Size-dependent vibration of laminated composite nanoplate with piezo-magnetic face sheets

Abstract

Laminated composites are extensively employed in many aerospace structures due to their excellent mechanical properties. In this paper, a size-dependent model on the basis of the nonlocal strain gradient theory is adopted to reveal the vibration behavior of laminated nanoplate with piezo-magnetic face sheets in its upper and lower surfaces. The governing equations are derived by employing the Hamilton’s principle and Mindlin plate theory. The validation of the present study is carried out by comparison with two previous works and good agreements are achieved. By comparing the vibrational frequencies of composite laminated core sandwich nanoplate with and without the piezo-magnetic face sheets, it is demonstrated that the upper and lower piezo-magnetic face sheets will extensively enhance the vibrational frequencies of laminated core sandwich nanoplates. Furthermore, a comprehensive numerical investigation is performed to examine the influence of the cross-ply laminated type, external electric and magnetic potentials, thickness ratio, size scale parameters, as well as aspect and width-to-thickness ratios on the vibration of the laminated core piezo-magnetic sandwich nanoplate. It is expected that the current work can provide some helpful guidelines for employing the piezo-magnetic surfaces as sensors and actuators to control their vibration behaviors of composite laminated nanostructures.

Appendices

Appendix A: Reduced material parameters of piezo-magnetic face sheets

$$\begin{gathered} \overline{C}_{11}^{f} = C_{11}^{f} - \frac{{\left( {C_{13}^{f} } \right)^{2} }}{{C_{33}^{f} }},\quad \overline{C}_{12}^{f} = C_{12}^{f} - \frac{{C_{13}^{f} C_{23}^{f} }}{{C_{33}^{f} }},\quad \overline{C}_{22}^{f} = C_{22}^{f} - \frac{{\left( {C_{23}^{f} } \right)^{2} }}{{C_{33}^{f} }}, \hfill \\ \quad \overline{C}_{44}^{f} = C_{44}^{f} ,\quad \overline{C}_{55}^{f} = C_{55}^{f} ,\quad \overline{C}_{66}^{f} = C_{66}^{f} , \hfill \\ \end{gathered}$$

(A.1)

$$\overline{e}_{31} = e_{31} - \frac{{C_{13}^{f} e_{33} }}{{C_{33}^{f} }},\quad \overline{e}_{32} = e_{32} - \frac{{C_{23}^{f} e_{33} }}{{C_{33}^{f} }},\quad \overline{e}_{24} = e_{24} ,\quad \overline{e}_{15} = e_{15} ,$$

(A.2)

$$\overline{g}_{31} = g_{31} - \frac{{C_{13}^{f} g_{33} }}{{C_{33}^{f} }},\quad \overline{g}_{32} = g_{32} - \frac{{C_{23}^{f} g_{33} }}{{C_{33}^{f} }},\quad \overline{g}_{24} = g_{24} ,\quad \overline{g}_{15} = g_{15} ,$$

(A.3)

$$\overline{q}_{11} = q_{11} ,\quad \overline{q}_{22} = q_{22} ,\quad \overline{q}_{33} = q_{33} + \frac{{\left( {e_{33} } \right)^{2} }}{{C_{33}^{f} }},$$

(A.4)

$$\overline{d}_{11} = d_{11} ,\quad \overline{d}_{22} = d_{22} ,\quad \overline{d}_{33} = d_{33} + \frac{{q_{33} e_{33} }}{{C_{33}^{f} }},$$

(A.5)

$$\overline{\kappa }_{11} = \kappa_{11} ,\quad \overline{\kappa }_{22} = \kappa_{22} ,\quad \overline{\kappa }_{33} = \kappa_{33} + \frac{{\left( {q_{33} } \right)^{2} }}{{C_{33}^{f} }}.$$

(A.6)

Appendix B: Rigidity coefficients

$$\left\{ {A_{{ij}} ,B_{{ij}} ,D_{{ij}} } \right\} = \int_{{ - h/2}}^{{ - h_{c} /2}} {\bar{C}_{{ij}}^{f} } \left\{ {1,z,z^{2} } \right\}{\text{d}}z +\sum\limits_{{k = 1}}^{N} {\int_{{z_{k} }}^{{z_{{k + 1}} }} {\bar{C}_{{ij}}^{{c} {(k)}} \left\{ {1,z,z^{2} } \right\}} } {\text{d}}z + \int_{{h_{c} /2}}^{{h/2}} {\bar{C}_{{ij}}^{f} \left\{ {1,z,z^{2} } \right\}} {\text{d}}z,$$

(B.1)

$$\begin{gathered} \left\{ {E_{31} ,G_{31} ,E_{32} ,G_{32} } \right\} = \int_{{h_{c} /2}}^{{h_{f} + h_{c} /2}} {\frac{\pi }{{h_{f} }}\sin \left( {\frac{{\pi z^{u} }}{{h_{f} }}} \right)} \left\{ {\overline{e}_{31} ,\overline{g}_{31} ,\overline{e}_{32} ,\overline{g}_{32} } \right\}{\text{d}}z \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \int_{{ - h_{f} - h_{c} /2}}^{{ - h_{c} /2}} {\frac{\pi }{{h_{f} }}\sin \left( {\frac{{\pi z^{b} }}{{h_{f} }}} \right)} \left\{ {\overline{e}_{31} ,\overline{g}_{31} ,\overline{e}_{32} ,\overline{g}_{32} } \right\}{\text{d}}z, \hfill \\ \end{gathered}$$

(B.2)

$$\begin{gathered} \left\{ {\tilde{E}_{31} ,\tilde{G}_{31} ,\tilde{E}_{32} ,\tilde{G}_{32} } \right\} = \int_{{h_{c} /2}}^{{h_{f} + h_{c} /2}} {\left( {\frac{\pi z}{{h_{f} }}} \right)\sin \left( {\frac{{\pi z^{u} }}{{h_{f} }}} \right)\left\{ {\overline{e}_{31} ,\overline{g}_{31} ,\overline{e}_{32} ,\overline{g}_{32} } \right\}} {\text{d}}z \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \int_{{ - h_{f} - h_{c} /2}}^{{ - h_{c} /2}} {\left( {\frac{\pi z}{{h_{f} }}} \right)\sin \left( {\frac{{\pi z^{b} }}{{h_{f} }}} \right)\left\{ {\overline{e}_{31} ,\overline{g}_{31} ,\overline{e}_{32} ,\overline{g}_{32} } \right\}} {\text{d}}z, \hfill \\ \end{gathered}$$

(B.3)

$$\begin{gathered} \left\{ {E_{15} ,G_{15} ,E_{24} ,G_{24} } \right\} = \int_{{h_{c} /2}}^{{h_{f} + h_{c} /2}} {\cos \left( {\frac{{\pi z^{u} }}{{h_{f} }}} \right)\left\{ {\overline{e}_{15} ,\overline{g}_{15} ,\overline{e}_{24} ,\overline{g}_{24} } \right\}} {\text{d}}z \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \int_{{ - h_{f} - h_{c} /2}}^{{ - h_{c} /2}} {\cos \left( {\frac{{\pi z^{b} }}{{h_{f} }}} \right)\left\{ {\overline{e}_{15} ,\overline{g}_{15} ,\overline{e}_{24} ,\overline{g}_{24} } \right\}} {\text{d}}z, \hfill \\ \end{gathered}$$

(B.4)

$$\begin{gathered} \left\{ {Q_{11} ,Q_{22} ,\tilde{D}_{11} ,\tilde{D}_{22} ,K_{11} ,K_{22} } \right\} = \int_{{h_{c} /2}}^{{h_{f} + h_{c} /2}} {\left[ {\cos \left( {\frac{{\pi z^{u} }}{{h_{f} }}} \right)} \right]^{2} \left\{ {\overline{q}_{11} ,\overline{q}_{22} ,\overline{d}_{11} ,\overline{d}_{22} ,\overline{\kappa }_{11} ,\overline{\kappa }_{22} } \right\}} {\text{d}}z \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \int_{{ - h_{f} - h_{c} /2}}^{{ - h_{c} /2}} {\left[ {\cos \left( {\frac{{\pi z^{b} }}{{h_{f} }}} \right)} \right]^{2} \left\{ {\overline{q}_{11} ,\overline{q}_{22} ,\overline{d}_{11} ,\overline{d}_{22} ,\overline{\kappa }_{11} ,\overline{\kappa }_{22} } \right\}} {\text{d}}z, \hfill \\ \end{gathered}$$

(B.5)

$$\begin{gathered} \left\{ {Q_{33} ,\tilde{D}_{33} ,K_{33} } \right\} = \int_{{h_{c} /2}}^{{h_{f} + h_{c} /2}} {\left[ {\left( {\frac{\pi }{{h_{f} }}} \right)\sin \left( {\frac{{\pi z^{u} }}{{h_{f} }}} \right)} \right]^{2} \left\{ {\overline{q}_{33} ,\overline{d}_{33} ,\overline{\kappa }_{33} } \right\}} {\text{d}}z \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \int_{{ - h_{f} - h_{c} /2}}^{{ - h_{c} /2}} {\left[ {\left( {\frac{\pi }{{h_{f} }}} \right)\sin \left( {\frac{{\pi z^{b} }}{{h_{f} }}} \right)} \right]^{2} \left\{ {\overline{q}_{33} ,\overline{d}_{33} ,\overline{\kappa }_{33} } \right\}} {\text{d}}z. \hfill \\ \end{gathered}$$

(B.6)

Appendix C: Elements in stiffness and mass matrices

$$\begin{gathered} k_{11} = \beta_{l} \left( { - A_{11} \chi_{m}^{2} - A_{66} \chi_{n}^{2} } \right),k_{12} = k_{21} = \beta_{l} \left( { - A_{12} \chi_{m} \chi_{n} - A_{66} \chi_{m} \chi_{n} } \right),k_{13} = k_{31} = 0,k_{14} = k_{41} = \beta_{l} \left( { - B_{11} \chi_{m}^{2} - B_{66} \chi_{n}^{2} } \right), \hfill \\ k_{15} = k_{24} = k_{42} = k_{51} = \beta_{l} \left( { - B_{12} \chi_{m} \chi_{n} - B_{66} \chi_{m} \chi_{n} } \right),k_{16} = E_{31} \chi_{m} ,k_{17} = G_{31} \chi_{m} ,k_{22} = \beta_{l} \left( { - A_{22} \chi_{n}^{2} - A_{66} \chi_{m}^{2} } \right),k_{23} = k_{32} = 0, \hfill \\ k_{25} = k_{52} = \beta_{l} \left( { - B_{22} \chi_{n}^{2} - B_{66} \chi_{m}^{2} } \right),k_{26} = E_{32} \chi_{n} ,k_{27} = G_{32} \chi_{n} , \hfill \\ k_{33} = - \beta_{l} k_{s} A_{44} \chi_{m}^{2} - \beta_{l} k_{s} A_{55} \chi_{n}^{2} - \beta_{ea} \left( {N_{xx}^{E} + N_{xx}^{M} } \right)\chi_{m}^{2} - \beta_{ea} \left( {N_{yy}^{E} + N_{yy}^{M} } \right)\chi_{n}^{2} , \hfill \\ k_{34} = k_{43} = \beta_{l} \left( { - k_{s} A_{44} \chi_{m} } \right),k_{35} = k_{53} = \beta_{l} \left( { - k_{s} A_{55} \chi_{n} } \right),k_{36} = \left( {k_{s} E_{15} \chi_{m}^{2} + k_{s} E_{24} \chi_{n}^{2} } \right), \hfill \\ k_{37} = \left( {k_{s} G_{15} \chi_{m}^{2} + k_{s} G_{24} \chi_{n}^{2} } \right),k_{44} = \beta_{l} \left( { - D_{11} \chi_{m}^{2} - D_{66} \chi_{n}^{2} - k_{s} A_{44} } \right), \hfill \\ k_{45} = k_{54} = \beta_{l} \left( { - D_{12} \chi_{m} \chi_{n} - D_{66} \chi_{m} \chi_{n} } \right),k_{46} = \left( {\tilde{E}_{31} + k_{s} E_{15} } \right)\chi_{m} ,k_{47} = \left( {\tilde{G}_{31} + k_{s} G_{15} } \right)\chi_{m} , \hfill \\ k_{55} = \beta_{l} \left( { - D_{22} \chi_{n}^{2} - D_{66} \chi_{m}^{2} - k_{s} A_{55} } \right),k_{56} = \left( {\tilde{E}_{32} + k_{s} E_{24} } \right)\chi_{n} ,k_{57} = \left( {\tilde{G}_{32} + k_{s} G_{24} } \right)\chi_{n} \hfill \\ k_{61} = \beta_{l} E_{31} \chi_{m} ,k_{62} = \beta_{l} E_{32} \chi_{n} ,k_{63} = \beta_{l} \left( {E_{15} \chi_{m}^{2} + E_{24} \chi_{n}^{2} } \right),k_{64} = \beta_{l} \left( {E_{15} \chi_{m} + \tilde{E}_{31} \chi_{m} } \right) \hfill \\ k_{65} = \beta_{l} \left( {E_{24} \chi_{n} + \tilde{E}_{32} \chi_{n} } \right),k_{66} = Q_{11} \chi_{m}^{2} + Q_{22} \chi_{n}^{2} + Q_{33} ,k_{67} = k_{76} = \tilde{D}_{11} \chi_{m}^{2} + \tilde{D}_{22} \chi_{n}^{2} + \tilde{D}_{33} \hfill \\ k_{71} = \beta_{l} G_{31} \chi_{m} ,k_{72} = \beta_{l} G_{32} \chi_{n} ,k_{73} = \beta_{l} \left( {G_{15} \chi_{m}^{2} + G_{24} \chi_{n}^{2} } \right),k_{74} = \beta_{l} \left( {G_{15} \chi_{m} + \tilde{G}_{31} \chi_{m} } \right), \hfill \\ k_{75} = \beta_{l} \left( {G_{24} \chi_{n} + \tilde{G}_{32} \chi_{n} } \right),k_{77} = K_{11} \chi_{m}^{2} + K_{22} \chi_{n}^{2} + K_{33} , \hfill \\ \end{gathered}$$

(C.1)

$$m_{11} = m_{22} = m_{33} = - \beta_{ea} I_{0} ,\quad m_{14} = m_{41} = m_{25} = m_{52} = - \beta_{ea} I_{1} ,\quad m_{44} = m_{55} = - \beta_{ea} I_{2} ,$$

(C.2)

where

$$\beta_{l} = 1 + l_{0}^{2} \left( {\chi_{m}^{2} + \chi_{n}^{2} } \right),\quad \beta_{ea} = 1 + (e_{0} a)^{2} \left( {\chi_{m}^{2} + \chi_{n}^{2} } \right).$$

(C.3)