Nonlinear low-velocity impact on damped and matrix-cracked hybrid laminated beams containing carbon nanotube reinforced composite layers

Abstract

This paper investigates the low-velocity impact response of a shear deformable laminated beam which contains both carbon nanotube reinforced composite (CNTRC) layers and carbon fiber reinforced composite (CFRC) layers. The effect of matrix cracks is considered, and a refined self-consistent model is selected to describe the degraded stiffness caused by the damage. The beam including damping effects rests on a two-parameter elastic foundation in thermal environments. Based on a higher-order shear deformation theory and von Kármán nonlinear strain–displacement relationships, the motion equations of the beam and impactor are established and solved by means of a two-step perturbation approach. The material properties of both CFRC layers and CNTRC layers are assumed to be temperature-dependent. To assess engineering application of this hybrid structure, two conditions for outer CNTRC layers and outer CFRC layers are compared. Besides, the effects of the crack density, volume fraction of carbon nanotube, temperature variation, the foundation stiffness and damping on the nonlinear low-velocity impact behavior of hybrid laminated beams are also discussed in detail.

Appendix

In Eqs. (41) and (42)

$$\begin{aligned} g_{30}= & {} -\gamma _{17} +m^{2}\left( {\gamma _{18} +\gamma _{19} } \right) \frac{\gamma _{21} m^{2}-\gamma _{23} }{\gamma _{22} m^{2}+\gamma _{23} } \nonumber \\&-\left( {\gamma _{29} +\gamma _{28} \frac{\gamma _{21} m^{2}-\gamma _{23} }{\gamma _{22} m^{2}+\gamma _{23} }} \right) \frac{\gamma _{12} m^{4}}{\gamma _{22} m^{2}+\gamma _{23} } \end{aligned}$$

(43)

$$\begin{aligned} g_{31}= & {} m^{4}\left[ {\gamma _{11} -\gamma _{12} \frac{\gamma _{21} m^{2}-\gamma _{23} }{\gamma _{22} m^{2}+\gamma _{23} }} \right] +(K_1 +K_2 m^{2}) \nonumber \\&+\,2m^{2}\pi \left[ {\gamma _{15} -\gamma _{14} \frac{\gamma _{21} -\gamma _{23} }{\gamma _{22} +\gamma _{23} }} \right] {\varPhi } \nonumber \\&+\,2\pi C_1 \left[ {\gamma _{15} -\gamma _{14} \frac{\gamma _{21} m^{2}-\gamma _{23} }{\gamma _{22} m^{2}+\gamma _{23} }} \right] {\varPhi } \end{aligned}$$

(44)

$$\begin{aligned} g_{32}= & {} 2m^{3}\pi \left[ {\gamma _{15} -\gamma _{14} \frac{\gamma _{21} m^{2}-\gamma _{23} }{\gamma _{22} m^{2}+\gamma _{23} }} \right] \nonumber \\&+\,\frac{3}{4}\pi ^{2}C_2 \gamma _{13} {\varPhi } \end{aligned}$$

(45)

$$\begin{aligned} g_{33}= & {} \frac{m^{4}\pi ^{2}}{4}\gamma _{13} \end{aligned}$$

(46)

$$\begin{aligned} g_c= & {} c_w +c_\psi \frac{m^{4}\gamma _{12} (\gamma _{21} m^{2}-\gamma _{23} )}{(\gamma _{22} m^{2}+\gamma _{23} )^{2}} \end{aligned}$$

(47)

$$\begin{aligned} g_q= & {} \frac{\hbox {2}k_c bL^{5/2}}{\pi ^{3}\overline{{D}}_{11} }\sin \frac{m}{2}\pi \end{aligned}$$

(48)

$$\begin{aligned} g_i= & {} -\frac{k_c \rho _0 L^{5/2}}{\pi ^{2}ME_0 } \end{aligned}$$

(49)

In Eqs. (44) and (45), $C_{1}$ and $C_{2}$ is dependent on the value of m. When m = 1, $C_{1}$ and $C_{2}$ are both equaled to be 1. In other case, $C_{1}=C_{2} = 0$.