Fujiu Ke

Numerical Simulation of Rock Failure and Earthquake Process on Mesoscopic Scale

Abstract—On the basis of the lattice model of Mora and Place, Discrete Element Method, and Molecular Dynamics approach, another kind of numerical model is developed. The model consists of a 2-D set of particles linked by three kinds of interactions and arranged into triangular lattice. After the fracture criterion and rules of changes between linking states are given, the particle positions, velocities and accelerations at every time step are calculated using a finite-difference scheme, and the configuration of particles can be gained step by step. Using this model, realistic fracture simulations of brittle solid (especially under pressure) and simulation of earthquake dynamics are made.n

Evolution of Localized Damage Zone in Heterogeneous Media

Evolution of localized damage zone is a key to catastrophic rupture in heterogeneous materials. In the present article, the evolutions of strain fields of rock specimens are investigated experimentally. The observed evolution of fluctuations and autocorrelations of strain fields under uniaxial compression demonstrates that the localization of deformation always appears ahead of catastrophic rupture. In particular, the localization evolves pronouncedly with increasing deformation in the rock experiments. By means of the definition of the zone with high strain rate and likely damage localization, it is found that the size of the localized zone decreases from the sample size at peak load to an eventual value. Actually, the deformation field beyond peak load is bound to suffer bifurcation, namely an elastic unloading part and a continuing but localized damage part will co-exist in series in a specimen. To describe this continuous bifurcation and localization process observed in experiments, a model on continuum mechanics is developed. The model can explain why the decreasing width of localized zone can lead stable deformation to unstable, but it still has not provided the complete equations governing the evolution of the localized zone.

Evolution-induced Catastrophe and its Predictability

Both earthquake prediction and failure prediction of disordered brittle media are difficult and complicated problems and they might have something in common. In order to search for clues for earthquake prediction, the common features of failure in a simple nonlinear dynamical model resembling disordered brittle media are examined. It is found that the failure manifests evolution-induced catastrophe (EIC), i.e., the abrupt transition from globally stable (GS) accumulation of damage to catastrophic failure. A distinct feature is the significant uncertainty of catastrophe, called sample-specificity. Consequently, it is impossible to make a deterministic prediction macroscopically. This is similar to the question of predictability of earthquakes. However, our model shows that strong stress fluctuations may be an immediate precursor of catastrophic failure statistically. This might provide clues for earthquake forecasting.

Adaptive Mesh Refinement FEM for Damage Evolution of Heterogeneous Brittle Media

Damage evolution of heterogeneous brittle media involves a wide range of length scales. The coupling between these length scales is the underlying mechanism of damage evolution and rupture. However, few of previous numerical algorithms consider the effects of the trans-scale coupling effectively. In this paper, an adaptive mesh refinement finite element method (FEM) algorithm is developed to simulate this trans-scale coupling. The adaptive serendipity element is implemented in this algorithm, and several special discontinuous base functions are created to avoid the incompatible displacement between the elements. Both the benchmark and a typical numerical example under quasi-static loading are given to justify the effectiveness of this model. The numerical results reproduce a series of characteristics of damage and rupture in heterogeneous brittle media.

Evolution Induced Catastrophe in a Nonlinear Dynamical Model of Material Failure

In order to study the failure of disordered materials, theensemble evolution of a nonlinear chain model was examined by using astochastic slice sampling method. The following results were obtained.(1) Sample-specific behavior, i.e. evolutions are different from sampleto sample in some cases under the same macroscopic conditions, isobserved for various load-sharing rules except in the globally meanfield theory. The evolution according to the cluster load-sharing rule,which reflects the interaction between broken clusters, cannot bepredicted by a simple criterion from the initial damage pattern and eventhen is most complicated. (2) A binary failure probability, itstransitional region, where globally stable (GS) modes andevolution-induced catastrophic (EIC) modes coexist, and thecorresponding scaling laws are fundamental to the failure. There is asensitive zone in the vicinity of the boundary between the GS and EICregions in phase space, where a slight stochastic increment in damagecan trigger a radical transition from GS to EIC. (3) The distribution ofstrength is obtained from the binary failure probability. This, likesample-specificity, originates from a trans-scale sensitivity linkingmeso-scopic and macroscopic phenomena. (4) Strong fluctuations in stressdistribution different from that of GS modes may be assumed as aprecursor of evolution-induced catastrophe (EIC).

Control dynamics of severe acute respiratory syndrome transmission

Severe acute respiratory syndrome (SARS) is a serious disease with many puzzling features. We present a simple, dynamic model to assess the epidemic potential of SARS and the effectiveness of control measures. With this model, we analysed the SARS epidemic data in Beijing. The data fitting gives the basic case reproduction number of 2.16 leading to the outbreak, and the variation of the effective reproduction number reflecting the control effect. Noticeably, our study shows that the response time and the strength of control measures have significant effects on the scale of the outbreak and the lasting time of the epidemic.

Catastrophic Rupture Induced Damage Coalescence in Heterogeneous Brittle Media

In heterogeneous brittle media, the evolution of damage is strongly influenced by the multiscale coupling effect. To better understand this effect, we perform a detailed investigation of the damage evolution, with particular attention focused on the catastrophe transition. We use an adaptive multiscale finite-element model (MFEM) to simulate the damage evolution and the catastrophic failure of heterogeneous brittle media. Both plane stress and plane strain cases are investigated for a heterogeneous medium whose initial shear strength follows the Weibull distribution. Damage is induced through the application of the Coulomb failure criterion to each element, and the element mesh is refined where the failure criterion is met. We found that as damage accumulates, there is a stronger and stronger nonlinear increase in stress and the stress redistribution distance. The coupling of the dynamic stress redistribution and the heterogeneity at different scales result in an inverse cascade of damage cluster size, which represents rapid coalescence of damage at the catastrophe transition.

Characteristic dimensionless numbers in multi-scale and rate-dependent processes

Abstract Multi-scale modeling of materials properties and chemical processes has drawn great attention from science and engineering. For these multi-scale and rate-dependent processes, how to characterize their trans-scale formulation is a key point. Three questions should be addressed:n • How do multi-sizes affect the problems? • How are length scales coupled with time scales? • How to identify emergence of new structure in process and its effect? For this sake, the macroscopic equations of mechanics and the kinetic equations of the microstructural transformations should form a unified set that be solved simultaneously. As a case study of coupling length and time scales, the trans-scale formulation of wave-induced damage evolution due to mesoscopic nucleation and growth is discussed. In this problem, the trans-scaling could be reduced to two independent dimensionless numbers: the imposed Deborah number De*=(ac*)/(LV*) and the intrinsic Deborah number D* = (nN* c*5)/V*, where a, L, c*, V* and nN* are wave speed, sample size, microcrack size, the rate of microcrack growth and the rate of microcrack nucleation density, respectively. Clearly, the dimensionless number De*=(ac*)/(LV*) includes length and time scales on both meso- and macro- levels and governs the progressive process. Whereas, the intrinsic Deborah number D* indicates the characteristic transition of microdamage to macroscopic rupture since D* is related to the criterion of damage localization, which is a precursor of macroscopic rupture. This case study may highlight the scaling in multi-scale and rate-dependent problems. Then, more generally, we compare some historical examples to see how trans-scale formulations were achieved and what are still open now. The comparison of various mechanisms governing the enhancement of meso-size effects reminds us of the importance of analyzing multi-scale and rate-dependent processes case by case. For multi-scale and rate-dependent processes with chemical reactions and diffusions, there seems to be a need of trans-scale formulation of coupling effect of multi-scales and corresponding rates. Perhaps, two trans-scale effects may need special attention. One is to clarify what dimensionless group is a proper trans-scale formulation in coupled multi-scale and rate-dependent processes with reactions and diffusion. The second is the effect of emergent structures and its length scale effect.

Experimental Evidence of Critical Sensitivity in Catastrophe

The paper presents an experimental study on critical sensitivity in rocks. Critical sensitivity means that the response of a system to external controlling variable may become significantly sensitive as the system approaches its catastrophic rupture point. It is found that the sensitivities measured by responses on three scales (sample scale, locally macroscopic scales and mesoscopic scale) display increase prior to catastrophic transition point. These experimental results do support the concept that critical sensitivity might be a common precursory feature of catastrophe. Furthermore, our previous theoretical model is extended to explore the fluctuations in critical sensitivity in the rock tests.