Effect of anisometry of a platelike nanofiller on the elastic constants of a transversely isotropic composite

Analysis results for the elastic properties of a composite with a small amount of coplanarly arranged platelike filler particles are presented. The geometrical form of the particles is described by an oblate ellipsoid of revolution. The calculations are performed by formulas obtained by using the Eshelby approach for media with a low concentration of inclusions. The effect of anisometry of the ellipsoidal particles and of the ratio between the elastic moduli of the filler and matrix on the effective elastic constants of the composite is discussed. Calculation results are compared with experimental data for the elastic moduli of a nanocomposite containing completely exfoliated particles of an unmodified montmorillonite. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 4, pp. 493–504, July–August, 2008.     

Effective moduli of particulate solids: Lubrication approximation method

To efficiently calculate the effective properties of a composite, which consists of rigid spherical inclusions not necessarily of the same sizes in a homogeneous isotropic elastic matrix, a method based on the lubrication forces between neighbouring particles has been developed. The method is used to evaluate the effective Lamé moduli and the Poisson's ratio of the composite, for the particles in random configurations and in cubic lattices. A good agreement with experimental results given by Smith (1975) for particles in random configurations is observed, and also the numerical results on the effective moduli agree well with the results given by Nunan & Keller (1984) for particles in cubic lattices.     

Deformation of composites with arbitrarily oriented orthotropic fibers under matrix microdamages

In the present work, a model of nonlinear deformation of stochastic composites under microdamaging is developed for the case of a composite with orthotropic inclusions, when microdefects are accumulated in the matrix. The composite is treated as an isotropic matrix strengthened by triaxial arbitrarily oriented ellipsoidal inclusions with orthotropic symmetry of the elastic properties. It is assumed that the process of loading leads to accumulation of damage in the matrix. Fractured microvolumes are modeled by a system of randomly distributed quasispherical pores. The porosity balance equation and relations for determining the effective elastic modules in the case of orthotropic components are taken as basic relations. The fracture criterion is specified as the limiting value of the intensity of average shear stresses acting in the intact part of the material. On the basis of the analytic and numerical approach, we propose an algorithm for the determination of nonlinear deformation properties of the investigated material. The nonlinearity of composite deformations is caused by the finiteness of deformations. By using the numerical solution, the nonlinear stress–strain diagrams are predicted and discussed for an orthotropic composite material for various cases of orientation of inclusions in the matrix.     

Deformation of orthotropic composites with unidirectional ellipsoidal inclusions under matrix microdamages

In the present paper, a model of deformation of stochastic composites under microdamaging is developed for the case of orthotropic composite, when the microdamages are accumulated in the matrix. The composite is treated as an isotropic matrix strengthened by three-axial ellipsoidal inclusions with orthotropic symmetry of elastic properties. It is assumed that the loading process leads to accumulation of damages in the matrix. Fractured microvolumes are modeled by a system of randomly distributed quasispherical pores. The porosity balance equation and relations for determining the effective elastic moduli for the case of a composite with orthotropic components are taken as the basic relations. The fracture criterion is assumed to be given as the limit value of the intensity of average shear stresses occurring in the undamaged part of the material. Based on the analytical and numerical approach, an algorithm for the determination of nonlinear deformation properties of such a material is constructed. The nonlinearity of composite deformations is caused by the accumulation of microdamages in the matrix. Using the numerical solution, nonlinear stress-strain diagrams for the orthotropic composite in the case of biaxial extension are obtained. Published in Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 121–130, January–March, 2008.     

Effective moduli of elasticity of a composite composed of anisotropic layers

Relations are obtained for the effective moduli of elasticity and Poisson's ratios of a laminated fiber-reinforced composite, each layer of which has at least orthorhombic symmetry. The elastic properties of the composite in terms of the elastic constants of the layer are expressed exactly, and the elastic constants of the individual layer in terms of the values for the fiber and the matrix are expressed approximately. Two approximations are considered: one corresponds to the Hashin-Shtrikman variational approach, while in the second the comparison material is assigned elastic properties equal to the Voigt or Reuss means of the values for each layer. A numerical example is worked for the combination boron fibers-epoxy resin. The results of the calculation are compared with the exact solution of the problem for a composite composed of alternating layers of boron and epoxy resin.     

Nonlinear deformation of Three-Component Composites

The model of nonlinear deformation of stochastic composites under microdamaging is developed for the case of threecomponent composite, when the microdamages are accumulated in the matrix. The composite is treated as isotropic matrix strengthened by two different types of spheroidal inclusions with transversally-isotropic symmetry of elastic properties. Fractured microvolumes are modeled by a system of randomly distributed quasispherical pores. The porosity balance equation and relations for determining the effective elastic modules for the case of transversally-isotropic components are taken as basic relations. The fracture criterion is assumed to be given as the limit value of the intensity of average shear stresses occurring in the undamaged part of the material. The algorithm for determination of nonlinear deformative properties of such a material is constructed. The nonlinear stress-strain diagrams for three-component concrete for the case of uniaxial tension are obtained. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effective elastic moduli and stress concentrator factors in random structure aligned fiber composites

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically uniform random set of aligned fibers. Effective elastic moduli as well as the stress concentrator factors in the components are estimated. The micromechanical approach is based on the Green’s function technique as well as on the generalization of the “multiparticle effective field method” (MEFM, see for references, Buryachenko [1]). The refined version of the MEFM takes into account the variation of the effective fields acting on each pair of fibers. The dependence of effective elastic moduli and stress concentrator factors on the radial distribution function of the fiber locations is analyzed. Received: October 20, 2004     

Effective bulk moduli of materials containing stochastically distributed nano-inhomogeneities with surface stresses

In the present contribution, a mathematical model for the investigation of the effective properties of a material with randomly distributed nano-particles is proposed. The surface effect is introduced via Gurtin-Murdoch equations describing properties of the matrix/nano-particle interface. They are added to the system of stochastic differential equations formulated within the framework of linear elasticity. The homogenization problem is reduced to finding a statistically averaged solution of the system of stochastic differential equations. These equations are based on the fundamental equations of linear elasticity, which are coupled with surface/interface elasticity accounting for the presence of surface tension. Using Green's function this system is transformed to a system of statistically non-linear integral equations. It is solved by the method of conditional moments. Closed-form expressions are derived for the effective moduli of a composite consisting of a matrix with randomly distributed spherical inhomogeneities. The radius of the nano-particles is included in the expression for the bulk moduli. As numerical examples, nano-porous aluminum and nano-porous gold are investigated assuming that only the influence of the interface effects on the effective bulk modulus is of interest. The dependence of the normalized bulk moduli of nano-porous aluminum on the pore volume fraction (for certain radii of nano-pores) are compared to and discussed in the context of other theoretical predictions. The effective Young's modulus of nano-porous gold as a function of pore radius (for fixed void volume fraction) and the normalized Young's modulus vs. the pore volume fraction for different pore radii are analyzed. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非完美界面弹性复合材料中的微分几何方法

首次用微分几何方法计算了含一般旋转椭球体嵌入相的非完美界面弹性复合材料的有效模量·用内蕴几何量表出了能量泛函中的全部界面积分项，由此得到了这种统一嵌入相模型的复合材料有效模量的上下界限·在三种极限情况，即球、盘和针状嵌入相下，本文的结果将退化到Ｈａｓｈｉｎ（１９９２）的结果·     

Effect of interphase layers on the elastic properties of a carbon-nanotube-reinforced composite

A variant of a stepwise analysis of the elastic properties of a carbon-nanotube-reinforced composite with account of the effect of interphase layers between the nanotubes and the polymer matrix is reported. The preliminary calculation of the elastic constants of a structural element incorporating a nanotube and an interphase layer and the subsequent calculation of independent elastic constants of a composite with such transversely isotropic structural elements oriented in one direction are both performed by using the Mori–Tanaka theory of an equivalent medium. The calculations are carried out for a wide range of ratios between the elastic moduli of the interphase layer and matrix. The elastic constants of a composite with randomly oriented nanotubes are obtained by using the method of orientational averaging.     

Incremental Scheme to Homogenize Anisotropic Elastic Properties of Multi-Phase Composites

An incremental homogenization scheme for the prediction of elastic properties of composites is reviewed. Similar to the differential scheme, the inclusions are included step-by-step. This approach accounts for high volume fractions of inclusion of different shape and elastic properties. A numerical example for a composite consisting of a polymeric matrix, glass fibers and voids is shown. The fiber distribution is chosen equivalently to a distribution in an injection molded short-fiber reinforced composite. The volume fraction of the voids is varied. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiscale modeling of multiphase "in situ" composite

The elastic-plastic behaviour of rapidly solidified Al based (FeSi)-enriched alloys containing intermetallic compounds is considered. A new multilevel mechanical model for the “in situ” composite is proposed considering the aluminium matrix as a micropolar elastic plastic Cosserat material and the hardening phases as pure elastic ones. A two steps homogenization procedure is applied to obtain the overall properties of the multiphase “in situ“ composite, taking into account the existence of different sizes of intermetallic inclusions. A variational approach is applied to evaluate the equivalent stress on macro level at the transition from micro to macro scale. The model is developed using information provided by microstructural investigations and EDX analysis. The multistage bulk material manufacturing process from rapid solidified powders or ribbons is simulated using the Finite Element Method. The model is implemented as user subroutines into the FE code MARC. Numerical simulations are provided, corresponding to different values of metal forming parameters. The influence of the different inclusions sizes on the hardening behavior is discussed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of the method of conditional moments to investigating the deformation properties of orthotropic composites with fiber microdamages

A model of deformation of stochastic composites subjected to microdamage is developed for the case of orthotropic materials with microdamages accumulating in the fibers. The composite is treated as a matrix strengthened with elliptic fibers with orthotropic elastic properties. The fractured microvolumes are modeled by a system of randomly distributed quasi-spherical pores. The porosity balance equation and relations for determining the effective elastic moduli for the case of a fibrous composite with orthotropic components are used as the fundamental relations. The fracture criterion is given as a limit value of the intensity of average shear stresses occurring in the undamaged part of the material, which is assumed to be a random function of coordinates and is described by the Weibull distribution. Based on an analytical and numerical approach, the algorithm for determining the nonlinear deformation properties of such a material is constructed. The nonlinearity of composite deformations is caused by the accumulation of microdamages in the fibers. By using a numerical solution, the nonlinear stress–strain diagrams for an orthotropic composite in uniaxial tension are obtained. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 1, pp. 17–30, January–February, 2009.     

A generalized self-consistent method for composites with random elastic properties of inclusions

The generalized self-consistent method is extended to the problems of statistical mechanics of composites with random elastic properties of inclusions. This approach makes it possible to reduce the problem of predicting the effective elastic properties of composites with random structures to a sequence of simpler homogenized boundary-value problems for solitary inclusions with inhomogeneous elastic transition layers in a homogeneous effective elastic medium and with the corresponding boundary conditions. The elastic properties of a solitary inclusion for the gth homogenized problem are found from the solutions of the gth and (g+1)th homogenized problems. The elastic properties and sizes of the transition layers account for the random distribution, random sizes, and random elastic properties of inclusions in the composite. A test problem of predicting the effective elastic properties of a transversely isotropic layer composite with random elastic properties of some layers is solved by using the method proposed. The solution obtained coincides with the known exact solution [1].Perm State Technical University, Perm, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 785–796, November–December, 1999.     

Creep of plastics with spherical inclusions

A new variant of the theory of creep of plastics with spherical inclusions or pores is proposed on the basis of approximate equations for the integral parameters and the Volterra principle. Rabotnov's theory of viscoelasticity is used to describe linear creep of the matrix. The remaining components of the composite are assumed to be elastic. The complete system of operator equations of the linear viscoelasticity of plastics with spherical inclusions is obtained on the basis of the hypothesis of elastic deformation of the composite and hydrostatic pressure. Sample calculations are performed. A. A. Blagonravov Institute of Mechanical Engineering, Russian Academy of Sciences, Moscow. Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 668–675, September–October, 1996.     

Shear of orthogonally reinforced spherofibrous composites

The problem of determining the shear characteristics and interphase stress concentration of fibrous composites with spherical inclusions is examined on the basis of a three-phase model. Stress fields caused by diffusion interaction of phases are neglected. The elastic moduli of the composite are investigated and compared with those obtained from a two-phase model. The general formula for determination of the shear modulus of triorthogonally reinforced compsites is derived using previously investigated relationships for averaged stress fields. The matrix of these compsites contained spherical cavities. The dependence of integral characteristics of three-phase composites on their bulk phase concentration was investigated. The stresses between phases were studied as a function of composite structure.A. A. Blagonravov Machine-Science Institute, Russian Academy of Sciences, Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, No. 1, 104–111, January–February, 1997.     

Polydisperse structures of composites with random elastic properties of inclusions

A generalized self-consistent method [1, 2] is developed and applied to the boundary-value problems of composites with random elastic properties of inclusions. The approach suggested makes it possible to allow for a random mutual arrangement, statistical dispersion of elastic properties and sizes of the inclusions, and their mutual correlation in terms of special homogenized indicator functions. For comparison, the analytical solutions and those obtained from a corresponding sequence of H+1 (H=0,1,…) linked homogenized problems of the self-consistent method for the strain distribution in the inclusions and for the tensor of effective elastic properties of the composite are given. A numerical calculation of the effective transversely isotropic elastic characteristics for a unidirectional polydisperse fibrous composite is also presented. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 1, pp. 33–58, January–February, 2000.     

Exact relations for effective tensors of composites: Necessary conditions and sufficient conditions

Typically the elastic and electrical properties of composite materials are strongly microstructure dependent. So it comes as a nice surprise to come across exact formulae for effective moduli that are universally valid no matter what the microstructure. Such exact formulae provide useful benchmarks for testing numerical and actual experimental data and for evaluating the merit of various approximation schemes. They can also be regarded as fundamental invariances existing in a given physical context. Classic examples include Hill's formulae for the effective bulk modulus of a two‐phase mixture when the phases have equal shear moduli, Levin's formulae linking the effective thermal expansion coefficient and effective bulk modulus of two‐phase mixtures, and Dykhne's result for the effective conductivity of an isotropic two‐dimensional polycrystalline material. Here we present a systematic theory of exact relations embracing the known exact relations and establishing new ones. The search for exact relations is reduced to a search for matrix subspaces having a structure of special Jordan algebras. One of many new exact relations is for the effective shear modulus of a class of three‐dimensional polycrystalline materials. We present complete lists of exact relations for three‐dimensional thermoelectricity and for three‐dimensional thermopiezoelectric composites that include all exact relations for elasticity, thermoelasticity, and piezoelectricity as particular cases. © 2000 John Wiley & Sons, Inc.     

Non-linear deformation properties of materials with stochastically distributed anisotropic inclusions

In the present contribution, the problem of non-linear deformation of materials with stochastically distributed anisotropic inclusions is considered on the basis of the methods of mechanics of stochastically non-homogeneous media. The homogenization model of materials of stochastic structure with physically non-linear components is developed for the case of a matrix which is strengthened by unidirectional ellipsoidal inclusions. It is assumed that the matrix is isotropic, deforms non-linearly; inclusions are linear-elastic and have transversally-isotropic symmetry of physical and mechanical properties. Stochastic differential equations of physically non-linear elasticity theory form the underlying equations. Transformation of these equations into integral equations by using the Green's function and application of the method of conditional moments allow us to reduce the problem to a system of non-linear algebraic equations. This system of non-linear algebraic equations is solved by the Newton-Raphson method. On the analytical as well as the numerical basis, the algorithm for determination of the non-linear effective characteristics of such a material is introduced. The non-linear behavior of such a material is caused by the non-linear matrix deformations. On the basis of the numerical solution, the dependences of homogenized Poisson's coefficients on macro-strains and the non-linear stress-strain diagrams for a material with randomly distributed unidirectional ellipsoidal pores are predicted and discussed for different volume fractions of pores. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Identifying the elastic moduli of composite plates by using high-order theories. 2. Theoretical-experimental approach

The identification of elastic properties of laminated composite plates from measured eigenfrequencies has been performed. The elastic moduli of the laminates were determined by using a multilevel modeling and a two-step identification procedure. At the first step, based on a genetic algorithm, the Young’s and shear moduli were found, but at the second one, by minimizing an error function, the values of transverse moduli were refined. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 207–216, March–April, 2008.