Some basic ideas for the reconstruction of statistically similar microstructures for multiscale simulations

In this contribution we propose a method for the generation of statistically similar representative volume elements (RVE's). These RVE's are obtained by assuming that given real microstructures may be represented by periodic ones. Then an optimization problem has to be solved where the side condition of equal spectral density of the real microstructure and the statistically similar RVE is taken into account. First numerical results show that the proposed method works in principle and leads to more efficient direct multiscale simulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Construction of Statistically Similar Representative Volume Elements Based on the Lineal-Path Function

In this contribution we propose a method for the construction of statistically similar representative volume elements (SSRVEs) which are characterized by a much less complexity than usual random RVEs and which represent quite accurately the mechanical response of the real material. For the design of such SSRVEs an objective function is minimized taking into account least-square functionals based on suitable statistical measures, that characterize the inclusion morphology. Here, we find that the incorporation of the lineal-path function leads to promising results in FE²-simulations of macroscopically homogeneous boundary value problems as well as inhomogenous ones. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Identification of the lamination stack's behavior simulated with a multi-scale homogenization using a progressive contact formulation

The influence of the lamination stack on the mechanical behavior of an electrical motor is significant, especially for lightweight designs. Information about the stack's behavior is relevant for a proper calculation of the whole motor. This behavior is dependent on the contact between the single sheets resulting from the roughness and the behavior of the roughness peaks. The normal behavior is an elasto-plastic and progressive one, which results mainly of the number of roughness peaks in contact. Whereas the tangential micro-slip behavior is dependent on the single contacts and the friction coefficient. Both behaviors can be described by models and thus it is possible to calculate the behavior of a whole stack using an homogenization based on representative volume elements. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Enhancement of the micro mechanical basis for local approach cleavage models

The present study is concerned with the investigation of the micro mechanisms of micro defect nucleation in bainitic steels in order to provide an enhanced basis for probabilistic cleavage models. By a micro mechanical modelling of the cleavage initiation process the effects and the interactions of the relevant parameters can be identified. For this purpose Representative Volume Elements (RVE) of the micro structure are utilised, accounting for both, the grain structure as well as the brittle particles at the grain boundaries. The RVE's are loaded based on the local mechanical field quantities determined numerically at the cleavage origins of different fracture mechanics specimens. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Kloosterman Sums and a Problem of D. H. Lehmer

A classical problem of D. H. Lehmer suggests the study of distributions of elements of $\bZ/p\bZ$ of opposite parity to the multiplicative inverse mod $p$. Zhang initiated this problem and found an asymptotic evaluation of the number of such elements. In this paper, an asymptotic formula for the fourth moment of the error term of Zhang is proved, from which one may see that Zhang''s error term is optimal up to the logarithm factor. The method also applies to the case of arbitrary positive integral moments.     

Effective parameters of cancellous bone

Cancellous bone is a two–component structure consisting of the bone frame and interstitial blood marrow. In the scope of this presentation, the multiscale finite element method is used for its modeling. This method results from a combination of homogenization theory and the theory of finite elements and is based on the calculation of effective material parameters by investigating representative volume elements (RVEs). For the particular kind of material considered here, a cubic two–phase RVE is assumed where the dry skeleton is modeled in different ways. Apart from the variations of the geometry, the influence of the usage of different types of finite elements is studied in this context. Note that the presence of a liquid phase requires dynamic investigation including the viscous phenomena. To this end, acoustic excitation and an analysis in the complex domain are chosen. The method permits calculation of the effective material parameters such as Young's modulus, bulk modulus and Poisson's ratio and furthermore the simulation of the behaviour of the complete bone or of its parts. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A counterexample to the generalization of Sperner's theorem

It has been conjectured that the analog of Sperner's theorem on non-comparable subsets of a set holds for arbitrary geometric lattices, namely, that the maximal number of non-comparable elements in a finite geometric lattice is max w(k), where w(k) is the number of elements of rank k. It is shown in this note that the conjecture is not true in general. A class of geometric lattices, each of which is a bond lattice of a finite graph, is constructed in which the conjecture fails to hold.     

Experimental Validation of RVE Based Failure Simulations of Macro-Porous Materials

Connections between inhomogeneities and the failure behavior of brittle material may be investigated by finite element simulations of representative volume elements. Representative volume elements are typically subjected to periodic boundary conditions. Moreover, representative volume elements are often chosen as planar, i. e., two dimensional in order to reach reasonable statistics with regard to random distributions of inhomogeneities. The significance of such strongly simplified simulations needs to be validated, especially if the matrix failure is potentially dominated by defects, as is the case, e. g., in macro-porous ceramics. We propose a quasi-periodic concept to design specimens with cylindrical pores, which reproduce the stress state in a two dimensional representative volume element. This is achieved by a partial periodic replication of the region of interest. We suggest that material models used in simulations can be assessed by comparison between simulated and experimentally observed failure. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Simple Model for the Simulation of Delamination in Fiber-ReinforcedComposite Laminates under Mixed-Mode Loading Conditions

For a reliable prediction of the mechanical behavior of unidirectional fiber-reinforced composite laminates (FRCL), it is inevitable to take into account various damage and fracture mechanisms. In this work, delamination under arbitrary mixedmode loading conditions is examined in the framework of the finite element method. Delamination is assumed to be caused by failure of the resin-rich area in the interface between two layers of FRCL's. In this work, a cohesive interface elementin terms of natural stress-strain relationships which allows to describe the interlaminar mechanical behavior of FRCL's is introduced. The proposed model prevents the restoration of cohesion in the interface. The interpenetration of the crack faces is avoided by incorporating a simple contact algorithm. A representative numerical example shows the applicability of the proposed concept. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical modelling of regular structures with randomly perturbed structural elements

The problem of the effective elastic properties of regular composites with randomly perturbed geometric and mechanical parameters is formulated and solved numerically. Mean sample values and standard deviations of compliances are used to characterize the elastic properties. Compliance data are obtained by solving reduction problems for each of a set of realizations of random perturbations, and here the number of realizations is increased until the values of the statistical means become stable (within a given tolerance). Calculations for each realization are carried out by numerical solution of the complex hypersingular boundary integral equations obtained for a doubly periodic structure. The principal cell of this structure, containing a fairly large number of perturbed elements, is identified with a representative volume when a further increase in the number of perturbed elements do not alter the statistical means (again, within a given tolerance). Calculations are carried out for square and triangular grids with different densities of circular inclusions or holes, the centre coordinates of which are given random perturbations (weak, medium and strong). The results of the calculations are summed up in tables showing the effective compliances with an accuracy to at least three significant digits. An analysis of the values obtained for the holes shows’ that, with a tolerance of 5%, the principal cell of a square grid with four holes determines the representative volume for all the geometric parameter combinations investigated. For rigid inclusions this cell is the representative volume at a considerably greater tolerance than for compliant inclusions (4% as against 0.9%). Data on the effective properties of perturbed structures indicate that the difference between their compliance and that of the initial regular structures depends substantially on the relative stiffness of the inclusions. It is most marked for holes and rigid inclusions (9.5 and 12.6% respectively). It is established that, for a square grid, random perturbations have a stronger effect on the normal components of the compliance than on the shear component, and the opposite for a regular triangular grid – perturbations have a greater effect on the shear compliance. Calculations also show that symmetrical perturbations of holes (rigid inclusions) along one of the coordinates lead to a marked increase (reduction) in compliance in the orthogonal direction. The established dependence of the additional effective compliance on the amplitude of the perturbation enables the inverse problem to be solved: to find the parameters of the perturbed structure using data on its effective statistical properties.     

Strong liftings with application to measurable cross sections in locally compact groups

There are two principal theorems. The adjustment theorem asserts that a lifting may be changed on a set of measure zero so as to become slightly stronger. In conjunction with the standard lifting theorem, it yields generalizations (with shorter proofs) of a number of known results in the theory of strong liftings. It also inspires a characterization of strong liftings, when the measure is regular, by the fact that they induce upon every open set an artificial “closure” of that set which differs from it by a set of measure zero. The projection theorem asserts that, in the presence of a strict disintegration, a strong lifting may be transferred or “projected” from one topological measure space onto another. In conjunction with Losert's example, it yields regular Borel, measures, carried on compact Hausdorff spaces of arbitrarily large weight, which everywhere fail to have the strong lifting property. It also provides the final link needed to obtained, with no separability assumptions, a measurable cross section (or right inverse) for the canonical map Ω:G→G/H, whereG is an arbitrary locally compact group, and whereH is an arbitrary closed subgroup ofG.     

Stabilized Mixed Tetrahedrals with Volume and Area Bubble Functions at Large Deformations

In this contribution, stabilized mixed finite tetrahedral elements are presented in order to avoid volume locking and stress oscillations. Geometrically non-linear elastic problems are addressed. The mixed method of incompatible modes is considered. As a key idea, volume and area bubble functions are used for the method of incompatible modes [1], thus giving it the interpretation of a mixed finite element method with stabilization terms. Concerning non-linear problems these are non-linearly dependent on the current deformation state, however, linearly dependent stabilization terms are used. The approach becomes most attractive for the numerical implementation, since the use of quantities related to the previous Newton iteration step is completely avoided. The variational formulation for the standard two-field method, the method of incompatible modes in finite deformation problems is derived for a hyper elastic Neo-Hookean material. In the representative examples Cook's membrane problem and a block under central pressure illustrate the good performance of the presented approaches compared to existing finite element formulations. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stabilized mixed triangular finite elements at large deformations using area bubble functions

In this paper stabilized mixed triangular finite elements are presented in order to avoid volume locking and to damp stress oscillations. Geometrically non-linear elastic problems are addressed. The mixed method of incompatible modes and the mixed method of enhanced strains are considered as special cases. As a key idea, volume and area bubble functions are used for the method of incompatible modes and the enhanced strain method [1], thus giving both the interpretation of a mixed finite element method with stabilization terms. Concerning non-linear problems these are non-linearly dependent on the current deformation state, however, linearly dependent stabilization terms are used [1]. The approach becomes most attractive for the numerical implementation, since the use of quantities related to the previous Newton iteration step is completely avoided. The variational formulation for the standard two-field method, the method of incompatible modes and the enhanced strain method in finite deformation problems is derived for a hyper elastic Neo-Hookean material. In the representative example Cook's membrane problem illustrates the good performance of the presented approaches compared to existing finite element formulations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Use of Halley's Method in the Nonlinear Finite Element Analysis

Halley's method is a higher order iteration method for the solution of nonlinear systems of equations. Unlike Newton's method, which converges quadratically in the vicinity of the solution, Halley's method can exhibit a cubic order of convergence. The equations of Halley's method for multiple dimensions are derived using Padé approximants and inverse one-point interpolation, as proposed by Cuyt. The investigation of the performance of Halley's method concentrates on eight-node volume elements for nonlinear deformations using Staint Venant-Kirchhoff's constitutive law, as well as a geometric linear theory of von Mises plasticity. The comparison with Newton's method reveals the sensibility of Halley's method, in view of the radius of attraction but also demonstrates the advantages of Halley's method considering simulation costs and the order of convergence. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非均匀变厚度梁的动力响应的一般解

在本文中提出一个新方法——阶梯折算法来研究在任意载荷下任意非均匀和任意变厚度伯努利-欧拉梁的动力响应问题.研究了自由振动和强迫振动.新方法需要将区间离散为一定数目的元素,每个元素可看作是均匀和等厚度的.因此均匀、等厚度梁的一般解可在每个元素上应用.然后用初参数表示的整个梁的一般解使之满足相邻二元素间的物理和几何连续条件,这样就可以得到解析形式的自由振动的频率方程和解析形式的强迫振动的最终解,它化为求解二元线性代数方程,与离散元素的数目无关.现在的方法可推广应用至任意非均匀及任意变厚度有粘滞性和其他种类的梁以及其他结构元件问题上去.     

Flow-induced anisotropic viscosity in short fiber reinforced polymers

The commonly used flow models for fiber reinforced polymers often neglect the flow induced mechanical anisotropy of the suspension. With an increasing fiber volume fraction, this plays, however, an important role. There are some models which count on this effect, they are, however, phenomenological and require a fitted model parameter. In this paper, a micromechanically based constitutive law is proposed which considers the flow induced anisotropic viscosity of the fiber suspension. The introduced viscosity tensor can handle arbitrary anisotropy of the fluid-fiber mixture depending on the actual fiber orientation distribution. A homogenization method for unidirectional structures in contribution with orientation averaging is used to determine the effective viscosity tensor. The motion of rigid ellipsoidal fibers induced by the flow of the matrix material is described by Jeffery's equation. A numerical implementation of the introduced model is applied to representative flow modes. The calculated stress values are analyzed in transient and stationary flow cases. They show a less pronounced anisotropic viscous behaviour in every investigated case compared to the results obtained by the use of the Dinh-Armstrong constitutive law. The reason for the qualitative difference is that the presented model depends on the complete orientation information, while the other one is linear in the fourth-order fiber orientation tensor. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local probabilistic homogenization schemes for assessment of material uncertainties in solid foams

The present study is concerned with a numerical scheme for prediction of the effective properties of solid foams considering their uncertainty. The approach is based on an analysis of a large-scale, statistically representative volume element which is subdivided into small-scale testing volume elements. Application of a standard homogenization scheme to the testing volume elements together with a stochastic evaluation yields a complete probabilistic characterization of the material which may be used for a random field definition of the material behaviour in a macroscopic effective field analysis of foam structures. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vizing's conjecture: a survey and recent results

Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw‐free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 46–76, 2012     

Two Remarks on Partitions of ω with Finite Blocks

We prove that all algebras P(w)/I_R, where the I_R-'s are ideals generated by partitions of W into finite and arbitrary large elements, are isomorphic and homogeneous. Moreover, we show that the smallest size of a tower of such partitions with respect to the eventually-refining preordering is equal to the smallest size of a tower on ω.