INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2007; 72:111–126 Published online 13 February 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2017 A new formulation and C 0 -implementation of dynamically consistent gradient elasticity Harm Askes 1, ∗ ,† , Terry Bennett 1 and Elias C. Aifantis 2, 3 1 Department of Civil and Structural Engineering, University of Sheffield, Sheffield S1 3JD, U.K. 2 Laboratory of Mechanics and Materials, Polytechnic School, Aristotle University of Thessaloniki, 54006 Thessaloniki, Greece 3 Department of Mechanical Engineering, Center for Mechanics of Materials and Instabilities, Michigan Technological University, Houghton, MI 49931, U.S.A. SUMMARY In this article a special form of gradient elasticity is presented that can be used to describe wave dispersion. This new format of gradient elasticity is an appropriate dynamic extension of the earlier static counterpart of the gradient elasticity theory advocated in the early 1990s by Aifantis and co-workers. In order to capture dispersion of propagating waves, both higher-order inertia and higher-order stiffness contributions are included, a fact which implies (and is denoted as) dynamic consistency. The two higher-order terms are accompanied by two associated length scales. To facilitate finite element implementations, the model is rewritten such that C 0 -continuity of the interpolation is sufficient. An auxiliary displacement field is introduced which allows the original fourth-order equations to be split into two coupled sets of second- order equations. Positive-definiteness of the kinetic energy requires that the inertia length scale is larger than the stiffness length scale. The governing equations, boundary conditions and the discretized system of equations are presented. Finally, dispersive wave propagation in a one-dimensional bar is considered in a numerical example. Copyright 2007 John Wiley & Sons, Ltd. Received 1 November 2006; Revised 11 January 2007; Accepted 11 January 2007 KEY WORDS: gradient elasticity; higher-order continuum; C 0 -continuity; wave dispersion; higher-order inertia; length scale ∗ Correspondence to: Harm Askes, Department of Civil and Structural Engineering, University of Sheffield, Sheffield S1 3JD, U.K. † E-mail: h.askes@sheffield.ac.uk Contract/grant sponsor: Engineering and Physical Sciences Research Council; contract/grant number: EP/D041368/1 Copyright 2007 John Wiley & Sons, Ltd.