M. OrtizUCSD 01/08

Discrete Dislocation Dynamics(D3)

M. Ortiz1,,P. Ariza22, A. Ramasubramanian33

11California Institute of Technology22University of Seville33Princeton University

Symposium on Multiscale Dislocation DynamicsLa Jolla, January 19-20, 2008

M. OrtizUCSD 01/08

Strength – Multiscale modeling

Lattice defects, EoS

Dislocation dynamics

Subgrainstructures

length

time

mmnm µm

µsns

ms

Polycrystals

Validationtests (SCS)

M. OrtizUCSD 01/08

Dislocation dynamics – Limitations• Dislocation energy/length diverges as ~log(R/r0)• Core cut-off radius r0 is extraneous parameter• No natural separation between slip planes• No dislocation core structure accounted for• Evaluation of Peach-Koehler forces ~ O(N2)• Self-interaction of segments entails difficulty• Finite bodies, general geometries, difficult• Isotropic elasticity is often used in calculations• Topological transitions (e.g., junctions, Orowan

loops, annihilation…) are difficult to account for• Dislocation nucleation is difficult to account for• Mobility law needs to be supplied

M. OrtizUCSD 01/08

Discrete Dislocation Dynamics (D3)• Discrete dislocations in discrete crystal lattice

– No logarithmic divergence– Dislocation lines have full core structure– Gamma-surface

• Empirical potentials (EAM, Finis-Sinclair, …)• 3D anisotropic dislocation interactions,

reactions• Topological transitions automatically included• Computationally efficient:

– O(N logN) complexity– Nonlinear equilibrium calculations restricted to small

sets (cores, junctions, …)

• Amenable to analysis (e.g., Γ-convergence)

M. OrtizUCSD 01/08

Discrete calculus on crystal lattices0-cells

1-cells2-cells

3-cell

BCC crystallite

movie

(P. Ariza and M. Ortiz, ARMA, 2005)

M. OrtizUCSD 01/08

BCC lattice complex: 1-cells

M. OrtizUCSD 01/08

BCC lattice complex: 2-cells

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BCC lattice: Lattice complex

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BCC lattice – Differential operators

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The discrete differential complex

(discrete Helmholtz-Hodge decomposition)

M. OrtizUCSD 01/08

Continuum crystal plasticity

Lattice-invariant deformations(no elastic

energy cost!)

M. OrtizUCSD 01/08

Discrete crystal plasticity

force constants

lattice-invariantshears

dd mm

bbee11

integer-valued!

M. OrtizUCSD 01/08

Discrete dislocations

Elementarydislocation“loopons”and Burgers circuits

M. OrtizUCSD 01/08

Discrete crystal plasticity – Stored energy

Interaction energy

betweenelementarydislocation

loops

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γ-energy – Comparison to first principles

Shear stress-strain curves in FCC Al computedfrom EAM, OFDFT and CASTEP Kohn-Sham DFT(Fago, Hayes, Carter and Ortiz, SIAM Multiscale

Model. Simul., 4 (2005) p. 359)

Eigendefs

M. OrtizUCSD 01/08

Discrete dislocations – Core structure

Displacement-difference maps

Ab initio Mo screw easy core (S. Ismail-Beigi and T. Arias,

PRL, 84:1499, 2000)

Discrete Mo screw core(Ramasubramanian,Ariza and Ortiz’ 05)

b

Bcc screwdisocation

M. OrtizUCSD 01/08

Discrete dislocations – Core energy

Energy of Mo screw core(Xu & Moriarty, Phys. Rev.

B 54, 1996)

Energy of Ta screw core(Yang et al. Phil.Mag.

A 81, 2001)

Energy per unit length of screw dislocationin concentric cylinder of radius R

Mo Ta

M. OrtizUCSD 01/08

Discrete dislocation dynamics – D3

mobility law

storedenergy

applied forcing

M. OrtizUCSD 01/08

D3 – Solution procedure

Yes

No

M. OrtizUCSD 01/08

D3 – Solution procedure

• Unconstrained problem local in Fourier space

• FFT algorithm FFTW (Johnson & Frigo)

• Force update: tabulate stress field of spin

• Easy decomposition over processors

• Minimal inter-processor communication

• Nonlinear equilibration restricted to dislocation cores and junctions: small set!

Point of dilatation

unbalanced1-cells

unconstrainedsolution

M. OrtizUCSD 01/08

D3 – Dilatation point – BCC vanadium

1 million atomsin unit cell

27 million atomsin unit cell

64 million atomsin unit cell

Periodic array of dilatation points in vanadium

(Ramasubramanian, Ariza and Ortiz ’05)

M. OrtizUCSD 01/08

D3 – Dilatation point – BCC vanadium

xD6Slip system D6:

plane (1 1 0)direction [1 1 1]

Slip system C3:plane (1 0 1)direction [1 1 1]

xA6

M. OrtizUCSD 01/08

Crystal plasticity – Hardening

Uniaxialtension test

Copper tensile test(Franciosi and Zaoui ’82)

Irreversible accommodation of shear by crystallographic slip

M. OrtizUCSD 01/08

100 axis

110 axis

Uniaxial tension test - Vanadium

M. OrtizUCSD 01/08

100 axis

110 axis

Uniaxial tension test - Vanadium

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Molybdenum

Systems C5 & A6

Uniaxial tension test - Vanadium

Junction formation

initial deformed

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Stored energy - Dilute limit

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Stored energy – Continuum limit

M. OrtizUCSD 01/08

Stored energy – Dilute limit

pinningpoints

M. OrtizUCSD 01/08

D3 – Outlook

• D3 combines realism of discrete models:– Discrete dislocations on crystal lattices– Dislocation cores, reactions, topological transitions…

with convenience of continuum models– Analytical interaction law– Dimensional reduction (dislocation lines, junctions…)– Analytical tractability (continuum limit, dilute limit…)

• Extensions in progress:– Implementation of dilute limit (discrete line tension)– Application to forest hardening…

• Future extensions:– Fully anharmonic treatment (Gallego R and Ortiz M,

“A Harmonic-Anharmonic Energy Partition Method for Lattice Statics Computations” Model. Simul. Mater. Sci. Engr., 1 (4): 417-436, 1993)

Discrete Dislocation Dynamics�(D3)�Strength – Multiscale modelingDislocation dynamics – LimitationsDiscrete Dislocation Dynamics (D3)Discrete calculus on crystal latticesBCC lattice complex: 1-cellsBCC lattice complex: 2-cellsBCC lattice: Lattice complexBCC lattice – Differential operatorsThe discrete differential complexContinuum crystal plasticityDiscrete crystal plasticityDiscrete dislocationsDiscrete crystal plasticity – Stored energyγ-energy – Comparison to first principles Discrete dislocations – Core structureDiscrete dislocations – Core energyDiscrete dislocation dynamics – D3 D3 – Solution procedureD3 – Solution procedureD3 – Dilatation point – BCC vanadiumD3 – Dilatation point – BCC vanadiumCrystal plasticity – HardeningUniaxial tension test - VanadiumUniaxial tension test - VanadiumUniaxial tension test - VanadiumStored energy - Dilute limitStored energy – Continuum limitStored energy – Dilute limitD3 – Outlook