This paper proposes a phase field model for fracture in poroelastic media. The porous medium is modeled based on the classical Biot poroelasticity theory and the fracture behavior is controlled by ...the phase field model. Moreover, the fracture propagation is driven by the elastic energy where the phase field is used as an interpolation function to transit fluid property from the intact medium to the fully broken one. We use a segregated (staggered) scheme and implement our approach in Comsol Multiphysics. The proposed model is verified by a single-phase solid subjected to tension and a 2D specimen subjected to an increasing internal pressure. We also compare our results with analytical solutions. Finally, we show 2D and 3D examples of internal fluid injection to illustrate the capability of the proposed approach.
•A phase-field modeling approach of fracture propagation in poroelastic media is proposed.•The fracture propagation is driven by the elastic energy and no stress threshold is set.•2D and 3D examples are presented.
•A phase field framework is applied to investigate hydraulic fracturing in layered media.•No penetration criteria are required atmaterial interfaces.•Hydraulic fractures in the soft to stiff and ...stiff to soft configurations are investigated.•Different inclination angles of the layer interface are included.•The penetration, singly-deflected, and doubly-deflected scenarios can be predicted by phase field modeling.
In the hydraulic fracturing of natural rocks, understanding and predicting crack penetrations into the neighboring layers is crucial and relevant in terms of cost-efficiency in engineering and environmental protection. This study constitutes a phase field framework to examine hydraulic fracture propagation in naturally-layered porous media. Biot's poroelasticity theory is used to couple the displacement and flow field, while a phase field method helps characterize fracture growth behavior. Additional fracture criteria are not required and fracture propagation is governed by the equation of phase field evolution. Thus, penetration criteria are not required when hydraulic fractures reach the material interfaces. The phase field method is implemented within a staggered scheme that sequentially solves the displacement, phase field, and fluid pressure. We consider the soft-to-stiff and the stiff-to-soft configurations, where the layer interface exhibits different inclination angles θ. Penetration, singly-deflected, and doubly-deflected fracture scenarios can be predicted by our simulations. In the soft-to-stiff configuration, θ=0° exhibits penetration or symmetrical doubly-deflected scenarios, and θ=15° exhibits singly-deflected or asymmetric doubly-deflected scenarios. Only the singly-deflected scenario is obtained for θ=30°. In the stiff-to-soft configuration, only the penetration scenario is obtained with widening fractures when hydraulic fractures penetrate into the soft layer.
Machine learning (ML) methods have shown powerful performance in different application. Nonetheless, designing ML models remains a challenge and requires further research as most procedures adopt a ...trial and error strategy. In this study, we present a methodology to optimize the architecture and the feature configurations of ML models considering a supervised learning process. The proposed approach employs genetic algorithm (GA)-based integer-valued optimization for two ML models, namely deep neural networks (DNN) and adaptive neuro-fuzzy inference system (ANFIS). The selected variables in the DNN optimization problems are the number of hidden layers, their number of neurons and their activation function, while the type and the number of membership functions are the design variables in the ANFIS optimization problem. The mean squared error (MSE) between the predictions and the target outputs is minimized as the optimization fitness function. The proposed scheme is validated through a case study of computational material design. We apply the method to predict the fracture energy of polymer/nanoparticles composites (PNCs) with a database gathered from the literature. The optimized DNN model shows superior prediction accuracy compared to the classical one-hidden layer network. Also, it outperforms ANFIS with significantly lower number of generations in GA. The proposed method can be easily extended to optimize similar architecture properties of ML models in various complex systems.
In this paper, a deep collocation method (DCM) for thin plate bending problems is proposed. This method takes advantage of computational graphs and backpropagation algorithms involved in deep ...learning. Besides, the proposed DCM is based on a feedforward deep neural network (DNN) and differs from most previous applications of deep learning for mechanical problems. First, batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries. A loss function is built with the aim that the governing partial differential equations (PDEs) of Kirchhoff plate bending problems, and the boundary/initial conditions are minimised at those collocation points. A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters. In Kirchhoff plate bending problems, the C1 continuity requirement poses significant difficulties in traditional mesh-based methods. This can be solved by the proposed DCM, which uses a deep neural network to approximate the continuous transversal deflection, and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.
Filling materials such as clay or sand widely exist in natural rock joints and work as weak bonds between the joint surfaces. The fillings affect rock deformation and failure behavior, and show ...different influences in terms of single crack or multiple cracks. While most of the literature have focused on unfilled cracks in brittle materials, this study aims to investigate various filling materials on the crack behavior, e.g., initiation, secondary cracks and peak strength. In this paper, the crack propagation in rock-like specimens with double-filled and unfilled cracks are investigated experimentally and numerically. Uniaxial compression tests were conducted and the experimental observations indicate that the peak stress and first crack initiation stress of the specimens vary with different geometries and different filling materials, while the crack initiation location and the pattern of crack coalescence show similar behavior between filled and unfilled cracks. In parallel to the experimental tests, numerical simulations were carried out using a modified phase field model (PFM) to complement the experiments and provide a new perspective. The PFM is found to produce consistent stress–strain curve, strength, and crack patterns with those observed in the experimental tests for both unfilled and filled cracks.
The fracture energy is a substantial material property that measures the ability of materials to resist crack growth. The reinforcement of the epoxy polymers by nanosize fillers improves ...significantly their toughness. The fracture mechanism of the produced polymeric nanocomposites is influenced by different parameters. This paper presents a methodology for stochastic modelling of the fracture in polymer/particle nanocomposites. For this purpose, we generated a 2D finite element model containing an epoxy matrix and rigid nanoparticles surrounded by an interphase zone. The crack propagation was modelled by the phantom node method. The stochastic model is based on six uncertain parameters: the volume fraction and the diameter of the nanoparticles, Young’s modulus and the maximum allowable principal stress of the epoxy matrix, the interphase zone thickness and its Young’s modulus. Considering the uncertainties in input parameters, a polynomial chaos expansion surrogate model is constructed followed by a sensitivity analysis. The variance in the fracture energy was mostly influenced by the maximum allowable principal stress and Young’s modulus of the epoxy matrix.
This paper proposes a phase field model (PFM) for describing hydraulic fracture propagation in transversely isotopic media. The coupling between the fluid flow and displacement fields is established ...according to the classical Biot poroelasticity theory, while the phase field model characterizes the fracture behavior. The proposed method applies a transversely isotropic constitutive relationship between stress and strain as well as anisotropy in fracture toughness and permeability. We add an additional pressure-related term and an anisotropic fracture toughness tensor in the energy functional, which is then used to obtain the governing equations of strong form via the variational approach. In addition, the phase field is used to construct indicator functions that transit the fluid property from the intact domain to the fully fractured one. Moreover, the proposed PFM is implemented using the finite element method where a staggered scheme is applied to solve the displacement, fluid pressure, and phase field sequentially. Afterward, two examples are used to initially verify the proposed PFM: a transversely isotropic single-edge-notched square plate subjected to tension and an isotropic porous medium subjected to internal fluid pressure. Finally, numerical examples of 2D and 3D transversely isotropic media with one or two interior notches subjected to internal fluid pressure are presented to further prove the capability of the proposed PFM in 2D and 3D problems.
We present a stochastic deep collocation method (DCM) based on neural architecture search (NAS) and transfer learning for heterogeneous porous media. We first carry out a sensitivity analysis to ...determine the key hyper-parameters of the network to reduce the search space and subsequently employ hyper-parameter optimization to finally obtain the parameter values. The presented NAS based DCM also saves the weights and biases of the most favorable architectures, which is then used in the fine-tuning process. We also employ transfer learning techniques to drastically reduce the computational cost. The presented DCM is then applied to the stochastic analysis of heterogeneous porous material. Therefore, a three dimensional stochastic flow model is built providing a benchmark to the simulation of groundwater flow in highly heterogeneous aquifers. The performance of the presented NAS based DCM is verified in different dimensions using the method of manufactured solutions. We show that it significantly outperforms finite difference methods in both accuracy and computational cost.
The mechanical behavior of rock is strongly dependent on its embedded discontinuities such as cracks and joints. Natural rock joints are found to exist mostly with filling materials such as sand or ...clay as weak bond. The filling has been observed in engineering practice to have influence on rock failure behavior. To investigate this topic, the crack propagation behavior of the filled and unfilled crack is studied and compared by testing rock-like specimens subjected to uniaxial compression. A qualitative analysis of the crack propagation paths is described where crack is classified into four types, namely the original, secondary, wing and anti-wing cracks. The experiments indicate the crack initiation time, initiation location and propagation behavior are different between filled and unfilled joints. The experimental results also showed that the peak stress for filled joint is higher than for the unfilled. Numerical tests simulating the experimental process are carried out using the extended finite element method (XFEM) to explore complementary explanations and provide proofs to the experiments.
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