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

Sunil Bhat · S. Narayanan

A computational model and experimental validation of shielding and amplifying effects at a crack tip near perpendicular 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 interface body in a bimetallic system is influenced when the crack tip yield or cohesive zone spreads to the interface body. The concept of crack tip stress intensity parameter, K tip , is therefore employed in fracture analysis of the bimetallic body. A computational model to determine K tip is reviewed in this paper. The model, based upon i) Westergaard’s complex potentials coupled with Kolosov–Muskhelishvili’s relations between a crack tip stress field and complex potentials and ii) Dugdale’s representation of the cohesive zone clearly indicates shielding or amplifying effects of strength mismatch across the interface, depending upon the direction of the strength gradient, over the crack tip. The model is successfully validated by conducting series of high cycle fatigue tests over Mode I cracks advancing towards various strength-mismatched interfaces in bimetallic compact tension specimens prepared by electron beam welding of elastically identical weak ASTM 4340 alloy and strong MDN 250 maraging steels. List of symbols a Distance of crack tip from interface a∗ Radial coordinate of point near crack tip A Parent body/parent steel containing crack B Interface body/back up steel b Length of cohesive zone across interface c Crack length cc Crack length ahead of load axis cmin Crack length required for linear elastic regime C Paris constant e, eav Percent difference between theoretical and experimental result, average of percent differences E Modulus of elasticity f (θ ) A function of angle w.r.t. crack axis F Applied load √ i Imaginary quantity, −1 S. Bhat (B) · S. Narayanan School of Mechanical and Building Sciences, Vellore Institute of Technology, Vellore, TN 632014, India E-mail: [email protected] S. Narayanan E-mail: [email protected]

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K applied KC K C(bimetallic) K IC KL K tip l L m n N p∞ r t T u v W Y z Zf Zr  κ μ ν ξ σ σeff σi j σx σy δi j τx y ϕ, φ, φ1 , φ2 , ψ, 1 , 2

Applied stress intensity parameter Plane stress fracture toughness of homogenous body Plane stress fracture toughness of bimetallic body Plane strain fracture toughness of homogeneous body Stress intensity parameter over cohesive zone in interface material Stress intensity parameter at crack tip Extension of cohesive zone into interface body Distance between load axis and left end of specimen Paris constant No. of data points Number of fatigue cycles Far field applied stress Cohesive zone length in homogeneous parent body Specimen thickness T stress Displacement in x direction in cohesive zone Displacement in y direction from crack axis in cohesive zone Weld Yield strength Complex variable Distance from specimen right end to front weld interface Distance from specimen right end to rear weld interface Parameter under cyclic or fatigue load A material constant Shear modulus Poisson’s ratio A variable Cohesive stress Effective cohesive stress Crack tip stress field Stress in x direction Stress in y direction Kronecker delta Shear stress in xy plane Complex potentials

Superscripts A B max min W *

Parent body/parent steel Interface body/back up steel Maximum value Minimum value Weld Value at fracture

1 Introduction Class A composites, i.e. fibre-reinforced plastics (FRPs) and homogeneous metal resin (MMCs) are characterized by significant elastic mismatch between the constituents and poor strength and plasticity of at least one of the constituents. For instance, the ratio of shear modulus between carbon and epoxy, boron and epoxy and aluminium 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 of 1–6%. Fibres too exhibit a poor percent elongation. Fracture aspects of such composites are therefore mainly influenced by material elastic properties. A crack tip on touching the interface of the constituents in these composites experiences a sharp jump or discontinuity in stresses in the direction of the load to fulfil the condition of continuity of displacement and strain across the interface. Consequently, the stress intensity parameter at a

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261

crack tip also exhibits a change. It increases sharply when the crack tip advancing from the compliant material side 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 new separate class of non-homogeneous, externally welded metal–metal composites are marked by marginal or nil elastic but appreciable strength mismatch between the constituents coupled with good plasticity of at least one of them. For example, the ratio of shear modulus between steel and aluminum, steel and titanium, steel and copper, boron and aluminum, glass and aluminum is as low as 3, 2, 1.6, 4.5 and 1, respectively. But percent elongation of the stated metals and their alloys is high, which ranges from 25–45%. Strength mismatch between them is also significant. As a result, material strength and plasticity properties primarily influence the fracture characteristics of these composites. With the advent of solid state and fusion welding processes like diffusion bonding, explosive cladding, friction welding, electron and laser beam welding, etc., which are capable of joining dissimilar metals with relatively clean and strong weld, non-homogeneous bimetallic composites have been fabricated and successfully tested as functionally graded, material optimized units with life-enhanced features in applications pertaining to nuclear pressure vessels, oil pipelines, aerospace, gas turbine discs, tubes and barrels, etc. Efficient coating technologies like physical and chemical vapour deposition, etc., have also helped in producing strong and wear resistant films and coatings over substrates of different materials. In view of the above, the study of fatigue and fracture aspects of bimetallic composites gets importance especially in load applications when the crack is near or at the interface of the constituents because failure of the interface renders the composite ineffective by degrading its functional capabilities. Since two steels are easy to weld, substantial work has been reported over the fracture characteristics of class 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 ones in chronological order who have successfully investigated a Mode I crack near the interface of strength-mismatched steels or other plastically mismatched materials. All of them in one form or another have reported shielding or amplifying effects of strength mismatch across the interface over the crack tip—a shielding effect when the crack tip in the weaker material approaches the interface of the stronger material and an amplifying effect when the crack tip in the stronger material approaches the interface of the weaker material. Their work was, 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 provided the solution for crack tip opening displacement and the latter the solution for a J integral at the crack tip. It is seen from the literature survey that sufficient theoretical solutions for life prediction of bimetallic composites are either not available or are not fully validated by experiments. The work reported in this paper therefore first reviews a computational model to obtain K tip in monotonic and fatigue load regimes, since K tip adequately defines the state at the crack tip near the bimetallic interface, and then validates the model with the help of the results obtained from series of high cycle fatigue tests conducted over Mode I cracks in bimetallic compact tension specimens prepared by electron beam welding of weak ASTM 4340 alloy and strong MDN 250 maraging steels. Thick ultra-strong weld between elastically identical steels results in two strength-mismatched interfaces, viz., front weld interface between cracked parent steel and weld and the rear or 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 up by maraging steel, whereas in Type II specimen, the crack in the parent maraging steel grows perpendicularly towards the weld backed up by alloy steel. Quantification by a computational model about the magnitude of effects 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 plastic or yield zone across the interface is modelled in this work as Dugdale’s cohesive zone [12] for elastic-ideally plastic material which is then analysed with the help of basic principles of Westergaard’s complex potentials and Kolosov–Muskhelishvili’s relations between a crack tip stress field and complex potentials. The mechanical behaviour of Dugdale’s zone is characterised by constant closing cohesive stresses acting over it. These stresses are generated due to an elastic constraint exerted by surrounding non-yielded material over the cohesive zone and are considered as the function of material yield strength to fulfil the condition of zero stress singularity at the tip of the cohesive zone. They are equal to the material yield strength in a plane stress condition and are √ approximated as 3 times the material yield strength in the plane strain case.

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y,v

x,u

Axis (y=0) At Tip

(x = 0)

Far field applied stress, p∞

p∞

σ

σB

Interface body, B

A



Crack

a

l

Interface

σA

Parent body, A

Axis

c

Tip

Cohesive zone r

b

Crack

c

(a)

(b)

Stage I (Cohesive zone in parent body)

Stage II (Extension of cohesive zone into interface body)

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 an elastically identical but different strength interface body, B, in stages I and II. In stage I, Fig. 1a, the crack tip is far away from the interface body and the entire cohesive zone of size r is contained in the parent body. r by Dugdale’s criterion under the action of applied stress intensity parameter, K applied , and cohesive stress, σ A , is   K applied 2 given by π8 . Refer to Section I of the Appendix. For a crack of length c with its cohesive zone yet A σ 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 stress condition,   ∂v 2 = lim φ(z) − φ(z) . ∂x Ei y→0

(1)

The solution of Eq. (1) requires a complex potential, φ(z). The potentials i) φ1 (z) which considers the effect of monotonic far field applied stress, p∞ , and K applied 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. The potential, φ1 (z), is derived as  φ1 (z) =

 K applied p∞ − . √ 4 2 2π(z − r )

(2)

Refer to Section III of the Appendix. The potential, φ2 (z), is obtained as −i φ2 (z) = √ 2π r − z

r 0

√ σA σA r −ξ dξ − . ξ −z 4

(3)

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263

The final potential is written by the principle of superposition as φ(z) = φ1 (z) + φ2 (z). Therefore, r A √ K applied p∞ i σA σ r −ξ − − dξ − , φ(z) = √ √ 4 ξ −z 4 2π r − z 2 2π(z − r ) 0

K applied p∞ i lim φ(z) = √ − − √ y→0 4 2π r − x 2i 2π(r − x)

r 0

√ σA σA r −ξ dξ − ξ −x 4

(4)

(5)

and K applied p∞ i − + lim φ(z) = − √ √ y→0 4 2π r − x 2i 2π(r − x)

r 0

√ σA σA r −ξ dξ − . ξ −x 4

Substitution of Eqs. (5) and (6) in Eq. (1) results in ⎧ ⎫ r A √ 1 2 ⎨ K applied ∂v σ r −ξ ⎬ + √ =− dξ . √ ⎭ ∂x E ⎩ 2π(r − x) π r − x ξ −x

(6)

(7)

0

On integrating Eq. (7), the expression for v as the function of distance x from the crack tip is obtained as ⎡ x ⎤  x r A √ K applied dx σ r −ξ ⎦ 2 ⎣ v(x) = − dx + dξ . (8) √ √ E ξ −x π r −x 2π(r − x) 0

0

0

Stage I is valid till the distance of the crack tip from the interface, a, fulfills the condition, a ≥ r . The effect of 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 of the interface with its cohesive zone having spread to the interface body, B, by distance l such that a < r which marks the beginning of the effect of the interface body or the onset of stage II. The length of the cohesive zone across the interface, b, is (a + l) which is different from r. Since E A = E B = E the following expression is written for v(x) under the simultaneous effect of cohesive stresses σ A and σ B over cohesive volumes in parent and interface bodies, respectively, with the help of Eq. (8), ⎡ x ⎤  x a A √ x b B √ K applied 2 ⎣ dx σ b−ξ dx σ b−ξ ⎦ v(x) = − dx + dξ + dξ . √ √ √ E ξ −x ξ −x 2π(b − x) π b−x π b−x 0

0

0

0

a

(9) Equation (9) involves singular integrals. Their evaluation by Ukadgaonker et al. [13] results in ⎧   √  ⎫ √ √ (a+l)−√(a+l−x) 2 σA ⎪ ⎪ √ ⎪ ⎪ K x ln − 2 (a + l − x) + (a + l)(a + l − x) applied π ⎪ ⎪ π (a+l)+ (a+l−x) ⎬ ⎨ 2       √  √   . (10) v(x) = √ √  (a+l−x)− l  √ ⎪ E⎪ σ B −σ A ⎪ ⎪ √(a+l−x)+√l  + x ln √ √ a ln − 2 + l(a + l − x) ⎪ ⎪   ⎭ ⎩ π (a+l−x)+ l  (a+l−x)− l 

Solutions of v(x) in Eqs. (8) and (10) indicate that the entire effect of K applied is felt by the crack tip in stage I but not in stage II due to plasticity-induced load transfer either towards the parent or interface body depending upon the direction of the strength gradient parameter, σ B − σ A , across the interface. To maintain continuity of displacements and strain, a larger portion of applied energy is consumed towards cohesive plasticity in the interface body if the interface body is stronger than the parent body, thereby inducing a shielding effect at the crack tip. Conversely, an additional driving force or amplifying effect is imparted at the crack tip in case the parent body is stronger than the interface body. Consequently, it is inferred that the stress state at the crack tip in the stage II of a bimetallic body differs from that at a crack tip in the homogeneous parent body alone. As such, the concept of a stress intensity parameter at the crack tip, K tip , is introduced which necessitates v(x) to be stated in terms of K tip instead of K applied . Using v(0) = crack tip (x = 0) is written as

2 K tip

2Eσ A

in plane stress condition, Eq. (10) at the

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S. Bhat, S. Narayanan

2 K tip

4σ A

=

⎧ ⎨ ⎩

 K applied

⎤⎫ √ ⎞ √ ⎬  ⎣a ln ⎝  (a + l) + √l  ⎠ − 2 l(a + l)⎦ . √   ⎭  (a + l) − l  ⎡

σA

2 (a + l) − [2(a + l)] + π π

σB

−σA π



(11) K 2 1−ν

! 2

tip E for plane strain condition in Eq. (10). Equation (11) holds good E is replaced by 1−ν 2 and v(0) by 2Eσ A in plane strain condition as well, only the values of σ A and σ B being different from those employed under plane stress condition. The difference between K tip and K applied depends upon the mismatch between σ A and σ B and the value of a. At fracture, Eq. (10) changes to ⎧   √ ∗ √ ∗  ⎫ √ A a+l −√a+l −x 2 ⎪ ∗ − x) + σ ∗ ) (a + l ∗ − x) ⎪ √ ⎪ ⎪ K x ln − 2 + l + l (a (a ⎪ ⎪ π a+l ∗ + a+l ∗ −x ⎬ ⎨ C(bimetallic) π 2      √  ∗ √   . v (x) = √ √ ∗ ∗  (a+l −x)− l  √∗ ∗ ∗ ⎪ E⎪ σ B −σ A ∗ − x) ⎪ ⎪ √(a+l −x)+√l  + x ln √ √ a ln − 2 + l l + (a ⎪ ⎪  ⎭ ⎩ π ∗ ∗ (a+l ∗ −x)+ l ∗

 (a+l −x)− l 

(12) Since K tip = K CA and v ∗ (0) =

!2 K CA at 2Eσ A

fracture, Eq. (12) takes the form

"  !2 !$ K CA σA # 2 ∗) − = K + l 2 a + l∗ (a C(bimetallic) A 4σ π π ⎡ ⎛ ⎤⎫ √ ⎞ √ ⎬ ∗) + l ∗  σB −σA ⎣ + l (a ∗ (a + l ∗ )⎦ . ⎠  a ln ⎝ √ + l − 2 √   ⎭ π  (a + l ∗ ) − l ∗ 

(13)

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

!2 K CA , 2Eσ A

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

fracture toughness of the bimetallic body, K C(bimetallic) , differs from the fracture toughness of the parent body, K CA . 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 K tip  = K applied . It is evident from Eqs. (11) and (13) that for a particular value of a, K tip < K applied , K C(bimetallic) > K CA when σ B > σ A , Jinterface being +ve. Similarly, K tip > K applied , K C(bimetallic) < K CA when σ B < σ A , Jinterface  being −ve. Also, the higher the magni σ B − σ A , the more is the value of  K applied − K tip  or tude of strength mismatch represented by parameter    K C(bimetallic) − K A . The trend continues with increasing intensity as a reduces with crack growth till the C crack 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 around the interface may not exceed the yield strength of the interface body. However, load transfer to the interface body still continues elastically due to its higher yield limit than the weak parent body. The high elastic strain zone in the interface body subjected to stresses less than its yield strength is replaced by the much smaller cohesive zone subjected to higher cohesive stresses for the realization of similar effects upon application of the stated theoretical model.

2.1 Validation of the solution Solutions of stage I and stage II are checked as follows: Stage I  % x K applied 2 The solution of integral − 0 √2π(r d x in Eq. (8) is K applied π (r − x). The lower limit is disregarded −x) since the value of K applied at the upper limit acts over the specific location in the cohesive zone. The singular

A computational model and experimental validation

265

 √  √ %x % r A √r −ξ A −√r −x + (r − x) integral − 0 π √drx−x 0 σ ξ −x dξ is evaluated [13, p. 703] as − σπ (r − ξ ) ln √rr −ξ −ξ + r −x √       r √ √ √ √ √ √ √ r − r −x r + r −x +√r −x σA √ √ √ √ ln √rr −ξ − 2 r ln + (r − x) ln r − ξ r − x which simplifies to − π −ξ − r −x r + r −x  r − r −x 0   $ √ √ 2 2r 2σ A r −2 r r − x . v(x) at x = 0, i.e., v(0) is obtained as E K applied π − π . On applying Dugdale’s   K applied 2 (K applied )2 , v(0) reduces to 2Eσ which is the well-known solution for the half crack tip criterion, r = π8 A σA   K applied 2 opening displacement in a homogenous body A in the linear elastic regime. Further, a = r = π8 A σ and l = 0 at the end of stage I which when substituted in Eq. (11) results in K tip = K applied which is true.   K C(bimetallic) 2 Similarly, substitution of a = r ∗ = π8 and l = 0 in Eq. (13) results in K CA = K C(bimetallic) which A σ also holds good in stage I. Stage II ! K2 Jtip , expressed as Etip , is equal to 2σ A v(0), whereas Jinterface is equal to 2 σ B − σ A v(a). Since Japplied is given by

2 K applied E ,

the following equation is obtained: 2 K applied

E

  = 2σ A v(0) + 2 σ B − σ A v(a).

(14)

Equation (10) leads to the following expressions: & "  √  √   σA σB −σA 2 2 (a + l) + l K applied (a + l) − a ln √ v(0) = [2(a + l)] + √ − 2 l(a + l) E π π π (a + l) − l and 2 v(a) = E

"

 K applied

  √ & √   σB −σA σA 2 (a + l) − l l+ a ln √ (2l) . √ − 2 (a + l)(l) − π π π (a + l) + l

The applied stress intensity parameter [11, p. 404], is written as    2l  A 2(a + l) B A +2 σ −σ . K applied = 2σ π π ' !2 !2 l ! 8 σ A (a+l) R.H.S. of Eq. (14) results in the expression E1 +8 σB −σA π π + which equals

2 K applied E

(15) 16σ A σ B −σ A π

!

√ l(a + l)

(

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 cyclic values, i.e., r, v(x), K applied and K tip by r, v(x), K applied and K tip , respectively. The cohesive stresses σ A and σ B are replaced by 2σ A and 2σ B , respectively. K tip = K applied in stage I. In stage II, Eqs. (11) and (15) change to the following form: "  2 K tip 2σ A 2 = K (a + l) − [2(a + l)] applied 8σ A π π & √ ! √   2 σB −σA (a + l) + l (16) a ln √ + √ − 2 l(a + l) , π (a + l) − l    2l  A 2(a + l) B A K applied = 4σ +4 σ −σ . (17) π π

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S. Bhat, S. Narayanan

Parent body

Interface body

F 16

Notch tip (crack)

Notch

60

6.25

a

c

Load axis

Interface 30 Hole dia. =12.5

F Dimensions in mm (Figure not to scale)

L = 50

12.5

Fig. 2 A standard bimetallic compact tension specimen

A computer programme solves Eq. (17) by a numerical iterative convergence scheme. Known input values to the code are σ A , σ B , K applied and a. A value of l is initially assumed. In each iteration, the value of l is suitably incremented or decremented till the actual K applied value and the one obtained from the equation converge. With output value of l, K tip is obtained from Eq. (16). The value of a is reduced in every new set of computation to estimate the change in the effect of approaching the interface body over the advancing crack. Refer to Fig. 2. When a standard bimetallic compact tension specimen comprising elastically matched bodies is subjected to a tension–tension cycle carrying maximum load, F max , and minimum load, F min , K applied [14] is written as        c − 12.5 0.5 c − 12.5 1.5 c − 12.5 2.5 F max − F min 29.6 − 185.5 + 655.7 K applied = L L L t L 0.5   3.5 4.5  c − 12.5 c − 12.5 . (18) −1017.0 + 639.8 L L K applied is found at various positions of growing fatigue crack. K tip is subsequently determined at each position. 3 Experimental work Two Type I (numbered 1 and 2) and one Type II bimetallic compact tension specimens were fabricated for experimental work. The specimens conformed to the overall dimensions of Fig. 2. Weak ASTM 4340 alloy steel and strong MDN 250 maraging steel plates with lateral thickness, t, of 10 mm were used. The yield strength of alloy and maraging steel ranged from 460 to 500 MPa and 1,750 to 1,800 MPa, respectively. The steels were chosen because of their high strength mismatch in order to generate a pronounced effect over the crack tip. The plates were joined by electron beam welding resulting in 1.5- mm-wide ultra-strong weld. No filler material was 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. Nearly identical coefficients of the thermal expansion of steels also minimized the chances of development of residual stresses in them during welding. All these factors made the conditions conducive for the intended examination of 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 tip at the notch was obtained by cutting with a wire of 0.3 mm size. A welded and machined bimetallic specimen is displayed in Fig. 3. The front weld interface was at the distance of 48.5 and 44.35 mm from the right or notch end of Type I and Type II specimens, respectively. The vickers hardness at different positions ahead of the notch tip on top and bottom faces of the specimens was measured, the average of which provided the local material yield strength value for use in the computational model. Conversion tables between Vickers hardness and ultimate tensile strength for steels were referred. Yield strengths of alloy and maraging steels

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267

Load, F F max Load cycle

F min Back up steel, B

Time

Parent steel, A t =10

Weld, W Rear Front weld weld interface interface

Specimen type Type I bimetallic

Load cycle details Zf ,

Zr

48.5 mm , 50 mm

F max

F min

14.7 kN

0.98 kN

Frequency 20 cps

Notch Crack

Left end

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

30.5 1.5

Right end

zf zr

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

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 the weld comprised maraging steel as confirmed by its micro-structural examination, its yield strength in each specimen was extrapolated from the yield strength of maraging steel. Plain specimens of alloy and maraging steels with similar geometry and notch configuration were also prepared to compare their results with those of bimetallic specimens and to evaluate the effect of interfaces. All the specimens were subjected to tension–tension fatigue cycles of constant amplitude at ambient environment in a ±250 kN capacity fatigue test rig till they fractured. A Mode I crack was generated at the notch tip in each specimen. Details of the load cycles are provided at Fig. 3. It was ensured with selected load values that 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 right end of the specimen. The number of cycles, N, required for incremental crack growth were noted. The crack growth rate, ddNc , was computed. The number of cycles applied till specimen fracture was also recorded. Various metallurgical investigations were undertaken on fractured specimens. The microstructure of alloy steel, weld and maraging steel was examined by an optical microscope. Miniature polished and etched samples were prepared for this purpose. Non-uniform grains of ferrite and pearlite were observed in alloy steel. Maraging steel and weld displayed a similar low carbon iron–nickel martensitic structure. Fatigue crack surfaces were examined by scanning electron microscope to confirm the location of critical cracks. The residual stresses in alloy steel, developed during manufacture of plates followed by their welding with maraging steel, were measured by X-ray diffraction equipment in y direction, i.e., perpendicular to the crack surface at different positions 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 K applied vs. length, c, of the crack in parent alloy steel, A, advancing towards weld, W, are at Fig. 5. The crack remained stable up to longer lengths in bimetallic specimens sustaining an increased K applied value when compared to the crack in a plain alloy steel specimen which hinted at enhanced fatigue life of bimetallic specimens. The bimetallic specimens failed soon after the crack crossed the front weld interface, at a length just exceeding 48.5 mm, whereas a plain alloy steel specimen failed earlier at crack length of 42.4 mm. Specimens 1 and 2 required 29,442 and 23,542 cycles, respectively, for fracture which were more than 18,309 cycles consumed by

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S. Bhat, S. Narayanan

Fractured Type I bimetallic specimen

Critical crack on crossing front weld interface

Fractured plain alloy steel specimen

Critical crack at 42.4 mm length

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

a plain specimen. As expected, K applied and ddNc values in each bimetallic specimen increased with the crack growth in stage I. A sudden drop in ddNc marked termination of stage I and the beginning of stage II or shielding effect of weld over the crack tip. Ideally, in a specimen subjected to uniform symmetrical load, stage II begins   K applied 2 when the crack tip is at distance, r , from the front weld interface where r is equal to π8 . The 2σ A estimated r under such condition was 4.07 and 3.98 mm at crack lengths at 44.43 and 44.52 mm in specimens 1 and 2, respectively. But it was noticed during experiments that stage II commenced earlier at crack lengths of 38.0 and 37.80 mm. This was attributed to small thermal mismatch between the materials and the load transfer effect induced by an eccentric bending load over the specimen. As the bending moment was applied to the specimen, tensile stresses developed near the crack tip and compressive stresses at the left end of the specimen. The zero stress or neutral point, known as the rotation centre, moved far away from the crack tip soon after the crack growth initiation [3, p. 355] and located itself on the other side of the front weld interface because of very high yield strengths of weld and maraging steel. This caused premature reduction in the elastic constraint 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, K tip (Experimental), were determined in stage II # $m using measured ddNc values in the Paris law of the form ddNc = C K tip (Experimental) , 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. K tip (Experimental) values are shown in Fig. 5. These values clearly deviated from K applied with the onset of stage II. In specimen 1, when the crack tip in stage II was at the √ front weld interface, i.e., c = 48.5 mm, K tip (Experimental) was found to have reduced to 46.06 MPa m from the K applied value of 147 MPa √ m because σ W  σ A . The ddNc value at the crack length of 38 mm in stage I was 0.0055 mm/cycle which dropped to 0.002 mm/cycle at the crack length of 48.5 mm. Likewise in specimen 2, when √ the crack tip touched the front weld interface, K tip (Experimental) was found to have reduced to 59.76 MPa m from the K applied √ value of 147 MPa m. The ddNc 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 ddNc value at a critical crack length of 42.4 mm in a plain √ alloy steel specimen was much higher at 0.02 mm/cycle under a K applied or K CA value of 82.42 MPa m—plane stress condition existing for 10 mm thickness. Refer images of fatigue crack surfaces of bimetallic and plain specimens at Fig. 6. The micrograph of the alloy steel surface at position B in bimetallic specimens, Fig. 7, when the specimens had not fractured was different from the micrograph of the fractured surface of plain alloy steel specimen at position F, Fig. 8, which confirmed delayed fracture of bimetallic specimens. Therefore, there was a clear evidence of load transfer towards stronger materials on the other side of the front weld interface in bimetallic specimens, initially due to eccentric load over the specimens and later on due to material plasticity effects which shielded the crack tip in weak alloy steel, thereby allowing it to grow longer without becoming critical. The tensile nature of residual stresses

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269

Type I bimetallic (Specimen No. 1)

170

Series 1 - ΔK applied

Weld, W

Series 2 - ΔK tip ( Experimental ) Series 3 - ΔK tip (Theoretica l )

Alloy steel, A

Maraging steel, B

Fracture zone of bimetallic specimen K w IC Fracture zone of plain alloy steel specimen

KC

Stage II

A

50.5 46.5 42.5 Crack length, c (mm)

110 MPa m

90

38.5

34.5

Series 3 - ΔK tip (Theoretical )

Alloy steel, A

Series3

170

Series 2 - ΔK tip ( Experimental )

Maraging steel, B

Series2

30 30.5

Series 1 - ΔK applied

Weld , W

K IC w KC A

Fracture toughness data

K C A = 82.42 MPa m

150

K IC W ≈ K IC B = 100.8 MPa m

130

Yield strength Alloy steel 490 MPa Weld 2315 MPa Maraging steel 1760 MPa

Average hardness 209 HV 420 HV 319 HV

110 MPa m

90 70

Gain in fatigue life of alloy steel by 5233 cycles

Stage II

50.5

48.5

46.5

42.5

38.5

Series1

Stage I

50 Crack growth

54.5

Average hardness 205 HV 435 HV 317 HV

Series1

Type I bimetallic (Specimen no. 2)

58.5

Yield strength Alloy steel 480 MPa Weld 2400 MPa Maraging steel 1750 MPa

50 Crack growth

62.5

130

Stage I

48.5

54.5

K IC W ≈ K IC B = 100.8 MPa m

70

Gain in fatigue life of alloy steel by 11133 cycles

58.5

K C A = 82.42 MPa m

150

Front weld interface

Rear weld interface

62.5

Fracture toughness

34.5

Series2 Series3

30 30.5

Crack length, c (mm)

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

Type I bimetallic (Specimen no. 1)

Type I bimetallic (Specimen no. 2)

Pla in a l l o y s te e l s pe c im e n

Crack length, c (mm) Weld

Crack front

Weld

Crack front

Crack front

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 K applied did not change. The crack growth rate min could, however, have marginally increased due to an increase in the load ratio, FFmax . But the effect of the load ratio is pronounced at threshold values of K applied and not at high values employed in the present work. As such, the dip in the crack growth rates was solely due to strength mismatch between alloy steel and the weld.

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S. Bhat, S. Narayanan

Micrograph at A

Crack front

c=0

Type I bimetallic specimen Fatigue crack growth path

Weld

F

B

Alloy steel

C Rear weld interface

c = 50 mm

A

Notch face

Right end of specimen

Crack front

Maraging steel

D

Stable fatigue crack growth

Notch face

Alloy steel

Alloy steel

c = 30.5 mm

Front weld interface

c = 48.5 mm

Micrograph at C (Fracture initiation)

Micrograph at B

Front weld interface

Weld

Alloy steel Weld

Fractured surface

Micrograph at D

Rear weld interface

Fractured surface

Maraging steel

Fractured surface

Weld

Micrographs of Specimen no. 1 Fig. 7 Micrographs of fatigue surfaces of Type I bimetallic specimens

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Micrograph at B

Micrograph at A

Crack front Front weld interface Notch face Stable fatigue crack growth Alloy steel

Weld Alloy steel

Alloy steel

Micrograph at C (Fracture initiation)

Micrograph at D

Rear weld interface

Fractured surface

Weld

Fractured surface

Fractured surface Maraging steel

Weld

Micrographs of Specimen no. 2 Fig. 7 continued Micrograph at F (Fracture initiation)

Micrograph at A

Crack front Fractured surface Stable fatigue crack growth

Alloy steel

Alloy steel

Alloy steel

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

) The shielding effect over the crack tip in each bimetallic specimen, quantified by K applied − K tip (Experimental)}, increased as the crack tip neared the front weld interface. Since the weld was made of marW , for 10 mm thickness of each specimen, was considered aging steel, its plain strain fracture toughness, K IC

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S. Bhat, S. Narayanan

Fractured Type II bimetallic specimen

Critical crack near rear weld interface

Fractured maraging steel specimen

Critical crack at 46 mm length

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 succeeding Section, ignoring the change in fracture toughness of weld in the specimens due to a slight deviation of weld strengths. The crack tip after penetrating into the weld faced the rear weld interface of less stronger maraging steel. This terminated the shielding effect of weld which caused K tip (Experimental) to increase and assume √ W , triggered the value of K applied , of the order of 147 MPa m. K tip (Experimental), being higher than K IC weld fracture followed by unstable crack propagation in back up maraging steel. The strength mismatch effect across the rear weld interface did not come into play as the crack had already become critical before reaching there. 4.2 Type II specimen Fractured Type II bimetallic and plain maraging steel specimens are shown in Fig. 9. A crack in bimetallic specimen was seen to curve while propagating through maraging steel, A, into √ ultra-strong weld, W, and became critical at the length of 45.85 mm under a K applied 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 at A value of 100.8 MPa m—plane strain condition a slightly higher length of 46 mm under a K applied or K IC existing for 10 mm thickness. To confirm this critical observation, additional plain maraging steel specimens were tested. Cracks failed at lengths even higher than 46 mm in all these specimens. This demonstrated the amplifying effect of approaching a weak alloy steel over the crack tip in ultra-strong weld which caused fracture of the bimetallic specimen at lesser crack length when compared to the crack in the plain maraging steel specimen. The effect of interfaces over the crack tip was highly localized due to the plane strain condition because of high yield strength of maraging steel and weld through which the crack grew resulting in very small r values. Consequently, a sufficient number of data points could not be collected during experiments to precisely monitor crack growth rates under the influence of approaching interfaces. A failure analysis of the bimetallic 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 the specimens Case 1: Crack tip away from the front weld interface Configurational 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 propagation to a large extent. The crack while growing in a material follows the path of least resistance to fulfil the principle of maximum dissipation and stability. Refer to Figs. 4 and 9. In Type I and plain alloy steel specimens, the cracks 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

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273

material effects or the notch axis not being exactly perpendicular to the applied load. The crack continued to curve further with the onset of mixed mode loading. Case 2: Crack tip near the front weld interface The conditions possible other than the stable crack growth in this case are as follows: i) Crack curving substantially before defecting into interface K T stress in conventional singular crack tip stress field solution, defined as σi j = √ tip ∗ f (θ ) + T δ1i δ1 j , 2πa * K holds importance. T stress has been determined by Jiang et al. [7, p. 263] using T = √ tip∗ (geometry) πa * * where (geometry) is a dimensionless shape factor. The relation between (geometry) and a crack length parameter is provided by [15,16]. Cotterell and Rice [17] suggested for a Mode I crack that when T > 0 the crack becomes unstable and curves and vice versa. However, these conditions were not found to be valid under high load level in the material combination of mild and stainless steels [7, pp. 264–265]. In the present work too, a high driving force or high K tip value and small pre-curved lengths of cracks at their juncture of touching the front weld interfaces led to their penetration into the welds in Type I specimens. In a Type II specimen also the crack penetrated into the weld again due to high K tip levels and the fact that the curved crack 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 CA W K IC

1.

K CA W K IC

≈ 1 supported crack penetration.

ii) Crack branching or bifurcating Pippan 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 K tip values, of the order of 18 MPa m and under a small toughness value of the interface material. High K tip levels in the presence of reasonably strong and tough weld 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 K tip was equal to K applied in stage I which did not necessitate any analysis. The minimum fatigue crack length, cmin , required to realize lin  K tip (Experimental) 2 ear elastic or high cycle fatigue condition at a stage II data point was considered as 2.5 2Y [19], where the yield strength, Y, corresponded to the parent material containing the crack tip. The actual crack length ahead of the load axis, cc , given by (c − 12.5) mm had to be larger than cmin . Plane stress or plane strain condition was also checked in materials across interfaces at each stage II data point. In the parent material,     K tip (Experimental) 2 K tip (Experimental) 2 fulfilment of conditions t ≤ 2.5 and t > 2.5 indicated plane 2Y 2Y stress and plane strain conditions, respectively, at the selected data point. For a material of yield strength, Y, 2 2   L L and t > 2.5 K hinted at plane on the other side of the interface, fulfilment of conditions t ≤ 2.5 K 2Y 2Y stress and plane strain conditions in the interface material. The stress intensity parameter over the cohesive zone in the interface material, K L , differed from K tip (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 specimens 1 and 2. Plane stress condition was nearly realized in alloy steel, A, whereas plane strain condition existed in weld, W. Term σ A supposed to be equal to Y A in plane stress condition was replaced by the effective cohesive A , to account for the premature effect of the weld due to eccentric loading over the specimen. stress term, σeff + ,2 K applied A was obtained from the relation r = π , where r was measured experimentally. Values σeff A 8 2σeff √ A in specimens 1 and 2 were found as 288 and 234.8 MPa, respectively. σ W was considered as 3YW . of σeff The computational model was executed over data points of the crack tip advancing towards the front weld interface to obtain theoretical values of the crack tip stress intensity parameter, K tip (Theoretical). The value of K tip (Theoretical) was found to be slightly higher than K tip (Experimental) at the first data point in both the specimens due to a larger shielding effect by the weld which was not considered in the computational model based on plasticity effects alone. A small correction factor, introduced to make K tip (Theoretical)

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S. Bhat, S. Narayanan

Type II bimetallic specimen Weld, W

Rear weld interface

Fracture toughness

330

Series 1 - ΔK applied Series 2 - ΔK tip (Theoretical)

K IC A ≈ K ICW = 100.8 MPa m K C B = 82.42 MPa m

280

Front weld interface

Maraging steel Weld Alloy steel

230 Alloy steel, B

Maraging steel, A

Yield strength 1775 MPa 2418 MPa 498 MPa

Average hardness 322 HV 438.5 HV 213 HV

MPa m

180 130

X

80

Series1

Crack growth

Series2 62.5

58.5

54.5

50.5 46.5 42.5 Crack length, c (mm)

38.5

34.5

30 30.5

330

Series 1 - ΔK applied Series 2 - ΔK tip (Theoretical) Rear weld interface

Weld, W

280 Front weld interface

230 MPa m

180 Fracture zone of plain maraging steel specimen

Stage II

130

Fracture zone of bimetallic specimen

K IC A ≈ K IC w Stage I

Stage I

80

Magnified view at X

Stage II

45.8

44.28

45.85

46

44.35

45.5

45 44.5 Crack length, c (mm)

44

30 43.5

Series1 Series2

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

equal to K tip (Experimental) at the first data point, was used throughout the computation. Computation revealed that the maximum extent of spread of the cohesive zone was restricted to the weld without crossing over into maraging steel. The values of l at crack lengths of 48.5 mm were found to be 0.12 and 0.13 mm in specimens 1 and 2 which were less than the weld width of 1.5 mm. The values of K tip (Theoretical) were in good agreement with those of K tip (Experimental) which validated the accuracy of the computational model. The percent difference between K tip (Theoretical) and K tip (Experimental), defined by mod|K tip (Theoretical)−K tip (Experimental)| e = × 100, was computed at each stage II data point. For n K tip (Experimental) * number of data points in the specimen, the average % difference, eav , was determined by eav = n1 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 steel was near the front weld interface and secondly when the crack tip in the weld was near the rear weld interface. 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 plane strain and plane stress conditions were√ fulfilled in weld√and alloy steel, B, respectively, at second stage II data A points. Therefore, the relations σ = 3 Y A , σ W = 3 Y W and σ B = Y B were used in the computational

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275

model which was executed separately over the cases of the crack tip near front and rear weld interfaces. The ultra-strong weld began to influence the approaching crack tip in comparatively weaker parent maraging steel at a crack length of 44.28 mm. Till then, the crack tip was in stage I as distance of the crack tip from front weld interface, a, which was more than the cohesive zone length, r . Both a and r were 0.07 mm at the crack length of 44.28 mm. Beyond the crack length of 44.28 mm, the condition a < r was fulfilled which implied stage II or development of a cohesive zone into weld causing the crack tip to experience the shielding effect of weld. K tip (Theoretical) dropped below K applied because σ W > σ A . The effect continued till the crack 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 weld interface of weak alloy steel, B. During a crack length of 44.35–45.8 mm, the crack tip was in stage I and did not feel the effect of approaching alloy steel. a and r were equal to 0.05 mm at a crack length of 45.8 mm. The crack grew under amplifying effect of alloy steel, in stage II, at a length beyond 45.8 mm. The effect caused K tip (Theoretical) to increase above K applied because σ B σ W . The position of the crack when it became critical was between 45.8 and 45.85 mm. The exact location could not be identified due to sudden failure of the specimen. For a hypothetical case of a crack at the rear weld interface (crack length of 45.85 mm), when the amplifying effect on the crack tip could have been maximum, K applied and corresponding K tip (Theoretical) √ √ obtained from the computational model were 96.39 MPa m and 280 MPa m, respectively, with the extension of the cohesive zone in alloy steel, l, being 3.7 mm. K tip (Theoretical) clearly exceeded the fracture toughness 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-mismatched interfaces over a nearby crack tip is obtained from the work presented in the paper. Fatigue tests conducted over bimetallic compact tension specimens fabricated by welding weak ASTM 4340 alloy and strong MDN 250 maraging steels confirm the crack tip experiencing a shielding effect when it is near the interface of stronger steel and an amplifying effect when it is in the vicinity of the interface of weaker steel, both the effects influencing K tip . There is dip in K tip from K applied with consequent drop in ddNc in the case of crack tip shielding. Trends reverse during crack tip amplification. As a result, fatigue life of a bimetallic specimen differs from that of plain specimen. Micrographic examinations of a fatigue crack surfaces convincingly support the findings. Arrest of a fatigue crack in weak parent steel near the interface of stronger steel is possible if K tip drops to the value below the fatigue threshold of the parent steel. Life of a bimetallic composite can be predicted if K tip is known. A computational model for this purpose is reviewed and validated by experiments. Values of K tip from the computational model are in good agreement with 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 stresses are non-uniform [20] and decrease as the crack tip opening displacement increases upon application of load ii) experimental errors in the form of inconsistency in material properties which could have affected crack growth rates and therefore the accuracy of Paris constants. Despite such errors, the trends and inferences are well substantiated. Acknowledgments Support received from the School of Mechanical and Building Sciences, V.I.T., Vellore, India, during the course 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). Differentiating Eq. (A.1) results in + , ∂u ∂v 2μ +i = κϕ (z) − zϕ (z) − ϕ (z) − ψ (z). ∂x ∂x

(A.1)

(A.2)

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On substituting the relation by Westergaard [22], ψ (z) = −z ϕ (z), in Eq. (A.2), one obtains , + ∂u ∂v +i = κϕ (z) − (z − z)ϕ (z) − ϕ (z) 2μ ∂x ∂x or , + ∂u ∂v +i = κϕ (z) − 2iy ϕ (z) − ϕ (z). 2μ ∂x ∂x On replacing ϕ (z) by φ(z) and ϕ (z) by φ (z) in Eq. (A.4), one obtains , + ∂u ∂v +i = κφ(z) − 2iyφ (z) − φ(z). 2μ ∂x ∂x Overall, the conjugate of Eq. (A.5) gives + , ∂u ∂v 2μ −i = κφ(z) − 2iyφ (z) − φ(z). ∂x ∂x

(A.3)

(A.4)

(A.5)

(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   ∂v (A.7) = lim κφ(z) − κφ(z) + φ(z) − φ(z) 4μi y→0 ∂x or  . ∂v (A.8) = lim (κ + 1) φ(z) − φ(z) . 4μi y→0 ∂x On using κ = 3−ν 1+ν and μ = stress condition as

E [2(1+ν)]

in Eq. (A.8), one obtains the expression for the displacement, v, in plane   ∂v 2 = lim φ(z) − φ(z) . ∂x Ei y→0

Similarly on using κ = 3 − 4ν and μ = strain condition:

E [2(1+ν)]

(A.9)

in Eq. (A.8), the following expression is obtained in plane

!   2 1 − ν2 ∂v = lim φ(z) − φ(z) . y→0 ∂x Ei

(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 in an infinite homogenous parent body, A, under the sole action of far field applied stress, p∞ , of the form  z p∞ z 2 − ξ 2 1 p∞ dξ − (A.11) φ1 (z) = √ 2 2 z − ξ 4 2π z − c −z

or φ1 (z) =

Using Green’s

%z p∞ √z+ξ √ Equation, dξ z−ξ −z

z

1

√ 2π z 2 − c2

−z

√ p∞ z + ξ p∞ . dξ − √ 4 z−ξ

(A.12)

= p∞ (π z), Eq. (A.12) is written as

φ1 (z) =





1 z2

− c2

p∞ (π z) −

p∞ . 4

(A.13)

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277

To study the behaviour close to the tip of the crack, the limit z → c is introduced to obtain φ1 (z) =

1 p∞ p∞ (πc) − . √ 4 2π (2c)(z − c)

(A.14)

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

K applied p∞ φ1 (z) = √ − . 4 2 2π(z − c)

(A.15)

On differentiating Eq. (A.13) after applying the limit z → c, one obtains φ1 (z) = −

p∞ c 2

!3 .

(A.16)

p∞ − zφ1 (z). 2

(A.17)

p∞ p∞ c 2 z + !3 . 2 2 z 2 − c2 2

(A.18)

2 z 2 − c2

2

Another potential is written as

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

1 (z) =

The potentials φ1 (z) and 1 (z) must satisfy the following boundary conditions: Set I: At far field, where z → ∞, σ y = p∞ ; σx = 0; τx y = 0, Set II: At crack axis where y = 0 and x ≤ 0, σ y = 0; τx y = 0. The following stress field equations [21, p. 114] are used to verify whether the selected potentials satisfy the boundary conditions or not: σx + σ y = 4Re [φ1 (z)] , # $ σ y − σx + 2iτx y = 2 z¯ φ1 (z) + 1 (z) .

(A.19) (A.20)

On using Eqs. (A.13), (A.19) takes the form ' σx + σ y = 4Re

( p∞ p∞ z − . √ 4 2 z 2 − c2

On using Eqs. (A.16), (A.18) in Eq. (A.20), one obtains ⎡ ⎤ 2 2z p z ¯ p p c c ∞ ∞ ∞ ⎦ σ y − σx + 2iτx y = 2 ⎣− !3 + 2 + !3 . 2 2 2 2 2 z −c 2 z − c2 2 At far field where z → ∞, Eqs. (A.21) and (A.22) reduce to p p∞  ∞ − = p∞ σx + σ y = 4 2 4

(A.21)

(A.22)

(A.23)

and σ y − σx + 2iτx y = p∞ .

(A.24)

Equation (A.24) implies σ y − σ x = p∞ The solution of Eqs. (A.23) and (A.25) yields

and τx y = 0.

(A.25)

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σ y = p∞ and σx = 0. Set I of boundary conditions is fulfilled. At y = 0 and at x ≤ 0, Eq. (A.21) changes to p p∞  ∞ σx + σ y = 4Re (imag) − 2 4 or σ x + σ y = − p∞ .

(A.26)

Similarly, at y = 0 and at x ≤ 0, Eq. (A.22) changes to σ y − σx + 2iτx y = p∞ .

(A.27)

Equation (A.27) implies σ y − σ x = p∞

and τx y = 0.

(A.28)

On solving Eqs. (A.26) and (A.28), one obtains σ y = 0; τx y = 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   K applied p∞ φ1 (z) = − , (A.29) √ 4 2 2π(z − r ) because the same K applied 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 zone of length r in the parent body, A, is written as r A  2 1 σA σ ξ − r2 φ2 (z) = dξ − . (A.30) √ ξ −z 4 2πi z 2 − r 2 0

As z and ξ → r , Eq. (A.30) can be written as 1 φ2 (z) = √ 2πi z − r

r 0

√ σA σA ξ −r dξ − ξ −z 4

(A.31)

or −i φ2 (z) = √ 2π r − z

r 0

√ σA r −ξ σA dξ − ξ −z 4

for ξ and z < r.

(A.32)

Another potential is written as σA − zφ2 (z). 2 The potentials must satisfy the following boundary conditions: Set III: At y = 0 and 0 < x < r in the cohesive zone, σ y = −σ A ; σx = 0; τx y = 0 Using Kolosov–Mushkelishvili’s equations, one obtains

2 (z) = −

σx + σ y = 4Re [φ2 (z)] = −σ A and σ y − σx + 2iτx y = 2

#

$

z φ2 (z) + 2 (z)

' ( σA = 2 (z − z)φ2 (z) − . 2

(A.33)

(A.34)

(A.35)

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

and τx y = 0.

(A.36)

The solution of Eqs. (A.34) and (A.36) results in σ y = −σ A ; σx = 0; τx y = 0. Set III of boundary conditions is fulfilled.

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