tutorial ii: constitutive models for crystalline solids

IHPC-IMS Program onAdvances & Mathematical Issues

in Large Scale Simulation(Dec 2002 - Mar 2003 & Oct - Nov 2003)

Tutorial II:Constitutive Models for Crystalline Solids

Alberto M. CuitiñoMechanical and Aerospace Engineering

Rutgers UniversityPiscataway, New [email protected]

Institute of High Performance Computing Institute for Mathematical Sciences, NUS

Singapore 2003 cuitiño@rutgers

• Bill Goddard• Marisol Koslowski• Stephen Kuchnicki• Michael Ortiz• Raul Radovitzky • Laurent Stainier• Alejandro Strachan• Zisu Zhao

Collaborators


Hierarchy of Scales

length

time

mmnm µm

ms

µs

ns

Phase stability, elasticityEnergy barriers, pathsPhase-boundary mobility

Microstructures

Grains

Direct FE simulationPolycrystals

Single crystals

SCS test

Force Field


General Framework

Incremental Field Equations


General Framework

Additive Decomposition of the Free Energy

Multiplicative Decomposition of Deformation Gradient

HARDENINGELASTIC RESPONSE

ELASTIC RESPONSE

with

and

Temperature-dependent elastic constants for Ta

Absolute Temperature

Internal Variables


Multiplicative Decomposition

Initial Configuration F = Fe Fp Deformed Configuration

Intermediate Configuration

ELASTIC EFFECTS Lattice deformation and

rotation

PLASTIC EFFECTS Dislocation Slip

Fp Fe


General FrameworkHARDENING

Applied Resolved Shear Stress

Flow Rule

In rate form where

Microscopic Description

Piola-Kirchoff Stress Tensor

Slip Direction

Slip-plane Normal

Micro/Macro Relations

Current flow Stress

LOCAL CONSTITUTIVE LAW


Dislocation Mobility (double-kink formation and thermally activated motion of

kinks)

FORMATION ENTHAPY FOR DOUBLE KINK

We consider the thermally activated motion of dislocations within an obstacle-free slip plane. Under these

conditions, the motion of the dislocations is driven by an applied resolved shear stress and is hindered by the lattice resistance, which is weak enough that it may be overcome by thermal activation. The lattice resistance is presumed to be well-described by a Peierls energy function.

Energy formation of kink pair. Estimated by atomistic calculations of the order of 1 eV

(Xu and Moriarty, 1998)

Kink proliferation is expected at

Then,

Estimated by atomistic calculations of the order of few Gpa (Xu and Moriarty, 1998)


Dislocation Mobility (double-kink formation and thermally activated motion of

kinks)

From Orowan’s equation and transition state theory,

with

STRAIN-RATE AND TEMPERATURE DEPENDENT EFFECTIVE PEIERLS STRESS

where= dislocation densityb = Burgers vectorl = mean free-path of kinksD = Debye Frequency


Forest Hardening (close-range interactions between primary and forest)

In the forest-dislocation theory of hardening, the motion of dislocations, which are the agents of plastic deformation in crystals, is impeded by secondary -or forest- dislocations crossing the slip plane. As the moving and forest dislocations intersect, they form jogs or junctions of varying strengths which, provided the junction is sufficiently short, may be idealized as point obstacles. Moving dislocations are pinned down by the forest dislocations and require a certain elevation of the applied resolved shear stress in order to bow out and bypass the pinning obstacles. The net effect of this mechanism is macroscopic hardening.

STRENGTH OF OBSTACLE PAIR

IN BCC CRYSTALS

Bow-out mechanism in BCC crystals

Energetic condition for bow-out process

Retaining dominant terms,


NOTE 2: It is interesting to note that the probability density of obstacle-pair strengths for BCC differs markedly from FCC crystals. This difference owes to the different bow-out configurations for the two crystal classes and the comparatively larger values of the Peierls stress.

Forest Hardening (distribution function of obstacle-pair strength )

We assume that the point obstacles are randomly distributed over the slip plane with a mean density nof obstacles per unit area. We also assume that the obstacle pairs spanned by dislocation segments are nearest-neighbors in the obstacle ensemble.

PROBABILITY DENSITY FUNCTION

ASSOCIATED DISTRIBUTION FUNCTION

NOTE 1: The function just derived provides a complete description of the distribution of the obstacle-pair strengths when the point obstacles are of infinite strength and, consequently, impenetrable to the dislocations.


for FCC


Forest Hardening (distribution function of obstacle-pair strength)

We extend the preceding analysis to include point obstacle with finite strength.


STRENGTH OF OBSTACLE FORMED BY DISLOCATIONS OF SYSTEMS AND

Heaviside Function

ASSOCIATED DISTRIBUTION FUNCTION

where Probability that the weakest of two

obstacles forming a pair be of type

with Number of obstacles of type per unit area of the slip plane which

is estimated as


Forest Hardening (percolation motion of dislocations through a random

array)From non-equilibrium statistical concepts, we obtain

KINETIC EQUATION OF EVOLUTION

where

Then, the incremental plastic strain driven by and increment in the resolved shear stress can be expressed by

AVERAGE DISTANCE BETWEEN OBSTACLES

AVERAGE NUMBER OF JUMPS BEFORE DISLOCATION SEGMENTS ATTAIN STABLE POSITIONS


Forest Hardening (hardening relations in rate form)

PARTICULAR CASE: OBSTACLES OF UNIFORM STRENGTH

where,HARDENING MODULUS

CHARACERISTIC STRAIN


Dislocation Intersection(jog formation energy)

After intersection

Before intersection

Reaction coordinate

JOG FORMATION ENERGY

Favorable Junction

Unfavorable Junction

Details of Intersection Process



Slip Plane

Larger population of

SCREW SEGMENTSScrew Segment

Ed

ge

Se

gm

en

tHigher mobility of

EDGE SEGMENTS INITIAL

FINAL

ENERGY FORMATION OF A EDGE SEGMENT INTERSECTING A SCREW SEGMENT

Further assuming

where r is the ratio between edge and screw energies



Example of normalized jog-formation energy for r = 1.77 Computed value of r for Ta from atomistic calculations (Wang et al., 2000)


Dislocation Intersection(obstacle strength)

By equating the energy expended in forming jogs with the potential energy released as a result of the motion of dislocation, we obtain that the forest obstacles become transparent to the motion of primary dislocations when

Obstacle strength at zero temperature

Invoking transition state theory concepts,

OBSTACLE STRENGTH

where


Dislocation Evolution

DISLOCATION PRODUCTION by breeding by double cross-slip

where

Dislocation length emitted by source prior to saturation

Rate of generation of dynamic sources induced by

cross-slip

Energy Barrier for cross slip

Mean-free path between cross-slip events

Length of screw segment effecting cross

slip

Assuming the the mean-free path is inversely proportional to the square root of the dislocation density, we obtain

where

DISLOCATION ANNIHILATION by cross-slip

where



DISLOCATION MULTIPLICATION RATE

or

This rate equation expresses a competition between the dislocation multiplication and annihilationmechanisms. For small slip strains, the multiplication term dominates and the dislocation density grows as a quadratic function of the slip rate. By contrast, for large strains, the rates of multiplication and annihilation balance out and saturation sets in. After saturation is attained, the dislocation density remains essentially unchanged. It should be carefully noted, that the saturation slip strain is a function of temperature and strain rate.



TOTAL DISLOCATION MULTIPLICATION

DISLOCATION PRODUCTION by Frank-Reed sources

PRODUCTION

DISLOCATION PRODUCTION by breeding by double cross-slip = +

Thermally activated process with activation energy Ecross


Trapped Dislocation Annihilation

Escaping Dislocation No Annihilation


Minimum Annihilation

Radius

Annihilation Radius (T,Strain Rate)

Imposed velocity

ANNIHILATION Maximum Annihilation

Radius

INTERACTION FORCE


Comparison with Experiment

ORIENTATION DEPENDENCE

Sensitivity to misalignment


Comparison with Experiment(Theory and Experiment)

TEMPERATURE DEPENDENCE


Comparison with Experiment(Theory and Experiment)

STRAIN-RATE DEPENDENCE


Microscopic PredictionsSlip Strains



Microscopic PredictionsDislocation Densities



Cor

e en

ergy

(eV

/b)

1/2a<111> edge dislocationin (110) plane

1/2a<111> screw dislocation

Volume (1/V0)

540 GPa

MD Data

Application of EoS

Find Jacobian of F

(=det(F))

Use EoS to find

hydrostatic pressure

Evaluate core energies

and elastic moduli

Data from MP group

Goddard, Strachan, Cagin, Wu

Pressure Dependency


[213] Ta single crystal;

Strain speed: 10-3 /s;

Without core energy data

With core energy data

Pressure and Plasticity


Closer examination of core energy effects

Elastic range shows

expected behavior

Plastic behavior converges

with increasing pressure

Pressure in Core Energy + EoS


Material Parameters

MATERIAL PROPERTYFITTED FROM EXPERIMENT

Lkink/b 13

Ekink [eV] 0.70

Uedge/ b2 0.200

Uedge/ Uscrew 1.77

Ecross [eV]0.65

b = 2.86e-10 m ; T_Debye = 340




Mitchell and Spitzig, 1965 EXPERIMENT THEORY



Mitchell and Spitzig, 1965

EXPERIMENT THEORY



Material Parameters

MATERIAL PROPERTY

FITTED FROM EXPERIMENT

COMPUTED BY ATOMISTICS

Lkink/b 13 17

Ekink [eV] 0.70 0.73

Uedge/ b2 0.200 0.216

Uedge/ Uscrew 1.77 1.77

Ecross [eV]0.65 -

ATOMISTICS from WANG, STRACHAN, CAGIN and GODDARD

MATERIAL PROPERTY

FITTED FROM EXPERIMENT

Lkink/b 13

Ekink [eV] 0.70

Uedge/ b2 0.200

Uedge/ Uscrew 1.77

Ecross [eV]0.65


Multiscale Modeling Final Remarks

• Multiscale modeling leads to material parameters which quantify well-defined physical entities

• The material parameters for Ta have been determined independently in two ways:

• Both approaches have yielded ostensibly identical material parameters!

• Same agreement with experiment would have been obtained if the parameters had been determined directly by simulation in the absence of data.

• This provides validation of modeling and simulation paradigm (as a complement to experimental science).

Fitting Atomistic calculations


Another Study Case

ab initio QMEoS of various phasesTorsional barriersVibrational frequencies

Force Fields and MDElastic, dielectric constantsNucleation BarrierDomain wall and interface mobilityPhase transitionsAnisotropic Viscosity

Meso- Macro-scaleNanostructure-properties relationshipsConstitutive Laws

Direct problem

Inverse problem


Features: - Utilizes multiplicative decomposition of the deformation gradient into elastic-piezoelectric and phase transitional parts,

-Accounts for amorphous + several orientation of crystalline phases tracking mass concentration of each phase

-Phase transformation is thermodynamically driven Electric Gibbs free energy for one component,

-Stress and electric displacement are that of the volume average of components ,

-Weak (integral) formulation based on generaized principle of virtual work

peFFF

1 210 EEyεεCεEe iii 2

1

2

1iiG

ii

i

iii

i

V

VSSS

ii

i

iii

i

V

VDDD

i

iiGG

00

0`: 0101001 B B nnB not

dSdVdV ηtηfηP 000

01001 B nB n dSqdV ξξD

Full-Field Coupled Electromechanical


Macroscale simulation: version

Initial conditionNon-polar

Load

Mechanically driven non-polar (T3G) to polar (all-trans) transformation

ALLOWS FOR ARBITRARY SHAPES AND GENERAL

ELECTROMECHANICAL BC IN 2D and 3D

Complex nucleation of polar phase

Undeformed Deformed

(T3G)

(all-trans)


Macroscale simulation: StressBlue (0) = T3G Red (1) = All Trans

(T3G) (all-trans)

Transformed Region

Normal stress along chains



(T3G) (all-trans)

Transformed Region

Normal stress perpendicular to chains



(T3G) (all-trans)

Transformed Region

Shear


Macroscale simulation:Electric Displacement

Blue (0) = T3G Red (1) = All Trans

(T3G) (all-trans)

Transformed Region

Along the Chains



(T3G) (all-trans)

Transformed Region

Normal to the Chains

Macroscale simulation:Electric Displacement


Macroscale simulation: Electric Potential


(T3G) (all-trans)

Transformed Region

tutorial ii: constitutive models for crystalline solids

Documents

double kink

micromechanical unit

following unit processes

energy formation of

double cross

macroscopic forces

lattice resistance

unit processesthe analysis