Ahmed Abouelregal - Department of Mathematics, Faculty of Science, Mansoura University, Mansoura ۳۵۵۱۶, Egypt
Ashraf Zenkour - Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah ۲۱۵۸۹, Saudi Arabia|Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh ۳۳۵۱۶, Egypt

In this work, the vibrations of viscoelastic functionally graded Euler–Bernoulli nanostructure beams are investigated using the fractional-order calculus. It is assumed that the functionally graded nanobeam (FGN) is due to a periodic heat flux. FGN can be considered as nonhomogenous composite structures; with continuous structural changes along the thick- ness of the nanobeam usually, it changes from ceramic at the bottom of the metal at the top. Based on the nonlocal model of Eringen, the complete analytical solution to the problem is established using the Laplace transform method. The effects of different parameters are illustrated graphically and discussed. The effects of fractional order, damping coefficient, and periodic frequency of the vibrational behavior of nanobeam was investigated and discussed. It also provides a conceptual idea of the FGN and its distinct advantages compared to other engineering materials. The results obtained in this work can be applied to identify of many nano-structures such as nano-electro mechanical systems (NEMS), nano-actuators, etc.

Viscoelastic, fractional derivatives, FG nanobeam, periodic heat flux