Computation of Energy Release Rates for a Nearly Circular Crack

Abstract

This paper deals with a nearly circular crack, Ω in the plane elasticity. The problem of finding the resulting shear stress can be formulated as a hypersingular integral equation over a considered domain, Ω 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, Ω, as in Figure 1, located on the Cartesian coordinate (𝑥,𝑦,𝑥₃) with origin 𝑂, and Ω lies in the plane 𝑥₃=0. Let the radius of the crack, Ω be 𝑎 and Ω={(𝑟,𝜃)∶0≤𝑟<𝑎,−𝜋≤𝜃<𝜋}.

Figure 1 

Stresses acting on a circular crack.

If the equal and opposite shear stresses in the 𝑥 and 𝑦 directions, 𝑞₁(𝑥,𝑦) and 𝑞₂(𝑥,𝑦), respectively, are applied to the crack plane, and it is assumed that the 𝑥₃ direction is traction free, then in the view of shear load, the entire plane, must free from the normal stress, that is 𝜏₃₃𝑥,𝑦,𝑥₃=0for𝑥₃=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, 𝑥₃=0: 𝜏₁₃𝑥,𝑦,𝑥₃=𝜇1−𝜈𝑞¹𝜏(𝑥,𝑦),(𝑥,𝑦)∈Ω,₂₃𝑢𝑥,𝑦,𝑥₃=1−𝜇𝜈𝑞²(𝑥,𝑦),(𝑥,𝑦)∈Ω,₁𝑥,𝑦,𝑥₃=𝑢₂𝑥,𝑦,𝑥₃=0,(𝑥,𝑦)∈Γ⧵Ω,(2.2) where 𝜏_𝑖𝑗 is stress tensor, 𝜇 is shear modulus, 𝜈 is denoted as Poisson's ratio, and Γ is the entire 𝑥₃=0. Also, the problem satisfies the regularity conditions at infinity 𝑢_𝑖𝑥,𝑦,𝑥₃=𝑂1𝑅,𝜏^𝑖𝑗𝑥,𝑦,𝑥₃=𝑂𝑅1,𝑖,𝑗=1,2,3,𝑅→∞,(2.3) where 𝑅 is the distance 𝑅=𝑥−𝑥₀²+𝑦−𝑦₀²,𝑥₀,𝑦₀∈Ω.(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 18𝜋_Ω(2−𝜈)𝑤(𝑥,𝑦)+3𝜈𝑒^2𝑗Θ𝑤(𝑥,𝑦)𝑅³𝑑Ω=𝑞𝑥₀,𝑦₀,𝑥₀,𝑦₀∈Ω,(2.5) where 𝑤(𝑥,𝑦)=[𝑢₁(𝑥,𝑦)]+𝑗[𝑢₂(𝑥,𝑦)] is the unknown crack opening displacement, 𝑞(𝑥₀,𝑦₀)=𝑞₁(𝑥₀,𝑦₀)+𝑗𝑞₂(𝑥₀,𝑦₀), 𝑗²=√−1, the 𝑤(𝑥,𝑦)=[𝑢₁(𝑥,𝑦)]−𝑗[𝑢₂(𝑥,𝑦)], and the angle Θ is defined by 𝑥−𝑥₀=𝑅cosΘ,𝑦−𝑦₀=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 𝑤=0 on 𝜕Ω where 𝜕Ω is boundary of Ω. For the constant shear stress in 𝑥 direction, we have 𝜏₂₃=0 and [𝑢₂(𝑥,𝑦)]=0, hence, (2.5) becomes 18𝜋_Ω2−𝜈+3𝜈𝑒^2𝑗Θ𝑅³𝑤(𝑥,𝑦)𝑑Ω=𝑞𝑥₀,𝑦₀,𝑥₀,𝑦₀∈Ω.(2.7)

Polar coordinates (𝑟,𝜃) and (𝑟₀,𝜃₀) are chosen so that the loadings 𝑞(𝑥,𝑦) and 𝑞(𝑥₀,𝑦₀) can be written as a Fourier series 𝑞(𝑥,𝑦)=^∞_𝑛_=−∞𝑞_𝑛𝑟𝑎𝑒^𝑗𝑛𝜃,𝑞𝑥₀,𝑦₀=^∞_𝑛_=−∞𝑞_𝑛𝑎𝑟⁰₀𝑒^{𝑗𝑛𝜃0},(2.8) where the Fourier components 𝑞_𝑛 are 𝑗-complex. The 𝑗-complex crack opening displacement, 𝑤(𝑥,𝑦) and 𝑤(𝑥₀,𝑦₀), have similar expressions 𝑤(𝑥,𝑦)=^∞_𝑛_=−∞𝑤_𝑛𝑟𝑎𝑒^𝑗𝑛𝜃,𝑤𝑥₀,𝑦₀=^∞_𝑛_=−∞𝑤_𝑛𝑎𝑟⁰₀𝑒^{𝑗𝑛𝜃0}.(2.9) Without loss of generality, we consider 𝑎=1. Using Guidera and Lardner [20], the dimensionless function 𝑞_𝑛 and 𝑤_𝑛 can be expressed as 𝑞_𝑛(𝑟)=𝑟^∞|𝑛|_𝑘=0𝑄^𝑛𝑘Γ|𝑛|+12Γ𝑘+32(|𝑛|+𝑘)!√1−𝑟²𝐶^|𝑛|+1_2𝑘+12𝑤_𝑛√1−𝑟²,(𝑟)=𝑟^∞|𝑛|_𝑘=0𝑊^𝑛𝑘Γ(|𝑛|+1/2)𝑘!Γ(|𝑛|+𝑘+3/2)𝐶^{|𝑛|+1/22𝑘+1}√1−𝑟²,(2.10) where the 𝑗-complex coefficients 𝑄^𝑛𝑘 are known, 𝑊^𝑛𝑘 are unknown, and 𝐶^𝜆𝑚(𝑥) is an orthogonal Gegenbauer polynomial of degree 𝑚 and index 𝜆, which is defined recursively by [21] (𝑚+2)𝐶^𝜆𝑚+2(𝑥)=2(𝑚+𝜆+1)𝑥𝐶^𝜆𝑚+1(𝑥)−(2𝜆+𝑚)𝐶^𝜆𝑚(𝑥),(2.11) with the initial values 𝐶^𝜆0(𝑥)=1 and 𝐶^𝜆1(𝑥)=2𝜆𝑥. For a constant shear loading, 𝑞(𝑥,𝑦)=−𝜏, the solution for a circular crack is obtainable.

3. Nearly Circular Crack

Let Ω be an arbitrary shaped crack of smooth boundary with respect to origin 𝑂, such that Ω is defined as Ω={(𝑟⋅𝜃)∶0≤𝑟<𝜌(𝜃),−𝜋≤𝜃<𝜋},(3.1) where the boundary of Ω, 𝜕Ω is given by 𝑟=𝜌(𝜃). Let 𝜁=𝜉+𝑖𝜂=𝑠𝑒^𝑖𝜑 with |𝜁|<1 such that the unit disc is 𝐷≡{(𝑠,𝜑)∶0≤𝑠<1,−𝜋≤𝜑<𝜋}.(3.2) By the properties of Reimann mapping theorem [22], a circular disc 𝐷 is mapped conformally onto Ω using 𝑧=𝑎𝑓(𝜁). This approach works for a general smooth star-shaped domain, Ω. For a particular application, let 𝑓 be an analytic function, simply connected in the domain Ω, |𝑓^(𝜁)| is nonzero and bounded for all |𝜁|<1, 𝑓(𝜁)=𝜁+𝑐𝑔(𝜁)with𝑔(𝜁)=𝜁^𝑚+1,(3.3) which maps a unit circle, 𝐷 in the 𝜁-plane into a nearly circular domain Ω in the 𝑧-plane where 𝑐 is a real parameter and 𝑟=𝜌(𝜃) is the boundary of Ω. This domain has a smooth, regular boundary for 0≤(𝑚+1)|𝑐|<1. As (𝑚+1)|𝑐|→1 one or more cusps develop; see Figure 2 with various choices of 𝑐.

Figure 2 

The domain Ω for 𝑓(𝜁)=𝜁+𝑐𝜁𝑚+1 at different choices of 𝑐, 𝑚=2.

Let 𝑧−𝑧₀=𝑎𝑓(𝜁)−𝑓𝜁₀=Re^𝑖Θ,(3.4) and define 𝑆 and Φ as 𝑑Ω=𝑑𝑥𝑑𝑦=𝑎₂𝜁−𝜁₀=𝑆𝑒^𝑖Φ,||𝑓^(𝜁)||²𝑑𝜉𝑑𝜂=𝑎²||𝑓^(𝜁)||²𝑠𝑑𝑠𝑑𝜑,(3.5) where 𝑥=𝑎𝑢(𝜉,𝜂) and 𝑦=𝑎𝑣(𝜉,𝜂) so that 𝑓=𝑢+𝑖𝑣. Next, we define 𝛿 and 𝛿₀ as 𝑓^(𝜁)=||𝑓^(𝜁)||𝑒^𝑖𝛿,𝑓^𝜁₀=||𝑓^𝜁₀||𝑒^𝑖𝛿0.(3.6) Set 𝑞𝑥𝑤𝜁(𝑥(𝜁),𝑦(𝜁))=𝑎||𝑓^(𝜁)||^−1/2𝑒^𝑗𝛿𝑊(𝜉,𝜂),(3.7)₀,𝑦𝜁₀=𝑎||𝑓^𝜁₀||^−3/2𝑒^𝑗𝛿0𝑄𝜉₀,𝜂₀.(3.8) Substituting (3.5), (3.6), (3.7), and (3.8) into (2.7) gives2−𝜈+3𝜈𝑒^2𝑗Θ8𝜋_𝐷𝑊(𝜉,𝜂)𝑆³𝑑𝜉𝑑𝜂+2−𝜈8𝜋_𝐷+3𝜈𝑊(𝜉,𝜂)𝐾⁽¹⁾𝜁,𝜁₀𝑑𝜉𝑑𝜂8𝜋_𝐷𝑊(𝜉,𝜂)𝐾⁽²⁾𝜁,𝜁₀𝑑𝜉𝑑𝜂=𝑄𝜉₀,𝜂₀,𝜉₀,𝜂₀∈𝐷,(3.9) where the kernel 𝐾⁽¹⁾(𝜁,𝜁₀) and 𝐾⁽²⁾(𝜁,𝜁₀) are [17]𝐾⁽¹⁾𝜁,𝜁₀=||𝑓^(𝜁)||^3/2||𝑓^𝜁₀||^3/2||𝑓(𝜁)−𝑓𝜁₀||³𝑒^{𝑗(𝛿−𝛿0)}−1||𝜁−𝜁₀||³𝐾,(3.10)₍₂₎𝜁,𝜁₀=||𝑓^(𝜁)||^3/2||𝑓^𝜁₀||^3/2||𝑓(𝜁)−𝑓𝜁₀||³𝑒^{𝑗(2Θ−𝛿−𝛿0)}−1||𝜁−𝜁₀||³𝑒^2𝑗Φ.(3.11) This hypersingular integral equation over a circular disc 𝐷 is to be solved subject to 𝑊=0 on 𝑠=1, and the 𝐾⁽¹⁾(𝜁,𝜁₀) is a Cauchy-type singular kernel with order 𝑆⁻², and the kernel 𝐾⁽²⁾(𝜁,𝜁₀) is weakly singular with 𝑂(𝑆⁻¹), as 𝜁→𝜁₀ (see the appendix).

We are going to solve (3.9) numerically. Write 𝑊(𝜉,𝜂) as a finite sum 𝑊(𝜉,𝜂)=_𝑛,𝑘𝑊^𝑛𝑘𝐴^𝑛𝑘(𝑠,𝜑),(3.12) where 𝐴^𝑛𝑘(𝑠,𝜑) is defined by 𝐴^𝑛𝑘(𝑠_𝑛,𝑘,𝜑)=𝑠^|𝑛|𝐶^{|𝑛|+1/22𝑘+1}√1−𝑠²𝑒^𝑗𝑛𝜑,=^𝑁1_{𝑛=−𝑁1}^𝑁2_𝑘=0,𝑁₁,𝑁₂∈ℤ.(3.13) Introduce𝐿^𝑚ℎ(𝑠,𝜑)=𝑠^|𝑚|𝐶^{|𝑚|+1/22ℎ+1}√1−𝑠²cos𝑚𝜑,(3.14) where 𝑚,ℎ∈ℤ. The relationship between these two functions, 𝐴^𝑛𝑘(𝑠,𝜑), and 𝐿^𝑚ℎ(𝑠,𝜑) can be expressed as _Ω𝐴^𝑛𝑘(𝑠,𝜑)𝐿^𝑚ℎ(𝑠,𝜑)𝑠𝑑𝑠𝑑𝜑√1−𝑠²=𝐵^𝑛𝑘𝛿_𝑘ℎ𝛿_𝑚𝑛,(3.15) where 𝛿_𝑖𝑗 is Kronecker delta and 𝐵^𝑛𝑘=⎧⎪⎨⎪⎩2𝜋4𝑘+3,𝜋𝑛=0,₂Γ(2𝑘+2𝑛+2)2^2𝑛+1(2𝑘+𝑛+3/2)(2𝑘+1)![Γ(𝑛+1/2)]²,𝑛≠0.(3.16) Both functions 𝐴^𝑛𝑘(𝑠,𝜑) and 𝐿^𝑚ℎ(𝑠,𝜑) have square-root zeros at 𝑠=1.

Krenk [23] showed that 14𝜋_Ω𝐴^𝑛𝑘(𝑠,𝜑)𝑅³𝑑Ω=−𝐸^𝑛𝑘𝐴^𝑛𝑘𝑠₀,𝜑₀1−𝑠²⁰,(3.17) where 𝐸^𝑛𝑘=Γ(|𝑛|+𝑘+3/2)Γ(𝑘+3/2)(|𝑛|+𝑘)!𝑘!.(3.18)

Substituting (3.17) and (3.12) into (3.9) yields _𝑛,𝑘ℱ^𝑛𝑘𝑠₀,𝜑₀𝑊^𝑛𝑘=𝑄𝜉₀𝑠₀,𝜑₀,𝜂₀𝑠₀,𝜑₀,(3.19) where ℱ^𝑛𝑘𝑠₀,𝜑₀=−𝐸^𝑛𝑘2−𝜈+3𝜈𝑒^2𝑗Θ𝐴^𝑛𝑘𝑠₀,𝜑₀21−𝑠²⁰+2−𝜈8𝜋_𝐷+3𝜈𝐴^𝑛𝑘(𝑠,𝜑)𝐾⁽¹⁾𝜁,𝜁₀𝑑𝜉𝑑𝜂8𝜋_𝐷𝐴^𝑛𝑘(𝑠,𝜑)𝐾⁽²⁾𝜁,𝜁₀𝑑𝜉𝑑𝜂;0≤𝑠≤1,0≤𝜑<2𝜋.(3.20)

Next, define 𝑊^𝑛𝑘=−𝑊^𝑛𝑘𝐺^{|𝑛|+1/22𝑘+1}𝐸^𝑛𝑘𝐵^𝑛𝑘,(3.21) where 𝐺^{|𝑛|+1/22𝑘+1}=(2𝑛+2𝑘+1)!/(2𝑘+1)!(2𝑛)!. Multiply (3.19) by 𝐿^𝑚ℎ(𝑠₀,𝜑₀), integrate over 𝐷 and using (3.15), (3.19) becomes _𝑛,𝑘𝑊^𝑛𝑘−2−𝜈+23𝜈𝑒^2𝑗Θ𝛿^ℎ𝑘𝛿_|𝑚||𝑛|+𝑆^{𝑚𝑛ℎ𝑘}=𝑄^𝑚ℎ,−𝑁₁≤𝑚≤𝑁₁,0≤ℎ≤𝑁₂,(3.22) where 𝑆^{𝑚𝑛ℎ𝑘}=18𝜋√𝐸^𝑛𝑘𝐵^𝑛𝑘√𝐸^𝑚ℎ𝐵^𝑚ℎ𝑇_{𝑚𝑛ℎ𝑘}𝑇^{𝑚𝑛ℎ𝑘},𝑄=_𝑚ℎ_𝐷𝐿^𝑚ℎ𝜁₀_𝐷𝐴^𝑛𝑘(𝜁)𝐻𝜁,𝜁₀𝑑𝜁𝑑𝜁₀,=1√𝐸^𝑚ℎ𝐵^𝑚ℎ_𝐷𝐻𝜁,𝜁₀𝐿^𝑚ℎ𝜁₀𝑄𝜁₀𝑑𝜁₀,=(2−𝜈)𝐾⁽¹⁾𝜁,𝜁₀+3𝜈𝐾⁽²⁾𝜁,𝜁₀.(3.23)

In (3.22), we have used the following notation: 𝜁₀=𝜁₀(𝑠₀,𝜑₀), 𝑑𝜁₀=𝑠₀𝑑𝑠₀𝑑𝜑₀, and 𝑄(𝜁₀)=𝑄(𝜉₀,𝜂₀)=𝑄(𝑠₀cos𝜑₀,𝑠₀sin𝜑₀).

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,𝜑) and (𝑠₀,𝜑₀) defined as follows: 𝑠_𝑖=𝜋4+𝜋4^𝑀1_𝑖=1𝑊(𝑖),𝑠_0𝑖=𝜋4+𝜋4^𝑀1_𝑖=1𝜑𝑊₀(𝑖),_𝑗=∑^𝑀2_𝑗=1𝑗𝜋𝑀₂,𝜑_0𝑗=∑^𝑀2_𝑗=1(𝑗+0.5)𝜋𝑀₂,(3.24) where 𝑊(𝑖) and 𝑊₀(𝑖) are abscissas for 𝑠_𝑖 and 𝑠_0𝑖, 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)×(2𝑁₁+1)(𝑁₂+1) system of linear equations A𝑊=𝑏̃,(3.25) where 𝐴 is a square matrix, and 𝑊 and 𝑏̃ are vectors, 𝑊 to be determined.

4. Energy Release Rate

The energy release rate (measured in 𝐽𝑀⁻²), 𝐺(𝜑) by Irwin's relation subject to shear load is defined as [7, 8] 𝐺(𝜑)=1−𝜈²𝐸𝐾^𝐼𝐼(𝜑)²+(1+𝜈)𝐸𝐾^𝐼𝐼𝐼(𝜑)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 𝜇, is 𝜈=𝐸2𝜇−1,(4.2) and 𝐾_𝐼𝐼(𝜑) and 𝐾_𝐼𝐼𝐼(𝜑), the sliding and tearing mode stress intensity factor, respectively, are defined as [5, 7, 8] 𝐾_𝑗(𝜑)=lim_𝑟→𝑎𝑉_𝑗2𝜋𝑎−𝑟𝑤(𝑥,𝑦),𝑗=𝐼𝐼,𝐼𝐼𝐼,(4.3) where 𝑉_𝑗 are constants.

Let 𝑎(𝜑)=|𝑓(𝑒^𝑖𝜑)|, 𝑟=|𝑓(𝑠𝑒^𝑖𝜑)|, and as 𝑠 close to 1, (4.3) leads to 𝐾_𝑗(𝜑)=lim_𝑠→1−𝑉_𝑗2𝜋(1−𝑠)||𝑓^(𝑒^𝑖𝜑)||𝑤(𝑥,𝑦),𝑗=𝐼𝐼,𝐼𝐼𝐼.(4.4) Therefore, substituting (3.7) into (4.4) and simplifying gives 𝐾_𝑗(𝜑)=𝑉_𝑗||𝑓^𝑒^𝑖𝜑||⁻¹_𝑛,𝑘𝑊^𝑛𝑘√𝐸^𝑛𝑘𝐵^𝑛𝑘𝑌^𝑛𝑘(𝜑),𝑗=𝐼𝐼,𝐼𝐼𝐼,(4.5) where 𝑌^𝑛𝑘(𝜑)=𝐷^{|𝑛|+1/22𝑘+1}(0)cos(𝑛𝜑), and 𝐶^{|𝑛|+1/22𝑘+1}(√1−𝑠²)=√1−𝑠²𝐷^{|𝑛|+1/22𝑘+1}(√1−𝑠²), where 𝐷^𝜆𝑚(𝑥) is defined recursively by 𝑚𝐷^𝜆𝑚(𝑥)=2(𝑚+𝜆−1)𝑥𝐷^𝜆𝑚−1(𝑥)−(𝑚+2𝜆−2)𝐷^𝜆𝑚−2(𝑥),𝑚=2,3,…,(4.6) with 𝐷^𝜆0(𝑥)=2𝜆 and 𝐷^𝜆1(𝑥)=2𝜆𝑥.

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.

Table 1 

Numerical convergence for the energy release rate, 𝐺(𝜑) for 𝑓(𝜁)=𝜁+𝑐𝜁³ when 𝑐=0.1.

Figures 3, 4, 5, and 6 show the variations of 𝐺 against 𝜑 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(𝜑)=±1 or sin(𝜑)=±1. Similar behavior can be observed for the solution of 𝐺(𝜑) 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 𝑚=2 are 3.6066×10⁻⁶, 4.7064×10⁻⁵, 5.3503×10⁻⁵, and 9.0000×10⁻⁵ for 𝑐=0.001, 𝑐=0.01, 𝑐=0.10, and 𝑐=0.30, respectively.

Figure 3 

The energy release rate, 𝐺(𝜑) for 𝑓(𝜁)=𝜁+0.001𝜁3.

Figure 4 

The energy release rate, 𝐺(𝜑) for 𝑓(𝜁)=𝜁+0.01𝜁3.

Figure 5 

The energy release rate, 𝐺(𝜑) for 𝑓(𝜁)=𝜁+0.1𝜁3.

Figure 6 

The energy release rate, 𝐺(𝜑) for 𝑓(𝜁)=𝜁+0.3𝜁3.

Figure 7 

The energy release rate, 𝐺(𝜑) for 𝑓(𝜁)=𝜁+𝑐𝜁3 at various 𝑐.

Figure 8 

The energy release rate, 𝐺(𝜑) for 𝑓(𝜁)=𝜁+0.1𝜁3 at various choices of 𝜈 with 𝜇=1.

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].

Appendix

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

At 𝜁=𝜁₀, we have 𝑓(𝜁)−𝑓𝜁₀=𝜁−𝜁₀𝑓^𝜁₀+𝜁−𝜁⁰²𝑓^𝜁₀2+⋯.(A.1) Differentiate 𝑓(𝜁) with respect to 𝜁, we have 𝑓^(𝜁)=𝑓^𝜁₀+𝜁−𝜁₀𝑓^𝜁₀+𝜁−𝜁⁰²𝑓^𝜁₀2+⋯.(A.2) Let 𝐹₁=𝜁−𝜁₀𝑓^𝜁₀𝑓^𝜁₀=𝑢¹𝐹+𝑖𝑣₁=𝑂(𝑆),(A.3)₂=𝜁−𝜁₀²𝑓^𝜁₀2𝑓^𝜁₀=𝑢²+𝑖𝑣₂=𝑂𝑆²as𝑆⟶0,(A.4) where 𝑢₁, 𝑢₂, 𝑣₁, and 𝑣₂ are real. As 𝐹₁=𝑂(𝑆) and 𝐹₂=𝑂(𝑆²) as 𝑆→0, we see that 𝑢_𝑖 and 𝑣_𝑖 are 𝑂(𝑆^𝑖) as 𝑆→0 (𝑖=1,2).

Hence, (A.1) becomes 𝑓(𝜁)−𝑓𝜁₀=𝑓^𝜁₀𝜁−𝜁₀1+𝐹¹2+⋯.(A.5) Substituting (A.3) into (A.2) gives 𝑓^(𝜁)=𝑓^𝜁₀1+𝐹₁+⋯,𝑓^𝜁₀=𝑓^(𝜁)1−𝐹₁−⋯.(A.6)

As 𝑆→0 and truncate (A.1) at second order, then (A.6) can be written as 𝑓^(𝜁)≃𝑓^𝜁₀1+𝐹₁,𝑓^𝜁₀≃𝑓^(𝜁)1−𝐹₁,(A.7) respectively. Now, consider 𝐾⁽¹⁾(𝜁,𝜁₀). Let 𝛿−𝛿₀≃𝑣₁=𝑂(𝑆) where 𝛿 and 𝛿₀ defined in (3.6), then, from (3.6), we have 𝑒^{𝑖(𝛿−𝛿0)}=𝑓^(𝜁)||𝑓^𝜁₀||𝑓^𝜁₀||𝑓^(𝜁)||.(A.8)

Apply 𝑧𝑧=|𝑧|² leads to 1+𝐹₁||1+𝐹₁||=11++𝐹𝑢¹₁×1−𝑢¹1−𝑢₁≃≃11++𝑖𝑣𝐹₁₁1−𝑢₁.(A.9) Hence, 𝑒^{𝑗(𝛿−𝛿0)}≃1+𝑗𝑣₁.(A.10)

Martin [24] showed that ||1+𝛼𝐹₁+𝛽𝐹₂||^𝜆=|1+𝛼𝑢₁+𝛽𝑢₂+𝑖𝛼𝑣₁+𝛽𝑣₂|^𝜆/2≃1+𝛼𝜆𝑢₁+12𝜆(𝜆−1)𝛼²𝑢²¹+𝛼²𝑣²¹+2𝛽𝑢₂,(A.11) where 𝛼, 𝛾, and 𝛽 are constants and ||𝑓^(𝜁)||^3/2||𝑓^𝜁₀||^3/2||𝑓(𝜁)−𝑓𝜁₀||³−1||𝜁−𝜁₀||³=1𝑆³|1+𝐹¹+𝐹₂|^3/2||1+(1/2)𝐹₁+(1/3)𝐹₂||³−1=𝑂𝑆⁻¹,(A.12) as 𝑆→0.

Next, using (A.12), (A.11), and (A.10), we obtain 𝐾⁽¹⁾𝜁,𝜁₀=||𝑓^(𝜁)||^3/2||𝑓^𝜁₀||^3/2||𝑓(𝜁)−𝑓𝜁₀||³1+𝑗𝑣₁−1||𝜁−𝜁₀||³=1𝑆³||1+𝐹¹||^3/2||1+𝐹₁/2||³−1+1𝑆³||1+𝐹¹||^3/2||1+𝐹₁/2||³𝑗𝑣₁,(A.13) where ||1+𝐹₁||^3/2||1+𝐹₁/2||³−1≃38𝑣²¹−𝑢²¹,||||1+𝐹¹2||||³⟶1.(A.14) Thus, 𝐾⁽¹⁾(𝜁,𝜁₀) reduces to 𝐾⁽¹⁾𝜁,𝜁₀=38𝑆³𝑣²¹−𝑢²¹+𝑗𝑆³𝑣₁.(A.15) Since 𝐹₁=𝑢₁+𝑖𝑣₁, then 𝑢₁+𝑖𝑣₁²=𝑢²¹−𝑣²¹+2𝑖𝑢₁𝑣₁,Re𝐹₁=−𝑣²¹−𝑢²¹,(A.16) so, (A.3) leads to Re𝐹²¹=Re𝑒^2𝑗Φ𝑓^𝜁₀²𝑓^𝜁₀²≃𝐷𝜁⁰,Φ𝑆,𝑆⟶0,(A.17) where 𝐷(𝜁₀,Φ)=Re{𝑒^2𝑗Φ((𝑓^(𝜁₀))²/(𝑓^(𝜁₀))²)} and 𝜁−𝜁₀=𝑆𝑒^𝑖Φ defined in (2.7). Thus, 𝐾⁽¹⁾𝜁,𝜁₀=𝑂𝑆⁻¹+1𝑆³𝑗𝑣₁≃𝑗𝑣₁𝑆⁻³=𝑂𝑆⁻².(A.18) Therefore, 𝐾⁽¹⁾(𝜁,𝜁₀)≃𝑗𝑣₁𝑆⁻³, that is, 𝐾⁽¹⁾(𝜉,𝜉₀)≃𝑂(𝑆⁻²) as 𝑆→0.

For 𝐾⁽²⁾(𝜁,𝜁₀), expand 𝑓(𝜁) at 𝜁=𝜁₀, and truncating at second order, (3.4) gives Re^𝑖Θ=𝑎𝜁−𝜁₀𝑓^𝜁₀1+𝐹¹2,(A.19) where 𝑒^{𝑖(Θ−Φ−𝛿0)}≃1+𝐹¹2||||1+𝐹2¹||||⁻¹=1+𝑖𝑣²2.(A.20) Next, substituting (A.5) and (A.7) into (3.4) gives Re^𝑖Θ=𝑎𝜁−𝜁₀𝑓^(𝜁)1−𝐹₁1+𝐹¹2,(A.21) where 𝑒^{𝑖(Θ−Φ−𝛿)}≃1−𝐹¹2||||1−𝐹2¹||||⁻¹=1+𝑖𝑣²2.(A.22) Using (A.22) and (A.20) yields 𝑒^{𝑖(2Θ−2Φ−𝛿−𝛿0)}≃1+𝑖²14𝑣²²+𝑖𝑣₂=1+𝑂𝑆².(A.23) Hence, as 𝑆→0, then 𝑒^{𝑖(2Θ−2Φ−𝛿−𝛿0)}≃1+𝑂(𝑆²). It is not difficult to see that 𝑅≃𝑎|𝑓^(𝜁₀)|𝑆, Θ≃Φ+𝛿₀, 𝑅≃𝑎|𝑓^(𝜁)|𝑆, and Θ≃Φ+𝛿, respectively; then (3.11) becomes 𝐾⁽²⁾𝜁,𝜁₀=||𝑓^(𝜁)||^3/2||𝑓^𝜁₀||^3/2||𝑓(𝜁)−𝑓𝜁₀||³𝑒^2𝑗Φ−1|𝜁−𝜁₀|³𝑒^2𝑗Θ.(A.24) Applying similar procedures as in 𝐾¹(𝜁,𝜁₀) gives 1||𝜁−𝜁₀||³||𝑓^(𝜁)||^3/2||𝑓^𝜁₀||^3/2||𝑓(𝜁)−𝑓𝜁₀||³=1||𝜁−𝜁₀||³−1𝑆³|1+𝐹¹|^3/2||1+𝐹₁/2||³≃3−18𝑣²¹≃3−𝑢²¹8𝐹𝜁⁰,Φ𝑆as𝑆⟶0.(A.25) Thus, 𝐾⁽²⁾𝜁,𝜁₀=𝑒^2𝑗Φ38𝐹𝜁⁰,Φ=𝑂𝑆₋₁𝑆as𝑆⟶0.(A.26)

Acknowledgments

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.