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A phase field model is used to study dendritic growth in a media with impurities. The model consists of a square lattice where a parameter $\psi$ can take values between $0$ and $1$ at each site. A site is in the solid phase for $\psi >1$, in the liquid phase for $\psi =0$, and the solid-liquid interface is expressed by $0<\psi <1$. A fraction of the sites are considered impurities that cannot be solidified, i.e.$\psi$ is fixed and taken as zero. These impurities are distributed randomly. As the probability $p$ of impure sites in the lattice increases, the growth loses its dendritic characteristic. It is shown that the perimeter of the growing solid goes from quadratic to a linear function with time. It was also found that as the probability of impurities reaches $p=0.004$, the solid undergoes a transition from anisotropic to isotropic growth.

Consider a binary substitutional solid solution not in equilibrium with a fluid rich in either solid component. At the interface, depending on the chemical energy, the solid may selectively lose or gain atoms from the fluid. Any atom exchange upsets the nominal composition; thereby stresses are generated in the solid by simultaneous action of lattice dilation and interdiffusion. As stresses build up, the local changing volume tends to split up into islands in an effort to escape from stresses, while curvature tends to amalgamate the islands in an effort to minimize surface energy. This competition could lead to substantial corrugations of the solid-fluid interface. These corrugations and the thresholds for noticeable competition between stresses and curvatures had an old tradition in chemo-mechanical models of sharp interfaces beginning with Srolovitz (1989). However, stresses were generally considered as a mere cause of external loads, so a pure solid was tacitly asserted. This paper couples elasticity with a conserved-phase field model to mainly investigate the effect of compositional strain on the interface instability process between a binary solid selectively interacting with a fluid. The chemical energy barrier is considered, however, uncoupled from stress, and the mobility quadratically dependent on the gradient of composition. Under initial enforced sinusoidal perturbations, results of simulations showed that the compositional strain parameter can dramatically alter the effect of the chemical energy barrier on the critical wavelength of perturbations that trigger interface instability.

In order to study a one-dimensional analogue of the spontaneous curvature model for two-component lipid bilayer membranes, we consider planar curves that are made of a material with two phases. Each phase induces a preferred curvature to the curve, and these curvatures as well as phase boundaries may lead to the development of kinks. We introduce a family of energies for smooth curves and phase fields, and we show that these energies Γ-converge to an energy for curves with a finite number of kinks. The theoretical result is illustrated by some numerical examples.

In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with non-homogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility of the phase field variable which results in a constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces acting on the boundary which play the role of the control variable in the considered model. To this end, we prove existence of minimizers and study a family of "local" approximations via adapted cost functionals.

Laser Powder Bed Fusion (LPBF) is an additive manufacturing method that manufactures high density and quality metal products. We present a coupled grain growth and heat transfer modeling technique to understand the materials microstructure evolution in metals during the cooling process of LPBF. The phase-field model is combined with a transient heat transfer equation to simulate the solidification and crystallization of the melt pool simultaneously. Specifically, the variable domain and driving force of the order parameters in the phase-field calculation are defined using current temperature distribution. Additionally, the latent heat generated by crystallization is introduced as a heat source to affect temperature evolution in the cooling process. The finite element method with a staggering strategy is employed to solve the coupled governing equations on an irregular computational domain. The computational framework is verified in a one-dimensional solidification problem by comparing the velocity of the fluid-solid interface. The two-way coupling solution of solidification and crystallization is studied in an example of LPBF of Aluminum alloys.

The commercial finite element software is usually used to analyze the failure modes of metal structures. In this work, we present a generalized ductile phase field model to solve the fracture problems of three-dimensional metal structures. This model can be easily implemented in Abaqus software. The isotropic hardening model and exponential hardening law were used to describe plastic behavior of metals. The different element types were introduced to mesh the structures conveniently. The ductile phase field governing equations were given and solved by the Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton monolithic algorithm. Moreover, an efficient and accurate adaptive analytic method of the third-order real symmetric matrix was proposed to implement the tension–compression decomposition. Some typical specimens commonly used in engineering were designed and manufactured. The corresponding static and fatigue tests were carried out, and the simulation results were compared with the experimental ones. The proposed model can predict the crack initiation and propagation of arbitrary three-dimensional metal structures under tensile, shear, torsional and fatigue loading.

Ferroelectric nanostructures are attracting considerable attention due to their unusual physical properties and potential applications in memory devices and nanoelectromechanical systems. It has been found that low-dimensional ferroelectrics, such as ferroelectric nanodots, ferroelectric nanotubes and ferroelectric thin films, exhibit polarization vortices or vortex-like domain structures due to the strong depolarization field and the size effect. The polarization vortex is regarded as a new toroidal order in ferroelectrics which is different from the rectilinear order of polarization. The vortex states of polarization are bistable and can be switched from one state to the other, which holds the potential application in next generation ferroelectric memories. This paper briefly reviews the recent work on the phase field studies of polarization vortex in ferroelectric nanostructures. The homogeneous bulk thermodynamics of ferroelectrics is first introduced based on the Landau–Devonshire theory. To describe the inhomogeneous polarization distribution in ferroelectrics, the phase field model including interface thermodynamics is then presented in the form of time-dependent Ginzburg–Landau equations.

The domain wall structure of ferroelectric/ paraelectric superlattices can be much more complex due to the influence of the superlattice stacking structure, the in-plane strain induced by the substrate and environmental temperature. In this study, we employed a phase field model to investigate the domain wall state of the SrTi${\mathrm{\text{O}}}_3$/BaTi${\mathrm{\text{O}}}_3$ superlattice structure. The domain wall thickness for the SrTi${\mathrm{\text{O}}}_3$/BaTi${\mathrm{\text{O}}}_3$ layer was measured using a hyperbolic function. Based on the simulation results, here, we show a domain wall state diagram to distinguish the hard and soft domain states. The polarization profiles across hard/ soft domain walls were illustrated and analyzed. Our simulation results offer a useful concept for the control of the domain wall state in the ferroelectric superlattice.

Numerical modeling of concrete fractures is of prime importance in the durability assessment of civil engineering structures. The phase field model has been demonstrated as a promising framework to simulate crack propagation in brittle material while using the many existing techniques. In this paper, we discuss choosing the most appropriate phase field model for describing the fracture behavior of concrete. More specifically, we present a detailed analysis of the existing models, which have been created by combining different spectral decompositions and crack density functions. The numerical simulation predictions are confronted with the experimental observation of a benchmark problem from the literature. The obtained results showed that the extensive/ compressive decomposition and the quadratic crack density function are the most suitable models to study concrete cracking behavior. The investigation’s size effects demonstrated heterogeneities played an important role in concrete’s post-cracking behavior and softening branches, especially for the small concrete structure.