Estimating parameters of a multipartite loglinear graph model via the EM algorithm

We will amalgamate the Rash model (for rectangular binary tables) and the newly introduced $\alpha$-$\beta$ models (for random undirected graphs) in the framework of a semiparametric probabilistic graph model. Our purpose is to give a partition of the vertices of an observed graph so that the generated subgraphs and bipartite graphs obey these models, where their strongly connected parameters give multiscale evaluation of the vertices at the same time. In this way, a heterogeneous version of the stochastic block model is built via mixtures of loglinear models and the parameters are estimated with a special EM iteration. In the context of social networks, the clusters can be identified with social groups and the parameters with attitudes of people of one group towards people of the other, which attitudes depend on the cluster memberships. The algorithm is applied to randomly generated and real-word data

Towards a unified framework for modeling fault zone evolution: From particles comminution to secondary faults branching

The brittle portion of the crust contains structural features such as faults, jogs, joints, bends, and cataclastic zones that span a wide range of length scales. These features may have a profound effect on earthquake nucleation, propagation, and arrest. Incorporating these existing features in modeling and the ability to spontaneously generate new one in response to earthquake loading is crucial for predicting seismicity patterns, distribution of aftershocks and nucleation sites, earthquakes arrest mechanisms, and topological changes in the seismogenic zone structure. Here, we report on our efforts in modeling two important mechanisms contributing to the evolution of fault zone topology: (i) Grain comminution at the submeter scale, and (ii) Secondary faults generation at the scale of few to hundreds of meters. We model grain comminution within the framework of Shear Transformation Zone theory, a nonequilibrium statistical thermodynamic framework for modeling plastic deformation in amorphous materials. We postulate, based on energy balance, an equation for the grain size reduction as a function of the applied work rate and pressure. We show that grain breakage is a potential weakening mechanism at high strain rate. It promotes strain localization and may explain the long-term persistence of shear bands in natural faults. To model secondary faults generation we developed a nested fault scheme using the finite element software PyLith. As the dynamic rupture propagates on the main fault the stress state changes and eventually the off-fault shear stress is high enough to overcome the pressure-dependent rock strength defined by the Mohr–Coulomb failure envelope. If the Mohr–Coulomb failure criterion is satisfied, a new secondary fault is generated. The angle of the secondary fault with respect to the main fault is taken to be equal to the angle of the critical shear plane. This procedure is repeated until there is no need to add new faults (i.e., stresses everywhere are below the failure threshold). The secondary faults relax the medium contributing to slip and energy partitioning. They also lead to wave diffraction, slip heterogeneity, and slowing down of the rupture on the main fault. They provide potential nucleation site for future ruptures promoting complexity in earthquake cycle simulation. Under repeated earthquake ruptures, regions in the vicinity of primary slip surfaces become heavily fragmented. These regions are modeled using STZ theory. Incorporating the microscale granular model within the macroscopic finite element simulation provide a physics-based multiscale description for damage accumulation. The model provides insight into the dynamic evolution of fault zone topology coupled within the different phases of the seismic cycle. This is crucial for better evaluation of seismic hazard and risk

Scaling laws for stress and energy for an interface with strong rate weakening friction

Strong rate weakening friction has been hypothesized to play an important role in the dynamics of fault zones at seismic slip rates. Here, we explore its implications on the scaling of different mechanical quantities. These include the average prestress, the average stress drop, the frictional dissipation, and magnitude number statistics. We idealize the frictional interface as a series of sliders that are being pulled from one edge under displacement controlled boundary conditions. The friction for each block decreases inversely with the block slip rate. The system response is governed by three parameters: the system elastic modulus, the loading rate, and the rate of frictional weakening. We have found that the average stress before rupture as well as the frictional dissipation decrease with increasing rupture size. We have also observed a transition from a macroscopic plastic response to a macroscopic brittle response as the system parameters vary. We discuss the implications of our results on size effects in nominal strength for solids with frictional interfaces

Crack Propagation in Bone on the Scale of Mineralized Collagen Fibrils : Role of Polymers with Sacrificial Bonds and Hidden Length

Sacrificial bonds and hidden length (SBHL) in structural molecules provide a mechanism for energy dissipation at the nanoscale. It is hypothesized that their presence leads to greater fracture toughness than what is observed in materials without such features. Here, we investigate this hypothesis using a simplified model of a mineralized collagen fibril sliding on a polymeric interface with SBHL systems. A 1D coarse-grained nonlinear spring-mass system is used to model the fibril. Rate-and-displacement constitutive equations are used to describe the mechanical properties of the polymeric system. The model quantifies how the interface toughness increases as a function of polymer density and number of sacrificial bonds. Other characteristics of the SBHL system, such as the length of hidden loops and the strength of the bonds, are found to influence the results. The model also gives insight into the variations in the mechanical behavior in response to physiological changes, such as the degree of mineralization of the collagen fibril and polymer density in the interfibrillar matrix. The model results provide constraints relevant for bio-mimetic material design and multiscale modeling of fracture in human bone