LOAD SEQUENCE INFLUENCE ON LOW CYCLE FATIGUE · PDF fileLoad sequence influence on low cycle...

193Load sequence influence on low cycle fatigue life

TECHNICAL SCIENCESAbbrev.: Techn. Sc., No 8, Y. 2005

LOAD SEQUENCE INFLUENCE ON LOWCYCLE FATIGUE LIFE

Sylwester K³yszFaculty of Technical Sciences

University of Warmia and Mazury in Olsztyn

K e y w o r d s: low cycle fatigue (LCF) tests, durability, load history.

A b s t r a c t

The study reviews basic issues relating to low cycle fatigue of metals and methods ofdescribing the results of tests in this respect. Particularly examined are relationships be-tween fatigue life and the strain range applied in tests, material strength properties, size ofthe hysteresis loop as well as the load sequence. Algorithms are presented to calculate thearea of cyclic hysteresis loop registered in low cycle fatigue tests. The results of research arean element of expanding knowledge on construction element durability estimates.

WP£YW OBCI¥¯EÑ NA PRZEBIEG NISKOCYKLOWEJTRWA£O�CI ZMÊCZENIOWEJ

Sylwester K³yszKatedra Materia³ów Funkcjonalnych i Nanotechnologii

Uniwersytet Warmiñsko-Mazurski w Olsztynie

S ³ owa k lu c zowe: badania zmêczeniowe niskocyklowe (LCF), trwa³o�æ, historia obci¹¿eñ.

S t r e s z c z e n i e

Zaprezentowano przegl¹d podstawowych zagadnieñ niskocyklowego zmêczenia metalii metod opisu wyników badañ w tym zakresie. W szczególno�ci rozpatrzono zwi¹zki miêdzytrwa³o�ci¹ zmêczeniow¹ a zakresem odkszta³ceñ zadanych w testach, w³a�ciwo�ciami wy-trzyma³o�ciowymi materia³u, wielko�ci¹ pêtli histerezy oraz sekwencj¹ obci¹¿eñ. Przedsta-wiono algorytm do obliczania pól powierzchni pêtli histerezy cyklicznej rejestrowanychw próbach niskocyklicznego zmêczenia. Wyniki badañ s¹ elementem poszerzania wiedzyz zakresu szacowania trwa³o�ci elementów konstrukcji.

194 Sylwester K³ysz

Introduction

Knowledge about the fatigue characteristics of construction materials,among others such as durability under low cycle fatigue conditions, is signi-ficant with respect to estimating the life of construction elements at thestructure design stage, during operation as well as in analysis of construc-tion overhaul life assessment and possibilities of its extension (MISHNAEVSKY

1997, FUCHS et al. 1980, KOCAÑDA, SZALA 1991, Problemy badañ... 1993). Thehysteresis loop of material subject to cyclic loads includes valuable informa-tion on the details on the cyclic behavior of material and its resistance tofatigue. The shape of the hysteresis loop registered during low cycle fatiguetests and its characteristic sizes when stabilized depend on the type of ma-terial and load conditions � e.g. its width at a stress level of zero is equal tothe plastic strain range Depl, which determines durability in a low cyclefatigue tests.

Review of relationships describingthe low cycle fatigue of metals

The basic equation describing the behavior of metals with respect tolow cycle fatigue is an experimental relationship formulated by Manson andCoffin (KOCAÑDA et al. 1989) associating the number of cycles to destructionNf with the plastic strain range Deapl:

CN aplkf =De (1)

where:k, C � material constants.For low-carbon and low-alloyed steel as well as stainless austenitic steel

of strength amounting to Rm < 700 MPa, the exponent k @ 0.5.The constant C, characterizing the degree of steel plasticity, is determi-

ned by the following relationship:

ZeC rz −

==100

10021

21

ln (2)

where:erz � real strain,Z � reduction of area at static fracture, expressed in %.The widest application in this area is the Morrow's formula (KOCAÑDA et

al. 1989):

( ) ( )cff

bf

faplasac NN

E22 '

'

εσ

εεε +=+= (3)


where:s'f and b � fatigue strength coefficient and exponent;e'f and c � cyclic plastic strain coefficient and exponent.A similar composition to formula (3) is the equation proposed by Man-

son, based on data from a static tension test, in which the assumed fixedexponent is b = -0.12 and c = -0.6:

( ) ( ) 6012053 ...

−− +=∆+∆=∆ frzfm

aplasac NNE

Rεεεε (4)

A simplification of formula (3) useful for engineering calculations is theLanger's formula (KOCAÑDA et al. 1989):

( ) E

Z

ZN fasaplac

150 100

100

4

1 −+−

=+= ln.

εεε (5)

The first term of equation (5) is derived from the second term of equ-ation (3), in which Langer assumed constants e'f = 0.35e'rz and c = -0.5.

The low correlation between the stress amplitude sa and the number ofcycles Nf inclined Langer to replace sa with the fatigue limit Z-1 for thesymmetrical cycle. According to conducted research (MACHUTOV 1981), the va-lue of Z-1 may be determined by the following relationship:

mRZ ⋅=− γ1 (6)where:

g � steel characteristic.For Rm £ 700 MPa, coefficient g = 0.4¸0.55 g = 0.4 is generally applied.The study (MACHUTOV 1981) proposes the following formula to calculate

the entire strain:

( ) sf

u

fasaplac k

NE

R

ZN⋅

⋅+

−⋅

⋅=+= 4350

100100

4

150

.ln.

εεε (7)

where:Ru � fracture stress,ks = 0.09¸0.12. ks = 0.1 is generally applied.Formulas (3), (5) and (7) refer to the symmetric cycle at fixed load.MACHUTOV (1981) proposed � for nonsymmetrical cycles, for which the

plastic strain amplitude is lower than in symmetrical cycles � applying thefollowing average plastic strain amplitude in formula (5):

ε

εεεR

Raplrapl −

+=

11

�, (8)


max

min

εε

ε =R (9)

where:emin, emax � minimum and maximum cycle strain.The decreasing of the fatigue limit in nonsymmetrical load cycles is

taken into account by introducing into the second term of equation (5) � thefollowing relationship:

( )

ε

εε

RR

RZ

Rf

m −+

⋅+=

−11

1

1

1 (10)

The Langer's formula (5), after taking into account relationships (8) and(10), adopts the following form in nonsymmetrical cycles:

( )

−+

⋅++

−⋅

−+

+=

−

−

ε

εε

εε

RR

RZ

E

Z

ZRR

Nm

f

ac

11

1100

100

11

4

1

1

1

50ln

. (11)

Formula (11) may be presented as the following in a general entry:

( ) ( ) ( )εεεε RfE

ZNRfNf ffrzac ⋅+⋅⋅= −1

41

, (12)

where:

( )

−+

⋅+

=− 50

11

2501

1

..

,

f

f

NRR

NRf

ε

εε

(13)

According to equation (13), the influence of the cycle's asymmetry Re onthe amplitude of the elasto-plastic strain within the range of number ofcycles N = 5×102 to N = 5×105 is not large.

In recommendations (MACHUTOV et al. 1987), the following should be en-tered instead of Z-1 in formula (11): for steel of Rm £ 1200 MPa for Z £ 30% �Zx = Z-1 while for steel of Z > 30% � Zx = Z-1/2+15.


Analysis of shape and properties of the cyclichysteresis loop

Macroscopic changes occurring in the metal during cyclic loads are pre-sented in the shape of a cyclic hysteresis loop, for this reason the descrip-tion of the loop and cyclic strain curve is an issue of fundamental meaningupon analyzing low cycle fatigue.

The cyclic strain curve, generated by combining peaks of stabilized hy-steresis loop respective for various strain ranges, is determined by the Ram-berg � Osgood formula:

'

'

n

KE

1

+= σσε (14)

where:K' and n' � respectively cyclic strength coefficient and cyclic strengthe-

ning exponent.

The cyclic strengthening exponent n' for steel of 8020 .. ≤mR

R may be

adopted, according to (MACHUTOV et al. 1987), as equal to the strengtheningexponent at static tension, in accordance with the following formula:

⋅+

−

⋅=

−220

20

1020

100100

750

.

lnlg

lg

.

.

.

ER

Z

RR

n

u

(15)

Masing's description the hysteresis loop assumes that loop branches aredescribed based on the static tension curve drawn on a 2:1 scale. The cyclicstrain curve was used in other propositions for this purpose � moving thepeaks of stabilized loops in tension or compression halfcycles to a commonpoint. The equation describing the Masing's curve is similar to formula (14),taking into account scale transformation:

n

KE′

′∆+∆=∆

1

22

σσε (16)

as such the ascending and descending hysteresis loop branches are as fol-lows:


nrrr KE

′

′

−+

−=−

1

22

σσσσεε (17)

nrrr KE

′

′

−+

−=−

1

22

σσσσεε (18)

Subsequent load recurrences in Masing's model are executed in accor-dance with these relations, while the material remembers the coordinates ofthe beginning of the current and previous recurrences, whose loops are out-side the currently executed hysteresis loop. When the strain and stress inthe current recurrence reach a value at which level the recurrence of theprevious began, the current hysteresis loop closes and further strain takesplace according to the same relation as prior to commencing the presentlyclosed loop.

The analysis of influence of material property changes under conditionsof cyclic loads and variable sequence loads (including overloads) on the fati-gue life and shape of the hysteresis loop, are also a significant factor takeninto account upon estimating the durability of structural components, andare subject of many presentations at conferences (e.g. KALETA 1996, LEE etal. 1987, MROZIÑSKI 1998). However, most frequently described in professio-nal literature studies on the influence of load history on the fatigue charac-teristics, relate to fatigue crack propagation and less frequently to low cycleresearch. An extensive analysis of fatigue properties in combination withchanges in the shape of the hysteresis loop under various testing and opera-ting conditions is presented in monographs (KOCAÑDA et al. 1989, GOSS 1982,GOSS 2004). The results of the above-mentioned analyses and the low cyclecharacteristics of materials as well as load parameters are an essential ele-ment of methodology for assessing the durability of construction elements(SOBCZYKIEWICZ 1983) � mainly with respect to methods of summarizing dama-ges, as well as when disregarding these methods (GASSNER et al. 1961, SZALA

1980), e.g. by comparing the fatigue characteristics determined under con-stant amplitude and random load. They have already made their mark ininternational standards relating to supervision and control over the techni-cal state of construction elements (RTO/AGARD 1999).

The study (K£YSZ 2000) presents the results of low cycle fatigue tests on18G2A and St3SY steel under De = const load conditions applied in variousconfigurations (variable load sequence or with overload cycles � tension orcompression). The nature of the changes was assessed with respect to theshape of the hysteresis loop compared to typical courses.

In the case of a standard test for the strain range De = const, the regi-stered hysteresis loops in selected load cycles is presented in Fig. 1, while


the change in load amplitude (minimum and maximum stresses) in subsequ-ent cycles is presented in Fig. 2. Typical characteristics are visible for thetested material in this type of test:� symmetrical changes in load amplitudes (minimum and maximum values),� the hardening/softening of material in subsequent load cycles (in this case

the softening of material in the first few hundred load cycles, followed by thehardening of material in the subsequent thousands of cycles),

� the occurrence of minimum amplitude value after a few hundred cycles(a certain type of stabilization) as well as its sudden drop immediately priorto specimen destruction,

� clear deformation of the hysteresis loop's shape in cycles immediately priorto specimen destruction.

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

600

500

400

300

200

1000

-100

-200

-300

-400

-500

-600

s(M

Pa)

e (%)

Fig. 1. Course of a hysteresis loop in selected cycles for a specimen tested in a fixed strainrange e = ±0.25% � steel 18G2A

600

500

400

300

200

100

0

-100

-200

-300

-400

-500

-600

s(M

Pa)

10 100 1000

N (cycle)

10 000

Fig. 2. Course of maximum and minimum stress values in subsequent hysteresisloops from Fig. 1 � steel 18G2A


Such processes are typical for metals. Only the proportions betweenparticular curve fragments are subject to change, depending on the range ofstrain that the specimens are subject to, i.e. on their durability to destruc-tion.

The following tests were applied to test the behavior of material underlow cycle fatigue conditions in various load sequences (strain ranges):

test A � load with respect to strain ±e=const on subsequent levels ofstrain (e.g. ±0.10%, ±0.13%, ±0.16%, ±0.19%, ±0.22%, ±0.25%,±0.28%, ±0.31%, ±0.34%, ±0.37%, ±0.40%) for a given numberof cycles on each level,

test B � single overload in the first load cycle and further as in test A,test C � single underload in the first load cycle and further as in test A,test D � load with respect to strain range ±e = const on subsequent le-

vels of strain in reverse order to test A (i.e. ±0.40%, ±0.37%,±0.34%, ... etc.) for a given number of cycles on each level.

Examples of hysteresis loop courses at subsequent strain levels of test Aare presented in Fig. 3. A clear change in stress amplitude in subsequentcycles is visible for the first smallest strain level (0.1%). The plotted loops forsubsequent levels of strain reflect the beginning and end of each test stage.In these cases, changes in stress amplitudes in hysteresis loops are insignifi-cant. Very well reflected test conditions, symmetry and regularity are visiblein the generated hysteresis loops at all strain levels � it is a classic result forthis type of tests. The gradual deformation of the hysteresis loop's shape isalso visible at the final stage of the test (for strain level 0.4%).

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

600

500

400

300

200

1000

-100

-200

-300

-400

-500

-600

s(M

Pa)

e (%)

-

-

Fig. 3. Course of a hysteresis loop at subsequent stages of test A (at the beginning and endof each stage) � steel 18G2A


Figure 4 presents a change in stress value (minimum and maximum)respective for two identical A tests. The symmetry and regularity generatedin the results is typical also in this case. At subsequent stages, the stressrange increases for increasing strain ranges. The loading force decreased inthe first three stages (at strain levels of 0.10%, 0.13%, 0.16%) � the mate-rial softened during the cycles. In subsequent stages of tests (for strain ran-ges from 0.19%) the force would generally increase (except, possibly, a veryshort period at the beginning of each stage) � the material would harden.Similarly with regard to minimum forces. It may be stated that the mate-rial responds to the load applied in a typical manner.

600

500

400

300

200

100

0

-100

-200

-300

-400

-500

-600

s(M

Pa)

10 100 1000

N (cycle)

10 000

sample 56G

sample 22G

sample 22G

sample 56G

Fig. 4. Change in minimum and maximum stress values for a specimentested according to test A � steel 18G2A

The hysteresis loops for specimens tested in B and C tests are presen-ted in Fig. 5a (specimen in the first overload cycle) and 5b (specimen in thefirst underload cycle). The changes in stress amplitudes registered in boththese tests are presented in Fig. 6. As shown, the applied overload or un-derload cycle introduced asymmetry to the generated results compared withthe results of test A (Fig. 3) � in the first cycles for the lowest strain rangeas well as for the remaining stages of the test. In these cases, changes instress amplitudes in subsequent load cycles clearly pertain to one (maxi-mum or minimum value), while the other one changes much less.

A decrease in maximum loads with a nearly stable minimum load levelwas observed on lower levels of strain for a specimen initially overloaded,while specimens initially underloaded experienced a decrease in minimumloads � with a nearly stable maximum load level.

In the case of the specimen initially overloaded, the maximum stresslevel in the first 4 stages of the test was higher than in the case of the


specimen in test A, which resulted in durability reduction, despite the factthat the stress range was greater for the specimen in test A. With regardto the specimen initially underloaded (test C), the minimum stress level(compression) was higher throughout the entire test than in test A, whilemaximum stress levels were initially significantly lower than in test A, butin subsequent stages they quickly increased and exceeded those in test A.Ultimately the durability of the specimen in test C was near the durabilityof the specimen in test A.

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

600

500

400

300

200

1000

-100

-200

-300

-400

-500

-600

s(M

Pa)

e (%)-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

600

500

400

300

200

1000

-100

-200

-300

-400

-500

-600

s(M

Pa)

e (%)

Fig. 5a. Course of a hysteresis loopat subsequent stages of test B

(at the beginning and end of each stage)� steel 18G2A

Fig. 5b. Course of a hysteresis loopat subsequent stages of test B

(at the beginning and end of each stage)� steel 18G2A

600

500

400

300

200

100

0

-100

-200

-300

-400

-500

-600

s(M

Pa)

10 100 1000

N (cycle)

10 000

test Atest B

test C

test C

test A

test B

Fig. 6. Change in minimum and maximum stress values for specimens tested accordingto tests A, B and C � steel 18G2A


The course of registered hysteresis loops examined in test D is presen-ted in Fig. 7. In subsequent stages of the test, despite a decreasing strainrange, the stress level remained at a nearly unchanged level (Fig. 8) (thisconfirmed the occurrence of material memory effect) and did not demon-strate characteristics that were typical of test A. As a result, the durabilityof the specimen decreased compared to the specimen from test A and thetest was not performed on all strain levels such as it was performed in testA � the test was completed at strain level ±0.22%.

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

600

500

400

300

200

1000

-100

-200

-300

-400

-500

-600

s(M

Pa)

e (%)

600

500

400

300

200

100

0

-100

-200

-300

-400

-500

-600

s(M

Pa)

10 100 1000

N (cycle)

10 000

Fig. 7. Course of a hysteresis loopat subsequent stages of test D (at the

beginning and end of each stage)� steel 18G2A

Fig. 8. Change in minimum and maximumstress values for a specimen testedaccording to test D � steel 18G2A

Algorithm to cyclic hysteresis loop area calculation

Another parameter associated with low cycle tests, which includes infor-mation on the process of material fatigue as well as the amount of energyrequired until destruction, is the hysteresis loop area. Strains at a microlevel are irreversible plastic deformations, which is connected with energydissipation, believed to be the main factor resulting in material damage andformation of fatigue micro cracks. The basis of most energy criteria appliedto describe fatigue life as well as the cumulative fatigue damage hypothesisis the assumption that plastic deformation energy absorbed by the materialvolume unit during one load cycle is equal to the hysteresis loop area (KO-CAÑDA et al. 1989, GOSS 1982, POLÁK 1991, KUJAWIÑSKI 1991). The analysismethodology within this respect as well as the results of calculations for18G2A and St3SY steel are presented below.

Every low cycle fatigue test registers n hysteresis loops, each of a speci-fied number of load cycles Nj. They are registered in the form of points (ei,si), where i=1,2,...,k-1, with k as the number of measuring points of a givenhysteresis loop (Fig. 9).


If the trapezoid method is adopted to calculate the areas of hysteresisloops, fragments of the loop's area included between two experimental po-ints (ei, si), (ei+1, si+1) and the axis of coordinate system, are determined bythe following formula:

( ) ( )iiiii absabsP εεσσ −⋅+⋅= ++ 1150. (19)

On the basis of the analysis of a typical hysteresis loop shape (Fig. 9) indomains I�IV, in which fields Pi are added to (domains II and IV) or deduc-ted from (domains I and III) the cumulative field P of the entire hysteresisloop, the final formula for the entire hysteresis loop area may be written asfollows:

∑−

=++ ⋅+⋅⋅−=

1

111 50

k

iiiiii PP ))(.sgn()sgn( σσεε (20)

Typical hysteresis loop coordinate properties were taken into account(the '+' and '-' symbols in the Table under the Figure signify that the statedvalues are positive or negative in a given domain, while the field Pi isadded or subtracted upon summing domains under/over hysteresis loopcurves).

e

s

II

III

I

IV

sisi+1

ei ei+1

Fig. 9. Diagram of a hysteresis loop to calculated area

niamoD

eulaVI II III VI

ei 1+ � ei + + � �

5.0 ×(σi 1+ + si) � + + �

aerA � + � +


Fig. 10 presents examples of changes in the hysteresis loop areas insubsequent load cycles for selected specimens in various strain rangesDe = const, with the load ratio R = -1. The hysteresis loop areas do not chan-ge considerably for a significant part of the test, but the size of these chan-ges increase along with an increase in strain range (size of hysteresis loop).More significant changes in hysteresis loop areas � mainly their decline,occur at the final stage of tests.

10 100 1000

N (cycle)

10 0001 100 000

20

15

10

5

0

loop's

are

a(M

J/m

)3

18G2A

St3SY

Fig. 10. Change in the hysteresis loop area in subsequent load cycles for selected specimensof St3SY and 18G2A steel tested in various strain ranges and R = �1

Similar changes in the sum of hysteresis loop areas as a function of thenumber of cycles to destruction is presented in Fig. 11 (in two coordinatesystems). Curves with similar courses were obtained for particular strainranges.

Table 1 presents the results of calculations relating to the size of hyste-resis loop areas P (for the 2nd and 3rd cycle reflecting the mid-life values ofa given specimen and the last prior to specimen destruction) as well as thesum of hysteresis loop areas SP for the specimens tested in the study. Fi-gure 12 presents relationships between the determined sizes of hysteresisloop areas P and fatigue until destruction Nf of the specimens.

Comparing product hysteresis loop areas P from Table 1 and the num-ber of cycles to destruction Nf for particular specimens with a determinedarea sum SP, it is possible to assess the consistency of both sizes, whichwould signify that the area of a given hysteresis may be a parameter ena-bling the estimation of the final fatigue life of the specimen. Table 2 pre-sents the values of the relative error of such estimation, determined asd = Nf × P/SP.


1elbaT

saerapoolsiseretsyhfoeziS iP selcycdaoldetcelesni

nemicepS Nfselcyc

Pelcycdn2

m/JM( 3)

Pelcycdr3

m/JM( 3)

P2/1 Nf elcyc

m/JM( 3)

Pelcyclanif

m/JM( 3)

SPm/JM( 3)

G9 233 16.21 84.31 73.31 23.9 40.8734

G21 863 02.51 22.61 58.51 39.9 92.3375

G01 583 09.51 30.61 05.51 58.11 38.0395

G8 164 42.61 89.51 44.51 04.11 79.7607

G42 9011 04.6 40.7 48.6 - 28.8847

G52 9311 69.6 02.7 58.6 94.4 85.1677

G12 2935 78.2 08.2 55.2 - 07.24631

G1 1355 46.3 77.2 93.2 34.2 39.15231

G5 3755 21.2 36.2 54.2 26.1 92.84531

G92 8656 44.3 96.2 15.2 - 92.00561

G62 15111 15.1 74.1 92.1 50.0 58.32241

G6 95741 03.1 54.2 32.1 51.0 57.71081

G31 30071 10.2 27.1 61.1 - 64.10391

10 100 1000

N (cycle)100001 100000

sum

of

are

a(M

J/m

)3

18G2A St3SY

10 000

1 000

100

10

1

sum

of

are

a(M

J/m

)3 20 000

15 000

10 000

5 000

110 100 1000 100001 100000

N (cycle)

25 000

De=1%

De

De

De

=0.5%

=0.25%

=0.167%

De=1%

De

De

De

=0.5%

=0.25%

=0.167%

18G2A St3SYDe=1%

De

De

De

=0.5%

=0.25%

=0.167%

De=1%

De

De

De

=0.5%

=0.25%

=0.167%

Fig. 11. Change in the sum of hysteresis loop areas for selected specimens of St3SY and18G2A steel tested in various strain ranges and R = �1

The best estimation is obtained for hysteresis loop areas correspondingto the mid-life cycle of specimens � the error does not exceed 2.5%. Theestimation error may be significant in the remaining cases. The conclusionfrom these comparisons would be optimistic if not for the fact that the know-ledge of the hysteresis loop area in a cycle reflecting specimen mid-life valu-es is impossible to achieve until its durability is known. It would be morebeneficial to be able to estimate the final durability of the specimen on the


basis of, for example, hysteresis loops from the 2nd or 3rd load cycles, sincethis would indicate a significant decrease in testing time and lower costs. Thepresented relationships are material characteristics of a given material typeand a given a load type � analogous to, for example, the Morrow's curve.

loop's

are

a(M

J/m

)3

10 000 100000N (cycle)

18G2A – 2 loop

Y=-8.4523*log(N)+35.4598

18G2A – 3 loop

18G2A – middle loop

18G2A – final loop

Y=-8.7700*log(N)+36.7448

Y=-8.8776*log(N)+36.7199

Y=-6.8541*log(N)+27.8534

18

15

12

9

6

3

0

Fig. 12. Relationship P = f(Nf) between the size of the hysteresis loop area P and durabilityuntil specimen destruction Nf

2elbaT

rorreetamitseevitalerfoeziS d

nemicepSd

elcycdn2d

elcycdr3d

2/1 Nf elcycd

elcyclanif

G9 69.0 20.1 10.1 17.0

G21 89.0 40.1 20.1 46.0

G01 30.1 40.1 10.1 77.0

G8 60.1 40.1 10.1 47.0

G42 59.0 40.1 10.1 -

G52 20.1 60.1 10.1 66.0

G12 31.1 11.1 10.1 -

G1 25.1 61.1 00.1 10.1

G5 78.0 80.1 10.1 76.0

G92 73.1 70.1 00.1 -

G62 81.1 51.1 10.1 40.0

G6 60.1 10.2 10.1 21.0

G31 77.1 25.1 20.1 -


Fig. 13 presents the correlation between the area sum SP as a functionof the number of cycles to destruction Nf for particular specimens of bothtested steels, along with respective regression equations.

sum

of

are

a(M

J/m

)3

20 000

15 000

10 000

5 000

1

10000 100000N (cycle)

25 000

18G2A

St3SYY=7574.03*log(N)–14294.2

Y=8427.12*log(N)–17134.7

Fig. 13. Relationship of the area sum SP as a function of the number of cycles until destruc-tion Nf for particular specimens and steel

Conclusions

The problem of metal low cycle fatigue is mathematically widely descri-bed with relationships connecting basic material characteristics and fatiguetest parameters. They may be useful to apply in practical fatigue analysis aswell as be a basis for comparing the properties of various materials.

Changes in the shape and size of hysteresis loop areas registered duringlow cycle fatigue tests indicate a wide range of regularities that characterizetest conditions and include information on the destruction process. Hystere-sis loop areas, or their sum in subsequent load cycles during fatigue tests,may be correlated with the durability of specimens tested to destruction.

The sum of hysteresis loop areas is a critical parameter in analyzingthe fatigue life of tested material. The presumption based on the size ofhysteresis loops reflecting cycles equal to mid-life values seems to generatethe smallest error. Adopting hysteresis loop area sizes for analysis from the2nd or 3rd load cycle may generate durability estimations with accuracy ofaround 30�50%, but exceptions to this rule should be taken into account,which compared with the significant benefits of this approach may be worthconsidering.

An important fact resulting from the presented relationships betweenthe sum of hysteresis loop areas and the number of cycles is that theyadopt values from a range of 4 orders of magnitude, while the correlation


between strain range and the number of cycles on Manson-Coffin graphshave a range of 2 orders of magnitude, signifying a 100-fold higher sensitivi-ty to the benefit of the former. Greater precision of all estimates based onthe analysis of hysteresis loop areas should be expected.

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Translated by Aleksandra Poprawska Accepted for print 2005.04.29


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