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Modeling of wear of solids at different scales
I.G. Goryacheva / Physical Mesomechanics 10 56 (2007) 247254
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Modeling of wear of solids at different scales I.G. Goryacheva Institute for Problems in Mechanics RAS, Moscow, 119526, Russia The paper puts forward an approach to the analysis of stress state and fracture of surface layers in deformed solids. The approach is based on the consideration of contact interaction processes at several scale levels determined by characteristic structural parameters of the surface and subsurface layers of contacting bodies (micro- and mesoscales) and by their macrogeometry. The approach can be used to estimate the influence of geometrical, mechanical and physical characteristics of materials of contacting bodies and the intermediate medium on the performance of friction pairs. Methods of increasing the wear resistance and service life of the pairs and reducing energy loss owing to the control of wear and friction processes in thin surface layers can be developed.
1. Introduction Wear is a process of progressive loss of material from the operating surfaces of solids arising from the surface fracture. The dimensions of the body and its shape vary in this process. The study of wear shows that there are different causes for surface fracture. First of all, it is caused by high stresses in subsurface layers in the friction process which lead to the crack formation and to the detachment of the material particles (wear debris) from the rubber surface. This widespread type of wear is classified as mechanical wear, and often the word mechanical is omitted. Since the mechanical wear of two bodies in friction interaction is determined by the stresses in the subsurface layers, modeling of the surface fracture process is based on the methods used in contact mechanics and fracture mechanics. However, such modeling does not amount to the use of the results from those fields of the mechanics of solids. The process of wear has some peculiarities setting it as a special form of fracture. 2. The main stages in modeling First of all the specific feature of wear implies that this process is not critical for the operation of junction. Usually, admissible limit wear of moving parts of junctions is much more than the typical size of wear debris. Thus, repeated particle detachment from the rubber surface can occur during the life of the parts. Repeated unit fracture acts are the distinctive feature of the wear process, as opposed to the bulk material process. After removal of surCopyright © 2007 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV. All rights reserved. doi:10.1016/j.physme.2007.11.003
face material due to wear, the subsurface layer enters the contact. The characteristics of this layer, including those that determine wear intensity (microgeometry, damage of material, etc.), depend on the entire history of the frictional interaction. Thus, wear can be considered as a process of hereditary type. Besides, wear is a feedback process. Selforganization and equilibrium structure formation in wear occur as a result of the feedback action. In wear modeling it is first of all necessary to determine the stress and temperature distributions arising in the subsurface layers under given contact conditions, mechanical properties of contacting bodies, and surface macro- and microgeometry (contact state). At the macroscale these are the average (nominal) contact characteristics determined based on the given macroshape of the contacting bodies and contact conditions. At the microscale these are the real contact pressure, real internal stress and temperature distributions in the subsurface layer, which were calculated taking into account surface microgeometry and variation of mechanical and tribological characteristics of the subsurface layer in wear. Based on this analysis, the most probable wear mechanism is recognized. Use of the fracture criterion corresponding to this mechanism allows us to determine the onset of failure and the surface shape after particle detachment (or particle transfer to the counter-body). The determined surface microgeometry characteristics are used to calculate the stress and temperature distributions at the next instant of time, etc. The described stages of wear modeling are schematically shown in Fig. 1.
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In the majority of cases when wear is analyzed, only macrocharacteristics of the process (body shape variation at the instants of time which are much more then the intervals between the individual acts of fracture, the maximum linear wear at the surface, etc.) are of interest. The microgeometry variation after wear particle detachment, the shape of each particle and the other characteristics related to the calculation of stresses and temperature distribution at the microscale (sizes comparable to the typical diameter of the contact spot) are therefore the internal parameters of the model. In fact, for applications the main task of the wear modeling is the prediction of the macrostate (macroshape) evolution under given external conditions (load, velocity). It is possible to develop the wear models at two scales: macroscale models studying the macroshape variation in wear process, and microscale ones describing the elementary acts of wear particle detachment. The repeated calculations at the microscale allow us to evaluate the variation of the contact characteristics at the macroscale (macroshape variation, approach of the contacting bodies due to their wear, etc.) and to calculate the lifetime of junction in respect to the wear resistance criterion. The application of the described algorithm to the analysis of the frictional pairs is very time consuming. The analysis of processes at the microscale makes possible to develop a phenomenological wear model for a frictional pair at the macroscale, in which the wear rate dw* dt as a function of the macrocharacteristics of the junction (pressure p, velocity V, relative displacements of the contacting surfaces, etc.) is included (Fig. 1). This function may be constructed through modeling the fracture process at the microscale and elementary acts of wear particle detachment. 3. Modeling of surface fracture at the microscale The subject of wear analysis at the microscale is a thin surface layer whose thickness is comparable to the contact spot size. Wear modeling includes the determination of the crack initiation instant and trajectory of crack propagation up to particle detachment, transportation of the detached particles from the contact zone, and description of the sur-
Fig. 1. The main stages in wear modeling and their mutual relations
face microgeometry variation and properties of the thin subsurface layer in the process of its fracture. In wear modeling at the microscale (lower part of the scheme presented in Fig. 1) it is necessary to determine the microscale contact characteristics following from the macroscale, to determine the physical mechanism of an elementary act of fracture and to choose an appropriate fracture criterion; to calculate stresses, temperature distribution and other functions involved in the fracture criterion; to model particle detachment; to determine new characteristics of the surface layer after particle detachment (contact and internal stress and temperature distributions, etc.) and the next instant when failure occurs. The first stage consists of the wear mechanism analysis and determining the fracture criterion corresponding to this mechanism. As a rule, the fracture criterion depends on the absolute or amplitude value of stresses, temperature, mechanical characteristics of the materials and so on. Note that the wear mechanism itself depends on the level of stresses and temperature in an active surface layer. The next stage is to calculate functions involved in the fracture criterion and to determine the place and instant of crack initiation. For this purpose the stress and temperature distributions in the active layer are calculated based on the solutions of contact mechanics problems for rough surfaces. The methods of fracture mechanics are used to determine the onset of failure and to model particle detachment based on the fracture criterion and state of the active layer. The shape and size of particles detached from the surface can be evaluated as well. So, the modeling of wear at the microscale should involve contact mechanics problems taking into account the macro- and microgeometry of the contacting bodies, inhomogeneity of mechanical properties of the subsurface layer and nonuniform temperature distribution in this layer as well as fracture mechanics problems used to describe the particle detachment from the surface. In our opinion, the choice of the fracture criterion is the most difficult problem in modeling, because the processes that cause the wear particle detachment can be of different kind. The type of wear depends on the mechanical properties of the contacting bodies, loading and kinematics conditions, existence and nature of the lubricant, and other conditions. However, there are some general peculiarities of surface fracture (wear) which set wear as a special form of fracture. First of all, a high stress concentration determined by the loading conditions, microgeometry and friction coefficient occurs near contact spots in friction interaction of rough surfaces. Frictional heating near the contact spots leads to a considerable increase of temperature at the surface layer. Finally, the stress and temperature fields in the subsurface layer arise due to migration of contact spots in relative motion of the contacting bodies.
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Modeling the wear process is a very complicated problem. Hence, quite simple models are usually proposed in tribology, which take into account in detail only a limited number of features of the wear process. 4. Fatigue wear The experimental results prove that surface fracture can be very often explained by the concept of fatigue. When two rough surfaces move along each other, an inhomogeneous cyclic stress field with high amplitude values of stresses occurs in the subsurface layer and causes damage accumulation near the surface. The analysis of the surface fracture process in friction interaction of rough surfaces is based in [15] on a macroscopic approach to the development of the contact fatigue model. As known [6], it involves the construction of the positive function Q(M, t) not decreasing in time, characterizing the measure of material damage at the point M and depending on the amplitude values of stresses at this point. Failure occurs when this function reaches a threshold level. This concept of fatigue is applied to the investigation of surface failure as well as bulk failure of materials. There are many different physical approaches to the damage concept in which the damage accumulation rate is considered as a function of stresses, temperature and other parameters depending on the fracture mechanism, material type and so on. We present here the results obtained in [2, 3], where a thermokinetic model [7] is used for the description of damage accumulation in friction interaction of two rough surfaces. In accordance with this model, the rate of damage accumulation is given by the relation
q ( x, y , z , t )
§ U JV( x, y, z , t ) · 1 exp ¨ ¸, kT ( x, y, z, t ) ¹ ©
(1)
where U is the activation energy, - and J are the material characteristics, k is the Boltzmann coefficient, and V(x, y, z, t) is the characteristic of the stress field at the point (x, y, z) within the deformable body at the instant t. Using various stress field characteristics as V(x, y, z, t) in (1), we can reproduce different types of fracture in the framework of the considered approach. Note that the thermokinetic model of damage accumulation implies the consideration of thermal effects in an explicit form. Since the temperature field T(x, y, z, t) in subsurface layers is essentially inhomogeneous, its calculation must be carried out with high accuracy. It is therewith impossible to use averaged temperature characteristics. The damage accumulation function Q(x, y, z, t) at an arbitrary instant of time t is calculated from the formula
Q ( x, y , z , t )
t
³ q( x, y, z, W) dW Q0 ( x, y, z),
(2)
0
where q(x, y, z, W) is determined by equation (1) and the function Q0 ( x, y, z ) describes the initial damage distribution. In calculations presented in [2] the principal shear
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stress W max ( x, y, z, t ) was used as the stress field characteristic. In summary, the following parameters determine the damage accumulation process in the framework of the described model (the subscripts i = 1, 2 correspond to contacting bodies): initial shape of the bodies in contact f i ( x, y ); dependence of external load on time P(t); elastic characteristics of the bodies Ei , Q i ; parameters of the worn body: heat conductivity, heat capacity, density; characteristics describing damage accumulation inside the worn body: U, - and J; coefficient of thermal flux distribution K; friction coefficient P; relative sliding velocity V; environmental temperature. The model allows describing the kinetics of fatigue wear of rough surfaces and determining its characteristics: wear rate, size and shape of particles detached from the surface, surface microgeometry and contact pressure evolution. The analysis of the model shows that the size of the detached particles depends substantially on the friction coefficient. Its change influences both stress and temperature distribution. The increase of the friction coefficient causes an increase in heat generation and the displacement of the point where the principal shear stress is maximum towards the surface. Both these factors decrease the size of the detached particles. Some calculation results are shown in Fig. 2. The analysis of the dependencies of the total wear w and root-meansquare deviation V* on time t (all characteristics are presented in the dimensionless form) for two different values of the thermal flux distribution coefficient K makes it possible to conclude that the wear process includes an incubation period, i.e. the time interval between the beginning of interaction and fracture origination. It is a typical feature of the fatigue wear. The wear intensity during this period is zero. The incubation period becomes shorter if the rate of damage accumulation increases, i.e. if there is an increase of temperature and stresses in the subsurface layer. This can be caused by an increase of load, friction coefficient or the quantity of heat absorbed by the worn body. Note that the factors that lead to a shortening of the incubation period also cause an increase in the wear rate. The comparison of curves 1 and 2 in Fig. 2 shows that with heating being neglected (K = 0, curves 2) the incubation period is longer. The wear is lower in this case, but it results from the detachment of larger particles and increases surface roughness compared to the case when heating is taken into account (K = 1, curves 1). As a result of wear, the root-mean-square deviation of the profile can increase or decrease in comparison with the initial one (see Fig. 2(b)). Both the situations were observed experimentally [8].
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b
Fig. 2. Characteristics of the wear process: total wear (a) and root-meansquare deviation of the profile (b) versus time for P = 0.2 and K = 1 (1), 0 (2)
The analysis of the model allows us to conclude that: wear can increase or decrease the initial roughness (the root-mean-square deviation of the profile is the measure of roughness in the analysis); the friction coefficient growth increases wear intensity and reduces the size of particles detached from the surface in wear; the load increase rises wear intensity and increases the size of detached particles if there is no heating or it is negligibly low; otherwise, the size of detached particles decreases due to the intensive heating of subsurface layers. Some conclusions following from the analysis of the model coincide with the results obtained for a one-dimensional model describing the delamination process in fatigue wear (see [1]). The conclusions that evidently reflect the common features of the fatigue wear are: the incubation period in the wear process, two types of surface fracture taking place simultaneously in the wear process, such as continuous surface wear accompanied by the detachment of finite-sized particles at discrete instants of time, and the steady-state regime of wear under definite external conditions. Modeling of the wear process at the microscale allows reducing the wear equation to the dependence of wear rate on nominal pressure, relative velocity of contacting surfaces and other parameters. This equation is used to calculate the variation of contact characteristics of junctions at the macroscale. 5. Modeling of the wear process for tribological junctions When calculating the lifetime of junctions, it is necessary to take into account the variation of the mechanical properties of surface layers and irreversible changes of the
macroshape of contacting bodies in friction interaction. The formulation of wear contact problems at the macroscale allows us to calculate the temporal evolution of the worn surface shape, nominal contact pressure, the approach of junction elements in wear as well as to evaluate the time of running-in wear when intensive macrogeometry variation occurs, i.e. to answer the main questions that arise in the prediction of wear of moving junctions. The first formulation and solution of the wear problem were given by Pronikov [9, 10] on the assumption that the contacting bodies are rigid or there is a power dependence of normal surface displacement on contact pressure. The contact problem for elastic bodies which takes into account surface shape variation during wear was formulated by Korovchinskii [11]. Galin [1216] developed the theory of wear contact problems and suggested mathematical methods to solve them. The irreversible shape changes of bodies in contact arising from wear of their surfaces are taken into consideration for the mathematical formulation of wear contact problems. The linear wear value w (change of the linear body dimension in the direction perpendicular to the rubbing surface) is often used to describe wear quantitatively. Generally, the surfaces are worn nonuniformly and hence the linear wear is the function of the surface point coordinates (x, y) and time t, i.e. w* w ( x, y , t ). The wear equation, i.e. the dependence of the wear rate ww ( x, y, t ) wt on pressure, sliding velocity, temperature and so on, is determined by the wear mechanism (fatigue wear, abrasive, etc.). These dependencies are established based on the empirical and theoretical study. In many cases, the power law dependence of wear rate on pressure p and sliding velocity V is used for the wear analysis [8]: ww K w p DV E, (3) wt where K w is the wear coefficient, and D and E are parameters. The values K w , D and E depend on frictional properties and surface microgeometry of the contacting bodies, temperature and so on. In wear contact problem formulation it is admissible to assume that the irreversible surface displacement due to wear w(x, y, t) is small and comparable to the elastic displacement u z ( x, y , t ), in the direction perpendicular to the rubbing surface. For the determination of the stress-strain state of the contacting bodies boundary conditions are set on the underformed surface, so that the contact pressure p(x, y, t) and elastic displacement u z ( x, y , t ) for an arbitrary instant of time are related by operator A, which is analogous to the operator relating pressure and elastic displacement in a corresponding contact problem when no wear occurs, i.e. u z ( x, y , t ) A[ p( x, y , t )]. (4) For example, equation (4) has the following form for the frictionless contact of an axisymmetric punch and elastic half-space:
I.G. Goryacheva / Physical Mesomechanics 10 56 (2007) 247254
u z (U, t )
4(1 Q 2 ) SE
a (t )
³ 0
p (Uc, t )Uc § 2 UUc · K¨ ¸ dUc, ¨ U Uc ¸ U Uc © ¹
(5)
0 d U d a (t ), where K(U) is the complete elliptic integral of the first kind, x2 y2 . In particular, it follows from (5) that the operator A depends on the shape and size of the contact region :. If the size of the contact region does not change during the wear process, i.e. a(t) = a(0), the operator A for any instant of time coincides with the operator at the initial instant t = 0. This occurs, e.g., in the contact problem for the punch with a flat face that wears the elastic foundation. Otherwise, the unknown boundary * of the contact region : should be obtained at each instant of time from the complementary condition, e.g., from the condition of continuous pressure distribution at the boundary *: U
p( x, y, t ) ( x , y )*
0.
It should be noted that the requirement of a small value of the linear wear w(x, y, t) follows from the functional restrictions for components operating, e.g., in precision junctions. However, for some frictional components the value of linear wear is comparable to the characteristic size of the worn element. This takes place, e.g., when wear of a thin elastic layer is studied and the displacements due to wear are comparable with the coating thickness. In this case, the relation between the elastic displacement and pressure becomes nonlinear and depends on the geometry of the worn bodies [17, 18]. To complete the system of equations (3) and (4), the contact condition is used which takes the following form in a general case: u z ( x, y, t ) w ( x, y , t ) ) ( x, y, t ), (6) where the function )(x, y, t) is given as a rule. This function depends on the macroshape of the contacting bodies and the type of relative motion. For example, in wearing of the elastic half-space by a sliding cylindrical punch whose shape in the contact region is described by the function g(x) the function )(x, y, t) has the following forms: )(x, t) = = D(t) g(x) if the punch slides along its generatrix (the coordinate axis Oxyz is connected with the boundary of the elastic half-space, the axis Oz coincides with the normal to the half-space boundary which passes through the point O of the initial contact, D(t) is the displacement of the cylinder along the axis Oz due to elastic deformations and wear of the half-space surface); )(x, t) = D(t) g(x Vt) if the punch slides with constant velocity V perpendicularly to its generatrix [5]. If the function D(t) is unknown, the system of equations (3), (4) and (6) is completed by an equilibrium equation relating the contact stresses and external forces applied to the contacting bodies. For the frictionless contact the equilibrium equation has the form:
³³ p( x, y, t ) I( x, y )dxdy
251
P (t ),
(7)
: (t )
where P(t) is the normal load, I(x, y) is the geometrical multiplier connected with the contact geometry. For example, I(x, y) = 1 for the flat punch and I(x, y) = cos x for the cylindrical punch. The solution of the system of equations allows us to determine the unknown contact pressure p(x, y, t), elastic displacements u z ( x, y , t ), the shape of the worn surface determined by the function w(x, y, t) and the approach D(t) of the contacting bodies due to their deformation and wear. The extensive literature on the formulations and methods of wear contact problem solutions is reviewed in [19]. As a rule, the system of equations (3) (7) is reduced to linear or nonlinear integral or differential equations depending on the models used in contact problem formulations. The methods developed are used for the analysis of the wear process of journal bearings, abrasive and cutting tools, wheels and rails, and other elements of frictional junctions used in different applications.
6. Steady-state wear For definite conditions the system of equations (3) (6) has a steady-state solution that corresponds to the steadystate wear process. The necessary condition for the existence of the steady-state solution is the stabilization in time of all external parameters of the problem (the approach rate of the contacting bodies lim D (t ) Df or normal load
lim P(t )
t o f
t o f
Pf , the relative sliding velocity lim V ( x, y, t )
Vf ( x, y ), the contact region lim : (t ) t o f
t o f
: f and so on).
Under these conditions and on the assumption that the operator A (4) is time independent, the sufficient conditions for the existence of the asymptotically stable steady-state wear were formulated in [5, 20] and the expression for the determination of the asymptotically stable pressure distribution pf ( x, y ) was derived: pf ( x, y ) lim p( x, y, t ) t of
1D
ª º Df (8) « » . E ¬ K w ( x , y )V ( x , y ) ¼ Here, K w ( x, y ) is the wear coefficient which is the function of coordinates in the general case. The dependence of the wear coefficient on coordinates takes place, in particular, in studying the wear process of heterogeneous and locally hardened surfaces. If Df 0 or Pf 0, the contact pressure tends to zero, i.e. pf ( x, y ) 0. Substituting the contact pressure pf ( x, y ) into (3), (4), (6) and (7) gives the asymptotic values of u z , w, P and thus allows us to determine the shape of the worn surface and wear rate in the steady-state wear. Note that the steady-state pressure distribution (8) is determined by the form of the wear equation and features
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of the sliding motion of the contacting bodies, whereas the steady-state shape of the worn surface essentially depends on the form of the operator A. For illustration, we reproduce here the solution of a 2D wear contact problem for a flat punch and locally hardened elastic body obtained in [21]. The scheme of the contact between the punch and elastic half-space hardened periodically inside strips ( nl a d x ( n 1)l , f < y < + f) is shown in Fig. 3. The punch is under the normal load P l (l is the period) and slides back and forth along the axis Oy in the plane z = 0 connected with the punch base. Due to local hardening of the elastic body surface the wear coefficient is the step function
K w1 , x [nl , a nl ], (9) ® ¯ K w 2 , x [nl , a nl ], where K w1 and K w 2 are the wear coefficients outside and inside the hardened zones [nl a, (n 1)l ] , respectively ( K w1 ! K w 2 ). The elastic characteristics of the half-space E and Q, which are as a rule structure-insensitive, are considered as constant. In the case under consideration, the operator A has the following form: 2(1 Q 2 ) A[ p ( x, t )] u SE 1 S( xc x ) u ³ p ( x c, t ) ln 2 sin dxc. (10) l 0 Kw
The steady-state pressure distribution in this problem is presented by the step function. The substitution of this function in (10) permits us to calculate the shape of the worn surface for the steady-state stage of the wear process which can be represented by the following series [21]: 2(1 Q 2 ) Pm u f f ( x) 2 S E (1 a~m ) f sin( § § 2x ·· Sna~) cos ¨¨ Sn ¨ a~ ¸ ¸¸ , u¦ (11) 2 l ¹¹ n n 1 © © 1D where a~ a l ; m 1 ( K K ) . w2
w1
Figure 4 illustrates the shapes of the worn surface in the steady-state stage calculated from formula (11) for two
Fig. 3. Scheme of contact of a flat punch and elastic half-space hardened inside strips
different values of the geometrical and tribological parameters of hardening. Analysis of the solutions of periodical contact problems for the elastic half-space hardened inside strips or circular domains presented in [5, 21, 22] suggest that during wear the surface becomes wavy. The geometrical characteristics of the worn surface depend on the ratio of the wear coefficients of the hardened and unhardened zones and their characteristic dimensions. The parameters of hardening also influence wear rate. The appearance of such waviness under definite conditions improves the performance of friction pairs. The operation of junctions in the steady-state stage of the wear process is most desirable, since this stage is characterized by the stabilization of pressure and its all other characteristics (pressure is a controlling parameter in this case). Based on the models developed, it is possible to formulate and solve an inverse problem: to formulate definite requirements to the surface (its macroshape and structure parameters, etc.) which optimize the wear process based on the criterions formulated from the practical requirements. 7. The problems of controlling the wear process The study of the wear contact problems makes it possible to develop methods of controlling the wear process through: decreasing the running-in time and saving the material by making the initial surface shape approach the steadystate shape f f ( x, y ) calculated from models (problem 1), providing the desired characteristics of the junction in the steady-state wear by approaching the steady-state shape f f ( x, y ) of the surface to the optimal shape f s ( x , y ) (problem 2). If there are strong requirements to the configuration, material properties of the contacting bodies and wear conditions, the steady-state shape f f ( x, y ) is completely determined. If the function f f ( x, y ) is admissible for the operation of the junction, it is possible to formulate and solve the problem of running-in time minimization by making the initial shape f 0 ( x, y ) as close as possible to the steady-state shape f f ( x, y ) (problem 1). The running-in time is zero and the steady-state wear occurs during junction operation if the initial shape satisfies the condition
Fig. 4. The steady-state shape of the worn surface for m = 0.5, a = 0.2 (solid line) and m = 0.3, a = 0.6 (dashed line)
I.G. Goryacheva / Physical Mesomechanics 10 56 (2007) 247254 a
253 b
Fig. 5. Abrasive tool surface with inclusions (à) and variation of the inclusion density versus radius providing the flat surface of the tool during the wear process (b)
f0 ( x, y ) = f f ( x, y ) . For instance, the steady-state shape of the worn surface of the cylindrical punch base of radius R, which moves reciprocally on the surface of the unworn (or wearing uniformly) elastic half-space surface, can be determined by substituting the steady-state contact pressure (8) in expression (5). Since at a constant wear coefficient the steady-state pressure is distributed uniformly within the contact region, the initial shape f0 (r ), providing the steadystate wear during operation is determined by the following expression: 4(1 Q2 ) P ª § r · º 1 , f 0 (r ) E (12) S2 RE «¬ ¨© R ¸¹ »¼ where E(t) is the complete second-kind elliptical integral and P is the normal load applied to the cylinder. The problem of stabilizing the optimal shape fs ( x, y ) of the worn surface (problem 2) can be formulated if there are some parameters J i ( x, y ), characterizing the properties of the junction or its elements, which admit variation within the definite class of functions [23]. To solve this problem, we put ff ( x, y ) fs ( x, y) and consider the following integral equation in respect of the one or several unknown functions J i ( x, y ) : fs ( x, y )
A[ pf ( x , y ), J i ( x, y )],
(13)
As a rule, the exact solution of this equation cannot be derived in practice. Usually the parameters J i ( x, y ) have practical limitations imposed by the technology of producing heterogeneous surfaces. Therefore, instead of solving equation (13), the problem of minimizing the functional F, which is a metric in some space, is considered: F
§~
1
~
·
³³ ¨¨ f ( x, y) mes: ³³ f ( xc, yc)dxcdyc ¸¸
2
dxdy, (14) : © ¹ ~ where f ( x, y ) f f ( x, y ) f s ( x, y ), and mes : is the area of the contact region :. During wear of the locally hardened surface discussed above, m and a~ are the governing parameters that characterize hardening intensity and relative thickness of the hardened strips. The variation of these parameters admissible :
in the technology gives an opportunity of controlling the tribological and geometrical characteristics of the wavy surface ff ( x) (11) formed in the wear process. The wear coefficient is not a unique parameter that influences the steady-state shape of the worn surface. For a discrete contact the steady-state shape essentially depends on the distribution of contact spots and their mutual influence. The solution of problem 2 for the abrasive tool surface is obtained in [20]. A method of calculating an optimal distribution of abrasive inclusions on the tool surface which provide its uniform wear is proposed based on this solution. The function N(r) characterizing the contact density of inclusions in the point (x, y) was used to control the wear process. Since it is impossible to manufacture a tool with the density N(r) varying continuously, the solution was sought in the class of step functions. The calculation results for the tool operation surface in the form of a ring rotating with constant angular velocity on the boundary of an elastic half-space are illustrated in Fig. 5. The function N(r) guarantees the surface to be practically flat during wear. This function consists of three different parts because densities different by less than 10 % were considered to be indistinguishable for a technological reason. Thus, the wear modeling makes it possible both to predict the main features of the wear process at the micro- and macroscales for various friction conditions and to control this process providing an optimal surface shape of friction pair elements and minimizing the running-in time. This saves the material used for friction pairs and increases the lifetime of junctions. The work has been supported by the Russian Foundation for Basic Research (Grants Nos. 06-08-01105, 07-0100282) and the Grant of the RF President for the support of leading scientific schools No. NSh-1245.2006.1. References [1] I.G. Goryacheva and O.G. Chekina, Surface fatigue damage model, Sov. J. Frict. Wear, 11, No. 3 (1990) 1. [2] O.G. Chekina, The Modelling of Fracture of Surface Layers in Contact of Rough Bodies, in Strength and Plasticity, Moscow, Nauka, V. 1 (1996) 186 (in Russian).
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[3] I.G. Goryacheva and O.G. Chekina, Wear of surfaces: From microfracture modeling to analysis of macroshape variations, Izv. RAS, Mech. Solids, No. 5 (1999) 131. [4] V.M. Bogdanov, A.P. Goryachev, I.G. Goryacheva, M.N. Dobychin, S.M. Zakharov, V.G. Krivonogov, I.A. Soldatenkov, and O.G. Chekina, Study of wheel-rail contact, wear and cumulative damage, Sov. J. Frict. Wear, 17, No. 1 (1996) 12. [5] I.G. Goryacheva, Contact Mechanics in Tribology, Kluwer Academic Publishers, DordrechtBostonLondon (1998). [6] V.N. Ionov and P.M. Ogibalov, Strength of Three-Dimensional Structural Elements, Nauka, Moscow, 1972 (in Russian). [7] R.V. Regel, A.I. Slutsker, and E.E. Tomashevsky, Kinetics Foundation of Strength of Solids, Nauka, Moscow, 1974 (in Russian). [8] I.V. Kragelsky, M.N. Dobychin, and V.S. Kombalov, Friction and Wear: Calculation Methods, Pergamon Press, Oxford (1982). [9] A.S. Pronikov, Classification and Calculation of Wear of Elements of Junctions, in Friction and Wear in Machines, Izd-vo AN SSSR, Moscow, No. 11 (1956) 121 (in Russian). [10] A.S. Pronikov, Contact Problem for Contacting Surfaces of Machine Elements, in Friction and Wear in Machines, Izd-vo AN SSSR, Moscow, No. 15 (1962) 375 (in Russian). [11] M.V. Korovchinskii, Local Contact of Elastic Bodies with Wear of their Surfaces, Nauka, Moscow (1971) 130 (in Russian). [12] L.A. Galin, Contact problems of the theory of elasticity in the presence of wear, J. Appl. Math. Mech., 40, No. 6 (1976) 981. [13] L.A. Galin and I.G. Goryacheva, Axisymmetric contact problem of the theory of elasticity in the presence of wear, J. Appl. Math. Mech., 41, No. 5 (1977) 807.
[14] L.A. Galin and I.G. Goryacheva, Contact Problems of the Theory of Elasticity in the Presence of Wear, in Theory of Friction, Wear and the Problems of Standards, Priokskoe Izd-vo, Bryansk (1978) 251 (in Russian). [15] L.A. Galin and I.G. Goryacheva, Contact problems and their applications to the theory of friction and wear, Sov. J. Frict. Wear, 1, No. 1 (1980) 105. [16] L.A. Galin, Contact Problems of the Theory of Elasticity and Viscoelasticity, Nauka, Moscow, 1980 (in Russian). [17] I.G. Goryacheva and I.A. Soldatenkov, Theoretical study of running-in and the steady-state wear regime of solid lubricant coatings, Sov. J. Frict. Wear, 4, No. 3 (1983) 420. [18] I.A. Soldatenkov, Wear of thin elastic coating with a varying contact area, Sov. J. Frict. Wear, 6, No. 2 (1985) 247. [19] I.G. Goryacheva and I.A. Soldatenkov, Wear Contact Problems, in Contact Mechanics, Ed. by I.I. Vorovich and V.M. Alexandrov, Fizmatlit, Moscow (2001) 438 (in Russian). [20] I.G. Goryacheva, Wear contact problems for the system of punches, Izv. USSR Acad. Sci., Solid Mech., No. 6 (1987) 62. [21] I.G. Goryacheva and M.N. Dobychin, Wear of nonuniformly hardened surfaces, Sov. J. Frict. Wear, 7, No. 6 (1986) 985. [22] I.G. Goryacheva and E.V. Torskaya, Wear contact problems with wear intensity variable at the surface, Sov. J. Frict. Wear, 13, No. 1 (1992) 185. [23] I.G. Goryacheva and O.G. Chekina, Control of surface shape variation during wear process, Sov. J. Frict. Wear, 10, No. 1 (1989) 5.
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Modeling of wear of solids at different scales I.G. Goryacheva Institute for Problems in Mechanics RAS, Moscow, 119526, Russia The paper puts forward an approach to the analysis of stress state and fracture of surface layers in deformed solids. The approach is based on the consideration of contact interaction processes at several scale levels determined by characteristic structural parameters of the surface and subsurface layers of contacting bodies (micro- and mesoscales) and by their macrogeometry. The approach can be used to estimate the influence of geometrical, mechanical and physical characteristics of materials of contacting bodies and the intermediate medium on the performance of friction pairs. Methods of increasing the wear resistance and service life of the pairs and reducing energy loss owing to the control of wear and friction processes in thin surface layers can be developed.
1. Introduction Wear is a process of progressive loss of material from the operating surfaces of solids arising from the surface fracture. The dimensions of the body and its shape vary in this process. The study of wear shows that there are different causes for surface fracture. First of all, it is caused by high stresses in subsurface layers in the friction process which lead to the crack formation and to the detachment of the material particles (wear debris) from the rubber surface. This widespread type of wear is classified as mechanical wear, and often the word mechanical is omitted. Since the mechanical wear of two bodies in friction interaction is determined by the stresses in the subsurface layers, modeling of the surface fracture process is based on the methods used in contact mechanics and fracture mechanics. However, such modeling does not amount to the use of the results from those fields of the mechanics of solids. The process of wear has some peculiarities setting it as a special form of fracture. 2. The main stages in modeling First of all the specific feature of wear implies that this process is not critical for the operation of junction. Usually, admissible limit wear of moving parts of junctions is much more than the typical size of wear debris. Thus, repeated particle detachment from the rubber surface can occur during the life of the parts. Repeated unit fracture acts are the distinctive feature of the wear process, as opposed to the bulk material process. After removal of surCopyright © 2007 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV. All rights reserved. doi:10.1016/j.physme.2007.11.003
face material due to wear, the subsurface layer enters the contact. The characteristics of this layer, including those that determine wear intensity (microgeometry, damage of material, etc.), depend on the entire history of the frictional interaction. Thus, wear can be considered as a process of hereditary type. Besides, wear is a feedback process. Selforganization and equilibrium structure formation in wear occur as a result of the feedback action. In wear modeling it is first of all necessary to determine the stress and temperature distributions arising in the subsurface layers under given contact conditions, mechanical properties of contacting bodies, and surface macro- and microgeometry (contact state). At the macroscale these are the average (nominal) contact characteristics determined based on the given macroshape of the contacting bodies and contact conditions. At the microscale these are the real contact pressure, real internal stress and temperature distributions in the subsurface layer, which were calculated taking into account surface microgeometry and variation of mechanical and tribological characteristics of the subsurface layer in wear. Based on this analysis, the most probable wear mechanism is recognized. Use of the fracture criterion corresponding to this mechanism allows us to determine the onset of failure and the surface shape after particle detachment (or particle transfer to the counter-body). The determined surface microgeometry characteristics are used to calculate the stress and temperature distributions at the next instant of time, etc. The described stages of wear modeling are schematically shown in Fig. 1.
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In the majority of cases when wear is analyzed, only macrocharacteristics of the process (body shape variation at the instants of time which are much more then the intervals between the individual acts of fracture, the maximum linear wear at the surface, etc.) are of interest. The microgeometry variation after wear particle detachment, the shape of each particle and the other characteristics related to the calculation of stresses and temperature distribution at the microscale (sizes comparable to the typical diameter of the contact spot) are therefore the internal parameters of the model. In fact, for applications the main task of the wear modeling is the prediction of the macrostate (macroshape) evolution under given external conditions (load, velocity). It is possible to develop the wear models at two scales: macroscale models studying the macroshape variation in wear process, and microscale ones describing the elementary acts of wear particle detachment. The repeated calculations at the microscale allow us to evaluate the variation of the contact characteristics at the macroscale (macroshape variation, approach of the contacting bodies due to their wear, etc.) and to calculate the lifetime of junction in respect to the wear resistance criterion. The application of the described algorithm to the analysis of the frictional pairs is very time consuming. The analysis of processes at the microscale makes possible to develop a phenomenological wear model for a frictional pair at the macroscale, in which the wear rate dw* dt as a function of the macrocharacteristics of the junction (pressure p, velocity V, relative displacements of the contacting surfaces, etc.) is included (Fig. 1). This function may be constructed through modeling the fracture process at the microscale and elementary acts of wear particle detachment. 3. Modeling of surface fracture at the microscale The subject of wear analysis at the microscale is a thin surface layer whose thickness is comparable to the contact spot size. Wear modeling includes the determination of the crack initiation instant and trajectory of crack propagation up to particle detachment, transportation of the detached particles from the contact zone, and description of the sur-
Fig. 1. The main stages in wear modeling and their mutual relations
face microgeometry variation and properties of the thin subsurface layer in the process of its fracture. In wear modeling at the microscale (lower part of the scheme presented in Fig. 1) it is necessary to determine the microscale contact characteristics following from the macroscale, to determine the physical mechanism of an elementary act of fracture and to choose an appropriate fracture criterion; to calculate stresses, temperature distribution and other functions involved in the fracture criterion; to model particle detachment; to determine new characteristics of the surface layer after particle detachment (contact and internal stress and temperature distributions, etc.) and the next instant when failure occurs. The first stage consists of the wear mechanism analysis and determining the fracture criterion corresponding to this mechanism. As a rule, the fracture criterion depends on the absolute or amplitude value of stresses, temperature, mechanical characteristics of the materials and so on. Note that the wear mechanism itself depends on the level of stresses and temperature in an active surface layer. The next stage is to calculate functions involved in the fracture criterion and to determine the place and instant of crack initiation. For this purpose the stress and temperature distributions in the active layer are calculated based on the solutions of contact mechanics problems for rough surfaces. The methods of fracture mechanics are used to determine the onset of failure and to model particle detachment based on the fracture criterion and state of the active layer. The shape and size of particles detached from the surface can be evaluated as well. So, the modeling of wear at the microscale should involve contact mechanics problems taking into account the macro- and microgeometry of the contacting bodies, inhomogeneity of mechanical properties of the subsurface layer and nonuniform temperature distribution in this layer as well as fracture mechanics problems used to describe the particle detachment from the surface. In our opinion, the choice of the fracture criterion is the most difficult problem in modeling, because the processes that cause the wear particle detachment can be of different kind. The type of wear depends on the mechanical properties of the contacting bodies, loading and kinematics conditions, existence and nature of the lubricant, and other conditions. However, there are some general peculiarities of surface fracture (wear) which set wear as a special form of fracture. First of all, a high stress concentration determined by the loading conditions, microgeometry and friction coefficient occurs near contact spots in friction interaction of rough surfaces. Frictional heating near the contact spots leads to a considerable increase of temperature at the surface layer. Finally, the stress and temperature fields in the subsurface layer arise due to migration of contact spots in relative motion of the contacting bodies.
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Modeling the wear process is a very complicated problem. Hence, quite simple models are usually proposed in tribology, which take into account in detail only a limited number of features of the wear process. 4. Fatigue wear The experimental results prove that surface fracture can be very often explained by the concept of fatigue. When two rough surfaces move along each other, an inhomogeneous cyclic stress field with high amplitude values of stresses occurs in the subsurface layer and causes damage accumulation near the surface. The analysis of the surface fracture process in friction interaction of rough surfaces is based in [15] on a macroscopic approach to the development of the contact fatigue model. As known [6], it involves the construction of the positive function Q(M, t) not decreasing in time, characterizing the measure of material damage at the point M and depending on the amplitude values of stresses at this point. Failure occurs when this function reaches a threshold level. This concept of fatigue is applied to the investigation of surface failure as well as bulk failure of materials. There are many different physical approaches to the damage concept in which the damage accumulation rate is considered as a function of stresses, temperature and other parameters depending on the fracture mechanism, material type and so on. We present here the results obtained in [2, 3], where a thermokinetic model [7] is used for the description of damage accumulation in friction interaction of two rough surfaces. In accordance with this model, the rate of damage accumulation is given by the relation
q ( x, y , z , t )
§ U JV( x, y, z , t ) · 1 exp ¨ ¸, kT ( x, y, z, t ) ¹ ©
(1)
where U is the activation energy, - and J are the material characteristics, k is the Boltzmann coefficient, and V(x, y, z, t) is the characteristic of the stress field at the point (x, y, z) within the deformable body at the instant t. Using various stress field characteristics as V(x, y, z, t) in (1), we can reproduce different types of fracture in the framework of the considered approach. Note that the thermokinetic model of damage accumulation implies the consideration of thermal effects in an explicit form. Since the temperature field T(x, y, z, t) in subsurface layers is essentially inhomogeneous, its calculation must be carried out with high accuracy. It is therewith impossible to use averaged temperature characteristics. The damage accumulation function Q(x, y, z, t) at an arbitrary instant of time t is calculated from the formula
Q ( x, y , z , t )
t
³ q( x, y, z, W) dW Q0 ( x, y, z),
(2)
0
where q(x, y, z, W) is determined by equation (1) and the function Q0 ( x, y, z ) describes the initial damage distribution. In calculations presented in [2] the principal shear
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stress W max ( x, y, z, t ) was used as the stress field characteristic. In summary, the following parameters determine the damage accumulation process in the framework of the described model (the subscripts i = 1, 2 correspond to contacting bodies): initial shape of the bodies in contact f i ( x, y ); dependence of external load on time P(t); elastic characteristics of the bodies Ei , Q i ; parameters of the worn body: heat conductivity, heat capacity, density; characteristics describing damage accumulation inside the worn body: U, - and J; coefficient of thermal flux distribution K; friction coefficient P; relative sliding velocity V; environmental temperature. The model allows describing the kinetics of fatigue wear of rough surfaces and determining its characteristics: wear rate, size and shape of particles detached from the surface, surface microgeometry and contact pressure evolution. The analysis of the model shows that the size of the detached particles depends substantially on the friction coefficient. Its change influences both stress and temperature distribution. The increase of the friction coefficient causes an increase in heat generation and the displacement of the point where the principal shear stress is maximum towards the surface. Both these factors decrease the size of the detached particles. Some calculation results are shown in Fig. 2. The analysis of the dependencies of the total wear w and root-meansquare deviation V* on time t (all characteristics are presented in the dimensionless form) for two different values of the thermal flux distribution coefficient K makes it possible to conclude that the wear process includes an incubation period, i.e. the time interval between the beginning of interaction and fracture origination. It is a typical feature of the fatigue wear. The wear intensity during this period is zero. The incubation period becomes shorter if the rate of damage accumulation increases, i.e. if there is an increase of temperature and stresses in the subsurface layer. This can be caused by an increase of load, friction coefficient or the quantity of heat absorbed by the worn body. Note that the factors that lead to a shortening of the incubation period also cause an increase in the wear rate. The comparison of curves 1 and 2 in Fig. 2 shows that with heating being neglected (K = 0, curves 2) the incubation period is longer. The wear is lower in this case, but it results from the detachment of larger particles and increases surface roughness compared to the case when heating is taken into account (K = 1, curves 1). As a result of wear, the root-mean-square deviation of the profile can increase or decrease in comparison with the initial one (see Fig. 2(b)). Both the situations were observed experimentally [8].
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b
Fig. 2. Characteristics of the wear process: total wear (a) and root-meansquare deviation of the profile (b) versus time for P = 0.2 and K = 1 (1), 0 (2)
The analysis of the model allows us to conclude that: wear can increase or decrease the initial roughness (the root-mean-square deviation of the profile is the measure of roughness in the analysis); the friction coefficient growth increases wear intensity and reduces the size of particles detached from the surface in wear; the load increase rises wear intensity and increases the size of detached particles if there is no heating or it is negligibly low; otherwise, the size of detached particles decreases due to the intensive heating of subsurface layers. Some conclusions following from the analysis of the model coincide with the results obtained for a one-dimensional model describing the delamination process in fatigue wear (see [1]). The conclusions that evidently reflect the common features of the fatigue wear are: the incubation period in the wear process, two types of surface fracture taking place simultaneously in the wear process, such as continuous surface wear accompanied by the detachment of finite-sized particles at discrete instants of time, and the steady-state regime of wear under definite external conditions. Modeling of the wear process at the microscale allows reducing the wear equation to the dependence of wear rate on nominal pressure, relative velocity of contacting surfaces and other parameters. This equation is used to calculate the variation of contact characteristics of junctions at the macroscale. 5. Modeling of the wear process for tribological junctions When calculating the lifetime of junctions, it is necessary to take into account the variation of the mechanical properties of surface layers and irreversible changes of the
macroshape of contacting bodies in friction interaction. The formulation of wear contact problems at the macroscale allows us to calculate the temporal evolution of the worn surface shape, nominal contact pressure, the approach of junction elements in wear as well as to evaluate the time of running-in wear when intensive macrogeometry variation occurs, i.e. to answer the main questions that arise in the prediction of wear of moving junctions. The first formulation and solution of the wear problem were given by Pronikov [9, 10] on the assumption that the contacting bodies are rigid or there is a power dependence of normal surface displacement on contact pressure. The contact problem for elastic bodies which takes into account surface shape variation during wear was formulated by Korovchinskii [11]. Galin [1216] developed the theory of wear contact problems and suggested mathematical methods to solve them. The irreversible shape changes of bodies in contact arising from wear of their surfaces are taken into consideration for the mathematical formulation of wear contact problems. The linear wear value w (change of the linear body dimension in the direction perpendicular to the rubbing surface) is often used to describe wear quantitatively. Generally, the surfaces are worn nonuniformly and hence the linear wear is the function of the surface point coordinates (x, y) and time t, i.e. w* w ( x, y , t ). The wear equation, i.e. the dependence of the wear rate ww ( x, y, t ) wt on pressure, sliding velocity, temperature and so on, is determined by the wear mechanism (fatigue wear, abrasive, etc.). These dependencies are established based on the empirical and theoretical study. In many cases, the power law dependence of wear rate on pressure p and sliding velocity V is used for the wear analysis [8]: ww K w p DV E, (3) wt where K w is the wear coefficient, and D and E are parameters. The values K w , D and E depend on frictional properties and surface microgeometry of the contacting bodies, temperature and so on. In wear contact problem formulation it is admissible to assume that the irreversible surface displacement due to wear w(x, y, t) is small and comparable to the elastic displacement u z ( x, y , t ), in the direction perpendicular to the rubbing surface. For the determination of the stress-strain state of the contacting bodies boundary conditions are set on the underformed surface, so that the contact pressure p(x, y, t) and elastic displacement u z ( x, y , t ) for an arbitrary instant of time are related by operator A, which is analogous to the operator relating pressure and elastic displacement in a corresponding contact problem when no wear occurs, i.e. u z ( x, y , t ) A[ p( x, y , t )]. (4) For example, equation (4) has the following form for the frictionless contact of an axisymmetric punch and elastic half-space:
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u z (U, t )
4(1 Q 2 ) SE
a (t )
³ 0
p (Uc, t )Uc § 2 UUc · K¨ ¸ dUc, ¨ U Uc ¸ U Uc © ¹
(5)
0 d U d a (t ), where K(U) is the complete elliptic integral of the first kind, x2 y2 . In particular, it follows from (5) that the operator A depends on the shape and size of the contact region :. If the size of the contact region does not change during the wear process, i.e. a(t) = a(0), the operator A for any instant of time coincides with the operator at the initial instant t = 0. This occurs, e.g., in the contact problem for the punch with a flat face that wears the elastic foundation. Otherwise, the unknown boundary * of the contact region : should be obtained at each instant of time from the complementary condition, e.g., from the condition of continuous pressure distribution at the boundary *: U
p( x, y, t ) ( x , y )*
0.
It should be noted that the requirement of a small value of the linear wear w(x, y, t) follows from the functional restrictions for components operating, e.g., in precision junctions. However, for some frictional components the value of linear wear is comparable to the characteristic size of the worn element. This takes place, e.g., when wear of a thin elastic layer is studied and the displacements due to wear are comparable with the coating thickness. In this case, the relation between the elastic displacement and pressure becomes nonlinear and depends on the geometry of the worn bodies [17, 18]. To complete the system of equations (3) and (4), the contact condition is used which takes the following form in a general case: u z ( x, y, t ) w ( x, y , t ) ) ( x, y, t ), (6) where the function )(x, y, t) is given as a rule. This function depends on the macroshape of the contacting bodies and the type of relative motion. For example, in wearing of the elastic half-space by a sliding cylindrical punch whose shape in the contact region is described by the function g(x) the function )(x, y, t) has the following forms: )(x, t) = = D(t) g(x) if the punch slides along its generatrix (the coordinate axis Oxyz is connected with the boundary of the elastic half-space, the axis Oz coincides with the normal to the half-space boundary which passes through the point O of the initial contact, D(t) is the displacement of the cylinder along the axis Oz due to elastic deformations and wear of the half-space surface); )(x, t) = D(t) g(x Vt) if the punch slides with constant velocity V perpendicularly to its generatrix [5]. If the function D(t) is unknown, the system of equations (3), (4) and (6) is completed by an equilibrium equation relating the contact stresses and external forces applied to the contacting bodies. For the frictionless contact the equilibrium equation has the form:
³³ p( x, y, t ) I( x, y )dxdy
251
P (t ),
(7)
: (t )
where P(t) is the normal load, I(x, y) is the geometrical multiplier connected with the contact geometry. For example, I(x, y) = 1 for the flat punch and I(x, y) = cos x for the cylindrical punch. The solution of the system of equations allows us to determine the unknown contact pressure p(x, y, t), elastic displacements u z ( x, y , t ), the shape of the worn surface determined by the function w(x, y, t) and the approach D(t) of the contacting bodies due to their deformation and wear. The extensive literature on the formulations and methods of wear contact problem solutions is reviewed in [19]. As a rule, the system of equations (3) (7) is reduced to linear or nonlinear integral or differential equations depending on the models used in contact problem formulations. The methods developed are used for the analysis of the wear process of journal bearings, abrasive and cutting tools, wheels and rails, and other elements of frictional junctions used in different applications.
6. Steady-state wear For definite conditions the system of equations (3) (6) has a steady-state solution that corresponds to the steadystate wear process. The necessary condition for the existence of the steady-state solution is the stabilization in time of all external parameters of the problem (the approach rate of the contacting bodies lim D (t ) Df or normal load
lim P(t )
t o f
t o f
Pf , the relative sliding velocity lim V ( x, y, t )
Vf ( x, y ), the contact region lim : (t ) t o f
t o f
: f and so on).
Under these conditions and on the assumption that the operator A (4) is time independent, the sufficient conditions for the existence of the asymptotically stable steady-state wear were formulated in [5, 20] and the expression for the determination of the asymptotically stable pressure distribution pf ( x, y ) was derived: pf ( x, y ) lim p( x, y, t ) t of
1D
ª º Df (8) « » . E ¬ K w ( x , y )V ( x , y ) ¼ Here, K w ( x, y ) is the wear coefficient which is the function of coordinates in the general case. The dependence of the wear coefficient on coordinates takes place, in particular, in studying the wear process of heterogeneous and locally hardened surfaces. If Df 0 or Pf 0, the contact pressure tends to zero, i.e. pf ( x, y ) 0. Substituting the contact pressure pf ( x, y ) into (3), (4), (6) and (7) gives the asymptotic values of u z , w, P and thus allows us to determine the shape of the worn surface and wear rate in the steady-state wear. Note that the steady-state pressure distribution (8) is determined by the form of the wear equation and features
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of the sliding motion of the contacting bodies, whereas the steady-state shape of the worn surface essentially depends on the form of the operator A. For illustration, we reproduce here the solution of a 2D wear contact problem for a flat punch and locally hardened elastic body obtained in [21]. The scheme of the contact between the punch and elastic half-space hardened periodically inside strips ( nl a d x ( n 1)l , f < y < + f) is shown in Fig. 3. The punch is under the normal load P l (l is the period) and slides back and forth along the axis Oy in the plane z = 0 connected with the punch base. Due to local hardening of the elastic body surface the wear coefficient is the step function
K w1 , x [nl , a nl ], (9) ® ¯ K w 2 , x [nl , a nl ], where K w1 and K w 2 are the wear coefficients outside and inside the hardened zones [nl a, (n 1)l ] , respectively ( K w1 ! K w 2 ). The elastic characteristics of the half-space E and Q, which are as a rule structure-insensitive, are considered as constant. In the case under consideration, the operator A has the following form: 2(1 Q 2 ) A[ p ( x, t )] u SE 1 S( xc x ) u ³ p ( x c, t ) ln 2 sin dxc. (10) l 0 Kw
The steady-state pressure distribution in this problem is presented by the step function. The substitution of this function in (10) permits us to calculate the shape of the worn surface for the steady-state stage of the wear process which can be represented by the following series [21]: 2(1 Q 2 ) Pm u f f ( x) 2 S E (1 a~m ) f sin( § § 2x ·· Sna~) cos ¨¨ Sn ¨ a~ ¸ ¸¸ , u¦ (11) 2 l ¹¹ n n 1 © © 1D where a~ a l ; m 1 ( K K ) . w2
w1
Figure 4 illustrates the shapes of the worn surface in the steady-state stage calculated from formula (11) for two
Fig. 3. Scheme of contact of a flat punch and elastic half-space hardened inside strips
different values of the geometrical and tribological parameters of hardening. Analysis of the solutions of periodical contact problems for the elastic half-space hardened inside strips or circular domains presented in [5, 21, 22] suggest that during wear the surface becomes wavy. The geometrical characteristics of the worn surface depend on the ratio of the wear coefficients of the hardened and unhardened zones and their characteristic dimensions. The parameters of hardening also influence wear rate. The appearance of such waviness under definite conditions improves the performance of friction pairs. The operation of junctions in the steady-state stage of the wear process is most desirable, since this stage is characterized by the stabilization of pressure and its all other characteristics (pressure is a controlling parameter in this case). Based on the models developed, it is possible to formulate and solve an inverse problem: to formulate definite requirements to the surface (its macroshape and structure parameters, etc.) which optimize the wear process based on the criterions formulated from the practical requirements. 7. The problems of controlling the wear process The study of the wear contact problems makes it possible to develop methods of controlling the wear process through: decreasing the running-in time and saving the material by making the initial surface shape approach the steadystate shape f f ( x, y ) calculated from models (problem 1), providing the desired characteristics of the junction in the steady-state wear by approaching the steady-state shape f f ( x, y ) of the surface to the optimal shape f s ( x , y ) (problem 2). If there are strong requirements to the configuration, material properties of the contacting bodies and wear conditions, the steady-state shape f f ( x, y ) is completely determined. If the function f f ( x, y ) is admissible for the operation of the junction, it is possible to formulate and solve the problem of running-in time minimization by making the initial shape f 0 ( x, y ) as close as possible to the steady-state shape f f ( x, y ) (problem 1). The running-in time is zero and the steady-state wear occurs during junction operation if the initial shape satisfies the condition
Fig. 4. The steady-state shape of the worn surface for m = 0.5, a = 0.2 (solid line) and m = 0.3, a = 0.6 (dashed line)
I.G. Goryacheva / Physical Mesomechanics 10 56 (2007) 247254 a
253 b
Fig. 5. Abrasive tool surface with inclusions (à) and variation of the inclusion density versus radius providing the flat surface of the tool during the wear process (b)
f0 ( x, y ) = f f ( x, y ) . For instance, the steady-state shape of the worn surface of the cylindrical punch base of radius R, which moves reciprocally on the surface of the unworn (or wearing uniformly) elastic half-space surface, can be determined by substituting the steady-state contact pressure (8) in expression (5). Since at a constant wear coefficient the steady-state pressure is distributed uniformly within the contact region, the initial shape f0 (r ), providing the steadystate wear during operation is determined by the following expression: 4(1 Q2 ) P ª § r · º 1 , f 0 (r ) E (12) S2 RE «¬ ¨© R ¸¹ »¼ where E(t) is the complete second-kind elliptical integral and P is the normal load applied to the cylinder. The problem of stabilizing the optimal shape fs ( x, y ) of the worn surface (problem 2) can be formulated if there are some parameters J i ( x, y ), characterizing the properties of the junction or its elements, which admit variation within the definite class of functions [23]. To solve this problem, we put ff ( x, y ) fs ( x, y) and consider the following integral equation in respect of the one or several unknown functions J i ( x, y ) : fs ( x, y )
A[ pf ( x , y ), J i ( x, y )],
(13)
As a rule, the exact solution of this equation cannot be derived in practice. Usually the parameters J i ( x, y ) have practical limitations imposed by the technology of producing heterogeneous surfaces. Therefore, instead of solving equation (13), the problem of minimizing the functional F, which is a metric in some space, is considered: F
§~
1
~
·
³³ ¨¨ f ( x, y) mes: ³³ f ( xc, yc)dxcdyc ¸¸
2
dxdy, (14) : © ¹ ~ where f ( x, y ) f f ( x, y ) f s ( x, y ), and mes : is the area of the contact region :. During wear of the locally hardened surface discussed above, m and a~ are the governing parameters that characterize hardening intensity and relative thickness of the hardened strips. The variation of these parameters admissible :
in the technology gives an opportunity of controlling the tribological and geometrical characteristics of the wavy surface ff ( x) (11) formed in the wear process. The wear coefficient is not a unique parameter that influences the steady-state shape of the worn surface. For a discrete contact the steady-state shape essentially depends on the distribution of contact spots and their mutual influence. The solution of problem 2 for the abrasive tool surface is obtained in [20]. A method of calculating an optimal distribution of abrasive inclusions on the tool surface which provide its uniform wear is proposed based on this solution. The function N(r) characterizing the contact density of inclusions in the point (x, y) was used to control the wear process. Since it is impossible to manufacture a tool with the density N(r) varying continuously, the solution was sought in the class of step functions. The calculation results for the tool operation surface in the form of a ring rotating with constant angular velocity on the boundary of an elastic half-space are illustrated in Fig. 5. The function N(r) guarantees the surface to be practically flat during wear. This function consists of three different parts because densities different by less than 10 % were considered to be indistinguishable for a technological reason. Thus, the wear modeling makes it possible both to predict the main features of the wear process at the micro- and macroscales for various friction conditions and to control this process providing an optimal surface shape of friction pair elements and minimizing the running-in time. This saves the material used for friction pairs and increases the lifetime of junctions. The work has been supported by the Russian Foundation for Basic Research (Grants Nos. 06-08-01105, 07-0100282) and the Grant of the RF President for the support of leading scientific schools No. NSh-1245.2006.1. References [1] I.G. Goryacheva and O.G. Chekina, Surface fatigue damage model, Sov. J. Frict. Wear, 11, No. 3 (1990) 1. [2] O.G. Chekina, The Modelling of Fracture of Surface Layers in Contact of Rough Bodies, in Strength and Plasticity, Moscow, Nauka, V. 1 (1996) 186 (in Russian).
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