An exact solution for the three-phase thermo-electro-magneto-elastic cylinder model and its application to piezoelectric–magnetic fiber composites

A three-phase cylindrical model for analyzing fiber composite subject to in-plane mechanical load under the coupling effects of multiple physical fields (thermo, electric, magnetic and elastic) is presented. By introducing an eigenstrain corresponding to the thermo-electro-magnetic-elastic effect, the complex multi-field coupling problem can be reduced to a formal in-plane elasticity problem for which an exact closed form solution is available. The present three-phase model can be applied to fiber/interphase/matrix composites, such that a lot of interesting thermo-electro-magnetism and stress coupling phenomena induced by the interphase layer are revealed. The present model can also be applied to fiber/matrix composites, in terms of which a generalized self-consistent method (GSCM) is developed for predicting the effective properties of piezoelectric–magnetic fiber reinforced composites. The effective piezoelectric, piezomagnetic, thermoelectric and magnetoelectric moduli can be expressed in compact explicit formulae for direct references and applications. A comparison of the predictions by the GSCM with available experimental data is presented, and interesting magnification effects and peculiar product properties are discussed. As a theoretical basis for the GSCM, the equivalence of the three sets of different average field equations in predicting the effective properties are proved, and this fact provides a strong evidence of mathematical rigor and physical realism in the formulation.     

Propagation of axial shear magneto–electro-elastic waves in piezoelectric–piezomagnetic composites with randomly distributed cylindrical inhomogeneities

The integral equations of the scattering problem for piezoelectric–piezomagnetic composites with an inhomogeneity are derived. In the long-wave limit, the solutions of these integral equations for the composites containing a single inhomogeneous fiber are solved in close forms. The total scattering cross-section for the one-fiber composites is also obtained. By the so-called effective field method, the multi-fiber scattering problem is simplified to the one-fiber scattering problem, and the analytical expressions of magneto–electro-elastic fields for the multi-fiber composites are obtained in the long-wave limit. These solved magneto–electro-elastic fields are then used to solve the expressions of the static effective moduli, effective wave velocity and attenuation factor of piezoelectric–piezomagnetic composites with randomly distributed cylindrical inhomogeneities. Through numerical examples, it concludes that, if the random set of fiber cross-sections is homogeneous and isotropic, the effective field method is coincident with the Mori–Tanaka mean field method when the static effective moduli of piezoelectric–piezomagnetic composites are looked for. Moreover, the rules of the effective wave velocity versus the volume fraction of fibers are investigated for specific materials.     

Effect of negative permeability on elastic wave propagation in magnetoelastic multilayered composites

With the advent of left-handed magnetic materials, it is desirable to develop high-performance wave devices based on their novel properties of wave propagation. This letter reports the special properties of elastic wave propagation in magnetoelastic multilayered composites with negative permeability as comparecd to those in counterpart structures with positlve permeability. These novel properties of elastic waves are discerned from the diversified dispersion curves, which represent the propagation and attenuation characteristics of elastic waves. To compute these dispersion curves, the method of reverberation-ray matrix is extended for the analysis of elastic waves in magnctoelastic multilayered composites. Although only the results of a single piezomagnetic and a binary magnetoelastic layers with mechanically free and magnetically short surfaces as well as pelrfect interface are illustrated in the numerical examples, the analysis is applicable lo magnetoelastic multilayered structures with other kinds of boundaries/interfaces.     

Effective elastic properties of matrix composites with transversely-isotropic phases

The present work addresses the problem of calculation of the macroscopic effective elastic properties of composites containing transversely isotropic phases. As a first step, the contribution of a single inhomogeneity to the effective elastic properties is quantified. Relevant stiffness and compliance contribution tensors are derived for spheroidal inhomogeneities. The limiting cases of spherical, penny-shaped and cylindrical shapes are discussed in detail. The property contribution tensors are used to derive the effective elastic moduli of composite materials formed by transversely isotropic phases in two approximations: non-interaction approximation and effective field method. The results are compared with elastic moduli of quasi-random composites.     

Evaluation of the effective elastic moduli of tetragonal fiber-reinforced composites based on Maxwell’s concept of equivalent inhomogeneity

Maxwell’s concept of an equivalent inhomogeneity is employed for evaluating the effective elastic properties of tetragonal, fiber-reinforced, unidirectional composites with isotropic phases. The microstructure induced anisotropic effective elastic properties of the material are obtained by comparing the far-field solutions for the problem of a finite cluster of isotropic, circular cylindrical fibers embedded in an infinite isotropic matrix with that for the problem of a single, tetragonal, circular cylindrical equivalent inhomogeneity embedded in the same isotropic matrix. The former solutions precisely account for the interactions between all fibers in the cluster and for their geometrical arrangement. The solutions to several example problems that involve periodic (square arrays) composites demonstrate that the approach adequately captures microstructure induced anisotropy of the materials and provides reasonably accurate estimates of their effective elastic properties.     

Closed-form formulas for the effective properties of random particulate nanocomposites with complete Gurtin–Murdoch model of material surfaces

The objective of this work is to present an approach allowing for inclusion of the complete Gurtin–Murdoch material surface equations in methods leading to closed-form formulas defining effective properties of particle-reinforced nanocomposites. Considering that all previous developments of the closed-form formulas for effective properties employ only some parts of the Gurtin–Murdoch model, its complete inclusion constitutes the main focus of this work. To this end, the recently introduced new notion of the energy-equivalent inhomogeneity is generalized to precisely include all terms of the model. The crucial aspect of that generalization is the identification of the energy associated with the last term of the Gurtin–Murdoch equation, i.e., with the surface gradient of displacements. With the help of that definition, the real nanoparticle and its surface possessing its own distinct elastic properties and residual stresses are replaced by an energy-equivalent inhomogeneity with properties incorporating all surface effects. Such equivalent inhomogeneity can then be used in combination with any existing homogenization method. In this work, the method of conditional moments is used to analyze composites with randomly dispersed spherical nanoparticles. Closed-form expressions for effective moduli are derived for both bulk and shear moduli. As numerical examples, nanoporous aluminum is investigated. The normalized bulk and shear moduli of nanoporous aluminum as a function of residual stresses are analyzed and evaluated in the context of other theoretical predictions.     

Non-linear constitutive relations for magnetostrictive materials

In this paper, non-linear deformation behavior of magnetostrictive materials is studied and three magnetoelastic coupling constitutive models are developed. The standard square (SS) constitutive model is developed by means of truncating the polynomial expansion of the Gibbs free energy. The hyperbolic tangent (HT) constitutive equations, which involve a hyperbolic tangent magnetic-field dependence, are proposed to model the magnetic-field-induced strain saturation of magnetostrictive materials in the region of intense magnetic fields. A new model based on density of domain switching (DDS) is established in terms of the basic truth that magnetic domain switching underlies magnetostrictive deformation. In this model, it is assumed that the relation between density of domain switching, defined by the quantity of magnetic domains switched by per unit magnetic field and magnetic field can be described by a density function with normal distribution. The moduli in these constitutive models can be determined by a material function that is proposed to describe the dependence of the peak piezomagnetic coefficient on the compressive pre-stress for one-dimensional cases based on the experimental results published. The accuracy of the non-linear constitutive relations is evaluated by comparing the theoretical values with experimental results of a Terfenol-D rod operated under both compressive pre-stress and bias magnetic field. Results indicate that the SS constitutive equations can accurately predict the experimental results under a low or moderate magnetic field while the HT model can, to some extent, reflect the trend of saturation of magnetostrictive strain under a high magnetic field. The model based on DDS, which is more effective in simulating the experimental curves, can capture the main characteristics of the mechanism of magnetoelastic coupling deformation of a Terfenol-D rod, such as the notable dependence of magnetoelastic response on external stress and the saturation of magnetostrictive strain under intense magnetic fields. In addition, the SS constitutive relation for a general three-dimensional problem is discussed and an approach to characterize the modulus tensors is proposed.     

Two-Dimensional Magnetoelastic Problem for a Multiply Connected Piezomagnetic Body

A general method based on complex variable theory is proposed to determine the magnetic and elastic fields of a piezomagnetic body. This method is used to derive the basic relations for complex potentials in the two-dimensional problem of magnetoelasticity, their general representations for a multiply connected domain, expressions for stresses, displacements, vectors of magnetic field intensity and magnetic flux density, and magnetic field potential. A closed-form solution is obtained for a body with an elliptic (circular) hole or crack subjected at infinity to the action of a constant magnetoelastic field. Numerical results for a piezomagnetic plate with a circular hole are presented __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 10, pp. 64–74, October 2005.     

Analytical procedure for determining the linear and nonlinear effective properties of the elastic composite cylinder

In this work we consider a cylindrical structure composed of a nonlinear core (inhomogeneity) surrounded by a different nonlinear shell (matrix). We elaborate a technique for determining its linear elastic moduli (second order elastic constants) and the nonlinear elastic moduli, which are called Landau coefficients (third order elastic constants). Firstly, we develop a nonlinear perturbation method which is able to turn the initial nonlinear elastic problem into a couple of linear problems. Then, we prove that only the solution of the first linear problem is necessary to calculate the linear and nonlinear effective properties of the heterogeneous structure. The following step consists in the exact solution of such a linear problem by means of the complex elastic potentials. As result we obtain the exact closed forms for the linear and nonlinear effective elastic moduli, which are valid for any volume fraction of the core embedded in the external shell.     

Equivalent inhomogeneity method for evaluating the effective elastic properties of unidirectional multi-phase composites with surface/interface effects

A new technique is presented for evaluating the effective properties of linearly elastic, multi-phase unidirectional composites. Various effects on the fiber/matrix interfaces (perfect bond, homogeneously imperfect interfaces, uniform interphase layers) are allowed. The analysis of nano-composite materials based on the Gurtin and Murdoch model of material surface is also included. The basic idea of the approach is to construct a circular inhomogeneity in an infinite plane whose effects on the displacements and stresses at distant points are the same as those of a finite cluster of inhomogeneities (fibers of circular cross-section) arranged in a pattern representative of the composite material in question. The elastic properties of the equivalent inhomogeneity then define the effective elastic properties of the material. The volume ratio of the composite material is found after the size of the equivalent circular inhomogeneity is defined in the course of the solution procedure. This procedure is based on a semi-analytical solution of a problem of an infinite plane containing a cluster of non-overlapping circular inhomogeneities subjected to loading at infinity. The method works equally well for periodic and random composites and – importantly – eliminates the necessity for averaging either stresses or strains. New results for nano-composite materials are presented.     

Iteration method in linear elasticity of random structure composites containing heterogeneities of noncanonical shape

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of heterogeneities of arbitrary shape. The general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure where the notion of a perturbator is introduced and statistical averages are obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. Effective elastic moduli and the first statistical moments of stresses in the heterogeneities are estimated for statistically homogeneous composites with the general case of both the shape and inhomogeneity of the heterogeneities moduli. The explicit new representations of the effective moduli and stress concentration factors are built by the iteration method in the framework of the quasicristallite approximation but without basic hypotheses of classical micromechanics such as both the EFH and “ellipsoidal symmetry” assumption. Numerical results are obtained for some model statistically homogeneous composites reinforced by aligned identical homogeneous heterogeneities of noncanonical shape. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.     

Size-dependent effective elastic moduli of particulate composites with interfacial displacement and traction discontinuities

This work aims at estimating the size-dependent effective elastic moduli of particulate composites in which both the interfacial displacement and traction discontinuities occur. To this end, the interfacial discontinuity relations derived from the replacement of a thin uniform interphase layer between two dissimilar materials by an imperfect interface are reformulated so as to considerably simplify the characteristic expressions of a general elastic imperfect model which is adopted in the present work and include the widely used Gurtin–Murdoch and spring-layer interface models as particular cases. The elastic fields in an infinite body made of a matrix containing an imperfectly bonded spherical particle and subjected to arbitrary remote uniform strain boundary conditions are then provided in an exact, coordinate-free and compact way. With the aid of these results, the elastic properties of a perfectly bonded spherical particle energetically equivalent to an imperfectly bonded one in an infinite matrix are determined. The estimates for the effective bulk and shear moduli of isotropic particulate composites are finally obtained by using the generalized self-consistent scheme and discussed through numerical examples.     

Eshelby problem of polygonal inclusions in anisotropic piezoelectric full- and half-planes

This paper presents an exact closed-form solution for the Eshelby problem of polygonal inclusion in anisotropic piezoelectric full- and half-planes. Based on the equivalent body-force concept of eigenstrain, the induced elastic and piezoelectric fields are first expressed in terms of line integral on the boundary of the inclusion with the integrand being the Green's function. Using the recently derived exact closed-form line-source Green's function, the line integral is then carried out analytically, with the final expression involving only elementary functions. The exact closed-form solution is applied to a square-shaped quantum wire within semiconductor GaAs full- and half-planes, with results clearly showing the importance of material orientation and piezoelectric coupling. While the elastic and piezoelectric fields within the square-shaped quantum wire could serve as benchmarks to other numerical methods, the exact closed-form solution should be useful to the analysis of nanoscale quantum-wire structures where large strain and electric fields could be induced by the misfit strain.     

Micromechanics predictions of the effective moduli of magnetoelectroelastic composite materials

In this paper, the self-consistent, generalized Mori–Tanaka and dilute micromechanics theories are extended to study the coupled magnetoelectroelastic composite materials. The heterogeneous inclusion problem of magnetoelectroelastic behavior is formulated in terms of five interaction tensors related to the Green's functions for an infinite three-dimensional transversely isotropic magnetoelectroelastic solid. These tensors are then used to predict the effective moduli of the magnetoelectroelastic solid based on the self-consistent, Mori–Tanaka and the dilute approaches. Numerical results are obtained for various types of inclusions. These results are employed to study the effects of the inclusion properties, such as moduli, volume fractions, shapes, etc., on the effective moduli of magnetoelectroelastic composites, in particular, the related magnetic properties. The results obtained using the self-consistent model, the generalized Mori–Tanaka's model and the dilute approach are compared with the existing experimental and theoretical results.     

Thermo-elastic properties of heterogeneous materials with imperfect interfaces: Generalized Levin's formula and Hill's connections

We study the thermo-elastic properties of heterogeneous materials containing spherical particles or cylindrical fibres. The interface between the matrix and second-phase inhomogeneity is imperfect with either the displacement or the stress experiencing a jump across it. We relate the effective coefficient of thermal expansion (CTE) to the effective elastic moduli and thereby generalize Levin's formula, and reveal two connections among the effective elastic moduli, thereby generalizing Hill's connections. In contrast to the classical results, the effective CTE in the presence of an imperfect interface is strongly dependent on the size of the inhomogeneity, besides the interface elastic and thermo-elastic properties. This size dependence has been accurately captured by simple scaling laws.     

Solution of two-dimensional scattering problem in piezoelectric/piezomagnetic media using a polarization method

Using a polarization method, the scattering problem for a two-dimensional inclusion embedded in infinite piezoelectric/piezomagnetic matrices is investigated. To achieve the purpose, the polarization method for a two-dimensional piezoelectric/piezo-magnetic "comparison body" is formulated. For simple harmonic motion, kernel of the polarization method reduces to a 2-D time-harmonic Green's function, which is ob-tained using the Radon transform. The expression is further simplified under condi-tions of low frequency of the incident wave and small diameter of the inclusion. Some analytical expressions are obtained. The analytical solutions for generalized piezoelec-tric/piezomagnetic anisotropic composites are given followed by simplified results for piezoelectric composites. Based on the latter results, two numerical results are provided for an elliptical cylindrical inclusion in a PZT-5H-matrix, showing the effect of different factors including size, shape, material properties, and piezoelectricity on the scattering cross-section.     

Size-dependent effective thermoelastic properties of nanocomposites with spherically anisotropic phases

Composites made of semi-crystalline polymers and nanoparticles have a spherulitic microstructure which can be reasonably represented by a spherically anisotropic volume element. Due to the high surface-to-volume ratio of a nanoparticle, the particle-matrix interface stress, usually neglected in determining the effective elastic moduli of particle-reinforced composites, may have a non-negligible effect. To account for the latter in estimating the effective thermoelastic properties of a composite consisting of nanoparticles embedded in a semi-crystalline polymeric matrix, this work adopts a coherent interface model for the nanoparticle-matrix interface and proposes an extended version of the classical generalized-self consistent method. In particular, Eshelby's formulae widely used to calculate the elastic energy change of a homogeneous medium due to the introduction of an inhomogeneity are extended to the thermoelastic case. The nanoparticle size effect on the effective thermoelastic moduli of the composite are theoretically shown and numerically illustrated.     

A continuum theory for isotropic two-phase elastic composites

The paper presents the effective stiffness theory for isotropie two-phase elastic composites. The theory predicts dispersion of longitudinal and transverse plane time-harmonic travelling waves. The limiting phase velocities at vanishing wave numbers serve in the determination of the elastic moduli of the equivalent homogeneous isotropic medium. These elastic moduli are compared with the effective moduli defined statically.     

PREDICTION OF THE VISCOELASTIC PROPERTIES OF THE EQUIVALENT PARTICLE FOR THE INTERCALATED MULTI-LAYER STACK OF NANOPLASTICS

The aim of this paper is to apply the asymptotic homogenization method to deter- mining analytically and numerically the transversely isotropic viscoelastic relaxation moduli of the equivalent particle for the intercalated multi-layer stack of intercalated type nanoplastics.A two-phase multilayered material containing n layers is considered.The matrix is assumed to be an isotropic viscoelastic standard linear body and the reinforcement is assumed to be an isotropic elastic body.Final explicit analytical formulae for the effective elastic moduli of the multilay- ered material are derived first;and then the correspondence principle is employed to obtain the homogenized relaxation moduli of the equivalent intercalated particle.A numerical example is given.Final explicit analytical formulae in the time domain derived here make it convenient to estimate the influence of all the particle parameters of micro-structural details on the effective properties of the equivalent intercalated particle.The results of this paper can also be applied to multi-layer composites.     

Effective elastic moduli of inhomogeneous solids by embedded cell model

An embedded cell model is presented to obtain the effective elastic moduli for three-dimensional two-phase composites which is an exact analytic formula without any simplified approximation and can be expressed in an explicit form. For the different cells such as spherical inclusions and cracks surrounded by sphere and oblate ellipsoidal matrix, the effective elastic moduli are evaluated and the results are compared with those from various micromechanics models. These results show that the present model is direct, simple and efficient to deal with three-dimensional two-phase composites. The project supported by the National Natural Science Foundation of China (No. 19704100) and the National Natural Science Foundation of Chinese Academy of Sciences (No. KJ951-1-201)