Dynamic Bending Characterization of Delaminated Epoxy/Glass Fiber Based Hybrid Composite Plate Reinforced with Multi-walled Carbon Nanotubes

Abstract

Modeling and Methods

This work is devoted to the numerical dynamic bending characterization of delaminated multi-walled carbon nanotubes (MWCNTs) reinforced epoxy/glass fiber-based hybrid composite plate based on finite element method with displacement fields derived from classical laminated plate theory (CLPT).

Verification

The accuracy and performance of the developed method were confirmed by a validation study with reference solutions available in the literature.

Results

Various parametric studies were carried out to study the influence of MWCNTs content, delamination location, delamination interface, ply orientation, and end condition on the dynamic bending characteristics of the intact and delaminated hybrid laminated composite plate.

Conclusion

The weight fraction of MWCNTs, delamination location, and clamping condition significantly influences the natural frequency. The natural frequencies also depended upon the ply configuration and delamination interface irrespective of the MWCNT loadings. However, anomalous trends were observed as far as the influence of end conditions is concerned.

Appendices

Appendix

Shape Functions

$$N_{i} = \frac{1}{4lb}(l + x_{i} \,x)\;(b + y_{i} \,y)$$

$$N_{{w\,0{\kern 1pt} {\kern 1pt} i}} = \frac{1}{{8\,l^{3} b^{3} }}\;(l + x_{i} \,x)\;(b + y_{i} \,y)\;(2\,l^{2} b^{2} + lb^{2} {\kern 1pt} x_{i} \,x + l^{2} b{\kern 1pt} y_{i} \,y - b^{2} x^{2} - l^{2} y^{2}$$

$$N_{{\theta \,x{\kern 1pt} i}} = \frac{1}{{8{\kern 1pt} l^{2} b}}\,(lx_{i} + x)\;(x^{2} - l^{2} )\;(b + y_{i} \,y)$$

$$N_{{\theta \,y{\kern 1pt} i}} = \frac{1}{{8{\kern 1pt} lb^{2} }}\;(l + x_{i} \,x)\;(by_{i} + y)\;(y^{2} - b^{2} )$$

$$\left[ {N_{i} (x,y)} \right] = \left[ {\begin{array}{*{20}c} {N_{i} } & 0 & 0 & 0 & 0 \\ 0 & {N_{i} } & 0 & 0 & 0 \\ 0 & 0 & {N_{{w_{0} i}} } & {N_{\theta xi} } & {N_{\theta yi} } \\ 0 & 0 & {\frac{{\partial N_{{w_{0} i}} }}{\partial x}} & {\frac{{\partial N_{\theta xi} }}{\partial x}} & {\frac{{\partial N_{\theta yi} }}{\partial x}} \\ 0 & 0 & {\frac{{\partial N_{{w_{0} i}} }}{\partial y}} & {\frac{{\partial N_{\theta xi} }}{\partial y}} & {\frac{{\partial N_{\theta yi} }}{\partial y}} \\ \end{array} } \right]$$

where$N_{{}}$ is the shape function used to interpolate in-plane displacements which vary linearly along each side of the element; $N_{{w\,0{\kern 1pt} {\kern 1pt} }}$,$N_{{\theta \,x{\kern 1pt} }}$ and $N_{{\theta \,y{\kern 1pt} }}$ denote the shape functions used to interpolate transverse deflection and rotational displacements about x and y axis, respectively.

Strain Displacement Matrices

$$\begin{gathered} \left\{ \chi \right\} = \left[ {\overline{B}(x,y)} \right]\left\{ d \right\}; \hfill \\ \{ \chi \} = \left\{ {\begin{array}{*{20}c} {\frac{{\partial u_{m} }}{\partial x}} & {\frac{{\partial v_{m} }}{\partial y}} & {\frac{{\partial u_{m} }}{\partial y} + \frac{{\partial v_{m} }}{\partial x}} & { - \frac{{\partial^{2} w_{m} }}{{\partial x^{2} }}} & { - \frac{{\partial^{2} w_{m} }}{{\partial y^{2} }}} & { - 2\frac{{\partial^{2} w_{m} }}{\partial x\partial y}} \\ \end{array} } \right\}^{{\text{T}}} \hfill \\ \left\{ {\,d} \right\} = \left\{ {\;u_{m1} ,v_{m1} ,w_{m1} , - \theta_{y1} ,\theta_{x1} ...\;u_{m4} ,v_{m4} ,w_{m4} , - \theta_{y4} ,\theta_{x4} } \right\}\;^{T} \hfill \\ \end{gathered}$$

$$[\overline{B}_{i} (x,y)] = \left[ {\begin{array}{*{20}c} {\frac{{\partial N_{i} }}{\partial x}} & 0 & 0 & 0 & 0 \\ 0 & {\frac{{\partial N_{i} }}{\partial y}} & 0 & 0 & 0 \\ {\frac{{\partial N_{i} }}{\partial y}} & {\frac{{\partial N_{i} }}{\partial x}} & 0 & 0 & 0 \\ 0 & 0 & { - \frac{{\partial^{2} N_{w0i} }}{{\partial x^{2} }}} & { - \frac{{\partial^{2} N_{\theta yi} }}{{\partial x^{2} }}} & { - \frac{{\partial^{2} N_{\theta xi} }}{{\partial x^{2} }}} \\ 0 & 0 & { - \frac{{\partial^{2} N_{w0i} }}{{\partial y^{2} }}} & { - \frac{{\partial^{2} N_{\theta yi} }}{{\partial y^{2} }}} & { - \frac{{\partial^{2} N_{\theta xi} }}{{\partial y^{2} }}} \\ 0 & 0 & { - 2\frac{{\partial^{2} N_{w0i} }}{\partial x\partial y}} & { - 2\frac{{\partial^{2} N_{\theta yi} }}{\partial x\partial y}} & { - \,2\frac{{\partial^{2} N_{\theta xi} }}{\partial x\partial y}} \\ \end{array} \,{\kern 1pt} } \right]$$