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A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia 1 , Sami Holopainen 1 , Timo Saksala 1 and Andrew Roiko 2 1 Tampere University of Technology Department of Mechanical Engineering and Industrial Systems 2 VTT Technical Research Centre of Finland Partially funded by TEKES - the National Technology Agency of Finland Project SCarFace, number 40205/12 14 th European Mechanics of Materials Conference August 27–29, 2014, Gothenburg, Sweden
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Page 1: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

A multiaxial high-cycle transversely isotropicfatigue model

Reijo Kouhia1, Sami Holopainen1, Timo Saksala1 and Andrew Roiko2

1Tampere University of TechnologyDepartment of Mechanical Engineering and Industrial Systems

2VTT Technical Research Centre of Finland

Partially funded by TEKES - the National Technology Agency of FinlandProject SCarFace, number 40205/12

14th European Mechanics of Materials ConferenceAugust 27–29, 2014, Gothenburg, Sweden

Page 2: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Outline

• Motivation and background

• Isotropic model

• Transversely isotropic model

• Parameter estimation

• Results

• Conclusions and futuredevelopments

0 20 40 60 80−200

0

200

400

600

800

stress

[MPa]

80 0 20 40 60 800

0.005

0.01

0.015

0.02

Damage

EMMC-14, August 27–29, 2014 2

Page 3: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Motivation and background

Certain materials exhibit transversely isotropic symmetry

• unidirectionally reinforcedcomposites

• forged metals

– elasticity isotropic– fatigue properties transversely

isotropic

Figures from http://aciers.free.fr

EMMC-14, August 27–29, 2014 3

Page 4: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Background - Fatigue models

Either stress, strain or energy based.

Stress based criteria are commonly used in high-cycle fatigue

• stress invariant criteria, Sines 1955, Crossland 1956, Fuchs 1979

• critical plane criteria, Findley 1959, Dang Van 1989, McDiarmid 1990

• average stress criteria, Grubisic and Simburger 1976, Papadopoulos 1997.

Cumulative damage theories.

A more fundamental approach using evolution equations.

EMMC-14, August 27–29, 2014 4

Page 5: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Continuum approach

Proposed by Ottosen, Stenstrom and Ristinmaa in 2008.

Endurance surface postulated as

β =1

σoe

(σ + AI1 − σoe),

where

σ =√

3J2(s − α) =√

32(s − α) : (s − α),

I1 = trσ.

Back stress and damage evolution eqs.

α = C(s − α)β,

D = g(β,D)β = K exp(Lβ)β.

I1

‖s‖

β > 0β < 0

β =0

β =0

α 6= 0

σ1

σ2 σ3

α’

α

dα’

A

ds

B

β < 0

β > 0

EMMC-14, August 27–29, 2014 5

Page 6: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Transversely isotropic model

The stress is decomposed as

σ = σL + σT ,

whereσT = PσP , P = I −B,

where B = b ⊗ b is the structural tensor and b is the unit vector normal to thetransverse isotropy plane.

Integrity basis of a transversely isotropic solid

I1 = trσ, I2 =12tr (σ

2), I3 =13tr (σ

3), I4 = tr (σB), I5 = tr (σ2B).

EMMC-14, August 27–29, 2014 6

Page 7: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Endurance surfacePresent transversely isotropic formulation

β ={

σ +ALIL1

+ATIT1

−[

(1− ζ)ST+ ζS

L

]}

/ST= 0,

where

σ =√

3J2(s−α), IL1 = trσL = I4, IT1 = trσT = I1 − I4,

and

ζ =(

σL : σL

σ : σ

)n

=

(

2I5 − I242I2

)n

.

In uniaxial loading σ = σn⊗ n the ζ-factor has the form

ζ = (2 cos2ψ − cos4ψ)n,

where ψ is the angle between n and b.

EMMC-14, August 27–29, 2014 7

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Shape in the π-plane and ζ-factor

,

σ1

σ2 σ3

n = 2n = 1

n = 0.5

ψ

ζ

0 π/8 π/4 3π/8 π/2

1

0.8

0.6

0.4

0.2

0

SL/ST = 1 dotted black line, 1.5 dashed blue line, 2 red lineAL = 0.225, AT = 0.275 b = (0, 0, 1)T

EMMC-14, August 27–29, 2014 8

Page 9: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Evolution equations for α and D

Damage and the back-stress evolves only when moving away from theendurance surface

D =K

1−Dexp(Lβ)β, α = C(s−α)β.

time

1

2

3

4

5σ1

σ2

σ3

σ4

α1 -- α2 = α3

- α4 = α5

EMMC-14, August 27–29, 2014 9

Page 10: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Estimation of the parameters

Five material parameters in the endurance surface SL, ST , AL, AT and n.

Three material parameters in the evolution equations for the back-stress anddamage C,K and L.

Data from tests with forged 34CrMo6 steel. Due to the lack of data in theintermediate directions we have chosen n = 1.

SL = 447MPa, ST = 360MPa, AL = 0.225, AT = 0.300,

C = 33.6, K = 12.8 · 10−5, L = 4.0

EMMC-14, August 27–29, 2014 10

Page 11: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Fatigue strengths σm = 0

300

400

500

600

104 105 106 107

σa[M

Pa]

!"

!"

!"

!"

!"

!"

!"

!"

102

103

104

105

106

0

0.2

0.4

0.6

0.8

1

Damage

N N

△ denotes experimental results, • model predictions

EMMC-14, August 27–29, 2014 11

Page 12: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Effect of mean stress

σx = σxm + σxa sin(ωt) σy = σym + σya sin(ωt)

longitudinal transverse

0.5

0.6

0.7

0.8

0.9

1.0

0 0.5 1.0 1.5 2.0σxm/σxa

σxa(σ

xm)/σxa(0)

!"

!"

!"

0.5

0.6

0.7

0.8

0.9

1.0

0 0.5 1.0 1.5 2.0σym/σya

σya(σ

ym)/σya(0)

!"

!"

△ denotes experimental results from McDiarmid 1985 (34CrNiMo6), • model predictions

EMMC-14, August 27–29, 2014 12

Page 13: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Effect of mean shear stress

τxy = τxym + τxya sin(ωt)

!"

!"

!"

!"

!"

0.7

0.8

0.9

1.0

0 0.5 1.0 1.5 2.0τxym/τxya

τ xya(τ

xym)/τ x

ya(0)

100 1000 10000 500000

0.2

0.4

0.6

0.8

1 .0 1.0

NDamage

EMMC-14, August 27–29, 2014 13

Page 14: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Effect of phase shift

σx = σxm + σxa sin(ωt) σx = σxa sin(ωt)

σy = σxm + σxa sin(ωt − φy) τxy = 12σxa sin(ωt − φxy)

(a) (b)

0.6

0.7

0.8

0.9

1.0

1.1

0 45 90 135 180

φy

σxa(φ

y)/σxa(0)

0.9

1.0

1.1

1.2

1.3

0 30 60 90

φxy

σxa(φ

xy)/σxa(0)

EMMC-14, August 27–29, 2014 14

Page 15: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Effect of frequency differencemodel based on isotropic AISI SAE 4340 transversely isotropic 34CrMo6

exp. results shown 25CrMo4 (Liu & Zenner) 34CrNiMo6 (McDiarmid)σx = σxa sin(ωxt) σx = σxa sin(ωxt)

τxy = 12σxa sin(ωxyt) σy = σxa sin(ωyt)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

10−1 100 101

ωxy/ωx

σxa(ω

xy)/σxa(1)

!

!

!

!

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1 2 3 4 5 6 7 8

ωy/ωx

σxa(φ

y)/σxa(0)

"#

"#

"#

EMMC-14, August 27–29, 2014 15

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Conclusions and future developments

• Transversally isotropiccontinuum based HCF-model

• More tests needed

• Microstructurally basedanisotropic damage model

• Constitutive model with anisotropicdamage

• Implementation into a FE code Alexander Roslin: Lady with the veil, 1768

Thank you for your attention!

EMMC-14, August 27–29, 2014 16

Page 17: A multiaxial high-cycle transversely isotropic fatigue model · 2014-08-28 · A multiaxial high-cycle transversely isotropic fatigue model Reijo Kouhia1, Sami Holopainen1, Timo Saksala1

Deviatoric invariants

Deviatoric invariants and max shear in the longitudinal and in the isotropy plane

J2 =12tr (s

2), J4 = tr (sB), J5 = tr (s2B).

τmax(σT) =

J2 +14J

24 − J5, τmax(σL

) =√

J5 − J24 .

EMMC-14, August 27–29, 2014 17


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