Two-phase flow analysis based on a phase-field model using orthogonal basis bubble function finite element method

In this study, a finite element method based on a phase-field model for gas–liquid two-phase flow is proposed. MINI element based on a bubble function element stabilisation method is employed for the incompressible Navier–Stokes equations. The Cahn–Hilliard equation is employed to estimate the interface of gas and liquid. The orthogonal basis bubble function element is used to solve the Cahn–Hilliard equation. In particular, a detailed explanation for solving the Cahn–Hilliard equation based on a finite element method is given.     

Effect of elastic strain energy on grain growth and texture in AZ31 magnesium alloy by phase-field simulation

A phase-field model is modified to investigate the grain growth and texture evolution in AZ31 magnesium alloy during stressing at elevated temperatures. The order parameters are defined to represent a physical variable of grain orientation in terms of three angles in spatial coordinates so that the grain volume of different order parameters can be used to indicate the texture of the alloy. The stiffness tensors for different grains are different because of elastic anisotropy of the magnesium lattice. The tensor is defined by transforming the standard stiffness tensor according to the angle between the (0001) plane of a grain and the direction of applied stress. Therefore, different grains contribute to different amounts of work under applied stress. The simulation results are well-explained by using the limited experimental data available, and the texture results are in good agreement with the experimental observations. The simulation results reveal that the applied stress strongly influences AZ31 alloy grain growth and that the grain-growth rate increases with the applied stress increasing, particularly when the stress is less than 400 MPa. A parameter (△d) is introduced to characterize the degree of grain-size variation due to abnormal grain growth; the △d increases with applied stress increasing and becomes considerably large only when the stress is greater than 800 MPa. Moreover, the applied stress also results in an intensive texture of the 〈0001〉 axis parallel to the direction of compressive stress in AZ31 alloy after growing at elevated temperatures, only when the applied stress is greater than 500 MPa.     

二元合金多晶粒的枝晶生长的等温相场模型

基于Ginzburg-Landau理论和单晶粒的枝晶生长模型,发展了一个单相场控制的多个晶粒的枝晶生长模型. 采用相场和溶质场耦合的方法,以Al-2%Cu合金为例模拟了二元合金等温凝固过程中多个晶粒的生长过程. 结果表明,这个模型的计算结果展现了多个晶粒枝晶的竞争生长,能较真实的再现凝固过程中的枝晶的生长过程. 关键词： 相场法 多晶粒 等温凝固 二元合金     

Phase-field simulation of the coupled evolutions of grain and twin boundaries in nanotwinned polycrystals

Nanotwinned polycrystals exhibit an excellent strength-ductility combination due to nanoscale twins and grains. However, nanotwin-assisted grain coarsening under mechanical loading reported in recent experiments may result in strength drop based on the Hall-Petch law. In this paper, a phase-field model is developed to investigate the effect of coupled evolutions of twin and grain boundaries on nanotwin-assisted grain growth. The simulation result demonstrates that there are three pathways for coupled motions of twin and grain boundaries in a bicrystal under the applied loading, dependent on the amplitude of applied loading and misorientation of the bicrystal. It reveals that a large misorientation angle and a large applied stress promote the twinning-driven grain boundary migration. The resultant twin-assisted grain coarsening is confirmed in the simulations for the microstructural evolutions in twinned and un-twinned polycrystals under a high applied stress.     

各向异性相态场模型的Gibbs—Thompson关系

本文利用奇异摄动的内、外解匹配方法,分析了各向异性时在相态场模型的边界层上表面张菌、法向速度、平均曲率和各向异性函数的影响,得到了各向异性时的Ｇｉｂｂｓ－Ｔｈｏｍｐｓｏｎ关系,以及边界层所满足的方程。     

Discontinuous Galerkin finite element method applied to the coupled unsteady Stokes/Cahn-Hilliard equations

Two-phase flows driven by the interfacial dynamics are studied by tracking implicitly interfaces in the framework of the Cahn-Hilliard theory. The fluid dynamics is described by the Stokes equations with an additional source term in the momentum equation taking into account the capillary forces. A discontinuous Galerkin finite element method is used to solve the coupled Stokes/Cahn-Hilliard equations. The Cahn-Hilliard equation is treated as a system of two coupled equations corresponding to the advection-diffusion equation for the phase field and a nonlinear elliptic equation for the chemical potential. First, the variational formulation of the Cahn-Hilliard equation is presented. A numerical test is achieved showing the optimal order in error bounds. Second, the variational formulation in discontinuous Galerkin finite element approach of the Stokes equations is recalled, in which the same space of approximation is used for the velocity and the pressure with an adequate stabilization technique. The rates of convergence in space and time are evaluated leading to an optimal order in error bounds in space and a second order in time with a backward differentiation formula at the second order. Numerical tests devoted to two-phase flows are provided on ellipsoidal droplet retraction, on the capillary rising of a liquid in a tube, and on the wetting drop over a horizontal solid wall.     

Phase-field study of spinodal decomposition under effect of grain boundary

Grain boundary directed spinodal decomposition has a substantial effect on the microstructure evolution and properties of polycrystalline alloys. The morphological selection mechanism of spinodal decomposition at grain boundaries is a major challenge to reveal, and remains elusive so far. In this work, the effect of grain boundaries on spinodal decomposition is investigated by using the phase-field model. The simulation results indicate that the spinodal morphology at the grain boundary is anisotropic bicontinuous microstructures different from the isotropic continuous microstructures of spinodal decomposition in the bulk phase. Moreover, at grain boundaries with higher energy, the decomposed phases are alternating α/β layers that are parallel to the grain boundary. On the contrary, alternating α/β layers are perpendicular to the grain boundary.     

Maximal attractors for the phase-field equations of penrose-fife type

In this paper, the dynamics for the phase-field equations of Penrose-Fife type arising from the study of phase transitions is investigated. One of important features of this problem is that the metric space H we work with is incomplete.     

Investigation of polyamide 12 isothermal crystallization through the application of the phase-field model

Crystallization of polyamide 12 (PA12) is an essential process requiring thorough investigation for evaluating the mechanical properties after the polymer parts are manufactured. The change in crystallization temperature results in different crystallization behaviors for PA12. Hence, the crystal morphology of PA12 achieved provides important information about crystallization behavior, especially for those produced through additive manufacturing due to its heterogenous cooling rate in a single print bed. Considering the need of investigating PA12 crystallization using phase-field modeling, this paper aims to simulate the spherulite morphology of PA12 undergoing isothermal crystallization. This model is compared with the spherulite morphologies obtained from the optical microscopy test. The model shows that PA12 spherulites have thicker dendrites when the isothermal temperature is higher. The present phase-field model can determine the spherulite morphologies of bulk printed PA12 based on the crystallization condition and be used to evaluate the properties of the printed part.     

Elasto-plastic phase-field simulation of martensitic transformation in lath martensite steels

Phase-field simulation of lath martensite formation in Fe–0.1C mass% steel was carried out based on the two types of slip deformation (TTSD) model, which is recently developed as a result of analytical solution for the martensitic transformation being composed of Bain deformation and plastic deformation but without lattice rotation matrix. The simulation results reveal that the plastic deformation along the two types of slip system is complementary. The simulation result of the relationship between the two types of slip deformation is consistent with the analytical result calculated by TTSD model, indicating the validity of TTSD model for explaining the formation of lath martensite.