Physical and Constructive (Limiting) Criterions of Gear Wheels Wear

We suggest using a generalized model of friction - the model of elastic-plastic deformation of the body element, which is located on the surface of the friction pairs. This model is based on our new engineering approach to the problem of friction-triboergodynamics. Friction is examined as transformative and dissipative process. Structural-energetic interpretation of friction as a process of elasto-plastic deformation and fracture contact volumes is proposed. The model of Hertzian (heavy-loaded) friction contact evolution is considered. The least wear particle principle is formulated. It is mechanical (nano) quantum. Mechanical quantum represents the least structural form of solid material body in conditions of friction. It is dynamic oscillator of dissipative friction structure and it can be examined as the elementary nanostructure of metal’s solid body. At friction in state of most complete evolution of elementary tribosystem (tribocontact) all mechanical quanta (subtribosystems) with the exception of one, elasticity and reversibly transform energy of outer impact (mechanic movement). In these terms only one mechanical quantum is the lost – standard of wear. From this position we can consider the physical criterion of wear and the constructive (limiting) criterion of gear teeth and other practical examples of tribosystems efficiency with new tribology notion – mechanical (nano) quantum.


Introduction
We suggest using a generalized model of friction the model of elastic-plastic deformation of the body element, which is located on the surface of the friction pairs. This model is based on our new engineering approach to the problem of friction -triboergodynamics. Triboergodynamics [1] is an extension (one of its parts) of general Ergodynamics of deformable bodies [2][3][4]. Ergodynamics is a synthesis to the problem of deformation most general laws of thermodynamics for non-reversable processes, molecular kinetics and dislocation theory in their mutual, dialectical tie on the basis of a most general law of nature -the law of energy conservation at its transformations. Triboergodynamics is based on modern knowledge of friction too: 1. Friction is a phenomenon of resistance to relative motion between two bodies, originating at their surfaces contact area; 2. Friction is the process of transformation and dissipation of energy of external movement into other kinds of energy; 3. Friction is the process of elasto-plastic deformation localized in thin surface layers of rubbing materials. Thus, within the framework of triboergodynamics the model of elastic-plastic deformation of contact volumes is examined as a generalized mechanism of transformation and dissipation energy and determines essence of resistance to surfaces displacement. The major distinction of triboergodynamics from general Ergodynamics of deformed solids is «scale factor» which exhibits itself in existence of critical friction volume. This volume determines the limit friction parameters and separate, in essence, the surface deformation from the traditional volume deformation.

Structural model
Deformable body is considered as an open, multicomponent, substantially nonhomo-geneous and nonequilibrium system with hierarchy of different levels (from submicro-to macrolevel) of metastable structural elements (defects and damages) which are statistically uniformly distributed in the volume. Some of these elements are virtual sources and sinks of elementary defects (vacancies, dislocations, etc.), the others are a barrier to their motion. The structure state is defined by the basic parameters [4]:   is overstress factor of interatomic bonds which evaluates nonuniformity of external stresses  distribution in the bonds   specifies a relationship between theoretical   and actual  S strength of a solid body.

Physical model and structural-energetic interpretation of the process
Macroscopic phenomenon -plastic deformation and fracture of the body element is considered as a cooperation of a huge number of microscopic elementary acts of atomic-molecular regroupings under external force field (mechanical, thermal, electrical, etc.) which are activated by the thermal energy fluctuations. From the thermodynamic point of view, all the mechanisms and structural levels of the process are divided into two most characteristic groups of adaptive and dissipative (relaxation) types. They differ in physical nature and kinetic behavior. The simple acts controlling generation and accumulation of unit defects in deformed body element (damage) are classified as the first group. The specific (referred to unit volume) pumping power of excessive (latent) energy e u  is an overall characteristic of the processes rate The mechanisms and simple acts controlling relaxation (dissipative) processes of plastic deformation are classified as the second group. The specific power of thermal effect q  of plastic deformation is overall characteristic of the processes As a parameter of damage (scattered fracture) we shall take the density of internal energy stored in the deformed volume. The energy is defined as a sum of two components: potential (latent) energy e u and kinetic (thermal) energy T u that is, The energy is related to static ( e u  ) and dynamic ( T u  ) damages and distortions of crystal lattice in deformed body. Consequently, it is responsible for scattered fracture (damage). The body element is looked upon as fractured if at least in one local volume responsible for fracture the internal energy density reaches the critical (ultimate) value  u . This value corresponds to the loss of stability «in great» by crystal lattice. At this instant the cracks of critical size (after Griffith-Orowan-Irwin) and sharp localization of the process at the crack tip occur in a local volume. The thermodynamic condition of local fracture is written as Here   0 ,  r u  -density of internal energy of the material in initial (before deformation, -specific power of internal energy sources in local macrovolume of the material responsible for fracture;  r  -parameter characterizing coordinates ( ) -of the local volume responsible for fracture.

Thermodynamic criterion of fracture
According to structural-energetic analogy between mechanical fracture and melting of metals and alloys [5] and experimental data [2], the critical value of internal energy  u in the local volume responsible for fracture agrees with known thermodynamic characteristic of material s H  (enthalpy of melting)  . Relationship (14) generalizes the known proposition of dislocation theory on mutual relation between the flow stress s  and the density of latent (stored) energy e u [6] for the case of combined stress state. The material damage e u in the local volume responsible for fracture becomes critical, therefore, relationship (14) makes it possible to estimate the actual strength S of the material.
Here  , E -elasticity modulus and Poisson's ratio.

Structural-energetic interpretation of friction process
It is known friction is characterized a product of frictional forces F by friction distance  , that is, the work f  , expended on overcoming frictional force: Here  ) and (20) should be rewritten as: These equations show, that from thermodynamic point of view, the work f  of friction forces, (friction power f   ) is related to plastic deformation of the contact volumes. The work f  may be divided conventionally into two specific parts. The first part is related to variation of the latent (potential) energy ( 1 e u  and 2 e u  ) in deformed (contact) volumes. This is the energy of various simple defects and damages which are generated and accumulated in the bulk. This energy is unique and the total characteristic of submicro-and microstructural variations occurring in plastically deformed volumes [2,3].This is a measure of strain hardening and damage of material. , as well as 1 q and 2 q are defined by physico-chemical properties of the materials of the friction pair, their structure and friction conditions. Since the contact volumes (not unit sizes) of the materials forming a friction couple become strained by friction (figure 2), equations (1) and (2) can be rewritten in the form , where Solving equations (23) and (24) for the frictional force F , one obtains: where l and v are the friction path and the slip velocity. Dividing equations (25) and (26) by the normal force N gives generalized equations for the friction coefficient: Therefore, the friction is generally described by the energy balance equation and with thermodynamical point of view [1][2][3] is the process of two interrelated, oppositely directed and concurrent trends operating in a strained contact. According to the energy balance scheme (figure 1) for plastic deformation and fracture [2] presented above, equations for friction work f W , frictional force F and friction coefficient  (without lubrication) has view: where -is the rate of latent energy Thus, viewed thermodynamically, the work done by friction forces f W (the friction power f W  ), the friction force F and the friction coefficient  may be classified conventionally into two specific components with different kinetic behavior [3]. The first component is associated with microscopic mechanisms of adaptive type and relates to the change of latent (potential) energy ( 2 1 , e e u u   ) of various elementary defects and damages that are generated and accumulate in the deformable volumes of materials friction pair (figure 1). This energy is a unique and integral characteristic of the submicroand microstructural transformations that occur in plastically strained materials [2][3][4]. This energy is a measure of strain hardening and damageability of materials. The second component is associated with microscopic mechanisms of dissipative type and relates to dynamic recovery processes in which latent energy is released and heat effect of friction ( 2 1 , q q ) take place. This energy originates in the motion and destruction of various elementary defects of opposite signs, the egress of these defects to the surface, the healing of reversible submicroscopic discontinuities, etc. The ratios of the components 1 e u  and 2 e u  as well as 2 1 q q of the balance vary over a wide range, depending on the physical, chemical, and structural properties of the materials that comprise the friction couple and the friction process conditions. Thus, the thermodynamic analysis of friction (plastic deformation and fracture) has led to generalized (two-term) relations for the force F and coefficient of friction  , which agrees with current concepts of the nature of friction [7,8] -molecular-mechanical theory (32) and deformable-adhesion theory (34). But it is more correct to speak about the adaptive-dissipative nature (model) of friction (33). Relationships (21)-(28) which generalize the mechanism of energy dissipation at friction allow to classify the tribosystem states. According to ergodynamics of deformed solids (relationships ) and equations (23)-(24) may be transformed to: As follows from equations of energy balance (35), (36), all exhibitions of friction and wear may be reduced conventionally at least to two basically different states: the first state defines all types of damage and wear, the second -the so-called "wearless" condition. The state of damage and wear is characterized by the components of energy balance (35), (36), which are responsible for accumulation of internal energy in deformed volumes , i.e. the process is irreversible. The "wearless" state is characterized by the components responsible for dynamic dissipation (reversibility) of strain energy into elastic and structural dissipated energy of friction contact . In its turn, the first state may be classified depending on the relation between potential e u  and kinetic T u  components of internal energy. It is subdivided conventionally into mechanical damage and wear (due to so-called structure activation) and thermal damage and wear (due to thermal activation). For instance, let the thermal component of internal energy In the most general case, the energy balance at dry friction (23) should be written as: In the special case, where the friction is localized into volume of the "third body" (figure 2) equation (37) develops into:

Energy interpretation of Leonardo da Vinci (Amonton's) friction coefficient
According to thermodynamic theory of strength [2], the structure parameter should be related to the portion of the accumulated plastic deformation that is responsible for strain hardening. This portion is uniquely and integrally defined by the density of the potential component of internal energy (that is, the latent energy density e u  ) of various defects and damages that accumulate in a plastically strained material. With this in mind, if we neglect the heat effect Q of friction, one will infer from the thermodynamic analysis of friction of equations (27)-(28) that the Amonton (Leonardo da Vinci) friction coefficient is Consequently, the coefficient of friction has a very deep physical sense. On the one hand, it is the parameter which generally characterizes the resistance of relative displacement (movement) of surfaces, for it reflects the portion of energy, which «is done by friction away» as accumulated latent energy e U  , by relation to parameter of external forces work Nl   (energy of external relative movement). On the other hand, it is the generalized characteristic of damage, for it is defined of the latent energy density e u  as integral characteristic of the structure defectiveness measure, because this energy is the generalized parameter of damage. Here too, coefficient of friction generally reflects the structural order (disorder) of deforming contact volume, since the parameter Therefore, coefficient of friction is a true and generalized parameter of tribosystem state. From this conclusion we can say that the analysis of the evolution of the states of a tribosystem is primarily an analysis of the latent deformation energy accumulated within the contact friction volumes.

Generalized experimental friction curves
The dependences obtained for the friction coefficient  are in agreement with experimental curves

Structural-energy regularities of rubbing surfaces evolution
We propose an energetic interpretation of the experimental friction curves ) , ( v N    ( figure 6). According to our concept [1,11], the ascending portion of the friction coefficient curve  is mainly controlled by processes associated with the accumulation of latent energy e U  in various structural defects and damages. Here the increase in  is due to the increasing density of latent (potential) energy e u  and the increasing adaptive friction volume f V . The descending portion of the friction curve is mainly controlled by processes associated with the release and dissipation of energy . Here the decrease in  is due to the decrease in latent energy density within the friction volume f V or (which is virtually the same) to the decrease of the adaptive friction volume   Figure 6. Structural-energy diagram for evolution of rubbing surfaces [1,14].
An ideal evolution of tribosystem is symmetrical. The process starts and finishes within areas of elastic behavior. A plastic maximum (a superactivated condition) exists between them as a condition of selforganisation and adaptation. In the most general case evolution (adaptation) regularities of tribosystems may be presented as

The limit (point 4) of this stage is characterized by a full transformation of adaptive critical volume
The volumes mentioned above characterize different regularities of transforming energy of outer mechanical movement at friction. Adaptive volume adapt V is connected with non-reversible absorption of deformation energy. It is in this volume where latent deformation energy e u  accumulates and where the centres of destruction initially emerge (birth). Dissipative volume dis V is capable of reversible transformation (dissipate) of outer movement energy. It doesn't accumulate latent deformation energy owing to reversible elastic-viscous-plastic deformation. Suggested theoretical and calculation evaluation [1,11] showed that dissipative friction volume performs reversible elastic energy transformation of outer mechanical movement with density  q  equal to critical density of latent energy  e u . Culmination of tribosystem evolution is its final and limited condition of point 4 -a state of anomalously low friction and wearlessness (maximum efficient). A schematic evolution of the contact volume of friction in diagram's points 1-5 is presented in figure  7.  The result of ideal elementary tribosystem (contact) evolution is forming of unique nanostructure -a mechanical (nano) quantum. Strict notions about mechanical quantum have been obtained [1] considering equation of quasiideal solid body for point 4 of diagram of friction evolution - 1 Where k -Boltzmann constant; W -condition probability; U S -configuration entropy of friction, contact volume. Tribosystem always tends to some optimal condition, characterized, i.e. to a most probable condition for the given friction conditions.
Analysis and solution of these equations [1] allows to demonstrate the principle of constant probability value (parameter of tribosystem condition (order)) W for the whole range of compatible friction : where MQ R -universal constant of deformation at friction.  This mechanical quantum constitutes a minimum number of atoms capable to provide such a configurational distribution (structure) which obtains the property of reversibly taking and dissipating (recovering) energy of outer mechanical movement. It also constitutes minimum structure in conditions of plastic deformation and it is formed at tribosystem transition (deformed volume) through an ultimately activated (critical) condition (see figure 6) due to development of selforganisational tribosystem adaptation processes. Mutual rotation-oscillation movement of these mechanical quanta in respect of each other within elementary tribosystem (contact) determines condition of most perfect dissipative friction structure. Properly speaking, such condition is described by equation of quasiideal solid body condition (41), a condition when interaction between structural elements (mechanical In these terms (point 4) only one mechanical quantum [1,[11][12][13][14] is the lost -standard wear. The tribosystem (friction contact) has the ideal damping properties -«wearlessness». The principle of mechanical quantum determines nanoquantum levels of all friction parameters of compatible (optimal) tribosystems and other.
4. Nanoquantum models of tribosystem maximum capacity for work

Gear wear calculation principle
The all parameters of compatibility (optimal) friction have to be in quanta levels -commensurable with the parameters of the one mechanical quantum -standard of wear. So, all heavy-loaded tribosystems it is necessary to examine with position of tribosystem ideal evolution. This ideal state of tribocontact is true indicator of tribosystem state for practical examples of tribology. It is the standard of maximum tribosystems efficiency -anomalously low friction and wearlessness. The state of friction contact under its most full evolution is the characteristic with exploitation of hard loaded Hertzian contact, for example, on the surfaces of gear wheels teeth and systems of wheel-rail and other. We can examine the active surface of gear wheel, which consist of equilibrium spherical form asperities after run-in. During one revolution of gear wheel each asperity of gear wheel teeth is loaded one time too. Under it the loss of one friction contact is equal to one mechanical (nano) quantum. Therefore, the whole contact volume is fatigue failured during about 63 millions cycles.  Figure 9. Model of an active surface of gear wheel with equilibrium spherical form asperities.
Thus, an elementary nano-structure of deformed solids may examine as the standard of wear and to apply with optimization the life time of real hard pressed Hertzian contact systems.

Estimation of bearing capacity for work of internal combustion engines
Take the engine with a frequency of This speed limit is determined by the principle of filling the entire nominal friction sliding system area with elementary tribosystems, damping process. Above this speed happens full unloading tribosystem, detachment of wheels from the surface of the rail as distorts the principle of minimum resistance to movement (the principle of one elementary tribosystem or irreversibility). In this case, all mechanical quantuns of elementary tribosystem will repel the wheel. There will be no quantum which activates a process to maintain the system in an excited state. The calculation will be performed in the following order. Denote elementary nominal contact area. By definition [1], on elementary, nominal area of tribosystem can accommodate and work This result is close to modern speed of h km 574, 8 (TGV, France).

Conclusions
5.1. Structural-energy analysis of the friction process allows us to examine the friction process as the evolution process; 5.2. From the energy balance equations of friction follows that the evolution of tribosystem (contact) has an adaptive-dissipative character. 5.3. Experimental friction curves of ) , ( v N    type may be examined as generalized friction experimental curves; 5.4. The fuller evolution of tribosystem has symmetrical view -the friction process is started and finished within elastic area. 5.5. Under fuller evolution of friction contact (elementary tribosystem) the unique nanostructure is formed; the basis of this structure is the mechanical (nano) quantum and the contact (material point of mechanics) consists of about 8 10 63 , 0  such quantums. 5.6. We can examine the mechanical quantum as the least structural form of solid material body and the standard of wear. 5.7. All parameters of compatibility (optimal) friction have to be in quanta levels -commensurable with the parameters of the one mechanical quantum. 5.8. Interaction between nanoquantums is nature the net elasticity. The value of the coefficient of friction between mechanical quantums has order 8 10 587 , 1    MQ  . 5.9. Exploitation of gear wheels and other heavy-loaded tribosystems (Hertzian contact) are subjected to model of nanoquanta damping, when one mechanical quantum is the standard of wear (subtribosystem).