Scaling of Structural Failure

Bažant, Zdeněk P.; Chen, Er-Ping

doi:10.1115/1.3101672

Cited by 293 publications

(126 citation statements)

References 191 publications

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“…3b). That formula, originally derived by energy arguments of fracture mechanics (34)(35)(36), was in turn shown to fit closely the size effect tests of mean flexural strength of various concretes and polymer-fiber composites collected from the literature (Fig. 3b) and also to agree closely with finite element simulations based on the nonlocal Weibull theory (39).…”

Section: Mean Size Effect Curvesupporting

confidence: 50%

Mechanics-based statistics of failure risk of quasibrittle structures and size effect on safety factors

Bažant

Pang

2006

Proc. Natl. Acad. Sci. U.S.A.

106

108

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In mechanical design as well as protection from various natural hazards, one must ensure an extremely low failure probability such as 10 ؊6 . How to achieve that goal is adequately understood only for the limiting cases of brittle or ductile structures. Here we present a theory to do that for the transitional class of quasibrittle structures, having brittle constituents and characterized by nonnegligible size of material inhomogeneities. We show that the probability distribution of strength of the representative volume element of material is governed by the Maxwell-Boltzmann distribution of atomic energies and the stress dependence of activation energy barriers; that it is statistically modeled by a hierarchy of series and parallel couplings; and that it consists of a broad Gaussian core having a grafted far-left power-law tail with zero threshold and amplitude depending on temperature and load duration. With increasing structure size, the Gaussian core shrinks and Weibull tail expands according to the weakest-link model for a finite chain of representative volume elements. The model captures experimentally observed deviations of the strength distribution from Weibull distribution and of the mean strength scaling law from a power law. These deviations can be exploited for verification and calibration. The proposed theory will increase the safety of concrete structures, composite parts of aircraft or ships, microelectronic components, microelectromechanical systems, prosthetic devices, etc. It also will improve protection against hazards such as landslides, avalanches, ice breaks, and rock or soil failures.cohesive fracture ͉ extreme value statistics ͉ activation energy ͉ Maxwell-Boltzmann ͉ scaling

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Section: Mean Size Effect Curvesupporting

confidence: 50%

Mechanics-based statistics of failure risk of quasibrittle structures and size effect on safety factors

Bažant

Pang

2006

Proc. Natl. Acad. Sci. U.S.A.

106

108

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“…1 applies to many geometries. It was verified experimentally and justified theoretically for a surprisingly broad range of many different materials and structures (9,(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23). The law was derived by asymptotic approximations of equivalent LEFM (9,15,18,24), of the cohesive crack model (9,12) and of J-integral (25) and was verified by numerical simulations with the crack band model, nonlocal-damage model, and various types of randomparticle or lattice models.…”

Section: Scaling Laws and Their Asymptotic Supportmentioning

confidence: 74%

“…The size-effect that bridges types 1 and 2 (i.e., for short notches or short cracks comprised within the boundary layer) is more complex. Using similar asymptotic matching procedures, one can construct the universal size-effect law (17). Fig.…”

mentioning

confidence: 99%

Scaling theory for quasibrittle structural failure

Bažant

2004

Proc. Natl. Acad. Sci. U.S.A.

232

147

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This inaugural article has a twofold purpose: (i) to present a simpler and more general justification of the fundamental scaling laws of quasibrittle fracture, bridging the asymptotic behaviors of plasticity, linear elastic fracture mechanics, and Weibull statistical theory of brittle failure, and (ii) to give a broad but succinct overview of various applications and ramifications covering many fields, many kinds of quasibrittle materials, and many scales (from 10 ؊8 to 10 6 m). The justification rests on developing a method to combine dimensional analysis of cohesive fracture with second-order accurate asymptotic matching. This method exploits the recently established general asymptotic properties of the cohesive crack model and nonlocal Weibull statistical model. The key idea is to select the dimensionless variables in such a way that, in each asymptotic case, all of them vanish except one. The minimal nature of the hypotheses made explains the surprisingly broad applicability of the scaling laws. Discovery of the concept of stress and strength by Galileo (1) may, in retrospect, be regarded as the first scaling theory of solid mechanics. This theory, now known to apply only to elastoplastic behavior, captures the fact that, under controlled load P, geometrically similar structures of different sizes D fail at the same nominal stress N , defined as the maximum stress in the structure (if no stress singularities exist) or simply as N ϭ P͞D 2 or P͞bD for three-or two-dimensional scaling (b ϭ structure thickness). Any departure from such scaling came to be known as the ''size effect.'' About 350 years ago, Mariotte (2) pointed out that a size effect must arise because the local material strength is random and its minimum encountered in a structure decreases with D. Nevertheless, proper mathematical formulation of this idea had to wait until 1939. That year, Weibull (3) experimentally demonstrated that N for brittle structures has the probability distribution that came to bear his name, although already in 1928 this distribution was derived by Fischer and Tippett (4) as the only possible limiting distribution with a threshold for the minimum of a set of n independent random variables for n 3 ϱ. The tail of this distribution is a power law, from which Weibull deduced that the statistical size effect is also a power law, the exponent of which is a function of the coefficient of variation of material strength.For half a century afterward, whenever a size effect was observed, it was generally attributed to Weibull theory (3, 5, 6), which was amply confirmed for fatigued metals and fine-grained ceramics. However, beginning with studies of concrete for nuclear reactors during 1970-1985, it gradually transpired that Weibull theory does not apply to materials now termed ''quasibrittle'' (7-9). These are materials in which the fracture process zone (FPZ) is not negligible compared with the cross-section dimension D (and may even encompass the entire cross section).Depending on the scale of observation or application, quasib...

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“…It was tried to explain the reduced exponent value by notches of a finite angle, which however is objectionable for two reasons: (i) notches of a finite angle cannot propagate (rather, a crack must emanate from the notch tip), (ii) the singular stress field of finite-angle notches gives a zero flux of energy into the notch tip. Same as Weibull theory, Leicester's power law also implied nonexistence of a characteristic length (see [3], Equations (1)- (3)), which cannot be the case for concrete due to the large size of its inhomogeneities. More extensive tests of notched geometrically similar concrete beams of different sizes were carried out by Walsh [45,46].…”

Section: Quasibrittle Size Effect Bridging Plasticity and Lefm And Imentioning

confidence: 97%

“…For an in-depth review with several hundred literature references, the recent article by Ba~ant and Chen [3] and Ba2ant's book [40] may be consulted. A full exposition of most of the material reviewed here is found in the recent book by Ba~ant and Planas [4].…”

Section: Introductionmentioning

confidence: 99%

RILEM TC QFS ‘quasibrittle fracture scaling and size effect’-final report

Burtscher¹,

Chiaia²,

Dempsey³

et al. 2004

Mat. Struct.

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The report attempts a broad review of the problem of size effect or scaling of failure, which has recently come to the forefront of attention because of its importance for concrete and geotechnical engineering, geomechanics, arctic ice engineering, as well as in designing large loadbearing parts made of advanced ceramics and composites, e.g. for aircraft or ships. First the main results of Weibtlll statistical theory of random strength are briefly summarized and its applicability and limitations described. In this theory as well as plasticity, elasticity with a strength limit, and linear elastic fracture mechanics (LEFM), the size effect is a simple power law because no characteristic size or length is present. Attention is then focused on the deterministic size effect in quasibrittle materials which, because of the existence of a non-negligible material length characterizing the size of the fracture process zone, represents the bridging between the simple power-law size effects of plasticity and of LEFM. The energetic theory of quasibrittle size effect in the bridging region is explained and then a host of recent refinements, extensions and ramifications are discussed. Comments on other types of size effect, including that which might be associated with the fractal geometry of fracture, are also made. The historical development of the size effect theories is outlined and the recent trends of research are emphasized.

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Scaling of Structural Failure

Cited by 293 publications

References 191 publications

Mechanics-based statistics of failure risk of quasibrittle structures and size effect on safety factors

Mechanics-based statistics of failure risk of quasibrittle structures and size effect on safety factors

Scaling theory for quasibrittle structural failure

RILEM TC QFS ‘quasibrittle fracture scaling and size effect’-final report

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