Multiaxial fatigue of an alumina particle reinforced aluminum alloy

Int. J. Fatigue Vol. 20. No. 1, pp. 51-56, 0

1998

1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0142-I 123/98/$19.00+.00

ELSEVIER

PII: SO142-1123(97)00114-X

Multiaxial fatigue of an alumina reinforced aluminum alloy

particle

2. Xia and F. ElIyin* Department of Mechanical Canada T6G 2G8

Engineering

University

of Alberta

Edmonton,

Alberta,

Multiaxial fatigue tests were performed on thin-walled tubular specimens made of a particulate reinforced metal matrix composite-22% vol. Al,O,, 6061 Al with T6 heat treatment condition. Four types of strain-controlled, fully-reversed cyclic tests were investigated. They included: uniaxial; pure shear; equibiaxial in-phase; and 90” out-of-phase loading paths. It is found that the conventional equivalent stress or equivalent strain criteria can only correlate individually the fatigue life of this metal matrix composite for each type of loading path. However, these parameters cannot be used for general multiaxial loading cases. In contrast, a total strain energy density parameter does correlate, of all four types of loading paths by a unique strain energy-fatigue Science Ltd.

very well, with the test results life curve. 0 1998 Elsevier

(Keywords: alumina particles; aluminum alloy; metal matrix composite; multiaxial fatigue; strain energy)

stress. The latter is due to the stress concentration especially near the area with large size of particles and clusters”. It is found that at the high-cycle fatigue regime, the composite behaviour is generally equal or superior than the matrix materials. However, at the low-cycle fatigue region, both inferior and superior results have been reported*,“. The comparative results also depend on the chosen parameters. For example, when the strain range is used, the strain-life curves of the composites are mostly lower than that of the matrix materials, whereas when the stress range is used, the stress-life curves of the composites can be higher than that of the matrix materials’. A stress-strain product was also used to compare the performance in Ref. s. It should be noted that all the above mentioned studies are confined to the uniaxial cyclic loading case. Investigations of the behaviour of PMMCs under multiaxial cyclic loading conditions have scarcely been reported”. To the best of the author’s knowledge, no multiaxial fatigue life data have been published in the open literature. Under service conditions, most load carrying components are subjected to loads of a multiaxial and cyclic nature. Therefore, the multiaxial fatigue properties of the PMMCs are required to further the use of these composites in the industrial application. In this paper, results of multiaxial fatigue tests on thin-walled tubular specimens made of a particulate reinforced metal matrix composite, 22% vol. A1203p 6061 Al with T6 heat treatment condition will be presented. Four types of strain-controlled fully reversed cyclic tests are investigated: uniaxial; pure shear; equibiaxial in-phase; and 90” out-of-phase loading paths.

INTRODUCTION Particulate reinforced metal matrix composites (PMMCs) are an important class of engineered materials having emerged during the last decade. They show substantial advantages such as increased stiffness, strength, creep and wear resistance, and have superior performance at elevated temperatures. In contrast to long fiber reinforced metal matrix composites with highly directional properties, the PMMCs generally have isotropic properties and are much easier to pro-

duce by standard metallurgical methods (power metallurgy, casting, forging, rolling and extrusion)‘-“. There is increasing interest in application of these materials, predominantly in the sports equipment, automobile and high speed machinery industries4. Some examples are diesel pistons, automotive drive shafts, brake rotor disks, bicycle frames, etc. A good understanding of the fatigue behaviour and the corresponding fatigue life data of the PMMCs are important for further extension of their applications. There is a relatively extensive literature concerned with the mechanisms of fatigue-crack initiation and growth properties’- ‘. The introduction of ceramic particulates into metals (matrix) may have both beneficial and detrimental effects on their fatigue performance. The beneficial effect has been ascribed to delayed initiation of fatigue crack’ or to a short crack trapping mechanism’O. The main disadvantages are attributed to a decrease of ductility, and an increased local maximum *Author

for correspondence.

51

Z. Xia and F. Ellyin

52

external pressures (!I, and I>~) and an axial force. F. The axial. circumferential (hoop) and the radial stresse\ are determined from:

R40

(1)

Different parameters are explored to correlate the multiaxial fatigue life data. It will be shown that a total strain energy density parameter is the most appropriate among the ones investigated here. By using this parameter, all the test results of the four different loading paths can be correlated by a single energy densitylife curve. MATERIAL.

SPECIMENS

AND

TEST

SET-UP

The material tested was WD-22A of Duralcan USA. a 6061 aluminum alloy reinforced with 22% volume fraction of alumina particulate. The specimens were cut from extruded tubes with 45 mm outside diameter and 2 mm thickness. The specimens were heat treated with T6 condition, that is, solution heat treated at 530°C for 4.5 min. cold water quenched, natural aging for 24 h, and then precipitation heat treated at 175°C for 8 h. After the treatment, the specimens were fine machined to the final dimensions as shown in Figuw 1. The tests were performed on a servo-controlled electro-hydraulic closed-loop machine. a modified MTS system. A schematic diagram of the test set-up for loading of thin-walled tubular specimens is shown in Figuw 2. The specimen was subjected to internal and

where (I, and rl, are external and internal diamcterx ot‘ the gauge length Uld the specimen at A = ~(03 -- c/f)/4 is the cross-sectional area. The \pecimen thickness is \‘cry small in comparison K ith the diameter. and therefore the radial stress. CJ,.is much smaller than the axial and hoop stresses. Thn\ it can be assumed that the specimen is essentially sub,jcctetl to a biaxial stress state under the above loading COIIdition. A pair of grips wet-c designed (not shown in FigL/w 2) to hold the specimen. The LILLY grip assembly is 01‘a floating typ e and thus prevents introducing bending stresses caused by misalignment in the axial direction. All tests were performed under a struincontrolled mode. The input signals were gencratcd by a computer program. Through the feedback systems, and axial load were ad,justecl the internal pressure (external pressure was constant during the test) to maintain the value of axial and diametral strain\ ready ing to those of the initial input signals (see F‘i,qlr~r~3. Data were recorded in a real time mode on I’OUI channels (axial and diametral strains. internal prcss~~rc and axial load) by a computer, and the hy\tcresi\ loops in axial and hoop directions were plotted on the monitor. EXPERIMENTAL<

PROCEDURES

AND

All the tests wcrc conducted at room four types of controlled cyclic strain strain-hoop strain plane, are shown three biaxial cyclic strain paths, can

RESUL.TS

temperature. The paths. in the axial in Fig~rr~~ 3. The be expressed as:

(7) in which for (b = 180”, it corresponds to a pure shear state in a plane of 45” inclined to the two principal strain directions; for 4 = O”, the applied strain path is an equi-biaxial in-phase (proportional) cycling; whereas

Et Feedback

I

L Feedback

Axial Extensometer

E

Figure3 Figure 2

A schematic

A

diagram

of the test set-up

shear,

CD;

Four types of cyclic strain paths: cqui-hiaxial. uniaxial, EF: 90” out-of-phase, cIrclc

,\H:

pun

Multiaxial

fatigue

of an alumina

when 4 = 90”, a circular (non-proportional) cyclic strain path, is obtained. The fatigue tests are automatically terminated when an initiated crack penetrates through the wall of the specimens. A through the wall crack causes the oil to flow between the internal and external pressure systems, and an error is detected by the feedback system because the required pressure difference cannot be sustained. In practice this detection was found to be very sensitive. and fairly small cracks were detected. The uniaxial fatigue tests were performed on both solid and thin-walled specimens in order to confirm that the results from the thin-walled specimens are consistent with the solid ones. For the thin-walled specimens, a small amount of internal pressure was applied ( I IO MPa in hoop stress) in order to detect the crack in the uniaxial fatigue tests. The deformation behaviour of this composite material under biaxial cyclic loading has been reported in another paper by the authors”. It was noted that with the T6 heat treatment, the material showed a very small amount of cyclic hardening during the first few cycles, and thereafter a stable response was observed during the cyclic loading process, see Figures 7, 14 and 17 in Ref. 12. Due to the space limitation, similar figures are not produced in the present paper. In total 36 specimens were tested. For a given strain range of each loading path, generally two tests were carried out to ascertain the reliability of the results. The test results are summarized in Tcrhlr 1.

particle

reinforced

aluminum

TO

Various fatigue failure criteria for the multiaxial stress state have been suggested in the past. ElIyin” has summarized these criteria by classifying them into three categories: stress-based; strain-based; or energy-based criteria. In the following four commonly used parameters will be chosen to fit the multiaxial fatigue data of the PMMC. Maximum

rquivalrrzt

stress

In the stress-based criteria, von Mises stress is the most popular parameter used for metals and alloys. For the biaxial stress state, the maximum equivalent stress is defined as

Muximum

eyui\lnlrnt

sttzjll

Here the maximum value of von Mises equivalent strain is used as a parameter to correlate the fatigue data. It is defined as: 5

ma\

=

( 2e,, e,,/3)$,

(4)

where t’,, are the deviatoric strain components. using Equation (2), Equation (4) reduces to: 2( I + V) 3( 1 - 5) 0 for

By

dj = 0”

I \ 2( 1 + V)

(5)

For the uniaxial cyclic loading. %. ,,,a\ =

3

0

In the above i is an effective be calculated from’?

(6) Poisson’s ratio. It can

where I’~ is the elastic Poisson’s ratio. ZJ,,is the plastic Poisson’s ratio ( = 0.5 for von Mises plastic materials), <, & are elastic parts of the total strains. E:,, E,,. Note that V = v, if E:,= <. E,,= G; and V = I’,, if <; = 6; = 0. The values of cc, ,,,i,\ are listed in the sixth column of Tuhle I. The data points are shown in Figure 5. Four best fit lines for the four different types of loading paths are also shown in Figure 5. The order of the four lines from the low to the high is the same as that in the Figure 4. If a single curve is used to fit all the data points, the equation of the best tit curve would be, Ed,,,,ilh= 0.902N,

where u,, (J? are principal stress values and a, = 0. For each fatigue test the value of Us,milxis listed in the fifth column of TLzble 1. The listed values of a,., ,,,.,x, as well as others in the table (E,, ,~ax, 7*, AW) are based on the recorded stress-strain response of the two-hundredth cycle, N = 200. The stress-response of this material with T6 heat treatment condition were relatively stable during the cycling process. Therefore, these values listed in Table I, are also nearly constant for each of the fatigue tests. The correlation of the cre, n,i,xwith the fatigue life of the specimens is shown in Figure 4. It is seen that the data points are spread over a relatively wide range in the figure and they cannot be correlated by using a single curve. However, for each individual type of loading path, the data points

53

can be approximately fitted by a linear curve as shown in Figure 4. These lines are obtained through the least square method. It is noted that the curve for the 90” out-of-phase non-proportional loading is the lowest one, that is, for a specified equivalent stress, the life is the shortest. The sequence of severity thereafter are: equibiaxial in-phase; uniaxial; and pure shear. Furthermore, the three lower lines are nearly parallel to one another. These trends are consistent with the experimental data of steels and other metallic alloys”.

2( 1 + V) FITTING SEVERAL FATIGUE PARAMETERS THE MULTIAXIAL TEST RESULTS

alloy

” Ia5

(8)

This curve is shown in FiRwe 5 with a dashed line. It can be seen that the scatter band around this best fit line is relatively large.

Critical plane models proposed by Brown and Milleris and other investigators are argued on an interpretation of the fatigue process under simple loadings whereby cracks generally grow on particular planes, termed as critical planes. One of the most often used parameters in correlating multiaxial fatigue data is expressed as: Y” = Ym:,,+ k%,

(9)

Z. Xia and F. El/yin

54 Table 1

Summary

NO.

of tee

results

6 (“)

2 3 4 5 6 7 x 9 IO II 12 13 14 15 16 17 18 I’) 20 21 22 23 24 25 26 27 2X 29 30 31 32 33 34 35 36

0

( IO

‘1

I 80

0.20

I80 I80 I80 I80 I X0 I80 I80 I80 I80 0 0 0 0 0 0 0 0 0 YO YO 90 90 90 90 90 90 uni. uni.

0.25 0.15 0.30 0.275 0.33 0.15 0.10 0.20 0.30 0.20 0.25 0.15 0.10 0.25 0.20 0.15 0.10 0.065 0.20 0.25 0.15 0.10 0.25 0.15 0.20 0.075 0.30 0.20 0.10 0.30 0.15 0.395 0.206 0.405 0.28 I

uni. uni. uni. uni. uni. uni. uni.

N,

(cycles)

t, ,,,./\( IO

(‘c. ,/>.,\ (MPa)

53 47s 23 0.53 86551 so13 27X6 855 I78 295 > I 500 000 49 178 5542 2325 357 IOYII 358 912 291 3940 37 645 426 542 > I 500 000 16523 586 43 218 147 957 1561 38 301 29.538 > I 500 000 I574 35 000 > I so0 000 1472 999 033 6154 99 00 I 3029 XX2

II

267.X 31 I.1 NY.3 346,s 338.3 359.3 198.7 125.5 254.2 343.6 2XY.2 206.5 2 3 3 ..3 144,s 302.7 267.5 217,s 147.X 101.X 21 I.6 264.X 169.9 130.9 250.3 160. I 219.9 83.1 308.2 219.x 107.2 306.0 16 I .O 267.X 201.1 307.6 2.59.0

y!: (IO

?I

2)

0.23 I

0.200

0.289 0. I73 0.346 0.3 IX 0.3x I 0. I73 0. I I6 0.23 I 0.346 0.277 0.377 0.205 0. I.32 0.373 0.2x7 0. I99 0. I32 0.0860 0. IY6 0.255 0. I48 0.0990 0.254 0. I40 0. I x7 0.0702 0.269 0. I78 0.0887 0.268 0. I.13 0.350 0.183 0.378 0.353

0.250 0. I SO 0.300 0.275 0.330 0. I so 0. IO0 0.200 0.300 0.2 I I 0.293 0. I66 0.0996 0.290 0.220 0. I49 0.0996 0.0637 0.172 0.219 0. I29 0.0X64 0.21x 0. I26 0. I63 0.063 I 0.216 0. I44 0.07 IS 0.2 16 0.10x 0.2X3 0. I47 0.286 0.203

AL+’ (MJ 11,

4, phase lag, Equation (2); LI, strain amplitude, Equation (2); N,, number of cycles tO failure; cr,,,,,,,,, maximum equivalent stress, Equation E,,,,,,,, maximum equivalent strain, Equations (5) and (6); y*, multiaxial fatigue parameter hascd on critical plane, Equation (Y): AM”, multiaxial fatigue parameter based on strain energy, Equation (I 9).

‘I

(3):

400

300

i

k

c L,

hl z,

j

E ,,_

\

= 0.902 Nf-

200

@=I80

00000

?oooop=L1 ?

100

e

nnnnnp=90 00000 unioxial,~tibuior thH~f&

uniaxial,solld

OWrr IO2

IO3

Number Figure 4 multiaxial

H

OG1VOO uniaxial,tubulor f-&k-k+ unioxial,solid IO4

of

Cycles

Correlation of the maximum fatigue life data

IO5

to

IO6

10:

Failure,

equivalent

stress with

N,

; 3

I~,

III7

’ ’“““’ ’ “““’ IO3

PJumber

the

Figure 5 multiaxial

10 4

of

’ “““’ ’ “‘mrl

Cycles

Correlation of the maximum fatigue life data

lo5

to

IC”

Failure.

equivalent

strain

“‘T

IO’

N, with

the

Multiaxial

fatigue

of an alumina

where ymax is the maximum shear strain, E, is the normal strain value on the plane of the maximum shear strain and k is a material constant. Equation (2) gives 1 - 3v 2(1-V)a

(I + if)

Y,nax=

2(1-i$==

for 4 = 0”

(10) ,,‘I + i.J2 Ymnx=

reinforced

Multiaxial theor?:

for 4 = 90”

(11)

6, =

for 4=

0

180”

(12)

1+ P - 3 2 a 9 En =!’ 2 n

parameter

based nn strain energy

Ellyin et al. have developed a fatigue damage theory, in which a parameter based on strain energy is correlated with multiaxial fatigue life data of materials. The suggested function relation is expressed as”:

In the above, AIV’ is the plastic strain energy per cycle and is calculated from:

The following function relation is adopted to fit the experimental data y” = 3/l,,;,,+ ke, =

KM;’

(14)

The best fit result to the data noints with the corresponding values of the constants- is y” = ylnn\ + 0.346” = 0.73Ofvf

‘).‘33

(15)

The results are shown in Figure 6. Comparing this figure with Figure 5, one would conclude that no significant improvement in the correlation is obtained

8i b

Y = Y,,

+ 0.34en

u;, de;

= 0.730

where a,; is the stress tensor and EF;is the plastic part of the strain tensor. AW’ + is the elastic energy associated with positive (tensile) stress and is calculated from:

N,+.“’

AM/“+ =

H(a,)H(dc)a; i CYClC

y*

[Equation

(9)]

with the

dr;‘

(18)

where a, (i = 1, 2, 3) are principal stresses and r;’ are elastic part of the principal strains, and H(x) is the Heaviside function, that is, H(x) = 1 if x > 0 and H(x) = 0 if x 5 0. fi is called multiaxial constraint factor (MCF). K, a and C on the right-hand-side of Equation (15) are material constants, where C is a non-damaging energy associated with a material’s fatigue limit. The MCF, fi, is associated with the surface constraint where cracks initiate in most metallic alloys in low cycle fatigue. The MCF is determined by considering different surface constraint conditions for different biaxial stress states. For uniaxial stress state, 6 = 1. It varies from fi = 1 + V (pure shear) to fi = 1 - V (equi-biaxial loading). In Ref. I’, it is pointed out that for high cycle fatigue the MCF is not necessary due to the highly localized nature of the fatigue damage. For the particulate reinforced metal matrix composites the damage process has an intrinsic characteristic of a high localization due to the existence of reinforced particles. The fatigue crack initiation can take place at particle-matrix interface, processing related intermetallic inclusions. large particles fracture during manufacturing, and crack also initiate at clusters of reinforcement. Therefore, the influence of the surface constraint may not be dominant compared to the above characteristics. Furthermore, the material tested in the present study was heat treated with T6 condition. For most tests the response was elastic or nearly elastic with a small plastic deformation. Based on the above consideration, the total strain energy parameter will be used to correlate the multiaxial fatigue data for the PMMCs. That is, the MCF is assumed to be unity (p = 1) for all stress states AW’=AW’+AW+

Correlation of the parameter fatigue life data

(17)

for uniaxial (13)

multiaxial

fatigue

(16)

and

Figure 6

55

~ a,

ymax= a,

.

alloy

by using the parameter y*. This is in spite the fact that constant k in Equation (14) was determined from a best fit to data.

AW’=

YnUX =

aluminum

2(1 - V)

l-2v-2 ~~ u 2(1 - V)L’l + r;

E, =

particle

=KN;Y+C

(19)

Z. Xia and F. Ellyin 1 _) $7 E =

_: i

'i' j AW' = AWP + AW"

= 12.2NY9

+ 0.05

loading when correlating with microscopic damage parameters. This macroscopic mechanical parameter encompasses both the physics of the damage mechanisms and the empirical life correlation. Therefore. it is a suitable multiaxial fatigue parameter to bc used in prediction of fatigue life of PMMCs.

ACKNOWLEDGEMENTS This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC strategic Grant STR 0149082). The authors would like to thank the staff of the machine shop. A. Muir and D. Fuhr for preparation of specimens and B. Faulkner. Technician, in maintaining the multiaxial testing facility.

REFERENCES Figure 7 (l9)j

Correlation of the lotal strain energy with the multiaxial latigue life data

density

[Equation

2

The values of AMr calculated based on the experiment data and Equations (17) and ( 18) are listed in the last column of Tcrhlr I. The best tit result can be expressed as AM/’= AWI’ + SW’+

= 12.2N,

” “” + 0.05

I

(20)

Figurr 7 shows the total strain energy density-life curve and the 36 experimental data points. It can be seen that the scatter band around the curve is small compared to Figuws 4-6. For further quantitative comparison of the fitting results, it is possible to calculate the coefficients of correlation, p, for each best fit equation corresponding to parameters: E,, ,,,i,x, y* and AMr [Equation (8), Equations (15) and (20)]. The values of the correlation coefficients are 0.756, 0.742 and 0.909, respectively. Thus. it is seen that the parameter 7:‘: with p = 0.742. provides no improvement in the predictive results compared to the maximum equivalent strain. 6,. ,,,.,\, (p = 0.756) while the total strain energy density, AM/” has the best correlation with the fatigue life data of the composite material investigated (p = 0.909).

3

1

5

0

7

X

Y

IO

II CONCLUDING

REMARKS

Carefully designed multiaxial fatigue tests were performed on an alumina particle reinforced aluminum alloy composite. The stress-. strain- and strain energy-based criteria were used to correlate the fatigue life data. It is found that the total strain energy criterion provides the best correlation for various cyclic loading paths including uniaxial and multiaxial, proportional and nonproportional loading paths. The essential interaction between stress and strain and hence the path dependence is inherently included in the strain energy parameter. A combination of plastic energy, Aw and elastic energy, Aw’ + has extended the applicability of the criterion to both low and high cycle fatigue. It is worth mentioning that in a recent numerical correlation analysis for the PMMC, Li and Ellyin17 have found that the applied strain energy parameter provides a unique curve for the equibiaxial loading and uniaxial

I2

13 I4 IS

I6

17