In this study, the discontinuous contact problem between a functionally graded (FG) layer, which is loaded symmetrically with point load P through a rigid block, and a homogeneous half-space was solved using the theory of elasticity and integral transform techniques. The shear modulus and density of the layer addressed in the problem vary with an exponential function along with its height. The half-space is homogeneous, and no binder exists on the contact surface containing the FG layer. In the solution, the body force of the FG layer was considered, whereas that of the homogeneous half-space was neglected. The Poisson’s ratios of both the FG layer and homogeneous half-space were assumed to remain constant. Additionally, all the surfaces addressed in the problem were assumed to be frictionless. Using the theory of elasticity and integral transform techniques, the discontinuous contact problem was reduced to two integral equations, wherein the contact stress under the rigid block and the slope of the separation, which occurred at the interface of the FG layer and homogeneous half-space, are unknown. These integral equations were solved numerically for the flat condition of the rigid block profile using the Gauss–Chebyshev integration formula. Consequently, the stress distributions, start–end points of the separation region, and separation displacements between the FG layer and homogeneous half-space were obtained for various dimensionless quantities.

在这项研究中，功能梯度（FG）层之间的不连续接触问题，点载荷P对称地加载通过一个刚性块，并使用弹性理论和积分变换技术求解了均匀的半空间。该问题解决的层的剪切模量和密度随其高度随指数函数而变化。半空间是均匀的，并且在包含FG层的接触表面上不存在粘合剂。在解决方案中，考虑了FG层的力，而忽略了均匀半空间的力。假设FG层和均质半空间的泊松比均保持恒定。另外，假设解决该问题的所有表面都是无摩擦的。利用弹性理论和积分变换技术，将不连续接触问题简化为两个积分方程，其中，在FG层和均匀半空间的界面处发生的刚性块下的接触应力和分离斜率是未知的。使用Gauss–Chebyshev积分公式，对刚性块轮廓的平坦条件进行了数值积分求解。因此，对于各种无量纲的量，获得了应力分布，分离区域的起点和终点，以及FG层和均匀半空间之间的分离位移。