计算复合材料有效弹性模量的重心有限元方法

采用几何法构造出任意边数多边形单元的重心插值形函数, 应用Galerkin法提出了求解弹性力学问题的重心有限元方法。用重心有限元方法对SiC/Ti和B/Al 2种纤维复合材料横向截面的有效弹性模量进行了预报。计算模型取纤维呈六边形排列且为各向同性的代表性单胞, 对其杨氏模量、 剪切模量和体积模量在较大的体积分数范围内进行了数值模拟。通过与解析公式和传统有限元的计算结果对比, 重心有限元方法的计算结果符合解析公式解的趋势, 与传统有限元的计算结果吻合较好。与传统有限元方法相比, 重心有限元方法的单元划分不受三角形或四边形的形状限制, 能够再现材料的真实结构。由于单元较大且数目较少, 本文方法具有很高的计算效率。      

The effect of crack face contact on the anisotropic effective moduli of microcrack damaged media

The generalized self-consistent method (GSCM) in conjunction with a computational finite element method is used to calculate the anisotropic effective moduli of a medium containing damage consisting of microcracks with an arbitrary degree of alignment. Since cracks respond differently under different external loads, the moduli of the medium subjected to tension, compression and an initially stress-free state are evaluated and shown to be significantly different, which will further affect the wave speed inside the damaged media. There are four independent material moduli for a 2-D plane stress orthotropic medium in tension or compression, and seven independent material moduli for a 2-D plane stress orthotropic cracked medium, which is initially stress free. When friction exists, it further changes the effective moduli. Numerical methods are used to take into account crack face contact and friction. The wave slowness profiles for microcrack damaged media are plotted using the predicted effective material moduli.     

夹杂问题中的自洽有限元法和复合材料的平均弹性性能

本文提出了处理夹杂问题的自洽有限元法。它以自洽模型为基础,利用有限元迭代法计算有效介质的平均弹性性能。作为应用,本文利用该方法计算了单向短纤维增强复合材料的平均弹性模量,并比较了纤维的几何形状对复合材料平均弹性性能的影响。数值结果与实验值吻合较好。      

Analysis on the variances of material and structural properties based on random field theory

Based on the random field theory (RFT) and the stochastic finite element method (SFEM), the variances of the mechanical properties of materials and structures are studied. Manufacturing processes can easily lead to the spatial variations of the load and the material properties such as moduli and density. Characterizing the elastic moduli, load and density with one-dimensional random fields, the analytical solutions for the coefficient of variations (COVs) of effective material moduli, displacement and natural frequencies of beams are obtained. Then, with the fiber and matrix properties, volume fraction modeled by two-dimensional random fields and the fiber angle as a single random variable, a Monte Carlo simulation (MCS) is performed to generate the variances of effective modulus of fiber-reinforced composite laminar plate. Compared with the previous numerical conclusions, the present results reveal that the variances of effective material properties and structural displacement are greatly dependent on both the random fields and the sizes of structures in theory.     

Predicting Effective Elastic Moduli and Strength of Ternary Blends with Core–Shell Structure by Second–Order Two–Scale Method

Core–shell particle–filled PA6/EPDM–g–MA/HDPE ternary blend has excellent mechanical properties. In this paper, effective elastic properties and tensile yield strength of the ternary blend are predicted by the second–order two– scale method, to investigate the relationship between morphology and mechanical properties. The method and the limit analysis for predicting mechanical properties of random heterogeneous materials are briefly introduced. Realistic morphology of the ternary blend including both core–shell particles and pure particles is simulated, and finite element mesh is generated. The unified strength theory is embedded in the method for the convenience of selecting a suitable yield criterion. The effective elastic moduli and tensile yield strength predicted by the method in this paper are compared with analytical and experimental results. Finally, effect of shell thickness in the core–shell particles on the effective elastic moduli and tensile yield strength is investigated.     

The use of boundary strip method (BSM) for evaluation of the transverse mechanical properties of fibrous composites

The boundary strip method (BSM) is applied for evaluation of the transverse mechanical properties of fibrous composites with random and periodical fiber distributions. This special semi numerical method helps find the link between the microscopic behavior of the composite material and its macroscopic response in a rather detailed manner, enabling definition of stress and strain magnitudes at each point of the cross section. Here, specifically statistical model based on the boundary strip method, is used for assessment of the transverse effective moduli of fibrous composites. Random fiber distributions are compared with periodic fiber distributions having square or hexagonal array arrangements. Those are the common models used nowadays and modeled by the finite element or the boundary element. A comparison with the bounds of the polarization extremum principles is conducted too. The influence of the randomly distributed fibers on the transverse effective moduli is investigated and a good correlation is found between the results of the present model and the lower bound of the polarization extremum principles.     

基于响应面方法的支架结构模型修正研究

为减小有限元建模中不可避免的误差,以支架结构模型为研究对象,选取结构不同部件的三个弹性模量和三个密度参数作为设计变量。根据中心复合设计,建立试验样本空间,利用样本参数的显著性分析结果筛选修正参数。结合线性回归方法,建立支架模态频率与修正参数的二阶多项式。以试验模态分析结果为依据,完成支架结构模型修正。响应面方法修正后的模态频率与试验模态频率具有一致性,且避免重复调用有限元模型,大大提高修正效率,具有修正响应快速性和工程实用性。     

A generalized plane strain technique for estimating effective properties of particulate metal matrix composites using FEM

A procedure to estimate the effective elastic moduli and coefficient of thermal expansion (CTE) of particulate-reinforced metal matrix composites (MMCs) using a two-dimensional finite element method is presented. The actual microstructural geometry of the composites with randomly distributed second-phase particles is incorporated in the model. A generalized plane strain technique, realistically to describe the three-dimensional behaviour, is also incorporated in the model. The elastic moduli and the CTE, estimated using this model, agree favourably with the experimental data. The technique is shown to be superior compared to the conventional two-dimensional plane stress and plane strain approximations. Also, the results indicate that the effect of the shape of the randomly distributed second-phase particles on the effective elastic moduli is insignificant. Although the procedure is demonstrated for particulate MMCs, it can be easily extended to many other materials as well.     

Overall elastoplastic property for micropolar composites with randomly oriented ellipsoidal inclusions

Overall linear and non-linear properties for micropolar composites containing 3D and in-plane randomly oriented inclusions are examined with an analytical micromechanical method. This method is based on Eshelby solution for a general ellipsoidal inclusion in a micropolar media and secant moduli method. The influence of inclusion’s shape, size and orientation on the classical effective moduli, yielding surface and non-linear stress and strain relation are examined. The results show that the effective moduli and non-linear stress–strain curves are always higher for micropolar composites than the corresponding classical composites. When the inclusion’s size is sufficiently large, the classical results can be recovered.     

Boundary element analysis of nanoinhomogeneities of arbitrary shapes with surface and interface effects

In this paper, a boundary element method (BEM) is proposed to analyze the stress field in nanoinhomogeneities with surface/interface effect. To consider this effect, the continuity conditions along the internal interfaces between the matrix and inhomogeneities are modeled by the well-known Gurtin–Murdoch constitutive relation. In the numerical analysis, the interface elastic moduli and the geometry of the nanoscale inhomogeneity are varied to show their influence on the induced stress field. The interaction between nanoscale inhomogeneities and the effect of different geometric shapes of inhomogeneities, including ellipse, triangle, and square are also investigated for different interface material parameters. It is shown that the elastic field can be greatly influenced by the interfacial energy and geometry of nanoscale inhomogeneities. The proposed BEM formulation is very general, including the complete Gurtin–Murdoch model and is further convenient for arbitrary shapes of inhomogeneity.     

Numerical analysis of interaction and coalescence of numerous microcracks

The deformation and failure behaviors of brittle or quasi-brittle solids are closely related to interaction and propagation of stochastically distributed microcracks. The influence of microcrack interaction and evolution on the mechanical properties of materials presents a problem of considerable interest, which has been extensively argued but has not been resolved as yet. In the present paper, a novel numerical method is used to calculate the effective elastic moduli and the tensile strength, and to simulate the failure process of brittle specimens containing numerous microcracks. The influences of some crack distribution parameters reflecting the non-uniform spatial concentration, size and orientation distributions are examined. The effective elastic moduli and the tensile strength of brittle materials exhibit different dependences on microcrack interaction. For example, two microcrack distributions that lead to the identical effective elastic moduli may cause a pronounced difference in the tensile strengths and failure behaviors of materials. By introducing two criteria for microcrack growth and coalescence in terms of Griffith’s energy release rate, the above numerical method is extended to simulate the coalescence process of microcracks that results in a fatal crack and the final rupture of a specimen.     

Effective Elastic and Plastic Properties of Interpenetrating Multiphase Composites

In interpenetrating phase composites, there are at least two phases that are each interconnected in three dimensions, constructing a topologically continuous network throughout the microstructure. The dependence relation between the macroscopically effective properties and the microstructures of interpenetrating phase composites is investigated in this paper. The effective elastic moduli of such kind of composites cannot be calculated from conventional micromechanics methods based on Eshelby's tensor because an interpenetrating phase cannot be extracted as dispersed inclusions. Using the concept of connectivity, a micromechanical cell model is first presented to characterize the complex microstructure and stress transfer features and to estimate the effective elastic moduli of composites reinforced with either dispersed inclusions or interpenetrating networks. The Mori–Tanaka method and the iso-stress and iso-strain assumptions are adopted in an appropriate manner of combination by decomposing the unit cell into parallel and series sub-cells, rendering the calculation of effective moduli quite easy and accurate. This model is also used to determine the elastoplastic constitutive relation of interpenetrating phase composites. Several typical examples are given to illustrate the application of this method. The obtained analytical solutions for both effective elastic moduli and elastoplastic constitutive relations agree well with the finite element results and experimental data.     

基于共焦点椭球构型的复合材料有效性质预测

介绍了共焦点椭球构型,给出了基于该构型对空间任意取向复合材料模量的解析计算公式,并将其同Mori-Tanaka(MT)法、Ponte-Castaneda-Willis (PCW)方法以及Hashin-Shtrikman(HS)界限进行了比较。数值结果显示,基于该构型的预测处于MT法和PCW法所预测的值之间,并且与MT法所预测的值接近。此外,还对纤维不同的角度平均方法对有效性质的影响做了讨论。     

Size-dependent effective electroelastic moduli of piezoelectric nanocomposites with interface effect

The problem of nanocomposite materials under far-field antiplane mechanical load and inplane electric load is investigated. Based on the theory of Gurtin?CMurdoch surface/interface model, an exact solution is obtained for the inhomogeneity/matrix/equivalent medium model, in terms of which a generalized self-consistent approach is proposed for predicting the effective electroelastic moduli of nanocomposites. A closed-form solution of the effective electroelastic moduli is presented. The numerical results reveal that the effective electroelastic moduli are size dependent when the size of the inhomogeneity is on the order of nanometer. With the increase in the size of the inhomogeneity, the present solution approaches to the classical results obtained in the classical theory.     

Analysis of damaged material containing periodically distributed elliptical microcracks by using the homogenization method based on the superposition method

The presence of multiple microcracks in a structural component causes material degradation such as reduction in the stiffness or reduction in the fracture toughness of the component. In this paper, the homogenization method is used to evaluate mechanical properties of the damaged material. The adaptation of the superposition method to the homogenization method is also presented. The proposed method makes use of the finite element solution of uncracked solid and the analytical solution. The effective elastic moduli of damaged materials containing lattice-distribution microcracks are estimated by the proposed method. Furthermore, the stress fields and the stress intensity factors of the elliptical microcracks in the damaged material at a micro-mechanics scale are evaluated to illustrate microscopic behavior such as crack interaction.     

Effective elastic moduli of porous solids

The principles of continuum mechanics can be extended to porous solids only if the effective moduli are known. Although the effective bulk modulus has already been determined by approximating the geometry of a porous solid to be a hollow sphere, bounds could only be established for the other moduli. This problem of indeterminacy of the moduli is solved in this study using a particular model from the variation of the effective Poisson's ratio. In addition to this, the results are extended for the hollow sphere to real geometry by introducing a porositydependent factor. These results are compared with experimental data and the agreement is found to be good. As the effective Poisson's ratio cannot be determined accurately using experiments, the derived equation is verified using finite element analysis.     

轮对振动和噪声的分析

以轮轨表面粗糙度为激励,利用车辆-轨道多刚体耦合振动模型计算轮轨作用力.利用有限元理论建立轮对的有限元分析模型,以轮轨作用力为激励进行轮对的振动频响分析.以振动响应分析结果作为边界条件,利用边界元理论建立轮对边界元声学分析模型,对轮对振动声学特性进行了计算分析.其结果与公认的模型和软件的计算结果相比具有较好的一致性,证明本文做法的正确性.     

Evaluation of the in-plane effective elastic moduli of single-layered graphene sheet

In this paper, an equivalent continuum-structural mechanics approach is used to characterize the mechanical behaviour of nanostructured graphene. The in-plane elastic deformation of armchair graphene sheets is simulated by using finite element modelling. The model is based on the assumption that force interaction among carbon atoms can be modelled by load-carrying beams in a representative two-dimensional honeycomb lattice structure. The elastic properties of beam elements are determined by equating the energies of the molecular structure and the continuum beam model subjected to small strain deformation. Then an equivalent continuum technique is adopted to estimate effective elastic moduli from which elastic constants are extracted. A comparison of elastic constants obtained from current modelling concur with results reported in literature. With the multifunctional properties of graphene sheets as manifested in a broad range of industrial applications, determination of their elastic moduli will facilitate a better design of the corresponding materials at macroscopic level.     

复合材料圆管构件等效模量的计算方法

针对任意壁厚的复合材料圆管构件的等效弹性模量和剪切模量提出了高阶理论计算方法, 它考虑了构件的横向剪切效应以及层合材料的三维本构关系, 并在相同壁厚条件下对三种缠绕方式( [0°/(±θ)_n ]S, [ (±θ)_n ]S 和[ 90°/(±θ)_n ]S) 的等效模量进行了预测, 并且与经典层合板理论的预测结果进行了比较, 该方法可用于复合材料杆件结构的设计中。      

Nonlinear large deflection buckling analysis of compression rod with different moduli

Many novel materials exhibit a property of different elastic moduli in tension and compression. One such material is graphene, a wonder material, which has the highest strength yet measured. Investigations on buckling problems for structures with different moduli are scarce. To address this new problem, first, the nondimensional expression of the relation between offset of neutral axis and deflection curve is derived based on the phased integration method, and then using the energy method, load–deflection relation of the rod is determined; second, based on the improved constitutive model for different moduli, large deformation finite element formulations are developed, and combined with the arc-length method, finite element iterative program for rods with different moduli is established to obtain buckling critical loads; third, material mechanical properties testing of graphite, which is the raw material of graphene, is performed to measure the tensile and compressive elastic moduli; moreover, buckling tests are also conducted to investigate the buckling behavior of this kind of graphite rod. By comparing the calculation results of the energy method and finite element method with those of laboratory tests, the analytical model and finite element numerical model are demonstrated to be accurate and reliable. The results show that it may lead to unsafe results if the classic theory was still adopted to determine the buckling loads of those rods composed of a material having different moduli. The proposed models could provide a novel approach for further investigation of nonlinear mechanical behavior for other structures with different moduli.