Fracture initiation and propagation in intact rock—A review

The initiation and propagation of failure in intact rock are a matter of fundamental importance in rock engineering. At low confining pressures, tensile fracturing initiates in samples at 40%-60% of the uniaxial compressive strength and as loading continues, and these tensile fractures increase in density, ultimately coalescing and leading to strain localization and macro-scale shear failure of the samples. The Griffith theory of brittle failure provides a simplified model and a useful basis for discussion of this process. The Hoek-Brown failure criterion provides an acceptable estimate of the peak strength for shear failure but a cutoff has been added for tensile conditions. However, neither of these criteria adequately explains the progressive coalition of tensile cracks and the final shearing of the specimens at higher confining stresses. Grain-based numerical models, in which the grain size distributions as well as the physical properties of the component grains of the rock are incorporated, have proved to be very useful in studying these more complex fracture processes.

Markov Jump Processes in Estimating Sharing of Identity by Descent

Identity by descent(IBD)sharing is a very important genomic feature in population genetics which can be used to reconstruct recent demographic history.In this paper we provide a framework to estimate IBD sharing for a demographic model called two-population model with migration.We adopt the structured coalescent theory and use a continuous-time Markov jump process{X(t),t≥0}to describe the genealogical process in such model.Then we apply Kolmogorov backward equation to calculate the distribution of coalescence time and develop a formula for estimating the IBD sharing.The simulation studies show that our method to estimate IBD sharing for this demographic model is robust and accurate.

Reconstruct recent multi-population migration history by using identical-by-descent sharing

Identical-by-descent(IBD)is a fundamental genomic characteristic in population genetics and has been widely used for population history reconstruction.However,limited by the nature of IBD,which could only capture the relationship between two individuals/haplotypes,existing IBD-based history inference is constrained to two populations.In this study,we propose a framework by leveraging IBD sharing in multipopulation and develop a method,MatrixiBD,to reconstruct recent multi-population migration history.Specifically,we employ the structured coalescent theory to precisely model the genealogical process and then estimate the IBD sharing across multiple populations.Within our model,we establish a theoretical connection between migration history and IBD sharing.Our method is rigorously evaluated through simulations,revealing its remarkable accuracy and robustness.Furthermore,we apply MatrixiBD to Central and South Asia in the Human Genome Diversity Project and successfully reconstruct the recent migration history of three closely related populations in South Asia.By taking into account the IBD sharing across multiple populations simultaneously,MatrixlBD enables us to attain clearer and more comprehensive insights into the history of regions characterized by complex migration dynamics,providing a holistic perspective on intricate patterns embedded within the recent population migration history.