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1. Random field realization and fracture simulation of rocks with angular bias for fracture strength

   Garrard, J. M.; Abedi, Reza; Clarke, P. L. (January 2018, 52nd U.S. Rock Mechanics/Geomechanics Symposium)

   Realistic fracture simulations in rock as a heterogeneous brittle material with significant inherent randomness require the use of models that incorporate its inhomogeneities and statistical variability. The high dependence of their fracture progress on microstructural defects results in wide scatter in their ultimate strength and the so-called size effect. This paper proposes an approach based on statistical volume elements (SVEs) to characterize rock fracture strength at the mesoscale. The use of SVEs ensures that the material randomness is maintained upon averaging of microscale features. Because the fracture strength varies not just spatially, but also by the angle of loading, this work includes angular variability to properly model a heterogeneous rock domain. Two different microcrack distributions, one angularly uniform and one angularly biased towards a specific angle, are used to show that implementing angle into the random field provides the most realistic fracture simulation. An adaptive asynchronous spacetime discontinuous Galerkin (aSDG) finite element method is used to perform the dynamic fracture simulations. 

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2. Modeling of rock inhomogeneity and anisotropy by explicit and implicit representation of microcracks

   Abedi, Reza; Clarke, Philip L. (January 2018, Proceedings of ARMA 2018)

   Fracture in rock as a heterogeneous brittle material, having significant inherent randomness, requires including probabilistic considerations at different scales. Crack growth in rocks is generally associated with complex features such as crack path oscillations, microcrack and crack branching events. Two methods will be presented to address rock inhomogeneity and anisotropy. First, microcracks are explicitly realized in a domain based on specific statistics of crack length and location. Second, a statistical model is used to implicitly represent an inhomogeneous field for fracture strength. Both approaches can be used for rocks in which the natural fractures are oriented in a specific angle, i.e. an aspect for modeling bedding planes in sedimentary rocks. 

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3. Comparison of Interfacial and Continuum Models for Dynamic Fragmentation Analysis

   https://doi.org/10.1115/IMECE2018-88294  

   Bahmani, Bahador; Clarke, Philip; Abedi, Reza (November 2018, Proceedings of ASME 2018 International Mechanical Engineering Congress and Exposition)

   The microstructural design has an essential effect on the fracture response of brittle materials. We present a stochastic bulk damage formulation to model dynamic brittle fracture. This model is compared with a similar interfacial model for homogeneous and heterogeneous materials. The damage models are rate-dependent, and the corresponding damage evolution includes delay effects. The delay effect provides mesh objectivity with much less computational efforts. A stochastic field is defined for material cohesion and fracture strength to involve microstructure effects in the proposed formulations. The statistical fields are constructed through the Karhunen-Loeve (KL) method. An advanced asynchronous Spacetime Discontinuous Galerkin (aSDG) method is used to discretize the final system of coupled equations. Application of the presented formulation is shown through dynamic fracture simulation of rock under a uniaxial compressive load. The final results show that a stochastic bulk damage model produces more realistic results in comparison with a homogenizes model. 

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4. Modeling of rock inhomogeneity and anisotropy by explicit and implicit representation of microcracks

   Abedi, R.; Clarke, P.L. (January 2018, Proceeding 52th U.S. Rock Mechanics/Geomechanics Symposium)

   Fracture patterns experienced under a dynamic uniaxial compressive load are highly sensitive to rock microstructural defects due to its brittleness and the absence of macroscopic stress concentration points. We propose two different approaches for modeling rock microstructural defects and inhomogeneity. In the explicit realization approach, microcracks with certain statistics are incorporated in the computational domain. In the implicit realization approach, fracture strength values are sampled using a Weibull probability distribution. We use the Mohr-Coulomb failure criterion to define an effective stress in the context of an interfacial damage model. This model predicts crack propagation at angles ±ɸch = ±(45 − ɸ/2) relative to the direction of compressive load, where ɸ is the friction angle. By using appropriate models for fracture strength anisotropy, we demonstrate the interaction of rock weakest plane and ɸch. Numerical results demonstrate the greater effect of strength anisotropy on fracture pattern when an explicit approach is employed. In addition, the density of fractures increases as the angle of the weakest planes approaches ±ɸch. The fracture simulations are performed by an h-adaptive asynchronous spacetime discontinuous Galerkin (aSDG) method that can accommodate crack propagation in any directions. 

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5. Approximating Fracture Paths in Random Heterogeneous Materials: A Probabilistic Learning Perspective

   https://doi.org/10.1061/JENMDT.EMENG-7617  

   Quek, Ariana; Yi_Yong, Jin; Guilleminot, Johann (June 2024, Journal of Engineering Mechanics)

   Approximation frameworks for phase-field models of brittle fracture are presented and compared in this work. Such methods aim to address the computational cost associated with conducting full-scale simulations of brittle fracture in heterogeneous materials where material parameters, such as fracture toughness, can vary spatially. They proceed by combining a dimension reduction with learning between function spaces. Two classes of approximations are considered. In the first class, deep learning models are used to perform regression in ad hoc latent spaces. PCA-Net and Fourier neural operators are specifically presented for the sake of comparison. In the second class of techniques, statistical sampling is used to approximate the forward map in latent space, using conditioning. To ensure proper measure concentration, a reduced-order Hamiltonian Monte Carlo technique (namely, probabilistic learning on manifold) is employed. The accuracy of these methods is then investigated on a proxy application where the fracture toughness is modeled as a non-Gaussian random field. It is shown that the probabilistic framework achieves comparable performance in the 𝐿2 sense while enabling the end-user to bypass the art of defining and training deep learning models. 

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