the evolution and appication of three- dimensional stress ... · appication of three-dimensional...

THETHE EVOLUTIONEVOLUTION ANDAND APPICATIONAPPICATION OFOF THREETHREE--DIMENSIONALDIMENSIONAL STRESSSTRESS--INTENSITYINTENSITY FACTORSFACTORS

J. C. Newman, Jr.Mississippi State University

Starkville, MS

I. S. RajuNASA Langley Research Center

Hampton, VA

S. A. FawazU. S. Air Force Academy

Colorado Springs, CO

The George R. Irwin Centennial ConferenceUniversity of Maryland

18-20 March 2007College Park, MD

Irwin 100th Conference - # 2

George Rankin IrwinGeorge Rankin Irwin


OUTLINE OF PRESENTATIONOUTLINE OF PRESENTATION

• Embedded Elliptical Crack

• Methods of Solution for Finite-Body Problems

• The Surface-Crack Problem• The Boundary-Layer Effect• Surface and Corner Crack(s) at a Hole• Application to Fatigue-Crack Growth• Application to Fracture• Concluding Remarks


EMBEDDED ELLIPTICAL CRACK TO ANEMBEDDED ELLIPTICAL CRACK TO ANAPPROXIMATE SURFACE CRACK SOLUTIONAPPROXIMATE SURFACE CRACK SOLUTION

πGreen & Sneddon (1950) Irwin (1962)

fφ


METHODS OF SOLUTION FOR FINITEMETHODS OF SOLUTION FOR FINITE--BODY PROBLEMSBODY PROBLEMS

• Engineering Estimates • Alternating Methods• Line-Spring Model• Boundary-Element Methods• Finite-Element Methods

COD methodsJ-Integral or energy methodsNodal-force method


THE SURFACETHE SURFACE--CRACK PROBLEMCRACK PROBLEM

2w


SEMISEMI--CIRCULAR SURFACE CRACK UNDER CIRCULAR SURFACE CRACK UNDER REMOTE TENSIONREMOTE TENSION

a / t0.0 0.2 0.4 0.6 0.8 1.0

0.9

1.0

1.1

1.2

1.3

1.4

Irwin

Anderson-Holms-Orange

Paris-Sih

Raju-Newman

KobayashiSmith

Shah-Kobayashi

Kobayashi-Moss

Newman

Hellen-Blackburn

Newman-Raju

a / c = 1φ = π / 2

K

S √πa/Q

Newman (1979)


SEMISEMI--ELLIPTICAL SURFACE CRACK UNDERELLIPTICAL SURFACE CRACK UNDERREMOTE TENSIONREMOTE TENSION

a / t0.0 0.2 0.4 0.6 0.8 1.0

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4Newman

Smith

Raju-Newman

Rice-Levy

Masters et.al

Kobayashi

IrwinParis-Sih

Anderson et.al

Shah-Kobayashi

Kobayashi-Moss

Smith-Sorensen

Newman-Rajua / c = 0.2φ = π / 2

K

S √πa/Q

Newman (1979)


THE BOUNDARYTHE BOUNDARY--LAYER EFFECTLAYER EFFECT

Lose of square-root singularity Free surface

Hartranft & Sih (1970)Benthem & Koiter (1973)

Crack


EFFECT OF FE MESH REFINEMENT ON EFFECT OF FE MESH REFINEMENT ON NORMALIZED STRESSNORMALIZED STRESS--INTENSITY FACTORSINTENSITY FACTORS

2 φ / π0.0 0.2 0.4 0.6 0.8 1.0

1.00

1.05

1.10

1.15

1.20

8 Wedge model10 Wedge model14 Wedge model

Semi-circular crack:a / c = 1a / t = 0.2

K

S √πa/Q

Estimatedboundaylayer

Raju & Newman (1979)


CRACK CONFIGURATIONS ANALYZED WITH FEACRACK CONFIGURATIONS ANALYZED WITH FEAUNDER REMOTE TENSION OR BENDING LOADSUNDER REMOTE TENSION OR BENDING LOADS

2r

2r

w

2w

2w2w

2w

Raju & Newman (1979-1986)


SURFACE CRACK AT A HOLE UNDER TENSION SURFACE CRACK AT A HOLE UNDER TENSION

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

K

S √πa/Q

2 φ / π

a / c = 1

a / c = 0.2

Shah

Raju-Newman

Raju-Newman

Shah

Newman-Raju Equation

r / t = 1a / t = 0.5

Newman & Raju (1981)

K = S (πa/Q)1/2 F(φ, a/c, a/ t, c/ r, c/ w)


ILLILL--SHAPED ELEMENT MESH PROBLEM SHAPED ELEMENT MESH PROBLEM CORNER CRACK AT A HOLE UNDER TENSION CORNER CRACK AT A HOLE UNDER TENSION

φ, degrees

0 30 60 900.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Model A (ill-shaped elements)Model B (new mesh)

Model C

Newman-Raju Equation

Force Method:

VCCT Method:

K

S √πa/Q

r / t = 1a / c = 1a / t = 0.2

Tan et al (1988)


STRESSSTRESS--INTENSITY FACTORS FOR QUARTERINTENSITY FACTORS FOR QUARTER--ELLIPTICELLIPTICCORNER CRACKSCORNER CRACKS

Parametric angle, φ , degrees

0 30 60 902.0

2.5

3.0

3.5

BEM

Equation(Newman-Raju)

WFM(3D)FEM(DIM)

Plate surface Hole surface

FADD(3D)

FEAM

FEM(GIL,J)

5%

r/t = 2; r/w = 0.2a/c = 0.8; a/t = 0.2

K

S √πa/Q

Bakuckas (1999)


CORNER CRACK(S) AT AN OPENCORNER CRACK(S) AT AN OPEN--HOLE UNDER REMOTEHOLE UNDER REMOTETENSION AND BENDING LOADSTENSION AND BENDING LOADS

• Raju and Newman (1979-86)FEA (h-version)~10,000 dof (0.5 < r / t < 2)

• Fawaz and Andersson (2000-04)FEA (p-version)100,000+ dof (0.1 < r / t < 10)

K = S (πa/Q)1/2 F(φ, a/c, a/ t, c/ r, c/ w)

2w


Fawaz & Andersson Crack Configurations Analyzed and Fawaz & Andersson Crack Configurations Analyzed and SelectedSelected Values for Tension and Bending EvaluationsValues for Tension and Bending Evaluations

Hole-radius-to-plate-thickness (r/ t) ratio:0.1, 0.111, 0.125, 0.143, 0.166, 0.2, 0.25, 0.333,0.5, 0.666, 0.75, 0.8, 1.0, 1.25, 1.333, 1.5, 1.666,2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0

Crack-depth-to-crack-length (a/c) ratio:0.1, 0.111, 0.125, 0.143, 0.166, 0.2, 0.25, 0.333, (0.4),0.5, 0.666, 0.75, 1.0, 1.333, 1.5, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0

Crack-depth-to-plate-thickness (a/ t) ratio:0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99

Note: a/ t < 0.1 needs further FE analyses (like a/ t = 0.01 & 0.05)Red: Considered by Raju & Newman


Corner Crack at Hole under Tension: a/c = 1 and Corner Crack at Hole under Tension: a/c = 1 and φφ = 0 & 90= 0 & 90oo

a / t0.0 0.2 0.4 0.6 0.8 1.0

F(0)or

F(90)

0

1

2

3

4

5

NR φ = 0 NR φ = 90 degFA φ = 0 FA φ = 90 deg

Corner crack at hole - Tensionr / t = 1 and a / c = 1 Major

discovery

w = 6 r

w = 400 r

}

}


Corner Crack at Hole under Bending: a/c = 1 and Corner Crack at Hole under Bending: a/c = 1 and φφ = 0 & 90= 0 & 90oo

a / t0.0 0.2 0.4 0.6 0.8 1.0

F(0)or

F(90)

-2

-1

0

1

2

3

NR φ = 0 NR φ = 90 degFA φ = 0 FA φ = 90 deg

Corner crack at hole - Bendingr / t = 1 and a / c = 1

Majordiscovery

w = 6 r

w = 400 r

}

}


Corner Crack at Hole under Tension: a/c = 0.25 and a/t = 0.1 Corner Crack at Hole under Tension: a/c = 0.25 and a/t = 0.1

2φ / π0.0 0.2 0.4 0.6 0.8 1.0

Ft

0

1

2

3

4

5

0.1250.51.02.08.0

Corner cracks at hole - Tensionr / w = 0; a / c = 0.25; a / t = 0.1Fawaz and Andersson (2000) r / t

Newman and Raju (1986)NRFA Equations



2φ / π0.0 0.2 0.4 0.6 0.8 1.0

Ft

0

1

2

3

4

5

0.1250.51.02.08.0

Corner cracks at hole - Tensionr /Fawaz and Andersson (2000) r / t


w = 0; a / c = 0.5; a / t = 0.1



2φ / π0.0 0.2 0.4 0.6 0.8 1.0

Ft

0

1

2

3

4

5

6

0.1250.51.02.08.0

Corner cracks at hole - Tensionr / w = 0; a / c = 1.0; a / t = 0.5

Fawaz and Andersson (2000) r / t

NRFA EquationsNewman and Raju (1986)



2φ / π0.0 0.2 0.4 0.6 0.8 1.0

Ft

0

1

2

3

4

5

6

0.1250.51.02.08.0


Fawaz and Andersson (2000)


r / t


Corner Crack at Hole under Tension: a/c = 4 and a/t = 0.95 Corner Crack at Hole under Tension: a/c = 4 and a/t = 0.95

2φ / π0.0 0.2 0.4 0.6 0.8 1.0

Ft

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.1250.51.02.08.0


Fawaz and Andersson (2000) r / t



APPLICATION TO FATIGUEAPPLICATION TO FATIGUE--CRACK GROWTH CRACK GROWTH

Plane-stress behavior Free surface

Jolles & Tortoriello (1983)Newman & Raju (1984)

Plane-strain behavior

Crack


PLANEPLANE--STRESSSTRESS--TOTO--PLANEPLANE--STRAIN CONVERSIONSTRAIN CONVERSION

Stress ratio, R0.0 0.2 0.4 0.6 0.8 1.0

βR

0.0

0.2

0.4

0.6

0.8

1.0

1.2Jolles & Tortoriello (1983)

βR = 0.9 + 0.2 R2 - 0.1 R4

βRβR

Newman & Raju (1984)

∆Kfs = βR ∆K


OFFSET ANGLES TO AVOID BOUNDARY LAYER OFFSET ANGLES TO AVOID BOUNDARY LAYER

Parametric angle, φ , degrees

0 30 60 902.0

2.5

3.0

3.5

Plate surface Hole surface

K

S √πa/Q

FASTRAN

AFGROW

NASGRO

FASTRAN (βR)

AFGROW &NASGRO(βR)

Typical solution

Offset angles


PREDICTION OF SURFACEPREDICTION OF SURFACE--CRACKCRACK--ATAT--HOLE SHAPE HOLE SHAPE AND CRACKAND CRACK--GROWTH BEHAVIORGROWTH BEHAVIOR

N x 10-3, cycles

0 5 10 15 20 25

a , cmm

0

1

2

3

4

5

6

7 McGee & Hsu (1981) 7075-T651 t = 6.3 mm r = 4 mm R = 0.1

Test (a)

Predicted

Test (c)

a / t0.0 0.2 0.4 0.6 0.8 1.0

a / c

0.0

0.5

1.0

1.5

2.0

2.5

McGee & Hsu (1981) 7075-T651 t = 6.3 mm r = 4 mm R = 0.1

Test

Predicted

2r


APPLICATION TO FRACTUREAPPLICATION TO FRACTURE(Surface crack in D6ac steel under bending loads) (Surface crack in D6ac steel under bending loads)

Parametric angle, φ, deg.

0 30 60 90

Parameters (αh, Fb, Fb αh)

0.0

0.5

1.0

1.5

2.0

2.5

Fb

αh

Fb αh

D6ac steela / c = 0.86a / t = 0.72c / w = 0.1Sb / σo = 0.87

Critical location (φc = 14.4 deg.)

Test range


FRACTURE OF SURFACE AND THROUGH CRACKS FRACTURE OF SURFACE AND THROUGH CRACKS

Constraint, αh

1.0 1.5 2.0 2.5 3.0

(Kφ)Ic

0

20

40

60

80

100

+20%

-20%

(Kφ)Ic αh = 140 MPa-m1/2

Three-point bend

D6AC steel (Reuter and Panontin):Surface crack - tensionSurface crack - bending


CONCLUDING REMARKSCONCLUDING REMARKS

1. Advancements in computers and highly-refined finite-element models have been used to develop more accurate stress-intensity factors for three-dimensional crack configurations – but more analyses and improved equations are needed over a wide range of loading and crack configuration parameters (such as very shallow and very deep cracks).

2. The Newman-Raju equations have been found to be fairly accurate over a wide range in crack configurations, but the new Fawaz-Anderssonfinite-element solutions for a corner-crack-at-a-hole under remote tension or bending loads have resulted in more accurate equations.

3. Three-dimensional stress-intensity factor solutions have improved the fatigue-crack growth predictions for complex crack configurations.

4. Three-dimensional stress-intensity factor solutions and local crack-front constraint variations have allowed the correlation of fracture for surface and through cracks under both tension and bending loads.

Thank you, Dr. Irwin …Thank you, Dr. Irwin …

the evolution and appication of three- dimensional stress ... · appication of three-dimensional...

Documents