Pergmon Mechanics Research Communications, Vol. 23, No. 1, pp. 35-40, 1996 Copyright O 1996 Elsevier Science Ltd Printed in the USA. All rights reserved 0093-6413/96 $12.00 + .00 0093-6413(95")00074-7 LONGITUDINAL VIBRATIONS OF A BEAM: A GRADIENT ELASTICITY APPROACH B. S.ALTAN +, H. A.Evensen+ and E. C. Aifantis *+ +Center for Mechanics of Materials and Instabilities, Michigan Tech, Houghton, MI 49931, USA *+Laboratory of Mechanics, Aristotle Univ. Thessaloniki, Thessaloniki. 54006, Greece (Received 21 August 1995; accepted for print 6 October 1995) 1. INTRODUCTION Generalization of elasticity theory by incorporating the effect of higher gradients of the displacement field into the strain energy density function was systematically studied by Cosserat and Cosserat [1], although the main idea can be traced all the way back to the works of Bernoulli and Canchy. For almost five decades this type of generalization was almost ignored but the question of "oriented medium" or "couple stress the- ory" reopened in the early 1960's in connection with the theory of defects in continua. A modem system- atic treatment of elasticity with gradients was given by Truesdell and Toupin [2] which was completed later by Mindlin and Tiersten [3] and Toupin [4]. The gradient dependent theory of elasticity has since been improved and extended by Green and Rivlin [5], Toupin [6] and Mindlin [7]. The common feature of all these studies is that they relate the higher gradients of the displacement field to higher order stresses instead of directly introducing higher gradients into the constitutive equation for the conventional stress (i.e. force per unit area). An exception is the study by Triantafyllidis and Aifantis [8] in which the second gradient of the displacement field was incorporated into the strain energy function, but no higher order stresses were assumed. The gradient elasticity developed in this study was applied to analyze the pre- and post-localization behavior of the deformation bands. It was shown that the width and direction of these bands can be described without loss of the ellipticity of the governing equations, in contrast to previous results. The linearized version of this theory, applied to the mode III crack by Altan and Aifantis [9], pro- duced a smooth crack opening, i.e. firfite strains at the crack tip and finite dislocation distribution emitted from the crack tip. But, since the field equations of this theory become fourth order differential equations, it is clear that some boundary conditions are required in addition to the classical ones. The main purpose of this study is to analyze the consequences of gradient theory on the simplest problems possible. First, the strain field in a straight bar subject to uniform tension is analyzed under various extra boundary conditions. It is shown that the strain distribution in the bar is not homogeneous even when the tension is homogeneous, in contrast to the classical case. Then the longitudinal vibration of a bar under homogeneous boundary conditions is analyzed and it is shown that the vibration modes contain not only travelling waves but also exponential end-effect terms. In the first section the field equations and the boundary value problems of gradient elasticity are defined briefly. The subsequent section is devoted to the analysis of the strain field in a bar subject to uniform ten- sion under various combinations of extra boundary conditions. Then, in Section 4, the longitudinal vibra- tion of a straight bar under homogeneous boundary conditions is analyzed. The final section contains conclusions and some comments concerning further studies. 35