W. A. Curtin

A. Needleman, C. Briant, S. Kumar, H. Kumar, J. Arata, W. Xuan

Brown University, Providence, RI 02906

1. Cohesive zones; scaling and heterogeneity

2. Fracture in Nanolamellar Ti-Al

3. Modeling of Complex Microstructures

Show some on-going directions of research (incomplete)

Emphasize Computational Mechanics Methods

Intersection of Heterogeneity, Materials, Mechanics

OUTLINE

Supported by the NSF MRSEC “Micro and Nanomechanics of Electronic and Structural Materials” at Brown

Fracture in Heterogeneous Materials

Cohesive Zone Model:

Cohesive Zone Model (CZM) contains several key features:Maximum stress , followed by softening

Material separates naturallyNucleation without pre-existing cracks

= inherent strength of material

max

maxmax

Work of Separation = Energy to create new surface

contains all energy/dissipation physically occuring within material

0

)( duuT(follows from work/energy arguments, e.g. J-integral)

T

Replace localized non-linear deformation zone by an equivalent set of tractions that this material exerts on the surrounding elastic material

)(uT u

If uc << all other lengths in problem: “small scale yielding”: stress intensity is a useful fracture parameter fracture is governed by critical Kc or details of T vs. u are irrelevant, only is important

Scaling: Cohesive Zone Model introduces a LENGTH uc

2max

Euc

E=Elastic modulus of bulk material

uc= Characteristic Length of Cohesive Zone at Failure

uc

u*

If uc ~ other lengths in problem: “large-scale bridging”:fracture behavior is geometry and scale-dependent

If uc >> all other lengths in problem: fracture controlled by maxScale of heterogeneity vs. Scale of decohesion is important

Form T vs. u specific to physics and mechanics of decohesion:

Polymer crazing: “Dugdale Model”Polymer craze material drawn out of bulk at ~ constant stress

Atomistic separation: “Universal Binding”

uu

euu

eT max

u

Fiber bridging: “Sliding/Pullout Model”

Fiber/matrix interface sliding friction, fiber fracture, pullout

0

5

10

15

20

25

30

35

40

0 0.05 0.1 0.15

Half crack opening, (mm)

Stre

ss in

Tub

e, (G

Pa)

.

X X

X

t=195 MPat=170 MPa

t=40 MPa

Gb/2 t=100 MPam=3c=20 GPa

0

5

10

15

20

25

30

35

40

0 0.05 0.1 0.15

Half crack opening, (mm)

Stre

ss in

Tub

e, (G

Pa)

.

X X

X

t=195 MPat=170 MPa

t=40 MPa

Gb/2 t=100 MPam=3c=20 GPa

First-Principles Quantum Calculations:

Increasing H concentration

mMPa

muMPac

1~K ; m100~u

GPa; 5E ;1~* ;50~

ICc m

m

mMPam

muMPac

200~K ; m5~u

GPa; 100E ;10~* ;200~

ICc

m

mMPan

nmuGPac

4.1~K ; m10~u

GPa; 100E ;1~* ;10~

ICc

Distributed, Nucleated Damage: difficult to model in brittle systemsCracks form at cohesive strength ; difficult to stopc

Imagine local stress concentration that nucleates crack; will crack stop if it encounters a region of higher toughness?

Crack stops if:2/1

1

co

c

uL

Stress concentration is huge

or Length scale of heterogeneity is small

Can’t stop “typical” nucleated crack in brittle materials

uc

Multiscale Modeling of Fracture in Ti-Al:

“Colonies” of lamellae

Ti3Al

Toughening: Occurs at Colony Boundaries

Fracture in Ti-Al: preferentially along lamellar direction

20 mm

Microcracks

Ti-Al: Alternating nanoscale layers of TiAl and Ti3Al

1 microns

500 microns

Where are cracks: TiAl, Ti3Al, or at TiAl/Ti3Al interface?Microcracking at scales >> lamellar spacing Why?

Does small-scale fracture toughness depend on lamellar structure?

How much toughening due to boundary misorientation?Does microcracking enhance toughness?

Modeling across scales to address issues, guide optimal material design

Questions to answer about real material:

Role of microstructure and heterogeneity at various scales

Atomistics of fracture in nanolamellar Ti-Al

Toughness vs. TiAl Lamellar Thickness (Cohesive Zones)

Model of realistic colony microstructure

Cohesive Zones reflecting varying TiAl widths

1

Realistic models of colony boundary damage

Multiscale Modeling of Ti-Al:

Continuum models

Prediction of damage evolution, toughening vs. microstructure

10 um

Analytic selection of likely planes for microcracking

Crack growth in TiAl lamella between two Ti3Al lamellae:

Dislocation emission followed by crack cleavage; depends on microstructure

Atomistic Simulations: Derive Toughness vs. Nanolamellar Structure

Applied KI vs. Crack Growth (R-curve):

60 nm

50 nm40 nm30 nm

Fracture toughness increases with increasing lamellar thickness vs. Iapp cleaveK t : linear scaling

Fracture/Dislocation Model predicts this behavior:

1 2 1 21 1

2 1 2 1( )

( ) ( ) ( ) ( )Ic IIc Ic IIc

Iapp cleave Ica K a K a K a KK t a K ff f f f

Scales with square root of lamellar thickness; Thicker is tougher

Toughness:

Implications for fracture in Ti-Al nanolaminates:

Thin TiAl lamellae are “weak link” in Ti-Al nanolaminates

Fracture strongly preferred along lamellar direction

• renucleate across boundary in thin TiAl layers

• microcrack in thin TiAl layers

Cracks inhibited at “colony” boundaries preferentially

thin TiAl

Mesoscale Model of Fracture Across Colony Boundaries

“Real” microstructure• Lamellar misorientation

• Low-toughness lamellae modeled by cohesive zones

• Heterogeneity in toughness due to variations in lamellar thickness

• 1 um low-toughness lamellar spacing

• Elastic matrix w/ fracture via cohesive zones

Model of lamellar colony boundary:

Heterogeneous Toughnesses

Initial Crack

Low-toughness planes

1 mm

1cK2cK3cK4cK5cK

Computational microstructure:

Where, when do cracks nucleate? Interplay of heterogeneity, length scales?

1.0

2.02.53.5

2.51.52.01.5 1.0

Two microcracks on weak planes away from main crack

2.03.04.03.02.01.5 1.0

Microcrack on weak plane away from main crack

Microcrack Nucleation: critical stress needed over some distance

Numerical Results on Fracture in Heterogeneous Lamellar System:Impose range of low-toughness values; explore microcrack nucleation

1.52.53.5 1.5 2.01.51.0

Microcrack on weak plane near main crack

1 um

Heterogeneity can drive distributed microcracking 2.64I nucleationK 2.79I nucleationK 2.79I nucleationK

Microstructural Model for Fracture in Ti-Al:

“Real” microstructure• Lamellar misorientation

• Colony boundary layer modeled by cohesive zones

• Low-toughness lamellae modeled by cohesive zones

• 20 um low-toughness lamellar spacing: weakest lamellae

• Elastic/plastic matrix w/ fracture via cohesive zones

Computational microstructure:

Scale of weak planes set by heterogeneity, not lamellar scale (real microstructural-specific models not included yet)

Fracture through Polycolony Lamellar Ti-Al:

Toughening as crack crosses colony boundaries

Modified Colony 5 Orientation:

Orientation highly unfavorable for cracking

Multiple microcracking

Experiment

Only slight decrease in toughening

Decrease in Matrix Yield Stress More damage, higher toughness850 y MPa 425 y MPa

Microcrack closure (reversible cohesive zone)

Microstructural models capture range of physical phenomena

Subtle interplay between toughnesses of various phases and boundaries, and plastic behavior

Small changes in microscopic quantitities can lead to large changes in macroscopic modes of cracking and toughening

Optimization of material for engineering requires understanding of

Microscopic Details (alloying to harden/strengthen)

Control of Microstructure (colony size, distribution)

Summary for Ti-Al

• Cohesive Zone Model: powerful technique for nucleation and crack growth naturally: derive from smaller-scale input

• Ti-Al: Nano/micro scale structure determines lamellar toughness

• CZMs shows heterogeneity in microstructure at sub-micronscale can drive microcracking on larger scales

• CZMs at microstructural scale capture physical phenomena,competition between toughness, plasticity, microstructure

• Coupled Multiscale Models may guide optimization of microstructures for mechanical performance

• 3d Fracture is important: extend CZ and Microstructure models

• Experimental optimization of microstructures could beguided by insight from computations

• Failure behavior is controlled by undesirable features;computations could identify such features -

what should experimentalists look for?help avoid unexpected failure?

Modeling of Complex Microstructures

Goal: Identify global and local microstructural correlation functions that influence flow, hardening, failure;

Use knowledge to guide experimental microstructural design

• Generate a family of microstructures “statistically similar” to a real system• Computationally test microstructures• Probe dependence of performance on microstructure• Investigate optimum classes of microstructures• Compare simulated performance to experimental results• Guide fabrication toward optimal microstructures

Microstructure Reconstruction:

Initial Digitized

1. Digitized microstructure of “parent”; label each pixel by phase; Calculate P2(r) and L2(r) by scanning along horizontal, vertical lines

2. Generate initial reconstructed microstructure; Fix volume fraction = parent value;

Compute the P2(r) and L2(r)

3. Calculate the “energy” E (mean square difference) between parent, synthetic microstructure.

4. Evolve E through Simulated Annealing: Consider exchange of two sites Compute energy change Accept exchange with probability P

T=“temperature”: decrease by ad-hoc annealing schedule.

2)2()2(

2)2()2( )()()()(

N

i ipisN

i ipis rLrLrPrPE

0 if )/exp(0 if 1

ETEPEP

(Yeong + Torquato)

Initial

After 60 steps

After 45 steps

Sample Evolution Path

Final

Parent

Key Features of Reconstruction Method:• Simple to implement for arbitrary systems• Unbiased treatment of microstructures• Can incorporate a variety of correlation functions

(limited only by simulated annealing time)• 3d structures can be generated using correlation functions

obtained from 2d images• Multiple realizations of the same parent microstructure

can be generated and tested• Microstructures already naturally in a form suitable for

numerical computations via FEM (pixel = element)• Can construct NEW structures from hypothetical

correlation functions• Microstructures can be built around “defects”

or “hot spots” of interest to probe them

Cut Along the yz-plane

Cut Along the xz-plane

Cut Along the xy-plane

2D image of Parent microstructure

Real, Complex Microstructures: Ductile Iron

Parent

Child #1

Child #2

Child #3

Correlation Functions

P2

L2

Carbon Iron

Finite Element Analysis: Elastic/Plastic MatrixStress-Strain ResponseUniaxial Tension

Parent, Children essentially identical !

Matrix only

Microstructure-induced Hardening

Low-order correlations: excellent description of non-linear response

Fe matrix 210 0.30C particles 15 0.26

E (GPa)

What microstructural features trigger LOCALIZATION?

Local Onset of Instability: Sample-to-Sample Variations (of course)O

nset

Inst

abili

ty

U=0.150; =847 MPa

U=0.200; =855 MPa

U=0.141; =856 MPa U=0.118; =829 MPa U=0.109; =808 MPa

U=0.234; =857 MPa U=0.207; =858 MPa U=0.204; =855 MPa

Parent Child #1 Child #2 Child #3

What is characteristic “weak” feature driving localization?

Identify hot spot; Choose test box

Insert into new reconstruction

(Grandchild)

Test new microstructures Analyze hot spot behavior Vary test box size and retest

“Genetic” Methodology for Identification of Hot Spots:

Extract test box microstructure

Build new microstructures around box

Window = 20 X 20

Strain = 11.60 % Stress = 795 MPa

Strain = 25.00 % Stress = 856 MPaOnset often at same location, same stress and strain range

Strain=15.10% Stress=847MPa Strain=19.59 % Stress=855MPa

Window = 15 X 15

Onset mostly at another location, much higher stress and strain rangeGrandchild

Grandchild

Analyze worst of the children (statistical tail):

Strain = 11.90 % Stress= 798 MPa Strain = 20.15 % Stress= 855 MPa

Window = 30 X 30

Onset mostly at same location, similar stress and strain

Computational identification of “characteristic” weak-link microstructure

Grandchild

15 x 15

20 x 20

30 x 30

Not similar to Child

Often similar to Child

Mostly similar to Child847 15.16

Quantitative Evaluation of Hot Spot Damage Nucleation

Characteristic size & structure consistently drives low-stress localization event

Summary• “Reconstruction” Methodology

– Method can establish sizes for statistical similarity (representative volume elements)

– Method can identify, represent anisotropy– Current method has difficulty with isotropic, elongated structures

• Examples demonstrated– Stress-strain behavior controlled by low-order structural correlations!– Localization is microstructure-specific (not surprising)

• Quantitatively analyze hot spots driving failure– Successive generations allow weak-links to be isolated– Example calculations show characteristic hot-spot size

Future Work

• Further pursue 3-d reconstruction algorithm • Cohesive zones for fracture initiation, propagation• Extend hot-spot analysis methods

statistical characterization? • Validate model quantitatively vs. experiments• Methods for optimization?• Hard work still ahead ……