Improved dynamical response of functionally graded GPL-reinforced sandwich beams subjected to external excitation via nonlinear dispersion pattern

Abstract

This paper investigates sandwich beams' forced vibrations reinforced with graphene platelets (GPLs) based on higher-order shear deformation theory. A novel nonlinear reinforcement distribution based on the power-law method is proposed to model functionally graded (FG) graphene platelet reinforced composite face sheets. Due to the manufacturing constraints, the face sheets are considered laminated so that the weight fraction of GPLs in each layer is constant and varies functionally along with the thickness of the beam. Also, the effective material properties of each layer have been calculated using the Halpin Tsai micro-mechanical model. The nonlinear partial equations of motion are derived using Hamilton's principle in the third-order laminated beam model framework. Afterward, the governing equations of motion are discretized to a system of ordinary nonlinear equations by applying Galerkin's method and solved using an analytical approach. Numerical results obtained from the presented work are compared and validated with the literature. The results show that the proposed model can predict the desired GPLs distribution pattern in the face sheets according to the sandwich layers thickness ratio, the amount of the external force, and the total weight fraction of the GPLs. Also, it is found that increasing the concentration of GPLs in the outer layers can significantly increase the vibration frequency of the system and reduce its nonlinearity. However, the core-to-beam thickness ratio is highly effective in how the reinforcement pattern affects the system behavior. Besides, the external excitation characteristics have notable effects on the system's behavior, and multivaluedness in response will be observed depending on the frequency of excitation.

Appendix A

The coefficients $g_{{11}}$, $g_{{12}}$, … in Eqs. (17)–(19) can be defined as

$$ g_{{11}} = \frac{{n^{2} \pi ^{2} \lambda _{{11}} }}{{2L}}, $$

(38)

$$ g_{{12}} = \frac{{n^{3} \pi ^{3} \lambda _{{13}} }}{{2L^{2} }}, $$

(39)

$$ g_{{13}} = \frac{{\left( {1 + \left( { - 1} \right)^{{n + 1}} } \right)n^{2} \pi ^{2} \lambda _{{11}} }}{{3L^{2} }}, $$

(40)

$$ g_{{14}} = \frac{{n^{2} \pi ^{2} \lambda _{{12}} }}{{2L}}, $$

(41)

$$ g_{{21}} = \frac{{n^{2} \pi ^{2} \lambda _{{21}} }}{{2L}}, $$

(42)

$$ g_{{22}} = \frac{{n^{3} \pi ^{3} \lambda _{{23}} }}{{2L^{2} }} - \frac{{n\pi \lambda _{{24}} }}{2}, $$

(43)

$$ g_{{23}} = \frac{{\left( {1 + \left( { - 1} \right)^{{n + 1}} } \right)n^{2} \pi ^{2} \lambda _{{21}} }}{{3L^{2} }}, $$

(44)

$$ g_{{24}} = \frac{{n^{2} \pi ^{2} \lambda _{{22}} }}{{2L}} - \frac{{L\lambda _{{24}} }}{2}, $$

(45)

$$ g_{f} = \frac{{\left( {1 + \left( { - 1} \right)^{{n + 1}} } \right)L}}{{n\pi }}, $$

(46)

$$ g_{{31}} = \frac{{n^{3} \pi ^{3} \lambda _{{31}} }}{{2L^{2} }}, $$

(47)

$$ g_{{32}} = \frac{{n^{4} \pi ^{4} \lambda _{{33}} }}{{2L^{3} }} - \frac{{n^{2} \pi ^{2} \lambda _{{34}} }}{{2L}} $$

(48)

$$ g_{{33}} = \frac{{n^{2} \pi ^{2} \left( { - 1 + \left( { - 1} \right)^{n} + 4\left( {2 + \left( { - 1} \right)^{n} } \right){\text{Sin}}^{4} \left( {\frac{{n\pi }}{2}} \right)} \right)\lambda _{{35}} }}{{3L^{2} }}, $$

(49)

$$ g_{{34}} = \frac{{4\left( { - 2 + \left( { - 1} \right)^{{n + 1}} } \right)n^{3} \pi ^{3} {\text{Sin}}^{4} \left( {\frac{{n\pi }}{2}} \right)\lambda _{{31}} }}{{3L^{3} }}, $$

(50)

$$ g_{{35}} = - \frac{{3n^{4} \pi ^{4} \lambda _{{35}} }}{{16L^{3} }}, $$

(51)

$$ g_{{36}} = \frac{{n^{3} \pi ^{3} \lambda _{{32}} }}{{2L^{2} }} - \frac{{n\pi \lambda _{{34}} }}{2}, $$

(52)

$$ g_{{37}} = \frac{{n^{2} \pi ^{2} \left( { - 1 + \left( { - 1} \right)^{n} + 4\left( {2 + \left( { - 1} \right)^{n} } \right){\text{Sin}}^{4} \left( {\frac{{n\pi }}{2}} \right)} \right)\lambda _{{36}} }}{{3L^{2} }}, $$

(53)

$$ g_{{38}} = \frac{{n\pi \lambda _{{38}} }}{2}, $$

(54)

$$ g_{{39}} = \frac{{n^{2} \pi ^{2} \lambda _{{310}} }}{{2L}} - \frac{{L\lambda _{{37}} }}{2}, $$

(55)

$$ g_{{310}} = \frac{{n\pi \lambda _{{39}} }}{2}. $$

(56)