Optimal scaling laws in ductile fracture
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1 Optimal scaling laws in ductile fracture M. Ortiz California Institute of Technology In collaboration with: S. Conti (Uni-Bonn), L. Fokoua (Caltech), S. Heyden (Caltech), B. Li (Case Western), K. Weinberg (Uni-Siegen) IUTAM Symposium on Micromechanics of Defects in Solids University of Seville, Spain, June 9-13, 2014
2 Background on ductile fracture void sheet Photomicrograph of a copper disk tested in a gas-gun experiment showing the formation of voids and their coalescence into a fracture plane Heller, A., How Metals Fail, Science & Technology Review Magazine, Lawrence Livermore National Laboratory, pp , July/August, 2002
3 Scope Micro-macro relations for ductile fracture (Universal) scaling relations in ductile fracture? Application of optimal scaling to ductile fracture Results for metals and polymers
4 Background on ductile fracture brittle ductile (Courtesy NSW HSC online) Ductile fracture in metals occurs by void nucleation, growth and coalescence Fractography of ductilefracture surfaces exhibits profuse dimpling, vestige of microvoids Ductile fracture entails large amounts of plastic deformation (vs. surface energy) and dissipation. Fracture surface in SA333 steel, room temp., dε/dt= s -1 (S.V. Kamata, M. Srinivasa and P.R. Rao, Mater. Sci. Engr. A, 528 (2011) )
5 Background on ductile fracture Charpy energy of A508 steel (Tanguy et al., Eng. Frac. Mechanics, 2005) A number of ASTM engineering standards are in place to characterize ductile fracture properties (J-testing, Charpy test) The Charpy test data reveals a brittle-to-ductile transition temperature In general, the specific fracture energy for ductile fracture is greatly in excess of that required for brittle fracture
6 Micromechanics of ductile fracture Objective: Elucidate microstructure/property relations (voids to specific fracture energy) Traditional micromechanics approach: Select a specific microscale model (crystal plasticity, porous plasticity, strain-gradient plasticity ) Select a representative microstructure (void in periodic cell, shear/damage localization band ) Perform unit-cell calculations, parametric studies Critique: Pros: Calculations exact (within numerical precision) Cons: Model-specific results, non-optimal static microstructures, numerical (vs. epistemic) results Alternative: Analysis (e.g., optimal scaling)
7 Scaling laws in science A broad variety of physical phenomena obey power laws over wide ranges of parameters Scale invariance: If y = C x α, then (x,y) iff (λx, λ α y), law of corresponding states Universality: Exponents are material-independent ( universal ) Systems displaying identical scaling behavior are likely to obey the same fundamental dynamics Experimental master curves, data collapse Examples: Critical phenomena (second-order transitions) Materials science (Taylor, Hall-Petch, creep laws ) Continuum mechanics (hydrodynamic, fracture )
8 Optimal scaling Originally applied to branched microstructures in martensite (Kohn-Müller 92, 94; Conti 00) Applications to micromagnetics (Choksi-Kohn- Otto 99), thin films (Belgacem et al 00)...
9 Naïve model: Local plasticity Ti sub-linear energy growth! Eleiche & Campbell (1974)
10 Naïve model: Local plasticity Energies with sublinear growth relax to 0. For hardening exponents in the range of experimental observation, local plasticity yields no useful information regarding ductile fracture properties of materials! Need additional physics, structure
11 Strain-gradient plasticity N. Fleck et al. (1993) The yield stress of metals is observed to increase in the presence of strain gradients Deformation theory of straingradient plasticity: W. Nix & H. Gao (1998) Strain-gradient effects may be expected to oppose localization Growth of W with respect to the second deformation gradient?
12 Strain-gradient plasticity Fence structure in copper (J.W. Steeds, Proc. Roy. Soc. London, A292, 1966, p. 343) Dislocation wall
13 Strain-gradient plasticity & fracture Mathematical model: Minimize Heller, A., How Metals Fail, Science & Technology Review Magazine, Lawrence Livermore National Laboratory, pp , July/August, 2002 For metals, local plasticity exhibits sub-linear growth, strain-gradient plasticity linear growth Question: Can ductile fracture be understood as the result of a competition between sublinear growth and strain-gradient plasticity?
14 Optimal scaling Ductile fracture
15 Optimal scaling Ductile fracture
16 Sketch of proof Upper bound void sheet Heller, A., Science & Technology Review Magazine, LLNL, pp , July/August, 2002 void sheet
17 Sketch of proof Upper bound void void growth!
18 Optimal scaling Ductile fracture Optimal (matching) upper and lower bounds: Bounds apply to classes of materials having the same growth, specific model details immaterial Energy scales with area (L 2 ): Fracture scaling! Energy scales with power of opening displacement (δ): Cohesive behavior! Lower bound degenerates to 0 when the intrinsic length (l) decreases to zero Bounds on specific fracture energy:
19 Fracture of polymers T. Reppel, T. Dally, T. and K. Weinberg, Technische Mechanik, 33 (2012) Crazing in 800 nm polystyrene thin film (C. K. Desai et al., 2011) Polymers undergo entropic elasticity and damage due to chain stretching and failure Polymers fracture by means of the crazing mechanism consisting of fibril nucleation, stretching and failure The free energy density of polymers saturates in tension once the majority of chains are failed: p=0! Crazing mechanism is incompatible with straingradient elasticity
20 Fracture of polymers
21 Sketch of proof Upper bound crazing Crazing in 800 nm polystyrene thin film (C. K. Desai et al., 2011) craze sheet
22 Sketch of proof Upper bound
23 Optimal scaling Crazing Optimal (matching) upper and lower bounds: Fractional strain-gradient elasticity supplies bounded energies for crazing mechanism Energy scales with area (L 2 ): Fracture scaling! Energy scales with power of opening displacement (δ): Cohesive behavior! Lower bound degenerates to 0 when the intrinsic length (l) decreases to zero Bounds on specific fracture energy:
24 Concluding remarks Ductile fracture can indeed be understood as the result of the competition between sublinear growth and (possibly fractional) strain-gradient effects Optimal scaling laws are indicative of a well-defined specific fracture energy, cohesive behavior, and provide a (multiscale) link between macroscopic fracture properties and micromechanics (intrinsic micromechanical length scale, void-sheet and crazing mechanisms ) Ductile fracture can be efficiently implemented through material-point erosion schemes
25 Concluding remarks Thanks!
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