# Closed-form solutions for tilted three-part piecewise-quadratic half-plane contacts

## TL;DR

This paper derives closed-form solutions for a complex half-plane contact problem with three-piece piecewise-quadratic geometry, accounting for moments and shear forces, to better understand asymmetrical dovetail contact behaviour in gas turbine blades.

## Contribution

It extends existing symmetrical contact solutions to asymmetrical cases, providing a general analytical approach for complex three-part geometries in contact mechanics.

## Key findings

- Closed-form solutions for asymmetric three-part contact problems.
- Analysis of contact pressure variations at edges under different forces.
- Mathematica code provided for practical problem solving.

## Abstract

A general half-plane contact problem in which the geometry is specified in a piecewise-quadratic sense over three segments is solved in closed form. This includes the effects of a moment applied sufficient to introduce separation of one segment and the application of a shearing force sufficient or insufficient to cause sliding. Extending existing solutions to asymmetrical problems is necessary in order to broaden our understanding of the behaviour of dovetail roots of gas turbine fan blades. In previous studies symmetrical contacts have often been used to represent a dovetail flank contact. In the asymmetrical case, the contact pressure may be considerably higher at one of the contact edges compared to the corresponding symmetrical case. Exploiting the generality provided with the solution presented in this study, several simpler indenter problems are investigated making use of an algebraic manipulator. The Mathematica code is made available for download.