基于均匀化理论韧性复合材料塑性极限分析

运用细观力学中的均匀化方法分析了韧性复合材料的塑性极限承载能力.从反映复合材料细观结构的代表性胞元入手,将均匀化理论运用到塑性极限分析中,计算由理想刚塑性、Mises组分材料构成的复合材料的极限承载能力.运用机动极限方法和有限元技术,最终将上述问题归结为求解一组带等式约束的非线性数学规划问题,并采用一种无搜索直接迭代算法求解.为复合材料的强度分析提供了一个有效手段.     

确定复合材料宏观屈服准则的细观力学方法

运用细观力学中的均匀化方法，分析了含周期性微结构复合材料的宏观屈服准则，并对Hill-Tsai准则进行了修正。从基于复合材料细观结构的代表性胞元入手，运用塑性极限理论中的机动分析以及有限元方法，计算了细观结构的极限载荷域。通过宏细观尺度对应关系，得到复合材料的宏观屈服准则。     

Negative Poisson's ratios in composites with star-shaped inclusions: a numerical homogenization approach

Summary Materials with specific microstructural characteristics and composite structures are able to exhibit negative Poisson's ratio. This result has been proved for continuum materials by analytical methods in previous works of the first author, among others [1]. Furthermore, it also has been shown to be valid for certain mechanisms involving beams or rigid levers, springs or sliding collars frameworks and, in general, composites with voids having a nonconvex microstructure.Recently microstructures optimally designed by the homogenization approach have been verified. For microstructures composed of beams, it has been postulated that nonconvex shapes with re-entrant corners are responsible for this effect [2]. In this paper, it is numerically shown that mainly the shape of the re-entrant corner of a non-convex, star-shaped, microstructure influences the apparent (phenomenological) Poisson's ratio. The same is valid for continua with voids or for composities with irregular shapes of inclusions, even if the individual constituents are quite usual materials. Elements of the numerical homogenization theory are reviewed and used for the numerical investigation. Accepted for publication 10 September 1996     

HOMOGENIZATION—BASED TOPOLOGY DESIGN FOR PURE TORSION OF COMPOSITE SHAFTS

In conjunction with the homogenization theory and the finite element method, the mathematical models for designing the corss-section of composite shafts by maximizing the torsion rigidity are developed in this paper. To obtain the extremal torsion rigidity, both the cross-section of the macro scale shaft and the representative microstructure of the composite material are optimized using the new models. The micro scale computational model addresses the problem of finding the periodic microstructures with extreme shear moduli. The optimal microstructure obtained with the new model and the homogenization method can be used to improve and optimize natural or artificial materials. In order to be more practical for engineering applications, cellular materials rather than ranked materials are used in the optimal process in the existence of optimal bounds for the elastic properties. Moreover, the macro scale model is proposed to optimize the cross-section of the torsional shaft based on the tailared composites. The validating optimal results show that the models are very effective in obtaining composites with extreme elastic properties, and the cross-section of the composite shaft with the extremal torsion rigidity. The project supported by the National Natural Science Foundation of China (10172078 and 10102018)     

Effective thermoelastic properties of composites with periodicity in cylindrical coordinates

The aim of this work is to study composites that present cylindrical periodicity in the microstructure. The effective thermomechanical properties of these composites are identified using a modified version of the asymptotic expansion homogenization method, which accounts for unit cells with shell shape. The microscale response is also shown. Several numerical examples demonstrate the use of the proposed approach, which is validated by other micromechanics methods.     

Prediction of elastic properties of heterogeneous materials with complex microstructures

The phase-field microelasticity (PFM) is adapted into a homogenization process to predict all the effective elastic constants of three-dimensional heterogeneous materials with complex microstructures. Comparison between the PFM approach and the Hashin-Shtrikman variational approach is also given. Using 3D images of two-phase heterogeneous media with regular and irregular microstructures, results indicate that the PFM approach can accurately take into account the effects of both elastic anisotropy and inhomogeneity of materials with arbitrary microstructure geometry, such as complex porous media with suspended inclusions.     

Design of microstructure-sensitive properties in elasto-viscoplastic polycrystals using multi-scale homogenization

Evolution of properties during processing of materials depends on the underlying material microstructure. A finite element homogenization approach is presented for calculating the evolution of macro-scale properties during processing of microstructures. A mathematically rigorous sensitivity analysis of homogenization is presented that is used to identify optimal forging rates in processes that would lead to a desired microstructure response. Macro-scale parameters such as forging rates are linked with microstructure deformation using boundary conditions drawn from the theory of multi-scale homogenization. Homogenized stresses at the macro-scale are obtained through volume-averaging laws. A constitutive framework for thermo-elastic–viscoplastic response of single crystals is utilized along with a fully-implicit Lagrangian finite element algorithm for modelling microstructure evolution. The continuum sensitivity method (CSM) used for designing processes involves differentiation of the governing field equations of homogenization with respect to the processing parameters and development of the weak forms for the corresponding sensitivity equations that are solved using finite element analysis. The sensitivity of the deformation field within the microstructure is exactly defined and an averaging principle is developed to compute the sensitivity of homogenized stresses at the macro-scale due to perturbations in the process parameters. Computed sensitivities are used within a gradient-based optimization framework for controlling the response of the microstructure. Development of texture and stress–strain response in 2D and 3D FCC aluminum polycrystalline aggregates using the homogenization algorithm is compared with both Taylor-based simulations and published experimental results. Processing parameters that would lead to a desired equivalent stress–strain curve in a sample poly-crystalline microstructure are identified for single and two-stage loading using the design algorithm.     

基于迭代法的非线性弹性均质化研究

微观结构对复合材料的宏观力学性能具有至关重要的影响, 通过合理设计复合材料微观结构可以得到期望的宏观性能. 均质化方法作为一种有效的设计方法, 它从微观结构的角度出发, 利用均匀化的概念, 实现了对复合材料宏观力学性能的预测和设计. 而当考虑非线性因素, 均质化的实现就非常困难. 本文利用双渐近展开方法, 将位移按照宏观位移和微观位移展开, 推导了非线性弹性均质化方程. 通过直接迭代法, 对非线性弹性均质化方程进行了求解, 并给出了具体的迭代方法和实现步骤. 本文基于迭代步骤和非线性弹性均质化方程编写MATLAB 程序, 对3种典型本构关系的周期性多孔材料平面问题进行了计算, 对比细致模型的应变能、最大位移和等效泊松比, 对程序及迭代方法的准确性进行了验证. 之后对一种三元橡胶基复合材料进行多尺度均质化, 将其分为芯丝尺度和层间尺度. 用线弹性的均质化方法得到了芯丝尺度的等效弹性参数, 并将其作为层间尺度的材料参数. 在层间尺度应用非线性弹性均质化方法对结构进行计算, 得到材料的宏观等效性能, 并以实验结果为基准进行评价.      

The homogenization method for the study of variation of Poisson's ratio in fiber composites

Summary Materials with specific microstructural characteristics and composite structures are able to exhibit negative Poisson's ratio. This fact has been shown to be valid for certain mechanisms, composites with voids and frameworks and has recently been verified for microstructures optimally designed by the homogenization approach. For microstructures composed of beams, it has been postulated that nonconvex shapes (with reentrant corners) are responsible for this effect. In this paper, it is numerically shown that mainly the shape, but also the ratio of shear-to-bending rigidity of the beams do influence the apparent (phenomenological) Poisson's ratio. The same is valid for continua with voids, or for composites with irregular shapes of inclusions, even if the constituents are quite usual materials, provided that their porosity is strongly manifested. Elements of the numerical homogenization theory and first attempts towards an optimal design theory are presented in this paper and applied for a numerical investigation of such types of materials. Received 11 March 1997; accepted for publication 12 September 1997     

Bounds for nonlinear composites via iterated homogenization

Improved estimates of the Hashin–Shtrikman–Willis type are generated for the class of nonlinear composites consisting of two well-ordered, isotropic phases distributed randomly with prescribed two-point correlations, as determined by the H-measure of the microstructure. For this purpose, a novel strategy for generating bounds has been developed utilizing iterated homogenization. The general idea is to make use of bounds that may be available for composite materials in the limit when the concentration of one of the phases (say phase 1) is small. It then follows from the theory of iterated homogenization that it is possible, under certain conditions, to obtain bounds for more general values of the concentration, by gradually adding small amounts of phase 1 in incremental fashion, and sequentially using the available dilute-concentration estimate, up to the final (finite) value of the concentration (of phase 1). Such an approach can also be useful when available bounds are expected to be tighter for certain ranges of the phase volume fractions. This is the case, for example, for the “linear comparison” bounds for porous viscoplastic materials, which are known to be comparatively tighter for large values of the porosity. In this case, the new bounds obtained by the above-mentioned “iterated” procedure can be shown to be much improved relative to the earlier “linear comparison” bounds, especially at low values of the porosity and high triaxialities. Consistent with the way in which they have been derived, the new estimates are, strictly, bounds only for the class of multi-scale, nonlinear composites consisting of two well-ordered, isotropic phases that are distributed with prescribed H-measure at each stage in the incremental process. However, given the facts that the H-measure of the sequential microstructures is conserved (so that the final microstructures can be shown to have the same H-measure), and that H-measures are insensitive to length scales, it is conjectured that the new bounds may hold for more general classes of microstructures with prescribed volume fractions and H-measures (independent of the separation of length scales hypotheses that was made in the derivation of the result using iterated homogenization).     

多相材料微结构多目标拓扑优化设计

在采用多尺度均匀化方法求解微结构等效特性的基础上，提出了多相材料 微结构的多目标优化设计模型. 以组分材料用量为约束，采用周长控制消除棋盘格，结合有 限元方法和对偶凸规划求解技术，对两相和三相材料微结构多项等效模量的组合进行了优化 设计. 研究比较了微结构网格粗细、材料组分以及三相材料微结构优化中的两相实体材料弹 性模量相对比例不同对优化结果的影响. 数值算例验证了优化模型和优化算法的有效性，表 明了相关因素对优化结果的影响.     

A LOWER BOUND LIMIT ANALYSIS OF DUCTILE COMPOSITE MATERIALS

The plastic load-bearing capacity of ductile composites such as metal matrix composites is studied with an insight into the microstructures. The macroscopic strength of a composite is obtained by combining the homogenization theory with static limit analysis, where the temperature parameter method is used to construct the self-equilibrium stress field. An interface failure model is proposed to account for the effects of the interface on the failure of composites. The static limit analysis with the finite-element method is then formulated as a constrained nonlinear programming problem, which is solved by the Sequential Quadratic Programming （SQP） method. Finally, the macroscopic transverse strength of perforated materials, the macroscopic transverse and off-axis strength of fiber-reinforced composites are obtained through numerical calculation. The computational results are in good agreement with the experimental data.     

A fully automated numerical tool for a comprehensive validation of homogenization models and its application to spherical particles reinforced composites

This paper presents a fully automated numerical tool for computing the accurate effective properties of two-phase linearly elastic composites reinforced by randomly distributed spherical particles. Virtual microstructures were randomly generated by an algorithm based on molecular dynamics. Composites effective properties were computed using a technique based on Fast Fourier Transforms (FFT). The predictions of the numerical tool were compared to those of analytical homogenization models for a broad range of phases mechanical properties contrasts and spheres volume fractions. It is found that none of the tested analytical models provides accurate estimates for the whole range of contrasts and volume fractions tested. Furthermore, no analytical homogenization models stands out of the others as being more accurate for the investigated range of volume fractions and contrasts. The new fully automated tool provides a unique means for computing, once and for all, the accurate properties of composites over a broad range of microstructures. In due course, the database generated with this tool might replace analytical homogenization models.     

复合材料周期性线弹性微结构的拓扑优化设计

提出复合材料周期性线弹性微结构拓扑优化设计的模型，模型1设计具有极值弹性特性的复合材料，模型2设计工况最刚微结构单胞。通过该模型和均匀化技术可以获得优化的微结构单胞，进而改善或者得到最优宏观特性的复合材料。为了便于制造和应用，用胞体材料而不是多相材料来得到复合材料的极值弹性特性和最大刚度。优化结果表明，该模型与数值方法相结合可以有效地实现微结构的拓扑优化设计。     

Assessment of material uncertainties in solid foams based on local homogenization procedures

The present study is concerned with a numerical prediction and assessment of uncertainties in the macroscopic material properties of solid foams. The material properties are determined by means of a homogenization analysis considering a large scale representative volume element. The microstructure for the representative volume element is determined randomly using a Voronoï tesselation in Laguerre geometry with prescribed cell size distribution. For assessment of the scatter in the effective material response, the homogenization scheme is applied to subsets of the large scale representative volume element. By this means, an interrelation between the local microstructural characteristics and the local mesoscopic material response is established. In a first approach, the individual cells of the foam microstructure are employed as testing volume elements. As an alternative, a moving window technique is applied. The sets of homogenization results obtained by both approaches are evaluated by stochastic methods. For the local effective properties, a distinct scatter is observed. The results in both cases reveal that the local porosity is the most important influence parameter. For the microstructures investigated, only weak local correlations of the effective stiffnesses with a rapid spatial decay of the correlation is observed.     

Material microstructure optimization for linear elastodynamic energy wave management

We describe a systematic approach to design material microstructures to achieve desired energy propagation in a two-phase composite plate. To generate a well-posed topology optimization problem we use the relaxation approach which requires homogenization theory to relate the macroscopic material properties to the microstructure, here a sequentially ranked laminate. We introduce an algorithm whereby the laminate layer volume fractions and orientations are optimized at each material point. To resolve numerical instabilities associated with the dynamic simulation and constrained optimization problem, we filter the laminate parameters. This also has the effect of generating smoothly varying microstructures which are easier to manufacture. To demonstrate our algorithm we design microstructure layouts for tailored energy propagation, i.e. energy focus, energy redirection, energy dispersion and energy spread.     

Micromechanics of particle-reinforced elasto-viscoplastic composites: Finite element simulations versus affine homogenization

The micromechanics of elasto-viscoplastic composites made up of a random and homogeneous dispersion of spherical inclusions in a continuous matrix was studied with two methods. The first one is an affine homogenization approach, which transforms the local constitutive laws into fictitious linear thermo-elastic relations in the Laplace–Carson domain so that corresponding homogenization schemes can apply, and the temporal response is computed after a numerical inversion of Laplace transform. The second method is the direct numerical simulation by finite elements of a three-dimensional representative volume element of the composite microstructure. The numerical simulations carried out over different realizations of the composite microstructure showed very little scatter and thus provided – for the first time – “exact” results in the elasto-viscoplastic regime that can be used as benchmarks to check the accuracy of other models. Overall, the predictions of the affine homogenization model were excellent, regardless of the volume fraction of spheres, of the loading paths (shear, uniaxial tension and biaxial tension as well as monotonic and cyclic deformation), particularly at low strain rates. It was found, however, that the accuracy decreased systematically as the strain rate increased. The detailed information of the stress and strain microfields given by the finite element simulations was used to analyze the source of this difference, so that better homogenization methods can be developed.     

Micromechanical analysis of fibrous piezoelectric composites with imperfectly bonded adherence

In this work, two-phase parallel fiber-reinforced periodic piezoelectric composites are considered wherein the constituents exhibit transverse isotropy and the cells have different configurations. Mechanical imperfect contact at the interface of the piezoelectric composites is studied via linear spring model. The statement of the problem for two-phase piezoelectric composites with mechanical imperfect contact is given. The local problems are formulated by means of the asymptotic homogenization method, and their solutions are found using complex variable theory. Analytical formulae are obtained for the effective properties of the composites with spring imperfect type of contact and different rhombic cells. Using the concept of a representative volume element (RVE), a finite element model is created, which combines the angular distribution of fibers and imperfect contact conditions (spring type) between the phases. Periodic boundary conditions are applied to the RVE, so that effective material properties can be derived. The fibers are distributed in such a way that the microstructure is characterized by a rhombic cell. The presented numerical homogenization technique is validated by comparing results with theoretical approach reported in the literature. Some studies of particular cases, numerical examples, and comparisons between the two aforementioned methods with other theoretical results illustrate that the model is efficient for the analysis of composites with presence of rhombic cells and the aforementioned imperfect contact.