Modélisation polycristalline du comportement élasto-viscoplastique des aciers inoxydables austénitiques 316L(N) sur une large gamme de chargements : application à l'étude du comportement cyclique à température élevée

HAL Id: tel-02325387

https://tel.archives-ouvertes.fr/tel-02325387

Submitted on 22 Oct 2019

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Modélisation polycristalline du comportement

élasto-viscoplastique des aciers inoxydables austénitiques

316L(N) sur une large gamme de chargements :

application à l’étude du comportement cyclique à

température élevée

Diogo Goncalves

To cite this version:

Diogo Goncalves. Modélisation polycristalline du comportement élasto-viscoplastique des aciers

in-oxydables austénitiques 316L(N) sur une large gamme de chargements : application à l’étude du

comportement cyclique à température élevée. Mécanique des solides [physics.class-ph]. Sorbonne

Université, 2018. Français. �NNT : 2018SORUS089�. �tel-02325387�

𝑑𝐺𝐵

𝑑𝐵𝐺

𝒂

𝜸

𝜸

𝑺𝑭𝑬

𝒃

𝝓

𝑲

𝑯𝑷

𝜿

𝒃

𝒗

𝑫

𝑹

𝑻

𝑸

𝑽

𝒕

𝒕

𝒉

𝒕

𝒄𝒍

𝒕

𝒘

𝒕

𝒂

𝜹

𝟎

𝒂

𝐂

𝟏𝟏

, 𝐂

𝟏𝟐

, 𝐂

𝟒𝟒

𝚼

𝝁

𝝂

𝒇

𝜶

𝒍

𝜶

𝒏

𝐦

𝒊

𝑖

𝐧

𝒊

𝑖

𝝈

𝚺

𝝈

𝒅

𝚺

𝐝

𝚺

𝐞𝐪

𝜮

𝒎

𝜮

𝒂

𝜮

𝒎𝒂𝒙

𝜮

𝒎𝒊𝒏

𝜮

𝒓𝒆𝒍𝒂𝒙

𝚺

𝒚

𝚺

𝟎

𝚺

𝒇

𝝐

𝑬, 𝑬

𝒕𝒐𝒕

𝑬

𝟎

𝑬

𝒂

𝝐

𝒑

_{, 𝝐}

𝒗𝒑

𝑬

𝒑

, 𝑬

𝒗𝒑

𝑬

_𝒂

𝒑

, 𝑬

_𝒂

𝒗𝒑

𝑬

𝒓

𝝐

_𝒆𝒒

𝒗𝒑

𝑬

𝒆𝒒

𝒗𝒑

𝑬

_𝒄𝒖𝒎

𝒗𝒑

𝜼

𝑵

𝜸

𝜸

_𝒊

𝒑

, 𝜸

_𝒊

𝒗𝒑

i

𝑖

𝜸

_𝒊,𝒆

𝒗𝒑

, 𝜸

_𝒊,𝒔

𝒗𝒑

𝑒

𝑠

𝑒

𝑠

𝝉

𝒊

𝑖

𝑖

𝝉

𝒆𝒇𝒇

𝑹

𝑹

𝒄𝒓𝒊𝒕

𝝉

𝟎

𝝉

𝒄

𝝉

∗

𝝉

𝒑𝒓𝒊𝒎𝒂𝒓𝒚

𝝉

𝒔𝒆𝒄𝒐𝒏𝒅𝒂𝒓𝒚

𝝉

𝒑

𝝉

𝒄𝒍

𝝉

𝒏𝒐𝒓𝒎

𝝉

𝒆𝒍𝒂𝒔

𝒙

𝑿

𝑸

𝑽

𝑨

𝜶, 𝑪

𝑴

̅

𝝆, 𝝆

𝒕

𝝆

𝒐

𝝆

𝒆

𝝆

𝒔

𝝆

𝒅

𝝆

𝑷𝑳

𝝆

𝒎

𝝆

𝒄

𝝆

𝒘

𝜶

̅

𝒎𝒐𝒏

𝜶

̅

_𝒄𝒚𝒄

𝒉

𝒊𝒋

𝒉

𝒐

𝒉

𝟏

, 𝒉

𝒄𝒐𝒍𝒍

𝒉

𝟐

, 𝒉

𝑯𝒊𝒓𝒕𝒉

𝒉

𝟑

, 𝒉

𝒄𝒐𝒑𝒍𝒂

𝒉

𝟒

, 𝒉

𝑮𝒍𝒊𝒔𝒔

𝒉

𝟓

, 𝒉

𝑳𝒐𝒎𝒆𝒓

𝒒

𝑳

𝒚

𝒆

, 𝒚

𝒔

(𝑒)

(𝑠)

(𝑒)

(𝑠)

𝒉

𝒎𝒂𝒙

𝒉

𝜿

𝒔𝒊𝒏𝒈𝒍𝒆

𝜿

𝒎𝒖𝒍𝒕𝒊

𝜿

𝒔𝒆𝒍𝒇

𝜿

𝒄𝒐𝒑𝒍𝒂

𝜿

𝒋𝒐𝒏𝒄𝒕𝒊𝒐𝒏

𝑬

𝒗𝒂𝒄

𝝉

𝑰𝑰𝑰

, 𝝉

𝒊𝒏𝒕

𝑨

𝟏

, 𝑨

𝟐

, 𝜶

𝟏

, 𝜶

𝟐

𝒅

𝟎

𝜽

𝑳

𝟏

, 𝑳

𝟐

𝒆

𝑩𝑮

𝒇

𝑩𝑮

𝒅

𝑩𝑮

𝜸

_{𝑩𝑮,𝒄𝒓𝒊𝒕}

𝒑

𝑪

𝒔𝒐𝒍

𝑪

𝒔𝒂𝒕

𝑪

𝟎

𝑼

𝒄𝒓

𝑫

𝑪𝒓

𝑫

𝒐

𝑪𝒓

𝑸

𝜸

𝑪𝒓

𝒗

̅

𝒗

𝒄𝒍

𝛌

̅

𝒇

𝑫𝑺𝑨

𝝐

𝒂

𝑲

𝒔𝒊𝒎

𝒅

𝒄𝒆𝒍𝒍

𝒄

𝒋

𝑫

𝒔𝒅

𝑫

_𝟎

𝒔𝒅

𝐐

𝐬𝐝

𝑎

𝛾

= 0,3595 𝑛𝑚

𝑎

𝛾

= 0,3595 𝑛𝑚

53 𝜇𝑚

50 𝜇𝑚

𝜌

𝑜

~ 10

10

𝑚

−2

𝜌

𝑜

= 3,4 . 10

12

𝑚

−2

𝑖

|𝜏

𝑖

_|,

_𝜏

0

𝜏

𝑖

= 𝜎. 𝑓

𝑖

𝑓

𝑖

_{= 𝑐𝑜𝑠 𝛼}

𝑛

. 𝑐𝑜𝑠 𝛼

𝑙

𝛼

𝑙

𝛼

𝑛

𝑙⃗

𝑛⃗⃗

𝑓

𝑖

_𝑖

𝑖

𝜏

0

𝛾

𝑆𝐹𝐸

𝛾

𝑆𝐹𝐸

𝜇

𝑏

𝛾

𝑆𝐹𝐸

𝛾

𝑆𝐹𝐸

𝛾

𝑆𝐹𝐸

= 25,7 + 2 ∙ %𝑁𝑖 + 410 ∙ %𝐶 − 0,9 ∙ %𝐶𝑟 − 77 ∙ %𝑁 − 13 ∙ %𝑆𝑖

− 1,2 ∙ %𝑀𝑛

𝜸

𝑺𝑭𝑬

(𝒎𝑱/𝒎²)

𝜸

𝑺𝑭𝑬

/𝝁𝒃 (𝟏𝟎

−𝟑

)

𝛾

𝑆𝐹𝐸

/𝜇𝑏

𝛾

𝑆𝐹𝐸

/𝜇𝑏

𝛾

𝑆𝐹𝐸

𝜌

𝜏 = 𝛼̅𝜇𝑏 √𝜌

𝛼̅

𝛼̅ = 0.35 ±

0.15

𝑖

𝑗

Système

Ab

Ac

Af

Bb

Bd

Be

Ca

Cc

Ce

Da

Dd

Df

Plan

Direction

ℎ

𝑜

ℎ

3

= ℎ

𝑐𝑜𝑝𝑙𝑎

ℎ

2

= ℎ

𝐻𝑖𝑟𝑡ℎ

ℎ

4

= ℎ

𝑔𝑙𝑖𝑠

ℎ

5

= ℎ

𝐿𝑜𝑚𝑒𝑟

ℎ

1

= ℎ

𝑐𝑜𝑙𝑙

Ab Ac

Af

Bb Bd Be Ca

Cc

Ce

Da Dd Df

Ab

Ac

Af

Bb

Bd

Be

Ca

Cc

Ce

Da

Dd

Df

𝐸̇

𝐸̇

vp

= 𝐸̇

0

𝑒𝑥𝑝 (_𝑄 𝑅𝑇

⁄

)

𝑄

𝑅

𝑇

𝑄

Q

BT

= 133 kJ.mol

-1

Q

γ

C

= 138 kJ.mol

-1

Q

HT

= 278 kJ.mol

-1

Q

γ

Cr

=243 kJ.mol

-1

𝐸

𝑎

= ±0,6%

𝐶

𝜎

𝑖𝑗

= 𝐶

𝑖𝑗𝑘𝑙

∙ 𝜀

𝑘𝑙

𝐶

𝑖𝑗𝑘𝑙

𝑎 =

2 C

44

C

11

− C

12

𝑎 = 1

𝑎 = 3,3

𝐶

11

= 197,5 𝐺𝑃𝑎 𝐶

12

= 125 𝐺𝑃𝑎

𝐶

44

= 122 𝐺𝑃𝑎

𝑎

Υ

𝜈

𝛶

𝛶

~𝜇/2000

100

120

140

160

180

200

220

0

200

400

600

800

1000

1200

316LN [RCC-MR 2002]

316LN - SQ [Gaudin, 2002]

316LN - AVESTA [Gentet, 2010]

316L [Alain, 1997]

316 [ASM Hanbook, 1994]

316 [Barnby, 1965]

316 [Letbetter, 1981]

Température,

M

o

d

u

le

d'

Yo

u

n

g,

𝛾

𝑆𝐹𝐸

/𝜇𝑏

𝐸

𝑝

> 15%

[Single glide] 0 100 200 300 400 500 600 700 800 0 0.2 0.4 0.6 Cristal [100] Cristal [111] Cristal [123] Monocristaux 316L SS Essai de traction -T=293K Déformation, C on trai nte, Stade I

𝐸

𝑝

= 3%

𝐸

𝑝

_{= 1 𝑒𝑡 3%}

𝐸

𝑝

_{= 1 𝑎𝑛𝑑 3%}

[100] [110]

[111]

Single-slip (tensile loading, Ep=1.0%, 𝐸̇=3.10-4 s-1) Main dislocation patterns (tensile loading, Ep=3.0%,𝐸̇=3.10-4 s-1) 316L(N) SS Dipole walls Polarized dislocations

𝑥

|𝜏 − 𝑥| = 𝜏

𝑒𝑓𝑓

𝑥 ≠

0,

𝛾

𝑆𝐹𝐸

/𝜇𝑏

𝜏

𝑝

𝑥

𝜏

0

, 𝑥

𝜏

𝑜

≈ 𝑥

𝜏

𝜏

_𝑜

𝐸̇ = = 4.10

−3

_𝑠

−1

𝐸

𝑎

𝑝

= ±0.1%

𝐸

𝑎

𝑃

> ±0.5%

= ±0.1%

= ±0.5%

= ±0.1%

= ±0.5%

T=293K

= ±0.2%

= ±0.2%

= ±0,5%

=

±0,5%

T=873K

𝛴

𝑚

▪

▪

▪

𝑑𝐸

𝑟

/𝑑𝑁

𝛴

𝑚

~ 50𝑀𝑃𝑎

𝛴

𝑚

= 50 𝑀𝑃𝑎

𝑑𝐸

𝑟

/𝑑𝑁

𝑑𝐸

𝑟

/𝑑𝑡

𝛴

𝑚𝑎𝑥

𝛴

𝑚

= 50𝑀𝑃𝑎

𝑑𝐸

𝑟

/𝑑𝑁

𝛴

𝑚

𝑑𝐸

𝑟

/𝑑𝑡

𝛴

𝑚𝑎𝑥

𝛴

𝑚

= 50𝑀𝑃𝑎



𝑚

= 50 𝑀𝑃𝑎

𝛾

𝑆𝐹𝐸

/𝜇𝑏

𝛾

𝑆𝐹𝐸

/𝜇𝑏

𝛾

𝑆𝐹𝐸

/𝜇𝑏

𝚺𝒎𝒂𝒙 𝒅𝑬 𝒓 / 𝒅𝑵

𝛴 = 420 𝑀𝑃𝑎

𝛴 = −280 𝑀𝑃𝑎

𝛴 = 70 ± 350 𝑀𝑃𝑎

𝛴 = 420 𝑀𝑃𝑎

𝛴 = −280 𝑀𝑃𝑎

𝛴 = 70 ± 350 𝑀𝑃𝑎

𝛴 = 150 ± 150 𝑀𝑃𝑎

𝛴̇

𝛴 = 150 ± 150 𝑀𝑃𝑎

𝛴̇

𝑬(%)

𝑬

(%

)

Σ

a

𝐸

𝑎

= ±0.3%

A

m

p

li

tu

d

e

d

e

c

o

n

tr

a

in

te

,

)

Nombre de cycles

823K

Σ

𝑚𝑎𝑥,𝑡ℎ

/Σ

𝑚𝑎𝑥,0

Σ

𝑚𝑎𝑥,𝑡ℎ

/Σ

𝑚𝑎𝑥,0

> 1

Σ

𝑚𝑎𝑥,𝑡ℎ

/Σ

𝑚𝑎𝑥,0

< 1

𝛴

𝑚𝑎𝑥,𝑡ℎ

𝛴

𝑚𝑎𝑥,0

𝛴

𝑚𝑎𝑥,𝑡ℎ

𝛴

𝑚𝑎𝑥,0 R a p p o rt d e la c o n tr a in te e n t ra c ti o n , Temps de maintien (𝒕𝒕𝒉) 823K _873K Temps de maintien ( ) Rapp ort de la con traint e en tractio n,

𝑡

𝑡ℎ

= 1 𝑚𝑖𝑛

𝑡

𝑡ℎ

= 13.3 𝑚𝑖𝑛

𝑡

𝑡ℎ

= 30 𝑚𝑖𝑛

̇

̇

•

𝛾

𝑆𝐹𝐸

/𝜇𝑏

•

•

•

•

•

𝛾

𝑆𝐹𝐸

/𝜇𝑏

•

•

𝚼

𝑣

𝜎̇ = Σ̇ + 2𝜇(1 − 𝛽) (𝐸̇

𝑣𝑝

_{− 𝜖̇}

𝑣𝑝

_{) + 2𝜇(1 − 𝛽) 𝜂}

−1

_(Σ

d

_{− 𝜎}

𝑑

₎

𝛽 =

2(4−5𝜈)

15(1−𝜈)

𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 ⟹ 𝜂 =

Σ̇

11

𝐸̈

₁₁

𝑣𝑝

𝑠𝑒𝑐𝑎𝑛𝑡 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 ⟹ 𝜂 =

Σ

11

𝐸̇

₁₁

𝑣𝑝

𝜎̇

𝜖̇

𝑣𝑝

Σ̇

𝐸̇

𝑣𝑝

𝜎

𝑑

Σ

d

𝜇

𝜂

−1

= 0

𝜂

−1

_{≠ 0}

𝜏

𝑖

𝜎,

𝑖 = 1,12

𝜏

𝑖

= 𝜎: (m

𝑖

⨂ n

𝑖

)

m

𝑖

_n

𝑖

𝑖

𝑖

𝜏

𝑖

∗

𝑇

𝑉

𝑄

κ

𝐴

𝛾̇

_𝑖

𝑣𝑝

= 𝐴 ∙ 𝑒𝑥𝑝 (−

𝑄

𝜅

𝑏

𝑇

) ∙ 𝑠𝑖𝑛ℎ (

𝑉𝜏

𝑖

∗

𝜅

𝑏

𝑇

)

𝜏

𝑖

∗

= |𝜏

𝑖

| − 𝜏

0

≥ 0

𝜏

_𝑖

∗

= |𝜏

𝑖

− 𝑥

𝑖

| − 𝜏

𝑐,𝑖

− 𝜏

0

≥ 0

𝑥

𝑖

𝑥̇

𝑖

= 𝛼𝐶𝛾̇

𝑖

𝑝

_{− 𝛼𝑥}

𝑖

|𝛾̇

𝑖

𝑣𝑝

|

(𝜏

𝑐,𝑖

)

𝜏̇

𝑐,𝑖

= ℎ

0

∑ 𝑎

𝑖,𝑗

|𝛾̇

𝑗

𝑣𝑝

|

12

𝑗=1

, 𝑤𝑖𝑡ℎ {

𝑎

_𝑎

𝑖,𝑗

= 𝑞

𝑖,𝑖

= 1

ℎ

0

𝜏

0

𝑞

𝐶

𝛼

0 0.2 0.4 0.6 0.8 1 0.0 0.5 1.0 1.5 C umulat iv e prob ab il it y

Normalized equivalent plastic strain

Kroner-Secant (500 grains) Kroner-Secant (300 grains) Kroner-Secant (100 grains) Kroner-Secant (50 grains) Evp_{≈ 1} -2 Kröner-Secant 0 200 400 600 800 1,000 1,200 1,400 0.00 0.02 0.04 Kroner-Secant - 500 grains Kroner-Secant - 300 grains Kroner-Secant - 100 grains Kroner-Secant - 50 grains Kröner-Secant S tress, Strain,

0 0.2 0.4 0.6 0.8 1 0.0 0.5 1.0 1.5 2.0 C umulat iv e prob ab il it y

Normalized equivalent plastic strain

FE Computations - 343 grains and 343 FE/grain FE Computations - 125 grains and 343 FE/grain FE Computations - 125 grains and 125 FE/grain Evp_{≈ 1} -2 FE Computations 0 200 400 600 800 1,000 1,200 1,400 0.00 0.01 0.02 0.03 0.04 0.05

FE Computations - 343 grains and 343 FE/grain FE Computations - 343 grains and 125 FE/grain FE Computations - 125 grains and 729 FE/grain FE Computations - 125 grains and 343 FE/grain FE Computations - 125 grains and 125 FE/grain

FE Computations

S

tres

s,

Υ

𝑣

𝜏

0

𝑉

𝑄

𝐴

𝐴

𝑄

𝐴 = 𝜌𝜈

𝐷

𝑏²

𝜌~10

13

𝑚

−2

𝚼

𝒗

𝝉

𝟎

𝑨

𝑸

𝑽

𝒃

𝒃 = 𝟐. 𝟓𝟔 Å

𝐸 = 5%

𝐸̇ = (10

−3

_{, 10}

−5

_{, 10}

−7

_{) 𝑠}

−1

𝐸 = 5%

𝐸̇ = 10

−3

_𝑠

−1

𝛴 = (200 − 600) 𝑀𝑃𝑎

Σ̇ = 100 𝑀𝑃𝑎. 𝑠

−1

𝐸 = 5%

𝐸̇ = (10

−3

_{, 10}

−5

_{, 10}

−7

_{) 𝑠}

−1

𝐸 = 5%

𝐸̇ = 10

−3

_𝑠

−1

𝛴 = (100 − 250)

Σ̇ = 100 𝑀𝑃𝑎. 𝑠

−1

𝐸 = 5%

𝐸̇ = (10

−3

_{, 10}

−5

_{, 10}

−7

_{) 𝑠}

−1

𝐸 = 5%

𝐸̇ = 10

−3

_𝑠

−1

𝛴 = (80 − 120) 𝑀𝑃𝑎

Σ̇ = 100 𝑀𝑃𝑎. 𝑠

−1

𝑉 = 15000𝑏³

𝐸̇ = 10

−7

𝑠

−1

𝑉 = 20000𝑏³)

𝐸̇ = 10

−9

𝑠

−1

𝜏

𝑜

𝑀

̅

𝑀

̅

𝑀

̅

𝑀

̅

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 5.0 10.0 15.0 FE Computations (2.65±0.05) Kroner (3.04±0.12) Kroner-Tangent (2.03±0.08) Kroner-Secant (2.88±0.10) Berveiller-Zaoui (*) Hill-Hutchinson (*)

Rate-insensitivity limit

Tay lor upper bound = 3.06

Reuss lower bound = 2.01

Ta

yl

o

r

Fa

ct

o

r,

𝑴̅

=

𝚺

/𝝉

𝟎

𝑬

_{. /𝟐𝝉}

𝟎

𝑉 = 30𝑏³

𝑉 = 100𝑏³

𝑉 = 300𝑏³

𝑽 = 𝟑𝟎𝒃³

𝑽 = 𝟏𝟎𝟎𝒃³

𝑽 = 𝟑𝟎𝟎𝒃³

𝑬 ̇(𝒔

−𝟏

₎

₁₀

−3

₁₀

−5

₁₀

−7

₁₀

−3

₁₀

−5

₁₀

−7

₁₀

−3

₁₀

−5

₁₀

−7 0 200 400 600 800 1000 1200 1400 1600 0 0.01 0.02 0.03 0.04 0.05 FE Computations Kroner Kroner-Tangent Kroner-Secant

Str

e

s

s

, 𝚺

(M

Pa

)

Strain,

𝑬

= 30 ³, 𝐸̇ = 10

−3

_𝑠

−1 0 100 200 300 400 500 600 0 0.01 0.02 0.03 0.04 0.05 FE Computations Kroner Kroner-Tangent Kroner-Secant

Str

e

s

s

, 𝚺

(M

Pa

)

Strain,

𝑬

= 100 ³, 𝐸̇ = 10

−3

_𝑠

−1 0 50 100 150 200 250 0 0.01 0.02 0.03 0.04 0.05 FE Computations Kroner Kroner-Tangent Kroner-Secant

Str

e

s

s

, 𝚺

(M

Pa

)

Strain,

𝑬

= 300 ³, 𝐸̇ = 10

−3

_𝑠

−1

𝐸̇

𝑣𝑝

=

Σ̇

𝐴

+ 𝜂

−1

_(Σ

d

₎

A = 2𝜇(1 − 𝛽)

𝜂 =

Σ̇

11

𝐸̈

₁₁𝑣𝑝

𝜂 =

Σ

₁₁

𝐸̇

₁₁𝑣𝑝

𝑉 = 300𝑏³

𝑉 = 30𝑏³

10

−3

_𝑠

−1

_.

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

1E-6 1E-5 1E-4 1E-3 1E-2

Kroner-Secant Kroner-Tangent Viscoplastic strain V=300b³, dE/dt=10-3_s-1 C o n tr ib u ti o n to th e v is c o p la s ti c s tr a in r a te Viscoplastic contribution Thermo-elastic contribution Tangent moduli Evp_=5.2x10-5 Secant moduli Evp_=1.5x10-4 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

1E-5 1E-4 1E-3 1E-2

Kroner-Secant Kroner-Tangent C o n tr ib u ti o n to th e v is c o p la s ti c s tr a in Viscoplastic strain V=30b³, dE/dt=10-3_s-1 Viscoplastic contribution Thermo-elastic contribution Secant moduli Evp=1.2x10-3 Tangent moduli Evp=2.0x10-4

〈𝝐

𝒆𝒒 𝒗𝒑

〉

𝒈𝒓

〈𝝈〉

𝒈𝒓

𝐸

_𝑒𝑞

𝑣𝑝

= [〈

𝝐

𝒗𝒑

〉𝒑𝒐𝒍𝒚]

𝒆𝒒

Σ = 〈

𝝈

〉𝒑𝒐𝒍𝒚

.

𝐸

𝑣𝑝

≈ 10

−4

𝐸

𝑣𝑝

≈ 10

−3

𝐸

𝑣𝑝

≈ 10

−2

10

−3

_𝑠

−1

_{𝑉 = 30𝑏³}

_{𝑉 =}

300𝑏³

𝐸

𝑣𝑝

_{≈ 10}

−4

0 0.2 0.4 0.6 0.8 1 0.0 1.0 2.0 3.0 C u m u la ti v e p ro b a b il it y

Normalized equivalent viscoplastic strain

FE Computations Kroner Kroner-Tangent Kroner-Secant

V=30b³, dE/dt=10

-3

_s

-1

E

vp

_{≈ 1}

-4 0 0.2 0.4 0.6 0.8 1 0.0 1.0 2.0 3.0 C u m u la ti v e p ro b a b il it y

Normalized equivalent viscoplastic strain

FE Computations Kroner Kroner-Tangent Kroner-Secant

V=30b³, dE/dt=10

-3

_s

-1

E

vp

_{≈ 1}

-2 0 0.2 0.4 0.6 0.8 1 0.0 1.0 2.0 3.0 C u m u la ti v e p ro b a b il it y

Normalized equivalent viscoplastic strain

FE Computations Kroner Kroner-Tangent Kroner-Secant

V=300b³, dE/dt=10

-3

_s

-1

E

vp

_{≈ 1}

-4 0 0.2 0.4 0.6 0.8 1 0.0 1.0 2.0 3.0 C u m u la ti v e p ro b a b il it y

Normalized equivalent viscoplastic strain

FE Computations Kroner Kroner-Tangent Kroner-Secant

V=300b³, dE/dt=10

-3

_s

-1

E

vp

_{≈ 1}

-2 0 0.2 0.4 0.6 0.8 1 0.0 1.0 2.0 3.0 C u m u la ti v e p ro b a b il it y

Normalized equivalent viscoplastic strain

FE Computations Kroner Kroner-Tangent Kroner-Secant

V=30b³, dE/dt=10

-3

_s

-1

E

vp

_{≈ 1}

-3 0 0.2 0.4 0.6 0.8 1 0.0 1.0 2.0 3.0 C u m u la ti v e p ro b a b il it y

Normalized equivalent viscoplastic strain

FE Computations Kroner Kroner-Tangent Kroner-Secant

V=300b³, dE/dt=10

-3

s

-1

E

vp

_{≈ 1}

-3

10

−3

𝑉 = 300𝑏³,

𝐸

𝑣𝑝

≈ 10

−2

𝑉 = 30𝑏³

𝑉 = 300𝑏³

10

−3

𝑠

−1

𝑉 =

300𝑏³

0 0.2 0.4 0.6 0.8 1 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 C u m u la ti v e p ro b a b il it y Normalized stress FE Computations Kroner Kroner-Tangent Kroner-Secant Ev p_{≈ 1} -2 V=300b³, dE/dt=10-3_s-1 0 0.2 0.4 0.6 0.8 1 0.0 0.5 1.0 1.5 2.0 2.5 C u m u la ti v e p ro b a b il it y Normalized stress FE Computations Kroner Kroner-Tangent Kroner-Secant Ev p_{≈ 1} -3 V=300b³, dE/dt=10-3_s-1 0 0.2 0.4 0.6 0.8 1 0.0 0.5 1.0 1.5 2.0 2.5 C u m u la ti v e p ro b a b il it y Normalized stress FE Computations Kroner Kroner-Tangent Kroner-Secant Ev p_{≈ 1} -4 V=30b³, dE/dt=10-3_s-1 0 0.2 0.4 0.6 0.8 1 0.0 0.5 1.0 1.5 2.0 2.5 C u m u la ti v e p ro b a b il it y Normalized stress FE Computations Kroner Kroner-Tangent Kroner-Secant Ev p_{≈ 1} -3 V=30b³, dE/dt=10-3_s-1 0 0.2 0.4 0.6 0.8 1 0.0 0.5 1.0 1.5 2.0 2.5 C u m u la ti v e p ro b a b il it y Normalized stress FE Computations Kroner Kroner-Tangent Kroner-Secant Ev p_{≈ 1} -2 V=30b³, dE/dt=10-3_s-1 0 0.2 0.4 0.6 0.8 1 0.0 0.5 1.0 1.5 2.0 2.5 C u m u la ti v e p ro b a b il it y Normalized stress FE Computations Kroner Kroner-Tangent Kroner-Secant Ev p_{≈ 1} -4 V=300b³, dE/dt=10-3_s-1

Σ

𝑟𝑒𝑙𝑎𝑥

𝑡

ℎ

10

−3

𝑠

−1

𝐸 = 5%

𝑡

ℎ

Σ

th

Σ

o

Σ

𝑟𝑒𝑙𝑎𝑥

= Σ

th

− Σ

o

.

𝑉 = 30𝑏³

𝑉 = 100𝑏³

𝑉 = 300𝑏³

-1400 -1200 -1000 -800 -600 -400 -200 0 0.0 2.0 4.0 6.0 8.0 FE Computations Kroner Kroner-Tangent Kroner-Secant S tr e s s r e la x a ti o n , 𝚺 𝐞 (M P a ) Holding time, ( ),(𝒔) = 30 ³, 𝐸̇ = 10−3 _𝑠−1 -450 -400 -350 -300 -250 -200 -150 -100 -50 0 0.0 2.0 4.0 6.0 8.0 FE Computations Kroner Kroner-Tangent Kroner-Secant = 100 ³, 𝐸̇ = 10−3 _𝑠−1 S tr e s s r e la x a ti o n , 𝚺𝐞 (M P a ) Holding time, ( ),(𝒔) -160 -140 -120 -100 -80 -60 -40 -20 0 0.0 2.0 4.0 6.0 8.0 FE Computations Kroner Kroner-Tangent Kroner-Secant = 300 ³, 𝐸̇ = 10−3 _𝑠−1 S tr e s s r e la x a ti o n , 𝚺𝐞 (M P a ) Holding time, ( ),(𝒔)

𝑉 = 100𝑏³ 𝐸̇ = 10

−3

_𝑠

−1

_{𝐸 = 5%}

-600 -500 -400 -300 -200 -100 0

0.0E+0 1.0E+7 2.0E+7 3.0E+7

Holding time, 𝒕𝒉(𝒔) S tre s s re la x a ti o n , 𝝈 𝐞 ( ) FE Computations = 100 ³, 𝐸̇ = 10−3 _𝑠−1 -500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0

0.0E+0 1.0E+7 2.0E+7 3.0E+7

Holding time, 𝒕𝒉(𝒔) Kröner = 100 ³, 𝐸̇ = 10−3 _𝑠−1 S tr e ss r e la x a ti on , 𝝈 𝐞 ( ) -350 -300 -250 -200 -150 -100 -50 0

0.0E+0 1.0E+7 2.0E+7 3.0E+7

Holding time, 𝒕𝒉(𝒔) Kröner-Tangent = 100 ³, 𝐸̇ = 10−3 _𝑠−1 R e la x e d s tr e s s , 𝝈 𝐞 ( ) -500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0

0.0E+0 1.0E+7 2.0E+7 3.0E+7

Holding time, 𝒕𝒉(𝒔) Kröner-Secant = 100 ³, 𝐸̇ = 10−3 _𝑠−1 S tr e ss r e la x a ti on , 𝝈 𝐞 ( )

𝑉 = 100𝑏³ 𝐸̇ =

10

−3

_𝑠

−1

_{𝐸 = 5%}



0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.0E+0 1.0E+7 2.0E+7 3.0E+7

Holding time, FE Computations V isc o p las tic strai n , 0 0.01 0.02 0.03 0.04 0.05 0.06

0.0E+0 1.0E+7 2.0E+7 3.0E+7

Holding time, Kröner V isc o p las tic strai n , 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

0.0E+0 1.0E+7 2.0E+7 3.0E+7

Holding time, 𝒕𝒉(𝒔) Kröner-Tangent = 100 ³, 𝐸̇ = 10−3 _𝑠−1 V is c o p la s ti c s tra in , 𝝐 0 0.01 0.02 0.03 0.04 0.05 0.06

0.0E+0 1.0E+7 2.0E+7 3.0E+7

Holding time, V isc o p las tic strai n , Kröner-Secant

𝑉 = 30𝑏³

𝛴 = 400 𝑀𝑃𝑎

𝑉 = 100𝑏³

𝛴 = 150 𝑀𝑃𝑎

𝑉 = 300𝑏³

𝛴 = 100 𝑀𝑃𝑎

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.0E+0 5.0E+6 1.0E+7

FE Computations Kroner Kroner-Tangent Kroner-Secant Holding time, (𝒔) = 30 ³, Σ = 400 𝑀𝑃𝑎 V is c o p la s ti c s tr a in , 𝑬 𝒗𝒑 0.0E+00 5.0E-03 1.0E-02 1.5E-02 2.0E-02 2.5E-02

0.0E+0 5.0E+6 1.0E+7

FE Computations Kroner Kroner-Tangent Kroner-Secant Holding time, (𝒔) = 100 ³, Σ = 150 𝑀𝑃𝑎 V is c o p la s ti c s tr a in , 𝑬 𝒗𝒑 0.000 0.002 0.004 0.006 0.008 0.010

0.0E+0 5.0E+6 1.0E+7

FE Computations Kroner Kroner-Tangent Kroner-Secant Holding time, (𝒔) V is c o p la s ti c s tr a in , 𝑬 𝒗𝒑 = 300 ³, Σ = 100 𝑀𝑃𝑎

𝑉 = 30𝑏³

𝑉 = 100𝑏³

𝑉 = 300𝑏³

𝑉 = 100 − 200𝑏³

𝑄 = 1.5 − 2.0 𝑒𝑉

1.0E-11 1.0E-10 1.0E-9 1.0E-8 1.0E-7 1.0E-6 1.0E-5 1.0E-4 1.0E-3 0 500 1000 FE Computations Kroner Kroner-Tangent Kroner-Secant Stress, 𝚺 (MPa) V is c o p la s ti c s tr a in r a te , 𝑬 ̇ 𝒗𝒑 𝒔 − 𝟏

= 30 ³

1.0E-11 1.0E-10 1.0E-9 1.0E-8 1.0E-7 1.0E-6 1.0E-5 1.0E-4 1.0E-3 0 100 200 300 400 FE Computations Kroner Kroner-Tangent Kroner-Secant Stress, 𝚺 (MPa) V is c o p la s ti c s tr a in r a te , 𝑬̇ 𝒗𝒑 𝒔 − 𝟏

= 100 ³

1.0E-12 1.0E-11 1.0E-10 1.0E-9 1.0E-8 1.0E-7 1.0E-6 1.0E-5 1.0E-4 1.0E-3 1.0E-2 0 50 100 150 200 FE Computations Kroner Kroner-Tangent Kroner-Secant Stress, 𝚺 (MPa) V is c o p la s ti c s tr a in r a te , 𝑬 ̇ 𝒗𝒑 𝒔 − 𝟏 = 300 ³

𝚼

𝒗

𝜿

𝒃

𝑨

𝑸

𝑽

𝜏

0

𝐻

0

𝐻

0

= 1 𝑀𝑃𝑎

𝑎

𝑖,𝑗

= 𝑞 = 1.4 (𝑖 ≠ 𝑗

𝑎

𝑖,𝑖

= 1

𝝉

𝟎

𝑯

𝟎

𝒒

𝜶

𝑪

𝐸 = ± 0.15%

𝐸 = ± 1.3%

𝐸 = 0.5 ± 0.5%

𝐸 = 0.15 − 1.3%

𝛴 = 50 ± 250 𝑀𝑃𝑎

𝛴 = 50 ± 270 𝑀𝑃𝑎

𝛴 = 50 ± 290 𝑀𝑃𝑎

𝛴 = 70 ± 250 𝑀𝑃𝑎

𝛴 = 30 ± 290 𝑀𝑃𝑎

𝛴 = 125 ± 275 𝑀𝑃𝑎

𝑬̇

𝜮̇

𝐸 = 0.5 ± 0.5%

𝑅

𝑋

𝐸

𝑣𝑝

= 10

−4

Σ

𝑦

-400 -300 -200 -100 0 100 200 300 400 500 0 0.005 0.01 FE Computations Kröner Kröner-Secant Str e s s , 𝚺 (M Pa ) Strain, 𝑬 Bauschinger Effect 𝑉 = 150𝑏³, 𝐸̇ = 10−3 _𝑠−1 _{, 𝐸 = 0.5 ± 0.5%} N=0.25 N=1.25 N=0.75

𝐸

𝑣𝑝

_{= 10}

−4

_𝛴

𝑦

Σ

𝑦+0

Σ

𝑚𝑎𝑥+0

Σ

𝑦−0

R

−0

= |

Σ

𝑚𝑎𝑥+0

− Σ

𝑦−0

2

|

𝑋

−0

_{= Σ}

𝑦 −0

_{+ R}

−0

Σ

𝑚𝑖𝑛−0

Σ

𝑦+1

Σ

𝑚𝑎𝑥+1

R

+1

= |

Σ

𝑚𝑖𝑛−0

− Σ

𝑦+1

2

|

𝑋

+1

_{= Σ}

𝑦 +1

_{+ R}

+1

+0

−0

𝚺

𝒚−0

R

−0

𝚺

𝒚−0

𝑋

−0

Σ

𝑚𝑖𝑛−0

R

+1

𝑋

+1

𝐸 = ±0.5%

𝐸 = ±1.3%

-500 -400 -300 -200 -100 0 100 200 300 400 500 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 FE Computations Kroner Kroner-Secant Str e s s , 𝚺 (M Pa ) Strain, 𝑬

Stress-strain hysteresis loop

𝑉 = 150𝑏3_{,𝐸̇ = 10}−3 _𝑠−1_{, 𝐸 = ±0.5%} -800 -600 -400 -200 0 200 400 600 800 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 FE Computations Kroner Kroner-Secant Str e s s , 𝚺 (M Pa ) Strain, 𝑬

Stress-strain hysteresis loop

𝑉 = 150𝑏3_{,𝐸̇ = 10}−3 _𝑠−1_{, 𝐸 = ±1.1%} 200 250 300 350 400 450 500 550 600 650

0.0E+0 2.5E-3 5.0E-3 7.5E-3 1.0E-2

FE Computations Kroner Kroner-Secant S tress ampli tud e, (M P a)

Viscoplastic strain amplitude,

Cyclic stress-strain curve

𝐸

𝑎

𝑣𝑝

< 2.10

−3

𝐸

_𝑎

𝑣𝑝

> 5.10

−3

𝐸

_𝑎

𝑣𝑝

= 10

−2

𝐸

𝑟

𝐸

𝑟

=

𝐸

𝑚𝑎𝑥

𝑣𝑝

+ 𝐸

𝑚𝑖𝑛

𝑣𝑝

2

𝐸

𝑚𝑎𝑥

𝑣𝑝

𝐸

𝑚𝑖𝑛

𝑣𝑝

Σ

𝑚𝑎𝑥

Σ

𝑚

𝛴

𝑚𝑎𝑥

𝛴

𝑚𝑎𝑥

𝛴

𝑚𝑎𝑥

𝛴

𝑚

𝛴

𝑚 0.0E+0 5.0E-4 1.0E-3 1.5E-3 2.0E-3 2.5E-3 3.0E-3 3.5E-3 4.0E-3 0 25 50 FE Computations Kröner Kröner-Secant N (Cycles) R atch eti n g strain, Ratcheting 0.0E+0 1.0E-3 2.0E-3 3.0E-3 4.0E-3 5.0E-3 6.0E-3 7.0E-3 8.0E-3 9.0E-3 0 25 50 FE Computations Kröner Kröner-Secant N (Cycles) R atch eti n g strain Ratcheting 0.0E+0 2.0E-3 4.0E-3 6.0E-3 8.0E-3 1.0E-2 1.2E-2 1.4E-2 1.6E-2 1.8E-2 0 25 50 FE Computations Kröner Kröner-Secant N (Cycles) Ra tche ti ng str ain, Ratcheting 0.0E+0 2.0E-3 4.0E-3 6.0E-3 8.0E-3 1.0E-2 1.2E-2 0 25 50 FE Computations Kröner Kröner-Secant N (Cycles) R atche ti ng str ain,

Ratcheting

0.0E+0 1.0E-3 2.0E-3 3.0E-3 4.0E-3 5.0E-3 6.0E-3 0 25 50 FE Computations Kröner Kröner-Secant N (Cycles) R atche ti ng str ain,

Ratcheting

𝐸

𝑎

𝑣𝑝

𝑑𝐸

𝑟

/𝑑𝑁

𝐸

_𝑎

𝑣𝑝

𝐸

𝑎

𝑣𝑝

𝐸

𝑎

𝑣𝑝

𝛴 = 50 ± 250

1.0E-5 2.0E-5 3.0E-5 4.0E-5 5.0E-5 6.0E-5 7.0E-5 8.0E-5 9.0E-5 1.0E-4 0 25 50 FE Computations Kröner Kröner-Secant N (cycles) V isc op las ti c str ain ampli tud e ,

Ratcheting

1.0E-6 1.0E-5 1.0E-4 1.0E-3 0 25 50 FE Computations Kröner Kröner-Secant N (cycles) R atche ti ng str ain rat e,

Ratcheting

𝛴 = 125 ± 275𝑀𝑃𝑎

0.E+0 1.E-3 2.E-3 3.E-3 4.E-3 5.E-3 6.E-3 7.E-3 8.E-3 0 5 10 15 20 FE Computations Kroner Kroner-Secant R a c th e ti n g s tr a in , 𝑬𝒓 N (cycle) Ratcheting Σ = 125 ± 275𝑀𝑃𝑎, Σ̇ = 50𝑀𝑃𝑎. 𝑠−1 0.0E+0 5.0E-3 1.0E-2 1.5E-2 2.0E-2 2.5E-2 0 20 40 60 80 100

FE Computation (linear extrapolation) FE Computations Kroner Kroner-Secant 𝑬 𝒓 N (cyles) Ratcheting 𝑉 = 150𝑏3_{,Σ = 125 ± 275𝑀𝑃𝑎, 𝐸̇ = 50𝑀𝑃𝑎. 𝑠}−1

Υ

inc

= 350𝐺𝑃𝑎

ν

inc

=

0.287

Υ

mat

_{= 150𝐺𝑃𝑎}

_ν

mat

_{= 0.3}

𝐸̇

𝑒𝑞

= 𝐶. 𝛴

𝑒𝑞𝑛

Σ

yy

= 220 𝑀𝑃𝑎

𝚺

30 𝑀𝑃𝑎 < Σ

𝑚

< 70 𝑀𝑃𝑎

β

β

0 100 200 300 400 500 600 700 800 0 10 20 30 40 50 FE Computations Kroner Kroner-Secant s tr e s s ,𝝈 (M Pa ) temps (s) 0 100 200 300 400 500 600 700 800 0 200 400 600 800 1000 FE Computations Kroner Kroner-Secant s tr e s s ,𝝈 (M Pa ) temps (s)

•

•

•

•

E

a

vp

> 5.10

−3

•

Σ

𝑚

>

125 𝑀𝑃𝑎)

β

β

β

𝜇

𝜈

𝜂

𝜂

𝑡𝑎𝑛

= Σ̇

11

/𝐸̈

11

𝑣𝑝

𝜂

𝑠𝑒𝑐

= Σ

11

/𝐸̇

11

𝑣𝑝

Σ

𝑚𝑖𝑛

< −150 𝑀𝑃𝑎

𝐸

𝑎

𝑣𝑝

> 0.5%

𝜏

𝑖

𝜎,

𝑖

𝑡ℎ

𝜏

𝑖

= 𝜎: (m

𝑖

⨂ n

𝑖

)

m

𝑖

n

𝑖

𝜏

𝑖

𝜏

𝑐,𝑖

𝜏

𝑖

∗

𝜏

𝑖

∗

= |𝜏

𝑖

| − 𝜏

𝑐,𝑖

≥ 0

𝜏

𝑐,𝑖

𝜏

𝑐,𝑖

= 𝜏

0

+ 𝜇𝑏√∑ ℎ

𝑖𝑗

𝜌

𝑗

12

𝑗=1

𝜏

0

𝜇

ℎ

𝑖𝑗

𝛾̇

_𝑖

𝑝

𝜅

𝑏

𝑣

𝐷

𝜌

𝑒

𝑖

𝜌

𝑠

𝑖

𝛾̇

_𝑖,𝑒

𝑝

= 2𝑣

𝐷

𝑏

2

𝜌

𝑒

𝑖

∙ 𝑒𝑥𝑝 (−

𝑄

𝜅

𝑏

𝑇

) ∙ 𝑠𝑖𝑛ℎ (

𝑉𝜏

𝑖

∗

𝜅

𝑏

𝑇

)

𝛾̇

_𝑖,𝑠

𝑝

= 2𝑣

𝐷

𝑏

2

𝜌

𝑠

𝑖

∙ 𝑒𝑥𝑝 (−

𝑄

𝜅

𝑏

𝑇

) ∙ 𝑠𝑖𝑛ℎ (

𝑉𝜏

𝑖

∗

𝜅

𝑏

𝑇

)

𝜏

0

𝜏

0

≈ 𝑄/𝑉

𝜌̇

𝑑

𝜌̇

𝑒

𝑖

=

2

𝑏𝐿

∙ 𝛾̇

𝑠

𝑖

_{∙ 𝑠𝑖𝑔𝑛 (}

𝛾̇

𝑠

𝑖

𝛾

𝑠

𝑖

) −

2

𝑒

𝑏

∙ 𝜌

𝑒

𝑖

_{∙ |𝛾̇}

𝑒

𝑖

| − 𝜌̇

𝑑

𝑖

𝜌̇

𝑠

𝑖

=

2

𝑏𝐿

∙ 𝛾̇

𝑒

𝑖

_{∙ 𝑠𝑖𝑔𝑛 (}

𝛾̇

𝑒

𝑖

𝛾

𝑒

𝑖

) −

2

𝑠

𝑏

∙ 𝜌

𝑠

𝑖

_{∙ |𝛾̇}

𝑠

𝑖

|

𝜌̇

𝑑

𝑖

= 2𝜌

𝑒

𝑖

∙

ℎ

𝑚𝑎𝑥

−

𝑒

𝑏

∙ |𝛾̇

𝑒

𝑖

_{| −}

2

𝑒

𝑏

∙ 𝜌

𝑑

𝑖

_{∙ |𝛾̇}

𝑒

𝑖

|

𝑒

𝑠

ℎ

𝑚𝑎𝑥

𝛾

𝑆𝐹𝐸

15 𝑚𝐽. 𝑚

−2

30 𝑚𝐽. 𝑚

−2

𝐿

𝐿 =

1

𝜙

+

𝜅

√𝜌

𝑡

𝜌

𝑡

𝜙

𝐿 = min (𝜙,

𝜅

√𝜌

𝑡

)

𝜅

𝜅

𝜅

𝑠𝑖𝑛𝑔𝑙𝑒

= 300)

𝜅

𝑚𝑢𝑙𝑡𝑖

= 30)

𝜅

𝑠𝑖𝑛𝑔𝑙𝑒

𝜅

𝑚𝑢𝑙𝑡𝑖

𝑒

𝑠

(

𝑒

𝑒𝑥𝑝

)

𝑒

𝑒𝑥𝑝

𝛾

𝑆𝐹𝐸

/𝜇𝑏

𝑒𝑒𝑥𝑝

𝜸

𝑺𝑭𝑬

/𝛍𝐛(𝟏𝟎

−𝟑

)

𝒚

𝒆

(𝐧𝐦)

𝑒

𝐶𝑎𝑙𝑐𝑢𝑙

=

𝜇𝑏

4

2𝜋(1 − 𝜈)𝐸

𝑣𝑎𝑐

𝐸

𝑣𝑎𝑐

𝑏 =

2.54 10

−10

_𝑚

_{𝜇 = 75 𝐺𝑃𝑎}

_𝐸

𝑣𝑎𝑐

= 1.4𝑒𝑉

𝑒

𝐶𝑎𝑙𝑐𝑢𝑙

𝑒

𝐶𝑎𝑙𝑐𝑢𝑙

= 0.38 −

0.63 nm

𝑒

𝑒𝑥𝑝

= 1.2 nm

𝑠

𝑒𝑥𝑝

𝑠

𝑒𝑥𝑝

𝛾

𝑆𝐹𝐸

/𝜇𝑏

𝑠

𝑒𝑥𝑝

= 15 nm

𝑒

ℎ

𝑚𝑎𝑥

y

s

= 4.2546 (γ

SFE

/μb)^1.5397

R² = 0.9728

1

10

100

1000

0

5

10

15

20

25

Cu-10%Al (Franciosi et al., 1980)

316LN (Feaugas, 1999)

Cu-2%Al (Franciosi et al., 1980)

Cu (Essmann and Mugrhabi, 1979)

Al (Jaoul, 1964)

𝒚

𝒔

(

𝒏𝒎

)

𝜸

𝑺𝑭𝑬

𝝁𝒃

(𝟏𝟎

−𝟑

)

𝑒

ℎ

𝑚𝑎𝑥

ℎ

𝑚𝑎𝑥

=

𝜇𝑏

8𝜋(1 − 𝑣)𝜏

𝑐,𝑖

h

_max

_Annihilation

_y

e

Dipole

𝜎̇ = Σ̇ + 2𝜇(1 − 𝛽) (𝐸̇

𝑣𝑝

− 𝜖̇

𝑣𝑝

) + 2𝜇(1 − 𝛽) 𝜂

−1

(Σ

d

− 𝜎

𝑑

)

𝛽 =

2(4−5𝜈)

15(1−𝜈)

𝜂 =

Σ

11

𝐸̇

₁₁𝑣𝑝

𝜎̇

𝜖̇

𝑣𝑝

Σ̇

𝐸̇

𝑣𝑝

𝜎

𝑑

_Σ

d

_{𝜂 =}

Σ

11

𝐸̇

₁₁𝑣𝑝

𝜂

−1

= 0

𝚼

𝚼

𝝓

μ

μ

μ

μ

μ

ℎ

𝑖𝑗

,

𝑒

𝑠

𝜌

0

𝜿

𝒃

1.38 10

−23

𝐽. 𝐾

−1

𝒗

𝑫

10

13

𝑠

−1

𝒃

2.54 10

−10

𝒗

𝒚

𝒆

𝒚

𝒔

𝒉

𝟎

𝒉

𝒄𝒐𝒑𝒍𝒂

𝒉

𝑳𝒐𝒎𝒆𝒓

𝒉

𝑯𝒊𝒓𝒕𝒉

𝒉

𝒈𝒍𝒊𝒔

𝒉

𝒄𝒐𝒍𝒍

𝝆

𝟎

10

10

𝑚

−2

𝜅

𝑠𝑖𝑛𝑔𝑙𝑒

𝜅

𝑚𝑢𝑙𝑡𝑖

𝑉

𝑄

𝜏

0

τ − τ

0

τ

0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 0.02 0.04 0.06 Karaman et al., 2001 Simulation -τ=90MPa Simulation -τ=180MPa Simulation -τ=45MPa

Single crystal [123]

K=400

plastic strain, 0 10 20 30 40 50 60 0 0.005 0.01 0.015 0.02 Karaman et al., 2001 Simulation -τ=90MPa Simulation -τ=180MPa Simulation -τ=45MPa plastic strain,

Single crystal [100]

K=25

0 10 20 30 40 50 60 70 80 0 0.01 0.02 0.03 0.04 Karaman et al., 2001 Simulation -τ=90MPa Simulation -τ=45MPa Simulation -τ=180MPa platic strain,

Single crystal [111]

K=25

𝜅

𝑠𝑖𝑛𝑔𝑙𝑒

= 400

𝜅

𝑚𝑢𝑙𝑡𝑖

= 25

𝛾

𝑝

= 2%

𝜏

𝑝𝑟𝑖𝑚𝑎𝑟𝑦

𝜏

𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦

𝑅 = 𝜏

𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦

/𝜏

𝑝𝑟𝑖𝑚𝑎𝑟𝑦

𝑅

𝑅

𝑐𝑟𝑖𝑡

𝑅

𝑐𝑟𝑖𝑡

𝑅

𝑐𝑟𝑖𝑡

𝑅

𝑐𝑟𝑖𝑡

= 0.97

|𝜏

𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦

/𝜏

𝑝𝑟𝑖𝑚𝑎𝑟𝑦

|

𝑉

𝑄

𝜏

0

[100] [110] [111] 𝜏 𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦 𝜏 𝑝𝑟𝑖𝑚𝑎𝑟𝑦 = 1.00 𝜏 𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦 𝜏 𝑝𝑟𝑖𝑚𝑎𝑟𝑦 = 0.72 𝑝

for FCC crystals

Color step =0.03

Single-slip (tensile loading,

E

p

=1.0%, 𝐸̇=3.10

-4

s

-1

)

Main dislocation patterns (tensile

loading, E

p

=3.0%,

𝐸̇=3.10

-4

s

-1

)

316L(N) SS

Dipole

walls

Polarized

dislocations

𝑄/𝑉,

𝑸 𝑽

= 𝟏𝟔𝟑. 𝟎 𝑴𝑷𝒂

𝑄 𝑉

= 112.4 𝑀𝑃𝑎

0 50 100 150 200 250 300 350 400 0 0.005 0.01 Simulation - dE/dt=1E-03 /s Yu et al., 2012 - dE/dt=1E-03 /s Simulation - dE/dt=1E-04 /s Yu et al., 2012 - dE/dt=1E-04 /s Simulation - dE/dt=1E-05 /s Yu et al., 2012 - dE/dt=1E-05 /s Str ess, (M Pa) Strain, 316L(N) SS 0 50 100 150 200 250 300 350 400 0 0.005 0.01 Simulation - T=293K Gaudin, 2002 - T=293K Simulation - T=373K Gaudin, 2002 - T=373K S tress, (M P a) Strain, 316L(N) SS 0 50 100 150 200 250 300 0 0.005 0.01 0.015 0.02 Simulation - dE/dt=2E-3/s Kang et al., 2010 - dE/dt=2E-3/s Simulation - dE/dt=3E-4/s Portier et al., 1999 - dE/dt=3E-4/s

S tres s, (MP a) Strain, 316L SS 0 50 100 150 200 250 300 0 0.005 0.01 0.015 0.02 Simulation - T=293K Portier et al., 1999 - T=293K Simulation - T=523K Portier et al., 1999 - T=523K S tress, (M P a) Strain, 316L SS

𝐸̇ = 3.10

−4

_𝑠

−1

Σ

0

= 250 𝑀𝑃𝑎

Σ

0

= 200 𝑀𝑃𝑎

1.0E+12 1.0E+13 1.0E+14 1.0E+15

1.0E-03 1.0E-02 1.0E-01

Gaudin, 2002 Simulation Viscoplastic strain, 𝑬𝒗𝒑 D lo o d y ρ -2)

Total dislocation density (edge, screw and dipole) 316L(N) SS

𝐸 =

±0.6%

𝐸̇ = 3.10

−4

_𝑠

−1 0 50 100 150 200 250 300 0 20 40 60 80 100

Exp. Lemaitre & Chaboche, 1988 -Σo=250MPa Simulation -Σo=250MPa

Exp. Lemaitre & Chaboche, 1988 -Σo=200MPa Simulation -Σo=200MPa Holding time(h) S tress, (M P a) 316L SS -400 -300 -200 -100 0 100 200 300 400

-7E-3 0E+0 7E-3

Simulation Gentet, 2010 316L(N) SS - N=1 Strain, Str ess,

𝐸 =

±0.6%

𝐸̇ = 3.10

−4

_𝑠

−1

𝐸

_𝑐𝑢𝑚

𝑣𝑝

=

4. 𝑁. 𝐸

𝑎

𝑣𝑝

𝜌

𝑐𝑎𝑙𝑐𝑢𝑙

= 3.6 10

14

𝑚

−2

𝜌

𝑒𝑥𝑝

= 2.4 10

14

𝑚

−2

𝐸 =

±0.6%

𝐸̇ = 3.10

−4

_𝑠

−1 0 50 100 150 200 250 300 350 400 0 50 100 150 Simulation - T=293K Gentet , 2010 - T=293K Simulation - T=323K Gentet, 2010 - T=323K Simulation - T=473K Gentet, 2010 - T=473K 316L(N) SS S tress ampl it ud e, (M P a) N (cycles) 1E+11 1E+12 1E+13 1E+14 1E+15 0 1.2 2.4 3.6 4.8 6 Total dislocations Edge dislocations Screw dislocations Edge-dipole dislocations Dis loc a tion de ns ity (m -2)

Cumulative viscoplastic strain, N (cycle) 600 750 450 300 150 0 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0 1.2 2.4 3.6 4.8 6 Edge dislocations Screw dislocations Edge-dipole dislocations F ra c ti o n o f d is lo c a ti o n p o p u la ti o n d e n s it y

Cumulative viscoplastic strain, 𝑬𝒄𝒖𝒎𝒗𝒑

N (cycle) 600 750 450 300 150 0

𝐸

_𝑎

𝑣𝑝

= ±0.1%

𝐸

_𝑎

𝑣𝑝

= ±0.8%

𝐸 = ±0.5%; 𝐸 = ±0.65%; 𝐸 =

±0.8%

0 100 200 300 400 500 600

1.0E-4 5.1E-3 1.0E-2 1.5E-2

Exp. Gaudin, 2002 Simulation 316L(N) SS S tress ampli tu d e, (M P a)

Viscoplastic strain amplitude,

0 100 200 300 400 500 600

1.0E-4 1.0E-3 1.0E-2 1.0E-1

Exp. Gaudin, 2002 Simulation 316L(N) SS S tress ampl it ud e, (M P a)

Viscoplastic strain amplitude,

-400 -300 -200 -100 0 100 200 300 400 -0.01 -0.005 0 0.005 0.01

Simulation - T=293K - E=0.5% Portier, 1999 - E=0.5% Simulation - T=293K - E=0.65% Portier, 1999 - E=0.65% Simulation - T=293K - E=0.8% Portier, 1999 - E=0.8%

316L SS - Stable cycle

Stress,

(M

Pa)

𝑉 = 96𝑏³

𝑄 = 1.55 𝑒. 𝑉

𝑉 = 136𝑏³

𝑄 = 1.6 𝑒. 𝑉

𝐸

𝑟

𝑣𝑝

𝐸

𝑟

Σ

a

= ±250𝑀𝑃𝑎

𝛴

𝑚

= 50𝑀𝑃𝑎

−150𝑀𝑃𝑎 < 𝛴

𝑚𝑖𝑛

< 0 𝑀𝑃𝑎

Σ

a

= ±275 𝑀𝑃𝑎

𝛴

𝑚

= 125 𝑀𝑃𝑎

𝛴 = 50 ± 250 𝑀𝑃𝑎

0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 0 100 200 300 400 500 Gaudin, 2002 -Σ = 50 250 MPa Simulation -Kröner Simulation- Kröner-Secant 316L(N) SS R atc h etin g strai n , N (cycles)

𝛴 = 125 ± 275 𝑀𝑃𝑎

𝛴

𝑚

𝛴

𝑚𝑎𝑥

𝛴

𝑚

𝛴

𝑚 0.0E+0 1.0E-2 2.0E-2 3.0E-2 4.0E-2 5.0E-2 6.0E-2 7.0E-2 8.0E-2 0 20 40 60 80 100 Exp. CEA/SRMA Kroner Kroner-Secant N (Cycles) R a tc h e ti n g s tr a in

316L(N) SS

T = 293 , Σ = 125 ± 275 𝑀𝑃𝑎, Σ̇ = 50𝑀𝑃𝑎. 𝑠−1_{, 𝜙 = 50 𝜇𝑚} 0% 5% 10% 15% 20% 25% 0 100 200 300 400 500

Simul. Kroner -Σ=50 480 MPa Gaudin, 2002 -Σ = 50 480 MPa Simul. Kroner -Σ=50 400 MPa Gaudin, 2002 -Σ = 50 400 MPa Simul. Kroner -Σ=50 250 MPa Gaudin, 2002 -Σ = 50 250 MPa

R atch eti n g str ai n , N (cycles) 316L(N) SS 0.0% 0.5% 1.0% 1.5% 2.0% 0 20 40 60 80 100

Simul. Kroner -Σ=125 200 MPa Yu et al., 2012 -Σ=125 200MPa Simul. Kroner -Σ=125 175 MPa Yu et al., 2012 - Σ=125 175MPa Simul. Kroner -Σ=125 150 MPa Yu et al., 2012 - Σ=125 150MPa

M axi mu n axi al str ai n , N (cycles) 316L(N) SS

Σ = 50 ± 250 𝑀𝑃𝑎

𝛴

𝑚𝑎𝑥

𝛴

𝑚𝑎𝑥 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 0 100 200 300 400 500

Simul. Kroner -Σ=80 270 MPa Gaudin, 2002 -Σ = 80 270 MPa

Simul. Kroner -Σ=175 175 MPa Gaudin, 2002 -Σ = 175 175 MPa

R atch eti n g str ai n , N (cycles) 316L(N) SS 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 0 100 200 300 400 500

Simul. Kroner -Σ=80 370 MPa Gaudin, 2002 -Σ = 80 370 MPa

Simul. Kroner -Σ=20 430 MPa Gaudin, 2002 -Σ = 20 430 MPa

N (cycles) R atch eti n g str ai n , 316L(N) SS

𝛴 = 50 ± 250 𝑀𝑃𝑎

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

1.0E+0

1E+11

1E+12

1E+13

1E+14

0

100

200

300

400

500

Total dislocations

Edge dislocations

Screw dislocations

Edge-dipole dislocations

dEr/dN

D

is

lo

c

a

ti

o

n

d

e

n

s

it

y

(

m

-2

)

N (cycle)

R

a

tc

h

e

tin

g

s

tr

a

in

r

a

te

,

𝒅

𝑬

𝒓

_/

𝒅𝑵

316L(N) SS

𝑇 = 293 𝐾, Σ = 20 ± 250 𝑀𝑃𝑎

0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% 0 20 40 60 80 100

Simul. Kroner - dΣ/dt=10MPa/s Yu et al., 2012 - dΣ/dt=10MPa/s Simul. Kroner - dΣ/dt=100MPa/s Yu et al., 2012 - dΣ/dt=100MPa/s Simul. Kroner - dΣ/dt=1000MPa/s Yu et al., 2012 - dΣ/dt=1000MPa/s

M axi mu n axi al str ai n , N (cycles) 316L(N) SS

Σ=225±225 MPa

𝛴 = 100 ± 140 𝑀𝑃𝑎

𝛴 = 60 ± 100 𝑀𝑃𝑎

0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 9.0% 10.0% 0 100 200 300 400 Gaudin, 2002 Simulation - Kroner R atcheti ng str ai n, N (cycles) 316L(N) SS T=293K 1) 2) 3) 1 2 3 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 9.0% 10.0% 0 100 200 300 400 Gaudin, 2002 Simulation - Kroner R atcheti ng str ai n, N (cycles) 1 2 3 316L(N) SS T=293K 1) 2) 3) 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% 1.6% 0 50 100 Portier, 1999 Simul. Kroner R atcheti ng str ai n, N (cycles)

316L SS

0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6% 0.7% 0.8% 0.9% 1.0% 0 50 100 Portier, 1999 Simul. Kroner R atcheti ng str ai n, N (cycles)

316L SS

(𝐸̇ = 3.10

−4

𝑠

−1

𝐸 = ±0.6% 𝐸̇ =

3.10

−4

_𝑠

−1

_{(Σ = 50 ± 250 𝑀𝑃𝑎 𝐸̇ = 3.10}

−4

_𝑠

−1

𝑬 = 𝟑%)

𝝆

𝟎

→ (𝟏𝟎

𝟗

; 𝟏𝟎

𝟏𝟏

) 𝒎

−𝟐

−0.03% ; 0.16%

−0.01% ; 0.22%

−0.46% ; 0.39%

𝒚

𝒆

→ (𝟎. 𝟔 ; 𝟏. 𝟖) 𝒏𝒎

0.11% ; −0.21%

4.51% ; −4.37%

−4.41% ; 10.53%

𝒚

𝒔

→ (𝟕. 𝟓 ; 𝟐𝟐. 𝟓) 𝒏𝒎

0.19% ; −0.11%

10.79% ; −1.18%

−7.50% ; 0.94%

𝑲

𝒔𝒊𝒏𝒈𝒍𝒆

→ (𝟐𝟎𝟎 ; 𝟔𝟎𝟎)

+0.00% ; −0.00%

0.35% ; −0.10%

−0.07% ; 0.06%

𝑲

𝒎𝒖𝒍𝒕𝒊

→ (𝟏𝟐. 𝟓 ; 𝟑𝟕. 𝟓)

8.79% ; −3.08%

3.89% ; −1.68%

−15.06% ; 7.50%

𝜌

0

𝜌

0

~3 10

12

m

−2

10

12

_𝑚

−2

₁₀

13

_𝑚

−2

10

12

_𝑚

−2

0 50 100 150 200 250 300 350 400 450 500 0.00 0.01 0.02 0.03 Gaudin, 2002 Rho0=10^10 m² (reference) Rho0=10^9 m² Rho0=10^11 m² Rho0=10^12 m² Rho0=10^13 m² Effect of S tress, (M P a) Strain, 0 100 200 300 400 500 0 50 100 150 200 Gentet, 2010 Rho0=10^10 m² (reference) Rho0=10^09 m² Rho0=10^11 m² Rho0=10^12 m² Rho0=10^13 m² Effect of S tress ampli tud e, (M P a) N (cycle) 0.000 0.002 0.004 0.006 0.008 0.010 0 100 200 300 400 500 Gaudin, 2002 Rho0=10^10 m² (reference) Rho0=10^9 m² Rho0=10^11 m² Rho0=10^12 m² Rho0=10^13 m² Effect of Ratchet ing st rai n, N (cycle)