Application of the Theory of Extreme Values in Fracture Problems

Epstein, Benjamin

doi:10.2307/2280277

Cited by 16 publications

(8 citation statements)

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“…The generation of trueF¯V(σ) from F¯(σ) is based on the independence of the fundamental volume elements and is consistent with “weakest-link” arguments, in which a chain (component) is only as strong as its weakest link (element) (if the links are independent, the strength of the chain is not altered by the presence or absence of links that are stronger than the weakest link.) Independence and weakest link ideas were used by Weibull and Epstein in early considerations of strength variability [36-39], consistent with dilute or “isolated” populations of flaws (i.e., c << Δ V 1/3 ) and the absence of “non-local” effects in brittle fracture strengths [32, 40, 41]. The cdf of a strength distribution is complementary to the ccdf, and thus the cdf for the group of component strengths, F V ( σ ), is given by FV(σ)=1−trueF¯V(σ)=1−F¯(σfalse)k…”

Section: Fracture Probability Analysismentioning

confidence: 99%

Material Flaw Populations and Component Strength Distributions in the Context of the Weibull Function

Cook

DelRio

2018

Exp Mech

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A clear relationship between the population of brittle-fracture controlling flaws generated in a manufactured material and the distribution of strengths in a group of selected components is established. Assumptions regarding the strength-flaw size relationship, the volume of the components, and the number in the group, are clarified and the contracting effects of component volume and truncating effects of group number on component strength empirical distribution functions highlighted. A simple analytical example is used to demonstrate the forward prediction of population → distribution and the more important reverse procedure of empirical strength distribution → underlying flaw population. Three experimental examples are given of the application of the relationships to state-of-the-art micro- and nano-scale strength distributions to experimentally determine flaw populations: two on etched microelectromechanical systems (MEMS) structures and one on native and oxidized silicon nanowires. In all examples, the minimum threshold strength and conjugate maximum flaw size are very well estimated and the complete flaw population, including the minimum flaw size, are very poorly estimated, although etching, bimodal, and oxidation effects were clearly discernible. The results suggest that the best use of strength distribution information for MEMS manufacturers and designers might be in estimation of the strength threshold.

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Section: Fracture Probability Analysismentioning

confidence: 99%

Material Flaw Populations and Component Strength Distributions in the Context of the Weibull Function

Cook

DelRio

2018

Exp Mech

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“…The basic physical idea is to treat failure as revealing the weakest microstructural feature of the system. Starting with da Vinci's early efforts to quantify the strength of ropes (26), followed by Mariotte's more quantitative work (27), fracture became a popular application of extreme value statistics (e.g., see 28,29). Describing the overall statistics of the results of mechanical testing does not, however, expose the underlying connection to a material's microstructure or to the still more fundamental behavior of large populations of dislocations.…”

Section: Weakest Microstructural Featurementioning

confidence: 99%

Polycrystal Plasticity: Comparison Between Grain - Scale Observations of Deformation and Simulations

Pokharel

Lind²,

Kanjarla³

et al. 2014

Annu. Rev. Condens. Matter Phys.

142

106

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The response of polycrystals to plastic deformation is studied at the level of variations within individual grains, and comparisons are made to theoretical calculations using crystal plasticity (CP). We provide a brief overview of CP and a review of the literature, which is dominated by surface observations. The motivating question asks how well CP represents the mesoscale behavior of large populations of dislocations (as carriers of plastic strain). The literature shows consistently that only moderate agreement is found between experiment and calculation. We supplement this with a current example of microstructure evolution in the interior of a copper sample subjected to tensile deformation. Nondestructive measurements of orientation fields were performed using the near-field highenergy X-ray diffraction microscopy (nf-HEDM) technique at the Advanced Photon Source (APS). Starting at highly ordered grains, a single two-dimensional slice of microstructure containing ∼150 grains was followed through multiple strain states, where it tracked lattice rotations and defect accumulation of up to 14% elongation. In accord with the literature, at the scale of individual grains, comparison of observations with CP models indicates reasonable qualitative agreement but significant variations between simulation and experiment are apparent. The conclusion is that in order to be able to quantify the effects of microstructure on the distributions of slip, orientation change, and damage accumulation, the empirically derived constitutive relations used in continuumscale simulations need to be improved. Equally important will be the development of large-scale simulations of polycrystals that directly model dislocations. 317Annu. Rev. Condens. Matter Phys. 2014.5:317-346. Downloaded from www.annualreviews.org by University of Texas -San Antonio on 08/26/14. For personal use only.

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“…An asymptotic development allows expressing r into z [41][42][43] r ¼ m À s ffiffiffiffiffiffiffiffiffi ffi 2 ln i p À ln ln i þ ln 4p…”

Section: Example Iii: Brittle Strength Homogeneitymentioning

confidence: 99%

Stochastic heterogeneity as fundamental basis for the design and evaluation of experiments

Stroeven

Chen

2008

Cement and Concrete Composites

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Application of the Theory of Extreme Values in Fracture Problems

Cited by 16 publications

References 0 publications

Material Flaw Populations and Component Strength Distributions in the Context of the Weibull Function

Material Flaw Populations and Component Strength Distributions in the Context of the Weibull Function

Polycrystal Plasticity: Comparison Between Grain - Scale Observations of Deformation and Simulations

Stochastic heterogeneity as fundamental basis for the design and evaluation of experiments

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