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.
Similar content being viewed by others
Abbreviations
- \(\underline{d}^s\) :
-
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,
- LEFM:
-
Linear elastic fracture mechanics,
- FE:
-
Finite element,
- X-FEM:
-
Extended finite element method,
- POD:
-
Proper orthogonal decomposition.
References
Araújo J, Nowell D (1999) Analysis of pad size effects in fretting fatigue using short crack arrest methodologies. Int J Fatigue 21:947–956
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 75:1763–1778
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 29:95–107
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 47:31–43
Creager M, Paris PC (1967) Elastic field equations for blunt cracks with reference to stress corrosion cracking. Int J Fract Mech 3:247–252
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 134:267–285
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 39:1158–1165
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 62:194–209
Fouvry S, Nowell D, Kubiak K, Hills D (2008) Prediction of fretting crack propagation based on a short crack methodology. Eng Fract Mech 75:1605–1622
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 28:1071–1085
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 46:2955–2968
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 23:561–572
Giner E, Sabsabi M, Ródenas JJ, Fuenmayor FJ (2014) Direction of crack propagation in a complete contact fretting-fatigue problem. Int J Fatigue 58:172–180
Giner E, Sukumar N, Denia F, Fuenmayor F (2008) Extended finite element method for fretting fatigue crack propagation. Int J Solids Struct 45:5675–5687
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 76:1402–1418
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 42:1269–1275
Hills D, Nowell D (1994) Mechanics of fretting-fatigue. Kluwer Academic Publishers, Dordrecht
Khoei AR (2014) Extended finite element method: theory and applications. Wiley, Hoboken
Ladevèze P (1999) Nonlinear computational structural mechanics—new approaches and non-incremental methods of calculation. Springer Verlag, Berlin
Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150
Montebello C (2015) Analysis of the stress gradient effect in fretting-fatigue through a description based on nonlocal intensity factors, PhD thesis, Université Paris-Saclay
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 82:188–198
Munoz S, Proudhon H, Dominguez J, Fouvry S (2006) Prediction of the crack extension under fretting wear loading conditions. Int J Fatigue 28:1769–1779
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 339:532–551
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 32:899–915
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 196:3230–3247
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 31:2041–2048
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 190:6183–6200
Sukumar N, Moës N, Moran B, Belytschko T (2000) Extended finite element method for three-dimensional crack modelling. Int J Numer Meth Eng 48:1549–1570
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Velocity field and spatial reference fields computation
Appendix: Velocity field and spatial reference fields computation
As presented in Sect. 2.1, the velocity field in the vicinity of the contact edges can be partitioned into a product of spatial reference fields associated to the local geometry of the problem and nonlocal intensity factors associated to the macroscopic loads, Eq. (1). In this case, the velocity field is computed with respect to a reference frame attached to the contact edges, Fig. 21.
For the sake of notation simplicity, the velocity field computed with respect to the contact edges \(\underline{v}_{R'}(t)\) appears only as \(\underline{v}\) in this work.
As mentioned in Sect. 2.1, the spatial modes \(\underline{d}^s\) and \(\underline{d}^{a}\) are computed extracting the velocity field (relative to the contact edge) in some strategic time steps of the load history depicted in Fig. 5. In this case:
Note that \(t_s\) is the time instant where a small perturbation is introduced to the normal load P while the tangential load Q is null. On the other hand, \(t_a\) is the time step where the tangential load Q starts being reversed (full stick condition) whereas the normal load is kept constant. Doing so, the effects of the normal and tangential loads can be taken into account separately. It is worth mentioning that when these fields are computed in polar coordinates (Eqs. 6, 7), they are further normalized in order to be comparable to the displacement field obtained at the crack tip during a elastic loading phase either \(K_I = \) 1 MPa \(\sqrt{m}\) or \(K_{II} = \) 1 MPa \(\sqrt{m}\), Figs. 6 and 7. A 2D distribution of those spatial reference fields extracted from a FE computation can be found in Fig. 22. The coordinate system, normalized with respect to the contact semi-width, is expressed in the reference system fixed at the contact edge. The problem solved is depicted in Fig. 5. The normal load applied to the pad was \(P = 227~\hbox {N/mm}\) and the tangential load \(Q = 169~\hbox {N/mm}\). The small perturbation in the normal load (time instant \(t_s\)) was equal to 0.05P. The contact parts are elastically similar with the following material properties: \(E = 200\) MPa and \(\nu = 0.3\). The friction coefficient in the slip zones was assumed as \(\mu = 0.9\). Linear elastic regime was always assumed.
Rights and permissions
About this article
Cite this article
Cardoso, R.A., Néron, D., Pommier, S. et al. An enrichment-based approach for the simulation of fretting problems. Comput Mech 62, 1529–1542 (2018). https://doi.org/10.1007/s00466-018-1577-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-018-1577-6