Phase Field Research Articles

Orientation-dependent deformation and failure of micropillar shape memory ceramics: A 3D Phase-Field Study

Some microscopic samples of zirconia-based shape memory ceramics (SMCs) have shown full martensitic phase transformation (MPT) over multiple loading cycles without cracking. However, the occurrence of MPT is strongly influenced by grain orientation. Depending on the specific grain orientation relative to the loading direction, alternative mechanisms such as plastic slip and fracture may emerge. This study introduces a phase-field (PF) based framework that integrates a PF-MPT model, a PF fracture model, and a crystal viscoplasticity model to investigate the effects of grain orientation on MPT, plastic slip, and fracture mechanisms in SMC micropillars. Single crystal micropillars are created to distinguish the orientations that facilitate each mechanism. A wide range of grain orientations are found to predominantly exhibit MPT. Micropillars with grain orientations close to the (100) and (001) directions primarily experience fracture, with minimal plastic slip. Additionally, samples oriented along the (110) direction show a significant amount of plastic slip. A pole figure is constructed to elucidate the interplay between MPT, cracking, and slip under compressive loading conditions. This research provides valuable insights into the intricate behavior of SMCs under different loading scenarios, crucial for optimizing their performance in practical applications.

Self-folding of two-dimensional thin templates into pyramidal microstructures by a liquid drop: a numerical model

We present a numerical framework bringing together a structural and a surface-energy minimization model to compute nano- and microorigami self-folding processes, that are three-dimensional in nature. A liquid drop, initially at rest on the template, triggers the spontaneous folding and deforms dynamically with the template. As application, the self-folding of thin two-dimensional templates into pyramidal microstructures is simulated. Each template is composed of a fixed regular base connected to rigid triangular side panels by two elastic hinges. We presently determine the condition, at which the transition from partial to full drop encapsulation occurs. The present model makes use of phase fields to numerically represent the drop and the template. This allows to: (i) develop an efficient computational method, which incorporates time-derivative terms and (ii) study three-dimensional complex self-folding processes.

Integrated experimental and thermodynamic modelling study of phase equilibria in the “CuO0.5”-PbO-CaO system in equilibrium with Cu/Pb metal

Phase equilibria studies were undertaken on the “CuO0.5”-PbO-CaO system in equilibrium with Cu/Pb metal as part of the integrated experimental and thermodynamic modelling study of phase equilibria in the Cu-Pb-Zn-Fe-Ca-Si-Al-Mg-O-S-(As, Sn, Sb, Bi, Ag, Au, Ni, Cr, Co and Na) gas/oxide liquid/matte/speiss/metal/solids system in support of the development and optimisation of pyrometallurgical processes. The equilibration and quenching technique followed by the electron probe X-ray microanalysis (EPMA) was used in the present study. The primary phase fields of massicot (PbO), copper plumbite (Cu2PbO2), cuprite (Cu2O) and lime (CaO) were experimentally characterised. The experimental data obtained in the present study, as well as available literature data on the phase equilibria and thermodynamics of the “CuO0.5”-PbO-CaO system in equilibrium with Cu/Pb were used for the optimisation of the model parameters of the discussed system in agreement with the Cu-Pb-Zn-Fe-Ca-Si-Al-Mg-O-S-(As, Sn, Sb, Bi, Ag, Au, Ni, Cr, Co and Na) gas/oxide liquid/matte/speiss/metal/solids system to obtain a self-consistent set of parameters of the thermodynamic model for all phases. The predicted liquidus surface of the “CuO0.5”-PbO-CaO system was presented for the first time in the complete range of temperatures and compositions.

Unconditional Energy Stable Runge-Kutta Schemes for a Phase Field Model for Diblock Copolymers

Unconditional Energy Stable Runge-Kutta Schemes for a Phase Field Model for Diblock Copolymers

Far-field reactivation of natural fractures by stress shadow effect

Hydraulic fracturing is the most important means to achieve the reservoir stimulation of tight formations like shale, where the reservoir permeability is enhanced after the interaction between hydraulic fracture (HF) and natural fractures (NFs). Apart from the widely known direct intersection behaviors between HF and NFs (i.e., the HF meets the NFs), the NFs may also be reactivated by the stress shadow effect of HF, but the failure mechanism and reactivation modes are still unclear. To address this, the far-field reactivation of NFs under the stress shadow effect is described based on a hybrid phase field model (PFM) with the frictional contact criterion, where the open or slip conditions can be quantitatively calculated using the Mohr-Coulomb yield function along the contact interface of NFs. The zone with reactivated NFs is divided into mode II and mixed mode dominated sub-zones respectively, and the phase diagram of reactivation behaviors is summarized as variations of the location and angle of the NF. The NF reactivation modes are discussed under different geological conditions (strength of NFs, distribution angle of NFs, and initial stress difference). It could be found that the higher initial stress ratio leads to more mode II reactivated NFs, and NFs strengths and angles influence both mode II and mixed mode zones notably. The NF reactivation tends to take place with lower strength, higher injection rate, higher initial stress ratio, and special angle (e.g., HF are perpendicular to NFs), during which the permeability enhancement and reservoir stimulation are evaluated by the phase field model.

Modelling of fracture-involved large strain behaviors of amorphous glassy polymers via a unified physically-based constitutive model coupled with phase field method

To promote the application of amorphous glassy polymers in structural components, a reliable prediction of the deformation and the potential fracturing behaviors is in demand. This work aims to provide a simple and feasible computational method to analyze the large strain behaviors, including elasticity, viscoplasticity, and the subsequent fracture, of amorphous glassy polymers. This is achieved by incorporating a physically-based constitutive model coupled with the fracture phase field method into the commercial finite element software Abaqus/Explicit. Inside the constitutive model, shear-yielding, crazing, and disentangling are considered as the underlying mechanisms for viscoplastic deformation and damage initiation. It is noteworthy that a unified craze-initiation criterion with a clear physical meaning is proposed, distinguishing this work from the previous in the literature. Moreover, a relatively user-friendly numerical implementation is suggested by exploiting the built-in features of Abaqus/Explicit. Taking the typical amorphous glassy polymers for example, i.e., polycarbonate (PC) and poly-methyl-methacrylate (PMMA), various experiments from the literature have been simulated. The proposed approach has been validated, since an acceptable agreement between the simulated and experimental results is realized.

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

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

Fracture as an Emergent Phenomenon

The mechanics of fracture propagation provides essential knowledge for the risk tolerance design of devices, structures, and vehicles. Techniques of free energy minimization provide guidance, but have limited applicability to material systems evolving away from equilibrium. Experimental evidence shows that the material response depends on driving forces arising from mechanical fields. Recent years have witnessed the development of new methods for modeling complex dynamic and quasistatic fracture. New approaches may differ remarkably from previous ones, as they involve implicit coupling between damaged and undamaged states, allowing fracture to be modeled as emergent phenomena.The focus of this workshop is on the most advanced techniques for modeling fracture, represented by eigenerosion methods, variational approaches, phase field fracture models, and non-local approaches. Technical progress is contingent on the further development of the mathematical framework underlying these techniques. This is necessary for accurate and reliable computational modeling of fracture for multiple freely propagating cracks. The objective of this workshop is to mathematically identify and discuss open issues related to fracture modeling and to highlight recent advances. Addressing fundamental issues will foster exchange between the different communities, essential for advancing the field.

A microstructural thermo‐elastoplastic analysis of the mushy zone during laser beam welding

AbstractAs a modern method of contact free fusion, laser beam welding has gained importance in recent years. This is due to both the possibiliy of a high advance speed of the laser and low thermal distortion of the components compared to other welding processes, see Dal and Fabbro. These beneficial process properties follow from a focused laser, which thus results in a precise energy input. In addition, laser beam welding processes can be automated which makes the process attractive for a wide range of applications, cf. Dilthey. Nevertheless, various influences, like the temperature gradient, the chemical composition or process parameters, may have an effect on the occurrence of so‐called solidification cracks. Solidification cracks arise during solidification and occur in the area of the mushy zone, which is located behind the melt pool. The material behavior in the mushy zone is governed by a dendritic morphology. Due to this, liquid phase inclusions may occur in some parts of the mushy zone, see Hills et al.. When these trapped areas solidify, no further liquid phase can follow and a negative pressure is created. The resulting stresses can then lead to solidification cracks. In further studies, the geometry of the dendritic morphology will be generated on the basis of previous phase field simulations. These describe the dendrite growth controlled by the chemical concentration. The resulting geometry will then be simulated using the possibilities of a thermo‐elastoplastic material model, which will enable temperature‐driven imitation of growth. This contribution addresses the evolving state of the dendritic microstructure in a simplified manner. Therefore, for the first study, a simplified morphology is used, that is, no secondary arms and an elliptical geometry of the dendrites is assumed. Thermal as well as elastoplastic effects are already taken into account, in order to represent the inherent stress and strain states in the microstructure. These may be related to critical states leading to failure in the future.

Mechanics of Materials: Multiscale Design of Advanced Materials and Structures

Materials can now be designed and architectured like structural components for targeted mechanical and physical properties. Structures and microstructures should not be studied independently and their design will benefit from a multiscale approach combining nonlinear continuum mechanics approaches and physical descriptions of elasticity, viscoplasticity, phase transformations and damage of microstructures, at various scales. The aim of the workshop was to gather outstanding junior and senior researchers in the various branches of mathematics, physics and engineering sciences suited to address the question of design of materials and structures by means of multiscale discrete and continuum approaches to their constitutive behavior. Examples include atomic or macroscopic lattices, random or periodic cellular materials, smart materials like shape memory alloys, 3D woven composites, acoustic and electromagnetic metamaterials, etc. Modern continuum mechanics relies on sophisticated constitutive laws for anisotropic materials exhibiting elastoviscoplastic behavior, still a field of intense research with new mathematical concepts. In particular size-dependent properties are addressed by resorting to generalized continua such as gradient or micromorphic and phase field models. The latter are attractive for the simulation of microstructure evolution coupled with mechanics, due to thermodynamic and metallurgical processes and damage. Scale transition and homogenization methods for continuous and discrete systems are required for the determination of effective material and structural behavior. Metamaterials are architectured materials specifically designed to achieve certain propagation and dispersion properties of elastic and plastic waves. Optimization strategies for the design of optimal architectures are involved in the design process. Target functions for optimization are now based on multicriteria (stiffness, strength, thermal expansion, transport properties, anisotropy etc.).

Stress-strain state of dental immediate prosthesis and associated tissue under the influence of external force

Purpose: to analyze the stress-strain state (SSS) of a removable plate immediate denture with a printed dentition made of polymethyl methacrylate (PMMA) and a polyethylene terephthalate (PET) base under multifactor loading conditions based on the principles of mathematical modeling. Materials and methods: For mathematical modeling of SSS, Comsol Multiphysics 5.6 software (Comsol Inc., USA) was used, which provided numerical data for analytical interpretation. We considered a flat 2D model consisting of a printed monolithic dentition and a PMMA bonding layer, a PET base, the mucous membrane of the prosthetic field and jaw bone tissue. Results: The fields of von Mises stresses (VM) and deformations under the influence of normal, oblique and tangential loads (up to 500 N), the localization of their maximum values and the average value inside the prosthetic structure, mucous membrane and bone tissue were determined. The maximum values of VM in the mucous membrane are compared with the pain threshold based on known literature data on algesiometry of the oral mucosa. An assessment was made of the critical length of microcracks (CLM) according to Griffiths, which lead to the formation of a main crack followed by brittle fracture of the structure base. Using the phase field method, the probability of crack formation in the considered model of an orthopedic structure under the influence of static loads of various magnitudes and directions was calculated. Conclusions: Using the method of mathematical modeling of SSS, it was shown that failures when using immediate dentures with printed teeth made of PMMA and a PET base can be caused by the effect of overloading the areas of the base and caused by the deformation process inside the prosthetic structure. The maximum stress in the prosthetic elements was 79.7 MPa and deformation – 0.47 mm. The pain threshold of the oral mucosa is higher than the highest stress value in the prosthetic bed according to Mises (0.002 MPa) for normal load of 500 N. The SSS heterogeneity coefficient of the removable plate immediate prosthesis model is 8.6. The critical length of a microcrack according to Griffiths is lcrit = 1.9...2.7 µm. Using the Comsol Multiphysics phase field method, it was established that the probability of the formation of a large crack in the base of the prosthesis was 6·10–3 %, 0.36 %,0.73 % for a normal, inclined and shear load of 500 N, which is a sufficient value for the operation of removable laminar dentures with printed teeth from PMMA and a PET base during the warranty period.

CFD analysis of pore morphology, gravity, and fluid characteristics influences on water flooding process

In this research, computational fluid dynamics (CFD) was employed to examine the two-phase flow of water and oil in a porous medium. For this purpose, the Navier-Stockes equations, which describe fluid motion, and the Cahn-Hillard phase field, which defines the interface between two phases, are coupled. The numerical discretization system of equations was solved using the finite element method (FEM) with the COMSOL software. To validate the results and the phase-field model (PFM), the Lucas and Washburn equation was used together with the experimental data. Through this study, the effective parameters were evaluated in two parts. In the first part, seven models with different pore morphologies were designed, and the impact of petrophysical parameters of the reservoir, including the shape of pores, connectivity of pores with or without throat lines, porous media heterogeneity, and relative permeability, on oil recovery was investigated. The second part was devoted to performing sensitivity analysis on the effect of fluid properties, including interfacial tension, wettability, viscosity ratio, injected fluid flow, and gravity, upon enhanced oil recovery (EOR) in different morphologies. Owing to the uniform distribution of capillary pressure in the patterns with the throat lines, the sweep efficiency of the injected fluid was found to be better, and thereby oil production increased. The results of the present work proved the significant influence of gravity on EOR, so that by applying gravity to the solution domain, the breakthrough time and oil recovery factor increased by 20 minutes and 28.6 %, respectively. Moreover, the velocity of the injected fluid as a representative of the flow rate had the greatest effect on EOR, with a 35 % increase in production.

A novel semi-explicit numerical algorithm for efficient 3D phase field modelling of quasi-brittle fracture

Phase field models have become an effective tool for predicting complex crack configurations including initiation, propagation, branching, intersecting and merging. However, several computational issues have hindered their utilisation in engineering practice, such as the convergence challenge in implicit algorithms, numerical stability issues in explicit methods and significant computational costs. Aiming to providing a more efficient numerical algorithm, this work integrates the explicit integral operator with the recently developed neighbored element method, for the first time, to solve the coupled governing equations in phase field models. In addition, the damage irreversibility can be ensured automatically, avoiding the need to introduce extra history variable for the maximum driving force in traditional algorithms. Six representative fracture benchmarks with different failure modes are simulated to verify the effectiveness of the proposed method, including the multiple cracks in heterogeneous concrete at mesoscale. It is found that this semi-explicit numerical algorithm yields consistent crack profiles and load capacities for all examples to the available experimental data and literature. In particular, the computational cost is significantly reduced when compared to the traditional explicit modelling. Therefore, the presented numerical algorithm is highly attractive and promising for phase-field simulations of complicated 3D solid fractures in structural-level engineering practices.

Integrating Phase Field Modeling and Machine Learning to Develop Process-Microstructure Relationships in Laser Powder Bed Fusion of IN718

Integrating Phase Field Modeling and Machine Learning to Develop Process-Microstructure Relationships in Laser Powder Bed Fusion of IN718

Crack patterns in masonry structures using phase field method

This work aims to study the crack pattern in masonry-like structures by a simplified description of the mediumusing a variational damage model. In literature, when modeling fractures in masonry structures, capturingboth fracture types, cracks in units, and cracks at the interface simultaneously, usually requires a complicated scheme. The phase-field method has been widely used recently because of its robustness in capturing variousfracture types. In this work, we consider the mortar and the interface (between the mortar and units) as a thickinterface layer with pseudo-material properties. We conduct numerical tests on a bending beam and a tensilewall using strategies of material properties with a phase field model. As it is not distinguished the crack in themortar and the brick-mortar interface, it is shown in this work that both types of fracture, in the thick interfacelayer and the unit, can be captured just using the phase field method and the simplified micro model.

Phase Field Study of Discontinuous Precipitation in a Miscibility Gap System

AbstractA phase field model is developed to study the discontinuous (cellular) precipitation (DP) reaction in a hypothetical A-B miscibility gap system. Unlike previous treatments, the model employs an interaction energy term which couples the composition and the phase field variable to enable the necessary grain boundary movement. The influence of factors such as interaction energy, interfacial mobility and grain boundary diffusivity on the transformation rate and overall microstructure evolution are discussed.

Oil-in-water emulsion stabilized by hydrolysed black soldier fly larvae proteins: Reproduction of experimental data via phase-field modelling

A phase field model based on the Cahn-Hilliard equation was validated experimentally by comparison of the numerical data with experimental data of emulsions of oil in water stabilized with Black Soldier Fly Larvae Proteins (BSFLP) and protein hydrolysates. Among the model parameters, the surface tension was determined by the pendant drop method, and the mobility by two different experiments, one based on the Turbiscan Stability Index, and another one based on the history of the average d3,2, both of them throughout a 48 h period. The initial condition was built from an experimental droplet size distribution measured prior to phase separation. Results show that the model is able to quantitatively reproduce the phase separation kinetics, and provide an intuitive graphical representation of the droplet growth. Application of this procedure to other systems will allow the generalization of phase field modelling as a predictive tool for food applications.

Accelerated calculation of phase-variable for numerical simulation of multiphase flows

In this manuscript, we unveil an innovative acceleration technique for the simulation of multiphase flows, which builds upon the foundation laid by our previously devised multiphase flow solver. The cornerstone of this method lies in augmenting computational efficiency through the selective updating of variables solely within domains characterized by significant gradients of the phase variable. This tactic diminishes the dimensionality of the system of linear equations, thereby hastening the computational process. To pinpoint the regions warranting focused attention, a judicious criterion is indispensable to strike an optimal balance between efficiency and precision. This criterion affords a marked decrement in computational expenditure while preserving the fidelity of the original methodology. Rigorous validations corroborate that this acceleration mechanism can enhance computational efficiency by a minimum of 49.5% in resolving the phase field equation and by at least 70.2% in the computation of pragmatic two-phase flows, without compromising accuracy. Moreover, the efficacy of this acceleration technique is inversely proportional to the rate of interface evolution, becoming increasingly efficient as the interface evolves more slowly.

A phase field crystal model for real-time grain boundary formation and motion in complex concentration alloy

The complex concentration alloys exhibit the excellent mechanical property due to the dual roles of their heterogeneous compositions and microstructures. However, the formation and motion of grain boundaries to significantly regulate the mechanical properties remain unknown at several microsecond and nanoscale in the complex concentration alloys. Here, we utilize a phase field crystal model to investigate the real-time grain boundary formation and motion in complex concentration alloys, specifically, in comparison to traditional alloys, using the example of AlCu alloys. Our findings reveal that the low concentration alloys exhibit segregation primarily at grain boundaries with minimal compositional fluctuations over a long period of grain growth, and the complex concentration alloys display a high concentration gradient within the grains to cause rapid grain rotation and merging in a short time frame. In the complex concentration alloys, the large component fluctuations not only cause the local lattice distortion to hinder dislocation movement during the deformation process, but also lead to the occurrence from dislocation locking to self-unlocking, which is beneficial to improve the strength and ductility. The present work provides a real-time atomic-scale understanding of grain boundary evolution using in situ atomic-resolution simulations.

Homotopy trust-region method for phase-field approximations in perimeter-regularized binary optimal control

We consider optimal control problems that have binary-valued control input functions and a perimeter regularization. We develop and analyze a trust-region algorithm that solves a sequence of subproblems in which the regularization term and the binarity constraint are relaxed by a non-convex energy functional. We show how the parameter that controls the distinctiveness of the resulting phase field can be coupled to the trust-region radius updates and be driven to zero over the course of the iterations in order to obtain convergence to points that satisfy a first-order optimality condition of the limit problem under suitable regularity assumptions. Finally, we highlight and discuss the assumptions and restrictions of our approach and provide the first computational results for a motivating application in the field of control of acoustic waves in dissipative media.