Variational Phase-field Model Research Articles

On the energy decomposition in variational phase-field models for brittle fracture under multi-axial stress states

AbstractPhase-field models of brittle fracture are typically endowed with a decomposition of the elastic strain energy density in order to realistically describe fracture under multi-axial stress states. In this contribution, we identify the essential requirements for this decomposition to correctly describe both nucleation and propagation of cracks. Discussing the evolution of the elastic domains in the strain and stress spaces as damage evolves, we highlight the links between the nucleation and propagation conditions and the modulation of the elastic energy with the phase-field variable. In light of the identified requirements, we review some of the existing energy decompositions, showcasing their merits and limitations, and conclude that none of them is able to fulfil all requirements. As a partial remedy to this outcome, we propose a new energy decomposition, denoted as star-convex model, which involves a minimal modification of the volumetric-deviatoric decomposition. Predictions of the star-convex model are compared with those of the existing models with different numerical tests encompassing both nucleation and propagation.

Phase-field modeling of fracture for ferromagnetic materials through Maxwell’s equation

Electro-active materials are classified as electrostrictive and piezoelectric materials. They deform under the action of an external electric field. Piezoelectric material, as a special class of active materials, can produce an internal electric field when subjected to mechanical stress or strain. In return, there is the converse piezoelectric response, which expresses the induction of the mechanical deformation in the material when it is subjected to the application of the electric field. This work presents a variational-based computational modeling approach for failure prediction of ferromagnetic materials. In order to solve this problem, a coupling between magnetostriction and mechanics is modeled, then the fracture mechanism in ferromagnetic materials is investigated. Furthermore, the failure mechanics of ferromagnetic materials under the magnetostrictive effects is studied based on a variational phase-field model of fracture. Phase-field fracture is numerically challenging since the energy functional may admit several local minima, imposing the global irreversibility of the fracture field and dependency of regularization parameters related discretization size. Here, the failure behavior of a magnetoelastic solid body is formulated based on the Helmholtz free energy function, in which the strain tensor, the magnetic induction vector, and the crack phase-field are introduced as state variables. This coupled formulation leads to a continuity equation for the magnetic vector potential through well-known Maxwell’s equations. Hence, the energetic crack driving force is governed by the coupled magneto-mechanical effects under the magneto-static state. Several numerical results substantiate our developments.

A generally variational phase field model of fracture

Although the phase field method has been widely used in the field of fracture, various models exist, such as the second-order and fourth-order phase field model in the field of fracture mechanics, and the phase field evolution model in the field of physics, and it is not yet possible to establish these models under a unified framework. In this paper, a thermodynamically generalized framework of variational phase field model of fracture is presented, which can effectively unify different models under this framework. Based on this variational framework, not only the current second-order and fourth-order phase field model can be unified, but also new second-order, fourth-order, sixth-order, and eighth-order phase field model can be obtained from this framework. In addition, a method is proposed to convert the lower-order model into a higher-order model, which can convert a phase field distribution function of the lower-order model into a high-order phase field distribution function, and then obtain a higher-order phase field model. Finally, the new model is verified by some representative examples, and the second-order and fourth-order phase field models are compared, which shows the effectiveness of building a new phase field model from the variational framework.

A framework to model the hydraulic fracturing with thermo-hydro-mechanical coupling based on the variational phase-field approach

In this research, a numerical framework for hydraulic fracturing has been formulated, incorporating thermo-hydro-mechanical (THM) coupled effects. While previous studies have reported various hydraulic fracturing models based on the phase-field method, a THM coupling scheme grounded in the variational phase-field approach remains unexplored. The THM coupling is of paramount importance for understanding underground fracture propagation. This study’s principal innovation lies in its extension of the variational phase field model, introducing it to a thermoporoelastic medium, thereby integrating fluid flow and heat transfer elements in fractured materials. In our advanced model, the fluid flow and heat transfer in the matrix and fracture are modeled independently. Subsequently, employing a set of reasonable assumptions, we introduce unified flow and heat transfer equations derived from phase-field calculus. A pivotal contribution of our work is the development of an iterative solution algorithm based on the fixed stress method, pivotal for tackling the THM coupled system. The proposed equations are discretized employing the FEM. Additionally, to address the numerical oscillations in advection-dominant issues that manifest in the heat transfer model, we have applied a generalized Streamline Upwind Petrov Galerkin method for 8-node elements. The accuracy of fracture propagation was assessed by comparing the numerical results of the evolution of fracture length, fracture width at the injection point, and the injection pressure under the K regime with the corresponding analytical solution. Furthermore, the process of heat transfer within a fractured domain, often referred to as the Barends problem, has been studied by proposed model. Additionally, some numerical experiments were conducted to gain insights into the intricate interplay between the thermal–hydraulic–mechanical system during hydraulic fracturing, thereby further emphasizing the significance of our advanced model.

On poroelastic strain energy degradation in the variational phase-field models for hydraulic fracture

Though a number of formulations have been proposed for phase-field models for hydraulic fracture in a fully saturated porous medium, the definition of the degraded poroelastic strain energy varies from one model to another. This study explores previously proposed forms of the poroelastic strain energy with diffused fracture and assesses their ability to recover the explicit fracture opening aperture. We then propose a new form of degraded poroelastic strain energy derived from micromechanical analyses. Unlike the previously proposed models, our poroelastic strain energy degradation depends not only on the phase-field variable (damage) but also on the type of strain energy decomposition. Comparisons against closed form solutions suggest that our proposed model can recover crack opening displacement more accurately irrespective of Biot’s coefficient or the pore-pressure distribution. We then verify our model against the plane strain hydraulic fracture propagation, known as the KGD fracture, in the toughness dominated regime. Finally, we demonstrate the model’s ability to handle complex hydraulic fracture interactions with a pre-existing natural fracture.

Variational quantitative phase-field modeling of nonisothermal sintering process.

Phase-field modeling has become a powerful tool in describing the complex pore-structure evolution and the intricate multiphysics in nonisothermal sintering processes. However, the quantitative validity of conventional variational phase-field models involving diffusive processes is a challenge. Artificial interface effects, like the trapping effects, may originate at the interface when the kinetic properties of two opposing phases are different. On the other hand, models with prescribed antitrapping terms do not necessarily guarantee the thermodynamics variational nature of the model. This issue has been solved for liquid-solid interfaces via the development of the variational quantitative solidification phase-field model. However, there is no related work addressing the interfaces in nonisothermal sintering, where the free surfaces between the solid phase and surrounding pore regions exhibit strong asymmetry of mass and thermal properties. Also, additional challenges arise due to the conserved order parameter describing the free surfaces. In this work, we present a variational and quantitative phase-field model for nonisothermal sintering processes. The model is derived via an extended nondiagonal phase-field model. The model evolution equationshave naturally cross-coupling terms between the conserved kinetics (i.e., mass and thermal transfer) and the nonconserved one (grain growth). These terms are shown via asymptotic analysis to be instrumental in ensuring the elimination of interface artifacts, while also examined to not modify the thermodynamic equilibrium condition (characterized by a dihedral angle). Moreover, we demonstrate that the trapping effects and the existence of surface diffusion in conservation laws are direction-dependent. An anisotropic interpolation scheme of the kinetic mobilities that differentiates between the normal and tangential directions along the interface is discussed. Numerically, we demonstrate the importance of the cross-couplings and the anisotropic interpolation by presenting thermal-microstructural evolutions.

A quantitative variational phase field framework

The finite solid–liquid interface width in phase field models results in non-equilibrium effects, including solute trapping. Prior phase field modeling has shown that this extra degree of freedom, when compared to sharp-interface models, results in solute trapping that is well captured when realistic parameters, such as interface width, are employed. However, increasing the interface width, which is desirable for computational reasons, leads to artificially enhanced trapping thus making it difficult to model departure from equilibrium quantitatively. In the present work, we develop a variational phase field model that guarantees a temporal decrease in the free energy with independent kinetic equations for the solid and liquid phases. Separate kinetic equations for the phase concentrations obviate the assumption of point wise equality of diffusion potentials, as is done in previous works. Non-equilibrium effects such as solute trapping, drag and interface kinetics can be introduced in a controlled manner in the present model. In addition, the model parameters can be tuned to obtain “experimentally-relevant” trapping while using significantly larger interface widths than prior efforts. A comparison with these other phase field models suggests that interface width of about three to twenty-five times larger than current best-in-class models can be employed depending upon the material system at hand leading to a speed-up by a factor of W(d+2), where W and d denote the interface width and spatial dimension, respectively. Finally the capacity to model non-equilibrium phenomena is demonstrated by simulating oscillatory instability leading to the formation of solute bands.

Variational modeling of hydromechanical fracture in saturated porous media: A micromechanics-based phase-field approach

This paper presents a novel variational phase-field model for different fracture processes in fully saturated porous media. As a key feature, the model employs a micromechanics-based theory for the description of brittle-tensile and compressive–ductile fracture. As such, the field variables are linked to physical mechanisms at the microcrack level, with damage emerging as the consequence of microcrack growth. Similarly, plasticity emerges as a consequence of the frictional sliding of closed microcracks. In this way, the evolution of opening microcracks in tension leads to (mode I) brittle fracture, while the evolution of closed microcracks in compression/shear leads to (mode II) ductile fracture. These failure mechanisms are coupled to fluid flow, resulting in a Darcy–Biot–type hydromechanical model. Therein, in the tensile regime, plasticity naturally vanishes, while damage is driven by poroelastic energy, accounting for the pressure in fluid-filled opening microcracks. On the other hand, in the compressive/shear regime, the plastic driving force naturally follows as a Terzaghi-type effective stress in terms of the local stress field acting on the microcrack surfaces, while damage is solely driven by the frictionally blocked free energy. As another important feature, the model includes a non-associative frictional plasticity law. Nevertheless, a thermodynamically consistent variational framework is employed, for which different energetic principles are discussed. Finally, the numerical simulations show that the model captures relevant hydromechanical coupling effects in benchmark problems, including mechanically induced shear fracture and hydraulically induced tensile fracture.

Approximation of surface diffusion flow: A second-order variational Cahn–Hilliard model with degenerate mobilities

This paper addresses the approximation of surface diffusion flow using Cahn–Hilliard-type models. We introduce and analyze a new variational phase field model that associates the classical Cahn–Hilliard energy with two degenerate mobilities and guarantees a second-order accuracy in the approximation of the sharp limit. We also introduce simple and efficient numerical schemes to approximate the solutions to three Cahn–Hilliard-type models for surface diffusion flow, including the one we propose, and we provide 2D and 3D numerical experiments that illustrate the advantages of our approach.

Second-order phase-field formulations for anisotropic brittle fracture

We address brittle fracture in anisotropic materials featuring two-fold and four-fold symmetric fracture toughness. For these two classes, we develop two variational phase-field models based on the family of regularizations proposed by Focardi (2001), for which Γ-convergence results hold. Since both models are of second order, as opposed to the previously available fourth-order models for four-fold symmetric fracture toughness, they do not require basis functions of C1-continuity nor mixed variational principles for finite element discretization. For the four-fold symmetric formulation we show that the standard quadratic degradation function is unsuitable and devise a procedure to derive a suitable one. The performance of the new models is assessed via several numerical examples that simulate anisotropic fracture under anti-plane shear loading. For both formulations at fixed displacements (i.e. within an alternate minimization procedure), we also provide some existence and uniqueness results for the phase-field solution.

A micromechanics-based variational phase-field model for fracture in geomaterials with brittle-tensile and compressive-ductile behavior

This paper presents a framework for modeling failure in quasi-brittle geomaterials under different loading conditions. A micromechanics-based model is proposed in which the field variables are linked to physical mechanisms at the microcrack level: damage is related to the growth of microcracks, while plasticity is related to the frictional sliding of closed microcracks. Consequently, the hardening/softening functions and parameters entering the free energy follow from the definition of a single degradation function and the elastic material properties. The evolution of opening microcracks in tension leads to brittle behavior and mode I fracture, while the evolution of closed microcracks under frictional sliding in compression/shear leads to ductile behavior and mode II fracture. Frictional sliding is endowed with a non-associative law, a crucial aspect of the model that considers the effect of dilation and allows for realistic material responses with non-vanishing frictional energy dissipation. Despite the non-associative law, a variationally consistent formulation is presented using notions of energy balance and stability, following the energetic formulation for rate-independent systems. The material response of the model is first described, followed by the numerical implementation procedure and several benchmark finite element simulations. The results highlight the ability of the model to describe tensile, shear, and mixed-mode fracture, as well as responses with brittle-to-ductile transition. A key result is that, by virtue of the micromechanical arguments, realistic failure modes can be captured, without resorting to the usual heuristic modifications considered in the phase-field literature. The numerical results are thoroughly discussed with reference to previous numerical studies, experimental evidence, and analytical fracture criteria.

Variational phase-field model based on lower-dimensional interfacial element in FEM framework for investigating fracture behavior in layered rocks

The fracture characteristics of layered shale rock determine the efficiency of shale gas production through hydraulic fracturing. An effective and robust numerical method may play a significant role in predicting the crack propagation path in layered shale. In this study, a lower-dimensional interfacial element and tangential derivative variable are developed to describe the evolution of the phase-field at the interface within the framework of the phase-field approximation. The variational phase-field model is derived based on a new energetic framework. The lower-dimensional interface is discretized together with the rock matrix. A separated coupling strategy is adopted to solve the coupled system, and the equations are solved sequentially during each time step. The model is validated by conducting two 2D classical benchmark tests and a 3D three-point-bending experiment. Further, the presented approach is applied to modeling layered shale failure in 2D notched square shale specimens subjected to tension. The stiff-to-stiff, soft-to-stiff, and stiff-to-soft configurations are designed to investigate the fracture behavior in layered rocks with perfectly-bonded and weakly-bonded interfaces. The results indicate that the proposed method exhibits extreme robustness and can be applied to evaluate the interfacial fracture and crack-interface interaction.

A phase-field thermomechanical framework for modeling failure and crack evolution in glass panes under fire

This paper presents a novel phase-field thermomechanical modeling framework for predicting complicated behaviors of thermal cracking in glass panes under fire. The main idea is to incorporate the proposed mathematical model, which calculates the exact deformation of the mesh elements, into the variational phase-field model to simulate the thermal fracture behavior in glass panes in an effective manner. The developed model improves upon previous attempts to predict thermal cracking in the following ways: (1) in a major departure from the classical phase-field simulation of thermomechanical fracture, crack evolution can be predicted using only temperature distributions; the phase-field formulations are kept fixed to overcome mesh dependency and convergency; (2) the new modeling framework directly transforms temperature variations into thermal strains (rate of loading) using fewer mesh elements and a larger time step, thus substantially reducing the computational effort; and (3) the proposed model can simultaneously predict multiple cracks distributed in any arbitrary space in the glass panes more realistically than the previous numerical models, regardless of glass pane type and size, fixation method, and thermal loading variation. The proposed coupling model is validated through comparisons against experimental observations and ANSYS simulations. Moreover, the validated model is used to examine for the first time the effect of real engineering influential conditions, namely the heating rate, glass pane size ratio under non-uniform thermal loading, and glass pane fixation with a frame on three sides, on thermal cracking behavior.

A variational phase-field model For ductile fracture with coalescence dissipation

A novel phase-field model for ductile fracture is presented. The model is developed within a consistent variational framework in the context of finite-deformation kinematics. A novel coalescence dissipation introduces a new coupling mechanism between plasticity and fracture by degrading the fracture toughness as the equivalent plastic strain increases. The proposed model is compared with a recent alternative where plasticity and fracture are strongly coupled. Several representative numerical examples motivate specific modeling choices. In particular, a linear crack geometric function provides an “unperturbed” ductile response prior to crack initiation, and Lorentz-type degradation functions ensure that the critical fracture strength remains independent of the phase-field regularization length. In addition, the response of the model is demonstrated to converge with a vanishing phase-field regularization length. The model is then applied to calibrate and simulate a three-point bending experiment of an aluminum alloy specimen with a complex geometry. The effect of the proposed coalescence dissipation coupling on simulations of the experiment is first investigated in a two-dimensional plane strain setting. The calibrated model is then applied to a three-dimensional calculation, where the calculated load-deflection curves and the crack trajectory show excellent agreement with experimental observations. Finally, the model is applied to simulate crack nucleation and growth in a specimen from a recent Sandia Fracture Challenge.

Nucleation under multi-axial loading in variational phase-field models of brittle fracture

Phase-field models of brittle fracture can be regarded as gradient damage models including an intrinsic internal length. This length determines the stability threshold of solutions with homogeneous damage and thus the strength of the material, and is often tuned to retrieve the experimental strength in uniaxial tensile tests. In this paper, we focus on multiaxial stress states and show that the available energy decompositions, introduced to avoid crack interpenetration and to allow for unsymmetric fracture behavior in tension and compression, lead to multiaxial strength surfaces of different but fixed shapes. Thus, once the length scale is tailored to recover the experimental tensile strength, it is not possible to match the experimental compressive or shear strength. We propose a new energy decomposition that enables the straightforward calibration of a multi-axial failure surface of the Drucker-Prager type. The new decomposition, which hinges upon the theory of structured deformations, encompasses the volumetric-deviatoric and the no-tension models as special cases. Preserving the variational structure of the model, it includes an additional free parameter that can be calibrated based on the experimental ratio of the compressive to the tensile strength (or, if possible, of the shear to the tensile strength), as successfully demonstrated on two data sets taken from the literature.

Numerical Discretization of Variational Phase Field Model for Phase Transitions in Ferroelectric Thin Films

Phase field methods have been widely used to study phase transitions and polarization switching in ferroelectric thin films. In this paper, we develop an efficient numerical scheme for the variational phase field model based on variational forms of the electrostatic energy and the relaxation dynamics of the polarization vector. The spatial discretization combines the Fourier spectral method with the finite difference method to handle three-dimensional mixed boundary conditions. It allows for an efficient semi-implicit discretization for the time integration of the relaxation dynamics. This method avoids explicitly solving the electrostatic equilibrium equation (a Poisson equation) and eliminates the use of associated Lagrange multipliers. We present several numerical examples including phase transitions and polarization switching processes to demonstrate the effectiveness of the proposed method.

An accelerated staggered scheme for variational phase-field models of brittle fracture

There is currently an increasing interest in developing efficient solvers for variational phase-field models of brittle fracture. The governing equations for this problem originate from a constrained minimization of a non-convex energy functional, and the most commonly used solver is a staggered solution scheme. This is known to be robust compared to the monolithic Newton method, however, the staggered scheme often requires many iterations to converge when cracks are evolving. The focus of our work is to accelerate the solver through a scheme that sequentially applies Anderson acceleration and over-relaxation, switching back and forth depending on the residual evolution, and thereby ensuring a decreasing tendency. The resulting scheme takes advantage of the complementary strengths of Anderson acceleration and over-relaxation to make a robust and accelerating method for this problem. The new method is applied as a post-processing technique to the increments of the solver. Hence, the implementation merely requires minor modifications to already available software. Moreover, the cost of the acceleration scheme is negligible. The robustness and efficiency of the method are demonstrated through numerical examples.

Variational phase-field continuum model uncovers adhesive wear mechanisms in asperity junctions

Wear is well known for causing material loss in a sliding interface. Available macroscopic approaches are bound to empirical fitting parameters, which range several orders of magnitude. Major advances in tribology have recently been achieved via Molecular Dynamics, although its use is strongly limited by computational cost. Here, we propose a study of the physical processes that lead to wear at the scale of the surface roughness, where adhesive junctions are formed between the asperities on the surface of the materials. Using a brittle formulation of the variational phase-field approach to fracture, we demonstrate that the failure mechanisms of an adhesive junction can be linked to its geometry. By imposing specific couplings between the damage and the elastic energy, we further investigate the triggering processes underlying each failure mechanism. We show that a large debris formation is mostly triggered by tensile stresses while shear stresses lead to small or no particle formation. We also study groups of junctions and discuss how microcontact interactions can be favored in some geometries to form macro-particles. This leads us to propose a classification in terms of macroscopic wear rate. Although based on a continuum approach, our phase-field calculations are able to effectively capture the failure of adhesive junctions, as observed through discrete Molecular Dynamics simulations.

Evaluation of variational phase-field models for dynamic brittle fracture

During the last decades various phase-field model have been developed in the variational approach to model dynamic fracture of brittle solids. Complex fracture phenomena such as crack branching, dynamic crack instability and fragmentation, etc., can be captured with a minimal list of physical parameters. However, an objective evaluation of existing results is missing and in particular, the issues of mesh size dependence and length scale sensitivity have been rarely studied. Aiming to bridge the above gap, this paper presents an extensive evaluation of commonly used variational phase-field models for dynamic fracture to address the question: how well can we predict dynamic fracture with current PFMs. Several popular PFMs in the literature are considered, implemented within a single code platform and used to simulate six dynamic fracture problems of homogeneous brittle solids. Mesh convergence and length scale sensitivity are systematically analyzed. Quantitative evaluation of the branching angle, crack tip position and/or velocity with respect to available experiments, and comparison against the results of other approaches, are both presented.

On crack opening computation in variational phase-field models for fracture

Phase-field models for fracture have gained exceptional popularity in the last couple of decades and have been extended into areas well beyond brittle quasi-static fracture propagation to ductile, dynamic, or hydraulic fracturing. Despite the significant theoretical advancements in these more complex physical settings, little attention has been paid to the quantification of crack opening displacement to date as most applications do not explicitly require the crack opening displacement for the morphological evolution of cracks. However, one of the exemptions would be hydraulic fracturing where the crack propagation is driven by the fluid pressure which strongly depends on the crack opening displacement.In this study, we look into two known approaches, a line integral and a level-set method, for crack opening computation mainly from an implementation point of view. Firstly, we derive an approximation of a discontinuous function field in the variational phase-field setting which is then applied to obtain the crack opening (displacement jump) and verify the approximation against a closed solution. We then propose a “certain distance from the crack” required for the level-set function using a one-dimensional analysis. Finally, we compare these approaches under several different conditions such as crack alignment to the mesh or under loading (asymmetric displacement field) and investigate the convergence with respect to the mesh size.