hydrostatic stress effects in low cycle fatiguechriswilson/phillip_allen_abaqus_umat/phillip... ·...
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Hydrostatic Stress Effects inLow Cycle Fatigue
A Dissertationby
Phillip Allen
Tennessee Technological UniversityNovember 13, 2002
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 2
Outline• Technical Background• Mechanical Testing• Constitutive Model Development• Finite Element Model Development• Finite Element Results• Conclusions and Recommendations
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 3
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 4
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 5
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 6
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 7
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 8
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 9
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 10
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 11
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?
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 12
Outline• Technical Background• Mechanical Testing• Constitutive Model Development• Finite Element Model Development• Finite Element Results• Conclusions and Recommendations
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 13
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 14
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 15
2024-T851 NRB Tension
•6 variations of ρ
•0.005 in. – 0.120 in.
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 16
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 17
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 18
Outline• Technical Background• Mechanical Testing• Constitutive Model Development• Finite Element Model Development• Finite Element Results• Conclusions and Recommendations
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 19
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 20
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 21
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 22
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 23
Outline• Technical Background• Mechanical Testing• Constitutive Model Development• Finite Element Model Development• Finite Element Results• Conclusions and Recommendations
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 24
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 25
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 26
Notched Round Bar FEM•Two symmetry planes
•Axisymmetric elements
•6 notch geometries
Uniform Displacement
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 27
•One symmetry plane
•Plane strain elements
•IN100
Nodal Displacement
Equal-Arm Bend FEM
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 28
Outline• Technical Background• Mechanical Testing• Constitutive Model Development• Finite Element Model Development• Finite Element Results• Conclusions and Recommendations
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 29
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 30
UMAT Program Verification
•NRB (ρ = 0.040 in.)
•Multilinear isotropic hardening
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 31
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 32
2024-T851 NRB (ρ = 0.080 in.) Tensile
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 33
NRB (ρ = 0.120 in.) First Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 34
NRB (ρ = 0.120 in.) Fifth Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 35
NRB (ρ = 0.120 in.) Tenth Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 36
Equal-Arm Bend First Cycle
Small strain analysis
Strain Gage
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 37
Equal-Arm Bend Second Cycle
Small strain analysis
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 38
Equal-Arm Bend Third Cycle
Small strain analysis
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 39
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 40
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 41
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 42
Additional Information
Extra Slides
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 43
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 44
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 45
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 46
2024-T851 Tension and CompressionTension
Compression
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 47
IN100 Tension and CompressionTension
Compression
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 48
NRB (ρ = 0.120 in.) First Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 49
NRB (ρ = 0.120 in.) Second Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 50
NRB (ρ = 0.120 in.) Third Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 51
NRB (ρ = 0.120 in.) Fourth Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 52
NRB (ρ = 0.120 in.) Fifth Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 53
NRB (ρ = 0.120 in.) Tenth Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 54
NRB (ρ = 0.120 in.) Twentieth Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 55
NRB (ρ = 0.120 in.) Thirtieth Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 56
IN100 Tension and Compression
•E – 2.5% higher for compression
•0.2% offset σy –approx. same
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 57
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 58
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 59
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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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 60
Smooth Compression Cylinder FEM
•Two symmetry planes
•Axisymmetric elements
Uniform Displacement
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 61
NRB (ρ = 0.005in.) Mesh Convergence
Effective stress across the neck at maximum load
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 62
NRB (ρ = 0.005in.) Mesh Convergence
Mean stress across the neck at maximum load
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 63
NRB (ρ = 0.005in.) Mesh Convergence
Radial stress across the neck at maximum load
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 64
NRB (ρ = 0.005in.) Mesh Convergence
Equivalent plastic strain across the neck at maximum load
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 65
UMAT Program Verification
•NRB (ρ = 0.040 in.)
•Bilinear Kinematic hardening
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 66
UMAT Program Verification
•Smooth Compression Cylinder
•Vary β values
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 67
2024-T851 Smooth Tensile
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 68
2024-T851 Smooth Compression
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 69
2024-T851 NRB (ρ = 0.005 in.) Tensile
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 70
2024-T851 NRB (ρ = 0.010 in.) Tensile
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 71
2024-T851 NRB (ρ = 0.020 in.) Tensile
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 72
2024-T851 NRB (ρ = 0.040 in.) Tensile
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 73
2024-T851 NRB (ρ = 0.120 in.) Tensile
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 74
NRB (ρ = 0.040 in.) First Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 75
NRB (ρ = 0.040 in.) Third Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 76
NRB (ρ = 0.040 in.) Ninth Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 77
NRB (ρ = 0.080 in.) First Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 78
NRB (ρ = 0.080 in.) Fifth Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 79
NRB (ρ = 0.080 in.) Tenth Cycle
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 80
IN100 Smooth Tensile
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 81
IN100 Smooth Compression
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 82
Equal-Arm Bend First Cycle
Large strain analysis
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 83
Equal-Arm Bend Second Cycle
Large strain analysis
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 84
Equal-Arm Bend Third Cycle
Large strain analysis
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Hydrostatic Stress Effects in Low Cycle Fatigue
Allen 85
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