A Proposed Verification Check when Developing Stress Intensity Solutions

Marcus Stanfield September 10 2019

1 Overview

. Validation vs. Verification . Previous work . Benchmark case . J vs. K . Material sensitivity . Radius of integration . Mesh dependency . AFGROW solutions

2 Background

Validation – “How bad do I suck at Physics?”

FCL-Pretty Good FCL-Terrible

3 Background

Verification – “How bad do I suck at Math?”

10 x 10 = 103 Would you buy this calculator?

Different levels of accuracy required for validation and verification

4 StressCheck Verification

. Increasing polynomial order to converge energy

p-level DOF Total Potential Energy Convergence Rate % Error 1 4323 -1.85E-07 0 5.48 2 30189 -1.86E-07 0.72 1.36 3 97142 -1.86E-07 1.21 0.33 4 224730 -1.86E-07 1.51 0.09 5 432501 -1.86E-07 1.33 0.04 6 740003 -1.86E-07 1.33 0.02

. K (CIM) is super convergent . Necessary, but not sufficient

5 Previous Work

“Benchmarking Problems in Fatigue Crack Growth Analyses” Pilarczyk et. al. AFGROW Workshop 2016

. Edge/Center Cracked Plate – Handbook is not dependent on t . 2D (Plane Strain) Comparisons – Max Error ~0.5% . Beasy is not symmetric

. 3D Comparison – Max Error ~5% . StressCheck 3D consistently higher than FRANC3D

6 Benchmark Case

Quarter circular crack at hole . Looking at damage tolerance: a = c = 0.05 inch . Width = 5 inches . Height = 6 inches . Diameter = 0.5 inch . Thickness = 0.125 inch . Young’s Modulus = 10.3e6 psi . Poisson’s Ratio = 0.33 . Tensile Traction = 1 psi Handbook solution along crack front Newman & Raju, Forman, AFGROW DTDHandbook Determining both K and J

7 K vs. J

Stress Intensity Factor (K) vs. Strain Energy Release Rate (J) How do these compare? =

𝐾𝐾𝐼𝐼 𝐽𝐽𝐼𝐼𝐸𝐸′ Where = ′ 1 𝐸𝐸 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸 � 𝐸𝐸 2 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 Plane-strain error: − 𝑣𝑣 = 1 . . = 1 0.33 =5.93%2 −0 5 increase 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 2 −−0 𝑣𝑣5 − 8 Quarter Circular Crack at Hole

StressCheck vs. FRANC3D

0.9

0.85

0.8 @ Theta = 45 0.75 Difference = 3.11% FRANC3D 0.7 StressCheck CIM

0.65

Stress Intensity Intensity (psi*in^0.5) Stress 0.6 0 45 90 Theta (degrees) StressCheck is generally higher 9 Quarter Circular Crack at Hole

StressCheck vs. FRANC3D vs. Handbook

0.9

0.85

0.8

0.75 FRANC3D StressCheck CIM 0.7 Handbook 0.65

Stress Intensity Intensity (psi*in^0.5) Stress 0.6 0 45 90 Theta (degrees) This doesn’t help… yet 10 Quarter Circular Crack at Hole

StressCheck vs. FRANC3D vs. Handbook + SC J-Plane Strain

0.9

0.85

0.8 FRANC3D 0.75 StressCheck CIM 0.7 StressCheck Plane-Strain Handbook 0.65

Stress Intensity Intensity (psi*in^0.5) Stress 0.6 0 45 90 Theta (degrees) J-Plane Strain is closer to FRANC3D 11 Quarter Circular Crack at Hole

StressCheck vs. FRANC3D vs. Handbook + SC J-Plane Strain + SC J-Plane Stress

0.9

0.85

0.8 FRANC3D 0.75 StressCheck CIM StressCheck Plane-Stress 0.7 StressCheck Plane-Strain 0.65 Handbook

Stress Intensity Intensity (psi*in^0.5) Stress 0.6 0 45 90 Theta (degrees) J-Plane Stress is close to the Handbook solution 12 Converting J to K

“The conversion… is valid for linear elastic materials under plane stress or plane strain conditions. For these cases it can be shown that the conversion of J to K produces the same result as directly computing K using the Contour Integral Method (CIM). However, for 3D part-thru (corner) cracks that are not purely plane stress or strain, the J to K conversion… produces values of K that may differ from those computed by CIM. Numerical studies have shown that the difference is problem-dependent but is typically limited to 5% or less”

StressCheck Master Guide, Advanced Guide, Chapter 3: Fracture Mechanics, Note about Converting J to K

Should plane strain and plane stress bound the CIM?

13 Quarter Circular Crack at Hole

StressCheck vs. SC J-Plane Strain vs. SC J-Plane Stress

9.00E-01

8.50E-01

8.00E-01 @ Theta = 45 7.50E-01 StressCheck CIM Difference = 2.08% StressCheck Plane-Stress 7.00E-01 StressCheck Plane-Strain 6.50E-01

Stress Intensity Intensity (psi*in^0.5) Stress 6.00E-01 0 45 90 Theta (degrees) CIM is outside the Plane Stress and Strain bounds 14 Material Sensitivity

Modulus of Elasticity 8.50E-01

8.00E-01

“Stress7.50E-01 Intensity Factors are not dependent on the material” –SIAG-2010-10884 E = 10e6 psi 7.00E-01 “Procedure for Developing Stress Intensity Factor (SIF or K) E = 30e6 psi Solutions in StressCheck” 6.50E-01 Stress Intensity Intensity (psi*in^0.5) Stress 6.00E-01 0 45 90 Theta (degrees) No change for Modulus of Elasticity 15 Material Sensitivity

Poisson’s ratio 9.00E-01

8.50E-01

8.00E-01

7.50E-01 v = 0.33 7.00E-01 v = 0.0001

6.50E-01

Stress Intensity Intensity (psi*in^0.5) Stress 6.00E-01 0 45 90 Theta (degrees) Sensitive to Poisson’s Ratio 16 Material Sensitivity

Poisson’s ratio, J – Plane-Stress, Handbook 9.00E-01

8.50E-01

8.00E-01 v = 0.33 7.50E-01 v = 0.0001 7.00E-01 StressCheck Plane-Stress Handbook 6.50E-01

Stress Intensity Intensity (psi*in^0.5) Stress 6.00E-01 0 45 90 Theta (degrees) Solutions collapse onto a single curve close to Handbook 17 USAF/BAMF Ground Rules

USAF: Layers = 3, To = 0.0225*c, Ttotal = 0.2475*c, Krad = 0.06*c BAMF: Layers = 2, To = 0.0225*c, Ttotal = 4*To, Krad = 1.3*To 9.00E-01 8.50E-01 8.00E-01 USAF CIM 7.50E-01 BAMF CIM 7.00E-01

(psi*in^0.5) USAF Plane-Strain @ Theta = 45 Stress Intensity Intensity Stress BAMF Plane-Strain 6.50E-01 USAF difference = 2.08% 6.00E-01 BAMF difference = 1.50% 0 45 90 Theta (degrees) Similar behavior between USAF and BAMF 18 Radius of Integration

“In general for 3D crack problems the contour integral is not path-independent, in particular for curved cracked fronts. … the radius of integration includes the effect of adjacent points… Using an integration radius as small as possible will decrease the influence of adjacent points when computing SIFs… the SIFs must be evaluated in the limit when the radius of integration goes to zero… Ideally, the radius should be the smallest possible radius outside the first layer of refinement… Numerical evidence suggests that an integration radius in the range R/a = 0.025 – 0.030 is enough to obtain accurate results” when Ri = 0.0225*a

StressCheck Master Guide, Advanced Guide, Chapter 3: Fracture Mechanics, A note about the use of the CIM in 3D

19 Radius of Integration

Convergence

K J 7.25E-01 4.60E-08 7.20E-01 4.50E-08 7.15E-01 4.40E-08 7.10E-01 4.30E-08 7.05E-01 y = 2.46E-01x + 6.98E-01 4.20E-08 7.00E-01 R² = 9.88E-01 4.10E-08 y = 2.17E-08x + 4.07E-08 6.95E-01 R² = 9.98E-01 - lbf/in^2) (in J1p K1 (psi*sqrt(in)) 6.90E-01 4.00E-08 6.85E-01 3.90E-08 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 R/a R/a Both K and J decrease linearly with decreasing radius 20 Radius of Integration

Error K J 7% 9% 6% 8% 7% 5% 6% 4% 5% 3% 4% 3% 2% Difference (%) Difference (%) Difference 2% 1% 1% 0% 0% 0 20 40 60 80 0 20 40 60 80 Angle (degrees) Angle (degrees) 2-3% error for K, 2-4% error for J 21 Radius of Integration

Limit as radius of integration goes to zero 0.9

0.85

0.8 @ Theta = 45 FRANC3D CIM vs J Difference = 1.18% 0.75 StressCheck CIM CIM vs FRANC3D = 1.29% StressCheck Plane-Stress 0.7 StressCheck Plane-Strain 0.65 Handbook Stress Intensity Intensity (psi*in^0.5) Stress 0.6 0 45 90 Theta (degrees) Spread is decreasing, noise is increasing 22 StressCheck Ground Rules

Elliptical Crack Meshing Guidelines Rev 3 (May 2019) . Additional curve refinement along crack front . Higher mesh density, To can be decreased to 0.15*0.15*0.15*c

23 StressCheck Ground Rules

Standard vs. Fine Mesh Difference 0.9 Radius\Mesh Standard Fine 0.85 1.25 1.43% 0.62% 0 0.89% 0.32% 0.8 FRANC3D 0.75 StressCheck CIM 0.7 StressCheck Plane-Strain Handbook 0.65 Stress Intensity Intensity (psi*in^0.5) Stress Stress Intensity Intensity (psi*in^0.5) Stress @ Theta = 45 0.6 Handbook Difference = 5.84% 0 45 90 Theta (degrees) Improved difference with new ground rules 24 AFGROW Solutions Difference Classic and Advanced Solution C A 0.9 Classic -3.32% -4.27% 0.85 Advanced -0.87% -0.52% 0.8 FRANC3D StressCheck CIM 0.75 StressCheck Plane-Strain 0.7 Handbook AFGROW Classic 0.65 AFGROW Advanced Stress Intensity Intensity (psi*in^0.5) Stress 0.6 0 45 90 Theta (degrees) Advanced solution is closest to FEM solutions 25 So What?

AFGROW used with different levels of uncertainty on select T-38 locations

Uncertainty Beta Uncertainty = 3% Beta Life 0.800 3.00% 10.00% 0.700 1.50% 4.00% 0.600 0.50% 1.70% 0.500 0.400 Implications on: 0.300 Crack growth rate 0.200 Crack Length (in) Crack shape 0.100 SOLR correlations 0.000 Flight Hours

26 Summary/Conclusions

. Verification errors should be less than Validation errors . K solutions are dependent on material . Plane-strain conditions at crack front due to stress triaxiality . Handbook solutions do not account for material . K solution is dependent on radius of integration . Current USAF/BAMF ground rules result in 3% uncertainty . StressCheck’s meshing guidelines produce the least uncertainty . Small beta uncertainty can lead to large life uncertainty

. Extra Verification: Compare to J integral derived SIF

27 Recommendations

USAF Revise SIAG-2010-10884 “Procedure for Developing Stress Intensity Factor (SIF or K) Solutions in StressCheck” . Solution is dependent on Poisson’s ratio . Update meshing ground rules with StressCheck’s recent guidelines . Include information on radius of integration

ESRD . Incorporate automated radius of integration convergence . Implement calculation of K from J integral

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