Currently the strength reduction parts of safety factors are considered as fixed, determined purely from statistics of test data. However, this is adequate only for purely brittle or ductile limiting cases. Here the key idea is that, for the transitional range of quasibrittle structures (concrete structures, fiber composites, rigid foams, ceramics, rocks, ice, soils, wood, etc., and those on approach to nanoscale), the safety factor must depend on structure size and geometry, and that to ensure tolerable failure probability, such as one in a million, the safety factor must be determined from extreme-value statistics based on stress dependence of activation energy of interatomic bonds. The strength probability distributions of material elements and of structure, deviations from power-law size effect, kinks on strength histograms, and dependence on temperature, loading rate and duration, are derived and exploited for verification and calibration. Relevance and practical applications are demonstrated by stochastic numerical analysis of documented structural disasters. Major implications for design codes and practices for quasibrittle structures, as well as reliability of micro-nano-scale thin films and MEMS, are identified.
The strength and lifetime of structures are inevitably random. Their statistical distributions must be known to design structures safely and economically. While traditionally the safety factors in design, defined as the ratio structural strength to the design load, have been set empirically (making structures lighter and lighterm, and more and more daring, until disaster occurred), modern design requires that the required safety factor be computed from the structural geometry and material properties. Great success in this direction has already occurred in the design of ductile structures and brittle structures. In the former, the load capacity remains constant at increasing deflection after the structural strength has been reached, In the latter, the material failure process localizes into a tiny zone, a point-wise fracture process zone, which propagates through the structures, as described by classical fracture mechanics. For such structures, the statistics of strength and lifetime is reasonably well known, and generally leads to the so-called Weibull distribution. The situation is more difficult for the so-called quasibrittle materials, in which material heterogeneity engenders limited ductility and the failure occurs by propagation of a relatively large fracture process zone process zone whose size is not negligible compared to structure dimensions. The quasibrittle behavior is characteristic of concretes, especially modern high-performance concretes, modern tough ceramics and fiber composites, rocks, mortars, see ice, bone and various biomaterials, and most materials on the micrometer scale as in MEMS or computer chips. For quasibrittle materials the correct statistical distribution of material and structural strength and lifetime has not been known, and the safety factors have been purely empirical. In some cases this has led to a higher frequency of failure than acceptable (as in large concrete structures, as historically documented), and in others to uneconomic of timid designs. The present investigation has changed that. It led to the formulations of nano-mechanics based theory of strength and lifetime statistics of quasibrittle structures. By contrast to perfectly brittle structures, it was found theoretically, as well on the basis of test results, the probability distribution of strength of a small material element (precisely, the representative volume element) is mostly Gaussian (or normal), with a remote power-law tail. Most interestingly, with increasing structure size, this distribution was found to gradually change, with the tail growing into the core of distribution in the Weibull form while the Gaussian part is shrinking until, for structures comprising more than 1.E4 representative volume elements, the entire distribution becomes Weibullian. The same was found to be true of the structural lifetime. The consequences are profound. The safety factors should not be considered to be independent of the structure size (and shape), as they are for brittle and ductile structures, but should be varied with the structure size, according to a calculation method developed (based on the weakest-link model with a finite number of links). This new finding affect large concrete structures as well as the new designs of light fully composite airframes for commercial aircraft. Proposals to introduce these new ideas into the structural design practice and codes have been made, and computer methods to carry out this kind of probabilistic design are pursued as an extension of this project. An interesting spin-off was found for the desing oh the new computer chips with high-k dielectrics, about 5 nm thick, which attempt to extend the famous Moore's law. Although physically the problem is totally different, mathematically is it completely analogous, with same kind of statistical distribution describing the electronic breakdown. Various honest broadreach efforts of the usual kind have been made. .
