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

- 103 Downloads
- 1 Citations

## 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).

## References

- 1.Araújo J, Nowell D (1999) Analysis of pad size effects in fretting fatigue using short crack arrest methodologies. Int J Fatigue 21:947–956CrossRefGoogle Scholar
- 2.Araújo J, Susmel L, Taylor D, Ferro J, Ferreira J (2008) On the prediction of high-cycle fretting fatigue strength: theory of critical distances vs. hot-spot approach. Eng Fract Mech 75:1763–1778CrossRefGoogle Scholar
- 3.Araújo J, Susmel L, Taylor D, Ferro J, Mamiya E (2007) On the use of the theory of critical distances and the modified Wöhler curve method to estimate fretting fatigue strength of cylindrical contacts. Int J Fatigue 29:95–107CrossRefGoogle Scholar
- 4.Baietto M-C, Pierres E, Gravouil A, Berthel B, Fouvry S, Trolle B (2013) Fretting fatigue crack growth simulation based on a combined experimental and XFEM strategy. Int J Fatigue 47:31–43CrossRefGoogle Scholar
- 5.Creager M, Paris PC (1967) Elastic field equations for blunt cracks with reference to stress corrosion cracking. Int J Fract Mech 3:247–252CrossRefGoogle Scholar
- 6.de Pannemaecker A, Fouvry S, Buffiere J (2015) Reverse identification of short–long crack threshold fatigue stress intensity factors from plain fretting crack arrest analysis. Eng Fract Mech 134:267–285CrossRefGoogle Scholar
- 7.Dini D, Nowell D, Dyson IN (2006) The use of notch and short crack approaches to fretting fatigue threshold prediction: Theory and experimental validation. Tribol Int 39:1158–1165CrossRefGoogle Scholar
- 8.Fouvry S, Gallien H, Berthel B (2014) From uni-to multi-axial fretting-fatigue crack nucleation: development of a stress-gradient-dependent critical distance approach. Int J Fatigue 62:194–209CrossRefGoogle Scholar
- 9.Fouvry S, Nowell D, Kubiak K, Hills D (2008) Prediction of fretting crack propagation based on a short crack methodology. Eng Fract Mech 75:1605–1622CrossRefGoogle Scholar
- 10.Fuenmayor F, Giner E, Tur M (2005) Extraction of the generalized stress intensity factor in gross sliding complete contacts using a path-independent integral. Fatigue Fract Eng Mater Struct 28:1071–1085CrossRefGoogle Scholar
- 11.Giannakopoulos A, Lindley T, Suresh S (1998) Aspects of equivalence between contact mechanics and fracture mechanics: theoretical connections and a life-prediction methodology for fretting-fatigue. Acta Mater 46:2955–2968CrossRefGoogle Scholar
- 12.Giannakopoulos A, Lindley T, Suresh S, Chenut C (2000) Similarities of stress concentrations in contact at round punches and fatigue at notches: implications to fretting fatigue crack initiation. Fatigue Fract Eng Mater Struct 23:561–572CrossRefGoogle Scholar
- 13.Giner E, Sabsabi M, Ródenas JJ, Fuenmayor FJ (2014) Direction of crack propagation in a complete contact fretting-fatigue problem. Int J Fatigue 58:172–180CrossRefGoogle Scholar
- 14.Giner E, Sukumar N, Denia F, Fuenmayor F (2008) Extended finite element method for fretting fatigue crack propagation. Int J Solids Struct 45:5675–5687CrossRefzbMATHGoogle Scholar
- 15.Giner E, Sukumar N, Fuenmayor F, Vercher A (2008) Singularity enrichment for complete sliding contact using the partition of unity finite element method. Int J Numer Meth Eng 76:1402–1418MathSciNetCrossRefzbMATHGoogle Scholar
- 16.Giner E, Tur M, Vercher A, Fuenmayor F (2009) Numerical modelling of crack-contact interaction in 2d incomplete fretting contacts using X-FEM. Tribol Int 42:1269–1275CrossRefGoogle Scholar
- 17.Hills D, Nowell D (1994) Mechanics of fretting-fatigue. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
- 18.Khoei AR (2014) Extended finite element method: theory and applications. Wiley, HobokenCrossRefzbMATHGoogle Scholar
- 19.Ladevèze P (1999) Nonlinear computational structural mechanics—new approaches and non-incremental methods of calculation. Springer Verlag, BerlinzbMATHGoogle Scholar
- 20.Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150CrossRefzbMATHGoogle Scholar
- 21.Montebello C (2015)
*Analysis of the stress gradient effect in fretting-fatigue through a description based on nonlocal intensity factors*, PhD thesis, Université Paris-SaclayGoogle Scholar - 22.Montebello C, Pommier S, Demmou K, Leroux J, Meriaux J (2016) Analysis of the stress gradient effect in fretting-fatigue through nonlocal intensity factors. Int J Fatigue 82:188–198CrossRefGoogle Scholar
- 23.Munoz S, Proudhon H, Dominguez J, Fouvry S (2006) Prediction of the crack extension under fretting wear loading conditions. Int J Fatigue 28:1769–1779CrossRefGoogle Scholar
- 24.Pierres E, Baietto M-C, Gravouil A (2011) Experimental and numerical analysis of fretting crack formation based on 3d X-FEM frictional contact fatigue crack model. Comptes Rendus Mécanique 339:532–551CrossRefzbMATHGoogle Scholar
- 25.Pommier S, Lopez-Crespo P, Decreuse P (2009) A multi-scale approach to condense the cyclic elastic–plastic behaviour of the crack tip region into an extended constitutive model. Fatigue Fract Eng Mater Struct 32:899–915CrossRefGoogle Scholar
- 26.Ribeaucourt R, Baietto-Dubourg M-C, Gravouil A (2007) A new fatigue frictional contact crack propagation model with the coupled X-FEM/latin method. Comput Methods Appl Mech Eng 196:3230–3247MathSciNetCrossRefzbMATHGoogle Scholar
- 27.Rossino LS, Castro F, Filho WWB, Araújo J (2009) Issues on the mean stress effect in fretting fatigue of a 7050–t7451 al alloy posed by new experimental data. Int J Fatigue 31:2041–2048CrossRefGoogle Scholar
- 28.Sukumar N, Chopp DL, Moës N, Belytschko T (2001) Modeling holes and inclusions by level sets in the extended finite-element method. Comput Methods Appl Mech Eng 190:6183–6200MathSciNetCrossRefzbMATHGoogle Scholar
- 29.Sukumar N, Moës N, Moran B, Belytschko T (2000) Extended finite element method for three-dimensional crack modelling. Int J Numer Meth Eng 48:1549–1570CrossRefzbMATHGoogle Scholar