Analysis of an interface crack with multiple electric boundary conditions on its faces in a one-dimensional hexagonal quasicrystal bimaterial

Abstract

An interface crack between dissimilar one-dimensional hexagonal quasicrystals with piezoelectric effect under anti-plane shear and in-plane electric loadings is considered. Mixed boundary conditions at the crack faces are studied. Using special representations of field variables via sectionally analytic vector-functions, a homogeneous combined Dirichlet–Riemann boundary value problem and a Hilbert problem are formulated. Exact analytical solutions of both these problems are obtained, and analytical expressions for the phonon and phason stresses and the electric field as well as for the derivative jumps of the phonon and phason displacements and also the electrical displacement jump along the bimaterial interface are derived. The field intensity factors are determined as well. The dependencies of the mentioned values on the magnitude and direction of the external electric loading and different ratios of electrically conductive and electrically permeable crack face zone lengths are demonstrated in graph and table forms.

Appendices

Appendix A

Consider a bimaterial compound, which consists of two piezoelectric quasicrystalline half-spaces $x_{2} > 0$ and $x_{2} < 0$ of different material properties. We assume, that the stresses of the phonon and phason fields, and the tangential component of the electric field are continuous across the whole bimaterial interface. This means that the boundary conditions in the plane $x_{2} = 0$ are

$$ \left\langle {\sigma_{32} (x_{1} )} \right\rangle = 0,\quad \left\langle {H_{32} (x_{1} )} \right\rangle = 0,\quad \left\langle {E_{1} (x_{1} )} \right\rangle = 0,\quad {\text{for}}\;x_{1} \in ( - \infty ,\infty ). $$

(A1)

To construct the representations, which satisfy the interface conditions, we use Eqs. (10) and (11) for the upper ($m = 1$) and lower $(m = 2)$ half-spaces, which can be written in the form

$$ {\mathbf{v}}^{{(m)}} = {\mathbf{M}}^{{(m)}} {\mathbf{f}}^{{\prime (m)}} (z) + {\boldsymbol{\bar{M}}}^{{(m)}} {\boldsymbol{\bar{f}}}^{{\prime (m)}} (\bar{z}) $$

(A2)

$$ {\mathbf{p}}^{{(m)}} = {\mathbf{N}}^{{(m)}} {\mathbf{f}}^{{\prime (m)}} (z) + {\boldsymbol{\bar{N}}}^{{(m)}} {\boldsymbol{\bar{f}}}^{{\prime (m)}} (\bar{z}), $$

(A3)

where the arbitrary vector-functions ${\mathbf{f}}^{{(1)}} (z)$ and ${\mathbf{f}}^{{(2)}} (z)$ are analytic in the upper and the lower half-spaces, respectively.

According to the interface conditions (A1) and the relations (A3), we get

$$ {\mathbf{N}}^{{(1)}} {\mathbf{f}}^{{\prime (1)}} (x_{1} + i0) - {\boldsymbol{\bar{N}}}^{{(2)}} {\boldsymbol{\bar{f}}}^{{\prime (2)}} = N^{{(2)}} f^{{\prime (2)}} (x_{1} - i0) - \bar{N}^{{(1)}} \bar{f}^{{\prime (1)}} (x_{1} - i0)\;{\text{for}}\;x_{1} \in ( - \infty ,\infty ) $$

(A4)

The left-hand side of equation (A4) is the boundary value of a vector-function being analytic in the domain $x_{2} > 0$, and the right-hand side of this equation is the boundary value of another vector-function being analytic in the domain $x_{2} < 0$. Hence, both vector-functions can be analytically continued into the whole complex plane, i.e., they are equal to an arbitrary vector-function, defined as

$$ {\mathbf{J}}(z) = \left\{ {\begin{array}{*{20}c} {{\mathbf{N}}^{{(1)}} {\mathbf{f}}^{{\prime (1)}} (z) - {\bar{\mathbf{N}}}^{{(2)}} {\bar{{\bf f}}}^{{\prime (2)}} (z){\text{ for }}x_{2} > 0} \\ {{\mathbf{N}}^{{(2)}} {\mathbf{f}}^{{\prime (2)}} (z) - {\bar{\mathbf{N}}}^{{(1)}} {\bar{\mathbf{f}}}^{{\prime (1)}} (z){\text{ for }}x_{2} < 0} \\ \end{array} } \right. , $$

(A5)

which is analytic in the whole complex plane, including points along all bimaterial interface.

Taking into account that the phonon and phason stresses and the electric field are bonded at infinity, it follows from equation (A3) that ${\mathbf{J}}(\infty ) = {\mathbf{J}}^{\infty }$, where ${\mathbf{J}}^{\infty }$ is a constant vector. But according to Liouville’s theorem, this means that ${\mathbf{J}}(z) = {\mathbf{J}}^{\infty }$ holds true in the whole complex plane. Thus from equation (A5) it follows

$$ \begin{aligned} & {{\bar{\bf f}}}^{{\prime (2)}} (z) = \left( {\overline{N}^{(2)} } \right)^{ - 1} N^{(1)} {\mathbf{f}}^{{\prime (1)}} (z) - \left( {\overline{N}^{(2)} } \right)^{ - 1} J^{\infty } \;{\text{for}}\;x_{2} > 0, \\ & {\mathbf{f}}^{{{\prime }(2)}} (z) = \left( {N^{(2)} } \right)^{ - 1} \overline{N}^{(1)} {\boldsymbol{\bar{f}}}^{{{\prime }(1)}} (z) - \left( {N^{(2)} } \right)^{ - 1} J^{\infty } \;{\text{for}}\;x_{2} < 0. \\ \end{aligned} $$

(A6)

Since $f^{\prime (1)} (z)$ and $f^{\prime (2)} (z)$ are arbitrary functions, one can set $J^{\infty } = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} } \right]^{T}$, and equations (A6) get the form

$$ \begin{aligned} & {{\bar{\bf f}}}^{{\prime (2)}} (z) = \left( {\overline{N}^{(2)} } \right)^{ - 1} N^{(1)} {\mathbf{f}}^{{\prime (1)}} (z)\;{\text{for}}\;x_{2} > 0, \\ & {\mathbf{f}}^{{\prime (2)}} (z) = \left( {N^{(2)} } \right)^{ - 1} \overline{N}^{(1)} {\bar{\mathbf{f}}}^{{\prime (1)}} (z)\;{\text{for}}\;x_{2} < 0. \\ \end{aligned} $$

(A7)

Consider further the expressions

$$ \left\langle {v(x_{1} )} \right\rangle = v^{(1)} (x_{1} + i0) - v^{(2)} (x_{1} - i0) $$

for the derivatives of the jumps of phonon and phason displacements and electrical displacement jump over the bimaterial interface, which, in view of (A2) and (A7), takes the form

$$ \left\langle {v(x_{1} )} \right\rangle = {\mathbf{D}}{\mathbf{f}}^{{\prime (1)}} (x_{1} ) + {\bar{\mathbf{D}}}{\bar{\mathbf{f}}}^{{\prime (1)}} , $$

where ${\mathbf{D}} = {\mathbf{M}}^{{(1)}} - {\boldsymbol{\bar{M}}}^{{(2)}} \left( {{\boldsymbol{\bar{N}}}^{{(2)}} } \right)^{{ - 1}} {\mathbf{N}}^{{(1)}} .$

Furthermore, we assume that the part $L$ of the bimaterial interface is mechanically and electrically bounded, i.e., the boundary conditions at this part of the bimaterial interface are or

$$ \left\langle {v(x_{1} )} \right\rangle = 0\;{\text{for}}\;x_{1} \in L $$

$$ {\mathbf{D}} = {\mathbf{M}}^{{(1)}} (x_{1}) = - \bar{\mathbf{D}}\bar{\mathbf{f}}^{{\prime (1)}} (x_{1} )\;{\text{for}}\;x_{1} \in L. $$

(A8)

Continuity of the phonon and phason displacements and the electrical displacement across the bonded bimaterial interface, as inferred from (A8), implies that an arbitrary vector-function defined as

$$ {\mathbf{W}}(z) = \left\{ {\begin{array}{*{20}c} {{\mathbf{D}}{\mathbf{f}}^{{\prime (1)}} (z){\text{ for }}x_{2} > 0} \\ { - {\bar{\mathbf{D}}}{\bar{\mathbf{f}}}^{{\prime (1)}} (z){\text{ for }}x_{2} < 0} \\ \end{array} } \right. $$

is analytic in the whole complex plane cut along $( - \infty ,\infty )\backslash L$. Then, the field variables at the bimaterial interface can be expressed via the boundary values of the function ${\mathbf{W}}{\text{(z)}}$ in such a way that

$$ \left\langle {v(x_{1} )} \right\rangle = {\mathbf{W}}^{ + } (x_{1} ) - {\mathbf{W}}^{ - } (x_{1} ), $$

(A9)

$$ {\mathbf{p}}(x_{1} ,0) = {\mathbf{GW}}^{ + } (x_{1} ) - {\boldsymbol{\bar{G}W}}^{ - } (x_{1} ) , $$

(A10)

where ${\mathbf{G}} = {\mathbf{N}}^{{(1)}} \left( {\mathbf{D}} \right)^{{ - 1}}$ and the superscripts ‘ + ’ and ‘−‘ indicate the limit values at the bimaterial interface taken from the upper and the lower half-spaces, respectively.

Equations (A9) and (A10) can be used for the analysis of a bimaterial compound, which consists of dissimilar one-dimensional hexagonal piezoelectric QCs with cracks at their bimaterial interface.

Appendix B

The general solution of the homogeneous combined Dirichlet–Riemann boundary value problem (28) and (29) can be presented in the form [41]

$$ \Phi_{1} (z) = X(z)\left[ {P(z) + iY(z)Q(z)} \right], $$

(B1)

where

$$ X(z) = \frac{{e^{i\phi (z)} }}{{(z - d)\sqrt {(z - a_{1} )(z - a_{2} )} }},\;Y(z) = \sqrt {\frac{{(z - a_{1} )(z - a_{2} )}}{{(z - b_{1} )(z - b_{2} )}}} , $$

$$ \phi (z) = - Z(z)\left( {\varepsilon_{1} \int\limits_{{a_{1} }}^{{a_{2} }} {\frac{{{\text{d}}t}}{{Z^{ + } (t)(t - z)}} \, + \, i\int\limits_{{b_{1} }}^{{a_{1} }} {\frac{{h_{1} (t){\text{d}}t}}{{Z^{ + } (t)(t - z)}} + i\int\limits_{{a_{2} }}^{{b_{2} }} {\frac{{h_{2} (t){\text{d}}t}}{{Z^{ + } (t)(t - z)}}} } } } \right),\;\varepsilon_{1} = \frac{{\ln \gamma_{1} }}{2\pi }, $$

$$ Z(z) = \sqrt {(z - a_{1} )(z - a_{2} )(z - b_{1} )(z - b_{2} )} ,\;h_{1} (x_{1} ) = n^{*} ,\;h_{2} (x_{1} ) = \left\{ {\begin{array}{*{20}c} {1, \, x_{1} \in (a_{2} ,d)} \\ {0, \, x_{1} \in (d,b_{2} )} \\ \end{array} } \right., $$

$n^{*}$ is an integer, and $d \in (a_{2} , \, b_{2} )$ is an unknown pole of the function $X(z)$.

The function $\varphi (z)$ can be represented via elliptic integrals as

$$ \phi (z) = \frac{ - 2}{{\sqrt {(b_{2} - a_{1} )(a_{2} - b_{1} )} }}\left\{ {\varepsilon_{1} \sqrt {\frac{{(z - a_{2} )(z - b_{2} )}}{{(z - a_{1} )(z - b_{1} )}}} \varphi_{1} (z)} \right. + $$

$$ \left. { + n^{*} \sqrt {\frac{{(z - a_{1} )(z - a_{2} )}}{{(z - b_{1} )(z - b_{2} )}}} \varphi_{2} (z) - \sqrt {\frac{{(z - b_{1} )(z - b_{2} )}}{{(z - a_{1} )(z - a_{2} )}}} \varphi_{3} (z)} \right\}, $$

where

$$ \phi_{1} (z) = (a_{1} - b_{1} )\Pi (p_{1} ,q) + (z - a_{1} )K(q),\;p_{1} = p_{1}^{*} \frac{{z - b_{1} }}{{z - a_{1} }},\;p_{1}^{*} = \frac{{a_{2} - a_{1} }}{{a_{2} - b_{1} }}, $$

$$ \phi_{2} (z) = (b_{1} - b_{2} )\Pi (p_{2} ,r) + (z - b_{1} )K(r),\;p_{2} = p_{2}^{*} \frac{{z - b_{2} }}{{z - b_{1} }},\;p_{2}^{*} = \frac{{b_{1} - a_{1} }}{{b_{2} - a_{1} }}, $$

$$ \phi_{3} (z) = (a_{2} - a_{1} )\Pi (\mu ,p_{3} ,r) + (z - a_{2} )F(\mu ,r),\;p_{3} = p_{3}^{*} \frac{{z - a_{1} }}{{z - a_{2} }},\;p_{3}^{*} = \frac{{b_{2} - a_{2} }}{{b_{2} - a_{1} }}, $$

$q = \sqrt {\frac{{(a_{2} - a_{1} )(b_{2} - b_{1} )}}{{(b_{2} - a_{1} )(a_{2} - b_{1} )}}} ,\;r = \sqrt {\frac{{(b_{2} - a_{2} )(a_{1} - b_{1} )}}{{(b_{2} - a_{1} )(a_{2} - b_{1} )}}} ,\;\mu = \arcsin \sqrt {\frac{{(b_{2} - a_{1} )(d - a_{2} )}}{{(b_{2} - a_{2} )(d - a_{1} )}}} ,$ Here $F(\mu ,r)$ and $\Pi (\mu ,p,r)$ are incomplete elliptic integrals of the first and third kind, while $K(r)$ and $\Pi (p,r)$ are complete elliptic integrals of the first and third kind.

The expansion of the function $\varphi (z)$ at infinity has the form

$$ \left. {\varphi (z)} \right|_{z \to \infty } = A_{1} z + \left( {A_{2} + \xi_{1} A_{1} } \right) + \left( {A_{3} + \xi_{1} A_{2} + \xi_{2} A_{1} } \right)z^{ - 1} + O\left( {z^{ - 2} } \right), $$

where

$$ A_{j} = \varepsilon_{1} \int\limits_{{a_{1} }}^{{a_{2} }} {\frac{{t^{j - 1} {\text{d}}t}}{Z(t)} \, + \, i\int\limits_{{b_{1} }}^{{a_{1} }} {\frac{{t^{j - 1} h_{1} (t){\text{d}}t}}{{Z^{ + } (t)}} \, + \, i\int\limits_{{a_{{2}} }}^{{b_{{2}} }} {\frac{{t^{j - 1} h_{2} (t){\text{d}}t}}{{Z^{ + } (t)}}} } } ,\;j = 1,2,3. $$

The integer $n^{*}$ and the pole $d$ can be found from the condition of finite values at infinity of the function $\varphi (z)$ as

$$ - \varepsilon_{1} \frac{K(q)}{{K(r)}} < n^{*} < 1 - \varepsilon_{1} \frac{K(q)}{{K(r)}} $$

$d = \frac{{a_{1} \left( {b_{2} - a_{2} } \right)sn^{2} \left( {\omega , \, r} \right) - a_{2} \left( {b_{2} - a_{1} } \right)}}{{\left( {b_{2} - a_{2} } \right)sn^{2} \left( {\omega , \, r} \right) - \left( {b_{2} - a_{1} } \right)}}$

where $sn\left( {\omega , \, r} \right)$ is the Jacobi elliptic function and $\omega = \varepsilon_{1} K(q) + n^{*} K(r)$.

The polynomials $P(z)$ and $Q(z)$, appearing in the solution (B1), have the form

$$ P(z) = C_{0} + C_{1} z + C_{2} z^{2} ,\;Q(z) = D_{0} + D_{1} z + D_{2} z^{2} , $$

where the coefficients

$$ C_{0} = - C_{1} \left( {d - \frac{\chi }{{\chi^{*} }}} \right) - {\text{d}}C_{2} \left( {d - \frac{2\chi }{{\chi^{*} }}} \right) - \frac{{\chi^{2} }}{{\chi^{*} }}\left( {D_{1} + 2{\text{d}}D_{2} } \right), $$

$$ D_{0} = \frac{1}{{\chi^{*} }}\left( {C_{1} + 2{\text{d}}C_{2} } \right) - D_{1} \left( {d + \frac{\chi }{{\chi^{*} }}} \right) - {\text{d}}D_{2} \left( {d + \frac{2\chi }{{\chi^{*} }}} \right), $$

$$ C_{1} = \alpha_{1} D_{2} - \nu_{1} C_{2} , $$

$$ D_{1} = - \left( {\nu_{1} + \eta_{1} } \right)D_{2} - \alpha_{1} C_{2} , $$

$$ C_{2} = \frac{{h_{13} E_{1}^{\infty } }}{{1 + \gamma_{1} }}\cos \alpha_{0} + \frac{{ - h_{11} \sigma_{32}^{\infty } - H_{32}^{\infty } }}{{1 + \gamma_{1} }}\sin \alpha_{0} , $$

$$ D_{2} = \frac{{ - h_{11} \sigma_{32}^{\infty } - H_{32}^{\infty } }}{{1 + \gamma_{1} }}\cos \alpha_{0} - \frac{{h_{13} E_{1}^{\infty } }}{{1 + \gamma_{1} }}\sin \alpha_{0} $$

are determined by the condition at infinity (31) for the function $\Phi_{1} (z)$ as well as the condition of the phonon and phason displacement uniqueness and the absence of an electric charge in the crack region. In the above formulas

$$ \chi = \sqrt {\frac{{(d - a_{1} )(d - a_{2} )}}{{(d - b_{1} )(b_{2} - d)}}} , $$

$$ \chi^{*} = \frac{1}{2\chi }\left[ {\frac{{(2d - a_{1} - a_{2} )(d - b_{1} )(b_{2} - d) + (2d - b_{1} - b_{2} )(d - a_{1} )(d - a_{2} )}}{{(d - b_{1} )^{2} (b_{2} - d)^{2} }}} \right], $$

$$ \eta_{1} = - \frac{1}{2}\left( {a_{1} + a_{2} - b_{1} - b_{2} } \right),\;\nu_{1} = \frac{{a_{1} + a_{2} }}{2} + d,\;\alpha_{0} = A_{2} ,\;\alpha_{1} = A_{3} + \xi_{1} A_{2} . $$

Further, taking into account the expression $F_{1} (z) = i\Phi_{1} (z)$, we get

$$ F_{1} (z) = iX(z)\left[ {P(z) + iY(z)Q(z)} \right]. $$

(B2)