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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
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/
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
• 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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
ATOMISTIC MODELING OF GRAIN BOUNDARY BEHAVIOR
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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Far field normal displacement (A)
Ave
rage
norm
alst
ress
(Gpa
)
0 5 10 15 20
0
3
6
9
12
ATOMISTIC MODELING OF GRAIN BOUNDARY BEHAVIOR
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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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)
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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)
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
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
0.500002 Equivalent Stress (MPa)0.01064660.009936850.009227080.00851730.007807530.007097750.006387980.00567820.004968430.004258650.003548880.00283910.002129330.001419550.000709775
SIMULATION OF GB FAILURE IN BICRYSTALSRESPONSE IN TENSION(PURE OPENING)
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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)
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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) )
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Far field normal displacement (A)
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)
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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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?
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Far field normal displacement (A)
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
Far field normal displacement (A)
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
LAWS FOR DIFFERENT LOADING REGIMES
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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.
LAWS FOR DIFFERENT LOADING REGIMES
Far field normal displacement (A)
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)
Nor
mal
stre
ss(G
Pa)
3 6 9
-2
0
2
4
6
8
10
12
14
16
18
20
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Far field normal displacement (A)
Nor
mal
stre
ss(G
Pa)
3 6 9
-2
0
2
4
6
Far field normal displacement (A)
Nor
mal
stre
ss(G
Pa)
3 6 9
-2
0
2
4
6
Far field normal displacement (A)
Nor
mal
stre
ss(G
Pa)
3 6
-2
0
2
4
6
LAWS FOR DIFFERENT LOADING REGIMES
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
LAWS FOR DIFFERENT LOADING REGIMES
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
Far field normal displacement (A)
Nor
mal
stre
ss(G
Pa)
0 3 60
2
4
6
8
10
12
14
16
18
20
22
Far field normal displacement (A)
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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
TRIPLE JUNCTION
LAWS FOR DIFFERENT LOADING REGIMES
Far field normal displacement (A)
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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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?)
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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