Experimental and numerical investigation on the dynamic characteristics of thick laminated plant fiber-reinforced polymer composite plates

Abstract

Although considerable attention has been devoted to the experimental research works of plant fiber-reinforced composites, theoretical determination of their structural characteristics is also much needed one in their evolution. But it is still considered as a difficult process due to the several factors such as short and discontinuous fiber form, plant fiber’s intrinsic characteristics, fiber anisotropy and porosity. In this study, vibration characteristics of woven jute/epoxy and woven aloe/epoxy laminated composite plate structures are investigated theoretically and experimentally. The elastic constants of the woven jute/epoxy and woven aloe/epoxy laminated composites are obtained through experimental testing. The governing differential equations of motion for the uniform woven jute/epoxy and woven aloe/epoxy laminated thick composite plates are carried out using the p-version finite element method based on higher-order shear deformation theory. The effectiveness of the developed finite element formulations is demonstrated by comparing the natural frequencies obtained using the present finite element method with those obtained from the experimental measurements. Also a comparative study is carried out between the results of h-version FEM and p-version FEM to check their accuracies and efficiencies. The effects of aspect ratio and angle of fiber orientation under various end conditions on the free vibration responses of the woven jute/epoxy laminated composite plate are also investigated. The forced vibration response of the woven jute/epoxy laminated composite plate under the harmonic force excitation is carried out under various end conditions.

Appendices

Appendix A

Shape functions

1. h Version FEM

1.1 Four-node rectangular element

$$\begin{aligned} N_1 =\frac{1}{4}(1-\xi )(1-\eta ),\\ N_2 =\frac{1}{4}(1+\xi )(1-\eta ),\\ N_3 =\frac{1}{4}(1+\xi )(1+\eta ),\\ N_4 =\frac{1}{4}(1-\xi )(1+\eta )\\ \end{aligned}$$

1.2 Nine-node rectangular element

$$\begin{aligned} N_1= & {} (\xi -1)(\eta -1)\xi \eta /4, \\ N_2= & {} (\xi +1)(\eta -1)\xi \eta /4, \\ N_3= & {} (\xi +1)(\eta +1)\xi \eta /4, N_4 =(\xi -1)(\eta +1)\xi \eta /4, \\ N_5= & {} -(1-\xi ^{2})(1-\eta )\eta /2,\\ N_6= & {} -(1+\xi )(\eta ^{2}-1)/2, \\ N_7= & {} -(\xi ^{2}-1)(1+\eta )\eta /2,\\ N_8= & {} -(\xi -1)(\eta ^{2}-1)\xi /2,\\ N_9= & {} (1-\xi ^{2})(1-\eta ^{2}) \end{aligned}$$

2. p Version FEM

2.1 Shape functions of a bar element up to the polynomial order of 8

Order p	Shape function
1	$P_0 (\xi )=\frac{1}{2}\left( {1-\xi } \right)$ and $P_1 (\xi )=\frac{1}{2}\left( {1+\xi } \right)$
2	$P_2 (\xi )=\left( {1-\xi ^{2}} \right)$
3	$P_3 (\xi )=2\xi \left( {\xi ^{2}-1} \right)$
4	$P_4 (\xi )=-\frac{15}{4}\xi ^{4}+\frac{9}{2}\xi ^{2}-\frac{3}{4}$
5	$P_5 (\xi )=7\xi ^{5}-10\xi ^{3}+3\xi$
6	$P_6 (\xi )=-\frac{105}{8}\xi ^{6}+\frac{175}{8}\xi ^{4}-\frac{75}{8}\xi ^{2}+\frac{5}{8}$
7	$P_7 (\xi )=\frac{99}{4}\xi ^{7}-\frac{189}{4}\xi ^{5}+\frac{105}{4}\xi ^{3}-\frac{15}{4}\xi$
8	$P_8 (\xi )=-\frac{3003}{64}\xi ^{8}+\frac{1617}{16}\xi ^{6}-\frac{2005}{32}\xi ^{4}+\frac{245}{16}\xi ^{2}-\frac{35}{64}$

Appendix B

1.1 Element stiffness and mass matrices

The element stiffness matrix $[k^\mathrm{e}]$ can be expressed as

$$\begin{aligned}{}[k^\mathrm{e}]=[k^\mathrm{b}]+[k^\mathrm{s}] \end{aligned}$$

where $[k^\mathrm{b}]=\sum _i^{p+1} {\sum _j^{p+1} {F_i F_j [B^\mathrm{b}(\xi ,\eta )]^\mathrm{T}[W^\mathrm{b}][B^\mathrm{b}(\xi ,\eta )]\left| J \right| } }$ and

$$\begin{aligned}{}[k^\mathrm{s}]=\sum _i^{p+1} {\sum _j^{p+1} {F_i F_j [B^\mathrm{s}(\xi ,\eta )]^\mathrm{T}[W^\mathrm{s}][B^\mathrm{s}(\xi ,\eta )]\left| J \right| } } \end{aligned}$$

The mass $[m^{e}]$ matrix can be expressed as

$$\begin{aligned}{}[m^{e}]=\sum _i^{p+1} {\sum _j^{p+1} {F_i F_j } } [N_i ]^\mathrm{T}[I][N_i ]\left| J \right| \end{aligned}$$

where $\left| J \right|$ is the determinant of the Jacobian matrix, p is the polynomial order, $F_i$ and $F_j$ are the weighing factors of Gaussian quadrature. The superscript b and s denotes the bending and shear associated terms.

The bending $\left[ {W^\mathrm{b}} \right]$ and shear $\left[ {W^\mathrm{s}} \right]$stiffness matrices of the laminated composite plate can be expressed as

$$\begin{aligned} \left[ {W^\mathrm{b}} \right] =\left[ {{\begin{array}{lllll} {\left[ A \right] }&{} {\left[ B \right] }&{} {\left[ E \right] } \\ {\left[ B \right] }&{} {\left[ D \right] }&{} {\left[ F \right] } \\ {\left[ E \right] }&{} {\left[ F \right] }&{} {\left[ H \right] } \\ \end{array} }} \right] ; \quad \left[ {W^\mathrm{s}} \right] =\left[ {{\begin{array}{lllll} {\left[ {A^\mathrm{s}} \right] }&{} {\left[ {B^\mathrm{s}} \right] } \\ {\left[ {B^\mathrm{s}} \right] }&{} {\left[ {D^\mathrm{s}} \right] } \\ \end{array} }} \right] \end{aligned}$$

The matrices $\left[ {B^\mathrm{b}(\xi ,\eta )} \right]$ and $\left[ {B^\mathrm{s}(\xi ,\eta )} \right]$ can be expressed as

$$\begin{aligned} \left[ {B^\mathrm{b}(\xi ,\eta )} \right] =\sum _{i=1}^n {\left[ {{\begin{array}{lllllll} {N_{i,\xi } }&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ 0&{} {N_{i,\eta } }&{} 0&{} 0&{} 0&{} 0&{} 0 \\ {N_{i,\eta } }&{} {N_{i,\xi } }&{} 0&{} 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} {N_{i,\xi } }&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0&{} {N_{i,\eta } }&{} 0&{} 0 \\ 0&{} 0&{} 0&{} {N_{i,\eta } }&{} {N_{i,\xi } }&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} {-\,c_1 N_{i,\xi } }&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} {-\,c_1 N_{i,\eta } } \\ 0&{} 0&{} 0&{} 0&{} 0&{} {-\,c_1 N_{i,\eta } }&{} {-\,c_1 N_{i,\xi } } \\ \end{array} }} \right] } \end{aligned}$$

where $c_1 =\frac{4}{3h^{2}}$

$$\begin{aligned} \left[ {B^\mathrm{s}(\xi ,\eta )} \right] =\sum _{i=1}^n {\left[ {{\begin{array}{lllllll} 0&{} 0&{} {N_{i,\eta } }&{} 0&{} {N_i }&{} 0&{} 0 \\ 0&{} 0&{} {N_{i,\xi } }&{} {N_i }&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} {-\,c_2 N_i } \\ 0&{} 0&{} 0&{} 0&{} 0&{} {-\,c_2 N_i }&{} 0 \\ \end{array} }} \right] } \end{aligned}$$

where $c_2 =\frac{4}{h^{2}}$ and n is sum of nodal and hierarchical shape functions of an element. $N_i$ is the shape function and $N_{i,\xi }$ and $N_{i,\eta }$ are the first-order derivatives of $\xi$ and $\eta$, respectively.

The inertia matrix can be expressed as

$$\begin{aligned} I=\left[ {{\begin{array}{lllllll} {I_0 }&{}\quad 0&{}\quad 0&{}\quad {I_1 }&{}\quad 0&{}\quad {-\,c_1 I_3 }&{}\quad 0 \\ 0&{}\quad {I_0 }&{}\quad 0&{}\quad 0&{}\quad {I_1 }&{}\quad 0&{}\quad {-\,c_1 I_3 } \\ 0&{}\quad 0&{}\quad {I_0 }&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ {I_1 }&{}\quad 0&{}\quad 0&{}\quad {2I_2 }&{}\quad 0&{}\quad {-\,c_1 I_4 }&{}\quad 0 \\ 0&{}\quad {I_1 }&{}\quad 0&{}\quad 0&{}\quad {2I_2 }&{}\quad 0&{}\quad {-\,c_1 I_4 } \\ {-\,c_1 I_3 }&{}\quad 0&{}\quad 0&{}\quad {-\,c_1 I_4 }&{}\quad 0&{}\quad {c_1^2 I_6 }&{}\quad 0 \\ 0&{}\quad {-\,c_1^2 I_6 I_3 }&{}\quad 0&{}\quad 0&{}\quad {-\,c_1 I_4 }&{}\quad 0&{}\quad {c_1^2 I_6 } \\ \end{array} }} \right] \end{aligned}$$

where $c_1 =\frac{4}{3h^{2}}$

The inertia terms $I_i \left( {i=0,1,2,3,4,6} \right)$ in the inertia matrix are defined by

$$\begin{aligned}&\left( {I_0 ,I_1 ,I_2 ,I_3 ,I_4 ,I_6 } \right) =\iint \limits _{-h/2}^{h/2} {\rho (1,z,z^{2},z^{3},z^{4},z^{6})} \mathrm{d}z\\&\left[ {N_i } \right] =\left[ {{\begin{array}{lllllll} {N_i } &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} {N_i } &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} {N_i } &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} {N_i } &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} {N_i } &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {N_i } &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {N_i } \\ \end{array} }} \right] \\&\left[ {\mathop U\limits ^{\bullet } _i } \right] =\left[ {{\begin{array}{lllllll} {\partial u_{0i} /\partial t} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} {\partial v_{0i} /\partial t} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} {\partial w_{0i} /\partial t} &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} {\partial \phi _{1i} /\partial t} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} {\partial \phi _{2i} /\partial t} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {\partial \theta _{1i} /\partial t} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {\partial \theta _{2i} /\partial t} \\ \end{array} }} \right] \end{aligned}$$