An analytical solution for static stability of multi-scale hybrid nanocomposite plates

Abstract

An analytical answer to the buckling problem of a composite plate consisted of multi-scale hybrid nanocomposites is presented here for the first time. In other words, the constituent material of the structure is made of an epoxy matrix which is reinforced by both macro- and nanosize reinforcements, namely, carbon fiber (CF) and carbon nanotube (CNT). The effective material properties such as Young’s modulus or density are derived utilizing a micromechanical scheme incorporated with the Halpin–Tsai model. To present a more realistic problem, the plate is placed on a two-parameter elastic substrate. Then, on the basis of an energy-based Hamiltonian approach, the equations of motion are derived using the classical theory of plates. Finally, the governing equations are solved analytically to obtain the critical buckling load of the system. Afterward, the normalized form of the results is presented to emphasize the impact of each parameter on the dimensionless buckling load of composite plates. It is worth mentioning that the effects of various boundary conditions are covered, too. To show the efficiency of presented modeling, the results of this article are compared to those of former attempts.

Appendix

The corresponding k_ij arrays of stiffness matrix introduced in Eq. (36) can be computed as

$$\begin{aligned} k_{11} & = A_{11} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & + A_{66} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial X_{m} (x)}}{\partial x}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & + 2A_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$

$$\begin{aligned} k_{12} & = \left( {A_{12} + A_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial X_{m} (x)}}{\partial x}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & + A_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x{\text{d}}y} } \\ & + A_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$

$$\begin{aligned} k_{13} & = - B_{11} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & - \left( {B_{12} + 2B_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial X_{m} (x)}}{\partial x}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & - 3B_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x{\text{d}}y} } \\ & - B_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$

$$\begin{aligned} k_{21} & = \left( {A_{12} + A_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x{\text{d}}y} } \\ & + A_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & + A_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial X_{m} (x)}}{\partial x}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$

$$\begin{aligned} k_{22} & = A_{66} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x{\text{d}}y} } \\ & + 2A_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial X_{m} (x)}}{\partial x}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & + A_{22} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$

$$\begin{aligned} k_{23} & = - B_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & - \left( {B_{12} + 2B_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x{\text{d}}y} } \\ & - 3B_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial X_{m} (x)}}{\partial x}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & - B_{22} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$

$$\begin{aligned} k_{31} & = B_{11} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{4} X_{m} (x)}}{{\partial x^{4} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & + \left( {B_{12} + 2B_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & + 3B_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & + B_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$

$$\begin{aligned} k_{32} & = B_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & + \left( {B_{12} + 2B_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & + 3B_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } \\ & + B_{22} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{4} Y_{n} (y)}}{{\partial y^{4} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$

$$\begin{aligned} k_{33} & = - D_{11} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{4} X_{m} (x)}}{{\partial x^{4} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & - 2\left( {D_{12} + 2D_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & - 4D_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } - 4D_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } \\ & - D_{22} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{4} Y_{n} (y)}}{{\partial y^{4} }}{\text{d}}x{\text{d}}y} } - k_{w} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & + \left( {k_{p} - N^{b} } \right)\left( {\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y){\text{d}}x{\text{d}}y} } + \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } } \right). \\ \end{aligned}$$

(A1)