a multiscale approach to crack growth

A MULTISCALE APPROACH TO CRACK GROWTH

R. Jones, C. Wallbrink, Monash UniversityL. Molent, S. Pitt, Air Vehicles Division, DSTO

Background courtesy of Paul White and Simon Barter Air Vehicles Division, DSTO, Melbourne,

Conclusion• This presentation reveals that the science base underpinning

the fatigue characterisation of materials behaviour at low crack growth rates is invalid

• We also see that short crack growth follows the generalised Frost-Dugdale crack growth law

Why is this important

• The scientific basis all of the current fatigue crack growth laws, AFGROW, FASTRAN, NASGRO, etc, is invalid in Region I.

• Region I is the region in which a crack will spend most of its life.

• The current similitude based methodology is non conservative forsmall sub mm cracks in highly stressed components, i.e. it is both incorrect and unconservative.

• Corrects the science base needed to predict delamination growth in composites.

• The USAF funded research in the late 50’s

H. W. Liu, “Crack propagation in thin metal sheet under repeated loading”, WADC TN 59-383, 1959.

An extensive test program using centre cracked panels revealed that for short cracks

da/dN α a N α ln(a)- ln(ainitial)

• Frost and Dugdale at Sheffield and the National Engineering Labs, in East Kilbride, Scotland.

N. E. Frost and D. S. Dugdale, “The Propagation of Fatigue Cracks in Test Specimens”, Journal Mechanics and Physics of Solids, 6, pp 92-110, 1958.

Performed an extensive test program that revealed that for short cracks da/dN α a

Early crack growth concepts:

• Same conclusions reached as part of a USAF/General Dynamics evaluation of small crack growth

J. M. Potter and B. G . W. Yee, “Use of Small Crack Data to Bring About and Quantify Improvements to Aircraft Structural Integrity”, Proceedings AGARD Specialists Meeting on Behavior of Short Cracks in Airframe Structure, Toronto , Canada , 1982.

Subsequently confirmed byL. Molent, R. Jones, S. Barter, and S. Pitt, “Recent Developments In Fatigue Crack Growth Assessment”, International Journal of Fatigue, In press.Polak J., Zezulka P., “ Short crack growth and fatigue life in austenitic-ferritic duplex stainless steel”, Fatigue Fract Engng Mater Struct 28, 923–935, 2005.S. A. Barter, L. Molent, N. Goldsmith and R. Jones, “An experimental evaluation of fatigue crack growth”, Journal Engineering Failure Analysis, Vol 12, 1, pp 99-128, 2005.Harkegard G., Denk J., Stark K., “ Growth of naturally initiated fatigue cracks in ferritic gas turbine rotor steels”, International Journal of Fatigue, (2005), pp 1–12.Murakamia Y., Miller K. J., “ What is fatigue damage? A view point from the observation of low cycle fatigue process”, International Journal of Fatigue, pp 1–15, 2005. Kawagoishi, N. Chen Q. and Nisitani H., “ Significance of the small crack growth law and its practical application”, Metallurgical and Materials Transactions A, Volume 31A, pp 2005-2013, 2000.R. Jones, S. Barter, L. Molent, and S. Pitt, “A multi-scale approach to crack growth”, Chapter in the Book, Multiscaling in molecular and continuum mechanics: biology, electronics and material science, Edited by G. C. Sih, Springer, March 31, 2006. Caton M. J., Jones JW, Boileau J. M, and Allison J. E., “ The effect of solidification rate on the growth of small fatigue cracks in a cast 319-type aluminum alloy”, Metallurgical and Materials Transactions A, Volume 30A,pp 3055-3068, 1999. Nisitani, H, Goto M. and Kawagoishi, N., “A small-crack growth law and its related phenomena”, Engineering Fracture Mechanics, Vol. 41, No. 4, pp. 499-513, 1992.H. W. Liu, “A review of fatigue crack growth analyses”, Theoretical and Applied Fracture Mechanics 16 (1991), pp 91-108, Tomkins, B.,“ Fatigue crack propagation—an analysis ”, Phil. Magazine , 18, 1041–1066, (1968).

The da/dN v ∆K concept (similitude) Paris et al, and Liu

• P.C. Paris, M.P. Gomez and W.E. Anderson, A rational analytic theory of fatigue, Trend in Engineering, 13 (January 1961) 9 14.

First paper to link da/dN to Kmax

• H.W. Liu, Fatigue crack propagation and applied stress range, ASME Trans., Journal of. Basic Engineering, 85D (1) (March 1963pp 116-122.

Linked da/dN to (∆K)

Built on Paris’ concept of similitude so that the need to have a linear relationship between ln(a) and N forced Liu to set m =2. However, this gave an incorrect stress dependency.

• P.C. Paris and F. Erdogan, “A critical analysis of crack propagation laws”, ASME Trans., J. Basic Engineering, 85D (4) (December 1963) pp 528. Same general formulae as Liu (1963) but stated

da/dN α (∆K)4 and da/dN α a2

This infers that crack growth is merely a function of ∆K and Kmax. Two different cracks with the same (effective) stress intensity factor range ∆K and K max will grow at the same rate.

This is termed similitude.

• Whilst this gave an improved stress dependency it was inconsistent with, and hence ignored, Liu’s and Fost and Dugdale’s earlier experimental results for the crack growth history of centre cracked panels that revealed:

N α ln(a)- ln(ainitial)

Current approaches are built on the generally accepted hypothesis for crack growth:

• Commonly accepted hypothesis: Crack growth is uniquely characterised by the stress intensity factor.

Two different cracks with the same (effective) stress intensity factor range ∆K and Kmax will grow at the same rate.

• In the mid 1970’s Pearson, RAE Farnborough UK, showed that fatigue crack growth laws, determined for macroscopic crack growth data, could not be used to predict the growth of small sub-millimetre cracks, and that the constants in the crack growth law were a function of the size of the crack.

Pearson S., “Initiation of fatigue cracks in commercial aluminium alloys and the subsequent propagation of very short cracks”, Engineering Fracture Mechanics, 1975. Vol. 7, pp. 235-247.

The scientific method

• An hypothesis can be disproved by presenting a single case where it is false.

• Such a case is called a counter example, and it disproves the hypothesis.

1st example: Crack growth v ∆K data for L6B-2 cast steel

Data courtesy of B. Chen, Rail CRC

Traditional way of characterising material behaviour

In this region growth is non similitude

In this case analysis of the data shows that da/dN = C (∆K)3 /a1/2

Here we see a linear relationship between da/dN and ∆K3/a1/2,

i.e. the crack growth rate is a function of both ∆K and the crack length a.

This expression for da/dN also yields the experimentally observed log-linear relationship

Note: The axes have a linear scale not log scales

2nd counter example: Grade 1 Austempered Ductile Iron

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1 10 100∆K

da/d

N m

m/c

ycle

Spec3Spec4

In this region similitude is invalid

2nd counter example: Grade 1 Austempered Ductile Iron

Data courtesy of B. Chen, Rail CRC

Here we again see that the crack growth rate is a function of both ∆K and the crack length a.


3rd and 4th Counter example: Wheel steels, Rail CRC data from B. Chen.

similitude is invalid

Table 1 Conventional wheel steels Class C Si Mn P S Ni Cr Mo Al Cu Nb Sn Ti V B 0.64 0.25 0.75 0.009 0.019 0.1 0.11 0.02 0.009 0.23 0.003 0.012 0.005 0.002C 0.74 0.27 0.79 0.008 0.027 0.08 0.08 0.02 0.006 0.22 0.003 0.02 0.005 0.002MB 0.65 0.87 0.77 0.008 0.019 0.08 0.11 0.04 0.01 0.21 0.002 0.01 0.007 0.089MC 0.74 0.81 0.84 0.007 0.019 0.06 0.07 0.02 0.007 0.12 0.002 0.006 0.006 0.082

Wheel steels

Here we again see that the crack growth rate is a function of both ∆K and the crack length a.


5th Counter example: Martensitic steels, with various amounts of matensite; H = high, = intermediate, L = low, Rail CRC data from B. Chen.


Here we again see that the crack growth rate is a function of both ∆K and the crack length a and that growth is not governed by similitude.

Here we see a linear relationship between da/dNand ∆K3/a1/2.

6th Counter example: Crack growth rate for D16 Aluminium under a range of stress amplitudes and R ratio’s

y = 1.94E-06x1.06

R2 = 0.96

1.00E-05

1.10E-04

2.10E-04

3.10E-04

4.10E-04

5.10E-04

6.10E-04

7.10E-04

8.10E-04

9.10E-04

0 1000 2000 3000 4000 5000 6000 7000 8000 9000∆Kd

3 /a1/2

da/d

N (m

m/c

ycle

)R=0 Stress= 64 MPaR =0.33 Stress = 64 MPaR = 0.0 Stress = 140 MPaR=0.75 Stress =35 MPa

J. Schijve, M. Skorupa, A. Skorupa, T. Machniewicz, P. Gruszczynski , “Fatigue crack growth in the aluminium alloy D16 under constant and variable amplitude loading”, International Journal of Fatigue 26, 1–15, 2004.

∆Kd = ∆Ktot(1-p)Kmax,tot

p , where p =0.25, i.e. growth is governed by the Unigro formulation

Note: The axes have a linear scale and not log scales

7th Counter examples: Scott C. Forth, Christopher W. Wright, and William M. Johnston, Jr., “7075-T6 and 2024-T351 Aluminum Alloy

Fatigue Crack Growth Rate Data”, NASA/TM-2005-213907, August 2005

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1 10 100∆K MPa √m

da/d

N m

/cyc

le

AL-7-23AL-7-21AL-7-22

Al 7075-T6 R=0.1

In this region growth is not self-similar

7th Counter example: Scott C. Forth, Christopher W. Wright, and William M. Johnston, Jr., “7075-T6 and 2024-T351 Aluminum Alloy


y = 5.90E-08x - 5.92E-05R2 = 9.91E-01

1.00E-06

1.01E-04

2.01E-04

0 1000 2000 3000 4000 5000

∆Κ 3 /a1/2 MPa3 m

da/d

N m

m/c

ycle

Al-7-23 7075-T6

In Region I there is a linear relationship between da/dN and ∆K3/a1/2 i.e. non self similar growth




1.00E-05

1.00E-04

1.00E-03

1.00E-02

1 10 100∆ K MPa √m

da/d

N m

m/c

ycle

AL-2-26 2024-T351AL-2-27 2024-T351AL-2-28 2024-T351

Examine crack growth in this region



y = 1.22E-08x - 1.76E-04R2 = 9.97E-01

1.00E-05

6.00E-05

1.10E-04

1.60E-04

2.10E-04

2.60E-04

3.10E-04

15000 20000 25000 30000 35000 40000∆K3 /a1/2

da/d

N (m

/cyc

le)

Al 2024-T351 40 N

In Region I there is a linear relationship between da/dN and ∆K3/a1/2 i.e. similitude is invalid


9th Counter example C.M. Hudson, “Effect of Stress Ratio on Fatigue-Crack Growth in Aluminum-Alloy 7075-T6 and 2024-T3 Specimens”, NASA Technical

Note, NASA TN D-5390, August 1969.

y = 3.36E-08x - 8.09E-05R2 = 9.69E-01

1.000E-06

2.010E-04

4.010E-04

6.010E-04

8.010E-04

1.001E-03

0 10000 20000 30000 40000

(∆K0.75 Kmax0.25)3/a1/2

da/d

n m

m/c

ycle

21 R=0.521 R=0.812 R=0.719 R=0.821 R=0.7Linear (21 R=0.8)


Conclusion• The experimental data which covers three quite different materials viz:

i) six (rail) cast steels, ii) an austempered ductile iron, iii) 3 aircraft quality aluminium alloy’s with a range of R

ratio’s.

and was obtained by a variety of researchers, Rail CRC, NASA, DSTO, USAF, etc, using a variety of specimen configurations, i.e. ASTM compact tension, ASTM middle tension, etc, shows that the science base underpinning crack growth models AFGROW, FASRAN, NASGRO, etc in Region I is invalid.

• In these instances the hypothesis of similitude is incorrect.

• This is also true for delamination growth in composites.

Fatigue Behaviour Revisited

• Experimental evidence reveals that crack growth exhibits the same characteristics for both short and long cracks

Experimental results when plotted as log(a) v cycles (blocks/flights hours) show that short and long cracks have the same relationship to load history, viz:

da/dN = (a)1-γ/2 C (∆Keff) γ

= (a)1-γ/2 C ( ∆Ktot(1-p)(Kmax,tot)p ) γ

For short cracks this yields da/dN α ( ∆σtot

(1-p)(σmax,tot)p ) γ a

So that a = a0 e ϖN/ σ‘

where σ‘ = (∆σtot(1-p)(σmax,tot)p ) γ

Simple Master Curve For Fatigue Crack Growth From Small (microns) to Macro Sized Cracks (4-10 mm’s)

For small cracks this yields the following relationship, viz:

ln(a) – ln(ai) α B; da/dB α a

Thenln(a/af) – ln(ai /af) α B/Bf

{da/dB/(da/dB)f } α a/af

Here Bf is the life corresponding to af which is a macro crack size, not necessarily the crack size at failure.

Growth of near nano scale cracks: Ritchie, et al

y = 24.4e0.188x

R2 = 0.966

y = 18.974e0.3723x

R2 = 0.9557

1.00E+01

1.00E+02

0 0.5 1 1.5 2 2.5Num ber of Cycles x 1010

Cra

ck L

engt

h a

(nm

)Ritchie

Muhls tein

Crack growth at the nano-scale in 2 µm thick polysilicon, adapted from Ritchie et al

Y. Murakamia K. J. Miller, What is fatigue damage? A view point from the observation of low cycle fatigue processInt. Journal of Fatigue, (2005) 1–15.

y = 38.49e3.4503x

R2 = 0.9911

y = 188.19e2.7169x

R2 = 0.993

1

10

100

1000

10000

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

N/Nf

a (m

icro

ns)

510 MPa d = 40 microns294 MPa d = 40 microns373 MPa d = 40 microns412 MPa d = 40 microns373 MPa d =200 microns412 MPa d= 200 microns530 MPa d = 200 microns510 MPa d = 200 microns

Simple Master Curve For Fatigue Crack Growth From Small (microns) to Macro Sized Cracks (4-10 mm’s)

For small cracks this yields the following relationship, viz:

ln(a) – ln(ai) α B; da/dB α a

Thenln(a/af) – ln(ai /af) α B/Bf

{da/dB/(da/dB)f } α a/af

Here Bf is the life corresponding to af which is a macro crack size, not necessarily the crack size at failure.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Normalised crack length a/af

Nor

mal

ised

cra

ck g

row

th r

ate

da/d

b/(d

a/db

)f

A7F/A-18 IFOSTPAlohoaF/A-18 Bulkhead test1969 F111 wing testMirage III fatigue test, spar crackSwiss Mirage III fatigue test

Proof: 5 full scale aircraft tests, 2 real aircraft flight data: From microns to 10’s mm’s

D. Angelova and R. Akid, “A normalization of corrosion fatigue behaviour: an exampleusing an offshore structural steel in chloride environments”, Fatigue Fract

Engng Mater Struct, 22, 409–420, 1999.

20 micron < a < 2 mm Note that in the later stages the crack is tearing.

y = 0.0042e2.553x

R2 = 0.9679

y = 0.0124e2.9944x

R2 = 0.9833

0.001

0.010

0.100

1.000

0 0.2 0.4 0.6 0.8 1N/Nf

a/a f

Major defect sizesMinor defect sizes

i

Proof: Boeing 767 and 757 Materials Characterisation Data Base Boeing constant amplitude crack growth data,

y = 0 .1 5 9 e 1 .2 7 xR 2 = 1 .0

y = 0 .1 5 4 e 1 .4 3 xR 2 = 1 .0

y = 0 .1 6 0 e 1 .1 8 xR 2 = 0 .9 9y = 0 .1 3 7 e 1 .4 9 x

R 2 = 0 .9 70 .1

1

0 0 .2 0 .4 0 .6 0 .8 1N o rm a l is e d fa tig u e l i fe

Nor

mal

ised

cra

ck le

ngth

2 3 2 4 - T3 9 3 .2 mm th ic k w = 4 0 6 mm7 1 5 0 - T6 5 1 3 .2 mm th ic k7 1 5 0 - T6 5 1 1 2 .7 mm th ic k 2 0 2 4 - T3 6 .3 mm

Yet more Proof: Boeing crack growth data for tests under various transport load spectra

0.01

0.1

1

0 0.2 0.4 0.6 0.8 1Normalised fatigue life

Nor

mal

ised

cra

ck le

ngth

2324-T39 Spectra A2024-T351Spectra A7075-T651 Spectra C7150-T651 Spectra C7150-T651 Spectra B7075-T651 Spectra B

M. Miller, V. K. Luthra, and U. G. Goranson, “Fatigue Crack Growth Characterization of Jet Transport Structures”, Presented at 14th Symposium of the International Conference onAeronautical Fatigue (ICAF), Ottawa, Canada, June 10-12, 1987.

Comparison of F-16 wing test, P3C full scale test, F/A-18 Centre Barrel (Final) & the 1969 F-111 wing test crack growth rates per flight hour

y = 0.0006x0.98

R2 = 0.99

y = 0.0004x0.99

R2 = 0.99

0.000001

0.001001

0.002001

0.003001

0.004001

0.005001

0.006001

0.007001

0.008001

0.001 10.001 20.001 30.001 40.001Crack length (mm)

Cra

ck g

row

th ra

te d

a/dF

light

hou

rs

P3C full scale test

F/A-18 Final

1969 F111 wing test

F-16 wing test crack B28

F16 wing test, wing skin crack P10

F16 wing test, crack 12L spar 6

DSTO F/A-18 Fatigue Test Program FINAL

F/A-18 Bulkhead

Y488 Bulkhead Failure From ~30 micron corrosion pits

NEi-NASTRAN Finite Element Model of the Bulkhead

Sub Section

NEi-NASTRAN Von Mises Stresses

Experimental results when plotted as log(a) v cycles (blocks/flights hours) show that short and long cracks have the same relationship to load history, viz:

da/dN = (a)1-γ/2 C (∆Keff) γ

= (a)1-γ/2 C (∆Ktot(1-p)(Kmax,tot)p ) γ

γ = 3.36, p = 0.25, C = 2.94 10-12

Comparison with FASTRAN

• Comparison of experiment with predictions and with FASTRAN IIpredictions using DSTO and US crack growth data as input

Note: Large errors when using FASTRAN/AFGROW to try and predict growth from small 3 micron defect

• Same conclusions reached as part of a USAF/General Dynamics evaluation of small crack growth


• No discernable short crack - long crack effect

The 1969-70 F111 Full Scale Fatigue Test

Small 0.008 inch flaw on inside surface of steel plate

Finite element (NEi-NASTRAN) 3D model of F-111 wing with ~ 91,000 nodes

Flaw placed here

Location of a 0.008” semi-elliptical surface flaw

Technical Details

• Analysis uses 3D (generalised) weight functions for semi elliptical, quarter elliptical flaws at an arbitrary notch.

• Total fatigue life analysis time ~ 2 minutes on an Intel P4 2 GHz processor.

• Uses uncracked finite element model only.• No crack modeling or special mesh refinement needed.• Analysis directly interfaced with NEi-NASTRAN• Arbitrary load spectra, geometry, flaw geometries, etc.

Life Prediction: Crack growth in F-111 wing model under General Dynamics (Lockheed) test spectra and comparison with 1970’s crack

growth data

Why is this important ?• The scientific basis for all of the current fatigue crack growth laws, AFGROW,

FASTRAN, NASGRO, etc, is invalid in Region I.

• Region I is the region in which a crack will spend most of its life.

• For small cracks growth will generally initially lie in Region I.

• Cracks arising due to corrosion damage will generally initially lie in Region I.

• The current similitude based methodology is non conservative for small sub mm cracks in highly stressed components, i.e. it is both incorrect and unconservative.

• Methodology used to evaluate materials fatigue characterisation tests for crack growth in Region I is invalid.

• The conclusions reached as part of a USAF/General Dynamics evaluation of small crack growth were correct.


• No discernable short crack - long crack effect.

• However, the current similitude laws for long cracks are invalid.

H. Chen, K. Shivakumar and F. Abali, "A comparison of total fatigue life models for composite laminates", Fatigue Fractures of Engineering Materials & Structures, 29, 31–39, 2006.

Mode I delaminationgrowth in an E-glass DCB specimen

Block loading DCB, E-glasswoven roving

0

50

100

150

200

0 10000 20000 30000 40000 50000 60000 700Cycles

a (m

m)

Fig 11Equivalent block prediction

H. Chen, K. Shivakumar and F. Abali, "A comparison of total fatigue life models for composite laminates", Fatigue Fract Engng Mater Struct 29, 31–39, 2006.

da/dN= 8.5 10-3 (Gmax/G1C)1.5/a0.5

Mean growth rate ~ 5/3 10-3 mm/cycle

M. T. Yu and T. H. Topper, The effects of materials strength, stress ratio, acompressive overload in the threshobehavior of a SAE 1045 steel. J. Engng Mater. Technol., Trans. ASME lM, 19-(1985).

y = 1.42E-05x1.02

y = 5.50E-05x1.04

y = 2.80E-04x1.07

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-01 1.00E+00 a (mm)

da/d

N (m

m/c

ycle

)

1045 steel 400 MPa1045 steel 510 MPa1045 steel 640 MPa

0.025 mm < a < 0.7 mm

More Proof Near Micron size flaws:M. T. Yu and T. H. Topper, The effects of materials strength, stress ratio, and compressive overload in the threshold behaviour of SAE 1045 steel. J. Engng Mater Technol, Trans. ASME, 19-25 (1985).

y = 1.42E-05x1.02

0.0000001

0.000001

0.00001

0.0001

0.001

1E+11 1E+12 1E+13 1E+14 1E+15

σ5.4a (mm)

da/d

N (m

m/c

ycle

) 1045 steel 400 MPa1045 steel 510 MPa1045 steel 640 MPa

0.025 mm < a < 0.7 mm

F/A-18 Coupon test Results, APOL load sequence

y = 7E-06x1.05

R2 = 0.9153

0.000001

0.00001

0.0001

0.001

1 10 100

(∆K) γ a1-γ/2/100

da/d

t

324.1 MPa

396.5 MPa

428.9 MPa

358.5 MPa

γ= 3.36

p= 0.25

Crack lengths from 3 microns to 3 mm’s

F/A-18 Coupon test Results, Mini-Falstaff load sequence

y = 0.0014x1.06

R2 = 0.93

0.001

0.01

0.1

1 10 100

(∆K)γ a1-γ/2/100

da/d

bloc

k (m

m/b

lock

)

Mini Falstaff Data

γ= 3.36

p= 0.25

a multiscale approach to crack growth

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