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Hydrostatic Stress Effects inLow Cycle Fatigue
A Dissertationby
Phillip Allen
Tennessee Technological UniversityNovember 13, 2002
Hydrostatic Stress Effects in Low Cycle Fatigue
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Outline• Technical Background• Mechanical Testing• Constitutive Model Development• Finite Element Model Development• Finite Element Results• Conclusions and Recommendations
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Classical Metal Plasticity• Bridgman’s experiments• Yield – independent of
hydrostatic or mean stress, σm
• Internal hydrostatic pressure –notched or cracked geometries
( )zzyyxxm I σσσσ ++==31
31
1
131 Ip m −=−= σ
• Determine three quantities:1. Yield Function2. Hardening Rule3. Flow Rule
Hydrostatic Pressure
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Yield Function
σ1
p
k
σ2
σ3
• Yield function, f
• f < 0; Elastic material behavior
• f = 0; Yielding occurs, plastic material behavior
( )321 ,, σσσff =von Mises yield function
( ) 222 kJJf −=
•Independent of hydrostatic stress
•Almost always used in computational plasticity
•Recommended for finite element analysisk – Yield strength in pure shear
23Jeff =σ
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Hardening Rule
σ1 σ2
σ3
k1
k2 a
b
σ1 σ2
σ3
a
b
+
Isotropic Hardening
Combined Hardening
=
Kinematic Hardening
Yield surface – change size, location, or both
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Flow Rule• Relates stresses and plastic strain increments• Analogous to Hooke’s law for elastic behavior
φσ
ε dgdij
plij ∂
∂=
φσ
ε dfdij
plij ∂
∂=
General flow rule
Associated flow, f = g
g – Plastic potential function
dφ – Positive constant
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Richmond, Spitzig, and Sober’s Experiments
I1
σeff = (3J2)1/2
a
σc
σeff = Effective stressσc = Theoretical cohesive strengtha = Sloped = Modified yield strengthσeff = d - aI1
Yield Function daIJJIf −+= 1221 3),(
d
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Drucker-Prager Yield Function
σ1
p
k
σ2
σ3
von Mises
Drucker-Prager
•Dependent on hydrostatic stress•Often used in compacted soil calculations•Seldom recommended for metal plasticity calculations
daIJJIf −+= 1221 3),(
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Low Cycle Fatigue
εf
2Nf
100 1072Nt
cb
ElasticPlastic
Total = Elastic and PlasticLo
g Sc
ale
Δε/2
'
εf/E'•Strain dependent material response
•Total Strain, ple εεε Δ+Δ=Δ
•Sum of elastic and plastic lines
•Coffin-Manson Equation,
( ) ( )cff
bf
f NNE
222
εσε ′+
′=
Δ
222pl
Eεσε Δ
+Δ=Δor
σ
ε
E
Δσ
ΔεΔε
Δεpl
el
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Low Cycle Fatigue ResearchTopics• Empirical formulas and
modifications of Coffin-Manson equation
• Use of stress or strain invariants
• Use of the critical plane
Researchers• Mowbray (1980)• Kalluri and Bonacuse
(1993)• Lefebvre (1989)• Brown and Miller (1973)• Lohr and Ellison (1980)
Pressure dependent yield function in low cycle fatigue?
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Research ProgramGoal: Use FEA to simulate first few cycles of LCF tests
using a pressure-dependent yield function
Experimental Program
•Two materials
-2024-T851
-Inconel 100 (IN100)
•Material properties
•LCF test data
Analytical Program
•Pressure-dependent constitutive model with combined hardening
•Finite element models
•Compare FEA and experimental results
Hydrostatic stress effect in low cycle fatigue?
Hydrostatic Stress Effects in Low Cycle Fatigue
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Outline• Technical Background• Mechanical Testing• Constitutive Model Development• Finite Element Model Development• Finite Element Results• Conclusions and Recommendations
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Mechanical Tests• Alignment• Elastic constants• Smooth uniaxial tensile• Smooth uniaxial compression• NRB tensile• NRB low cycle fatigue• Smooth round bar LCF
All per ASTM standards
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2024-T851 Tension and Compression•L direction
•E – 5% higher for compression
•0.2% offset σy –approx. same
•L-T direction, similar results
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2024-T851 NRB Tension
•6 variations of ρ
•0.005 in. – 0.120 in.
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2024-T851 NRB Low Cycle Fatigue
•3 variations of ρ, ρ = 0.040, 0.080, 0.120 in.
•Strain control
•±0.004 in. gage displacement
•0.040 in., Nf ≈ 10 cycles
•0.080 in., Nf ≈ 27 cycles
•0.120 in., Nf ≈ 44 cyclesρ = 0.080 in.
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2024-T851 Smooth Round Bar LCF
•Strain control
•±0.015 in. gage strain
•Nf ≈ 37 cycles
•Specimen AD01 -interrupted after 10 cycles, then monotonically loaded to failure
•Transitional cyclic stress-strain curve
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Outline• Technical Background• Mechanical Testing• Constitutive Model Development• Finite Element Model Development• Finite Element Results• Conclusions and Recommendations
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Constitutive Model Development
Why?• ABAQUS built-in models
– Multilinear isotropic and bilinear kinematic hardening for von Mises– Only multilinear isotropic hardening for Drucker-Prager
• No linear combined hardening models
What?• Pressure-dependent
(Drucker-Prager) constitutive model
• Combined multilinear kinematic and multilinear isotropic hardening
How?• ABAQUS user subroutine
(UMAT) function• Code written in FORTRAN
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Drucker-Prager UMAT
( ) 03 =−−= pleqyseff apf εσσ
Ι+= pijqpl
ij n εεε &v
&&31
•Yield function –
Split plastic strain rate into 2 parts –
Numerical integration – backward Euler method, implicit
Incremental solution, 5 equations and 5 unknowns: p, σeff, Δεp, Δεq, pl
eqεΔ
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Bilinear Hardeningσ
ε
E
Et
1
1σys0
t
t
EEEE
H−
= •Yield (effective) stress – linear function of equivalent plastic strain
•Commonly seen in literature and FEA code
σys
εeqpl
0σys
H
1
σys n
σys n+1
plεeq n Δεeqpl
pl
Δσys
εeq n+1
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Multilinear Hardening
•H not a constant
•Flexibility in modeling complex stress-strain behaviors
•Possibly multiple values of Hfor a given increment in plastic strain
•Yield stress – piecewise linear function of equivalent plastic strain
•Increases complexity of constitutive model equations
σys
εeqpl
0σys
σys n
σys n+1
pl εeq n Δεeqpl
εeq n+1pl
Δσys
1
1
1
11
H1
H2
H3
H4H5
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Outline• Technical Background• Mechanical Testing• Constitutive Model Development• Finite Element Model Development• Finite Element Results• Conclusions and Recommendations
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Finite Element Analysis Overview• Two constitutive models: von Mises and Drucker-
Prager• UMAT with combined multilinear hardening• Large strain analyses• Q4 elements with full integration• Displacement control• Symmetry utilized• Three levels of mesh refinement: coarse, medium,
fine• Medium mesh used for all analyses
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2024-T851 Material Property Inputs
•Elastic constants from mechanical test results
•uniaxial tensile data −σ – εpl table
•Power law to complete table
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Notched Round Bar FEM•Two symmetry planes
•Axisymmetric elements
•6 notch geometries
Uniform Displacement
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•One symmetry plane
•Plane strain elements
•IN100
Nodal Displacement
Equal-Arm Bend FEM
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Outline• Technical Background• Mechanical Testing• Constitutive Model Development• Finite Element Model Development• Finite Element Results• Conclusions and Recommendations
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UMAT ProgramVerification• Check accuracy of elasto-plasitc equations and
corresponding FORTRAN code• Compare UMAT with built-in ABAQUS models• Compare solutions for different values of the
combined hardening parameter, β• Use NRB with ρ = 0.040 in. and smooth
compression geometries• Use 2024-T851 material properties
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UMAT Program Verification
•NRB (ρ = 0.040 in.)
•Multilinear isotropic hardening
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UMAT Program Verification
•NRB (ρ = 0.040 in.)
•Vary β values
•Pure isotropic, β = 1.0
•Pure kinematic, β = 0.0
•Linear combination of kinematic and isotropic, 0.0 > β > 1.0
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2024-T851 NRB (ρ = 0.080 in.) Tensile
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NRB (ρ = 0.120 in.) First Cycle
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NRB (ρ = 0.120 in.) Fifth Cycle
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NRB (ρ = 0.120 in.) Tenth Cycle
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Equal-Arm Bend First Cycle
Small strain analysis
Strain Gage
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Equal-Arm Bend Second Cycle
Small strain analysis
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Equal-Arm Bend Third Cycle
Small strain analysis
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Overall Conclusions• Drucker-Prager solutions more accurately
predicted the behavior of the test specimens for first few cycles
• Once the stable material response was reached, neither the Drucker-Prager nor von Mises results were entirely satisfactory
• Neither solution truly captures the shapes of the hysteresis loops
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Supporting Conclusions1. Careful experimental testing - a necessity when
attempting to model a specimen’s LCF behavior2. UMAT - useful as a building block for future
studies in LCF behavior3. Drucker-Prager constitutive model - superior to
the von Mises model for simulating tensile monotonic test behavior
4. Drucker-Prager model - superior to the von Mises model for simulating the first few LCF cycles
5. Simulating the equal-arm bend three-cycle proof test - both models performed equally well
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Recommendations1. Develop new method to determine the
Drucker-Prager constant, a.2. Develop the pressure-dependent Jacobian. 3. Modify existing pressure-dependent
UMAT - improve the modeling capability after the first few cycles.
4. Generate additional LCF data from notched specimens.
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Additional Information
Extra Slides
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Bridgman’s ExperimentsEarly Tests (1930’s)• Smooth tensile bars• Variety of metals• External hydrostatic
pressures up to 450 ksi• No significant influence on
yield until highest pressures
• Reported by early plasticity researchers
Later Tests (1940’s)• More precise
instrumentation• Yield was dependent on
hydrostatic pressure• Not mentioned in plasticity
textbooks
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Richmond, Spitzig, and Sober’s Experiments
• Tests in the 1970’s and early 1980’s• Tension and compression under hydrostatic
pressure up to 160 ksi• Maraging steel, HY-80, AISI 4310, 4330• Grade 1100 Aluminum• Polyethylene and polycarbonate
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Recent Developments• Larrson, Royal Inst. Tech. – 1996-1998• Lu, Cambridge - 1998• Pan, et al., Univ. of Michigan – 1996 - Present• Lissenden, et al., Penn. State Univ.– 1999 – Present• Wilson & Allen, Tenn. Tech. Univ.– 1997 - Present
Drucker-Prager yield theory to model pressure sensitivity of non-geological materials
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2024-T851 Tension and CompressionTension
Compression
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IN100 Tension and CompressionTension
Compression
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NRB (ρ = 0.120 in.) First Cycle
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NRB (ρ = 0.120 in.) Second Cycle
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NRB (ρ = 0.120 in.) Third Cycle
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NRB (ρ = 0.120 in.) Fourth Cycle
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NRB (ρ = 0.120 in.) Fifth Cycle
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NRB (ρ = 0.120 in.) Tenth Cycle
Hydrostatic Stress Effects in Low Cycle Fatigue
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NRB (ρ = 0.120 in.) Twentieth Cycle
Hydrostatic Stress Effects in Low Cycle Fatigue
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NRB (ρ = 0.120 in.) Thirtieth Cycle
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IN100 Tension and Compression
•E – 2.5% higher for compression
•0.2% offset σy –approx. same
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IN100 Low Cycle Fatigue•Equal-arm bend specimen
•Three-cycle proof test
•3% strain to 1.3 % strain
•Conducted by Pratt and Whitney
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IN100 Material Property Inputs
•Elastic constants from mechanical test results
•uniaxial tensile data −σ – εpl table
•Linear fit to complete table
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Smooth Tensile Bar FEMUniform Displacement
•Two symmetry planes
•Axisymmetric elements
•0.25 in. and 0.35 in. diameter tensile bars modeled
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Smooth Compression Cylinder FEM
•Two symmetry planes
•Axisymmetric elements
Uniform Displacement
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NRB (ρ = 0.005in.) Mesh Convergence
Effective stress across the neck at maximum load
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NRB (ρ = 0.005in.) Mesh Convergence
Mean stress across the neck at maximum load
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NRB (ρ = 0.005in.) Mesh Convergence
Radial stress across the neck at maximum load
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NRB (ρ = 0.005in.) Mesh Convergence
Equivalent plastic strain across the neck at maximum load
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UMAT Program Verification
•NRB (ρ = 0.040 in.)
•Bilinear Kinematic hardening
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UMAT Program Verification
•Smooth Compression Cylinder
•Vary β values
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2024-T851 Smooth Tensile
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2024-T851 Smooth Compression
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2024-T851 NRB (ρ = 0.005 in.) Tensile
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2024-T851 NRB (ρ = 0.010 in.) Tensile
Hydrostatic Stress Effects in Low Cycle Fatigue
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2024-T851 NRB (ρ = 0.020 in.) Tensile
Hydrostatic Stress Effects in Low Cycle Fatigue
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2024-T851 NRB (ρ = 0.040 in.) Tensile
Hydrostatic Stress Effects in Low Cycle Fatigue
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2024-T851 NRB (ρ = 0.120 in.) Tensile
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NRB (ρ = 0.040 in.) First Cycle
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NRB (ρ = 0.040 in.) Third Cycle
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NRB (ρ = 0.040 in.) Ninth Cycle
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NRB (ρ = 0.080 in.) First Cycle
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NRB (ρ = 0.080 in.) Fifth Cycle
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NRB (ρ = 0.080 in.) Tenth Cycle
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IN100 Smooth Tensile
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IN100 Smooth Compression
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Equal-Arm Bend First Cycle
Large strain analysis
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Equal-Arm Bend Second Cycle
Large strain analysis
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Equal-Arm Bend Third Cycle
Large strain analysis
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Supporting Conclusions II1. Careful experimental testing - a necessity when
attempting to model a specimen’s LCF behavior2. UMAT - useful as a building block for future studies in
LCF behavior3. Drucker-Prager constitutive model - superior to the von
Mises model for simulating tensile monotonic test behavior
4. Simulating monotonic compressive behavior - unclear which model is superior
5. Drucker-Prager model - superior to the von Mises model for simulating the first few LCF cycles
6. Simulating the equal-arm bend three-cycle fatigue test -both models performed equally well