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.     

随机孔隙缺陷对3D编织复合材料力学性能的影响研究

3D编织复合材料经RTM工艺成型将难以避免产生孔隙缺陷,孔隙缺陷主要包括纤维束内干斑和富树脂区孔隙。经分析RTM工艺孔隙缺陷的产生机理,提出了随机孔隙缺陷分布模型,建立了基于细观胞元的含孔隙缺陷有限元分析模型,并应用概率统计方法,分析了缺陷的随机分布和孔隙率对材料等效弹性性能的影响,可为3D编织复合材料力学性能的设计提供依据。     

Towards gigantic RVE sizes for 3D stochastic fibrous networks

The size of representative volume element (RVE) for 3D stochastic fibrous media is investigated. A statistical RVE size determination method is applied to a specific model of random microstructure: Poisson fibers. The definition of RVE size is related to the concept of integral range. What happens in microstructures exhibiting an infinite integral range? Computational homogenization for thermal and elastic properties is performed through finite elements, over hundreds of realizations of the stochastic microstructural model, using uniform and mixed boundary conditions. The generated data undergoes statistical treatment, from which gigantic RVE sizes emerge. The method used for determining RVE sizes was found to be operational, even for pathological media, i.e., with infinite integral range, interconnected percolating porous phase and infinite contrast of properties.     

A procedure for estimating Young’s modulus of textured polycrystalline materials

In this study, a procedure for estimating Young’s modulus of textured and non-textured polycrystalline materials was examined based on finite element analyses, which were performed using three-dimensional polycrystalline finite element models of a random structure, generated using the Voronoi tessellation. Firstly, the local stress/strain distribution and its influence on macroscopic elastic properties were evaluated. Then, the statistical relationship between Young’s modulus obtained from the finite element analyses and averaged Young’s modulus of all grains evaluated based on Voigt’s or Reuss’ model was investigated. It was revealed that the local stress/strain in the polycrystalline body is affected by crystal orientation and deformation constraint caused by adjacent grains, whereas only the crystal orientation affects Young’s modulus of the polycrystalline body when the number of grains is large enough. It was also shown that Young’s modulus correlates well with the averaged Young’s modulus of all grains, in which the size of grains is considered in the averaging. Finally, a procedure for estimating Young’s modulus of textured and non-textured materials was proposed. Young’s modulus of various materials can be estimated from the elastic constants of single crystal and the distribution of crystal orientation and size of grains, which can be obtained by using electron backscatter diffraction (EBSD).     

Nonlinear viscoelastic multi-scale repetitive unit cell model of 3D woven composites with damage evolution

Three-dimensional (3D) textile composites have great potential applications to aircrafts and high speed vehicles because of the high strength/weight ratios and the capabilities of manufacturing complex, net-shape preforms. This paper reports the nonlinear viscoelastic responses and damage mechanisms of one kind of 3D textile composites, named as 3D orthogonal woven composite (3DOWC) under quasi-static tensile loading based on a micro/meso-scale repetitive unit cells (RUCs) model. In the RUCs model, the resin is described with a nonlinear viscoelastic material and the fibers/tows with an elastic material. The damage initiation and propagation in resin are simulated by the post-damage constitutive models with maximum principal theory failure criteria. The fibers/tows impregnated with resin are defined by elastic transverse-isotropic material model with ultimate strengths failure of ‘expanded smeared crack’ both along and perpendicular to fibers/tows axis direction. The engineering parameters and ultimate strengths of homogenized fibers/tows filled with matrix in meso-RUCs model are transferred from the numerical analysis of the micro-RUCs. The results are compared with experimental and theoretical values of RUC deformation and damage initiation and propagation under monotonic axial loading. The methodology of establishing the nonlinear visco-elastic multi-scale model of 3D textile composites without introducing the real fabric architecture in finite element analyses is explained. With the multi-scale RUCs model, the mechanical behaviors of other kinds of 3D textile composites can also be predicted.     

Numerical investigations of field-defect interactions in piezoelectric ceramics

The interactions of electro-elastic fields with voids and cracks contained in a piezoelectric solid are addressed both in closed form and numerically. The numerical algorithm, based on finite elements, is used to generate results that compare very well with available analytical solutions. Through the numerical results, it is shown that the simplified impermeable and the exact electric boundary conditions lead to the same result when the hole is open, but differ significantly when the hole is rather slender or becomes a mathematical crack. For the latter situations, the finite element meshes must be very structured and progressive, and special care must be taken with the resolution. It is also shown that the geometry of the void, the nature of the materials inside and outside the void, and the assumed electric boundary conditions play a significant role in the quantitative and qualitative nature of the elastic and electric field distributions in the neighborhood of the void’s tip, thus all being important in the design of electromechanical devices and structures. The development of the finite elements described in the article goes far beyond its usage as a validation tool. Rather, we show that it is an effective and accurate tool to address problems far complicated to be solved in closed form.     

一阶和二阶金字塔点阵材料等效弹性参数研究

本文通过分析一阶和二阶金字塔点阵夹芯结构的受力特点,研究了它们的等效弹性参数,得到了一阶和二阶芯体在受到3方向简单压缩及1、2方向单向剪切载荷时的等效弹性参数的解析解.此外,还分别建立了三维一阶和二阶点阵夹芯结构的有限元模型,并将有限元解与相同条件下的理论解进行了比较,验证了本文理论计算的正确性.     

考虑粘结面厚度的局部粘结单元法

传统无厚度粘结单元法CFEM (Cohesive finite element method)在模拟脆性材料断裂方面具有很强的优势,但也存在很大问题.一是单元尺寸增大,收敛性变差;二是单元尺寸变小,模型刚度发生折减.为了克服这两个问题,发展了考虑厚度的局部粘结单元法,即在裂纹可能扩展区插入具有一定厚度的粘结面单元.粘结面单元采用拓展虚内键本构(Augmented virtual internal bond)描述.由于考虑了厚度,粘结面交叉处会形成多边形空缺.为了弥补这一空缺,将其看作多边形键元胞,采用离散虚内键模型(Discretized virtual internal bond)对其建模,保证了模型的几何完整性.模拟结果表明,本文方法有效,克服了传统CFEM方法的刚度折减问题,提高了计算稳定性和收敛性.     

Representative elementary volume analysis of polydisperse granular packings using discrete element method

The representative elementary volume (REV) for three-dimensional polydisperse granular packings was determined using discrete element method simulations. Granular mixtures of various sizes and particle size distributions were poured into a cuboid chamber and subjected to uniaxial compression. Findings showed that the minimum REV for porosity was larger compared with the REV for parameters such as coordination number, effective elastic modulus, and pressure ratio. The minimum REV for porosity and other parameters was found to equal 15, 10, and 5 times the average grain diameter, respectively. A study of the influence of sample size on energy dissipation in random packing of spheres has also confirmed that the REV size is about 15 times the average grain diameter. The heterogeneity of systems was found to have no effect on the REV for the parameters of interest for the narrow range of coefficient of uniformity analyzed in this paper. As the REV approach is commonly applied in both experimental and numerical studies, determining minimum REV size for polydisperse granular packings remains a crucial issue.     

边界元法应用的若干近期研究

边界元法是在有限元法等其它数值方法的推动和竞争下发展的,作为一种数值方法,发展其实际应用至关重要,本文扼要介绍作者们近期近作若干工作,其中包括：二维弹性体移动接触和滚动接触,网络要群环境下边界元并行计算,以及二维三维边界元实用软件及应用等。     

Computing elastic moduli on 3-D X-ray computed tomography image stacks

A numerical task of current interest is to compute the effective elastic properties of a random composite material by operating on a 3D digital image of its microstructure obtained via X-ray computed tomography (CT). The 3-D image is usually sub-sampled since an X-ray CT image is typically of order 1000³ voxels or larger, which is considered to be a very large finite element problem. Two main questions for the validity of any such study are then: can the sub-sample size be made sufficiently large to capture enough of the important details of the random microstructure so that the computed moduli can be thought of as accurate, and what boundary conditions should be chosen for these sub-samples? This paper contributes to the answer of both questions by studying a simulated X-ray CT cylindrical microstructure with three phases, cut from a random model system with known elastic properties. A new hybrid numerical method is introduced, which makes use of finite element solutions coupled with exact solutions for elastic moduli of square arrays of parallel cylindrical fibers. The new method allows, in principle, all of the microstructural data to be used when the X-ray CT image is in the form of a cylinder, which is often the case. Appendix A describes a similar algorithm for spherical sub-samples, which may be of use when examining the mechanical properties of particles. Cubic sub-samples are also taken from this simulated X-ray CT structure to investigate the effect of two different kinds of boundary conditions: forced periodic and fixed displacements. It is found that using forced periodic displacements on the non-geometrically periodic cubic sub-samples always gave more accurate results than using fixed displacements, although with about the same precision. The larger the cubic sub-sample, the more accurate and precise was the elastic computation, and using the complete cylindrical sample with the new method gave still more accurate and precise results. Fortran 90 programs for the analytical solutions are made available on-line, along with the parallel finite element codes used.     

Finite element modeling of porous titanium

The porous microstructures of metallic foams cause microscopic stress and strain localization under deformation which reduces the damage tolerance and therefore limits application of the materials. In this paper, the deformation of a relatively low porosity porous titanium is examined using two-dimensional (2D) plane strain and three-dimensional (3D) finite element models to identify the accuracy and limitations of such simulations. To generate the finite element models, a simulated microstructure was created based on micrographs of an experimental material. Compared to the 2D models, the 3D models require smaller model size to obtain convergent results. The macroscopic responses predicted by the 3D models are in reasonable agreement with experimental results while the 2D models underestimated the response. In addition, 3D models predicted more uniform microscopic field variable distributions. 2D models predicted higher probability of Von Mises stress and equivalent plastic strain exceeding a certain value and therefore overestimate the failure probability of the material.     

THE EFFECTIVE ELECTROMECHANICAL PROPERTIES OF CELLULAR PIEZOELECTRET FILM:FINITE ELEMENT MODELING

Cellular space-charge polymer film,also called cellular piezoelectret,has very large piezoelectric effect due to their unique microvoid structure.In this article,the cellular piezoelectret film is considered to be a periodic composite material with closed-cell microvoids aligned periodically.Three dimensional finite element modeling is carried out to obtain the effective elastic modulus and piezoelectric coefficients.Sensitivity analysis was presented by modeling the effective electromechanical properties with different individual variable,including material constants and void shape parameters.By assuming a relation between void shape and void volume fraction,the finite element model can simulate quite well the inflation experiments of the voided charged Polypropylene film published in literature.Finally,the finite element model is used to explore the voided charge polymer film with non-uniform distribution of the trapped charges on the internal surface of voids.It was found that the resultant overall piezoelectric coefficients will be more significant if charges are closely gathered in the central area,and sparse in the around area of the internal surface of voids.     

Two-dimensional modeling of heterogeneous structures using graded finite element and boundary element methods

In the present work, graded finite element and boundary element methods capable of modeling behaviors of structures made of nonhomogeneous functionally graded materials (FGMs) composed of two constituent phases are presented. A numerical implementation of Somigliana’s identity in two-dimensional displacement fields of the isotropic nonhomogeneous problems is presented using the graded elements. Based on the constitutive and governing equations and the weighted residual technique, effective boundary element formulations are implemented for elastic nonhomogeneous isotropic solid models. Results of the finite element method are derived based on a Rayleigh–Ritz energy formulation. The heterogeneous structures are made of combined ceramic–metal materials, in which the material properties vary continuously along the in-plane or thickness directions according to a power law. To verify the present work, three numerical examples are provided in the paper.     

Influences of initial porosity,stress triaxiality and Lode parameter on plastic deformation and ductile fracture

Local mechanical properties in aluminum cast components are inhomogeneous as a consequence of spatial distribution of microstructure,e.g.,porosity,inclusions,grain size and arm spacing of secondary dendrites.In this work,the effect of porosity is investigated.Cast components contain voids with different sizes,forms,orientations and distributions.This is approximated by a porosity distribution in the following.The aim of this paper is to investigate the influence of initial porosity,stress triaxiality and Lode parameter on plastic deformation and ductile fracture.A micromechanical model with a spherical void located at the center of the matrix material,called the representative volume element(RVE),is developed.Fully periodic boundary conditions are applied to the RVE and the values of stress triaxiality and Lode parameter are kept constant during the entire course of loading.For this purpose,a multi-point constraint(MPC)user subroutine is developed to prescribe the loading.The results of the RVE model are used to establish the constitutive equations and to further investigate the influences of initial porosity,stress triaxiality and Lode parameter on elastic constant,plastic deformation and ductile fracture of an aluminum die casting alloy.     

基于三维细观力学模型的开孔泡沫弹性性能预测

泡沫材料的宏观力学性能主要取决于基体材料的力学特性及其微细观结构特征,基于细观力学模型的分析方法是泡沫材料力学性能研究的重要途径。文中基于Matlab语言和Abaqus软件构建了描述中等孔隙率开孔弹性泡沫材料微结构特征的三维随机分布球形泡孔模型,并采用有限元方法对弹性泡沫压缩变形进行了模拟,并计算给出了不同孔隙率弹性泡沫材料弹性模量、剪切模量、体积模量以及泊松比的分布,建立了相应的唯象表达式。与理论模型及测试结果的比较表明,本文基于三维随机泡孔模型模拟结果构建的唯象表达式能够对弹性泡沫材料的弹性力学性能给出很好的预测。     

Nonlocal integral elasticity: 2D finite element based solutions

A finite element based method, theorized in the context of nonlocal integral elasticity and founded on a nonlocal total potential energy principle, is numerically implemented for solving 2D nonlocal elastic problems. The key idea of the method, known as nonlocal finite element method (NL-FEM), relies on the assumption that the postulated nonlocal elastic behaviour of the material is captured by a finite element endowed with a set of (cross-stiffness) element’s matrices able to interpret the (nonlocality) effects induced in the element itself by the other elements in the mesh. An Eringen-type nonlocal elastic model is assumed with a constitutive stress–strain law of convolutive-type which governs the nonlocal material behaviour. Computational issues, as the construction of the nonlocal element and global stiffness matrices, are treated in detail. Few examples are presented and the relevant numerical findings discussed both to verify the reliability of the method and to prove its effectiveness.     

二维三轴编织复合材料的弹性性能分析

提出了二维三轴编织复合材料的几何模型,模型考虑了纤维束的弯曲扭转状态及空间交错特性等几何元素。基于体积平均法,建立了预测二维三轴编织复合材料弹性性能的理论分析模型;通过引入更普遍适用的周期性位移边界条件,结合二维三轴编织复合材料的细观实体结构,建立了分析其力学性能的有限元模型。两种模型预测结果均与试验结果吻合,证明了方法的合理有效性。分析了材料受载下的细观应力分布,并讨论了编织参数对材料性能的影响。研究表明,二维三轴编织复合材料轴向性能得到了增强,应力分布更均匀,编织角以及纤维体积含量对材料弹性性能影响较大。     

Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique

This paper presents a strategy for computation of super-convergent solutions of multi-dimensional problems in the finite element method (FEM) by recursive application of the one-dimensional (1D) element energy projection (EEP) technique. The main idea is to conceptually treat multi-dimensional problems as generalized 1D problems, based on which the concepts of generalized 1D FEM and its consequent EEP formulae have been developed in a unified manner. Equipped with these concepts, multi-dimensional problems can be recursively discretized in one dimension at each step, until a fully discretized standard finite element (FE) model is reached. This conceptual dimension-by-dimension (D-by-D) discretization procedure is entirely equivalent to a full FE discretization. As a reverse D-by-D recovery procedure, by using the unified EEP formulae together with proper extraction of the generalized nodal solutions, super-convergent displacements and first derivatives for two-dimensional (2D) and three-dimensional (3D) problems can be obtained over the domain. Numerical examples of 3D Poisson’s equation and elasticity problem are given to verify the feasibility and effectiveness of the proposed strategy.     

Nanoscale finite element models for vibrations of single-walled carbon nanotubes: atomistic versus continuum

By the atomistic and continuum finite element models, the free vibration behavior of single-walled carbon nanotubes （SWCNTs） is studied. In the atomistic finite element model, the bonds and atoms are modeled by the beam and point mass elements, respectively. The molecular mechanics is linked to structural mechanics to determine the elastic properties of the mentioned beam elements. In the continuum finite element approach, by neglecting the discrete nature of the atomic structure of the nanotubes, they are modeled with shell elements. By both models, the natural frequencies of SWCNTs are computed, and the effects of the geometrical parameters, the atomic structure, and the boundary conditions are investigated. The accuracy of the utilized methods is verified in comparison with molecular dynamic simulations. The molecular structural model leads to more reliable results, especially for lower aspect ratios. The present analysis provides valuable information about application of continuum models in the investigation of the mechanical behaviors of nanotubes.