Fracture in Heterogeneous Materials
W. A. Curtin
A. Needleman, C. Briant, S. Kumar, H. Kumar, J. Arata, W. Xuan
Brown University, Providence, RI 02906
Supported by the NSF MRSEC “Micro and Nanomechanics of
Electronic and Structural Materials” at Brown
OUTLINE
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
Cohesive Zone Model:
Replace localized non-linear deformation zone by an equivalent set of
tractions that this material exerts on the surrounding elastic material
T
T (u )
u
Cohesive Zone Model (CZM) contains several key features:
Maximum stress  max , followed by softening
 max
Material separates naturally
Nucleation without pre-existing cracks
 max = inherent strength of material
Work of Separation

= Energy to create new surface
 contains all energy/dissipation
physically occuring within material

   T (u )du
0
(follows from work/energy arguments, e.g. J-integral)

Scaling: Cohesive Zone Model introduces a LENGTH uc
uc 
E

2
max
E=Elastic modulus of bulk material
uc= Characteristic Length of Cohesive Zone at Failure
u*
uc
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
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
 max
Scale of heterogeneity vs. Scale of decohesion is important
Form T vs. u specific to physics and mechanics of decohesion:
Polymer crazing: “Dugdale Model”
Fiber bridging: “Sliding/Pullout Model”
t=195 MPa
t=170 MPa
40
Fiber/matrix interface
sliding friction, fiber
fracture, pullout
 c ~ 50MPa; u* ~ 1mm; E  5 GPa;
35
Stress in Tube, (GPa) .
Polymer craze material
drawn out of bulk at ~
constant stress
30
X X
25
t=40 MPa
20
X
15
10
t=100 MPa
m=3
c=20 GPa
Gb /2
5
0
0
u c ~ 100mm ; K IC ~ 1MPa m
0.05
0.1
0.15
 c ~ 200MPa; u* ~ 10mm; E  100GPa;
Half crack opening, (mm)
u c ~ 5mm ; K IC ~ 200MPa m
Atomistic separation: “Universal Binding” First-Principles Quantum Calculations:
T e
 max
u


ue
u
Increasing H
concentration
u
u
 c ~ 10GPa; u* ~ 1nm; E  100GPa;
u c ~ 10nm ; K IC ~ 1.4MPa m
Distributed, Nucleated Damage: difficult to model in brittle systems
Cracks form at cohesive strength
c
; difficult to stop
Imagine local stress concentration that nucleates crack;
will crack stop if it encounters a region of higher toughness?
uc
1/ 2
Crack stops if:
c 1  L 
  
 o   uc 
Stress concentration is huge
or
Length scale of heterogeneity is small
Can’t stop “typical” nucleated crack in brittle materials
Multiscale Modeling of Fracture in Ti-Al:
500 microns
Ti-Al: Alternating
nanoscale layers of
TiAl and Ti3Al
Ti3Al
“Colonies”
of lamellae
1 microns
Fracture in Ti-Al: preferentially
along lamellar direction
Microcracks
20 m m
Toughening: Occurs at
Colony Boundaries
Questions to answer about real material:
Role of microstructure and heterogeneity at various scales
How much toughening due to boundary misorientation?
Does microcracking enhance toughness?
Microcracking at scales >> lamellar spacing Why?
Where are cracks: TiAl, Ti3Al, or at TiAl/Ti3Al interface?
Does small-scale fracture toughness depend on lamellar structure?
Modeling across scales to address issues, guide optimal material design
Multiscale Modeling of Ti-Al:
Cohesive
Zones
reflecting
varying TiAl
widths
1
Atomistics of fracture in
nanolamellar Ti-Al
Toughness vs. TiAl Lamellar
Thickness (Cohesive Zones)
Realistic models of colony boundary damage
10 um
Continuum models
Prediction of damage evolution, toughening vs.
microstructure
Analytic selection of likely
planes for microcracking
Model of realistic colony microstructure
Atomistic Simulations: Derive Toughness vs. Nanolamellar Structure
Crack growth in TiAl lamella between two Ti3Al lamellae:
Dislocation emission followed by crack cleavage; depends on microstructure
Fracture toughness increases with increasing lamellar thickness
Applied KI vs. Crack Growth (R-curve):
K Iapp cleave vs.
t
: linear scaling
60 nm
50 nm
40 nm
30 nm
Fracture/Dislocation Model predicts this behavior:
Toughness:
K Iapp cleave
 a1K Ic  a2 K IIc 
a1K Ic  a2 K IIc 
 t
a1K Ic  f1 ( )


f
(

)

f
(

)
f
(

)

f
(

)
 2


1
2
1
Scales with square root of lamellar thickness; Thicker is tougher
Implications for fracture in Ti-Al nanolaminates:
Fracture strongly preferred along lamellar direction
Thin TiAl lamellae are “weak link” in Ti-Al nanolaminates
Cracks inhibited at “colony” boundaries preferentially
• renucleate across boundary in thin TiAl layers
• microcrack in thin TiAl layers
thin TiAl
Mesoscale Model of Fracture Across Colony Boundaries
Kc1
Model of lamellar colony boundary:
Kc2
Low-toughness planes
Kc3
Kc4
Kc5
Initial Crack
1 mm
“Real” microstructure
Heterogeneous
Toughnesses

Computational 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
Where, when do cracks nucleate? Interplay of heterogeneity, length scales?
1.0
Numerical Results on Fracture in Heterogeneous Lamellar System:
Impose range of low-toughness values; explore microcrack nucleation
1 um
Microcrack on weak
plane near main crack
K I nucleation  2.64
2.0
3.0
4.0
3.0
2.0
1.5
1.0
1.5
2.5
3.5
1.5
2.0
1.5
1.0
Microcrack on weak plane
away from main crack
K I nucleation  2.79
2.0
2.5
3.5
2.5
1.5
2.0
1.5
1.0
Two microcracks on
weak planes away from
main crack
K I nucleation  2.79
Heterogeneity can drive distributed microcracking
Microcrack Nucleation: critical stress needed over some distance
Microstructural Model for Fracture in Ti-Al:
Scale of weak planes set by heterogeneity, not lamellar scale
(real microstructural-specific models not included yet)
“Real” microstructure
Computational 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
Fracture through Polycolony Lamellar Ti-Al:
Toughening as
crack crosses
colony boundaries
Modified Colony 5 Orientation:
Orientation highly
unfavorable for cracking
Multiple microcracking
Only slight decrease
in toughening
Experiment
Decrease in Matrix Yield Stress  More damage, higher toughness
 y  850 MPa
Microcrack closure
(reversible cohesive
zone)
 y  425 MPa
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-micron
scale 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
Modeling of Complex Microstructures
• Experimental optimization of microstructures could be
guided 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?
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:
(Yeong + Torquato)
1. Digitized microstructure of “parent”;
label each pixel by phase;
Calculate P2(r) and L2(r) by scanning
along horizontal, vertical lines
Initial
Digitized
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.




E   i Ps( 2) (ri )  Pp( 2) (ri )   i L(s2) (ri )  L(p2) (ri )
N
2
P  1 if E  0
P  exp(E / T ) if E  0
N
2
Sample Evolution Path
Initial
After 45 steps
After 60 steps
Parent
Final
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
2D image of Parent microstructure
Cut Along the xy-plane
Cut Along the xz-plane
Cut Along the yz-plane
Real, Complex Microstructures: Ductile Iron
Correlation Functions
Child #1
P2
Parent
Child #2
L2
Carbon
Iron
Child #3
Finite Element Analysis: Elastic/Plastic Matrix
Uniaxial Tension
Stress-Strain Response
Parent, Children
essentially identical !
Microstructureinduced
Hardening
Matrix
only
Fe matrix
C particles
E (GPa)

210
15
0.30
0.26
Low-order correlations: excellent description of non-linear response
What microstructural features trigger LOCALIZATION?
Local Onset of Instability: Sample-to-Sample Variations (of course)
Child #1
Child #2
Child #3
Onset
Parent
U=0.141; =856 MPa
U=0.200; =855 MPa
U=0.234; =857 MPa
U=0.118; =829 MPa
U=0.109; =808 MPa
Instability
U=0.150; =847 MPa
U=0.207; =858 MPa
U=0.204; =855 MPa
What is characteristic “weak” feature driving localization?
“Genetic” Methodology for Identification of Hot Spots:
Identify hot spot;
Choose test box
Test new microstructures
Analyze hot spot behavior
Vary test box size and retest
Build new microstructures
around box
Extract test box
microstructure
Insert into new
reconstruction
(Grandchild)
Analyze worst of the children (statistical tail):
Window = 15 X 15
Grandchild
Strain=15.10% Stress=847MPa
Strain=19.59 % Stress=855MPa
Onset mostly at another location, much higher stress and strain range
Window = 20 X 20
Grandchild
Strain = 11.60 % Stress = 795 MPa
Strain = 25.00 % Stress = 856 MPa
Onset often at same location, same stress and strain range
Window = 30 X 30
Grandchild
Strain = 11.90 % Stress= 798 MPa
Strain = 20.15 % Stress= 855 MPa
Onset mostly at same location, similar stress and strain
Computational identification of
“characteristic” weak-link microstructure
Quantitative Evaluation of Hot Spot Damage Nucleation
15 x 15
Not
similar
to
Child
20 x 20
Often
similar
to Child
30 x 30
Mostly
similar
to Child
847
15.16
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 ……