2D and 3D finite element modelling of the ploughing part of friction M. BARGE*-P. GILLES**-M. DURSAPT*-J.-M. BERGHEAU* * Laboratoire de Tribologie et Dynamique des Systèmes 58, rue Jean Parot 42100 Saint-Etienne (France)
[email protected] [email protected] [email protected] ** FRAMATOME-ANP Division EE Tour Framatome 92084 Paris-La Défense (France)
[email protected] Understanding and quantification of ploughing resistance and associated deformation is of primary importance for materials undergoing abrasive wear. Most studies led to the view that abrasive wear could be expressed as the sum of three processes : cutting, cracking and ploughing. The aim of our study is to understand mechanical phenomenon induced by a ploughing process. Numerical simulation is of great interest for this study since it allows the variation of many parameters like penetration depth and indenter geometry. In this paper the finite element method is used to analyse the effect of the sliding of indenters on a flat surface. 2D and 3D simulations have been performed and have shown a great correlation between proposed models and our results. A first study of indentation is provided in this paper and its results are compared to Hertz and Meyer theory for respectively elastic and elastoplastic models. Then mechanical phenomena (strain, stresses and forces) involved in sliding are analysed to understand the ploughing process. ABSTRACT :
KEY WORDS : abrasive
1.
wear , ploughing , numerical simulation , friction model
Introduction
During Pressurized Water Reactor exploitation, wear appears on some primary and secondary circuits structures componants and may led to incidents. Most of these componants are streamlined tubular structures maintained in support with play and subjected to vibration induced by external flows. These structures come into contact with their supports, chocs and friction which occur develop wear called “impact/sliding wear”[1].
FE modelling of the ploughing part of friction
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Most studies dealing with wear converge on the classification into five fundamental wear modes : adhesion, tribochemical, fatigue, abrasive and erosive wear. Our study focuses on abrasive wear mode which can be expressed as the sum of three processes : cutting, cracking and ploughing [2]. From different experimental researches [3,4,5], it has been shown that ploughing wear was a rapidly steadied process, a groove is formed which has ridges on both sides and no wear debris is formed in single sliding. In this paper, the influence of indenter geometry and penetration depth was studied during a ploughing process. This test was modeled with a three-dimensional implicit finite element code: SYSTUS+ . ®
2.
Finite Element Model
2.1. Material In this study, the material is caracterised using large deformation and elastoplastic theory. In particular, the material used is a plastic von Mises criterion coupled with a Pragger kinematical strain hardening law. This material is representative of AISI 304L stainless steel used in several primary and secondary circuit mechanisms of PWRs. Its caracteristics for a 350°C temperature, found experimentaly, is a Young modulus E=172000 MPa, a Poisson ratio ν=0.3 and a power strain hardenning law σ=550 εt0.23 . The indenter is considered perfectly rigid. 2.2. Geometry and meshes The indenter is modeled as a sphere or cylinder in order to study its shape effect. Meshes are designed so that each meshing is very fine near the indenter (in order to resolve the contact conditions and allow for accurate contact area determination), but also are sufficiently large to approximate a semi-infinite solid [6][7]. The contact between indenter and the workpiece is assumed frictionless and loading is achieved by means of control of the indenter quasi-static displacement (dynamic effect are neglected) which is first pushed vertically into the workpiece for the study of indentation but also pushed horizontally for the study of ploughing.
Indentation mesh
Ploughing mesh for cylindrical indenter
Figure 1. Different meshes used
Ploughing mesh for spherical indenter
FE modelling of the ploughing part of friction
3.
3
Results and discussion
3.1. Indentation 3.1.1.
Elastic material
Hertzian contact is a classical application to test the trouble-free operation of a contact algorithm. A reference solution can be found analitically in literature dealing with this type of contact [8][9]. Hertz (1881) has shown that when a rigid sphere or cylinder is pressed against the surface of an elastic solid, the action of the force Fn, the radius aH of the contact, the radius R of the indenter and the elastic penetration δH could be linked together. Figure 2 shows the comparison between Hertz theory and the computed contact radius-penetration depth curves for each indenter. Continuous curves represent equation [1] and [2] representative of theory. It must be noticed that the steps obtained in FE results are the fact of a graduated coming into contact of the nodes.
aH2 R
For a spherical indenter :
δH =
[1]
For a cylindrical indenter :
a δH = H 2R
2
0,8
⎛ 1 ⎛ ⎜ + ln ⎜ 2 R ⎜ a ⎜ 3 ⎝ H ⎝
⎞ ⎞ ⎟⎟ ⎟ ⎟ ⎠ ⎠
[2]
0,7
contact radius (mm)
0,6
0,5
0,4
0,3
0,2
Hertz theory FE results with a cylindrical indenter FE results with a spherical indenter
0,1
0 0
0,01
0,02
0,03
0,04
0,05
0,06
0,07
0,08
0,09
0,1
Penetration depth (mm)
Figure 2. Verification of contact radius/penetration depth relationship Regarding the distorted material, we noticed that the “sphere profile” was closer to the indenter than the “cylinder profile”. Indeed we observed a high stress concentration rate closer to the indenter for the sphere than for the cylinder. In compensation, the cylinder gives rise to a larger stress field but with lower maximal values what give us some ideas of material behavior in accordance with the indenter shape.
FE modelling of the ploughing part of friction
3.1.2.
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Elasto-plastic material
The analysis of the indentation of elasto-plastic materials is based on the analysis of the Brinell hardness test [11][12]. In a study of indentation of metals by spheres, Meyer found that the hardness, H, followed a simple power law. O’Neill and then Tabor have completed this study to lead to equation [3]:
F a ⎞ ⎛ H = n2 = 2.8 κ ⎜ 0.2 H ⎟ π aH R ⎠ ⎝
m
[3]
This equation is the one we compared to our results (fig.3). Constants κ and m are determined from power strain hardening law. This theory has been checked valid for predominating plasticity what can explain differences observed at the beginning of indentation. Nevertheless for large deformation finite element results seem to converge on the theoretical curve given by Meyer’s law. The observed steps are due to progressive coming of the nodes into contact with the workpiece. mean pressure (Mpa)
900
800
700
600
500
400
300
theoretical law (eq.(7)) FE results for mean pressure
200
100
aH/R 0 0
0, 05
0, 1
0, 15
0, 2
0, 25
0, 3
0, 35
0, 4
Figure 3. Comparison of FE results with Meyer’s law Regarding workpiece deformed profile (fig.4) we have the corroboration of what we observed for an elastic material. In terms of stress, the localisation of an higher concentration near the indenter for the spherical indenter leads to a closer ridge. Furthermore the larger stress field observed for cylindrical indenter gives rise to a spreading out of the ridge. 0,1
0,05
0
-0,05
-0,1
Figure 4. Distorted material
Sphere profile Cylinder profile
FE modelling of the ploughing part of friction
3.1.3.
5
Conclusion on indentation:
The great correlation obtained between theory and finite element results contribute to the finite elements discretisation validation. In the elastoplastic case, few theoretical results are to our disposal but the study is more realistic and could be compared to experimental results. These results are positive since they have already disclosed the influence of indenter shape on strains and stresses. One of the most important aspect of abrasive wear is ridges forming since it is the first step of chip formation. The aim of the next simulation is to study ridges forming in sliding contact. 3.2. Ploughing 3.2.1.
General analysis of the results.
The general appearance is the same for each simulated test (whatever the indenter or penetration depth is). We will focus on each test specifities later, in this section we underline common phenomena observed for all tests. The distorted material is presented figure 5. Deformation amplitude (mm)
0, 04
S l i d i n g D i r e c t i o n
0, 03
0, 02
Unsteady State
0, 01
0 0
10
20
-‐0, 01
-‐0, 02
Steady State
-‐0, 03
-‐0, 04
-‐0, 05
D is to rted M aterial
Ind enter after s lid ing
ind enter after ind entatio n
Figure 5. Distorted material after sliding We can notice from the observation of the distorted material that sliding is caracterized by the appearance of ridges behind the indenter (unsteady state). Then the phenomenen leads to a steady state with a residual depth. Figure 6 give rise to an explanation of these phenomena. During a first step of the unsteady state, we can notice a stress concentration in front of the indenter what can lead to the view that material is pushed on the front. This can explain the growth of the frontal ridge with few effects on the rear. A second step corresponds to stabilisation of the stress concentration in front of the indenter but with a growth of stresses at the rear. This lead to the view that the material flows from the front to the rear of the indenter. In fact there is a crush of the frontal ridge leading to a material transfer from the front to the rear and so a rear ridge growth. Nevertheless ploughing compensate cruching of the frontal ridge leading to its stabilisation. The rear ridge formation has never been observed during experiments because of the difficulties to obtain a frictionless contact during ploughing. Nevertheless it could be usefull to plough a well lubricated surface to observe the rear ridge formation.
FE modelling of the ploughing part of friction Frontal ridge growth
6
0,04
0,03
0,02
0,01
0 0
1
2
3
4
0
1
2
3
4
5
6
7
8
-0,01
-0,02
-0,03
-0,04
-0,05
0,04
0,03
0,02
0,01
0 5
6
7
8
-0,01
-0,02
-0,03
Rear ridge growth
-0,04
-0,05
Figure 6. Ridges growth 3.2.2.
Penetration depth influence
We have noticed a growth of the unsteady state (in terms of amplitude and length) with penetration depth. This can be explained from the growth of displaced material which leads to a higher tranfer of material from the front to the rear of the indenter. This can explain the growth of frontal ridge too (fig.7). Concerning the steady state, the growth of the residual depth of the groove can be explained by the growth of plastification with penetration depth. 3.2.3.
Indenter shape influence
If we focus on frontal ridge evolution with sliding length (fig.7), we can underline the indenter shape influence. Indeed we can notice a stabilisation of frontal ridge with the sphere due to the capability of the material to flow on the sides of the indenter. In the cylinder case this flow can not take place leading to a constant growth of the frontal ridge with sliding length. 0,04
Frontal ridge amplitude (mm)
Frontal ridge amplitude (mm)
0,04
0,035
0,03
0,025
0,02
0,035
0,03
0,025
0,02
0 , 0 15
0 , 0 15
0,01
0,01
Cylinder
0,005
S l i d i n g l e n g t h ( m m )
0 0
2
4
6
8
Sphere
0,005
10
s l i d i n g l e n g t h ( m m )
0 0
1
2
3
4
5
Figure 7. Influence of penetration depth and indenter shape on frontal ridge growth We recover this shape effect in terms of distorted material. Indeed we can notice on figure 8 a drop of unsteady state phase length due to the capability of side flow
FE modelling of the ploughing part of friction
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Deformation amplitude (mm)
but also a growth of the groove residual depth for the spherical indenter in comparison with the cylinder. This can be explained from the higher stress concentration under the sphere (what had been mentioned for indentation case) leading to a more important plastification and so to a higher groove residual depth. 0,04
0,03
side ridge
0,02
0,01
0
-0,01
-0,02
Cylinder -0,03
Sphere -0,04
-0,05
Figure 8. Indenter shape effect on the distorted material 3.2.4.
Comparison to Bowden and Tabor model
Most studies have led to the view that friction could be expressed as the sum of two terms[12]: the force to shear interfacial junctions plus the force to plough the asperities on the harder surface through the softer surface. The ploughing ter may be estimated reliably from Bowden and Tabor’s model. It consist in estimating the ploughing friction coefficient by the ratio of tangential projected contact area with normal projected contact area. So we have compared this model to the computed ratio of horizontal force with vertical force which give for a frictionless contact the ploughing part of friction coefficient as shown on figure 9 and table 1. Indenter Radius (mm) 5 CYLINDER Indentation Depth 0,05mm 0,1 mm Fx 399,86N/mm 588,42 N/mm Fy SYSTUS+ 29,73N/mm 60,04 N/mm 0,074 0,102 µ lab=Fy/Fx Bowden and Tabor Model 0,071 0,100
SPHERE 0,05 mm 0,1 mm 508,98N 1020,30 N 31,09N 85,68 N 0,061 0,084 0,060 0,085
Table 1. FE results comparison to Bowden and Tabor’s model
Frear
Even if the theory neglected the area in contact with the pile up and the rear of the indenter (fig.9), our results are very close to model’s one. In fact it seems that pile up contact area and rear contact area counterbalance their Fpile up actions and that this model given for a sphere could be generalized to cylindrical indenter.
Fmodel
Figure 9. Diagram of the contact
FE modelling of the ploughing part of friction
4.
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Conclusion
In this work, we have been able to show that the finite element technique may effectively be used to understand and characterize material behavior during indentation or ploughing. Numerical simulation is an helpfull tool which give informations difficult to obtain experimentally (contact area, stresses and strains fields...) and permit easy variation of many parameters. Results presented here have been compared successfully with theory and have shown the influence of penetration depth or indenter shape. They opened up new ways of research particulary on friction caracterization and abrasion study. 5.
Acknoledgements
EE Framatome-ANP division and P. Saillard (ESI Group) are acknowleged for many usefull discussion involving this work. 6.
References
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[2]
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[3]
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[4]
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[5]
KATO K. , HOKKIRIGAWA K. – “An experimental and theoretical investigation of ploughing, cutting and wedge formation during abrasive wear” – International Journal of Tribology. 1988 , vol. 21 , pp. 51-57.
[6]
LAURSEN T.A. , SIMO J.C. – “A study of micoindentation using finite elements” – Journal of Material Research. 1992 , vol. 7 , N° 3 , pp. 618-626.
[7]
BUCAILLE J.L. , FELDER E. , HOCHSTETTER G. – “Mechanical analysis of the scratch test on elastic perfectly plastic materials with the three dimensionnal finite element modeling” – Wear. 2001 , vol. 249 , pp. 422-432.
[8]
ROARK R. , YOUNG W. – “ Formulas for stress and strain” – Ed. Mc Graw & Hill. 1975 , 624p.
[9]
TIMOSHENKO S.P. , GOODIER J.N. – “Theory of elasticity” – Ed. Mc Graw & Hill. 1970 , 567p.
[10] MATTHEWS J.R. – “Indentation hardness and hot pressing” – Acta metallurgica. 1980 , vol.28, pp. 311-318. [11] ADLER T.A. – “Elastic-plastic indentation of hard, brittle materials with spherical indenters” – Journal of American Ceramic Society. 1994 , Vol.77 , N°12 , pp. 3177-85.
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