**5. Results and discussions**

The newly advanced algorithm is validated in this section by comparison with existing closed-form solutions for the contact of similarly elastic materials. The results are then extended by simulating the three-dimensional contact between dissimilarly elastic materials, proving the capacity of the program to solve a large variety of problems incorporating the slip-stick processes in elastic mechanical contacts.

Numerical Simulation of Slip-Stick Elastic Contact 147

lim

small. Although some perturbations still appear, the large number of increments used in the

The Hertz frictionless theory for the similarly elastic contact scenario predicts a central pressure 1.996 *Hp GPa* and a contact radius 0.489 *Ha mm* , which are used as normalizers. Shear tractions profiles corresponding to equal loading levels but laying on different trajectories are depicted in Fig. 5, proving that the current state depends not only on the present load level, but also on its past evolution, thus showing a hysteretic behaviour or memory effect. Only profiles corresponding to states on AD are shown in Fig. 5. States on the DF trajectory can be obtained from those on AD by symmetry with respect to the 1 *x* -

axis, and states on FH overlap those on BD corresponding to the same loading level.

**Figure 5.** Profiles of shear tractions in the plane 2 *x* 0 corresponding to different points on the

As depicted in Fig. 6, the force-displacement curve forms a hysteretic loop, also referred to as a fretting loop. The rigid-body tangential displacement is normalized by its value

corresponding to lim *T* . The analytical model predicts that the states corresponding to points B and F on the loading curve should overlap, which is also obtained through numerical

From a mechanical point of view, regions undergoing the most severe stresses are of interest, leading to prediction of yield inception or crack nucleation in the contacting bodies. An algorithm to assess stresses due to known, but otherwise arbitrarily distributed contact tractions is readily available (Liu & Wang, 2002). A typical distribution of von Mises

path, is depicted in Fig. 7. Evolution of magnitude and of depth of von Mises maximum

stress in a fretting loop is depicted in Figs. 8-9, for various frictional coefficients.

*p* in the plane 2 *x* 0 , corresponding to point B on the loading

loading curve,

0.1

equivalent stress *VM H*

simulation, within an imposed precision.

current simulations 700 *N* led to well converged numerical solutions.

#### **5.1. Simulation of a fretting loop**

The newly proposed algorithm is first validated by comparison with the closed-form solution advanced in (Johnson, 1985) for the spherical contact undergoing a fretting loop. A steel ball of radius *R mm* 18 is pressed against an elastic half-space having the same elastic properties, with a normal force *W kN* 1 . In this case, the NC is uncoupled from the TC, as shown by Eq. (32). A tangential force 1 *T* oscillating between two limiting values lim *T* and lim *T* , where lim *T W* 0.9 , is subsequently applied. During a fretting contact process, it is expected that friction vary on the contact area, as well as with accumulation of debris particles resulted from additional wear. However, for validation purposes, a frictional coefficient uniform over all contact area and constant during load application is assumed in this study. The numerical approach can equally handle mapped distributions of , if such information is available. The loading history for fretting processes is depicted in Fig. 4, in which the load is applied incrementally in 700 *N* equal steps.

**Figure 4.** The loading history in simulation of a fretting loop; all moments are assumed to vanish

*N* is chosen differently in the following simulations. When studying the contact between similarly elastic materials, accurate (i.e. concurring with the existing closed-form solution) results can be obtained with even a small number of increments ( 42 *N* increments in simulations depicted in Figs. 5 and 6). Moreover, it is found that every different state on the loading path can be obtained by simulating only the states when the tangential load changes its sign of, e.g. every state M on the trajectory DF require computation of three states: the ones corresponding to points B and D, as well as the current one (corresponding to point M). This is not the case when simulating the contact between dissimilarly elastic materials. It is found that a large number of increments is required to obtain the detailed contact behaviour, due to non-overlapping stick regions from one loading step to the next. This feature was also observed by Gallego et al. (Gallego et al., 2010), who pointed out that a waved shear tractions profile is predicted numerically when the number of increments is small. Although some perturbations still appear, the large number of increments used in the current simulations 700 *N* led to well converged numerical solutions.

146 Numerical Simulation – From Theory to Industry

**5.1. Simulation of a fretting loop** 

lim *T* , where lim *T W* 0.9

slip-stick processes in elastic mechanical contacts.

extended by simulating the three-dimensional contact between dissimilarly elastic materials, proving the capacity of the program to solve a large variety of problems incorporating the

The newly proposed algorithm is first validated by comparison with the closed-form solution advanced in (Johnson, 1985) for the spherical contact undergoing a fretting loop. A steel ball of radius *R mm* 18 is pressed against an elastic half-space having the same elastic properties, with a normal force *W kN* 1 . In this case, the NC is uncoupled from the TC, as shown by Eq. (32). A tangential force 1 *T* oscillating between two limiting values lim *T* and

expected that friction vary on the contact area, as well as with accumulation of debris particles resulted from additional wear. However, for validation purposes, a frictional coefficient uniform over all contact area and constant during load application is assumed in

information is available. The loading history for fretting processes is depicted in Fig. 4, in

this study. The numerical approach can equally handle mapped distributions of

**Figure 4.** The loading history in simulation of a fretting loop; all moments are assumed to vanish

*N* is chosen differently in the following simulations. When studying the contact between similarly elastic materials, accurate (i.e. concurring with the existing closed-form solution) results can be obtained with even a small number of increments ( 42 *N* increments in simulations depicted in Figs. 5 and 6). Moreover, it is found that every different state on the loading path can be obtained by simulating only the states when the tangential load changes its sign of, e.g. every state M on the trajectory DF require computation of three states: the ones corresponding to points B and D, as well as the current one (corresponding to point M). This is not the case when simulating the contact between dissimilarly elastic materials. It is found that a large number of increments is required to obtain the detailed contact behaviour, due to non-overlapping stick regions from one loading step to the next. This feature was also observed by Gallego et al. (Gallego et al., 2010), who pointed out that a waved shear tractions profile is predicted numerically when the number of increments is

which the load is applied incrementally in 700 *N* equal steps.

, is subsequently applied. During a fretting contact process, it is

, if such The Hertz frictionless theory for the similarly elastic contact scenario predicts a central pressure 1.996 *Hp GPa* and a contact radius 0.489 *Ha mm* , which are used as normalizers. Shear tractions profiles corresponding to equal loading levels but laying on different trajectories are depicted in Fig. 5, proving that the current state depends not only on the present load level, but also on its past evolution, thus showing a hysteretic behaviour or memory effect. Only profiles corresponding to states on AD are shown in Fig. 5. States on the DF trajectory can be obtained from those on AD by symmetry with respect to the 1 *x* axis, and states on FH overlap those on BD corresponding to the same loading level.

**Figure 5.** Profiles of shear tractions in the plane 2 *x* 0 corresponding to different points on the loading curve, 0.1

As depicted in Fig. 6, the force-displacement curve forms a hysteretic loop, also referred to as a fretting loop. The rigid-body tangential displacement is normalized by its value lim corresponding to lim *T* . The analytical model predicts that the states corresponding to points B and F on the loading curve should overlap, which is also obtained through numerical simulation, within an imposed precision.

From a mechanical point of view, regions undergoing the most severe stresses are of interest, leading to prediction of yield inception or crack nucleation in the contacting bodies. An algorithm to assess stresses due to known, but otherwise arbitrarily distributed contact tractions is readily available (Liu & Wang, 2002). A typical distribution of von Mises equivalent stress *VM H p* in the plane 2 *x* 0 , corresponding to point B on the loading path, is depicted in Fig. 7. Evolution of magnitude and of depth of von Mises maximum stress in a fretting loop is depicted in Figs. 8-9, for various frictional coefficients.

Numerical Simulation of Slip-Stick Elastic Contact 149

**Figure 9.** Dimensionless depth of maximum von Mises equivalent stress versus loading level (indicated

It is found through numerical simulation that shear tractions have a weak impact on the

competition between the in-depth maximum, which fluctuates around its position in the frictionless contact (corresponding to point A in Fig. 9), and a second extremum developing on the surface, at the trailing edge of the contact. The latter can seriously diminish the loadcarrying capacity of the contact, especially in case of surfaces with poorly controlled surface

Program validation is subsequently performed in case of a spherical contact undergoing torsion applied simultaneously to a normal constant force. Based on the results of this author, (Mindlin, 1949), Johnson (Johnson, 1985) presents the closed-form solution for this contact scenario when a partial slip regime is established on the contact area (i.e. when the

later reviewed and enhanced for the case of viscoelastic materials by Dintwa et al. (Dintwa

A spherical indenter of radius *R mm* 18 is pressed with a normal force *W kN* 1 against an elastic half-space, having the same elastic parameters, and subsequently an increasing torsional moment *M M* 3 3 lim is applied, leading to shear tractions 1 *q* and 2 *q* depicted in Figs. 10a and b, respectively. The norm of these tractions 1 1 **q**(,) (,) (,) *ij q ij q ij* is

Figure 12 depicts the dimensionless stick radius when the torsional moment *M*3 is varied in the domain corresponding to partial slip. In all investigated cases, a good agreement between analytical results and numerical predictions is found, giving further confidence in

. However, larger friction processes lead to a

). The solution is

torsional moment *M*3 is smaller than a limiting value 3 lim 3 16 *M WaH*

compared in Fig. 11 with analytical results, using cylindrical coordinates.

quality, due to superimposition of microtopography-induced stress perturbations.

by the corresponding point on the loading curve)

maximum von Mises stress when 0.2

**5.2. Simulation of torsional contact** 

the newly advanced computer program.

et al., 2005).

**Figure 6.** The force-displacement curve in the fretting contact between similarly elastic materials

**Figure 7.** Contour lines of von Mises equivalent stress *VM H p* in the plane 2 *x* 0 , corresponding to point B on the loading path; position of maximum is denoted by the symbol "x"

**Figure 8.** Magnitude of maximum von Mises equivalent stress versus loading level (indicated by the corresponding point on the loading curve)

**Figure 9.** Dimensionless depth of maximum von Mises equivalent stress versus loading level (indicated by the corresponding point on the loading curve)

It is found through numerical simulation that shear tractions have a weak impact on the maximum von Mises stress when 0.2 . However, larger friction processes lead to a competition between the in-depth maximum, which fluctuates around its position in the frictionless contact (corresponding to point A in Fig. 9), and a second extremum developing on the surface, at the trailing edge of the contact. The latter can seriously diminish the loadcarrying capacity of the contact, especially in case of surfaces with poorly controlled surface quality, due to superimposition of microtopography-induced stress perturbations.

### **5.2. Simulation of torsional contact**

148 Numerical Simulation – From Theory to Industry

**Figure 6.** The force-displacement curve in the fretting contact between similarly elastic materials

**Figure 8.** Magnitude of maximum von Mises equivalent stress versus loading level (indicated by the

*p* in the plane 2 *x* 0 , corresponding to

**Figure 7.** Contour lines of von Mises equivalent stress *VM H*

corresponding point on the loading curve)

point B on the loading path; position of maximum is denoted by the symbol "x"

Program validation is subsequently performed in case of a spherical contact undergoing torsion applied simultaneously to a normal constant force. Based on the results of this author, (Mindlin, 1949), Johnson (Johnson, 1985) presents the closed-form solution for this contact scenario when a partial slip regime is established on the contact area (i.e. when the torsional moment *M*3 is smaller than a limiting value 3 lim 3 16 *M WaH* ). The solution is later reviewed and enhanced for the case of viscoelastic materials by Dintwa et al. (Dintwa et al., 2005).

A spherical indenter of radius *R mm* 18 is pressed with a normal force *W kN* 1 against an elastic half-space, having the same elastic parameters, and subsequently an increasing torsional moment *M M* 3 3 lim is applied, leading to shear tractions 1 *q* and 2 *q* depicted in Figs. 10a and b, respectively. The norm of these tractions 1 1 **q**(,) (,) (,) *ij q ij q ij* is compared in Fig. 11 with analytical results, using cylindrical coordinates.

Figure 12 depicts the dimensionless stick radius when the torsional moment *M*3 is varied in the domain corresponding to partial slip. In all investigated cases, a good agreement between analytical results and numerical predictions is found, giving further confidence in the newly advanced computer program.

Numerical Simulation of Slip-Stick Elastic Contact 151

The numerical program advanced in this study is subsequently used to simulate the fretting contact between dissimilarly elastic materials. To our best knowledge, an analytical solution to this contact process has not been advanced yet. The ball in the previous example is considered rigid, and the loading history is simulated using 700 *N* equal increments.

**Figure 13.** Shear tractions profiles in the fretting contact between dissimilarly elastic materials,

**Figure 14.** Shear tractions profiles in the fretting contact between dissimilarly elastic materials,

0.6

0.3

corresponding to different points on the loading path,

corresponding to different points on the loading path,

**5.3. Extension of results** 

**Figure 10.** Distribution of shear tractions in torsional contact, 0.1 , 3 3 lim *M M* 0.9

**Figure 11.** Profiles of radial shear tractions, 0.1

**Figure 12.** Dimensionless stick radius versus torsional moment

#### **5.3. Extension of results**

150 Numerical Simulation – From Theory to Industry

**Figure 10.** Distribution of shear tractions in torsional contact,

**Figure 11.** Profiles of radial shear tractions,

**Figure 12.** Dimensionless stick radius versus torsional moment

0.1 0.1 , 3 3 lim *M M* 0.9

The numerical program advanced in this study is subsequently used to simulate the fretting contact between dissimilarly elastic materials. To our best knowledge, an analytical solution to this contact process has not been advanced yet. The ball in the previous example is considered rigid, and the loading history is simulated using 700 *N* equal increments.

**Figure 13.** Shear tractions profiles in the fretting contact between dissimilarly elastic materials, corresponding to different points on the loading path, 0.3

**Figure 14.** Shear tractions profiles in the fretting contact between dissimilarly elastic materials, corresponding to different points on the loading path, 0.6

Numerical simulations predict that the investigated fretting contact exhibits a unique behaviour in the first two loading cycles, after which it stabilizes to a fixed path. Figures 13 and 14 suggest that the shear tractions profiles corresponding to the BD and FH trajectories no longer match, as in case of similarly elastic materials. However, states D and H overlap, beside a few perturbations induced by the former boundaries of the stick area. Presumably, these perturbations are related to the discretization error, and the D and H states can be considered as concurring. Subsequent oscillating loading is expected to lead to states following the same fixed path, as in case of similarly elastic materials. An analogous behaviour was observed by Wang et al. (Wang et al., 2010) when studying numerically the partial slip contact of elastic layered half-spaces.

Numerical Simulation of Slip-Stick Elastic Contact 153

of contact failure through yield inception or crack nucleation. Study of partial slip elasticplastic contact is anticipated for future contributions, by addition of the residual term,

*Department of Mechanics and Technologies, "Stefan cel Mare" University of Suceava, Romania* 

This paper was supported by the project "Progress and development through post-doctoral research and innovation in engineering and applied sciences – PRiDE - Contract no. POSDRU/89/1.5/S/57083", project co-funded from European Social Fund through Sectorial

Allwood, J. M. (2005). Survey and Performance Assessment of Solution Methods for Elastic Rough Contact Problems. *ASME Journal of Tribology*, Vol. 127, No. 1, pp. 10-23, ISSN:

Boussinesq, J. (1969). *Application des potentiels á l'etude de l'equilibre et du mouvement des solides* 

Cattaneo, C. (1938). Sul contatto di due corpi elastici: distribuzione locale degli sforzi, *Accademia Nazionale Lincei, Rendiconti, Ser. 6*, Vol. 27, pp. 342–348, 434–436, 474–478,

Cerruti, V. (1882). Ricerche intorno all'equilibrio de'corpi elastici isotropi. *Reale Accademia dei* 

Chen, W. W. & Wang Q. J. (2008). A Numerical Model for the Point Contact of Dissimilar Materials Considering Tangential Tractions. *Mechanics of Materials*, Vol. 40, No. 11, pp.

Dintwa, E., Zeebroeck, M. V., Tijskens, E., & Ramon, H. (2005). Torsion of Viscoelastic Spheres in Contact. *Granular Matter*, Vol. 7, No. 2, pp. 169–179, ISSN: 1434-5021. Gallego, L., Nélias, D., & Deyber, S. (2010). A Fast and Efficient Contact Algorithm for Fretting Problems Applied to Fretting Modes I, II and III, *Wear*, Vol. 258, No. 1-2, pp.

Goodman, L. E. (1962). Contact Stress Analysis of Normally Loaded Rough Spheres. *ASME*

Hertz, H., (1895). Uber die Beruhrung fester elasticher Korper. *Gesammelte Werke*, Bd. 1,

Hills, D. A. Nowell, D., & Sackfield, A. (1993). *Mechanics of Elastic Contacts*, Butterworth

*Journal of Applied Mechanics*, Vol. 29, No. 3, pp. 515-522, ISSN: 0021-8936.

Heinemann Ltd., ISBN: 978-0750605403, Oxford, United Kingdom.

*élastiques*, Reed. A. Blanchard, ISBN: 978-2853670715, Paris.

related to development of plastic strains.

Sergiu Spinu and Dumitru Amarandei

Operational Program Human Resources 2007-2013.

*Lincei*, *Ser. 3*, Vol. 13, pp. 81–122, Roma.

**Author details** 

**Acknowledgement** 

**7. References** 

0742-4787.

ISSN: 0392-7881.

936-948, ISSN: 0167-6636.

208-222, ISSN: 0043-1648.

Leipzig, pp. 155-173.
