a facility for simulating the dynamic response of materialsscuola normale di pisa september 19,...
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Michael OrtizPisa 09/06
Multiscale modeling of materials: (2) Dislocation structures → polycrystals
M. OrtizM. OrtizCalifornia Institute of Technology
Scuola Normale di Pisa September 19, 2006
Michael OrtizMRS 11/04
Metal plasticity − Multiscale hierarchy
Lattice defects, EoS
Dislocation dynamics
Subgrainstructures
length
time
mmnm µm
µsns
ms
Polycrystals
Engineeringapplications
Ultimate goal: Ascertain macroscopic behavior from first principles
Continuum
Quantum mechanical
Discrete
Michael OrtizMRS 11/04
Continuum models of crystal plasticity
Michael OrtizMRS 11/04
General linear elastic dislocations
elastic deformationplastic deformation
dislocation loop
slip plane
Michael OrtizMRS 11/04
General dislocations − Energy
Michael OrtizMRS 11/04
Straight dislocations – Dissipation
Kink
lattice friction
Michael OrtizMRS 11/04
Obstacles − Topological obstructions
(Humphreys and Hirsch ’70)Impenetrable obstacles
pinningpoints
Michael OrtizMRS 11/04
The standard continuum model
strain energy stored energy
Michael OrtizMRS 11/04
The standard continuum model
Michael OrtizMRS 11/04
The standard continuum model
strain energy plastic work core energy
Michael OrtizMRS 11/04
Standard model − Local
Michael OrtizMRS 11/04
Standard model − Local
(Ortiz and Repetto, JMPS, 47(2) 1999, p. 397)
Michael OrtizMRS 11/04
Standard model − Relaxation
Michael OrtizMRS 11/04
Standard model − Relaxation
Michael OrtizMRS 11/04
Standard model − Relaxation
Michael OrtizMRS 11/04
Standard model – Relaxation
(Crone and Shield, JMPS, 2002)
(Rice, Mech. Mat., 1987)
Ideal plasticity Slip-line energy
Michael OrtizMRS 11/04
Relaxation and computation
Indentation of [001] surface of FCC crystal(Hauret and Ortiz, 2005)
Michael OrtizMRS 11/04
Relaxation and computation
Michael OrtizMRS 11/04
Relaxation and computationfo
rce
relaxed
bubbleenrichmentunrelaxed
elastic
Indentation of [001] FCC surface
(Hauret and Ortiz, 2005)
displacement
Michael OrtizMRS 11/04
1
2
3
4
Relaxation and computation
averagedeformation
localdeformation
averagestress
localstress
Microstructures generatedat quadrature points on the fly
Michael OrtizMRS 11/04
Model boundary-value problem
grain
matrixinterface
d
Michael OrtizMRS 11/04
Optimal scaling laws
• Upper bounds determined by construction• Lower bounds: Rigidity estimates, ansatz-free lower
bound inequalities (Kohn and Müller ’92, ’94; Conti ’00)
Michael OrtizMRS 11/04
Optimal scaling – Laminate construction
parabolic hardening +Hall-Petch scaling
dislocation wallsboundary layer
grain
Michael OrtizMRS 11/04
Optimal scaling – Branching construction
boundary pile-up
Michael OrtizMRS 11/04
Optimal scaling – Microstructures
LiF impact(Meir and Clifton´86)
Laminate BranchingShocked Ta(Meyers et al ´95)
Dislocation structures corresponding to the lamination and branching constructions
Michael OrtizMRS 11/04
Optimal scaling – Phase diagram
ElasticRigid
Lamellar Branching
T = dislocation energyG = shear modulusγ = deformationb = Burgers vectord = grain size μ = GB strength
Michael OrtizMRS 11/04
Non-locality and computation
Wrong picture!