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Temperature effects on adhesive wear in dry sliding contacts
Wear 268 (2010) 968–975
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Temperature effects on adhesive wear in dry sliding contacts A. Gåård ∗ , N. Hallbäck, P. Krakhmalev, J. Bergström Department of Mechanical and Materials Engineering, Karlstad University, SE-651 88 Karlstad, Sweden
a r t i c l e
i n f o
Article history: Received 3 April 2009 Received in revised form 25 November 2009 Accepted 7 December 2009 Available online 16 December 2009 Friction Adhesion Sliding wear Thermal effects Galling
a b s t r a c t In many metal forming operations, frictional heating occurs at the interface due to a sliding contact. Generally, the controlling wear mechanism in the tribological system is attributed to adhesive wear. Understanding of the influence of temperature on wear mechanisms is needed for the development of materials and for optimization of the forming process. Dry sliding tests were conducted at different sliding velocities and, hence, different surface temperatures due to frictional heating. A significant influence of temperature on adhesion was observed and increasing temperature led to a higher tendency for initiation of severe adhesive wear. The results were compared to atomic force microscopy force curve measurements, which show that the adhesive force increases with temperature. A very good agreement between the results was observed, which suggests that the controlling mechanism for the observations in the present work is temperature-induced high adhesion. © 2009 Elsevier B.V. All rights reserved.
1. Introduction A sliding contact between metal surfaces involves several tribological phenomena, e.g. friction and frictional heating, at the interface. The interaction between the surfaces occurs at the real area of contact, which is influenced by normal load and mechanical properties. If the surfaces are not separated, for instance by a lubricant, adhesion occurs across the interface, which influences both friction and wear. Elevated temperature has several effects on a tribosystem. If a shear displacement is added to a static normal load, the real area of contact, generally, increases due to junction growth by plastic flow. If the interface is stronger than the cohesive strength of the materials, the size of the junctions are determined by the ductility. For metals, the yield strength decreases and ductility increases at high temperatures and, hence, both the real area of contact and adhesion increase under loading [1,2]. Several authors have reported higher adhesive wear rates at elevated temperatures, particularly for open tribosystems [3,4] where either one, or both, of the surfaces in contact are always new. Sheet metal forming (SMF) is a typical application, corresponding to an open tribosystem, in which the controlling wear mechanism, generally, is attributed to adhesive wear [5–8]. During sliding, sheet material is transferred to the tools surface which, subsequently, leads to scratching of the sheets. It is well known that forming of high-strength steel sheets are more demanding from a wear point of view and material transfer occurs more rapidly. The latter is often
∗ Corresponding author. Tel.: +46 0 54 7001262; fax: +46 0 54 7001449. E-mail address: [email protected] (A. Gåård). 0043-1648/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2009.12.007
attributed to increased contact pressures due to higher hardness of the sheet. However, the consequences of high pressures on wear in SMF are not fully understood. The amount of adhesive wear is generally proportional to the normal load and inversely proportional to the hardness of the softer material, i.e. the real area of contact for a plastic contact [1]. In analogue to SMF, introducing of higher-strength sheets do increase the contact pressure, but, at the same time the hardness of the sheet is increased as well. Hence, it is not obvious that the higher amount of wear observed for forming of high-strength sheets are due to higher contact pressure. Another possible origin is frictional heating which depends on the contact stresses. Possibly, the elevated temperatures lead to softening and growth of the real area of contact resulting in higher adhesion. Nonetheless, in addition to temperature effects on mechanical properties, other phenomena may be of importance as temperature increases. Recent results using atomic force microscopy (AFM), indicate that the adhesive force depends on temperature, with increasing adhesion as temperature rises [9,10]. In [9], the pull-off force, which represents the interfacial adhesion, was measured for several metallic materials against Si using atomic force microscopy. For all investigated materials, adhesion started to rise at approximately 125–150 ◦ C. Hence, the implication is that temperature-induced high adhesion may be an additional mechanism to the increased wear observed at elevated temperatures. The objective of this study is to investigate the effect of frictional heating on the adhesive wear mechanism. Tribological tests were conducted at dry sliding test conditions at different contact pressures and velocities using a Slider-On-Flat-Surface tribometer (SOFS). The temperatures during the sliding were calculated using FE-analysis.
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Fig. 1. (a) The test principle of the disc sliding against the plate. (b) Overview of the SOFS test equipment and (c) close-up of the disc holder with load actuator C and normal and friction force gauges A and B. Table 1 The velocities used in the SOFS experiment for the different normal loads and calculated maximum sheet surface temperatures. P [N]
v1 [mm/s]
v2 [mm/s]
v3 [mm/s]
v4 [mm/s]
v5 [mm/s]
250 500 Tmax [◦ C]
13 8 46
27 17 72
82 50 160
137 83 238
281 167 409
Table 2 Flow stress versus effective strain for the carbon steel sheet. p
[MPa]
εeff
1175 1234 1326 1344
0 0.0042 0.0138 0.0237
2. Experimental 2.1. Tribological evaluation Tribological tests were conducted using the SOFS tribometer, Fig. 1, which is described in detail in [6]. A disc-shaped tool with radii 25 and 5 mm was forced against a sheet material at a constant normal load (P) of 250 and 500 N and slid for a total distance of 900 mm. Tests were conducted at different sliding velocities (v1 to v5 ) to obtain increasing surface temperatures, generated by frictional heating. The velocities, as seen in Table 1, were selected based on FE-analysis to achieve similar surface temperatures for each of the applied loads. The maximum calculated temperatures are shown in Table 1. The tests were repeated twice for each velocity. The tool material was an AISI D2 tool steel (wt.% 1.5C, 0.01N, 12Cr, 0.9Mo, 0.8V) with a hardness of 60 HRC and polished to Ra = 0.05 m. The sheet was a 1.5 mm thick martensitic carbon steel (wt.% 0.11C, 0.2Si, 1.2Mn, 0.04Al), with material properties as seen in Tables 2 and 3. Tests were performed in air atmosphere at dry sliding conditions, meaning that the sheets were washed with a degreasing agent and acetone prior to testing. Results from the SOFS tests were extracted from the recorded friction and normal force data and represented as coefficient of friction versus sliding distance diagrams. The tool and sheet surfaces Table 3 Material properties for the tool steel and the carbon steel sheet.
E [GPa]
v [kg/m3 ] k [W/m K] c [J/kg K]
Tool steel
Steel sheet
210 0.3 7700 20 460
213 0.3 7812 46 480
were investigated using optical profilometry (OP) Veeco NT3300 and scanning electron microscopy LEO 1530. 2.2. Finite element calculations 2.2.1. Material description The tool steel disc was assumed to behave linearly isotropic elastic with Young’s modulus Et and Poisson’s ratio t , while the sheet was considered to behave elastic–plastic. The elastic part of the deformation was governed by an isotropic linear elastic model with Young’s modulus Ec and Poisson’s ratio c , while the plastic deformation was assumed to obey the von Mises theory of plasticity with isotropic hardening. Based on tensile testing, the hardening function for the sheet was approximated by a piecewise linear function from the yield stress to the ultimate tensile stress, after which the flow stress was assumed constant. The data points for the piecewise function is seen in Table 2, where f denotes the von Mises p flow stress and εeff the effective plastic strain. Heat conduction was supposed to obey the Fourier law of heat conduction with thermal conductivities kt and kc , specific heats ct and cc and densities t and c , for the tool and the carbon steel, respectively. A coefficient of friction of = 0.4 was used, based on experimental data. The material properties, found in Table 3, were assumed constant, using RT values, independent of temperature. 2.2.2. Analyses The aim of the analyses was to compute the steady state temperature distribution during sliding as function of the sliding velocity. Hence, the following procedure was adopted: 1. Calculate the contact pressure and the contact area for static contact for both normal loads used in the SOFS experiments. 2. Perform a heat conduction analysis where the tool and the sheet metal are held in contact over a contact area as derived from the first step. Assume perfect heat conduction across the contact and zero heat conduction elsewhere. Prescribe downstream heat convection with velocity v in the sheet metal. Apply a heat flux distribution given by the coefficient of friction, the velocity and the Herzian pressure distribution over the contact area. Continue the analysis until a steady state temperature distribution is obtained. 2.2.3. Contact area and pressure analyses using Hertz theory of contact The contact area for non-conforming surfaces in contact forms an ellipse, with major and minor axes a and b, respectively. The contact stress distribution, p(x, y), may be expressed as [11]:
p(x, y) = p0
1−
x 2 a
−
y 2 b
(1)
In this particular case, with one of the surfaces being flat, the ratio between the major and minor axes could be approximately
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Fig. 2. The dotted line illustrates a FEM contour plot of the contact pressure distribution at 500 N normal load and the solid line illustrates the Herzian contact pressure.
derived as b = a
R −2/3 1
(2)
R2
where R1 = 25 mm and R2 = 5 mm are the principal radii of curvature of the disc. This gives b/a ≈ 0.342. The maximum contact pressure is derived as
p0 =
3P = 2ab
6PE ∗2 3 R1 R2
1/3 R1 F3
R2
(3)
where
E∗ =
1 − t2 1 − c2 + Et Ec
−1 (4)
and the shape factor F3 ≈ 1.07. Hence, for the applied force P = 500 N, the maximum contact pressure becomes p0 = 2340 MPa. Combining Eqs. (2) and (3) gives a = 0.546 mm and b = 0.187 mm. In the same way, P = 250 N gives p0 = 1857 MPa, a = 0.434 mm and b = 0.148 mm. The above solution is not valid if plastic deformation is initiated in the sheet. Bounds for the pressure at incipient yielding p0y are given by [11] 1.6y ≤ p0y ≤ 1.79y
(5)
which implies that plastic deformation at macro-level is initiated at a normal force P =259–363 N. Hence, P = 500 N leads to some amount of plastic deformations and a more detailed analysis of the contact pressure was necessary. 2.2.4. Finite element analysis of the static contact In order to verify the applicability of the Hertzian contact pressure distribution, FE-analysis was performed of the setup. In this analysis the effect of shear stresses due to friction and sliding was neglected. It is well known that the contact pressure is essentially unaffected by shear stresses due to frictional sliding as long as the material behaves elastic and the bodies have similar elastic properties [11]. The model comprised 18669 tetrahedral elements with quadratic shape functions. It was found that only relatively small sub-surface plastic deformation occurred with a maximum value of 0.212%. Plastic deformation was initiated at 305 N, which is within the bounds given by Eq. (5). A contour plot of the pressure over one quarter of the contact surface is shown in Fig. 2 along with the pressure distribution along the major and minor axis of the contact surface. The FEM-result is shown by circles, while the Hertzian contact pressure according to Eq. (1) is shown by solid lines. As seen, despite some deviations at the center of the contact, possibly due to sub-surface plastic deformation, the Hertzian contact pressure distribution was in good agreement to the FEM-solution. Hence, the Hertzian solution was regarded as a sufficiently accurate approximation to the pressure distribution in the SOFS tribometer for the particular cases.
Fig. 3. (a) Mesh used in the heat conduction analyzes. (b) Close-up of the contact region showing the steady state temperature distribution for Pe = 2.6.
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Fig. 4. Temperature evolution at the sheet surface as function of sliding distance for (a) Pe = 0.38 and (b) Pe = 3.72.
2.2.5. Heat conduction analysis 2.2.5.1. Finite element modeling and analysis. The finite element mesh used in the heat conduction analysis is shown in Fig. 3. Similar to the static analysis, a coordinate system was introduced at the center of the contact, with the x-axis pointing in the sliding direction. Due to symmetry, half of the assembly was modeled. The wheel was modeled with 3522 tetrahedral heat conduction (diffusion) elements, while the sheet was modeled with 6144 hexahedral convection/diffusion elements. Linear interpolation was adopted for all elements. By using convection/diffusion elements in the sheet, the motion of the sheet relative to the wheel was represented by a prescribed mass flow rate in the negative x-direction, equal to the density of the carbon steel sheet times the velocity of the disc. Contact between the disc and the sheet was enforced over an elliptic contact area, as derived from the Hertzian theory of contact. Perfect heat conduction was assumed across the contact, implying equal temperature on opposite sides of the contact. A distributed heat flux q(x, y) was prescribed over the contact area by aid of the user subroutine (DFLUX) facility available in ABAQUS. The heat flux distribution was computed from the contact pressure according to Eq. (1) as q(x, y) = p(x, y)v
(6)
All other boundaries of the assembly were assumed to be perfectly insulated. The initial temperature of both the wheel and the sheet was set to 20 ◦ C. 3. Results 3.1. Finite element analyses
is seen that, as the Peclet number (i.e. the velocity) increases, the temperature in front of the wheel decreases at the same time as the temperature in the sheet track behind the wheel increases. In both cases, however, the temperature decreases rapidly behind the contact. s , was essentially reached at Steady state conditions, i.e. ˆ c,max s/a ≈ 6. Since the disc is finite it is well recognized that steady state conditions only prevail in an approximate sense due to overall heating of the wheel. The effect increases as the Peclet number s , the maximum temdecreases. As a representative value of ˆ c,max perature at s = 100 mm (corresponding to s/a = 183) was chosen. Compared to the highest temperature at s = 900 mm, the maximum difference was 2%, which occurred for the lowest Pe under consideration. In most other cases, the difference was well below 1%. s Fig. 5 shows ˆ c,max as function of the Peclet number. The solid line represents a least squares curve fit to the numerical results. This relationship was used to calculate the velocities in Table 1 for the two normal forces under consideration. In this way the contact pressure could be varied, while keeping the temperature constant. A convergence check of the result was made by repeating the analysis with a finite element mesh containing roughly half as many elements adjacent to the contact. For Pe = 3.72 this gave a s difference in ˆ c,max of around 2.5%. For lower Peclet numbers the difference was smaller. This was expected since high Peclet numbers are more demanding from a numerical point of view, due to the higher temperature gradients encountered in those cases. Presumably, further mesh refinement would give even smaller differences. In summary, the numerical results were considered accurate enough for the purpose of the present research.
Using dimensional analysis (see Appendix A) it can be shown that the maximum steady state temperature at the contact can be expressed as Tmax = T0 +
p0 va ˆ s c,max kc
c
t
,
kc a , , Pe kt b
(7)
s where Pe = va/2 c is the Peclet number and ˆ c,max is the maximum non-dimensionalised steady state temperature increase in the sheet. The range of Peclet numbers used in the experiments was 0.18 < Pe < 5 and numerical analyses were conducted for eight Peclet numbers within this range. As the sliding distance, s, increases the temperature distribution in the contact approaches steady state. The evolution of the temperature in the carbon steel sheet (expressed as non-dimensionalised temperature increase, ˆ c , see Appendix A), as function of x/a for different sliding distances, s/a, are shown in Fig. 4 for the cases Pe = 0.378 and Pe = 3.72. It
Fig. 5. Maximum non-dimensionalised steady state temperature increase versus Pe .
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3.2. Tribological testing 3.2.1. Friction diagrams Coefficient of friction versus sliding distance diagrams for all test conditions are illustrated along with the maximum calculated tem-
peratures for the 250 and 500 N normal loads in Fig. 6. Based on the FE-calculations, the sliding velocities were selected so that equal temperatures were obtained for both loads. Two test series were performed for each velocity and as seen, the friction data showed good repeatability. The frictional behavior was found dependent
Fig. 6. Coefficient of friction versus sliding distance diagrams for the 250 and 500 N normal loads at increasing velocity with corresponding maximum temperatures.
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Fig. 7. (a) A typical SEM image of the sheet track illustrating a flake-like morphology, typical for adhesive wear and (b–f) OP images at 500 N and increasing velocity, showing a typical surface morphology of the tool with adhered sheet material.
on velocity. For both normal loads, increasing sliding velocity led to a transition in friction, at a specific sliding distance, sr , where the coefficient of friction increased from approximately 0.4 to 0.8. The distance until the increase in friction, sr , was determined as the distance until highest friction occurred, as shown in Fig. 6 for the 250 N load at a sliding velocity of 82 mm/s. During sliding at the lowest velocity, for both normal loads, a relatively constant coefficient of friction was observed during the entire test interval with a value of approximately 0.4. Notably, the sliding distance sr did not decrease further when slid at the highest velocity, on the contrary, a slight increase was observed for both loads. 3.2.2. Tool and sheet surface morphology Optical profilometry of the tool surfaces showed that sheet material transfer occurred at all test conditions due to severe adhesive wear of the sheet, Fig. 7a. Typical tool surface morphologies are illustrated in Fig. 7b–f for the 500 N normal load at increasing sliding velocity. In agreement to the friction diagrams, the amount of adhered material increased with increasing velocity and at the highest velocity, the adhered layer was approximately five times thicker, as compared to the lowest velocity.
4. Discussion In the present investigation, dry sliding tests were conducted using the SOFS tribometer at varying sliding velocities at two different normal loads. The surface temperature increase due to frictional heating was calculated by means of FE-analyzes and, based on the FE-calculations, sliding velocities were selected to achieve similar surface temperatures, ranging from 40 to 400 ◦ C, for both normal loads. During the sliding, the coefficient of friction increased at a specific sliding distance, sr in Fig. 6, due to initiation of severe adhesive wear of the sheet, Fig. 7a, for both normal loads. Typically, the sliding distance sr decreased as the velocity and calculated temperature, increased. However, as seen in Fig. 6, sliding at the two lowest velocities did not have a large impact on the frictional behavior, possibly due to the low temperature rise. As the temperature exceeded about 80 ◦ C, a significant deterioration of the performance was observed. At 230 ◦ C, the sliding distance until the transition in friction was only between 13 and 20% of the total sliding distance used in the experiments. However, additional increase of the temperature did not lead to a further decrease of the critical sliding distance. On the contrary, both loads
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A. Gåård et al. / Wear 268 (2010) 968–975
Fig. 8. The diagram illustrates AFM measured adhesive force in a silicon–carbon steel contact and the sliding distance until severe adhesive wear for both normal loads versus temperature.
showed slightly longer sliding distances until the frictional transition at 400 ◦ C. Notably, the transition in friction occurred at similar sliding distances for comparable calculated temperatures, although the normal load and sliding speed was different. Good agreement was found between the friction diagrams and the tool surface morphology after testing. Typically, the shorter sliding distance to the transition to severe adhesive wear, Fig. 6, the more sheet material was transferred to the tools surface, Fig. 7. The results are in good agreement to previous observations, where several authors have reported temperature effects on wear and adhesion at both macro- and nanoscales. In [3,4], macrolevel tribological tests were conducted at elevated temperatures and the results showed that increasing tool bulk temperature led to increasing tool damage due to adhesive wear. On the nanoscale, atomic force microscopy measurements imply that adhesion is influenced by temperature. In [9], the pulloff force, which reflects the interfacial adhesion, for metal–silicon couples was found to increase as temperature increased. The measurements were performed in ultra-high vacuum at a temperature range of approximately 25–400 ◦ C. Adhesion started to increase as the temperature reached approximately 125–200 ◦ C, depending on material couple. A similar trend was observed in [10], where the pull-off force for SiC–SiC couples was measured and in that case, adhesion started to increase at about 300 ◦ C. The nanoscale measurements for carbon steel obtained in [9] are compared to the macroscopic measurements in the present work in Fig. 8. An adhesive wear index Iw = sr /stot , where stot denotes the total sliding distance, was defined to quantify the propensity for severe adhesive wear in the present investigation. As seen, the trend in the diagrams agrees very well. The AFM data indicated that adhesion starts to increase after approximately 100 ◦ C and as seen for the present results, the adhesive wear index Iw decreased at about the same temperature. Further, the AFM observations illustrated that the adhesive force decreased slightly in a temperature interval of 277–377 ◦ C. A similar behavior was seen in the present research, where an increase of the temperature from approximately 300 to 400 ◦ C, led to somewhat higher Iw . Several material properties, related to temperature, influence the adhesive contact problem presently addressed. The FEM temperature calculations applied includes temperature dependant properties which, for simplicity, were assumed at RT levels and independent of temperature. In the temperature range up to 400 ◦ C, they may vary considerably for the present materials. Considering the type of low carbon steel used in the experiments, Young’s modulus, thermal expansion, yield stress [12], specific heat capacity, thermal conductivity and density as supplied by the sheet manufacturer, can change from RT with −15%, +40% and −40%, +5%, −13%, −2%, respectively. The thermal properties of the tool steel grade may also vary in the test temperature range, while the mechanical
strength is more stable. The combined effect, for instance on estimated surface temperatures, introducing temperature dependent properties needs further investigation. It is well known that temperature may influence the adhesive wear mechanism. Generally, the latter is attributed to decreasing yield strength and increased ductility at elevated temperatures, which increases the real area of contact between the surfaces. A decrease in yield stress and Young’s modulus is anticipated in the low carbon steel reaching 400 ◦ C, tending to increase the real area of contact and contribute to the adhesive effect. Nevertheless, the very high sensitivity of adhesive wear to temperature already at lower temperatures reflected in Fig. 8 seems unlikely to depend on change in mechanical properties alone. Hence, given the strong correlation to adhesive force measured on the nanoscale by AFM microscopy, Fig. 8, a temperature-induced high adhesion mechanism is proposed adding to the temperature dependence in macroscopic adhesive wear. 5. Conclusions Dry sliding tests were conducted using the SOFS tribometer at various sliding velocities at 250 and 500 N normal loads, which led to increasing surface temperature due to frictional heating. The sliding speeds were estimated by use of FE-analyzes to achieve a similar temperature range for both loads and a range of approximately 40–400 ◦ C was investigated. Friction diagrams demonstrated that adhesive wear is influenced by temperature, as increasing temperature led to shorter sliding distances until the onset of severe adhesive wear, distinguished as an increase of the coefficient of friction. The transition to severe adhesive wear occurred at similar sliding distances for similar temperatures, although the normal loads and velocities were different. It was assumed that the very high sensitivity of adhesive wear to temperature, already at lower temperatures, was unlikely to depend on change in mechanical properties. Instead, a strong correlation to adhesive forces measured on the nanoscale, using atomic force microscopy in a similar temperature range, suggests a temperature-induced high adhesion mechanism, adding to the temperature dependence in macroscopic adhesive wear. Appendix A. Scaling of heat conduction equations A.1. Governing equations Heat conduction in the wheel is governed by t t =
∂t ∂t
where t =
kt t ct
(8)
Within the Eulerian framework adopted for the sheet, the partial time derivative is replaced by a material time derivative. The governing equation for the sheet thus becomes [13]: c c =
∂c ∂c −v ∂t ∂x
where c =
kc c cc
(9)
In Eqs. (8) and (9)t and c denotes the change in temperature in the wheel and the sheet, respectively, from the initial temperature. A.2. Boundary conditions At the contact area there are two boundary conditions that must be fulfilled. Perfect heat conduction and balance of heat flow across
A. Gåård et al. / Wear 268 (2010) 968–975
the contact implies that: t = c
and
q = −kt
temperature in the sheet then becomes
∂t ∂c + kc ∂z ∂z
(10)
respectively, where q is given by Eq. (6). All other boundaries are assumed to be perfectly insulated. A.3. Dimensionless variables and equations Dimensionless variables were introduced according to xˆ =
x , a
yˆ =
y , a
zˆ =
z , a
tˆ =
t t0
and
ˆ = 0
(11)
where t0 =
a2 c
and 0 =
q0 a kc
(12)
The reference heat flux in the second expression above is set to q0 = p0 v. Introducing the dimensionless variables into the governing Eqs. (8) and (9) gives:
ˆ t =
c ∂ˆ t t ∂tˆ
and ˆ c + 2Pe
∂ˆ c ∂ˆ c = ∂ˆx ∂tˆ
(13)
where Pe = va/2 c is the Peclet number. The boundary conditions transforms into:
ˆ t = ˆ c
1 − xˆ 2 −
and
a 2 1/2 b
yˆ 2
=−
kt ∂ˆ t ∂ˆ c + kc ∂ˆz ∂ˆz
(14)
where also Eq. (1) has been utilized. The dependence of v enters in the scaling of the temperature, but there is also an implicit dependence of v via the Peclet number in the second of the Eq. (13). To summarize, it can be concluded that the temperature increase in the sheet could be expressed as c =
p0 va ˆ c kc
xˆ , yˆ , zˆ , tˆ,
975
c kc a , , , Pe t kt b
(15)
When steady state conditions are reached in the sheet, the time derivative in the second of Eq. (13) vanishes and the dimensionless
c kc a ˆ c = ˆ cs xˆ , yˆ , zˆ , , , , Pe t kt b
(16)
Given the geometry of the wheel and the thermal properties of the wheel and sheet the above scaling can for instance be used to predict the effect from a change in the applied normal force F. This is particularly simple as long as the contact is almost elastic so that the change of a due to a change in F could be assessed by Hertz theory of contact. References [1] B. Bhushan, Introduction to tribology, John Wiley and Sons, New York, 2001. [2] S. Jacobson, S. Hogmark, Tribologi; Friction, smörjning och nötning, Liber utbildning AB, 1996. [3] P. Groche, G. Nitzsche, Influence of temperature on the initiation of adhesive wear with respect to deep drawing of aluminum-alloys, J. Mater. Process. Technol. 191 (2007) 314–316. [4] B. Persson, Sliding Friction, Springer, Germany, 2000. [5] A. Gåård, P. Krakhmalev, J. Bergström, N. Hallbäck, Galling resistance and wear mechanisms—cold work tool materials sliding against carbon steel sheets, Tribol. Lett. 26 (2006) 67–72. [6] A. Gåård, P. Krakhmalev, J. Bergström, Tribotest 14 (2008) 1–9. [7] A. Määttä, P. Vuoristo, T. Mäntylä, Friction and adhesion of stainless steel strip against tool steels in unlubricated sliding with high contact load, Tribol. Int. 34 (2001) 779–786. [8] T. Herai, M. Ejima, K. Yoshida, K. Miyauchi, H. Ike, Frictional surface damage and its mechanism in metal sheets, Sci. Paper Inst. Phys. Chem. Res. 72 (1978) 1–13. [9] A. Gåård, P. Krakhmalev, J. Bergström, J.H. Grytzelius, H.M. Zhang, Experimental study of the relationship between temperature and adhesive forces for lowalloyed steel, stainless steel, and titanium using atomic force microscopy in ultrahigh vacuum, J. Appl. Phys. 103 (2008), 124301–1-4. [10] K. Miyosi, Considerations in vacuum tribology (adhesion, friction, wear, and solid lubrication in vacuum), Tribol. Int. 32 (1999) 605–616. [11] K. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, UK, 1985. [12] M.F. Rothman, High Temperature Property Data: Ferrous Alloys, ASM International, 1989. [13] H. Carslaw, J. Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford, UK, 1959.
Contents lists available at ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Temperature effects on adhesive wear in dry sliding contacts A. Gåård ∗ , N. Hallbäck, P. Krakhmalev, J. Bergström Department of Mechanical and Materials Engineering, Karlstad University, SE-651 88 Karlstad, Sweden
a r t i c l e
i n f o
Article history: Received 3 April 2009 Received in revised form 25 November 2009 Accepted 7 December 2009 Available online 16 December 2009 Friction Adhesion Sliding wear Thermal effects Galling
a b s t r a c t In many metal forming operations, frictional heating occurs at the interface due to a sliding contact. Generally, the controlling wear mechanism in the tribological system is attributed to adhesive wear. Understanding of the influence of temperature on wear mechanisms is needed for the development of materials and for optimization of the forming process. Dry sliding tests were conducted at different sliding velocities and, hence, different surface temperatures due to frictional heating. A significant influence of temperature on adhesion was observed and increasing temperature led to a higher tendency for initiation of severe adhesive wear. The results were compared to atomic force microscopy force curve measurements, which show that the adhesive force increases with temperature. A very good agreement between the results was observed, which suggests that the controlling mechanism for the observations in the present work is temperature-induced high adhesion. © 2009 Elsevier B.V. All rights reserved.
1. Introduction A sliding contact between metal surfaces involves several tribological phenomena, e.g. friction and frictional heating, at the interface. The interaction between the surfaces occurs at the real area of contact, which is influenced by normal load and mechanical properties. If the surfaces are not separated, for instance by a lubricant, adhesion occurs across the interface, which influences both friction and wear. Elevated temperature has several effects on a tribosystem. If a shear displacement is added to a static normal load, the real area of contact, generally, increases due to junction growth by plastic flow. If the interface is stronger than the cohesive strength of the materials, the size of the junctions are determined by the ductility. For metals, the yield strength decreases and ductility increases at high temperatures and, hence, both the real area of contact and adhesion increase under loading [1,2]. Several authors have reported higher adhesive wear rates at elevated temperatures, particularly for open tribosystems [3,4] where either one, or both, of the surfaces in contact are always new. Sheet metal forming (SMF) is a typical application, corresponding to an open tribosystem, in which the controlling wear mechanism, generally, is attributed to adhesive wear [5–8]. During sliding, sheet material is transferred to the tools surface which, subsequently, leads to scratching of the sheets. It is well known that forming of high-strength steel sheets are more demanding from a wear point of view and material transfer occurs more rapidly. The latter is often
∗ Corresponding author. Tel.: +46 0 54 7001262; fax: +46 0 54 7001449. E-mail address: [email protected] (A. Gåård). 0043-1648/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2009.12.007
attributed to increased contact pressures due to higher hardness of the sheet. However, the consequences of high pressures on wear in SMF are not fully understood. The amount of adhesive wear is generally proportional to the normal load and inversely proportional to the hardness of the softer material, i.e. the real area of contact for a plastic contact [1]. In analogue to SMF, introducing of higher-strength sheets do increase the contact pressure, but, at the same time the hardness of the sheet is increased as well. Hence, it is not obvious that the higher amount of wear observed for forming of high-strength sheets are due to higher contact pressure. Another possible origin is frictional heating which depends on the contact stresses. Possibly, the elevated temperatures lead to softening and growth of the real area of contact resulting in higher adhesion. Nonetheless, in addition to temperature effects on mechanical properties, other phenomena may be of importance as temperature increases. Recent results using atomic force microscopy (AFM), indicate that the adhesive force depends on temperature, with increasing adhesion as temperature rises [9,10]. In [9], the pull-off force, which represents the interfacial adhesion, was measured for several metallic materials against Si using atomic force microscopy. For all investigated materials, adhesion started to rise at approximately 125–150 ◦ C. Hence, the implication is that temperature-induced high adhesion may be an additional mechanism to the increased wear observed at elevated temperatures. The objective of this study is to investigate the effect of frictional heating on the adhesive wear mechanism. Tribological tests were conducted at dry sliding test conditions at different contact pressures and velocities using a Slider-On-Flat-Surface tribometer (SOFS). The temperatures during the sliding were calculated using FE-analysis.
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Fig. 1. (a) The test principle of the disc sliding against the plate. (b) Overview of the SOFS test equipment and (c) close-up of the disc holder with load actuator C and normal and friction force gauges A and B. Table 1 The velocities used in the SOFS experiment for the different normal loads and calculated maximum sheet surface temperatures. P [N]
v1 [mm/s]
v2 [mm/s]
v3 [mm/s]
v4 [mm/s]
v5 [mm/s]
250 500 Tmax [◦ C]
13 8 46
27 17 72
82 50 160
137 83 238
281 167 409
Table 2 Flow stress versus effective strain for the carbon steel sheet. p
[MPa]
εeff
1175 1234 1326 1344
0 0.0042 0.0138 0.0237
2. Experimental 2.1. Tribological evaluation Tribological tests were conducted using the SOFS tribometer, Fig. 1, which is described in detail in [6]. A disc-shaped tool with radii 25 and 5 mm was forced against a sheet material at a constant normal load (P) of 250 and 500 N and slid for a total distance of 900 mm. Tests were conducted at different sliding velocities (v1 to v5 ) to obtain increasing surface temperatures, generated by frictional heating. The velocities, as seen in Table 1, were selected based on FE-analysis to achieve similar surface temperatures for each of the applied loads. The maximum calculated temperatures are shown in Table 1. The tests were repeated twice for each velocity. The tool material was an AISI D2 tool steel (wt.% 1.5C, 0.01N, 12Cr, 0.9Mo, 0.8V) with a hardness of 60 HRC and polished to Ra = 0.05 m. The sheet was a 1.5 mm thick martensitic carbon steel (wt.% 0.11C, 0.2Si, 1.2Mn, 0.04Al), with material properties as seen in Tables 2 and 3. Tests were performed in air atmosphere at dry sliding conditions, meaning that the sheets were washed with a degreasing agent and acetone prior to testing. Results from the SOFS tests were extracted from the recorded friction and normal force data and represented as coefficient of friction versus sliding distance diagrams. The tool and sheet surfaces Table 3 Material properties for the tool steel and the carbon steel sheet.
E [GPa]
v [kg/m3 ] k [W/m K] c [J/kg K]
Tool steel
Steel sheet
210 0.3 7700 20 460
213 0.3 7812 46 480
were investigated using optical profilometry (OP) Veeco NT3300 and scanning electron microscopy LEO 1530. 2.2. Finite element calculations 2.2.1. Material description The tool steel disc was assumed to behave linearly isotropic elastic with Young’s modulus Et and Poisson’s ratio t , while the sheet was considered to behave elastic–plastic. The elastic part of the deformation was governed by an isotropic linear elastic model with Young’s modulus Ec and Poisson’s ratio c , while the plastic deformation was assumed to obey the von Mises theory of plasticity with isotropic hardening. Based on tensile testing, the hardening function for the sheet was approximated by a piecewise linear function from the yield stress to the ultimate tensile stress, after which the flow stress was assumed constant. The data points for the piecewise function is seen in Table 2, where f denotes the von Mises p flow stress and εeff the effective plastic strain. Heat conduction was supposed to obey the Fourier law of heat conduction with thermal conductivities kt and kc , specific heats ct and cc and densities t and c , for the tool and the carbon steel, respectively. A coefficient of friction of = 0.4 was used, based on experimental data. The material properties, found in Table 3, were assumed constant, using RT values, independent of temperature. 2.2.2. Analyses The aim of the analyses was to compute the steady state temperature distribution during sliding as function of the sliding velocity. Hence, the following procedure was adopted: 1. Calculate the contact pressure and the contact area for static contact for both normal loads used in the SOFS experiments. 2. Perform a heat conduction analysis where the tool and the sheet metal are held in contact over a contact area as derived from the first step. Assume perfect heat conduction across the contact and zero heat conduction elsewhere. Prescribe downstream heat convection with velocity v in the sheet metal. Apply a heat flux distribution given by the coefficient of friction, the velocity and the Herzian pressure distribution over the contact area. Continue the analysis until a steady state temperature distribution is obtained. 2.2.3. Contact area and pressure analyses using Hertz theory of contact The contact area for non-conforming surfaces in contact forms an ellipse, with major and minor axes a and b, respectively. The contact stress distribution, p(x, y), may be expressed as [11]:
p(x, y) = p0
1−
x 2 a
−
y 2 b
(1)
In this particular case, with one of the surfaces being flat, the ratio between the major and minor axes could be approximately
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Fig. 2. The dotted line illustrates a FEM contour plot of the contact pressure distribution at 500 N normal load and the solid line illustrates the Herzian contact pressure.
derived as b = a
R −2/3 1
(2)
R2
where R1 = 25 mm and R2 = 5 mm are the principal radii of curvature of the disc. This gives b/a ≈ 0.342. The maximum contact pressure is derived as
p0 =
3P = 2ab
6PE ∗2 3 R1 R2
1/3 R1 F3
R2
(3)
where
E∗ =
1 − t2 1 − c2 + Et Ec
−1 (4)
and the shape factor F3 ≈ 1.07. Hence, for the applied force P = 500 N, the maximum contact pressure becomes p0 = 2340 MPa. Combining Eqs. (2) and (3) gives a = 0.546 mm and b = 0.187 mm. In the same way, P = 250 N gives p0 = 1857 MPa, a = 0.434 mm and b = 0.148 mm. The above solution is not valid if plastic deformation is initiated in the sheet. Bounds for the pressure at incipient yielding p0y are given by [11] 1.6y ≤ p0y ≤ 1.79y
(5)
which implies that plastic deformation at macro-level is initiated at a normal force P =259–363 N. Hence, P = 500 N leads to some amount of plastic deformations and a more detailed analysis of the contact pressure was necessary. 2.2.4. Finite element analysis of the static contact In order to verify the applicability of the Hertzian contact pressure distribution, FE-analysis was performed of the setup. In this analysis the effect of shear stresses due to friction and sliding was neglected. It is well known that the contact pressure is essentially unaffected by shear stresses due to frictional sliding as long as the material behaves elastic and the bodies have similar elastic properties [11]. The model comprised 18669 tetrahedral elements with quadratic shape functions. It was found that only relatively small sub-surface plastic deformation occurred with a maximum value of 0.212%. Plastic deformation was initiated at 305 N, which is within the bounds given by Eq. (5). A contour plot of the pressure over one quarter of the contact surface is shown in Fig. 2 along with the pressure distribution along the major and minor axis of the contact surface. The FEM-result is shown by circles, while the Hertzian contact pressure according to Eq. (1) is shown by solid lines. As seen, despite some deviations at the center of the contact, possibly due to sub-surface plastic deformation, the Hertzian contact pressure distribution was in good agreement to the FEM-solution. Hence, the Hertzian solution was regarded as a sufficiently accurate approximation to the pressure distribution in the SOFS tribometer for the particular cases.
Fig. 3. (a) Mesh used in the heat conduction analyzes. (b) Close-up of the contact region showing the steady state temperature distribution for Pe = 2.6.
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Fig. 4. Temperature evolution at the sheet surface as function of sliding distance for (a) Pe = 0.38 and (b) Pe = 3.72.
2.2.5. Heat conduction analysis 2.2.5.1. Finite element modeling and analysis. The finite element mesh used in the heat conduction analysis is shown in Fig. 3. Similar to the static analysis, a coordinate system was introduced at the center of the contact, with the x-axis pointing in the sliding direction. Due to symmetry, half of the assembly was modeled. The wheel was modeled with 3522 tetrahedral heat conduction (diffusion) elements, while the sheet was modeled with 6144 hexahedral convection/diffusion elements. Linear interpolation was adopted for all elements. By using convection/diffusion elements in the sheet, the motion of the sheet relative to the wheel was represented by a prescribed mass flow rate in the negative x-direction, equal to the density of the carbon steel sheet times the velocity of the disc. Contact between the disc and the sheet was enforced over an elliptic contact area, as derived from the Hertzian theory of contact. Perfect heat conduction was assumed across the contact, implying equal temperature on opposite sides of the contact. A distributed heat flux q(x, y) was prescribed over the contact area by aid of the user subroutine (DFLUX) facility available in ABAQUS. The heat flux distribution was computed from the contact pressure according to Eq. (1) as q(x, y) = p(x, y)v
(6)
All other boundaries of the assembly were assumed to be perfectly insulated. The initial temperature of both the wheel and the sheet was set to 20 ◦ C. 3. Results 3.1. Finite element analyses
is seen that, as the Peclet number (i.e. the velocity) increases, the temperature in front of the wheel decreases at the same time as the temperature in the sheet track behind the wheel increases. In both cases, however, the temperature decreases rapidly behind the contact. s , was essentially reached at Steady state conditions, i.e. ˆ c,max s/a ≈ 6. Since the disc is finite it is well recognized that steady state conditions only prevail in an approximate sense due to overall heating of the wheel. The effect increases as the Peclet number s , the maximum temdecreases. As a representative value of ˆ c,max perature at s = 100 mm (corresponding to s/a = 183) was chosen. Compared to the highest temperature at s = 900 mm, the maximum difference was 2%, which occurred for the lowest Pe under consideration. In most other cases, the difference was well below 1%. s Fig. 5 shows ˆ c,max as function of the Peclet number. The solid line represents a least squares curve fit to the numerical results. This relationship was used to calculate the velocities in Table 1 for the two normal forces under consideration. In this way the contact pressure could be varied, while keeping the temperature constant. A convergence check of the result was made by repeating the analysis with a finite element mesh containing roughly half as many elements adjacent to the contact. For Pe = 3.72 this gave a s difference in ˆ c,max of around 2.5%. For lower Peclet numbers the difference was smaller. This was expected since high Peclet numbers are more demanding from a numerical point of view, due to the higher temperature gradients encountered in those cases. Presumably, further mesh refinement would give even smaller differences. In summary, the numerical results were considered accurate enough for the purpose of the present research.
Using dimensional analysis (see Appendix A) it can be shown that the maximum steady state temperature at the contact can be expressed as Tmax = T0 +
p0 va ˆ s c,max kc
c
t
,
kc a , , Pe kt b
(7)
s where Pe = va/2 c is the Peclet number and ˆ c,max is the maximum non-dimensionalised steady state temperature increase in the sheet. The range of Peclet numbers used in the experiments was 0.18 < Pe < 5 and numerical analyses were conducted for eight Peclet numbers within this range. As the sliding distance, s, increases the temperature distribution in the contact approaches steady state. The evolution of the temperature in the carbon steel sheet (expressed as non-dimensionalised temperature increase, ˆ c , see Appendix A), as function of x/a for different sliding distances, s/a, are shown in Fig. 4 for the cases Pe = 0.378 and Pe = 3.72. It
Fig. 5. Maximum non-dimensionalised steady state temperature increase versus Pe .
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3.2. Tribological testing 3.2.1. Friction diagrams Coefficient of friction versus sliding distance diagrams for all test conditions are illustrated along with the maximum calculated tem-
peratures for the 250 and 500 N normal loads in Fig. 6. Based on the FE-calculations, the sliding velocities were selected so that equal temperatures were obtained for both loads. Two test series were performed for each velocity and as seen, the friction data showed good repeatability. The frictional behavior was found dependent
Fig. 6. Coefficient of friction versus sliding distance diagrams for the 250 and 500 N normal loads at increasing velocity with corresponding maximum temperatures.
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Fig. 7. (a) A typical SEM image of the sheet track illustrating a flake-like morphology, typical for adhesive wear and (b–f) OP images at 500 N and increasing velocity, showing a typical surface morphology of the tool with adhered sheet material.
on velocity. For both normal loads, increasing sliding velocity led to a transition in friction, at a specific sliding distance, sr , where the coefficient of friction increased from approximately 0.4 to 0.8. The distance until the increase in friction, sr , was determined as the distance until highest friction occurred, as shown in Fig. 6 for the 250 N load at a sliding velocity of 82 mm/s. During sliding at the lowest velocity, for both normal loads, a relatively constant coefficient of friction was observed during the entire test interval with a value of approximately 0.4. Notably, the sliding distance sr did not decrease further when slid at the highest velocity, on the contrary, a slight increase was observed for both loads. 3.2.2. Tool and sheet surface morphology Optical profilometry of the tool surfaces showed that sheet material transfer occurred at all test conditions due to severe adhesive wear of the sheet, Fig. 7a. Typical tool surface morphologies are illustrated in Fig. 7b–f for the 500 N normal load at increasing sliding velocity. In agreement to the friction diagrams, the amount of adhered material increased with increasing velocity and at the highest velocity, the adhered layer was approximately five times thicker, as compared to the lowest velocity.
4. Discussion In the present investigation, dry sliding tests were conducted using the SOFS tribometer at varying sliding velocities at two different normal loads. The surface temperature increase due to frictional heating was calculated by means of FE-analyzes and, based on the FE-calculations, sliding velocities were selected to achieve similar surface temperatures, ranging from 40 to 400 ◦ C, for both normal loads. During the sliding, the coefficient of friction increased at a specific sliding distance, sr in Fig. 6, due to initiation of severe adhesive wear of the sheet, Fig. 7a, for both normal loads. Typically, the sliding distance sr decreased as the velocity and calculated temperature, increased. However, as seen in Fig. 6, sliding at the two lowest velocities did not have a large impact on the frictional behavior, possibly due to the low temperature rise. As the temperature exceeded about 80 ◦ C, a significant deterioration of the performance was observed. At 230 ◦ C, the sliding distance until the transition in friction was only between 13 and 20% of the total sliding distance used in the experiments. However, additional increase of the temperature did not lead to a further decrease of the critical sliding distance. On the contrary, both loads
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Fig. 8. The diagram illustrates AFM measured adhesive force in a silicon–carbon steel contact and the sliding distance until severe adhesive wear for both normal loads versus temperature.
showed slightly longer sliding distances until the frictional transition at 400 ◦ C. Notably, the transition in friction occurred at similar sliding distances for comparable calculated temperatures, although the normal load and sliding speed was different. Good agreement was found between the friction diagrams and the tool surface morphology after testing. Typically, the shorter sliding distance to the transition to severe adhesive wear, Fig. 6, the more sheet material was transferred to the tools surface, Fig. 7. The results are in good agreement to previous observations, where several authors have reported temperature effects on wear and adhesion at both macro- and nanoscales. In [3,4], macrolevel tribological tests were conducted at elevated temperatures and the results showed that increasing tool bulk temperature led to increasing tool damage due to adhesive wear. On the nanoscale, atomic force microscopy measurements imply that adhesion is influenced by temperature. In [9], the pulloff force, which reflects the interfacial adhesion, for metal–silicon couples was found to increase as temperature increased. The measurements were performed in ultra-high vacuum at a temperature range of approximately 25–400 ◦ C. Adhesion started to increase as the temperature reached approximately 125–200 ◦ C, depending on material couple. A similar trend was observed in [10], where the pull-off force for SiC–SiC couples was measured and in that case, adhesion started to increase at about 300 ◦ C. The nanoscale measurements for carbon steel obtained in [9] are compared to the macroscopic measurements in the present work in Fig. 8. An adhesive wear index Iw = sr /stot , where stot denotes the total sliding distance, was defined to quantify the propensity for severe adhesive wear in the present investigation. As seen, the trend in the diagrams agrees very well. The AFM data indicated that adhesion starts to increase after approximately 100 ◦ C and as seen for the present results, the adhesive wear index Iw decreased at about the same temperature. Further, the AFM observations illustrated that the adhesive force decreased slightly in a temperature interval of 277–377 ◦ C. A similar behavior was seen in the present research, where an increase of the temperature from approximately 300 to 400 ◦ C, led to somewhat higher Iw . Several material properties, related to temperature, influence the adhesive contact problem presently addressed. The FEM temperature calculations applied includes temperature dependant properties which, for simplicity, were assumed at RT levels and independent of temperature. In the temperature range up to 400 ◦ C, they may vary considerably for the present materials. Considering the type of low carbon steel used in the experiments, Young’s modulus, thermal expansion, yield stress [12], specific heat capacity, thermal conductivity and density as supplied by the sheet manufacturer, can change from RT with −15%, +40% and −40%, +5%, −13%, −2%, respectively. The thermal properties of the tool steel grade may also vary in the test temperature range, while the mechanical
strength is more stable. The combined effect, for instance on estimated surface temperatures, introducing temperature dependent properties needs further investigation. It is well known that temperature may influence the adhesive wear mechanism. Generally, the latter is attributed to decreasing yield strength and increased ductility at elevated temperatures, which increases the real area of contact between the surfaces. A decrease in yield stress and Young’s modulus is anticipated in the low carbon steel reaching 400 ◦ C, tending to increase the real area of contact and contribute to the adhesive effect. Nevertheless, the very high sensitivity of adhesive wear to temperature already at lower temperatures reflected in Fig. 8 seems unlikely to depend on change in mechanical properties alone. Hence, given the strong correlation to adhesive force measured on the nanoscale by AFM microscopy, Fig. 8, a temperature-induced high adhesion mechanism is proposed adding to the temperature dependence in macroscopic adhesive wear. 5. Conclusions Dry sliding tests were conducted using the SOFS tribometer at various sliding velocities at 250 and 500 N normal loads, which led to increasing surface temperature due to frictional heating. The sliding speeds were estimated by use of FE-analyzes to achieve a similar temperature range for both loads and a range of approximately 40–400 ◦ C was investigated. Friction diagrams demonstrated that adhesive wear is influenced by temperature, as increasing temperature led to shorter sliding distances until the onset of severe adhesive wear, distinguished as an increase of the coefficient of friction. The transition to severe adhesive wear occurred at similar sliding distances for similar temperatures, although the normal loads and velocities were different. It was assumed that the very high sensitivity of adhesive wear to temperature, already at lower temperatures, was unlikely to depend on change in mechanical properties. Instead, a strong correlation to adhesive forces measured on the nanoscale, using atomic force microscopy in a similar temperature range, suggests a temperature-induced high adhesion mechanism, adding to the temperature dependence in macroscopic adhesive wear. Appendix A. Scaling of heat conduction equations A.1. Governing equations Heat conduction in the wheel is governed by t t =
∂t ∂t
where t =
kt t ct
(8)
Within the Eulerian framework adopted for the sheet, the partial time derivative is replaced by a material time derivative. The governing equation for the sheet thus becomes [13]: c c =
∂c ∂c −v ∂t ∂x
where c =
kc c cc
(9)
In Eqs. (8) and (9)t and c denotes the change in temperature in the wheel and the sheet, respectively, from the initial temperature. A.2. Boundary conditions At the contact area there are two boundary conditions that must be fulfilled. Perfect heat conduction and balance of heat flow across
A. Gåård et al. / Wear 268 (2010) 968–975
the contact implies that: t = c
and
q = −kt
temperature in the sheet then becomes
∂t ∂c + kc ∂z ∂z
(10)
respectively, where q is given by Eq. (6). All other boundaries are assumed to be perfectly insulated. A.3. Dimensionless variables and equations Dimensionless variables were introduced according to xˆ =
x , a
yˆ =
y , a
zˆ =
z , a
tˆ =
t t0
and
ˆ = 0
(11)
where t0 =
a2 c
and 0 =
q0 a kc
(12)
The reference heat flux in the second expression above is set to q0 = p0 v. Introducing the dimensionless variables into the governing Eqs. (8) and (9) gives:
ˆ t =
c ∂ˆ t t ∂tˆ
and ˆ c + 2Pe
∂ˆ c ∂ˆ c = ∂ˆx ∂tˆ
(13)
where Pe = va/2 c is the Peclet number. The boundary conditions transforms into:
ˆ t = ˆ c
1 − xˆ 2 −
and
a 2 1/2 b
yˆ 2
=−
kt ∂ˆ t ∂ˆ c + kc ∂ˆz ∂ˆz
(14)
where also Eq. (1) has been utilized. The dependence of v enters in the scaling of the temperature, but there is also an implicit dependence of v via the Peclet number in the second of the Eq. (13). To summarize, it can be concluded that the temperature increase in the sheet could be expressed as c =
p0 va ˆ c kc
xˆ , yˆ , zˆ , tˆ,
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c kc a , , , Pe t kt b
(15)
When steady state conditions are reached in the sheet, the time derivative in the second of Eq. (13) vanishes and the dimensionless
c kc a ˆ c = ˆ cs xˆ , yˆ , zˆ , , , , Pe t kt b
(16)
Given the geometry of the wheel and the thermal properties of the wheel and sheet the above scaling can for instance be used to predict the effect from a change in the applied normal force F. This is particularly simple as long as the contact is almost elastic so that the change of a due to a change in F could be assessed by Hertz theory of contact. References [1] B. Bhushan, Introduction to tribology, John Wiley and Sons, New York, 2001. [2] S. Jacobson, S. Hogmark, Tribologi; Friction, smörjning och nötning, Liber utbildning AB, 1996. [3] P. Groche, G. Nitzsche, Influence of temperature on the initiation of adhesive wear with respect to deep drawing of aluminum-alloys, J. Mater. Process. Technol. 191 (2007) 314–316. [4] B. Persson, Sliding Friction, Springer, Germany, 2000. [5] A. Gåård, P. Krakhmalev, J. Bergström, N. Hallbäck, Galling resistance and wear mechanisms—cold work tool materials sliding against carbon steel sheets, Tribol. Lett. 26 (2006) 67–72. [6] A. Gåård, P. Krakhmalev, J. Bergström, Tribotest 14 (2008) 1–9. [7] A. Määttä, P. Vuoristo, T. Mäntylä, Friction and adhesion of stainless steel strip against tool steels in unlubricated sliding with high contact load, Tribol. Int. 34 (2001) 779–786. [8] T. Herai, M. Ejima, K. Yoshida, K. Miyauchi, H. Ike, Frictional surface damage and its mechanism in metal sheets, Sci. Paper Inst. Phys. Chem. Res. 72 (1978) 1–13. [9] A. Gåård, P. Krakhmalev, J. Bergström, J.H. Grytzelius, H.M. Zhang, Experimental study of the relationship between temperature and adhesive forces for lowalloyed steel, stainless steel, and titanium using atomic force microscopy in ultrahigh vacuum, J. Appl. Phys. 103 (2008), 124301–1-4. [10] K. Miyosi, Considerations in vacuum tribology (adhesion, friction, wear, and solid lubrication in vacuum), Tribol. Int. 32 (1999) 605–616. [11] K. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, UK, 1985. [12] M.F. Rothman, High Temperature Property Data: Ferrous Alloys, ASM International, 1989. [13] H. Carslaw, J. Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford, UK, 1959.