Solvability via viscosity solutions for a model of phase transitions driven by configurational forces

This article is concerned with an initial-boundary value problem for an elliptic-parabolic coupled system arising in martensitic phase transition theory of elastically deformable solid materials, e.g., steel. This model was proposed in Alber and Zhu (2007) [4], and investigated in Alber and Zhu (2006) [3] the existence of weak solutions which are defined in a standard way, however the key technique used in Alber and Zhu (2006) [3] is not applicable to multi-dimensional problem. One of the motivations of this study is to solve this multi-dimensional problem, and another is to investigate the sharp interface limits. Thus we define weak solutions in a way, which is different from Alber and Zhu (2006) [3], by using the notion of viscosity solution. We do prove successfully the existence of weak solutions in this sense for one-dimensional problem, yet the multi-dimensional problem is still open.     

Solutions to a model with Neumann boundary conditions for phase transitions driven by configurational forces

We study an initial boundary value problem of a model describing the evolution in time of diffusive phase interfaces in solid materials, in which martensitic phase transformations driven by configurational forces take place. The model was proposed earlier by the authors and consists of the partial differential equations of linear elasticity coupled to a nonlinear, degenerate parabolic equation of second order for an order parameter. In a previous paper global existence of weak solutions in one space dimension was proved under Dirichlet boundary conditions for the order parameter. Here we show that global solutions also exist for Neumann boundary conditions. Again, the method of proof is only valid in one space dimension.     

Solutions to models for phase evolution driven by configurational forces

This work is concerned with the models for phase evolution driven by configurational forces in alloys. We investigate mathematically the validity of the models for the non-conserved case and the conserved case, however only for the one space dimensional case. Those models were proposed by Prof. H.-D. Alber in a previous article. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a numerical scheme for curved crack propagation based on configurational forces and maximum dissipation

A numerical scheme is presented to predict crack trajectories in two dimensional components. First a relation between the curvature in mixed–mode crack propagation and the corresponding configurational forces is derived, based on the principle of maximum dissipation. With the help of this, a numerical scheme is presented which is based on a predictor–corrector method using the configurational forces acting on the crack together with their derivatives along real and test paths. With the help of this scheme it is possible to take bigger than usual propagation steps, represented by splines. Essential for this approach is the correct numerical determination of the configurational forces acting on the crack tip. The methods used by other authors are shortly reviewed and an approach valid for arbitrary non–homogenous and non–linear materials with mixed–mode cracks is presented. Numerical examples show, that the method is a able to predict the crack paths in components with holes, stiffeners etc. with good accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Persistence of undercompressive phase boundaries for isothermal Euler equations including configurational forces and surface tension

The persistence of subsonic phase boundaries in a multidimensional Van der Waals fluid is analyzed. The phase boundary is considered as a sharp free boundary that connects liquid and vapor bulk phase dynamics given by the isothermal Euler equations. The evolution of the boundary is driven by effects of configurational forces as well as surface tension. To analyze this problem, the equations and trace conditions are linearized such that one obtains a general hyperbolic initial boundary value problem with higher‐order boundary conditions. A global existence theorem for the linearized system with constant coefficients is shown. The proof relies on the normal mode analysis and a linear form in suitable spaces that is defined using an associated adjoint problem. Especially, the associated adjoint problem satisfies the uniform backward in time Kreiss–Lopatinski? condition. A new energy‐like estimate that also includes surface energy terms leads finally to the uniqueness and regularity for the found solutions of the problem in weighted spaces. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal boundary controls for a phase field model

The phase field model is a nonlinear system of parabolic equationswhich describes the phase transitions between two differentphases, e.g. solid and liquid. In this paper, we consider ageneral optimal boundary control problem which is governed bythis model. The existence of the solutions of the phase fieldmodel is established by a rigorous analysis of the method oflines. The existence of the optimal solutions and the necessaryconditions for optimality are proved. For a special unconstrainedboundary control problem, we also prove some results concerningthe uniqueness of the optimal solutions. For a special constrainedboundary control problem, we obtain a result concerning thebang-bang principle.     

Application of configurational forces in the context of a kinked crack

The application of the configurational force approach in crack problems is often used in order to establish fracture criteria that are adapted to a specific material behaviour. The tangential component of the calculated vectorial quantity that acts at the crack tip is a generalisation of the conventional J-integral and can be interpreted as the energy release rate when the crack extends in this direction. However, the interpretation of nontangential components in the same way, and hence the interpretation of this vectorial quantity as the crack driving force, is not consistent with established kink criteria in the special case of linear elastic fracture mechanics. As a classical example, an in-plane loaded crack in a homogeneous isotropic linear elastic material is considered under the small strain assumption. Using the expansion of stress intensity factors at the extended crack tip, nontangential components of the configurational force can be interpreted as sensitivities to crack deflection. This perspective has the potential of generalisation which can be applied to more complex situations in order to study the interplay between mechanical fields in the vicinity of the crack tip and the microstructural influence within the process zone. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A phase field model for fracture

The variational formulation of brittle fracture as formulated for example by Francfort and Marigo in [1], where the total energy is minimized with respect to any admissible crack set and displacement field, allows the identification of crack paths, branching of preexisting cracks and even crack initiation without additional criteria. For its numerical treatment a continuous approximation of the model in the sense of Γ-convergence has been presented by Bourdin in [2]. In the regularized Francfort–Marigo model cracks are represented by an additional field variable (secondary variable) s∈[0,1] which is 0 if the material is cracked and 1 if it is undamaged. In this work, we reinterpret the crack variable as a phase field order parameter and address cracking as a phase transition problem. The crack growth is governed by the evolution equation of the order parameter which resembles the Ginzburg–Landau equation. The numerical treatment is done by finite elements combined with an implicit Euler scheme for the time integration. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An error estimate for a finite-element scheme for a phase field model

In this paper we propose a fully discrete finite-element schemeto solve a non-linear system of parabolic equations for a phasefield model and demonstrate an error estimate of optimal orderin L² for this scheme. This error estimate conforms with thenumerical results presented at the end of this paper.     

On a phase transition model

An Allen–Cahn phase transition model with a periodic nonautonomous term is presented for which an infinite number of transition states is shown to exist. A constrained minimization argument and the analysis of a limit problem are employed to get states having a finite number of transitions. A priori bounds and an approximation procedure give the general case. Decay properties are also studied and a sharp transition result with an arbitrary interface is proved.     

On a phase transition model in ferromagnetism

This paper deals with the limiting behavior of a phase transition model in ferromagnetism. The model describes the three-dimensional evolution of both thermodynamic and electromagnetic properties of the ferromagnetic material. We are concerned with the passage from 3D to 2D in the theory of the paramagnetic-ferromagnetic transition. We identify the limit problem by using the so-called energy method.     

On a nonlocal phase separation model

An alternative to the Cahn-Hilliard model of phase separation for two-phase systems in a simplified isothermal case is given. The model is derived from a free energy with a nonlocal interacting term and allows reasonable bounds for the concentrations. Using the free energy as Lyapunov functional the asymptotic state of the system is investigated and characterized by a variational principle.     

Optimum control laws in a non-autonomous dynamic model of a gas field

A model of gas field development described as a nonlinear optimum control problem with an infinite planning horizon is considered. The Pontryagin maximum principle is used to solve it. The theorem on sufficient optimumity conditions in terms of constructions of the Pontryagin maximum principles is used to substantiate the optimumity of the extremal solution. A procedure for constructing the optimum solution by dynamic programming is described and is of some methodological interest. The obtained optimum solution is used to construct the Bellman function. Reference is made to a work containing an economic interpretation of the problem.     

A thermodynamically consistent numerical method for a phase field model of solidification

A discretization is presented for the initial boundary value problem of solidification as described in the phase-field model developed by Penrose and Fife (1990) [1] and Wang et al. (1993) [2]. These are models that are completely derived from the laws of thermodynamics, and the algorithms that we propose are formulated to strictly preserve them. Hence, the discrete solutions obtained can be understood as discrete dynamical systems satisfying discrete versions of the first and second laws of thermodynamics. The proposed methods are based on a finite element discretization in space and a midpoint-type finite-difference discretization in time. By using so-called discrete gradient operators, the conservation/entropic character of the continuum model is inherited in the numerical solution, as well as its Lyapunov stability in pure solid/liquid equilibria.     

A phase field model for martensitic transformations with a temperature-dependent separation potential

Metallic materials often exhibit a complex microstructure with varying material properties in the different phases. Of major importance in mechanical engineering is the evolution of the austenitic and martensitic phases in steel. The martensitic transformation can be induced by heat treatment or by plastic surface deformation at low temperatures. A two dimensional elastic phase field model for martensitic transformations considering several martensitic orientation variants to simulate the phase change at the surface is introduced in [1]. However here, only one martensitic orientation variant is considered for the sake of simplicity. The separation potential is temperature dependent. Therefore, the coefficients of the Landau polynomial are identified by results of molecular dynamics (MD) simulations for pure iron [1]. The resulting separation potential is applied to analyse the mean interface velocity with respect to temperature and load. The interface velocity is computed by use of the dissipative part to the configurational forces balance as suggested in [3]. The model is implemented in the finite element code FEAP using standard 4-node elements with bi-linear shape functions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非守恒相场模型解的blow-up

讨论了一类非守恒相场模型解的性态,证明当ａ２ｐ － １ ＜ ０ 及初值充分大时解在有限时刻 ｂｌｏｗ ｕｐ．     

Finite dimensional exponential attractor for the phase field model

We consider the phase field equations in arbitrary space dimension. We show that the corresponding boundary value problems are well-posed when assuming that the initial data is square integrable and prove the existence of a maximal attractor and of an inertial set.     

A finite element phase field model for relaxor ferroelectrics

A mechanically coupled phase field model is presented for the domain evolution and mesoscopic response of relaxor ferroelectrics. In the model the spontaneous polarization is treated as order parameter. The model is derived from thermodynamic analysis including the material force theory. Random field theory is adopted to take the disorder of relaxor ferroelectrics into account. Results show that the model is capable of reproducing relaxor features, such as domain miniaturization, small remnant polarization and large piezoelectric response. Dependence of these features on the random field strength is discussed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)