Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 2, pp. 421-430, Warsaw 2015
DOI: 10.15632/jtam-pl.53.2.421

FATIGUE LIFE OF ALUMINIUM ALLOY 6082 T6 UNDER CONSTANT

AND VARIABLE AMPLITUDE BENDING WITH TORSION

Aleksander Karolczuk, Marta Kurek, Tadeusz Łagoda

Opole University of Technology, Faculty of Mechanical Engineering, Opole, Poland

e-mail: a.karolczuk@po.opole.pl; ma.kurek@po.opole.pl; t.lagoda@po.opole.pl

The paper presents the comparison of experimental and calculated fatigue lives for EN
AW-6082T6aluminiumalloy.Hour-glass shaped specimens havebeen subjected to constant
andvariable amplitude uniaxial andmultiaxial loadings, i.e. planebending, torsionand their
proportional combinations with zero mean values. Three multiaxial fatigue criteria based
on the critical plane approach have been verified being the linear combination of shear
and normal stresses on the critical plane. For the variable-amplitude loading, the rainflow
cycle countingmethod andPalmgren-Miner hypothesis have been applied. The best fatigue
criteria are pointed in the final conclusions.

Keywords: variable amplitude loading, multiaxial fatigue, aluminium alloy, critical plane
approach

1. Introduction

Numerous components of machines and devices work under variable operational loadings that
may damage the component and interrupt the work process. The phenomenon of fatigue of
materials and structures is a significant issue inmany sectors of the industry.Oneof the research
objectives on material fatigue is to identify a proper method of estimating the fatigue strength
and life on the stage of design of structural elements. Literature of the subject offers numerous
reportswhoseauthors (Carpinteri andSpagnoli, 2001;Kardas et al., 2008;KarolczukandKluger,
2014; Karolczuk andMacha, 2005; Karolczuk et al., 2008, Kluger andŁagoda, 2013; Łagoda and
Ogonowski, 2005; Niesłony et al., 2014; Skibicki and Pejkowski, 2012; Walat et al., 2012; Walat
and Łagoda, 2014) reviewed and verified multiaxial fatigue criteria. Among many multiaxial
fatigue criteria, one group that is based on the determination of an equivalent stress in the
critical plane can be distinguished as the group of increasing popularity. The popularity comes
from awide range of applicability and effectivenessof this approach. However, the critical plane
approach still requires experimental validation. The main idea of the critical plane approach
is reducing the multiaxial state of stress to the uniaxial (called equivalent) stress state by an
appropriate fatiguehypothesis and critical planeorientation.Theorientation of the critical plane
and the location of fatigue failure plane are relative to the type of material used and loading
conditions (Carpinteri et al., 2002; Walat and Łagoda, 2010).
The main aim of this paper is verification of three multiaxial fatigue criteria based on the

critical plane approach undermultiaxial constant and variable-amplitude loadings of aluminium
alloy ENAW-6082 T6 (PN-PA4).

2. Experimental research

The analysis has been conducted using the results of experimental studies under constant- and
variable-amplitude loading of EN AW-6082 T6 aluminium alloy. The chemical composition of



422 A. Karolczuk et al.

the alloy is provided in Table 1. Basic mechanical parameters of the analysedmaterial are listed
in Table 2. Cyclic properties of the analysed material are provided in Table 3.

Table 1.Chemical composition of ENAW-6082 [in weight %, EN 573-3: 2009] (the rest Al)

Cu Mg Mn Si Fe Zr+Ti Zn Cr

< 0.1 0.6-1.2 0.4-1.0 0.7-1.3 < 0.5 < 0.1 < 0.2 < 0.25

Table 2.Basic mechanical parameters of ENAW-6082 aluminium alloy

Rp0.2 [MPa] Rm [MPa] A12.5 [%] E [GPa] ν [–]

365 385 27.2 77 0.32

Table 3.Cyclic properties of ENAW-6082 aluminium alloy

Tension-compression Plane bending Torsion

εpa =(σa/K
′)1/n

′

εa =(σ
′

f/E)(2Nf)
b+ε′f(2Nf)

c logNf =Cσ logNf =Cτ
K′ n′ σ′f ε

′

f b c Aσ mσ σaf Aτ mτ τaf
[MPa] [–] [MPa] [–] [–] [–]

526 0.0651 651 1.292 −0.0785 −1.0139 23.8 8.0 154 21.4 7.7 91

where: εpa – amplitude of plastic strain; εa – total strain amplitude; σa,τa – stress amplitudes;

Nf – number of cycles to failure; σaf =σa(2 ·10
6),τaf = τa(2 ·10

6) – theoretical fatigue limits

Cσ =Aσ −mσ logσa,Cτ =Aτ −mτ logτa

Fig. 1. Fatigue life test stands: (a) cyclic loading, (b) variable-amplitude loading

Tests under constant-amplitude loadings have been performed usingMZGS-100 fatigue test
stand (Achtelik and Jamroz, 1982), in which the electric engine drives the wheel, whose centre
of mass is shifted with respect to axis of rotation (Fig. 1a). The frequency of loading was
equal to 28Hz. The fatigue failure condition of specimen is increased of vibration amplitude by
around 30%.Tests under variable-amplitude loadingswere performed on a prototype test stand,
inwhich the force was induced by an electromagnetic actuator (Fig. 1b). In this case, the failure
definition is total separation of specimen. The frequency of loading depending on the loading
level) was in the range 5-10Hz.

The tests were performed on smooth specimens of circular cross section with hour-glass
shape, as illustrated in Fig. 2.

In the case of the analysed constant- and variable amplitude loading, the samples were
subjected to: (i) plane bending, (ii) plane bending with torsion and (iii) torsion. The results of
fatigue tests of constant-amplitude bending and torsion are illustrated in the S-N diagram in a
log-log scale plot (Fig. 3), according to ASTM (ASTM, 1999).



Fatigue life of aluminium alloy 6082 T6... 423

Fig. 2. Geometry of the tested specimen

Fig. 3. Fatigue diagram for constant amplitude plane bending and torsion of aluminium alloy 6082 T6
(where τa and σa denote stress amplitudes of torsion and bending moments, respectively)

The results show that the bending and torsion S-N curves are nearly parallel (mσ = 8.0 ≈
mτ = 7.7). In the case of analysis under variable-amplitude loading, the driving signal is a
sinusoidal wave of variable amplitude. The control system randomly generated amplitudes of
the driving force on the basis of the Rayleigh distribution. However, due to limited capacity
of the test stand, the actual operational distribution differs from the generated one (Fig. 4c).
Figure 4 illustrates the specific loading characteristics of the given loading level, i.e.: (a) recorded
forceF; (b) forceF in the time range of 10s<t<15s; (c) histogramof amplitudes of the forceF;
(d) standard deviationFa,std of force amplitudes. The parameters of variable amplitude loadings
recalculated to parameters of nominal stresses on the assumption of the elastic strain range are
presented in Table 4.

Table 4.Parameters of variable amplitude loadings

λ No.
σa,m τa,m σa,std τa,std σa,max τa,max
[MPa] [MPa] [MPa] [MPa] [MPa] [MPa]

Bending 0 12 129-202 – 55-73 – 264-330 –

Torsion ∞ 13 – 68-104 – 30-42 – 198-139

Bending-torsion 0.25 12 110-160 27-39 51-66 12-16 303-357 74-87

Bending-torsion 0.50 12 85-124 42-62 38-52 19-26 248-262 124-131

Bending-torsion 1.00 12 53-75 54-76 24-30 25-31 128-150 131-154

The table contains five cases of loading described by:λ= τa,m/σa,m – ratio ofmean values of
stress amplitudes; No. – number of specimens; σa,m,τa,m –mean values of stress amplitudes for
bending and torsion loadings, respectively;σa,std,τa,std – standard deviation of stress amplitudes
for bending and torsion loadings, respectively; σa,max,τa,max –maximum values of stress ampli-
tudes for bending and torsion loadings, respectively. Exemplary photos of the fracture surface
are shown in Fig. 5.



424 A. Karolczuk et al.

Fig. 4. Example of loading characteristics of plane bending: (a) recorded force F(t); (b) force F(t) in
the time range of 10s<t<15s; (c) histogram of amplitudes of the force F with selected statistical

parameters; (d) changes in the standard deviation of force amplitudes

Fig. 5. Example of the fracture surface: (a) plane bending; (b) bending-torsion, λ=0.25;
(c) bending-torsion, λ=1.0

3. The algorithm of fatigue life evaluation

Figure 6 illustrates the scheme for estimating the fatigue life used in the calculations. The first
block refers to calculating components of stress for the high cyclic fatigue regime (HCF) (elastic
strain range). The next step involves determination of the critical plane orientation onwhich all
the stress tensor components are transformed. The specified critical plane orientation depends
on the selected fatigue criterion.

Calculations have been done (step 3) using threemultiaxial fatigue criteria, based on the cri-
tical plane concept. The coefficients in the formulas for equivalent stress are calculated according



Fatigue life of aluminium alloy 6082 T6... 425

Fig. 6. The scheme of fatigue life calculation

to typical fatigue limits. The general form of the equivalent stress according to the criteria on
the critical plane is expressed as

σeq(t)=Bτηs(t)+Kση(t) (3.1)

whereB,K are constants used for selection of specific form of the equivalent stress (Łagoda and
Ogonowski, 2005), ση(t), τηs(t) are, respectively, the normal and shear stresses in the selected
plane.
Depending on the plane orientation, the following are applied in the study:

C1 – criterion in themaximumnormal stress plane, for which B is the constant dependent on
the material type, andK =1;

C2 – criterion in themaximum shear stress plane, for which B is the ratio of fatigue limits for
bending and torsion limits σaf/τaf, andK =2−B;

C3 – criterion in the plane rotated in respect to direction ofmaximumnormal stress proposed
by Carpinteri and Spagnoli (2001), for which

B=
1

sin2β

[σaf
τaf
(1+cos2β)−2cos2β

]

K =
2+B sin2β

2cos2β
(3.2)

If the idea proposed by Carpinteri is used, the orientation of the critical plane will depend on
the fatigue limits ratio and be inclined against the maximum normal stress plane by the angle

β=
3

2

[

1−
(τaf
σaf

)2]

45◦ (3.3)

For the variable-amplitude loading, the next step involves calculation of cycles and half-cycles.
This is done with the rainflow-counting algorithm developed by Dowling (1972). The fati-
gue damage degree is determined only for the variable-amplitude loading using the Palmgren-
-Miner linear damage hypothesis, according to which the degree of damage S(T0) over the ob-
servation time T0 of the equivalent stress is calculated as follows

S(T0)=











k
∑

i=1

ni
N0(σaf/σeqa,i)

mσ
for σeqa,i ­ aσaf

0 for σeqa,i <aσaf

(3.4)

where S(T0) is the degree of damage of the material over time T0 according to Palmgren-Miner
hypothesis, T0 – observation time, k – number of class intervals of the amplitude histogram



426 A. Karolczuk et al.

(i < k), a – coefficient that allows including amplitudes below σaf, N0 – number of cycles
corresponding to the theoretical fatigue limit σaf (N0 = 2 · 10

6), ni – number of cycles with
σeqa,i equivalent stress amplitude (two identical half-cycles form one cycle, and one half-cycle
equals half damage corresponding to the full cycle).
Following the determination of the degree of damage S(T0) according to formula (3.4), the

next step for the algorithm is the calculation of the fatigue life

Tcal =
T0
S(T0)

(3.5)

In the case of cyclic loadings, the fatigue life is calculated using Basquin’s law (Kurek and
Łagoda, 2011; Kurek et al., 2014; Niesłony andKurek, 2012; Niesłony et al., 2012) of fatigue, in
accordance with the ASTM standard, expressed as

Ncal =10
Aσ−mσ logσeqa (3.6)

4. Verification of the proposed criterion

Figures 7-9 illustrate the comparison of the calculated and experimental number of cycles to
failure with the use of the discussed method of calculation for constant-amplitude loadings. In
the case of the criterion inmaximum normal stress plane (C1) for proportional loading,B may
take any value, since the shear stress value is 0.

Fig. 7. Comparison of the calculated number of cycles with the experimental number of cycles to failure
for aluminium alloy 6082 T6 under proportional loading, according to C1 criterion

Analysing the results of calculation and experimental tests (Figs. 7-9), it is determined that
for the material in question, the highest conformity of results would be achieved by using the
criterion in the maximum shear stress plane (C2 criterion) and the criterion C3. Figures 10-13
illustrate the comparison of obtained calculation strengths with experimental strengths for the
variable-amplitude loading, calculated for different values of the coefficient a, using Palmgren-
-Miner hypothesis, for a=0.5 and a=1, respectively.

5. Analysis of the obtained results

In order to assess the analysed criteria, the procedureproposed in (Walat andŁagoda, 2014) has
been applied, where the parameters of scatter between the experimental and calculation fatigue
lives are calculated according to the following correlations



Fatigue life of aluminium alloy 6082 T6... 427

Fig. 8. Comparison of the calculated number of cycles with the experimental number of cycles to
failurefor aluminium alloy 6082 T6 under proportional loading, according to C2 criterion

Fig. 9. Comparison of the calculated number of cycles with the experimental number of cycles to failure
for aluminium alloy 6082 T6 under proportional loading, according to C3 criterion

Fig. 10. Comparison of the calculated fatigue life with the experimental fatigue life for aluminium alloy
6082 T6 under random loadings, according to C2 criterion for a=0.5



428 A. Karolczuk et al.

Fig. 11. Comparison of the calculated fatigue life with the experimental fatigue life for aluminium alloy
6082 T6 under random loadings, according to C2 criterion for a=1

Fig. 12. Comparison of the calculated fatigue life with the experimental fatigue life for aluminium alloy
6082 T6 under random loadings, according to C3 criterion for a=0.5

Fig. 13. Comparison of the calculated fatigue life with the experimental fatigue life for aluminium alloy
6082 T6 under random loadings, according to C3 criterion for a=1



Fatigue life of aluminium alloy 6082 T6... 429

E =

√

√

√

√

1

n

n
∑

i=1

log2
Nexp
Ncal

(5.1)

where n is the number of samples collected for analysis.

The final parameter for assessing the criterion is

T =10E (5.2)

Table 5 lists the results of calculations of the parameter T for bending (experimental scatter
with respect to the calculated characteristic (3.6)) and other forms of loading.

Table 5.Results of calculations of parameter T , Eq. (5.2), for fatigue life scatter

Constant – Variable
amplitude test amplitude test

τ =0 (plane bending) 1.94 1.43

Criterion in the maximum
9.67 –

normal stress plane (C1)

Criterion in the maximum
2.67

for a=0.5 2.85
shear stress plane (C2) for a=1 2.87

Modified criterion of
3.21

for a=0.5 2.63
Carpinteri (C3) for a=1 2.64

6. Conclusions

On the basis of analysis of the experimental tests and performed calculations, the following
conclusions have been formulated:

• If the criterion in the maximum normal stress plane (C1) is used, the calculated fatigue
lives for aluminium alloy 6082 T6 will be incorrect.

• When the criterion in the maximum shear stress plane and the criterion (C3) are used,
the calculated fatigue lives will approach the values of experimental ones for constant- and
variable amplitude loadings.However, forboth criteria, thedeterminedscatter coefficientT
is higher than the scatter in the experimental tests. For the constant-amplitude loading,
higher conformity has been obtained by using the criterion in the maximum shear stress
plane (C2), and for variable-amplitude loading the lowest scatter has been obtained by
using the criterion (C3).

• For different values of the coefficient a in the Palmgren-Miner hypothesis, the results do
not differ significantly. For coefficient a=0.5, slightly better calculation results have been
obtained.

Project financed by the National Science Centre. Decision number: 2011/01/B/ST8/06850.

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Manuscript received September 12, 2014; accepted for print November 20, 2014