Finite element simulations of dynamics of multivariant martensitic phase transitions based on Ginzburg–Landau theory

A finite element approach is suggested for the modeling of multivariant stress-induced martensitic phase transitions (PTs) in elastic materials at the nanoscale for the 2-D and 3-D cases, for quasi-static and dynamic formulations. The approach is based on the phase-field theory, which includes the Ginzburg–Landau equations with an advanced thermodynamic potential that captures the main features of macroscopic stress–strain curves. The model consists of a coupled system of the Ginzburg–Landau equations and the static or dynamic elasticity equations, and it describes evolution of distributions of austenite and different martensitic variants in terms of corresponding order parameters. The suggested explicit finite element algorithm allows decoupling of the Ginzburg–Landau and elasticity equations for small time increments. Based on the developed phase-field approach, the simulation of the microstructure evolution for cubic-tetragonal martensitic PT in a NiAl alloy is presented for quasi-statics (i.e., without inertial forces) and dynamic formulations in the 2-D and 3-D cases. The numerical results show the significant influence of inertial effects on microstructure evolution in single- and polycrystalline samples, even for the traditional problem of relaxation of initial perturbations to stationary microstructure.     

Ginzburg–Landau Vortex Dynamics with Pinning and Strong Applied Currents

We study a mixed heat and Schrödinger Ginzburg–Landau evolution equation on a bounded two-dimensional domain with an electric current applied on the boundary and a pinning potential term. This is meant to model a superconductor subjected to an applied electric current and electromagnetic field and containing impurities. Such a current is expected to set the vortices in motion, while the pinning term drives them toward minima of the pinning potential and “pins” them there. We derive the limiting dynamics of a finite number of vortices in the limit of a large Ginzburg–Landau parameter, or \({\varepsilon \to 0}\) , when the intensity of the electric current and applied magnetic field on the boundary scale like \({|{\rm log} \, \varepsilon|}\) . We show that the limiting velocity of the vortices is the sum of a Lorentz force, due to the current, and a pinning force. We state an analogous result for a model Ginzburg–Landau equation without magnetic field but with forcing terms. Our proof provides a unified approach to various proofs of dynamics of Ginzburg–Landau vortices.     

Plasticity,internal structure and phase field model

The model presented here is based on the assumption that the plastic phase is due to a variation of the strain generating a dislocation migration, which in turn implies a different material response. That is, the transition from elastic to plastic behavior is related with two different internal or mesoscopic structures. So, within the Landau theory on phase transitions, and via the notion of order parameter, we suggest a model for a hardening plasticity by a second order phase transition, able to describe the elastic–plastic transformation. The differential problem related with this new variable will be represented by the Ginzburg–Landau equation. The model is supplemented by a differential constitutive equation among the strain, the stress, and the order parameter. By this system, we are able to obtain the classical behavior of hardening plastic phase diagrams.     

Bridging techniques in a multi-scale modeling of pattern formation

Bridging techniques between microscopic and macroscopic models are discussed in the case of wrinkling analysis. The considered macroscopic models are related to envelope equations of Ginzburg–Landau type, but generally, they are not valid up to the boundary. To this end, a multi-scale approach is considered: the reduced model is implemented in the bulk while the full model is applied near the boundary and these two models are coupled with the Arlequin method (Ben Dhia, 1998). This paper focuses on the definition of the coupling model and the transition between two scales. Especially, a new nonlocal bridging technique is presented and compared with another recent one (Hu et al., 2011). The present method can also be seen as a guide for coupling techniques involving other reduced order models.     

Long waves between a subsonic gas flow-film interface flowing down an inclined plate

The present work deals with temporal stability properties of a falling liquid film down an inclined plane in the presence of a parallel subsonic gas flow. The waves are described by evolution equation previously derived as a generalization of the model for the Newtonian liquid. We confirm linear stability results of the basic flow using the Orr–Sommerfeld analysis to that obtained by long wave approximation analysis. The non-linear stability criteria of the model are discussed analytically and stability branches are obtained. Finally, the solitary wave solutions at the liquid–gas interface are discussed, using specially envelope transform and direct ansatz approach to Ginzburg–Landau equation. The influence of different parameters governing the flow on the stability behavior of the system is discussed in detail.     

TDGL and mKdV equations for car-following model considering driver’s anticipation

Car-following models are proposed to describe the jamming transition in traffic flow on a highway. In this paper, a new car-following model considering the driver’s forecast effect is investigated to describe the traffic jam. The nature of the model is studied using linear and nonlinear analysis method. A thermodynamic theory is formulated to describe the phase transition and critical phenomenon in traffic flow and the time-dependent Ginzburg–Landau (TDGL) equation is derived to describe the traffic flow near the critical point. It is also shown that the modified Korteweg-de Veris (mKdV) equation is derived to describe the traffic jam. The connection between the TDGL and the mKdV equations is given.     

Nonlocal damage model using the phase field method: Theory and applications

A new nonlocal, gradient based damage model is proposed for isotropic elastic damage using the phase field method in order to show the evolution of damage in brittle materials. The general framework of the phase field model (PFM) is discussed and the order parameter is related to the damage variable in continuum damage mechanics (CDM). The time dependent Ginzburg–Landau equation which is also termed the Allen–Cahn equation is used to describe the damage evolution process. Specific length scale which addresses the interface region in which the process of changing undamaged solid to fully damaged material (microcracks) occurs is defined in order to capture the effect of the damaged localization zone. A new implicit damage variable is proposed through the phase field theory. Details of the different aspects and regularization capabilities are illustrated by means of numerical examples and the validity and usefulness of the phase field modeling approach is demonstrated.     

Competition of spiral waves in heterogeneous CGLE systems

Nonlinear Dynamics - The effects of spatial heterogeneity on a two-dimensional complex Ginzburg–Landau equation model are studied. In general, the interaction of a pair of spiral waves with a...     

A new Fourier-related double scale analysis for instability phenomena in sandwich structures

In this paper, we present a new Fourier-related double scale analysis to study instability phenomena of sandwich structures. By using the technique of slowly variable Fourier coefficients, a zig–zag theory based microscopical sandwich model is transformed into a macroscopical one that offers three numerical advantages. Firstly, only the envelopes of instability patterns are evaluated and this leads to a significant improvement on computational efficiency, especially when dealing with high wavenumber wrinkling phenomena. Secondly, the proposed macroscopical model allows one to select modal wavelength, which makes easy to control non-linear calculations. Thirdly, in contrast to Landau–Ginzburg envelope equations, it may also remain valid away from the bifurcation point and the coupling between global and local instabilities can be accounted for. The established non-linear system is solved by asymptotic numerical method (ANM), which is more reliable and less time consuming than other iterative classical methods. The proposed double scale analysis yields accurate results with a significant reduced computational cost.     

A real-space phase field model for the domain evolution of ferromagnetic materials

A real-space phase field model based on the time-dependent Ginzburg–Landau (TDGL) equation is developed to predict the domain evolution of ferromagnetic materials. The phase field model stems from a thermodynamic theory of ferromagnetic materials which employs the strain and magnetization as independent variables. The phase field equations are shown to reduce to the common micromagnetic model when the magnetostriction is absent and the magnitude of magnetization is constant. The strain and magnetization in the equilibrium state are obtained simultaneously by solving the phase field equations via a nonlinear finite element method. The finite-element based phase field model is applicable for the domain evolution of ferromagnetic materials with arbitrary geometries and boundary conditions. The evolution of magnetization domains in ferromagnetic thin film subjected to external stresses and magnetic fields are simulated and the magnetoelastic coupling behavior is investigated. Phase field simulations show that the magnetization vectors form a single magnetic vortex in ferromagnetic disks and rings. The configuration and size of the simulated magnetization vortex are in agreement with the experimental observation, suggesting that the phase field model is a powerful tool for the domain evolution of ferromagnetic materials.     

The time-dependent Ginzburg–Landau equation for car-following model considering anticipation-driving behavior

In this paper, a car-following model considering anticipation-driving behavior is considered to describe the traffic jam. The nature of the model is investigated using linear and nonlinear analysis method. A thermodynamic theory is formulated for describing the phase transitions and critical phenomena, and the time-dependent Ginzburg–Landau equation is derived to describe traffic flow near the critical point.     

Ginzburg–Landau Vortices Driven by the Landau–Lifshitz–Gilbert Equation

A simplified model for the energy of the magnetization of a thin ferromagnetic film gives rise to a version of the theory of Ginzburg–Landau vortices for sphere-valued maps. In particular, we have the development of vortices as a certain parameter tends to 0. The dynamics of the magnetization are ruled by the Landau–Lifshitz–Gilbert equation, which combines characteristic properties of a nonlinear Schrödinger equation and a gradient flow. This paper studies the motion of the vortex centers under this evolution equation.     

The Effect of Freezing and Discretization to the Asymptotic Stability of Relative Equilibria

In this paper we prove nonlinear stability results for the numerical approximation of relative equilibria of equivariant parabolic partial differential equations in one space dimension. Relative equilibria are solutions which are equilibria in an appropriately comoving frame and occur frequently in systems with underlying symmetry. By transforming the PDE into a corresponding PDAE via a freezing ansatz [2] the relative equilibrium can be analyzed as a stationary solution of the PDAE. The main result is the fact that nonlinear stability properties are inherited by the numerical approximation with finite differences on a finite equidistant grid with appropriate boundary conditions. This is a generalization of the results in [14] and is illustrated by numerical computations for the quintic complex Ginzburg Landau equation.      

A shape memory alloy model by a second order phase transition

Motivated by the experimental results of the paper [1] and unlike the general theories of shape memory alloys (SMAs), in this paper we suggest for such materials a phase field model by a second order phase transition. So that, with this new system we obtain a simulation of phase dynamics very convenient to describe the natural behavior of these materials. The differential system is governed by the motion equation, the heat equation and the Ginzburg–Landau (GL) equation and by a constitutive law between the phase field, the temperature, the strain and the stress. The use of this new model is characterized by new potentials of the GL equation and by a new dependence on the temperature in the constitutive equation. Using this new model, we obtain simulations in better agreement with experimental data and respect to previous work [2].     

Parametric disorder effects on a subcritical stationary bifurcation

For a spatial modulation of the control parameter which describes, for instance, major effects of a rough container boundary in Rayleigh–Bénard convection, the threshold value of the bifurcation from a homogeneous basic state to a spatially periodic state is provided analytically and numerically, taking the one-dimensional cubic–quintic complex Ginzburg–Landau equation with real coefficients as an example. Above the threshold, using the Poincaré–Lindstedt expansion, we show that the quintic term affects both the stationary nonlinear solution and the Nusselt number.     

TDGL equation in lattice hydrodynamic model considering driver’s physical delay

Based on an extended lattice hydrodynamic model considering the delay of the driver’s response in sensing headway, we get the time-dependent Ginzburg–Landau (for short, TDGL) equation to describe the transition and critical phenomenon in traffic flow by applying the reductive perturbation method. The corresponding solutions are obtained. Numerical simulation is carried out to examine the performance of the model and the results show coincidence with the analysis results.     

The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. Droplet Arrangement via the Renormalized Energy

This is the second in a series of papers in which we derive a Γ-expansion for the two-dimensional non-local Ginzburg–Landau energy with Coulomb repulsion known as the Ohta–Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small “droplets” of the minority phase in a “sea” of the majority phase. In our previous paper, we computed the Γ-limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order Γ-limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg–Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Thus, without appealing to the Euler–Lagrange equation, we establish for all configurations which have “almost minimal energy” the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of Γ-equivalence, the obtained results also yield an expansion of the minimal energy and a characterization of the zero super-level sets of the minimizers for the original Ohta–Kawasaki energy. This leads to the expectation of seeing triangular lattices of droplets as energy minimizers.     

Phase-field modeling of crack propagation in piezoelectric and ferroelectric materials with different electromechanical crack conditions

We present a family of phase-field models for fracture in piezoelectric and ferroelectric materials. These models couple a variational formulation of brittle fracture with, respectively, (1) the linear theory of piezoelectricity, and (2) a Ginzburg–Landau model of the ferroelectric microstructure to address the full complexity of the fracture phenomenon in these materials. In these models, both the cracks and the ferroelectric domain walls are represented in a diffuse way by phase-fields. The main challenge addressed here is encoding various electromechanical crack models (introduced as crack-face boundary conditions in sharp models) into the phase-field framework. The proposed models are verified through comparisons with the corresponding sharp-crack models. We also perform two dimensional finite element simulations to demonstrate the effect of the different crack-face conditions, the electromechanical loading and the media filling the crack gap on the crack propagation and the microstructure evolution. Salient features of the results are compared with experiments.     

Interaction between phase transformations and dislocations at the nanoscale. Part 1. General phase field approach

Thermodynamically consistent, three-dimensional (3D) phase field approach (PFA) for coupled multivariant martensitic transformations (PTs), including cyclic PTs, variant–variant transformations (i.e., twinning), and dislocation evolution is developed at large strains. One of our key points is in the justification of the multiplicative decomposition of the deformation gradient into elastic, transformational, and plastic parts. The plastic part includes four mechanisms: dislocation motion in martensite along slip systems of martensite and slip systems of austenite inherited during PT and dislocation motion in austenite along slip systems of austenite and slip systems of martensite inherited during reverse PT. The plastic part of the velocity gradient for all these mechanisms is defined in the crystal lattice of the austenite utilizing just slip systems of austenite and inherited slip systems of martensite, and just two corresponding types of order parameters. The explicit expressions for the Helmholtz free energy and the transformation and plastic deformation gradients are presented to satisfy the formulated conditions related to homogeneous thermodynamic equilibrium states of crystal lattice and their instabilities. In particular, they result in a constant (i.e., stress- and temperature-independent) transformation deformation gradient and Burgers vectors. Thermodynamic treatment resulted in the determination of the driving forces for change of the order parameters for PTs and dislocations. It also determined the boundary conditions for the order parameters that include a variation of the surface energy during PT and exit of dislocations. Ginzburg–Landau equations for dislocations include variation of properties during PTs, which in turn produces additional contributions from dislocations to the Ginzburg–Landau equations for PTs. A complete system of coupled PFA and mechanics equations is presented. A similar theory can be developed for PFA to dislocations and other PTs, like reconstructive PTs and diffusive PTs described by the Cahn–Hilliard equation, as well as twinning and grain boundaries evolution.     

Existence of 3-Dimensional Tori for 1D Complex Ginzburg–Landau Equation Via a Degenerate KAM Theorem

In this paper, we consider complex Ginzburg–Landau equation in one space dimension and rigorously show the existence of 3-dimensional tori. The proof is based on degenerate infinite-dimensional KAM theory and normal form technique.