Size dependent effects of two phase viscoelastic medium on damping vibrations of smart nanobeams: an efficient implementation of GDQM

Abstract

Abstract This paper studies the dynamics of nonlocal piezo-magnetic nanobeams (PMNBs) embedded in the local/nonlocal viscoelastic medium through the consistent and paradox-free model of the nonlocal theory. Besides, to perform the dynamic analysis, an exact solution and an efficient approach of generalized differential quadrature method (GDQM) are introduced. Since the size-dependency of the uniform loads is wrongly neglected by the nonlocal elasticity in differential form, the size-dependency of piezo-magnetic load is applied through the two-phase theory. Also, size dependency of the viscoelastic medium is accurately applied and examined through the solutions presented employing the differential two-phase theory and satisfying the constitutive boundary conditions. In this regard, the two-phase resultant equations of motions together with boundary conditions including the constitutive ones related to two-phase PMNB and the two-phase medium are attained. To confirm the credibility and efficiency of the extracted equations as well as presented solution procedures, several analogical studies are accomplished, and it is shown that the results obtained from the differential relations are reliable and consistence with those extracted from the integral nonlocal relations. It is shown that the present approach of the GDQM simplifies the solution procedures of the nonlocal problems and improves the precisions in the cases close to the pure nonlocal state. The presented results emphasize that the size-dependency of viscoelastic medium, external electric, and magnetic loads play significant roles on the vibration characteristics, and therefore it must be considered based on two-phase theory. The available results can be helpful to achieve an excellent design of smart nanobeams embedded in viscoelastic medium.