Abstract
In this chapter some electroelastic problems in more complex materials with defects are discussed. It is pointed out that the electroelastic analysis for electrostrictive materials, the entire system including the dielectric medium, its environment, and their common boundary should be considered together. So the Maxwell stress should be considered. The theory illustrated in this chapter is an important complement for the present theory published in literatures. The electroelastic analyses of an infinite isotropic electrostrictive material containing an elliptic hole, containing a crack with and without local saturation electric field near the crack tip, are carried out. The basic theory of the thermo-electro-elastic analysis is given. An elliptic hole in a homogeneous pyroelectric material, interface crack in dissimilar pyroelectric material, point heat source, and its interaction with cracks are discussed. The electroelastic analyses of a functionally graded piezoelectric material are also introduced. These analyses are useful in engineering applications.
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Keywords
- Electroelastic analysis
- Electrostrictive material
- Maxwell stress
- Pyroelectric material
- Functionally graded piezoelectric material
5.1 Isotropic Electrostrictive Material
5.1.1 Governing Equations
Some polyurethane elastomers and perovskite-type ceramics can produce large deformation under applied electric field. Their strains are proportional to the square of electric field and larger than \( {10^{-4 }}{{\left( {{{\mathrm{ m}} \left/ {\mathrm{ m}\mathrm{ V}} \right.}} \right)}^2}{E^2} \). The electrostrictive effect can occur in all dielectric, such as the electrostrictive ceramic PMN-PT, electrostrictive polymer EPs, and polyurethane PUE. The constitutive equation has been discussed in Sects. 2.2 and 2.6. In this section we only discuss the isotropic electrostrictive material occupying the region \( S \). The environment occupies \( {S^c} \). According to Eq. (2.27b) the constitutive equation with independent variables \( (\boldsymbol{\varepsilon}, \boldsymbol{ E}) \) is
where \( {a_1},{a_2} \) are electrostrictive coefficients. For electrostrictive materials the entire system including the dielectric medium, its environment, and their common boundary should be considered together, as shown in Sect. 2.2. The governing equations are
where \( \boldsymbol{ S} \) is the pseudo total stress (Jiang and Kuang 2003, 2004). In isotropic case S, σ and σ M are all symmetric. The boundary conditions are
The interface conditions are
For the ceramic material the difference between \( \boldsymbol{ S} \) and \( \boldsymbol{\sigma} \) is small, but for the electrostrictive polymer \( \epsilon \) and \( {a_m}{\varepsilon_{ij }} \) may be in the same order, and the difference between \( \boldsymbol{ S} \) and \( \boldsymbol{\sigma} \) may not be small.
In the case of small strain, it is usually assumed that the electric field is approximately independent to the displacement, i.e., the terms containing strains in \( \boldsymbol{ D} \) in Eq. (5.1) can be neglected, but the stress field is related to the electric field. So the electric field is decoupled with the elastic field and can be solved independently (Knops 1963; Smith and Warren 1966; McMeeking 1989; Jiang and Kuang 2003, 2004). Assuming the air is charge free, from \( \boldsymbol{\nabla} \cdot \boldsymbol{ D}={ 0} \), it is known that \( \varphi \) is a harmonic function, so it can be expressed by the real (or imaginary) part of a complex analytic function \( w(z) \), i.e.,
where \( A \) is called the stream function. Comparing Eqs. (3.83) and (5.5), it is found that \( w(z)=2\phi (z) \). These two expressions of the complex electric potential can all be found in literatures. It is noted that in this section \( \phi (z) \) denotes the complex stress function. Using Cauchy-Riemann condition \( {{{\partial \varphi }} \left/ {{\partial {x_1}}} \right.}={{{\partial A}} \left/ {{\partial {x_2}}} \right.} \), \( {{{\partial \varphi }} \left/ {{\partial {x_2}}} \right.}=-{{{\partial A}} \left/ {{\partial {x_1}}} \right.} \) yields
So the solution of the electric field is reduced to seek a function analytic in the region \( S \).
On a boundary we have
where a trivial integral constant is omitted.
For a plane strain problem, we have \( {\varepsilon_{i3 }}=0,\quad \left( {i=1,2,3} \right) \); the constitutive equation expressed by the pseudo total stress \( \boldsymbol{ S} \) is
Using \( {\varepsilon_{{\gamma \gamma }}}={{{\left[ {{S_{{\gamma \gamma }}}+\left( {a-2b} \right){E_{\gamma }}{E_{\gamma }}} \right]}} \left/ {{2\left( {\lambda +G} \right)}} \right.} \), Eq. (5.8) can also be written as
where \( Y=2G(1+v) \) is the elastic modulus and \( v \) is the Poisson ratio. Substitution of Eq. (5.9) into the compatible equation \( 2{\varepsilon_{12,12 }}={\varepsilon_{11,22 }}+{\varepsilon_{22,11 }} \) finally yields
Let \( \tilde{U} \) denote the pseudo total stress function satisfying the equilibrium equation automatically:
Substituting Eq. (5.11) into Eq. (5.10), after some manipulation, yields
Using the Muskhelishvili’s formulas (1975), the general solution of Eq. (5.12) is
where \( \left( {{{{\kappa}} \left/ {4} \right.}} \right)w(z)\overline{w(z)} \) is the special solution; \( \phi (z),\chi (z) \) are two analytic functions of z. Equation (5.11) yields
From Eqs. (5.2) and (5.6), it is known that
The mechanical stresses are
and displacements are
The stress boundary condition is
where \( A,B \) are two points on the boundary; \( {P_1},{P_2} \) are pseudo resultant forces; \( \tilde{T}_i={S_{ij }}{n_j} \).
5.1.2 An Impermeable Elliptic Hole in an Isotropic Electrostrictive Material
Let an isotropic electrostrictive material with an elliptic hole of semiaxes \( a \) and \( b \) directed along the material principle axes \( {x_1} \) and \( {x_2} \), respectively, filled by air. The uniform generalized stresses \( {{\boldsymbol{\sigma}}^{\infty }},{{\boldsymbol{ E}}^{\infty }} \) are applied at infinity, but the boundary of the hole is free; see Fig. 5.1. A further assumption is that the electric field in the air will be neglected due to the small permittivity comparing with the electrostrictive material. Therefore, in this simple case the Maxwell stress in the hole is neglected, and the electrostrictive material can be studied alone (Jiang and Kuang 2003; Kuang and Jiang 2006). The boundary conditions are
A 2D plane electrostrictive material with an elliptic hole or inclusion: (a) physical plane \( z \); (b) mapping plane \( \zeta \)
Electric field The mapping function method is used to solve this problem. The mapping function \( z=\omega \left( \varsigma \right) \) shown in Eq. (3.82a) is still adopted. In \( \varsigma \) plane the general solution of \( w\left( \varsigma \right) \) can be written as
\( w(\varsigma ) \) expressed by Eq. (5.20) satisfies the boundary at infinity and on the interface. In fact on the interface, we have
Stress field The general solutions of complex potentials \( \phi \left( \varsigma \right),\psi \left( \varsigma \right) \) can be assumed as
where \( {\phi_0}\left( \varsigma \right),\;{\psi_0}\left( \varsigma \right) \) are undetermined functions analytic in \( S \); \( {\varGamma_1},{\varGamma_2} \) are determined by
Because the boundary of the hole is free, the boundary condition Eq. (5.18) becomes
Substitution of Eqs. (5.20) and (5.21) into Eq. (5.23) yields
Multiplying Eq. (5.24) and its conjugate equation by \( {{{\mathrm{ d}\sigma }} \left/ {{\left[ {2\pi \mathrm{ i}\left( {\sigma -\varsigma } \right)} \right]}} \right.} \) and using the Cauchy integral formulas we find
Substitution of Eqs. (5.21), (5.22), and (5.25) into Eq. (5.16) yields the stresses
Asymptotic fields near the end of a narrow elliptic hole under the electric load As in Sect. 3.4.6 the asymptotic stress fields near the end of a narrow elliptic hole only under an electric load in the local coordinate system with the origin at the focus of the ellipse are
The electric asymptotic field is
where \( {\rho_0}={{{{b^2}}} \left/ {2a } \right.} \), \( c=\sqrt{{{a^2}-{b^2}}} \).
5.1.3 The Permeable Elliptic Hole
For a permeable elliptic hole, the electric connective conditions in Eq. (5.19) are changed to
According to the previous knowledge, it is assumed prior that the electric field in the air is constant (Smith and Warren 1966, 1968; Gao et al. 2010), and the complex electric potential in the media \( w(z) \) is in the following form:
where \( {w_0}(z) \) is an unknown function analytic in \( S \). Substituting Eqs. (5.5), (5.6), and (5.29) into Eq. (5.28) and using Eq. (5.7) we get
Substituting \( {x_1}={{{a\left( {\sigma +{\sigma^{-1 }}} \right)}} \left/ {2} \right.},\;{x_2}={{{\mathrm{ i}b\left( {\sigma -{\sigma^{-1 }}} \right)}} \left/ {2} \right.} \) and multiplying \( {{{\mathrm{ d}\sigma }} / {{\left[ {2\pi \mathrm{ i}\left( {\sigma -\varsigma } \right)} \right]}} } \) to two sides of Eq. (5.30) and then integrating the result identity we get
Equation (5.31) yields
So, using \( D_1^{\infty }=\epsilon E_1^{\infty }, D_1^{c }={\epsilon^c} E_1^{c }\) we obtain
Especially for a crack (\( b=0 \)) we have
It means that the electric fields are homogeneous in the crack, but with different constant values. When \( E_1^{\infty }=0 \), the electric asymptotic field near the right crack tip is
where \( \bar{{\delta}}={{{{{\bar{\epsilon}}^{\mathrm{ c}}}}} \left/ {\bar{b}} \right.}={{{\left( {{\epsilon^{\mathrm{ c}}}a} \right)}} \left/ {{\left( {\epsilon b} \right)}} \right.} \) is an important parameter; \( r,\theta \) are polar coordinates in the local coordinate system with the origin at the focus of the ellipse.
The complex stress functions are still expressed by Eq. (5.21). For \( E_1^{\infty }=0 \) finally we find
The stress is obtained by Eq. (5.16).
5.1.4 A Rigid Elliptic Conduction Inclusion
In this section we shall discuss a rigid elliptic conducting inclusion with boundary L in an isotropic electrostrictive material (Jiang and Kuang 2004). In this case the problem can be discussed independently in the material region \( \varOmega \) and the boundary conditions are assumed:
where \( {\omega^{\mathrm{ c}}} \) is the rotation angle about axis x 3 of the inclusion. The pseudo total moment \( \tilde{M} \), Maxwell stress moment \( {M^e} \), and mechanical moment \( M \) are, respectively,
When there are no body force and free charge, the stress complex potential can be assumed as
where \( {w_0}(z),{\phi_0}(z),{\psi_0}(z) \) are complex functions analytic in the region \( S \). \( {{\tilde{T}}_i} \) is the generalized concentrate force, which is zero in present case, so the terms containing \( \ln z \) will be omitted in later.
The conformal mapping method is used to solve the problem. The mapping function is shown in Eq. (3.82). It is easy to prove that the electric field in \( S \) can be obtained by changing \( \alpha \) to \( (-\alpha ) \) in Eq. (5.20) discussed in Sect. 5.1.2, i.e.,
Using Eq. (5.17) the displacement boundary condition in Eq. (5.36) can be expressed as
On the mapping plane Eq. (5.40) becomes
where \( \varLambda =-K=-3+4\nu \) and \( \phi (\varsigma ) \) and \( \psi (\varsigma ) \) are given in Eq. (5.38). Noting
The future process to solve the problem is fully similar to that in Sect. 5.1.2. Finally we obtain
If \( {\omega^{\mathrm{ c}}} \) is given, the mechanical moment acting on the inclusion can be determined by Eq. (5.37), or
In Eq. (5.44) points \( A \) and \( B \) are the same point, so only multiple value terms containing \( \ln \varsigma \) are not zero, i.e., only should keep terms containing \( \chi \left( \varsigma \right) \) and \( {\varOmega_1}\left( \varsigma \right) \). From the second equation in Eq. (5.43) we get
So we have
Noting \( {\varGamma_1},m,\varLambda, G,{\omega^{\mathrm{ c}}} \) are all real, Eq. (5.47) can be reduced to
If there is no moment acting on the inclusion, the \( {\omega^{\mathrm{ c}}} \) is determined by the following equation:
For a conductor ball \( m=0 \), from Eq. (5.49), it is seen that \( {\omega^{\mathrm{ c}}}=0 \), i.e., there is no rotation. It is also noted that for \( \beta ={{{n\pi }} \left/ {2} \right.},n=1,2,3,4 \), \( {\omega^{\mathrm{ c}}}=0 \) for pure electric loading. \( {\omega^{\mathrm{ c}}} \) is proportional to the square of the electric field and linear of the stress at infinity. Substituting \( {\omega^c} \) into Eq. (5.43), the stress potentials are obtained and then the stresses are all obtained. The asymptotic field near the right end of a narrow rigid elliptic inclusion under an electric field at infinity is
Jiang and Kuang (2005, 2007) discussed a general elliptic inclusion. Liang et al. (1995) discussed piezoelectric materials with a general elliptic inclusion.
5.2 Cracked Infinite Electrostrictive Plate with Local Saturation Electric Field
5.2.1 The Constitutive Equations and Boundary Conditions
For an electrostrictive ceramic with a crack under external high electric field, the mechanical state near the crack tip is elastic, but the electric field may be saturated. Jiang and Kuang (2006) discussed an infinite plate with a central crack of length \( 2a \), subjected to the electric field \( {E^{\infty }}=E_1^{\infty }+\mathrm{ i}E_2^{\infty } \) at infinity. It is assumed that the electric field in the region \( {S_0} \) of the plate is linear, but two zones \( {S_{\mathrm{ R}}} \) and \( {S_{\mathrm{ L}}} \) near the right and left crack tips are local small-scale saturated (Fig. 5.2). The constitutive equations for an isotropic electrostrictive material are
An infinite plane with a central crack located at \( \left( {-a,a} \right) \): (a) local small-scale saturation model at crack tips; (b) linear model
where \( D(E) \) is the uniaxial dielectric response in the absence of stress. Here it is assumed
where \( {D_{\mathrm{ c}}} \) and \( {E_{\mathrm{ c}}} \) are the saturation electric displacement and saturation electric field, respectively. For linear case \( \hat{\epsilon}=\epsilon \) is constant, but for the nonlinear case \( \hat{\epsilon} \) may be dependent to electric field. If the electric field is linear, \( \boldsymbol{\sigma} \) in Eq. (5.51a) can also be expressed by
The boundary condition of the problem is
5.2.2 The Electric Field in an Electrostrictive Material with an Impermeable Crack
1. The electric field in a linear plate without local saturation region
According to Eq. (5.6) and approximately taking \( \boldsymbol{ D}=\epsilon \boldsymbol{ E} \) we have
where \( w(z) \) is a complex potential shown in Eq. (5.5). On the crack surface
Equation (5.54) yields
This is a standard Hilbert problem. Noting Eq. (5.52) its solution is
The asymptotic field near the crack tip \( z=a \) is
where \( {K_{\mathrm{ e}}}=E_2^{\infty}\sqrt{{\pi a}} \) is the electric field intensity factor, \( {z-a=re^{i\theta}}\).
2. The electric field in a plate with local saturation region
The local saturation model of the electric field at the crack tip is similar to III-type yielding model in an elastoplastic material, so the method used in elastoplastic analysis can also be used here (Cherepanov 1979). The asymptotic solution near a tip of a central crack is the same as that in a semi-infinite crack problem. A local coordinate system \( O{y_i} \) with the origin located at the crack tip (Fig. 5.3) is also used. A point in it is denoted by \( y={y_1}+\mathrm{ i}{y_2}=z-a \). The boundary value problem is
The local saturation zone near crack tip: (a) physical plane \( z \); (b) mapping plane
where the origin \( O \) is not included in \( {S_R} \). Let
According to Eqs. (5.57) and (5.58), it yields (Fig. 5.3)
According to Eqs. (5.58) and (5.60), the crack boundary \( {y_2}=0,\;{y_1}< 0 \) in the \( y \) plane is transformed to \( \theta =\pm {\pi \left/ {2} \right.} \) in the \( \varsigma \) plane. Let \( R\left( \theta \right) \) be the boundary of the saturation zone \( {S_{\mathrm{ R}}} \); a point \( t \) on the boundary of \( {S_{\mathrm{ R}}} \) can be expressed as
According to Eq. (5.60) in the \( \varsigma \) plane, the boundary of \( {\varOmega_{\mathrm{ R}}} \) is \( {{\mathrm{ e}}^{{-\mathrm{ i}\theta }}}=\bar{\sigma} \). In order to simplify the problem, the hodograph transform method is used. The boundary value problem in the \( \varsigma \) plane is
In the \( \varsigma \) plane Eq. (5.62) shows that the zone \( {\varOmega_{\mathrm{ R}}} \) is constituted of a unit semicircle and a line segment \( -1\leq {\xi_2}\leq 1 \) on the image axis. The zone inside \( {\varOmega_{\mathrm{ R}}} \) is corresponding to the zone outside \( {S_{\mathrm{ R}}} \). Now we shall solve the problem, Eq. (5.62), in the \( \varsigma \) plane. Let
where \( f\left( {\bar{\sigma}} \right) \) is an unknown function. Because \( R\left( \theta \right) \) is real, so
It is considered that the linear asymptotic solution Eq. (5.57) can approximately be used in the present problem, i.e., outside \( {S_{\mathrm{ R}}} \) the following relation is held:
Equation (5.65) also satisfies the condition, \( \varsigma =0 \), when \( y\to \infty \).
From Eqs. (5.64) and (5.65), it is derived that outside the saturation zone we have
The boundary of the saturation zone \( {S_{\mathrm{ R}}} \) in the \( \varsigma \) plane is
Equation (5.67) shows that the saturation zone \( {S_{\mathrm{ R}}} \) in the \( y \) plane is a circle with radius \( \rho \). Equation (5.67) can also be obtained if in Eq. (5.57) let \( E_1^2+E_2^2=E_{\mathrm{ c}}^2 \).
From Eq. (5.66) it is found that the linear field in \( {S_0} \) for a material with a saturation zone near the tip is the same as that in a material without a saturation zone, if we use the effective crack length \( {a_{\mathrm{ eff}}} \) instead of the real crack length \( a \). It is just the method used in the elastoplastic fracture mechanics. The effective crack length is
Using the above theory the electric field in \( {S_0} \) for a central crack problem is
On the boundary of \( {S_{\mathrm{ R}}} \) we have \( z=a+\rho +\rho {{\mathrm{ e}}^{{\mathrm{ i}\varTheta }}}\left( {\varTheta =2\theta } \right) \) (Fig. 5.3a). Substituting it into Eq. (5.69) yields
It is seen that on the interface the limit values of the electric field taken from \( {S_0} \) and \( {S_R} \) are equal in the accuracy of \( {\rho \left/ {{\left( {a+\rho } \right)}} \right.} \). Usually \( {\rho \left/ {{\left( {a+\rho } \right)}} \right.}\ll 1 \), so the above solution is reasonable.
5.2.3 The Stress in an Impermeable Crack with Local Saturation
1. Stress in linear zone \( {S_0} \) This problem in \( {S_0} \) is similar to that in Sect. 5.1.2. Let \( b=0,m=1,R={a \left/ {2} \right.} \) and use the effective crack length instead of the real crack length; the solution of the central crack problem can be obtained from the solution of an elliptic hole problem. Equations (5.21), (5.22), (5.23), and (5.24) are still appropriate here, but it should be used the electric field Eq. (5.69) instead of Eq. (5.20). According to above discussions in the \( \varsigma \) plane, the stress potentials are determined by the following equations:
where \( z=\omega \left( \varsigma \right) \) is shown in Eq. (5.20) with \( m=1 \). \( {\varGamma_1},{\varGamma_2} \) are shown in Eq. (5.22).
Multiplying the first equation in Eq. (5.70) and its conjugate equation by \( {{{\mathrm{ d}\sigma }} \left/ {{\left[ {2\pi \mathrm{ i}\left( {\sigma -\varsigma } \right)} \right]}} \right.} \) and using the Cauchy integral formulas we find
Near the crack tip let \( z=a+r{{\mathrm{ e}}^{{\mathrm{ i}\theta }}} \) (Fig. 5.4); through tedious calculation the pseudo total asymptotic stresses are
Division regions near the crack tip
2. Stress in saturation zone \( {S_{\mathrm{ R}}} \) In the saturation zone \( {S_{\mathrm{ R}}} \), the electric displacements are finite; the asymptotic stresses near the crack tip will possess singular behavior like \( {1 \left/ {{\sqrt{r}}} \right.} \) and relate to the size of the saturation zone, so it is assumed
Because the electric displacements are continuous on the interface from \( {S_{\mathrm{ R}}} \) and \( {S_0} \), so the Maxwell stress and mechanical and pseudo total stresses are all continuous. So \( h_{ij}^{(1)}\left( \theta \right),h_{ij}^{(2)}\left( \theta \right) \) can be obtained from these continuous conditions:
where \( R\left( \theta \right)=2\rho \cos \theta \) and on the interface \( \varTheta =2\theta \), \( {l_0}=\left| {R\left( \theta \right){{\mathrm{ e}}^{{\mathrm{ i}\theta }}}-\rho } \right|=\rho \). If \( {\rho \left/ {a} \right.}\ll 1 \), \( {{{\left( {a+\rho } \right)}} \left/ {{\left| {R\left( \theta \right){{\mathrm{ e}}^{{\mathrm{ i}\theta }}}-\rho } \right|}} \right.}\approx {a \left/ {\rho } \right.} \). Comparing the coefficients before \( \sqrt{R} \) and \( {1 \left/ {\rho } \right.} \) yields
It is easy to prove that on the interface, the limit values of the stresses taken from \( {S_0} \) and \( {S_{\mathrm{ R}}} \) are equal in the accuracy of \( {1 \left/ {{\sqrt{r}}} \right.} \) and \( {1 \left/ {\rho } \right.} \) which is consistent of the electric field.
3. Division region near the crack tip According to Eqs. (5.72) and (5.73), the stress can be divided into four regions (Fig. 5.4).
Region I: Region I is located in \( {S_{\mathrm{ R}}} \) and very near the crack tip, where \( \sqrt{{{a \left/ {r} \right.}}}\gg {a \left/ {\rho } \right.} \) and the stresses possess the singularity \( {1 \left/ {{\sqrt{r}}} \right.} \). Under \( \sigma_{22}^{\infty },E_2^{\infty } \) at infinity we have
Region II: Region II is located in \( {S_{\mathrm{ R}}} \) and \( \sqrt{{{a \left/ {r} \right.}}}\sim {a \left/ {\rho } \right.} \). The stresses should be calculated by Eq. (5.73). The terms containing \( \sqrt{{{a \left/ {r} \right.}}},{a \left/ {\rho } \right.} \) all should be considered.
Region III: Region III is in \( {S_0} \) but neighboring \( {S_{\mathrm{ R}}} \) and \( \sqrt{{{a \left/ {r} \right.}}}\sim {a \left/ {{\left| {r{{\mathrm{ e}}^{{\mathrm{ i}\theta }}}-\rho } \right|}} \right.} \). The stresses should be calculated by Eq. (5.75).
Region IV: Region IV is in \( {S_0} \) and \( \sqrt{{{a \left/ {r} \right.}}}\gg {a \left/ {{\left| {r{{\mathrm{ e}}^{{\mathrm{ i}\theta }}}-\rho } \right|}} \right.} \). Terms containing \( {a \left/ {{\left| {r{{\mathrm{ e}}^{{\mathrm{ i}\theta }}}-\rho } \right|}} \right.} \) can be neglected. If \( {r \left/ {a} \right.} \) is still small, the stresses can be calculated from Eq. (5.76) also.
5.2.4 Conducting Crack
For the conducting crack or the soft electrode, the boundary conditions are
1. The electric field in a linear piezoelectric plate without local saturation zone
According to Eq. (5.6) on the electrode, we have
For the central crack \( \left( {-a,a} \right) \) from Eq. (5.78), it can be derived
The asymptotic solution near the crack tip \( z=a \) is
2. The electric field in a plate with local saturation zone
Similar to the impermeable crack the boundary value problem in \( {y} \) plane is
where \( R=R\left( \theta \right) \) is the boundary of the saturation zone in \( y \) plane. The hodograph transform method is used. Let
According to Eq. (5.80) the electric displacements in the saturation zone is assumed as
Obviously Eq. (5.83) satisfies Eq. (5.77). Repeating the discussion in Sect. 5.2.2, the boundary and the radius of the saturation zone are, respectively,
The remaining discussion is fully similar to Sect. 5.2.3 and omitted here.
5.3 Asymptotic Analysis of a Crack Subjected to Electric Loading
Yang and Suo (1994) and Hao et al. (1996) discussed the ceramic actuators caused by electrostriction; Beom et al. (2006) discussed the asymptotic analysis of an impermeable crack subjected to electric loading. The crack extension criterion in plane strain is mainly determined by the stress field near the crack tip, so they adopted the linear asymptotic solution of a semi-infinite crack as the boundary condition of the asymptotic analysis at infinity (Fig. 5.5). In this analysis the
Asymptotic analysis sketch of a crack with local saturation
Maxwell stress is not considered. Analogous to Eq. (5.57) we have
where \( {K_{\mathrm{ D}}} \) is the electric displacement intensity factor. Now we discuss an infinite piezoelectric material with an impermeable crack subjected to electric loading as shown in Eq. (5.85). As shown in Eqs. (5.57) and (5.69), the approximate solutions of the electric displacement can be taken as
where \( \rho ={{{K_{\mathrm{ D}}^2}} \left/ {{\left( {2\pi D_{\mathrm{ c}}^2} \right)}} \right.} \) is the radius of the saturation zone, \( {w(z)} \) represents electric displacement complex potential, \( {\varOmega_{{0}}} \) denotes the linear zone, and \( {\varOmega_{\mathrm{ s}}} \) denotes the saturation zone. Equation (5.86) satisfies the boundary condition on the crack surface and Eq. (5.85) at infinity. On the interface between \( {\varOmega_0} \) and \( {\varOmega_s} \), \( {\xi_0}=\rho {{\mathrm{ e}}^{{\mathrm{ i}\varTheta }}} \). The constitutive equation is shown in Eq. (5.9), but here the slight different form is used:
where \( Y \) is elastic modulus and \( \nu \) is Poisson ratio, Q and q are the electrostrictive coefficients. Apply the superposition method to solve this problem: Problem (1) is that a plate without crack is subjected to the above electric displacement fields. In this problem on the artificial cut corresponding to the original crack we can get the tractions \( \sigma_{22}^{\mathrm{ c}},\sigma_{21}^{\mathrm{ c}} \). Problem (2) is that the artificial cut is subjected tractions \( -\sigma_{22}^{\mathrm{ c}},-\sigma_{21}^{\mathrm{ c}} \). The solution of the original problem is the sum of solutions of these two problems.
According to Eqs. (5.86) and (5.87), the strains in the saturation zone induced by the saturation electric displacements are
The strains in Eq. (5.88) satisfy the compatible equation automatically, so they do not produce stresses. Neglecting the rigid displacements the displacements corresponding to these strains are
Analogous to Eqs. (5.6), (5.16), (5.17), and (5.18) in the linear zone we have
where
In the saturation zone we have
The two group solutions shown in Eqs. (5.90) and (5.92) should satisfy the continuity conditions of displacements and stresses on the interface between linear and saturation zones. The solution in the linear zone should also satisfy the boundary conditions at infinity.
Solution of problem (1) Assume the solutions are:
Substitution of Eq. (5.93) into Eqs. (5.90) and (5.92) yields
From the continuity conditions of displacements and resultant forces on the interface we have
where \( {\xi_0}=\rho {{\mathrm{ e}}^{{\mathrm{ i}\varTheta }}} \) is the value of \( \xi \) on the interface. Assuming the displacements vanish at infinity, by the standard analytic continuation theory from Eq. (5.95) we find
Analogously from the continuity conditions of resultant forces \( {i}(P_1+iP_2) \) on the interface we have
Assuming the displacements vanish at infinity, by the standard analytic continuation theory from Eq. (5.97)
Finally we have
Solution of problem (2) Eqs. (5.90) and (5.99) yield
When the crack surface is subjected to \( -\sigma_{22}^{\mathrm{ c}} \), the solution is
Solution of the original problem Superposing solutions of problems (1) and (2), finally we get the following. In the linear zone \( {\Omega_0} \),
In the saturation zone \( {\Omega_s} \),
It is also found that in the saturation zone the stresses at the real crack tip has the singularity \( {1} {{\sqrt{r}}} \) and at the effective crack tip (\( y=\rho \)) has the logarithmic singularity. Because accuracy of the electric field is of the order \( {\rho \left/ {a} \right.} \), the accuracy of solutions of the mechanical stresses is still in the same order.
Following the elastoplastic fracture mechanics, Beom et al. (2006) also discussed the modified boundary layer theory, i.e., replaced Eq. (5.85) by
where \( T \) is a finite electric displacement parallel to the crack surface.
Beom (1999)discussed the singular behavior near a crack tip in an electrostrictive material with the elastic behavior shown in Eq. (5.87), and for the electric behavior, he took the Ramberg-Osgood type constitutive equation
where \( k \) and \( n \) are material constants; \( E=f(D) \) is the uniaxial dielectric response in the absence of stress. In this case he got \( \boldsymbol{\sigma} \propto {r^{{-{1 \left/ {2} \right.}}}},\boldsymbol{ D}\propto {r^{{-{1 \left/ {{\left( {n+1} \right)}} \right.}}}} \).
5.4 Pyroelectric Material
5.4.1 Generalized Two-Dimensional Linear Thermo-electro-elastic Problem
In engineering the extensive applied governing equation is Eq. (2.89) with independent variables \( (\boldsymbol{\varepsilon}, \boldsymbol{ E},\vartheta ),\quad \vartheta =T-{T_0} \) for the pyroelectric materials:
The thermal conduction and the entropy equations are
The mechanical, electric, and thermal boundary conditions are
where \( {T^{*}},{\sigma^{*}},q_0^{*} \) are the traction, electric charge per area, and normal heat flow per area.
The continuity conditions on the interface are
The governing equations in \( (\boldsymbol{ u},\varphi, \vartheta ) \) are
For a multiply connected domain, the displacement and electric potential must satisfy the uniqueness condition Eq. (3.7).
The thermo-electro-elastic fundamental theory of the pyroelectric material was studied a long time (Tiersten 1971; Mindlin 1974). For a static problem with stationary temperature, from Eq. (5.110) we get
From Eq. (5.111) it is seen that the generalized displacements are dependent to the temperature, but the temperature is independent to the generalized displacements. So the temperature can be solved independently (Hwu 1992; Shen and Kuang 1998). Because \( \vartheta \) is real, it is assumed that
Substitution of Eq. (5.112) into the third equation in Eq. (5.111) yields
As in Sect. 3.2.1, from Eq. (5.113) we get a pair of conjugate complex roots \( {\mu_T},{{\bar{\mu}}_T} \) with \( \operatorname{Im}{\mu_T}> 0 \):
where \( \alpha \) is real. Using Eq. (5.114) the Fourier’s law can be written as
By using Eq. (3.27) the total heat flow \( \hat{q} \) through a line segment from \( {z_0} \) to \( z \) is
When \( \vartheta \) is solved, the terms in the right side of the first and second equations in Eq. (5.111) become known. The special solution introduced by the temperature \( \vartheta \) can be assumed as
where a subscript in upper case \( P \) takes the value 1,2,3, or 4 and a subscript in lower case \( i,j,\ldots \) takes the value 1, 2, or 3, as shown in Sect. 3.2.1. Substitution of Eq. (5.117) into the first and second equations in Eq. (5.111) yields the equations to determine \( \boldsymbol{ c}={{[{c_1},{c_2},{c_3},{c_4}]}^{\mathrm{ T}}} \):
where \( \boldsymbol{ Q},\boldsymbol{ R},\boldsymbol{ T} \) are expressed in Eq. (3.13). The generalized stress introduced by temperature is
The solution for the thermo-electro-elastic analysis in pyroelectric material is the sum of the special solution and the general solution of the corresponding homogeneous equations. For the stationary temperature the general solution is
The stress can be expressed as
where \( \boldsymbol{ F}({z_j})={\boldsymbol{ f}}^{\prime}({z_j}) \). Introduce the stress function \( \boldsymbol{\varPhi} \):
Equations (5.112), (5.115), (5.120), and (5.122) are the general solutions of the thermo-electro-elastic analysis in the pyroelectric material. Combining these equations and the appropriate boundary conditions, we can solve all the thermo-electro-elastic problems. For the multi-connected region the generalized displacement and temperature should satisfy the uniqueness condition.
5.4.2 A Thermal Impermeable Elliptic Hole in a Pyroelectric Material
As an example in this section, we discuss a generalized 2D problem of a pyroelectric material that occupied the region S with an elliptic hole that occupied the region S c filled with air under uniform generalized stresses (\( {{\boldsymbol{\sigma}}^{\infty }},{{\boldsymbol{ D}}^{\infty }} \)) and heat flow \( {{\boldsymbol{ q}}^{\infty }} \) (see Fig. 3.3). The interface \( L \) between the material and the hole is free of generalized forces and is thermal insulated (Lu et al. 1998; Gao et al. 2002). The boundary conditions are
Temperature field in the piezoelectric material with a thermal insulated hole As shown in Sect. 5.4.1, the temperature can be solved independently. As in Sect. 3.4 the transform method is used to solve this problem. The mapping function for \( {z_T} \) plane to \( {\varsigma_T} \) plane is similar to \( {z_j} \) plane to \( {\varsigma_j} \) plane in Eq. (3.86), but \( {\mu_j} \) is replaced by \( {\mu_T} \), i.e.,
The interface \( L \) in \( z \) plane is mapped to \( \varGamma \) in \( \varsigma \) plane. The temperature field can be chosen as
where \( {\beta_T} \) is a complex constant and \( {{{\mathrm{ g}}^{\prime}_0}}\left( {{\varsigma_T}} \right) \) is holomorphic outside the unit circle in \( {\varsigma_T} \) plane. Equations (5.112), (5.115), and (5.125) yield
Because the interface is thermal insulated, Eqs. (5.116) and (5.126) yield
where \( \sigma \) is the value of \( {\varsigma_T} \) on \( \varGamma \). Multiplying Eq. (5.127) by \( \int\nolimits_L {\left[ {{{{\mathrm{ d}\sigma }} \left/ {{\left( {\sigma -\varsigma } \right)}} \right.}} \right]} \) and using the Cauchy integral formula we get
From Eqs. (5.125) and (5.128) in \( z \) plane, we get
where \( {\beta_T}{z_T} \) represents the complex potential of a uniform heat flow \( {{\boldsymbol{ q}}^{\infty }} \) in an infinite material without hole.
Superposition method By means of superposition, the solution of the original problem can be obtained as the sum of the following three problems:
(1) A pyroelectric material with an elliptic hole under boundary conditions
Problem (1) can be reduced to the following problem: a piezoelectric material, with an elliptic hole, subjected generalized stresses at infinity under constant temperature, which has been discussed in Sect. 3.4.
(2) A pyroelectric material without elliptic hole under boundary conditions
The solution is
This temperature field does not affect the generalized stress field, because a linear temperature field always satisfies the strain compatible equation.
(3) A pyroelectric material with an elliptic hole under boundary and single valued conditions
Now we discuss the solution of the problem (3) Subtracting the solution of problem (2) from Eq. (5.129), the temperature potential in \( \varsigma \) plane of problem (3) can be obtained:
The electric field inside the hole filled with air is fully the same as that in Sect. 3.4.2 and Eqs. (3.81), (3.82a), (3.82b), (3.83), (3.84), and (3.85) are still held. The complex potential \( \phi (\varsigma ) \) is still expressed by Eq. (3.85), i.e.,
From Eq. (5.120) it is seen that \( \boldsymbol{ f}({z_P}) \) and \( {\rm g}({z_T}) \) have the similar role in the generalized displacements, so \( \boldsymbol{ f}({\varsigma_P}) \) in S can be assumed in the following form:
Substitution of Eqs. (5.132) and (5.133) into Eq. (5.120) yields
In Eqs. (5.132), (5.133), (5.134), and (5.135) functions \( \mathrm{ g}\left( \varsigma \right),\boldsymbol{ f}\left( \varsigma \right),\phi \left( \varsigma \right) \) are all the functions of \( \varsigma \), but in Eq. (5.130c) we need their derivatives with \( s \) and \( n \) on the \( L \) in the \( z \) plane, so the following relations are needed. Eq. (3.82) yields
Using Eq. (5.136) it is easy to get
Substituting Eqs. (5.135), (5.137), and (5.122) into the connective conditions on the interface \( \varGamma \) and the single valued condition in Eq. (5.130c) and then comparing the coefficients of the corresponding terms on both sides in result equations, we get
where \( {\delta_{P4 }} \) is Kronecker delta. Solving undetermined coefficients finally yields
It is seen from Eq. (5.139) that \( {\mathrm{ g}}^{\prime}\left( {{\varsigma_T}} \right),{{f^{\prime}_P}}({\varsigma_P})\to 0 \) when \( \left| {{\varsigma_P}} \right|,\;\left| {{\varsigma_T}} \right|\to \infty \). So the boundary conditions at infinity are satisfied also.
In Eq. (5.139) \( {\varphi^{\mathrm{ c}}}\left( \varsigma \right) \) can also be rewritten as
Therefore, the electric field in the elliptic hole varies linearly with the coordinates.
5.5 Interface Crack in Dissimilar Pyroelectric Material
5.5.1 General Discussion
The fundamental theory of the pyroelectric material has been discussed in Sect. 5.4. Now the interface crack in dissimilar pyroelectric material (see Fig. 4.2) will be discussed (Shen and Kuang 1998; Gao and Wang 2001). The general solutions \( \boldsymbol{ U}\left( {{z_j},{z_T}} \right) \), \( \boldsymbol{\varPhi} \left( {{z_j},{z_T}} \right) \), and \( \vartheta \) are shown in Eqs. (5.120), (5.122), and (5.112), respectively. The boundary conditions are assumed:
where \( \hat{\boldsymbol{ d}} \) is the displacement disconnected value between crack surfaces. Equation (5.141) shows that on whole axis \( {x_1} \) we have
From Equation (5.115) it is known that \( {q_2}=-\mathrm{ i}\alpha \mathrm{ g}^{\prime \prime}({z_T})+\mathrm{ i}\alpha \bar{\mathrm{ g}}^{\prime\prime} ({{\bar{z}}_T}) \), where \( {z_T},\alpha \) are shown in Eqs. (5.112) and (5.114), respectively. Equation (5.142) yields
Analogous to Eq. (4.22) from Eq. (5.143) we have
It is assumed that the temperature satisfies the same equation:
Equations (5.112) and (5.145) yield
Analogously from Eqs. (5.142), (5.145), and (5.122) we get
Equations (5.120) and (5.147) yield
5.5.2 The Solution of Temperature
Using Eq. (5.146) and \( {\vartheta_{\mathrm{ I}}}\left( {{x_1}} \right)={\vartheta_{\mathrm{ I}\mathrm{ I}}}\left( {{x_1}} \right) \) on the connective surface yields
So we can construct a function \( \theta \left( {{z_T}} \right) \) analytic in whole \( {z_T} \) plane except \( {L_{\mathrm{ c}}} \):
The heat flow on the crack surface is
So the boundary condition of the heat flow on the crack surface is reduced to
Its solution is
where \( C({z_T}) \) is the polynomial degree \( n \) of \( {z_T} \).
5.5.3 The Solution of Generalized Stress
Because on \( L-{L_{\mathrm{ c}}} \) \( i{\boldsymbol{ d}}^{\prime}({x_1})=0 \), so
where \( \boldsymbol{ H}={{\boldsymbol{ Y}}_{\mathrm{ I}}}+{{\bar{\boldsymbol{ Y}}}_{\mathrm{ I}\mathrm{ I}}},\;{{\boldsymbol{ Y}}_{\alpha }}=\mathrm{ i}{{\boldsymbol{ A}}_{\alpha }}\boldsymbol{ B}_{\alpha}^{-1}\;\left( {\alpha =\mathrm{ I},\mathrm{ II}} \right) \). So we can construct a function \( \boldsymbol{ h}(z) \) analytic in whole \( z \) plane except \( {L_{\mathrm{ c}}} \):
Using Eqs. (5.145), (5.147), and (5.155), Eq. (5.122) can be reduced to
Substituting Eq. (5.156) into the generalized stress boundary condition in (5.141) yields
Equation (5.157) is identical with (4.28) except using \( {{\tilde{\boldsymbol{\varSigma}}}_0}({x_1}) \) instead of \( {{\boldsymbol{\varSigma}}_0}({x_1}) \), so its solution is still expressed by Eqs. (4.41a) and (4.9):
From Eq. (5.157) it is seen that its homogeneous equation is fully identical with (4.29) and does not relate to the temperature, so the eigenvalues and eigenvectors of both equations are also the same. Therefore, \( \boldsymbol{ Q}(z) \) and \( \boldsymbol{\varOmega} \) in Eq. (5.158) are still expressed by Eq. (4.37).
On the connective surface \( {{\boldsymbol{ h}}^{+}}({x_1})={{\boldsymbol{ h}}^{-}}({x_1})=\boldsymbol{ h}({x_1}),\quad {{\theta^{\prime}}^{+}}({x_1})={{\theta^{\prime}}^{-}}({x_1})={\theta}^{\prime}({x_1}) \), so we have
The open displacement disconnected value \( \hat{\boldsymbol{ d}} \) behind the crack tip is
5.5.4 A Single Interface Crack
In the case of a crack of length \( 2a \), we have \( {Z_0}({z_T})=\sqrt{{z_T^2-a^2}} \). If only the normal heat flow q 0 on the crack surface, Eq. (5.153) yields
where \( {C_1}=0 \) due to \( \boldsymbol{ q}\cdot \boldsymbol{ n}=\mathbf{0} \) at infinity, and \( {C_0}=0 \) due to the temperature single value condition \( \int\nolimits_{-a}^a {\left[ {{{{\theta^{\prime\prime}}}^{+}}\left( {{x_1}} \right)-{{{\theta^{\prime\prime}}}^{-}}\left( {{x_1}} \right)} \right]} \mathrm{ d}{x_1}=0 \). Equation (5.150) yields
Because Eq. (5.158) is decoupling, on the crack surface, for normalized Ω we have
In Eq. (5.163) the integrated function containing \( \sqrt{{{z^2}-{a^2}}} \), so when use Eq. (4.18), \( {\mathrm{ g}^{*}}=-1 \) should be used due to \( \mathop{\lim}\limits_{{z\to {x^{-}}}}\sqrt{{{z^2}-{a^2}}}=-\mathop{\lim}\limits_{{z\to {x^{+}}}}\sqrt{{{z^2}-{a^2}}} \). These integrals are
Using Eq. (5.164), Eq. (5.163) is reduced to
At infinity, \( \boldsymbol{ Q}(z)\to \boldsymbol{ I}/z \), \( {\theta}^{\prime}({z_T})\to 0 \), \( {{\boldsymbol{\varSigma}}_2}({x_1})=\mathbf{0} \), from Eqs. (5.159) and (5.165) we get
Substitution of Eq. (5.166) into Eq. (5.165) yields
\( {{\boldsymbol{ C}}_0} \) is determined by the single value condition, and according to Eq. (5.160) it is equivalent to
On the crack surface there has \( \left\langle {Y_0^{{\left( j \right)-}}({x_1})} \right\rangle =-\left\langle {{{\mathrm{ e}}^{{2\pi {\varepsilon_{j}}}}}Y_0^{{\left( j \right)+}}({x_1})} \right\rangle \) or \( {{\boldsymbol{ Q}}^{+}}-{{\boldsymbol{ Q}}^{-}}=\left\langle {1+{{\mathrm{ e}}^{{2\pi {\varepsilon_{j}}}}}} \right\rangle {{\boldsymbol{ Q}}^{+}} \). Using the following equation (Shen and Kuang 1998)
and noting \( \int\nolimits_{-a}^a {\sqrt{{x_1^2-{a^2}}}\mathrm{ d}{x_1}=} \pm \mathrm{ i}\pi {a^2}/2 \), from the single valued condition we get
The stress intensity is
where \( {{\boldsymbol{\varSigma}}_2}({x_1}) \) is determined by Eq. (5.159).
For a homogeneous material \( {{\boldsymbol{ A}}_{\mathrm{ I}}}={{\boldsymbol{ A}}_{\mathrm{ I}\mathrm{ I}}}=\boldsymbol{ A} \ 1 \), and \( \boldsymbol{ H}=\bar{\boldsymbol{ H}},\ {{\boldsymbol{ C}}_0}=\mathbf{0}, Y_0^{(j) }={1 \left/ {{\sqrt{{z_j^2-{a^2}}}}} \right.} \). So the solution is
And the asymptotic stress field near the crack tip \( {x_1}=a \) is
The stress intensity factor at \( {x_1}=a \) is
5.6 Point Heat Source and Interaction with Cracks
5.6.1 Point Heat Source in Piezoelectric and Bi-piezoelectric Material
Hwu (1990) discussed the thermal stress in an anisotropic elastic material. Shen et al. (1995) and Shen and Kuang (1998) discussed the thermal stress in a pyroelectric material, the point heat source, and their interactions.
1. Heat source in a homogeneous material For a point heat source, the temperature \( \vartheta =T-{T_0} \) can be expressed as
According to Eq. (5.116) for a point heat source with strength \( M \) located at \( {z_0}({x_{10 }},{x_{20 }}) \) in an infinite homogeneous pyroelectric material, \( c \) is determined by the following equation:
So finally the solution of the temperature in an infinite homogeneous pyroelectric material is
2. Heat source in a bimaterial The solving method of a heat source in a bimaterial is analogous to that in Paragraph 3.6.2. Let the point heat source with strength \( M \) be located at \( {z_0}({x_{10 }},{x_{20 }}) \) in material \( \mathrm{ II} \) that occupied \( {S^{-}},{x_2}< 0 \). The solution can be assumed as
Because heat flow and temperature are continuous in whole axis \( {x_1} \), so according to Eqs. (5.115) and (5.112) it yields
If \( \mathbf{q}\to \mathbf{0},T\to 0 \) when \( \left| z \right|\to \infty \), like Eqs. (3.161), (3.162), (3.163), (3.164), (3.165) or (4.22), (4.23) we have
Equations (5.178), (5.179), and (5.180) yield
On the interface \( {x_2}=0 \) we have
Because the generalized stress and displacement are continuous on whole axis \( {x_1} \), according to Eqs. (3.161), (3.162), (3.163), (3.164), and (3.165), we can derive
and
From Eqs. (5.183) and (5.184), the stress functions are
According to Eq. (5.121) on \( {x_2}=0 \) we have
5.6.2 The Point Heat Source Located at the External of an Elliptic Inclusion
Let an infinite piezoelectric material \( \mathrm{ II} \) occupied region \( {\varOmega^{-}} \) with an elliptic inclusion \( \mathrm{ I} \), occupied region \( {\varOmega^{+}} \) of major semiaxis \( a \) and minor axis \( b \) directed along the material principle axes \( {x_1} \) and \( {x_2} \), respectively. The interface of \( {\varOmega^{-}} \) and \( {\varOmega^{+}} \) is denoted by \( L \), its normal is denoted by \( \boldsymbol{ n} \) directed the inside of the inclusion or the outside of the piezoelectric material. At infinity \( {{\boldsymbol T}_{\mathrm{ II}}}=\mathbf{0},\;{q_n}=0 \) and the connective conditions on the \( L \) are
Let a point heat source at \( {z_0} \) with strength \( M \) be located in the piezoelectric material. We shall use the transform method to solve this problem (Qin 1998, 1999). The transform function from \( z \) plane to \( \varsigma \) plane is shown in Eqs. (3.82) and (3.86). The point \( {\varsigma_0} \) in \( \varsigma \) plane is corresponding to point \( {z_0} \) in \( z \) plane. \( L \) is transformed to \( \varGamma \). Let \( {{\mathrm{ g}}_0}({\varsigma_T}) \) be the fundamental solution in the \( \varsigma \) plane when the piezoelectric material occupies the whole space and as in Sect. 5.6.1 we take
Obviously \( {{\mathrm{ g}}_0}({\varsigma_T}) \) is analytic in the inclusion \( {\varOmega^{+}} \). Assume the solution of the problem is
where \( {{{\mathrm{ g}}^{\prime}_{\mathrm{ I}}}}({\varsigma_T}) \) and \( {{{\mathrm{ g}}^{\prime}_{\mathrm{ II}}}}({\varsigma_T}) \) are analytic functions in \( {\varOmega^{+}}-{\varOmega_0} \) and \( {\varOmega^{-}} \), and \( {\varOmega_0} \) is the region \( \rho \leq {\rho_0}=\sqrt{m},0\leq \theta < 2\pi \) and on \( {\varOmega_0} \) \( \phi ({\rho_0}{{\mathrm{ e}}^{{\mathrm{ i}\psi }}})=\phi ({\rho_0}{{\mathrm{ e}}^{{-\mathrm{ i}\psi }}}) \) (see Sect. 3.4.2).
According to Eqs. (5.175) and (5.176), the continuity conditions of temperature \( \vartheta \) and heat \( {q_n}\mathrm{ d}s \) through a differential arc on \( \varGamma \) can be reduced to
It is noted that \( {{{\mathrm{ g}}^{\prime}_{\mathrm{ I}}}}(\sigma ) \) is analytic only in an annular region \( {\varOmega^{+}}-{\varOmega_0} \). Similar to Eqs. (3.84) and (3.85), it yields
So Eq. (5.190) can be reduced to
From Eq. (5.192) it is known that the functions at the left side in Eq. (5.192) are analytic in the region \( {\varOmega^{+}} \), whereas those on the right side are analytic in the region \( {\varOmega^{-}} \), and they are continuous on \( \varGamma \). So these functions are analytic in whole plane and must be constants. So we have
If there are no generalized external forces acting at infinite, these constants must be zero, i.e., \( {\theta_1}\left( \infty \right)={\theta_2}\left( \infty \right)=0 \), so \( {\theta_1}\left( \varsigma \right)={\theta_2}\left( \varsigma \right)=0 \) and from Eq. (5.193) we get
Solving Eq. (5.194) yields
Solving \( {d_k},{{{\mathrm{ g}}^{\prime}_{\mathrm{ II}}}}(\varsigma ) \) and using \( {\varsigma_T} \) instead of \( \varsigma \), from Eq. (5.189), \( {\mathrm{ g}}^{\prime}({\varsigma_T}) \) is obtained:
5.6.3 Interaction of an Impermeable Crack with a Singularity in a Piezoelectric Bimaterial
Let a mechanical singular generalized load with strength \( (\boldsymbol{ b},\boldsymbol{ p}) \) be located at \( {z_0} \) in material II that occupied the lower half-plane \( {\varOmega^{-}},{x_2}< 0 \). An insulated crack \( (-a,a) \) is located on the interface \( {x_2}=0 \). According to Eqs. (3.165), (3.166), and (3.160), the generalized stress on the interface introduced by the singularity in a piezoelectric material is
The original problem can be simply solved by the superposition method: The singularity in a piezoelectric material without crack and an external force \( -{{\boldsymbol{\varSigma}}_2}({x_1}) \) applies on the crack surface. The last problem has been solved in Sect. 4.2.4.
From Eq. (5.197) it is seen that the effect of a mechanical singularity is equivalent to adding an external force \( -{{\boldsymbol{\varSigma}}_2}({x_1}) \) on the crack surface, and it does not affect the heat flow.
A point heat source with strength \( M \) located at \( {z_0} \) in material II, \( {\varOmega^{-}},{x_2}< 0 \) will produce the heat flow, as shown in Eq. (5.82), and generalized surface traction, as shown in Eq. (5.186), i.e., the point heat source affects both the stress and temperature fields. A point heat source in a bimaterial is equivalent to a point heat source in an infinite homogeneous piezoelectric material II and on the crack surface superposed the following loads:
where \( {{{\mathrm{ g}}^{\prime}_0}}({x_1}),{{{\mathrm{ g}}^{\prime}_{{_{\mathrm{ I}}}}}}({x_1}),{{\boldsymbol{ F}}_{\mathrm{ I}}}({x_1}) \) are calculated from Eqs. (5.175), (5.178), and (5.185), respectively.
As an example we discuss the interaction of the above point heat source with a single crack located at \( \left( {-a,a} \right) \) (Shen and Kuang 1998). It is assumed that the boundary conditions are
Substitution of Eqs. (5.198) and (5.178) into Eq. (5.153) yields
where \( {Z_0}({z_T})={{\left( {z_T^2-{a^2}} \right)}^{{-{1 \left/ {2} \right.}}}} \). The integral in Eq. (5.200) can be integrated. At first we discuss the contour integral
where \( \varLambda \) is shown in Fig. 5.6. Inside the contour there are three poles: \( {z_T},{z_0},{{\bar{z}}_0} \). Using the residual theorem the \( \varPhi \) is reduced to
Integral path \( \varLambda \) for the integral \( \varPhi \)
On the other hand it is easy to prove that the integral \( \varPhi \) on the path \( DEFGH \) vanishes whereas on the path \( HJD \) equals
From the condition at infinity in Eq. (5.199) and the single valued condition of temperature we find \( C({z_T})=0 \). So Eq. (5.200) is reduced to
Using \( T=0 \) at infinity finally we get
Substituting Eq. (5.202) into (5.158) \( {{\bar{\boldsymbol{\varOmega}}}^{\mathrm{ T}}}\boldsymbol{ h}(z) \) can be obtained:
From Eq. (5.155) \( {{\boldsymbol{ F}}_{\mathrm{ I}}}\left( {{z_j}} \right) \) and \( {{\boldsymbol{ F}}_{\mathrm{ II}}}\left( {{z_j}} \right) \) can be obtained.
Gao and Wang (2001) discussed the permeable crack problem, Herrmann and Loboda (2003) discussed the contact zone model in pyroelectric material, and Norris (1994) discussed the dynamic Green function in piezoelectric material.
5.7 Functionally Graded Piezoelectric Material
5.7.1 Fundamental Equations in Antiplane Shear Problem
Functionally graded piezoelectric material (FGPM) is a kind of material with continuously varying properties (Wu et al. 1996) which is very useful as a transit layer instead of the bonding agent in order to avoid the very large stresses near the interface. Li and Weng (2002) discussed the antiplane crack problem (Fig. 5.7) with varied material constants for a transversely material:
An antiplane crack in FGPM
where \( C_{44}^0,e_{15}^0,\epsilon_{11}^0 \) are the values at \( {x_2}=0 \) and \( C_{44}^h,e_{15}^h,\epsilon_{11}^h \) are the values at \( {x_2}=\pm h \); \( k \) and \( \alpha \) are material constants. It is assumed that the geometry, material behavior, and applied loading are symmetric about the \( {x_2} \)-axis, so we only need to study the part of \( {x_1}\geq 0,{x_2}\geq 0 \) and \( \left| {{x_2}} \right|={x_2} \). The fundamental equations (4.238) and (4.239) of antiplane shear problem discussed in Sect. 4.8.1 are still held in a FGPM, but the material constants are functions of coordinates.
Substitution of Eq. (5.204) into Eq. (4.239) yields
where \( {\nabla^2}={\partial \left/ {{\partial x_1^2}} \right.}+{\partial \left/ {{\partial x_2^2}} \right.} \). In general case \( {{\left( {e_{15}^0} \right)}^2}+C_{44}^0\epsilon_{11}^0\ne 0 \), so we also have
The boundary and connective conditions on x 2 = 0 are
where the right superscript “c” means that the related variable is in the air. The boundary conditions on x 2 = h are divided into two forms dependent to giving D 2 or E 2:
where \( {D_0} \) and \( {E_0} \) are the external electric displacement and field, respectively; \( {\tau_0} \) is the stress at zero electric loading, \( C_{44}^{h* }=C_{44}^h+{{{{{{\left( {e_{15}^h} \right)}}^2}}} \left/ {{\epsilon_{11}^h}} \right.} \).
5.7.2 Solution of the Antiplane Shear Problem
Considering the symmetry about \( {x_2} \)-axis, Li and Weng (2002) used the Fourier cosine transforms to solve this problem. Let
where \( \beta ={{{\left( {k-1} \right)}} \left/ {2} \right.} \); \( {{\mathrm{ I}}_{\beta }} \) and \( {{\mathrm{ K}}_{\beta }} \) are the first and second kind of modified Bessel’s functions, respectively; \( {A_i}(s) \) and \( {B_i}(s) \) are undetermined functions; \( {a_1},{b_1} \) are real constants. Equation (5.208) yields
where \( {{{\mathrm{ I}}^{\prime}_{\beta }}},{{{\mathrm{ K}}^{\prime}_{\beta }}} \) are the derivatives of \( {{\mathrm{ I}}_{\beta }},{{\mathrm{ K}}_{\beta }} \).
In the air between the crack surfaces, we have
Its solution can be assumed as
where \( C(s) \) is an unknown function. Using the boundary conditions on x 2 = h yields
Substituting \( {E_1}\left( {{x_1},0} \right),\varphi \left( {{x_1},0} \right),E_1^{\mathrm{ c}}\left( {{x_1},0} \right) \) into the corresponding boundary conditions in Eq. (5.207) yields the following dual integral equation:
If let
where \( {J_0} \) is the zero-order Bessel function of the first kind, then the second equation in Eq. (5.215) is satisfied automatically and the first equation in Eq. (5.215) requires \( \varPhi \left( \eta \right)=0 \). So it is easy to obtain \( {B_1}(s)=0 \) and then straightly \( {B_2}(s)=0 \).
Substituting \( {\sigma_{32 }}\left( {{x_1},0} \right),{u_3}\left( {{x_1},0} \right) \) into the corresponding boundary conditions in Eq. (5.207) and noting \( {B_1}(s)={B_2}(s)=0 \), the following dual integral equation is obtained:
where
The solution of Eq. (5.217) can be written as
Equation (5.219) satisfies the second equation in Eq. (5.217) automatically. In order to satisfy the first equation in Eq. (5.217), \( \Psi \left( \eta \right) \) should be satisfied by the following Fredholm integral equation of the second kind:
5.7.3 The Generalized Stress Asymptotic Fields Near the Crack Tip
The singular generalized stress fields are determined by the behavior of the solution when \( s\to \infty \). Using integration by parts to decompose Eq. (5.219) into singular and regular parts,
where \( {J_1} \) is the first-order Bessel function of the first kind. The integral in Eq. (5.221) is finite at the crack tip \( {x_1}=\pm a \), and the singular behavior is determined by the term containing \( \Psi (1) \). It is noted that the modified Bessel functions have the following behaviors:
After complex derivation we obtain
where
where the meanings of \( r,{r_1},{r_2},\theta, {\theta_1},{\theta_2} \) can be seen in Fig. 5.7. Let \( \theta \to 0,{\theta_2}\to 0 \), \( {r_2}\to 2a,r\to a \) from Eq. (5.223) we get
It is found that for the functional gradient piezoelectric material, the asymptotic fields of the generalized stress still have the singularity \( {1 \left/ {{\sqrt{r}}} \right.} \). Because \( {a_1},{b_1} \) is enclosed in \( \hat{C}_{44}^0 \) (see Eq. (5.219)), the generalized stress intensity factors are different for two different electric boundary conditions. It is also found that the electric field does not have singularity at the crack tip.
Yang et al. (2004) also discussed the electric field gradient effects in antiplane problems of polarized ceramics.
5.7.4 Plane Strain Problem
The constitutive equations of the in-plane problem are
It is assumed that the material properties are one dimensional dependent to \( {x_3} \) as
where \( \beta \) is a material constant. The equilibrium equations in terms of generalized displacements are
In the air between the crack surfaces, the governing equations are still shown in Eq. (5.212).
As in Sect. 5.7.1 it is assumed that the geometry, material behavior, and applied loading are all symmetric about the \( {x_3} \)-axis, so we only need to study the part of \( {x_1}\geq 0,{x_3}\geq 0 \) and \( \left| {{x_2}} \right|={x_2} \). The boundary conditions on the crack and connective surfaces are
The boundary conditions on the edge x 3 = h are divided into two forms:
where \( {D_0} \) and \( {E_0} \) are the external electric displacement and field, respectively, and \( {\sigma_0} \) is the stress at zero electric loading, \( C_{33}^{0* }=C_{33}^0+{{{{{{\left( {e_{33}^0} \right)}}^2}}} \left/ {{\epsilon_{33}^0}} \right.} \).
The single valued condition of the generalized displacements is
where \( \psi \left( {{x_1}} \right) \) is the generalized dislocation density and on the connective surface \( {\psi =0} \).
Ueda (2005) adopted the Fourier integral transform techniques to solve this problem. Let
where \( {A_j}(s) \) is undetermined function and \( {a_0},{b_0} \) are unknown constants; \( {\gamma_j}(s),{a_j}(s),{b_j}(s) \) are known functions. \( {\gamma_j}(s) \) is the root of the following equation:
For convenience let \( \operatorname{Re}{\gamma_j}(s)< \operatorname{Re}{\gamma_{j+1 }}(s),j=1-5 \). \( {a_j}(s),{b_j}(s) \) are determined by
where
Let
where \( B(s) \) is undetermined function.
Using the dislocation density \( \psi \left( {{x_1}} \right) \), \( {\sigma_{33 }}\left( {{x_1},0} \right)=0,\ 0\leq {x_1}< a \), other boundary conditions, and Eq. (5.235), finally we can get the following singular integral equation:
The expressions of \( M_1(t,x_1) \) and \( M_2(t,x_1) \) are omitted here.
Equations (5.236) and (5.231) form a singular integral equation system. Let
Substitute Eq. (5.237) into (5.236) and then use the Gauss-Jacobi numerical integral technique to solve the integral equation. The generalized stress intensity factors are
A lot of literatures studied the functional graded piezoelectric materials, such as Zhou and Chen (2008), Chen et al. (2003), and Wang and Zhang (2004).
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© 2014 Shanghai Jiao Tong University Press, Shanghai and Springer-Verlag Berlin Heidelberg
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Kuang, ZB. (2014). Some Problems in More Complex Materials with Defects. In: Theory of Electroelasticity. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36291-0_5
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