MODELING THE EFFECT OF STOCHASTIC DEFECTS FORMED IN PRODUCTS DURING MACHINING ON THE LOSS OF THEIR FUNCTIONAL DEPENDENCIES

Modeling the effect of stochastic defects formed in products during machining on the loss of their functional dependencies. The article investigates the influence of hereditary defects formed in the surface layer of products from metals of heterogeneous structure on the quality of surfaces treated with finishing methods. The research is based on an integrated approach based on the results of the deterministic theory of defect development and methods of probability theory. The treated layer of the product is considered as a medium weakened by random defects that do not interact with each other, namely: structural changes, cracks, inclusions, the parameters of which are random variables with known laws of their probability distribution. The causes of structural changes, crack formation on the treated surface product depending on different types of probability distribution of dimensions are investigated: length, depth of defects, and their orientation. From these positions, technological possibilities of their elimination by definition of branch of combinations of the technological parameters providing necessary quality of the processed surfaces are considered. Modeling of thermomechanical processes in the treated surface containing hereditary defects is carried out based on thermoelastic equations with discontinuous boundary conditions in the places of their accumulation. The research used the apparatus of boundary value problems of mathematical physics equations, the method of singular integral equations for solving problems of fracture mechanics, Fourier-Laplace integral transformations for obtaining exact solutions, the method of constructing discontinuous functions. The dependences determining the intensity of stresses in the vertices of hereditary defects are obtained. A method for predicting the nature of crack formation depending on the probability distribution of defects, the values of heat flux entering the surface layer of the processed product has been developed. It is established that the increase in the homogeneity of the material leads to an increase in the value of heat flux, which corresponds to a fixed probability of failure.


Introduction
Purpose and objectives of the study This study aims to develop the theory and recommendations on technological methods of significantly reducing grinding defects such as burns and cracks in processing parts made of materials and alloys, the surface layer of which has hereditary defects of structural or technological origin.
Achieving this goal requires setting and solving the following tasks: 1. To study the mechanism of formation of defects in the surface layer of parts made of materials and alloys prone to defect formation during their processing by grinding, considering previous operations and hereditary inhomogeneities that occur.
2. To develop a mathematical model that describes the thermomechanical processes in the surface layer during the grinding of parts of materials and alloys, considering their inhomogeneities that affect the formation of technological defects and determining the criteria for defect formation. 3. Check the adequacy of the results obtained on products made of materials prone to defects during processing by finishing methods.

Materials and methods of research
Theoretical research is carried out based on scientific bases of technology of mechanical engineering and thermophysics of mechanical and physical and technical processing processes, theories of thermoelasticity, the complex approach of modern deterministic theories of fracture mechanics, and methods of probability theory. In addition, the research uses the apparatus of boundary value problems of equations of mathematical physics and the method of singular integral equations for solving destruction problems.
First, we need to build the first part of the model describing thermomechanical processes and the impact of stochastic defects formed in products on finishing operations on the loss of their functional properties, namely the equations describing thermomechanical processes in the treated layer of products on finishing operations.
When choosing and substantiating the mathematical model, it was taken into account that both thermal and mechanical phenomena accompany the process of grinding parts. However, the predominant effect on the stress-strain state of the surface layer has temperature fields. Given that the bulk of the surface layer of the metal during grinding is in the elastic state, we can use the model of the thermoelastic body, which reflects the relationship of mechanical and thermal phenomena at the final values of heat fluxes. Since information on the propagation of temperatures and stresses along the depth and direction of movement of the tool is essential for the study of the thermomechanical state of polished surfaces, a flat problem is considered [4].
The influence of inhomogeneities in phase transformations of unstable structures, intergranular films, contour boundaries of hereditary austenitic grains, carbide stitching, nonmetallic inclusions, shells, flocs, and other defects arising in the surface layer and defects in the form of conditional cracks, which has the form (Fig. 1).
The system of equations that determine the thermal and stress-strain state when grinding the surface of parts with coatings, the upper layer of which has inhomogeneities such as inclusions and microcracks, contains: The equation of nonstationary thermal conductivity: . τ Lame elasticity equation in displacements: where ( , , ) T x y τ is the temperature at the point with coordinates ( , ) x y and at any time τ ; a is the thermal conductivity of the material; t a is a temperature coefficient of linear expansion; , G µ are La-mé constants; , v ν are components of the displacement vector of the point ( , ) x y ; The initial conditions for this task can be taken as: ( , , 0) 0.
Boundary conditions for temperature and deformation fields, taking into account heat transfer from the surface outside the area of contact of the tool with the part and intense heat dissipation in the processing area, are: ( , ) ; 0; ; where ( , ) q y τ is the intensity of heat flux, which is formed as a result of the interaction of the tool with the part; λ is the coefficient of thermal conductivity of the material to be ground; * 2a is the length of the contact zone of the tool with the surface to be treated; γ is the heat transfer coefficient with the environment; , x xy σ τ are normal and tangential stresses.
The influence of the design parameters of the tool on the thermomechanical state of the surface layer is determined using the boundary conditions in the form of: v v t is the grinding modes, * 2a is the length of the arc of contact of the tool with the part; * l is the distance between the cutting grains. The maximum values of the instantaneous temperature M T , from single grains to the constant componentk T , were theoretically and experimentally confirmed. For the layer being processed and having structural and technological inhomogeneities, the discontinuity conditions of the solution, depending on the type of defect, will be: : : where , , σ , τ The problem's solution was carried out by the method of discontinuous solutions [21]. These solutions satisfy the Fourier equations of thermal conductivity and Lame elasticity everywhere except the defect boundaries. When crossing the boundary of the field of displacements and stresses suffer jumps of the first kind, i.e., their jumps , , σ , τ The problems of thermoelasticity (1) -(8) were solved using integral Fourier transforms on the variable γ and Laplace on τ to the functions ( , , ) T x y τ , , ( ) , x x y σ τ . Classical strength criteria evaluated the equilibrium state of the deformable surface layer. Of the available failure criteria that consider the local physical and mechanical properties of inhomogeneous materials, the most appropriate is the criteria of the force approach associated with the use of the concept of stress intensity factor (SIF). When the load leads to the fact that the stress intensity I K becomes equal to the limit value Ic K , the crack-like defect turns into the main crack [19,20,21,22]. The development of technological criteria for the control of defect-free grinding is carried out based on the established functional relationships between the physical and mechanical properties of the processed materials and the main technological parameters.
The quality of the treated surfaces will be ensured if with the help of control technological parameters to choose such processing modes, lubricating and cooling media and tool characteristics that the current values of grinding temperature ( , , ) T x y τ and heat flux ( , ) q y τ , stresses M σ and grinding forces , x y P P coefficient Ic K do not exceed their limits. Implementation of the system of limiting inequalities by the values of the temperature and the depth of its distribution in the form of [9, 10]: which avoids the formation of grinding burns and can be the basis for the design of grinding cycles by thermal criteria. Processing of materials and alloys without grinding cracks can be provided to limit the limit values of stresses formed in the zone of intensive cooling of stress: In the case of the dominant influence of hereditary inhomogeneity on the intensity of the formation of grinding cracks, it is necessary to use criteria, the structure of which includes determining relationships of technological parameters and properties of the inhomogeneities themselves. As such, we can use the limitations of the stress intensity factor [23,24,25]: or providing with the help of technological parameters of the limiting value of heat flux, which maintains the balance of structural defects: Defective grinding conditions can be implemented using information about the material structure being processed. Thus, in the case of the predominant nature of structural imperfections of length 2l of their regular location relative to the contact zone of the tool with the part, you can use as a criterion the equilibrium condition of the defect in the form: where , , k p q t ν ν are the grinding modes; , D C are tool parameters; , a λ are thermophysical characteristics of the treated coating; c K is the crack resistance of this coating; G is the modulus of elasticity; ν is the Poisson's ratio; t a is the temperature coefficient of linear expansion; l is the characteristic linear size of the structural parameter (structure defect).
These inequalities correlate the longitudinal characteristics of the temperature and force fields with the control and technological parameters. Furthermore, they specify the range of combinations of these parameters that meet the obtained thermomechanical criteria. At the same time, the properties of the processed material are taken into account, and product quality assurance is guaranteed.
Based on the received criterion ratios, it is possible to provide the quality of a superficial layer of details at grinding, taking into account the maximum processing productivity.
Next, we need to build the second part of the model, which describes the thermomechanical processes and the impact of stochastic defects formed in products at finishing operations on the loss of their functional properties, namely to calculate the probabilities and criteria of equilibrium of stochastically distributed defects in the surface layer.
Modern materials used in technology have a complex structure formed by interacting particles. Depending on the scale of consideration (structural level), such particles can be atoms of different elements, vacancies, dislocations, and their networks, then -crystals, blocks of crystals, grains, polycrystalline aggregates, radicals, fibers, lamellar or three-dimensional inclusions, micro-and macrocracks.
Particles may differ in chemical composition, physical properties, geometry, and relative position. Larger particles in an adequately ordered structure may contain more minor defects.
As a result, there is a tremendous local heterogeneity of authentic materials. The reduction of the theoretical strength of authentic materials to the technical level is a consequence of defects -gaps in the continuity and homogeneity of the building, which arise from the formation of materials and their products (structural and technological defects).
The complexity of deformation and destruction that occur in the microvolumes of the body does not allow taking into account the impact on these processes of all the imperfections characteristic of this material. These defects refer to different structural equations. Therefore, their effect on strength and fracture is different.
In studying the causes of cracking during the grinding of materials, the most dangerous defects are considered -cracks (actually cracks, crevices, elongated sharp cavities) and sharp rigid inclusions. Such defects include extraneous elastic inclusions with very small or substantial elastic and strength characteristics compared to the characteristics of the base material (matrix). For example, low-strength graphite plate inclusions in the ferrite matrix of cast iron with some approximation can be interpreted as cracks.
The detailed introduction of defects, given by their defining parameters, is the first feature of the material model under consideration. The second feature is related to the static nature of the defect risk distribution.
denote the geometric parameters of defects of a specific r-th grade; r n is the number of defining parameters for a given defect, where r is the type of defect. They determine the defects' size, configuration, and location (orientation). For example, for isolated flat elliptical cracks or rigid inclusions, we have five independent parameters (two ellipse axes and three angular orientation parameters for circular defects -three (radius and two angular parameters)).
As the strength of the continuum, take the value of the resistance of the material of development The appearance of these functions depends on the structure and technology of the material. Determinants can be stochastically independent or dependent. It, in particular, may be a consequence of the manufacturing technology (for example, in the thermo-machining of the alloy Alnico 8HC (R1-1-13) (Al -7%; Ni -14%; Co -38%; Cu -3%; Ti -8%; Fe -other), between size and orientation and inclusions, there is a specific correlation). In the case of stochastic independence of defect parameters, the total distribution of parameters is equal to the product of partial distributions of each parameter separately: Since, as accepted [26,27], the ultimate load for a body is equal to the ultimate load of its least strong element (weak link), the distribution function of the ultimate load 1 , ( ) n F P η for bodies of volume V can be found according to the formula for the distribution of the minimum member of the samples consisting of n elements of the general set of elements described by the function 1 The same result is obtained by determining the probability of destruction of at least one element in the set of 0 0 V n V defective elements, and the probability of destruction of each of them separately is equal to 1 1 , , ( ) F P η ξ at fixed 1 , , P η ξ . Formula (18) is used in many works on the statistical theory of brittle strength, based on the hypothesis of a weak link. The value of the function 1 , ( , ) n F P η ξ is equal to the probability P of local destruction of a body of size V under the action of a given homogeneous complex stress field: 1 1 ( , , ) ( , , ) . P P Fn P η ξ = η ξ (19) Determining this probability is one of the main tasks of choosing the technological parameters of defect-free processing. which leads to a Weibull-type distribution [26,27]: where 0 C > , 0 m > are limitations, which depend on the number of defects of the value determined experimentally for a given type of material and load. Determining these values on the basis of the function 1 ( ) n F P makes it possible to establish the parameters on which they depend, in particular, to establish their explicit relationship with the characteristics of material defects and the type of stress state that causes failure.
The above scheme of determining the probability of destruction of the body under the action of a complex homogeneous stress state is valid in the case when defects of one kind weaken the body material. Moreover, if the body is weakened by defects of different varieties that do not interact with each other, the result is easily generalized.
Let the material be weakened by defects ( ) γ r of different varieties. The average number of 0 r n defects of each variety per unit size 0 V is assumed to be known, where 1, 2, 3, , i = … γ . Then, similar to the previous one, we can determine the function 1 1 , ( , ) R F P η ξ of the strength distribution of elements weakened by one defect of each variety separately. We need to know the appropriate deterministic conditions for destroying an element with a defect of each variety and the probability distributions of the defining parameters of defective elements of each variety. Since the value of 1r F determines the probability of destruction of an element with one defect of this variety, and  (  ( , , ) , Let us consider an example of the calculation of statistical parameters of destruction at thermal influence. Let the half-plane weakened by uniformly scattered random cracks that do not interact with each other be under the action of uniform heat flux of intensity q (Fig. 1). The laws of the joint distribution of the half-length k l and the angle of orientation k ϕ will be known. At a specific value of heat flux (let us call it the limited value * q ) develops at least one crack, i.e., the process of destruction begins. The condition for the development of a single crack with given geometric parameters is established in [26,27] by formula (22): We determine the probability distribution of the limiting heat flux and some of its statistical characteristics.
From formula (22) it is seen that due to the randomness of the length l and the orientation of the cracks ϕ , which vary within some limits 0 1 ≤ ≤ α , π / 2 ϕ ≤ , the value of * q the limiting heat flux on the element, the half-plane with one crack is also random. The function * 1 ( ) F q of the probability distribution is found based on formula (20): where R is a two-dimensional domain of possible values of random variables and l in which the rela- is the density distribution of probabilities of length l and orientation φ of cracks.
Assuming the values of l and ϕ are statistically independent ( , ) ( ) ( ) f l f l ϕ = ϕ ⋅ we obtain the formula: Here the Lϕ is the integration domain of all possible values of π 2   ϕ ϕ ≤     , which executes the relation: We assume that the distribution of cracks in orientation is uniform, ie, 3 π ( 1 ) f ϕ = , and in length has the form [29,30]: where S is the crack shape parameter that determines the structural heterogeneity of the material (the larger S , the more likely small cracks, ie the material is more homogeneous), a is the scale parameter or [28]: where r is the fracture parameter of the material (the larger r , the more likely small cracks).
The distribution function ( ) 3 F l will be written: Substituting the expressions for 3 ( ) F l and 3 ( ) f ϕ in formula (23) we obtain: For distribution (27), when 0 1 d ≤ ≤ , 3 2 min q Aa − = and relation (28) holds for π / 2 ϕ ≤ ϕ ≤  , where: In this case, the function * 1 ( ) F q can be represented as: If the half-plane contains n cracks, we assume that the limiting heat flux for the half-plane is equal to the smallest value of the limiting heat fluxes of its elements (weak link hypothesis) [29,30]. Then the distribution function of the limiting heat flux of the half-plane with n cracks is determined by the following formula: The graphs show that the increase in the homogeneity of the material leads to an increase in the value of heat flux, corresponding to a fixed probability of failure. On the other hand, increasing S or r decreases the probability of failure, which corresponds to a given value of heat flux. An increase in the value of S or r leads to an elongation of the area of change of random heat flux, for which the probability of destruction is low.
The magnitude of stress intensity coefficients for defects such as cracks is influenced by the size and orientation of these defects, the depth of their occurrence and mutual location, the magnitude of heat flux. The stochastic model of crack formation during the grinding of metals of heterogeneous structure is based on an integrated approach based on the results of the deterministic theory of the development of individual defects and methods of probability theory. The surface layer is considered as a medium weakened by random non-interacting defects -cracked inclusions, which determine the parameters of which are random variables with known laws of their probability distribution. The probability of destruction of the surface layer depends on different types of the probability distribution of dimensions/length, depth/defects, and orientation. The probabilistic characteristics of the limiting heat flux are considered from these positions. It is established that the increase in the homogeneity of the material leads to an increase in the value of heat flux, which corresponds to a fixed probability of failure.

Conclusions
The scientific problem of establishing the calculated dependences that determine the impact of hereditary defects from previous operations on the quality of the surface layer during grinding to create optimal technological processing conditions taking into account the accumulated defects and inhomogeneities in the surface layer of materials and alloys.
The following results were obtained. 1. The influence of technological and structural heterogeneity of materials on the mechanism of origin and development of defects under the influence of thermomechanical phenomena accompanying diamond-abrasive processing is established.
2. The analytical model of the definition of a thermomechanical condition at grinding of details in which working surfaces contain inhomogeneities of a hereditary origin is developed. Based on this model, the functional relationships of quality criteria with the technological parameters that control are determined.
3. The stochastic model of the process of crack formation during heterogeneous structure materials grinding of is constructed based on which dependences of the probability of cracks on modes and other parameters of the process of grinding are received. It has been calculated that an increase in the homogeneity of the material leads to a decrease in the probability of defects such as cracks and, consequently, an increase in the grinding operation while maintaining the required quality.
In combination with experimental studies, the obtained dependencies allow to theoretically determine the areas of combinations of technological parameters that provide the required quality of the treated surfaces.