A new non-linear continuum damage mechanics model for fatigue life prediction under variable loading/Naujas netiesinio koontiniumo pazeidimo mechaninis modelis nuovargio trukmes prie kintamu apkrovu numatymui.
Yuan, Rong ; Li, Haiqing ; Huang, Hong-Zhong 等
1. Introduction
The mechanical components of major equipment including many
operating in aviation, power generating, automotive industry and other
industries are usually subjected to variable cyclic loading and fatigue.
Therefore, fatigue is one of the main failure forms of these components.
Fatigue failure is a damage accumulation process in which material
property deteriorates continuously under fatigue loading and the damage
depends on the size of stress and strain [1]. With the accumulation of
fatigue damage, some accidents occur for these components. Research
shows that a reliable lifetime prediction method is particularly
important in the design, safety assessments, and optimization of
engineering components and structures [2-5]. Thus, it is important to
formulate an accurate method to evaluate the fatigue damage accumulation
and effectively predict the fatigue life of these components.
Damage accumulation in materials is very important, but very
challenging to characterize in a meaningful and reliable way. Until now,
dozens of methods have been developed to predict the fatigue life. In
general, fatigue damage accumulation theories can be classified into two
categories: linear damage accumulation theories and non-linear damage
accumulation theories. The linear damage rule (LDR), also called the
Palmgreen-Miner rule (just Miner's rule for short) [6], is commonly
used to calculate the cumulative fatigue damage based on the following
assumptions [7]:
1. the rate of damage accumulation remains constant over each
loading cycle;
2. fatigue damage occurs and accumulates only when the loading
stress is higher than its fatigue limit;
3. the cycles be extracted and arranged in ascending order of
magnitude without any regard for its order of occurrence.
According to these assumptions, fatigue life of components under
variable amplitude loading can be estimated by:
D = [k.summation over (i=1)] [n.sub.i]/[N.sub.i], (1)
where [n.sub.i] is the number of loading cycles at a given stress
level [[sigma].sub.i], [N.sub.i] is the number of cycles to failure at
[[sigma].sub.i], D is the total damage (it is usually assumed to be one
at the point of fatigue failure life) as shown in Fig. 1.
[FIGURE 1 OMITTED]
However, Eq. (1) neglects the damage contribution of the loading
stress which is lower than the fatigue limit. According to some
experimental results such as: Lu and Zheng [8-10], Sinclair [11], and
Makajima et al. [12], it has shown that the damage of low amplitude
loads is one of the main reasons for prediction errors. Moreover, the
influence of loading sequence effects on fatigue life is ignored in
Miner's rule. That is to say, it is not sensitive to the loading
sequence effects for the linear damage accumulation theory. Thus,
Miner's rule often leads to a discrepancy of up an order of
magnitude between the predicted and experimental life. According to the
author's previous work [13], a new linear damage accumulation rule
is put forward to consider the strengthening and damaging of low
amplitude loads with different sequences using fuzzy sets theory.
Consequently, Miner's rule has been undergone many modifications in
an attempt to successfully apply this simple rule. And many researchers
have tried to modify the Miner's rule, but due to its intrinsic
deficiencies, no matter which version is used, life prediction based on
this rule is often unsatisfactory [14].
[FIGURE 2 OMITTED]
Accordingly, lots of non-linear damage accumulation methods have
been proposed to consider the loading sequence effects, which can be
depicted as shown in Fig. 2. In Fig. 2 the x-axis is the ratio of
loading cycle and the y-axis is the cumulative damage. In Fig. 2, a is
the cumulative damage under high-low and Fig. 2, b is that under
low-high loading respectively. Fig. 2, c is the damage accumulation
curve of Miner's rule. As is well known, a nonlinear damage
accumulation equation proposed by Marco and Starkey [1] is:
D = [k.summation over (i=1)] [([n.sub.i]/[N.sub.i])[m.sub.i], (2)
where [m.sub.i] is a coefficient depends on the i-th level of load.
It considers the effects of loading sequences. However, some research
showed that only in some case for some materials, the fatigue lives
predicted by Marco and Starkey's model shows a good agreement with
the experimental results. In addition, it is difficult to determine the
corresponding coefficient. Therefore, these equations have limited use
in practical engineering application.
Recently, another approach, based on damage mechanics of continuous
media, has been proposed since Kachanov presented the concepts of
"continuum factor" and "effective stress" firstly
[15]. This theory deals with the mechanical behavior of a deteriorating
medium at the continuum scale. Chaboche and Lemaitre [16] applied these
principles to formulate a non-linear damage evolution equation:
dD = f (...) dN, (3)
where the variables in the function f may be the stress, total
strain, plastic strain, damage variable, temperature and/or hardening
variables etc. In order to describe non-linear damage accumulation and
loading sequence effects, in addition, the variables and the loading
parameters in the function are inseparable. This model can calculate the
damage provoked by the cycles below the fatigue limit and considers the
effects of mean stress. Many forms of fatigue damage equation have been
derived based on Chaboche's work [17-19].
In this paper, according to the varying characteristic of fatigue
ductility, a modified non-linear uniaxial fatigue damage accumulation
model is proposed on the basis of the continuum damage mechanics theory.
And it can be used to predict failure of specimens and describe the
whole process of fatigue damage accumulation.
2. Non-linear fatigue damage accumulation theory
For the fatigue problem, it should consider the following aspects
[20]:
1. existing microinitiation and micropropagation;
2. nonlinear cumulative effects under two-stress level loading or
multi-block loading;
3. existing fatigue limit which decreases after prior damage;
4. effects of mean stress on the fatigue limit or the S-N curve.
Based on the above mentioned characteristics of fatigue damage, for
the uniaxial fatigue loading problems, the fatigue damage is defined, as
originally proposed by Chaboche, by the following differential equation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)
where the function [alpha] depends on the loading parameters
([[sigma].sub.Max], [[sigma].sub.m]), and [alpha]([[sigma].sub.Max],
[[sigma].sub.m]) is the function in the nonlinear continuum damage
model.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where [M.sub.0], a, b, b' and [beta] are material constants.
[[sigma].sub.-1] is the fatigue limit under fully reversed condition,
[[sigma].sub.l] ([[sigma].sub.m]) is the fatigue limit under non-zero
mean stress loading condition. Symbol < > is defined as <m>
= 0 if m<0, <m> = m if m>0.
Integrating Eq. (4) for constant [[sigma].sub.Max] and
[[sigma].sub.m], between D = 0 and D = 1 leads to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)
This model can be extended from damage mechanics using the
effective stress concept. And it assumes that damage does not exist
before the half-life of materials, and the damage can only be measured
at the last part of life, which cannot comprehensively reflect the
fatigue damage mechanism [21]. Thus, it is important to define an
adaptive fatigue damage variable which can establish a fatigue damage
accumulation model and can comprehensively reflect the fatigue damage
behaviour.
3. Establishment of a new non-linear fatigue damage accumulation
model
Fatigue damage is a process of the micro-cracks and the micro-hole
continually initiating and propagating due to the irreversible evolution
of material microscopic structure. This irreversible evolution process
directly affects the macro-property of materials. If the characteristic
of fatigue ductility is combined with the continuum damage mechanics
theory, we can rewrite Eq. (3) as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)
Integrating Eq. (8) from D = 0 to D = 1, the number of cycles to
failure and the damage evolution equation expressed in a function of
[n.sub.i]/[N.sub.F] can be obtained, respectively:
[N.sub.F] = (1 - [alpha])[[[[[[sigma].sub.Max] -
[[sigma].sub.m]]/M([[sigma].sub.m])]].sup.-[beta]], (9)
D = 1 - [(1 - [n.sub.i]/[N.sub.F]).sup.1/1-[alpha]], (10)
where the exponent [alpha] depends on the loading function
[alpha]([[sigma].sub.Max], [[sigma].sub.m]) = 1 - 1/a
lg([[sigma].sub.Max/[[sigma].sub.-1]([[sigma].sub.m]))], and
[[sigma].sub.Max] is the maximum stress.
If the loading parameters are the strains, the stress may be
transformed to the strain by means of the cyclic stress-strain
relationship:
[DELTA][sigma]/2 = c[([DELTA][epsilon]/2).sup.n], (11)
where c and n are the material constants, which can be obtained
from uniaxial cyclic loading tests.
Therefore, Eqs. (8) and (9) can be rewritten as follows,
respectively:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (12)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)
For two-stress level loading, the specimen is firstly loaded at
stress [[sigma].sub.1] for [n.sub.1] cycles and then at stress
[[sigma].sub.2] for [n.sub.2] cycles up to failure. In order to make use
of equivalence of damage for different loading conditions, it is
possible to establish an equivalent number of cycles [n.sub.2] applied
with stress amplitude [[sigma].sub.2] which would cause the same amount
of damage as caused by [n.sub.1] cycles at [[sigma].sub.1]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)
For the case of high-low loading sequence ([[sigma].sub.1] >
[[sigma].sub.2]), it follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)
Then fatigue cumulative damage under high-low loading sequence is
as follows:
[n.sub.1]/[N.sub.F1] + [n.sub.2]/[N.sub.F2] <
[n.sub.1]/[N.sub.F1] + 1 - [n.sub.1]/[N.sub.F1] = 1. (16)
For the high-low loading conditions, the cumulative damage is less
than unit. In the same way, it may be proven, for the low-high loading
conditions, the cumulative damage is more than unit. For the same
two-level stress loading [[alpha].sub.1] = [[alpha].sub.2], then:
[n.sub.2]/[N.sub.F2] = 1 - [n.sub.1]/[N.sub.F1] [??]
[n.sub.1]/[N.sub.F1] + [n.sub.2]/[N.sub.F2] = 1. (17)
It is reduced to the Miner rule. Similarly, under multi-stress
level loading condition and through sequential calculation, it is easy
to get a fatigue damage cumulative formula using an auxiliary variable
V:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)
The integration is pursued until [V.sub.i] = 1, which corresponds
to the fatigue life.
The proposed formulation for characterizing the damage evolution of
metals is consistent with the physical significance of fatigue damage:
1. The damage variable is satisfied with the boundary conditions
[22]:
n = 0, D = 0, n = [N.sub.F], D = 1; (19)
2. Fatigue damage is an irreversible process of material
degradation and it increases monotonically with the applied cycles [23]:
[partial derivative]D/[partial derivative]n > 0; (20)
3. The higher applied loading stress often leads to the larger
fatigue damage:
[[partial derivative].sup.2]D/[partial derivative]n[partial
derivative][sigma] > 0. (21)
4. Experimental verifications of the proposed model
The experimental data of the normalized 45 steel in [22] are used
to verify the proposed model. For the normalized 45 steel, the
mechanical properties are as follows: yield strength is [[sigma].sub.y]
= 37L7 MPa, ultimate tensile strength is [[sigma].sub.b] = 598.2 MPa.
The damage variable D is obtained by measuring the static relative
ductility change of material. The comparison between the damage
evolution curves and the experimental results are shown in Figs. 3 and
4, it should be noted that the results are satisfactory.
Since the life analysis under two-stress level loading is one of
the basic random loading analysis, many fatigue damage accumulation
models are based on the two-stress level loading experiments. Therefore,
in order to verify the application of Eq. (16) under multi-stress level
loading, two categories of experimental data of 30CrMnSiA [23] and
30NiCrMoV12 are used. The tests of 30CrMnSiA were performed by two
groups. For the first group, the mean stress is 250 MPa and the maximum
stress is 732-836 MPa. For the second group, the mean stress is 450 MPa
and the maximum stress is 797-940 MPa. The results of 30CrMnSiA between
experiment and prediction are listed in Table 1. Moreover, the
experimental data of the 30NiCrMoV12 under two-stress level loading are
used to verify the proposed model. The mechanical properties of
30NiCrMoV12 are as follows: yield strength [[sigma].sub.y] = 755 MPa,
ultimate tensile strength [[sigma].sub.b] = 1035 MPa [24]. It is shown
that in Figs. 5-7 the predicted results using the proposed model are
satisfactory compared with the experimental results, which is better
than Miner's rule.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
Though the proposed model has the same basic principles as that of
Chaboche and Lemaitre, it is easily applied, and the damage variable is
directly related with the ductility which can be measured using a simple
experimental procedure. In addition, it is verified using experimental
data of three kinds of materials, and good results are obtained.
However, the proposed model suits only for the fatigue life prediction
of ductile material, and further validation on different materials are
required. Moreover, application of the proposed method to full scale
components under complex loading conditions, such as multiaxial fatigue
loading, also needs further study.
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
5. Conclusions
By using the Chaboche's continuum damage theory as the
starting point, a uniaxial non-linear fatigue damage accumulation model
is proposed, which takes the effects of loading interaction and loading
sequences into account. In addition, according to the equivalence of
damage, the recurrence formula under multi-stress level loading is also
derived, and the comparison between experimental data available from
literature with predicted results showed a good agreement, which stated
clearly that the proposed model has a reasonable descriptive ability of
fatigue damage accumulation.
Nomenclature
D - damage variable; [[sigma].sub.Max] - maximum stress;
[[sigma].sub.m] - mean stress; M (), a () - functions in the non-linear
continuum damage model; [M.sub.0], a, b, [beta] - coefficients of the
nonlinear continuum damage model; [[alpha].sub.-1] - fatigue limit for
fully reversed condition; [[sigma].sub.1] ([[sigma].sub.m]) - fatigue
limit for a nonzero mean stress; < > - defined as <m> = 0 if
<m> < 0, <m> = m if m>0 ; [[sigma].sub.b] - ultimate
tensile strength; [[sigma].sub.y] - yield strength.
Received August 29, 2012
The authors would like to acknowledge the partial supports provided
by the National Natural Science Foundation of China under the contract
number 11272082 and the Fundamental Research Funds for the Central
Universities under the contract number E022050205.
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Rong Yuan, University of Electronic Science and Technology of
China, Chengdu, Sichuan, 611731, P.R. China, E-mail: yuanrong27@126.com
Haiqing Li, University of Electronic Science and Technology of
China, Chengdu, Sichuan, 611731, P.R. China, E-mail: lihaiqing27@163.com
Hong-Zhong Huang, University of Electronic Science and Technology
of China, Chengdu, Sichuan, 611731, P.R. China, E-mail:
hzhuang@uestc.edu.cn
Shun-Peng Zhu, University of Electronic Science and Technology of
China, Chengdu, Sichuan, 611731, P.R. China, E-mail:
zspeng2007@uestc.edu.cn
Yan-Feng Li, University of Electronic Science and Technology of
China, Chengdu, Sichuan, 611731, P.R. China, E-mail: lyfmoon@gmail.com
cross ref http://dx.doi.org/10.5755/j01.mech.19.5.5541
Table 1
Experiment and prediction comparison of two-stress level test results
for 30CrMnSiA
Stress, Loading Cycles Cycle ratio Experiment
MPa sequence [n.sub.1] [n.sub.1]/[[N.sub.f1] [n.sub.2]
732-836 Low-high 13000 0.233 6602
Low-high 15000 0.269 6501
Low-high 25000 0.448 5400
Low-high 35000 0.628 4428
Low-high 45000 0.807 3254
High-low 1200 0.167 36911
836-732 High-low 1800 0.208 32450
High-low 3000 0.417 16002
High-low 5000 0.694 6969
797-940 Low-high 100000 0.288 3625
Low-high 200000 0.576 2862
High-low 1000 0.292 106320
940-797 High-low 1700 0.497 44821
High-low 2400 0.702 22236
High-low 3200 0.936 2432
Experiment Miner rule
Stress, Loading Cycles [n.sub.2]/ [n.sub.2]/
MPa sequence [n.sub.1] [N.sub.f2] [N.sub.f2]
732-836 Low-high 13000 0.917 0.767
Low-high 15000 0.903 0.731
Low-high 25000 0.750 0.552
Low-high 35000 0.615 0.372
Low-high 45000 0.425 0.193
High-low 1200 0.662 0.833
836-732 High-low 1800 0.582 0.792
High-low 3000 0.287 0.583
High-low 5000 0.125 0.306
797-940 Low-high 100000 1.060 0.712
Low-high 200000 0.837 0.424
High-low 1000 0.306 0.708
940-797 High-low 1700 0.129 0.503
High-low 2400 0.064 0.298
High-low 3200 0.007 0.064
Hashin's rule Prediction
Stress, Loading Cycles [n.sub.2]/ [n.sub.2]/
MPa sequence [n.sub.1] [N.sub.f2] [N.sub.f2]
732-836 Low-high 13000 0.954 0.888
Low-high 15000 0.938 0.863
Low-high 25000 0.817 0.756
Low-high 35000 0.627 0.628
Low-high 45000 0.365 0.460
High-low 1200 0.570 0.680
836-732 High-low 1800 0.524 0.610
High-low 3000 0.338 0.310
High-low 5000 0.158 0.085
797-940 Low-high 100000 0.984 0.923
Low-high 200000 0.854 0.767
High-low 1000 0.297 0.327
940-797 High-low 1700 0.182 0.108
High-low 2400 0.096 0.020
High-low 3200 0.011 0.001
