Propagation Criterion Research Articles

A model scaling approach for fracture and size effect simulations in solids: Cohesive zone, smeared crack band and phase-field models

Despite many noteworthy contributions, the computational modeling of arbitrary fracture configurations and in particular, the quantification of size effect in quasi-brittle solids, still remains a challenging issue. In this work we proposed a model scaling approach for fracture and size effect simulations in solids. It employs only the geometric transformation relations and thus applies to any material model for fracture. Those popular models, i.e., the cohesive zone model, the smeared crack model and the phase-field cohesive zone model, are considered. With the geometric scaling factor intrinsically incorporated in the formulation, the resulting scaled models are able to predict the energetic size effect of a series of geometrically similar structures by parametric numerical simulations of the reference one. The 1-D analytical results of all the three scaled models are coincident with those from the underlying unscaled counterparts, validating the proposed model scaling approach. As it is independent of the mesh discretization and insensitive to the length scale, the scaled phase-field cohesive zone model (PF-CZM) is adopted for application to several representative benchmark tests of concrete and ice with mode-I and mixed I+II failure modes. The numerical results demonstrate that, owing to the strength-based crack nucleation criterion, the energy-based crack propagation criterion and the variational principle based path chooser, as well as the geometric scaling factor, are all incorporated into a standalone framework, the scaled PF-CZM is able to predict both the type II size effect law of pre-cracked quasi-brittle structures and the type I size effect law of intact ones. Moreover, those laborious manual manipulations in the numerical modeling of a large size range of geometrically similar structures, e.g., spatial discretization, enforcement of loading and boundary conditions, postprocessing of output data, etc., need to be done only once on the reference structure. These advantages make the scaled PF-CZM rather promising in the quantification of size effect in quasi-brittle structures.

A robust finite element‐finite volume strategy for viscosity‐dominated hydraulic fracture propagation using an asymptotic tip enrichment

AbstractFluid‐driven fracture propagation is widely observed in various geological processes and crucial to many applications of geological engineering. Developing robust and accurate numerical strategies has significance in advancing the scientific understanding and engineering applications related with fluid‐driven fracture propagation. We present a finite element‐finite volume strategy using asymptotic fracture tip enrichment to model the fluid‐driven fracture propagation in three‐dimensional Cartesian meshes under the viscosity‐dominated regime, in which the fluid viscosity‐related process is the dominant energy dissipation mechanism. We use the finite element method to discretize the balance of linear momentum equation for the deformed solid and the finite volume method to discretize the Reynolds equation that governs the fluid flow. In order to track the evolving fracture front in heterogeneous media, we extend the implicit level set approach originally proposed for the displacement discontinuity method. Through this process, a signed distance‐based fracture propagation criterion naturally emerges and is suitable for the viscosity‐dominated regime when solid toughness becomes irrelevant. Critically, we enrich the fluid volume treatment near the fracture front using the tip asymptotic solution. This enrichment strategy is crucial to overcome the mesh nonconformity caused by the arbitrary intersections between propagating fracture front and underlying Cartesian meshes. We compare the numerical results with analytical solutions of the KGD problem and the penny‐shape problem, and illustrate the mesh size and time step‐insensitivity of the numerical results due to the tip enrichment technique. Also, we demonstrate the capabilities of the proposed method to model fluid‐driven fracture propagation in various heterogeneous media.

Analysis of crack propagation in onshore wind turbine foundations using the double-K fracture model

The load-bearing mechanism of the embedded steel ring to foundation connection used for onshore wind turbines causes high stress concentrations in the concrete around the base flange. Cracks initiate and propagate through the foundation from the base flange. The crack growth under normal operating and extreme loading conditions reduces the stiffness and load-bearing capacity of the tower to foundation connection. This in turn affects power production and could lead to economic losses if failure of the foundation occurs. This paper investigated the stability of a horizontal crack propagation in a typical onshore wind turbine foundation using the double-K fracture propagation criterion. The study involved experiments on two sub-part models and numerical simulation of a large-diameter onshore wind turbine foundation. Using the numerical results, the stress intensity factors at the tip of a horizontal crack emanating from the outer edge of the base flange were calculated at different stages of loading. The stress intensity factors were compared with the initial and unstable fracture toughness of concrete to assess the stability of the crack propagation. It was found that for vertical displacements of the embedded steel ring exceeding 9 mm, the crack propagation became unstable. Based on the crack propagation behaviour, a method was proposed to assess the safety of onshore wind turbine foundations using the vertical displacements of the embedded steel ring.

A toughness-based crack initiation-propagation criterion using a modified fracture energy parameter

Using the energy balance relationship, a toughness-based crack initiation-propagation criterion using a modified fracture energy parameter is proposed in this paper for quasi-brittle materials. Validation of the proposed crack initiation-propagation criterion is carried out by performing three-point bending beam tests through finite element analysis. The classical crack initiation-propagation criterion with the fracture energy parameter is directly used, and better consistencies between the mechanical responses (P-Delta curves, P-CMOD curves, FPZ length variation, KR curves) obtained by the modified fracture energy-based crack propagation criterion and those obtained by experimental means are found. It should be emphasized that the ratio of elastic energy has an important inluence on the value of fracture energy involved in the crack initiation-propagation criterion. By this paper, it is found that introduction of the modified fracture energy in the toughness-based crack initiation-propagation criterion is necessary for getting better estimations of mechanical responses for crack initiation and growth analysis in quasi-brittle materials such as concretes.

Phase-field modeling of interfacial debonding in multi-phase materials via an adaptive isogeometric-meshfree approach

The phase-field modeling of fracture in multi-phase materials is conducted via an adaptive isogeometric-meshfree approach. In this modeling, the crack and interface phase fields are introduced for the regularized representations of crack surfaces and interfaces, respectively, and a convenient approach is proposed for implementing the regularized energy related to both crack surfaces and interfaces without requiring the line integration along them. Numerical implementations of the present modeling are performed via the isogeometric-meshfree approach that constructs the equivalence between isogeometric and meshfree approximations. The developed approach can conveniently achieve mesh refinement around interfaces or cracks, and dynamically track cracks without requiring crack propagation criteria. Moreover, behaviors including the crack initiation, propagation, coalescence, interfacial debonding and matrix cracking of multi-phase materials can be efficiently captured. Finally, simulations are conducted for analyzing two- and three-dimensional fracture of multi-phase materials to demonstrate the reliability of the developed approach.

Continuum damage analysis of delamination in composite laminates using a stress-based layerwise plate model

This contribution addresses the modelling of delamination in a stress-based layerwise plate model. Such models benefit from a good representation of the local 3D stress field, especially inter-laminar stresses which are natural generalized stress unknowns arising in their derivation. We extend their construction to situations involving onset and propagation of interfacial delamination. We propose a variational formulation relying on a continuous damage description of the opening and sliding interfacial degradation processes using two damage variables. A mixed-mode delamination propagation criterion can be obtained through proper coupling between both variables. Finally, the model predictions are validated against classical delamination tests in pure and mixed modes.

Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation

We establish sharp criteria for the instantaneous propagation of free boundaries in solutions to the thin-film equation. The criteria are formulated in terms of the initial distribution of mass (as opposed to previous almost-optimal results), reflecting the fact that mass is a locally conserved quantity for the thin-film equation. In the regime of weak slippage, our criteria are at the same time necessary and sufficient. The proof of our upper bounds on free boundary propagation is based on a strategy of “propagation of degeneracy” down to arbitrarily small spatial scales: We combine estimates on the local mass and estimates on energies to show that “degeneracy” on a certain space-time cylinder entails “degeneracy” on a spatially smaller space-time cylinder with the same time horizon. The derivation of our lower bounds on free boundary propagation is based on a combination of a monotone quantity and almost optimal estimates established previously by the second author with a new estimate connecting motion of mass to entropy production.

Crack propagation in fine grained graphites under mode I and mixed-mode loading, as observed in situ by microtomography

Stable crack growth under mode I and mixed-mode loading of two fine grained advanced graphites, SNG742 and T220, has been studied within small (18 mm diameter) specimens of the diametral compression disc geometry to investigate the criterion for crack propagation. Cracks were propagated from a central notch, inclined at 0° or 30° to the loading axis, as the tests were observed in situ by X-ray computed microtomography. The three-dimensional (3D) displacement fields were measured by digital volume correlation. The crack shape, tip location and crack opening displacements were determined by 3D phase congruency edge detection of full field displacements. Linear elastic 3D finite element simulations calculated the J-integral and mode I, mode II and mode III stress intensity factors (SIFs) acting on the crack tip, using the full field displacements as boundary conditions. For SNG742 and T220 specimens with notch angle of 0°, nearly pure mode I crack propagation was obtained. For specimens loaded with the 30° notch angle, the SIFs changed from mixed mode I/II towards pure mode I as the crack propagated. The mode II SIF was less than 20% of the mode I SIF, and as the crack extended over a distance ∼3 mm, the critical mode I SIF (fracture toughness) was insensitive to the mode II SIF, with SNG742 graphite having higher fracture toughness than T220 graphite.

Analysis of the hydraulic fracturing mechanism and fracture propagation law with a new extended finite element model for the silty hydrate reservoir in the South China Sea

The existing hydrate exploitation technology is characterized by low gas production in single wells and a small exploitation range, owing to the low porosity and low permeability of submarine hydrate reservoirs; therefore, commercial exploitation is difficult to achieve. There is an urgent need to improve exploitation efficiency by means of reservoir reconstruction. Hydraulic fracturing technology may be useful for low-permeability reservoirs such as silty hydrate reservoirs in the South China Sea (SCS). In this study, we established a new hydraulic fracturing model of a horizontal single well for gas-hydrate-bearing-sediments (GHBS) using the cohesive zone method (CZM) based on the extended finite element method (XFEM). To this end, we utilized field measured data of GHBS at site GMGS3-W19. The effect of fluid-solid coupling was considered, and the fracture propagation criterion was set as the maximum principal stress criterion to analyze the hydraulic fracture propagation law of hydrate reservoirs in the SCS. First, we illustrated the effectiveness of the CZM based on XFEM in simulating the fracture morphology of GHBS hydraulic fracturing. A Parametric study showed that the hydrate saturation, horizontal geo-stress coefficient, perforation angle, and injection rate play crucial roles in the fracture propagation path and characteristic parameters of GHBS hydraulic fractures. The fracture development pattern and propagation law of a silty hydrate reservoir within the SCS were theoretically obtained. The results of the present study partly provide theoretical support for hydrate exploitation engineering in the SCS.

A three-dimensional Generalized Finite Element Method for simultaneous propagation of multiple hydraulic fractures from a wellbore

In this article, a 3-D methodology for the simulation of hydraulic fracture propagation using the Generalized Finite Element Method (GFEM) is extended for the simultaneous propagation and interaction of multiple hydraulic fractures. A 3-D isotropic elastic material for the rock and Reynolds lubrication theory for the fluid flow in the fractures are assumed. The elastic solid governing equation is discretized in space with a quadratic GFEM and the equation for the flow in the fractures is discretized in space with a quadratic FEM. With the GFEM, the solid mesh does not need to fit the fracture and accuracy is improved around the fracture front with the use of singular enrichment functions. Discretization error is further controlled by employing mesh adaptivity around the fracture front. The injected fluid partitioning among fractures is computed by modeling the wellbore, where the flow is assumed to be governed by the Hagen–Poiseuille relation. The pressure losses between wellbore and hydraulic fractures are modeled with the sharp-edged orifice equation and with the use of 1-D connecting elements. A linear FEM is adopted for the spatial discretization of the equation governing the flow in the wellbore and the connection between wellbore and hydraulic fracture. A propagation criterion based on a regularization of Irwin’s criterion is adopted and a methodology to automatically estimate the time step that leads to the propagation of fractures based on linear interpolation/extrapolation is presented. Three verification problems are solved and compared with results from the literature. A problem with initial fractures not aligned with the in situ stresses is analyzed. Complex final fracture geometries are observed due to this misalignment.

Hydraulic fracture propagation at weak interfaces between contrasting layers in shale using XFEM with energy-based criterion

The propagation behavior of hydraulic fractures (HFs) in unconventional shale formation is influenced by anisotropic fracture toughness and interaction with existing nature fractures (NFs) or bedding planes (BPs). This complex propagation behavior has not been fully understood. We propose a novel energy-based fracture propagation criterion for predicting HFs' propagation at weak interfaces between contrasting layers. Based on He-Hutchinson's theory, our proposed criterion takes HF's directional variation of energy release rate into account, thus providing a detailed analysis of HFs' possible crack paths. The deflection, crossing, and offset behaviors of HFs' interacting with weak interfaces can be determined. Besides, the stress singularity at the fracture tip is solved by applying practical enrichment functions through the extended finite element method (XFEM). Then, the stress intensity factors (SIFs) are calculated directly by considering all components of the fracture tip enrichment functions without extra post-processing. This technique guarantees the accuracy of the calculation of the SIFs, which is significant to determine the propagation direction in the proposed criterion. Typically, for HFs propagation at the weak interfaces with arbitrary direction, the influence of intersection angle, fracture toughness ratio, and rock property on the crack behavior are investigated and discussed meticulously. By combining theoretical and numerical methods, our model can further reveal the interaction mechanism between HFs and different types of weak interfaces in shale, which can help to optimize the design of hydraulic fracturing means and facilitate the generation of complex fracture networks.

Crack nucleation and propagation in the phase-field cohesive zone model with application to Hertzian indentation fracture

Hertzian indentation fracture (Hertz, 1896) is an intriguing problem with no pre-defects. Recent studies indicated that those popular phase-field models for brittle fracture might be restrictive in dealing with this problem — crack nucleation can be considered only for the failure strength within a rather limited range. In this work, this problem is analyzed by the phase-field cohesive zonde model (PF-CZM), focusing on the whole failure process of crack nucleation and propagation. Analytical and numerical results show that for any realistic failure strength, the ‘spontaneous’ crack nucleation in an initially defect-free surface, the subsequent small but rapid vertical extension and finally the stable cone-shaped crack propagation, etc., can all be qualitatively and quantitatively captured by the PF-CZM with no modification. This competence is credited to treating the failure strength and the fracture energy (toughness) as two independent material properties. Accordingly, the phase-field length scale is a numerical parameter of no further constraint, which can be made as small as possible such that the convergence to the Barenblatt (1959) cohesive zone model is guaranteed even for short cracks. Moreover, the seamless incorporation of the strength-based crack nucleation criterion and the energy-based crack propagation criterion endows the PF-CZM with the capability of tackling crack nucleation and propagation in pristine solids.

Impact Analysis of Initial Cracks’ Angle on Fatigue Failure of Flange Shafts

A fatigue test on the failure mode of flange shafts was conducted. The propagation characteristics of the initial crack at the junction between the shaft and the flange as well as its angle effect were studied. This study developed an analysis program of fatigue crack propagation, based on the APDL (ANSYS Parametric Design Language). It obtained the effective angle interval within which the initial crack is able to propagate. The fitting calculation formula was derived and the results showed that: (1) The initial crack at the junction between the shaft and the flange would propagate in the radial and axial directions; the unstable crack propagation would cause an abrupt fracture of the cross-section, failing connection; and the angle of initial crack was uncertain. (2) The crack followed the I-II-III mixed mode, which was dominated by mode I. An initial crack with a larger angle showed more noticeable II-III characteristics; KII and KIII affected the crack’s propagation angle in the radial and axial directions and they also affected the structure’s surface direction. (3) The deepest point A of the crack was located at the junction between the shaft and the flange. Its crack propagation can be divided into three stages: rapid growth (stage 1), steady decline (stage 2, buffer stage), and instability (stage 3). The initial crack angle not only affected the propagation rate at stage 1 but also influenced the fatigue life distribution of the structure during propagation. The larger the initial crack angle was, the smaller the proportion of buffer stage in the total fatigue life would be. Moreover, the propagation of crack with a larger initial angle reached instability faster after stage 1, which would cause an abrupt fracture of the cross-section. This was unfavorable for deciding the crack detection time or carrying out maintenance and reinforcement. (4) The crack propagation at the junction between the shaft and the flange was determined by the size relation between ΔKI and ΔKth, instead of the effective stress intensity factor. The effective stress intensity factor can partly reflect the law of crack propagation, but cannot serve as the only criterion for crack propagation; it must be combined with the effective angle interval, which was negatively correlated with the crack’s shape ratio, to determine whether the crack would propagate.

An elastoplastic constitutive damage model based on peridynamics formulation

Peridynamic Theory based models allow simulating the initiation and growth of cracks in solid materials, without the aid of additional methods commonly employed in the conventional finite element formulation. Within this context, a new elastoplastic damage model is proposed to use with the Peridynamic Theory. This proposed damage model combines Von Mises plasticity-based theory with a smeared cracking approach enabling damage prediction within an energy-based framework. The formulation incorporates a mixed-mode propagation criterion to account for the effect of both axial and shear stresses in the simulation, which in turn allows prediction of damage progression in ductile materials under multiaxial loading without knowing a priori the mode mixity ratio. This proposed damage modeling approach can be used within any constitutive peridynamic model, by relying on the displacement field obtained in the simulation.

Measurement and approximation by simulation of the instantaneous breakdown voltage of lightning discharge in the presence of protection

PurposeThe paper aims to estimate the instantaneous breakdown voltage of the lightning discharge from simulated figures in an energized rod-plane configuration protected by a lightning rod. The same configuration of electrodes has been the subject of experimental investigations for the measurement of the instantaneous breakdown voltage using oscillographic monitoring. This study validates the simulation model by making a comparison with experimentation and involves the role of the inception field of the upward discharge in the propagation of this last one.Design/methodology/approachThe research methodology is based on the development of a fractal lightning protection model based on real physical conditions of the discharge propagation, such as the downward discharge and the upward one emanating from protection. The voltage drop and the randomness character of the lightning discharge are also taken into account. The electrical field is an important parameter in discharge development; it is considered in this work at each step of the discharge propagation by the finite element method. The instantaneous breakdown voltage is measured and estimated by both empirical equations and simulated figures of lightning dischargeFindingsThe established model that allows estimating the instantaneous breakdown voltage from simulated discharges and empirical equations gives results in a good agreement with experimentation. The involvement of the upward discharge inception field emanating from the lightning rod in the evolution of electrical discharge is illustrated.Practical implicationsThe work presented in this paper aims to develop a new fractal lightning protection model taking into consideration physical phenomena intervening in the development of the lightning discharge.Originality/valueThe originality of this work consists of the combination between fractals modelling of the electrical discharge and the protection against lightning, in addition, to use one of the characteristics of the electrical discharge, which is the instantaneous breakdown voltage, to prove the importance of the inception field emanating from the upward discharge in the propagation criterion of this last one.

A Numerical Method for Applying Cohesive Stress on Fracture Process Zone in Concrete Using Nonlinear Spring Element.

Aiming at the numerical simulation of the entire crack propagation process in concrete, a numerical method is proposed, in which cohesive stress on the fracture process zone (FPZ) is simulated and applied by a nonlinear spring element. Using displacement control, the cohesive stress values on the FPZ are obtained from solving a system of nonlinear equations through an iterative process. According to a crack propagation criterion based on initial fracture toughness, the approach adds the spring elements to finite element analysis when simulating mode I crack propagation in standard three-point bending notched concrete beams with different strengths, initial crack ratios , and depths (D). The simulated load versus displacement (P-Delta) curves are performed to recalculate the fracture energy and verify the accuracy of cohesion in the proposed method. The simulated load versus crack mouth opening displacement (P-CMOD) curves are consistent with the previous experimental results. Subsequently, the variations of the FPZ length and the crack extension resistance ( curves are studied according to the proposed iterative approach. Compared with the existing methods using a noniterative process, the iterative approach generates a larger maximum FPZ length and curve where the FPZ length is mainly determined by the fracture energy, tensile strength, and geometry shape of the beam, and the curve is primarily determined by the fracture energy and FPZ length. The significant differences in numerical results indicate that the applying cohesion is essential in numerical simulation. It is reasonable to conclude that the proposed nonlinear spring element is more applicable and practical in the numerical simulation of the concrete mode I crack propagation process by improving the accuracy of the cohesion applied on the FPZ.

Determination of Breakdown Voltage Along the Surface of a Cylindrical Insulator

This article investigates the electric breakdown along the surface of a cylindrical dielectric, part of a coaxial transmission line. The particular geometry of the insulator is very common, as it is used at the input or output of most high-power, high-voltage (HV) generators. The streamer propagation criterion is used to calculate the flashover voltage. Studies of the dependence of the flashover voltage on the distance between the electrodes obtained in tests with dc voltage and also by applying voltage impulses with different durations and rise times are reported. The highest difference between the estimated values and the experimental data is less than 15%, which proves the applicability of the criterion for determining the surface breakdown voltage within the geometry under study. In particular, the results of these studies are used in the design of a compact 0.5-MV pulsed power system currently under development.

Analysis of full-scale burst tests by FE modelling using constant CTOA fracture criterion

Several backfilled full-scale burst tests were recently performed by TC Energy on CSA Z245.1 Grade 550 Category II pipe steels. This paper presents a new interpretation of the recent full scale burst test data using the crack tip opening angle (CTOA) as fracture propagation criterion. ASTM E3039 was used to determine the CTOA of the pipe steels used in the tests. Correlations of fracture velocity with CTOA and with Charpy absorbed energy CVN were compared. A finite element (FE) model with shell elements and a constant CTOA fracture criterion was implemented for the simulation of dynamic fracture propagation. The commercial FE code ABAQUS was used to generate the models and perform the analyses. The CTOAs determined from ASTM E3039 were used as input to the model and the fracture velocities found from the simulations were in good agreement with experimental values. The polyurethane foam coating used in the tests was incorporated in the model and shown to have a significant effect on the predicted fracture velocity. The fracture resistance curves (pressure vs. velocity) from the present model were compared with those proposed in the HLP-Sumitomo model and shown to be in approximate agreement.

Simulation of Deformation Process Failure of Jointed Rock Masses Based on the Numerical Manifold Method

On the basis of the numerical manifold method, this work introduces the concept of stress intensity factor at the crack tip in fracture mechanics and proposes the utilisation of artificial joint technology to ensure the accuracy of joint geometric dimensions in the element generation of the numerical manifold method. The contour integral method is used to solve the stress intensity factor at the joint tip, and the failure criterion and direction of crack propagation at the joint tip are determined. Element reconstruction and crack tracking are implemented in crack propagation, and a simulation programme of the entire process of deformation, failure, propagation and coalescence of jointed rock masses is developed. The rationality of the proposed method is verified by performing the typical uniaxial compression test and direct shear test.

An XFEM-based numerical strategy to model three-dimensional fracture propagation regarding crack front segmentation

Under mixed I-III loading, a crack frequently segments into several echelon-shaped daughter cracks rotated towards the direction of principal stress. This phenomenon is called crack front segmentation and has been widely observed at different length scales in both laboratory and nature. In this paper, based on the extended finite element method (XFEM), a three-dimensional numerical strategy is proposed, validated, and adopted to model quasi-static fracture propagation considering crack front segmentation. A new crack propagation criterion based on the weighted average principal stress in conjunction with a cubic spline curve fitting of front vertexes is introduced to update the crack geometry. Further, to get a higher resolution of the stress field around the crack front, a robust and efficient local mesh refinement scheme fully integrated with the XFEM is proposed. Besides, a partitioning scheme of enriched elements containing segmentation points is presented to achieve an accurate numerical integration. Several examples are presented to illustrate the effectiveness and robustness of the proposed strategy. The hydraulic fracturing example indicates that ignoring fracture front segmentation for a mixed-mode I-III fluid-driven fracture will severely underestimate the injection pressure due to the stress shadow effects between adjacent overlapped segments.