A computational model and experimental validation of shielding and amplifying effects at a crack tip near perpendicular strength-mismatched interfaces

Acta Mech 216, 259–279 (2011)DOI 10.1007/s00707-010-0365-y

Sunil Bhat · S. Narayanan

A computational model and experimental validationof shielding and amplifying effects at a crack tip nearperpendicular strength-mismatched interfaces

Received: 29 September 2009 / Revised: 14 June 2010 / Published online: 25 July 2010© Springer-Verlag 2010

Abstract The stress field around the crack tip near an elastically matched but strength-mismatched interfacebody in a bimetallic system is influenced when the crack tip yield or cohesive zone spreads to the interfacebody. The concept of crack tip stress intensity parameter, Ktip, is therefore employed in fracture analysis of thebimetallic body. A computational model to determine Ktip is reviewed in this paper. The model, based upon i)Westergaard’s complex potentials coupled with Kolosov–Muskhelishvili’s relations between a crack tip stressfield and complex potentials and ii) Dugdale’s representation of the cohesive zone clearly indicates shieldingor amplifying effects of strength mismatch across the interface, depending upon the direction of the strengthgradient, over the crack tip. The model is successfully validated by conducting series of high cycle fatigue testsover Mode I cracks advancing towards various strength-mismatched interfaces in bimetallic compact tensionspecimens prepared by electron beam welding of elastically identical weak ASTM 4340 alloy and strong MDN250 maraging steels.

List of symbolsa Distance of crack tip from interfacea∗ Radial coordinate of point near crack tipA Parent body/parent steel containing crackB Interface body/back up steelb Length of cohesive zone across interfacec Crack lengthcc Crack length ahead of load axiscmin Crack length required for linear elastic regimeC Paris constante, eav Percent difference between theoretical and experimental result,

average of percent differencesE Modulus of elasticityf (θ) A function of angle w.r.t. crack axisF Applied loadi Imaginary quantity,

√−1

S. Bhat (B) · S. NarayananSchool of Mechanical and Building Sciences,Vellore Institute of Technology,Vellore, TN 632014, IndiaE-mail: [email protected]

S. NarayananE-mail: [email protected]

260 S. Bhat, S. Narayanan

Kapplied Applied stress intensity parameterKC Plane stress fracture toughness of homogenous bodyKC(bimetallic) Plane stress fracture toughness of bimetallic bodyKIC Plane strain fracture toughness of homogeneous bodyKL Stress intensity parameter over cohesive zone in interface materialKtip Stress intensity parameter at crack tipl Extension of cohesive zone into interface bodyL Distance between load axis and left end of specimenm Paris constantn No. of data pointsN Number of fatigue cyclesp∞ Far field applied stressr Cohesive zone length in homogeneous parent bodyt Specimen thicknessT T stressu Displacement in x direction in cohesive zonev Displacement in y direction from crack axis in cohesive zoneW WeldY Yield strengthz Complex variableZ f Distance from specimen right end to front weld interfaceZr Distance from specimen right end to rear weld interface� Parameter under cyclic or fatigue loadκ A material constantμ Shear modulusν Poisson’s ratioξ A variableσ Cohesive stressσeff Effective cohesive stressσi j Crack tip stress fieldσx Stress in x directionσy Stress in y directionδi j Kronecker deltaτxy Shear stress in xy planeϕ, φ, φ1, φ2, ψ, 1, 2 Complex potentials

SuperscriptsA Parent body/parent steelB Interface body/back up steelmax Maximum valuemin Minimum valueW Weld* Value at fracture

1 Introduction

Class A composites, i.e. fibre-reinforced plastics (FRPs) and homogeneous metal resin (MMCs) are charac-terized by significant elastic mismatch between the constituents and poor strength and plasticity of at least oneof the constituents. For instance, the ratio of shear modulus between carbon and epoxy, boron and epoxy andaluminium and epoxy is as high as 300, 135 and 25, respectively. But percent elongation of resins like epoxy,vinyl ester, phenolic, etc., which are the common components in stated composites, is just of the order of1–6%. Fibres too exhibit a poor percent elongation. Fracture aspects of such composites are therefore mainlyinfluenced by material elastic properties. A crack tip on touching the interface of the constituents in these com-posites experiences a sharp jump or discontinuity in stresses in the direction of the load to fulfil the conditionof continuity of displacement and strain across the interface. Consequently, the stress intensity parameter at a

A computational model and experimental validation 261

crack tip also exhibits a change. It increases sharply when the crack tip advancing from the compliant materialside touches the interface of the stiffer material and vice versa.

On the other hand, Class B composites, i.e. homogeneous metal–metal, metal–ceramic (MMCs) and a newseparate class of non-homogeneous, externally welded metal–metal composites are marked by marginal or nilelastic but appreciable strength mismatch between the constituents coupled with good plasticity of at least oneof them. For example, the ratio of shear modulus between steel and aluminum, steel and titanium, steel andcopper, boron and aluminum, glass and aluminum is as low as 3, 2, 1.6, 4.5 and 1, respectively. But percentelongation of the stated metals and their alloys is high, which ranges from 25–45%. Strength mismatch betweenthem is also significant. As a result, material strength and plasticity properties primarily influence the fracturecharacteristics of these composites. With the advent of solid state and fusion welding processes like diffusionbonding, explosive cladding, friction welding, electron and laser beam welding, etc., which are capable ofjoining dissimilar metals with relatively clean and strong weld, non-homogeneous bimetallic composites havebeen fabricated and successfully tested as functionally graded, material optimized units with life-enhancedfeatures in applications pertaining to nuclear pressure vessels, oil pipelines, aerospace, gas turbine discs, tubesand barrels, etc. Efficient coating technologies like physical and chemical vapour deposition, etc., have alsohelped in producing strong and wear resistant films and coatings over substrates of different materials. In viewof the above, the study of fatigue and fracture aspects of bimetallic composites gets importance especially inload applications when the crack is near or at the interface of the constituents because failure of the interfacerenders the composite ineffective by degrading its functional capabilities.

Since two steels are easy to weld, substantial work has been reported over the fracture characteristics ofclass B, bimetallic, steel-steel composites. Suresh et al. [1], Sugimura et al. [2], Kikuchi [3], Kim et al. [4,5],Pippan et al. [6], Jiang et al. [7], Wang and Siegmund [8], Predan et al. [9], etc., are the few important onesin chronological order who have successfully investigated a Mode I crack near the interface of strength-mis-matched steels or other plastically mismatched materials. All of them in one form or another have reportedshielding or amplifying effects of strength mismatch across the interface over the crack tip—a shielding effectwhen the crack tip in the weaker material approaches the interface of the stronger material and an amplifyingeffect when the crack tip in the stronger material approaches the interface of the weaker material. Their workwas, however, numerical or experimental in nature. Only Wappling et al. [10] and Reimelmoser and Pippan[11] have reported theoretical work on Mode I crack near a strength-mismatched interface. The former pro-vided the solution for crack tip opening displacement and the latter the solution for a J integral at the cracktip. It is seen from the literature survey that sufficient theoretical solutions for life prediction of bimetalliccomposites are either not available or are not fully validated by experiments. The work reported in this papertherefore first reviews a computational model to obtain Ktip in monotonic and fatigue load regimes, sinceKtip adequately defines the state at the crack tip near the bimetallic interface, and then validates the modelwith the help of the results obtained from series of high cycle fatigue tests conducted over Mode I cracksin bimetallic compact tension specimens prepared by electron beam welding of weak ASTM 4340 alloy andstrong MDN 250 maraging steels. Thick ultra-strong weld between elastically identical steels results in twostrength-mismatched interfaces, viz., front weld interface between cracked parent steel and weld and the rearor inner weld interface between weld and back up steel. Two types of specimens, Type I and Type II, are tested.In Type I specimen, the crack in the parent alloy steel advances perpendicularly towards the weld backed upby maraging steel, whereas in Type II specimen, the crack in the parent maraging steel grows perpendicularlytowards the weld backed up by alloy steel. Quantification by a computational model about the magnitude ofeffects exerted by different interfaces is found to be in good agreement with experimental findings.

2 Theoretical review

The spread of crack tip plasticity into the interface body influences the crack tip near the interface. The plasticor yield zone across the interface is modelled in this work as Dugdale’s cohesive zone [12] for elastic-ideallyplastic material which is then analysed with the help of basic principles of Westergaard’s complex potentials andKolosov–Muskhelishvili’s relations between a crack tip stress field and complex potentials. The mechanicalbehaviour of Dugdale’s zone is characterised by constant closing cohesive stresses acting over it. These stressesare generated due to an elastic constraint exerted by surrounding non-yielded material over the cohesive zoneand are considered as the function of material yield strength to fulfil the condition of zero stress singularity atthe tip of the cohesive zone. They are equal to the material yield strength in a plane stress condition and areapproximated as

√3 times the material yield strength in the plane strain case.


(b) Stage II

(Extension of cohesive zone into interface body)

(a) Stage I

(Cohesive zone in parent body)

Far field applied stress, ∞p

Interface body, B

Parent body, A

σA

Interface

Cohesive zone cr

∞p

σAσB

Crack Tip

ca l

b

Axis

At Tip

(x = 0)

x,u Axis (y=0)

Crack

y,v

⇐

Fig. 1 Stages of crack advancement towards the strength-mismatched interface

Refer to Fig. 1. A Mode I crack in the parent body, A, is shown to grow under monotonic load towards anelastically identical but different strength interface body, B, in stages I and II. In stage I, Fig. 1a, the crack tipis far away from the interface body and the entire cohesive zone of size r is contained in the parent body. r byDugdale’s criterion under the action of applied stress intensity parameter, Kapplied, and cohesive stress, σ A, is

given by π8

(Kapplied

σ A

)2. Refer to Section I of the Appendix. For a crack of length c with its cohesive zone yet

to cross the interface, the following expression is obtained for the crack opening or load line displacement, v,in y direction from the crack axis in the cohesive zone as the function of potential, φ(z), under plane stresscondition,

∂v

∂x= 2

Eilimy→0

[φ(z)− φ(z)

]. (1)

The solution of Eq. (1) requires a complex potential, φ(z). The potentials i) φ1(z) which considers the effectof monotonic far field applied stress, p∞, and Kapplied and ii) φ2(z) which accounts for the influence of σ A

are determined independently and then superimposed to obtain φ(z). Refer to Section II of the Appendix. Thepotential, φ1(z), is derived as

φ1(z) ={

Kapplied

2√

2π(z − r)− p∞

4

}. (2)

Refer to Section III of the Appendix. The potential, φ2(z), is obtained as

φ2(z) = −i

2π√

r − z

r∫

0

σ A√r − ξ

ξ − zdξ − σ A

4. (3)


The final potential is written by the principle of superposition as φ(z) = φ1(z)+ φ2(z). Therefore,

φ(z) = Kapplied

2√

2π(z − r)− p∞

4− i

2π√

r − z

r∫

0

σ A√r − ξ


4, (4)

limy→0

φ(z) = Kapplied

2i√

2π(r − x)− p∞

4− i

2π√

r − x

r∫

0

σ A√r − ξ

ξ − xdξ − σ A

4(5)

and

limy→0

φ(z) = − Kapplied

2i√

2π(r − x)− p∞

4+ i

2π√

r − x

r∫

0

σ A√r − ξ

ξ − xdξ − σ A

4. (6)

Substitution of Eqs. (5) and (6) in Eq. (1) results in

∂v

∂x= − 2

E

⎧⎨⎩

Kapplied√2π(r − x)

+ 1

π√

r − x

r∫

0

σ A√r − ξ

ξ − xdξ

⎫⎬⎭ . (7)

On integrating Eq. (7), the expression for v as the function of distance x from the crack tip is obtained as

v(x) = − 2

E

⎡⎣

x∫

0

Kapplied√2π(r − x)

dx +x∫

0

dx

π√

r − x

r∫

0

σ A√r − ξ

ξ − xdξ

⎤⎦ . (8)

Stage I is valid till the distance of the crack tip from the interface, a, fulfills the condition, a ≥ r . The effectof the interface body is not felt by the crack tip in this stage.

Refer to Fig. 1b. The crack and its cohesive zone have advanced further. The crack is now in the vicinity ofthe interface with its cohesive zone having spread to the interface body, B, by distance l such that a < r whichmarks the beginning of the effect of the interface body or the onset of stage II. The length of the cohesive zoneacross the interface, b, is (a + l) which is different from r. Since E A = E B = E the following expression iswritten for v(x) under the simultaneous effect of cohesive stresses σ A and σ B over cohesive volumes in parentand interface bodies, respectively, with the help of Eq. (8),

v(x) = − 2

E

⎡⎣

x∫

0

Kapplied√2π(b − x)

dx +x∫

0

dx

π√

b − x

a∫

0

σ A√b − ξ

ξ − xdξ +

x∫

0

dx

π√

b − x

b∫

a

σ B√b − ξ

ξ − xdξ

⎤⎦ .

(9)

Equation (9) involves singular integrals. Their evaluation by Ukadgaonker et al. [13] results in

v(x) = 2

E

⎧⎪⎪⎪⎨⎪⎪⎪⎩

Kapplied

√2π(a + l − x)+ σ A

π

[x ln

(√(a+l)−√

(a+l−x)√(a+l)+√

(a+l−x)

)− 2

√(a + l)(a + l − x)

]

+σ B−σ A

π

[a ln

( √(a+l−x)+√

l∣∣∣√(a+l−x)−√l∣∣∣

)+ x ln

(∣∣∣√(a+l−x)−√l∣∣∣

√(a+l−x)+√

l

)− 2

√l(a + l − x)

]

⎫⎪⎪⎪⎬⎪⎪⎪⎭. (10)

Solutions of v(x) in Eqs. (8) and (10) indicate that the entire effect of Kapplied is felt by the crack tip in stage Ibut not in stage II due to plasticity-induced load transfer either towards the parent or interface body dependingupon the direction of the strength gradient parameter, σ B − σ A, across the interface. To maintain continuityof displacements and strain, a larger portion of applied energy is consumed towards cohesive plasticity in theinterface body if the interface body is stronger than the parent body, thereby inducing a shielding effect at thecrack tip. Conversely, an additional driving force or amplifying effect is imparted at the crack tip in case theparent body is stronger than the interface body. Consequently, it is inferred that the stress state at the crack tipin the stage II of a bimetallic body differs from that at a crack tip in the homogeneous parent body alone. Assuch, the concept of a stress intensity parameter at the crack tip, Ktip, is introduced which necessitates v(x)

to be stated in terms of Ktip instead of Kapplied. Using v(0) = K 2tip

2Eσ A in plane stress condition, Eq. (10) at thecrack tip (x = 0) is written as


K 2tip

4σ A=

⎧⎨⎩Kapplied

√2

π(a + l)− σ A

π[2(a + l)] + σ B − σ A

π

⎡⎣a ln

⎛⎝

√(a + l)+ √

l∣∣∣√(a + l)− √l∣∣∣

⎞⎠ − 2

√l(a + l)

⎤⎦⎫⎬⎭ .

(11)

E is replaced by E1−ν2 and v(0) by

K 2tip

(1−ν2)

2Eσ A for plane strain condition in Eq. (10). Equation (11) holds good

in plane strain condition as well, only the values of σ A and σ B being different from those employed underplane stress condition. The difference between Ktip and Kapplied depends upon the mismatch between σ A andσ B and the value of a. At fracture, Eq. (10) changes to

v∗(x) = 2

E

⎧⎪⎪⎪⎨⎪⎪⎪⎩

KC(bimetallic)

√2π(a + l∗ − x)+ σ A

π

[x ln

(√a+l∗−√

a+l∗−x√a+l∗+√

a+l∗−x

)− 2

√(a + l∗) (a + l∗ − x)

]

+σ B−σ A

π

[a ln

( √(a+l∗−x)+√

l∗∣∣∣√(a+l∗−x)−√l∗

∣∣∣

)+ x ln

(∣∣∣√(a+l∗−x)−√l∗

∣∣∣√(a+l∗−x)+√

l∗

)− 2

√l∗ (a + l∗ − x)

]

⎫⎪⎪⎪⎬⎪⎪⎪⎭.

(12)

Since Ktip = K AC and v∗(0) =

(K A

C

)2

2Eσ A at fracture, Eq. (12) takes the form

(K A

C

)2

4σ A=

{KC(bimetallic)

√2

π(a + l∗)− σ A

π

[2(a + l∗

)]

+σB − σ A

π

⎡⎣a ln

⎛⎝

√(a + l∗)+ √

l∗∣∣∣√(a + l∗)− √l∗∣∣∣

⎞⎠ − 2

√l∗ (a + l∗)

⎤⎦⎫⎬⎭ . (13)

The crack grows when the critical half crack tip opening displacement, v∗(0), achieves the value of(K A

C

)2

2Eσ A , which is a material property and independent of the influence of the interface body. Therefore, the

fracture toughness of the bimetallic body, KC(bimetallic), differs from the fracture toughness of the parentbody, K A

C . Conservation of the energy release rate criterion requires the condition Japplied = Jtip + Jinterface,where Jinterface = 2

(σ B − σ A

)v(a), to be satisfied. Since Jinterface has a finite +ve or −ve value, Jtip �=

Japplied, which in turn implies that Ktip �= Kapplied. It is evident from Eqs. (11) and (13) that for a par-ticular value of a, Ktip < Kapplied, KC(bimetallic) > K A

C when σ B > σ A, Jinterface being +ve. Similarly,Ktip > Kapplied, KC(bimetallic) < K A

C when σ B < σ A, Jinterface being −ve. Also, the higher the magni-tude of strength mismatch represented by parameter

∣∣σ B − σ A∣∣, the more is the value of

∣∣Kapplied − Ktip∣∣ or∣∣KC(bimetallic) − K A

C

∣∣. The trend continues with increasing intensity as a reduces with crack growth till thecrack tip touches the interface body.

It is possible with a very strong interface body that in initial phases of stage II the stress field at and aroundthe interface may not exceed the yield strength of the interface body. However, load transfer to the interfacebody still continues elastically due to its higher yield limit than the weak parent body. The high elastic strainzone in the interface body subjected to stresses less than its yield strength is replaced by the much smallercohesive zone subjected to higher cohesive stresses for the realization of similar effects upon application ofthe stated theoretical model.

2.1 Validation of the solution

Solutions of stage I and stage II are checked as follows:Stage I

The solution of integral − ∫ x0

Kapplied√2π(r−x)

dx in Eq. (8) is Kapplied

√2π(r − x). The lower limit is disregarded

since the value of Kapplied at the upper limit acts over the specific location in the cohesive zone. The singular


integral −∫ x0

dxπ

√r−x

∫ r0σ A√

r−ξξ−x dξ is evaluated [13, p. 703] as −σ A

π

[(r − ξ) ln

(√r−ξ−√

r−x√r−ξ+√

r−x

)+ (r − x)

ln(√

r−ξ+√r−x√

r−ξ−√r−x

)− 2

√r − ξ

√r − x

]r

0which simplifies to −σ A

π

[r ln

(√r−√

r−x√r+√

r−x

)+ (r − x) ln

(√r+√

r−x√r−√

r−x

)

−2√

r√

r − x]. v(x) at x = 0, i.e., v(0) is obtained as 2

E

{Kapplied

√2rπ

− 2σ Arπ

}. On applying Dugdale’s

criterion, r = π8

(Kapplied

σ A

)2, v(0) reduces to

(Kapplied)2

2Eσ A which is the well-known solution for the half crack tip

opening displacement in a homogenous body A in the linear elastic regime. Further, a = r = π8

(Kapplied

σ A

)2

and l = 0 at the end of stage I which when substituted in Eq. (11) results in Ktip = Kapplied which is true.

Similarly, substitution of a = r∗ = π8

(KC(bimetallic)

σ A

)2and l = 0 in Eq. (13) results in K A

C = KC(bimetallic) which

also holds good in stage I.Stage II

Jtip, expressed asK 2

tipE , is equal to 2σ Av(0), whereas Jinterface is equal to 2

(σ B − σ A

)v(a). Since Japplied

is given byK 2

appliedE , the following equation is obtained:

K 2applied

E= 2σ Av(0)+ 2

(σ B − σ A

)v(a). (14)

Equation (10) leads to the following expressions:

v(0) = 2

E

{Kapplied

√2

π(a + l)− σ A

π[2(a + l)] + σ B − σ A

π

[a ln

(√(a + l)+ √

l√(a + l)− √

l

)− 2

√l(a + l)

]}

and

v(a) = 2

E

{Kapplied

√2

πl + σ A

π

[a ln

(√(a + l)− √

l√(a + l)+ √

l

)− 2

√(a + l)(l)

]− σ B − σ A

π(2l)

}.

The applied stress intensity parameter [11, p. 404], is written as

Kapplied = 2σ A

√2(a + l)

π+ 2

(σ B − σ A

)√2l

π. (15)

R.H.S. of Eq. (14) results in the expression 1E

[8(σ A

)2(a+l)π

+ 8(σ B − σ A

)2 ( lπ

) + 16σ A(σ B−σ A

)π

√l(a + l)

]

which equalsK 2

appliedE i.e., L.H.S, thereby validating the solution.

2.2 Computational model under fatigue load

The equations of monotonic load are modified for fatigue load by replacing the parameters with their cyclicvalues, i.e., r, v(x), Kapplied and Ktip by �r,�v(x),�Kapplied and �Ktip, respectively. The cohesive stressesσ A and σ B are replaced by 2σ A and 2σ B , respectively. �Ktip = �Kapplied in stage I. In stage II, Eqs. (11)and (15) change to the following form:

�K 2tip

8σ A=

{�Kapplied

√2

π(a + l)− 2σ A

π[2(a + l)]

+2(σ B − σ A

)

π

[a ln

(√(a + l)+ √

l√(a + l)− √

l

)− 2

√l(a + l)

]}, (16)

�Kapplied = 4σ A

√2(a + l)

π+ 4

(σ B − σ A

)√2l

π. (17)


Parent body

Interface body

Interface 30

L = 50

16

Notch tip (crack)

F

12.5

6.2560

c

Hole dia. =12.5

F

Dimensions in mm(Figure not to scale)

Notch

a Load axis

Fig. 2 A standard bimetallic compact tension specimen

A computer programme solves Eq. (17) by a numerical iterative convergence scheme. Known input valuesto the code are σ A, σ B,�Kapplied and a. A value of l is initially assumed. In each iteration, the value of l issuitably incremented or decremented till the actual �Kapplied value and the one obtained from the equationconverge. With output value of l,�Ktip is obtained from Eq. (16). The value of a is reduced in every newset of computation to estimate the change in the effect of approaching the interface body over the advancingcrack.

Refer to Fig. 2. When a standard bimetallic compact tension specimen comprising elastically matched bod-ies is subjected to a tension–tension cycle carrying maximum load, Fmax, and minimum load, Fmin,�Kapplied[14] is written as

�Kapplied = Fmax − Fmin

t L0.5

[29.6

{c − 12.5

L

}0.5

− 185.5

{c − 12.5

L

}1.5

+ 655.7

{c − 12.5

L

}2.5

−1017.0

{c − 12.5

L

}3.5

+ 639.8

{c − 12.5

L

}4.5]. (18)

�Kapplied is found at various positions of growing fatigue crack. �Ktip is subsequently determined at eachposition.

3 Experimental work

Two Type I (numbered 1 and 2) and one Type II bimetallic compact tension specimens were fabricated forexperimental work. The specimens conformed to the overall dimensions of Fig. 2. Weak ASTM 4340 alloy steeland strong MDN 250 maraging steel plates with lateral thickness, t, of 10 mm were used. The yield strength ofalloy and maraging steel ranged from 460 to 500 MPa and 1,750 to 1,800 MPa, respectively. The steels werechosen because of their high strength mismatch in order to generate a pronounced effect over the crack tip. Theplates were joined by electron beam welding resulting in 1.5- mm-wide ultra-strong weld. No filler materialwas used. Since welding was carried out in vacuum, atmospheric contaminations in the weld were eliminated.Heat-affected zones were reduced due to highly focused and low heat energy input to the specimen. Nearlyidentical coefficients of the thermal expansion of steels also minimized the chances of development of residualstresses in them during welding. All these factors made the conditions conducive for the intended examinationof the effect of strength mismatch alone across the weld interfaces over the crack tip.

A notch, 30.5 mm long and 6.25 mm wide at the mouth, was machined perpendicular to the weld. A fine tipat the notch was obtained by cutting with a wire of 0.3 mm size. A welded and machined bimetallic specimenis displayed in Fig. 3. The front weld interface was at the distance of 48.5 and 44.35 mm from the right ornotch end of Type I and Type II specimens, respectively. The vickers hardness at different positions aheadof the notch tip on top and bottom faces of the specimens was measured, the average of which provided thelocal material yield strength value for use in the computational model. Conversion tables between Vickershardness and ultimate tensile strength for steels were referred. Yield strengths of alloy and maraging steels


Load cycle details Specimen type fZ , rZ Frequency

Type I bimetallic 48.5 mm , 50 mm 14.7 kN 0.98 kN 20 cps

Parent steel, A : Alloy steel ; Back up steel, B : Maraging steel, Intermediate weld, W

Type II bimetallic 44.35 mm, 45.85 mm 13.0 kN 0.98 kN 20 cps

Parent steel, A : Maraging steel ; Back up steel, B : Alloy steel, Intermediate weld, W

F max F min

Parent steel, ABack up steel, B

Front weld interface

Crack

30.5

1.5

Left end

Right end

Notch

t =10

Weld, W Rear weld interface

fz

rz

F max

F min

Load, F

Time

Load cycle

Fig. 3 A welded and machined bimetallic specimen

were considered as 0.73 and 0.977 times the values of their ultimate tensile strengths, respectively. Since theweld comprised maraging steel as confirmed by its micro-structural examination, its yield strength in eachspecimen was extrapolated from the yield strength of maraging steel. Plain specimens of alloy and maragingsteels with similar geometry and notch configuration were also prepared to compare their results with those ofbimetallic specimens and to evaluate the effect of interfaces.

All the specimens were subjected to tension–tension fatigue cycles of constant amplitude at ambient envi-ronment in a ±250 kN capacity fatigue test rig till they fractured. A Mode I crack was generated at the notchtip in each specimen. Details of the load cycles are provided at Fig. 3. It was ensured with selected load valuesthat a limit load or plastic collapse situation was not reached in the specimens under the effect of load transfer.Length, c, of a crack advancing towards weld interfaces was measured at different positions from the rightend of the specimen. The number of cycles, N, required for incremental crack growth were noted. The crackgrowth rate, dc

dN, was computed. The number of cycles applied till specimen fracture was also recorded.

Various metallurgical investigations were undertaken on fractured specimens. The microstructure of alloysteel, weld and maraging steel was examined by an optical microscope. Miniature polished and etched sampleswere prepared for this purpose. Non-uniform grains of ferrite and pearlite were observed in alloy steel. Mar-aging steel and weld displayed a similar low carbon iron–nickel martensitic structure. Fatigue crack surfaceswere examined by scanning electron microscope to confirm the location of critical cracks. The residual stressesin alloy steel, developed during manufacture of plates followed by their welding with maraging steel, weremeasured by X-ray diffraction equipment in y direction, i.e., perpendicular to the crack surface at differentpositions near the front weld interface. The stresses were found to be tensile in nature.

4 Results and discussions

4.1 Type I specimens

Fractured Type I bimetallic and plain alloy steel specimens are displayed in Fig. 4. Plots of �Kapplied vs.length, c, of the crack in parent alloy steel, A, advancing towards weld, W, are at Fig. 5. The crack remainedstable up to longer lengths in bimetallic specimens sustaining an increased �Kapplied value when comparedto the crack in a plain alloy steel specimen which hinted at enhanced fatigue life of bimetallic specimens. Thebimetallic specimens failed soon after the crack crossed the front weld interface, at a length just exceeding48.5 mm, whereas a plain alloy steel specimen failed earlier at crack length of 42.4 mm. Specimens 1 and 2required 29,442 and 23,542 cycles, respectively, for fracture which were more than 18,309 cycles consumed by


Critical crack on crossing front weld interface

Fractured Type I bimetallic specimen

Critical crack at 42.4 mm length

Fractured plain alloy steel specimen

Fig. 4 Fractured Type I bimetallic and plain alloy steel specimens

a plain specimen. As expected,�Kapplied and dcdN

values in each bimetallic specimen increased with the crack

growth in stage I. A sudden drop in dcdN

marked termination of stage I and the beginning of stage II or shieldingeffect of weld over the crack tip. Ideally, in a specimen subjected to uniform symmetrical load, stage II begins

when the crack tip is at distance, �r , from the front weld interface where �r is equal to π8

(�Kapplied

2σ A

)2. The

estimated�r under such condition was 4.07 and 3.98 mm at crack lengths at 44.43 and 44.52 mm in specimens1 and 2, respectively. But it was noticed during experiments that stage II commenced earlier at crack lengthsof 38.0 and 37.80 mm. This was attributed to small thermal mismatch between the materials and the loadtransfer effect induced by an eccentric bending load over the specimen. As the bending moment was appliedto the specimen, tensile stresses developed near the crack tip and compressive stresses at the left end of thespecimen. The zero stress or neutral point, known as the rotation centre, moved far away from the crack tipsoon after the crack growth initiation [3, p. 355] and located itself on the other side of the front weld interfacebecause of very high yield strengths of weld and maraging steel. This caused premature reduction in the elasticconstraint over the cohesive zone leading to its enhanced length which triggered an early effect of the weld.Experimental values of crack tip stress intensity parameter,�Ktip(Experimental), were determined in stage II

using measured dcdN

values in the Paris law of the form dcdN

= C[�Ktip(Experimental)

]m , where constants C

and m were obtained from stage I data. C, m values of specimen 1 and 2 were 10−9.1, 3.85 and 10−18.1, 8.5,respectively. The variation in Paris constants was due to varying grain sizes in alloy steel of the specimens.�Ktip(Experimental) values are shown in Fig. 5. These values clearly deviated from�Kapplied with the onsetof stage II.

In specimen 1, when the crack tip in stage II was at the front weld interface, i.e., c = 48.5 mm,�Ktip(Experimental) was found to have reduced to 46.06 MPa

√m from the �Kapplied value of 147 MPa√

m because σW � σ A. The dcdN

value at the crack length of 38 mm in stage I was 0.0055 mm/cycle whichdropped to 0.002 mm/cycle at the crack length of 48.5 mm. Likewise in specimen 2, when the crack tip touchedthe front weld interface,�Ktip(Experimental)was found to have reduced to 59.76 MPa

√m from the�Kapplied

value of 147 MPa√

m. The dcdN

value at a crack length of 37.8 mm was 0.0011 mm/cycle which instead of

increasing was found to be 0.001 mm/cycle at a crack length of 48.5 mm. On the other hand, the dcdN

valueat a critical crack length of 42.4 mm in a plain alloy steel specimen was much higher at 0.02 mm/cycle undera �Kapplied or �K A

C value of 82.42 MPa√

m—plane stress condition existing for 10 mm thickness. Referimages of fatigue crack surfaces of bimetallic and plain specimens at Fig. 6. The micrograph of the alloy steelsurface at position B in bimetallic specimens, Fig. 7, when the specimens had not fractured was different fromthe micrograph of the fractured surface of plain alloy steel specimen at position F, Fig. 8, which confirmeddelayed fracture of bimetallic specimens. Therefore, there was a clear evidence of load transfer towards stron-ger materials on the other side of the front weld interface in bimetallic specimens, initially due to eccentricload over the specimens and later on due to material plasticity effects which shielded the crack tip in weakalloy steel, thereby allowing it to grow longer without becoming critical. The tensile nature of residual stresses


Crack length, c (mm)

30

50

70

90

110

130

150

170

30.534.538.542.546.550.554.558.562.5

Series1

Series2

Series3

Alloy steel, A

Maraging steel, B

Series 1 - appliedKΔ

Series 2 - )( alExperimentKtipΔ

Series 3 - )( lTheoreticaKtipΔ

Fracture zone of bimetallic specimen


mMPa

Weld, W

Fracture zone of plain alloy steel

specimen A

CK

wICK

Gain in fatigue life of alloy steel by 11133 cycles

Rear weld interface

Crack growth

Stage II

Type I bimetallic (Specimen No. 1) Fracture toughness

ACK = 82.42 mMPa

BIC

WIC KK ≈ = 100.8 mMPa

Yield strength Average hardnessAlloy steel 480 MPa 205 HV Weld 2400 MPa 435 HV Maraging steel 1750 MPa 317 HV

48.5

30

50

70

90

110

130

150

170

30.534.538.542.546.550.554.558.562.5

Series1

Series2

Series3

ACK

wICK

Gain in fatigue life of alloy steel by 5233 cycles


Series 2 - )( alExperimentKtipΔ

Series 3 - )( lTheoreticaKtipΔ Alloy steel, A

Maraging steel, B

Weld , W

mMPa


Crack growth

Stage I Stage II

Fracture toughness dataA

CK = 82.42 mMPa

BIC

WIC KK ≈ = 100.8 mMPa

Yield strength Average hardnessAlloy steel 490 MPa 209 HV Weld 2315 MPa 420 HV Maraging steel 1760 MPa 319 HV

Type I bimetallic (Specimen no. 2)

48.5

Stage I

Fig. 5 Plots of applied and crack tip stress intensity parameter in Type I bimetallic specimens

Crack length, c (mm) Crack front Weld Weld

Crack front Crack

front

Type I bimetallic (Specimen no. nemicepsleetsyollanialP)1 Type I bimetallic (Specimen no. 2)

Fig. 6 Fatigue crack surfaces of Type I bimetallic and plain alloy steel specimens

in alloy steel did not affect the crack growth rate because �Kapplied did not change. The crack growth rate

could, however, have marginally increased due to an increase in the load ratio, Fmin

Fmax . But the effect of the loadratio is pronounced at threshold values of �Kapplied and not at high values employed in the present work. Assuch, the dip in the crack growth rates was solely due to strength mismatch between alloy steel and the weld.


Rear weld interface

Weld

Fractured surface

Fractured surface

Maraging steel

Micrograph at D

Micrographs of Specimen no. 1

A B

C Alloy steel

Maraging steel

Notch face

Crack front

Fatigue crack growth path Right end of specimen


Rear weld interface

Weld

c=0

c = 30.5 mm

c = 48.5 mm c = 50 mm

Type I bimetallic specimen

Crack front

Notch face Stable fatigue crack growth

Alloy steel

Alloy steel


Alloy steel Weld Weld

Fractured surface

Micrograph at A

Micrograph at C (Fracture initiation)Micrograph at B

D F

Fig. 7 Micrographs of fatigue surfaces of Type I bimetallic specimens


Crack front

Notch face

Stable fatigue crack growth

Alloy steel

Alloy steel


Alloy steel

Weld

Micrograph at A Micrograph at B

Micrographs of Specimen no. 2

Weld

Fractured surface

Rear weld interface

Weld

Fractured surface

Fractured surface

Maraging steel

Micrograph at DMicrograph at C (Fracture initiation)

Fig. 7 continued

Crack front

Stable fatigue crack growth

Alloy steel

Alloy steel

Alloy steel

Fractured surface

Micrograph at F (Fracture initiation)Micrograph at A

Fig. 8 Micrographs of fatigue surface of plain alloy steel specimen

The shielding effect over the crack tip in each bimetallic specimen, quantified by{�Kapplied −�Ktip

(Experimental)}, increased as the crack tip neared the front weld interface. Since the weld was made of mar-aging steel, its plain strain fracture toughness, �K W

IC , for 10 mm thickness of each specimen, was considered


Critical crack near rear weld interface Critical crack at

46 mm length

Fractured Type II bimetallic specimen Fractured maraging steel specimen

Fig. 9 Fractured Type II bimetallic and plain maraging steel specimens

equal to that of back up maraging steel. The value was taken as 100.8 MPa√

m, as discussed in the succeedingSection, ignoring the change in fracture toughness of weld in the specimens due to a slight deviation of weldstrengths. The crack tip after penetrating into the weld faced the rear weld interface of less stronger maragingsteel. This terminated the shielding effect of weld which caused�Ktip(Experimental) to increase and assumethe value of�Kapplied, of the order of 147 MPa

√m.�Ktip(Experimental), being higher than�K W

IC , triggeredweld fracture followed by unstable crack propagation in back up maraging steel. The strength mismatch effectacross the rear weld interface did not come into play as the crack had already become critical before reachingthere.

4.2 Type II specimen

Fractured Type II bimetallic and plain maraging steel specimens are shown in Fig. 9. A crack in bimetallic spec-imen was seen to curve while propagating through maraging steel, A, into ultra-strong weld, W, and becamecritical at the length of 45.85 mm under a �Kapplied value of 96.39 MPa

√m on reaching a near rear weld

interface of alloy steel, B. On the other hand, the crack in the plain maraging steel specimen became critical ata slightly higher length of 46 mm under a�Kapplied or�K A

IC value of 100.8 MPa√

m—plane strain conditionexisting for 10 mm thickness. To confirm this critical observation, additional plain maraging steel specimenswere tested. Cracks failed at lengths even higher than 46 mm in all these specimens. This demonstrated theamplifying effect of approaching a weak alloy steel over the crack tip in ultra-strong weld which caused frac-ture of the bimetallic specimen at lesser crack length when compared to the crack in the plain maraging steelspecimen. The effect of interfaces over the crack tip was highly localized due to the plane strain conditionbecause of high yield strength of maraging steel and weld through which the crack grew resulting in verysmall �r values. Consequently, a sufficient number of data points could not be collected during experimentsto precisely monitor crack growth rates under the influence of approaching interfaces. A failure analysis of thebimetallic specimen is therefore undertaken with the help of the computational model in Sect. 5.

4.3 Crack stability aspects

Two major cases are identified as follows for the study of stability and direction of crack propagation in thespecimens

Case 1: Crack tip away from the front weld interfaceConfigurational parameters (depending upon the material micro-structure and magnitude of external and

body forces) and the type of loading over the crack (e.g., mixed mode) affect the direction of crack propagationto a large extent. The crack while growing in a material follows the path of least resistance to fulfil the principleof maximum dissipation and stability. Refer to Figs. 4 and 9. In Type I and plain alloy steel specimens, thecracks showed negligible unstability and nearly followed a straight line path. However, in Type II specimen,curving of the crack in maraging steel was seen ahead of the notch tip which could have been due to the


material effects or the notch axis not being exactly perpendicular to the applied load. The crack continued tocurve further with the onset of mixed mode loading.

Case 2: Crack tip near the front weld interfaceThe conditions possible other than the stable crack growth in this case are as follows:

i) Crack curving substantially before defecting into interface

T stress in conventional singular crack tip stress field solution, defined as σi j = Ktip√2πa∗ f (θ) + T δ1iδ1 j ,

holds importance. T stress has been determined by Jiang et al. [7, p. 263] using T = Ktip√πa∗

∑(geometry)

where∑

(geometry) is a dimensionless shape factor. The relation between∑

(geometry) and a crack lengthparameter is provided by [15,16]. Cotterell and Rice [17] suggested for a Mode I crack that when T > 0the crack becomes unstable and curves and vice versa. However, these conditions were not found to be validunder high load level in the material combination of mild and stainless steels [7, pp. 264–265]. In the presentwork too, a high driving force or high �Ktip value and small pre-curved lengths of cracks at their junctureof touching the front weld interfaces led to their penetration into the welds in Type I specimens. In a Type IIspecimen also the crack penetrated into the weld again due to high �Ktip levels and the fact that the curvedcrack length was not high enough for a deflection of the crack into the interface. Another condition [18] for

the crack to deflect into the interface is�K A

C�K W

IC 1.

�K AC

�K WIC

≈ 1 supported crack penetration.

ii) Crack branching or bifurcatingPippan et al. [6, pp. 229–233] observed bifurcation or branching of a crack in weak ARMCO iron near

the interface of strong SAE 4340 steel but at low �Ktip values, of the order of 18 MPa√

m and under a smalltoughness value of the interface material. High �Ktip levels in the presence of reasonably strong and toughweld prevented the occurrence of such a phenomenon in both Type I and II specimens.

5 Validation of the computational model

Stage II data was chosen for the validation of the computational model since �Ktip was equal to �Kapplied instage I which did not necessitate any analysis. The minimum fatigue crack length, cmin, required to realize lin-

ear elastic or high cycle fatigue condition at a stage II data point was considered as 2.5[�Ktip(Experimental)

2Y

]2

[19], where the yield strength, Y, corresponded to the parent material containing the crack tip. The actual cracklength ahead of the load axis, cc, given by (c −12.5)mm had to be larger than cmin. Plane stress or plane straincondition was also checked in materials across interfaces at each stage II data point. In the parent material,

fulfilment of conditions t ≤ 2.5[�Ktip(Experimental)

2Y

]2and t > 2.5

[�Ktip(Experimental)

2Y

]2indicated plane

stress and plane strain conditions, respectively, at the selected data point. For a material of yield strength, Y,

on the other side of the interface, fulfilment of conditions t ≤ 2.5[�KL2Y

]2and t > 2.5

[�KL2Y

]2hinted at plane

stress and plane strain conditions in the interface material. The stress intensity parameter over the cohesivezone in the interface material, �KL, differed from �Ktip(Experimental) due to the load transfer effect.

5.1 Type I specimens

Refer to Fig. 5. Liner elastic/high cycle fatigue condition was fulfilled at all stage II data points in both speci-mens 1 and 2. Plane stress condition was nearly realized in alloy steel, A, whereas plane strain condition existedin weld, W. Term σ A supposed to be equal to Y A in plane stress condition was replaced by the effective cohesivestress term, σ A

eff , to account for the premature effect of the weld due to eccentric loading over the specimen.

σ Aeff was obtained from the relation �r = π

8

(�Kapplied

2σ Aeff

)2

, where �r was measured experimentally. Values

of σ Aeff in specimens 1 and 2 were found as 288 and 234.8 MPa, respectively. σW was considered as

√3YW .

The computational model was executed over data points of the crack tip advancing towards the front weldinterface to obtain theoretical values of the crack tip stress intensity parameter,�Ktip(Theoretical). The valueof�Ktip(Theoretical) was found to be slightly higher than�Ktip(Experimental) at the first data point in boththe specimens due to a larger shielding effect by the weld which was not considered in the computationalmodel based on plasticity effects alone. A small correction factor, introduced to make �Ktip(Theoretical)


30

80

130

180

230

280

330

30.534.538.542.546.550.554.558.562.5

Series1

Series2


Series 2 - tipKΔ (Theoretical)

Front weld interface Rear weld interface

Alloy steel, B

Maraging steel, A

Weld, W

X


mMPa

Crack growth

Type II bimetallic specimen Fracture toughnessW

ICA

IC KK ≈ = 100.8 mMPa

BCK = 82.42 mMPa

Yield strength Average hardnessMaraging steel 1775 MPa 322 HV Weld 2418 MPa 438.5 HV Alloy steel 498 MPa 213 HV

30

80

130

180

230

280

330

43.54444.54545.546

Series1

Series2

Fracture zone of bimetallic specimen

Magnified view at X

mMPa

Front weld interface Rear weld interface

wIC

AIC KK ≈

Fracture zone of plain maraging steel specimen

Weld, W


44.35 45.85 45.8


Series 2 - tipKΔ (Theoretical)

44.28

Stage I

Stage II

Stage I

Stage II

Fig. 10 Plots of applied and crack tip stress intensity parameter in Type II bimetallic specimen

equal to �Ktip(Experimental) at the first data point, was used throughout the computation. Computationrevealed that the maximum extent of spread of the cohesive zone was restricted to the weld without crossingover into maraging steel. The values of l at crack lengths of 48.5 mm were found to be 0.12 and 0.13 mmin specimens 1 and 2 which were less than the weld width of 1.5 mm. The values of �Ktip(Theoretical)were in good agreement with those of �Ktip(Experimental) which validated the accuracy of the compu-tational model. The percent difference between �Ktip(Theoretical) and �Ktip(Experimental), defined by

e = mod|�Ktip(Theoretical)−�Ktip(Experimental)|�Ktip(Experimental) × 100, was computed at each stage II data point. For n

number of data points in the specimen, the average % difference, eav, was determined by eav = 1n

∑n1 (e) j .

eav was obtained as 16.75 and 12.38 in specimens 1 and 2, respectively.

5.2 Type II specimen

Refer to Fig. 10. Stage II existed at two instances in the specimen. First, when the crack tip in maraging steelwas near the front weld interface and secondly when the crack tip in the weld was near the rear weld inter-face. The linear elastic/high cycle fatigue condition was fulfilled at both first and second stage II data points.Plane strain condition existed in maraging steel, A, and weld, W, at first stage II data points, whereas planestrain and plane stress conditions were fulfilled in weld and alloy steel, B, respectively, at second stage II datapoints. Therefore, the relations σ A = √

3 Y A, σW = √3 Y W and σ B = Y B were used in the computational


model which was executed separately over the cases of the crack tip near front and rear weld interfaces. Theultra-strong weld began to influence the approaching crack tip in comparatively weaker parent maraging steelat a crack length of 44.28 mm. Till then, the crack tip was in stage I as distance of the crack tip from frontweld interface, a, which was more than the cohesive zone length, �r . Both a and �r were 0.07 mm at thecrack length of 44.28 mm. Beyond the crack length of 44.28 mm, the condition a < �r was fulfilled whichimplied stage II or development of a cohesive zone into weld causing the crack tip to experience the shieldingeffect of weld. �Ktip(Theoretical) dropped below �Kapplied because σW > σ A. The effect continued till thecrack tip touched the front weld interface at a crack length of 44.35 mm. On crossing the front weld interface,the shielding effect over the crack tip was over as the tip now in parent ultra-strong weld faced the rear weldinterface of weak alloy steel, B. During a crack length of 44.35–45.8 mm, the crack tip was in stage I and didnot feel the effect of approaching alloy steel. a and�r were equal to 0.05 mm at a crack length of 45.8 mm. Thecrack grew under amplifying effect of alloy steel, in stage II, at a length beyond 45.8 mm. The effect caused�Ktip(Theoretical) to increase above�Kapplied because σ B σW . The position of the crack when it becamecritical was between 45.8 and 45.85 mm. The exact location could not be identified due to sudden failure of thespecimen. For a hypothetical case of a crack at the rear weld interface (crack length of 45.85 mm), when theamplifying effect on the crack tip could have been maximum,�Kapplied and corresponding�Ktip(Theoretical)obtained from the computational model were 96.39 MPa

√m and 280 MPa

√m, respectively, with the exten-

sion of the cohesive zone in alloy steel, l, being 3.7 mm. �Ktip(Theoretical) clearly exceeded the fracturetoughness of weld and back up alloy steel resulting in specimen fracture.

6 Conclusions

Conclusive evidence about shielding or an amplifying effect exerted by elastically matched but strength-mis-matched interfaces over a nearby crack tip is obtained from the work presented in the paper. Fatigue testsconducted over bimetallic compact tension specimens fabricated by welding weak ASTM 4340 alloy andstrong MDN 250 maraging steels confirm the crack tip experiencing a shielding effect when it is near theinterface of stronger steel and an amplifying effect when it is in the vicinity of the interface of weaker steel,both the effects influencing �Ktip. There is dip in �Ktip from �Kapplied with consequent drop in dc

dNin the

case of crack tip shielding. Trends reverse during crack tip amplification. As a result, fatigue life of a bime-tallic specimen differs from that of plain specimen. Micrographic examinations of a fatigue crack surfacesconvincingly support the findings. Arrest of a fatigue crack in weak parent steel near the interface of strongersteel is possible if �Ktip drops to the value below the fatigue threshold of the parent steel.

Life of a bimetallic composite can be predicted if�Ktip is known. A computational model for this purpose isreviewed and validated by experiments. Values of�Ktip from the computational model are in good agreementwith experimental values. A marginal difference between the two is attributed to i) modelling assumptions,e.g., use of Dugdale’s model which allows constant cohesive stresses, whereas in reality the cohesive stressesare non-uniform [20] and decrease as the crack tip opening displacement increases upon application of loadii) experimental errors in the form of inconsistency in material properties which could have affected crackgrowth rates and therefore the accuracy of Paris constants. Despite such errors, the trends and inferences arewell substantiated.

Acknowledgments Support received from the School of Mechanical and Building Sciences, V.I.T., Vellore, India, during thecourse of this work is gratefully acknowledged.

Appendix

Section I

The relation given by Kolosov–Muskhelishvili [21] between the displacements, u and v, in the cohesive zone is

2μ(u + iv) = κϕ(z)− zϕ′(z)− ψ ′(z). (A.1)

Differentiating Eq. (A.1) results in

2μ

(∂u

∂x+ i

∂v

∂x

)= κϕ′(z)− zϕ′′(z)− ϕ′(z)− ψ ′′(z). (A.2)


On substituting the relation by Westergaard [22], ψ ′′(z) = −z ϕ′′(z), in Eq. (A.2), one obtains

2μ

(∂u

∂x+ i

∂v

∂x

)= κϕ′(z)− (z − z)ϕ′′(z)− ϕ′(z) (A.3)

or

2μ

(∂u

∂x+ i

∂v

∂x

)= κϕ′(z)− 2iy ϕ′′(z)− ϕ′(z). (A.4)

On replacing ϕ′(z) by φ(z) and ϕ′′(z) by φ′(z) in Eq. (A.4), one obtains

2μ

(∂u

∂x+ i

∂v

∂x

)= κφ(z)− 2iyφ′(z)− φ(z). (A.5)

Overall, the conjugate of Eq. (A.5) gives

2μ

(∂u

∂x− i

∂v

∂x

)= κφ(z)− 2iyφ′(z)− φ(z). (A.6)

On subtracting Eqs. (A.5) and (A.6) and applying the limit y → 0 on the line ahead of the crack, one obtains

4μi∂v

∂x= lim

y→0

[κφ(z)− κφ(z)+ φ(z)− φ(z)

](A.7)

or

4μi∂v

∂x= lim

y→0

[(κ + 1)

{φ(z)− φ(z)

}]. (A.8)

On using κ = 3−ν1+ν and μ = E

[2(1+ν)] in Eq. (A.8), one obtains the expression for the displacement, v, in planestress condition as

∂v

∂x= 2

E ilimy→0

[φ(z)− φ(z)

]. (A.9)

Similarly on using κ = 3 − 4ν and μ = E[2(1+ν)] in Eq. (A.8), the following expression is obtained in plane

strain condition:

∂v

∂x= 2

(1 − ν2

)

E ilimy→0

[φ(z)− φ(z)

]. (A.10)

Section II

Sedov [23] defined the general solution of a semi-centre crack of length 2c or semi-edge crack of length c inan infinite homogenous parent body, A, under the sole action of far field applied stress, p∞, of the form

φ1(z) = 1

2π√

z2 − c2

z∫

−z

p∞√

z2 − ξ2

z − ξdξ − p∞

4(A.11)

or

φ1(z) = 1

2π√

z2 − c2

z∫

−z

p∞√

z + ξ√z − ξ

dξ − p∞4. (A.12)

Using Green’s Equation,z∫

−z

p∞√

z+ξ√z−ξ dξ = p∞(π z), Eq. (A.12) is written as

φ1(z) = 1

2π√

z2 − c2p∞(π z)− p∞

4. (A.13)


To study the behaviour close to the tip of the crack, the limit z → c is introduced to obtain

φ1(z) = 1

2π√(2c)(z − c)

p∞(πc)− p∞4. (A.14)

Using Kapplied = p∞√πc for an infinite body, Eq. (A.14) changes to

φ1(z) = Kapplied

2√

2π(z − c)− p∞

4. (A.15)

On differentiating Eq. (A.13) after applying the limit z → c, one obtains

φ′1(z) = − p∞c2

2(z2 − c2

) 32

. (A.16)

Another potential is written as

′1(z) = p∞

2− zφ′

1(z). (A.17)

On using Eq. (A.16) in (A.17), one obtains

′1(z) = p∞

2+ p∞c2z

2(z2 − c2

) 32

. (A.18)

The potentials φ1(z) and ′1(z) must satisfy the following boundary conditions:

Set I: At far field, where z → ∞, σy = p∞; σx = 0; τxy = 0,

Set II: At crack axis where y = 0 and x ≤ 0, σy = 0; τxy = 0.

The following stress field equations [21, p. 114] are used to verify whether the selected potentials satisfy theboundary conditions or not:

σx + σy = 4Re [φ1(z)] , (A.19)

σy − σx + 2iτxy = 2[z̄φ′

1(z)+ ′1(z)

]. (A.20)

On using Eqs. (A.13), (A.19) takes the form

σx + σy = 4Re

[p∞z

2√

z2 − c2− p∞

4

]. (A.21)

On using Eqs. (A.16), (A.18) in Eq. (A.20), one obtains

σy − σx + 2iτxy = 2

⎡⎣− z̄ p∞c2

2(z2 − c2

) 32

+ p∞2

+ p∞c2z

2(z2 − c2

) 32

⎤⎦ . (A.22)

At far field where z → ∞, Eqs. (A.21) and (A.22) reduce to

σx + σy = 4[ p∞

2− p∞

4

]= p∞ (A.23)

and

σy − σx + 2iτxy = p∞. (A.24)

Equation (A.24) implies

σy − σx = p∞ and τxy = 0. (A.25)

The solution of Eqs. (A.23) and (A.25) yields


σy = p∞ and σx = 0. Set I of boundary conditions is fulfilled.At y = 0 and at x ≤ 0, Eq. (A.21) changes to

σx + σy = 4Re[ p∞

2(imag)− p∞

4

]

or σx + σy = −p∞. (A.26)

Similarly, at y = 0 and at x ≤ 0, Eq. (A.22) changes to

σy − σx + 2iτxy = p∞. (A.27)

Equation (A.27) implies

σy − σx = p∞ and τxy = 0. (A.28)

On solving Eqs. (A.26) and (A.28), one obtains σy = 0; τxy = 0. Set II of boundary conditions is fulfilled.As the cohesive zone of size r alone is being modelled in the present case, the potential, φ1(z), is written as

φ1(z) ={

Kapplied

2√

2π(z − r)− p∞

4

}, (A.29)

because the same Kapplied shall act over the small cohesive zone as well.

Section III

The potential, φ2(z), due to cohesive stresses, σ A, acting alone in −ve y direction over the semi-cohesive zoneof length r in the parent body, A, is written as

φ2(z) = 1

2π i√

z2 − r2

r∫

0

σ A√ξ2 − r2


4. (A.30)

As z and ξ → r , Eq. (A.30) can be written as

φ2(z) = 1

2π i√

z − r

r∫

0

σ A√ξ − r


4(A.31)

or

φ2(z) = −i

2π√

r − z

r∫

0

σ A√r − ξ


4for ξ and z < r. (A.32)

Another potential is written as

′2(z) = −σ

A

2− zφ′

2(z). (A.33)

The potentials must satisfy the following boundary conditions:

Set III: At y = 0 and 0 < x < r in the cohesive zone, σy = −σ A; σx = 0; τxy = 0

Using Kolosov–Mushkelishvili’s equations, one obtains

σx + σy = 4Re [φ2(z)] = −σ A (A.34)

and

σy − σx + 2iτxy = 2[z φ′

2(z)+ ′2(z)

] = 2

[(z − z)φ′

2(z)− σ A

2

]. (A.35)

At y = 0, Eq. (A.35) reduces to

σy − σx = −σ A and τxy = 0. (A.36)

The solution of Eqs. (A.34) and (A.36) results in σy = −σ A; σx = 0; τxy = 0. Set III of boundary conditionsis fulfilled.


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Documents

A computational model and experimental validation of shielding and amplifying effects at a crack tip near perpendicular strength-mismatched interfaces