Nomenclature
W,                                        out-of-plane displacement component;
a,                                          wedge radius;
C₄₄, C₄₅, C₅₅,                 elastic constants;
τ,                                          stress component;
θ₁,                                      apex angle of the upper wedge;
θ₂,                                      apex angle of the lower wedge;
U,                                         arbitrary complex function;
r,                                          radial component of the polar coordinate;
θ,                                         tangential component of the polar coordinate;
c₁,                                      radial distance of the nearest crack tip to the wedge apex;
c₂,                                      radial distance of the furthest crack tip to the wedge apex;
Z,                                         complex variable;
S,                                         complex transform parameter;
U₁^∗k/U₂^∗k,                    complex integral transform of the first/second kind;
H,                                        complex arbitrary function;
f,                                          density function of the screw dislocation;
γ,                                         arbitrary variable;
a₀,                                      semi-crack length;
c₀,                                      distance of the center of the crack with respect to the apex;
K,                                        Mode III stress intensity factor (SIF). 1 Introduction

The stress intensity factor (SIF) is an important concept in fracture mechanics analysis. Crack vastly occurs in bonded structures and composite laminates. The debonding between laminas is one of the most important reasons causing failings in these structures. Wedges are familiar and popular geometries. Engineers and scientists always use the elasticity theory to solve the problems involving wedges. Various applications of wedges have encouraged them to enter the world of wedges. The usages of lap joints in industry are only a few benefits of wedges. Wedge-like geometries are able to be modeled in various angles from 0 ^◦ to 360 ^◦.

The stresses in cracked wedges have been analyzed by various investigators. Kargarnovin et al.^[1] accomplished the stress analysis in a finite radius wedge under anti-plane shear loading. Shahani^[2] studied the anisotropic finite wedges subjected to anti-plane shear loading with defining some complex integral transformations. The circular boundary of the wedge was subjected to the traction-free condition,and three kinds of boundary conditions were applied on the radial edges,i.e.,displacement-displacement,and traction-traction,and traction-displacement.

The stress analysis in a wedge with an infinite radius has also been considered by various investigators. Shahani and Adibnazari^[3] used the Mellin transform to solve their cracked wedge problem imposed by shear deformation. The achieved equations which were the singular integral equations were solved by Muskhelishvili’s method,and then the derived SIFs along the crack were plotted. Shahani^[4] considered the anti-plane shear deformation of some wedges with a crack in their interface. He analytically extracted the SIFs of rounded shafts with edge cracks, bonded half planes including an interfacial edge crack,bonded wedges having an interfacial edge crack,and double cantilever beams with various boundary conditions. The responses were derived for simple isotropic materials,anisotropic materials,and bonded dissimilar materials. It was depicted that the same terms were achieved for the SIF with similar geometries but different aforementioned material characteristics. Faal et al.^[5] studied a non-finite wedge with isotropic properties damaged by a screw dislocation. They used a numerical method to consider the angle effects of the wedge and the location and orientation of the crack on the SIFs of straight line cracks. The dislocation solution is a general solution to answer cracked wedge problems. Lin and Ma^[6] employed the Mellin transform and image method to achieve two-dimensional fullfield responses of the non-similar isotropic circular wedges imposed by anti-plane non-external forces and screw dislocations. The composite wedges were bi-material wedges having the same angles on their apexes which were bonded together. Clear results were accessible in forms of summations for traction and displacement boundary conditions. The elucidation procedure of finding the SIFs was done for a semi-infinite cracked wedge and a circular composite cracked disc. Shahani^[7] studied the deformation of a bi-material wedge with a finite radius and various boundary conditions. Shahani^[8] studied a problem which contained two edge-bonded nonsimilar wedges. Faal et al.^[9] analyzed the isotropic finite wedge subjected to various boundary conditions weakened by multiple cavities with the image method. Shahani^[10] investigated the isotropic finite wedge under anti-plane shear loading,and extracted the closed form solution for the stress distribution. In the special case,the SIFs in a circular shaft with an edge crack subjected to various boundary conditions were obtained. Chen et al.^[11] investigated the composite finite wedge under anti-plane loading. The fixed-fixed,free-free,and free-fixed conditions were considered for the radial edges,and anti-plane loading was prescribed on the circular arc. Shahani and Ghadiri^[12] studied the bonded finite wedges with a crack in their middle. The circular boundary of the wedge was free from any traction. Traction-traction boundary conditions were applied to the edges which were radial. The new finite complex transforms were employed to solve the governing partial differential equations (PDEs). Shahani and Ghadiri^[13] considered an anisotropic sector with a radial crack,and obtained the SIFs at both ends of the crack numerically. Ghadiri and Shahani^[14] studied the problem of non-isotropic cracked wedges with finite length,and plotted the SIFs at both ends of the crack under various conditions.

In the present article,the antiplane stress problem of two anisotropic finite wedges with arbitrary radii and apex angles bonded together along a common edge is considered. The radial boundaries of the wedge are under displacement-displacement boundary conditions,and the crack faces is free from any traction. The displacement boundary condition is a general term of the Taylor series expansion of displacement prescribed on the radial edge of the wedge. The applied boundary condition on the crack zone is expressed as a singular integral equation by use of the complex integral transforms devised by Shahani^[2]. The standard singular integral equation is extracted in this paper by describing an exact analytical method. The resultant singular integral equations are solved numerically,and the obtained results including the SIFs at the crack tips are plotted. The obtained results in the special cases show a complete agreement in comparison with those cited in the literature. According to the form of the displacement boundary condition,every kind of displacement boundary conditions can be analyzed based on the obtained results from the foregoing general displacement condition. The obtained results in the special cases are compared with the results obtained by the finite element analysis (FEA). 2 Statement of problem and solution

A finite radius wedge made from two bonded anisotropic wedges having two angles θ₁ and θ₂ at their apexes with non-finite length in the direction perpendicular to the plane of the wedge is investigated (see Fig. 1). Because of non-perfect bonding,there is a crack at the length of the general edge. The crack lies along θ = 0 between the radii r = c₁ and r = c₂. The composite wedge is applied to anti-plane shear deformation. There are also displacement-displacement boundary conditions on the edges though a non-traction condition is imposed over the crack. With these settings,the only non-zero displacement term is the out-of-plane term,W(r,θ), which is a function of r and θ. The non-zero stress terms are τ_{r z}(r,θ) and τ_θz(r,θ).

The fundamental equations of anisotropic materials imposed by anti-plane deformation are stated as

where C₄₄,C₄₅,and C₅₅ are the wedge elastic constants,and k = 1,2.

Since there is no body force,the equilibrium equation in the expressions of displacement changes to

According to Fig. 1,the following boundary conditions may be considered:

where n is a real constant. The differential equation (3) must be analyzed under the above boundary conditions. It is noteworthy to indicate that the boundary condition in (4) may be the general term of the Taylor series expansion of the displacement prescribed on the edge OB about r = 0.

The boundary condition is the traction free condition on the circular segment of the wedge circumference,i.e.,

The relations between the stress polar components,τ_rz and τ_θz,and the stress rectangular components,τ_xz and τ_yz,are expressed as follows:

For solving the governing equation (3),define the complex plane as

such that

where U(Z) is the complex conjugate of the arbitrary function U(Z),and p is a parameter whose value depends on the elasticity constants. Substituting (12) into (3) yields

whose solution provides two roots for p. One root is

and the other root is

Replacing (1) and (2) into (10) and using (12) yield

where

Equation (9) together with (15) yields

Therefore,(18) is the boundary condition on the circular boundary of the wedge.

Now,for solving the mixed boundary value problem,two complex integral transforms are employed which were presented by Shahani^[2] . These transforms are

where k = 1,2,S is a complex parameter,and

Now,the path of integration is selected along a path of fixed θ. Thus,(20) becomes

where

The complex conjugate of (22) becomes

We should define the first and second kind new transformations for solving the problem,i.e.,

Analogous to what has been done on (20),we select the integration path in (25) and (26) to be along a path of fixed θ. Then,we have

where

Similar to (24),the complex conjugate relations can be written for (27) and (28). Partly integrating (27) and (28) yields

Applying the boundary condition (18) on (30) and (31) yields

provided that

Again,integrating (32) and (33) by parts,we obtain

provided that

Multiplying (3 6 ) by S and adding it to (35) yield

which is similar to (22) if we define

Taking the second kind Mellin transform from (12),and then substituting (22) and (24) into it,we obtain

Now,multiplying (32) by S,adding it to (33),and then applying the resultant relation and its complex conjugate on (15) and (16),we obtain

where

The unknown terms U^*k₂ (S) and U^*k₂(S) in (40) and (42) are obtained with substituting the boundary conditions.

The found solution in the complex Mellin transform domain can be used to obtain the displacement and stress distribution by the following inverse formula^[3]:

Additionally,the strip of regularity,which specifies a proper value range for the real quantity C in (44),can be obtained by the expressed conditions in (34) and (37). 3 Extraction of singular integral equation

Writing relations with (40) and (42) for the upper and lower wedges,we obtain

Now,applying the second kind Mellin transform (22) on (4) and (5),and then replacing the obtained results into (45) and (46),we can write

Substituting (7) on (47) and (48) yields

Rewriting (49) and (50),we obtain

Now,substituting (52) and (53) into (51) and performing some mathematical manipulations yield

Replacing (54) into (53) yields

In (52),(54),and (55),the unknown function U^*1₂(S) is obtained by applying the rest boundary condition.

Now,we define the following unknown function by use of (6)^[3] :

Then,the continuity condition of displacement outside the crack becomes

Also,for the single-valuedness condition of displacement,we can write

Now,applying the inverse transformto (45) and (46),and then substituting the obtained relationsinto (56),we have

Taking the Mellin transform of the second kind from (59) yields

Substituting (52),(54),and (55) into (45) and (46) and then applying θ = 0 into the resulting expressions yield the following results:

Replacing foregoing expressions into (60) and facilitating the obtained result yield

Now,substituting (63) into (52) yields

Substituting (63) and (64) into (47) and applying the inverse transform to the obtained relation yield

Now,a closed path is used to calculate the integral in the light of the residual theorem. Hence, the integration path is completed with an infinite radius semi-circular arc. This semi-circular arc can be the negative or positive part of the real axis . To compute the poles ,we write

in which

The new parameter ψ depends on the wedge apex angle and material constants,and it is called the transformed apex angle of the anisotropic wedge. Then,(66) together with (14) yields

From (68) and (23),we have

where

Now,substituting the foregoing expressions into (65) and facilitating,we have

with putting the denominator of the first term of (72) to be equal to zero . Then,the equation of the poles can be written as

It is worth mentioning that in a wedge where the transformed angle ψ₁ − ψ₂ ≠ mπ/n (m ∈ Z), all singularities of the integrand are simple poles. According to the above relation,the strip of regularity becomes

Rewriting (72) for θ = 0 and then applying the boundary condition (8),we obtain

By using the residue theorem and applying some mathematical operations,we have^[2]

in which

Performing some mathematical manipulations on (76),we obtain

where u(ν − r) is the Heaviside step function. Let

Define

Then,we can show (78) in the form of a singular integral equation according to Ref. ^[15] as follows:

where

Also,the single-valuedness condition of displacement,(58),by use of (79) and (80),can be shown in the following form:

Equations (81) and (83) represent the double singular integral equations for the cracked anisotropic finite wedge with the displacement-displacement boundary conditions on the radial edges of the wedge. The double singular integral equations can be solved numerically to obtain the dens ity functio n f(r),and thus the stress field can be calculated.

For an anisotropic finite wedge without crack (c₁ = c₂),in the case of θ₁ = α and θ₂ = 0, performing some mathematical manipulations on (72) yields

The above calculated relation is in agreement with that previously presented in Ref. ^[2] which in turn confirms the validation of the results rendered in the current investigation. 4 Solution of singularint egral equation

To obtain the numerical solution of singular integral equations,(81) and (83) are converted to linear algebraic equations according to Ref. ^[16] as follows:

where

In the above equations,F(η) is enclosed in the closed interval [−1,1],and η_k and χ_r are the roots of the Gauss-Chebyshev functions of the first and second kinds (T_n(η_k),U(χ_r)),respectively.

After determining F(η),the SIFs may be obtained by the following relations^[3]:

where τ¹_θz (r = 0,θ = 0) can be computed by (72).

In order to insert the length and the location of the crack,we may write the crack length and the crack center location in terms of the crack tips coordinates as follows:

5 Comparison of analytical and numerical results

In order to verify the assumption considered at the start of this paper,“the anti-plane shear deformation”,a finite element method-based numerical analysis is performed in ABAQUS simulation software. The problem is modeled in a 3D form in the software. The obtained results are then compared with those of the analytical approach.

Figure 2 shows the finite element meshes in the ABAQUS environment. In this model,for the crack tip,the singular element C3D15 is utilized; for the remainder of the body,the 3D secondorder element C3D20 is employed. The displacement-displacement boundary conditions radial edge of the wedge are prescribed,and the circular segment of the wedge is considered to be traction free. The parameters of the wedge are

Figures 3 and 4 depict the comparison of the analytical and numerical results for the shear stress τ_θz in the radial direction θ = 0 with n = 1 and n = 1/2 ,respectively. Based on Figs. 3 and 4,it is observed that the results of both analytical and numerical approaches are approximately identical. Consequently,the considered hypothesis (anti-plane shear deformation) is correct and valid for the planar solution of the considered problem.

It is worth mentioning that the displacement and the stress field become unbounded in the whole region as ψ₁ − ψ₂ = mπ/n . The integrand in (72) has a double pole at ,and the other singularities are simple poles. Because of some mathematical complexity,the effect of the double pole is neglected in the integral calculation by means of the residual theorem (see Ref. ^[2] for more details). Thus,according to Fig. 3 and 4,we can say that the difference between the stresses obtained by the analytical solution and the finite element method is due to the neglect of the double pole. 6 Results and discussion

Figure 5 shows the variations of the SIFs versus the relative crack distance with different values of the parameter n in the case of a/a₀= 5 and θ₁ = π,θ₂ = π/2 in an anisotropic wedge with C₄₄/ C₅₅ = 1 and C₄₅/C₅₅ = 0.5. According to Fig. 5,it is observed that the SIFs in both apexes of crack decrease with the increase in n.

Figures 6 and 7 show the variations of the SIFs versus the relative crack distance in different wedge apex angles in the case of a a₀= 5 and n = 2 in an anisotropic wedge with C₄₄C₅₅ = 1 and C₄₅/C₅₅ = 0.5. The considered apex angles of the wedge in Fig. 6 are in these three cases: θ₁ = π/2 ,θ₂ = π/6 (for which ψ₁ − ψ₂ = 2.464 5); θ₁ = 2π/3 ,θ₂ = π/4 (for which ψ₁ − ψ₂ = 2.988 1); and θ₁ = 3π/4 ,θ₂ = π/6 (for which ψ₁ − ψ₂ = 2.988 1). In Fig. 7,the apex angles of the wedge are considered as follows: θ₁ = π,θ₂ = π/2 (for which ψ₁ − ψ₂ = 4.188 8); θ₁ = 3π/4 ,θ₂ = π/2 (for which ψ₁ − ψ₂ = 3.665 2); and θ₁ = 2π/3 ,θ₂ = π/2 (for which ψ₁ − ψ₂ = 3.511 7).

In Fig. 6,θ₁ + θ₂ < π,while in Fig. 7,θ₁ + θ₂ > π. According to (73),when the parameter (ψ₁ − ψ₂) is greater than π,the stress field at the wedge apex becomes singular. It is worth mentioning that according to (71) and (73),the order of the stress singularity (λ = 1 − S₁) depends on the material property as well as the apex angle of the wedge.

In Fig. 6,within all cases,the value of (ψ₁ − ψ₂) is less than π. Thus,the singularity of the wedge apex disappears. It is found that by approaching the crack to the wedge apex c₀/a₀ → 1, the SIF increases. The reason of increasing the SIF in the wedge apex,where there is no stress singularity,is due to the displacement condition on the radial edge of the wedge. Because the exerted displacement on the radial edge is equivalent to the exerted force which has created the displacement,in small angles of the wedge apex,this kind of boundary conditions is nearer to the crack tips. Therefore,when c₀/a₀ → 1,by decreasing the angle of the wedge apex,the SIF increases. On the other hand,by going away from the wedge apex,the SIF decreases. By approaching the crack to the circular segment of the wedge,the SIF at the second crack tip tends to zero because the second crack tip approaches the free traction circular boundary of the wedge.

It is observed from Fig. 7 that the SIFs reduce as the relative crack distance c₀/a₀ increases. On the other hand,the SIFs at the first crack tip increase as the crack tip approaches the wedge apex,i.e.,c₀/a₀ → 1. It is seen that K(c₁) becomes infinitely large when ac 00 tends to unity. Nevertheless,Shahani^[17] showed that although K(c₁) has an increasing trend as c₀/a₀ → 1 (or the crack tip r = c₁ approaches the wedge apex),it becomes zero when c₀/a₀ = 1 (or c₁ = 0). Also,it is observed that by approaching the crack to the circular boundary of the wedge,the SIF at the second crack tip tends to zero. Another important point that can be observed from Figs. 6 and 7 is that the values of the SIFs in small angles of the wedge apex where the stress singularity disappears are greater than the values of the SIFs in large angles of the wedge apex where the stress singularity appears.

Figure 8 shows the variations of the SIFs versus the relative crack distance in the case of a/a₀ = 5 and n = 1 for an apex angle of θ₁ = π,θ₂ = 2π/3 in different composite wedges,i.e., different material properties. It is observed that in all of the cases,K(c₁) and K(c₂) tend to finite values as c₀/a₀ increases,K(c₁) → ∞ as ac₀ → 1 (or c₁ → 0),whereas K(c₂) is finite for c₀/a₀ = 1 (or c₁ = 0). In addition,it is seen that the SIF depends on the material property of the wedge.

7 Conclusions

In the present article,the antiplane stress analysis of two anisotropic finite wedges with arbitrary radii and apex angles that are bonded together along a common edge is investigated. The wedge radial boundaries are under displacement-displacement boundary conditions,and the circular boundary of the wedge is free from any traction. The new finite complex transforms are employed to solve the problem. These finite complex transforms have complex analogies to both kinds of standard finite Mellin transforms. The traction free condition on the crack faces is expressed as a singular integral equation by use of the exact analytical method. The explicit terms for the strength of singularity are extracted. This shows the dependence of the order of stress singularity on the wedge angle,material constants,and boundary conditions. The resultant singular integral equations are then solved numerically. The displacement boundary condition may be a general term of the Taylor series expansion of displacement prescribed on the radial edge of the wedge. Thus,the analysis of every kind of displacement boundary conditions can be obtained by the achieved results from the foregoing general displacement boundary condition. The obtained results show that the order of the stress singularity (λ = 1 − S₁ ) depends upon the material property,the apex angle of the wedge,and all kinds of boundary conditions. It is observed that as the radial crack approaches the traction-free boundary,the SIF at the second crack tip tends to zero. Also,it is found that the values of the SIFs in small angles of the wedge apex where the stress singularity disappears are greater than the values of the SIFs in large angles of the wedge apex where the stress singularity appears. It is seen that as the material of the wedges changes,the values of the SIFs change. Also,it is observed that the analytical and numerical results for the shear stress τ_θz in the radial direction θ = 0 are approximately identical.