Research Papers

Scaling of Static Fracture of Quasi-Brittle Structures: Strength, Lifetime, and Fracture Kinetics

[+] Author and Article Information
Jia-Liang Le1

 Assistant ProfessorDepartment of Civil Engineering,University of Minnesota, Minneapolis, MN 55455

Zdeněk P. Bažant2

 McCormick Institute Professor and W.P. Murphy Professorof Civil Engineering and Materials Science,Northwestern University,2145 Sheridan Rd., CEE, Evanston, IL 60208 e-mail: z-bazant@northwestern.edu


Formerly Graduate Research Assistant, Northwestern University.


Corresponding author.

J. Appl. Mech 79(3), 031006 (Apr 05, 2012) (10 pages) doi:10.1115/1.4005881 History: Received July 01, 2011; Revised January 13, 2012; Posted February 09, 2012; Published April 04, 2012; Online April 05, 2012

The paper reviews a recently developed finite chain model for the weakest-link statistics of strength, lifetime, and size effect of quasi-brittle structures, which are the structures in which the fracture process zone size is not negligible compared to the cross section size. The theory is based on the recognition that the failure probability is simple and clear only on the nanoscale since the probability and frequency of interatomic bond failures must be equal. The paper outlines how a small set of relatively plausible hypotheses about the failure probability tail at nanoscale and its transition from nano- to macroscale makes it possible to derive the distribution of structural strength, the static crack growth rate, and the lifetime distribution, including the size and geometry effects [while an extension to fatigue crack growth rate and lifetime, published elsewhere (Le and Bažant, 2011, “Unified Nano-Mechanics Based Probabilistic Theory of Quasibrittle and Brittle Structures: II. Fatigue Crack Growth, Lifetime and Scaling,” J. Mech. Phys. Solids, 1322–1337), is left aside]. A salient practical aspect of the theory is that for quasi-brittle structures the chain model underlying the weakest-link statistics must be considered to have a finite number of links, which implies a major deviation from the Weibull distribution. Several new extensions of the theory are presented: (1) A derivation of the dependence of static crack growth rate on the structure size and geometry, (2) an approximate closed-form solution of the structural strength distribution, and (3) an effective method to determine the cumulative distribution functions (cdf’s) of structural strength and lifetime based on the mean size effect curve. Finally, as an example, a probabilistic reassessment of the 1959 Malpasset Dam failure is demonstrated.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Facture of a nanoscale element (a) disordered nanoparticle network and (b) atomic lattice block [19]

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Figure 2

Approximation of grafted Weibull-Gaussian cdf of strength by the Taylor series expansion

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Figure 3

Mean size effects on structural strength and lifetime

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Figure 4

Optimum fits of strength and lifetime histograms

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Figure 5

Simplified 2D model of the Malpassets Dam

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Figure 6

Size effect on the mean nominal strength of the Malpassets Dam

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Figure 7

Size effect on the cdf of nominal strength of the Malpassets Dam




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