Stochastic Structural Model of Rock and Soil Aggregates by Continuum-Based Discrete Element Method

This paper first presents a stochastic structural model to describe the random geometrical features of rock and soil aggregates. The stochastic structural model uses mixture ratio, rock size and rock shape to construct the microstructures of aggregates,and introduces two types of structural elements (block element and jointed element) and three types of material elements (rock element, soil element, and weaker jointed element)for this microstructure. Then, continuum-based discrete element method is used to study the deformation and failure mechanism of rock and soil aggregate through a series of loading tests. It is found that the stress-strain curve of rock and soil aggregates is nonlinear, and the failure is usually initialized from weaker jointed elements. Finally, some factors such as mixture ratio, rock size and rock shape are studied in detail. The numerical results are in good agreement with in situ test. Therefore, current model is effective for simulating the mechanical behaviors of rock and soil aggregates.

Bayesian model selection for time series using Markov chain Monte Carlo

We present a stochastic simulation technique for subset selection in time series models, based on the use of indicator variables with the Gibbs sampler within a hierarchical Bayesian framework. As an example, the method is applied to the selection of subset linear AR models, in which only significant lags are included. Joint sampling of the indicators and parameters is found to speed convergence. We discuss the possibility of model mixing where the model is not well determined by the data, and the extension of the approach to include non-linear model terms.

Influence of stochastic mesoscopic structure on macroscopic mechanical behavior of brittle material

A newly developed numerical code, MFPA(2D) (Material Failure Process Analysis), is applied to study the influence of stochastic mesoscopic structure on macroscopic mechanical behavior of rock-like materials. A set of uniaxial compression tests has been numerically studied with numerical specimens containing pre-existing crack-like flaw. The numerical results reveal the influence of random mesoscopic structure on failure process of brittle material, which indicates that the variation of failure mode is strongly sensitive to the local disorder feature of the specimen. And the patterns of the crack evolution in the specimens are very different from each other due to the random mesoscopic structure in material. The results give a good explanation for various kinds of fracture modes and peak strength variation observed in laboratory studies with specimens made from the same rock block being statistically homogenous in macro scale. In addition, the evolution of crack is more complicated in heterogeneous cases than in homogeneous cases.

Statistical and stochastic description of microvoid evolution observed in dynamic failure of ductile materials

A brief review is presented of statistical approaches on microdamage evolution. An experimental study of statistical microdamage evolution in two ductile materials under dynamic loading is carried out. The observation indicates that there are large differences in size and distribution of microvoids between these two materials. With this phenomenon in mind, kinetic equations governing the nucleation and growth of microvoids in nonlinear rate-dependent materials are combined with the balance law of void number to establish statistical differential equations that describe the evolution of microvoids' number density. The theoretical solution provides a reasonable explanation of the experimentally observed phenomenon. The effects of stochastic fluctuation which is influenced by the inhomogeneous microscopic structure of materials are subsequently examined (i.e. stochastic growth model). Based on the stochastic differential equation, a Fokker-Planck equation which governs the evolution of the transition probability is derived. The analytical solution for the transition probability is then obtained and the effects of stochastic fluctuation is discussed. The statistical and stochastic analyses may provide effective approaches to reveal the physics of damage evolution and dynamic failure process in ductile materials.

Convergence of attractors

The system of coupled oscillators and its time-discretization (with constant stepsize h) are considered in this paper. Under some conditions, it is showed that the discrete systems have one-dimensional global attractors l(h) converging to l which is the global attractor of continuous system.

A theoretical analysis on stochastic behaviour of short cracks and fatigue damage of metals

In this paper, the closed form of solution to the stochastic differential equation for a fatigue crack evolution system is derived. and the relationship between metal fatigue damage and crack stochastic behaviour is investigated. It is found that the damage extent of metals is independent of crack stochastic behaviour ii the stochastic deviation of the crack growth rate is directly proportional to its mean value. The evolution of stochastic deviation of metal fatigue damage in the stage close to the transition point between short and long crack regimes is also discussed.

Convergence of the auxiliary particle implementation of the PHD filter

Optimal Bayesian multi-target filtering is in general computationally impractical owing to the high dimensionality of the multi-target state. The Probability Hypothesis Density (PHD) filter propagates the first moment of the multi-target posterior distribution. While this reduces the dimensionality of the problem, the PHD filter still involves intractable integrals in many cases of interest. Several authors have proposed Sequential Monte Carlo (SMC) implementations of the PHD filter. However, these implementations are the equivalent of the Bootstrap Particle Filter, and the latter is well known to be inefficient. Drawing on ideas from the Auxiliary Particle Filter (APF), a SMC implementation of the PHD filter which employs auxiliary variables to enhance its efficiency was proposed by Whiteley et. al. Numerical examples were presented for two scenarios, including a challenging nonlinear observation model, to support the claim. This paper studies the theoretical properties of this auxiliary particle implementation. $\mathbb{L}_p$ error bounds are established from which almost sure convergence follows.

A stochastic jump and deterministic dynamics model of impact failure evolution with rate effect

Motivated by the observation of the rate effect on material failure, a model of nonlinear and nonlocal evolution is developed, that includes both stochastic and dynamic effects. In phase space a transitional region prevails, which distinguishes the failure behavior from a globally stable one to that of catastrophic. Several probability functions are found to characterize the distinctive features of evolution due to different degrees of nucleation, growth and coalescence rates. The results may provide a better understanding of material failure.