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Paper XIV (iii) A low cycle fatigue wear model and its application to layered systems
329
Paper XIV (iii)
A low cycle fatigue wear model and its application to layered systems A.G. Tangena
Wear is a very complex subject. Wear research i s therefore mostly restricted to the description ofphenomena. In practice, however one is more interested in the prediction of wear. A wear model based on low cycle fatigue has been developed that describes the influence of plastic deformation on wear. Used on layered systems, this model makes it possible to predict the influence of the substrate, the normal load and the contact geometry on wear. 1. INTRODUCTION Whenever we want to write on a piece of paper we note how important the supporting base is. Paper on a soft support will,tear, while on a hard table we can easily write on it. The shape of the pen is also important. With a sharp pencil we will more easily penetrate the paper than with a large brush. Even the environment is important. Just try to write on paper in the rain. Obviously writing on a piece of paper depends on a complete system. There is an analogy in wear. It is now wellknown that wear is not a material property, but i s characterized by the complete tribological system (Czichos [I]). Such a system consists of two bodies, an intermediate medium and the environment. Adhesion, geometry, roughness and the contact load determine the forces transmitted from one surface to the other. Wear in turn depends on the response of the contact materials to these loads, in other words on their deformation and fracture behaviour. Because of the complexity of such a tribological system most researchers confine themselves to the description of wear phenomena, without attempting to construct wear models that could serve as the basis for rational design procedures. An overview of wear models, relevant calculation procedures and the physics of the wear process can be found in Kragelskii [Z] and Tabor [3]. The most well-known wear equation is Archard's [1].He assumed that the volume of a wear particle was proportional to the third power of the contact radius between the two bodies. In that way he obtained: k Fn x distance fravelled Wearvolume = -
H
(1)
where k is the wear constant [5,6], f , the contact load and H the hardness. The wear equation is determined by the nature of the roughness distribution [7]. For very smooth surfaces, or in cases where plastic deformation becomes predominant, equation (1) is no longer valid. For layered systems the hardness H is not easy to define, since
it depends on the geometry of the system [7].' Evidently equation (1) has its limitations. A wear model that was often used i n the sixties is the 'zero wear' model of Bayer and Ku [8]. In this model fatigue is postulated as the main cause for the formation of wear particles. The zero wear theory states that wear can be controlled by limiting the maximurn shear stress in or under the contact area or, more specifically: wear can be at a practically zero level for a certain number of passes n when the shear stress T is smaller than or equal to r(n) x T,,, , where fln) is a function depending on the lubrication regime of the system. The zero wear model uses the elastic Hertz formulas and Palmgren's empirical relation between life and load for ball-bearings [9]. So the zero wear model defines wear-no-wear situations. In many cases, however, we are interested in situations where measurable wear occurs.
2. LOW CYCLE FATIGUE WEAR MODEL In our opinion a wear model must be based upon the physics of the wear process, taking into account the complete tribological system. If we confine ourselves to reasonably ductile materials we note that wear depends on the response of materials to cyclic loads, i.e. wear is concerned primarily with cyclic plastic deformation, crack initiation and crack growth (Glardon [lo], Jahanmir [ I l l ) . These are also the driving processes in low cycle fatigue (Challen [12]). Because of this analogy with the wear process a model was constructed that is based on a formulation analoguous to the well-known fatigue relation of Coffi n-Manson: b
x Nc = cc = constant.
(2)
In this formula AcP is the plastic strain per cycle and cc are the critical number of cycles and the critical strain necessary to originate fracture, while the superscript b is a fatigue constant with a value of approximately 0.5.
N, and
330
If we apply equation (2) to the wear process and the number of wear cycles i s larger than the critical number N,, there will be wear. The wear volume per unit distance travelled W is taken proportional to the number of wear cycles n divided by the critical number of cycles N,:
(3) where k , is an arbitrary wear constant and V the de-formed volume. Combination of (2) and (3) gives: 1 ._ b
W = k, V n [ % ] The amount of plastic deformation per cycle AE, i s associated with the applied von Mises stress (true stress) if through a cyclic stress-strain curve [13]:
F
=
n'
K' AcP,
where n' is the cyclic strain-hardening coefficient and K' a constant. The values of K' and n' depend on the number of cycles. The coefficient n' changes from some initial value to a value between 0.1 and 0.2 at large numbers of cycles for most metals [13]. If finally we combine equations (4) and (5),the wear volume per unit distance travelled W becomes:
Typical values of I/bn' are around 10. The large value of this coefficient indicates that one system parameter, the von Mises stress in the stressed volume, is dominant i n the wear process and that the response to this parameter determines the wear i n a low cycle fatigue process. The magnitude of the von Mises stress depends on the contact geometry, the mechanical properties of the materials, the friction and the externally applied forces. So this wear model indicates that the total tribological system determines the wear behaviour. 3. TESTING OF THE WEAR MODEL For testing the wear model we choose a wide range of experimental circumstances (see Table 1). Such a wide range can be created relatively easily if we use a layered system (Antler [14], Tangena [IS]). Variation of the substrate, the normal load and the geometry of the contact gives different von Mises stresses in the layer. In conformity with equation (6) the wear of the layer should also be different. Details of these experiments can be found i n the author's thesis [I61 and in reference [1'7]. The main points will be explained here. For the contact geometry a ball (pin) on a flat was chosen. The ball was made of ball-bearing chromium steel (DIN 5041). The flat consisted either of a substrate of soft annealed copper (OFHC-CU 99.9%, H, 50) or a hard glass (H, 4800) with several
layers on top. In a number of cases a hard sulphamate nickel layer was electrodeposited on the copper substrate. The top layer consisted of pure gold (99.9 %). This layer was applied either by evaporation i n vacuum or by electrodeposition. The pin was made of ball-bearing steel, so that we only had to account for the wear of the layered system. In this investigation the stresses i n the layer were determined by finite element calculations. The calculated stress was then correlated with the wear volume according to equation (6). 4. FINITE ELEMENT CALCULATIONS If we want to calculate the stresses in a complicated contact geometry we face the limitations of temporary finite element packages. 1. Our finite element package can only calculate contact situations with a rigid indentation body. This means that we have to choose contact situations where one body has a much larger Young's modulus and yield stress than the counter body, so that one body can be considered rigid. When the substrate i s hard as i n the case of the glass substrate, the radius of the indenter must be adjusted i n the calculations, since then the pin deforms elastically as well. 2. In a wear experiment the ball slides over the surface. Since it i s very difficult to calculate the actual 3-D stress situation, because of the amount of memory and computer time necessary, we have to approximate the stress situation i n the layered system. When the friction coefficient is low we do not make too large a mistake when we approximate a sliding situation with a n axially symmetric indentation [16]. In the experiments the friction coefficient was always around 0.1, so this approximation is justified. 3. The materials used i n the layered system deform not only elastically but also plastically. Mechanical behaviour i n the plastic region i s mostly expressed i n terms of a true stress-strain curve, where the von Mises stress at which yielding occurs is expressed as a function of the effective strain. This stress-strain curve i s mostly determined i n a uniaxial tensile strength test. For thin films such a test is virtually impossible. Therefore an alternative method was developed to obtain the stress-strain curve from a Brinell indentation test [18, 161. Even though both types of gold were applied via different processes, the mechanical properties proved to be practically the same. The finite element mesh used was axially symmetric. The gold layer was represented by 132 and the substrate by 396 elements. The indenter was simulated by 65 so-called gap elements. More details can be found in reference [17]. Because wear occurs primarily i n the top layer, we evaluated the stresses i n this layer. Since equation (6) shows the importance of the von Mises stress this stress is given i n Table 1. The stress during the first indentation was taken as an approximation for the stress during the wear process, so "shake-down" was not taken into account. The contact situations calculated are given in Table 1 : four normal loads, four different pin radii, two substrates, a nickel intermediate layer and two
33 1 Table 1. Results of FE calculations and wear experiments. - 7a diu: Au Substrate Ni pin aye1 aye1 rmm] -
2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 4.0 4.0 4.0 4.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5
3.0 3.0
3.0 3.0 3.0 3.0 3.0 3.0 3.0
3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 1.0 1.0 1.0 1.0
3.0 3.0 3.0 3.0 1.0 1.0 1.o 1.0
0.0 0.0 0.0 0.0 4.0
4.0 4.0 4.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
4.0 4.0 4.0 4.0 0.0
0.0 0.0 0.0
4.0 4.0 4.0 4.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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28 68 87 113 33 59 77 98 38 66 87
75 118 122 129 77 78 106 120 64
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0.5 1.5 2.5 4.0 0.5 1.5 2.5 4.0 0.5 1.5 2.5 4.0 0.5 1.5 2.5 4.0 0.5 I .5 2.5 4.0 0.5 1.5 2.5 4.0
thicknesses for the gold layer. We will not discuss all situations separately, but instead indicate the influ-. ence of important parameters. As a rule, an increasing normal load gives an increase in the von Mises stress, while the deformations in the layer can change from purely elastic to plastic. The substrate has a very strong influence on the stresses in the gold layer (Figure 1). The three vertical lines in this figure indicate contact radii. The horizontal solid line i s the yield stress of the gold (114 N/mm2).We note that with a soft copper substrate the stress and also the contact radius are large. Using a hard nickel layer between the gold and the copper substrate reduces the stresses, especially i n the centre of the contact. The contact radius is only slightly smaller than without the nickel layer. The
113
40 66 87
110
35 63 82 103 45 73 91 117 42 70 89 113 23 35 40 47 23 35 42 49 40 59 70 82 42 GI 73 84
110 117
122 59 104 114 118 38 47 74 100 34 59 80 101 19 34 46 60 5G 56 55 63 42 69 86 101
12 20 26 32 ia
29 35 42
Wear waporated Au l/rn721
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299 689
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51 185 289 679 24 102 iai
63 123 300 528 17 98 199 402
22 78 140 174
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hard glass substrate reduces the contact radius considerably, while the von Mises stress is low over tho entire contact area. The contact radius increases slightly and the vori Mises stress decreases with increasing pin radius. The influence of the layer thickness of the gold was investigated for a 1 prn and a 3 prn gold layer on a glass substrate. A thicker layer will give lower stresses. This i s understandable since both the gold layer and the glass substrate remain elastic. In that case the gold layer is no longer softer than the glass substrate. On the contrary, because the gold layer has a higher Young’s modulus than the glass = 79 G P a , €,/, = 70 GPa), the system behaves as a system with a hard layer on a soft substrate. The thicker the hard layer, the lower the stresses.
332
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The wear results of the layered system are indicated in Table 1. The cross-section of the wear track increased with increasing normal load. The substrate had a large influence on the wear of the gold film. A soft copper substrate gave considerable wear. A hard nickel layer in between the gold and the copper reduced wear, while a glass substrate reduced wear to practically zero. Increasing the radius of the pin resulted in a wider wear track, but despite this effect the cross-section of the track decreased with increasing radius. The thickness of the gold film had little influence on the wear. Both types of gold, the evaporated and the electrodeposited, showed the same trends in wear.
150
Radio1 position ( p m )
Figure 1. The von Mises stress in the gold top layer as a function of the radial position for an axially symmetric system, indented by a rigid ball with a radius of 12.5 mm, at a normal load of 4 N . The layered system consists of a 3.0 pm thick gold film on a copper substrate with or without a hard nickel intermediate layer and on a glass substrate.
5. WEAR EXPERIMENTS A precision wear apparatus was constructed in order to do accurate wear experiments. The apparatus consisted of a flat, that was linearly sliding under a spherical pin. Both the flat and the pin were airborne. Details can be found in reference [17]. The normal load, the position of the plate in a horizontal plane (x,y) and the speed were controlled, while the normal load, the shear force and the height of the pin were measured by a microcomputer. The data were evaluated on a mainframe computer. The wear tracks were measured with a modified profilometer. Special software was written, that calculated the cross-section and the depth of a wear track. The width of the wear tracks measured with an optical microscope correlated well with the width measured using the profilometer. The microscope was also used to check for transfer to and wear of the pin. Wear experiments were performed on two types of pure gold. One type was applied by evaporation in vacuum, and one type by electrodeposition. The evaporated layer was tested with a soft copper and a hard glass substrate, while the electrodeposited layer was used with a soft copper substrate with or without a 4 pm-thick hard nickel layer. The wear experiments were performed with several radii of the pin at 4 different loads. This resulted in 32 contact situations for the evaporated gold and 24 for the electrodeposited gold (see Table 1). During wear the contact area increased, since one of the contact bodies was a hard wear-resistant pin. This resulted i n an equilibrium situation, in which the layered system did not wear any more. For all situations the cross-section of the wear track was measured as the difference in cross-section between 1 and 250 cycles to compensate for pure plastic deformation of the substrate. A modified Talysurf profilometer was used. As the results were reproducible within 10 %, it was considered sufficient to measure every situation twice. The wear tracks had a grooved appearance, indicating that small wear particles were generated, which subsequently ploughed the surface (see Figure 2 ) .
Figure 2. The wear track after 250 cycles for a 3.0 pm thick evaporated gold layer on a copper substrate. The normal load was 4.0 N , the radius.of the pin 2 . 5 m m , and the width of the wear track 305 pm. 6. COMPARISON WITH THE THEORETICAL MODEL
The results of the wear experiments were first evaluated with the aid of equation ( I ) , the Archard equation. In figure 3.a the cross-section of the wear track in pm2 as a function of the normal load divided by the hardness for both the evaporated and the electrodeposited gold are represented. For the hardness we have taken the hardness of the substrate. This figure shows that the correlation of the experiments with equation (1) is not good. This is caused by the fact that every different radius of the indenter and every change in the substrate (Ni layer) gives a different k-factor in equation (1). So the use.fulness of equation (1) is limited. We have also evaluated the results of the wear experiments with the aid of equation (6). For this purpose we plotted in Figure 3.b the cross-section of the wear track as a function of the calculated von Mises stress averaged over the layer thickness and the contact radius ( in N / m m 2 ) .In this figure the yield stress is represented by the vertical line at 114 N/mm2.We notice the strong correlation between the von Mises stress and the wear. When the average
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Figure 3.a. Cross-section of the wear track for both evaporated and electrodeposited gold layers as a function of the normal load divided by the hardness for 56 different contact situations.
50
100
1
v. Mises stress (N/mmz) Figure 3.b. Cross-section of the wear track for both evaporated and electrodeposited gold layers as a function o f the von Mises stress averaged over the depth and the contact radius for 56 different contact situations. Tlie solid line is obtained from a low cycle fatigue wear model. von Mises stress reaches the yield stress and the whole layer deforms plastically, wear increases enormously. The influence of the normal load, the substrates and the different radii of the pin is very well represented by the influence of the average von Mises stress in the gold layer. The 56 different contact situations from Table 1 fit very well in Figure 3.b. We notice that the same correlation is found for both the evaporated and the electrodeposited gold.
If we apply the theoretical model (in accordance with equation (6)) to the layered system, we have to know the mechanical properties of the gold layer. Since the number of cycles is low for fatigue, we approximate n' with the value from the true stress-strain curve [17], i.e. 0.31. The value of the coefficient l / b n ' then becomes 6.45. The large value of this coefficient indicates that the von Mises stress is extremely dominant in the wear process. In a layered system there i s no simple relation between the deformed volume and the von Mises stress. The variation in the deformed volume, however, is small compared with the variation caused by the von Mises stress to the power of 6.45. The experimental results in Figure 3.b also show the relative unimportance of the deformed volume V. In this figure wear was plotted against the average von Mises stress without taking the deformed volume into account. For simplicity we therefore take the volume V as a constant. Now we can construct a theoretical curve in Figure 3.b if we assume a value for k p x V (in this case 1.5 10 I f [N/mm2]-645~m2]). This value is small because K' i n equation (5) is large. In Figure 3.b the theoretical curve according to equation (6) is sketched as a solid line. We note that the theorPtical curve is very close to the experimental points. The good correlation between theoretical curve and experimental values shows that low cycle fatigue i s an important parameter in this wear process. We note that even for the experimental points of Figure 3.b in the elastic regime (average von Mises stress in the layer elastic !!) there can be plastic deformation on a local scale (see Figure 1). In this paper we have not gone into detail about the actual wear mechanism. In fact we have calculated a macroscopic stress field and have not taken microscopic effects from for instance roughness asperities into account. A more refined wear model should also account for these effects. We note in Figure 3.b that for low von Mises stress the experimental values are somewhat larger than the theoretical ones. This discrepancy could be caused by surface asperities influencing the macroscopic stress field or the ploughing of the particles generated. Improvements on existing finite element packages, like the possibility to have two deforming bodies and calculations for 3-D situations, could widen the applicability of the proposed model. For the systems investigated, being rather ductile layers, the amount of plastic deformation and
334
thus the von Mises stress is a good criterion. The combination of calculations and experiments could also be applied to hard and brittle surface layers like the carbides, nitrides o r oxides. The von Mises criterion should then be replaced by a tensile stress criterion or a critical stress intensity factor 1191.
7. CONCLUSIONS It was possible to predict the wear of a layered system with reasonable accuracy by means of a low cycle fatigue wear model. This model gave the relation between the wear and the von Mises stress. The experimental contact situation could be modelled to calculate the von Mises stress. Using these calculations the influence of the normal load, the substrate and the radius of the indenter on the wear of a layered system could be described. In other words the influence of the tribological system parameters on wear could be determined via the von Mises stress, introduced i n equation (6). ACKNOWLEDGEMENT Dr. Ir. E.A. Muijderman and Ing. P.J.M. Wijnhoven contributed significantly to the present work. Their help is gratefully acknowledged.
REFERENCES CZICHOS H. ’Importance of properties of solids to friction and wear behaviour’ in: , Int. Conf., NASA Tribology i n the ~ O ’ S Proc. Lewis Research Center Cleveland, Ohio, April 18-21, 1983, p 71. KRAG ELSKl I I.V., MARCHENKO E.A. ’Wear of machine components’ Trans. ASME, J. Lub. Techn., 104, jan. 1984, p. 1. TABOR D. ’Wear-A critical synoptic view‘ Trans. ASME, J. Lub. Techn., Oct. 1977, p. 387. ARCHARD J.F. ‘Contact and rubbing of flat surfaces’ J. Appl. Phys., 24, 1953, p. 981. RABINOWICZ E. ’The wear coefficient magnitude, scatter, uses‘ Trans. ASME, J. Lub. Techn., 103, April 1981, p. 188. VERBEEK H. wear factors’
’Tribological systems and Wear, 56 (1979), p. 81.
HALLING J. ’Towards a mechanical wear equation’ Trans. ASME, J. Lub. Techn., April 1983, 105, p. 213. BAYER R.G., KU T.C. ‘Handbook of analytical design for wear’ Plenum Press, New York, 1964. PALMGREN ’Grundlagen Waelzlagertechnik’ Stuttgart, 1964.
Der
GLARDON R . , FlNNlE I. ‘A review of the recent literature on the unlubricated sliding wear of dissimilar metals’ Trans. ASME, J. Eng. Mat. Techn., Oct. 1981, 103, p. 333. JAHANMIR S. ’On the wear mechanisms and the wear equations‘ in: Fundamentals of Tribology, Suh N.P., Saka N. editors, MIT Press, 1978, p. 455. CHALLEN J.M., OXLEY P.L.B., HOCKENHULL B.S. ’Prediction of Archard’s wear coefficient for metallic slidi ng friction assuming a low cycle fatigue wear mechanism Wear, 111 (1986) p.
275.
’Manual on Low Cycle Fatigue testing’ American Society for Testing and Materials, 1969, p. 1. ANTLER M. ’Sliding wear of metallic contacts’ IEEE Trans. Comp. Hybr. and Manuf. Techn., March 1981, CHMT-4 p.15. TANGENA A.G., WlJNtiOVEN P.J.M. ’The correlation between mechanical stresses and wear i n a layered system’ Wear, 121 (1988) p. 27. TANGENA A.G. ’Tribology of thin film systems’ Ph. D. Thesis University of Technology Eindhoven, April 1987. TANGENA A.G., WIJNHOVEN P.J.M., MUIJDERMAN E.A. ’The role of plastic deformation i n wear of thin films: A comparison between FEM calculations and experiments’ Trans. ASME, J. Tribology, October 1988, p. 602. TANGENA A.G., HURKX G.A.M. ‘The determination of mechanical properties of thin metal films using indentation tests’ Trans. ASME, J. Eng. Mat. Techn., July 1986, 108, p. 230. TANGENA A.G., FRANKLIN S., FRANSE J. ’Scratch tests of hard layers’ Proc. Leeds-Lyon Symposium, September 1989.
Paper XIV (iii)
A low cycle fatigue wear model and its application to layered systems A.G. Tangena
Wear is a very complex subject. Wear research i s therefore mostly restricted to the description ofphenomena. In practice, however one is more interested in the prediction of wear. A wear model based on low cycle fatigue has been developed that describes the influence of plastic deformation on wear. Used on layered systems, this model makes it possible to predict the influence of the substrate, the normal load and the contact geometry on wear. 1. INTRODUCTION Whenever we want to write on a piece of paper we note how important the supporting base is. Paper on a soft support will,tear, while on a hard table we can easily write on it. The shape of the pen is also important. With a sharp pencil we will more easily penetrate the paper than with a large brush. Even the environment is important. Just try to write on paper in the rain. Obviously writing on a piece of paper depends on a complete system. There is an analogy in wear. It is now wellknown that wear is not a material property, but i s characterized by the complete tribological system (Czichos [I]). Such a system consists of two bodies, an intermediate medium and the environment. Adhesion, geometry, roughness and the contact load determine the forces transmitted from one surface to the other. Wear in turn depends on the response of the contact materials to these loads, in other words on their deformation and fracture behaviour. Because of the complexity of such a tribological system most researchers confine themselves to the description of wear phenomena, without attempting to construct wear models that could serve as the basis for rational design procedures. An overview of wear models, relevant calculation procedures and the physics of the wear process can be found in Kragelskii [Z] and Tabor [3]. The most well-known wear equation is Archard's [1].He assumed that the volume of a wear particle was proportional to the third power of the contact radius between the two bodies. In that way he obtained: k Fn x distance fravelled Wearvolume = -
H
(1)
where k is the wear constant [5,6], f , the contact load and H the hardness. The wear equation is determined by the nature of the roughness distribution [7]. For very smooth surfaces, or in cases where plastic deformation becomes predominant, equation (1) is no longer valid. For layered systems the hardness H is not easy to define, since
it depends on the geometry of the system [7].' Evidently equation (1) has its limitations. A wear model that was often used i n the sixties is the 'zero wear' model of Bayer and Ku [8]. In this model fatigue is postulated as the main cause for the formation of wear particles. The zero wear theory states that wear can be controlled by limiting the maximurn shear stress in or under the contact area or, more specifically: wear can be at a practically zero level for a certain number of passes n when the shear stress T is smaller than or equal to r(n) x T,,, , where fln) is a function depending on the lubrication regime of the system. The zero wear model uses the elastic Hertz formulas and Palmgren's empirical relation between life and load for ball-bearings [9]. So the zero wear model defines wear-no-wear situations. In many cases, however, we are interested in situations where measurable wear occurs.
2. LOW CYCLE FATIGUE WEAR MODEL In our opinion a wear model must be based upon the physics of the wear process, taking into account the complete tribological system. If we confine ourselves to reasonably ductile materials we note that wear depends on the response of materials to cyclic loads, i.e. wear is concerned primarily with cyclic plastic deformation, crack initiation and crack growth (Glardon [lo], Jahanmir [ I l l ) . These are also the driving processes in low cycle fatigue (Challen [12]). Because of this analogy with the wear process a model was constructed that is based on a formulation analoguous to the well-known fatigue relation of Coffi n-Manson: b
x Nc = cc = constant.
(2)
In this formula AcP is the plastic strain per cycle and cc are the critical number of cycles and the critical strain necessary to originate fracture, while the superscript b is a fatigue constant with a value of approximately 0.5.
N, and
330
If we apply equation (2) to the wear process and the number of wear cycles i s larger than the critical number N,, there will be wear. The wear volume per unit distance travelled W is taken proportional to the number of wear cycles n divided by the critical number of cycles N,:
(3) where k , is an arbitrary wear constant and V the de-formed volume. Combination of (2) and (3) gives: 1 ._ b
W = k, V n [ % ] The amount of plastic deformation per cycle AE, i s associated with the applied von Mises stress (true stress) if through a cyclic stress-strain curve [13]:
F
=
n'
K' AcP,
where n' is the cyclic strain-hardening coefficient and K' a constant. The values of K' and n' depend on the number of cycles. The coefficient n' changes from some initial value to a value between 0.1 and 0.2 at large numbers of cycles for most metals [13]. If finally we combine equations (4) and (5),the wear volume per unit distance travelled W becomes:
Typical values of I/bn' are around 10. The large value of this coefficient indicates that one system parameter, the von Mises stress in the stressed volume, is dominant i n the wear process and that the response to this parameter determines the wear i n a low cycle fatigue process. The magnitude of the von Mises stress depends on the contact geometry, the mechanical properties of the materials, the friction and the externally applied forces. So this wear model indicates that the total tribological system determines the wear behaviour. 3. TESTING OF THE WEAR MODEL For testing the wear model we choose a wide range of experimental circumstances (see Table 1). Such a wide range can be created relatively easily if we use a layered system (Antler [14], Tangena [IS]). Variation of the substrate, the normal load and the geometry of the contact gives different von Mises stresses in the layer. In conformity with equation (6) the wear of the layer should also be different. Details of these experiments can be found i n the author's thesis [I61 and in reference [1'7]. The main points will be explained here. For the contact geometry a ball (pin) on a flat was chosen. The ball was made of ball-bearing chromium steel (DIN 5041). The flat consisted either of a substrate of soft annealed copper (OFHC-CU 99.9%, H, 50) or a hard glass (H, 4800) with several
layers on top. In a number of cases a hard sulphamate nickel layer was electrodeposited on the copper substrate. The top layer consisted of pure gold (99.9 %). This layer was applied either by evaporation i n vacuum or by electrodeposition. The pin was made of ball-bearing steel, so that we only had to account for the wear of the layered system. In this investigation the stresses i n the layer were determined by finite element calculations. The calculated stress was then correlated with the wear volume according to equation (6). 4. FINITE ELEMENT CALCULATIONS If we want to calculate the stresses in a complicated contact geometry we face the limitations of temporary finite element packages. 1. Our finite element package can only calculate contact situations with a rigid indentation body. This means that we have to choose contact situations where one body has a much larger Young's modulus and yield stress than the counter body, so that one body can be considered rigid. When the substrate i s hard as i n the case of the glass substrate, the radius of the indenter must be adjusted i n the calculations, since then the pin deforms elastically as well. 2. In a wear experiment the ball slides over the surface. Since it i s very difficult to calculate the actual 3-D stress situation, because of the amount of memory and computer time necessary, we have to approximate the stress situation i n the layered system. When the friction coefficient is low we do not make too large a mistake when we approximate a sliding situation with a n axially symmetric indentation [16]. In the experiments the friction coefficient was always around 0.1, so this approximation is justified. 3. The materials used i n the layered system deform not only elastically but also plastically. Mechanical behaviour i n the plastic region i s mostly expressed i n terms of a true stress-strain curve, where the von Mises stress at which yielding occurs is expressed as a function of the effective strain. This stress-strain curve i s mostly determined i n a uniaxial tensile strength test. For thin films such a test is virtually impossible. Therefore an alternative method was developed to obtain the stress-strain curve from a Brinell indentation test [18, 161. Even though both types of gold were applied via different processes, the mechanical properties proved to be practically the same. The finite element mesh used was axially symmetric. The gold layer was represented by 132 and the substrate by 396 elements. The indenter was simulated by 65 so-called gap elements. More details can be found in reference [17]. Because wear occurs primarily i n the top layer, we evaluated the stresses i n this layer. Since equation (6) shows the importance of the von Mises stress this stress is given i n Table 1. The stress during the first indentation was taken as an approximation for the stress during the wear process, so "shake-down" was not taken into account. The contact situations calculated are given in Table 1 : four normal loads, four different pin radii, two substrates, a nickel intermediate layer and two
33 1 Table 1. Results of FE calculations and wear experiments. - 7a diu: Au Substrate Ni pin aye1 aye1 rmm] -
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3.0 3.0
3.0 3.0 3.0 3.0 3.0 3.0 3.0
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3.0 3.0 3.0 3.0 1.0 1.0 1.o 1.0
0.0 0.0 0.0 0.0 4.0
4.0 4.0 4.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
4.0 4.0 4.0 4.0 0.0
0.0 0.0 0.0
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thicknesses for the gold layer. We will not discuss all situations separately, but instead indicate the influ-. ence of important parameters. As a rule, an increasing normal load gives an increase in the von Mises stress, while the deformations in the layer can change from purely elastic to plastic. The substrate has a very strong influence on the stresses in the gold layer (Figure 1). The three vertical lines in this figure indicate contact radii. The horizontal solid line i s the yield stress of the gold (114 N/mm2).We note that with a soft copper substrate the stress and also the contact radius are large. Using a hard nickel layer between the gold and the copper substrate reduces the stresses, especially i n the centre of the contact. The contact radius is only slightly smaller than without the nickel layer. The
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hard glass substrate reduces the contact radius considerably, while the von Mises stress is low over tho entire contact area. The contact radius increases slightly and the vori Mises stress decreases with increasing pin radius. The influence of the layer thickness of the gold was investigated for a 1 prn and a 3 prn gold layer on a glass substrate. A thicker layer will give lower stresses. This i s understandable since both the gold layer and the glass substrate remain elastic. In that case the gold layer is no longer softer than the glass substrate. On the contrary, because the gold layer has a higher Young’s modulus than the glass = 79 G P a , €,/, = 70 GPa), the system behaves as a system with a hard layer on a soft substrate. The thicker the hard layer, the lower the stresses.
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The wear results of the layered system are indicated in Table 1. The cross-section of the wear track increased with increasing normal load. The substrate had a large influence on the wear of the gold film. A soft copper substrate gave considerable wear. A hard nickel layer in between the gold and the copper reduced wear, while a glass substrate reduced wear to practically zero. Increasing the radius of the pin resulted in a wider wear track, but despite this effect the cross-section of the track decreased with increasing radius. The thickness of the gold film had little influence on the wear. Both types of gold, the evaporated and the electrodeposited, showed the same trends in wear.
150
Radio1 position ( p m )
Figure 1. The von Mises stress in the gold top layer as a function of the radial position for an axially symmetric system, indented by a rigid ball with a radius of 12.5 mm, at a normal load of 4 N . The layered system consists of a 3.0 pm thick gold film on a copper substrate with or without a hard nickel intermediate layer and on a glass substrate.
5. WEAR EXPERIMENTS A precision wear apparatus was constructed in order to do accurate wear experiments. The apparatus consisted of a flat, that was linearly sliding under a spherical pin. Both the flat and the pin were airborne. Details can be found in reference [17]. The normal load, the position of the plate in a horizontal plane (x,y) and the speed were controlled, while the normal load, the shear force and the height of the pin were measured by a microcomputer. The data were evaluated on a mainframe computer. The wear tracks were measured with a modified profilometer. Special software was written, that calculated the cross-section and the depth of a wear track. The width of the wear tracks measured with an optical microscope correlated well with the width measured using the profilometer. The microscope was also used to check for transfer to and wear of the pin. Wear experiments were performed on two types of pure gold. One type was applied by evaporation in vacuum, and one type by electrodeposition. The evaporated layer was tested with a soft copper and a hard glass substrate, while the electrodeposited layer was used with a soft copper substrate with or without a 4 pm-thick hard nickel layer. The wear experiments were performed with several radii of the pin at 4 different loads. This resulted in 32 contact situations for the evaporated gold and 24 for the electrodeposited gold (see Table 1). During wear the contact area increased, since one of the contact bodies was a hard wear-resistant pin. This resulted i n an equilibrium situation, in which the layered system did not wear any more. For all situations the cross-section of the wear track was measured as the difference in cross-section between 1 and 250 cycles to compensate for pure plastic deformation of the substrate. A modified Talysurf profilometer was used. As the results were reproducible within 10 %, it was considered sufficient to measure every situation twice. The wear tracks had a grooved appearance, indicating that small wear particles were generated, which subsequently ploughed the surface (see Figure 2 ) .
Figure 2. The wear track after 250 cycles for a 3.0 pm thick evaporated gold layer on a copper substrate. The normal load was 4.0 N , the radius.of the pin 2 . 5 m m , and the width of the wear track 305 pm. 6. COMPARISON WITH THE THEORETICAL MODEL
The results of the wear experiments were first evaluated with the aid of equation ( I ) , the Archard equation. In figure 3.a the cross-section of the wear track in pm2 as a function of the normal load divided by the hardness for both the evaporated and the electrodeposited gold are represented. For the hardness we have taken the hardness of the substrate. This figure shows that the correlation of the experiments with equation (1) is not good. This is caused by the fact that every different radius of the indenter and every change in the substrate (Ni layer) gives a different k-factor in equation (1). So the use.fulness of equation (1) is limited. We have also evaluated the results of the wear experiments with the aid of equation (6). For this purpose we plotted in Figure 3.b the cross-section of the wear track as a function of the calculated von Mises stress averaged over the layer thickness and the contact radius ( in N / m m 2 ) .In this figure the yield stress is represented by the vertical line at 114 N/mm2.We notice the strong correlation between the von Mises stress and the wear. When the average
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v. Mises stress (N/mmz) Figure 3.b. Cross-section of the wear track for both evaporated and electrodeposited gold layers as a function o f the von Mises stress averaged over the depth and the contact radius for 56 different contact situations. Tlie solid line is obtained from a low cycle fatigue wear model. von Mises stress reaches the yield stress and the whole layer deforms plastically, wear increases enormously. The influence of the normal load, the substrates and the different radii of the pin is very well represented by the influence of the average von Mises stress in the gold layer. The 56 different contact situations from Table 1 fit very well in Figure 3.b. We notice that the same correlation is found for both the evaporated and the electrodeposited gold.
If we apply the theoretical model (in accordance with equation (6)) to the layered system, we have to know the mechanical properties of the gold layer. Since the number of cycles is low for fatigue, we approximate n' with the value from the true stress-strain curve [17], i.e. 0.31. The value of the coefficient l / b n ' then becomes 6.45. The large value of this coefficient indicates that the von Mises stress is extremely dominant in the wear process. In a layered system there i s no simple relation between the deformed volume and the von Mises stress. The variation in the deformed volume, however, is small compared with the variation caused by the von Mises stress to the power of 6.45. The experimental results in Figure 3.b also show the relative unimportance of the deformed volume V. In this figure wear was plotted against the average von Mises stress without taking the deformed volume into account. For simplicity we therefore take the volume V as a constant. Now we can construct a theoretical curve in Figure 3.b if we assume a value for k p x V (in this case 1.5 10 I f [N/mm2]-645~m2]). This value is small because K' i n equation (5) is large. In Figure 3.b the theoretical curve according to equation (6) is sketched as a solid line. We note that the theorPtical curve is very close to the experimental points. The good correlation between theoretical curve and experimental values shows that low cycle fatigue i s an important parameter in this wear process. We note that even for the experimental points of Figure 3.b in the elastic regime (average von Mises stress in the layer elastic !!) there can be plastic deformation on a local scale (see Figure 1). In this paper we have not gone into detail about the actual wear mechanism. In fact we have calculated a macroscopic stress field and have not taken microscopic effects from for instance roughness asperities into account. A more refined wear model should also account for these effects. We note in Figure 3.b that for low von Mises stress the experimental values are somewhat larger than the theoretical ones. This discrepancy could be caused by surface asperities influencing the macroscopic stress field or the ploughing of the particles generated. Improvements on existing finite element packages, like the possibility to have two deforming bodies and calculations for 3-D situations, could widen the applicability of the proposed model. For the systems investigated, being rather ductile layers, the amount of plastic deformation and
334
thus the von Mises stress is a good criterion. The combination of calculations and experiments could also be applied to hard and brittle surface layers like the carbides, nitrides o r oxides. The von Mises criterion should then be replaced by a tensile stress criterion or a critical stress intensity factor 1191.
7. CONCLUSIONS It was possible to predict the wear of a layered system with reasonable accuracy by means of a low cycle fatigue wear model. This model gave the relation between the wear and the von Mises stress. The experimental contact situation could be modelled to calculate the von Mises stress. Using these calculations the influence of the normal load, the substrate and the radius of the indenter on the wear of a layered system could be described. In other words the influence of the tribological system parameters on wear could be determined via the von Mises stress, introduced i n equation (6). ACKNOWLEDGEMENT Dr. Ir. E.A. Muijderman and Ing. P.J.M. Wijnhoven contributed significantly to the present work. Their help is gratefully acknowledged.
REFERENCES CZICHOS H. ’Importance of properties of solids to friction and wear behaviour’ in: , Int. Conf., NASA Tribology i n the ~ O ’ S Proc. Lewis Research Center Cleveland, Ohio, April 18-21, 1983, p 71. KRAG ELSKl I I.V., MARCHENKO E.A. ’Wear of machine components’ Trans. ASME, J. Lub. Techn., 104, jan. 1984, p. 1. TABOR D. ’Wear-A critical synoptic view‘ Trans. ASME, J. Lub. Techn., Oct. 1977, p. 387. ARCHARD J.F. ‘Contact and rubbing of flat surfaces’ J. Appl. Phys., 24, 1953, p. 981. RABINOWICZ E. ’The wear coefficient magnitude, scatter, uses‘ Trans. ASME, J. Lub. Techn., 103, April 1981, p. 188. VERBEEK H. wear factors’
’Tribological systems and Wear, 56 (1979), p. 81.
HALLING J. ’Towards a mechanical wear equation’ Trans. ASME, J. Lub. Techn., April 1983, 105, p. 213. BAYER R.G., KU T.C. ‘Handbook of analytical design for wear’ Plenum Press, New York, 1964. PALMGREN ’Grundlagen Waelzlagertechnik’ Stuttgart, 1964.
Der
GLARDON R . , FlNNlE I. ‘A review of the recent literature on the unlubricated sliding wear of dissimilar metals’ Trans. ASME, J. Eng. Mat. Techn., Oct. 1981, 103, p. 333. JAHANMIR S. ’On the wear mechanisms and the wear equations‘ in: Fundamentals of Tribology, Suh N.P., Saka N. editors, MIT Press, 1978, p. 455. CHALLEN J.M., OXLEY P.L.B., HOCKENHULL B.S. ’Prediction of Archard’s wear coefficient for metallic slidi ng friction assuming a low cycle fatigue wear mechanism Wear, 111 (1986) p.
275.
’Manual on Low Cycle Fatigue testing’ American Society for Testing and Materials, 1969, p. 1. ANTLER M. ’Sliding wear of metallic contacts’ IEEE Trans. Comp. Hybr. and Manuf. Techn., March 1981, CHMT-4 p.15. TANGENA A.G., WlJNtiOVEN P.J.M. ’The correlation between mechanical stresses and wear i n a layered system’ Wear, 121 (1988) p. 27. TANGENA A.G. ’Tribology of thin film systems’ Ph. D. Thesis University of Technology Eindhoven, April 1987. TANGENA A.G., WIJNHOVEN P.J.M., MUIJDERMAN E.A. ’The role of plastic deformation i n wear of thin films: A comparison between FEM calculations and experiments’ Trans. ASME, J. Tribology, October 1988, p. 602. TANGENA A.G., HURKX G.A.M. ‘The determination of mechanical properties of thin metal films using indentation tests’ Trans. ASME, J. Eng. Mat. Techn., July 1986, 108, p. 230. TANGENA A.G., FRANKLIN S., FRANSE J. ’Scratch tests of hard layers’ Proc. Leeds-Lyon Symposium, September 1989.