Field distributions in cracked periodically layered electromagnetothermoelastic composites

Abstract

A novel approach is employed for the prediction of stress, electrical, and magnetic fields generated by applying combined loadings on damaged composites that consist of periodic electromagnetothermoelastic layers. The damage may represent cavities and cracks and must be localized in the sense that its effect on the remote-loaded boundaries of the composite is negligible. This approach is based on the combined use of the representative cell method, the higher-order theory, and the high-fidelity generalized method of cells micromechanical model. In the framework of the representative cell method, the problem for a periodic composite that is discretized into numerous identical cells is reduced to a problem of a single cell in the discrete Fourier transform domain. In the framework of the higher-order theory, the resulting governing equations and interfacial conditions in the transform domain are solved by dividing the single cell into subcells and imposing the latter in an average (integral) sense. The high-fidelity generalized method of cells is utilized for the prediction of the proper far-field boundary conditions, which are based on the unperturbed effective properties of the composite. The inverse of the Fourier transform provides the real elastic field at any point of composite with localized effects. The damage existence is modeled by introducing fictitious unknown eigenfields that are computed by an iterative procedure. This modeling is verified by a comparison with five analytical solutions of cavities and cracks embedded in piezoelectric and electromagnetoelastic materials. Several applications of cracked layered composites are given.

Keywords

Smart composites piezoelectric piezomagnetic cracking effects representative cell method higher-order theory high-fidelity generalized method of cells

Introduction

Electromagnetoelastic composites form an important class of smart materials. This is due to the existing coupling between the electric, magnetic, thermal, and mechanical effects that can be utilized for sensing, actuation, and detection. Furthermore, an additional electromagnetic coupling in these composite exists that is absent in the monolithic electromagnetoelastic phases that can be also utilized for sensing. Electromagnetoelastic materials can be also employed in multilayer structures and laminated composites for sensing and actuation. Thus, Sosa and Pourki (1993), for example, presented a method for the analysis of electromechanical behavior of multilayer structures where the main components are piezoelectric materials. Gopinathan et al. (2000), on the other hand, published an extensive discussion and review of theories for piezoelectric laminates where classical and shear deformation theories are discussed, in conjunction with sensor and actuator problems.

It is obviously important to predict the effective properties of these composites. To this end, several micromechanical models have been employed, for example, Carman et al. (1995) (who employed the concentric cylinder model); Li and Dunn (1998), Wu and Huang (2000), and Li (2000) (who employed the Mori–Tanaka method); and Aboudi (2001) (who employed a homogenization technique for periodic composites).

The piezoelectric and piezomagnetic phases of the electromagnetoelastic composites are usually brittle (e.g. lead zirconate titanate (PZT) is a piezoelectric material, which is very stiff and brittle). Hence, fracture may take place in these composites during fabrication or service. Several types of flaws in multilayer actuators have been described by Winzer et al. (1989). The introduction of cracks in composites to be analyzed by standard micromechanical models like the Mori–Tanaka method is not possible (since these methods are based on the average stresses in the phases). Similarly, the introduction of cracks in composites with periodic microstructure is not possible because the crack would be incorporated into the analysis in a periodic manner, which is not a realistic situation. When a single crack or several cracks occur in a periodic composite, the periodicity assumption is lost because it is not possible to identify a repeating unit cell that represents the multiphase composite that can be analyzed. Therefore, these micromechanical models are not applicable anymore.

In a recent investigation, Aboudi and Ryvkin (2012) presented an analysis for a distributed damage over a confined region within fiber-reinforced composites. It is based on the combination of the representative cell method (Ryvkin and Nuller, 1997), the high-fidelity generalized method of cells (HFGMC) micromechanical model (Aboudi, 2004), and the higher-order theory (Aboudi et al., 1999). In the first approach, the discrete Fourier transform is applied on the periodic composite in which the distributed damage effects have been included. As a result, a representative cell problem is obtained in the transform domain. The formulation of the specific boundary conditions that should be imposed in this problem requires the knowledge of the effective elastic moduli of the undamaged periodic composite. These elastic effective moduli are established by the HFGMC micromechanic analysis that forms the second approach. The solution of the governing equations in the transform domain is achieved by employing the higher-order theory that was originally developed for the analysis of functionally graded material. In the framework of this theory, the representative cell is divided into several subcells in which a second-order expansion of the displacements in the transform domain is employed and the governing equations and the interfacial and boundary conditions in this domain are imposed in the average (integral) sense. Once the solution of the representative cell problem has been established for all Fourier harmonics, an inverse transform is employed to obtain the actual elastic field in the composite. The effect of the localized damage appears in the constitutive equations in the form of eigenstresses. These field variables are not known in advance, and therefore, an iterative procedure was employed that establishes the requested elastic field in the composite to a preassigned degree of accuracy.

The above-mentioned approach is presently applied to investigate the mechanics and electric and magnetic field distributions caused by a single or a system of cracks in periodically layered electromagnetothermoelastic composites that are subjected to combined loadings. In addition, the analysis of Aboudi and Ryvkin (2012) is presently generalized for the modeling of inclusions embedded in this type of composites. It is shown that this approach is capable of analyzing stiff as well as soft inclusions with respect to the surrounding matrix (including cavities).

The proposed approach is verified by comparisons with five cases in which analytical solutions are available. These include inclusions and cracks in piezoelectric and electromagnetoelastic materials under combined loadings.

The present article is organized as follows. In sections “Governing equations” and “Method of solution,” the governing equations and the methods of solution are presented. The “Verifications” section presents the five verifications, which is followed by the “Application” section. Finally, the “Conclusion” section concludes the article.

Governing equations

Consider electromagnetoelastic linear materials in which the axis of symmetry is oriented in the x₃-direction. Piezoelectric materials are obtained as a special case in which the magnetic effects are ignored. These materials are governed by the following constitutive equations

{\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{23} \\ σ_{13} \\ σ_{12} \end{matrix}} = [\begin{matrix} c_{11} & c_{12} & c_{13} & 0 & 0 & 0 \\ c_{11} & c_{13} & 0 & 0 & 0 \\ c_{33} & 0 & 0 & 0 \\ c_{44} & 0 & 0 \\ c_{44} & 0 \\ sym . & \frac{1}{2} (c_{11} - c_{12}) \end{matrix}] {\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ 2 ε_{23} \\ 2 ε_{13} \\ 2 ε_{12} \end{matrix}} - [\begin{matrix} 0 & 0 & e_{31} \\ 0 & 0 & e_{31} \\ 0 & 0 & e_{33} \\ 0 & e_{15} & 0 \\ e_{15} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] {\begin{matrix} E_{1} \\ E_{2} \\ E_{3} \end{matrix}} - [\begin{matrix} 0 & 0 & q_{31} \\ 0 & 0 & q_{31} \\ 0 & 0 & q_{33} \\ 0 & q_{15} & 0 \\ q_{15} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] {\begin{matrix} H_{1} \\ H_{2} \\ H_{3} \end{matrix}}

(1)

{\begin{matrix} D_{1} \\ D_{2} \\ D_{3} \end{matrix}} = [\begin{matrix} 0 & 0 & 0 & 0 & e_{15} & 0 \\ 0 & 0 & 0 & e_{15} & 0 & 0 \\ e_{31} & e_{31} & e_{33} & 0 & 0 & 0 \end{matrix}] {\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ 2 ε_{23} \\ 2 ε_{13} \\ 2 ε_{12} \end{matrix}} + [\begin{matrix} κ_{11} & 0 & 0 \\ 0 & κ_{11} & 0 \\ 0 & 0 & κ_{33} \end{matrix}] {\begin{matrix} E_{1} \\ E_{2} \\ E_{3} \end{matrix}} + [\begin{matrix} a_{11} & 0 & 0 \\ 0 & a_{11} & 0 \\ 0 & 0 & a_{33} \end{matrix}] {\begin{matrix} H_{1} \\ H_{2} \\ H_{3} \end{matrix}}

(2)

{\begin{matrix} B_{1} \\ B_{2} \\ B_{3} \end{matrix}} = [\begin{matrix} 0 & 0 & 0 & 0 & q_{15} & 0 \\ 0 & 0 & 0 & q_{15} & 0 & 0 \\ q_{31} & q_{31} & q_{33} & 0 & 0 & 0 \end{matrix}] {\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ 2 ε_{23} \\ 2 ε_{13} \\ 2 ε_{12} \end{matrix}} + [\begin{matrix} a_{11} & 0 & 0 \\ 0 & a_{11} & 0 \\ 0 & 0 & a_{33} \end{matrix}] {\begin{matrix} E_{1} \\ E_{2} \\ E_{3} \end{matrix}} + [\begin{matrix} μ_{11} & 0 & 0 \\ 0 & μ_{11} & 0 \\ 0 & 0 & μ_{33} \end{matrix}] {\begin{matrix} H_{1} \\ H_{2} \\ H_{3} \end{matrix}}

(3)

In these equations, $σ_{ij}$ , $ε_{ij}$ , $D_{i}$ $E_{i}$ , $B_{i}$ , and $H_{i}$ denote the components of the stress, strain, electric displacement, electric field, magnetic flux, and magnetic field, respectively. The elements $c_{ij}$ , $e_{ij}$ , $κ_{ij}$ , $q_{ij}$ , $μ_{ij}$ , and $a_{ij}$ represent the elastic stiffnesses; piezoelectric, dielectric, piezomagnetic, and magnetic permeability; and magnetoelectric coefficients; respectively. For a monolithic material $a_{ij} = 0$ , and therefore, there are 15 independent material constants.

The components of the small strain tensor $ε_{ij}$ are expressed in terms of the displacement components $u_{i}$ by

ε_{ij} = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}), i, j = 1, 2, 3

(4)

The components of the electric field $E_{i}$ are obtained from the electric potential $ξ$ via

E_{i} = - \frac{\partial ξ}{\partial x_{i}}

(5)

Similarly, the components of the magnetic field $H_{i}$ are obtained from the magnetic potential $η$ via

H_{i} = - \frac{\partial η}{\partial x_{i}}

(6)

The earlier constitutive equations are applicable in the isothermal case. In order to incorporate the thermal effects, let us define the following vector

Ω = [α_{1}, α_{2}, α_{3}, 0, 0, 0, 0, 0, α_{9}, 0, 0, α_{12}]

(7)

where $α_{1}$ , $α_{2}$ , and $α_{3}$ are the thermal expansion coefficients in the three directions, respectively, and $α_{9}$ and $α_{12}$ are the associated pyroelectric and pyromagnetic constants, respectively. Let us also define the vectors $X$ and $Y$ as follows

X = [ε_{11}, ε_{22}, ε_{33}, 2 ε_{23}, 2 ε_{13}, 2 ε_{12}, - E_{1}, - E_{2}, - E_{3}, - H_{1}, - H_{2}, - H_{3}]

(8)

Y = [σ_{11}, σ_{22}, σ_{33}, σ_{23}, σ_{13}, σ_{12}, D_{1}, D_{2}, D_{3}, B_{1}, B_{2}, B_{3}]

(9)

Consequently, equations (1) to (3) can be written in the presence of thermal effects described by a temperature deviation $θ$ from a reference temperature in the following compact matrix form

Y = Z : X - Γ θ

(10)

where the square 12th-order symmetric matrix of coefficients $Z$ has the following form

Z = [\begin{matrix} C & e^{t} & q^{t} \\ e & - κ & - a \\ q & - a & - μ \end{matrix}]

(11)

and

Γ = Z : Ω

(12)

In equation (11), $C$ is the sixth-order stiffness matrix, $e^{t}$ denotes the transpose of the rectangular 3 × 6 piezoelectric matrix, $q^{t}$ denotes the transpose of the rectangular 3 × 6 piezomagnetic matrix, $κ$ is the square dielectric matrix of order 3, $a$ is a square matrix of the third order that represents the magnetoelectric coefficients, and $μ$ is the permeability square matrix of the third order.

In order to include damage effects in the considered electromagnetoelastic material, damage variables $0 \leq d_{pq} \leq 1$ , $p, q = 1, \dots, 12$ , are associated with every one of the independent elements of $Z_{pq}$ (for a monolithic material, there are 15 independent constants, and therefore, there are 15 corresponding damage variables). Thus, equation (10) takes the form

Y_{p} = \sum_{q = 1}^{12} Z_{pq} (1 - d_{pq}) X_{q} - \sum_{q = 1}^{12} Z_{pq} (1 - d_{pq}) Ω_{pq} θ, p = 1, \dots, 12

(13)

As in Aboudi and Ryvkin (2012), let us write this equation in the form

Y = Z : X - Y^{e}

(14)

where of $Y^{e}$ is electromagnetoelastic eigenfield vector whose components are given by

Y_{p}^{e} = \sum_{q = 1}^{12} [Z_{pq} d_{pq} X_{q} + Z_{pq} Ω_{pq} θ - Z_{pq} d_{pq} Ω_{pq} θ], p = 1, \dots, 12

(15)

It will be shown in the following that for a monolithic electromagnetoelastic material, 15 different damage variables will enable the modeling of cracks and the analysis of stiff and soft inclusions (including cavities) with respect to the material within which they are embedded. For elastic isotropic materials, just two independent damage variables would be needed (cf. Ju, 1990).

In the present investigation, we consider a periodically layered composite in which one of its electromagnetoelastic layers is damaged by the appearance of a transverse crack perpendicular to the layering direction together with two symmetric interfacial cracks, thus forming an H-crack. The transverse crack is assumed to exist in the weak layer (denoted by m) whose width is $t_{m}$ , whereas the width of the stiff layer (denoted by f) is $t_{f}$ . The length of the interfacial cracks is denoted by $t_{c}$ . As result of the localized damage that appears in the form of transverse and interfacial cracks, thus, forming an H-crack flaw, the periodicity of the layered composite is lost, and it is not possible to identify a repeating unit cell anymore. As has been mentioned earlier, the axis of symmetry (poling) is taken in the x₃-direction. Therefore, two situations exist. In Figure 1(a), the leading edges of the cracks are perpendicular to the axis of symmetry, whereas in Figure 1(d), the leading edges are parallel to this axis. It should be noted that a similar configuration of a transverse crack together with debonding along the interfaces of (conventional) ceramic layered composites has been investigated by Chan et al. (1993), for example. This configuration simulates a crack branching along a weak interface as a result of which a configuration in the form of H-crack appears. The stress distributions caused by these cracks in standard (nonsmart) ceramic matrix composites have been extensively investigated by Ryvkin and Aboudi (in press).

Figure 1.

(a) A periodically layered composite with transverse and interfacial cracks (thus forming an H-crack) whose leading edges are perpendicular to the poling direction $x_{3}$ , subjected to constant remote stress ${\bar{σ}}_{22}$ at infinity. (b) A rectangular domain $2 H \times 2 L$ of the layered composite is divided into repeating cells. These cells are labeled by $(K_{2}, K_{3})$ with $- M_{2} \leq K_{2} \leq M_{2}$ and $- M_{3} \leq K_{3} \leq M_{3}$ , and the size of each is $2 h \times 2 l$ (the figure is shown for $M_{2} = M_{3} = 2$ ). (c) A characteristic cell $(K_{2}, K_{3})$ in which local coordinates $(x'_{2}, x'_{3})$ are introduced whose origin is located at the center. (d–f) are analogous to (a–c) for cracks whose leading edges are parallel to the poling direction $x_{3}$ .

The response of the considered damaged layered composite can be determined by satisfying the static equilibrium equation that, in the absence of body forces, is given by

\nabla \cdot σ = 0

(16)

Furthermore, in the absence of volume charges, the following Maxwell’s equation

\nabla \cdot D = 0

(17)

must be satisfied, and similarly

\nabla \cdot B = 0

(18)

In addition, the interfacial conditions that require the continuity of displacements $u$ , tractions t , electric potential $ξ$ , normal electric displacements D , magnetic potential $η$ , and normal magnetic flux density B between the stiff and soft layers must be imposed. Finally, the far-field boundary conditions that are applied on the composite should be incorporated.

Method of solution

Far away from the damaged layers, the periodically bi-layered electromagnetoelastic composite behavior is governed by its effective moduli that can be determined by an appropriate micromechanical analysis (e.g. Aboudi, 2001). In the following, we derive the method of solution for the configuration shown in Figure 1(a) since a similar formulation is applicable to the one shown by Figure 1(d).

To this end, let us consider a rectangular domain $- H \leq X_{2} \leq H$ , $- L \leq X_{3} \leq L$ of the composite that includes the damaged region. Although this region includes the localized damage, it is assumed that it is extensive enough such that the electromagnetoelastic field at its boundaries is not influenced by the damage existence, and therefore, the micromechanical analysis prediction is applicable. Consequently, the boundary conditions that are applied on $X_{2} = \pm H$ and $X_{3} = \pm L$ are referred to as the far-field boundary conditions. According to the representative cell method (Ryvkin and Nuller, 1997), this region is divided into $(2 M_{2} + 1) \times (2 M_{3} + 1)$ cells (see Figure 1(b) for $M_{2} = M_{3} = 2$ ). Every cell is labeled by $(K_{2}, K_{3})$ with $K_{2} = - M_{2}, \dots, M_{2}$ and $K_{3} = - M_{3}, \dots, M_{3}$ . In each cell, local coordinates $(x'_{2}, x'_{3})$ are introduced whose origins are located at its center (see Figure 1(c)).

The governing equations (16) to (18) of the materials within the cell $(K_{2}, K_{3})$ take the form

\nabla \cdot σ^{(K_{2}, K_{3})} = 0

(19)

\nabla \cdot D^{(K_{2}, K_{3})} = 0

(20)

\nabla \cdot B^{(K_{2}, K_{3})} = 0

(21)

The constitutive equation in the cell, equation (14), can be written as

Y^{(K_{2}, K_{3})} = Z : X^{(K_{2}, K_{3})} - Y^{e (K_{2}, K_{3})}

(22)

where the components of $Y^{e (K_{2}, K_{3})}$ take the form

Y_{p}^{e (K_{2}, K_{3})} = \sum_{q = 1}^{12} [Z_{pq} d_{pq} X_{q}^{(K_{2}, K_{3})} + Z_{pq} Ω_{pq} θ^{(K_{2}, K_{3})} - Z_{pq} d_{pq} Ω_{pq} θ^{(K_{2}, K_{3})}], p = 1, \dots, 12

(23)

In order to formulate the continuity conditions that the various variables should fulfill, let us define the vectors $V_{r}^{(K_{2}, K_{3})}$

V_{r}^{(K_{2}, K_{3})} = {[u_{1 r}, u_{2 r}, u_{3 r}, σ_{1 r}, σ_{2 r}, σ_{3 r}, ξ, D_{r}, η, B_{r}]}^{(K_{2}, K_{3})}, r = 2, 3

(24)

These vectors assemble the components of the displacements, tractions, electric displacements, magnetic flux densities as well as the electric and magnetic potentials on a plane perpendicular to the x_r-axis at the cell $(K_{2}, K_{3})$ .

The continuity of displacements, tractions, electric potential, electric displacements, magnetic potential, and magnetic flux densities between adjacent cells should be imposed. Thus

[V_{2} (h, x'_{3})]^{(K_{2}, K_{3})} - [V_{2} (- h, x'_{3})]^{(K_{2} + 1, K_{3})} = 0

(25)

where $K_{2} = - M_{2}, \dots, M_{2} - 1$ , $K_{3} = - M_{3}, \dots, M_{3}$ , $- l \leq x'_{3} \leq l$ , and

[V_{3} (x'_{2}, l)]^{(K_{2}, K_{3})} - [V_{3} (x'_{2}, - l)]^{(K_{2}, K_{3} + 1)} = 0

(26)

where $K_{2} = - M_{2}, \dots, M_{2}$ , $K_{3} = - M_{3}, \dots, M_{3} - 1$ , and $- h \leq x'_{2} \leq h$ .

In the following, the appropriate form of the far-field boundary conditions that specify the electromagnetoelastic field at the opposite sides $X_{2} = \pm H$ , $X_{3} = \pm L$ of the rectangle of Figure 1(b) is presented. The tractions, electric displacements, and magnetic flux densities at the opposite sides of the rectangular domain are equal

{[σ_{2 j}]}^{(M_{2}, q)} (h, x'_{3}) - {[σ_{2 j}]}^{(- M_{2}, q)} (- h, x'_{3}) = 0, q = - M_{3}, \dots, M_{3}, j = 1, 2, 3

(27)

{[D_{2}]}^{(M_{2}, q)} (h, x'_{3}) - {[D_{2}]}^{(- M_{2}, q)} (- h, x'_{3}) = 0, q = - M_{3}, \dots, M_{3}

(28)

{[B_{2}]}^{(M_{2}, q)} (h, x'_{3}) - {[B_{2}]}^{(- M_{2}, q)} (- h, x'_{3}) = 0, q = - M_{3}, \dots, M_{3}

(29)

as well as

{[σ_{3 j}]}^{(p, M_{3})} (x'_{2}, l) - {[σ_{3 j}]}^{(p, - M_{3})} (x'_{2}, - l) = 0, p = - M_{2}, \dots, M_{2}, j = 1, 2, 3

(30)

{[D_{3}]}^{(p, M_{3})} (x'_{2}, l) - {[D_{3}]}^{(p, - M_{3})} (x'_{2}, - l) = 0, p = - M_{2}, \dots, M_{2}

(31)

{[B_{3}]}^{(p, M_{3})} (x'_{2}, l) - {[B_{3}]}^{(p, - M_{3})} (x'_{2}, - l) = 0, p = - M_{2}, \dots, M_{2}

(32)

The displacements, electric potential, and magnetic potential at the opposite sides, on the other hand, differ by certain jumps as follows

u^{(M_{2}, q)} (h, x'_{3}) - u^{(- M_{2}, q)} (- h, x'_{3}) = δ_{2}, q = - M_{3}, \dots, M_{3}

(33)

ξ^{(M_{2}, q)} (h, x'_{3}) - ξ^{(- M_{2}, q)} (- h, x'_{3}) = δ_{ξ_{2}}, q = - M_{3}, \dots, M_{3}

(34)

η^{(M_{2}, q)} (h, x'_{3}) - η^{(- M_{2}, q)} (- h, x'_{3}) = δ_{η_{2}}, q = - M_{3}, \dots, M_{3}

(35)

and

u^{(p, M_{3})} (x'_{2}, l) - u^{(p, - M_{3})} (x'_{2}, - l) = δ_{3}, p = - M_{2}, \dots, M_{2}

(36)

ξ^{(p, M_{3})} (x'_{2}, l) - ξ^{(p, - M_{3})} (x'_{2}, - l) = δ_{ξ_{3}}, p = - M_{2}, \dots, M_{2}

(37)

η^{(p, M_{3})} (x'_{2}, l) - η^{(p, - M_{3})} (x'_{2}, - l) = δ_{η_{3}}, p = - M_{2}, \dots, M_{2}

(38)

where $δ_{2}$ and $δ_{3}$ denote the vectors of the far-field displacement differences that are given by

δ_{2 j} = 2 H {\bar{ε}}_{2 j}, δ_{3 j} = 2 L {\bar{ε}}_{3 j}, j = 1, 2, 3

(39)

and ${\bar{ε}}_{2 j}$ and ${\bar{ε}}_{2 j}$ are the average strains of the unperturbed periodic composite. Similarly

δ_{ξ_{2}} = - 2 H {\bar{E}}_{2}, δ_{η_{2}} = - 2 H {\bar{H}}_{2}

(40)

and

δ_{ξ_{3}} = - 2 L {\bar{E}}_{3}, δ_{η_{2}} = - 2 L {\bar{H}}_{3}

(41)

where ${\bar{E}}_{r}$ and ${\bar{H}}_{r}$ , for $r = 2, 3$ , are the average electric and magnetic field components of the unperturbed periodic composite.

The far-field values ${\bar{ε}}_{2 j}$ , ${\bar{E}}_{2}$ , ${\bar{H}}_{2}$ , ${\bar{ε}}_{3 j}$ , ${\bar{E}}_{3}$ , and ${\bar{H}}_{3}$ are determined from the HFGMC micromechanical analysis of Aboudi (2001) that provides the effective properties of the composite in the following form (cf. equation (10))

\bar{Y} = Z^{*} : \bar{X} - Γ^{*} θ

(42)

with

Γ^{*} = Z^{*} : Ω^{*}

(43)

where $Z^{*}$ and $Ω^{*}$ assemble the corresponding predicted effective constants. Thus, for a given far-field electromagnetothermomechanical loading, one can readily determine from equations (42) and (43) the values of $δ_{2 j}$ , $δ_{ξ_{2}}$ , $δ_{η_{2}}$ , $δ_{3 j}$ , $δ_{ξ_{3}}$ , and $δ_{η_{2}}$ that are needed to impose on the rectangular boundaries.

The double discrete Fourier of the displacement vector $u^{(K_{2}, K_{3})}$ , for example, is defined by

\hat{u} (x'_{2}, x'_{3}, ϕ_{p}, ϕ_{q}) = \sum_{K_{2} = - M_{2}}^{M_{2}} \sum_{K_{3} = - M_{3}}^{M_{3}} u^{(K_{2}, K_{3})} (x'_{2}, x'_{3}) exp [i (K_{2} ϕ_{p} + K_{3} ϕ_{q})]

(44)

where

ϕ_{p} = \frac{2 π p}{2 M_{2} + 1}, p = 0, \pm 1, \pm 2, \dots, \pm M_{2}, ϕ_{q} = \frac{2 π q}{2 M_{3} + 1}, q = 0, \pm 1, \pm 2, \dots, \pm M_{3}

The application of this transform to the boundary problem (19) to (38) for the rectangular domain $- H < X_{2} < H$ , $- L < X_{3} < L$ , divided into $(2 M_{2} + 1) \times (2 M_{3} + 1)$ cells, converts it into the problem for the single representative cell $- h < x'_{2} < h$ , $- l < x'_{3} < l$ with respect to the complex valued transforms. The field equations obtained from the equilibrium, Maxwell and constitutive equations have the following form

\nabla \cdot \hat{σ} = 0

(45)

\nabla \cdot \hat{D} = 0

(46)

\nabla \cdot \hat{B} = 0

(47)

and

\hat{Y} = Z : \hat{X} - {\hat{Y}}^{e}

(48)

where the components of the transformed eigenfield vector are given by

{\hat{Y}}_{p}^{e} = \sum_{q = 1}^{12} [Z_{pq} d_{pq} {\hat{X}}_{q} + Z_{pq} Ω_{pq} \hat{θ} - Z_{pq} d_{pq} Ω_{pq} \hat{θ}], p = 1, \dots, 12

(49)

The conditions relating the opposite boundaries of the representative cell in the perfectly elastic problem were derived by Aboudi and Ryvkin (2012). By generalizing their approach, one obtains from equations (25) to (38) that

{\hat{σ}}_{2 j} (h, x'_{3}) = exp (- i ϕ_{p}) {\hat{σ}}_{2 j} (- h, x'_{3}), - l \leq x'_{3} \leq l, j = 1, 2, 3

(50)

{\hat{D}}_{2} (h, x'_{3}) = exp (- i ϕ_{p}) {\hat{D}}_{2} (- h, x'_{3}), - l \leq x'_{3} \leq l

(51)

{\hat{B}}_{2} (h, x'_{3}) = exp (- i ϕ_{p}) {\hat{B}}_{2} (- h, x'_{3}), - l \leq x'_{3} \leq l

(52)

\hat{u} (h, x'_{3}) = exp (- i ϕ_{p}) \hat{u} (- h, x'_{3}) + δ_{0, q} (2 M_{3} + 1) δ_{2} exp (i ϕ_{p} M_{2}), - l \leq x'_{3} \leq l

(53)

\hat{ξ} (h, x'_{3}) = exp (- i ϕ_{p}) \hat{ξ} (- h, x'_{3}) + δ_{0, q} (2 M_{3} + 1) δ_{ξ_{2}} exp (i ϕ_{p} M_{2}), - l \leq x'_{3} \leq l

(54)

\hat{η} (h, x'_{3}) = exp (- i ϕ_{p}) \hat{η} (- h, x'_{3}) + δ_{0, q} (2 M_{3} + 1) δ_{η_{2}} exp (i ϕ_{p} M_{2}), - l \leq x'_{3} \leq l

(55)

and

{\hat{σ}}_{3 j} (x'_{2}, l) = exp (- i ϕ_{q}) {\hat{σ}}_{3 j} (x'_{2}, - l), - h \leq x'_{2} \leq h, j = 1, 2, 3

(56)

{\hat{D}}_{3} (x'_{2}, l) = exp (- i ϕ_{q}) {\hat{D}}_{3} (x'_{2}, - l), - h \leq x'_{2} \leq h

(57)

{\hat{B}}_{3} (x'_{2}, l) = exp (- i ϕ_{q}) {\hat{B}}_{3} (x'_{2}, - l), - h \leq x'_{2} \leq h

(58)

\hat{u} (x'_{2}, l) = exp (- i ϕ_{q}) \hat{u} (x'_{2}, - l) + δ_{0, p} (2 M_{2} + 1) δ_{3} exp (i ϕ_{q} M_{3}), - h \leq x'_{2} \leq h

(59)

\hat{ξ} (x'_{2}, l) = exp (- i ϕ_{q}) \hat{ξ} (x'_{2}, - l) + δ_{0, p} (2 M_{2} + 1) δ_{ξ_{3}} exp (i ϕ_{q} M_{3}), - h \leq x'_{2} \leq h

(60)

\hat{η} (x'_{2}, l) = exp (- i ϕ_{q}) \hat{η} (x'_{2}, - l) + δ_{0, p} (2 M_{2} + 1) δ_{η_{3}} exp (i ϕ_{q} M_{3}), - h \leq x'_{2} \leq h

(61)

where $p = 0, \dots, \pm M_{2}; q = 0, \dots, \pm M_{3} .$

In these equations, $δ_{p, q}$ denotes the Kronecker delta.

The representative cell boundary value problem (45) to (61) (together with the standard continuity conditions at the interfaces between the phases) is solved by employing the higher-order theory (Aboudi et al., 1999) to which the electromagnetic equations and constitutive relations should be incorporated. According to this theory, the domain $- h \leq x'_{2} \leq h$ , $- l \leq x'_{3} \leq l$ (the representative cell) is divided into several rectangular subcells. The transformed displacement vector is expanded into a second-order polynomial, and the equilibrium equations, interfacial and boundary conditions are imposed in the average (integral) sense.

Once the solution in the transform domain has been established, the actual electromagnetoelastic fields can be readily determined at any point in the desired cell $(K_{2}, K_{3})$ of the considered rectangular region $- H \leq X_{2} \leq H$ , $- L \leq X_{3} \leq L$ by the inverse transform formula whose form for the displacements, for example, is

u^{(K_{2}, K_{3})} (x'_{2}, x'_{3}) = \frac{1}{(2 M_{2} + 1) (2 M_{3} + 1)} \times \sum_{p = - M_{2}}^{M_{2}} \sum_{q = - M_{3}}^{M_{3}} \hat{u} (x'_{2}, x'_{3}, ϕ_{p}, ϕ_{q}) exp [- i (K_{2} ϕ_{p} + K_{3} ϕ_{q})]

(62)

In the application of this theory, the eigenfield vector ${\hat{Y}}^{e}$ to be used in equation (48) is not known. Hence, an iterative solution has to be employed as follows:

Start by assuming that ${\hat{Y}}^{e} = 0$ and solve the earlier equations in the transform domain.

Apply the inverse transform formula to compute the stress field. The latter can be employed to compute the current eigenfield $Y^{e (K_{2}, K_{3})}$ in the actual space.

Compute the transform of $Y^{e (K_{2}, K_{3})}$ by employing equation (49).

Solve again the equations in the transform domain.

This procedure should be continued until a convergence to a desired degree of accuracy is achieved.

The computational efficiency of the present approach can be illustrated by considering the case with H-cracks. In order to ensure that the applied remote loading is sufficiently far away from the crack, the region has been divided into $21 \times 21$ cells (i.e. $M_{2} = M_{3} = 10$ ). The representative cell has been divided in the framework of the higher-order theory into $40 \times 40$ subcells (which has been found to provide accurate results). The higher-order analysis requires the solution of 15 unknowns in each subcell (i.e. 30 unknowns in the transform domain). Hence, this discretization requires for each combination of $ϕ_{p}$ and $ϕ_{q}$ , the solution of a sparse system of $48, 000$ algebraic equations. A direct numerical solution with the same number of degrees of freedom would require solving a system of $40 \times 40 \times 30 \times 21 \times 21 \approx 21 \times 10^{6}$ equations that is very large.

Verifications

There are several solutions that provide analytical expressions that can be employed to verify the accuracy of the results obtained by the present approach. These verifications are illustrated in the following five examples.

It should be noted that in both the “Verifications” and “Application” sections, the order of magnitudes of the values of the applied far-field are about $1 MV / m$ for the electric field, $0.01 - 0.001 C / m$ for the electric displacements, and $1 - 50 MPa$ for the stresses (see Pak, 1992; Sosa, 1991). As to the applied magnetic field, the value of about $10^{4} A / m$ keeps the response in the linear region (e.g. see Culshaw, 1996).

Circular inclusion in piezoelectric material under antiplane mechanical loading

For a circular piezoelectric inclusion embedded in an infinite piezoelectric medium that is subjected to remote antiplane mechanical loading and in-plane electric one, closed-form solution of the electro-elastic field is available (Pak, 1992). The axis of the inclusion is parallel to the poling direction, that is, it is oriented in the x₃-direction, and the configuration shown by Figure 1(d) to (f) is applicable. The closed-form solution, formulated in polar coordinates $(r, ψ)$ , has been established by expanding the displacement and electric potential within the matrix and inclusion in infinite series of $r^{n} \sin n ψ$ terms. The application of the interfacial and far-field conditions yields the exact solution.

To this end, consider $BaTi O_{3}$ and cadmium selenide piezoelectric materials whose properties are given in Tables 1 and 2 (Li and Dunn, 1998; Qin, 2001). Here, the following three different cases are studied:

Table 1.

Elastic properties.

$Material$	$C_{11} (GPa)$	$C_{12} (GPa)$	$C_{13} (GPa)$	$C_{33} (GPa)$	$C_{44} (GPa)$
${BaTiO}_{3}$	166	77	78	162	43
Cadmium selenide	74.1	45.2	39.3	83.6	13.2
$PZT - 7 A$	148	76.2	74.2	131	25.4
${CoFe}_{2} O_{4}$	286	173	170	269.5	45.3

PZT: lead zirconate titanate.

Table 2.

Electric properties.

$Material$	$e_{15} (C / m^{2})$	$e_{31} (C / m^{2})$	$e_{33} (C / m^{2})$	$κ_{11} (10^{- 12} C / Vm)$	$κ_{33} (10^{- 12} C / Vm)$
${BaTiO}_{3}$	11.6	−4.4	18.6	11,200	12,600
Cadmium selenide	0.138	−0.16	0.347	82.6	90.3
$PZT - 7 A$	9.2	−2.1	9.5	4070	2070
${CoFe}_{2} O_{4}$	0	0	0	80	93

PZT: lead zirconate titanate.

A stiff inclusion ( $BaTi O_{3}$ ) embedded in the soft (cadmium selenide) matrix: To simulate this situation, the entire medium is filled with the stiff $BaTi O_{3}$ material and the following 10 damage variables $1 - d_{pq}$ that correspond to the material properties of Tables 1 and 2 are $74.1 / 166$ , $45.2 / 77$ , $39.3 / 78$ , $83.6 / 162$ , $13.2 / 43$ , $0.138 / 11.6$ , $0.16 / 4.4$ , $0.347 / 18.6$ , $82.6 / 11200$ , and $90.3 / 12600$ . These damage variables are applied in the matrix region as a result of which the properties of the latter match to the desired values of the cadmium selenide parameters. Within the inclusion of cell $(K_{1} = 0 and K_{2} = 0)$ , no damage takes place, that is, $d_{pq} = 0$ .

A soft inclusion (cadmium selenide) embedded in the stiff matrix ( $BaTi O_{3}$ ): Here, the entire medium is again filled with the stiff $BaTi O_{3}$ , but the above-mentioned values of damage variables are applied only within the inclusion region of cell $(K_{1} = 0 and K_{2} = 0)$ , whereas in the matrix, $d_{pq} = 0$ .

A cavity within the stiff matrix ( $BaTi O_{3}$ ): In this case, the entire region is filled with the $BaTi O_{3}$ material, but the 10 damage variables within the inclusion of cell $(K_{1} = 0 and K_{2} = 0)$ are equal to 1 ( $d_{pq} = 1$ , complete damage).

In all these three cases, the far-field loading is given by $σ_{23}^{\infty} = 50 MPa$ and $E_{2}^{\infty} = 1 MV / m$ .

Figure 2 exhibits comparisons between the resulting stresses and electrical displacements obtained from the closed-form solution and the present approach. It can be noted that very good agreements exist. The particular low order of magnitude of the scale of Figure 2(b) shows a slight difference between the two solutions within the inclusion. It is obvious that more cases could be considered (e.g. a cavity in the soft cadmium selenide matrix) and verified by a comparison with the closed-form solution.

Figure 2.

Comparisons between the exact solution and present approach for the variation along $x_{1}$ of the stress and electrical displacement $D_{2}$ in (a and b) stiff piezoelectric inclusion ( $BaTi O_{3}$ ) embedded in an infinite soft piezoelectric medium (cadmium selenide), (c and d) soft piezoelectric inclusion (cadmium selenide) embedded in an infinite stiff piezoelectric medium ( $BaTi O_{3}$ ), and (e and f) a cavity embedded in an infinite piezoelectric medium ( ${BaTiO}_{3}$ ). The mechanical and electrical fields are generated by the application of the remote loading: $σ_{23}^{\infty} = 50 MPa$ and $E_{2}^{\infty} = 1 MV / m$ .

Crack in piezoelectric material under antiplane mechanical loading

A closed-form solution for a crack of length $2 a$ embedded in an infinite piezoelectric material and subjected to an antiplane mechanical loading and in-plane electric field has been derived by Pak (1990). The leading edge of the crack is parallel to the poling direction $x_{3}$ (see Figure 1(d) for configuration). The method of solution is based on the observation that the governing equations for the displacement and electric potential are satisfied if the latter are harmonic functions in the complex plane. The stresses and electric displacements are determined from these functions. The resulting solution can be expressed in the form

σ_{23} + i σ_{13} = c_{44} \frac{Az}{\sqrt{z^{2} - a^{2}}} - e_{15} \frac{Bz}{\sqrt{z^{2} - a^{2}}}

D_{2} + i D_{1} = e_{15} \frac{Az}{\sqrt{z^{2} - a^{2}}} + κ_{11} \frac{Bz}{\sqrt{z^{2} - a^{2}}}

(63)

where $z = x_{1} + i x_{2}$ and $A$ and $B$ are constants that can be determined from the applied remote boundary conditions.

In our present approach, the crack region is modeled by vanishing damage variables $d_{pq} = 0$ in cell $(K_{1} = 0 and K_{2} = 0)$ . Figure 3 shows a comparison between this closed-form expression and the present approach for a crack of length $2 a / 2 d = 0.5$ embedded in PZT-7A piezoelectric material (Tables 1 and 2) with an applied remote field: $σ_{23}^{\infty} = 50 MPa$ and $E_{2}^{\infty} = 1 MV / m$ . The two solutions are indistinguishable.

Figure 3.

A comparison between the exact solution and present approach for the variation of the stress and electrical displacement along the crack’s line $x_{1}$ generated in piezoelectric PZT-7A material by the application of the remote loading: $σ_{23}^{\infty} = 50 MPa$ and $E_{2}^{\infty} = 1 MV / m$ .

Circular cavity in piezoelectric material under in-plane loadings

Exact solution for a elliptical cavity embedded in a piezoelectric material that is subjected to remote in-plane mechanical loading has been presented by Sosa (1991). Here, the axis of the cavity is perpendicular to the poling direction $x_{3}$ . The method of solution is based on Lekhnitskii’s formalism wherein the mechanical and electric variables are expressed in terms of complex potentials (see Qin, 2001 for a brief summary). This exact solution is utilized herein to compare its prediction for a piezoelectric material ( $BaTi O_{3}$ ) with an embedded circular cavity. The remote in-plane loading is given by $σ_{33}^{\infty} = 10 MPa$ and $D_{3}^{\infty} = 0.01 C / m^{2}$ (see Figure 1(a) for configuration). Figure 4 shows this comparison for the variations of $σ_{33}$ and $D_{3}$ along $x_{2}$ , exhibiting that the two solutions are indistinguishable.

Figure 4.

A comparison between the exact and present solution for the variation of the (a) stress and (b) electrical displacement along $x_{2}$ generated in a piezoelectric material ( $BaTi O_{3}$ ) with an embedded cavity. The remote in-plane loading is given by $σ_{33}^{\infty} = 10 MPa$ and $D_{3}^{\infty} = 0.01 C / m^{2}$ .

Crack in piezoelectric material under in-plane loadings

The closed-form solution of Sosa (1991) for elliptical cavity provides, as a special case, the mechanical and electrical fields in a piezoelectric medium with an embedded crack. Here, the leading edge of the crack is perpendicular to the poling direction $x_{3}$ (see Figure 1(a) for configuration). For a crack extending along $- a \leq x_{2} \leq a$ , the resulting stress and electric displacement variation along the crack’s line $x_{2}$ are (Sosa, 1992)

σ_{33} (x_{2}, 0) = σ_{33}^{\infty} \frac{| x_{2} |}{\sqrt{x_{2}^{2} - a^{2}}}, D_{3} (x_{2}, 0) = D_{3}^{\infty} \frac{| x_{2} |}{\sqrt{x_{2}^{2} - a^{2}}}, | x_{2} | \geq a

(64)

The veracity of the present method of solution can be verified by considering two cases in which the remote loading is parallel and perpendicular to the poling direction of a piezoelectric material (PZT-7A). In the first case, the crack extends along $- a \leq x_{2} \leq a$ , whereas in the second case, it extends along $- a \leq x_{3} \leq a$ (i.e. along the poling direction). The variations of the normal stresses along the crack’s lines in these two cases are shown in Figure 5(a) and (b). Figure 5(a) results from a remote loading of $σ_{33}^{\infty} = 10 MPa$ and $D_{3}^{\infty} = 0.01 C / m^{2}$ , whereas Figure 5(b) results from $σ_{22}^{\infty} = 10 MPa$ and $D_{2}^{\infty} = 0.01 C / m^{2}$ . Here, again, the two solutions are indistinguishable from the closed-form expressions.

Figure 5.

Comparisons between the exact and present solution for the variation of the stress and electrical displacement in a cracked PZT-7A material along the crack’s lines. (a) The remote in-plane loading is given by $σ_{33}^{\infty} = 10 MPa$ and $D_{3}^{\infty} = 0.01 C / m^{2}$ (parallel to the poling direction) and (b) the remote in-plane loading is given by $σ_{22}^{\infty} = 10 MPa$ and $D_{2}^{\infty} = 0.01 C / m^{2}$ (perpendicular to the poling direction).

In the aforementioned solutions, the surfaces of the cavity and crack were assumed to be electrically impermeable, namely, the normal electric displacement component is zero. Permeable solution for an elliptical cavity in a piezoelectric material has been presented by Sosa and Khutoryansky (1996). Comparisons of the present approach with this latter exact solution also revealed excellent agreements. However, the solutions of Sosa (1991) and Sosa and Khutoryansky (1996) for impermeable and permeable crack boundary conditions are very close as long as the stress and electric field variations are investigated.

Square inclusion in electromagnetoelastic material under antiplane mechanical loadings

Thus far, the predicted results from the present approach have been verified in the case of piezoelectric materials under various types of loading. It is possible to employ the analytical solution that has been presented by Wang and Shen (2003) for a rectangular inclusion whose sides are parallel to the directions $x_{1}$ and x₂, embedded in an infinite electromagnetoelastic material (see Figure 1(d) for configuration). The loading is expressed in the form of antiplane mechanical and in-plane electric and magnetic eigenfield that is applied within the inclusion in the form

Π^{e} = [ε_{23}^{e} + i ε_{13}^{e}, - \frac{1}{2} (E_{2}^{e} + {iE}_{1}^{e}), - \frac{1}{2} (H_{2}^{e} + {iH}_{1}^{e})]

(65)

The method of solution is based on Ru’s method (Ru, 1999), which employs analytical continuation and conformal mapping. The resulting analytical solution is valid within the rectangular inclusion. It is in fact an approximate solution since it is based on a conformal mapping to a unit circle of the rectangular inclusion that includes infinite number of terms, the truncation of which results in an approximate solution (see Ru, 1999).

Since the electromagneto-coupling term is included in the considered analytical solution, the present verification has been carried out by considering a unidirectional fiber-reinforced piezoelectric/piezomagnetic ${BaTiO}_{3} / CoF e_{2} O_{4}$ composite with a constituent volume ratio of $0.5$ , whose effective properties have been predicted by the HFGMC micromechanical method (Aboudi, 2001). The properties of the constituents are given in Tables 1 to 3, and the relevant predicted effective constants are $c_{44}^{*} = 49.6 GPa$ , $e_{15}^{*} = 0.15 C / m^{2}$ , $κ_{11}^{*} = 0.23 \times 10^{- 9} C / Vm$ , $q_{15}^{*} = 172 N / Am$ , $μ_{11}^{*} = - 0.18 \times 10^{- 3} N s^{2} / C^{2}$ , and $a_{11}^{*} = 0.58 \times 10^{- 11} N s / VC$ . The applied eigenfield within the square inclusion $0.5 \leq x_{1} / 2 d \leq 05$ , $0.5 \leq x_{2} / 2 h \leq 05$ , ( $d = h$ ) is $ε_{13}^{e} = ε_{23}^{e} = 0.01 %$ , $E_{1}^{e} = E_{2}^{e} = 1 MV / m$ , and $H_{1}^{e} = H_{2}^{e} = 10 kA / m$ .

Table 3.

Magnetic properties.

$Material$	$q_{15} (N / Am)$	$q_{31} (N / Am)$	$q_{33} (N / Am)$	$μ_{11} (10^{- 6} N s^{2} / C^{2})$	$μ_{33} (10^{- 6} N s^{2} / C^{2})$
${BaTiO}_{3}$	0	0	0	5	10
${CoFe}_{2} O_{4}$	550	580.3	699.7	−590	157

Figure 6(a) to (c) shows a comparison between the approximate analytical solution and the present approach for the stresses within the square inclusion. Satisfactory agreements and tendencies can be observed. These stresses, as predicted by the present solution, are shown in Figure 6(c) and (d) within and outside the inclusion along x₁-axis. Finally, Figure 7(a) to (d) shows the corresponding variations of the electrical displacements and magnetic inductions as predicted by the present approach.

Figure 6.

A square inclusion embedded within an infinite unidirectional electromagnetoelastic ${BaTiO}_{3} / CoF e_{2} O_{4}$ composite within which eigenfield is imposed. (a and b) Stress comparisons between the approximate analytical and present predictions within the inclusion along $x_{1}$ , (c and d) stress variations along $x_{1}$ as predicted by the present approach.

Figure 7.

A square inclusion embedded within an infinite unidirectional electromagnetoelastic ${BaTiO}_{3} / CoF e_{2} O_{4}$ composite within which eigenfield is imposed. (a–d) Electric displacements and magnetic inductions variations along $x_{1}$ .

Applications

In the previous section, results have been presented under circumstances where analytical solutions to be compared and verified with are available. In the present section, three applications of the proposed approach are given under more complicated circumstances. These include a cavity embedded in electromagnetoelastic medium, a layered piezoelectric composite with a transverse and H-crack, a layered electromagnetoelastic composite with an H-crack, and a layered piezoelectric composite with an H-crack under elevated temperature.

A circular cavity embedded in electromagnetoelastic medium

The first application considers a circular cavity of radius $a$ embedded in electromagnetoelastic infinite material. The material is ${CoFe}_{2} O_{4}$ whose properties are given in Tables 1 to 3. The magnetic permeability $μ_{11}$ is negative and therefore the solution procedure did not converge. The same difficulty has been encountered by Wang and Mai (2007) who suggested using the positive value, that is, $μ_{11} = 157 \times 10^{- 6} N s^{2} / C^{2}$ . The remote in-plane boundary conditions are $σ_{33}^{\infty} = 10 MPa$ , $E_{3}^{\infty} = 1 MV / m$ , and $H_{3}^{\infty} = 10^{4} A / m$ , that is, they are applied in the electric and magnetic direction of the poling (see Figure 1(a) for configuration). The axis of the cavity, on the other hand, is perpendicular to the poling direction. Electric and magnetic permeable boundary conditions have been applied on the rim of the cavity, namely

σ_{rr} = 0, D_{r} = κ_{0} E_{r} B_{r} = μ_{0} H_{r}, r = a

(66)

where $κ_{0} = 8.85 \times 10^{- 12} C^{2} / N m^{2}$ and $μ_{0} = 1.26 \times 10^{- 6} N s^{2} / C^{2}$ being the dielectric permittivity and magnetic permeability of air, respectively. These interfacial boundary conditions can be realized by assuming that the entire medium and cavity posses the values of the parameters for the ${CoFe}_{2} O_{4}$ given by Tables 1 to 3 (with $μ_{11} = 157 \times 10^{- 6} N s^{2} / C^{2}$ ), but within the cavity, the damage variables that correspond to $κ_{11}$ , $κ_{33}$ , $μ_{11}$ , and $μ_{33}$ are given by $1 - κ_{0} / κ_{11}$ , $1 - κ_{0} / κ_{33}$ , $1 - μ_{0} / μ_{11}$ , and $1 - μ_{0} / μ_{33}$ , respectively. The resulting mechanical, electric, and magnetic field variations along $x_{2}$ are shown in Figure 8.

Figure 8.

A cylindrical cavity embedded within an infinite ${CoFe}_{2} O_{4}$ electromagnetoelastic material. The in-plane boundary conditions are in the electric and magnetic poling direction $x_{3}$ , whereas the axis of the cavity is perpendicular to this direction. Permeable electric and magnetic boundary conditions have been applied on the rim of the cavity. The figure shows the variation along the x₂-axis of the stress, electric displacement, and magnetic induction.

Periodically layered piezoelectric composite with a broken layer

Consider a periodically layered BaTiO₃/cadmium selenide piezoelectric composite whose constituent properties are given in Tables 1 and 2 (see Figure 1(a) for configuration with $t_{f} = t_{m}$ ). It is assumed that a single transverse crack has been developed in one of the weaker (cadmium selenide) material layers. Two situations can be investigated.

In the first one, the layered composite is loaded in the x₂-direction with the crack of length $t_{m}$ extending along the x₁-direction (Figure 1(d)) with the leading edge of the crack in the poling direction. The in-plane applied loading is $σ_{22}^{\infty} = 10 MPa$ and $D_{2}^{\infty} = 0.01 C / m^{2}$ . In the second one, the layered composite is loaded in the x₃-direction with the crack of length $t_{m}$ extending along the x₂-direction (Figure 1(a)), with the leading edge of the crack perpendicular to the poling direction. Here, the in-plane applied loading is $σ_{33}^{\infty} = 10 MPa$ and $D_{3}^{\infty} = 0.01 C / m^{2}$ in the poling direction. The resulting stress distributions along the cracks lines are shown in Figure 9(a) and (b). In both cases, the crack’s surfaces are permeable, and the effect of the cracks rapidly decays as the observation point moves away from the cracked region. It is interesting that the stress variations in the vicinity of the cracks in these two cases are not similar. In addition, since both solutions were generated by employing the same level of discretization, it appears that the intensity of the stress at the vicinity of the crack’s tip is higher in the second case.

Figure 9.

Transverse cracks in a periodically layered piezoelectric BaTiO₃/cadmium selenide composite. The cracks are located in the cadmium selenide layer. (a) The stress variations along the crack’s line $x_{1}$ , caused by the remote loading: $σ_{22}^{\infty} = 10 MPa$ and $D_{2}^{\infty} = 0.01 C / m^{2}$ . (b) The stress variations along the crack’s line $x_{2}$ , caused by the remote loading: $σ_{33}^{\infty} = 10 MPa$ and $D_{3}^{\infty} = 0.01 C / m^{2}$

Periodically layered piezoelectric composite with an H-crack

Consider presently the same configurations that have been discussed in the previous application but this time with an H-crack in a cadmium selenide layer. The results from the two cases discussed earlier are presently generated and shown in Figure 10(a) and (b). In both cases, $t_{c} / 2 h = t_{c} / 2 l = 1$ , with $h = l$ . Thus, Figure 10(a) exhibits the distribution of $σ_{22} / σ_{22}^{\infty}$ in the vicinity of the H-crack, whereas Figure 10(b) shows the distribution of $σ_{33} / σ_{33}^{\infty}$ . A comparison between these figures reveals that the effect of the H-crack is more pronounced in the second case in which the loading is oriented in the poling direction $x_{3}$ . The decay of this effect away from the crack can be clearly observed. Also observed is the stress concentration in the vicinity of the four tips of the longitudinal cracks.

Figure 10.

H-crack in a periodically layered piezoelectric BaTiO₃/cadmium selenide composite. The cracks are located in the cadmium selenide layer. (a) The stress $σ_{22} / σ_{22}^{\infty}$ distribution caused by the remote loading: $σ_{22}^{\infty} = 10 MPa$ and $D_{2}^{\infty} = 0.01 C / m^{2}$ , and (b) the stress $σ_{33} / σ_{33}^{\infty}$ distribution caused by the remote loading: $σ_{33}^{\infty} = 10 MPa$ and $D_{3}^{\infty} = 0.01 C / m^{2}$ .

Periodically layered electromagnetoelastic composite with an H-crack

The periodically layered composite presently consists of ${CoFe}_{2} O_{4} / BaTi O_{3}$ , alternating electromagnetoelastic layers of equal widths $t_{f} = t_{m}$ . An H-crack is located within one of the weaker ${BaTiO}_{3}$ material layers (see Figure 1(a)) with $t_{c} / 2 l = 1$ . The composite is subjected to an in-plane remote combined mechanical, electrical, and magnetic loadings: $σ_{33}^{\infty} = 10 MPa$ , $E_{3}^{\infty} = 1 MV / m$ , and $H_{3}^{\infty} = 10 kA / m$ in the electrical and magnetic poling direction. The leading edge of the H-crack is perpendicular to the poling direction. The resulting $σ_{33} / σ_{33}^{\infty}$ and $D_{3}$ distributions are shown in Figure 11(a) and (b), respectively (the effect of the cracking on the magnetic induction is negligible). It is interesting that whereas the effect of the cracking on the stress distribution is quite localized within the cracked layer, the distribution of the electrical displacement in the neighboring layers to the crack is significantly affected.

Figure 11.

H-crack in a periodically layered electromagnetoelastic ${BaTiO}_{3} / CoF e_{2} O_{4}$ composite. The cracks are located in the ${BaTiO}_{3}$ layer. The remote applied mechanical, electrical, and magnetic loadings are $σ_{33}^{\infty} = 10 MPa$ , $E_{3}^{\infty} = 1 MV / m$ , and $H_{3}^{\infty} = 10 kA / m$ , respectively. (a) The stress $σ_{33} / σ_{33}^{\infty}$ distribution and (b) the electrical displacement $D_{3} (C / m^{2})$ distribution.

Applied temperature in periodically layered piezoelectric composite with an H-crack

Thus far, all results were obtained under isothermal conditions, that is, $θ = 0$ . Let an applied temperature deviation of $θ = 100 K$ be uniformly applied in a periodically layered piezoelectric BaTiO₃/cadmium selenide composite ( $t_{f} = t_{m}$ ). The thermal properties of these constituents are given in Table 4 (Ashida et al., 1994; Qin, 2001). The H-crack is oriented as shown in Figure 1(a) with $t_{c} / 2 l = 1$ . Except for the applied thermal loading, the mechanical and electrical applied far-fields are zero. The resulting stress $σ_{33}$ and electrical displacement $D_{3}$ distributions, generated by the thermoelectromechanical mismatch properties of the layers, are shown in Figure 12. It can be observed that whereas the cracking of the cadmium selenide layer significantly reduces the stress in this layer, its effect on the electrical displacement field is not pronounced.

Table 4.

Thermal properties.

$Material$	$α_{1} (10^{- 6} K^{- 1})$	$α_{3} (10^{- 6} K^{- 1})$	$α_{9} (10^{5} N / CK)$
${BaTiO}_{3}$	8.53	1.99	0.133
Cadmiumselenide	4.49	2.26	0.253

Figure 12.

H-crack in a periodically layered piezoelectric BaTiO₃/cadmium selenide composite. The cracks are located in the cadmium selenide layer. The applied uniform thermal loading is $θ = 100 K$ . (a) The stress $σ_{33} (MPa)$ distribution and (b) the electrical displacement $D_{3} (C / m^{2})$ distribution.

Conclusion

A methodology has been presented that is capable to establish the field distributions of periodically layered electromagnetothermoelastic composites with damage in the form of cavities and cracks. The composites are subjected to arbitrary combined electric, magnetic, and mechanical loadings that must be sufficiently remote from the localized perturbed regions where cavities and cracks exist. This methodology is based on the combination of three distinct theories namely, the representative cell theory, the higher-order theory, and the HFGMC micromechanical model. The localized effects of the cavities and cracks appear in the constitutive equations of electromagnetothermoelastic material in the form of unknown eigenfield that necessitates the application of an iterative procedure until a convergence of the solution is achieved.

In addition, the present methodology enables the modeling of inclusions of any type (irrespective whether they are stiff or soft with respect to the surrounding matrix). This is based on the fact that the damage variables should always reduce their corresponding moduli (see equation (13)). Therefore, it is always required to define the given configuration with the largest moduli and apply the appropriate value of the corresponding damage variable that should reduce the modulus to its desired value in the considered region.

The method has been verified by a comparison with analytical solutions in five special cases. Several applications are presented, which show, in particular, the effect of H-cracks on periodically layered composites. The effect of these cracks is seen to be rapidly decaying. Thus, the theoretical assumption that the composite consists of infinite number of periodic layers is not a limitation, and the present approach can be utilized in practical situations.

Generalizations of the present method for the analysis of several localized interacting systems of cavities, cracks, and H-cracks are possible.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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