Lifetime simulation under multiaxial random loading with regard to the microcrack growth

Lifetime simulation under multiaxial random loading

with regard to the microcrack growth

A. Ahmadi a, H. Zenner b,*

a Volkswagen AG, Wolfsburg, Germanyb Institute for Plant Engineering and Fatigue Analysis, Clausthal University of Technology, Leibnizstr. 32, D-38678 Clausthal-Zellerfeld, Germany.

Received 1 October 2004; received in revised form 8 August 2005; accepted 12 September 2005

Available online 5 January 2006

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 the most cases, a non-proportional cyclic multiaxial state of stress occurs. Compared to the in-

phase loading, a multiaxial loading with a phase shift between the stress components and a load ratio of ta/saz0.5 between tension/compression

and torsion leads to a significant influence on the fatigue lifetime. The reason is the changing direction and rotation of the principal stresses during

one cycle. 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 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; In-phase; Out-of-phase; 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. The influence of multiaxial,

in-phase or out-of-phase stresses and strains on the fatigue life

becomes more and more important for fatigue analysis using

numerical concepts [1,2]. This investigation was aimed to build

an experimental database for the verification of a new concept

of lifetime estimation with the application of the simulation of

microcrack growth [3–7]. This fatigue damage model requires

to be expressed in terms of cracks.

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].

0142-1123/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijfatigue.2005.09.015

* Corresponding author. Address. Tel.: C49 5323 72 2201; fax: C49 5323

72 3264.

E-mail address: [email protected] (H. Zenner).

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.

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 modelled as a structure of two-

dimensional hexagonal elements. The program mCracksim

offers to simulate also the possibility irregular grains up to

real crystalline structures. Individual slip systems are active in

each grain with a randomised crystallographic orientation u,

Fig. 2. Metallographic slip planes beyond that are not

International Journal of Fatigue 28 (2006) 954–962

www.elsevier.com/locate/ijfatigue

http://www.elsevier.com/locate/ijfatigue

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 28 (2006) 954–962 955

considered. The stress state in the slip plane of each grain is

dependent on its orientation and the externally applied loading.

By the following simulations in this paper only the material

surface with its plane stress conditions is considered, where the

crack grows at the material surface as shear and tensile crack.

In the case of biaxial loads s1Zs2 cracks start to grow into the

material, for this opportunity provide the simulation model

some assumption for microcrack growth simulation [6].

The nucleation point of microcracks is 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 stages I to 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 phases of stages I and 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. The microcrack

growth equation is

da

dN

� �I

ZAðDtuÞaðdKaÞ 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

A. Ahmadi, H. Zenner / International Journal of Fatigue 28 (2006) 954–962956

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

ZBðDsuÞbacKC 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 coalescence [6]. In

the following simulations the final crack length is 500 mm.

The simulations in this paper do not consider the crack

growth in the depth direction of the material. Furthermore, the

deformation behaviour of microstructure, the cyclic hardening

and softening of the material, specially the out of phase

hardening and the crack opening effects have to be taken into

account in further improvement steps.

The major advantage of the presented microcrack concept

over other possible multiaxial models is to have a simplified

model with only five material parameters, which could be

delivered by the parameter identification tool in mCracksim

from uniaxial/multiaxial S–N curves of the material or

component. The verification of the simulations and exper-

iments specially for multiaxial load situations in Section 3

confirms the suitability of these damage accumulation model.

The state of damage is described as the length of maximum

micro crack.

2.3. The parameter identification tool in mCracksim

The parameter identification tool is capable of identifying

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 [4].

The essential reason to integrate a parameter identification

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

was only determined in the case of uniaxial loading and

afterward the simulation of fatigue crack growth is applied for a

random load sequence in the case of variable amplitude loading

and for constant amplitudes.

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. Fig. 3 shows how the simulation software mCracksim

operates.

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 (Ck 15) is illustrated in Fig. 4.

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. 4.

3.1. Results of the fatigue tests with variable amplitude loading

The investigated materials in this study are a SAE 1015

(Ck 15) steel (314 MPa yield stress, 456 MPa tensile stress)

and a SAE 1042 (42CrMo4V) low alloyed steel (743 MPa yield

stress, 920 MPa tensile stress). The specimen geometries are

shown in Figs. 5 and 6.

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=saz0:5Þ. For this evaluation, a Gaussian

random sequence of amplitudes (normal distribution) [11]

with a sequence length of H0Z1!104, RZK1 and the

irregularity factor IZ0.99 was used. The failure criterion was

the macroscopic crack initiation of the specimen.

Fig. 7 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 (Ck 15) [6]. The comparison with the test results for

proportional random loading confirms the results of the

simulation.

Multiaxial random fatigue tests with notched specimens of

SAE 1042 (42CrMo4V), notch factor for bending, KtBZ2.0,

and torsion KtTZ1.6, were carried out load controlled under

combined bending, torsion and proportional loading with a

ratio of ta=saz0:5. A Gaussian random sequence of

amplitudes (normal distribution) with a sequence length of

H0Z1!106, RZK1 and the irregularity factor IZ0.99 was

used for the fatigue tests [12,13]. The failure criterion was

defined to be the failure of the specimens and the crack

initiation. Fig. 8 shows the results of simulation at RZK1 for

combined variable amplitude in-phase loading (proportional

Fig. 3. Simulation software mCracksim.

Fig. 4. Simulated crack growth for variable amplitude loading, SaZ450 MPa, steel SAE 1015 (Ck 15), smooth specimens, real grain structure (dz45 mm).

Fig. 5. Geometry of the smooth specimens of SAE 1015 (Ck 15) and the multiaxial MTS-deformation transducer.


Fig. 9. Fatigue test results for variable amplitudes under non-correlated loading

(non-proportional loading) and proportional loading, load controlled, notched

specimen, simulation and experiment.

Fig. 6. Geometry of the notched specimens of SAE 1042 (42CrMo4V) (notch

factor for bending, KtBZ2.0, and torsion KtTZ1.6).

Fig. 7. Fatigue test results under combined variable amplitude loading

(proportional loading), strain controlled, smooth specimen, simulation and

experiment.


loading). 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 or loading

cases with small changing of first principal stress directions by

using the application of the simulation of microcrack growth

with the software mCracksim or other conventional hypotheses

is still satisfying.

Fig. 8. Fatigue test results under multiaxial proportional loading, load

controlled, notched specimen, simulation and experiment.

Fig. 9 shows a comparison of experimentally determined

and simulated results for tests with proportional and non-

correlated (non-proportional) loads. The simulated results

correspond to a large extent with the experimental data.

Compared to the experimental data for a non-correlated

multiaxial loading with a load ratio of ta/saZ0.5, the

simulation results lead to an increase of the fatigue life (factor

of 2 compared with proportional loads). From these results, it

can be stated that due to the out of phase loading, the effective

shear stress amplitude in Eq. (1) is reduced compared to the in-

phase loading. For the uncorrelated load case most multiaxial

damage concepts lead to an overestimation of the fatigue life.

The results of the simulation for all test series are plotted in

Fig. 10 and compared with the experimental data. The

simulated results show a good correlation with the experi-

mental data.

In the case of proportional loading, the effective shear stress

amplitude tu in the slip planes is increased compared to the

non-proportional loading. On the other hand, due to out of

phase loading, different deformation processes because of out

of phase hardening effects and different material behaviour

lead to a reduction or increasing of fatigue life depend on

Fig. 10. Comparison of fatigue life, simulation mCracksim and experiments,

steel SAE 1042 (42CrMo4V), notched specimens, D is damage sum.

Table 1

Material behaviour under in-phase and out-phase loading, smooth specimens (dZ908 phase shift between shear and normal stress/strain)

Material Experimental procedure NdZ908/NdZ08 Reference

saZconst 3aZconst

0.39% C steel saZconst 0.1 Tipton

StE 460 saZconst 3aZconst 2–3 0.3 Sonsino

SAE-1045 saZconst 0.4 Sonsino

SAE-1045 saZconst 3aZconst 5 0.6 Pan

25 CrMo 4 saZconst 0.6 Grun

30 CrNiMo 8 saZconst 3aZconst 1.0 0.4 Sonsino

X6CrNiTi 1810 saZconst 3aZconst 3.8 0.5 Hug

X10CrNiTi 189 saZconst 3aZconst 1.0 0.5 Sonsino

AlMg4-5Mn 3aZconst 0.8 Vormwald

Al2O36061 Al T6 3aZconst 0.2 Xia

AlMgSi 1 3aZconst 0.5 Zenner

AZ 91 3aZconst 3–4 Zenner


different experimental procedure [14]. This effect can be shown

clearly in Table 1.

According to the results taken from Table 1, load controlled

tests with smooth specimens of ductile materials, for example

Fig. 11. Modification of the Palmgren-Miner rule for taking into account

damage below the endurance limit.

Fig. 12. Statistical analysis of the cumulative damage sum and comparison of

different modification of Miner’s rule with the simulation of microcrack

growth.

of steel SAE 1045 (Ck45) under out of phase loading, reveal an

increase of fatigue life (factor of about 5), but strain controlled

a decrease of fatigue life (factor of about 0.6). Components

have always notches and notches behave strain controlled state

at the critical area, only results obtained under strain controlled

smooth specimens are relevant and realistic for the develop-

ment of practically applicable damage models. For future

developments fatigue damage models which ignore these

different material behaviour of fatigue life should not be used

for fatigue life calculations.

Note that another misconception by the fatigue cycle

counting causes an unsuccessful calculation of the fatigue

life. It is necessary to analyse the different deformation

processes during one load cycle and to quantify the failure

process [15,16]. In addition to that due to the changing of first

principal stress directions in the case of out of phase loading

more crack plane with regard to microcrack growth are active.

More active planes means higher crack density in the case of

changing first principal stress directions. As a result, a

microcrack experiences more stress cycles during one load

cycle.

Fig. 13. Comparison of fatigue life, simulation mCracksim and experiment,

notched specimens, variable amplitude loading.

Table 2

Mean values and scatter ranges of damage sums for simulation of microcrack

growth and nominal stress concept with different modifications of Miner’s rule,

notched specimens, variable amplitude loading

69 test results (DABEF) Deff TD

Simulation mcracksim 0.99 3.2

Miner elementary 0.49 6.1

Miner modified (Haibach) 0.32 5.4

Miner consistent 0.32 6.0

Miner modified (Liu-Zenner) 0.85 2.9

Fig. 14. Frequency distribution of microcracks stage I versus c


Please note that for notched specimens the notch region is

not modelled. But the simulation takes into account of different

strain/stress gradients around the notch by identifying the

material parameters from the component S–N curves, see

above Section 2.3.

3.2. Reliability of fatigue life estimation

For validation of the simulation concept, 11 various steel

materials with 69 experimental horizons taken from DABEF

database [17] (failure criterion fracture) were evaluated.

rack orientation in experiment and simulation, SAE 1015.


All test series were carried out with different notched

specimens and a Gaussian random sequence of amplitudes

with the irregularity factor IZ0.99 was used for these

investigations. The determination of effective damage sums

and scatter band TD (log. normal distribution) are given by the

following equations:

Deff ZNexperiment; 50%

Nsimulation; 50%(3)

and

TD ZD90%

D10%(4)

Fig. 12 illustrates the statistical analysis of the cumulative

damage sum for the simulation of microcrack growth and

comparison with the nominal stress concept by different

modifications of Miner’s rule, Miner elementary, Miner

modified (Haibach), Miner consistent and Miner modified by

Liu-Zenner [18]. In the modification of Miner consistent is

assumed that the endurance limit decreases continuously by

increasing the damage [20]. Fig. 11 indicates clearly the

different modifications of Miner’s rule based on the nominal

stress concept. The major object of the modifications is the

assumption that how the amplitudes below the endurance limit

are damaging.

It can be shown that the simulation software mcracksim gives

a more accurate assessment of fatigue life (TDsimZ3.18) than

the nominal stress approaches with the known modifications of

Miner’s rule, with the exception of the modification Liu-

Zenner. As a reason for that it can be supposed that the damage

accumulation with regard to the microcrack growth is not

linear and the simulation model is capable to describe the

effects of various loading sequences, multiaxial stress states as

well as overloads on the fatigue lifetime quantitatively good

(Fig. 13).

Table 2 gives a overview about the mean values and scatter

ranges of damage sums for the simulation of microcrack

growth with the software mCracksim and for different

modifications of Miner’s rule.

3.3. 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 steel SAE 1015 (Ck 15) 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 are based on surface replica studies [19].

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. Despite

that, the results of the simulation are in a qualitatively good

agreement with experimental results.

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 microcrack growth

from uniaxial tests with constant amplitudes and to predict the

results of experimental tests subjected to uniaxial and

multiaxial loading with constant and variable amplitudes

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. In the

future, the simulation software should be extended for

achieving higher accuracy and a field for wider application in

the assessment of 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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Documents

Lifetime simulation under multiaxial random loading with regard to the microcrack growth