Phase-field Model Research Articles

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.

Elucidating the role of Cr migration in Ni-Cr exposed to molten FLiNaK via multiscale characterization

Structural materials used in nuclear reactor environments are subjected to coupled extremes such as high temperature, irradiation, and corrosion which act in concert to degrade their functional performance. Connecting alloy microstructure such as grain boundaries, and accumulating point defects with corrosion attack and pore morphology is critical to understanding underlying mechanisms. We uncover the compositional variations and morphology at multiple length scales in corrosion-damaged Ni-Cr alloy after exposure to oxidants in molten fluoride salts. A complex network of dense corrosion pores is detected by surface-level SEM observations. The corrosion pores take on a 1-dimensional morphology and are enriched with Ni and depleted of Cr 1–5 µm from the pore surface. STEM-EDS and 4D-STEM strain maps acquired simultaneously highlight the local variations in composition and structure of a ≤ 200 nm Cr-rich layer identified from a cross-section taken at the bottom of an isolated corrosion pore between the Ni-Cr alloy matrix and the embedded salt. However, the absence of an observed interface between the Ni-Cr alloy matrix and the FCC-structured Cr-rich layer suggests that Cr plating from the salt did not transpire. These findings support a proposed Cr lattice diffusion mechanism rather than Cr-precipitation from the salt to accommodate temperature transient conditions during sample cooling. Through the development of a 1D phase field model, these results are rationalized by formation energies for the Ni- and Cr-oxidation into the molten salt. This study reveals the locally altered microstructure caused by high temperature corrosion in non-steady-state molten salt nuclear reactor environments.

Asymptotic Behavior and Numerical Simulations of a Conservative Phase-Field Model with Two Temperatures

One of the types of problem that has attracted the attention of mathematicians in recent years is the phase field system. The field of application includes materials science, where phenomena such as phase separation in alloys, crystal formation and thermal welding are legion. Among these phase transition systems, the family of conservative systems is very popular with industry. Indeed, minimising losses in production systems is a major issue for their profitability. In this paper, we study the well-posedness of the formulation and the asymptotic behaviour of the solutions, by proving the existence of a finite-dimensional global attractor for a conservative variant of the two-temperature phase field system with homogeneous Neumann boundary conditions. The inclusion of two temperatures in the definition of the enthalpy of the system is a necessity in the case of a non-simple material. In a simple material, once the phase-change temperature has been reached, the temperature of the system remains constant until the material has completely changed state. This is not true in the case of a non-simple material, where an increase in the temperature of the system is observed even after the phase-change temperature has been reached. To conclude the work, we present a method for numerically approximating the solution and carry out some numerical tests.

Analyzing the Effects of Cr and Mo on the Pearlite Formation in Hypereutectoid Steel Using Experiments and Phase Field Numerical Simulations.

In this study, we quantitatively investigate the impact of 1.4 wt.% chromium and 1.4 wt.% molybdenum additions on pearlitic microstructure characteristics in 1 wt.% carbon steels. The study was carried out using a combination of experimental methods and phase field simulations. We utilized MatCalc v5.51 and JMatPro v12 to predict transformation behaviors, and electron microscopy for microstructural examination, focusing on pearlite morphology under varying thermal conditions. Phase field simulations were carried out using MICRESS v7.2 software and, informed by thermodynamic data from MatCalc v5.51 and the literature, were conducted to replicate pearlite formation, demonstrating a good agreement with the experimental observations. In this work, we introduced a semi-automatic reliable microstructural analysis method, quantifying features like lamella dimensions and spacing through image processing by Fiji ImageJ v1.54f. The introduction of Cr resulted in longer, thinner, and more homogeneously distributed cementite lamellae, while Mo led to shorter, thicker lamellae. Phase field simulations accurately predicted these trends and showed that alloying with Cr or Mo increases the density and circularity of the lamellae. Our results demonstrate that Cr stabilizes pearlite formation, promoting a uniform microstructure, whereas Mo affects the morphology without enhancing homogeneity. The phase field model, validated by experimental data, provides insights into the morphological changes induced by these alloying elements, supporting the optimization of steel processing conditions.

A mixed-element phase field method for the fracture analysis of beams

The prediction of fracture in thin structures is crucial for the safety assessments in various fields of engineering. Modeling methods such as the phase-field method typically assume damage to be constant across the thickness of structural models like beams, plates and shells which, especially in load cases coupling stretching and bending, is an imperfect approximation. In this contribution, fracture behaviors along the thickness direction of Euler–Bernoulli beam considering axial deformation are addressed with a phase-field modeling approach. To achieve this, a mixed-element beam model is introduced, which combines 1D beam elements representing the displacement field with 2D rectangular elements describing a crack phase-field. The nonlinear coupled governing equations are solved using a staggered scheme. The transfer of variables between the coupled fields occurs at the quadrature points which in turn need to have corresponding geometric locations. Numerical examples ranging from the problems of fracture in standalone beams to lattice structures show the applicability of the approach, and detailed comparisons with a standard 2D model confirm its accuracy and efficiency.

Smoothed point interpolation methods for phase-field modelling of pressurised fracture

The problem of hydraulic fracturing is of great relevance to various areas and is characterised by the occurrence of complex crack patterns with bifurcations and branches. For this reason, an interesting approach is the modelling of hydraulic fracture using a phase-field model. In addition to the discretisation using the Finite Element Method (FEM), some works have already explored the discretisation of the phase-field model with meshfree methods, including the Smoothed Point Interpolation Methods (SPIM) family. Seeking to take advantage of the good convergence results of SPIM for phase-field modelling of brittle fractures, this paper proposes the use of SPIM for phase-field modelling of pressurised fractures. In order to limit the computational cost, a prescribed SPIM-FEM coupling is employed, with the purpose of concentrating the meshless discretisation only in the regions of expected crack propagation. The model is characterised by a constant internal pressure load along the fracture that is applied indirectly from the formulation of the phase-field model. A series of numerical simulations is presented. The aim is to evaluate the proposed model, verify the results and point out characteristics of the phase-field model with internal pressure.

Grain orientation and shape evolution of ferroelectric ceramic thick films simulated by phase-field method

The orientation and shape of ceramics grains was always neglected, resulting in a lot of information during sintering has not been excavated. In this study, a modified phase-field model in order to express the anisotropy of grain boundary energy is developed. The effects of the anisotropy of grain boundary energy on the grain orientation and shape evolution are investigated in detail. The ferroelectric ceramic thick films are prepared by tape casting. The comparison of experiment and simulation results shows that the anisotropy of grain boundary energy results in uneven grain orientation and bimodal grain size distribution. The quantitative analysis of grain microstructures helps to establish a relationship with the degree of anisotropy of grain boundary energy. Our findings provide a new way to judge the degree of anisotropy by calculating the relevant parameters in the SEM images of ceramics materials.

What is the physical origin of the gradient flow structure of variational fracture models?

We investigate a physical characterization of the gradient flow structure of variational fracture models for brittle materials: a Griffith-type fracture model and an irreversible fracture phase field model. We derive the Griffith-type fracture model by assuming that the fracture energy in Griffith's theory is an increasing function of the crack tip velocity. Such a velocity dependence of the fracture energy is typically observed in polymers. We also prove an energy dissipation identity of the Griffith-type fracture model, in other words, its gradient flow structure. On the other hand, the irreversible fracture phase field model is derived as a unidirectional gradient flow of a regularized total energy. We have considered the time relaxation parameter a mathematical approximation parameter, which we should choose as small as possible. In this research, however, we reveal the physical origin of the gradient flow structure of the fracture phase field model (F-PFM) and show that the small time relaxation parameter is characterized as the rate of velocity dependence of the fracture energy. It is verified by comparing the energy dissipation properties of those two models and by analysing a travelling wave solution of the irreversible F-PFM. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Integrated Framework to Model Microstructure Evolution and Decipher the Microstructure-Property Relationship in Polymeric Porous Materials.

Unraveling the microstructure-property relationship is crucial for improving material performance and advancing the design of next-generation structural and functional materials. However, this is inherently challenging because it requires both the comprehensive quantification of microstructural features and the accurate assessment of corresponding properties. To meet these requirements, we developed an efficient and comprehensive integrated modeling framework, using polymeric porous materials as a representative model system. Our framework generates microstructures using a physics-based phase-field model, characterizes them using various average and localized microstructural features, and evaluates microstructure-aware properties, such as effective diffusivity, using an efficient Fourier-based perturbation numerical scheme. Additionally, the framework incorporates machine learning methods to decipher the intricate microstructure-property relationships. Our findings indicate that the connectivity of phase channels is the most critical microstructural descriptor for determining effective diffusivity, followed by the domain shape represented by curvature distribution, while the domain size has a minor impact. This comprehensive approach offers a novel framework for assessing microstructure-property relationships in polymer-based porous materials, paving the way for the development of advanced materials for diverse applications.

Multiscale modelling of microstructure evolution, and local solidification behaviours of the AlSi10Mg build component in laser powder bed fusion process

Laser powder bed fusion (L-PBF) process is an additive manufacturing process, in which rapid heating and cooling takes place. The microstructure and the solidification behaviour of L-PBF components have significant impacts on their mechanical properties. The solidification takes place in a very short duration which is difficult to capture experimentally, therefore the finite-element (FE) simulation is the best feasible solution. In this paper, to study the evolution of the AlSi10Mg alloy’s microstructure during the L-PBF process, a model composed of macroscopic finite element (FE) model and microscopic phase field (PF) model was built. This model is suitable to predict a more realistic thermal behaviour involving the possible physical phenomena and the thermal behaviour for various heat inputs was calculated. The PF model was then updated with these results in order to simulate the chemical compositions’ concentration field and solidified microstructure. According to the simulation’s results, the primary dendrite arm spacing (PDAS) is significantly influenced by the laser line energy. To validate the FE model along with the Phase field (PF) model, the results were compared with the analytical and experimental findings; and the FE model and PF results exhibited good agreement with the experimental results. Additionally, the solute distribution analysis demonstrated that during the solidification process, micro-segregation developed and the process parameters affected the micro-segregation. Finally, the specimens were subjected to metallographic and element distribution analyses using an energy-dispersive spectrometer (EDS) to compare the results with those from the PF simulation and establish the relationship between the process parameters.

Cracking mechanism of CMAS-corroded thermal barrier coatings based on a coupled thermo-chemo-mechanically phase-field fracture model

A thermo-chemo-mechanically coupled theoretical framework is proposed to describe the calcium-magnesium-alumina-silicate (CMAS) corrosion process during the cooling process. A phase-field fracture model is developed to investigate the effect of cooling temperature and CMAS concentration on the degree of corrosion reaction, the stress evolution and the crack initiation and propagation. σ11 concentrates in the region beneath the overlay of CMAS and σ22 appears at the interface between top ceramic coating (TC) and bond coating (BC). The higher stress concentration of σ11 and σ22 contribute to the formation of both vertical and transverse cracks. Transverse cracks first emerge at the interface between TC and BC in the edge region, followed by the formation of vertical cracks in the CMAS-coated region. Vertical cracks propagate to the interface and deflect into transverse cracks. The transverse cracks at the interface further propagate and merge, ultimately leading to the coating delamination. The higher initial cooling temperature and CMAS concentration contribute to the accelerated development of vertical cracking and the increase of the quantity and length of transverse and vertical cracks. The model provides a significant advantage in predicting the failure of TBCs during the cooling stage of CMAS corrosion.

Phase-Field Simulation of Precipitation and Grain Boundary Segregation in Fe-Cr-Al Alloys under Irradiation.

A phase-field model for the precipitation of Fe-Cr-Al alloy is established incorporating grain boundary (GB) effects and irradiation-accelerated diffusion. The radiation source and grain boundary effect are incorporated to broaden the applicability of the Fe-Cr-Al precipitated phase-field model. The model is firstly employed to simulate the precipitation of the Cr-rich α' phase in a single-crystal alloy. The precipitation rate and the size distribution of the precipitated phase were analyzed. Subsequently, the model is utilized to simulate segregation at GBs in a double-crystal system, analyzing the enrichment of Cr and depletion of Al near these boundaries. The simulation results are consistent with experimental observations reported in the references. Finally, the model is applied to simulate the precipitation in a polycrystalline Fe-Cr-Al system. The simulated results revealed that the presence of GBs induces the formation of localized regions with enhanced Cr and Al content as well as depleted zones adjacent to these boundaries. GBs also diminish both the quantity and precipitation rate of the formed phase within the grains.

Design of polar boundaries enhancing negative electrocaloric performance by antiferroelectric phase-field simulations

Electrocaloric refrigeration which is environmentally benign has attracted considerable attention. In distinction to ferroelectric materials, which exhibit an extremely high positive electrocaloric effect near the Curie temperature, antiferroelectric materials represented by PbZrO3 have a specific negative electrocaloric effect, i.e., electric field decreases the temperature of the materials. However, the explanation of the microscopic mechanism of the negative electrocaloric effect is still unclear, and further research is still needed to provide a theoretical basis for the negative electrocaloric effect enhancement. Herein, the antiferroelectric phase-field model has been proposed to design polar boundaries enhancing antiferroelectric negative electrocaloric performance in PbZrO3-based materials. Based on this, we have simulated the polarization response and domain switching process of the temperature and electric field-induced antiferroelectric—ferroelectric phase transition. It is shown that the temperature range tends to increase as the density of polar boundaries increases from the antiferroelectric stripe domain, polymorphic domain to the nanodomain. Among them, the peak adiabatic temperature change of antiferroelectric nanodomains can reach −13.05 K at 84 kV/cm, and a wide temperature range of about 75 K can be realized at 42 kV/cm. We expect these discoveries to spur further interest in the potential applications of antiferroelectric materials for next-generation refrigeration devices.

Phase-field modeling of thermal shock fracture in functionally graded materials

In this paper, the thermo-elastic fracture problems of functionally graded materials (FGMs) are thoroughly investigated based on a modified phase field model. In this model, the material constants and fracture toughness vary along a specified direction, the thermal conductivity and stiffness constants in the damaged regions are degraded by the phase-field variable, and the crack evolution is driven by the variation of elastic energy induced by the thermo-mechanical loading. Therefore, the temperature, mechanical and damage fields are coupled with each other. The validation of the present approach has been checked by comparing the present results with the analytical, numerical, and experimental results in literature. The examples of thermo-elastic tensile fracture and thermal shock cracking of FGMs are performed to investigate the influence of material heterogeneity and external thermal loading. Furthermore, the experimental phenomenon of crack deflection in FGMs has been captured by the present simulations. The present study provides a guidance for understanding the fracture mechanism, material design and safety assessment of FGM structures in engineering practice.

A thermodynamically consistent machine learning-based finite element solver for phase-field approach

In this article, a thermodynamics-based data-driven approach utilizing machine learning is proposed to accelerate multiscale phase-field simulations. To obtain training data, the interface propagation kinetics, integrated into a physics-based phase-field model, are monolithically solved using a finite element method-based code developed within the Python-based open-source platform FEniCS. The admissible sets of internal state variables (e.g., stress, strain, order parameter, and its gradient) are extracted from the simulations and then utilized to identify the deformation fields of the microstructure at a given state in a thermodynamics-based artificial neural network. Finally, the high performance of the proposed machine learning-enhanced solver is illustrated through detailed comparisons with nanostructural calculations at the nanoscale. Unlike previous methods, the current analysis is not restricted by specific morphologies and boundary conditions, given the length and time scales required to reproduce these results.

Mechanical property evaluation of 3D multi-phase cement paste microstructures reconstructed using generative adversarial networks

This study proposes an artificial intelligence based framework for reconstructing the 3D multi-phase cement paste microstructure to evaluate its mechanical properties using simulation. The reconstruction of cement paste microstructures is performed using modified generative adversarial networks (GANs) based on microstructural images from micro-CT. For computational efficiency, 2D microstructures are first reconstructed and then extended to 3D microstructures. The reconstructed microstructures exhibit the same microstructural features as the original microstructures when characterized by probability functions. Mechanical properties such as stiffness and tensile strength are evaluated for the original and reconstructed specimens using a phase-field fracture model, and similar behaviors are observed. The results confirm that the reconstructed virtual microstructures can be used to supplement the real microstructures in evaluating the mechanical properties of 3D multi-phase cement paste. This approach thus provides a critical element of a data-driven approach to correlating its microstructure and properties.

Modelling and simulation of natural hydraulic fracturing applied to experiments on natural sandstone cores

Under in-situ conditions, natural hydraulic fractures (NHF) can occur in permeable rock structures as a result of a rapid decrease of pore water accompanied by a local pressure regression. Obviously, these phenomena are of great interest for the geo-engineering community, as for instance in the framework of mining technologies. Compared to induced hydraulic fractures, NHF do not evolve under an increasing pore pressure resulting from pressing a fracking fluid in the underground but occur and evolve under local pore-pressure reductions resulting in tensile stresses in the rock material. The present contribution concerns the question under what quantitative circumstances NHF emerge and evolve. By this means, the novelty of this article results from the combination of numerical investigations based on the Theory of Porous Media with a tailored experimental protocol applied to saturated porous sandstone cylinders. The numerical investigations include both pre-existing and evolving fractures described by use of an embedded phase-field fracture model. Based on this procedure, representative mechanical and hydraulic loading scenarios are simulated that are in line with experimental investigations on low-permeable sandstone cylinders accomplished in the Porous Media Lab of the University of Stuttgart. The values of two parameters, the hydraulic conductivity of the sandstone and the critical energy release rate of the fracture model, have turned out essential for the occurrence of tensile fractures in the sandstone cores, where the latter is quantitatively estimated by a comparison of experimental and numerical results. This parameter can be taken as reference for further studies of in-situ NHF phenomena and experimental results.

Surface energies control the anisotropic growth of [formula omitted]-Ni(OH)2 nanoparticles in stirred reactors

Nickel hydroxides are an important family of electrode materials in the field of batteries and electrochemical energy storage. Two polymorphs, α-Ni(OH)2 and β-Ni(OH)2 have been identified, with intermediate structures also reported. However, the synthesis of the β-phase precipitants that are ideal for electrochemical applications is not trivial. The growth and morphology of the final products can widely vary with the pH. The electrochemical performance of the β-phase is sensitive to its structure and morphology, which are also sensitive to the reaction conditions. In order to better understand the initial nucleation, growth and morphological evolution of the β-phase, we present a combined experimental and theoretical study including a multiscale phase-field model for the evolution of the morphology of β-Ni(OH)2. Surface indices and energies for the phase-field modeling were obtained from density functional calculations (DFT). The developed phase-field model can reproduce the growth of β-Ni(OH)2 phase in different conditions. The model shows that the basal planes of β-Ni(OH)2 can grow in high pH but at the same time its growth is limited by other high energy prismatic surfaces.

A micromagnetic-mechanically coupled phase-field model for fracture and fatigue of magnetostrictive alloys

Magnetostrictive alloys are usually brittle materials with micromagnetic structures. Their structural reliability and durability depend on the complex micromagnetic-mechanical coupling at smaller length scales encompassing the evolution of micromagnetic structures. Herein we propose a micromagnetic-mechanically coupled phase-field model for fracture and fatigue behavior of magnetostrictive alloys with evolution of the micromagnetic structure. The thermodynamically-consistent model is derived from microforce theory, laws of thermodynamics, and Coleman–Noll analysis. The evolution of crack phase-field and magnetization-vector order parameters that are fully coupled is governed by history field dependent Allen–Cahn and Landau–Lifshitz–Gilbert equations, respectively. The model is extended to fatigue by introducing a degradation prefactor for the fracture energy as a function of positive elastic energy. One-dimensional analyses are then presented to anatomize the crack driving forces in terms of fully coupled micromagnetic-mechanical and pure mechanical driving force. We demonstrate the model capabilities by finite-element numerical studies on the micromagnetic domain evolution during the crack propagation and the influence of external magnetic field for type-I, type-II, and three-point bending fracture, as well as for the fracture of a single-edge notched specimen with an elliptical inclusion. The simulation result shows that depending on how micromagnetic domains are switched under micromagnetic-mechanical coupling, the magnetic field can enhance or decrease the critical load. In the presence of inclusion with larger fracture toughness, a crack is found to nucleate in the tri-junction of multi-domain micromagnetic structure owing to the high elastic strain around the tri-junction point. It is further found that a suitable magnetic field promoting magnetization-vector rotation around the crack tip could remarkably improve the fracturing load and fatigue life. The results demonstrate the model promising for the study of micromagnetic-mechanically coupled fracture and fatigue in magnetostrictive alloys.

The role of Sn grain orientation in Cu depletion of Sn-based solders

The miniaturization of solder bumps in flip-chip packages results in large variability in their microstructure making the prediction of joint lifetime challenging. This is of particular importance in Sn-rich solders due to the anisotropic diffusion behavior of Sn and the presence of fast diffusion pathways, such as grain boundaries and twins that vary from joint to joint. We developed a multi-physics phase-field model to predict the electromigration-enhanced evolution of intermetallic compounds in Cu/Sn system with different grain structures. We present simulations of solder joints with polycrystalline Sn with grains with different orientations to observe the effect of Sn anisotropy and grain boundaries on electromigration failure. The simulations predict Cu depletion in the cathode and the formation and accumulation of intermetallic compounds along Sn grain boundaries dependent on the Sn grain orientation. Additionally, Cu6Sn5 clusters are formed in the Sn matrix when high diffusion rates at the interfaces are considered.