**1. Introduction**

28 Will-be-set-by-IN-TECH

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Fretting defines a condition in which mechanical contacts are subjected to alternating tangential displacements, small compared to dimensions of contact area, due to oscillating loading conditions. Fretting wear and fretting fatigue are between the most important factors responsible for contact failure, especially when high loads are transmitted through non-conforming contacts, leading to highly localized stress concentrators in the vicinity of the contact region. Prediction of life span of machine elements working in such conditions requires assessment of stress and strain in the contacting bodies, which is the main subject of Contact Mechanics. Although fretting is intrinsically a multidisciplinary process, involving adhesion, oxidation, abrasion and pitting, modern approach suggests that contact stresses play a chief role.

While analytic solutions in this research field lead to complex mathematical models, many without closed-form solution, numerical approach reveals itself as a useful engineering tool, capable of extending the few existing analytical results to technologically important contact scenarios. A numerical study may advance the understanding of fretting contact and provide assistance to the design of contacts with improved load-carrying capacity.

Elastic contact analysis considering interfacial friction and slip-stick behaviour originated in the works of Cattaneo (Cattaneo, 1938) and Mindlin (Mindlin, 1949). They proved independently that, even when the contacting bodies are globally sticking, a peripheral region of slip is to be assumed in order to remain in the frame of Linear Theory of Elasticity and to obey the Coulomb's law of friction. Based on these results, Johnson (Johnson, 1985) advanced the closed-form solution for the contact between similarly elastic materials undergoing a fretting loop.

In case of dissimilarly elastic materials, when the effects of normal and tangential tractions are coupled, an iterative solution has been achieved (Hills et al., 1993) only for the plane (i.e. cylindrical) contact. Many authors employ the so-called Goodman approximation

<sup>© 2012</sup> Spinu and Amarandei, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

(Goodman, 1962) when dealing with this type of contact, which neglects the influence of shear tractions on pressure, but retains that of pressure on tangential tractions. As proved in (Hills et al., 1993), this approximation is satisfactory in case of plane (cylindrical) contacts if Poisson's ratio is large enough, but the inaccuracy introduced in the simulation of the three dimensional contact between dissimilarly elastic materials cannot be a priori assessed.

Numerical Simulation of Slip-Stick Elastic Contact 131

1 2 (,) *<sup>j</sup>* **<sup>q</sup>** *q q* in the tangential

, with *i j* 1,2,3, 1,2 .

*<sup>i</sup> u* , and

(or flexing) moments 1 2 *M* , *M* and the torsional moment *M*<sup>3</sup> . Superscripts denote the contacting body, and subscripts are used for the direction of the referred quantity. Under load, the bodies deform unless assumed rigid, leading to elastic displacements ( )*<sup>j</sup>*

In Contact Mechanics, it is also assumed that contact area dimensions are small compared to extents of the contacting bodies, and therefore stresses in the contact region are independent of other boundary conditions. This assumption is well verified in case of non-conforming contacts, when stresses induced by the contact process are highly localized in the vicinity of

Once a contact area is established, the imposed forces and moments lead to contact tractions,

direction. The latter appears only if interfacial friction is assumed, leading to three possible cases, in relation to the magnitude of the tangential load: full stick, partial slip (or slip-stick), or gross slip. The latter case is trivial, as shear tractions are related to pressure through Coulomb's law on all contact area. On the other hand, the works of Cattaneo (Cattaneo, 1938) and Mindlin (Mindlin, 1949) prove that the full-sticking contact cannot be solved in the Frame of Linear Theory of Elasticity, as it leads to infinite stresses at the boundary of the contact area. The study of the partial slip contact, which is found in fretting contact processes, concluding with assessment of contact tractions, is the main goal of this work.

Based on the works developed in (Johnson, 1985; Polonsky & Keer, 1999), the model for the

() ( , ,) ;

<sup>2</sup> 12 112

12 12 312 3 1 2 2 1 12 ( , , ) ( , ) ( , , ) ( ) ( ) ( ) , ( , ) ( ). *<sup>C</sup> h x x t hi x x u x x t t t x t x x x t* 

> ( , , ) 0 ( , , ) 0, ( , ) ( ); ( , , ) 0 ( , , ) 0, ( , ) ( ).

*px x t hx x t x x t px x t hx x t x x t* 

The temporal dimension *t* is included in this model along the spatial dimensions 1 2 *x x*, to provide basis for reproduction of the loading history. Consequently, ( ) *<sup>C</sup> t* denotes the

1 2 1 2 1 2 1 2 1 2 1 2

*W t p x x t dx dx*

12 12

( )

 

*<sup>C</sup> t*

(1)

() ( , ,) .

*C C*

(2)

(3)

(4)

*M t p x x t x dx dx*

contact in the normal direction consists in the following equations and inequalities:

( )

*<sup>C</sup> t*

1 12 212

*M t p x x t x dx dx*

() ( , ,) ;

( )

*<sup>C</sup> t*

2. The geometrical condition of deformation:

3. The contact complementarity conditions:

*i* and rotations ( )*<sup>j</sup>*

*i* 

i.e. pressure ( )*<sup>j</sup> p* in the normal direction and shear traction ( )

**2.1. The contact model in the normal direction** 

1. The static force equilibrium:

move as rigid-bodies with translations ( )*<sup>j</sup>*

the contact region.

In order to overcome this obstacle, recent works aimed to solve the problem numerically, using a method derived from the boundary element method, also referred to as semianalytical (SAM) in a review paper by Renauf et al. (Renauf et al., 2011). The strong point of this technique is that only a small region of the boundary of the contacting bodies, enclosing the contact area, is to be meshed, leading to a dramatic decrease in computational complexity compared to finite element method, in which discretization of the entire bulk is required.

Chen and Wang (Chen & Wang, 2008) advanced an algorithm for the non-conforming contact of dissimilarly elastic materials, and predicted the additional effect of an increasing tangential loading. Wang, Meng, Xiao, and Wang (Wang et al., 2011) investigated numerically the supplementary effect of a torsional moment, while Wang et al. (Wang et al., 2010) applied the algorithm advanced in (Chen & Wang, 2008) to contact of elastic layered half-spaces. However, the loading history was not accounted for in these studies, i.e. the full load was applied in one step.

Gallego, Nélias, and Deyber (Gallego et al., 2010) applied numerical analysis in an incremental approach to study different fretting modes, and concluded that assumptions adopted in existing analytical models lead to arguably inaccurate results. It is asserted in (Gallego et al., 2010) that, due to irreversibility of friction, which is a dissipative process, loading history should be considered although a purely elastic contact analysis is intended.

An incremental iterative algorithm for the fully coupled elastic contact with slip and stick is advanced in this work. Existing algorithms for the uncoupled normal or tangential contact problems are adapted for modeling of transient contact, and combined in an iterative approach based on the mutual adjustment between contact tractions, resulting in a three level nested loop algorithm. The use of modern numerical methods allows for a fine discretization in both spatial and temporal domain, leading to well converged numerical solutions.
