A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory

Amine Zemri , Mohammed Sid Ahmed Houari , Abdelmoumen Anis Bousahla , Abdelouahed Tounsi

Abstract

This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler–Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton\'s principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

Key Words

nanobeam; nonlocal elasticity theory; bending; buckling; vibration; functionally graded materials

Address

Amine Zemri — Material and Hydrology Laboratory, Faculty of Technology, Civil Engineering Department, University of Sidi Bel Abbes, Algeria
Mohammed Sid Ahmed Houari — Departement de Genie Civil, Universite de Mascara, Algeria; Laboratoire des Structures et Materiaux Avances dans le Genie Civil et Travaux Publics, Universite de Sidi Bel Abbes, Faculte de Technologie, Departement de genie civil, Algeria
Abdelmoumen Anis Bousahla — Material and Hydrology Laboratory, Faculty of Technology, Civil Engineering Department, University of Sidi Bel Abbes, Algeria; Laboratoire de Modelisation et Simulation Multi-echelle, Departement de Physique, Faculte des Sciences Exactes, Universite de Sidi Bel Abbes, Algeria
Abdelouahed Tounsi — Material and Hydrology Laboratory, Faculty of Technology, Civil Engineering Department, University of Sidi Bel Abbes, Algeria; Laboratoire de Modelisation et Simulation Multi-echelle, Departement de Physique, Faculte des Sciences Exactes, Universite de Sidi Bel Abbes, Algeria; Laboratoire de Modelisation et Simulation Multi-echelle, Departement de Physique, Faculte des Sciences Exactes, Universite de Sidi Bel Abbes, Algeria

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