Phase-field modeling for anisotropic ductile damage of magnesium alloys at finite deformations

The damage anisotropy of an extruded ZK60 Mg alloy is characterized using tensile tests and scanning electronic microscopy.The accumulation of anisotropic deformations leads to the great differences of the dimple evolution and strains at fracture along different loading directions.To introduce the anisotropic deformation information into the damage constitutive relationship,a thermodynamically consistent phase-field model of ductile damage fully coupled with elastoplastic finite deformations is developed in this study.Using the user-defined constitutive relationship and displacement-temperature coupling element,the finite element simulations are conducted.The results show that:(1)ZK60 Mg alloys presents clear R-value difference in 0°,45°,and 90°tests of intact specimens.The 45°test possesses the greatest R-value(1.50)and the greatest strain at fracture,however,the R-value for 0°is less than 1,indicating the thinning is preferential.(2)The higher ultimate stress leads to a larger average dimension of the dimples,whereas the higher density correlates with a larger elongation ratio at the fracture.The disappearance of the stress-bearing area indicates that the phase-field assumption on stress degradation is completely compatible with the dimple analysis on fractography.(3)The simulation results of the stress-strain relationships and damage paths correlate well with the experimental ductile damage of magnesium alloys at 200◦C.Slight errors are basically attributed to the modeling parameters and finite element iteration algorithm.The proposed model presents fine applicability and reliability for the predictions of plastic deformations,ductile damage,and fracture of anisotropic Mg alloys.

Phase-field lattice-Boltzmann study on fully coupled thermal-solute-convection dendrite growth of Al-Cu alloy

Dendrite growth is a complex liquid-solid phase transition process involving multiple physical factors.A phase-field lattice-Boltzmann method was developed to simulate the two-and three-dimension dendrite growth of Al-Cu alloy.The effect of fully coupled thermal-solute-convection interaction on the dendrite growth was investigated by incorporating a parallel-adaptive mesh refinement algorithm into the numerical model.By accurately reproducing the latent heat release,solute diffusion and convective transport behaviors at the liquidsolid interface,the interaction mechanism among thermal-solute-convection transport as well as their coupling effects on the dendrite growth dynamics were discussed.The simulation results show that the release of latent heat slows down the dendrite growth rate,and both natural and forced convection disrupt the symmetrical growth of dendrites.Their combination makes the growth of dendrites more complex,capturing important physical aspects such as recalescence,dendrite tip splitting,dendrite tilting,dendrite remelting,and solute plume in the simulation case.Based on the robustness and powerful ability of the numerical model,the formation mechanisms of these physical aspects were revealed.

Comparative study on phase transition behaviors of fractional molecular field theory and random-site Ising model

Fractional molecular field theory(FMFT)is a phenomenological theory that describes phase transitions in crystals with randomly distributed components,such as the relaxor-ferroelectrics and spin glasses.In order to verify the feasibility of this theory,this paper fits it to the Monte Carlo simulations of specific heat and susceptibility versus temperature of two-dimensional(2D)random-site Ising model(2D-RSIM).The results indicate that the FMFT deviates from the 2D-RSIM significantly.The main reason for the deviation is that the 2D-RSIM is a typical system of component random distribution,where the real order parameter is spatially heterogeneous and has no symmetry of space translation,but the basic assumption of FMFT means that the parameter is spatially uniform and has symmetry of space translation.

Mechanisms of fracture propagation from multi-cluster using a phase field based HMD coupling model in fractured reservoir

Natural fractures(NFs)are common in shale and tight reservoirs,where staged multi-cluster fracturing of horizontal wells is a prevalent technique for reservoir stimulation.While NFs and stress interference are recognized as significant factors affecting hydraulic fracture(HF)propagation,the combined influence of these factors remains poorly understood.To address this knowledge gap,a novel coupled hydromechanical-damage(HMD)model based on the phase field method is developed to investigate the propagation of multi-cluster HFs in fractured reservoirs.The comprehensive energy functional and control functions are established,while incorporating dynamic fluid distribution between multiple perforation clusters and refined changes in rock mechanical parameters during hydraulic fracturing.The HMD coupled multi-cluster HF propagation model investigates various scenarios,including single HF and single NF,reservoir heterogeneity,single HF and NF clusters,and multi-cluster HFs with NF clusters.The results show that the HMD coupling model can accurately capture the impact of approach angle(θ),stress difference and cementation strength on the interaction of HF and NF.The criterion of the open and cross zones is not fixed.The NF angle(a)is not a decisive parameter to discriminate the interaction.According to the relationship between approach angle(θ)and NF angle(a),the contact relationship of HF can be divided into three categories(θ=a,θ<a,andθ>a).The connected NF can increase the complexity of HF by inducing it to form branch fracture,resulting in a fractal dimension of HF as high as2.1280 at angles of±45°.Inter-fracture interference from the heel to the toe of HF shows the phenomenon of no,strong and weak interference.Interestingly,under the influence of NFs,distant HFs from the injection can become dominant fractures.However,as a gradually increases,inter-fracture stress interference becomes the primary factor influencing HF propagation,gradually superseding the dominance of NF induced fractures.

Bifurcation Analysis Reveals Solution Structures of Phase Field Models

The phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems.Here,we develop a new analytical approach based on the bifurcation analysis to explore the mathematical solution structure of phase field models.Revealing such solution structures not only is of great mathematical interest but also may provide guidance to experimentally or computationally uncover new morphological evolution phenomena in materials undergoing electronic and structural phase transitions.To elucidate the idea,we apply this analytical approach to three representative phase field equations:the Allen-Cahn equation,the Cahn-Hilliard equation,and the Allen-Cahn-Ohta-Kawasaki system.The solution structures of these three phase field equations are also verified numerically by the homotopy continuation method.

Stress-corrosion coupled damage localization induced by secondary phases in bio-degradable Mg alloys:phase-field modeling

In this study,a phase-field scheme that rigorously obeys conservation laws and irreversible thermodynamics is developed for modeling stress-corrosion coupled damage(SCCD).The coupling constitutive relationships of the deformation,phase-field damage,mass transfer,and electrostatic field are derived from the entropy inequality.The SCCD localization induced by secondary phases in Mg is numerically simulated using the implicit iterative algorithm of the self-defined finite elements.The quantitative evaluation of the SCCD of a C-ring is in good agreement with the experimental results.To capture the damage localization,a micro-galvanic corrosion domain is defined,and the buffering effect on charge migration is explored.Three cases are investigated to reveal the effect of localization on corrosion acceleration and provide guidance for the design for resistance to SCCD at the crystal scale.

Simulation of Corrosion-Induced Cracking of Reinforced Concrete Based on Fracture Phase Field Method

Accurate simulation of the cracking process caused by rust expansion of reinforced concrete(RC)structures plays an intuitive role in revealing the corrosion-induced failure mechanism.Considering the quasi-brittle fracture of concrete,the fracture phase field driven by the compressive-shear term is constructed and added to the traditional brittle fracture phase field model.The rationality of the proposed model is verified by a mixed fracture example under a shear displacement load.Then,the extended fracture phase model is applied to simulate the corrosion-induced cracking process of RC.The cracking patterns caused by non-uniform corrosion expansion are discussed for RC specimens with homogeneous macroscopically or heterogeneous with different polygonal aggregate distributions at the mesoscopic scale.Then,the effects of the protective layer on the crack propagation trajectory and cracking resistance are investigated,illustrating that the cracking angle and cracking resistance increase with the increase of the protective layer thickness,consistent with the experimental observation.Finally,the corrosion-induced cracking process of concrete specimens with large and small spacing rebars is simulated,and the interaction of multiple corrosion cracking is easily influenced by the reinforcement spacing,which increases with the decrease of the steel bar interval.These conclusions play an important role in the design of engineering anti-corrosion measures.The fracture phase field model can provide strong support for the life assessment of RC structures.

Diagnosing ratio of electron density to collision frequency of plasma surrounding scaled model in a shock tube using low-frequency alternating magnetic field phase shift

A non-contact low-frequency(LF)method of diagnosing the plasma surrounding a scaled model in a shock tube is proposed.This method utilizes the phase shift occurring after the transmission of an LF alternating magnetic field through the plasma to directly measure the ratio of the plasma loop average electron density to collision frequency.An equivalent circuit model is used to analyze the relationship of the phase shift of the magnetic field component of LF electromagnetic waves with the plasma electron density and collision frequency.The applicable range of the LF method on a given plasma scale is analyzed.The upper diagnostic limit for the ratio of the electron density(unit:m^(-3))to collision frequency(unit:Hz)exceeds 1×10^(11),enabling an electron density to exceed 1×10^(20)m^(-3)and a collision frequency to be less than 1 GHz.In this work,the feasibility of using the LF phase shift to implement the plasma diagnosis is also assessed.Diagnosis experiments on shock tube equipment are conducted by using both the electrostatic probe method and LF method.By comparing the diagnostic results of the two methods,the inversion results are relatively consistent with each other,thereby preliminarily verifying the feasibility of the LF method.The ratio of the electron density to the collision frequency has a relatively uniform distribution during the plasma stabilization.The LF diagnostic path is a loop around the model,which is suitable for diagnosing the plasma that surrounds the model.Finally,the causes of diagnostic discrepancy between the two methods are analyzed.The proposed method provides a new avenue for diagnosing high-density enveloping plasma.

Phase field model for electric-thermal coupled discharge breakdown of polyimide nanocomposites under high frequency electrical stress

In contrast to conventional transformers, power electronic transformers, as an integral component of new energy power system, are often subjected to high-frequency and transient electrical stresses, leading to heightened concerns regarding insulation failures. Meanwhile, the underlying mechanism behind discharge breakdown failure and nanofiller enhancement under high-frequency electrical stress remains unclear. An electric-thermal coupled discharge breakdown phase field model was constructed to study the evolution of the breakdown path in polyimide nanocomposite insulation subjected to high-frequency stress. The investigation focused on analyzing the effect of various factors, including frequency, temperature, and nanofiller shape, on the breakdown path of Polyimide(PI) composites. Additionally, it elucidated the enhancement mechanism of nano-modified composite insulation at the mesoscopic scale. The results indicated that with increasing frequency and temperature, the discharge breakdown path demonstrates accelerated development, accompanied by a gradual dominance of Joule heat energy. This enhancement is attributed to the dispersed electric field distribution and the hindering effect of the nanosheets. The research findings offer a theoretical foundation and methodological framework to inform the optimal design and performance management of new insulating materials utilized in high-frequency power equipment.

Optimal parameter space for stabilizing the ferroelectric phase of Hf_(0.5)Zr_(0.5)O_(2) thin films under strain and electric fields

Hafnia-based ferroelectric materials, like Hf_(0.5)Zr_(0.5)O_(2)(HZO), have received tremendous attention owing to their potentials for building ultra-thin ferroelectric devices. The orthorhombic(O)-phase of HZO is ferroelectric but metastable in its bulk form under ambient conditions, which poses a considerable challenge to maintaining the operation performance of HZO-based ferroelectric devices. Here, we theoretically addressed this issue that provides parameter spaces for stabilizing the O-phase of HZO thin-films under various conditions. Three mechanisms were found to be capable of lowering the relative energy of the O-phase, namely, more significant surface-bulk portion of(111) surfaces, compressive c-axis strain,and positive electric fields. Considering these mechanisms, we plotted two ternary phase diagrams for HZO thin-films where the strain was applied along the in-plane uniaxial and biaxial, respectively. These diagrams indicate the O-phase could be stabilized by solely shrinking the film-thickness below 12.26 nm, ascribed to its lower surface energies. All these results shed considerable light on designing more robust and higher-performance ferroelectric devices.

Exploration of the coupled lattice Boltzmann model based on a multiphase field model:A study of the solid-liquid-gas interaction mechanism in the solidification process

A multiphase field model coupled with a lattice Boltzmann(PF-LBM)model is proposed to simulate the distribution mechanism of bubbles and solutes at the solid-liquid interface,the interaction between dendrites and bubbles,and the effects of different temperatures,anisotropic strengths and tilting angles on the solidified organization of the SCN-0.24wt.%butanedinitrile alloy during the solidification process.The model adopts a multiphase field model to simulate the growth of dendrites,calculates the growth motions of dendrites based on the interfacial solute equilibrium;and adopts a lattice Boltzmann model(LBM)based on the Shan-Chen multiphase flow to simulate the growth and motions of bubbles in the liquid phase,which includes the interaction between solid-liquid-gas phases.The simulation results show that during the directional growth of columnar dendrites,bubbles first precipitate out slowly at the very bottom of the dendrites,and then rise up due to the different solid-liquid densities and pressure differences.The bubbles will interact with the dendrite in the process of flow migration,such as extrusion,overflow,fusion and disappearance.In the case of wide gaps in the dendrite channels,bubbles will fuse to form larger irregular bubbles,and in the case of dense channels,bubbles will deform due to the extrusion of dendrites.In the simulated region,as the dendrites converge and diverge,the bubbles precipitate out of the dendrites by compression and diffusion,which also causes physical phenomena such as fusion and spillage of the bubbles.These results reveal the physical mechanisms of bubble nucleation,growth and kinematic evolution during solidification and interaction with dendrite growth.

Pharmacological intervention for chronic phase of spinal cord injury

Spinal cord injury is an intractable traumatic injury. The most common hurdles faced during spinal cord injury are failure of axonal regrowth and reconnection to target sites. These also tend to be the most challenging issues in spinal cord injury. As spinal cord injury progresses to the chronic phase, lost motor and sensory functions are not recovered. Several reasons may be attributed to the failure of recovery from chronic spinal cord injury. These include factors that inhibit axonal growth such as activated astrocytes, chondroitin sulfate proteoglycan, myelin-associated proteins, inflammatory microglia, and fibroblasts that accumulate at lesion sites. Skeletal muscle atrophy due to denervation is another chronic and detrimental spinal cord injury–specific condition. Although several intervention strategies based on multiple outlooks have been attempted for treating spinal cord injury, few approaches have been successful. To treat chronic spinal cord injury, neural cells or tissue substitutes may need to be supplied in the cavity area to enable possible axonal growth. Additionally, stimulating axonal growth activity by extrinsic factors is extremely important and essential for maintaining the remaining host neurons and transplanted neurons. This review focuses on pharmacotherapeutic approaches using small compounds and proteins to enable axonal growth in chronic spinal cord injury. This review presents some of these candidates that have shown promising outcomes in basic research(in vivo animal studies) and clinical trials: AA-NgR(310)ecto-Fc(AXER-204), fasudil, phosphatase and tensin homolog protein antagonist peptide 4, chondroitinase ABC, intracellular sigma peptide,(-)-epigallocatechin gallate, matrine, acteoside, pyrvate kinase M2, diosgenin, granulocyte-colony stimulating factor, and fampridine-sustained release. Although the current situation suggests that drug-based therapies to recover function in chronic spinal cord injury are limited, potential candidates have been identified through basic research, and these candidates may be subjects of clinical studies in the future. Moreover, cocktail therapy comprising drugs with varied underlying mechanisms may be effective in treating the refractory status of chronic spinal cord injury.

Low‑Temperature Oxidation Induced Phase Evolution with Gradient Magnetic Heterointerfaces for Superior Electromagnetic Wave Absorption

Gradient magnetic heterointerfaces have injected infinite vitality in optimizing impedance matching,adjusting dielectric/magnetic resonance and promoting electromagnetic(EM)wave absorption,but still exist a significant challenging in regulating local phase evolution.Herein,accordion-shaped Co/Co_(3)O_(4)@N-doped carbon nanosheets(Co/Co_(3)O_(4)@NC)with gradient magnetic heterointerfaces have been fabricated via the cooperative high-temperature carbonization and lowtemperature oxidation process.The results indicate that the surface epitaxial growth of crystal Co_(3)O_(4) domains on local Co nanoparticles realizes the adjustment of magnetic-heteroatomic components,which are beneficial for optimizing impedance matching and interfacial polarization.Moreover,gradient magnetic heterointerfaces simultaneously realize magnetic coupling,and long-range magnetic diffraction.Specifically,the synthesized Co/Co_(3)O_(4)@NC absorbents display the strong electromagnetic wave attenuation capability of−53.5 dB at a thickness of 3.0 mm with an effective absorption bandwidth of 5.36 GHz,both are superior to those of single magnetic domains embedded in carbon matrix.This design concept provides us an inspiration in optimizing interfacial polarization,regulating magnetic coupling and promoting electromagnetic wave absorption.

Ageing process of pre-precipitation phase in Ni_(0.75)Al_(0.05)Fe_(0.2) alloy based on phase field method

By utilizing phase field method combined with analysis on free energy and interatomic potentials, pre-precipitation phase formation and transformation process of Ni0.75Al0.05Fe0.2 alloy in early precipitation stage during the ageing process under 1 000 K were studied. And free energy, microstructures, compositions and volume fractions of pre-precipitation phase and equilibrium phase were analyzed. The simulation results indicate that nonstoichiometric Llo pre-precipitation phase formed first, and then would gradually transform into L12 equilibrium phase. It is discovered that the phase transformation process was closely related to free energy and interatomic potentials. Additionally, it is revealed that free energy of Llo pre-precipitation phase was higher and interatomic potential was smaller than that of L12 equilibrium phase. Therefore, it is concluded that Llo phase was unstable, and phase transformation would occur to L12 which was more stable.

Phase field model for strong interfacial energy anisotropy of HCP materials

Based on the Karma model and the Eggleston regularization technique of the strong interfacial energy anisotropy, a phase-field model was established for HCP materials. An explicit finite difference numerical method was used to solve phase field model and simulate the dendrite growth behaviors of HCP materials. Results indicate that the dendrite morphology presents obvious six-fold symmetry, and discontinuity in the variation of interface orientation occurs, resulting in a fact that the corners were formed at the tips of the main stem and side branches. When the interfacial energy anisotropy strength is lower than the critical value（1/35）, the steady-state tip velocity of dendrite increases with anisotropy as expected. As the anisotropy strength crosses the critical value, the steady-state tip velocity drops down by about 0.89%. With further increase in anisotropy strength, the steady-state tip velocity increases and reaches the maximum value at anisotropy strength of 0.04, then decreases.

Effect of forced lamina flow on microsegregation simulated by phase field method quantitatively

The influence of supercooled melt forced lamina flow on microsegregation was investigated. The concentration distribution at solid-liquid boundary of binary alloy Ni-Cu was simulated using phase field model coupled with flow field. The microsegregation, concentration maximum value, boundary thickness of concentration near upstream dendrite and normal to flow dendrite, and downstream dendrite were studied quantitatively in the case of forced lamia flow. The simulation results show that solute field and flow field interact complexly. Compared with melt without flow, in front of upstream dendrite tip, the concentration boundary thickness is the lowest and the concentration maximum value is the smallest for melt with flow. However, in front of downstream dendrite tip, the results are just the opposite. The zone of poor Cu in upstream dendrite where is the most severely microsegregation and shrinkage cavity is wider and the concentration is lower for melt with flow than that without flow.

Thermodynamic consistent phase field model for sintering process with multiphase powders

A thermodynamic consistent phase field model is developed to describe the sintering process with multiphase powders. In this model, the interface region is assumed to be a mixture of different phases with the same chemical potential, but with different compositions. The interface diffusion and boundary diffusion are also considered in the model. As an example, the model is applied to the sintering process with Fe-Cu powders. The free energy of each phase is described by the well-developed thermodynamic models, together with the published optimized parameters. The microstructure and solute distribution during the sintering process can both be obtained quantitively.

Phase field calculation of interface mobility in a ternary alloy

A novel approach based on the quantitative phase field model was proposed to calculate the interface mobility and applied to the α/β interface of a ternary Ti-6Al-4V alloy.Phase field simulations indicate that the higher interface mobility leads to the faster transformation rate,but only a unique value of interface mobility matches the diffusion equation under the diffusion-controlled condition.By comparing the transformation kinetics from phase field simulations with that from classical diffusion equation,the interface mobility at different temperatures can be obtained.The results show that the calculated interface mobility increases with increasing temperature and accords with Arrhenius equation very well.

Phase field method simulation of faceted dendrite growth with arbitrary symmetries

A numerical simulation based on a regularized phase field model is developed to describe faceted dendrite growth morphology. The effects of mesh grid, anisotropy, supersaturation and fold symmetry on dendrite growth morphology were investigated, respectively. These results indicate that the nucleus grows into a hexagonal symmetry faceted dendrite. When the mesh grid is above 640×640, the size has no much effect on the shape. With the increase in the anisotropy value, the tip velocities of faceted dendrite increase and reach a balance value, and then decrease gradually. With the increase in the supersaturation value, crystal evolves from circle to the developed faceted dendrite morphology. Based on the Wulff theory and faceted symmetry morphology diagram, the proposed model was proved to be effective, and it can be generalized to arbitrary crystal symmetries.

Electromagnetic holographic sensitivity field of two-phase flow in horizontal wells

Electromagnetic holographic data are characterized by two modes, suggesting that image reconstruction requires a dual-mode sensitivity field as well. We analyze an electromagnetic holographic field based on tomography theory and Radon inverse transform to derive the expression of the electromagnetic holographic sensitivity field （EMHSF）. Then, we apply the EMHSF calculated by using finite-element methods to flow simulations and holographic imaging. The results suggest that the EMHSF based on the partial derivative of radius of the complex electric potential φ is closely linked to the Radon inverse transform and encompasses the sensitivities of the amplitude and phase data. The flow images obtained with inversion using EMHSF better agree with the actual flow patterns. The EMHSF overcomes the limitations of traditional single-mode sensitivity fields.