Continuum Fracture Research Articles

Applications of fractured continuum model to enhanced geothermal system heat extraction problems

This paper describes the applications of the fractured continuum model to the different enhanced geothermal systems reservoir conditions. The capability of the fractured continuum model to generate fracture characteristics expected in enhanced geothermal systems reservoir environments are demonstrated for single and multiple sets of fractures. Fracture characteristics are defined by fracture strike, dip, spacing, and aperture. The paper demonstrates how the fractured continuum model can be extended to represent continuous fractured features, such as long fractures, and the conditions in which the fracture density varies within the different depth intervals. Simulations of heat transport using different fracture settings were compared with regard to their heat extraction effectiveness. The best heat extraction was obtained in the case when fractures were horizontal. A conventional heat extraction scheme with vertical wells was compared to an alternative scheme with horizontal wells. The heat extraction with the horizontal wells was significantly better than with the vertical wells when the injector was at the bottom.

Continuum Fracture Analysis and Molecular Dynamic Study on Crack Initiation and Propagation in Nanofilms

Crack initiation and propagation in a nanostructured nickel film were studied by molecular dynamic simulation as well as an interatomic‐potential‐based continuum approach. In the molecular dynamic simulation, the interatomic potential was described by using Embedded Atom Method (EAM), and a reduced 2D plane model was employed to simulate the mechanical behavior of nanofilms. Atomistic simulation shows that the reduced plane model in this paper can not only reveal the physical nature of crack initiation clearly but also give the critical time of crack initiation accurately as the continuum fracture analysis does. The normal stress and average atom energy at the crack tip which resulted from atomistic simulation at the time of crack initiation agree well with the analytical results. On the other hand, the crack propagation in nanofilms was studied by interatomic‐potential‐based continuum fracture mechanics analysis based on Griffith criterion. The coupled continuum‐atomic analysis can predict the crack initiation and atomic stress accurately. Continuum analysis with material property parameters determined by interatomic potential is proved to be another promising way of solving failure problem on nanoscale.

A grid generation algorithm for reservoir domains with discrete fracture networks

In this paper, development of an algorithm for generation of unstructured grids applicable to combined discrete fracture networks (DFNs) and continuum reservoir media is described. Automatic grid refinement, grid size control, grid distribution, and grid quality enhancement are among the algorithm’s capabilities. Grid refinement is based on a combination of point insertion and cell subdivision methods. Edge/face-based and cell-based connectivity matrices along with grid size control, grid distribution, and grid quality enhancement measures were automatically employed to refine unstructured grids. Computational unstructured grids are generated in two and three-dimensional fractured reservoir models to demonstrate the algorithm applications.

Micromechanics modeling for fatigue damage analysis designed for fabric reinforced ceramic matrix composites

A micromechanics analysis modeling method was developed to analyze the damage progression and fatigue failure of fabric reinforced composite structures, especially for the brittle ceramic matrix material composites. A repeating unit cell concept of fabric reinforced composites was used to represent the global composite structure. The thermal and mechanical properties of the repeating unit cell were considered as the same as those of the global composite structure. The three-phase micromechanics, the shear-lag, and the continuum fracture mechanics models were integrated with a statistical model in the repeating unit cell to predict the progressive damages and fatigue life of the composite structures. The global structure failure was defined as the loss of loading capability of the repeating unit cell, which depends on the stiffness reduction due to material slice failures and nonlinear material properties in the repeating unit cell. The present methodology is demonstrated with the analysis results evaluated through the experimental test performed with carbon fiber reinforced silicon carbide matrix plain weave composite specimens.

Energy-based mechanistic approach for damage characterization of pre-flawed visco-elasto-plastic materials

Damage characterization plays a significant role in producing durable and high performance structural materials. However, it is somewhat complicated because of the particular characteristics of many materials, such as pre-existing flaws, time-dependent behaviors, and coexistence of cracking and permanent deformation. This kind of materials is pre-flawed visco-elasto-plastic material. In order to characterize damage in such materials, this paper proposes an energy-based mechanistic (EBM) approach that provides a complete solution to these problems. As typical pre-flawed visco-elasto-plastic materials, asphalt mixtures are selected to demonstrate the principles and applications of the EBM approach.When an asphalt mixture is not damaged, the pre-existing flaws are air voids, characterized by the average air voids size and number of air voids calculated by the EBM approach. The calculated values are more accurate than those measured by the X-ray Computed Tomography system. Due to the increased accuracy, it is discovered that the air voids becomes smaller when the mixture is aged, which serves as an evidence of the change of the internal structure of the material due to aging.When an asphalt mixture is damaged, the damage includes cracking and permanent deformation. The cracking damage is a multitude of randomly distributed cracks. A new concept, distributed continuum fracture (DCF), is introduced to model the distributed cracks in the EBM approach. Development of cracking damage is quantified by the evolution of damage density, average crack size and number of cracks. The damage densities of eight different mixtures are proven to correctly reflect the effect of mixture composition and aging. New features of number of cracks discovered lead to new definitions of cracking history in pre-flawed materials. The energy for permanent deformation is separated from that expended for cracking in the same asphalt mixture. Such a separation acknowledges the fact that cracking and permanent deformation always occur simultaneously. The separated energy for cracking is used to define a cracking energy dissipation rate, a direct indicator of cracking susceptibility of asphalt mixtures.In a word, the EBM approach is able to characterize damage in asphalt mixtures under various conditions using one type of test on one specimen. It requires simple inputs: stress, strain, and time, and all the calculations are performed automatically by the Excel. Using this approach to analyze the test data is more efficient than some alternative methods because of less testing effort and more informative results with improved accuracy.

Opto-Mechanical Scission of Polymer Chains in Photosensitive Diblock-Copolymer Brushes

In this paper we report on an opto-mechanical scission of polymer chains within photosensitive diblock-copolymer brushes grafted to flat solid substrates. We employ surface-initiated polymerization of methylmethacrylate (MMA) and t-butyl methacrylate (tBMA) to grow diblock-copolymer brushes of poly(methylmethacrylate-b-t-butyl methacrylate) following the atom transfer polymerization (ATRP) scheme. After the synthesis, deprotection of the PtBMA block yields poly(methacrylic acid) (PMAA). To render PMMA-b-PMAA copolymers photosensitive, cationic azobenzene containing surfactants are attached to the negatively charged outer PMAA block. During irradiation with an ultraviolet (UV) interference pattern, the extent of photoisomerization of the azobenzene groups varies spatially and results in a topography change of the brush, i.e., formation of surface relief gratings (SRG). The SRG formation is accompanied by local rupturing of the polymer chains in areas from which the polymer material recedes. This opto-mechanically induced scission of the polymer chains takes place at the interfaces of the two blocks and depends strongly on the UV irradiation intensity. Our results indicate that this process may be explained by employing classical continuum fracture mechanics, which might be important for tailoring the phenomenon for applying it to poststructuring of polymer brushes.

Crackling versus Continuumlike Dynamics in Brittle Failure

We study how the loading rate, specimen geometry, and microstructural texture select the dynamics of a crack moving through an heterogeneous elastic material in the quasistatic approximation. We find a transition, fully controlled by two dimensionless variables, between dynamics ruled by continuum fracture mechanics and crackling dynamics. Selection of the latter by the loading, microstructure, and specimen parameters is formulated in terms of scaling laws on the power spectrum of crack velocity. This analysis defines the experimental conditions required to observe crackling in fracture. Beyond failure problems, the results extend to a variety of situations described by models of the same universality class, e.g., the dynamics in wetting or of domain walls in amorphous ferromagnets.

Predicting residual stiffness of cracked composite laminates subjected to multi-axial inplane loading

This is a contribution to the exercise that aims to benchmark and validate the current continuum damage and fracture mechanics methodologies used for predicting the mechanical behaviour of fibre-reinforced plastic composites under complex loadings. The paper describes an analytical approach to predict the effect of intra- (matrix cracking and splitting) and inter-laminar (delamination) damage on the residual stiffness properties of the laminate, which can be used in the post-initial failure analysis, taking full account of damage mode interaction. The approach is based on a two-dimensional shear lag stress analysis and the equivalent constraint model of the damaged laminate with multiple damaged plies. The application of the approach to predicting degraded stiffness properties of a multidirectional laminate with multilayer intra- and inter-laminar damage is demonstrated for [Formula: see text] and [Formula: see text] cross-ply laminates made from a specific glass/epoxy system under in-plane uniaxial and biaxial loading damaged by transverse and longitudinal matrix cracks and crack-induced transverse and longitudinal delamination.

Design life assessment of the blade root of a gas turbine unit under thermomechanical loading

The semianalytic finite element method with the use of approaches of continuum fracture mechanics was employed to assess the design life of the blade root of a gas turbine unit under thermomechanical loading. The results of computations, obtained with two- and three-dimensional problems, were compared. For design life assessment in creep, the nonuniform temperature distribution over the root cross-section should be taken into account.

A continuum voxel approach to model flow in 3D fault networks: A new way to obtain up-scaled hydraulic conductivity tensors of grid cells

Modelling transfers in fractured media remains a challenging task due to the complexity of the system geometry, high contrasts and large uncertainties on flow and transport properties. In the literature, fractures are classically modelled by equivalent properties or are explicitly represented. The new Fracture Continuum Voxel Approach (FCVA), is a continuum approach partly able to represent fracture as discrete objects; the geometry of each fracture is represented on a regular meshing associated with a heterogeneous field of equivalent flow properties. The mesh-identification approach is presented for a regular grid. The derivation of equivalent voxel parameters is developed for flow simulated with a Mixed Hybrid Finite Element (MHFE) scheme. The FCVA is finally validated and qualified against some reference cases. The resulting method investigates multi-scaled fracture networks: a small scale homogenised by classical methods and large discrete objects as that handled in the present work.

Fracture Analysis of Brittle Materials Based on Nonlinear FEM and Application in Arch Dam with Fractures

Current fracture analysis models based on fracture mechanics or continuum damage mechanics are still limited in the application to three-dimensional structure. Based on deformation reinforcement theory coming from elastoperfect plastic theory, unbalanced force is proposed to predict initiation and propagation of cracks. Unbalanced force is the driving force of time-dependent deformation according to Perzyna’s viscoplasticity theory. It is also related to the damage driving force in viscoplastic damage model. The distribution of unbalanced force indicates cracks initiation area, while its direction predicts possible cracks propagation path. Uniaxial compression test of precrack specimen is performed as verification to this method. The trend and distribution of cracks are in good agreement with numerical results, proving that unbalanced force is feasible and effective for fracture analysis. The method is applied in fracture analysis of Xiaowan high arch dam, which is subjected to some cracks in dam due to the temperature control program. The results show that the deformation and stress of cracks and the stress characteristics of dam are insensitive to grouting of cracks. The existing cracks are stable and dam heel is still the most possible cracking position.

Numerical Evaluation of Effective Unsaturated Hydraulic Properties of Fractured Rocks using a Stochastic Continuum Approach

This study builds on past studies that have investigated the effective properties of unsaturated porous media and merges stochastic continuum concepts to develop a numerical procedure for estimating the effective flow parameters that are independent of pressure head or saturation. The procedure involves the development of a numerical permeameter based on randomly generated, spatially correlated fields of fundamental fracture properties such as spacing and aperture. Local values of hydraulic conductivity, and the van Genuchten fitting parameters α and n in this random field are computed from local values of mechanical aperture and spacing based on the cubic law and relationships established from pore‐scale fracture simulations. By considering multiple realizations of these spatially correlated fields and by running the numerical permeameter over a range of effective saturations and flow rates, outputs are generated that can be fit with the standard van Genuchten model to estimate block effective values of saturated hydraulic conductivity and the van Genuchten unsaturated hydraulic parameters of a fracture continuum at the field scale. These parameters can then be incorporated to conventional numerical simulators originally developed for porous media. The study shows that reasonable approximations to the results obtained through computationally demanding stochastic simulations can be developed based on estimates of the mean fracture spacing and aperture. Although the example application assumes the presence of an impermeable rock matrix, the hydraulic properties estimated for the fracture continuum with this methodology can be combined with those measured or similarly estimated for a matrix continuum in dual‐permeability continuum models.

Damage driven crack initiation and propagation in ductile metals using XFEM

Originally Continuum Damage Mechanics and Fracture Mechanics evolved separately. However, when it comes to ductile fracture, an unified approach is quite beneficial for an accurate modelling of this phenomenon. Ductile materials may undergo moderate to large plastic deformations and internal degradation phenomena which are well described by continuum theories. Nevertheless in the final stages of failure, a discontinuous methodology is essential to represent surface decohesion and macro-crack propagation. In this work, XFEM is combined with the Lemaitre ductile damage model in a way that crack initiation and propagation are governed by the evolution of damage. The model was built under a finite strain assumption and a non-local integral formulation is applied to avoid pathological mesh dependence. The efficiency of the proposed methodology is evaluated through various numerical examples.

Crack propagation in pneumatic tires: Continuum mechanics and fracture mechanics approaches

In this paper, the fatigue life of pneumatic tires is investigated with particular focus on the phenomenon of belt separation. This phenomenon yields the propagation of fatigue cracks in the rubber separating the belts; not the separation of the rubber at the interfaces with the belts. Cracks at the belts edges are related among others to the high stress concentration due to the very different stiffness of the two materials. A steady-state rolling finite element model of the full tire is used to evaluate the stress and strain fields in the rubber at the edges of the belts. Suitable life predictors are defined and computed for the identification of the zones with a higher expectation of failure. For the evaluation of crack propagation in the bulk rubber material between different belts, in principle, two different approaches are employed associated with crack nucleation and crack growth. The two approaches are applied to the study of crack propagation in a truck tire and the results of different predictors are compared to crack propagation found in a real tire.

Simulation of Non-Darcy Porous Media Flow According to the Barree and Conway Model

Non-Darcy porous media flow has been traditionally handled using the Forchheimer equation. However, recent experimental studies have shown that the Forchheimer model is unable to fit laboratory results at high flow rates. On the other hand, the non-Darcy flow model, proposed by Barree and Conway, is capable of describing the entire range of relationships between flow rate and potential gradient from low- to high-flow rates through proppant packs. In this paper, we present a numerical model by incorporating the Barree and Conway model into a general-purpose reservoir simulator for modeling single-phase and multiphase non-Darcy flow in porous and fractured media. The numerical formulation is based on the TOUGH2 methodology, i.e., spatial integral-finite-difference discretization, leading to an unstructured grid, followed by time discretization carried out with a backward, first-order, finite-difference method. The final discrete nonlinear equations are handled fully implicitly by Newton iteration. In the numerical approach, flow through fractured rock is handled using a general multi-continuum approach, applicable to both continuum and discrete fracture conceptual models. In an effort for model validation, we use analytical solutions to verify our numerical model results for both single-phase and multiphase non-Darcy flow. In addition, the numerical model is applied for well testing analysis of transient non-Darcy flow toward a well.

A multiple-continuum model for simulating single-phase and multiphase flow in naturally fractured vuggy reservoirs

The existence of vugs or cavities in naturally fractured reservoirs has long been observed. Even though these vugs are known for their large attribution to reserves of oil, natural gas, or groundwater, few quantitative investigations of fractured vuggy reservoirs have been conducted. In this paper, a multiple-continuum conceptual model is presented, based on geological data and observations of core samples from carbonate formations in China, to investigate single-phase and multiphase flow behavior in such vuggy fractured reservoirs. The conceptual model has been implemented into a three-dimensional, three-phase reservoir simulator with a generalized multiple-continuum modeling approach. The conceptual model considers fractured vuggy rock as a triple- or multiple-continuum medium, consisting of (1) highly permeable and well-connected fractures, (2) low-permeability rock matrix, and (3) various-sized vugs. The matrix system may contain a large number of small or isolated cavities, whereas vugs are larger cavities, indirectly connected to fractures through small fractures, microfractures or matrix. Similar to the conventional double-porosity model, the fracture continuum is primarily responsible for the occurrence of global flow, while vuggy and matrix continua, providing storage space, are locally connected to each other and interacting with globally connecting fractures. In addition, flow in fractured vuggy reservoirs may be further complicated by occurrence of non-Darcy's and other nonlinear flow behavior, because of large pore space and high-permeability flow channels. To account for such complicated flow regime, our model formulation includes non-Darcy flow using the multiphase extension of the Forchheimer equation as well as flow according to parallel-wall fracture and tube models, based on solutions of flow through a parallel-wall, uniform fracture and Hagen–Poiseuille tube flow.

Theoretical analysis and field evidence of reciprocity gaps during interference pumping tests

We address the question of the reciprocity of drawdowns observed during a sequence of interference pumping tests performed within a porous medium, in the context of different conceptual models of subsurface flow. We provide a generalization of the work by Bruggeman (1972) [5], extending his results obtained in 1972 for Darcian flows in an unbounded, heterogeneous porous medium. We then analyze reciprocity within a dual-continuum conceptualization, where the medium is viewed as being composed of two overlapping continua, representing the porous matrix and embedded fractures, respectively. We show theoretically, and demonstrate numerically, that only drawdowns associated with the fracture continuum display reciprocity under transient flow conditions. Conversely, non-flowing matrix and fracture continua display reciprocity behavior under steady-state conditions. We then provide field evidence of the insurgence of reciprocity gaps by analyzing interference test data from the karstic limestone aquifer of the Hydrogeological Experimental Site (HES) in Poitiers, France. On the basis of our theoretical results and experimental observations, we discuss different interpretations of the observed reciprocity gaps. These include (a) non-linear dependencies of local hydraulic parameters, (b) occurrence of internal boundaries within the domain, (c) inertial effects that develop through open conduits within the rock matrix, (d) modifications of the aquifer properties between subsequent pumping tests, and (e) significant contribution of the matrix pressure in monitored wells when the behavior of the aquifer is conceptualized by a dual-continuum approach.

Interface Toughness of Carbon Nanotube Reinforced Epoxy Composites

Traditional single-fiber pull-out type experiments were conducted on individual multiwalled carbon nanotubes (MWNT) embedded in an epoxy matrix using a novel technique. Remarkably, the results are qualitatively consistent with the predictions of continuum fracture mechanics models. Unstable interface crack propagation occurred at short MWNT embedments, which essentially exhibited a linear load-displacement response prior to peak load. Deep embedments, however, enabled stable crack extension and produced a nonlinear load-displacement response prior to peak load. The maximum pull-out forces corresponding to a wide range of embedments were used to compute the nominal interfacial shear strength and the interfacial fracture energy of the pristine MWNT-epoxy interface.

Study on external water pressure of water conveyance tunnel for the pumped storage power stations in Huizhou City

A 3D numerical model of groundwater flow is established according to the geological conditions. Fractured rock mass is consist of rocks matrix, which is represented using the equivalent continuum model and fracture network, which is described with a discrete-fractured network model. The two models are coupled with continuum groundwater table and flux equilibrium, and solution of coupling model is deducted. The comparison is conducted between the measured and simulated external water pressure, which shows that the coupled model is feasible. Also different drainage schemes are designed and discuss the distribution of external water pressure of the schemes.

A numerical study on the impact resistance of composite shells using an energy based failure model

This paper presents a numerical study on the impact resistance of composite shells laminates using an energy based failure model. The damage model formulation is based on a methodology that combines stress based, continuum damage mechanics (CDM) and fracture mechanics approaches within a unified procedure by using a smeared cracking formulation. The damage model has been implemented as a user-defined material model in ABAQUS FE code within shell elements. Experimental results obtained from previous works were used to validate the damage model. Finite element models were developed in order to investigate the pressure and curvature effects on the impact response of laminated composite shells.