Experimental verification of some elastic properties of unidirectional composites

The first part of the paper deals with homogenization models of unidirectional composites, in which each phase of the material is bounded by parallel cylindrical surfaces. For a GFRP with epoxy resin and glass fibres, five elastic constants for six models of the composite are calculated. In the second part, the results of strain gauge tests, photoelasticity investigations, and scanning electron inspection are discussed. With these data, some elastic constants of the composite in tension and compression are found. A comparison of experimental and analytical results is presented. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 195–206, March–April, 2008.     

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.     

Numerical-experimental estimation of the mechanical properties of an epoxy nanocomposite

The possible reasons for the violation of additivity laws for the density and elastic modulus of polymer nanocomposites (epoxy resin filled with silicon oxide nanoparticles) are considered. The fact that each nanoparticle is surrounded by a distinctive boundary layer is used to describe this phenomenon. The thickness and density of the layer are determined by measuring the density and elastic modulus of a polymer with different volume fractions of filler. A model for determining the strength of an epoxy nanocomposite by using the theory of short fibers is proposed. This model allowed us to describe the nonmonotonic relationship between the ultimate strength of the epoxy nanocomposite and the volume fraction of filler. It is shown that the filling of epoxy resin with silicon oxide nanoparticles beyond 5 vol. % decreases the strength of the composite in comparison with that of pure epoxy resin.     

Calculation of Dependent Elastic Constants of Plastic Foams with a Pronounced Strut-Like Structure

Seven dependent elastic constants of monotropic plastic foams with an expressed strut-like structure are calculated. For this purpose, the basic results of the previously elaborated mathematical model for light-weight plastic foams is used. The model includes a model cell of local structure for monotropic/isotropic plastic foams and an ensemble of structural elements, which allows one to calculate the seven dependent elastic constants, taking into account the pronounced polydispersity of the structure of plastic foams. The numerical values of the constants are compared with the available experimental data, and a satisfactory agreement is found to exist. As a final result, a full set of general expressions and numerical values are obtained for all 12 elastic constants of monotropic plastic foams.     

Problems of cuts in a composite elastic wedge

Problems of strip and elliptical cuts (tensile cracks) in the middle of a three-layer elastic wedge are investigated in a three-dimensional formulation. Free or rigid clamping conditions or the stress-free condition are stipulated on the outer surfaces of the composite wedge. The problems are assumed to be symmetrical about the plane of the cut. The wedge-shaped layer containing the cut is incompressible and hinged along both faces with two other layers. The integral equations of the problems with respect to the opening of the cut are derived. Inverse operators are obtained for the operators occurring in the kernels of these equations. The relation between problems on cuts and the corresponding contact problems for a composite wedge of half the aperture angle is used. The method of paired integral equations is used for the case of a strip cut emerging from the edge of the wedge. The problems are reduced to Fredholm integral equations of the second kind in certain auxiliary functions, in terms of the values of which the normal stress intensity factors are expressed. A regular asymptotic solution is constructed for the case of an elliptic cut.     

On the uniform stress state inside an inclusion of arbitrary shape in a three-phase composite

The stress field inside a two-dimensional arbitrary-shape elastic inclusion bonded through an interphase layer to an infinite elastic matrix subjected to uniform stresses at infinity is analytically studied using the complex variable method in elasticity. Both in-plane and anti-plane shear loading cases are considered. It is shown that the stress field within the inclusion can be uniform and hydrostatic under remote constant in-plane stresses and can be uniform under remote constant anti-plane shear stresses. Both of these uniform stress states can be achieved when the shape of the inclusion, the elastic properties of each phase, and the thickness of the interphase layer are properly designed. Possible non-elliptical shapes of inclusions with uniform hydrostatic stresses induced by in-plane loading are identified and divided into three groups. For each group, two conditions that ensure a uniform hydrostatic stress state are obtained. One condition relates the thickness of the interphase layer to elastic properties of the composite phases, while the other links the remote stresses to geometrical and material parameters of the three-phase composite. Similar conditions are analytically obtained for enabling a uniform stress state inside an arbitrary-shape inclusion in a three-phase composite loaded by remote uniform anti-plane shear stresses.     

Comparison of Mechanical Properties and Effects in Micro- and Nanocomposites with Carbon Fillers (Carbon Microfibers,Graphite Microwhiskers,and Carbon Nanotubes)

The mechanical properties and effects in fibrous composite materials are compared. The materials are based on the same matrix (EPON-828 epoxy resin) and differ in the type of fibers: Thornel-300 carbon microfibers, graphite microwhiskers, carbon zigzag nanotubes, and carbon chiral nanotubes. Two material models are considered: a model of elastic medium (macrolevel model) and a model of elastic mixture (micro-nanolevel model). Mechanical constants of 40 materials (4 types + 10 modifications) are calculated and compared. The theoretical ultimate compression strength along the fibers is discussed. The effects accompanying the propagation of longitudinal waves in the fiber direction are investigated.     

Nonlinear vibrations of embedded multi-walled carbon nanotubes using a variational approach

The present work deals with the problem of the nonlinear vibrations of multi-walled carbon nanotubes embedded in an elastic medium. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the van der Waals interlayer force. The variational iteration method (VIM) is adopted to obtain the amplitude–frequency curves for large-amplitude vibrations of single-, double- and triple-walled carbon nanotubes. The influences of changes in material constants of the surrounding elastic medium and the geometric parameters on the vibration characteristics of multi-walled carbon nanotubes are investigated. The results from the VIM solution are compared and shown to be in excellent agreement with the available solutions from the open literature. The capability of the present analytical technique is clarified in terms of numerical accuracy as well as computational efficiency.     

三角形弹性夹杂对裂纹的影响

研究了三角形弹性夹杂和裂纹之间的相互影响问题。应用Chau和Wang导出的面力边值问题的边界积分方程为基本方程,用夹杂和基体交界面上的面力和位移的连续性条件为补充方程,从而得到了一组能够解决夹杂和裂纹相互影响问题的方程,最后的方程组用一种新的边界单元法求解。计算了各种不同的夹杂和基体的材料常数以及夹杂和基体之间不同距离情况下裂纹尖端的应力强度因子。文中结果对研究新型复合材料有一定的应用价值。     

Anisotropy of elasticity of a composite with irregularly oriented anisometric filler particles

The effects of orientation and shape of filler particles on the elastic properties of composites have been analyzed. The elastic constants of a composite with irregularly oriented filler particles were calculated by using the method of orientational averaging of the properties of a representative structural element. The elastic constants of the structural element were found according to a known generalized Eshelby solution for a finite concentration of ellipsoidal inclusions. The diagrams of elasticity anisotropy for a transversely isotropic structural element and an orthotropic composite with irregularly oriented inclusions are presented. A quantitative estimate for the degree of anisotropy of elastic properties of composites is suggested. Data on the influence of shape anisometry of inclusions on the anisotropy coefficient of filled composites are also reported.     

New mathematical model for nonuniform deformation of composites

A new mathematical model is suggested for nonuniform deformation of composite materials valid for arbitrary external load gradients. As a basis stochastic equations are suggested for the statics of an elastic microinhomogeneous two-component material with volumetric forces differing from zero. Expressions are obtained for all coefficients in terms of elastic constants of the components and geometric parameters of the structure.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October, 1995).Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 3, pp. 310–318, May–June, 1995.     

Displacement potential solution of stiffened composite struts subjected to eccentric loading

The elastic behavior of stiffened struts of orthotropic composite material is investigated under an eccentric loading using displacement potential formulation. Solutions are obtained for different aspect ratios of the strut where the fibers are assumed to be directed along the axis of the struts. Solutions of stresses and displacements of boron/epoxy composite struts are presented numerically in the form of graphs. The effects of material orthotropy and different stiffeners on the elastic field are also investigated. Finally, some of the results of the present solution are compared with the corresponding finite element predictions, which, in turn, establishes the soundness as well as reliability of the present analysis of the composite struts.     

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·     

Analysis of effective elastic properties of unidirectional boron fiber-reinforced plastic by a generalized self-consistent method

A new generalized self-consisrtent method is developed for the statistical mechanics of composites which makes it possible to reduce the problem of predicting the effective elastic properties of composites with random structures to solution of a simpler averaged problem of an inclusion with a transitional layer in a material with the effective elastic properties sought. The typical size of the transition layer is determined by the correlation radius of the random structure, and its elastic properties are considered as both the close order of the mutual position and the variation of inclusion dimensions in terms of a special averaged indicator function of the structure. A numerical calculation is presented by the generalized self-consistent method for the average indicator function and the transversely-isotropic tensor for effective elastic properties of unidirectional boron fiber-reinforced plastic based on different models for actual random structure in the plane of isotropy. Analysis of the numerical results compared with experimental data and known solutions of other authors demonstrates the high accuracy of the generalized self-consistent method for a broad class of random composite structures.Perm State Mechanical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 6, pp. 747–758, November–December, 1996.     

Three-phase inclusions of arbitrary shape with internal uniform hydrostatic thermal stresses

We investigate the internal thermal stress field of a three-phase inclusion of arbitrary shape which is bonded to an infinite matrix through an interphase layer. The three phases have different thermoelastic constants. It is found that the internal thermal stress field induced by a uniform change in temperature can be uniform and hydrostatic within an inclusion of elliptical or hypotrochoidal shape when the thickness of the interphase layer is properly designed for given material parameters of the three-phase composite. Several examples are presented to demonstrate the solution. The thermal stress analysis of a (Q + 2)-phase inclusion of arbitrary shape with Q ≥ 2 is also carried out under the assumption that all the phases except the internal inclusion share the same elastic constants. It is found that the irregular inclusion shape permitting internal uniform hydrostatic thermal stresses becomes really arbitrary if a sufficiently large number of interphase layers are added between the inclusion and the matrix.     

The shear modulus of a composite material with a transversely isotropic matrix and a fibre

A relation is proposed for determining of the shear modulus of a fibrous composite material with a transversely isotropic matrix and a fibre as a function of the elastic constants of the matrix and the fibre as well as the volume fraction of each of them in the composite material. The isotropy planes of the matrix and fibre coincide and are perpendicular to the fibre axis. Two boundary value problems are solved in order to obtain the required relation: the problem of the longitudinal shear of a transversely isotropic solid cylinder that simulates the fibrous composite material and the problem of the combined longitudinal shear of a hollow and solid cylinder that simulate the matrix material and the fibre material respectively. Calculations using the proposed formula are compared with the available experimental data.     

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.     

Multifibre polymer composites: Prediction of deformation and thermophysical properties

Conclusions A theoretical and experimental investigation was carried out to examine the possibilities of a structural approach for prediction of elastic constants, creep functions and thermophysical characteristics of hybrid polymer composites reinforced with anisotropic fibres of several types. The theoretical solutions were obtained by generalizing the self-consistent method for the case of a three phase model. The effects of brittle fibre breakdown under tension in the direction of reinforcement of a unidirectional hybrid composite were studied under conditions of a short-term loading and a long-term creep. It has been shown that a creep of viscoelastic fibres plays a principal role in creep of the hybrid composite. It is just this creep that significantly increases the fibre damage during creep of the composite.A variant of the solution has been proposed for predicting the thermorheologically complex behavior of hybrid composites containing not only elastic but also viscoelastic thermorheologically simple components with different temperature-time shift factors. The peculiarities of thermal expansion of hybrid composites and the possibilities for a purposeful control of thermal expansion coefficients by hybridization were studied. The considered thermal interval included a region of transition of the polymer matrix from a glass state into a viscoelastic one.The control tests were performed for specimens of organic/glass, organic/carbon, glass/carbon and organic/boron polymer composites with different ratios of fibre volume contents. On the whole, the obtained accuracy of predicting the characteristics of the examined hybrid composites may be considered as acceptable for engineering applications.Published in Mekhanika Kompozitnykh Materialov, Vol. 30, No. 3, pp. 299–313, May–June, 1994.     

Non-linear near-resonance oscillations of an elastic incompressible layer

Plane one-dimensional waves of small amplitude, propagating transverse to an incompressible elastic layer and reflected successively from its boundaries, are considered. The oscillations are caused by small periodic (or close to periodic) external action on one of the layer boundaries, when the period of the external action is close to the period of natural oscillations of the layer. One of the boundaries of the elastic layer is fixed, while the other performs small specified two-dimensional motion in its plane. In such a near-resonance situation, non-linear effects occur which may build up over time. A system of equations is obtained which describes the slow change in the functions characterizing the oscillations of the medium in each period of the external action. It is assumed that all the quantities depend both on real time, any change of which in the approach considered is limited to one period, and on “slow” time, for which one period of real time serves as a small quantity. It is assumed that the evolution of the solution occurs when the slow time changes, while the role of real time is similar to the role of a spatial variable. This system of equations is obtained by the method of averaging over a period of the quantities representing nonlinear terms and the effect of the boundary conditions in the equations. It contains derivatives with respect to the real and slow times and also values of the functions characterizing the solution averaged over a period of the real time. If the averaged values are known, the equations have a hyperbolic form and their solutions can be both continuous and contain weak and strong discontinuities.