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Experimental and numerical analysis of friction and wear behavior of polycarbonate
Wear 251 (2001) 1541–1556
Experimental and numerical analysis of friction and wear behavior of polycarbonate J.H. Lee, G.H. Xu, H. Liang∗ Department of Mechanical Engineering, University of Alaska, Fairbanks, AK 99775, USA
Abstract Static and kinetic contact of a steel ball against a polycarbonate half-plane was experimentally examined to study its friction and wear behavior. Changes in friction were correlated with the apparent wear volume. The finite element method, using a physically based visco-plastic constitutive model, simulated the initial stage of sliding contact through indentation, sliding and unloading stages. Detailed deformation history was examined to understand the friction and wear mechanisms for the glassy polymeric material. The experimental and numerical results correlated well, qualitatively, in the coefficient of friction and the deformed surface profile. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Polycarbonate; Polymer; Friction
1. Introduction Many polymeric materials have an excellent strengthto-weight ratio. They often are easy to manufacture. Some polymers also possess excellent tribological properties [1]. Polycarbonate is a widely used thermoplastic polymer for its high hardness and toughness either as a single phase material or as the matrix of a composite material. The friction and wear of polycarbonate has been the subject of many studies [2–4], mostly experimental. There has been fewer theoretical studies on the friction and wear behavior of polymers in general, and polycarbonate in particular. Wear of polymers is more complicated than that of metals. Studies have shown that polymer wear undergoes two main mechanisms, deformation wear and interfacial wear [5]. Deformation wear involves abrasive and fatigue wear while the interfacial wear involves adhesive or transfer wear. So far, there is no agreement in the literature that when deformation wear and interfacial wear should be separated. Many researchers have found a particular surface roughness at which the dominance of the two wear modes changes [6–8]. Different pairs of rubbing materials have different roughness. In this initial study, we focus on the deformation behavior of a glassy polymer, i.e. polycarbonate. Whereas a general deformation map for polycarbonate can be constructed [2], a detailed mechanistic understanding of the friction and wear of polycarbonate is still yet to ∗ Corresponding author. Tel.: +1-907-474-6135; fax: +1-907-474-6141. E-mail address: [email protected] (H. Liang).
emerge. A large number of variables can contribute to different regimes of the deformations of polymers and there is a possibility of transitions from one regime to another. In addition, it would be useful if one can link the mechanistic results with the experimental ones towards a better, validated understanding of the phenomenon. Modeling of the tribology of polymers is often done using constitutive equations developed for metals [9]. The physical mechanisms of the plastic deformations of polymers are very much different from those of metals. Consequently, using physically-based constitutive equations for the modeling of the friction and wear of polymers may gain a better understanding of their behavior. Glassy polymers are usually different from metals in aspects, such as a larger elastic zone, pressure and strain-rate sensitivity, strain-hardening due to the orientation of polymer chains (orientation hardening), strain-softening and a more viscous behavior. In this paper, we present numerical and experimental results of our initial attempt towards a better understanding of the friction and deformation wear behavior of polycarbonate. For numerical modeling we concentrate on the initial stage of single-pass sliding contact between a hard and a soft surface, the latter modeled as a half-plane made of polycarbonate. The experiments used a pin-on-disk tester to conduct sliding tests. We also used a surface profilometer and an optical microscope to obtain wear volume and topography of wear tracks. In the following, we first present experimental procedures and results on the sliding contact of a steel ball against a polycarbonate disk. We then present a summary of the
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constitutive equations used in the finite element model. Numerical modeling procedures are then described and detailed results on significant mechanical variables as well as the topography of wear tracks during the initial indentation, the subsequent sliding and the final unloading stages are presented in detail. Finally, we discuss the significance of the numerical results in relation to the experimental ones.
2. Experimental procedure 2.1. Materials and preparation Disks with a dimension of ∅ 160 mm × 12 mm were machined from a polycarbonate sheet. A 52100 steel ball with a diameter of 9.5 mm and surface finish of 0.02 m (Ra) was used as counterbody for contact against the polycarbonate disk. A schematic drawing of the contact configuration is shown in Fig. 1. Before the experiments were conducted, specimens were cleaned with alcohol and then dried with an air blow dryer. 2.2. Experimental method and equipment A series of sliding wear tests were conducted on a pin-on-disk tester. A steel ball was fixed in a sample holder which was mounted on a loading level arm used to apply a normal load of 19.6 N. The polycarbonate disk was adhered on the rotating disk that was driven by a variable speed motor. The rotating speed was 23 rpm and the wear track diameter was 20 mm. The steel ball slides over the polycarbonate disk at a linear velocity of 48 mm/s. The friction force developed was measured by a force sensor and displayed on a control panel. The experiment was finished after sliding a distance of 86.7 m within 30 min, at a room temperature of 19.1◦ C. Before and after wear tests, the surface roughness was measured and wear track profiles of the polycarbonate disk were mapped with an automated 3D stylus profilometer system. The system was equipped with a diamond cone stylus with a spherical tip of 5 m in radius. The mean surface roughness of the polycarbonate was 0.04m. Before mapping the wear track, the polycarbonate disk was cleaned with alcohol again so that the wear particles could be cleared away
Fig. 1. Experimental sliding contact configuration.
from the examined surface. As the wear track was a circle, its topography was profiled along a length of 12.5 mm in the tangential and radial directions, respectively. Four measurement at least were carried out randomly along the wear track. With the wear profile measured in the tangential direction, the elastic or plastic deformation of the polycarbonate around the contact area could be studied. And by obtaining the topography of the wear track in the radial direction, the wear characteristics of the polycarbonate could also be investigated. The wear volume of polycarbonate was calculated approximately by the formula, v = (2π r 2 h/3), r is radius of steel ball, h the depth of wear track. It should be noted that the wear track measured in the tangential direction corresponds closer to the numerical model to be discussed later. 2.3. Experimental results The evolution of the friction coefficient with time under conditions mentioned above is given in Fig. 2. It can be seen that the friction coefficient increased rapidly at the beginning of the experiment. After sliding for 4 min, the friction coefficient increased slowly and then showed another rapid increase period. The value reached a maximum of 0.31 when the counterbodies rubbed against each other for 9 min, equal to a sliding distance of 26 m. Then the friction coefficient decreased quickly and reached the steady-state value of 0.28 at 12 min. The variation of wear volume of the polycarbonate as a function of time is shown in Fig. 3. It can be concluded that severe wear took place during the initial period. Then the steady stage was reached after sliding for 8 min, corresponding to the results of friction coefficient. When the experiment would finish, the wear volume showed a little increase due to the abrasion of the wear debris. The profile map of the wear trace in the tangential direction is shown in Fig. 4(a). Accordingly, only the marked range, located at the deepest area of the wear trace, should be recognized as the contact spot to which the largest contact pressure applied. So the marked range of the profile was expanded in order to show the deformation behavior of
Fig. 2. The evolution of experimental friction coefficient with sliding time.
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Fig. 3. The variation of wear volume as a function of time.
polycarbonate around the contact area (Fig. 4(b)). In other words, the contact configuration could be approximately regarded as a steel ball sliding along the polycarbonate in a direct line, instead of around a circle. It can be seen that the maximum penetration depth was 28 m. When the
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sliding motion continued, the material in front of the steel ball was extruded forward to form a surface peak which was reduced to a height of 10 m due to the slider pushing. Subsequently, the steel ball penetrated the surface of the polycarbonate again with a depth of 15 m. After climbing over a small peak, the steel ball extruded the material and produced a peak with a height of about 25 m. Fig. 5 showed the topography of the wear track of polycarbonate in the radial direction. It showed clearly that there was a sharp peak to the left of the wear scar with a depth of 185 m and a width of 400 m. The distance between the peak with the height of 165 m and the wear scar was 450 m. So, the peak could be the extruded materials or the heaped wear particles produced by the sliding motion of the counterbodies. Four sliding wear experiments were performed under the same testing conditions. The difference in the results for the experiments was very small. The error in friction coefficient was lower than 6%. So, it was concluded that the sliding contact experiments were well controlled and repeatable.
Fig. 4. (a) The topography of the wear track in the tangential direction; (b) the expanded typography of the marked range of (a).
Fig. 5. The topography of the wear track in the radial direction.
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3. Finite element modeling
According to this model, the total stress Tij has two components
In the following, we first present the constitutive equations used in the model. The finite element analysis procedure is described next. The results of the simulations are then discussed.
Tij = Tij∗ + Bij
3.1. Physically-based visco-plastic model The plastic deformations of glassy polymers have several important characteristics that are more pronounced than or not found in metals. These characteristics include strain softening, pressure-dependency [10], higher strain-rate dependency and orientation hardening [11] which can be attributed to different physical mechanisms of plastic deformations. We will use the model developed in [12] which is a generalization of the one by [11]. It should be noted that although elastic deformation is fully taken into account, for simplicity, this type of model is often classified as visco-plastic rather than visco-elastoplastic. The model can be approximately broken down into two parts. The first part assumes that there is an isotropic resistance to plastic flow due to molecular chain segment rotation [13]. This resistance is manifested in rate and pressure dependency, and strain softening. The second part assumes that there is an anisotropic resistance to molecular chain alignment which is similar to rubber elasticity [11,14]. This anisotropic resistance is manifested in the (orientation) hardening of the material. In the following, index notations will be used where repeated indices are summed. The isotropic plastic shear resistance γ˙ p is τ 5/6 A˜s γ˙ p = γ˙0 exp − 1− (1) Θ s˜ where γ˙0 is the pre-exponential shear strain rate factor, s˜ the modified athermal shear resistance, Θ the absolute temperature, A˜s the zero stress activation free energy divided by Boltzmann’s constant and τ the second invariant of the ‘driving’ stress tensor Tij∗ to be defined below. The (pressure) modified athermal shear resistance s˜ is αp s˜ = s 1 + s where s is the athermal shear resistance, α the pressure dependence constant, and p is the hydrostatic pressure. The evolution of s˙ is given by s γ˙ p s˙ = h 1 − sss where h is the slope of the yield drop with respect to the plastic strain (strain-softening), sss the value of s reaching a steady-state with an initial value of 0.077(µ/1 − ν), where µ is the shear modulus, ν the Poisson’s ratio.
(2)
where Bij is a back stress (responsible for orientation hardening) and Tij∗ the driving, or, reduced stress (responsible for other effects, such as pressure-dependency, etc.). The principal components of the back stress tensor are defined as N
λj λL N −1 λN 1 i −1 (3) Bi = CR λi L − λN jL 3 λL 3 λL j
where λN i is the principal network stretch component (the principal plastic stretch), λL the tensile locking network stretch (the natural draw ratio), and CR the rubbery modulus, L−1 the inverse Langevian function, the Langevian function for a parameter βi is defined as L(βi ) = coth(βi ) −
1 βi
The general structure of the equations is similar to that of a kinematic hardening model with a few important differences. Firstly, the back stress is obtained as principal values which are then transformed into the directions of the coordinate system. Secondly, unlike conventional kinematic hardening models, the back stress is not defined in a rate form but in terms of the principal plastic stretches. The von Mises stress is defined with respect to the total stresses as σ = 23 Tij Tij (4) The second invariant of Tij∗ , τ , is then defined as τ = 21 Tij∗ Tij∗
where Tij∗ is the deviatoric stress of Tij∗ , such that
Tij∗ = Tij∗ + (1/3)Tij∗ δij , and δij the Kronecker delta. Finally, the visco-plastic strain rate is defined as
1 vp ε˙ ij = γ˙ p √ Tij∗ 2τ
(5)
Note that, according to this definition, the visco-plastic strain rate is only deviatoric, i.e. without a hydrostatic component. We also define the equivalent visco-plastic strain rate as vp vp vp ε˙ = 21 ε˙ ij ε˙ ij (6) The total equivalent visco-plastic strain is then defined as vp ε vp = ε˙ dt (7) t
For finite element calculations, the constitutive equations are linearized and implemented as user subroutines [15] in the general purpose finite element program ABAQUS [16].
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3.2. Description of the numerical method We analyzed the sliding of a hard asperity over an initially flat substrate to simulate the sliding contact of a single asperity over the other one, similar to [17]. The soft asperity is initially flat assuming the applied load is high enough to have flattened it. The hard asperity is assumed to be rigid and the soft one is polycarbonate. To reduce the level of complexity in modeling, we assume that the deformation state is plane strain rather than three-dimensional as is the case for pin-on-disk type of tests. The finite element mesh, shown in Fig. 6, has 1921 elements and 5824 nodes with a dimension of 0.6 mm in the horizontal (sliding) direction and 0.3 mm in the vertical direction. The polycarbonate substrate is modeled by 1800 8-node plane strain elements with reduced integration (CPE8R). The 8-node elements with reduced integration are commonly used in large strain plasticity due to their good performance and efficiency. The rigid asperity is modeled
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using the rigid analytic surface option of ABAQUS which generates 121 rigid contact elements internally. The contact surfaces consist of the rigid contact elements and the elements on the top row of the model. The density of the mesh is graded in the vertical direction to accommodate the higher gradient of parameters near the contact surface. The left and right side of the model are restrained horizontally with the bottom fixed. The diameter of the asperity is 100 m and its center is located initially at 0.15 mm from the left boundary. The asperity will also be referred to as the cylinder in ensuing discussions. The constants used for the calculations are for the materials previously studied in the cold drawing of polycarbonate bars [15,18,19]. The material properties used have a good fit with the experimental data and are as follows, Θ = 300 K, α = 0.2, γ˙0 = 2.463 × 106 /s, A = 253.6 K/MPa, ν = 0.38, n = 5, h = 0.4 GPa, sss = 63 MPa, CR = 8.57 MPa. Here s0 , the initial value of the athermal shear resistance s, is 72 MPa. Two Young’s moduli are used to represent the initial portion of the non-linear stress–strain curve, E = 1.174 GPa(ε ≤ 0.021),
E = 0.37 GPa(ε > 0.021)
The behavior of the rigid elements is prescribed by using a reference node (shown as a cross in Fig. 6). Throughout the simulation, the rigid body is constrained to have zero rotation. The simulations are conducted in three stages, indentation, sliding and unloading. The rigid asperity is first indented onto the polycarbonate half-plane to a prescribed depth and then slides over towards the right. The indentation is under displacement control where the rigid body is also constrained to have zero horizontal displacement. This constraint is needed as the contact algorithm is only approximate which may satisfy a contact condition by introducing an undesirable rotation in the rigid elements. For the sliding stage, the vertical reaction of the reference node at the end of the indentation stage is used as a constant concentrated load. The sliding is controlled by displacement. At the end of sliding, unloading is accomplished by gradually decreasing the force at the reference node to zero. 3.3. Numerical results
Fig. 6. The finite element mesh.
We first present results where the initial penetration depth is 30 m. We assume no adhesion between the asperities such that the macroscopic friction is only due to the plastic deformation of the substrate. The vertical applied force for the sliding process is 12.16 N. The simulation spans a time period of about 68 s over the 300 m sliding distance which resulted in an average speed of 4.4 m/s (0.0044 mm/s). The analysis is highly nonlinear involving large strains, large deformations, visco-plasticity, strain softening and contact. It takes quite a bit of trial and error to come up with the proper size of time steps to maintain the stability of the analysis with good results. The final simulation took about 3 h CPU time on an SGI Octane workstation.
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Fig. 7 shows the evolution of the von Mises stress (Eq. (4)) at four stages of deformations. Fig. 7(a) is for a sliding distance of 15.4 m, shortly after the completion of the indentation stage. Fig. 7(b) is for a a sliding distance of 108 m, or a distance about the diameter of the asperity. Fig. 7(c) is for results at the end of the sliding stage, a distance of 300 m, or three times the diameter
of the asperity. Fig. 7(d) is at the end of the unloading process. For an initial indentation of 30 m, the maximum von Mises stress is around 100 MPa, which induces substantial plastic deformations. During sliding, the maximum von Mises stress typically occurs below the top surface of the model, tangential to and ahead of the cylinder. Fig. 7(b)
Fig. 7. The distribution of the von Mises stress at: (a) initial sliding; (b) intermediate sliding; (c) end of sliding; and (d) end of unloading.
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Fig. 7 (Continued ).
shows the distribution of residual stresses behind the cylinder. The fully developed residual stress zone in the same area is seen clearly in Fig. 7(c), with a maximum value around 40 MPa. The magnitude and shape of this residual stress zone is by and large not changed even after unloading. Fig. 8 shows the evolution of the equivalent plastic strain (Eq. (7)) at the same four stages of deformations as Fig. 7. Fig. 8(a) shows three areas of high plastic strains. The first is near the surface and ahead of the cylinder, which is created
by the initial indentation process and has a maximum value around 36.6%. The second, also caused by the initial indentation process, is beneath the cylinder and towards its left side. The shape of this second area was horizontal but now rotated clockwise caused by the cylinder that is still moving downward. The third area is slightly below and almost tangential to the cylinder and is caused by the cylinder sliding forward. Fig. 8(b) shows the maximum equivalent plastic strain, about 43.3%, occurring near the third area described in Fig. 8(a). The first two areas described in Fig. 8(a) have
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Fig. 8. The distribution of the total equivalent plastic strain at: (a) initial sliding; (b) intermediate sliding; (c) end of sliding; and (d) end of unloading.
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Fig. 8 (Continued ).
been fully developed and are left behind by the moving cylinder. Fig. 8(c), at the end of the sliding process, shows a large plastically deformed region between the initial indentation area and where the cylinder is currently working. The depth of the zone is almost the radius of the cylinder with a maximum value around 67.5%. Fig. 8(d), at the end of unloading, shows that the plastic zone in Fig. 8(c) is almost unchanged. Whereas the equivalent plastic strain describes the current accumulated permanent deformation, the distribution of the instantaneous plastic shear strain rate (1 s1 ) (Eq. (6))
can be illuminating on the location and area of active plastic deformation. Fig. 9(a) shows that, after some initial sliding, plastic deformation is being actively developed ahead, and almost tangential to the cylinder with a maximum value around 8.3%. The zone that has the second highest plastic strain rate is located behind the underside of the cylinder with a maximum value around 5%. The figure shows a transition of the plastic deformations from the initial indentation mode to the current sliding mode. Fig. 9(b) shows that active plastic deformation occurs not only ahead, and almost tangential to, the cylinder with a maximum value of
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Fig. 9. The distribution of the plastic strain rate at: (a) initial sliding; (b) intermediate sliding; (c) end of sliding.
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Fig. 9 (Continued )
about 50% but also occurs at locations slightly farther from the top surface. A similar trend is observed at the end of the sliding process shown in Fig. 9(c). At the end of unloading, the plastic strain rate is practically zero everywhere as there should not be any additional observable plastic deformation. As the plastic deformation of the material is pressuredependent, we also show in Fig. 10(a), at the end of the sliding, the distribution of the mean stress. As expected, the maximum compressive mean stress (about 130 MPa), occurring below the cylinder, is much higher than the tensile one (about 33.4 MPa) occurring on the surface. This indicates that the material follows more the compressive stress–strain curve than the tensile one as shown in Fig. 11. In addition, the hardening of the material model is attributed primarily to the back stress. We also show the distribution of one of the principal back stresses (B1 ) in Fig. 10(b) which has a maximum magnitude of about 48.3 MPa and has a significant contribution to the total stress. It is also interesting to see the wear track developed by the sliding process. When loaded, the deformed profile can be seen from Fig. 7(c). The peak of the profile occurs near the left side of the model and has a magnitude of 9.52 m. The valley of the profile occurs below the cylinder and has a value of 39.5 m, which is 9.5 m more than the initial indentation depth. The height from peak to valley is about 49 m. After unloading, Fig. 7(d) shows a peak of 8.53 m,
about 1 m smaller than the value before unloading, possibly due to viscous effects. The valley is reduced to 14 m which is primarily due to elastic unloading. With a height from peak to valley about 22.5 m, the unloading process restores about half of the initial height. This illustrates the effect of a larger elasticity and the time-dependent nature of the visco-plastic behavior of polycarbonate. We also conducted simulations of the sliding process by using a smaller initial indentation depth of 16 m, corresponding to a vertical force of 7.086 N, while keeping everything else the same as the previous case. The general trend shown above also holds except for the maximum values of parameters and the extent of plastic deformations. For example, the maximum von Mises stress is between 99.4 and 109 MPa during loading and is 49.5 MPa after unloading. At the end of the indentation process, there is very little plastic deformation directly beneath the indentor, with a small amount of plastic strain (0.84%) occurring tangential to and on both sides of the cylinder. The maximum magnitude of the equivalent plastic strain increases to 21.1% at the end of sliding. An indentation of 16 m, comparing with 30 m, also gives a shallower profile. At the end of the sliding process, the peak of the profile is 8.27 m and the valley has a magnitude of 21.78 m which constitutes a total height of about 30 m. After unloading, the peak of the profile is 7.73 m and the valley is 5.97 m with a total height of 13 m.
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Fig. 10. At the end of sliding, the distribution of: (a) the mean stress; (b) the first principal back stress.
Finally, it is of interest to estimate the coefficient of friction (µ) due to asperity contact as modeled here. An average coefficient of friction can be calculated, at selected time steps, by dividing the horizontal reaction at the bottom of the model by the vertical applied force. It should be noted that there is a gradual transition between the indentation and the sliding process as discussed in conjunction with Fig. 9(a). This transition is marked by the value of zero horizontal reaction, at the bottom of the model, which
subsequently increases. The coefficients of friction for the 30 and 16 m indentation cases are shown in Fig. 12 as a function of sliding distance normalized against the diameter of the asperity. The origin of the horizontal axis is measured not respect to when the cylinder starts to move horizontally but with respect to the time when the horizontal reaction becomes zero. Fig. 12 shows µ increases first, and after reaching a plateau, starts to decrease. But at the end of simulations, µ has not yet reached a steady-state
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Fig. 11. The stress–strain curves of polycarbonate in tension and compression.
Fig. 12. Coefficient of friction vs. sliding distance/asperity diameter at two different loads.
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Fig. 13. (a) The depth of the maximum von Mises stress vs. sliding distance; (b) maximum von Mises stress vs. sliding distance.
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value. It should also be noted that this µ is a result of plastic deformation alone as no adhesion is present in the model.
4. Discussion The behaviors we observed in the numerical results are quite different from the results for metals in [17]. With an initial indentation depth of approximately 0.047D (D is the diameter of the cylinder) and a sliding distance of 1.5D, their simulation created two permanent grooves. One is near where the initial indentation started and the other is near where the sliding ended. In comparison, the sliding distance of our model is 3.0D and the initial indentation depth is 0.16 and 0.3D for the two cases presented, respectively. Instead of a groove near the initial indentation spot for metals, our results showed a ‘plateau-like upheaval’ formation of the surface above the initial level, similar to that reported by [20] at the beginning of their scan-scratch test on polycarbonate and further studied by [3]. It is interesting to note that this ‘bump’ was formed for both indentation cases. For the 16 m initial indentation, there was very little initial plastic deformations near the indentor as discussed previously. On the other hand, for the 30 m initial indentation situation, large subsurface plastic deformations have occurred. In other words, the deformation of the former is elastic, whereas the latter is plastic in nature. As discussed in conjunction with Fig. 8 previously, the bump was created primarily by the rotation of the material due to the movement of the cylinder, irrespective of the regimes of deformations (elastic versus plastic). The bumps remain in place even after unloading mainly due to the presence of residual stresses (cf. Fig. 7(d)). For both cases, the mechanisms described in Fig. 9(b) and (c) in [3], crack formation in the bumps followed by its propagation and accumulation, could very possibly occur with repeated sliding. Another difference between our results and those in [17] is in the magnitude of strains and the failure mechanisms that could lead to wear. Almost unreasonably large strain, around six, was obtained in [17]. In our case, although large plastic strains did occur, they were not high enough to cause failure and none of the plastic strains have reached the limiting stretch λL (which is approximately 2.23 for the material used in the model) even after deeper indentation and longer sliding distance than those in [17]. It should be noted that breaking of polymer chains, as a precursor of failure, could occur when λL is reached. This be speaks of the wear-resistant nature of polycarbonate. We also note that the maximum von Mises stress during sliding does not vary a lot (Fig. 7). The extent of plastic zones, however, does. A large plastic zone may lead to higher damage of the material under repeated loading. As an aid to visualize this behavior, we plot the variation of the normal distance of the maximum von Mises stress to the cylinder versus the sliding displacement in Fig. 13(a), and the von Mises stress versus sliding displacement in Fig. 13(b). It is
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clearly seen that, after certain sliding distance, the normal distance continues to increase whereas the maximum von Mises stress remains more or less constant. It is assumed that this normal distance is an approximate measure of the damage of the material under repeated loading and can lead to failure and wear. Previous studies have correlated the surface topography features with single-pass abrasive war in the simplest abrasive wear model [5,21]: φ V = KW tan H where V is the wear volume per unit sliding distance, K the abrasive wear coefficient, W the normal load, φ the average slope of the asperities, and H the hardness of a polymer. We will investigate the relationship between the plastic deformation and wear volume in future studies. It should be noted that the experiments reported here have different conditions than the modeling work. The experiments were conducted at a much faster speed, with a higher load, and over a much longer time frame than the simulations. Consequently, we do not expect to have good quantitative correlations between them. However, we do note that the coefficients of friction at the initial stage of sliding (Fig. 2 and 13) are similar, i.e. 0.12 for experiments and 0.16 for numerical simulations. The average height between the peak and valley of experiments (Fig. 4) is about 30 m, which is close to the 22.5 m in the simulation (with 30 m initial indentation). The similarities between experiments and modeling could be attributed to the fact that, after multiple passes, steady-state conditions may be achieved eventually. Since the wear of the material (Fig. 4) correlates well with the coefficient of friction (Fig. 3), we feel optimistic in correlating our numerical modeling with experiments in the future for multi-pass situations.
5. Conclusions We conducted preliminary investigations of the sliding contact of a steel ball against a polycarbonate half-plane using a combined numerical and experimental approach. The numerical simulation used more realistic polymer-specific constitutive equations. An initial bump, elevated above the original surface, was found and its geometric origin was elucidated. Detailed history of important variables such as the von Mises stress, the equivalent plastic strain and the shear plastic strain rate were obtained to gain insight into the formation of surface profiles and the accumulation of plastic deformation which may lead to damage under repeated sliding. Despite differences in experimental and numerical conditions, our experimental data did show similarity in the surface profile of the wear track and the coefficient of friction with numerical results. We also found a possible correlation between the profile of the wear track and the depth of the maximum von Mises stress.
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Acknowledgements The authors wish to thank the Arctic Regions Supercomputing Center (ARSC) for computing resources. Part of this work was supported by the President’s Special Project Fund and Alaska Space Grant Program at UAF. We also acknowledge the assistance of Dr. T. Zhang in the preparation of the manuscript.
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[8] N.S. Eiss, J.H. Warren, The effect of surface finish on the friction and wear of PCTFE plastic on mild steel, Paper no. IQ75-125, Soc. Manu. Eng., Detroit, MI, 1975. [9] K. Sadeghipour, W.Chen, G. Baran, Spherical micro-indentation process of polymer-based materials: a finite element study, J. Phys. D: Appl. Phys. 27 (1994), 1300-1310. [10] J.H. Lee, Characterization of strain hardening for a simple pressuresensitive plasticity model, Acta Mech. 77 (1989) 133–151. [11] R.N. Haward, G. Thackray, The use of a mathematical model to describe isothermal stress–strain curves in glassy thermoplastics, Proc. Roy. Soc. A. 302 (1968) 453–472. [12] M.C. Boyce, D.M. Parks, A.S. Argon, Large inelastic deformation of glassy polymers. Part 1. Rate-dependent constitutive model, Mech. Mater. 7 (1988) 15–33. [13] A.S. Argon, A theory for the low-temperature plastic deformation of glassy polymers, Phil. Mag. 28 (1973) 39. [14] L.R.G. Treloar, The Physics of Rubber Elasticity, 3rd Edition, Clarendon Press, Oxford, 1975. [15] D.J. Choi, Finite element modeling of photoplastic processes in the cold drawing of polycarbonate bars, Ph.D. Dissertation, 2001, submitted for publication. [16] Hibbet, Karlsson & Sorense, Inc., ABAQUS User Manual, 1998. [17] Y.M. Chen, L.K. Ives, J.W. Dally, Numerical simulation of sliding contact over a half-plane, Wear 185 (1995) 83–91. [18] J.H. Lee, M. Wan, Strain distributions in the cold drawing of cylindrical polymeric bars using photoplasticity, Exp. Mech. 30 (1990) 145–151. [19] J.H. Lee, D.F. Simon, D.J. Choi, Photoplastic studies of strain distributions in the cold drawing of polycarbonate cylindrical bars: the rate effect, Exp. Mech. 35 (1995) 182–191. [20] R. Kaneko, E. Hamada, Microwear process of polymer surface, Wear 162–164 (1993) 370–377. [21] S.B. Ratner, I. Farberova, O.V. Radyerkevich, E.G. Lur.s, Connection between wear-resistance of plastics and other mechanical properties, in: D.S. James (Ed.), Abrasion of Rubber, Macaren, London, 1967, p. 145.
Experimental and numerical analysis of friction and wear behavior of polycarbonate J.H. Lee, G.H. Xu, H. Liang∗ Department of Mechanical Engineering, University of Alaska, Fairbanks, AK 99775, USA
Abstract Static and kinetic contact of a steel ball against a polycarbonate half-plane was experimentally examined to study its friction and wear behavior. Changes in friction were correlated with the apparent wear volume. The finite element method, using a physically based visco-plastic constitutive model, simulated the initial stage of sliding contact through indentation, sliding and unloading stages. Detailed deformation history was examined to understand the friction and wear mechanisms for the glassy polymeric material. The experimental and numerical results correlated well, qualitatively, in the coefficient of friction and the deformed surface profile. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Polycarbonate; Polymer; Friction
1. Introduction Many polymeric materials have an excellent strengthto-weight ratio. They often are easy to manufacture. Some polymers also possess excellent tribological properties [1]. Polycarbonate is a widely used thermoplastic polymer for its high hardness and toughness either as a single phase material or as the matrix of a composite material. The friction and wear of polycarbonate has been the subject of many studies [2–4], mostly experimental. There has been fewer theoretical studies on the friction and wear behavior of polymers in general, and polycarbonate in particular. Wear of polymers is more complicated than that of metals. Studies have shown that polymer wear undergoes two main mechanisms, deformation wear and interfacial wear [5]. Deformation wear involves abrasive and fatigue wear while the interfacial wear involves adhesive or transfer wear. So far, there is no agreement in the literature that when deformation wear and interfacial wear should be separated. Many researchers have found a particular surface roughness at which the dominance of the two wear modes changes [6–8]. Different pairs of rubbing materials have different roughness. In this initial study, we focus on the deformation behavior of a glassy polymer, i.e. polycarbonate. Whereas a general deformation map for polycarbonate can be constructed [2], a detailed mechanistic understanding of the friction and wear of polycarbonate is still yet to ∗ Corresponding author. Tel.: +1-907-474-6135; fax: +1-907-474-6141. E-mail address: [email protected] (H. Liang).
emerge. A large number of variables can contribute to different regimes of the deformations of polymers and there is a possibility of transitions from one regime to another. In addition, it would be useful if one can link the mechanistic results with the experimental ones towards a better, validated understanding of the phenomenon. Modeling of the tribology of polymers is often done using constitutive equations developed for metals [9]. The physical mechanisms of the plastic deformations of polymers are very much different from those of metals. Consequently, using physically-based constitutive equations for the modeling of the friction and wear of polymers may gain a better understanding of their behavior. Glassy polymers are usually different from metals in aspects, such as a larger elastic zone, pressure and strain-rate sensitivity, strain-hardening due to the orientation of polymer chains (orientation hardening), strain-softening and a more viscous behavior. In this paper, we present numerical and experimental results of our initial attempt towards a better understanding of the friction and deformation wear behavior of polycarbonate. For numerical modeling we concentrate on the initial stage of single-pass sliding contact between a hard and a soft surface, the latter modeled as a half-plane made of polycarbonate. The experiments used a pin-on-disk tester to conduct sliding tests. We also used a surface profilometer and an optical microscope to obtain wear volume and topography of wear tracks. In the following, we first present experimental procedures and results on the sliding contact of a steel ball against a polycarbonate disk. We then present a summary of the
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constitutive equations used in the finite element model. Numerical modeling procedures are then described and detailed results on significant mechanical variables as well as the topography of wear tracks during the initial indentation, the subsequent sliding and the final unloading stages are presented in detail. Finally, we discuss the significance of the numerical results in relation to the experimental ones.
2. Experimental procedure 2.1. Materials and preparation Disks with a dimension of ∅ 160 mm × 12 mm were machined from a polycarbonate sheet. A 52100 steel ball with a diameter of 9.5 mm and surface finish of 0.02 m (Ra) was used as counterbody for contact against the polycarbonate disk. A schematic drawing of the contact configuration is shown in Fig. 1. Before the experiments were conducted, specimens were cleaned with alcohol and then dried with an air blow dryer. 2.2. Experimental method and equipment A series of sliding wear tests were conducted on a pin-on-disk tester. A steel ball was fixed in a sample holder which was mounted on a loading level arm used to apply a normal load of 19.6 N. The polycarbonate disk was adhered on the rotating disk that was driven by a variable speed motor. The rotating speed was 23 rpm and the wear track diameter was 20 mm. The steel ball slides over the polycarbonate disk at a linear velocity of 48 mm/s. The friction force developed was measured by a force sensor and displayed on a control panel. The experiment was finished after sliding a distance of 86.7 m within 30 min, at a room temperature of 19.1◦ C. Before and after wear tests, the surface roughness was measured and wear track profiles of the polycarbonate disk were mapped with an automated 3D stylus profilometer system. The system was equipped with a diamond cone stylus with a spherical tip of 5 m in radius. The mean surface roughness of the polycarbonate was 0.04m. Before mapping the wear track, the polycarbonate disk was cleaned with alcohol again so that the wear particles could be cleared away
Fig. 1. Experimental sliding contact configuration.
from the examined surface. As the wear track was a circle, its topography was profiled along a length of 12.5 mm in the tangential and radial directions, respectively. Four measurement at least were carried out randomly along the wear track. With the wear profile measured in the tangential direction, the elastic or plastic deformation of the polycarbonate around the contact area could be studied. And by obtaining the topography of the wear track in the radial direction, the wear characteristics of the polycarbonate could also be investigated. The wear volume of polycarbonate was calculated approximately by the formula, v = (2π r 2 h/3), r is radius of steel ball, h the depth of wear track. It should be noted that the wear track measured in the tangential direction corresponds closer to the numerical model to be discussed later. 2.3. Experimental results The evolution of the friction coefficient with time under conditions mentioned above is given in Fig. 2. It can be seen that the friction coefficient increased rapidly at the beginning of the experiment. After sliding for 4 min, the friction coefficient increased slowly and then showed another rapid increase period. The value reached a maximum of 0.31 when the counterbodies rubbed against each other for 9 min, equal to a sliding distance of 26 m. Then the friction coefficient decreased quickly and reached the steady-state value of 0.28 at 12 min. The variation of wear volume of the polycarbonate as a function of time is shown in Fig. 3. It can be concluded that severe wear took place during the initial period. Then the steady stage was reached after sliding for 8 min, corresponding to the results of friction coefficient. When the experiment would finish, the wear volume showed a little increase due to the abrasion of the wear debris. The profile map of the wear trace in the tangential direction is shown in Fig. 4(a). Accordingly, only the marked range, located at the deepest area of the wear trace, should be recognized as the contact spot to which the largest contact pressure applied. So the marked range of the profile was expanded in order to show the deformation behavior of
Fig. 2. The evolution of experimental friction coefficient with sliding time.
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Fig. 3. The variation of wear volume as a function of time.
polycarbonate around the contact area (Fig. 4(b)). In other words, the contact configuration could be approximately regarded as a steel ball sliding along the polycarbonate in a direct line, instead of around a circle. It can be seen that the maximum penetration depth was 28 m. When the
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sliding motion continued, the material in front of the steel ball was extruded forward to form a surface peak which was reduced to a height of 10 m due to the slider pushing. Subsequently, the steel ball penetrated the surface of the polycarbonate again with a depth of 15 m. After climbing over a small peak, the steel ball extruded the material and produced a peak with a height of about 25 m. Fig. 5 showed the topography of the wear track of polycarbonate in the radial direction. It showed clearly that there was a sharp peak to the left of the wear scar with a depth of 185 m and a width of 400 m. The distance between the peak with the height of 165 m and the wear scar was 450 m. So, the peak could be the extruded materials or the heaped wear particles produced by the sliding motion of the counterbodies. Four sliding wear experiments were performed under the same testing conditions. The difference in the results for the experiments was very small. The error in friction coefficient was lower than 6%. So, it was concluded that the sliding contact experiments were well controlled and repeatable.
Fig. 4. (a) The topography of the wear track in the tangential direction; (b) the expanded typography of the marked range of (a).
Fig. 5. The topography of the wear track in the radial direction.
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3. Finite element modeling
According to this model, the total stress Tij has two components
In the following, we first present the constitutive equations used in the model. The finite element analysis procedure is described next. The results of the simulations are then discussed.
Tij = Tij∗ + Bij
3.1. Physically-based visco-plastic model The plastic deformations of glassy polymers have several important characteristics that are more pronounced than or not found in metals. These characteristics include strain softening, pressure-dependency [10], higher strain-rate dependency and orientation hardening [11] which can be attributed to different physical mechanisms of plastic deformations. We will use the model developed in [12] which is a generalization of the one by [11]. It should be noted that although elastic deformation is fully taken into account, for simplicity, this type of model is often classified as visco-plastic rather than visco-elastoplastic. The model can be approximately broken down into two parts. The first part assumes that there is an isotropic resistance to plastic flow due to molecular chain segment rotation [13]. This resistance is manifested in rate and pressure dependency, and strain softening. The second part assumes that there is an anisotropic resistance to molecular chain alignment which is similar to rubber elasticity [11,14]. This anisotropic resistance is manifested in the (orientation) hardening of the material. In the following, index notations will be used where repeated indices are summed. The isotropic plastic shear resistance γ˙ p is τ 5/6 A˜s γ˙ p = γ˙0 exp − 1− (1) Θ s˜ where γ˙0 is the pre-exponential shear strain rate factor, s˜ the modified athermal shear resistance, Θ the absolute temperature, A˜s the zero stress activation free energy divided by Boltzmann’s constant and τ the second invariant of the ‘driving’ stress tensor Tij∗ to be defined below. The (pressure) modified athermal shear resistance s˜ is αp s˜ = s 1 + s where s is the athermal shear resistance, α the pressure dependence constant, and p is the hydrostatic pressure. The evolution of s˙ is given by s γ˙ p s˙ = h 1 − sss where h is the slope of the yield drop with respect to the plastic strain (strain-softening), sss the value of s reaching a steady-state with an initial value of 0.077(µ/1 − ν), where µ is the shear modulus, ν the Poisson’s ratio.
(2)
where Bij is a back stress (responsible for orientation hardening) and Tij∗ the driving, or, reduced stress (responsible for other effects, such as pressure-dependency, etc.). The principal components of the back stress tensor are defined as N
λj λL N −1 λN 1 i −1 (3) Bi = CR λi L − λN jL 3 λL 3 λL j
where λN i is the principal network stretch component (the principal plastic stretch), λL the tensile locking network stretch (the natural draw ratio), and CR the rubbery modulus, L−1 the inverse Langevian function, the Langevian function for a parameter βi is defined as L(βi ) = coth(βi ) −
1 βi
The general structure of the equations is similar to that of a kinematic hardening model with a few important differences. Firstly, the back stress is obtained as principal values which are then transformed into the directions of the coordinate system. Secondly, unlike conventional kinematic hardening models, the back stress is not defined in a rate form but in terms of the principal plastic stretches. The von Mises stress is defined with respect to the total stresses as σ = 23 Tij Tij (4) The second invariant of Tij∗ , τ , is then defined as τ = 21 Tij∗ Tij∗
where Tij∗ is the deviatoric stress of Tij∗ , such that
Tij∗ = Tij∗ + (1/3)Tij∗ δij , and δij the Kronecker delta. Finally, the visco-plastic strain rate is defined as
1 vp ε˙ ij = γ˙ p √ Tij∗ 2τ
(5)
Note that, according to this definition, the visco-plastic strain rate is only deviatoric, i.e. without a hydrostatic component. We also define the equivalent visco-plastic strain rate as vp vp vp ε˙ = 21 ε˙ ij ε˙ ij (6) The total equivalent visco-plastic strain is then defined as vp ε vp = ε˙ dt (7) t
For finite element calculations, the constitutive equations are linearized and implemented as user subroutines [15] in the general purpose finite element program ABAQUS [16].
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3.2. Description of the numerical method We analyzed the sliding of a hard asperity over an initially flat substrate to simulate the sliding contact of a single asperity over the other one, similar to [17]. The soft asperity is initially flat assuming the applied load is high enough to have flattened it. The hard asperity is assumed to be rigid and the soft one is polycarbonate. To reduce the level of complexity in modeling, we assume that the deformation state is plane strain rather than three-dimensional as is the case for pin-on-disk type of tests. The finite element mesh, shown in Fig. 6, has 1921 elements and 5824 nodes with a dimension of 0.6 mm in the horizontal (sliding) direction and 0.3 mm in the vertical direction. The polycarbonate substrate is modeled by 1800 8-node plane strain elements with reduced integration (CPE8R). The 8-node elements with reduced integration are commonly used in large strain plasticity due to their good performance and efficiency. The rigid asperity is modeled
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using the rigid analytic surface option of ABAQUS which generates 121 rigid contact elements internally. The contact surfaces consist of the rigid contact elements and the elements on the top row of the model. The density of the mesh is graded in the vertical direction to accommodate the higher gradient of parameters near the contact surface. The left and right side of the model are restrained horizontally with the bottom fixed. The diameter of the asperity is 100 m and its center is located initially at 0.15 mm from the left boundary. The asperity will also be referred to as the cylinder in ensuing discussions. The constants used for the calculations are for the materials previously studied in the cold drawing of polycarbonate bars [15,18,19]. The material properties used have a good fit with the experimental data and are as follows, Θ = 300 K, α = 0.2, γ˙0 = 2.463 × 106 /s, A = 253.6 K/MPa, ν = 0.38, n = 5, h = 0.4 GPa, sss = 63 MPa, CR = 8.57 MPa. Here s0 , the initial value of the athermal shear resistance s, is 72 MPa. Two Young’s moduli are used to represent the initial portion of the non-linear stress–strain curve, E = 1.174 GPa(ε ≤ 0.021),
E = 0.37 GPa(ε > 0.021)
The behavior of the rigid elements is prescribed by using a reference node (shown as a cross in Fig. 6). Throughout the simulation, the rigid body is constrained to have zero rotation. The simulations are conducted in three stages, indentation, sliding and unloading. The rigid asperity is first indented onto the polycarbonate half-plane to a prescribed depth and then slides over towards the right. The indentation is under displacement control where the rigid body is also constrained to have zero horizontal displacement. This constraint is needed as the contact algorithm is only approximate which may satisfy a contact condition by introducing an undesirable rotation in the rigid elements. For the sliding stage, the vertical reaction of the reference node at the end of the indentation stage is used as a constant concentrated load. The sliding is controlled by displacement. At the end of sliding, unloading is accomplished by gradually decreasing the force at the reference node to zero. 3.3. Numerical results
Fig. 6. The finite element mesh.
We first present results where the initial penetration depth is 30 m. We assume no adhesion between the asperities such that the macroscopic friction is only due to the plastic deformation of the substrate. The vertical applied force for the sliding process is 12.16 N. The simulation spans a time period of about 68 s over the 300 m sliding distance which resulted in an average speed of 4.4 m/s (0.0044 mm/s). The analysis is highly nonlinear involving large strains, large deformations, visco-plasticity, strain softening and contact. It takes quite a bit of trial and error to come up with the proper size of time steps to maintain the stability of the analysis with good results. The final simulation took about 3 h CPU time on an SGI Octane workstation.
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Fig. 7 shows the evolution of the von Mises stress (Eq. (4)) at four stages of deformations. Fig. 7(a) is for a sliding distance of 15.4 m, shortly after the completion of the indentation stage. Fig. 7(b) is for a a sliding distance of 108 m, or a distance about the diameter of the asperity. Fig. 7(c) is for results at the end of the sliding stage, a distance of 300 m, or three times the diameter
of the asperity. Fig. 7(d) is at the end of the unloading process. For an initial indentation of 30 m, the maximum von Mises stress is around 100 MPa, which induces substantial plastic deformations. During sliding, the maximum von Mises stress typically occurs below the top surface of the model, tangential to and ahead of the cylinder. Fig. 7(b)
Fig. 7. The distribution of the von Mises stress at: (a) initial sliding; (b) intermediate sliding; (c) end of sliding; and (d) end of unloading.
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Fig. 7 (Continued ).
shows the distribution of residual stresses behind the cylinder. The fully developed residual stress zone in the same area is seen clearly in Fig. 7(c), with a maximum value around 40 MPa. The magnitude and shape of this residual stress zone is by and large not changed even after unloading. Fig. 8 shows the evolution of the equivalent plastic strain (Eq. (7)) at the same four stages of deformations as Fig. 7. Fig. 8(a) shows three areas of high plastic strains. The first is near the surface and ahead of the cylinder, which is created
by the initial indentation process and has a maximum value around 36.6%. The second, also caused by the initial indentation process, is beneath the cylinder and towards its left side. The shape of this second area was horizontal but now rotated clockwise caused by the cylinder that is still moving downward. The third area is slightly below and almost tangential to the cylinder and is caused by the cylinder sliding forward. Fig. 8(b) shows the maximum equivalent plastic strain, about 43.3%, occurring near the third area described in Fig. 8(a). The first two areas described in Fig. 8(a) have
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Fig. 8. The distribution of the total equivalent plastic strain at: (a) initial sliding; (b) intermediate sliding; (c) end of sliding; and (d) end of unloading.
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Fig. 8 (Continued ).
been fully developed and are left behind by the moving cylinder. Fig. 8(c), at the end of the sliding process, shows a large plastically deformed region between the initial indentation area and where the cylinder is currently working. The depth of the zone is almost the radius of the cylinder with a maximum value around 67.5%. Fig. 8(d), at the end of unloading, shows that the plastic zone in Fig. 8(c) is almost unchanged. Whereas the equivalent plastic strain describes the current accumulated permanent deformation, the distribution of the instantaneous plastic shear strain rate (1 s1 ) (Eq. (6))
can be illuminating on the location and area of active plastic deformation. Fig. 9(a) shows that, after some initial sliding, plastic deformation is being actively developed ahead, and almost tangential to the cylinder with a maximum value around 8.3%. The zone that has the second highest plastic strain rate is located behind the underside of the cylinder with a maximum value around 5%. The figure shows a transition of the plastic deformations from the initial indentation mode to the current sliding mode. Fig. 9(b) shows that active plastic deformation occurs not only ahead, and almost tangential to, the cylinder with a maximum value of
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Fig. 9. The distribution of the plastic strain rate at: (a) initial sliding; (b) intermediate sliding; (c) end of sliding.
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Fig. 9 (Continued )
about 50% but also occurs at locations slightly farther from the top surface. A similar trend is observed at the end of the sliding process shown in Fig. 9(c). At the end of unloading, the plastic strain rate is practically zero everywhere as there should not be any additional observable plastic deformation. As the plastic deformation of the material is pressuredependent, we also show in Fig. 10(a), at the end of the sliding, the distribution of the mean stress. As expected, the maximum compressive mean stress (about 130 MPa), occurring below the cylinder, is much higher than the tensile one (about 33.4 MPa) occurring on the surface. This indicates that the material follows more the compressive stress–strain curve than the tensile one as shown in Fig. 11. In addition, the hardening of the material model is attributed primarily to the back stress. We also show the distribution of one of the principal back stresses (B1 ) in Fig. 10(b) which has a maximum magnitude of about 48.3 MPa and has a significant contribution to the total stress. It is also interesting to see the wear track developed by the sliding process. When loaded, the deformed profile can be seen from Fig. 7(c). The peak of the profile occurs near the left side of the model and has a magnitude of 9.52 m. The valley of the profile occurs below the cylinder and has a value of 39.5 m, which is 9.5 m more than the initial indentation depth. The height from peak to valley is about 49 m. After unloading, Fig. 7(d) shows a peak of 8.53 m,
about 1 m smaller than the value before unloading, possibly due to viscous effects. The valley is reduced to 14 m which is primarily due to elastic unloading. With a height from peak to valley about 22.5 m, the unloading process restores about half of the initial height. This illustrates the effect of a larger elasticity and the time-dependent nature of the visco-plastic behavior of polycarbonate. We also conducted simulations of the sliding process by using a smaller initial indentation depth of 16 m, corresponding to a vertical force of 7.086 N, while keeping everything else the same as the previous case. The general trend shown above also holds except for the maximum values of parameters and the extent of plastic deformations. For example, the maximum von Mises stress is between 99.4 and 109 MPa during loading and is 49.5 MPa after unloading. At the end of the indentation process, there is very little plastic deformation directly beneath the indentor, with a small amount of plastic strain (0.84%) occurring tangential to and on both sides of the cylinder. The maximum magnitude of the equivalent plastic strain increases to 21.1% at the end of sliding. An indentation of 16 m, comparing with 30 m, also gives a shallower profile. At the end of the sliding process, the peak of the profile is 8.27 m and the valley has a magnitude of 21.78 m which constitutes a total height of about 30 m. After unloading, the peak of the profile is 7.73 m and the valley is 5.97 m with a total height of 13 m.
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Fig. 10. At the end of sliding, the distribution of: (a) the mean stress; (b) the first principal back stress.
Finally, it is of interest to estimate the coefficient of friction (µ) due to asperity contact as modeled here. An average coefficient of friction can be calculated, at selected time steps, by dividing the horizontal reaction at the bottom of the model by the vertical applied force. It should be noted that there is a gradual transition between the indentation and the sliding process as discussed in conjunction with Fig. 9(a). This transition is marked by the value of zero horizontal reaction, at the bottom of the model, which
subsequently increases. The coefficients of friction for the 30 and 16 m indentation cases are shown in Fig. 12 as a function of sliding distance normalized against the diameter of the asperity. The origin of the horizontal axis is measured not respect to when the cylinder starts to move horizontally but with respect to the time when the horizontal reaction becomes zero. Fig. 12 shows µ increases first, and after reaching a plateau, starts to decrease. But at the end of simulations, µ has not yet reached a steady-state
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Fig. 11. The stress–strain curves of polycarbonate in tension and compression.
Fig. 12. Coefficient of friction vs. sliding distance/asperity diameter at two different loads.
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Fig. 13. (a) The depth of the maximum von Mises stress vs. sliding distance; (b) maximum von Mises stress vs. sliding distance.
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value. It should also be noted that this µ is a result of plastic deformation alone as no adhesion is present in the model.
4. Discussion The behaviors we observed in the numerical results are quite different from the results for metals in [17]. With an initial indentation depth of approximately 0.047D (D is the diameter of the cylinder) and a sliding distance of 1.5D, their simulation created two permanent grooves. One is near where the initial indentation started and the other is near where the sliding ended. In comparison, the sliding distance of our model is 3.0D and the initial indentation depth is 0.16 and 0.3D for the two cases presented, respectively. Instead of a groove near the initial indentation spot for metals, our results showed a ‘plateau-like upheaval’ formation of the surface above the initial level, similar to that reported by [20] at the beginning of their scan-scratch test on polycarbonate and further studied by [3]. It is interesting to note that this ‘bump’ was formed for both indentation cases. For the 16 m initial indentation, there was very little initial plastic deformations near the indentor as discussed previously. On the other hand, for the 30 m initial indentation situation, large subsurface plastic deformations have occurred. In other words, the deformation of the former is elastic, whereas the latter is plastic in nature. As discussed in conjunction with Fig. 8 previously, the bump was created primarily by the rotation of the material due to the movement of the cylinder, irrespective of the regimes of deformations (elastic versus plastic). The bumps remain in place even after unloading mainly due to the presence of residual stresses (cf. Fig. 7(d)). For both cases, the mechanisms described in Fig. 9(b) and (c) in [3], crack formation in the bumps followed by its propagation and accumulation, could very possibly occur with repeated sliding. Another difference between our results and those in [17] is in the magnitude of strains and the failure mechanisms that could lead to wear. Almost unreasonably large strain, around six, was obtained in [17]. In our case, although large plastic strains did occur, they were not high enough to cause failure and none of the plastic strains have reached the limiting stretch λL (which is approximately 2.23 for the material used in the model) even after deeper indentation and longer sliding distance than those in [17]. It should be noted that breaking of polymer chains, as a precursor of failure, could occur when λL is reached. This be speaks of the wear-resistant nature of polycarbonate. We also note that the maximum von Mises stress during sliding does not vary a lot (Fig. 7). The extent of plastic zones, however, does. A large plastic zone may lead to higher damage of the material under repeated loading. As an aid to visualize this behavior, we plot the variation of the normal distance of the maximum von Mises stress to the cylinder versus the sliding displacement in Fig. 13(a), and the von Mises stress versus sliding displacement in Fig. 13(b). It is
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clearly seen that, after certain sliding distance, the normal distance continues to increase whereas the maximum von Mises stress remains more or less constant. It is assumed that this normal distance is an approximate measure of the damage of the material under repeated loading and can lead to failure and wear. Previous studies have correlated the surface topography features with single-pass abrasive war in the simplest abrasive wear model [5,21]: φ V = KW tan H where V is the wear volume per unit sliding distance, K the abrasive wear coefficient, W the normal load, φ the average slope of the asperities, and H the hardness of a polymer. We will investigate the relationship between the plastic deformation and wear volume in future studies. It should be noted that the experiments reported here have different conditions than the modeling work. The experiments were conducted at a much faster speed, with a higher load, and over a much longer time frame than the simulations. Consequently, we do not expect to have good quantitative correlations between them. However, we do note that the coefficients of friction at the initial stage of sliding (Fig. 2 and 13) are similar, i.e. 0.12 for experiments and 0.16 for numerical simulations. The average height between the peak and valley of experiments (Fig. 4) is about 30 m, which is close to the 22.5 m in the simulation (with 30 m initial indentation). The similarities between experiments and modeling could be attributed to the fact that, after multiple passes, steady-state conditions may be achieved eventually. Since the wear of the material (Fig. 4) correlates well with the coefficient of friction (Fig. 3), we feel optimistic in correlating our numerical modeling with experiments in the future for multi-pass situations.
5. Conclusions We conducted preliminary investigations of the sliding contact of a steel ball against a polycarbonate half-plane using a combined numerical and experimental approach. The numerical simulation used more realistic polymer-specific constitutive equations. An initial bump, elevated above the original surface, was found and its geometric origin was elucidated. Detailed history of important variables such as the von Mises stress, the equivalent plastic strain and the shear plastic strain rate were obtained to gain insight into the formation of surface profiles and the accumulation of plastic deformation which may lead to damage under repeated sliding. Despite differences in experimental and numerical conditions, our experimental data did show similarity in the surface profile of the wear track and the coefficient of friction with numerical results. We also found a possible correlation between the profile of the wear track and the depth of the maximum von Mises stress.
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Acknowledgements The authors wish to thank the Arctic Regions Supercomputing Center (ARSC) for computing resources. Part of this work was supported by the President’s Special Project Fund and Alaska Space Grant Program at UAF. We also acknowledge the assistance of Dr. T. Zhang in the preparation of the manuscript.
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