A simple but precise method for solving axisymmetric contact problems involving elastically graded materials

TL;DR

The paper introduces an efficient, exact method for solving axisymmetric frictionless contact problems involving elastically graded materials, extending the method of dimensionality reduction to handle complex geometries and adhesion effects.

Contribution

It presents a generalized, easy-to-apply mapping technique that reduces 3D contact problems with arbitrary profiles and adhesion to 1D models, validated by examples.

Findings

Exact solutions match 3D contact theories

Applicable to arbitrary axisymmetric profiles

Incorporates adhesion via generalized JKR-theory

Abstract

An efficient method is presented for solving axisymmetric, frictionless contact problems between a rigid punch and an elastically non-homogeneous, power-law graded half-space. Provided that the contact area is simply-connected profiles of arbitrary shape can be considered. Moreover, adhesion in the framework of the generalized JKR-theory can be taken into account. All results agree exactly with those given by three-dimensional contact theories. The method uses the fact that three-dimensional contact problems can be mapped to one-dimensional ones with a properly defined Winkler-foundation; hence, the method is to be understood as an extension of the method of dimensionality reduction. A prerequisite of its applicability forms the generality of contact stiffness regardless of the geometry of the axisymmetric profile, which is proved. All the necessary mapping rules are derived and their ease of use explained by solving contact problems based on actual examples and up to now unsolved problems. ----- Es wird eine sehr effiziente Methode zur exakten L\"osung des axialsymmetrischen, reibungsfreien Kontaktproblems zwischen einem starren Indenter und einem elastisch inhomogenen Halbraum vorgestellt, dessen Elastizit\"smodul mit der Tiefe nach einem Potenzgesetz zunimmt. Unter der Voraussetzung einer einfach zusammenh\"angenden Kontaktfl\"ache k\"onnen beliebig geformte Profile angenommen und auch Kontakte mit Adh\"asion gel\"ost werden. Die Methode nutzt die Tatsache aus, dass dreidimensionale Kontaktprobleme auf Kontaktprobleme mit einer eindimensionalen Winklerschen Bettung abgebildet werden k\"onnen und ist daher als Erweiterung der Methode der Dimensionsreduktion zu verstehen. In diesem Beitrag werden alle notwendigen Abbildungsregeln hergeleitet und ihre einfache Anwendung zur L\"osung von Kontaktproblemen anhand von aktuellen Beispielen und bisher ungel\"osten Problemen erl\"autert.