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

W. A. Curtin A. Needleman, C. Briant, S. Kumar, H. Kumar, J. Arata, W. Xuan Brow www.phwiki.com

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

Coppola, Chris, Executive Editor has reference to this Academic Journal, PHwiki organized this Journal W. A. Curtin A. Needleman, C. Briant, S. Kumar, H. Kumar, J. Arata, W. Xuan Brown University, Providence, RI 02906 Cohesive zones; scaling in addition to heterogeneity Fracture in Nanolamellar Ti-Al 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 in addition to Nanomechanics of Electronic in addition to Structural Materials” at Brown Fracture in Heterogeneous Materials Cohesive Zone Model: Cohesive Zone Model (CZM) contains several key features: (follows from work/energy arguments, e.g. J-integral) Replace localized non-linear de as long as mation zone by an equivalent set of tractions that this material exerts on the surrounding elastic material u Scaling: Cohesive Zone Model introduces a LENGTH uc E=Elastic modulus of bulk material uc= Characteristic Length of Cohesive Zone at Failure Scale of heterogeneity vs. Scale of decohesion is important

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Form T vs. u specific to physics in addition to mechanics of decohesion: Polymer crazing: “Dugdale Model” Polymer craze material drawn out of bulk at ~ constant stress Atomistic separation: “Universal Binding” u Fiber bridging: “Sliding/Pullout Model” Fiber/matrix interface sliding friction, fiber fracture, pullout First-Principles Quantum Calculations: Distributed, Nucleated Damage: difficult to model in brittle systems Imagine local stress concentration that nucleates crack; will crack stop if it encounters a region of higher toughness Crack stops if: 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: Toughening: Occurs at Colony Boundaries Fracture in Ti-Al: preferentially along lamellar direction Ti-Al: Alternating nanoscale layers of TiAl in addition to Ti3Al 1 microns 500 microns

Modeling across scales to address issues, guide optimal material design Questions to answer about real material: Role of microstructure in addition to heterogeneity at various scales Multiscale Modeling of Ti-Al: Prediction of damage evolution, toughening vs. microstructure 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): Fracture toughness increases with increasing lamellar thickness : linear scaling Fracture/Dislocation Model predicts this behavior: Scales with square root of lamellar thickness; Thicker is tougher Toughness: Implications as long as fracture in Ti-Al nanolaminates: 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: Computational microstructure: Where, when do cracks nucleate Interplay of heterogeneity, length scales

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 Heterogeneity can drive distributed microcracking Microstructural Model as long as 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:

Modified Colony 5 Orientation: Orientation highly unfavorable as long as cracking Decrease in Matrix Yield Stress More damage, higher toughness Microcrack closure (reversible cohesive zone) Microstructural models capture range of physical phenomena Subtle interplay between toughnesses of various phases in addition to boundaries, in addition to plastic behavior Small changes in microscopic quantitities can lead to large changes in macroscopic modes of cracking in addition to toughening Optimization of material as long as engineering requires underst in addition to ing of Microscopic Details (alloying to harden/strengthen) Control of Microstructure (colony size, distribution)

Summary as long as Ti-Al Cohesive Zone Model: powerful technique as long as nucleation in addition to 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 as long as mechanical per as long as mance 3d Fracture is important: extend CZ in addition to Microstructure models Generate a family of microstructures “statistically similar” to a real system Computationally test microstructures Probe dependence of per as long as mance on microstructure Investigate optimum classes of microstructures Compare simulated per as long as mance to experimental results Guide fabrication toward optimal 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 as long as help avoid unexpected failure Modeling of Complex Microstructures Goal: Identify global in addition to local microstructural correlation functions that influence flow, hardening, failure; Use knowledge to guide experimental microstructural design Microstructure Reconstruction: 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. (Yeong + Torquato)

Initial After 60 steps After 45 steps Sample Evolution Path Final Parent Key Features of Reconstruction Method: Simple to implement as long as 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 in addition to tested Microstructures already naturally in a as long as m suitable as long as 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

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Real, Complex Microstructures: Ductile Iron Parent Child 1 Child 2 Child 3 Correlation Functions P2 L2 Carbon Iron Finite Element Analysis: Elastic/Plastic Matrix Stress-Strain Response Uniaxial Tension Parent, Children essentially identical ! Matrix only Microstructure-induced Hardening Low-order correlations: excellent description of non-linear response What microstructural features trigger LOCALIZATION Local Onset of Instability: Sample-to-Sample Variations (of course) Onset Instability U=0.150; s=847 MPa U=0.200; s=855 MPa U=0.141; s=856 MPa U=0.118; s=829 MPa U=0.109; s=808 MPa U=0.234; s=857 MPa U=0.207; s=858 MPa U=0.204; s=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 (Gr in addition to child) Test new microstructures Analyze hot spot behavior Vary test box size in addition to retest “Genetic” Methodology as long as 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 MPa Onset often at same location, same stress in addition to strain range Strain=15.10% Stress=847MPa Strain=19.59 % Stress=855MPa Window = 15 X 15 Onset mostly at another location, much higher stress in addition to strain range Gr in addition to child Gr in addition to child 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 in addition to strain Computational identification of “characteristic” weak-link microstructure Gr in addition to child

15 x 15 20 x 20 30 x 30 Not similar to Child Often similar to Child Mostly similar to Child 847 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 as long as 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 as long as fracture initiation, propagation Extend hot-spot analysis methods statistical characterization Validate model quantitatively vs. experiments Methods as long as optimization Hard work still ahead

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