Quantification of stochastically stable representative volumes for random heterogeneous materials

This paper details a procedure to determine lower bounds on the size of representative volume elements (RVEs) by which the size of the RVE can be quantified objectively for random heterogeneous materials. Here, attention is focused on granular materials with various distributions of inclusion size and volume fraction of inclusions. An extensive analysis of the RVE size dependence on the various parameters is performed. Both deterministic and stochastic parameters are analysed. Also, the effects of loading mode and the parameter of interest are studied. As the RVE size is a function of the material, some material properties such as Young's modulus and Poisson's ratio are analysed as factors that influence the RVE size. The lower bound of RVE size is found as a function of the stochastically distributed volume fraction of inclusions; thus the stochastic stability of the obtained results is assessed. To this end a newly defined concept of stochastic stability (DH-stability) is introduced by which stochastic effects can be included in the stability considerations. DH-stability can be seen as an extension of classical Lyapunov stability. As is shown, DH-stability provides an objective tool to establish the lower bound nature of RVEs for fluctuations in stochastic parameters.     

Constitutive response of idealized granular media under the principal stress axes rotation

This paper is dedicated to the understanding of the phenomena, which give rise to anisotropy and non-coaxiality in granular materials. In achieving three-dimensional numerical simulation under static condition of granular media, granular element method (GEM) is adopted in this study. The method has been incorporated into the so-called mathematical homogenization theory for quasi-static equilibrium problems, which enables us to obtain the macroscopic/phenomenological inelastic deformation response of a representative volume element (RVE). To examine the anisotropic macroscopic deformation properties of the assumed RVE, which is solved by granular element method (GEM), a series of numerical experiments involving the pure rotation of the principal stress axes are carried out, and its results are discussed in relation to induced anisotropy and non-coaxiality.     

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.     

考虑内部胞元能量等效的代表体元法

具有周期性胞元的超轻质材料在制造和应用过程中,不可避免地会出现基体材料、微结构拓扑和尺寸的随机性变化.此时,评价材料的等效弹性性能需要借助基于均匀化方法(周期性边界条件)或代表体元法(周期性边界条件,均匀应力或均匀应变边界条件等)的蒙特卡洛模拟.该文首先通过算例分析和比较了不同边界条件下的数值结果,讨论了结果的尺度效应和对胞元选取的依赖性.为了提高和改善Dirichlet边界条件下的计算效率和结果,提出了一种考虑内部胞元能量等效的代表体元法.该方法能够有效削弱边界条件和胞元选取的影响,从而实现了采用较小的代表体元得到更好的结果.数值算例验证了方法在预测确定性材料和随机性材料等效模量时的有效性.     

基于偶应力理论的格栅材料等效介质模型

考察了结构最小尺寸与材料特征长度量级相当的格栅材料等效性能,建议了基于偶应力理论的格栅材料等效介质模型以及确定等效模量的代表体元模型,给出了相应的位移边界条件. 在此基础上导出了正交各向异性偶应力介质的特征长度表达式和偶应力介质梁的抗弯刚度表达式,定义了偶应力影响因子\delta以表征梁的偶应力效应. 具体计算了几种典型的格栅材料的等效偶应力模量以及格栅梁在一定工况下的挠曲线,并与相应的有限元离散解进行对比,结果表明,等效结果具有较高精度,且当宏观结构的尺寸和微结构尺寸相差不大时,宏观结构表现出强烈的偶应力效应.偶应力介质的特征长度表征了偶应力效应的强弱,进而分析了格栅材料的相对密度,单胞尺寸以及几何构型对等效介质特征长度的影响.      

金属材料的强度与应力-应变关系的球压入测试方法

压入法获取材料单轴应力-应变关系和抗拉强度对服役结构完整性评价有重要的基础意义.假定材料均匀连续、各向同性、应力应变关系符合Hollomon律,基于能量等效假定,即代表性体积单元(representativevolume element, RVE)的vonMises等效和有效变形域内能量中值等效假定,本文提出了关联材料载荷、深度、球压头直径和Hollomon律的四参数半解析球压入(semi-analyticalspherical indentation,SSI)模型.通过球压入载荷-深度试验关系获得材料的应力-应变关系和抗拉强度.考虑压入过程中的损伤效应,针对金属材料提出了用于球压入测试的材料弹性模量修正模型.对11种延性金属材料完成了球压入试验,采用本文提出的球压入试验方法测到的弹性模量、应力-应变关系和抗拉强度与单轴拉伸试验结果吻合良好.      

桁架板等效刚度分析

桁架材料的连续介质等效模型的研究已有相当基础,而工程中桁架材料往往以类板结构形式出现,其变形表现出明显的弯曲特征。将类板桁架材料采用弯曲板模型模拟,研究合理的方法确定等效板模型的刚度具有重要意义。本文在基于Kirchhoff假定的小挠度薄板弹性理论框架下,研究了类板桁架材料的等效弯曲薄板模型,提出了确定薄板模型等效刚度的基于Dirichlet位移边界条件的代表体元法,给出了确定各刚度系数所对应的代表体元的边界位移形式。具体计算了几种典型形式桁架板的等效刚度,并采用有限元离散模型和实验技术分析了桁架板在一定的边界约束和荷载作用下的响应,并与等效板模型的分析结果进行了对比。结果表明,在响应分析中,具有等效刚度的薄板模型可准确模拟类板桁架材料;连续介质板等效刚度计算的积分法不能给出准确的桁架板等效刚度,而基于Dirichlet位移边界条件的代表体元法获得的等效板的刚度具有很高的精度。     

Numerical analysis of effective elastic properties of geomaterials containing voids using 3D random fields and finite elements

The purpose of the study is to investigate the influence of porosity and void size on effective elastic geotechnical engineering properties with a 3D model of random fields and finite element. The random field theory is used to generate models of geomaterials containing spatially random voids with controlled porosity and void size. A “tied freedom” analysis is developed to evaluate the effective Young’s modulus and Poisson’s ratio in an ideal block material of finite elements. To deliver a mean and standard deviation of the elastic parameters, this approach uses Monte-Carlo simulations and finite elements, where each simulation leads to an effective value of the property under investigation. The results are extended to investigate an influence of representative volume element (RVE). A comparison of the effective elastic stiffness of 2D and 3D models is also discussed.     

Scaling function, anisotropy and the size of RVE in elastic random polycrystals

This article is focused on the identification of the size of the representative volume element (RVE) in linear elastic randomly structured polycrystals made up of cubic single crystals. The RVE is approached by setting up stochastic Dirichlet and Neumann boundary value problems consistent with the Hill(-Mandel) macrohomogeneity condition. Within this framework we introduce a scaling function that relates the single crystal anisotropy to the scale of observation. We derive certain exact characteristics of the scaling function and postulate others based on detailed calculations on copper, lithium, tantalum, magnesium oxide and antimony-yttrium. In deriving the above, we make use of the fact that cubic crystals and polycrystals have a uniquely determined scale-independent bulk modulus. It turns out that the scaling function is exact in the single crystal anisotropy. A methodology to develop a material selection diagram that clearly separates the microscale from the macroscale is proposed. The proposed scaling function not only bridges the length scales but also unifies the treatment of a wide spectrum of cubic crystals. Although the scope of this article is restricted to aggregates made up of cubic-shaped and cubic-symmetry single crystals, the concept of the scaling function can be generalized to other crystal shapes and classes as well as to scaling of other elastic/inelastic properties.     

Generalized self-consistent estimation of the apparent isotropic elastic moduli and minimum representative volume element size of heterogeneous media

Under investigation is a heterogeneous material consisting of an elastic homogeneous isotropic matrix in which layered elastic isotropic inclusions or pores are embedded. The generalized self-consistent model (GSCM) is extended so as to be capable of estimating the apparent elastic properties of a finite-size specimen smaller than a representative volume element (RVE). The kinematical or static apparent shear modulus is determined as a root of a cubic polynomial equation instead of a quadratic polynomial equation as in the classical GSCM of Christensen and Lo [Christensen, R.M., Lo, K.H., 1979. Solutions for effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Solids 27, 315–330]. It turns out that the extended GSCM establishes a link between the composite sphere assemblage model (CSAM) of Hashin [Hashin, Z., 1962. The elastic moduli of heterogeneous materials. J. Appl. Mech. 29, 143–150] and the classical GSCM. Demanding that the normalized distance between the kinematical and static apparent moduli of a finite-size specimen be smaller than a certain tolerance, the minimum RVE size is estimated in a closed form.     

A novel implementation algorithm of asymptotic homogenization for predicting the effective coefficient of thermal expansion of periodic composite materials

Asymptotic homogenization (AH) is a general method for predicting the effective coefficient of thermal expansion (CTE) of periodic composites. It has a rigorous mathematical foundation and can give an accurate solution if the macrostructure is large enough to comprise an infinite number of unit cells. In this paper, a novel implementation algorithm of asymptotic homogenization (NIAH) is devel-oped to calculate the effective CTE of periodic composite materials. Compared with the previous implementation of AH, there are two obvious advantages. One is its implemen-tation as simple as representative volume element (RVE). The new algorithm can be executed easily using commercial finite element analysis (FEA) software as a black box. The detailed process of the new implementation of AH has been provided. The other is that NIAH can simultaneously use more than one element type to discretize a unit cell, which can save much computational cost in predicting the CTE of a complex structure. Several examples are carried out to demonstrate the effectiveness of the new implementation. This work is expected to greatly promote the widespread use of AH in predicting the CTE of periodic composite materials.     

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.     

The elastic response of a cohesive aggregate—a discrete element model with coupled particle interaction

A model is presented for the deformation of a cohesive aggregate of elastic particles that incorporates two important effects of large-sized inter-particle junctions. A finite element model is used to derive a particle response rule, for both normal and tangential relative deformations between pairs of particles. This model agrees with the Hertzian contact theory for small junctions, and is valid for junctions as large as half the nominal particle size. Further, the aggregate model uses elastic superposition to account for the coupled force–displacement response due to the simultaneous displacement of all of the neighbors of each particle in the aggregate. A particle stiffness matrix is developed, relating the forces at each junction to the three displacement degrees of freedom at all of the neighboring-particle junctions. The particle response satisfies force and moment equilibrium, so that the model is properly posed to allow for rigid rotation of the particle without introducing rotational degrees of freedom. A computer-simulated sintering algorithm is used to generate a random particle packing, and the stiffness matrix is derived for each particle. The effective elastic response is then estimated using a mean field or affine displacement calculation, and is also found exactly by a discrete element model, solving for the equilibrium response of the aggregate to uniform-strain boundary conditions. Both the estimate and the exact solution compare favorably with experimental data for the bulk modulus of sintered alumina, whereas Hertzian contact-based models underestimate the modulus significantly. Poisson's ratio is, however, accurately determined only by the full equilibrium discrete element solution, and shown to depend significantly on whether or not rigid particle rotation is permitted in the model. Moreover, this discrete element model is sufficiently robust, so it can be applied to problems involving non-homogeneous deformations in such cohesive aggregates.     

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.     

Homogenization of geomaterials containing voids by random fields and finite elements

The paper describes the use of random fields and finite elements to assess the influence of porosity and void size on the effective elastic stiffness of geomaterials. A finite element model is developed involving “tied freedoms” that allows analysis of an ideal block of materials leading to direct evaluation of the effective Young’s modulus and Poisson’s ratio. The influence of block size and representative volume elements (RVE) are discussed. The use of random fields and Monte-Carlo simulations deliver a mean and standard deviation of the elastic parameters that lead naturally to a probabilistic interpretation. The methodology is extended to a foundation problem involving a footing on an elastic foundation containing voids. The approach enables estimates to be made of the probability of excessive settlement.     

Mechanical properties of nanostructure of biological materials

Natural biological materials such as bone, teeth and nacre are nanocomposites of protein and mineral with superior strength. It is quite a marvel that nature produces hard and tough materials out of protein as soft as human skin and mineral as brittle as classroom chalk. What are the secrets of nature? Can we learn from this to produce bio-inspired materials in the laboratory? These questions have motivated us to investigate the mechanics of protein-mineral nanocomposite structure. Large aspect ratios and a staggered alignment of mineral platelets are found to be the key factors contributing to the large stiffness of biomaterials. A tension-shear chain (TSC) model of biological nanostructure reveals that the strength of biomaterials hinges upon optimizing the tensile strength of the mineral crystals. As the size of the mineral crystals is reduced to nanoscale, they become insensitive to flaws with strength approaching the theoretical strength of atomic bonds. The optimized tensile strength of mineral crystals thus allows a large amount of fracture energy to be dissipated in protein via shear deformation and consequently enhances the fracture toughness of biocomposites. We derive viscoelastic properties of the protein-mineral nanostructure and show that the toughness of biocomposite can be further enhanced by the viscoelastic properties of protein.     

Measurement of Temperature-Dependent Young’s Modulus at a Strain Rate for a Molding Compound by Nanoindentation

The mechanical properties of a molding compound on a packaged integrated circuit (IC) were measured by spherical nanoindentation using a 50 μm radius diamond tip. The molding compound is a heterogeneous material, consisting of assorted diameters of glass beads embedded in an epoxy. Statistical analysis was conducted to determine the representative volume element (RVE) size for a nanoindentation grid. Nanoindentation was made on the RVE to determine the effective viscoelastic properties. The relaxation functions were converted to temperature-dependent Young’s modulus at a given strain rate at several elevated temperatures. The Young’s modulus values at a given strain rate from nanoindentation were found to be in a good agreement with the corresponding data obtained from tensile samples at or below 90 °C. However, the values from nanoindentation were significantly lower than the data obtained from tensile samples when the temperature was near or higher than 110 °C, which is near the glass transition. The spatial distribution of the Young’s modulus at a given strain rate was determined using nanoindentation with a Berkovich tip. The spatial variation of the Young’s modulus at a given strain rate is due to the difference in nanoindentation sites (glass beads, epoxy or the interphase region). A graphical map made from an optical micrograph agrees reasonably well with the nanoindentation results.     

Effective couple-stress continuum model of cellular solids and size effects analysis

The purpose of this paper is to develop a homogeneous, orthotropic couple-stress continuum model to take the place of the periodic heterogeneous cellular solids. Through generalizing the definition of the characteristic length for isotropic couple-stress continuum, four characteristic lengths are introduced as material engineering constants for such kind of continuum. In order to determine the effective moduli and the characteristic lengths of the effective couple-stress continuum, a Representative Volume Element (RVE) method is constructed. The effective properties are obtained based on the response of the RVE under prescribed boundary conditions, and our results agree with the analytical solutions in literature. In addition, the influences of the relative density, the topology, the size, and the properties of the solid material of cellular materials on the effective moduli as well as the characteristic lengths are discussed, respectively. Furthermore, the size effects in cellular solid beams are investigated using our effective couple-stress continuum model. The results show that the developed continuum model in this paper can precisely capture the size effects in cellular solids.     

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.     

Effective pressure-sensitive elastoplastic behavior of particle-reinforced composites and porous media under isotropic loading

The composite under investigation consists of an elastoplastic matrix reinforced by elastic particles or weakened by pores. The material forming the matrix is pressure-sensitive. The Drucker–Prager yield criterion and a one-parameter non-associated flow rule are employed to formulate the yield behavior of the matrix. The objective of this work is to estimate the effective elastoplastic behavior of the composite under isotropic tensile and compressive loadings. To achieve this objective, the composite sphere assemblage model of Hashin [Z. Hashin, The elastic moduli of heterogeneous materials, ASME J. Appl. Mech. 29 (1962) 143–150] is used. Exact solutions are thus derived as estimations for the effective secant and tangent bulk moduli of the composite. The effects of the loading modes and phase properties on the effective elastoplastic behavior of the composite are analytically and numerically evaluated.