A Phase Field Approach for Martensitic Transformations in Elastoplastic Materials

A multivariant phase field model for martensitic transformations in elastoplastic materials is introduced which is in mathematical terms the regularization of a sharp interface approach. The evolution of microstructure is assumed to follow a time dependent Ginzburg-Landau equation. The coupled problem of the mechanical balance equation and the evolution equations is solved using finite elements and an implicit time integration scheme. In this work, plasticity is considered for the austenitic phase which influences the martensitic evolution. With aid of the model these interactions are studied in detail. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Phase Field Model for Martensitic Transformations

The martensitic transformation is described using a phase field model which is in mathematical terms the regularization of a sharp interface approach. In this work, up to two martensitic orientation variants are considered. The evolution of microstructure is assumed to follow a time dependent Ginzburg-Landau equation. The coupled problem of the mechanical balance equation and the evolution equations is solved using finite elements and an implicit time integration scheme. In this work, the global energy evolution during the martensitic transformation and the influence of external loads on the formation of the different martensitic phases are studied in 2d. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Regularized Sharp Interface Model for Martensitic Phase Transformations at Large Strains

The macroscopic mechanical behavior of many functional materials crucially depends on the formation and evolution of their microstructure. When considering martensitic shape memory alloys, this microstructure typically consists of laminates with coherent twin boundaries. We suggest a variational-based phase field model for the dissipative evolution of microstructure with coherence-dependent interface energy and construct a suitable gradient-extended incremental variational framework for the proposed dissipative material. We use our model to predict laminate microstructure in martensitic CuAlNi. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Phase Field Approach for Dynamic Fracture

In phase field fracture models cracks are indicated by the value of a scalar field variable which interpolates smoothly between broken and undamaged material. The evolution equation for this crack field is coupled to the mechanical field equations in order to model the mutual interaction between the crack evolution and mechanical quantities. In finite element simulations of crack growth at comparatively slow loading velocities, a quasi-static phase field model yields reasonable results. However, the simulation of fast loading or the nucleation of new cracks challenges the limits of such a formulation. Here, the quasi-static phase field model predicts brutal crack extension with an artificially high crack speed. In this work, we analyze to which extend a dynamic formulation of the mechanical part of the phase field model can overcome this paradox created by the quasi-static formulation. In finite element simulations, the impact of the dynamic effects is studied, and differences between the crack propagation behavior of the quasi-static model and the dynamic formulation are highlighted. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Phase Field Modeling of Laminate Deformation Microstructure in Finite Gradient Crystal Plasticity

The macroscopic mechanical behavior of many materials crucially depends on the formation and evolution of their microstructure. In this work, we consider the formation and evolution of laminate deformation microstructure in plasticity. Inspired by work on the variational modeling of phase transformation [5] and building on related work on multislip gradient crystal plasticity [9], we present a new finite strain model for the formation and evolution of laminate deformation microstructure in double slip gradient crystal plasticity. Basic ingredients of our model are a nonconvex hardening potential and two gradient terms accounting for geometrically necessary dislocations (GNDs) by use of the dislocation density tensor and regularizing the sharp interfaces between different kinematically coherent plastic slip states. The plastic evolution is described by means of a nonsmooth dissipation potential for which we propose a new regularization. We formulate a continuous gradient-extended rate-variational framework and discretize it in time to obtain an incremental-variational formulation. Discretization in space yields a finite element formulation which is used to demonstrate the capability of our model to predict the formation and evolution of laminate deformation microstructure in f.c.c. Copper with two active slip systems in the same slip plane. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heat Conduction in a Phase Field Model for Martensitic Transformation

We study the martensitic transformation with a phase field model, where we consider the Bain transformation path in a small strain setting. For the order parameter, interpolating between an austenitic parent phase and martensitic phases, we use a Ginzburg-Landau evolution equation, assuming a constant mobility. In [1], a temperature dependent separation potential is introduced. We use this potential to extend the model in [2], by considering a transient temperature field, where the temperature is introduced as an additional degree of freedom. This leads to a coupling of both the evolution equation of the order parameter and the mechanical field equations (in terms of thermal expansion) with the heat equation. The model is implemented in FEAP as a 4-node element with bi-linear shape functions. Numerical examples are given to illustrate the influence of the temperature on the evolution of the martensitic phase. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一种非经典晶体塑性本构模型及其应用

根据不可逆变形过程中材料微结构的储能特性，采用由弹簧和塑性阻尼器构成的简单机械模型建立了不采用屈服判据的单晶本构关系·在此基础上形成了与ＫＢＷ自洽理论相应的多晶计算格式·计算格式中无需对滑移系的开动和滑移方向进行搜索，使计算过程大为简化·在多晶体分析中，提出了一种基于正２０面体各面内取向随机分布单晶响应的高斯平均和在空间完全均匀分布的２０个方向上算术平均的混合平均方案，与通常的纯高斯积分平均方案相比，在计算精度和效率上都有较大提高·用所发展的模型和算法分析了３１６不锈钢在具有代表性的路径下的循环塑性，得到了与实验相一致的结果·     

Phase Field Approach to Multiphase Flow Modeling

We review the phase field (otherwise called diffuse interface) model for fluid flows, where all quantities, such as density and composition, are assumed to vary continuously in space. This approach is the natural extension of van der Waals?? theory of critical phenomena both for one-component, two-phase fluids and for partially miscible liquid mixtures. The equations of motion are derived, assuming a simple expression for the pairwise interaction potential. In particular, we see that a non-equilibrium, reversible body force appears in the Navier-Stokes equation, that is proportional to the gradient of the generalized chemical potential. This, so called Korteweg, force is responsible for the convection that is observed in otherwise quiescent systems during phase change. In addition, in binary mixtures, the diffusive flux is modeled using a Cahn-Hilliard constitutive law with a composition-dependent diffusivity, showing that it reduces to Fick??s law in the dilute limit case. Finally, the results of several numerical simulations are described, modeling, in particular, a) mixing, b) spinodal decomposition, c) nucleation, d) enhanced heat transport, e) liquid-vapor phase separation.     

A Crystal Growth Approach for Topographical Global Optimization

A new approach for topographical global minimization of a function f(x), x A Rⁿ by using sampled points in A is presented. The globally sampled points are firstly obtained by uniform random sampling or uniform sampling with threshold distances. The point with the lowest function value is used as the nucleus atom to start a crystal growth process. A first triangular nucleus includes the nucleus atom and two nearest points. Sequential crystal growth is continued for which a point next closest to the nucleus atom is bonded to the crystal by attaching to two nearest solidified points. A solidified point will be marked during the crystal growth process if any of two connected points has a lower function value. Upon completion of entire crystal growth process, all unmarked points are then used as starting points for subsequent local minimizations. Extension of the topographical algorithms to constrained problems is exercised by using penalty functions. Formulas for estimation on the number of sampled points for problems with an assumed number of local minima are provided. Results on three global minimization problems by two topographical algorithms are discussed.     

A Variational Approach for One-Dimensional Scalar Field Problems

In this paper, we are interested in the existence of infinitely many weak solutions for a onedimensional scalar field problem. By using variational methods, in an appropriate functional space which involves the potential V, we determine intervals of parameters such that our problem admits either a sequence of weak solutions strongly converging to zero provided that the nonlinearity has a suitable behavior at zero or an unbounded sequence of weak solutions if a similar behavior occurs at infinity.     

获得非线性演化方程Backlund变换的一种新的途径

本文给出一种求非线性演化方程Backlund变换的方法,应用于非线性演化方程时,得到了与WTC方法一致的Backlund变换,避开了WTC方法涉及到的递推关系和截尾的讨论.     

A Penalty Approach to the Rate Problem in Small-strain Plasticity

Variational aspects of the rate problem in plasticity are discussed,the emphasis being on a unified treatment. Prominence is givento the penalty method; after showing its relationship to othervariational approaches, the question of convergence of penaltyapproximations is addressed, both in a Hilbert space (for theoriginal problem) and in a finite-dimensional subspace, thelatter relating to finite-element approximations. Finally, anexample is given which shows the viability of the penalty methodas a tool for the numerical solution of problems in plasticity.     

Error Analysis of SAV Finite Element Method to Phase Field Crystal Model

In this paper, we construct and analyze an energy stable scheme by combining the latest developed scalar auxiliary variable (SAV) approach and linear finite element method (FEM) for phase field crystal (PFC) model, and show rigorously that the scheme is first-order in time and second-order in space for the $L^2$ and $H^{-1}$ gradient flow equations. To reduce efficiently computational cost and capture accurately the phase interface, we give a simple adaptive strategy, equipped with a posteriori gradient estimator, i.e., $L^2$ norm of the recovered gradient. Extensive numerical experiments are presented to verify our theoretical results and to demonstrate the effectiveness and accuracy of our proposed method.     

A Unified Approach to Outliers,Inuence, and Transformations in Discriminant Analysis

This article extends the analysis of multivariate transformations to linear and quadratic discriminant analysis. It shows that the standard application of deletion diagnostic techniques for validating a particular transformation suffers from masking and so may fail if several outliers are present. We therefore suggest a simple and powerful method which is based on a forward search algorithm. This robust diagnostic procedure orders the observations from those most in agreement with the suggested model to those least in agreement with it. It provides a unified approach to the detection of inuential observations and outliers in discriminant analysis. Simulated and real data are used to show the necessity of considering multivariate transformations in discriminant analysis. The examples demonstrate the power of the suggested approach in revealing the correct structure of the data when this is obscured by outliers.     

A Distributional Approach to Multiple Stochastic Integrals and Transformations of the Poisson Measure

We study a class of ‘nonpoissonian’ transformations of the configuration space and the corresponding transformations of the Poisson measure. For some class of Poisson measures we find conditions which are sufficient for the transformed measure (which in general is nonpoissonian) to be absolutely continuous with respect to the initial Poisson measure and get the expression for the corresponding Radon–Nikodym derivative. To solve this problem we use a distributional approach to Poisson multiple stochastic integrals.     

A Phase Field Model for Ductile to Brittle Failure Mode Transition

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in dynamic problems with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. We outline a conceptual framework for phase field models of crack propagation in brittle elastic and ductile elastic-plastic solids under dynamic loading and investigate the ductile to brittle failure mode transition observed in the experiment performed by Kalthoff and Winkeler [3]. We develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. To this end, we define energy storage and dissipation functions for the plastic flow including the fracture phase field. The introduction of local history fields that drive the evolution of the crack phase field inspires the construction of robust operator split schemes. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On an Algebro-Geometric Approach to Ribaucour Transformations and Bianchi Cubes

Mathematical Notes - In this paper, we develop an algebro-geometric approach to Ribaucour transformations and Bianchi cubes of orthogonal nets. Explicit transformations of algebro-geometric data...