Modeling functionally graded materials containing multiple heterogeneities

The effective material properties of functionally graded materials are predicted using a modified Mori–Tanaka and a self-consistent method. The proposed micromechanics model accounts for both multi-phase heterogeneity and a number of layers. The influence of geometries and distinct elastic material properties of each constituent and voids on the effective elastic properties of FGM is investigated. Numerical examples of different functionally graded materials are presented. The predicted elastic properties obtained from the current model agree well with experimental results from the literature.     

Finite element modeling of effective thermomechanical properties of Al–B<Subscript>4</Subscript>C metal matrix composites

The present work aims to investigate the influences of thermal residual stresses and material properties on the thermomechanical deformation behavior of Al–B₄C composites. Boron carbide-reinforced aluminum matrix composites having 4, 8, and 12 vol% boron carbide were fabricated using squeeze liquid stir casting method for experimental characterization of their microstructure, effective elastic moduli and effective CTEs at room temperature as well as elevated temperatures. Next, the thermomechanical behavior of fabricated composites was investigated using finite element modeling. The effects of thermal residual stresses on the effective material properties were examined by simulating the cooling process of MMCs from processing temperature to room temperature. The effective elastic moduli and the effective CTEs were predicted considering linear elastic as well as elastoplastic deformation of aluminum matrix, and the results obtained were compared with the experimental values. The effects of voids on effective material behavior are studied by simulating the void growth and nucleation using Gurson–Tvergaard–Needleman model.     

Mathematical investigation of interfacial property in fiber reinforced model composites

In the current paper, we have investigated the dependence of the effective elastic properties of a composite material on the fiber/matrix interface elastic property. The model composite consists of a single cylindrical fiber embedded in a concentric cylindrical matrix material. A three dimensional mathematical method has been developed connecting the interface properties with the effective axial Young’s modulus of the composite structure. Special effort has been devoted to decode information about the quality of the interface by exploiting the information provided by the elastic effective parameters. In particular, the effective modulus vs. stiffness coefficient curves have been generated for Ti/SiC composites. The aforementioned curves reveal the characteristics of the transition from the regime of perfect interface to the realm of complete debonding.     

Interface energy on effective elastic constant of piezoelectric composites with spherically anisotropy nano-particles under external uniform strain

The surfaces/interfaces effect is vital in nanocomposites. The electro-elastic surface/interface theory is introduced to predict the size-dependent effective elastic constants of piezoelectric composites with spherically anisotropy nano-particles under external uniform strain, and the analytical solution is obtained. New boundary conditions governing the surface piezoelectricity are used to analyze the surface piezoelectricity effect. The average electro-elastic coupling field of the randomly distributed nano-particles with surface/interface effect is derived by using the effective field method. In the numerical examples, the interface effect under different material constituents is analyzed. It is found that the interface effect on the effective elastic constants is significantly related to the material properties of nano-particles. The elastic constants and piezoelectric constants show different effects on the effective elastic constants.     

On the Possibility of Approximation of Irregular Porous Microstructure by Isolated Spheroidal Pores

Calculation of the effective elastic constants of a material with irregularly shaped porous space is discussed. It is shown that isolated pores of irregular shapes may be approximated, with good accuracy, by the spheroidal ones. Procedure of evaluation of the average aspect ratio from the photomicrograph is described. Several commonly used approximate schemes are applied to predict effective Young’s and shear moduli of a material with spheroidal pores. Comparisons of these predictions with the experimentally measured elastic constants for completely sintered hydroxyapatite exhibit reasonably good accuracy. The best agreement is given by self-consistent and effective field methods.     

On stiffness and strength reduction of solids due to crack development

Modification of the effective elastic and plastic constants of initially homogeneous and isotropic material with regularly distributed cracks is considered in the paper. The stress-strain relation for linearly elastic range is formulated as a tensor function with two independent variables: the stress tensor and damage tensor describing the current state of the cracked solid. This equation made it possible to evaluate all the elastic constants and is a starting point in the analysis of the plastic behavior of the damaged material. The appropriate yield criterion is derived in the form of an isotropic scalar function with the same variables as in the elastic range. To choose the most important terms of the general representation of this function, the energy of the elastic strain was calculated for homogenized equivalent material. This was done employing the stress-strain relation of elasticity for damaged solid proposed in the paper. The theoretical considerations were verified experimentally. To this end the material constants determined theoretically in the elastic and plastic ranges were compared with those measured experimentally for the models simulating the damaged material.     

A multi-scale framework for effective elastic properties of porous materials

This paper presents a statistical micromechanics-based multi-scale material modeling framework to predict the effective elastic moduli of porous materials. The present formulation differs from most of the existing theoretical models in that the interaction effects among the pores are directly accounted for by considering the pair-wise interaction and the statistical information of pore distribution is included by applying the ensemble volume averaging process. The theory of average fields is employed to derive the stress and strain concentration factor tensors that relate the local average fields to the global averages. Closed-form and analytical explicit expressions for the effective elastic moduli of porous materials are obtained in terms of the mechanical properties of the matrix material and porosity. The dependence of effective elastic properties on the porosity is investigated. Comparison of our theoretical prediction with the results of the published experimental data and other existing theoretical models is performed to illustrate the predictive capability of the proposed framework for porous materials.     

考虑混杂效应时纤维混杂缠绕筒三维等效弹性模量的理论估算和试验研究

给出了多种纤维混杂多向缠绕筒三维等效弹性模量的理论估算方法, 应用该方法计算了玻璃-碳纤维混杂缠绕筒的三维等效弹性模量。计及混杂效应的影响, 考虑了碳纤维体积分数、 铺层方式、 纤维分散度等因素, 对玻璃-碳纤维混杂缠绕结构, 引入混杂效应系数对该方法进行修正。试验结果表明, 该方法预测的三维等效弹性模量的精度较高。采用该方法可将复杂的纤维混杂缠绕结构等效为具有各向异性性质的均质单一材料, 极大降低了应力分析的工作量。     

基于复合材料力学模型的生态复合墙体研究

生态复合墙体作为生态复合墙结构的主要受力构件,它是由生态复合墙板与隐形外框组成的墙肢或墙段.复合墙体构造上的特殊性及多种材料的使用,使其弹性常数不易确定,进而影响墙体抗侧刚度的计算.基于前期试验研究,针对墙体独特构造,结合复合材料力学理论,建立适用于生态复合墙体的弹性阶段模型--双向纤维单层复合材料模型,推导出墙体弹性模量及剪切模量实用计算公式,为生态复合墙体的实用抗侧刚度计算公式提供必要的参数;利用弹性力学对各向异性等效弹性板的抗侧刚度进行推导,理论计算与试验结果的对比分析表明:在生态复合墙体抗侧刚度计算中,复合材料力学简化模型具有实用性和有效性.     

Ultrasonic wave propagation in two-phase media: Spherical inclusions

The scattering theory, recently developed via the extended method of equivalent inclusion, is used to study the propagation of time-harmonic waves in two-phase media of elastic matrix with randomly distributed elastic spherical inclusion materials. The elastic moduli and mass density of the composite medium are determined as functions of frequencies when given properties and concentration of the spheres and the matrix. Velocities and attenuation of ultrasonic waves in two-component media are determined. An averaging theorem that requires the equivalence of the strain energy and the kinetic energy between the effective medium and the original matrix with inhomogeneities is employed to derive the effective moduli and mass density. The functional dependency of these quantities upon frequencies and concentration provides a method of data analysis in ultrasonic evaluation of material properties. Numerical results for effective moduli, velocity and/or attenuation as functions of concentration of spherical inclusion material, or porosity, are graphically displayed.     

Novel approach for measuring the effective shear modulus of porous materials

A new approach is proposed for the experimental study of the effective shear modulus of porous elastic materials using the uniaxial tension test. The idea is to measure strains at a few points surrounding a cluster of holes that represents the structure of the material. The representative cluster is placed in the material with the same elastic properties as those of the matrix. The measured strains lead to the properties of the equivalent circular inhomogeneity, which would produce the same elastic fields as from the cluster. An aluminum plate containing a cluster of seven circular or hexagonal holes was used. The effective shear modulus obtained from the strain data was compared with theoretical predictions and various bounds, and it was shown that the laboratory estimated values are quite accurate. The experimental technique can be used for the determination of the effective Poisson’s ratio of porous media and/or cellular solids if more detailed strain data are obtained.     

Determination of plastic yield stress from spherical indentation slope curve

The indentation slope curve from a spherical indentation on elastic-plastic materials is examined. By comparing it with that of an linear elastic material of the same elastic properties, we found that the start point of plastic yielding for an elastic-plastic material can be easily located from the indentation slope curve. Based on this analysis, a simple but effective method is proposed to measure the plastic yield stress of very small samples from a spherical nano-indentation slope curve.     

Similarity of Stress-Singularity Problems for Materials with Linear and Bilinear Behaviors

An analogy is established between the solutions of the problems of singularities of stresses in linear and bilinear elastic isotropic media. It is shown that the distributions of stresses and displacements in the vicinity of singular points on the boundary of the body (characterized by the singularities of stresses) are described, in both cases, by the same functional dependences on the space coordinates but with different characteristics of the material. We deduce expressions for the effective moduli of elasticity and Poisson's ratio of the bielastic medium including the parameter of hardening of the material. The solution of the problem of singularities of stresses in bilinear materials is obtained from the solution of the corresponding problem for the linear elastic medium by replacing the elastic constants with the corresponding effective values depending on the parameter of hardening of the material. The cases of wedge-shaped notches (for various boundary conditions imposed on their edges), two-component wedges, plane wedge-shaped cracks, and circular conic notches or rigid inclusions in the bielastic space are studied in detail.     

A numerical method for homogenization in non-linear elasticity

In this work, homogenization of heterogeneous materials in the context of elasticity is addressed, where the effective constitutive behavior of a heterogeneous material is sought. Both linear and non-linear elastic regimes are considered. Central to the homogenization process is the identification of a statistically representative volume element (RVE) for the heterogeneous material. In the linear regime, aspects of this identification is investigated and a numerical scheme is introduced to determine the RVE size. The approach followed in the linear regime is extended to the non-linear regime by introducing stress–strain state characterization parameters. Next, the concept of a material map, where one identifies the constitutive behavior of a material in a discrete sense, is discussed together with its implementation in the finite element method. The homogenization of the non-linearly elastic heterogeneous material is then realized through the computation of its effective material map using a numerically identified RVE. It is shown that the use of material maps for the macroscopic analysis of heterogeneous structures leads to significant reductions in computation time.     

The effect of honeycomb relative density on its effective in-plane elastic moduli: An experimental study

Honeycombs are discrete materials at the macro-scale level but their mechanical properties need to be calculated as a continuum material in order to simplify their design in engineering applications. The effective mechanical behavior of hexagonal honeycombs was studied by analytical means and correlated with experimental results for aluminum honeycombs. In particular, the effective in-plane elastic moduli of the honeycombs were studied as a function of their relative density. The effect of the bending, shear and axial deformations on various existing beam models was analyzed for both in-plane honeycomb directions. An experimental program was performed with honeycombs of densities ranging from the commercial low-density ones to the solid construction material. It is shown experimentally that the beam models describe well the material response in the direction of the honeycomb double wall. However, it is concluded that the effective elastic moduli for honeycombs with low relative densities are not similar in the two in-plane directions as predicted by previous studies.     

A theory for physically nonlinear elastic fiber-reinforced composites

A three-dimensional constitutive relationship for the fiber-reinforced composite with a nonlinearly elastic matrix material is derived. The composite is modeled by a medium consisting of thin nonlinear matrix layers alternating with effective linearly elastic fibrous layers. Shear wave propagations are investigated. The formation of shock wave, its stability and growth behaviors are discussed.     

Explicit relations between elastic and conductive properties of materials containing annular cracks

The impact of annular cracks on the effective elastic and conductive properties of a material is analysed. The compliance contribution tensor of an annular crack - the quantity that determines the increase in compliance of a solid due to introduction of such a crack - is derived analytically. The resistivity contribution tensor of an annular crack is calculated numerically. It is shown that an effective circular crack, i.e. a crack which yields the same change in elastic/conductive properties of a material as the given annular crack, can be chosen to match both of these tensors. Using this result, the explicit relation between elastic and conductive properties of a material containing annular cracks is obtained. The relation is derived using a non-interaction approximation. Applicability of the derived formulae to real materials (to plasma-sprayed coatings, in particular) is discussed.     

On the J-integral in periodically layered composites

When a heterogeneous elastic material is represented by an effective homogeneous elastic solid, average stress and strain fields are used. The meaning of the J-integral in the effective homogeneous solid is investigated. A periodically layered medium is considered. The relation between the J-integrals in the original layered medium and the effective medium is derived.     

非均匀复合材料板中剪切波传播的研究

基于弹性界层中弹性波干涉理论,采用有效介质法,研究了剪切波在非均匀、纤维随机分布复合材料板中的传播,得到了非均匀弹性介质内的有效波数。通过满足弹性有界层的上、下边界条件,得到了非均匀界层中的频散方程。作为特例,绘出了不同参数下板中的前四阶频散曲线。可以看出,非均匀弹性有界层中的频散曲线和均匀界层中的有很大不同。最后分析了纤维和基体特性比、纤维的体积份数以及板厚与纤维半径比对频散曲线的影响。     

Calculation of effective Young's modulus distribution from electron backscatter diffraction results for stochastic analyses in aerospace applications

In reliability analyses of safety‐relevant components it is necessary to consider the variations in load, geometry and material properties. The scatter in load, geometry and global material properties is nowadays addressed in some stochastic analyses [ 1 , 2 ], whereas the scatter in local material properties is not adequately regarded due to difficulties in determining meaningful and stochastically relevant local material properties. In this paper a method for estimating the local distribution of the effective elastic behavior, belonging to a defined component direction, is presented. First, electron backscatter diffraction (EBSD) analyses of relevant cross sections were performed. Second, the effective elastic behavior perpendicular to the cross section was calculated for each scan point from the determined Euler angles. By using direct tensor rotation and generalized Hooke's law, the method can be applied to all kind of crystal structures if the elastic constants of the single crystal material are known. The experimental procedure, the mathematical background und the numerical realization are described in detail. Two examples of applying the procedure to materials for aerospace components as well as a stochastic finite element simulation considering the determined scatter of the elastic properties are shown.