Band structures of transverse waves in nanoscale multilayered phononic crystals with nonlocal interface imperfections by using the radial basis function method

Abstract

A radial basis function collocation method based on the nonlocal elastic continuum theory is developed to compute the band structures of nanoscale multilayered phononic crystals. The effects of nonlocal imperfect interfaces on band structures of transverse waves propagating obliquely or vertically in the system are studied. The correctness of the present method is verified by comparing the numerical results with those obtained by applying the transfer matrix method in the case of nonlocal perfect interface. Furthermore, the influences of the nanoscale size, the impedance ratio and the incident angle on the cut-off frequency and band structures are investigated and discussed in detail. Numerical results show that the nonlocal interface imperfections have significant effects on the band structures in the macroscopic and microscopic scale.

Appendix

The matrices and vectors in Eq. (24) are as follows

$$\begin{aligned} {\varvec{\alpha }} _i= & {} \left( {{\varvec{\alpha }} _i^1 ,\hbox { }{\varvec{\alpha }} _i^2 ,\hbox { }\ldots ,\hbox { }{\varvec{\alpha }} _i^n } \right) , \end{aligned}$$

(A1)

$$\begin{aligned} {{\varvec{A}}}_i= & {} -D_i^2 \sin ^{2}\theta \cdot \left[ {\phi _i^s (\gamma _i^l )} \right] _{\left( {n-2} \right) \times n} , \end{aligned}$$

(A2)

$$\begin{aligned} {{\varvec{B}}}_i= & {} q_i \cdot \left[ {\phi _i^s (\gamma _i^l )} \right] _{\left( {n-2} \right) \times n} +D_i^2 \cdot \left[ {\phi _i^{s~{\prime }{\prime }}(\gamma _i^l )} \right] _{\left( {n-2} \right) \times n} , \end{aligned}$$

(A3)

$$\begin{aligned} {{\varvec{C}}}_i= & {} \frac{c_{Ti}^2 }{c_{T1}^2 }\cdot \left[ {\phi _i^{s~{\prime }{\prime }}(\gamma _i^l )} \right] _{\left( {n-2} \right) \times n} \hbox {, }l=2,3,\ldots ,n-1,\hbox { }s=1,2,\ldots ,n,\nonumber \\ \end{aligned}$$

(A4)

$$\begin{aligned} {{\varvec{M}}}= & {} \left[ {{\begin{array}{ll} {\left[ {{\begin{array}{ll} {\left[ {{\begin{array}{ll} {\varvec{0}}&{} {{\varvec{E}}_{n\times n} } \\ {{\varvec{C}}_1 }&{} {-{\varvec{B}}_1 } \\ \end{array} }} \right] }&{} {\hbox { }\left[ 0 \right] _{(2n-2)\times 2n} } \\ {\left[ 0 \right] _{(2n-2)\times 2n} }&{} {\hbox { }\left[ {{\begin{array}{ll} {\varvec{0}}&{} {{\varvec{E}}_{n\times n} } \\ {{\varvec{C}}_2 }&{} {-{\varvec{B}}_2 } \\ \end{array} }} \right] } \\ \end{array} }} \right] } \\ {{\begin{array}{ll} {{\begin{array}{ll} {{\begin{array}{ll} {\left[ {\exp (\hbox {i}{\varvec{k}}_x d)\phi _1^s \left( {\gamma _1^1 } \right) } \right] }&{} {\left[ 0 \right] _{1\times n} } \\ \end{array} }} \\ {{\begin{array}{ll} {\left[ {\phi _1^s \left( {\gamma _1^n } \right) } \right] }&{} {\hbox { }\left[ 0 \right] _{1\times n} } \\ \end{array} }} \\ \end{array} }}&{} {{\begin{array}{ll} {{\begin{array}{ll} {\left[ {-\phi _2^s \left( {\gamma _2^n } \right) } \right] }&{} {\left[ 0 \right] _{1\times n} } \\ \end{array} }} \\ {{\begin{array}{ll} {\left[ {-\phi _2^s \left( {\gamma _2^1 } \right) } \right] }&{} {\left[ 0 \right] _{1\times n} } \\ \end{array} }} \\ \end{array} }} \\ \end{array} }} \\ {{\begin{array}{ll} {{\begin{array}{ll} {\left[ {\exp (\hbox {i}{\varvec{k}}_x d)c_{44}^{(1)} V_1 } \right] }&{} {\hbox { }\left[ 0 \right] _{1\times n} } \\ \end{array} }}&{} {{\begin{array}{ll} {\left[ {-c_{\hbox {44}}^{\hbox {(2)}} \hbox {V}_{2} } \right] \hbox { }}&{} {\left[ \hbox {0} \right] _{{1\times n}} } \\ \end{array} }} \\ \end{array} }} \\ {{\begin{array}{ll} {{\begin{array}{ll} {\left[ {c_{44}^{(1)} J_1 } \right] }&{} {\hbox { }\left[ 0 \right] _{1\times n} } \\ \end{array} }}&{} {{\begin{array}{ll} {\left[ {-c_{44}^{(2)} J_2 } \right] \hbox { }}&{} {\left[ 0 \right] _{1\times n} } \\ \end{array} }} \\ \end{array} }} \\ \end{array} }} \right] , \end{aligned}$$

(A5)

$$\begin{aligned} {{\varvec{N}}}= & {} \left[ {{\begin{array}{ll} {\left[ {{\begin{array}{ll} {\left[ {{\begin{array}{ll} {{{\varvec{E}}}_{n\times n} }&{} \mathbf{0} \\ \mathbf{0}&{} {{\varvec{A}}_i } \\ \end{array} }} \right] }&{} {\hbox { }\left[ 0 \right] _{(2n-2)\times 2n} } \\ {\left[ 0 \right] _{(2n-2)\times 2n} }&{} {\hbox { }\left[ {{\begin{array}{ll} {{{\varvec{E}}}_{n\times n} }&{} \mathbf{0} \\ \mathbf{0}&{} {{{\varvec{A}}}_i } \\ \end{array} }} \right] } \\ \end{array} }} \right] } \\ {{\begin{array}{ll} {\left[ 0 \right] _{4\times 2n} }&{} {\hbox { }\left[ 0 \right] _{4\times 2n} } \\ \end{array} }} \\ \end{array} }} \right] , \end{aligned}$$

(A6)

$$\begin{aligned} {\varvec{V}}_1= & {} \left[ {\int _{\varOmega _1 } {\frac{1}{2D_1 }} \hbox {exp}(\frac{-\left| {\gamma _1^1 -\gamma _1 } \right| }{D})\phi _1^{s~{\prime }}(\gamma _1 )d\gamma _1 } \right] _{1\times n} ,\nonumber \\ {\varvec{V}}_2= & {} \left[ {\int _{\varOmega _2 } {\frac{1}{2D_2 }} \hbox {exp}(\frac{-\left| {\gamma _2^n -\gamma _2 } \right| }{D_2 })\phi _2^{s~{\prime }}(\gamma _2 )d\gamma _2 } \right] _{1\times n} ,\hbox { }s=1,2,\ldots ,n, \nonumber \\\end{aligned}$$

(A7)

$$\begin{aligned} {{\varvec{J}}}_1= & {} \left[ {\int _{\varOmega _1 } {\frac{1}{2D_1 }} \hbox {exp}(\frac{-\left| {\gamma _1^n -\gamma _1 } \right| }{D_1 })\phi _1^{s~{\prime }}(\gamma _1 )d\gamma _1 } \right] _{1\times n} ,\nonumber \\ {\varvec{J}}_2= & {} \left[ {\int _{\varOmega _2 } {\frac{1}{2D_2 }} \hbox {exp}(\frac{-\left| {\gamma _2^1 -\gamma _2 } \right| }{D_2 })\phi _2^{s~{\prime }}(\gamma _2 )d\gamma _2 } \right] _{1\times n} ,\hbox { }s=1,2,\ldots ,n,\nonumber \\ \end{aligned}$$

(A8)