Simulation of microcrack growth for different load sequences and comparison with experimental...
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Simulation of microcrack growth for different load sequences
and comparison with experimental results
A. Ahmadia, H. Zennerb,*
aVolkswagen AG, Wolfsburg, GermanybInstitute for Plant Engineering and Fatigue Analysis, Clausthal University of Technology, Leibnizstr. 32, D-38678 Clausthal-Zellerfeld, Germany
Received 21 July 2004; received in revised form 7 February 2005; accepted 19 February 2005
Abstract
Generally, most machine parts are loaded with a combination of different variable forces and moments which often causes a state of
multiaxial stress in the fatigue critical areas of the parts. In this paper, a model designed to simulate the damage process based on the growth
of microcracks under the influence of cyclic loading is presented. The crack growth is initially dominated by shear stresses leading to
microstructurally short cracks (stage I) and continues to grow under the influence of normal stresses (physically short cracks). The model is
applied to tubular specimens of the aluminum alloy AlMgSi1, steel SAE 1015 and notched specimens of 42CrMo4V which are subjected to
multiaxial random loading and loading sequences of high–low/low–high stress or a consecutive loading of torsion–tension/compression and
tension/compression–torsion for various loading sequences. In terms of the microcrack distribution as a function of their orientation, the
simulated crack growth behaviour reveals a close match with the experimental results. The results of the lifetime estimation generated by
means of the new concept on the basis of microcrack growth are compared and verified with those experiences obtained from multiaxial
fatigue testing.
q 2005 Elsevier Ltd. All rights reserved.
Keywords: Microcrack growth; Crack growth simulation; Multiaxial stress state; High–low; Low–high; Consecutive loading; Lifetime
1. Introduction
Many parts of structures and engineering components are
subjected to multiaxial loading which often cause a multiaxial
stress in the fatigue critical areas. Inadequate predictions
appear particularly when the loading sequence involves effects
with high–low/low–high changes, overloads or mixture and
changing of mean stresses. If these loading effects are
combined with multiaxial stress states, most lifetime calcu-
lation concepts fail. This investigation was aimed to build an
experimental data base for the verification of a new concept of
lifetime estimation with the application of the simulation of
microcrack growth [1–7]. This fatigue damage model requires
to be expressed in terms of cracks.
0142-1123/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijfatigue.2005.02.005
* Corresponding author. Tel.: C49 5323 72 3263/220; fax: C49 5323 72
3263.
E-mail addresses: [email protected] (A. Ahmadi),
[email protected] (H. Zenner).
In the following paper, a damage accumulation model to
describe physically the material damage mechanisms will
be presented. The rate of fatigue damage accumulation
should be physically expressed in terms of fatigue crack
growth rates [8].
2. Calculation of fatigue life
2.1. The simulation software mCracksim
The program mCracksim designed with Visualbasic 6.0,
Visual CCC and Delphi can be operated with Windows and
was developed to simulate the material damage process
caused by microcrack growth. Our first aim was to predict
the microcrack distribution as a function of their orientation
as well as the simulated crack growth behaviour up to the
defined final crack length of about 500 mm during a fatigue
test. Therefore, for the verification with experimental data
an adequate parameter identification is necessary. Fig. 1
shows the flow chart of the simulation software mCracksim.
International Journal of Fatigue 27 (2005) 853–861
www.elsevier.com/locate/ijfatigue
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Fig. 1. Flow chart of the simulation software mCracksim.
Fig. 2. Microstructure, stress state and crack growth of simulation.
A. Ahmadi, H. Zenner / International Journal of Fatigue 27 (2005) 853–861854
2.2. Model concept and microcrack simulation
The model must be capable to describe microcrack
growth in metallic materials subjected to cyclic loading. The
polycrystalline material is modeled as a structure of two-
dimensional hexagonal elements. The program mCracksim
(Fig. 4) offers to simulate also the possibility irregular grains
up to real crystalline structures. Individual slip systems are
active in each grain with a randomized crystallographic
orientation u (Fig. 2). The stress state in the slip plane of
each grain is dependent on its orientation and the externally
applied loading. Only the material surface with its plane
stress state is considered.
The nucleation point of microcracks are determined by a
random generator. Note, also that the starting point of
cracks is anywhere in the grain. The shape of the
microcrack seed is a point with no spatial extension,
denoting an initial crack length of zero. It is assumed that
the points of crack nucleation are given at the beginning of
the simulation and that the crack growth starts with the first
load cycle.
The growth of microstructurally short cracks (MSC, stage
I) is simulated starting from the points of microcrack
nucleation using the equations for crack growth rate
developed by Miller [9]. Further assumptions consider the
additional effect of the normal stress on the shear crack stage I
as well as the coalescence of neighbouring microcracks.
Finally after a specific crack length has been attained a
diversion of the crack takes place from stage I to stage
II—physically small cracks (PSC). The initiating cause for
crack growth is the stress state in the direction of the
appropriate slip plane. It is assumed, that the microcrack
growth can be divided into the phases of stage I and stage II
crack growth. During the stage-I crack growth, the cracks are
driven by the cyclic shear stress, which occur in the slip
planes of the polycrystalline material. The crack growth rate
depends on the magnitude of shear stress amplitude and on
the distance (dKa) between the crack tips and the dominant
microstructural barriers, in this case the grain boundary.
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Fig. 3. Axial strain amplitude 3a and shear strain amplitude ga versus number of cycles N, experimental data and simulation.
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The microcrack growth equation is
da
dN
� �I
Z AðDtuÞa$ðd KaÞ mm=cycle (1)
where Dtu is the shear stress amplitude, d is the grain size and
a the length of the microcrack. (dKa) is the crack tip distance
to the next barrier, and A and a are material parameters. This
relationship states that the crack growth rate decreases with
increasing crack length (decreasing distance between crack
tip and barrier). Eq. (1) assumes that the shear cracks
propagate (stage I). In the current model, the grain boundary
is regarded to be the dominant material barrier. When the
stage I-crack is sufficiently long to permit an opening of the
crack front, the development of stage II (tensile) crack
occurs. At this point, the influence of the microstructure is
limited, and crack growth can be described by continuum
mechanics. Stage II crack growth can be described by the
equation proposed by Hobson, Brown and de los Rios [10]
da
dN
� �II
Z BðDsuÞbac KC mm=cycle (2)
where Dsu represents the tensile stress perpendicular to the
crack plane, and b, c, B and C are experimentally determined
material parameters. In the transition zone, the crack growth
is calculated by using the higher value between Eqs. (1) and
(2). In addition to this continuous crack growth, there is a
sudden discontinuous crack growth through crack coalesc-
ence. In the following simulations, the final crack length is
500 mm.
The simulation does not yet consider the crack growth in
the depth direction of the material. Furthermore, the
deformation behavior of microstructure, the cyclic hardening
and softening of the material and the crack opening effects
have to be taken into account in further improvement steps.
2.3. The parameter identification tool in mCracksim
The parameter identification tool is capable of identify-
ing material parameters for the simulation of microcracks to
predict the number of cycles to failure (final crack length of
aiZ500 mm). As a result of the parameter identification the
simulated S–N curves performed for uniaxial loading show
a close match with the experimental results [2].
The essential reason to integrate a parameter identifi-
cation tool in the simulation software is to describe the
effects of various loading sequences, multiaxial stress states
as well as overloads on the fatigue lifetime.
In the case of multiaxial loading, material parameters
must be determined for tension/compression loading as well
as for torsional loading. A successful identification of
parameters allows for a good comparability between the
simulations and the experiments.
The cyclic material parameters, uniaxial S–N curve data
of the investigated material and also synthetic S–N curve
data can be used as entry data for the identification tool.
Note, that for the following evaluations, the adaption of the
material constants were only determined in the case of
uniaxial loading and afterward the simulation of fatigue
crack growth is applied for a random load spectrum in the
case of variable amplitude loading.
The next step is the damage accumulation by using the
application of simulation of microcrack growth. Finally the
fatigue life is given by a defined technical crack length of
about 500 mm. In Fig. 3, as an example the axial strain
amplitude 3a and shear strain amplitude ga are plotted versus
the number of load cycles for Cast-AlMgSi1. It can be seen,
that the feature parameter identification tool in the simulation
software mCracksim is capable to predict the material
constants and also to describe as well the different failure
processes (bending or torsion) in the case of uniaxial loading
(Fig. 4).
3. Simulation results and discussion
As an example, the simulated crack growth behaviour for
variable amplitude loading subjected to smooth specimens
of steel SAE 1015 is illustrated in Fig. 5 [4].
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Fig. 4. shows how the simulation software mCracksim operates.
Fig. 5. Simulated crack growth for variable amplitude loading, SaZ450 MPa, steel SAE 1015, smooth specimens, real grain structure.
Fig. 6. Geometry of the smooth specimens of SAE 1015 and the multiaxial deformation transducer.
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Fig. 7. Geometry of the notched specimens of 42CrMo4V (notch factor for
bending, KtBZ2.0, and torsion KtTZ1.6).
A. Ahmadi, H. Zenner / International Journal of Fatigue 27 (2005) 853–861 857
It can be observed, that the microcracks occur in the
zone of maximum shear stress and change their growth
direction at a specific crack size in the direction
perpendicular to the first principal stress vector, as it
can be seen in the experiments. The simulated crack
length a (mm) versus the number of load cycles is plotted
in Fig. 5.
3.1. Results of the fatigue tests with variable amplitude
loading
The investigated materials in this study are a SAE 1015
steel (314 MPa yield stress, 456 MPa tensile stress) and a
42CrMo4V steel (743 MPa yield stress, 920 MPa tensile
stress). The specimen geometries are shown in Figs. 6 and 7.
The fatigue tests with hollow cylindrical specimens of
steel SAE 1015 were carried out strain controlled under
combined axial and torsional loading with a ratio of
ga=3aZ1:33ðta=sa z0:5Þ. For this evaluation, A Gaussian
random sequence of amplitudes (normal distribution) [11]
with a sequence length of H0Z1!104 and the irregularity
Fig. 8. Fatigue test results under combined variable amplitude loading
(proportional loading), strain controlled, smooth specimen, simulation and
experiment.
factor IZ0.99 was used. The failure criterion was the crack
initiation of the specimen.
Fig. 8 presents the test results obtained for variable
amplitude in-phase loading (proportional loading) and the
simulated fatigue-life curve subjected to smooth specimens
of SAE 1015 [7]. The comparison with the test results for
proportional random loading confirms the results of the
simulation.
Multiaxial random fatigue tests with notched speci-
mens of 42CrMo4V (notch factor for bending, KtBZ2.0,
and torsion KtTZ1.6) were carried out load controlled
under combined bending, torsion and proportional load-
ing with a ratio of ta=sa z0:5. A Gaussian random
sequence of amplitudes (normal distribution) with a
sequence length of H0Z1!106 and the irregularity
factor IZ0.99 was used for the fatigue tests [12].
The failure criterion was defined to be the failure of
the specimens and the crack initiation. Fig. 9 shows
the results of the simulation at RZK1 for combined
variable amplitude in-phase loading (proportional load-
ing). The simulated fatigue-life curve shows similar
results to those of the experiments.
It can be observed, that the accuracy of the lifetime
calculation in the case of proportional loading also for
notched specimens (components like) by using the appli-
cation of the simulation of microcrack growth with the
software mCracksim is still satisfying. The reason for that is
successfully identifying of the material constants for the
model.
3.2. Results of two-step tests high–low and low–high
Fig. 10 shows the results of the simulation for increasing
and decreasing load amplitudes achieved from a two step
loading experiment.
In the case of a high–low sequence the damage sum is
D!1, whereas in the case of low–high sequence DO1. The
comparison with two-step tension/compression carried out
Fig. 9. Fatigue test results under multiaxial proportional loading, load
controlled, notched specimen, simulation and experiment.
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Fig. 10. Results of the simulation for two-step tests, high–low and low–high, 42CrMo4 steel.
A. Ahmadi, H. Zenner / International Journal of Fatigue 27 (2005) 853–861858
with smooth specimens experimental data taken from
Schulze et al. [13] also confirms as well as results of the
simulation.
Fig. 11 shows the simulated SN-curve and the fatigue life
curve for the Gassner 8-block-program test for different
types of load spectra. The load spectrum with a block size of
H0Z1000 and a shape of nZ2 (normal distribution) is
shown in the diagram.
It can also be observed that there is no effect of the
sequential loading on the lifetime, if the block size has been
repeated several times.
Fig. 12 shows a comparison of the simulated lifetime for
different sizes of repeated stress blocks and the experimental
data, Lipp [14]. The results for 8-block program test applied
to specimens of the steel SAE 5140 show a shortening of the
lifetime with a decreasing of the block sizes. This effect can
also be seen in the simulation.
3.3. Results of consecutive load tests
The results of the simulation are plotted in Fig. 13,
whereas Miner’s linear damage accumulation is represented
by the straight line.
The results show that D!1 for torsion–tension loading
sequences (t/s) is and for tension–torsion loading (s/t)
sequences it turns out that DO1. The found behavior
corresponds to a large extent with the experimental data,
Harada and Endo [15].
Fig. 11. Simulated lifetime for the Gassner 8-block program test with
different types of the load spectra, block size H0Z1000 and shape nZ2.
3.4. Comparison between simulation and experimental
results
The orientation, length and density of microcracks
depend on the magnitude and type of loading. The density
and orientation of microcracks taken from experimental
results with aluminum AlMgSi1 compared to the density
and orientation of microcracks obtained from simulation
results are shown in Fig. 14 for uniaxial and multiaxial
loading cases. These observations being based on surface
replica studies [16].
In addition to the crack density, the variation of normal
and shear stress amplitudes acting on the crack plane are
also plotted, respectively. In all loading cases, the
maximum crack density appears in the direction of
maximum shear planes. In contrast, the experiment yields
an irregular distribution of microcracks affected by
anisotropy of the material structure. The anisotropy might
be caused by the rolling process. The results of the
simulation are in a qualitatively good agreement with
experimental results.
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Fig. 12. Simulated lifetime for different sizes of repeated stress block and comparison with experimental results.
A. Ahmadi, H. Zenner / International Journal of Fatigue 27 (2005) 853–861 859
4. Conclusions
A microcrack simulation software is presented in this
paper. A useful feature of the software is the ability to
determine the material parameters for the simulation of
Fig. 13. Damage accumulation for tension/compression–tors
microcrack growth and to predict the results of experimental
tests subjected to multiaxial random loading with a
reasonable accuracy. From the study described in this
paper, reliable statements are obtained about the fatigue life
estimation by using the simulation software mcracksim.
ion sequences, experiment and simulation, 1045 steel.
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Fig. 14. Frequency distribution of microcracks stage I versus crack direction in experiment and simulation, Cast-AlMgSi1.
A. Ahmadi, H. Zenner / International Journal of Fatigue 27 (2005) 853–861860
In the future, the simulation software should be extended for
achieving higher accuracy in the prediction of the fatigue
life for structural components.
Acknowledgements
The authors would like to thank the German Research
Council ‘Deutsche Forschungsgemeinschaft DFG’ for the
financial support of the research program.
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