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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.

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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.

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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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Correspondence to David Néron.

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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.

$$\begin{aligned} \underline{v}_{R'}(t) = \underline{v}_{R}(t) - \underline{v}_{R'R}(t) \end{aligned}$$
(22)

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.

Fig. 21
figure 21

Reference frame velocity field computation

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:

$$\begin{aligned} \begin{aligned} \underline{d}^s(\underline{x}) = \underline{v}(\underline{x},t_s) \\ \underline{d}^a(\underline{x}) = \underline{v}(\underline{x},t_a) \end{aligned} \end{aligned}$$
(23)
Fig. 22
figure 22

Spatial fields extracted from FE modelling: a \(\underline{d}^s\) on the x direction, b \(\underline{d}^s\) on the y direction, c \(\underline{d}^a\) on the x direction, \(\underline{d}^a\) on the y direction

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.

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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

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