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Analysis of a Kinked Crack in an Anisotropic Material Under Antiplane Deformation† H

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Analysis of a Kinked Crack in an Anisotropic Material Under Antiplane Deformation† H Journal of Mechanical Science and Technology 26 (2) (2012) 411~419 www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-1025-4 Analysis of a kinked crack in an anisotropic material under antiplane deformation† H. G. Beom*, J. W. Lee and C. B. Cui Department of Mechanical Engineering, Inha University, 253 Yonghyun-dong, Incheon, 402-751, Korea (Manuscript Received July 21, 2011; Revised September 23, 2011; Accepted September 23, 2011) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract This paper analyzes the asymptotic problem of a kinked crack in an anisotropic material under antiplane deformation. Using the linear transformation method proposed in this paper, a solution to the asymptotic problem of a kinked crack in an anisotropic material can be obtained from the solution of the corresponding isotropic kinked crack problem. The exact solution of the stress intensity factor for the kinked crack in the anisotropic material is obtained from the solution of the isotropic problem. The effect of the kink angle and two ani- sotropic parameters on the stress intensity factor is discussed for the inclined orthotropic material as well as the anisotropic material. In order to verify the exact solution of the stress intensity factor, numerical calculations are performed by using finite element analysis. Keywords: Kinked crack; Anisotropic material; Antiplane deformation; Linear transformation method; Stress intensity factor ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- based on the Stroh formalism, and numerically obtained the 1. Introduction stress intensity factors. Kinking of a crack is observed in an elastic material that is The purpose of this study is to analyze the asymptotic prob- subjected to mixed-mode loading. The problem of a kinked lem of a kinked crack in an anisotropic material under anti- crack in elastic material has thus been investigated to predict plane deformation. Our attention is focused on the onset of the direction of crack propagation. The results of studies by crack kinking. A linear transformation method is proposed to many researchers on kinked cracks in elastic material under solve the anisotropic antiplane problem. Once the solution of inplane deformation were presented in Refs. [1, 2]. The analy- the transformed isotropic problem is determined, the complete sis of kinked cracks in isotropic material under antiplane de- fields of the antiplane displacement and stress for the original formation has also been carried out. Sih [3] solved the prob- anisotropic problem can be evaluated based on the linear lem of a kinked crack under antiplane shear loading. Subse- transformation method. The asymptotic problem of the kinked quently, studies on kinked cracks under antiplane deformation crack in isotropic material is solved by using the Schwarz- have been undertaken by several researchers. Smith [4] exam- Christoffel transformation. A closed-form solution of the ined crack-forking in antiplane deformation. Wu [5] used the stress intensity factor for the kinked crack in anisotropic mate- conformal mapping method to solve the problem of a slender rial is obtained. It is shown that the stress intensity factor for Z-crack. Choi and Earmme [6] explored the problem of a the kinked crack is significantly affected by the anisotropic kinked crack in antiplane shear by using the Mellin transform parameters as well as the kink angle. In order to validate the and the Wiener-Hopf technique. Some antiplane problems for exact solution of the stress intensity factor obtained in this an anisotropic material have been investigated. Shahani [7] paper, numerical calculations are performed. The stress inten- considered an anisotropic wedge under antiplane deformation. sity factors are obtained numerically from the J integral. The J He employed the standard finite Mellin transforms to obtain integral is evaluated in an auxiliary post-processing program closed-form expressions for the elastic fields. Li and Lee [8] after finite element analysis. solved a crack in an orthotropic strip by using the Fourier series method. Blanco et al. [9] and Yang and Yuan [10] 2. Formulation solved the kinked crack in anisotropic material under both 2.1 Kinked crack in an anisotropic material inplane deformation and antiplane deformation. They formu- lated the problem in the form of singular integral equations Consider a homogeneous linear elastic solid under antiplane † This paper was recommended for publication in revised form by Editor deformation. The material considered in this study is anisot- Jai Hak Park ropic material with the x3 = 0 plane corresponding to mate- *Corresponding author. Tel.: +82 32 860 7310, Fax.: +82 32 868 1716 E-mail address: [email protected] rial-property symmetry, so that the inplane and antiplane de- © KSME & Springer 2012 formations are decoupled. The antiplane displacement, u3 , 412 H. G. Beom et al. / Journal of Mechanical Science and Technology 26 (2) (2012) 411~419 ⎡ ⎤ depends on the inplane coordinates x1 and x2 . The constitu- i 2z u3 = Re⎢ K III ⎥ tive equation of the anisotropic material under antiplane de- ⎣μ π ⎦ formation can be written in the following form: x2 p r x θ ⎧⎫⎡γ SS ⎤⎧⎫σ 1 31 55 45 31 p p b ⎨⎬= ⎢⎥ ⎨⎬ (1) x2 θ ⎩⎭⎣γ 32SS 45 44 ⎦⎩⎭σ 32 ω x1 where σ31 and σ32 are the shear stresses, γ 31 and γ 32 are the shear strains, and S44 , S45 , and S55 are the components of the conventional compliance matrix. The general solution of antiplane displacement in anisot- ropic material satisfying the equilibrium equation and the cor- responding stress components can be written in terms of one Fig. 1. Asymptotic problem of a semi-infinite kinked crack in an ani- analytic function as [11] sotropic material. ⎡⎤1 angle ω in an anisotropic material, as shown in Fig. 1. The uif(z)3 = 2Re⎢⎥ ⎣⎦μ main crack lies on the negative x1 axis. The kink angle ω is measured from the positive x1 axis and has a positive sign ⎧⎫σ31 ⎡⎤⎧⎫p ⎨⎬= 2Re⎢⎥ ⎨⎬f (z)′ for the counterclockwise direction. The crack surface is as- σ −1 ⎩⎭32 ⎣⎦⎢⎥⎩⎭ sumed to be traction free, and the remote load is given by the Tf(z)3 = 2Re⎣⎦⎡⎤. (2) singular field of the corresponding crack without a kink. The boundary conditions on the crack surface are Here, T3 is the resultant force over an arc of the material. Re denotes the real part, ( )' indicates the derivative with σ3θ = 0 , r > 0 , θπ= ± ± respect to the associate argument, and μ is the equivalent σ3θ = 0 , 0 < rb< , θω= (6) shear modulus given by where r and θ are cylindrical coordinates centered at the 1 μ = . (3) main crack tip, as shown in Fig. 1. The analytic function 2 SS44 55− S 45 f ()z that generates the remote fields of the asymptotic prob- lem can be expressed as The function f(z) is a holomorphic function, z is a com- z plex variable defined by zx=+12 px, and p is f(z)=− KIII , as z →∞ (7) 2π pi=+λη (4) where K III is the mode III stress intensity factor for the main where λ and η are the non-dimensional parameters given crack without a kink. It is noted that KIII is the applied stress by intensity factor at infinity. From Eqs. (2) and (7), the dis- placement and stress field at infinity for the asymptotic prob- 2 SS44 55− S 45 lem is given by λ = S55 ⎡ iz2 ⎤ S45 uK=−Re η = . (5) 3 ⎢ III ⎥ ⎢ μπ ⎥ S55 ⎣ ⎦ ⎧⎫σ ⎡ K ⎧ p ⎫⎤ 31 =−Re III . (8) The parameters λ and η measure the anisotropy of the ⎨ ⎬⎨⎬⎢ ⎥ ⎩⎭σ32 ⎣⎢ 2πz ⎩⎭−1 ⎦⎥ material. The positive definiteness of the strain energy density requires that λ > 0 . It is noted that λ =1 and η = 0 for an 2.2 The linear transformation method isotropic material. We now consider a kinked crack in an anisotropic material. A linear transformation method is proposed to solve the ani- Our attention is focused on the initiation of the crack kink; sotropic antiplane problem. Using the linear transformation thus, the kink length considered in this study is assumed to be method, the solution of the asymptotic problem of a kinked small compared to all of the inplane geometric lengths. There- crack in an anisotropic material can be obtained from the solu- fore, this situation can be considered as an asymptotic problem tion of the corresponding isotropic kinked-crack problem. We of a semi-infinite kinked crack with a kink of length b and introduce a linear transformation defined by H. G. Beom et al. / Journal of Mechanical Science and Technology 26 (2) (2012) 411~419 413 xxˆ11=+ηx 2 and the boundary is separated into Su and St , where the displacement and traction are specified, respectively. When xˆ22= λx . (9) the boundary conditions of the transformed problem are given Under transformation Eq. (9), point ( x1 , x2 ) of the anisot- as ropic solid is mapped into point ( xˆ1 , xˆ2 ) of the transformed solid. It is easily seen from Eq. (9) that ˆ uuˆ33= % , on Su TTˆ = % , on Sˆ (16) zzˆ = (10) 33 t ˆ ˆ where zˆ is the complex variable defined by zxixˆ =+ˆˆ12. where Su and St are the transformed boundaries in which The transformed solid is assumed to be composed of an iso- the displacement and traction are specified, respectively. The tropic material with shear modulus μˆ . For the isotropic mate- solution of the transformed problem with the boundary condi- rial, Eq. (2) can be rewritten as tions Eq. (16) satisfies Eq. (12). Using Nanson’s formula [12], it can be shown that ⎡⎤1 ˆ ˆ uif(z)3 = 2Re⎢⎥ˆ ˆ 22 ⎣⎦μ dsˆ =+−()(λη n121 n n ) ds . (17) ˆ ⎡⎤ ⎧⎫σ31 ⎧⎫i ˆ ⎨⎬= 2Re⎢⎥ ⎨⎬f (z)′ ˆ ˆ Here, ni is the component of the unit outward vector normal ⎩⎭σ32 ⎣⎦⎢⎥⎩⎭−1 to the boundary surface, while dsˆ and ds are the infini- Tf(z)ˆ = 2Re⎡⎤ˆ ˆ (11) ˆ 3 ⎣⎦ tesimal arc lengths along St and St , respectively.
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