Nonlinear deformation and stress responses of a graded carbon nanotube sandwich plate structure under thermoelastic loading

Abstract

The large deformation and stresses (normal and shear) of the graded nanotube-reinforced sandwich structure are numerically investigated under the influence of mechanical loading and a uniform temperature field. A higher-order nonlinear finite element model in conjunction with the direct iterative technique has been adopted for the solution purpose. Also, the structural distortion was modeled via the full-scale geometrical nonlinearity (Green–Lagrange strain) in the framework of higher-order displacement functions. Further, to replicate the actual operational conditions, the temperature-dependent properties of the individual material constituents (i.e., carbon nanotube and polymer) have been implemented in the current material modeling steps. The final deflections and stress values are evaluated via an own developed computer code using the currently proposed nonlinear mathematical formulation. The model accuracy and solution stability are checked by comparing the responses (deflection and stress) with available published results. Lastly, a variety of numerical examples is solved for different design parameters and deliberated in detail.

Appendices

Appendix A

$$\begin{aligned} \left\{ \lambda \right\}&=\left\{ {u v w} \right\} ^{T}, \nonumber \\ \left\{ {\lambda _0 } \right\}&=\left[ {u_0 \qquad v_0 \qquad w_{0} \qquad \varphi _x \qquad \varphi _y \qquad \psi _x \qquad \psi _y \qquad \theta _y \qquad \theta _x } \right] ^{T}, \nonumber \\ [H]&=\left[ {{\begin{array}{lllllllll} 1&{}\qquad 0&{}\qquad 0&{}\qquad z&{}\qquad 0&{}\qquad {z^{2}}&{}\qquad 0&{}\qquad {z^{3}}&{}\qquad 0 \\ 0&{}\qquad 1&{}\qquad 0&{}\qquad 0&{}\qquad z&{}\qquad 0&{}\qquad {z^{2}}&{}\qquad 0&{}\qquad {z^{3}} \\ 0&{}\qquad 0&{}\qquad 1&{}\qquad 0&{}\qquad 0&{}\qquad 0&{}\qquad 0&{}\qquad 0&{}\qquad 0 \\ \end{array} }} \right] , \end{aligned}$$

$$\begin{aligned}&\varepsilon _x^0 =u_{,x} , \varepsilon _y^0 =v_{,y} , \gamma _{xy}^0 =u_{,y} +v_{,x} , \gamma _{xz}^0 =\varphi _x +w_{,x} , \gamma _{yz}^0 =\varphi _y +w_{,y} , k_x^1 =\varphi _{x,x} , k_y^1 =\varphi _{y,y} ,\\&k_{xy}^1 =\varphi _{x,y}+\varphi _{y,x} , k_{zx}^1 =2\psi _x , k_{yz}^1 =2\psi _y , k_x^2 =\psi _{x,x} , k_y^2 =\psi _{y,y} , k_{xy}^2=\psi _{x,y} +\psi _{y,x} ,\\&k_{zx}^2 =3\theta _x , k_{yz}^2 =3\theta _y , k_x^3 =\theta _{x,x} , k_y^3 =\theta _{y,y} ,\\&k_{xy}^3 =\theta _{x,y} +\theta _{y,x} , \\&k_{zx}^3 =-\theta _x , \\&k_{yz}^3 =-\theta _y , \\&\varepsilon _x^4 =\left[ {\left( {u_{,x} } \right) ^{2}+\left( {v_{,x} } \right) ^{2}+\left( {w_{,x} } \right) ^{2}} \right] ,\\&\left( {\varepsilon _y^4 } \right) =\left[ {\left( {u_{,y} } \right) ^{2}+\left( {v_{,y} } \right) ^{2}+\left( {w_{,y} } \right) ^{2}} \right] , \\&\gamma _{xy}^4 =2\left[ {u_{,x} u_{,y} +v_{,x} v_{,y} +w_{,x} w_{,y} } \right] , \\&\gamma _{zx}^4 =2\left[ {\varphi _x u_{,x} +\varphi _y v_{,x} } \right] , \\&\gamma _{zx}^4 =2\left[ {\varphi _x u_{,x} +\varphi _y v_{,x} } \right] , \\&\gamma _{zx}^4 =2\left[ {\varphi _x u_{,x} +\varphi _y v_{,x} } \right] ,\\&\gamma _{yz}^4 =2\left[ {\varphi _x u_{,y} +\varphi _y v_{,y} } \right] , \\&\left( {k_x^5 } \right) =2\left[ {\varphi _{x,x} u_{,x} +\varphi _{y,x} v_{,x} } \right] , \\&\left( {k_y^5 } \right) =2\left[ {\varphi _{x,y} u_{,y} +\varphi _{y,y} v_{,y} } \right] , \\&\left( {k_{xy}^5 } \right) =2\left[ {\varphi _{x,y} u_{,x} +\varphi _{x,x} u_{,y} +2\varphi _{y,x} v_{,y} +\varphi _{y,y} v_{,x} } \right] , \\&\left( {k_{zx}^5 } \right) =2\left[ {\varphi _x \varphi _{x,x} +2\psi _x u_{,x} +\varphi _y \varphi _{y,x} +2\psi _y v_{,x} } \right] , \\&\left( {k_{yz}^5 } \right) =2\left[ {\varphi _x \varphi _{x,y} +2\psi _x u_{,y} +\varphi _y \varphi _{y,y} +2\psi _y v_{,y} } \right] , \left( {k_x^6 } \right) =\left[ {\varphi _{x,x}^2 +\varphi _{y,x}^2 +2\psi _{x,x} u_{,x} +2\psi _{y,x} v_{,x} } \right] , \\&\left( {k_y^6 } \right) =\left[ {\varphi _{x,y}^2 +\varphi _{y,y}^2 +2\psi _{x,y} u_{,y} +2\psi _{y,y} v_{,y} } \right] , \\&\left( {k_{xy}^6 } \right) =2\left[ {\varphi _{x,x} \varphi _{x,x} +\varphi _{y,x} \varphi _{y,y} +\psi _{x,x} u_{,y} +\psi _{x,x} u_{,y} +\psi _{y,x} v_{,y} +\psi _{y,y} v_{,x} } \right] , \\&\left( {k_{zx}^6 } \right) =2\left[ {\psi _{x,x} \varphi _x +\psi _{y,x} \varphi _y +2\varphi _{x,x} \psi _x +2\varphi _{y,x} \psi _y +3u_{,x} \theta _x +3v_{,x} \theta _y } \right] , \\&\left( {k_{yz}^6 } \right) =2\left[ {\psi _{x,y} \varphi _x +\psi _{y,y} \varphi _y +2\varphi _{x,y} \psi _x +2\varphi _{y,y} \psi _y +3u_{,y} \theta _x +3v_{,y} \theta _y } \right] \quad \\&\left( {k_x^7 } \right) =2\left[ {u_{,x} \theta _{x,x} +v_{,x} \theta _{y,x} +\varphi _{x,x} \psi _{x,x} +\varphi _{y,x} \psi _{y,x} } \right] , \\&\left( {k_y^7 } \right) =2\left[ {u_{,y} \theta _{x,y} +v_{,y} \theta _{y,y} +\varphi _{x,y} \psi _{x,y} +\varphi _{y,y} \psi _{y,y} } \right] , \\&\left( {k_{xy}^7 } \right) =2\left[ {u_{,x} \theta _{x,y} +u_{,y} \theta _{x,x} +v_{,x} \theta _{y,y} +v_{,y} \theta _{y,x} +\varphi _{x,x} \psi _{x,y} +\varphi _{x,y} \psi _{x,x} +\varphi _{y,x} \psi _{y,y} +\varphi _{y,y} \psi _{y,x} } \right] , \\&\left( {k_{zx}^7 } \right) =2\left[ {\theta _{x,x} \varphi _x +\theta _{y,x} \varphi _y +2\psi _{x,x} \psi _x +2\psi _{y,x} \psi _y +3\varphi _{x,x} \theta _x +3\varphi _{y,x} \theta _y } \right] ,\\&\left( {k_{yz}^7 } \right) =2\left[ {\varphi _x \theta _{x,y} +\varphi _x \theta _{y,y} +2\psi _x \psi _{x,y} +2\psi _y \psi _{y,y} +3\theta _x \varphi _{x,y} +3\theta _y \varphi _{y,y} } \right] , \\&\left( {k_x^8 } \right) =\left[ {\psi _{x,x}^2 +\psi _{y,x}^2 +2\varphi _{x,x} \theta _{x,x} +2\varphi _{y,x} \theta _{y,x} } \right] , \\&\left( {k_y^8 } \right) =\left[ {\psi _{x,y}^2 +\psi _{y,y}^2 +2\varphi _{x,y} \theta _{x,y} +2\varphi _{y,y} \theta _{y,y} } \right] , \\&\left( {k_{xy}^8 } \right) =\left[ \psi _{x,x} \psi _{x,y} +\psi _{y,x} \psi _{y,y} +2\theta _{x,x} \varphi _{x,y} \right. \\&\left. +2\theta _{y,x} \varphi _{y,y} +2\theta _{x,x} \varphi _{x,x} +2\theta _{y,x} \varphi _{y,x} \right] , \\&\left( {k_{zx}^8 } \right) =2\left[ {2\psi _x \theta _{x,x} +2\psi _y \theta _{y,x} +3\theta _x \psi _{x,x} +3\theta _y \psi _{y,x} } \right] , \\&\left( {k_{yz}^8 } \right) =2\left[ {2\psi _x \theta _{x,y} +2\psi _y \theta _{y,y} +3\theta _x \psi _{x,y} +3\theta _x \psi _{x,y} } \right] , \\&\left( {k_x^9 } \right) =2\left[ {\psi _{x,x} \theta _{x,x} +\psi _{y,x} \theta _{y,x} } \right] , \\&\left( {k_y^9 } \right) =2\left[ {\psi _{x,y} \theta _{x,y} +\psi _{y,y} \theta _{y,y} } \right] , \\&\left( {k_{xy}^9 } \right) =2\left[ {\psi _{x,x} \theta _{x,y} +\psi _{y,x} \theta _{y,y} +\theta _{x,x} \psi _{x,y} +\theta _{y,x} \psi _{y,y} } \right] , \\&\left( {k_{zx}^9 } \right) =2\left[ {3\theta _x \theta _{x,x} +3\theta _y \theta _{y,x} } \right] , \\&\left( {k_{yz}^9 } \right) =\left[ {6\left( {\theta _x \theta _{x,y} +\theta _y \theta _{y,y} } \right) } \right] , \\&\left( {k_x^{10} } \right) =\left[ {\theta _{x,x}^2 +\theta _{y,x}^2 } \right] , \\&\left( {k_y^{10} } \right) =\left[ {\theta _{x,y}^2 +\theta _{y,y}^2 } \right] , \\&\left( {k_y^{10} } \right) =\left[ {\theta _{x,y}^2 +\theta _{y,y}^2 } \right] , \left( {k_{xy}^{10} } \right) =2\left[ {\theta _{x,x} \theta _{x,y} +\theta _{y,x} \theta _{y,y} } \right] , \\&k_{zx}^{10} =0, k_{yz}^{10} =0. \end{aligned}$$

Appendix B

Some coupled terms in the above equations are:

$$\begin{aligned}&\displaystyle u_{,x} =\frac{\partial u_0 }{\partial x}, \quad v_{,y} =\frac{\partial v_0 }{\partial y} v_{,y} =\frac{\partial v_0 }{\partial y}, \quad v_{,y} =\frac{\partial v_0 }{\partial y}, \nonumber \\&\displaystyle u_{,y} =\frac{\partial u_0 }{\partial y}, \quad v_{,x} =\frac{\partial v_0 }{\partial x}, \quad w_{,x} =\frac{\partial w_0 }{\partial x}, \quad w_{,y} =\frac{\partial w_0 }{\partial y}, \quad \varphi _{y,y} =\frac{\partial \varphi _y }{\partial y}, \nonumber \\&\displaystyle \varphi _{x,y} =\frac{\partial \varphi _x }{\partial y}, \quad \varphi _{y,x} =\frac{\partial \varphi _y }{\partial x}, \quad \psi _{x,x} =\frac{\partial \psi _x }{\partial x}, \nonumber \\&\displaystyle \psi _{y,y} =\frac{\partial \psi _y }{\partial y}, \quad \psi _{x,y} =\frac{\partial \psi _x }{\partial y}, \nonumber \\&\displaystyle \psi _{y,x} =\frac{\partial \psi _y }{\partial x}, \quad \theta _{x,x} =\frac{\partial \theta _x }{\partial x}, \quad \theta _{y,y} =\frac{\partial \theta _y }{\partial y}, \nonumber \\&\displaystyle \theta _{x,y} =\frac{\partial \theta _x }{\partial y}, \quad \theta _{y,x} =\frac{\partial \theta _y }{\partial x}. \end{aligned}$$

Appendix C

Linear and nonlinear thickness coordinate matrix

$$\begin{aligned} \left[ {H_L } \right] =\left[ {{\begin{array}{llllllllllllllllllll} 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}}&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}}&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}}&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}} \\ \end{array} }} \right] , \nonumber \\ \begin{array}{l} [H_{NL} ]=\left[ {{\begin{array}{llllllllllllllllllll} 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}}&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}}&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}}&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}} \\ \end{array} }} \right. \\ \qquad \qquad \qquad \left. {{\begin{array}{lllllllllllllll} {z^{4}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{5}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{6}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad {z^{4}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{} \quad {z^{5}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{6}}&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad {z^{4}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{5}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{6}}&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad {z^{4}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{5}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{6}}&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{4}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{5}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{6}} \\ \end{array} }} \right] , \\ \end{array} \end{aligned}$$

Appendix D

$$\begin{aligned} \left[ {D_1 } \right]&=\sum _{k=1}^3 {\mathop {\int }\limits _{Z_{k-1} }^{Z_k } {\left[ {T_L } \right] ^{T}} \left[ {\bar{{Q}}} \right] \left[ {T_L } \right] \text {d}z} ,\quad \left[ {D_2 } \right] =\sum _{k=1}^3 {\mathop {\int }\limits _{Z_{k-1} }^{Z_k } {\left[ {T_L } \right] ^{T}} \left[ {\bar{{Q}}} \right] \left[ {T_{NL} } \right] \text {d}z}, \nonumber \\ \left[ {D_3 }\right]&=\sum _{k=1}^3 {\mathop {\int }\limits _{Z_{k-1} }^{Z_k } {\left[ {T_{NL} } \right] ^{T}} \left[ {\bar{{Q}}} \right] \left[ {T_L } \right] \text {d}z}\quad \hbox {and}\quad \left[ {D_4 } \right] =\sum _{k=1}^3 {\mathop {\int }\limits _{Z_{k-1} }^{Z_k } {\left[ {T_{NL} } \right] ^{T}} \left[ {\bar{{Q}}} \right] \left[ {T_{NL} } \right] \text {d}z} . \end{aligned}$$