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PHYSICAL REVIEW E 85, 055102(R) (2012)

Dynamic fracture in irregularly structured systems

Xiaodan RenSchool of Civil Engineering, Tongji University, Shanghai 200092, China

Jie LiThe State Key Laboratory on Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China

(Received 19 February 2012; published 7 May 2012)

Although commonly used materials are composed of irregular microstructures, most existing numericalmethods for fracture dynamics are developed via regular discretizations. In the present Rapid Communicationwe investigate the dynamic fracture numerically by irregular domain discretizations. To explore the relationshipbetween microscopic crack branching and the macroscopic instability of fracture dynamics, we simulate detaileddiagrams for the crack branching and also calculate the crack speeds by varying the parameters of crack-tipcohesion. In particular, an equation to describe the relation between the crack speed and the fracture energyis proposed based on the simulation results. The present results indicate that the irregularities of mesoscopicstructure contribute to the intrinsic instabilities of dynamic fracture and eventually to the crack speed. And thesingle-crack continuum theory should be at least carefully modified to describe the dynamic fracture governedby the complex branching and fluctuations.

DOI: 10.1103/PhysRevE.85.055102 PACS number(s): 62.20.mm, 62.20.mt, 89.75.Kd

The propagation of cracks within brittle solids evolves asthe foundation of many scientific disciplines. The continuumtheory of crack propagation is developed based on theassumption of a smooth crack route and predicts that the crackpropagates in the limit speed of a Rayleigh wave [1]. However,experimental results [2–4] indicate that a crack surface couldevolve into an extremely rough pattern with certain branchessplitting from the main crack, and the irregular distributionsof energy dissipation and crack speed are triggered during thedynamic fracturing. These features violate the assumption ofmirror-like crack surfaces in the continuum fracture theoryand therefore cannot be well tackled within the frameworkof continuum mechanics. On the other side, the invalidationof the continuum theory suggests that the material structuresand properties on the micro- or mesoscale may criticallyinfluence the fracturing behaviors on the macroscale. Thusthe appropriate descriptions are required for the analysisof dynamic fracture problems on the micro- or mesoscale.Two main approaches have been proposed by pioneeringresearchers, i.e., the atomistic simulation (MD) and thecohesive element method (CM).

Atomistic simulations were introduced to crack-propagation problems by Ashurst and Hoover [5] early on.Abraham et al. [6] suggested the instability of dynamic fractureby large-scale atomistic simulations with the advent of scalableparallel computers. Buehler et al. [7] and Buehler and Gao[8] investigated the influences of hyperelasticity and largedeformation at the crack tip by massively parallel large-scaleatomistic simulations. And recently, Heizler and coworkers [9]simulated the fracture in amorphous material by introducingthe random network model. By taking a microscopic insightof the cracking process, the atomistic simulations are ableto reproduce some unexplained features of continuum theory.However, even with the help of large-scale parallel computingsystems, the time and length scales in the atomistic simulationare significantly different from typical laboratory experiment[6]. The real-time atomistic simulations of solids still remain

extraordinarily challenging nowadays. Some other researcherstried to explore the dynamic fracture on the mesoscale. Thecrack branching was simulated with cohesive surface decohe-sion formulation by Xu and Needleman [10,11]. Accordingto these investigations, the cohesive elements which representthe potential crack path within solids are embedded in betweenthe boundaries of volume finite elements. Since the initiationand propagation of cracks have to be along the finite elementborder, sufficient degrees of freedom should be provided forarbitrary crack paths by the refinement of the discretization.The numerical results should reproduce a number of typicalphenomena observed by experiments, such as crack branching,the dependence of crack speed on loading rate, and the abruptcrack arrest.

When we take a close observation at the commonlyused materials, most of them are of irregular mesoscopicmaterial structures. As we know, the dynamic fracturingsystem is an evolving hyperbolic system within which the localfluctuations might spread out over the whole system. And theexperiments carried out by Livne et al. [12] also suggestedthat the mesoscopic irregularities could play a critical rolefor the unstable dynamic fracture. However, according to theexisting investigations, the corresponding numerical modelswere mostly developed via regular discretization patterns,for example, the triangle equilateral pattern [7,8] and thecross triangle quadrilateral pattern [11]. Actually, althoughthe dependency between the crack instability and the ma-terial structure have been perceived by some researchers,the relevant analysis or simulations are rare and most ofthem are developed by using MD [9]. Therefore in thepresent Rapid Communication we investigate the dynamicsof fracture of the irregularly structured solid by the cohesiveelement method. Our study reveals that the irregularitiesof material structures could significantly influence the pat-terns of crack branching. Moreover, the cohesive fractureenergy at crack tips governs the cracking speed in dynamicfractures.

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XIAODAN REN AND JIE LI PHYSICAL REVIEW E 85, 055102(R) (2012)

(a)

(b)

FIG. 1. (Color online) Irregular discretization of a 2D rectangularsolid. (a) A 2D rectangular solid discretized by the random point set.(b) Delaunay triangulation based on the random point set.

As depicted in Fig. 1, a set of random points which representthe irregular material structures are distributed in the 2Ddomain as well as the corresponding boundaries. Then thedomain spanned by the random point set is decomposed bythe Delaunay triangulation scheme. Thereafter, each Delaunaytriangle is directly modeled by a linear displacement basedfinite element. These elements are connected by the cohesiveelements which represent the potential crack paths within thesolid. Due to the strong nonlinearities induced by the dynamiccracking procedure, we chose the explicit solution scheme forthe temporal integration of the dynamic system.

Sharon et al. performed a series of experimental researcheson the dynamic fracture of PMMA [3,4]. As shown in Fig. 2,the upper and lower boundaries of the testing specimen werefixed, and the prescribed stress field was applied in the y

direction. At the beginning of the test, a sharp initial crack

FIG. 2. (Color online) A specimen configuration for the dynamicfracture simulation.

nT

nδ

tf

tδO

fG

FIG. 3. (Color online) Linear decay cohesive law.

was introduced by a razor blade. Then the follow-up crackpropagation would be driven by the strain energy stored withinthe block. In the present work, the numerical model is devel-oped based on Sharon’s experiments. Considering the balancebetween the simulating precision and the computational cost,a reduced model (l = 16 mm, h = 4 mm, and a = 2 mm) isadopted (Fig. 2), which is of the same model dimensions asproposed by Zhang et al. [13].

The material parameters are taken to be E = 3.24 GPa,ν = 0.35, and ρ = 1190 kg/m3, which yield the Rayleighwave speed cR = 939 m/s. The fracture behaviors are de-scribed by the linear decay cohesive law (Fig. 3), with whichthe cohesion on the crack tips can be solely determined bythe maximum cohesive stress ft and the cohesive energyGf . The numerical system, which is developed based onthe aforementioned irregular discretization scheme, containsmore than 6 × 104 volumetric elements and 9 × 104 cohesiveelements. As the average mesh length is 50 μm, the proposedmodel would not experience mesh dependencies in the senseof cracking speed according to the mesh convergence studiesperformed by Zhang et al. [13].

A series of numerical experiments are carried out bysystematically varying the cohesion parameters ft and Gf , andrather different crack patterns and crack speeds are observedby the simulating results. Figure 4 shows the changing of crackpatterns subjected to different cohesive energies. It is observedthat the centered crack propagates starting from the tip ofthe initial notch within the block. The stochastic recursivebranching is formed during the simulation. As is shown,large cohesive energy keeps the propagation of dynamicalcracks along a smooth or even mirror-like surface [Fig. 4(a)],which is usually considered by the conventional theory offracture dynamics [1]. As the cohesive energy reduces, a mistcrack band is formed by the local branches nearby the crackpath [Fig. 4(b)]. In this stage, the local branching generatesmore surfaces to dissipate the input energies of the dynamiccracking system. As the branches are arrested nearby themajor crack path, we can define that as weakly unstablecrack propagation. Figure 4(c) indicates the strongly unstablecrack propagation subjected to very small cohesive energy.As is shown, the branches on both sides of the major crackkeep propagating into the hinterland of the solid even withsubbranches of different levels. The fragmentation is inducedby the crack tree developed in the overall domain of thesolid. It can be also observed from Fig. 4 that the irregulardiscretization introduced in the present communication brings

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DYNAMIC FRACTURE IN IRREGULARLY STRUCTURED . . . PHYSICAL REVIEW E 85, 055102(R) (2012)

(a)

(b)

(c)

FIG. 4. Crack patterns. (a) Smooth crack (ft = 130 MPa, Gf =1540 N/m, vf = 206 m/s). (b) Rough crack (ft = 130 MPa, Gf =600 N/m, vf = 709 m/s). (c) Branching crack (ft = 130 MPa, Gf =120 N/m, vf = 759 m/s).

in some uncertainties for the branching direction at the cracktip. Furthermore, the stochastic branching patterns offer morerealistic simulation of the dynamic cracking observed byexperimentations than the conventional regular discretized MDor CM based methods.

The energy balance of the dynamic fracturing system yields

U = −WL + UE + US + UK, (1)

where U is the total energy, WL is the external work, UE is thestrain energy, US is the energy dissipated by generating newsurfaces, and UK is the kinetic energy. The first three termson the right-hand side of Eq. (1) describe the energy balanceof static fracture, which was proposed by Griffith [14]. Later,Mott [15] introduced the kinetic energy term UK to considermoving cracks. Considering crack propagation within solids,the thermodynamical equilibrium yields [14,15]

dU

dc= 0, (2)

where c is the crack length defined by the forefront branch.Substituting Eq. (1) in Eq. (2), we have

G − 2� = dUK

dc, (3)

where the static energy release rate G = − d(−WL+UE )dc

and thesurface energy 2� = dUS

dc. Define the dynamic fracture energy

release rate Gd = G − dUK

dc. Then we have

Gd = 2� = dUS

dc. (4)

As the branching of cracks leads to large oscillations in theinstantaneous crack speed and fracture energy release rate, thecorresponding averaged quantities are usually calculated for

0.0 0.2 0.4 0.6 0. .00.0

0.2

0.4

0.6

0.8

1.0

1-β

Gf / G

d

v f / c R

ft=130MPa ft=200MPa ft=80MPa Proposed equation

FIG. 5. (Color online) Crack speed as a function of cohesiveenergy.

the investigations. We define the averaged quantities by thefinite differences as follows:

Gd = dUS

dc≈ �US

�dc, vf = dc

dt≈ �c

�t. (5)

As observed experimentally [3,4], Gf /Gd corresponds tothe increase of the fracture surfaces due to microbranching andcan be defined as an index of fracture morphology. A singlestraight crack gives Gf /Gd = 1 and branching cracks yieldGf /Gd < 1. Strong dependencies between the crack velocityand the crack branching patterns have been indicated in ournumerical results. The simulated curve of the dimensionlesscrack speed vf /cR versus the dimensionless fracture energyGf /Gd is shown in Fig. 5. As Gf → 0, the crack speedvf approaches its maximum, which is usually less than theRayleigh wave speed cR . As the cohesive energy increases,the crack speed decreases slowly in the beginning stage butrapidly after a turning point, while the crack pattern alsoreduces from global crack tree to local rough crack and finallysingle major crack. If the cohesive energy is large enough,the input energy can be totally dissipated by the nonlinearcrack-tip field rather than crack propagation, whereas the crackspeed reduces to zero. The crack branching and crack speedindicate two competing mechanisms in dynamic fracture. Inthe stable fracture stage characterized by a single crack, therate of energy dissipation is controlled by the crack speed.When the crack speed becomes large enough, the crack intendsto dissipate energy by generating more surfaces via morebranches rather than faster propagations. This also agrees wellwith the experimental investigations [3].

The equation of energy balance at the crack tip can beexpressed as follows [1,16]:

Gf = GdA(vf ) ≈ Gd

(1 − vf

cR

). (6)

By inverting Eq. (6), we obtain the equation of motion for apropagating crack:

vf

cR

= 1 − Gf

Gd

. (7)

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XIAODAN REN AND JIE LI PHYSICAL REVIEW E 85, 055102(R) (2012)

Equation (7) works well to describe the motion of a singlecrack in an effectively infinite medium. Marder [17] proposedan equation of motion for a crack in an infinite strip as follows:

Gf

Gd

= 1 − α1[

1 − ( vf

CR

)2]2 , (8)

where the dimensionless acceleration α = bv̇

c2l

was assumed to

be very small due to the requirement of the perturbation theory.Both Eqs. (7) and (8) are developed for the single prop-

agating cracks so that they fail to qualitatively describe themultiple crack state invoked by the fast propagation. Onthe other hand, they also suggests that the governing equationof the dynamic fracture may be expressed by vf

cRand Gf

Gd. Thus

we propose a fitting equation of the simulated resultsas follows:

vf

cR

= 1 − β

√Gf

Gd

, 0 <Gf

Gd

� 1. (9)

The result of the proposed equation (9) is also plotted in Fig. 5.It is indicated that the parameter (1 − β) defines the thresholdof the fracture stability. As we can see from the expression

vf

cR

� 1 − β ⇒ Gf

Gd

= 1, (10)

the dynamic fracture energy Gd equals the static fractureenergy Gf , which yields the stable crack propagation withoutany branching. On the other side, if we have

vf

cR

> 1 − β ⇒ Gf

Gd

< 1, (11)

one obtains Gd > Gf , which means the propagation ofmultiple cracks and unstable fracture state.

In summary, we have developed a simulation methodfor dynamic crack propagation by an irregular structured

system. The irregular discretization enables us to considerthe stochastic material structures for dynamic fracture. Ourmethod changes the conventional simulation method relyingon regular discretization. And the stochastic branching dia-grams simulated by the proposed method can help to explaina range of controversial experimental results [2–4]. Basedon the systematic simulation results, the phenomenologicalrelation between crack speed and dynamic fracture energy isproposed with the parameters determined from comparisonwith numerical results. The proposed equation teaches us thatthe transition between the stable and the unstable dynamicfractures is not governed by a single threshold of crack speed.Besides the irregularity of microstructure introduced in thepresent communication, another random perturbation whichcould have a large effect on brittle fracture is the thermal noise[18]. The comparative study of the two types of randomnessleads to the future development of the present work.

One may argue that the theoretical derivation of theproposed equation may be more interesting because thecomprehension of the crack-limiting speed could be furtherclarified. However, the authors believe that the investigationson unstable dynamic fracture may rely mainly on numericaland experimental approaches. By analogy with the systemof turbulence in fluid dynamics, the qualitative results forthe unstable dynamic fracture system may not be easilydeveloped through theoretical approaches due to its endowednonidentities and uncertainties. Thus the proposed numericalmethod, which considers the constitution of the mesostruc-ture, offers a possibility to promote investigations in thisfield.

We acknowledge financial support from the NationalNatural Science Foundation of China (NSFC), Grant No.90715033. We would also like to thank the referee for manyvaluable suggestions and corrections.

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