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Influence of geometry and material on the stress intensity of an interfacial crack propagating from a bi-material notch
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Influence of geometry and material on the stress intensity of an interfacial crack propagating from a bi-material notch Changzheng Cheng∗, Wei Pan, Yifan Huang, Jialin Sun, Zhongrong Niu Department of Engineering Mechanics, School of Civil and Hydraulic Engineering, Hefei University of Technology, Hefei 230009, China
a r t i c l e
i n f o
Keywords: V-notch Semicircle notch Interfacial crack Stress intensity factor Boundary element method
a b s t r a c t The stress intensity factor (SIF) is an important parameter to characterize the stress intensity near a crack tip. In order to successfully evaluate the SIFs for the interfacial crack emanating from a bi-material notch, a method coupling the boundary element method with the singularity asymptotic expansion technique is introduced. Then, the crack initiating from a sharp V-notch tip and the one emanating from a semicircle notch root are respectively taken into consideration, to investigate the influence of notch geometry shape and material property on the stress intensity for the interfacial crack from a bi-material notch. It is found that the notch opening angle has great influence on the SIFs of an interfacial short crack, while it has little influence on the SIFs of an interfacial long crack propagating from a bi-material V-notch. The value of KI for the interfacial crack initiating from a V-notch tip increases with the notch depth, while KII decreases with it. The value of KI and absolute value of KII for the interfacial crack initiating from a V-notch tip increase with the elasticity modulus ratio. The elasticity modulus ratio has little influence on KI , while it has important influence on KII for the interfacial crack emanating from a semicircle notch root. The value of KI and absolute value of KII for the interfacial crack initiating from a semicircle notch root increase with the notch radius.
1. Introduction The geometry discontinuity in the engineering structure will generate a structure style named the notch [1], which can be classified into the sharp V-notch whose radius is zero and blunt notch including the semicircle notch and elliptical one. Because of the sudden change of geometry shape, the stress becomes singular or seriously concentrated near the vertex or root of a notch [2]. The crack is easy to be initiated from the notch tip due to the stress concentration [3,4]. Some research work was focused on the crack emanating from a plane notch [5]. A short crack approach was developed by Ranganathan et al. [6] to determine the fatigue crack initiation life at a notch tip. Basing on the averaged strain energy density approach, the crack initiation life in notched steel bars under torsional fatigue was studied by Campagnolo et al. [7]. The initiation and crack growth behaviour of a crack from notch was studied with the direct current potential drop technique by Kolitsch et al. [8]. By applying the finite element method, the fatigue crack initiation and growth from the notch under combined loading were analysed by Ding et al. [9]. The other research work was contributed to the crack propagating from the anti-plane or threedimensional notch. A crack emanating from the apex of anisotropic wedge under anti-plane shear was investigated by Beom and Jang [10]. The edge crack in front of the anisotropic wedge tip interacting with an
∗
anti-plane singularity was studied by Shen et al. [11]. The exact solution for the stress field ahead of the crack initiating from a sharp notch under anti-plane shear was derived in a close form by Salviato et al. [12]. The difference of the potential energy in an elastic 3D V-notch with and without a small crack was provided by Mittelman and Yosibash [13]. The material discontinuity will generate the bi-material interface edge, which is another stress singularity source [14]. An interface crack usually initiates from the interface edge. A systematic experimental study was conducted to investigate the interaction between the crack, notch and interface by Arun and Xu [15]. The crack propagation under flexure in layered ceramics designed with strong interface was investigated by Nahlik et al. [16]. Different criteria for the direction of crack propagation were studied for the fracture initiated at a corner between two different isotropic materials by Grenestedt and Hallstrom [17]. As mentioned before, both the geometry discontinuity and material discontinuity will individually generate the stress singularity in the engineering structures. The stress singularity will be very complicated if these two kinds of discontinuities occur at the same time, because they will affect each other. The effect of notch geometry on the propagation of fatigue crack emanating from a sharp V-notch was investigated by Benedetti et al. [18]. The influence of geometric and structural notch on the fatigue notch factor in steel welded joint was studied by Lagoda
Corresponding author. E-mail address: [email protected] (C. Cheng).
https://doi.org/10.1016/j.enganabound.2019.10.016 Received 7 August 2019; Received in revised form 11 October 2019; Accepted 30 October 2019 Available online xxx 0955-7997/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: C. Cheng, W. Pan and Y. Huang et al., Influence of geometry and material on the stress intensity of an interfacial crack propagating from a bi-material notch, Engineering Analysis with Boundary Elements, https://doi.org/10.1016/j.enganabound.2019.10.016
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et al. [19]. The finite element method was applied to analyze the behaviour of a crack emanating from semicircular notch root growing in an interface by Ouinas et al. [20]. The propagation of the crack emanating from a homogenous V-notch tip under fatigue loading was studied by Cheng et al. [21]. According to the knowledge of authors, the main scientific interests are focused on the stress intensity for the crack from a homogeneous V-notch, while the effect of geometry discontinuity on the stress intensity for the interfacial crack emanating from a bi-material notch has rarely been investigated. The boundary element method can reduce the dimensions of the problem by one and it can describe the singular field more accurate compared with the finite element method [22,23]. Herein, a semi-analytical method which couples the boundary element method and the singularity asymptotic expansions [24] is introduced to evaluate the stress intensity factors for the crack emanating from a bi-material notch. Then, a sharp V-notch and a semicircle notch are respectively taken, to analyze the influence of geometry characteristic and material property on the stress intensity for the crack initiating from a bi-material notch. The rule with respect to the influence of notch geometry and material property on the stress intensity for the interfacial crack from a notch is obtained.
Part 3
Material 1 Part 1
y
r o
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ρ
θ x
Part 2
Part 4
Material 2
(b)
(a)
Fig. 1. (a) The interfacial crack initiating from a bi-material V-notch tip; (b) The cracked structure is departed into four parts.
σ0
2. Determination of SIFs for the crack emanating from a bi-material V-notch
Material 1
h
The stress intensity factors (SIFs) characterize the speed of a crack growth. When a crack is propagated from a notch, it receives a driving force by the stress field created by the notch. A method should be first proposed to determine the SIFs for the crack emanating from a bimaterial notch. Let’s consider an interfacial crack emanating from a bi-material Vnotch tip shown in Fig. 1a. The cracked structure is departed into four parts as shown in Fig. 1b, which includes the singular stress semicircle zone Part 1 and Part 2, whose radii are 𝜌, and the non-singular stress zone Parts 3 and 4. The stress singularity asymptotic expressions governing in Parts 1 and 2 will be coupled with the boundary integral equations established on Parts 3 and 4 by the interfacial continuous conditions to yield the stress field of the whole structure.
γ Δl
l a
h
Material 2 σ0 w
Fig. 2. Geometry of an interfacial crack propagating from a bi-material V-notch tip.
Fig. 3. Variation of (a) KI and (b) KII with the crack propagating length under different notch opening angles for interfacial cracks propagating from a bi-material V-notch tip when l = 4 and E1 /E2 = 10. 2
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0.6
12 11
0.2 0.0
8
-0.2
KII
9
7
-0.4
6
-0.6
5
-0.8
4 0.0
0.5
1.0
1.5
2.0
l=4 l=6 l=8
0.4
l=4 l=6 l=8
10
KI
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2.5
3.0
3.5
4.0
4.5
-1.0 0.0
5.0
0.5
1.0
1.5
2.0
2.5
Δl
Δl
(a)
(b)
3.0
3.5
4.0
4.5
5.0
Fig. 4. Variation of (a) KI and (b) KII with the crack propagating length under different notch depths for interfacial cracks propagating from a V-notch tip when 𝛾 = 60° and E1 /E2 = 10.
8
1.2
7
E2/E1=2
KII
KI
6
1.0
E2/E1=2
0.8
E2/E1=4
0.6
E2/E1=20
E2/E1=4
4
E2/E1=20
0.0
E2/E1=50
-0.2
0.2
E2/E1=10
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
E2/E1=50
0.4
5
3 0.0
E2/E1=10
-0.4 0.0
5.0
0.5
1.0
1.5
2.0
2.5
Δl
Δl
(a)
(b)
3.0
3.5
4.0
4.5
5.0
Fig. 5. Variation of (a) KI and (b) KII with the crack propagating length under different elasticity modulus ratios for interfacial cracks propagating from a V-notch tip when l = 4 and 𝛾 = 120°.
The singular eigen value 𝜆 for an interfacial crack can be written as [25,26] 𝜆 = 0.5 + 𝑖𝜀
𝑢𝑗𝑞 =
𝑁 ∑ 𝑘=1
{ [ ] 𝑗 𝑗 𝜌1∕2 𝐴𝑘R 𝑈̃ 𝑞𝑘 cos (𝜀 ln 𝜌) − 𝑈̃ 𝑞𝑘 sin(𝜀 ln 𝜌) R I
[ ] } 𝑗 𝑗 − 𝐴𝑘I 𝑈̃ 𝑞𝑘 sin (𝜀 ln 𝜌) − 𝑈̃ 𝑞𝑘 cos (𝜀 ln 𝜌) R I
(1)
(4a)
where i is the imaginary unit, and 𝜀=
1 ln 2π
(
1−𝛽 1+𝛽
)
𝑡𝑗𝑞 =
(2)
𝜇1 (𝜅2 − 1) − 𝜇2 (𝜅1 − 1) 𝜇1 (𝜅2 + 1) + 𝜇2 (𝜅1 + 1)
𝑘=1
{ [ ] 𝑗 𝑗 𝜌−1∕2 −𝐴𝑘R 𝑇̃𝑞𝑘 cos (𝜀 ln 𝜌) + 𝑇̃𝑞𝑘 sin (𝜀 ln 𝜌) R I
[ ]} 𝑗 𝑗 + 𝐴𝑘I 𝑇̃𝑞𝑘 sin (𝜀 ln 𝜌) − 𝑇̃𝑞𝑘 cos (𝜀 ln 𝜌) R I
with 𝛽=
𝑁 ∑
(4b)
where q = 1, 2, N is the truncated asymptotic expansion terms. 𝑈̃ and 𝑇̃ respectively are the combinations of displacement and stress eigen angular functions, which can be determined by the material property in the bi-material structure [27]. AkR and AkI respectively are the real part and imaginary part of amplitude coefficients in the asymptotic expansions, which are determined by the external loading. The displacement and traction on the boundary of the jth part (j = 3, 4) can be calculated by the displacement boundary integral
(3)
in which 𝜅 j = (3 − vj )/(1 + vj ) (j = 1, 2) for a plane stress problem and 𝜅 j = 3 − 4vj for a plane strain problem, vj and 𝜇 j are the Poisson’s ratio and shear modulus for the jth material, respectively. The displacement and singular traction components on the arc boundary of the jth part (j = 1, 2) can be written as [24] 3
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notch can be determined by the equation as follows √ [ ]| 𝐾I + 𝑖𝐾II = lim 2π𝑟−1∕2−𝑖𝜀 𝜎𝜃 (𝑟, 𝜃) + 𝑖𝜎𝑟𝜃 (𝑟, 𝜃) || 𝑟→0 |𝜃=0◦
σ0
σ0
h
Material 1
As shown in Fig. 2, a bi-material lateral sharp V-notch formed by the assembly of two different materials is considered, in which half height h = 60, width w = 40, thickness t = 1. The plate is subjected to an uniaxial load 𝜎 0 = 1 in the vertical direction. The plane stress condition is assumed. A crack of length Δl is supposed to emanate from the V-notch tip along the interface. The notch opening angle and notch depth are respectively denoted as 𝛾 and l, and the total length of the V-notch and crack from its tip is denoted as a = l + Δl. There are 48 uniform segments divided on the arc boundary of the sector dug out from the crack tip in the singularity analysis. The remote structure without the singular stress area is discretized into 228 quadratic boundary elements. First, the notch depth is set at l = 4 and the elasticity modulus ratio of two materials is set at E1 /E2 = 10, where E1 and E2 respectively are the elasticity modulus of material 1 and material 2, to investigate the influence of notch opening angle on the SIFs of the interfacial crack emanating from a V-notch tip. The results are shown in Fig. 3, from which it can be seen that KI increases with the increment of propagating crack length Δl, while the variation rule for KII is on the contrary. The difference of the SIFs between different opening angle is significant when the propagating crack length Δl is small, however, this difference is approaching to zero when the propagating crack length becomes larger. It means that the notch opening angle has great influence on the short crack propagating from a bi-material V-notch tip, while it has little influence on the long crack. Then, the notch opening angle and elasticity modulus ratio are respectively set at 𝛾 = 60° and E1 /E2 = 10, to investigate the behavior of the crack propagating from a bi-material V-notch under different notch depth. The results are shown in Fig. 4, from which it can be found that KI increases with the notch depth significantly, while KII decreases with the notch depth. The difference for KII under different notch depth is small for the short crack, while this difference is significant for the long crack. Next, the notch depth and notch opening angle are respectively set at l = 4 and 𝛾 = 120°, to investigate the influence of elasticity modulus ratio on the SIFs of the interfacial crack from a bi-material V-notch tip. The results are shown in Fig. 5, from which it can be observed that, KI
b
Δl
h
h
Material 2
Material 2
σ0
σ0 w
w
(a)
(b)
Fig. 6. (a) Geometry of an interfacial crack propagating from a bi-material semicircle notch root; (b) Geometry of a lateral pure interfacial crack.
equation as follows [28], 𝐶𝑝𝑞 (𝐲)𝑢𝑗𝑞 (𝐲) =
∫Γ𝑗
𝑗 𝑈𝑝𝑞 (𝐱, 𝐲)𝑡𝑗𝑞 (𝐱)dΓ(𝐱) −
∫Γ𝑗
𝑗 𝑇𝑝𝑞 (𝐱, 𝐲)𝑢𝑗𝑞 (𝐱)dΓ(𝐱) (𝑗 = 3, 4) (5)
where p, q = 1, 2, y is the source point and x is the field point, Cpq (y) is the singularity coefficient which depends on the geometry boundary at point y, 𝑢𝑗𝑞 and 𝑡𝑗𝑞 respectively are the displacement and traction com𝑗 ponents on the boundary of the jth part. 𝑈𝑝𝑞 (𝐱, 𝐲) is the Kelvin funda𝑗 𝑗 mental solution, and 𝑇𝑝𝑞 (𝐱, 𝐲) is the linear combination of 𝑈𝑝𝑞 (𝐱, 𝐲) and its derivatives with respect to coordinate xj . By using the interfacial continuous conditions between the boundaries of four parts shown in Fig. 1b, the boundary unknowns together with the amplitude coefficients AkR and AkI in Eq. (4) can be evaluated. Then, the stresses 𝜎 𝜃 and 𝜎 r𝜃 near the crack tip can be yielded by the asymptotic expansions. Then, mode I type stress intensity factor 𝐾I and mode II type stress intensity factor KII of the interfacial crack emanating from a bi-material
12
8 7
10
6 8
KI
KI
5 4 3
Crack from semicircular notch root Pure laterial crack
2 1 4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
(6)
3. Interfacial crack propagating from a sharp V-notch vertex
h
Material 1
ρn
b
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6 4 2 8.0
9.0
Crack from semicircular notch root Pure laterial crack 8.5
9.0
9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0
b
b
(a)
(b)
Fig. 7. Variation of KI with the crack length for cracks emanating from a homogeneous semicircle notch root: (a)𝜌n = 4, (b)𝜌n = 8. 4
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8
1.2
7
1.0
E2/E1=2 E2/E1=5
0.8
6
E2/E1=10
0.6
5
E2/E1=2
KII
KI
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4
E2/E1=10
0.2
3
E2/E1=20
0.0
2 1 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
E2/E1=50
0.4
E2/E1=5
E2/E1=50
E2/E1=20
-0.2 4.5
-0.4 0.0
5.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Δl
Δl
(b)
(a)
12
1.6
10
1.2
8
0.8
6
KII
KI
Fig. 8. Variation of (a) KI and (b) KII with the crack propagating length under different elasticity modulus ratios for interfacial cracks emanating from a semicircle notch root when 𝜌n = 4.
ρn=6
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
ρn=4 ρn=6
-0.4
ρn=8
2 0.0
0.0
ρn=4
4
0.4
-0.8 0.0
5.0
ρn=8 0.5
1.0
1.5
2.0
2.5
Δl
Δl
(a)
(b)
3.0
3.5
4.0
4.5
5.0
Fig. 9. Variation of (a) KI and (b) KII with the crack propagating length underunder different notch radii for interfacial cracks propagating from a semicircle notch root when E2 /E1 = 10.
notch root. The SIFs for 𝜌n = 4 and 𝜌n = 8 are plotted in Fig. 7, which are compared with the ones from a lateral pure crack. It can be observed that there is a great difference at the beginning between two lines in each graph in Fig. 7, which means that the SIFs of short crack emanating from a semicircle notch root are quite different with the ones corresponding to a lateral pure crack, because the small crack is located in the stress concentration field of the non-cracked semicircle notch. It can also be observed from Fig. 7 that, the SIFs are approaching to the same values with the increase of crack length, which means that the geometry influence of semicircle notch on the SIFs of long crack emanating from its root disappears gradually. Then, the notch radius is set at 𝜌n = 4, to study the influence of elasticity modulus ratio E2 /E1 of the bi-material couple on the variation of the SIFs for the interfacial crack emanating from a semicircular notch root. The results are presented in Fig. 8, from which it can be concluded that the elasticity modulus ratio has little influence on KI , while it affects KII strongly. KII changes sign passing from the negative to the positive with the increase of crack length, while its absolute value increases with elasticity modulus ratio from the beginning to the end.
increases with elasticity modulus ratio, while the influence of elasticity modulus ratio on KI is not significant. However, the influence of elasticity modulus ratio on KII is drastic, and the absolute value of KII increases with elasticity modulus ratio. 4. Interfacial crack propagating from a semicircle notch root The kinetic at the crack tip emanating from a bi-material semicircular notch is compared to that of the crack without presence of the geometrical defect. Let’s consider a bi-material elastic plate with a semicircular lateral notch whose radius is denoted as 𝜌n shown in Fig. 6a, where an interfacial crack with length Δl propagating from the notch root. The total length of notch radius and interfacial crack is signed as b = 𝜌n + Δl. Fig. 6b gives out the geometry of a lateral pure crack of length b along the material interface for reference. The mesh discretization is the same with the model of an interfacial crack propagating from a bi-material V-notch tip. If two materials in Fig. 6 are identical, it is changed to the problem with respect to the crack emanating from a homogeneous semicircle 5
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Next, the elasticity modulus ratio is set at 𝐸2 ∕𝐸1 = 10, to investigate the influence of notch radius on the strength of the crack from a semicircle notch root. Fig. 9 illustrates the variation of the SIFs as a function of crack length Δl for various notch radius, from which it can be seen that the values of KI corresponding to three notch radii are close to each other for short crack, while their difference becomes larger with the increase of crack length. In addition, it can be concluded from Fig. 9 that the value of KI and the absolute value of KII increase with the radius of the semicircle notch.
[2] Zhang JM, Dong YQ, Ju CM, Lin WC. A new singular element for evaluating stress intensity factors of V-shaped notches under mixed-mode load. Eng Anal Bound Elem 2018;93:161–6. [3] Nahlik L, Stegnerova K, Hutar P. Estimation of critical applied stress for crack initiation from a sharp V-notch. Theor Appl Fract Mech 2018;93:247–62. [4] Leguillon D, Yosibash Z. Crack onset at a V-notch. influence of the notch tip radius. Int J Fract 2003;122:1–21. [5] Verreman Y, Limodin N. Fatigue notch factor and short crack propagation. Eng Fract Mech 2008;75:1320–35. [6] Ranganathan N, Aldroe H, Lacroix F, Chalon F, Leroy R, Tougui A. Fatigue crack initiation at a notch. Int J Fatigue 2011;33:492–9. [7] Campagnolo A, Meneghetti G, Berto F, Tanaka K. Crack initiation life in notched steel bars under torsional fatigue: synthesis based on the averaged strain energy density approach. Int J Fatigue 2017;100:563–74. [8] Kolitsch S, Gänser HP, Pippan R. Experimental and calculation challenges of crack propagation in notches–from initiation to end of lifetime. Int J Fatigue 2017;103:395–404. [9] Ding ZY, Gao ZL, Ma CC, Wang XG. Modeling of i + ii mixed mode crack initiation and growth from the notch. Theor Appl Fract Mech 2016;84:129–39. [10] Beom HG, Jang HS. A wedge crack in an anisotropic material under antiplane shear. Int J Eng Sci 2011;49:867–80. [11] Shen MH, Lin CP, Hung SY. Edge crack in front of anisotropic wedge interacting with anti-plane singularity. Theor Appl Fract Mech 2012;58:1–8. [12] Salviato M, Zappalorto M, Maragoni L. Exact solution for the mode III stress fields ahead of cracks initiated at sharp notch tips. Eur J Mech A Solids 2018;72:88–96. [13] Mittelman B, Yosibash Z. Asymptotic analysis of the potential energy difference because of a crack at a V-notch edge in a 3D domain. Eng Fract Mech 2014;131:232–56. [14] Shiah YC, Hwu CB, Yao JJ. Boundary element analysis of the stress intensity factors of plane interface cracks between dissimilarly adjoined anisotropic materials. Eng Anal Bound Elem 2019;106:68–74. [15] Arun K, Xu LR. An experimental study on the crack initiation from notches connected to interfaces of bonded bi-materials. Eng Fract Mech 2013;111:65–76. [16] Nahlik L, Sestakova L, Hutar P, Bermejo R. Prediction of crack propagation in layered ceramics with strong interfaces. Eng Fract Mech 2010;77:2192–9. [17] Grenestedt JL, Hallstrom S. Crack initiation from homogeneous and bimaterial corners. J Appl Mech 1997;64:811–18. [18] Benedetti M, Beghini M, Fontanari V, Monelli B. Fatigue cracks emanating from sharp notches in high-strength aluminium alloys: the effect of loading direction, kinking, notch geometry and microstructure. Int J Fatigue 2009;31:1996–2005. [19] Lagoda T, Bilous P, Blacha L. Investigation on the effect of geometric and structural notch on the fatigue notch factor in steel welded joints. Int J Fatigue 2017;101:224–31. [20] Ouinas D, Bouiadjra BB, Serier B, Vina J. Influence of bimaterial interface on kinking behaviour of a crack growth emanating from notch. Comput Mater Sci 2008;41:508–14. [21] Cheng CZ, Recho N, Niu ZR, Zhou HL. Effect of notch dimension on the fatigue life of V-notched structure. Nucl Eng Des 2011;241:3573–9. [22] Gu Y, Fan CM, Xu RP. Localized method of fundamental solutions for large-scale modelling of two-dimensional elasticity problems. Appl Math Lett 2019;93:8–14. [23] Zhang AX, Gu Y, Hua QS, Chen W, Zhang CZ. A regularized singular boundary method for inverse Cauchy problem in three-dimensional elastostatics. Adv Appl Math Mech 2018;10(6):1459–77. [24] Cheng CZ, Niu ZR, Recho N. Analysis of the stress singularity for a bi-material V-notch by the boundary element method. Appl Math Model 2013;37:9398–408. [25] Williams ML. The stresses around a fault or crack in dissimilar media. Bull Seismol Soc Am 1959;49(2):199–204. [26] Yuuki R, Xu JQ. Stress based criterion for a interface crack kinking out of the interface in dissimilar materials. Eng Fract Mech 1992;41(5):635–44. [27] Niu ZR, Ge DL, Cheng CZ, Ye JQ, Recho N. Evaluation of the stress singularities of plane V-notches in bonded dissimilar materials. Appl Math Model 2009;33:1776–92. [28] Luo JF, Liu YJ, Berger EJ. Analysis of two-dimensional thin structures (from micro- to nano-scales) using the boundary element method. Comput Mech 1998;22:4040–412.
5. Conclusion The stress intensity factor (SIF) is a main parameter to evaluate the strength of a cracked component. Due to the mutual effect of geometry and material discontinuity, the calculation of the SIFs for the crack initiating from a bi-material notch is complicated. Firstly, the SIFs for the crack emanating from a bi-material notch are evaluated by the boundary element method coupled with the singularity asymptotic expansion technique. Then, the influences of geometry characteristic and material property on the SIFs of the crack propagating from a notch are investigated. The interfacial cracks from a V-notch tip and a semicircle notch root are respectively taken into consideration. Some conclusions are made as follows: The notch opening angle has great influence on the SIFs of the interfacial short crack propagating from a bi-material V-notch, while it has little influence on the SIFs of the interfacial long crack. Mode I type SIF of interfacial crack initiating from a V-notch tip increases with the notch depth, while mode II type SIF decreases with it. The value of KI and absolute value of KII for the crack from a V-notch tip increase with the elasticity modulus ratio. The elasticity modulus ratio has little influence on KI for the interfacial crack emanating from a semicircle notch root, while it has great influence on the absolute value of KII , which increases with the elasticity modulus ratio. The value of KI and absolute value of KII for the interfacial crack initiating from a semicircle notch root increase with the notch radius. Acknowledgment This project was supported by the National Natural Science Foundation of China (No. 11772114) and the Fundamental Research Funds for the Central Universities of China (PA2019GDQT0016). References [1] Berto F, Kotousov A, Lazzarin P, Pegorin F. On a coupled mode at sharp notches subjected to anti-plane loading. Eur J Mech A Solids 2013;38:70–8.
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Influence of geometry and material on the stress intensity of an interfacial crack propagating from a bi-material notch Changzheng Cheng∗, Wei Pan, Yifan Huang, Jialin Sun, Zhongrong Niu Department of Engineering Mechanics, School of Civil and Hydraulic Engineering, Hefei University of Technology, Hefei 230009, China
a r t i c l e
i n f o
Keywords: V-notch Semicircle notch Interfacial crack Stress intensity factor Boundary element method
a b s t r a c t The stress intensity factor (SIF) is an important parameter to characterize the stress intensity near a crack tip. In order to successfully evaluate the SIFs for the interfacial crack emanating from a bi-material notch, a method coupling the boundary element method with the singularity asymptotic expansion technique is introduced. Then, the crack initiating from a sharp V-notch tip and the one emanating from a semicircle notch root are respectively taken into consideration, to investigate the influence of notch geometry shape and material property on the stress intensity for the interfacial crack from a bi-material notch. It is found that the notch opening angle has great influence on the SIFs of an interfacial short crack, while it has little influence on the SIFs of an interfacial long crack propagating from a bi-material V-notch. The value of KI for the interfacial crack initiating from a V-notch tip increases with the notch depth, while KII decreases with it. The value of KI and absolute value of KII for the interfacial crack initiating from a V-notch tip increase with the elasticity modulus ratio. The elasticity modulus ratio has little influence on KI , while it has important influence on KII for the interfacial crack emanating from a semicircle notch root. The value of KI and absolute value of KII for the interfacial crack initiating from a semicircle notch root increase with the notch radius.
1. Introduction The geometry discontinuity in the engineering structure will generate a structure style named the notch [1], which can be classified into the sharp V-notch whose radius is zero and blunt notch including the semicircle notch and elliptical one. Because of the sudden change of geometry shape, the stress becomes singular or seriously concentrated near the vertex or root of a notch [2]. The crack is easy to be initiated from the notch tip due to the stress concentration [3,4]. Some research work was focused on the crack emanating from a plane notch [5]. A short crack approach was developed by Ranganathan et al. [6] to determine the fatigue crack initiation life at a notch tip. Basing on the averaged strain energy density approach, the crack initiation life in notched steel bars under torsional fatigue was studied by Campagnolo et al. [7]. The initiation and crack growth behaviour of a crack from notch was studied with the direct current potential drop technique by Kolitsch et al. [8]. By applying the finite element method, the fatigue crack initiation and growth from the notch under combined loading were analysed by Ding et al. [9]. The other research work was contributed to the crack propagating from the anti-plane or threedimensional notch. A crack emanating from the apex of anisotropic wedge under anti-plane shear was investigated by Beom and Jang [10]. The edge crack in front of the anisotropic wedge tip interacting with an
∗
anti-plane singularity was studied by Shen et al. [11]. The exact solution for the stress field ahead of the crack initiating from a sharp notch under anti-plane shear was derived in a close form by Salviato et al. [12]. The difference of the potential energy in an elastic 3D V-notch with and without a small crack was provided by Mittelman and Yosibash [13]. The material discontinuity will generate the bi-material interface edge, which is another stress singularity source [14]. An interface crack usually initiates from the interface edge. A systematic experimental study was conducted to investigate the interaction between the crack, notch and interface by Arun and Xu [15]. The crack propagation under flexure in layered ceramics designed with strong interface was investigated by Nahlik et al. [16]. Different criteria for the direction of crack propagation were studied for the fracture initiated at a corner between two different isotropic materials by Grenestedt and Hallstrom [17]. As mentioned before, both the geometry discontinuity and material discontinuity will individually generate the stress singularity in the engineering structures. The stress singularity will be very complicated if these two kinds of discontinuities occur at the same time, because they will affect each other. The effect of notch geometry on the propagation of fatigue crack emanating from a sharp V-notch was investigated by Benedetti et al. [18]. The influence of geometric and structural notch on the fatigue notch factor in steel welded joint was studied by Lagoda
Corresponding author. E-mail address: [email protected] (C. Cheng).
https://doi.org/10.1016/j.enganabound.2019.10.016 Received 7 August 2019; Received in revised form 11 October 2019; Accepted 30 October 2019 Available online xxx 0955-7997/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: C. Cheng, W. Pan and Y. Huang et al., Influence of geometry and material on the stress intensity of an interfacial crack propagating from a bi-material notch, Engineering Analysis with Boundary Elements, https://doi.org/10.1016/j.enganabound.2019.10.016
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et al. [19]. The finite element method was applied to analyze the behaviour of a crack emanating from semicircular notch root growing in an interface by Ouinas et al. [20]. The propagation of the crack emanating from a homogenous V-notch tip under fatigue loading was studied by Cheng et al. [21]. According to the knowledge of authors, the main scientific interests are focused on the stress intensity for the crack from a homogeneous V-notch, while the effect of geometry discontinuity on the stress intensity for the interfacial crack emanating from a bi-material notch has rarely been investigated. The boundary element method can reduce the dimensions of the problem by one and it can describe the singular field more accurate compared with the finite element method [22,23]. Herein, a semi-analytical method which couples the boundary element method and the singularity asymptotic expansions [24] is introduced to evaluate the stress intensity factors for the crack emanating from a bi-material notch. Then, a sharp V-notch and a semicircle notch are respectively taken, to analyze the influence of geometry characteristic and material property on the stress intensity for the crack initiating from a bi-material notch. The rule with respect to the influence of notch geometry and material property on the stress intensity for the interfacial crack from a notch is obtained.
Part 3
Material 1 Part 1
y
r o
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ρ
θ x
Part 2
Part 4
Material 2
(b)
(a)
Fig. 1. (a) The interfacial crack initiating from a bi-material V-notch tip; (b) The cracked structure is departed into four parts.
σ0
2. Determination of SIFs for the crack emanating from a bi-material V-notch
Material 1
h
The stress intensity factors (SIFs) characterize the speed of a crack growth. When a crack is propagated from a notch, it receives a driving force by the stress field created by the notch. A method should be first proposed to determine the SIFs for the crack emanating from a bimaterial notch. Let’s consider an interfacial crack emanating from a bi-material Vnotch tip shown in Fig. 1a. The cracked structure is departed into four parts as shown in Fig. 1b, which includes the singular stress semicircle zone Part 1 and Part 2, whose radii are 𝜌, and the non-singular stress zone Parts 3 and 4. The stress singularity asymptotic expressions governing in Parts 1 and 2 will be coupled with the boundary integral equations established on Parts 3 and 4 by the interfacial continuous conditions to yield the stress field of the whole structure.
γ Δl
l a
h
Material 2 σ0 w
Fig. 2. Geometry of an interfacial crack propagating from a bi-material V-notch tip.
Fig. 3. Variation of (a) KI and (b) KII with the crack propagating length under different notch opening angles for interfacial cracks propagating from a bi-material V-notch tip when l = 4 and E1 /E2 = 10. 2
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0.6
12 11
0.2 0.0
8
-0.2
KII
9
7
-0.4
6
-0.6
5
-0.8
4 0.0
0.5
1.0
1.5
2.0
l=4 l=6 l=8
0.4
l=4 l=6 l=8
10
KI
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2.5
3.0
3.5
4.0
4.5
-1.0 0.0
5.0
0.5
1.0
1.5
2.0
2.5
Δl
Δl
(a)
(b)
3.0
3.5
4.0
4.5
5.0
Fig. 4. Variation of (a) KI and (b) KII with the crack propagating length under different notch depths for interfacial cracks propagating from a V-notch tip when 𝛾 = 60° and E1 /E2 = 10.
8
1.2
7
E2/E1=2
KII
KI
6
1.0
E2/E1=2
0.8
E2/E1=4
0.6
E2/E1=20
E2/E1=4
4
E2/E1=20
0.0
E2/E1=50
-0.2
0.2
E2/E1=10
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
E2/E1=50
0.4
5
3 0.0
E2/E1=10
-0.4 0.0
5.0
0.5
1.0
1.5
2.0
2.5
Δl
Δl
(a)
(b)
3.0
3.5
4.0
4.5
5.0
Fig. 5. Variation of (a) KI and (b) KII with the crack propagating length under different elasticity modulus ratios for interfacial cracks propagating from a V-notch tip when l = 4 and 𝛾 = 120°.
The singular eigen value 𝜆 for an interfacial crack can be written as [25,26] 𝜆 = 0.5 + 𝑖𝜀
𝑢𝑗𝑞 =
𝑁 ∑ 𝑘=1
{ [ ] 𝑗 𝑗 𝜌1∕2 𝐴𝑘R 𝑈̃ 𝑞𝑘 cos (𝜀 ln 𝜌) − 𝑈̃ 𝑞𝑘 sin(𝜀 ln 𝜌) R I
[ ] } 𝑗 𝑗 − 𝐴𝑘I 𝑈̃ 𝑞𝑘 sin (𝜀 ln 𝜌) − 𝑈̃ 𝑞𝑘 cos (𝜀 ln 𝜌) R I
(1)
(4a)
where i is the imaginary unit, and 𝜀=
1 ln 2π
(
1−𝛽 1+𝛽
)
𝑡𝑗𝑞 =
(2)
𝜇1 (𝜅2 − 1) − 𝜇2 (𝜅1 − 1) 𝜇1 (𝜅2 + 1) + 𝜇2 (𝜅1 + 1)
𝑘=1
{ [ ] 𝑗 𝑗 𝜌−1∕2 −𝐴𝑘R 𝑇̃𝑞𝑘 cos (𝜀 ln 𝜌) + 𝑇̃𝑞𝑘 sin (𝜀 ln 𝜌) R I
[ ]} 𝑗 𝑗 + 𝐴𝑘I 𝑇̃𝑞𝑘 sin (𝜀 ln 𝜌) − 𝑇̃𝑞𝑘 cos (𝜀 ln 𝜌) R I
with 𝛽=
𝑁 ∑
(4b)
where q = 1, 2, N is the truncated asymptotic expansion terms. 𝑈̃ and 𝑇̃ respectively are the combinations of displacement and stress eigen angular functions, which can be determined by the material property in the bi-material structure [27]. AkR and AkI respectively are the real part and imaginary part of amplitude coefficients in the asymptotic expansions, which are determined by the external loading. The displacement and traction on the boundary of the jth part (j = 3, 4) can be calculated by the displacement boundary integral
(3)
in which 𝜅 j = (3 − vj )/(1 + vj ) (j = 1, 2) for a plane stress problem and 𝜅 j = 3 − 4vj for a plane strain problem, vj and 𝜇 j are the Poisson’s ratio and shear modulus for the jth material, respectively. The displacement and singular traction components on the arc boundary of the jth part (j = 1, 2) can be written as [24] 3
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notch can be determined by the equation as follows √ [ ]| 𝐾I + 𝑖𝐾II = lim 2π𝑟−1∕2−𝑖𝜀 𝜎𝜃 (𝑟, 𝜃) + 𝑖𝜎𝑟𝜃 (𝑟, 𝜃) || 𝑟→0 |𝜃=0◦
σ0
σ0
h
Material 1
As shown in Fig. 2, a bi-material lateral sharp V-notch formed by the assembly of two different materials is considered, in which half height h = 60, width w = 40, thickness t = 1. The plate is subjected to an uniaxial load 𝜎 0 = 1 in the vertical direction. The plane stress condition is assumed. A crack of length Δl is supposed to emanate from the V-notch tip along the interface. The notch opening angle and notch depth are respectively denoted as 𝛾 and l, and the total length of the V-notch and crack from its tip is denoted as a = l + Δl. There are 48 uniform segments divided on the arc boundary of the sector dug out from the crack tip in the singularity analysis. The remote structure without the singular stress area is discretized into 228 quadratic boundary elements. First, the notch depth is set at l = 4 and the elasticity modulus ratio of two materials is set at E1 /E2 = 10, where E1 and E2 respectively are the elasticity modulus of material 1 and material 2, to investigate the influence of notch opening angle on the SIFs of the interfacial crack emanating from a V-notch tip. The results are shown in Fig. 3, from which it can be seen that KI increases with the increment of propagating crack length Δl, while the variation rule for KII is on the contrary. The difference of the SIFs between different opening angle is significant when the propagating crack length Δl is small, however, this difference is approaching to zero when the propagating crack length becomes larger. It means that the notch opening angle has great influence on the short crack propagating from a bi-material V-notch tip, while it has little influence on the long crack. Then, the notch opening angle and elasticity modulus ratio are respectively set at 𝛾 = 60° and E1 /E2 = 10, to investigate the behavior of the crack propagating from a bi-material V-notch under different notch depth. The results are shown in Fig. 4, from which it can be found that KI increases with the notch depth significantly, while KII decreases with the notch depth. The difference for KII under different notch depth is small for the short crack, while this difference is significant for the long crack. Next, the notch depth and notch opening angle are respectively set at l = 4 and 𝛾 = 120°, to investigate the influence of elasticity modulus ratio on the SIFs of the interfacial crack from a bi-material V-notch tip. The results are shown in Fig. 5, from which it can be observed that, KI
b
Δl
h
h
Material 2
Material 2
σ0
σ0 w
w
(a)
(b)
Fig. 6. (a) Geometry of an interfacial crack propagating from a bi-material semicircle notch root; (b) Geometry of a lateral pure interfacial crack.
equation as follows [28], 𝐶𝑝𝑞 (𝐲)𝑢𝑗𝑞 (𝐲) =
∫Γ𝑗
𝑗 𝑈𝑝𝑞 (𝐱, 𝐲)𝑡𝑗𝑞 (𝐱)dΓ(𝐱) −
∫Γ𝑗
𝑗 𝑇𝑝𝑞 (𝐱, 𝐲)𝑢𝑗𝑞 (𝐱)dΓ(𝐱) (𝑗 = 3, 4) (5)
where p, q = 1, 2, y is the source point and x is the field point, Cpq (y) is the singularity coefficient which depends on the geometry boundary at point y, 𝑢𝑗𝑞 and 𝑡𝑗𝑞 respectively are the displacement and traction com𝑗 ponents on the boundary of the jth part. 𝑈𝑝𝑞 (𝐱, 𝐲) is the Kelvin funda𝑗 𝑗 mental solution, and 𝑇𝑝𝑞 (𝐱, 𝐲) is the linear combination of 𝑈𝑝𝑞 (𝐱, 𝐲) and its derivatives with respect to coordinate xj . By using the interfacial continuous conditions between the boundaries of four parts shown in Fig. 1b, the boundary unknowns together with the amplitude coefficients AkR and AkI in Eq. (4) can be evaluated. Then, the stresses 𝜎 𝜃 and 𝜎 r𝜃 near the crack tip can be yielded by the asymptotic expansions. Then, mode I type stress intensity factor 𝐾I and mode II type stress intensity factor KII of the interfacial crack emanating from a bi-material
12
8 7
10
6 8
KI
KI
5 4 3
Crack from semicircular notch root Pure laterial crack
2 1 4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
(6)
3. Interfacial crack propagating from a sharp V-notch vertex
h
Material 1
ρn
b
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6 4 2 8.0
9.0
Crack from semicircular notch root Pure laterial crack 8.5
9.0
9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0
b
b
(a)
(b)
Fig. 7. Variation of KI with the crack length for cracks emanating from a homogeneous semicircle notch root: (a)𝜌n = 4, (b)𝜌n = 8. 4
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1.2
7
1.0
E2/E1=2 E2/E1=5
0.8
6
E2/E1=10
0.6
5
E2/E1=2
KII
KI
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4
E2/E1=10
0.2
3
E2/E1=20
0.0
2 1 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
E2/E1=50
0.4
E2/E1=5
E2/E1=50
E2/E1=20
-0.2 4.5
-0.4 0.0
5.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Δl
Δl
(b)
(a)
12
1.6
10
1.2
8
0.8
6
KII
KI
Fig. 8. Variation of (a) KI and (b) KII with the crack propagating length under different elasticity modulus ratios for interfacial cracks emanating from a semicircle notch root when 𝜌n = 4.
ρn=6
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
ρn=4 ρn=6
-0.4
ρn=8
2 0.0
0.0
ρn=4
4
0.4
-0.8 0.0
5.0
ρn=8 0.5
1.0
1.5
2.0
2.5
Δl
Δl
(a)
(b)
3.0
3.5
4.0
4.5
5.0
Fig. 9. Variation of (a) KI and (b) KII with the crack propagating length underunder different notch radii for interfacial cracks propagating from a semicircle notch root when E2 /E1 = 10.
notch root. The SIFs for 𝜌n = 4 and 𝜌n = 8 are plotted in Fig. 7, which are compared with the ones from a lateral pure crack. It can be observed that there is a great difference at the beginning between two lines in each graph in Fig. 7, which means that the SIFs of short crack emanating from a semicircle notch root are quite different with the ones corresponding to a lateral pure crack, because the small crack is located in the stress concentration field of the non-cracked semicircle notch. It can also be observed from Fig. 7 that, the SIFs are approaching to the same values with the increase of crack length, which means that the geometry influence of semicircle notch on the SIFs of long crack emanating from its root disappears gradually. Then, the notch radius is set at 𝜌n = 4, to study the influence of elasticity modulus ratio E2 /E1 of the bi-material couple on the variation of the SIFs for the interfacial crack emanating from a semicircular notch root. The results are presented in Fig. 8, from which it can be concluded that the elasticity modulus ratio has little influence on KI , while it affects KII strongly. KII changes sign passing from the negative to the positive with the increase of crack length, while its absolute value increases with elasticity modulus ratio from the beginning to the end.
increases with elasticity modulus ratio, while the influence of elasticity modulus ratio on KI is not significant. However, the influence of elasticity modulus ratio on KII is drastic, and the absolute value of KII increases with elasticity modulus ratio. 4. Interfacial crack propagating from a semicircle notch root The kinetic at the crack tip emanating from a bi-material semicircular notch is compared to that of the crack without presence of the geometrical defect. Let’s consider a bi-material elastic plate with a semicircular lateral notch whose radius is denoted as 𝜌n shown in Fig. 6a, where an interfacial crack with length Δl propagating from the notch root. The total length of notch radius and interfacial crack is signed as b = 𝜌n + Δl. Fig. 6b gives out the geometry of a lateral pure crack of length b along the material interface for reference. The mesh discretization is the same with the model of an interfacial crack propagating from a bi-material V-notch tip. If two materials in Fig. 6 are identical, it is changed to the problem with respect to the crack emanating from a homogeneous semicircle 5
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Next, the elasticity modulus ratio is set at 𝐸2 ∕𝐸1 = 10, to investigate the influence of notch radius on the strength of the crack from a semicircle notch root. Fig. 9 illustrates the variation of the SIFs as a function of crack length Δl for various notch radius, from which it can be seen that the values of KI corresponding to three notch radii are close to each other for short crack, while their difference becomes larger with the increase of crack length. In addition, it can be concluded from Fig. 9 that the value of KI and the absolute value of KII increase with the radius of the semicircle notch.
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5. Conclusion The stress intensity factor (SIF) is a main parameter to evaluate the strength of a cracked component. Due to the mutual effect of geometry and material discontinuity, the calculation of the SIFs for the crack initiating from a bi-material notch is complicated. Firstly, the SIFs for the crack emanating from a bi-material notch are evaluated by the boundary element method coupled with the singularity asymptotic expansion technique. Then, the influences of geometry characteristic and material property on the SIFs of the crack propagating from a notch are investigated. The interfacial cracks from a V-notch tip and a semicircle notch root are respectively taken into consideration. Some conclusions are made as follows: The notch opening angle has great influence on the SIFs of the interfacial short crack propagating from a bi-material V-notch, while it has little influence on the SIFs of the interfacial long crack. Mode I type SIF of interfacial crack initiating from a V-notch tip increases with the notch depth, while mode II type SIF decreases with it. The value of KI and absolute value of KII for the crack from a V-notch tip increase with the elasticity modulus ratio. The elasticity modulus ratio has little influence on KI for the interfacial crack emanating from a semicircle notch root, while it has great influence on the absolute value of KII , which increases with the elasticity modulus ratio. The value of KI and absolute value of KII for the interfacial crack initiating from a semicircle notch root increase with the notch radius. Acknowledgment This project was supported by the National Natural Science Foundation of China (No. 11772114) and the Fundamental Research Funds for the Central Universities of China (PA2019GDQT0016). References [1] Berto F, Kotousov A, Lazzarin P, Pegorin F. On a coupled mode at sharp notches subjected to anti-plane loading. Eur J Mech A Solids 2013;38:70–8.
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