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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
E
ucE=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”
u
u
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)
Str
ess
in T
ub
e, (
GP
a)
.
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)
Str
ess
in T
ub
e, (
GP
a)
.
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:
2/11
co
c
u
L
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 nm
40 nm
30 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 IIcIapp cleave Ic
a K a K a K a KK t a K f
f 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 toughness
850 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 ipis
N
i ipis rLrLrPrPE
0 if )/exp(
0 if 1
ETEP
EP
(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
abil
ity
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 range
Grandchild
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 ……