Particle modeling of dynamic fragmentation-I: theoretical …publish.illinois.edu/martinos/files/0000/cms_i.20121101... · 2012. 11. 1. · small discrete solid physical particle

Computational Materials Science 33 (2005) 429–442

www.elsevier.com/locate/commatsci

Particle modeling of dynamic fragmentation-I:theoretical considerations

G. Wang a, M. Ostoja-Starzewski a,b,*

a Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada H3A 2K6b McGill Institute for Advanced Materials, McGill University, Montreal, Quebec, Canada H3A 2K6

Received 27 February 2004; received in revised form 4 August 2004; accepted 24 August 2004

Abstract

This paper series adopts particle modeling (PM) to simulation of dynamic fracture phenomena in homogeneous and

heterogeneous materials, such as encountered in comminution processes in the mining industry. This first paper is con-

cerned with the setup of a lattice-type particle model having the same functional form as the molecular dynamics (MD)

model (i.e., the Lennard–Jones potential), yet on centimeter length scales. We formulate four conditions to determine

four key parameters of the PM model (also of the Lennard–Jones type) from a given MDmodel. This leads to a number

of properties and trends of resulting Young�s modulus in function of these four parameters. We also investigate the

effect of volume, at fixed lattice spacing, on the resulting Young modulus. As an application, we use our model to revisit

the dynamic fragmentation of a copper plate with a skew slit [J. Phys. Chem. Solids, 50(12) (1989) 1245].

� 2004 Elsevier B.V. All rights reserved.

PACS: 68.45.Kg; 61.43.Bn; 62.50.+p; 46.30.My; 46.30.Nz; 83.80.Nb; 83.20.�dKeywords: Dynamics; Structural modeling; Shocks; Fracture mechanics; Cracks; Minerals; Constitutive relations

1. Introduction

Attaining a better understanding of the commi-

nution of rocks, such as commonly taking place in

the mining industry (e.g. [1]), is the primary moti-

0927-0256/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.commatsci.2004.08.008

* Corresponding author. Tel.: +514 398 7394; fax: +514 398

7365.

E-mail address: [email protected] (M. Ostoja-

Starzewski).

vation of the present study. Comminution involves

complex crushing and fragmentation processes,

which, from a basic engineering science perspec-

tive, are complex dynamic fractures of multi-phase

materials. Thus, there is a need to simulate such

processes from basic principles. The first tool that

comes to mind is the continuum-type dynamic

fracture mechanics. That approach, however, iswell suited for analysis of well defined boundary-

initial-value problems with simple geometries,

ed.

mailto:[email protected]

Nomenclature

Fa interaction force (F = �G/rp + H/rq) perpair atoms

/a interaction potential energy (/ = ��Fdr)per pair atoms

Ga parameter G in atomic structure

Ha parameter H in atomic structure

Sa stiffness S0ð¼ ðd2/=dr2Þr¼r0Þ in atomic

structure

Ea Young�s modulus (E = S0/r0) in atomicstructure

ra equilibrium position in atomic struc-

ture, e.g., 2.46A for copper

pa exponential parameter in atomic

structure

qa exponential parameter in atomicstructure

ma mass of each atom (g)

i max total quasi-particle number in x–

direction

j max total quasi-particle number in y–

direction

k max total quasi-particle number in z–

directionA length of material specimen (cm)

B width of material specimen (cm)

C height of material specimen (cm)

430 G. Wang, M. Ostoja-Starzewski / Computational Materials Science 33 (2005) 429–442

e.g. [2]. When more complex shapes are involved,

numerical methods based on this approach—usu-

ally involving finite element methods—are neces-

sary, and, with increasing problem complexity, tend

to be unwieldy. Thus, when multiple cracks occur

in minerals of arbitrary shapes with complex and

disordered microstructures, yet another method

is needed. To this end, we propose here a powerfulmodeling technique involving a uniform lattice dis-

cretization of the material domain, providing the

lattice spacing is smaller (or much smaller) than

the single heterogeneity of interest, e.g. [3,4]. The

heterogeneity is, say, a gold particle, whose libera-

tion from the mineral is of major industrial interest

in the fragmentation process. While the case of quasi-

static fracture/damage phenomena was researchedin the past decade, the intrinsically dynamic com-

minution requires a fully dynamic lattice-type

model with nonlinear constitutive responses.

In this paper series we adopt the so-called parti-

cle modeling (PM), developed by Greenspan [5–8]

as an alternative to computational continuum

physics methods in problems which become either

hopelessly intractable or very expensive (time con-suming) in atomistic and multi-scale solid and fluid

systems. Since the method has its roots in molecu-

lar dynamics, it is sometimes called quasi-mole-

cular modeling or discrete modeling. In essence,

particle modeling is a dynamic simulation that uses

small discrete solid physical particle (or quasi-

molecular particles) as a representation of a given

fluid or solid.

The two basic rules in the model set-up on lar-

ger-than-atomistic-scales are the conservation of

mass and the conservation of equilibrium energy

between the quasi-particle system and the atomis-

tic material structure. Interaction between any twoneighbors in PM involves a potential of the same

type as the interatomic potential—here typically

one of a Lennard–Jones type. Particle modeling

can handle very complicated interactions in solid

and fluid mechanics problems, also with compli-

cated boundary and/or initial conditions; an exam-

ple of the latter is the dynamic free surface

generation in solids� fracture. In fact, due to theseadvantages, particle modeling has recently found

increasing use in mineral and mining research

especially in the studies of tumbling mills [9,10].

Research of the existing literature in PM shows

that the following questions still remain open:

1. How does the choice of parameters in the inter-

action potential affect the resulting Young mod-ulus and the effective strength of the material to

be modeled?

2. How does the choice of volume of the simulated

material, at fixed lattice spacing of quasi-parti-

cles, affect the resulting Young modulus?

G. Wang, M. Ostoja-Starzewski / Computational Materials Science 33 (2005) 429–442 431

3. How does the choice of parameters in the inter-

action potential affect fracture of a copper plate

(an example problem in studies [6,7])?

4. What are the combined effects of all three fac-

tors above?5. What would be a proper, or optimal, choice of

a? (a is a parameter introduced by Greenspan

defining particle interaction as ‘‘small relative

to gravity’’.)

In this paper we address the above issues, and

use the same basic set-up of a copper plate with

an initial slit (crack) undergoing dynamic fractureas that in the aforementioned works of Greenspan.

We abandon Greenspan�s definition of particle

interaction as small ‘‘relative to gravity’’ in his dy-

namic equations that actually result in a pseudo-

dynamic solution. In so doing, we also modify

the proper time increment in PM. In essence, we

set up rules for the formulation of models of frac-

ture of multi-phase materials, which are thenimplemented in practical comminution problems

in Part II of this paper series.

2. Theoretical introduction

2.1. Classical molecular dynamics (MD)

In molecular dynamics (MD), the motion of a

system of atoms or molecules is governed by

classical molecular potentials and Newtonian

mechanics. In our study, following [7], a 6-12 Len-

nard–Jones potential of copper is adopted

/ðrÞ ¼ � 1:398068� 10�10

r6þ 1:55104� 10�8

r12erg

ð1Þ

Here r is measured in angstroms (A). From Eq.(1), it follows that the interaction force F bet-

ween two copper atoms at r A apart is ap-

proximately

F ðrÞ ¼ � d/ðrÞdr

¼ � 8:388408� 10�2

r7þ 18:61246

r13dyne ð2Þ

The minimum of / results when F(r) = 0, which

occurs at ra ¼ 2:46 A, and we have

/ð2:46AÞ ¼ �3:15045� 10�13 erg ð3ÞIn [11] Ashby and Jones present a simple method

to evaluate Young�s modulus E of the material

from /(r), namely

E ¼ S0

r0ð4Þ

where

S0 ¼d2/dr2

� �r¼r0

ð5Þ

With this method, we obtain Young�s modulus of

copper as 152.94GPa, a number that closely

matches the physical property of copper and cop-

per alloys valued at 120–150GPa.

Ashby & Jones [11] also defined the continuum-

type tensile stress r(r) as

rðrÞ ¼ NF ðrÞ ð6Þwhere N is the number of bonds/unit area, equal to

1=r20. Tensile strength, rTS, results when dF ðrÞdr ¼ 0,

that is, at rd = 2.73A, the bond damage spacing,

and yields

rTS ¼ NF ðrdÞ ¼ 462:84MN=m2 ð7ÞThis number falls within the range of values re-

ported for actual copper and copper-based alloys:

250–1000MN/m2.

2.2. Particle modeling (PM)

In particle modeling (PM), the interaction force

is also considered only between nearest-neighbor

(quasi-)particles and assumed to be of the sameform as in MD:

F ¼ �Grp

þ Hrq

ð8Þ

Here G, H, p and q are positive constants, and

q > p to obtain the repulsive effect that is necessar-ily (much) stronger than the attractive one.

The four parameters G through q are yet to be

determined. If p, q and r0 (the equilibrium spacing

between two quasi-particles) are given, then, by

conditions of mass and energy conservation, G


and H can be derived. Consequently, Young�smodulus is evaluated by Eq. (4) and tensile

strength by Eq. (7). To represent an expected

material property such as copper in this study,

we would have to do many sets of testing until aunique (p,q) is found to match both Young�s mod-

ulus and tensile strength of the material. Obvi-

ously, on the other hand, even though the MD

energy equation for copper is kept, changing

(p,q) in Eq. (8) can result in different material

properties, say, ductile or brittle. In this paper,

for simplicity, we map out a range of different

materials according to this idea.Just as in MD, the dynamical equation of mo-

tion for each particle Pi of the system is then given

by [7]

mid2~ridt2

¼ aX

�Gi

rpijþ Hi

rqij

!~rjirij

" #; i 6¼ j ð9Þ

where mi and~rji are mass of Pi and the vector from

Pj to Pi; a is a normalizing constant for Pi obtained

from

aj � Gi=Dp þ Hi=Dqj < ð0:001Þ � 980mi ð10Þwhere D is distance of local interaction parameter,

1.7r0cm taken in this paper, where r0 is the equilib-

rium spacing of the quasi-particle structure. The

reason for introducing the parameter a by Green-

span was to define the interaction force between

two particles as local in the presence of gravity.

Henceforth, in contradistinction to [7], we set

a = 1.0. Indeed, after conducting many numericaltests, we find that setting a via Eq. (10) would re-

sult in a ‘‘pseudo-dynamic’’ solution. This is an

important correction to Greenspan�s theory.Note that if an equilateral triangular lattice

structure is adopted in 2-D, the resulting Poisson

ratio equals 1/4 (or 1/3) when a 3-D (respectively,

plane) elasticity formulation is adopted [13].

Fig. 1. Meshing system for a 2-D plate. Circles are positions of

void.

3. Coordinate-system setup

3.1. 2-D Plate

For the sake of a comparison, we follow the

example of Greenspan [7]: 2713 particles in two

dimensions (2-D) are used to simulate a rectangu-

lar copper plate 8cm · 11.43cm. The equilibrium

spacing is chosen at r0 = 0.2cm. The correspond-

ing mesh system is built via the following 1-D stor-

age method:

xð1Þ ¼ �3:9; yð1Þ ¼ �5:71576764

xð41Þ ¼ �4:0; yð41Þ ¼ �5:54256256

xðiþ 1Þ ¼ xðiÞ þ r0; yðiþ 1Þ ¼ yð1Þ;

i ¼ 1; 2; . . . ; 39

xðiþ 1Þ ¼ xðiÞ þ r0; yðiþ 1Þ ¼ yð41Þ;

i ¼ 41; 42; . . . ; 80

xðiÞ ¼ xði� 81Þ; yðiÞ ¼ yði� 81Þ þ 2r0 sin 60�;

i ¼ 82; 83; . . . ; 2713

Void particles are numbered as i = 1070 + 41k,

k = 0,1,2, . . . , 14. The corresponding mesh system

is shown in Fig. 1.


3.2. 3-D rectangular block

We now construct a 40 · 67 · 20 mesh system in

x, y, and z directions, respectively, to approxi-

mately represent a domain 8 · 11.43 · 3.1cm of acopper rectangular block. Similarly, the equilib-

rium spacing is also chosen as r0 = 0.2cm. We

choose a face centered cubic (f.c.c) lattice structure

for the three-dimensional packing structure since

many common metals (e.g., Al, Cu and Ni) have

this structure type [11].

First, in the x–y plane, a 40 · 67 mesh system

is constructed in a similar way as in the 2-Dcase. The left-corner point is, x(1,1,1) = �3.9,

y(1,1,1) = �5.71576764, z(1,1,1) =� 1.5513435038.

It is built up by:

The coordinate of first row:

xði; 1; 1Þ ¼ xð1; 1; 1Þ þ r0 � ði� 1Þyði; 1; 1Þ ¼ yð1; 1; 1Þzði; j; 1Þ ¼ zð1; 1; 1Þ i ¼ 1; 2; 3; . . . ; 40

Then,

(i) for odd rows:

xði; j; 1Þ ¼ xði; 1; 1Þyði; j; 1Þ ¼ yði; 1; 1Þ þ r0 � sin 60� � ðj� 1Þzði; j; 1Þ ¼ zð1; 1; 1Þ j ¼ 1; 3; 5; . . .(ii) for even rows:

xði; j; 1Þ ¼ xði; j� 1; 1Þ � r0 � cos 60�

yði; j; 1Þ ¼ yði; j� 1; 1Þ þ r0 � sin 60�

zði; j; 1Þ ¼ zð1; 1; 1Þ j ¼ 2; 4; 6; . . .

In the z direction, for even sections, the displace-

ment increments in x, y and z are Dx = r0 cos

60�, Dy = r0 sin 60� and Dz ¼ r0ffiffiffi6

p=3. The mesh

system is obtained by

(i) for odd sections:

xði; j; kÞ ¼ xði; j; 1Þ � Dxyði; j; kÞ ¼ yði; j; 1Þ þ Dy=3zði; j; kÞ ¼ zði; j; 1Þ þ Dz� ðk � 1Þ k ¼ 1; 3; 5; . . .(ii) for even sections:

xði; j; kÞ ¼ xði; j; 1Þyði; j; kÞ ¼ yði; j; 1Þzði; j; kÞ ¼ zði; j; 1Þ þ Dz� ðk � 1Þ k ¼ 2; 4; 6; . . .

Fig. 2 shows this 3-D mesh system. We can cal-

culate the total number N* of atoms in this plate

as

N � ¼ 8� 108

raþ 1

� �� 11:43� 108

ra sin 60�þ 1

� �

� 3:1� 108

raffiffiffi6

p=3

þ 1

!ffi 2:6952� 1025 ð11Þ

Since the mass of a copper atom is 1.0542 ·10�22g, the total mass M of all the copper atoms

in our plate is M � 2.841 · 103g. By the mass con-

servation–meaning that the total mass of the atom-

istic structure (i.e., the MD system) must be equal

to that of the PM system–we determine each

(quasi-)particle�s mass: m � 5.3 · 10�2g.

4. Numerical methodology

Just as in molecular dynamics, there are two

commonly used numerical schemes in particle

modeling: ‘‘completely conservative method’’ and

‘‘leapfrog method.’’ The first scheme is exact but

requires a very costly solution of a large algebraicproblem, while the second one is approximate.

Since in most problems, both in MD and in PM,

one needs thousands of particles to adequately

represent a simulated body, the completely conser-

vative method is unwieldy and, therefore, com-

monly abandoned in favor of the leapfrog

method [12]. In the following, we employ the latter

one.

4.1. Leapfrog method

The leapfrog formulas relating position, veloc-

ity and acceleration for particles Pi(i = 1,2, . . . ,N)

[7] are

~V i;1=2 ¼ ~V i;0 þðDtÞ2

~ai;0 ðstarter formulaÞ ð12Þ

~V i;kþ1=2 ¼ ~V i;k�1=2 þ ðDtÞ~ai;k; k ¼ 1; 2; 3; . . . ð13Þ

~ri;kþ1 ¼~ri;k þ ðDtÞ~V i;kþ1=2; k ¼ 0; 1; 2; . . . ð14Þ

where ~V i;k,~ai;k and~ri;k are the velocity, accelerationand position vectors of particle i at time tk = kDt,

Fig. 2. Meshing system for a 3-D material body.


Dt is the time step. ~V i;kþ1=2 stands for the velocity

of particle i at time tk = (k + 1/2)Dt, and so on.

Notably, the leapfrog method is of second-order

accuracy: O((Dt)2) [7,12].

4.2. Stability of leapfrog method

To ensure that numerical errors do not growrapidly in time, the time step has to satisfy a stabil-

ity condition. In PM, the safe time step is obeyed

by the same formulas as those in MD, wherein it

is dictated by the root locus method [12]:

XDt � 2; X ¼ 1

mdFdr

��max

� �1=2

; ð15Þ

Now, observe from (8) that, as r! 0, dF/dr ! 1,

resulting in Dt ! 0. Since this may well cause

problems in computation, one thus introduces a

smallest distance between two particles:

(i) For a stretching problem of a plate/beam, wetake ðdF =drmaxÞ � ðdF =drÞr¼r0

.

(ii) For an impact problem, we need to set a min-

imum distance limiting the spacing between

any two nearest-neighbor particles, e.g., rmin =

0.1 · r0. This means that within rmin, the inter-

particle force remains equal to that at rmin. It is

easy to see from Fig. 7(b) that, in this case, this

suitable time increment is greatly reduced

because of a rapid increase in X. Note, if a

kinematic boundary condition is adopted for

the impact case, then the safe time step should

also be constrained by Dt · u�r0. Obviously,it will break correct physical boundary condi-

tion if u exceeds r0 in one time step. Generally

speaking, impact cases require much smaller

Dt than stretching cases [14].

From Eq. (15), we find that, Dt � 10�7 � 10�6 s.

We observe that, even within the domain of stabletime increments, adopting a smaller time step can

result in smoother results. Therefore, Dt = 10�7 s

is applied for stretching/tensile cases and Dt =10�8 s for impact/compression cases in our study.

There also exists another criterion for stability

and convergence [1]: DT < 2ffiffiffiffiffiffiffiffiffim=k

p, where m is

the smallest mass to be considered, k is the stiffness

that is the same as S0 in Eq. (5). In effect, there isnot much quantitative difference between both cri-

teria in case of tensile loadings.


5. Important rules for application of PM model

5.1. Passage from MD model to PM model via

equivalence of mass and energy

As we have seen from Eq. (8), different (p,q)

values result in different material properties, such

as Young�s modulus E. It is not difficult to see that

changing the equilibrium spacing r0 and volume of

the simulated material, V(=A · B · C) will addi-

tionally influence Young�s modulus. Therefore, in

general, we have some functional dependence

E ¼ f ðp; q; r0; V Þ ð16ÞAt this point, let us note that the PM model is de-

rived from the MD model based on the conserva-

tion (or equivalence) of mass and energy between

both systems. The ensuing derivation generalizes

basic ideas outlined by Greenspan [7]. We choose

a face centered cubic (f.c.c) lattice for both atomicand quasi-particle structures, and consider

qa > pa > 1.

First, for the atomic structure (MD model), we

have: interaction force [dynes]:

F a ¼ �Ga

rpaþ H a

rqað17Þ

interaction potential energy [ergs]:

/a ¼Gar1�pa

1� paþ H ar1�qa

1� qa

� �� 10�8 ð18Þ

stiffness:

Sa ¼ � dF a

dr

� �r¼ra

� 10�8 ð19Þ

Young�s modulus [GPa]:

Ea ¼Sa

ra

� �� 106 ð20Þ

total number of atoms:

N � ¼ A� 108

raþ 1

� �� B� 108

ra sin 60�þ 1

� �

� C � 108

raffiffiffi6

p=3

þ 1

!ð21Þ

Next, for the quasi-particle structure (PM model),

we have: interaction force [dynes]:

F ¼ �Grp

þ Hrq; q > p ð22Þ

interaction potential energy [ergs]:

/ ¼ Gr1�p

1� pþ Hr1�q

1� q; for p > 1 ð23Þ

/ ¼ G ln r þ Hr1�q

1� q; for p ¼ 1 ð24Þ

stiffness:

S0 ¼ � dFdr

� �r¼r0

ð25Þ

Young�s modulus [GPa]:

E ¼ S0

r0ð26Þ

total number of quasi-particles:

N ¼ imax � jmax � kmax ð27ÞWe now postulate the equivalence of MD and PM

models. From the mass conservation, we calculate

the mass of each quasi-particle M:

M ¼ N � � ma=N ð28ÞFrom the energy conservation, we have:

ðN � /Þr¼r0¼ ðN � � /aÞr¼ra

ð29Þ

under the requirement:

F ðr0Þ ¼ 0 ð30ÞFrom Eqs. (29), (30), we now derive Young�s mod-

ulus E: for p = 1:

G ¼ Hr1�qo ; H ¼

ðN � � /aÞr¼rað1� qÞ

Nð1� qÞr1�q0 ln r0 � r1�q

0

ð31Þ

E ¼ �Gr�30 þ qHr�q�2

0 ð32Þfor p > 1:

G ¼ Hr1�q0 ; H ¼

ðN � � /aÞr¼rað1� pÞð1� qÞ

Nðp � qÞ rq�10

ð33Þ

E ¼ �pGr�p�20 þ qHr�q�2

0 ð34ÞAt this stage we introduce two additional condi-

tions: equality of Young�s modulus (E) and ten-

sile strength (rTS) in the PM and MD models.

Table 1

G, H and E corresponding to different (p,q) chocies

(p,q) 3–5 5–10 7–14

G 2.473 · 107 1.781 · 106 1.102 · 105

H 9.892 · 105 5.698 · 102 1.411

E (GPa) 15.457 69.557 150.7062


Evidently, the four parameters (p,q), r0 and V af-

fect E and rTS, and, in the following, we discuss

those dependencies in detail.

5.2. Effect of changing (p,q) at fixed r0 and

volume V

Using the mesh system shown in Fig. 2 (r0 =

0.2cm, V = 8.0 · 11.43 · 3.1cm3), we can simulate

a range of different materials with (p,q) pairs

drawn from p = 1,2, . . . , 14 and q = 2,3, . . . , 15,

Fig. 3. Young�s modulus and tensile strength by (p,q) in

interaction force equation as r0 = 0.2cm: (a) Young�s modulus

and (b) tensile strength.

providing q > p. First, Fig. 3(a) shows clearly that,

in general, under a fixed r0 and a fixed volume V,

the larger the p and q values are adopted, the lar-

ger is Young�s modulus.

Fig. 4. Time-dependent fracture of a slotted plate, with

r0 = 0.2cm, (p,q) = 3,5 DT = 10�7 s, stretching rate = 20cm/s.

(a) T = 0.0s, (b) T = 2.6 · 10�2 s, (c) T = 3.007 · 10�2 s and (d)

T = 3.0 · 10�2 s.

rTS (MN/m ) 86.205 263.570 441.534


Next, we shall mainly take these three pairs as

our principal study cases: (p,q) = (3,5); (5,10)

and (7,14). Table 1 shows the different G, H and

Young�s modulus, E, values under this change of

(p,q). In particular, compared with the physicalYoung modulus of copper, it is obvious that

(p,q) = (7,14) is most suitable. This suggests a rule

for choosing a suitable (p,q) in the PM model:

based on fixed r0 and V, we can do a series of com-

putations on (p,q) and then, select that (p,q) pair

which results in E matching a given E. In general,

that there are several (p,q) pairs that result in E

and material strength (very) close to the desired

Fig. 5. Time-dependent fracture of a slotted plate with

r0 = 0.2cm, (p,q) = 5, 10. DT = 10�7 s, stretching rate = 20cm/

s. (a) T = 0.0s, (b) T = 1.477 · 10�2 s, (c) T = 1.485 · 10�2 s and

(d) T = 1.489 · 10�2 s.

value, but have differing toughnesses. Thus, we

actually have some degree of freedom in choosing

those PM parameters, which offer more or less

toughness, depending on the given material being

modeled.The larger the (p,q) values are, the more rapid

the fracture process. To have full freedom in choos-

ing toughness, one would have to use a more com-

plicated potential (a composite one) having five

parameters. Continuing this line of thinking, one

would need yet another parameter to model mate-

rials with a 3-D Poisson ratio different from 1/4,

and so on when more precise modeling is desired.

Fig. 6. Time-dependent fracture of a slotted plate with

r0 = 0.2cm, (p,q) = 7,14. DT = 10�7 s, stretching rate = 20cm/

s. (a) T = 0.0s, (b) T = 1.024 · 10�2 s, (c) T = 1.026 · 10�2 s and

(d) T = 1.030 · 10�2 s.


Figs. 4–6 show time-dependent vector fracture

results when the above three choices are used,

respectively, at a fixed r0 (0.2cm) and V

(8.0 · 11.43cm2); also Dt = 1.0 · 10�7 s. A kine-

matical boundary condition is used at the topand the bottom edges: they are stretched outward

at a constant velocity of 20.0cm/s. A zero-traction

condition is applied at side edges.

Fig. 7. Potential energy and interaction force of PM under

r0 = 0.2cm, V = 8.0 · 11.43 · 3.1cm3. (a) Potential energy and

(b) interaction force.

A comparison of these three pictures shows

that, for smaller (p,q) values (and hence, smaller

Young�s modulus E), the fracture process is slower

than at larger (p,q). By virtue of the formulas

of Section 5.1, with E decreasing, there is an in-crease of toughness, Fig. 7. We also note from

Figs. 4–6 that the higher is Young�s modulus, E,

the more pieces is the material fragmented into:

brittleness increases. This conclusion provides a

stepping-stone to PM simulation of crushing

processes.

5.3. Effect of changing r0 and (p,q) at

fixed-volume V

In some situations, when the size of the simu-

lated material is fixed, within the satisfaction of

engineering need, we often hope to get rapid result

by an adoption of as big equilibrium spacing as

possible. So the question is, what is the relation-

ship between changing (p,q), r0 and E? The answeris shown by Fig. 8, in which V = 8.0 · 11.43 ·3.1cm3 and r0 is changed from 0.1 to �0.5cm.

It shows that for the cases of p = 1, the larger

change with r0 is adopted, the bigger E is obtained.

On the contrary, for p 5 1 cases, the larger change

with r0, the smaller E is resulted. But, generally

speaking, this increase or decrease does not change

Fig. 8. Inter-relationship between E,r0 and (p,q) at a fixed

volume of material V = 8.0 · 11.43 · 3.1cm3.


very much (<30GPa in maximum for r0 = 0.1–

0.5cm). Similarly, small (p,q) compositions pro-

duce small E, and the differences between them

are also small. This observation may be very useful

in cases of modest computer facilities—for in-

Fig. 9. Young�s modulus, E, for varying r0, volume V and (p,q). (a)

(e) (p, q) = 5, 10 and (f) (p, q) = 7, 10.

stance, a choice of r0 = 0.2cm instead of 0.1cm

in a 3-D problem could reduce the number of par-

ticles by a factor of 23, but the difference in E be-

tween these two adoptions would be smaller than

10GPa.

(p, q) = 1.5, (b) (p, q) = 1, 10, (c) (p, q) = 1, 14, (d) (p, q) = 3, 5,


5.4. Effect of changing r0, volume V and (p,q)

When r0, volume V and (p,q) are all varied, the

dependence on p = 1 is qualitatively different from

that at p5 1, Fig. 9. The results are drawn here bygradually changing the initial volume V (=8.0 ·11.43 · 3.1cm3) as well as the distance r0. For

p = 1, if V fixed, E gets larger as r0 increases,

but, if r0 is fixed, a small increase in V can result

in a larger E. Following this, E keeps constant

even if a continuous increase of r0 is carried out.

For all the p 5 1 cases, as opposed to p = 1,

with an increase of r0, E decreases. Also, for eachr0, E increases only a little with the volume up to a

certain volume, and then remains constant. Thus,

it is impractical to increase E by setting up a large

volume. To sum up, high (p,q) values result in

large E.

5.5. Effect of changing volume V and (p,q) at

fixed r0

We now turn to this question: For a 3-D lattice

structure, at a fixed r0, what is the effect of increas-

ing volume by an enlargement in the X, Y, and Z

directions? First, Fig. 10 illustrates the effect of

(p,q)–taking values (1,5), (1,10), (1,14), (3,5)

Fig. 10. Young�s modulus for different (p,q), increasing in X

direction as a fixed amplifier of 25 in both Y and Z directions.

(5,10) up to (7,14)–on Young�s modulus. Here

the X, Y and Z directions are all amplified from

the reference spacing r0 = 0.2cm. It is clearly seen

that E increases quite rapidly within a small

enlargement of length in X direction and then con-verges to a constant for all different (p,q) combina-

tions. This results in a very important hint for a

PM user: to choose a �safe� computation domain

in case of a fixed r0, and we see that E does not

change much beyond the size 25 · r0.

Since all the cases of (p,q) have the same effect

on E, here we may focus on (p,q) = (7,14) for

further discussion. Fig. 11 shows how the chang-ing of volume in all three directions will affect E

at (p,q) = (7,14). From two different angles of

view, we see that E is quite small when a small

computation domain is adopted—below, say, five

times larger than the grid spacing–but increases

quite rapidly afterwards, and it asymptotes to a

constant beyond an enlargement factor of 25

times.

6. Conclusions

This paper series employs particle modeling

(PM) approach for simulation of dynamic fracture

phenomena in homogeneous and heterogeneous

materials, such as encountered in comminutionprocesses in the mining industry. This first paper

is concerned with the setup of a particle model

having the same functional form as the molecular

dynamics (MD) model (i.e., the Lennard–Jones

potential), yet on centimeter length scales.

Basically, we have four conditions to determine

four unknown variables, G, H, p and q of the PM

model: (i) the same mass in the PM and MD mod-els, (ii) the same elastic energy in the PM and MD

models, (iii) equality of Young�s modulus in the

PM and MD models, (iv) the same tensile strength

in the PM and MD models. We have derived

the equations for G, H, p and q, and carried out a

parametric study to find the differing effects on p,

q, V and r0. Overall, we have found the following

rules:

(i) The larger the values of (p,q) are adopted, the

larger is E generated. This is typically associ-

Fig. 11. Young�s modulus, E, generated by increasing volume V in 3 axis with a fixed r0 = 0.2cm and (p,q)t = 7,14. (a) Angle of view I

and (b) Angle of view II.


ated with the material becoming more brittlethan ductile, albeit there is a range of tough-

ness to choose from. Also, with E going up,

there is a fragmentation into a larger number

of pieces.

(ii) In the case of p = 1, the larger r0 spacing isadopted, the higher is Young�s modulus of

the PM material. On the contrary, in the spe-

cial case of p 5 1, there is an opposite trend.

In any case, this increase or decrease does not


change very much (i.e. it is below 30GPa) at

r0 = 0.1–0.5cm.

(iii) In the case of p 5 1, while keeping the vol-

ume fixed, an increase of r0 produces a

decrease of Young�s modulus. The situationis again opposite in the case of p = 1.

(iv) A uniform augmentation of volume V by dila-

tion in all three directions (XYZ), at any

(p,q) combination, results in Young�s modu-

lus increasing first strongly and then leveling

off.

We have also used our model to revisit the pre-viously studied case of dynamic fragmentation of a

copper plate with a skew slit, and we have modi-

fied the previous results of Greenspan in two ways.

First, we corrected the ‘‘local interaction parame-

ter’’ concept in his dynamic equations that render

his theory a pseudo-dynamic model, so as to have

the full dynamics. Secondly, we studied the influ-

ence of four parameters G, H, p and q of the PMmodel according to the new equivalence of the

MD and PM models.

Acknowledgments

We have benefited from correspondence with

Prof. D. Greenspan (University of Texas atArlington). Discussions with the staff at COREM,

Quebec City, have proved helpful in orienting

this research. The work reported herein has been

made possible through support of the Canada Re-

search Chairs program and the funding from

NSERC.

References

[1] T.J. Napier-Munn, S. Morrell, R.D. Morrison and T.

Kojovic, Mineral Comminution Circuits—Their Operation

and Optimisation, Julius Kruttschnitt Mineral Research

Centre, The University of Queensland, 1999.

[2] L.B. Freund, Dynamic Fracture Mechanics, Cambridge

University Press, Cambridge, 1990.

[3] M. Ostoja-Starzewski, P.Y. Sheng, I. Jasiuk, Damage

patterns and constitutive response of random matrix-

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[8] D. Greenspan, New approaches and new applications for

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dae Mathematicae 71 (2002) 279–313.

[9] L.K. Nordell, A.V. Potapov, Comminution simulation

using discrete element method (DEM) approach–from

single particle breakage to full-scale SAG mill operation,

SAG, 2001.

[10] D.D. Zhang, Simulation Techniques for Discrete Element

Models, Ph.D. Thesis, University of Queensland, 1998.

[11] M.F. Ashby, D.R.H. Jones, Engineering Materials 1: An

Introduction to Their Properties and ApplicPergamon

Press, Pergamon Press, Oxford, 1980.

[12] R.W. Hockney, J.W. Eastwood, Computer Simulation

Using Particles, McGraw Hill, New York, 1999.

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Appl. Mech. Rev. 55 (1) (2002) 35–60.

[14] M. Moore, J. Wilhelms, Collision detection and response

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