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Stress Analysis on Layered Materials in Point Elastohydrodynamic-Lubricated Contacts

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Abstract

For multilayered or coated substrates in elastohydrodynamic-lubricated (EHL) contacts, the subsurface stress distributions under a normal load combined with shear traction have been analyzed in this article through computer simulations. The Papkovich-Neuber potentials and Fourier transform are adopted to deduce the pressure–displacement, pressure–stress, and shear traction–stress response functions in frequency domain for the coated substrates, and to calculate distributions of pressure and subsurface stress. The results from the analysis of EHL contacts on coated substrates are compared with those from dry contact model in which shear traction is assumed to obey Coulomb’s law. Effects of the Young’s modulus of coatings, the properties of lubricants, and the magnitude of traction are discussed. Similar to the results in dry contacts, hard coatings in lubricated cases tend to increase the von Mises stress, whereas soft coatings decrease the stress. Shear traction makes the max von Mises stress increasing and moving closer to surface. However, the changes in subsurface stress due to shear traction are less obvious in lubricated contacts. Comparison between EHL and dry contact models reveals that lubrication can reduce the von Mises stress in the coating layer due to smaller shear traction. The analyses show that pressure, film thickness, and subsurface stress distributions are influenced by surface coatings, sliding velocity, rheological models, and pressure–viscosity behaviors.

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Abbreviations

a :

Hertzian contact radius (mm)

\( \tilde{C}_{p}^{{u_{z}^{\left( 1 \right)} }} \) :

The frequency response function of pressure–displacement in frequency domain

\( \tilde{C}_{p}^{{{\varvec{\upsigma}}^{\left( k \right)} }} \) :

The frequency response functions of pressure–stress in frequency domain

\( \tilde{C}_{q}^{{{\varvec{\upsigma}}^{\left( k \right)} }} \) :

The frequency response functions of shear traction–stress in frequency domain

E 1 :

Young’s modulus for the coating (GPa)

E 2 :

Young’s modulus for the substrate (GPa)

G 1 :

Shear modulus for the coating (GPa)

G 2 :

Shear modulus for the substrate (GPa)

h :

Coating thickness (mm)

h f :

Oil film thickness (mm)

h 0 :

Body separation between two surfaces (mm)

h s :

Original geometry of contacting bodies (mm)

J 2 :

The second invariant of the deviatoric stress tensor (MPa)

\( \sqrt {J_{2} } \) :

The von Mises stress (MPa)

m, n:

Frequency domain correspond to x, y, respectively

p :

Pressure (MPa)

q :

Shear traction (MPa)

p h :

Maximum Hertzian pressure (MPa)

R :

Ball radius (mm)

u x , u y , u z :

Displacements of three directions (mm)

U :

Surface velocity (mm/s)

V :

Surface deformation (mm)

w :

Applied load (N)

x, y, z:

Space coordinates (mm)

α:

Pressure–viscosity coefficient (GPa−1)

γS :

Pressure coefficient corresponding to the friction coefficient in boundary lubrication

\( \dot{\gamma } \) :

Shear strain rate of lubricant film (s−1)

γL :

The pressure coefficient

δ1, δ2 :

The roughness amplitudes of surface 1 and 2, respectively (mm)

δ ij :

Kronecker delta

η:

Viscosity (Pa s)

η0 :

Viscosity at normal temperature and pressure (Pa s)

ν1 :

Poisson’s ratio of the coating

ν2 :

Poisson’s ratio of the substrate

ρ:

Lubricant’s density (kg/m3)

ρ0 :

Lubricant’s density at normal temperature and pressure (kg/m3)

σ ij :

Stress (MPa)

τc :

Shear stress of boundary film (MPa)

τS0 :

Initial shear strength of boundary film (MPa)

τl :

Shear stress of lubricant film (MPa)

τL :

Limiting shear stress of lubricant film (MPa)

τL0 :

Initial limiting shear stress (MPa)

τmax :

Maximum shear stress (MPa)

ϕ, ψ1, ψ2, ψ3 :

Papkovich-Neuber potentials

Tilde(~) or FT:

The Fourier transform

IFFT:

The inverse discrete Fourier transform

k = 1, 2:

Means in the coating and substrate, respectively

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Acknowledgments

The authors would like to thank the support of the National Basic Research Program of China (973 Program), under Grant No. 2006CB705403, and National Science Foundation of China, under Grant Nos. 50675111 and 50721004.

Author information

Authors and Affiliations

  1. State Key Laboratory of Tribology, Tsinghua University, Beijing, People’s Republic of China

    Zhan-jiang Wang, Hui Wang & Yuan-zhong Hu

  2. School of Mechanical Vehicular Engineering, Beijing Institute of Technology, Beijing, People’s Republic of China

    Wen-zhong Wang

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  1. Zhan-jiang Wang
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Correspondence to Yuan-zhong Hu.

Appendix

Appendix

1.1 Response Functions for Stresses and Surface Normal Displacement in Layered Elastic Half-Space due to a Normal Traction

$$ \tilde{C}_{p}^{{u_{z}^{\left( 1 \right)} }} \left( {m,n,0} \right) = \frac{1}{{2G_{1} }}\left[ { - \alpha \left( {A^{\left( 1 \right)} - \bar{A}^{\left( 1 \right)} } \right) - \left( {3 - 4\nu_{1} } \right)\left( {C^{\left( 1 \right)} + \bar{C}^{\left( 1 \right)} } \right)} \right] $$

or

$$ \frac{{\nu_{1} - 1}}{{G_{1} }}\left( {1 + 4\alpha h\kappa \theta - \lambda \kappa \theta^{2} } \right)\alpha R $$
$$ \tilde{C}_{p}^{{\sigma_{xx}^{\left( k \right)} }} \left( {m,n,z_{k} } \right) = - m^{2} \left( {A^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{A}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + 2\alpha \nu_{k} \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) - z_{k} m^{2} \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right), $$
$$ \tilde{C}_{p}^{{\sigma_{yy}^{\left( k \right)} }} \left( {m,n,z_{k} } \right) = - n^{2} \left( {A^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{A}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + 2\alpha \nu_{k} \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) - z_{k} n^{2} \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right), $$
$$ \tilde{C}_{p}^{{\sigma_{zz}^{\left( k \right)} }} \left( {m,n,z_{k} } \right) = \alpha^{2} \left( {A^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{A}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + 2\alpha \left( {1 - \nu_{k} } \right)\left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + z_{k} \alpha^{2} \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right), $$
$$ \tilde{C}_{p}^{{\sigma_{xy}^{\left( k \right)} }} \left( {m,n,z_{k} } \right) = - mn\left( {A^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{A}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) - z_{k} mn\left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right), $$
$$ \tilde{C}_{p}^{{\sigma_{xz}^{\left( k \right)} }} \left( {m,n,z_{k} } \right) = - i\left[ {m\alpha \left( {A^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{A}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + m\left( {1 - 2\nu_{k} } \right)\left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + z_{k} m\alpha \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right)} \right], $$
$$ \tilde{C}_{p}^{{\sigma_{yz}^{\left( k \right)} }} \left( {m,n,z_{k} } \right) = - i\left[ {n\alpha \left( {A^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{A}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + n\left( {1 - 2\nu_{k} } \right)\left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + z_{k} n\alpha \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right)} \right], $$

where G k is the Shear modulus and ν k is the Poisson’s ratio, k = 1 means in the coating, and k = 2 means in the substrate.

$$ G = \frac{{G_{1} }}{{G_{2} }},\quad \alpha = \sqrt {m^{2} + n^{2} } ,\quad \lambda = 1 - \frac{{4\left( {1 - \nu_{1} } \right)}}{{1 + G\left( {3 - 4\nu_{2} } \right)}},\quad \kappa = \frac{G - 1}{{G + 3 - 4\nu_{1} }},\quad \theta = e^{ - 2\alpha h} , $$
$$ R = - \frac{1}{{\left[ {1 - \left( {\lambda + \kappa + 4\kappa \alpha^{2} h^{2} } \right)\theta + \lambda \kappa \theta^{2} } \right]\alpha^{2} }}, $$
$$ A^{\left( 1 \right)} = R\left\{ { - \left( {1 - 2\nu_{1} } \right)\left[ {1 - \left( {1 - 2\alpha h} \right)\kappa \theta } \right] + \frac{1}{2}\left( {\kappa - \lambda - 4\kappa \alpha^{2} h^{2} } \right)\theta } \right\}, $$
$$ \bar{A}^{\left( 1 \right)} = R\theta \left\{ {\left( {1 - 2\nu_{1} } \right)\kappa \left( {1 + 2\alpha h - \lambda \theta } \right) + \frac{1}{2}\left( {\kappa - \lambda - 4\kappa \alpha^{2} h^{2} } \right)} \right\}, $$
$$ A^{\left( 2 \right)} = - \frac{1}{2}R\sqrt \theta \left\{ {\left( {3 - 4\nu_{2} } \right)\left( {1 - \lambda } \right)\left[ {1 - \left( {1 - 2\alpha h} \right)\kappa \theta } \right] + \left( {\kappa - 1} \right)\left( {1 + 2\alpha h - \lambda \theta } \right)} \right\}, $$
$$ \bar{A}^{\left( 2 \right)} = 0, $$
$$ \bar{C}^{\left( 2 \right)} = 0, $$
$$ C^{\left( 1 \right)} = \left[ {1 - \left( {1 - 2\alpha h} \right)\kappa \theta } \right]\alpha R, $$
$$ \bar{C}^{\left( 1 \right)} = \left( {1 + 2\alpha h - \lambda \theta } \right)\kappa \theta \alpha R, $$
$$ C^{\left( 2 \right)} = \left( {1 - \lambda } \right)\sqrt \theta C^{\left( 1 \right)} = \left[ {1 - \left( {1 - 2\alpha h} \right)\kappa \theta } \right]\left( {1 - \lambda } \right)\alpha R\sqrt \theta . $$

1.2 Response Functions for Stresses in Layered Elastic Half-Space due to a Tangential Traction

$$ \begin{gathered} \tilde{C}_{q}^{{\sigma_{xx}^{\left( k \right)} }} \left( {m,n,z_{k} } \right) = - m^{2} \left( {A^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{A}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + 2\alpha \nu_{k} \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) - z_{k} m^{2} \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) \hfill \\ + 2im\left( {\nu_{k} - 2} \right)\left( {B^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{B}^{\left( k \right)} e^{{\alpha z{}_{k}}} } \right) + im^{3} z_{k} \alpha^{ - 1} \left( {B^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{B}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) - im^{2} \left( {B_{,m}^{\left( 1 \right)} e^{{ - \alpha z_{k} }} + \bar{B}_{,m}^{\left( 1 \right)} e^{{\alpha z_{k} }} } \right), \hfill \\ \end{gathered} $$
$$ \begin{gathered} \tilde{C}_{q}^{{\sigma_{yy}^{\left( k \right)} }} \left( {m,n,z_{k} } \right) = - n^{2} \left( {A^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{A}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + 2\alpha \nu_{k} \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) - z_{k} n^{2} \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) \hfill \\ - 2im\nu_{k} \left( {B^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{B}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + imn^{2} z_{k} \alpha^{ - 1} \left( {B^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{B}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) - in^{2} \left( {B_{,m}^{\left( 1 \right)} e^{{ - \alpha z_{k} }} + \bar{B}_{,m}^{\left( 1 \right)} e^{{\alpha z_{k} }} } \right), \hfill \\ \end{gathered} $$
$$ \begin{gathered} \tilde{C}_{q}^{{\sigma_{zz}^{\left( k \right)} }} \left( {m,n,z_{k} } \right) = \alpha^{2} \left( {A^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{A}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + 2\left( {1 - \nu_{k} } \right)\alpha \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + z_{k} \alpha^{2} \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) \hfill \\ + 2im\left( {1 - \nu_{k} } \right)\left( {B^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{B}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) - imz_{k} \alpha \left( {B^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{B}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + i\alpha^{2} \left( {B_{,m}^{\left( 1 \right)} e^{{ - \alpha z_{k} }} + \bar{B}_{,m}^{\left( 1 \right)} e^{{\alpha z_{k} }} } \right), \hfill \\ \end{gathered} $$
$$ \begin{gathered} \tilde{C}_{q}^{{\sigma_{xy}^{\left( k \right)} }} \left( {m,n,z_{k} } \right) = - mn\left( {A^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{A}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) - z_{k} mn\left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) \hfill \\ - 2imn\left( {1 - \nu_{k} } \right)\left( {B^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{B}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + im^{2} nz_{k} \alpha^{ - 1} \left( {B^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{B}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) - imn\left( {B_{,m}^{\left( 1 \right)} e^{{ - \alpha z_{k} }} + \bar{B}_{,m}^{\left( 1 \right)} e^{{\alpha z_{k} }} } \right), \hfill \\ \end{gathered} $$
$$ \begin{gathered} \tilde{C}_{q}^{{\sigma_{xz}^{\left( k \right)} }} \left( {m,n,z_{k} } \right) = - i\left[ {m\alpha \left( {A^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{A}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + m\left( {1 - 2\nu_{k} } \right)\left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + z_{k} m\alpha \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right)} \right] \hfill \\ - m^{2} z_{k} \left( {B^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{B}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + \left[ {2\alpha \left( {1 - \nu_{k} } \right) + m^{2} \alpha^{ - 1} } \right]\left( {B^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{B}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + m\alpha \left( {B_{,m}^{\left( 1 \right)} e^{{ - \alpha z_{k} }} - \bar{B}_{,m}^{\left( 1 \right)} e^{{\alpha z_{k} }} } \right), \hfill \\ \end{gathered} $$
$$ \begin{gathered} \tilde{C}_{q}^{{\sigma_{yz}^{\left( k \right)} }} \left( {m,n,z_{k} } \right) = - i\left[ {n\alpha \left( {A^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{A}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + n\left( {1 - 2\nu_{k} } \right)\left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} + \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right) + z_{k} n\alpha \left( {C^{\left( k \right)} e^{{ - \alpha z_{k} }} - \bar{C}^{\left( k \right)} e^{{\alpha z_{k} }} } \right)} \right] \hfill \\ + mn\left( {\alpha^{ - 1} - z_{k} } \right)B^{\left( k \right)} e^{{ - \alpha z_{k} }} - mn\left( {\alpha^{ - 1} + z_{k} } \right)\bar{B}^{\left( k \right)} e^{{\alpha z_{k} }} + n\alpha \left( {B_{,m}^{\left( 1 \right)} e^{{ - \alpha z_{k} }} - \bar{B}_{,m}^{\left( 1 \right)} e^{{\alpha z_{k} }} } \right), \hfill \\ \end{gathered} $$

where G k is the Shear modulus and ν k is the Poisson’s ratio, k = 1 means in the coating, and k = 2 means in the substrate.

$$ G = \frac{{G_{1} }}{{G_{2} }},\quad \alpha = \sqrt {m^{2} + n^{2} } ,\quad \lambda = 1 - \frac{{4\left( {1 - \nu_{1} } \right)}}{{1 + G\left( {3 - 4\nu_{2} } \right)}},\quad \kappa = \frac{G - 1}{{G + 3 - 4\nu_{1} }},\quad \theta = e^{ - 2\alpha h} , $$
$$ S_{0} = \left[ {G + \left( {3 - 4\nu_{1} } \right)} \right]\left( {1 - \kappa \theta } \right), $$
$$ \bar{B}^{\left( 1 \right)} = - \frac{{\left( {G - 1} \right)\theta }}{{\left( {1 + G} \right) + \left( {1 - G} \right)\theta }} \cdot \frac{1}{{2\alpha \left( {1 - \nu_{1} } \right)}}, $$
$$ B^{\left( 1 \right)} = \bar{B}^{\left( 1 \right)} - \frac{1}{{2\alpha \left( {1 - \nu_{1} } \right)}}, $$
$$ B^{\left( 2 \right)} = - \frac{{2\left( {1 - \nu_{1} } \right)}}{{1 - \nu_{2} }} \cdot \frac{\sqrt \theta }{{\left( {1 + G} \right) + \left( {1 - G} \right)\theta }} \cdot \frac{1}{{2\alpha \left( {1 - \nu_{1} } \right)}}, $$
$$ C^{\left( 1 \right)} = \left( {1 - \lambda } \right)S_{0} R_{c} /\left\{ {4\left( {1 - \nu_{1} } \right)\left( {G + 3 - 4\nu_{1} } \right)\left[ {1 - \left( {\lambda + \kappa + 4\kappa \alpha^{2} h^{2} } \right)\theta + \lambda \kappa \theta^{2} } \right]} \right\} $$
$$ \bar{C}^{\left( 1 \right)} = \left[ {2\left( {G - 1} \right)\alpha h\theta C^{\left( 1 \right)} + \theta R_{a} } \right]/S_{0} , $$
$$ A^{\left( 1 \right)} = \left[ { - \left( {3 - 4\nu_{1} } \right)C^{\left( 1 \right)} + \bar{C}^{\left( 1 \right)} + \alpha \left( {R_{1} + R_{2} } \right)} \right]/\left( {2\alpha } \right), $$
$$ \bar{A}^{\left( 1 \right)} = \left\{ {\left( {1 - \lambda } \right)\theta C^{\left( 1 \right)} + \left[ {\left( {3 - 4\nu_{1} } \right)\left( {1 - \theta } \right) - 2\alpha h} \right]\bar{C}^{\left( 1 \right)} + \theta R_{d} } \right\}/\left[ {2\alpha \left( {1 - \theta } \right)} \right], $$
$$ C^{\left( 2 \right)} = \left\{ {\left[ {4\left( {1 - \nu_{1} } \right)\left( {1 - \lambda } \right)\sqrt \theta } \right]C^{\left( 1 \right)} + \left( {1 - \lambda } \right)\alpha \sqrt \theta \left( {R_{3} - R_{4} + R_{5} - R_{6} } \right)} \right\}/\left[ {4\left( {1 - \nu_{1} } \right)} \right], $$
$$ A^{\left( 2 \right)} = \left\{ {2\alpha h\sqrt \theta \left[ {S_{0} - \left( {G - 1} \right)\left( {1 - \theta } \right)} \right]C^{\left( 1 \right)} - \left[ {\left( {3 - 4\nu_{2} } \right)S_{0} } \right]C^{\left( 2 \right)} + \alpha \sqrt \theta S_{0} \left( {R_{1} + R_{2} - R_{3} - R_{4} } \right) - \sqrt \theta \left( {1 - \theta } \right)R_{a} } \right\}/\left( {2\alpha S_{0} } \right), $$

where

$$ - \alpha^{2} R_{1} = im\left( {B^{\left( 1 \right)} - \bar{B}^{\left( 1 \right)} } \right) + i\alpha^{2} \left( {B_{,m}^{\left( 1 \right)} - \bar{B}_{,m}^{\left( 1 \right)} } \right), $$
$$ - \alpha^{2} R_{2} = 2im\left( {1 - \nu_{1} } \right)\left( {B^{\left( 1 \right)} + \bar{B}^{\left( 1 \right)} } \right) + i\alpha^{2} \left( {B_{,m}^{\left( 1 \right)} + \bar{B}_{,m}^{\left( 1 \right)} } \right), $$
$$ - \alpha^{2} R_{3} = i\left( {m - m\alpha h} \right)B^{\left( 1 \right)} - i\left( {m + m\alpha h} \right)\theta^{ - 1} \bar{B}^{\left( 1 \right)} - im\theta^{{ - \frac{1}{2}}} B^{\left( 2 \right)} + i\alpha^{2} B_{,m}^{\left( 1 \right)} - i\alpha^{2} \theta^{ - 1} \bar{B}_{,m}^{\left( 1 \right)} - i\alpha^{2} \theta^{{ - \frac{1}{2}}} B_{,m}^{\left( 2 \right)} , $$
$$ - \alpha^{2} R_{4} = \left[ {2i\left( {1 - \nu_{1} } \right)m - im\alpha h} \right]B^{\left( 1 \right)} + \left[ {2i\left( {1 - \nu_{1} } \right)m + im\alpha h} \right]\theta^{ - 1} \bar{B}^{\left( 1 \right)} - \left[ {2i\left( {1 - \nu_{2} } \right)m} \right]\theta^{{ - \frac{1}{2}}} B^{\left( 2 \right)} + i\alpha^{2} B_{,m}^{\left( 1 \right)} + i\alpha^{2} \theta^{ - 1} \bar{B}_{,m}^{\left( 1 \right)} - i\alpha^{2} \theta^{{ - \frac{1}{2}}} B_{,m}^{\left( 2 \right)} , $$
$$ - \alpha^{2} R_{5} = - im\alpha hB^{\left( 1 \right)} + im\alpha h\theta^{ - 1} \bar{B}^{\left( 1 \right)} + i\alpha^{2} B_{,m}^{\left( 1 \right)} + i\alpha^{2} \theta^{ - 1} \bar{B}_{,m}^{\left( 1 \right)} - iG\alpha^{2} \theta^{{ - \frac{1}{2}}} B_{,m}^{\left( 2 \right)} , $$
$$ - \alpha^{2} R_{6} = i\left( {m - m\alpha h} \right)B^{\left( 1 \right)} - i\left( {m + m\alpha h} \right)\theta^{ - 1} \bar{B}^{\left( 1 \right)} - iGm\theta^{{ - \frac{1}{2}}} B^{\left( 2 \right)} + i\alpha^{2} B_{,m}^{\left( 1 \right)} - i\alpha^{2} \theta^{ - 1} \bar{B}_{,m}^{\left( 1 \right)} - iG\alpha^{2} \theta^{{ - \frac{1}{2}}} B_{,m}^{\left( 2 \right)} , $$
$$ R_{a} = \left( {G - 1} \right)\alpha \left( {R_{1} + R_{2} } \right) - G\alpha \left( {R_{3} + R_{4} } \right) + \alpha \left( {R_{5} + R_{6} } \right), $$
$$ R_{b} = \alpha \left( {R_{2} - R_{1} } \right) + \alpha \theta \left( {R_{3} - R_{4} } \right), $$
$$ R_{c} = \frac{{4\left( {1 - \nu_{1} } \right)}}{1 - \lambda }\left( {\frac{2\alpha h\theta }{{S_{0} }}R_{a} + R_{b} } \right), $$
$$ R_{d} = \alpha \left( {R_{1} - R_{2} - R_{3} + R_{4} } \right) + \frac{{\left( {1 - \lambda } \right)\alpha }}{{4\left( {1 - \nu_{1} } \right)}}\left( {R_{3} - R_{4} + R_{5} - R_{6} } \right). $$

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Wang, Zj., Wang, Wz., Wang, H. et al. Stress Analysis on Layered Materials in Point Elastohydrodynamic-Lubricated Contacts. Tribol Lett 35, 229–244 (2009). https://doi.org/10.1007/s11249-009-9452-4

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