On the contact problem of a moving rigid cylindrical punch sliding over an orthotropic layer bonded to an isotropic half plane

Abstract

In this study, the frictional moving contact problem for an orthotropic layer bonded to an isotropic half plane under the action of a sliding rigid cylindrical punch is considered. Boundary conditions of the problem include the normal and tangential forces applied to the layer with a cylindrical punch moving on the surface of the layer in the lateral direction at a constant velocity V. It is assumed that the contact area is subjected to the sliding condition where Coulomb’âŁ™s law is used to relate the tangential traction to the normal traction. Using the Fourier integral transform technique and Galilean transformation, the plane contact problem is reduced to a singular integral equation in which the unknowns are the contact stress and the contact width. The singular integral equation is solved numerically using Gauss–Jacobi integration formulae. Numerical results for the contact widths and the contact stresses are given as a solution.

Keywords

Sliding contact orthotropic coating isotropic substrate singular integral equations friction

1. Introduction

Contact mechanics studies have great importance in the field of tribology and have an impact on the design of components for industries, such as the automotive and aerospace industries. The usage of composite materials in these industries is getting higher and higher as the weight savings compared with the required stiffness are much better than conventional or isotropic materials. Some applications of coatings that might be treated as orthotropic will be given in the following discussion.

Thermal barrier coatings used in heat engine applications, often fabricated from zirconia or zirconia-based compound composites, are applied to the metal substrate to protect the engine from corrosion. These coatings are usually manufactured using a physical vapour deposition process. This manufacturing process introduces anisotropy and these coatings are usually modelled as orthotropic coatings [1]. Usually the failure mechanisms of these coatings are spallation and delamination [2].

Titanium alloys are also used in the automotive aerospace and shipping industries [3]. However, titanium alloys have low surface hardness and their tribological properties are poor. To improve the wear strength and enhance surface performance of these alloys, they are coated with nanostructured coatings [4] that might be treated as orthotropic.

Other applications of orthotropic coatings are the polytetrafluoroethylene coatings used in air-conditioning and refrigeration compressors [5]. These coatings offer significant improvement on the tribological performance of compressors and are usually modelled as orthotropic coatings.

Since the failure in the components occurs near highly stressed regions and these regions are commonly seen near contact regions, it is highly important to study the contact behaviour and accurately calculate the contact stresses.

There are a great many studies in the literature of the contact mechanics of isotropic materials, usually conducted under the assumption of quasistatic loading conditions. These studies include several different kinds of contact, such as fully sliding, partial slip, fretting, receding and tractive rolling. Since the scope of this study is limited to sliding contact, a brief literature review of the sliding contact problems related to orthotropic materials will be given.

In most of the literature on contact mechanics studies, the contact problem is modelled using the assumption that the inertial effects are negligible and therefore the problem can be treated as a quasistatic problem. However, there are problems that arise in practice in which the speed of one body relative to the other is quite large, and therefore the dynamic character of the contact problem must be considered [6]. The moving contact problem between rigid punches and an orthotropic half plane is solved by Zhou et al. [7 –9] using the Fourier transform and the Galilean transformation. Punches with flat and cylindrical [7] or parabolic and triangular profiles [8] are assumed to move at a constant velocity on the surface of an orthotropic half plane. Çömez [10] studied the frictional dynamic contact problem of a rigid cylindrical stamp that moves on an isotropic layer at a constant velocity. The dynamic contact problem between flat or triangular punches and an isotropic layer bonded to the half plane is studied by Balci and Dag [11]. Çömez [12] investigated the frictionless contact problem of a functionally graded layer indented by a moving rigid cylindrical punch. The moving contact problem between the flat or cylindrical punches and the magneto-electro-elastic half plane is examined by Zhou and Lee [13, 14]. The stresses and displacements of an orthotropic half plane have been examined under a line load moving at a constant subsonic, transonic or supersonic velocity over its surface [15].

Although the moving contact problems of a monoclinic half plane have been investigated by Zhou and his coworkers, the moving contact problem of an orthotropic layer bonded to a half plane has not been studied [8]. To fill this gap in the literature, in this paper, the frictional dynamic contact problem between a rigid cylindrical punch and a half plane coated with an orthotropic layer is considered. Using the Fourier transform and Galilean transformation, general expressions for the stresses and the displacements for both the orthotropic layer and the isotropic half plane are derived. Applying the boundary conditions of the problem, a second-kind singular integral equation is obtained. Then, the singular integral equation is solved numerically using the Gauss–Jacobi integration formula.

2. Formulation of the problem

Consider the plane strain contact problem illustrated in Figure 1. An orthotropic layer of thickness h is bonded to an isotropic half plane. The loading is provided by a rigid cylindrical punch with radius R subjected to a concentrated normal force P and a tangential force Q. The punch moves on the layer in the X-direction at a constant velocity V. Friction between the layer and the punch is taken into account. The rectangular coordinates $(X, Z)$ are fixed in the orthotropic layer and the translating coordinates $(x, z)$ are attached to the rigid cylindrical punch.

Figure 1.

Geometry of the moving contact problem.

2.1. Displacement and stress expressions for the orthotropic layer

The wave equations of elastodynamics can be written as:

\frac{\partial σ_{xi}}{\partial X} + \frac{\partial τ_{xzi}}{\partial Z} = ρ_{i} \frac{\partial^{2} u_{i}}{\partial t^{2}}

(1a)

\frac{\partial τ_{xyi}}{\partial X} + \frac{\partial τ_{yzi}}{\partial Z} = ρ_{i} \frac{\partial^{2} v_{i}}{\partial t^{2}}

(1b)

\frac{\partial τ_{xzi}}{\partial X} + \frac{\partial σ_{zi}}{\partial Z} = ρ_{i} \frac{\partial^{2} w_{i}}{\partial t^{2}}

(1c)

where $u_{i}$ , $v_{i}$ and $w_{i}$ are the X-, Y- and Z-components of the displacement vector and t denotes the time variable.

For the plane contact problem under consideration, Hooke’s law for the orthotropic layer can be written as:

σ_{x 1} = C_{11} \frac{\partial u_{1}}{\partial X} + C_{13} \frac{\partial w_{1}}{\partial Z},

(2a)

σ_{y 1} = C_{12} \frac{\partial u_{1}}{\partial X} + C_{23} \frac{\partial w_{1}}{\partial Z},

(2b)

σ_{z 1} = C_{13} \frac{\partial u_{1}}{\partial X} + C_{33} \frac{\partial w_{1}}{\partial Z}

(2c)

τ_{xz 1} = C_{55} (\frac{\partial u_{1}}{\partial Z} + \frac{\partial w_{1}}{\partial X}),

(2d)

τ_{yz 1} = C_{44} \frac{\partial v_{1}}{\partial Z},

(2e)

τ_{xy 1} = C_{66} \frac{\partial v_{1}}{\partial X}

(2f)

Owing to constant speed, it is tractable to introduce Galilean transformation:

x = X - Vt, y = Y, z = Z

(3)

Substituting equation (3) into the wave equations and using Hooke’s law, the equations of equilibrium in terms of the displacements are obtained as:

(C_{11} - V^{2} ρ_{1}) \frac{\partial^{2} u_{1}}{\partial x^{2}} + C_{55} \frac{\partial^{2} u_{1}}{\partial z^{2}} + (C_{13} + C_{55}) \frac{\partial^{2} w_{1}}{\partial x \partial z} = 0

(4a)

(C_{66} - V^{2} ρ_{1}) \frac{\partial^{2} v_{1}}{\partial x^{2}} + C_{44} \frac{\partial^{2} v_{1}}{\partial z^{2}} = 0

(4b)

(C_{13} + C_{55}) \frac{\partial^{2} u_{1}}{\partial x \partial z} + (C_{55} - V^{2} ρ_{1}) \frac{\partial^{2} w_{1}}{\partial x^{2}} + C_{33} \frac{\partial^{2} w_{1}}{\partial z^{2}} = 0

(4c)

Using the Fourier transform technique, the following expressions may be written:

{u_{i} (x, z), v_{i} (x, z), w_{i} (x, z)} = \int_{- \infty}^{\infty} {{\tilde{u}}_{i} (ξ, z), {\tilde{v}}_{i} (ξ, z), {\tilde{w}}_{i} (ξ, z)} e^{- i ξ x} d ξ

(5)

where ${\tilde{u}}_{i} (ξ, z)$ , ${\tilde{v}}_{i} (ξ, z)$ and ${\tilde{w}}_{i} (ξ, z)$ are Fourier transforms of $u_{i} (x, z)$ , $v_{i} (x, z)$ and $w_{i} (x, z)$ , respectively, $ξ$ is the transform variables and $I = \sqrt{- 1}$ . Substituting equation (5) into equation (4), the following ordinary differential equation system can be obtained:

- ξ^{2} (C_{11} - V^{2} ρ_{1}) {\tilde{u}}_{1} + C_{55} \frac{d^{2} {\tilde{u}}_{1}}{d z^{2}} + (- i ξ) (C_{13} + C_{55}) \frac{d^{2} {\tilde{w}}_{1}}{d z} = 0

(6a)

- ξ^{2} (C_{66} - V^{2} ρ_{1}) {\tilde{v}}_{1} + C_{44} \frac{d^{2} {\tilde{v}}_{1}}{d z^{2}} = 0

(6b)

(- i ξ) (C_{13} + C_{55}) \frac{d {\tilde{u}}_{1}}{d z} - ξ^{2} (C_{55} - V^{2} ρ_{1}) {\tilde{w}}_{1} + C_{33} \frac{d^{2} {\tilde{w}}_{1}}{d z^{2}} = 0

(6c)

The displacement components ${\tilde{u}}_{1} (ξ, z)$ , ${\tilde{v}}_{1} (ξ, z)$ and ${\tilde{w}}_{1} (ξ, z)$ can be obtained by solving equation (6). Substituting the displacement components into equation (5), the expressions of the displacements yield

u_{1} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} (\sum_{j = 1}^{4} A_{j} e^{n_{1 j} ξ z}) e^{- i ξ x} d ξ

(7a)

v_{1} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} (\sum_{j = 5}^{6} A_{j} e^{n_{1 j} ξ z}) e^{- i ξ x} d ξ

(7b)

w_{1} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} (I \sum_{j = 1}^{4} m_{1 j} A_{j} e^{n_{1 j} ξ z}) e^{- i ξ x} d ξ

(7c)

where

n_{11} = \sqrt{- \frac{L_{2} + \sqrt{L_{2}^{2} - 4 L_{1} L_{3}}}{2 L_{3}}},

(8a)

n_{12} = \sqrt{- \frac{L_{2} - \sqrt{L_{2}^{2} - 4 L_{1} L_{3}}}{2 L_{3}}},

(8b)

n_{13} = - n_{11},

(8c)

n_{14} = - n_{12},

(8d)

n_{15} = \sqrt{\frac{C_{66} - V^{2} ρ_{1}}{C_{44}}},

(8e)

n_{16} = - n_{15},

(8f)

L_{1} = (C_{11} - V^{2} ρ_{1}) (C_{55} - V^{2} ρ_{1}),

(8g)

L_{2} = C_{13}^{2} - C_{11} C_{33} + 2 C_{13} C_{55} + (C_{33} + C_{55}) V^{2} ρ_{1},

(8h)

L_{3} = C_{33} C_{55},

(8i)

m_{1 j} = \frac{n_{1 j} (- C_{11} C_{33} + {(C_{13} + C_{55})}^{2} + C_{33} C_{55} n_{1 j}^{2} + C_{33} V^{2} ρ_{1})}{(C_{55} - V^{2} ρ_{1}) (C_{13} + C_{55})} .

(8j)

Substituting equation (7) into equation (2), the stress components for the orthotropic layer can be obtained as:

σ_{x 1} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} [- i ξ \sum_{j = 1}^{4} A_{j} (C_{11} - C_{13} m_{1 j} n_{1 j}) e^{n_{1 j} ξ z}] e^{- i ξ x} d ξ

(9a)

σ_{y 1} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} [- i ξ \sum_{j = 1}^{4} A_{j} (C_{12} - C_{23} m_{1 j} n_{1 j}) e^{n_{1 j} ξ z}] e^{- i ξ x} d ξ

(9b)

σ_{z 1} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} [- i ξ \sum_{j = 1}^{4} A_{j} (C_{13} - C_{33} m_{1 j} n_{1 j}) e^{n_{1 j} ξ z}] e^{- i ξ x} d ξ

(9c)

τ_{yz 1} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} [ξ \sum_{j = 5}^{6} A_{j} C_{44} n_{1 j} e^{n_{1 j} ξ z}] e^{- i ξ x} d ξ

(9d)

τ_{xz 1} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} [ξ \sum_{j = 1}^{4} A_{j} C_{55} (m_{1 j} + n_{1 j}) e^{n_{1 j} ξ z}] e^{- i ξ x} d ξ

(9e)

τ_{xy 1} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} [- i ξ \sum_{j = 5}^{6} A_{j} C_{66} e^{n_{1 j} ξ z}] e^{- i ξ x} d ξ

(9f)

where $A_{j} (j = 1, \dots, 6)$ are the unknowns, which will be determined from the boundary conditions of the moving contact problem.

3. Displacement and stress expressions for the isotropic half plane

The stress–strain relation for the isotropic half plane under plane strain conditions can be written as:

σ_{x 2} = \frac{μ}{κ - 1} [(κ + 1) \frac{\partial u_{2}}{\partial X} + (3 - κ) \frac{\partial w_{2}}{\partial Z}],

(10a)

σ_{y 2} = \frac{(3 - κ)}{κ - 1} μ [\frac{\partial u_{2}}{\partial X} + \frac{\partial w_{2}}{\partial Z}],

(10b)

σ_{z 2} = \frac{μ}{κ - 1} [(3 - κ) \frac{\partial u_{2}}{\partial X} + (κ + 1) \frac{\partial w_{2}}{\partial Z}],

(10c)

τ_{yz 2} = μ \frac{\partial v_{2}}{\partial Z},

(10d)

τ_{xz 2} = μ [\frac{\partial u_{2}}{\partial Z} + \frac{\partial w_{2}}{\partial X}],

(10e)

τ_{xy 2} = μ \frac{\partial v_{2}}{\partial X},

(10f)

where $κ = 3 - 4 ν$ , $ν$ and $μ$ are the Poisson ratio and shear modulus of the isotropic half plane, respectively. Substituting equation (10) into equation (1) and using equations (5) and (3), the following ordinary differential equation system can be obtained:

(κ - 1) \frac{d^{2} {\tilde{u}}_{2}}{d z^{2}} - 2 i ξ \frac{d {\tilde{w}}_{2}}{d z} - ξ^{2} [1 + κ + (1 - κ) \frac{V^{2} ρ_{2}}{μ}] {\tilde{u}}_{2} = 0,

(11a)

\frac{d^{2} {\tilde{v}}_{2}}{d z^{2}} - ξ^{2} (1 - \frac{V^{2} ρ_{2}}{μ}) {\tilde{v}}_{2} = 0,

(11b)

(κ + 1) \frac{d^{2} {\tilde{w}}_{2}}{d z^{2}} - 2 i ξ \frac{d {\tilde{u}}_{2}}{d z} - ξ^{2} (κ - 1) [1 - \frac{V^{2} ρ_{2}}{μ}] {\tilde{w}}_{2} = 0 .

(11c)

Solving equation (11) and using equation (5), the expressions of the displacements for the isotropic half plane can be obtained as:

u_{2} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} (\sum_{j = 1}^{4} B_{j} e^{n_{2 j} | ξ | z}) e^{- i ξ x} d ξ,

(12a)

v_{2} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} (\sum_{j = 5}^{6} B_{j} e^{n_{2 j} | ξ | z}) e^{- i ξ x} d ξ,

(12b)

w_{2} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} (Sign (ξ) I \sum_{j = 1}^{4} B_{j} m_{2 j} e^{n_{2 j} | ξ | z}) e^{- i ξ x} d ξ,

(12c)

where

n_{21} = \sqrt{1 - \frac{V^{2} ρ_{2}}{μ}},

(13a)

n_{22} = \sqrt{1 - (\frac{κ - 1}{κ + 1}) \frac{V^{2} ρ_{2}}{μ}},

(13b)

n_{23} = - n_{21},

(13c)

n_{24} = - n_{22},

(13d)

n_{25} = n_{21},

(13e)

n_{26} = n_{22},

(13f)

m_{21} = \frac{1}{n_{21}},

(13g)

m_{22} = n_{22},

(13h)

m_{23} = - \frac{1}{n_{21}},

(13i)

m_{24} = - n_{22} .

(13j)

The stress components for the half plane can be obtained by substituting equation (12) into equation (10), as:

σ_{x 2} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} [- \frac{μ}{κ - 1} i ξ \sum_{j = 1}^{4} B_{j} [(κ + 1) + m_{2 j} n_{2 j} (κ - 3) Sign (ξ)] e^{n_{2 j} | ξ | z}] e^{- i ξ x} d ξ,

(14a)

σ_{y 2} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} [μ \frac{κ - 3}{κ - 1} i ξ \sum_{j = 1}^{4} B_{j} [1 - m_{2 j} n_{2 j} Sign (ξ)] e^{n_{2 j} | ξ | z}] e^{- i ξ x} d ξ,

(14b)

σ_{z 2} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} [\frac{μ}{κ - 1} i ξ \sum_{j = 1}^{4} B_{j} [(κ - 3) + m_{2 j} n_{2 j} (κ + 1) Sign (ξ)] e^{n_{2 j} | ξ | z}] e^{- i ξ x} d ξ,

(14c)

τ_{yz} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} [μ | ξ | \sum_{j = 5}^{6} B_{j} e^{n_{2 j} | ξ | z}] e^{- i ξ x} d ξ,

(14d)

τ_{xz} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} [μ ξ \sum_{j = 1}^{4} B_{j} [m_{2 j} + n_{2 j} Sign (ξ)] e^{n_{2 j} | ξ | z}] e^{- i ξ x} d ξ,

(14e)

τ_{xy} (x, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} [- μ i ξ \sum_{j = 5}^{6} B_{j} e^{n_{2 j} | ξ | z}] e^{- i ξ x} d ξ,

(14f)

where $B_{j}$ are the unknowns, which will be determined from the boundary conditions of the given problem. Note that the displacement and stresses for the half plane should tend to zero for $\sqrt{x^{2} + z^{2}} \to \infty$ . Thus, the non-zero unknowns $B_{3}$ , $B_{4}$ and $B_{6}$ must be extracted from the expressions of displacement and stresses.

4. Boundary conditions and the singular integral equation

In the coordinate system $(x, z)$ , the governing equations of the problem are subjected to the following boundary conditions:

σ_{z 1} (x, 0) = (\begin{matrix} - p (x), & - a < x < b \\ 0, & x \leq - a, x \geq b \end{matrix}

(15a)

τ_{xz 1} (x, 0) = (\begin{matrix} - η p (x), & - a < x < b \\ 0, & x \leq - a, x \geq b \end{matrix}

(15b)

τ_{yz 1} (x, 0) = 0 (- \infty \leq x < \infty),

(15c)

u_{1} (x, - h) = u_{2} (x, - h) (- \infty \leq x < \infty),

(15d)

v_{1} (x, - h) = v_{2} (x, - h) (- \infty \leq x < \infty),

(15e)

w_{1} (x, - h) = w_{2} (x, - h) (- \infty \leq x < \infty),

(15f)

σ_{z 1} (x, - h) = σ_{z 2} (x, - h) (- \infty \leq x < \infty),

(15g)

τ_{xz 1} (x, - h) = τ_{xz 2} (x, - h) (- \infty \leq x < \infty),

(15h)

τ_{yz 1} (x, - h) = τ_{yz 2} (x, - h) (- \infty \leq x < \infty),

(15i)

where $p (x)$ is the unknown contact stress between the rigid punch and the orthotropic layer on the contact area $(- a, b)$ .

On taking the Fourier transform of the boundary conditions given by equation (15), the unknowns $A_{j}$ and $B_{j}$ and, consequently, the displacements and the stresses may be determined in terms of the unknown contact stress $p (x)$ . The unknowns $A_{j}$ and $B_{j}$ can be obtained in the form:

A_{j} = C_{66} \int_{- a}^{b} p (ω) e^{i ξ ω} (A_{j}^{p} + η A_{j}^{q}) d ω,

(16a)

B_{j} = μ \int_{- a}^{b} p (ω) e^{i ξ ω} (B_{j}^{p} + η B_{j}^{q}) d ω .

(16b)

An additional condition may be written in the form of a derivative to eliminate rigid-body displacement and to ensure continuity of normal displacements, as:

\frac{d w_{1} (x, 0)}{d x} = \frac{x}{R}, - a < x < b

(17)

Substituting unknowns $A_{j}$ into the mixed boundary condition (equation (17)) for the cylindrical punch, the following singular integral equation may be obtained:

η \frac{φ_{2}}{φ_{1}} p (x) + \frac{1}{π} \int_{- a}^{b} p (ω) d ω [\frac{1}{ω - x} + k_{1} (x, ω) + η k_{2} (x, ω)] = \frac{4}{φ_{1}} C_{66} \frac{x}{R}, (- a < x < b)

(18)

where

k_{1} (x, ω) = \frac{1}{φ_{1}} \int_{0}^{\infty} (M_{1} (ξ) - φ_{1}) \sin ξ (w - x) d ξ,

(19a)

k_{2} (x, ω) = \frac{1}{φ_{1}} \int_{0}^{\infty} (M_{2} (ξ) - φ_{2}) \cos ξ (ω - x) d ξ,

(19b)

M_{1} (ξ) = i ξ \sum_{j = 1}^{4} A_{j}^{p} m_{1 j},

(19c)

M_{2} (ξ) = ξ \sum_{j = 1}^{4} A_{j}^{q} m_{1 j},

(19d)

φ_{1} = lim_{ξ \to \infty} M_{1} (ξ),

(19e)

φ_{2} = lim_{ξ \to \infty} M_{2} (ξ) .

(19f)

In the equation (18), in addition to the contact stress $p (x)$ , the contact widths a and b are also unknown. The equilibrium condition gives a required extra equation to solve the contact widths:

\int_{- \infty}^{\infty} σ_{z} (x, 0) d x = - \int_{- a}^{b} p (ω) d ω = - P

(20)

5. On the solution of the singular integral equation

The contact stress $p (x)$ is zero at the end of the contact area because of the smooth contact at the end points. Therefore, the index of the integral of equation (18) is $- 1$ [16]. To solve equation (18) numerically, the interval $(- a, b)$ is normalized to $(- 1, 1)$ by defining:

ω = \frac{a + b}{2} r + \frac{b - a}{2},

(21a)

x = \frac{a + b}{2} s + \frac{b - a}{2},

(21b)

ϕ (s) = \frac{p (s)}{C_{66}}

(21c)

Thus, the singular integral equation may be expressed in the following form:

η \frac{φ_{2}}{φ_{1}} ϕ (s) + \frac{1}{π} \int_{- 1}^{1} ϕ (r) d r [\frac{1}{r - s} + k_{1}^{*} (s, r) + η k_{2}^{*} (s, r)] = H (s)

(22)

where

k_{1}^{*} (s, r) = \frac{a + b}{2 h} k_{1} (x, ω),

(23a)

k_{2}^{*} (s, r) = \frac{a + b}{2 h} k_{2} (x, ω),

(23b)

H (s) = \frac{1}{φ_{1}} \frac{1}{R / h} (\frac{a + b}{2 h} s + \frac{b - a}{2 h}) .

(23c)

Similarly, the equilibrium condition (equation (20)) becomes

\frac{a + b}{2 h} \int_{- 1}^{1} ϕ (r) d r = \frac{P}{C_{66} h},

(24)

Equation (22) is of the second kind because of the presence of the first term in equation (22). The solution of the integral equation may be expressed as

ϕ (r) = g (r) ψ (r)

(25)

where $ψ (r)$ is the weight function of $ϕ (r)$ . By applying the Gauss–Chebyshev integration formulae [16] for the bounded functions $g (r)$ , the integral equation can be converted into a system of algebraic equations, as:

\sum_{m = 1}^{N} W_{m}^{N} g (r_{i}) [\frac{1}{r_{m} - s_{k}} + k_{1} (s_{k}, r_{m}) + η k_{2} (s_{k}, r_{m})] = H (s_{k}), k = 1, 2, \dots, N + 1

(26)

Similarly, the equilibrium condition (equation (24)) can be written in a discretized form as

\frac{b + a}{2 h} \sum_{m = 1}^{N} W_{m}^{N} g (r_{m}) = \frac{1}{π} \frac{P}{C_{66} h}

(27)

where

ψ (r) = (1 - r)^{α} (1 + r)^{β},

(28a)

α = \frac{1}{2 π i} \ln [\frac{η φ_{2} / φ_{1} - i}{η φ_{2} / φ_{1} + i}] + N_{0},

(28b)

β = - \frac{1}{2 π i} \ln [\frac{η φ_{2} / φ_{1} - i}{η φ_{2} / φ_{1} + i}] + M_{0} .

(28c)

where $N_{0}$ and $M_{0}$ are arbitrary integers and must be determined using the inequality $0 < Re [α, β] < 1$ for the index $- 1$ , $r_{m}$ and $s_{k}$ are the roots of related Jacobi polynomials and $W_{m}^{N}$ is the weighting constant:

P_{N}^{(α, β)} (r_{m}) = 0, m = 1, 2, \dots, N,

(29a)

P_{N + 1}^{(- α, - β)} (s_{k}) = 0, k = 1, 2, \dots, N + 1,

(29b)

W_{m}^{N} = - \frac{1}{π} \frac{2 N + α + β}{(N + 1)!} \frac{Γ (N + α + 1) Γ (N + β + 1)}{Γ (N + α + β + 1)} \frac{2^{(α + β)}}{P_{N}^{(α, β)'} (r_{m}) P_{N + 1}^{(α, β)} (r_{m})} .

(29c)

Equations (26) and (27) are to be solved simultaneously to determine the unknowns $g (r_{m})$ and a and b. Note that there are $N + 1$ equations to determine the N unknowns, $g (r_{m})$ in equation (26). The additional equation is equivalent to the consistency condition of the singular integral equation and should be extracted from equation (26) to determine a and b with the equilibrium condition (equation (27)). Thus, equations (26) and (27) provide $N + 2$ algebraic equations to determine $N + 2$ unknowns, which are $g (r_{m})$ $(m = 1, \dots, N)$ , and the contact areas ‘a, b’. Since the system of equations is nonlinear in a and b, an iterative scheme based on the Newton–Raphson method is used to obtain the unknown contact areas. The iteration starts with the initial guessed values of contact widths. With these initial values of the contact widths, equation (26) is solved and $g (r_{i})$ determined. Then the extracted equation and the equilibrium condition (equation (27)) are checked to satisfy the tolerance limit. The iteration continues until the desired accuracy is obtained in the control equations. The desired accuracy is selected to be $10^{- 6}$ in this study.

Once the contact stress and contact areas are obtained, the surface stress components in dimensionless form can be computed using the following relations:

\frac{σ_{z 1} (x, 0)}{C_{66}} = - \frac{p (x)}{C_{66}},

(30a)

\frac{τ_{xz 1} (x, 0)}{C_{66}} = - η \frac{p (x)}{C_{66}} .

(30b)

Similarly, the in-plane stress at the surface of the orthotropic layer can be expressed as

\frac{σ_{x 1} (x, 0)}{C_{66}} = {\begin{matrix} φ_{3} p (x) / C_{66} + T (x), - a < x < b \\ T (x), x \leq - a, x \geq b \end{matrix}}

(31)

where

T (x) = \frac{1}{π C_{66}} \int_{- a}^{b} p (ω) [φ_{4} \frac{η}{ω - x} + k_{3} (x, ω) + η k_{4} (x, ω)] d ω,

(32a)

k_{3} (x, ω) = \int_{0}^{\infty} (M_{3} (α) - φ_{3}) \cos ξ (ω - x) d ξ,

(32b)

k_{4} (x, ω) = \int_{0}^{\infty} (M_{4} (ξ) - φ_{4}) \sin ξ (ω - x) d ξ,

(32c)

M_{3} (ξ) = - i ξ \sum_{j = 1}^{4} A_{j}^{p} (C_{11} - C_{13} m_{1 j} n_{1 j})

(32d)

M_{4} (ξ) = ξ \sum_{j = 1}^{4} A_{j}^{q} (C_{11} - C_{13} m_{1 j} n_{1 j})

(32e)

φ_{3} = lim_{ξ \to \infty} M_{3} (ξ),

(32f)

φ_{4} = lim_{ξ \to \infty} M_{4} (ξ) .

(32g)

6. Numerical results

In this study, the moving contact problem for a graded orthotropic layer and isotropic substrate system is considered under dynamic loading conditions. To ensure the correctness of the formulation of the graded problem, the results of this study are validated with three different studies available in the literature. The first and second comparisons are made with the studies conducted by Guler [17] and Balci and Dag [18] for the contact problem between a rigid cylindrical punch and an isotropic coating or isotropic half plane in the static and moving punch cases, respectively. This validation was made possible by reducing the orthotropic material properties to the isotropic counterparts by choosing the following material properties:

C_{11} = C_{33} = \frac{μ_{1} (κ_{1} + 1)}{κ_{1} - 1}, C_{13} = \frac{μ_{1} (3 - κ_{1})}{κ_{1} - 1}, C_{44} = C_{55} = C_{66} = μ_{1} .

(33)

Figure 2 shows the comparison of the results of this study with those of Guler [17] and Balci and Dag [18]. There is an excellent agreement between the results of this study and the benchmark study of Guler [17]. However, there is a discrepancy with the results of Balci and Dag [18] and Guler [17] on the distribution of contact pressure, $σ_{z 1} (x, 0)$ . In the next comparison, we compared the results of this study with those of Balci and Dag [18] under the moving punch case (see Figure 3). Again, a good agreement was observed in the comparison between contact pressure; however, there is a discrepancy for the contact area in the in-plane stress comparison. The main reason for this discrepancy might be that a different method was used to solve the singular integral equation, that is, while Balci and Dag [18] used a Jacobi integration formula, the Gauss–Jacobi integration formula was used in this study. Based on the comparison shown in Figure 2, it is believed that the results of this study are more accurate than the results of Balci and Dag [18].

Figure 2.

Comparison of stress distribution on the surface of the isotropic layer with results of Guler [17] and Balci and Dag [18] $(V = 0, μ^{*} = μ / C_{66} = 1, η = 0.3, (b + a) / R = 0.02, R / h = 50, ν_{1} = ν = 0.3$ ).

Figure 3.

Comparison of moving contact problem stress distribution on the surface of the isotropic layer with results of Balci and Dag [18] $(μ^{*} = μ / C_{66} = 10, ρ_{1} / ρ = 1 / 8, η = 0.3, (b + a) / R = 0.01, R / h = 100, ν_{1} / ν = 0.8,$ $c_{1}^{2} = V^{2} ρ_{1} / C_{66}$ ).

A convergence study was conducted to investigate the effect of the collocation point, N, as shown in Table 1 and Figure 4. Note that the material properties of polymer and metal matrix unidirectional composites used in the simulations hereinafter are obtained from Binienda and Pindera [19] and listed in Table 2. It can be seen from Table 1 that $N = 4$ is enough to find the contact width within a desired accuracy. While the contact width convergence is not affected even if the collocation point is increased further, the in-plane stress convergence is achieved when $N = 100$ . Thus, the collocation point number is selected as $N = 100$ in this study. In the final comparison, the results of this study were compared with the results of Yilmaz et al. [20] under quasistatic loading conditions ( $V = 0$ ). Contact pressure and in-plane stress distribution along the surface of the orthotropic coating are presented in Figure 5 for various values of the punch radius to thickness ratios, $R / h$ . Again, there is an excellent agreement between the results of the current study and those of Yilmaz et al. [20].

Table 1.

Variation of contact width for different values of N, $(V^{*} = V^{2} ρ_{1} / C_{66} = 0.4$ ( $P / (C_{66} h) = 0.01$ , $η = 0.4$ , $μ / C_{66} = 2$ , $R / h = 100$ , $ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

N	$a / h$	$b / h$
2	0.7641387	0.8513740
4	0.7626963	0.8499541
20	0.7626963	0.8499541
100	0.7626963	0.8499541
200	0.7626963	0.8499541

Figure 4.

Effect of the collocation point, N, on the contact stresses ( $V^{*} = V^{2} ρ_{1} / C_{66} = 0.4$ , $(P / (C_{66} h) = 0.01, η = 0.4, μ / C_{66} = 2, R / h = 100, ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

Table 2.

Material properties of polymer and metal matrix unidirectional composites [19].

Material property	Gl/Ep	Gr/Ep	Gr/Ep	B/Al	Gr/Al
		(T300/934)	(P75/934)
	Material A	Material B	Material C	Material D	Material E
$E_{x}$ (GPa)	42.7	144.8	243	227.5	402.6
$E_{y}$ (GPa)	11.7	10.3	7.2	137.9	24.1
$E_{z}$ (GPa)	11.7	10.3	7.2	137.9	24.1
$ν_{xy}$	0.27	0.3	0.33	0.24	0.29
$ν_{xz}$	0.27	0.3	0.33	0.24	0.29
$ν_{yz}$	0.55	0.5	0.49	0.4	0.45
$G_{xy}$ (GPa)	8.238	5.515	3.929	55.152	16.75
$G_{xz}$ (GPa)	8.238	5.515	3.929	55.152	16.75
$G_{yz}$ (GPa)	3.778	3.447	2.406	49.244	8.34

Figure 5.

Comparison of the stress distribution on the surface of the orthotropic layer (solid curves) with results of Yilmaz et al. [20] (symbols) $(V = 0, (P / h) / E_{x} = 2.34 \times 10^{- - 3}, η = 0.4, E / E_{x} = 0.585, ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

We now present a parametric study to investigate the effect of several parameters on the contact stresses. For easier understanding of the results, herein, we define the following non-dimensional parameters for moving velocity:

V^{*} = \frac{V^{2} ρ_{1}}{C_{66}},

(34)

for the applied external load,

P^{*} = \frac{P}{C_{66} h},

(35)

for the stiffness ratio,

μ^{*} = \frac{μ}{C_{66}},

(36)

and for the density ratio,

ρ^{*} = \frac{ρ_{2}}{ρ_{1}},

(37)

where the Poisson ratios of the orthotropic layer and the substrate are each selected as $ν = 0.25$ .

To capture the effect of the moving velocity of the contact and in-plane stress distribution, for different values of moving velocity, $V^{*}$ , the contact and in-plane stress distribution figures are plotted for both the frictionless $(η = 0)$ and the frictional case $(η = 0.4)$ in Figure 6. As expected, as the velocity increases, the contact stress decreases and the contact length increases in the frictionless case. As far as the in-plane stress is concerned, there is a similar trend, except that there is a tensile peak just outside the contact area. This tensile peak is more pronounced as the velocity is increased. Both contact and in-plane stress distributions are symmetrical for the frictionless case. As the friction is introduced to the system, the contact stress distributions are no longer symmetrical and slant towards the trailing edge of the contact. For the in-plane stress distribution, there is a tensile peak at the trailing edge and the magnitude of this peak increases as the velocity is increased. For quasistatic or small values of velocity, the in-plane stress distribution is all compressive in the leading edge of the contact. However, when the velocity is increased, the tensile peak is also seen in the leading edge of the contact. Hereinafter, contact width values corresponding to each figure from Figures 3 –9 will be listed in Tables 3 –9 respectively.

Figure 6.

Contact stress and in-plane stress distribution for different values of moving velocity: $V^{*} = V^{2} ρ_{1} / C_{66}$ ( $P^{*} = P / (C_{66} h) = 0.01$ , $μ^{*} = μ / C_{66} = 2$ , $R / h = 100$ , $ρ^{*} = ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

Table 3.

Variation of contact width for different values of moving velocity, $V^{*} = V^{2} ρ_{1} / C_{66}$ ( $P^{*} = P / (C_{66} h) = 0.01$ , $η = 0.4$ , $μ^{*} = μ / C_{66} = 2$ , $R / h = 100$ , $ρ^{*} = ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

$V^{*} = V^{2} ρ_{1} / C_{66}$	$a / h$	$b / h$
0.0	0.6673	0.7390
0.4	0.7627	0.8500
0.8	0.9893	1.1014

Table 4.

Variation of contact width for different values of friction coefficient, $η$ , $V^{*} = V^{2} ρ_{1} / C_{66} = 0.4$ ( $P / (C_{66} h) = 0.01$ , $η = 0.4$ , $μ / C_{66} = 2$ , $R / h = 100$ , $ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

$η$	$a / h$	$b / h$
0.0	0.8042	0.8042
0.4	0.7627	0.8500
0.8	0.7259	0.8993

Table 5.

Variation of contact width for different values of external load, $P^{*} = P / (C_{66} h)$ ( $V^{*} = V^{2} ρ_{1} / C_{66} = 0.4$ , $η = 0.4$ , $μ / C_{66} = 2$ , $R / h = 100$ , $ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

$P^{*} = P / (C_{66} h)$	$a / h$	$b / h$
0.005	0.5571	0.6236
0.010	0.7627	0.8500
0.020	1.0337	1.1507

Table 6.

Variation of contact width for different values of punch radius, $R / h$ ( $V^{*} = V^{2} ρ_{1} / C_{66} = 0.4$ , $P / (C_{66} h) = 0.01$ , $η = 0.4$ , $μ / C_{66} = 2$ , $R / h = 100$ , $ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

$R / h$	$a / h$	$b / h$
50	0.5571	0.6236
100	0.7627	0.8500
200	1.0336	1.1507

Table 7.

Variation of contact width for different values of stiffness ratio, $μ / C_{66}$ $(V^{*} = V^{2} ρ_{1} / C_{66} = 0.4$ , ( $P / (C_{66} h) = 0.01$ , $η = 0.4$ , $μ / C_{66} = 2$ , $R / h = 100$ , $ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

$μ / C_{66}$	$a / h$	$b / h$
0.75	0.8640	1.0407
1.00	0.8134	0.9484
2.00	0.7627	0.8500

Table 8.

Variation of contact width for different values of density ratio, $ρ_{2} / ρ_{1}$ , $(V^{*} = V^{2} ρ_{1} / C_{66} = 0.4$ ( $P / (C_{66} h) = 0.01$ , $η = 0.4$ , $μ / C_{66} = 2$ , $R / h = 100$ , $ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

$Γ_{2} = ρ_{2} / ρ_{1}$	$a / h$	$b / h$
0.2	0.7074	0.7838
1.0	0.7079	0.7866
5.0	0.7105	0.8114

Table 9.

Variation of contact width for different material types $(V^{*} = V^{2} ρ_{1} / C_{66} = 0.4$ ( $P / (C_{66} h) = 0.01$ , $η = 0.4$ , $P / (G_{xz} h) = 0.01$ , $μ / G_{xz} = 2$ , $R / h = 100$ , $G_{xz} = 8.238 \times 10^{9}$ ).

Material	$a / h$	$b / h$
GI/Ep	0.7628	0.8500
Gr/Ep (T300/934)	0.8748	0.9266
Gr/Ep (P75/934)	1.0765	1.1156
B/Al	0.2410	0.3027
Gr/Al	0.5236	0.5685

The coefficient of friction has a great effect on the tensile peak in the in-plane stress distribution observed at the trailing edge of the contact (see Figure 7). These peaks are usually responsible for crack initiation and subsequently lead to fatigue failure of the components. Note that for the frictionless case there are also tensile peaks at each end of the contact.

Figure 7.

Contact stress and in-plane stress distribution for different values of friction coefficient, $η$ ( $V^{*} = V^{2} ρ_{1} / C_{66} = 0.4$ , $P / (C_{66} h) = 0.01$ , $μ / C_{66} = 2$ , $R / h = 100$ , $ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

The effects of the external load and the punch radius on the stress profiles are depicted in Figures 8 and 9, respectively. Both quantities have a similar effect, that is, as the load is increased both contact pressure and in-plane stress levels are also increased.

Figure 8.

Contact stress and in-plane stress distribution for different values of the external load $P^{*} = P / (C_{66} h)$ ( $V^{*} = V^{2} ρ_{1} / C_{66} = 0.4$ , $η = 0.4$ , $μ / C_{66} = 2$ , $R / h = 100$ , $ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

Figure 9.

Contact stress and in-plane stress distribution for different values of punch radius $R^{*} = R / h$ ( $V^{*} = V^{2} ρ_{1} / C_{66} = 0.4$ , $η = 0.4$ , $P / (C_{66} h) = 0.01$ , $μ / C_{66} = 2$ , $ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

The effect of stiffness ratio, $μ^{*} = μ / C_{66}$ , on the contact stresses can be seen in Figure 10. It is clear that, as the stiffness increases, the contact pressure and the tensile peak at the trailing edge of the contact increase. The effect of density ratio, $ρ^{*} = ρ_{2} / ρ_{1}$ , on the behaviour of contact stresses is shown in Figure 11. There seems to be very little effect of $ρ^{*}$ on the contact pressure; however, the effect is much more pronounced in in-plane stress distribution.

Figure 10.

Contact stress and in-plane stress distribution for different values of the stiffness ratio $μ^{*} = μ / C_{66}$ ( $V^{*} = V^{2} ρ_{1} / C_{66} = 0.4$ , $η = 0.4$ , $P / (C_{66} h) = 0.01$ , $R / h = 100$ , $ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

Figure 11.

Contact stress and in-plane stress distribution for different values of density ratios $ρ^{*} = ρ_{2} / ρ_{1}$ $(V^{*} = V^{2} ρ_{1} / C_{66} = 0.4, η = 0.4, P / (C_{66} h) = 0.01, μ / C_{66} = 2, R / h = 100$ , Gl/Ep).

To investigate the dynamic contact problem of a moving rigid punch on composite materials, a further study was carried out. Three out of five of these composite materials are polymer (glass/epoxy, graphite/epoxy (T300/934) and graphite epoxy (P75/934)) and the remainder is metal matrix (boron/aluminium, graphite/aluminium). Their material properties are obtained from the study conducted by Binienda and Pindera [19]. The frictional contact stress distributions for these five different composite materials are given in Figure 12. Overall, the highest stiffness in the z-direction is observed in boron/aluminium (B/Al) ( $E_{zz} = 137.9$ GPa) and the lowest one in graphite/epoxy (Gr/Ep P75/934) ( $E_{zz} = 7.2$ GPa). Hence, the highest contact pressure and the lowest contact width is observed in Br/Al, whereas the smallest contact pressure and the highest contact width are observed in Gr/Ep P75/934. Since the distributions of the in-plane stresses are also closely related to the stiffness in the x- and y-directions (i.e. $E_{xx}$ and $E_{yy}$ ), it is difficult to discuss the behaviour of in-plane stress distribution given in Fig. 12(b).

Figure 12.

Contact stress and in-plane stress distribution for different material types ( $V^{*} = V^{2} ρ_{1} / C_{66} = 0.4$ , $η = 0.4$ , $P / (G_{xz} h) = 0.01$ , $μ / G_{xz} = 2$ , $R / h = 100$ , $G_{xz} = 8.238 GPa)$ .

Finally, the variation of the contact width with the external load, $P^{*} = P / (C_{66} h)$ , was examined for various values of moving velocity $V^{*} = V^{2} / (ρ_{1} C_{66})$ (Figure 13) for glass/epoxy composite material. It can be concluded that moving velocity has a great impact on the size of the contact width. As the velocity increases, the contact width also increases.

Figure 13.

Variation of the contact width with the external load $P^{*} = P / (C_{66} h)$ , for various values of moving velocity $V^{*} = V^{2} ρ_{1} / C_{66}$ $(η = 0.4, μ / C_{66} = 2, R / h = 100, ρ_{2} / ρ_{1} = 1$ , Gl/Ep).

7. Conclusions

In this study, the frictional dynamic contact problem between an orthotropic coating bonded to an isotropic substrate is considered in the framework of linear elasticity theory. The contact problem is solved under the boundary conditions and the plane contact problem is reduced to the singular integral equation of the second kind. Using Gauss–Jacobi integration formulae, the integral equation is solved numerically. Generated results include five different composite materials, of which three are polymer matrix and two are metal matrix. From this study, the following conclusions can be outlined.

When the punch moves faster, the contact stress decreases and the contact length increases. The effect of punch speed is much more pronounced for the in-plane stress distribution. There are tensile peaks at both ends of the punch for the frictionless case. However, the peaks at the trailing edge have larger magnitudes than those at the leading edge of the contact in the case of frictional contact. Tensile peaks seen in the in-plane stress distributions are important in terms of crack initiation.

Although the effect of the friction on the contact stress is slight, its effect on the in-plane stress is very significant.

Increasing the value of the stiffness ratio results in an increase in both contact stress and the in-plane stresses, as expected.

As the punch radius decreases, both contact and in-plane stresses increase.

Footnotes

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iD

I Çömez

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