A generic homogenization model for magnetoelectric multiferroics

Abstract

A generic homogenization modeling framework which incorporates crystallographic domain features is introduced and computationally implemented for magnetoelectric multiferroics of all symmetries. The homogenization, mathematically applicable to heterogeneous media with contrasts in physical properties, replaces the heterogeneity of the multiferroics by an equivalent effective medium with uniform physical characteristics. A statistically representative unit-cell is proposed to encompass all forms of multiferroics and their composites in bulk. The variational formulation of the coupled magneto-electromechanical problem reveals the nature of interaction between mechanical, electrical, and magnetic fields of a multiferroic at a microscopic scale with high resolution. Furthermore, the mathematical homogenization theory of the multiferroic is implemented in finite element method by solving the coupled equilibrium electrical, magnetic, and mechanical fields. A “multiferroic finite element” is conceived for this purpose. The model is applied to a two-phase multiferroic magnetoelectric composite to demonstrate its validity by characterizing the equivalent physical properties.

Keywords

Magnetoelectric effect multiferroics micromechanical modeling microstructure finite element method

Introduction

Materials simultaneously possessing two or more of the ferroic order parameters—ferromagnetism, ferroelectricity, ferroelasticity, and ferrotorroidicity—in the same phase (Schmid, 1994, 2008) are referred to as multiferroics. Recently, the term multiferroics is meant predominantly for the coexistence of magnetism and ferroelectricity and has shown great potential for information storage and sensor technologies (Eerenstein et al., 2006; Gajek et al., 2007; Martin et al., 2010). There are 31 point groups that allow a spontaneous magnetization $M_{s}$ , and 31 that allow a spontaneous electrical polarization $P_{s}$ . 13 point groups intersect both these sets permitting both spontaneous polarization $P_{s}$ and spontaneous magnetization $M_{s}$ in the same phase (Hill, 2000; Schmid, 2008; Shuvalov and Belov, 1962). The degree of complexity of domain patterns and, for that matter, the degree of experimental difficulties (in synthesizing single-domain crystals) increase with the number of possible domain states. In single-phase multiferroics such as bismuth ferrite (BiFO₃), magnetic and ferroelectric domains coexist closely in bulk and thin film forms (Chu et al., 2008; Ke et al., 2010; Lebeugle et al., 2008).

A key feature of ferroic materials is the formation of domains consequent to the appearance of a ferroic order at the phase transition. The lesser the symmetry of the ferroic phase, the more the number of possible domain states. For instance, the less symmetric monoclinic multiferroic crystal with space group m and having $P_{s}$ and $M_{s}$ in (110)_c possesses 12 and 24 ferroelectric and ferromagnetic domain states, respectively. Yet the more symmetric orthorhombic 2-mm crystals with $P_{s} ∥ M_{s} ∥ {[100]}_{c}$ have only 6 ferroelectric and 12 ferromagnetic domain states. (Here, the orientations of $P_{s}$ and $M_{s}$ , i.e., ${[hkl]}_{c}$ , are given relative to the cubic reference system.) However, the domain architecture differs in various multiferroics (Lebeugle et al., 2008; Meier et al., 2009). Domain orientation essentially is the manifestation of the orientation of underlying unit-cell. It should be emphasized here that the cell referred to here is a true unit-cell whose definition involves the magnetic structure of the multiferroic crystal which may be different from pure crystallographic unit-cell. Crystallographic orientation can be characterized by a set of Euler angles $(ϕ, θ, ϑ)$ (Goldstein, 1978). Developments in the direction of controlled manipulation of multiferroic domains could open opportunities for novel device architectures.

Theoretical treatment of equivalent or effective properties of complex material systems would often be sought as a means for material characterization where experiments would be difficult to be accomplished. The costly and delicate process of synthesis of crystals of fairly large size and associated deleterious effects such as depoling and twinning affects the accuracy of the measurement of physical properties (Ahart et al., 2008). Recently, continuum models incorporating domain-scale information such as orientation and its distribution have been used to describe the macroscopic features of coupled crystalline materials (Jayachandran et al., 2010; Li et al., 2010). An efficient and convenient way with affordable computational times is the use of homogenization technique. It has been demonstrated that the figures of merit of multiferroic thin films, where the electrical, magnetic, and mechanical fields are coupled, can be tuned by the design of crystal morphology at the microstructure level (Martin et al., 2010).

Here, we have developed a generic homogenization method theoretically for multiferroics irrespective of its crystallographic symmetry. A two-scale asymptotic analysis combined with a variational formulation of the underlying mechanical, electrical, and magnetic fields is carried out to unveil their interaction in the microscopic scale. Furthermore, the numerical solution of the coupled magneto-electromechanical problem is sought using the finite element method (FEM) to eventually compute the homogenized (effective) properties of multiferroics. FEM is a feasible computational approach and a natural choice of numerical approximation in the homogenization procedure. Subsequently, an example is shown by applying the method to a two-phase multiferroic composite of a piezoelectric and a piezomagnetic material. The variational asymptotic homogenization used in this study provides no direct simulation of magnetic or electric domains. This framework instead models the piezoelectric and magnetoelectric (ME) coupling in a continuum setting (where piezomagnetism is treated analogous to piezoelectricity), where the scales involved are of the order of micrometers.

Since there are very few single-phase multiferroic materials (Hill, 2000; Khomskii, 2009), and most of them show very weak ME coupling at room temperature, ME composites present an alternative for engineering ME coupling (Eerenstein et al., 2006). Most of the theoretical developments (Benveniste, 1995; Benveniste and Milton, 2003; Li and Dunn, 1998; Nan et al., 2001, 2005; Srinivasan et al., 2001) are based on the strain-mediated two-phase multilayer composites of magnetostrictive and piezoelectric phases proposed by Harshe et al. (Avellaneda and Harshe, 1994; Harshe et al., 1993). Experimental and theoretical developments in these classes of materials are compiled in recent reviews (Bichurin et al., 2010; Nan et al., 2008; Srinivasan, 2010). FEMs (Blackburn et al., 2008; Galopin et al., 2008) and micromechanical methods (Bravo-Castillero et al., 2008; Corcolle et al., 2008; Tang and Yu, 2008) have been used separately to model the heterogeneity of the ME composites. Nonetheless, most of the theoretical developments are limited to two-phase ME composites which form only a subset of the multiferroics that encompasses a very diverse class of materials. The homogenization framework implemented in this article provides a unique platform to treat linear interactions of all forms of multiferroics, namely, single-phase multiferroic, single crystals, polycrystals, as well as multiferroic composites. The representative microstructure (Figure 1) is composed of domains in single crystals, while crystallites with subgranular domain structure make the polycrystal microstructure. For composites, the microstructure is formed by single or polycrystalline magnetic and ferroelectric materials. The accompanying FEM implementation can encompass all these features by proper choice of finite elements.

Figure 1.

A schematic picture of the microstructure (representative unit-cell) of a multiferroic material.

Constitutive laws

We consider a multiferroic body occupying a volume $Ω$ and that physical properties of the material change periodically and the period is equal to the dimension of an elementary cell $Y$ . A linear but not necessarily homogeneous material is assumed here. The nonlinear ME effect, resulted from the coupling interaction between magnetostriction (nonlinear magneto-mechanical effect) and piezoelectricity, is not considered in this model. A hierarchical schematic picture is shown in Figure 2, showing the various coordinates involved as well as the composition of the multiferroic macrostructure and the microstructure. For small deformation, the linear constitutive laws of multiferroics in the absence of heat flux are given by

σ_{ij} = C_{ijkl}^{EH} ϵ_{kl} - e_{kij} E_{k} - e_{kij}^{M} H_{k}

(1)

D_{i} = e_{ijk} ϵ_{jk} + κ_{ij}^{ϵ H} E_{j} + α_{ij} H_{j}

(2)

B_{i} = e_{ijk}^{M} ϵ_{jk} + α_{ji} E_{j} + μ_{ij}^{ϵ E} H_{j}

(3)

and the field equations are given by

σ_{ij, j} + b_{i} = ρ {\overset{\cdot\cdot}{u}}_{i}

(4)

\begin{matrix} D_{i, i} = 0 \\ B_{i, i} = 0 \end{matrix}}

(5)

Here, $σ$ , $ϵ$ , $u$ , $b$ , $ρ$ , $E$ , $D$ , $H$ , and $B$ are stress, strain, displacement, body force, mass density, electrical field vector, electric displacement vector, magnetic field vector, and magnetic flux density, respectively. $C^{EH}$ , $e$ , $e^{M}$ , $κ^{ϵ H}$ , $μ^{ϵ E}$ , and $α$ are stiffness, strain to (electric, magnetic) field coupling constants (or piezoelectric and piezomagnetic coefficients), permittivity (dielectric), (magnetic) permeability, and ME coupling constant, respectively. Equation (5) resulted from the quasi-static assumption on the steady-state Maxwell’s equations for electromagnetic phenomena. In addition to the above equations, we have the strain–mechanical displacement relations and the electrical (magnetic) field–electrical (magnetic) potential relations

ϵ_{ij} = \frac{1}{2} (u_{i, j} + u_{j, i})

(6)

\begin{matrix} E_{i} = - φ_{, i} \\ H_{i} = - ψ_{, i} \end{matrix}}

(7)

where $φ$ and $ψ$ are scalar potentials. A comma in the subscript denotes spatial differentiation, and summation is applied to repeated indices. (We have used Einstein summation convention, that repeated indices are implicitly summed over, throughout this text.) Obviously, the above system of equations is to be juxtaposed with the boundary and initial conditions for one to seek the solution of the multiferroic constitutive equations.

Figure 2.

A schematic picture showing the progression of different coordinate transformations involved in the homogenization of multiferroics.

In homogenization theory, it is assumed that the material is locally formed by the spatial repetition of very small microstructure (representative volume element (RVE) or unit-cell) when compared with the overall macroscopic dimensions. The homogenization, mathematically applicable to heterogeneous media with reasonable contrasts in physical properties, replaces the heterogeneity by an equivalent effective medium with uniform physical characteristics (Sanchez-Palencia, 1980). The constitutive laws in equations (1) to (3), therefore, can be rewritten with the effective properties and average fields as

〈 σ_{ij} 〉 = {\tilde{C}}_{ijkl}^{EH} 〈 ϵ_{kl} 〉 - {\tilde{e}}_{kij} 〈 E_{k} 〉 - {\tilde{e}}_{kij}^{M} 〈 H_{k} 〉

(8)

〈 D_{i} 〉 = {\tilde{e}}_{ijk} 〈 ϵ_{jk} 〉 + {\tilde{κ}}_{ij}^{ϵ H} 〈 E_{j} 〉 + {\tilde{α}}_{ij} 〈 H_{j} 〉

(9)

〈 B_{i} 〉 = {\tilde{e}}_{ijk}^{M} 〈 ϵ_{jk} 〉 + {\tilde{α}}_{ji} 〈 E_{j} 〉 + {\tilde{μ}}_{ij}^{ϵ E} 〈 H_{j} 〉

(10)

where $〈 ■ 〉$ denotes volume averages, namely, $((1 / | V |) \int_{V} dV)$ , and the coefficients with tilde, namely, ${\tilde{C}}_{ijkl}^{EH}, {\tilde{e}}_{ijk}^{M}, \dots$ , refer to the homogenized magneto-electromechanical properties. The indices $i, j, k, l = 1, \dots, d$ , where $d$ is the dimension of the space (physically $d = 2$ or $d = 3$ ).

Homogenization theory of multiferroics

Magneto-electromechanical problem of multiferroics

Let $Ω$ be bounded region of space $R^{3}$ of coordinates $x (or x_{i})$ (Figure 2). For a homogeneous material, the physical properties do not depend on $x$ (the global frame of reference). Nonetheless, if the material is heterogeneous (not homogeneous), the magneto-electromechanical properties, $C^{EH}$ , $e$ , $e^{M}$ , $κ^{ϵ H}$ , $μ^{ϵ E}$ , and $α$ , effectively depend on $x$ , such that $C^{EH} \equiv C^{EH} (x)$ , $e \equiv e (x)$ , $e^{M} \equiv e^{M} (x)$ , $κ^{ϵ H} \equiv κ^{ϵ H} (x)$ , $μ^{ϵ E} \equiv μ^{ϵ E} (x)$ , and $α \equiv α (x)$ . But for materials with a periodic structure, they are periodic functions of space variables. In certain cases, the length of the period is very small with respect to the other lengths appearing in the problem. In such cases, for instance, the present case of multiferroics, the solution is approximately same as the corresponding solution for a homogenized material with constant matrices, ${\tilde{C}}^{EH}, \tilde{e}, {\tilde{e}}^{M}, {\tilde{κ}}^{ϵ H}, {\tilde{μ}}^{ϵ E}$ , and $\tilde{α}$ .

We shall consider $Ω$ as a heterogeneous body obtained by the translation of microstructure of edges $y_{i}^{0}$ of size Y, as shown in Figure 1. The material properties, $C^{EH}$ , $e$ , $e^{M}$ , $κ^{ϵ H}$ , $μ^{ϵ E}$ , and $α$ , are all Y-periodic functions, such that we restrict attention to the case of periodic media having a very small spatial period $ε$ , that is

{\begin{matrix} C_{ijkl}^{EH ε} (x) = C_{ijkl}^{EH ε} (x, y); e_{kij}^{ε} (x) = e_{kij}^{ε} (x, y) \\ e_{kij}^{M ε} (x) = e_{kij}^{M ε} (x, y); κ_{ij}^{ϵ H ε} (x) = κ_{ij}^{ϵ H ε} (x, y) \\ μ_{ij}^{ϵ E ε} (x) = μ_{ij}^{ϵ E ε} (x, y); α_{ij}^{ε} (x) = α_{ij}^{ε} (x, y) \end{matrix}

(11)

with $y (= x / ε) \in Y$ . Here, $ε$ is a real positive parameter and is the characteristic length of the spatial period. The magneto-electromechanical property tensors satisfy the usual symmetry and positivity properties as detailed in Appendix 1. Dependence of properties on $y$ means that they vary within a very small region with dimensions much smaller than those of macroscopic level. $y$ is a magnified frame of reference with respect to $x$ . The microscopic coordinate $y$ is magnified by a factor $1 / ε$ considering the fact that $y = x / ε$ and $ε < < 1$ . In other words, $x$ is a scaled down coordinate system compared to $y$ by a factor of $ε$ . Thus, $y$ can trace out more details and information of the trajectory of the evolution of a function compared to $x$ .

Let the surface $Γ$ of the body $Ω$ be subjected to prescribed surface traction $\bar{t_{k}}$ and surface charge per unit area $\bar{σ}$ and normal magnetic flux density $\bar{B_{n}}$ . The energy functional (Qin and Yang, 2009) for a ME multiferroic can be written as

\begin{matrix} G (u^{ε}, φ^{ε}, ψ^{ε}) = \int_{ε} [\frac{1}{2} C_{ijkl}^{EH ε} (x, y) ϵ_{ij}^{ε} ϵ_{kl}^{ε} - e_{ikl}^{ε} (x, y) E_{i}^{ε} ϵ_{kl}^{ε} - e_{ikl}^{M ε} (x, y) H_{i}^{ε} ϵ_{kl}^{ε} - \frac{1}{2} κ_{ij}^{ϵ H ε} (x, y) E_{i}^{ε} E_{j}^{ε} - α_{ij}^{ε} (x, y) E_{i}^{ε} H_{j}^{ε} - \frac{1}{2} μ_{ij}^{ϵ E ε} (x, y) H_{i}^{ε} H_{j}^{ε}] d ε \\ - \int_{Γ} \bar{t_{k}} u_{k}^{ε} d Γ + \int_{Γ} \bar{σ} φ^{ε} d Γ + \int_{Γ} \bar{B_{n}} ψ^{ε} d Γ - \int_{ε} \frac{1}{2} ρ^{ε} (x, y) {({\overset{\cdot}{u}}_{k}^{ε})}^{2} d ε - \int_{ε} b_{k}^{ε} (x, y) u_{k}^{ε} d ε \end{matrix}

(12)

Here, $\bar{B_{n}}$ is the prescribed value of the normal component of magnetic flux density on the boundary, $ρ^{ε} (\equiv ρ (x, y))$ is the density and $b (\equiv b (x, y))$ is the body force. Here, surface traction $\bar{t_{k}}$ , surface charge density $\bar{σ}$ , and normal magnetic flux density $\bar{B_{n}}$ are assumed to be independent of the spatial scale $ε$ .

Homogenization of multiferroics

We describe in this section the formal asymptotic procedure for solving equation (12), as $ε \to 0$ . One seeks a formal solution to the problem in the form of a standard two-scale asymptotic expansion up to the first-order variation terms

\begin{matrix} u^{ε} & = & u^{0} (x, y) + ε u^{1} (x, y) \\ φ^{ε} & = & φ^{0} (x, y) + ε φ^{1} (x, y) \\ ψ^{ε} & = & ψ^{0} (x, y) + ε ψ^{1} (x, y) \end{matrix}}, y = x / ε

(13)

where $u^{1} (x, y)$ , $φ^{1} (x, y)$ , and $ψ^{1} (x, y)$ are functions to be determined, which are Y-periodic in $y$ . It can be shown that the functions $u^{0} (x, y)$ , $φ^{0} (x, y)$ , and $ψ^{0} (x, y)$ are constants in $y$ (Sanchez-Palencia, 1980). This points out that $u^{0} (x),$ $φ^{0} (x)$ , and $ψ^{0} (x)$ are the homogeneous part of the solution and are functions in $x$ alone and varies in the “slow” scale $x$ . The $u^{1} (x, y)$ , $φ^{1} (x, y)$ , and $ψ^{1} (x, y)$ are the local variations describing the heterogeneous part of the solution and are associated with $y = x / ε$ in the “fast” changing scale. Also the field perturbations, $u^{1} (x, y)$ , $φ^{1} (x, y)$ , and $ψ^{1} (x, y)$ (denoted by the trailing terms in asymptotic expansion equation (13)) must be $Y$ -periodic, and this condition plays the role of the boundary conditions. Based on this notion, the gradients of various fields from equation (13) can be written as

{\begin{matrix} \frac{\partial u_{i}^{ε}}{\partial x_{j}} = \frac{\partial {u_{i}}^{0} (x)}{\partial x_{j}} + ε \frac{\partial u_{i}^{1} (x, y)}{\partial x_{j}} + \frac{\partial u_{i}^{1} (x, y)}{\partial y_{j}} \\ \frac{\partial φ^{ε}}{\partial x_{j}} = \frac{\partial φ^{0} (x)}{\partial x_{j}} + ε \frac{\partial φ^{1} (x, y)}{\partial x_{j}} + \frac{\partial φ^{1} (x, y)}{\partial y_{j}} \\ \frac{\partial ψ^{ε}}{\partial x_{j}} = \frac{\partial ψ^{0} (x)}{\partial x_{j}} + ε \frac{\partial ψ^{1} (x, y)}{\partial x_{j}} + \frac{\partial ψ^{1} (x, y)}{\partial y_{j}} \end{matrix}

(14)

when $y = x / ε$ . We would get the variation $δ G$ by taking the variation of the energy functional in equation (12), as shown in Appendix 2.

Collecting coefficients of $δ u_{i}^{0}, δ u_{i}^{1}, \dots$ of the variational equation (41) in Appendix 2, considering that they are independent and also considering that

lim_{ε \to 0} \int_{ε} ϕ (x, x / ε) d ε = \frac{1}{| Y |} \int_{ε} \int_{Y} ϕ (x, y) d y d ε

(15)

will lead to a set of macroscopic (equations involving the variables $δ u_{i}^{0}, δ φ^{0}, and δ ψ^{0}$ ) and microscopic (equations involving the variables $δ u_{i}^{1}, δ φ^{1}, and δ ψ^{1}$ ) equations. Due to linearity of the problem, and assuming separation of variables, we denote

\begin{array}{l} u_{k}^{1} (x, y) & = χ_{k}^{m n} (x, y) ϵ_{m n} (u^{0} (x)) + Φ_{k}^{m} (x, y) \\ \times \frac{\partial φ^{0} (x)}{\partial x_{m}} + Γ_{k}^{m} (x, y) \frac{\partial ψ^{0} (x)}{\partial x_{m}} \\ φ^{1} (x, y) & = η^{m n} (x, y) ϵ_{m n} (u^{0} (x)) + R^{m} (x, y) \\ \times \frac{\partial φ^{0} (x)}{\partial x_{m}} + Ψ^{m} (x, y) \frac{\partial ψ^{0} (x)}{\partial x_{m}} \\ ψ^{1} (x, y) & = λ^{m n} (x, y) ϵ_{m n} (u^{0} (x)) + Θ^{m} (x, y) \\ \times \frac{\partial φ^{0} (x)}{\partial x_{m}} + Q^{m} (x, y) \frac{\partial ψ^{0} (x)}{\partial x_{m}} \end{array}}

(16)

Here, $χ$ , $R$ , and $Q$ are characteristic material, electric, and magnetic displacements. $Φ, Γ, η, Ψ, λ$ , and $Θ$ are characteristic coupled functions. The material functions (electromechanical property tensors such as $C^{EH}$ , $e$ , and $e^{M}$ ) are assumed to be functions of class $L^{\infty} (Y)$ , which means they belong to the Lebesgue class of integrals, in the sense that an integrable function is summable as well. Moreover, the characteristic functions introduced in equation (16) are required to belong to subspaces of the form $H_{per} (Y) = {v \in H^{1} (Y)}$ such that $v$ takes equal values on opposite sides of Y and $H_{per} (Y, R^{3}) = {v = (v_{i}) | v_{i} \in H_{per} (Y)}$ , where $H^{1}$ is a Sobolev space. Here, $v$ is any arbitrary function.

Taking the spatial derivatives of the functions in equation (16) with respect to the microscopic coordinate $y$ and substituting into the microscopic equations resulted from equation (41) in Appendix 2 and making use of the periodic boundary conditions and the identity in equation (15), we obtained the microscopic equations as

\frac{1}{| Y |} \int_{Y} [C_{ijkl}^{EH} (x, y) (δ_{im} δ_{jn} + \frac{\partial χ_{i}^{mn}}{\partial y_{j}}) + e_{ikl} (x, y) \frac{\partial η^{mn}}{\partial y_{i}} + e_{ikl}^{M} (x, y) \frac{\partial λ^{mn}}{\partial y_{i}}] \times ϵ_{kl} (δ u^{1} (x, y)) d Y = 0 \forall δ u^{1} \in H_{per} (Y, R^{3})

(17)

\begin{matrix} \frac{1}{| Y |} \int_{Y} [e_{ikl} (x, y) (δ_{im} + \frac{\partial R^{m}}{\partial y_{i}}) + C_{ijkl}^{EH} (x, y) \times \frac{\partial Φ_{i}^{m}}{\partial y_{j}} + e_{ikl}^{M} (x, y) \frac{\partial Θ^{m}}{\partial y_{i}}] \times ϵ_{kl} (δ u^{1} (x, y)) d Y = 0, \forall δ u^{1} \in H_{per} (Y, R^{3}) \end{matrix}

(18)

\frac{1}{| Y |} \int_{Y} [e_{ikl}^{M} (x, y) (δ_{im} + \frac{\partial Q^{m}}{\partial y_{i}}) + C_{ijkl}^{EH} (x, y) \times \frac{\partial Γ_{i}^{m}}{\partial y_{j}} + e_{ikl} (x, y) \frac{\partial Ψ^{m}}{\partial y_{i}}] \times ϵ_{kl} (δ u^{1} (x, y)) d Y = 0, \forall δ u^{1} \in H_{per} (Y, R^{3})

(19)

\frac{1}{| Y |} \int_{Y} [e_{ikl} (x, y) (δ_{km} δ_{\ln} + \frac{\partial χ_{k}^{mn}}{\partial y_{l}}) - κ_{ij}^{ϵ H} (x, y) \frac{\partial η^{mn}}{\partial y_{j}} - α_{ij} (x, y) \frac{\partial λ^{mn}}{\partial y_{j}}] \times (\frac{\partial δ φ^{1} (x, y)}{\partial y_{i}}) d Y = 0, \forall δ φ^{1} \in H_{per} (Y)

(20)

\begin{matrix} \frac{1}{| Y |} \int_{Y} [e_{ikl} (x, y) \frac{\partial Φ_{k}^{m}}{\partial y_{l}} - κ_{ij}^{ϵ H} (x, y) (δ_{jm} + \frac{\partial R^{m}}{\partial y_{j}}) - α_{ij} (x, y) \frac{\partial Θ^{m}}{\partial y_{j}}] \times (\frac{\partial δ φ^{1} (x, y)}{\partial y_{i}}) d Y = 0, \forall δ φ^{1} \in H_{per} (Y) \end{matrix}

(21)

\frac{1}{| Y |} \int_{Y} [e_{ikl} (x, y) \frac{\partial Γ_{k}^{m}}{\partial y_{l}} - κ_{ij}^{ϵ H} (x, y) \frac{\partial Ψ^{m}}{\partial y_{j}} - α_{ij} (x, y) (δ_{jm} + \frac{\partial Q^{m}}{\partial y_{j}})] \times (\frac{\partial δ φ^{1} (x, y)}{\partial y_{i}}) d Y = 0, \forall δ φ^{1} \in H_{per} (Y)

(22)

\frac{1}{| Y |} \int_{Y} [e_{ikl}^{M} (x, y) (δ_{km} δ_{\ln} + \frac{\partial χ_{k}^{mn}}{\partial y_{l}}) - α_{ij} (x, y) \frac{\partial η^{mn}}{\partial y_{i}} - μ_{ij}^{ϵ E} (x, y) \frac{\partial λ^{mn}}{\partial y_{j}}] \times (\frac{\partial δ ψ^{1} (x, y)}{\partial y_{i}}) d Y = 0, \forall δ ψ^{1} \in H_{per} (Y)

(23)

\frac{1}{| Y |} \int_{Y} [e_{ikl}^{M} (x, y) \frac{\partial Φ_{k}^{m}}{\partial y_{l}} - α_{ij} (x, y) \times (δ_{im} + \frac{\partial R^{m}}{\partial y_{i}}) - μ_{ij}^{ϵ E} (x, y) \frac{\partial Θ^{m}}{\partial y_{j}}] \times (\frac{\partial δ ψ^{1} (x, y)}{\partial y_{i}}) d Y = 0, \forall δ ψ^{1} \in H_{per} (Y)

(24)

\frac{1}{| Y |} \int_{Y} [e_{ikl}^{M} (x, y) \frac{\partial Γ_{k}^{m}}{\partial y_{l}} - α_{ij} (x, y) (\frac{\partial Ψ^{m}}{\partial y_{i}}) - μ_{ij}^{ϵ E} (x, y) (δ_{im} + \frac{\partial Q^{m}}{\partial y_{i}})] \times (\frac{\partial δ ψ^{1} (x, y)}{\partial y_{i}}) d Y = 0, \forall δ ψ^{1} \in H_{per} (Y)

(25)

where $ϵ_{kl} (δ u^{1} (x, y)) = (1 / 2) (\partial δ u_{i}^{1} (x, y) / \partial y_{j} + \partial δ u_{j}^{1} (x, y) / \partial y_{i})$ . The solution of the microscopic equations (equations (17) to (25)) would yield the unknown periodic functions entering into equation (16). And this set of problems is called local problems. Substitution of the solution of the local equations back into equation (16) and subsequently into equation (13) would unveil the characteristics of the local fields, namely, the displacements, magnetic, and electrical potentials of the multiferroic material.

After performing analogous mathematical operations, as did for obtaining the microscopic equations (Equations (17) to (25)), one can arrive at the set of macroscopic equations. Having solved the local problems, we can arrive at the homogenized (effective) moduli as

{\tilde{C}}_{klmn}^{EH} = \frac{1}{| Y |} \int_{Y} [C_{ijpq}^{EH} (x, y) (δ_{im} δ_{jn} + \frac{\partial χ_{i}^{mn}}{\partial y_{j}}) + e_{ipq} (x, y) \frac{\partial η^{mn}}{\partial y_{i}} + e_{ipq}^{M} (x, y) \frac{\partial λ^{mn}}{\partial y_{i}} \times (δ_{pk} δ_{ql} + \frac{\partial χ_{p}^{kl}}{\partial y_{q}}) d Y

(26)

{\tilde{e}}_{klm} = \frac{1}{| Y |} \int_{Y} [e_{ipq} (x, y) (δ_{im} + \frac{\partial R^{m}}{\partial y_{i}}) + C_{ijpq}^{EH} (x, y) \frac{\partial Φ_{i}^{m}}{\partial y_{j}} + e_{ipq}^{M} (x, y) \times \frac{\partial Θ^{m}}{\partial y_{i}}] (δ_{pk} δ_{ql} + \frac{\partial χ_{p}^{kl}}{\partial y_{q}}) d Y

(27)

{\tilde{e}}_{klm}^{M} = \frac{1}{| Y |} \int_{Y} [e_{ipq}^{M} (x, y) (δ_{im} + \frac{\partial Q^{m}}{\partial y_{i}}) + C_{ijkl}^{EH} (x, y) \frac{\partial Γ_{i}^{m}}{\partial y_{j}} + e_{ikl} (x, y) \times \frac{\partial Ψ^{m}}{\partial y_{i}}] (δ_{pk} δ_{ql} + \frac{\partial χ_{p}^{kl}}{\partial y_{q}}) d Y

(28)

{\tilde{κ}}_{nm}^{ϵ H} = \frac{1}{| Y |} \int_{Y} [e_{pkl} (x, y) \frac{\partial Φ_{k}^{m}}{\partial y_{l}} - κ_{pj}^{ϵ H} (x, y) \times (δ_{jm} + \frac{\partial R^{m}}{\partial y_{j}}) - α_{pj} (x, y) \frac{\partial Θ^{m}}{\partial y_{j}}] \times (δ_{np} + \frac{\partial R^{n}}{\partial y_{p}}) d Y

(29)

{\tilde{μ}}_{ij}^{ϵ E} = \frac{1}{| Y |} \int_{Y} [e_{pkl}^{M} (x, y) \frac{\partial Γ_{k}^{m}}{\partial y_{l}} - μ_{pj}^{ϵ E} (x, y) \times (δ_{jm} + \frac{\partial Q^{m}}{\partial y_{j}}) - α_{pj} (x, y) \frac{\partial Ψ^{m}}{\partial y_{j}}] \times (δ_{np} + \frac{\partial Q_{n}}{\partial y_{p}}) d Y

(30)

{\tilde{α}}_{ij} = \frac{1}{| Y |} \int_{Y} [e_{pkl} (x, y) \frac{\partial Γ_{k}^{m}}{\partial y_{l}} - κ_{pj}^{ϵ H} (x, y) \frac{\partial Ψ^{m}}{\partial y_{j}} - α_{pj} (x, y) (δ_{jm} + \frac{\partial Q^{m}}{\partial y_{j}})] \times (δ_{np} + \frac{\partial R^{n}}{\partial y_{p}}) d Y

(31)

Rigorously, the two-scale asymptotic analysis leads directly to the strong formulation of the local problems depicted in equations (17) to (25). Consequently, the material functions that appear in those equations must be more regular (Glaka et al., 1992). Nevertheless, the point of departure is the variational (weak) formulation of the problem shown in Appendix 2, which would lead to the local problems shown above. This would essentially map the underlying magneto-electromechanical fields of a multiferroic material from a macroscopic coordinates into a local coordinate system possessing more resolutions.

Rotation of multiferroics

The expressions for homogenized material coefficients that appear in section “Homogenization of multiferroics” could be used for finding out the effective properties of polycrystalline and composite multiferroics. While dealing with both cases, we must take into account the significant impact of the orientation effect of the underlying constituent domains (or crystals). In fact, multiferroic domains are volume elements with uniform orientation of constituent crystallographic unit-cells (see Figure 2 in order to distinguish between the representative unit-cell or the microstructure used for sampling the macroscopic multiferroic material and the crystallographic unit-cell).

The physical quantities that appear in the homogenization equations, namely, $C_{ijkl}^{EH} (x, y)$ , $e_{ijk} (x, y)$ , $e_{ijk}^{M} (x, y)$ , $κ_{ij}^{ϵ H} (x, y)$ , $μ_{ij}^{ϵ E} (x, y)$ , and $α_{ij} (x, y)$ , are the magneto-electromechanical properties of the crystallite which constitutes the microscopic structure and can be described in microscopic coordinates $y$ as

C_{ijkl}^{EH} (x, y) = a_{ip} a_{jq} a_{kr} a_{ls} C_{pqrs}^{EH'} (x, y, z)

(32a)

e_{ijk} (x, y) = a_{ip} a_{jq} a_{kr} e_{pqr}' (x, y, z)

(32b)

e_{ijk}^{M} (x, y) = a_{ip} a_{jq} a_{kr} e_{pqr}^{M'} (x, yz)

(32c)

κ_{ij}^{ϵ H} (x, y) = a_{ip} a_{jq} κ_{pq}^{ϵ H'} (x, y, z)

(32d)

μ_{ij}^{ϵ E} (x, y) = a_{ip} a_{jq} μ_{pq}^{ϵ E'} (x, y, z)

(32e)

α_{ij} (x, y) = a_{ip} a_{jq} α_{ij}' (x, y, z)

(32f)

where $a_{ij}$ are the Euler transformation matrix components from crystallographic coordinate system $(x, y, z)$ to the local microscopic coordinates $y$ . Here, the primed moduli are the ones expressed in crystallographic coordinate system. Also, the ME response is determined along an arbitrary crystallographic direction determined by the Euler angles $(ϕ, θ, ϑ)$ with respect to the reference frame of the microstructure, that is, $y$ . (Here, the microstructure refers to the RVE of the multiferroic single crystal used for the homogenization.) Euler angles $ϕ$ , $θ$ , and $ϑ$ can completely specify the orientation of the crystallographic coordinate system embedded in domains and thereby the orientation relative to a fixed Cartesian coordinate system. Figure 2 explains how the Euler angles map the rotation of a typical perovskite.

The transformation matrix from the crystallographic coordinate system $(x, y, z)$ to the local coordinate system $y$ is given by (Goldstein, 1978)

a_{ij} = (\begin{matrix} \cos ϑ \cos ϕ - \cos θ \sin ϕ \sin ϑ & \cos ϑ \sin ϕ + \cos θ \cos ϕ \sin ϑ & \sin ϑ \sin θ \\ - \sin ϑ \cos ϕ - \cos θ \sin ϕ \cos ϑ & - \sin ϑ \sin ϕ + \cos θ \cos ϕ \cos ϑ & \cos ϑ \sin θ \\ \sin θ \sin ϕ & - \sin θ \cos ϕ & \cos θ \end{matrix})

(33)

Before being used in the homogenization of polycrystals and composites of multiferroics, the transformation equations (32) and (33) could well be used for characterizing the transformed magneto-electromechanical moduli of rotated multiferroic single crystals. Here, we assume that the multiferroic single crystal to be monodomain. Nevertheless, it is possible to model the rotation of real poly-domain single crystals using the methodology described in the following section but with the knowledge of the proper domain orientation distribution function. Symmetric reduction should be applied according to the single-crystal symmetry. Knowledge of the full set of magneto-electromechanical moduli along the spontaneous polarization/magnetization direction of a multiferroic single crystal is necessary to have a complete picture of the orientational effect. Nonetheless, due to the paucity of the full set of measured data of the multiferroics, single-crystal orientation analysis is not possible.

Polycrystalline multiferroics

Polycrystalline ceramics present ease in manufacture and in compositional modifications compared to single crystals. As-grown polycrystalline multiferroic material is an aggregate of single crystalline grains (or crystallites) with randomly oriented (spontaneous) electric (or magnetic) polarizations. Each grain in a polycrystalline material is assumed to be made of a single, pinned, chemically homogeneous domain. External magnetic or electrical loading can reorient either the polarization axis or the magnetization easy axis or both of most of the domains. Figure 3 shows the schematics of domain switching in a multiferroic polycrystal due to electrical poling. However, electric and magnetic states can be reversibly controlled by an electrical field alone, which simultaneously switches the electrical polarization inside a ferroelectric domain and controls the equivalent magnetic domains (Lee et al., 2008). Hence, the material could be transformed from a randomly oriented sample into a mostly oriented (textured) one. In a randomly oriented multiferroic polycrystal, the orientations of domains with reference to a fixed coordinate system (in this study, it is the microscopic coordinates $y$ ) would be different for each domain, that is, the distribution of Euler angles ( $ϕ, θ, ϑ$ ) subtended by the domains would fall in a uniform distribution for an as-grown crystal (Figure 3(a)). Nonetheless, as the external load is applied, the domains get aligned and their orientation distribution slowly deviates from the uniform distribution (Figure 3(b)). One of the possibilities is that the resultant orientation distribution after loading could be a Gaussian one with the mean points toward a preferred direction fully dictated by the crystallographic symmetry of the multiferroic (Lee et al., 2008; Schmid, 1994). The probability distribution function (pdf) of Gaussian distribution is defined by

f (α | μ, σ) = \frac{1}{(σ \sqrt{2 π})} \exp - [\frac{{(α - μ)}^{2}}{2 σ^{2}}]

(34)

where $μ$ and $σ$ are the parameters of the distribution, namely, the mean and the standard deviation, respectively. $α$ stands for the Euler angles $(ϕ, θ, ϑ)$ .

Here, the Gaussian distribution can encompass both extremes of the polycrystal configuration: the perfectly textured as well as the random sample. It can be explained in the following way: as the standard deviation $σ \to 0$ , the distribution becomes narrow and one can approach perfect texture close to single crystals. While $σ \to \infty$ , the distribution becomes flatter (close to uniform distribution) and generates a randomly oriented polycrystal with little net polarization/magnetization, if not none. Euler angles generated according to the above equation (34) could be used in equations (32) and (33) to find out the transformed moduli of each domain/grain in the polycrystal. These transformed moduli could consequently be plugged into equations (26) to (31) for the homogenization of the polycrystals.

Figure 3.

Possible switching of domains that reorient the polarization vectors (arrows) from unpoled (a) to poled (b) configuration due to electrical field (E) loading in polycrystal multiferroic material. We have not shown the equivalent magnetic domains or the corresponding magnetization here.

Numerical results and discussion

In order to solve the microscopic system of equations given through equations (17) to (25) in section “Homogenization of multiferroics,” we developed a finite element formulation. (Further details of finite element formulation are given in Appendix 3.) A simple, three-dimensional (3D) “multiferroic finite element” is conceived with the simplest nodal arrangement for a brick element employing the vertices with 5 degrees of freedom (DOFs) (three spatial and one each for magnetic and electrical potentials). In the present context, 8-noded isoparametric elements with $2 \times 2 \times 2$ Gauss point integration are used to obtain solutions. The use of the $2 \times 2 \times 2$ Gauss point integration has been proved to be optimal and adequate since it represents the best sampling point for the values of derivatives such as the electrical/magnetic fields (Zienkiewicz, 1971). Substituting equation (42) in Appendix 3 into the microscopic equations (17) to (25) in section “Homogenization of multiferroics” and assembling the individual equations for each finite element, one can obtain the following global system of equations for each load case $mn or m$

[\begin{matrix} K_{uu} & K_{u φ} & K_{u ψ} \\ K_{u φ}^{t} & - K_{φ φ} & - K_{φ ψ} \\ K_{u ψ}^{t} & - K_{φ ψ}^{t} & - K_{ψ ψ} \end{matrix}] {\begin{matrix} χ^{mn} \\ η^{mn} \\ λ^{mn} \end{matrix}} = {\begin{matrix} F_{1}^{mn} \\ F_{2}^{mn} \\ F_{3}^{mn} \end{matrix}}

(35)

[\begin{matrix} K_{uu} & K_{u φ} & K_{u ψ} \\ K_{u φ}^{t} & - K_{φ φ} & - K_{φ ψ} \\ K_{u ψ}^{t} & - K_{φ ψ}^{t} & - K_{ψ ψ} \end{matrix}] {\begin{matrix} Φ^{m} \\ R^{m} \\ Θ^{m} \end{matrix}} = {\begin{matrix} f_{1}^{m} \\ f_{2}^{m} \\ f_{3}^{m} \end{matrix}}

(36)

[\begin{matrix} K_{uu} & K_{u φ} & K_{u ψ} \\ K_{u φ}^{t} & - K_{φ φ} & - K_{φ ψ} \\ K_{u ψ}^{t} & - K_{φ ψ}^{t} & - K_{ψ ψ} \end{matrix}] {\begin{matrix} Γ^{m} \\ Ψ^{m} \\ Q^{m} \end{matrix}} = {\begin{matrix} f_{1}^{m} \\ f_{2}^{m} \\ f_{3}^{m} \end{matrix}}

(37)

where the superscript $t$ indicates transpose of the corresponding matrix. The globally assembled matrices $K_{uu}, K_{u φ}, K_{u ψ}, K_{φ φ}, K_{φ ψ}, and K_{ψ ψ}$ in equations (35) to (37) are the mechanical stiffness matrix, the piezoelectric stiffness matrix, the piezomagnetic stiffness matrix, the dielectric stiffness matrix, the “magnetoelectric stiffness” matrix, and the “diamagnetic stiffness” matrix, respectively, and $F_{i}^{mn}, f_{i}^{m}$ , and $f_{i}^{m}$ , where $i = 1, 2, 3$ , are the magneto-electromechanical load vectors. $χ^{mn}, η^{mn}, λ^{mn}, Φ^{m}, R^{m}, Θ^{m}, Γ^{m}, Ψ^{m}$ , and $Q^{m}$ are the globally assembled vectors of characteristic functions of the multiferroic medium. All the load cases are solved imposing periodic boundary conditions.

As the representative unit-cell is expected to capture the response of the entire multiferroic system where the macroscopic material is constructed by adding, contiguously to the representative unit-cell, a large number of other identical unit-cells, particular care is taken to ensure that the deformation across the boundaries of the representative unit-cell is compatible with the deformation of adjacent unit-cells. This requirement is applicable to the electrical and magnetic potentials as well. Hence, all the load cases are solved by enforcing periodic boundary conditions in the unit-cell for the displacements, electrical, and magnetic potentials.

The multiferroics offer the possibility of fast low-power electrical write operation and nondestructive magnetic read operation. One of the order parameters, either electronic or magnetic, is, in general, a weak property resulting from a complex phase transformation, orbital ordering, geometric frustration, and so on in materials (Martin et al., 2010). Nevertheless, composite multiferroics offer extraordinary coupling at room temperature and above. Here, we apply the methodology to a simple two-phase multiferroic ME composite due to the rarity of single-phase single-crystal data (Eerenstein et al., 2006) to model an ideal multiferroic. Laminated two-phase composites of a magnetostrictive (or piezomagnetic) and electrostrictive (piezoelectric) material (Figure 4) introduce indirect ME coupling mediated through strain (Eerenstein et al., 2006; Ryu et al., 2001). The system used here is a composite of cobalt ferrite (magnetostrictive) and barium titanate (electrostrictive). We have estimated the effective magneto-electromechanical properties of the laminate as a function of the volume fraction of the electrostrictive (barium titanate, BaTiO₃) medium, with the magnetostrictive (cobalt ferrite, CoFe₂O₄) medium as the base material.

Figure 4.

Schematic diagram of a composite two-phase multiferroic magnetoelectric laminate.

The set of linear equations in equations (35) to (37) is coupled by the matrices $K_{u φ}, K_{u ψ}$ , and $K_{φ ψ}$ . These equations can be uncoupled into piezoelectric finite element equations and piezomagnetic finite element equations by the following way. First, if we let the piezoelectric stress tensor $e_{ijk} = 0$ , then this would imply that $K_{u φ} = 0$ . Next, if we let the piezomagnetic tensor $e_{ijk}^{M} = 0$ , the corresponding stiffness matrix $K_{u ψ} = 0$ . Moreover, the dielectric permittivity

κ_{ij}^{ϵ H} = 0 \Rightarrow K_{φ φ} = 0

for the former case and magnetic permeability

μ_{ij}^{ϵ E} = 0 \Rightarrow K_{ψ ψ} = 0

for the latter case. The ME coupling $α_{ij}$ and the corresponding $K_{φ ψ}$ would obviously vanish in both instances. Since our model is general in the sense that it encompasses the ME multiferroic material as a whole, we apply the above notion to treat the magnetostrictive and electrostrictive components of the two-phase composite.

The material properties of the individual phases of the composite used for the simulation taken from the Pan and Heyliger (2002) are listed in Table 1. The homogenized magneto-electromechanical properties calculated using the present model are shown in Figures 5 to 9. The results compare well with the micromechanical calculation on laminated BaTiO₃−CoFe₂O₄ composite by Li and Dunn (1998). This is a proof of the validity of the present model. Experiments on the particular geometry, that is, the 2-2 laminar ME composite using materials CoFe₂O₄ and BaTiO₃ are scant in the literature for direct comparison (Nan et al., 2008). Thus, a quantitative comparison is difficult due to lack of information regarding the ME coupling tensor pertaining to the CoFe₂O₄−BaTiO₃ laminate. However, geometries such as particulates, powder composites, and 1-3 fiber nanocomposites, heterostructure consisting of nanopillars of the ferro/ferrimagnetic phase (CoFe₂O₄) embedded in a ferroelectric matrix (BaTiO₃) are reported (Zheng et al., 2004). Nevertheless, laminate composites are potentially useful because of the large coupling between electrical and magnetic properties due to large area of contact between phases. Yet the self-assembly of the nanocomposite occurs at volume fractions (0.35CoFe₂O₄−0.65BaTiO₃) (Zheng et al., 2004) close to the volume fraction (≈ 0.7BaTiO₃) in our model at which the laminate exhibits maximum longitudinal ME coupling (see Figure 9).

Table 1.

Material properties of BaTiO₃ and CoFe₂O₄ used for simulation (Pan and Heyliger, 2002).

Physical property	BaTiO₃	CoFe₂O₄
C ₁₁	166	286
C ₁₂	77	173
C ₁₃	78	170.5
C ₃₃	162	269.5
C ₄₄	43	45.3
C ₆₆	44.5	56.5
e ₃₁	−4.4	0
e ₃₃	18.6	0
e ₁₅	11.6	0
$κ_{11}^{ε H}$	11.2	0.08
$κ_{33}^{ε H}$	12.6	0.093
$e_{31}^{M}$	0	580.3
$e_{33}^{M}$	0	699.7
$e_{15}^{M}$	0	550
$μ_{11}^{ε E}$	5	−590
$μ_{33}^{ε E}$	10	157

Here, $C_{μ ν}$ in units of $10^{9}$ N/m², $e_{j ν}$ in C/m², $κ_{ij}^{ϵ H}$ in $10^{- 9}$ C²/Nm², $e_{j ν}^{M}$ in N/A m, and $μ_{ij}^{ϵ E}$ in $10^{- 6}$ Ns²/C².

Figure 5.

The homogenized elastic stiffnesses ${\tilde{C}}_{μ ν}^{EH}$ of the laminated multiferroic magnetoelectric composite BaTiO₃−CoFe₂O₄ versus the volume fraction $v_{f}$ of BaTiO₃: (a) the homogenized ${\tilde{C}}_{11}^{EH}, {\tilde{C}}_{12}^{EH}$ , and ${\tilde{C}}_{13}^{EH}$ and (b) ${\tilde{C}}_{33}^{EH}, {\tilde{C}}_{44}^{EH}$ , and ${\tilde{C}}_{66}^{EH}$ .

Figure 6.

The homogenized piezoelectric stress coefficients ${\tilde{e}}_{i μ}$ of the laminated multiferroic magnetoelectric composite BaTiO₃−CoFe₂O₄ versus the volume fraction $v_{f}$ of BaTiO₃.

Figure 7.

The homogenized piezomagnetic coefficients ${\tilde{e}}_{i μ}^{M}$ of the laminated multiferroic magnetoelectric composite BaTiO₃−CoFe₂O₄ versus the volume fraction $v_{f}$ of the piezoelectric BaTiO₃.

Figure 8.

The homogenized magnetoelectric coefficient ${\tilde{α}}_{11}$ of the laminated multiferroic magnetoelectric composite BaTiO₃−CoFe₂O₄ versus the volume fraction $v_{f}$ of the piezoelectric BaTiO₃.

Figure 9.

The homogenized magnetoelectric coefficient ${\tilde{α}}_{33}$ of the laminated multiferroic magnetoelectric composite BaTiO₃−CoFe₂O₄ versus the volume fraction $v_{f}$ of piezoelectric BaTiO₃.

Another important aspect to be noted in this study is that neither the piezoelectric nor the magnetostrictive phase exhibits the ME effect manifested by the ME coupling ${\tilde{α}}_{ij}$ . Nonetheless, the composite phase derives this property and the nonvanishing components, namely, ${\tilde{α}}_{11}$ and ${\tilde{α}}_{33}$ , exhibit contrast in their dependence on the piezoelectric component fraction as shown in Figures 8 and 9. ${\tilde{α}}_{11}$ shows a maximum (absolute value) at around $v_{f} = 0.5$ of BaTiO₃ (Figure 8). However, the longitudinal component of ME coefficient ${\tilde{α}}_{33}$ shows the maximum (absolute value) at around $v_{f} = 0.8$ of BaTiO₃ (Figure 9). Moreover, from the simulation results, we observe that they display a symmetry such that ${\tilde{α}}_{11} \equiv {\tilde{α}}_{22}$ . Only two (i.e. ${\tilde{α}}_{11} (\equiv {\tilde{α}}_{22})$ and ${\tilde{α}}_{33}$ ) of the nine components of ${\tilde{α}}_{ij}$ are found to be independent and nonvanishing in BaTiO₃−CoFe₂O₄ composite.

Conclusion

A rigorous theoretical model, in a generic setting, is developed for the homogenization of multiferroics for treating single-phase multiferroic single crystals as well as polycrystals and for multiferroic composites. The microstructural features such as orientation and texture are incorporated fully into the modeling framework. A multiferroic microstructure (or representative unit-cell) is conceived in the local scale. This unit-cell could encompass the features of single crystals possessing domain structure, polycrystals possessing crystallites with subgranular domain structure, and composites formed by single or polycrystalline magnetic and ferroelectric materials. The homogenization expressions for entire magneto-electromechanical properties are derived assuming the macroscopic body is build by the contiguous juxtaposition of the representative unit-cell in 3D space. The proposed modeling approach can treat multiferroics irrespective of their crystallographic symmetry, multiferroic polycrystals, and multiphase ME composites of any geometry and configuration, sandwich structures, and so on.

The variational formulation of the magneto-electromechanical problem enables us to derive a set of local problems which essentially unravel the relationships among the underlying microscopic fields. Homogenization is implemented in FEM by solving these local coupled equilibrium electrical, magnetic, and mechanical fields to enable computation of the effective properties. A “multiferroic finite element” is conceived with 5 DOFs (three for the spatial coordinates and one each for the electrical and magnetic potential) per node.

The transformation of the multiferroic’s physical properties due to the crystal rotation is characterized through Euler angles ( $ϕ, θ, ϑ$ ). The correspondence between various scales and frames of references involved in this study is demonstrated (see Figure 2). The incorporation of the aggregate texture (orientation distribution) of a multiferroic polycrystal into the present model is shown with a simple example of a Gaussian distribution function. As an example case, the present model is applied to a two-phase multiferroic ME composite to evaluate its homogenized physical properties. The general multiferroic FEM is used for the solution by “turning off” (uncouple) the magnetic DOF to treat the electrostrictive phase, and the electrical DOF is “turned off” to treat the magnetostrictive phase. The evolution of the homogenized physical properties with the varying volume fraction of the electrostrictive phase compares well with other models. The nonvanishing components of ME coupling ${\tilde{α}}_{ij}$ tensor is obtained for the product phase (i.e., magnetoelectric composite phase) although none of the constituent phases possess them individually. Hence, the model captures the coupling between magnetic and ferroelectric subsystems in this case mediated via strain.

The size-effect and the nonlinear behavior of the “responses” (i.e. the magnetization $M$ in an electrical field $E$ and the inverse effect of polarization $P$ generated by the application of the magnetic field $H$ ) are not accounted in the present model. Nevertheless, the size-effect scales down to crystallographic cell size (few Angstroms) in ferroic materials that warrant an atomistically enriched continuum model to address the issue. A plausible approach to address both these deficiencies might be to develop the energy functional with nonlinear terms which contain force constants to be evaluated from atomistic potentials. A modeling framework in this direction that ensures the accuracy of the atomistic models and the robustness and economy of the continuum mechanics are in progress.Yet the homogenization model in the present work covers all the forms of multiferroics (and its composites) and captures their linear equivalent (effective) behaviour and offer computational efficiency besides unveiling the nature of the underlying microscopic field characteristics.

Footnotes

Appendix 1

Appendix 2

Appendix 3 Acknowledgements

We would like to acknowledge G. Srinivasan and K.P. Surendran for insightful discussions. K.P.J. acknowledges the award of Ciência 2007 from Fundação para a Ciência e a Tecnologia (FCT), Portugal.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This research project is partially funded by FCT (Fundação para a Ciência e a Tecnologia, Portugal) through project PTDC/EME-PME/120630/2010.

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