Stress-Strain-Diffusion

Interactions in Solids

J. Svoboda1 and F.D. Fischer2

1Institute of Physics of Materials, Brno,Czech Republic

2Institute of Mechanics, Montanuniversität Leoben, Austria

DSL 2014

CONTENT

1. Introduction

2. Generalized Manning theory for diffusion

3. Solution of 1-D example – two assembled sheets

4. Simulation of system evolution

5. Summary/conclusions

1. Introduction

- A solid state system often involves defects, which can significantly influence its kinetics at elevated temperatures. - Defects can influence both effective diffusion coefficients as well as activity of sources and sinks for vacancies causing the Kirkendall effect and internal stress development with the feed back on diffusion kinetics. - Manning’s theory considering the vacancy wind effect has been generalized to account also for influence of sources and sinks for vacancies and the stress field, providing the evolution of the chemical composition coupled with deformation state of the system.- Motivation: to demonstrate the generalized Manning’s theory on a simple example and to show the influence of stress-strain-diffusion interactions on the system evolution kinetics.

2. Theory

System with substitutional components

Site fractions

Volume corresponding to 1 mole of lattice positions

Chemical potential of vacancies

Chemical potential of components

01

1n

kk

y y

0 00 1

1n n

k k k kk k

y y y

0 0 0ln eqgR T y y

0 lnk k g kR T y

*0 0

*0k k

n

0 HCoupling terms

0k H

Diffusion of substitutional components – Manning’s concept

for fcc alloys, for bcc alloys

01

n

kk

j j

*

1

, 1,...,n

k ik ii

L grad k n

j

1

1, , 1,...,i k

ik k ik n

jj

f A AL A i k n

f A

0

0 0

, 1,....,kk keq

g

y DA y k n

y R T

0.7815f 0.7272f

Conservation laws

Generation/annihilation of vacancies at non-ideal sources/sinks

Generalized creep (Fischer, Svoboda: Int. J. Plasticity 27, 1384-1390, 2011)

0 , 1,...,k k ky y k n j

0 0 0 01

1n

kk

y y y

j

*0

bvK

0

0 02

eqg

bv

fy R TK

y DaH

1

1 1

1n n

ii i

i ii

yD f f y D

D

H I s

*0

010 0

2

3 15 3 1

ncr

k kkbv bvK K y

I

I j s

3. Solution of 1-D example – two assembled sheets

Interface at

Axial (z-axis) symmetry

Elastic strain components

Creep strain rate components

Total strain components

0z 1 2h z h r z z 0z

2 3H r 3r rs s 2 3z rs

1 , 2el elr r z rv E z v E z

el crr r r el cr

z z z

,0

10

2 1

3 45 3 1

nk zcr r

r kkbv

j

K y z

,0

10

4 1

3 45 3 1

nk zcr r

z kkbv

j

K y z

Total thickness of the specimen

No force and no moment acting on the outer boundary of the specimen

2

1

1 dh

z

h

h z

0 1, ,r z t U t U t z h 0 1, ,1

crr r

Ez t U U z h z t

2

1

2 2

1 1

10

0 , 1 d

1 d , 1 d1

h

r z

h

h hcr

z r z

h h

P z t z

UEU h z z z t z

h

2

1

2 2 2

1 1 1

210

0 , 1 d

1 d 1 d , 1 d1

h

r z

h

h h hcr

z z r z

h h h

M z t z z

UEU z z z z z t z z

h

P M

Volume elements of actual configuration________

Curvature of the deformed system obtaining the shape of a spherical shell

Shift of the Kirkendall plane measured from the lower end of the system

Solution of equations provides time evolution of- profiles of diffusive fluxes inclusive those of vacancies- profile of rate of vacancy generation/annihilation - profiles of site fractions of components inclusive vacancies- profiles of creep rates - profiles of strains and stresses- Kirkendall shift and curvature

11 d drR z U h

1

0

dK z

h

z

4. Simulation of system evolution

Material parameters (I)

Starting values

1110 PaE 0.3 0.7272f

30 10eqy

31 10 mh 3

2 10 mh

5 3 -11 10 m mol 5 3 -1

2 0.9 10 m mol 5 3 -13 0.8 10 m mol

01 10 gR T

-18000JmolgR T 02 20 gR T 0

3 30 gR T 14 2 -1

1 10 m sD 14 2 -12 3 10 m sD 14 2 -1

3 5 10 m sD

102.5 10 ma 13 18 -310 ,10 mH

1 ,0 :z h 0 0eqy y 1 0.5y 2 0.3y 3 00.2 eqy y

20, :z h 0 0eqy y 1 0.6y 3 00.3 eqy y 2 0.1y

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.98

0.99

1.00

1.01

1.02

H=1017m-3

t=0s

t=105s

t=106s

t=3x104s

t=3x104s

t=3x104sy 0/

y 0eq

z/m

-0.0010 -0.0005 0.0000 0.0005 0.00100.97

0.98

0.99

1.00

1.01

1.02

H=1014m-3

t=3x105s

t=106s

t=107s

t=3x107s

t=0s

t=3x106s

t=3x104s

t=105s

y 0/y 0eq

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.50

0.52

0.54

0.56

0.58

0.60

H=1017m-3

t=3x104s

t=3x107s

t=3x104s

t=3x104s

t=107st=106st=105s

t=0s

y 1

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.50

0.52

0.54

0.56

0.58

0.60

H=1014m-3

t=3x104st=3x105st=3x106s t=3x107s

t=107st=106st=105s

t=0s

y 1

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.10

0.15

0.20

0.25

0.30

H=1017m-3t=105s

t=3x105s

t=3x104s

t=107s

t=3x106s

t=3x107s

t=106s

t=0sy 2

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.10

0.15

0.20

0.25

0.30

H=1014m-3t=107s

t=106s

t=105s

t=0s

t=3x106s

t=3x107s

t=3x105s

t=3x104s

y 2

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.20

0.22

0.24

0.26

0.28

0.30

H=1017m-3

t=0s

t=105s

t=3x105s

t=106s

t=3x106s

t=107s

t=3x107s

t=3x104s

y 3

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.20

0.22

0.24

0.26

0.28

0.30

H=1014m-3

t=3x107s

t=3x105s t=3x106s

t=3x104s

t=107s

t=106s

t=105s

t=0s

y 3

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

-80

-60

-40

-20

0

20

40

60

80

H=1014m-3

t=107s

t=105st=106s

t=0st=3x106s

t=3x105st=3x104s

r/MP

a

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010-40

-20

0

20

40

H=1017m-3

t=3x106st=106s

t=3x105st=105s

t=3x104s

r/Mpa

z/m

-0.0010 -0.0005 0.0000 0.0005 0.00100.97

0.98

0.99

1.00

1.01

1.02

1.03

H=1017m-3

H=1018m-3H=1016m-3

H=1015m-3

H=1014m-3

H=1013m-3

t=107s

y 0/y 0e

q

z/m

-0.0010 -0.0005 0.0000 0.0005 0.00100.50

0.52

0.54

0.56

0.58

0.60

H<1014m-3

H>1016m-3

H=1015m-3

t=107s

y 1

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.14

0.16

0.18

0.20

0.22

0.24

0.26

t=107s

y 2

z/m

-0.0010 -0.0005 0.0000 0.0005 0.00100.23

0.24

0.25

0.26

0.27

H=1015m-3

H=1016m-3

H>1017m-3

H=1014m-3

H<1013m-3

t=107s

y 3

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010-20

-10

0

10

20

t=107sH=1018m-3

H=1017m-3

H=1016m-3

H=1015m-3

H=1014m-4

H=1013m-3 r/M

pa

z/m

0 1x107 2x107 3x107-20

-15

-10

-5

0

H=1018m-3

H=1017m-3

H=1016m-3

H=1015m-3

H=1014m-3

H<1013m-3

K/

m

t/s

0 1x107 2x107 3x1070

5

10

15H>1017m-3

H=1016m-3

H=1015m-3

H=1014m-3

H<1013m-3

/m-1

t/s

Material parameters (II)

Starting values

1110 PaE 0.3 0.7272f

30 10eqy

31 10 mh 3

2 10 mh

5 3 -11 10 m mol 5 3 -1

2 0.9 10 m mol 5 3 -13 0.8 10 m mol

01 10 gR T

-18000JmolgR T 02 20 gR T 0

3 30 gR T 16 2 -1

1 5 10 m sD 15 2 -1

2 10 m sD 14 2 -13 5 10 m sD

102.5 10 ma 13 20 -310 ,10 mH

1 ,0 :z h 0 0eqy y 1 0.5y 2 0.3y 3 00.2 eqy y

20, :z h 0 0eqy y 1 0.6y 3 00.3 eqy y 2 0.1y

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.90

0.95

1.00

1.05

1.10

H=1019m-3

t=106s

t=105s

t=0s, t>107s

t=3x106st=3x105s

t=3x104sy 0/

y 0eq

z/m

-0.0010 -0.0005 0.0000 0.0005 0.00100.8

0.9

1.0

1.1

1.2

H=1014m-3

t=3x105s

t=3x106s t=3x104s

t=105s

t=106st=107s

t=0s

t=3x107sy 0/

y 0eq

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.50

0.52

0.54

0.56

0.58

0.60

t=0s

H=1014m-3

t=106s

t=107s

t=3x106s

t=3x105s

t=3x107s

y 1

z/m

-0.0010 -0.0005 0.0000 0.0005 0.00100.45

0.50

0.55

0.60

0.65

H=1019m-3

t=3x104st=3x105s

t=3x106s

t=3x107s

t=0s

t=105s t=106st=107sy 1

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.10

0.15

0.20

0.25

0.30

H=1019m-3

t=107s

t=106st=105s

t=3x105st=3x104s

t=0s

t=3x106s

t=3x107s

y 2

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.10

0.15

0.20

0.25

0.30

H=1014m-3

t=3x104st=3x105s

t=3x106s

t=0s

t=105s

t=106s

t=3x107s

t=107sy 2

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.20

0.22

0.24

0.26

0.28

0.30

H=1014m-3t=0s

t=105s

t=107s

t=106s

t=3x105s

t=3x104s

t=3x107s

t=3x106s

y 3

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.20

0.22

0.24

0.26

0.28

0.30

H=1019m-3

t=105s

t=106s

t=107s

t=0st=3x104s

t=3x105st=3x106s

t=3x107s

y 3

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

-0.04

-0.02

0.00

0.02

0.04H=1019m-3

t=3x104s

t=3x105s

t=105st=0s

t=106s

t=3x106s

t=3x107s

t=107s

r

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010-150

-100

-50

0

50

100

150

H=1014m-3

t=3x104s t=3x105s

t=3x106s

t=105s

t=3x107s

t=0s

t=107s

t=106s r/M

pa

z/m

-0.0010 -0.0005 0.0000 0.0005 0.00100.8

0.9

1.0

1.1

1.2

H=1019m-3

H=1018m-3

H=1017m-3

H=1016m-3

H=1015m-3

H=1020m-3

H<1014m-3

t=106sy 0/

y 0eq

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.48

0.50

0.52

0.54

0.56

0.58

0.60

0.62

t=106s

H=1016m-3

H=1017m-3

H=1018m-3

H>1019m-3

y 1

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.10

0.15

0.20

0.25

0.30

t=106s

H=1018m-3

H>1019m-3

H<1017m-3y 2

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

0.20

0.22

0.24

0.26

0.28

0.30

H=1020m-3, t=105s

H=1018m-3

H<1017m-3

H>1019m-3

t=106s

y 3

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010-150

-100

-50

0

50

100

150

t=106s

H=1020m-3

H=1019m-3

H=1017m-3

H=1018m-3

H=1016m-3

H=1015m-3

H<1014m-3

r/Mpa

z/m

-0.0010 -0.0005 0.0000 0.0005 0.0010

-0.04

-0.02

0.00

0.02

0.04

t=106s

H<1015m-3

H=1016m-3

H=1017m-3

H=1019m-3

H=1018m-3

H=1020m-3

r

z/m

0 1x107 2x107 3x107

0

5

10

15

20

H>1019m-3

H=1018m-3

H=1017m-3

H=1016m-3

H<1014m-3

H=1015m-3

K/

m

t/s

0 1x107 2x107 3x107

-25

-20

-15

-10

-5

0H<1014m-3

H>1018m-3

H=1017m-3

H=1016m-3

H=1015m-3

/m-1

t/s

5. Summary/conclusions- Generalized Manning theory for diffusion of substitutional

components accounting for sources and sinks for vacancies and interaction with developing internal stress field are presented.

- The theory is demonstrated on simulation of evolution of a diffusion couple consisting of two assembled sheets; activity of sources and sinks for vacancies and diffusion coefficients are taken as system parameters.

- For nearly the same diffusion coefficients of all components the influence of activity of sources and sinks for vacancies on system evolution kinetics is not significant.

- For significantly different diffusion coefficients the influence of activity of sources and sinks for vacancies is evident. In such a case the measurement of diffusion coefficients MUST be completed by the characterization of the microstructure! If this is NOT done, the measured diffusion coefficients loose their credibility.