A bidimensional phase-field model with convection for change phase of an alloy

The article analyzes a two-dimensional phase-field model for a non-stationary process of solidification of a binary alloy with thermal properties. The model allows the occurrence of fluid flow in non-solid regions, which are a priori unknown, and is thus associated to a free boundary value problem for a highly non-linear system of partial differential equations. These equations are the phase-field equation, the heat equation, the concentration equation and a modified Navier-Stokes equations obtained by the addition of a penalization term of Carman-Kozeny type which accounts for the mushy effects. A proof of existence of weak solutions for such system is given. The problem is firstly approximated and a sequence of approximate solutions is obtained by Leray-Schauder fixed point theorem. A solution is then found by using compactness argument.     

Hysteresis operators in phase-field models of Penrose-fife type

Phase-field systems as mathematical models for phase transitions have drawn a considerable attention in recent years. However, while they are suitable for capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming of existing phase-field theories, the authors have recently proposed a new approach to phase-field models which is based on the mathematical theory of hysteresis operators developed in the past fifteen years. Well-posedness and thermodynamic consistency were proved for a phase-field system with hysteresis which is closely related to the model advanced by Caginalp in a series of papers. In this note the more difficult case of a phase-field system of Penrose-Fife type with hysteresis is investigated. Under slightly more restrictive assumptions than in the Caginalp case it is shown that the system is well-posed and thermodynamically consistent.     

The phase-field model with an auto-calibrated degradation function based on general softening laws for cohesive fracture

Phase-field models have become popular to simulate cohesive failure problems because of their capability of predicting crack initiation and propagation without additional criteria. In this paper, a new phase-field damage model coupled with general softening laws for cohesive fracture is proposed based on the unified phase-field theory. The commonly used quadratic geometric function in the classical phase-field model is implemented in the proposed model. The modified degradation function related to the failure strength and length scale is used to obtain the length scale insensitive model. Based on the analytical solution of a 1-D case, general softening laws in cohesive zone models can be considered. Parameters in the degradation function can be calibrated according to different softening curves and material properties. Numerical examples show that the results obtained by the proposed model have a good agreement with experimental results and the length scale has a negligible influence on the load-displacement curves in most cases, which cannot be observed in classical phase-field model.     

Error Bounds for Uniform Asymptotic Expansions—Modified Bessel Function of Purely Imaginary Order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals.By using a class of rational functions,they express these quantities in terms of Cauchy-type integrals;these expressions are natural generalizations of integral representations of the coe？cients and the remainders in the Taylor expansions of analytic functions.By using the new representation,a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.     

Weak solutions to a model with elasticity energy for electrochemically induced phase transitions in Lithium-Ion batteries

Using olivine LiFePO₄ as a model system, we study the existence of global solutions to a phase-field model with elasticity energy for Lithium-Ion batteries, which consists of a linear elasticity sub-system and nonlinear evolution equations for the order parameter and the lithium concentration. This model can be described the evolving microstructure for electrochemically induced phase transitions in electrochemical storage. Our numerical experiments are carried out to simulate the evolutions of lithium concentration and of phase interfaces for the model.     

Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions

In this paper a nonlocal phase-field model for non-isothermal phase transitions with a non-conserved order parameter is studied. The paper extends recent investigations to the non-isothermal situation, complementing results obtained by H. Gajewski for the non-isothermal case for conserved order parameters in phase separation phenomena. The resulting field equations studied in this paper form a system of integro-partial differential equations which are highly nonlinearly coupled. For this system, results concerning global existence, uniqueness and large-time asymptotic behaviour are derived. The main results are proved using techniques that have been recently developed by P. Krej?í and the authors for phase-field systems involving hysteresis operators.     

First-Order Phase Transitions and Hysteresis

A simple phase-field model for first-order phase transitions with hysteresis is proposed. It describes both temperature- and stress-induced transitions between austenitic and (oriented) martensitic regimes in a shape memory alloy (SMA). Finally, numerical simulations of local paths of the system are performed in the (ε,σ) and (ε,θ) planes, respectively, when either stress or temperature cyclic processes are considered and phase diffusion is neglected.     

Modeling dendritic solidification of Al–3%Cu using cellular automaton and phase-field methods

We compared a cellular automaton (CA)–finite element (FE) model and a phase-field (PF)–FE model to simulate equiaxed dendritic growth during the solidification of cubic crystals. The equations of mass and heat transports were solved in the CA–FE model to calculate the temperature field, solute concentration, and the dendritic growth morphology. In the PF–FE model, a PF variable was used to identify solid and liquid phases and another PF variable was considered to determine the evolution of solute concentration. Application to Al–3.0 wt.% Cu alloy illustrates the capability of both CA–FE and PF–FE models in modeling multiple arbitrarily-oriented dendrites in growth of cubic crystals. Simulation results from both models showed quantitatively good agreement with the analytical model developed by Lipton–Glicksman–Kurz (LGK) in the tip growth velocity and the tip equilibrium liquid concentration at a given melt undercooling. The dendrite morphology and computational time obtained from the CA–FE model are compared to those of the PF–FE model and the distinct advantages of both methods are discussed.     

On dynamic finite element analysis of viscoplastic thin-walled structures with non-local damage: A phase-field approach

[EN] In this work, a nonlocal damage model is proposed for dynamic analysis of viscoplastic shell structures using the phase-field approach. A phase-field variable on the mid surface is introduced to characterize the nonlocal damage as well as the transition between undamaged and damaged phase. The total free energy in [1] is modified as a sum of Helmholtz free-energy and Ginzburg-Landau one. The latter is defined as a function of the phase-field variable and its corresponding gradient. This enhancement gives rise to an introduction of gradient parameters in terms of a substructure-related intrinsic length-scale. The evolution of the phase-field based damage variable can be found from the minimum principle of the dissipation potential [3]. The performance of the proposed model is demonstrated through numerical results of a plate with a circular hole. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermal Shock Crack Propagation of Alumina Simulated With the Phase-Field Method Under Temperature-Dependent Damage Criteria北大核心CSCD

氧化铝陶瓷材料的力学性能受温度影响显著,因此使用相场法模拟热冲击裂纹的扩展时有必要考虑损伤判据的温度相关性.在现有热力学相场模型的基础上通过引入温度相关性损伤判据,修正了相场模型的控制方程.利用该模型对氧化铝陶瓷热冲击实验进行有限元模拟,并将模拟结果与氧化铝热冲击实验结果和不考虑温度相关性损伤判据的有限元模拟结果进行对比.结果表明,通过引入温度相关性损伤判据,可实现对热冲击裂纹的萌生和扩展过程更合理的模拟.     

Rupture in soft biological tissues modeled by a phase-field method

We present an application of the phase-field method of fracture to the simulation of artery rupture at large strains. To achieve this, the crack driving force function associated with the evolution of the crack phase-field is modified to account for the inherent anisotropy of the soft biological tissues. The phase-field methods present a promising and innovative approach to the thermodynamically consistent modeling of fracture. A key advantage lies in the prediction of the complex crack topologies where the cohesive zone approaches to fracture are known to suffer. A regularized crack surface functional is introduced that Γ-converges to a sharp crack topology for vanishing length scale parameter. The evaluation of the phase-field follows the minimization of this crack surface functional. The phase-field variable can be treated as a geometric quantity whose evolution is coupled to the anisotropic bulk response in a modular format in terms of a crack driving state function. A stress-based anisotropic failure criterion is introduced whose maximum value from the deformation history drives the irreversible crack phase-field. The formulation is verified by the finite element based simulation of a real arterial cross-section undergoing rupture in a two-dimensional setting when subjected to inflation pressure. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite volume methods for degenerate chemotaxis model

A finite volume method for solving the degenerate chemotaxis model is presented, along with numerical examples. This model consists of a degenerate parabolic convection-diffusion PDE for the density of the cell-population coupled to a parabolic PDE for the chemoattractant concentration. It is shown that discrete solutions exist, and the scheme converges.     

Continuum Thermodynamics and Phase-Field Models

Phase transitions between two phases are modelled as space regions where a phase-field changes smoothly. The two phases are separated by a thin transition layer, the so-called diffuse interface. All thermodynamic quantities are allowed to vary inside this layer, including the pressure and the mass density. A thermodynamic approach is developed by allowing for the nonlocal character of the continuum. It is based on an extra entropy flux which is proved to be non vanishing inside the transition layer, only. The phase-field is regarded as an internal variable and the kinetic or evolution equation is viewed as a constitutive equation of rate type. Necessary and sufficient restrictions placed by thermodynamics are derived for the constitutive equations and, furthermore, a general form of the evolution equation for the phase-field is obtained within the schemes of both a non-conserved and a conserved phase-field. Based on the thermodynamic restrictions, a phase-field model for the ice-water transition is established which allows for superheating and undercooling. A model ruling the liquid-vapor phase transition is also provided which accounts for both temperature and pressure variations during the evaporation process. The explicit expression of the Gibbs free enthalpy, the Clausius-Clapeyron formula and the customary form of the vapor pressure curve are recovered.     

Analysis of weak solutions for the phase-field model for lithium-ion batteries

We study a phase-field model for lithium-ion batteries of olivine LiFePO₄. During electrochemical cycling the fundamental behavior of the crystal is the diffusion of Li which controls the movement of the phase boundary without changing the olivine topology. This model with diffusive phase interfaces consists of two nonlinear parabolic equations of second order. We first prove the existence of global solutions to an initial-boundary value problem of this model. Numerical experiments of the model are then performed to simulate the evolution of lithium concentration and of phase interfaces.     

Well-posedness of a stochastic phase-field problem with multiplicative noises

In this paper, we prove the existence and uniqueness of the solution of a stochastic phase-field problem with multiplicative noises. Phase-field models are typically used to describe melting and solidification processes. We consider here the case of multiplicative noises induced by a Q-Brownian motion.     

Interpolation error estimates of a modified 8-node serendipity finite element

Summary. Interpolation error estimates for a modified 8-node serendipity finite element are derived in both regular and degenerate cases, the latter of which includes the case when the element is of triangular shape. For defined over a quadrilateral K, the error for the interpolant is estimated as , where in the regular case and in the degenerate case, respectively. Thus, the obtained error estimate in the degenerate case is of the same quality as in the regular case at least for . Results for some related elements are also given. Received June 2, 1997 / Published online March 16, 2000     

Quasilinear systems with memory kernels

We consider a quasilinear integrodifferential system in non-normal form. Such a system is a generalization of a phase-field model with memory and includes, as a particular case, the system describing the combustion of a material with memory. In this paper, we study both the direct and the inverse problems. Our fundamental tools are: the theory of analytic semigroups, optimal regularity results and fixed point arguments.     

On a system of nonlinear PDE's for phase transitions with vector order parameter and convective effect

The paper deals with a system of nonlinear PDE's which describes a phase-field model with convection and temperature dependent constraint to the vector order parameter. Existence of solutions for the system under consideration is proved by the method of Yosida approximation and fixed point arguments.     

From phase field to sharp cracks: Convergence of quasi-static evolutions in a special setting

We consider the quasi-static evolution of a straight crack within the recently developed phase-field approach and the classical sharp crack approach, and we show a strong correlation between the outcomes from the two approaches: the corresponding energies, minimizers, energy release rates and quasi-static evolutions converge as the internal length parameter of the phase-field model tends to zero. A crucial point in the proof is a novel representation of the energy release rate, which allows one to pass to the limit under weak convergence of the strains.