Exterior statistics based boundary conditions for representative volume elements of elastic composites

Statistically equivalent representative volume elements or SERVEs are representations of the microstructure that are used for micromechanical simulations to generate homogenized material constitutive responses and properties (Swaminathan et al., 2006a, Ghosh, 2011). Typically, a SERVE is generated from the parent microstructure as a statistically equivalent region, whose size is determined from the requirements of convergence of macroscopic properties. Standard boundary conditions, such as affine transformation-based displacement boundary conditions (ATDBCs), uniform traction boundary conditions (UTBCs) or periodic boundary conditions (PBCs) are conventionally applied on the SERVE boundary for micromechanical simulations. However, when the microstructure is characterized by arbitrary, nonuniform distributions of heterogeneities, these simple boundary conditions do not represent the effect of regions exterior to the SERVE. Improper boundary conditions can result in significantly larger than optimal SERVE domains, needed for converged properties. In an attempt to overcome the limitations of the conventional boundary conditions on the SERVE, this paper explores the effect of boundary conditions that incorporate the statistics of the exterior region on the SERVE of elastic composites. Using Green's function based interaction kernels, coupled with statistical functions of the microstructural characteristics like one-point and two-point correlation functions, a novel exterior statistics-based boundary condition or ESBC is derived for the SERVE. The advantages of the ESBC are established by comparing with results of simulations using conventional boundary conditions. Results of the SERVE simulations subjected to ESBCs are also compared with those from other popular methods like statistical volume element (SVE) and weighted statistical volume element (WSVE). The proposed ESBCs offer significant advantages over other methods in the SERVE-based analysis of heterogeneous materials.     

New boundary conditions for the computation of the apparent stiffness of statistical volume elements

We present a new auxiliary problem for the determination of the apparent stiffness of a Statistical Volume Element (SVE). The SVE is embedded in an infinite, homogeneous reference medium, subjected to a uniform strain at infinity, while tractions are applied to the boundary of the SVE to ensure that the imposed strain at infinity coincides with the average strain over the SVE. The main asset of this new auxiliary problem resides in the fact that the associated Lippmann–Schwinger equation involves without approximation the Green operator for strains of the infinite body, which is translation-invariant and has very simple, closed-form expressions. Besides, an energy principle of the Hashin and Shtrikman type can be derived from this modified Lippmann–Schwinger equation, allowing for the computation of rigorous bounds on the apparent stiffness. The new auxiliary problem requires a cautious mathematical analysis, because it is formulated in an unbounded domain. Observing that the displacement is irrelevant for homogenization purposes, we show that selecting the strain as main unknown greatly eases this analysis. Finally, it is shown that the apparent stiffness defined through these new boundary conditions “interpolates” between the apparent stiffnesses defined through static and kinematic uniform boundary conditions, which casts a new light on these two types of boundary conditions.     

Periodization of random media and representative volume element size for linear composites

Several existing numerical studies show that the effective linear properties of random composites can be accurately estimated using small volumes subjected to periodic boundary conditions – more suitable than homogeneous strain or stress boundary conditions – providing that a sufficient number of realizations are considered. Introducing the concept of periodization of random media, this Note gives a new definition of representative volume element which leads to estimates of its minimum size in agreement with existing theoretical results. A qualitative convergence criterion for the numerical simulations is proposed and illustrated with finite element computations. To cite this article: K. Sab, B. Nedjar, C. R. Mecanique 333 (2005).     

On selection of repeated unit cell model and application of unified periodic boundary conditions in micro-mechanical analysis of composites

Most micro-mechanical analyses for composites are based on repeated unit cell models (RUCs) by assuming a periodical distribution of the reinforcing phase. In this paper, the uniqueness of solution by applying unified displacement-difference periodic boundary conditions on the RUCs has been proved. Further it is deduced that (1) selection of the RUCs for a fixed periodic array may not be unique, however, the solution is independent on the choice of the different RUCs; (2) boundary traction continuity conditions can be guaranteed by the application of the proposed unified displacement-difference periodic boundary conditions. Illustrative examples are presented and advantages of applying this type of unified periodic boundary conditions are discussed.     

Determination of the size of the representative volume element for random quasi-brittle composites

A representative volume element (RVE) is related to the domain size of a microstructure providing a “good” statistical representation of typical material properties. The size of an RVE for the class of quasi-brittle random heterogeneous materials under dynamic loading is one of the major questions to be answered in this paper. A new statistical strategy is thus proposed for the RVE size determination. The microstructure illustrating the methodology of the RVE size determination is a metal matrix composite with randomly distributed aligned brittle inclusions: the hydrided Zircaloy constituting nuclear claddings. For a given volume fraction of inclusions, the periodic RVE size is found in the case of overall elastic properties and of overall fracture energy. In the latter case, the term “representative” is discussed since the fracture tends to localize. A correlation factor between the “elastic” RVE and the “fracture” RVE is discussed.     

A Quantitative study of minimum sizes of representative volume elements of cubic polycrystals—numerical experiments

The concept of representative volume element (RVE) plays a key role in correlating the properties of microscopically heterogeneous materials with those of their macroscopically homogenized ones. However, up to now little quantitative knowledge is available about RVE scales or sizes of various engineering materials, which have been becoming a necessity due to the rapid development of, for instance, microelectromechanical systems. A new and convenient definition of the minimum RVE size is introduced. Then more than 500 kinds of cubic polycrystalline material in the planar stress state are numerically tested. The major finding from these numerical experiments is that the RVE size for the effective shear modulus (as well as the Young's modulus) depends roughly linearly upon the anisotropy degree of the single crystal, while the effective area modulus does not. For the latter observation a theoretical proof is also given. With a maximum relative error 5%, all the materials tested (with one exception) have a minimal RVE size of 20 or less times as large as the grain size.     

Determination of the size of the representative volume element for random composites: statistical and numerical approach

The representative volume element (RVE) plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties. A quantitative definition of its size is proposed in this work. A RVE size can be associated with a given precision of the estimation of the wanted overall property and the number of realizations of a given volume V of microstructure that one is able to consider. It is shown to depend on the investigated morphological or physical property, the contrast in the properties of the constituents, and their volume fractions. The methodology is applied to a specific random microstructure, namely a two-phase three-dimensional Voronoı̈ mosaic. Finite element simulations of volumes of different sizes are performed in the case of linear elasticity and thermal conductivity. The volumes are subjected to homogeneous strain, stress or periodic boundary conditions. The effective properties can be determined for large volumes and a small number of realizations. Conversely, smaller volumes can be used providing that a sufficient number of realizations are considered. A bias in the estimation of the effective properties is observed for too small volumes for all types of boundary conditions. The variance of computed apparent properties for each volume size is used to define the precision of the estimation. The key-notion of integral range is introduced to relate this error estimation and the definition of the RVE size. For given wanted precision and number of realizations, one is able to provide a minimal volume size for the computation of effective properties. The results can also be used to predict the minimal number of realizations that must be considered for a given volume size in order to estimate the effective property for a given precision. The RVE sizes found for elastic and thermal properties, but also for a geometrical property like volume fraction, are compared.     

A unified approach to direct and inverse boundary layer solutions

A unified approach is presented for solving the two-dimensional incompressible boundary layer equations. Solutions are obtained for direct and inverse options using the same equation formulation by a simple interchange of boundary conditions. A modified form of the mechul function scheme obtains inverse solutions with specification of transformed wall shear, skin friction coefficient or displacement thickness distributions. Direct solutions may be obtained without altering the block tridiagonal structure of the system by simply requiring no corrections on the streamwise pressure gradient parameter. Fourth-order spline discretization approximates normal derivatives with two- and three-point backward differences approximating streamwise derivatives, yielding a fully implicit solution method. The resulting spline/finite difference equations are solved by Newton-Raphson iteration together with partial pivoting. The results of the study demonstrate the importance of proper linearization of all equations. The successful use of spline discretization is also tied to the use of strong two-point boundary conditions at the wall for cases involving reversed flow. Numerical solutions are presented for several non-similar flows and compared with published results.     

复合材料周期结构数学均匀化方法的一种新型单胞边界条件

数学均匀化方法是计算周期复合材料结构的有效方法之一,单胞边界条件施加的合理性直接决定了影响函数控制方程的计算效率和精度,进而影响均匀化弹性参数和摄动位移的计算精度.本文首先将单胞影响函数作为虚拟位移处理,给出了单胞在结构中真实的边界条件,结果表明,四边固支适合作为二维结构单胞边界条件;其次,针对二维结构提出了超单胞周期边界条件,有效提高了影响函数的计算精度,并使用与虚拟位移相对应的虚拟势能泛函验证超单胞周期边界条件的有效性;最后,利用数值分析验证多尺度渐进展开方法的计算精度,强调了二阶摄动的必要性.     

A numerical method for coupling three dimensional nonlinear finite elements with elastic boundary elements

This paper systematically deals with the following three problems: (1) Some numerical schemes in coupling FEM- and BEM: including condensation of the boundary integral equation, symmetrization of equivalent stiffness matrix and treatment of traction discontinuity, (2) Coupling of elastoplastic finite elements to elastic boundary elements, (3) Coupling of elasto-viscoplastic finite elements to elastic boundary elements and numerical stability condition.     

Mixed boundary elements for laminar flows

This paper presents a mixed boundary element formulation of the boundary domain integral method (BDIM) for solving diffusion–convective transport problems. The basic idea of mixed elements is the use of a continuous interpolation polynomial for conservative field function approximation and a discontinuous interpolation polynomial for its normal derivative along the boundary element. In this way, the advantages of continuous field function approximation are retained and its conservation is preserved while the normal flux values are approximated by interpolation nodal points with a uniquely defined normal direction. Due to the use of mixed boundary elements, the final discretized matrix system is overdetermined and a special solver based on the least squares method is applied. Driven cavity, natural and forced convection in a closed cavity are studied. Driven cavity results at Re=100, 400 and 1000 agree better with the benchmark solution than Finite Element Method or Finite Volume Method results for the same grid density with 21×21 degrees of freedom. The average Nusselt number values for natural convection 10³≤Ra≤10⁶ agree better than 0.1% with benchmark solutions for maximal calculated grid densities 61×61 degrees of freedom. Copyright © 1999 John Wiley & Sons, Ltd.     

Residual stresses in random elastic composites: nonlocal micromechanics-based models and first estimates of the representative volume element size

Random elastic composites with residual stresses are examined in this paper with the aim of understanding how the prestress may influence the overall mechanical properties of the composite. A fully non-local effective response is found in perfect analogy with the un-prestressed case examined in (Drugan and Willis, J. Mech. Phys. Solids 44(4):497–524, 1996). The second gradient approximation is considered and the impact of the residual stresses on the estimate of the RVE size is studied whenever the local response is used to describe the mechanical properties of the heterogeneous medium. To this aim, total and incremental formulations are worked out in this paper and the influence of both uniform and spatially varying prestresses are studied. Among other results, it is shown how rapid oscillations of relatively “small” residual stresses in most cases may result in the impossibility of describing the overall behavior of the composite with a local constitutive equation. On the other hand, prestresses with relatively high amplitudes and slow spatial oscillations may even reduce the RVE size required for approximating the mechanical properties of un-prestressed heterogeneous media with a local constitutive equation.     

A certain boundary value problem and its applications

Summary In the present paper a linear second order partial differential system arising in mathematical physics is solved with the help of Laplace transforms, involving more general boundary conditions, in terms of Mathieu functions. Four physical problems are presented and it is indicated that their solutions can be deduced as a particular case.     

自由边界条件轴对称薄圆柱壳体的混合有限元数值解

基于文「８」发展的混合结构问题稳定方法的框架,针对自由边界条件轴对称薄圆柱壳体问题,给出了其新型混合有限元计算模型。应用线性元,已证明该模型是强制的,其对称正定代数方程组的条件数Ｏ（ｈ＾－２）。     

Implementation of slip boundary conditions in the finite volume method: new techniques

Two different techniques for the implementation of the linear and nonlinear slip boundary conditions into a finite volume method based numerical code are presented. For the linear Navier slip boundary condition, an implicit implementation in the system of equations is carried out for which there is no need for any relaxation, especially when handling high slip coefficients. For three different nonlinear slip boundary conditions, two different methods are devised, one based on solving a transcendental equation for the boundary and the other on the linearization of the slip law. For assessment purposes, comparison is made between these new methods and the usual iterative process. With these new methods, the convergence difficulties, typical of the iterative procedure, are eliminated, and for some of the test cases, the convergence rate even increased with the slip velocity. The details of these implementations are given first for a simple geometry using orthogonal meshes and Cartesian coordinates followed by their generalization to non‐Cartesian coordinates and nonorthogonal meshes. The developed code was tested in the benchmark slip‐stick and 4:1 contraction flows, evidencing the robustness of the proposed procedures. Copyright © 2013 John Wiley & Sons, Ltd.