Mechanic-Stochastic Model for the Simulation of Polycrystals

A mechanic-stochastic model for the mechanics of polycrystal material based on the single crystal orientations is presented. With this method, the distributions of strain and stress tensor components can be obtained in practice. Specific examples of computed distributions are shown, together with the distributions of the FEM-calculated mechanical responses of a polycrystal. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the asymptotic behavior of variable exponent power–law functionals and applications

We study, via Γ-convergence, the asymptotic behavior of several classes of power–law functionals acting on fields belonging to variable exponent Lebesgue spaces and which are subject to constant rank differential constraints. Applications of the Γ-convergence results to the derivation and analysis of several models related to polycrystal plasticity arising as limiting cases of more flexible power–law models are also discussed.     

A thin film model

Using the theory of fiber bundles we construct a new model of a thin polycrystal film. The quantum equations of motion thereby obtained are used to determine the regularity of the influence of substructural ordering on the physical characteristics of the film.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 36–40.     

Application of the Green Tensor to the Modeling of Solution-Precipitation Creep

Our aim is to present a mechanical model for solution-precipitation creep, a diffusive deformation process occurring in polycrystalline and granular materials. Within the scope of the model, a Lagrangian consisting of the elastic power and dissipation is proposed. The former is kept in a standard form typical for linear material behavior while the latter is assumed as a surface integral depending on the velocity of the material transport and the velocity of the motion of the boundary. The minimization with respect to the total deformations leads to the equilibrium equation which is solved analytically, by using Green's function. The evolution equations are obtained as the result of the minimization with respect to the internal variables and are solved via finite element method. The contribution closes with a numerical example showing the deformation of a polycrystal consisting of hexagonal crystals. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of solutions to the Kobayashi–Warren–Carter system

In this paper, a coupled system of two parabolic type initial-boundary value problems is considered. The system is known as the Kobayashi–Warren–Carter model of grain boundary motion in a polycrystal. Kobayashi–Warren–Carter model is derived as a gradient descent flow of an energy functional, which is called “free-energy”, with respect to two unknown variables and it involves a weighted-unknown dependent total variation term. The main goal of this paper is to obtain existence of solutions to this system. We solve the problem by means of a time-discretization of a relaxed system and a highly non-trivial passage to the limit. We point out that our time-discretization method is effective not only for the original Kobayashi–Warren–Carter system but also for its relaxed versions. Therefore, we provide a uniform approach for obtaining solutions to systems associated with this model.     

Polycrystal vs. Classical Continuum Models for Structures

In this work a comparison of polycrystal and classical continuum models illustrated by examples of full structures is investigated. The general idea is to represent the averaged distribution of displacement, stress and strain fields for statistically randomized realizations of discrete structures. A technique for averaging fields in the FEM program ABAQUS is proposed and implemented. To improve the computation efficiency mesh dependence was investigated. Results of simulation are given for a rectangular plate and for the Kirsch's problem for both examples elastic as well as inelastic material behavior. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cosserat continuum modelling of grain size effects in metal polycrystals

The global response of polycrystalline aggregates is investigated, in order to simulate grain size effects in IF ferritic steels. The mechanics of generalized continua is used to describe the studied phenomena. The polycrystal is regarded as a heterogeneous Cosserat medium, and the overall properties are estimated using a specific homogenization technique. To illustrate the capabilities of the model, some simple bidimensionnal computations are presented for different grain sizes. Afterwards tridimensionnal computations are shown in order to extract the global effect on the mechanical behaviour for grain sizes ranging from 5 µm to 120 µm. The finite element response is harder for the smallest grain size, but the model still underestimate the grain size effect on the tensile response. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Phasan software package as a Windows-95 application

We study a mathematicalmodel of X-ray analysis of polycrystal mixtures. We describe a new version of the Phasan program for phase analysis, based on a multi-document interface. Two figures. Bioliography: 6 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 245–255.     

A laminate-based framework for switching and microstructure evolution in polycrystalline ferroelectrics

This contribution deals with the numerical modelling of polycrystalline ferroelectric materials considering a sequential laminate-based approach established for tetragonal single-crystal ferroelectrics. The particular model [1] is considered and extended to predict the material behaviour of poly-crystal tetragonal ferroelectric ceramics. The derived laminate-based model is implemented in a finite element environment to simulate the time-dependent domain evolution and switching response of a bulk polycrystalline ferroelectric ceramic. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiscale Modeling of Nanocrystalline Materials: A Variational Approach

This paper presents a variational multi-scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc-material is modeled as a two-phase material consisting of a grain interior (GI) phase and a grain boundary (GB) phase. A rate-independent isotropic porous plasticity model is employed to describe the GB phase, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the GI phase. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor-type homogenization. It is shown that the proposed model is able to capture the inverse Hall-Petch effect. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of Size Effects in Ferroelectric Materials using a Phase Field Model

An electro-mechanically coupled phase field model for domain evolution in ferroelectric materials is presented. The inner length scale introduced by the model gives rise to size effects, especially in the context of the poling behavior of polycrystals. Such size effects are investigated by 2D numerical simulations for barium titanate polycrystals. Ferroelectric hysteresis curves and coercive fields are calculated for two different transition conditions for the order parameter at the grain boundaries. The results show that there is a significant size effect for the investigated polycrystal systems. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An anisotropic flow law for incompressible polycrystalline materials

New and explicit anisotropic constitutive equations between the stretching and deviatoric stress tensors for the two- and three-dimensional cases of incompressible polycrystalline materials are presented. The anisotropy is assumed to be driven by an Orientation Distribution Function (ODF). The polycrystal is composed of transversally isotropic crystallites, the lattice orientation of which can be characterized by a single unit vector. The proposed constitutive equations are valid for any frame of reference and for every state of deformation. The basic assumption of this method is that the principle directions of the stretching and of the stress deviator are the same in the isotropic as well as in the anisotropic case. This means that the proposed constitutive laws are able to model the effects of anisotropy only via a change of the fluidity due to a change of the ODF. Such an assumption is justified to guarantee that, besides knowledge of the parameters involved in the isotropic constitutive equation, the anisotropic material response is completely characterized by only one additional parameter, a type of enhancement factor. Explicit comparisons with experimental data are conducted for Ih–ice. Dedicated to Prof. L.W. Morland on the occasion of his 70^th birthday Received: July 6, 2004; revised: November 8, 2004     

An anisotropic flow law for incompressible polycrystalline materials

New and explicit anisotropic constitutive equations between the stretching and deviatoric stress tensors for the two- and three-dimensional cases of incompressible polycrystalline materials are presented. The anisotropy is assumed to be driven by an Orientation Distribution Function (ODF). The polycrystal is composed of transversally isotropic crystallites, the lattice orientation of which can be characterized by a single unit vector. The proposed constitutive equations are valid for any frame of reference and for every state of deformation. The basic assumption of this method is that the principle directions of the stretching and of the stress deviator are the same in the isotropic as well as in the anisotropic case. This means that the proposed constitutive laws are able to model the effects of anisotropy only via a change of the fluidity due to a change of the ODF. Such an assumption is justified to guarantee that, besides knowledge of the parameters involved in the isotropic constitutive equation, the anisotropic material response is completely characterized by only one additional parameter, a type of enhancement factor. Explicit comparisons with experimental data are conducted for Ih–ice.     

Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities

The heat transfer problem in a polycrystal with nonlinear jump conditions on the grain boundaries will be homogenized using the method of stochastic two-scale convergence developed by Zhikov and Pyatnitskii [V.V. Zhikov and A.L. Pyatnitskii, Homogenization of random singular structures and random measures, Izv. Math. 70(1) (2006), pp. 19–67] and recently extended by the author [M. Heida, An extension of stochastic two-scale convergence and application, Asympt. Anal. (2010) (in press)]. It will be shown that for monotone Lipschitz jump conditions differentiable in 0, the nonlinearity vanishes in the limit. Additionally, existing Poincaré inequalities will be extended to more general geometric settings with the only restriction of local C ¹-interfaces with finite intensity. In particular, the result can now be applied to the Poisson–Voronoi tessellation.     

On the Lipschitz constant of the RSK correspondence

We view the RSK correspondence as associating to each permutation π∈Sn a Young diagram λ=λ(π), i.e. a partition of n. Suppose now that π is left-multiplied by t transpositions, what is the largest number of cells in λ that can change as a result? It is natural refer to this question as the search for the Lipschitz constant of the RSK correspondence.We show upper bounds on this Lipschitz constant as a function of t. For t=1, we give a construction of permutations that achieve this bound exactly. For larger t we construct permutations which come close to matching the upper bound that we prove.     

Adjacency on the order polytope with applications to the theory of fuzzy measures

In this paper we study the adjacency structure of the order polytope of a poset. For a given poset, we determine whether two vertices in the corresponding order polytope are adjacent. This is done through filters in the original poset. We also prove that checking adjacency between two vertices can be done in quadratic time on the number of elements of the poset. As particular cases of order polytopes, we recover the adjacency structure of the set of fuzzy measures and obtain it for the set of p-symmetric measures for a given indifference partition; moreover, we show that the set of p-symmetric measures can be seen as the order polytope of a quotient set of the poset leading to fuzzy measures. From this property, we obtain the diameter of the set of p-symmetric measures. Finally, considering the set of p-symmetric measures as the order polytope of a direct product of chains, we obtain some other properties of these measures, as bounds on the volume and the number of vertices on certain cases.     

Properties of the probability density function of the non-central chi-squared distribution

In this paper we consider the probability density function (pdf) of a non-central χ² distribution with arbitrary number of degrees of freedom. For this function we prove that can be represented as a finite sum and we deduce a partial derivative formula. Moreover, we show that the pdf is log-concave when the degrees of freedom is greater or equal than 2. At the end of this paper we present some Turán-type inequalities for this function and an elegant application of the monotone form of l'Hospital's rule in probability theory is given.     

Enhancing discretized formulations: the knapsack reformulation and the star reformulation

Discretized formulations have proved to be useful for modeling combinatorial optimizations. The main focus of this work is on how to strengthen the linear programming relaxation of a given discretized formulation. More precisely, we will study and strengthen subproblems that arise in these formulations. In one case we will focus on the so-called knapsack reformulation which is based on viewing these models as the intersection of two polyhedra, one of them being a specialized knapsack problem. We will show that strong inequalities used in previous works are a special case of inequalities implied by the knapsack formulation. In the second case we will analyze a star-like subproblem and will provide an extended formulation for this problem as well as a set of inequalities on the original space, implied by the inequalities of the extended formulation. We will use a generalization of the Degree Constrained Spanning Tree problem as a setting for this study. In the present work, besides contextualizing these enhancements in terms of discretized models presented in previous works, we also compare and combine together them, for the first time.