INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2012; 89:671–685 Published online 9 August 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3255 Coupling of nonlocal and local continuum models by the Arlequin approach Fei Han * ,† and Gilles Lubineau Physical Science and Engineering Division, COHMAS, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia SUMMARY The objective of this work is to develop and apply the Arlequin framework to couple nonlocal and local con- tinuum mechanical models. A mechanically-based model of nonlocal elasticity, which involves both contact and long-range forces, is used for the ‘fine scale’ description in which nonlocal interactions are considered to have non-negligible effects. Classical continuum mechanics only involving local contact forces is introduced for the rest of the structure where these nonlocal effects can be neglected. Both models overlap in a coupling subdomain called the ‘gluing area’ in which the total energy is separated into nonlocal and local contri- butions by complementary weight functions. A weak compatibility is ensured between kinematics of both models using Lagrange multipliers over the gluing area. The discrete formulation of this specific Arlequin coupling framework is derived and fully described. The validity and limits of the technique are demonstrated through two-dimensional numerical applications and results are compared against those of the fully nonlocal elasticity method. Copyright © 2011 John Wiley & Sons, Ltd. Received 2 April 2011; Revised 7 June 2011; Accepted 9 June 2011 KEY WORDS: multimodel; multiscale; nonlocal continuum; Arlequin method 1. INTRODUCTION The classical continuum theory, fully based on the assumption of contact forces, has been widely and successfully used for all sorts of structural problems. However, some observed experimental phe- nomena remain unexplained within the classical local continuum theory because of strong nonlocal effects at the mesoscale or microscale [1, 2]. Nonlocal models can also result from homogenization of nonlinear damage models, resulting in a dramatic increase of the computation cost [3, 4]. In such situations, alternative models have to be used to fully account for these nonlocal effects. Popular approaches are, for example, molecular mechanics or molecular dynamics [5]. However, discrete atomistic simulations still remain out of reach with current computing facilities for many engineering and physical problems. Accordingly, improved continuum theories that can capture the main part of these nonlocal effects have attracted considerable attention in the last decade [6–9]. A recent improvement is the mechanically-based model of nonlocal elasticity developed by Di Paola et al. [10, 11]. They introduced in a consistent framework not only contact forces between adjacent volumes, but also long-range central forces between nonadjacent volumes. These inter- actions depend on the relative displacements of the volumes along the central direction and on a proper distance-decay function. Contrary to discrete atomistic models, this consistent nonlocal elas- ticity model is still in the category of the continuum theory. This means that individual atoms need not be modeled, and that interatomic potential need not be known. *Correspondence to: Fei Han, Physical Science and Engineering Division, COHMAS, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia. † E-mail: harrisonhanfei@gmail.com Copyright © 2011 John Wiley & Sons, Ltd.