Non-Equilibrium, Finite Temperature Quasicontinuum Analysis of

Plastic Deformations in Metals

Kevin G. Wang, Mauricio Ponga, and Michael Ortiz California Institute of Technology

USNCCM 12 Raleigh, NC, July 24th, 2013

MOTIVATION

Non-equilibrium thermomechanics in extreme environments

- challenge in modeling and simulation: vastly disparate scales

• atomic level rate-limiting processes: thermal vibrations, lattice defects,

atom hopping, …

• macroscopic processes of interest: ductile fracture, grain boundary

embrittlement, fatigue, irradiation damage, …

hypervelocity perforation pipeline explosion

multiscale models: bridging the scales

OVERVIEW

Non-equilibrium, finite-temperature quasicontinuum (“Hot QC”)

- provides a time-averaged and coarse-grained description of non

equilibrium thermodynamic systems

- can be coupled with (discrete) kinetic laws to solve

thermo/chemo/mechanical problems

- capable of non-equilibrium processes, large samples, at atomic scale

iT QCiT

time-averaged position and momentum, atomic temperature, atomic frequency

references: Y. Kulkarni et al. (2008), M.P. Ariza et al. (2012)

UPSCALING IN TIME

Non-equilibrium statistical thermodynamics

- assumption: separation of time scales

• fine scale: thermal vibration (~fs) “modeled”

• coarse scale: global relaxation (>ps) “solved for”

probability: 𝒒 , 𝒑 ~ 𝑝 𝒒 , *𝒑+

- maximum entropy principle: the least biased distribution maximizes

system entropy, subject to all imposed constraints

max𝑝

𝑆 𝑝 = −𝑘𝐵⟨log 𝑝⟩ (information entropy)

𝑓 = 𝑓 𝒒 , 𝒑 𝑝 𝒒 , 𝒑 𝑑𝒒𝑑𝒑Γ

with

subject to: 1 = 1

𝐻 = 𝐸

= 𝑕𝑖( 𝒒 , 𝒑 )𝑁

𝑖<1 Hamiltonian: 𝐻 = 𝐻 𝒒 , 𝒑

𝑕𝑖 = 𝑒𝑖 𝑖 = 1, … , 𝑁 (atomic internal energy)

- solve the constrained optimization problem using Lagrange multipliers:

𝐿 𝑝, 𝛼, 𝛽 = 𝑆 𝑝 − 𝑘𝐵𝛼 1 − 1 − 𝑘𝐵 𝛽 𝑇( 𝑕 − 𝑒 ) (free entropy)

- enforcing stationarity with respect to 𝑝 yields

NON-EQUILIBRIUM THERMODYNAMICS

Maximum entropy principle

𝑝 𝒒, 𝒑 =1

Ξe; 𝛽 𝑇 ℎ( 𝒒 , 𝒑 ) with Ξ( 𝛽 ) =

1

h3NN! 𝑒; 𝛽 𝑇 ℎ 𝒒 , 𝒑 𝑑𝒒𝑑𝒑𝑅3𝑁×𝑅3𝑁

- thermodynamic potentials

𝑇𝑖 =1

𝑘𝐵𝛽𝑖

𝑆 = −𝑘𝐵 log 𝑝 = 𝑘𝐵 log Ξ + 𝛽 𝑇 𝑕 = 𝑘𝐵(log Ξ + 𝛽 𝑇 𝑒 )

Φ = sup𝑝

𝐿 𝑝, 𝛽 = 𝑘𝐵 log Ξ (free entropy)

𝛽𝑖 =1

𝑘𝐵

𝜕𝑆

𝜕𝑒𝑖

atomic temperature:

(entropy)

- 𝑝, 𝑆, and Φ all depend on the explicit form of 𝑕𝑖 = 𝑕𝑖( 𝒑 , 𝒒 ). In general, the closed-form formulation of thermodynamic potentials is intractable

𝑕𝑖 = −

𝒑 − 𝒑𝟎2

2𝑚+ 𝑉𝑖(*𝒒+) (e.g. )

NON-EQUILIBRIUM THERMODYNAMICS

Meanfield approximation (Yeomans 1992)

- given a class of trial Hamiltonian parameterized by 𝝅 : 𝑕0𝑖( 𝒑 , 𝒒 ; 𝝅 ), the trial probability density function can be defined as

- theorem (bounding principle for the free entropy):

- for fixed Lagrange multipliers 𝛽 , the optimal probability of the trial

space *𝑝0∗+ is obtained by maximizing the constrained entropy with

respect to 𝝅 , i.e.

𝑝0∗ = argmax

*𝝅+𝐿, 𝑝0 𝜋 , 𝛽 -

Φ *𝛽+ ≥ maxℎ0

Φ( 𝑕0 , 𝛽 )

the maximum free entropy of the trial space provides a lower bound

of the actual free entropy

𝑝0 𝒒, 𝒑 =1

Ξ0e; 𝛽 𝑇 ℎ0( 𝒒 , 𝒑 ;*𝝅+)

𝑝0∗ converges to 𝑝∗ as 𝑕0 → 𝑕

NON-EQUILIBRIUM THERMODYNAMICS

Meanfield model: an example

- in practice there is always a trade-off between accuracy and

computability

- we define the trial Hamiltionian as

• meanfield parameters: 𝒒 𝑖 (mean atomic position), 𝒑 𝒊 (mean atomic

momentum), 𝜔𝑖 (mean atomic frequency)

𝑝0∗ = argmax

𝒑 𝑖 , 𝒒 𝒊 , *𝜔𝑖+𝐿,𝑝0, 𝛽 -

Gaussian distribution

“uncoupled harmonic oscillators”

𝑕0𝑖 𝒒 , 𝒑 ; 𝒒 , 𝒑 , 𝜔 =1

2𝑚𝒑𝑖 − 𝒑 𝑖

2 +𝑚𝜔𝑖

2

2𝒒𝑖 − 𝒒 𝑖

2

*𝝅+

𝑝0(*𝒒+, *𝒑+) =1

Ξ𝑒𝑥𝑝 − 𝛽𝑖

1

2𝑚𝒑𝑖 − 𝒑 𝑖

2 +𝑚𝜔𝑖

2

2𝒒𝑖 − 𝒒 𝑖

2

𝑁

𝑖<1

𝑕𝑖 = −𝒑 − 𝒑𝟎

2

2𝑚+ 𝑉𝑖(*𝒒+) explicit formulations of 𝐸, 𝑆, Φ, 𝐹, 𝑒𝑡𝑐.

- motivation

COARSE-GRAINING IN SPACE

Quasicontinuum reduction

crack propagation in Al 6061 (courtesy of K. Ravi-Chandar, UT Austin)

• plastic core surrounded by large elastic zone

• multiple plastic cores interact through elastic

region

adaptive mesh refinement

- key ideas

- reference:

atom

represent -ative atom

summation cluster

• finite element approximation

• cluster summation rule

• adaptive mesh refinement

• qcmethod.org

THERMO-MECHANICAL COUPLING

Governing equations

- local equilibrium conditions

- atomic heat conduction equation

Staggered solution procedure from 𝑇𝑛 to 𝑇𝑛:1

𝑆 𝑖 −

𝜅

𝑇𝑖∆𝑙𝑜𝑔

𝑇𝑖

𝑇0= 0

- dynamic equation of motion (for dynamic analysis)

𝑚𝒒 𝑖 + 𝐹𝑖𝑖𝑛𝑡 𝒒 i, 𝜔𝑖 , 𝑇𝑖 = 𝐹𝑒𝑥𝑡(𝒒 𝑖), where 𝐹𝑖

𝑖𝑛𝑡 𝒒 i, 𝜔𝑖 , 𝑇𝑖 =𝜕Φ

𝜕𝒒 𝑖

𝒕 = 𝑻𝒏

given:

𝒒 𝑛, 𝑻𝑛, 𝝎𝑛

solve heat

conduction equation

𝒒 𝑛

𝑻𝑛:1

𝝎𝑛

solve

equation of motion

𝒒 𝑛:1

𝑻𝑛:1

𝝎𝑛

enforce

equilibrium

conditions

𝒕 = 𝑻𝒏:𝟏

𝒒 𝑛:1,

𝑻𝑛:1, 𝝎𝑛:1

𝜕Φ

𝜕𝜔𝑖= 0;

𝜕Φ

𝜕𝒒 𝑖= 0;

more details on thermo-mechanical coupling

Thermodynamics and Quasicontinuum

Models of Equilibrium and Non-

Equilibrium Thermomechanics of Alloys

Speaker: Ignacio Romero

July 25, Friday, 10:00AM

MS 8.2

MODAL VALIDATION

Specific heat at constant volume (𝐶𝑣)

Max-ent (EAM)

Classical (Dulong-

Petit Law): Cv = 3kB

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 200 400 600 800 1000

temperature (Kelvin)

ch

an

ge i

n i

nte

rnal

en

erg

y

(eV

)

- minimize free entropy (Φ) with respect to atomic

frequency (𝜔𝑖)

- validates the thermodynamic models

- also validates the empirical interatomic potentials

𝐶𝑣 =1

𝑁

𝜕𝐸

𝜕𝑇 𝑉<𝑐𝑜𝑛𝑠𝑡

≈1

𝑁 𝐸 𝑞 0, 𝑇1, 𝜔1 − 𝐸 𝑞 0, 𝑇0, 𝜔0

𝑇1 − 𝑇0

computational domain for Mg (hcp lattice), NVT

Mg, EAM potential by X-Y Liu et al. (1996)

Cu, EAM-FS potential by Sutton and Chen (1990)

Thermal expansion

- solution approach

for 𝑇 = 100𝐾, 200𝐾, 300𝐾, 𝑒𝑡𝑐;

for 𝑎 = 𝑎1, 𝑎2, 𝑎3, 𝑒𝑡𝑐 (pre-selected sample points)

find 𝛷𝑎 = 𝛷 𝑎, 𝑇 = 𝑚𝑖𝑛 𝜔 𝛷(𝑎, 𝜔 , 𝑇)

fit *(𝑎, 𝐹𝑎)+ by a polynomial function 𝑓(𝑎)

find 𝑎𝑇 = 𝑎𝑟𝑔𝑚𝑖𝑛 𝑓(𝑎)

approximate 𝛼𝑉(𝑇) by finite difference

MODAL VALIDATION

- minimize free entropy (Φ) with respect to atomic frequency (𝜔𝑖) and

mean atomic position (𝒒 𝑖)

𝒒 = argmin𝒒 ,*𝜔+

Φ( 𝒒 , 𝜔 , 𝑇)

𝛼𝑉 𝑇 =1

𝑉0

𝜕𝑉

𝜕𝑇≈

1

𝑉0

𝑉1 − 𝑉0

𝑇1 − 𝑇0

(volumetric thermal expansion coefficient)

Thermal expansion coefficient (volumetric): Mg single crystal

THERMAL EXPANSION

temperature

(K) simulation

(× 10;6) experimental

(× 10;6) data source

rel. error

(%)

90 - 190 64.5 57.9 Goens & Schmid (1936) 10.23

270 - 430 81.7 81.9 Raynor & Hume-Rothery (1939) 0.24

290 - 470 81.3 82.2 Goens & Schmid (1936) 1.11

320 - 520 82.4 84.7 Hanawalt & Frevel (1938) 2.79

430 - 580 88.2 82.8 Raynor & Hume-Rothery (1939) 6.12

580 - 725 81.2 93 Raynor & Hume-Rothery (1939) 14.53

APPLICATION

Nanovoid growth under tension at finite temperature

- problem setup

• objective: study the evolution of nanovoid in a single FCC crystal of Cu

• representative volume: 26nm x 26nm x 26nm (~ 106 atoms)

• initial void diameter: 2nm

• initial atomic zone: 5nm x 5nm x 5nm (12a0 x 12a0 x 12a0)

• interatomic potential: EAM-Mishin (Y. Mishin et al. 2001)

• uniaxial and triaxial loading with prescribed boundary displacement - temperature (K): 150, 300, 450, 600

- strain rate (1/s): 105, 107, 108, 109, 1010

cut-view of initial FE mesh QC approximation

atomic zone

void

26nm

5nm

NANOVOID GROWTH

Uniaxial loading, T = 300K, 𝜖 = 10101/𝑠

* centro-symmetric deviation:

𝐶𝑆𝐷 =

𝑞𝑖 + 𝑞𝑖:

𝑁2

2

𝑎0

𝑁2

𝑖<1

𝜖 = 6.6% (emission of dislocation)

𝜖 = 6.8% (leading

Shockley partials)

𝜖 = 7.0% (trailing

Shockley partials)

1/6,1 1 2-

dislocation core identified by CSD*

1/6,11 2-

1/6,112-

1/6,1 12- *111+

atomic temperature

before dislocation emission after dislocation emission

NANOVOID GROWTH

Triaxial loading, T = 300K, 𝜖 = 105, 108,109 , 1010 1/𝑠

dislocation core

colored by CSD

(𝜖 = 10101/𝑠) engineering strain

vir

ial str

ess (

GP

a)

0

2

4

6

8

10

12

14

0 2 4 6 8 10

𝜖 = 1010, MD

- virial stress (Allen and Tildesley, 1989)

0

2

4

6

8

10

12

14

0 2 4 6 8 10

𝜖 = 1010, MD

𝜖 = 1010, HotQC, dynamic

0

2

4

6

8

10

12

14

0 2 4 6 8 10

𝜖 = 1010, MD

𝜖 = 1010, HotQC, dynamic

𝜖 = 109, HotQC, dynamic

𝜖 = 108, HotQC, dynamic

𝜖 = 105, HotQC, quasi-static

stress-strain response

𝜎𝑖𝑗 =1

𝑉Ω −𝑚𝑘 𝑞 𝑖

𝑘 − 𝑞 𝑖𝑘 𝑞 𝑗

𝑘 − 𝑞 𝑗𝑘 +

1

2𝑞𝑖

𝑙 − 𝑞𝑖𝑘 𝑓𝑗

𝑘𝑙

𝑘∈Ω

DEFORMATION TWINNING

Mobility of twin boundary in Magnesium

- objective of study: twin boundary mobility

• intermediate temperatures and

strain rates

• velocity vs. shear stress, strain rate,

temperature, etc.

- challenge: multiple scales in time/space

• MD is limited to extreme strain rate

(>108 1/s) and small sample size

- why Mg?

• low density (“lightest useful metal”)

• high strength-to-weight ratio

c

a3 a1

a2

(101 2)

,1 011-

migration of *101 2+ twin boundary

(Serra, A. et al, 2007)

- mechanical applications

• automobile engine blocks

• helmet and body armor

• etc.

- future goals: stronger; more ductile; corrosion resist; better formability

*101 2+ TWIN BOUNDARY MIGRATION

Problem setup

- quasi-static analysis with heat conduction (for Mg, 𝜅 = 156 𝑊𝑚;1𝐾;1)

- twin boundary introduced in initial sample

- prescribed displacement at each loading step: ∆𝑞 = 0.025𝑎0

- strain rate 𝜖 = 1.0 × 1081/𝑠, 1.0 × 106 1/𝑠

- using the EAM potential of X-Y Liu et al. (1996)

y

z basal plane

twin boundary

unit cell

2D projection onto the shear plane

twin crystal

matrix crystal

twin boundary

basal plane

periodic

periodic

fixed

prescribed displacement

x y

z

,1 011- ,12 10-

,𝟏 𝟎𝟏𝟏-

SIMULATION RESULT

Simulation results (𝜖 = 108 1/𝑠)

2D projection on shear plane

prescribed displacement

-0.9160

-0.9158

-0.9156

-0.9154

-0.9152

-0.9150

-0.9148

-0.9146

-0.9144

-0.9142

0 10 20 30 40 50 60 70

evolution of internal energy

loading step no.

inte

rnal energ

y (

E)

per

ato

m (

eV

)

3D view

-0.9160

-0.9158

-0.9156

-0.9154

-0.9152

-0.9150

-0.9148

-0.9146

-0.9144

-0.9142

0 10 20 30 40 50 60 70

twin boundary migration

ONGOING WORK

Ongoing work

- verification and validation strain rate (1/s)

dynamic HotQC

compare with MD

quasi-static HotQC

compare with

dynamic HotQC and MD

quasi-static HotQC

compare with

experiments

1010 108 107 105 (or less)

- extension to Mg alloys with mass transport

Thermodynamics and Quasicontinuum

Models of Equilibrium and Non-

Equilibrium Thermomechanics of Alloys

Speaker: Ignacio Romero

July 25, Friday, 10:00AM

MS 8.2