On the minimum size of representative volume element: An experimental investigation

In this paper, the minimum size of the representative volume element (RVE) of a heterogeneous material is determined experimentally using the digital image correlation technique. Experiments of uniaxial compression and thermal expansion were conducted on PBS 9501, a high explosive simulant material. The minimum size of the RVE of the PBS 9501 heterogeneous material, where the average crystal diameter of the material is of the order of 100 μm, was determined to be approximately 1.5 mm. This result is consistent with numerical calculations on polycrystalline materials and some other composites.     

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).     

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.     

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.     

Theory of the elastic moduli of a heterogeneous medium

It is shown that the theory of random functions permits the expansion of the effective tensor X~jkl for the elastic moduli with respect to correlation functions and that it leads in the second approximation in the Voigt-Reuss scheme to values that lie to one side of the Xijkl, while in the third approximation it brackets the latter. The analysis is used to refine the Hashin limits to the elastic moduli for a mechanical mixture of isotrcpic components and polycrystalline aggregates of cubic structure.There are two methods for calculating the effective elastic moduli of heterogeneous solids: virial expansion [2] (as a power series in the concentration of one of the components) and the method of correlation functions [2] (expansion with respect to relative fluctuation of the elastic moduli). Identical results should be obtained in the two cases if all terms are incorporated, but great mathematical difficulties restrict one to the lowest approximations. The first approximation in the virial method gives better results when the concentration of one component is low, while the method of correlation functions gives better results when the fluctuations in the elastic moduli are small and the concentrations are similar.Methods have been developed for determining the upper and lower bounds in both approaches, and various schemes of averaging are used for this purpose in the correlation-function method. The upper bound is established by renormalizing the equation of equilibrium, while the lower one is found by renormalizing the equation of incompatibility. The range of the bracketing can be reduced by means of higher approximations. The range can be reduced in the limit to zero, which implies passing from an approximate effective tensor to the true one, which relates the means in stress and strain over the material. Here we show that the two methods of renormalization give identical results when all terms of the series are summed.If the tensor has a Gaussian distribution, the moment functions of odd order are zero, while the even ones are expressed via combinations of the binary functions [3]. However, a mechanical mixture of several components is not Gaussian, and the odd moments are not zero. Splitting of the higher-order correlation functions is possible also for mechanical mixtures having determinate phase interfaces, but this involves various simplifying assumptions. A derivation is given for a moment of arbitrary order, which allows one to formulate the conditions under which such splitting is possible. The results are used in calculating the exact value of the effective bulk modulus for a medium with a homogeneous shear modulus.We are indebted to V. V. Bolotin for a discussion.     

The elastic moduli estimation of the solid-water mixture

Conceptually, the undrained elastic constants estimated by the poroelasticity theory should be identical to the effective moduli of the two-phase composite of a porous material saturated with pore water. Here we show numerically that the undrained elastic constants determined by an effective moduli estimate are almost identical with those calculated by poroelasticity theory, and if pore shapes are not exactly known and the porosity is around 50%, estimating the elastic constant as the average value of its Voigt and Reuss bounds is reasonably accurate. This is the situation in bone and dentin, the materials that are our primary intended application. This result will hold for situations in which the totally enclosed water phase is constrained to small deformations by virtue of its confinement. Importantly, in this work we assume that water is an isotropic elastic solid with a shear modulus that is 10^?4 times the bulk modulus of the water. Note that it is compressible, but almost incompressible with a Poisson’s ratio of 0.4999.     

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.     

Evaluation of the elastic moduli of a transversely isotropic aggregate of cubic crystals

The crystals and the aggregate have the same bulk modulus. The other three overall elastic moduli in the simplified stress-strain relations of Walpole (1985) are placed between upper and lower, Voigt and Reuss, bounds and some exact calculations are given for particular fibre textures.     

Bounds on the elastic moduli of statistically isotropic multicomponent materials and random cell polycrystals

Minimum energy and complementary energy principles are used to derive the upper and lower bounds on the effective elastic moduli of statistically isotropic multicomponent materials in d ($d=2$ or 3) dimensions. The trial fields, involving harmonic and biharmonic potentials, and free parameters to be optimized, lead to the bounds containing, in addition to the properties and volume proportions of the material components, the three-point correlation information about the microgeometries of the composites. The relations and restrictions among the three-point correlation parameters are explored. The upper and lower bounds are specialized to symmetric cell materials and asymmetric multi-coated spheres, which are optimal or even converge in certain cases. New bounds for random cell polycrystals are constructed with particular results for random aggregates of cubic crystals.     

Surface and interfacial waves of arbitrary form in isotropic elastic media

A method due to Friedlander of accommodating disturbances of arbitrary form into the theory of surface waves in a semi-infinite isotropic elastic body is extended and shown to yield a simple closed form solution for the displacement field. An analogous treatment of interfacial waves of arbitrary form at a plane contact discontinuity separating different isotropic elastic materials is also given.

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Résumé On développe une méthode, conçue par Friedlander, qui fait entrer les perturbations de forme arbitraire dans la théorie des ondes de surface dans un corps élastique isotropique semi-infini, et on montre qu'elle permet d'obtenir une solution simple et exacte pour le champ de déplacement. Les ondes de forme arbitraire qui existent dans le plan à la frontière de materiaux élastiques isotropiques differents sont traitées de façon analogue.

     

Upscaling of conductivity of heterogeneous formations: General approach and application to isotropic media

The numerical simulation of flow through heterogeneous formations requires the assignment of the conductivity value to each numerical block. The conductivity is subjected to uncertainty and is modeled as a stationary random space function. In this study a methodology is proposed to relate the statistical moments of the block conductivity to the given moments of the continuously distributed conductivity and to the size of the numerical blocks. After formulating the necessary conditions to be satisfied by the flow in the upscaled medium, it is found that they are obeyed if the mean and the two-point covariance of the space averaged energy disspation function over numerical elements in the two media, of point value and of upscaled conductivity, are identical. This general approach leads to a systematic upscaling procedure for uniform average flow in an unbounded domain. It yields the statistical moments of upscaled logconductivity that depend only on those of the original one and on the size and shape of the numerical elements.The approach is applied to formations of isotropic heterogeneity and to isotropic partition elements. After a general discussion based on dimensional analysis, the procedure is illustrated by using a first-order approximation in the logconductivity variance. The upscaled logconductivity moments (mean, two-point covariance) are computed for two and three dimensional flows, isotropic heterogeneous media and elements of circular or spherical shape. The asymptotic cases of elements of small size, which preserve the point value conductivity structure on one hand, and of large blocks for which the medium can be replaced by one of deterministic effective properties, on the other hand, are analyzed in detail. The results can be used in order to generate the conductivity of numerical elements in Monte Carlo simulations.Nomenclature C covariance - e rate of dissipation of mechanical energy per unit weight of fluid - E total rate of energy dissipation in the flow domain - H overlap function - K hydraulic conductivity - K _G geometrical mean of conductivity - I integral scale - J=P mean head gradient - L characteristic size of - l characteristic size of also diameter of circle and sphere - n number of dimensions - P pressure head - Q total fluid discharge - S _A ,S _B inlet and outlet boundaries of flow domain - v velocity - Y logconductivity - characteristic scale of flow nonuniformity - autocorrelation function - ² variance - flow domain - partition element Overlining space averaged over - Ã upscaled quantity - â Fourier transform ofa     

Evolution of the fronts of quasi-compressional and quasi-shear discontinuous waves in inhomogeneous transversely isotropic elastic media

The propagation and evolution of the fronts of discontinuous waves in inhomogeneous transversely isotropic elastic media are studied. A method to draw evolving rays and fronts is proposed. Geometrical singularities on the fronts are studied for different parameters of anisotropy and inhomogeneity     

Assessment of existing and introduction of a new and robust efficient definition of the representative volume element

Accurate numerical homogenization necessitates the thorough determination of the Representative Volume Element (RVE). There exists several seminal works on the notion of the RVE in homogenization, its definitions and methods of determination for efficient computation of composite effective properties. The objective of the current work is to assess the ability of numerical RVE determination methods to deliver accurate effective properties of composite materials. This paper demonstrates that common and well-established RVE determination methods, based on studying the convergence rate of the effective properties with respect to the volume element size, are invalid for the case of composites reinforced by randomly oriented fibers and yield erroneous estimates of their effective properties. Following the failure of traditional RVE determination methods, we proposed a new RVE determination criterion that is not based on the average property stability, but its statistical variations. Our new proposed criterion has been shown to be more accurate than other criteria in computing the effective properties of composites for aspect ratios up to 60. Moreover, the proposed criterion does not necessitate a convergence study over the volume element size, hence reducing considerably the RVE determination cost. Finally, our work questions the validity of many published works dealing with composites including heterogeneities of high aspect ratios.     

高导无氧铜杆的冲击拉伸断裂系列实验以及基于样本体积单元的分析

采用Hopkinson装置和一种基于一级气体炮的高速冲击拉伸断裂装置,研究了无刻槽高导无氧铜 (OFHC)杆在一系列冲击拉伸速度下的断裂。当冲击拉伸速度大于40m/s时,断裂位置总在冲击拉伸端附 近,此速度被确定为OFHC的实验临界冲击拉伸速度。一种受单轴冲击拉伸荷载的、中心含椭球空穴的样本 体积单元被用于数值模拟所含空穴的增长与失稳的过程。OFHC的J-C与Z-A 本构关系用于描述基体材料 的动态响应。讨论了空穴失稳条件并提出以空穴形状演化为判据,比较了空穴失稳时的样本体积单元平均径 向应变与无刻槽杆的冲击断裂应变。也用这种样本体积单元模型分析了OFHC的实验临界冲击拉伸速度。