A DCAMM seminar will be presented by
Professor Zdenek P. Bažant
Civil and Mechanical Engineering and Materials Science
Northwestern University, USA
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 inhomogeneity 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. Kramer’s rule of transition rate theory for the frequency or probability of nano-crack jumps, and some new simple rules for 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. The structure size effect is based on the weakest-link statistics for a chain whose length is not infinite but finite. 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.
The theory is further extended to the size dependence of Paris’ law and Basquin’s law for fatigue fracture and statistics of fatigue lifetime. Based only on a few common hypotheses, the theory describes well the existing experimental results on the monotonic strength, static and fatigue crack growth rate, and static and fatigue lifetimes, including their distributions and size effects on the distributions. 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 static and fatigue lifetimes can be predicted from tests of size effect on the mean short-time strength and of crack growth rate. An interesting mathematical analogy predicting the lifetime of new nano-scale high-k dielectrics is pointed out. Finally, the implications for computer analysis of quasibrittle structures are outlined.