Two analytical models for the study of periodic fibrous elastic composite with different unit cells

In this work, the effective elastic moduli of two-phase fibrous periodic composites are obtained by means of the Asymptotic Homogenization Method (AHM) and eigenfunction expansion-variational method (EEVM), for different types of parallelogram cells. The constituents exhibit transversely isotropic properties. A doubly periodic parallelogram array of cylindrical inclusions under longitudinal shear is considered. The behavior of the shear elastic coefficient for different geometry arrays of the cell related to the angle of the fibers is studied. Some numerical examples and comparisons with other theoretical results demonstrate that both methods (AHM and EEVM) are efficients for the analysis of composites with presence of rhombic cell. The effect of the configuration of the cells on the shear effective property is observed.     

Analysis of fibrous elastic composites with nonuniform imperfect adhesion

In most composites, the fiber–matrix adhesion is imperfect; the continuity conditions for stresses and displacements are not satisfied. In this contribution, effective elastic moduli are obtained by means of the asymptotic homogenization method (AHM), for three-phase fibrous composites (matrix/mesophase/fiber) with parallelogram periodic cell. Interaction between fiber and matrix is considered, and this is called the mesophase model where the nonuniform mesophase is studied. Besides, there is another type of matrix–fiber contact which is called nonuniform spring imperfect contact. In this case, the contrast or jump of the displacements in the boundary of each phase is proportional to the corresponding component of the tension in the interface in terms of a parameter given by a certain function that depends on the position. The constituents of the composites exhibit transversely isotropic properties. A doubly periodic parallelogram array of cylindrical inclusions under longitudinal shear is considered. The three-phase model is validated by the Finite Element Method and the AHM both approaches applied to two-phase composites with nonuniform spring imperfect contact. Comparisons with theoretical and experimental results verified that the present model is efficient for the analysis of composites with presence of nonuniform imperfect interface and parallelogram cell. The effect of the nonuniform imperfection on the shear effective property is observed. The present method can provide benchmark results for other numerical and approximate methods.     

Mechanical properties of hybrid composites using finite element method based micromechanics

A micromechanical analysis of the representative volume element of a unidirectional hybrid composite is performed using finite element method. The fibers are assumed to be circular and packed in a hexagonal array. The effects of volume fractions of the two different fibers used and also their relative locations within the unit cell are studied. Analytical results are obtained for all the elastic constants. Modified Halpin–Tsai equations are proposed for predicting the transverse and shear moduli of hybrid composites. Variability in mechanical properties due to different locations of the two fibers for the same volume fractions was studied. It is found that the variability in elastic constants and longitudinal strength properties was negligible. However, there was significant variability in the transverse strength properties. The results for hybrid composites are compared with single fiber composites.     

双周期涂层纤维增强复合材料反平面剪切问题

研究了双周期含涂层纤维增强复合材料在远场反平面载荷作用时的问题 , 利用 Eshelby等效夹杂方法和 Laurent 级数展开技术 , 并结合双准周期 Riemann边值问题理论 , 获得了其全场解析解 , 得到了应力场和有效模量表达式。与有限元结果的对照显示出本方法的效率和精度。考察了涂层参数对复合材料细观应力场和宏观有效性能的影响。当涂层刚度较大时 , 涂层内存在高的应力集中 , 且涂层刚度越大、 涂层相对厚度越小 , 应力集中系数越大。纤维刚度对复合材料有效模量的影响也取决于涂层性能 , 非常软或非常硬的涂层都大大限制了纤维刚度对复合材料有效模量的贡献。      

微观结构对复合材料弹性有效性能的影响

在渐近均匀化方法的基础上,用ANSYS参数设计语言建立了周期性边界条件,用ANSYS有限元程序对单胞进行求解,得到了复合材料的有效性能。分析了不同微观结构对材料有效性能的影响,并与实验和其它理论结果进行比较。得到了不同方向的方形纤维对于材料的有效模量和有效泊松比的影响。     

Micromechanics based analysis of randomly distributed fiber reinforced composites using simplified unit cell model

The geometry of the simplified unit cell (SUC) model [Aghdam MM, Smith DJ, Pavier MJ. Finite element micromechanical modeling of yield and collapse behavior of metal matrix composites. J Mech Phys Solids 48 (2000) 499–528] is extended to study effects of random fiber arrangement on the mechanical and thermal characterizations of unidirectional composites. The representative volume element (RVE) considered in the model consists of an r × c unit cells in which fibers are surrounded randomly by matrix cells. The presented model is general and can be used to predict the behavior of a fibrous composite subjected to thermal and mechanical, normal and shear, loading. The model also is capable of analyzing various combinations of these loading conditions such as off-axis test of unidirectional coupons. Both random and repeating fiber arrays can be considered in the model. Results for the overall thermal and elastic properties of a SiC/Ti metal matrix composite (MMC) show good agreement with both the finite element and other analytical models with repeating fiber arrays. Results of transverse properties also revealed that hexagonal array assumption for fiber arrangement is more realistic than square array assumption.     

Micromechanical analysis of fiber-reinforced composites on account of influence of fiber coatings

The new asymptotic method for the analysis of inhomogeneous composite materials on account of the micromechanical influence of fiber coatings is proposed in the present paper. The problem of longitudinal shear of the fiber-reinforced composites with the square and hexagonal lattices of periodically distributed parallel cylindrical fibers is examined. The asymptotic homogenization method is applied and the relevant unit cell problem is solved with the aid of the method of perturbation of boundary shape. The asymptotic analytical solutions are found for the effective longitudinal shear moduli and for the local stresses occurring in the composites on the microlevel. Local shear stresses along the fiber-coating and the matrix-coating interfaces are calculated. The influence of properties of coatings on the maximum local shear stresses on the interfaces of constituents is analysed. The obtained results are suitable for any values of stiffness and volume fractions of constituents, including the limiting case of absolutely rigid fibers converging to contact.     

Stiffness reduction of cracked solids

A method for the determination of the effective moduli of elastic solids containing a doubly periodic rectangular array of cracks is given. The derivation is based on the analysis of a unit cell in which the displacement vector is expanded to a second order in the distances from centerlines. The equilibrium equations in conjunction with the continuity conditions for the displacements and tractions, give a system of equations for the elastic field variables. The determination of the elastic internal energy provides the requested effective moduli of the cracked body. The method is applied to predict the loss of stiffness of cracked isotropic solids and unidirectional composites, as well as cracked cross-ply laminates.     

均质化法在分析非连续碳纳米管增强复合材料微观应力分布规律中的应用

利用具有精确周期性边界条件的均质化理论, 用宏微观有限元法分析了非连续碳纳米管呈规则和交错2 种排列情况下, 纳米管沿管长方向的应力分布规律。为保证传统的连续力学理论的适用性, 本文中的碳纳米管采用了用分子动力学方法简化的等效纤维模型。规则排列所得结果与应用Cox 剪滞理论及Lauke、Fu 等经典理论得出的结果比较发现: 除了经典理论中指出的碳纳米管长径比及纳米管体积含量2 个因素外, 纳米管形状及在基体中的排列方式对材料的力学性质也有较大影响。交错排列的纳米管在复合材料中有较高效率的应力转化和传递能力, 碳纳米管的端部间距(2 T_f ) 对应力的分布有较大的影响。结果显示出碳纳米管作为材料增强相的特殊性, 证明了均质化理论分析碳纳米管增强复合材料应力分布规律的可行性。      

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.     

In-plane problem for wave propagation through elastic solids with a periodic array of cracks

Summary In the context of wave propagation in damaged (elastic) solids, an analytical method previously introduced for scalar problems, is now applied to study the (vector) problem for normal penetration of a longitudinal plane wave into a periodic array of collinear cracks. Reduced the problem to an integral equation holding over the openings, an approximation of one-mode type leads to analytical solutions and then to explicit representations for the wave fields and the scattering parameters. Some graphs will finally compare our results with the numerical ones by other authors.     

双周期电磁弹性纤维增强复合材料的纵向剪切问题

利用等效夹杂法并结合双 (准) 周期解析函数理论,为双周期电磁弹性圆截面纤维复合材料的纵向剪切问题发展了实用有效的解析方法,获得了全场的级数解,给出了应力、电位移和磁感应强度的显式结果。利用数值算例讨论了该类非均匀材料微结构对其电磁弹性性能的影响。本文方法为电磁弹性复合材料的性能分析和微结构设计提供了一个实用有效的计算工具。      

An eigenfunction expansion-variational method based on a unit cell in analysis of a generally doubly periodic array of cracks

A new variational functional for a unit cell of a heterogeneous solid with periodic microstructures is constructed by incorporating the quasi-periodicity of the displacement field and the periodicity of the stress and strain fields into the strain energy functional. The functional can accommodate a broad class of periodic structures including the case where symmetry or antisymmetry properties of the unit cell may not exist. Then the functional is applied to deal with a doubly periodic array of cracks under plane and anti-plane loading. By combining with the eigenfunction expansions of the complex potentials satisfying the traction-free condition on the crack surfaces, an eigenfunction expansion-variational method based on a unit cell is developed. Numerical examples are presented and compared with existing results to demonstrate the high accuracy and efficiency, and wide application scope of the present method. Some interesting phenomena of multi-crack interaction, which do not occur in the case of symmetrical arrays of cracks, are revealed and discussed.     

二维机织复合材料弹性常数的有限元法预测

为了预测二维机织复合材料的弹性性能,建立了有限元力学分析模型。基于二维机织复合材料的几何特征,建立了参数化的单胞模型;考虑了织物纤维束呈现出的各向异性材料特征,将有限元中材料主方向转化到纤维屈曲方向,建立其力学分析有限元模型;分析了单胞边界面保持平面假设的不足,提出了对于二维机织复合材料通用的周期边界条件,获得了更为准确的二维机织复合材料的工程弹性常数。结果表明:织物衬垫单胞边界面,在单向拉伸载荷和纯剪切载荷下,呈凹凸翘曲变形,即为周期边界;应用给出的织物参数化几何建模方法与有限元求解方法,可以精确地获得工程弹性常数,数值计算结果与实验值吻合较好。      

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.     

Physical Deposition Improved SERS Stability of Morphology Controlled Periodic Micro/Nanostructured Arrays Based on Colloidal Templates

An effective and inexpensive method is developed to fabricate periodic arrays by sacrificial colloidal monolayer template route by chemical deposition and further physical deposition. By a colloidal template induced precursor solution dipping strategy, different periodic arrays of semi‐hollow sphere array, inverse opal with monolayer pore arrays and hole arrays are obtained under different conditions. After magnetron sputtering deposition, their morphologies are changed to novel micro/nanostructured arrays of honeycomb‐shaped arrays, hollow cavity arrays, and regular network arrays due to multiple direction deposition of sputtering deposition and shadow effect. After coating a gold thin layer, these periodic micro/nanostructured arrays are used as SERS active substrates and demonstrate a very stable SERS performance compared with periodic arrays achieved by direct colloidal template‐induced chemical deposition. Additionally, a honeycomb‐shaped array displays better SERS enhancement than that of a hollow cavity array or a regular network array. After optimization of honeycomb‐shaped arrays with different periodicities, an array with periodicity of 350 nm demonstrates much stronger SERS enhancement and possesses a low detection limit of 10^?11 M R6G. Such stable SERS performance is useful for practical application in portable Raman detecting devices to detect organic molecules.     

石墨烯/铜复合材料剪切性能的分子动力学模拟

采用分子动力学方法系统地研究了石墨烯/铜复合材料的剪切力学性能,包括剪切弹性模量、剪切屈服强度、剪切破坏强度及剪切变形机制。研究发现,与单晶铜的剪切模拟相比较,石墨烯的加入显著增强了石墨烯/铜复合材料的剪切强度,并且剪切强度随着石墨烯体积分数的增大而提高。复合材料中的石墨烯层与铜层产生了协同作用,即石墨烯层阻碍了铜的位错扩展,铜层限制了石墨烯的结构屈曲。对含球形缺陷的石墨烯/铜复合材料的剪切性能也进行了研究。结果表明,不同位置和数量的球形小缺陷对复合材料的剪切性能影响不大,小缺陷石墨烯/铜复合材料仍具有较好的性能和使用价值。但随着缺陷直径的增大,复合材料的剪切强度明显减小。     

Elastodynamic response of a coplanar periodic layer of elastic spherical inclusions

Reflection and transmission spectra of a plane longitudinal wave normally incident on a coplanar periodic array of identical spherical elastic inclusions in a solid (polyester) matrix are measured at wavelengths that are comparable to the inter-particle distance. These spectra exhibit pronounced Wood's anomalies i.e., a sharp variations in amplitude at the onset of a shear wave diffraction order as well as a drop in the transmission and a corresponding maximum in the reflection coefficients due to an enhancement of the resonance of the particles (for the case reported herein, this is the rigid body translational “dipole” resonance). An approximate low frequency (ka<1) self-consistent model is developed in which multiple scattering is explicitly taken into account by adding appropriate terms to the well-known solution for the scattering of a plane longitudinal wave by a single spherical particle in an unbounded matrix. The results of the numerical calculations show excellent agreement with the experimental data.     

Crack patterns in brittle thin films

The physical system studied is a brittle elastic film bonded to an elastic substrate with different elastic properties; a residual tensile stress is presumed to exist in the film. The focus of the study is the influence of the mismatch in elastic properties on patterns of crack formation in the film. The stress intensity factor and crack driving force for growth of a periodic array of cracks in the direction normal to the interface under two-dimensional conditions are determined for any crack depth and any mismatch in elastic parameters. It is found that, even for a relatively stiff film material, the stress intensity factor of each crack as a function of crack depth exhibits a local maximum. The driving force for crack extension in the direction parallel to the interface is then determined on the basis of these two-dimensional results, and the equilibrium spacing of crack arrays is estimated for given residual stress. The results of the calculations are used as a basis for qualitative arguments to explain the crack patterns which have been observed in GaN films on Si substrates.     

Composite materials whose phases have partially equal elastic moduli

Hill [J. Mech. Phys. Solids 11 (1963) 357, 12 (1964) 199] discovered that, regardless of its microstructure, a linearly elastic composite of two isotropic phases with identical shear moduli is isotropic and has the effective shear modulus equal to the phase ones. The present work generalizes this result to anisotropic phase composites by showing and exploiting the fact that uniform strain and stress fields exist in every composite whose phases have certain common elastic moduli. Precisely, a coordinate-free condition is given to characterize this specific class of elastic composites; an efficient algebraic method is elaborated to find the uniform strain and stress fields of such a composite and to obtain the structure of the effective elastic moduli in terms of the phase ones; sufficient microstructure-independent conditions are deduced for the orthogonal group symmetry of the effective elastic moduli. These results are applied to elastic composites consisting of isotropic, transversely isotropic and orthotropic phases.