are organized

s follows: in Sections 2 and 3 , the way to incorporate the contact rela-

ionship model and the visco-elastic constitutive model in the NMM is

127

resented, respectively; in Section 4 , simulation examples of meso-scale

tructure PBX models under tensile and compressive loadings with an

mphasis to explore the influences of the PVF and the explosive particle

eometrical distribution (PGD) are presented; finally, main coclusions

re drawn in Section 5 .

G. Kang et al. Engineering Analysis with Boundary Elements 96 (2018) 123–137

Fig. 12. The macro-equivalent stress–strain curves of the PBX with different

PVFs under uniaxial tensile loading.

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. Incorporation of the bilinear cohesive contact relationship

BCCR) model in the NMM

.1. Development of a linear contact relatioship (LCR) model in the NMM

Here, a linear contact relationhsip model is incorporated in the NMM

rst. As for the simulation of physical discontinuities, the kinematic

onstraints of no tension and no penetration between the discontinu-

us interfaces are imposed by the penalty method in the NMM [37 , 38] .

ontacts emerge on the boundaries of two adjacent physical regions.

ach physical region, which is discretized into manifold elements, is de-

ned as a block, and the boundary of each physical region is defined as

block loop. More concepts and the fundamentals of the NMM can be

ound in references [21] , etc. Similar to that in the DDA method [21] , in

he NMM, contacts on discontinuity surfaces are mainly classified into

hree types: vertex-to-vertex, vertex-to-edge, and edge-to-edge.

In this study, in terms of the contact cases of vertex-to-vertex and

ertex-to-edge, it is assumed that there is no tension and no cohesion

ecause of “zero ” length of the contact surfaces. As for the contact case

f edge-to-edge shown in Fig. 2 , it falls into two conditions, namely,

uperposition and malposition of two edges, where i and j are manifold

lements, and segments P 1 P 2 and P 3 P 4 are edges of element i and j on

he opposite sides of a discontinuity, respectively.

In fact, the edge-to-edge contact is always treated as to two converted

ertex-to-edge contacts. Therefore, in Fig. 2 (a), it has two vertex-to-edge

ontacts: P 1 to P 3 P 4 , and P 2 to P 3 P 4 , respectively, and the half length

f segment P 1 P 2 or P 3 P 4 is defined as the contact length for each con-

erted contact. Similarly, in Fig. 2 (b), the two converted vertex-to-edge

ontacts are P 1 to P 3 P 4 and P 3 to P 1 P 2 , respectively, and the half length

f overlap segment P 1 P 3 is the contact length for each converted con-

act. Accordingly, the lengths of segment P 1 P 2 or P 3 P 4 in Fig. 2 (a) and

egment P 1 P 3 in Fig. 2 (b) are the exact contact lengths along the dis-

ontinuity between element i and j , respectively.

The NMM calculates the static as well as dynamic mechnaical be-

aviors of materials with a time marching schedule. In an initial NMM

odel, for the contacts of co-edges along a discontinuity with zero

idth, the contact state is “lock ”, and a tensile strength and a cohesive

trength are implemented to the co-edges. As the time step proceeds un-

er loading conditions, these locked contacts may turn into the states

f “open ” or “sliding ” based on the given strength values. If the friction

ngle of the co-edges is not zero and the normal contact displacement is

egative, the Mohr–Coulomb criterion is considered for the sliding con-

ition. Taking the converted contact of P 1 to P 3 P 4 from the edge-to-edge

128

ontact of P 1 P 2 to P 3 P 4 in Fig. 2 as an example, for tensile contacts, the

ermitted maximum normal contact displacement e 1 can be derived as:

1 =

𝑡 1 ⋅ 𝑜 6 𝑔 0

(1)

here t 1 is the tensile strength of the discontinuity, g 0 is the penalty

pring stiffness in the normal direction of the discontinuity interface, o 6 s the half length of segment P 3 P 4 in Fig. 2 (a), or the half length of seg-

ent P 1 P 3 for the case in Fig. 2 (b), namely, the length of contact. The

ormal and tangential contact displacements of vertex P 1 to edge P 3 P 4 ,

enoted as o 0 and o 1 , respectively, are calculated in the NMM program

nder loading conditions. If o 0 > e 1 , the state for the contact of P 1 to

3 P 4 will become “open ” from the previous state of “lock ”, which indi-

ates tensile failure happens and the tensile strength for the converted

ontact of P 1 to P 3 P 4 will disappear. For shear contacts, by employing

he Mohr–Coulomb criterion, the permitted tangential contact displace-

ent e 2 can be calculated as:

2 = tan 𝜃 ⋅ 𝑜 0 +

𝑡 2 ⋅ 𝑜 6 𝑔 0

(2)

here g 0 and o 6 have the same meanings as above, t 2 is the cohesive

trength of the discontinuity, and 𝜃 is the friction angle of the disconti-

uity interface. If e 2 > o 1 / h 2 , in which h 2 is the ratio between the normal

pring stiffness and tangential spring stiffness [21] , generally, h 2 = 2.5,

he contact state will keep as “lock ”. When e 2 < o 1 / h 2 is satisfied, the

ontact state will become “sliding ”, which indicates shear failure hap-

en and the cohesive strength will disappear.

After the above LCR model is incorporated into the NMM code, it

s verified by the two-block model shown in Fig. 3 (a). Two kinds of

eshing methods are used to generate NMM covers: independent cover

eshing method ( Fig. 3 (b)), in which each physical block is a manifold

lement, and the finite cover meshing method ( Fig. 3 (c)), in which each

hysical block is divided into many manifold elements. Sometimes it is

omputationally economical to adopt the independent cover meshing

ethod, because of the small contact and element numbers involved.

n the contrary, with finite cover meshing method, there exist many

ontacts along a discontinity and the contact condition is more compli-

ated, and also many elements in each physical block domain. However,

y both the two meshing methods, the total length of the contact co-edge

an be accurately calculated the same as 1 m for the model in Fig. 3 .

The modulus of the upper block and base block are set as 20 GPa

nd 100 GPa, respectively. The friction angle, tensile strength, and the

ohesive strength between the upper block and the base are set as 30°,

MPa, and 1 MPa, respectively. The normal contact spring stiffness g 0 s 200 GPa. According to h 2 , the tangential contact spring stiffness g s s calculated to be 80 GPa. The base is fixed, and upwards vertical or

orizontal displacement loading is applied to the upper block for ten-

ile or shear condition, respectively. Moreover, in the shear condition,

downward vertical volume force F V = 400 kN is applied to the upper

lock to compress the upper block on the base during horizontal sliding.

t is a tiny deformation problem for the two blocks and the two different

eshing methods should get nearly the same calculation results of the

nterface behaviors.

Under tensile condition, before the failure takes place, the tensile

raction force F tr between the two blocks equals to the total force F n f the normal contact spring forces along the block interface. When

ensile failure takes place, the critical value for F tr can be calcu-

ated as F tr c = 1 MPa × 1 m = 1000 kN, and the corresponding

heoretical critical normal contact displacement can be calculated as

0 c = F n / g 0 = F tr

c / g 0 = 0.005 mm.

Similarly, under shear condition, before the failure takes place, the

hear traction force F sr between the two blocks equals to the total force

s of the shear contact spring forces. When shear failure takes place, the

ritical value for F sr can be calculated as F sr c = F f + F c , where F f is the

tatic friction force and F c is the cohesive force, which can be calculated

s F c = 1 MPa × 1 m = 1000 kN, and F f = F N × tan 𝜃 = 230.94 kN,

G. Kang et al. Engineering Analysis with Boundary Elements 96 (2018) 123–137

Fig. 13. Meso-scale deformation and fracturing process of the PBX with different PVFs under uniaxial compressive loading.

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espectively. Then, F sr c can be derived as 1230.94 kN, and the corre-

ponding theoretical critical tangential contact displacement can be cal-

ulated as o 1 c = F s /( g 0 / h 2 ) = F sr

c /( g 0 / h 2 ) = 0.015386 mm.

The NMM simulation results of the total normal or shear spring

orces (contact forces) against the loading displacements with the

ndependent cover meshing method and the finite cover meshing

129

ethod are plotted in Fig. 5 . It can be found that those curves agree

ell with the corresponding theoretical ones. The maximum differ-

nce of the critical loading displacements between the NMM results

nd the theoretical results is smaller than 0.1%. The effectiveness

f the incorporated LCR model in the NMM is validated. Then, the

MM can be used to calculate the brittle failures along discontinuity

G. Kang et al. Engineering Analysis with Boundary Elements 96 (2018) 123–137

Fig. 14. The macro-equivalent stress–strain curves of the PBX with different

PVFs under uniaxial compressive loading.

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nterfaces under the tensile strength criterion and the Mohr–Coulomb

riterion.

.2. Incorporation of the BCCR model

In the following, a bilinear cohesive contact relationship (BCCR)

odel will be further incorporated in the NMM to describe the mechan-

cal behaviors of particle-binder interface in the PBX.

The BCCR model, shown in Fig. 6 , will be applied not only in the

ormal direction but also shear direction of an edge-to-edge contact,

here d is the open or sliding displacement of the contact. Variable F ,

escribed as F = T 0 ∙o 6 , is the tensile or cohesive force of the contact,

n which T 0 is the tensile or cohesive strength of the contact and o 6 s the contact length. As mentioned above, when the friction angle is

ot zero and the normal contact displacement is a negative value, the

ohr–Coulomb criterion is employed for sliding problems. In the model,

0 and k 1 are the spring stiffness at the ascending and the descending

tage, respectively, d 0 is the contact displacement corresponding to the

ensile or shear strength, and d c is the maximum open or sliding dis-

lacement of the contact. A damage variable D is introduced to describe

he damage behavior of the contact when d exceeds d 0 , as shown in

ig. 6 , and the spring stiffness k ’ of the damaged contact can be ob-

ained by k ’ = (1 − D ) k 0 . When d exceeds d 0 , if unloading takes place, k ’

ill be used in the unloading and the upcoming loading stage until D

ets a new value. In addition, as shown in Fig. 6 , the area of the trian-

le below the k 0 ascending line and the k 1 descending line is defined as

he fracture energy G F of interface. In the present research, d c is given

nstead of giving k 1 .

Under tensile condition, d 0 , d c can be derived as:

𝑑 0 =

𝑡 1 ⋅𝑜 6 𝑔 0

𝑑 𝑐 =

2 ⋅𝐺 𝐹 ⋅𝑜 6 𝑡 1 ⋅𝑜 6

(3)

here t 1 , g 0 , o 6 have the same meanings as in Eq. (1) . Then, the damage

alue D can be derived as:

𝐷 = 0 , 𝑜 0 < = 𝑑 0 𝐷 =

( 𝑜 0 − 𝑑 0 ) ⋅𝑑 𝑐 ( 𝑑 𝑐 − 𝑑 0 ) ⋅𝑜 0

, 𝑑 0 < 𝑜 0 < = 𝑑 𝑐

𝐷 = 1 , 𝑜 0 > 𝑑 𝑐

(4)

here o 0 is the normal displacement of the contact derived by the NMM.

he contact state will keep locked until D reaches 1.

Under shear condition, taking the Mohr–Coulomb into account, d 0 ,

can be derived as:

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𝑑 0 = tan 𝜃 ⋅ 𝑜 0 +

𝑡 2 ⋅𝑜 6 𝑔 0

𝑑 𝑐 = tan 𝜃 ⋅ 𝑜 0 +

2 ⋅𝐺 𝐹 ⋅𝑜 6 𝑡 2 ⋅𝑜 6 ⋅ℎ 2

(5)

here t 1 , g 0 , o 6 , t 2 , 𝜃, G F have the same meanings as above. Then, the

amage value D can be derived as:

𝐷 = 0 , 𝑜 1 ∕ ℎ 2 < = 𝑑 0 𝐷 =

( 𝑜 1 ∕ ℎ 2 − 𝑑 0 ) ⋅𝑑 𝑐 ( 𝑑 𝑐 − 𝑑 0 ) ⋅𝑜 1 ∕ ℎ 2

, 𝑑 0 < 𝑜 1 ∕ ℎ 2 < = 𝑑 𝑐

𝐷 = 1 , 𝑜 1 ∕ ℎ 2 > 𝑑 𝑐

(6)

here o 1 is the tangential displacement of the contact, and h 2 is the ratio

f normal spring stiffness to tangential spring stiffness [21] . Normally,

2 takes a constant value of 2.5. The contact state will keep locked until

reaches 1.

The geometric model in Fig. 3 is adopted again to verify the im-

lemented BCCR model. The geometry, material and computation pa-

ameters are the same as that used in previous simulations. The facture

nergy G F is given as 100 J/m. Then, d 0 , d c can be calculated according

o Eq. (3) or Eq. ( 5 ).

With the BCCR model, the force analysis under tensile and shear con-

itions are the same as that in Fig. 4 . However, with this contact relation-

hip model, when the normal or tangential contact displacements exceed

0 which corresponds to the tensile or shear strength of the interface, the

pring forces decrease slowly to zero as the loading proceeds, instead of

n instant drop to zero as that in the LCR model. Moreover, in the BCCR

odel, the maximum contact displacement d c corresponds to tensile or

hear separation of the discontinuity interface. With G F = 100 J/m, un-

er the tensile condition, the theoretical maximum normal contact dis-

lacement can be calculated as d c = 2 G F / t 1 = 0.2 mm, and under the

hear condition, the theoretical maximum tangential contact displace-

ent is d c = F f / g s + 2 G F / t 2 = 0.20288673 mm. The NMM simulation

esults of the contact forces against the loading displacements under

he tensile or shear condition are plotted Fig. 7 . It is found that the nu-

erically derived contact displacements d 0 and d c agree well with the

orresponding theoretical values. However, there are small differences

etween the numerical and theoretical values for the contact forces. As

entioned above, in the BCCR, the displacement field and force field

long the discontinuity interface are constrained to be continuous. When

he damage variable D becomes larger than 0 under certain moment, the

udden change of the spring stiffness from k 0 to k ’ may lead to the sud-

en change of the displacement on the interface, and result in the sudden

hanges of the contact forces as shown in Fig. 7 (a) and (b). However, the

aximum differences between numerical and theoretical values are less

han 5%. Therefore, it is interpreted that the BCCR model implemented

n the NMM is ready to be applied to simulate the interface cohesive

roperties of the PBX.

. Incorporation of the visco-elastic constitutive model of prony

eries in the NMM

The polymer binder is an important constituent in the PBX and its

roperties have an obvious influence on the PBX performances [3] . Gen-

rally, the binder has good chemical and thermal stability. Mechanical

xperiment studies demonstrate that the binder is a visco-elastic mate-

ial which is extremely sensitive to both the strain rate and the tem-

erature [5,6] . Here, a visco-elastic constitutive model which combines

everal Maxwell elements with relaxation times 𝜏1 , 𝜏2 , …, 𝜏N is intro-

uced and implemented in the NMM. As described in references [35,36] ,

he shear relaxation modulus in the generalized Maxwell model can be

xpressed by the following relaxation time distribution:

( 𝑡 ) = 𝐺 𝑒 +

𝑁 ∑𝑛 =1

𝐺 𝑛 𝑒 − 𝑡 ∕ 𝑡 𝑛 (7)

here t n is the n th relaxation time, G n is the n th shear relaxation mod-

lus, and G , if different from zero, is called the equilibrium modulus.

e

G. Kang et al. Engineering Analysis with Boundary Elements 96 (2018) 123–137

Fig. 15. Meso-scale deformation and fracturing process of the PBX with different PGDs under uniaxial tensile loading.

G

r

L

𝐿

i

t

b

enerally, N is 22 and G e is 0 in this model [36] . The calculation of the

elaxation time is introduced in detail in reference [35] . A Williams–

andell–Ferry (WLF) equation is presented as follows:

𝑜𝑔( 𝛼𝑇 ) = −6 . 5 ( 𝑇 − 𝑇 0 )

120 + 𝑇 − 𝑇 (8)

0 𝜏

131

n which T 0 , identical to 19 °C, is a reduced temperature, and 𝛼T is the

ime-temperature shift factor. The relaxation time 𝜏n can be calculated

y:

= 1 . 5 ⋅ 𝛼 10 (7− 𝑛 ) (9)

𝑛 𝑇

G. Kang et al. Engineering Analysis with Boundary Elements 96 (2018) 123–137

Table 1

The shear relaxation modulus G n , the Young’s relaxation modulus E n , and the relaxation time 𝜏n .

G e G 1 G 2 G 3 G 4 G 5 G 6 G 7

Shear relaxation Prony terms(/MPa) 0 0.00417 0.00741 0.00159 0.0380 0.0676 0.0891 0.0056

G 8 G 9 G 10 G 11 G 12 G 13 G 14 G 15

0.1622 0.2218 0.4753 2.104 2.618 12.882 52.481 223.872

G 16 G 17 G 18 G 19 G 20 G 21 G 22

436.516 457.088 346.737 251.188 177.83 117.489 75.8

Normal relaxation Prony terms(/MPa) E e E 1 E 2 E 3 E 4 E 5 E 6 E 7 0 0.0125 0.0222 0.0475 0.114 0.203 0.2673 0.0168

E 8 E 9 E 10 E 11 E 12 E 13 E 14 E 15

0.4866 0.6654 1.426 6.312 7.854 38.647 157.44 671.616

E 16 E 17 E 18 E 19 E 20 E 21 E 22

1309.54 1371.26 1040.21 753.56 533.48 352.46 227.573

Relaxation time Prony terms(/s) 𝜏1 𝜏2 𝜏3 𝜏4 𝜏5 𝜏6 𝜏7 𝜏8

7.355e 5 7.355e 4 7.355e 3 7.355e 2 7.355e 1 7.355 7.355e − 1 7.355e − 2

𝜏9 𝜏10 𝜏11 𝜏12 𝜏13 𝜏14 𝜏15 𝜏16

7.355e − 3 7.355e − 4 7.355e − 5 7.355e − 6 7.355e − 7 7.355e − 8 7.355e − 9 7.355e − 10

𝜏17 𝜏18 𝜏19 𝜏20 𝜏21 𝜏22

7.355e − 11 7.355e − 12 7.355e − 13 7.355e − 14 7.355e − 15 7.355e − 16

Fig. 16. The macro-equivalent stress–strain curves of the PBX with different

PGDs under uniaxial tensile loading.

t

𝐸

i

i

m

𝐸

t

r

l

i

E

𝜎

w ∑

t

𝑆

i

c

H

a

[

w

𝑆

𝑆

𝑆

𝑆

i

t

𝑡

a

𝑡

a

Δ

i

t

i

Δ

𝑡

a

𝑡

For the case of uniaxial stress condition, the relationship between

he shear modulus and the Young’s modulus is:

( 𝑡 ) = 2 ⋅ 𝐺( 𝑡 ) ⋅ (1 + 𝑣 ) (10)

n which, 𝜈 is the Poisson’s ratio, and for the polymer binder of the PBX,

t is 0.495, generally. Therefore, the Young’s modulus can be approxi-

ately expressed as:

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

According to reference [35] and the above equations, at the room

emperature (25 °C), the terms for the Prony series defining the shear

elaxation modulus G n , the Young’s relaxation modulus E n and the re-

axation time 𝜏n are listed in Table 1 .

The visco-elastic constitutive model above can be further expressed

n the following Prony series form, in which an additional elastic term

0 𝜀 is further considered:

= 𝐸 0 𝜀 +

𝑁 ∑𝑛 =1

𝐸 𝑛 ∫𝑡

0

∙𝜀 𝑒

− 𝑡 − 𝜏𝜏𝑛 𝑑𝜏 (12)

here E 0 is the Young’s modulus, N equals 22, and

𝑁

𝑛 =1 𝐸 𝑛 ∫ 𝑡

0 ∙𝜀 𝑒

− 𝑡 − 𝜏𝜏𝑛 𝑑𝜏is the visco-elastic term. Extend Eq. (12) to

he two dimensional state as shown below:

𝑖𝑗 = 𝐸 0 [ 𝐀 ] 𝐄 𝑘𝑙 +

𝑁 ∑𝑛 =1

𝐸 𝑛 ∫𝑡

0 [ 𝐀 ]

𝜕 𝐄 𝑘𝑙

𝜕𝜏𝑒 − 𝑡 − 𝜏

𝜏𝑛 𝑑𝜏 (13)

132

n which, S ij is the component of the Kirchhoff stress tensor, and E kl is a

olumn matrix consisting of the components of the Green strain tensor.

ere, for the plane stress or plane strain, matrix [ A ] can be expressed

s below, respectively:

𝐀 ] =

1 1 − 𝜈2

⎡ ⎢ ⎢ ⎣ 1 𝜈 0 𝜈 1 0 0 0 1− 𝜈

2

⎤ ⎥ ⎥ ⎦ , [ 𝐀 ] =

1 − 𝜈

(1 + 𝜈)(1 − 2 𝜈)

⎡ ⎢ ⎢ ⎢ ⎣ 1 𝜈

1− 𝜈 0 𝜈

1− 𝜈 1 0 0 0 1−2 𝜈

2(1− 𝜈)

⎤ ⎥ ⎥ ⎥ ⎦ (14)

here 𝜈 is Poisson’s ratio of the material.

Define variables 𝑆 𝐸 𝑖𝑗

, 𝑆 𝑀 1 𝑖𝑗

, 𝑆 𝑀 2 𝑖𝑗

… 𝑆 𝑀 𝑁

𝑖𝑗 :

𝐸 𝑖𝑗 = 𝐸 0 [ 𝐀 ] 𝐄 𝑘𝑙

𝑀 1 𝑖𝑗

= 𝐸 1 ∫ 𝑡

0 [ 𝐀 ] 𝜕 𝐄 𝑘𝑙 𝜕𝜏

𝑒 − 𝑡 − 𝜏

𝜏1 𝑑𝜏

𝑀 2 𝑖𝑗

= 𝐸 2 ∫ 𝑡

0 [ 𝐀 ] 𝜕 𝐄 𝑘𝑙 𝜕𝜏

𝑒 − 𝑡 − 𝜏

𝜏2 𝑑𝜏

...

𝑀 𝑁

𝑖𝑗 = 𝐸 𝑁

∫ 𝑡

0 [ 𝐀 ] 𝜕 𝐄 𝑘𝑙 𝜕𝜏

𝑒 − 𝑡 − 𝜏𝜏𝑁 𝑑𝜏

(15)

n which 𝑆 𝐸 𝑖𝑗

is elastic term, and 𝑆 𝑀 1 𝑖𝑗

, 𝑆 𝑀 2 𝑖𝑗

, …, 𝑆 𝑀 𝑁

𝑖𝑗 are the visco-elastic

erms.

As for the elastic term 𝑆 𝐸 𝑖𝑗

, at time t ,

𝑆 𝐸 𝑖𝑗 = 𝐸 0 [ 𝐀 ] 𝑡 𝐄 𝑘𝑙 (16)

nd at time t + Δt ,

+Δ𝑡 𝑆 𝐸 𝑖𝑗 = 𝐸 0 [ 𝐀 ] 𝑡 +Δ𝑡 𝐄 𝑘𝑙 (17)

Therefore, from time t to t + Δt , increment form of 𝑆 𝐸 𝑖𝑗

can be derived

s:

𝑆 𝐸 𝑖𝑗 =

𝑡 +Δ𝑡 𝑆 𝐸 𝑖𝑗 −

𝑡 𝑆 𝐸 𝑖𝑗 = 𝐸 0 [ 𝐀 ]Δ𝐄 𝑘𝑙 (18)

n which E kl is a column matrix consisting of the components of

he Green strain tensor, ΔE kl is the column matrix composed of the

ncrements of the components of the Green strain tensor, namly,

E kl = [ ΔE 11 , ΔE 22 , ΔE 12 ] T .

As for the arbitrary visco-elastic term 𝑆 𝑀 𝑛

𝑖𝑗 , at time t ,

𝑆 𝑀 𝑛

𝑖𝑗 = 𝐸 𝑛 ∫

𝑡

0 [ 𝐀 ]

𝜕 𝑡 𝐄 𝑘𝑙

𝜕𝜏𝑒 − 𝑡 − 𝜏

𝜏𝑛 𝑑𝜏 (19)

nd at time t + Δt ,

+Δ𝑡 𝑆 𝑀 𝑛

𝑖𝑗 = 𝐸 𝑛 ∫

𝑡 +Δ𝑡

0 [ 𝐀 ]

𝜕 𝑡 +Δ𝑡 𝐄 𝑘𝑙

𝜕𝜏𝑒 − 𝑡 +Δ𝑡 − 𝜏

𝜏𝑛 𝑑𝜏 (20)

G. Kang et al. Engineering Analysis with Boundary Elements 96 (2018) 123–137

Fig. 17. Meso-scale deformation and fracturing process of the PBX with nearly the same PGDs under uniaxial compressive loading.

s

As to Eq. (20) ,

𝑡 + Δ𝑡 𝑆 𝑀 𝑛

𝑖𝑗 = 𝐸 𝑛 ∫

𝑡 +Δ𝑡

0 [ 𝐀 ]

𝜕 𝑡 +Δ𝑡 𝐄 𝑘𝑙

𝜕𝜏𝑒 − 𝑡 +Δ𝑡 − 𝜏

𝜏𝑛 𝑑𝜏

= 𝐸 𝑛 ∫𝑡

0 [ 𝐀 ]

𝜕 𝑡 +Δ𝑡 𝐄 𝑘𝑙

𝜕𝜏𝑒 − 𝑡 +Δ𝑡 − 𝜏

𝜏𝑛 𝑑 𝜏 + 𝐸 𝑛 ∫𝑡 +Δ𝑡

𝑡

[ 𝐀 ] 𝜕 𝑡 +Δ𝑡 𝐄 𝑘𝑙

𝜕𝜏𝑒 − 𝑡 +Δ𝑡 − 𝜏

𝜏𝑛 𝑑 𝜏

133

= 𝐸 𝑛 ∫𝑡

0 [ 𝐀 ]

𝜕 𝑡 +Δ𝑡 𝐄 𝑘𝑙

𝜕𝜏𝑒 − 𝑡 − 𝜏

𝜏𝑛 𝑑 𝜏 ⋅ 𝑒 − Δ𝑡 𝜏𝑛 + 𝐸 𝑛 ∫

𝑡 +Δ𝑡

𝑡

[ 𝐀 ] 𝜕 𝑡 +Δ𝑡 𝐄 𝑘𝑙

𝜕𝜏𝑒 − 𝑡 +Δ𝑡 − 𝜏

𝜏𝑛 𝑑 𝜏

(21)

When the time increment Δt is very small, it is interpreted that the

train rates at time t and time t + Δt are identical, i.e.:

𝜕 𝑡 𝐄 𝑘𝑙

𝜕𝜏=

𝜕 𝑡 +Δ𝑡 𝐄 𝑘𝑙

𝜕𝜏=

Δ𝐄 𝑘𝑙

Δ𝑡 (22)

G. Kang et al. Engineering Analysis with Boundary Elements 96 (2018) 123–137

Fig. 18. The macro-equivalent stress–strain curves of the PBX with different

PGDs under uniaxial compressive loading.

𝑡

(

Δ

s

Δ

)

)

⎧⎪⎨⎪⎩

i

t

[

t

⎧⎪⎨⎪⎩

∐

i

a

𝑆

i

𝑆

a

−

i

−

v

i

a

e

Therefore, Eq. (21) can be further written as:

+ Δ𝑡 𝑆 𝑀 𝑛

𝑖𝑗 = 𝐸 𝑛 ∫

𝑡

0 [ 𝐀 ]

𝜕 𝑡 𝐄 𝑘𝑙

𝜕𝜏𝑒 − 𝑡 − 𝜏

𝜏𝑛 𝑑 𝜏 ⋅ 𝑒 − Δ𝑡 𝜏𝑛 + 𝐸 𝑛 [ 𝐀 ]

Δ𝐄 𝑘𝑙

Δ𝑡 ∫𝑡 +Δ𝑡

𝑡

𝑒 − 𝑡 +Δ𝑡 − 𝜏

𝜏𝑛 𝑑 𝜏

= 𝐸 𝑛 ∫𝑡

0 [ 𝐀 ]

𝜕 𝑡 𝐄 𝑘𝑙

𝜕𝜏𝑒 − 𝑡 − 𝜏

𝜏𝑛 𝑑 𝜏 ⋅ 𝑒 − Δ𝑡 𝜏𝑛 + 𝐸 𝑛 [ 𝐀 ]

Δ𝐄 𝑘𝑙

Δ𝑡 ∫𝑡 +Δ𝑡

𝑡

𝑒 − 𝑡 +Δ𝑡 − 𝜏

𝜏𝑛 𝑑 𝜏

=

𝑡 𝑆 𝑀 𝑛

𝑖𝑗 ⋅ 𝑒

− Δ𝑡 𝜏𝑛 + 𝐸 𝑛 [ 𝐀 ]

Δ𝐄 𝑘𝑙

Δ𝑡 𝜏1

(

1 − 𝑒 − Δ𝑡 𝜏𝑛

)

(23)

Then, the increment form of 𝑆 𝑀 𝑛

𝑖𝑗 can be calculated from Eqs. (19) ,

23 ):

𝑆 𝑀 𝑛

𝑖𝑗 =

𝑡 +Δ𝑡 𝑆 𝑀 𝑛

𝑖𝑗 −

𝑡 𝑆 𝑀 𝑛

𝑖𝑗 = 𝐸 𝑛 [ 𝐀 ]

Δ𝐄 𝑘𝑙

Δ𝑡 𝜏1

(

1 − 𝑒 − Δ𝑡 𝜏𝑛

)

−

𝑡 𝑆 𝑀 1 𝑖𝑗

(

1 − 𝑒 − Δ𝑡 𝜏𝑛

)

(24)

Combining Eqs. (18) and (24) , in time increment Δt , the Kirhhoff

tress increment can be expressed as:

𝑆 𝑖𝑗 = Δ𝑆 𝐸 𝑖𝑗 + Δ𝑆 𝑀 1

𝑖𝑗 + Δ𝑆 𝑀 2

𝑖𝑗 + ⋯ + Δ𝑆 𝑀 𝑁

𝑖𝑗

= 𝐸 0 [ 𝐀 ]Δ𝐄 𝑘𝑙 +

𝑁 ∑𝑛 =1

{

𝐸 𝑛 [ 𝐀 ] Δ𝐄 𝑘𝑙

Δ𝑡 𝜏𝑛

(

1 − 𝑒 − Δ𝑡 𝜏𝑛

)

−

𝑡 𝑆 𝑀 𝑛

𝑖𝑗

(

1 − 𝑒 − Δ𝑡 𝜏𝑛

) }(25

In order to apply Eq. (25) to into the NMM, it is expanded as follows:

Δ𝑆 𝑖𝑗 = 𝐸 0 [ 𝐀 ]Δ𝐄 𝑘𝑙 +

𝑁 ∑𝑛 =1

𝐸 𝑛 [ 𝐀 ] Δ𝐄 𝑘𝑙

Δ𝑡 𝜏𝑛

(

1 − 𝑒 − Δ𝑡 𝜏𝑛

)

−

𝑁 ∑𝑛 =1

𝑡 𝑆 𝑀 𝑛

𝑖𝑗

(

1 − 𝑒 − Δ𝑡 𝜏𝑛

)

= 𝐸 0 [ 𝐀 ]Δ𝐄 𝑘𝑙 +

𝑁 ∑𝑛 =1

𝐸 𝑛 𝜏𝑛

(

1 − 𝑒 − Δ𝑡 𝜏𝑛

)

Δ𝑡 [ 𝐀 ]Δ𝐄 𝑘𝑙 −

𝑁 ∑𝑛 =1

𝑡 𝑆 𝑀 𝑛

𝑖𝑗

(

1 − 𝑒 − Δ𝑡 𝜏𝑛

)

=

⎛ ⎜ ⎜ ⎝ 𝐸 0 +

𝑁 ∑𝑛 =1

𝐸 𝑛 𝜏𝑛 1 − 𝑒

− Δ𝑡 𝜏𝑛

Δ𝑡

⎞ ⎟ ⎟ ⎠ [ 𝐀 ]Δ𝐄 𝑘𝑙 −

𝑁 ∑𝑛 =1

𝑡 𝑆 𝑀 𝑛

𝑖𝑗

(

1 − 𝑒 − Δ𝑡 𝜏𝑛

)

(26

Write it in the matrix form:

134

Δ𝑆 11 Δ𝑆 22 Δ𝑆 12

⎫ ⎪ ⎬ ⎪ ⎭ =

⎛ ⎜ ⎜ ⎝ 𝐸 0 +

𝑁 ∑𝑛 =1

𝐸 𝑛 𝜏𝑛 1 − 𝑒

− Δ𝑡 𝜏𝑛

Δ𝑡

⎞ ⎟ ⎟ ⎠ [ 𝐀 ] ⎧ ⎪ ⎨ ⎪ ⎩ Δ𝐸 11 Δ𝐸 22 Δ𝐸 12

⎫ ⎪ ⎬ ⎪ ⎭ −

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝑁 ∑𝑛 =1

𝑡 𝑆 𝑀 𝑛

11 (1 − 𝑒 − Δ𝑡 𝜏𝑛 )

𝑁 ∑𝑛 =1

𝑡 𝑆 𝑀 𝑛

22 (1 − 𝑒 − Δ𝑡 𝜏𝑛 )

𝑁 ∑𝑛 =1

𝑡 𝑆 𝑀 𝑛

12 (1 − 𝑒 − Δ𝑡 𝜏𝑛 )

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (27)

n which, 𝑡 𝑆 𝑀 𝑛

𝑖𝑗 ( i = 1, 2; j = 1, 2) are stress values reserved at the end of

he last time step, and Δt is the time interval of the current time step.

By assuming:

𝐏 ] =

⎛ ⎜ ⎜ ⎝ 𝐸 0 +

𝑁 ∑𝑛 =1

𝐸 𝑛 𝜏𝑛 1 − 𝑒

− Δ𝑡 𝜏𝑛

Δ𝑡

⎞ ⎟ ⎟ ⎠ [ 𝐀 ] , [ 𝐐 ] =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

−

𝑁 ∑𝑛 =1

𝑡 𝑆 𝑀 𝑛

11 (1 − 𝑒 − Δ𝑡 𝜃𝑛 )

−

𝑁 ∑𝑛 =1

𝑡 𝑆 𝑀 𝑛

22 (1 − 𝑒 − Δ𝑡 𝜃𝑛 )

−

𝑁 ∑𝑛 =1

𝑡 𝑆 𝑀 𝑛

12 (1 − 𝑒 − Δ𝑡 𝜃𝑛 )

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (28)

hen Eq. (27) can be written as:

Δ𝑆 11 Δ𝑆 22 Δ𝑆 12

⎫ ⎪ ⎬ ⎪ ⎭ = [ 𝐏 ]

⎧ ⎪ ⎨ ⎪ ⎩ Δ𝐸 11 Δ𝐸 22 Δ𝐸 12

⎫ ⎪ ⎬ ⎪ ⎭ + [ 𝐐 ] (29)

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

𝑒

= ∬ 𝐴

1 2 (Δ𝐸 11 Δ𝐸 22 Δ𝐸 12

)⎛ ⎜ ⎜ ⎝ Δ𝑆 12 Δ𝑆 12 Δ𝑆 12

⎞ ⎟ ⎟ ⎠ 𝑑 𝑥𝑑 𝑦

= ∬ 𝐴

1 2 (Δ𝐸 11 Δ𝐸 22 Δ𝐸 12

)⎧ ⎪ ⎨ ⎪ ⎩ [ 𝑃 ] ⎧ ⎪ ⎨ ⎪ ⎩ Δ𝐸 11 Δ𝐸 22 Δ𝐸 12

⎫ ⎪ ⎬ ⎪ ⎭ + [ 𝑄 ]

⎫ ⎪ ⎬ ⎪ ⎭ 𝑑 𝑥𝑑 𝑦

= ∬ 𝐴

1 2 (Δ𝐸 11 Δ𝐸 22 Δ𝐸 12

)[ 𝑃 ] ⎧ ⎪ ⎨ ⎪ ⎩ Δ𝐸 11 Δ𝐸 22 Δ𝐸 12

⎫ ⎪ ⎬ ⎪ ⎭ 𝑑 𝑥𝑑 𝑦

+ ∬ 𝐴

1 2 (Δ𝐸 11 Δ𝐸 22 Δ𝐸 12

)[ 𝑄 ] 𝑑 𝑥𝑑 𝑦

=

1 2 [ 𝐷 𝑒 ] 𝑇 𝑆 𝑒 [ 𝐵 𝑒 ] 𝑇 [ 𝑃 ][ 𝐵 𝑒 ][ 𝐷 𝑒 ] +

1 2 [ 𝐷 𝑒 ] 𝑇 𝑆 𝑒 [ 𝐵 𝑒 ] 𝑇 [ 𝑄 ] (30)

n which, S e is the area of the manifold element, [ B e ] = ( B e (1) B e (2) B e (3) ),

nd e ( i ) is the i th physical cover of the manifold element, i = 1, 2, 3.

Therefore, the following stiffness matrix for the manifold element

𝑒 [ 𝐵 𝑒 ] 𝑇 [ 𝐏 ][ 𝐵 𝑒 ] = 𝑆 𝑒

⎧ ⎪ ⎨ ⎪ ⎩ [ 𝐵 𝑒 (1) ] 𝑇 [ 𝐵 𝑒 (2) ] 𝑇 [ 𝐵 𝑒 (3) ] 𝑇

⎫ ⎪ ⎬ ⎪ ⎭ [ 𝐏 ]([ 𝐵 𝑒 (1) ] [ 𝐵 𝑒 (2) ] [ 𝐵 𝑒 (3) ] ) (31)

s added to the global coefficient matrix of the NMM:

𝑒 [ 𝐵 𝑒 ( 𝑟 ) ] 𝑇 [ 𝐏 ][ 𝐵 𝑒 ( 𝑠 ) ] →[𝐊 𝑒 ( 𝑟 ) 𝑒 ( 𝑠 )

], 𝑟, 𝑠 = 1 , 2 , 3 (32)

nd the following loading matrix for the manifold element

1 2 𝑆 𝑒 [ 𝐵 𝑒 ] 𝑇 [ 𝐐 ] = −

1 2 𝑆 𝑒

⎧ ⎪ ⎨ ⎪ ⎩ [ 𝐵 𝑒 (1) ] 𝑇 [ 𝐵 𝑒 (2) ] 𝑇 [ 𝐵 𝑒 (3) ] 𝑇

⎫ ⎪ ⎬ ⎪ ⎭ [ 𝐐 ] (33)

s added to the global loading matrix of the NMM:

1 2 𝑆 𝑒 [ 𝐵 𝑒 ( 𝑟 ) ] 𝑇 [ 𝐐 ] →

{

𝐅 𝑒 ( 𝑟 ) }

, 𝑟 = 1 , 2 , 3 (34)

A single manifold element model, shown in Fig. 8 , is simulated to

erify the validity of above Prony series constitutive model, which is

mplemented in the NMM. The length and width of the element model

re 1 m and 0.1 m, respectively. The left end is fixed, and at the right

nd, a tensile displacement load with different velocities of 0.5 mm/s,

G. Kang et al. Engineering Analysis with Boundary Elements 96 (2018) 123–137

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mm/s, 50 mm/s, 500 mm/s, 5000 mm/s is applied, respectively. The

orresponding strain rates are 0.0005 s − 1 , 0.005 s − 1 , 0.05 s − 1 , 0.5 s − 1 ,

s − 1 . The length/width ratio of 10 for the element model guarantees

ne-dimensional deformation of the element to a large extent.

Two different temperatures of 25 °C and 40 °C are considered.

he corresponding relaxation times can be calculated by the WLF Eqs.

8) and (9) . The shear relaxation modulus and the Young’s relaxation

odulus can be found in Table 1 . The Young’s modulus of elastic term

0 is taken as 1 MPa according to reference [16] , and the Poisson’s ratio

s 0.495.

Generally, at a certain loading velocity, the strain state is a constant.

hen Eq. (12) can be further deduced as:

= 𝐸 0 𝜀 +

𝑁 ∑𝑛 =1

𝐸 𝑛 ∫𝑡

0

∙𝜀 𝑒

− 𝑡 − 𝜏𝜏𝑛 𝑑𝜏

= 𝐸 0 𝜀 +

𝑁 ∑𝑛 =1

𝐸 𝑛

∙𝜀 ∫

𝑡

0 𝑒 − 𝑡 − 𝜏

𝜏𝑛 𝑑𝜏 (35)

n which, the term ∫ 𝑡

0 𝑒 − 𝑡 − 𝜏

𝜏𝑛 𝑑𝜏 can be analytically integrated as follows:

𝑡

0 𝑒 − 𝑡 − 𝜏

𝜏𝑛 𝑑𝜏 = 𝜏𝑛 (1 − 𝑒 − 𝑡 𝜏𝑛 ) (36)

Then,

= 𝐸 0 𝜀 +

𝑁 ∑𝑛 =1

[ 𝐸 𝑛

∙𝜀 𝜏𝑛 (1 − 𝑒

− 𝑡 𝜏𝑛 )] (37)

Eq. (37) is the theoretical one-dimensional stress–strain relationship

f the Prony series constitutive model at constant strain rates. The theo-

etical and NMM simulation results of the stress–strain relationships of

he single element model in Fig. 8 at the two different temperatures are

lotted in Fig. 9 .

It can be found that the agreement between the theoretical and simu-

ation results is satisfactory. The validity of the Prony series constitutive

odel incorporated in the NMM is verified, and this model will be ap-

lied to simulate the thermo-mechanical behavior of the PBX binder in

ection 4 .

. NMM simulations of meso-mechanical behaviors of the PBX

The meso-mechanical responses of the PBX, especially the damage

ehaviors, such as particle/binder interface debonding and crack evolu-

ion in the particle or the binder, have strong influences on the rigidity,

tructural strength, ignition and detonation performances of the PBX

5 , 6] . Some studies [16 , 17] indicate that, many factors, such as the ex-

losive PVF and PGD, the explosive particle, binder and interface prop-

rties, the loading rates and environmental temperatures, and the initial

amages, et al., may have influences on the meso-mechanical behaviors

f the PBX. In the present paper, only two factors including the PVF and

he PGD are taken into account. Two groups of PBX meso-scale structure

odels with different PVF and different PGD are established. One of the

odel is shown in Fig. 10 . The size of the model is 1.5 mm × 1.5 mm.

he number of the manifold elements is 9376. A fixing board is set below

he model to constrain the x- and y- direction displacements of bottom

oundary under the tensile condition, or to constrain the y -direction dis-

lacement under the compressive condition. The left and right bound-

ries of the model are set to be free. A loading board is set above the

odel and a upward or downward displacement loading with the ve-

ocity of 0.4 mm/s will be applied. Accordingly, the macro-equivalent

train rate of the PBX model is 0.267 s − 1 . A measurement board is placed

etween the loading board and the PBX model. It is assumed that the

easurement board is in a force equilibrium state during the loading

rocess. The force imposed on the measurement board through the load-

ng board can be regarded the same as that applied to the PBX model.

herefore, the measurement board can be used to record the macro-

quivalent mechanical response of the PBX meso-structure.

135

In the NMM simulation, the explosive particles are elastic. The

aterial parameters for the three constituents of the PBX, which are

aken from references [3 , 25] , are listed in Table 2 . The parameters in

he Prony series constitutive model for the polymer binder shown in

able 1 are used. As mentioned above, the NMM is a unified continuous-

iscontinuous numerical method, and an open-close iteration process is

xecuted to achieve the convergence of the contact status at the discon-

inuity interfaces in each time step. Some numerical control parameters,

uch as the time step, displacement ratio and the spring stiffness of the

ontact, have extremely important influences on the convergence of the

MM computations. Through massive numerical experiments, the time

tep, displacement ratio and spring stiffness are all selected to be 2e-5 s,

e-4 and 300 GPa, respectively, for all the PBX simulation examples in

he present paper. In the following, the influences of the PVF and the

GD on the meso-scale deformation and damage behaviors of the PBX

ill be numerically investigated.

.1. The particle volume fraction (PVF)

Three different PVFs, namely, 76.95%, 80.35%, 83.18%, are con-

idered. These PVF values are obviously lower than that in real PBX

aterials which can reach 90–95%. This is due to the fact that it is dif-

cult to construct a two-dimensional PBX meso-scale model with very

igh PVF, especially when irregular particle shape is considered.

Fig. 11 shows the deformation and fracturing process of the three

odels under the tensile loading. Because the particle is much stronger

han the binder, it can be found that the binder deforms much more than

he particles during the whole loading process. Generally, it is also found

hat the particle/binder interface debonding and the polymer binder

racturing are the major damage mode.

In detail, it can be observed that fractures appear along the parti-

le/binder interface firstly. This is because the interface is the weakest

art in the PBX meso-structure, which is extremely easy to deform and

racture under tension. As the loading proceeds, the fractures along the

article/binder interface, accompanying with some occasional fractures

enetrating the binder, gradually form some main cracks, as marked by

he arrows in the figures. The direction of the main crack is also gener-

lly perpendicular to the loading direction. This is a typical phenomenon

asy to be observed in meso-scale experiment study of the PBX [5 , 6] .

oreover, it can be found that the initial fracturing time decreases as

he PVF increases. This could be attributed that in the model with lower

VF, the larger volume fraction of the binder provides greater capability

o bear the deformation; meanwhile, with a higher PVF, there is a larger

mount of particle/binder interfaces, which provides more chances for

ebonding failures.

Fig. 12 shows the corresponding macro-equivalent stress–strain

urves of the simulations in Fig. 11 . It can be found that the curve climbs

aster as the PVF increases, indicating a larger maco-equivalent elas-

ic modulus of the PBX. The increase of the PVF also helps to improve

he macro-equivalent tensile strength of the PBX. The maximum macro-

quivalent strength derived from the simulation is 0.90 MPa, lower than

he experimental value of 1.21 MPa [36] . This is because the maximum

VF of the simulation models is only 83%, not as high as that in the real

BX material.

Fig. 13 shows the deformation and fracturing process of the three

odels under the compressive loading. It can be found that, similar to

hat under the tensile loading condition, the initial fractures also appear

long the particle/binder interface, however, nearly parallel to the load-

ng direction. Generally, the initial particle/binder interface debonding

s more likely to be generated on the boundaries of bigger particles [5] ,

s marked by arrows in the figures. As the loading proceeds, the inter-

ace debonding along with some fractures penetrating the binder grad-

ally forms the shear bands in the inclined direction, as marked by the

ed dotted lines. Under the uniaxial compressive condition, the parti-

le/binder interface debonding and the polymer binder fracturing are

lso the major damage mode. However, different from that under the

G. Kang et al. Engineering Analysis with Boundary Elements 96 (2018) 123–137

Table 2

Material parameters of the PBX.

Constituents Density (g/cm

− 3 ) Tensile modulus Compressive modulus Possion’s ratio Friction angle Cohesive strength Tensile strength Fracture energy

(MPa) (MPa) ( °) (MPa) (MPa) (J/m)

Particle 1.9 13,375 13,375 0.322 30 30 10 /

Binder 1.28 0.37 100 0.495 25 30 2 /

Interface / / / / 15 1.66 1.66 81

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niaxial tension, the explosive particle fracturing also can be observed

nder compression, as shown in Fig. 13 (c). Moreover, the increase of

he PVF also leads to early appearance of the initial fractures, and leads

o early macro-failure of the PBX under the compressive displacement

oading as well.

Fig. 14 shows the corresponding macro-equivalent stress–strain

urves of the simulations in Fig. 13 . Similar to that in the tension

ases, as the PVF increases, the curve climbs faster and a larger macro-

quivalent elastic modulus of the PBX is obtained. The increase of the

VF also improves the macro-equivalent compressive strength, and the

aximum macro-equivalent strength derived from the simulation is

bout 10.8 MPa, very close to the experimental result of 11.11 MPa

n reference [39] .

.2. The particle geometrical distribution (PGD)

Figs. 15 and 16 show the meso-scale deformation and fracturing pro-

ess and the maco-equivalent stress–strain curves of three PBX meso-

tructure models with approximately the same PVF but different PGDs

nder uniaxial tension, respectively. From Fig. 15 , it can be found the

nitial particle/binder interface debonding is generally produced on the

oundaries of bigger particles. The final fracturing bands in the three

odels all lie in the near-horizontal direction, however, in different lo-

ations of the model. With different PGDs, the macro-equivalent stress–

train curves of the three models do not differ greatly. The difference

f the curves could be largely attributed to the slight PVF differences

etween the models, such as between 80.35% and 80.80%. Gener-

lly, higher PVF brings faster climbing of the curve and larger macro-

quivalent failure strength, agreeing with the conclusions derived in

ection 4.1 . In general, different PGDs bring different specific fracturing

outes, however, have little influence on the macro-mechanical response

f the PBX meso-scale structures.

Figs. 17 and 18 shows the meso-scale deformation and fractur-

ng process and the maco-equivalent stress–strain curves of the above

hree PBX meso-structure models under unaxial compression, respec-

ively. Similar to the corresponding tension cases, initial particle/binder

nterface debonding appears on the boundaries of bigger particles.

hear main cracks are formed in the inclined directions due to inter-

ace debonding and the fracturing of the binder. Occasional fracturing

lso takes place to particles, especially to the small ones, as shown in

ig. 17 (c). The particle fracturing under compressive conditions should

ot be ignored in numerical simulations as this is a common phe-

omenon observed in the experiments [40] . From Fig. 18 , it also can

e found that the three curves differ slightly from each other, and this

lso should be attributed to the difference of the PVF, rather than the

GD.

. Conclusions

In this study, the NMM is extended by incorporating a visco-elastic

onstitutive model with 22 parameters for PBX matrix and a bilinear

ontact relationship model with 3 parameters for PBX particle/matrix

nterface. After validations of these two models, the PBX meso-scale

eformation and damage process of microcrack initiation, crack prop-

gation and formation of crack bands under tensile or compressive

oading conditions are simulated using the extended NMM. Results

ndicate that, during the initial stage, the polymer binder bears a large

136

ontent of the deformation, and interface debonding, especially around

igger particles, is the major damage format of the PBX meso-structure

odels. As the displacement loading proceeds, interface debonding

ogether with fracturing of the binder gradually form the crack bands

erpendicular to the loading direction under tensile condition, or form

nclined shear bands under compressive condition. Under compression,

ccasional fracturing of the explosive particle also takes place. Gen-

rally, interface debonding and binder fracturing are the main failure

odes of PBX meso-structures. Moreover, a larger PVF brings a larger

acro-equivalent modulus as well as larger failure strength, while the

DG mainly influence the meso-scale damage specifics, but has no

bvious influence on the macro-equivalent mechanical response of the

BX. The numerically derived macro-equivalent strengths are also very

lose to those values in references.

It is proved that the NMM is a robust numerical tool for the simula-

ions of the meso-scale mechanical behaviors of particle-filled compos-

tes like the PBX from the aspect of the numerical method. For particle-

lled composites of other constituents, different forms of the contact

elationships and material constitutive models can be incorporated to

he NMM. Finally, since the polymer binder in the PBX is a visco-elastic

aterial, investigations of the temperature and loading rates along with

ther factors on the meso-scale deformation and damage of PBX will be

arried out through NMM simulation more deeply in the future.

cknowledgments

This research is financially supported by the Science and Technol-

gy Program Project of Sichuan Province (Grant no. 2017JY0128 ), the

pen Project of the State Key Laboratory of Coal Resources and Safe

ining, CUMT (Grant no. SKLCRSM16KF08 ), the NSAF project (Grant

o. U1330202 ), and the Open Project of the State Key Laboratory of

xplosion Science and Technology, BIT (Grant no. KFJJ13-11 M ).

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