Calculation of stress intensity factor using weight function method for a patched crack with debonding region

Engineering Fracture Mechanics 67 (2000) 303±310

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Technical Note

Calculation of stress intensity factor using weight function method for a patched crack with debonding region J.H. Kim 1, S.B. Lee * Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Science Town, Taejon 305-701, South Korea Received 25 January 2000; received in revised form 25 April 2000; accepted 1 June 2000

Abstract A patched crack with a debonded region is treated with a crack-bridging model: assuming continuous distribution of springs acting between crack surfaces. Adopting weight function method, the stress intensity factor for the patched crack within in®nite plate is successfully obtained by numerical calculation. As the crack length a increases, the restraint on the relative displacement of the crack faces for a given value r0 is increased. The constant relative displacement (or crack surface displacement) did explain the reason for the constant stress intensity factors being independent of the crack length. A simple asymptotic solution was proposed for estimating the reduction of patching eciency due to debonding and compared with numerical solution obtained by weight function method. Ó 2000 Published by Elsevier Science Ltd. Keywords: Debonding; Crack-bridging model; Weight function method; Stress intensity factor

1. Introduction Recently, the use of adhesive bonding as a repair process in aircraft structure is an accepted means to enhance the remaining service life of damaged components. Especially, the development of high-strength ®bers and adhesives has accelerated the repair technique of bonding composite patch to the damaged structure. Using the linear elastic fracture mechanics approach, Rose [1] showed that for a center-cracked plate bonded to an reinforcing patch, the stress intensity factor did not increase in®nitely with increasing crack length, as it would if there was no reinforcement. He also explained that the reason for this asymptotic behavior was due to the crack-bridging mechanism of the reinforcing patches, such that the applied load could be fully transmitted across the crack with only a ®nite relative displacement between the crack faces. Meanwhile, the stress intensity factor for reinforced cracks in solids or structures were obtained by assumption of a continuous distribution of springs acting between the crack faces [2,3]. *

Corresponding author. Tel.: +82-42-869-3029; fax: +82-42-869-3210. E-mail addresses: [email protected] (J.H. Kim), [email protected] (S.B. Lee). 1 Tel.: +82-42-869-5033; fax: +82-42-869-3095. 0013-7944/00/$ - see front matter Ó 2000 Published by Elsevier Science Ltd. PII: S 0 0 1 3 - 7 9 4 4 ( 0 0 ) 0 0 0 5 8 - 8

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Some researchers have evaluated the e€ect of debond, with increasing the crack growth for a center crack subjected to an uniform loading [4,5]. However, it is not easy to obtain the stress intensity factor for the through-thickness cracks with or without reinforcing patches within a ®nite body, especially under arbitrary loading. The weight function method proposed by Bueckner [6] and Rice [7] has proved to be a very useful and versatile method of calculating stress intensity factors for cracks subjected to non-uniform stress ®elds, such as residual stress or thermal loading. Using the weight function method, Fett [8] and Cox et al. [9] evaluated the relation between crack opening displacement and stress intensity factor and explained the crack-bridging phenomenon. This paper also show how the weight function method can be applied to calculate the stress intensity factor for a center crack with a patch partially debonded on one side in an in®nite plate. 2. Basic problem formulation The problem being considered is an in®nite center-cracked plate, with crack length 2a and debond of length 2b subjected to a remote uniform tensile stress r1 as shown in Fig. 1(a) and (b). The assumption of a parallel debond is not unrealistic because the assumed region of debond in front of the crack tip will have little in¯uence on reinforcement eciency [10]. The problem is to determine the stress intensity factor Kr in the repaired plate. Subscripts, P, R, A will be used to identify parameters corresponding to the plate, the reinforcing patch or the adhesive layer, respectively. Thus, EP , ER will denote the YoungÕs modulus of the plate and the reinforcement; GA , the shear modulus of the adhesive, and tP , tR , tA , the respective thickness. Here are some assumptions: (i) the plate and the reinforcement are both isotropic and have the same Poisson ratio m…ˆ mP ˆ mR † and all deformations are linearly elastic, (ii) there is no out-of-plane bending due to the one-sided reinforcement and no residual thermal stress induced by bonding process, and (iii) the reinforced plate ignores any variation across the thickness. For the cross-section con®guration shown in Fig. 1(b), it was assumed that the redistribution of stress in uncracked plate without debond or with debond 2b was almost same. Therefore, the stress obtained explicitly from the one-dimensional theory of bonded joints was used [1]. The reduced stress is expressed as r0 ˆ r1 =…1 ‡ S†, where S ˆ ER0 tR =EP0 tP represents the sti€ness ratio between the plate and the reinforcement. Here E0 ˆ E=…1 ÿ m2 † is YoungÕs modulus for plane strain condition, E0 ˆ E for plane stress condi-

Fig. 1. Repair con®guration: (a) patched crack with reinforcement with debond 2b, (b) cross-section along A±A0 .

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Fig. 2. Distributed springs model for a patched crack.

tion. Then, it is assumed that distributed linear spring act between the crack faces as shown in Fig. 2. The boundary conditions are described by ryy ˆ r0 ;

as x2 ‡ y 2 ! 1;

ryy ˆ kEP0 uy … x†;

…1a†

at j xj < a; y ˆ 0;

…1b†

where k is the spring constant. Under plane stress condition, the appropriate value of k can be determined from the stress±displacement relation for the overlap shear joint [1]. The total crack opening displacement of the repaired plate as shown in Fig. 3 is given by r1 b; …2† uP ˆ d ‡ e R b ˆ d ‡ ER where the term d is due to shear strain in the adhesive as shown in Appendix A, and the other term is due to the strain in the reinforcement over the debond region. Finally, the spring constant is calculated using Eq. (1b) as r0 r0 r0 : …3† ˆ  ˆ kˆ EP uP EP …d ‡ eR b† EP d ‡ r1 b ER

3. Results and discussion If the remote stress r0 (x) as well as spring stresses rj acting on n segments are applied to the crack surfaces, the resultant crack surface displacement uy …xi † at xi is expressed by weight function as [8]

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Fig. 3. Cross-section for overlap shear joints.

uy …xi † ˆ

1 EP0

Z

a xi

Z 0

a

 n X ÿ
r0 … x†m… x; a†dx m…xi ; a†da ÿ r j g xi ; xj ;

…4†

jˆ1

where g…xi ; xj † is de®ned as the in¯uence function, i.e., the crack surface displacement at xi due to a uniform stress rj acting on a segment 2w located at xj as shown in Fig. 4. In Eq. (4) rj is given as EP0 kuy …xj † de®ned in

Fig. 4. Model for the crack surface displacement at xi due to a uniform stress rj acting on a segment 2w of the crack surface located at xj .

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307

Table 1 Physical dimensions and material properties of a typical repair Layer

YoungÕs modulus (GPa)

PoissonÕs ratio

Thickness (mm)

Plate Reinforcement Adhesive

70 200 0.7a

0.33 0.33 0.33

3.0 1.0 0.2

a

AdhesiveÕs shear modulus GA .

Eq. (1b). Then, the numerical solution for the linear system of Eq. (4) can be obtained from the Gauss± Seidel iterative method as proposed by Newman [11]. Using the obtained crack surface displacement uy …xi †, the stress intensity factor Kr …a† is represented ®nally as [8] Z a n Z xi ‡w X r0 …x†m…x; a†dx ÿ EP0 kuy …xi †m…x; a†dx: …5† Kr …a† ˆ 0

iˆ1

xi ÿw

To obtain the stress intensity factor, the weight function m(x, a) for a center crack in an in®nite plate is used [7]. The width of a segment in Eq. (5), 2w ˆ 0:1 mm is used to calculate the stress intensity factor Kr . The dimensions and material properties of the cracked plate, reinforcement and the adhesive layer are summarized in Table 1. For a center crack with a patch fully bonded on one side in an in®nite plate, Rose [1] obtained the upper bound and the analytical approximate solution with physical parameter K ˆ 1=p k de®ned in Appendix A as p …6† Kc ˆ r0 pK; s paK : Kr ˆ r0 …a ‡ K†

…7†

On the other hand, the stress intensity factors for a patched crack with partially debonded region as shown in Fig. 1(a) is obtained by inserting Eq. (3) into Eqs. (6) and (7) as s r 1 EP … ER d ‡ r 1 b† ; …8† ˆ r0 Kc ˆ r0 k ER r0 s r ap ap ˆ r0 : Kr ˆ r 0 r0 apk ‡ 1 ap E …EERd‡r ‡1 b† P

R

…9†

1

For a patched crack without debonding, Fig. 5 shows the comparison between the numerical solution obtained from Eq. (5) and RoseÕs solution Eq. (7), and indicates that two solutions are almost same. Upper bound Kc , and K0 for a center crack subjected to the reduced remote stress r0 are also shown. When the crack length a is very small, the reduction in stress intensity factor Kr is mainly due to the stress reduction from r1 to r0 . As the crack length a increases, the stress intensity factor Kr approaches the constant value. This indicates the increase of the restraint on the relative displacement of the crack faces for a given value r0 . Fig. 6 represents the crack surface displacement obtained from Eq. (4) and the crack surface displacement for a center crack subjected to remote stress r0 . The crack surface opening displacement along the crack surface is constant and ®nite due to the bridging of the reinforcement, and explains that the constraint on the crack surface displacement increases. Thus, the reason for the constant stress intensity factor being independent on crack length can also be explained.

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Fig. 5. Variation Kr with crack length (debonding length b ˆ 0 mm).

Fig. 6. Crack surface displacement uy …x† versus distance from crack mouth.

To estimate the e€ect of the debonded region, Fig. 7 shows the stress intensity factors Kr for a patched crack with debonding length b ˆ 0, 10 and 20 mm, respectively. In case of b ˆ 0 mm, the stress intensity factor becomes the RoseÕs solution, Eq. (7). The comparison between the numerical solution obtained from Eq. (5) and the analytical solution (9) indicates that two solutions are almost same. As the length of debonded region b increases, the stress intensity factor approaches to a higher limiting value (8) which is the same as the equation proposed by Baker [10].

4. Conclusion For a reinforced center crack with debonding region subjected to remote uniform stress in an in®nite plate, the stress intensity factor was calculated simply by weight function method, and compared with the

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Fig. 7. Variation Kr for a patched crack with debonding length b ˆ 0, 10 and 20 mm.

extended approximate solution based on RoseÕs solution. Finally the agreement of two solutions showed that the weight function method could be e€ectively used to obtain the stress intensity factor for the reinforced cracks subjected to non-uniform stress ®eld such as residual stress or thermal loading. Appendix A For the overlap shear joint shown in Fig. 3 without debonding region, Rose [1] obtained the displacement as dˆ

r0 tP tA b : GA

…A:1†

And he also de®ned the spring constant as 1=pK and calculated using Eq. (1b) as kˆ

1 r0 GA ˆ : ˆ pK EP d EP tP tA b

…A:2†

And here b is denoted as s     s GA 1 1 GA 1 ‡ 1‡ : ˆ bˆ tA EP tP ER tR tP tA EP S

…A:3†

References [1] Rose LRF. Theoretical analysis of crack patching. In: Baker AA, Jones R, editors. Bonded repair of aircraft structures. Martinus Nijho€, Dordrecht, The Netherlands, 1988. p. 77±106 [Chapter 5]. [2] Marshall DB, Cox BN, Evans AG. The mechanics of matrix cracking in brittle-matrix ®ber composites. Acta Metall 1985; 33(11):2013±21. [3] Rose LRF. Crack reinforcement by distributed springs. J Mech Phys Solids 1987;35(4):383±405. [4] Rose LRF. In¯uence of debonding on the eciency of crack patching. Theor Appl Fract Mech 1987;7:125±32. [5] Cox BN, Rose LRF. Time- or cycle-dependent crack bridging. Mech Mater 1994;19:39±57.

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[6] Bueckner HF. A novel principle for the computation of stress intensity factors. Z Angew Math Mech 1970;50:529±46. [7] Rice JR. Some remarks on elastic crack-tip stress ®elds. Int J Solids Struct 1972;8:751±8. [8] Fett T. Evaluation of the bridging relation from crack-opening-displacement measurements by use of the weight function. J Am Ceram Soc 1995;78(4):945±8. [9] Cox BN, Marshall DB. Stable and unstable solutions for bridged cracks in various specimens. Acta Metall Mater 1991;39(4): 579±89. [10] Baker AA. Repair eciency in fatigue-cracked aluminium components reinforced with boron/epoxy patches. Fatig Fract Engng Mater Struct 1993;16(7):753±65. [11] Newman Jr JC. A crack-closure model for predicting fatigue crack growth under aircraft spectrum loading. ASTM STP 748 1982;53±84.