Application of Gradient Elasticity to Benchmark Problems of Beam Vibrations

Application of Gradient Elasticity to Benchmark Problems of Beam Vibrations

K.M. Kateb1, K.A. Alnefaie1, N.H. Abu-Hamdeh1,*, and E.C. Aifantis1,2

1Dept. of Mechanical Engineering, King Abdulaziz University Jeddah 21589, Saudi Arabia

2Lab of Mechanics and Materials, Polytechnic School, Aristotle University, Thessaloniki, 52124, Greece

Abstract

The Gradient approach, specifically gradient elasticity theory, is adopted to revisit certain typical configurations on mechanical vibrations. New results on size effects and scale-dependent behavior not captured by classical elasticity are derived, aiming at illustrating the usefulness of this approach to applications in advanced technologies. In particular, elastic prismatic straight beams in bending are discussed using two different governing equations: the gradient elasticity bending moment equation (fourth order) and the gradient elasticity deflection equation (sixth order). Different boundary/support conditions are examined. One problem considers the free vibrations of a cantilever beam loaded by an end force. A second problem is concerned with a simply supported beam disturbed by a concentrated force in the middle of the beam. Both problems are solved analytically. Exact free vibration frequencies and mode shapes are derived and presented. The difference between the gradient elasticity solution and its classical counterpart is revealed. The size ratio c/L (c denotes internal length and L is the length of the beam) induces significant effects on vibration frequencies. For both beam configurations, it turns out that as the ratio c/L increases, the vibration frequencies decrease, a fact which implies lower beam stiffness. Numerical examples show this behavior explicitly and recover the classical vibration behavior for vanishing size ratio c/L.

Introduction

The topic of size effect is a subject of increasing interest due to the fact that current applications in modern technology involve a variety of length scales ranging from a few centimeters (sheet metal forming) down to few nanometers (thin film technology). This range of scales and related necessity for modeling and experiment has revealed that a connection between the various length scales involved and the corresponding mechanical response of

*Corresponding author Tel +966-545993930; email: nabuhamdeh@kau.edu.sa

different size, but otherwise geometrically similar specimens, can be established through gradient theory, as advanced by Aifantis and co-workers (see for example [1-13]).

The implication of the theory of gradient elasticity was not as obvious as in the theory of gradient plasticity where pattern-forming deformation instabilities (dislocation