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 OUTLINE Supported by the NSF MRSEC “Micro and Nanomechanics of Electronic and Structural Materials” at Brown Fracture in Heterogeneous Materials
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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.
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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
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
• “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 ……