Fatigue Limits of Steels and Stress Gradient
На основании обобщения результатов многочисленных экспериментальных исследований предложены и обоснованы эмпирические соотношения между пределами выносливости сталей и градиентом напряжений, в которых учитывается накопление усталостного повреждения в гладких образцах и образцах с концентраторами н...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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irk-123456789-1127812017-01-28T03:02:53Z Fatigue Limits of Steels and Stress Gradient Troshchenko, V.T. Khamaza, L.A. Научно-технический раздел На основании обобщения результатов многочисленных экспериментальных исследований предложены и обоснованы эмпирические соотношения между пределами выносливости сталей и градиентом напряжений, в которых учитывается накопление усталостного повреждения в гладких образцах и образцах с концентраторами напряжений. Предложен метод расчета параметров этих соотношений с учетом механических свойств сталей. Получено хорошее соответствие между предложенными соотношениями и экспериментальными результатами. На основі узагальнення результатів численних експериментальних досліджень запропоновано і обґрунтовано емпіричні співвідношення між границями витривалості сталей і градієнтом напружень, у яких враховується накопичення втомного пошкодження в гладких зразках та зразках із концентратором напружень. Запропоновано метод розрахунку параметрів цих співвідношень з урахуванням механічних властивостей сталей. Отримано хорошу збіжність між запропонованими співвідношеннями й експериментальними результатами. 2011 Article Fatigue Limits of Steels and Stress Gradient / V.T. Troshchenko, L.A. Khamaza // Проблемы прочности. — 2011. — № 4. — С. 74-86. — Бібліогр.: 35 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/112781 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Troshchenko, V.T. Khamaza, L.A. Fatigue Limits of Steels and Stress Gradient Проблемы прочности |
| description |
На основании обобщения результатов многочисленных экспериментальных исследований предложены и обоснованы эмпирические соотношения между пределами выносливости сталей и
градиентом напряжений, в которых учитывается накопление усталостного повреждения в
гладких образцах и образцах с концентраторами напряжений. Предложен метод расчета
параметров этих соотношений с учетом механических свойств сталей. Получено хорошее
соответствие между предложенными соотношениями и экспериментальными результатами. |
| format |
Article |
| author |
Troshchenko, V.T. Khamaza, L.A. |
| author_facet |
Troshchenko, V.T. Khamaza, L.A. |
| author_sort |
Troshchenko, V.T. |
| title |
Fatigue Limits of Steels and Stress Gradient |
| title_short |
Fatigue Limits of Steels and Stress Gradient |
| title_full |
Fatigue Limits of Steels and Stress Gradient |
| title_fullStr |
Fatigue Limits of Steels and Stress Gradient |
| title_full_unstemmed |
Fatigue Limits of Steels and Stress Gradient |
| title_sort |
fatigue limits of steels and stress gradient |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2011 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112781 |
| citation_txt |
Fatigue Limits of Steels and Stress Gradient / V.T. Troshchenko, L.A. Khamaza // Проблемы прочности. — 2011. — № 4. — С. 74-86. — Бібліогр.: 35 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT troshchenkovt fatiguelimitsofsteelsandstressgradient AT khamazala fatiguelimitsofsteelsandstressgradient |
| first_indexed |
2025-07-08T04:37:39Z |
| last_indexed |
2025-07-08T04:37:39Z |
| _version_ |
1837052158063673344 |
| fulltext |
UDC 539.4
Fatigue Limits of Steels and Stress Gradient*
V. T. Troshchenko
1
and L. A. Khamaza
2
Pisarenko Institute of Problems of Strength, National Academy of Sciences of Ukraine,
Kiev, Ukraine
1 vtt@ipp.kiev.ua
2 lah2@ipp.kiev.ua
ÓÄÊ 539.4
Ïðåäåëû âûíîñëèâîñòè ñòàëåé è ãðàäèåíò íàïðÿæåíèé
Â. Ò. Òðîùåíêî, Ë. À. Õàìàçà
Èíñòèòóò ïðîáëåì ïðî÷íîñòè èì. Ã. Ñ. Ïèñàðåíêî ÍÀÍ Óêðàèíû, Êèåâ, Óêðàèíà
Íà îñíîâàíèè îáîáùåíèÿ ðåçóëüòàòîâ ìíîãî÷èñëåííûõ ýêñïåðèìåíòàëüíûõ èññëåäîâàíèé ïðåä-
ëîæåíû è îáîñíîâàíû ýìïèðè÷åñêèå ñîîòíîøåíèÿ ìåæäó ïðåäåëàìè âûíîñëèâîñòè ñòàëåé è
ãðàäèåíòîì íàïðÿæåíèé, â êîòîðûõ ó÷èòûâàåòñÿ íàêîïëåíèå óñòàëîñòíîãî ïîâðåæäåíèÿ â
ãëàäêèõ îáðàçöàõ è îáðàçöàõ ñ êîíöåíòðàòîðàìè íàïðÿæåíèé. Ïðåäëîæåí ìåòîä ðàñ÷åòà
ïàðàìåòðîâ ýòèõ ñîîòíîøåíèé ñ ó÷åòîì ìåõàíè÷åñêèõ ñâîéñòâ ñòàëåé. Ïîëó÷åíî õîðîøåå
ñîîòâåòñòâèå ìåæäó ïðåäëîæåííûìè ñîîòíîøåíèÿìè è ýêñïåðèìåíòàëüíûìè ðåçóëüòàòàìè.
Êëþ÷åâûå ñëîâà: ìíîãîöèêëîâàÿ óñòàëîñòü, ïðåäåë âûíîñëèâîñòè, ãðàäèåíò
íàïðÿæåíèé.
Introduction. Numerous experimental investigations on high-cycle fatigue of
metals and alloys show a significant increase in the fatigue limit with an increase in
the stress gradient that characterizes the nonuniformity of the stress distribution
over the specimen cross section.
Thus, the fatigue limits in bending are considerably higher than those under
axial loading, the local stresses at the stress concentrator tip that correspond to the
fatigue limits are higher than the fatigue limits in the uniform stressed state.
The generalization of the investigation results is done either by constructing
the empirical relationships that relate the fatigue limit value to the stress gradient
[1], or by constructing the models that are based on certain hypotheses.
It is supposed in [2] that the difference in the fatigue limits in the uniform and
nonuniform stressed states is governed by the difference between the nominal
stresses calculated on the assumption of elastic deformation and the actual ones
calculated taking into account the inelastic cyclic deformation in the nonuniform
stressed state.
In the studies described in [3, 4, and others], the stress gradient effect is
explained in terms of statistical strength theories.
© V. T. TROSHCHENKO, L. A. KHAMAZA, 2011
74 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 4
* Report on International Colloquium “Mechanical Fatigue of Metals” (13–15 September 2010,
Opole, Poland).
In the investigations of [5, 6, and others], the explanation of this effect is
given in terms of critical distance theories, which assume that either stresses at
some distance from the surface or averaged stresses in the surface layer are
responsible for the fatigue fracture.
In [7–10, and others], the stress gradient effect is due to the difference in the
inelastic deformation behavior of surface layers of the metal under conditions of
uniform and nonuniform stressed states.
All the mechanisms that underlie the above hypotheses take place, to a certain
extent, in real materials during fatigue fracture. With the use of the relationships
that are based on one of these hypotheses, the consideration of the effect of other
factors is achieved by the empirical selection of parameters in the chosen model.
Empirical relationships that relate the fatigue limit to the stress gradient are
the most simple to use because there is no need to determine the parameters
corresponding to different hypotheses, which is not an easy task.
At the same time, empirical relationships, many of which have been formulated
quite a long time ago, need to be further improved taking into account new
experimental data, with new approaches to generalizing experimental results being
developed.
Starting from the generalization of the results taken from the literature and
original experimental investigations performed under axial loading and in bending
[11–30], this paper proposes and justifies the relationships for determining the
fatigue limits of steels in the nonuniform stressed state, including those in the
presence of the stress concentration, that are based on the use of the parameters of
the mechanical properties of the material under study, relative stress gradient,
theoretical stress concentration factor and fatigue characteristics in the uniform
stressed state.
Investigation Results and Their Analysis. A characteristic of the nonuniform
stressed state is either the stress gradient
�
�
�
d
dx
,
where � is the stresses and x is the geometric size, or the relative stress gradient
for the most stressed point that takes place on the specimen or stress concentrator
surface under elastic deformation
�
�
�
�
�
1
0max
.
d
dx x
In the case of elastic deformation in bending, the relative stress gradient is
��
2
h
or ��
2
d
,
where h is the height of the flat specimen and d is the diameter of the circular
specimen.
Fatigue Limits of Steels and Stress Gradient
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 4 75
In the stress concentrator, under axial loading, the relative stress gradient is
given as
�
�
�
2
,
and in bending it is given as
�
�
� �
2 2
h
or �
�
� �
2 2
d
,
where � is the curvature radius in the concentrator.
In further analysis, the fatigue limits at107 cycles with a symmetric load cycle
for smooth specimens under axial loading ��1 and in bending ( )��1 b , as well as
the fatigue limits (nominal stresses) of notched specimens under axial loading ��1
nth
and in bending ( )��1 b
nth were considered.
The effective stress concentration factor was determined by the formulas:
K
nth�
�
�
�
�
�
1
1
or K
b
b
nth�
�
�
�
�
�
( )
( )
.
1
1
The ratio of the fatigue limits for smooth specimens in bending to those under axial
loading is denoted by �K � .
In the analysis of the results of experimental investigations, as was shown in
[31, 32, and others], account was taken of the fact that for the overwhelming
majority of steels, cyclic inelastic strains at the stresses equal to the fatigue limit at
107 cycles to failure are not large, and the difference between the nominal and
actual stresses is small, not exceeding 2–3%, which gives grounds to calculate
stresses using formulas of elasticity theory. A similar picture is observed in the
stress concentrator. Higher cyclic inelastic strains occur in high-ductility austenitic
steels, which requires special consideration.
In the analysis of the relationship between fatigue limit and stress gradient, it
is necessary to take into account the different nature of the stress distribution over
the height of smooth specimens and those with stress concentrators. In concentrators,
in areas of high stresses where the maximum stress gradients take place, the
relative volume of the material is lower than in smooth specimens in bending,
which leads to an increase in the local characteristics of the limit state.
Critical distance theories [5, 6, and others], which have been used in recent
years to explain the effect of stress concentration on the fatigue strength, are based
on the consideration of this factor.
In the process of the above generalization, we used 156 fatigue curves
obtained experimentally and their corresponding fatigue limits for 40 grades of
carbon and alloy steels at 107 cycles to failure in the nonuniform stressed state,
including the fatigue limits in bending of smooth specimens with the relative stress
gradients varying within �� 0.2 to 2.0; the fatigue limits in tension–compression
of specimens with stress concentrators with the relative stress gradients and
theoretical stress concentration factors varying within �� 0.34 to 15.5 and
V. T. Troshchenko and L. A. Khamaza
76 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 4
� �1 56. to 3.35, respectively, and, finally, the fatigue limits in bending of
specimens with stress concentrators with the relative stress gradients and theoretical
stress concentration factors varying within �� 1.352 to 242.0 and � � 1.25 to
6.3, respectively. Experimental results for flat specimens with circular holes and
side notches and cylindrical specimens provided with a circular recess were
analyzed. All fatigue curves under analysis were obtained during the tests with a
fully-reversed load cycle in the absence of manufacturing residual stresses in
surface layers of both smooth and notched specimens. The diameter and thickness
of specimens varied within 5 to 40 mm.
The fatigue limits found experimentally for some of the investigated steels in
the nonuniform stressed state ( )��1
nth
e are presented selectively as an example in
Table 1 using the data given in [13, 25, 26]. This same table gives the values of the
theoretical and effective stress concentration factors [ � , ( )K e� ], the relative
stress gradient (�), and also those of both the fatigue limits ( )��1
nth
c and the
effective stress concentration factor ( )K c� calculated using the method given
below.
The ratio of the values of the fatigue limit to the effective stress concentration
factor in tension is determined by relationship
� � � ��� � �� �1 1 1
* ( ) ,nth f whence K
fnth�
��
�
�
� �
�
�
1
1
( )
, (1)
and in bending is determined by relationship
� � � ��� � �� �1 1 1
* ( ) ( ) ,b
nth f whence K
f
b
nth�
��
�
�
� �
�
�
1
1( ) ( )
, (2)
where ��1
* is the fatigue limit (local stresses at the concentrator tip), f ( )� is the
function defining the effect of the stress gradient, and � is the theoretical stress
concentration factor.
As a result of the generalization of the experimental data and the use of
relations (1) and (2), the empirical relationships were obtained for the function
f ( ),� fatigue limits and effective stress concentration factors for the specimen
geometry and modes of loading under consideration (Table 2).
The formulas given in Table 2 make it possible to determine the fatigue limits
and effective stress concentration factors for specimens with stress concentrators in
bending using not only the values of fatigue limits under axial loading, but also the
bending test results.
Relationships (12) and (13) in Table 2 are derived by substituting the ��1
values found from (6) and (7) into relations (10) and (11).
Numerical values of the parameters c1 and c2 were determined from the
formulas for the fatigue limit given in Table 1 provided that the fatigue limits
found experimentally and the calculated ones are equal.
Based on this analysis, it was assumed that for smooth specimens in bending
c1 1 0� . for �
10. and c1 0 7� . for ��10. .
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 4 77
Fatigue Limits of Steels and Stress Gradient
78 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 4
V. T. Troshchenko and L. A. Khamaza
Ò a b l e 1
Calculation Results Based on Some Literature Data.
Fatigue Limits of Steels in the Nonuniform Stressed State
Material,
steel grade,
� �0 2. b
Type of
loading
� ,
mm�1
� ( ) ,��1 e
( ) ,��1
nth
e
MPa
( ) ,��1
nth
c
MPa
( )K e� ( )K c� ( ) ,c e1
( )c e2
( ) ,c c1
( )c c2
1 2 3 4 5 6 7 8 9 10
CSN 12010 [13]
(normalization)
� �0 2 0 634. .b �
T–C 0 1.00 203.0 – 1.000 1.000 – –
0.340 2.18 110.1 105.0 1.844 1.933 0.680 0.466
1.000 2.18 114.7 112.7 1.770 1.801 0.520 0.466
1.000 1.56 173.1 157.6 1.173 1.288 0.770 0.466
1.000 1.95 133.3 126.0 1.523 1.611 0.640 0.466
1.000 2.26 115.0 108.8 1.765 1.866 0.640 0.466
3.040 2.68 104.5 102.0 1.943 1.990 0.520 0.466
7.300 2.75 112.7 111.0 1.801 1.829 0.490 0.466
15.500 2.85 138.6 119.9 1.465 1.693 0.700 0.466
B 2.000 1.00 290.0 286.4 1.430 1.410 0.740 0.700
1.000 1.00 270.0 264.7 1.330 1.300 0.770 0.700
0.500 1.00 270.0 265.2 1.330 1.310 1.100 1.000
0.200 1.00 250.0 244.2 1.230 1.200 1.150 1.000
CSN 12010 [13]
(heat-treated)
� �0 2 0 718. .b �
T–C 0 1.00 290.0 – 1.000 1.000 – –
0.340 2.18 160.6 167.4 1.806 1.732 0.800 1.000
3.040 2.68 151.1 141.8 1.919 2.045 0.540 0.411
7.800 2.75 160.0 154.5 1.813 1.877 0.460 0.411
B 1.000 1.00 360.0 378.1 1.241 1.304 0.540 0.700
40Kh [26]
� �0 2 0 407. .b �
B 0.400 1.00 315.0 – 1.000 1.000 – –
5.400 2.05 180.0 189.1 1.750 1.666 0.535 0.634
1.650 1.32 250.0 251.6 1.260 1.252 0.620 0.634
0.933 1.20 285.0 288.0 1.105 1.094 0.960 1.000
0.667 1.10 295.0 302.1 1.068 1.043 0.900 1.000
0.100 1.00 275.0 – 1.000 1.000 – –
5.100 3.65 97.0 102.4 2.835 2.686 0.525 0.634
1.350 2.05 160.0 154.1 1.719 1.785 0.765 0.634
0.413 1.32 230.0 232.8 1.196 1.181 0.940 1.000
0.050 1.00 255.0 – 1.000 1.000 – –
2.550 3.65 85.0 89.6 3.000 2.846 0.510 0.634
1.300 2.55 113.0 118.7 2.257 2.148 0.500 0.634
0.675 2.05 150.0 151.8 1.700 1.680 0.950 1.000
0.206 1.32 210.0 210.6 1.214 1.211 0.980 1.000
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 4 79
Fatigue Limits of Steels and Stress Gradient
Continued Table 1
1 2 3 4 5 6 7 8 9 10
45 [25]
� �0 2 0 598. .b �
B 0.266 1.000 309.0 – 1.000 1.000 – –
0.683 1.207 256.0 280.4 1.210 1.102 0.640 1.000
1.330 1.444 214.0 212.3 1.480 1.455 0.510 0.491
2.647 1.698 182.0 175.3 1.920 1.763 0.575 0.491
3.599 1.931 160.0 159.3 2.190 1.940 0.500 0.491
5.266 2.146 144.0 142.4 2.570 2.170 0.510 0.491
7.673 2.414 128.0 127.7 3.020 2.420 0.495 0.491
10.260 2.452 126.0 115.4 3.490 2.678 0.650 0.491
0.133 1.000 286.0 – 1.000 1.000 – –
0.667 1.355 211.0 223.0 1.480 1.283 0.770 1.000
1.331 1.713 167.0 159.6 1.920 1.792 0.620 0.491
2.133 2.000 143.0 136.0 2.360 2.103 0.620 0.491
2.633 2.270 126.0 127.7 2.570 2.240 0.460 0.491
3.467 2.383 120.0 117.6 2.880 2.432 0.540 0.491
5.133 2.698 106.0 104.7 3.400 2.732 0.52 0.491
8.133 3.147 91.0 88.2 4.300 3.243 0.550 0.491
35 [26]
� �0 2 0 47. .b �
B 0.400 1.000 245.0 – 1.000 1.000 – –
1.650 1.320 185.0 192.2 1.324 1.275 0.485 0.585
0.100 1.000 225.0 – 1.000 1.000 – –
20.100 6.300 60.0 59.3 3.750 3.794 0.610 0.585
2.600 2.550 103.0 107.2 2.184 2.099 0.500 0.585
1.350 2.050 120.0 124.0 1.875 1.815 0.500 0.585
0.050 1.000 205.0 – 1.000 1.000 – –
1.300 2.550 90.0 93.8 2.278 2.186 0.470 0.585
0.206 1.320 160.0 169.3 1.281 1.211 0.700 1.000
38Kh2N2MA
[26]
� �0 2 0 5. .b �
B 0.400 1.000 320.0 – 1.000 1.000 – –
20.400 3.650 120.0 129.1 2.667 2.479 0.460 0.562
10.400 2.550 155.0 164.8 2.065 1.942 0.460 0.562
5.400 2.050 190.0 185.6 1.684 1.724 0.610 0.562
0.100 1.000 290.0 – 1.000 1.000 – –
5.100 3.650 110.0 104.4 2.636 2.778 0.680 0.562
2.600 2.550 140.0 136.9 2.071 2.118 0.620 0.562
1.350 2.050 170.0 158.6 1.706 1.828 0.780 0.562
0.050 1.000 265.0 – 1.000 1.000 – –
2.550 3.650 95.0 90.4 2.789 2.931 0.685 0.562
1.300 2.550 130.0 120.4 2.038 2.201 0.800 0.562
0.675 2.050 165.0 157.8 1.606 1.679 1.200 1.000
Note. “T–C” – tension–compression, “B” – bending; subscripts e and c correspond to the
experimental and calculated values, respectively.
The value of c2 is defined by relation
c b b2 0 2 0 2
21 0 25� � �� � � �. .. ( ) , (14)
where � 0 2. and � b are the 0.2 offset yield stress and the ultimate strength of the
material under study, respectively.
Figure 1 presents the values of c2 as a function of the ratio � �0 2. b for the
materials under consideration for different test types.
As follows from the above, the c2-values which determine the dependence of
the fatigue limit on the stress gradient for specimens with stress concentrators, are
significantly lower than the c1-values, which determine the dependence of the
fatigue limit on the stress gradient for smooth specimens in bending.
The main difference of the relations in Table 2 from those given in the
literature is that the function f ( ),� which defines the dependence of the ��1
*
values on the stress gradient, is not the same for smooth specimens and those with
stress concentrators. This difference is defined by the difference in the parameters
c1 and c2 .
Figure 2 illustrates the dependences f ( )� �� shown as solid lines for
smooth specimens in bending (1) and specimens with stress concentrators (2) that
are calculated from the formulas in Table 2, together with the experimental data for
different steels, for which the investigation results are presented as points.
80 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 4
V. T. Troshchenko and L. A. Khamaza
Ò a b l e 2
Relationships for Determining the Fatigue Limits of Steels in the Nonuniform Stressed State
Test conditions f ( )� Fatigue limit Effective stress
concentration factor
Bending of smooth
specimens
1 1� c � (3) ( )�� �1 b
� ��� �1 11 c (6)
� � ��
�
K
b
�
�
�
( )1
1
� �1 1c � (7)
Tension–compression
of specimens with a
stress concentrator
1 2� c � (4) �� �1
nth
�
�
��
�
�
1
21 c
(8)
K
nth�
�
�
� ��
�
1
1
�
�
�
�
1 2c
(9)
Bending of specimens
with a stress concentrator
(calculation of the
fatigue limit of smooth
specimens in tension–
compression)
1 2� c � (4) ( )�� �1 b
nth
�
�
��
�
�
1
21 c
(10)
K
b
nth�
�
�
� ��
�
1
1( )
�
�
�
�
1 2c
(11)
Bending of specimens
with a stress concentrator
(calculation of the
fatigue limit of smooth
specimens in bending)
1
1
2
1
�
�
c
c
�
�
(5)
( )�� �1 b
nth
�
�
�
�( )�
�
��
1
2
1
1
1
b
c
c
(12)
K
b
b
nth�
�
�
� ��
�
( )
( )
1
1
�
�
�
�
�
� 1
1
1
2
c
c
(13)
The presented results show that the calculated dependences f ( )� �� for the
same material (which is used in smooth specimens in bending and specimens with
stress concentrators) differ appreciably. Experimental points obtained in the above-
mentioned studies correspond to the calculated dependences.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 4 81
Fatigue Limits of Steels and Stress Gradient
Fig. 1 Dependence of experimental values of the coefficient c2 on the magnitude of the ratio � �0 2. b
for the materials under investigation: (1) experimental points for specimens with stress concentrators
in tension–compression; (2) experimental points for specimens with stress concentrators in bending;
(3) approximating curve according to relationship (14).
Fig. 2. Calculated dependences f ( )� �� for smooth specimens in bending (1) and specimens with
stress concentrators (2); experimental data (I, II) for different steels: (à) steel CSN 12 010 (�� �1 203
ÌPà, � �0 2. b � 0.634) [13]; (b) steel 40Kh [( )�� �1 b 255; 275; and 315 ÌPà, � �0 2. b � 0.407]
[26]; (c) steel 35 [( )�� �1 b 205; 225; and 245 ÌPà, � �0 2. b � 0.47] [26]; (d) steel 45 [( )�� �1 b 286;
309 ÌPà, � �0 2. b � 0.598] [25].
a b
c d
It also follows from the plots given in Fig. 2 that the rate of increase in the
function f ( )� with an increase in � is higher for smooth specimens than for
specimens with stress concentrators, which is confirmed by the experimental data.
The analysis of the relationships presented in Table 2 shows that, according to
them, the effective stress concentration factor increases with the ratio � �0 2. b .
Figure 3 presents the calculated dependences K b� � �� 0 2. for the stress
concentrator in bending shown in this figure for various values of the theoretical
stress concentration factor and their corresponding relative stress gradients. It
follows from this figure that the K � -value increases with the ratio � �0 2. b .
This nature of the K � versus � �0 2. b dependence was observed in many
experimental investigations [33, 34].
Figure 4 shows a comparison between the experimental and calculated
values of the fatigue limits in the nonuniform stressed state determined by formulas
(6), (8), (10), and (12). It is seen that the experimental and calculated values of the
fatigue limits correspond to a single correlation relationship with a high correlation
coefficient R � 0.983 to 0.997.
The fact that the above dependences describe equally well the results obtained
in bending and tensile tests for flat specimens with stress concentrators when a
linear stressed state takes place at their tips, and for cylindrical specimens with an
annular groove in which a plane stressed state takes place at the concentrator tip
can be explained by that, as was shown in [35], the equivalent stresses in these
concentrators, according to the von Mises strength theory describing steel test
results fairly well, do not much differ from the maximum normal stresses.
The results shown in Fig. 4 testify that the proposed relationships (6), (8),
(10), and (12) are in good agreement with the experimental data and can be directly
applied in the practice provided that the type of loading, values of the relative
stress gradients and theoretical stress concentration factors, and certain mechanical
properties of the material are known.
Conclusions. The experimental relationships between the fatigue limits of
steels and effective stress concentration factors versus the stress gradient have been
proposed and validated.
82 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 4
V. T. Troshchenko and L. A. Khamaza
Fig. 3. Calculated dependences K b�� �0 2. in bending for different values of �� and �.
In contrast to similar relationships available in the literature, the nature of the
dependence of the fatigue limit on the stress gradient is assumed to be different for
smooth specimens in bending and specimens with stress concentrators. A method
for determining the parameters of the proposed relationships that takes into account
the mechanical properties of steels under study has been validated. Good agreement
has been shown between the proposed relationships and the experimental data.
Ð å ç þ ì å
Íà îñíîâ³ óçàãàëüíåííÿ ðåçóëüòàò³â ÷èñëåííèõ åêñïåðèìåíòàëüíèõ äîñë³ä-
æåíü çàïðîïîíîâàíî ³ îá´ðóíòîâàíî åìï³ðè÷í³ ñï³ââ³äíîøåííÿ ì³æ ãðàíèöÿ-
ìè âèòðèâàëîñò³ ñòàëåé ³ ãðà䳺íòîì íàïðóæåíü, ó ÿêèõ âðàõîâóºòüñÿ íàêîïè-
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 4 83
Fatigue Limits of Steels and Stress Gradient
a b
c
Fig. 4. Comparison of the experimental and calculated values of the fatigue limits determined by
formulas (6), (8), (10), and (12): (à) smooth specimens in bending; (b) specimens with stress
concentrators in tension; (c) specimens with stress concentrators in bending.
÷åííÿ âòîìíîãî ïîøêîäæåííÿ â ãëàäêèõ çðàçêàõ òà çðàçêàõ ³ç êîíöåíòðàòîðîì
íàïðóæåíü. Çàïðîïîíîâàíî ìåòîä ðîçðàõóíêó ïàðàìåòð³â öèõ ñï³ââ³äíîøåíü ç
óðàõóâàííÿì ìåõàí³÷íèõ âëàñòèâîñòåé ñòàëåé. Îòðèìàíî õîðîøó çá³æí³ñòü
ì³æ çàïðîïîíîâàíèìè ñï³ââ³äíîøåííÿìè é åêñïåðèìåíòàëüíèìè ðåçóëüòàòàìè.
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Received 04. 04. 2011
86 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 4
V. T. Troshchenko and L. A. Khamaza
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/KOR <FEFFc7740020c124c815c7440020c0acc6a9d558c5ec0020b370c2a4d06cd0d10020d504b9b0d1300020bc0f0020ad50c815ae30c5d0c11c0020ace0d488c9c8b85c0020c778c1c4d560002000410064006f0062006500200050004400460020bb38c11cb97c0020c791c131d569b2c8b2e4002e0020c774b807ac8c0020c791c131b41c00200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000410064006f00620065002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002e>
/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken voor kwaliteitsafdrukken op desktopprinters en proofers. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
/NOR <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>
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/ENU (Use these settings to create Adobe PDF documents for quality printing on desktop printers and proofers. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /NoConversion
/DestinationProfileName ()
/DestinationProfileSelector /NA
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure true
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /NA
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|