Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 741075 |

Nik Mohd Asri Nik Long, Lee Feng Koo, Zainidin K. Eshkuvatov, "Computation of Energy Release Rates for a Nearly Circular Crack", Mathematical Problems in Engineering, vol. 2011, Article ID 741075, 17 pages, 2011.

Computation of Energy Release Rates for a Nearly Circular Crack

Academic Editor: Jerzy Warminski
Received04 Aug 2010
Revised06 Dec 2010
Accepted14 Jan 2011
Published08 Mar 2011


This paper deals with a nearly circular crack, Ξ© in the plane elasticity. The problem of finding the resulting shear stress can be formulated as a hypersingular integral equation over a considered domain, Ξ© and it is then transformed into a similar equation over a circular region, 𝐷, using conformal mapping. Appropriate collocation points are chosen on the region 𝐷 to reduce the hypersingular integral equation into a system of linear equations with (2𝑁+1)(𝑁+1) unknown coefficients, which will later be used in the determination of energy release rate. Numerical results for energy release rate are compared with the existing asymptotic solution and are displayed graphically.

1. Introduction

The determination of energy release rate, a measurement of energy necessary for crack initiation in fracture mechanics, has stirred a huge interest among researchers, and different approaches have been applied. Williams and Isherwood [1] proposed an approximate method in terms of a mean stress to approximate the strain-energy release rates of finite plates. Sih [2] proposed the energy density theory as an alternative approach for fracture prediction. Hayashi and Nemat-Nasser [3] modelled the kink as a continuous distribution of infinitesimal edge dislocations to obtain the energy release rate at the onset of kinking of a straight crack in an infinite elastic medium subjected to a predominantly Mode I loading. Further, a similar method to [3] has also been adopted by Hayashi and Nemat-Nasser [4] to obtain the energy release rate for a kinked from a straight crack under combined loading based on the maximum energy release rate criterion. Gao and Rice [5] extended Rice's work [6] in finding the energy release rate for a plane crack with a slightly curved front subject to shear loading. While, Gao and Rice [7] and Gao [8] considered a penny-shaped crack as a reference crack in solving the energy release rate for a nearly circular crack subject to normal and shear loads. Jih and Sun [9] employed the finite element method based on crack-closure integral in calculating the strain energy release rate elastostatic and elastodynamic crack problems in finite bodies whereas Dattaguru et al. [10] adopted the finite element analysis and modified crack closure integral technique in evaluating the strain energy release rate. Poon and Ruiz [11] applied the hybrid experimental-numerical method for determining the strain energy release rate. Wahab and de Roeck [12] evaluated the strain energy release rate from three-dimensional finite element analysis with square-root stress singularity using different displacement and stress fields based on the Irwin's crack closure integral method [13]. Guo et al. [14] used the extrapolation approach in order to avoid the disadvantages of self-inconsistency in the point-by-point closed method to determine the energy release rate of complex cracks. Xie et al. [15] applied the virtual crack closure technique in conjunction with finite element analysis for the computation of energy release rate subject to kinked crack, while interface element based on similar approach also adopted by Xie and Biggers [16] in calculating the strain energy release rate for stationary cracks subjected to the dynamic loading.

In this paper, we focus our work on obtaining the numerical results for energy release rate for a nearly circular crack via the solution of hypersingular integral equation and compare our computational results with Gao's [8].

2. Formulation of the Problem

Consider the infinite isotropic elastic body containing a flat circular crack, Ξ©, as in Figure 1, located on the Cartesian coordinate (π‘₯,𝑦,π‘₯3) with origin 𝑂, and Ξ© lies in the plane π‘₯3=0. Let the radius of the crack, Ξ© be π‘Ž and Ξ©={(π‘Ÿ,πœƒ)∢0β‰€π‘Ÿ<π‘Ž,βˆ’πœ‹β‰€πœƒ<πœ‹}.

If the equal and opposite shear stresses in the π‘₯ and 𝑦 directions, π‘ž1(π‘₯,𝑦) and π‘ž2(π‘₯,𝑦), respectively, are applied to the crack plane, and it is assumed that the π‘₯3 direction is traction free, then in the view of shear load, the entire plane, must free from the normal stress, that is 𝜏33ξ€·π‘₯,𝑦,π‘₯3ξ€Έ=0forπ‘₯3=0,(2.1) and the stress field can be found by considering the above problem subjected to the following mixed boundary condition on its surface, π‘₯3=0: 𝜏13ξ€·π‘₯,𝑦,π‘₯3ξ€Έ=πœ‡π‘ž1βˆ’πœˆ1𝜏(π‘₯,𝑦),(π‘₯,𝑦)∈Ω,23ξ€·π‘₯,𝑦,π‘₯3ξ€Έ=πœ‡π‘ž1βˆ’πœˆ2𝑒(π‘₯,𝑦),(π‘₯,𝑦)∈Ω,1ξ€·π‘₯,𝑦,π‘₯3ξ€Έ=𝑒2ξ€·π‘₯,𝑦,π‘₯3ξ€Έ=0,(π‘₯,𝑦)βˆˆΞ“β§΅Ξ©,(2.2) where πœπ‘–π‘— is stress tensor, πœ‡ is shear modulus, 𝜈 is denoted as Poisson's ratio, and Ξ“ is the entire π‘₯3=0. Also, the problem satisfies the regularity conditions at infinity 𝑒𝑖π‘₯,𝑦,π‘₯3ξ€Έξ‚€1=𝑂𝑅,πœπ‘–π‘—ξ€·π‘₯,𝑦,π‘₯3ξ€Έξ‚€1=𝑂𝑅,𝑖,𝑗=1,2,3,π‘…β†’βˆž,(2.3) where 𝑅 is the distance 𝑅=ξ€·π‘₯βˆ’π‘₯0ξ€Έ2+ξ€·π‘¦βˆ’π‘¦0ξ€Έ2,ξ€·π‘₯0,𝑦0ξ€ΈβˆˆΞ©.(2.4) Martin [17] showed that the problem of finding the resultant force with condition (2.2) can be formulated as a hypersingular integral equation 1ξ€²8πœ‹Ξ©(2βˆ’πœˆ)𝑀(π‘₯,𝑦)+3πœˆπ‘’2π‘—Ξ˜π‘€(π‘₯,𝑦)𝑅3ξ€·π‘₯𝑑Ω=π‘ž0,𝑦0ξ€Έ,ξ€·π‘₯0,𝑦0ξ€ΈβˆˆΞ©,(2.5) where 𝑀(π‘₯,𝑦)=[𝑒1(π‘₯,𝑦)]+𝑗[𝑒2(π‘₯,𝑦)] is the unknown crack opening displacement, π‘ž(π‘₯0,𝑦0)=π‘ž1(π‘₯0,𝑦0)+π‘—π‘ž2(π‘₯0,𝑦0), 𝑗2=βˆšβˆ’1, the 𝑀(π‘₯,𝑦)=[𝑒1(π‘₯,𝑦)]βˆ’π‘—[𝑒2(π‘₯,𝑦)], and the angle Θ is defined by π‘₯βˆ’π‘₯0=𝑅cosΘ,π‘¦βˆ’π‘¦0=𝑅sinΘ.(2.6) The cross on the integral means the hypersingular, and it must be interpreted as a Hadamard finite part integral [18, 19]. Equation (2.5) is to be solved subject to 𝑀=0 on πœ•Ξ© where πœ•Ξ© is boundary of Ξ©. For the constant shear stress in π‘₯ direction, we have 𝜏23=0 and [𝑒2(π‘₯,𝑦)]=0, hence, (2.5) becomes 1ξ€²8πœ‹Ξ©2βˆ’πœˆ+3πœˆπ‘’2π‘—Ξ˜π‘…3ξ€·π‘₯𝑀(π‘₯,𝑦)𝑑Ω=π‘ž0,𝑦0ξ€Έ,ξ€·π‘₯0,𝑦0ξ€ΈβˆˆΞ©.(2.7)

Polar coordinates (π‘Ÿ,πœƒ) and (π‘Ÿ0,πœƒ0) are chosen so that the loadings π‘ž(π‘₯,𝑦) and π‘ž(π‘₯0,𝑦0) can be written as a Fourier series π‘ž(π‘₯,𝑦)=βˆžξ“π‘›=βˆ’βˆžπ‘žπ‘›ξ‚€π‘Ÿπ‘Žξ‚π‘’π‘—π‘›πœƒξ€·π‘₯,π‘ž0,𝑦0ξ€Έ=βˆžξ“π‘›=βˆ’βˆžπ‘žπ‘›ξ‚΅π‘Ÿ0π‘Ž0ξ‚Άπ‘’π‘—π‘›πœƒ0,(2.8) where the Fourier components π‘žπ‘› are 𝑗-complex. The 𝑗-complex crack opening displacement, 𝑀(π‘₯,𝑦) and 𝑀(π‘₯0,𝑦0), have similar expressions 𝑀(π‘₯,𝑦)=βˆžξ“π‘›=βˆ’βˆžπ‘€π‘›ξ‚€π‘Ÿπ‘Žξ‚π‘’π‘—π‘›πœƒξ€·π‘₯,𝑀0,𝑦0ξ€Έ=βˆžξ“π‘›=βˆ’βˆžπ‘€π‘›ξ‚΅π‘Ÿ0π‘Ž0ξ‚Άπ‘’π‘—π‘›πœƒ0.(2.9) Without loss of generality, we consider π‘Ž=1. Using Guidera and Lardner [20], the dimensionless function π‘žπ‘› and 𝑀𝑛 can be expressed as π‘žπ‘›(π‘Ÿ)=π‘Ÿβˆž|𝑛|ξ“π‘˜=0π‘„π‘›π‘˜Ξ“ξ‚€1|𝑛|+2Γ3π‘˜+2ξ‚βˆš(|𝑛|+π‘˜)!1βˆ’π‘Ÿ2𝐢|𝑛|+122π‘˜+1ξ‚€βˆš1βˆ’π‘Ÿ2,𝑀𝑛(π‘Ÿ)=π‘Ÿβˆž|𝑛|ξ“π‘˜=0π‘Šπ‘›π‘˜Ξ“(|𝑛|+1/2)π‘˜!𝐢Γ(|𝑛|+π‘˜+3/2)|𝑛|+1/22π‘˜+1ξ‚€βˆš1βˆ’π‘Ÿ2,(2.10) where the 𝑗-complex coefficients π‘„π‘›π‘˜ are known, π‘Šπ‘›π‘˜ are unknown, and πΆπœ†π‘š(π‘₯) is an orthogonal Gegenbauer polynomial of degree π‘š and index πœ†, which is defined recursively by [21] (π‘š+2)πΆπœ†π‘š+2(π‘₯)=2(π‘š+πœ†+1)π‘₯πΆπœ†π‘š+1(π‘₯)βˆ’(2πœ†+π‘š)πΆπœ†π‘š(π‘₯),(2.11) with the initial values πΆπœ†0(π‘₯)=1 and πΆπœ†1(π‘₯)=2πœ†π‘₯. For a constant shear loading, π‘ž(π‘₯,𝑦)=βˆ’πœ, the solution for a circular crack is obtainable.

3. Nearly Circular Crack

Let Ξ© be an arbitrary shaped crack of smooth boundary with respect to origin 𝑂, such that Ξ© is defined as Ξ©={(π‘Ÿβ‹…πœƒ)∢0β‰€π‘Ÿ<𝜌(πœƒ),βˆ’πœ‹β‰€πœƒ<πœ‹},(3.1) where the boundary of Ξ©, πœ•Ξ© is given by π‘Ÿ=𝜌(πœƒ). Let 𝜁=πœ‰+π‘–πœ‚=π‘ π‘’π‘–πœ‘ with |𝜁|<1 such that the unit disc is 𝐷≑{(𝑠,πœ‘)∢0≀𝑠<1,βˆ’πœ‹β‰€πœ‘<πœ‹}.(3.2) By the properties of Reimann mapping theorem [22], a circular disc 𝐷 is mapped conformally onto Ξ© using 𝑧=π‘Žπ‘“(𝜁). This approach works for a general smooth star-shaped domain, Ξ©. For a particular application, let 𝑓 be an analytic function, simply connected in the domain Ξ©, |π‘“ξ…ž(𝜁)| is nonzero and bounded for all |𝜁|<1, 𝑓(𝜁)=𝜁+𝑐𝑔(𝜁)with𝑔(𝜁)=πœπ‘š+1,(3.3) which maps a unit circle, 𝐷 in the 𝜁-plane into a nearly circular domain Ξ© in the 𝑧-plane where 𝑐 is a real parameter and π‘Ÿ=𝜌(πœƒ) is the boundary of Ξ©. This domain has a smooth, regular boundary for 0≀(π‘š+1)|𝑐|<1. As (π‘š+1)|𝑐|β†’1 one or more cusps develop; see Figure 2 with various choices of 𝑐.

Let π‘§βˆ’π‘§0ξ€·ξ€·πœ=π‘Žπ‘“(𝜁)βˆ’π‘“0ξ€Έξ€Έ=Reπ‘–Ξ˜,(3.4) and define 𝑆 and Ξ¦ as πœβˆ’πœ0=𝑆𝑒𝑖Φ,𝑑Ω=𝑑π‘₯𝑑𝑦=π‘Ž2||π‘“ξ…ž||(𝜁)2π‘‘πœ‰π‘‘πœ‚=π‘Ž2||π‘“ξ…ž||(𝜁)2,π‘ π‘‘π‘ π‘‘πœ‘(3.5) where π‘₯=π‘Žπ‘’(πœ‰,πœ‚) and 𝑦=π‘Žπ‘£(πœ‰,πœ‚) so that 𝑓=𝑒+𝑖𝑣. Next, we define 𝛿 and 𝛿0 as π‘“ξ…ž||𝑓(𝜁)=ξ…ž||𝑒(𝜁)𝑖𝛿,π‘“ξ…žξ€·πœ0ξ€Έ=||π‘“ξ…žξ€·πœ0ξ€Έ||𝑒𝑖𝛿0.(3.6) Set ||𝑓𝑀(π‘₯(𝜁),𝑦(𝜁))=π‘Žξ…ž||(𝜁)βˆ’1/2π‘’π‘—π›Ώπ‘žξ€·π‘₯ξ€·πœπ‘Š(πœ‰,πœ‚),(3.7)0ξ€Έξ€·πœ,𝑦0||𝑓=π‘Žξ…žξ€·πœ0ξ€Έ||βˆ’3/2𝑒𝑗𝛿0π‘„ξ€·πœ‰0,πœ‚0ξ€Έ.(3.8) Substituting (3.5), (3.6), (3.7), and (3.8) into (2.7) gives2βˆ’πœˆ+3πœˆπ‘’2π‘—Ξ˜ξ€²8πœ‹π·π‘Š(πœ‰,πœ‚)𝑆3π‘‘πœ‰π‘‘πœ‚+2βˆ’πœˆξ€§8πœ‹π·π‘Š(πœ‰,πœ‚)𝐾(1)ξ€·πœ,𝜁0ξ€Έ+π‘‘πœ‰π‘‘πœ‚3πœˆξ€œ8πœ‹π·π‘Š(πœ‰,πœ‚)𝐾(2)ξ€·πœ,𝜁0ξ€Έξ€·πœ‰π‘‘πœ‰π‘‘πœ‚=𝑄0,πœ‚0ξ€Έ,ξ€·πœ‰0,πœ‚0ξ€Έβˆˆπ·,(3.9) where the kernel 𝐾(1)(𝜁,𝜁0) and 𝐾(2)(𝜁,𝜁0) are [17]𝐾(1)ξ€·πœ,𝜁0ξ€Έ=||π‘“ξ…ž||(𝜁)3/2||π‘“ξ…žξ€·πœ0ξ€Έ||3/2||ξ€·πœπ‘“(𝜁)βˆ’π‘“0ξ€Έ||3𝑒𝑗(π›Ώβˆ’π›Ώ0)βˆ’1||πœβˆ’πœ0||3𝐾,(3.10)(2)ξ€·πœ,𝜁0ξ€Έ=||π‘“ξ…ž||(𝜁)3/2||π‘“ξ…žξ€·πœ0ξ€Έ||3/2||ξ€·πœπ‘“(𝜁)βˆ’π‘“0ξ€Έ||3𝑒𝑗(2Ξ˜βˆ’π›Ώβˆ’π›Ώ0)βˆ’1||πœβˆ’πœ0||3𝑒2𝑗Φ.(3.11) This hypersingular integral equation over a circular disc 𝐷 is to be solved subject to π‘Š=0 on 𝑠=1, and the 𝐾(1)(𝜁,𝜁0) is a Cauchy-type singular kernel with order π‘†βˆ’2, and the kernel 𝐾(2)(𝜁,𝜁0) is weakly singular with 𝑂(π‘†βˆ’1), as πœβ†’πœ0 (see the appendix).

We are going to solve (3.9) numerically. Write π‘Š(πœ‰,πœ‚) as a finite sum ξ“π‘Š(πœ‰,πœ‚)=𝑛,π‘˜π‘Šπ‘›π‘˜π΄π‘›π‘˜(𝑠,πœ‘),(3.12) where π΄π‘›π‘˜(𝑠,πœ‘) is defined by π΄π‘›π‘˜(𝑠,πœ‘)=𝑠|𝑛|𝐢|𝑛|+1/22π‘˜+1ξ‚€βˆš1βˆ’π‘ 2ξ‚π‘’π‘—π‘›πœ‘,𝑛,π‘˜=𝑁1𝑛=βˆ’π‘1𝑁2ξ“π‘˜=0,𝑁1,𝑁2βˆˆβ„€.(3.13) IntroduceπΏπ‘šβ„Ž(𝑠,πœ‘)=𝑠|π‘š|𝐢|π‘š|+1/22β„Ž+1ξ‚€βˆš1βˆ’π‘ 2cosπ‘šπœ‘,(3.14) where π‘š,β„Žβˆˆβ„€. The relationship between these two functions, π΄π‘›π‘˜(𝑠,πœ‘), and πΏπ‘šβ„Ž(𝑠,πœ‘) can be expressed as ξ€œΞ©π΄π‘›π‘˜(𝑠,πœ‘)πΏπ‘šβ„Ž(𝑠,πœ‘)π‘ π‘‘π‘ π‘‘πœ‘βˆš1βˆ’π‘ 2=π΅π‘›π‘˜π›Ώπ‘˜β„Žπ›Ώπ‘šπ‘›,(3.15) where 𝛿𝑖𝑗 is Kronecker delta and π΅π‘›π‘˜=⎧βŽͺ⎨βŽͺ⎩2πœ‹πœ‹4π‘˜+3,𝑛=0,2Ξ“(2π‘˜+2𝑛+2)22𝑛+1[](2π‘˜+𝑛+3/2)(2π‘˜+1)!Ξ“(𝑛+1/2)2,𝑛≠0.(3.16) Both functions π΄π‘›π‘˜(𝑠,πœ‘) and πΏπ‘šβ„Ž(𝑠,πœ‘) have square-root zeros at 𝑠=1.

Krenk [23] showed that 1ξ€²4πœ‹Ξ©π΄π‘›π‘˜(𝑠,πœ‘)𝑅3𝑑Ω=βˆ’πΈπ‘›π‘˜π΄π‘›π‘˜ξ€·π‘ 0,πœ‘01βˆ’π‘ 20,(3.17) where πΈπ‘›π‘˜=Ξ“(|𝑛|+π‘˜+3/2)Ξ“(π‘˜+3/2).(|𝑛|+π‘˜)!π‘˜!(3.18)

Substituting (3.17) and (3.12) into (3.9) yields 𝑛,π‘˜β„±π‘›π‘˜ξ€·π‘ 0,πœ‘0ξ€Έπ‘Šπ‘›π‘˜ξ€·πœ‰=𝑄0𝑠0,πœ‘0ξ€Έ,πœ‚0𝑠0,πœ‘0,ξ€Έξ€Έ(3.19) where β„±π‘›π‘˜ξ€·π‘ 0,πœ‘0ξ€Έ=βˆ’πΈπ‘›π‘˜ξ€·2βˆ’πœˆ+3πœˆπ‘’2π‘—Ξ˜ξ€Έπ΄π‘›π‘˜ξ€·π‘ 0,πœ‘0ξ€Έ21βˆ’π‘ 20+2βˆ’πœˆξ€œ8πœ‹π·π΄π‘›π‘˜(𝑠,πœ‘)𝐾(1)ξ€·πœ,𝜁0ξ€Έ+π‘‘πœ‰π‘‘πœ‚3πœˆξ€œ8πœ‹π·π΄π‘›π‘˜(𝑠,πœ‘)𝐾(2)ξ€·πœ,𝜁0ξ€Έπ‘‘πœ‰π‘‘πœ‚;0≀𝑠≀1,0β‰€πœ‘<2πœ‹.(3.20)

Next, define π‘Šπ‘›π‘˜ξ‚‹π‘Š=βˆ’π‘›π‘˜πΊ|𝑛|+1/22π‘˜+1ξƒŽπΈπ‘›π‘˜π΅π‘›π‘˜,(3.21) where 𝐺|𝑛|+1/22π‘˜+1=(2𝑛+2π‘˜+1)!/(2π‘˜+1)!(2𝑛)!. Multiply (3.19) by πΏπ‘šβ„Ž(𝑠0,πœ‘0), integrate over 𝐷 and using (3.15), (3.19) becomes 𝑛,π‘˜ξ‚‹π‘Šπ‘›π‘˜ξ‚΅βˆ’2βˆ’πœˆ+3πœˆπ‘’2π‘—Ξ˜2π›Ώβ„Žπ‘˜π›Ώ|π‘š||𝑛|+π‘†π‘šπ‘›β„Žπ‘˜ξ‚Ά=π‘„π‘šβ„Ž,βˆ’π‘1β‰€π‘šβ‰€π‘1,0β‰€β„Žβ‰€π‘2,(3.22) where π‘†π‘šπ‘›β„Žπ‘˜=1√8πœ‹πΈπ‘›π‘˜π΅π‘›π‘˜βˆšπΈπ‘šβ„Žπ΅π‘šβ„Žπ‘‡π‘šπ‘›β„Žπ‘˜,π‘‡π‘šπ‘›β„Žπ‘˜=ξ€œπ·πΏπ‘šβ„Žξ€·πœ0ξ€Έξ€œπ·π΄π‘›π‘˜ξ€·(𝜁)𝐻𝜁,𝜁0ξ€Έπ‘‘πœπ‘‘πœ0,π‘„π‘šβ„Ž=1βˆšπΈπ‘šβ„Žπ΅π‘šβ„Žξ€œπ·πΏπ‘šβ„Žξ€·πœ0ξ€Έπ‘„ξ€·πœ0ξ€Έπ‘‘πœ0,π»ξ€·πœ,𝜁0ξ€Έ=(2βˆ’πœˆ)𝐾(1)ξ€·πœ,𝜁0ξ€Έ+3𝜈𝐾(2)ξ€·πœ,𝜁0ξ€Έ.(3.23)

In (3.22), we have used the following notation: 𝜁0=𝜁0(𝑠0,πœ‘0), π‘‘πœ0=𝑠0𝑑𝑠0π‘‘πœ‘0, and 𝑄(𝜁0)=𝑄(πœ‰0,πœ‚0)=𝑄(𝑠0cosπœ‘0,𝑠0sinπœ‘0).

In evaluating the multiple integrals in (3.22), we have used the Gaussian quadrature and trapezoidal formulas for the radial and angular directions, with the choice of collocation points (𝑠,πœ‘) and (𝑠0,πœ‘0) defined as follows: 𝑠𝑖=πœ‹4+πœ‹4𝑀1𝑖=1π‘Š(𝑖),𝑠0𝑖=πœ‹4+πœ‹4𝑀1𝑖=1π‘Š0πœ‘(𝑖),𝑗=𝑀2βˆ‘π‘—=1π‘—πœ‹π‘€2,πœ‘0𝑗=𝑀2βˆ‘π‘—=1(𝑗+0.5)πœ‹π‘€2,(3.24) where π‘Š(𝑖) and π‘Š0(𝑖) are abscissas for 𝑠𝑖 and 𝑠0𝑖, respectively, 𝑀1 and 𝑀2 is the number of collocation points in radial and angular directions, respectively. This effort leads to the (2𝑁1+1)(𝑁2+1)Γ—(2𝑁1+1)(𝑁2+1) system of linear equations Aξ‚‹Μƒπ‘Š=𝑏,(3.25) where 𝐴 is a square matrix, and ξ‚‹π‘Š and ̃𝑏 are vectors, ξ‚‹π‘Š to be determined.

4. Energy Release Rate

The energy release rate (measured in π½π‘€βˆ’2), 𝐺(πœ‘) by Irwin's relation subject to shear load is defined as [7, 8] 𝐺(πœ‘)=1βˆ’πœˆ2𝐸𝐾𝐼𝐼(πœ‘)2+(1+𝜈)𝐸𝐾𝐼𝐼𝐼(πœ‘)2,(4.1) where 𝐸, Young's modulus, a measurement of the stiffness of an isotropic elastic material and the relationship of 𝐸, 𝜈 and πœ‡, is 𝐸𝜈=2πœ‡βˆ’1,(4.2) and 𝐾𝐼𝐼(πœ‘) and 𝐾𝐼𝐼𝐼(πœ‘), the sliding and tearing mode stress intensity factor, respectively, are defined as [5, 7, 8] 𝐾𝑗(πœ‘)=limπ‘Ÿβ†’π‘Žπ‘‰π‘—ξ‚™2πœ‹π‘Žβˆ’π‘Ÿπ‘€(π‘₯,𝑦),𝑗=𝐼𝐼,𝐼𝐼𝐼,(4.3) where 𝑉𝑗 are constants.

Let π‘Ž(πœ‘)=|𝑓(π‘’π‘–πœ‘)|, π‘Ÿ=|𝑓(π‘ π‘’π‘–πœ‘)|, and as 𝑠 close to 1, (4.3) leads to 𝐾𝑗(πœ‘)=lim𝑠→1βˆ’π‘‰π‘—ξƒŽ2πœ‹||𝑓(1βˆ’π‘ )ξ…ž(π‘’π‘–πœ‘)||𝑀(π‘₯,𝑦),𝑗=𝐼𝐼,𝐼𝐼𝐼.(4.4) Therefore, substituting (3.7) into (4.4) and simplifying gives 𝐾𝑗(πœ‘)=𝑉𝑗||π‘“ξ…žξ€·π‘’π‘–πœ‘ξ€Έ||βˆ’1𝑛,π‘˜ξ‚‹π‘Šπ‘›π‘˜βˆšπΈπ‘›π‘˜π΅π‘›π‘˜π‘Œπ‘›π‘˜ξƒ°(πœ‘),𝑗=𝐼𝐼,𝐼𝐼𝐼,(4.5) where π‘Œπ‘›π‘˜(πœ‘)=𝐷|𝑛|+1/22π‘˜+1(0)cos(π‘›πœ‘), and 𝐢|𝑛|+1/22π‘˜+1(√1βˆ’π‘ 2√)=1βˆ’π‘ 2𝐷|𝑛|+1/22π‘˜+1(√1βˆ’π‘ 2), where π·πœ†π‘š(π‘₯) is defined recursively by π‘šπ·πœ†π‘š(π‘₯)=2(π‘š+πœ†βˆ’1)π‘₯π·πœ†π‘šβˆ’1(π‘₯)βˆ’(π‘š+2πœ†βˆ’2)π·πœ†π‘šβˆ’2(π‘₯),π‘š=2,3,…,(4.6) with π·πœ†0(π‘₯)=2πœ† and π·πœ†1(π‘₯)=2πœ†π‘₯.

Table 1 shows that our numerical scheme converges rapidly at a different point of the crack with only a small value of 𝑁=𝑁1=𝑁2 are used.

𝑁 𝐺 ( 0 . 0 0 ) 𝐺 ( πœ‹ / 4 ) 𝐺 ( πœ‹ / 2 ) 𝐺 ( 3 πœ‹ / 4 ) 𝐺 ( πœ‹ )

0 7 . 8 6 7 6 𝐸 βˆ’ 1 0 9 . 0 1 2 3 𝐸 βˆ’ 1 0 1 . 6 0 6 7 𝐸 βˆ’ 0 9 9 . 0 1 2 3 𝐸 βˆ’ 1 0 7 . 8 6 7 6 𝐸 βˆ’ 1 0
1 7 . 2 7 2 4 𝐸 βˆ’ 1 0 8 . 9 3 9 2 𝐸 βˆ’ 1 0 1 . 3 1 5 9 𝐸 βˆ’ 0 9 8 . 9 3 9 2 𝐸 βˆ’ 1 0 7 . 2 7 2 4 𝐸 βˆ’ 1 0
2 9 . 2 6 6 8 𝐸 βˆ’ 0 7 7 . 4 6 5 2 𝐸 βˆ’ 1 0 1 . 5 6 4 9 𝐸 βˆ’ 0 9 7 . 4 6 5 2 𝐸 βˆ’ 1 0 9 . 2 6 6 8 𝐸 βˆ’ 0 7
3 0 . 0 0 0 0 𝐸 + 0 0 0 . 0 0 0 0 𝐸 + 0 0 6 . 3 5 1 7 𝐸 βˆ’ 1 0 0 . 0 0 0 0 𝐸 + 0 0 0 . 0 0 0 0 𝐸 + 0 0
4 1 . 1 8 5 9 𝐸 βˆ’ 0 5 7 . 4 0 4 1 𝐸 βˆ’ 1 9 4 . 6 7 0 9 𝐸 βˆ’ 0 9 7 . 4 0 4 1 𝐸 βˆ’ 1 9 1 . 1 8 5 9 𝐸 βˆ’ 0 5
5 3 . 1 4 2 9 𝐸 βˆ’ 0 3 8 . 8 2 1 1 𝐸 βˆ’ 0 4 9 . 2 5 2 8 𝐸 βˆ’ 0 6 8 . 8 2 1 1 𝐸 βˆ’ 0 4 3 . 1 4 2 9 𝐸 βˆ’ 0 3
6 3 . 0 4 2 1 𝐸 βˆ’ 0 3 8 . 7 9 0 8 𝐸 βˆ’ 0 4 9 . 5 7 9 1 𝐸 βˆ’ 0 4 8 . 7 9 0 8 𝐸 βˆ’ 0 4 3 . 0 4 2 1 𝐸 βˆ’ 0 3
7 1 . 5 7 9 4 𝐸 βˆ’ 0 3 8 . 4 3 0 8 𝐸 βˆ’ 0 4 9 . 2 9 4 5 𝐸 βˆ’ 0 4 8 . 4 3 0 8 𝐸 βˆ’ 0 4 1 . 5 7 2 1 𝐸 βˆ’ 0 3
8 9 . 7 5 5 7 𝐸 βˆ’ 0 4 1 . 1 9 0 3 𝐸 βˆ’ 0 3 9 . 5 0 0 1 𝐸 βˆ’ 0 4 1 . 1 9 0 3 𝐸 βˆ’ 0 3 9 . 7 5 5 7 𝐸 βˆ’ 0 4
9 9 . 7 5 5 7 𝐸 βˆ’ 0 4 1 . 1 9 0 3 𝐸 βˆ’ 0 3 9 . 5 0 0 1 𝐸 βˆ’ 0 4 1 . 1 9 0 3 𝐸 βˆ’ 0 3 9 . 7 5 5 7 𝐸 βˆ’ 0 4
10 9 . 7 5 5 7 𝐸 βˆ’ 0 4 1 . 1 9 0 3 𝐸 βˆ’ 0 3 9 . 5 0 0 1 𝐸 βˆ’ 0 4 1 . 1 9 0 3 𝐸 βˆ’ 0 3 9 . 7 5 5 7 𝐸 βˆ’ 0 4

Figures 3, 4, 5, and 6 show the variations of 𝐺 against πœ‘ for 𝑐=0.001, 𝑐=0.01, 𝑐=0.10, and 𝑐=0.30, respectively. It can be seen that the energy release rate has local extremal values when the crack front is at cos(πœ‘)=Β±1 or sin(πœ‘)=Β±1. Similar behavior can be observed for the solution of 𝐺(πœ‘) for a different 𝑐 and 𝜈 at 𝑐=0.1, displayed in Figures 7 and 8. Our results agree with those obtained asymptotically by Gao [8], with the maximum differences for π‘š=2 are 3.6066Γ—10βˆ’6, 4.7064Γ—10βˆ’5, 5.3503Γ—10βˆ’5, and 9.0000Γ—10βˆ’5 for 𝑐=0.001, 𝑐=0.01, 𝑐=0.10, and 𝑐=0.30, respectively.

5. Conclusion

In this paper, the hypersingular integral equation over a nearly circular crack is formulated. Then, using the conformal mapping, the equation is transformed into hypersingular integral equation over a circular crack, which enable us to use the formula obtained by Krenk [23]. By choosing the appropriate collocation points, this equation is reduced into a system of linear equations and solved for the unknown coefficients. The energy release rate for the mentioned crack subject to shear load is presented graphically. Our computational results seem to agree with the asymptotic solution obtained by Gao [8].


The Singularity of the Kernel 𝐾(1)(𝜁,𝜁0) and 𝐾(2)(𝜁,𝜁0)

At 𝜁=𝜁0, we have ξ€·πœπ‘“(𝜁)βˆ’π‘“0ξ€Έ=ξ€·πœβˆ’πœ0ξ€Έπ‘“ξ…žξ€·πœ0ξ€Έ+ξ€·πœβˆ’πœ0ξ€Έ2π‘“ξ…žξ…žξ€·πœ0ξ€Έ2+β‹―.(A.1) Differentiate 𝑓(𝜁) with respect to 𝜁, we have π‘“ξ…ž(𝜁)=π‘“ξ…žξ€·πœ0ξ€Έ+ξ€·πœβˆ’πœ0ξ€Έπ‘“ξ…žξ…žξ€·πœ0ξ€Έ+ξ€·πœβˆ’πœ0ξ€Έ2π‘“ξ…žξ…žξ…žξ€·πœ0ξ€Έ2+β‹―.(A.2) Let 𝐹1=ξ€·πœβˆ’πœ0ξ€Έπ‘“ξ…žξ…žξ€·πœ0ξ€Έπ‘“ξ…žξ€·πœ0ξ€Έ=𝑒1+𝑖𝑣1𝐹=𝑂(𝑆),(A.3)2=ξ€·πœβˆ’πœ0ξ€Έ2π‘“ξ…žξ…žξ…žξ€·πœ0ξ€Έ2π‘“ξ…žξ€·πœ0ξ€Έ=𝑒2+𝑖𝑣2𝑆=𝑂2ξ€Έasπ‘†βŸΆ0,(A.4) where 𝑒1, 𝑒2, 𝑣1, and 𝑣2 are real. As 𝐹1=𝑂(𝑆) and 𝐹2=𝑂(𝑆2) as 𝑆→0, we see that 𝑒𝑖 and 𝑣𝑖 are 𝑂(𝑆𝑖) as 𝑆→0 (𝑖=1,2).

Hence, (A.1) becomes ξ€·πœπ‘“(𝜁)βˆ’π‘“0ξ€Έ=π‘“ξ…žξ€·πœ0ξ€Έξ€·πœβˆ’πœ0𝐹1+12ξ‚Ή+β‹―.(A.5) Substituting (A.3) into (A.2) gives π‘“ξ…ž(𝜁)=π‘“ξ…žξ€·πœ0ξ€Έξ€Ί1+𝐹1ξ€»+β‹―,π‘“ξ…žξ€·πœ0ξ€Έ=π‘“ξ…žξ€Ί(𝜁)1βˆ’πΉ1ξ€»βˆ’β‹―.(A.6)

As 𝑆→0 and truncate (A.1) at second order, then (A.6) can be written as π‘“ξ…ž(𝜁)β‰ƒπ‘“ξ…žξ€·πœ0ξ€Έξ€½1+𝐹1ξ€Ύ,π‘“ξ…žξ€·πœ0ξ€Έβ‰ƒπ‘“ξ…žξ€½(𝜁)1βˆ’πΉ1ξ€Ύ,(A.7) respectively. Now, consider 𝐾(1)(𝜁,𝜁0). Let π›Ώβˆ’π›Ώ0≃𝑣1=𝑂(𝑆) where 𝛿 and 𝛿0 defined in (3.6), then, from (3.6), we have 𝑒𝑖(π›Ώβˆ’π›Ώ0)=π‘“ξ…ž||𝑓(𝜁)ξ…žξ€·πœ0ξ€Έ||π‘“ξ…žξ€·πœ0ξ€Έ||π‘“ξ…ž(||𝜁).(A.8)

Apply 𝑧𝑧=|𝑧|2 leads to 1+𝐹1||1+𝐹1||=1+𝐹11+𝑒1Γ—1βˆ’π‘’11βˆ’π‘’1≃1+𝐹1ξ€Έξ€·1βˆ’π‘’1≃1+𝑖𝑣1.(A.9) Hence, 𝑒𝑗(π›Ώβˆ’π›Ώ0)≃1+𝑗𝑣1.(A.10)

Martin [24] showed that ||1+𝛼𝐹1+𝛽𝐹2||πœ†=|1+𝛼𝑒1+𝛽𝑒2ξ€·+𝑖𝛼𝑣1+𝛽𝑣2ξ€Έ|πœ†/2≃1+π›Όπœ†π‘’1+12πœ†ξ€½(πœ†βˆ’1)𝛼2𝑒21+𝛼2𝑣21+2𝛽𝑒2ξ€Ύ,(A.11) where 𝛼, 𝛾, and 𝛽 are constants and ||π‘“ξ…ž||(𝜁)3/2||π‘“ξ…žξ€·πœ0ξ€Έ||3/2||ξ€·πœπ‘“(𝜁)βˆ’π‘“0ξ€Έ||3βˆ’1||πœβˆ’πœ0||3=1𝑆3|1+𝐹1+𝐹2|3/2||1+(1/2)𝐹1+(1/3)𝐹2||3ξƒͺξ€·π‘†βˆ’1=π‘‚βˆ’1ξ€Έ,(A.12) as 𝑆→0.

Next, using (A.12), (A.11), and (A.10), we obtain 𝐾(1)ξ€·πœ,𝜁0ξ€Έ=||π‘“ξ…ž||(𝜁)3/2||π‘“ξ…žξ€·πœ0ξ€Έ||3/2||ξ€·πœπ‘“(𝜁)βˆ’π‘“0ξ€Έ||3ξ€·1+𝑗𝑣1ξ€Έβˆ’1||πœβˆ’πœ0||3=1𝑆3||1+𝐹1||3/2||1+𝐹1||/23ξƒͺ+1βˆ’1𝑆3||1+𝐹1||3/2||1+𝐹1||/23𝑗𝑣1ξƒͺ,(A.13) where ||1+𝐹1||3/2||1+𝐹1||/233βˆ’1≃8𝑣21βˆ’π‘’21ξ€Έ,||||𝐹1+12||||3⟢1.(A.14) Thus, 𝐾(1)(𝜁,𝜁0) reduces to 𝐾(1)ξ€·πœ,𝜁0ξ€Έ=38𝑆3𝑣21βˆ’π‘’21ξ€Έ+𝑗𝑆3𝑣1.(A.15) Since 𝐹1=𝑒1+𝑖𝑣1, then 𝑒1+𝑖𝑣1ξ€Έ2=𝑒21βˆ’π‘£21+2𝑖𝑒1𝑣1𝐹,Re1𝑣=βˆ’21βˆ’π‘’21ξ€Έ,(A.16) so, (A.3) leads to 𝐹Re21𝑒=Re2π‘—Ξ¦ξ€·π‘“ξ…žξ…žξ€·πœ0ξ€Έξ€Έ2ξ€·π‘“ξ…žξ€·πœ0ξ€Έξ€Έ2ξƒ°β‰ƒπ·ξ€·πœ0ξ€Έ,Φ𝑆,π‘†βŸΆ0,(A.17) where 𝐷(𝜁0,Ξ¦)=Re{𝑒2𝑗Φ((π‘“ξ…žξ…ž(𝜁0))2/(π‘“ξ…ž(𝜁0))2)} and πœβˆ’πœ0=𝑆𝑒𝑖Φ defined in (2.7). Thus, 𝐾(1)ξ€·πœ,𝜁0𝑆=π‘‚βˆ’1ξ€Έ+1𝑆3𝑗𝑣1≃𝑗𝑣1π‘†βˆ’3𝑆=π‘‚βˆ’2ξ€Έ.(A.18) Therefore, 𝐾(1)(𝜁,𝜁0)≃𝑗𝑣1π‘†βˆ’3, that is, 𝐾(1)(πœ‰,πœ‰0)≃𝑂(π‘†βˆ’2) as 𝑆→0.

For 𝐾(2)(𝜁,𝜁0), expand 𝑓(𝜁) at 𝜁=𝜁0, and truncating at second order, (3.4) gives Reπ‘–Ξ˜ξ€·=π‘Žπœβˆ’πœ0ξ€Έπ‘“ξ…žξ€·πœ0𝐹1+12ξ‚Ό,(A.19) where 𝑒𝑖(Ξ˜βˆ’Ξ¦βˆ’π›Ώ0)≃𝐹1+12ξ‚Ά||||𝐹1+12||||βˆ’1𝑣=1+𝑖22.(A.20) Next, substituting (A.5) and (A.7) into (3.4) gives Reπ‘–Ξ˜ξ€·=π‘Žπœβˆ’πœ0ξ€Έπ‘“ξ…žξ€·(𝜁)1βˆ’πΉ1𝐹1+12ξ‚Ό,(A.21) where 𝑒𝑖(Ξ˜βˆ’Ξ¦βˆ’π›Ώ)≃𝐹1βˆ’12ξ‚Ά||||𝐹1βˆ’12||||βˆ’1𝑣=1+𝑖22.(A.22) Using (A.22) and (A.20) yields 𝑒𝑖(2Ξ˜βˆ’2Ξ¦βˆ’π›Ώβˆ’π›Ώ0)≃1+𝑖214𝑣22+𝑖𝑣2𝑆=1+𝑂2ξ€Έ.(A.23) Hence, as 𝑆→0, then 𝑒𝑖(2Ξ˜βˆ’2Ξ¦βˆ’π›Ώβˆ’π›Ώ0)≃1+𝑂(𝑆2). It is not difficult to see that π‘…β‰ƒπ‘Ž|π‘“ξ…ž(𝜁0)|𝑆, Ξ˜β‰ƒΞ¦+𝛿0, π‘…β‰ƒπ‘Ž|π‘“ξ…ž(𝜁)|𝑆, and Ξ˜β‰ƒΞ¦+𝛿, respectively; then (3.11) becomes 𝐾(2)ξ€·πœ,𝜁0ξ€Έ=||π‘“ξ…ž||(𝜁)3/2||π‘“ξ…žξ€·πœ0ξ€Έ||3/2||ξ€·πœπ‘“(𝜁)βˆ’π‘“0ξ€Έ||3𝑒2π‘—Ξ¦βˆ’1|πœβˆ’πœ0|3𝑒2π‘—Ξ˜.(A.24) Applying similar procedures as in 𝐾1(𝜁,𝜁0) gives 1||πœβˆ’πœ0||3||π‘“ξ…ž||(𝜁)3/2||π‘“ξ…žξ€·πœ0ξ€Έ||3/2||ξ€·πœπ‘“(𝜁)βˆ’π‘“0ξ€Έ||3||πœβˆ’πœ0||3ξƒͺ=1βˆ’1𝑆3|1+𝐹1|3/2||1+𝐹1||/23ξƒͺ≃3βˆ’18𝑣21βˆ’π‘’21≃38πΉξ€·πœ0ξ€Έ,Φ𝑆asπ‘†βŸΆ0.(A.25) Thus, 𝐾(2)ξ€·πœ,𝜁0ξ€Έ=𝑒2𝑗Φ38πΉξ€·πœ0ξ€Έ,Φ𝑆𝑆=π‘‚βˆ’1ξ€Έasπ‘†βŸΆ0.(A.26)


The authors would like to thank the reviewers for their very constructive comments to improve the quality of the paper. This project is supported by Ministry of Higher Education Malaysia for the Fundamental Research Grant scheme, project no. 01-04-10-897FR and the second author received a NSF scholarship.


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Copyright © 2011 Nik Mohd Asri Nik Long et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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