Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements

In this work, a modeling of electroelastic composite materials is proposed. The extension of the heterogeneous inclusion problem of Eshelby for elastic to electroelastic behavior is formulated in terms of four interaction tensors related to Eshelby’s electroelastic tensors. Analytical formulations of interaction tensors are presented for ellipsoidal inclusions. These tensors are basically used to derive the self-consistent model, Mori–Tanaka and dilute approaches. Numerical solutions are based on numerical computations of these tensors for various types of inclusions. Using the obtained results, effective electroelastic moduli of piezoelectric multiphase composites are investigated by an iterative procedure in the context of self-consistent scheme. Generalised Mori–Tanaka’s model and dilute approach are re-formulated and the three models are deeply analysed. Concentration tensors corresponding to each model are presented and relationships of effective coefficients are given. Numerical results of effective electroelastic moduli are presented for various types of piezoelectric inclusions and for various orientations and compared to existing experimental and theoretical ones.     

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.     

Solution of Eshelby's inclusion problem with a bounded domain and Eshelby's tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory

A solution for Eshelby's inclusion problem of a finite homogeneous isotropic elastic body containing an inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). An extended Betti's reciprocal theorem and an extended Somigliana's identity based on the SSGET are proposed and utilized to solve the finite-domain inclusion problem. The solution for the disturbed displacement field is expressed in terms of the Green's function for an infinite three-dimensional elastic body in the SSGET. It contains a volume integral term and a surface integral term. The former is the same as that for the infinite-domain inclusion problem based on the SSGET, while the latter represents the boundary effect. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is not considered. The problem of a spherical inclusion embedded concentrically in a finite spherical elastic body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. This Eshelby tensor depends on the position, inclusion size, matrix size, and material length scale parameter, and, as a result, can capture the inclusion size and boundary effects, unlike existing Eshelby tensors. It reduces to the classical Eshelby tensor for the spherical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing as the inclusion becomes large enough, and the boundary effect is vanishing as the inclusion volume fraction gets sufficiently low.     

Eshelby tensors for a spherical inclusion in microstretch elastic fields

In the present work, microelastic and macroelastic fields are presented for the case of spherical inclusions embedded in an infinite microstretch material using the concept of Green’s functions. The Eshelby tensors are obtained for a spherical inclusion and it is shown that their forms for microelongated, micropolar and the classical cases are the proper limiting cases of the Eshelby tensors of microstretch materials.     

Inhomogeneity of the first and second statistical moments of stresses inside the heterogeneities of random structure matrix composites

In this paper linearly thermoelastic composite media are treated, which consist of a homogeneous matrix containing a statistically homogeneous random set of heterogeneities. Effective properties (such as compliance, thermal expansion, stored energy) as well as the first statistical moments of stresses in the phases are estimated for the general case of nonhomogeneity of the thermoelastic inclusion properties. The micromechanical approach is based on the generalization of the “multiparticle effective field” method (MEFM, see for references Buryachenko, Appl. Mech. Rev. (2001), 54, 1–47), previously proposed for the estimation of stress field averages in the phases. The method exploits as a background the new general integral equation proposed by the author before and makes it possible to abandon the use of the central concept of classical micromechanics such as effective field hypothesis as well as their satellite hypothesis of “ellipsoidal symmetry”. The implicit recursion representations of the effective thermoelastic properties and stress concentration factor are expressed through some building blocks described by numerical solutions for both the one and two inclusions inside the infinite medium subjected to the inhomogeneous effective fields evaluated from subsequent self-consistent estimations. One also estimates the inhomogeneous statistical moments of local stress fields which are extremely useful for understanding the evolution of nonlinear phenomena such as plasticity, creep, and damage. Just at some additional assumptions (such as an effective field hypothesis) the involved tensors can be expressed through the Green function, Eshelby tensor and external Eshelby tensor. These estimated inhomogeneities of effective fields lead to the detection of fundamentally new effects for the local stresses inside the heterogeneities.     

Eshebly tensors for a finite spherical domain with an axisymmetric inclusion

In recent papers the finite Eshelby tensors for a concentrically placed spherical inclusion in a finite spherical domain have been computed and applied to numerous micromechanical problems. The present work is the extension of the computation of finite Eshelby tensors to general inclusions that are axisymmetric with respect to enclosing spherical domain. The problem of finding the finite Eshelby tensors is transformed into the integral equation. It is shown in the paper that the integral equation has a unique solution. Existence of the solution is proved by exploiting the symmetry of the problem which induce invariant subspaces of the integral equation. In the particular case for a excentrically placed spherical inclusion the problem is explicitly solved. Using computer algebra the solution is found in a closed form up to the second order.     

Prediction of thermal conductivity of composites with spherical fillers by successive embedding

In this study, a new model to predict the thermal conductivity of composites with spherical fillers is proposed. The original Eshelby model is extended to a finite filler volume fraction by successively embedding small filler volume fraction. The predicted results by the present model are compared with bounds such as parallel, series, and Hashin–Shtrikman models, results by modified Eshelby model, generalized self-consistent model, and effective medium theory, and the experimental results from the literature. It is found that the present model always lies between the bounds and shows better agreement with the experimental results than the other models for various filler volume fractions and thermal conductivity ratios.     

Solution of the Eshelby problem in gradient elasticity for multilayer spherical inclusions

We consider gradient models of elasticity which permit taking into account the characteristic scale parameters of the material. We prove the Papkovich–Neuber theorems, which determine the general form of the gradient solution and the structure of scale effects. We derive the Eshelby integral formula for the gradient moduli of elasticity, which plays the role of the closing equation in the self-consistent three-phase method. In the gradient theory of deformations, we consider the fundamental Eshelby–Christensen problem of determining the effective elastic properties of dispersed composites with spherical inclusions; the exact solution of this problem for classical models was obtained in 1976.     

Influence of interfaces on effective properties of nanomaterials with stochastically distributed spherical inclusions

The method of conditional moments is generalized to include evaluation of the effective elastic properties of particulate nanomaterials and to investigate the size effect in those materials. Determining the effective constants necessitates finding a stochastically averaged solution to the fundamental equations of linear elasticity coupled with surface/interface conditions (Gurtin–Murdoch model). To obtain such a solution the system of governing stochastic differential equations is first transformed to an equivalent system of stochastic integral equations. Using statistical averaging, the boundary-value problem is then converted to an infinite system of linear algebraic equations. A two-point approximation is considered and the stress fluctuations within the inclusions are neglected in order to obtain a finite system of algebraic equations in terms of component-average strains. Closed-form expressions are derived for the effective moduli of a composite consisting of a matrix and randomly distributed spherical inhomogeneities. As a numerical example a nanoporous material is investigated assuming a model in which the interface effects influence only the bulk modulus of the material. In that model the resulting shear modulus is the same as for the material without surface effects. Dependence of the bulk moduli on the radius of nanopores and on the pore volume fraction is analyzed. The results are compared to, and discussed in the context of other theoretical predictions.     

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.     

Objective Tensor Rates and Applications in Formulation of Hyperelastic Relations

Following Ogden, a class of objective (Lagrangian and Eulerian) tensors is identified among the second-rank tensors characterizing continuum deformation, but a more general definition of objectivity than that used by Ogden is introduced. Time rates of tensors are determined using convective rates. Sufficient conditions of objectivity are obtained for convective rates of objective tensors. Objective convective rates of strain tensors are used to introduce pairs of symmetric stress and strain tensors conjugate in a generalized sense. The classical definitions of conjugate Lagrangian (after Hill) and Eulerian (after Xiao et al.) stress and strain tensors are particular cases of the definition of conjugacy of stress and strain tensors in the generalized sense used in the present paper. Pairs of objective stress and strain tensors conjugate in the generalized sense are used to formulate constitutive relations for a hyperelastic medium. A family of objective generalized strain tensors is introduced, which is broader than Hill’s family of strain tensors. The basic forms of the hyperelastic constitutive relations are obtained with the aid of pairs of Lagrangian stress and strain tensors conjugate after Hill (the strain tensors in these pairs belong to the family of generalized strain tensors). A method is presented for generating reduced forms of the constitutive relations with the aid of pairs of Lagrangian and Eulerian stress and strain tensors conjugate in the generalized sense which are obtained from pairs of Lagrangian tensors conjugate after Hill by mapping tensor fields on one configuration of a deformable body to tensor fields on another configuration.      

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.     

Ultrasonic attenuation in polycrystals using a self-consistent approach

The self-consistent method of averaging elastic moduli to define the effective medium of a polycrystal is used to investigate the dynamic problem of wave propagation. An alternative covariance tensor describing the elastic moduli fluctuations of the polycrystal containing self-consistent elastic properties is derived and found to be significantly smaller than the covariance tensor formed through traditional Voigt averaging. Attenuation curves are generated using the self-consistent elastic moduli and covariance tensors and these results are compared with previous Voigt-averaged estimates. The second-order polycrystalline dispersion relation for the self-consistent scheme is found for cases of low and high crystallite anisotropy. The attenuation coefficients and dispersion relations derived through the self-consistent scheme are considerably different than previous estimates. Experimentally measured longitudinal attenuation coefficients support the use of the self-consistent scheme for estimation of attenuation.     

A unified model for piezocomposites with non-piezoelectric matrix and piezoelectric ellipsoidal inclusions

In this paper, the closed-form solutions of the electroelastic Eshelbys tensors of a piezoelectric ellipsoidal inclusion in an infinite non-piezoelectric matrix are obtained via the Greens function technique. Based on the generalized Budianskys energy-equivalence framework and the closed-form solutions of the electroelastic Eshelbys tensors, a unified model for multiphase piezocomposites with the non-piezoelectric matrix and piezoelectric inclusions is set up. The closed-form solutions of the effective electroelastic moduli of piezocomposites are also obtained. The unified model has a rigorous but simple form, which can describe the multiphase piezocomposites with different connectivities, such as 0–3, 1–3, 2–2, 2–3, 3–3 connectivities, etc. It can also describe the effects of non-interaction and interaction among the inclusions. As examples, the closed-form solutions of the effective electroelastic moduli are given by means of the dilute solution for the 0–3 piezocomposite with transversely isotropic piezoelectric spherical inclusions and by means of the dilute solution and the Mori–Tanakas method for the 1–3 piezocomposite with two kinds of transversely isotropic piezoelectric cylindrical inclusions. The predicted results are compared with experimental data, which shows that the theoretical curves calculated by means of the Mori–Tanakas method agree quite well with the experimental values, but the theoretical curves obtained by the dilute solution agree well with the experimental values only when the volume fraction of the ceramic inclusion is less than 0.3. The results in this paper can be used to analyze and design the multiphase piezocomposites.     

Mechanical modeling of growth considering domain variation—Part II: Volumetric and surface growth involving Eshelby tensors

The growth of biological tissues is here described at the continuum scale of tissue elements. Relying on a previous work in Ganghoffer and Haussy (2005), the rephrasing of the balance laws for tissue elements under growth in terms of suitable Eshelby tensors is done in the present contribution, considering successively volumetric and surface growth. Balance laws for volumetric growth are written in both compatible and incompatible configurations, highlighting the material forces for growth associated to Eshelby tensors. Evolution laws for growth are written from the expression of the local dissipation in terms of a relation linking the growth velocity gradient to a growth-like Eshelby stress, in the spirit of configurational mechanics. Surface growth is next envisaged in terms of phenomena taking place in a varying reference configuration, relying on the setting up of a surface potential depending upon the surface transformation gradient and to the normal to the growing surface. The balance laws resulting from the stationnarity of the potential energy are expressed, involving surface Eshelby tensors associated to growth. Simulations of surface growth in both cases of fixed and moving generating surfaces evidence the interplay between diffusion of nutrients and the mechanical driving forces for growth.     

Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress

The fundamental framework of micromechanical procedure is generalized to take into account the surface/interface stress effect at the nano-scale. This framework is applied to the derivation of the effective moduli of solids containing nano-inhomogeneities in conjunction with the composite spheres assemblage model, the Mori-Tanaka method and the generalized self-consistent method. Closed-form expressions are given for the bulk and shear moduli, which are shown to be functions of the interface properties and the size of the inhomogeneities. The dependence of the elastic moduli on the size of the inhomogeneities highlights the importance of the surface/interface in analysing the deformation of nano-scale structures. The present results are applicable to analysis of the properties of nano-composites and foam structures.     

Electromechanical response of (3–0, 3–1) particulate,fibrous, and porous piezoelectric composites with anisotropic constituents: A model based on the homogenization method

An analytical framework based on the homogenization method has been developed to predict the effective electromechanical properties of periodic, particulate and porous, piezoelectric composites with anisotropic constituents. Expressions are provided for the effective moduli tensors of n-phase composites based on the respective strain and electric field concentration tensors. By taking into account the shape and distribution of the inclusion and by invoking a simple numerical procedure, solutions for the electromechanical properties of a general anisotropic inclusion in an anisotropic matrix are obtained. While analytical forms are provided for predicting the electroelastic moduli of composites with spherical and cylindrical inclusions, numerical evaluation of integrals over the composite microstructure is required in order to obtain the corresponding expressions for a general ellipsoidal particle in a piezoelectric matrix. The electroelastic moduli of piezoelectric composites predicted by the analytical model developed in the present study demonstrate excellent agreement with results obtained from three-dimensional finite-element models for several piezoelectric systems that exhibit varying degrees of elastic anisotropy.     

Special properties of Eshelby tensor for a regular polygonal inclusion

When studying the regular polygonal inclusion in 1997, Nozaki and Taya discovered numerically some remarkable properties of Eshelby tensor: Eshelby tensor at the center and the averaged Eshelby tensor over the inclusion domain are equal to that of a circular inclusion and independent of the orientation of the inclusion. Then Kawashita and Nozaki justified the properties mathematically. In the present paper, some other properties of a regular polygonal inclusion are discovered. We find that for an N-fold regular polygonal inclusion except for a square, the arithmetic mean of Eshelby tensors at N rotational symmetrical points in the inclusion is also equal to the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion. Furthermore, in two corollaries, we point out that Eshelby tensor at the center, the averaged Eshelby tensor over the inclusion domain, and the line integral average of Eshelby tensors along any concentric circle of the inclusion are all identical with the arithmetic mean.The project supported by the National Natural Science Foundation of China (10172003 and 10372003) The English text was polished by Keren Wang.     

基于细观力学方法的混凝土热膨胀系数预测

建立混凝土材料的有效性质与微结构参数之间的关系,是混凝土材料优化设计的基础。本文用细观力学方法对复合材料宏观有效热膨胀系数进行研究,得到了含有一球形夹杂物的无限大介质在均匀变温作用下的应力场。假定混凝土为由骨料和砂浆基质组成的二相复合材料,根据混凝土宏观体积热膨胀量与组成混凝土的各相介质细观体积热膨胀量相等的原则,采用基于Mori-Tanaka方法的混凝土宏观有效剪切模量,推导出混凝土有效热膨胀系数的解答。对稀疏解法、自洽方法和有限单元数值试验结果的比较说明,本文提出的基于自洽方法的混凝土宏观有效热膨胀系数的理论公式能够较好的描述混凝土的热学特性,该方法可以推广到多相复合材料宏观有效热膨胀系数的预测中。     

On the generalized virial theorem and Eshelby tensors

The formal relationships between the scalar and tensorial virials and Eshelby tensors have been presently investigated. The key idea is to evaluate the Eshelby stress from discrete or atomistic simulations for a structured body, conceived as a numerical homogenization method to reconstitute the macroscopic continuum behavior in multiscale modelling approaches. Extending first the writing of the scalar virial to a material format, it is shown that the average of the elaborated scalar material virial is the trace of the (material) Eshelby stress. The spatial and material virials are further related to eachother in the framework of hyperelasticity, and a tensorial extension of the material virial is provided. Interpretation of those results from the microscopic point of view shows that Eshelby stress may be identified and calculated at the discrete level from the average of the virial tensor. Consideration of the material version of the virial theorem further leads to express Eshelby stress versus the average of the internal tensorial material virial and of the kinetic energy. The average scalar virial is further identified to the grand potential in a thermodynamic context. A definition of the material scalar virial for a second order continuum is lastly proposed, based on the identification of a second order Eshelby stress and in line with the second order Cauchy–Born rule.