Materials Process Design and Control Laboratory MOLECULAR DYNAMICS APPROACH FOR INVESTIGATION OF GRAIN BOUNDARY RESPONSE WITH APPLICATIONS TO CONTINUUM.

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


MOLECULAR DYNAMICS APPROACH FOR INVESTIGATION OF GRAIN BOUNDARY RESPONSE WITH APPLICATIONS TO CONTINUUM SIMULATION OF FAILURE IN NANO-CRYSTALLINE MATERIALS

Baskar Ganapathysubramanian, Veeraraghavan Sundararaghavan and Nicholas Zabaras

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801Email: [email protected]

URL: http://mpdc.mae.cornell.edu/




ACKNOWLEDGEMENTSACKNOWLEDGEMENTS

FUNDING SOURCES: Air Force Research Laboratory

Air Force Office of Scientific Research National Science Foundation (NSF)

ALCOA Army Research Office

COMPUTING SUPPORT: Cornell Theory Center (CTC)




OVERVIEW

– Motivation– Problem definition– Molecular dynamics

simulation– Cohesive model: ISV

method– Conclusions– Scope for further

work Far field normal displacement (A)

Nor

mal

stre

ss(G

Pa)

0 1 2 3 4 5-2

0

2

4

6

8




MOTIVATION

– Design polycrystalline materials with tailored properties

– Accurate design of processes to obtain tailored properties in crystals

– Simulation of response/failure of crystalline materials

– Simulation of grain boundary failure

– Enhanced models of grain boundary (GB) separation and sliding mechanisms based on quasi-static MD simulations in a bicrystal

Meshing

GB PropertiesMolecular

dynamics

Control loads

FEM models

Intra-granular: Crystal plasticity models

Grain boundary (Cohesive zone models)




Cohesive zone models are used to describe the grain boundary response and allow for natural initiation of intergranular cracks.

The tool can be used for metallic polycrystal systems including nanostructured materials.

Reducing the characteristic length scale of the grains, closes the gap between meso-scale simulations and atomistic simulations. Allows calibration of constitutive models using MD simulations

MOTIVATION

Kumar, Ritchie, Gao (llnl/ucb)

Sethna, Cornell




Modelling grain boundary response:

1) Simple isotropic hardening rule (Anand-Staroselsky (1998), Fu et al (2004)): no fracture with slip inside grains

2) Use of cohesive zones: Espinosa, Ingraffea, Anand, Mcdowell, Ortiz ect

3) Zavattieri & Espinosa (2001): simple cohesive law with rate dependence accounting for different Tmax, weibull distribution to account for uncertainty in the form of misorientations, no plastic effects

4) McDowell (2004): simple cohesive law with rate independent form, No plastic effects

5) Anand(2004): Cohesive law based on state variable with hardening of grain boundary, reversible law with elasto plastic decomposition of displacement jump

6) Bower(2004) : Arrhenius law to calculate strain rates based on activation energy of grain boundary

LITERATURE




GRAIN BOUNDARY MODEL

1

1

1( )

2

( ) ( )

( ) ( )

a a a

n

a aa

n

a aa

a a a

x x x

x s x N s

s x N s

x x x

Grain boundary

x+

x-

a

a

a

1. Integration is carried out over the centerline of the element.

2. The displacement jumps are interpolated using shape functions of the centerline element

4 noded cohesive element

e p

Finite Element Method




• EAM potentials for quantitative studies

• Choosing geometries

• Goal: Find general law for strength of grain boundaries, depending on geometry, temperature and strain rate

• Limitations: computionally intensive, large domain/long time, need a compromise

ATOMISTIC MODELING OF GRAIN BOUNDARY BEHAVIOR

Pull grain apart with constrained atoms – measure stress at each step

Motivation: feed cohesive laws to FEM simulations

Bicrystal arrangement





METHODOLOGY

Initialize atoms

Set boundary conditions

Randomize velocities, set temperature

Thermalize for 5 ps (NVE with velocity rescaling)

Set to NVT canonical

Pull boundary atoms at prescribed rate

Set force on Boundary atoms to zero

Calculate average stress of mobile atom

FCC Cu Bi-crystal with a 45º misorientation

20262 atoms

Strain rate of 1 A/ps

EAM potential




Far field normal displacement (A)

Ave

rage

norm

alst

ress

(Gpa

)

0 5 10 15 20

0

3

6

9

12


L x B x H: 40 x 20 x 40 atomic planes

Normal stress-displacement response.

Dominant peak with associated peak stress.

Peak stress at 5.8 Å

Compared with Spearot et. al (Mech . Mat. 36 (2004))

Smaller domain size

Peak stress 5.6 Å




COHESIVE ELEMENT FORMULATION -1

2( ) ( )

1

2( ) ( )

1

(1)

(2)

Yield surface: 0,

0

p i i

i

e

e p

ij j

i

N

N

m

t K

s h

t s

t s

GB Constitutive law

p

e

(Anand’s Model)

0 5 10 15-2

0

2

4

6

8

10

12

14

16

Far field displacement (A)

Nor

mal

Str

ess

(GP

a)

13 parameter l2 norm fit

MD results

Some continuum scale parameters:

Interface friction = 0.004

Interface normal stiffness = 22.6076 GPa/nm

Exponent of 0.5 for initial hardening




INTRAGRANULAR SLIP MODEL

Crystallographic slip and re-orientation of crystals are assumed to be the primary

mechanisms of plastic deformation

Evolution of various material configurations for a single crystal as needed in the integration of the

constitutive problem.

Evolution of plastic deformation gradient

The elastic deformation gradient is given by

Incorporates thermal effects on shearing rates and slip

system hardening(Ashby; Kocks; Anand)




X

Y

0.99999 1 1.00001 1.00002

0.499997

0.499998

0.499999

0.5

0.500001

0.500002

0.500003

0.500004

Equivalent Stress (MPa)9.32E-038.71E-038.10E-037.49E-036.89E-036.28E-035.67E-035.06E-034.45E-033.84E-033.23E-032.63E-032.02E-031.41E-038.01E-04

Pure grain boundary sliding separation

XY

0.99999 1 1.00001 1.00002

0.499997

0.499998

0.499999

0.5

0.500001

0.500002

0.500003

0.500004

Equivalent Stress (MPa)9.32E-038.71E-038.10E-037.49E-036.89E-036.28E-035.67E-035.06E-034.45E-033.84E-033.23E-032.63E-032.02E-031.41E-038.01E-04

SIMULATION OF GB FAILURE IN BICRYSTALSRESPONSE IN SHEAR(PURE SLIDING)




XY

0.5 0.75 1 1.250.499999

0.5

0.5

0.500001

0.500001

0.500002

0.500002 Equivalent Stress (MPa)0.01064660.009936850.009227080.00851730.007807530.007097750.006387980.00567820.004968430.004258650.003548880.00283910.002129330.001419550.000709775

X

Y

0.5 0.75 1 1.250.499999

0.5

0.5

0.500001

0.500001

0.500002


X

Y

0.5 0.75 1 1.250.499999

0.5

0.5

0.500001

0.500001

0.500002


SIMULATION OF GB FAILURE IN BICRYSTALSRESPONSE IN TENSION(PURE OPENING)




X

Y

0 2.5E-06 5E-06 7.5E-06

4.9999E-06

5.0000E-06

5.0001E-06

5.0002E-06

X

Y

0 2.5E-06 5E-06 7.5E-06

4.9999E-06

5.0000E-06

5.0001E-06

5.0002E-06

X

Y

0 2.5E-06 5E-06 7.5E-06

4.9999E-06

5.0000E-06

5.0001E-06

5.0002E-06

X

Y

0 2.5E-06 5E-06 7.5E-06

4.9999E-06

5.0000E-06

5.0001E-06

5.0002E-06

SIMULATION OF GB FAILURE IN BICRYSTALSRESPONSE IN TENSION(MIXED MODE)




SIMULATION OF GB FAILURE IN BICRYSTALSRESPONSE IN TENSION

Response of crystal is still in elastic regime

GB produces a plastic response at low strains

Length of simulation tailored to MD simulation

Boundary conditions:

Pulled along y

Compressed in x




TEMPERATURE COMPENSATED STRAIN RATE

Equivalence of the effects of change in strain rates and in temperature upon the stress strain relation in metals

Intended for investigation of behavior of steels at very high deformation rates. Obtained by tests at moderate strain rates at low temperatures

Use the equivalence relation to extract behavior at low strain rates by simulating high- strain rate deformation at higher temperatures.

Isothermal deformation

Zener and Hollomon: J. Applied Physics (1943)

exp( / )Z Q RT

Z is the Zener-Hollomon parameter

έ is the strain rate

R is the universal gas constant

Q is the activation energy

Q shown to be the equal to the self-diffusion for pure metals ( Kuper et. al Physical Review 96 (1954) )




TEMPERATURE COMPENSATED STRAIN RATE

1

2 2 1

exp( / )

1 1ln ( )

Z Q RT

Q

R T T

Q for copper is 213 KJ/mole

R: 8.3144 J/mole/K

Look at realistic pulling rate 0.1 mm/s

Ratio of strain rates 10-6

0.1 Å/ps at 300 K equivalent to 10 Å/ps at 317 K

0.1 mm/s at 300 K equivalent to 100 m/s at 381 K


Nor

mal

stre

ss(G

Pa)

3 6 9

-2

0

2

4

6

8

10

12

14

16

18

20


Nor

mal

stre

ss(G

Pa)

3 6 9

-2

0

2

4

6

8

10

12

14

16

18

20

Temp compensated pulling rate of 0.1 mm/s

Pulling rate of 100 m/s




LAWS FOR DIFFERENT LOADING REGIMES

Strain rate dependence

Temperature dependence

Deformation mode dependence?

Orientation dependence

Size effects

Triple points and other grain junctions?

Material dependence

PARAMETRIC STUDIES Effect of temperature variation on peak stress and magnitude of deformation at peak stress

Effect of strain rate variation on peak stress

Vary mis-orientation angle. How does deformation and failure proceed?





Nor

mal

stre

ss(G

Pa)

3 6 9

-2

0

2

4

6

8

10

12

14

16

18

20

Far field normal displacement (A)N

orm

alst

ress

(GP

a)

3 6 9

-2

0

2

4

6

8

10

12

14

16

18

20


Nor

mal

stre

ss(G

Pa)

3 6 9

-2

0

2

4

6

8

10

12

14

16

18

20

Copper bi-crystal

Tension test at different temperatures

T = 300 K, 400 K, 500 K

Slope constant with temperature

Displacement associated with peak stress decreases

Peak stress decreases

TEMPERATURE EFFECTS





MISORIENTATION EFFECTS

Misorientation has effect on peak stress

Displacement associated with peak stress not sensitive

Slope remains constant

At higher temperatures thermalization leads to diffusion of the grain boundary.



Nor

mal

stre

ss(G

Pa)

3 6 9

-2

0

2

4

6

8

10

12

14

16

18

20


Nor

mal

stre

ss(G

Pa)

3 6 9

-2

0

2

4

6

8

10

12

14

16

18

20




Aluminium bi-crystal

Tension test at different strain rates

v = 10 A/ps, 1 A/ps, 0.1 A/ps

Displacement associated with peak stress increases

Peak stress increases with strain rate

STRAIN RATE EFFECTS


Nor

mal

stre

ss(G

Pa)

3 6 9

-2

0

2

4

6


Nor

mal

stre

ss(G

Pa)

3 6 9

-2

0

2

4

6


Nor

mal

stre

ss(G

Pa)

3 6

-2

0

2

4

6






SIZE EFFECTS

The simulation domain size affects the magnitude of the peak stress and the displacement associated with it.

Can we extract the asymptotic limit?

Finite size scaling

A(L) = Ao + c/Ln


Nor

mal

stre

ss(G

Pa)

0 3 60

2

4

6

8

10

12

14

16

18

20

22


Nor

mal

stre

ss(G

Pa)

0 3 60

2

4

6

8

10

12

14

16

18

20

22

Yield stress decreases with increasing size

Ductility increases with size




0

( )

0 0

exp( ) 0

activation energy = f( , )

e

th t

t th

t

t t

t t

t K

s s s

t t s

t

Gt s

k

G t s

Espinosa (2001): Rate and temperature dependent law:

Bower (2004):

Misorientation dependence:

Weibull distribution (Espinosa 2002) ?

COHESIVE ELEMENT FORMULATION -2

Conrad and Narayan (1999):

exp[ / ]vN Ab G kT




TRIPLE JUNCTION



Nor

mal

stre

ss(G

Pa)

3 6-4

-2

0

2

4

6

8

10

12

14

16

18

20

Trijunction (T: 30, 60)

Peak stress lower, displacement lesser




SEVERAL OPEN ISSUES

Open issues:

Should triple points cohesive zones have special constitutive models (using MD)?

Can the response be generalized considering large complexity since a space of misorientations of 3 different grains need to be explored?

Triple point element quad junction element

Complexity:

Orientation space x Grain junctions x Deformation mode dependence x Temperature dependence x Strain rate dependence x Material type = Large data set that needs to be explored (Statistical learning?)




SCHEMATIC OF A GB DATABASE

Strain rate dependence

Temperature dependence

Orientation dependence

Deformation mode dependence

Triple points and other grain junctions

Material

Multiphase

Continuum cohesive law trained using gradient optimization

State variable evolution, traction separation law

Sethna et al




Extend to complex interfaces

Look at other failure/deformation mechanisms

Simulation of larger length scales

A database of grain boundary properties

Design of processes to tailor properties

SCOPE FOR FUTURE WORK




INFORMATIONINFORMATION

RELEVANT PUBLICATIONSRELEVANT PUBLICATIONS

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801Email: [email protected]

URL: http:/mpdc.mae.cornell.edu/

Prof. Nicholas Zabaras

CONTACT INFORMATIONCONTACT INFORMATION

V. Sundararaghavan and N. Zabaras, "A dynamic material library for the representation of single phase polyhedral microstructures", Acta Materialia, Vol. 52/14, pp. 4111-4119, 2004

S. Ganapathysubramanian and N. Zabaras, "Modeling the thermoelastic-viscoplastic response of polycrystals using a continuum representation over the orientation space", International Journal of Plasticity, Vol. 21/1 pp. 119-144, 2005

Materials Process Design and Control Laboratory MOLECULAR DYNAMICS APPROACH FOR INVESTIGATION OF GRAIN BOUNDARY RESPONSE WITH APPLICATIONS TO CONTINUUM.

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