The influence of the cohesive process zone in hydraulic fracturing modelling

This paper studies the importance of the cohesive zone in the modelling of a fluid driven fracture under plain strain conditions. The fracture is driven by pumping of an incompressible viscous fluid at the fracture inlet. Rock deformation is modeled for linear elastic and poroelastic solids. Fluid flow in the fracture is modeled by lubrication theory. The cohesive zone approach is used as the fracture propagation criterion. Finite element analysis was used to compute the solution for the crack length, the fracture opening and propagation pressure as a function of the time and distance from the wellbore. It is demonstrated that the crack profiles and the propagation pressures are larger in the case of elastic-softening cohesive model compared to the results of the rigid-softening cohesive model for both elastic and poroelastic cohesive solids. It is found that the results are affected by the slope of the loading branch of the cohesive model and they are nearly unaffected from the exact form of the softening branch. Furthermore, the size of the process zone, the fracture geometry and the propagation pressure increase with increasing confining stresses. These results may explain partially the discrepancies in net-pressures between field measurements and conventional model predictions.     

A cohesive model for fatigue failure of polymers

A cohesive failure model is proposed to simulate fatigue crack propagation in polymeric materials. The model relies on the combination of a bi-linear cohesive failure law used for fracture simulations under monotonic loading and an evolution law relating the cohesive stiffness, the rate of crack opening displacement and the number of cycles since the onset of failure. The fatigue component of the cohesive model involves two parameters that can be readily calibrated based on the classical log-log Paris failure curve between the crack advance per cycle and the range of applied stress intensity factor. The paper also summarizes a semi-implicit implementation of the cohesive model into a cohesive-volumetric finite element framework, allowing for the simulation of a wide range of fatigue fracture problems.     

An embedded atom hyperelastic constitutive model and multiscale cohesive finite element method

Based on embedded atom method (EAM), an embedded atom hyperelastic (EAH) constitutive model is developed. The proposed EAH constitutive model provides a multiscale formalism to determine mesoscale or macroscale material behavior by atomistic information. By combining the EAH with cohesive zone model (CZM), a multiscale embedded atom cohesive finite element model (EA-cohesive FEM) is developed for simulating failure of materials at mesoscale and macroscale, e.g. fracture and crack propagation etc. Based on EAH, the EA-cohesive FEM applies the Cauchy-Born rule to calculate mesoscale or macroscale material response for bulk elements. Within the cohesive zone, a generalized Cauchy-Born rule is applied to find the effective normal and tangential traction-separation cohesive laws of EAH material. Since the EAM is a realistic semi-empirical interatomic potential formalism, the EAH constitutive model and the EA-cohesive FEM are physically meaningful when it is compared with experimental data. The proposed EA-cohesive FEM is validated by comparing the simulation results with the results of large scale molecular dynamics simulation. Simulation result of dynamic crack propagation is presented to demonstrate the capacity of EA-cohesive FEM in capturing the dynamic fracture.     

Continuum coupled cohesive zone elements for analysis of fracture in solid bodies

Embedding cohesive surfaces into finite element models is a widely used technique for the numerical simulation of material separation (i.e. crack propagation). Typically, a traction-separation law is specified that relates the magnitude of the cohesive traction to the distance between the separating surfaces. Thus the characterization of fracture in such models is not directly coupled to the bulk constitutive response, in the sense that the cohesive traction does not explicitly depend on material stretching in the plane of the fracture surface. In this work, an initially-rigid cohesive-traction formulation that is coupled to the surrounding continuum is introduced as a further development of the cohesive zone idea. In this model, the traction-separation law - and therefore the fracture phenomenology - derives directly from the bulk constitutive law. The immediate goal is an improved cohesive zone framework that naturally and logically initiates cohesive separation behavior, and couples its evolution to the material state in the region of the crack tip. A cohesive element based on this model is implemented in an explicit three-dimensional finite element code. Proof-of-concept analyses using both linear elastic and Gurson void growth constitutive relations are presented. A three-point bend simulation is found to give good agreement with experimental results.     

Analysis of mixed-mode fracture in concrete using interface elements and a cohesive crack model

The paper presents a model, based on nonlinear fracture mechanics, for analysing crack propagation in quasi-brittle materials, such as concrete. The work is limited to two-dimensions, and therefore, the fracture modes of interest are mode I (pure tension) and mode II (pure shear). The constitutive model has been implemented in the context of the finite element method using interface elements. The fracture is simulated through a discrete crack represented by the interface with a cohesive crack stress-separation relation derived from the model, which is based on a fracture criterion, together with a flow rule and a softening law. The model is used for simulating results from an experimental study on beams with centric and eccentric notches of high and normal strength concretes, and explaining other test results available in the literature.     

Pore pressure cohesive zone modeling of hydraulic fracture in quasi-brittle rocks

Hydraulic fracturing technology has been widely applied in the petroleum industry for both waste injection and unconventional gas production wells. The prevailing analytical solutions for hydraulic fracture mainly depend on linear elastic fracture mechanics. These methods can give reasonable prediction for hard rock, but are ineffective in predicting hydraulic fractures in quasi-brittle materials, such as ductile shale and sandstone. One of the reasons is that the fracture process zone ahead of the crack tip and the softening effect should not be neglected for quasi-brittle materials. In the current work, a set of chevron-notch three point bending tests were performed on sandstone samples from an oil field in Ordos Basin, Shaanxi province, China, and the results were compared with the cohesive zone method based on finite element analysis. The numerical results fit the experimental data well and it shows that the cohesive zone model and the Traction-Separation law used in the model are effective in modeling fracture nucleation and propagation in sandstone without considering the porous effect. A 3D pore pressure cohesive zone model was developed to predict nucleation and propagation of a penny-shaped fluid-driven fracture. The predictions were compared with the analytical asymptotic solutions and a field minifrac test from the literature; it shows that the proposed method can not only predict the length and aperture of hydraulic fracture well, but also predict the bottomhole pressure with reasonable accuracy. Based on analytical asymptotic and computational solutions, parametric studies were conducted to investigate the effects of different parameters on the fracture aperture and fracture length, fracture process zone and bottomhole pressure.     

Multiple cohesive crack growth in brittle materials by the extended Voronoi cell finite element model

This paper is aimed at modeling the propagation of multiple cohesive cracks by the extended Voronoi cell finite element model or X-VCFEM. In addition to polynomial terms, the stress functions in X-VCFEM include branch functions in conjunction with level set methods and multi-resolution wavelet functions in the vicinity of crack tips. The wavelet basis functions are adaptively enriched to accurately capture crack-tip stress concentrations. Cracks are modeled by an extrinsic cohesive zone model in this paper. The incremental crack propagation direction and length are adaptively determined by a cohesive fracture energy based criterion. Numerical examples are solved and compared with existing solutions in the literature to validate the effectiveness of X-VCFEM. The effect of cohesive zone parameters on crack propagation is studied. Additionally, the effects of morphological distributions such as length, orientation and dispersion on crack propagation are studied.     

Simulation of cup-cone fracture using the cohesive model

The cohesive model is used for the prediction of the crack path during stable crack extension in ductile materials. The problem of crack-path deviation is investigated by means of simulation of crack propagation in a round tensile bar. The respective phenomenon is known as cup-cone fracture. It is shown that the model is able to predict the failure mechanism, which consists of normal fracture in the center and combined normal/shear fracture in the so-called “shear lips” at the specimen’s rim. The damage evolution and crack path predicted by the model are presented. The effect of the normal and shear failure parameters on the crack-deflection point and several aspects of the finite element mesh are discussed.     

A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials

Numerical investigations are conducted to simulate high-speed crack propagation in pre-strained PMMA plates. In the simulations, the dynamic material separation is explicitly modeled by cohesive elements incorporating an initially rigid, linear-decaying cohesive law. Initial attempts using a rate-independent cohesive law failed to reproduce available experimental results as numerical crack velocities consistently overestimate experimental observations. As proof of concept, a phenomenological rate-dependent cohesive law, which bases itself on the physics of microcracking, is introduced to modulate the cohesive law with the macroscopic crack velocity. We then generalize this phenomenological approach by establishing a rate-dependent cohesive law, which relates the traction to the effective displacement and rate of change of effective displacement. It is shown that this new model produces numerical results in good agreement with experimental data. The analysis demonstrates that the simulation of high-speed crack propagation in brittle structures necessitates the use of rate-dependent cohesive models, which account for the complicated rate-process of dynamic fracture at the propagating crack tip.     

Modeling end notched flexure tests to establish cohesive element Mode II fracture parameters

The objective of this work was to establish Mode II fracture parameters for cohesive elements that can be further utilized to evaluate Mode II interfacial fracture strength of polyurea/AISI 4340 steel composite structures. To obtain the fracture parameters, end notched flexure (ENF) tests were conducted to validate proposed finite element models. The fracture behavior observed from the tests was highly nonlinear and large plastic deformations were involved during crack formation and propagation. A strain incompatibility model was introduced to describe the nonlinearity prior to fracture. This nonlinear and plastic behavior made Linear Elastic Fracture Mechanics (LEFM) approaches not applicable to approximate the fracture parameters. As a result of these experimental observations, finite element analyses of the ENF tests were performed to develop the necessary fracture parameters for cohesive elements selected to replicate the failure modes. Good agreement between the selected numerical models and experimental data was observed.     

Finite volume analysis of dynamic fracture phenomena – II. A cohesive zone type methodology

A truly predictive dynamic fracture model would require detailed information about the local fracture process. For instance, recent experimental evidence has conclusively demonstrated that rapid crack propagation (RCP) in brittle polymers such as PMMA is accompanied by the nucleation, growth, interaction and coalescence of microcracks and the advancing macroscopic fracture. Some insights into this phenomenon are offered and a novel computational model of this aspect of dynamic fracture is proposed. The procedure is based upon local material (cohesive) strength considerations. Here, the initial development of such procedures is presented. Application is at this stage restricted to single crack problems so that the effects of various geometrical, but more importantly, cohesive parameters on predicted fractures may be examined. Extension of the cohesive computational procedures of this work to multiple crack problems is proposed to be straightforward and without the need for extensive reconstructing of the computational procedures outlined.     

A micromechanical model for a viscoelastic cohesive zone

A micromechanical model for a viscoelastic cohesive zone is formulated herein. Care has been taken in the construction of a physically-based continuum mechanics model of the damaged region ahead of the crack tip. The homogenization of the cohesive forces encountered in this region results in a damage dependent traction-displacement law which is both single integral and internal variable-type. An incrementalized form of this traction-displacement law has been integrated numerically and placed within an implicit finite element program designed to predict crack propagation in viscoelastic media. This research concludes with several example problems on the response of this model for various displacement boundary conditions.     

An extended finite element method with analytical enrichment for cohesive crack modeling

A recent approach to fracture modeling has combined the extended finite element method (XFEM) with cohesive zone models. Most studies have used simplified enrichment functions to represent the strong discontinuity but have lacked an analytical basis to represent the displacement gradients in the vicinity of the cohesive crack. In this study enrichment functions based upon an existing analytical investigation of the cohesive crack problem are proposed. These functions have the potential of representing displacement gradients in the vicinity of the cohesive crack and allow the crack to incrementally advance across each element. Key aspects of the corresponding numerical formulation and enrichment functions are discussed. A parameter study for a simple mode I model problem is presented to evaluate if quasi‐static crack propagation can be accurately followed with the proposed formulation. The effects of mesh refinement and mesh orientation are considered. Propagation of the cohesive zone tip and crack tip, time variation of the cohesive zone length, and crack profiles are examined. The analysis results indicate that the analytically based enrichment functions can accurately track the cohesive crack propagation of a mode I crack independent of mesh orientation. A mixed mode example further demonstrates the potential of the formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

A boundary integral method for a dynamic, transient mode I crack problem with viscoelastic cohesive zone

We consider the problem of the dynamic, transient propagation of a semi-infinite, mode I crack in an infinite elastic body with a nonlinear, viscoelastic cohesize zone. Our problem formulation includes boundary conditions that preclude crack face interpenetration, in contrast to the usual mode I boundary conditions that assume all unloaded crack faces are stress-free. The nonlinear viscoelastic cohesive zone behavior is motivated by dynamic fracture in brittle polymers in which crack propagation is preceeded by significant crazing in a thin region surrounding the crack tip. We present a combined analytical/numerical solution method that involves reducing the problem to a Dirichlet-to-Neumann map along the crack face plane, resulting in a differo-integral equation relating the displacement and stress along the crack faces and within the cohesive zone.     

Freeze-thaw degradation of FRP-concrete interface: Impact on cohesive fracture response

Results from an experimental investigation into the influence of freeze-thaw action on the FRP-concrete interface fracture properties are presented. The FRP-concrete bond behavior is investigated using a direct shear test. The cohesive stress transfer between FRP and concrete during debonding is determined from spatially continuous measurements of surface strains obtained at different stages of the debonding load response. The non-linear material law for the interface shear fracture, which provides a relation between the interface shear stress as a function of relative slip between the FRP and concrete, is established for specimens subjected to different levels of damage associated with freezing and thawing action. The influence of freeze-thaw action on the cohesive stress transfer during crack propagation, and on the cohesive interface fracture parameters is evaluated using a statistical hypothesis testing method. A larger percentage decrease in the interface fracture energy due to freeze-thaw cycles compared to the corresponding decrease in the ultimate nominal stress at debonding was noted. A decrease in the length of the cohesive stress transfer zone and the maximum interface cohesive stress were also observed with freeze-thaw cycling.     

Extended isogeometric analysis for cohesive fracture

The objective of this study is to present an extended isogeometric formulation for cohesive fracture. The approach exploits the higher order interelement continuity property of nonuniform rational B-splines (NURBS), in particular the higher accuracy that results for the stress prediction, which yields an improved estimate for the direction of crack propagation compared to customary Lagrangian interpolations. Shifting is used to ensure compatibility with the surrounding discretization, where, different from extended finite element methods, the affected elements stretch over several rows perpendicular to the crack path. To avoid fine meshes around the crack tip in case of cohesive fracture, a blending function is used in the extension direction of the crack path. To comply with standard finite element data structures, Bézier extraction is used. The absence of the Kronecker-delta property in the higher order interpolations of isogeometric analysis impedes the enrichment scheme and compatibility enforcement. These issues are studied comprehensively at the hand of several examples, while crack propagation analyses show the viability of the approach.     

The cohesive band model: a cohesive surface formulation with stress triaxiality

In the cohesive surface model cohesive tractions are transmitted across a two-dimensional surface, which is embedded in a three-dimensional continuum. The relevant kinematic quantities are the local crack opening displacement and the crack sliding displacement, but there is no kinematic quantity that represents the stretching of the fracture plane. As a consequence, in-plane stresses are absent, and fracture phenomena as splitting cracks in concrete and masonry, or crazing in polymers, which are governed by stress triaxiality, cannot be represented properly. In this paper we extend the cohesive surface model to include in-plane kinematic quantities. Since the full strain tensor is now available, a three-dimensional stress state can be computed in a straightforward manner. The cohesive band model is regarded as a subgrid scale fracture model, which has a small, yet finite thickness at the subgrid scale, but can be considered as having a zero thickness in the discretisation method that is used at the macroscopic scale. The standard cohesive surface formulation is obtained when the cohesive band width goes to zero. In principle, any discretisation method that can capture a discontinuity can be used, but partition-of-unity based finite element methods and isogeometric finite element analysis seem to have an advantage since they can naturally incorporate the continuum mechanics. When using interface finite elements, traction oscillations that can occur prior to the opening of a cohesive crack, persist for the cohesive band model. Example calculations show that Poisson contraction influences the results, since there is a coupling between the crack opening and the in-plane normal strain in the cohesive band. This coupling holds promise for capturing a variety of fracture phenomena, such as delamination buckling and splitting cracks, that are difficult, if not impossible, to describe within a conventional cohesive surface model.     

A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks

A two-scale model is developed for fluid flow in a deforming, unsaturated and progressively fracturing porous medium. At the microscale, the flow in the cohesive crack is modelled using Darcy’s relation for fluid flow in a porous medium, taking into account changes in the permeability due to the progressive damage evolution inside the cohesive zone. From the micromechanics of the flow in the cavity, identities are derived that couple the local momentum and the mass balances to the governing equations for an unsaturated porous medium, which are assumed to hold on the macroscopic scale. The finite element equations are derived for this two-scale approach and integrated over time. By exploiting the partition-of-unity property of the finite element shape functions, the position and direction of the fractures are independent from the underlying discretization. The resulting discrete equations are nonlinear due to the cohesive crack model and the nonlinearity of the coupling terms. A consistent linearization is given for use within a Newton–Raphson iterative procedure. Finally, examples are given to show the versatility and the efficiency of the approach. The calculations indicate that the evolving cohesive cracks can have a significant influence on the fluid flow and vice versa.     

Modelling of crack propagation of gravity dams by scaled boundary polygons and cohesive crack model

Crack propagation in concrete gravity dams is investigated using scaled boundary polygons coupled with interface elements. The concrete bulk is assumed to be linear elastic and is modelled by the scaled boundary polygons. The interface elements model the fracture process zone between the crack faces. The cohesive tractions are modelled as side-face tractions in the scaled boundary polygons. The solution of the stress field around the crack tip is expressed semi-analytically as a power series. It reproduces the singular and higher-order terms in an asymptotic solution, such as the William’s eigenfunction expansion when the cohesive tractions vanish. Accurate results can be obtained without asymptotic enrichment or local mesh refinement. The stress intensity factors are obtained directly from their definition and provide a convenient and accurate means to assess the zero-K condition, which determines the stability of a cohesive crack. The direction of crack propagation is determined from the maximum circumferential stress criterion. To accommodate crack propagation, a local remeshing algorithm that is applicable to any polygon mesh is augmented by inserting cohesive interface elements between the crack surfaces as the cracks propagate. Three numerical benchmarks involving crack propagation in concrete gravity dams are modelled. The results are compared to the experimental and other numerical simulations reported in the literature.     

Instability during cohesive zone growth

Tensile microcracking of quasi-brittle materials is studied by means of micromechanics, based on (i) an elasto-damaging cohesive zone model accounting for cohesive softening and (ii) a dilute distribution of non-interacting microcracks of uniform orientation and size. Considering virgin microcracks (initially without cohesive zones), macroscopic tensile load increase results in growth of cohesive zones ahead of stationary (non-propagating) cracks and, subsequently, in crack propagation which, notably, will be encountered before the cohesive zones are fully developed i.e. onset of instable cohesive zone growth will be encountered at a load level (i) at which tractions are still transmitted across the inner edges of the cohesive zones and (ii) at which the separation at the inner edges of the cohesive zones is smaller than its critical value. Focusing on onset of instable cohesive zone growth, the chosen approach allows for accessing quantities characterizing the stability limit (e.g., the intensity of the macroscopic loading and the opening at the inner edges of the cohesive zones), without raising the need for non-linear Finite Element analyses. It is shown that the tensile macrostrength of materials containing virgin microcracks is larger than the one related to cracks with already initially fully developed cohesive zones, and related strength differences are quantified for a wide class of cohesive softening behavior. The proposed model is validated by comparing model predictions with an exact solution (available for the special case of constant cohesive tractions) and with results from reliable Finite Element analyses. The paper will be of interest for engineers involved in testing and/or in modeling of quasi-brittle media including cementitious materials and rock.