Dynamic fracture mechanics analysis of failure mode transitions along weakened interfaces in elastic...

$: Dynamic fracture mechanics analysis of failure mode transitions along weakened interfaces in elastic solids$
Engineering Fracture Mechanics 73 (2006) 1597–1614

www.elsevier.com/locate/engfracmech

Dynamic fracture mechanics analysis of failure modetransitions along weakened interfaces in elastic solids

L. Roy Xu *, Ping Wang

Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, USA

Received 18 April 2005; received in revised form 4 January 2006; accepted 6 February 2006Available online 27 April 2006

Abstract

Dynamic fracture mechanics theory was employed to analyze the crack deflection behavior of dynamic mode-I crackspropagating towards inclined weak planes/interfaces in otherwise homogenous elastic solids. When the incident mode-Icrack reached the weak interface, it kinked out of its original plane and continued to propagate along the weak interface.The dynamic stress intensity factors and the non-singular T-stresses of the incident cracks were fitted, and then dynamicfracture mechanics concepts were used to obtain the stress intensity factors of the kinked cracks as functions of kinkingangles and crack tip speeds. The T-stress of the incident crack has a small positive value but the crack path was quite sta-ble. In order to validate fracture mechanics predictions, the theoretical photoelasticity fringe patterns of the kinked crackswere compared with the recorded experimental fringes. Moreover, the mode mixity of the kinked crack was found todepend on the kinking angle and the crack tip speed. A weak interface will lead to a high mode-II component and a fastcrack tip speed of the kinked mixed-mode crack.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Crack kinking; Dynamic stress intensity factors; Failure mode transition; The T-stress; Mode mixity

1. Introduction

When cracks propagate in homogenous and brittle solids, they can only do so under locally mode-I con-ditions and at sub-Rayleigh wave speeds typically below the crack branching speed [1–3]. Even if the appliedloading is asymmetric, the dynamically growing crack will curve and follow the path that will result in locallyopening (mode-I) conditions. The situation is entirely different if a crack is constrained to propagate along aweak preferred path in an otherwise homogenous solid. In this case and depending on the bond strength,the weak crack path or bond often traps the crack, suppresses any tendency of branching or kinking out ofthe weak plane and permits very fast crack growth much beyond the speeds observable in monolithic solids[4,5]. Indeed, when mode-I cracks propagate in both isotropic and orthotropic solids containing weak crackpaths [6], they can reach speeds as high as the Rayleigh wave speed of the solid. On the other hand, when

0013-7944/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfracmech.2006.02.009

* Corresponding author. Fax: +1 615 322 3365.E-mail address: [email protected] (L.R. Xu).

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1598 L.R. Xu, P. Wang / Engineering Fracture Mechanics 73 (2006) 1597–1614

mode-II cracks are made to propagate along such weak paths, they tend to go at even faster speeds that areclearly within the intersonic regime of the solid [4,7]. Although the extreme mode-I and mode-II cases haverecently been studied experimentally and theoretically, very little is known about the dynamic mixed-modecrack growth along weak paths, a situation that has only recently been analyzed by Geubelle and Kubair[8] about the transition of an incident dynamic mode-I crack into a mixed-mode crack as it encounters a weakplane or interface. Recently, Xu et al. [9] examined the incidence of dynamically growing cracks at inclinedinterfaces of various strengths. Interesting phenomena on mixed-mode crack growth along an interface wereobserved. They tested weakly bonded systems composed of identical constituents so that the resulting materialremains constitutively homogenous. However, the existence of a weak bond (bond of lower fracture tough-ness) made this material inhomogeneous regarding its fracture resistance behavior. Therefore, the complica-tion of the stiffness property and wave speed mismatch across the interface was avoided while retaining theessential properties of a weak interface or bond whose strength could be experimentally varied and analyticallymodeled.

As shown in Fig. 1(a), a novel wedge-loaded Homalite-100 plate is employed to produce a single, straightdynamic crack propagating towards the weakly bonded, inclined interface. The specimen sizes were largeenough such that the major stress waves reflected from free boundaries entered the field of view, 20 ls afterthe incident crack reached the interface. Inclined interfaces included several characteristic interfacial anglesof 10�, 30�, 45�, 60� and 90�. To provide different interfacial strengths and fracture toughnesses, two kindsof adhesives were used to bond the interfaces and to create weak interfaces of toughness less than that ofmonolithic Homalite (brittle polymer with well-known dynamic fracture behavior, see [10]). Fig. 1 shows aseries of dynamic photoelasticity images of the crack deflection process at a weak interface (interfacial angle30�). The vertical line appearing in every image is the camera streak line, which was used for positioning andreference purposes. Another inclined thin line reveals the position of the interface. A dark circular spot, at theleft-hand bottom, is a scaling mark. In Fig. 1(b), a dynamic mode-I crack (featuring symmetric fringe patterns)is seen to propagate towards the interface. Around 164 ls after impact, we notice that the crack tip fringe pat-tern has already lost some of its symmetry. Around 170 ls (Fig. 1(d)), this mode-I incident crack has alreadytransitioned into a mixed-mode crack at the interface, whose fringe pattern at the crack tip is clearly asymmet-ric with respect to its propagation direction. In fact, a close look at this pattern reveals that its line of sym-metry was still parallel to the horizontal line although the crack propagates along the inclined weakinterface. Also, the caustic (or shadow spot surrounding the crack tip, see [11,12]) size at the crack tip is sig-nificantly reduced in comparison to the caustic sizes in Fig. 1(b) and (c). As the interfacial crack quickly movedout of the field of view, the horizontal crack faces of the original mode-I crack were seen to experience clearfrictional contact as evidenced from Fig. 1(f). The abruptness of the transition behavior between a mode-Iincident crack and a mixed-mode interfacial crack could be graphically witnessed by the impressive jump incrack speeds across the interface. Fig. 2(a) shows the total crack length history as the incident mode-I crackdeveloped and transitioned into a mixed-mode interfacial crack. The interfacial crack length used in Fig. 2(a)is defined as the total instantaneous arc length measured along the non-straight crack path. Differentiation ofthe crack length record furnished the tangential crack tip speeds before and after crack deflection. Since thedifferentiation process is based on a three-point fitting of the crack length history, the exact crack speed at theinterface could not be obtained. Before crack deflection, the crack tip speed was approximately 400 m/s, whichwas a speed very close to the branching speed of Homalite-100. After crack deflection, the speed jumped to asmuch as 800–1000 m/s and then decreased as it propagated further along the interface. However, otherdynamic fracture mechanics parameters such as the dynamic stress intensity factors and the mode mixity ofthe kinked interfacial crack were not further analyzed.

In this investigation, we will fit the isochromatic fringe patterns around the incident dynamic crack toobtain the dynamic stress intensity factors, and the non-singular T-stress. Then, the relation of the stress inten-sity factors for the kinked crack and the incident crack will be developed. As validation, experimental fringepatterns of the kinked interface crack will be directly compared to the theoretical patterns predicted usingdynamic fracture mechanics theory. Moreover, the change of important mode mixity of the kinked crack willbe analyzed. Such results will be very useful to investigate complicated dynamic failure mode transition in bi-materials and composite materials such as the transition of matrix cracking and delamination [13–18].

Fig. 1. Dynamic crack deflection process at a weak interface (interfacial angle 30�, from [9]).

L.R. Xu, P. Wang / Engineering Fracture Mechanics 73 (2006) 1597–1614 1599

cdp30i384-3

30

40

50

60

70

80

90

100 120 140 160 180 200 220Time (microseconds)

Cra

ck le

ngth

(mm

)

Mode I incident crack

Mixed-mode interfacial crack

cdp30i384-3

0200400600800

1000120014001600

100 120 140 160 180 200

Time (microseconds)

Cra

ck s

peed

(m/s

)

mode I incident crack

mixed-mode interfacial crack

Interface

(a)

(b)

Fig. 2. Crack length history (a) and crack speed history (b) before and after crack deflection at a weak interface (specimenscode cdp30i384-Crack Deflection experiments using photoelasticity, interfacial angle 30�, interfacial bonding using Loctitle 384adhesive).


2. Dynamic fracture mechanics analysis

Fig. 3 shows a schematic diagram describing the geometry relevant to the dynamic crack deflection/kinkingproblem. Two identical homogeneous and isotropic elastic solids are bonded along an interface indicated hereby the dashed line. The Young’s and shear moduli, Poisson’s ratio and mass density are denoted by E, l, t andq, respectively. Before reaching the interface, a dynamic steady mode-I crack propagates within a homogenousand elastic solid as shown in Fig. 3(a). The angle between the incident crack plane and the inclined interface isdenoted by b. In this investigation, we mainly analyze the dynamic failure mechanics governing the transitionof an incident mode-I crack to a mixed-mode interfacial crack as shown in Fig. 3(b).

2.1. Stress field around the tip of a dynamically propagating mode-I crack

Stress field of a steady mode-I crack is given by a well-known form (see [19,1]):

rIij ¼

KIðtÞffiffiffiffiffiffiffi2prp RI

ijðh; vÞ þ T di1dj1 þOð1Þ ði; j ¼ 1; 2Þ ð1Þ

where KI(t) is the dynamic stress intensity factor of the mode-I crack as a function of time t; T is a non-singularterm, which is called ‘‘the T-stress’’ or rox [20]; O(1) represents higher order terms; the functions RI

ijðh; vÞ thatrepresent the angular variation of stress components for an instantaneous crack tip speed v are listed inAppendix A.

x1

x2

v1

Interface

Incident crack

x1

x2 y xv2

Deflected crack

(a)

(b)

β

β

Fig. 3. Schematic diagram showing a mode-I crack arriving (a) and subsequently deflecting at a weak interface between two identicalhomogenous solids (b).


2.2. Stress field around the tip of a dynamically propagating mixed-mode crack

Similarly, the asymptotic stress field of a steady mode-II crack can be expressed by

rIIij ¼

KIIðtÞffiffiffiffiffiffiffi2prp RII

ij ðh; vÞ þOð1Þ ði; j ¼ 1; 2Þ ð2Þ

KII(t) is the dynamic stress intensity factor of the mode-II crack as a function of time t. There is no T-stressinvolved in a pure mode-II crack stress expression. The functions RII

ij ðh; vÞ that represent the angular variationof stress components for an instantaneous crack tip speed v are given in Appendix A.

Based on Eqs. (1) and (2), the stress field of a mixed-mode crack can be obtained using linear superpositionprinciple:

rij ¼ rIij þ rII

ij ¼KIðtÞffiffiffiffiffiffiffi

2prp RI

ijðh; vÞ þ T di1dj1 þKIIðtÞffiffiffiffiffiffiffi

2prp RII

ij ðh; vÞ þOð1Þ ði; j ¼ 1; 2Þ ð3Þ

In order to evaluate these stress fields, photoelasticity technique is employed to generate isochromatic fringepatterns, which are directly related to the dynamic stress fields.

2.3. Isochromatic fringe patterns of dynamic cracks

Recall the maximum in-plane shear stress sm is related to the three in-plane stress components by

s2m ¼

r11 � r22

2

� �2

þ r212 ð4Þ

and the governing equation for the isochromatic fringe pattern is [21]:

sm ¼Nf r

2hð5Þ

where N is the fringe order, fr is the material fringe constant and h is the specimen thickness. From Eqs. (4)and (5), we get

Nf r

2h

� �2

� r11 � r22

2

� �2

� r212 ¼ 0 ð6Þ


Substitution of Eq. (3) into Eq. (6) leads to an equation, which describes the shape of the dynamic isochro-matic pattern [22]:

Nf r

ffiffiffiffiffiffiffi2prp

2h

!2

� KIB1B2 þ KIIB4B5 þTffiffiffiffiffiffiffi2prp

2

!2

� ðKIB1B3 þ KIIB6Þ2 ¼ 0 ð7Þ

where B1, B2, B3, B4, B5, B6 are functions defined by

B1 ¼1þ a2

s

DB2 ¼ ð1þ a2

dÞcos 1

2hdffiffiffiffiffi

cd

p � 4asad

1þ a2s

cos 12hsffiffiffiffi

cs

p ð8Þ

B4 ¼ �2as=D B3 ¼ 2ad

sin 12hdffiffiffiffiffi

cd

p �sin 1

2hsffiffiffiffi

cs

p� �

ð9Þ

B5 ¼ ð1þ a2dÞ

sin 12hdffiffiffiffiffi

cd

p � ð1þ a2s Þ

sin 12hsffiffiffiffi

cs

p B6 ¼1

D4asad

cos hd

2ffiffiffiffifficd

p � ð1þ a2s Þ

2 cos hs

2ffiffiffiffics

p !

ð10Þ

The N � K relation given in Eq. (7) is non-linear in terms of the three unknown parameters KI, KII and T. Thereare several approaches to solve the non-linear equation. In this investigation, we mainly use the over-determin-istic method [22] to obtain KI, KII and T. From Eq. (7), we can define a governing function as follows:

f ðKI;KII; T Þ ¼Nf r

ffiffiffiffiffiffiffi2prp

2h

!2

� KIB1B2 þ KIIB4B5 þTffiffiffiffiffiffiffi2prp

2

!2

� ðKIB1B3 þ KIIB6Þ2 ¼ 0 ð11Þ

For a specific fringe order N, we can measure several data points as long as their distances to the crack tip aremore than half specimen thickness as suggested by Rosakis and Ravi-Chandar [23]. Substituting these datapoints into Eq. (11), we can get a series of equations to determine three unknown parameters KI, KII, T usingthe least squares method.

For a crack kinking problem, the T-stress is quite important. As noted by Cotterell and Rice [24], staticcrack kinking or deflection is directly related to the sign of the T-stress. In order to compare the relative valuesof the T-stress for different cases, a so-called ‘‘biaxial ratio’’ was introduced by [25]:

BR ¼ Tffiffiffiffiffiffipap

KI

ð12Þ

Although the biaxial ratio was initially employed for static cracks only, it will be used to characterize the dy-namic crack case in this investigation using the dynamic stress intensity factors, the T-stress values and thecrack lengths. In the following sections, the history of the dynamic stress intensity factors and the T-stressas well as the biaxial ratio of the incident crack will be analyzed.

2.4. Relation of an incident mode-I crack and a kinked mixed-mode crack

As seen in Fig. 1, an incident crack is mode-I dominated, as evidenced by its symmetric fringe patterns, whilethe kinked interfacial crack is indeed a mixed-mode crack with un-symmetric patterns. The interesting fringepattern transition is a result of the stress intensity factor and mode-mixity changes at the interface. Fig. 3 showsa schematic diagram describing the geometry relevant to the dynamic crack deflection/kinking problem.

Under certain circumstances, the dynamic crack stress intensity factor KdI can be related to its static coun-

terpart KsI through a ‘‘universal function of crack tip speed,’’ kI(v) [1]:

KdI ¼ kIðvÞKs

I ð13Þ
Similarly,
KdII ¼ kIIðvÞKs

II ð14Þ
where
kIðvÞ ffi1� v=cRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� v=cd

p ; kIIðvÞ ffi1� v=cRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� v=cs

p ð15Þ


On the other hand, let KskI , Ksk

II denote static mode-I and mode-II stress intensity factors for the deflected(kinked) mixed-mode crack, and they are related to the static stress intensity factors of the incident dynamiccracks as a function of the kinking angle b (see [26,27]):

KskI ¼ c11Ks

I þ c12KsII ð16Þ

KskII ¼ c21Ks

I þ c22KsII ð17Þ

where the coefficients are:

c11 ¼3

4cos

b2þ 1

4cos

3b2

� �c12 ¼ �

3

4sin

b2þ sin

3b2

� �ð18Þ

c21 ¼1

4sin

b2þ 1

4sin

3b2

� �c22 ¼

1

4cos

b2þ 3

4cos

3b2

� �ð19Þ

Let v2 denote the speed of the deflected crack tip at the instant right after deflection, and let KdkI , Kdk

II be dy-namic mode-I and mode-II stress intensity factors for the deflected (kinked) mixed-mode crack. We still as-sume that the universal relation between the dynamic and static stress intensity factors also holds for thedeflected crack tip, i.e.,

KdkI ¼ kIðv2ÞKsk

I ð20ÞKdk

II ¼ kIIðv2ÞKskII ð21Þ

where v2 is the crack tip speed of the kinked crack. Based on the above relations, if we know the dynamic stressintensity factors, the crack tip speed of the incident crack, the kinking angle as well as the crack tip speed ofthe kinked crack, we can get the dynamic stress intensity factors of the kinked crack and hence the crack tipstress fields around the deflected crack using Eq. (3). Also, based on Eqs. (4) and (5), we can predict the fringepatterns of the interfacial crack. It should be noticed that the above relations are only applicable to a specialsituation: an interfacial crack initiated right after the incident crack reached the interface. In some situations,the interfacial crack initiated before the incident crack reached the interface as recorded by Xu and Rosakis[28]. Then, different fracture theory should be employed.

Besides the crack tip stress fields of the mixed-mode interfacial crack, mode mixity is one of the most impor-tant parameters in interface fracture mechanics analysis [26], which is defined by the non-dimensional ratio ofthe dynamic mode-II stress intensity factor over its mode-I counterpart:

tgU ¼ KdkII

KdkI

¼ kIIðv2ÞKskII

kIðv2ÞKskI

ð22Þ

Substituting Eqs. (15)–(17) into Eq. (22), we obtain:

tgU ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� v2=cd

psin b

2þ sin 3b

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� v2=cs

p3 cos b

2þ cos 3b

2

ð23Þ

It is not surprising to see that the dynamic mode mixity is a function of the kinking angle and the crack tipspeed, which is different from the static crack kinking or deflection.

Here we should pay great attention to the T-stress change after crack kinking. The T-stress is the non-sin-gular, constant term in William’s series solutions. It acts along the crack surface and is determined only by thefar-field load [29–32]. According to the stress tensor decomposition principle, there will be three non-singularconstant stress components acting on the path of kinked crack. Indeed, there were no convincing results on theT-stress change at crack kinking in previous investigations [33,34]. However, the effect of the T-stress on crackkinking is quite significant [35–39]. Very recently, Li and Xu [40] extensively discussed the T-stresses acrossstatic crack kinking. Analytical results on the T-stress change across dynamic crack kinking, are still notavailable.


3. Results and discussion

3.1. Stress intensity factors and the T-stress for the incident crack

In this section, we mainly analyze the dynamic fracture parameters of the crack deflection experimentsreported by Xu et al. [9]. Fig. 1 showed a series of dynamic photoelasticity images of the incident crack prop-agating towards the inclined interface (interfacial angle 30�). The incident crack speed was approximately400 m/s. Using the over-deterministic method to fit these fringes led to the history of the stress intensity factorand the T-stress of the incident crack as shown in Figs. 4 and 5. It is not surprising to see that KI is approx-imately 0.5 MPa m1/2 and KII is close to zero since the incident crack was indeed a mode-I crack. The aboveresults verified the wedge-loading mechanism for controlling an opening crack [9]. However, the change of theT-stress with time and crack length was quite large (see Fig. 5) and the T-stress ranged from 0.5 to 1.5 MPa.For the same material and interface bond, another experiment was conducted using a higher projectile impactspeed. Figs. 6 and 7 show the history of the stress intensity factors and the T-stress for this case. ComparingFig. 4 with Fig. 6, it was obvious that a high impact speed led to a high mode-I stress intensity factor (from0.5 MPa m1/2 to 0.8 MPa m1/2) and the T-stress (related to the stress along the crack path). Fig. 8 comparesthe history of two biaxial ratios for these two cases. The biaxial ratio is related to the normalized T-stress andit is expected to be a geometry-independent parameter. Obviously, for different load cases but the same mate-rial and interface, it is hard to find common features in Fig. 8.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

44 46 48 50 52 54 56 58 60Time (microseconds)

Stre

ss In

tens

ity F

acto

rs (M

Pa*m

1/2)

KK

I

II

cdp30i384

Fig. 4. Dynamic stress intensity factors (SIFs) as a function of time of the incident crack (interfacial angle 30�).

0

0.5

1

1.5

2

120 130 140 150 160Time (microseconds)

T (M

Pa)

T

cdp30i384

Fig. 5. Non-singular T-stress as a function of time of the incident crack (interfacial angle 30�).

-0.2

0

0.2

0.4

0.6

0.8

1

110 120 130 140 150 160Time (microseconds)

Stre

ss In

tens

ity F

acto

rs (M

Pa*m

1/2)

KK

I

II

k30hm384

Fig. 6. Dynamic stress intensity factors (SIFs) as a function of time of the incident crack from a higher speed impact (specimens codek30hm384-crack Kink experiment using Homalite, interfacial angle 30�, interfacial bonding using Loctitle 384 adhesive).

k30hm384

0

0.5

1

1.5

2

2.5

3


T (M

Pa)

Fig. 7. T-stress as a function of time of the incident crack from a higher speed impact (interfacial angle 30�).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


BR

cdp30i384k30hm384

Fig. 8. Biaxiality ratio as a function of time of the incident crack for different speed impact (interfacial angle 30�).


One interesting issue is the influence of the T-stress on the crack path stability. According to Cotterell andRice [24], the crack growth is stable if T < 0 and unstable if T > 0. For the present dynamic crack propagationcase, our results indicate that this statement might be modified. In our dynamic experiments as shown in


Fig. 8, the T-stress was always greater than zero for the incident mode-I cracks, but their crack paths (as indi-cated by curves macroscopically) were pretty stable as shown in Fig. 1. Recently, some researchers also showthe sign of the T-stress was not enough to judge the crack stability and other parameters should be introduced.Melin [41] showed that when T > 0, the crack path was still stable. Richardson and Goree [42] also observedthat in PMMA specimens of different dimensions, the crack did not kink immediately if the T-stress becamepositive.

In order to verify the fitting process, we plotted one recovered fringe pattern using the fitted stress intensityfactors and compared with the experimental pattern. As shown in Fig. 9, it can be seen that two fringes agreedwell. Similarly, Figs. 10 and 11 showed history of the stress intensity factors and the T-stress for the case ofinterfacial angle 45�. For the case of interface angle 60�, we analyzed two different interfacial bondingstrengths, and the history of the stress intensity factors and the T-stresses were shown in Fig. 12. Based onthe history of the stress intensity factors and the T-stresses in different load and interface cases, we find thatthe same impact speed leads to the same level of dynamic stress intensity factors for the mode-I incident crack.Furthermore, for different interfacial angles, the stress intensity factors of incident cracks are almost the samesince the interfacial angle only affects the kinked crack. All these results indicated that the wedge loadingmechanism was a good way to produce a stable mode-I crack and control the crack tip speed.

recovered fringecrack plane

Fig. 9. The comparison of the experimental fringe and recovered fringe.

-0.1

0

0.1

0.2

0.3

0.4

0.5


Stre

ss In

tens

ity F

acto

r (M

Pa*m

1/2 )

KK

I

II

cdp45i384

Fig. 10. Dynamic stress intensity factors (SIFs) as a function of time of the incident crack (specimens code cdp45i384-Crack Deflectionexperiments using photoelasticity, interfacial angle 45�, interfacial bonding using Loctitle 384 adhesive).

0

0.5

1

1.5

2


T (M

Pa)

T

cdp45i384

Fig. 11. Non-singular T-stress as a function of time of the incident crack (interfacial angle 45�).

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

110 120 130 140 150Time (microseconds)

Stre

ss In

tens

ity F

acto

rs (M

Pa*m

1/2 )

K (strong interface)K (weak interface)K (strong interface)K (weak interface)

I

I

II

II

cdp60i384&cdp60wd10

0

0.5

1

1.5

2

2.5

110 115 120 125 130 135 140 145 150Time (microseconds)

T (M

Pa)

T (strong interface) T (weak interface)

cdp60i384&cdp60wd10

(b)

(a)

Fig. 12. History of (a) the stress intensity factors (SIFs) and (b) the T-stress of two incident cracks (specimens code cdp60i384-CrackDeflection experiments using photoelasticity, interfacial angle 60�, interfacial bonding using Loctitle 384 adhesive; specimens codecdp60iwd10-Crack Deflection experiments using photoelasticity, interfacial angle 60�, interfacial bonding using Weldon 10 adhesive).


3.2. Predicted fringe patterns of the kinked interfacial crack

After fitting the stress intensity factors KdI , Kd

II and the T-stress of the incident crack, we can predict thekinked interfacial crack using fracture mechanics theory. Our first step is to obtain the static counterpartsof the dynamic stress intensity factors using Eq. (13). After crack deflection, the dynamic stress intensity fac-tors of the kinked crack can be calculated using Eqs. (16) and (17) and Eqs. (20) and (21). Then, the fringepatterns of the interfacial crack at the moment of crack deflection can be predicted using Eqs. (3) and (5).


Because it is very hard to record the exact moment of crack kinking at the interface in dynamic fracture exper-iments, average values of the stress intensity factors of the incident crack were used to calculate the stressintensity factors of the kinked interfacial crack.

Fig. 13 showed the predicted fringe pattern of a kinked interfacial crack (interfacial angle 30�). The coor-dinate origin is located at the intersection point of the incident crack and the kinked crack, and its x-axis is

°

Fig. 13. Comparison of (a) experimental fringe (strong interface) and (b) predicted fringe pattern (v1 = 400 m/s, v2 = 766 m/s, N = 1) forinterfacial angle 30�.


along the interface. For this case, the crack tip speed of the incident crack was around 400 m/s. Right aftercrack kinking, the interfacial crack tip speed was about 800 m/s [9]. Since the T-stress of the incident crackis around 1 MPa, and there are no convincing results for the T-stress of the kinked crack, the T-stress ofthe kinked crack was assumed to be zero in all our predictions. In order to highlight our comparison ofthe predicted and experimental fringe patterns, only fringe order 1 was plotted. Fig. 13(a) presents the exper-imental fringe showing the transition from an incident crack to an interface crack (the horizontal line was the

-2 -1 1

-4

-3

-2

-1

1

2

3

x (mm)

y (mm)

Incident crack

Interface

°45

(a)

(b)



interface). Fig. 13(b) showed the predicted fringe and the two kinds of photoelasticity fringes were very sim-ilar. Figs. 14 and 15 showed the experimental and predicted fringe patterns for interfacial angles of 45� and60�, respectively. All these cases indicated that the predicted fringes and the experimental fringes generallyagreed well and they demonstrated that our dynamic fracture mechanics modeling and assumptions were rea-sonable. However, some discrepancy is also noticed because (a) it is very hard to take one photo at the righttime and right position as the theoretically predicted one. (b) T-stress has significant influence on these fringepatterns. But no one reported the T-stress after dynamic crack kinking.

One interesting observation is the large concave wedge effect. As seen in Fig. 15(a), two caustic spots (onecaused by the kinked crack tip and the other due to a large concave wedge) were clearly observed when amode-I incident crack kinked along a weak interface with a large kinking angle (60�). In most previous crack

-2 -1 1

-3

-2

-1

1

2

x (mm)

y (mm)

Incident crack

Interface

°60

(a)

(b)



kinking analyses, researchers only considered the singular stress field due to a kinked daughter crack andignored the singular stress field of a concave wedge. Interestingly, William’s classical solution of wedge stresssingularities [43] is the foundation of the full-field stress field of a traction-free crack in linear elastic fracturemechanics [44]. Indeed, Cotterell and Rice’s [24] classical work mainly deals with a slightly kinked crack, not alarge kinking angle case. To authors’ knowledge, only Azhdari and Nemat-Nasser [45] provided a simpleexplanation to this phenomenon for a static crack kinking case.

3.3. Mode mixity of the kinked interfacial crack

Mode mixity is an important parameter in interface fracture analysis. It is the ratio measure of the mode-IIstress intensity factor/energy release rate over its mode-I counterparts. There were numerous studies on modemixity in static fracture cases but very few results were reported in dynamic fracture investigation [46,47,5].Indeed, mode mixity is a key parameter in controlling failure mode transitions along interfaces. In this inves-tigation, when the incident mode-I crack reached the interface, it kinked along the interface and became onemixed mode crack. Based on Eq. (23), for the kinked interfacial crack, its mode mixity depends on the kinkedcrack tip speed and the interfacial or kinking angle. Obviously, the dependence on the dynamic mode mixityon the crack tip speed is a special phenomenon in dynamic fracture mechanics.

The variations of the mode mixity with the interfacial angle and the kinked crack tip speed are plotted inFig. 16. It is not surprising to see that when the crack tip speed of the interfacial crack remained constant,mode mixity increased with the increase of the kinking angle. In other words, the larger of the interfacial angle,the larger is the mode-II component for the mixed mode crack. This result is similar to the common conclusionin static crack kinking analysis [27]. As a special feature of dynamic crack kinking, the mode mixity increaseswith the increasing kinked crack speed if the interface angle is fixed. In Fig. 16(a), for a fixed kinking orinterfacial angle 50�, the mode mixity for a high crack tip speed (90% of the shear wave speed of the matrix

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70 80 90 100Interfacial angle (degree)

Mod

e-M

ixity

V/Cs=0V/Cs=0.3V/Cs=0.6V/Cs=0.9

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9V/Cs

Mod

e-M

ixity

Interfacial angle=30Interfacial angle=45Interfacial angle=60°

°

(a)

(b)

°

Fig. 16. Dynamic facture mode mixity as functions of (a) interfacial angle and (b) crack tip speed.


material) is almost doubled compared to the mode mixity for a static kinked crack. Here, we should noticethat the crack tip speed of the kinked crack is related to the interfacial bonding strength [9]. A weak interfacewill lead to a fast interfacial crack tip and a high mode-II component as a result. Fig. 16(b) shows the modemixity dependence on the crack tip speed for different kinking angles. It is interesting to see that each curve hasa similar shape and is shifted by some amount for a different kinking angle. In this investigation, the kinkingor interfacial angle is limited to 0–90�. Recently, Rousseau and Rosakis [48] examined important crack kink-ing behavior for very large interfacial angle (greater than 90�). One important difference is that their incidentcrack was an intersonic shear crack along a weak path rather than a slow mode-I crack as in our investigation.In order to suppress possible crack branching, our incident crack speed was controlled to be less than the crackbranching speed (around 30–40% of the shear wave speed for Homailte-100). Chalivendra and Rosakis [49]used the same wedge-induced crack but along a weak path such that the incident mode-I crack was closeto the Rayleigh wave speed (around 90% of the shear wave speed, see [50]). In all these investigations, themode mixity of the kinked interfacial crack was found to depend on not only the kinking angle but alsoon the crack tip speed.

4. Conclusion

Using fitted dynamic stress intensity factors and the non-singular T-stresses of the incident cracks, we makeuse of dynamic fracture mechanics concepts obtain the stress intensity factors of the kinked cracks as functionsof kinking angles and crack tip speeds. The T-stress of the incident crack has a small positive value but thecrack path is still quite stable. In order to validate fracture mechanics predictions, the theoretical photoelas-ticity fringe patterns of the kinked cracks were compared with the recorded experimental fringes. Moreover,the mode-mixity of the kinked interfacial crack was found to depend on the kinking angle and the crack tipspeed. A weak interface (interfacial strength or fracture toughness much less than that of the bulk material)will lead to a high mode-II component and a fast crack tip speed of the kinked mixed-mode crack.

Acknowledgements

The authors gratefully acknowledge the support from the Office of Naval Research Young InvestigatorProgram (N00014-03-1-0505, Dr. Roshdy G.S. Barsoum, Program Officer), and the National Science Foun-dation (CMS-0456807, CMS-0409665).

Appendix A

Functions RIijðh; vÞ that represent the angular variation of stress components for an instantaneous crack tip

speed v are given by [1]:

RI11 ¼

1

Dð1þ a2

s Þð1þ 2a2d � a2

s Þcos 1

2hdffiffiffiffiffi

cd

p � 4asad


cs

p� �

ðA:1Þ

RI12 ¼

2adð1þ a2s Þ

Dsin 1

2hdffiffiffiffiffi

cd

p �sin 1

2hsffiffiffiffi

cs

p� �

ðA:2Þ

RI22 ¼ �

1

Dð1þ a2

s Þ2 cos 1

2hdffiffiffiffiffi

cd

p � 4asad


cs

p� �

ðA:3Þ

where

cd ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðv sin h=cdÞ2

q; tan hd ¼ ad tan h ðA:4Þ

cs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðv sin h=csÞ2

q; tan hs ¼ as tan h ðA:5Þ

D ¼ 4asad � ð1þ a2s Þ

2 ðA:6Þ


as ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðv=csÞ2

q; ad ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðv=cdÞ2

qðA:7Þ

cs ¼ffiffiffilq

r; cd ¼

ffiffiffiffiffiffiffiffiffiffiffijþ 1

j� 1

rcs ðA:8Þ

j ¼3� 4t ðplane strainÞ3�t1þt ðplane stressÞ

(ðA:9Þ

where cs and cd are the shear wave and dilatational wave speeds of the material. Similarly,

RII11 ¼ �

2as

Dð1þ 2a2

d � a2s Þ

sin hd

2ffiffiffiffifficd

p � ð1þ a2s Þ

sin hs

2ffiffiffiffics

p( )

ðA:10Þ

RII12 ¼

1

D4asad

cos hd

2ffiffiffiffifficd

p � ð1þ a2s Þ

2 cos hs

2ffiffiffiffics

p( )

ðA:11Þ

RII22 ¼

2adð1þ a2s Þ

Dsin hd

2ffiffiffiffifficd

p �sin hs

2ffiffiffiffics

p( )

ðA:12Þ

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