State-of-the-art report: Fracture mechanics of concrete: concepts, models and determination of material properties. Why fracture mechanics? Essential results from linear elastic fracture mechanics. Nonlinear fracture models with softening zone. Special nonlinear fracture models based on adaptions of LEFM. Size effect and brittleness of structures. Experimental or analytical determination of material fracture parameters. Factors influencing fracture parameters. Effect of reinforcement. Crack systems. Concluding remarks. References and bibliography. Appendix 1- derivations of some formulas. Conference papers. Material models for concrete fracture. Damage modelling. Numerical analysis of concrete fracture. Experimental methods and determination of fracture characteristics. Measurements of damage and size effect. Dynamic fracture. Fracture under shear. Fracture of reinforced concrete. Interaction between concrete and reinforcement. Fatigue and rate effects. Environment effects [temperature, shrinkage, corrosion]. Index.

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Fracture mechanics of concrete structures

This article attempts to review the progress achieved in the understanding of scaling and size ef­ fect in the failure of structures. Particular emphasis is placed on quasi brittle materials for which the size etTect is important and complicated. After reflections on the long history of size effect studies, attention is focused on three main types of size effects, namely the statistical size effect due to randomness of strength, the energy release size effect, and the possible size effect due to fractality of fracture or microcracks. Definitive conclusions on the applicability of these theories are drawn. Subsequently, the article discusses the application of the known size effect law for the measurement of material fracture properties, and the modeling of the size effect by the cohesive crack model, non local finite element models and discrete element models. Extensions to com­ pression failure and to the rate-dependent material behavior are also outlined. The damage con­ stitutive law needed for describing a microcracked material in the fracture process zone is dis­ cussed. Various applications to quasibrittle materials, including concrete, sea ice, fiber compos­ ites, rocks and ceramics are presented. There are 377 references included in this article.

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Scaling of Structural Failure

This paper extends the theoretical framework presented in the preceding Part I to the lifetime distribution of quasibrittle structures failing at the fracture of one representative volume element under constant amplitude fatigue. The probability distribution of the critical stress amplitude is derived for a given number of cycles and a given minimum-to-maximum stress ratio. The physical mechanism underlying the Paris law for fatigue crack growth is explained under certain plausible assumptions about the damage accumulation in the cyclic fracture process zone at the tip of subcritical crack. This law is then used to relate the probability distribution of critical stress amplitude to the probability distribution of fatigue lifetime. The theory naturally yields a power-law relation for the stress-life curve (S-N curve), which agrees with Basquin's law. Furthermore, the theory indicates that, for quasibrittle structures, the S-N curve must be size dependent. Finally, physical explanation is provided to the experimentally observed systematic deviations of lifetime histograms of various ceramics and bones from the Weibull distribution, and their close fits by the present theory are demonstrated.

http://www.civil.northwestern.edu/people/bazant/PDFs/Papers/500-501.pdf

Unified nano-mechanics based probabilistic theory of quasibrittle and brittle structures: I. Strength, static crack growth, lifetime and scaling

A fractal analysis of size effect on fatigue crack growth

"Size effects on fracture and localization: Aper,cu of recent advances and their extension to simultaneous fatigue and rate sensitivity" in Fracture Pro

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Fracture Processes in Concrete. Rock and Ceramics

Crack growth caused by load repetitions in geometrically similar notched concrete specimens of various sizes is measured by means of the compliance method. It is found that the Paris law, which states that the crack length increment per cycle is a power function of the stress intensity factor amplitude, is valid only for one specimen size (the law parameters being adjusted for that size) or asymptotically, for very large specimens. To obtain a general law, the Paris law is combined with size-effect law for fracture under monotonic loading, proposed previously by Bazant. This leads to a size-adjusted Paris law, which gives the crack length increment per cycle as a power function of the amplitude of size-adjusted stress intensity factor. The crack growth is also characterized in terms of the nominal stress amplitude.

Kangming Xu

Papers

Size Effect in Fatigue Fracture of Concrete