Simulation of microcrack growth for different load sequences and comparison with experimental...

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

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 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.

Fig. 3. Axial strain amplitude 3a and shear strain amplitude ga versus number of cycles N, experimental data and simulation.

A. Ahmadi, H. Zenner / International Journal of Fatigue 27 (2005) 853–861 855

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].

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.


Fig. 7. Geometry of the notched specimens of 42CrMo4V (notch factor for

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


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.

Fig. 10. Results of the simulation for two-step tests, high–low and low–high, 42CrMo4 steel.


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.

Fig. 12. Simulated lifetime for different sizes of repeated stress block and comparison 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

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.

Fig. 14. Frequency distribution of microcracks stage I versus crack direction in experiment and simulation, Cast-AlMgSi1.


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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