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Energy-based method of fatigue damage cumulation
Accepted Manuscript Energy-based method of fatigue damage cumulation Stanisław Mroziński PII: DOI: Reference:
S0142-1123(18)30680-7 https://doi.org/10.1016/j.ijfatigue.2018.12.008 JIJF 4922
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
16 October 2018 4 December 2018 5 December 2018
Please cite this article as: Mroziński, S., Energy-based method of fatigue damage cumulation, International Journal of Fatigue (2018), doi: https://doi.org/10.1016/j.ijfatigue.2018.12.008
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Energy-based method of fatigue damage cumulation Stanisław MROZIŃSKI * * University of Science and Technology, Faculty of Mechanical Engineering, Bydgoszcz, Poland, e-mail: [email protected] Abstract The study presents the results of low-cycle fatigue tests on AW-2024 aluminium alloy specimens under constant amplitude and multistage loading conditions. As expected, significant hardening of the material was observed in both cases. The tests confirmed that cyclic properties of AW-2024 alloy under multistage loading conditions can be represented based on the results of tests under constant amplitude conditions for the same alloy. The basis for a new fatigue damage cumulation method was formulated based on a comparative analysis of the test results. The main feature of the proposed method consists in including into the analysis the cyclic hardening of AW-2024 alloy during fatigue life calculation. An experimental verification of the method proved its practical applicability. A comparative analysis of the results for a standard Palmgren-Miner hypothesis and the new approach demonstrated the advantages of using the new method. Keywords: Low-cycle fatigue, strain, stress, plastic strain energy, hysteresis, Palmgren-Miner hypothesis, fatigue calculations, cyclic softening, cyclic hardening 1. INTRODUCTION Fatigue life calculations for structural components are related with the notion of fatigue damage development. Since 1924, an estimated 40 different fatigue damage cumulation hypotheses have been proposed, which are listed and compared in [1-3]. The first developed and the least complicated is a linear Palmgren-Miner (PM) approach (Palmgren, 1924, Miner, 1945) [4,5]). A fatigue damage cumulation based on the PM hypothesis can be performed using different variables (stress strain and energy Wpl). The analysis of the fatigue characteristics shows that the energy-based fatigue characteristic is more comprehensive than stress- or strain-based, since it allows for the stress-strain interaction. Several fatigue damage growth hypotheses have been formulated as alternatives to the PM model for the energy-based characteristic [6-11]. The hypotheses concerning fatigue damage are most often based on the concept of constant damage (isodamage) lines, which are usually straight lines. However, studies are available where constant damage lines are not straight [1,12,13]. In general, the hypotheses based on the concept of constant damage lines can be classified into two groups. The first one includes approaches allowing for the residual fatigue life [1,14], while the second group includes hypotheses based on the existence of constant fatigue damage lines [15,16]. The difference between both groups consists in the position of the pivot point of the damage line [1]. A comparative analysis of the various damage cumulation methods carried out in [3] indicates that the linear PM hypothesis can be interpreted based on residual durability lines and permanent fatigue failure lines. For the sake of explanation, Figure 1 shows in a schematic way damage cumulation resulting from the hypothesis related to the concept based on solid damage lines [15,16] (Fig. 1a) and in accordance with the PM hypothesis (Fig. 1b). On the basis of Figure 1, it can be concluded that the key problem in the line of constant damage hypothesis is to define the position of the intersection point "P" of the damage line. When the "P" point in the concept of permanent fault lines tends to stresses a=0, the lines of permanent faults will be parallel and the description of fatigue damage cumulation will correspond to the PM hypothesis.
1
a)
b) 5'
5
a 1
Stress amplitude a
3 1
1' 3 3' 4 4' 2
4
Crack
1'
1
n Fatigue curve
P
Se
5 5'
Crack
2
2'
Isodamages line
5
Stress amplitude a
Crack
Load program
Fatigue curve
3 3' 4' Se P
Pivot point
P
4
a=0 2
2' Isodamages lines
Ne
Number of cycles N
Ne
N
Number of cycles N
N
Fig. 1. Isodamage lines: a) hypothesis of Hashin Z., Rotem A. [15, 16], b) PM hypothesis as hypothesis of isodamage line [3]. Despite using new criterion variables, the fatigue life calculation methods are still below the expected efficiency level due to several reasons. One of them is related to the fact that the energy Wpl is calculated after the test. The energy Wpl is controlled throughout the test (Wpl =const) only in a small number of studies [17]. Also, similar to other hysteresis loop parameters, the energy is not completely stabilized under constant amplitude loading conditions, as verified by numerous studies in the low-cycle metal fatigue range [18-21]. Therefore, the efficiency of fatigue life calculations cannot be improved by assuming that the energy Wpl is in a stable state under constant amplitude loading conditions. It seems more effective to accept in the fatigue life calculations that no stabilization occurs. Another issue is the course of changes of cyclic properties under operational loading and their predictability. The test results presented in [22] demonstrate the ability to characterize the changes in cyclic properties of 30HGSA alloy steel subjected to irregular loading, on the basis of the data from the constant-amplitude tests. The present study is a continuation of the previous research aiming to develop new damage cumulation analysis methods for the low-cycle fatigue range. Its main purpose is to formulate the foundations this methods and to experimentally verify the energy-based fatigue damage cumulation hypothesis which allows to include changes in cyclic properties of the material. 2. FATIGUE LIFE CALCULATIONS 2.1. Analytical characteristics of hysteresis loop An analytical characteristic of the hysteresis loop is required as a part of energy-based fatigue life calculations. It often utilises a relationship between stress and strain . The stress-strain curve can be obtained based on the experimental data by connecting the peaks of stabilized hysteresis loops at different strain levels. The analytical characteristic of the curve may be based on different models. The most commonly used is the Ramberg - Osgood model [23]: 1
n ' 2 2E 2K '
(1)
Parameters n’ and K’ are determined in low-cycle fatigue tests, whereas Young modulus E is determined in monotonic tensile tests. When processing the fatigue test results it is assumed that stress is a power function of strain ap [24]: 2
ap K ' 2 2
n'
(2)
The literature also includes different characteristics of the cyclic strain curve using single or twoparameter models [25]. The study presented in [25] focuses on n’ and K’ parameters of the model appearing in relationship (1). The equation describing the ascending branch of a hysteresis loop can be obtained by multiplying the relationship (1) by 2: 1
n ' 2 E 2K '
(3)
The descending branches of hysteresis loops can be obtained using equation (3) after transforming the coordinate system to a top peak of the hysteresis loop. The methods are discussed in detail in the literature [6]. The above equations are used to characterize the hysteresis loop for materials showing Masing behaviour [26]. For non-Masing behaviour, a special plot is used, formed by the upper and lower branches of the hysteresis loop [1]. When estimating the fatigue life, the hysteresis loop energy Wpl is calculated after the tests under controlled stress (a=const) or controlled strain (a=const, ap=const). With defined n' and K' parameters, the energy Wpl can be obtained for any level of and ap. Figure 2 shows the methodology for the energy Wpl calculation procedure [27].
1
n ' 2 E 2K '
F
Wpl F
0
Fig. 2. Procedure performed during plastic deformation energy calculation Wpl. According to the diagram shown in Figure 2, the dependence on the energy of plastic deformation Wpl can be saved as: 1 n ' Wpl 2 F 2 2 d E 2 K ' 0
(4)
The final form of the energy equation Wpl can be written as follows:
W pl
2 E
1
4
n '
1
2 K ' n1' 1 1 n'
(5)
3
As mentioned before, as a result of changes in cyclic properties (material hardening or softening), other changes of criterion value - energy Wpl in function of the number of load cycles [18-21] may occur. Therefore, the energies Wpl calculated from (4) apply to a single fatigue life, for which n' and K’ parameters were determined. If no stabilization occurs, the fatigue life is usually n/N=0.5 [28]. In consequence, for fatigue life n/N< 0.5 and n/N>0.5, energies Wpl from tests and calculations may be significantly different. 2.2. Energy-based fatigue damage cumulation analysis To calculate fatigue life of structural elements, an energy-life curve, a loading program and a damage accumulation hypothesis are required. The study uses a criterion variable, i.e. a plastic strain energy Wpl. The measure of the energy at a specific loading level is the area of the hysteresis loop. This area is calculated based on the strain and force recorded during the test. The fatigue life results are usually approximated in a log Wpl - log N coordinate system with a regression line [18]:
Wpl K p (N )
(6)
The PM hypothesis states that under constant-amplitude loading, each loading cycle contributes to the damage accumulation to the same degree. It means that the fatigue damage Di is a linear function of the number of cycles and can be defined as:
Di
ni Ni
(7)
For a multi-stage loading cycle, the specimen failure will occur if the following condition is met: k
Di i 1
ni 1 Ni
(8)
The cumulation of damage is based on the PM hypothesis using energy Wpl, as shown in Fig. 3. Fig. 3 presents the example lines L1, L2, Li parallel to the energy - life curve, also referred to as the constant damage lines [1-3]. Damage accumulation along those lines yields changes in loading levels (in this case, energy Wpl). a)
b) logWpl
Energy life curve
n1
Energy calculation Wpl , equation (5)
n3 n2
W pl K p (N ) n1
Wpl1
n3
Wpl3
N1 N3 n2
Wpl2
L2 L1 Damage line
n
1
N2
Li
log N
Fig. 3. Energy-based calculations of fatigue life based on the PM hypothesis: a) loading program, b) damage cumulation. 4
The energy Wpl (equation 5) for subsequent block loading stages was calculated based on the material data (n' and K') determined using equation (2). In this procedure, during fatigue calculations, the values are constant at the same strain level until condition (8) is met. It is a significant simplification of the actual course of the energy Wpl, which may be one of the reasons for observed discrepancies between calculated and measured fatigue life results. 3. EXPERIMENTAL TESTS a) constant-amplitude loading Fatigue test specimens were made of AW-2024 aluminium alloy as per [28]. The main reason for choosing this particular alloy for testing was that it is widely used in machine design and shows strong hardening under cyclic loading. Strength properties of the alloy are as follows: Rm=520 MPa, Re=354.8 MPa, E=72500 MPa, A5=15.5 %. Fig. 4 shows the geometry of the test specimen. 15 10h8
R40
25
R40
Ra0,32
1x45o
A 0,03 A
1x45o
200
Fig. 4. Test specimen. Low-cycle fatigue tests were carried out both under constant-amplitude and multistage loading conditions. Constant-amplitude loading was applied using five levels of controlled total strain (ac=0.5;0.65;0.8;1.0;1.5%). The tests were performed under controlled strain conditions ( ac=const) as per [28]. Three tests were carried out at each strain level. The 5% decrease in the maximum loading force for a half-cycle of the tensile test for two subsequent loading cycles was used as a criterion for specimen failure during the test. b) two-stage loading Strain levels for a two-stage loading program were determined based on the analysis of the constant-amplitude test results. The levels were ac(1)=0.65% and ac(2)=0.8%, respectively. The change from ac(1) to ac(2) or from ac(2) to ac(1) occurred at different damage levels D of the specimen at the first level defined as:
D
n N
(9)
where n denotes the number of constant-amplitude loading cycles at level εac(i), at which the strain changed from ac(1) to ac(2) or from ac(2) to ac(1), while N is the number of constant-amplitude loading cycles at level εac(i) until failure (based on N - Wpl curve.) Two sequences of strain level changes were used: - low strain to high strain - Lo-Hi sequence, - high strain to low strain - Hi-Lo sequence. Fig. 5 show the diagrams and parameters of the loading programs.
5
b)
ac2=0,8%
t
n2
n1
n1=0.25N1; n1=0.5N1; n1=0.75N1
%
%
n1=0.25N1; n1=0.5N1; n1=0.75N1
% ac1=0.65%
ac2=0.8%
%
n2
n1
ac1=0.65%
a)
t
Fig. 5. Two-stage loading: a) Lo-Hi, b) Hi-Lo. Loading frequency under two-stage tests was identical as in the constant-amplitude tests (0.2 Hz.) Momentary force and strain values for all loading cycles were recorded as a part of two-stage tests. c) multi-stage loading The tests under multi-stage loading were carried out in similar conditions as the tests under constant amplitude at ac=const. Programmed loading was applied as blocks of n0=100 cycles at an irregular sequence of stages. Fig. 6 shows the maximum stain acmax at subsequent stages aci, and the number of cycles for a specific program block stage. a)
b) 4
1.5 1
6
1
10
5
9
3 7
0
n
ac5
ac, %
0.5
acmax=1.5 %
8
2
-0.5 -1 n5
-1.5
no
no
Level
nj
1 2 3 4 5 6 7 8 9 10
14 19 8 2 15 9 2 5 9 17
0.175 0.28 0.07 0.35 0.14 0.245 0.035 0.315 0.105 0.21
acmax, acj, % 0.5 0.25 0.4 0.1 0.5 0.2 0.35 0.05 0.45 0.15 0.3
0.8 0.4 0.65 0.16 0.8 0.32 0.56 0.08 0.72 0.24 0.48
1.0 0.5 0.8 0.2 1.0 0.4 0.7 0.1 0.9 0.3 0.6
1.5 0.75 1.2 0.3 1.5 0.6 1.05 0.15 1.35 0.45 0.9
Fig. 6. Multi-stage loading parameters: a) program diagram, b) program parameters. All constant-amplitude and multi-stage loading cycles were oscillating cycles (R=-1). The loading program was characterized by the maximum total strain acmax and strain duty cycle calculated from the following equation: k n (10) acj j n0 j 1 ac max For the loading program, the duty cycle was 0.56. For the tests under multi-stage loading, the constant strain increment rate of 1%/s was used. Constant-amplitude and multi-stage loading tests were carried out using Instron 8501 hydraulic testing machine. Specimen strains under constant-amplitude loading and multi-stage loading conditions were measured using an extensometer with a gauge length of 10 mm and a measuring range of ±1 mm. Under constant amplitude loading conditions, instantaneous loading force and specimen strain were recorded for the selected loading cycles, while under multi-stage loading conditions, the parameter values for the entire loading blocks (n0=100 cycles) were collected. 6
4. TEST RESULTS AND DISCUSSION 4.1. Constant-amplitude loading tests Loading force and specimen strain recorded under constant amplitude loading conditions were used to calculate basic parameters of the hysteresis loop. Fig. 7 shows plots of changes in basic loop parameters as functions of fatigue life n/N at five strain levels ac. As expected, the tested material is characterized by significant hardening. It manifests itself in increasing a with a number of stress cycles (Fig. 7b), and decreasing the plastic strain ap (Fig. 7c) and energy Wpl (Fig. 7d) at constant total strain ac.
Wpl
ac=1.5% ac=1.0%
max
550
a, MPa
ac=0.8%
min
ap
L10-n10',K10'
L5-n5',K5'
600
L2-n2',K2'
b) L1-n1',K1'
a)
500 ac=0.65%
450 400
ac=0.5%
350
0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
n/N
L10-n10',K10'
L5-n5',K5'
0.01
L2-n2',K2'
d) L1-n1',K1'
c)
0.008
14
ac=1.5%
12
Wpl, MPa
ap, mm·mm-1
ac=1.5%
16
10
0.006 0.004
ac=1.0%
8 ac=1.0%
6 4
0.002
ac=0.8% ac=0.65%
ac=0.5%
2
0.2
0.4
0.6
n/N
0.8
ac=0.5%
ac=0.65%
0
0 0
ac=0.8%
1.0
0
0.2
0.4
0.6
n/N
Fig. 7. Changes in loop parameters as functions of n/N: a) loop parametersb) a , c) ap, d) Wpl. Comparative analysis of the above plots shows that among the three mentioned parameters, energy Wpl shows the least variability. This validates the results presented in [18, 22]. Due to the lack of stabilization, the material data for AW-2024 alloy corresponding to the fatigue life of approx. 0.5N was used. Figure 8 presents the hysteresis loops for that period in the - (Fig. 8a) and -(Fig. 8b) coordinate systems.
7
a)
b)
600
MPa
ac=1.0%
MPa
ac=0.65% ac=0.5%
0 -1
1000
ac=0.8%
300
-2
1200
ac=1.5%
0
1
-300
ac=1.5%
800
ac=1.0% ac=0.8%
600
ac=0.65%
400
2
ac=0.5%
200 0 0
-600
1
2
3
4
Fig. 8. Hysteresis loops for fatigue life n/N=0.5 in: a) - ; b) - coordinate system. The position of the hysteresis loops indicates that the tested material shows Masing behaviour [6], due to the common upper branch of the hysteresis loop being attached at the beginning of the coordinate system. The fatigue test results were used to plot the life curve in a bi-logarithmic coordinate system: energy Wpl - number of cycles N (Fig. 9). The fatigue life results were approximated with a straight line (6). To illustrate the simplification used at the stage of plotting the energy-life curve, Fig. 6 shows the additional plots for energy Wpl as a function of fatigue life n/N=0.1 and n/N=0.9. Different positions of the plots result from changes in the cyclic properties as functions of the number of load cycles, observed in Fig. 7. Different positions of the plots also prove that no relationship exists between stress a and strain ap (2) for AW-2024 alloy. Graph
1 2 3
100 ac=1.5%
Wpl, MJm-3
Wpl , MJm-3 Kp, MJm-3
1 cycle n=0.5N n=N
Wpl=Kp N
ac=1.0%
10
ac=0.8%
-0.9844 -1.6718 -1.7109
1 2 3
ac=0.65%
1
563.73 6159.9 13221.0
0.1
ac=0.5%
Wpl
0.01 0.001 1
10
100
1000
10000
N
Fig. 9. Energy life curve. To define the changes in n’ and K’ parameters from equation (2), 10 different fatigue life values n/N were determined. The periods for determining the data were included in Fig. 7 as lines L1 L2..... L10. Ten sets of the hysteresis loop parameters ( ap, a) were obtained for different fatigue life values n/N. In the plot shown in Fig. 7, lines L1 L2..... L10 can be considered as the constant damage lines [2]. Values of ap and a for each fatigue life were approximated in a logarithmic coordinate system with the regression lines [2]. The procedure for determining n' and K' parameters was discussed in detail in [29]. Figure 10a shows the results in the form of the a - ap relationship, while Fig. 10b presents the graphical representation of n' and K' values as functions of fatigue life n/N. 8
a)
b) 1000
0.084
2
840
3
a, MPa
0.08
820
1
a n' K' K’ MPa n’ 758.2 0.0698 825.3 0.0854 839.5 0.0760
n 0.1N 0.5N 0.9N
100 0.000001 0.00001 0.0001
0.001
ap, mm/mm
n'
ap
Graph 1 2 3
860
0.076
K' n'
0.072
800
K'=f(n/N) n'=f(n/N)
780 760 740
n/N=0.5 0.068
720 0
0.01
K' , MPa
1
K'0,5n/N
n'0,5n/N
0.2
0.4
0.6
0.8
1.0
n/N
Fig. 10. Results: a) a - ap relationship, b) n’ and K’ as functions of fatigue life n/N. The positions of the plots demonstrate that the determined n’ and K’ parameters depend on fatigue life n/N. The parameter values are the lowest at the beginning of the fatigue life and increase slightly with the increase in fatigue life n/N. Parameters n’ and K’ for n/N=0.5 were indicated in Fig. 10b to illustrate the simplification. The energy Wpl (equation 4) was calculated for 10 different fatigue life n/N values at 5 total strain ac levels. Fig. 11 shows example energy plots Wpl at strain levels ac=0.8 and 1.0% from the tests and calculations using n' and K' parameters at 10 different fatigue life periods. For comparison, also the hysteresis loop energy Wpl for n’ and K’ values determined for n/N=0.5 is plotted in Fig. 11. 7
ac=1.0%
5 Wpl
Wpl
4
new proposal
n/N=0.5
Wpl - Exp
3
ac=0.8%
Wpl, MJ·m-3
6
2 1 0
0.2
0.4
0.6
0.8
1.0
n/N Fig. 11. Changes in energy Wpl values from tests and calculations. The comparative analysis of the positions of energy Wpl plots from tests and calculations shows that the accuracy of energy representation depends on the fatigue life n/N at which n’ and K’ parameters are determined. Using data for a single fatigue life (e.g. n/N=0.5) to describe the hysteresis loop results in an average energy Wpl for the entire fatigue life period.
9
4.2. Two-stage loading As in the constant-amplitude tests, changes in basic hysteresis loop parameters and no clear stabilization period were observed also under two-stage loading. Fig. 12 shows example strain energy Wpl curves as functions of relative fatigue life n/N under constant-amplitude and two-stage loading. a)
b) 4
4
ac2=0.8%=const
3.5
3
Lo-Hi
Wpl, MJ/m3
Wpl, MJ/m3
3 2.5 2 n1/N1=0,5
1.5
ac1=0.65%=const (constant amplitude test)
1 0.5
ac2=0.8%=const (constant amplitude test)
3.5
(constant amplitude test)
2.5 2
ac1=0.65%=const (constant amplitude test)
1.5
Hi-Lo
1 0.5
0
0 0
0.2
0.4
n/N
0.6
0.8
1.0
0
0.2
0.4
n/N
0.6
0.8
1.0
Fig. 12. Wpl under constant-amplitude and two-stage loading (n1/N1=0.5): a) Lo-Hi, b) Hi-Lo. Until the change in strain level, the energy Wpl curve is identical as in the constant-amplitude loading. It results in the strong hardening of AW2024 alloy that can be observed as a significant reduction in Wpl with the increase in the number of cycles n1 at the first stage. After the change in strain level, the aluminium alloy is further hardened. The comparative analysis of the energy curves under constantamplitude and two-stage loading shows quantitative similarity of the energies in the same fatigue life periods and the nature of changes in energy curves, which validates the results in [22]. The influence of the loading program on fatigue life was observed during the tests. As expected, higher fatigue life values were observed for the Lo-Hi program. As also expected, with an increase in the number of cycles n1 at the first stage, the number of cycles n2 at the second stage decreased. 4.3. Multi-stage loading For the subsequent loading program blocks, basic parameters of the hysteresis loop, i.e. stress a, plastic strain ap and energy Wpl were determined. Figure 13 presents the example plots of changes in strain ap and energy Wpl for selected multi-stage loading blocks (acmax= 0.8%). a)
b) 0.0025
3
0.0015
1 5
10
15
0.001
Wpl, MJ/m3
ap, mm/mm
0.002 2
1 5 10 15
1
0.0005 0
0
0
20
40
60
n
80
100
0
20
40
n
60
80
100
Fig. 13. Changes in loop parameters for the loading program block : a) ap, b) Wpl. 10
The analysis of the plots ap and Wpl in subsequent program blocks demonstrates that irrespective of the strain level, AW-2024 is subject to cyclic hardening, indicated by reduced strain ap and energy Wpl at the same stages for subsequent loading blocks. The study includes a comparative analysis of strain a, stress ap and energy Wpl at selected programmed and constant-amplitude loading stages. Due to a limited volume of this study, the results are presented for a single strain level only (ac2=0.65%) - Fig. 14. a)
b) 480 460
%
a, MPa
ac=0.65%
1
440 420
n2
1- constant amplitude 2- programmable
400
n
2
380 0
c)
0.2
0.4
n/N
0.6
0.8
1
d) 0.001
2
0.0008
Wpl, MJ/m3
2.5
ap, mm/mm
0.0012
1- constant amplitude 2- programmable
0.0006 0.0004
1
1- constant amplitude 2- programmable
1.5
2
1
1
2
0.5
0.0002 i
0 0
0.2
0
0.4
0.6
n/N
0.8
1
0
0.2
0.4
0.6
0.8
1
n/N
Fig. 14. The loop parameters during programmable and constant amplitude loading stage at amplitude ac=0.65%: a) loading program, b) changes in a, c) changes in ap , d) changes in Wpl . The comparative study of the plots shown in Fig. 14 reveals a qualitative and quantitative similarity of the analysed parameters during constant-amplitude and programmable loading. The analysis results indicate that irrespective of the strain level, instantaneous loop parameter values (a, ap, Wpl) for the same fatigue life n/N are comparable. The hysteresis loop parameters obtained at the final cycles of programmable loading stages at amplitude ac=0.65% for the same damage rate reach values close to the level observed during constant-amplitude loading. The plots show that despite the stabilization being disturbed by the changes in strain at a subsequent stage, the material “remembers” the process observed during constant-amplitude loading. The diagrams of changes in three parameters (a, ap, Wpl) during multistage loading show that the direction of plots is similar to the direction of respective plots during constant-amplitude loading. It has a major practical application by validating the ability to predict changes in cyclic properties of material during operational loading on the basis of their course during constant-amplitude loading. AW-2024 alloy properties determined under constant-amplitude and programmable loading conditions validate the results in [22] for 30HGSA grade steel specimens. 11
5. VERIFICATION OF NEW DAMAGE CUMULATION METHOD 5.1. Two-stage loading The study also involved the experimental verification of the original Palmgren-Miner hypothesis, also referred to as the PM(O) hypothesis. The representative energies at two strain levels, Wpl1 and Wpl2, were determined at the half of the fatigue life [28]. The method to determine Wpl1 and Wpl2 allows the values to be informally extrapolated to all loading cycles at the first stage. Fig. 15 shows the diagram of the method used. 100
log Wpl
Wpl1', Wpl1''- calculation (Eq.5)
ac=0,8%
10
n2
Wpl2 Wpl1' Wpl1''
n 1'
Wpl1
n',K'=const
N2 N1' N1''
n1''
n1'''
N1 ac=0,65%
0,1 n1=0,5N1
0,01 1
10
100
1000
10000
log N Fig. 15. Damage accumulation for Lo-Hi (n1=0.5N1) program. In accordance with the PM(O) hypothesis, the specimen should fail at the cumulative damage D equal to 1, which, for the Lo-Hi program, can be expressed as:
D
n1 n2 1 N1 N 2
(11)
After n1=0.5N1 cycles of the first loading stage, the cumulative fatigue damage D is:
D
n1 0,5 N1
(12)
It is a simplification, since in accordance with Fig. 15 for all the loading cycles at the first stage of LoHi program, Wpli is higher than Wpl1 used as a representative value for this strain level. The actual cumulative damage D* after n1 cycles will be higher than 0.5: n1
D* i 1
ni 0,5 Ni
(13)
The cumulative damage D* at the first stage is affected by the alloy hardening (changes in energy). It is different at the second stage. For all cycles of the second stage, Wpli is lower than Wpl2 used as a representative value for this strain level. As a result, the cumulative damage D after n2 cycles calculated using the standard method will be higher than the actual cumulative damage and can be expressed as:
12
n2 n n2 i N 2 i 1 N i
(14)
It shows that regarding for the changes in energy during damage cumulation will also, to a certain degree, allow for including into the analysis the loading history. To calculate the actual cumulative damage D*, Wpli values for all cycles of the first stage have to be known, as well as n' and K' values for all loading cycles (dashed line in Fig. 10). If the mathematical description of changes in n' and K' is complicated, it can be simplified and the solid line can be replaced with a broken line (Fig. 15). This method is shown in Fig. 11. Using the denotations in Fig. 15, the cumulative damage D1* after n1 cycles of the first loading stage can be calculated:
D1*
n1, n1,, n1,,, N 1, N 1'' N 1
(15)
A similar analysis can be carried out for the second stage of the loading program. The cumulative damage was calculated as a part of the study using the original damage cumulation PM (O) method where changes in energy are not allowed for, and using the modified method PM (M), where changes in Wpli at subsequent loading cycles are included. Values of Wpli for all loading cycles were used to verify the results. Results for Wpli for subsequent cycles were calculated based on momentary loading force and strain recorded during the test. Values of were calculated as a ratio of the momentary force to the initial cross-section area of the specimen. m 1 1 W pli [ ( i i 1 )( i 1 i )] ( m 1 )( 1 m ) i 1 2
(16)
In the above equation symbol m denotes the number of recorded momentary force and strain values for a single loading cycle (m=200 points). Sum of damage D until the crack ocurred is presented in Figure 16a. For comparison of the durability results obtained from the calculations (Ncal) and tests (Nexp) in Figure 16b presented are Ncal/Nexp values for the implemented load programs. b)
Cumulative damage D Load n1=0.75N1 n1=0.25N1 n1=0.5N1 PM PM(M) PM (O) PM(O) PM(M) PM(O) (M) Lo-Hi 1.18 1.24 1.20 1.12 1.10 1.33 Hi-Lo 0.81 0.79 0.80 0.87 0.86 0.75 PM(O) - original PM hypothesis, PM(M) - modified PM hypothesis
Ncal/Nexp
1.3 1.2
Ncal -PM(M) Ncal -PM(O)
1.33 1.23
1.27
1.16
1.25 1.15
1.1
Ncal=Nexp
1.0 0.9
0.91 0.85 0.81 0.75
0.8
0.89 0.83
Hi-Lo
1.4
Lo-Hi
a)
0.7 0.25N1
0.5N1
0.75N1
Fig. 16. Results of calculation of Lo-Hi and Hi-Lo programs: a) damage sum D, b) Ncal/Nexp. As expected, various fatigue durabilities were obtained for the same sequences of Lo-Hi and Hi-Lo load programs. Based on the analysis of sums D summarized in the form of a table in Figure 16b and the Ncal/Nexp values shown as graphs in Figure 16b, it can be concluded that for the Lo-Hi program 13
sequence fatigue life obtained from calculations is lower than the durability obtained from the tests (NcalNexp). On the basis of the comparative analysis of damage sum D and Ncal/Nexp values obtained with the method PM (O) and PM(M), it can be concluded that taking into account accumulative fatigue changes of energy Wpl (PM(M) method) reduces the diversification of fatigue results obtained between calculations and tests. 5.2. Multi-stage loading In the PM(M) hypothesis, the lines of constant n' and K' parameters (lines L1, L2, Li in Fig. 3) were used as a special case of constant fatigue damage lines utilised in the fatigue damage cumulation hypotheses [1,2]. Figure 17 shows the damage cumulation procedure for multi-stage loading. The data required to calculate fatigue life include: Wpl -N curve (1), n’ and K’ as functions of relative fatigue life n/N (Fig. 10b) and the loading program (Fig. 6). a)
b)
ac 1 n1 3 2
n3
c) 1
n ' 2 E 2K '
a E
logWpl
K'
Wpl1 Wpl3
n2
Energy life curve
1
a n'
n1
n3
N3 N2
n2
Wpl2
Wpl
Lines of fixed values n’ i K’
N1
L1
n
1 n1=nb(1) nb(3) nb(2)
L2 L3
log N
Fig. 17. Energy-based damage cumulation: a) loading program, b) Wpl calculations for strain , c) damage cumulation diagram. Values of n' and K' expressed as curves shown in Fig. 10 (broken lines) were used to verify the PM (M) hypothesis. The n' and K’ values required to calculate Wpl(j=1) for the first cycle of the loading program were determined for the fatigue life within the range of 00.1 is met. From that point on, n’ and K’ values for calculating Wpl in the subsequent cycles will be taken from the fatigue life period between 0.1
nj
j 1
Nj
D
1
(17)
The fatigue life N under multi-stage loading conditions is equal to the total number of cycles at each level carried out until fatigue failure:
14
k
N nij
(18)
i 1 j 1
Similar assumptions can be formulated for the fatigue life calculations under random loading conditions, where changes in cyclic properties are allowed for after each loading cycle. In this case, it seems necessary to approximate the changes in n’ and K’ using polynomials n'=f(n/N) and K'=f(n/N) (Fig. 10b). For the experimental verification of the PM(M) damage cumulation model, the results obtained during multi-stage loads were used. The base characteristic for the durability calculation was the fatigue graph in the logarithmic coordinate system N-Wpl described by equation (6) obtained for loop parameters from durability n/N=05. The calculations were carried out for five load programs with a spectral load factor differing in the maximum strain level in the block (Figure 2). The results of calculations using the new and classical method are presented in Figure 18 in the form of fatigue charts in the logarithmic coordinate system N - Wpl(max). For comparison, Fig. 18 contains also the energy-life curve for constantamplitude tests.
Graph 1 2 3 4
Kp 6165.9 2591.1 923.3 873.6
-1.6718 -1.0005 -0.8744 -0.8486
Wpl=Kp(N)
10 1
0.1 acmax
Wpl(max) MJ/m3
100
Kind results From tests (constant amplitude test) From tests (block load) From calculations, PM(M) From calculations, PM(O)
j=1
0.01
n n0=100
0.001 10
100
1000
10000 100000 1000000
N Fig. 18. Fatigue life test and calculation results. The analysis of the position of the fatigue curves shows that the calculated fatigue life for the highest strain (energy) levels is lower than indicated by the test results. However, at the lowest strain levels, the opposite is true. This applies both to the standard and new method. The position of the fatigue curve based on calculations shows slight differences in fatigue life results in the high strain region. The effect of the new method on the calculation result is more significant in the low strain region. The comparison of the calculation and test results indicates that the fatigue life calculations are closer to the experimental results if the changes in cyclic properties (n' and K’) are considered. For example, for a program with strain amplitude acmax=0.35%, the results of calculations carried out using variable n’ and K’ values are almost 50% lower than the results obtained using constant data for the fatigue life of n/N= 0.5.
15
6. SUMMARY The study shows qualitative and quantitative similarity of the AW-2024 alloy hardening process under constant-amplitude and multistage loading conditions. Changes in basic parameters of the hysteresis loop under multistage loading conditions can be predicted based on the constant-amplitude loading test results. The results of the presented research validate the conclusions presented in [22]. The classical analytical model of the hysteresis loop uses the parameters determined under variable load (n’ and K’). However, for cyclically unstable materials these parameters change with the number of loading cycles. As a result, such analytical description of the loop provides a good characteristic of the hysteresis loop only for a period from which the parameters characterizing the cyclic properties were taken. The material data for the analytical model used to characterize the hysteresis loop (n’, K’) allow to reproduce changes in energy Wpl of AW-2024 alloy under constant amplitude loading conditions. The highest accuracy of reproducing the hysteresis loop is available at the highest total strain ac levels, since at these levels, changes in the actual energy Wpl are the lowest (Fig. 7). As expected, the material data used in the model characterizing the hysteresis loop do not affect the fatigue life. Seemingly minor differences in material data n’ and K’ and energy Wpl may lead to significant discrepancies in fatigue life values from tests and calculations. The effect of a new calculation PM(M) method on the results is minor in the high strain region. This effect becomes more pronounced with the decrease in strain (energy Wpl ) level throughout the loading program. It is due to the repeated use of material data in the fatigue life calculations, and thus the cumulation of the differences between the actual energy Wpl from tests and calculations. The results of the study have a practical value, since they validate the need to take into account the changes in cyclic properties during calculations. This can be particularly important when designing devices made of materials characterized by pronounced changes in cyclic properties. The proposed PM(M) method can be applied in the design of engineering structures used at elevated temperatures, since changes in cyclic properties are significantly higher at increased temperatures than at ambient temperature [30, 31].
This publication is financed by the National Science Centre as part of the research project no. 2017/25/B/ST8/02256. REFERENCES [1] Mansson S.S., Halford G.R.: Re - Examination of Cumulative Fatigue Damage Analysis - an Engineering Perspective, Engineering Fracture Mechanics, Vol 25 No 5/6, 1986. [2] Fatemi A., Yang L.: Cumulative Fatigue Damage and Life Prediction Theories: A Survey of the State of the Art for Homogeneous Materials. International Journal of Fatigue 20(1), (1998) 9-34. [3] Szala J.: Hipotezy sumowania uszkodzeń zmęczeniowych. Wydawnictwa Uczelniane ATR w Bydgoszczy, Bydgoszcz 1998 (in Polish). [4] Palmgren A.: Die Lebensdauer von Kugellagern, Verfahrenstechnik, Berlin, 68 (1924) 339-341. [5] Miner M.A.:Cumulative Damage in Fatigue, J. Appl.. Mech.,67 (A159-A164) 1945. [6] Ellyin F., Kujawski D.: Plastic strain energy in fatigue failure, J. Pressure Vessel Technology, Trans. ASME 106 (1984) 342-347. [7] Gołoś K. Ellyin F.: Generalisation of cumulative damage criterion to multilevel cyclic loading, Theoretical and Applied Fracture Mechanics 7(1987) 169-176. [8] Kujawski D.,Ellyin F.: A Cumulative damage theory for fatigue crack initiation and propagation, International Journal of Fatigue 6(2) (1984) 83-88. [9] Smith K.N., Watson P., Topper T.H.: A stress-strain function for the fatigue of metals, Journal of Materials, Vol. 5(4) (1970) 767-776.
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[10] Mroziński S. Topoliński T.: New Energy Model of Fatigue Damage Accumulation and its verification for 45-steel, Journal of Theoretical and Applied Mechanics, 2, 37 (1999) 223-239. [11] Gołoś K., Ellyin F.: A total strain energy theory for cumulative fatigue damage. Transaction ASME, Journal of Pressure Vessel Technology, Vol. 110 (1988) 35-41. [12] Pavlou D.G.: The theory of the S-N fatigue damage envelope: generalization of linear, double linear, and non-linear fatigue damage models. International Journal of Fatigue, 110 (2018) 204-214. [13] Mroziński S.: A fatigue damage accumulation method in low cycle fatigue zone, Journal of Theoretical and Applied Mechanics, 4,38 (2000) 767-780. [14] Schott G.: Lebensdauerberechnung für Schwingbelastungen auf der Grundlage von Folgenwöhlerkurwen. Maschinenbautechnik 7, 1981. [15] Hashin Z., Rotem A.: A Cumulative Damage Theory of Fatigue Failure, Mater. Sci. Eng., 34(2) (1978) 147160. [16] Subramanyan S.: A Cumulative Damage Rule Based on the Knee Point ofthe S-N Curve. Transactions of the ASME, Journal of Engineering Materials and Technology. 98(4), (1976) 316-321. [17] Mroziński S., Boroński D.: Metal tests in conditions of controlled strain energy density, Theoretical and Applied Mechanics, Volume 45, Issue 4, (2007) 773-784.
Journal of
[18] B. Atzori, G. Meneghetti, M. Ricotta.: Unified material parameters based on full compatibility for low-cycle fatigue characterisation of as-cast and austempered ductile irons, International Journal of Fatigue 68, (2014) 111–122. [19] Mroziński S., Golański G.: Fatigue life of GX12CrMoVNbN9-1 cast steel in the energy-based approach. Advanced Materials Research. Vol. 396-398, (2012) 446-449. [20] Li D.M., Kim K.W., Lee C.S.: Low cycle fatigue data evaluation for a high-strength spring steel. International Journal of Fatigue 19(8-9), 1997, 607-612. [21] Mroziński S., Egner H., Piotrowski M.: Effects of fatigue testing on low-cycle properties of P91 steel, International Journal of Fatigue, 120 (2019) 65-72. [22] Mroziński S.: The Influence of Loading Program on the Course of Fatigue Damage Cumulation, Journal of Theoretical and Applied Mechanics, Volume 49, Issue 1, (2011) 83-95. [23] Ramberg W.,Osgood W.R.: Description of stress-strain curves by three parameters, NACA, Tech.Note, No 402, 1943. [24] Morrow J.D.W.: Cyclic Plastic Strain Energy and Fatigue of Metals, Internal Friction, Damping and Cyclic plasticity, ASMT STP-378 Philadelphia, 1965, 45-87. [25] Kaleta J.: Doświadczalne podstawy formułowania energetycznych hipotez zmęczeniowych. Oficyna Wydawnicza Politechniki Wrocławskiej, Monografia nr 24, (1998) (in Polish). [26] Masing G.,: Eigenspannungen und Verfestigung beim Messing. Proc. 2 nd Inter. Congress of Appl. Mechanics, Zurich, (1962) 332-335. [27] Halford G.R.: The energy required for fatigue, Journal of Materials, Vol. 1,1 (1966) 3-18. [28] ASTM E606-92: Standard Practice for Strain -Controlled Fatigue Testing. [29] Mroziński, S.; Lipski, A.: Method for processing of the results of low-cycle fatigue tests, Materials Science, Volume 48, Issue 1, (2012) 83-88. [30] De-Long Wu, Peng Zhao, Qiong-Qi Wang, Fu-Zhen Xuan,: Cyclic behavior of 9–12% Cr steel under different control modes in low cycle regime: A comparative study, International Journal of Fatigue 70, (2015) 114–122. [31] Mroziński S., Skocki R., Influence of temperature on the cyclic properties of martensitic cast steel. Materials Science Forum Vol. 726 (2012)150-155.
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Nomenclature i - loading program block repetition, j - loading program stage number, k - number of loading program stages, no - number of cycles in the loading program block, nj - number of cycles in the j-th loading program stage, Nj - number of cycles until fatigue failure at the j-th program stage, - number of loading program block repetitions until fatigue failure, E - Young’s modulus, K' - cyclic strength coefficient, n - number of applied cycles, n' - cyclic strain hardening exponent, N - number of cycles to failure, n/N - relative number of cycles, - energy per cycle exponent, Kp - energy per cycle coefficient, Wpl - plastic strain energy in a unit volume of material per cycle ( area of the hysteresis loop), Wpl(max)- hysteresis loop energy for the loading program stage with the highest strain amplitude, - stress in general terms, a - amplitude of stress, - strain in general terms, ac - amplitude of total strain, ap - amplitude of plastic strain, ae - amplitude of elastic strain, - cyclic total strain range, ap - cyclic plastic strain range, - cyclic stress range.
18
T h e st u d y p r e s e nt s t h e r e su lt s o f lo w - c yc le fat ig u e t e st s o n AW - 2 0 2 4 , T h e t est s co n f ir me d t hat c yc l ic p r o p er t ie s u n d er mu lt ist ag e lo ad i n g co n d it io n s New fat ig u e d a ma g e cu mu lat io n met ho d E x p er i me n t a l v er i f ic at io n o f t h e ne w met ho d A co mp ar at iv e a n a l y s is o f t h e r e su lt s fo r a st a nd ar d P a l mg r e n - M in e r hypot hesis
19
S0142-1123(18)30680-7 https://doi.org/10.1016/j.ijfatigue.2018.12.008 JIJF 4922
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
16 October 2018 4 December 2018 5 December 2018
Please cite this article as: Mroziński, S., Energy-based method of fatigue damage cumulation, International Journal of Fatigue (2018), doi: https://doi.org/10.1016/j.ijfatigue.2018.12.008
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Energy-based method of fatigue damage cumulation Stanisław MROZIŃSKI * * University of Science and Technology, Faculty of Mechanical Engineering, Bydgoszcz, Poland, e-mail: [email protected] Abstract The study presents the results of low-cycle fatigue tests on AW-2024 aluminium alloy specimens under constant amplitude and multistage loading conditions. As expected, significant hardening of the material was observed in both cases. The tests confirmed that cyclic properties of AW-2024 alloy under multistage loading conditions can be represented based on the results of tests under constant amplitude conditions for the same alloy. The basis for a new fatigue damage cumulation method was formulated based on a comparative analysis of the test results. The main feature of the proposed method consists in including into the analysis the cyclic hardening of AW-2024 alloy during fatigue life calculation. An experimental verification of the method proved its practical applicability. A comparative analysis of the results for a standard Palmgren-Miner hypothesis and the new approach demonstrated the advantages of using the new method. Keywords: Low-cycle fatigue, strain, stress, plastic strain energy, hysteresis, Palmgren-Miner hypothesis, fatigue calculations, cyclic softening, cyclic hardening 1. INTRODUCTION Fatigue life calculations for structural components are related with the notion of fatigue damage development. Since 1924, an estimated 40 different fatigue damage cumulation hypotheses have been proposed, which are listed and compared in [1-3]. The first developed and the least complicated is a linear Palmgren-Miner (PM) approach (Palmgren, 1924, Miner, 1945) [4,5]). A fatigue damage cumulation based on the PM hypothesis can be performed using different variables (stress strain and energy Wpl). The analysis of the fatigue characteristics shows that the energy-based fatigue characteristic is more comprehensive than stress- or strain-based, since it allows for the stress-strain interaction. Several fatigue damage growth hypotheses have been formulated as alternatives to the PM model for the energy-based characteristic [6-11]. The hypotheses concerning fatigue damage are most often based on the concept of constant damage (isodamage) lines, which are usually straight lines. However, studies are available where constant damage lines are not straight [1,12,13]. In general, the hypotheses based on the concept of constant damage lines can be classified into two groups. The first one includes approaches allowing for the residual fatigue life [1,14], while the second group includes hypotheses based on the existence of constant fatigue damage lines [15,16]. The difference between both groups consists in the position of the pivot point of the damage line [1]. A comparative analysis of the various damage cumulation methods carried out in [3] indicates that the linear PM hypothesis can be interpreted based on residual durability lines and permanent fatigue failure lines. For the sake of explanation, Figure 1 shows in a schematic way damage cumulation resulting from the hypothesis related to the concept based on solid damage lines [15,16] (Fig. 1a) and in accordance with the PM hypothesis (Fig. 1b). On the basis of Figure 1, it can be concluded that the key problem in the line of constant damage hypothesis is to define the position of the intersection point "P" of the damage line. When the "P" point in the concept of permanent fault lines tends to stresses a=0, the lines of permanent faults will be parallel and the description of fatigue damage cumulation will correspond to the PM hypothesis.
1
a)
b) 5'
5
a 1
Stress amplitude a
3 1
1' 3 3' 4 4' 2
4
Crack
1'
1
n Fatigue curve
P
Se
5 5'
Crack
2
2'
Isodamages line
5
Stress amplitude a
Crack
Load program
Fatigue curve
3 3' 4' Se P
Pivot point
P
4
a=0 2
2' Isodamages lines
Ne
Number of cycles N
Ne
N
Number of cycles N
N
Fig. 1. Isodamage lines: a) hypothesis of Hashin Z., Rotem A. [15, 16], b) PM hypothesis as hypothesis of isodamage line [3]. Despite using new criterion variables, the fatigue life calculation methods are still below the expected efficiency level due to several reasons. One of them is related to the fact that the energy Wpl is calculated after the test. The energy Wpl is controlled throughout the test (Wpl =const) only in a small number of studies [17]. Also, similar to other hysteresis loop parameters, the energy is not completely stabilized under constant amplitude loading conditions, as verified by numerous studies in the low-cycle metal fatigue range [18-21]. Therefore, the efficiency of fatigue life calculations cannot be improved by assuming that the energy Wpl is in a stable state under constant amplitude loading conditions. It seems more effective to accept in the fatigue life calculations that no stabilization occurs. Another issue is the course of changes of cyclic properties under operational loading and their predictability. The test results presented in [22] demonstrate the ability to characterize the changes in cyclic properties of 30HGSA alloy steel subjected to irregular loading, on the basis of the data from the constant-amplitude tests. The present study is a continuation of the previous research aiming to develop new damage cumulation analysis methods for the low-cycle fatigue range. Its main purpose is to formulate the foundations this methods and to experimentally verify the energy-based fatigue damage cumulation hypothesis which allows to include changes in cyclic properties of the material. 2. FATIGUE LIFE CALCULATIONS 2.1. Analytical characteristics of hysteresis loop An analytical characteristic of the hysteresis loop is required as a part of energy-based fatigue life calculations. It often utilises a relationship between stress and strain . The stress-strain curve can be obtained based on the experimental data by connecting the peaks of stabilized hysteresis loops at different strain levels. The analytical characteristic of the curve may be based on different models. The most commonly used is the Ramberg - Osgood model [23]: 1
n ' 2 2E 2K '
(1)
Parameters n’ and K’ are determined in low-cycle fatigue tests, whereas Young modulus E is determined in monotonic tensile tests. When processing the fatigue test results it is assumed that stress is a power function of strain ap [24]: 2
ap K ' 2 2
n'
(2)
The literature also includes different characteristics of the cyclic strain curve using single or twoparameter models [25]. The study presented in [25] focuses on n’ and K’ parameters of the model appearing in relationship (1). The equation describing the ascending branch of a hysteresis loop can be obtained by multiplying the relationship (1) by 2: 1
n ' 2 E 2K '
(3)
The descending branches of hysteresis loops can be obtained using equation (3) after transforming the coordinate system to a top peak of the hysteresis loop. The methods are discussed in detail in the literature [6]. The above equations are used to characterize the hysteresis loop for materials showing Masing behaviour [26]. For non-Masing behaviour, a special plot is used, formed by the upper and lower branches of the hysteresis loop [1]. When estimating the fatigue life, the hysteresis loop energy Wpl is calculated after the tests under controlled stress (a=const) or controlled strain (a=const, ap=const). With defined n' and K' parameters, the energy Wpl can be obtained for any level of and ap. Figure 2 shows the methodology for the energy Wpl calculation procedure [27].
1
n ' 2 E 2K '
F
Wpl F
0
Fig. 2. Procedure performed during plastic deformation energy calculation Wpl. According to the diagram shown in Figure 2, the dependence on the energy of plastic deformation Wpl can be saved as: 1 n ' Wpl 2 F 2 2 d E 2 K ' 0
(4)
The final form of the energy equation Wpl can be written as follows:
W pl
2 E
1
4
n '
1
2 K ' n1' 1 1 n'
(5)
3
As mentioned before, as a result of changes in cyclic properties (material hardening or softening), other changes of criterion value - energy Wpl in function of the number of load cycles [18-21] may occur. Therefore, the energies Wpl calculated from (4) apply to a single fatigue life, for which n' and K’ parameters were determined. If no stabilization occurs, the fatigue life is usually n/N=0.5 [28]. In consequence, for fatigue life n/N< 0.5 and n/N>0.5, energies Wpl from tests and calculations may be significantly different. 2.2. Energy-based fatigue damage cumulation analysis To calculate fatigue life of structural elements, an energy-life curve, a loading program and a damage accumulation hypothesis are required. The study uses a criterion variable, i.e. a plastic strain energy Wpl. The measure of the energy at a specific loading level is the area of the hysteresis loop. This area is calculated based on the strain and force recorded during the test. The fatigue life results are usually approximated in a log Wpl - log N coordinate system with a regression line [18]:
Wpl K p (N )
(6)
The PM hypothesis states that under constant-amplitude loading, each loading cycle contributes to the damage accumulation to the same degree. It means that the fatigue damage Di is a linear function of the number of cycles and can be defined as:
Di
ni Ni
(7)
For a multi-stage loading cycle, the specimen failure will occur if the following condition is met: k
Di i 1
ni 1 Ni
(8)
The cumulation of damage is based on the PM hypothesis using energy Wpl, as shown in Fig. 3. Fig. 3 presents the example lines L1, L2, Li parallel to the energy - life curve, also referred to as the constant damage lines [1-3]. Damage accumulation along those lines yields changes in loading levels (in this case, energy Wpl). a)
b) logWpl
Energy life curve
n1
Energy calculation Wpl , equation (5)
n3 n2
W pl K p (N ) n1
Wpl1
n3
Wpl3
N1 N3 n2
Wpl2
L2 L1 Damage line
n
1
N2
Li
log N
Fig. 3. Energy-based calculations of fatigue life based on the PM hypothesis: a) loading program, b) damage cumulation. 4
The energy Wpl (equation 5) for subsequent block loading stages was calculated based on the material data (n' and K') determined using equation (2). In this procedure, during fatigue calculations, the values are constant at the same strain level until condition (8) is met. It is a significant simplification of the actual course of the energy Wpl, which may be one of the reasons for observed discrepancies between calculated and measured fatigue life results. 3. EXPERIMENTAL TESTS a) constant-amplitude loading Fatigue test specimens were made of AW-2024 aluminium alloy as per [28]. The main reason for choosing this particular alloy for testing was that it is widely used in machine design and shows strong hardening under cyclic loading. Strength properties of the alloy are as follows: Rm=520 MPa, Re=354.8 MPa, E=72500 MPa, A5=15.5 %. Fig. 4 shows the geometry of the test specimen. 15 10h8
R40
25
R40
Ra0,32
1x45o
A 0,03 A
1x45o
200
Fig. 4. Test specimen. Low-cycle fatigue tests were carried out both under constant-amplitude and multistage loading conditions. Constant-amplitude loading was applied using five levels of controlled total strain (ac=0.5;0.65;0.8;1.0;1.5%). The tests were performed under controlled strain conditions ( ac=const) as per [28]. Three tests were carried out at each strain level. The 5% decrease in the maximum loading force for a half-cycle of the tensile test for two subsequent loading cycles was used as a criterion for specimen failure during the test. b) two-stage loading Strain levels for a two-stage loading program were determined based on the analysis of the constant-amplitude test results. The levels were ac(1)=0.65% and ac(2)=0.8%, respectively. The change from ac(1) to ac(2) or from ac(2) to ac(1) occurred at different damage levels D of the specimen at the first level defined as:
D
n N
(9)
where n denotes the number of constant-amplitude loading cycles at level εac(i), at which the strain changed from ac(1) to ac(2) or from ac(2) to ac(1), while N is the number of constant-amplitude loading cycles at level εac(i) until failure (based on N - Wpl curve.) Two sequences of strain level changes were used: - low strain to high strain - Lo-Hi sequence, - high strain to low strain - Hi-Lo sequence. Fig. 5 show the diagrams and parameters of the loading programs.
5
b)
ac2=0,8%
t
n2
n1
n1=0.25N1; n1=0.5N1; n1=0.75N1
%
%
n1=0.25N1; n1=0.5N1; n1=0.75N1
% ac1=0.65%
ac2=0.8%
%
n2
n1
ac1=0.65%
a)
t
Fig. 5. Two-stage loading: a) Lo-Hi, b) Hi-Lo. Loading frequency under two-stage tests was identical as in the constant-amplitude tests (0.2 Hz.) Momentary force and strain values for all loading cycles were recorded as a part of two-stage tests. c) multi-stage loading The tests under multi-stage loading were carried out in similar conditions as the tests under constant amplitude at ac=const. Programmed loading was applied as blocks of n0=100 cycles at an irregular sequence of stages. Fig. 6 shows the maximum stain acmax at subsequent stages aci, and the number of cycles for a specific program block stage. a)
b) 4
1.5 1
6
1
10
5
9
3 7
0
n
ac5
ac, %
0.5
acmax=1.5 %
8
2
-0.5 -1 n5
-1.5
no
no
Level
nj
1 2 3 4 5 6 7 8 9 10
14 19 8 2 15 9 2 5 9 17
0.175 0.28 0.07 0.35 0.14 0.245 0.035 0.315 0.105 0.21
acmax, acj, % 0.5 0.25 0.4 0.1 0.5 0.2 0.35 0.05 0.45 0.15 0.3
0.8 0.4 0.65 0.16 0.8 0.32 0.56 0.08 0.72 0.24 0.48
1.0 0.5 0.8 0.2 1.0 0.4 0.7 0.1 0.9 0.3 0.6
1.5 0.75 1.2 0.3 1.5 0.6 1.05 0.15 1.35 0.45 0.9
Fig. 6. Multi-stage loading parameters: a) program diagram, b) program parameters. All constant-amplitude and multi-stage loading cycles were oscillating cycles (R=-1). The loading program was characterized by the maximum total strain acmax and strain duty cycle calculated from the following equation: k n (10) acj j n0 j 1 ac max For the loading program, the duty cycle was 0.56. For the tests under multi-stage loading, the constant strain increment rate of 1%/s was used. Constant-amplitude and multi-stage loading tests were carried out using Instron 8501 hydraulic testing machine. Specimen strains under constant-amplitude loading and multi-stage loading conditions were measured using an extensometer with a gauge length of 10 mm and a measuring range of ±1 mm. Under constant amplitude loading conditions, instantaneous loading force and specimen strain were recorded for the selected loading cycles, while under multi-stage loading conditions, the parameter values for the entire loading blocks (n0=100 cycles) were collected. 6
4. TEST RESULTS AND DISCUSSION 4.1. Constant-amplitude loading tests Loading force and specimen strain recorded under constant amplitude loading conditions were used to calculate basic parameters of the hysteresis loop. Fig. 7 shows plots of changes in basic loop parameters as functions of fatigue life n/N at five strain levels ac. As expected, the tested material is characterized by significant hardening. It manifests itself in increasing a with a number of stress cycles (Fig. 7b), and decreasing the plastic strain ap (Fig. 7c) and energy Wpl (Fig. 7d) at constant total strain ac.
Wpl
ac=1.5% ac=1.0%
max
550
a, MPa
ac=0.8%
min
ap
L10-n10',K10'
L5-n5',K5'
600
L2-n2',K2'
b) L1-n1',K1'
a)
500 ac=0.65%
450 400
ac=0.5%
350
0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
n/N
L10-n10',K10'
L5-n5',K5'
0.01
L2-n2',K2'
d) L1-n1',K1'
c)
0.008
14
ac=1.5%
12
Wpl, MPa
ap, mm·mm-1
ac=1.5%
16
10
0.006 0.004
ac=1.0%
8 ac=1.0%
6 4
0.002
ac=0.8% ac=0.65%
ac=0.5%
2
0.2
0.4
0.6
n/N
0.8
ac=0.5%
ac=0.65%
0
0 0
ac=0.8%
1.0
0
0.2
0.4
0.6
n/N
Fig. 7. Changes in loop parameters as functions of n/N: a) loop parametersb) a , c) ap, d) Wpl. Comparative analysis of the above plots shows that among the three mentioned parameters, energy Wpl shows the least variability. This validates the results presented in [18, 22]. Due to the lack of stabilization, the material data for AW-2024 alloy corresponding to the fatigue life of approx. 0.5N was used. Figure 8 presents the hysteresis loops for that period in the - (Fig. 8a) and -(Fig. 8b) coordinate systems.
7
a)
b)
600
MPa
ac=1.0%
MPa
ac=0.65% ac=0.5%
0 -1
1000
ac=0.8%
300
-2
1200
ac=1.5%
0
1
-300
ac=1.5%
800
ac=1.0% ac=0.8%
600
ac=0.65%
400
2
ac=0.5%
200 0 0
-600
1
2
3
4
Fig. 8. Hysteresis loops for fatigue life n/N=0.5 in: a) - ; b) - coordinate system. The position of the hysteresis loops indicates that the tested material shows Masing behaviour [6], due to the common upper branch of the hysteresis loop being attached at the beginning of the coordinate system. The fatigue test results were used to plot the life curve in a bi-logarithmic coordinate system: energy Wpl - number of cycles N (Fig. 9). The fatigue life results were approximated with a straight line (6). To illustrate the simplification used at the stage of plotting the energy-life curve, Fig. 6 shows the additional plots for energy Wpl as a function of fatigue life n/N=0.1 and n/N=0.9. Different positions of the plots result from changes in the cyclic properties as functions of the number of load cycles, observed in Fig. 7. Different positions of the plots also prove that no relationship exists between stress a and strain ap (2) for AW-2024 alloy. Graph
1 2 3
100 ac=1.5%
Wpl, MJm-3
Wpl , MJm-3 Kp, MJm-3
1 cycle n=0.5N n=N
Wpl=Kp N
ac=1.0%
10
ac=0.8%
-0.9844 -1.6718 -1.7109
1 2 3
ac=0.65%
1
563.73 6159.9 13221.0
0.1
ac=0.5%
Wpl
0.01 0.001 1
10
100
1000
10000
N
Fig. 9. Energy life curve. To define the changes in n’ and K’ parameters from equation (2), 10 different fatigue life values n/N were determined. The periods for determining the data were included in Fig. 7 as lines L1 L2..... L10. Ten sets of the hysteresis loop parameters ( ap, a) were obtained for different fatigue life values n/N. In the plot shown in Fig. 7, lines L1 L2..... L10 can be considered as the constant damage lines [2]. Values of ap and a for each fatigue life were approximated in a logarithmic coordinate system with the regression lines [2]. The procedure for determining n' and K' parameters was discussed in detail in [29]. Figure 10a shows the results in the form of the a - ap relationship, while Fig. 10b presents the graphical representation of n' and K' values as functions of fatigue life n/N. 8
a)
b) 1000
0.084
2
840
3
a, MPa
0.08
820
1
a n' K' K’ MPa n’ 758.2 0.0698 825.3 0.0854 839.5 0.0760
n 0.1N 0.5N 0.9N
100 0.000001 0.00001 0.0001
0.001
ap, mm/mm
n'
ap
Graph 1 2 3
860
0.076
K' n'
0.072
800
K'=f(n/N) n'=f(n/N)
780 760 740
n/N=0.5 0.068
720 0
0.01
K' , MPa
1
K'0,5n/N
n'0,5n/N
0.2
0.4
0.6
0.8
1.0
n/N
Fig. 10. Results: a) a - ap relationship, b) n’ and K’ as functions of fatigue life n/N. The positions of the plots demonstrate that the determined n’ and K’ parameters depend on fatigue life n/N. The parameter values are the lowest at the beginning of the fatigue life and increase slightly with the increase in fatigue life n/N. Parameters n’ and K’ for n/N=0.5 were indicated in Fig. 10b to illustrate the simplification. The energy Wpl (equation 4) was calculated for 10 different fatigue life n/N values at 5 total strain ac levels. Fig. 11 shows example energy plots Wpl at strain levels ac=0.8 and 1.0% from the tests and calculations using n' and K' parameters at 10 different fatigue life periods. For comparison, also the hysteresis loop energy Wpl for n’ and K’ values determined for n/N=0.5 is plotted in Fig. 11. 7
ac=1.0%
5 Wpl
Wpl
4
new proposal
n/N=0.5
Wpl - Exp
3
ac=0.8%
Wpl, MJ·m-3
6
2 1 0
0.2
0.4
0.6
0.8
1.0
n/N Fig. 11. Changes in energy Wpl values from tests and calculations. The comparative analysis of the positions of energy Wpl plots from tests and calculations shows that the accuracy of energy representation depends on the fatigue life n/N at which n’ and K’ parameters are determined. Using data for a single fatigue life (e.g. n/N=0.5) to describe the hysteresis loop results in an average energy Wpl for the entire fatigue life period.
9
4.2. Two-stage loading As in the constant-amplitude tests, changes in basic hysteresis loop parameters and no clear stabilization period were observed also under two-stage loading. Fig. 12 shows example strain energy Wpl curves as functions of relative fatigue life n/N under constant-amplitude and two-stage loading. a)
b) 4
4
ac2=0.8%=const
3.5
3
Lo-Hi
Wpl, MJ/m3
Wpl, MJ/m3
3 2.5 2 n1/N1=0,5
1.5
ac1=0.65%=const (constant amplitude test)
1 0.5
ac2=0.8%=const (constant amplitude test)
3.5
(constant amplitude test)
2.5 2
ac1=0.65%=const (constant amplitude test)
1.5
Hi-Lo
1 0.5
0
0 0
0.2
0.4
n/N
0.6
0.8
1.0
0
0.2
0.4
n/N
0.6
0.8
1.0
Fig. 12. Wpl under constant-amplitude and two-stage loading (n1/N1=0.5): a) Lo-Hi, b) Hi-Lo. Until the change in strain level, the energy Wpl curve is identical as in the constant-amplitude loading. It results in the strong hardening of AW2024 alloy that can be observed as a significant reduction in Wpl with the increase in the number of cycles n1 at the first stage. After the change in strain level, the aluminium alloy is further hardened. The comparative analysis of the energy curves under constantamplitude and two-stage loading shows quantitative similarity of the energies in the same fatigue life periods and the nature of changes in energy curves, which validates the results in [22]. The influence of the loading program on fatigue life was observed during the tests. As expected, higher fatigue life values were observed for the Lo-Hi program. As also expected, with an increase in the number of cycles n1 at the first stage, the number of cycles n2 at the second stage decreased. 4.3. Multi-stage loading For the subsequent loading program blocks, basic parameters of the hysteresis loop, i.e. stress a, plastic strain ap and energy Wpl were determined. Figure 13 presents the example plots of changes in strain ap and energy Wpl for selected multi-stage loading blocks (acmax= 0.8%). a)
b) 0.0025
3
0.0015
1 5
10
15
0.001
Wpl, MJ/m3
ap, mm/mm
0.002 2
1 5 10 15
1
0.0005 0
0
0
20
40
60
n
80
100
0
20
40
n
60
80
100
Fig. 13. Changes in loop parameters for the loading program block : a) ap, b) Wpl. 10
The analysis of the plots ap and Wpl in subsequent program blocks demonstrates that irrespective of the strain level, AW-2024 is subject to cyclic hardening, indicated by reduced strain ap and energy Wpl at the same stages for subsequent loading blocks. The study includes a comparative analysis of strain a, stress ap and energy Wpl at selected programmed and constant-amplitude loading stages. Due to a limited volume of this study, the results are presented for a single strain level only (ac2=0.65%) - Fig. 14. a)
b) 480 460
%
a, MPa
ac=0.65%
1
440 420
n2
1- constant amplitude 2- programmable
400
n
2
380 0
c)
0.2
0.4
n/N
0.6
0.8
1
d) 0.001
2
0.0008
Wpl, MJ/m3
2.5
ap, mm/mm
0.0012
1- constant amplitude 2- programmable
0.0006 0.0004
1
1- constant amplitude 2- programmable
1.5
2
1
1
2
0.5
0.0002 i
0 0
0.2
0
0.4
0.6
n/N
0.8
1
0
0.2
0.4
0.6
0.8
1
n/N
Fig. 14. The loop parameters during programmable and constant amplitude loading stage at amplitude ac=0.65%: a) loading program, b) changes in a, c) changes in ap , d) changes in Wpl . The comparative study of the plots shown in Fig. 14 reveals a qualitative and quantitative similarity of the analysed parameters during constant-amplitude and programmable loading. The analysis results indicate that irrespective of the strain level, instantaneous loop parameter values (a, ap, Wpl) for the same fatigue life n/N are comparable. The hysteresis loop parameters obtained at the final cycles of programmable loading stages at amplitude ac=0.65% for the same damage rate reach values close to the level observed during constant-amplitude loading. The plots show that despite the stabilization being disturbed by the changes in strain at a subsequent stage, the material “remembers” the process observed during constant-amplitude loading. The diagrams of changes in three parameters (a, ap, Wpl) during multistage loading show that the direction of plots is similar to the direction of respective plots during constant-amplitude loading. It has a major practical application by validating the ability to predict changes in cyclic properties of material during operational loading on the basis of their course during constant-amplitude loading. AW-2024 alloy properties determined under constant-amplitude and programmable loading conditions validate the results in [22] for 30HGSA grade steel specimens. 11
5. VERIFICATION OF NEW DAMAGE CUMULATION METHOD 5.1. Two-stage loading The study also involved the experimental verification of the original Palmgren-Miner hypothesis, also referred to as the PM(O) hypothesis. The representative energies at two strain levels, Wpl1 and Wpl2, were determined at the half of the fatigue life [28]. The method to determine Wpl1 and Wpl2 allows the values to be informally extrapolated to all loading cycles at the first stage. Fig. 15 shows the diagram of the method used. 100
log Wpl
Wpl1', Wpl1''- calculation (Eq.5)
ac=0,8%
10
n2
Wpl2 Wpl1' Wpl1''
n 1'
Wpl1
n',K'=const
N2 N1' N1''
n1''
n1'''
N1 ac=0,65%
0,1 n1=0,5N1
0,01 1
10
100
1000
10000
log N Fig. 15. Damage accumulation for Lo-Hi (n1=0.5N1) program. In accordance with the PM(O) hypothesis, the specimen should fail at the cumulative damage D equal to 1, which, for the Lo-Hi program, can be expressed as:
D
n1 n2 1 N1 N 2
(11)
After n1=0.5N1 cycles of the first loading stage, the cumulative fatigue damage D is:
D
n1 0,5 N1
(12)
It is a simplification, since in accordance with Fig. 15 for all the loading cycles at the first stage of LoHi program, Wpli is higher than Wpl1 used as a representative value for this strain level. The actual cumulative damage D* after n1 cycles will be higher than 0.5: n1
D* i 1
ni 0,5 Ni
(13)
The cumulative damage D* at the first stage is affected by the alloy hardening (changes in energy). It is different at the second stage. For all cycles of the second stage, Wpli is lower than Wpl2 used as a representative value for this strain level. As a result, the cumulative damage D after n2 cycles calculated using the standard method will be higher than the actual cumulative damage and can be expressed as:
12
n2 n n2 i N 2 i 1 N i
(14)
It shows that regarding for the changes in energy during damage cumulation will also, to a certain degree, allow for including into the analysis the loading history. To calculate the actual cumulative damage D*, Wpli values for all cycles of the first stage have to be known, as well as n' and K' values for all loading cycles (dashed line in Fig. 10). If the mathematical description of changes in n' and K' is complicated, it can be simplified and the solid line can be replaced with a broken line (Fig. 15). This method is shown in Fig. 11. Using the denotations in Fig. 15, the cumulative damage D1* after n1 cycles of the first loading stage can be calculated:
D1*
n1, n1,, n1,,, N 1, N 1'' N 1
(15)
A similar analysis can be carried out for the second stage of the loading program. The cumulative damage was calculated as a part of the study using the original damage cumulation PM (O) method where changes in energy are not allowed for, and using the modified method PM (M), where changes in Wpli at subsequent loading cycles are included. Values of Wpli for all loading cycles were used to verify the results. Results for Wpli for subsequent cycles were calculated based on momentary loading force and strain recorded during the test. Values of were calculated as a ratio of the momentary force to the initial cross-section area of the specimen. m 1 1 W pli [ ( i i 1 )( i 1 i )] ( m 1 )( 1 m ) i 1 2
(16)
In the above equation symbol m denotes the number of recorded momentary force and strain values for a single loading cycle (m=200 points). Sum of damage D until the crack ocurred is presented in Figure 16a. For comparison of the durability results obtained from the calculations (Ncal) and tests (Nexp) in Figure 16b presented are Ncal/Nexp values for the implemented load programs. b)
Cumulative damage D Load n1=0.75N1 n1=0.25N1 n1=0.5N1 PM PM(M) PM (O) PM(O) PM(M) PM(O) (M) Lo-Hi 1.18 1.24 1.20 1.12 1.10 1.33 Hi-Lo 0.81 0.79 0.80 0.87 0.86 0.75 PM(O) - original PM hypothesis, PM(M) - modified PM hypothesis
Ncal/Nexp
1.3 1.2
Ncal -PM(M) Ncal -PM(O)
1.33 1.23
1.27
1.16
1.25 1.15
1.1
Ncal=Nexp
1.0 0.9
0.91 0.85 0.81 0.75
0.8
0.89 0.83
Hi-Lo
1.4
Lo-Hi
a)
0.7 0.25N1
0.5N1
0.75N1
Fig. 16. Results of calculation of Lo-Hi and Hi-Lo programs: a) damage sum D, b) Ncal/Nexp. As expected, various fatigue durabilities were obtained for the same sequences of Lo-Hi and Hi-Lo load programs. Based on the analysis of sums D summarized in the form of a table in Figure 16b and the Ncal/Nexp values shown as graphs in Figure 16b, it can be concluded that for the Lo-Hi program 13
sequence fatigue life obtained from calculations is lower than the durability obtained from the tests (Ncal
b)
ac 1 n1 3 2
n3
c) 1
n ' 2 E 2K '
a E
logWpl
K'
Wpl1 Wpl3
n2
Energy life curve
1
a n'
n1
n3
N3 N2
n2
Wpl2
Wpl
Lines of fixed values n’ i K’
N1
L1
n
1 n1=nb(1) nb(3) nb(2)
L2 L3
log N
Fig. 17. Energy-based damage cumulation: a) loading program, b) Wpl calculations for strain , c) damage cumulation diagram. Values of n' and K' expressed as curves shown in Fig. 10 (broken lines) were used to verify the PM (M) hypothesis. The n' and K’ values required to calculate Wpl(j=1) for the first cycle of the loading program were determined for the fatigue life within the range of 0
nj
j 1
Nj
D
1
(17)
The fatigue life N under multi-stage loading conditions is equal to the total number of cycles at each level carried out until fatigue failure:
14
k
N nij
(18)
i 1 j 1
Similar assumptions can be formulated for the fatigue life calculations under random loading conditions, where changes in cyclic properties are allowed for after each loading cycle. In this case, it seems necessary to approximate the changes in n’ and K’ using polynomials n'=f(n/N) and K'=f(n/N) (Fig. 10b). For the experimental verification of the PM(M) damage cumulation model, the results obtained during multi-stage loads were used. The base characteristic for the durability calculation was the fatigue graph in the logarithmic coordinate system N-Wpl described by equation (6) obtained for loop parameters from durability n/N=05. The calculations were carried out for five load programs with a spectral load factor differing in the maximum strain level in the block (Figure 2). The results of calculations using the new and classical method are presented in Figure 18 in the form of fatigue charts in the logarithmic coordinate system N - Wpl(max). For comparison, Fig. 18 contains also the energy-life curve for constantamplitude tests.
Graph 1 2 3 4
Kp 6165.9 2591.1 923.3 873.6
-1.6718 -1.0005 -0.8744 -0.8486
Wpl=Kp(N)
10 1
0.1 acmax
Wpl(max) MJ/m3
100
Kind results From tests (constant amplitude test) From tests (block load) From calculations, PM(M) From calculations, PM(O)
j=1
0.01
n n0=100
0.001 10
100
1000
10000 100000 1000000
N Fig. 18. Fatigue life test and calculation results. The analysis of the position of the fatigue curves shows that the calculated fatigue life for the highest strain (energy) levels is lower than indicated by the test results. However, at the lowest strain levels, the opposite is true. This applies both to the standard and new method. The position of the fatigue curve based on calculations shows slight differences in fatigue life results in the high strain region. The effect of the new method on the calculation result is more significant in the low strain region. The comparison of the calculation and test results indicates that the fatigue life calculations are closer to the experimental results if the changes in cyclic properties (n' and K’) are considered. For example, for a program with strain amplitude acmax=0.35%, the results of calculations carried out using variable n’ and K’ values are almost 50% lower than the results obtained using constant data for the fatigue life of n/N= 0.5.
15
6. SUMMARY The study shows qualitative and quantitative similarity of the AW-2024 alloy hardening process under constant-amplitude and multistage loading conditions. Changes in basic parameters of the hysteresis loop under multistage loading conditions can be predicted based on the constant-amplitude loading test results. The results of the presented research validate the conclusions presented in [22]. The classical analytical model of the hysteresis loop uses the parameters determined under variable load (n’ and K’). However, for cyclically unstable materials these parameters change with the number of loading cycles. As a result, such analytical description of the loop provides a good characteristic of the hysteresis loop only for a period from which the parameters characterizing the cyclic properties were taken. The material data for the analytical model used to characterize the hysteresis loop (n’, K’) allow to reproduce changes in energy Wpl of AW-2024 alloy under constant amplitude loading conditions. The highest accuracy of reproducing the hysteresis loop is available at the highest total strain ac levels, since at these levels, changes in the actual energy Wpl are the lowest (Fig. 7). As expected, the material data used in the model characterizing the hysteresis loop do not affect the fatigue life. Seemingly minor differences in material data n’ and K’ and energy Wpl may lead to significant discrepancies in fatigue life values from tests and calculations. The effect of a new calculation PM(M) method on the results is minor in the high strain region. This effect becomes more pronounced with the decrease in strain (energy Wpl ) level throughout the loading program. It is due to the repeated use of material data in the fatigue life calculations, and thus the cumulation of the differences between the actual energy Wpl from tests and calculations. The results of the study have a practical value, since they validate the need to take into account the changes in cyclic properties during calculations. This can be particularly important when designing devices made of materials characterized by pronounced changes in cyclic properties. The proposed PM(M) method can be applied in the design of engineering structures used at elevated temperatures, since changes in cyclic properties are significantly higher at increased temperatures than at ambient temperature [30, 31].
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[18] B. Atzori, G. Meneghetti, M. Ricotta.: Unified material parameters based on full compatibility for low-cycle fatigue characterisation of as-cast and austempered ductile irons, International Journal of Fatigue 68, (2014) 111–122. [19] Mroziński S., Golański G.: Fatigue life of GX12CrMoVNbN9-1 cast steel in the energy-based approach. Advanced Materials Research. Vol. 396-398, (2012) 446-449. [20] Li D.M., Kim K.W., Lee C.S.: Low cycle fatigue data evaluation for a high-strength spring steel. International Journal of Fatigue 19(8-9), 1997, 607-612. [21] Mroziński S., Egner H., Piotrowski M.: Effects of fatigue testing on low-cycle properties of P91 steel, International Journal of Fatigue, 120 (2019) 65-72. [22] Mroziński S.: The Influence of Loading Program on the Course of Fatigue Damage Cumulation, Journal of Theoretical and Applied Mechanics, Volume 49, Issue 1, (2011) 83-95. [23] Ramberg W.,Osgood W.R.: Description of stress-strain curves by three parameters, NACA, Tech.Note, No 402, 1943. [24] Morrow J.D.W.: Cyclic Plastic Strain Energy and Fatigue of Metals, Internal Friction, Damping and Cyclic plasticity, ASMT STP-378 Philadelphia, 1965, 45-87. [25] Kaleta J.: Doświadczalne podstawy formułowania energetycznych hipotez zmęczeniowych. Oficyna Wydawnicza Politechniki Wrocławskiej, Monografia nr 24, (1998) (in Polish). [26] Masing G.,: Eigenspannungen und Verfestigung beim Messing. Proc. 2 nd Inter. Congress of Appl. Mechanics, Zurich, (1962) 332-335. [27] Halford G.R.: The energy required for fatigue, Journal of Materials, Vol. 1,1 (1966) 3-18. [28] ASTM E606-92: Standard Practice for Strain -Controlled Fatigue Testing. [29] Mroziński, S.; Lipski, A.: Method for processing of the results of low-cycle fatigue tests, Materials Science, Volume 48, Issue 1, (2012) 83-88. [30] De-Long Wu, Peng Zhao, Qiong-Qi Wang, Fu-Zhen Xuan,: Cyclic behavior of 9–12% Cr steel under different control modes in low cycle regime: A comparative study, International Journal of Fatigue 70, (2015) 114–122. [31] Mroziński S., Skocki R., Influence of temperature on the cyclic properties of martensitic cast steel. Materials Science Forum Vol. 726 (2012)150-155.
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Nomenclature i - loading program block repetition, j - loading program stage number, k - number of loading program stages, no - number of cycles in the loading program block, nj - number of cycles in the j-th loading program stage, Nj - number of cycles until fatigue failure at the j-th program stage, - number of loading program block repetitions until fatigue failure, E - Young’s modulus, K' - cyclic strength coefficient, n - number of applied cycles, n' - cyclic strain hardening exponent, N - number of cycles to failure, n/N - relative number of cycles, - energy per cycle exponent, Kp - energy per cycle coefficient, Wpl - plastic strain energy in a unit volume of material per cycle ( area of the hysteresis loop), Wpl(max)- hysteresis loop energy for the loading program stage with the highest strain amplitude, - stress in general terms, a - amplitude of stress, - strain in general terms, ac - amplitude of total strain, ap - amplitude of plastic strain, ae - amplitude of elastic strain, - cyclic total strain range, ap - cyclic plastic strain range, - cyclic stress range.
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T h e st u d y p r e s e nt s t h e r e su lt s o f lo w - c yc le fat ig u e t e st s o n AW - 2 0 2 4 , T h e t est s co n f ir me d t hat c yc l ic p r o p er t ie s u n d er mu lt ist ag e lo ad i n g co n d it io n s New fat ig u e d a ma g e cu mu lat io n met ho d E x p er i me n t a l v er i f ic at io n o f t h e ne w met ho d A co mp ar at iv e a n a l y s is o f t h e r e su lt s fo r a st a nd ar d P a l mg r e n - M in e r hypot hesis
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