This book presents an experimentally validated probabilistic strength theory of structures made of concrete, composites, ceramics and other quasibrittle materials.

The statistics of safety factors and structural reliability has so far been generally regarded as independent of mechanics, but further progress requires the cumulative distribution function (cdf) of structural strength to be derived from the thermomechanics of bond breaks. The cdf of strength of quasibrittle structures of positive geometry is modelled as series coupling of representative volume elements (RVE), each of which is statistically represented by a hierarchical model, connecting to the nano-scale of atomic lattice. Based on Maxwell-Boltzmann distribution of thermal energies of atoms, the cdf of strength of a nano-scale connection is deduced from the stress dependence of the interatomic activation energy barriers, and is expressed as a function of absolute temperature and stress-duration (or loading rate). The power-law tail exponent, which is 1 on the atomistic scale, is raised by the hierarchical statistical model to exponent m = 10 to 50, representing the Weibull modulus on the structural scale. Its physical meaning is shown to be the minimum number of cuts needed to separate the hierarchical model into two separate parts, which should be equal to the number of dominant cracks needed to break the RVE, governed by the packing of inhomogeneities. On the RVE scale, the model yields a broad Gaussian core, onto which a short power-law tail of exponent m is grafted. The model predicts how the grafted cdf depends on T and t* and provides a physical proof that, on a large enough scale, quasibrittle structures must follow Weibull distribution with a zero threshold. Experimental histograms with kinks, which were previously thought to require the use of a finite threshold, are shown to be fitted much better by the present chain-of-RVEs model. For not too small structures, the model is also shown to be essentially a discrete equivalent of the previously developed nonlocal Weibull theory. The theory indicates the need to refine the reliability indices and identify the common errors in applying Weibull statistical theory to scatter and size effect of quasibrittle structures. Stochastic finite element method is proposed to conduct computer simulations of extreme value statistics, crucial for safe design of structures.

This latest volume in the Foundations & Philosophy of Science & Technology series provides an account of probabilistic functional analysis and shows its applicability in the formulation of the behaviour of discrete media with the inclusion of microstructural effects. Although quantum mechanics have long been recognized as a stochastic theory, the introduction of probabilistic concepts and principles to classical mechanics has in general not been attempted. In this study the author takes the view that the significant field quantities of a discrete medium are random variables or functions of s ...

Well-written introduction covers probability theory from two or more random variables, reliability of such multivariable structures, theory of random function, Monte Carlo methods for problems incapable of exact solution, more.

Most industrial and natural materials exhibit a macroscopic behaviour which results from the existence of microscale inhomogeneities. The influence of such inhomogeneities is commonly modelled using probabilistic methods. Most of the approaches to the evaluation of the safety of structures according to probabilistic criteria are somewhat scattered. This text attempts to present such material in a coherent and up-to-date form, and also defines the great tasks that must be tackled in coming years.