Particle detachment in elastically loaded contacts: a continuum mechanics based model

Tribology Research: From Model Experiment to Industrial Problem G. Dahnaz et al. (Editors) 9 2001 Elsevier Science B.V. All rights reserved.

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P a r t i c l e d e t a c h m e n t in elastically l o a d e d c o n t a c t s 9 a c o n t i n u u m m e c h a n i c s b a s e d m o d e l D. Clair a, L. Baillet a, Y. Berthier a and M. Zbinden b a Laboratoire de M6canique des Contacts, UMR INSA-CNRS 5514, Institut Europ6en de Tribologie, Institut National des Sciences Appliqu6es de Lyon, 20 Av. A. Einstein 69621 Villeurbanne, Cedex, FR. b Laboratoire Usure et Tribologie, Electricit6 de France (EdF), 77250 Moret sur Loing, FR. ABSTRACT This paper presents a model for the prediction of particle detachment in elastically loaded contacts. Its originality relies on a coupling between the fatigue damage of the material and its elastic behavior. A 2D finite element model enables the calculation of the stress history at every point of two contacting bodies during a loading cycle, i. e. a fretting cycle. Multiaxial fatigue criteria are then used to predict the loci and number of cycles needed for the first appearance of fatigue damage. The elastic behavior of the material is locally modified and a new finite elements calculation is performed on the damaged structure taking into account the possible opening and closure of defects. This elementary loop is repeated and the micro-propagation of defects is predicted. This model predicts the degradation regimes transition (cracking to particle detachment) encountered in fretting conditions. 1. INTRODUCTION Fretting is a common type of tribological loading which has been extensively studied from theoretical and experimental points of view [1-2]. The experimental work based on fretting maps [3-5] has led to the definition of fretting regimes implying different type of degradation of the contacting materials. One of the main conclusions of these studies is that depending on sticking, partial slip or gross slip of the contact, the material degradation is respectively no damage, crack nucleation or material removal after particle detachment. Nevertheless, the transitions between these degradation types are continuous and real life fretting contacts may exhibit both crack initiation and particle detachment. This continuity is not handled by theoretical models which have been developed separately to predict crack nucleation [6] or to quantify material removal [7-8]. The tribological behaviour or "response" of materials to fretting loading is complex and the associated physical mechanisms may be numerous (shear band formation, dynamic recrystallization, phase transition,...) [9] and presently not completely understood. Therefore an effective quantitative modelling is not reachable yet. Nevertheless solid mechanics based models have proved to be quite effective in predicting some aspects of tribological degradation. For "heavily loaded contacts" involving large plastic deformation, Merwin and Johnson [10],

Kapoor and Johnson [11], and several other authors have developed models including cyclic plasticity that can predict shakedown or ratchetting regime for rolling and/or sliding hertzian contacts. For "lightly loaded contacts" where plasticity may be confined to very small areas leading to fatigue phenomena, recent studies have been carried out using multiaxial fatigue criteria [6]. The purpose of this paper is to present a model capable of predicting location, size and evolution of tribological degradations of bodies subjected to contact loading. This model, developed for "lightly loaded contacts", relies on a coupling between a finite element model for the calculation of elastic stresses and strains and a multiaxial fatigue criterion for the fatigue damage assessment. The fatigue damage is used for estimating the material degradation. The elastic behaviour of the damaged material is locally modified at the micro-defects and a new finite element calculation is performed on the damaged structure taking into account the possible "opening" and "closure" of defects. This elementary loop is repeated and the micro-propagation of defects is predicted. The originality of the present model is to consider simultaneously the degradation of both contacting bodies without assumptions for the contact pressure distribution. Investigations into the fretting wear mechanism of the contact of a rod in a conforming hole are presented with special

608 emphasis on the influence of fretting regimes. The results are shown to be in good agreement with classical results of previous experimental studies. 2. MODEL

2. 1. Principles and assumptions The present model aims at evaluating the capabilities and the limits of a new approach based on a coupling between the fatigue damage of the material and its elastic behaviour in predicting tribological degradation induced by fretting loading. This model applies to "lightly loaded contacts" which require a large number of cycles for the first appearance of tribological degradation (typically greater than 104). It is also limited to small scale defects which justifies the assumption that no instability rupture of the material exists and covers the physics corresponding to nucleation and micropropagation of defects. The nucleation is determined by the fatigue calculation and the micro-propagation of defects is associated with the stress redistribution caused by the nucleation. As the goal of this first model is to test the feasibility and validity of the undertaken approach, the assumption of isotropic material is chosen. The initial behaviour of the virgin constitutive material of contacting bodies is considered as isotropic linear elastic. The material affected by fatigue behaves as an isotropic non-linear elastic material. The nonlinearity is linked with the activation of fatigue micro-defects. The material data required are the elastic properties (Young's modulus E and Poisson's ratio v) and fatigue S-N curves. With the fatigue criterion used, the fatigue data needed are ~ - I ( N ) , X-l(N) and o0 (N), corresponding respectively to a reversed tensile test, a reversed torsion test and a zero to maximum tensile test.

2.2 Finite Element Method The estimation of fatigue damage requires computing the stress history at every integration point of the mesh of contacting bodies during a fretting cycle. As fatigue damage is very sensitive to the loading path, this step of the calculation process has to be precise which implies an accurate solution of normal contact and friction. Furthermore, as degradation of both contacting bodies is of interest, contact calculation between deformable bodies is required. This problem is affordable using an

appropriate finite element method avoiding a priori assumptions for the pressure field. The choice of a contact algorithm is relevant for a precise contact solution. Lagrange multipliers based methods [1213] appear to be more efffective as far as contact mechanics application are concerned because they allow the application of exact constraints between surfaces of contacting bodies. Moreover, contrary to penalisation methods [14], they do not include extrinsic user defined parameters which can lead to spurious results. For example, using penalisation for fretting-fatigue analysis, Petiot et al. [6] have reported "a spring effect when sliding starts at all nodes" which they attribute to "a numerical origin". The model presented here has been implemented using the multipurpose finite element code ABAQUS/Standard that allows both Lagrange multipliers contact solution and integration of user's programs [ 15].

2.3 Fatigue criterion The choice of a fatigue criterion is determined by the nature of the stress tensor within contact areas which is both multiaxial and non-proportional during a fretting cycle. Multiaxial fatigue criteria can be split up into two categories: global criteria and critical plane criteria. Critical plane criteria assume that the fatigue behaviour of the material depends on the direction of greatest damage. Global criteria are based either on an invariant of the stress tensor as Sines [16] or Crossland [17] criteria or upon the integration of a fatigue damage indicator calculated in every plane of space as Fogue-Bahuaud criterion [18]. As the model presented here is isotropic, the Fogue-Bahuaud criterion which has proved to be effective for non-proportional loading is chosen for the fatigue life assessment.The formulation and validation of the Fogue-Bahuaud criterion are detailed in [ 18-19]. It is presented here in its finite fatigue lives form, which enables one to predict the number of identical multiaxial cycles after which the nucleation of a fatigue induced defect occurs. The Fogue-Bahuaud criterion computation is divided into three main steps. First a fatigue indicator Eh is calculated over a large number of physical planes of unit normal vector h describing spatial plane orientation around the point n where the fatigue damage has to be determined : Eh = a(N)Xha +b(N)crhha + d(N)o'hhm

~_~(N)

(1)

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where: Tha is the shear stress amplitude, 13hha is the normal stress amplitude, 13hhm is the mean normal stress during the cycle, a(N), b(N) and d(N) are criterion parameters calculated using 13-1(N), x-1 (N) and 130(N). Then the fatigue function is determined as the RIMS value of the fatigue indicator calculated at the point n 9

EFB=

I 1 IE2hdS -Ss

(2)

Finally the number of cycles N for the nucleation of a fatigue induced defect is determined by solving the implicit equation" (3)

The fatigue damage dc corresponding to a cycle of loading is simply determined as the inverse of the number of cycle N for the nucleation of a fatigue induced defect" 1

dc = - N

(4)

2.4. Damaged material b e h a v i o u r For the isotropic model defined here the fatigue damage is a scalar traducing the remaining life of the material: d-0 for the virgin material and d-1 for the totally damaged material. There is no physical link between the fatigue damage of the material and the evolution of its elastic properties unless the fatigue damage equals one which means the appearance of a micro-defect and a sudden fall of elastic properties to zero. Translated in terms of isotropic elastic properties, the behaviour of the material remains unchanged until fatigue damage reaches one. The elastic properties of the material are then equal to zero providing that the micro-defects are active which is defined by a condition of activation. The damaged material behaviour can be expressed using the following equation : Dd = (1- H ( d - 1))D

For the case of 2D plane strain the stress/strain relation is expressed as:

o=D~orlo22

=

[.1312

where S is the area of the sphere of unit radius used for the calculation Of Eh.

EFB =1

where D is the elastic stiffness matrix of the virgin material, Do is the current elastic stiffness matrix, d the fatigue damage at the material point, H is defined by, H(x) = 1 if x>0, H(x)=0 if x<0.

(5)

L 0

k+21a 01l~22~ (6) 0

and t333 = 13(1311 + 1322)

~t_][)'12J (6')

where 13 is the stress tensor, g is the strain tensor, 13E E )~ = (1 + u)(1- 213) and g = 2(1 + u) As the applied loading are cyclic the material behaviour model has to take into account the possible "opening" or "closing" of micro-defects. This phenomenon is accounted for through an unilateral damage condition or condition of activation [20]. The condition of activation can be written in terms of either stress or strain. Taking into account anisotropic material behaviour, Chaboche [21] has def'med a strain based condition. In the isotropic case presented here, it appears more judicious to use a stress based condition and an invariant of the stress tensor for the formulation of the condition of activation. The chosen formulation is the following: Dd = D, if p _>0, Dd = (1- H ( d - 1))D if p < 0 (7) where p is the hydrostatic pressure, p =--~1 Tr(13). The choice of the hydrostatic pressure as the parameter for damage activation is motivated by the fact that the value of hydrostatic pressure reflects the global state of tension or compression of the material. The obtained material law is consequently

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non-linear and the symmetry condition of the elastic tensor is respected. 2.5. Coupling procedure Figure 1 presents the flow chart of the coupling procedure. The finite element calculation is carried out first in order to obtain the stress history at every point of the mesh during the cycle of loading. The computed stress history is used to calculate the fatigue damage. Then the points reaching a full damage state (din= 1) in the minimum number Nm~nof cycles are detected. This number of cycle is added to the total number of applied cycles Nt leading to the updated value of Nt: Nt ~-- Nt + Nmln (8) The fatigue damage at every point n of the structure is cumulated using a Miner rule and is given by: dtn +-- dtn + Nmindcn

(9)

where dtn is the total damage, den the current damage at the point n calculated for one cycle. Finally, the totally damaged points are given the damaged material behaviour. Then a new loop is started by a finite element calculation on the modified structure. This elementary loop is repeated as long as the applied number of cycles Nt is lower than a number Ntest fixed by the user or if the maximum of coupling loops allowed is reached. S T A R T (dtn=0, Nt=0, Nt~st) [--~ F . E . M c~leulation Fatigue calculation

Stress tustory [on (t)] at every point n for a cycle of loachng Elementary damage den at every point n

D e t e r m i n a t i o n of Nm,,, Nmm = mm(Nmm n ) with Nmm n -

1-dtn dcn

o

O ,.m

U p d a t i i g of Nt U p d a t i n g of

o

dtn

Nt r Nt +Nmm dtn ~-- dtn+ dcnNmm at every point n

o

i , n U p d a t i n g of material properties

Figure 1. Flow chart of the coupling procedure.

3. APPLICATION TO THE FRETTING OF ROD IN A CONFORMING HOLE 3. 1. Presentation The goal of this study is to estimate the possible transitions of degradation mechanisms induced during fretting of a cylindrical rod in a conforming hole of an annular support (fig. 2). This case study is interesting because this type of contact geometry is very common among power plants components, eg. controls rods assembly or steam generator, subjected to fretting wear [22]. As the aim of this paper is to present the model and to evaluate its capability, only several examples of possible calculation are detailed. The influence of the mesh will not be studied here. A complete investigation will be presented separately later. 3. 2. Finite element model The geometry and material properties for the finite element model are presented in fig. 2 and fig. 3. The mesh used is made of 5315 four-nodes isoparametric plane strain elements having four integration points. For reason of computational cost, the presented model is only applied to the part of the bodies surrounding the contacting surfaces. This represents 1570 elements (6280 integration points). The size of the mesh in the contact area is about 15gm ensuring thirty elements in contact. The applied loading are displacements Ux and Uy of the centre Or of the rod. The normal displacement Uy is kept constant for all calculations. The external boundary of the support is considered as embedded (fig. 2). In order to describe accurately the stress history, the maximum magnitude of a load increment against the maximum displacement is fixed at 10% during normal loading and at 1% during tangential loading. This point is extremely important because of non-linearities induced by both contact and damaged material properties. The fatigue S-N curves used for the present calculation correspond to a low carbon steel. The fatigue limit at 2.106 cycles are the following: ~-1 = 252MPa, z-1 =183MPa and G0 =397MPa. The friction between the rod and the support is modelled with a Coulomb law. The local coefficient of friction g is given a constant value (f=0.8).The typical CPU time for an elementary coupling loop is about 30 minutes on a HP (180 MHz) workstation.

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rod/support contact and if corresponding damages can be predicted by the present model six different fretting amplitudes 5y were studied. The results gathered in table 1 are analysed in terms of defect nucleation (after the first fatigue calculation) and micro-propagation (after twelve coupling loops).

In order to be used for the fatigue calculation, the stress history due to the loading cycle has to be representative of the stabilised behaviour. In the case of fretting the normal displacement Uy is applied first and is followed by a reciprocating tangential displacement u• of amplitude Sxalong Ox. The starting point of loading corresponds to the position where the rod and the support are tangent. The initial stress state before the first tangential cycle is applied, is solely due to the normal loading (Ux = 2.5 ~m, Fn-73N per mm axial length). It is therefore necessary to apply a first cycle of reciprocating displacement before getting a stress history corresponding to a stabilised fretting cycle. 3. 3 Results

As previously stated, one of the main conclusions of studies concerning fretting is that depending on sticking, partial slip or gross slip of the contact, the material response is respectively no damage, crack nucleation or particle detachment. In order to know if such behaviour exists for the

Defect nucleation The maximum of damage is always located at the surface of contacting bodies. This can be related to the relatively high value of friction used for the calculations implying maximum stresses near the surface of contacting bodies. At very low fretting amplitudes ( 8y < 1.3gm ), as only a small part of the contact area is in sliding conditions no damage occurs. Increasing the tangential displacement amplitude to 1.5gm leads to the appearance of two damaged zone localised at the limit of the contact zone (fig. 4). If fretting amplitude is further increased the width of the damaged zone increases (fig. 5 and 6). By comparing figures 4, 5 and 6, it appears clearly that the position of the maximum damage moves from the frontier of the contact zone obtained for a zero value of the tangential displacement toward the centre of the contact zone for increasing values of the tangential displacement amplitude. A comparison of the damaged zone of the two bodies shows that the maximum value of damage is lower in the rod than in the support and that the width of the damaged zone of the rod is also smaller than in the support for every displacement amplitude tested. Figure 7 represents the number of cycles for the initial nucleation of the micro-defects as a function of the tangential displacement amplitude 8y ( 0gin < 8y < 10gm ). The vertical asymptote defines the minimum value of 5y for the onset of degradation. The horizontal asymptote represents the limit of the number of cycles for the onset of degradation in the gross slip regime. This diagram shows clearly that the number of cycles for the first appearance of defects decreases drastically as 5y exceeds the value corresponding to the transition between partial slip and gross slip regimes. Micro-propagation The way the nucleated defects propagate within the support varies considerably with the value of the imposed tangential displacement 5y. At very low

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amplitudes of the tangential displacement (By _<2gm), the damaged zone grows essentially toward the depth of the support. In this range, the rate of propagation of the defects toward the depth of the support increases with By. A damage propagation depth of about 50gm is obtained in 2.1 105 cycles with 8 y - 1.5gm whereas it requires only 4.10 4 cycles with 8y =2gm. At higher amplitudes of the tangential displacement (By > 2gm), as the contact is in gross slip, the damage propagates diffusely. In these cases the damaged zones do not present any special orientation. The propagation of defects within the rod is somewhat different. At very low amplitudes of the tangential displacement (By _<2gm ), the damaged zone tends to grow toward the depth of the bulk but at a much smaller propagation rate than in the support. At higher amplitudes of the tangential displacement (By > 2gm), the evolution of the damaged zone is quite similar to the one observed within the support.

b. After 12 coupling loops. Figure 4. Damage distribution ( 8y =l.5gm)

b. After 12 coupling loops. Figure 6. Damage distribution ( 8y =8gm)

613

. . . . . ~y ................. contact'reg;ime ....... Number of Cycles for first Number of Cyciesafter 12 Degradation regime loops ..................................... (gm) . . . . . . . . . . . . . . . . . . . . .nucleation . . . . . . . . . . . .of . . defects . . . . . . . . . . . . . . . . . . . . . .coupling ... 0.3 Partial slip No damage No damage No degradation 1.5 Partial slip 2.7 105 4.8 105 Single crack 2 Gross slip 1.1 105 1.5 105 Single crack 3 Gross slip 7.5 104 1.0 105 Multi-cracks 5 Gross slip 5.104 6.0 104 Multi-cracks or particle detachment 8 Gross slip 3.6 104 4.5 104 Multi-cracks or particle detachment .... ~ .........Tai~ie i. c'aiculations results: number of CYcies,contact and degradation regimes. ~....

Figure 7. Number of fretting cycle for the first appearance of defects as a function 5y. 4. DISCUSSION The aim of the presented model was to extend the ability of multiaxial fatigue criterion to the study of nucleation and micro-propagation of fatigue induced defects. To be as close as possible to the real case the model considers both contacting bodies as deformable and sensitive to fatigue damage. The analysis of defect nucleation shows that the width of the damaged zone, the number of cycles needed for the appearance of defects and the location of the maximum of damage are strongly dependant on the applied tangential amplitude and, on the contact regime. The micro-propagation also provides interesting results. It shows how, by increasing the tangential displacement, damage growth turns from a very oriented nature to a more diffused one. When the micro-propagation is oriented, it leads to the formation of a crack of sufficient size to possibly enter a propagation phase in a fracture mechanics sense. When the micro-propagation is diffused several small cracks may be formed leading to possible particle detachment. The establishment of one of these two regimes can be related to the

gradient of damage among the damaged zone. If the gradient is great which means that the damage is localised then the propagation will be strongly oriented by the redistribution of stresses due to the apparition of the micro-defects. On the other hand, if the damage is quite homogeneous among the damaged zone, the effect of stresses redistribution is averaged and the damage progress becomes diffused. This behaviour fits correctly the degradation regimes of a fretting contact ranging from crack nucleation to particle detachment. For two possible reasons, the calculations reveal the same kind of behaviour as in classical fretting studies performed with a cylinder on plane or sphere on plane contact geometry. Firstly the radius difference between the rod and the support is quite important and the applied load is relatively low. As already stated, this lead to a normal contact configuration which can be correctly approximated by a hertzian approach [22-23]. Secondly, the applied tangential displacement amplitudes are small and the inherent geometrical effect is negligible. 5. CONCLUSIONS Calculations presented here show only some examples of the capability of the model in predicting the onset of tribological induced degradations. Further calculations including studies of the influence of other operating parameters such as normal load and friction coefficient have yet to be performed in order to complete the analysis and establish theoretical maps relating the contact operating conditions with the induced tribological degradations. The implementation of an anisotropic model using critical plane fatigue criteria is under development. This new model will enable a more precise description of tribological degradation and will also provide initial data for fracture mechanics

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analysis. Modelling of tribological degradation is a challenging task. The present model provides a phenomenological description of fretting wear mechanism. Further improvements are needed to enable an accurate description of the material behaviour in the particle detachment regime associated to plastic deformation and other physical modification of the material [9]. REFERENCES

1. Waterhouse, R.B., 1972, "Fretting Corrosion", Pergamon, Oxford. 2. Hurricks, P.L., 1972, "The fretting wear of mild steel from room temperature to 200~ '', Wear, Vol. 19 pp.207-229. 3. Vingsbo, O. and Soderberg, D., 1988, "On fretting maps", Wear, Vol. 126 pp. 131-137. 4. Berthier, Y., Dubourg, M.-C., Godet, M. and Vincent, L., 1990, "Wear data what can be made of it ? Simulation tuning", Proc of the 18th Leeds-Lyon Symposium on Tribology, D. Dowson et al. (Editors), Lyon, pp. 161-173. 5. Vincent, L., Berthier, Y. and Dubourg, M-C., 1992, "Mechanics and material in fretting", Wear, Vol. 153, pp. 135-148. 6. Petiot, C., Vincent, L., Dang-Van, K., Maouche, N., Foulquier, J. and Joumet, B., 1995, "An analysis of fretting fatigue failure combined with numerical calculations to predict crack nucleation", Wear, Vol. 181-183, pp. 101-111. 7. Mohrbaker, H., Blanpain, B., Celis, J.P., Roos, J. R., 1995, "The influence of humidity on the fretting behaviour of PVD TiN coatings", Wear, Vol. 180, pp. 43-52. 8. Fouvry, S., 1997, "Etude quantitative des d6gradations en fretting", These de doctorat, Ecole Centrale de LYON, LYON, France. 9. Rigney, D. A., 1997, "Comments on the sliding wear of metals", Tribology International, Vol. 30, n~ pp. 361-367. 10. Merwin, J.E., Johnson, K.L., 1963, "An analysis of plastic deformation in rolling contact", Proc. I. Mech. I., Vol. 177, pp. 676-685. 11. Kapoor, A. and Johnson, K.L., 1995 "Plastic ratchetting as a mechanism of erosive wear", Wear, Vol. 186-187 pp. 86-91. 12. Hughes, TH. J. R., Taylor, R. L., Sackman ,J. L., Cumier, A., Kanoknukulchai, W., 1976, "A finite element method for a class of contact-impact problems". Computer Methods in Applied Mechanics and Engineering, 8,249-276.

13. Carpenter, N.J, Taylor, R.L and Katou, M.G., 1991, "Lagrange constraints for transient finite element surface contact", Int. Journal for numerical Methods in Engineering, 32, 103-128. 14. Wriggers, P., T. Vu Van, T. and Stein, E., 1990, "Finite element formulation of large deformation impact-contact problems with friction". Computers and Structures, 37, 319-331. 15. Abaqus, 1998, "User's and Theory manuals", Hibbit, Karlsson & Sorensen, Inc., Pawtuckett. 16. Sines, G., 1955, "Failure of materials under combined repeated stresses with superimposed static stress", (N.A.C.A), Washington, Tech.Note 3495, p.69. 17. Crossland, B., 1956, "Effect of large hydrostatic pressure on the torsional fatigue strength of an alloy steel", Proc. of the Int. Conf. On fatigue of metals. Inst. Mech. Eng., London, pp. 138-149. 18. Fogue, M., 1987, "Crit6re de fatigue ~ longue dur6e de vie pour des 6tats multiaxiaux de contraintes sinusoYdale en phase et hors phase", Th6se de doctorat, INSA LYON, LYON, France. 19. Robert, J. L., Fogue, M., and Bahuaud, J., 1994, "Fatigue life prediction under periodical or random multiaxial stresses state". Automation in Fatigue and Fracture : Testing and analysis, ASTM STP 1231, Amzallag, C. (ed.), ASTM, Philadelphia, pp.369-387. 20. Lemaitre, J., 1996, "A course on damage mechanics", Springer-Verlag, Berlin Heidelberg. 21. Chaboche, J.L., 1992, "Une nouvelle condition unilat6rale pour d6crire le comportement des mat6riaux avec dommage anisotrope", Cr. A c. des Sc. Paris, t.314, Serie II, pp. 1395-1401. 22. Ko, P.L., 1985, "The significance of shear and normal force components on tube wear due to fretting and periodic impacting", Wear, Vol. 106 pp. 261-281. 23. Persson, P., 1964, "On the stress distribution of cylindrical elastic bodies in contact", Dissertation, Chalmers Tek. Hogskola, Goteborg. 24. Johnson, K. L., 1985, "Contact Mechanics", Cambridge University Press, London. ACKNOWLEDGEMENTS

This work is supported by Electricit6 de France (E.d.F). The authors are grateful to Pak L. Ko (NRCC) for his comments of the manuscript. Dr Jean-Louis Robert and Bastien Weber (Laboratory of Solid Mechanics of INSA LYON) are also gratefully acknowledged.