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Crack Path Determination for Non-Proportional Mixed-Mode Fatigue
Shelby HighsmithDoctoral Thesis Defense
13 March 2009
Introduction: Crack path in damage tolerance risk assessment
•Crack path prediction is essential to accurate life prediction & risk assessment
‣ Crack growth rate through varying stress fields
‣ Energy of fracture ligament in rotating turbomachinery
•Current crack growth models still need development for complex mixed-mode loading
Project objectives
•Assess the ability to extend current knowledge of crack path selection and crack growth mode to non-proportional fatigue crack growth
•Validate new specimen design for testing a broad range of mixed-mode loading
•Contribute to material database for aerospace superalloy Inconel 718 under mixed-mode crack growth
•Enhance crack path prediction capabilities of linear elastic fracture mechanics (LEFM)-based fracture modeling software FRANC3D
Background: Mixed-mode crack path
•Crack growth largely considered in terms of Mode I (opening loads)
•Microstructure, geometry, load transients can perturb crack angle or applied load
‣ Addition of Mode II
•What is expected behavior of crack trajectory?
Δθ?
KI
KII
τrθ
Early models of fracture extension angle
•Erdogan & Sih (1963) - max tangential stress (σθθ) criterion (MTSC)
•Hussain et al (1974) - max strain energy release rate G (“Griffith” or MERR criterion)
•Sih (1974) - min strain energy density S
θ σθθ
σrrr
•All generally predict same crack deflection as function of mode mixity Φ (0° ~ Mode I, 90° ~ Mode II)
Prediction of symmetric fracture
Δθ
φ0°(KI)
90°(KII)
0°
-45°
-90°
Gmax
Smin
MTSC
! = tan!1
!KII
KI
"
ϕ
Beyond the inherent assumption of symmetric fracture
•Any criterion for non-zero angle crack branching under non-zero Mode II loading will predict stable Mode I growth
•Under high KII, stable mixed-mode or Mode II growth is observed
‣Mode I to Mode II-dominated transition (to max shear plane) around Φ = 40° in 7075-T6 (Hallback & Nilsson, 1994)
‣ Transition to shear crack growth in range of 68° < Φ < 75° in 2024-T3 (Amstutz et al, 1995)
‣ Transition to co-planar at Φ = 45° in HY130 steel at RT (Maccagno & Knott, 1992)
Critical stress criteria
• Chao & Liu (1997) argue that crack propagation occurs along most critical mode
‣ Follows plane where either τ or σ first goes criticalσθθ
σθθ
θ*
σrr
τrθr
τrθσrr
rθ**
Critical stress criteria
• Chao & Liu (1997) argue that crack propagation occurs along most critical mode
‣ Follows plane where either τ or σ first goes critical
• Competition b/w MTSC and max shear stress criterion (MSSC) based on loading path (mixity)
‣ Transition between the two is based on ratio of material props τcrit/σcrit
MTSC
Expected bimodal data
Angle θ** of max τrθ
Angle θ* of max σθθφcrit
Problem: Non-proportional loads
•Different points of waveform have different mixities φ
•Which parameters can be used to predict θ?
•Can we do so using only LEFM / K?
Max tension of constant torque or
OOP test
Max torque of constant tension or
OOP test
Research in non-proportional crack growth direction
•Much work in rolling contact fatigue (RCF) for rail application
‣Sequential KI-KII sub-cycles
‣Generally studying conditions leading to co-planar mixed-mode growth or Mode I crack branching
‣KII usually fully reversed -- not a clear case for current study
‣Elastic-plastic analysis of residual crack opening displacements & closure/interference
•Significant opportunities for expanding modal transition criteria
Experimental approach
•Development & validation of novel specimen design
‣ Inclined through-crack round (ITCR) specimen
‣Developed for 3-mode testing, enhanced 2-mode testing
•NASA testing of conventional notched thin-walled tube
•Four fatigue load cases tested
‣Proportional in-phase (baseline)
‣KI/ΔKII and ΔKI/KII
‣ 180° out-of-phase (sinusoidal) ΔKI/ΔKII
2mmEDM
thru-slot
Tension-torsion notched thin-walled tube
• Inconel 718 per AMS 5662
• 17 specimens tested at NASA Marshall Space Flight Center
‣ Added to this research program during development of ITCR specimen
• Uniaxial compression then tension pre-cracking
• Measure initial deflection angle from pre-crack upon mixed-mode loading
• Examine fracture surface morphology
OD 30.5mm ID 25.4mm
254mm
70mm
SIFs for tubular T-T specimen
•FRANC3D linear elastic boundary element model
SIFs for tubular T-T specimen
•FRANC3D linear elastic boundary element model
•Each specimen precrack geometry modeled using fracture surface measurements
SIFs for tubular T-T specimen
•FRANC3D linear elastic boundary element model
•Each specimen precrack geometry modeled using fracture surface measurements
•Tension & torsion applied individually and together to verify separability
NASA Mixed mode test plan
•Some initial tests prior to collaboration
•Baseline in-phase tests over range of mixity
•Constant tension (KI) / cyclic torsion (KII)
•Constant torsion (KII) / cyclic tension (KI)
•180° out-of-phase
Results: NASA in-phase tests
MSSC
MTSC
Mixity
Fractographsfollowing…
• In-phase deflections follow Max Tangential Stress criterion as expected up to critical φ value, then see transition to Max Shear Stress
• Torque limitations prevented further MSS testing
In-phase fractography - 500x
• Clear morphology difference reinforces transition in crack path deflection mechanism‣ Mode I transgranular crystallographic cracking observed previously (Mercer 1999)‣ Mode II appearance ~ slip-enhanced transgranular cleavage
Tensile crack (MTS) deflectionθ = -27°
Shear crack (MSS) deflectionθ = 18°
In-phase fractography - 2000x
• Fine microstructural features on shear crack flats suggest they are not the product of crack face contact
• In-phase transition as expected
Tensile crack (MTS) deflectionθ = -55°
Shear crack (MSS) deflectionθ = 18°
Effect of pre-crack shape, N09
• Bifurcation between branch modes at mid-thickness corresponds to change in due to irregular crack shape‣ The reason for Appendix A
Results: Constant KI / Cyclic KII
•Two distinct groups of crack path deflection•No clear indicator of transition criterion
Max loadcondition
Mixity
Fractography - Constant Tension / Cyclic Torsion 500x
• More pronounced morphological difference but similar to in-phase• What are the respective driving forces?
Shear crack (MSS) deflectionθ = +2°
Tensile crack (MTS) deflectionθ = -41°
Local kink tip reference frame
•Local stress intensity factors (SIFs) at small kink, k1 & k2
• In first order approximation, identical to considering primary crack tip stresses
KI
KII
k1, k2
k1 =!
2!r "!! = cos#
2
!KI cos2
#
2" 3
2KII sin #
"
k2 =!
2!r "r! =12
cos#
2
!KI sin # + KII(3 cos # " 1)
"
Kink tip SIFs, N13
•Tensile crack branch between k1max and local Δk1max
•Shear crack branch between k2max and Δk2max
•This pattern evident for all branch angles
k1maxk2max
•Start from (da/dN)max approach to incorporate mean stress/R-ratio effect in two parameter crack growth law
‣Prior implementations not directly applicable
•Rearrange Walker equation
‣
•Apply in terms of kink SIFs k1 and k2 and maximize
‣What is w?
Effective SIF range
da
dN= C0!K
m
!K =!K
(1!R)1!w= !KwK1!w
max
Fitting Walker ΔK
• Initial estimate of w=0.3 from prior superalloy data
‣ Good correlation b/w Δk1 and tensile branches (lower square data points)
‣ k1max stronger driver than Δk1
• Shear branches follow more closely to Δk2max than k2max
‣ Swap exponents on Δk2 and k2max in Walker form (upper triangle data pts)
‣ w1=0.3, w2=0.7
‣ Mode-dependent weighting not without precedent
!k1
!k2
!K = !KwK1!wmax
Perfectprediction
Defining a modal transition
•Using w1=0.3, w2=0.7 in this approach also allows for a clear demarcation of branch mode transition
‣ Critical ratio of Δk2max to Δk1max denoted by slope of lines from origin
‣ Shear & tensile branches (open & solid symbols) separate on either side for w1=0.3, w2=0.7 (red squares); not for other cases
Results: Constant KII/cyclic KI
•Single cluster of deflection angles, presumably tensile‣ “Tilted” by steady torque-induced crack tip tensile stress at -70°
Max loadcondition
Fitting Walker ΔK -constant KII / cyclic KI
•Unlike previous case, Δk1 stronger driver than k1max
‣w1=0.7 better fit
‣Was 0.3 for prior case
•No modal transition for comparison
!K = !KwK1!wmax
Results: 180° out-of-phase
• Apparently alternating deflection behavior with increasing KII/KI based on traditional framework‣ Positive angle, to negative, to divergent with increasing KII
• Driving forces not as expected
ϕ(KIImax/KImax)
Results: 180° out-of-phase
• Highest and lowest KII/KI tests produce tensile branches even at positive angles (w1=0.3), and a second local maximum for N18
• Middle KII/KI test turns out to have highest Δk2max (w2=1)
N19 - Fractography
•Ambiguous fracture surface appearance likely contact-smeared tensile branch features
500x 2,000x
N18 (right) - Fractography
•Highly faceted crystallographic cracking appearance as previous tensile branches (500x)
ITCR Specimen Development
ITCR specimen design
• Inconel 718 per AMS 5662
•4 specimens fabricated in-house (β = 0°)
‣ 4 β = 30° also fabricated but not tested
•Solid bar with through-thickness EDM slot pre-flaw
ITCR specimen design
• Inconel 718 per AMS 5662
• Originally designed for combined Mode I-II-III testing (inclined crack)
• 4 specimens fabricated in-house (β = 0°)
‣ 4 β = 30° also fabricated but not tested
• Solid bar with through-thickness EDM slot pre-flaw
• Generates range of mode mixity as KII increases with radius (with torsion)
b
2b
SIF solution for ITCR
• FRANC3D modeling of multiple specimen configurations a/r, β‣ Essentially same boundary
conditions as NASA model
• FRANC3D modeling of multiple specimen configurations a/r, β‣ Essentially same boundary
conditions as NASA model
• Linear fits across crack front
SIF solution for ITCR
KI =!o!
"a cos2 #!0.8162 + 0.8286
a
r
"
± $o!
"a sin # cos #!"0.018 + 0.759
a
r
"
KII =x
b
!!o!
"a cos #
"0.7114 + 2.3439
a
r" 4.6816
#a
r
$2%&
KIII =!o!
"a sin# cos #!0.8513 + 0.3933
a
r
"
± $o!
"a#sin2 # " cos2 #
$ !"0.0199 + 0.6846
a
r
"
ITCR specimen test matrix
•Proportional in-phase
‣Match NASA’s highest KII/KI at mid-span
ITCR specimen test matrix
•Proportional in-phase
‣Match NASA’s highest KII/KI at mid-span
•Constant KI/ΔKII
‣Match 2 NASA tests x/b ~ 0.4-0.6
ITCR specimen test matrix
•Proportional in-phase
‣Match NASA’s highest KII/KI at mid-span
•Constant KI/ΔKII
‣Match 2 NASA tests x/b ~ 0.4-0.6
•Constant KII/ΔKI
ITCR specimen test matrix
•Proportional in-phase
‣Match NASA’s highest KII/KI at mid-span
•Constant KI/ΔKII
‣Match 2 NASA tests x/b ~ 0.4-0.6
•Constant KII/ΔKI
•180° out-of-phase
Crack angle measurement
•Thin-walled tube crack angles directly observed
• ITCR specimen measured by incremental grinding of resin replicas of fractured specimen along x/b axis
ITCR specimen results: in-phase
ITCR specimen results: in-phase
• Generally follows standard MTS-MSS criteria & transition (Φ~55°-58°)‣ Slight offset in the direction of diminished KII influence
- NASA specimens similar shift in tensile branches
Increasing KII
ITCR results: Constant KI/cyclic KII
ITCR results: Constant KI/cyclic KII
• Use of same Δk formulation (w1=0.3, w2=0.7) as NASA KI/ΔKII data works reasonably well for direction‣ Mode II improves with higher w2
‣ Transition condition ambiguous as data falls in a narrow band of Δk2/Δk1
(w1=0.3, w2=0.7)
!K = !KwK1!wmax
ITCR results: Constant KII/cyclic KI
(Note lip at onset of fatigue cracking)
ITCR results: Constant KII/cyclic KI
• Small lip of initial kink angle bound by w1=0.7 employed in NASA data• Higher w1=0.95 for stable crack angle indicates diminished influence
of k1max for ITCR compared to thin-walled tube
!K = !KwK1!wmax
ITCR3-bottom
Rotation
Up
Up
Down
Down
Final FractureLigament
Final FractureLigament
ITCR results: 180° out-of-phase
ITCR results: 180° out-of-phase
• Even more challenging than NASA OP test data ‣ Requires assumption of full-range Δk1
‣ Crack face contact under negative k1 reducing Mode II-enhanced closure
2
Summary / conclusions
• Crack branch modal transition observed in Inconel 718
‣ In-phase, KI/ΔKII, 180° out-of-phase (thin-walled only)
‣ Verified by clear fracture surface morphological differences
• Walker effective SIF range at kink tip is a very good predictor of crack angle within most mixed-mode fatigue load cases
‣ Though relative influence of kmax & Δk varies by load condition
• Constraint delays modal transition from ϕ ~ 44° in plane-stress (thin-walled tube) to ~55°-58° in solid bar ITCR
• Check your specimen pre-crack shapes
Contributions
• Generated necessary mixed-mode crack branching data for Inconel 718 under multiple conditions
‣ Provided an initial basis for modeling mixed-mode crack path in LEFM framework
‣ Showed the varying influence of LEFM parameters and moved the research question from determining crack branch angles to refining the physical basis for the varying crack-driving forces
‣ Expanded envelope of mixed-mode FCG data beyond the focused realm of RCF research
‣ Generated a library of mixed-mode fractography
• Analyzed & validated a novel specimen design for generating a wide range of data in a single test, producing 3-mode load conditions, and adding different constraint data to standard tubular specimen data
Recommendations
•Full 3-D FEM analysis of specimens should compare in-plane and transverse constraints and their influence on transition
•Elastic-plastic analysis of crack tip deformations by load path may physically justify the varying SIF contributions to crack driving force in the Walker effective SIF framework
‣ Including residual crack opening displacements as they affect closure/interference
• Incorporation of KIII may refine quantification of modal transition criteria through contribution to shear stresses
‣Non-zero β ITCR specimens can contribute significant data
Acknowledgements• My advisor, Dr. Steve Johnson, for his patience & acceptance
• My committee, Dr. Rick Neu, Dr. Jianmin Qu, Dr. Tom Sanders, Dr. Naresh Thadhani, for being ready when I finally was
• MPRL’s Rick Brown & Robert Cooper, for keeping my hands attached & load frames running just long enough
• Dr. Rick Pettit at Pratt & Whitney for conceiving & guiding this whole project, Dr. Greg Swanson & Dr. Tarek Sayyah at NASA MSFC for adding test data & insight, and Michael Middlemas for the SEM fractography library
• My parents for their constant support in every way
• My friends in Atlanta for sharing in research, politics, apartments