Improved Temperature Dependence of

the Material Parameters in a Visco-

plastic Chaboche Law for an Accurate

Cyclic Hardening Modelling

Laurent Duchêne, Hélène Morch, Anne Marie Habraken

Context: Solar power plant

2

Solar receivers: extreme thermo-mechanical

conditions

Khi Solar One power plant (South Africa)

Context: Solar receiver

3

Solar receiver (source : W.B.Stine, R.W.Harrigan,

Solar Energy Systems Design)

Panel of tubes manufactured from

nickel alloy sheet (Haynes 230) (source : CMI Solar)

Working fluid

(from 330°C to

565°C)

Tube:

Ø 50 mm

t=1.5 mm

4

► Fatigue + creep

► Extreme Thermo-mechanical loading

(Haynes 230)

► Advanced model

Working fluid

(molten salt)

T≈560°C

Solar

radiation

Temperature distribution in a tube (Lagamine FE code)

Context: The tubes

T

(°C)

Advanced Chaboche model

5

Constitutive equations :

•Von Mises criterion (yield locus) 1 parameter

•Norton’s viscosity function 2 parameters

Isotropic hardening

2 parameters

Kinematic hardening 2 parameters

Mean stress evolution

2 parameters

Static recovery 2 parameters

Cyclic hardening

5 parameters

Influence of maximum

temperature 2 parameters

→ 40 parameters

X 3 back-stresses

Parameter identification

6

► From isothermal experimental tests

► Validation on anisothermal tests

► Significant number of parameters:

Uniqueness of the solution not guaranteed

Physics-based manual identification

+ Optimization algorithm

Temperature dependence

7

1st method: multi-linear

One set of parameters at each test t°: P

→ 40 x nt° parameters

(=200)

Temperature dependence

8

1st method: multi-linear

Problem: lack of continuity

Temperature dependence

9

1st method: multi-linear

Problem: lack of continuity

Kinematic hardening

2ˆ ˆ( )3

vp

i i i i iX C X Y p

Temperature dependence

10

2nd method: Exponential T1 expP p

P

P A BC

E. Hosseini, S. R. Holdsworth, I. Kühn, and

E. Mazza, Mater. High Temp., vol. 32, no.

4, pp. 404–411, 2015.

P

→ 40 x 3 parameters

(=120)

Temperature dependence

11

2nd method: Exponential

P

Temperature dependence

12

2nd method: Exponential

Problem: poor accuracy of the cyclic hardening

Temperature dependence

13

2nd method: Exponential

Barrett, P. R., Ahmed, R., Menon, M,

Hassan, T., International Journal of Solids

and Structures, 88–89, pp. 146–164, 2016

Temperature dependence

14

3rd method: Double exponential

P

→ 40 x 5 parameters

(=200)

T1 exp

T 1 exp

P p

P

P p

P

P A BC

A DE

Temperature dependence

15

T1 exp

T 1 exp

P p

P

P p

P

P A BC

A DE

Cyclic hardening

0

0

( )

i

i i i i

c q

i i i

D p

a b e

3rd method: Double exponential

Temperature dependence

16

3rd method: Double exponential

Conclusions

17

Multi-linear

P

Single

exponential

Double

exponential P P

► Lack of continuity

► 40 x 5 parameters

(=200)

► Only monotonic

► 40 x 3 parameters

(=120)

► Good accuracy

► 40 x 5 parameters

(=200)

19

Modèle de base

Surface de plasticité

Contrainte visqueuse

Loi de viscosité

:

: :

él vp th

él

él él

E

E E

0 0f X R

n

vpK

2: :

3

vp vpAvec p

0v f

Evolution du module d’Young

max(1 )

( )

E E T

S

E E E E

E f E f E

f b f f p

Influence sur l’écrouissage

cyclique

max( )T

i D i iD b D D p

Ecrouissage

isotrope

( )R b Q R p

Ecrouissage

cinématique

1

ˆ ˆ

2ˆ ˆ( )3

nAF

i

i

vp

i i i i i

X X

X C X Y p

Restauration statique

1ˆ ˆ... ir

i i i iX b X X

Influence de la

température

maximale

Variations de température

1ˆ ˆ... ii i

i

CX TX

C T

Evolution de la contrainte moyenne

, ,

ˆ3

2

irii b i st i i i

i

XY Y Y X

X

Ecrouissage cyclique

0

0

( )

i

i i i i

c q

i i i

D p

a b e