This paper investigates the frictional receding contact problem between a functionally graded (FG) layer resting on a homogeneous coated half-plane when the system indented by a rigid cylindrical punch. The shear modulus of the upper graded layer is assumed to vary exponentially in the depth direction. Upon loading, the advancing contact and receding contact occur between the FG layer and rigid punch and between the FG layer and coating, respectively. Under the assumptions of sliding contact, the shear and normal contact stresses are related through Coulomb’s law of friction. The contact problem is converted analytically using Fourier integral transforms and the proper boundary conditions into a system of two singular integral equations. The unknowns of these integral equations consist of the contact stresses under the punch and between the FG layer and coating as well as the dimensions of the contact zones. The Gauss–Jacobi quadrature collocation method is then employed to transform the singular integral equations into a system of nonlinear equations which are solved using an appropriate iterative algorithm to compute the contact stresses and the dimensions of the contact areas. The main purpose of this paper is to examine the influence of several parameters on the upper and lower contact stresses, which include the material inhomogeneity parameter, friction coefficient, punch radius, applied load, thickness of homogeneous layer, shear modulus of homogeneous layer and shear modulus of homogeneous half-plane. One of the most important conclusions from this study is that the contact stresses and contact areas can be controlled with the addition of the homogeneous layer.

中文翻译：

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均质涂层半平面上的功能梯度层的摩擦后退接触问题

本文研究了当系统由刚性圆柱冲头压入时，位于均匀涂层半平面上的功能梯度（FG）层之间的摩擦后退接触问题。假定上部渐变层的剪切模量沿深度方向呈指数变化。加载时，前进接触和后退接触分别发生在FG层和刚性冲头之间以及FG层和涂层之间。在滑动接触的假设下，剪切应力和法向接触应力通过库仑摩擦定律关联。使用傅里叶积分变换和适当的边界条件将接触问题解析地转换为两个奇异积分方程组。这些积分方程的未知数由冲头下方以及FG层和涂层之间的接触应力以及接触区的尺寸组成。然后，采用高斯-雅各比正交配置方法将奇异积分方程式转换为非线性方程式系统，然后使用适当的迭代算法对其进行求解，以计算接触应力和接触区域的尺寸。本文的主要目的是研究几个参数对上下接触应力的影响，包括材料不均匀性参数，摩擦系数，冲头半径，施加的载荷，均质层的厚度，均质层的剪切模量和剪切力。均匀半平面的模数