Stress concentrations in the particulate composite with transversely isotropic phases

The accurate series solution have been obtained of the elasticity theory problem for a transversely isotropic solid containing a finite or infinite periodic array of anisotropic spherical inclusions. The method of solution has been developed based on the multipole expansion technique. The basic idea of method consists in expansion the displacement vector into a series over the set of vectorial functions satisfying the governing equations of elastic equilibrium. The re-expansion formulae derived for these functions provide exact satisfaction of the interfacial boundary conditions. As a result, the primary spatial boundary-value problem is reduced to an infinite set of linear algebraic equations. The method has been applied systematically to solve for three models of composite, namely a single inclusion, a finite array of inclusions and an infinite periodic array of inclusions, respectively, embedded in a transversely isotropic solid. The numerical results are presented demonstrating that elastic properties mismatch, anisotropy degree, orientation of the anisotropy axes and interactions between the inclusions can produce significant local stress concentration and, thus, affect greatly the overall elastic behavior of composite.     

Neo-Hookean fiber-reinforced composites in finite elasticity

The response of a transversely isotropic fiber-reinforced composite made out of two incompressible neo-Hookean phases undergoing finite deformations is considered. An expression for the effective energy-density function of the composite in terms of the properties of the phases and their spatial distribution is developed. For the out-of-plane shear and extension modes this expression is based on an exact solution for the class of composite cylinder assemblages. To account for the in-plane shear mode we incorporate an exact result that was recently obtained for a special class of transversely isotropic composites. In the limit of small deformation elasticity the expression for the effective behavior agrees with the well-known Hashin-Shtrikman bounds. The predictions of the proposed constitutive model are compared with corresponding numerical simulation of a composite with a hexagonal unit cell. It is demonstrated that the proposed model accurately captures the overall response of the periodic composite under any general loading modes.     

Some qualitative properties of incompressible hyperelastic spherical membranes under dynamic loads

Some nonlinear dynamic properties of axisymmetric deformation are ex- amined for a spherical membrane composed of a transversely isotropic incompressible Rivlin-Saunders material. The membrane is subjected to periodic step loads at its inner and outer surfaces. A second-order nonlinear ordinary differential equation approximately describing radially symmetric motion of the membrane is obtained by setting the thick- ness of the spherical structure close to one. The qualitative properties of the solutions are discussed in detail. In particular, the conditions that control the nonlinear periodic oscillation of the spherical membrane are proposed. In certain cases, it is proved that the oscillating form of the spherical membrane would present a homoclinic orbit of type ＂∞＂, and the amplitude growth of the periodic oscillation is discontinuous. Numerical results are provided.     

Stress Concentration at Interacting Spherical Inclusions in a Transversely Isotropic Body

An elastic-equilibrium problem is rigorously solved for a transversely isotropic body containing a finite number of arbitrarily arranged and oriented transversely isotropic spherical inclusions or an infinite periodic array of such inclusions. The solution is found based on the superposition principle for general solutions of single-particle problems and the addition theorems for partial vector solutions of the elastic equilibrium equations. The numerical results presented allow assessing the influence of matrix anisotropy on the stress concentration between inclusions     

横观各向同性超弹性球壳的有限振动

应用有限变形动力学理论研究了一种横观各向同性不可压超弹性材料球壳在表面突加均布拉伸载荷作用下的有限振动问题．给出了球壳振动的振幅和外加载荷之间的关系,得到了,球壳振动的相同和近似的周期,讨论了球壳振动的振幅、相图及振动的周期和材料各向异性程度的关系．     

Dynamical formation of cavity in transversely isotropic hyper-elastic spheres

The cavity formation in a radial transversely isotropic hyper-elastic sphere of an incompressible Ogden material, subjected to a suddenly applied uniform radial tensile boundary dead-load, is studied fllowing the theory of finite deformation dynamics. A cavity forms at the center of the sphere when the tensile load is greater than its critical value. It is proved that the evolution of the cavity radius with time follows that of nonlinear periodic oscillations. The project supported by the National Natural Science Foundation of China (10272069) and Shanghai Key Subject Program     

Overall longitudinal shear elastic modulus of a 1–3 composite with anisotropic constituents

The overall properties of a binary elastic periodic fiber-reinforced composite, with transversely isotropic constituents in an anti-plane-strain deformation state, are studied here for a cell periodicity of square type. This analysis considers four different orientations of the axis of transverse isotropy of constituents with respect to the direction of fibers. Each case is characterized by very simple closed-form expressions for the effective coefficients, which were obtained using the asymptotic homogenization method. Local problems defined on a periodic square unit cell are solved using Weierstrassian and Natanzon’s functions and perturbation theory relative to small anisotropy. In the isotropic limit, comparison with rigorous bounds and some well-known mixing rules are made. Also, comparisons with finite element calculations show that the derived closed-form formulae provide excellent results even for large anisotropy.     

An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites

The present work deals with the modeling of 1–3 periodic composites made of piezoceramic (PZT) fibers embedded in a soft non-piezoelectric matrix (polymer). We especially focus on predicting the effective coefficients of periodic transversely isotropic piezoelectric fiber composites using representative volume element method (unit cell method). In this paper the focus is on square arrangements of cylindrical fibers in the composite. Two ways for calculating the effective coefficients are presented, an analytical and a numerical approach. The analytical solution is based on the asymptotic homogenization method (AHM) and for the numerical approach the finite element method (FEM) is used. Special attention is given on definition of appropriate boundary conditions for the unit cell to ensure periodicity. With the two introduced methods the effective coefficients were calculated for different fiber volume fractions. Finally the results are compared and discussed.     

Indentation of punches into a piezoceramic body: two-dimensional contact problem of electroelasticity

The paper establishes the relationship between the solutions of the static contact problems of elasticity (no friction) for an isotropic half-plane and problems of electroelasticity for a transversely isotropic piezoelectric half-plane with the boundary perpendicular to the polarization axis. This allows finding the contact characteristics in the electroelastic case from the known elastic solution, without the need to solve the electroelastic problem. The contact problems of electroelasticity for different types of wedge-shaped punches (flat punch with rounded one or two edges, half-parabolic punch, and a periodic system of punches) are solved as examples Translated from Prikladnaya Mekhanika, Vol. 44, No. 11, pp. 55–70, November 2008.     

A transversely isotropic composite with a statistical distribution of spheroidal inclusions: a geometrical approach to overall properties

In the literature, the determination of global elastic properties of composites with ellipsoidal inclusions is based on the averaged stress, strain and elastic-energy fields (e.g. Compos. Sci. Technol. 27 (1986) 111). These are related to the local fields of the inclusion, the matrix, and the inclusion-matrix interface. In this study, we propose a method to obtain the global elastic properties of any transversely isotropic composite directly from the elastic properties of the matrix and the inclusions. Thus, it is not necessary to refer to the stress and strain applied globally or generated locally. The inclusions can have any transversely isotropic probability distribution of orientation. The problem is entirely geometrized and is treated in terms of averages of Walpole's (Adv. Appl. Mech. 21 (1981) 169) components of the fourth-order tensors describing the problem. We give a general numerical solution for any transversely isotropic statistical distribution of orientation, and also provide a validation of our method by applying it to some known cases and by retrieving the known exact solutions from the literature.