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On adaptive refinement techniques in multi-field problems including cohesive fracture
Abstract
This paper presents a generalized finite element formulation which incorporates solid and fluid phases together with a temperature field. The model is developed to obtain time-dependent solutions of complex 2-D cases, such as concrete gravity dams subjected to loading–unloading cycles, non-homogeneous specimens subjected to thermo-mechanical effects, etc. The solid behaviour incorporates a fully coupled cohesive-fracture discrete model, which includes thermal and hydraulic loads and the resulting crack nucleation and propagation is fully described. The evolution of fractures leads to continuous topological changes of the domain and these are handled by systematic local remeshing of the domain and a corresponding change of fluid and thermal boundary conditions. Optimality of the size of automatically generated finite elements is controlled, and the mesh density is adaptively adjusted on the basis of an a posteriori error estimation. For the process zone an element threshold number is introduced to obtain mesh independent results. The presented applications demonstrate the efficiency of the procedure and the importance of mesh refinement in multi-physics problems.
... Improving the research work in hydraulic fracturing, a model was proposed with a tip velocity provided as part of the solution algorithm. Hydraulic fractures were successfully propagated in an unknown path that may enucleate everywhere, depending only on the stress and pressure fields [20]. More recently, research works investigated in detail the influence of the constitutive cohesive zone characteristics on the size of process zone and consequently on the obtained results in hydraulic fracturing modeling. ...
... It is then clear that with increase of the in situ stress difference, the plastic zones will increase. Equation (20) suggests that larger plastic zones will be generated with increasing formation pore pressure. ...
... In this research work, we investigated all the mentioned characteristics (influence of stress deviator and formation pressure) that affect the plastic zone development according to the augmented Equation (20) that describes the plastic zone scaling by conducting parametric analysis in a poroelastoplastic model to determine the influence of the pore pressure parameter on the aforementioned scaling. ...
... Of the fracture criteria, the initiation criterion answers the question of whether or not the displacement discontinuity should initiate or advance. The length by which the crack increases during a time (load) step is often taken to be fixed (e.g., a predetermined length as in [18,19] or the length of a typical element in the crack-tip area as in [5,6]) or, alternatively, determined based on an extrinsic advancement rule e.g., by using a Paris-type law as in [20,21] or enforcing a zero cohesive potential condition at the new crack tip as in [22]. The angle of crack propagation can be determined from a criterion such as that of maximum circumferential stress [23] or taken a priori along a predetermined crack path (e.g., a material interface) or restricted to the union of element edges in the mesh. ...
... Time-discontinuity also introduces challenges in obtaining convergence of implicit time integrators during solution of the system (1) due to the interference of crack propagation criteria with Newton iterations [32]. Many researchers have sought to circumvent these challenges either by using explicit time integrators or by keeping the cracks fixed within the computations of the Newton solver, see e.g., [5,18,19,25,33] among many others. With explicit time stepping, numerical results have been shown to suffer from over-activation of interfaces and non-physical velocity fields emanating from internal shocks induced at initiation of failure due to the discontinuity [24,26,29,32]. ...
... As a matter of fact, convergence of crack-tip velocities in this class of methods has been disputed in recent years [35]. Non-convergent behavior of this nature has been reported for various problems in the literature; see, for example, Figure 6 in [19], Figure 19 in [18], and Figure 23 in [36], to name a few. ...
We highlight the ability of a proposed energy-based cohesive interface method to produce stable and convergent solutions where methods based on failure criteria at similar discretization levels fail. The key feature of the method is the smooth transition from “uncracked” to “cracked” states, i.e., internal forces remain continuous functions of the deformation at initiation of failure. This property is missing in methods based on stress criteria. In explicit time stepping calculations, lack of continuity gives rise to spurious crack opening velocity fields. This issue is particularly significant in multiphysics problems such as hydraulic fracturing due to the coupling of the unknown fields and may lead to instability of the computational algorithm. In implicit time stepping calculations, lack of continuity introduces challenges in obtaining convergence of Newton iterations. Often the issue is circumvented by keeping cracks fixed within the iterative solver; the configuration of cracks is only updated at the end of a time step in such algorithms. This approach leads to the dependence of the crack-tip velocity on temporal and spatial discretization parameters. We present various simulation results to show that the energy approach is free of all such undesirable behaviors.
... The final example has been selected to demonstrate how well the suggested phase-field methodology performs when dealing with a more complex boundary value problem, such as a concrete gravity dam subject to hydrostatic pressure caused by rising water levels in the reservoir and the weight of the dam itself. The dam's structure resembles that of the ICOLD benchmark exercise A2 modelled in [56]. Previous studies by Schrefler et al. [56] and Khoei et al. [23,57] employ quasi-static cohesive fracture analysis, dynamic analysis of cohesive fracture propagation, and extended finite element analysis, respectively. ...
... The dam's structure resembles that of the ICOLD benchmark exercise A2 modelled in [56]. Previous studies by Schrefler et al. [56] and Khoei et al. [23,57] employ quasi-static cohesive fracture analysis, dynamic analysis of cohesive fracture propagation, and extended finite element analysis, respectively. In the current simulation, the dynamic phase-field framework is implemented to assess the crack growth pattern in the dam's concrete foundation, and the obtained results are compared to existing literature. ...
This paper presents a comprehensive study on phase-field modelling in COMSOL Multi-Physics for simulating dynamic hydraulic fracturing in porous media based on Biot's poro-elasticity theory. The focus is on addressing the challenges associated with crack width estimation in this context. A new strain-based crack width formulation is proposed, offering improved accuracy in predicting fracture permeability and ease of implementation in numerical approaches. The model's capabilities are extended to consider dynamic crack propagation by incorporating the kinetic energy in the governing coupled hydro-mechanical-damage equations. The numerical implementations in COMSOL MultiPhysics are thoroughly explained, providing insights into the techniques used to solve the governing equations. Verification examples, including the benchmark KGD verification, are presented to demonstrate the model's capabilities in simulating hydraulic fractures in porous media and validate its accuracy and reliability. A final numerical example focusing on the dynamics of crack propagation in a gravity dam is simulated, allowing for a comprehensive examination of the model's performance. The proposed strain-based crack width formulation and consideration of dynamic crack propagation contribute to improved accuracy in predicting fracture permeability.
... While most mechanical models of HF treat the rock material as isotropic (for example, Detournay and Cheng [4], Detournay [5], Zimmermann et al. [6], Schrefler et al. [7] set the plane strain fractures in isotropic rocks), the reservoir rocks are commonly anisotropic [8], in the reality especially for shale and the evidence for this can been seen in many experimental tests [9][10][11]. ...
... In addition, an additional pressure-related term and an anisotropic fracture toughness tensor are introduced in the energy functional, which is then used to achieve the governing the fracture because the fracture opening cannot be directly extracted in a PFM. Therefore, future research will incorporate more reasonable permeability models [70] into PFMs with reference to more benchmark examples [7,71]. In addition, in future research, the proposed PFM should be extended to more complex cases such as inelastic, partially saturated, or heterogeneous transversely isotropic porous media. ...
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 uses a transversely isotropic constitutive relationship between stress and strain as well as anisotropy in fracture toughness and permeability. An additional pressure-related term and an anisotropic fracture toughness tensor are added 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 and the displacement and fluid pressure are monolithically solved in a staggered step. Afterwards, two examples are tested 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.
... A disadvantage of interface elements is that they require an a priori knowledge of the crack extension direction. Remeshing was introduced as the remedy for the arbitrary crack propagation 22,23,24 , also in saturated porous media 25,26 . ...
... This causes an unwanted opening of the domain in front of the crack. This can be avoided by adopting a Heaviside step function, seeFigure 5. Employing shifting and blending Equations(24)(25) become:̇ ...
An extended isogeometric analysis (XIGA) approach is proposed for modelling fracturing in a fluid-saturated porous material. XIGA provides a definition of the discontinuity independent of the underlying mesh layout, which obviates the need of knowing the crack extension direction a priori. Unlike Lagrange shape functions used in the standard finite element approach, Non-Uniform Rational B-Splines (NURBS) provide a higher-order interelement continuity which leads to a continuous fluid flow also at element boundaries, thereby satisfying the local mass balance. It also leads to an improved estimate of the crack path due to a smoother stress distribution. The NURBS basis functions are cast in finite element data structure using Bézier extraction. To model the discontinuity, the Heaviside sign function is utilised within the displacement and the pressure fields, complemented by the shifting and the blending techniques to enforce compatibility perpendicular and parallel to the crack path, respectively. Different aspects of the approach are assessed through examples comprising straight and curved crack paths for stationary and propagating discontinuities.
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... Several authors have reported negative pressure in front of the fracture tip (A. Mikelić et al., 2015a;Salimzadeh & Khalili, 2015;Salimzadeh et al., 2017;Schrefler et al., 2006;Secchi & Schrefler, 2012). In Mikelić et al. (2015a), negative pressure is observed from the start of simulations and before fracture propagation. ...
... In Mikelić et al. (2015a), negative pressure is observed from the start of simulations and before fracture propagation. In Schrefler et al. (2006), Secchi and Schrefler (2012), Salimzadeh and Khalili (2015), and Salimzadeh et al. (2017), negative pressure is observed at the crack tip during fracture propagation. The authors reason that the fluid lag in filling the created fractures leads to negative pressure. ...
Hydraulic fracturing of tight subsurface formations has mainly been conducted using water. Waterless hydraulic fracturing by CO2 may have advantages over fracturing by water. The challenge has been numerical simulation of the process. In this work, we conduct a comprehensive study on 2D hydraulic fracturing simulation by water, CO2, and nitrogen. The simulations are based on the phase‐field; the results are compared with the lab data. We first present advances in simulation of fluid exchange between the fractures and the rock matrix. The conventional nodal‐based finite element method may give rise to unphysical negative pressure around the tip and fracture notch. We demonstrate that the mixed hybrid finite element method gives non‐negative pressure distribution. Next, we examine the effect of inertial term on fracture configuration and observe branching by CO2 under dynamic formulation with a low critical energy release rate. Lastly, we find from our simulations that in shale rocks, the critical energy release rate is the lowest for CO2, followed by nitrogen, and then water. Consequently, CO2 provides the highest fracture surface area and fracture intensity. In a sandstone sample we find nitrogen has a lower energy release rate than water. As a result, the fracture surface area is higher for nitrogen than water. Our simulation results confirm that the phase‐field method may incorporate fluid‐rock surface free energy without the need for various parameter adjustments.
... Thus, the overall density can be expressed as ρ = α n α ρ αR . A comprehensive theoretical and applications review on the porous materials are provided in the works [58,93,99,114,283,214]. The fluid volume fraction (porosity) n F is linked to the fluid volume ratio θ (fluid content) per unit volume of the undeformed reference configuration B via ...
... For the modeling of fluid transport at the microscale, we assume laminar flow described by the Stokes equation which can be linked to Darcy's law in porous media. A review on the foundations and applications of porous materials can be seen in the pioneering works [58,93,99,114,283,214]. The coupling between those porous-media models and the DP elasto-plasticity formulations at failure are based on an energetic and a dissipative response functions for plasticity, fluid flow and fracture. ...
The underlying Habilitation aims to contribute to the research on fracture mechanics of solids across the scales. This active research field is driven by the investigation and development of new methods, processes and technologies applicable to engineering problems with complex material behavior of solids at fracture. It includes mathematically precise formulations of theoretical and computational models with emphasis on continuum physics as well as the development of variation methods and efficient numerical implementations tools. In particular, two directions will be considered in this contribution: (i) the construction of advanced multi-scale techniques and (ii) modern element technologies. On the multi-scale techniques, a robust and efficient Global-Local approach for numerically solving fracture-mechanics problems is developed in the first part of this contribution. This method has the potential to tackle practical field problems in which a large-structure might be considered and fracture propagation is a localized phenomenum. In this regard, failure is analyzed on a lower (Local) scale, while dealing with a purely linear problem on an upper (Global) scale. The modeling of crack formation at the Local scale is achieved in a convenient way by continuum phase-field formulations to fracture, which are based on the regularization of sharp crack discontinuities. For this purpose, a predictor-corrector scheme is designed in which the local domains are dynamically updated during the computation. To cope with different element discretizations at the interface between the two nested scales, a non-matching dual mortar method is formulated. Hence, more regularity is achieved on the interface. The development of advanced discretization schemes accounting for meshes with highly irregular shaped elements and arbitrary number of nodes is the main focus in the second part of this work. To this end, a relatively new method - the virtual element method (VEM) - will be presented here that leads to an exceptional efficient and stable formulation for solving a wide range of boundary value problems in science and engineering. The structure of VEM comprises a term in the weak formulation or the potential density functional in which the unknowns, being sought are replaced by their projection onto a polynomial space. This results in a rank-deficient structure, therefore it is necessary to add a stabilization term to the formulation. The performance of the virtual element method is comparable to using finite elements of higher order. It is even more robust than FEM in case of a severe distortion of the element.
... During the past decades, based on different numerical approaches like the finite element method (FEM) [3,4], the displacement discontinues method (DDM) [5], the phase field method [6,7], the distinct element method (DEM) [8], the extended finite element method (XFEM) [9][10][11][12], and the proper generalized decomposition method (PGD) [13], a variety of numerical models have been established and applied by researchers to study hydraulic fracture propagation in consideration of different kinds of influence factors. Among these studies, Schrefler et al. [3] proposed an adaptive refinement technique to simulate hydraulic fracturing problems based on the generalized finite element formulation. ...
... During the past decades, based on different numerical approaches like the finite element method (FEM) [3,4], the displacement discontinues method (DDM) [5], the phase field method [6,7], the distinct element method (DEM) [8], the extended finite element method (XFEM) [9][10][11][12], and the proper generalized decomposition method (PGD) [13], a variety of numerical models have been established and applied by researchers to study hydraulic fracture propagation in consideration of different kinds of influence factors. Among these studies, Schrefler et al. [3] proposed an adaptive refinement technique to simulate hydraulic fracturing problems based on the generalized finite element formulation. Song et al. [4] performed a series of numerical simulations and examined the influence of some key factors on hydraulic fracturing propagation using RFPA2D-Flow. ...
Field data suggests that carbonate reservoirs contain abundant natural fractures and cavities. The propagation mechanisms of hydraulic fractures in fracture-cavity reservoirs are different from conventional reservoirs on account of the stress concentration surrounding cavities. In this paper, we develop a fully coupled numerical model using the extended finite element method (XFEM) to investigate the behaviors and propagation mechanisms of hydraulic fractures in fracture-cavity reservoirs. Simulation results show that a higher lateral stress coefficient can enhance the influence of the natural cavity, causing a more curved fracture path. However, lower confining stress or smaller in-situ stress difference can reduce this influence, and thus contributes to the penetration of the hydraulic fracture towards the cavity. Higher fluid viscosity and high fluid pumping rate are both able to attenuate the effect of the cavity. The frictional natural fracture connected to the cavity can significantly change the stress distribution around the cavity, thus dramatically deviates the hydraulic fracture from its original propagation direction. It is also found that the natural cavity existing between two adjacent fracturing stages will significantly influence the stress distribution between fractures and is more likely to result in irregular propagation paths compared to the case without a cavity.
... Several numerical algorithms have been designed to concurrently track two independent moving boundaries, following the evolution of the lag in time: the singular dislocation solutions, either for the case of semi-infinite crack propagating at constant velocity [30,39,35] or for the plane strain case [40]; finite elements [28,41,42]; XFEM [43,44,45,46,47,48]. The present work aims at establishing a set of governing equations that result in a novel numerical scheme, capable to accurately describe the evolution of a crack filled by a viscous Newtonian fluid in an infinite impermeable elastic medium. ...
... If the onset of crack propagation is equivalently written in the form ϕ(s, t) = K I (s, t) − K C I , from normality condition (40) one obtains that v(s, t) equals the plastic multiplieṙ λ(s, t). Thermodynamic consistency K I (s, t)v(s, t) ≥ 0 (42) in therefore satisfied in view of positive definiteness of K I and complementarity laws (41). Plugging eq. ...
Propagation of a fluid-driven crack in an impermeable linear elastic medium under axis-symmetric conditions is investigated in the present work. The fluid exerting the pressure inside the crack is an incompressible Newtonian one and its front is allowed to lag behind the propagating fracture tip. The tip cavity is considered as filled by fluid vapors under constant pressure having a negligible value with respect to the far field confining stress. A novel algorithm is here presented, which is capable of tracking the evolution of both the fluid and the fracture fronts. Particularly, the fracture tracking is grounded on a recent viscous regularization of the quasi-static crack propagation problem as a standard dissipative system. It allows a simple and effective approximation of the fracture front velocity by imposing Griffith's criterion at every propagation step. Furthermore, for each new fracture configuration, a non linear system of integro-differential equations has to be solved. It arises from the non local elastic relationship existing between the crack opening and the fluid pressure, together with the non linear lubrication equation governing the flow of the fluid inside the fracture.
... Method (XFEM) [6][7][8][9][10], Partition-of-Unity Finite Element Method (PUFEM) [11,12], Embedded Discontinuity Finite Element Method (EDFEM) [13][14][15][16][17], Finite element methods with adaptive remeshing techniques [8,[18][19][20][21], Central force lattice model [8,22,23], Phase field model [24][25][26][27] or Discrete lattice model [28][29][30][31][32]. The solid phase-pore fluid interaction is further extended to account for the temperature effects in the saturated porous media. ...
... 18 Comparison of vertical overloads ...
This thesis studies the issue of the overall safety of structures built of heterogeneous and pore-saturated materials under extreme loads in application to fluid-structure interaction problems, such as the dam-reservoir interaction. We propose a numerical model of interaction capable of predicting main tendencies and overall behavior of pore-saturated dam structure interacting with the reservoir in failure analyses of practical interest. The proposed numerical model is first presented in two-dimensional (2D) framework and later extended to three-dimensional (3D) framework. We consider the structure built of porous cohesive material. We assume that the external fluid in interaction with the structure acts as a source of pore saturation. We model the response of the pore-saturated structure with the coupled discrete beam lattice model based on Voronoi cell representation of domain with inelastic Timoshenko beam finite elements enhanced with additional kinematics in terms of embedded strong discontinuities acting as cohesive links. The coupling between the solid phase and the pore fluid is handled with Biot’s porous media theory, and Darcy’s law governing the pore fluid flow. The numerical consideration of internal coupling results with an additional pressure-type degree of freedom placed at each node of the Timoshenko beam finite element, which is later used at the fluidstructure interface. The confined conditions met for external fluid placed in the reservoir enable the modeling of external fluid motion with the acoustic wave theory. For the numerical representation of the external fluid limited to small (irrotational) motion, we choose a Lagrangian formulation and the mixed displacement/pressure based finite element approximation. The end result are the displacement and pressure degrees of freedom per node of external fluid finite elements, which allows for the issue of the fluid-structure interface to be solved in an efficient and straightforward manner by directly connecting the structure and external fluid finite elements at common nodes. As a result, all computations can be performed in a fully monolithic manner. All numerical implementations and computations are performed with the research version of the computer code FEAP (Finite Element Analysis Program). The proposed numerical models of structure, external fluid and ultimately numerical model of interaction are validated in the linear elastic regime of structure response by comparing computed results against reference values obtained either with analytical solutions or continuum models. The numerical simulations in the nonlinear regime of structure response are performed with the aim to demonstrate the proposed coupled discrete beam lattice model capabilities to capture complete macro-scale response and failure mechanisms in pore-saturated structures. Finally, the proposed numerical model of interaction ability to deal with the progressive localized failure of a dam structure built of porous cohesive material under damreservoir interaction for a particular loading program was tested. To account for the temperature effects, the thermal coupling is introduced in the numerical model of the structure.
... So far, the existing literature on Hydraulic Fracturing (HF) has not answered firmly whether the irregular or stepwise fracture propagation in porous media is a real phenomenon or a numerical artefact. We focus our attention on the cohesive fracture model because it is with such a model that the problem of fracture advancement appeared first in the mechanics community (Schrefler et al., 2006). In this context, we also address the problem of consistency in modelling fracture advancement with a cohesive model in saturated homogeneous porous media, especially when the dynamic effects are not negligible. ...
There is growing evidence that fracture advancement in saturated and dry porous media may be smooth or stepwise. In the stepwise behaviour, there are also pressure oscillations in saturated porous materials, as predicted by Biot’s theory. The type of behavior depends on the specifications of the problem and material prop-
erties. Not all the adopted numerical models are capable of capturing stepwise behaviour. While non-local fracturing models are perfectly adapted to capture the stepwise behaviour, it is shown that cohesive models are also capable to model such a behaviour. In fact, it is necessary to satisfy a consistency condition for the numerical
solution; in other words, the fracture advancement/time-stepping algorithm must not impose a constraint on the tip advancement speed. This is peculiar to cohesive models because there the unit of fracture advancement is specified beforehand, while in other models, such as peridynamics, it is an outcome of the analysis. In inhomogeneous media, it can be expected to capture irregular results from hydraulic fracturing (HF); however, homogeneous media can also show such behaviour as presented herein. In this study, the eXtended Finite Element Method (XFEM) is used in conjunction with a cohesive crack model to investigate the stepwise or irregular behaviour of HF in homogeneous saturated porous media both under quasistatic and dynamic conditions. The results clearly show that the extent of irregularity depends directly on the intensity of dynamic effects in the problem. Moreover, fracture forerunning is one of the reasons of the stepwise fracture growth in the solutions. Although the results are obtained by XFEM, the conclusions are not restricted to this particular numerical method, and the findings can provide useful insights into the necessary aspects of the numerical algorithms for dynamic HF modelling.
... The typical continuum-based approach mainly adopts the finite element method (FEM) to cooperate with the fracture/damage computational techniques and models. For instance, Schrefler et al. [33] and Secchi et al. [34] applied the adaptive refinement technique in the FEM to analyze cohesive hydraulic fracture problems. Sarris and Papanastasiou [35] and Yao et al. [36] adopted the cohesive zone model to investigate the fracture propagation in the poro-elastoplastic continuum. ...
... A variety of numerical tools have been developed to simulate the dynamic hydraulic fracturing in saturated media. A fully coupled cohesive-fracture discrete model was combined to a generalized finite element formulation in [17] for solving the dynamic fracture problems in porous media in thermal-hydromechanical coupled context. Further, the three-dimensional version of the model was presented in [18] and applied to deal with the problem of hydraulic fracturing in a concrete dam in quasi-static conditions. ...
In this paper, a novel hybrid FEM and Peridynamic modeling approach proposed in Ni et al. (2020) is used to predict the dynamic solution of hydro-mechanical coupled problems. A modified staggered solution algorithm is adopted to solve the coupled system. A one-dimensional dynamic consolidation problem is solved first to validate the hybrid modeling approach, and both -convergence and -convergence studies are carried out to determine appropriate discretization parameters for the hybrid model. Thereafter, dynamic fracturing in a rectangular dry/fully saturated structure with a central initial crack is simulated both under mechanical loading and fluid-driven conditions. In the mechanical loading fracture case, fixed surface pressure is applied on the upper and lower surfaces of the initial crack near the central position to force its opening. In the fluid-driven fracture case, the fluid injection is operated at the centre of the initial crack with a fixed rate. Under the action of the applied external force and fluid injection, forerunning fracture behavior is observed both in the dry and saturated conditions.
... These approaches are potentially appropriate for irregular body shapes with heterogeneous material properties and nonlinear behavior. In [11,12,13], the adaptive re-meshing technique is used to follow the crack surfaces during the hydraulic fracturing process. Re-meshing is usually computationally expensive and it is difficult to develop robust schemes considering mass and momenta conservation [13]. ...
This paper presents a hybrid modeling approach for simulating hydraulic fracture propagation in saturated porous media: ordinary state-based peridynamics is used to describe the behavior of the solid phase, including the deformation and crack propagation, while FEM is used to describe the fluid flow and to evaluate the pore pressure. Classical Biot poroelasticity theory is adopted. The proposed approach is first verified by comparing its results with the exact solutions of two examples. Subsequently, a series of pressure- and fluid-driven crack propagation examples are solved and presented. The phenomenon of fluid pressure oscillation is observed in the fluid-driven crack propagation examples, which is consistent with previous experimental and numerical evidences. All the presented examples demonstrate the capability of the proposed approach in solving problems of hydraulic fracture propagation in saturated porous media.
... The phenomenon has important economic relevance in fracking operations [11] and it would be a pity if computational continuum models would not be able to simulate it properly. But this is not the case as shown in [9,[24][25][26][27][28][29][30][31][32][33][34]. These authors used either standard Finite Elements, XFEM, Finite Volume, Finite Differences on originally continuum models. ...
Comments to K.M. Pervaiz Fathima, Ren\'e de Borst, Implications of single or multiple pressure degrees of freedom at fracture in fluid saturated porous media, Engineering Fracture Mechanics, 213 (2019), 1-20.
... Lecampion et al. (2018) and Chen et al. (2022) conducted comprehensive reviews on the advances and limitations of different modeling approaches. On the contrary, a few simulation studies reported the intermittent or stepwise fracture advancing phenomenon using different numerical methods (Schrefler et al., 2006, Secchi and Schrefler, 2014, Milanese et al., 2016, Cao et al., 2018. Peruzzo et al. (2019) stressed that stepwise fracture tip advancement does exist and has been proved with laboratory experiments and indirect field observations (e.g., wellbore pressure), which cannot be ignored in numerical models. ...
Characterizing the fluid-driven fracture tip advancing process presents a significant challenge due to the difficulty of replicating real-world conditions in laboratory experiments and the lack of precise field measurements. However, recent advances in low-frequency distributed acoustic sensing (LF-DAS) technology offer new opportunities to investigate the dynamics of propagating hydraulic fractures. In this study, we propose an iterative inversion method to characterize fracture-tip advancing behaviors using LF-DAS data. A forward geomechanical model is developed using the three-dimensional displacement discontinuity method, and the optimization is realized by a conjugate gradient method. The performance of the inversion algorithm is demonstrated using a synthetic case, in which the fracture half-length evolution and propagation velocity match well with the reference solutions. Additionally, the averaged fracture cross-section area, fracture volume, and fracturing fluid efficiency can also be estimated, showing good agreements with true values of the synthetic case under reasonable assumptions. Then a field case with a single-cluster hydraulic fracturing treatment from the Hydraulic Fracturing Test Site 2 project (HFTS2) is studied. Our analysis of the inversion results reveal that the fracture propagates intermittently, as evidenced by the fracture half-length evolution. This unique field evidence can guide modeling efforts to incorporate this important physical behavior into fracture models, and the secondary information gathered from the study, including fracture cross-section area and volume, can help evaluate and optimize fracturing efficiency.
... In the reservoir domain, the fluid flow is regarded as Darcy-type flow, while in the cracked domain, the fluid flow is modeled as a Darcy-type flow equivalent to the Plane Poiseuille-type flow [8,9]. Alternatively, the equivalent permeability of the cracked domain is dependent on the crack width [24]. The width of phase-field crack is linearly related to the characteristic length of a single element perpendicular to the crack [8]. ...
... It is well known that the cohesive zone model (CZM) can be used to study the onset and propagation of delamination, and it is also convenient to implement into finite element models. Currently, CZM has been widely developed to investigate multi-physics problems [43][44][45][46]. For example, the thermo-mechanical CZMs were presented to allow for discontinuities in temperature and displacement, and then it could analyze the change of the heat flow pattern and local stress [47][48][49]. ...
... In recent decades, finite element method (FEM) [34,35], extended finite element method(XFEM) [36,37] and boundary element method (BEM) [38] are the widely used method to model hydraulic fracturing. However, fracture propagation in these methods is generally under the assumption of linear elastic fracture mechanics (LEFM), which uses a singular stress distribution at the fracture tip as fracture propagation criteria. ...
Understanding hydraulic fracture propagation behaviours is vital for optimizing fracturing design in oil and gas production. It has been observed that the stress and pore pressure distribution are much more complex during fracturing treatment as a result of long-term fluid injection and production in the reservoir. When rock is subjected to internal hydraulic pressure and external mechanical loading, the closing, opening, or other interactions of pre-existing weaknesses or induced new fractures will be altered. In this paper, a number of hydraulic simulation scenarios are performed to investigate the effect of pore pressure distribution on fracture initiation and propagation using the unified pipe-interface element method (UP-IEM). The proposed method for predicting the hydraulic fracture path in the permeable porous medium with non-uniform pore pressure field is validated with experimental results. Considering the arrangement of well patterns and natural fractures in the rock, the problems of hydraulic fracture propagation behaviours with wellbore interaction and pore pressure induced natural fracture activation are simulated. The influence of pore pressure saturation and the injection sequence of wellbores on hydraulic fracturing are researched. A sensitivity analysis is carried out to investigate the parameter effects on pre-existing fracture tortuosity, opening width and breakdown pressure under pore pressure. These numerical examples provide guidance to engineers to estimate the probability of danger or possibilities to improve the injection and production of wells.
... Remeshing can however remove this limitation and has been used successfully in the simulation of fracturing in a porous medium. 17,1 For the interpolation of the field variables several basis functions have been employed in this context. Lagrange basis functions have been used frequently, 15,16 due to their simplicity and ease of implementation. ...
This paper addresses fluid‐driven crack propagation in a porous medium. Cohesive interface elements are employed to model the behaviour of the crack. To simulate hydraulic fracturing, a fluid pressure degree of freedom is introduced inside the crack, separate from the fluid degrees of freedom in the bulk. Powell‐Sabin B‐splines, which are based on triangles, are employed to describe the geometry of the domain and to interpolate the field variables: displacements and interstitial fluid pressure. Due to their C1$\mathcal {C}^1$‐continuity, the stress and pressure gradient are smooth throughout the whole domain, enabling a direct assessment of the fracture criterion at the crack tip and ensuring local mass conservation. Due to the use of triangles, crack insertion and remeshing are straightforward and can be done directly in the physical domain. During remeshing a mapping of the state vector (displacement and interstitial fluid pressure) is required. For this, a new methodology is exploited based on a least‐square fit with the energy balance and mass conservation as constraints. The accuracy to model free crack propagation is demonstrated by two numerical examples, including crack propagation in a plate with two notches.
... Remeshing was introduced to remove this restriction, and has been used for arbitrary crack propagation in saturated porous media. 24,25 Early works on discrete crack propagation utilized Lagrange basis functions for the approximation of the field variable. 14,15 Since the stress is then discontinuous at element boundaries, including at the crack tip, the accuracy in the stress predication is lower, which is particularly important for determining the crack initiation and the prediction of the crack propagation direction. ...
Powell‐Sabin B‐splines are employed to model progressive fracturing in a fluid‐saturated porous medium. These splines are defined on triangles and are C1‐continuous throughout the domain, including the crack tips, so that crack initation can be evaluated directly at the tip. On one hand, the method captures stresses and fluid fluxes more accurately than when using standard Lagrange elements, enabling a direct assessment of the fracture criterion at the crack tip and ensuring local mass conservation. On the other hand, the method avoids limitations for discrete crack analysis which adhere to isogeometric analysis. A crack is introduced directly in the physical domain. Due to the use of triangles, remeshing and crack path tracking are straightforward. During remeshing transfer of state vectors (displacement, fluid pressure) is required from the old to the new mesh. The transfer is done using a new approach which exploits a least‐squares fit with the energy balance and conservation of mass as constraints. The versatility and accuracy to simulate free crack propagation are assessed for mode‐I and mixed‐mode fracture problems.
... Recently, the aperture asymptote of Dontsov and Peirce [19] provided a single asymptote capable of predicting propagation throughout the toughness-viscous propagation regime [20]. Such a simple strategy avoids the computational cost incurred when relying exclusively on a single (toughness) asymptote [21]; which otherwise requires employing sophisticated mesh generation and transfer schemes [22,23], and/or parallelisations [24][25][26][27][28] for an accurate prediction of propagation. The Dontsov and Peirce aperture asymptote [19,20], herein termed HFM asymptote, provides an additional feature that is lacking within methods that solely rely on the toughness asymptote; the numerical solution for the aperture can be used to predict the fracture's propagation velocity. ...
The finite element method is a powerful and general numerical method used to simulate subsurface processes. In this paper, we take recent hydraulic fracturing propagation algorithms and assess their performance when used within the finite element framework. In particular, we evaluate aperture and energy-based methodologies that are capable of extracting the propagation velocity of a hydraulic fracture propagating throughout the toughness and viscous regime. Such algorithms have the benefit of a quicker convergence on the fracture front. The aperture-based methodology consists of the multi-scale aperture asymptote that is yet to be applied with finite elements. On the other-hand, the energy-based methodology consists of a recently developed procedure for predicting the propagation velocity from the energy release rate, which is calculated using a J-integral devised for hydraulic fracturing. A comparison of the accuracy and the number of iterations required to converge on the fracture length is undertaken, and found to produce similar results for both methods. Consequently, we conclude that the higher accuracy of energy-based methods in extracting stress intensity factors does not immediately translate to a higher accuracy in extracting propagation velocities, most notably in the toughness-dominated propagation regime. Given the similar performance of the methods, and the simplicity of the aperture-based approach, we then extend the evaluation of the multi-scale aperture asymptote to the case of buoyancy-driven propagation. As a result, the aperture asymptote is shown to be a simple and efficient method for the simulation of subsurface processes using a finite element framework.
... In parallel, Hutchinson [13] proved that J-integral can be viewed as a nonlinear, stress-intensity parameter as well as an indicator of the energy release rate. With the aid of the high-performance computational tools in recent years a range of complex mechanisms in the context of fracture mechanics have has implemented numerically using a variety of FEM [14,15], XFEM [16,17], Mesh-free Galerkin [18,19], Lattice Spring [20], and Phase-field [21] approaches. ...
Mechanical characterization of fractures, i.e., identifying their characteristic parameters such as energy release rate, is crucial to assess the safety and stability of structural members. This is generally achieved using a combination of finite element analysis and optimization. Machine learning models are increasingly used to characterize engineering problems. While such models have shown impressive performance on smooth data, their performance diminishes significantly on data with discontinuities and sharp gradients. For fractures, this issue is more severe due to the singular solutions in the vicinity of the fracture tips. To resolve this difficulty, leveraging classical fracture mechanics, we construct custom functions, using neural networks, containing fundamental solutions of fracture mechanics. Through a series of examples, we demonstrate the proposed framework not only captures the singular solution and characteristic parameters accurately on both noise-free and noisy data, but also is highly efficient due to its simplicity and small number of parameters.
... ). Source: Reprinted fromSchrefler et al. (2006), Copyright (2006, with permission from Elsevier.Figure 6.43 Zoom near the fracture for maximum principal stress contour. Source: Reprinted from Schrefler et al. (2006), Copyright (2006), with permission from Elsevier.Figure 6.44 Zoom for pressure distribution within the crack and fluid lag. ...
This chapter deals with the solution of static and quasi‐static problems in which dynamic (inertial) effects are negligible and with hydraulic fracturing where inertia effects may be included. It is concerned with the deformation and movement of the soil or of its associated foundation. The chapter illustrates how the procedures can be used to solve seepage problems. Water seepage is encountered, for instance, in a study of earth dams or concrete dams, or in the analysis of foundations or slope stability to mention just a few examples. The chapter is devoted to various problems concerning consolidation, including a 3‐D example with adaptivity in time. It addresses pressure‐driven fracture in fully saturated porous media, both in quasi‐static and dynamic situations. Fracture propagation in porous media, such as a fluid‐driven fracture propagating in rock to enhance the recovery of hydrocarbons from underground reservoirs, is a common problem in geomechanics.
... Remeshing was introduced as the remedy for the arbitrary crack propagation [28,29,27], also in saturated porous media [116,118]. ...
... In the context of cohesive extrinsic fracture, these are zero displacement jump solutions at the time of crack initiation which frequently occur throughout the domain. Other authors, for example [24], recognizing the issues presented by implicit solutions of extrinsic models, opted for an explicit solution of the quasistatic problem or circumvented difficulties of an implicit solution by either keeping the cracks fixed during Newton's iterations [25,26] or employing an ''elastic-cohesive'' (''intrinsic'') model [27]. Implicit methods have been studied in the context of peridynamics in [28]. ...
A method for quasistatic cohesive fracture is introduced that uses an alternating direction method of multipliers (ADMM) to implement an energy approach to cohesive fracture. The ADMM algorithm minimizes a non-smooth, non-convex potential functional at each strain increment to predict the evolution of a cohesive-elastic system. The optimization problem bypasses the explicit stress criterion of force-based (Newtonian) methods, which interferes with Newton iterations impeding convergence. The model is extended with an extrapolation method that significantly reduces the computation time of the sequence of optimizations. The ADMM algorithm is experimentally shown to have nearly linear time complexity and fast iteration times, allowing it to simulate much larger problems than were previously feasible. The effectiveness, as well as the insensitivity of the algorithm to its numerical parameters is demonstrated through examples. It is shown that the Lagrange multiplier method of ADMM is more effective than earlier Nitsche and continuation methods for quasistatic problems. Close spaced minima are identified in complicated microstructures and their effect discussed.
... In the context of cohesive extrinsic fracture, these are zero displacement jump solutions at the time of crack initiation which frequently occur throughout the domain. Other authors, for example [24], recognizing the issues presented by implicit solutions of extrinsic models, opted for an explicit solution of the quasistatic problem or circumvented difficulties of an implicit solution by either keeping the cracks fixed during Newton's iterations [25,26] or employing an ''elastic-cohesive'' (''intrinsic'') model [27]. Implicit methods have been studied in the context of peridynamics in [28]. ...
A method for quasistatic cohesive fracture is introduced that uses an alternating direction method of multipliers (ADMM) to implement an energy approach to cohesive fracture. The ADMM algorithm minimizes a non-smooth, non-convex potential functional at each strain increment to predict the evolution of a cohesive-elastic system. The optimization problem bypasses the explicit stress criterion of force-based (Newtonian) methods, which interferes with Newton iterations impeding convergence. The model is extended with an extrapolation method that significantly reduces the computation time of the sequence of optimizations. The ADMM algorithm is experimentally shown to have nearly linear time complexity and fast iteration times, allowing it to simulate much larger problems than were previously feasible. The effectiveness, as well as the insensitivity of the algorithm to its numerical parameters is demonstrated through examples. It is shown that the Lagrange multiplier method of ADMM is more effective than earlier Nitsche and continuation methods for quasistatic problems. Close spaced minima are identified in complicated microstructures and their effect discussed.
... Several remeshing techniques, in particular the h-adaptive remeshing [134], have been proposed till date, see e.g. [135][136][137][138][139][140][141]. H-adaptive remeshing modifies only the element geometry, shape and sizes based on a posteriori error estimator, but keeps the element formulations unchanged [142]. ...
Thin unidirectional-tape and woven fabric-reinforced composites are widely utilized in the aerospace and automotive industries due to their enhanced fatigue life and impact damage resistance. The increasing industrial applications of such composites warrants a need for high-fidelity computational models to assess their structural integrity and ensure robust and reliable designs. Damage detection and modelling is an important aspect of overall design and manufacturing lifecycle of composite structures. In particular, in thin-ply composites, the damage evolves as a result of coupled in-plane (membrane) and out-of-plane (bending) deformations that often arise during critical events, e.g., bird strike/ hail impact or under in-flight service loads. Contrary to metallic structures, failure in composites involves complex and mutually interacting damage patterns, e.g., fibre breakage/ pullout/ bridging, matrix cracking, debonding and delamination. Providing high-fidelity simulations of intra-laminar damage is a challenging task both from a physics and a computational perspective, due to their complex and largely quasi-brittle fracture response. This is manifested by matrix cracking and fibre breakage, which result in a sudden loss of strength with minimum crack openings; subsequent fibre pull-outs result in a further, although gradual, strength loss. To effectively model this response, it is necessary to account for the cohesive forces evolving within the fracture process zone. Furthermore, the interaction of the failure mechanisms pertinent to both the fibres and the matrix necessitate the definition of anisotropic damage models. In addition, the failure in composites extends across multiple scales; it initiates at the fibre/ matrix-level (micro-scale) and accumulates into larger cracks at the component/ structural level (macro-scale). From a simulation standpoint, accurate prediction of the structure’s critical load bearing capacity and its associated damage thresholds becomes a challenging task; accuracy necessitates a fine level of resolution, which renders the corresponding numerical model computationally expensive. To this point, most damage models are applied at the meso-scale based on local stress-strain estimates, and considering material heterogeneity. Such damage models are often computationally expensive and practically inefficient to simulate the failure behaviour in real-life composite structures. Moreover at the macro-scale, the effect of local stresses is largely minimised, which necessitates definition of a homogenised failure criterion based on global macro-scale stresses. This thesis presents a phase field based MITC4+ (Mixed Interpolation of Tensorial Components) shell element formulation to simulate fracture propagation in thin shell structures under coupled membrane and bending deformations. The employed MITC4+ approach renders the element shear- and membrane- locking free, hence providing high-fidelity fracture simulations in planar and curved topologies. To capture the mechanical response under bending-dominated fracture, a crack-driving force description based on the maximum strain energy density through the shell-thickness is considered. Several numerical examples simulating fracture in flat and curved shell structures which display significant transverse shear and membrane locking are presented. The accuracy of the proposed formulation is examined by comparing the predicted critical fracture loads against analytical estimates. To simulate diverse intra-laminar fracture modes in fibre reinforced composites, an anisotropic cohesive phase field model is proposed. The damage anisotropy is captured via distinct energetic crack driving forces, which are defined for each pertinent composite damage mode together with a structural tensor that accounts for material orientation dependent fracture properties. Distinct 3-parameter quasi-quadratic degradation functions based on fracture properties pertinent to each failure mode are used, which result in delaying or suppressing pre-mature failure initiation in all modes simultaneously. The degradation functions can be calibrated to experimentally derived strain softening curves corresponding to relevant failure modes. The proposed damage model is implemented in Abaqus and is validated against experimental results for woven fabric-reinforced and unidirectional composite laminates. Furthermore, a dynamic explicit cohesive phase field model is proposed to capture the significantly nonlinear damage evolution behaviour pertinent to impact scenarios. A strategy is presented to combine the phase field and the cohesive zone models to perform full composite-laminate simulations involving both intra-laminar and inter-laminar damage modes. Finally, the developed phase field model is employed within the framework of a multiscale surrogate modelling technique. The latter is proposed to perform fast and efficient damage simulation involving different inherent scales in composites. The technique is based on a multiscale FE2 (Finite Element squared) homogenisation approach, however the computationally expensive procedure of solving the meso- and macro-scale models simultaneously is avoided by using a robust surrogate model. The meso-scale is defined as a unit-cell representative volume element (RVE) model, which is analysed under a large number of statistically randomised mixed-mode macro-strains, applied with periodic boundary conditions. The complex damage mechanisms occurring at the meso-scale are captured using the anisotropic cohesive phase field model, and the homogenised stress-strain responses post-damage evolution are obtained. These anisotropic meso-scale fracture responses are used to train the Polynomial Chaos Expansion (PCE) and Artificial Neural Network (ANN) based surrogate models, which are interrogated at the macro-scale using arbitrary macro-strain combinations. The accuracy of the surrogate model is validated against high-fidelity phase field simulations for a set of benchmarks.
... Secchi and Schrefler [282] developed a full 3D hydraulic fracturing model based on their previous 2D models [280,283,302,303]. Biot's consolidation is introduced to simulate the rock deformation and fluid flow in reservoir. ...
Along with horizontal drilling techniques, multi-stage hydraulic fracturing has improved shale gas production significantly in past decades. In order to understand the mechanism of hydraulic fracturing and improve treatment designs, it is critical to conduct modelling to predict stimulated fractures. In this paper, related physical processes in hydraulic fracturing are firstly discussed and their effects on hydraulic fracturing processes are analysed. Then historical and state of the art numerical models for hydraulic fracturing are reviewed, to highlight the pros and cons of different numerical methods. Next, commercially available software for hydraulic fracturing design are discussed and key features are summarised. Finally, we draw conclusions from the previous discussions in relation to physics, method and applications and provide recommendations for further research.
... A number of continuum or discontinuumbased numerical methods have also been developed to simulate hydraulic fracture propagation. Early finite element method (FEM) is usually combined with adaptive meshing methods [36,37], representing fracture by a separation of elements along common edges in conjunction with singular crack tip elements. Later, the partition of unity model was introduced to the extended finite element method (XFEM) [38][39][40][41][42] and generalized finite element method (GFEM) [43], allowing for fracture growth through finite elements without the need of remeshing. ...
Understanding the hydro-mechanical (HM) coupling grouting process in fractured rock masses is vital for improving grouting effects. In this paper, a hybrid formulation of the unified-pipe network method coupled with zero-thickness interface elements is developed for simulating fracture grouting, named the unified pipe-interface element method (UP-IEM). The UP-IEM can naturally capture both the initiations and propagations of cracks without enrichments or crack tracking algorithms. Additionally, the grout flow in both the porous medium and fractures is solved in one solution system. A semi-explicit frame is built for solving the hydro-mechanical coupling problems, avoiding the potential numerical instability of implicit solutions and long computation time of explicit solutions. The proposed UP-IEM for simulating curved fracture paths in the near-hole region is validated by several numerical tests. Then, a parametric study is carried out to investigate the parameter effects on grout-driven fracture tortuosity and injection pressure. Finally, a grouting model with multiple rows of grouting holes is established according to actual grouting engineering and the simultaneous fracture grouting and sequential fracture grouting processes are analyzed. The results show that the UP-IEM is potentially useful for optimizing the fracture grouting process.
... This unique type of loading, which results in the simultaneous presence of consolidation and dynamic loading, imposes an extra challenge in the simulation of the cargo response, particularly in selecting appropriate time step sizes to avoid numerical instability. It may also be noted that existing adaptive time marching solution schemes require extensive modification to be used for this type of analysis (Schrefler et al., 2006). This is because they are either Fig. 1. ...
Over the last 40 years, there have been several ship losses carrying granular mineral ores that are believed to have been caused by the sudden shifting of the cargo as a result of liquefaction. It is known that the initial moisture content during loading is important for cargo types which are considered susceptible to liquefaction. To limit the moisture content, the degree of saturation must be below 80% in compaction tests designed to reproduce the energy associated with loading cargo into a ship's hold. During transport, ships can experience storms causing cyclic loading which leads to densification of the cargo and increases in the degree of saturation. There can also be transient increases in pore pressure and associated reductions in the cargo material's resistance , which when the ship heels increase the likelihood of the cargo shifting and in the extreme, the ship capsizing. In this paper, a recently developed fully coupled dynamic finite element analysis and constitutive model for unsaturated soil is used to perform a parametric study to explore the capabilities of the numerical code and the mechanics of the problem. The numerical analyses are shown to be capable of running for up to 2500 cycles and simulating both cyclic loading and simultaneous consolidation and drainage, a computationally very demanding task. Outcomes of the analyses are used to discuss the factors influencing cargo stability and vessel instability.
... The finite element method (FEM) is commonly known as a robust numerical simulation tool for solving continuous hydro-mechanical problems, that is, in the absence of fracturing. To simulate fracture propagation process, various remeshing technique 6 in conjunction with or without cohesive zone models 7 and crack paths a priori defined are required for FEM. Alternatively, various local discontinuous enrichment functions have been successfully introduced into conventional finite elements to simulate hydraulic fractures [8][9][10][11] though challenges for complex three-dimensional crack patterns remain. ...
In the present manuscript fracture propagation in a saturated porous medium is
modeled based on the classical Biot theory, where solid skeleton and fluid flow
are represented by separate two layers. The non-ordinary state-based peridynamics (NOSBPD) layer is employed to capture deformation including fracturing of
the solid skeleton, while the fluid flow is controlled by the finite element method
(FEM) layer. The interaction between the layers is realized by considering the
effects of pore pressure from the FEM layer on the NOSBPD layer and, vice versa,
the effect of the volumetric strain, porosity, and permeability variations from the
NOSBPD layer on the FEM layer. The coupling terms retain their parent characteristics, that is, the interaction term in the momentum balance equation is
approximated by the local FEM formulation whereas the interaction term in the
mass balance equation is approximated by the nonlocal NOSBPD formulation.
By doing so, the model retains the flexibility of coupling two independent discretizations. The coupled system is solved by a fully implicit solution scheme.
The accuracy of the proposed method has been verified against available closedform solutions and published numerical approaches for the pressure- and fluiddriven facture propagation problems.
... Other applications of hydraulic fracturing include geothermal energy production [3,4], block caving of mines [5], and decontamination of soils [6], to name a few. Complexities in the study of hydraulic fracturing treatments arise due to thermo-hydro-mechanical coupling [7,8], presence of proppants [9], non-Darcican flow [10], fluid-lag [11], anisotropy [12], material non-linearity [13][14][15], leakage [16,17], multi-zone/multi-stage fracturing [18][19][20], crack tip singularity [21,22], etc. Numerical simulation has proven to be a formidable tool in tackling the intricate fracturing problem [23][24][25]. Among advanced computational frameworks that have demonstrated promising capabilities in the simulation of the fracturing process one can name XFEM [26][27][28][29]12], Meshless methods [30,31], Phase-field method [32][33][34][35][36], Lattice Spring method [37,38], discontinuous Galerkin method [39,40], and displacement discontinuity method [41,42]. ...
In this study, the evolution of hydraulically driven fractures in naturally layered and fractured media is investigated. The non-differentiable energy minimization approach to cohesive fracture is employed with a Discontinuous Galerkin (DG) discretization in which every element edge in the mesh is a potential site of cracks. The coupled hydro-mechanical model accounts for the flow continuity equation governing the hydro-fractures inflow in which the opening-dependent permeability of the hydro-fractures is modelled based upon the well-known cubic law. The proposed framework provides great flexibility in modeling multiple fractures and is applied to study the interaction between hydro-fractures and material interfaces in naturally fractured layered domains. Results indicate robustness and versatility of the algorithm and exemplify several simulation aspects there were eluded in many of the works in previous literature.
... The fracturing process involves the injection of a highly pressurized fluid into a rock formation to generate permeable fractures throughout the hydrocarbon bearing formations. Over the past few decades, numerical models have proven to be the most rigorous and cost-effective tool to tackle the complex nature of hydraulic fracturing treatments due to their great flexibility in adapting to the non-uniformities of geological formations [2][3][4][5]. A major step forward in modelling hydraulic fracturing is the mesh independent descriptions of fracture growth, such as XFEM [6][7][8][9][10][11][12][13][14][15][16][17], meshfree methods [18,19], and the phase-field approach [20][21][22][23], which all offer superior computational performance. ...
A novel fully coupled hydro-mechanical model is used to assess the effect of fluid loss on the efficiency of the fracturing treatment within saturated porous media. In the context of XFEM, the pore pressure is defined independently on either side of the hydro-fracture using the Heaviside enrichment function. Meanwhile, independent pressure degrees of freedom are employed to develop a generalized model for the hydro-fracture inflow. In this way, the pressure jump due to the formation of a bedding layer of settled proppants and/or additive chemicals in the vicinity of the hydro-fracture faces can be taken into account (technically referred to as the cake layer effect). On the other hand, the reduction in the hydraulic permeability of zones adjacent to those areas affected by the very high fracturing pressure, also known as filtrate effects, is taken into account through a series of experimental correlations. Finally, the ability of the developed framework to tackle the modelling of hydraulic fracturing, particularly in medium to low permeability formations, is illustrated by means of numerical simulations.
... Thus, the overall density can be expressed as ρ = ∑ α n α ρ αR . A comprehensive theoretical and applications review on the porous materials is provided in the works [59][60][61][62][63][64]. The fluid volume fraction (porosity) n F is linked to the fluid volume ratio θ (fluid content) per unit volume of the undeformed reference configuration ...
In this work, phase-field modeling of hydraulic fractures in porous media is extended towards a Global–Local approach. Therein, the failure behavior is solely analyzed in a (small) local domain. In the surrounding medium, a simplified and linearized system of equations is solved. Both domains are coupled with Robin-type interface conditions. The fractures inside the local domain are allowed to propagate and consequently, both subdomains change within time. Here, a predictor–corrector strategy is adopted, in which the local domain is dynamically adjusted to the current fracture pattern. The resulting framework is algorithmically described in detail and substantiated with some numerical tests.
Characterizing the fluid-driven fracture-tip advancing process presents a significant challenge due to the difficulty of replicating real-world conditions in laboratory experiments and the lack of precise field measurements. However, recent advances in low-frequency distributed acoustic sensing (LF-DAS) technology offer new opportunities to investigate the dynamics of propagating hydraulic fractures. In this study, we propose an iterative inversion method to characterize fracture-tip advancing behaviors using LF-DAS data. A forward geomechanical model is developed using the three-dimensional displacement discontinuity method, and the optimization is realized by a conjugate gradient method. The performance of the inversion algorithm is demonstrated using a synthetic case, in which the fracture half-length evolution and propagation velocity match well with the reference solutions. In addition, the averaged fracture cross-section area, fracture volume, and fracturing fluid efficiency can also be estimated, showing good agreements with true values of the synthetic case under reasonable assumptions. Then, a field case with a single-cluster hydraulic fracturing treatment from the Hydraulic Fracturing Test Site 2 project (HFTS-2) is studied. Our analysis of the inversion results reveals that the fracture propagates intermittently, as evidenced by the fracture half-length evolution. This unique field evidence can guide modeling efforts to incorporate this important physical behavior into fracture models, and the secondary information gathered from the study, including fracture cross-section area and volume, can help evaluate and optimize fracturing efficiency.
A overview is given of different approaches to simulate and predict crack propagation in quasi-brittle materials, pointing out their merits, disadvantages and complementarity. Discrete crack approaches, smeared crack approaches and Molecular Dynamics are considered from a historical perspective, and it is discussed in which circumstances they are best used. For the various versions of the discrete and smeared approaches it is shown how the methods within each class have evolved, or why new methods have developed due to shortcomings of earlier versions.
In this paper, we propose a new numerical approach abbreviated as Cos-SDA for analyzing strain localization problems of geomaterials. The Cos-SDA is achieved by implanting the strong discontinuity approach (SDA) into the computational framework of the Cosserat continuum finite element approach (Cos-FEA). Through two numerical examples of plane strain compression test and slope stability, it is demonstrated that the Cos-SDA model can effectively simulate the entire progressive failure process of geomaterials from weak discontinuity to strong discontinuity. Cos-SDA can effectively alleviate the influence of mesh distortion in shear zone, and the numerical solution can still maintain convergence even under large deformation. Cos-FEA and Cos-SDA have much stiffer mechanical response than SDA-FEA since an internal length scale is introduced into the governing equations. In contrast, the SDA-FEA shows more conservative as the unmature bifurcation occurs and weak discontinuous stage is ignored in the numerical simulation, while Cos-FEA numerical results show an over-stiff mechanical response in the stage of post failure. Cos-SDA inherits the advantages of both SDA-FEA and Cos-FEA and neutralizes their mechanical responses, so it is more reasonable in simulating the progressive failure process from weak discontinuity deformation (strain localization) to strong discontinuity deformation (slip) occurring in geomaerials.
Computational Fracture Mechanics (CFM) is key to several branches in science and engineering, such as Solid and Structural Mechanics, Geomechanics, Aerospace Engineering, Structural Health Monitoring and Damage Tolerant Design. Among the different numerical techniques that are being employed and developed in the framework of Computational Fracture Mechanics, the extended Finite Element Method (X-FEM) is one of the most powerful and versatile. By introducing enrichment functions along with standard Finite Element shape functions, the X-FEM enables very accurate simulation of fields with discontinuities and fields that feature singularities such as crack opening displacements and stresses at the vicinity of a crack tip respectively. This chapter aims to highlight the effectiveness of the X-FEM method for fracture simulation and structural integrity assessment through two state of the art applications namely; (1) Hydraulic Fracture Propagation in Naturally Fractured Porous Media, and (2) Coupling X-FEM with Peridynamics (PD) for dynamic fracture propagation in brittle materials. A brief introduction to Computational Fracture Mechanics with emphasis on the X-FEM method history, development and relevant literature is included.
Strain localization analysis based on finite element method (FEM) usually requires an intensive computation to capture an accurate shear band and the limit stress, especially in the heterogeneous problems, and thus costs much computational resource and time. This paper proposes an adaptive multiscale FEM (AMsFEM) to improve the computational efficiency of strain localization analysis in heterogeneous solids. In the multiscale analysis, h- and p-adaptive strategies are proposed to update the fine and coarse meshes, respectively. The problem of mesh dependence is handled by the Cosserat continuum theory. In the fine-scale adaptive procedure, triangular elements are taken to discretize the fine-scale domain, and the newest vertex bisection is utilized for refinement based on the gradient of displacement. In the coarse-scale adaptive procedure, a multi-node coarse element technique is considered. By introducing a probability density function for each side of a coarse element, the optimal positions of the newly added coarse nodes can be determined. With the proposed adaptive multiscale procedure, the computational DOFs are reduced smartly and massively. Three representative heterogeneous examples demonstrate that the proposed method can accurately capture shear bands, with an improved computational efficiency and robust convergence.
The numerical simulation of a 3D complex reservoir has been an important but difficult problem. One of the biggest challenges is how to deal with the large number of natural fractures with consideration of hydromechanical coupling effect in those fractures. To address this problem, a 3D hydromechanical coupled element partition method (3D-EPM) is developed. The 3D-EPM allows a fracture to run through an element without any extra degree of freedom. It can embed any number of fractures in a background mesh without mesh modification. This brings great convenience to 3D reservoir simulation. To represent the permeability of a cracked element, the equivalent permeability of this cracked element is derived based on the cubic law. To account for the hydromechanical coupling effect in a fracture, the coupled equation of 3D-EPM is developed. It is verified by the analytical model. The simulation results suggest that this method can simulate the 3D hydraulic fracture propagation, its interaction with natural fractures and the impact of in-situ stresses. This paper provides an alternative method for the 3D hydraulic fracture simulation in a complex reservoir with consideration of hydromechanical coupling effect in a fracture.
A refined model is presented which numerically resolves the fluid flow within a fracture inside a saturated poroelastic material. At the discontinuity, the mass balance of the fluid is solved using the velocity profile inside the fracture, with the velocity profile being determined numerically from the momentum balance in the integration points at the discontinuity. The resolution of the mass balance and the velocity profile in the integration points at the discontinuity is coupled to surrounding poroelastic material model in a two-scale approach. The resulting monolithic scheme allows for complex fluid behaviour to be included, which is demonstrated via the inclusion of the fluid inertia and the use of a Carreau fluid. The governing equations are discretised using T-splines, while the fracture is represented by spline-based interface elements, and an implicit time discretisation scheme is used. Mesh refinement studies are carried out for a typical case, which contains a pressurised and propagating fracture. A coarse macro-scale mesh is sufficient to obtain the correct propagation velocities, but a finer mesh is needed to prevent pressure oscillations. Time step refinement studies show the capabilities of the model to capture pressure waves within the fracture. Finally, it is shown that if inertial effects are limited and a Newtonian fluid is used, a fracture scale discretisation using a single element is sufficient. However, for a Carreau fluid or when the problem is inertia-dominated, smaller elements are needed to correctly represent the velocity profile.
In this paper, a novel hybrid FEM and Peridynamic modeling approach proposed in [1] is used to predict the dynamic solution of hydro-mechanical coupled problems. A modified staggered solution algorithm is adopted to solve the coupled system. A one-dimensional dynamic consolidation problem is solved first to validate the hybrid modeling approach, and both δ−convergence and m r −convergence studies are carried out to determine appropriate discretization parameters for the hybrid model. Thereafter, dynamic fracturing in a rectangular dry/fully saturated structure with a central initial crack is simulated both under mechanical loading and fluid-driven conditions. In the mechanical loading fracture case, fixed surface pressure is applied on the upper and lower surfaces of the initial crack near the central position to force its opening. In the fluid-driven fracture case, the fluid injection is operated at the centre of the initial crack with a fixed rate. Under the action of the applied external force and fluid injection, forerunning fracture behavior is observed both in the dry and saturated conditions.
In this study, an energy based hydro-mechanical model and computational algorithm for the problem of hydraulically driven fracture networks developing in naturally fractured impermeable media is developed. The model is based on non-differentiable energy minimization for the dynamic deformation and fracture of the body coupled with mass balance of fluid flow within the hydro-fractures. Time-discontinuity induce spurious crack opening velocity fields which lead to nonphysical solutions for the coupled fluid pressure field defined locally along the crack faces. The use of a time-continuous fracture model, such as the present non-differentiable energy minimization approach, is crucial for the numerical soundness and stability of the hydraulic fracture propagation algorithm. A discontinuous Galerkin finite element formulation is implemented, in which every element edge in the mesh is a potential site of hydro-fracture initiation and propagation. The presence of pre-existing natural fractures, as a common challenge in nearly all geological formations, are modelled with desirable flexibility by simply assigning different fracture properties to the element edges defining the natural fractures. Using the graph theory principles, a search algorithm is proposed to identify, among all, the sub-set of cracked interfaces that form the interconnected hydraulically loaded fracture network. Robustness of the proposed computational algorithm and its versatility in the study of hydraulic fracturing are shown through several numerical simulations.
Computationally solving the equations of elasticity is a key component in many materials science and mechanics simulations. Phenomena such as deformation-induced microstructure evolution, microfracture, and microvoid nucleation are examples of applications for which accurate stress and strain fields are required. A characteristic feature of these simulations is that the problem domain is simple (typically a rectilinear representative volume element (RVE)), but the evolution of internal topological features is extremely complex. Traditionally, the finite element method (FEM) is used for elasticity calculations; FEM is nearly ubiquituous due to (1) its ability to handle meshes of complex geometry using isoparametric elements, and (2) the weak formulation which eschews the need for computation of second derivatives. However, variable topology problems (e.g. microstructure evolution) require either remeshing, or adaptive mesh refinement (AMR) - both of which can cause extensive overhead and limited scaling. Block-structured AMR (BSAMR) is a method for adaptive mesh refinement that exhibits good scaling and is well-suited for many problems in materials science. Here, it is shown that the equations of elasticity can be efficiently solved using BSAMR using the finite difference method. The boundary operator method is used to treat different types of boundary conditions, and the “reflux-free” method is introduced to efficiently and easily treat the coarse-fine boundaries that arise in BSAMR. Examples are presented that demonstrate the use of this method in a variety of cases relevant to materials science: Eshelby inclusions, fracture, and microstructure evolution. Reasonable scaling is demonstrated up to ∼4000 processors with tens of millions of grid points, and good AMR efficiency is observed.
The authors start with an introduction to the concepts involved in physics giving the equations of flow through porous media and the deformation characteristics of soils and rocks. Succeeding chapters deal with the practical implications of these phenomena and explain the application of theory in both experimental and field work. Details are given of actual incidents, such as the subsidence experienced in Venice and Ravenna. The authors have also formulated a consolidation code, which is detailed at the end of the book, and provide instructions on how to modify the given program.
The focus of this paper is on constructing the solution for a semi-infinite hydraulic crack for arbitrary toughness, which accounts for the presence of a lag of a priori unknown length between the fluid front and the crack tip. First, we formulate the governing equa-tions for a semi-infinite fluid-driven fracture propagating steadily in an impermeable linear elastic medium. Then, since the pressure in the lag zone is known, we suggest a new inversion of the integral equation from elasticity theory to express the opening in terms of the pressure. We then calculate explicitly the contribution to the opening from the loading in the lag zone, and reformulate the problem over the fluid-filled portion of the crack. The asymptotic forms of the solution near and away from the tip are then dis-cussed. It is shown that the solution is not only consistent with the square root singularity of linear elastic fracture mechanics, but that its asymptotic behavior at infinity is actually given by the singular solution of a semi-infinite hydraulic fracture constructed on the assumption that the fluid flows to the tip of the fracture and that the solid has zero toughness. Further, the asymptotic solution for large dimensionless toughness is derived, including the explicit dependence of the solution on the toughness. The intermediate part of the solution (in the region where the solution evolves from the near tip to the far from the tip asymptote) of the problem in the general case is obtained numerically and relevant results are discussed, including the universal relation between the fluid lag and the toughness.
ffith’s criterion for crack propagation is generalized to consider penny-shaped cracks in a spatially uniform, but otherwise arbitrary, state of stress. A further generalization incorporates the effect of interfacial friction for cracks closed in normal compression. Various criteria for crack propagation are discussed in terms of recent experimental evidence for process zones in which the region around a crack tip is seen to be permeated by smaller microcracks.
This paper presents a mathematical model for the analysis of cohesive fracture propagation through a non-homogeneous porous medium. Governing equations are stated within the frame of Biot's theory, accounting for the flow through the solid skeleton, along the fracture and across its sides toward the surrounding medium. The numerical solution is obtained in a 2D context, exploiting the capabilities of an efficient mesh generator, and requires continuous updating of the domain as the fractures enucleate and propagate. It results that fracture paths and their velocity of propagation, usually assumed as known, are supplied directly by the model without introducing any simplifying assumption.
This analysis investigates the steady-state propagation of a hydraulic fracture in an infinite isotropic fluid-saturated elastic porous medium in the limiting cases of slow and rapid crack growth. In the limiting case of slow crack propagation, it is found that the governing field equations become decoupled, and that a closed-form solution for the stress and pore pressure components can be obtained by solving these equations subject to the appropriate boundary conditions. In the limiting case of rapid crack propagation, it is found that the field equations reduce to a degenerate form, and that the stress and pore pressure components given by the solution of the equations do not satisfy the boundary conditions imposed at the crack faces and at the crack tip. This suggests the presence of stress and pore pressure boundary layers along the surfaces and at the tip of the propagating crack. Closed-form solutions for stress and pore pressure components in the boundary layer are then obtained by formulating a singular perturbation problem which is subsequently investigated by the method of matched asymptotic expansions.
The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.
In the paper we present a postprocessed type of a posteriori error estimate and a h-version adaptive procedure for the semidiscrete finite element method in dynamic analysis. In space the super-convergent patch recovery technique is used for determining higher-order accurate stresses and, thus, a spatial error estimate. In time a postprocessing technique is developed for obtaining a local error estimate for one step time integration schemes (the HHT-α method). Coupling the error estimate with a mesh generator, a h-version adaptive finite element procedure is presented for two-dimensional dynamic analysis. It updates the spatial mesh and time step automatically so that the discretization errors are controlled within specified tolerances. Numerical studies on different problems are presented for demonstrating the performances of the proposed adaptive procedure.
The eect of temperature on the propagation paths of subinterfacial cracks is studied. An aluminum/polymethyl-methacrylate bimaterial specimen with large mechanical and thermal mismatches is used to investigate the eect of combined thermal and mechanical loads. The stress intensity factor (SIF) generated by mechanical loading in the presence of a temperature gradient is obtained by means of experiment and analyzed using the ®nite element method. A full-®eld optical shearing interferometry technique was used to measure the crack tip stress state. The results show that thermal eects can alter the SIFs suciently large enough to change the fracture behavior of subinterfacial cracks.
A procedure for numerical approximation to two-dimensional, hydraulically-driven fracture propagation in a poroelastic material is described. The method uses a partitioned solution procedurè to solve a finite element approximation to problems described by the theory of poroelasticity, in conjunction with a finite difference approximation for modelling fluid flow along the fracture. An equilibrium fracture model based on a generalized, Dugdale–Barenblatt concept is used to determine the fracture dimensions. An important feature is that the fracture length is a natural product of the solution algorithm. Two example problems verify the accuracy of the numerical procedure and a third example illustrates a fully-coupled simulation of fracture propagation. Photographs taken from a high-performance engineering workstation provide insight into the nature of the coupling among the physical phenomena.
In this paper an adaptive method for the analysis of thermomechanical coupled multi-body contact problems is presented. The method is applied to non-linear elastic solids undergoing finite (thermal) deformations. The contact model considers non-linear pressure-dependent heat flux as well as frictional heating in the interface. A time–space-finite element discretization of the governing equations is formulated including unilateral constraints due to contact. A staggered solution algorithm has been constructed that allows an independent spatial discretization of the coupled subproblems. A posteriori projection-based error estimators, which enforce implicitly the special boundary conditions due to thermal contact, are used to control the spatial discretization as well as the adaptive time stepping. Numerical examples are presented to corroborate the applicability of the adaptive algorithm to the considered problem type. Copyright © 2004 John Wiley & Sons, Ltd.
An effective h-version finite element adaptive strategy combined with mesh regeneration is presented. This is based on the error estimator developed in Reference 1. The rate of convergence of the adaptive procedure has been tested for some examples and very strong convergence observed. Unlike some existing h-version adaptive procedures, a nearly optimal mesh of predicted accuracy can be obtained in one or two adaptive process steps.
The paper presents a fully-coupled numerical model for the analysis of fracture initiation and propagation in a two dimensional non-homogeneous elastic medium driven by mechanical loads and transient thermal fields. Cohesive crack behaviour is assumed for the solid. The solution of the coupled problem is obtained by using the finite element method without using special approximation techniques nor interface elements. Evolution of the process zone results in continuous changes of the domain topology. This is accounted for by updating the boundary geometry and successive remeshing of the domain. Optimality of the shape of the finite elements generated is controlled and the mesh density is adjusted adaptively on the basis of an error estimator. Two numerical applications are presented, which demonstrate the effectiveness of the proposed procedure. In the first, comparison is made with a laboratory experiment, whereas the second handles a problem with crack path completely unknown.
A Lagrangian finite element method of fracture and fragmentation in brittle materials is developed. A cohesive-law fracture model is used to propagate multiple cracks along arbitrary paths. In axisymmetric calculations, radial cracking is accounted for through a continuum damage model. An explicit contact/friction algorithm is used to treat the multi-body dynamics which inevitably ensues after fragmentation. Rate-dependent plasticity, heat conduction and thermal coupling are also accounted for in calculations. The properties and predictive ability of the model are exhibited in two case studies: spall tests and dynamic crack propagation in a double cantilever beam specimen. As an example of application of the theory, we simulate the experiments of Field (1988) involving the impact of alumina plates by steel pellets at different velocities. The calculated conical, lateral and radial fracture histories are found to be in good agreement with experiment.
A method is presented in which fracture mechanics is introduced into finite element analysis by means of a model where stresses are assumed to act across a crack as long as it is narrowly opened. This assumption may be regarded as a way of expressing the energy adsorption GC in the energy balance approach, but it is also in agreement with results of tension tests. As a demonstration the method has been applied to the bending of an unreinforced beam, which has led to an explanation of the difference between bending strength and tensile strength, and of the variation in bending strength with beam depth.RésuméUne méthode est présentée, par laquelle la méchanique des ruptures est introduite dans l'analyse des éléments finis à l'aide d'un modèle, où les contraintes sont supposées d'opérer sur les côtés d'une fissure tant que cette fissure est étroite.Cette hypothèse peut être considérée comme un moyen d'exprimer l'absorption Gc d'énergie en usant l'approche de l'équilibre d'énergie. Cette hypothèse est aussi justifiée par les résultats des essais de tension.Pour en prouver la validité,, cette méthode a été appliquée au fléchissement d'une poutre non armée et fournit une explication de la différence entre la résistance au moment de flexion et la résistance à l'effort de tension, ainsi que de la variation de la résistance au moment de flexion en fonction de la profondeur de la poutre.
This paper presents a procedure for the discretization of 2D domains using a Delaunay triangulation. Improvements over existing similar methods are introduced, proposing in particular a multi-constraint insertion algorithm, very effective in the presence of highly irregular domains, and the topological structure used together with its primitives. The method obtained requires limited input and can be applied to a wide class of domains. Quadrilateral subdivisions with a control of the aspect ratio of the generated elements can also be reached. Further it is suitable for evolutionary problems, which require continuous updating of the discretization. Presented applications and comparisons with other discretization methods demonstrate the effectiveness of the procedure.
In this paper, two brittle fracture problems are numerically simulated: the failure of a ceramic ring under centrifugal loading and crack branching in a PMMA strip. A three-dimensional finite element package in which cohesive elements are dynamically inserted has been developed. The cohesive elements' strength is chosen to follow a modified weakest link Weibull distribution. The probability of introducing a weak cohesive element is set to increase with the cohesive element size. This reflects the physically based effect according to which larger elements are more likely to contain defects. The calculations illustrate how the area dependence of the Weibull model can be used to effectively address mesh dependency. On the other hand, regular Weibull distributions have failed to reduce mesh dependency for the examples shown in this paper.
The ceramic ring calculations revealed that two distinct phenomena appear depending on the magnitude of the Weibull modulus. For low Weibull modulus, the fragmentation of the ring is dominated by heterogeneities. Whereas many cracks were generated, few of them could propagate to the outer surface. Monte Carlo simulations revealed that for highly heterogeneous rings, the number of small fragments was large and that few large fragments were generated.
For high Weibull modulus, signifying that the ring is close to being homogeneous, the fragmentation process was very different. Monte Carlo simulations highlighted that a larger number of large fragments are generated due to crack branching. Copyright
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