Hygrothermal modeling of the buckling behavior of sandwich plates with nanocomposite face sheets resting on a Pasternak foundation

Abstract

In this work we investigate the buckling response of sandwich plates with a polymeric core and two face sheets reinforced by carbon nanotubes (CNTs). The problem is tackled analytically by means of a higher-order sandwich plate theory, where the face sheets are modeled according to a classical plate theory and modified strain gradient theory with temperature-dependent and moisture-dependent material properties. A Mori–Tanaka method is applied to determine the mechanical properties associated with the face sheets, while considering the agglomeration effect of CNTs. The governing equations of the problem are derived from the Hamilton’s principle, whose solutions are recovered by means of a Navier–Stokes method. A thorough sensitivity study of the structural response to different parameters includes the agglomeration and volume fraction of CNTs, the material length scale parameter, the side and aspect ratios, together with the temperature variation and humidity conditions. The sandwich plates are assumed to be immersed within an orthotropic Pasternak foundation, whose normal and shear moduli can affect the overall buckling response of the structure. Numerical experiments show that sandwich plates with nanocomposite face sheets, resting on orthotropic elastic foundations, feature an increased stiffness, where the proposed formulation yields accurate results, due to the possibility of considering the variation in temperature, humidity and agglomeration of the reinforcing CNTs within the solution.

Appendix

For the top face sheet

$$\begin{aligned}&\begin{array}{l} -\rho _{\mathrm{t}} {h}_{\mathrm{t}} \frac{\partial ^{2}{u}_{\mathrm{0t}} }{\partial {t}^{2}}+{A}_{11}^{\mathrm{t}} \left( {\frac{\partial ^{2}{u}_{\mathrm{0t}} }{\partial {x}^{2}}-\frac{\partial ^{2}\alpha _{11}^{\mathrm{t}} }{\partial {x}}\Delta {T}} \right) +{A}_{12}^{\mathrm{t}} \left( {\frac{\partial ^{2}{v}_{\mathrm{0t}} }{\partial {x}\partial {y}}-\frac{\partial \alpha _{22}^{\mathrm{t}} }{\partial {x}}\Delta {T}} \right) +{A}_{66}^{\mathrm{t}} \left( {\frac{\partial ^{2}{u}_{\mathrm{0t}} }{\partial {x}^{2}}-\frac{\partial ^{2}{v}_{\mathrm{0t}} }{\partial {x}\partial {y}}} \right) \\ -{B}_{11}^{\mathrm{t}} \frac{\partial ^{3}{w}_{\mathrm{t}} }{\partial {x}^{3}}-{B}_{12}^{\mathrm{t}} \frac{{\partial }^{3}{w}_{\mathrm{t}} }{\partial {x}\partial {y}^{2}}-2{B}_{66}^{\mathrm{t}} \frac{{\partial }^{3}{w}_{\mathrm{t}} }{\partial {x}\partial {y}^{2}}-2{G}_{12} {l}_{0}^{2} {h}_{\mathrm{t}} \left( {\frac{{\partial }^{4}{u}_{0{\mathrm{t}}} }{\partial {x}^{4}}+\frac{\partial ^{4}{v}_{0{\mathrm{t}}} }{\partial {x}^{3}\partial {y}}+\frac{{\partial }^{4}{u}_{0{\mathrm{t}}} }{\partial {x}^{2}\partial {y}^{2}}+\frac{{\partial }^{4}{v}_{0{\mathrm{t}}} }{\partial {x}\partial {y}^{3}}} \right) \\ -2{G}_{12} {l}_{0}^{2} {h}_{\mathrm{t}} \left( {\frac{4}{5}\frac{{\partial }^{4}{u}_{0{\mathrm{t}}} }{\partial {x}^{4}}+\frac{4}{15}\frac{{\partial }^{4}{v}_{0{\mathrm{t}}} }{\partial {x}^{3}\partial {y}}+\frac{4}{3}\frac{{\partial }^{4}{u}_{0{\mathrm{t}}} }{\partial {x}^{2}\partial {y}^{2}}+\frac{4}{15}\frac{{\partial }^{4}{v}_{0{\mathrm{t}}} }{\partial {x}\partial {y}^{3}}+\frac{8}{15}\frac{{\partial }^{4}{u}_{0{\mathrm{t}}} }{\partial {y}^{4}}} \right) \\ -\frac{1}{4}{G}_{12} {l}_{2}^{2} {h}_{\mathrm{t}} \left( {\frac{{\partial }^{4}{u}_{\mathrm{ot}} }{\partial {x}^{2}\partial {y}^{2}}-\frac{{\partial }^{4}{v}_{\mathrm{ot}} }{\partial {x}^{3}\partial {y}}+\frac{{\partial }^{4}{u}_{\mathrm{ot}} }{\partial {y}^{4}}-\frac{\partial ^{4}{v}_{\mathrm{ot}} }{\partial x\partial {y}^{3}}} \right) -\lambda _{\mathrm{xt}} =0 \end{array} \end{aligned}$$

(A1)

$$\begin{aligned}&\begin{array}{l} -\rho _{\mathrm{t}} {h}_{\mathrm{t}} \frac{{\partial }^{2}{v}_{\mathrm{ot}} }{\partial {t}^{2}}+{A}_{22}^{\mathrm{t}} \left( {\frac{{\partial }^{2}{v}_{\mathrm{ot}} }{\partial {y}^{2}}-\frac{\partial \alpha _{22}^{\mathrm{t}} }{\partial {y}}\Delta \,{T}} \right) +{A}_{12}^{\mathrm{t}} \left( {\frac{\partial ^{2}{u}_{\mathrm{ot}} }{\partial {x}\partial {y}}-\frac{\partial \alpha _{11}^{\mathrm{t}} }{\partial {y}}\Delta \,{T}} \right) +{A}_{66}^{\mathrm{t}} \left( {\frac{{\partial }^{2}{u}_{\mathrm{ot}} }{\partial {x}\partial {y}}-\frac{{\partial }^{2}{v}_{\mathrm{ot}} }{\partial {x}^{2}}} \right) \\ -{B}_{22}^{\mathrm{t}} \frac{{\partial }^{3}{w}_{\mathrm{t}} }{\partial {y}^{3}}-{B}_{21}^{\mathrm{t}} \frac{{\partial }^{3}{w}_{\mathrm{t}} }{\partial {x}^{2}\partial {y}}-2{B}_{66}^{\mathrm{t}} \frac{{\partial }^{3}{w}_{\mathrm{t}} }{\partial {x}^{2}\partial {y}}-2{G}_{12} {l}_{0}^{2} {h}_{\mathrm{t}} \left( {\frac{{\partial }^{4}{v}_{\mathrm{ot}} }{\partial {x}\partial {y}^{3}}+\frac{\partial ^{4}{v}_{\mathrm{ot}} }{\partial {y}^{4}}+\frac{{\partial }^{4}{u}_{\mathrm{ot}} }{\partial {x}^{3}\partial {y}}+\frac{{\partial }^{4}{v}_{\mathrm{ot}} }{\partial {x}^{2}\partial {y}^{2}}} \right) \\ -2{G}_{12} {l}_{1}^{2} {h}_{\mathrm{t}} \left( {\frac{4}{15}\frac{{\partial }^{4}{v}_{\mathrm{ot}} }{\partial {x}\partial {y}^{3}}+\frac{4}{5}\frac{{\partial }^{4}{v}_{\mathrm{ot}} }{\partial {y}^{4}}+\frac{4}{3}\frac{{\partial }^{4}{v}_{\mathrm{ot}} }{\partial {x}^{2}\partial {y}^{2}}+\frac{4}{15}\frac{{\partial }^{4}{u}_{\mathrm{ot}} }{\partial {x}^{3}\partial {y}}+\frac{8}{15}\frac{{\partial }^{4}{v}_{\mathrm{ot}} }{\partial {x}^{4}}} \right) \\ -\frac{1}{4}G_{12} l_{2}^{2} h_{\mathrm{t}} \left( {\frac{\partial ^{4}v_{ot} }{\partial x^{2}\partial y^{2}}-\frac{\partial ^{4}u_{ot} }{\partial x^{3}\partial y}+\frac{\partial ^{4}v_{ot} }{\partial x^{4}}-\frac{\partial ^{4}u_{ot} }{\partial x\partial y^{3}}} \right) -\lambda _{xt} =0 \\ \end{array} \end{aligned}$$

(A2)

$$\begin{aligned}&{\begin{array}{l} -\rho _{\mathrm{t}} h_{\mathrm{t}} \frac{\partial ^{2}w_{\mathrm{t}} }{\partial t^{2}}+\frac{1}{12}\rho _{\mathrm{t}} h_{\mathrm{t}}^{3} \left( \frac{\partial ^{4}w_{\mathrm{t}} }{\partial x^{2}\partial t^{2}}+\frac{\partial ^{4}w_{\mathrm{t}} }{\partial y^{2}\partial t^{2}} \right) +B_{11}^{t} \left( \frac{\partial ^{3}u_{ot} }{\partial x^{3}}-\frac{\partial \alpha _{11}^{t} }{\partial x^{2}}\Delta T \right) +B_{12}^{t} \left( \frac{\partial ^{3}v_{ot} }{\partial x^{2}\partial y}-\frac{\partial \alpha _{22}^{t} }{\partial x^{2}}\Delta T \right) \\ +B_{22}^{t} \left( \frac{\partial ^{3}v_{ot} }{\partial y^{3}}-\frac{\partial ^{2}\alpha _{22}^{t} }{\partial y^{2}}\Delta T \right) +B_{21}^{t} \left( \frac{\partial ^{3}u_{ot} }{\partial x\partial y^{2}}-\frac{\partial ^{2}\alpha _{11}^{t} }{\partial x^{2}}\Delta T \right) +2B_{66}^{t} \left( \frac{\partial ^{3}u_{ot} }{\partial x\partial y^{2}}+\frac{\partial ^{3}v_{ot} }{\partial x^{2}\partial y} \right) \\ -4D_{66}^{t} \frac{\partial ^{4}w_{\mathrm{t}} }{\partial x^{2}\partial y^{2}}-D_{11}^{t} \frac{\partial ^{4}w_{\mathrm{t}} }{\partial x^{4}}-D_{12}^{t} \frac{\partial ^{4}w_{\mathrm{t}} }{\partial x^{2}\partial y^{2}}-D_{22}^{t} \frac{\partial ^{4}w_{\mathrm{t}} }{\partial y^{4}}-D_{21}^{t} \frac{\partial ^{4}w_{\mathrm{t}} }{\partial x^{2}\partial y^{2}} \\ -{G}_{12} {l}_{0}^{2} {h}_{{t}} \left( 2\frac{{\partial }^{4}{w}_{{t}} }{\partial {y}^{4}}-\frac{1}{2}{h}_{{t}}^{2} \frac{{\partial }^{6}{w}_{{t}} }{\partial {x}^{4}\partial {y}^{2}}-\frac{1}{6}{h}_{{t}}^{2} \frac{{\partial }^{6}{w}_{{t}} }{\partial {x}^{6}}-\frac{1}{6}{h}_{{t}}^{2} \frac{{\partial }^{6}{w}_{{t}} }{\partial {y}^{6}}-\frac{1}{2}{h}_{{t}}^{2} \frac{{\partial }^{6}{w}_{{t}} }{\partial {x}^{2}\partial {y}^{4}}+2\frac{{\partial }^{4}{w}_{{t}} }{\partial {x}^{4}}+4\frac{{\partial }^{4}{w}_{{t}} }{\partial {x}^{2}\partial {y}^{2}} \right) \\ -{G}_{12} {l}_{1}^{2} {h}_{\mathrm{t}} \left( \frac{8}{15}\frac{{\partial }^{4}{w}_{\mathrm{t}} }{\partial {y}^{4}}-\frac{1}{5}{h}_{\mathrm{t}}^{2} \frac{{\partial }^{6}{w}_{\mathrm{t}} }{\partial {x}^{4}\partial {y}^{2}}-\frac{1}{15}{h}_{\mathrm{t}}^{2} \frac{{\partial }^{6}{w}_{\mathrm{t}} }{\partial {y}^{6}}-\frac{1}{5}{h}_{\mathrm{t}}^{2} \frac{{\partial }^{6}{w}_{\mathrm{t}} }{\partial {x}^{2}\partial {y}^{4}}+\frac{8}{15}\frac{{\partial }^{4}{w}_{\mathrm{t}} }{\partial {x}^{4}}+\frac{16}{15}\frac{{\partial }^{4}{w}_{\mathrm{t}} }{\partial {x}^{2}\partial {y}^{2}} \right) \\ -{G}_{12} {l}_{2}^{2} {h}_{\mathrm{t}} \left( \frac{{\partial }^{4}{w}_{\mathrm{t}} }{\partial {y}^{4}}+\frac{{\partial }^{4}{w}_{\mathrm{t}} }{\partial {x}^{4}}+2\frac{{\partial }^{4}{w}_{\mathrm{t}} }{\partial {x}^{2}\partial {y}^{2}} \right) -{\bar{{N}}}_{\mathrm{xx}}^{\mathrm{t}} \left( \frac{{\partial }^{2}{w}_{\mathrm{t}} }{\partial {x}^{2}}+\mu ^{\mathrm{t}}\frac{{\partial }^{2}{w}_{\mathrm{t}} }{\partial {y}^{2}}\right) - \frac{1}{2}{h}_{\mathrm{t}} \left( \frac{{\partial }\lambda _{\mathrm{xt}}}{\partial {x}}+\frac{{\partial }\lambda _{\mathrm{yt}}}{\partial {y}}\right) -\lambda _{\mathrm{zt}} = 0\\ \\ \end{array}} \end{aligned}$$

(A3)

For the core

$$\begin{aligned}&-{\rho }_{\mathrm{c}} {h}_{\mathrm{c}} \frac{{\partial }^{2}{u}_{\mathrm{oc}} }{\partial {t}^{2}}-\frac{1}{12}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{3}\frac{{\partial }^{2}{u}_{2{\mathrm{c}}} }{\partial {t}^{2}}+{\lambda }_{\mathrm{xt}} -{\lambda }_{\mathrm{xb}} =0 \end{aligned}$$

(A4)

$$\begin{aligned}&\begin{array}{l} -\frac{1}{12}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{3}\frac{{\partial }^{2}{u}_{1{\mathrm{c}}} }{\partial {t}^{2}}-\frac{1}{80}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{5}\frac{{\partial }^{2}{u}_{3{\mathrm{c}}} }{\partial {t}^{2}}-{A}_{55}^{\mathrm{C}} \left( {{u}_{1{\mathrm{c}}} +\frac{\partial {w}_{0{\mathrm{c}}} }{\partial {x}}} \right) \\ -{B}_{55}^{\mathrm{C}} \left( {2{u}_{2{\mathrm{c}}} +\frac{\partial {w}_{1{\mathrm{c}}} }{\partial {x}}} \right) -{D}_{55}^{\mathrm{C}} \left( {3{u}_{3{\mathrm{c}}} +\frac{\partial {w}_{2{\mathrm{c}}} }{\partial {x}}} \right) -\frac{1}{2}{h}_{\mathrm{c}} \left( {{\lambda }_{\mathrm{xt}} +{\lambda }_{\mathrm{xb}} } \right) =0 \\ \end{array} \end{aligned}$$

(A5)

$$\begin{aligned}&\begin{array}{l} -\frac{1}{12}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{3}\frac{{\partial }^{2}{u}_{0{\mathrm{c}}} }{\partial {t}^{2}}-\frac{1}{80}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{5}\frac{{\partial }^{2}{u}_{2{\mathrm{c}}} }{\partial {t}^{2}}-2{B}_{55}^{\mathrm{C}} \left( {{u}_{1{\mathrm{c}}} +\frac{\partial {w}_{0{\mathrm{c}}} }{\partial {x}}} \right) \\ -2{D}_{55}^{\mathrm{C}} \left( {2{u}_{2{\mathrm{c}}} +\frac{\partial {w}_{1{\mathrm{c}}} }{\partial {x}}} \right) -2{F}_{55}^{\mathrm{C}} \left( {3{u}_{3{\mathrm{c}}} +\frac{\partial {w}_{2{\mathrm{c}}} }{\partial {x}}} \right) +\frac{1}{4}{h}_{\mathrm{c}}^{2}\left( {{\lambda }_{\mathrm{xt}} -{\lambda }_{\mathrm{xb}} } \right) =0 \\ \end{array} \end{aligned}$$

(A6)

$$\begin{aligned}&\begin{array}{l} -\frac{1}{80}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{5}\frac{{\partial }^{2}{u}_{1{\mathrm{c}}} }{\partial {t}^{2}}-\frac{1}{448}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{7}\frac{{\partial }^{2}{u}_{3{\mathrm{c}}} }{\partial {t}^{2}}-3{D}_{55}^{\mathrm{C}} \left( {{u}_{1{\mathrm{c}}} +\frac{\partial {w}_{0{\mathrm{c}}} }{\partial {x}}} \right) \\ -3{F}_{55}^{\mathrm{C}} \left( {2{u}_{2{\mathrm{c}}} +\frac{\partial {w}_{1{\mathrm{c}}} }{\partial {x}}} \right) -3{L}_{55}^{\mathrm{C}} \left( {3{u}_{3{\mathrm{c}}} +\frac{\partial {w}_{2{\mathrm{c}}} }{\partial {x}}} \right) \frac{1}{8}{h}_{\mathrm{c}}^{3}\left( {{\lambda }_{\mathrm{xt}} +{\lambda }_{\mathrm{xb}} } \right) =0\\ \end{array} \end{aligned}$$

(A7)

$$\begin{aligned}&-{\rho }_{\mathrm{c}} {h}_{\mathrm{c}} \frac{{\partial }^{2}{v}_{1{\mathrm{c}}} }{\partial {t}^{2}}-\frac{1}{12}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{3}\frac{{\partial }^{2}{v}_{1{\mathrm{c}}} }{\partial {t}^{2}}+{\lambda }_{\mathrm{yt}} -{\lambda }_{\mathrm{yb}} =0 \end{aligned}$$

(A8)

$$\begin{aligned}&\begin{array}{l} -\frac{1}{12}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{3}\frac{{\partial }^{2}{v}_{1{\mathrm{c}}} }{\partial {t}^{2}}-\frac{1}{80}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{5}\frac{{\partial }^{2}{v}_{3{\mathrm{c}}} }{\partial {t}^{2}}-{A}_{44}^{\mathrm{C}} \left( {{v}_{1{\mathrm{c}}} +\frac{\partial {w}_{0{\mathrm{c}}} }{\partial {y}}} \right) \\ -{B}_{44}^{\mathrm{C}} \left( {2{v}_{2{\mathrm{c}}} +\frac{\partial {w}_{1{\mathrm{c}}} }{\partial {y}}} \right) -{D}_{44}^{\mathrm{C}} \left( {3{v}_{3{\mathrm{c}}} +\frac{\partial {w}_{2{\mathrm{c}}} }{\partial {y}}} \right) -\frac{1}{2}{h}_{\mathrm{c}} \left( {{\lambda }_{\mathrm{yt}} +{\lambda }_{\mathrm{yb}} } \right) =0 \\ \end{array} \end{aligned}$$

(A9)

$$\begin{aligned}&\begin{array}{l} -\frac{1}{12}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{3}\frac{{\partial }^{2}{v}_{0{\mathrm{c}}} }{\partial {t}^{2}}-\frac{1}{80}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{5}\frac{{\partial }^{2}{v}_{2{\mathrm{c}}} }{\partial {t}^{2}}-2{B}_{44}^{\mathrm{C}} \left( {{v}_{1{\mathrm{c}}} +\frac{\partial {w}_{0{\mathrm{c}}} }{\partial {y}}} \right) \\ -2{D}_{44}^{\mathrm{C}} \left( {2{v}_{2{\mathrm{c}}} +\frac{\partial {w}_{1{\mathrm{c}}} }{\partial {y}}} \right) -2{F}_{44}^{\mathrm{C}} \left( {3{v}_{3{\mathrm{c}}} +\frac{\partial {w}_{2{\mathrm{c}}} }{\partial {y}}} \right) +\frac{1}{4}{h}_{\mathrm{c}}^{2} \left( {{\lambda }_{\mathrm{yt}} -{\lambda }_{\mathrm{yb}} } \right) =0 \\ \end{array} \end{aligned}$$

(A10)

$$\begin{aligned}&\begin{array}{l} -\frac{1}{80}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{5}\frac{{\partial }^{2}{v}_{1{\mathrm{c}}} }{\partial {t}^{2}}-\frac{1}{448}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{7}\frac{{\partial }^{2}{v}_{3{\mathrm{c}}} }{\partial {t}^{2}}-3{D}_{44}^{\mathrm{C}} \left( {{v}_{1{\mathrm{c}}} +\frac{\partial {w}_{0{\mathrm{c}}} }{\partial {y}}} \right) \\ -3{F}_{44}^{\mathrm{C}} \left( {2{v}_{2{\mathrm{c}}} +\frac{\partial {w}_{1{\mathrm{c}}} }{\partial {y}}} \right) -3{L}_{44}^{\mathrm{C}} \left( {3{v}_{3{\mathrm{c}}} +\frac{\partial {w}_{2{\mathrm{c}}} }{\partial {y}}} \right) -\frac{1}{8}{h}_{\mathrm{c}}^{3} \left( {{\lambda }_{\mathrm{yt}} +{\lambda }_{\mathrm{yb}} } \right) =0 \\ \end{array} \end{aligned}$$

(A11)

$$\begin{aligned}&\begin{array}{l} -{\rho }_{\mathrm{c}} {h}_{\mathrm{c}} \frac{{\partial }^{2}{w}_{0{\mathrm{c}}} }{\partial {t}^{2}}-\frac{1}{12}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{3}\frac{{\partial }^{2}{w}_{2{\mathrm{c}}} }{\partial {t}^{2}}+{A}_{55}^{\mathrm{C}} \left( {\frac{\partial {u}_{1{\mathrm{c}}} }{\partial {x}}+\frac{\partial {w}_{0{\mathrm{c}}} }{\partial {x}^{2}}} \right) +{A}_{44}^{\mathrm{C}} \left( {\frac{\partial {v}_{1{\mathrm{c}}} }{\partial {y}}+\frac{{\partial }^{2}{w}_{0{\mathrm{c}}} }{\partial {y}^{2}}} \right) \\ +{B}_{55}^{\mathrm{C}} \left( {2\frac{\partial {u}_{2{\mathrm{c}}} }{\partial {x}}+\frac{\partial ^{2}{w}_{1{\mathrm{c}}} }{\partial {x}^{2}}} \right) +{B}_{44}^{\mathrm{C}} \left( {2\frac{\partial {v}_{2{\mathrm{c}}} }{\partial {y}}+\frac{{\partial }^{2}{w}_{1{\mathrm{c}}} }{\partial {y}^{2}}} \right) +{D}_{55}^{\mathrm{C}} \left( {3\frac{\partial {v}_{3{\mathrm{c}}} }{\partial {x}}+\frac{{\partial }^{2}{w}_{2{\mathrm{c}}} }{\partial {x}^{2}}} \right) \\ +{D}_{44}^{\mathrm{C}} \left( {3\frac{\partial {v}_{3{\mathrm{c}}} }{\partial {y}}+\frac{\partial ^{2}{w}_{2{\mathrm{c}}} }{\partial {y}^{2}}} \right) +{\lambda }_{\mathrm{zt}} -{\lambda }_{\mathrm{zb}} =0 \\ \end{array} \end{aligned}$$

(A12)

$$\begin{aligned}&\begin{array}{l} -\frac{1}{12}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{3}\frac{{\partial }^{2}{w}_{1{\mathrm{c}}} }{\partial {t}^{2}}-{A}_{33}^{\mathrm{C}} \left( {{w}_{1{\mathrm{c}}} -{\alpha }_{33}^{\mathrm{c}} \Delta \,{T}} \right) -2{B}_{33}^{\mathrm{C}} {w}_{2{\mathrm{c}}} +{B}_{55}^{\mathrm{C}} \left( {2\frac{\partial {u}_{1{\mathrm{c}}} }{\partial {x}}+\frac{{\partial }^{2}{w}_{0{\mathrm{c}}} }{\partial {x}^{2}}} \right) \\ +{B}_{44}^{\mathrm{C}} \left( {2\frac{\partial {v}_{1{\mathrm{c}}} }{\partial {y}}+\frac{\partial ^{2}{w}_{1{\mathrm{c}}} }{\partial {y}^{2}}} \right) +{D}_{55}^{\mathrm{C}} \left( {2\frac{\partial {u}_{2{\mathrm{c}}} }{\partial {x}}+\frac{{\partial }^{2}{w}_{1{\mathrm{c}}} }{\partial {x}^{2}}} \right) +{D}_{44}^{\mathrm{C}} \left( {2\frac{\partial {v}_{2{\mathrm{c}}} }{\partial {y}}+\frac{{\partial }^{2}{w}_{1{\mathrm{c}}} }{\partial {y}^{2}}} \right) \\ +{F}_{55}^{\mathrm{C}} \left( {3\frac{\partial {u}_{3{\mathrm{c}}} }{\partial {x}}+\frac{\partial ^{2}{w}_{2{\mathrm{c}}} }{\partial {x}^{2}}} \right) +{F}_{44}^{\mathrm{C}} \left( {3\frac{\partial {v}_{3{\mathrm{c}}} }{\partial {y}}+\frac{{\partial }^{2}{w}_{2{\mathrm{c}}} }{\partial {y}^{2}}} \right) -\frac{1}{2}{h}_{\mathrm{c}} \left( {{\lambda }_{\mathrm{zt}} +{\lambda }_{\mathrm{zb}} } \right) =0 \\ \\ \end{array} \end{aligned}$$

(A13)

$$\begin{aligned}&\begin{array}{l} -\frac{1}{12}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{3}\frac{{\partial }^{2}{w}_{0{\mathrm{c}}} }{\partial {t}^{2}}-\frac{1}{80}{\rho }_{\mathrm{c}} {h}_{\mathrm{c}}^{5}\frac{{\partial }^{2}{w}_{2{\mathrm{c}}} }{\partial {t}^{2}}+2{B}_{33}^{\mathrm{C}} \left( {{w}_{1{\mathrm{c}}} -{\alpha }_{33}^{\mathrm{c}} \Delta \,{T}} \right) -4{D}_{33}^{\mathrm{C}} {w}_{2{\mathrm{c}}} +{D}_{55}^{\mathrm{C}} \left( {2\frac{\partial {u}_{1{\mathrm{c}}} }{\partial {x}}+\frac{{\partial }^{2}{w}_{0{\mathrm{c}}} }{\partial {x}^{2}}} \right) \\ +{D}_{44}^{\mathrm{C}} \left( {2\frac{\partial {v}_{1{\mathrm{c}}} }{\partial {y}}+\frac{\partial ^{2}{w}_{0{\mathrm{c}}} }{\partial {y}^{2}}} \right) +{F}_{55}^{\mathrm{C}} \left( {2\frac{\partial {u}_{2{\mathrm{c}}} }{\partial {x}}+\frac{{\partial }^{2}{w}_{1{\mathrm{c}}} }{\partial {x}^{2}}} \right) +{F}_{44}^{\mathrm{C}} \left( {2\frac{\partial {v}_{2{\mathrm{c}}} }{\partial {y}}+\frac{{\partial }^{2}{w}_{1{\mathrm{c}}} }{\partial {y}^{2}}} \right) \\ +{L}_{55}^{\mathrm{C}} \left( {3\frac{\partial {u}_{3{\mathrm{c}}} }{\partial {x}}+\frac{\partial ^{2}{w}_{2{\mathrm{c}}} }{\partial {x}^{2}}} \right) +{L}_{44}^{\mathrm{C}} \left( {3\frac{\partial {v}_{3{\mathrm{c}}} }{\partial {y}}+\frac{{\partial }^{2}{w}_{2{\mathrm{c}}} }{\partial {y}^{2}}} \right) +\frac{1}{4}{h}_{\mathrm{c}}^{2}\left( {{\lambda }_{\mathrm{zt}} -{\lambda }_{\mathrm{zb}} } \right) =0 \\ \end{array} \end{aligned}$$

(A14)

For bottom face sheet

$$\begin{aligned}&\begin{array}{l} -{\rho }_{\mathrm{b}} {h}_{\mathrm{b}} \frac{{\partial }^{2}{u}_{\mathrm{ob}} }{\partial {t}^{2}}+{A}_{11}^{\mathrm{b}} \left( {\frac{{\partial }^{2}{u}_{\mathrm{ob}} }{\partial {x}^{2}}-\frac{\partial ^{2}{\alpha }_{11}^{\mathrm{b}} }{\partial {x}}\Delta \,{T}} \right) +{A}_{12}^{\mathrm{b}} \left( {\frac{{\partial }^{2}{v}_{\mathrm{ot}} }{\partial {x}\partial {y}}-\frac{\partial \alpha _{22}^{\mathrm{b}} }{\partial {x}}\Delta \,{T}} \right) \\ +{A}_{66}^{\mathrm{b}} \left( {\frac{{\partial }^{2}{u}_{\mathrm{ob}} }{\partial {y}^{2}}-\frac{{\partial }^{2}{v}_{\mathrm{ob}} }{\partial {x}\partial {y}}} \right) -{B}_{11}^{\mathrm{b}} \frac{{\partial }^{3}{w}_{\mathrm{b}} }{\partial {x}^{3}}-{B}_{12}^{\mathrm{b}} \frac{\partial ^{3}{w}_{\mathrm{b}} }{\partial {x}\partial {y}^{2}}-2{B}_{66}^{\mathrm{b}} \frac{{\partial }^{3}{w}_{\mathrm{b}} }{\partial {x}\partial {y}^{2}} \\ -2{G}_{12} {l}_{0}^{2} {h}_{\mathrm{b}} \left( {\frac{{\partial }^{4}{u}_{\mathrm{ob}} }{\partial {x}^{4}}+\frac{{\partial }^{4}{v}_{\mathrm{ob}} }{\partial {x}^{3}\partial {y}}+\frac{\partial ^{4}{u}_{\mathrm{ob}} }{\partial {x}^{2}\partial {y}^{2}}+\frac{{\partial }^{4}{v}_{\mathrm{ob}} }{\partial {x}\partial {y}^{3}}} \right) \\ -2{G}_{12} {l}_{1}^{2} {h}_{\mathrm{b}} \left( {\frac{4}{5}\frac{{\partial }^{4}{u}_{\mathrm{otb}} }{\partial {x}^{4}}+\frac{4}{15}\frac{{\partial }^{4}{v}_{\mathrm{ob}} }{\partial {x}^{3}\partial {y}}+\frac{4}{3}\frac{{\partial }^{4}{u}_{\mathrm{ob}} }{\partial {x}^{2}\partial {y}^{2}}+\frac{4}{15}\frac{{\partial }^{4}{v}_{\mathrm{ob}} }{\partial {x}\partial {y}^{3}}+\frac{8}{15}\frac{{\partial }^{4}{u}_{\mathrm{ob}} }{\partial {y}^{4}}} \right) \\ -\frac{1}{4}{G}_{12} {l}_{2}^{2} {h}_{\mathrm{b}} \left( {\frac{{\partial }^{4}{u}_{\mathrm{ob}} }{\partial {x}^{2}\partial {y}^{2}}-\frac{{\partial }^{4}{v}_{\mathrm{ob}} }{\partial {x}^{3}\partial {y}}+\frac{{\partial }^{4}{u}_{\mathrm{ob}} }{\partial {y}^{4}}-\frac{\partial ^{4}{v}_{\mathrm{ob}} }{\partial {x}\partial {y}^{3}}} \right) +{\lambda }_{\mathrm{xb}} =0 \\ \end{array} \end{aligned}$$

(A15)

$$\begin{aligned}&\begin{array}{l} -{\rho }_{\mathrm{t}} {h}_{\mathrm{t}} \frac{{\partial }^{2}{v}_{\mathrm{ob}} }{\partial {t}^{2}}+{A}_{22}^{\mathrm{b}} \left( {\frac{{\partial }^{2}{v}_{\mathrm{ob}} }{\partial {y}^{2}}-\frac{\partial \alpha _{22}^{\mathrm{b}} }{\partial {y}}\Delta \,{T}} \right) +{A}_{12}^{\mathrm{b}} \left( {\frac{\partial ^{2}{u}_{\mathrm{ob}} }{\partial {x}\partial {y}}-\frac{\partial {\alpha }_{11}^{\mathrm{b}} }{\partial {y}}\Delta \,{T}} \right) \\ +{A}_{66}^{\mathrm{b}} \left( {\frac{{\partial }^{2}{u}_{\mathrm{ob}} }{\partial {x}\partial {y}}-\frac{{\partial }^{2}{v}_{\mathrm{ob}} }{\partial {x}^{2}}} \right) -{B}_{22}^{\mathrm{b}} \frac{{\partial }^{3}{w}_{\mathrm{b}} }{\partial {y}^{3}}-{B}_{21}^{\mathrm{b}} \frac{\partial ^{3}{w}_{\mathrm{b}} }{\partial {x}^{2}\partial {y}}-2{B}_{66}^{\mathrm{b}} \frac{{\partial }^{3}{w}_{\mathrm{b}} }{\partial {x}^{2}\partial {y}} \\ -2{G}_{12} {l}_{0}^{2} {h}_{\mathrm{b}} \left( {\frac{{\partial }^{4}{v}_{\mathrm{ob}} }{\partial {x}\partial {y}^{3}}+\frac{{\partial }^{4}{v}_{\mathrm{ob}} }{\partial {y}^{4}}+\frac{\partial ^{4}{u}_{\mathrm{ob}} }{\partial {x}^{3}\partial {y}}+\frac{{\partial }^{4}{v}_{\mathrm{ob}} }{\partial {x}^{2}\partial {y}^{2}}} \right) \\ -2{G}_{12} {l}_{1}^{2} {h}_{\mathrm{b}} \left( {\frac{4}{15}\frac{{\partial }^{4}{v}_{\mathrm{ob}} }{\partial {x}\partial {y}^{3}}+\frac{4}{5}\frac{{\partial }^{4}{v}_{\mathrm{ob}} }{\partial {y}^{4}}+\frac{4}{3}\frac{{\partial }^{4}{v}_{\mathrm{ob}} }{\partial {x}^{2}\partial {y}^{2}}+\frac{4}{15}\frac{{\partial }^{4}{u}_{\mathrm{ob}} }{\partial {x}^{3}\partial {y}}+\frac{8}{15}\frac{{\partial }^{4}{v}_{\mathrm{ob}} }{\partial {x}^{4}}} \right) \\ -\frac{1}{4}G_{12} l_{2}^{2} h_{\mathrm{b}} \left( {\frac{\partial ^{4}v_{ob} }{\partial x^{2}\partial y^{2}}-\frac{\partial ^{4}u_{ob} }{\partial x^{3}\partial y}+\frac{\partial ^{4}v_{ob} }{\partial x^{4}}-\frac{\partial ^{4}u_{ob} }{\partial x\partial y^{3}}} \right) +\lambda _{yb} =0 \\ \end{array} \end{aligned}$$

(A16)

$$\begin{aligned}&{\begin{array}{l} -\rho _{\mathrm{t}} h_{\mathrm{t}} \frac{\partial ^{2}w_{\mathrm{b}} }{\partial t^{2}}+\frac{1}{12}\rho _{\mathrm{t}} h_{\mathrm{t}}^{b} \left( \frac{\partial ^{4}w_{\mathrm{b}} }{\partial x^{2}\partial t^{2}}+\frac{\partial ^{4}w_{\mathrm{b}} }{\partial y^{2}\partial t^{2}} \right) +B_{11}^{b} \left( \frac{\partial ^{3}u_{ob} }{\partial x^{3}}-\frac{\partial \alpha _{11}^{b} }{\partial x^{2}}\Delta T \right) +B_{12}^{b} \left( \frac{\partial ^{3}v_{ob} }{\partial x^{2}\partial y}-\frac{\partial \alpha _{22}^{b} }{\partial x^{2}}\Delta T \right) \\ +B_{22}^{t} \left( \frac{\partial ^{3}v_{ob} }{\partial y^{3}}-\frac{\partial ^{2}\alpha _{22}^{b} }{\partial y^{2}}\Delta T \right) +B_{21}^{b} \left( \frac{\partial ^{3}u_{ob} }{\partial x\partial y^{2}}-\frac{\partial ^{2}\alpha _{11}^{b} }{\partial x^{2}}\Delta T \right) +2B_{66}^{t} \left( \frac{\partial ^{3}u_{ob} }{\partial x\partial y^{2}}+\frac{\partial ^{3}v_{ob} }{\partial x^{2}\partial y} \right) \\ -4D_{66}^{b} \frac{\partial ^{4}w_{\mathrm{b}} }{\partial x^{2}\partial y^{2}}-D_{11}^{b} \frac{\partial ^{4}w_{\mathrm{b}} }{\partial x^{4}}-D_{12}^{b} \frac{\partial ^{4}w_{\mathrm{b}} }{\partial x^{2}\partial y^{2}}-D_{22}^{b} \frac{\partial ^{4}w_{\mathrm{b}} }{\partial y^{4}}-D_{21}^{b} \frac{\partial ^{4}w_{\mathrm{b}} }{\partial x^{2}\partial y^{2}} \\ -G_{12} l_{0}^{2} h_{\mathrm{b}} \left( 2\frac{\partial ^{4}w_{\mathrm{b}} }{\partial y^{4}}-\frac{1}{2}h_{\mathrm{b}}^{2} \frac{\partial ^{6}w_{\mathrm{b}} }{\partial x^{4}\partial y^{2}}-\frac{1}{6}h_{\mathrm{b}}^{2} \frac{\partial ^{6}w_{\mathrm{b}} }{\partial x^{6}}-\frac{1}{6}h_{\mathrm{b}}^{2} \frac{\partial ^{6}w_{\mathrm{b}} }{\partial y^{6}}-\frac{1}{2}h_{\mathrm{b}}^{2} \frac{\partial ^{6}w_{\mathrm{b}} }{\partial x^{2}\partial y^{4}}+2\frac{\partial ^{4}w_{\mathrm{b}} }{\partial x^{4}}+4\frac{\partial ^{4}w_{\mathrm{b}} }{\partial x^{2}\partial y^{2}} \right) \\ -G_{12} l_{1}^{2} h_{\mathrm{b}} \left( \frac{8}{15}\frac{\partial ^{4}w_{\mathrm{b}} }{\partial y^{4}}-\frac{1}{5}h_{\mathrm{b}}^{2} \frac{\partial ^{6}w_{\mathrm{b}} }{\partial x^{4}\partial y^{2}}-\frac{1}{15}h_{\mathrm{b}}^{2} \frac{\partial ^{6}w_{\mathrm{b}} }{\partial y^{6}}-\frac{1}{5}h_{\mathrm{b}}^{2} \frac{\partial ^{6}w_{\mathrm{b}} }{\partial x^{2}\partial y^{4}}+\frac{8}{15}\frac{\partial ^{4}w_{\mathrm{b}} }{\partial x^{4}}+\frac{16}{15}\frac{\partial ^{4}w_{\mathrm{b}} }{\partial x^{2}\partial y^{2}} \right) \\ -G_{12} l_{2}^{2} h_{\mathrm{b}} \left( \frac{\partial ^{4}w_{\mathrm{b}} }{\partial y^{4}}+\frac{\partial ^{4}w_{\mathrm{b}} }{\partial x^{4}}+2\frac{\partial ^{4}w_{\mathrm{b}} }{\partial x^{2}\partial y^{2}} \right) -{\bar{{N}}}_{{xx}}^{{b}} \left( \frac{{\partial }^{2}{w}_{{b}} }{\partial {x}^{2}}+\mu ^{{b}}\frac{{\partial }^{2}{w}_{{b}} }{\partial {y}^{2}}\right) - \frac{1}{2}{h}_{{b}} \left( \frac{{\partial }\lambda _{{xb}}}{\partial {x}}+\frac{{\partial }\lambda _{{yb}}}{\partial {y}}\right) +\lambda _{{zb}}\\ +k_{g\mu } \left( \cos ^{2}\theta \frac{{\partial }^{2}{w}_{{b}} }{\partial {x}^{2}} +2 \cos \theta \sin \theta \frac{{\partial }^{2}{w}_{{b}} }{\partial {x}\partial {y}} +\sin ^{2}\theta \frac{{\partial }^{2}{w}_{{b}} }{\partial {y}^{2}} \right) \\ +k_{g\eta } \left( \sin ^{2}\theta \frac{{\partial }^{2}{w}_{{b}} }{\partial {x}^{2}} -2 \cos \theta \sin \theta \frac{{\partial }^{2}{w}_{{b}} }{\partial {x}\partial {y}} +\cos ^{2}\theta \frac{{\partial }^{2}{w}_{{b}} }{\partial {y}^{2}} \right) =0\\ \\ \end{array}} \end{aligned}$$

(A17)

$$\begin{aligned}&\mu ^{i}=\frac{\bar{N}_{yy}^{j}}{\bar{N}_{xx}^{j}}, \quad j=(t,b), \mu ^{t} = \mu ^{b} = \mu \end{aligned}$$

(A18)

$$\begin{aligned}&A_{kl}^{j},B_{kl}^{j},D_{kl}^{j} =\int \limits _{{-h} h/ 2}^{h \big / 2}{Q_{kl}^{j} \left( {1,z_{j},z_{j}^{2} } \right) \mathrm{d}z_{j} } \quad \left( {k,l=1,2,6} \right) \quad \left( {j=t,b} \right) \end{aligned}$$

(A19)

$$\begin{aligned}&A_{kl}^{c},B_{kl}^{c},D_{kl}^{c},F_{kl}^{c},L_{kl}^{c} =\int \limits _{{-h} \big / 2}^{h \big / 2} {c_{kl} \left( {1,z_{j},z_{j}^{2} ,z_{j}^{3},z_{j}^{4} } \right) \mathrm{d}z_{\mathrm{c}} } \quad \left( {k,l=3,4,5} \right) . \end{aligned}$$

(A20)