A new paradigm for fatigue analysis - evolution …rmseura.tkk.fi/rm50/rm50_presentations/RM50_Kouhia.pdfA new paradigm for fatigue analysis - evolution equation based continuum approach

A new paradigm for fatigue analysis -evolution equation based continuum

approach

Joonas Jussila1, Terhi Kaarakka2, Sami Holopainen3,Reijo Kouhia3, Jari Makinen, Heikki Orelma3,

Niels Saabye Ottosen4, Matti Ristinmaa4 and Timo Saksala3

1AVANT Techno Oy2Tampere University of Technology, Mathematics

3Tampere University of Technology, Civil Engineering

4Lund University, Division of Solid Mechanics

50 years anniversary seminar of Rakenteiden Mekaniikka, Vaasa, August 24-25, 2017

1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

Content

1 Introduction - fatigue models

2 Evolution equation based HCF model

3 Endurance surface

4 LCF-HCF approach

5 Stochastic analysis

6 Gradient effects

7 Concluding remarks and future work

A new paradigm for fatigue analysis – RK RM50, 2017 2/22

1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions



3 Endurance surface

4 LCF-HCF approach


6 Gradient effects



1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

Introduction - fatigue models

Problems in fatigue analyses:

Multiaxiality

Damage accumalation rules

Low-cycle- and high-cycle -fatigue regimes are treated separately

Mostly based on well defined cycles.

A more fundamental approach for HCF based on evolution equationsproposed by Ottosen, Stenstrom and Ristinmaa in IJF 2008.

It provides a well defined and consistent approach for multiaxial

fatigue analysis which can be “easily” extended to anisotropic and

stochastic appraches and in which the gradient effects can be included.

In addition, the LCF and HCF regimes can be treated in a uniform

manner.


1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions



3 Endurance surface

4 LCF-HCF approach


6 Gradient effects



1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

Evolution equation based HCF model

Key ingredients are:

Endurance surface

β(σ, {α}; parameters) = 0,

evolution equations for damageD and the internal variables {α}

{α} = {G}(σ, {α})β,

andD = g(β,D)β.

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 �= 0

σ1

σ2 σ3

α’

α

dα’

dα

A

ds

B

β < 0

β > 0

EMMC-14, August 27–29, 2014 5


1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

Conditions for evolution

94 CHAPTER 6. Fatigue

σ1

σ2 σ3

α

dαs

dsβ > 0β ≥ 0α = 0D ≥ 0

(a)

σ1

σ2 σ3

α

s

dsβ > 0β < 0

α = 0D = 0

(b)

Figure 6.9: Ottosen’s HCF model. (a) Movement of the endurance surface and damagegrowth when the stress is outside the endurance surface and moving away from it. (b)When the stress is outside the endurance surface, damage and back stress does not evolve.

Version February 26, 2016


1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions



3 Endurance surface

4 LCF-HCF approach


6 Gradient effects



1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

Endurance surface

Original formulation by Ottosen et al. for isotropic fatigue

β =1

σ−1

[√3J2 +AI1 − σ−1

]= 0,

where J2 = 12 tr (s −α)2, I1 = trσ, A = σ−1/σ0 − 1, and

σ−1 = σaf,R=−1

σ0 = σaf,R=0

σa

σm

σ−1

A

1


1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

Effect of mean stress

The model describes well the mean stress effect in cyclic tensionas well as the non-linear effect on mean shear stress on thefatigue strength.

Transversely isotropic case: forged 34CrMo6 steel.

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)

b

b

b

ut

ut

ut

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)

b

b

b

ut

ut

(a) (b)

Figure 10: Effect of mean stress on fatigue life of 106 cycles under longitudinal (left) and transverse(right) uniaxial cyclic tension. The x-coordinate direction is parallel with the preferred longitudinaldirection. Experimental data for EN24T steel depicted by the markers △ is taken from ?.

5

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)

b

b

b

ut

ut

ut

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)

b

b

b

ut

ut

Figure 10: Effect of mean stress on fatigue life of 106 cycles under longitudinal (left) and transverse(right) uniaxial cyclic tension. The x-coordinate direction is parallel with the preferred longitudinaldirection. Experimental data for EN24T steel depicted by the markers △ is taken from ?.

0.7

0.8

0.9

1.0

0 0.5 1.0 1.5 2.0

τxym/τxya

τ xya(τ

xym)/τ x

ya(0)

bb

b

100 1000 10000 500000

0.2

0.4

0.6

0.8

1 .0 1.0

Dam

age

Figure 11: (Left) Effect of mean shear stress on the fatigue strength as the number of cycles isN = 106 and N = 5 · 104. (Right) Damage evolution during 5 · 104 cycles as τxym = 0 (solid),τxym = τxya (dash-and-dot), and τxym = 2τxya (dashed).

5

cyclic normal stress in mean shear stress effectlongitudinal and transverse directions on fatigue stress


1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

Anisotropic case

So far transversely isotropic and orthotropic symmetry has beenconsidered. Formulation based on structural tensors.

Denoting: τ−1 = ησ−1, consider biaxial loading

σx = σm + σa sinωt, σy = σm + σa sin(ωt+ φ)

ψ

φ

π/23π/8π/4π/80

1.2

1

0.8

0.6

0.4

0.2

0

-0.2

1

Figure 8: Fatigue strength ratio φ = (σ−ψ − σ−T)/(σ−L

− σ−T) as a function of the

angle ψ for the transversely isotropic model with different values of the shear fatiguestrength τ−L

. From the bottom τ−L/σ−L

= 0.475, 0.5, 0.525, 0.55, 0.575, 0.6 and 0.625.The markers indicate some test data at 45◦ direction: top 34CrMo6 Holopainen et al.(2016), middle 25MnCrSiVB6 Pessard et al. (2012) and bottom Splitasco Pessardet al. (2012).

exp. St35exp. 34CrMoexp. 34Cr4

η = 0.8η = 1/

√3

η = 0.4

φ

σa(φ)/σa(0)

π3π/4π/2π/40

1.1

1

0.9

0.8

0.7

0.6

1

Figure 9: Influence of the shear fatigue strength on the fatigue amplitude and as afunction of the phase difference (η = τ−1σ−1). Experimental data from ...

15


1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

Some results - HCF industrial test caseTransversely isotropc HCF-analysis of a forged 34CrMo6 steel fillet. The fatigue model is implemented in Abaqus FEprogram using the UMAT subroutine. Colour shows the value of damage D.

Sami Holopainen, Reijo Kouhia, Timo Saksala

Department of Mechanical Engineering and Industrial Systems, Tampere University of Technology, Tampere, Finland

Evolutionary models for fatigue analysis

Motivation

The model

The key idea of the model is a moving endurance surface

ᵦ. Movement of the surface is described by by a back-stress

type tensor α. Development of α and damage D is governed by evolution

equations. Transverse isotropy is accounted by splitting the stress

tensor as The transverse part is obtained from where the projector with a unit vector b indicating the preferred direction. Model equations:

Results

Figure 2. The model predicts well the mean stress effect in uniaxial cyclic test in (a) longitudinal and (b) transverse direction. Test data from McDiarmid 1989 marked with triangles

Conclusions A general evolutionary macroscopic multiaxial HCF-

model for transversely isotropic fatigue is developed. The model is implemented as UMAT-subroutine in the

Abaqus FEA-program. Extension of the model to unify the HCF and LCF

regimes is under development.

Table 1. Calibrated for forged 34CrMo6 and isotropic AISI-SAE 4340 steel grades, n = k = 1.

Most multiaxial HCF-models based on a static criterion and the damage accumulation rule on cycle-counting.

In this work, the evolutionary approach proposed by Ottosen, Stenström and Ristinmaa, IJF, 2008, is extended to transversely isotropic fatigue modelling.

Benefits are: (i) uniaxial and multiaxial stress states modelled in a unified manner for arbitrary loading histories, (ii) cycle-counting techniques need not to be applied.

Figure 3. (LHS) Influence of phase shift under two cyclic normal stresses R = 0.05. Dashed line indicates isotropic model prediction. (RHS) Effect of frequency shift for cyclic shear and normal stresses. Test data marked by triangles from Liu and Zenner 2003.

Figure 1. Endurance surface on the meridian and on the deviatoric plane, SL /ST = 1 (black dotted line), 1.5 (blue dashed line), 2 (red solid lline). Preferred direction coincides with the 3-axis.

Figure 4. Damage field of a real life test case. Isotropic case on the LHS and transversely isotropic on the RHS. ST/SL=0.8, AL=0.225, AT=0.285. In isotropic case the longitudinal values used. Preferred direction b=(cos(45o),0,-sin(45o))T.

Loading and mesh by Dr. Juho Konno, Watrsila Finland Oy


1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions



3 Endurance surface

4 LCF-HCF approach


6 Gradient effects



1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

LCF-HCF approach

Evolution equation for the α-tensor

α = C(s −α)β

and for damage

D = K exp[Lexp(−ξεp)β +M〈sgn(f)〉εp]β

Plasticity model based on Armstrong-Frederick model

f(σ,X , R) =√

32 (s −X ) : (s −X )− (σy +R) = 0

R = γR∞ (1−R/R∞) ˙εp

X = 23X∞εp − γ ˙εpX

εp = λ∂f

∂σ


1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

Illustration in deviatoric plane

σ1

σ2 σ3

α

dαs

ds

X

β > 0

β ≥ 0

α 6= 0

D ≥ 0

(a)

1


1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

∆ε-N curve in LCF-regime - AISI 4340

ASTM Handbool (Coffin-Manson + Basquin):

∆ε

2= 0.58(2Nf)

−0.57 + 0.0062(2Nf)−0.09


1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions



3 Endurance surface

4 LCF-HCF approach


6 Gradient effects



1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

Stochastic analysis

We have considered stress processes as Ornstein-Uhlenbeckprocess (a stationary Gauss-Markov process depending onparameters λ,µ and η)

dσ(t) = λ(µ− σ(t))dt+ ηdW (t)

Process W (t) is a Wiener process (Brownian motion)

It is a stochastic differential equation, solution can be found as

σ(t) = µ+ (σ0 − µ) exp(−λt) + η

∫ t

0

exp(−λ(t− s))dW(s),

where the integral is the so called Ito integral wrt the Wienerprocess.


1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions



3 Endurance surface

4 LCF-HCF approach


6 Gradient effects



1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

Gradient effects

Simply substitute

σ−1,corr = σ−1(1 +√As),

where

s =∇σeff · ∇σeff

σeff

where σeff is the standard von Mises stress.

A is the only additional material parameter (the Neuberparameter)


1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions



3 Endurance surface

4 LCF-HCF approach


6 Gradient effects



1 Introduction

2 Fatigue model

3 Endurance surface

4 LCF-HCF

5 Stochastic

6 Stochastic

7 Conclusions

Concluding remarks and future work

Consistent unified approachCan be “easilily” extended toanisotropy, stochastic analysis,gradient effect can be includedParameter estimationMicromechanical motivation ofthe evolution equations.

Watercolor by

Pia Erlandsson

Acknowledgements: The work was partially funded by TEKES - The

National Technology Foundation of Finland, project MaNuMiES.

Thank you for your attention!A new paradigm for fatigue analysis – RK RM50, 2017 22/22

Documents

A new paradigm for fatigue analysis - evolution …rmseura.tkk.fi/rm50/rm50_presentations/RM50_Kouhia.pdfA new paradigm for fatigue analysis - evolution equation based continuum approach