Stochastic Fracture Model Research Articles

Advanced tensile fracture analysis of alumina ceramics: Integrating hybrid finite-discrete element modeling with experimental insights

This study offers novel insights into the Mode I tensile response of an alumina ceramic through the use of computational modeling and the flattened Brazilian disk (FBD) experiments. A modified hybrid finite-discrete element method (HFDEM) is developed, integrating a coupled damage and friction cohesive model and a microscopic stochastic fracture model with a Weibull strength distribution by Monte Carlo simulation. The model is used to simulate direct tensile failure processes under quasi-static loading conditions, providing qualitative and quantitative predictions of direct tensile failure processes of an alumina ceramic. Concurrently, quasi-static flattened Brazilian disk tests (indirect tensile tests) are performed on a standard MTS machine coupled with a high-speed camera. The modified HFDEM model is also applied to reproduce the FBD experiments, and our simulated tensile strength is consistent with the experimental results. The results of the modified HFDEM model show three kinds of phenomena (i.e., “underestimation”, “reasonable estimation”, and “overestimation” of the indirect tensile strength) and four different types of associated fracture and fragment patterns of FBD testing. The integration of simulation and experimental results reveal relationships between fracture patterns, fragment geometry, tensile strength, and indirect tensile strength. The fracture and fragmentation patterns derived from our modified HFDEM model can be utilized to analyze the “tensile strength” measured in BD testing. Overall, this research offers important insights and direction for future Brazilian disk experiments and tensile strength assessments, enhancing our comprehension of ceramic failure mechanisms under tensile loadings.

A Unified concrete damage model for monotonic loading and fatigue loading

AbstractIn this work, a unified damage model is proposed for describing the mechanical behavior of concrete under monotonic loading and fatigue loading. Based on the micro‐meso stochastic fracture model (MMSF), the multiscale energy dissipation within the microspring is investigated. The rate process theory and a crack hierarchy model are employed to describe the nanocrack growth rate and crack number, respectively. It is found out that the fracture strain of the microspring can be derived from the energy dissipation analysis. The expression of the stochastic damage evolution law is changed to “energy dissipation threshold” instead of “fracture strain.” The new model can describe different damage evolution forms under monotonic and fatigue loading with the same parameters. The mechanisms of different damage evolution laws under different loading modes are analyzed in detail. Finally, a numerical example is presented to verify the proposed model.

A unified stochastic damage model for concrete under monotonic and fatigue loading

This study proposes a unified concrete damage model that can be applied to both monotonic and fatigue loading scenarios. The proposed model adopts the framework of stochastic damage mechanics and considers the micro-spring in the micro-meso stochastic fracture model (MMSF) as an energy dissipation element. To describe the multiscale energy dissipation process of the micro-spring, the rate process theory and a crack hierarchy model are employed. It is found that the energy dissipation threshold is linked to the elastic strain energy of the micro-spring, and the derived fracture strain of the micro-spring in MMSF enables the proposed model to describe the damage evolution laws for both monotonic loading and fatigue loading. The underlying damage mechanisms associated with the two loading processes are analyzed in detail. Additionally, the energy dissipation analysis enables the definition of the corresponding micro-damage to the micro-spring, which leads to the demonstration of the nonlinear properties of the micro-spring's constitutive law. Finally, several numerical examples are presented to validate the model.

Identification of probabilistic distribution parameters for the mesoscopic stochastic fracture model

As a kind of multiphase composite material, the basic mechanical behaviors of concrete are randomness and nonlinearity. The mesoscopic stochastic fracture model (MSFM) which can reflect the coupling effect of randomness and nonlinearity, has been widely used for the nonlinear analysis of concrete structures. In this paper, we proposed a new stochastic modeling principle to identify the probabilistic distribution parameters of MSFM. In order to reduce the modeling works, a dimension-reduced algorithm is proposed as well. In this paper, an overview of MSFM is firstly presented to introduce the background of the research. Then the stochastic harmonic function (SHF) representation is introduced to express the random field mentioned in the MSFM, and the probability density evolution method (PDEM) is applied to obtain the theoretical probability density function (PDF) of the stress–strain relationships. Furthermore, a stochastic modeling principle is proposed, in which minimizing the Kullback–Leibler divergence (KLD) is taken as the optimization object. Based on the framework of genetic algorithm, a dimension-reduced algorithm is proposed to identify the parameters with reference to the data from tested complete curves of uniaxial compressive and uniaxial tensile stress–strain relationship of concrete. The results indicate that the proposed principle and algorithm can be used to identify the parameters of MSFM accurately and efficiently.

Numerical Simulation and Analysis of Tunnel Failure Mode by Stochastic Fracture Model

The dip angle, length, spacing, and fracture distance of rock fissure affect the morphology of roadway after collapse. The numerical simulation software CDEM is used to simulate the morphology of roadway collapse. The Monte Carlo model is used to simulate different types of crack models in two-dimensional plane and generate different crack models. The effects of crack angle, crack length, fracture distance, and spacing on the deformation of surrounding rock are analyzed. The influence of different rock burst on the failure strap-fall modes of fissure roadway and roadway in different sections is analyzed, and the stability law of roadway is studied. Under the condition of high stress, the roadway shape has little influence on the distribution of the principal stress difference of surrounding rock, but the equivalent excavation radius determines the distribution of the plastic zone of surrounding rock. The larger the ineffective reinforcement zone is, the larger the deformation around the roadway will be. The decrease of the angle between the structural plane and the vertical stress increases the failure range of the roadway under the gravity burst pressure. Under the horizontal tectonic stress type rock burst, when the structural plane inclination angle is 0°, the two-sided caving body fills the roadway and the roof caving range becomes smaller.

A two-scale stochastic damage model for concrete under fatigue loading

The microscopic stochastic fracture model is extended by introducing a concept of microscale damage, whose evolution law is derived by an energy analysis on the propagation of the cracks from nano- to microscale. A material parameter termed nano-to-micro comprehensive fracture energy is found to play a key role in the random and nonlinear behavior of concrete. It is shown that the proposed model can simulate the mean and standard deviation of the complete stress–strain curve of concrete very well. Moreover, the fatigue life, stiffness degradation and their inherent randomness under various loading conditions can be well predicted by the model.

Prediction of Charpy impact toughness of steel weld heat-affected zones by combined micromechanics and stochastic fracture model – Part II : Model validation by experiment

The present paper presents the results of Charpy impact testing for a variety of simulated heat-affected zone (HAZ) samples having different chemical compositions and thermal cycles. The target microstructure was coarse-grained and predominantly composed of upper bainite with martensite-austenite (MA) constituents. Microstructural parameters, including volume fractions of the MA and grain boundary (GB) ferrite and the statistical distributions of the MA particle thickness, lath width and GB ferrite thickness, were measured. All these data were fed into the computational model to predict the Charpy impact absorbed energy transition behavior, presented in Part I. The predicted results agreed well with the experimental results. The present model does not rely on data from multiple specimens, but the Charpy absorbed energy together with its scatter can be predicted from the microstructural parameters and tensile properties only.

Prediction of Charpy impact toughness of steel weld heat-affected zones by combined micromechanics and stochastic fracture model – Part I: Model presentation

The present paper proposes a new computational model in which the Charpy impact property of steels is predicted from the microstructural information and tensile properties. During cleavage fracture, the Charpy impact property is controlled by the presence of a martensite-austenite constituent (MA) in the predominantly upper bainite microstructure in the weld heat-affected zone and is based on the weakest link mechanism and the micromechanics of the cleavage fracture. The cleavage fracture strength equation has been modified to statistically account for microcrack nucleation and propagation. An example calculation reveals that the proposed model reproduces the Charpy absorbed energy transition curve with a scatter in toughness. The validation of the model with experimental data is presented in the companion paper.

Physical stochastic damage model for concrete subjected to fatigue loading

A unified damage model applicable to both monotonic and fatigue loading is developed for concrete in the framework of mesoscopic stochastic fracture models. Besides traditional strain-based failure criteria, a new kind of energy-based failure criteria for meso-elements is established relying on nanoscale mechanics. Verifications show that the proposed model can not only reasonably reproduce the randomness in fatigue lifetime but also well describe the nonlinearity in fatigue damage evolution of concrete. In addition, the physical mechanisms lying behind these randomness and nonlinearity are clearly demonstrated for the first time.

A multi-scale stochastic fracture model for characterizing the tensile behavior of 2D woven composites

In this paper, a multi-scale stochastic fracture model is presented to study the nonlinear mechanical behaviors of two-dimensional (2D) woven C/SiC composites under uniaxial tension. This model considers the randomly distributed intrinsic flaws and the complex braided structure of composites. In the model, the micro-scale representative volume element (RVE) is developed to compute the effective properties of yarns, which are then used in the meso-scale RVE to capture the macroscopic stress response of woven composites. Weibull distribution law is applied to the material elements to reflect the random flaws. The failure mechanism of composites is identified by analyzing the damage rates and the fracture morphologies of fiber, matrix, and interface. The effects of fiber volume fraction and temperature on fracture strength are predicted and the results demonstrate that the failure location varies as the temperature increase due to the relief of thermal residual stress. The presented model provides an efficient tool for evaluating the mechanical properties of textile composites.

A physically motivated model for fatigue damage of concrete

The fatigue problem of concrete is still a challenging topic in the researches and applications of concrete engineering. This paper aims to develop a fatigue damage evolution law based model for concrete motivated by the analysis of physical mechanism. In this model, the fatigue energy dissipation process at microscale is investigated with rate process theory. The concept of self-similarity is employed to bridge the scale gap between microscale cracking and mesoscale dissipative element. With the stochastic fracture model, the crack avalanches and macro-crack nucleation processes from mesoscale to macroscale are simulated to obtain the behaviors of macroscope damage evolution of concrete. In conjunction with continuum damage mechanics framework, the fatigue damage constitutive model for concrete is then proposed. Numerical simulations are carried out to verify the model, revealing that the proposed model accommodates well with physical mechanism of fatigue damage evolution of concrete whereby the fatigue life of concrete structures under different stress ranges can be predicted.

A methodology for pseudo-genetic stochastic modeling of discrete fracture networks

Stochastic simulation of fracture systems is an interesting approach to build a set of dense and complex networks. However, discrete fracture models made of planar fractures generally fail to reproduce the complexity of natural networks, both in terms of geometry and connectivity. In this study a pseudo-genetic method is developed to generate stochastic fracture models that are consistent with patterns observed on outcrops and fracture growth principles. The main idea is to simulate evolving fracture networks through geometric proxies by iteratively growing 3D fractures. The algorithm defines heuristic rules in order to mimic the mechanics of fracture initiation, propagation, interaction and termination. The growth process enhances the production of linking structure and impacts the connectivity of fracture networks. A sensitivity study is performed on synthetic examples. The method produces unbiased fracture dip and strike statistics and qualitatively reproduces the fracture density map. The fracture length distribution law is underestimated because of the early stop in fracture growth after intersection.

Characterizing and modeling natural fracture networks in a tight carbonate reservoir in the Middle East: A methodology

Fractured reservoirs are challenging to handle because of their high level of heterogeneity. In particular, natural fractures have a significant impact on well performance and water production. Therefore, understanding their significance through fracture characterization is helpful in well placement and field development. This paper presents a best practice methodology for building a 3D stochastic fracture model using a Middle Eastern tight carbonate field example. This model is generated through the analysis and integration of data including cores, borehole images (BHI), logs, mud losses, production logs, well test data and 3D seismic data. The impact of lithology on fracture occurrence was quantified based on rock-typing. Rock-types are distributed in a 3D geological model using a high resolution sequence stratigraphic framework. The length, dip angle and orientation of fractures, together with the shale content of the facies where they occur, were defined to sort the tectonic fractures from the non-tectonic (diagenetic) ones. It was found that multiple sets of tectonic diffuse fractures are generally associated with cleaner limestone units. Altogether, three sets of diffuse fractures were identified from BHI data: NE-SW, EW and NW-SE. In addition, large-scale fracture corridors, including sub-seismic faults identified from seismic analysis, were detected and calibrated with cores and BHI. The final model incorporates two scales of tectonic fractures with a direct bearing on field production behavior: diffuse fractures and large fracture corridors. Fracture calibration was performed using production logs and well production data. Permeability at wells was computed in the 3D fracture model and matched with real build-up data. These data were then used to propagate 3D fracture properties (fracture porosity, fracture permeability and equivalent block size) in the upscaled geological model, for constructing a full-field reservoir simulation model. Few changes of the fracture properties were needed to obtain a good history match, indicating that the fracture model produced is robust.

A Simple Formula To Estimate 2D Fracture Connectivity

Summary A simple expression is derived for average fracture intersections per fracture, λ, as the product of fracture density, average length, and angular standard deviation in 2D fracture models. The calculated λ value quite accurately estimates the observed λ values for a variety of cases including one- or two-fracture sets, truncated or not-truncated fractures, and log-normal or power law length distribution. The formula fails to estimate the number of fracture intersections accurately when fractures are clustered. Fracture clusters could either be fault-related fracture corridors or highly fractured layers. In either case, it is necessary to calculate λ for each fracture corridor or each fractured layer with a different fracture density. The relationship between λ and fracture connectivity was investigated using several stochastic fracture models. λ seems to be a reasonably good estimator of fracture connectivity, which we define as the percentage of fractures within the largest interconnected fracture aggregate. The relationship between the number of fracture intersections and percolation threshold is, however, weak because percolation threshold is dependent on the relative length of fractures with respect to the distance between two opposite ends of a fractured medium.

Comparison of deterministic with stochastic fracture models in water-flooding numerical simulations

Determination of multiphase flow properties considering the variation of fracture patterns (i.e., number of fracture sets, their orientation, length distribution, spacing, and in-situ aperture) remains a key challenge in reservoirs. In reservoir engineering, one way is by studying outcrop analogs with comparable petrophysical properties and a similar geological history, and incorporating these data into model building, discretization, and numerical simulation. The limitation of directly incorporating attributes measured on outcrops is that this method is error prone because of postburial processes. Mineralized fracture (vein) attributes are good candidates to use as analogs for open fractures formed under in-situ conditions, to establish the relationship between fracture length and aperture and help to reveal the conditions at the time of their formation, and to quantify fracture-induced porosity in rock masses. Vein attributes determined from scan lines and window samples were combined to condition the stochastic generation of fractures using the discrete fracture network code FracMan. Comparison of water breakthrough time and oil saturation at breakthrough was then determined by applying a constant pressure gradient for each realization to simulate water-flooding numerical simulation using the combined finite element–finite volume method. The different stochastic realizations were compared with discrete fracture and matrix models, and we show how the uncertainty in these fracture attributes affects multiphase flow behavior in naturally fractured rocks. Uncertainty in quantifying these attributes has a profound impact for predicting the oil recovery and water breakthrough time based on limited information from boreholes.

Probabilistic predictions regarding key blocks using stochastic discrete fracture networks - example from a rock cavern in south-east Sweden

The natural variation of the fracture/joint network geometry and limited data access are the main sources of uncertainties in key block predictions. To make the uncertainties easily quantifiable the predicting variables should be presented as probability density functions. This paper illustrates the implementation of a stochastic discrete fracture network to predict the amount and size of key blocks around the rock cavern of the Centralt Storage Facility for Spent Nuclear Fuel (CLAB-2, Central Lager Anvant Bransle) in south-eastern Sweden. The data used in the study were effectively limited to fracture mapping in boreholes. The stochastic fracture model was generated with a FracMan discrete fracture simulator by adopting random fracture locations. Subsequently, the key block statistics along a simulated tunnel positioned inside the fracture model were generated. To illustrate the value of the predictions made, block statistics were undertaken for two different tunnel orientations. The methodology presented offers the potential to optimize the excavation design.

A model for complex aftershock sequences

The decay rate of aftershocks is commonly very well described by the modified Omori law, n(t) ∝ t−p, where n(t) is the number of aftershocks per unit time, t is the time after the main shock, and p is a constant in the range 0.9 < p < 1.5 and usually close to 1. However, there are also more complex aftershock sequences for which the Omori law can be considered only as a first approximation. One of these complex aftershock sequences took place in the eastern Pyrenees on February 18, 1996, and was described in detail by Correig et al. [1997]. In this paper, we propose a new model inspired by dynamic fiber bundle models to interpret this type of complex aftershock sequences with sudden increases in the rate of aftershock production not directly related to the magnitude of the aftershocks (as in the epidemic‐type aftershock sequences). The model is a simple, discrete, stochastic fracture model where the elements (asperities or barriers) break because of static fatigue, transfer stress according to a local load‐sharing rule and then are regenerated. We find a very good agreement between the model and the Eastern Pyrenees aftershock sequence, and we propose that the key mechanism for explaining aftershocks, apart from a time‐dependent rock strength, is the presence of dynamic stress fluctuations which constantly reset the initial conditions for the next aftershock in the sequence.

Analysis of compressive fracture in rock using statistical techniques: Part II. Effect of microscale heterogeneity on macroscopic deformation

Abstract We have performed a parameter-sensitivity analysis to evaluate the relative importance of different types of grain-scale heterogeneity on fracture processes and compressive strength in simulated compression tests of brittle, heterogeneous materials such as rock. This was done using a non-linear, rule-based model described in a companion paper. Results presented here indicate that heterogeneity in local stress field (due to grain shape and loading) has a first-order effect on macroscopic properties and is much more important than heterogeneity in site strength or location. In particular, increasing local stress heterogeneity lowers the mean ultimate strength following an inverse power law. Increasing heterogeneity in the lattice-site locations (i.e. irregular lattice spacing) decreases crack localization and decreases normalized crack strain energy. This result is consistent with the postulate that systems with increasing disorder require more energy to break. Heterogeneity in site-strength distribution had a relatively minor effect on macroscopic behavior. Peak strength is dependent on the mean site strength, not on the width of the site-strength distributions. This study also revealed that percolation thresholds are much lower than those predicted from stochastic fracture models. Consequently, statistical models for rock fracture must consider alternative percolation algorithms such as directed-bond percolation, because the standard percolation models may not be appropriate for analyzing systems where crack interaction dominates behavior at a low fraction of sites broken.

A New Approach to Fracture Modelling in Reservoirs Using Deterministic, Genetic and Statistical Models of Fracture Growth: ABSTRACT

It is increasingly acknowledged that in order to understand and forecast the behavior of fracture influenced reservoirs we must attempt to reproduce the fracture system geometry and use this as a basis for fluid flow calculation. This article aims to present a recently developed fracture modelling prototype designed specifically for use in hydrocarbon reservoir environments. The prototype {open_quotes}FRAME{close_quotes} (FRActure Modelling Environment) aims to provide a tool which will allow the generation of realistic 3D fracture systems within a reservoir model, constrained to the known geology of the reservoir by both mechanical and statistical considerations, and which can be used as a basis for fluid flow calculation. Two newly developed modelling techniques are used. The first is an interactive tool which allows complex fault surfaces and their associated deformations to be reproduced. The second is a {open_quotes}genetic{close_quotes} model which grows fracture patterns from seeds using conceptual models of fracture development. The user defines the mechanical input and can retrieve all the statistics of the growing fractures to allow comparison to assumed statistical distributions for the reservoir fractures. Input parameters include growth rate, fracture interaction characteristics, orientation maps and density maps. More traditional statistical stochastic fracture models are also incorporated. FRAME ismore » designed to allow the geologist to input hard or soft data including seismically defined surfaces, well fractures, outcrop models, analogue or numerical mechanical models or geological {open_quotes}feeling{close_quotes}. The geologist is not restricted to {open_quotes}a priori{close_quotes} models of fracture patterns that may not correspond to the data.« less

Size-frequency relation of earthquakes in load-transfer models of fracture

Abstract. Using Monte Carlo simulations of the process of breaking in arrays of elements with load-transfer rules, we have obtained the size- frequency relation of the avalanches occurring in 1- and 2-dimensional stochastic fracture models. The resulting power-law behaviour resembles the Gutenberg-Richter law for the relation between the size (liberated energy) of earthquakes and their number frequency. The value of the power law exponent is calculated as a function of the degree of stress dissipation present in the model. The degree of dissipation is implemented in a straightforward and simple way by assuming that only a fraction of the stress is transferred in each breaking event. The models are robust with respect to the degree of dissipation and we observe a consistent power-law behaviour for a broad range of dissipation values, both in ID and 2D. The value of the power-law exponent is similar to the phenomenological b- value (0.8 < b < 1.1) for intermediate magnitude earthquakes.