Postbuckling analysis of meta-nanocomposite beams by considering the CNTs’ agglomeration

Abstract

Enhancement of the structural elements’ stiffness as well as reducing their weight can be made possible by arranging nanocomposites in an auxetic form. Motivated by this reality, this work undergoes with the postbuckling characteristics of thin beams made from auxetic carbon nanotube-reinforced nanocomposites for the first time. A bi-stage micromechanical homogenization technique is implemented to attain the effective modulus of such meta-nanocomposites. In this method, the effects of CNT agglomerates on the modulus estimation will be captured. Next, the von Kármán strain–displacement relations will be hired as well as Euler–Bernoulli beam theory to find the nonlinear strain of the continuous system. Using the principle of virtual work, the nonlinear governing equation of the problem will be gathered. Then, Galerkin’s analytical method will be employed to find the nonlinear buckling load of the auxetic nanocomposite beams with simply supported and clamped ends. After proving the validity of the presented modeling, illustrative case studies are provided for reference. The highlights of this article indicate on the fact that the meta-nanocomposite beam will be strengthened against buckling-mode failure if small auxeticity angles are selected. Also, it is demonstrated that the structure fails under smaller buckling loads if a wide auxetic lattice is employed. Furthermore, it is shown that how can the buckling resistance of the auxetic nanocomposite beam be affected by the agglomeration phenomenon.

Appendix

The stiff variables implemented in Eqs. (5) and (6) can be defined as below [32]:

$$ \alpha_{r} = \frac{{3\left( {K_{m} + G_{m} } \right) + k_{r} + l_{r} }}{{3\left( {G_{m} + k_{r} } \right)}} $$

(34)

$$ \beta_{r} = \frac{1}{5}\left( {\frac{{4G_{m} + 2k_{r} + l_{r} }}{{3\left( {G_{m} + k_{r} } \right)}} + \frac{{4G_{m} }}{{G_{m} + p_{r} }} + \frac{{2\left( {G_{m} \left( {3K_{m} + G_{m} } \right) + G_{m} \left( {3K_{m} + 7G_{m} } \right)} \right)}}{{G_{m} \left( {3K_{m} + G_{m} } \right) + m_{r} \left( {3K_{m} + 7G_{m} } \right)}}} \right) $$

(35)

$$ \delta_{r} = \frac{1}{3}\left( {n_{r} + 2l_{r} + \frac{{\left( {2k_{r} + l_{r} } \right)\left( {3K_{m} + G_{m} - l_{r} } \right)}}{{G_{m} + k_{r} }}} \right) $$

(36)

$$ \eta_{r} = \frac{1}{5}\left( {\frac{2}{3}\left( {n_{r} - l_{r} } \right) + \frac{{8G_{m} p_{r} }}{{G_{m} + p_{r} }} + \frac{{\left( {2k_{r} - l_{r} } \right)\left( {2G_{m} + l_{r} } \right)}}{{3\left( {G_{m} + k_{r} } \right)}} + \frac{{8m_{r} G_{m} \left( {3K_{m} + 4G_{m} } \right)}}{{3K_{m} \left( {m_{r} + G_{m} } \right) + G_{m} \left( {7m_{r} + G_{m} } \right)}}} \right) $$

(37)

In the above relations, subscripts “m” and “r” denote matrix and reinforcing nanofillers, respectively. Also, it must be mentioned that $k_{r}$, $l_{r}$, $m_{r}$, $n_{r}$, and $p_{r}$ are the Hill’s elastic constants of the CNTs used in the modeling. These constants can be found by referring to Table 1 of present paper.