Phase-field Model Research Articles

A phase-field model for the brittle fracture of Euler–Bernoulli beams coupling stretching and bending

Damage gradient models seek to simulate fracture mechanics by modulating the material stiffness. Within this framework, a singular scalar field representing damage is commonly utilized to globally decrease elastic energy. However, when considering structural models like beams and plates, this approach often fails to adequately capture important aspects due to the interaction between stretching and bending contributions. In this work, we propose a model for planar Euler–Bernoulli beams that incorporates the following features: firstly, we utilize two phase-field damage parameters to describe material damaging, specifically addressing the ‘erosion’ occurring above and below the original, undamaged beam; secondly, we assume a simple linear dependence of the axial stress field on the thickness coordinate, along with linear dependence on the axial force and bending moment. By appropriately identifying the constitutive response, our model effectively considers the coupling between stretching and bending induced during through-the-thickness damage, demonstrating good agreement with 2D observations.

Quasi-Static and Dynamic Crack Propagation by Phase Field Modeling: Comparison with Previous Results and Experimental Validation

In this paper, experimental tensile tests for pre-cracked high Carbon steel ‘C90’ specimens were performed for quasi-static and dynamic loading. High loading velocity affects the crack patterns by preventing deflection. On the other hand, an efficient numerical tool based on the phase field model was developed and validated to predict brittle fracture trajectories. A staggered numerical scheme was adopted to solve the displacement and damage fields separately. Implementation efficiency in initiating and propagating cracks, even from an undamaged microstructure, was proved. The effect of the critical fracture energy density Gc on the crack path was tested; with smaller Gc, the crack patterns become more complex. In addition, the impact of loading velocities was examined, and earlier and faster crack formation and greater crack branching is observed with higher impact velocity. In this study, bidimensional plane stress cases were treated. The phase field model with hybrid formulation was able to predict crack pattern and especially crack arrest and branching found in the literature. The developed model accurately determined the transition zone of the crack path topology that has been observed experimentally.

Gradient elasticity in Swift–Hohenberg and phase-field crystal models

The Swift–Hohenberg (SH) and phase-field crystal (PFC) models are minimal yet powerful approaches for studying phenomena such as pattern formation, collective order, and defects via smooth order parameters. They are based on a free-energy functional that inherently includes elasticity effects. This study addresses how gradient elasticity (GE), a theory that accounts for elasticity effects at microscopic scales by introducing additional characteristic lengths, is incorporated into SH and PFC models. After presenting the fundamentals of these theories and models, we first calculate the characteristic lengths for various lattice symmetries in an approximated setting. We then discuss numerical simulations of stress fields at dislocations and comparisons with analytic solutions within first and second strain-gradient elasticity. Effective GE characteristic lengths for the elastic fields induced by dislocations are found to depend on the free-energy parameters in the same manner as the phase correlation length, thus unveiling how they change with the quenching depth. The findings presented in this study enable a thorough discussion and analysis of small-scale elasticity effects in pattern formation and crystalline systems using SH and PFC models and, importantly, complete the elasticity analysis therein. Additionally, we provide a microscopic foundation for GE in the context of order-disorder phase transitions.

Effect of Ni4Ti3 precipitates on the functional properties of NiTi shape memory alloys: A phase field study

Ni4Ti3 precipitates, which generally exist in aged Ni-rich NiTi shape memory alloys (SMAs), can have a profound effect on material properties. However, the fundamental insights into the effects of Ni4Ti3 precipitates on the functional properties, including the superelasticity (SE), elastocaloric effect (eCE), and shape memory effects (SMEs), are not well understood yet, especially those originating from the B2-B19′ martensite transformation (MT). In this work, a phase field model coupling the precipitation of Ni4Ti3 and the B2-B19′ MT is proposed, where the thermo-mechanical coupling effect and grain size effect are considered. The precipitate-dependent SE, eCE, one-way SME (OWSME), and stress-assisted two-way SME (SATWSME) of single-crystal, polycrystalline, and gradient-nanograined NiTi SMAs are simulated. The effects of the precipitate density, grain orientation range (texture), and gradient-distributed precipitate are examined, and the underlying microscopic mechanisms are revealed. The simulation results and new findings not only contribute to a more comprehensive insight into the effect of Ni4Ti3 precipitates on the MT, martensite reorientation, and functional properties of NiTi SMAs but also provide a reference for the development of excellent SMA-based solid refrigerants or SMA smart materials with designable functional properties.

Prediction of Dynamic Ni Morphology Changes in Patterned Ni-YSZ Anode with Physics-Informed Neural Networks

Abstract Nickel (Ni) film on patterned Ni- yttria-stabilized zirconia (YSZ) anode shows dynamic spreading and splitting during solid oxide fuel cell (SOFC) operation, where wettability of Ni on YSZ is greatly enhanced (Z. Jiao, N. Shikazono, J. Power Sources 396 119–123, 2018). In the present study, a physics-informed neural network (PINN) constrained by Cahn-Hilliard equation of phase field model is proposed to estimate the unknown parameters for predicting dynamic Ni movements of the patterned Ni-YSZ anode. The unknown parameters such as interface thickness and mobility are inversely inferred by PINN using top-view images obtained from the operando experiments. Obtained excess surface diffusivity values were three to four orders of magnitude larger than the values reported for surface diffusion in the literature. It is therefore considered that Ni spreading and splitting of patterned anode cannot be simply explained by surface diffusion, and other mechanisms should be introduced.

On fracture mechanism of additively manufactured triply periodic minimal surface structures using an explicit phase field model

Triply periodic minimal surface (TPMS) structures have been extensively studied for their exceptional mechanical characteristics. However, numerical analysis of their fracture behaviour remains insufficient due to the complexity of the fracture mechanisms. This study aims to utilise a new phase field model to predict the mechanical responses and analyse the fracture mechanism of TPMS gyroid and primitive structures. Firstly, the gyroid (G) and primitive (P) structures were additively manufactured using Ti-6Al-4V titanium and tested under both axial and oblique compression. Secondly, an explicit phase field model was developed by incorporating the Bao-Wierzbicki fracture model to capture damage initiations. It was found that the explicit phase field enables accurate reproduction of experimental force-displacement responses, deformation modes, crack initiations and propagations for G and P structures under both loading conditions. It is noted that medium stress triaxiality tension was the dominant stress state to trigger material damage, regardless of structure and loading conditions. Thirdly, compared with axial compression, it is found that oblique loading introduced a more non-proportional loading history, leading damage initiation points to being far away from the fracture locus. Further, in comparison with the G structure, the P structure involved more medium and high stress triaxiality tension induced fracture initiations, resulting in more damaged material points. This study offers valuable insight into the fracture mechanism of triply periodic minimal surface structures, which is beneficial to improving the design of those structures.

Study of droplet formation in parallel flow focusing microchannel under electrostatic field control

To achieve efficient preparation of emulsion droplet, this paper uses a two-dimensional mathematical model coupled with a phase field model and an electrostatic field model to calculate the droplet formation in parallel flow focusing multichannel, and analyzes the effect of electric field strength on the droplet formation process. In addition, the effects of different capillary numbers and dispersed phase viscosity on droplet formation were also analyzed. The results show that the emulsion in the channel always generates droplet by dripping regime, and the droplet have the best mono-dispersity. Applying appropriate electric field strength can realize droplet formation frequency and size regulation. Increasing the dispersed phase viscosity increases the breakup thread of the dispersed phase fluid, and the droplet size obtained gradually decreases, the difference in droplet size can be slowed down by applying the electric field. Increasing the electrostatic force leads to a higher droplet deformation rate. The droplet shape classification is plotted using the capillary number (Ca) and electrostatic number (CaE). It is found that the droplet flow pattern in the channel remains a circle only when Ca and CaE are sufficiently small.

Low-cycle fatigue crack growth in brittle materials: Adaptive phase-field modeling with variable-node elements

This work presents an adaptive phase field framework for the simulation of crack nucleation and propagation in brittle materials subject to low-cycle loading. We introduce a fatigue history strain parameter within the phase-field framework to capture the fatigue effect. The resulting coupled differential equations are solved using a staggered iteration scheme. In order to improve the computational efficiency of the phase-field model, an adaptive refinement scheme is introduced. This scheme utilizes an artificial threshold that takes into consideration of both phase-field variables and cumulative fatigue history field variables to determine the elements to be refined. The handling of the hanging nodes resulting from mesh refinement is accomplished through the application of the variable-node element technique, offering a flexible and efficient mesh integration technique. We validate our proposed numerical scheme through five representative numerical examples: single-edge cracked specimen, compact tension specimen, double-edge cracked panel, plate with a hole and plate with multi-cracks and hole. The results demonstrate that the adaptive mesh refinement scheme significantly reduces computational costs without compromising the accuracy of the numerical predictions.

Growth mechanism of intermetallic compound in SnAg/Cu interface of micro-bumps under extremely low-temperature by phase field method

Scientific investigation will increasingly focus on deep space exploration. Spacecraft will be subjected to extremely low-temperature, changing internal microstructure and micro-bumps mechanics, reducing reliability. Thus, micro-bumps must be tested in extremely low temperatures. This research examines the impact of low-temperature storage on IMC (Intermetallic Compound) growth and micro-bumps crystallography at −196 °C. Secondly, a diffusion-weak coupled phase field theory is summarized, and the COMSOL multi-physics simulation platform simulates the growth of IMC layer in micro-bumps developing during extremely low-temperature storage. Finally, microhardness and elastic modulus of various phases were studied. Experimental data indicate that holding Sn at −196 °C for 1000 h produces recrystallization, resulting in tiny grains with diverse orientations. Micro-bumps with Sn/IMC interfaces have highest von Mises equivalent stress. Strain gradient thickens IMC layer at low temperatures with storage time, and edge IMC layer is thinner than axis IMC layer. The diffusion-force weak coupling phase field model's driving force curve indicates that stress gradient, not phase separation, drives IMC growth at extremely low temperatures. Low-temperature storage enhances Sn microhardness and elastic modulus through compressive stress and grain refining. IMC microhardness and elasticity decrease with tensile stress.

Accurate prediction of aging effects on microstructure evolution and related mechanical strength of Mg-Zn alloys via multiscale simulations

Age strengthening is a common method to achieve superior mechanical properties employed in magnesium (Mg) alloys. However, the experimental approaches to examine the aging effects on mechanical properties are primarily based on trial and error, posing challenges for the design and development of high-strength Mg alloys. A promising solution to address this challenge is to establish an accurate prediction model for the aging effects on Mg alloys, covering the evolution of microstructure and corresponding mechanical properties. This paper introduces a multiscale simulation approach to accurately predict the aging effects on the Mg-2.3Zn (at.%) alloy. To begin, we establish a phase field model to simulate the microstructure evolution at 150 ℃, demonstrating precipitate growth during aging in alignment with experimental observations. Secondly, based on the microstructures obtained from phase field simulations, the corresponding yield strength is determined through finite element method simulations. The simulation successfully captures the peak-aged behavior of yield strength during aging, resulting in a yield strength of 167 MPa at the peak-aged state, with a deviation of less than 5 % from experimental results. In-depth analysis indicates that the peak value of total yield strength arises from the interplay between solid solution strengthening and precipitate strengthening. Furthermore, precipitate strengthening reaches a peak value at a specific aging time, deviating from the total yield strength. This deviation is primarily attributed to the effect of precipitate size on shear strengthening during aging. This study presents a general method for predicting the microstructure evolution and corresponding mechanical properties of alloys through multiscale simulations, offering valuable insights for designing and predicting the performance of metallic alloys.

Phase-field approach assisted solution heat treatment efficient design for novel multicomponent Co-based superalloys

Fast homogenization without incipient melting is the key to designing solution heat treatment (SHT) process for single crystal superalloys. In this paper, a multicomponent phase-field model coupled with solidus temperature is developed for dynamically tracking incipient melting temperature, TIM, of alloy during SHT. The simulation process runs completely automatically only by inputting the composition information of as-cast specimen. Based on the obtained TIM-time profile, the holding temperature and the time of SHT are determined. Applying this approach to Co-30Ni-11Al-4W-1Ta-4Ti-5Cr single crystal specimen, a gradual increased TIM from 1247°C to 1293°C within 72000 s is obtained, and thus a stepwise SHT process with a short time of less than 20 h is designed. The experimental verification shows that after SHT, the segregation coefficient of the slowest diffusing element W decreased from more than 2 to 1.065, the content of harmful secondary phase has decreased to below 0.1%, and no incipient melting occurs.

A coupled high-accuracy phase-field fluid–structure interaction framework for Stokes fluid-filled fracture surrounded by an elastic medium

In this work, we couple a high-accuracy phase-field fracture reconstruction approach iteratively to fluid–structure interaction. The key motivation is to utilize phase-field modelling to compute the fracture path. A mesh reconstruction allows a switch from interface-capturing to interface-tracking in which the coupling conditions can be realized in a highly accurate fashion. Consequently, inside the fracture, a Stokes flow can be modelled that is coupled to the surrounding elastic medium. A fully coupled approach is obtained by iterating between the phase-field and the fluid–structure interaction model. The resulting algorithm is demonstrated for several numerical examples of quasi-static brittle fractures. We consider both stationary and quasi-stationary problems. In the latter, the dynamics arise through an incrementally increasing given pressure.

Phase-field simulation and coupled criterion link echelon cracks to internal length in antiplane shear

This paper provides a comprehensive numerical analysis of daughter crack localization in pure antiplane shear. Although antiplane shear fracture is important in various industrial applications, understanding the morphology of the resulting fragmentation remains challenging. The paper develops innovative phase-field models to induce the facets using a small spatial variation in the toughness field and examines the impact of numerical and material parameters on the newly formed daughter cracks’ shape and spacing. Through meticulous comparison to the coupled criterion, the paper reveals a compelling connection between the internal length-scale of damage regularization, Irwin’s length and the facet crack spacing. Furthermore, the effect of Poisson’s ratio on the crack form and spacing is investigated: the results reveal a significant influence and showcase comparable initiation distances between the numerical simulations and experimental measurements in pure antiplane loading.

A three-dimensional phase-field model for studying the orientation-dependent interface evolution in stress-induced martensitic phase transformation

In this study, we address the challenging task of predicting the motion of interfaces in stress-driven martensitic phase transformations within shape memory alloys. The novelty of our approach lies in the introduction of a monolithically solved thermodynamic-based phase-field method, specifically tailored for large strain conditions at the nanoscale for three-dimensional problems. To achieve this, we have developed a custom finite element software seamlessly integrated into the FEniCS open-source framework, providing a sophisticated and efficient tool for the examination of nanostructure evolution. Our investigation delves into five distinct and complex scenarios under diverse loading conditions for two– and three-dimensional problems, each presenting unique challenges: (i) a straightforward uniaxial tension scenario featuring a square domain with a pre-existing martensitic nucleus; (ii) the evolution of interfaces in a square sample incorporating a circular central nanovoid under biaxial tensile stress; (iii) the analysis of a rectangular beam subjected to horizontal and vertical compressive loads; (iv) the assessment of an initially voided rectangular beam experiencing mixed loading conditions; and (v) the three-dimensional simulations of cubic-to-tetragonal phase transformation using various orientation of the habit plane. In contrast to prior studies, our analysis not only explores the standard factors of habit plane reorientation and strain transformation values but also delves into the impact of nanovoids on austenite–martensite interface evolution.

Emergence of rapid solidification microstructure in additive manufacturing of a Magnesium alloy

Abstract Bioresorbable Mg-based alloys with low density, low elastic modulus, and excellent biocompatibility are outstanding candidates for temporary orthopedic implants. Coincidentally, metal additive manufacturing (AM) is disrupting the biomedical sector by providing fast access to patient-customized implants. Due to the high cooling rates associated with fusion-based AM techniques, they are often described as rapid solidification processes. However, conclusive observations or rapid solidification in metal AM — attested by drastic microstructural changes induced by solute trapping, kinetic undercooling, or morphological transitions of the solid-liquid interface — are scarce. Here we study the formation of banded microstructures during laser powder-bed fusion (LPBF) of a biomedical-grade Magnesium-rare earth alloy, combining advanced characterization and state-of-the-art thermal and phase-field modeling. Our experiments unambiguously identify microstructures as the result of an oscillatory banding instability known from other rapid solidification processes. Our simulations confirm that LPBF-relevant solidification conditions strongly promote the development of banded microstructures in a Mg-Nd alloy. Simulations also allow us to peer into the sub-micrometer nanosecond-scale details of the solid-liquid interface evolution giving rise to the distinctive banded patterns. Since rapidly solidified Mg alloys may exhibit significantly different mechanical and corrosion response compared to their cast counterparts, the ability to predict the emergence of rapid solidification microstructures (and to correlate them with local solidification conditions) may open new pathways for the design of bioresorbable orthopedic implants, not only fitted geometrically to each patient, but also optimized with locally-tuned mechanical and corrosion properties.

Evaluation and associated mechanism of mitigating Cr depletion by vanadium in modified 9Cr-1Mo steel: Experiment and phase-field modelling

The 9Cr-1Mo steel is extensively utilized in nuclear heat exchangers due to its exceptional corrosion resistance properties. The Chromium (Cr) element plays a pivotal role in enhancing the corrosion resistance. The Cr accumulates at grain boundaries (GBs) where defects and mismatches are located during long-term usage, leading to the depletion of Cr within the grain interior. This results in the Cr-rich carbide precipitates form along GBs, with a corresponding the depletion zone residing in the adjacent regions. In this study, we employed aberration-corrected high-resolution electron microscopy in high-angle annular dark field microscopy (HAADF) mode, equipped with energy dispersive spectroscopy (EDS), to analyse carbides formation along GBs and obtain elemental distribution information. Our investigation reveals that the Cr-rich carbide promotes the formation of chromium-depletion zone (CDZ) by consuming the surrounding Cr in the matrix, and the presence of vanadium-rich carbide inhibits the formation of (CDZ) from atomic scale.A phase-field modelling approach is employed to investigate the diffusive formation process in a polycrystalline system. Our modelling results are consistent with experimental observations. Cr preferentially diffuses towards grain boundaries (GBs) and accumulates, facilitating the formation of Cr-rich phases by consuming adjacent Cr in the surrounding region. The bilateral region of GBs serves as a source for supplying Cr to these newly formed phases while depleting Cr content, ultimately leading to the development of CDZ.By employing experimental techniques and phase-field method, this research provides valuable insights into the role of Cr, carbide precipitates, influence of vanadium, and CDZ formation.

Phase-field damage simulation of subloop loading in TiNi SMA

In practical applications, TiNi shape memory alloys (SMAs) exhibit behavior that can pose a challenge with current constitutive models and their implementations in finite element method (FEM) software. TiNi SMA devices typically operate in the forward or reverse martensitic transformation regime, which is known as subloop loading. During such cyclic loading–unloading, the hysteresis stress–strain loop changes because of material damage, which can be considered the fatigue of TiNi SMAs. During both the loading and unloading processes, the stress plateau decreases. At the same time, the accumulated (residual) martensitic transformation strain increases. In this study, the experimental investigation results and observations of the aforementioned phenomena are presented. Next, the phase-field damage model is employed, along with a modified Lagoudas constitutive model, to simulate the change in stress–strain hysteresis. Furthermore, a fatigue function is used to simulate the accumulation of martensitic transformation strain. The experimental stress–strain response is compared with the simulation results, and good quantitative and qualitative agreement is obtained. The damage and martensitic volume fraction with respect to strain are discussed for full-loop and subloop loading. The observations and conclusions, as well as open questions, are presented. Possible directions for future research are provided.

Fourth-order phase-field modeling for brittle fracture in piezoelectric materials

Fourth-order phase-field modeling for brittle fracture in piezoelectric materials

A phase-field model for simulating the propagation behavior of mixed-mode cracks during the hydraulic fracturing process in fractured reservoirs

A phase-field model for simulating the propagation behavior of mixed-mode cracks during the hydraulic fracturing process in fractured reservoirs

Phase field model for multi-material shape optimization of inextensible rods

We derive a model for the optimization of the bending and torsional rigidities of non-homogeneous elastic rods. This is achieved by studying a sharp interface shape optimization problem with perimeter penalization, that treats both rigidities as objectives. We then formulate a phase field approximation of the optimization problem and show the convergence to the aforementioned sharp interface model via $\Gamma$-convergence. In the final part of this work we numerically approximate minimizers of the phase field problem by using a steepest descent approach and relate the resulting optimal shapes to the development of the morphology of plant stems.