Influence of parallelogram cells in the axial behaviour of fibrous composite

Effective longitudinal shear moduli closed-form analytical expressions of two-phase fibrous periodic composites are obtained by means of the asymptotic homogenization method (AHM) for a parallelogram array of circular cylinders. This work is an extension of previous reported results, where elastic, piezoelectric and magneto-electro-elastic composites for square and hexagonal arrays with perfect contact were considered. The constituents exhibit transversely isotropic properties. A doubly period-parallelogram array of cylindrical inclusions under longitudinal shear is studied. The behaviour of the anisotropic shear elastic coefficients is studied for several cell geometry arrays. Numerical examples and comparisons with other theoretical results demonstrate that the present model is efficient for the analysis of composites in which the periodic cell is rectangular, rhombic or a parallelogram. The effect of the arrangement of the cells on the shear effective property is discussed. The present method can provide benchmark results for other numerical and approximate methods.     

An eigenfunction expansion-variational method for the anti-plane electroelastic behavior of three-phase fiber composites

The anti-plane electroelastic behavior of three-phase piezoelectric composites (fiber/interphase/matrix) with doubly periodic microstructures is dealt with. A new variational functional for a unit cell is constructed by incorporating the periodic boundary conditions into the energy functional. Then, by combining with the eigenfunction expansions of the complex potentials satisfying the fiber-interphase-matrix interfacial conditions, an eigenfunction expansion-variational method based on a unit cell is developed. The numerical results of the effective electroelastic moduli show a rapid convergence of the present method. A unified first-order approximation formula is also provided, where an equivalent parameter matrix reflecting the overall influence of the electroelastic properties of the fiber and interphase on the effective properties, is found. The equivalent parameter matrix can greatly simplify the complicated relation of the effective electroelastic properties to the internal structure of a three-phase fiber composite. Though the equivalent parameter matrix is extracted in the first-order approximation formula, its validity is also verified in the high-order numerical results.     

A meso-scale finite element model for simulating free-edge effect in carbon/epoxy textile composite

Textile composites are well known for their excellent through thickness properties and impact resistance. In this study, a representative unit cell model of a triaxial braided composite is developed based on the composite fiber volume ratio, specimen thickness and microscopic image analysis. A meso-scale finite element (FE) mesh is generated based on the detailed unit cell dimensions and fiber bundle geometry parameters. The fiber bundles are modeled as unidirectional fiber reinforced composites. A micromechanical finite element model was developed to predict the elastic and strength material properties of each unidirectional composite by imposing correct boundary conditions that can simulate the actual deformation within the braided composite. These details are then applied in the meso-mechanical finite element model for a 0°/+60°/−60° triaxially braided T700s/E862 carbon/epoxy composite. Model correlations are conducted by comparing numerical predicted and experimental measured axial tension and transverse tension response of a straight-sided, single-layer (one ply thick) coupon. By applying a periodic boundary condition in the loading direction, the meso model captures the local damage initiation and global failure behavior, as well as the periodic free-edge warping effect. The failure mechanisms are studied using the field damage initiation contours and local stress history. The influence of free-edge effect on the failure behaviors is investigated. The numerical study results reveal that this meso model is capable of predicting free-edge effect and allows identification of its impact on the composite response.     

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。     

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.     

复合材料粘弹性本构关系与热应力松弛规律研究Ⅱ:数值分析

给出了预测复合材料粘弹性松弛模量、等效热应力松弛系数和等效时变热膨胀系数的均匀化方法的有限元数值实现步骤, 研究了单向纤维复合材料随温度变化的粘弹性本构关系, 以及热应力松弛规律和热膨胀系数的时变特征。单向纤维复合材料的一维热变形分析数据显示了热应变对时间的强烈依赖关系;以数值形式给出的等效热应力松弛模量对时间的依赖关系表明, 等效的热应力松弛模量对时间的依赖性较弱, 其冲击模量和渐近模量只相差0.4 %。     

On numerical evaluation of effective material properties for composite structures with rhombic fiber arrangements

This paper deals with unidirectional fiber reinforced composites with rhombic fiber arrangements. It is assumed, that there is a periodic structure on micro level, which can be taken by homogenization as a representative volume element (RVE) for the composite, where the composite phases have isotropic or transversely isotropic material characterizations. A special procedure is developed to handle the primary non-rectangular periodicity with common numerical homogenization techniques based on FE-models. Due to appropriate boundary conditions applied to the RVE elastic effective macroscopic coefficients are derived. Results are listed and compared with other publications and good agreements are shown. Furthermore new results are presented, which exhibit the special orthotropic behavior of such composites caused by the rhombic fiber arrangement.     

纤维聚合物复合材料的有效湿扩散系数

根据纤维聚合物复合材料的微观结构, 建立了基于复合材料单胞模型的湿扩散计算方法, 研究了不同温度和不同体积分数下纤维聚合物复合材料的湿扩散性能。假设纤维是不可渗透的, 并在聚合物基体中均匀分布, 计算了不同温度不同体积分数下复合材料的有效湿扩散系数。结果表明: 复合材料的有效湿扩散系数随温度的升高而增大, 随纤维体积分数的增大而减小; 在相同温度、 相同体积分数下, 正六边形排列的纤维复合材料的湿扩散系数比正方形的略大。计算结果及经验公式与Gueribiz曲线基本一致, 说明用单胞模型计算复合材料的湿扩散性能是非常有效的, 有助于理解纤维复合材料的湿扩散机制和性能。      

圆环形截面压电纤维复合材料有效电弹性性能研究

研究含周期分布压电纤维的压电复合材料的有效电弹性性能。通过在材料代表性体积单元边界上施加位移和电势周期边界条件,利用有限元法求得了代表性体积单元内的电弹性场。由平均电弹性场和压电复合材料有效电弹性性能定义,预测了圆环形截面压电纤维复合材料的有效电弹性系数。通过算例,比较了相同压电材料体积分数下圆环形截面压电纤维复合材料与圆截面压电纤维复合材料有效电弹性性能的差异,讨论了圆环形截面压电纤维内部非压电填充物的力学性质对有效压电系数的影响。该文结论可为高灵敏度压电复合材料设计提供 参考。     

粘弹性复合材料的有效三维阻尼矩阵预报与能耗计算

提出了一种计算纤维增强复合材料粘弹性阻尼和能耗的新方法。在层片的层次上,基于各向异性粘弹性理论导出以阻尼矩阵表示的材料能耗计算公式。在层板的层次上由能量等效原理得到有效阻尼矩阵。由此,可由层板的有效应力和有效应变计算出其能量耗散进而得到比阻尼。其中的关键点是有效阻尼矩阵的推导,最终得到的有效阻尼矩阵是各单层阻尼系数、体积分数、刚度系数以及层合板有效刚度系数的函数。用有效阻尼矩阵计算能耗可以适用于任意复合材料的应力应变状态,这对于粘弹性复合材料结构能耗的数值分析尤其重要。最后将该理论应用于几种情况下的复合材料的能耗和比阻尼分析,算例表明与其它的理论预测和试验相比吻合良好,证实了该方法的合理性。      

A dual homogenization and finite element approach for material characterization of textile composites

A novel procedure for predicting the effective nonlinear elastic moduli of textile composites through a combined approach of the homogenization method and the finite element formulation is presented. The homogenization method is first applied to investigate the meso-microscopic material behavior of a single fiber yarn based on the properties of the constituent phases. The obtained results are compared to existing analytical and experimental results to validate the homogenization method. Very good agreements have been obtained. A unit cell is then built to enclose the characteristic periodic pattern in the textile composites. Various numerical tests such as uni-axial and bi-axial extension and trellising tests are performed by 3D finite element analysis on the unit cell. Characteristic behaviors of force versus displacement are obtained. Meanwhile, trial mechanical elastic constants are imposed on a four-node shell element with the same outer size as the unit cell to match the force–displacement curves. The effective nonlinear mechanical stiffness tensor is thus obtained numerically as functions of elemental strains. The procedure is exemplified on a plain weave glass composite and is validated by comparing to experimental data. Using the proposed approach, the nonlinear behavior of textile composites can be anticipated accurately and efficiently.     

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.     

Coefficients of thermal expansion for single crystal piezoelectric fiber composites

Piezoelectric fiber composites were developed to overcome drawbacks of typical monolithic piezoceramic (PZT) actuators. Although piezoelectric fiber composites had many improvements over the monolithic PZT, there are still improvements. Thus, the single crystal piezoelectric fiber composite actuator is proposed. Single crystal piezoelectric materials such as PMN-PT have larger piezoelectric strain constants, higher bandwidth and higher energy density than polycrystalline counterparts. Piezoelectric fiber composites can improve the performance of various structures, and can be subject to wide temperature range where the thermoelastic behavior is important. Therefore, this paper studies the coefficients of thermal expansion (CTE) for single crystal piezoelectric fiber composites. The Macro Fiber Composite (MFC) as the piezoelectric fiber composite is considered. To calculate the effective properties of two orthotropic layers of the MFC, PMN-PT(or PZT)/epoxy and copper/epoxy layers, the rule of mixture is adopted. With the effective properties known for each layers, the two CTE of the MFC actuator are obtained from the classical lamination theory considering thermal effects. The difference of the CTE between the single crystal MFC and the standard MFC is studied.     

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.     

Homogenization of fibrous piezoelectric composites with general imperfect interfaces under anti-plane mechanical and in-plane electrical loadings

The present paper aims mainly to estimate the size-dependent effective properties of fibrous piezoelectric composites with general imperfect interfaces. The interface model used states that the displacement, traction, electric potential, and normal electric displacement all suffer jumps across an interface. In addition, it can degenerate into the well-known special ones by employing appropriate high-contrast interfacial parameters. To achieve our objective, an auxiliary inhomogeneity problem of a circular fiber embedded in an infinite cylindrical reference phase via general imperfect interface under anti-plane mechanical and in-plane electrical boundary conditions is analytically solved. This solution allows us to apply the well-known micromechanical schemes such as the dilute, Mori–Tanaka to obtain the closed-form expressions for the size-dependent overall properties of composites under consideration. Some numerical examples are provided for illustrating the features of the obtained general results.     

基于三维流场模型的含孔隙复合材料弹性常数有限元预测模型

为预测含孔隙复合材料单向层合板的有效弹性常数, 基于孔隙周边纤维分布和形态与三维Rankine椭圆体绕流流场的相似性, 提出了一种基于三维Rankine椭圆体绕流流场比拟的含孔隙复合材料弹性常数计算模型与方法。建立了含孔隙复合材料的有限元单胞计算模型, 用流场的速度变化比拟单胞内纤维体积分数的变化, 用流线形状比拟孔隙周边纤维的形态。通过对单胞施加周期性边界条件, 结合孔隙形态的概率分布模型和刚度平均法, 计算了含孔隙复合材料单向层合板的弹性常数。计算结果与实验数据有较好的一致性, 数值计算可以有效反映孔隙对复合材料单向层合板弹性常数的影响。      

Micromechanical analysis of magneto-electro-thermo-elastic composite materials with applications to multilayered structures

The method of asymptotic homogenization was used to analyze a periodic magnetoelectric smart composite structure consisting of piezoelectric and piezomagnetic phases. The asymptotic homogenization model is derived, the governing equations are determined and subsequently general expressions called unit-cell problems that can be used to determine the effective elastic, piezoelectric, piezomagnetic, thermal expansion, dielectric, magnetic permeability, magnetoelectric, pyroelectric and pyromagnetic coefficients are presented. The latter three sets of coefficients are particularly interesting in the sense that they represent product or cross-properties; they are generated in the macroscopic composite via the interaction of the different phases, but may be absent from the constituents themselves. The derived expressions pertaining to the unit-cell problems and the resultant effective coefficients are very general and are valid for any 3-D geometry of the unit cell. The model is illustrated by means of longitudinally-layered smart composites consisting of piezoelectric (Barium Titanate) and piezomagnetic (Cobalt Ferrite) constituents. Closed-form expressions for the effective properties are derived and the results are plotted vs. the volume fraction of the piezoelectric phase. Pertaining to the product properties of this particular magnetoelectric laminate, it is observed that the effective pyroelectric and pyromagnetic coefficients attain a maximum value at a BaTiO₃ volume fraction of 0.5 and maximum values for the magnetoelectric coefficients at a BaTiO₃ volume fraction of 0.4. Likewise, the maximum value of a magnetoelectric figure of merit (characterizing efficiency of energy conversion in longitudinal direction) is also attained at a volume fraction of 0.4.     

纤维复合材料的热膨胀系数

提出了一种利用压电光声技术测量材料热膨胀系数的实验方法,并测试了单向复合材料C/C、C/Al的横向、纵向的热膨胀系数。根据已有的理论计算方法与实验结果对该方法的测试结果进行验证,证明了该检测方法的可靠性,进而又测量了C/C、C/Al材料在任一方向上的热膨胀系数。这种方法克服了理论计算过程复杂以及常规手段无法测量任一方向上热膨胀系数的缺陷。     

Magnetoelectric coupling and cross-property connections in a square array of a binary composite

Effective elastic, dielectric, magnetic, piezoelectric, piezomagnetic and magnetoelectric properties for the anti-plane shear magnetoelectroelastic state are determined. This properties are calculated by means of the Asymptotic Homogenization Method for a composite material with unidirectional cylindrical fiber periodically distributed in a square array. Closed-form formulae are obtained for the effective properties. The formulae exhibit explicitly the dependence on (i) the geometry through fiber volume fraction, (ii) the periodicity of the array through its lattice sums, and finally (iii) the material properties of the phases. Anti-plane property calculations are carried out for BaTiO₃/CoFe₂O₄ composite. The local problem is solved using potential methods of a complex variable. The solution involves double periodic Weierstrass elliptic and related functions. The shear modulus experiments a stiffening due to the coupling piezoelectric and piezomagnetic effect. Both exact and empirical cross-property connections are found for these composites. For the empirical case, the knowledge of the anti-plane effective dielectric property, ε₁₁, experimentally or otherwise, yields the remaining ones, “plane and anti-plane”, with good accuracy within a wide range of fiber volume fractions. The magnetoelectric effect results from the mechanical interaction between constituents. Hence, in order to maximize the magnetoelectric coefficient during composite design, matrix plane shear modulus must be as higher as possible than plane bulk modulus.