Abstract
In this chapter, there are mainly two kinds of problems discussed. The first kind of problems is the 3D electroelastic problems: the potential function method, the solutions of the penny-shaped crack and elliptic inclusions. The second kind of problems is the applied electroelastic problems which are used in engineering: simple electroelastic problems, laminated piezoelectric plates containing classical and higher-order theories and piezoelectric composite shells. A unified first-order approximate theory of an electro-magneto-elastic thin plate derived by the physical variational principle is given when the electromagnetic induction effect can be neglected.
Access provided by Autonomous University of Puebla. Download chapter PDF
Keywords
7.1 Potential Function Methods in Transversely Isotropic Piezoelectric Materials
7.1.1 Governing Equations
The governing equations of transversely isotropic piezoelectric materials have been discussed in previous chapters. In the material principle coordinates, the constitutive equations are
The generalized momentum equations are
where \( i=1,2,3 \). Substitution of Eq. (7.1) into Eq. (7.2) yields
7.1.2 General Solution of the Static Problem (I)
Wang and Zheng (1995) discussed the general solution of (7.3) for the static problem by introducing potential functions. They assume
where \( {k_1} \) and \( {k_2} \) are undetermined constants and \( \psi \) and \( \chi \) are potential functions. Substitution of Eq. (7.4) into Eq. (7.3) yields
In order to have nontrivial solution of Eq. (7.6), the following relations must be held:
Eliminating \( {k_1} \) and \( {k_2} \), a cubic algebra equation of \( \lambda \) is obtained:
Assume root \( {\lambda_1} \) is positive real and \( {\lambda_2} \) and \( {\lambda_3} \) are either a pair of conjugate complex roots with positive real parts or positive real roots. Corresponding to each \( {\lambda_i} \), a potential function \( {\psi_j} \) in Eq. (7.6) can be obtained:
Substituting \( {\lambda_j} \) into Eq. (7.7), \( {k_{1j}} \) and \( {k_{2j}} \) can be obtained. So the general solution of Eq. (7.3) can be expressed in potential functions:
Usually the numerical method is used to solve \( {\lambda_j} \) in Eq. (7.8) due to its complex roots. As an example for material PZT-6B with material constants,
The solved \( \boldsymbol{\lambda} \) is \( {\lambda_1}=3.92,\quad {\lambda_2}=0.73+0.87\mathrm{i},\quad {\lambda_3}=0.73-0.87\mathrm{i} \).
7.1.3 General Solution of the Dynamic Problem
Ding et al. (1996) discussed the dynamic problem. Let
where \( \psi \) and \( \chi \) are potential functions, but their meanings are different with that in Sect. 7.1.2. Substituting Eq. (7.11) into the first two equations in Eq. (7.3) yields
Let \( A={H_{,2}},\quad B={H_{,1}} \),and Eq. (7.12) is reduced to \( {\nabla^2}H=0 \). One particular solution is \( H=\mathrm{constant} \). Adopt a particular solution
Using this result, substituting Eq. (7.11) into the last two equations in Eq. (7.3) and listing the results with Eq. (7.13) together we get
Introduce a new function \( F \) and let \( \left| \boldsymbol{D} \right|F=0 \) or
where
After solving \( F \), it can be proved that the three group solutions of \( \chi, {u_3},\varphi \) are
where \( {A_{ij}} \) in Eq. (7.18) is the algebraic complement of \( \left| \boldsymbol{D} \right| \), i.e.,
By substitution of Eq. (7.18) into Eq. (7.11), the general solution is
The general solution of an axial-symmetric problem can be obtained from Eq. (7.22) if let \( \psi =0 \) and \( F \) is independent of \( \theta \).
7.1.4 General Solution of the Static Problem (II)
Let all the potential functions be independent of the time the general solution of the static problem can be obtained from the results in Sect. 7.1.3. Equation (7.14) yields
Equation (7.16) can be reduced to
where \( s_i^2 \) is the root of the following equation:
No loss of generality let \( {s_1} \) be real and assume \( \operatorname{Re}\left( {{s_i}} \right)> 0 \). It is easy to prove that \( {F_i} \) satisfying the following equation is the solution of Eq. (7.24):
The general solutions of Eq. (7.24) are
-
1.
$$ s_1^2\ne s_2^2\ne s_3^2;\quad F={F_1}+{F_2}+{F_3} $$(7.27)
-
2.
$$ s_1^2\ne s_2^2=s_3^2\quad F={F_1}+{F_2}+{x_3}{F_3} $$(7.28a)
-
3.
$$ s_1^2=s_2^2=s_3^2;\quad F={F_1}+{x_3}{F_2}+x_3^2{F_3}; $$(7.28b)
From Eq. (7.26), it is obtained that \( {\nabla^2}=-{{{{\partial^2}}} \left/ {{\partial z_i^2}} \right.},\;{\partial \left/ {{\partial {x_3}}} \right.}={s_i}{\partial \left/ {{\partial {z_i}}} \right.} \). Substituting these results into Eq. (7.20) yields
The general solution Eq. (7.22) can be rewritten as
If let \( {\alpha_{1i}}{s_i}{{{{\partial^3}{F_i}}} \left/ {{\partial z_i^3}} \right.}={\psi_i},{\psi_0}=-\psi \), Eq. (7.30) can be reduced to
7.2 A Penny-Shaped Crack in Transversely Isotropic Material
7.2.1 Governing Equations
Consider a transversely isotropic piezoelectric material weakened by a flat impermeable crack of radius \( a \) occupied region \( S \) in the plane \( {x_3}=0 \), subjected to distributed pressure \( -p\left( {{x_1},{x_2}} \right) \) and surface electric charge \( q\left( {{x_1},{x_2}} \right) \) (Fig. 7.1). In Fig. 7.1 only a pair of concentrated force and electric charge is shown. The Cartesian coordinates \( \left( {{x_1},{x_2},{x_3}} \right) \) and cylindrical coordinates \( \left( {r,\theta, z} \right) \) are adopted simultaneously. Using the symmetry with respect to the crack surface, this problem can be reduced to a mixed boundary value problem for a half space subjected to the following boundary conditions:
A penny-shaped crack in a transversely isotropic piezoelectric material
Chen and Shioya (1999) extended the method proposed by Fabrikant (1989) in the elasticity, to solve above problem. Introduce notation \( \varLambda ={\partial \left/ {{\partial {x_1}}} \right.}+\operatorname{i}{\partial \left/ {{\partial {x_2}}} \right.} \) and complex displacement \( U={u_1}+\operatorname{i}{u_2} \). Let \( w={x_3} \), the generalized momentum equations in complex displacement is
where \( \bar{U},\bar{\varLambda} \) mean the conjugate value of \( U,\varLambda \). By using the complex displacement, the general solution in potential functions, Eq. (7.31), become
The generalized stresses become
where \( {s_i} \) is the root of the Eq. (7.25).
7.2.2 Potential Theory Method of Crack Problem
The solution satisfying the boundary conditions in Eq. (7.32) can be expressed by two harmonic functions \( G \) and \( H \):
where \( {c_i},{d_i} \) are undetermined constants. \( G \) and \( H \) can be expressed by two potentials of a simple layer:
where \( \hat{u}(N)=w({x_1},{x_2},0) \) and \( \hat{\varphi}(N)=\hat{\varphi}({x_1},{x_2},0) \) are undetermined displacement and electric potential on the crack surface, respectively. \( N(r_1,\theta_1, 0) \) is a point on \( S \), \( M(r,\theta, z) \) is a certain point in the material, and \( \rho (M,N) \) is the distance between \( N \) and \( M \). According to the property of the potential of a simple layer, the boundary conditions \( w=\varphi =0 \) outside the crack in Eq. (7.37) are satisfied automatically. Inside the crack we have
Equations (7.34), (7.36), and (7.38) yield
The boundary conditions of \( {\sigma_{31}},{\sigma_{32}} \) in Eq. (7.32) demand
Combining Eqs. (7.39) and (7.40) yields
\( \hat{u}(N) \) and \( \hat{\varphi}(N) \) can be determined from the first two boundary conditions in Eq. (7.32):
where \( {N_0}({r_0},{\theta_0},0),N(r_1,\theta_1, 0)\in S \) and the integral is over all points on \( S \). Equation (7.42) yields
Equation (7.43) can be applied for a crack with any shape.
7.2.3 The Solution of a Circular Penny-Shaped Crack
For a circular penny-shaped crack of diameter \( 2a \), the solution of Eq. (7.43) is
Substituting Eq. (7.44) into (7.37) yields
The kernel function \( K \) in Eq. (7.45) is
Using the relation \( {{{\partial K}} \left/ {{\partial z}} \right.}=-\left[ {{{{2\pi}} \left/ {{\rho (M,{N_0})}} \right.}} \right]\arctan \left[ {{h \left/ {{\rho (M,{N_0})}} \right.}} \right] \) (Fabrikant 1989), the derivatives of \( G \) and \( H \) in Eq. (7.45) are
So for arbitrary polynomial distributed loadings \( p \) and \( q \), all the generalized stresses can be expressed by elementary functions.
7.2.4 A Circular Penny-Shaped Crack Subjected to Generalized Concentrate Loading
Assume the penny-shaped crack is subjected to a pair of normal concentrated generalized loading \( \left( {-P,Q} \right) \) at point \( \left( {{r_0},{\theta_0},{0^{\pm}}} \right),{r_0}< a \) (Fig. 7.1). By using the general solution in the above section, after some manipulation, the generalized displacements and stresses can be obtained as
where
The generalized stress intensity factors are
For the distributed loadings \( \left( {-p,q} \right) \), the generalized stress intensity factors are
For homogeneous distributed loadings \( \left( {-{p_0},{D_{30}}} \right) \), the generalized stress intensity factors are
From Eq. (7.51), it is seen that the stress intensity factor is determined independently by the mechanical loading and the electric displacement intensity factor is determined independently by the electric loading. Equation (7.51) obviously can be used to the case where homogeneous generalized stresses \( {\sigma_{33}} \) and \( -{D_3} \) are applied at infinity.
There are many papers to discuss the penny-shaped crack, such as Huang (1997) and Wang (1992).
7.2.5 A Conducting Penny-Shaped Crack
Chen and Lim (2005) discussed the conducting penny-shaped crack. For a conducting crack, it adopts \( \varphi =0 \) instead of \( {D_3}=q({x_1},{x_2}) \) in Eq. (7.32). The boundary conditions are
According to \( \varphi =0, - \infty <({x_1},{x_2})<\infty \) in Eq. (7.52), it is concluded that \( H=0 \) in Eqs. (7.36) and (7.37). Equation (7.42) becomes
Equation (7.45) is reduced to
where \( K\left( {r,\theta, z;{r_0},{\theta_0}} \right) \) is still expressed by Eq. (7.46). The generalized displacements and stresses for a circular penny-shaped crack subjected to a concentrated force \( -P \) are
The functions in Eq. (7.55) are still expressed by Eq. (7.48b). The generalized stress intensity factors are
where \( \beta =-\sum\nolimits_{i=1}^3 {{{{{\gamma_{i1}}{c_i}}} \left/ {{{\eta_1}}} \right.}} \). Above results are assumed that Eq. (7.27) or Eq. (7.24) has three different roots, \( {s_i},i=1,2,3 \). Chen and Lim (2005) also discussed other cases.
7.2.6 Solve an Impermeable Penny-Shaped Crack by Hankel Transform
In order to discuss the results of the Vickers indentation cracking of experiments, Jiang and Sun (2001) gave a solution of an impermeable penny-shaped crack with boundary conditions as shown in Eq. (7.32). They discussed an axisymmetric piezoelectric body under axisymmetric loading by using the Hankel transform method in cylindrical coordinates. In cylindrical coordinates, the constitutive equations are
The generalized geometric equations are
where \( u,w \) are the displacements along \( r,z \) directions, respectively.
The generalized equilibrium equations are
or
Equation (7.32) becomes
Introduce the Hankel transform pair with \( {J_k} \); \( {J_k} \) is the Bessel’s function of the first kind of order \( k \):
Applying the Hankel transform, Eq. (7,62) to Eq. (7,60) with \( k=1 \) for \( u \) and \( k=0 \) for \( w \) and \( \varphi \) yields
where a prime indicates the derivative with respect to z. The solutions are assumed in the forms
Substituting Eq. (7,64) into Eq. (7,63) yields
In order to have nontrivial solutions for \( \hat{u},\;\hat{w} \) and \( \hat{\varphi} \), it should obtain the characteristic equation
where \( {B_1},{B_2} \) and \( {B_3} \) are coefficients constituted of material constants. Since coefficients in Eq. (7.66) are real, in general it has six roots. Because the upper half space (\( z\geq 0 \)) is discussed, without loss of generality, we take eigenvalue \( {\eta_1} \) is a real positive number and \( {\eta_2},{\eta_3} \) are, in general, a pair of complex conjugates with positive real part. One form of the corresponding eigenvectors is
Let \( {{\hat{w}}_i}(\xi )=({1 \left/ {{{\eta_i}\xi}} \right.}){f_i}(\xi ) \). Through the inverse transform, the general solutions are
Using relations \( {{\left[ {{J_1}(\xi r)} \right]}_{,r}}=\xi \left[ {{J_0}(\xi r)-({1 \left/ {2} \right.}){J_1}(\xi r)} \right],\;{{\left[ {{J_0}(\xi r)} \right]}_{,r}}=-\xi {J_1}(\xi r) \), the generalized stresses are
For discussed penny-shaped crack (Fig. 7.1), since \( {\tau_{rz}}\left( {r,z} \right)0=0 \) due to symmetry, Eq. (7.69) yields
Eliminating \( {f_3} \) from Eq. (7.68), (7.69), and (7.70) and then substituting the results into Eq. (7.61), two pairs of integral equations are obtained:
where
The solution of the dual integral equations in Eq. (7.71) is
When uniform pressure \( {p_0} \) is applied over the area of radius \( r=c< a \) and uniform charge \( {q_0} \) applied over the area of \( r=b< a \), we get
The solution for a point force \( -{P_0} \) and point charge \( {Q_0} \) acted at the crack center \( (r=0) \) can be obtained by using the limiting procedure \( \mathop{\lim}\limits_{{c\to 0}}\pi {c^2}{p_0}={P_0} \) and \( \mathop{\lim}\limits_{{b\to 0}}\pi {b^2}{q_0}={Q_0} \).
The generalized stress intensity factors for uniform loadings \( {p_0} \) and \( {q_0} \) applied over \( r=a \) are the same as shown in Eq. (7.51). For the point loadings are
A modified stress intensity factor \( K_{\mathrm{I}}^{*} \) for a semicircular surface crack in a homogeneous isotropic elastic material given by Cherepanov (1979) is
In the Vicker’s indentation cracking of experiments, a semicircular surface crack is located in an isotropic plane \( ({x_1},{x_2}) \), the stress intensity factor can approximately adopt Eq. (7.76), but it should take \( 2{P_0} \) instead of \( {P_0} \) in \( {K_{\mathrm{I}}} \).
There were many literatures discussing the penny-shaped crack, such as Ueda (2007) which discussed a penny-shaped crack in a functionally graded piezoelectric strip under thermal loading.
7.3 Ellipsoidal Inclusion and Inhomogeneity
7.3.1 Basic Concept of Electroelastic Green Functions
A subdomain with prescribed eigenstrain in a matrix is usually called the inclusion, and material properties of the inclusion are the same with the matrix. A subdomain with different material properties from the matrix is usually called the inhomogeneity. The eigenstrain may be introduced by many physical phenomena, such as phase transformation strains, thermal strains, and plastic strains. The eigenstrain will produce self-equilibrium stresses in a constrained matrix. Eshelby’s theory (1957) of the inclusion and inhomogeneity problems is important in the analyses of piezoelectric composite materials.
For convenience the notations given in Eq. (3.8) in Sect. 3.1 are adopted. A subscript in upper case takes the value 1,2,3, and 4 and a subscript in lower case takes the value 1,2, and 3. Figure 7.2 shows an ellipsoid inclusion occupied region \( \varOmega^{in} \) in a 3D space \( {R^3} \). In \( \varOmega^{in} \) there is generalized eigenstrain \( {{\boldsymbol{Z}}^{*}}\left( {{Z_{ij}}=\varepsilon_{ij}^{*},{Z_{4j}}=-E_j^{*};\;i,j=1,2,3;\;\mathrm{without}\;{Z_{44}}} \right) \). The constitutive equations with eigenstrain are
An elliptic inclusion
where \( {Z_{Kl}}={U_{K,l}},\boldsymbol{U}={{\left[ {{u_k},\varphi} \right]}^{\mathrm{T}}} \). The generalized equilibrium equations are
where \( {f_1},{f_2},{f_3} \) are components of the body force and \( {f_4}=-{\rho_{\mathrm{e}}} \) is the body electric charge density. From Eq. (7.78), it is seen that the role of \( {E_{iJKl}}Z_{Kl,i}^{*}(\boldsymbol{x}) \) is analogous to the body force.
The Green function \( {G_{KR,il}}(\boldsymbol{x}-{\boldsymbol{x}}^{\prime}) \) in an infinite body is defined as
where \( {\delta_{JR}} \) is the generalized Kronecker delta and \( \delta (\boldsymbol{x}-{\boldsymbol{x}}^{\prime}) \) is the three-dimensional Dirac delta function. Except \( \boldsymbol{x}={\boldsymbol{x}}^{\prime},\delta (\boldsymbol{x}-{\boldsymbol{x}}^{\prime})=0 \), and for a regular function \( f(\boldsymbol{x}) \),
The Green function defined in Eq. (7.79) satisfies the generalized equilibrium equation. \( {G_{IJ}}(\boldsymbol{x}-{\boldsymbol{x}}^{\prime}) \) and its derivative approach zero when \( \left| {\boldsymbol{x}-{\boldsymbol{x}}^{\prime}} \right|\to \infty \).
The electroelastic Green function \( {G_{IJ}}(\boldsymbol{x}-{\boldsymbol{x}}^{\prime}) \) is extensively applied to study the inclusion and inhomogeneity problems in piezoelectric materials. \( {G_{ij}}(\boldsymbol{x}-{\boldsymbol{x}}^{\prime}) \) denotes the elastic displacement at \( \boldsymbol{x} \) in the \( {x_i} \) direction due to a unit point force at \( {\boldsymbol{x}}^{\prime} \) in the \( {x_j} \) direction; \( {G_{i4}}(\boldsymbol{x}-{\boldsymbol{x}}^{\prime}) \) denotes the elastic displacement at \( \boldsymbol{x} \) in the \( {x_i} \) direction due to a unit point charge at \( {\boldsymbol{x}}^{\prime} \); \( {G_{4j}}(\boldsymbol{x}-{\boldsymbol{x}}^{\prime}) \) denotes the electric potential at \( \boldsymbol{x} \) due to a unit point force at \( {\boldsymbol{x}}^{\prime} \) in the \( {x_j} \) direction; and \( {G_{44}}(\boldsymbol{x}-{\boldsymbol{x}}^{\prime}) \) denotes the electric potential at \( x \) due to a unit point charge at \( {\boldsymbol{x}}^{\prime} \).
7.3.2 Fourier Transform Method
Assume that the generalized eigenstrains and displacements can be expressed as (Mura 1987)
Analogous to elasticity, substituting Eq. (7.81) into (7.78) and neglecting the body force yield
where \( {\varpi_{JMN}} \) is the permutation tensor and \( {N_{MJ}} \) is the algebraic complement of \( {\varPi_{MJ}} \) in the matrix \( \boldsymbol{\varPi} \). For the general case, the Fourier integral transform is used (Mura 1987; Wang 1992):
Analogous to elasticity, substituting Eq. (7.83) into (7.78) and using (7.82) yield
where \( {G_{MJ,i}}(\boldsymbol{x}-{\boldsymbol{x}}^{\prime})={{{\partial {G_{MJ}}(\boldsymbol{x}-{\boldsymbol{x}}^{\prime})}} \left/ {{\partial {x_i}}} \right.}=-{{{\partial {G_{MJ}}(\boldsymbol{x}-{\boldsymbol{x}}^{\prime})}} \left/ {{\partial {{{x^{\prime}_i}}}}} \right.} \). If \( {G_{MJ}}(\boldsymbol{x}-{\boldsymbol{x}}^{\prime}) \) is a Green function in a finite region, then
7.3.3 Radon Transform Method
The Radon integral transform plays a fundamental role in the tomography, such as CT scanning in medicine. At first the basic concepts are introduced as follows (Deans 1983):
An arbitrary function \( f({x_1},{x_2}) \) is defined on some domain \( \varOmega \) of a 2D plane \( {{\boldsymbol{R}}^2} \). Let \( L \) be any line in the plane, then the mapping defined by the projection or line integral of \( f \) along all possible lines \( L \) is the 2D Radon transform of \( f \), i.e.,
where \( \omega =\boldsymbol{\xi} \cdot \boldsymbol{x}={x_1}\cos \phi +{x_2}\sin \phi \) is the perpendicular distance from the origin to \( L \), \( \boldsymbol{\xi} =\left( {\cos \phi, \sin \phi} \right) \) is a unit vector along \( \boldsymbol{\omega} \) which defines the orientation of \( L \), \( \boldsymbol{\eta} =(-\sin \phi, \cos \phi )\bot \boldsymbol{\xi} \) is along \( L \), \( \tau \) is determined by \( \boldsymbol{x}=\omega \boldsymbol{\xi} +\tau \boldsymbol{\eta} \), and \( \phi \) is the angle between \( \boldsymbol{\xi} \) and positive \( {x_1} \)-axis (Fig. 7.3). The last equation in Eq. (7.86) is easy extended to three- and higher-dimensional space. If \( \bar{f}(\omega, \boldsymbol{\xi} ) \) is known for all \( \omega \) and \( \phi \), then \( \bar{f}(\omega, \boldsymbol{\xi} ) \) is a 2D Radon transform of \( f({x_1},{x_2}) \). In an \( n \)-dimensional space, \( L \) represents \( (n-1) \)-dimensional hyperplane. Especially for a 3D space, \( L \) represents a plane. For \( n \)-dimensional space, the inversion Radon transform is
A sketch of Radon transform
where \( {{\boldsymbol{\varDelta}}_x} \) is the Laplacian operator. Especially for 3D space, Eq. (7.87) is reduced to
Deeg (1980), Dunn and Taya (1993), and Dunn (1994) pointed out that the Radon transform can also be used to the electroelastic Green function \( {G_{IJ}}(\boldsymbol{x}-{\boldsymbol{x}}^{\prime}) \), i.e.,
where \( \boldsymbol{\xi}, \omega \) are variables in the transform space and the integral domain is a 2D plane \( \boldsymbol{\xi} \cdot \boldsymbol{x}=\omega \). In the inverse Radon transform, the integral domain is the surface of a unit sphere \( \left| \boldsymbol{\xi} \right|=1 \). Using the Radon transform to Eq. (7.79), after some manipulation, yields
where \( {K_{JM}}(\boldsymbol{\xi} )={E_{iJMn}}{\xi_i}{\xi_n} \). Using the inversion Radon transform from Eq. (7.90) yields
where \( {K_{JM}}(\boldsymbol{\xi} )K_{MR}^{-1}(\boldsymbol{\xi} )={\delta_{JR}} \) and \( \boldsymbol{t} \) is the unit vector along \( \boldsymbol{x}-{\boldsymbol{x}}^{\prime} \). Using the property of the Dirac delta, in an orthogonal coordinate system \( \boldsymbol{t}\text{-}\boldsymbol{m}\text{-}\boldsymbol{n} \), Deeg (1980) reduced the integral in Eq. (7.91) to the following contour integral:
where \( C \) is the contour produced by \( \left| \boldsymbol{\xi} \right|=1 \) in \( \boldsymbol{m}\text{-}\boldsymbol{n} \) plane normal to \( \boldsymbol{x}-{\boldsymbol{x}}^{\prime} \). Compared to Fourier transform method, Eq. (7.92) is a simpler effective method to seek the Green function. The second partial derivatives of the electroelastic Green function is
7.3.4 A Single Ellipsoidal Inclusion with Uniform Eigenstrains
Let the coordinates coincide with the principle axes of the material. For an ellipsoidal inclusion with uniform eigenstrains, the Eshelby’s method is used. At first the inclusion is cut off from the matrix, so the inclusion and matrix are all free. The eigenstrains of the inclusion in the free state is denoted by \( {{\boldsymbol{Z}}^{*}} \). The following generalized stresses are applied on the boundary of the inclusion:
Then put the inclusion subjected to above generalized stresses into the matrix. After this procedure, the original problem is transformed to a homogeneous material with a force \( -{T_J}={E_{iJMl}}Z_{Ml}^{*} \) acting on the counter which originally is the boundary of the inclusion. If \( {Z^{*}} \) is uniform, Eq. (7.94) yields (Deeg 1980)
Using Eq. (7.93) from Eq. (7.95) yields (Deeg 1980; Dunn and Taya 1993)
Analogous to the elastic inclusion problem, the generalized strains inside the ellipsoid induced by the uniform eigenstrains are also uniform and
where \( {S_{MnKl}} \) is called the “electroelastic Eshelby tensor,” but it is not a tensor, i.e., it does not obey the tensor transform rule under a coordinate transformation. The key point to solve \( {S_{MnKl}} \) is to calculate \( {I_{MJin}}(\boldsymbol{\xi} ) \). \( {I_{MJin}}(\boldsymbol{\xi} ) \) can be transformed to the following form (Mikata 2000):
For transversely isotropic piezoelectric materials, Mikata (2000) pointed out that in \( {I_{MJin}}\left( \boldsymbol{\xi} \right) \) only \( {I_{1212}},{I_{1313}},{I_{1314}},{I_{2323}},{I_{2324}} \) and \( {I_{11MJ}},{I_{22MJ}},{I_{33MJ}},MJ=11,22,33,44,34 \) are not zero, and the Eshelby tensor only has 36 components:
For an elliptic cylindrical inclusion along the \( {x_3} \)-axis, the Eshelby tensor with 22 components is
For a penny-shaped crack perpendicular to \( {x_3} \)-axis, the Eshelby tensor with 18 components is
7.3.5 Ellipsoid Inhomogeneity
Analogous to elastic inhomogeneity problem, the electroelastic inhomogeneity problem can be handled by the equivalent inclusion method. Let \( E_{iJKl}^{\mathrm{M}} \) and \( E_{iJKl}^{\mathrm{in}} \) denote the material constants in the matrix and inhomogeneity, respectively. The generalized stress \( \Sigma_{Mn}^0 \) is applied at infinity. Obviously for a homogeneous material, the generalized stress in material is also \( \Sigma_{Mn}^0 \) and the corresponding strain is \( Z_{Mn}^0={{\left( {E_{iJMn}^M} \right)}^{-1}}\Sigma_{Mn}^0 \). Assume the strain due to the inhomogeneity is \( {Z_{Mn}} \) and then the stress in the matrix and inhomogeneity are, respectively,
The key point of the equivalent inclusion method is that let the inhomogeneity possess the same material constants with the matrix, but the artificial eigenstrain \( Z_{Mn}^{*} \) is added. Then let
Using \( {Z_{Mn}}(\boldsymbol{x})={S_{MnKl}}Z_{Kl}^{*} \) in the inhomogeneity, so Eq. (7.101) yields
Solving \( Z_{Kl}^{*} \), the problem can be solved.
As an example an ellipsoidal piezoelectric sensor embedded in an elastic material is discussed (Fan and Qin 1995). The constitutive equations of the sensor (as an elliptic inclusion) and matrix are, respectively,
Comparing Eqs. (7.103) and (7.104), one can consider the terms \( e_{kij}^{\mathrm{in}}{E_k},e_{ikl}^{\mathrm{in}}{\varepsilon_{kl}} \) in Eq. (7.103) produced by some kind of eigenstrains. When the matrix is subjected to uniform generalized stresses \( \left( {\sigma_{ij}^0,E_i^0} \right) \) at infinity, the generalized stresses in the sensor are changed to
By using the equivalent inclusion method, the original inhomogeneity problem subjected to uniform generalized stresses at infinity can be decoupled into two equivalent inclusion problems:
1. The elastic equivalent inclusion problem. Equation (7.105) can also be written as
where \( \varepsilon_{kl}^{*} \) is the virtual eigenstrain. Using the Eshelby inclusion theory yields
2. The dielectric equivalent inclusion problem. Equation (7.106) can also be written as
where \( E_k^{*} \) is the virtual eigen electric field. Using the Eshelby inclusion theory yields
where \( {s_{kl}}={S_{4k4l}} \). Comparing Eqs. (7.105) and (7.107), it is concluded that
Comparing Eqs. (7.106) and (7.110), it is concluded that
Solving \( \varepsilon_{kl}^{**} \) and \( E_k^{**} \), the stress and electric fields are obtained in the sensor, i.e.,
The stress \( \sigma_{ij}^{\mathrm{out}} \) and electric fields \( {E^{\mathrm{out}}} \) in the matrix can be solved as follows:
The displacement \( \boldsymbol{u} \) and the surface traction \( \boldsymbol{\sigma} \cdot \boldsymbol{n}=\mathbf{0} \) across the interface of inclusion and matrix must be continuous, and jump of the displacement gradient \( \mathbf{\nabla}\otimes \boldsymbol{u} \) must be normal to the interface. The continuous conditions of the electric field and electric displacement \( \boldsymbol{E}\times \boldsymbol{n} \) and \( \boldsymbol{D}\cdot \boldsymbol{n} \) across the interface demand that the jump of \( \boldsymbol{E} \) must be normal to the interface. So
where \( \boldsymbol{\lambda}, \eta \) are proportional constants. Substituting the constitutive equations into Eq. (7.115), it is obtained:
Therefore, the stress \( \sigma_{ij}^{\mathrm{out}} \) and electric fields \( {E^{\mathrm{out}}} \) in the matrix are
7.4 Some Simpler Practical Problems
7.4.1 Extension of a Rod
Figure 7.4 shows a transversely isotropic piezoelectric long cylindrical rod with polarized \( {x_3} \)-axis. The two silver-coated end faces are used as electrodes and subjected to uniform normal traction \( p \).
An axial poled rod
Using the first kind of constitutive equation, the solutions of this problem are as follows:
-
1.
For shorted electrodes,
$$ \begin{array}{lll} {\sigma_{33}}=p,\quad \mathrm{all}\;\mathrm{other}\;{\sigma_{ij}}=0;\quad {\varepsilon_{33}}={s_{33}}p,\quad {\varepsilon_{11}}={\varepsilon_{22}}={s_{13}}p; \\ {E_3}={E_1}={E_2}=0,\quad {D_3}={d_{33}}p,\quad {D_1}={D_2}=0;\quad {{\mathfrak A}_s}={{{{\sigma_{33}}{\varepsilon_{33}}}} \left/ {2} \right.}={{{s_{33}}{p^2}}} \left/ {2} \right.\end{array} $$(7.118) -
2.
For open electrodes,
$$ \begin{array}{lll}{\sigma_{33}}&= p,\quad \mathrm{all}\;\mathrm{other}\;{\sigma_{ij}}=0;\quad {\varepsilon_{33}}={s_{33}}\left( {1-{{{d_{33}^2}} \left/ {{{\epsilon_{33}}{s_{33}}}} \right.}} \right)p,\quad {\varepsilon_{11}}={\varepsilon_{22}}={s_{13}}p+{d_{31}}{E_3}; \\{D_3}&= {D_1}={D_2}=0,\;\;{E_3}=-\left( {{{{{d_{33}}}} \left/ {{{\epsilon_{33}}}} \right.}} \right)p,\;\;{E_1}={E_2}=0;\;\;{U_o}=\left( {{{{{s_{33}}}} \left/ {2} \right.}} \right)\left( {1-{{{d_{33}^2}} \left/ {{{\epsilon_{33}}{s_{33}}}} \right.}} \right) p^2\end{array} $$(7.119)The rod appears to be stiffer for the open electrodes than sorted electrodes due to \( {{{d_{33}^2}} \left/ {{{\epsilon_{33}}{s_{33}}}} \right.}> 0 \).
-
3.
The longitudinal electromechanical coupling factor \( {k_{33}} \):
$$ k_{33}^2={{{({U_{\mathrm{s}}}-{U_{\mathrm{o}}})}} \left/ {{{U_{\mathrm{s}}}}} \right.}={{{d_{33}^2}} \left/ {{{\epsilon_{33}}{s_{33}}}} \right.} $$(7.120)
7.4.2 Torsion of a Piezoelectric Circular Cylinder
Figure 7.5 shows a transversely isotropic piezoelectric circular cylinder of length \( L \), inner radius \( a \), and outer radius \( b \) with polarized \( \theta \)-axis. The two silver-coated end faces are used as electrodes and subjected to a torque \( M \) and charge \( {Q_{\mathrm{e}}} \). Using the second kind of constitutive equation, the general solution of this problem is
A circular cylinder in torsion
-
1.
$$ \mathrm{Shorted}\ \mathrm{electrodes}:\quad B=0,\quad A={M \left/ {{{C_{44}}}} \right.}{I_{\mathrm{p}}} $$(7.122)
-
2.
$$ \begin{array}{lll}\mathrm{Open}\ \mathrm{electrodes}:\quad {Q_{\mathrm{e}}}=0,\ \quad \\ \quad A=M\left[ {{C_{44}} {{I_{\mathrm{p}}}+\frac{{e_{15}^2}}{{{\epsilon_{11}}}}{{{\left( {\frac{{2\pi}}{3}} \right)}}^2}\frac{{\left( {{b^3}-{a^3}} \right)}}{{\pi \left( {{b^2}-{a^2}} \right)}}}} \right]^{-1} \end{array}$$(7.123)
7.4.3 A Circular Hole Under Longitudinal Shear in an Infinite Piezoelectric Plate
Figure 7.6 shows a circular cylindrical hole of radius \( R \) in an unbounded transversely isotropic piezoelectric material with polarized \( {x_3} \)-axis under a uniform longitudinal shear stress \( {\sigma_{23}}={\tau_0} \) at \( {x_2}=\pm \infty \). The hole surface is a grounded electrode. The governing equations and boundary conditions for an electrically open case are
An infinite plate with a circular cylindrical hole under longitudinal shear
For the electrically open case, the solution is
where \( {k^2}={{{e_{15}^2}} \left/ {{{C_{44}}{\epsilon_{11}}}} \right.} \). For the electrically shorted case, the solution can also be obtained.
7.4.4 Thickness-Shear Vibration of a Quartz Plate
Figure 7.7 shows the sketch of a widely used piezoelectric resonator manufactured by rotated Y-cut quartz plate. Surfaces at \( {x_2}=\pm h \) are traction-free and electroded, with a driving voltage \( {V_0}{{\mathrm{e}}^{{\mathrm{i}\omega t}}} \). Let \( {u_3}=0 \) and \( {{{\partial {u_2}}} \left/ {{\partial {x_2}}} \right.}\approx 0 \) due to the plate is thin enough. So the displacement and potential fields can be assumed in the following forms:
An electrode quartz plate
Under above assumptions, the constitutive equations are
The stresses \( {\sigma_5}={C_{56}}{u_{1,2}}+{e_{25}}{\varphi_{,2}},\;{D_3}={e_{36}}{u_{1,2}}-{\epsilon_{23}}{\varphi_{,2}} \) are omitted because they are not used. The generalized momentum equation and boundary condition are
The general solutions are
From the boundary conditions, we can get two group equations:
1. Free vibration \( {V_0}=0 \) , symmetric modes From Eq. (7.131) it is obtained
where \( {\omega_n} \) is the nth order resonance frequency. In the same time \( A_2\ne 0,\ {A_1}=B=0 \). The corresponding symmetric modes are
2. Free vibration \( {V_0}=0 \) , antisymmetric modes In this case \( {A_1}\ne 0, B \ne 0, A_2=0 \). Nontrivial solutions may exist in Eq. (7.130) if
From Eq. (7.130), it is obtained \( {B_{\nu}}=-\left( {{{{C_{66}^{*}}} \left/ {{{e_{26}}}} \right.}} \right){A_1}{\lambda_{\nu}}\cos {\lambda_{\nu}}h \). In this case \( \sin \lambda h\ne 0 \), so \( {A_2}=0 \). The corresponding antisymmetric modes are
3. Forced vibration From Eq. (7.131) we have \( {A_2}=0 \). From Eq. (7.130) we get
Yang (2005) gave also some other interest problems except the above examples in this section.
7.5 Laminated Piezoelectric Plates
7.5.1 Basic Concepts and Governing Equations
In the earlier work, the piezoelectric actuator structure constituted of an elastic substrate (beam or bar), electroded piezoelectric elements, and finite-thickness bonding layers. For a pair actuators fixed on the upper and lower surfaces, if the same voltage is applied to both actuators, it results in pure extension, and if the opposite voltage is applied to both actuators, it results in bending (Fig. 7.8). In the present time, the “intelligent structure” may be a laminated piezoelectric beam, plate, shell and distributed actuator, sensor, and processor networks. In engineering the classical beam, plate and shell theory are commonly used, and sometimes will use the higher-order theories.
A sketch of a simple intelligent beam
Consider an \( N \)-layer laminated piezoelectric plate of dimensions \( {L_1} \) and \( {L_2} \) in \( {x_1} \) and \( {x_2} \) directions and total thickness \( H \) in \( {x_3} \)-direction, and the \( i \)th layer has thickness \( {h_i} \). The plate is polarized along \( {x_3} \)-axis. Except the global coordinate system, a local coordinate system \( \left( {{x_1},{x_2},{z_i}} \right) \) in the middle plane of \( i \)th layer is also adopted (Fig. 7.9). Layer 1 is the bottom layer and the layer \( N \) is the top layer. At each interface with perfect bonding between layers, continuity conditions of generalize displacements and tractions must be satisfied. For the \( i \)th interface between \( i \)th and \( (i+1) \)th layers in the local coordinate system, the continuity conditions are
A multiply laminated piezoelectric plate
It is also noted that sometimes a bond-line may be simulated by a layer of small thickness. For each interface, there are eight continuity conditions, so for a laminate plate with \( N \) layers, there are \( 8(N-1) \) continuity conditions; on the lower surface \( (z=-{{{{h_1}}} \left/ {2} \right.}) \) of the first layer and on the upper surface \( (z={{{{h_N}}} \left/ {2} \right.}) \) of the \( N \)th layer, there are four boundary conditions, respectively. Therefore, there are total \( 8N \) boundary conditions to determine \( 8N \) unknowns. For an orthotropic material layer, the constitutive equation is shown in Eq. (3.69). Analogous to Eq. (7.3), the motion equations in terms of generalized displacements are
7.5.2 Bending in Simply Supported Orthotropic Laminated Rectangular Plate
It is assumed that the bottom and lateral surfaces are free, and known normal traction and potential are imposed on the top surface:
For the simply supported orthotropic laminated rectangular plate, the solution in each layer is assumed (Heyliger 1997):
where \( {u_{i0}},{\varphi_0},s \) are undetermined constants and the superscript (i) is omitted. Substituting Eq. (7.140) into (7.138) yields
Equation (7.141) will have nontrivial solution if \( \left| \boldsymbol{\varLambda} \right|=0 \). From which we can obtained eight eigenvector \( {s_i} \), \( {s_i},i=1-8 \). Corresponding each \( {s_i} \), an eigenvector \( {{\boldsymbol{U}}_{0j}} \) (\( {u_{10i}},{u_{20i}},{u_{30i}},{\varphi_{0i}} \)) with one unknown \( {u_{10j}} \) is obtained. As shown in Sect. 7.5.1, the problem can be solved uniquely.
7.5.3 Free Vibration of Laminates in Cylindrical Bending
Let \( {L_1}\to \infty \) in \( {x_1} \) direction and all variables be independent with \( {x_1} \). Assuming \( {u_1}=0 \), from Eq. (7.138), the generalized motion equations are
The continuity conditions on interface are shown in Eq. (7.137). For the free vibration, the mechanical boundary conditions on the top and bottom surfaces are
The electrical boundary conditions have two different kinds:
For the cylindrical bending vibration in \( ({x_2},{x_3}) \) plane, the boundary conditions on the lateral surfaces are
For each layer, there are six unknowns and six continuity conditions due to \( {u_1}=0 \). There are also total six boundary conditions on the top and bottom surfaces. In order to satisfy Eq. (7.145) automatically, Heyliger and Brooks (1995) took the solution in the following form:
Substituting Eq. (7.146) into (7.142) yields
Setting the determinant of above matrix to zero for a nontrivial solution yields an eigen-equation. From the eigen-equation, we get six eigenvalues \( {s_i},i=1-6 \) for \( s \). Corresponding each \( {s_i} \), an eigenvector \( {{\boldsymbol{U}}_{0j}} \) (\( {u_{10i}},{u_{20i}},{u_{30i}},{\varphi_{0i}} \)) with one unknown \( {u_{10j}} \) can be obtained. As shown in Sect. 7.5.1, the problem can be solved uniquely.
7.5.4 A Mindlin-Type Plate Bending Theory
Consider an orthotropic piezoelectric plate of moderate thickness. Let \( ({x_1},{x_2}) \) be located on the middle surface. The basic assumptions of the Mindlin bending theory are:
-
1.
Straight lines normal to the \( {x_1}-{x_2} \) plane before deformation remain straight with unchanged length after deformation, but not compulsory normal to the mid-surface, i.e.,
$$ \begin{array}{lll}{\varepsilon_{11}}&= u_{1,1}^0+{x_3}{\psi_{1,1}},\quad {\varepsilon_{22}}=u_{2,2}^0+{x_3}{\psi_{2,2}},\quad {\gamma_{23}}=u_{3,2}^0+{\psi_2} \\{\gamma_{13}}&= u_{3,1}^0+{\psi_1},\quad {\gamma_{12}}=u_{1,2}^0+u_{2,1}^0+{x_3}({\psi_{1,2}}+{\psi_{2,1}})\end{array} $$(7.148)where \( {{\boldsymbol{u}}^0} \) is the displacement in the mid-surface and \( {\psi_1} \) and \( {\psi_2} \) are the absolute cross-sectional rotations.
-
2.
Stress \( {\sigma_{33}} \) can be neglected. So the constitutive equation can be written as
$$ \begin{array}{lll}{\sigma_1}&= {\sigma_{11}}={{\bar{C}}_{11}}{u_{1,1}}+{{\bar{C}}_{12}}{u_{2,2}}+{{\bar{e}}_{31}}{\varphi_{,3}},\quad {\sigma_2}={\sigma_{22}}={{\bar{C}}_{12}}{u_{1,1}}+{{\bar{C}}_{22}}{u_{2,2}}+{{\bar{e}}_{32}}{\varphi_{,3}} \\{\sigma_4}&= {\sigma_{23}}={{\bar{C}}_{44}}({u_{2,3}}+{u_{3,2}})+{{\bar{e}}_{24}}{\varphi_{,2}},\quad {\sigma_5}={\sigma_{31}}={{\bar{C}}_{55}}({u_{1,3}}+{u_{3,1}})+{{\bar{e}}_{15}}{\varphi_{,1}} \\{\sigma_6}&= {\sigma_{12}}={{\bar{C}}_{66}}({u_{2,1}}+{u_{1,2}}),\quad {D_1}={{\bar{e}}_{15}}({u_{1,3}}+{u_{3,1}})-{{\bar{\epsilon}}_{11}}{\varphi_{,1}} \\{D_2}&= {{\bar{e}}_{24}}({u_{2,3}}+{u_{3,2}})-{{\bar{\epsilon}}_{22}}{\varphi_{,1}},\quad {D_3}={{\bar{e}}_{31}}{u_{1,1}}+{{\bar{e}}_{32}}{u_{2,2}}-{{\bar{\epsilon}}_{33}}\varphi_{,3}\end{array} $$(7.149)where \( u_{3,3} \) has been eliminated by using \( \sigma_3 =0\), and
$$ {{\bar{C}}_{ij}}={C_{ij}}-{{{{C_{i3}}{C_{j3}}}} \left/ {{{C_{33}}}} \right.},\quad {{\bar{e}}_{ij}}={e_{ij}}-{{{{e_{33}}{C_{ji}}}} \left/ {{{C_{33}}}} \right.},\quad {{\bar{\epsilon}}_{ij}}={{\bar{\epsilon}}_{ij}}+{{{e_{33}^2{\delta_{i3}}{\delta_{j3}}}} \left/ {{{C_{33}}}} \right.} $$(7.150)are the reduced material coefficients.
Wang and Yang (2000) reviewed the higher-order theories of the piezoelectric plates. The equivalent single-layer models for the multiply layer plate (Krommer and Irschik 2000) are adopted here. Substituting Eq. (7.148) into (7.149) and integrating through the thickness yield the constitutive equation in the stress resultants and moments:
where \( {N_1},{N_2},{N_{12}} =N_6\) are the membrane forces, \( {M_1},{M_2},{M_{12}}=M_6 \) are the bending moments, and \( {q_1},{q_2} \) are the shear forces per unit length. The generalized stiffness in Eq. (7.151) are
where \( {\varPhi_{i}},{\varPhi_{j}} \) are shear factors which are determined by the shear stress distribution on the cross section and \( {N_{1e}},{M_{1e}},{N_{2e}},{M_{2e}},\;{N_{12e}},{M_{12e}},{q_{1e}},{q_{2e}} \) are introduced by piezoelectric effect. The electric variables will be studied layer by layer, and for each layer, there are two-type boundary conditions:
-
1.
Electrically open Given the electric charge density \( {\sigma^{(i)}} \) on the upper and lower surfaces of the layer. Usually \( D_1^{(i)} \) and \( D_2^{(i)} \) are neglected and \( D_3^{(i)} \) is reserved. So \( D_3^{(i)} \) is constant along the thickness direction due to Gauss equation \( D_{3,3}^{(i)}=0 \):
$$ D_j^{(i)}{n_j}=D_3^{(i)}=-{\sigma^{(i)}},\quad E_3^{(i)}={{{-{\sigma^{(i)}}}} \left/ {{\bar{\epsilon}_{33}^{(i)}}} \right.} $$(7.153) -
2.
Electrically shorted Given the potential \( {V^{(i)}} \) on the upper and lower surfaces of the layer. For convenience, assume the variation of the potential is linear along the thickness direction, so
$$ E_3^{(i)}={{{{V^{(i)}}}} \left/ {{{h_i}}} \right.},\quad D_3^{(i)}=\bar{\epsilon}_{33}^{(i)}{{{{V^{(i)}}}} \left/ {{{h_i}}} \right.};\quad E_1^{(i)}=E_2^{(i)}=0 $$(7.154)
In Mindlin theory, the motion equations are
where \( p \) is the transverse loading and
Equations (7.148), (7.149), (7.150), (7.151), (7.152), (7.153), (7.154), (7.155), and (7.156) are the complete governing equations.
For a symmetrically laminated, transversely isotropic and simply supported plate, bending and extension are decouple, and the following relations are held:
Eliminating the cross-sectional rotations \( \psi \) from Eqs. (7.151) and (7.155) for the bending vibration, a fourth-order partial differential equation for \( u_3^0 \) is obtained (Krommer and Irschik, 2000):
where \( {\nabla^2}u_3^0=u_{3,11}^0+u_{3,22}^0 \) and \( {\mathop{u_{3}}\limits^{\cdots\ 0}}={{{{\partial^4}u_3^0}} \left/ {{\partial {t^4}}} \right.} \). When the external loadings are \( p={p_0}{{\mathrm{e}}^{{\mathrm{i}\omega t}}} \) and \( {M_{1e}}={M_{10e}}{{\mathrm{e}}^{{\mathrm{i}\omega t}}} \), the frequency equation of the bending vibration is
where the common factor \( {{\mathrm{e}}^{{\mathrm{i}\omega t}}} \) is omitted and \( u_3^0=U_3^0{{\mathrm{e}}^{{\mathrm{i}\omega t}}} \).
7.5.5 Third-Order Shear Deformation Theory of Laminate Plate
Consider the linear piezoelectric material, so the Maxwell stress and the environment need not be considered. The variational principle \( \delta \varPi =0 \) in Eq. (2.7) is becomes
where the body force and body electric charge are neglected. For the orthotropic material of moderate thickness, Mitchell and Reddy (1995) adopted the displacements of the equivalent single-layer plate as
where \( \left( {u_1^0,u_2^0,u_3^0} \right) \) are the displacements of a point on the midplane and \( ({\psi_1},{\psi_2}) \) are the rotations of a transverse normal at \( {x_3}=0 \) on the midplane about the \( {x_2} \) and \( -{x_1} \) axes, respectively. \( h \) is the total thickness of the plate. The strains corresponding to Eq. (7.161) are
From Eqs. (7.161) and (7.162), it is known that for \( {\eta_1}({x_3}),{\eta_2}({x_3}) \) given in Eq. (7.161), the transverse shear strains \( {\gamma_{13}},{\gamma_{23}} \) are zeros on the upper and lower surfaces and vary quadratically through the thickness. It is also without the normal strain. The potential is modeled on a discrete layer approximation as
where \( N \) is the layer number of the laminate plate, \( m \) is the number of interpolation points in a layer, and \( {\varphi^{(k,j)}} \) is the potential at \( j \)th interpolation point of \( k \)th layer. \( {f_k}({x_3}) \) is the Lagrange interpolation function. It is noted that the potential on the upper surface of the \( k-1 \) layer must be equal to that on the lower surface of the \( k \) layer.
The middle plane is denoted by A and its boundary is L. Substituting Eqs. (7.161) and (7.163) into (7.160) and neglecting \( {\sigma_{33}} \) yield the following equations:
The variation of the mechanical energy:
where the Voigt notation has been used and
The variation of the electric energy:
The variation of the kinetic energy or inertial energy:
where
In Eq. (7.167) term \( -\int_L\left[ \left( I_3{\ddot u}_1^0+I_5{\ddot\psi}_1-I_6{\ddot u^0_{3,1}} \right)n_1+\left( I_3{\ddot u}_2^0 + I_5{\ddot \psi}_2 - I_6{\ddot u}^0_{3,2}\right)n_2 \right]\delta u_3^0 \) has been neglected. This term is difficult to explanation.
The variation of the work of the generalized external force is as follows: The mechanical force acted on the equivalent single-layer plate should be distinguished two parts—\( T_i^{*} \) is the equivalent traction on the midplane \( A \), and \( t_i^{*} \) is the equivalent resultant force on the lateral surface \( a \). The electric charge acted on the \( k \)th layer should also be distinguished two parts: \( {\sigma^{(k)*}} \) is the surface electric charge density of the \( k \)th layer, and \( {q^{(k)*}} \) is the surface electric charge on the lateral surface. Neglecting some secondary terms we can get
Substitution of Eqs. (7.163), (7.164), (7.165), (7.166), (7.167), (7.168), and (7.169) into Eq. (7.160) yields
and the natural boundary conditions are
It is also noted that on the boundaries given generalized displacements, we have
Mitchell and Reddy (1995) adopted Hamilton principle, \( \delta {\varPi_{\mathrm{H}}}=\delta {\varPi_{{\mathrm{H}1}}} \) in Eq. (2.32); their results are slightly different. It can be seen that this complex approximate theory is difficult to exactly discuss and give some new simplified postulations are needed.
The generalized forces can be obtained from Eqs. (7.165) and (7.149). Let
where the elastic variables are denoted by an over-bar and variables related to piezoelectric effect are denoted by a superscript “p,” and
where
Equations (7.170), (7.171), (7.172), (7.173a), (7.173b), (7.173c), (7.173d), and (7.174) are the complete governing equations.
7.5.6 Bending Theory of Timoshenko Beam
A narrow plate can be considered as a beam. The Timoshenko theory (Timoshenko and Woinowsky-Krieger 1959) considering the shear deformation of a beam of a moderate thickness can be obtained from the Mindlin theory. Let all variables be independent to \( {x_2} \) in Mindlin theory, Eq. (7.149) is reduced to
where \( Y,G \) are elastic coefficients. Corresponding to Eq. (7.148), the deformation in Timoshenko beam is assumed as
If \( \psi =-u_{3,1}^0 \), then \( {\gamma_{13}}=0 \),Timoshenko beam is reduced to Bernoulli-Euler beam. Corresponding to Eqs. (7.173) and (7.174), we have
Corresponding to Eq. (7.155), we have
where \( {\rho_0},{\rho_1},{\rho_2} \) are shown in Eq. (7.156) and electric displacements and electric fields are shown in Eq. (7.153) or (7.154).
7.5.7 Bending Model of Beams of Crawley
Crawley and De Luis (1987) proposed an extension-bending model to study the simple intelligent beam structure as shown in Fig. 7.8. They assumed that the strain is uniform through the actuator thickness, the beam obeys Bernoulli-Euler rule, and the adhesive layer transfers loads only through shear. The formulas obtained by this model are well for extension, but not bending, especially for a thin plate (Crawley and Aanderson 1989).
7.6 The First-Order Approximate Theory of an Electro-magneto-elastic Thin Plate
7.6.1 Basic Postulations
The nonlinear theory of an electroelastic thin plate is not well established, and different authors proposed different theories. In this section, a first-order approximate theory of the quasi-static electro-magneto-elastic thin plate is recommended for small deformation, when the electromagnetic induction effect can be neglected. Let the origin of the coordinate system axes \( {x_1},{x_2} \) be located on the midplane and \( {x_3} \) be upward normal to the midplane. The plate is bending upward. The role of \( {x_3} \) is not the same as that of \( ({x_1},{x_2}) \), so it will be discussed alone. For the present plate theory, three following basic postulations are assumed (Kuang 2011):
(1) \( {\sigma_{33}}\ll {\sigma_{{\alpha 3}}} \ll {\sigma_{{\alpha \beta}}},(\alpha =1,2) \), so \( {\sigma_{33}} \) is fully neglected and the effect of \( {\sigma_{{\alpha 3}}} \) is considered partly.
(2) The Kirchhoff assumption is adopted, i.e.,
where \( {{\mathbf{u}}^0} \) is the displacement on the middle surface and \( \mathbf{u} \) is the displacement at a certain point in the plate. According to Eq. (7.179), we have \( {u_{{3,\alpha}}}+{u_{{\alpha, 3}}}=0 \), but it is not appropriate for the free boundary and should be modified approximately as shown later.
(3) The electromagnetic field obeys the 3D theory, but in order to consistent with the classical plate theory, the resultant electromagnetic force is reduced to the middle plane \( S \) (\( \mathrm{d}S=\mathrm{d}{x_1}\mathrm{d}{x_2} \)) or to the contour \( L \) of the middle plane. Usually when we solve the electromagnetic field, the electric field due to the direct piezoelectric effect can be approximately neglected compared to the lager applied electric field.
7.6.2 Governing Equations Derived from the First Method
In engineering the piezoelectric plate is surrounded by air, so the plate has only the interface with the air and does not have its own independent boundary. In air the mechanical stresses can be neglected, so only the electromagnetic field should be considered. It is assumed that there is no body force and body electric charge. According to Eqs. (2.19) and (2.21), there are two methods to establish the thin plate theory. Similar to Eq. (2.36), the first alternative form of the PVP, Eq. (2.19), for the static electromagnetic problem is modified as
\( S_{kl}^{\mathrm{air}} \) has the similar expression. The mechanical part related to the plate of the PVP is
Therefore, the thin plate theory can apply the usual elastic plate theory, but \( \boldsymbol{\sigma} \) is replaced by \( \boldsymbol{S} \) and on the interface \( {{\boldsymbol{T}}^{{*\;\mathrm{int}}}} \) is replaced by \( {{\boldsymbol{T}}^{{*\;\mathrm{int}}}}+{{\boldsymbol{\sigma}}^{{\mathrm{M}\;\mathrm{air}}}}\cdot \boldsymbol{n} \). According to the textbook of elastic plate theory, the bending theory of elastic electromagnetic thin plate can be expressed as
Boundary conditions:
The notations in Eqs. (7.182) and (7.183) are
where 2h is the thickness of the plate, \( {q^{*}} \) is the distributed loading on the plate surface, and \( Q_n^{*} \) is the distributed loading on the lateral boundary of the midplane. At first the electromagnetic fields in plate and air are solved under the assumption that the elastic effect can be neglected. After solving the electromagnetic fields, the entire problem is reduced to a linear problem.
7.6.3 Governing Equations Derived from the Second Method
Similar to Eq. (2.36), the second alternative form of the PVP, Eq. (2.21), for the static electromagnetic problem is modified as
According to postulation (3), the electromagnetic field obeys the 3D theory, which has been discussed in Chap. 2. Here only the mechanical part related to the plate of the variational principle is discussed in detail, which is
Applying postulations (1) and (2) and noting \( {n_3}=1,{n_{\alpha}}=0 \) on the midplane, \( {n_3}=0 \) on the lateral surface, it is obtained:
where \( 2h \) is the thickness of the plate. The expression of \( \sigma_{ip}^{{\mathrm{M}\;\mathrm{air}}} \) is similar to \( \sigma_{ip}^{\mathrm{M}} \). The notations in Eq. (7.187) are
It is noted that when the Maxwell stresses on the upper and lower surfaces are reduced to the midplane, a distributed couple \( \int\nolimits_S {\left( {m_{{3\alpha}}^{\mathrm{M}}-m_{{3\alpha}}^{{\mathrm{M}\;\mathrm{env}}}} \right)\delta u_{{3,\alpha}}^0\mathrm{d}S}, \quad m_{{3\alpha}}^{\mathrm{M}}=\left[ {\sigma_{{3\alpha}}^{\mathrm{M}}(h)+\sigma_{{3\alpha}}^{\mathrm{M}}\left( {-h} \right)} \right]\left( {{h \left/ {2} \right.}} \right) \) may be produced, but this effect is neglected in Eq. (7.187) due to small \( h \). It is also noted that Eq. (7.179) is not fully appropriate for the free lateral boundary. In fact from the variational formula, \( {\sigma_{{\alpha 3}}} \) on the middle plane is approximately considered by \( P_3^{{\operatorname{int}\;*}} \), but \( {\sigma_{{\alpha 3}}} \) on the free boundary should not be considered. So on the boundary \( L \), a term \( {N_{{3\beta}}}=\int\nolimits_{-h}^h {{\sigma_{{\beta 3}}}\mathrm{d}{x_3}} \) should be added to the variational formula.
Substituting Eqs. (7.187) into Eq. (7.186), adding a term \( \int\nolimits_L {{N_{{3\beta}}}{n_{\beta}}\delta u_3^0\mathrm{d}L} \), and finishing the variational calculation, we finally get:
The mechanical governing equations of the plane problem are
The mechanical governing field equation for the bending problem is
In Eq. (7.190) terms \( \left( {{\rho_1}\ddot{u}_{{\alpha, \alpha}}^0-{\rho_2}u_{{3,\alpha \alpha}}^0} \right) \) and \( \left( {{\rho_1}\ddot{u}_k^0-{\rho_2}u_{3,k}^0} \right) \) can be neglected. The usual three boundary conditions for the plate bending can easily be derived from Eq. (7.190).
According to the assumption (3), the electromagnetic field is reduced to
In deriving the above equations, the constitutive equations were not used, so the governing equations can be used for all materials satisfying the basic postulations (1) to (3).
7.6.4 Some Discussions
The soft electromagnetic plate under a uniform transverse magnetic field can be bending or buckling when the magnetic field exceeds a critical value (Moon and Pao 1968; Pao and Yeh 1973). The natural frequency of a soft electromagnetic plate can be changed under a longitudinal magnetic field (Zhou and Miya 1998). Zhou and Zheng (1997) pointed out that there was not a unified theory to discuss the above two problems, and they proposed a variational method attempting to unified deal with these problems. The key problem is to get the electromagnetic force acting on the plate. Though for the electromagnetically static problem and the problem without magnetic field the theory discussed above is appropriate, but for a MQS system (\( {{{\partial \boldsymbol{D}}} \left/ {{\partial t}} \right.}=0,\ {{{\partial \boldsymbol{B}}} \left/ {{\partial t}} \right.}\ne 0 \)), such as vibration problem in a magnetic field, it should be modified, the motional electric force should be considered. As an example, the transverse vibration of an elastic electroconductive plate is subjected to the external uniform magnetic field \( {{\boldsymbol{H}}_0}={H_{01}}{{\mathbf{i}}_1}+{H_{02}}{{\mathbf{i}}_2} \) parallel to the \( ({x_1},{x_2}) \) plane only. The induced motional electric field \( \boldsymbol{e},\boldsymbol{b},\boldsymbol{j} \) in the plate due to the plate motion is (Librescu et al. 2004; Belubekyan et al. 2007)
The corresponding Maxwell stress is
Belubekyan et al. (2007) adopted the electromagnetic body force \( \boldsymbol{f}={\boldsymbol{\nabla}}\cdot {{\boldsymbol{\sigma}}^{\mathrm{M}}} \), but Librescu et al. (2004) adopted the Lorentz formula \( \boldsymbol{f}=\boldsymbol{j}\times {{\boldsymbol{B}}_0} \). The MQS problems should be further studied.
7.7 Piezoelectric Composite Shells
7.7.1 First-Order Shear Deformation Theory
Consider a finite, simply supported, \( N \)-layered laminated circular cylindrical shell of mean radius \( R \), length \( L \), and thickness \( H \). The shell is constituted of elastic orthotropic or radially polarized piezoelectric materials. The cylindrical coordinates \( (x,\theta, z) \) are adopted with \( (x,\theta ) \) spanning the mid-surface and \( z \) along the normal (radial) direction. Analogous to the Mindlin plate theory, Kapuria et al. (1998) assumed that the displacements can be approximated as
where \( {u^0},{{\boldsymbol v}^0},{w^0} \) are the displacement components on the mid-surface and \( {\psi_1},{\psi_2} \) are the rotations of its normal. The corresponding strains are
On the mid-surface, the resultant membrane forces \( {N_x},\;{N_{\theta}},\;{N_{{x\theta}}},\;{N_{{\theta x}}} \), transverse forces \( {Q_x},\;{Q_{\theta}} \), and resultant moments \( {M_x},\;{M_{\theta}},\;{M_{{x\theta}}},\;{M_{{\theta x}}} \) are
The constitutive equation is shown in Eq. (7.149). The equilibrium equations are
where \( {p_x},{p_{\theta}},{p_z},{m_x}{m_{\theta}} \) are the external forces and moments, respectively. The boundary conditions are defined:
where \( {\theta_0} \) is the span of the cylindrical panel. Usually \( {\sigma_z} \) can be neglected. Using Eqs. (7.149) and (7.150), the equilibrium equations in terms of displacements can be obtained. Here it is omitted. The above theory is easily extended to the combined multiply layer shell.
The electric potential \( \varphi \) is assumed to vary linearly across the actuated layer, and the electric field is computed as \( {E_x}=-{\varphi_{,x}},\ {E_{\theta}}=-{{{{\varphi_{{,\theta}}}}} \left/ {(R+z)} \right.},\ {E_z}=-{\varphi_{,z}} \).
For the classical shell theory, the transverse shear strains \( {\gamma_{zx}},{\gamma_{{\theta z}}} \) are neglected. Hence,
The above thin shell theories yield poor predictions of the transverse stress components \( {\sigma_{xz}},{\sigma_{{\theta z}}},{\sigma_z} \), so sometimes a post-processing technique is needed. The transverse stress can approximately be obtained from the 3D equilibrium equations (Kapuria et al. 1998):
where \( {c_i} \) is determined by the boundary conditions at the outer shell surface. Kapuria et al. (1998) compared the numerical results of the shell theory with that of the exact 3D theory and gave some comments. Saviz et al. (2007) proposed a layerwise model which is formulated by introducing piecewise continuous approximations through the thickness for each state variables. They showed that the results calculated by this model more consist with that from the 3D theory.
7.7.2 The Cylindrical Bending of a Laminated Infinitely Long Shell
The exact analytical solution of a cylindrical shell by 3D theory is difficult, but for some simpler cases, it is possible. Now discuss an infinitely long laminated orthotropic cylindrical shell with simple supported edges under purely cylindrical bending. The top and bottom layers are piezoelectric actuators, and the middle layer is an elastic orthotropic substrate. The cylindrical coordinates \( r,\theta, z \) are used, where \( r,\theta \) and \( z \) refer to the radial, circumferential, and axial directions, respectively, and \( {u_r},{u_{\theta}} \) and \( {u_z} \) are the corresponding displacements (Fig. 7.10).
Cylindrical bending of an infinitely long shell
Because the shell is infinitely long, variables can be considered as the functions of \( (r,\theta ) \) only. The equilibrium, geometric, and constitutive equations are, respectively,
where \( {C_{ij}} \) is the reduced material coefficients. The boundary conditions are
-
1.
Simply supported \( {u_r}={\sigma_{\theta}}={\sigma_{{\theta z}}}=0;\quad \varphi =0,\quad \mathrm{when}\quad \theta =0,\quad {\theta_0} \)
-
2.
$$ \mathrm{On}\, \mathrm{the}\, \mathrm{interfaces}\quad {u_r},\;{u_{\theta}},\;{\sigma_r},\;{\sigma_{{r\theta}}}\quad \mathrm{continuous}\ \mathrm{and}\quad \varphi =0 $$(7.204)
-
3.
Upper surface of the outer actuator \( {\sigma_r}={q_0}\sin p\theta, \quad {\sigma_{{r\theta}}}=0,\quad \varphi =V\sin p\theta \)
-
4.
Lower surface of the inner actuator \( {\sigma_r}={\sigma_{{r\theta}}}=0;\quad {D_r}=0 \)
where \( p={{{m\pi}} \left/ {{{\theta_0}}} \right.} \),\( m \) is an integer, and \( V \) and \( {q_0} \) are given values.
In order to satisfy the boundary conditions \( {u_r}={\sigma_{\theta}}=\varphi =0 \) on the edges, Chen et al. (1996) adopted the following generalized displacements for actuators:
Substitution of Eq. (7.205) into Eq. (7.201) for actuators yields
where \( {f}^{\prime}={f_{,r}},\quad {f}^{\prime\prime}={f_{,rr}} \) for any \( f \). Let
Substitution of Eq. (7.207) into Eq. (7.206) for actuators yields the homogeneous equation of \( {A_r},{A_{\theta}},{A_{\varphi}} \). In order to have nontrivial solutions for \( {A_r},{A_{\theta}},{A_{\varphi}} \), the coefficient determinate of them must be zero, so the following character equation is obtained:
From Eq. (7.208) \( s \) has 6 real roots \( {s_j},\;j=1-6 \) for piezoelectric material. For each \( {s_j} \), a group \( ({A_{rj}},{A_{{\theta j}}},{A_{{\varphi j}}}) \) with one unknown is obtained, so the general solution of Eq. (7.206) for each actuator is
The governing equations of the middle orthotropic composite matrix can be obtained if the electric variables in Eq. (7.206) are omitted. Let the generalized displacements in matrix be
The character equation of \( {a_r},{a_{\theta}} \) is
where the superscript M of \( C_{ij}^M \) is omitted. So the solution of the substrate is
There are 16 unknowns: 6 \( {A_j} \) of the outer actuator, 6 \( {A_j} \) of the inner actuator, and 4 \( {a_j} \) of the middle substrate. There are also 16 boundary conditions: 4 conditions on each interface, 3 conditions on upper surface, 3 conditions on lower surface, and \( {\sigma_{{\theta z}}}=0 \) at \( \theta =0,{\theta_0} \). Therefore, the problem is solved.
7.7.3 Approximate Theory of a Functionally Graded Shallow Piezoelectric Shell
Figure 7.11 shows a functionally graded shallow piezoelectric shell of thickness \( 2h \); \( ({\alpha_1},{\alpha_2}) \) are the orthogonal curvilinear coordinates on the mid-surface; and its corresponding Lamé parameters are \( {H_1},{H_2} \) and radii of curvatures are \( {R_1},{R_2} \). \( {\alpha_3} \) is a linear coordinate and normal to the mid-surface. For a thin shell, \( h\ll {R_i},(i=1,2) \), \( {R_i} \) is approximately independent of \( {\alpha_3} \). Let \( ({u_1},{u_2},{u_3},\varphi ) \) be the generalized displacements in the orthogonal curvilinear coordinates, then
Shallow piezoelectric shell
Let \( {C_{ijkl}}(z),{e_{kij}}(z),{\epsilon_{ij}}(z) \), \( z\equiv {\alpha_3} \). Wu et al. (2002) assumed
Substitution of Eq. (7.214) into Eq. (7.213) yields the generalized strains, then substitutes the strains into the variational formula Eq. (7.160). Approximately take \( 1+{z \left/ {{{R_1}}} \right.}\approx 1,\ 1+{z \left/ {{{R_2}}} \right.}\approx 1 \). Noting \( \mathrm{d}V={H_1}{H_2}\mathrm{d}{\alpha_1}\mathrm{d}{\alpha_2}\mathrm{d}z \) and finishing the variational calculation, the approximate equations of the thin shell can be obtained. The generalized momentum equation is
The natural boundary conditions are
Let \( T_i^{*+} \) and \( T_i^{*-} \) denote the traction on the upper and lower surfaces, respectively, and
It is noted that the generalized displacements must satisfy the boundary conditions when the variational formula Eq. (7.160) is used. Using the constitutive equation, the governing equations in terms of the generalized displacements are easily obtained.
7.7.4 Free Vibration of a Functionally Graded Piezoelectric Hollow Cylinder Filled with Compressible Fluid
Consider an orthotropic piezoelectric hollow cylinder of inner radius \( R \), thickness \( h \), and length \( L \). Chen et al. (2004) adopted the cylindrical coordinates \( r,\theta, z \) and adopted the state space method to analyze the free vibration of a functionally graded piezoelectric hollow cylinder filled with compressible fluid. Assumed all material constants and mass are the functions of \( r \). From the constitutive equations, geometric equations, and motion equations, the state equation can be obtained as
where \( \boldsymbol{Y} \) is the state vector and \( \boldsymbol{M} \) is \( 8\times 8 \) matrix. Chen et al. (2004) discussed the simply supported case with the boundary conditions:
In order to satisfy Eq. (7.219), they assumed that the solution of state vector can be expanded in double trigonometric series:
where \( {R_0}=R+{h \left/ {2} \right.} \), \( \eta ={r \left/ {{{R_0}}} \right.} \), and \( \varsigma ={z \left/ {L} \right.} \); \( m \) and \( n \) are integers; and \( \omega \) is the angular frequency. Variables at the outer cylindrical surface \( r=R+h \) are denoted by right superscript “out,” and at the inner surface \( r=R \) will be denoted by right superscript “inn.” Substitution of Eq. (7.220) into Eq. (7.218) yields
where \( \bar{\boldsymbol{Y}} \) is a constant vector and \( \boldsymbol{N} \) is a \( 8\times 8 \) matrix and is not constant, so the solution cannot be obtained directly from Eq. (7.221). The approximate laminated model, for which the cylinder is divided into \( N \) thin layers, is adopted. For \( i \)th layer, \( {{\boldsymbol{N}}_i} \) can be assumed constant and takes its value at midplane. Using the transfer matrix method as shown in Section 6.4.1, \( {{\boldsymbol{Y}}^{\mathrm{out}}}=\boldsymbol{T}{{\boldsymbol{Y}}^{\mathrm{inn}}} \) can be obtained, through the transfer matrix \( \boldsymbol{T} \), and the variables on the outer surface are expressed by the inner variables. The boundary condition on the inner cylindrical surface is solved by the fluid-solid coupling theory. For a nonviscous fluid, the connected conditions on the inner surface are
where \( {{\boldsymbol v}_{{\mathrm{f}r}}},{{\boldsymbol v}_r} \) are the radial components of the velocity of the fluid and solid, respectively, and \( {p_{\mathrm{f}}} \) is the fluid pressure. Finally, the boundary conditions on the inner and outer surfaces are
where, \( {\rho_{\mathrm{f}}},{\rho^{\mathrm{out}}},{c_{\mathrm{f}}} \) are the fluid density, solid density at \( r=R+h \), and the sound velocity in fluid, respectively.
The electrically boundary conditions on \( r=R,\;R+h \) are as follows:
Electrically open, \( {D_r}=0 \), or electrically shorted, \( \varphi =0 \), at
Substituting the boundary conditions at the inner and outer surfaces into Eq. (7.221), the frequency equation can be obtained. The frequency equations for the electrically shorted case and the electrically open case are different.
References
Belubekyan M, Ghazaryan K, Marzocca P, Cormier C (2007) Localized magnetoelastic bending vibration of an electroconductive elastic plate. J Appl Mech 74:1071–1077
Chen W-Q, Lim CW (2005) 3Dpoint force solution for a permeable penny-shaped crack embedded in an infinite transversely isotropic piezoelectric medium. Int J Solids Struct 131:231–246
Chen W-Q, Shioya T (1999) Fundamental solution for a penny-shaped crack in a piezoelectric media. Int J Solids Struct 47:1459–1475
Chen C-Q, Shen Y-P, Wang XM (1996) Exact solution of orthotropic cylindrical shell with piezoelectric layers under cylindrical bending. Int J Solids Struct 33:4481–4494
Chen W-Q, Bian ZG, Lv C-F, Ding HJ (2004) 3Dfree vibration analysis of a functionally graded piezoelectric hollow cylinder filled with compressible fluid. Int J Solids Struct 41:947–964
Cherepanov GP (1979) Mechanics of brittle fracture. McGraw-Hill, New York
Crawley EF, Aanderson E (1989) Detailed models of piezoceramic actuation of beams, In: Proceedings of the 30th AIAA/ASMI/ASCH/AHS/AHC structures, structural dynamics and materials conference, Washington, DC
Crawley EF, De Luis J (1987) Use of piezoelectric actuators as elements of intelligent structures. AIAA J 25:1373–1385
Deans SR (1983) The radon transform and some of its applications. Wiley, New York
Deeg WF (1980) The analysis of dislocation, crack and inclusion problems in piezoelectric solids. Ph.D. dissertation, Stanford University
Ding H-J, Chen B, Liang J (1996) General solutions for coupled equations for piezoelectric media. Int J Solids Struct 33:2283–2298
Dunn ML (1994) Electroelastic Green’s functions for transversely isotropic piezoelectric media and their application to the solution of inclusion and inhomogeneity problems. Int J Eng Sci 32:119–131
Dunn ML, Taya M (1993) An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities. Proc R Soc Lond A 443:265–287
Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond A 241:376–396
Fabrikant VI (1989) Applications of potential theory in mechanics, a selection of new results. Kluwer Academic Publishers, Dordrecht/Boston/London
Fan H, Qin S (1995) A piezoelectric sensor embedded in a non- piezoelectric matrix. Int J Eng Sci 33:379–388
Heyliger P (1997) Exact solutions for simply supported laminated piezoelectric plates. J Appl Mech 64:299–306
Heyliger P, Brooks S (1995) Free vibration of piezoelectric laminates in cylindrical bending. Int J Solids Struct 32:2945–2960
Huang JH (1997) A fracture criterion of a penny-shaped crack in transversely isotropic piezoelectric media. Int J Solids Struct 34:2631–2644
Jiang LZ, Sun CT (2001) Analysis of indentation cracking in piezoceramics. Int J Solids Struct 38:1903–11918
Kapuria S, Sengupta S, Dumir C (1998) Assessment of shell theories for hybrid piezoelectric cylindrical shell under electromechanical load. Int J Mech Sci 40:461–477
Krommer M, Irschik H (2000) A Reissner-Mindlin-type plate theory including the direct piezoelectric and the pyroelectric effect. Acta Mech 141:51–69
Kuang Z-B (2011) Physical variational principle and thin plate theory in electro-magneto-elastic analysis. Int J Solids Struct 48:317–325
Librescu L, Hasanyan D, Ambur DR (2004) Electromagnetically conducting elastic plates in a magnetic field: modeling and dynamic implications. Int J Nonlinear Mech 39:723–729
Mikata Y (2000) Determination of piezoelectric Eshelby tensor in transversely isotropic piezoelectric solids. Int J Eng Sci 38:605–641
Mitchell JA, Reddy JN (1995) A refined hybrid plate theory for composite laminates with piezoelectric laminae. Int J Eng Sci 32:2345–2367
Moon EC, Pao YH (1968) Magnetoelastic buckling of a thin plate. J Appl Mech 35:53–68
Mura T (1987) Micromechanics of defects in solids, 2nd rev. edn. Martinus-Nijhoff, Deventer
Pao YH, Yeh CS (1973) A linear theory for soft ferromagnetic elastic bodies. Int J Eng Sci 11:415–436
Saviz MR, Shakeri M, Yas MH (2007) Electroelastic fields in a layered piezoelectric cylindrical shell under dynamic load. Smart Mater Struct 16:1683–1695
Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-Hill Book Company, New York
Ueda S (2007) A penny-shaped crack in a functionally graded piezoelectric strip under thermal loading. Eng Fract Mech 74:1255–1273
Wang B (1992) Three-dimensional analysis of an ellipsoidal inclusion in a piezoelectric material. Int J Solids Struct 29:293–308
Wang J, Yang J (2000) Higher-order theories of piezoelectric plates and applications. Appl Mech Rev 53:87–99
Wang Z-K, Zheng B-L (1995) The general solution of three dimensional problems in piezoelectric media. Int J Solids Struct 32:105–115
Wu X, Chen C, Shen Y, Tian X (2002) A high order theory for functionally graded piezoelectric shells. Int J Solids Struct 39:5325–5344
Yang J (2005) An introduction to the theory of piezoelectricity. Springer, New York
Zhou Y-H, Miya K (1998) A theoretical prediction of natural frequency of a ferromagnetic beam plate with low susceptibility in an in-plane magnetic field. J Appl Mech 65:121–126
Zhou Y-H, Zheng X-J (1997) A general expression of magnetic force for soft ferromagnetic plates in complex magnetic field. Int J Eng Sci 35:1405–1417
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Shanghai Jiao Tong University Press, Shanghai and Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Kuang, ZB. (2014). Three-Dimensional and Applied Electroelastic Problems. In: Theory of Electroelasticity. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36291-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-36291-0_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36290-3
Online ISBN: 978-3-642-36291-0
eBook Packages: EngineeringEngineering (R0)