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A generalized supercell model of defect-introduced phononic crystal microplates

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Abstract

A generalized supercell model is established to predict band structures of phononic crystals (PnCs) microplate with or without defects. Microstructure effect in elastic flexural waves is described by the modified couple stress theory. The plane-wave expansion (PWE) method and the supercell technique are applied to solve the wave equations. The generalized supercell model contains P × Q square inclusions that can be adjusted according to different situations. The current model can be used to solve various defect state problems of multiphase materials (arbitrary) composite PnCs microplates. Single-point defect and multiple defects cases are calculated. The results demonstrate that the current model can well predict defect states of the PnCs microplates whether there is coupling between defects and supercell or not. This new model can provide help for the design and application of PnCs with multiple defects at the microscale in different fields.

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Funding

GYZ acknowledges the support by the National Natural Science Foundation of China [Grant # 12002086] and the Fundamental Research Funds for the Central Universities [Grant nos # 2242020R10027 and 2242022R40040]. STG acknowledges the support by the National Natural Science Foundation of China [Grant # 11672099]. The authors also would like to thank Prof. Shaofan Li and two anonymous reviewers for their encouragement and helpful comments on an earlier version of the paper.

Author information

Authors and Affiliations

  1. School of Civil Engineering, Chongqing University, Chongqing, 400044, China

    Wei Shen & Shuitao Gu

  2. Navier, Ecole Des Ponts, Univ Gustave Eiffel, CNRS, 77455, Marne-la-Vallée, France

    Wei Shen & Haiping Yin

  3. LMEE, University Paris-Saclay, Univ Evry, 91020, Evry, France

    Yu Cong

  4. Jiangsu Key Laboratory of Engineering Mechanics, School of Civil Engineering, Southeast University, Nanjing, 210096, China

    Gongye Zhang

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  1. Wei Shen
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  2. Yu Cong
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  3. Shuitao Gu
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  4. Haiping Yin
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  5. Gongye Zhang
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Correspondence to Shuitao Gu or Gongye Zhang.

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Appendices

Appendix A

For J-phase PnCs, \(\alpha _{{\mathbf{G}(M,N)}}\) can be expressed as follows:

$$\alpha_{{{\mathbf{G}}_{(M,N)} }} = \frac{1}{A}\iint_{\Omega } \alpha ({\mathbf{r}})e^{{ - i{\mathbf{G}}_{(M,N)} {\mathbf{r}}}} {\text{d}}{\mathbf{r}} = \frac{1}{A}\sum\limits_{j = 1}^{J} {\left( {\iint_{{\Omega_{\left( j \right)} }} {\alpha_{\left( j \right)} }e^{{ - i{\mathbf{G}}_{(M,N)} {\mathbf{r}}}} {\text{d}}{\mathbf{r}}} \right)}$$
(A.1)

when G = 0,

$$\begin{aligned} \alpha_{{{\mathbf{G}}_{(M,N)} }} & = \frac{1}{A}\sum\limits_{j = 1}^{J} {\left( {\iint_{{\Omega_{\left( j \right)} }} {\alpha_{\left( j \right)} }{\text{d}}{\mathbf{r}}} \right)} = \frac{1}{A}\sum\limits_{j = 1}^{J} {A^{\left( j \right)} \alpha_{\left( j \right)} } \\ & = \sum\limits_{j = 1}^{J} {\frac{{A^{\left( j \right)} }}{A}\alpha_{\left( j \right)} } = \sum\limits_{j = 1}^{J} {V_{f}^{\left( j \right)} \alpha_{\left( j \right)} } \\ & = \sum\limits_{j = 1}^{J - 1} {V_{f}^{\left( j \right)} \alpha_{\left( j \right)} } + \left( {1 - \sum\limits_{j = 1}^{J - 1} {V_{f}^{\left( j \right)} } } \right)\alpha_{\left( J \right)} \\ \end{aligned}$$
(A.2)

when G ≠ 0,

$$\begin{aligned} \alpha_{{{\mathbf{G}}_{(M,N)} }} & = \frac{1}{A}\sum\limits_{j = 1}^{J} {\left( {\iint_{{\Omega_{\left( j \right)} }} {\alpha_{\left( j \right)} }e^{{ - i{\mathbf{G}}_{(M,N)} {\mathbf{r}}}} {\text{d}}{\mathbf{r}}} \right)} \\ & = \frac{1}{A}\sum\limits_{j = 1}^{J - 1} {\left[ {\iint_{{\Omega_{\left( j \right)} }} {\left( {\alpha_{\left( j \right)} - \alpha_{\left( J \right)} } \right)}e^{{ - i{\mathbf{G}}_{(M,N)} {\mathbf{r}}}} {\text{d}}{\mathbf{r}}} \right]} + \alpha_{\left( J \right)} \iint_{{\Omega_{\left( 1 \right)} + \Omega_{\left( 2 \right)} + \cdots \Omega_{\left( J \right)} }} {e^{{ - i{\mathbf{G}}_{(M,N)} {\mathbf{r}}}} {\text{d}}{\mathbf{r}}} \\ & = \sum\limits_{j = 1}^{J - 1} {\left( {\alpha_{\left( j \right)} - \alpha_{\left( J \right)} } \right)F_{j} \left( {{\mathbf{G}}_{{\left( {M,N} \right)}} } \right)} + \alpha_{\left( J \right)} \iint_{\Omega } {e^{{ - i{\mathbf{G}}_{(M,N)} {\mathbf{r}}}} {\text{d}}{\mathbf{r}}} \end{aligned}$$
(A.3)

according to Zhang and Gao[45],

$$\iint_{\Omega } {e^{{ - i{\mathbf{G}}_{(M,N)} {\mathbf{r}}}} {\text{d}}{\mathbf{r}}} = \int_{{ - {\text{P}}a/2}}^{{{\text{P}}a/2}} {e^{{ - iG_{x} x}} } dx\int_{{ - {\text{Q}}a/2}}^{{{\text{Q}}a/2}} {e^{{ - iG_{y} y}} } dy = 0$$
(A.4)

thus, when G ≠ 0,

$$\alpha_{{G_{(M,N)} }} = \sum\limits_{j = 1}^{J - 1} {\left( {\alpha_{\left( j \right)} - \alpha_{\left( J \right)} } \right)F_{j} \left( {G_{{\left( {M,N} \right)}} } \right)} .$$
(A.5)

Appendix B

For the inclusion at position (p, q), the shape function can be expressed as follows:

$$\begin{aligned} & F_{(p,q)} \left( {{\text{G}}_{(M,N)} ,d_{{\left( {p,q} \right)}} } \right) = \frac{1}{A}\iint_{{\Omega_{{\rm I}} (p,q)}} {e^{{ - i{\mathbf{G}}_{{\left( {M,N} \right)}} \cdot {\mathbf{r}}}} }{\text{d}}{\mathbf{r}} \hfill \\ &\;\;\;\;\;\;\;\;\; = \frac{1}{{{\text{PQ}}a^{2} }}\int_{{q^{*} a - d_{{\left( {p,q} \right)}} /2}}^{{q^{*} a + d_{{\left( {p,q} \right)}} /2}} {e^{{ - iG_{y} y}} {\text{d}}y} \int_{{p^{*} a - d_{{\left( {p,q} \right)}} /2}}^{{p^{*} a + d_{{\left( {p,q} \right)}} /2}} {e^{{ - iG_{x} x}} } {\text{d}}x \hfill \\ &\;\;\;\;\;\;\;\;\; = \frac{1}{{{\text{PQ}}a^{2} }}e^{{iG_{x} }} \int_{{q^{*} a - d_{{\left( {p,q} \right)}} /2}}^{{q^{*} a + d_{{\left( {p,q} \right)}} /2}} {\left\{ {e^{{ - i\left[ {G_{x} (p^{*} a + d_{{\left( {p,q} \right)}} /2) + G_{y} y} \right]}} - e^{{ - i\left[ {G_{x} (p^{*} a - d_{{\left( {p,q} \right)}} /2) + G_{y} y} \right]}} } \right\}} {\text{ d}}y \hfill \\ &\;\;\;\;\;\;\;\;\; = \frac{1}{{{\text{PQ}}a^{2} }}e^{{iG_{x} }} e^{{ - iG_{x} p^{*} a}} \int_{{q^{*} a - d_{{\left( {p,q} \right)}} /2}}^{{q^{*} a + d_{{\left( {p,q} \right)}} /2}} {\left\{ {e^{{ - i\left[ {G_{x} \cdot \left( {d_{{\left( {p,q} \right)}} /2} \right) + G_{y} y} \right]}} - e^{{ - i\left[ {G_{x} \cdot \left( { - d_{{\left( {p,q} \right)}} /2} \right) + G_{y} y} \right]}} } \right\}} {\text{ d}}y \hfill \\ & \;\;\;\;\;\;\;\;\; = \frac{1}{{{\text{PQ}}a^{2} }}e^{{ - iG_{x} p^{*} a}} \int_{{q^{*} a - d_{{\left( {p,q} \right)}} /2}}^{{q^{*} a + d_{{\left( {p,q} \right)}} /2}} {{\text{d}}y} \int_{{ - d_{{\left( {p,q} \right)}} /2}}^{{d_{{\left( {p,q} \right)}} /2}} {e^{{ - i{\mathbf{G}}_{{\left( {M,N} \right)}} \cdot {\mathbf{r}}}} } {\text{d}}x \hfill \\ & \;\;\;\;\;\;\;\;\; = \frac{1}{{{\text{PQ}}a^{2} }}e^{{ - i\left( {G_{x} p^{*} + G_{y} q^{*} } \right) \cdot a}} \int_{{ - d_{{\left( {p,q} \right)}} /2}}^{{d_{{\left( {p,q} \right)}} /2}} {{\text{d}}y} \int_{{ - d_{{\left( {p,q} \right)}} /2}}^{{d_{{\left( {p,q} \right)}} /2}} {e^{{ - i{\mathbf{G}}_{{\left( {M,N} \right)}} \cdot {\mathbf{r}}}} } {\text{d}}x \hfill \\ &\;\;\;\;\;\;\;\;\; = \left[ {\cos \left( {G_{x} p^{*} a + G_{y} q^{*} a} \right) - i\sin \left( {G_{x} p^{*} a + G_{y} q^{*} a} \right)} \right]F_{{\left( {0,0} \right)}} \left( {{\mathbf{G}}_{{\left( {M,N} \right)}} ,d_{{\left( {p,q} \right)}} } \right) \hfill \\ \end{aligned}$$
(B.1)

where

$$p^{*} = \left\{ {\begin{array}{*{20}l} {p{\mkern 1mu} {\mkern 1mu} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{for}}\;{\text{P}}\;{\text{is}}\;{\text{odd}},} \hfill \\ {p - 0.5p/\left| p \right|{\mkern 1mu} {\mkern 1mu} \;\;\;\;\;{\text{for}}\;{\text{P}}\;{\text{is}}\;{\text{even}},} \hfill \\ \end{array} } \right.{\mkern 1mu} {\mkern 1mu}$$
(B.2)
$$q^{*} = \left\{ {\begin{array}{*{20}l} {q{\mkern 1mu} {\mkern 1mu} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{for Q is}}\;{\text{odd}},} \hfill \\ {q - 0.5q/\left| q \right|{\mkern 1mu} {\mkern 1mu} \;\;\;\;\;\;{\text{for}}\;{\text{Q}}\;{\text{is}}\;{\text{even}}.} \hfill \\ \end{array} } \right.{\mkern 1mu} {\mkern 1mu}$$
(B.3)

Appendix C

The matrices [K] and [M] in Eq. (11) are given by

$$\left[ {\mathbf{K}} \right] = \left[ {\begin{array}{c@{\quad}c@{\quad}c} {\left( {K_{11} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } & {\left( {K_{12} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } & {\left( {K_{13} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } \\ {\left( {K_{21} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } & {\left( {K_{22} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } & {\left( {K_{23} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } \\ {\left( {K_{31} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } & {\left( {K_{32} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } & {\left( {K_{33} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } \\ \end{array} } \right]$$
(C.1)
$$\left[ {\mathbf{M}} \right] = \left[ {\begin{array}{*{20}c} { - \left( {m_{0} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } & {} & {} \\ {} & { - \left( {m_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } & {} \\ {} & {} & { - \left( {m_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } \\ \end{array} } \right]$$
(C.2)

where

$$\begin{aligned} \left( {K_{11} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} &= - \left( {S_{1} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x} } \right) - \left( {S_{1} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y} } \right) \\ &\quad - \left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right)\left( {k_{y} + G_{y} } \right) \\ &\quad - \frac{1}{4}\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y} } \right)\left( {k_{y} + G_{y} } \right) \\ &\quad + \frac{1}{4}\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y} } \right)\left( {k_{y} + G_{y} } \right) \\ &\quad + \frac{1}{4}\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right)\left( {k_{x} + G_{x} } \right) \\ &\quad - \frac{1}{4}\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x} } \right)\left( {k_{x} + G_{x} } \right) \\ \end{aligned}$$
(C.3)
$$\begin{gathered} \left( {K_{12} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} = - i\left( {S_{1} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x} } \right) + \frac{1}{2}i\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right)\left( {k_{y} + G_{y} } \right) \\ - \frac{1}{4}i\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y} } \right)\left( {k_{y} + G_{y} } \right) \\ + \frac{1}{4}i\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x} } \right)\left( {k_{x} + G_{x} } \right) \\ \end{gathered}$$
(C.4)
$$\begin{gathered} \left( {K_{13} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} = - i\left( {S_{1} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y} } \right) + \frac{1}{2}i\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x} } \right)\left( {k_{y} + G_{y} } \right) \\ + \frac{1}{4}i\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y} } \right)\left( {k_{y} + G_{y} } \right) \\ - \frac{1}{4}i\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right)\left( {k_{x} + G_{x} } \right) \\ \end{gathered}$$
(C.5)
$$\begin{gathered} \left( {K_{21} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} = i\left( {S_{1} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right) - \frac{1}{2}i\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y} } \right) \\ + \frac{1}{4}i\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right) \\ - \frac{1}{4}i\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x} } \right) \\ \end{gathered}$$
(C.6)
$$\begin{aligned} \left( {K_{22} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} &= - \left[ {\left( {S_{3} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} + 2\left( {S_{4} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } \right]\left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x} } \right) \\ &\quad - \left( {S_{4} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y} } \right) - \left( {S_{1} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \\ &\quad - \frac{1}{2}\left( {S_{5} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y} } \right)\left( {k_{y} + G_{y} } \right) \\ &\quad - \frac{1}{2}\left( {S_{5} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right)\left( {k_{y} + G_{y} } \right) \\ &\quad - \left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y} } \right) - \frac{1}{4}\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x} } \right) \\ \end{aligned}$$
(C.7)
$$\begin{aligned} \left( {K_{23} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} &= - \left( {S_{3} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right) - \left( {S_{4} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y} } \right) \\ &\quad + \frac{1}{2}\left( {S_{5} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y} } \right)\left( {k_{y} + G_{y} } \right) \\ &\quad + \frac{1}{2}\left( {S_{5} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x} } \right)\left( {k_{y} + G_{y} } \right) \\ &\quad + \frac{1}{2}\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y} } \right) + \frac{1}{4}\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right) \\ \end{aligned}$$
(C.8)
$$\begin{aligned} \left( {K_{31} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} &= i\left( {S_{1} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right) - \frac{1}{2}i\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right) \\ &\quad - \frac{1}{4}i\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y} } \right) \\ &\quad + \frac{1}{4}i\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y} } \right) \\ \end{aligned}$$
(C.9)
$$\begin{aligned} \left( {K_{32} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} & = - \left( {S_{3} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y} } \right) - \left( {S_{4} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right) \\ &\quad + \frac{1}{2}\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right) \\ &\quad + \frac{1}{2}\left( {S_{5} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right)\left( {k_{x} + G_{x} } \right) \\ &\quad + \frac{1}{2}\left( {S_{5} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right)\left( {k_{y} + G_{y} } \right) \\ &\quad + \frac{1}{4}\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y} } \right) \\ \end{aligned}$$
(C.10)
$$\begin{aligned} \left( {K_{33} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} & = - \left[ {\left( {S_{3} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} + 2\left( {S_{4} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} } \right]\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y} } \right) \\ &\quad - \left( {S_{4} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x} } \right) - \left( {S_{1} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \\ &\quad - \frac{1}{2}\left( {S_{5} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x} } \right)\left( {k_{x} + G_{x} } \right) \\ &\quad - \frac{1}{2}\left( {S_{5} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{x} + G_{x} } \right)\left( {k_{y} + G_{y} } \right) \\ &\quad - \left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{x} + G_{x}^{\prime } } \right)\left( {k_{x} + G_{x} } \right) - \frac{1}{4}\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}} \left( {k_{y} + G_{y}^{\prime } } \right)\left( {k_{y} + G_{y} } \right) \\ \end{aligned}$$
(C.11)

in which

$$\alpha_{{{\mathbf{G}} - {\mathbf{G}}^{\prime } }} = \frac{1}{A}\iint_{\Omega } \alpha e^{{ - i\left( {{\mathbf{G}}_{(M,N)} - {\mathbf{G}}_{(m,n)}^{\prime } } \right) \cdot {\mathbf{r}}}} {\text{d}}{\mathbf{r}}$$
(C.12)

where \(\alpha_{{{\mathbf{G}} - {\mathbf{G}}^{\prime } }}\) represents \(\left( {S_{1} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}}\), \(\left( {S_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}}\), \(\left( {S_{3} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}}\), \(\left( {S_{4} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}}\), \(\left( {S_{5} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}}\), \(\left( {m_{0} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}}\) and \(\left( {m_{2} } \right)_{{{\mathbf{G}} - {\mathbf{G^{\prime}}}}}\).

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Shen, W., Cong, Y., Gu, S. et al. A generalized supercell model of defect-introduced phononic crystal microplates. Acta Mech 235, 1345–1360 (2024). https://doi.org/10.1007/s00707-023-03804-y

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