Damage accumulation and fatigue crack propagation in a squeeze-formed aluminium alloy

IntJ Fatigue11 No 4 (1989) pp 249-254

D age acc ulation and fatigue crack propagation in a squeezeformed a l u m i n i u m alloy A. Plumtree and B.P.D. O ' C o n n o r

Damage accumulation in strain cycled squeeze-formed 6066-T6 aluminium alloy was studied by measuring the ratio of cracked to original cross-sectional areas of round, unnotched specimens and by monitoring the apparent modulus changes after different life fractions. These methods were found to correlate well. Stage II fatigue crack growth commenced when the surface crack length reached a critical value of approximately 700 I~m. A linear relationship with crack depth was then established. Below this threshold, the crack depth extended to only one or two dendrites and damage was less than 1%. Over 60% of the life was spent in Stage I (or surface dominated) crack growth. From a practical viewpoint, detectable damage and hence Stage II crack propagation were considered to begin after this life fraction and a damage mechanics model was successfully applied to the experimental results. The damage accumulation rate may be described by a Paris-type equation. Tests conducted at low total cyclic strains involving a very small amount of plasticity (plastic to elastic strain ratios of the order of 1:750) were found to maintain low damage levels over longer life fractions compared with tests at higher cyclic strains. Damage accumulation in tests involving greater plastic strain components was found to be similar. Under these conditions Miner's linear summation rule may be applied to the results. Key words: aluminium alloy; crack propagation; strain cycling; elastic modulus changes

Damage mechanics has been successful in predicting the mechanical deterioration of structural materials under cyclic loading at high temperatures, l'z Studies have incorporated the constitutive equations for damage derived by Kachanov3 from thermodynamic principles. Damage may be presented as a continuous variable written in terms of stress and strain, and represented by the density of defects produced in the material on loading. For fatigue, damage is the initiation as well as the growth of microcracks, and ultimately the complete rupture of the material. Generally, an undamaged component contains no cracks whereas a fully damaged one can be defined either as completely ruptured or partially fractured to a critical level, D c. For metals the value of Dc lies in the range 0.2-0.8. 4 Lemaitre and Plumtrees applied mechanics principles to describe damage evolution quantitatively in a material subjected to cyclic straining, which may be expressed as D = 1-

1-~r )

(])

where D = damage occurring after N cycles, Nf --- number of cycles to produce a critical amount of damage, De, as outlined in Ref. 5, and p = damage exponent. On the other hand, a simple damage equation has been formulated6 to account for the cracked area, namely D = l-

2

-A

(2)

where ~1 is the effective area of the damaged material and A is the original cross-sectional area. Damage can also be expressed in terms of the modulus of elasticity e as n

=

1 -

(3)

where /~ is the apparent modulus of the damaged material and E is the modulus of the undamaged material. The purpose of the present work was to monitor experimentally the apparent modulus of elasticity and the extent of cracking during strain cycling of a squeeze-formed aluminium alloy. In this way, the damage quantified through Equations (2) and (3) may be evaluated and the effectiveness of the equations assessed when considering an alloy produced by a novel fabrication process.

Experimental details Materials and properties The composition of the alloy used in this investigation is given in Table 1. Squeeze-forming aluminium alloy 6066 resulted in a refined microstructure, consisting of cast primary aluminium rich dendrites and a second phase of aFeA1MnSi in the interdendritic regions, as shown in Fig. 1. No porosity was observed. Grain boundaries surrounded many dendrite arms indicating that the pressures involved in the squeeze-forming process were sufficient to break them

0142-1123/89/040249-06 $3.00 © 1989 Butterworth & Co (Publishers) Ltd Int J Fatigue July 1989

840

Table 1. M a t e r i a l composition Element (wt%)

Cu

Fe

Mg

Mn

Si

Ni

Ti

Zn

Cr

AI

1.14

0.21

1.43

0.80

1.64

0.01

0.05

0.06

0.25

Bal

Fig. 1 Microstructureof squeeze-formedalloy Table 2. Mechanical f o r m e d 6066-T6 Young's modulus (GPa)

properties

of squeeze-

0.2% proof Ultimate tensile stress stress (MPa) (MPa)

72.1

345

365

Strain to failure (%)

1.4

from the primary cores. This resulted in a relatively freegrained structure with an average dendrite spacing of 84 jam and dendrite size distribution ranging from 25 jam to 300 jam. Coring in primary dendrites produced during squeeze-forming persisted after the T6 heat treatment which was achieved by solutionizing at 530 °C for 4 h, water quenching and aging for 8 h at 175 °C. The average width of the precipitate free interdendritic region was 15 Jam. Second phase particles varied from 3-30 Jam in length. The strengthening precipitate in this system was ~' Mg2Si. Table 2 gives the static mechanical properties.

Procedure Cylindrical fatigue specimens were machined to 90 mm in

length with a 15 m m gauge length and a nominal diameter of 5 ram. The gauge section was polished with alumina powder to a 0.3 jam finish. All tests were conducted at room temperature on a closed loop servo-hydraulic testing system under strain control using a sinusoidal waveform. A deformation rate of 0.04 s -1 was employed for all tests. Axial strain was controlled by an extensometer with a gauge length of 10 ram. Total strain amplitudes, A~,/2, between 0.15% and 1.0% were imposed. Representative elastic and plastic strain components at half life are given in Table 3. Approximately 20 hysteresis loops were recorded at a frequency of 0.05 Hz on an X Y plotter during the life of each specimen. The slope of the unloading curve from maximum tension gave the damaged or apparent modulus /~. The undamaged modulus, E, was obtained from the slope on loading after maximum compression, taken from the hysteresis loop recorded during the first cycle, thereby allowing an assessment of damage to be made according to Equation (3). Having recorded a hysteresis loop the load was reduced to zero and the corresponding strain recorded. The extensometer was then removed and acetate replicating tape was applied over the gauge length. The dried replicas were examined using an optical stereomicroscope to locate the position of cracks. This relatively quick examination could detect cracks of approximately 200 ~tm in length. Upon detecting such cracks the extensorneter was replaced at zero load. The strain reading was set to that recorded prior to removal of the extensometer and the test was continued. The peak loads before and after this procedure corresponded exactly. After cyclic testing, examination of the dried replicas in the scanning electron microscope (SEM) allowed accurate crack length measurements to be made. The crack profile was examined by interrupting cyclic tests with strain amplitudes of 0.15-1.0% at various stages of the fatigue life. The cracks were opened by applying a tensile load of 150 kg and a dye was painted over the gauge length. The cracked specimens were then broken using an impact machine and the crack profile was examined optically. The crack profile was very slightly curved, approximating a semi-ellipse. This shape was found to exist over the full range of strains examined up to damage levels of 50% which was used as the life criterion in this study. Since small specimens were tested, the relatively large crack tip plastic zone must be taken into account. This was

Table 3. Elastic and plastic cyclic strain c o m p o n e n t s at half life for tests monitored for d a m a g e Strain amplitude, Azt/2

Elastic strain amplitude,

Plastic strain amplitude,

(%)

A~,/2 (%)

Azp/2 (%)

0.15 0.20 0.35 0.50

0.1498 0.1990 0.3429 0.4680

0.0002 0.0010 0.0071 0.0320

250

Elastic to plastic ratio

Cycles to failure,

Nf 749:1 199:1 49:1 15:1

1.37 1.33 5.95 1.33

x x x x

10• 105 103 103

Int J Fatigue July 1989

achieved by using the line integral (strain intensity factor, J-integral) method in order to measure the intensity of the elastic-plastic stress-strain field surrounding the crack tip. Hutchinson and Paris 7 presented theoretical justification for use of the J-integrai and other workers 1~'9 have used the change in J (AJ) during cycling to correlate fatigue crack propagation. AJ was employed as a correlation factor for the damage accumulation rate in the present work. Its derivation for a semi-eliiptical crack is outlined in the appendix and expressed by AJ -

3.941Q A o A % a + ~ A o A % a

(4)

where Ao = cyclic stress range, Ace = elastic strain range, Aev = plastic strain range, a = crack depth, and Q = crack profile correction factor (see appendix)

ship between crack depth and surface crack length, shown in Fig. 3, indicates that the crack profile was independent of strain level. This profile, once established, remained constant over the fatigue life. Below surface crack lengths of approximately 700 Ixm the corresponding depths were small (less than 100 Ixm) and inclined at an angle to the major stress axis, indicating that propagation was dominated by shear at the free surface (Stage I crack propagation). However once this threshold of about 700 Ixm was reached the crack changed direction to correspond with that for the principal stress (Stage II crack propagation). Using a semi-elliptical crack shape and the depth relationships described, the cracked area was calculated. Damage was determined according to Equation (2) and the corresponding plot of damage against normalized life, N/Nf, is shown in Fig. 4. Here, N¢ is defined as the number of cycles to cause 50% damage. It appears that damage is practically

Results

4.1:

Cyclic stresHtrain

and strain-life

data

Symbol13 A~t12(%) 0.2

In contrast to extruded 6066-T6 which cyclically hardened and then softened 10 the squeeze-formed alloy initially displayed slight hardening (5-10%) and then stabilized. The cyclic hysteresis loops at half life represented the stable behaviour observed during most of the life. Such loops enabled the cyclic stress--strain curve to be generated, expressed by A~t/2

=

Ao/2(72.1

x

10o)

+

Ao/2(475.1

x

• 3.0 E

U

Superposition of the strain-life curves for the elastic and plastic components of tests with strain amplitudes between 0.15% and 1.0% produced the strain-life relationship shown in Fig. 2, given by Act/2 = 0.009(2Nf) -°J°7 + 0.0276(2Nf) -°'°35

(6)

The first term on the right hand side gives the elastic component and the second, the plastic component. The term 2Nf gives the number of reversals to failure where Nf is the number of cycles to failure. from cracked

0.5

•

0.75

x

,o

/

I

J

"~'-- ° = (I-0" 7) x 0" 384 (mm)

=I .2, [ I I 4.0 6.0 8.0 Surface crack length, I (mm)

~

2.0

~

I

10.0

Fig. 3 Relationship between surface crack length and maximum crack depth 0.6 Strain amplitude (%) 0.15 [3 0.20 V 0.35 O 0.50

0.5

area

0.4

100

e~

0.4

0

1.0

0 '~ 0

For damage to be determined from cracked area measurements according to Equation (2), the crack profile must be known at all stages during the fatigue life. The linear relation-

~

•

100) 21.2

(5)

Damage

0.25 0.35

V

0.3 1:2 Elastic z~ Plastic 0 Total

10

o" C~

1 q, .

0.2

p = 9.3--~/

0.,

o

e-

ra

0.1

0.4 o o, I . . . . .

100

........

101

102

103 Reversals

104

........

105

to f a i l u r e ,

106

........

107

I

0.5

I

I

0.6

I

i

0.7

,

Normalized life,

.....

108

2Nf

Fig. 2 Strain-life properties with the elastic and plastic components

Int J Fatigue July 1 9 8 9

I

I

0.8

I

I

0.9

t

.0

N/Nf

Fig. 4 Plot of damage accumulation obtained from cracked area estimates for four strain amplitudes at different life fractions (data points), with damage exponents determined from a least squares fit using Equation (7)

251

80

Table 4. Damage exponents Strain amplitude, ,ts,/2

Damage exponent, p

(%)

Equation (7) fitted to Fig. 4

Equation (7) fitted to Fig. 6

0.15

69.5

54.3

0.20

( 9.3

13.8

0.35 Average = 1 2 . 3 {14.2

Average = 17.9{22.7

050

1,13.5

• Compliance method O Compliance method, Ref. 13 • Cracked area method [] Cracked area method, Ref. 13

70

60

e\ \

~" 50

\

g

(17.3

\

o

o

\

40-

\ \

g

\

30--

undetectable for all the strain levels examined up to 0.6 Nf. Thereafter, lower damage levels persisted over a longer life fraction at the lowest strain amplitude where the elastic to plastic strain ratio was high 750:1 for A~, = 0.15%, as given in Table 3). In general, surface crack lengths of 700 lam were attained at normalized life ratios, of about 0.6, indicating that below this life fraction the threshold length had not been reached. Very small amounts of damage (less than 1%) had been accumulated to this juncture. From an analytical viewpoint then, evolution of detectable damage is considered to start at 0.6 Nf. Equation (1) may be modified accordingly, 4 resulting in

\

20--

(eg

D

=

1 -

•

10 --

~ - r Z 0.6Nr

I

I

0

n

O O

I IIIIII

i

J

I

IIIIII

0.001

i

l

i Itlt

0.01

0.1

Plastic strain amplitude (~,) Fig. 5 Variation of damage exponent, p, with cyclic plastic strain amplitude 0.6 Strain amplitude (~)

(7)

Z~ 0.15 n 0.20 V 0.35 O 0.50

0.5

This equation was fitted to the experimental data by a least squares fit. The damage exponent, p, corresponding to the fitted curves is given in Table 4. The damage values over the last 10% of life dominated the curve fitting process and hence the value of the damage coefficient, p. This is reflected in the highest p value representing the lowest strain level which accumulated damage very late in life. By contrast, lower and approximately constant values o f p were recorded for larger amounts of plasticity, as shown in Fig. 5. Furthermore, in those tests with strain amplitudes of 0.20% and greater, the small variance of p suggests that the manner in which damage evolution takes place is independent of the plastic strain amplitude.

Damage from modulus changes Hysteresis loops were recorded during each strain controlled fatigue test. The apparent tensile modulus decreased whilst the compressive modulus remained constant throughout life, indicating the presence of a crack. Damage, calculated from Equation (3), is plotted in Fig. 6 against normalized life. Curves derived from Equation (7) are superimposed on this plot. Fitting Equation (7) to the experimental data given in Fig. 6 results in the same trend in p value with strain amplitude as that obtained using the cracked area method (see Table 4 and Fig. 5 for comparison).

Discussion The successful application of damage evolution expressed by Equation (1) to the data derived from Equations (2) and (3) indicates that both the apparent modulus changes and cracked area measurements are good indicators of damage (Figs 4 and 6). Apparent modulus measurements are attractive because the results can be determined quickly. However

252

•

•

0.0001

N- O.6Nf) ./(I+p) 1

•

o

\

N/Nf,

(

o \

O

0.4

O

Lu

O.3

r':

°0

0.2

_.~. j ~ 0=13.8 17.3--,,

n

,, 0 0.4

o

I 0.5

t

I 0.6

,

I 0.7

,

Normalized life,

I 0.8

i

I 0.9

i .0

N/Nf

Fig. 6 Plot of damage accumulation obtained from apparent modulus changes for four strain levels, Damage exponents are determined by fitting Equation (7) to data points

sections must be removed from the component, whereas the cracked area method uses non-destructive techniques to find the location of the crack and to estimate its size. In this latter case, the crack profile must be known to establish an accurate estimate of damage. Added complications such as the presence of multiple cracks are overcome by the modulus method because of its direct relationship to cracked area irrespective of the presence of one or more cracks. In the present investigation multiple cracking was observed early in life when the surface crack lengths were short, generally less than 200 I~m in size. Appreciable damage,

Int J Fatigue July 1989

however, developed later after these short cracks coalesced to produce a crack of about 700 ~tm in length (Fig. 3). At this point microstructure independent radial growth began. Previously the crack had been confined to one or two dendrites. The depth of the dominant crack was determined from Fig. 3 and with this information AJ was calculated from Equation (4). The rate at which damage accumulated was obtained from the slope of successive points on a damage plot, such as Fig. 4, and correlated to the strain intensity factor range (AJ). This relationship is shown in Fig. 7 which indicates that the damage accumulation rate (dD/dN) followed a Paris-type equation, ie

dD/dN

CAd"

=

(8)

where the values of C and n are given in Fig. 7. This equation may be integrated to give the life of the component. Besides life prediction made in terms of damage, it is possible to use crack depth, if convenient, since the two are related for a constant surface crack length to depth ratio. Life predictions may also be made once the amount of damage has been determined for a particular number of cycles in service, N, according to a rearranged version of Equation (7), namely

N/Nr

=

0.4 [ 1 - ( 1 - D ) ~+"] + 0.6

(9)

This investigation has shown that the strain level has a bearing on the value of damage exponent p when the plastic component is very small, corresponding to long life fatigue. For strains within the transition regions of the strainlife curve where the elastic to plastic strain ratios are less than approximately 750:1, damage accumulation is independ-

/

100

dD/dN= l x 1 0 -6(AJ)1.'85 10-2

"o

/

I0 -4 --

~

-

tO

_m 10 i=

ent of the level of plasticity. Thus once in the transition region of the strain-life curve, the strain level may be increased and residual life can be predicted accurately by applying Miner's rule. The damage accumulation curve for both strain levels will be the same since a single value of p may be used to describe damage evolution. The average value of p at these strain levels is 12.3 after monitoring cracked area, as given in Table 4. Referring to Fig. 4, if the total strain amplitude were changed from 0.20% (average p = 12.3) to 0.50% (average p = 12.3) after 90% of life (N/Nf = 0.9 when D = 0.09) then since there would be no shift in damage curves, the remaining life would be 0.10Nf at the higher strain level. However if the initial strain amplitude were increased from 0.15% (p = 69.5) after 90% of life (N/Nf = 0.9 when D = 0.02) to 0.50% (average p = 12.3), the remaining life would be 0.30Nf at this level, resulting in a Miner's linear summation of 1.20(0.90+0.30). Conversely, linear summation would give 0.91(0.90+0.01) for a change in strain amplitude from 0.50% after 0.9N/Nf to 0.15%. Similar deductions may be made from work on A533B steep where the crack length at a given life ratio was found to be independent of strain level within the transition region of the strain-life curve. The present work indicates that this effect is due to the dominance of Stage II crack propagation. On the other hand, for low strain fatigue, Stage I becomes dominant and low damage levels are maintained for longer fractions of the life. This is in accord with experimental results where the specimen subjected to a total strain amplitude of 0.15% spent 88% of its life in Stage I crack growth with a surface crack length less than about 700 I.tm. The corresponding value at a strain amplitude of 0.50% was 51%Nf. Whilst this may explain why the highest p value has been obtained for the elastic dominated test, it also indicates that damage at a given life fraction at low strains will be strain dependent since a longer Stage I period is required for lower strain levels. Damage accumulations would therefore occur at correspondingly later stages which would lead to even higher p values than observed.

Conclusions 1) oo

-6

2)

U U

gto 10-a

E Q

10-10

10-12 10 -4

3) t/

I 10-2

J

I 100

,

I 102

~J (kNrn-11 Fig. 7 Relationship between damage accumulation rate (dD/dN) and &J. Symbols for data points correspond to those in Figs 3, 4 and 6

Int J Fatigue July 1989

Fatigue damage may be assessed with reasonable accuracy by measuring the cracked area or by monitoring the apparent modulus changes over the life of a specimen. The present work shows good correlation between the two. The accumulation of damage is independent of cyclic strain for all but the lowest levels of strain where the elastic to plastic component exceeds about 750:1. Miner's linear summation rule may be applied successfully to those conditions involving greater plasticity (constant p value). Damage evolution is similar for the majority of strain levels and the difference in fatigue life at lower cyclic strain levels is due to a longer period of Stage I crack growth. By knowing the crack profile it is possible to relate damage accumulation rate to the strain intensity factor range. Damage accumulation rate is shown to follow a Paris-type relationship.

Acknowledgements The authors would like to express their thanks to GKN Technology, Wolverhampton, UK for supplying the material used in this work. The Natural Sciences and Engineering

253

Research Council of Canada is acknowledged for financial support through Grant A2770. Thanks are also due to Lorna Spencer for typing the manuscript.

Substituting for AK from Equation (A.1) and assuming that the same geometry correction factor applies to both components, the equation for AJ may be expressed as AJ = (1" 12) 2 2na[g(n)Ac0= + h(n)A0~p]

Q

Appendix

The strain hardening exponent, n, was determined as 0.0472, from which h(n) = 2.53.12 The Irwin plastic zone correction factor, g(n), was taken as unity. Using these values and the relations

Stress and strain intensity derivation Stress intensity factor range, A K

The stress intensity factor for a semi-elliptical crack in a flat plate is given al as AK = 1.12ME where

~

(A.4)

At0=-

(A.I)

Ac -~- = stress amplitude

A~ 2 2E

(A.5)

(A.6)

AO)p = Ao" Agp l+n AJ may be expressed as

a = crack depth M K = Kobayashi correction for free surface

AJ - 3.941 AoA~=a + 19.042 A ° A ~ p a

Q

Q

(A.7)

References ~y = yield stress

1.

, = y

2.

+

8

\c/

Since the present wore involved a round specimen the term c cannot be applied as above. However, the term a/c is the minor to major axis ratio and defines the profile of the crack. This shape could be closely approximated by assuming a minor to major ratio of 1/3.

3. 4.

Douglas, M.J. and Plumtree, A. "Fracture Mechanics" ASTM STP 677ed. C.W. Smith (American Society for Testing and Materials, 1979) pp 68-84 Lamaitre, J. and Chaboche, J.L. 'A non-linear model of creep-fatigue damage accumulation and interaction' Proceedings I.U.T.A.M. Symposium of Mechanics of Viscoelasticity Media and Bodies (Springer Verlag, Goteborg, Sweden, 1974) Kachanov, L.M. "On the time to failure under creep conditions" Izvest Akad Nauk, SSSR ONT8 (1958) pp 26-31 Lemaitre, J. and Chaboche, J.L. "Mechanique des Materiaux Solides" (Dunod, Paris, 1985)

Strain intensity factor range, A J

5.

Lemaitre, J. and Plumtree, A. J Engng Mater Technol 101 (1979) pp 284-292

The method of determining AJ is similar to that oudined by Dowling 9 where AJ was expressed by

6.

Plumtree, A. and Nihmon, J-O. Fatigue Fract Engng Mater Struct II (1988) pp 397-407

7.

Hutchinson, J.W. and Paris, P.C. ASTM STP 668 (American Society for Testing and Materials, 1979) p 37 Starkey, M.S. and Skelton, R.P. Fatigue Engng Mater Struct 6 (1982) pp 329-341 Dowling, N.E. "Cracks and Fracture" ASTM STP 601 (American Society for Testing and Materials, 1976) pp 19-32 O'Connor, B.P.D. Private communication Brock, D. "Elementary Engineering Fracture Mechanics" (Martinus Nijhoff Publishers, Boston. 1982) Shih, C.F. and Hutchinson, J.W. J Engng Mater Technol 98 (1976) pp 289-295

AJ =

1.12 x

2na[g(n)aAm= +

h(n)Am~]

(A.2)

where Am= and A~p are the elastic and plastic strain energies dissipated during a fatigue cycle, and g(n) and h(n) are functions of the strain hardening exponent, n. 12 The first bracketed term in Equation (A.2) represents a correction factor for the semi-circular profile observed in the work of Dowling. Equation (A.2) represents the sum of the elastic and plastic strain intensities, ie AJ = AJe + AJp where

AJ=-

9. 10. 11. 12. 13.

O'Connor, B.P.D. and Plumtree, A. "FractureMechanics: Nineteenth Symposium" T.A. Cruse (ed.) ASTM STP 969 (American Society for Testing Materials, 1988) pp 787-799

AK 2 E - 2nao~=

Authors

and

aJp = 2nah(n)op

254

(A.3)

8.

The authors are with the Mechanical Engineering Department, University of Waterloo, Waterloo, Ontario, Canada.

Int J Fatigue July 1989