Hydrostatic Stress Effects in Low Cycle Fatiguechriswilson/phillip... · Hydrostatic Stress Effects in Low Cycle Fatigue. Allen. 29. UMAT ProgramVerification • Check accuracy of

Hydrostatic Stress Effects inLow Cycle Fatigue

A Dissertationby

Phillip Allen

Tennessee Technological UniversityNovember 13, 2002

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


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


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


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


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


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


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),(


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


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?


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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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


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


Allen 15

2024-T851 NRB Tension

•6 variations of ρ

•0.005 in. – 0.120 in.


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.


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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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


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


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


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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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


Allen 25

2024-T851 Material Property Inputs

•Elastic constants from mechanical test results

•uniaxial tensile data −σ – εpl table

•Power law to complete table


Allen 26

Notched Round Bar FEM•Two symmetry planes

•Axisymmetric elements

•6 notch geometries

Uniform Displacement


Allen 27

•One symmetry plane

•Plane strain elements

•IN100

Nodal Displacement

Equal-Arm Bend FEM


Allen 28



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


Allen 30

UMAT Program Verification

•NRB (ρ = 0.040 in.)

•Multilinear isotropic hardening


Allen 31


•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


Allen 32

2024-T851 NRB (ρ = 0.080 in.) Tensile


Allen 33

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



Allen 38

Equal-Arm Bend Third Cycle



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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


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


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.


Allen 42

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


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


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


Allen 46

2024-T851 Tension and CompressionTension

Compression


Allen 47

IN100 Tension and CompressionTension

Compression


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Allen 49

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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Allen 53



Allen 54

NRB (ρ = 0.120 in.) Twentieth Cycle


Allen 55

NRB (ρ = 0.120 in.) Thirtieth Cycle


Allen 56

IN100 Tension and Compression

•E – 2.5% higher for compression

•0.2% offset σy –approx. same


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


Allen 58

IN100 Material Property Inputs

•Elastic constants from mechanical test results

•uniaxial tensile data −σ – εpl table

•Linear fit to complete table


Allen 59

Smooth Tensile Bar FEMUniform Displacement

•Two symmetry planes


•0.25 in. and 0.35 in. diameter tensile bars modeled


Allen 60

Smooth Compression Cylinder FEM

•Two symmetry planes


Uniform Displacement


Allen 61

NRB (ρ = 0.005in.) Mesh Convergence

Effective stress across the neck at maximum load


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Mean stress across the neck at maximum load


Allen 63


Radial stress across the neck at maximum load


Allen 64


Equivalent plastic strain across the neck at maximum load


Allen 65


•NRB (ρ = 0.040 in.)

•Bilinear Kinematic hardening


Allen 66


•Smooth Compression Cylinder

•Vary β values


Allen 67

2024-T851 Smooth Tensile


Allen 68

2024-T851 Smooth Compression


Allen 69

2024-T851 NRB (ρ = 0.005 in.) Tensile


Allen 70

2024-T851 NRB (ρ = 0.010 in.) Tensile


Allen 71

2024-T851 NRB (ρ = 0.020 in.) Tensile


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2024-T851 NRB (ρ = 0.040 in.) Tensile


Allen 73

2024-T851 NRB (ρ = 0.120 in.) Tensile


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Allen 75

NRB (ρ = 0.040 in.) Third Cycle


Allen 76

NRB (ρ = 0.040 in.) Ninth Cycle


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Allen 78



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Allen 80

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



Allen 84

Equal-Arm Bend Third Cycle



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

Documents

Hydrostatic Stress Effects in Low Cycle Fatiguechriswilson/phillip... · Hydrostatic Stress Effects in Low Cycle Fatigue. Allen. 29. UMAT ProgramVerification • Check accuracy of