Analytical approach to contact mechanics of functionally graded orthotropic layers with gravitational considerations

Contact problems involving deformable bodies are widespread in both industrial and everyday situations. They have a crucial impact on structural and mechanical systems, which has led to significant efforts in modeling and numerical simulations. These efforts aim to improve understanding and optimization in various engineering applications. This study examines the contact problem involving a functionally graded (FG) orthotropic layer resting on a rigid foundation, without considering frictional influences. A point load is applied to the layer through a rigid punch on its top surface. Additionally, the gravitational effects of the FG orthotropic layer are considered in the analyses. Material parameters and density of the FG orthotropic layer are presumed to exhibit exponential variations along the vertical axis. The resolution of the problem involves deriving stress and displacement expressions through the application of elasticity theory and integral transformation techniques. By imposing the pertinent boundary conditions onto these expressions, a singular integral equation is formulated, wherein the contact stress under the punch remains unknown. Employing the Gauss–Chebyshev integration method, this integral equation is subsequently numerically solved, particularly for a flat punch profile. The outcomes of this investigation encompass the determination of contact stresses under the punch, the critical separation load, and the critical separation point—marking the initial separation between the FG orthotropic layer and the rigid foundation. Additionally, the analysis yields dimensionless representations of normal stresses along the symmetry axis within the FG orthotropic layer, as well as shear stresses along a designated section proximate to the symmetry axis. Furthermore, it provides insights into the normal stresses along the x axis at the bottom surface of the FG orthotropic layer, contingent upon various parameters and distinct orthotropic material compositions.