2. Methods Titanium Nitride (TiN) is typically used in coating films due to their high elastic modulus E, low thermal conductivity k and low coefficient of thermal expansion α compared to steel or aluminum substrates. A normal force is applied in the form of Hertzian pressure that combined with the friction and displacement between the surfaces, generates a normal and a tangential mechanical and thermal loading, shown in Fig 1. Figure 1. Tribological pair under thermomechanical loading. The film coating was considered isotropic, following the laws of functional gradation according to Eqs (1) and (2) M βy 0 c c P (y ) Pe (1) 0 s M s k k ln y 1 β (2) where P is the property studied (α, k E), the index M = 0 is for HMG and M = 1 for FGM film structure, the subscripts s, c and 0 respectively identify the values at the interface between the film and the substrate, within the film and at the surface of the coating film and y is the direction across the film. The properties of the substrates, Poisson's ratio and the thermal diffusivity of the film coating were considered constant. It is assumed that the substrate is an elastic solid homogeneous, isotropic and dimensions enough large to be considered as infinite. The normal and tangential Hertzian pressure due to mechanical loading and the friction between the sliding surfaces generates normal and shear stress and heat. The normal pressure loads p(x), shear ptan(x) and heat flux are defined by Eqs (3) to (5) 2 0 p(x) p (1 x) (1 x) (3) p(x)μ p (x) tan (4) p(x)vμ q (x) p (5) where p0 is the load at x = 0, H is the Heaviside function, µ is the film friction coefficient, qp is the heat flux generated at the film surface and v is the load displacement velocity on x direction. The flexural tensile and shear stress at film and substrate interface, responsible for failure adhesion, are defined by Eqs (6) and (7)
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