=Paper= {{Paper |id=Vol-2485/paper59 |storemode=property |title=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 |pdfUrl=https://ceur-ws.org/Vol-2485/paper59.pdf |volume=Vol-2485 |authors=Vyacheslav Andreev,Olga Andreeva }} ==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== https://ceur-ws.org/Vol-2485/paper59.pdf

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 —
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; 𝜎∗ , 𝑁∗ 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 = 𝑙𝑔𝜎∗ −
2. Materials and method 𝑡𝑔(𝛼𝑊 )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 © 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 “dimension” 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 “conscious” 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 “dimension” 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
№ 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 1000С) 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 Nпр
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: sпр
tgaпр
𝜎reduced = −log(𝜎𝑅 ⁄𝜎∗ ), (1)

𝑁reduced = −log(𝑁𝐺 ⁄𝑁∗ ), (2) Fig. 4. Schematic representation of the surface of the given
parameters of fatigue resistance with points highlighted on it,
𝑡𝑔𝛼𝑊reduced = −log(𝑡𝑔𝛼𝑊 ), (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(𝑥)) − 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.
Nпр

sпр
tgaпр
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 = −log(𝜎𝑅 ⁄𝜎∗ ), (4) from the endurance limits, we can obtain dependences similar to
the generalized dependence of the reduced parameters of fatigue
𝑁reduced = −log(𝑁𝐺 ⁄𝑁∗ ), (5) resistance in terms of universality.
An examination of the family of such curves makes it
𝑡𝑔𝛼𝑊reduced = −log(𝑡𝑔𝛼𝑊 ), (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 = 𝐶 ∗ 𝑒𝑥𝑝(𝐷 ∗ 𝑡𝑔𝛼𝑊reduced ), (7)

𝜎reduced = 𝐴 ∗ exp(𝐵 ∗ 𝑡𝑔𝛼𝑊reduced ), (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 ∗ ln(𝑘) + 1,9383 , (9)

𝐷 = −0,0531 ∗ ln(𝑘) + 2,5247, (10)

𝐴 = 6,1287 ∗ 𝑘 2 – 5,5584 ∗ k + 5,868 , (11)

𝐵 = −0,7332 ∗ 𝑘 2 – 0,4453 ∗ k − 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 ∗ 𝑡𝑔𝛼𝑊reduced 4 − 10,831 ∗ 𝑡𝑔𝛼𝑊reduced 3 +
19,903 ∗ 𝑡𝑔𝛼𝑊reduced 2 − 𝑡𝑔𝛼𝑊reduced + 𝐹, (13)

𝐹 = 0,0455 ∗ 𝑘 2 − 0,4222 ∗ 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 № 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.