Quasi-molecular, particle modeling of crack generation and fracture

$Page 1: Quasi-molecular, particle modeling of crack generation and fracture$
W&7%9/86 53.00 4 .oO e 1986 Perpmon Press Ltd.

QUASI-MOLECULAR, PARTlCLE MODELING OF CRACK GENERATION AND FRACTURE

DONALD GREESSPAN Department of Mathematics. The University of Texas at Arlington, Arlington, TX 76019. U.S.A.

(Received I5 October 1984)

Abstract-In this paper a particle type approach is developed for the simulation of crack generation and fracture. The major objective is to develop a new modeling approach which will allow experiments in fracture phenomena directly on a digital computer. Rectangular plates are modeled as particle aggregates and then stressed by using molecular type formulas. Computer examples are described and discussed which relate to customary wedge and punched hole phenomena.

I. ISTRODKTION

The study of fracture and crack growth is of interest. for example, to engineers, mathematicians, and geologists (see. e.g. Refs. [l-27]). Most studies, whether experimental. theoretical, or numerical, have emphasized brittle materials, tensile forces, and stress concentration factors near crack tips. However, interest is also manifest in crack propagation rates, energy release rates, branching, tem- perature dependence, compressive and impulsive force effects, and effects of impurities and fatigue. In theoretical modeling, materials are usually assumed to behave linearly and to be continuous, ho- mogeneous and isotropic. In recent years, however, atomic and molecular type models have also emerged[2, 11, 121.

In this paper. we will initiate a new, computer oriented, quasi-molecular approach[28, 291 to the study of crack and fracture generation. The models will not require the severe constraints imposed on continuous models, will allow for highly nonlinear behavior, and will be more consistent with the knowledge that the phenomena to be studied occur on the atomic and molecular levels. Though the method will be described in complete generality, we will, however, apply it specifically to phenomena related to crack generation due to tensile forces. In the final section we will compare our approach with atomic models developed by others.

2. CLASSICAL MOLECULAR MECHANICS

For purposes of intuition, it will be important to review, first, how molecules interact. Within a larger body, molecules interact only locally, that is, only with their nearest neighbors. This interaction is of the following general nature[28]. If two molecules are pushed together they repel each other, if pulled apart they attract each other, and mutual repulsion is of a greater order of magnitude than is mutual attraction. Mathematically, this behavior is often formulated as follows. The magnitude F of the force F between two molecules which are locally r

units apart is of the form

F= -$+$, (2.1)

where, typically,

G > 0, H > 0, 9>pr7. (2.2)

The major problem in any simulation of a phys- ical body is that there are too many component molecules to incorporate into the model. The classical mathematical approach is to replace the large, but finite, number of molecules by an infinite set of points. In so doing, the rich physics of molecular interaction is lost because every point always has an infinite number of neighbors which are arbitrar- ily close. A viable computer alternative is to replace the large number of moleclrles by a much smaller number of particles and then readjust the parameters in (2.1) to compensate. It is this latter approach which we will follow.

3. A GENERAL COMPUTER ALGORITHM

The general idea outlined in Sec. 2 can be im- plemented easily in the following constructive fashion[28].

Consider N particles Pi. i = I, 2, . . . , N. For At > 0, let tk = kAt. k = 0, 1, 2, . . . . For each ofi= 1,2,... , N, let mi denote the mass of Pi, and let Pi at tr: be located at ri.k = (Xi,k, yi.k, Zi.k), have velocity viaA = (Vi.k.x. vi,k.?, ui,k.r), and have acceleration ai.p = (ai.P_rr (li.&..y, 9i.p.:). Let position, velocity, and acceleration be related by the recurs on formulas

vi. I/Z = vi.0 + 1 (Af)ai.o (starter formula),

(3.1)

v;.k+ 10 = vi.k- 112 + (Al)at,k, k = I, 2,3, . . . ,

(3.2)

b+ I = ri.k + (At)Vi.k+ l/z, k = 0, I, 2,. . . .

(3.3)

I055

1056 DONALD GREENSPAN

At tn. let the force acting on P, be Fi.x = (Fi,k,.,, F,,r.y. Fi.k,,). We relate force and acceleration by the dynamical equation

Fi.k = mia,.k. (3.4)

As soon as the precise structure of F;.k is given. the motion of each Pi will be determined explicitly and recursively by (3.1)-(3.4) from given initial data. The force Fi.k is described now as follows. Assume first that Pj is a neighbor of Pi. Then, let rii.k be the vector from Pi to Pj at time la. so that rij.p = 1 ri.k - rj.k 1 is the distance between the two particles. Then the force FQ.~ on Pi exerted by Pj at time tk is assumed to be

Fij.r, = [[

G H --++

hj.k.)p (6Y.k)’ 1 1

‘ii.lr

rti,k ’ (3.5)

in consistency with (2.1). Next, suppose that Pj is not a neighbor of Pie Then we define the force on Pi exerted by Pj to be

F,k = 0. (3.6)

Finally, the total force Fi.k on Pi at f/, is_d$ined by

Fi.k = $, Fu,k. j=l

j+i

(3.7)

For the convenience of the reader, a basic, two- dimensional, FORTRAN program of the above algorithm is given in the Appendix of Greenspan[30]. Extensions and modifications for the examples which follow can be developed easily from it.

4. EXAMPLES

Let us begin by considering a flat, rectangular sheet which is to be stressed on top and on bottom. We assume first that the sheet has a large interior

Fig. I. Particle configuration.

Fig. 2. Coordinate system.

hole, or punched area. As shown in Fig. 1, the sheet is approximated by a set of one hundred forty six particles P,-PI~~. The exact coordinates of the particles, in the XY system shown in Figure 2, are given in the Table. The distance between each particle and any adjaceat particle is unity, though this is not apparent from the figures because the units on the coordinate axes are slightly different. All particles are assigned unit masses and zero initial velocities.

The neighbors of any given particle are defined to be those particles which are adjacent to the given particle in the usual numerical sense. For example, in the given mosaic, the neighbors of PI are Pz and Pu; those of P3 are P2, P4, P16, PIT; those of PI5 are PI, Pz, PIN, Pzs, Pz9; those of PI7 are P,, Pq, pl6, PM, P3o, p31; those of P28 are PIS, P29, P4?; those of Pd7 are P3,, P34, P46, Pd8. P,; those of P6, are P48, P49, p62, I-54 ; those of P73 are P601 Fk. PBS, p06; thoseofPIlt areP97.P98rP~101P112,P124. PIz5;

and those of P14o are Pi& Plz7, Pi39, Pi4,. Throughout, the parameter choices in (3.1)-(3.5)

are p = 4, 4 = 6, G = H = I, At = 10-j. As indicated in Sec. 2, these are not Lennard-Jones parameters, the rationale of which is given in Sec. 5.

Since all initial data and parameters are now fixed, we proceed to apply tensile forces in the following way. At each time step, that is, every low4 time units, the particles Pi-Pi4 are relocated so that there X coordinates remain the same, their Y coordinates are decreased by O.OOOOOOl, and their velocities are set to zero. The particles P133-Pi46 are treated similarly with the exception that their Y coordinates are increased by O.OOOOOOl. In this fashion, the top and bottom rows are stretched. Each remaining particle reacts to this stretching process by interacting with its neighbors in accordance with (3.1)-(3.5). Finally, we impose an elastic limit L in

the following way. If and when a particle and one of its neighbors move a distance apart which is greater than L, then the force between these particles is set to zero and their bond is considered to

i x

Crack generation and fracture 1057

Table 1.

Y i X Y i X Y

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

:; 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

-6.50000 -4.330 -5.5OoOO -4.330 -4.50000 - 4.330 -3.5OoOO -4.330 - 2.50000 - 4.330 - l.SmOO -4.330 - 0.5OOoO - 4.330

0.50000 - 4.330 1.50000 - 4.330 2.5OooO - 4.330 3.5OoOO - 4.330 4.50000 - 4.330 5,5OmO -4.330 6.50000 -4.330

-6.OOOOO -3.464 -5.OOOOO -3.464 -4.OOooO -3.464 -3mOo0 - 3.464 -2.OOOOO -3.464 - l.OooOO -3.464

O.OOOOO - 3.464 l.OOOOO -3.464 2.Oooo0 - 3.464 3.OOooO -3.464 4.OOOo0 -3.464 5.OtnloO -3.464 6.OOOOO -3.464

- 6.50000 - 2.598 -5.50000 - 2.598 -4.50000 - 2.598 -3.sOooO - 2.598 -2.sOoOo - 2.598 - 1.5OoOO - 2.598 - o.moO - 2.598

0.50000 - 2.598 1.50000 - 2.598 2.5OoOO -2.598 3.5OooO - 2.598 4.5ooOO - 2.598 5.5aOoO - 2.598 6.5OOOQ - 2.598

-6.OOOOO - 1.132 -5.oooo0 - 1.732 -4.OOoOO - 1.732 -3.OOOOO - 1.732 -2.OooOO - 1.732 -l.ooooo - 1.732

O.OOOOO - 1.732 l.OOOOO - 1.732 2.OooOO - 1.732

51 52 53 54 55 56

:: 59 60 61 62 63 64 65 66 67 68 69 70

:: 73 74 15 76 77 78 79 80 81 82

:: 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

100

3.oG#OO 4.OOOOO 5.OOOOO 6.OOOOO

- 6.50000 -5.5OOOG -4.50000 - 3.5OOoO - 2.50000 - 1.50000

0.50000 1.50000 2.50000 3.50000 4.50000 5.50000 6.50000

- 6.00000 -5.OOOOO - 4.00000 -3.OOOOO -2.OOOOO -l.OOOOO

l.OOMIO 2.OOOOO 3.OOOOO 4.OOOOO 5.OOOOO 6.OQOOO

- 6.50000 -5.50000 -4.50000 - 3.50000 -2.50000 - 1.5OwO -0.50000

1.50000 2.50000 3.50000 4.50000 5.50000 6.50000

-6.O4lOOO -5.OOOOO -4.OOOOO - 3.00000 -2.00000 - 1.00000

O.ooooO l.OOOOO

- 1.732 - 1.732 - 1.732 - 1.732 -0.866 - 0.866 -0.866 - 0.866 - 0.866 - 0.866 - 0.866 - 0.866 - 0.866 - 0.866 - 0.866 -0.866 - 0.866

0.000

::E 0.000 0.000 0.000 0.000 0.000 O.ooO 0.000 0.000 0.000 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 1.732 1.732 1.732 1.732 1.732 1.732 1.732 1.732

101 2.OOoOO 1.732 102 3.OOOOO 1.732 103 4.OoOOO 1.732 104 5.OOOOO 1.732 105 6.OOOOO 1.732 106 - 6.50000 2.598 107 -5.50000 2.598 108 -4.50000 2.598 109 -3.50000 2.598 110 - 2.50000 2.598 111 - 1.5oooo 2.598 112 - 0.50000 2.598 113 0.50000 2.598 114 1.5OOoO 2.598 115 2.50000 2.598 116 3.50000 2.598 117 4.50000 2.598 118 5.50000 2.598 119 6.50000 2.598 120 -6.OOOOO 3.464 121 -5.OoOoO 3.464 122 -4.OoOOO 3.464 123 -3.OOoOO 3.464 124 -2.OOOoO 3.464 125 -l.OOOoO 3.464 126 O.OOOOO 3.464 127 l.OOOOO 3.464 128 2.OoooO 3.464 129 3.OOOOO 3.464 130 4.OoOOO 3.464 131 5.OOoOO 3.464 132 6.OOOOO 3.464 133 - 6.50000 4.330 134 -5.50000 4.330 135 -4.5OOoo 4.330 136 -3.50000 4.330 137 -2.5OoOO 4.330 138 - 1.5OOoO 4.330 139 -0.5OoOO 4.330 140 0.50000 4.330 141 1.50000 4.330 142 2.5OooO 4.330 143 3.5OoOo 4.330 144 4.5OOoO 4.330 145 5.50000 4.330 146 6.50000 4.330

be broken thereafter, thereby simulating the generation of a crack.

For L = 1.004, Figs. 3-7 show the developing force field throughout the sheet at the times T = 1,2,5, 11, 16, i.e. after 10000,20000,5OCtOO. 118000, 160000 time steps, respectively. In these figures, the force field is represented by vectors emanating from the centers of the particles. Figure 3 reveals the stress effect being transmitted to the interior of the sheet. Though this transmission occurs in a relatively short time, the fact that it is not instanta- neous is significant, as will be shown later in the application of an impulsive force. Figure 4 shows that at T = 2 the maximum stress is now at the ends

of the hole. By the time T = 5, as shown in Fig. 5, the greatest stress around the hole has now shifted to its sides. At T = 11, one now finds that the direc- tions of the forces around the sides of the hole are changing significantly. Finally, at T = 16, as shown in Fig. 7, one sees clearly from the force field that a hairline crack has now developed on the lower left and upper right sides of the hole. And, indeed, the computer output does show that the distances I P60h It I P60P73 I9 I P74P87 !, I P75P87 I are now greater than 1.004.

Figure 8 shows at T = 18 the relatively rapid growth of the hairline cracks from the central area of the configuration and, in addition, the develop-

1058 DON+LD GREENSPAV

Fig. 3. Example I at T = I.

Fig. 4. Example 1 at 7’ = 2.

Fig. 5. Example I at T = 5.

ment of additional hairline cracks at the lower left and upper right.

For our second example, let us reconsider the first example with only one change. Instead of moving the points PI-PI4 and PI))-P146 a distance 0.000 000 1 every time step, let us move them 0.00001. The resulting simulation corresponds to an impulsive force. that is, one which does not allow suf-

Fig. 6. Example I at T = 1 I.

Fig. 7. Example I at 7 = 16.

Fig. 8. Example I at T = 18.

ficient time for its effect to be transmitted to the interior. By the time T = 1.6, both the top and the bottom have been pulled off from the larger configuration. Figure 9 shows the force field at the time T = 1.5, just prior to the fracture.

For our third example. we will show that the concept of an impulsive force is relative by considering a more flexible material than that considered

Crack generation and fracture I059

Fig. 9. Example 2 at T = 1.5

in the two examples above. For this purpose, we consider the same parameters as in the second example, but with only one change, that is, the elastic limit is set to L = 1.3. With such a limit the sheet should show extensive stretching. Figures 10-12, at times 2.1, 4.1, and 5.6, respectively, show ex- actly this behavior. Moreover, the development of the force field is entirely similar to that shotifi in Figs. 3-7, with the newly developed cracks in Fig.

12 at the same places as in Fig. 7. It must be indicated immediately that the force field in Fig. 12 is approximately ten times greater in magnitude rhan that shown in Fig. 7, but that resealing for il- lustrative purposes has concealed this fact.

As a fourth, and final, example, we consider a modification of the sheet shown in Fig. 1. The three particles P,, P8, Pzl are reset to fill in the large slanted hole, thus resulting in a sheet with a wedge on the bottom, as shown in Fig. 13. The configuration is now stretched right and left by moving the right and left boundary particles a distance O.OOOOOO1 every time step. For the elastic limit L = 1.0015, Figs. 14-17 show the developing force field throughout the sheet at the respective times T

Fig. 10. Example 3 at T = 2.1

Fig. Il. Example 3 at T = 4.1.

Fig. 12. Example 3 at T = 5.6.

Fig. 13. Wedge configuration.

1060 DONALD GREENSPAX

Fig. 14. Example 4 at T = 2.

Fig. IS. Example 4 at T = 4.

Fig. 16. Example 4 at 7 = 5.

= 2, 4, 5, 5.2. Figures 17 and 18 show the force field and the crack development at T = 5.2.

5. REMARKS

First, let us compare our method with atomic methods proposed by others (see, e.g.. Refs. [2, 11, 121). The relative advantages and disadvantages of previously proposed methods have been summa- rized well by Dienes and Paskin[lZ]. Our method

Fig. 17. Example 4 at 7 = 5.2

Fig. 18. Example 4 at T = 5.2.

differs from these in that we do not use classical molecular force formulas, per se, whereas others do. The advantage of using a Lennard-Jones potential is that the parameters in (2.1) are known for various atoms and one can hope, thereby, to obtain quantitative results. The disadvantage is that in so doing the computer constraint of having to utilize relatively few atoms results in relatively large, volatile motions atypical of gross material motions. In any real solid, for example, the motions of the atoms, though volatile, are relatively small, so that the gross material effect is usually not noticeable. In our approach, we conserve mass and try to pre- serve gross motion by, essentially, clumping atoms into “particles”. These particles behave qualita- tively like atoms and molecules, but not in such a volatile fashion. This was accomplished by decreas- ing the exponents in (2.1) from those in the Len- nard-Jones potential. Whereas others have aimed at quantitative results, we have been more modest and have aimed at qualitative results. Indeed, we have tried only to show how stress patterns develop and where hairline cracks will first result. Thus far our approach has been made quantitative only for the one-dimensional study of stress waves in alu- minum bars[31]. The technique used there, and

Crack generation and fracture 1061

which we hope to extend to the present study, is to tit the parameters of (2. I) to experimental data. However, an alternative method for such parameter determination is to consider larger ranges of choices and run all possibilities on the computer. From these computer experiments one could then choose a viable parameter set by direct comparison with known experimental results and proceed to make predictions by studying new configurations and stresses.

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Quasi-molecular, particle modeling of crack generation and fracture