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for Probabilistic Nano-Mechanical Theory of Quasibrittle Structure Strength, Lifetime and Fatigue
=Zdenek Bazant, Ph.D.=

===April 7, 2011===

Probabilistic Nano-Mechanical Theory of Quasibrittle 
Structure Strength, Lifetime and Fatigue 

Abstract: The size effect on structural strength and its statistical distribution is a complex problem for quasibrittle materials because their failure behavior transits from quasi-plastic at small sizes to brittle at large sizes. These are heterogeneous materials with brittle constituents in which the inhomogeneinty size is not negligible compared to the structure size. They are exemplified by concrete, as the archetypical example, fiber composites, coarse-grained ceramics, rocks, sea ice, wood, bone, foam, masonry etc., and all brittle materials at the micro- or nano-scale. The lecture begins by reviewing the statistical and energetic size effect on the mean strength of quasibrittle structures. The frequency or probability of nano-crack jumps and the multiscale transition to material scale are used to show that the type of probability distribution of structural strength depends on the structure size and geometry. On the scale of the representative volume element of material, the probability distribution of strength is found to be Gaussian, with a remote Weibullian tail. For increasing structure size, the Weibullian portion gradually spreads into the Gaussian core and, for very large sizes, the distribution becomes purely Weibullian. Based on an atomistic derivation of the power law for creep crack growth, it is further shown that a similar change of distribution occurs for structure lifetime. Numerous experimental results and well as numerical simulations support the theory. One practical consequence is that the safety factors for large quasibrittle structures, e.g. concrete structures, airframes or ship hulls made of composites, and ceramic micro-devices, must depend on their size and shape. Another is that the lifetime can be predicted from tests of size effect on the mean short-time strength and of creep crack growth rate. The theory is also extended to the size dependence of Paris law and Basquin law for fatigue fracture and statistics of fatigue lifetime. An interesting mathematical analogy that facilitates assessment of lifetime of new nano-scale high-k dielectrics is pointed out. Finally, the implications for computer analysis of failure probability of large quasibrittle structures are pointed out.

BIO-SKETCH:  Born and educated in Prague (Ph.D. 1963), Bažant joined Northwestern in 1969, where he has been W.P. Murphy Professor since 1990 and simultaneously McCormick Institute Professor since 2002, and Director of Center for Geomaterials (1981-87). He was inducted to National Academy of Sciences, National Academy of Engrg.and Am. Acad. of Arts & Aci., as well as Italian Nat. Acad. (dei Lincei), Austrian Acad. of Sciences, Spanish Royal Acad. of Eng., Czech Acad. of Engrg., Istituto Lombardo and Eur. Acad. of Sci. & Arts. An Hon. Member of ASCE and ACI, and an Illinois Registered Structural Engineer, he received six honorary doctorates (Prague, Karlsruhe, Colorado, Milan, Lyon, Vienna), ASCE von Karman, Newmark, Croes Medals and Lifetime Achievement Award, SES Prager Medal, ASME Timoshenko, Warner and Nadai Medals, RILEM L'Hermite Medal, Torroja Medal, etc. He authored six books: Scaling of Structural Strength, Inelastic Analysis, Fracture and Size Effect, Stability of Structures, Concrete at High Temperatures, and Concrete Creep. With H-index 55 and >14,500 citations, he is one of to original top 100 ISI Highly Cited Scientists in engineering (www.ISIhighlycited.com). 

References:
    Bažant, Z.P. (2002).  Scaling of Structural Strength. 2nd updated ed., Elsevier, London 2005.
    Bažant, Z.P., and Planas, J. (1998).  Fracture and Size Effect in Concrete and Other Quasibrittle Materials. CRC Press. 
    Bažant, Z.P. (2004). “Scaling theory for quasibrittle structural failure.”  Proc., Nat. Acad. of Sciences 101 (37), 14000—14007.
    Bažant, Z.P., and Pang, S.-D. (2006). “Mechanics based statistics of failure risk of quasibrittle structures and size effect on safety factors.” Proc. of the National Academy of Sciences 103(25), 9434-9439.
    Bažant, Z.P., Le, J.-L., and Bazant, M.Z. (2008). “Scaling of strength and lifetime distributions based on atomistic fracture mechanics.”  Proc. of the Nat. Academy of Sciences, 106 (28), 11484-11489.
    Pang, S.-D., Bažant, Z.P., and Le, J.-L. (2009). Statistics of strength of ceramics: finite weakest-link model and necessity of zero threshold.” Int. J. of Fracture 154, 131-145.
    Le, J.-L., and Bažant, Z.P. (2009). “Strength Distribution of Dental Restorative Ceramics: Finite Weakest-Link Model with Zero Threshold.” Journal of Dental Materials, 25, 641--648.
    Le, J.-L., Bažant, Z.P. and Bazant, M.Z. (2009). “Subcritical Crack Growth Law and Its Consequences for Lifetime Statistics and Size Effect of Quasibrittle Structures. J. of Physics D: Applied Physics 42, 214008.
    Bažant, Z.P., and Le, J.-L. (2009). “Nano-mechanics based modeling of lifetime distribution of quasibrittle structures”. Engineering Failure Analysis 16, 2521-–2529.
    Le, J.-L., Bažant, Z.P., and Bazant, M.Z. (2009). “Lifetime of high-k gate dielectrics and analogy with strength of quasi-brittle structures”. J. of Applied Physics 106, paper 104119, 1—8.
    Le, J.-L., Bažant, Z.P., and Bazant, M.Z. (2011). “Unified Nano-Mechanics Based Probabilistic Theory of Quasibrittle and Brittle Structures”. J. of the Mechanics and Physics of Solids. in press.
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