Acta MechDOI 10.1007/s00707-013-0903-5

Isa Çömez

Contact problem of a functionally graded layer restingon a Winkler foundation

Received: 7 March 2013 / Revised: 6 May 2013© Springer-Verlag Wien 2013

Abstract The contact problem for a functionally graded layer supported by a Winkler foundation is consideredusing linear elasticity theory in this study. The layer is loaded by means of a rigid cylindrical punch that appliesa concentrated force in the normal direction. Poisson’s ratio is taken as constant, and the elasticity modulusis assumed to vary exponentially through the thickness of the layer. The problem is reduced to a Cauchy-typesingular integral equation with the use of Fourier integral transform technique and the boundary conditionsof the problem. The numerical solution of the integral equation is performed by using Gauss–Chebyshevintegration formulas. The effect of the material inhomogeneity, stiffness of the Winkler foundation and punchradius on the contact stress, the contact area and the normal stresses are given.

1 Introduction

Functionally graded materials (FGMs) are inhomogeneous composites whose material properties vary gradu-ally from one point to another in a component. As the application of FGMs has increased in modern industries,new methodologies have been developed to analyse the mechanical behaviour of structural elements made ofthese materials.

The contact mechanics of a functionally graded layer have been widely studied during the past two decades.Most of these studies are related to the frictional contact problem of an FGM-coated half-plane. The analyticalsolutions of the frictional plane strain contact problems are obtained for various stamps using the Fouriertransforms in the studies of Guler and Erdogan [1,2]. They examined the influence of the material inhomogenityconstant on the critical stresses. Liu and Wang [3] studied the axisymmetric frictionless contact problem forvarious indenters with the Hankel transform technique. In these studies, the shear modulus of the graded coatingis assumed to be exponential. A multi-layered model for the two-dimensional frictionless and frictional contactproblems is developed using the transfer matrix method and the Fourier transforms in the studies of Ke andWang [4,5]. They divided the coating into a series of sub-layers assuming the elastic modulus to vary linearly ineach sub-layer and being continuous on the sub-surfaces. Liu et al. [6] extended this model to the axisymmetricfrictionless contact problem by using Hankel transforms. The two-dimensional contact of a rigid cylinder onan elastic-graded substrate was presented by Giannakopoulos and Pallot [7]. The elastic modulus was assumedto vary with depth according to a power function. Chen and Chen [8] showed that stress concentration isrelatively weaker near the flat punch edges when the graded surface obeys a linear variation law than in thecase with an exponential graded law.

Guler and Erdogan [9] studied the frictional contact problem of two elastic cylinders with graded coatings.Yang and Ke [10] investigated the contact problem of a three-layer coating which consists of a homogeneous

I. Çömez (B)Civil Engineering Department, Karadeniz Technical University, 61080 Trabzon, TurkeyE-mail: isacomez@hotmail.comTel.: +904623774047

I. Çömez

coating, a graded layer and a substrate. They indicated that the exponential variation of the shear modulus wasmuch more effective both on the contact stress and the contact area than in the case of power law variation.El-Borgi et al. [11] and Rhimi et al. [12] studied plane and axisymmetric receding contact problems of an FGMlayer resting on a homogeneous substrate. Choi [13] investigated the contact problem of an FGM layer bondedto a rigid substrate. Bakirtas [14] studied the contact problem of a non-homogeneous half space indented by arigid punch.

In the Winkler foundation model, it is assumed that the foundation behaves like an infinite number ofindividual elastic springs and the foundation reaction is proportional to its displacement. Dempsey et al. [15]and [16] studied the plane and axisymmetric contact problems of a homogeneous elastic layer supportedby a Winkler foundation. They showed that the contact pressure distributions can be accurately determinedfrom the wrapping theory when the foundation is relatively flexible. Birinci and Erdol [17] investigated thecontinuous and discontinuous contact problems of two homogeneous elastic layers supported by a Winklerfoundation. Wozniak et al. [18] studied axisymmetric contact problems for a homogeneous layer resting onthe rigid base with a Winkler type excavitation. Matysiak and Pauk [19] studied the crack problem in a homo-geneous elastic layer, and Kadioglu et al. [20] studied the crack problem in an FGM layer resting on an elasticfoundation.

Although the contact problem of a homogeneous elastic layer has been studied by many researchers, afunctionally graded layer resting on a Winkler foundation has not been investigated yet. In this study, thefrictionless contact problem for a functionally graded layer resting on a Winkler foundation is consideredusing linear elasticity theory. The FG layer is indented by a rigid cylindrical stamp which is subjected to aconcentrated normal force. Poisson’s ratio is taken as constant, and the elasticity modulus is assumed to varyexponentially through the thickness of the layer. The problem is reduced to a Cauchy-type singular integralequation in which the contact stress function and the contact area are the unknowns by using Fourier transformsand the boundary conditions of the problem. The contact stress, the contact areas and the normal stresses arecalculated by solving the integral equation numerically.

2 Formulation of the problem

Consider the symmetric plane strain contact problem shown in Fig. 1. A functionally graded elastic layer ofthickness h is supported by an elastic foundation of Winkler type. A concentrated force P is applied to thelayer through a rigid cylindrical punch with radius R. Poisson’s ratio ν is taken as a constant, and the shearmodulus μ is assumed to vary exponentially through the thickness of the layer as follows:

μ(y) = μ0eγ y (1)

where μ0 is the shear modulus on the top surface of the layer and γ is a constant characterising the materialinhomogeneity.

y

P

-a a

h

R

x

μ(y)

Fig. 1 Geometry of the contact problem

Functionally graded layer

Assuming that the FG layer is isotropic at every point, Hooke’s law can be written as follows:

σx (x, y) = μ(y)

κ − 1

[(κ + 1)

∂u

∂x+ (3 − κ)

∂v

∂y

], (2.1)

σy(x, y) = μ(y)

κ − 1

[(3 − κ)

∂u

∂x+ (κ + 1)

∂v

∂y

], (2.2)

τxy = μ(y)

[∂u

∂y+ ∂v

∂x

](2.3)

where u, v are the x- and y- components of the displacement vector, and κ = 3 − 4ν for plane strain.Substituting Eqs. (2) into the equilibrium equations, the following equations in u(x, y) and v(x, y) are

obtained:

(κ + 1)d2u

dx2 + (κ − 1)d2u

dy2 + 2d2v

dxdy+ γ (κ − 1)

[du

dy+ dv

dx

]= 0, (3.1)

(κ − 1)d2v

dx2 + (κ + 1)d2v

dy2 + 2d2u

dxdy+ γ

[(3 − κ)

du

dx+ (κ + 1)

dv

dy

]= 0. (3.2)

By using the symmetry consideration and the Fourier transforms, the following expressions may be written:

u(x, y) = 2

π

∞∫0

φ(α, y) sin(αx)dα, (4.1)

v(x, y) = 2

π

∞∫0

ψ(α, y) cos(αx)dα (4.2)

where φ(α, y) and ψ(α, y) are inverse Fourier transforms of u(x, y) and v(x, y), respectively. Substituting(4.1, 2) into (3.1, 2), the following ordinary differential equations are obtained:

− (κ + 1)α2φ + (κ − 1)d2φ

dy2 − 2αdψ

dy+ γ (κ − 1)

[dφ

dy− αψ

]= 0, (5.1)

−(κ − 1)α2ψ + (κ + 1)d2ψ

dy2 + 2αdφ

dy+ γ

[(3 − κ)αφ + (κ + 1)

dψ

dy

]= 0. (5.2)

The solution of Eqs. (5) can be obtained as:

φ(y) =4∑

j=1

A j en j y, (6.1)

ψ(y) =4∑

j=1

A j m j en j y (6.2)

where A j are the unknown functions that will be determined from the boundary conditions of the problem,and n j ( j = 1, . . . , 4) satisfies the following characteristic equation:

n4j + 2γ n3

j + (γ 2 − 2α2)n2j − 2α2γ n j + α2

(α2 + γ 2 3 − κ

1 + κ

)= 0, (7)

I. Çömez

n1 = −1

2

⎛⎝γ +

√4α2 + γ 2 + 4iα |γ |

√3 − κ

κ + 1

⎞⎠ , (8.1)

n2 = −1

2

⎛⎝γ +

√4α2 + γ 2 − 4iα |γ |

√3 − κ

κ + 1

⎞⎠ , (8.2)

n3 = −1

2

⎛⎝γ −

√4α2 + γ 2 + 4iα |γ |

√3 − κ

κ + 1

⎞⎠ , (8.3)

n4 = −1

2

⎛⎝γ −

√4α2 + γ 2 − 4iα |γ |

√3 − κ

κ + 1

⎞⎠ (8.4)

and

m j =(n j + 2γ ν)

(−2n2

j (1 − ν)+ 2n jγ (1 − ν)− α2(3 − 2ν))

α(α2 − 4γ 2(−1 + ν)ν

) . (9)

Substituting Eqs. (4, 6) into Eqs. (2), the stress components for the layer are readily obtained as

σx (x, y)

μ0= 2

π

∞∫0

4∑j=1

A j

κ − 1

[(3 − κ)m j n j + α(κ + 1)

]e(n j +γ )y cos(αx)dα, (10.1)

σy(x, y)

μ0= 2

π

∞∫0

4∑j=1

A j

κ − 1

[(κ + 1)m j n j + α(3 − κ)

]e(n j +γ )y cos(αx)dα, (10.2)

τxy(x, y)

μ0= 2

π

∞∫0

4∑j=1

A j[(n j − αm j )

]e(n j +γ )y sin(αx)dα. (10.3)

3 The boundary conditions and the singular integral equation

The boundary conditions for the layer can be defined as follows:

σy(x, 0) ={−p(x) 0 < x < a

0 a ≤ x < ∞}, (11.1)

τxy(x, 0) = 0 (0 ≤ x < ∞), (11.2)

τxy(x,−h) = 0 (0 ≤ x < ∞), (11.3)

σy2(x,−h) = kwv(x,−h) (0 ≤ x < ∞), (11.4)

∂v(x, 0)

∂x= F(x) 0 < x < a (12)

where p(x) is the unknown contact stress between the rigid punch and the layer on the contact area (−a, a).kw is the stiffness of the elastic foundation, and F(x) is the derivative of the profile of the rigid stamp; onemay write

F(x) = x

R. (13)

Applying the boundary conditions given by (11), four of the unknown functions A j ( j = 1, . . . , 4), appearingin the stress and the displacement expressions for the layer, may be eliminated in terms of the unknown contact

Functionally graded layer

stress p(x). Substituting them into the rested boundary condition (12) and using the symmetry consideration,p(x) = p(−x), the following Cauchy-type singular integral equation is obtained:

1

π

a∫−a

p(t)

[1

t − x+ k(x, t)

]dt = μ0

β

x

R(14.1)

where

k(x, t) = 1

β

∞∫0

[M(α)− β] sin α(t − x)dα, (14.2)

M(α) =4∑

j=1

αm j A j , (14.3)

β = limα→∞ M(α) = −κ + 1

4. (14.4)

In the singular integral equation (14.1), the contact area a is also unknown, as well as the contact stress p(x). Tocomplete the solution of the problem, the contact stress p(x)must satisfy the following equilibrium condition:

a∫−a

p(t)dt = P. (15)

4 On the solution of the singular integral equation

The following normalised quantities are defined to solve the integral equation numerically:

s = x/a, t = r/a, (16.1)

φ(r) = p(r)

P/h. (16.2)

Using these normalised quantities, the integral equation (14) and the equilibrium condition (15) can be writtenas

1

π

1∫−1

φ(r)

[1

r − s+ a

hk(s, r)

]dr = β

R/h

μ0

P/h

a

hs, (17)

a

h

1∫−1

φ(r)dr = 1. (18)

Since there are smooth contacts at the end points, the index of the integral equation (17) is −1 (Erdogan [21]).The solution of the integral equation can be expressed as

φ(r) = g(r)√(1 − r2) (19)

where g(r) is a continuous and bonded function in the interval [−1,1]. Applying the conventional collocationtechnique (Erdogan [21]; Krenk [22]), the integral equation (17) can be transformed into the equivalent systemof algebraic equations as follows:

N∑i=1

W Ni

[1

ri − sk+ a

hk(sk, ri )

]g(ri ) = β

R/h

μ0

P/h

a

hsk, k = 1, . . . N + 1, (20)

I. Çömez

and the equilibrium condition becomes

a

h

N∑i=1

W Ni g(ri ) = 1

π(21)

where ri and sk are the roots of the related Chebyshev polynomials, and W Ni is the weighting constant:

ri = cos

(iπ

N + 1

)i = 1, . . . N , (22.1)

sk = cos

(π

2

2k − 1

N + 1

)k = 1, . . . N + 1, (22.2)

W Ni = 1 − r2

i

N + 1. (22.3)

Note that there are N + 1 equations to determine the N unknowns g(ri ) in Eq. (20). Since the extra equation isused to normalise the interval of integration, it is sufficient to choose only N of the N + 1 possible collocationpoints (Krenk [22]). Thus, Eqs. (20) and (21) give N + 1 equations to determine the N + 1 unknowns, whichare g(ri ) and a. The system of equations is linear in terms of the g(ri ) but highly nonlinear in variable a.Therefore, an iterative method is used to obtain the unknowns.

5 Numerical results

The influence of the material inhomogeneity, stiffness of the Winkler foundation and punch radius on thedistribution of the contact stress, the contact width and the normal stresses along the symmetry plane x = 0are given by solving the integral equation numerically. Since the values μ0, h and P are related to morethan one-dimensionless quantity, they should be considered fixed. Poisson’s ratio and the dimensionlessquantity related to the load are taken as ν = 0.2 and P/(μ0h) = 0.002 in the following analysis. Theinhomogeneity parameter γ h designates the shear modulus on the bottom surface of the layer μh , thatis, μh increases with increasing absolute values of negative γ h but decreases with increasing values ofpositive γ h.

Figure 2 and Table 1 show the effect of μh/μ0 and kw/μ0 on the contact width a. With decreasing valuesof kw/μ0, the elastic foundation becomes more flexible, so that the contact width a increases. Similarly, as

0 1 2 3 4 5

0.2

0.3

0.4

0.5

0.6

0.7

(1)

(2)

(3)

10/)3(

1/)2(

1.0/)1(

0

0

0

===

μμμ

w

w

w

k

k

k

ha /

0/ μμ h

Fig. 2 Variation of the contact width with μh/μ0 for various values of (kw/μ0) (ν = 0.2, P/(μ0h) = 0.002, R/h = 100)

Functionally graded layer

Table 1 Variation of the contact width with μh/μ0 and (kw/μ0) (ν = 0.2, P/(μ0h) = 0.002, R/h = 100)

γ h μh/μ0 kw = 0.1 kw = 1 kw = 10 kw → ∞(rigidfoundation)

4.60517 0.01 1.0394 0.8567 0.8334 0.83072.30258 0.1 0.6220 0.4734 0.4452 0.44150.01 1.001 0.3725 0.3304 0.3151 0.3116−2.30258 10 0.2779 0.2673 0.2617 0.2594−4.60517 100 0.2371 0.2338 0.2322 0.2311

-0.8 -0.4 0 0.4 0.8

0

0.5

1

1.5

2

2.5

hx /

1.0/....

2.0/...

0.1/_____

0.5/

10/..........

0

0

0

0

0

=−−−=−−−==−−−−=

μμμμμμμμμμ

h

h

h

h

h

hP

xp

/

)(

Fig. 3 Contact stress distribution under the rigid punch for various values of μh/μ0. (ν = 0.2, P/(μ0h) = 0.002, R/h =100, kw/μ0 = 0.1)

Fig. 4 Contact stress distribution under the rigid punch for various values of kw/μ0. (ν = 0.2, P/(μ0h) = 0.002, R/h =100, μh/μ0 = 0.2)

the μh/μ0 decrease, that is, the rigidity of the layer μ(y) decrease top to the bottom, the layer becomes softerand the contact width a increases. In the cases of μh/μ0 > 10 and kw/μ0 > 10, the contact width slightlydecreases even though the μh/μ0 and kw/μ0 increase considerably.

I. Çömez

-1.5 -1 -0.5 0 0.5 1 1.5

0

1.5

3

4.5

6

hx /

250/...

100/_____

50/

10/..........

=−−−

=

=−−−−

=

hR

hR

hR

hR

hP

xp

/

)(

Fig. 5 Contact stress distribution under the rigid punch for various values of R/h. (ν = 0.2, P/(μ0h) = 0.002, kw/μ0 =0.1, μh/μ0 = 0.2)

-4 -2 0 2 4 6 8

-1

-0.8

-0.6

-0.4

-0.2

0

0/),0( μσ yx

1.0/....

2.0/...

0.1/_____

0.5/

10/..........

0

0

0

0

0

=−−−=−−−==−−−−=

μμμμμμμμμμ

h

h

h

h

h

h

y

Fig. 6 Axial stress σx (0, y)/μ0 distribution along the symmetry plane x = 0 for various values ofμh/μ0. (ν = 0.2, P/(μ0h) =0.002, R/h = 100, kw/μ0 = 0.1)

The present study may be extended to the contact problem of a rigid cylindrical punch and an FG layerbonded to a rigid substrate by letting the stiffness of the Winkler foundation kw approach infinity. The calculatedcontact widths are given in Table 1 for this case. It can be seen from the Table that the minimum contact widthsoccur in the case of a rigid substrate.

The contact stress distribution is symmetrical, and the peak value of the contact stress occurs at the centreof the contact area (Figs. 3, 4, 5). When the contact width increases, the load is distributed over a larger area;thus, the peak value of the contact stress decreases. It can be observed from the figures that the contact stressesvanish towards the end of the contact area, which is in agreement with the boundary condition (11.1).

Figures 3 and 4 show the contact stress distribution along the contact area for various values of theμh/μ0 and kw/μ0, respectively. The maximum values of the contact stresses decrease as the layer and theelastic foundation become softer. This is because the contact widths increase so that the contact stresses can be

Functionally graded layer

-3 -2 -1 0 1

-1

-0.8

-0.6

-0.4

-0.2

0

h

y

0/),0( μσ yx

1.0/....

2.0/...

0.1/_____

10/

/..........

0

0

0

0

0

=−−−=−−−==−−−−

∞→

μμμμ

μ

w

w

w

w

w

k

k

k

k

k

Fig. 7 Axial stress σx (0, y)/μ0 distribution along the symmetry plane x = 0 for various values of kw/μ0. (ν = 0.2, P/(μ0h) =0.002, R/h = 100, μh/μ0 = 0.2)

-2.5 -2 -1.5 -1 -0.5 0

-1

-0.8

-0.6

-0.4

-0.2

0

h

y

0/),0( μσ yy

1.0/....

2.0/...

0.1/_____

0.5/

10/..........

0

0

0

0

0

=−−−=−−−==−−−−=

μμμμμμμμμμ

h

h

h

h

h

Fig. 8 Axial stress σy(0, y)/μ0 distribution along the symmetry plane x = 0 for various values ofμh/μ0. (ν = 0.2, P/(μ0h) =0.002, R/h = 100, kw/μ0 = 0.1)

distributed over a larger area in these cases. The contact stress is slightly affected by kw/μ0 when kw/μ0 > 10,and the maximum contact stress occurs in the case of a rigid foundation (kw/μ0 → ∞). It can be observedfrom Fig. 5 that the peak value of the contact stress decreases with an increasing radius of the stamp. ForR/h = 10, the high peak occurs at the contact stress since the contact width between the punch and the layeris too small and approaches the point contact of the concentrated load.

Figures 6 and 7 show the axial stress σx (0, y) distribution along the −y axis for various values of theμh/μ0 and kw/μ0, respectively. The maximum compressive stress appears on the top surface of the layer,while the maximum tensile stress occurs on the bottom surface of the layer. Varying μh/μ0 seems to havealmost no influence on the maximum compressive stress but a rather significant effect on the maximum tensilestress (Fig. 6). By increasing the rigidity of the bottom surface, the tensile stress increases considerably at that

I. Çömez

surface. This result is important for the crack initiation and propagation problems and can give guidance indesigning the inhomogeneity parameter. As can be seen in Fig. 7, both the compressive stress and the tensilestress decrease with increasing the kw/μ0. Note that the axial stress σx (0, y) is almost unaffected by increasingthe rigidity of the foundation after a certain value of kw/μ0.

The axial stress σy(0, y) distribution along the −y plane x = 0 for various values of the μh/μ0 is givenin Fig. 8. σy(0, y) is always compressive and takes its absolute maximum value at the top surface of the layer.As expected, the stress decreases from the loaded top surface to the bottom surface. The compressive stressunder the rigid punch increases with increasing the μh/μ0 which is in agreement with the results presented inFig. 3.

6 Conclusions

In this study, the frictionless plane contact problem for a functionally graded layer supported by a Winklerfoundation is considered using linear elasticity theory. Poisson’s ratio is taken as constant, and the elasticitymodulus is assumed to vary exponentially through the thickness of the layer. The material inhomogeneityparameter, which designates the rigidity of the layer along the thickness direction, has an important effect onthe contact width and the stress distributions. The contact width between the rigid punch and the FG layerincreases when the rigidity of the layer decreases from the loaded top surface to the supported bottom surface.The contact width also increases with increasing the radius of the punch and the flexibility of the foundation.The peak value of the contact stress can be reduced by the cases that increase the contact width. When thebottom surface of the layer becomes stiffer according to the loaded top surface, the tensile stress increase isconsiderable in the lower portion of the layer. This result may provide guidance to control the inhomogenityparameter in the crack initiation and propagation problems.

References

1. Guler, M.A., Erdogan, F.: Contact mechanics of graded coatings. Int. J. Solids Struct. 41, 3865–3889 (2004)2. Guler, M.A., Erdogan, F.: The frictional sliding contact problems of rigid parabolic and cylindrical stamps on graded

coatings. Int. J. Mech. Sci. 49, 161–182 (2007)3. Liu, T.J., Wang, Y.S.: Axisymmetric frictionless contact problem of a functionally graded coating with exponentially varying

modulus. Acta Mech. 199, 151–165 (2008)4. Ke, L.L., Wang, Y.S.: Two-dimensional contact mechanics of functionally graded materials with arbitrary spatial variations

of material properties. Int. J. Solids Struct. 43, 5779–5798 (2006)5. Ke, L.L., Wang, Y.S.: Two-dimensional sliding frictional contact of functionally graded materials. Eur. J. Mech.

A/Solids 26, 171–188 (2007)6. Liu, T.J., Wang, Y.S., Zhang, C.: Axisymmetric frictionless contact of functionally graded materials. Arch. Appl.

Mech. 78, 267–282 (2008)7. Giannakopoulos, A.E., Pallot, P.: Two-dimensional contact analysis of elastic graded materials. J. Mech. Phys.

Solids 48, 1597–1631 (2000)8. Chen, P., Chen, S.: Contact behaviors of a rigid punch and a homogeneous half-space coated with a graded layer. Acta

Mech. 223, 563–577 (2012)9. Guler, M.A., Erdogan, F.: Contact mechanics of two deformable elastic solids with graded coatings. Mech. Mater. 38,

633–647 (2006)10. Yang, J., Ke, L.L.: Two dimensional contact problem for a coating–graded layer–substrate structure under a rigid cylindrical

punch. Int. J. Mech. Sci. 50, 985–994 (2008)11. El-Borgi, S., Abdelmoula, R., Keer, L.: A receding contact problem between a functionally graded layer and a homogeneous

substrate. Int. J. Solids Struct. 43, 658–674 (2006)12. Rhimi, M., El-Borgi, S., Ben Said, W., Ben Jemaa, F.: A receding contact axisym-metric problem between a functionally

graded layer and a homogeneous substrate. Int. J. Solids Struct. 46, 3633–3642 (2009)13. Choi, H.J.: On the plane contact problem of a functionally graded elastic layer loaded by a frictional sliding flat punch.

J. Mech. Sci. Technol. 23, 2703–2713 (2009)14. Bakirtas, I.: The problem of a rigid punch on a nonhomogeneous elastic half space. Int. J. Eng. Sci. 18, 597–610 (1980)15. Dempsey, J.P., Zhao, Z.G., Minnetyan, L., Li, H.: Plane contact of an elastic layer supported by a Winkler foundation.

J. Appl. Mech. 57, 974–980 (1990)16. Dempsey, J.P., Zhao, Z.G., Li, H.: Axisymmetric indentation of an elastic layer supported by a Winkler foundation. Int.

J. Solids Struct. 27, 73–87 (1991)17. Birinci, A., Erdol, R.: A frictionless contact problem for two elastic layers supported by a Winkler foundation. Struct. Eng.

Mech. 15, 331–344 (2003)18. Wozniak, M., Hummel, A., Pauk, V.J.: Axisymmetric contact problems for an elastic layer resting on a rigid base with a

Winkler type excavitation. Int. J. Solids Struct. 39, 4117–4131 (2002)

Functionally graded layer

19. Matysiak, S.J., Pauk, V.J.: Edge crack in an elastic layer resting on a Winkler foundation. Eng. Fract. Mech. 70,2353–2361 (2003)

20. Kadioglu, S., Dag, S., Yahsi, S.: Crack problem for a functionally graded layer on an elastic foundation. Int. J. Fract. 94,63–77 (1998)

21. Erdogan, F.: Mixed boundary value problems in mechanics. In: Nemat-Nasser, S. Mechanics Today, vol. 4, PergamonPress, New York (1978)

22. Krenk, S.: On quadrature formulas for singular integral equations of the first and the second kind. Q. Appl. Math. 33,225–232 (1975)