ARTICLE IN PRESS

Finite Elements in Analysis and Design 46 (2010) 601–610

Contents lists available at ScienceDirect

Finite Elements in Analysis and Design

0168-87

doi:10.1

� Corr

E-m

(S. Bhat

journal homepage: www.elsevier.com/locate/finel

Dugdale cohesive zone modeling to evaluate J integral at the interface ofstrength mismatched steels: A simplified numerical approach

Sunil Bhat �, Vijay G. Ukadgaonker

Mechanical Engineering Department, Indian Institute of Technology, Bombay, Mumbai 400076, India

a r t i c l e i n f o

Article history:

Received 15 September 2008

Received in revised form

22 June 2009

Accepted 15 February 2010Available online 12 March 2010

Keywords:

Alloy steel

Bimetallic

Cohesive zone

Dugdale’s model

Interface

Load transfer

Maraging steel

Strength mismatched interface

4X/$ - see front matter & 2010 Elsevier B.V. A

016/j.finel.2010.02.007

esponding author. Tel.: +91 9743299255.

ail addresses: sbbhat@rediffmail.com, sbhat_7

), vgu@me.iitb.ac.in (V.G. Ukadgaonker).

a b s t r a c t

When the crack tip in ductile parent body reaches near the interface of another elastically matched but

strength mismatched ductile body, the plasticity or yield zone at the tip spreads to interface body which

triggers the effect of strength mismatch between the bodies over the tip by way of plasticity induced

load transfer. The interface effect is best quantified by the value of J integral, Jinterface, obtained over a

path around the interface of the bodies in the yield zone. The paper presents finite element analysis of

minute yield zone, formed at Mode I crack tip in linear elastic regime, across a strength mismatched

interface to determine Jinterface. The yield zone, treated as Dugdale’s cohesive zone, is isolated from the

bimetallic body domain and modeled alone under the action of cohesive stresses. Effect of far field load

at the interface is obtained separately. Such an uncomplicated approach is first tested successfully

over theoretical results of cohesive zone across a thin interface of weak ASTM 4340 alloy and strong

MDN 250 maraging steels under monotonic load. Subsequently, it is verified vis-�a-vis experimental

results of cyclic cohesive zone developed across the weld interface in bimetallic compact tension

specimen by subjecting the crack in alloy steel near the interface of ultra strong weld between the

stated steels to high cycle fatigue.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Bimaterials can be broadly classified into two types dependingupon the properties of their constituents (i) Type I in which theconstituents have significant elastic mismatch between them andboth or at least one of them have minimal ductility or plasticityviz., fibre reinforced thermosetting plastics (FRPs), metal–resincombination in homogeneous metal matrix composites (MMCs)etc. (ii) Type II in which the constituents possessing goodplasticity have strength mismatch but negligible or nil elasticmismatch viz., metal–metal combination in homogeneous MMCs,non-homogeneous bimetallic composites comprising metals ex-ternally welded or bonded with each other like composites ofsteel and titanium, steel and copper, aluminum and copper,different types of steels, etc. Solid state welding processes likediffusion bonding, explosive cladding and friction welding, whichare reported to join dissimilar metals with reasonably clean andstrong weld, have vastly enhanced the utility aspects of non-

ll rights reserved.

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homogeneous bimetallic composites in load bearing applications.In fact, study of fracture aspects of all the types of bimaterialsassumes importance especially when the crack is in the vicinity orat the interface of the constituents because failure of the interfacerenders the bimaterial ineffective by diminishing its load sharingcapability. Hitherto, finite element analysis has been employedas a reliable tool to investigate fracture characteristics ofbimaterials.

High elastic mismatch between the constituents influencesfracture characteristics of a Type I bimaterial to a large extent. AMode I crack terminated at the interface in such a bimaterial hasattracted number of numerical investigations because of devia-tion in strength of crack tip stress singularity, l, [1] from the valueof 0.5 as found in the case of a crack embedded in single parenthomogeneous body where the elastic stress field around the cracktip is of the form, sij ¼ fð2proÞ

�l Kappliedf ðyÞþhigher order nonsingular termsg. f(y) term for crack at bimaterial interface changesto f ðy;a;bÞ where Dundurs parameters a and b represent elasticproperties namely shear modulus and Poisson’s ratio of parentand interface bodies. A sharp jump or discontinuity in stresses inthe direction of the load along the interface occurs to satisfy thecondition of continuity of displacements and strain across theinterface. This changes the crack tip stress intensity parameter,Ktip, as well. Lin and Mar [2] reported a hybrid element in 1976which accounted for change in l of crack at the interface. Ahmad

ARTICLE IN PRESS

Nomenclature

a distance of crack tip from interfaceA cracked parent body, alloy steelb length of cohesive zone across interfaceB interface body, maraging steelc crack lengthC Paris constantC1 a constantdc/dN crack growth rateds displacement along pathE modulus of elasticityf(y) function of angle from crack axisF applied loadJinterface J integral at interfaceJapplied applied J integralJtip J integral at crack tipJ1, J2 components of Jinterface

Kapplied applied stress intensity parameterKtip stress intensity parameter at crack tipKC fracture toughness of homogenous bodyKC(bimetallic) fracture toughness of bimetallic bodyl spread of cohesive zone into interface body/weldL distance between load axis and right end of specimenm Paris constantnx unit vector in x directionny unit vector in y directionN number of fatigue cyclesP cyclic path near boundary of cohesive zoner size of cohesive zone in homogeneous bodyr3 radial coordinate of element near crack tip

Tx traction along path in x directionTy traction along path in y direction~u displacement in x directionv crack opening displacement from axisvc contribution of cohesive stress in load line displace-

ment~v displacement in y directionW weldWe strain energy densityY yield strengthD parameter under cyclic or fatigue loadl strength of stress singularitym shear moduluss cohesive stressseff effective cohesive stresssij stress field around crack tipsx stress in x directionsy stress in y directiontxy shear stress in xy plane

Superscripts

A cracked parent body, alloy steelB interface body, maraging steelW weld* critical value, i.e., at fracturemax maximum valuemin minimum valuen nodal value

S. Bhat, V.G. Ukadgaonker / Finite Elements in Analysis and Design 46 (2010) 601–610602

[3] analysed the crack normal to the interface in unidirectionalfibre composites by finite element analysis and used the results toformulate the criteria for assessment of cracking normal to thefibre, interface splitting and fibre pull out. Meguid et al. [4]developed a novel singular finite element in 1995 to obtain l.Their results confirmed a gradual drop in Ktip from applied value,Kapplied, when the crack in parent body, A, grew nearer towards theinterface of body, B under the condition, mB4mA. Ktip thenincreased sharply when the crack touched the interface. The ratiobetween Ktip and Kapplied was found to be as high as 6 and 12 forcentre and edge cracks terminated at the interface respectivelywhen mB=103mA. Reverse trends were observed when mBomA.Findings of similar nature have been reported by all those whohave worked on such bimaterials.

Stresses however don’t exhibit an appreciable jump ordiscontinuity when the crack tip touches the interface in Type IIbimaterials due to nearly identical elastic properties across theinterface. But due to sufficient plasticity of parent and interfacebodies, the effect of strength mismatch between them is felt bythe crack tip in parent body even before touching the interfacebody. Crack tip plasticity or yield zone spreads to the interfacebody as the crack tip closes towards the interface. As the result,the part of yield zone in interface body carries stresses differentfrom those acting over the portion of yield zone in parent body tofulfill continuity of displacements and strain. The plasticityinduced load transfer towards interface or parent body, depend-ing upon the direction of strength gradient across the interface,changes the stress field around the crack which induces shieldingor amplification effect at the crack tip. This phenomenon has beenconfirmed by various researchers with the help of finite element

analysis. Motivated by experimental findings of Suresh et al. [5],Sugimura et al. [6] in 1995 carried out finite element analysis ofthe crack approaching a bimetallic interface of weak ferritic andstrong austenitic steels. J integral over the path far away from thecrack tip around the interface provided the applied value of J

integral, Japplied, whereas the integral over the path near the cracktip in parent body without crossing the interface resulted in J

integral at the crack tip, Jtip. They found that Jtip is less or morethan Japplied depending upon the crack approaching a stronger orweaker material respectively. Kikuchi [7] obtained similar resultsby conducting finite element analysis of crack in front of andbehind the perpendicular weld line of strength mismatchedsteels. He determined crack tip stress field, sij, J integrals and Q

factor defined by fsij�sijðHRRÞ=YAg, HRR denoting Hutchinson,Rice and Rosengren solution for elastic plastic stress field aroundcrack tip in homogeneous parent body, A. Kim et al. [8] employedfinite element analysis to investigate the crack approaching ahomogeneous or compositionally graded interlayer/weld betweenparent and interface bodies of varying strengths. They found thatthe interlayer with homogeneous yield behaviour (i.e., when theyield strength of interlayer is the average of constituent metals)provides a greater shielding effect than a graded interlayer withinwhich the yield strength varies linearly from one end to another.Kim et al. [9] also examined numerically the global and localfracture under the influence of interfacial effects. Reimelmoseret al. [10] discussed cyclic plasticity at crack tip near the interfacewith the help of cohesive zone model. They presented finiteelement analysis of fatigue crack growing from ARMCO irontowards the interface of SAE 4340 steel. Gubeljak et al. [11]worked on the effect of material inhomogeneity in strength

ARTICLE IN PRESS

S. Bhat, V.G. Ukadgaonker / Finite Elements in Analysis and Design 46 (2010) 601–610 603

mismatched weld over approaching crack in base body with thehelp of finite element analysis. Predan et al. [12], in 2007,reported finite element analysis of crack in parent base metalgrowing towards over and under matched welds.

In the latest work reported over Type II bimaterials which isbased upon the effects of crack tip plasticity, Wangand Siegmund [13] used cohesive zone model to investigatefatigue growth of normal and interfacial crack at plasticallymismatched interfaces. They simulated irreversible materialseparation through load history dependent degradation ofcohesive strength. Ukadgaonker et al. [14] examinedcharacteristics of fatigue crack tip propagating normally in weakASTM 4340 alloy steel towards ultra strong weld between alloyand strong MDN 250 maraging steels. They modeled yield zoneacross the interface to estimate load transfer effect at the crack tipand validated the solution with experiments. Above two refer-ences confirm the importance of crack tip cohesive zone infracture aspects of Type II bimaterials. In any basic model, thecohesive zone is considered as an extended line region straightahead of the crack tip under the action of closing pressure orcohesive stress, s(v). s(v) changes with crack opening displace-ment, v, upon application of load. Refined constitutive models forde-cohesion have been presented by Ostlund and Nilsson [15] andXu and Needleman [16]. Former proposed a linear relation,s(v)=s(1�(v/v*)), where s is intrinsic strength property ofmaterial which depends upon elongation undergone by thematerial. Latter presented an exponential law. Dugdale [17] foran elastic–ideally plastic material described cohesive zone in itssimplest case as the one subjected to constant cohesive stressindependent of v, i.e., s(v)=s. He stated s to be equal to materialyield strength, Y, in plane stress condition. The value of s isapproximated as

ffiffiffi3p

Y in plane strain condition. Dugdale’s modelcaters for both small and large scale yielding and has beenextensively used in studies world wide with encouraging results.Since the model essentially relies upon superimposition ofsolutions of crack under remote tension and opposing cohesivestresses [18] for fulfillment of zero stress singularity at the tip ofthe cohesive zone, the finite element analysis of Dugdale’scohesive zone, at Mode I crack tip in linear elastic regime, acrossa strength mismatched interface is undertaken under the action ofstated loads in the present work. Adopting a new approach tosimplify the analysis, the cohesive zone is isolated from thebimetallic body domain and modeled alone under the effect ofcohesive stresses over cohesive volumes in parent and interface

Parent body, A Interface body, B

Cohesive zone

Z

Tip

O

Crack (Stage II)

Thin interface

Crack ti

Applied load

Axis

Fig. 1. Cohesive zone acros

bodies to determine integral J1 over a path around the interfacewith the help of output stress and displacement fields. Integral J2

representing the effect of far field load at the interface is foundseparately from a theoretical model. Evaluation of J2 in such amanner is justified by the fact that finite element analysis of crackunder far field load is a common exercise which previously hasbeen undertaken by many researchers with their results matchingclosely with theoretical estimations. J1 and J2 are finally super-imposed to obtain J integral at the interface, Jinterface. The methodis relatively fast, simple and reasonably accurate. A bimetallicbody constituting parent and interface bodies of weak ASTM 4340alloy and strong MDN 250 maraging steels respectively isemployed in analysis. The steels are ductile with nearly identicalelastic but different plastic and strength properties. Ultra strongweld obtained by joining the steels is also elastically identicalwith the steels. Since yield strengths of steels and weld are closeto their respective ultimate tensile strength values, they areassumed to be elastic–ideally plastic to make the problemamenable for treatment by Dugdale’s hypothesis. Initially, finiteelement model is validated by using theoretical results ofcohesive zone across a thin interface of stated steels undermonotonic load. Subsequently, the model is tested vis-a-visexperimental results of cyclic cohesive zone developed acrossthe weld interface in bimetallic compact tension specimen bysubjecting the crack in alloy steel near the interface of ultra strongweld between the steels to high cycle fatigue.

2. Theoretical review

In a Type II bimaterial of parent and interface bodies A and B

respectively, the effect of interface body is not felt by Mode I cracktip in parent body (stage I) as long as the distance of crack tipfrom the interface, a, fulfills the condition, aZr, where Dugdale’scohesive zone length, r, under monotonic load in linear elasticregime is equal to ðp=8ÞðKapplied=sAÞ

2 [18, p. 20]. As the crack growsand reaches near the interface body such that aor (stage II), thecohesive zone spreads to the interface body which starts loadtransfer effect at the crack tip causing Ktip to differ from Kapplied.Refer Fig. 1. The cohesive zone is shown to have crossed intointerface body by distance, l, with its total length, b, equal to (a+ l).Ukadgaonker et al. [14, p. 693] presented following solutions instage II for load line displacement, v(x), in cohesive zone andKtip under action of Kapplied and cohesive stresses sA and sB over

Magnified view at Z

p Cohesive zone

a l

b

v(a)

Interface

x(x = 0, y = 0)

Axis

y

O

v(0)

O

�A

�B

Path P

s bimetallic interface.

ARTICLE IN PRESS

S. Bhat, V.G. Ukadgaonker / Finite Elements in Analysis and Design 46 (2010) 601–610604

cohesive volumes in parent and interface bodies respectively withplane stress condition existing at the crack tip:

vðxÞ ¼2

E

Kapplied

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

pðaþ l�xÞ

rþsA

px ln

ffiffiffiffiffiffiffiffiffiaþ lp

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaþ l�xp

ffiffiffiffiffiffiffiffiffiaþ lp

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaþ l�xp

!�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaþ lÞðaþ l�xÞ

p" #

þsB�sA

pa ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaþ l�xÞ

pþ

ffiffilp

jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaþ l�xÞ

p�

ffiffilpj

!þx ln

jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaþ l�xÞ

p�

ffiffilpjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðaþ l�xÞp

þffiffilp

!�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilðaþ l�xÞ

p" #8>>>>><>>>>>:

9>>>>>=>>>>>;

ð1Þ

K2tip

4sA¼ Kapplied

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

p ðaþ lÞ

r�sA

p ½2ðaþ lÞ�þsB�sA

p a ln

ffiffiffiffiffiffiffiffiffiffiffiffiðaþ lÞ

pþ

ffiffilp

jffiffiffiffiffiffiffiffiffiffiffiffiðaþ lÞ

p�

ffiffilpj

!�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffilðaþ lÞ

p" #( )ð2Þ

where Kapplied in stage II [10, p. 404] is obtained by

Kapplied ¼ 2sA

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðlþaÞ

p

rþ2ðsB�sAÞ

ffiffiffiffiffi2l

p

rð3Þ

KtipoKapplied if sB4sA and vice versa. Difference between Ktip andKapplied depends upon, a, and mismatch between sA and sB. Eqs.(2) and (3) are solved by numerical iterative convergence scheme.Input values to the computer code are sA, sB, Kapplied and a, theoutput values being l and Ktip. With known value of l, thedisplacement at the interface, v(a), is obtained fromEq. (1). Jinterface is equal to 2(sB

�sA)v(a). At fracture, Eqs. (1)–(3)are rewritten by replacing v(x), l, Ktip and Kapplied by v�ðxÞ, l�, KA

C

and KC(bimetallic) respectively. Jinterface in such a condition changes to2ðsB�sAÞv�ðaÞ.

In fatigue regime, equations of monotonic load are modified bychanging the parameters with their cyclic values. The cohesivestresses sA and sB are replaced by 2sA and 2sB respectively. For astandard bimetallic compact tension specimen shown in Fig. 2,subjected to tension–tension fatigue cycle with maximum load,Fmax, and minimum load, Fmin, DKapplied [19] is written as

DKapplied ¼Fmax�Fmin

tL0:529:6

c�12:5

L

� �0:5

�185:5c�12:5

L

� �1:5"

þ655:7c�12:5

L

� �2:5

�1017:0c�12:5

L

� �3:5

þ639:8c�12:5

L

� �4:5#

ð4Þ

DKapplied is determined at various positions of fatiguecrack advancing towards the weld interface. Value of

Parentbody

Interfacebody

Weld

30

L = 50

16

F

12.5

c

Hole dia. = 12.5

6.25

F

NotchCrack (Notch tip)

Weldinterface

Load axis

Fig. 2. A standard bimetallic c

Dv(a) is found at each crack position. DJinterface equals4(sB�sA)Dv(a).

3. Finite element model—approach and validation

Cohesive zone under monotonic load in stage II is chosen forfinite element analysis because interface body influences crack tipin this stage. The zone is isolated from the bimetallic body domainand modeled alone under dual action of cohesive stresses overcohesive volumes in parent and interface bodies. The approach isvalidated vis-�a-vis theoretical model by considering the case of acrack terminated at the interface of body, B, i.e., final phase ofstage II. Eq. (2) enables to write the following at crack tip for sucha condition by using a=x=0

vð0Þ ¼2

EKapplied

ffiffiffiffiffiffi2

pl

r�

2sBl

p

( )ð5Þ

Contribution of cohesive stresses, vc(0), in v(0) is �4sBl/pE.Integral J1 due to cohesive stresses, given by 2sBvcð0Þ, is obtainedas

J1 ¼�8ðsBÞ

2l

pEð6Þ

By definition, J integral over cyclic path, P, around the interface incohesive zone is numerically found by the relation [20]

J1 ¼

ZP

We dy�Tx@ ~un

@xds�Ty

@ ~vn

@xds

� �ð7Þ

where Tx ¼ snx nxþtn

xyny and Ty ¼ snynyþtn

xynx. Since load linedisplacements in cohesive zone are very small, y is negligibleand strain energy term, We dy, is nearly zero. Traction in x

direction, Tx, and term @ ~un=@x are also less due to cohesivepressure loading, sB, in y direction. Therefore J1 effectively

Dimensions are in mm

60

F max

F min

Load, F

Time

Load cycle

ompact tension specimen.

ARTICLE IN PRESS

Table 1Results of case study

sA=YA=700 MPa, sB=YB=1800 MPa

EA=EB=E=2�105 MPa, uA=uB=0.3

Parameter Case I Case II

a (mm) 7.21 6.31

l* (mm) 1.22 1.53

KC(bimetallic) (MPaffiffiffiffiffimp

) 163.9 167.8

v*(a) (mm) 0.0099 0.01286

v*(0)=0.08 mm, KAC ¼ 150 MPa

ffiffiffiffiffimp

.

S. Bhat, V.G. Ukadgaonker / Finite Elements in Analysis and Design 46 (2010) 601–610 605

reduces to

J1 ¼

ZP�Ty

@ ~vn

@xds

� �ð8Þ

Path length is 2l. For sny ¼ sB, tn

xy ¼ sB=C1 where C141,Ty ¼ sB½1þ½1=C1��. On assuming constant @ ~vn=@x along the path,J1 takes a much simpler form

J1 ¼�2sB 1þ1

C1

� �l@ ~vn

@xð9Þ

Comparison between Eqs. (6) and (9) yields @ ~vn=@x¼ 4C1sB=

ðC1þ1ÞpE, which appears reasonably close to the standardrelation, @ ~un=@yþ@ ~vn=@x¼ tn

xy=m¼ 2:6sB=C1E, for material havingPoisson’s ratio of 0.3.

Half of cohesive zone is only modeled due to symmetry. v(x),tapering from v(0) at crack tip to v(a) at the interface, finallyreduces to zero at the tip of the cohesive zone as shown in Fig. 1.Minimal lateral dimensions of cohesive zone when compared toits length allows its height to be assumed constant as v(0) overdistance, a, in parent body and v(a) over distance, l, in interfacebody to facilitate quick modeling. 2D, 8 noded quadrilateral plane82 elements are chosen for meshing as shown in Fig. 3. Cohesivestresses, sA and sB, are applied in the form of pressure in –ve y

direction over boundary nodes of cohesive volumes in bodies A

and B respectively. Nodes on crack axis are constrained in y

direction.

3.1. A case study

Refer Table 1. Sample results, obtained by applying thetheoretical model to material combination of low strengthASTM 4340 alloy steel, A, and high strength MDN 250 maragingsteel, B, are considered for testing of finite element model as ademonstrative exercise. Material data are suitably chosen. Theresults presented as Case I and Case II represent fracture data ofMode I crack in alloy steel at different stage II positions near thininterface of maraging steel when subjected to monotonic load inlinear elastic regime under plane stress condition. Elasticallystrained zone in maraging steel is replaced by much smallercohesive zone subjected to higher cohesive stresses to realizesimilar effects upon application of stated theoretical model.

Parent body, A

Constraints

Crack axis

Boundary

�A

x, y = 0

Fig. 3. Finite element discret

Before undertaking the finite element analysis, the results areverified in the following manner:

Case I:

Jtip ¼ðKA

C Þ2

E¼ 112:5 N=mm

Japplied ¼K2

CðbimetallicÞ

E¼ 134:3 N=mm

Jinterface ¼ 2ðsB�sAÞ � v�ðaÞ ¼ 21:78 N=mm

Case II:

Jtip ¼ 112:5 N=mm

Japplied ¼ 140:78 N=mm

Jinterface ¼ 28:3 N=mm

The conservation of energy release rate criterion, Japplied=Jtip+Jinterfacce,is satisfied in both the cases.

Cohesive zones are modeled in Ansys 9 software. Plane stresswith thickness option is adopted. Elements as small as0.0009�0.01 mm are created. Number of elements and nodesgenerated are of the order of 70 000 and 200 000. Magnified nodaldisplacements and stress plots at the interface area in cohesivezone of Case I are presented in Figs. 4 and 5 respectively. Stresspatterns at various locations are illustrated in Fig. 6. The requiredvalues are mapped over chosen path around the interface toobtain various terms of integral, J1. Different cyclic paths are tried.Appropriate value of J1 is obtained on path P near the boundary

Boundary

Interface body, B

8 noded, Plane 82 solid element

1 2(1,-1)

3(1,1)4

5

6

7

8 0(-1,-1)

(-1,1)

�B

isation of cohesive zone.

ARTICLE IN PRESS

-0.307E-06-0.273E-06-0.239E-06-0.205E-06-0.170E-06-0.136E-06-0.102E-06-0.682E-07-0.341E-07

v n (m)~u n (m)~

Alloy steel

Maraging steel

Interface

Path, P

Crack axis

-0.265E-06-0.206E-06-0.147E-06-0.882E-07-0.294E-07

Crack tip

Fig. 4. Nodal displacement plots at interface area in cohesive zone of Case I.

-0.228E+10 -0.210E+10 -0.192E+10 -0.174E+10 -0.156E+10 -0.138E+10 -0.120E+10 -0.102E+10 -0.845E+09

-0.522E+09 -0.406E+09 -0.290E+09 -0.174E+09 -0.580E+08 0.580E+08 0.174E+09 0.290E+09 0.406E+09

-0.684E+09 -0.600E+09 -0.517E+09 -0.433E+09 -0.349E+09 -0.266E+09 -0.182E+09 -0.987E+08 0.151E+08

�xyn (N/m2)

�xn (N/m2) �y

n (N/m2)

Fig. 5. Nodal stress plots at interface area in cohesive zone of Case I.

S. Bhat, V.G. Ukadgaonker / Finite Elements in Analysis and Design 46 (2010) 601–610606

where displacement values are higher. Stress values on path P arewell within the material yield strength value which justifies use of

linear analysis. Values of @ ~vn=@x and @ ~un=@x are obtained byshifting the path suitably by an infinitesimally small distance.Results obtained for half of path P are path length=8.59 mm,R

We dy� 0 N=mm,R

Txð@ ~un=@xÞds¼ 0:21 N=mm and

RTyð@ ~v

n=

@xÞds¼ 38:7 N=mm. Integral J1 on complete path P=�77.82 N/mm. Refer Eq. (1). Contribution of far field load over load linedisplacement at the interface, v�ðaÞ, is given by the term

ð2=EÞKCðbimetallicÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2=pÞl�

p. Integral J2, equal to ð4=EÞKCðbimetallicÞ

ðsB�sAÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2=pÞl�

p, is +100.46 N/mm. Jinterface= J1+ J2=22.64 N/mm.

This value is in good agreement with 21.78 N/mm obtainedfrom the theoretical model. Percent difference is 3.79. Similarly

in Case II, half path length=8 mm,R

We dy� 0 N=mm,R

Txð@ ~un=@xÞ

ds¼ 0:24 N=mm andR

Tyð@ ~vn=@xÞds¼ 42:48 N=mm. J1=�85.44

N/mm and J2= +115.18 N/mm. Jinterface=29.74 N/mm. This valuetoo is in good agreement with the theoretical value of 28.3 N/mm.Percent difference is 4.84.

ARTICLE IN PRESS

-800

-700

-600

-500

-400

-300

-200

-100

00

-800

-700

-600

-500

-400

-300

-200

-100

00

y, mm

At y = 0.0792 mm on Path P

At y = 0.0792 mm on Path P

At y = 0.0792 mm on Path P

= 0.24e−2

∂x

∂un= 0.75e−2,∂x∂vn

= 0.18e−2∂x

∂un= 0.72e−2,∂x

∂vn

≈ 0∂x

∂un= 0.70e−2,∂x

∂vn

vn = −0.3e − 3 mm, un = −0.13e−3mm

vn = −0.28e − 3 mm, un = −0.11e−3mm

At y = 0.009 mm on Path P

≈ 0∂x

∂un= 0.1e−2,∂x

∂vn

vn = −0.6e −4 mm, un ≈ 0 At y = 0.009 mm on Path P

= 0.03e−2∂x

∂un= 0.09e−2,∂x

∂vn

vn = −0.5e −4 mm, un = −0.18e−5

At y = 0.009 mm on Path P

= 0.03e−2∂x

∂un= 0.1e−2,∂x

∂vn

vn = −0.61e −4 mm, un = −0.2e − 4mm

vn = −0.25e − 3 mm, un ≈ 0

Stre

ss (

MPa

)Values at x = 0.02 mm Values at x = 3.60 mm

(region: alloy steel)

Values at x = 7.19 mm(region: alloy steel)

(region: alloy steel)

Crack axis Boundary ofcohesive zone

-800

-600

-400

-200

0

200

400

-2000

-1600

-1200

-800

-400

0

Values at x = 7.82 mm(region: maraging steel)

0.0099

-2000

-1600

-1200

-800

-400

00

Values at x = 7.31 mm(region: maraging steel)

0.0099

-2000

-1500

-1000

-500

0

500

Values at x = 8.41 mm(region: maraging steel)

0.0099

y, mm

y, mm

y, mm

Stre

ss (

MPa

)

�y

�y

�y �y

�y

�y

�x

�x

�x

�x

�x

�x

�xy

�xy

�xy�xy ≈ 0

�xy

�xy ≈ 0

y, mm

Stre

ss (

MPa

)St

ress

(M

Pa)

Stre

ss (

MPa

)St

ress

(M

Pa)

0.002 0.004 0.006 0.008

0 0.002 0.004 0.006 0.008

0 0.002 0.004 0.006 0.008

0.02 0.04 0.06 0.08

0 0.02 0.04 0.06 0.08

y, mm

0.02 0.04 0.06 0.08

Fig. 6. Stress patterns at various locations in cohesive zone of Case I.

48.5 mm

Alloy steel, A Maraging steel, B

Weldinterface

30.5 mm

1.5 mm

Right end Left end

Crack

10 mm

Weld, W

Fig. 7. A welded and machined bimetallic specimen.

S. Bhat, V.G. Ukadgaonker / Finite Elements in Analysis and Design 46 (2010) 601–610 607

4. Experimental work

Refer Fig. 7. Two bimetallic compact tension specimens,numbered 1 and 2, conforming to dimensions of Fig. 2, werefabricated by electron beam welding of 10 mm thick plates ofweak, hot rolled, ASTM 4340 steel, A, and strong, solutionannealed, MDN 250 maraging steel, B. The steels were not heattreated further. 1.5 mm wide ultra strong weld, W, had itsinterface at the distance of 48.5 mm from the left end of thespecimen. Since welding was carried out in vacuum, atmosphericcontamination in the weld was eliminated. The heat affectedzones were reduced due to highly focused and low heat energyinput to the specimen. Chances of development of residualstresses around the weld, due to thermal mismatch between thesteels, were minimized because of nearly identical coefficients ofthermal expansion, a, of the steels. All these factors made theconditions conducive for examination of strength mismatch effectalone across the weld interface over the crack tip. A notch,30.5 mm long and 6.5 mm wide at the mouth, was machined inalloy steel side, perpendicularly with respect to the weldinterface. Its front fine tip was obtained by cutting with a wire

ARTICLE IN PRESS

Table 2Material properties of specimens.

Specimen no. Parent alloy steel, A Weld, W Interface maraging steel, B

Hardness (HV) Yield strength YA (MPa) Hardness (HV) Yield strength YW (MPa) Hardness (HV) Yield strength YB (MPa)

1 205 480 457 2454 335 1800

2 205 480 435 2400 317 1750

EA=2�105 MPa EBE2�105 MPa

aA=12�10�6/(1C) aB=9.9�10�6/(1C)

EAEEWEEB=E

0

20

40

60

80

100

120

30.5

Series1

Series2

Series3

Series4

Stage I

Series 3−ΔKtip (Theoretical)

Series 2−ΔKtip (Experimental)

amm

ΔKapplied

mMPa

Δr = 9.4mm

0

20

40

60

80

100

120

140

160Series1

Series2

Series3

Series4 Stage I

Series 3−ΔKtip(Theoretical)

Series 2−ΔKtip(Experimental)

amm

ΔKapplied

mMPa

Δr = 7 mm

Specimen 1 Specimen 2

ΔKti

p

Crack growth

Weld interface

Crack growth

Weld interface

ΔKti

p

c (mm)

32.5 34.5 36.5 38.5 40.5 42.5 44.5 46.5 48.5 30.5 32.5 34.5 36.5 38.5 40.5 42.5 44.5 46.5 48.5

c (mm)

Stage II

�effA = 210.37 MPa

Stage II

�effA = 288 MPa

Fig. 8. Theoretical and experimental values of crack tip stress intensity parameter in bimetallic specimens.

S. Bhat, V.G. Ukadgaonker / Finite Elements in Analysis and Design 46 (2010) 601–610608

of 0.3 mm size. Material properties of specimens are presented inTable 2. The specimens were subjected to constant tension–tension fatigue cycles in ambient environment till they fractured.The crack was generated at the notch tip. Details of the load cycleare: maximum load, Fmax=14.7 kN, minimum load, Fmin=0.98 kNand frequency=20 cycles per second. Requirements of high cycleand plane stress conditions at crack tip were fulfilled withselected load values upon considering the effects of load transfer.Length, c, of crack advancing towards the weld interface wasmeasured at various positions from left or notch end of thespecimen. These measurement positions represent data points forthe purpose of analysis. Number of cycles, N, required forincremental crack growth were noted. Crack growth rate, dc/dN,was computed.

5. Comparative analysis

5.1. Experimental and theoretical results

Specimen 1 fractured prematurely at crack length of 46.5 mmbefore the crack could reach the weld interface. Catastrophic failureof this nature reported during crack transition from weak to strongmaterial [21] is attributed to damaged areas in the weld linking backto the crack tip. Crack in specimen 2 grew up to the weld interfacebut became critical immediately upon penetration into weld. dc/dN

in each specimen increased with crack growth in stage I, suddendrop in it marked end of stage I and the beginning of stage II orshielding effect of approaching ultra strong weld over the crack tipin alloy steel. Experimental values of crack tip stress intensity

parameter, DKtip (Experimental), are determined in stage II using dc/dN values in Paris law of the form, dc=dN¼ C½DKtipðExperimentalÞ�m,with constants C and m found from stage I data. C, m values ofspecimens 1 and 2 are obtained as 10�12.35, 5.69 and 10�9.10, 3.85,respectively. Stage II commenced earlier than predicted. This wasdue to the effect of eccentric or bending load over the crack tip. Asthe bending moment was applied to the specimen, the tensilestresses developed near the crack tip and compressive stresses at theright end of the specimen. The zero stress point, known as therotation centre of the specimen, suddenly moved far away from thecrack tip soon after the crack growth initiation [7, p. 355] andlocated itself in weld or maraging steel because of their higher yieldstrengths than that of alloy steel. This reduced the elastic constraintover cohesive zone. As the result, cohesive stresses dropped leadingto increased length of cohesive zone which triggered an early effectof the weld. Dr at start of stage II was measured, first data point notbeing considered to ensure existence of this stage. Term sA isreplaced by effective cohesive stress term, sA

eff , obtained from theexpression, Dr¼ p=8ðDKapplied=2sA

eff Þ2. Plane stress conditions ex-

isted in 10 mm thick alloy steel. Since the weld acted as interfacebody for the advancing crack, the cohesive stress term sB intheoretical model is replaced by sW which in turn is equal to

ffiffiffi3p

YW

to fulfill plain strain condition in 10 mm thick weld. The theoreticalmodel is executed at data points of specimens 1 and 2 to obtaincrack tip stress intensity parameter, DKtip (Theoretical). Value of DKtip

(Theoretical) is found to be higher than DKtip (Experimental) at thefirst data point in both the specimens since the shielding effect ofweld over the crack tip, as noticed during experiments, was morethan expected. Such an observation has also been reported at [21]. Asmall correction factor introduced to make DKtip (Theoretical) equal

ARTICLE IN PRESS

S. Bhat, V.G. Ukadgaonker / Finite Elements in Analysis and Design 46 (2010) 601–610 609

to DKtip (Experimental) at the first data point is used throughout thecomputation. DKtip (Theoretical) and DKtip (Experimental) values inspecimens 1 and 2 are available in Fig. 8(a) and (b), respectively.They are close to each other.

5.2. Finite element analysis and results

Following results, presented in the form of data sets 1 and 2, arepicked for treatment by finite element analysis.

Data set 1 (specimen no. 1)Material data: sA

eff ¼ 210:37 MPa, sW=2454 MPa, E=2�105

MPa and t=10 mmAt c=46.5 mma=2 mm, DKapplied ¼ 118:82 MPa

ffiffiffiffiffimp

and DJapplied=70.59 N/mm

Experimental result: dc/dN=0.001 mm/cycle and DKtip

ðExperimentalÞ ¼ 43:9 MPaffiffiffiffiffimp

Theoretical result: l=0.15 mm,

Dv(0)=0.01533 mm, DKtip ðTheoreticalÞ ¼ 50:8 MPaffiffiffiffiffimp

, DJtip

ðTheoreticalÞ ¼ 12:9 N=mm, Dv(a)=0.0018 mm and DJinterface

(Theoretical)=16.15 N/mm.Data set 2 (specimen no. 2)Material data: sA

eff ¼ 288 MPa, sW=2400 MPa, E=2�105 MPaand t=10 mm

At c=47.6 mma=0.9 mm, DKapplied ¼ 132:84 MPa

ffiffiffiffiffimp

and DJapplied=88.23 N/mm

Experimental result: dc/dN=0.0031 mm/cycle and DKtip

ðExperimentalÞ ¼ 51:6 MPaffiffiffiffiffimp

Theoretical result: l=0.23 mm,

Dv(0)=0.0082 mm, DKtip ðTheoreticalÞ ¼ 43:7 MPaffiffiffiffiffimp

, DJtip (Theo-

retical)=9.54 N/mm, Dv(a)=0.0016 mm and DJinterface (Theoreti-

cal)=13.52 N/mm.DJapplied is found to be greater than the sum of DJtip (Theoretical)

and DJinterface (Theoretical) because part of applied energy wasconsumed in overcoming extra shielding effect exerted by the weldover the crack tip which was not accounted for in the theoretical

Alloy steel

Weld

Crack axis

-0.546E-07-0.486E-07-0.425E-07-0.364E-07-0.303E-07-0.243E-07-0.182E-07-0.121E-07-0.607E-08

Path, P

Weldinterface

Specimen 1

v n (m)~

Crack tip

Fig. 9. Nodal displacement plots around w

Table 3Comparison among theoretical, finite element (FE) and experimental results of crack t

Data set DKtip ðMPaffiffiffiffiffimpÞ

FE Experiments Theoretical

1 48.19 43.9 50.8

2 41.61 51.6 43.7

Average % difference

model based on crack tip plasticity criterion alone. This necessi-tated the use of correction factor as discussed earlier.

Finite element analysis of cyclic cohesive zone, spread acrossalloy steel and weld, is undertaken for specimens 1 and 2. Sinceextension of cohesive zone was restricted to the weld withoutcrossing over into maraging steel, i.e., lo1.5 mm (width of weld),maraging steel is not included in modeling. Monotonic cohesivepressure, sA

eff and sW, are applied over boundary nodes incohesive volumes of alloy steel and weld. J integral under fatigueload, DJ1, over cyclic path P is written as

DJ1 ¼ 4

ZP

We dy�Tx@ ~un

@xds�Ty

@ ~vn

@xds

� �:

Results obtained for half of path P near the boundary of

cohesive zone in specimen 1 are:R

We dy� 0 N=mm,R

Txð@ ~un= @xÞ

ds¼ 1:54 N=mm andR

Tyð@ ~vn=@xÞds¼ 9:302 N=mm. DJ1 on com-

plete path P=�86.74 N/mm. DJ2 ¼ 8=EðsW�sAeff ÞDKapplied

ffiffiffiffiffiffiffiffiffiffiffiffiffið2=pÞl

por DJ2= +104.18 N/mm. DJinterface=DJ1+DJ2=17.44 N/mm. Using

the condition, (DJtip+DJinterface) Theoretical=(DJtip+DJinterface) Finite

element analysis, DJtip (Finite element analysis)=11.61 N/mm and

DKtip ðFinite element analysisÞ ¼ 48:19 MPaffiffiffiffiffimp

. Similarly, results

obtained for half of path P in specimen 2 are:R

We dy� 0 N=mm,RTxð@ ~u

n=@xÞ ds¼ 1:4 N=mm andR

Tyð@ ~vn=@xÞds¼ 13:77 N=mm. DJ1

on complete path P=�121.36 N/mm. DJ2= +135.76 N/mm.

DJinterface=DJ1+DJ2=14.40 N/mm. DJtip (Finite element analysis)=

8.66 N/mm and DKtip ðFinite element analysisÞ ¼ 41:61 MPaffiffiffiffiffimp

.

Critical paramater in DJ1 is @ ~vn=@x. Plots of nodal displacements

in y direction, ~vn, in specimens 1 and 2 are displayed in Fig. 9(a)and (b), respectively.

Values of DKtip (Finite element analysis) are compared withthose of DKtip (Experimental) and DKtip (Theoretical) in Table 3.They are in good agreement with one another which validates theapproach and accuracy of finite element analysis.

-0.852E-07-0.757E-07-0.663E-07-0.568E-07-0.473E-07-0.379E-07-0.284E-07-0.189E-07-0.94E-08

Specimen 2

v n (m)~

eld interface in cyclic cohesive zones.

ip stress intensity parameter.

% difference between finite

element and experimental results

% difference between finite

element and theoretical results

8.9 5.13

19.3 4.78

14.1 4.95

ARTICLE IN PRESS

S. Bhat, V.G. Ukadgaonker / Finite Elements in Analysis and Design 46 (2010) 601–610610

6. Conclusions

Finite element analysis of minute crack tip cohesive zonespread across the interface of elastically matched but strengthmismatched ductile bodies in linear elastic regime is presented toexamine the effect of strength mismatch between the bodies overthe crack tip near the interface of the bodies. Dugdale’s cohesivezone model is used in the analysis. Jtip of crack in weak body nearthe interface of stronger body is expectedly found to be less thanJapplied. Inferences drawn from the work are as follows:

(i)

It is feasible to simplify the finite element analysis by isolatingthe cohesive zone from the bimetallic body domain andmodel it alone under dual action of cohesive stresses in parentand interface bodies to obtain Jinterface which quantifies theeffect of strength mismatch across the interface over the cracktip. A theoretical model is used in conjunction with finiteelement analysis to determine the effect of far field load at theinterface. The approach is fast, simple and reasonablyaccurate. The finite element model is well substantiatedvis-�a-vis theoretical and experimental results of cohesivezone, at Mode I crack tip, developed across various possiblestrength mismatched interfaces in material combination ofweak ASTM 4340 and strong MDN 250 maraging steels whensubjected to monotonic and high cycle fatigue loads.

(ii)

J1 is not found to be path independent. Appropriate value ofthis integral is obtained on path near the boundary ofcohesive zone where displacements are higher. On paths nearthe constrained crack axis, the displacements are less whichresult in reduced value of the integral. As the result, Jinterface isalso not path independent.

Acknowledgement

Support received from the Mechanical Engineering Depart-ment of the Indian Institute of Technology, Bombay, India, duringthe course of this work is gratefully acknowledged.

References

[1] A.R. Zak, M.L. Williams, Crack point stress singularities at a bi-materialinterface, Journal of Applied Mechanics 30 (1963) 142–143.

[2] K.Y. Lin, J.W. Mar, Finite element analysis of stress intensity factors for cracksat a bi-material interface, International Journal of Fracture 12 (1976)521–531.

[3] J. Ahmad, A micromechanics analysis of cracks in unidirectional fibercomposites, Journal of Applied Mechanics 58 (1991) 964–972.

[4] S.A. Meguid, M. Tan, Z.H. Zhu, Analysis of cracks perpendicular to bimaterialinterfaces using a novel finite element, International Journal of Fracture 73(1995) 1–23.

[5] S. Suresh, Y. Sugimura, E.K. Tschegg, The growth of a fatigue crackapproaching a perpendicularly-oriented, bimaterial interface, Scripta Metal-lurgica et Materialia 27 (1992) 1189–1194.

[6] Y. Sugimura, P.G. Lim, C.F. Shih, S. Suresh, Fracture normal to a bimaterialinterface: effects of plasticity on crack tip shielding and amplification, ActaMetallurgica et Materialia 43 (1995) 1157–1169.

[7] M. Kikuchi, Ductile crack growth behaviour of welded plate, InternationalJournal of Fracture 78 (1996) 347–362.

[8] A.S. Kim, S. Suresh, C.F. Shih, Plasticity effects on fracture normal to interfaceswith homogeneous and graded compositions, International Journal of Solidsand Structures 34 (1997) 3415–3432.

[9] A.S. Kim, J. Besson, A. Pineau, Global and local approaches to fracturenormal to interfaces, International Journal of Solids and Structures 36 (1999)1845–1864.

[10] F.O. Reimelmoser, R. Pippan, The J-integrals at Dugdale cracks perpendicularto interfaces of materials with dissimilar yield stresses, International Journalof Fracture 103 (2000) 397–418.

[11] N. Gubeljak, O. Kolednik, J. Predan, M. Oblak, Effect of strength mismatchinterface on the crack driving force, Key Engineering Materials 251–252(2003) 235–243.

[12] J. Predan, N. Gubeljak, O. Kolednik, On the local variation of the crack drivingforce in a double mismatched weld, Engineering Fracture Mechanics 74(2007) 1739–1757.

[13] B. Wang, T. Siegmund, Simulation of fatigue crack growth at plasticallymismatched bi-material interfaces, International Journal of Plasticity 22(2006) 1586–1609.

[14] V.G. Ukadgaonker, S. Bhat, M. Jha, P.B. Desai, Fatigue crack growth towardsthe weld interface of alloy and maraging steels, International Journal ofFatigue 30 (2008) 689–705.

[15] S. Ostlund, F. Nilsson, Cohesive modeling of process regions for cracks inlinear elastic structures—fundamental aspects, Fatigue and Fracture ofEngineering Materials and Structures 16 (1993) 215–235.

[16] X.P. Xu, A. Needleman, Numerical simulation of dynamic interfacial crackgrowth allowing for crack growth away from the bond line, InternationalJournal of Fracture 74 (1995) 253–275.

[17] D.S. Dugdale, Yielding of steel sheets containing slits, Journal of theMechanics and Physics of Solids 8 (1960) 100–104.

[18] K. Hellan, Introduction to Fracture Mechanics, McGraw-Hill Book Company,1985 p. 19.

[19] J.F. Knott, Fundamentals of Fracture Mechanics, Butterworths, 1981p. 132.

[20] J.R. Rice, A path independent integral and approximate analysis of strainconcentration by notches and cracks, Journal of Applied Mechanics 35 (1968)379–386.

[21] M.T. Milan, P. Bowen, Experimental and predicted fatigue crack growthresistance in Al 2124/Al 2124+35%SiC (3mm) bimaterial, InternationalJournal of Fatigue 25 (2003) 657.