Particle modeling of dynamic fragmentation-I: theoretical considerations

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,b,*

Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada H3A 2K6 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 concerned 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 MD model. This leads to a number of properties and trends of resulting YoungÕs 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.
d Keywords: Dynamics; Structural modeling; Shocks; Fracture mechanics; Cracks; Minerals; Constitutive relations

1. Introduction Attaining a better understanding of the comminution of rocks, such as commonly taking place in the mining industry (e.g. [1]), is the primary moti* Corresponding author. Tel.: +514 398 7394; fax: +514 398 7365. E-mail address: [email protected] (M. OstojaStarzewski).

vation of the present study. Comminution involves complex crushing and fragmentation processes, which, from a basic engineering science perspective, 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, is well suited for analysis of well defined boundaryinitial-value problems with simple geometries,

0927-0256/$ - see front matter
2004 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.08.008

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Nomenclature Fa /a Ga Ha Sa Ea ra pa

interaction force (F =
G/rp + H/rq) per pair atoms interaction potential energy (/ =
Fdr) per pair atoms parameter G in atomic structure parameter H in atomic structure stiffness S 0 ð¼ ðd2 /=dr2 Þr¼r0 Þ in atomic structure YoungÕs modulus (E = S0/r0) in atomic structure equilibrium position in atomic struc˚ for copper ture, e.g., 2.46 A exponential parameter in atomic structure

e.g. [2]. When more complex shapes are involved, numerical methods based on this approach—usually involving finite element methods—are necessary, 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 powerful modeling technique involving a uniform lattice discretization 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 liberation from the mineral is of major industrial interest in the fragmentation process. While the case of quasistatic fracture/damage phenomena was researched in the past decade, the intrinsically dynamic comminution requires a fully dynamic lattice-type model with nonlinear constitutive responses. In this paper series we adopt the so-called particle 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 consuming) in atomistic and multi-scale solid and fluid systems. Since the method has its roots in molecular dynamics, it is sometimes called quasi-molecular modeling or discrete modeling. In essence, particle modeling is a dynamic simulation that uses

exponential parameter in atomic structure 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– direction A length of material specimen (cm) B width of material specimen (cm) C height of material specimen (cm)

qa

small discrete solid physical particle (or quasimolecular particles) as a representation of a given fluid or solid. The two basic rules in the model set-up on larger-than-atomistic-scales are the conservation of mass and the conservation of equilibrium energy between the quasi-particle system and the atomistic material structure. Interaction between any two neighbors 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 complicated boundary and/or initial conditions; an example of the latter is the dynamic free surface generation in solidsÕ fracture. In fact, due to these advantages, 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 interaction potential affect the resulting Young modulus 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-particles, affect the resulting Young modulus?

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

3. How does the choice of parameters in the interaction potential affect fracture of a copper plate (an example problem in studies [6,7])? 4. What are the combined effects of all three factors 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 fracture as that in the aforementioned works of Greenspan. We abandon GreenspanÕs definition of particle interaction as small ‘‘relative to gravity’’ in his dynamic equations that actually result in a pseudodynamic 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 fracture of multi-phase materials, which are then implemented in practical comminution problems in Part II of this paper series.

431

The minimum of / results when F(r) = 0, which ˚ , and we have occurs at ra ¼ 2:46 A ˚ ¼
3:15045  10
13 erg /ð2:46 AÞ

ð3Þ

In [11] Ashby and Jones present a simple method to evaluate YoungÕs modulus E of the material from /(r), namely E¼

S0 r0

ð4Þ

where
2  d/ S0 ¼ dr2 r¼r0

ð5Þ

With this method, we obtain YoungÕs modulus of copper as 152.94 GPa, a number that closely matches the physical property of copper and copper alloys valued at 120–150 GPa. Ashby & Jones [11] also defined the continuumtype tensile stress r(r) as rðrÞ ¼ NF ðrÞ

ð6Þ

2. Theoretical introduction

where N is the number of bonds/unit area, equal to 1=r20 . Tensile strength, rTS, results when dFdrðrÞ ¼ 0, ˚ , the bond damage spacing, that is, at rd = 2.73 A and yields

2.1. Classical molecular dynamics (MD)

rTS ¼ NF ðrd Þ ¼ 462:84 MN=m2

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 Lennard–Jones potential of copper is adopted /ðrÞ ¼


1:398068  10
10 1:55104  10
8 þ erg r6 r12 ð1Þ

˚ ). From Eq. Here r is measured in angstroms (A (1), it follows that the interaction force F bet˚ apart is apween two copper atoms at r A proximately d/ðrÞ dr 8:388408  10
2 18:61246 ¼
þ dyne r13 r7

F ðrÞ ¼


ð2Þ

ð7Þ

This number falls within the range of values reported for actual copper and copper-based alloys: 250–1000 MN/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 same form as in MD: F ¼


G H þ rp rq

ð8Þ

Here G, H, p and q are positive constants, and q > p to obtain the repulsive effect that is necessarily (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

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and H can be derived. Consequently, YoungÕs modulus 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 a unique (p, q) is found to match both YoungÕs modulus and tensile strength of the material. Obviously, 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 motion for each particle Pi of the system is then given by [7] " ! # X d2~ Gi H i ~ ri rji mi 2 ¼ a
p þ q ; i 6¼ j ð9Þ dt rij rij rij rji are mass of Pi and the vector from where mi and ~ Pj to Pi; a is a normalizing constant for Pi obtained from aj
Gi =Dp þ H i =Dq j < ð0:001Þ  980mi

dimensions (2-D) are used to simulate a rectangular copper plate 8 cm · 11.43 cm. The equilibrium spacing is chosen at r0 = 0.2 cm. The corresponding mesh system is built via the following 1-D storage method: xð1Þ ¼
3:9;

yð1Þ ¼
5:71576764

xð41Þ ¼
4:0;

yð41Þ ¼
5:54256256

xði þ 1Þ ¼ xðiÞ þ r0 ;

yði þ 1Þ ¼ yð1Þ;

i ¼ 1; 2; . . . ; 39 xði þ 1Þ ¼ xðiÞ þ r0 ;

yði þ 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.

ð10Þ

where D is distance of local interaction parameter, 1.7r0 cm taken in this paper, where r0 is the equilibrium spacing of the quasi-particle structure. The reason for introducing the parameter a by Greenspan 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 numerical tests, we find that setting a via Eq. (10) would result in a ‘‘pseudo-dynamic’’ solution. This is an important correction to GreenspanÕs 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].

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

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

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3.2. 3-D rectangular block We now construct a 40 · 67 · 20 mesh system in x, y, and z directions, respectively, to approximately represent a domain 8 · 11.43 · 3.1 cm of a copper rectangular block. Similarly, the equilibrium spacing is also chosen as r0 = 0.2 cm. 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-D case. 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
1Þ yði; 1; 1Þ ¼ yð1; 1; 1Þ zði; j; 1Þ ¼ zð1; 1; 1Þ i ¼ 1; 2; 3; . . . ; 40 Then, (i) for odd rows: xði; j; 1Þ ¼ xði; 1; 1Þ yði; j; 1Þ ¼ yði; 1; 1Þ þ r0  sin 60  ðj
1Þ zði; 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 displacement increments in x, y and z pare ffiffiffi Dx = r0 cos 60, Dy = r0 sin 60 and Dz ¼ r0 6=3. The mesh system is obtained by (i) for odd sections: xði; j; kÞ ¼ xði; j; 1Þ
Dx yði; j; kÞ ¼ yði; j; 1Þ þ Dy=3 zði; j; kÞ ¼ zði; j; 1Þ þ Dz  ðk
1Þ (ii) for even sections: xði; j; kÞ ¼ xði; j; 1Þ yði; j; kÞ ¼ yði; j; 1Þ zði; j; kÞ ¼ zði; j; 1Þ þ Dz  ðk
1Þ

Fig. 2 shows this 3-D mesh system. We can calculate the total number N* of atoms in this plate as

 8  108 11:43  108 þ1  þ 1 N ¼ ra ra sin 60 ! 3:1  108 pffiffiffi  ð11Þ þ 1 ffi 2:6952  1025 ra 6=3 Since the mass of a copper atom is 1.0542 · 10
22 g, the total mass M of all the copper atoms in our plate is M  2.841 · 103 g. By the mass conservation–meaning that the total mass of the atomistic structure (i.e., the MD system) must be equal to that of the PM system–we determine each (quasi-)particleÕs mass: m  5.3 · 10
2 g.

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 algebraic problem, 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 conservative method is unwieldy and, therefore, commonly 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, velocity and acceleration for particles Pi(i = 1,2, . . . , N) [7] are ~i;1=2 ¼ V ~i;0 þ ðDtÞ ~ V ai;0 2

k ¼ 1; 3; 5; . . .

ðstarter formulaÞ

~i;kþ1=2 ¼ V ~i;k
1=2 þ ðDtÞ~ V ai;k ; ~i;kþ1=2 ; ~ ri;k þ ðDtÞV ri;kþ1 ¼ ~

k ¼ 2; 4; 6; . . .

433

k ¼ 1; 2; 3; . . . k ¼ 0; 1; 2; . . .

ð12Þ ð13Þ ð14Þ

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

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

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 grow rapidly in time, the time step has to satisfy a stability 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]:
  1=2 1 dF  XDt  2; X ¼ ; ð15Þ m  dr max 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, we take ðdF =drmax Þ  ðdF =drÞr¼r0 . (ii) For an impact problem, we need to set a minimum distance limiting the spacing between any two nearest-neighbor particles, e.g., rmin =

0.1 · r0. This means that within rmin, the interparticle 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 · ur0. Obviously, it will break correct physical boundary condition 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 stable time 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 pffiffiffiffiffiffiffiffiffi for stability and convergence [1]: DT < 2 m=k , 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 is not much quantitative difference between both criteria in case of tensile loadings.

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

G H þ ; rp rq

5. Important rules for application of PM model

F ¼


5.1. Passage from MD model to PM model via equivalence of mass and energy

interaction potential energy [ergs]:

As we have seen from Eq. (8), different (p, q) values result in different material properties, such as YoungÕs 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 additionally influence YoungÕs modulus. Therefore, in general, we have some functional dependence E ¼ f ðp; q; r0 ; V Þ

ð16Þ

At this point, let us note that the PM model is derived from the MD model based on the conservation (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 atomic and quasi-particle structures, and consider qa > pa > 1. First, for the atomic structure (MD model), we have: interaction force [dynes]: Ga H a Fa ¼
p þ q ra ra interaction potential energy [ergs]:
 Ga r1
pa H a r1
qa /a ¼ þ  10
8 1
pa 1
qa stiffness:
 dF a Sa ¼
 10
8 dr r¼ra YoungÕs modulus [GPa]:
 Sa Ea ¼  106 ra total number of atoms:

 A  108 B  108 N ¼ þ1  þ 1 ra ra sin 60 ! C  108 pffiffiffi þ 1  ra 6=3

ð17Þ

/¼

q>p

ð22Þ

Gr1
p Hr1
q þ ; 1
p 1
q

for p > 1

ð23Þ

Hr1
q ; 1
q

for p ¼ 1

ð24Þ

/ ¼ G ln r þ

stiffness:
 dF S0 ¼
dr r¼r0

ð25Þ

YoungÕs modulus [GPa]: E¼

S0 r0

ð26Þ

total number of quasi-particles: N ¼ imax  jmax  k max

ð27Þ

We 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

ð28Þ

From the energy conservation, we have: ðN  /Þr¼r0 ¼ ðN   /a Þr¼ra

ð18Þ

435

ð29Þ

under the requirement: F ðr0 Þ ¼ 0

ð30Þ

From Eqs. (29), (30), we now derive YoungÕs modulus E: for p = 1: ð19Þ

ð20Þ

G ¼ Hr1
q o ;

H¼

ðN   /a Þr¼ra ð1
qÞ N ð1
qÞr01
q ln r0
r1
q 0


q
2 E ¼
Gr
3 0 þ qHr 0

ð31Þ ð32Þ

for p > 1: G ¼ Hr1
q 0 ;

H¼

ðN   /a Þr¼ra ð1
pÞð1
qÞ q
1 r0 N ðp
qÞ ð33Þ

ð21Þ

Next, for the quasi-particle structure (PM model), we have: interaction force [dynes]:

E ¼
pGr0
p
2 þ qHr0
q
2

ð34Þ

At this stage we introduce two additional conditions: equality of YoungÕs modulus (E) and tensile strength (rTS) in the PM and MD models.

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

Evidently, the four parameters (p, q), r0 and V affect 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 = 0.2 cm, V = 8.0 · 11.43 · 3.1 cm3), 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. YoungÕs modulus and tensile strength by (p, q) in interaction force equation as r0 = 0.2 cm: (a) YoungÕs 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 larger is YoungÕs modulus. Table 1 G, H and E corresponding to different (p, q) chocies (p, q) G H E (GPa) rTS (MN/m2)

3–5

5–10 7

2.473 · 10 9.892 · 105 15.457 86.205

7–14 6

1.781 · 10 5.698 · 102 69.557 263.570

1.102 · 105 1.411 150.706 441.534

Fig. 4. Time-dependent fracture of a slotted plate, with r0 = 0.2 cm, (p, q) = 3,5 DT = 10
7 s, stretching rate = 20 cm/s. (a) T = 0.0 s, (b) T = 2.6 · 10
2 s, (c) T = 3.007 · 10
2 s and (d) T = 3.0 · 10
2 s.

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

437

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 YoungÕs modulus, E, values under this change of (p, q). In particular, compared with the physical Young 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 computations 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

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 choosing toughness, one would have to use a more complicated potential (a composite one) having five parameters. Continuing this line of thinking, one would need yet another parameter to model materials with a 3-D Poisson ratio different from 1/4, and so on when more precise modeling is desired.

Fig. 5. Time-dependent fracture of a slotted plate with r0 = 0.2 cm, (p, q) = 5, 10. DT = 10
7 s, stretching rate = 20 cm/ s. (a) T = 0.0 s, (b) T = 1.477 · 10
2 s, (c) T = 1.485 · 10
2 s and (d) T = 1.489 · 10
2 s.

Fig. 6. Time-dependent fracture of a slotted plate with r0 = 0.2 cm, (p, q) = 7,14. DT = 10
7 s, stretching rate = 20 cm/ s. (a) T = 0.0 s, (b) T = 1.024 · 10
2 s, (c) T = 1.026 · 10
2 s and (d) T = 1.030 · 10
2 s.

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

Figs. 4–6 show time-dependent vector fracture results when the above three choices are used, respectively, at a fixed r0 (0.2 cm) and V (8.0 · 11.43 cm2); also Dt = 1.0 · 10
7 s. A kinematical boundary condition is used at the top and the bottom edges: they are stretched outward at a constant velocity of 20.0 cm/s. A zero-traction condition is applied at side edges.

A comparison of these three pictures shows that, for smaller (p, q) values (and hence, smaller YoungÕs 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 increase of toughness, Fig. 7. We also note from Figs. 4–6 that the higher is YoungÕs 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 simulated 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 relationship between changing (p, q), r0 and E? The answer is shown by Fig. 8, in which V = 8.0 · 11.43 · 3.1 cm3 and r0 is changed from 0.1 to 0.5 cm. 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. 7. Potential energy and interaction force of PM under r0 = 0.2 cm, V = 8.0 · 11.43 · 3.1 cm3. (a) Potential energy and (b) interaction force.

Fig. 8. Inter-relationship between E,r0 and (p,q) at a fixed volume of material V = 8.0 · 11.43 · 3.1 cm3.

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

very much (