Cohesive Fracture Model Research Articles

A cohesive fracture-enhanced phase-field approach for modeling the damage behavior of steel fiber-reinforced concrete

This study introduces a novel computational framework based on the phase-field fracture/damage model (PFM), providing rapid and accurate predictions of the damage behavior of Steel Fiber Reinforced Concrete (SFRC) material. The framework integrates the interfacial cohesive fracture model with the PFM to depict the interaction between the cementitious matrix and fibers, marking the first demonstration that this model adequately captures both local fracture phenomena between fibers and concrete (such as fiber bridging and interfacial debonding) and mesoscopic stress–strain behavior. Additionally, an enhanced geometric approximation is established based on the spatial distribution density function of fibers and particles, allowing for the transformation of the three-dimensions (3-D) fiber-reinforced material structure into an equivalent two-dimensions (2-D) material. This approach enables the reduction of computational time, a significant limitation of the phase-field method, thereby allowing an in-house code to perform various computations with complex material structures such as SFRC. The effectiveness of the approach is demonstrated through four examples involving variations in the microgeometry and material properties of SFRC structures, ranging from the simplest configurations to real experimental material structures. Extensive Monte Carlo simulations of the model are also employed to provide results closer to reality, which are then compared with recent experimental data and numerical models, demonstrating the efficacy of the proposed computational model.

Microstructure impact on chemo-mechanical fracture of polycrystalline lithium-ion battery cathode materials

Anisotropic volume change of layered oxide cathode active material such as Ni-rich nickel–manganese–cobalt-oxide (NMC) during cycling significantly contributes to mechanical degradation, emerging as a primary factor leading to instability issues in these high-capacity cathode active materials for lithium-ion batteries. Despite their commendable energy density, these challenges hinder the broader application of NMC, underscoring the need for rational analysis and innovative solutions to address the associated mechanical instability. In this contribution, we utilize a chemo-mechanically coupled cohesive fracture model, complemented by its 3D finite element implementation, to simulate and analyze the performance and cracking behavior of polycrystalline high-Ni nickel–manganese–cobalt cathode active materials. Thereby the two-way coupling between mechanics and diffusion is regarded both in the bulk and at grain boundary. In particular, novel algorithms are introduced to model the electrolyte penetration along the inter-granular cracks, capturing its intricate impact on particle performance. Utilizing simulations on synthetic and reconstructed microstructure samples derived from experimental images, extensive parameter studies were conducted. These investigations unveiled the intricate influence of various microstructure aspects, encompassing grain morphology, grain/particle sizes, and texture. The findings indicate that secondary active material particles exhibiting smaller grains and radially elongated primary particles, coupled with radially oriented high-diffusivity planes, demonstrate enhanced resistance to inter-granular mechanical damage.

A phase-field formulation for cohesive fracture based on the Park–Paulino–Roesler (PPR) cohesive fracture model

Unlike most cohesive zone models (CZMs), the Park–Paulino–Roesler (PPR) cohesive fracture model has the inherent ability to control the softening shape of the traction–separation relationships, which makes it suitable to model fracture for a wide variety of materials. Like other CZMs, the PPR model is well-suited for problems where the crack path is known a priori but its implementation may become complex in situations where the crack path is not known beforehand. To overcome this limitation, we recast the PPR model within the framework of the phase field method, which enables the modeling of crack propagation problems with complex crack topologies using a straightforward multi-field finite element implementation. We use constitutive functions (i.e., crack geometric function and degradation function) consistent with the PPR model for the case of mode-I fracture, such that equivalent traction–separation relationships from the phase-field model approximate those from the original PPR model. Our choice of geometric and degradation functions is based upon those proposed by Wu (2017). We present several numerical examples to demonstrate the ability of the model to capture fracture of problems with different materials, geometries, and boundary conditions. Also, we show that the results from our model converge to those obtained with the original PPR model for problems where the crack path is known a priori. Being crucial in engineering design, we finally show that the model can capture size and boundary effects with satisfactory accuracy.

Computation of the crack opening displacement in the phase-field model

The phase-field model treats discrete cracks in a smeared sense by a regularisation technique. It holds attractive properties: there is no need to consider cracks as geometric discontinuities, and it avoids remeshing around crack tips. The method has been employed in the analysis of brittle and cohesive fracture problems. In the brittle fracture model, a Griffith-like energy functional is used in the simulation, while in the cohesive fracture model, the fracturing problem exploits a displacement jump governed energy functional. Obviously, the displacement jump is crucial in the cohesive fracture model and in certain other applications, e.g. for hydraulic fracturing. In the current study, the approximated form of the crack opening displacement is derived for the brittle and cohesive fracture models. In both models, the crack opening displacement is associated with a line integral normal to the crack, but different factors in front of the integral apply. The derived integrals are verified analytically in a one-dimensional setting and numerically in multi-dimensional set-ups, featuring straight and curved cracks.

Thermoplastic fracture characteristics of granite suffering thermal shocks

Thermal-shock-dependent thermoplastic fracturing is a prominent characteristic of hydraulic fracturing in high-temperature reservoirs, such as hot-dry-rock geothermal and deep oil–gas reservoirs. When cold water is injected, the rock surrounding the hydraulic fracture tip experiences sudden temperature changes, affecting the cohesive crack's tension-open in the fracture process zone (FPZ), i.e., the temperature-dependent plastic softening behavior. Based on our previously established thermoplastic cohesive fracture model, a description model of rock thermoplastic fracture properties related to thermal shock was proposed. Then, we performed mode-I fracture tests on high-temperature granite subjected to different thermal shock treatments and used digital image correlation (DIC) to characterize the fracture process (especially FPZ development). By comprehensively analyzing loading curves and DIC-based surface deformations, we quantificationally delineated seven thermoplastic fracture characteristics of thermal-shock-treated granite based on the proposed description model. We find that (1) cohesive tensile strength linearly decreases with increased thermal shock. (2) Critical COD and completely developed FPZ length linearly decrease only for temperature differences above 140 °C while they vary slightly for temperature differences from 0 °C to 140 °C (3) Plastic modulus, fracture energy, and accumulated dissipated energy change nonlinearly with the intensification of thermal shock. These three properties slightly vary for temperature differences from 0 °C to 140 °C but significantly reduce once the temperature difference exceeds 140 °C. (4) The temperature sensitivity modulus is mainly positive but follows different evolution laws with the temperature difference (≤140 °C: decreasing; >140 °C: increasing). For temperature differences of 0 °C and 140 °C, the temperature sensitivity modulus shifts from positive to negative only when COD is approximately at 0.5 times the critical value. The positive temperature sensitivity modulus implies that the yield surface of the cohesive crack contracts with enhanced thermal shocks. (5) We compared the linear elastic fracture parameter, energy release rate, with the thermoplastic fracture parameter, fracture energy. Our analysis reveals that the fracture energy is 3.34 ∼ 1.58 times larger than the energy release rate as the thermal shock intensifies, with a significantly different trend in the variability. This highlights the importance of considering thermoplastic fracture characteristics in hydraulic fracturing applications for high-temperature reservoirs. Furthermore, we put forward optimization recommendations based on our findings.

A hybridized continuum-discontinuum modeling for investigating the fracture and energy evolution characteristics of rock mass under sequential explosive detonation using a bilinear cohesive fracture model

Modelling the dynamic mechanical response of rock mass under sequential explosive detonation, especially the fracture and energy evolution characteristics, is the key to optimizing the spatial distribution and initiation time interval of explosives. A hybridized continuum-discontinuum element method and energy statistics algorithm are implemented in this study to accurately investigate the rock dynamic response induced by sequential detonation loading. Based on the rock fracture status and the stress–strain curve of bilinear cohesive fracture model, an energy statistics algorithm considering all energy components is first proposed, and its accuracy and robustness are verified. Then, the continuum-discontinuum element method and energy statistics algorithm are adopted to simulate the dynamic response of rock mass under sequential detonation loading. The results indicate that the area of cracked interfaces generated by each explosive first increases and then decreases, which caused by explosive 2 is the largest, accounting for 38.19%. Due to the damage to rock caused by the previous detonation loading, the rock around explosive 1 mainly undergoes shear fracture, while the rock around explosives 2 and 3 mainly undergoes tensile fracture. The change trends of various energy components are different, and the explosion energy is mainly converted into friction energy and element kinetic energy.

Experimental and numerical study on the mechanical behaviours of two rocks with circular openings

Abstract The presence of discontinuities in rocks can significantly compromise the performance of the rocks in terms of strength and stiffness, thereby threatening the reliability and safety of engineering structures built in or on the rocks, for example, tunnels, and slopes. In this paper, a comprehensive set of experimental tests, including Brazilian disc tests and uniaxial compression tests, were performed for two distinct rocks including granite and sandstone with seven patterns of opening holes, in addition to intact rock specimens. To investigate the micro-cracks initiation and propagation, a hybrid continuum-discrete element method incorporating a cohesive fracture model was adopted to simulate the laboratory testing. The stress–strain relations, micro-cracks initiation and propagation, Young’s modulus, compressive strength, tensile strength and micro-cracks rate and orientation were discussed. The results showed that the presence of the holes could significantly influence the tensile strength and compressive strength. However, its influence on the Young’s modulus was found to be relatively insignificant.

Study on the developmental characteristics and mechanism of shale FPZs

The fracture process and behavior of shale play a key role in the success of hydraulic fracturing stimulation in shale reservoirs. In this study, the central straight cracked Brazilian disc (CSCBD) splitting test in three fracture directions (arrester, divider, and short transverse) is conducted to reveal the propagation behavior and mechanism of closed symmetric fractures in laminated shales, with a focus on the fracture process zone (FPZ). The developmental characteristics and anisotropy of the FPZ near the crack tip of Mode I fracture of shale are identified and quantified based on the digital image correlation (DIC) method. By monitoring the crack tip opening displacement (CTOD) during the whole process, the critical CTOD, as one of the pivotal cohesive parameters, is obtained, and the development of the FPZ can be phased into three stages by CTOD evolution patterns. Based on double-K fracture criterion, the initiation fracture toughness and unstable fracture toughness of shale fractures at different laminar configurations are also analyzed. Additionally, the cohesive fracture model in the shale FPZ is derived based on the power function form using the energy release rate method.

A cohesive phase-field fracture model for chemo-mechanical environments: Studies on degradation in battery materials

In the context of computational modeling of fracture in chemo-mechanical environments, physically-sound and strong coupling between different fields is essential. Furthermore, our knowledge of the fracture in a purely mechanical setting should be extended to the new realm adequately. In this work, we apply the cohesive phase-field (CPF) fracture models to address damage initiation and progression in a chemo-mechanical coupled environment. Since CPF models are shown to be independent of the length scale parameter, such models allow a unified simulation framework for bulk and interface damages that concurrently and competitively occur in the battery materials. First, a thermodynamical framework is discussed to obtain all the possible coupling terms consistently. Through a systematic derivation from dissipation inequality and by performing various studies, we intend to comparatively demonstrate the role of different coupling terms and their impact on the obtained results. Specifically, we focus on (1) the influence of the stress field as well as the damage variable on the flux vector, (2) concentration-dependent fracture properties, (3) advantages of cohesive phase-field formulation in the multiphysics environment, and (4) influence of phase-transformation on the cracking mechanism in solids. Finally, we present some initial studies on a simplified system of a solid-state battery system, where cracking inside a single crystalline active material surrounded by a solid electrolyte is under focus. The results of the study show the dominant crack patterns. Such investigations open up opportunities for better design of battery microstructures and enhance their lifetime and performance at the cell level.

Modeling of both tensional-shear and compressive-shear fractures by a unified phase-field model

Many phase-field models have been developed in recent years to captue different fracture modes from tensional to shear, tensional-shear, and compressive-shear fractures. However, there seems no phase-field model that can simulate the tensional, shear, tensional-shear, and compressive-shear fractures at the same time under complex stress states. In this paper, a unified phase-field model is proposed in the framework of the original phase-field theory. A universal fracture criterion, that can predict both tensional-shear and compressive-shear fractures under complex stress states is embedded in the proposed phase-field, and the failure compression strength is introduced to consider the fracture under a compressive stress state. Therefore, the crack direction can be directly determined from the universal fracture criterion. The strain energy of undamaged configuration is decomposed into three parts, the tensional/compressive part, the shear part, and the rest part. The tensional/compressive and shear parts can be degraded by different degradation functions or the same degradation function. Cohesive fracture models with general softening laws and the classical brittle fracture model can be used in the proposed model, and the length scale has much less influence on the global response if cohesive fracture models with general softening laws are applied. Numerical examples show that the proposed model has the ability to simulate both the tensional-shear and compressive-shear fractures in rock-like materials and the results are in good agreement with the experiments.

Three-dimensional modelling on the impact fracture of glass using a GPGPU-parallelised FDEM

Due to the brittleness and the wide use of glass in modern engineering applications, its vulnerability to impact actions and the corresponding fracture behaviour attracted growing attentions from academics and engineers. In this study, impact fracture responses of glass have been modelled and simulated using a 3D GPGPU-parallelised hybrid finite-discrete element method, i.e., the FDEM. Glass is discretised into discrete elements where finite element formulation is incorporated, enabling accurate predictions on contact forces and structural deformation. A cohesive fracture model accounting for the rupture of glass is implemented, and numerical examples are presented and validated with results from literatures. The influence of impact velocity, boundary condition and projectile nose shape on the fracture of glass has been investigated. It is found that: (i) fracture pattern changes with the change of velocity; (ii) a rigid boundary support can be used should no damage occur in the edge of glass; (iii) under the same circumstance, a larger contact surface results in more severe damage. The GPGPU-parallelised FDEM provides a practical, efficient and robust computational approach in analysing the impact transient dynamic behaviour of glass in 3D.

A thermodynamically consistent phase-field regularized cohesive fracture model with strain gradient elasticity and surface stresses

Microstructure and surface stresses play a vital role in the micro- and nano-scale. In this paper, a thermodynamically consistent phase-field regularized cohesive fracture model (PF-CZM) is initially proposed to analyze crack propagation in the micro- and nano-scale, where strain gradient elasticity is introduced by higher-order elastic energy, and surface stresses are obtained using geometrical nonlinearity. After that, the higher-order theory is implemented through a mixed-type formulation finite element method. The specimen size effect of Mode I crack propagation is investigated. It is found that both strain gradient elasticity and surface stresses increase the critical load of crack nucleation, which becomes more pronounced as the size of the specimen decreases. The strain gradient elasticity increases the fracture toughness due to the reduction in stress singularity, while the maximum value of stress is still equal to the fracture strength during crack propagation. The surface stresses improve the local-fracture strength at the crack tip. Finally, the strain gradient elasticity further enhances the effect of surface stresses by changing the damage distribution at the crack tip when surface stresses are taken into account along with strain gradient elasticity. The presented framework has shown great potential for modeling crack propagation in the micro- and nano-scale.

Finite Element Modeling of FHWA-Accelerated Loading Facility Test Sections With Fatigue Damage Using a Nonlinear Viscoelastic Cohesive Zone Integrated With Gaussian Damage Evolution

This study presents a computational fracture modeling approach for predicting highly nonlinear viscoelastic cracking, such as fatigue damage, in asphalt mixtures and pavements. The modeling approach is presented and validated against field performance data from the FHWA-Accelerated Loading Facility (ALF) performance test sections. To that end, five mixtures differing either in the type of binder used or the amount of reclaimed asphalt pavement and reclaimed asphalt shingles were selected to assess their linear viscoelastic behavior, fracture properties, and field performance. A nonlinear viscoelastic cohesive zone fracture model was used along with a Gaussian distribution damage evolution to characterize the mixture fracture properties through a numerical-experimental calibration process. Individual mixture characteristics were then used as inputs to analyze the ALF pavement structure, and the fatigue response was predicted and compared with the field performance data for model validation. Although there are several model limitations to improve, the good agreement in performance rank order among test lanes demonstrates the capability and validity of the modeling approach. This implies that the computational modeling approach attempted in this study could potentially be used to analyze and design pavements in a mechanistic manner. This could be done with just a few laboratory tests for mixture properties such as viscoelastic dynamic modulus and cohesive zone fracture parameters.

A cohesive fracture model for predicting crack spacing and crack width in reinforced concrete structures

The cracking behavior analysis, in terms of crack spacing and crack width prediction, is essential to assess the damage and durability of a reinforced concrete (RC) structure. However, available numerical models cannot give satisfactory results in the cracking prediction due to the continuum representation of the concrete material often adopted in the numerical framework. Based on the cohesive finite element method, an interelement fracture model has been proposed here to investigate the cracking behavior, as well as its influence on the load-carrying capacity, of RC structural elements under static loading conditions. This fracture model is employed together with an embedded truss model for the rebars allowing them to be crossed by the cracks. Cracking analyses are performed on RC members subjected to tension and flexural loads. Comparisons with available experimental and numerical results have highlighted the effectiveness of the proposed model to investigate the cracking behavior in RC structures.

Transverse fracture toughness of transparent wood biocomposites by FEM updating with cohesive zone fracture modeling

Little is known about fracture toughness of monomer-impregnated wood biocomposites, with no data for transparent wood. For building applications, toughness is important, but the anisotropic nature of wood generates complex stress distributions that needs to be accounted for. A four-point bending fracture mechanics test is combined with digital image correlation (DIC) measurements of displacement fields and finite element model updating. The elastic parameters of an orthotropic composite and the parameters for a cohesive zone fracture model are determined from one transverse crack growth experiment. The fracture toughness for tangential-longitudinal (TL) cracks was lower than expected, as explained by local peeling fracture of the wood cell wall.

An anisotropic cohesive fracture model: Advantages and limitations of length-scale insensitive phase-field damage models

The goal of the current work is to explore direction-dependent damage initiation and propagation within an arbitrary anisotropic solid. In particular, we aim at developing anisotropic cohesive phase-field (PF) damage models by extending the idea introduced in Rezaei et al. (2021) for direction-dependent fracture energy and also anisotropic PF damage models based on structural tensors. The cohesive PF damage formulation used in the current contribution is motivated by the works of Lorentz et al. (2011), Wu and Nguyen (2018) and Geelen et al. (2019). The results of the such models are shown to be insensitive with respect to the length scale parameter for the isotropic case. This is because they manage to formulate the fracture energy as a function of diffuse displacement jumps in the localized damaged zone. In the present paper, we discuss numerical examples and details on finite element implementations where the fracture energy, as well as the material strength, are introduced as an arbitrary function of the crack direction. Using the current formulation for anisotropic cohesive fracture, the obtained results are converging with respect to the length scale parameter. This is achieved by including the direction-dependent strength of the material in addition to its fracture energy. Utilizing the current formulation, one can increase the mesh size which reduces the computational time significantly without any severe change in the predicted crack path and overall obtained load–displacement curves. We also argue that these models still lack to capture mode-dependent fracture properties. Open issues and possible remedies for future developments are finally discussed as well.

Cracking behavior analysis of reinforced concrete structures by using a cohesive fracture model

The simulation of cracking behavior in reinforced concrete structures is one of the most challenging tasks of engineering practice due to the highly complex phenomena which arise in the structural element under both ultimate and service loads such as the interaction between concrete and steel reinforcement, crack coalescence and branching, and multiple crack propagation. In the literature, the most known and well-established constitutive models for concrete are based on a continuum representation for this material, in which both kinematic and static variables (i.e. strains and stresses) possess regular spatial distributions. However, due to the discrete nature of the cracks, this hypothesis should be removed for a concrete cracking analysis to be correct. In this context, the present work aims to investigate the cracking behavior, in terms of crack width and crack spacing, in reinforced concrete structures by means of a discrete fracture approach implemented in a finite element framework. In particular, such a model relies on an inter-element cohesive model to simulate crack onset and propagation used in combination with an embedded truss model to reproduce the mechanical behavior of the steel rebars as well as their interaction with the surrounding concrete. The adopted integrated model has been employed to perform numerical simulations for predicting the load-carrying capacity and the related crack pattern of real-scale reinforced concrete structural elements. Comparisons with experimental outcomes, in terms of loading curve, crack width and crack spacing, have demonstrated the capabilities and effectiveness of the proposed model to investigate the cracking behavior in reinforced concrete structures providing more accurate crack patterns, than existing models.

A strain-rate cohesive fracture model of rocks based on Lennard-Jones potential

To characterize the dynamic mechanical response of rocks during the initiation and propagation of cracks at a high strain rate, a strain-rate cohesive fracture model is established based on the Lennard-Jones potential and multi-scale model of rocks. The newly proposed model explains the micro-mechanism of strain rate effect from the molecular scale and establishes the potential energy function and force function. First, it is proposed that the strain rate effect arises due to the change of microscopic properties of molecules at a high strain rate. Thereafter, the potential energy function and force function of the strain-rate cohesive fracture model corresponding to the dynamic tensile and shear processes are established. Finally, the accuracy of the strain-rate cohesive fracture model is verified through numerical simulations. The results indicate that the strain-rate cohesive fracture model can accurately simulate the dynamic tensile failure and shear failure of rocks at different strain rates. The dynamic tensile strength, dynamic compressive strength, and dynamic tensile fracture energy obtained by numerical simulations and laboratory tests are similar.

Optimization of perforation parameters for horizontal wells in shale reservoir

Volume fracturing of horizontal wells is the key technology for efficient development of shale reservoirs, and perforation is the basic condition of hydraulic fracturing. In this paper, based on the basic equation of fluid–structure coupling and the cohesive fracture model, a three-dimensional numerical model of spiral perforation hydraulic fracture propagation in near well bore horizontal wells was established. The pipe flow elements were introduced to characterize the automatic distribution of fluid and the pressure drop caused by friction. The results show that the vertical perforation initiation pressure is lower than the horizontal perforation, the vertical perforation is easier to release energy after initiation, the fracturing fluid is easier to flow into the vertical perforation, and eventually form 1–2 major fractures. The common rule among different pore densities is that the fracture distribution at both ends of the cluster is more, and the fracture distribution in the middle is less, and the gap between the two increases with the increase of injection. When the hole density is higher than 8 P/m, the uneven development of fractures is significant. If the hole density is 4 P/m, the number of effective fractures around the wellbore decreases. Therefore, it is suggested that the optimal hole density is 8 P/m. With the increase of perforation depth, the effective area of liquid pressure on the hole wall increases, and the initiation pressure decreases. With the increase of the diameter of the hole, the initiation pressure is lower and the initiation and extension of the fracture are easier. Compared with pore diameter, the effect of pore depth on fracture propagation is greater than that of pore diameter. Under the calculation conditions, it is recommended that 8 P/m, hole depth is 0.8 m, and aperture is 20 mm.

A consistent framework for chemo-mechanical cohesive fracture and its application in solid-state batteries

Damage and fracture can be induced not only by mechanical loading but also due to chemical interactions within a solid. On one hand, species concentration may embrittle or toughen the material and on the other hand, the mechanical state adds additional driving force for diffusion. We propose a chemo-mechanically coupled cohesive fracture model with several novel features. It distinguishes the mode-dependent damage progression and its influence on lithium transport. Coupled with mode-dependent cohesive zone damage, the model recaptures both the normal and tangential transport behavior of lithium at the interface. Moreover, it tackles concentration-dependent crack initiation, various softening behavior, as well as the cyclic damage accumulation. The thermodynamic consistency of the proposed model with the mentioned features is demonstrated. The model is numerically implemented with the finite element method. Numerical results, along with comparison with related experimental data, demonstrate that the model can be applied to study diffusion-induced fracture in general solid ionic conductors in Lithium-ion batteries. In particular, illustrative numerical results are presented for both the intergranular fracture inside active material or solid electrolyte and the interface fracture between active material and solid electrolyte. Furthermore, it is discussed how the solid electrolyte influences the dominant crack patterns. The current contribution is applicable to address similar problems on hydrogen-induced cracking and moister-dependent fracture.