1 Introduction

Solving the elastic problems for the semi-strips with transverse cracks is very important for the practical applications and the developments of theoretical methods.

Crack type defects exist widely in many engineering constructions, e.g., components of machines, basements of buildings, and sheathing of ships and cars. These defects noticeably reduce the structural strength during practical operations, and in some cases can lead to complete destruction. Therefore, it is essential to evaluate the stress concentration degree around cracks and to ascertain how their locations affect the possible destruction of the bodies. One of engineering structure elements is beam. The plane deformation beam can be modelled by an elastic semi-strip.

Two types of approaches are used for elastic problems, i.e., analytical/semi-analytical methods and numerical methods. Numerical methods, e.g., the finite element method and the integral boundary equation method, can be applied to elastic problems. However, when there are some discontinuous zones, the applications cannot provide accurate results. Therefore, it is important to work out the analytic or semi-analytic methods to solve the elastic problems with defects.

A short review of the approaches used to solve the plane elastic problems for the elastic semi-strips with cracks is given as follows. Savruk et al.^[1] developed the method of singular integral equations (SIEs) to solve the two-dimensional elastic problems with cracks. Morozov^[2] considered the two-dimensional static problems of the crack theory. Slepyanm^[3] examined the static, slowly growing, and dynamic cracks in solid bodies and the criterion of the crack growth. Savruk^[4] considered methods to solve the two-dimensional boundary problems of the mathematical crack theory for isotropic bodies, and examined the problems for semi-planes and strips with cracks. Mykhas'kiv et al.^[5] investigated the three-dimensional elastodynamic interaction between a penny-shaped crack and a thin elastic inter layer joining two elastic half-spaces by an improved boundary integral equation method or boundary element method. Hakobyan^[6] presented the development of the mathematical apparatus for mixed boundary value problems.

With the two-sided Laplace transformation, Borisovich^[7] reduced the problem of a semi-infinite crack parallel to a boundary of the semi-plane to the matrix Riemann's problem. Borisova^[8] solved the problem of the stress concentration near the tip of a finite crack in the semi-plane with a free boundary with the modified explosive method, which allowed the creation of a SIE. Stepanova^[9] analyzed the stress state near a stationary crack in the plastic body. Antipov et al. 10] considered the plane strain elastic half-plane problem of an edge crack lying along the interface of two perfectly bonded dissimilar quarter-planes, and subsequently reduced the mixed boundary value problem to a system of two functional equations of the Wiener-Hopf type. Liu and Zhou^[11] investigated the dynamic behavior of a rectangular crack in a three-dimensional (3D) orthotropic elastic medium under a harmonic stress wave based on the non-local theory. Protserov and Vaysfeld^[12] evaluated the stress intensity factor (SIF), and showed that the SIF depended on the crack's geometry for the problem of a multilayered elastic finite cylinder with a fixed bottom face. Nazemnezhad and Fahimi^[13] investigated the surface energy effects, including the surface shear modulus, the surface stress, and the surface density, on the free torsional vibration of nanobeams with a circumferential crack and various boundary conditions.

Chai^[14] and Erdogan and Arin^[15] considered the spreading of cracks in infinite strips. Chen^[16] reduced the boundary value problem of the strip with periodical cracks into a complex mixed problem, and solved the problem by use of the eigenfunction expansion of the variational method. Chiang^[17] developed a numerical procedure based on Williams's eigenfunctions for solving elastic fracture problems. Civelek and Erdogan^[18] reduced the problem for an infinite strip containing multiple cracks perpendicular to its boundaries to a system of SIEs. With the Fourier transforms, the problems of determining the SIFs in an infinite elastic strip containing two^[19] and three^[20] coplanar Griffith cracks were reduced to the problems of solving a set of triple and quadruple integral equations, corresponding to a cosine kernel and a weight function, respectively. Georgiadis and Brock^[21] solved the problem of cracked elastic strips under concentrated loads with a method based on the integral transform analysis, an exact kernel factorization, and the analytic function theory. Ignatieva^[22] solved the first boundary value problems in the part-homogenous semi-planes and strips with a strongly permeable crack perpendicular to the boundaries with the rolling method of Fourier's transformations. Antipov and Schiavone^[23] deduced the nonstandard boundary value problem for Laplace's equation in an infinite strip containing a finite crack with the Fourier transform first to an integral equation and then to a vector Riemann-Hilbert problem. Wu et al.^[24] reduced the boundary-value problem of a semi-infinite interfacial crack between two bonded dissimilar elastic strips with equal thickness by means of the conformal mapping technique to the standard Riemann-Hilbert problem. Liu and Erdogan^[25] formulated an infinite elastic strip containing arbitrarily oriented cracks and subjected to uniform tension and a pair of concentrated forces in terms of a system of SIEs. Gecit^[26] studied the elastostatic plane problem of an infinite strip containing a transverse crack. Goldstein et al.^[27] investigated the SIFs in an infinite elastic strip with a central transverse crack. Itou and Tani^[28] reduced the boundary value problem for an infinite elastic strip with a semi-infinite crack to a SIE by use of single and double layer potentials. Kal'muk et al.^[29] gave the formulae for the stress intensity coefficients around the cracks and rigid inclusions in an infinite strip. Lamzyuk et al.^[30] gave the solution of the problem for a loaded crack in an infinite strip by use of the superposition method. Fan^[31] suggested a procedure based on Muskhelishvili's complex potential formulation and combining the conformal mapping technique to solve the SIFs for an infinite crack in a strip. Yetmez and Gecit^[32] numerically solved the problem of a finite strip containing a transverse symmetrical crack at the midplane and rigid plates bonded to the ends of the strip by use of a general purpose finite element code family MSC.MARC. Dyskin et al.^[33] considered two configurations of modelling the interaction of a crack with parallel free boundaries.

There are significantly less investigations of semi-strips with cracks in comparison with the investigations of cracks in strips and semi-spaces. The reason for this is the necessity to satisfy the conditions at the boundaries and the cracks. Therefore, the investigation of the cracks in the semi-strips is especially relevant.

Li and Guo^[34] considered the effects of the nonhomogenity on the dynamic SIFs for an antiplane interface crack in a functionally graded material bonded to an elastic semi-strip. Alexandrov and Pozharskii^[35] studied the plane problem of a crack on the interface between an elastic strip and an elastic half-plane made of another material. Sebryakov et al.^[36] solved the odd-symmetric boundary value problem for a semi-strip, the lateral sides of which were reinforced by ribs, with the help of the biorthogonal functions and Lagrange expansions of the Fadle-Papkovich functions. Sebryakov et al.^[37] used the Lagrange expansion in terms of the Fadle-Papkovich functions to solve the boundary-value problem for a semi-strip.

The necessity to consider the fixed singularities arises in the kernels of the SIEs when solving problems with cracks. Duduchava^[38] proposed an approach considering the fixed singularities. Kryvyi^[39] proposed a special generalized method, considering the fixed singularities at the ends of the integration interval. Antipov^[40] analyzed a symmetric characteristic SIE with two fixed singularities at the endpoints in the class of functions bounded at the ends.

However, in most of the above-mentioned works, the used methods are based on the presentation of the equilibrium equation solution via the auxiliary functions such as harmonic functions and biharmonic functions, which may lead to additional and quite difficult calculations for determining the important mechanical characteristics.

The novelty of the proposed work is that the solutions are directly constructed for the equilibrium equations according to the method proposed in Ref. [41]. The essential distinction between this work and the previous works is also in the consideration of the solution fixed singularity in the kernel of the SIE. It allows numerical calculations when the crack is located close to the lateral sides.

2 Problem statement

An elastic semi-strip is considered (see Fig. 1), where g is the shear modulus, μ is Poisson's ratio, 0 < x < a, and 0 < y < ∞. By the lines

the conditions of a slide contact are fulfilled, i.e.,

(1)

At the edge y=0, b < x < c, the semi-strip is loaded by

(2)

where p(x) is a known function. At the edges

the fixing boundary conditions are

(3)

The displacements satisfy Lame's equations

(4)

where

and κ=3-4μ is the Muskchelishvili constant. On the line c₀ < x < c₁, y=B, the crack is situated, i.e.,

(5)

(6)

The boundary conditions on the semi-strip edge are reformulated in terms of the displacements as follows:

(7)

(8)

(9)

One needs to solve the boundary value problem (3)-(9) so as to estimate the stress state of the semi-strip factor and the SIF at the edges of the crack.

3 Problem simplification

Fourier's transformation is applied to the system of Lame's equation and the boundary conditions with the generalized scheme^[41].

Then, Eq. (4) is rewritten as follows:

(10)

Here, the following new unknown function is introduced:

As it can be seen

Therefore, the conditions (8) and (9) are satisfied automatically.

The problem (10) is reduced to the vector boundary problem^[42]

(11)

where

I is an identity matrix, and

The solution of the vector boundary problem has been constructed earlier as follows^[43-44]:

(12)

where Y₁(x) and Y₂(x) are the fundamental matrix solutions, c_i (i=1, 2, 3, 4) are constants found from the boundary conditions, and G(x, ξ) is Green's matrix function for the vector boundary problem of the structure

where V₀ and V₁ are the boundary functions defined by

The components of Eq. (12) can be written as follows:

where G^{i, j}(x, ξ) is Green's matrix function element in the ith row and jth column. The facts

are taken into consideration during the calculation. Therefore, the expressions for the displacements are modified as follows:

These formulae will be the final ones if the unknown functions χ(ξ), ψ₁(ξ), and ψ₂(ξ) are known. To find them, one must satisfy the boundary conditions (5)-(7). With this aim, the inverse integral transformation formulae should be applied to Eq. (12). Then, the representations for the displacement functions can be written as follows:

(13)

where f_u(x, y, ξ), f_v(x, y, ξ), g_ui(x, y, ξ), and g_vi(x, y, ξ) (i=1, 2) are shown in Appendix A.

It should be taken into consideration that the integrals in these correspondences are conditionally convergent integrals. Therefore, before differentiating the displacement expressions, at first, one must extract the weakly convergent parts of these integrals^[43]. After this procedure, one can differentiate the displacement expression, which satisfies the following conditions:

Then, one gets the system of three SIEs.

4 Solution to the system of three SIEs

Case Ⅰ The crack is far from the semi-strip lateral sides.

The system of three SIEs is solved when

The changing of the variable in integrals with the limits b and c is done for the passing to the integration interval I₁=[-1,1]. The changing of the variables

in integrals with the limits c₀ and c₁ is done for the passing to the integration interval I₁. The changing of the variable

in the first equation and

in the second and third equations is done. Then, we can write the system of three SIEs as follows:

(14)

where x ∈I₁, and

In the above equations, , , , and are the known regular functions when b ≠ 0, and c ≠ a.

Equation (14) can be solved approximately by use of the orthogonal polynomial method^[41], where the real singularities of the solution at the ends of the integration interval can be considered. Since the distances from the crack tips to the lateral sides of the semi-strip are large enough, there are no fixed singularities in the kernels of Eq. (14). Therefore, the functions , , and are expanded in series by the Chebyshev polynomials of the second type.

(15)

(16)

where U_n(x) is the Chebyshev polynomial. Substitute the above expressions into Eq. (14). Then, the standard scheme of the orthogonal polynomial method can be realized^[41], i.e.,

As a result, the infinite system of the linear algebraic equations relative to the unknown coefficients s_i^j (i=0, 1, 2, …, j=0, 1, 2) is obtained, i.e.,

(17)

where Sm=(s_m⁰    s_m¹    s_m²)^T, and the components of f_m=(f_m⁰    f_m¹    f_m²)^T and D_mn={d_mn^ij} are shown in Appendix B.

The system (17) is solved by the reduction method. Substitute the founded constants into Eqs. (15) and (16). With Eq. (13), we can complete the construction of the solution.

Case Ⅱ The crack is close to the semi-strip lateral sides.

Equation (14) is solved when

When the crack is close to the lateral sides, one has to consider the fixed singularities at the tips of the crack. After integrating by parts, Eq. (14) can be rewritten as follows:

(18)

where x ∈I₁, and

In the above equations, , , , and are known regular functions, and

The function is expanded in series by the Chebyshev polynomials of the second type.

The second and third equations are the partial cases of the equation with two fixed singularities considered in Ref. [38]. Then, we can find the roots λ_kⁱ of the symbols of the SIE.

The generalized method used in Ref. [39] is used to solve Eq. (18). According to it, the functions and are expanded in series in two intervals^[45], i.e.,

(19)

where

in which k=0, 1, …, , and i=1, 2, and

in which , , , …, N-1, and i=1, 2.

The segment [-1,1] is divided into 2N equal segments with the length . Equation (18) is considered when , and i=0, 1, 2, …, 2N-1.

(20)

where the components of P_mn = {p_mn^ij} (i, j=0, 1, 2) and g_m=(g_m⁰   gm¹  g_m²)^T are known constants.

The expression (20) is the system of 6N equations with regard to 6N unknown constants s_kⁱ (i=0, 1, 2, k=0, 1, 2, …, 2N-1). Substitute the found constants into Eqs. (15) and (19). Then, with Eq. (13), we can complete the construction of the solution in the case when the crack tips are close to the lateral sides of the semi-strip.

5 SIF 5.1 Calculation of the SIF for Case Ⅰ

The stress functions can be written as follows:

The subsequent substitution of the expressions for from Eq. (16) and the change of the summation and integration order reduces to

For the calculation of the SIF, we consider^[41]

Then, we have

(21)

where s_kⁱ (i=1, 2) are the constants found from Eq. (17).

5.2 Calculation of the SIF for Case Ⅱ

For Case Ⅱ, one can get the SIF in the generalized form as follows^[39]:

(22)

where s_kⁱ (i=1, 2) are the constants found from Eq. (20).

6 Numerical results and discussion

The calculations are done for the elastic semi-strip (g = 61.278 195 5 × 10⁹ Pa, μ=0.33) with the parameters p(x)=1 Pa, a=10 m, b=a/10, c=0.9a, and B=a.

It is necessary to estimate the geometric parameters, which allow the consideration of simpler cases when =0 and the establishment of the minimal δ_i(i=0, 1) and ε_i (i=0, 1), which allow the usage of Cases Ⅰ and Ⅱ correspondingly.

With this aim, the simplified case with the one displacement jump =ψ₂(x)≠0, =0 is considered when the crack is located in the center of the semi-strip.

Case A The crack is far from the semi-strip lateral sides.

In the case when

Eq. (14) and the infinite system of the linear algebraic equations (17) can be simplified. Therefore, the SIFs K_I± can be calculated by Eq. (21).

In Fig. 2, the dynamic changes of the SIF K_I are presented when the crack length decreases from d=c₁-c₀=8 a/10 to d=a/10, where the dashed line presents the case with one displacement jump =ψ₂(x)≠0, and the solid line presents the case with two displacement jumps =ψ₁(x)≠0 and =ψ₂(x)≠0, where 0 < δ₀ < c₀ < x < c₁ < a − δ₁ < a. From Fig. 2, we can see that the SIF values decrease to zero when the crack length decreases for both cases. Also, the values of the SIF for the case with one jump are similar to the values of the SIF for the case with two jumps when the crack is located in the center of the semi-strip and the distances between the crack tips and the lateral sides are more than ∆d=a/10. When the distances between the crack tips and the lateral sides are less than ∆ d=a/10, both displacement jumps should be considered. When the crack is close to the lateral sides, one should consider the fixed singularities in the SIE kernel.

Case B The crack is close to the semi-strip lateral sides.

In this case, when

one can get the expressions of the SIFs K_I± as Eq. (22).

In Fig. 3, the dynamic changes of the SIF K_I are presented when the crack length decreases from d=c₁-c₀=9 a/10 to d=a/10, where the dashed line presents the case with the displacement jump =ψ₂(x)≠0 and the solid line presents the case with two displacement jumps =ψ₁(x)≠0 and =ψ₂(x)≠0, where 0 < ε₀ < c₀ < x < c₁ < a − ε₁ < a, and 0 < ε_i < δ₁ (i = 0, 1). From the figure, we can see that the SIF values decrease to zero when the crack length decreases for both cases. Moreover, the values of the SIF for the case with one jump are similar to the values of the SIE for the case with two jumps when the crack is located in the center of the semi-strip and the distances between the crack tips and the lateral sides are more than ∆d=a/10. When the distances between the crack tips and the lateral sides are less than ∆d = a/10, both displacement jumps should be considered.

The SIF values in Figs. 2 and 3 are similar when the crack length is less than d=8 a/10. When the crack length is more than d=8 a/10, the case allowing the fixed singularities in the SIE kernel should be considered.

7 Conclusions

The solution of the problem for a semi-strip with a transverse crack is reduced to the problem of solving the system of three SIEs. Two cases are considered. In the first case, the system of three SIEs is solved by the orthogonal polynomial method, and in the second case, the system of three SIEs is solved with the fixed singularities in the kernel.

The research shows that the fixed singularity in the kernel should be considered when the distance of the crack tips to the lateral sides is less than ∆d=a/10 because the standard scheme is inefficient in this case.

The numerical analysis shows that the one displacement jump is inconsequential when the crack is located in the equal distances from the lateral sides, and these distances are more than ∆d=a/10 for Case A and more than ∆d=a/20 for Case B. It significantly simplifies the problem statement and calculation of the SIF. In the case when the crack is located asymmetrically, both the displacement jumps should be considered.

It is necessary to consider the fixed singularities of the unknown displacement function at the semi-strip edge in the case when the load is applied to the whole edge of the semi-strip.

Appendix A

The functions in Eq. (13) are as follows:

Appendix B

The coefficients in Eq. (17) are as follows:

where i = 0, 1, …, M, j = 0, 1,…, N, in which M and N are finite natural digits, and k_i, t_i, and r_j are known constants.

Acknowledgements Gratitude to S. DYKE for the editing of the article text.