Evaluation of the contact problem of functionally graded layer resting on rigid foundation pressed via rigid punch by analytical and numerical (FEM and MLP) methods

Abstract

In this paper, frictionless contact problem for a functionally graded (FG) layer is considered. The FG layer is subjected to load with a rigid punch and the FG layer is bonded on a rigid foundation. Analysis of this contact problem was carried out by analytical method, finite element method (FEM) and multilayer perceptron (MLP), comparatively. The main target of this study is to investigate the applicability of MLP analysis for frictionless contact problem of FG layer bonded on a rigid foundation. Analytical solution of the problem is based on the theory of elasticity and integral transform techniques. The physical contact problem is transformed to mathematical system of integral equation. The integral equation in which the contact pressures are unknown functions is numerically solved with the Gauss–Jacobi integration formulation. Finite element analysis of the problem is carried out with ANSYS software by using the two-dimensional modeling technique. Finally, MLP analysis has been used to obtain the contact distances of the problem. Three-layer MLP was used for this calculation. Material properties and loading conditions were created by giving examples of different values in MLP training and testing stages. Program code was rewritten in C++. As a result, average deviation values such as 1.67 and 0.885 were obtained for FEM and MLP, respectively. It has been determined that the contact areas and contact stresses obtained from FEM and MLP are quite compatible with the results obtained from the analytical method.

Appendices

Appendix 1

$$ \begin{aligned} A_{1} = & - (P(D_{2} m_{3} e^{{h\,n_{2} }} - D_{2} m_{4} e^{{h\,n_{2} }} - D_{3} m_{2} e^{h\,n3} + D_{3} m_{4} e^{{h\,n_{3} }} + D_{4} m_{2} e^{{h\,n_{4} }} - D_{4} m_{3} e^{{h\,n_{4} }} ))/\Delta \\ A_{2} = & (P(D_{1} m_{3} e^{{h\,n_{1} }} - D_{1} m_{4} e^{{h\,n_{1} }} - D_{3} m_{1} e^{h\,n3} + D_{4} m_{1} e^{{h\,n_{4} }} + D_{3} m_{4} e^{{h\,n_{3} }} - D_{4} m_{3} e^{{h\,n_{4} }} ))/\Delta \\ A_{3} = & - (P(D_{1} m_{2} e^{{h\,n_{1} }} - D_{2} m_{1} e^{{h\,n_{2} }} - D_{1} m_{4} e^{{h\,n_{1} }} + D_{2} m_{4} e^{{h\,n_{2} }} + D_{4} m_{1} e^{{hn_{4} }} - D_{4} m_{2} e^{{h\,n_{4} }} ))/\Delta \\ A_{4} = & (P(D_{1} m_{2} e^{{h\,n_{1} }} - D_{1} m_{3} e^{{h\,n_{1} }} - D_{2} m_{1} e^{{h\,n_{2} }} + D_{2} m_{3} e^{{h\,n_{2} }} + D_{3} m_{1} e^{{h\,n_{3} }} - D_{3} m_{2} e^{{h\,n_{3} }} ))/\Delta \\ \Delta A = & (e^{{h\,n_{1} }} e^{{h\,n_{2} }} (C_{1} D_{2} m_{3} - C_{2} D_{1} m_{3} - C_{1} D_{2} m_{4} + C_{2} D_{1} m_{4} ) \\ & \;\; + e^{{h\,n_{1} }} e^{{h\,n_{3} }} ( - C_{1} D_{3} m_{2} + C_{3} D_{1} m_{2} + C_{1} D_{3} m_{4} - C_{3} D_{1} m_{4} ) \\ & \;\; + e^{{h\,n_{1} }} e^{{h\,n_{4} }} (C_{1} D_{4} m_{2} - C_{4} D_{1} m_{2} - C_{1} D_{4} m_{3} + C_{4} D_{1} m_{3} ) \\ & \;\; + e^{{h\,n_{2} }} e^{{h\,n_{3} }} (C_{2} D_{3} m_{1} - C_{3} D_{2} m_{1} - C_{2} D_{3} m_{4} + C_{3} D_{2} m_{4} ) \\ & \;\; + e^{{h\,n_{2} }} e^{{h\,n_{4} }} ( - C_{2} D_{4} m_{1} + C_{4} D_{2} m_{1} + C_{2} D_{4} m_{3} - C_{4} D_{2} m_{3} ) \\ & \;\; + e^{{h\,n_{3} }} e^{{h\,n_{4} }} (C_{3} D_{4} m_{1} - C_{4} D_{3} m_{1} - C_{3} D_{4} m_{2} + C_{4} D_{3} m_{2} )) \\ \end{aligned} $$

Appendix 2

$$ \begin{aligned} N(x,t) = & \int\limits_{0}^{\infty } {\frac{\xi (\kappa - 1)}{{2\Delta A}}} \{ [e^{{\left( {n_{1} + n_{2} } \right)h}} \left( {m_{1} m_{4} D_{2} - m_{1} m_{3} D_{2} + m_{2} m_{3} D_{1} - m_{2} m_{4} D_{1} } \right) \\ & \;\; + e^{{\left( {n_{1} + n_{3} } \right)h}} \left( {m_{1} m_{2} D_{3} - m_{1} m_{4} D_{3} - m_{2} m_{3} D_{1} + m_{3} m_{4} D_{1} } \right) \\ & \;\; + e^{{\left( {n_{1} + n_{4} } \right)h}} \left( { - m_{1} m_{2} D_{4} + m_{1} m_{3} D_{4} + m_{2} m_{4} D_{1} - m_{3} m_{4} D_{1} } \right) \\ & \;\; + e^{{\left( {n_{2} + n_{3} } \right)h}} \left( { - m_{1} m_{2} D_{3} + m_{2} m_{4} D_{3} + m_{1} m_{3} D_{2} - m_{3} m_{4} D_{2} } \right) \\ & \;\; + e^{{\left( {n_{2} + n_{4} } \right)h}} \left( {m_{1} m_{2} D_{4} - m_{2} m_{3} D_{4} - m_{1} m_{4} D_{2} + m_{3} m_{4} D_{2} } \right) \\ & \;\; + e^{{\left( {n_{3} + n_{4} } \right)h}} \left( { - m_{1} m_{3} D_{4} + m_{2} m_{3} D_{4} + m_{1} m_{4} D_{3} - m_{2} m_{4} D_{3} } \right) + \frac{\kappa + 1}{8}]\} \sin \xi (t - x){\text{d}}\xi \\ \end{aligned} $$

Appendix 3

$$ \begin{aligned} k(s,r) = & \int\limits_{0}^{\infty } {\frac{{\frac{z}{h}(\kappa - 1)}}{2\Delta A}} \{ [e^{{\left( {n_{1} + n_{2} } \right)h}} \left( {m_{1} m_{4} D_{2} - m_{1} m_{3} D_{2} + m_{2} m_{3} D_{1} - m_{2} m_{4} D_{1} } \right) \\ & \;\; + e^{{\left( {n_{1} + n_{3} } \right)h}} \left( {m_{1} m_{2} D_{3} - m_{1} m_{4} D_{3} - m_{2} m_{3} D_{1} + m_{3} m_{4} D_{1} } \right) \\ & \;\; + e^{{\left( {n_{1} + n_{4} } \right)h}} \left( { - m_{1} m_{2} D_{4} + m_{1} m_{3} D_{4} + m_{2} m_{4} D_{1} - m_{3} m_{4} D_{1} } \right) \\ & \;\; + e^{{\left( {n_{2} + n_{3} } \right)h}} \left( { - m_{1} m_{2} D_{3} + m_{2} m_{4} D_{3} + m_{1} m_{3} D_{2} - m_{3} m_{4} D_{2} } \right) \\ & \;\; + e^{{\left( {n_{2} + n_{4} } \right)h}} \left( {m_{1} m_{2} D_{4} - m_{2} m_{3} D_{4} - m_{1} m_{4} D_{2} + m_{3} m_{4} D_{2} } \right) \\ & \;\; + e^{{\left( {n_{3} + n_{4} } \right)h}} \left( { - m_{1} m_{3} D_{4} + m_{2} m_{3} D_{4} + m_{1} m_{4} D_{3} - m_{2} m_{4} D_{3} } \right) + \frac{\kappa + 1}{8}]\} \sin z\left( {\frac{ar}{h} - \frac{as}{h}} \right){\text{d}}z \\ \end{aligned} $$