Phase-field Model Research Articles

Numerical analysis of a FE/SAV scheme for a Caginalp phase field model with mechanical effects in stereolithography

In this work, we propose a phase field model based on a Caginalp system with mechanical effects to study the underlying physical and chemical processes behind stereolithography, which is an additive manufacturing (3D printing) technique that builds objects in a layer-by-layer fashion by using an ultraviolet laser to solidify liquid polymer resins. Existence of weak solutions is established by demonstrating the convergence of a numerical scheme based on a first order scalar auxiliary variable temporal discretization and a finite element spatial discretization. We further establish uniqueness and regularity of solutions, as well as optimal error estimates for the Caginalp system that are supported by numerical simulations. We also present some qualitative two-dimensional simulations of the stereolithography processes captured by the model.

Application of the Phase Field Approach for Crack Propagation in Viscoelastic Solid Materials under Thermal Stress: A Case Study of Solder Fracturing

To date, solder has been a crucial component for interconnecting circuit boards (PCBs) and electronic components in the electronics industry. However, solder faces certain challenges, such as cracking due to thermal changes. This paper investigates solder cracking under thermal expansion. We employ a phase field model to study crack propagation under thermal stress in a square domain and in solder with a fillet shape. The model is based on those proposed by Takaishi-Kimura and Alfat, where the stress and strain tensors are modified to account for variations in the temperature field. In this study, we consider the solder material to be viscoelastic, while the other materials are treated as homogeneous and isotropic. A numerical example is computed using the adaptive mesh finite element method, with the code implemented in FreeFEM software. The results of this study are in good agreement with previous numerical and experimental findings.

Dislocation assisted coarsening of coherent precipitates: a phase field study

ABSTRACT Coarsening of precipitates in coherent systems is influenced by the elastic fields of the precipitates and the interfacial curvature. It is also known that if precipitates are connected by dislocations, coarsening is affected by the elastic fields of the dislocations and the pipe diffusivity. Although there is experimental evidence of accelerated coarsening in the presence of dislocations, these studies do not capture the effect of the elastic fields. There exist generic theoretical models that can predict the average sizes and size distributions of coarsening precipitates considering the coherency-related elastic stress fields. In this paper, we use a phase field model to study the coarsening of precipitates connected by dislocations, incorporating its elastic fields and pipe diffusivity. Specifically, we study the effects of misfit strain, elastic moduli mismatch, faster pipe mobility and the elastic fields of the dislocation on the morphology and kinetics of the coarsening precipitates. The dilatational component, associated with the edge character of the dislocations interacts with the precipitates in an elastically homogeneous system. In an elastically inhomogeneous system, the deviatoric elastic fields also interact with the precipitates influencing the morphology and kinetics of coarsening precipitates. The kinetics of coarsening is different as well, with hard precipitates coarsening faster as compared to soft precipitates, when connected by dislocations. Further, we note that the modified Gibbs–Thomson equation, which was originally derived for an isolated precipitate in an infinite matrix, is also applicable for coarsening precipitates at close proximity.

Crosslinking degree variations enable programming and controlling soft fracture via sideways cracking

Large deformations of soft materials are customarily associated with strong constitutive and geometrical nonlinearities that originate new modes of fracture. Some isotropic materials can develop strong fracture anisotropy, which manifests as modifications of the crack path. Sideways cracking occurs when the crack deviates to propagate in the loading direction, rather than perpendicular to it. This fracture mode results from higher resistance to propagation perpendicular to the principal stretch direction. It has been argued that such fracture anisotropy is related to deformation-induced anisotropy resulting from the microstructural stretching of polymer chains and, in strain-crystallizing elastomers, strain-induced crystallization mechanisms. However, the precise variation of the fracture behavior with the degree of crosslinking remains to be understood. Leveraging experiments and computational simulations, here we show that the tendency of a crack to propagate sideways in the two component Elastosil P7670 increases with the degree of crosslinking. We explore the mixing ratio for the synthesis of the elastomer that establishes the transition from forward to sideways fracturing. To assist the investigations, we construct a novel phase-field model for fracture where the critical energy release rate is directly related to the crosslinking degree. Our results demonstrate that fracture anisotropy can be modulated during the synthesis of the polymer. Then, we propose a roadmap with composite soft structures with low and highly crosslinked phases that allow for control over fracture, arresting and/or directing the fracture. The smart combination of the phases enables soft structures with enhanced fracture tolerance and reduced stiffness. By extending our computational framework as a virtual testbed, we capture the fracture performance of the composite samples and enable predictions based on more intricate composite unit cells. Overall, our work offers promising avenues for enhancing the fracture toughness of soft polymers.

A phase-field study to explore the nature of the morphological instability of Kirkendall voids in complex alloys

The present research explores theoretical and computational aspects of the morphological instability of Kirkendall voids induced by a directed flux of vacancies. A quantitative phase-field model is coupled with a multi-component diffusion model and CALPHAD-type thermodynamic and kinetic databases to obtain a meso-scale description of Kirkendall void morphologies under isothermal annealing. The material under investigation is a diffusion couple consisting of a multi-phase multi-component single-crystal Ni-based superalloy on one side and pure Ni on the other side. The flux of the fastest diffuser in the superalloy, Al, towards the pure Ni causes a strong flux of vacancies in the opposite direction. This directed flux of vacancies leads to morphologically instable growth of voids. Phase-field simulations are performed in two (2D) and three dimensions (3D) to understand these instabilities, and the results are compared with experimental observations obtained by synchrotron X-ray tomography. Finally, the simulation results are analyzed with respect to the Mullins–Sekerka linear stability criterion.

Phase Field Modelling of Failure in Thermoset Composites Under Cure-Induced Residual Stress

This study examines the residual stress induced by manufacturing and its effect on failure in thermosetting unidirectional composites under quasi-static loading, using Finite Element-based computational models. During the curing process, the composite material develops residual stress fields due to various phenomena. These stress fields are predicted using a constitutive viscoelastic model and subsequently initialized within a damage-driven Phase Field model. Structural tensors are used to modify the stress-based failure criteria to account for inherent transverse isotropy. This influence is incorporated into the crack phase field evolution equation, enabling a modular framework that retains all residual stress information through a heat-transfer analogy. The proposed coupled computational model is validated through a representative numerical case study involving L-shaped composite parts. The findings reveal that cure-induced residual stresses, in conjunction with discontinuities, play a critical role in matrix cracking and significantly affect the structural load-carrying capacity. The proposed coupled numerical approach provides an initial estimation of the influence of manufacturing defects and streamlines the optimization of cure profiles to enhance manufacturing quality. Among the investigated curing strategies, the three-dwell cure cycle emerged as the most effective solution.

The high burn-up structure evolution in polycrystalline UO2: Phase-field modeling investigation

Abstract Understanding the evolution of microstructure in nuclear fuels under high burn-up conditions is a critical concern aimed at extending fuel refueling cycles and enhancing nuclear reactor safety. In this study, a phase-field model has been proposed to examine the evolution of high burn-up structures in polycrystalline UO2. The formation and growth of recrystallized grains were initially investigated. It was demonstrated that recrystallization kinetics adhere to the KJMA equation, and recrystallization represents a process of free energy reduction. Subsequently, the microstructure evolution in UO2 was analyzed as burn-up increased. Gas bubbles acted as additional nucleation sites, thereby augmenting recrystallization kinetics, while the presence of recrystallized grains accelerated bubble growth by increasing the number of grain boundaries. The observed variations in recrystallization kinetics and porosity with burn-up closely align with experimental findings. Furthermore, the influence of grain size on microstructure evolution was deliberated upon. Larger grain sizes were found to decrease porosity and the occurrence of high burn-up structures.

Numerical investigation of bubble dynamics in ageing foams using a phase-field model

A novel numerical method for simulating liquid foam decay has been developed. This method is based on a phase-field model and captures gas pressure inside the bubbles. It employs an algorithm that includes the spontaneous rupture of foam separating films and coalescence of bubbles. We found that the microstructure evolution in liquid foam in the dry foam limit is predicted. The numerical results demonstrate that the foam ageing dynamics are mapped for the decay process due to successive coalescence events. Moreover, the method is well suited for large-scale microstructure simulations. This allows the investigation of statistical properties of foams, based on the structures’ characteristics at the bubble scale. In summary, the method is effective to gain insight into the impact of fundamental factors controlling the evolution and dynamics of decaying liquid foam.

Phase field fracture in elastoplastic solids: a stress-state, strain-rate, and orientation dependent model in explicit dynamics and its applications to additively manufactured metals

Phase field models have gained growing popularity in analysing fracture behaviour of materials. However, few have been explored to simulate dynamic ductile fracture to date. This study aims to develop a phase field framework considering strain rate, stress state, and orientation dependent ductile fracture under dynamic loading. Firstly, governing equations of displacement and phase fields are formulated in an explicit finite element framework. Secondly, constitutive relations are established with a hypoelastic-plasticity framework, encompassing the influence of material orientation and strain rate on both plasticity and fracture initiation. Stress states dependent fracture initiation is also considered. Thirdly, finite element implementation and corotational formulation for constitutive equations are derived. To validate the proposed model, finally, additively manufactured samples, involving material-level and crack propagation specimens are tested under dynamic loading conditions. Overall, the proposed phase field model can properly reproduce the experimental force-displacement curves and crack paths. Uniaxial tension tests reveal that a higher strain rate can lead to a higher hardening curve and lower ductility. Other material specimens further demonstrate the capability to predict stress state and orientation dependent dynamic fracture. To simulate dynamic crack paths accurately, it is necessary to consider anisotropic fracture initiation. Lastly, the phase field model was applied for the first time to predict the dynamic response of triply periodic minimal surface (TPMS) structures. Dynamic crack patterns were effectively captured, and the fracture mechanisms were thoroughly analysed. This study provides an explicit phase field framework for dynamic ductile fracture, with applications to additively manufactured materials and structures.

Designing brittle fracture-resistant structures:A tensile strain energy-minimized topology optimization

This research proposes a novel method for designing fracture-resistant structures. By analyzing the relationship between tensile strain energy and phase field brittle fracture, it has been found that minimizing tensile strain energy can delay fracture and enhance resistance. Capitalizing on this insight, a new topology optimization method is proposed. This method focuses on minimizing tensile strain energy to suppress the well-documented tension-dominated fracture behavior observed in phase field brittle fracture analysis. In contrast to traditional topology optimization methods based on von Mises stress, this method generates more robust structures under tension. Furthermore, the method can incorporate stress constraints to mitigate the potential for stress concentrations arising from geometric discontinuities. Numerical results demonstrate the effectiveness of the proposed method. Using phase-field modeling, the mechanical fracture properties of the optimized structures, including peak load, failure displacement, and absorbed elastic energy before fracture, are quantified. Furthermore, experimental tests are also conducted. Both numerical simulations and experimental results are consistently show that structures designed with minimized tensile strain energy exhibit superior fracture toughness. Furthermore, the method offers significant computational efficiency compared to conventional approaches due to its reliance solely on linear elasticity analysis within the optimization process.

Phase-field modeling of interfacial fracture in quasicrystal composites

Quasicrystals (QCs) have been used as a particle reinforcement phase in polymer or metal matrix composites to enhance the material strength, hardness and wear resistance while maintaining the lightweight advantages of the composites. In this paper, a phase-field fracture model (PFM) is proposed to predict crack propagation and interfacial debonding in QC composites. The phase-field and interface-field variables are introduced to regularize the cracks and interfaces in the composites, respectively. An equivalent critical energy release rate is introduced to characterize the influence of the interface on crack propagation. The present model is numerically implemented in Comsol Multiphysics based on the Weak Form PDE module. Several numerical examples are simulated to demonstrate the ability of the proposed model to predict crack propagation and interfacial failure of QC composites and to analyze the influence of QC reinforcement phase on fracture behaviors of QC composites. Numerical results indicate that the interface significantly influences the crack propagation paths, and the phason elastic field has a remarkable influence on the peak force and failure displacement in the fracture test of QC composites. The developed phase-field model and numeral implementation approach provide a convenient tool for predicting interfacial failure and assessing the safety of QC composites in engineering.

Simulating hydrogen-controlled crack growth kinetics in Al-alloys using a coupled chemo-mechanical phase-field damage model

Environmentally Assisted Cracking (EAC) of 7xxx series aluminium alloys involves interactions between multiple physical phenomena, which ultimately influence the in-service life of critical components. In this work, we present a new model to study EAC in 7xxx series alloys, which is implemented in the multiphysics simulation framework, DAMASK. The chemo-mechanical model couples crack tip hydrogen generation, resulting from surface oxidation, and transport, with crystal-plasticity-governed intergranular crack propagation, through the microstructural trapping of hydrogen at dislocations, grain boundaries (GB), and crack tip stress fields. Large-scale simulations with realistic grain structures have been performed to provide novel insight into the dominant rate-controlling processes associated with intergranular EAC in 7xxx series aluminium alloys. The model was able to reproduce experimentally measured crack velocities under different loading conditions. Parametric studies indicate that, in addition to the GB network morphology, the crack growth rate was controlled by hydrogen generation at the crack tip with long-range diffusion having negligible influence. Additionally, the total hydrogen generated through crack tip oxidation appears to be more significant than the peak generation rate.

Adaptive phase-field modeling for brittle fracture in isotropic/orthotropic piezoelectric materials using multi-patch isogeometric analysis

In this paper, an adaptive multi-patch isogeometric phase-field model is presented for brittle fracture in isotropic/orthotropic piezoelectric materials. The polynomial splines over hierarchical T-meshes (PHT-splines) functions are employed to represent both geometry and field variables. Nitsche’s method provides the continuity of all field variables, including displacement, electric potential, and phase-field at the coupling edge. The local refinement uses a mesh refinement strategy determined by a user-defined threshold for the phase-field variable. An approach that integrates hybrid formulation with staggered solving is utilized for the numerical implementation. The proposed method’s accuracy, reliability, and robustness are demonstrated using several fracture simulations, including tensile tests of rectangular and grooved ceramic plates with an edge crack, plate with a single hole, porous plate, and ceramic laminate with different material orientations.

Adaptive phase-field modeling of fracture propagation in layered media: Effects of mechanical property mismatches, layer thickness, and interface strength

Fracture propagation in layered media is investigated using an adaptive phase-field method. We focus on the interplay between cracks and interfaces, considering both perfectly and imperfectly bonded interfaces. For perfectly bonded interfaces, three-layer models are analyzed to study the effects of mechanical property mismatches, layer thickness, and confinement pressure on crack growth. Results reveal that critical energy release rate mismatch significantly influences the crack geometry, leading to single through-going fractures, middle layer fragmentation, or delamination. There is an inverse relationship between layer thickness and fragmentation, and between confinement pressure and delamination. For imperfectly bonded interfaces, a phase-field method incorporating an interface energy term is introduced and validated with benchmark examples. This model is used to study the combined effects of mechanical property mismatch and interface strength on crack growth. Our findings demonstrate that the interface strength strongly influences the dominant failure mechanism, with high strength favoring mechanical property mismatch-driven fracture and low strength leading to interfacial failure. Finally, the robustness of the proposed method is illustrated through a complex seven-layer model. This study provides valuable insights into the various factors influencing macroscopic failure mechanisms in layered materials.

Unified characterization of failure surfaces and golden-ratio ductile-to-brittle classification for isotropic materials

The failure surface, often referred to as the failure angle, defines the specific planar orientation within a material that reaches its load-carrying capacity, represented by material strengths. Analyzing this critical surface is elemental for material characterization, providing profound insights into ductility and brittleness. Despite the diversity of methodologies employed for determining the failure plane from various criteria, a universally accepted theory that systematically governs this characteristic across a broad spectrum of isotropic materials remains elusive. Therefore, this paper aims to develop a unified framework for predicting the failure surface for all homogeneous isotropic solids by considering the convergence of three key material constitutive models: elasticity, failure, and plasticity. A universal energy-based failure criterion is utilized to determine failure angles under fundamental loading scenarios, including uniaxial tension, uniaxial compression, pure shear, and biaxial tension–compression. The sliding, splitting, and crushing behaviors are obtained from the direct and shear strain increments, while the ratio of the two strain increments elucidates the dominant roles in ductile and brittle failure modes. For the first time, the developed theory links the failure angle to Poisson’s ratio, and uniaxial strength properties, unveiling a connection between intrinsic material parameters and extrinsic ductility and brittleness induced by external loadings. The failure angle representing ductile-to-brittle transition under the applied stresses in the principal stress coordinates is shown to be directly related to the golden ratio and independent of loading types. This research addresses longstanding mysteries by providing a deeper understanding of the physics of solids and suggesting potential applications with a phase-field model for predicting the evolving fracture direction.

Phase-Field Approach to Formation of Aluminum Oxyhydroxide Nanofibrils on a Liquid Aluminum Alloy Surface in Water-Containing Atmospheres

Formation of peculiar nanofibril structures on the surface of liquid aluminum alloys as a result of chemical reaction of aluminum with water vapors was well studied experimentally for various alloy components starting from mercury. However, the mechanism of nanofibril formation remains unclear to date, and the lack of an appropriate theoretical model does not allow evaluation of the structure parameters depending on the process conditions. We assume that the interface layer between the liquid metal phase and the atmosphere, containing the water vapor, develops certain inhomogeneities respect to the content of hydroxyl groups. That provides areas of suppressed and enhanced diffusion of the aluminum from the liquid phase toward the atmosphere contact. Phase-field modeling, taking into account their elastic interaction, demonstrates formation of the array of nano-islands, enriched with hydroxyl groups, which can be precursors for formation and growth of nanofibrils with elements of pre-boehmite structure.

Nucleation of Fracture: The First-Octant Evidence Against Classical Variational Phase-Field Models

Abstract As a companion work to [1], this article presents a series of simple formulae and explicit results that illustrate and highlight why classical variational phase-field models cannot possibly predict fracture nucleation in elastic brittle materials. The focus is on “tension-dominated” problems where all principal stresses are nonnegative, that is, problems taking place entirely within the first octant in the space of principal stresses.

Research on Ni-WC Coating and a Carbide Solidification Simulation Mechanism of PTAW on the Descaling Roll Surface

The descaling roll is a critical component in a hot-rolling production line. The operating conditions are significantly impacted by water with high-pressure and dynamic shocks caused by high-temperature steel slab descaling. Roll surfaces often experience wear and corrosion failures. This is attributed to a combination of high temperatures, intense wear, and repeated thermal, mechanical, and fluid stresses. Production costs and efficiency are significantly affected by the replacement of descaling rolls. Practice shows that the use of plasma cladding technology forms high-performance coatings. Conventional metal surface properties can be significantly improved. In this study, a Ni-WC composite coating was prepared on the descaling roll surface by plasma-transferred arc welding (PTAW) technology. The microstructure and phase composition of the welding overlay were analyzed by scanning electron microscopy (SEM) and X-ray diffraction (XRD). Results show that the WC hard phase added to the molten pool dissolves, and subsequently M7C3 and W2C phases are formed. To further explore the morphological evolution mechanism of the hard phase, numerical simulations were performed using a phase-field method to model M7C3 phase precipitation. The evolution from nucleation, rod-like growth, to eutectic structure formation was revealed. Experimental and simulation results show high consistency, validating the established phase-field model. In this study, a theoretical foundation for designing and preparing high-performance coatings is provided.

Chemo-mechanical benchmark for phase-field approaches

Abstract Phase-field approaches have gained increasing popularity as a consequence of their ability to model complex coupled multi-physical problems. The efficient modeling of migrating diffuse phase boundaries is a fundamental characteristic. A notable advantage of phase-field methods is their ability to account for diverse physical driving forces for interfacial motion due to diffusive, mechanical, electro-chemical, and other processes. As a result of this versatility, phase-field methods are frequently employed in the fields of materials science, mechanics, and physics, and are continually undergoing development. To test the accuracy of these developments, it is indispensable to establish standardized benchmark tests, to ensure the thermodynamic consistency of studies carried out. This work presents a series of such tests based on chemo-elastic equilibrium states for Fe-C binary alloys, benchmarking the performance of a phase-field model with chemo-elastic coupling based on the grand potential density. Use of parameters for the Fe-C system from a Calphad database allows for the determination of the Gibbs free energy, thereby enabling the quantification of chemical driving forces. For a circular inclusion, the capillary driving force is derived on a geometrically motivated basis using the lever rule and expressed as a function of the chemical potential. These simulations contribute to the development of standardized benchmark tests that validate chemical, capillary, and mechanical driving forces separately and in combination. The present study compares phase-field simulation results with results from the analytic solution of chemo-elastic boundary value problems and the generalized Gibbs–Thomson equation.

Fracture Dynamics in Silicon Anode Solid-State Batteries.

Solid-state batteries (SSBs) with silicon anodes could enable improved safety and energy density compared to lithium-ion batteries. However, degradation arising from the massive volumetric changes of silicon anodes during cycling is not well understood in solid-state systems. Here, we use operando X-ray computed microtomography to reveal micro- to macro-scale chemo-mechanical degradation processes of silicon anodes in SSBs. Mud-type channel cracks driven by biaxial tensile stress form across the electrode during delithiation. We also find detrimental cracks at the silicon/solid electrolyte interface that form due to local reaction competition between neighboring domains of different sizes. Continuum phase-field damage modeling quantifies stress-driven channel cracking and shows that the lithiated silicon stress state is critical for determining the extent of interfacial fracture. This work reveals the mechanisms that govern SSBs compared to conventional lithium-ion batteries and provides guidelines for engineering chemo-mechanically resilient electrodes for high-energy batteries.