Phase Field Theory Research Articles

A well-balanced lattice Boltzmann method based on quasi-incompressible phase-field theory

Compared with the lattice Boltzmann equation (LBE) model based on incompressible phase field theory, the LBE based on quasi-incompressible phase field theory has the advantage of local mass conservation. However, previous quasi-incompressible phase-field-based LBE model does not satisfy the well-balance property, resulting in spurious velocities in the vicinity of interface and density profiles inconsistent with thermodynamics.<br>To address this difficulty, a novel LBE model is developed based on the quasi-incompressible phase-field theory. First, numerical artifacts in the original LBE for the Cahn-Hilliard are analyzed. Based on this analysis, the equilibrium distribution function and source term are reformulated to remove the numerical artifacts, which makes the new LBE realize the well-balance property at discrete level.<br>The performance of the proposed LBE model is tested by simulating a number of typical two-phase systems. The numerical results of the planar interface and static droplet problems demonstrate that the present model can remove spurious velocities and achieve well-balanced state. Numerical results of the layered Poiseuille flow demonstrate the accuracy of the present model in simulating dynamic two-phase flow problems. The well-balance properties of the LBE model with two different formulations of surface tension ($\boldsymbol{F}_s=-\phi \nabla \mu$ and $\boldsymbol{F}_s=\mu \boldsymbol{\nabla} \phi$) are also investigated. It is found that the formulation of $\boldsymbol{F}_s=\mu \boldsymbol{\nabla} \phi$ cannot eliminate the spurious velocities, while the formulation of $\boldsymbol{F}_s=\mu \boldsymbol{\nabla} \phi$ can achieve the well-balance state. The influences of viscosity formulations of the fluid mixture are also compared. Particularly, four mixing rules are considered. It is shown that the use of step mixing rule gives more accurate results for the layered Poiseuille flow. Finally, we compare the performance of the present quasi-incompressible LBE model and the original fully incompressible LBE model by simulating the phase separation problem, and the results show that the present model can ensure the local mass conservation, while the fully incompressible LBE can yield quite different predictions.

An investigation of crack propagation in porous quasi-brittle structures using isogeometric analysis and higher-order phase-field theory

An investigation of crack propagation in porous quasi-brittle structures using isogeometric analysis and higher-order phase-field theory

Exploring the Influence of Heterogeneity on Determination of a RVE for Concrete Structures

The mechanical behavior of a heterogeneous material is strongly influenced by the response of its constituents (and their respective interactions) at lower scales. This fact accounts for the nonlinearities observed at the macroscale, and consequently, the challenges in formulating constitutive models capable of providing accurate predictions in a practical application context. Essentially, there are two primary ways to represent a heterogeneous material, either through Phenomenological Models at a Single Scale or through Multiscale Models. Multiscale models represent the most robust approach to depicting a heterogeneous material. In these models, the connection between macro and meso levels is typically made through the concept of a Representative Volume Element (RVE). The RVE is the smallest sample of the material that can statistically represent the mechanical behavior of the material. Typically, in multiscale model applications, it is assumed that an RVE exists and that its size is initially prescribed. However, the appropriate determination of an RVE for quasi-brittle materials remains a subject of study. This paper discusses the determination of RVEs for concrete using normalized graded aggregate curves. For this purpose, numerical simulations were performed using the Finite Element Method, employing constitutive models based on Phase Field theory. Throughout the study, issues related to the size of the RVE and the determination of ergodic properties are discussed. The influence of the aggregate gradation curve on the mechanical behavior of the RVE and the influence of the aggregate-mortar interface are also examined. Finally, the difficulties encountered in determining the RVE in a practical application context are presented.

Alternative wetting boundary condition for binary fluids based on phase-field lattice Boltzmann method

Based on the phase-field theory, a new wetting boundary condition (WBC) scheme is proposed to describe the fluid–solid interaction of binary fluids. Different from the common linear, cubic and sine form of surface energy wetting conditions, we adopt a mixed cubic and sine form of free energy in the present scheme. Two conditions are given to ensure that the spurious film at the solid surface disappears and the reasonable boundary condition is obtained. Combined with the wetting scheme and lattice Boltzmann (LB) method based on phase-field theory, numerical simulation of droplet spreading on a cylindrical surface is carried out to verify the performance of the present WBC. It is found that the present wetting scheme can offer considerable accuracy for predicting a static contact angle, which means that it can be used to study the wetting boundary problems of binary fluids.

Degradation mechanism for epoxy resins under combined electric, thermal and compressive stresses

Epoxy resins are widely used in equipment such as ultra-high voltage dry-type bushings, which are subjected to severe electric, thermal, and compressive stresses. Some physical mechanisms have already been proposed to explain the degradations generated by these different stresses, however their effects have not yet been quantified.. The degradation characteristics of epoxy resin under combined stresses of 8 kV/mm, 40 °C-120 °C, and 0–60 MPa were investigated in experiments. The results showed that the degradation characteristics turned significantly with the increase of compressive stress. With the increase in compressive stress, initially the partial discharge initiation voltage, tree initiation voltage and time to breakdown increased, and fractal dimension decreased. While the compressive stress exceeded the turning point, the characteristics were reversed. It can be suggested that opposing mechanisms exist. The free volume and phase field theories dominate at lower and higher stresses, respectively. A novel degradation model of the epoxy resin was proposed based on the theories above. When the compressive stress was low, the reduction of free volume played a dominant role in slowing down the degradation. When the compressive stress was high, the partial energy density concentration accelerated the degradation and played a dominant role. The molecular dynamics and finite element simulation of degradation process stress was carried out and proved consistent with experiments. It confirmed the proposed degradation model reliable and valid.

Stress triaxiality and Lode angle parameters driven phase field coupled finite deformation plasticity formulation of ductile fracture

The effect of stress triaxiality and Lode angle parameters on crack initiation and propagation is investigated using the phase field coupled elasto-plasticity formulation. To accomplish this, a multi-invariants, i.e., first invariant of stress tensor and second and third invariants of deviatoric stress tensor, dependent finite deformation hyperelasto-plasticity is coupled with phase field theory of ductile fracture. The novel ideas include a proposition of a phase field coupled multi-invariants dependent yield function by including the Hosford equivalent stress in the Drucker–Prager yield function in order to accurately predict the ductile response of the material prior to initiation of the failure and the postulation of the threshold energy in the phase field evolution equation defining a measure of the ductility of materials. The measure of ductility is reported in the form of damage initiation surface in terms of threshold plastic energy as a function of stress triaxiality and normalized Lode angle parameters using the Mohr–Coulomb fracture initiation criterion. The model parameters are calibrated based on the available experimental results in the literature considering the stress triaxiality and Lode angle parameters dependent responses of different specimens. The crack initiation and propagation paths predicted by the proposed model under the different states of stress triaxiality and Lode angle parameters, such as axisymmetric tension, plane strain condition, and axisymmetric compression, match with those predicted experimentally in the literature.

Phase field thermal shock analysis of rotating porous cracked pretwisted FGM microblade using exact shear correction factor

The present work is an attempt to develop a simple and accurate finite element formulation for the thermal shock analysis of the rotating porous cracked pretwisted functionally graded material (FGM) microblade using modified coupled stress theory in conjunction with phase-field and first-order shear deformation theory (FSDT). The physical neural surface is taken as the reference plane and the exact value of the shear correction factor is calculated from the shear stiffness. The elastic properties are assumed to be temperature-dependent and the upper ceramic layer is subjected to a high thermal shock whereas the bottom metallic layer is maintained at room temperature or is thermally insulated. The governing differential equation for the present analysis is derived using Hamilton’s principle and Newmark average acceleration method is used to obtain the transient response of the rotating porous cracked pretwisted FGM microblade subjected to thermal shock. The results obtained from the present finite element formulation are first validated with several benchmark examples available in the literature. New results are presented investigating the effect of crack depth, crack location, crack angle, rotational velocity and material scale ratio on the transient response of the cracked rotating porous pretwisted FGM microblade subjected to thermal shock. It is shown here that the parameters like crack depth, crack location and crack angle have a significant influence on the transient response of the rotating porous cracked pretwisted FGM microblade.

Phase-field modeling of lithium dendrite deposition process: When an internal short circuit occurs

Lithium dendrite growth is the main trigger for internal short circuits (ISC) of lithium-ion batteries. This paper investigates the lithium dendrite growth during the whole short circuit evolution process. In this paper, the lithium dendrite growth is divided into the initial deposition of the initial operation and the secondary deposition due to the internal short circuit. Firstly, a two-dimensional model based on the phase-field theory is developed and the growth in static electrolyte is investigated. Secondly, we summarize the whole evolution process and analyze the internal structural changes as well as the external features due to the internal short circuit. Based on this analysis, a model of secondary deposition in a flowing electrolyte is proposed. The model takes into account the effect of the flowing electrolyte on the dendrites. Finally, we investigate the growth of dendrites during secondary deposition. Due to an internal short circuit that damages the separator, a flowing electrolyte appears around the damaged region. This flowing electrolyte leads to uneven growth of dendrites, with “splitting” and “branch inhibition”. This phenomenon will accelerate the continued deposition of dendrites and induce a secondary internal short circuit. This work provides a quantitative simulation model for dendrite growth when an ISC occurs. Moreover, it also provides a reference for designing the structure of liquid batteries with high cycle times and safety.

Phase-field finite element modelling of creep crack growth in martensitic steels

Creep fracture presents a major concern for structural materials and critical components operating at elevated temperatures, thus requiring effective computational models. This study presents a phase-field framework for modelling creep crack growth and fracture behaviour of modern high-temperature materials such as Creep Strength Enhanced Ferritic (CSEF) martensitic steels. The model was formulated using the thermodynamic principles of the variational phase field theory of fracture and considering some physical aspects of creep fracture. Within the modelling framework, a dissipation potential dependent on creep damage is introduced to capture the effect of creep cavitation on the fracture energy and the creep crack growth resistance of the solid in a phenomenological manner. An elastoplastic power-law creep model is coupled to the phase-field formulations to account for the non-linear deformation processes ahead of the crack tip due to inelastic deformations, as well as their contribution to fracture at high temperatures. The capability of the proposed model is assessed against experimental data from compact tension (CT) creep tests conducted on P91 and P92 steels. Good agreement was obtained between the FE-predicted creep crack growth behaviour and the experimental measurements, showcasing the model’s feasibility. Further, numerical experiments were conducted using the proposed model to elucidate some key aspects influencing the fracture behaviour of martensitic steels. The proposed computational framework not only demonstrated good capability but was also able to offer improved mechanistic insights into the influence of material tendency to develop creep cavities on crack growth behaviour. This work contributes valuable insights into understanding the fracture process of CSEF steels at elevated temperatures and further demystifies the role of creep ductility.

The phase field method coupled with molecular dynamics parameter for simulating phase separation behavior of SBS modified asphalt

Polymer-modified asphalt is a thermodynamically unstable system, in which the polymer and asphalt are prone to phase separation. Due to the inadequacy of characterization methods for phase separation, this field has not been thoroughly understood or explored. This paper introduces an innovative methodology by employing a simulation approach that couples phase-field theory with molecular dynamics parameters. Using SBS polymer-modified asphalt as an example, validate the method's applicability and establish a more precise parameter library for the phase separation field model of SBS-modified asphalt. The phase separation process of SBS-modified asphalt was simulated. In combination with fluorescence microscopy experiments, the evolution of the micro and meso-phase states of the modified asphalt over time was established and tracked. Results indicate that higher light component content and lower heavy component content in asphalt improve its compatibility. Migration coefficients and interaction parameters for the phase field model were determined, and a more accurate correction factor for the entropy contribution of SBS-modified asphalt was estimated. The phase-field model shows that higher light component content and lower heavy component content result in slower and smaller phase separation, leading to better compatibility. The combined phase field and molecular dynamics simulations accurately reproduce experimental findings from fluorescence microscopy, providing theoretical references for studying phase separation in polymers.

A generalized phase-field cohesive zone model ([formula omitted]PF-CZM) for fracture

In this work a generalized phase-field cohesive zone model (μPF-CZM) is proposed within the framework of the unified phase-field theory for brittle and cohesive fracture. With the introduction of an extra dissipation function for the crack driving force, in addition to the geometric function for the phase-field regularization and the degradation function for the constitutive relation, theoretical and application scopes of the original PF-CZM are broadened greatly. These characteristic functions are analytically determined from the conditions for the length scale insensitivity and a non-shrinking crack band in a universal, optimal and rationalized manner, for almost any specific traction–separation law. In particular, with an optimal geometric function, the crack irreversibility can be considered without affecting the target traction–separation softening law. Not only concave softening behavior but also high-order cohesive traction, both being limitations of the previous works, can be properly dealt with. The global fracture responses are insensitive not only to the phase-field length scale but also to the traction order parameter, though the crack bandwidth might be affected by both. Despite the loss of variational consistency in general cases, the resulting μPF-CZM is still thermodynamically consistent. Moreover, the existing numerical implementation can be adopted straightforwardly with minor modifications. Representative numerical examples are presented to validate the proposed μPF-CZM and to demonstrate its capabilities in capturing brittle and cohesive fracture with general softening behavior. The insensitivity to both the phase-field length scale and the traction order parameter is also sufficiently verified.

Topology optimization with a finite strain nonlocal damage model using the continuous adjoint method

This study presents a unified formulation of topology optimization with a finite strain nonlocal damage model using the continuous adjoint method. For the primal problem to describe the material response including deterioration, we consider the standard Neo–Hookean constitutive model and incorporate crack phase-field theory for brittle fracture within the finite strain framework. For the optimization problem, the objective function is set to accommodate multiple objectives by weighting each sub-function, and the continuous adjoint method is employed to derive the sensitivity. Thus, both the governing equations for primal and adjoint problems are written as strong forms and hold at any moment and at any location in the continuum body or on its boundary. Accordingly, the proposed formulation is independent of the requirements from numerical implementation, such as element type or discretization method. In addition, the reaction–diffusion equation is used to update the design variable in an optimizing process, by which the continuous distribution of the design variable, as well as material properties, are realized. After the basic performance of the proposed formulation is demonstrated with a simple numerical setup, two-material (matrix and inclusion materials) and single-material (material and null) topology optimizations are presented, and discussions are made.

Deformation twinning as a displacive transformation: computational aspects of the phase-field model coupled with crystal plasticity

AbstractSpatially-resolved modeling of deformation twinning and its interaction with plastic slip is achieved by coupling the phase-field method and crystal plasticity theory. The intricate constitutive relations arising from this coupling render the resulting computational model prone to inefficiencies and lack of robustness. Accordingly, together with the inherent limitations of the phase-field method, these factors may impede the broad applicability of the model. In this paper, our recent phase-field model of coupled twinning and crystal plasticity is the subject of study. We delve into the incremental formulation and computational treatment of the model and run a thorough investigation into its computational performance. We focus specifically on evaluating the efficiency of the finite-element discretization employing various element types, and we examine the impact of mesh density. Since the micromorphic regularization is an important part of the finite-element implementation, the effect of the micromorphic regularization parameter is also studied.

Electro-Chemo-Mechanical Modeling of Composite Cathodes in All-Solid-State Li-Ion Batteries

A model-based understanding can assist and accelerate developing all-solid-state batteries (ASSB). In addition to chemo-mechanical influences within electrode particles (e.g., NMC) [1-2], a solid electrolyte (e.g., argyrodites) introduces additional interfacial interactions between electrode and electrolyte phases [3-5]. The present research derives and implements a coupled multi-physics finite-element model that captures electrochemical, transport, and structural behaviors of composite electrode structures. The models incorporate concentration-dependent and anisotropic material properties that are based on previously published combinations of experiment and density functional theory (DFT). These include stiffness and fracture toughness, porosity and crystallographic orientation, and operating conditions such as charge/discharge rates and external pressure.Figure 1 illustrates predicted stresses developed during electrode manufacturing. The relatively complex cathode microstructure is based on replicating scanning electron microscopy (SEM) images [6]. The composite electrode consists of electrode, electrolyte, and pore phases (Fig. 1a). As illustrated in Fig. 1b, the ASSB synthesis process involves applying and removing high compressive pressure, which causes plastic deformation and introduces residual stresses. Figure 1c shows residual von Mises stresses near electrode-electrolyte interfaces that can be on the order of a gigapascal. The synthesis-generated residual stresses serve as initial conditions for modeling the chemo-mechanics during battery cycling.During cell operation, spatially varying Li concentrations cause material deformation and associated stresses. Figure 1d shows predicted crack nucleation and growth during operation. Depending on the stress levels, crack nucleation and growth leads to cell degradation and capacity fade. The models predict interfacial fracture and phase separations using phase-field fracture theory. In phase-field formulation, the sharp cracks are approximated as diffuse cracks using a process-zone. High values of the phase parameter ξ (Fig. 1d) represent cracked surfaces. The structural disintegration and loss of active surface areas increase the tortuous path for Li/Li-ion transport, which eventually manifests as capacity-fade.The simulations are validated using published experimental work. The modeling approach, which combines phase-field and finite-element algorithms, is implemented using the COMSOL Multiphysics software. The models are expected to inform microstructure/manufacturing design and optimal operating conditions that improve cycling performance and limit/prevent mechanical damage.[1] K. Taghikhani, P.J. Weddle, J.R. Berger, and R.J. Kee. Modeling coupled chemo-mechanical behavior of randomly oriented NMC811 polycrystalline Li-ion battery cathodes. J. Electrochem. Soc., 168(8):080511, 2021.[2] R. Xu, Y. Yang, F. Yin, P. Liu, P. Cloetens, Y. Liu, F. Lin, and K. Zhao, Heterogeneous damage in Li-ion batteries: experimental analysis and theoretical modeling. J. Mech. Phys. Solids, 129, 160, 2019.[3] K. Taghikhani, P.J. Weddle, R.M. Hoffman, J.R. Berger, and R.J. Kee. Electro-chemo-mechanical finite-element model of single-crystal and polycrystalline NMC cathode particles embedded in an argyrodite solid electrolyte. Electrochim. Acta, 460:142585, 2023.[4] A. Bielefeld, D.A. Weber, R. Rueß, V. Glavas, and J. Janek. Influence of lithium ion kinetics, particle morphology and voids on the electrochemical performance of composite cathodes for all-solid-state batteries. J. Electrochem. Soc., 169(2):020539, 2022.[5] P. Minnmann, F. Strauss, A. Bielefeld, R. Ruess, P. Adelhelm, S. Burkhardt, S.L. Dreyer, E. Trevisanello, H. Ehrenberg, T. Brezesinski, F.H. Richter, and J. Janek. Designing cathodes and cathode active materials for solid-state batteries. Adv. Energy Mater., 12(35):2201425, 2022.[6] C. Doerrer, I. Capone, S. Narayanan, J. Liu, C.R.M. Grovenor, M. Pasta, and P.S. Grant. High energy density single-crystal NMC/Li6PS5Cl cathodes for all-solid-state lithium-metal batteries. ACS Appl. Mater. & interfaces, 13(31):37809–37815, 2021. Figure 1

Anisotropic damage evolution in solid fractures: A novel phase field approach with multiple failure criteria and directional-dependent structural tensor

This study proposes a novel phase-field fracture model based on unified phase field theory, aiming to overcome current limitations in simulating material complex fracture behaviors. Through this model, analytical solutions for two-dimensional bars subjected to tensile or compressive stresses are provided, enabling the coupling of multiple failure criteria and further proficient simulation of mode-I, mode-II, and mixed mode-I/II fractures, effectively addressing challenges faced in modelling materials with different or complex failure modes under various loading conditions. Furthermore, to account for the strong anisotropic failure behavior of materials, a novel directional-dependent structural tensor is proposed. The tensor correlates fracture energy with crack surface orientation, facilitating precise characterization of material damage evolution with multiple potential crack orientations. This tensor ensures the consistency of phase-field fracture evolution with predefined fracture patterns. The effectiveness of the proposed model is validated through case studies, emphasizing its robustness and superior predictive capability in capturing fracture behavior under various conditions. This research provides a more accurate and universally applicable approach for simulating material failure, particularly for complex or multiple failure mode material failure simulations.

Systematic and quantitative testing simulations and theories on phase coarsening by experiments

Microgravity experiments on phase coarsening in solid-liquid mixtures provided an ideal tool to closely and accurately explore the kinetics of phase coarsening because the sedimentation and convective melt flow are eliminated in the International Space Station. In this study, we employed phase-field simulations to systematically investigate the microstructure evolution during phase coarsening at various volume fractions. Simulated microstructure evolution during phase coarsening are compared quantitatively with the microstructure evolution archived from microgravity experiments. Furthermore, kinetics of phase coarsening in Pb-Sn solid-liquid mixtures at various volume fractions is studied theoretically and numerically, which is compared with microgravity experiments. In particular, particle size distribution, relative coarsening rate constants, and scaled maximum particle radii, are predicted from theories, and deduced from microgravity experiments, then calculated from phase-field simulations. This systematic and quantitative study of phase coarsening confirms the consistency to the results from phase-field simulation, microgravity experiments and theories at lower volume fractions, and stimulates more careful microgravity experiments at higher volume fractions (≥0.7).

Pore-Scale Numerical Simulation of Acid-Rock Reaction Processes in Carbonate Reservoirs.

In the process of acidizing carbonate reservoirs, dissolution is employed for reservoir modification to enhance recovery rates. This study establishes a numerical model at the pore scale for acid-rock reaction flow based on a microscopic continuum medium model, integrating phase-field theory and component transport models. Subsequently, the results of the Darcy-Brinkman-Stokes model are compared to those of the arbitrary Lagrange-Euler method to validate the accuracy of the model. Finally, the flow behavior of the acid solution at the pore scale and the complex dissolution mechanisms in carbonate reservoirs are analyzed. The research indicates that the microscopic pore-scale dissolution in carbonate reservoirs mainly manifests as five dissolution modes: uniform dissolution, compact dissolution, conical wormholes, dominant wormhole, and ramified wormholes. Different distributions of microfractures will alter the flow state of the acid solution and the rock-acid reaction process within the pores. Once the wormhole breakthrough occurs, there is an increased probability of acid flow through the wormhole to the outlet, leading to a decrease in the effectiveness of the acidizing carbonate reservoirs. A proper understanding of pore-scale acid-rock reaction laws is of great significance for the development of carbonate oil and gas reservoirs.

Phase-field anisotropic damage with bond-slip model for reinforced concrete beam under the service load

Concrete, as a quasi-brittle material, undergoes a latent damage accumulation preceding macroscopic crack formation, which is further complicated by the intricate interaction between steel reinforcements and concrete in reinforced concrete (RC) structures. Traditional approaches inadequately capture these nuances, especially when it comes to the seamless transition from numerous micro cracks to a visible macro cracks. Phase-field damage theory stands out as a sophisticated tool, integrating sharp cracks into a continuous damage field representation and detailing the entire process of micro-to-macroscopic cracking evolution. In this work, a phase-field damage model that accounts for bond-slip behavior is proposed, grounded in the principle of stress space decomposition to address the inherent tension–compression anisotropy. Newton’s method with a viscosity coefficient of 0.001 enhances simulation accuracy and robustness through optimal convergence. The comparison between rigid bond and bond slip mechanisms indicates a significant difference in their impact on crack modes, revealing the necessity of considering bond slip. As the steel reinforcement ratio increases, the RC beam’s load-bearing capacity increases incrementally, concurrently exhibiting a reduction in macroscopic cracks, steel reinforcement stress, and overall damage severity. However, the comparative advantage of rigid bond in improving RC beam’s resistance against cracking begins to wane as the reinforcement proportion rises. This study thus sheds light on the nuanced interplay between bond characteristics and the cracking behavior of RC beams, providing a comprehensive and advanced perspective.

Peridynamics contact model: Application to healing using phase field theory

This paper presents a contact algorithm in non-ordinary state-based peridynamics (NOSB-PD) framework and its application in healing phase field (HPF) theory. The idea here is to exploit the nonlocality in PD to model the contact between two deformable bodies. The proposed method does not require artificial short-range force or additional degrees of freedom, e.g., Lagrange multiplier, which are often used in the conventional classical and PD contact models. In the regions where contact is likely to take place, neighbor search is performed at every time step and extra particles entering the horizon are identified. These extra particles form artificial bonds which transfer forces only in compression. This feature is essential for rebound simulation. Envisioning surface-to-surface bonding as the opposite process of cracking, we propose an HPF theory to model high velocity impact induced bonding between two material bodies. Here an HPF based bonding criterion is postulated and integrated with the proposed contact algorithm. The solution of the governing equations is obtained using the fourth order Runge-Kutta method. The efficacy of the proposed contact algorithm is demonstrated by simulations of impact and rebound of two plates, two cylinders, and cylinder with rectangular prism and comparing the energy time histories with finite element solutions obtained using ANSYS®. The potential of the proposed HPF theory is demonstrated through simulation of impact of two plates where the plates rebound when the impact velocity is below a critical value, and above that velocity, the plates join with each other.

An extended gradient damage model for anisotropic fracture

This paper combines energy decomposition and an extended gradient damage (EGD) model to develop an anisotropic fracture framework with decoupling of tensile and shear cohesive laws. By introducing the shear-normal decomposition in the energy form, the driving force of the damage variable is established within the framework of the EGD model, which is then capable of capturing the traction-separation of potential crack surfaces in both shear and normal directions. The intrinsic correspondence between the cohesive law and the damage evolution enables the accurate prediction of anisotropic fracture behavior in the mixed form of Mode I and Mode II. Furthermore, the proposed model also addresses the damage unloading issue, which still remains a challenge in classic phase field theory or non-local damage theory. A number of numerical examples are presented as validation. Some cutting-edge benchmarks, such as complex mixed-mode fracture and perfect shear fracture, are well reproduced.