A robust Bezier based solution for nonlinear vibration and post-buckling of random checkerboard graphene nano-platelets reinforced composite beams

In the present study, an accurate Bezier based multi-step method is developed and implemented to find the nonlinear vibration and post-buckling configurations of Euler-Bernoulli composite beams reinforced with graphene nano-platelets (GnP). The GnP is assumed to be randomly and uniformly dispersed in the composite mix-proportion, with a random checkerboard configuration. Therefore, a probabilistic model together with an efficient simulation technique is proposed to find the effective moduli of a matrix reinforced GnP. It is worth noting that the presented micro-mechanics model found by the employed Monte-Carlo simulation matches exactly the experimental data and predicts the composite elastic constants more accurate than that found from other common methods, including the Halpin-Tsai theory. Also, for mathematical simplification, the composite beam in-plane inertia is neglected. The presented multi-step method is based on Burnstein polynomial basis functions while shows interesting potential to provide robust solutions for various initial and boundary value problems. It is found that adding a relatively low content of GnP would drastically increase the composite elastic constants, particularly in the transverse direction to fiber. In addition, the numerical results are compared with those provided by exact analytical solutions, where the stability of results suggests the effectiveness of the presented methodology.