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CRACK GROWTH MODEL USING JAAP SCHIJVE METHOD

cycle 1 :

Yes 1 :

LT 1 :

No 0 :

TL 0 :

Applied Stress, maximum

σ.max 17.4 ksi * :

QUESTIONS ON PAGES 2 and 7

Applied Stress, minimum

σ.min 0 ksi * :

R σ.min σ.max / : 0

Applied stress ratio.

Bearing stress ratio at loaded hole.

k.brg 1.93 :

Diameter of hole in sheet.

D 0.128 in * :

Thickness of sheet.

t 0.045 in * :

Pitch of fastener holes for rivet row (also used as width of strip being analyzed.

w 1.0 in * :

TYPE OF PROBLEM TO BE SOLVED:

PRIMARY CRACK FROM PRIMARY HOLE:

caseA Yes :

Length primary crack (Chicago ACO, Sec. 9.1)

a.A 0.05 in * :

Length of rogue flaw (AF Memo EN-SB-08-002)

a.B 0.05 in * :

ROGUE FLAWS AT ALL HOLES:

caseC No :

Length of rogue flaw (AF Memo EN-SB-08-002)

a.C 0.01 in * :

In Linear Elastic Fracture Mechanics, the standard crack definition is 1/2 of the length from tip to tip. If there is only one crack at the hole, then the crack length starts at the edge of the hole

caseA a.init D a.A a.B + (+ 2 / : a.init D a.C + 2 / : if

Initial crack length, two cracks originating at a circular hole. Initial crack length, one crack only.

a.init in 0.114

Fracture Mechanic Material Properties

The AFGROW version 4-9 software uses an older database of material properties , from NASGRO version 3.0. These values are needed when reproducing the results from an analysis using AFGROW version 4-9.

Select the material grain direction aligned with the primary stress (use "LT" for Longitidunal and "TL" for transverse).

GRAIN LT :

Select the source of data to be used (use "3" for AFGROW version 4-9 and "4" for current NASGRO data).

VERSION 3 :

GRAIN LT ≡ F.tu 61 ksi * : F.tu 60 ksi * : if

Ultimate Tensile Strength of 2024-T3 sheet, B-basis, 0.010-0.062 thk (ref. MMPDS-01, Table 3.2.3.0(e1) Select value based on L or T-L grain direction.

NASGRO Material Database Version 4.0 (ref. DOT/FAA AR-05/15) Al. 2024-T3 Clad sheet grain direction LT TL ---------------------------

NASGRO Material Database Version 3.0 (used by AFGROW 4-9) Al. 2024-T3 Clad sheet grain direction TL LT ---------------------------

Chicago ACO Table 1 (for Walker equation) Al. 2024-T3 Clad sheet grain direction L-T T-L ---------------------------

Material 1.5 1.5 1.5 1.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.5 1.5 1.5 1.5 0.25 0.25 0.5 0.5 1111 3.20 3.0 2.601 3.284 0.80 108 -^* 1.3 108 -^* 2.44 108 -^* 0.829 108 -^* 30.0 ksi * in sqrt * 27.0 ksi * in sqrt * 29.0 ksi * in sqrt * 33.0 ksi * in sqrt * 74.3 ksi * in sqrt * 67.0 ksi * in sqrt * 58.0 ksi * in sqrt * 66.0 ksi * in sqrt * 2.7 ksi * in sqrt * 2.7 ksi * in sqrt * 2.9 ksi * in sqrt * 2.9 ksi * in sqrt * 0.100 in * 0.090 in * 0.100 in * 0.100 in * 53 ksi * 48 ksi * 53 ksi * 48 ksi * 10.6 106^* psi * 10.6 106^* psi * 10.6 106^* psi * 10.6 106^* psi * 144 mat :

q.W 0.64647 :

p.W 3.71980 :

C.W 6.76125 1010 -^* :

HOW DO I ADD COLUMNS TO AN EXISTING MATRIX?

GRAIN 1 ≡ (VERSION 4 ≡ (& COLUMN 1 : GRAIN 2 ≡ (VERSION 4 ≡ (& COLUMN 2 : GRAIN 1 ≡ (VERSION 3 ≡ (& COLUMN 3 : COLUMN 4 : if if if

Select appropriate column of data.

α A.k B.k C.th p q n C K.1c K.c K.0 t.o F.ty E 141 mat Material 114 COLUMN COLUMN submatrix :

α 1.5

A.k 1

B.k 1

C.th 1.5

p 0.5

q 1

n 2.601

C 2.44 108 -^*

K.1c ksi in sqrt * 29

K.c ksi in sqrt * 58

K.0 ksi in sqrt * 2.9

F.ty ksi 53

E psi 1.06 107^*

F.tu ksi 61

The stress intensity factor limits (Ko and Kcrit) must be determined before using the NASGRO curve-fit equation to determine the crack growth rate.

Compare the flow stress to the applied stress. The NASGRO data was collected with test specimens stressed using Smax/So = 0.3.

σ.o F.tu F.ty + 2 / :

Flow stress

σ.o ksi 57

σ.max σ.o / 0.31

Ratio of applied stress to flow stress.

A sequence of coefficients must be solved to determine an appropriate value for "f", the crack opening function (ref. NASGRO reference manual, Eq. 2.3 through Eq. 2.7).

A.0 0.825 0.34 α * - 0.05 α 2^* + (π 2 / σ.max σ.o / * cos (1 α /^* : 0.395

A.1 0.415 0.071 α * - (σ.max σ.o / * : 0.094

A.3 2 A.0 * A.1 + 1 - : 0.116 -

A.2 1 A.0 - A.1 - A.3 - : 0.627

f R A.0 A.1 R * + A.2 R 2^* + A.3 R 3^* + 21 mat max :

Reference AFGROW User's Guide, Section 3.2.2.3. (applicable only when R ≥ 0)

f 0.39

Determine the boundaries of the NASGRO SIF curve, where the lower left convertes in the threshold SIF (Kth) and the upper right converges on the critical SIF (Kcrit).

Initial crack length

a.i 0.05 in * :

Reference crack length for threshold stress intensity factor.

a.o 0.0015 in * :

Threshold stress intensity factor range at R=0. (Afgrow, pg. 40)

ΔK.0 2.9 ksi * in sqrt * :

ΔK.th ΔK.0 a.i a.i a.o - / sqrt * 1 f - 1 A.0 - (1 R - (* / (1 C.th R * + (^/ :

Threshold stress intensity factor for any given R.

ΔK.th ksi in sqrt * 2.9

Plane strain fracture toughness of aluminum 2024-T3 sheet (NASGRO Data, DOT/FAA AR-05/15)

K.1c 27 ksi * in sqrt * :

Reference thickness for K1c.

t.o in 0.100

K.crit K.1c 1 B.k e A.k t t.o / * (2^- (^* + (* :

Critical stress intensity factor (confirm by chart B-7)

K.crit ksi in sqrt * 49.1

Determination of the applied stress intensity factors. Requires the sum of 3 conditions: K1 = SIF due to series of holes (cracks) in a loaded sheet K4 = SIF due to an applied load that opens a hole (crack) K5 = SIF due to a rivet load that is applied to a loaded sheet These SIF's can be added due to the principle of superposition. The average of K4 and K5 must be found before adding it to K1.

(Schijve eq. 5.8) (Schijve eq. 5.12) (Schijve eq. 5.17) (Schijve Sec. 5.5)

In Linear Elastic Fracture Mechanics, the standard crack definition is 1/2 of the total crack length from tip to tip.

a.init in 0.114

β.1 π a.init * w / tan π a.init * w / / sqrt :

Stress Intensity Ratio with cracks at primary hole, in strip of finite width. (ref. Schijve, Eq. 5.8, pg.89)

β.1 1.022

K.1 β.1 σ.max * π a.init (* sqrt * :

Stress intensity factor of primary crack.

K.1 ksi in sqrt * 10.6

P k.brg σ.max * D * :

Rivet load, per unit thickness.

P lbf in / 4298

K.4 P w / (π a.init (* π a.init (* w / cos / sqrt * :

Stress intensity factor for loaded sheet, using rivet load distributed across the sheet(Schijve Eq. 5.12).

K.4 ksi in sqrt * 2.7

K.5 P π a.init * sqrt / :

Stress intensity factor for load applied to a cracked hole (Schijve Eq. 5.17).

K.5 ksi in sqrt * 7.2

K.load K.4 K.5 + 2 / :

Stress intensity factor for a rivet hole with load transfer (Schijve Section 5.5)

K.load ksi in sqrt * 4.9

K.max K.1 K.4 K.5 + 2 / + :

Total Stress Intensity Factor.

K.max ksi in sqrt * 15.6

Determine the stress intensity range (only useful if R ≠ 0).

Applied stress ratio.

R 0.0

K.min K.max R * :

Minimum stress intensity factor.

K.min ksi in sqrt * 0

ΔK K.max K.min - :

Applied stress intensity range.

ΔK ksi in sqrt * 15.6

Walker Equation for crack growth rate (ref. Chicago ACO paper, Section 7.0)

da_dN.W C.W in * (1 R - (q.W (^ΔK ksi in sqrt * / * (p.W (^* :

da_dN.W in cycle / 1.84 105 -^*

NASGRO formula (ref. DOT/FAA AR-05/15, Equation 4.1)

δa_δN C in * ΔK ksi in sqrt * / 1 f - 1 R - / (* (n^* 1 ΔK.th ΔK / - (p^1 K.max K.crit / - (q^/ * :

WHAT IS THE PROBLEM?

δa_δN in cycle / #

"result is above max. allowed positive number"

...But when I copy-paste the expression below...

C in * ΔK ksi in sqrt * / 1 f - 1 R - / (* (n^* 1 ΔK.th ΔK / - (p^1 K.max K.crit / - (q^/ * in cycle / 1.10 105 -^*

this solves