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Applying a population growth model to simulate wear of rough surfaces during running-in
Wear 294-295 (2012) 356–363
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Applying a population growth model to simulate wear of rough surfaces during running-in Zhanjiang Wang n, Qinghua Zhou State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 February 2012 Received in revised form 19 July 2012 Accepted 25 July 2012 Available online 2 August 2012
A population growth model (PGM) is introduced to simulate wear during running-in, which is one of three typical wear states. The rough contact region is divided into several bar-shaped elements while the wear of every grid point has been considered as a PGM process. The wear volume in rough surface can be calculated during the wear processes and the proposed wear formula contains three items: initial point height, determine function and stochastic function. By employing a PGM, the changes of every local peak and valley were simulated. The results show that the height of local peak dynamically decreased and the decreasing speed becomes slow during running-in, whereas, the local valley shows inverse tendency. Because the stochastic function contains Brownian motion, if a point is higher than other points at time t, it may be lower than other points at time tþ dt. Furthermore, if the height of every discrete point conforms to a PGM, the expectation of the wear volume follows an exponential form. & 2012 Elsevier B.V. All rights reserved.
Keywords: Wear modeling Stochastic process Surface analysis Contact mechanics
1. Introduction In mechanical system, wear plays an important role in surface failure and further affects the reliability of mechanical components and the whole system. There are three typical wear states: running-in, mild wear and severe wear. The wear behavior is so complex that different kinds of models were established to analyze it. Basically, there are two types of models to deal with wear between two contact surfaces: deterministic and stochastic models. Archard [1] pioneered the development of the deterministic models for wear. He first used the wear coefficient to describe the relations between the wear volume and the applied load. For numerical simulation of deterministic model, the finite element method (FEM) [2] and the boundary element method (BEM) [3] can be employed. Moreover, the deterministic wear models or equations to describe the wear are quite different for different materials and working conditions [4]. Meanwhile, the stochastic models were commonly used for wear analysis. D’Acunto [5] used a double-well model based on a microscopic diffusive process to describe the wear volume rate. Hu et al. [6] employed a dynamic system theory to analyze the running-in process in wear. They believed that running-in can be regarded as a self-adapting process of a wear system, of which the dynamic behaviors were controlled by the geometrical, physical and chemical feedback. Wang et al. [7] analyzed the dynamic
n
Corresponding author. Tel.: þ86 138 1060 8964. E-mail address: [email protected] (Z. Wang).
0043-1648/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.wear.2012.07.028
reliability behavior for sliding wear of multiple ring–disk system. They found that the critical wear depth can be utilized to estimate the dynamic reliability. Goh et al. [8] treated cumulative damage as a stochastic process. When the cumulative damage exceeds a certain prescribed threshold, the wear-out failure occurs. Besides, the wear coefficient was modeled as a random variable and a stochastic process in Ref. [9]. Surface topography was found to have much influence on wear [10–12]. Some researchers investigated contact characteristics of rough surfaces for dry contact [13] and lubrication contact [14,15]. Unfortunately, they just focused on the changes of statistical parameters such as wear volume, surface height distribution, arithmetic mean, root mean square and skewness. The statistical parameters just reflect global characters of wear region and local characters describing the local wear behaviors were ignored in their studies. The relationship between local behaviors and global wear characters needs to be defined and wear in every local place in rough surfaces should be modeled. Therefore, equations need to be established to reflect the changes of every local peak and valley in a rough surface. Recently, Moerlooze et al. [16,17] presented a theoretical spring model to describe the asperities wear evolution in a plain contact. This study aims to establish a stochastic wear model on every local peak and valley. A population growth model (PGM) based on the stochastic differential equations [18] is introduced to describe a rough surface wear during the running-in process. The rough contact region was divided into several bar-shaped elements. The wear state in every discrete point was to be considered as a stochastic process, which contains three items: initial point
Z. Wang, Q. Zhou / Wear 294-295 (2012) 356–363
Nomenclature a ¼ g þ aW relative rate of growth bðt,XðtÞÞ drift coefficient of X BðtÞ standard Brownian motion E expectation gðtÞ determinate function m,n grid numbers along the x and y axes N0 given initial value N(t) size of the population at time t Q wear volume rðBðtÞÞ stochastic function which contains Brownian motion BðtÞ Rþ set of positive real numbers t time, hr
height, determine function and stochastic function. Then, the changes of rough surface profile can be simulated, which will provide a mathematical perspective for engineers to understand the wear behaviors during the running-in process.
2. Surfaces wear processes model 2.1. Mathematics model of wear process In general mechanical system, wear exists in two mating rough surfaces, as shown in Fig. 1, the system consists of two mating rough surfaces, which has three-dimensional roughness and lubricant between them. When the two contact surfaces move relative to each other, the wear will occur. The typical wear stages include running-in, mild wear and severe wear, as plotted in Fig. 2. Running-in is the initial stage of the wear process. During running-in the wear rate varies from high to low and surfaces become planarized. Mild wear is the steady-state of the wear process, where wear rate is relatively slow over a long period of time. After the mild wear stage, the wear is quite rapid at the severe wear stage and failure will occur. According to Archard wear model [1], the wear was related to the contact pressure. If a local asperity is higher than other asperities, the local pressure will be higher than other regions. Therefore the wear should be related to the profile of the rough surface. A mathematical model was established to simulate the changes of rough surfaces profile during running-in. To form the mathematical model, wear surface is discretized. The mathematical model is shown in Fig. 3. X–Y plane is the base plane of the rough surface, which is determined by wear conditions. In the X–Y
Fig. 1. General lubricated sliding wear processes.
357
u V0 W X,Y XðtÞ zB ZðtÞ g,a
velocity volume between base plane and the initial surface one-dimensional ‘‘white noise’’ space coordinates, mm stochastic process value of the base plane asperity height at time t parameters to reflect the population growth rate and the stochastic fluctuation m mean value s standard deviation sðt,XðtÞÞ diffusion coefficient of X cðzÞ probability density function
plane, rectangular grids are employed to discretize rough surface. The whole contact region is divided into several bar-shaped elements. If they have enough grids, the real rough surfaces can be replaced by discrete rough surface in the mathematical model. The total number of grid points in the X–Y plane is m n. Z i,j ðtÞði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ is the height of each grid point ði, jÞ (or distance from the based plane), and its variation can be treated as a stochastic process, here t A ½0,T and T is the time for running-in process. In present study, a stochastic differential equation is employed to describe Z i,j ðtÞ.
Fig. 2. Typical wear curve.
Fig. 3. A discrete rough surface (several small hexahedrons were employed to represent the volume of discrete rough surface).
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2.2. It¨ o formula and PGM
Fig. 4. Surface profile and its changes during the time interval t to tþ dt.
The dynamic properties of many important continuous time stochastic processes can be modeled by using stochastic differential equations. We can construct the sample paths of diffusion directly from Brownian motion and Ito¨ formula [19] can be employed to solve stochastic differential equations. PGM is the model to describe the changes in a population over time, and be defined as solutions to stochastic differential equations. The wear is a dynamic stochastic process, so PGM can be employed to describe the height of every grid point Z i,j ðtÞ during the running-in process; the Ito¨ formula is introduced and used to resolve stochastic differential equations. Let Bðo,tÞ, o A O, t A R þ be a standard Brownian motion. Let b and s be two functions from ½0,T O into R. The Ito¨ process is a stochastic process XðtÞ, t A R þ of the form Z t Z t XðtÞ ¼ Xð0Þ þ bðs,XðsÞÞ ds þ sðs,XðsÞÞ dBðsÞ ð1Þ 0
0
or in differential form dXðtÞ ¼ bðt,XðtÞÞ dt þ sðt,XðtÞÞ dBðtÞ
ð2Þ
If Y ¼ yðt,XðtÞÞ has continuous partial derivatives @Y=@t, @Y=@x and @2 Y=@x2 . Then ! @y @y s2 @2 y @y þb þ ð3Þ dYðtÞ ¼ dt þ s dBðtÞ @t @x @x 2 @x2
Fig. 5. The changes of probability density function for surface height during time interval t to tþ dt.
Therefore, the wear model proposed in this paper during running-in is based on the following assumptions: 1. The height variation of every grid point is considered as a stochastic process, which contains three items: initial height, determine function and stochastic function. 2. There exists a base plane for the rough surface, which is determined by wear conditions. The height of surface profile is relative to base plane. 3. The changes of the height in every grid point are related to the base plane. If the local surface height is higher than the base plane, it will decrease during running-in, but it always keeps higher than the base plane. Otherwise, if the local surface height is lower than the base plane, the profile will be increasing, and it always keeps lower than the base plane. 4. If a local asperity is higher than other asperities at time t, the height variation of this local asperity will be greater than others during t to tþ dt, likewise, if a local valley is lower than other valleys at time t, the corresponding height variation will be greater than others during t to t þ dt.
From assumptions above, the schematic diagram of changes for surface profile during the time interval t to t þ dt can be seen in Fig. 4. Here zB is the height of the base plane. The changes of probability density function for surface height during time interval t to t þ dt can be seen in Fig. 5. Let cðzÞ denote probability density function. zm ðtÞ is the arithmetic mean of surface height at time t, and zm ðt þ dtÞ is the arithmetic mean of surface height at time t þ dt. zB is the value of the base plane. During the wear process, the area dA1 is redistributed into dA2 and the area dA3 is to dA4. According to the characters of probability density function and assumption 3, we get dA1 þ dA3 ¼ dA2 þ dA4 and dA1 ¼ dA2, dA3 ¼ dA4.
is called the Ito¨ formula [19]. A PGM describes the population’s evolution NðtÞ as follows: ( dNðtÞ ¼ aNðtÞ dt ð4Þ Nð0Þ ¼ N0 where NðtÞ is the size of the population at time t; a is the relative rate of growth, which equals to g þ aW; Wis one-dimensional ‘‘white noise’’; N 0 is a given initial value. Let us assume that g ¼constant and a ¼constant. The PGM is described by the following linear system of stochastic differential equation: ( dNðtÞ ¼ gNðtÞ dt þ aNðtÞ dBðtÞ ð5Þ Nð0Þ ¼ N0 where t A ½0,T, and BðtÞ is Brownian motion. Applying Ito¨ formula, i.e., Eqs. (3) and (5) have the explicit solution [18] 2
NðtÞ ¼ N 0 eðgða
=2ÞÞt aBðtÞ
e
ð6Þ
where parameters g and a can reflect the population growth rate and the stochastic fluctuation. Note that very few specific stochastic differential equations have explicitly known solutions. If they have no explicit solution, numerical solution is needed [20]. From Eq. (6) we get the following: (i) If 2g 4 a2 then NðtÞ-1 as t-1, a.s. (ii) If 2g o a2 then NðtÞ-0 as t-1, a.s. (iii) If 2g ¼ a2 then NðtÞ will fluctuate between arbitrary large and arbitrary small values as t-1, a.s. And also we can get the expectation of NðtÞ h i 2 2 ENðtÞ ¼ E N0 eðgða =2ÞÞt eaBðtÞ ¼ N0 eðgða =2ÞÞt EeaBðtÞ Z þ1 1 2 2 ¼ N0 eðgða =2ÞÞt eax pffiffiffiffiffiffiffiffi eðx =2tÞ dx 2pt 1 Z þ1 2 1 pffiffiffiffiffiffiffiffi eððxatÞ =2tÞ dx ¼ N 0 egt ¼ N0 egt 2pt 1
ð7Þ
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A PGM employed in present study is a diffusion [21] which is widely existing in physical phenomenon. Diffusion can be constructed by using stochastic differential equations. A diffusion process is a continuous time Markov process X ¼ XðtÞ, t Z 0 , 8x A R, t Z0, e Z0 such that (i) lim 1h Pð9Xðt þhÞx9 4 e9XðtÞ ¼ xÞ ¼ 0 h-0 (ii) lim 1h E ðXðt þ hÞxÞ9XðtÞ ¼ x ¼ b o 1 h-0 h i (iii) lim 1h E ðXðt þ hÞxÞ2 9XðtÞ ¼ x ¼ s2 o 1 h-0
where b and s2 are measurable functions, b is drift coefficient and s is diffusion coefficient. Ito¨ stochastic differential equation is a diffusion stochastic differential equation if bðt,xÞ ¼ bðxÞ and sðt,xÞ ¼ sðxÞ. Note that these diffusion formulas are time-homogeneous: the functions bðxÞ and sðxÞ do not vary with time. In a PGM, we know bðxÞ ¼ gx, sðxÞ ¼ ax, so it is a diffusion.
2.3. Wear processes model In this section, a PGM is used to describe the running-in process. By using the discrete method, as described in section 2.1, the rough surface is meshed by using several bar-shaped elements. Z i,j ðtÞði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ is the height variable for the grid point ði,jÞ. In this model, Z i,j ðtÞ is regarded as a stochastic variable and the height change for each point Z i,j ðtÞ is mutual independence. By using PGM in every grid point we get dZ i,j ðtÞ ¼ gi,j Z i,j ðtÞdt þ ai,j Z i,j ðtÞdBi,j ðtÞ
ði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ ð8Þ
where Z i,j ð0Þ is the initial height. Letgij ¼ g ði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ j ¼ 1,2,:::,nÞ. Then, we can get dZ i,j ðtÞ ¼ gZ i,j ðtÞ dt þ aZ i,j ðtÞ dBðtÞ
and
aij ¼ aði ¼ 1,2,:::,m;
ði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ
ð9Þ
Here g o0 and a 4 0. Note that gij or aij may have different values for the different points. Parameters gij and aij can be determined by experiments. When gij ¼ 0 and aij ¼ 0, the height Z i,j will not change during the wear, which will be deeply discussed in section 4. Applying Ito¨ formula Eqs. (3) and (9) has the explicit solution 2 Z i,j ðtÞ ¼ Z i,j ð0Þeðgða =2ÞÞt eaBðtÞ
ði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ
359
3. Wear process simulations 3.1. A single point simulation In this section, the changes of a single point during running-in are considered. Select a point in a discrete rough surface and apply Eq. (10). Let T¼5 h which includes 1000 steps Brownian motion, and initial height Z 0 ¼1 mm. Consider four different conditions in which g is constant whereas a changes in each condition: (1) g ¼ 1, a ¼0.1; (2) g ¼ 1, a ¼0.2; (3) g ¼ 1, a ¼0.3; (4) g ¼ 1, a ¼0.4. Each condition is simulated five times. Fig. 6 shows the results of simulations for a single point. Furthermore, we consider the other three different conditions in which g changes in each condition whereas a is a constant: (5) g ¼ 0.5, a ¼0.2; (6) g ¼ 1.5, a ¼0.2 and (7) g ¼ 2, a ¼0.2. Likewise, each condition is simulated five times. The changes of the height for a single point are shown in Fig. 7. Parameter a represents the fluctuation ratio. When a 4 0, if a reduces, the fluctuation will decrease (see Fig. 6). On the other hand, parameter g reflects the rate of decline. When g o0, the curves become stiffer as g reduces (see Fig. 7). For a high wear resistant material, g needs to be a small value, while for a low wear resistant material, g has a large value. When stochastic fluctuation of surface is tremendous during wear, a should be a large value to capture the stochastic behavior. 3.2. A cross-sectional profile simulation The changes of a cross-sectional profile are simulated in this section. Select a cross-sectional profile which consists of 100 discrete points. A spacing distance of 1 mm between two adjacent discrete points is supposed. Apply Eq. (10), let m¼100, n¼1 and T¼5 h which includes 1000 steps Brownian motion. Initial height Z i,j ð0Þði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ is chosen from a normal distribution with mean m ¼0.25 mm and standard deviation s ¼0.4 mm2. Let Z i,j ð0Þ ¼ 0:75 mm, if Z i,j ð0Þ r0:75 mm. Let Z i,j ð0Þ ¼ 1:25 mm, if Z i,j ð0Þ Z 1:25 mm. So Z¼ 0.25 mm is the arithmetic mean of surface height before running-in. We suppose Z¼0 mm is the base plane, which can be determined by wear conditions. Consider g ¼ 0.5, a ¼0.2, then cross-sectional profiles at different time during wear are shown in Fig. 8. Stochastic item will influence the height of every point, as can be seen from Fig. 8, the value of the initial highest peak is changed from 1.1867 mm to 0.0950 mm, while another initial higher peak will become the highest peak during wear and its value is changed from 1.1102 mm to 0.1405 mm. Because the stochastic item contains Brownian motion, if a point is higher than others at time t, it may lower than others at time t þ dt.
ð10Þ 3.3. A discrete rough surface simulation
Because it is a PGM, every point is a one-dimensional diffusion. The expression of Z i,j ðtÞ which can be seen in Eq. (10) includes three items: initial height, exponential movement and geometric Brownian movement. Initial height is related to initial surface profile. Exponential movement represents the tendency of the movement. Geometric Brownian movement represents wave motion and it is a stochastic item. During the running-in process, shedding of peak material and bonding of valley material are treated as stochastic items. In the wear process, parameters g and a denote the decline rate of surface point and the stochastic fluctuation of surface point, and the values depend on surface velocity, lubricant’s viscosity, loading weight, working conditions, surface parameters, material property, etc. Further the expectation of Z i,j ðtÞ can be obtained as follows: E½Z i,j ðtÞ ¼ Z i,j ð0Þegt
ði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ
ð11Þ
Multi-point is considered to reflect the changes of a rough surface during running-in. One hundred points were employed to represent a discrete rough surface. The grid spacing along the x-axis or y-axis is given as 1 mm. Apply Eq. (10), and let m¼10, n¼10 and T¼5 h which are included in 1000 steps Brownian motion. Likewise, initial height Z i,j ð0Þði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ are chosen from a normal distribution with mean m ¼0.25 mm and standard deviation s ¼0.4 mm2. Let Z i,j ð0Þ ¼ 0:75 mm, if Z i,j ð0Þ r0:75 mm. Let Z i,j ð0Þ ¼ 1:25 mm, if Z i,j ð0Þ Z 1:25 mm. So Z¼0.25 mm is the arithmetic mean of surface height before running-in. We suppose Z¼0 mm is the base plane, which can be determined by wear conditions. Consider g ¼ 0.5, a ¼0.2, then the surface during wear can be shown in Fig. 9. The results show that a discrete rough surface height is descending with dynamic fluctuation while descending speed slows down during the running-in process.
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Fig. 6. A single point simulation with a constant g but various a0 s.
Fig. 7. A single point simulation with a constant a but various g0 s.
By using Eq. (11), we can get the derivative of E½Z i,j ðtÞ dE½Z i,j ðtÞ ¼ Z i,j ð0Þgegt dt
ði ¼ 1,2,:::,n; j ¼ 1,2,:::,mÞ
ð12Þ
By ignoring the stochastic item, the derivative of E½Z i,j ðtÞ depends on initial height Z i,j ð0Þ and parameter g. In the present PGM we assume that all surface points have the same parameter g, so if initial height of the point is high, then its variation will be
Z. Wang, Q. Zhou / Wear 294-295 (2012) 356–363
361
Fig. 8. A cross-sectional profile at different times during running-in. (a) The initial surface profile, (b) worn for T/5, (c) worn for 3T/5 and (d) worn for T.
Fig. 9. A rough surface at different times during running-in. (a) The initial surface profile, (b) worn for T/5, (c) worn for 2T/5, (d) worn for 3T/5 (e) worn for 4T/5 and (f) worn for T
high. The change of the height during the running-in process is proportional to the initial height.
The wear volume Q in rough surface can be expressed as Z Z m X n X Q ðtÞ ¼ ðZð0ÞZðtÞÞdS ðZ i,j ð0ÞZ i,j ðtÞÞSi,j : ð13Þ s
4. Results and discussion The changes of every discrete point can be achieved by using Eq. (10), furthermore wear volume Q in rough surface is deduced to verify the model. As shown in Fig. 3, X–Y plane is the base plane of the rough surface, which is determined by wear conditions. If they have enough small grids, the volume of discrete rough surface can be calculated by adding many small hexahedrons together. Wear volume Q is equal to initial volume minus remainder volume.
i¼1j¼1
Considering all points follow PGMs. Substituting Eq. (10) into Eq. (13), it gives Q ðtÞ
m X n X
2
ðZ i,j ð0ÞZ i,j ð0Þeðgða
=2ÞÞt aBðtÞ
e
ÞSi,j
i¼1j¼1
And the expectation of the Q ðtÞ can be expressed as 2 3 m X n X g ða2 =2ÞÞt aBðtÞ ð 4 ðZ i,j ð0ÞZ i,j ð0Þe e ÞSi,j 5 E½Q ðtÞ E i¼1j¼1
ð14Þ
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2
E½ð1eðgða
=2ÞÞt aBðtÞ
e
Þ
m X n X
Z i,j ð0ÞSi,j
i¼1j¼1 2
V 0 ð1E½eðgða
=2ÞÞt aBðtÞ
e
Þ V 0 ð1egt Þ
ð15Þ Pm
Pn
where g o0 is the rate of decline, and V 0 ¼ i ¼ 1 j ¼ 1 Z i,j ð0ÞSi,j is approximately equal to the volume between base plane and the initial surface. The base plane of the rough surface is determined by wear conditions, so V 0 is also related with wear conditions. But here we can deduce it from boundary conditions and get V 0 ¼ ðE½Q ðTÞ=1egT Þ, where T denotes the time at which running-in process is finished. From Eq. (15) we know the expectation of wear volume Q follows exponential change, which is plotted in Fig. 10. From Eq. (10), we know the changes of the surface height contain stochastic item. Because discrete points are plenty enough to replace a rough surface, stochastic item cannot change the distribution during running-in. If the profile of the initial surface has a Gaussian distribution, the changes of surface height distribution also obey Gaussian distribution (see Fig. 11). In Fig. 11, cðzÞ denotes probability density function of the surface height, and zB is the value of base plane. Time is denoted by t0, t1, t2 and t3, and t0 ot1 ot2 ot3. During running-in, the curve of surface height distribution becomes sharper. In real engineering system, running-in is a complex process [22], it involves the material composition, surface microstructure, contact mechanics, chemistry and so on. Although, many engineering surfaces follow Gaussian distribution, the surface may follow non-Gaussian trend after wear [23]. Because the peaks go lower during the interactions of the contact while the valleys do not change or just decrease a little due to the merging of wear debris particles. This condition can be simulated by setting different parameters g and a for the peaks and valleys, for example if we assumed g ¼ 0 and a ¼ 0 for the valleys, the valleys
Fig. 12. Wear profile after T, where g ¼ 0.5 and a ¼0.2 for the peaks and g ¼ 0 and a ¼ 0 for the valleys.
will not change during the wear. A cross-sectional profile as discussed in Section 3.2 is used, however by setting g ¼ 0.5 and a ¼0.2 for the peaks while g ¼ 0 and a ¼ 0 for the valleys. The profile at time T is shown in Fig. 12, which is like the result of the experiments (see Fig. 2 in Ref. [23]). They found that the surface roughness after wear follows Pearson distribution. Now we consider a general wear model, the general expression of Z i,j ðtÞ can be expressed as Z i,j ðtÞ ¼ Z i,j ð0Þg i,j ðtÞr i,j ðBðtÞÞ
ði ¼ 1,2,:::,n; j ¼ 1,2,:::,mÞ
ð16Þ
where Z i,j ðtÞ can be expressed by three items: initial height Z i,j ð0Þ, determinate function g i,j ðtÞ and stochastic item r i,j ðBðtÞÞ which contains Brownian motion BðtÞ. Initial height Z i,j ð0Þ is related to initial surface profile, and determinate function g i,j ðtÞ represents the tendency of the surface movement, while stochastic item represents the wave motion. Shedding of the peak material and bonding of the valley material can be considered as stochastic items. Generally, a high wear rate can be observed at the beginning of wear, thus a proper function g i,j ðtÞ needs to be selected to reflect the dramatic change of the surfaces in this period, while in the steady-state period, g i,j ðtÞ can be a linear function to reflect a low wear rate behavior. Because wear is a complex process, the functions g i,j ðtÞ and r i,j ðBðtÞÞ in Eq. (16) should be established according to physical and chemical conditions.
5. Summary
Fig. 10. Expectation of the wear volume during running-in.
A discrete rough surface is established by dividing the wear region into several bar-shaped elements. The height variation of every grid point has been considered as a stochastic process. A PGM is introduced to simulate the changes of the height. By using this model, the variations of the rough surface were simulated during the running-in process. A single point, a crosssectional profile and a three-dimensional rough surface have been simulated. The following results can be obtained:
Fig. 11. Probability density function of surface height at different times t0, t1, t2 and t3 during running-in.
(1) Although a point is higher than other points at time t, it may be lower than other points at time tþ dt due to Brownian motion. The simulation for a discrete rough surface shows that the peak declines with dynamic fluctuation and the declining speed decreases during running-in, whereas, the valley presents inverse tendency. (2) Suppose that the height of every discrete point conforms to a PGM, the expectation of the wear volume Q follows a exponential form. When all discrete points have the same g and a, if the initial surface follows a Gaussian distribution, the worn surface shows Gaussian distribution. When different parameters g and a are chosen for the peaks and valleys, the worn surfaces show non-Gaussian trend. (3) The general expression of surface height Z i,j ðtÞ includes three items: initial height, determinate function and stochastic
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function. The expression of Z i,j ðtÞ should be established according to physical and chemical conditions. How to select suitable parameters should be further studied.
Acknowledgments The authors would like to express their sincere gratitude to the support from the Fundamental Research Funds for the Central Universities under Grant no. CDJRC10280005 and the National Science Foundation of China under Grant no. 51105391. References [1] J.F. Archard, Contact and rubbing of flat surfaces, Journal of Applied Physics 24 (1953) 981–988. ~ [2] P. Podra, S. Andersson, Simulating sliding wear with finite element method, Tribology International 32 (1999) 71–81. [3] G.K. Sfantos, M.H. Aliabadi, Wear simulation using an incremental sliding Boundary Element Method, Wear 260 (2006) 1119–1128. [4] H.C. Meng, K.C. Ludema, Wear models and predictive equations: their form and content, Wear 181–183 (1995) 443–457. [5] M. D’Acunto, Wear and diffusive processes, Tribology International 36 (2003) 553–558. [6] Y.Z. Hu, N. Li, K. Tonder, A dynamic system model for lubricated sliding wear and running-in, ASME Journal of Tribology 113 (1991) 499–505. [7] K.S. Wang, C.S. Chen, J.J. Huang, Dynamic reliability behavior for sliding wear of carburized steel, Reliability Engineering and System Safety 58 (1997) 31–41.
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[8] C.J. Goh, L.C. Tang, S.C. Lim, Reliability modelling of stochastic wear-out failure, Reliability Engineering and System Safety 25 (1989) 303–314. [9] C. Dasilvajr, G. Pintaude, Uncertainty analysis on the wear coefficient of Archard model, Tribology International 41 (2008) 473–481. [10] Y.R. Jeng, Z.W. Lin, S.H. Shyu, Changes of surface topography during runningin process, ASME Journal of Tribology 126 (2004) 620–625. [11] G.Y. Zhou, M.C. Leu, D. Blackmore, Fractal geometry model for wear prediction, Wear 170 (1993) 1–14. [12] S. Ge, G. Chen, Fractal prediction models of sliding wear during the runningin process, Wear 231 (1999) 249–255. [13] J.H. Horng, M.L. Len, J.S. Lee, The contact characteristics of rough surfaces in line contact during running-in process, Wear 253 (2002) 899–913. [14] W. Wang, P.L. Wong, Wear volume determination during running-in for PEHL contacts, Tribology International 33 (2000) 501–506. [15] W. Wang, P.L. Wong, F. Guo, Application of partial elastohydrodynamic lubrication analysis in dynamic wear study for running-in, Wear 257 (2004) 823–832. [16] K. De Moerlooze, F. Al-Bender, H. Van Brussel, A generalised asperity-based friction model, Tribology Letters 40 (2010) 113–130. [17] K. De Moerlooze, F. Al-Bender, H. Van Brussel, A novel energy-based generic wear model at the asperity level, Wear 270 (2011) 760–770. [18] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 6th ed., Springer-Verlag, 2003. [19] K. Ito, ¯ On stochastic differential equations, Memoirs, American Mathematical Society 4 (1951) 1–51. [20] P.E. Kloeden, E. Platen, H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer-Verlag, 1994. [21] L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 2: Ito¨ Calculus, Cambridge University Press, Cambridge, 2000. [22] P.J. Blau, On the nature of running-in, Tribology International 38 (2005) 1007–1012. [23] C.H. Lee, M. Eriten, A.A. Polycarpou, Application of elastic–plastic static friction models to rough surfaces with asymmetric asperity distribution, ASME Journal of Tribology 132 (2010) 031602–031611.
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Wear journal homepage: www.elsevier.com/locate/wear
Applying a population growth model to simulate wear of rough surfaces during running-in Zhanjiang Wang n, Qinghua Zhou State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 February 2012 Received in revised form 19 July 2012 Accepted 25 July 2012 Available online 2 August 2012
A population growth model (PGM) is introduced to simulate wear during running-in, which is one of three typical wear states. The rough contact region is divided into several bar-shaped elements while the wear of every grid point has been considered as a PGM process. The wear volume in rough surface can be calculated during the wear processes and the proposed wear formula contains three items: initial point height, determine function and stochastic function. By employing a PGM, the changes of every local peak and valley were simulated. The results show that the height of local peak dynamically decreased and the decreasing speed becomes slow during running-in, whereas, the local valley shows inverse tendency. Because the stochastic function contains Brownian motion, if a point is higher than other points at time t, it may be lower than other points at time tþ dt. Furthermore, if the height of every discrete point conforms to a PGM, the expectation of the wear volume follows an exponential form. & 2012 Elsevier B.V. All rights reserved.
Keywords: Wear modeling Stochastic process Surface analysis Contact mechanics
1. Introduction In mechanical system, wear plays an important role in surface failure and further affects the reliability of mechanical components and the whole system. There are three typical wear states: running-in, mild wear and severe wear. The wear behavior is so complex that different kinds of models were established to analyze it. Basically, there are two types of models to deal with wear between two contact surfaces: deterministic and stochastic models. Archard [1] pioneered the development of the deterministic models for wear. He first used the wear coefficient to describe the relations between the wear volume and the applied load. For numerical simulation of deterministic model, the finite element method (FEM) [2] and the boundary element method (BEM) [3] can be employed. Moreover, the deterministic wear models or equations to describe the wear are quite different for different materials and working conditions [4]. Meanwhile, the stochastic models were commonly used for wear analysis. D’Acunto [5] used a double-well model based on a microscopic diffusive process to describe the wear volume rate. Hu et al. [6] employed a dynamic system theory to analyze the running-in process in wear. They believed that running-in can be regarded as a self-adapting process of a wear system, of which the dynamic behaviors were controlled by the geometrical, physical and chemical feedback. Wang et al. [7] analyzed the dynamic
n
Corresponding author. Tel.: þ86 138 1060 8964. E-mail address: [email protected] (Z. Wang).
0043-1648/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.wear.2012.07.028
reliability behavior for sliding wear of multiple ring–disk system. They found that the critical wear depth can be utilized to estimate the dynamic reliability. Goh et al. [8] treated cumulative damage as a stochastic process. When the cumulative damage exceeds a certain prescribed threshold, the wear-out failure occurs. Besides, the wear coefficient was modeled as a random variable and a stochastic process in Ref. [9]. Surface topography was found to have much influence on wear [10–12]. Some researchers investigated contact characteristics of rough surfaces for dry contact [13] and lubrication contact [14,15]. Unfortunately, they just focused on the changes of statistical parameters such as wear volume, surface height distribution, arithmetic mean, root mean square and skewness. The statistical parameters just reflect global characters of wear region and local characters describing the local wear behaviors were ignored in their studies. The relationship between local behaviors and global wear characters needs to be defined and wear in every local place in rough surfaces should be modeled. Therefore, equations need to be established to reflect the changes of every local peak and valley in a rough surface. Recently, Moerlooze et al. [16,17] presented a theoretical spring model to describe the asperities wear evolution in a plain contact. This study aims to establish a stochastic wear model on every local peak and valley. A population growth model (PGM) based on the stochastic differential equations [18] is introduced to describe a rough surface wear during the running-in process. The rough contact region was divided into several bar-shaped elements. The wear state in every discrete point was to be considered as a stochastic process, which contains three items: initial point
Z. Wang, Q. Zhou / Wear 294-295 (2012) 356–363
Nomenclature a ¼ g þ aW relative rate of growth bðt,XðtÞÞ drift coefficient of X BðtÞ standard Brownian motion E expectation gðtÞ determinate function m,n grid numbers along the x and y axes N0 given initial value N(t) size of the population at time t Q wear volume rðBðtÞÞ stochastic function which contains Brownian motion BðtÞ Rþ set of positive real numbers t time, hr
height, determine function and stochastic function. Then, the changes of rough surface profile can be simulated, which will provide a mathematical perspective for engineers to understand the wear behaviors during the running-in process.
2. Surfaces wear processes model 2.1. Mathematics model of wear process In general mechanical system, wear exists in two mating rough surfaces, as shown in Fig. 1, the system consists of two mating rough surfaces, which has three-dimensional roughness and lubricant between them. When the two contact surfaces move relative to each other, the wear will occur. The typical wear stages include running-in, mild wear and severe wear, as plotted in Fig. 2. Running-in is the initial stage of the wear process. During running-in the wear rate varies from high to low and surfaces become planarized. Mild wear is the steady-state of the wear process, where wear rate is relatively slow over a long period of time. After the mild wear stage, the wear is quite rapid at the severe wear stage and failure will occur. According to Archard wear model [1], the wear was related to the contact pressure. If a local asperity is higher than other asperities, the local pressure will be higher than other regions. Therefore the wear should be related to the profile of the rough surface. A mathematical model was established to simulate the changes of rough surfaces profile during running-in. To form the mathematical model, wear surface is discretized. The mathematical model is shown in Fig. 3. X–Y plane is the base plane of the rough surface, which is determined by wear conditions. In the X–Y
Fig. 1. General lubricated sliding wear processes.
357
u V0 W X,Y XðtÞ zB ZðtÞ g,a
velocity volume between base plane and the initial surface one-dimensional ‘‘white noise’’ space coordinates, mm stochastic process value of the base plane asperity height at time t parameters to reflect the population growth rate and the stochastic fluctuation m mean value s standard deviation sðt,XðtÞÞ diffusion coefficient of X cðzÞ probability density function
plane, rectangular grids are employed to discretize rough surface. The whole contact region is divided into several bar-shaped elements. If they have enough grids, the real rough surfaces can be replaced by discrete rough surface in the mathematical model. The total number of grid points in the X–Y plane is m n. Z i,j ðtÞði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ is the height of each grid point ði, jÞ (or distance from the based plane), and its variation can be treated as a stochastic process, here t A ½0,T and T is the time for running-in process. In present study, a stochastic differential equation is employed to describe Z i,j ðtÞ.
Fig. 2. Typical wear curve.
Fig. 3. A discrete rough surface (several small hexahedrons were employed to represent the volume of discrete rough surface).
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2.2. It¨ o formula and PGM
Fig. 4. Surface profile and its changes during the time interval t to tþ dt.
The dynamic properties of many important continuous time stochastic processes can be modeled by using stochastic differential equations. We can construct the sample paths of diffusion directly from Brownian motion and Ito¨ formula [19] can be employed to solve stochastic differential equations. PGM is the model to describe the changes in a population over time, and be defined as solutions to stochastic differential equations. The wear is a dynamic stochastic process, so PGM can be employed to describe the height of every grid point Z i,j ðtÞ during the running-in process; the Ito¨ formula is introduced and used to resolve stochastic differential equations. Let Bðo,tÞ, o A O, t A R þ be a standard Brownian motion. Let b and s be two functions from ½0,T O into R. The Ito¨ process is a stochastic process XðtÞ, t A R þ of the form Z t Z t XðtÞ ¼ Xð0Þ þ bðs,XðsÞÞ ds þ sðs,XðsÞÞ dBðsÞ ð1Þ 0
0
or in differential form dXðtÞ ¼ bðt,XðtÞÞ dt þ sðt,XðtÞÞ dBðtÞ
ð2Þ
If Y ¼ yðt,XðtÞÞ has continuous partial derivatives @Y=@t, @Y=@x and @2 Y=@x2 . Then ! @y @y s2 @2 y @y þb þ ð3Þ dYðtÞ ¼ dt þ s dBðtÞ @t @x @x 2 @x2
Fig. 5. The changes of probability density function for surface height during time interval t to tþ dt.
Therefore, the wear model proposed in this paper during running-in is based on the following assumptions: 1. The height variation of every grid point is considered as a stochastic process, which contains three items: initial height, determine function and stochastic function. 2. There exists a base plane for the rough surface, which is determined by wear conditions. The height of surface profile is relative to base plane. 3. The changes of the height in every grid point are related to the base plane. If the local surface height is higher than the base plane, it will decrease during running-in, but it always keeps higher than the base plane. Otherwise, if the local surface height is lower than the base plane, the profile will be increasing, and it always keeps lower than the base plane. 4. If a local asperity is higher than other asperities at time t, the height variation of this local asperity will be greater than others during t to tþ dt, likewise, if a local valley is lower than other valleys at time t, the corresponding height variation will be greater than others during t to t þ dt.
From assumptions above, the schematic diagram of changes for surface profile during the time interval t to t þ dt can be seen in Fig. 4. Here zB is the height of the base plane. The changes of probability density function for surface height during time interval t to t þ dt can be seen in Fig. 5. Let cðzÞ denote probability density function. zm ðtÞ is the arithmetic mean of surface height at time t, and zm ðt þ dtÞ is the arithmetic mean of surface height at time t þ dt. zB is the value of the base plane. During the wear process, the area dA1 is redistributed into dA2 and the area dA3 is to dA4. According to the characters of probability density function and assumption 3, we get dA1 þ dA3 ¼ dA2 þ dA4 and dA1 ¼ dA2, dA3 ¼ dA4.
is called the Ito¨ formula [19]. A PGM describes the population’s evolution NðtÞ as follows: ( dNðtÞ ¼ aNðtÞ dt ð4Þ Nð0Þ ¼ N0 where NðtÞ is the size of the population at time t; a is the relative rate of growth, which equals to g þ aW; Wis one-dimensional ‘‘white noise’’; N 0 is a given initial value. Let us assume that g ¼constant and a ¼constant. The PGM is described by the following linear system of stochastic differential equation: ( dNðtÞ ¼ gNðtÞ dt þ aNðtÞ dBðtÞ ð5Þ Nð0Þ ¼ N0 where t A ½0,T, and BðtÞ is Brownian motion. Applying Ito¨ formula, i.e., Eqs. (3) and (5) have the explicit solution [18] 2
NðtÞ ¼ N 0 eðgða
=2ÞÞt aBðtÞ
e
ð6Þ
where parameters g and a can reflect the population growth rate and the stochastic fluctuation. Note that very few specific stochastic differential equations have explicitly known solutions. If they have no explicit solution, numerical solution is needed [20]. From Eq. (6) we get the following: (i) If 2g 4 a2 then NðtÞ-1 as t-1, a.s. (ii) If 2g o a2 then NðtÞ-0 as t-1, a.s. (iii) If 2g ¼ a2 then NðtÞ will fluctuate between arbitrary large and arbitrary small values as t-1, a.s. And also we can get the expectation of NðtÞ h i 2 2 ENðtÞ ¼ E N0 eðgða =2ÞÞt eaBðtÞ ¼ N0 eðgða =2ÞÞt EeaBðtÞ Z þ1 1 2 2 ¼ N0 eðgða =2ÞÞt eax pffiffiffiffiffiffiffiffi eðx =2tÞ dx 2pt 1 Z þ1 2 1 pffiffiffiffiffiffiffiffi eððxatÞ =2tÞ dx ¼ N 0 egt ¼ N0 egt 2pt 1
ð7Þ
Z. Wang, Q. Zhou / Wear 294-295 (2012) 356–363
A PGM employed in present study is a diffusion [21] which is widely existing in physical phenomenon. Diffusion can be constructed by using stochastic differential equations. A diffusion process is a continuous time Markov process X ¼ XðtÞ, t Z 0 , 8x A R, t Z0, e Z0 such that (i) lim 1h Pð9Xðt þhÞx9 4 e9XðtÞ ¼ xÞ ¼ 0 h-0 (ii) lim 1h E ðXðt þ hÞxÞ9XðtÞ ¼ x ¼ b o 1 h-0 h i (iii) lim 1h E ðXðt þ hÞxÞ2 9XðtÞ ¼ x ¼ s2 o 1 h-0
where b and s2 are measurable functions, b is drift coefficient and s is diffusion coefficient. Ito¨ stochastic differential equation is a diffusion stochastic differential equation if bðt,xÞ ¼ bðxÞ and sðt,xÞ ¼ sðxÞ. Note that these diffusion formulas are time-homogeneous: the functions bðxÞ and sðxÞ do not vary with time. In a PGM, we know bðxÞ ¼ gx, sðxÞ ¼ ax, so it is a diffusion.
2.3. Wear processes model In this section, a PGM is used to describe the running-in process. By using the discrete method, as described in section 2.1, the rough surface is meshed by using several bar-shaped elements. Z i,j ðtÞði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ is the height variable for the grid point ði,jÞ. In this model, Z i,j ðtÞ is regarded as a stochastic variable and the height change for each point Z i,j ðtÞ is mutual independence. By using PGM in every grid point we get dZ i,j ðtÞ ¼ gi,j Z i,j ðtÞdt þ ai,j Z i,j ðtÞdBi,j ðtÞ
ði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ ð8Þ
where Z i,j ð0Þ is the initial height. Letgij ¼ g ði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ j ¼ 1,2,:::,nÞ. Then, we can get dZ i,j ðtÞ ¼ gZ i,j ðtÞ dt þ aZ i,j ðtÞ dBðtÞ
and
aij ¼ aði ¼ 1,2,:::,m;
ði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ
ð9Þ
Here g o0 and a 4 0. Note that gij or aij may have different values for the different points. Parameters gij and aij can be determined by experiments. When gij ¼ 0 and aij ¼ 0, the height Z i,j will not change during the wear, which will be deeply discussed in section 4. Applying Ito¨ formula Eqs. (3) and (9) has the explicit solution 2 Z i,j ðtÞ ¼ Z i,j ð0Þeðgða =2ÞÞt eaBðtÞ
ði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ
359
3. Wear process simulations 3.1. A single point simulation In this section, the changes of a single point during running-in are considered. Select a point in a discrete rough surface and apply Eq. (10). Let T¼5 h which includes 1000 steps Brownian motion, and initial height Z 0 ¼1 mm. Consider four different conditions in which g is constant whereas a changes in each condition: (1) g ¼ 1, a ¼0.1; (2) g ¼ 1, a ¼0.2; (3) g ¼ 1, a ¼0.3; (4) g ¼ 1, a ¼0.4. Each condition is simulated five times. Fig. 6 shows the results of simulations for a single point. Furthermore, we consider the other three different conditions in which g changes in each condition whereas a is a constant: (5) g ¼ 0.5, a ¼0.2; (6) g ¼ 1.5, a ¼0.2 and (7) g ¼ 2, a ¼0.2. Likewise, each condition is simulated five times. The changes of the height for a single point are shown in Fig. 7. Parameter a represents the fluctuation ratio. When a 4 0, if a reduces, the fluctuation will decrease (see Fig. 6). On the other hand, parameter g reflects the rate of decline. When g o0, the curves become stiffer as g reduces (see Fig. 7). For a high wear resistant material, g needs to be a small value, while for a low wear resistant material, g has a large value. When stochastic fluctuation of surface is tremendous during wear, a should be a large value to capture the stochastic behavior. 3.2. A cross-sectional profile simulation The changes of a cross-sectional profile are simulated in this section. Select a cross-sectional profile which consists of 100 discrete points. A spacing distance of 1 mm between two adjacent discrete points is supposed. Apply Eq. (10), let m¼100, n¼1 and T¼5 h which includes 1000 steps Brownian motion. Initial height Z i,j ð0Þði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ is chosen from a normal distribution with mean m ¼0.25 mm and standard deviation s ¼0.4 mm2. Let Z i,j ð0Þ ¼ 0:75 mm, if Z i,j ð0Þ r0:75 mm. Let Z i,j ð0Þ ¼ 1:25 mm, if Z i,j ð0Þ Z 1:25 mm. So Z¼ 0.25 mm is the arithmetic mean of surface height before running-in. We suppose Z¼0 mm is the base plane, which can be determined by wear conditions. Consider g ¼ 0.5, a ¼0.2, then cross-sectional profiles at different time during wear are shown in Fig. 8. Stochastic item will influence the height of every point, as can be seen from Fig. 8, the value of the initial highest peak is changed from 1.1867 mm to 0.0950 mm, while another initial higher peak will become the highest peak during wear and its value is changed from 1.1102 mm to 0.1405 mm. Because the stochastic item contains Brownian motion, if a point is higher than others at time t, it may lower than others at time t þ dt.
ð10Þ 3.3. A discrete rough surface simulation
Because it is a PGM, every point is a one-dimensional diffusion. The expression of Z i,j ðtÞ which can be seen in Eq. (10) includes three items: initial height, exponential movement and geometric Brownian movement. Initial height is related to initial surface profile. Exponential movement represents the tendency of the movement. Geometric Brownian movement represents wave motion and it is a stochastic item. During the running-in process, shedding of peak material and bonding of valley material are treated as stochastic items. In the wear process, parameters g and a denote the decline rate of surface point and the stochastic fluctuation of surface point, and the values depend on surface velocity, lubricant’s viscosity, loading weight, working conditions, surface parameters, material property, etc. Further the expectation of Z i,j ðtÞ can be obtained as follows: E½Z i,j ðtÞ ¼ Z i,j ð0Þegt
ði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ
ð11Þ
Multi-point is considered to reflect the changes of a rough surface during running-in. One hundred points were employed to represent a discrete rough surface. The grid spacing along the x-axis or y-axis is given as 1 mm. Apply Eq. (10), and let m¼10, n¼10 and T¼5 h which are included in 1000 steps Brownian motion. Likewise, initial height Z i,j ð0Þði ¼ 1,2,:::,m; j ¼ 1,2,:::,nÞ are chosen from a normal distribution with mean m ¼0.25 mm and standard deviation s ¼0.4 mm2. Let Z i,j ð0Þ ¼ 0:75 mm, if Z i,j ð0Þ r0:75 mm. Let Z i,j ð0Þ ¼ 1:25 mm, if Z i,j ð0Þ Z 1:25 mm. So Z¼0.25 mm is the arithmetic mean of surface height before running-in. We suppose Z¼0 mm is the base plane, which can be determined by wear conditions. Consider g ¼ 0.5, a ¼0.2, then the surface during wear can be shown in Fig. 9. The results show that a discrete rough surface height is descending with dynamic fluctuation while descending speed slows down during the running-in process.
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Fig. 6. A single point simulation with a constant g but various a0 s.
Fig. 7. A single point simulation with a constant a but various g0 s.
By using Eq. (11), we can get the derivative of E½Z i,j ðtÞ dE½Z i,j ðtÞ ¼ Z i,j ð0Þgegt dt
ði ¼ 1,2,:::,n; j ¼ 1,2,:::,mÞ
ð12Þ
By ignoring the stochastic item, the derivative of E½Z i,j ðtÞ depends on initial height Z i,j ð0Þ and parameter g. In the present PGM we assume that all surface points have the same parameter g, so if initial height of the point is high, then its variation will be
Z. Wang, Q. Zhou / Wear 294-295 (2012) 356–363
361
Fig. 8. A cross-sectional profile at different times during running-in. (a) The initial surface profile, (b) worn for T/5, (c) worn for 3T/5 and (d) worn for T.
Fig. 9. A rough surface at different times during running-in. (a) The initial surface profile, (b) worn for T/5, (c) worn for 2T/5, (d) worn for 3T/5 (e) worn for 4T/5 and (f) worn for T
high. The change of the height during the running-in process is proportional to the initial height.
The wear volume Q in rough surface can be expressed as Z Z m X n X Q ðtÞ ¼ ðZð0ÞZðtÞÞdS ðZ i,j ð0ÞZ i,j ðtÞÞSi,j : ð13Þ s
4. Results and discussion The changes of every discrete point can be achieved by using Eq. (10), furthermore wear volume Q in rough surface is deduced to verify the model. As shown in Fig. 3, X–Y plane is the base plane of the rough surface, which is determined by wear conditions. If they have enough small grids, the volume of discrete rough surface can be calculated by adding many small hexahedrons together. Wear volume Q is equal to initial volume minus remainder volume.
i¼1j¼1
Considering all points follow PGMs. Substituting Eq. (10) into Eq. (13), it gives Q ðtÞ
m X n X
2
ðZ i,j ð0ÞZ i,j ð0Þeðgða
=2ÞÞt aBðtÞ
e
ÞSi,j
i¼1j¼1
And the expectation of the Q ðtÞ can be expressed as 2 3 m X n X g ða2 =2ÞÞt aBðtÞ ð 4 ðZ i,j ð0ÞZ i,j ð0Þe e ÞSi,j 5 E½Q ðtÞ E i¼1j¼1
ð14Þ
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Z. Wang, Q. Zhou / Wear 294-295 (2012) 356–363
2
E½ð1eðgða
=2ÞÞt aBðtÞ
e
Þ
m X n X
Z i,j ð0ÞSi,j
i¼1j¼1 2
V 0 ð1E½eðgða
=2ÞÞt aBðtÞ
e
Þ V 0 ð1egt Þ
ð15Þ Pm
Pn
where g o0 is the rate of decline, and V 0 ¼ i ¼ 1 j ¼ 1 Z i,j ð0ÞSi,j is approximately equal to the volume between base plane and the initial surface. The base plane of the rough surface is determined by wear conditions, so V 0 is also related with wear conditions. But here we can deduce it from boundary conditions and get V 0 ¼ ðE½Q ðTÞ=1egT Þ, where T denotes the time at which running-in process is finished. From Eq. (15) we know the expectation of wear volume Q follows exponential change, which is plotted in Fig. 10. From Eq. (10), we know the changes of the surface height contain stochastic item. Because discrete points are plenty enough to replace a rough surface, stochastic item cannot change the distribution during running-in. If the profile of the initial surface has a Gaussian distribution, the changes of surface height distribution also obey Gaussian distribution (see Fig. 11). In Fig. 11, cðzÞ denotes probability density function of the surface height, and zB is the value of base plane. Time is denoted by t0, t1, t2 and t3, and t0 ot1 ot2 ot3. During running-in, the curve of surface height distribution becomes sharper. In real engineering system, running-in is a complex process [22], it involves the material composition, surface microstructure, contact mechanics, chemistry and so on. Although, many engineering surfaces follow Gaussian distribution, the surface may follow non-Gaussian trend after wear [23]. Because the peaks go lower during the interactions of the contact while the valleys do not change or just decrease a little due to the merging of wear debris particles. This condition can be simulated by setting different parameters g and a for the peaks and valleys, for example if we assumed g ¼ 0 and a ¼ 0 for the valleys, the valleys
Fig. 12. Wear profile after T, where g ¼ 0.5 and a ¼0.2 for the peaks and g ¼ 0 and a ¼ 0 for the valleys.
will not change during the wear. A cross-sectional profile as discussed in Section 3.2 is used, however by setting g ¼ 0.5 and a ¼0.2 for the peaks while g ¼ 0 and a ¼ 0 for the valleys. The profile at time T is shown in Fig. 12, which is like the result of the experiments (see Fig. 2 in Ref. [23]). They found that the surface roughness after wear follows Pearson distribution. Now we consider a general wear model, the general expression of Z i,j ðtÞ can be expressed as Z i,j ðtÞ ¼ Z i,j ð0Þg i,j ðtÞr i,j ðBðtÞÞ
ði ¼ 1,2,:::,n; j ¼ 1,2,:::,mÞ
ð16Þ
where Z i,j ðtÞ can be expressed by three items: initial height Z i,j ð0Þ, determinate function g i,j ðtÞ and stochastic item r i,j ðBðtÞÞ which contains Brownian motion BðtÞ. Initial height Z i,j ð0Þ is related to initial surface profile, and determinate function g i,j ðtÞ represents the tendency of the surface movement, while stochastic item represents the wave motion. Shedding of the peak material and bonding of the valley material can be considered as stochastic items. Generally, a high wear rate can be observed at the beginning of wear, thus a proper function g i,j ðtÞ needs to be selected to reflect the dramatic change of the surfaces in this period, while in the steady-state period, g i,j ðtÞ can be a linear function to reflect a low wear rate behavior. Because wear is a complex process, the functions g i,j ðtÞ and r i,j ðBðtÞÞ in Eq. (16) should be established according to physical and chemical conditions.
5. Summary
Fig. 10. Expectation of the wear volume during running-in.
A discrete rough surface is established by dividing the wear region into several bar-shaped elements. The height variation of every grid point has been considered as a stochastic process. A PGM is introduced to simulate the changes of the height. By using this model, the variations of the rough surface were simulated during the running-in process. A single point, a crosssectional profile and a three-dimensional rough surface have been simulated. The following results can be obtained:
Fig. 11. Probability density function of surface height at different times t0, t1, t2 and t3 during running-in.
(1) Although a point is higher than other points at time t, it may be lower than other points at time tþ dt due to Brownian motion. The simulation for a discrete rough surface shows that the peak declines with dynamic fluctuation and the declining speed decreases during running-in, whereas, the valley presents inverse tendency. (2) Suppose that the height of every discrete point conforms to a PGM, the expectation of the wear volume Q follows a exponential form. When all discrete points have the same g and a, if the initial surface follows a Gaussian distribution, the worn surface shows Gaussian distribution. When different parameters g and a are chosen for the peaks and valleys, the worn surfaces show non-Gaussian trend. (3) The general expression of surface height Z i,j ðtÞ includes three items: initial height, determinate function and stochastic
Z. Wang, Q. Zhou / Wear 294-295 (2012) 356–363
function. The expression of Z i,j ðtÞ should be established according to physical and chemical conditions. How to select suitable parameters should be further studied.
Acknowledgments The authors would like to express their sincere gratitude to the support from the Fundamental Research Funds for the Central Universities under Grant no. CDJRC10280005 and the National Science Foundation of China under Grant no. 51105391. References [1] J.F. Archard, Contact and rubbing of flat surfaces, Journal of Applied Physics 24 (1953) 981–988. ~ [2] P. Podra, S. Andersson, Simulating sliding wear with finite element method, Tribology International 32 (1999) 71–81. [3] G.K. Sfantos, M.H. Aliabadi, Wear simulation using an incremental sliding Boundary Element Method, Wear 260 (2006) 1119–1128. [4] H.C. Meng, K.C. Ludema, Wear models and predictive equations: their form and content, Wear 181–183 (1995) 443–457. [5] M. D’Acunto, Wear and diffusive processes, Tribology International 36 (2003) 553–558. [6] Y.Z. Hu, N. Li, K. Tonder, A dynamic system model for lubricated sliding wear and running-in, ASME Journal of Tribology 113 (1991) 499–505. [7] K.S. Wang, C.S. Chen, J.J. Huang, Dynamic reliability behavior for sliding wear of carburized steel, Reliability Engineering and System Safety 58 (1997) 31–41.
363
[8] C.J. Goh, L.C. Tang, S.C. Lim, Reliability modelling of stochastic wear-out failure, Reliability Engineering and System Safety 25 (1989) 303–314. [9] C. Dasilvajr, G. Pintaude, Uncertainty analysis on the wear coefficient of Archard model, Tribology International 41 (2008) 473–481. [10] Y.R. Jeng, Z.W. Lin, S.H. Shyu, Changes of surface topography during runningin process, ASME Journal of Tribology 126 (2004) 620–625. [11] G.Y. Zhou, M.C. Leu, D. Blackmore, Fractal geometry model for wear prediction, Wear 170 (1993) 1–14. [12] S. Ge, G. Chen, Fractal prediction models of sliding wear during the runningin process, Wear 231 (1999) 249–255. [13] J.H. Horng, M.L. Len, J.S. Lee, The contact characteristics of rough surfaces in line contact during running-in process, Wear 253 (2002) 899–913. [14] W. Wang, P.L. Wong, Wear volume determination during running-in for PEHL contacts, Tribology International 33 (2000) 501–506. [15] W. Wang, P.L. Wong, F. Guo, Application of partial elastohydrodynamic lubrication analysis in dynamic wear study for running-in, Wear 257 (2004) 823–832. [16] K. De Moerlooze, F. Al-Bender, H. Van Brussel, A generalised asperity-based friction model, Tribology Letters 40 (2010) 113–130. [17] K. De Moerlooze, F. Al-Bender, H. Van Brussel, A novel energy-based generic wear model at the asperity level, Wear 270 (2011) 760–770. [18] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 6th ed., Springer-Verlag, 2003. [19] K. Ito, ¯ On stochastic differential equations, Memoirs, American Mathematical Society 4 (1951) 1–51. [20] P.E. Kloeden, E. Platen, H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer-Verlag, 1994. [21] L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 2: Ito¨ Calculus, Cambridge University Press, Cambridge, 2000. [22] P.J. Blau, On the nature of running-in, Tribology International 38 (2005) 1007–1012. [23] C.H. Lee, M. Eriten, A.A. Polycarpou, Application of elastic–plastic static friction models to rough surfaces with asymmetric asperity distribution, ASME Journal of Tribology 132 (2010) 031602–031611.