# An enrichment-based approach for the simulation of fretting problems

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## Abstract

The aim of this work is to improve the performance of fretting simulations making use of an enrichment approach. The idea is to take advantage of the fact that the mechanical fields around the contact edges in cylindrical contact configurations under fretting conditions are similar to the ones found close to the crack tip in linear elastic fracture mechanics problems. This similarity makes attractive the idea of enriching finite element fretting simulations through the X-FEM framework, which enables us to work with coarser meshes while keeping a good accuracy. As it will be shown in this work, it is possible to work with meshes up to 10 times coarser than it should be if a conventional FE method was used allowing a strong improvement of the computational performances.

## Keywords

Fretting Crack analogy X-FEM Model reduction## List of symbols

- \(\underline{d}^s\)
Symmetric spatial reference field,

- \(\underline{d}^a\)
Antisymmetric spatial reference field,

- \(\underline{d}^c\)
Complementary spatial reference field,

- \(I^s\)
Intensity factor (symmetric part),

- \(I^a\)
Intensity factor (antisymmetric part),

- \(I^c\)
Intensity factor (complementary part),

- \(\underline{v}\)
Velocity field expressed in the reference frame attached to the contact edge,

*P*Normal force applied to the cylindrical pad,

*Q*Fretting tangential force applied to the cylindrical pad,

- \(F_b\)
Bulk fatigue load applied to the rectangular specimen,

- \(\underline{u}\)
Displacement field,

- \(u_{x}\)
Tangential displacement imposed on the cylindrical pad,

- \(u_{y}\)
Vertical displacement imposed on the cylindrical pad,

- \(u_{x,max}\)
Maximum tangential displacement applied to the cylindrical pad,

*f*(*r*)Radial evolution of the spatial reference field,

- \(\underline{g}\)
Tangential evolution of the spatial reference field,

- \(\mu \)
Coulomb friction coefficient,

- \(N_i\)
Finite element basis function,

- \(\psi \)
Enrichment function,

- \(r_e\)
Enrichment radius,

- \(\lambda \)
Singularity order,

*a*Semi-width contact zone,

*c*Semi-width contact stick zone,

- \(\tilde{\mu }\)
Nonlocal Coulomb friction coefficient,

## Abbreviations

- LEFM
Linear elastic fracture mechanics,

- FE
Finite element,

- X-FEM
Extended finite element method,

- POD
Proper orthogonal decomposition.

## Notes

### Acknowledgements

The authors would like to acknowledge the financial support of SAFRAN Aircraft Engines to this project in the context of the international research group COGNAC. Raphael A. Cardoso also would like to acknowledge the scholarship granted by the Brazilian National Council for Scientific and Technological Development (CNPq) and the Brazilian Aerospace Agency (AEB).

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