Åström, J.; Alava, Mikko; Timonen, Juho Crack dynamics and crack surfaces in elastic beam lattices

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Powered by TCPDF (www.tcpdf.org) This is an eectronic reprint of the origina artice. This reprint may differ from the origina in pagination and typographic detai. Åström, J.

1 Powered by TCPDF ( This is an eectronic reprint of the origina artice. This reprint may differ from the origina in pagination and typographic detai. Åström, J.; Aava, Mikko; Timonen, Juho Crack dynamics and crack surfaces in eastic beam attices Pubished in: Physica Review E DOI: 1.113/PhysRevE.57.R1259 Pubished: 1/1/1998 Document Version Pubisher's PDF, aso known as Version of record Pease cite the origina version: Åström, J., Aava, M., & Timonen, J. (1998). Crack dynamics and crack surfaces in eastic beam attices. Physica Review E, 57(2), R1259-R1262. DOI: 1.113/PhysRevE.57.R1259 This materia is protected by copyright and other inteectua property rights, and dupication or sae of a or part of any of the repository coections is not permitted, except that materia may be dupicated by you for your research use or educationa purposes in eectronic or print form. You must obtain permission for any other use. Eectronic or print copies may not be offered, whether for sae or otherwise to anyone who is not an authorised user.

2 Crack dynamics and crack surfaces in eastic beam attices Jan Åström, 1 Mikko Aava, 2,3 and Jussi Timonen 1 1 Department of Physics, University of Jyväskyä, P.O. Box 35, FIN-4351 Jyväskyä, Finand 2 NORDITA, Begdamsvej 17, DK-21 Copenhagen, Denmark 3 Hesinki University of Technoogy, Laboratory of Physics, P.O. Box 11, FIN-215 HUT, Finand Received 24 September 1997 The dynamics of propagating cracks is anayzed in eastic two-dimensiona attices of beams. At eary times, inertia effects and static stress enhancement combine so that the crack-tip veocity is found to behave as t 1/7. At ate times a minima crack-tip mode reproduces the numerica simuation resuts. With no disorder and for fast oading, a mirror-mist-mirror crack-surface pattern emerges. Introduction of disorder eads, however, to the formation of the mirror-mist-hacke type interface typica in many experimenta situations. S X PACS number s : 3.2. i, 62.2.Mk, 46.3.Nz RAPID COMMUNICATIONS PHYSICAL REVIEW E VOLUME 57, NUMBER 2 FEBRUARY 1998 By far the simpest way to break a piece of gass is to appy tensie stress to its surface e.g., by bending it. An imperfection at the surface eads to stress concentration, and for a high enough oad a crack appears. As the crack propagates it has been observed to create a pattern on the crack surface known as the mirror-mist-hacke 1 3. The mechanisms eading to this pattern are not fuy understood, and it may ook sighty different in different materias. Combining a few quaitative descriptions 1 3 gives the foowing picture: At first the crack eaves behind a more or ess smooth surface the mirror. As the veocity of the crack tip increases, the crack surface roughens visiby, and cose to but beow a haf of the veocity of transverse eastic waves, sidebranching becomes so dense that the surface ooks misty. Further increase of roughness with growing crack veocity eads to simutaneous propagation of severa macroscopic cracks. As these cracks merge, the surface of the fina crack is usuay diverted away from the initia crack pane, which resuts in the hacke. If the crack continues to propagate, it may finay bifurcate, and if the stress becomes very high, the gass wi be shattered 2,3. The dynamic propagation of cracks in britte soids has recenty attracted revived interest. This topic was introduced by Yoffe 4, who demonstrated the instabiity of propagating cracks. The experimenta fact that crack veocities seem to be imited to a fraction of the Rayeigh veocity, has induced severa studies on this topic Experiments and the corresponding theoretica modes have addressed phenomena such as crack branching 5 8, osciation of the crack-tip veocity 7,9,1, and spatia crack osciations 11. In this paper we study the veocity of a propagating crack in a trianguar attice of beams 17. This mode, if there is no disorder, has a divergent crack-tip veocity, and produces a mirror-mist-mirror pattern as externa strain is increased. If disorder is introduced, a mirror-mist-hacke pattern is found. The crack veocity is found to resut from a combination of crack-tip inertia effects and static-imit fracture-mechanics effects. The specific nature of the beam attice seems to pay a roe in the formation of crack-surface topoogy and in the crack veocity when it is determined by oca dynamics ony. The beam-attice mode which we anayze, represents a discretization of a britte two-dimensiona soid obeying Cosserat easticity 18,19. This means that arge scae rotations are possibe. We use a trianguar attice in which the attice bonds are sender eastic beams with a square cross-section w 2, ength, and Young s moduus E. The stiffness matrix of a beam is given by EA K EA 12EI 3 2 EA 2 4EI 12EI 3 2 EA 2 4EI 12EI EI 12EI EI, X/98/57 2 / /$ R The American Physica Society

3 R126 JAN ÅSTRÖM, MIKKO ALVA, AND JUSSI TIMONEN 57 FIG. 1. The oca geometry of a trianguar beam attice cose to a crack tip. The sites and bonds are numbered, and the directions of dispacements u 1,u 2,u 3 are indicated. where A w 2 and the moment of inertia is I w 4 /12. Inertia is introduced in the attice by having masses m moment of inertia i on the attice sites, whie the beams are assumed massess. The boundary conditions we impose are such that a strain (t) is appied in the vertica y direction i.e., perpendicuar to one of the principa bond directions by constraining the sites at the top and bottom of the attice to move a distance up and down, respectivey. The sites at the eft and right edges are ony aowed to move in the y direction to avoid a goba Poisson contraction of the attice. The resuts presented in this paper can be compared to those reported for different boundary conditions in Ref. 8. To create a notch, which can then act as a seed for a crack, a number of bonds N rem 1 in the midde of the eft edge are removed at t. The other bonds are assumed to break instanty and irreversiby if their strain exceeds a threshod vaue, which in our case is taken to be a constant ( b ). The dynamics of the attice is determined by a discrete form of Newton s equations of motion incuding a sma viscose dissipation term damping coefficient c and with zero initia veocities. The attice mode is described in more detai in Refs. 8,2. Simuations for a constant strain-increase rate, and for a strain that is constant after crack propagation begins, have been performed. The strain rate was of the order.1 b, which is severa orders of magnitude ess than the sound veocity. Disorder is introduced in the attice by assigning random Young s modui (E i ) to the attice bonds. The vaues of E i are taken from the distribution E i E (1 e i ), where e is the strength of the disorder, and i.5,.5 is a stochastic variabe. The specific vaues of the parameters used in the simuations are given beow. We first consider the crack veocity anayticay within a minima mode 16. The resut wi then be extended to the regime of sow crack propagation in the initia phase and compared to numerica simuations. In Fig. 1 we show the attice in the neighborhood of a crack tip. A crack is propagating from the eft, and stress is appied in the vertica direction. We first study the case in which the externay appied strain is cose to the faiure threshod ( b ) in the diagona bonds of the attice i.e., the bonds that are not horizonta in Fig. 1. In this case ony a sma distortion around the crack tip is needed to break the next bond, which means that the crack wi propagate fast. Breaking of a bond at the crack tip induces a net force on the sites numbered 1 and 2 in the figure. These sites wi therefore, as ong as the force appied on them can be considered constant, undergo constant acceeration. The crack wi propagate a haf of a attice unit when bond 1 breaks. This happens when u2 u3 ( b ) sin, where is the bond ength and is the in-pane ange of the diagona bonds; u2 and u3 are defined in Fig. 1. Notice that u2 and u3 are ony the dispacements of the sites due to the crack propagation. The sow dispacements induced by the externa strain are not incuded in them. The net force on bond 3 resuts from the bending of bond 2. When the externa strain is cose to the faiure threshod, a dispacements around the crack tip are sma, and, therefore, the net force on site 3 is negigibe as compared to the force on site 2. The atter force (F) is given by F Ew2 sin sin 2 w/ 2 cos 2, where Ew 2 / is the axia stiffness of a beam. Newton s equation of motion gives simpy u2 Ft2 2m for the dispacement u2. We have assumed that t when the bond between sites 1 and 2 breaks. Since u3 for cose to b i.e., arge, we obtain the crack-tip veocity in the form.5 2m v /2 b / Ew 2 sin 2 w/ 2 cos 2. 3 This equation impies that the crack-tip veocity diverges as ( b ).5 when b. Stricty speaking there are aso horizonta dispacements and anguar rotations of the sites, but these do not affect much the crack veocity, which is confirmed by the simuations resuts beow. Notice that Eq. 3 was derived under the assumption of arge strains, which is particuary appropriate for overoading, or for cases in which the avaiabe eastic energy exceeds the Griffith s energy. For ower strains we have to consider the crack-tip dynamics to the next order, in which the force resuting from bending of the horizonta bonds that are cosest to the crack tip, i.e., bonds 2 and 3 in Fig. 1, has to be taken into account. If T is the constant time between two consecutive bond breakings at the crack tip, we can cacuate the dispacements of sites 1 and 4 in the same way as above. Then we obtain the force on sites 2 and 3 through bending of bonds 2 and 3. The tota dispacements u 2 and u 3 are then given by u 2 GF 5T2 t 2 /2 Tt 3 t 4 /6 2m 2 Ft2 2m, u 3 GF T2 t 2 /2 Tt 3 /3 t 4 /6 2m 2, 5 where G Ew 4 / 3 is the bending stiffness of the bonds. Bond 1 wi break when t T. Using the breaking criterion above and soving for T gives us again the crack-tip veocity v /2 f 2 4m 2 28Gf 3m 2 f.5 2m, 6 14Gf 3m 2

57 CRACK DYNAMICS AND CRACK SURFACES IN... R1261 FIG. 2. The crack-tip veocity v as a function of ( b )/ for attices of size 4 2 and 8 2, with E 1., w 1., 1.15, and m.5. The ines are given by Eqs.

4 57 CRACK DYNAMICS AND CRACK SURFACES IN... R1261 FIG. 2. The crack-tip veocity v as a function of ( b )/ for attices of size 4 2 and 8 2, with E 1., w 1., 1.15, and m.5. The ines are given by Eqs. 3 and 6. where ( b )/, and f Ew 2 sin 2 (w/) 2 cos 2 /. Notice that this equation gives Eq. 3 in the imit. When the externa strain is cose to the faiure threshod of the diagona bonds, as above, the crack wi propagate fast, and there wi not be time for stress reaxation in the neighborhood of the crack tip. If we woud try to use Eq. 6 to determine the crack-tip veocity for sma externa strains i.e., arge, we woud obtain v( ).25. However, for sma, the crack wi initiay propagate sowy, and there wi appear stress enhancement around the crack tip just as in the static case. The static or quasistatic stress reaxation around a narrow crack of ength L produces a stress enhancement at a sharp tip ( t ) of the form t L 1. By combining the two scaing expressions, t L, v.25 t, with the fact that the crack-tip veocity is defined by v dl/dt, we can estimate the behavior of the crack-tip veocity at sma strains. If the crack begins to propagate with s at t t, the crack veocity is given by v t s 2/7 t t 1/7. Notice that this equation hods ony for just above s. We first tested Eqs. 3 and 6 using the simuation mode. We determined the crack-tip veocity in attices of size 4 2 and 8 2, with E 1., w 1., 1.15, and m.5. The externa strain was increased from zero up to b, and the crack-tip veocity was computed at each bond faiure. The resuts are shown in Fig. 2 together with those given by Eqs. 3 and 6. For arge strains i.e. for ( b )/.2, Eqs. 3 and 6 both give the correct veocity, whie Eq. 6 gives, as expected, a somewhat better resut for sma strains. For ( b )/.2 the simpe approach eading to Eqs. 3 and 6 fais. Stress enhancement around the crack tip is now important, and thus the veocity is satisfactoriy predicted by Eq. 7 cf. Fig. 3. Notice aso the system-size dependent cross over to oca crack-tip dynamics. In the case of constant externa strain we found that, if the strain was high enough for the crack to propagate, the crack 7 FIG. 3. The crack-tip veocity v as a function of ( s ) for the same attice as in Fig. 2. The ine is given by Eq. 7. veocity remained constant throughout the entire simuation, except for a short initia phase. Moreover, the veocity for a particuar strain was exacty the same as that for dynamic boundary conditions at the time when the dynamic strain was simiar to the constant strain. Whie one may find the behavior of the crack-tip veocity by oca considerations suppemented with static fracture mechanics, phenomena ike tip spitting or crack branching and surface roughening are at present ony accessibe by computer simuations. Using the same parameters as in Figs. 2 and 3, and by increasing the externa strain from zero up to b, the crack pattern of Fig. 4A is obtained. In the very beginning of crack propagation, it propagates sowy and the situation is very much ike in the quasistatic case. The crack propagates straightforward with no side branches, and a mirror region is formed. When dynamica effects first become important, the externa strain is sti quite far from b. This means that bending deformations of the horizonta bonds are sti quite arge cf. Eq. 6. Breaking of bonds by bending eads to the formation of arge sidebranches 6 8. These sidebranches form the mist region. When approaches b, bending is no onger important, and a mirror region reappears. At the same time the crack veocity increases without bounds. This is in contrast with experiments, in which the hacke region appears and the veocity is imited to a fraction of the Rayeigh veocity 21, uness the stress becomes so high that the whoe body is shattered. In our case the reap- FIG. 4. Crack patterns formed in a attice of size 4 12, with E 1., w 1., 1.15, m.5, i.1, c.2, t 5, and b.6. A, e. and B, e.1.

5 R1262 JAN ÅSTRÖM, MIKKO ALVA, AND JUSSI TIMONEN 57 pearance of the mirror seems to resut from the crack propagating in a soft direction of a beam attice. It has been suggested that branching is the mechanism that restricts the crack-tip veocity to a fraction of its theoretica vaue 5. Veocity fuctuations typica of the branchforming region can be seen in Figs. 2 and 3. There are, however, no visibe signs of a reduced crack-tip veocity in this region. Fuctuations are rather sensitive to the choice of attice parameters ike the moment of inertia of the sites and the width of the beam eements, which means that there might be 7,8 a range of parameter vaues for which the branches do affect the veocity. We did simuations aso for other parameter vaues but found no effect on the veocity. The branches aways seemed to form at east a few bond engths behind the crack tip. This is consistent with the resuts of Refs. 22, in which branching was shown not to be connected with veocity fuctuations. In contrast with the ordered case above, the hacke region wi appear if disorder is introduced in the mode. In Fig. 4B we use the same attice as in Fig. 4A but now the disorder strength is e.1. When the externa strain reaches a high enough vaue, macroscopic cracks begin to form ahead and on the sides of the main crack. This is anaogous to static crack propagation in two-dimensiona disordered media 23. As these cracks merge, the fina crack wi deviate from the initia crack pain and a rough crack surface is formed. The crack-tip veocity is, of course, strongy affected by disorder in the hacke region. It is difficut, however, to determine a unique veocity in this region as severa cracks propagate simutaneousy. In the mirror and mist regions there was no visibe effect of disorder on the crack-tip veocity. In summary, we have presented a simpe minima crack tip anaysis of the veocity of a crack as a function of externa strain. This is sufficient for understanding the acceeration of the crack tip in an eastic beam attice. As ong as a staticike stress enhancement around the tip is important, the crack wi acceerate as t 1/7. There is then a crossover to stricty oca behavior, dictated by the constitutive aws and inertia. If there is no disorder in the attice, a mirror-mistmirror pattern is formed on the crack surface with increasing externa strain. Introducing disorder in the mode is sufficient for the experimentay observed mirror-mist-hacke pattern to appear. This indicates that microscopic randomness such as vacancies and microcracks are important in the roughening and dynamics of cracks. This is expected to be found aso for sow fracture 24 in which it is the intrinsic disorder which seems to ead to the roughening of crack interfaces. 1 H. J. Herrmann, in Statistica Modes for the Fracture of Disordered Media, edited by H. J. Herrmann and S. Roux North Hoand, Amsterdam, J. J. Mechosky, in Fractography of Gass, edited by R. C. Bradt and R. E. Tasser Penum Press, New York, E. Guioteau, H. Arribart, and F. Creuzet, in Fracture Instabiity Dynamics, Scaing and Ductie/Britte Behavior, edited by R. B. Seinger, J. Mechoasky, E. R. Fuer, Jr., and A. Carsson Materia Research Society Symposium Proceedings Vo. 49 MRS, Pittsburgh, 1996, p E. H. Yoffe, Phios. Mag. 42, E. Sharon, S. P. Gross, and J. Fineberg, Phys. Rev. Lett. 74, M. Marder and X. Liu, Phys. Rev. Lett. 71, P. Heino and K. Kaski, Phys. Rev. B 54, J. Åström and J. Timonen, Phys. Rev. B 54, J. Fineberg, S. P. Gross, M. Marder, and H. L. Swinney, Phys. Rev. Lett. 67, E. Sharon, S. P. Gross, and J. Fineberg, Phys. Rev. Lett. 74, F. Abraham, D. Brodbeck, R. A. Rafey, and W. E. Rudge, Phys. Rev. Lett. 73, S. P. Gross, J. Fineberg, M. Marder, W. D. McCormick, and H. L. Swinney, Phys. Rev. Lett. 71, E. Sharon, S. P. Gross, and J. Fineberg, Phys. Rev. Lett. 76, J. S. Langer, Phys. Rev. Lett. 7, E. S. C. Ching, J. S. Langer, and H. Nakanishi, Phys. Rev. Lett. 76, B. L. Hoian, R. Bumenfed, and P. Gumbsch, Phys. Rev. Lett. 78, H. Furukawa, Prog. Theor. Phys. 9, W. Nowacki, Theory of Micropoar Easticity Springer- Verag, Udine, S. Roux, in Ref J. Åström, M. Keomäki, M. Aava, and J. Timonen, Phys. Rev. E 56, L. B. Freund, Dynamic Fracture Mechanics Cambridge Univ. Press, New York, J. F. Boudet, S. Ciiberto, and V. Steinberg, J. Phys. France II 6, ; J. F. Boudet, V. Steinberg, and S. Ciiberto, Europhys. Lett. 3, V. I. Räisänen, E. T. Seppää, M. J. Aava, and P. M. Duxbury unpubished. 24 P. Daugier, B. Nghiem, E. Bouchaud, and F. Creuzet, Phys. Rev. Lett. 78,

Åström, Jan; Kellomäki, Markku; Alava, Mikko; Timonen, Juho Propagation and kinetic roughening of wave fronts in disordered lattices

Åström, Jan; Kellomäki, Markku; Alava, Mikko; Timonen, Juho Propagation and kinetic roughening of wave fronts in disordered lattices Powered by TCPDF (www.tcpdf.org) This is an eectronic reprint of the origina artice. This reprint may differ from the origina in pagination and typographic detai. Åström, Jan; Keomäki, Markku; Aava, Mikko;