2014, Vol. 35
The Chinese Meteorological Society
Article Information
- M. GHADIRI, H. GHASEMI 2014.
- Numerical methods for solving singular integral equations obtained by fracture mechanical analysis of cracked wedge
- Appl. Math. Mech. -Engl. Ed., 35 (3) : 311–324
- http: //dx. doi. org/10.1007/s10483-014-1793-6
Article History
- Received 2012-11-27;
- in final form 2013-09-08
1 Introduction
A stress intensity factor (SIF) is an important concept in fracture mechanical analysis. Crack vastly occurs in bonded structures and composite laminates,and the de-bonding between laminas is one of the most important reasons causing failing in these structures. Wedges are familiar and popular geometries in the world of mathematics,and have many applications in engineering and industry fields such as lap joints. Problems with wedges are usually solved by elasticity theories. Wedge-shape geometries are able to be expanded from 0◦ to 360◦.
Many scientists have tried to analyze non-finite radius wedges. Shahani and Adibnazari[1] used the Mellin transform to solve their cracked wedge problem imposed by shear deformation. The extracted singular integral equations were solved by the Muskhelishvili method,and the derived SIFs along the crack were plotted. Shahani[2] considered the anti-plane shear deformation of some wedges with cracks in their interfaces. He extracted the SIFs of the rounded shafts,where the bonded half plane included an interfacial edge crack,and the bonded wedges had an interfacial edge crack analytically. The responses were derived for simple isotropic material,anisotropic material,and bonded dissimilar material. Some same terms were achieved for the SIFs with similar geometries but different aforementioned material characteristics. Faal et al.[3] studied a non-finite wedge with the isotropic property which was damaged by a screw dislocation. They used numerical methods to consider the wedge angles,the location,and the orientation of the crack on the SIFs. The dislocation solution was a general solution for cracked wedge problems. Lin and Ma[4] employed the Mellin transform and the image method to achieve two-dimensional full field responses of non-similar circular isotropic wedges imposed by the anti-plane concentrated force and screw dislocation. The composite wedges were bimaterial that were bonded together with the same angles on their apexes. The elucidated procedure of finding the SIFs was applied on a semi-infinite cracked wedge and a circular composite cracked disc. Shahani[5] considered the deformation of a bi-material wedge with a finite radius under various boundary conditions. Shahani[6] also studied a problem for two non-similar edge-bonded wedges. Shahani and Ghadiri[7] studied bonded finite wedges with a crack in the middle. Shahani and Ghadiri[8] also considered an anistropic sector with a radial crack. The SIFs at both tips of the crack were calculated numerically. Ghadiri and Shahani[9] studied the problem of a non-isotropic cracked wedge with a finite radius. The SIFs at both tips of the crack were plotted with various boundary conditions.
In this paper,the responses of singular integral equations are numerically achieved by analyzing an interfacial cracked wedge under three types of boundary conditions:
(i) a cracked wedge with a non-finite radius and the traction-traction boundary condition;
(ii) a cracked wedge with a finite radius and the traction-traction boundary condition;
(iii) a cracked wedge with a finite radius and the traction-displacement boundary condition. According to the boundary conditions,the extracted singular integral equations are different. The obtained three different integral equations are solved numerically,and the results are shown in figures. The Gauss-Legendre polynomial is used in approximating the responses of the singular integral equations obtained by the fracture mechanical problems of the wedges. Because there is an analytical response for Problem (i),comparing the results of the problem in analytical and numerical forms is possible. The results show that the Gauss-Legendre method is not an appropriate method in solving integral equations but it is an advised method for solving singular integral equations obtained by contact mechanical problems. 2 Problem formulation 2.1 Problem (i)[1]
A wedge composed of two bonded isotropic wedges with non-finite length in the direction perpendicular to the plane of the wedge is studied,as depicted in Fig. 1,where α is the apex angle and µ1 and µ2 are the shear moduli. Because of the imperfect bonding,there is a crack at the length of the shared edge. The crack lies along the plane α = 0 and sticks between the radii r = a and r = b. The anti-plane shear deformation is applied to composite the wedge,and there are also traction-traction boundary conditions on the edges. However,a non-traction condition is imposed on all over the crack. With these settings,the merely non-zero displacement term is the out-of-plane term W which is a function of r and α. Therefore,the non-zero stress terms are τrz(r,α) and ταz(r,α),respectively.
 |
| Fig. 1. Cracked wedge of Problem (i) |
The fundamental equations of isotropic material imposed by the anti-plane deformation are stated as
As there is no body force,the equilibrium equations in the expressions of the displacement change to
The differential equation (2) must be analyzed by the following boundary conditions:
where δ shows the Dirac-Delta function.
Finally,using the Mellin transform gives us the following singular integral equation:
The single valuedness condition is
where
2.2 Problem (ii)[8]
A finite radius wedge made from two bonded anisotropic wedges with non-finite length in the direction perpendicular to the plane of the wedge is investigated as shown in Fig. 2. Like the previous problem,the crack exists at θ1 = θ2 = 0 between the radii r = c1 and r = c2. The antiplane shear deformation is exerted on the sector. The traction-traction boundary conditions are applied on the radial edges of the sector,while the traction free conditions are applied to the faces of the crack. With these boundary conditions,the merely non-zero displacement term is out of the plane term W,which is related to the in-plane coordinate (r,θ). Therefore,the appeared stress components are τrz(r,θ) and τθz(r,θ).
 |
| Fig. 2. Cracked wedge of Problem (ii) |
The constitutive equations for anistropic material undergoing the anti-plane deformation reduce to
where C44,C45,and C55 are the elastic constants of the wedges,and k = 1,2.
Without the body force,the equilibrium equation in terms of the displacement changes to
The differential equation (13) must be solved under the following boundary conditions:
where δ denotes the Dirac-Delta function. It is worth mentioning that the choice of these two boundary conditions leads to the Green function solution for the problem. In Eqs. (15) and (16),h is the location of application of the concentrated traction which may vary from zero to r = a. Without loss of generality,it is supposed that c1 ≤ c2 ≤ h.
Substituting the modified Mellin transform to Eq. (14) yields the following singular integral equation:
where
The single valuedness condition is
where
2.3 Problem (iii)[9]
A wedge fabricated from two anisotropic wedges with the same matter characteristics is shown in Fig. 3. The wedges are bonded together over the length of the shared edge and have non-finite length in the direction perpendicular to the wedge plane. The non-perfect bonding causes a crack along the shared edge. The anti-plane shear deformation is applied to the wedge. As a result,the merely appeared displacement term is the out of plane component W(r,θ),and the existing stress terms are τrz(r,θ) and τθz(r,θ). The constitutive expressions of anisotropic material under the above mentioned conditions are
where C44,C45,and C55 are the elastic constants of the sector,and k = 1,2.
 |
| Fig. 3. Cracked wedge of Problem (iii) |
From Eqs. (26) and (27),the equilibrium equation in terms of the displacement may be expressed as
The differential equation (28) must be solved by the following boundary conditions:
We make the following changes in the variables:
Then,we can reach this singular integral equation
where
The single valuedness condition is
3 Solutions of singular integral equations 3.1 Analytical solution
Muskhelishvili[11] has solved such singular integral equations as Problem (i),and the obtained analytical solution is also the analytical solution to Eqs. (8) and (9),i.e.,
where
Then,
3.2 Numerical solutions 3.2.1 Problem (i)
(I) Gauss-Chebyshev polynomial
Erdogan and Gupta[12] have presented a very simple and effective numerical solution to singular integral equations. Here,we will first take the same way for the Gauss-Chebyshev polynomial in this part,and then repeat the same way for the Gauss-Legendre polynomial in the next part.
First of all,change the interval into [−1,1] as follows:
Substituting Eq. (43) into Eqs. (8) and (9) yields
To reach the weight function of the Gauss-Chebyshev polynomial,we must insert
Now,F(η) is bounded in the closed interval [−1,1]. With the method described by Erdogan and Gupta[12],Eqs. (44) and (45) are reduced to
where ηk and χr are the roots of the Gauss-Chebyshev functions of the first and second kinds, respectively,i.e.,
All of the above mentioned constant parameters are dimensionless ones,and have the same meanings as those mentioned in Ref. [8].
Figures 4,5,and 6 show the variations of 
versus 
which have been normalized
into [1, 3] with 
,where
 |
| Fig. 4. Responses of Gauss-Chebyshev approximation with different points for Problem (i) |
 |
| Fig. 5. Responses of Gauss-Legendre approximation with different points for Problem (i) |
 |
| Fig. 6. Responses of Gauss-Chebyshev and Gauss-Legendre approximations with 40 points and analytical solution for Problem (i) |
It is mentionable that,in Eq. (51),W and f(r) show the displacement and density functions in the crack,respectively. Therefore,if f(r) is calculated,the displacement will be achieved. From Eqs. (26) and (27),we can find the stress. As depicted in all the following figures,when the values of the vertical axis incline to infinity,the stress inclines to infinity. For more details, see Ref. [8] and the references therein.
As depicted in Fig. 4,40 points are enough for the approximation to approach to the infinite values in singular points,and the results coincide with the expected results.
(II) Gauss-Legendre polynomial
Equations (43)-(45) are the same for the Gauss-Chebyshev method and the Gauss-Legendre method. However,because the weight function of the Gauss-Legendre polynomial is unity,there is no need to use the variable change like Eq. (46). Therefore,
where ηk and χr are the roots of the Gauss-Legendre polynomials of the first and second kinds, respectively,i.e.,
in which
Figure 5 considers the responses of the Gauss-Legendre polynomial with approximations of various points. It is observed that the difference between the approximations with 30 points and 40 points is very small. Therefore,we can say that 40 points are enough for the approximation.
Figure 6 shows the analytical solution and the responses of the Gauss-Chebyshev and GaussLegendre approximations with 40 points. It is observed that the Gauss-Legendre and GaussChebyshev approximation responses with 40 points are in acceptable agreement with the analytical response. It is mentionable that the accuracy increases with the increase in the approximation points and 40 points are enough for cutting off the calculation process and reaching a rough response. It is also shown that in the singular points of the crack,the result of the Gauss-Chebyshev approximation is nearer to the infinite value. 3.2.2 Problem (ii)
All subsequent figures show the variations of 
versus which have been normalized
into [1, 3]. The initial and final points of the interval show singular points of our physics.
Moreover,
(I) Gauss-Chebyshev polynomial
Changing the interval into [−1,1] yields that Eqs. (20) and (23) transform to
Inserting the roots of the Gauss-Chebyshev functions of the first and second kinds in Eqs.(56) and (57) and using the variable change,we have
Figure 7 shows the responses of the Gauss-Chebyshev approximation with different points for Problem (ii). The results show that 40 points are enough for the calculation precision.
 |
| Fig. 7. Responses of Gauss-Chebyshev approximation with different points for Problem (ii) |
(II) Gauss-Legendre polynomial Similar to the calculation process in solving Problem (i),from Eqs. (20)-(23),we have
Figure 8 shows the responses of the Gauss-Legendre approximation with different points for Problem (ii). It can be seen that 40 points are enough for the calculation precision of the Gauss-Legendre polynomial.
 |
| Fig. 8. Responses of Gauss-Legendre approximation with different points for Problem (ii) |
Now,we want to compare the responses of the Gauss-Chebyshev and Gauss-Legendre approximations with 40 points. The results are shown in Fig. 9. Because there is no analytical solution for this problem,Fig. 9 shows just two kinds of numerical responses. It is mentionable that the convergence speed of the Gauss-Chebyshev approximation going toward an infinite value is faster than that of the Gauss-Legendre approximation.
 |
| Fig. 9. Responses of Gauss-Chebyshev and GaussLegendre approximations with 40 points for Problem (ii) |
(I) Gauss-Chebyshev polynomial
With the same procedure in Problem (i),Eqs. (36) and (39) change to
Figure 10 shows the responses of the Gauss-Chebyshev approximation and the GaussLegendre approximation with different orders for Problems (iii). The results show that 40 points are enough to reach an admissible response.
 |
| Fig. 10. Convergence of Gauss-Legendre approximation with different points for Problem (iii) |
(II) Gauss-Legendre polynomial
With the same procedure as the above,Eqs. (36) and (39) change to
As you can see in Fig. 11,by adding the points of approximation,infinite values in singular points are approached.
 |
| Fig. 11. Convergence of Gauss-Legendre approximation with different points for Problem (iii) |
Figure 12 shows the responses of the Gauss-Legendre and Gauss-Chebyshev approximations with 40 points. Figure 12 shows that the Gauss-Chebyshev approximation and the GaussLegendre approximation have similar responses. However,the speed of the Gauss-Chebyshev approximation in approaching to infinite values in singular points is higher than that of the Gauss-Legendre approximation.
 |
| Fig. 12. Convergence of Gauss-Chebyshev and Gauss-Legendre approximations with 40 points for Problem (iii) |
In this paper,three singular integral equations obtained by fracture mechanical analyses of a cracked wedge under three different boundary conditions are solved numerically by GaussChebyshev and Gauss-Legendre polynomials. It is the first time that the Gauss-Legendre approximation is used for solving the singular integral equations obtained by fracture mechanical analyses. The singular integral equations of Problem (i) have analytical solutions because of their zero kernels.
The convergence of the Gauss-Chebyshev or Gauss-Legendre polynomial is considered by adding the points of approximation. The results are then compared with each other by figures.
The calculations show that because the weight function of the Gauss-Legendre polynomial is unity,there is no need to change variables. Therefore,the most important advantage of the Gauss-Legendre approximation toward the Gauss-Chebyshev approximation is its simplicity. It is mentionable that the nodes,weights,and singular points of the Gauss-Legendre polynomial are obtained by the Newton-Raphson method because their functions ((54) and (55)) have no analytical response. The values of the nodes,weights,and singular points of the GaussChebyshev polynomial are obtained by trigonometric relations which are more precise than those of the Gauss-Legendre polynomial. Therefore,it is a negative point of the Gauss-Legendre approximation and may lead to errors in the results. As depicted in Figs. 4-12,the speed of the Gauss-Chebyshev approximation to the infinite values in singular points is higher than that of the Gauss-Legendre approximation.
Finally,we conclude that despite of the reported results[10] that “ the Gauss-Chebyshev approximation is not a suitable method for solving contact singular integral equations”,the Gauss-Chebyshev approximation is a as well as or even better (from some aspects) method in solving fracture mechanic singular integral equations than the Gauss-Legendre approximation.
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