Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 ·...
Transcript of Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 ·...
![Page 1: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/1.jpg)
Michael OrtizMULTIMAT08
M. OrtizCalifornia Institute of Technology
MULTIMAT closing meetingBonn, Germany, September 12, 2008
Multiscale models of metal plasticity
Part I: Dislocation dynamics to crystal plasticity
![Page 2: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/2.jpg)
Michael OrtizMULTIMAT08
Metal plasticity• Behavior of polycrystalline metals is too
complex to yield to ad hoc modeling:– Polycrystalline texture– Sub-grain microstructures– Dislocation dynamics– Atomistic defect cores– Pressure/temperature/rate dependency– Scaling and size effects…
• Alternative paradigm: Multiscale analysis– Derive rigorous models of effective behavior– Identify underlying microstructural mechanisms – Develop high-fidelity (but fast!) physics models…
![Page 3: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/3.jpg)
Michael OrtizMULTIMAT08
Metal plasticity − Multiscale analysis
Lattice defects, EoS
Dislocation dynamics
Subgrainstructures
length
time
mmnm µm
ms
µsns
Polycrystals
Engineeringapplications
Quantum mechanical or atomistic
Discrete or linear elastic
Continuum
Objective: Derive ansatz-free,physics-based, predictive models of macroscopic behavior
![Page 4: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/4.jpg)
Michael OrtizMULTIMAT08
Metal plasticity − Multiscale analysis
• A well-developed body of theory exists for energy minimization problems:– Relaxation– Γ-convergence– Optimal scaling
• This framework suffices for many applications: – Martensitic phase transitions– Micromagnetics– Complex polymers
• However: Plasticity involves dissipation, hysteresis, irreversible behavior
• Need a theoretical framework that extends CoVto dissipative problems…
![Page 5: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/5.jpg)
Michael OrtizMULTIMAT08
Classical rate variational problems
![Page 6: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/6.jpg)
Michael OrtizMULTIMAT08
Classical rate variational problems
![Page 7: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/7.jpg)
Michael OrtizMULTIMAT08
Metal plasticity − Multiscale hierarchy
Lattice defects, EoS
Dislocation dynamics
Subgrainstructures
length
time
mmnm µm
ms
µsns
Polycrystals
Engineeringapplications
Quantum mechanical or atomistic
Discrete or linear elastic
Continuum
![Page 8: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/8.jpg)
Michael OrtizMULTIMAT08
Energy-dissipation functionals
local?
![Page 9: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/9.jpg)
Michael OrtizMULTIMAT08
Dislocation dynamics to crystal plasticity
Lattice defects, EoS
Dislocation dynamics
Subgrainstructures
length
time
mmnm µm
ms
µsns
Polycrystals
Engineeringapplications
Quantum mechanical or atomistic
Discrete or linear elastic
Continuum
![Page 10: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/10.jpg)
Michael OrtizMULTIMAT08
LE dislocations - Kinematics
elastic deformation plastic deformation (currents)
dislocation loop
slip plane
![Page 11: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/11.jpg)
Michael OrtizMULTIMAT08
LE dislocations – Constrained theory
The 12 slip systems of fcc crystals (Schmidt and Boas nomenclature)
• Mobility:{closed-packed planes of lattice}
• Energy: = lattice-preserving deformation
• Energy: inaaaa ({shortest translation vectors of lattice})
core-cutoffmollifier
![Page 12: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/12.jpg)
Michael OrtizMULTIMAT08
LE dislocations - Kinematicsdislocation
loopslip
plane
![Page 13: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/13.jpg)
Michael OrtizMULTIMAT08
LE dislocations - Dissipation
moving dislocation
dislocation speed
dislocationline
![Page 14: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/14.jpg)
Michael OrtizMULTIMAT08
LE dislocations – Dissipation
Kink
Stainier, L., Cuitino, A. and Ortiz, M., JMPS, 50(2002) 1511-1554.
Wang, G. et al., J. Computer Aided Materials Design, 2001)
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Michael OrtizMULTIMAT08
15
LE dislocations – Dissipation
Stainier, L., Cuitino, A. and Ortiz, M., JMPS, 50(2002) 1511-1554.
(Wasserbach, 1986)
T (K)
∂ψ
(MPa
)
![Page 16: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/16.jpg)
Michael OrtizMULTIMAT08
LE dislocations – Energy-dissipation
Crystal plasticity!
Problems are open!
![Page 17: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/17.jpg)
Michael OrtizMULTIMAT08
LE dislocations − Energy models
strain energy stored energy
![Page 18: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/18.jpg)
Michael OrtizMULTIMAT08
LE dislocations − Energy models
nonlocal!
![Page 19: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/19.jpg)
Michael OrtizMULTIMAT08
LE dislocations − Energy models
local line tension approximation!
• True in 2D: M. Ponsiglione, SIAM J. Math. Anal. 39 (2008) no. 2, 449-469; A. Garroni, G. Leone, M. Ponsiglione, preprint (2008).
• True in 2.5D: Garroni, S. Müller, SIAM J. Math. Anal., 36 (2005) no. 6, 1943–1964; S. Cacace, A. Garroni, preprint (2008).
![Page 20: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/20.jpg)
Michael OrtizMULTIMAT08
LE dislocations – Energy models
0.2 0.3 0.4 0.5 0.60
0.5
1
1.5
TRUE STRAIN
β
BETA-INTBETA-DIFF
Logarithmic strain
non-s
tore
d e
ner
gy
frac
tion
Kolsky (split Hopkinson) pressure bar
Fraction of total work dissipated as heat (Rittel, D, Ravichandran, G, Lee, S, Mechanics of
Materials, 34 (2002) 627-642)
OFHC copper, 1000,15 −≈ sε&
stored energynon-negligible!
![Page 21: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/21.jpg)
Michael OrtizMULTIMAT08
LE dislocations − Dissipation models
(Humphreys and Hirsch ’70)Impenetrable obstacles
pinningpoints
(movie)
(LLNL)
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Michael OrtizMULTIMAT08
22
LE dislocations − Dissipation models
jog formation energy
reaction coordinate
afterbefore
favorablejunction
unfavorable junction
Dissipation at junctions and jogs (Stainier, L., Cuitino, A., Ortiz, M., JMPS, 50
(2002) 1511-1554)
![Page 23: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/23.jpg)
Michael OrtizMULTIMAT08
LE dislocations − Energy-dissipation models
Dislocation motion through random obstacle array(Foreman, A.J.E., Makin, M.J., Phil. Mag., 14 (1966) 911)
![Page 24: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/24.jpg)
Michael OrtizMULTIMAT08
• Forest hardening: Dislocations in secondary systems supply obstacles to the motion of dislocations on primary systems
• Dislocation multiplication: Dislocation density increases as a result of slip activity due to– Activation of fixed sources– Dynamic sources (cross slip)– Anihilation
• Strong latent hardening: High rate of hardening of primary system due to activity on secondary systems
LE dislocations − Energy-dissipation models
(movie)(LLNL)
![Page 25: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/25.jpg)
Michael OrtizMULTIMAT08
Junctions − Strong latent hardeningLattice
OrientationPrimary
TestLattice
OrientationSecondary
Test
PrimaryTest
SecondaryTest
Latent hardening experiments UF Kocks, Acta Metallurgica, 8 (1960) 345
UF Kocks, Trans. Metall. Soc. AIME, 230 (1964) 1160
• Strong latent hardening: Crystals much ‘prefer’ to activate a single slip system at each material point, though the active system may vary from point to point
![Page 26: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/26.jpg)
Michael OrtizMULTIMAT08
LE dislocations − Energy-dissipation models
Measured latent hardening ratios (Franciosi, P., Berveiller, M.,
and Zaoui, A., Acta Metall., 28(1980) 273-283)
Al
Cu
Latent hardening ratios predicted from line tension percolation
model, multiple slip and dislocation multiplication
(Cuitino, A. and Ortiz, M., ModellingSimul. Mater. Sci. Engr., 1 (1992) 225-263)
![Page 27: Multiscale models of metal plasticity Part I: Dislocation ... Part I.pdfSep 12, 2008 · Dislocation dynamics to crystal plasticity • The problem of deriving single crystal plasticity](https://reader035.fdocuments.in/reader035/viewer/2022081404/5f043eac7e708231d40d0621/html5/thumbnails/27.jpg)
Michael OrtizMULTIMAT08
Dislocation dynamics to crystal plasticity• The problem of deriving single crystal plasticity
from dislocation dynamics remains open in general
• Mathematical formulation of the problem: Relax dislocation-dynamics energy-dissipation functional, pass to the causal limit
• Many important features of crystal plasticity are understood qualitatively from experiments, microscopy and engineering models
• Most important among those features are:– Crystallographic nature of plastic slip in crystals– Strong latent hardening under multiple slip
• These features give rise to microstructure…