Mode-I penny-shaped crack problem in an infinite space of one-dimensional hexagonal piezoelectric quasicrystal: exact solutions

Abstract

This paper aims to study the Mode-I penny-shaped crack problem of an infinite body of one-dimensional hexagonal piezoelectric quasicrystal. The problem is transformed into a mixed-boundary value problem in the context of electro-elasticity of quasicrystals, and the corresponding integro-differential equations are analytically solved. Two extreme cases of electrically impermeable and permeable crack surface are considered. By virtue of the generalized potential theory method, the three-dimensional complete analytical solutions of three-dimensional crack problems under symmetric concentrated and uniform loads are expressed in terms of elementary functions. Important parameters in fracture mechanics are explicitly derived, such as crack surface displacements, the distributions of generalized stresses at the crack tip and the corresponding generalized stress intensity factors. The validity of the proposed solutions and the coupling effect of phonon-phason-electric fields are investigagted through numerical examples.

Appendices

Appendix A

This Appendix gives the expressions of constants in Section 2.

$$\begin{aligned}&{\lambda _{1i}} = {s_i}\left( {{c_1} - {b_1}s_i^2 + {a_1}s_i^4} \right) \nonumber \\ {}&{\lambda _{ki}} = {a_k}s_i^6-{b_k}s_i^4+{c_k}s_i^2 -{d_k},\;\;\left( {k = 2,3,4} \right)&\end{aligned}$$

(A.1a)

$$\begin{aligned}&{\alpha _{1i}} = {{{\lambda _{2i}}}/{{\lambda _{1i}}}},\;\;{\alpha _{2i}} = {{{\lambda _{3i}}} /{{\lambda _{1i}}}},\;\;{\alpha _{3i}} = {{{\lambda _{4i}}}/{{\lambda _{1i}}}}&\end{aligned}$$

(A.1b)

$$\begin{aligned}&{\gamma _{1i}} = {s_i}\left( {{c_{33}}{\alpha _{1i}} + {R_2}{\alpha _{2i}} + {e_{33}}{\alpha _{3i}}} \right) - {c_{13}}\nonumber \\ {}&{\gamma _{2i}} = {s_i}\left( {{R_2}{\alpha _{1i}} + {K_1}{\alpha _{2i}} + {{e'}_{33}}{\alpha _{3i}}} \right) - {R_1}\nonumber \\ {}&{\gamma _{3i}} = {s_i}\left( {{e_{33}}{\alpha _{1i}} + {{e'}_{33}}{\alpha _{2i}} + {\xi _{33}}{\alpha _{3i}}} \right) - {e_{31}}\nonumber \\&{\gamma _{4i}} = 2{s_i}\left( {{c_{13}}{\alpha _{1i}} + {R_1}{\alpha _{2i}} + {e_{31}}{\alpha _{3i}}} \right) - {c_{11}} - {c_{12}}&\end{aligned}$$

(A.1c)

$$\begin{aligned}&{\beta _{1i}} = {c_{44}}\left( {{\alpha _{1i}} + {s_i}} \right) + {R_3}{\alpha _{2i}} + {e_{15}}{\alpha _{3i}}\nonumber \\&{\beta _{2i}} = {R_3}\left( {{\alpha _{1i}} + {s_i}} \right) + {K_2}{\alpha _{2i}} + {{e'}_{15}}{\alpha _{3i}}\nonumber \\&{\beta _{3i}} = {e_{15}}\left( {{\alpha _{1i}} + {s_i}} \right) + {{e'}_{15}}{\alpha _{2i}} - {\xi _{11}}{\alpha _{3i}}&\end{aligned}$$

(A.1d)

$$\begin{aligned}&{\rho _1} = {c_{44}},\;\;{\rho _2} = {R_3},\;\;{\rho _3} = {e_{15}}&\end{aligned}$$

(A.1e)

where ${a_i}$, ${b_i}$, ${c_i}$, ${d_i}$ are detailded in Appendix $\textrm{A}$ of Li et al. (2014).

Appendix B

This section presents the expression in Eq. (22) for the potential functions ${\Psi _m}\left( z \right)$. Substituting Eq. (21) into Eq. (17), we get

$$\begin{aligned}&{F_m}\left( N_0 \right) =- \frac{1}{A}\sum \limits _{n = 1}^3 {{q_{nm}} {{{P_n}\delta \left( {{\rho _0} - {\rho _n}} \right) \delta \left( {{\phi _0} - {\phi _n}} \right) }/{{\rho _0}}}},\;\; \nonumber \\&\quad \left( {m = 1,2,3} \right) .&\end{aligned}$$

(B.1)

Substituting Eq. (B.1) into Eq. (19), we are led to

$$\begin{aligned}&{\Psi _m}\left( z \right) = \frac{1}{{2{\pi ^3}A}}\sum \limits _{n = 1}^3 {{q_{nm}}{P_n}}\int _0^{2\pi }\nonumber \\ {}&\quad \int _{0}^{a} {K\left( {M;{N_0}} \right) \delta \left( {{\rho _0} - {\rho _n}} \right) \delta \left( {{\phi _0} - {\phi _n}} \right) d{\rho _0}d{\phi _0}},\nonumber \\ {}&\quad \left( {m = 1,2,3} \right) . \end{aligned}$$

(B.2)

According to the selectivity of the Dirac-delta function, Eq. (B.2) are recast to

$$\begin{aligned}&{\Psi _m}\left( z \right) = \frac{1}{{2{\pi ^3}A}}\sum \limits _{n = 1}^3 {{q_{nm}}{P_n}K\left( {M;{N_n}} \right) },\;\; \left( {m = 1,2,3} \right)&\end{aligned}$$

(B.3)

which is identical to Eq. (22).

Appendix C

With the help of the generalized potential theory method (Fabrikant 1989), the partial derivatives of each order of the Green’s function $K\left( {M,{N_k}} \right)$ for the points $M \left( {r,\theta ,z} \right)$ and $N_k\left( { \rho _{k},\phi _k,0} \right)$ are as follows:

$$\begin{aligned}&\Lambda K\left( {M;{N_k}} \right) =2\pi {f_{1k}}\left( z_i \right) \nonumber \\&\quad = \frac{{2\pi }}{{{\bar{q}}}}\left[ {\frac{z_i}{{R_k\left( {M,{N_k}} \right) }}{{\tan }^{ - 1}}\frac{h_k}{{R_k\left( {M,{N_k}} \right) }}} \right. \nonumber \\&\quad \quad \left. { - \frac{{{{\left( {{a^2} - \rho _{k}^2} \right) }^{{1/2}}}}}{{{\bar{s}}}}{{\tan }^{ - 1}}\frac{{{\bar{s}}}}{{{{\left( {l_2^2 - {a^2}} \right) }^{{1/2}}}}}} \right] \end{aligned}$$

(C.1a)

$$\begin{aligned}&\frac{{\partial K\left( {M;{N_k}} \right) }}{{\partial {z_i}}} = 2\pi f_{2k}\left( {{z_i}} \right) \nonumber \\&\quad =- \frac{{2\pi }}{{R_k\left( {M,{N_k}} \right) }}\arctan \left[ {\frac{h_k}{{R_k\left( {M,{N_k}} \right) }}} \right] \end{aligned}$$

(C.1b)

$$\begin{aligned}&\frac{{{\partial ^2}K\left( {M;{N_k}} \right) }}{{\partial z_i^2}} = 2\pi f_{3k}\left( z_i \right) =2\pi \nonumber \\&\quad \left[ {\frac{z_i}{{R_{k}^3}}{{\tan }^{ - 1}}\frac{h_k}{R_k} - \frac{h_k}{{z_i\left( {R_{k}^2 + {h_k^2}} \right) }}\left( {\frac{{{r^2} - l_1^2}}{{l_2^2 - l_1^2}} - \frac{{{{z_i}^2}}}{{R_{k}^2}}} \right) } \right] \end{aligned}$$

(C.1c)

$$\begin{aligned}&{\Lambda ^2}K\left( {M;{N_k}} \right) =2\pi f_{4k}\left( z_i \right) \nonumber \\&\quad =2\pi \left\{ \frac{{\sqrt{{a^2} - \rho _{k}^2} }}{{{\bar{q}} {\bar{s}}}}\left( {\frac{2}{{{\bar{q}} }} - \frac{{{\rho _k}{e^\mathrm{{j}{\phi _k}}}}}{{{{{\bar{s}}}^2}}}} \right) \arctan { \frac{{{\bar{s}}}}{{\sqrt{l_2^2 - {a^2}} }}} \right. \nonumber \\&\quad \quad - \frac{{z_i\left( {3{R_k^2} - {{z_i}^2}} \right) }}{{{{{\bar{q}} }^2}{R _k^3}}}\arctan \frac{h_k}{R_k}\nonumber \\&\quad + \frac{{\sqrt{{a^2} - {r^2}} \sqrt{{l_2}^2 - {a^2}} {\rho _k}{e^\mathrm{{j}{\phi _k}}}}}{{{\bar{q}} {{{\bar{s}}}^2}\left[ {{l_2}^2 - r{\rho _k}{e^{ - \textrm{j}\left( {\theta - {\theta _k}} \right) }}} \right] }} \nonumber \\&\quad \quad \left. - \frac{{{z_i}{h_k}}}{{\left( {{R_k^2} + {h_k^2}} \right) }}\left[ {\frac{q}{{{\bar{q}} {R_k^2}}} - \frac{{{r^2}{e^\mathrm{{j}2\theta }}}}{{\left( {l_2^2 - l_1^2} \right) \left( {{l_2}^2 - {r^2}} \right) }}} \right] \right\} \end{aligned}$$

(C.1d)

$$\begin{aligned}&\Lambda \frac{\partial {K\left( {M;{N_k}} \right) } }{{\partial {z_i}}} = 2\pi f_{5k}\left( {{z_i}} \right) \nonumber \\&\quad =2\pi \left[ {\frac{q}{{{R_k^3}}}\arctan \frac{h_k}{R_k} + \frac{h_k}{{{R_k^2} + {h_k^2}}}\left( {\frac{{r{e^\mathrm{{j}\theta }}}}{{l_2^2 - l_1^2}} + \frac{q}{{{R_k^2}}}} \right) } \right] \end{aligned}$$

(C.1e)

where $\overline{\left( \cdot \right) }$ represents the conjugate of the indicated complex variable.

$$\begin{aligned}&{\bar{q}} = r{e^{ - \textrm{j}\theta }} - {\rho _k}{e^{ - \textrm{j}{\phi _k}}},\;\;\; {\bar{s}} = \sqrt{ {{a^2} - r{\rho _k}{e^{ - \textrm{j}\left( {\theta - {\phi _k}} \right) }}} }&\nonumber \;\;\;\\&h_k = \sqrt{ {{{\left( {{a^2} - l_1^2} \right) }}{{\left( {{a^2} - \rho _{k}^2} \right) }}}}/{a}&\nonumber \;\;\;\\&{l_{1}} = {l_{1}}(a) = \frac{1}{2}\left[ {{\sqrt{ {{{\left( {r + a} \right) }^2} + {{z_i}^2}} }} - {\sqrt{ {{{\left( {r - a} \right) }^2} + {{z_i}^2}} }}} \right]&\nonumber \;\;\;\\&{l_{2}} = {l_{2}}(a) = \frac{1}{2}\left[ {{\sqrt{ {{{\left( {r + a} \right) }^2} + {{z_i}^2}} }} + {\sqrt{ {{{\left( {r - a} \right) }^2} + {{z_i}^2}} }}} \right]&\nonumber \;\;\;\\&{R_k} = R\left( {M,{N_k}} \right) = \sqrt{ {{r^2} + \rho _{k}^2 - 2r{\rho _k}\cos \left( {\theta - {\phi _k}} \right) + {{z_i}^2}} }.&\end{aligned}$$

(C.2)

The expressions of $\left[ q\right]$, A and B are given as

${\textcircled {{1}}}$ the electrically impermeable case

$$\begin{aligned}&\left[ q \right] = \nonumber \\&\quad \left[ {\begin{array}{*{20}{c}} {{g_{22}}{g_{33}} - {g_{23}}{g_{32}}}&{}{ - \left( {{g_{21}}{g_{33}} - {g_{23}}{g_{31}}} \right) }&{}{{g_{21}}{g_{32}} - {g_{22}}{g_{31}}}\\ { - \left( {{g_{12}}{g_{33}} - {g_{13}}{g_{32}}} \right) }&{}{{g_{11}}{g_{33}} - {g_{13}}{g_{31}}}&{}{ - \left( {{g_{11}}{g_{32}} - {g_{12}}{g_{31}}} \right) }\\ {{g_{12}}{g_{23}} - {g_{13}}{g_{22}}}&{}{ - \left( {{g_{11}}{g_{23}} - {g_{13}}{g_{21}}} \right) }&{}{{g_{11}}{g_{22}} - {g_{12}}{g_{21}}} \end{array}} \right]&\nonumber \\&A = {g_{11}}\left( {{g_{22}}{g_{33}} - {g_{23}}{g_{32}}} \right) - {g_{12}}\left( {{g_{21}}{g_{33}} - {g_{23}}{g_{31}}} \right) \nonumber \\ {}&\qquad + {g_{13}}\left( {{g_{21}}{g_{32}} - {g_{31}}{g_{22}}} \right)&\end{aligned}$$

(C.3a)

${\textcircled {{2}}}$ the electrically permeable case

$$\begin{aligned}&\left[ q \right] = \left[ {\begin{array}{*{20}{c}} { {g_{22}}}&{}{-{g_{21}}}\\ {-{g_{12}}}&{}{ {g_{11}}} \end{array}} \right]&\nonumber \\&B = {g_{11}}{g_{22}} - {g_{12}}{g_{21}}.&\end{aligned}$$

(C.3b)