1 Introduction

Contact mechanics is a fundamental discipline to study and understand the deformation and wear mechanisms of solids that are interacting with each other due to a contacting region [27, 39]. For the most of the engineering applications, the loads are transferred or applied between the mating parts by contact areas with the presence of static and dynamic friction. Braking systems, coupling devices, clutches, bush and ball bearing, hinges, gaskets, mechanical linkages, gasket seals and electrical contacts are the most obvious examples for the field of contact mechanics [39]. For these applications, an engineer has to design contacting components by first doing a stress analysis of the contacting elements in order to determine the effect of friction and wear in stationary or dynamic contact. Depending on the type of the loading (static, fatigue loading, etc.), a life assessment of the engineering component might be necessary. For example, in a rail-wheel contact, the surface and the subsurface stresses should be found in order to alleviate the unwanted effects such as wear, fatigue, fracture and frictional heating resulting from rolling contact fatigue [39]. Therefore, determining the contact stress distribution in the contact zone is critical when designing these components in order to improve the lifespan of mating parts.

Regarding the literature on the theoretical and numerical investigations of the contact mechanics, problems involving isotropic materials are well investigated by [20, 28, 31, 35,36,37, 43, 44]. For the sliding contact, partial slip contact, rolling and rolling tractive contact, the reader should refer to the following studies [5, 7, 11,12,13,14,15, 17, 29, 30]. Also, for the frictional, elastodynamic and thermoelastic contact problems, one should read the following review paper by [8].

As opposed to contact mechanics problems involving isotropic materials, the studies on the contact mechanic problems of orthotropic materials are rather limited. We herein briefly summarize the work related to anisotropic materials. Shi et al. [42] presented a study to expand the development of indentation of the orthotropic half-space using Sveklo’s analysis. He used a rigid elliptic stamp on an orthotropic half-space in the absence of friction on the contact surfaces. Willis [48] presented the solution of the contact problem for a special class of orthotropic materials (cubic media having three independent material constants). Turner [47] provided the solution of contact problems for transversely isotropic materials. By combining the solution of [38] and [45, 46] calculated the contact stresses for layer orthotropic materials under transverse pressure loading. [26] used Green’s function to investigate the three-dimensional contact problem of transversely isotropic materials. They concluded that the contact stresses may be significantly reduced by appropriately adjusting the material anisotropy. [6] used a semi-analytical method based on the Green’s functions developed by [49] to simulate the contact behavior on anisotropic materials. They have proven that the semi-analytical method is computationally more efficient compared to the finite element method. They also showed that the stiffness of the material in the normal direction to the contact has a great influence on the contact pressure distribution. Erbas et al. [21] studied the plane contact problem for an elastic orthotropic strip using SIE approach. They used iterative method and direct asymptotic method for thick and thin strips, respectively. Rodriguez et al. [40] extended the procedure developed by [48] and [46] to study the viscoelastic behavior of orthotropic contact problems. They showed that the stiffness of the viscoelastic material mainly depends on the anisotropy conditions. Guler [23] obtained closed-form solution for an orthotropic medium that is in a frictional sliding contact with a rigid shaped stamp. The two-dimensional contact problem for an orthotropic half-plane is formulated using Fourier transformation technique, and the resulting SIE of the contact problem is solved analytically by using Jacobi polynomials. He found that orthotropic material properties and friction have an immense effect on the contact stresses. Kucuksucu et al. [33] illustrated a study on the closed-form solution technique for an orthotropic half-space loaded by a rigid wedge-shaped stamp. They utilized Krenk’s effective material parameters (see [32]) to investigate the effects of orthotropic material properties and coefficient of friction on the contact stress distributions and contact areas.

Frictionless contact behavior of orthotropic piezoelectric materials is studied by [50]. They have presented the effects of dimensionless sliding speed, geometrical and mechanical–electric loadings and stamp profile on the contact stress distributions. It is revealed that the stamp profile has a great effect on contact stress and surface electric charge.

Boundary Element Method (BEM) is also utilized to solve the generalized plane problems for anisotropic materials. Blazquez et al. [10] solved the anisotropic problem for a laminate composite under tension and compared their BEM code with the solution derived from the laminate theory. Rodrguez-Tembleque et al. [41] developed a numerical method to study the contact problem for anisotropic solids using BEM.

In the studies listed above, either half-plane (or half-space) or coatings bonded to the homogeneous substrates are considered. To the best of authors’ knowledge, the problem stated in this study has never been solved using integral equations approach. Moreover, the frictional contact problems that are related with material orthotropy are mostly handled with the effective engineering constants that depend on the Krenk’s formulation. The novelty of this study comes from using the Lekhnitskii’s formulation and also using the real elastic material constants while constituting the stiffness coefficients in the analytical method. The developed analytical method is based on SIEs which are obtained by taking the Fourier transform of the boundary conditions and solved numerically. The results from analytical solutions are directly compared with the numerical solutions that are obtained with finite element analysis. The structure of the paper is given as follows. In Sect. 2, the formulation of the problem is illustrated. The boundary conditions and numerical solution of the SIE are provided in Sects. 3 and 4. The finite element model and the details of the finite element solution procedure are presented in Sects. 5 and 6. Finally, the results and discussion and the most important outcomes of this study are described in Sects. 7 and 8.

Fig. 1
figure 1

Geometry of the contact problem

2 Formulation of the problem

The geometry of the plane frictional contact problem of an orthotropic layer bonded to a rigid foundation is depicted in Fig. 1. The layer is subjected to concentrated normal and tangential force by means of a rigid cylindrical stamp with radius R . The generalized plane strain conditions are assumed as follows: [34].

$$\begin{aligned} u\equiv u(x,z), v\equiv v(x,z), w\equiv w(x,z) \end{aligned}$$
(1)

where u, v and w are the \(-x\), \(-y\) and \(-z\) components of the displacement vector. Assuming that the layer is orthotropic, the Hooke’s law under generalized plane strain state can be written as follows:

$$\begin{aligned}&{{\sigma }_{xx}}={{C}_{11}}\frac{\partial u}{\partial x}+{{C}_{13}}\frac{\partial w}{\partial z}, {{\sigma }_{yy}}={{C}_{12}}\frac{\partial u}{\partial x}+{{C}_{23}}\frac{\partial w}{\partial z}, {{\sigma }_{zz}}={{C}_{13}}\frac{\partial u}{\partial x}+{{C}_{33}}\frac{\partial w}{\partial z} \end{aligned}$$
(2a)
$$\begin{aligned}&{{\tau }_{xz}}={{C}_{55}}\left( \frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\right) , {{\tau }_{yz}}={{C}_{44}}\frac{\partial v}{\partial z}, {{\tau }_{xy}}={{C}_{66}}\frac{\partial v}{\partial x} \end{aligned}$$
(2b)

Ignoring the body forces, the Navier equations can be written as follows: [16]

$$\begin{aligned}&{{C}_{11}}\frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}+{{C}_{55}}\frac{{{\partial }^{2}}u}{\partial {{z}^{2}}}+({{C}_{13}}+{{C}_{55}})\frac{{{\partial }^{2}}w}{\partial x\partial z}=0 \end{aligned}$$
(3a)
$$\begin{aligned}&{{C}_{66}}\frac{{{\partial }^{2}}v}{\partial {{x}^{2}}}+{{C}_{44}}\frac{{{\partial }^{2}}v}{\partial {{z}^{2}}}=0 \end{aligned}$$
(3b)
$$\begin{aligned}&({{C}_{13}}+{{C}_{55}})\frac{{{\partial }^{2}}u}{\partial x\partial z}+{{C}_{55}}\frac{{{\partial }^{2}}w}{\partial {{x}^{2}}}+{{C}_{33}}\frac{{{\partial }^{2}}w}{\partial {{z}^{2}}}=0, \end{aligned}$$
(3c)

Equation (3) can be reduced to the ordinary differential equations using integral transform technique [43]. The solution can be assumed that

$$\begin{aligned} \{\ u(x,z),\,v(x,z),\,w(x,z)\ \}=\int \limits _{-\infty }^{\infty }{\{\ \tilde{u}(\xi ,z),\,\tilde{v}(\xi ,z),\,\tilde{w}(\xi ,z)\ \}}\,{{e}^{-I\xi x}}d\xi \end{aligned}$$
(4)

where \(\tilde{u}(\xi ,z),\,\tilde{v}(\xi ,z),\,\tilde{w}(\xi ,z)\ \) are the Fourier transforms of the \(u(x,z),\,v(x,z),\,w(x,z)\ \), respectively. \(\xi \) is the transform variables, and \(I=\sqrt{-1}\). Applying Eq. (4) into Eq. (3) and operating the subsequent ordinary differential equation system, the following characteristic polynomial can be obtained

$$\begin{aligned} {{L}_{1}}+{{L}_{2}}{{n}^{2}}+{{L}_{3}}{{n}^{4}}=0 \end{aligned}$$
(5)

where

$$\begin{aligned} {{L}_{1}}={{C}_{11}}{{C}_{55}}, {{L}_{2}}=C_{13}^{2}-{{C}_{11}}{{C}_{33}}+2{{C}_{13}}{{C}_{55}}, {{L}_{3}}={{C}_{33}}{{C}_{55}} \end{aligned}$$
(6)

The roots of Eq. (5) can be found as

$$\begin{aligned} {{n}_{1,2}}=\sqrt{-\frac{{{L}_{2}}\pm \sqrt{L_{2}^{2}-4{{L}_{1}}{{L}_{3}}}}{2{{L}_{3}}}}, {{n}_{3,4}}=-\sqrt{-\frac{{{L}_{2}}\pm \sqrt{L_{2}^{2}-4{{L}_{1}}{{L}_{3}}}}{2{{L}_{3}}}} \end{aligned}$$
(7)

The solution of Eq. (3) can be obtained by depending on the roots as follows:

$$\begin{aligned} \tilde{u}(\xi ,z)=\sum \limits _{j=1}^{4}{{{A}_{j}}}\,{{e}^{{{n}_{j}}\xi z}}, \tilde{v}(\xi ,z)=\sum \limits _{j=5}^{6}{{{A}_{j}}}\,{{e}^{{{n}_{j}}\xi z}}, \tilde{w}(\xi ,z)=I\sum \limits _{j=1}^{4}{{{k}_{j}}{{A}_{j}}}\,{{e}^{{{n}_{j}}\xi z}} \end{aligned}$$
(8)

where

$$\begin{aligned} {{n}_{5,6}}=\pm \sqrt{\frac{{{C}_{66}}}{{{C}_{44}}}} \end{aligned}$$
(9)

where \({{A}_{j}}\) are the unknowns, which will be established depending on the boundary conditions of the given problem. The stress components relating to the layer may be obtained substituting Eq. 8 into Eq. 2a as follows:

$$\begin{aligned}&{{\sigma }_{xx}}(x,z)=\frac{1}{2\pi }\int \limits _{-\infty }^{\infty }\left[ -I\xi \sum \limits _{j=1}^{4}{{{A}_{j}}}({{C}_{11}}-{{C}_{13}}{{k}_{j}}{{n}_{j}})\,{{e}^{{{n}_{j}}\xi z}}\right] {{e}^{-I\xi x}}d\xi \end{aligned}$$
(10a)
$$\begin{aligned}&{{\sigma }_{yy}}(x,z)=\frac{1}{2\pi }\int \limits _{-\infty }^{\infty }\left[ -I\xi \sum \limits _{j=1}^{4}{{{A}_{j}}}({{C}_{12}}-{{C}_{23}}{{k}_{j}}{{n}_{j}})\,{{e}^{{{n}_{j}}\xi z}}\right] {{e}^{-I\xi x}}d\xi \end{aligned}$$
(10b)
$$\begin{aligned}&{{\sigma }_{zz}}(x,z)=\frac{1}{2\pi }\int \limits _{-\infty }^{\infty }\left[ -I\xi \sum \limits _{j=1}^{4}{{{A}_{j}}}({{C}_{13}}-{{C}_{33}}{{k}_{j}}{{n}_{j}})\,{{e}^{{{n}_{j}}\xi z}}\right] {{e}^{-I\xi x}}d\xi \end{aligned}$$
(10c)
$$\begin{aligned}&{{\tau }_{yz}}(x,z)=\frac{1}{2\pi }\int \limits _{-\infty }^{\infty }\left[ \xi \sum \limits _{j=5}^{6}{{{A}_{j}}{{C}_{44}}{{n}_{j}}\,{{e}^{{{n}_{j}}\xi z}}}\right] {{e}^{-I\xi x}}d\xi \end{aligned}$$
(11a)
$$\begin{aligned}&{{\tau }_{xz}}(x,z)=\frac{1}{2\pi }\int \limits _{-\infty }^{\infty }\left[ \xi \sum \limits _{j=1}^{4}{{{A}_{j}}}{{C}_{55}}({{k}_{j}}+{{n}_{j}})\,{{e}^{{{n}_{j}}\xi z}}\right] {{e}^{-I\xi x}}d\xi \end{aligned}$$
(11b)
$$\begin{aligned}&{{\tau }_{xy}}(x,z)=\frac{1}{2\pi }\int \limits _{-\infty }^{\infty }\left[ -I\xi \sum \limits _{j=5}^{6}{{{A}_{j}}{{C}_{66}}{{e}^{{{n}_{j}}\xi z}}}\right] {{e}^{-I\xi x}}d\xi , \end{aligned}$$
(11c)

where

$$\begin{aligned} {{k}_{j}}=\frac{{{n}_{j}}(-{{C}_{11}}{{C}_{33}}+{{({{C}_{13}}+{{C}_{55}})}^{2}}+{{C}_{33}}{{C}_{55}}n_{j}^{2})}{{{C}_{55}}({{C}_{13}}+{{C}_{55}})} \end{aligned}$$
(12)

3 The boundary conditions and singular integral equation

The contact problem depicted in Fig. 1 should be analyzed considering the following boundary conditions

$$\begin{aligned}&{{\sigma }_{zz}}(x,0)=\left\{ \begin{matrix} -p(x)\quad -a<x<-b \\ \ 0\quad \quad \quad x\le -a,\ x\ge b \\ \end{matrix} \right. \end{aligned}$$
(13a)
$$\begin{aligned}&{{\tau }_{xz}}(x,0)=\left\{ \begin{matrix} -\eta p(x)\quad -a<x-b \\ \ 0\quad \quad \quad x\le a,\ x\ge b \\ \end{matrix} \right. \end{aligned}$$
(13b)
$$\begin{aligned}&{{\tau }_{yz}}(x,0) = 0 \end{aligned}$$
(13c)
$$\begin{aligned}&u(x,-h)=0 \end{aligned}$$
(13d)
$$\begin{aligned}&v(x,-h)=0 \end{aligned}$$
(13e)
$$\begin{aligned}&w(x,-h)=0 \end{aligned}$$
(13f)

where \(\eta \) is the coefficient of friction. The rigid stamp and orthotropic layer are in contact with each other on the interval \([-a,b]\). Both the contact stress p(x) and the contact widths are unknown priorly.

Taking the Fourier transform of the boundary conditions given by Eq. (13) yields a set of the linear algebraic equations. Solving the algebraic equations, the unknowns \({{A}_{j}}\) can be determined in terms of the unknown contact stress p(x) as the following form

$$\begin{aligned} {{A}_{j}}=\int \limits _{-a}^{b}{p(t)}{{e}^{I\xi t}}(A_{j}^{p}+\eta \,A_{j}^{q})dt \end{aligned}$$
(14)

To solve the contact stress, p(x), the following condition should be performed

$$\begin{aligned} \frac{\partial w(x,0)}{\partial x}=\frac{x}{R} \end{aligned}$$
(15)

Substituting the unknowns \({{A}_{j}}\) into the mixed condition in Eq. (15), the following second kind singular integral equation can be obtained

$$\begin{aligned} \eta \frac{{{\phi }_{2}}}{{{\phi }_{1}}}p(x)+\frac{1}{\pi }\int \limits _{-a}^{b}{\left[ \frac{1}{t-x}+{{K}_{1}}(x,t)+\eta {{K}_{2}}(x,t) \right] p(t)dt}=\frac{{1}}{{{\phi }_{1}}}\frac{x}{R}C_{66},\quad -a<x<b \end{aligned}$$
(16)

where

$$\begin{aligned} {{K}_{1}}(x,t)= & {} \frac{1}{{{\phi }_{1}}}\int \limits _{0}^{\infty }{(\ {{M}_{1}}}(\xi )-{{\phi }_{1}})\sin \xi (t-x)d\xi \end{aligned}$$
(17a)
$$\begin{aligned} {{K}_{2}}(x,t)= & {} \frac{1}{{{\phi }_{1}}}\int \limits _{0}^{\infty }{(\ {{N}_{2}}}(\xi )-{{\phi }_{2}})\cos \xi (t-x)d\xi \end{aligned}$$
(17b)
$$\begin{aligned} {{M}_{1}}(\xi )= & {} I\xi \sum \limits _{j=1}^{4}{A_{j}^{p}}{{k}_{j}}, {{M}_{2}}(\xi )=\xi \sum \limits _{j=1}^{4}{A_{j}^{q}}{{k}_{j}} \end{aligned}$$
(17c)
$$\begin{aligned} {{\phi }_{1}}= & {} \underset{\xi \rightarrow \infty }{\mathop {\lim }}\,\ {{M}_{1}}(\xi ), {{\phi }_{2}}=\underset{\xi \rightarrow \infty }{\mathop {\lim }}\,\ {{M}_{2}}(\xi ) \end{aligned}$$
(17d)

In the singular integral equation defined in Eq. (16), the contact widths a and b are also unknown. The following equilibrium condition is needed to find the contact widths.

$$\begin{aligned} \int \limits _{-\infty }^{\infty }{{{\sigma }_{zz}}(x,0)dx}=-\int \limits _{-a}^{b}{p(t)dt}=-P \end{aligned}$$
(18)

4 Numerical solution of the singular integral equation

Due to the contact stress vanishing at \(x=-a\) and \(x=b\), the index of the integral equation is \(-1\) [22].

The following transformations are introduced for the numerical solution

$$\begin{aligned} t=\frac{a+b}{2}r+\frac{b-a}{2}, \quad x=\frac{a+b}{2}s+\frac{b-a}{2}, \quad \varphi (s)=\frac{p(s)}{P{/}h} \end{aligned}$$
(19)

Utilizing the dimensionless quantities, the integral equation can be written in the normalized form as follows:

$$\begin{aligned} \eta \frac{{{\phi }_{2}}}{{{\phi }_{1}}}\,\varphi (s)+\frac{1}{\pi }\int \limits _{-1}^{1}{\left[ \frac{1}{r-s}+k_{1}^{*}(s,r)+\eta \,k_{2}^{*}(s,r) \right] \varphi (r)dr}=\frac{1}{{{\phi }_{1}}}\left( \frac{a+b}{2h}s+\frac{b-a}{2h}\right) \frac{C_{66}}{{{P{/}h}_{}}} \end{aligned}$$
(20)

where

$$\begin{aligned} k_{1}^{*}(s,r)=\frac{a+b}{2h}{{k}_{1}}(x,t), \quad k_{2}^{*}(s,r)=\frac{a+b}{2h}{{k}_{2}}(x,t) \end{aligned}$$
(21)

Similarly, the equilibrium condition in Eq. (18) may be expressed in the normalized form as follows:

$$\begin{aligned} \frac{a+b}{2h}\int \limits _{-1}^{1}{\varphi (r)dr}=1 \end{aligned}$$
(22)

The solution of singular integral equation Eq. (20) is accepted as

$$\begin{aligned} \varphi (r)=g(r)\omega (r) \end{aligned}$$
(23)

where \(\omega (r)\) is the weight function of \(\varphi (r)\) such that

$$\begin{aligned} \omega (r)= & {} {{(1-r)}^{\alpha }}{{(1+r)}^{\beta }} \end{aligned}$$
(24a)
$$\begin{aligned} \alpha= & {} \frac{1}{2\pi I}\ln \left[ \frac{\eta \,{{\phi }_{2}}/{{\phi }_{1}}-I}{\eta \,{{\phi }_{2}}/{{\phi }_{1}}+I} \right] +{{N}_{0}} \end{aligned}$$
(24b)
$$\begin{aligned} \beta= & {} -\frac{1}{2\pi I}\ln \left[ \frac{\eta \,{{\phi }_{2}}/{{\phi }_{1}}-I}{\eta \,{{\phi }_{2}}/{{\phi }_{1}}+I} \right] +{{M}_{0}} \end{aligned}$$
(24c)

Note that \(N_0\) and \(M_0\) are integers and determined from the physics of the contact problem. In this problem, since the contact is smooth at the ends of the contact regions (\(r=-1\), \(r=1\)), the weight function has to be vanished at this points.

Using the Gauss–Jacobi quad rate formulas [22], the integral equation in Eq. (20) can be converted into a system of algebraic equations as follows:

$$\begin{aligned} \sum \limits _{i=1}^{N}{W_{i}^{N}}g({{r}_{i}})\left[ \frac{1}{{{r}_{i}}-{{s}_{k}}}+{{k}_{1}}({{s}_{k}},{{r}_{i}})+\eta \,{{k}_{2}}({{s}_{k}},{{r}_{i}}) \right] =\frac{1}{{{\phi }_{1}}}\left( \frac{a+b}{2h}{{s}_{k}}+\frac{b-a}{2h} \right) \frac{C_{66}}{{{P{/}h}_{}}},\quad k=1,2\ldots N+1 \end{aligned}$$
(25)

Similarly, the equilibrium condition in Eq. (21) becomes

$$\begin{aligned} \frac{b+a}{2h}\sum \limits _{i=1}^{N}{W_{i}^{N}}g({{r}_{i}})=\frac{1}{\pi } \end{aligned}$$
(26)

where \({{r}_{i}}\) and \({{s}_{k}}\) are the roots of the associated Jacobi polynomials and \(W_{i}^{N}\) is the weighting constant.

$$\begin{aligned} P_{N}^{(\alpha ,\beta )}({{r}_{i}})= & {} 0,\quad i=1,2,\ldots ,N \end{aligned}$$
(27a)
$$\begin{aligned} P_{N+1}^{(-\alpha ,-\beta )}({{s}_{k}})= & {} 0,\quad k=1,2,\ldots ,N+1 \end{aligned}$$
(27b)
$$\begin{aligned} W_{i}^{N}= & {} -\frac{1}{\pi }\frac{2N+\alpha +\beta }{(N+1)!}\frac{\Gamma (N+\alpha +1)\Gamma (N+\alpha +1)}{\Gamma (N+\alpha +\beta +1)}\frac{{{2}^{(\alpha +\beta )}}}{P_{N}^{{{(\alpha ,\beta )}^{'}}}({{r}_{i}})P_{N+1}^{(\alpha ,\beta )}({{r}_{i}})} \end{aligned}$$
(27c)

Equation (25) gives \(N+1\) equations to determine the N unknowns \(g({{r}_{i}})\). The additional equation is extracted from Eq. (29) and utilized to determine the a and b with the equilibrium condition in Eq. (26). Thus, Eq. (25) and Eq. (26) yield \(N+2\) algebraic equations, \(N+2\) unknowns which are the contact areas a and b and \(g({{r}_{i}})\)\((i=1,....,N)\) can be obtained by employing an iterative scheme.

After solving the contact stress and contact areas, the surface stress components can be determined as follows:

$$\begin{aligned} \frac{{{\sigma }_{zz}}(x,0)}{P{/}h}=-\frac{p(x)}{P{/}h}, \frac{{{\tau }_{xz}}(x,0)}{P{/}h}=-\eta \frac{p(x)}{P{/}h} \end{aligned}$$
(28)

Similarly, the in-plane stress at the surface of the orthotropic layer may be calculated as

$$\begin{aligned} \frac{{{\sigma }_{xx}}(x,0)}{P{/}h}=\left\{ \begin{matrix} {{\phi }_{3}}\frac{p(x)}{P{/}h}+H(x), &{} -a<x<b \\ H(x), &{} x\le -a,x\ge b \\ \end{matrix} \right. \end{aligned}$$
(29)

where

$$\begin{aligned} H(x)= & {} \frac{1}{\pi }\int \limits _{-a}^{b}{p(t)}\left[ {{\phi }_{4}}\frac{\eta }{t-x}+{{k}_{3}}(x,t)+\eta \,{{k}_{4}}(x,t) \right] dt \end{aligned}$$
(30a)
$$\begin{aligned} {{k}_{3}}(x,t)= & {} \int \limits _{0}^{\infty }{(\ {{M}_{3}}}(\xi )-{{\phi }_{3}})\cos \xi (t-x)d\xi \end{aligned}$$
(30b)
$$\begin{aligned} {{k}_{4}}(x,t)= & {} \int \limits _{0}^{\infty }{(\ {{M}_{4}}}(\xi )-{{\phi }_{4}})\sin \xi (t-x)d\xi \end{aligned}$$
(30c)
$$\begin{aligned} {{M}_{3}}(\xi )= & {} I[-I\xi \sum \limits _{j=1}^{4}{{{A}_{j}}}({{C}_{11}}-{{C}_{13}}{{k}_{j}}{{n}_{j}})\,{{e}^{{{n}_{j}}\xi z}}] \end{aligned}$$
(30d)
$$\begin{aligned} {{M}_{4}}(\xi )= & {} -I\xi \sum \limits _{j=1}^{4}{{{A}_{j}}}({{C}_{11}}-{{C}_{13}}{{k}_{j}}{{n}_{j}})\,{{e}^{{{n}_{j}}\xi z}} \end{aligned}$$
(30e)
$$\begin{aligned} {{\phi }_{3}}= & {} \underset{\xi \rightarrow \infty }{\mathop {\lim }}\,\ {{M}_{3}}(\xi ) \end{aligned}$$
(30f)
$$\begin{aligned} {{\phi }_{4}}= & {} \underset{\xi \rightarrow \infty }{\mathop {\lim }}\,\ {{M}_{4}}(\xi ) \end{aligned}$$
(30g)

5 Finite element model

In addition to the analytical formulation and numerical solution of the singular integral equation presented in previous sections, a numerical approach that depends on the finite element method (FEM) is also conducted. The finite element model used in computational approach is constituted by ANSYS Parametric Design Language (APDL) 2017 and solved by ANSYS Parametric Design Language Solver 2017.

It is quite clear that there are many studies in the open literature on the solution of contact mechanics problems with computational methods. We focus only on the literature dealing with the solution of the contact mechanics problems using finite element method. Dag et al. [18] used both analytical and finite element method to solve the sliding frictional contact problem of medium which is graded in the transverse direction. They employed a homogeneous finite element approach based on defining the elastic properties of each element at its centroid and achieved highly accurate results with their analytical formulation. Guler et al. [24] investigated the contact problem of thin films attached to the graded coatings to determine the effect of the loading type on the stress distributions and shear stress singularities. In their numerical computations, they used ABAQUS software and showed that adjusting the loading type and gradation can significantly reduce the resulting stress distributions. By using a mapped mesh technique and treating each layer as homogeneous, isotropic and linear elastic, [19] obtained the finite element solutions for the sliding frictional contact problem of functionally graded coated elastic members. Abhilash and Murthy [4] used thermal gradient approach by defining the non-homogeneous material properties as a function of temperature. To obtain the material properties, their procedure depended on solving the temperature field through the thickness of the graded coating. A recent study conducted by Guler et al. [25] also employed the homogeneous finite element approach to investigate the orthotropic sliding frictional contact problem for a graded half-space. They found out that, by using an appropriate degree of finite element mesh, one can easily generate the near-numerical results with analytical solutions.

In the analytical formulation of the problem, the orthotropic material properties are defined using elastic stiffness tensor, \(C_{ij}\), where \(C_{11}\), \(C_{12}\), \(C_{13}\), \(C_{33}\), \(C_{44}\), \(C_{55}\), \(C_{66}\) are expressed in Eq. (31) below.

$$\begin{aligned} C_{11}= & {} \frac{1-{{\nu }_{yz}}{{\nu }_{zy}}}{{{E}_{yy}}{{E}_{zz}}\Delta } \end{aligned}$$
(31a)
$$\begin{aligned} C_{12}= & {} \frac{{{\nu }_{yx}}+{{\nu }_{yz}}{{\nu }_{zx}}}{{{E}_{yy}}{{E}_{zz}}\Delta } \end{aligned}$$
(31b)
$$\begin{aligned} C_{13}= & {} \frac{{{\nu }_{zx}}+{{\nu }_{yx}}{{\nu }_{zy}}}{{{E}_{yy}}{{E}_{zz}}\Delta } \end{aligned}$$
(31c)
$$\begin{aligned} C_{33}= & {} \frac{1-{{\nu }_{xy}}{{\nu }_{yx}}}{{{E}_{xx}}{{E}_{yy}}\Delta } \end{aligned}$$
(31d)
$$\begin{aligned} C_{44}= & {} \frac{1}{{{\mu }_{yz}}}, C_{55}=\frac{1}{{{\mu }_{xz}}}, C_{66}=\frac{1}{{{\mu }_{xy}}} \end{aligned}$$
(31e)

where

$$\begin{aligned} \Delta =\left( 1-{{\nu }_{xy}}{{\nu }_{yx}}-{{\nu }_{yz}}{{\nu }_{zy}}-{{\nu }_{xz}}{{\nu }_{zx}}-2{{\nu }_{yx}}{{\nu }_{zy}}{{\nu }_{xz}} \right) /{{E}_{xx}}{{E}_{yy}}{{E}_{zz}} \end{aligned}$$
(32)

and three unknown Poisson’s ratios can be found using reciprocal Poisson’s ratio equations as in the following form.

$$\begin{aligned} \frac{{{\nu }_{xy}}}{{{E}_{xx}}}=\frac{{{v}_{yx}}}{{{E}_{yy}}}, \frac{{{\nu }_{xz}}}{{{E}_{xx}}}=\frac{{{v}_{xz}}}{{{E}_{zz}}}, \frac{{{\nu }_{yz}}}{{{E}_{yy}}}=\frac{{{v}_{zy}}}{{{E}_{zz}}} \end{aligned}$$
(33)

Conversely, in the numerical formulation, the orthotropic material properties are defined using elastic material constants. Therefore, it can be said that if the material property is implemented in ANSYS Parametric Design Language (APDL) 2017 using elastic stiffness tensor, \(C_{ij}\), one must use linear, elastic and anisotropic material models with choosing the stiffness form. In the same way, if the material property is coded in ANSYS Parametric Design Language (APDL) 2017 using elastic material constants, one must use linear, elastic and orthotropic material models. So, it can be concluded that both material models result in the same outputs.

Fig. 2
figure 2

Finite element model of the contact problem

Fig. 3
figure 3

Deformed geometry of the finite element model using different scales with the parameters of \(R{/}h=100.0\), \(\left( P{/}h \right) /{{E}_{xx}}=0.00234\), \(\eta =0.4\) (Note that, for the orthotropic material properties, Material C is used and the stresses, \(\sigma _{zz}\), are normalized using P / h)

The novelty of the finite element model comes from using simpler and innovative ANSYS Parametric Design Language (APDL) 2017 code in the finite element model. By using default ANSYS Parametric Design Language (APDL) 2017 program menu and building up the model from beginning, one needs nearly one hour to build up the finite element model and converge the solution using bottom-up approach. On the other hand, the new method developed in this study automatically generates the finite element model, solves it and post-processes the results in less than fifteen minutes (see Sect. 6). Therefore, it can be said that the near-numerical results are accurately and rapidly generated besides building up the code for different cases. For the element behavior, plane strain conditions are used and pure displacement element formulation is implemented. The boundary conditions for rigid target nodes are kept as program default and Augmented Method is settled while forming the contact algorithm. The contact stiffness is updated in every iteration in order to stabilize the contact quality between the rigid and target nodes in the finite element model [2]. The finite element model and the deformed geometry of the problem using different scales are presented in Figs. 2 and 3, respectively. The numerical model requires the following parameters which are, the orthotropic material properties, \({{E}_{xx}}\), \({{E}_{yy}}\), \({{E}_{zz}}\), \({{\nu }_{xy}}\), \({{\nu }_{yz}}\), \({{\nu }_{xz}}\), \({{G}_{xy}}\), \({{G}_{yz}}\), \({{G}_{xz}}\) (see Table 1 for the corresponding values of the orthotropic material properties for Materials A, B, C, D and E), dimensionless cylindrical stamp radius, R / h, dimensionless press force, \(\left( P{/}h \right) /{{E}_{xx}}\), and the coefficient of friction, \(\eta \). The selected materials are Gr/Ep P75/934, Tr/Ep T300/934, Gl/Ep, Gr/Al, B/Al corresponding to Material A, B, C, D and E, respectively. There are 517018 nodes and 172802 8-noded quadrilateral finite elements, 1600 contact line elements and 2 target elements in the finite element model [1]. In order to prevent the stress singularities at the edges of the contact region, the homogeneous orthotropic layer is assumed to be wide enough. For all the cases in the numerical solution procedure, the width of the layer is kept as fixed with a ratio of \(h/W=20.0\). For the finite element model, TARGE169, PLANE183 and CONTA172 elements are used to model the problem. The cylindrical stamp is assumed to be rigid in all numerical solutions and is modeled using TARGE169 elements. Temperature, displacement, force and pressure can be used with this target segment element. In pair-based contact manager, TARGE169 element associates with CONTA172 element to form a contact using a shared real constant set. PLANE183 element is a 2D 8-noded or 6-noded higher-order finite element and is used to model the orthotropic layer in the finite element model. It can be used as a plane element, and except for the axisymmetric torsion option, the element has two degrees of freedom at each node. CONTA172 element is used to model the contact behavior between the cylindrical rigid stamp and orthotropic layer. This element is located on the mid-side nodes of the 8-noded quadrilateral PLANE183 element [3].

Table 1 Orthotropic material properties for homogeneous orthotropic layer [9]
Table 2 Finite element mesh sensitivity for different numbers of nodes and elements

In order to alleviate the negative effects of the time step on the convergence of the numerical solutions, the solution is divided into equal length sub-steps. The stability is ensured by checking the mesh sensitivity, and the findings are depicted in Table 2. As the number of nodes and elements in the finite element model increases, the percentage error between the analytical and numerical solutions decreases. Nonetheless, if an appropriate degree of meshing is used, the change in the true percentage error between the analytical and numerical solutions is negligible.

6 Details of the finite element solution procedure

This section illustrates the details of the finite element solution procedure in a more comprehensive and effective way.

  • Numerical inputs:

    • Guess the height of the homogeneous orthotropic layer, h, then input the stamp radius and press force using R / h and \((P{/}h)/E_{xx}\)

    • Input the orthotropic material properties, \({{E}_{xx}}\), \({{E}_{yy}}\), \({{E}_{zz}}\), \({{\nu }_{xy}}\), \({{\nu }_{yz}}\), \({{\nu }_{xz}}\), \({{G}_{xy}}\), \({{G}_{yz}}\), \({{G}_{xz}}\) using linear, elastic and orthotropic material models corresponding to Material A, B, C, D and E, respectively.

    • Input the coefficient of friction, \(\eta \), between the homogeneous orthotropic layer and rigid cylindrical stamp

  • Numerical formulation:

    • Input the element types, PLANE183, CONTA172 and TARGE169

    • Choose the integration method and element formulation as Gauss \(4\times 4\) and Augmented Method, respectively

    • Input the element behavior and mesh the numerical model with a convenient element size

    • Use equal length sub-steps for frictionless and frictional model solutions

  • Normalization:

    • Solve the finite element model using ANSYS Mechanical APDL Solver (2017)

    • Read out the corresponding contact widths, a and b, and stresses, \(\sigma _{zz}\) and \(\sigma _{xx}\)

    • Normalize the contact widths and stresses by dividing h and P / h, respectively.

7 Results and discussion

In this section, the comparison of the normalized contact widths, a / h and b / h, normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), and in-plane stress, \(\sigma _{xx}(x,0)/(P{/}h)\), distributions are presented. To make the problem more user friendly and to improve the understanding of the reader, the normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), distribution contours through the thickness of the homogeneous orthotropic layer are also given. The dimensionless geometrical parameters, R / h, \(\left( P{/}h \right) /{{E}_{xx}}\), and coefficient of friction are chosen between the ranges of \(50.0 \le R{/}h \le 200.0\), \(0.00117 \le \left( P{/}h \right) /{{E}_{xx}} \le 0.00468\) and \(0.0 \le \eta \le 0.8\), respectively. The comparison of the normalized contact widths, a / h and b / h, and comparison of error calculations by differing R / h, \(\left( P{/}h \right) /{{E}_{xx}}\), \(\eta \) and orthotropic material properties are given in Tables 3, 4, 5 and 6. In Figs. 4, 5, 6, 7, 8 and 9, the normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), and in-plane stress, \(\sigma _{xx}(x,0)/(P{/}h)\), distributions and normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), distribution contours through the thickness of the homogeneous orthotropic layer are illustrated. As seen in tables and figures, there is an excellent match between the analytical formulation and FEM solutions.

Table 3 Normalized contact widths, a / h and b / h, and comparison of error calculations between analytical formulation and FEM solutions for various values of coefficient of friction, \(\eta \), using Material C
Table 4 Normalized contact widths, a / h and b / h, and comparison of error calculations between analytical formulation and FEM solutions for various values of dimensionless press force, \(\left( P{/}h \right) /{{E}_{xx}}\), using Material C

As given in Table 3, the coefficient of friction, \(\eta \), between the homogeneous orthotropic layer and rigid cylindrical stamp has an inconsiderable effect on the normalized contact widths, a / h and b / h. Yet, as it increases from 0.0 to 0.8, the leading edge of the contact, a / h decreases and the trailing edge of the contact, b / h increases. The change in the normalized contact widths, a / h and b / h, with respect to the dimensionless press force, \(\left( P{/}h \right) /{{E}_{xx}}\), is given in Table 4. As conjectured, the increase in \(\left( P{/}h \right) /{{E}_{xx}}\) results in bigger values of a / h and b / h. Moreover, the effect of the dimensionless cylindrical stamp radius, R / h, on the a / h and b / h is presented in Table 5. It is certain that bigger stamp radius results in more penetration on the surface of the homogeneous orthotropic layer. Therefore, one may observe bigger a / h and b / h values. In Table 6, the normalized contact width changes for different orthotropic materials (Gr/Ep P75/934, Tr/Ep T300/934, Gl/Ep, Gr/Al and B/Al) are given. As given in Table 1, because the Young’s modulus in \(-z\) direction is the lowest for Material A (Gr/Ep P75/934), biggest normalized contact width is observed. On the other hand, the smallest normalized contact width is observed for Material E (B/Al). Note that, for Tables 3, 4, 5 and 6, the highest error percentage is 4.01.

Table 5 Normalized contact widths, a / h and b / h, and comparison of error calculations between analytical formulation and FEM solutions for various values of dimensionless cylindrical stamp radius, R / h, using orthotropic material parameters for Material C
Table 6 Normalized contact widths, a / h and b / h, and comparison of error calculations between analytical formulation and FEM solutions for different orthotropic materials
Fig. 4
figure 4

Normalized contact stress, \(\sigma _{zz}/(P{/}h)\), and in-plane stress, \(\sigma _{xx}/(P{/}h)\), distributions at the surface of the homogeneous orthotropic layer loaded by a rigid cylindrical stamp for various values of coefficient of friction, \(\eta \), with the parameters of \(R{/}h=100.0\), \(\left( P{/}h \right) /{{E}_{xx}}=0.00234\) (Note that, dotted lines represent the results from analytical formulation, solid lines represent FEM solutions, Material C is used and \({E}_{xx}=42.7\) GPa)

Fig. 5
figure 5

Normalized contact stress, \(\sigma _{zz}/(P{/}h)\), and in-plane stress, \(\sigma _{xx}/(P{/}h)\), distributions at the surface of the homogeneous orthotropic layer loaded by a rigid cylindrical stamp for various values of dimensionless press force, \(\left( P{/}h \right) /{{E}_{xx}}\), with the parameters of\(R{/}h=100.0\), \(\eta =0.4\) (Note that, dotted lines represent the results from analytical formulation, solid lines represent FEM solutions, Material C is used, \({E}_{xx}=42.7\) GPa and \({G}_{xy}=11.7\) GPa)

Fig. 6
figure 6

Normalized contact stress, \(\sigma _{zz}/(P{/}h)\), and in-plane stress, \(\sigma _{xx}/(P{/}h)\), distributions at the surface of the homogeneous orthotropic layer loaded by a rigid cylindrical stamp for various values of dimensionless stamp radius, R / h, with the parameters of \(\left( P{/}h \right) /{{E}_{xx}}=0.00234\), \(\eta =0.4\) (Note that, dotted lines represent the results from analytical formulation, solid lines represent FEM solutions, Material C is used and \({E}_{xx}=42.7\) GPa)

Fig. 7
figure 7

Normalized contact stress, \(\sigma _{zz}/(P{/}h)\), and in-plane stress, \(\sigma _{xx}/(P{/}h)\), distributions at the surface of the homogeneous orthotropic layer loaded by a rigid cylindrical stamp with the parameters of \(R{/}h=100.0\), \(\left( P{/}h \right) /{{E}_{xx}}=0.00234\), \(\eta =0.4\) (Note that, dotted lines represent the results from analytical formulation, solid lines represent FEM solutions, Material C is used and \({E}_{xx}=42.7\) GPa)

Fig. 8
figure 8

Normalized contact width, \((a+b){/}h\), changes for different P / h values with the parameters of \(R{/}h=100.0\), \(\eta =0.4\) (Note that, dotted lines represent the results from analytical formulation and solid lines represent FEM solutions)

Fig. 9
figure 9

Normalized contact stress, \(\sigma _{zz}/(P{/}h)\) distribution contours through the thickness of the homogeneous orthotropic layer with the parameters of \(R{/}h=100.0\)a\(\left( P{/}h \right) /{{E}_{xx}}=0.00234\), \(\eta =0.0\)b\(\left( P{/}h \right) /{{E}_{xx}}=0.00234\), \(\eta =0.4\)c\(\left( P{/}h \right) /{{E}_{xx}}=0.00234\), \(\eta =0.8\)d\(\left( P{/}h \right) /{{E}_{xx}}=0.00117\), \(\eta =0.4\)e\(\left( P{/}h \right) /{{E}_{xx}}=0.00234\), \(\eta =0.4\)f\(\left( P{/}h \right) /{{E}_{xx}}=0.00468\), \(\eta =0.4\)

In Figs. 4, 5, 6, 7, 8 and 9, the normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), and in-plane stress, \(\sigma _{xx}(x,0)/(P{/}h)\), distributions are given. It can be seen from Fig. 1 that the normal force, P, has no dependency on the coefficient of friction. Therefore, the normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), distributions are slightly affected. On the other hand, the tangential force, \(Q=\eta P\), has a dependency on the coefficient of friction. For this reason, the coefficient of friction change greatly affects the normalized in-plane stress, \(\sigma _{xx}(x,0)/(P{/}h)\), distributions (see Fig. 4). Moreover, for \(\eta =0.0\) both normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), and in-plane stress, \(\sigma _{xx}(x,0)/(P{/}h)\), distributions are symmetric.

The effect of the dimensionless press force, \(\left( P{/}h \right) /{{E}_{xx}}\), on the normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), and in-plane stress, \(\sigma _{xx}(x,0)/(P{/}h)\), distributions are depicted in Fig. 5. Note that \(\left( P{/}h \right) /{{E}_{xx}}\) varies between 0.00117 and 0.00468. As \(\left( P{/}h \right) /{{E}_{xx}}\) increases, the peak values of the normalized contact and in-plane stresses increase because of the increase in the net pressure on the contact surface.

Furthermore, Fig. 6 presents the effect of the dimensionless stamp radius, R / h, on the normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), and in-plane stress, \(\sigma _{xx}(x,0)/(P{/}h)\), distributions. It is certain that the increase in the R / h results in an increase in the net surface area. Therefore, the peak values of both normalized contact and in-plane stresses decrease. So, it can be said that an increase in the net surface area results in an increase in the net pressure.

In Fig. 5, the effect of press force, \(\left( P{/}h \right) /{{E}_{xx}}\), on the contact and in-plane stress distributions is given for Material C (Gl/Ep). The figure shows that as the press force increases, the maximum values of the normalized values of the contact stresses decrease and the resulting contact width increases. The normalized in-plane stress distributions have the same behavior. One should expect that, as the press force increases, the contact stresses and resulting contact widths should increase. Note that, every contact and in-plane stress distribution plot in Fig. 5 is normalized by its corresponding press force value. Because of the normalized with respect to the press force, this cannot be seen in Fig. 5. In order to understand the behavior of the stresses, the same figure is redrawn without normalized (see Fig. 6). It can be clearly seen that, as the press force increases, the resulting stresses and contact widths increase.

The effect of the normalized stamp radius, R / h on the contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\) and in-plane stress, \(\sigma _{xx}(x,0)/(P{/}h)\) distributions is investigated in Fig. 7. As the normalized stamp radius increases from 50 to 200, the peak values of the contact and in-plane stresses decreases. Furthermore, because of the large load distribution on the contact surface, resulting contact width values increase as the normalized stamp radius increases.

The effect of the orthotropic material properties on the normalized the contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), and in-plane stress, \(\sigma _{xx}(x,0)/(P{/}h)\), distributions is illustrated in Fig. 7 (see Table 1 for the orthotropic material properties for five different materials). As the normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), is mainly dependent on the normal force, P, the most significant material property among all the orthotropic material properties is \(E_{zz}\). It is apparent that the penetration of the rigid cylindrical stamp on a stiff surface is very smaller than the penetration on a softer surface, and it is known that the increase in the contact surface results in lower normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), distributions. For this reason, the lowest peak-normalized contact stress occurs for Material A (Gr/Ep P75/934) and highest peak-normalized contact stress occurs for Material E (B/Al). Therefore, it can be summarized that, for an identical normalized press force, \(\left( P{/}h \right) /{{E}_{xx}}\), the contact stress outstretches on the surface of the homogeneous orthotropic layer as the stiffness decreases. Figure 8 presents the normalized contact width, \((a+b){/}h\) changes for different P / h values with the parameters of \(R{/}h=100.0\) and \(\eta =0.4\). Again, the Material A (\(E_{zz}=7.2 GPa\)) has the lowest stiffness in \(-z\) direction and the Material E (\(E_{zz}=137.9 GPa\)) has the highest stiffness in \(-z\) direction. So, the biggest normalized contact width can be observed for Material A and the smallest normalized contact width can be observed for Material E.

Fig. 10
figure 10

Normalized contact stress, \(\sigma _{zz}/(P{/}h)\), and in-plane stress, \(\sigma _{xx}/(P{/}h)\), distributions at the surface of the homogeneous layer loaded by a rigid cylindrical stamp with the parameters of \(R{/}h=100.0\), \(\left( P{/}h \right) /{{E}_{xx}}=0.00234\), \(\eta =0.4\) for two isotropic and one orthotropic (Material C) materials (Note that \({E}_{xx}=42.7\) GPa)

The normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), distribution contours through the thickness of the homogeneous orthotropic layer are displayed in Fig. 9. As the contact between the rigid cylindrical stamp and homogeneous orthotropic medium is frictional, the stress distribution contours are not symmetrical. The effect of the coefficient of friction, \(\eta \), on the normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), distribution contours is given in Fig. 9a–c. It can be said that, as \(\eta \) increases, the stress contours inclined to the same direction with the tangential force, \(Q=\eta P\). Moreover, Fig. 9d–f illustrates the effect of the dimensionless press force \(\left( P{/}h \right) /{{E}_{xx}}\). It can be summarized that, as \(\left( P{/}h \right) /{{E}_{xx}}\) increases, the normalized contact stresses also increases (see the red-blended contour changes in Fig. 9d–f).

One of the most important findings of this study is to successfully understand the effect of material anisotropy on the normalized stress distributions and on the normalized contact widths. An additional figure is presented where three different normalized stress distributions for two isotropic and one orthotropic material (Material C) are compared (see Fig. 10). It is apparent that for an isotropic material condition, the change in the elastic modulus, E, from 11.7 to 42.7 GPa greatly affects the normalized stress distributions (see red and blue solid lines in Fig. 10). Moreover, as the material property converted from isotropic to orthotropic, the overall stiffness of the homogeneous layer increases owing to the fact that \(E_{xx}\) changes from 11.7 to 42.7 GPa and this results in a smaller penetration (smaller normalized contact width) of the rigid cylindrical stamp on the surface of the homogeneous layer. It is known that there is an inversely proportional relation between the sizes of the normalized contact width and normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), distributions. Therefore, it can be concluded that making the material orthotropic results in lower normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), distributions (see red and dark green solid lines in Fig. 10). Furthermore, one can easily observe that the absolute peak in the normalized in-plane stress, \(\sigma _{xx}(x,0)/(P{/}h)\), distribution at the trailing edge of the contact increases as the material behavior changes from isotropic to orthotropic (see red and dark green solid lines in Fig. 10).

8 Conclusions

In this study, contact problem for a homogeneous orthotropic layer loaded by a rigid cylindrical stamp is considered. The governing integral equations are obtained using the Fourier integral transform technique and linear elasticity theory. The analytical formulation and FEM are used to solve the problem, and good agreement is obtained between these two methods. From this study, the following conclusions can be drawn.

  • As \(\eta \) increases from 0.0 to 0.8, the leading edge of the contact, a / h decreases and the trailing edge of the contact, b / h increases.

  • The increase in the surface area greatly decreases the net pressure. The smallest peak values for normalized contact stress, \(\sigma _{zz}(x,0)/(P{/}h)\), and in-plane stress, \(\sigma _{xx}(x,0)/(P{/}h)\), are observed for \(R{/}h=200.0\)

  • As \(\left( P{/}h \right) /{{E}_{xx}}\) increases, the net pressure increases. Therefore, the resulting normalized stress distributions, \(\sigma _{zz}(x,0)/(P{/}h)\) and \(\sigma _{xx}(x,0)/(P{/}h)\), increase.

  • As the most important material property among all the orthotropic material properties is \(E_{zz}\), for an identical \(\left( P{/}h \right) /{{E}_{xx}}\), the resulting normalized contact stress distribution sprawls across the surface of the homogeneous orthotropic layer for soft surfaces such as Material A (Gr/Ep P75/934).

  • As the material property converted from isotropic to orthotropic, the overall stiffness of the homogeneous layer increases owing to the fact that \(E_{xx}\) changes from 11.7 to 42.7 GPa (Material C) and this results in a smaller penetration.