World Tribology Congress 2013 Torino, Italy, September 8 – 13, 2013 Representation of Wear Data by Power Curves Abhro Choudhury 1) , Savio Sebastian 2)* 1) Department of Mechanical Engineering, PES Institute of Technology, BSK III Stage, Bangalore 560085, India 2) Centre of Research and Innovation, PES Institute of Technology, BSK III Stage, Bangalore 560085, India * Corresponding author:
[email protected] 1. Introduction Wear of machine elements with respect to runtime is generally assumed to be linear. This assumption leads to the use of a wear factor that is (rather indiscriminately) used for predicting the service life of the machines. However, a closer look into the wear literature would reveal that the linear trend usually refers to the volume or mass of material removed; not the wear depth. The classical work of Archard [1] is based on the proportionality of wear volume with respect to runtime. However, Archard’s contemporary workers have pointed out the nonlinear nature of wear curves [2, 3]. This behavior is not limited to metals. Wear depth of polymeric materials was clearly found to be varying nonlinearly with sliding distance [4]. Ceramic materials also follow similar trends [5]. With the advent of numerical models, the nature of wear curves was further explored and found to be nonlinear [6, 7]. Similar behavior has been noticed in a variety of wear modes and erosion scenarios [8-10]. Data from several engineering applications also show a similar trend of wear depth [11,12]. In the light of these observations, we take a critical look at the commonly used linear wear model and point out the pitfalls in its usage. Data generated by a simple pin-on-disc experiment is presented to show a typical wear curve. A simple power curve is fitted to this data as well as similar published data to bring out the common traits. Finally we suggest that the simple model be used for assessing the wear life and monitoring the health of engineering components for which useful life is limited by a critical wear depth. 2. Wear equations Wear equations are generally derived either from the mechanics of a particular mode of wear or invoke, in empirical relationships, a set of factors that affect the wear behavior. Contact mechanics equations based on a single mode of wear have limited applicability as most engineering products will have multiple modes of wear, e.g., adhesive + abrasive, operating simultaneously. Even if we were to combine the mechanics-based relationships for several modes of wear, the relative contributions from each mode are unknown and will probably vary during the evolution of wear. On the other hand, empirical relationships include a few main factors relevant to the problem at hand; but generally fail to account for the interactions between the factors. Interaction of factors is an important, but often overlooked, issue. A well-crafted Design of Experiments (DoE) can bring out the interaction terms, as shown by Liu, et al [13] and Selcuk [14]. Liu et al investigated the effect of contact pressure, surface roughness, sliding distance and sliding speed on the wear of a polymer blend, using an L27 orthogonal array, and found that the interaction term of all four factors is significant. Selcuk also considered four factors (load, time, hardness and speed) which were varied according to a central composite DoE and the response surfaces of wear mass loss were obtained for two grades of steel. The results clearly show that, apart from the main factors, the interaction terms of load speed are very significant. Ignoring the significant interaction terms is a risk inherent in the formulation and application of empirical equations. These issues suggest that there is little advantage (and potential risk) in using wear equations based on single modes of failure and multiple factors without interaction terms. Therefore we argue that wear equations for general engineering applications could benefit from a single, lumped-parameter approach. We would hasten to add that scientific exploration of wear phenomena should be excluded from this approach. Wear is generally characterized by a wear factor, K, which is the slope of an assumed linear relationship between the wear depth (or volume) and the sliding distance, under a specified set of test conditions. The wear factor is usually based on Archard’s empirical relationship. Values of the wear factor are widely available in handbooks and product literature. If we were to design a tribosystem for which the life is limited by a specified wear depth or clearance, the design practice would be to look up the appropriate value of the wear factor and use it to determine the sliding distance (life) for the limiting wear depth. This is depicted schematically in figure 1 as curve A. However, the tribology literature is replete with wear curves that show a pronounced nonlinear behavior, particularly evident in the initial running-in period (curve B). In such cases too, the wear rate is reported as a constant, derived from the apparent linear behavior after the initial running-in. In this over simplification of the wear behavior, the initial wear, that usually comprises a significant portion of the total wear depth, tends to be ignored. The underestimation of wear depth due to the unaccounted initial wear is shown (=hB-hA) schematically in figure 1. It also entails the loss of valuable information obtained from standard tests. Therefore it is evident that we need more than one constant to represent the nonlinear nature of wear data reasonably well. 2 Figure 1: Wear curves, schematic The values of measured wear depth are influenced by a number of parameters. Meng and Ludema [15] listed over 100 parameters, invoked for ~180 wear equations that emerged during the second half of 20 th century. Due to reasons stated earlier, we shall not look into equations derived from mechanics and empirical relationships sporting multiple factors; we focus our inquiry on simple empirical relationships with a single (lumped) factor and a minimum number of constants. In order to select an appropriate form of the relationship, we postulate that the form of all wear equations must satisfy a few basic requirements as follows, irrespective of the mechanism of wear: The slope of the wear curve must be positive at all points, as the wear rate is never negative or zero (it may tend to zero, or become too small for measurement; e.g., the ultra-mild wear regime in IC engines). The equation should accommodate the commonly observed behaviour of decreasing slope as well as the less common traits of constant or increasing slopes. The initial value of wear is zero; i.e., h=0 at t=0, where h is the wear depth and t is the runtime. These requirements suggest an equation with at least two constants. It may be noted that more than two constants and cubic terms will be required to accommodate the transitions due to end-of-life behavior or changes in operating conditions. However, in order to keep the relationship simple, we will limit the number of constants to two. This simplification may exclude the capability to model the effects due to change of operating conditions. Nevertheless, such an equation should be able to predict the useful lifetime (based on wear depth), provide an adequate baseline for monitoring a well-behaved tribosystem operation and give clear warning of deviations boding impending failure. Four forms of empirical equations are cited by Meng and Ludema [15] as typical for calculation of wear volume, v: v =at (1a) v = b a 1- e-at( ) (1b) v = beat (1c) v =KFaV btc (1d) where , , K, a ,b and c are constants and t is the runtime. F is the force and V is the sliding velocity in the fourth equation. In view of the limitations of a single-constant relationship (linear) mentioned earlier, we preclude equations of type (1a). Of the remaining three, eqn. (1c) appears to be simpler; but the wear volume does not reduce to zero at t=0. Kumar, et al [16] and Yang [17] have developed relationships derived from separate treatments of the transient and steady state wear regimes, ultimately resulting in an equation similar to (1b). This equation was successfully used in several applications. Despite the advantages, equations of this form are not the simplest and they cannot handling slopes >1. Here we propose a simplified version of the eqn. (1d) that will be shown to cover a large variety of wear data. The proposed equation is of the form nskh (2) where h is the wear depth, s is the sliding distance and k & n are fitted constant and index. Fig.2 shows the nature of this relationship for values of n < 1, n = 1 and n > 1. Similar shapes of wear curves have been experimentally obtained by Burwell and Strang [2] over 60 years ago. Figure 2: Shapes of proposed power curves 3. Experiments The wear experiments were carried out using a simple pin-on-disc tribometer, kindly made available by Ducom Instruments, Bangalore [18]. The discs were made of AISI 52100 steel, hardened to ~62 HRc. The pins were made of mild steel, with flat tips of different diameters ranging from 3 to 6 mm. Alignment of the flat tips to the disc surface was insured by initial running on an abrasive paper. The normal load was 9.81 N (1 kgf) for all the tests. The pv value was approximately 3.5 MPa.mm/s and the total distance was 1500 m for all the experiments. The wear depth of the pin was recorded by an LVDT in the tribometer. Figure 3 shows typical wear plots and the fitted coefficients generated from the tribotests. Sliding distance, s Wear depth, h n1 h = k×sn A B K K Wear depth, h Sliding distance, s hB-hA 3 Figure 3: (a) Experimental wear curves (b) Fitted coefficient k 4. Published wear data Apart from the data that we generated, a number of publications (~70), in which wear curves were readily available, were also looked into and the curves were digitized in order to obtain inputs for the regression algorithm to fit power curves of the form given by equation (2). Wear is reported in several units in the literature; mg, mm 3 , mm, etc. Here we use the wear depth (in mm), which is arguably more significant in engineering applications. Similarly, the duration is reported in terms of sliding distance, runtime or number of cycles. In order to be consistent across several sets of reported data, we use the sliding distance (in m). Figure 4 shows the histogram of fitted index n obtained from the published datasets. Figure 4: Histogram of the index n (Eqn.2) 5. Discussion The experimental results presented in section 3 further confirm the usual nature of wear curves. It is interesting to note that the origin of the nonlinear nature is not entirely due to the changing contact area (of spherical or cylindrical contacts). The apparent contact area of the flat-tipped pins is invariant in our experiments; but the wear curves are nonlinear. Similar behavior was reported for data obtained from a specially modified pin-on-disc tribometer in which the flat-tipped pins were mounted in a self aligning pin holder [19]. The extent of initial nonlinear behavior is generally much more than the changes in the original asperity heights. Therefore we may speculate that sub-surface modifications of material properties also contribute to this behavior. In the case of spherical or cylindrical contacts (like in a journal bearing), there is a progressive reduction in the mean contact stress. If we assume that the wear volume is proportional to the mean stress, it follows that the wear depth will show a decreasing trend. The coefficient, k, in equation (2) is indicative of the intensity of wear. This trend is evident in the behavior of the fitted values obtained from our experiments (fig.3b). The value of k would be a very useful measure to compare different surface treatments subjected to similar wear conditions, as in the case of a standard tribotest. Therefore we propose that the values of k in equation (2) be compared for a better assessment of treatments, rather than using the “constant” wear rates as practiced by many tribologists. The index of the wear equation, n, shows an interesting distribution (fig.4). In most of the cases the value of n is less than 1, indicating the decreasing trend of wear depth with respect to sliding distance or runtime. Further, the values of n seem to be more abundant around 0.5, suggesting a square-root dependency, h s, on sliding distance. In such a relationship, the curve appears to be linear as wear progresses, justifying a simple linear function if one were to ignore the initial part. However, as noted earlier, this over simplification would be risky in some cases and we lose valuable information when authors choose to report only the “steady-state” wear rates. The power curve is quite useful in engineering applications where machines run under nearly constant operating parameters over its lifetime. With the help of a historical dataset to which a simple power curve is fitted, one may easily detect anomalies in wear behavior as the current data point goes beyond the upper confidence limit of the fitted curve. This would indicate a change in the wear mechanism, leading to eventual failure. In cases where the life is limited by a predetermined wear depth, the power curve provides a better estimate of the wear life. 6. Conclusions Our experimental results and the consolidated set of published data provide substantial evidence to a general nature of wear evolution, which may be conveniently 0 10 20 30 40 0 100 200 300 400 500W e ar d e p th , m ic ro m e te rs Sliding distance, m (a) d=3m m 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 C o e ff ic ie n t k Mean Contact Stress, MPa (b) 0 2 4 6 8 10 12 14 16 Fr e q u e n cy o f n Bin 4 represented by a simple power curve. The simplest form of the power curve, given by equation (2), complies with the requirements suggested by the inherent nature of wear data and is flexible enough to accommodate different observed trends, when operating under constant conditions. However, this simple curve cannot represent accelerated wear observed near the end of life. The power curve representation of wear data retains valuable information on the initial phase of operation and avoids pitfalls in using a constant wear rate for wear life prediction. The simple power curve is useful for monitoring the wear progression of machines that run under steady state operating conditions by defining the expected wear and its upper boundary as a function of runtime. 7. References [1] Archard, J.F., “Contact and Rubbing of Flat Surfaces”, J.Appl.Phys., 24 (1953) 981-988. [2] Burwell, J.T, Strang, C.D., “On the Empirical Law of Adhesive Wear”, J.Appl.Phys., 23 (1952) 18-28. [3] Hirst, W, Lancaster, J.K., “Surface Film Formation and Metallic Wear”, J.Appl.Phys., 27 (1956) 1057-1065. [4] Uchiyama, Y, Tanaka, K, “Wear Laws for Polytetrafluoroethylene”, Wear, 58 (1980) 223-235. [5] Kloss, H., Wasche, R., “Analytical Approach for Wear Prediction of Metallic and Ceramic Materials in Tribological Applications”, Wear, 266 (2009) 476-481. [6] Sfantos, G.K., Aliabadi, M.H., “Wear Simulation Using an Incremental Sliding Boundary Element Method”, Wear, 260 (2006) 1119-1128. [7] Hegadekatte, V, et al, “A Predictive Modeling Scheme for Wear in Tribometers”, Tribol. Intnl., 41 (2008) 1020-1031. [8] Mulhearn, T.O., Samuels, L.E., “The abrasion of metals: a Model of the Process”, Wear, 5 (1962) 478-498. [9] DeBaets, P., et al, “A Mathematical, Geometrical Wear Model for Continuous Wear Measurements Based on the Normal Approach between a Spherical and a Flat Specimen Subject to Fretting”, Wear, 208 (1997) 50-56. [10] Ahmad, M, Casey, M, Surken, N, “Experimental Assessment of Droplet Impact Erosion Resistance of Steam Turbine Blade materials”, Wear, 267 (2009) 1605-1618. [11] Dawson, T.G., Kurfess, T.R., “Modeling the Progression of Flank Wear on Uncoated and ceramic-Coated Polycrystalline Cubic Boron Nitride Tools in Hard Turning”, Trans. ASME, J.Manuf.Sci.Eng, 128 (2006) 104-109. [12] Dickerel, D.J., Sawyer, W.G., “Evolution of Wear in a Two-Dimensional Bushing”, Tribol.Trans., 47 (2004) 257-262. [13] Liu, C.Z., et al, “Effects of Operating Parameters on the Lubricated Wear Behavior of a PA-6/UHMWPE Blend: a Statistical Analysis”, Wear, 253 (2002) 878-884. [14] Selcuk, B, “Feasibility of Utilizing Response Surface Methodology for Predicting Wear of Steels”, Proc.I.Mech.E:Pt.J.J.Eng.Tribol., 225 (2011) 277-287. [15] Meng, H.C., Ludema, K.C., “Wear Models and Predictive Equations: Their Form and Content”, Wear, 181-183 (1995) 443-457. [16] Kumar, R, Prakash, B, Sethuramiah, A, “A Systematic Methodology to Characterize the Running in and Steady State Wear Processes”, Wear, 252 (2002) 445-453. [17] Yang, L.J., “A Test Methodology for the Determination of Wear Coefficient”, Wear, 259 (2005) 1453-1461. [18] Ducom Instruments; http://www.ducom.com/Products/Wear-And-Fricti on-Monitor---TR-201.php [19] Gracia-Prieto, I, Faulkner, M.D., Alcock, J.R., “The Influence of Specimen Misalignment on Wear in Conforming Pin on Disc Tests”, Wear, 257 (2004) 157-166.