Micromechanical analysis of the finite elastic–viscoplastic response of multiphase composites

Nonlinear thermoelastic–viscoplastic constitutive equations for large deformations with isotropic and directional hardening, are incorporated into a micromechanical finite strain analysis. As a result of this analysis, which is based on the homogenization technique for periodic microstructures, a global thermoinelastic constitutive law is established that governs the overall response of multiphase materials under finite deformations. This constitutive law is expressed in terms of the instantaneous effective mechanical and thermal stress tangent tensors together with the instantaneous global inelastic stress tensor that represents the viscoplastic effects. Results for a thermoinelastic matrix reinforced by a hyperelastic compressible material are given that illustrate the response of fibrous and particulate composites to various types of loading.     

The effect of anisotropic damage evolution on the behavior of ductile and brittle matrix composites

Anisotropic damage evolution laws for ductile and brittle materials have been coupled to a micromechanical model for the prediction of the behavior of composite materials. As a result, it is possible to investigate the effect of anisotropic progressive damage on the macroscopic (global) response and the local spatial field distributions of ductile and brittle matrix composites. Two types of thermoinelastic micromechanics analyses have been employed. In the first one, a one-way thermomechanical coupling in the constituents is considered according to which the thermal field affects the mechanical deformations. In the second one, a full thermomechanical coupling exists such that there is a mutual interaction between the mechanical and thermal fields via the energy equations of the constituents. Results are presented that illustrate the effect of anisotropic progressive damage in the ductile and brittle matrix phases on the composite’s behavior by comparisons with the corresponding isotropic damage law and/or by tracking the components of the damage tensor.     

压电介质有限变形的增量变分方程

从连续介质力学的基本理论出发，导出了压电介质几种非线性率型变分方程，在定义了各种增量之后，由率型变分方程得到了四种增量变分方程，即压电介质的ＴｏｔａｌＬａｇｒａｎｇｅ和ＵｐｄａｔｅｄＬａｇｒａｎｇｅ变分方程，它们是导出压电介质非线性有限元方程的基础，也可用于推导其它的简化理论。     

Variational principles of nonlinear piezothermoelastic media

Through the phenomenological approach, the nonlinear constitutive equations coupling the electro/magnetic thermoelastic media are derived. Several nonlinear variational principles for piezothermoelastic continua are presented and employed to formulate the incremental variational principles which are of important significance in practical applications such as the nonlinear finite element analysis, the buckling, postbuckling and dynamic stability analyses of piezoelectric smart structures. The work is supported by the National Natural Science Foundation, the Doctoral Education Foundation and the Aerospace Foundation.     

Electroelastic behavior of piezoelectric composites with coated reinforcements: Micromechanical approach and applications

In this work, a micromechanical model for the estimate of the electroelastic behavior of the piezoelectric composites with coated reinforcements is proposed. The piezoelectric coating is considered as a thin layer with active electroelastic properties different from those of the inclusion and the matrix. The micromechanical approach based on the Green’s functions technique and on the interfacial operators is designed for solving the electroelasticity inhomogeneous coated inclusion problem. The effective properties of a piezoelectric composite containing thinly coated inclusions are obtained through the Mori–Tanaka’s model. Numerical investigations into electroelastic moduli responsible for the electromechanical coupling are presented as functions of the volume fraction and characteristics of the coated inclusions. Comparisons with existing analytical and numerical results are presented for cylindrical and elliptic coated inclusions.     

Magneto-electro-elastic coated inclusion problem and its application to magnetic-piezoelectric composite materials

In this work, a micromechanical model for the estimate of the magneto-electro-elastic behavior of the magnetic-piezoelectric composites with coated reinforcements is proposed. The coating is considered as a thin layer with properties different from those of the inclusion and the matrix. The micromechanical approach based on the Green’s functions techniques and on the interfacial operators is designed for solving the magneto-electro-elastic inhomogeneous coated inclusion problem. The effective magneto-electro-elastic properties of the composite containing thinly coated inclusions are obtained through the Mori–Tanaka’s model. Numerical investigations into magneto-electro-elastic moduli responsible for the magneto-electric coupling are presented as functions of the volume fraction and characteristics of the coated inclusions. Comparisons with existing models are presented for various shape and orientation of the coated inclusions.     

三维五向编织复合材料渐进损伤分析及强度预测

基于材料连续体细观结构单胞,提出了材料的三维渐进损伤分析模型,采用非线性有限元方法并结合均匀化平均思想,首次建立了三维五向编织复合材料的强度预测模型。经研究典型编织角材料在拉伸载荷作用下细观损伤的发生及演化过程,分析了材料的细观失效机理,获得了材料的宏观拉伸应力应变曲线和极限破坏强度,并详细探讨了主要工艺参数编织角对材料宏观力学性能的影响规律。     

Local effective thermoelastic properties of graded random structure matrix composites

Summary  We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of ellipsoidal uncoated or coated inclusions, where the concentration of the inclusions is a function of the coordinates (functionally graded material). Effective properties, such as compliance and thermal expansion coefficient, as well as first statistical moments of stresses in the components are estimated for the general case of inhomogeneity of the thermoelastic inclusion properties. The micromechanical approach is based on the Green function technique as well as on the generalization of the multiparticle effective field method (MEFM), previously proposed for the research of statistically homogeneous random structure composites. The hypothesis of effective field homogeneity near the inclusions is used; nonlocal effects of overall constitutive relations are not considered. Nonlocal dependences of local effective thermoelastic properties as well as those of conditional averages of the stresses in the components on the concentration of the inclusions are demonstrated. Received 11 November 1999; accepted for publication 4 May 2000     

Three-dimensional thermoelastic analysis of finite length laminated cylindrical panels with functionally graded layers

Based on the 3D thermoelasticity theory, the thermoelastic analysis of laminated cylindrical panels with finite length and functionally graded (FG) layers subjected to three-dimensional (3D) thermal loading are presented. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The variations of the field variables across the panel thickness are accurately modeled by using a layerwise differential quadrature (DQ) approach. After validating the approach, as an important application, two common types of FG sandwich cylindrical panels, namely, the sandwich panels with FG face sheets and homogeneous core and the sandwich panels with homogeneous face sheets and FG core are analyzed. The effect of micromechanical modeling of the material properties on the thermoelastic behavior of the panels is studied by comparing the results obtained using the rule of mixture and Mori–Tanaka scheme. The comparison studies reveal that the difference between the results of the two micromechanical models is very small and can be neglected. Then, the effects of different geometrical parameters, material graded index and also the temperature dependence of the material properties on the thermoelastic behavior of the FG sandwich cylindrical panels are carried out.     

Micromechanical characterizing elastic,thermoelastic and viscoelastic properties of functionally graded carbon nanotube reinforced polymer nanocomposites

In this work, elastic, thermoelastic and viscoelastic properties of functionally graded carbon nanotube reinforced polymer nanocomposites are investigated using a 3-dimensional micromechanics-based approach. The main advantage of the proposed micromechanical model is its ability to give closed-form formulation for predicting the effective properties of nanocomposites. In the micromechanical modeling, the interphase formed due to non-boned van der Waals interaction between the continuous CNT and polymer matrix is considered through employing an individual representative volume element. The validity of the model is examined by comparing its results with other theoretical approaches and experimental data available in the literature. The effects of various types of CNTs arrangement in the matrix, i.e. uniform distribution and different functionally graded distributions on the elastic, thermoelastic and viscoelastic properties of polymer nanocomposites are investigated in detail. Furthermore, random arrangement of CNTs in the matrix is modelled. The influences of CNT/polymer matrix interphase and CNT volume fraction on the effective properties of nanocomposites are also studied. Finally, the viscoelastic response of nanocomposites under multiaxial loading is extracted and interpreted.     

Nonlinear ferro-electro-elastic beam theory

Motivated by the uniqueness and potential of the nonlinear range of piezoelectric and ferroelectric smart materials and structures, a static physically nonlinear ferro-electro-elastic beam theory which takes the effect of domain switching into account is developed. The kinematic assumptions adopt the geometrically linear Bernoulli–Euler form for the mechanical components and a first-order theory for the electrical potential, and lay the basis for further augmentation to higher order theories. The beam theory includes the field equations that correspond to the static case, the boundary conditions and the constitutive equations of ferro-electro-elasticity. The general 3-D constitutive equations are reduced to comply with the beam theory and formulated as ordinary differential equations by means of a set of generalized electro-mechanical stiffnesses. A micromechanical constitutive model that accounts for the loading history and for the domain switching phenomenon is adopted and an iterative solution procedure that incorporates the micromechanical approach is suggested. A numerical example that demonstrates the impact of the domain switching on the nonlinear electromechanical static response of a ferro-electro-elastic beam is presented and discussed. The quantitative assessment of this behavior takes a step towards new structural applications that cope with or even take advantage of the nonlinear ferro-electro-elastic range.     

A micromechanical model for effective elastic properties of particulate composites with imperfect interfacial bonds

A micromechanical model for effective elastic properties of particle filled acrylic composites with imperfect interfacial bonds is proposed. The constituents are treated as three distinct phases, consisting of agglomerate of particles, bulk matrix and interfacial transition zone around the agglomerate. The influence of the interfacial transition zone on the overall mechanical behavior of composites is studies analytically and experimentally. Test data on particle filled acrylic composites with three different interfacial properties are also presented. The comparison of analytical simulation with experimental data demonstrated the validity of the proposed micromechanical model with imperfect interface. Both the experimental results and analytical prediction show that interfacial conditions have great influence on the elastic properties of particle filled acrylic composites.     

Nonlinear micromechanical formulation of the high fidelity generalized method of cells

The recent High Fidelity Generalized Method of Cells (HFGMC) micromechnical modeling framework of multiphase composites is formulated in a new form which facilitates its computational efficiency that allows an effective multiscale material–structural analysis. Towards this goal, incremental and total formulations of the governing equations are derived. A new stress update computational method is established to solve for the nonlinear material constituents along with the micromechanical equations. The method is well-suited for multiaxial finite increments of applied average stress or strain fields. Explicit matrix form of the HFGMC model is presented which allows an immediate and convenient computer implementation of the offered method. In particular, the offered derivations provide for the residual field vector (error) in its incremental and total forms along with an explicit expression for the Jacobian matrix. This enables the efficient iterative computational implementation of the HFGMC as a stand alone. Furthermore, the new formulation of the HFGMC is used to generate a nested local-global nonlinear finite element analysis of composite materials and structures. Applications are presented to demonstrate the efficiency of the proposed approach. These include the behavior of multiphase composites with nonlinearly elastic, elastoplastic and viscoplastic constituents.     

Piezoresistive fiber-reinforced composites: A coupled nonlinear micromechanical–microelectrical modeling approach

Piezoresistive composites are materials that exhibit spatial and effective electrical resistivity changes as a result of local mechanical deformations in their constituents. These materials have a wide array of applications from non-destructive evaluation to sensor technology. We propose a new coupled nonlinear micromechanical-microelectrical modeling framework for periodic heterogeneous media. These proposed micro-models enable the prediction of the effective piezoresistive properties along with the corresponding spatial distributions of local mechanical–electrical fields, such as stress, strain, current densities, and electrical potentials. To this end, the high fidelity generalized method of cells (HFGMC), originally developed for micromechanical analysis of composites, is extended for the micro-electrical modeling in order to predict their spatial field distributions and effective electrical properties. In both cases, the local displacement vector and electrical potential are expanded using quadratic polynomials in each subvolume (subcell). The equilibrium and charge conservations are satisfied in an average volumetric fashion. In addition, the continuity and periodicity of the displacements, tractions, electrical potential, and current are satisfied at the subcell interfaces on an average basis. Next, a one way coupling is established between the nonlinear mechanical and electrical effects, whereby the mechanical deformations affect the electrical conductivity in the fiber and/or matrix constituents. Incremental and total formulations are used to arrive at the proper nonlinear solution of the governing equations. The micro-electrical HFGMC is first verified by comparing the stand-alone electrical solution predictions with the finite element method for different doubly periodic composites. Next, the coupled HFGMC is calibrated and experimentally verified in order to examine the effective piezoresistivity of different composites. These include conductive polymeric matrices doped with carbon nano-tubes or particles. One advantage of the proposed nonlinear coupled micro-models is its ability to predict the local and effective electro-mechanical behaviors of multi-phase periodic composites with different conductive phases.     

Thermoelastic coupling effect on a micro-machined beam resonator

In this paper, the effect of thermoelastic coupling on a micro-machined resonator is studied. The calculated results show that the frequency shift ratio caused by thermoelastic coupling is of the order of 10⁻³, which is much larger than that of air-damping. Furthermore, the non-dimensional frequency is scale-dependent with thermoelastic coupling being considered. In contrast, the non-dimensional frequency only depends upon the ratio of thickness to length when the themoelastic coupling effect is disregarded.     

Elastoplastic modeling of circular fiber-reinforced ductile matrix composites considering a finite RVE

A micromechanical elastoplastic damage model considering a finite RVE is proposed to predict the overall elastoplastic damage behavior of circular fiber-reinforced ductile (matrix) composites. The constitutive damage model proposed in our preceding work (Kim and Lee, 2009) considering a finite Eshelby’s tensor (Li et al., 2005, Wang et al., 2005) is extended to accommodate the elastoplastic behavior of the composites. On the basis of the exterior-point Eshelby’s tensor for circular inclusions and the ensemble-averaged effective yield criterion, a micromechanical framework for predicting the effective elastoplastic damage behavior of ductile composites is derived. A series of numerical simulations are carried out to illustrate stress–strain response of the proposed micromechanical framework and to examine the influence of a Weibull parameter on the elastoplastic behavior of the composites. Furthermore, comparisons between the present predictions and experimental data available in the literature are made to further assess the predictive capability of the proposed model.     

Estimation of the effective thermoelastic moduli of fibrous nanocomposites with cylindrically anisotropic phases

Recent developments in nanotechnology make it possible to fabricate nanofibers and identify their mechanical fibers. In particular, nanofibers are used as reinforcement in composites. The present work concerns unidirectional nanofibrous composites with cylindrically anisotropic phases and aims to analytically estimate their effective thermoelastic moduli. This objective is achieved by extending the classical generalized self-consistent model to the setting of thermoelasticity, to the case of cylindrically anisotropic phases, and to the incorporation of interface stress effect. Analytical closed-form estimations are derived for all the effective thermoelastic moduli, showing that these moduli depend on the fiber cross-section size. Numerical examples are provided to illustrate this size-dependent effect.     

The effect of a fiber loss in periodic composites

Three types of analyses are combined to investigate the effect of missing fibers in periodic continuous fiber composites that are subjected to thermomechanical loadings. The representative cell method is employed in the first analysis for the construction of Green’s functions elastic fields for the fiber–matrix interfacial jumps problem. As a result, the infinite domain problem is reduced, in conjunction with the discrete Fourier transform, to a finite domain problem on which Born–von Karman type boundary conditions are applied. In the second analysis, the transformed elastic field is determined by a second-order expansion of the displacement vector in terms of local coordinates, and by imposing the equilibrium equations, the interfacial traction and displacement conditions, and the Born–von Karman type boundary conditions. The actual non-periodic elastic field at any point is obtained from the Fourier-transformed fields by a numerical inversion. In the third one, a micromechanical analysis for periodic continuous fiber composites in which all fibers are perfectly bonded to the surrounding matrix provides the elastic field within the phases. A superposition of the thermoelastic fields obtained from the first and third analysis provides the traction-free boundary conditions at the interface of the missing fibers. The accuracy of the offered approach is verified by comparison with analytical solutions that exist in some special cases. Results show the effect of a missing fiber in boron/epoxy and glass/epoxy composites that are subjected to various types of thermomechanical loadings.     

Micromechanical models to guide the development of synthetic ‘brick and mortar’ composites

This paper describes a micromechanical analysis of the uniaxial response of composites comprising elastic platelets (bricks) bonded together with thin elastic perfectly plastic layers (mortar). The model yields closed-form results for the spatial variation of displacements in the bricks as a function of constituent properties, which can be used to calculate the effective properties of the composite, including elastic modulus, strength and work-to-failure. Regime maps are presented which indicate critical stresses for failure of the bricks and mortar as a function of constituent properties and brick architecture. The solution illustrates trade-offs between elastic modulus, strength and dissipated work that are a result of transitions between various failure mechanisms associated with brick rupture and rupture of the interfaces. Detailed scaling relationships are presented with the goal of providing material developers with a straightforward means to identify synthesis targets that balance competing mechanical behaviors and optimize material response. Ashby maps are presented to compare potential brick and mortar composites with existing materials, and identify future directions for material development.     

高体积含量非线性黏弹复合材料有效性质

提出了一个细观力学模型,可用于预测高体积含量非线性黏弹复合材料有效性质.该模型基于广义割线模量法、双球法以及Laplace-Carson变换技术.所提出的模型对玻璃微珠填充高密度聚乙烯(GB/HDPE)复合材料的应力应变关系进行了预测,结果与文献实验结果吻合;计算结果还表明在高体积百分比下文中所提出的方法比基于MT方法预测的粘性效应明显减弱;最后还将所提方法与线黏弹框架下的均质化模型做了比较,结果表明GB/HDPE表现出明显的非线性,线黏弹本构无法描述应变率对其力学行为的影响.