Magneto-electro-elastic node-based smoothed point interpolation method for micromechanical analysis of natural frequencies of nanobeams

Abstract

As an addition to the traditional finite element method (FEM), the magneto-electro-elastic node-based smoothed point interpolation method (MEE-NS-PIM) with asymptotic homogenization method (AHM) is presented to solve the micromechanical problems of MEE nanobeams, which overcomes the deficiency of FEM and improves the accuracy of the calculation results. Firstly, the basic equations of MEE medium are derived. Secondly, AHM is adopted to calculate the property parameters of MEE materials under microcosmic situations, and the AHM model is illustrated. Then, the relative formulations of the discretized system used to calculate the frequency of MEE nanostructures are deduced based on MEE-NS-PIM. Moreover, several numerical examples are calculated, and the results of MEE-NS-PIM are compared with those of FEM, which proves the convergence, precision, and effectiveness of MEE-NS-PIM. Therefore, MEE-NS-PIM combined with AHM can be used to analyze the microcosmic MEE coupling problems and obtain a more accurate and reliable solution for MEE micromechanics.

Appendix

Substituting Eqs. (27) and (30) into Eq. (24) and equating like powers of $\eta$, we have

$$\begin{aligned} T_{ij}^{(0)}= & {} c_{ijkl} \left( {\frac{{\varvec{\partial }} u_{k}^{(0)} }{{\varvec{\partial }} x_{l} }+\frac{{\varvec{\partial }} u_{k}^{(1)} }{{\varvec{\partial }} y_{l} }} \right) -e_{kij} E_{k}^{(0)} -q_{kij} H_{k}^{(0)}, \end{aligned}$$

(A.1)

$$\begin{aligned} T_{ij}^{(1)}= & {} c_{ijkl} \left( {\frac{{\varvec{\partial }} u_{k}^{(1)} }{{\varvec{\partial }} x_{l} }+\frac{{\varvec{\partial }} u_{k}^{(2)} }{{\varvec{\partial }} y_{l} }} \right) -e_{kij} E_{k}^{(1)} -q_{kij} H_{k}^{(1)}. \end{aligned}$$

(A.2)

Then, substituting Eqs. (27) and (28) into Eq. (25), Eqs. (27) and (29) into Eq. (26), we asymptotically expand the electric displacement and magnetic induction:

$$\begin{aligned} D_{i}^{(0)}= & {} c_{ikl} \left( {\frac{{\varvec{\partial }} u_{k}^{(0)} }{{\varvec{\partial }} x_{l} }+\frac{{\varvec{\partial }} u_{k}^{(1)} }{{\varvec{\partial }} y_{l} }} \right) +\varepsilon _{ik} E_{k}^{(0)} +m_{ik} H_{k}^{(0)}, \end{aligned}$$

(A.3)

$$\begin{aligned} D_{i}^{(1)}= & {} c_{ikl} \left( {\frac{{\varvec{\partial }} u_{k}^{(1)} }{{\varvec{\partial }} x_{l} }+\frac{{\varvec{\partial }} u_{k}^{(2)} }{{\varvec{\partial }} y_{l} }} \right) +\varepsilon _{ik} E_{k}^{(1)} +m_{ik} H_{k}^{(1)}, \end{aligned}$$

(A.4)

$$\begin{aligned} B_{i}^{(0)}= & {} q_{jkl} \left( {\frac{{\varvec{\partial }} u_{k}^{(0)} }{{\varvec{\partial }} x_{l} }+\frac{{\varvec{\partial }} u_{k}^{(1)} }{{\varvec{\partial }} y_{l} }} \right) +m_{ik} E_{k}^{(0)} +\mu _{ik} H_{k}^{(0)}, \end{aligned}$$

(A.5)

$$\begin{aligned} B_{i}^{(1)}= & {} q_{jkl} \left( {\frac{{\varvec{\partial }} u_{k}^{(1)} }{{\varvec{\partial }} x_{l} }+\frac{{\varvec{\partial }} u_{k}^{(2)} }{{\varvec{\partial }} y_{l} }} \right) +m_{ik} E_{k}^{(1)} +\mu _{ik} H_{k}^{(1)}. \end{aligned}$$

(A.6)

The set of effective coefficients related to the MEE nanostructure is given as below:

$$\begin{aligned} \tilde{{c}}_{ijkl}= & {} \frac{1}{\left| Y \right| }\int _Y {\left( {c_{ijkl} ({\mathbf{y}})+c_{ijmn} ({\mathbf{y}})\frac{{\varvec{\partial }} N_{m}^{kl} }{{\varvec{\partial }} y_{n} }} \right) \text {d}\theta }, \end{aligned}$$

(A.7)

$$\begin{aligned} \tilde{{e}}_{ijk}= & {} \frac{1}{\left| Y \right| }\int _Y {\left( {e_{ijk} ({\mathbf{y}})-c_{ijmn} ({\mathbf{y}})\frac{{\varvec{\partial }} N_{m}^{i} }{{\varvec{\partial }} y_{n} }} \right) \text {d}\theta } =\frac{1}{\left| Y \right| }\int _Y {\left( {e_{ijk} ({\mathbf{y}})+e_{imn} ({\mathbf{y}})\frac{{\varvec{\partial }} N_{m}^{jk} }{{\varvec{\partial }} y_{n} }} \right) \text {d}\theta }, \end{aligned}$$

(A.8)

$$\begin{aligned} \tilde{{\varepsilon }}_{ij}= & {} \frac{1}{\left| Y \right| }\int _Y {\left( {\varepsilon _{ij} ({\mathbf{y}})+e_{imn} ({\mathbf{y}})\frac{{\varvec{\partial }} N_{m}^{j} }{{\varvec{\partial }} y_{n} }} \right) \text {d}\theta }, \end{aligned}$$

(A.9)

$$\begin{aligned} \tilde{{q}}_{ijk}= & {} \frac{1}{\left| Y \right| }\int _Y {\left( {\tilde{{q}}_{ijk} ({\mathbf{y}})-c_{jkmn} ({\mathbf{y}})\frac{{\varvec{\partial }} M_{m}^{i} }{{\varvec{\partial }} y_{n} }} \right) \text {d}\theta } =\frac{1}{\left| Y \right| }\int _Y {\left( {q_{ijk} ({\mathbf{y}})+q_{imn} ({\mathbf{y}})\frac{{\varvec{\partial }} N_{m}^{jk} }{{\varvec{\partial }} y_{n} }} \right) \text {d}\theta }, \end{aligned}$$

(A.10)

$$\begin{aligned} \tilde{{m}}_{ijk}= & {} \frac{1}{\left| Y \right| }\int _Y {\left( {m_{ij} ({\mathbf{y}})+e_{imn} ({\mathbf{y}})\frac{{\varvec{\partial }} M_{m}^{j} }{{\varvec{\partial }} y_{n} }} \right) \text {d}\theta } =\frac{1}{\left| Y \right| }\int _Y {\left( {m_{ij} ({\mathbf{y}})+q_{imn} ({\mathbf{y}})\frac{{\varvec{\partial }} N_{m}^{j} }{{\varvec{\partial }} y_{n} }} \right) \text {d}\theta }, \end{aligned}$$

(A.11)

$$\begin{aligned} \tilde{{\mu }}_{ij}= & {} \frac{1}{\left| Y \right| }\int _Y {\left( {\mu _{ij} ({\mathbf{y}})+q_{imn} ({\mathbf{y}})\frac{{\varvec{\partial }} M_{m}^{j} }{{\varvec{\partial }} y_{n} }} \right) \text {d}\theta }. \end{aligned}$$

(A.12)

The local functions $N_{m}^{kl}$, $N_{m}^{i}$, $M_{m}^{i}$, and $W_{m}^{k}$ in Eqs. (41)–(46) satisfy the relevant unit-cell problems and should be first calculated from the appropriate unit-cell problems and then used to obtain the effective coefficients.

The effective coefficients of MEE materials were determined based on the AHM as follows [10]:

$$\begin{aligned} \tilde{{c}}_{11}= & {} \left\langle {c_{11} } \right\rangle -\left\langle {c_{12}^{2} c_{22}^{-1} } \right\rangle +\left\langle {c_{12} c_{22}^{-1} } \right\rangle ^{2}\left\langle {c_{22}^{-1} } \right\rangle ^{-1}, \tilde{{c}}_{12} =\left\langle {c_{12} c_{22}^{-1} } \right\rangle \left\langle {c_{22}^{-1} } \right\rangle ^{-1}, \end{aligned}$$

(A.13)

$$\begin{aligned} \tilde{{c}}_{13}= & {} \left\langle {c_{13} } \right\rangle -\left\langle {c_{12} c_{23} c_{22}^{-1} } \right\rangle +\left\langle {c_{12} c_{22}^{-1} } \right\rangle \left\langle {c_{22}^{-1} } \right\rangle ^{-1}\left\langle {c_{23} c_{22}^{-1} } \right\rangle , \tilde{{c}}_{22} =\left\langle {c_{22}^{-1} } \right\rangle ^{-1}, \end{aligned}$$

(A.14)

$$\begin{aligned} \tilde{{c}}_{33}= & {} \left\langle {c_{\text {33}} } \right\rangle -\left\langle {c_{23}^{2} c_{22}^{-1} } \right\rangle +\left\langle {c_{23} c_{22}^{-1} } \right\rangle ^{2}\left\langle {c_{22}^{-1} } \right\rangle ^{-1}, \tilde{{c}}_{23} =\left\langle {c_{23} c_{22}^{-1} } \right\rangle \left\langle {c_{22}^{-1} } \right\rangle ^{-1}, \end{aligned}$$

(A.15)

$$\begin{aligned} \tilde{{c}}_{44}= & {} \left\langle {c_{44}^{-1} } \right\rangle ^{-1}, \tilde{{c}}_{55} =\left\langle {c_{55}^{-1} } \right\rangle ^{-1}, \tilde{{c}}_{66} =\left\langle {c_{66}^{-1} } \right\rangle ^{-1}, \end{aligned}$$

(A.16)

$$\begin{aligned} \tilde{{e}}_{31}= & {} \left\langle {e_{31} } \right\rangle -\left\langle {c_{12} e_{32} c_{22}^{-1} } \right\rangle +\left\langle {c_{12} c_{22}^{-1} } \right\rangle \left\langle {c_{22}^{-1} } \right\rangle ^{-1}\left\langle {e_{32} c_{22}^{-1} } \right\rangle , \tilde{{e}}_{32} =\left\langle {c_{22}^{-1} } \right\rangle ^{-1}\left\langle {e_{32} c_{22}^{-1} } \right\rangle , \end{aligned}$$

(A.17)

$$\begin{aligned} \tilde{{e}}_{33}= & {} \left\langle {e_{33} } \right\rangle -\left\langle {c_{23} e_{32} e_{22}^{-1} } \right\rangle +\left\langle {c_{23} c_{22}^{-1} } \right\rangle \left\langle {c_{22}^{-1} } \right\rangle ^{-1}\left\langle {e_{32} c_{22}^{-1} } \right\rangle , \tilde{{e}}_{24} =\left\langle {c_{44}^{-1} } \right\rangle ^{-1}\left\langle {e_{24} c_{44}^{-1} } \right\rangle , \end{aligned}$$

(A.18)

$$\begin{aligned} \tilde{{e}}_{15}= & {} \left\langle {e_{15} } \right\rangle , \end{aligned}$$

(A.19)

$$\begin{aligned} \tilde{{\varepsilon }}_{11}= & {} \left\langle {\varepsilon _{11} } \right\rangle , \tilde{{\varepsilon }}_{22} =\left\langle {\varepsilon _{22} } \right\rangle +\left\langle {e_{24}^{2} c_{44}^{-1} } \right\rangle -\left\langle {e_{24} c_{44}^{-1} } \right\rangle ^{2}\left\langle {c_{44}^{-1} } \right\rangle ^{-1}, \end{aligned}$$

(A.20)

$$\begin{aligned} \tilde{{\varepsilon }}_{33}= & {} \left\langle {\varepsilon _{33} } \right\rangle +\left\langle {e_{32}^{2} c_{22}^{-1} } \right\rangle -\left\langle {e_{32} c_{22}^{-1} } \right\rangle ^{2}\left\langle {c_{22}^{-1} } \right\rangle ^{-1}, \end{aligned}$$

(A.21)

$$\begin{aligned} \tilde{{q}}_{31}= & {} \left\langle {q_{31} } \right\rangle -\left\langle {c_{12} q_{32} c_{22}^{-1} } \right\rangle +\left\langle {c_{12} c_{22}^{-1} } \right\rangle \left\langle {c_{22}^{-1} } \right\rangle ^{-1}\left\langle {q_{32} c_{22}^{-1} } \right\rangle , \tilde{{q}}_{32} =\left\langle {c_{22}^{-1} } \right\rangle ^{-1}\left\langle {q_{32} c_{22}^{-1} } \right\rangle , \end{aligned}$$

(A.22)

$$\begin{aligned} \tilde{{q}}_{33}= & {} \left\langle {q_{33} } \right\rangle -\left\langle {c_{23} q_{32} c_{22}^{-1} } \right\rangle +\left\langle {c_{23} c_{22}^{-1} } \right\rangle \left\langle {c_{22}^{-1} } \right\rangle ^{-1}\left\langle {q_{32} c_{22}^{-1} } \right\rangle , \tilde{{q}}_{24} =\left\langle {c_{44}^{-1} } \right\rangle ^{-1}\left\langle {q_{24} c_{44}^{-1} } \right\rangle , \end{aligned}$$

(A.23)

$$\begin{aligned} \tilde{{q}}_{15}= & {} \left\langle {q_{15} } \right\rangle , \end{aligned}$$

(A.24)

$$\begin{aligned} \tilde{{m}}_{11}= & {} \left\langle {m_{11} } \right\rangle , \tilde{{m}}_{22} =\left\langle {m_{22} } \right\rangle +\left\langle {e_{24} q_{24} c_{44}^{-1} } \right\rangle -\left\langle {e_{24} c_{44}^{-1} } \right\rangle \left\langle {c_{44}^{-1} } \right\rangle ^{-1}\left\langle {q_{24} c_{44}^{-1} } \right\rangle , \end{aligned}$$

(A.25)

$$\begin{aligned} \tilde{{m}}_{33}= & {} \left\langle {m_{33} } \right\rangle +\left\langle {e_{32} q_{32} c_{22}^{-1} } \right\rangle -\left\langle {e_{32} c_{22}^{-1} } \right\rangle \left\langle {c_{22}^{-1} } \right\rangle ^{-1}\left\langle {q_{32} c_{22}^{-1} } \right\rangle , \end{aligned}$$

(A.26)

$$\begin{aligned} \tilde{{\mu }}_{11}= & {} \left\langle {\mu _{11} } \right\rangle , \tilde{{\mu }}_{22} =\left\langle {\mu _{22} } \right\rangle +\left\langle {q_{24}^{2} c_{44}^{-1} } \right\rangle -\left\langle {q_{24} c_{44}^{-1} } \right\rangle ^{2}\left\langle {c_{44}^{-1} } \right\rangle ^{-1}, \end{aligned}$$

(A.27)

$$\begin{aligned} \tilde{{\mu }}_{33}= & {} \left\langle {\mu _{33} } \right\rangle +\left\langle {q_{32}^{2} c_{22}^{-1} } \right\rangle -\left\langle {q_{32} c_{22}^{-1} } \right\rangle ^{2}\left\langle {c_{22}^{-1} } \right\rangle ^{-1} \end{aligned}$$

(A.28)

where the quantities in the angular bracket refer to averaged quantities.