Analytical and numerical analysis of 3D grid-reinforced orthotropic composite structures

A comprehensive micromechanical investigation of 3D periodic composite structures reinforced with a grid of orthotropic reinforcements is undertaken. Two different modeling techniques are presented; one is based on the asymptotic homogenization method and the other is a numerical model based on the finite element technique. The asymptotic homogenization model transforms the original boundary value problem into a simpler one characterized by effective coefficients which are shown to depend only on the geometric and material parameters of a periodicity cell. The model is applied to various 3D grid-reinforced structures with generally orthotropic constituent materials. Analytical formula for the effective elastic coefficients are derived, and it is shown that they converge to earlier published results in much simpler case of 2D grid reinforced structures with isotropic constituent materials. A finite element model is subsequently developed and used to examine the aforementioned periodic grid-reinforced orthotropic structures. The deformations from the finite element simulations are used to extract the elastic and shear moduli of the structures. The results of the asymptotic homogenization analysis are compared to those pertaining to their finite element counterparts and a very good agreement is shown between these two approaches. A comparison of the two modeling techniques readily reveals that the asymptotic homogenization model is appreciably faster in its implementation (without a significant loss of accuracy) and thus is readily amenable to preliminary design of a given 3D grid-reinforced composite structure. The finite element model however, is more accurate and predicts all of the effective elastic coefficients. Thus, the engineer facing a particular design application, could perform a preliminary design (selection of type, number and spatial orientation of the reinforcements) and then fine tune the final structure by using the finite element model.     

Computational micro‐mechanical model of flexible woven fabric for finite element impact simulation

This work presents a computational material model of flexible woven fabric for finite element impact analysis and simulation. The model is implemented in the non‐linear dynamic explicit finite element code LSDYNA. The material model derivation utilizes the micro‐mechanical approach and the homogenization technique usually used in composite material models. The model accounts for reorientation of the yarns and the fabric architecture. The behaviour of the flexible fabric material is achieved by discounting the shear moduli of the material in free state, which allows the simulation of the trellis mechanism before packing the yarns. The material model is implemented into the LSDYNA code as a user defined material subroutine. The developed model and its implementation is validated using an experimental ballistic test on Kevlar woven fabric. The presented validation shows good agreement between the simulation utilizing the present material model and the experiment. Copyright © 2001 John Wiley & Sons, Ltd.     

Multi-scale computational method for elastic bodies with global and local heterogeneity

A multi-scale computational method using the homogenization theory and the finite element mesh superposition technique is presented for the stress analysis of composite materials and structures from both micro- and macroscopic standpoints. The proposed method is based on the continuum mechanics, and the micro–macro coupling effects are considered for a variety of composites with very complex microstructures. To bridge the gap of the length scale between the microscale and the macroscale, the homogenized material model is basically used. The classical homogenized model can be applied to the case that the microstructures are periodically arrayed in the structure and that the macroscopic strain field is uniform within the microscopic unit cell domain. When these two conditions are satisfied, the homogenization theory provides the most reliable homogenized properties rigorously to the continuum mechanics. This theory can also calculate the microscopic stresses as well as the macroscopic stresses, which is the most attractive advantage of this theory over other homogenizing techniques such as the rule of mixture. The most notable feature of this paper is to utilize the finite element mesh superposition technique along with the homogenization theory in order to analyze cases where non-periodic local heterogeneity exists and the macroscopic field is non-uniform. The accuracy of the analysis using the finite element mesh superposition technique is verified through a simple example. Then, two numerical examples of knitted fabric composite materials and particulate reinforced composite material are shown. In the latter example, a shell-solid connection is also adopted for the cost-effective multi-scale modeling and analysis. This revised version was published online in June 2006 with corrections to the Cover Date.     

高体积含量颗粒增强复合材料的一个细观力学模型Ⅰ: 弹性分析与等效模量

提出了一种高体积含量颗粒增强复合材料的细观力学模型。该模型将颗粒简化为同质、同尺寸的弹性圆球, 两颗粒之间的粘接材料(基体) 简化为连接颗粒的一段圆柱体, 假设了圆柱形基体中的细观位移分布形式, 在此基础上分析了一对颗粒之间弹性的细观应力场和细观弹性系数, 将颗粒对的细观弹性系数在空间各个方向上平均, 得到材料的宏观弹性常数, 并建立了宏、细观分析之间的联系。最后用本模型分析了一种实际材料(两种体积含量) , 弹性常数的预测与实验吻合良好, 研究还发现颗粒的空间分布方式对材料宏观弹性常数的影响不大, 而对细观应力的影响显著。      

基于单胞解析模型的复合材料层合板渐进损伤数值分析

基于单胞解析模型,建立一种从复合材料细观组分到宏观层合板的渐进损伤分析模型。根据连续介质力学和均匀化方法构建细-宏观关联矩阵,通过该矩阵将细观组分材料的弹性和损伤性能传递到宏观复合材料中。该模型只需给出纤维和基体的材料属性及纤维体积含量无需层合板的弹性和强度参数,通过组分材料的损伤失效判据确定其是否损伤,如果发生损伤则用损伤因子折算成相应的刚度衰减。通过用户材料子程序UMAT 及VUMAT将单胞解析模型以及损伤理论嵌入到有限元软件ABAQUS 中,对开孔复合材料层合板的渐进损伤过程进行模拟,预测了层合板强度。结果表明:预报的强度与试验值吻合较好,验证了该方法的有效性。     

Fully coupled heat conduction and deformation analyses of nonlinear viscoelastic composites

This study presents an integrated micromechanical model-finite element framework for analyzing coupled heat conduction and deformations of particle-reinforced composite structures. A simplified micromechanical model consisting of four sub-cells, i.e., one particle and three matrix sub-cells is formulated to obtain the effective thermomechanical properties and micro–macro field variables due to coupled heat conduction and nonlinear thermoviscoelastic deformation of a particulate composite that takes into account the dissipation of energy from the viscoelastic constituents. A time integration algorithm for simultaneously solving the equations that govern heat conduction and thermoviscoelastic deformations of isotropic homogeneous materials is developed. The algorithm is then integrated to the proposed micromechanical model. A significant temperature generation due to the dissipation effect in the viscoelastic matrix was observed when the composite body is subjected to cyclic mechanical loadings. Heat conduction due to the dissipation of the energy cannot be ignored in predicting the factual temperature and deformation fields within the composite structure, subjected to cyclic loading for a long period. A higher creep resistant matrix material or adding elastic particles can lower the temperature generation. Our analyses suggest that using particulate composites and functionally graded materials can reduce the heat generation due to energy dissipation.     

Homogenized mechanical behavior of composite micro-structures including micro-cracking and contact evolution

The aim of this paper is to study the effects of micro-cracking on the homogenized constitutive properties of elastic composite materials. To this end a novel micro-mechanical approach based on homogenization techniques and fracture mechanics concepts, is proposed and an original J-integral formulation is established for composite micro-structures. Accurate non-linear macroscopic constitutive laws are developed for a uniaxial and a shear macro-strain path by taking into account changes in micro-structural configuration owing to crack growth and crack face contact. Numerical results, carried out by coupling a finite element formulation and an interface model, are applied to a porous composite with edge cracks and a debonded short fiber-reinforced composite. The composite micro-structure is controlled by the macroscopic strain and the micro-to-macro transition, settled in a variational formulation, is obtained for three types of boundary conditions, i.e. linear displacements, uniform tractions and periodic fluctuations and anti-periodic tractions. The accuracy of the determined macroscopic constitutive properties to represent the failure characteristics of locally periodic defected composites is also investigated in terms of energy release rate predictions, by comparisons between a direct analysis and homogenization approaches. Results highlight the dependence of the macroscopic constitutive law for a micro-structure with evolving defects on both the macro-strain path and the type of boundary conditions and the capability of the proposed model to provide a failure model for a composite material undergoing micro-cracking and contact.     

弹塑性多尺度分析的实现及其在颗粒增强复合材料中的应用

基于满足周期性假设和尺度分离假设的渐进展开均匀化原理,应用商业有限元软件ABAQUS实现了快速识别颗粒增强复合材料的等效弹性参数,及获取其宏-细观尺度下的非线性应力应变场信息。在细观尺度上,为了更好地逼近实际的复合材料结构,其增强颗粒采用不同直径和随机分布的球形进行近似。通过对不同颗粒含量的等效弹性参数的误差分析,证明了细观模型构造的合理性。此外,通过宏-细观尺度间的耦合机制,利用ABAQUS多个用户自定义子程序,实现了颗粒增强复合材料的非线性多尺度耦合分析,并提出了一套加速算法。最后据此研究了颗粒增强材料细观模型塑性演化过程对宏观力学性能的影响。由于编写的程序及分析的思路具有良好的通用性,这一工作为研究颗粒增强及其它复合材料的宏观力学性能提供了有益的参考。     

Determination of sensitivity coefficients of the elastic effective properties for periodic fiber-reinforced composites using the response function method

Derivation and implementation of the homogenization method including determination of sensitivity gradients of the effective elasticity tensor using combined numerical-analytical approach are addressed in this paper. This is possible thanks to an application of the numerical response function together with the effective moduli method known from classical homogenization theory. Computational procedure is implemented using 4-noded quadrilateral plane strain finite elements (program MCCEFF) and the symbolic computations system MAPLE. The sensitivity coefficients are determined on the basis of partial derivatives of the homogenized elasticity tensor calculated using the response function method with respect to all composite components’ elastic characteristics. They are further separately subjected to normalization procedure for a final comparison with each other. Such an enriched homogenization procedure is tested on the periodic fiber-reinforced two component composites; the results of computational analysis are compared to the results of the central finite difference approach applied before. Computational methodology proposed here may be further successively applied not only in the context of homogenization method but also to extend various discrete computational techniques like boundary/finite element, finite difference and volumes together with various meshless methods.     

特定弹性性能材料的细观结构设计优化

针对具有特定弹性性质的两相复合材料, 研究了特定性能材料优化设计问题的数学模型,提出了基于形状优化的材料设计方法。该方法利用形状优化技术, 设计两相复合材料的细观结构形式, 以使复合材料具有特定的弹性性质。材料的宏观性质由均匀化方法确定。最后给出了零泊松比材料的设计过程和结果。      

Numerical and experimental determination of in-plane elastic properties of 2/2 twill weave fabric composites

This paper presents the in-plane elastic properties of 2/2 twill weave, T300 carbon/epoxy, woven fabric composite plates, obtained by both finite element analysis and experiments. A micromechanical, three-dimensional (3D) finite element model of the single layer unit cell of a 2/2 twill weave fabric composite is built, and a homogenization process is implemented. A unit cell is chosen such that it encloses the characteristic periodic repeat pattern in the fabric weave. Detailed geometry together with construction procedures for this new model are developed by using ANSYS Parametric Design Language (APDL). In this respect, the scope for altering the weave and yarn parameters is facilitated. Standard tensile and rail shear tests with modifications are performed for this kind of woven fabric composite. Elastic mechanical properties determined by experiments are presented, and the finite element model is verified. Satisfactory correlation between the predicted and experimental results are obtained.     

复合材料热传导系数均匀化计算的实现方法

均匀化理论可以有效预测周期性结构复合材料的等效热传导系数,然而其控制方程的载荷项形式特殊,通用有限元软件中没有对应的载荷形式,难以直接求解.提出一种本构关系及场变量的类比方法,证明了在此类比下等效热传导系数均匀化方程与等效弹性模量均匀化方程是等价的.根据求解等效弹性模量均匀化方程的热应变法,提出一种新的等效热传导系数均匀化方程数值求解方法.以ABAQUS为平台,预测单向纤维复合材料以及金属蜂窝夹芯板的等效热传导系数,计算结果与参考值吻合良好.该方法为基于通用有限元软件的复合材料等效热传导系数的均匀化计算提供了简便途径.     

Algorithms for automated meshing and unit cell analysis of periodic composites with hierarchical tri‐quadratic tetrahedral elements

Unit cell homogenization techniques together with the finite element method are very effective for computing equivalent mechanical properties of composites and heterogeneous materials systems. For systems with very complicated material arrangements, traditional, manual mesh generation can be a considerable obstacle to usage of these techniques. This problem is addressed here by developing automated meshing techniques that start from a hierarchical quad‐tree (in 2D) or oc‐tree (in 3D) mesh of pixel or voxel elements. From the pixel/voxel mesh, algorithms are presented for successive element splitting and nodal shifting to arrive at final meshes that accurately capture both material arrangements and constituent volume fractions, and the material‐scale stress and strain fields within the composite under different modalities of loading. The performance and associated convergence behaviour of the proposed techniques are demonstrated on both densely packed fibre and particulate composites, and on 3D textile‐reinforced composites. Copyright © 2003 John Wiley Sons, Ltd.     

A numerical approach for determining the effective elastic symmetries of particulate-polymer composites

In this paper we deal with the problem of determining on the one hand the effective elastic properties of particulate-polymer composite materials and on the other hand the actual degree of symmetry of the resulting homogenised material. This twofold purpose has been accomplished by building a 2D as well as a 3D finite element model of the heterogeneous material and by using the strain-energy based numerical homogenisation technique. Both finite element models are able to reproduce with a good level of accuracy the real microstructure of the composite material by considering a random distribution of both particles and air bubbles (that are generated by the fabrication process). To assess the effectiveness of the proposed models, we present a numerical study to determine the effective elastic properties of the composite along with a comparison with the existing analytical and experimental results taken from literature and a sensitivity analysis in terms of the spatial distribution of the particles of the unit cell. Numerical results show that both models are able to provide the equivalent elastic properties with a very good level of accuracy when compared to experimental results and that the particulate-reinforced polymer composite could show, depending on the particles volume fraction and arrangement, an isotropic or a cubic elastic symmetry.     

The method of cells and the mechanical properties of textile composites

The method of cells has been gaining ground as a method for predicting the elastic properties of textile reinforced composite materials. This method deals away with the constant stress/strain assumption present in most analytical models allowing for a more accurate prediction of the elastic properties, with computational cost only a fraction in comparison with finite element analysis. A new implementation is presented here, which links this method to a mechanistic predictive geometry preprocessor allowing the presented model to deal with 2D and 3D textile reinforced composites. Numerical results for prediction of stiffness and strength of textile composites are generated and compared to other methods.     

Coupled heat conduction and thermal stress analyses in particulate composites

This study introduces two micromechanical modeling approaches to analyze spatial variations of temperatures, stresses and displacements in particulate composites during transient heat conduction. In the first approach, a simple micromechanical model based on a first order homogenization scheme is adopted to obtain effective mechanical and thermal properties, i.e., coefficient of linear thermal expansion, thermal conductivity, and elastic constants, of a particulate composite. These effective properties are evaluated at each material (integration) point in three dimensional (3D) finite element (FE) models that represent homogenized composite media. The second approach treats a heterogeneous composite explicitly. Heterogeneous composites that consist of solid spherical particles randomly distributed in homogeneous matrix are generated using 3D continuum elements in an FE framework. For each volume fraction (VF) of particles, the FE models of heterogeneous composites with different particle sizes and arrangements are generated such that these models represent realistic volume elements “cut out” from a particulate composite. An extended definition of a RVE for heterogeneous composite is introduced, i.e., the number of heterogeneities in a fixed volume that yield the same expected effective response for the quantity of interest when subjected to similar loading and boundary conditions. Thermal and mechanical properties of both particle and matrix constituents are temperature dependent. The effects of particle distributions and sizes on the variations of temperature, stress and displacement fields are examined. The predictions of field variables from the homogenized micromechanical model are compared with those of the heterogeneous composites. Both displacement and temperature fields are found to be in good agreement. The micromechanical model that provides homogenized responses gives average values of the field variables. Thus, it cannot capture the discontinuities of the thermal stresses at the particle-matrix interface regions and local variations of the field variables within particle and matrix regions.     

Prediction of stiffness and stresses in z-fibre reinforced composite laminates

The mechanical properties of z-pinned composite laminates were examined numerically. Finite element calculations have been performed to understand how the through-thickness reinforcement modifies the engineering elastic constants and local stress distributions. Solutions were found for four basic laminate stacking sequences, all having two percent volume fraction of z-fibres. For the stiffness analysis, a micro-mechanical finite element model was employed that was based on the actual geometric configuration of a z-pinned composite unit cell. The numerical results agreed very well with some published solutions. It showed that by adding 2% volume fraction of z-fibres, the through-thickness Young's modulus was increased by 22–35%. The reductions in the in-plane moduli were contained within 7–10%. The stress analysis showed that interlaminar stress distributions near a laminate free edge were significantly affected when z-fibres were placed within a characteristic distance of one z-fibre diameter from the free edge. Local z-fibres carried significant amount of interlaminar normal and shear stresses.     

Microstructure-based simulation of thermomechanical behavior of composite materials by object-oriented finite element analysis

While it is well recognized that microstructure controls the physical and mechanical properties of a material, the complexity of the microstructure often makes it difficult to simulate by analytical or numerical techniques. In this paper we present a relatively new approach to incorporate microstructures into finite element modeling using an object-oriented finite element technique. This technique combines microstructural data in the form of experimental or simulated microstructures, with fundamental material data (such as elastic modulus or coefficient of thermal expansion of the constituent phases) as a basis for understanding material behavior. The object-oriented technique is a radical departure from conventional finite element analysis, where a “unit-cell” model is used as the basis for predicting material behavior. Instead, the starting point of object-oriented finite element analysis is the actual microstructure of the material being investigated. In this paper, an introduction to the object-oriented finite element approach to microstructure-based modeling is provided with two examples: SiC particle-reinforced Al matrix composites and double-cemented WC particle-reinforced Co matrix composites. It will be shown that object-oriented finite element analysis is a unique tool that can be used to predict elastic and thermal constants of the composites, as well as salient effects of the microstructure on local stress state.     

Tight bounds for three-dimensional nonlinear incompressible elastic composites

Tighter variational bounds, in the whole range of inclusion volume fraction, that is to say, even near percolation, for the effective energy of nonlinear composites, in the special case of 3D two-phase incompressible elastic composites with isotropic constituents are presented. Following the methodology of Talbot, Willis and Ponte Castañeda, a linear comparison material with the same microgeometry as the nonlinear composite is employed. The asymptotic homogenization method (AHM) combined with a finite element analysis (FEM), is used to find the displacement field as well as the effective properties for the comparison material. An elastic composite with periodically distributed spherical inclusions in a cubic array is considered as an example. Various numerical examples are performed. Comparisons with others theories (i.e. variational bounds, self-consistent estimates, etc.) are shown. Coincidence of the AHM-FEM results with the universal bounds of Nemat-Nasser, Yu and Hori serves as a useful check to the numerical calculation.