Generalization of Experimental Information and Identification of Patterns in the Behavior of Metals and Alloys under Fatigue Loading Based on the Mapping of Fatigue Curves into Reduced Spaces V.V. Andreev1, O.V. Andreeva1 vyach.andreev@mail.ru|andreevaov@gmail.com 1 Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, Russia The paper presents a characteristic of a fairly large sample of data on the fatigue of metals and alloys, including the coordinates of the fatigue limit and a description of the conditions under which these results were obtained. Using the reduction procedure allows us to obtain a generalized dependence of the reduced parameters of fatigue resistance, on the basis of which it is possible to develop forecasting methods. Extension of the reduction procedure to limited endurance limits allows one to obtain a generalized surface of the presented indicators of fatigue resistance and to increase the reliability of the forecasts made of indicators of fatigue resistance. Keywords: metals, fatigue metal, endurance limit, given indicators, reduced dependence, generalized surface, forecasting. coordinates. In this case (under certain experimental conditions), 1. Introduction the multi-cycle region and the endurance region are represented by two segments of straight lines intersecting at the fracture Designing structures, making the right choice of point. The angle of inclination of the fatigue curve to the axis of manufacturing and processing methods, ensuring the safe the number of loading cycles is the so-called structurally operation of technical devices, extending the life of a device, or sensitive parameter, an indicator of the intensity of damage making a conclusion about the need for decommissioning β€” accumulation processes in a structural material under cyclic when solving all these problems, it is necessary to assess the state loading. In fig. 1 is a schematic representation of a multi-cycle of the structural material. The condition of the structural material region of a fatigue curve in a logarithmic coordinate system. is determined by its strength indicators, its ability to withstand The special nature and causes of the appearance of a fatigue external loads. In this case, the main type of structural material curve fracture in a logarithmic coordinate system provide an failure is fatigue. The experimental determination of fatigue opportunity for the implementation of various forecasting resistance indicators is a lengthy and expensive procedure. The methods. From the point of view of stability theory, the inflection desire to reduce the costs of design and production inevitably point on the fatigue curve is a bifurcation point, the appearance leads to the need to predict the properties of metals and alloys of which indicates a fundamentally different behavior of the based on previously obtained experimental data, combining them structural material in the right horizontal section, compared with as part of some integral models for describing the properties of the left section of the fatigue curve that is steeply inclined to the structural materials under unsteady loading. axis of the number of loading cycles. Despite its high cost, the numerous data on fatigue of metals and alloys obtained by different authors represent a poorly structured, multidimensional array of information. The study of such data, for the purpose of generalization, is difficult. Quite often it is impossible to combine and jointly review the results of individual studies. A large amount of data remains unknown to a specific specialist, and it is often impossible to use this information effectively due to the publication of limited information. Quite often, there remains only the possibility of an almost intuitive assessment of the influence of a particular factor from the totality of acting factors that only an expert can fulfill. Most of the empirical dependencies proposed in the literature, which relate the values of the sought-for indicators of metal fatigue resistance to various calculated or experimentally obtained parameters, have a limited range of definitions and are difficult to reconcile when trying to jointly account within a Fig. 1. Schematic representation of the multi-cycle region single information system. of the fatigue curve in a logarithmic coordinate system: P - Thus, the need to develop a procedure for converting fatigue limit (point of fracture of the fatigue curve); N - is the experimental data on the fatigue of metals and alloys, which number of loading cycles; 𝜎R is the stress (MPa) corresponding allows combining available poorly structured, heterogeneous to the endurance limit; 𝑁G is the abscissa of the point of fracture information, is obvious. It is necessary, using the analysis of of the fatigue curve (the number of loading cycles accumulated experimental data and existing methods of limited corresponding to the transition of the fatigue curve to a generalization of information for individual groups of metals and horizontal section); 𝛼w is a structurally sensitive parameter of alloys within separate sets of factors, to try to develop a universal metal fatigue resistance; πœŽβˆ— , π‘βˆ— are conditional, physically system for generalizing information on the fatigue of metal unrealizable values of stress and durability at which a materials and to synthesize a model that describes the behavior straightened fatigue curve intersects the coordinate axes, 1 and of metals and alloys under cyclic loading as a single class 2 are experimental points corresponding to the destruction of construction materials. the objects under study after a certain number of loading cycles 𝑁1 and 𝑁2 at given stressed levels 𝜎1 , 𝜎2 ; lg𝜎 = π‘™π‘”πœŽβˆ— βˆ’ 2. Materials and method 𝑑𝑔(π›Όπ‘Š )𝑙𝑔𝑁 equation of the left branch of the fatigue curve. There are various forms of graphical representation of For joint consideration, more than 1000 fatigue curves of experimental metal fatigue data. The most convenient option is metals and alloys were selected during the construction of which the representation of the fatigue curve in a system of logarithmic Copyright Β© 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). the endurance limits were experimentally determined. These options for the location of the fatigue curve. Some complex endurance limits (the intersection points of the left and right indicators of fatigue resistance are needed, which would take into sections of the fatigue curve in the multi-cycle region) are account indicators of both strength and durability, and the presented in fig. 2. Table 1 shows the results of the stability of the structure under cyclic loading. systematization of the selected experimental data. This systematization made it possible to determine the β€œdimension” of 3. Literature review the feature space characterizing the data selected for analysis. These signs are a description of the factors (conditions) under A huge number of scientific papers have been devoted to the study of the phenomenon of fatigue both in our country and which periodic loading of laboratory samples and field parts was abroad. Since the moment of the β€œconscious” study of the carried out. phenomenon of fatigue, which is associated with the studies of the German engineer Weller (1858), over 30,000 scientific papers have been published on this problem. Among the scientists who made a significant contribution to the study of the phenomenon of fatigue, I.A. Odinga, V.T. Troshchenko, V.S. Ivanov, V.P. Kogaev, V.M. Grebennik, I.V. Kudryavtseva, V.F. Terentyeva L.M. Akimova, N.V. Oleinika, Griffith A.A., Orowan E., Coffin L.F., Mott N.F., Cottrel A.H., Yokobori T. and others [3-5, 7-10]. The development of methods for accelerated determination of fatigue resistance indicators was the subject of research by Stepnova M.N., Troshchenko V.T., Evstratova S.P., Panteleeva V.N., Goltseva D.I., Ivanova V.S. and Gordienko L.K., Yatskevich S.I., Muratova L.V. [6] I.A. was engaged in a quantitative analysis of the influence of various factors on indicators of fatigue resistance and the development of Fig. 2. The position of the fracture points of the metal a system for accounting for their joint action. Oding, N.I. fatigue curves in the multi-cycle region considered in the study Kononchuk, L.M. Akimov, D.I. Shetulov, V.P. Kogaev and (𝜎R is the stress, MPa; 𝑁G is the number of loading cycles) others [1, 6, 7]. Long-term studies of the phenomenon of fatigue made it Table 1 shows the results of the systematization of the possible to accumulate significant amounts of information on the selected experimental data. This systematization made it possible experimental evaluation of the results of periodic effects of to determine the β€œdimension” of the feature space characterizing various kinds on structural material, however, as noted in the the data selected for analysis. These signs are a description of the literature, researchers have not brought to the final solution to the factors (conditions) under which periodic loading of laboratory problem of accurately predicting the results of fracture processes samples and field parts was carried out. in structural materials when this kind of operational impact. There is no complete understanding of the physical nature of Table 1 fatigue; there is no way to accurately determine the time of onset A generalized description of factors under the action of of failure. It is customary to talk about established patterns or which the results of fatigue tests of laboratory samples and field hypotheses of fatigue failure, implemented using various, for parts considered in the study were obtained example, energy, dislocation, statistical, or other scientific theories to explain the processes of material failure under cyclic β„– Name of factor Range of variation loading. (number of different The variety of accumulated experimental data confirms the options, levels, methods, complexity of the behavior of metals and alloys during fatigue, modes) does not allow us to unequivocally accept, as the only one that 1. Steel or alloy grade 204 corresponds to the observed results, none of these hypotheses. 2. Method of loading 17 At the same time, starting from the 70s of the last century, (scheme) among scientists (see, for example, the works of Ivanova V.S. 3. Test environment 54 and Terentyev V.F. [7]), the belief that further practically useful 4. Test temperature 66 (from -269 to 1000ο‚°Π‘) results in this field spread possible only on the basis of 5. Cross sectional shape 4 interdisciplinary research, including with the involvement of the 6. Scale factor 65 (from 1 to 111 mm) methodology of synergetics, information processing theory and (characteristic size) system analysis [9]. The basis for such conclusions was the work 7. Heat treatment mode 81 of Prigozhin I., Stengers I., Haken G., Stanley H., Klein U., 8. Surface treatment 28 which made it possible to justify the consideration of metal as a method complex dynamic system located at the time of destruction far 9. Surface finish 63 (from Ra 0,2 to Ra from the equilibrium state and to apply the methodology for 24,07) analyzing the stability of dynamic systems , the theory of 10. Cycle frequency 68 (from 2,5 to 20000 Hz) catastrophes, to more thoroughly study the jump-like transitions of systems from one state to another (bifurcation points). Despite its objectively high cost, the numerous data obtained A known problem arises when it is necessary to compare two different fatigue curves. Taking into account only the strength by various authors on the fatigue of metals and alloys are a poorly structured, multidimensional array of information, the study of characteristics (for example, the endurance limit value or the which, and the development of any recommendations based on values of the limited endurance limit) or the durability it, are difficult. The substantial part of the results of fatigue tests characteristics (for example, comparing the abscissa of the inflection points of two fatigue curves) or even comparing the is often due to the narrow tasks of a specific study. This leads to the impossibility of combining and jointly considering the results values of structurally sensitive parameters (the slope angles of of individual studies, and obtaining, on the basis of their analysis, the compared fatigue curves) does not allow get an unambiguous practically useful generalized information. A large amount of answer about the advantage of one or another of the considered data remains unknown to a specific specialist, and it is often NΠΏΡ€ impossible to use this information effectively due to the publication of limited information. Quite often, there remains only the possibility of an almost intuitive assessment of the influence of a particular factor from the totality of acting factors that only an expert expert can perform. 4. Development the space of reduced parameters To obtain preliminary experimental data on the fatigue of metals and alloys. In order to obtain relative parameters (given indicators of fatigue resistance), in the coordinates of each point belonging to the left branch of the stress curve, refer to the conditional values of the stress and the number of loading cycles in which the conditionally continued fatigue curve intersects the coordinate axes: sΠΏΡ€ tgaΠΏΡ€ 𝜎reduced = βˆ’log(πœŽπ‘… β„πœŽβˆ— ), (1) 𝑁reduced = βˆ’log(𝑁𝐺 β„π‘βˆ— ), (2) Fig. 4. Schematic representation of the surface of the given parameters of fatigue resistance with points highlighted on it, π‘‘π‘”π›Όπ‘Šreduced = βˆ’log(π‘‘π‘”π›Όπ‘Š ), (3) corresponding to the physical endurance limits of metals and alloys The proposed transformation allows you to translate any fatigue curve from a system of logarithmic coordinates to one Consideration of all presented in fig. 1 experimental fatigue surface in the three-dimensional space of the transformed curves in the space of reduced coordinates allows you to get the coordinates. Moreover, the generatrix of this surface is a curve following picture (fig. 5, 6). of the form 𝑦 = (log(π‘₯)) βˆ’ 𝐴, and the guide is parallel to the axis of the transformed angle of inclination of the fatigue curve. A schematic image of the surface under consideration is presented in fig. 3. NΠΏΡ€ sΠΏΡ€ tgaΠΏΡ€ Fig. 5. Generalized dependence of the given parameters of fatigue resistance of metallic materials Fig. 3. Schematic representation of the surface within which the normalized space of fatigue curves is stratified by an In other words, we got a new object of study - a transformed informative parameter analogue of the studied subject area - an image of fatigue curves in space of the given indicators of fatigue resistance. In [4], on the basis of processing the obtained dependences, We repeat that the generalized dependence was obtained the following approximation expressions for the projections of while considering the endurance limits obtained experimentally the generalized dependence were obtained: on various fatigue curves. Comparing with each other the points on the fatigue curves, which are the same number of times apart 𝜎reduced = βˆ’log(πœŽπ‘… β„πœŽβˆ— ), (4) from the endurance limits, we can obtain dependences similar to the generalized dependence of the reduced parameters of fatigue 𝑁reduced = βˆ’log(𝑁𝐺 β„π‘βˆ— ), (5) resistance in terms of universality. An examination of the family of such curves makes it π‘‘π‘”π›Όπ‘Šreduced = βˆ’log(π‘‘π‘”π›Όπ‘Š ), (6) possible to obtain a generalized surface of the reduced parameters of fatigue resistance. The visualization program for The confidence coefficient for dependencies (4-6) was not the generalized surface of the reduced parameters allows not only lower than 0.93. to study the generalized dependence from different observation Given the large number of points (fatigue curves) considered points, but also to accurately position the arbitrary fatigue curve to obtain these dependences, we can talk about some generalized in the space of reduced coordinates by setting the angle of dependences of the behavior of a metal structural material under inclination of the fatigue curve and the stressed level. the action of a cyclic load under different sets of factors. Fig. 6. The enlarged part of the generalized dependence of the given parameters, containing the bulk of the experimental points Expressions were obtained for the projections of the lines corresponding to the same values of the stressed level (the difference between the current stressed value and the value of the physical endurance limit): 𝑁reduced = 𝐢 βˆ— 𝑒π‘₯𝑝(𝐷 βˆ— π‘‘π‘”π›Όπ‘Šreduced ), (7) 𝜎reduced = 𝐴 βˆ— exp(𝐡 βˆ— π‘‘π‘”π›Όπ‘Šreduced ), (8) the coefficients A, B, C and D in which are related to the coefficient k, which shows how many times the current value of the endurance limit exceeds the physical endurance limit, in Fig. 7. The main elements of the space of reduced accordance with the following expressions: indicators of fatigue resistance of metals C = 0,4416 βˆ— ln(π‘˜) + 1,9383 , (9) 𝐷 = βˆ’0,0531 βˆ— ln(π‘˜) + 2,5247, (10) 𝐴 = 6,1287 βˆ— π‘˜ 2 – 5,5584 βˆ— k + 5,868 , (11) 𝐡 = βˆ’0,7332 βˆ— π‘˜ 2 – 0,4453 βˆ— k βˆ’ 1,1474. (12) The last two expressions are applicable when 0.3 <= k <= 1.15. If a priori it is assumed that the limited endurance limit exceeds the physical endurance limit by more than 15%, instead of calculating the coefficients A and B, you must use the expression: 𝜎reduced = 2,2499 βˆ— π‘‘π‘”π›Όπ‘Šreduced 4 βˆ’ 10,831 βˆ— π‘‘π‘”π›Όπ‘Šreduced 3 + 19,903 βˆ— π‘‘π‘”π›Όπ‘Šreduced 2 βˆ’ π‘‘π‘”π›Όπ‘Šreduced + 𝐹, (13) 𝐹 = 0,0455 βˆ— π‘˜ 2 βˆ’ 0,4222 βˆ— k + 7,3732. (14) For the coefficients C and D (in the case of calculating the dependence for Npr), the calculated expressions do not change even if the coefficient k falls outside the range indicated above. Fig. 8. View of the generalized dependence, the generalized In fig. 7 shows the main elements of the space of reduced surface of the reduced parameters of fatigue resistance and fatigue resistance indices: experimental points corresponding to experimental points corresponding to the region of small angles the physical endurance limits of fatigue curves obtained of inclination of the fatigue curves to the axis of the number of experimentally, a generalized dependence of the reduced fatigue loading cycles resistance indices, and a generalized surface of the reduced fatigue resistance indices whose equal level lines correspond to the same stressed value, i.e., the same number of times differs 5. Results from the value of the physical endurance limit. Using the above indicators of fatigue resistance, it was The study of the results of the parallel representation of possible to develop a method for joint consideration of fatigue curves in the traditional and reduced coordinate systems experimental data on the fatigue of metals and alloys. As a result allowed us to confirm the greater convenience of the reduced of this, a generalized dependence of the reduced parameters of coordinate system for quantifying the effect of various factors on fatigue resistance was obtained. The domain of determination of the fatigue resistance parameters. In particular, the dependences the obtained generalized dependence and the method for of the parameters of the generalized dependence were obtained predicting fatigue resistance indicators based on the use of this for various combinations of acting factors. generalized dependence are estimated. The forecasting method is quite universal, it is applicable to describe the behavior of a wide [7] Terentyev V.F. Fatigue of metallic materials. -M .: Nauka, range of steel and alloy grades under various combinations of 2003. -- 248 p. acting factors. At the same time, this method has a limitation - it [8] Terentyev V.F. Fatigue strength of metals and alloys. M .: is applicable only under the condition when the existence of a Intermet Engineering. – 2002. - 287 p. fracture point (physical endurance limit) is assumed in the multi- [9] Troshchenko V.T. Deformation and fracture of metals under cycle region of the fatigue curve. multi-cycle loading. – Kiev: Science. Dumka, 1981.- 343 p. [10] Troshchenko V.T., Sosnovsky L.A. Fatigue resistance of metals and alloys. – Kiev: Science. Dumka, 1987.- 1303 p. Fig. 9. Elements of the space of reduced coordinates near the transition to large angles of inclination of the fatigue curve to the axis of the number of loading cycles. The generalized dependence of the reduced parameters of fatigue resistance is represented by a consecutive series of points. Added lines of equal voltage level (make up a system of isolines uniform with the generalized dependence) and grid lines of the generalized surface corresponding to equal values of the reduced angle of inclination of fatigue curves (for the presented case, the step of changing the reduced angle of inclination when constructing the generalized surface is 0.2; when constructing the generalized dependence, the step less; the ratio of stress to the endurance limit varies from 0.3 to 1.3 in increments of 0.1. The generalized relationship corresponds to a ratio of 1). Comparison with each other of points of different fatigue curves that differ from endurance limits by the same number of times made it possible to construct a generalized surface of the reduced parameters of fatigue resistance. Using the generalized surface of the given parameters of fatigue resistance made it possible to compare and generalize, within the framework of a single, practically functional dependence, the data on fatigue tests of metals and alloys in the case when the tests were carried out at a base less than the abscissa of the inflection point of the fatigue curve. 6. Gratitudes The work was supported by RFBR, Grant β„– 19-07-00455. 7. References [1] Andreev V.V. The endurance limit of metals on the generalized dependence of the given parameters of fatigue resistance.- N.Novgorod: Nizhny Novgorod. Univ., 2003. [2] Andreeva O.V. Model and algorithms for assessing the damage to the surface microstructure of metals and alloys from images / Monograph. NSTU named after R.E. Alekseeva, N. Novgorod, 2018, 5 ps. [3] Ekobori T. Physics and mechanics of fracture and solid strength. M.: Metallurgy, 1971. - 264 p. [4] Ivanova V.S. Fatigue failure of metals. – M. Metallurgizdat, 1963.- 272 p. [5] Kogaev V.P. Some questions of the fatigue strength of steel.-M .:, 1953.- P.126-132. [6] Oleinik N.V., Sklyar S.P. Accelerated fatigue tests. - Kiev: Science. Dumka, 1985.- 304 p.