a facility for simulating the dynamic response of materialsscuola normale di pisa september 19,...

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!