The simulation exhibits a morphological transition between branching and non- ... with many degrees of freedom. ... other than a branching morphology.
Materials &'ience and Engineering, A176 (1994) 295-298
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Dynamical modelling of fracture propagation Y.-H. Taguchi
Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152 (Japan)
Abstract The numerical modelling of fracture propagation is proposed. Material is modelled as a triangular lattice whose bonds have short-range interaction. The interaction models the elastic properties of materials. Using a gaussian-type interaction, it is possible to describe fracture propagation without introducing a breakdown threshold. This enables us to simulate fracture propagation effectively. The simulation exhibits a morphological transition between branching and nonbranching fracture propagation.
1. Introduction
Recently, many physicists have become interested in fracture propagation [1]. It provides us with a good example to understand strong non-linear phenomena with many degrees of freedom. Although many models are proposed, they concentrate on dealing with static cases [2]. However, fracture propagation that we usually observe includes dynamical fractures. Breaking of a glass plate or a piece of paper are examples of this. One should also make dynamical models. Although there are theories corresponding to the ideal cases [3], it is also important to make numerical models that can deal with general and realistic situations. Recently, Mori et al. [4] have proposed such a model. Their model obeys the equation d-xi
dx i
w
m dt: = - r l d t + 2 7 F(xi-xj) J
(1)
where x i is the displacement of the ith site and the summation runs over neighbouring sites j of site i, m is the mass of the ith particle, and t/is the phenomenological viscosity. They employed a triangular lattice. F is the force between two sites that is linear in their model. F becomes zero once the distance between two sites exceeds the threshold value x c (that is, the bond breaks down). (Here in their modelling, each site does not represent an individual atom but corresponds to some phenomenological regions. Therefore their model is completely different from a conventional molecular dynamics study.) Although Mori et al.'s model can reproduce only a branching morphology, we can observe many other shapes including straight cracks in daily life. It is hoped that the model can also reproduce such morphologies. (1921-5093/94/$7.00 SSl)/(1921-5(193(93)(12544-1)
In this paper, we will propose a new model without bond breakdown, which is believed to be necessary to simulate fracture propagation. The model can be easily simulated numerically and reproduce straight, branching and many other crack patterns. This paper is organized as follows. In Section 2, we discuss the problems of conventional models with bond breakdown. Section 3 contains the definition of the new model and the application to a one-dimensional ring is shown in Section 4. In Sections 5 and 6, two specific cases in two dimensions are discussed. A discussion and summary are provided in Section 7.
2. The conventional model with threshold
The models for fracture propagation are mainly defined on a lattice whose bonds have a breakdown threshold. When the length of a bond exceeds a threshold value x c, the bond breaks down. Although one may think that this is an essential point of fracture propagation, it prevents us from studying crack shapes other than a branching morphology. When the external force is applied, each bond continues to be stretched until a bond breaks down. The breakdown suddenly releases the constraint at that point. Then neighbouring bonds start to relax quickly. This violent motion causes a shock wave that always induces branching (breakdown of neighbouring bonds). Therefore, one should suppress this sudden motion if one would like to observe crack shapes other than branching cracks. This requires a fracture propagation model without breakdown. Another problem is the reliability of the threshold model. Although physicists like these types of models because of their simplicity, they do not have any © 1994 - Elsevier Sequoia. All rights reserved
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theoretical support as models of cracks. Linear elasticity is valid only when the deformation is small. One cannot prove linearity when the deformation is large enough to induce breakdown: it is nothing but phenomenology. If so, we can introduce any other types of "reasonable" interactions instead of the conventional interaction: linear dependence with threshold.
3. The model For this purpose, we introduce a short-range elastic potential in the general form in the general form
U(x)- f dx kxUo(x, Xc)
(2)
where x = [xl. Uo(x,xc) must satisfy the following conditions. (1) Uo(x,Xc) is non-negative. (2) Uo(x,Xc)-~Owhenx,> Xc. (3) Uo(X,Xc)= 1 w h e n x ' ~ x c. These conditions are "reasonable". Conditions (1) and (2) are natural. Condition (3)is needed to recover linearity for small deformations. When we employ Uo(x,Xc)= O(x -xc) and require irreversibility, the conventional elastic potential with threshold value x c is recovered. Instead of such selection, we choose simple decaying functions such as exp[-(X/Xc)2], exp(-lx[/Xc) and 1/[l +(x/Xc)2] as Uo(x,Xc) functions. The model that we propose in this paper obeys eqn. (1) with F(x) = - kxU0(x , Xc). Here r/ plays two roles in the modelling. When r/= 0, this model conserves energy, r/ is necessary to cause energy dissipation that the threshold model has even when r/= 0. Otherwise this model cannot become a model of crack propagation. Next, r/suppresses high frequency oscillations. When we consider only small deformations, eqn. ( 1 ) becomes that of a damped oscillator by setting Uo(x,Xc)=l. Then the effective frequency is [2k-(?1/2)2] 1/2, which decreases as increases. Although one may think it is unrealistic, each particle does not correspond to each atom. k and ~/are phenomenological. The important thing is that the model has elasticity and energy dissipation, k and q is introduced to satisfy these requirements. In the next section, we will show how these functions can provide us with a model of fracture propagation without introducing a threshold value.
4. One-dimensional ring In this section, we consider a one-dimensional ring having N particles under periodic boundary condi-
tions. For simplicity, we ignore dynamics and consider only the behaviour of potentials. It shows that we can define "broken bond" even if we employ a continuous functional form of Uo(x,Xc). Suppose the ring is expanded by a distance d. Here we consider a specific form of deformations: x i+1 - x i = x for i = l to N - 1 and xl-XN=d-(N-1)x. Then the total potential energy of the ring is given by Utot(X) = ( N - 1) V(x)+ V [ d - ( X - 1)x]. When x =d/N, the ring is expanded uniformly. That is, no bond breaks down. If not, the deformation of the ring is not uniform. That is, the bond between sites 1 and N breaks. To find the potential minimum of Utot(X), we require d Utot(x)/dx= 0. Then we obtain x
d-(N-
_U0[d-(N1)x
1)x,~]
(3)
U0[x, Xc]
Uniform expansion x=d/N satisfies this condition. When we put x = 0 , eqn. (3) becomes Uo(d,xc)=O. Since Uo(d,xc)-~O for d~>xc because of condition (2) for U0, x = 0 can also satisfy this condition. We denote this value by x b in the following. This corresponds to the breakdown of the bond between sites 1 and N, because the other bond is expanded by
d-(N-
1) Xb ~> Xb.
So that Utot(d/N) does not take a local minimum, d 2 Utot(x)/dx 2 < 0 at x = d/N. This condition is equal to
1/OlnUo(d/N, Xc) de, Utot(X) takes a minimum value at x b. Therefore we can define the breakdown of the bond definitely.
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5. Tearing elastic plate In this section we show how the above model can have "cracks" not only in the static case but also in dynamical fracture propagation. We employ a triangular lattice following Mori et al. We use a square of size 25 x 25 with free boundary conditions. By pulling one edge into two opposite directions, we simulate the case where the elastic plates are torn. In the right-hand half the sites belonging to the edge have a fixed velocity of x component v0 and the other sites have the component of opposite sign - v0. In Fig. 2(a), we show the tearing an elastic plate. Since the deformation concentrates at a few bonds, the location of the crack is clear. With increasing velocity v0, the crack acquires a ramified pattern (Fig. 2(b)). Therefore, this model can reproduce a reasonable character of fracture: increasing external force causes a branching morphology. We should mention here that these calculations are executed on cheap IBM PC personal computers in a few minutes. The introduction of an elastic force without a threshold value enables us to do this. The method of integration is the Euler method with time step At that changes at each step. This scheme is essentially the same as that used by us for powder dynamics [5-7].
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(b) Fig. 2. Tearing elastic plate (xc = 0 . 5 , q = 0 . 1 , m = l, k = 1). T h e lowest row is pulled in two o p p o s i t e horizontal directions. (a) t,t, = 0.02; (b) vo = 0.03.
6. Morphological changes of a crack in a glass plate Finally, we apply this model to a specific case. Recently, Yuse and Sano, and Marder have observed morphological changes of the cracks in quenched glass plates [8]. They heat a thin glass plate and lower it into cool water. The crack grows upward from the initial crack that they make.
As the temperature difference between the hot plate and cool water increases, the shape of the cracks changes from straight to branching patterns. In Fig. 3, we show the results of the simulation of this experiment. For a smaller temperature difference, cracks
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Dynamical modelling of fracture propagation
This tells us that the model also can reproduce the results of actual experiments.
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7. Summaryand conclusion
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In this paper, we reported a new model to simulate dynamical fracture propagation numerically. Removal of "breakdown" of bonds makes the simulation of the model easy. The model has given a phenomenological threshold value where the uniform deformation becomes unstable, in spite of the continuous functional form of the elastic force. The simulation of model reveals clear cracks. It also exhibits a non-branching to branching morphological transition for a tearing elastic plate and a quenching glass plate. This turns out to be an effective numerical model for dynamical fracture propagation.
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Acknowledgments
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I thank Professor M. Sano and A. Yuse for sending me their work before publication. NEC Software is also acknowledged for allowing me to use EWS-4800/ 220. Professors H. Shiba and H. Nishimori are acknowledged since they let me use the NEWS-5000 that is provided by a Grant-in-Aid for Scientific Research on Priority Areas, "Computational Physics as a New Frontier in Condensed Matter Research", from the Ministry of Education, Science and Culture, Japan.
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References
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1 H.J. Herrmann and S. Roux (eds.), Statistical Models for the Fracture of Disordered Media, Elsevier, Amsterdam, 1990, p. 354. T. Vicsek, Fractal Growth Phenomena, World Scientific, Singapore, 2nd edn., 1992, p. 488. 2 H. Takayasu, Prog. Theor. Phys., 74(1985) 1343. P. M. Duxbury, P. D. Beale and P. L. Leath, Phys. Rev. Lett., 57 (1986) 1343. Y.-H. Taguchi, Physica A, 156 (1989) 1343. O. Morgenstern, I. M. Sokolov and A. Blumen, Europhys. Lett., 22 (1993) 487. 3 J. S. Langer, Phys. Rev. A, 46 (1992) 3123, and references cited therein. 4 Y. Mori, K. Kaneko and M. Wadachi, J. Phys. Soc. Jpn., 60 (1991)1591. 5 Y.-H. Taguchi, Phys. Rev. Lett., 69(1993) 1367. 6 Y.-H. Taguchi, J. Phys. I1, 2 (1992) 2103. 7 Y.-H. Taguchi, Fractals, in press. 8 A. Yuse and M. Sano, Nature (London), 362(1993) 329. M. Marder, Nature (London), 362 (1993) 295. 9 Y.-H. Taguchi, unpublished.
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(b) Fig. 3. Change in morphology of cracks in a quenched plate
(xc =0.15, ~/= 5.0). The velocity of lowering of the plate is 130. Quenching causes the natural length of spring to decreases by Ax. In these simulations, the region with. y coordinates less than about 40 is cooled. (a) Ax = 0.09 (weak quench); (b) Ax = 0.13 (strong quench).
grow in a straight pattern. The crack acquires branches when the temperature difference becomes larger. The details of this simulation will be published elsewhere [9]. Wavy patterns can also be reproduced.
