Multiscale models of metal plasticity Lecture II: Energetics of … · 2020. 9. 4. · M. Ortiz and...
Transcript of Multiscale models of metal plasticity Lecture II: Energetics of … · 2020. 9. 4. · M. Ortiz and...
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Michael OrtizROME0611
M. OrtizCalifornia Institute of Technology
Sixth Summer School in Analysis and Applied Mathematics
Rome, June 20-24, 2011
Multiscale models of metal plasticity
Lecture II: Energetics of dislocations
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Michael OrtizROME0611
Outline of Lecture #2
• The necessity of dislocations• The classical theory of linear-elastic dislocations• The dislocation core: The Peierls-Nabarro model• Extensions of the PN model to 3D• Semi-discrete 2½D phase-field model • The classical micro-macro connection• The dilute limit and dislocation line tension
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Michael OrtizROME0611
The necessity of dislocations
J.L. Eriksen, ARMA, 72 (1979) 1-13.
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Michael OrtizROME0611
Discreteness of crystallographic slip
Slip traces on crystal surface(AFM, C. Coupeau)
Slip occurs on discrete planes!
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Michael OrtizROME0611
The necessity of dislocations
I. Fonseca, ARMA, 97 (1987) 189-220. I. Fonseca, J. Math. Pures et Appl., 67 (1988) 175-195.
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Michael OrtizROME0611
The necessity of dislocations
slip plane
dislocationcore
(Orowan, Taylor, Polanyi, 1934)
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Michael OrtizROME0611
The necessity of dislocations
Burgerscircuit
Burgersvector
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Michael OrtizROME0611
The necessity of dislocations
Kink
A.M. Cuitino et al.,J. Comput.-Aided Mater., 8 (2001) 127-149.
L. Stainier, A.M. Cuitinoand M. Ortiz, JMPS, 50(2002) 1511-1554.
dislocation line
Peierls potential
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Michael OrtizROME0611
Micromechanics of plastic deformation: Dislocation theory
• Kinematics of crystallographic slip• The cut-surface problem of linear elasticity• The Peierls-Nabarro model of the core• Extensions of the PN model to 3D
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Michael OrtizROME0611
Dislocation theory - Kinematics
elastic deformation plastic deformation (currents)
slip plane
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Michael OrtizROME0611
Dislocation theory - Kinematicsdislocation
loopslip
plane
V. Volterra, Ann. École Normale Supérieure, 24 (1907), 401
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Michael OrtizROME0611
Dislocation theory – Constrained
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
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Michael OrtizROME0611
Dislocation theory – Linear elasticity
slip plane
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Michael OrtizROME0611
The elastic field of cut surfaces
D.J. Bacon, D.M. Barnett and R.O. Scattergood, , Progress in Materials Science, 23 (1979), 51-262
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Michael OrtizROME0611
The elastic field of cut surfaces
V. Volterra, Ann. École Normale Supérieure, 24 (1907), 401
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Michael OrtizROME0611
The elastic field of dislocations
T. Mura, Phil. Mag., 3 (1963) 625
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Michael OrtizROME0611
The elastic field of dislocations
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Michael OrtizROME0611
The Peierls-Nabarro dislocation core
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Michael OrtizROME0611
The Micro-to-Macro transition
• Formal methods• Macroscopic plastic deformation• Macroscopic elastic energy• Stored energy of plastic work
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Michael OrtizROME0611
The Micro-to-Macro transition
• The classical micro-macro connection• The dilute limit and dislocation line tension
0.2 0.3 0.4 0.5 0.60
0.5
1
1.5
TRUE STRAIN
β
BETA-INTBETA-DIFF
Logarithmic strain
heat
fra
ctio
n (Q
/W) OFHC copper, 1000,15 −≈ sε
stored energy(E/W)
Energy balance: he
at fra
ctio
n (Q
/W)
Kolsky pressure barD. Rittel, G. Ravichandran and S. Lee,
Mechanics of Materials, 34 (2002) 627-642.
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Michael OrtizROME0611
The Micro-to-Macro transition
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Michael OrtizROME0611
The Micro-to-Macro transition
M. Ortiz and E.P. Popov, Proc. Roy. Soc. Lond. A 379, 439-458 (1982)
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Michael OrtizROME0611
The Micro-to-Macro transition
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Michael OrtizROME0611
The Micro-to-Macro transition
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Michael OrtizROME0611
Summary – Outlook• A number of useful identities can be obtained
from linear elasticity and formal methods:– Elastic-plastic decomposition of macroscopic strain– Relation between macro and micro plastic strains– Additive decomposition of macroscopic energy– Dependence of stored energy on dislocation density
• Beyond formal methods? The dilute limit!
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Michael OrtizROME0611
Metal plasticity − Multiscale analysis
Lattice defects, EoS
Dislocation dynamics
Subgrainstructures
length
time
mmnm µm
ms
µsns
Polycrystals
Engineeringapplications
Lecture #2: Dislocation energies, the line-tension approximation
Lecture #3: Dislocation kinetics,the forest-hardening model
Objective: Derive ansatz-free,physics-based, predictive models of macroscopic behavior
Lecture #4: Subgraindislocation structures
Multiscale models of metal plasticity �Lecture II: Energetics of dislocationsOutline of Lecture #2The necessity of dislocationsDiscreteness of crystallographic slipThe necessity of dislocationsThe necessity of dislocationsThe necessity of dislocationsThe necessity of dislocationsMicromechanics of plastic deformation: Dislocation theoryDislocation theory - KinematicsDislocation theory - KinematicsDislocation theory – Constrained Dislocation theory – Linear elasticityThe elastic field of cut surfacesThe elastic field of cut surfacesThe elastic field of dislocationsThe elastic field of dislocationsThe Peierls-Nabarro dislocation coreThe Micro-to-Macro transitionThe Micro-to-Macro transitionThe Micro-to-Macro transitionThe Micro-to-Macro transitionThe Micro-to-Macro transitionThe Micro-to-Macro transitionSummary – OutlookMetal plasticity − Multiscale analysis