Numerical Prediction of Dynamically Propagating and

Copyright c© 2004 Tech Science Press CMC, vol.1, no.2, pp.191-204, 2004

Numerical Prediction of Dynamically Propagating and Branching Cracks UsingMoving Finite Element Method

S. Tchouikov1, T. Nishioka1 and T. Fujimoto1

Abstract: Phenomena of dynamic crack branching areinvestigated numerically from a macroscopic point ofview. Repetitive branching phenomena, interaction ofcracks after bifurcation and their stability, bifurcationinto two and three branches were the objectives of this re-search. For the analysis of dynamic crack branching, re-cently we developed moving finite element method basedon Delaunay automatic triangulation [Nishioka, Furu-tuka, Tchouikov and Fujimoto (2002)]. In this study thismethod was extended to be applicable for complicatedcrack branching phenomena, such as bifurcation of thepropagating crack into more than two branches, multi-ple crack bifurcation and so on. The switching methodof the path independent dynamic J integral, which wasdeveloped for the case of simple two cracks branchingphenomena, demonstrated it’s excellent applicability alsofor the case of complicated crack branching. The simu-lation results are discussed with consideration to the ex-perimental findings.

keyword: Dynamic crack bifurcation, dynamic frac-ture, crack propagation and arrest, moving finite elementmethod, dynamic J integral, fracture prediction criteria,multiple branching.

1 Introduction

When dynamic fracture occurs in brittle materials andconstructions it is rarely takes place as a propagation ofa single crack. Conversely very often, single crack bifur-cates into two or more cracks, which propagate simulta-neously. Some of them branch again, others propagate asa single crack or arrest. Several propagating cracks in-teract and influence each other. The prediction of brittlefracture is very important problem in dynamic fracturemechanics. The accurate prediction of such complicatedfailure process is extremely important not only for aca-

1 Simulation Engineering Laboratory, Faculty of Maritime Sciences,Kobe University, Higashinada-Ku, Kobe, 658-0022, Japan

demic interest, but also for safety design of constructionsor for controllable dynamic destruction.

The dynamic crack bifurcation problem still remains asone of the most important unsolved problems in dynamicfracture mechanics. The problem of governing condi-tion of dynamic crack branching is investigated in ourrecent experimental studies [Nishioka, Kishimoto, Onoand Sakakura (1999a, 1999b)]. The experiments on dy-namic crack branching phenomena in Homalite-911 andHomalite-100 revealed, that the energy flux per unit timeinto a propagating crack tip or into a fracture processzone governs the dynamic crack branching. Other ex-perimental investigations on evaluation of dynamic crackbranching by method of caustics have been also per-formed recently [Nishioka, Matsumoto, Fujimoto andSakakura (2003)].

There are some difficulties that must be overcome to per-form accurate numerical simulation of dynamic crackbifurcation. In previous studies, the authors [Nish-ioka, Furutuka, Tchouikov and Fujimoto (2002)] devel-oped a moving finite element method based on Delau-nay automatic triangulation [Sloan and Houlsby (1984),Taniguchi (1992)], which satisfied the numerical require-ments for simulation of two branches dynamic crackbifurcation phenomena. In present work this methodis extended to simulate the dynamic crack branchinginto more than two branches, propagation of interact-ing cracks, multiple crack branching phenomena etc. Inthis method, the moving singularities at the tips of dy-namically propagating cracks are treated accurately; evencomplicated fracture path is carefully generated and thefracture parameters, such as dynamic J integral and dy-namic stress intensity factors are accurately evaluatedeven immediately after the bifurcation.

The propagation direction of each individual crack iscontrolled by local symmetry criterion. According to thiscriterion it was possible to find the propagation directionfor two or three branches even immediately after crack

192 Copyright c© 2004 Tech Science Press CMC, vol.1, no.2, pp.191-204, 2004

Figure 1 : Specimen geometry

bifurcation. For crack growth predicting a unique depen-dence of dynamic fracture toughness on the crack speedonly is assumed. For predicting of branching instant, atwhich dynamically propagating crack bifurcates, the crit-ical amount of energy flux per unit time into a processzone is assumed.

2 Experimental Measurements

The experimental investigations of dynamic crackbranching [Nishioka, Kishimoto, Ono and Sakakura(1999a, 1999b); Nishioka, Matsumoto, Fujimoto andSakakura (2003)] were performed with major point ofinterest in under what conditions the crack branch-ing occurs. In these experiments transparent materialHomalite-911 was used. The material properties areas follows: Young’s modulus E=2.15GPa, Poisson ratioν=0.38 and mass density ρ=1281 kg/m 3. The geometryof a typical specimen used for fast crack branching testis shown in Fig.1. In the experiments crack initiates froma blunt notch, which length was set as 23mm from upperend of the specimen. The notch-root diameter is about

Figure 2 : Loading equipment

0.5 mm.

Two sets of displacement-controlled loads were appliedto the specimen through two pairs of loading pins asshown in Fig.2. First, the some load P2 was appliedto the lower loading system and held constant. It pro-duces a strong tensile stress field to supply enough en-ergy into the propagating crack tip for crack branching.Then the upper loading system was gradually loaded bythe L shape jigs as shown in Fig.2, until dynamic crackpropagation occurred. The crack tries to bifurcate nearthe lower loading line. Examples of the high-speed pho-tographs of the caustic patterns, which were recorded inthe experiments, using a high-speed camera (with max-imum framing rate two million frames per second) areshown in Fig.3. Measuring the crack-tip positions in eachframe each crack propagation history was described by apolynomial curve. Then, differentiating the polynomialswith time, the crack velocities were determined. Fromdimensions of the caustics patterns in photographs theexperimental K values can be obtained by the method ofcaustics.

Numerical Prediction of Dynamically Propagating and Branching Cracks 193

61µs 120µs 179µs

(a) straight crack

58µs 119µs 180µs

(b) crack bifurcation into two branches

59µs 120µs 180µs

(c) crack bifurcation into three branches

Figure 3 : High-speed photographs of dynamic fracture

3 Numerical Procedures

3.1 Moving Finite Element Method

Recently, for modeling of the crack propagation manyinnovative techniques have been developed, such as:the cell method [Ferretti (2003)], the novel non-

hypersingular time-domain traction boundary elementmethod [Zhang and Savaidis (2003)], the symmetricGalerkin boundary element method [Han and Atluri(2002)], the material point method [Nairn (2003)].

To simulate the crack propagation by the finite element


(a) (b) (c)

Figure 4 : An example of automatic mesh generation (a) exterior boundary points, (b) specified interior points, (c)generated mesh pattern

method, two different concepts of computational model-ing can be considered, i.e., the stationary element proce-dure (or fixed element procedure), and the moving el-ement procedure. [Nishioka and Atluri (1986), Nish-ioka (1994, 1997)]. Due to some disadvantages ofthe fixed element procedure we used the moving ele-ment procedure, which allowed us to accurately modelthe dynamically propagating [Nishioka, Tokudome, Ki-noshita and Nishida (1998); Nishioka, Tokudome andKinoshita (2001)] and branching crack [Nishioka, Furu-tuka, Tchouikov and Fujimoto (2002)].

In this study for mesh generation we used the modifiedDelaunay automatic triangulation [Taniguchi (1992)],which requires only exterior, interior boundary pointsand specified interior points (if they are necessary)(Fig.4). In consideration of the stress singularity eachpropagating crack tip is always surrounded by the speci-fied interior points.

At the Delaunay automatic mesh generation stage the twosurfaces of crack path have common nodal points, andthe crack surfaces are described by element boundaries.In order to distinguish both surfaces of crack after De-launay automatic mesh generation, dual nodes setting oncrack path is used, so that, the nodal points with the samecoordinates are have different numbers if there are ly-ing on opposite crack surfaces. Therefore, the total num-ber of nodal points increases and the element-nodes rela-tions are changed. During crack propagation, when cracklength is increased more than certain value, new nodal

points are placed on crack path behind the group of sur-rounding interior points. Furthermore, only an area of thegroup of specified interior points with its neighborhoodis actually re-meshing during crack propagation, the restof the mesh pattern is remaining fixed for more accuracyof analysis.

For the time integration of the finite element equationof motion the Newmark method is used. To fulfill theunconditionally stable condition the Newmark’s parame-ters are chosen to be β=1/4 and δ=1/2 [Bathe and Wilson(1976)]. For further details about time integration proce-dures please refer to [Nishioka, Furutuka, Tchouikov andFujimoto (2002)].

3.2 Evaluation of Fracture Mechanics Parameters

In this study, to evaluate various fracture mechanics pa-rameters for a dynamically propagating and branchingcracks the path independent dynamic J integral derivedby Nishioka and Atluri (1983) is used.

For most numerical analyses, considering dynamicallypropagating crack in an elastic solid, the global-axiscomponents of the dynamic J integral (J ′) can be eval-uated by the following expression:

J′k =∫

Γ+Γc

[(W +K)nk − tiui,k]dS

+∫

VΓ

[(ρui − fi)ui,k −ρuiui,k]dV (1)

where ui, ti, fi,nk and ρ denote the displacement, traction,


body force, outward direction cosine, and mass density,respectively. W and K are the strain and kinetic energydensities, respectively, and (),k = ∂()

/∂Xk. The integral

paths Γε,Γ and the Γc shown in Fig.5 denote a near-fieldpath, far-field path and crack surface path, respectively.VΓ is the region surrounded by Γ, while Vε is the regionsurrounded by Γε.

Figure 5 : Definition of integral paths

The crack-axis components of the dynamic J integral canbe evaluated by the following coordinate transformation:

J′0l = αlk(θ0)J′k, (2)

where αlk is the coordinate transformation tensor and θ0

is the angle between the global X1 and the crack axis x01.

For the elastodynamically propagating crack with veloc-ity C the dynamic J integral can be related to the instan-taneous stress intensity factors as in [Nishioka and Atluri(1983)]:

J′01 =

12µ

{AI(C)K2I +AII(C)K2

II +AIII(C)K2III} (3)

J′02 = −AIV (C)

µKIKII (4)

where µ is the shear modulus, and AI (C)−AIV (C) arefunctions of crack velocity C and given in [Nishioka andAtluri (1983)].

The dynamic J integral has following salient features:

(i) It physically represents the dynamic energy releaserate G [Nishioka and Atluri (1983)].

(ii) For the far-field path, it has the property of the path-independent integral [Nishioka and Atluri (1983)].

(iii) For the near-field path, it is practically invariant withthe shape of the infinitesimal near-field path [Nish-ioka (1994)].

(iv) The dynamic J integral includes the static J integralfor elastostatic fracture problems.

(v) The near-field path can be taken as the boundary of afracture process zone (if it is known a priori).

To accurately evaluate the inplane mixed-mode stress in-tensity factors from the dynamic J integral values, thecomponent separation method [Nishioka (1994)] wasproposed. If the energy release rate G or the crack-axiscomponent is obtained by Eq.(2), the formulae of thecomponent separation method can be expressed as:

KI = δI

{2µJ

′01 β2

AI(δ2I β2 +δ2

IIβ1)

}1/2

= δI

{2µGβ2

AI(δ2I β2 +δ2

IIβ1)

}1/2

(5)

KII = δII

{2µJ

′01 β2

AII(δ2I β2 +δ2

IIβ1)

}1/2

= δII

{2µGβ2

AII(δ2I β2 +δ2

IIβ1)

}1/2

(6)

Some of the features of the component separation methodare: (i) mixed-mode stress intensity factors can be evalu-ated by ordinary non-singular elements, and (ii) the signsof KIand KII are automatically determined by the signsof δI and δII , respectively.

Because of difficulty in setting far-field integral path sep-arately for each just bifurcated crack tip, a switchingmethod of the path independent dynamic J integral wasproposed [Nishioka, Furutuka, Tchouikov and Fujimoto(2002)]:

J′k =∫

Γ+ΓC

[(W +K)nk − tiui,k]sdS

+∫

VΓ

[{(ρui − f )ui,k −ρuiui,k}sdV

+σi jui,ks, j − (W +K) s,k]dV (7)


where Γ is a far-field integral path, that encloses allbranched crack tips and s is a continuous function de-fined in VΓ (Fig.6).

For calculation of the dynamic J integral for certain cracktip the s function is set as s=1 for the point at that cracktip and for the points in whole domain VΓ and s=0 for thepoints at the others crack tips. Equation (7) made possi-ble accurate evaluation of the dynamic J integral compo-nents for interacting branched crack tips.

Figure 6 : Integral path for branched crack tips

3.3 Fracture Path Prediction Criterion

Many propagation-direction criteria have been proposedbefore. They can be divided into two groups: explicitprediction theories and implicit prediction theories. Anexplicit prediction theory predicts the propagation direc-tion satisfying the postulated criterion based on a phys-ical quantity for the current crack tip. Contrary, an im-plicit prediction theory seeks the propagation directionthat satisfies the postulated criterion based on physicalquantity after the crack is advanced with a small crack-length increment. It is known that in general the implicitprediction theories are more accurate, although an iter-ative process usually is needed to find the propagationdirection.

In this study the propagation direction of each individualcrack is predicted accordingly to a local symmetry crite-

rion [Goldstein and Salganik (1974)], which falls into thegroup of implicit prediction theories. The fracture path ispredicted in an iterative manner as follows. At a generictime step n, as a first trial, the crack is advanced in thetangential direction θ(i)

n (i=1) equal to a crack tip direc-tion at the previous step n-1. If mode II stress intensityfactor KII is almost zero at the attempted crack-tip loca-tion, the crack is advanced in this direction θ (i)

n . If the KII

value is not zero, the crack is tentatively advanced in thedirection of θ(i+1)

n = θ(i)n + ∆θ and θ(i+2)

n = θ(i)n −∆θ as

the next two trials. Each time the satisfaction of the cri-terion at the trial crack tip location is checked. If the cri-terion is not satisfied in both cases the next trial directionis calculated accordingly to the square polynomial curvefitting all KII values versus θ(i)

n (i=1,2,. . . ) of previous tri-als. These procedures are repeated until the criterion issatisfied for each propagating crack.

Furthermore, when using the local symmetry principleit is possible to find propagation direction of each crackimmediately after bifurcation in case of two and threecracks branching.

3.4 Crack Growth Prediction Criterion

In fast fracture mechanics, for dynamically propagatingcracks under quasi-static as well as dynamic loading con-ditions, it has been suggested that

K = KD(T,C,C, ...) (8)

where KD is the dynamic propagation fracture tough-ness. Effects of crack acceleration and deceleration arestill not fully clarified and under investigation [Taka-hashi and Arakawa (1987), Nishioka, Syano and Fu-jimoto (2000)]. Furthermore, several works [Gates(1980), Kalthoff, Beinert and Winkler (1981), Nishioka,Kishimoto, Ono, Sakakura (1991a)] have reported thegeometry-dependence of KD.

According to the criterion given by Eq.(8), crack arrestoccurs when the stress intensity factor becomes smallerthan or equal to a critical value. This can be expressed as

K ≤ KD(0)≡ Kdyna ≡ KA (9)

where Kdyna or KA denotes the dynamic crack arrest

toughness. The superscript “dyn” is used to distinguish amaterial property from the so-called static arrest tough-ness Ka (Ksta

a ).


In this study, we assume a unique dependence of dy-namic fracture toughness on the crack tip speed only[Dally, Fourney and Irwin (1985)]. For establishingcrack growth prediction criterion, we choused three ex-perimental results, in which crack propagated straightlywithout branching [Nishioka, Matsumoto, Fujimoto andSakakura (2003)]. The dynamic propagation toughnessKD(C) in each case was different. The KA value for eachexperiment was different and was calculated by interpo-lating KD(C) on C=0. Nominal dynamic stress inten-sity factors are defined by KA for each experiment. Aphenomenological relation between nominal stress inten-sity factor and nominal crack tip speed was derived fromthree experimental results (see Fig.7).

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

C/Cs

Exp. No.1

Exp. No.2

Exp. No.3

Figure 7 : Dynamic fracture toughness for Homalite-911(C: crack tip speed, Cs: shear wave speed)

Using KA value for each of three experiments a phe-nomenological relation shown in Fig.7 was obtained.

In the experiments, due to specimen geometry, as loadP1 was meant to just initiate crack, the lower loading P2

value was most influential on the crack propagation andbranching. For experimental lower loading P2 and initialconditions of each experiment, calculating KQ – a modeI stress intensity factor for the initial crack under quasi-static loading for various upper loading P1, a relation be-tween P2

/P1 and KQ

/KA was obtained (Fig.8).

Fracture initiation for static loading configurations isstrongly dependent on the initial crack tip radius. In thisstudy, we are basing on the experiments in which the ini-tial notch-root diameter was about 0.5mm. Therefore, for

0

2

4

6

8

10

0 2.5 5 7.5 10 12.5 15

P2/P

1

Exp. No.1

Exp. No.2

Exp. No.3

P1

P2

Figure 8 : The relation between dynamic crack arresttoughness KA, initiation quasi-static KQ and loading con-ditions for used specimen geometry

initiation under quasi-static loading from a blunt notch,the KI must reach some critical quasi-static stress inten-sity KQ value, which is assumed as KQ = 1.75[Pa

√m].

In numerical analysis, first a certain load P2 and verysmall load P1 is statically applied to both loading sys-tems. Then KI , calculated for the initial crack tip, is com-pared with the assumed KQ value. If KI is less than KQ,upper load P1 is gradually increased until KI reaches KQ.When value of KI becomes more or equal KQ, knowingP2

/P1 and KQ, the dynamic crack arrest toughness KA

for choused specimen is determined from curve in Fig.8.At the static analysis under the fracture loads, the dis-placements at the loading points were evaluated and thesewere used as the prescribed displacement in the finite el-ement model after crack initiation. For each individualcrack tip, crack speed is predicted by iteration process,so that KI of crack tip propagating with velocity C is al-ways equal KD (C) (Fig.7).

3.5 Crack branching criterion

The determination of crack branching instant (i.e. whereand when propagating crack bifurcates) is very impor-tant problem in the dynamic fracture prediction studies.Branched cracks are often observed in brittle materialsand structures. Many attempts have been made in or-der to clarify the mechanism of crack branching. How-ever the governing condition for dynamic cracks branch-


Figure 9 : Deformed mesh pattern for straight crack propagation (P2=1.4kN)

ing had not been fully elucidated until our recent experi-mental studies [Nishioka, Kishimoto, Ono and Sakakura(1999a, 1999b)]. The experimental results revealed, thatthe energy flux per unit time into a propagating crack tipor into a fracture process zone, which can be expressedas:

Φtotal = J′1 ·C, (10)

governs the dynamic crack bifurcation. In this study weassumed crack branching to take place at a given criti-cal Φc

total=550 [J/ms]. Furthermore, we suggested twobranching cases: into two and three branches.

4 Results

The input data, such as specimen geometry, materialproperties, nodes number, values of critical fracture pa-rameters, was set the same for all numerical examples,except only load P2, which was set 1.4kN, 1.6kN, 1.8kNand 2.0kN. In each loading case two possibilities of crackbifurcation were suggested: two and three macro-cracksbranching.

In numerical simulations, first, the static analysis underthe fracture loads was performed. The displacements atthe loading points were evaluated and these values wereused as the prescribed displacement in the finite elementmodel after crack initiation. The time increment of ∆t =2µs was used.

As the upper loading device can only “push” the load-ing pins but not “pull”, the fixed boundary conditions atthe upper loading points in the numerical analysis maylead to a spurious deformation behavior of the specimen.Thus, the possibility of lack of contact of the L shapejigs with the loading pins should be taken in account.In present study, to solve this problem, the contact/non-contact boundary conditions [Nishioka, Perl and Atluri(1983); Nishioka and Atluri (1982); Nishioka, Furutuka,Tchouikov and Fujimoto (2002)] are employed in the nu-merical simulations.

The deformed mesh pattern for case of load P2=1.4kNis shown in Fig.9. The initial number of elements andnumber of nodes were 6756 and 3500 respectively.

The time variations of the dynamic stress intensity fac-


-0.5

0.0

0.5

1.0

1.5

2.0

0

100

200

300

400

500

600

0 50 100 150 200Time [µs]

C

KI

KII

Φtotal

critical Φtotal

Figure 10 : Variations of dynamic SIF, energy flux and crack velocity C for not bifurcated crack (P2=1.4kN)

tors, energy flux to the propagating crack tip per unittime and predicted crack speed are plotted in Fig.10.The Φtotal of propagating crack did not reached a criti-cal value, so there was just a straight crack propagationwithout bifurcation. After crack starts propagate the KI

value drops for about a half of KQ and then increases untilcrack reaches lower loading line area. The crack propa-gation speed was in interval 490-535m/s which is morethen 0.6Cs.

The deformed mesh patterns for load P2 equal 1.6kN,1.8kN and 2.0kN are shown in Fig.11. Two typesof crack branching was assumed: into two and threebranches. For loading of 1.6kN and 1.8kN crack branch-ing occurred only once. The location of crack branchingis closer to initial crack for larger lower loading. In caseof lower loading 2.0kN for both types a multiple crackbranching occurred. For case of two cracks branchingboth of the cracks branched again symmetrically and didnot stop. For three branches case only a central crackbranched again. Finally for branching into three cracksboth sides cracks arrested and only a central crack con-tinued propagation. Due to random mesh and piling ofcalculations a slight asymmetry effects can be noticed.Furthermore, the contact/non-contact problem for cracksurfaces was ignored in this study, so a small overlappingcan be seen for three branches case. For multiple threebranches case, shown in Fig.11(c), due to five crack tipspropagating simultaneously, the number of elements andnodes increased exceedingly and was 20038 and 10378

respectively.

The computed variations of input (E), strain (W), kinetic(K) and fracture (F) energies with time for two and threebranches with load 1.6kN are shown in Fig.12. It shouldbe noted that in the present procedure, each of the quan-tities E, W, K and F is calculated separately and directly.Thus, the fact that W+K+F is almost equal to E at alltimes is an inherent check on the accuracy of the calcu-lation. It also can be seen that for three branches case thefracture energy F is slightly larger than fracture energyfor two branches case.

Even immediately after crack branching the values of dy-namic J integral were accurately calculated. The path in-dependence of dynamic J integral for central branchedcrack tip in case of 1.6kN loading is shown in Fig.13.

The dynamic stress intensity factors for (a) two and(b) three branches with loading of 1.6kN are plotted inFig.14. Due to the local symmetry criterion choused forpath prediction, mode II stress intensity factors are al-most zero for all propagating cracks throughout the anal-ysis. The mode I SIF for central crack in Fig.14(b) ismuch larger then KI of two others cracks, especially atthe final stage of simulations. Finally all cracks on sidesarrested, and only the central crack continued propagat-ing.

The variations of energy flux to the propagating cracktips per unit time for 1.6kN loading are plotted in Fig.15.It can be seen, that when the energy flux for straightlypropagating crack becomes larger a critical value crack


1.6kN1.6kN

1.8kN1.8kN

2.0kN2.0kN

(a) lower loading P2=1.6kN

(b) lower loading P2=1.8kN

(c) lower loading P2=2.0kN

1.6kN1.6kN

arrestarrest

1.8kN1.8kN

arrest arrest

2.0kN2.0kN

arrest arrest

Figure 11 : Deformed mesh patterns for crack branching into two and three branches with different lower loading(180µs)


0.0

0.1

0.2

0.3

0.4

0.5

0 50 100 150 200Time [µs]

E (Input Energy)

F (Fracture Energy)

W (Strain Energy)

K (Kinetic Energy)

W+K+F

(a) two branches

0.0

0.1

0.2

0.3

0.4

0.5

0 50 100 150 200Time [µs]

E (Input Energy)

F (Fracture Energy)

W (Strain Energy)

K (Kinetic Energy)

W+K+F

(b) three branches

Figure 12 : Energy balance (P2=1.6kN)

90 µs

0

200

400

600

800

1000

1200

Path-No.

68 µs

70 µs74 µs

126 µs134 µs142 µs152 µs

164 µs

180 µs

1 2 3 4 5

Figure 13 : Dynamic J integral against path number(P2=1.6kN, central crack of three branches)

bifurcation occurred. In this loading case energy flux forall branched cracks did not reach a critical value again,so multiple branching did not occur. It also can be seen,that a sum of energy flux for branched cracks is muchlarger for three cracks branching at the early stage afterbranching.

-0.5

0.0

0.5

1.0

1.5

2.0

0 50 100 150 200Time [µs]

KII(cr.1)

KI(cr.1)

KI(cr.2)

KI(cr.3)

KII(cr.2,cr.3)

crack branchingcr.1

cr.2 cr.3

(a) two branches

-0.5

0.0

0.5

1.0

1.5

2.0

0 50 100 150 200Time [µs]

KII(cr.1)

KI(cr.1)

KI(cr.4)

KI(cr.2)

KII(cr.2,cr.3,cr.4)

crack branching

KI(cr.3)

cr.1

cr.2 cr.4cr.3

(b) three branches

Figure 14 : Variations of dynamic SIF (P2=1.6kN)

The energy flow through a contour Γ with normal vectorn j to the crack tip can be expressed as follows:

Φ = −∫

Γϕ jn jdS (11)

where

ϕ j = −C · [(W +K)δ1 j −σi jui,1] (12)


0

200

400

600

800

1000

1200

1400

0 50 100 150 200Time [µs]

crack branchingcr.1

cr.2 cr.3

Φtotal

(cr.1)

critical Φtotal

Φtotal

(cr.2,cr.3)

Φtotal

(sum)

(a) two branches

0

200

400

600

800

1000

1200

1400

0 50 100 150 200Time [µs]

crack branching

Φtotal

(cr.1)

critical Φtotal

Φtotal

(cr.2,cr.4)

Φtotal

(sum)

cr.1

cr.2 cr.4cr.3

Φtotal

(cr.3)

(b) three branches

Figure 15 : Variations of energy flux (P2=1.6kN)

The sign in Exp.(12) is chosen so that the vector ϕ j point-ing toward the crack tip corresponds to an energy fluxinto the crack tip. The orientation of the energy flux vec-tor for whole specimen and in the vicinity of the prop-agating cracks after bifurcation for loading case 2.0kN,three branches type is shown in Fig.16. The energy fluxin front of central crack is oriented toward the propagat-ing crack tip but in front of side cracks it is oriented toa central propagating crack, especially for crack betweencentral crack and farthest left side crack. This explainswhy finally all sides crack have arrested while centralcrack continued propagating.

The simulation results (for loading 1.6kN) in comparisonwith actually fractured specimens are shown in Fig.17.The predicted fracture paths are very similar with exper-

Figure 16 : The orientation of the energy flux vector(P2 = 2.0kN, multiple three branches)

imental ones. For three branches type as in experiments,both side cracks have arrested while central crack con-tinued propagating. For two branches type there was nocrack arrest.


experiment prediction

(a) two branches

experiment prediction

(b) three branches

Figure 17 : Examples of fractured specimen and predic-tion results

5 Conclusions

In this study, the moving finite element method basedon Delaunay automatic triangulation was further devel-oped for the numerical simulation of complicated dy-namic crack bifurcation phenomena, such as branchinginto two or three branches and multiple branching (i.e.each branch encounters branching and so on). Variousdynamic fracture parameters were accurately evaluatedby the switching method of the path independent dy-namic J integral even immediately after crack bifurca-tion.

The dynamic bifurcation phenomena observed experi-mentally was successfully predicted by the numericalsimulations with conjunction of the local symmetry cri-terion, the dynamic fracture toughness criterion and thecritical energy flux criterion.

Acknowledgement: This work was supported by thegrant from Japan Society for the Promotion of Science(No. 14011647), by the Grant-in-Aid for Scientific Re-search (No. 14205019) and by the Natural Science Grantfrom Mitsubishi Foundation.

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