The contact problemof indentation of a functionally graded coating by a convex punch undera predefinedload

Applied Mathematics, Mechanics and Physics


Аuthors

Bobylov A. A.1*, Belashova I. S.1**, Kuzmin S. D.2***

1. Moscow Automobile and Road Construction State Technical University (MADI), MADI, 64, Leningradsky Prospect, Moscow, 125319, Russia
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: abobylov@gmail.com
**e-mail: irina455@inbox.ru.
***e-mail: info@ecooil.ru

Abstract

The nonlinear plane contact problem of indenting a hard punch in an elastic body with functionally graded coating is considered. The main vector and the main moment of the external forces applied to a hard punch are assumed set. Unilateral contact conditions are given on potential contact area on which the elastic body and the hard punch may come in contact. Thickness of the coating is commensurable with the sizes of the spot of contact. Such scheme is applied at research of discrete contact of rough bodies.
The variational method is used to solve the problem. The variational formulation of the problem in the form of a variational inequality and an extreme problem areobtained. The variational inequality is the formulation of the variational principles of virtual work and the extreme problem is the formulation of the variational principles of minimum potential energy.
The discretization of the problem is made by an unstructured triangular finite element grid of the axisymmetric finite elements with triangular cross section. The stiffness matrix of the elastic body is a sparse block matrix. The block sparse row format is used for saving it. This scheme of storage of the stiffness matrix imposes minimum requirements to memory, which volume is proportional to the first degree of number of unknown, and allows to organize effectively calculation matrix-vector multiplication. Modifications of the conjugate gradients method is used for numerical solving of the obtained finite-dimensional nonlinear programming problems.
Developed computational algorithms are implemented as a separate module of an application package for solution of non-homogeneous elastic finite bodies contact problems.
The carried-out calculations showed that existence of functionally graded coating, which elasticity module exceeds the elasticity module of the main material, leads the maximum values of contact pressure to increase the maximum values of contact pressure and the punch penetration to reduce.

Keywords:

functionally graded coating, axisymmetric contact problem, variational inequality, finite elements method

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