Field intensity factors around inclusion corners in 0–3 and 1–3 composites subjected to thermo-mechanical loads

A super inclusion corner apex element for polygonal inclusions in 0–3 and 1–3 composites is developed by using numerical stress and displacement field solutions based on an ad hoc finite element eigenanalysis method. Singular stresses near the apex of inclusion corner under thermo-mechanical loads can be obtained by using a super inclusion corner apex element in conjunction with hybrid-stress elements. The validity and the applicability of this technique are established by comparing the present numerical results with the existing solutions and the conventional finite element solutions. As examples of applications, a square array of square inclusions in 0–3 composites and a rectangular array of rectangular inclusions in 1–3 composites are considered. All numerical examples show that the present numerical method yields satisfactory solutions with fewer elements and is applicable to complex problems such as multiple singular points or fields in composite materials.     

Two-dimensional and axisymmetric unit cell models in the analysis of composite materials

Unit cell models have been widely used for investigating fracture mechanisms and mechanical properties of composite materials assuming periodically arrangement of inclusions in matrix. It is desirable to clarify the geometrical parameters controlling the mechanical properties of composites because they usually contain randomly distributed particulate. To begin with a tractable problem this paper focuses on the effective Young’s modulus E of heterogeneous materials. Then, the effect of shape and arrangement of inclusions on E is considered by the application of FEM through examining three types of unit cell models assuming 2D and 3D arrays of inclusions. It is found that the projected area fraction and volume fraction of inclusions are two major parameters controlling effective elastic modulus of inclusions.     

Analysis of the interaction within a rectangular array of rectangular inclusions using a new hybrid finite element method

This paper deals with the interaction problem of a rectangular array of rectangular inclusions under longitudinal tension. A novel ad hoc hybrid finite element method is applied to a rectangular cell containing single or multiple inclusions. Generalized stress intensity factors at the corners of inclusions are systematically calculated with varying the material type, shape and arrangement of rectangular inclusions. Present numerical solutions are compared with existing results. The present method is found to be yield rapidly converging numerical solutions with high accuracy.     

用切口尖端单元分析各向异性材料中多边形孔奇异性应力场

该文提出了一种基于全数值方法的新型杂交元方法, 用于研究各向异性复合材料中多边形孔奇异性应力场干涉问题。该方法的建立分3 个步骤:首先, 用一维有限元方法求解各向异性材料切口尖端奇异性应力场数值特征解;然后, 采用杂交有限元列式构造一种超级切口尖端单元, 其中, 假设应力场和位移场是利用上述奇异性场数值特征解推导出来的;最后, 将上述超级切口尖端单元与传统4 结点杂交应力元组装, 得到新型杂交元方法。算例中, 将裂纹问题作为考核例, 并进一步考察双菱形孔和双矩形孔的奇异性应力干涉问题。算例表明:当前模型能降低单元数, 且精度好;与传统有限元法和积分方程方法相比, 该模型更具有通用性和高效性, 为各向异性材料的细观力学分析打下了基础。     

Combining self-consistent and numerical methods for the calculation of elastic fields and effective properties of 3D-matrix composites with periodic and random microstructures

The work is devoted to the calculation of static elastic fields in 3D-composite materials consisting of a homogeneous host medium (matrix) and an array of isolated heterogeneous inclusions. A self-consistent effective field method allows reducing this problem to the problem for a typical cell of the composite that contains a finite number of the inclusions. The volume integral equations for strain and stress fields in a heterogeneous medium are used. Discretization of these equations is performed by the radial Gaussian functions centered at a system of approximating nodes. Such functions allow calculating the elements of the matrix of the discretized problem in explicit analytical form. For a regular grid of approximating nodes, the matrix of the discretized problem has the Toeplitz properties, and matrix-vector products with such matrices may be calculated by the fast fourier transform technique. The latter accelerates significantly the iterative procedure. First, the method is applied to the calculation of elastic fields in a homogeneous medium with a spherical heterogeneous inclusion and then, to composites with periodic and random sets of spherical inclusions. Simple cubic and FCC lattices of the inclusions which material is stiffer or softer than the material of the matrix are considered. The calculations are performed for cells that contain various numbers of the inclusions, and the predicted effective constants of the composites are compared with the numerical solutions of other authors. Finally, a composite material with a random set of spherical inclusions is considered. It is shown that the consideration of a composite cell that contains a dozen of randomly distributed inclusions allows predicting the composite effective elastic constants with sufficient accuracy.     

Multiple holes,cracks, and inclusions in anisotropic viscoelastic solids

By using the elastic–viscoelastic correspondence principle, the problems with multiple holes, cracks, and inclusions in two-dimensional anisotropic viscoelastic solids are solved for the cases with time-invariant boundaries. Based upon this principle and the existing methods for the problems with anisotropic elastic materials, two different approaches are proposed in this paper. One is concerned with an analytical solution for certain specific cases such as two collinear cracks, collinear periodic cracks, and interaction between inclusion and crack, and the other is a boundary-based finite element method for the general cases with multiple holes, cracks, and inclusions. The former considers only specific cases in infinite domain and can be used as a reference for any other numerical methods, and the latter is applicable to any combination of holes, cracks and inclusions in finite domain, whose number, size and orientation are not restricted. Unlike the conventional finite element method or boundary element method which usually needs very fine meshes to get convergence solutions, in the proposed boundary-based finite element method no meshes are needed along the boundaries of holes, cracks and inclusions. To show the accuracy and efficiency of these two proposed approaches, several representative examples are implemented analytically and numerically, and they are compared with each other or with the solutions obtained by the finite element method.     

基于均匀化方法的多孔材料细观力学特性数值研究

本文把均匀化理论与有限元法相结合 ,应用于多孔材料的弹性本构数值模拟 ,利用位移渐进展开建立了均匀化有限元列式。通过对正方形孔洞蜂窝材料有效模量的计算比较 ,表明本文方法可得到较准确的有效模量 ;同时还考察了胞壁固体相的力学性能参数vs 对宏观力学性能的影响 ,得到了与一些理论公式相同的结论。最后 ,本文对胞壁中含有弹性增强相的多孔材料的力学性能进行了数值研究 ,并利用二次均匀化方法给出定量的计算结果     

Singular stress analyses of V-notched anisotropic plates based on a novel finite element method

A super singular wedge tip element whose stiffness matrix is based on numerical eigensolutions is incorporated into standard hybrid-stress finite elements to study singular stress fields around the vertex of anisotropic multi-material wedge. The numerical eigensolutions are obtained by an ad hoc finite element eigenanalysis method. To demonstrate the validity of the method, singular stresses for some typical anisotropic single-material/bimaterial wedges are investigated. All numerical results show present finite element method converges rapidly to available solutions with few elements. The present method is applicable to dealing with the problems with more complex geometries.     

Finite element analysis of piezoelectric corner configurations and cracks accounting for different electrical permeabilities

Based on eigenfunctions of asymptotic singular electro-elastic fields obtained from a kind of ad hoc finite element method [Chen MC, Zhu JJ, Sze KY. Finite element analysis of piezoelectric elasticity with singular inplane electroelastic fields. Engng Fract Mech 2006;73(7):855-68], a super corner-tip element model is established from the generalized Hellinger-Reissner variational functional and then incorporated into the regular hybrid-stress finite element to determine the coefficients of asymptotic singular electro-elastic fields near a corner-tip. The focus of this paper is not to discuss the well-known behavior of electrically impermeable and permeable (usually it means fully permeable, hereinafter the same) cracks but analyze the limited permeable crack-like corner configurations embedded in the piezoelectric materials, i.e., study the influence of a dielectric medium inside the corner on the singular electro-elastic fields near the corner-tip. The boundary conditions of the impermeable or permeable corner can be considered as simple approximations representing upper and lower bounds for the electrical energy penetrating the corner. Benchmark examples on the piezoelectric crack problems show that present method yields satisfactory results with fewer elements than existing finite element methods do. As application, a piezoelectric corner configuration accounting for the limited permeable boundary condition is investigated, and it is found that the limited permeable assumption is necessary for corners with very small notch angles.     

二步法三维编织复合材料弹性性能的有限元法预报

用有限元法预测了二步法三维编织复合材料的有效弹性性能。在二步法方型三维编织复合材料细观结构大单胞模型的基础上, 考虑复合材料中纤维束的连续性及其空间的交织效应, 用离散杆单元构成的桁架结构有限元模型等效代替复合材料承受单轴拉伸载荷时的受力响应。同时, 以轴向拉伸性能测试试样为对象, 应用有限元软件包MARC 的结构静力分析部分计算了轴向弹性模量和泊松比, 数值计算结果与实验结果一致性较好。参数分析结果表明, 轴向弹性模量随轴纱与编织纱线密度之比和节距长度的增加呈增加趋势。     

Boundary collocation method for multiple defect interactions in an anisotropic finite region

The modified mapping collocation method is extended for the solution of plane problems of anisotropic elasticity in the presence of multiple defects in the form of holes, cracks, and inclusions under general loading conditions. The approach is applied to examine the stress and strain fields in an anisotropic finite region including an elliptical and a circular hole, an elliptical flexible inclusion, and a line crack. It can be readily incorporated into micro-mechanics models, capturing the relative importance of the matrix, the fiber/matrix interface, and reinforcement geometry and arrangement while estimating the effective elastic properties of composite materials. The accuracy and robustness of this method is established through comparison with results obtained from finite element analysis.     

A polarization‐based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast

It is recognized that the convergence of FFT‐based iterative schemes used for computing the effective properties of elastic composite materials drastically depends on the contrast between the phases. Particularly, the rate of convergence of the strain‐based iterative scheme strongly decreases when the composites contain very stiff inclusions and the method diverges in the case of rigid inclusions. Reversely, the stress‐based iterative scheme converges rapidly in the case of composites with very stiff or rigid inclusions but leads to low convergence rates when soft inclusions are considered and to divergence for composites containing voids. It follows that the computation of effective properties is costly when the heterogeneous medium contains simultaneously soft and stiff phases. Particularly, the problem of composites containing voids and rigid inclusions cannot be solved by the strain or the stress‐based approaches. In this paper, we propose a new polarization‐based iterative scheme for computing the macroscopic properties of elastic composites with an arbitrary contrast which is nearly as simple as the basic schemes (strain and stress‐based) but which has the ability to compute the overall properties of multiphase composites with arbitrary elastic moduli, as illustrated through several examples. Copyright © 2011 John Wiley & Sons, Ltd.     

Boundary value problems defined on stochastic self‐similar multiscale geometries

A method for solving elasticity problems defined on composite bodies with a stochastic multiscale microstructure is presented. It is considered that the composite is made from two types of materials with different elastic moduli. One of these is taken as the matrix, while the other forms the inclusions. The inclusions form a stochastic fractal with a finite, but potentially large, number of scales and are randomly distributed within the matrix. The method presented here leads to the statistics of the solution, i.e. the mean and the variance of the stress and displacement fields. It is based on the stochastic finite element method (spectral approach, second‐order technique) and on scaling properties of the spatial distribution of inclusions over the problem domain. This scaling allows for a simple formulation of the multiscale problem and leads to significant computation cost savings, especially when the fractal has a large number of relevant scales. Several examples are presented and used to verify the proposed method against computationally intensive classical finite element models in which the mesh is refined down to the scale of the finest inclusions. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

Eigenstrain and Fourier series for evaluation of elastic local fields and effective properties of periodic composites

The elastic stress and strain fields and effective elasticity of periodic composite materials are determined by imposing a periodic eigenstrain on a homogeneous solid, which is constrained to be equivalent to the heterogeneous composite material through the imposition of a consistency condition. To this end, the variables of the problem are represented by Fourier series and the consistency condition is written in the Fourier space providing the system of equations to solve. The proposed method can be considered versatile as it allows determining stress and strain fields in micro-scale and overall properties of composites with different kinds of inclusions and defects. In the present work, the method is applied to multi-phase composites containing long fibers with circular transverse section. Numerical solutions provided by the proposed method are compared with finite element results for both unit cell containing a single fiber and unit cell with multiple fibers of different sizes.     

Composites with auxetic inclusions showing both an auxetic behavior and enhancement of their mechanical properties

Composite materials made of auxetic inclusions and giving rise overall to negative Poisson’s ratio are considered, adopting a two-steps micromechanical approach for the calculation of their effective mechanical properties. The inclusions consist of periodic beam lattices, whose equivalent mechanical properties are calculated by a discrete homogenization scheme in a first step. The hexachiral and hexagonal reentrant lattices are considered as representative of the two main deformation mechanisms responsible for auxeticity. In a second step, the equivalent properties of the composite are calculated from numerical homogenization using the finite element method. It is shown that both an auxetic behavior and enhanced moduli can be obtained for not too slender micro-beams.     

Homogenization of random heterogeneous media with inclusions of arbitrary shape modeled by XFEM

The paper deals with the homogenization of random heterogeneous media with arbitrarily shaped inclusions simulated with the extended finite element method (XFEM) coupled with Monte Carlo simulation (MCS). The implementation of XFEM is particularly suitable for this type of problems since there is no need to generate a new finite element mesh at each MCS. The inclusions are randomly distributed and oriented within the medium while their shape is implicitly modeled by the iso-zero of an analytically defined random level set function, which also serves as the enrichment function in the framework of XFEM. Homogenization is performed based on Hill’s energy condition and MCS. The homogenization involves the generation of a large number of random realizations of the microstructure geometry based on a given volume fraction of the inclusions and other parameters (shape, spatial distribution and orientation). The influence of the inclusion shape on the effective properties of the random media is highlighted. It is shown that the statistical characteristics of the effective properties can be significantly affected by the shape of the inclusions especially in the case of large volume fraction and stiffness ratio.     

波浪与弹性板作用的数值模拟

该文采用高阶有限元和边界元联合的方法求解波浪与弹性板的相互作用。其中流场采用边界元法求解,结构弹性响应方程采用基于Mindlin板理论的有限元方法求解,通过模态叠加技术实现了弹性板变形与流场相互作用的解耦。通过对一矩形板的计算,验证了该文方法与他人试验结果和数值模拟结果都吻合良好。利用这一模型进一步分析了波浪与弹性圆形板的作用问题,并对圆形板运动响应的收敛性进行了分析。     

Uncertainty quantification in homogenization of heterogeneous microstructures modeled by XFEM

An extended finite element method (XFEM) coupled with a Monte Carlo approach is proposed to quantify the uncertainty in the homogenized effective elastic properties of multiphase materials. The methodology allows for an arbitrary number, aspect ratio, location and orientation of elliptic inclusions within a matrix, without the need for fine meshes in the vicinity of tightly packed inclusions and especially without the need to remesh for every different generated realization of the microstructure. Moreover, the number of degrees of freedom in the enriched elements is dynamically reallocated for each Monte Carlo sample run based on the given volume fraction. The main advantage of the proposed XFEM‐based methodology is a major reduction in the computational effort in extensive Monte Carlo simulations compared with the standard FEM approach. Monte Carlo and XFEM appear to work extremely efficiently together. The Monte Carlo approach allows for the modeling of the size, aspect ratios, orientations, and spatial distribution of the elliptical inclusions as random variables with any prescribed probability distributions. Numerical results are presented and the uncertainty of the homogenized elastic properties is discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

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。