Comput Mech DOI 10.1007/s00466-007-0222-6 ORIGINAL PAPER A cohesive finite element for quasi-continua Xiaohu Liu · Shaofan Li · Ni Sheng Received: 18 May 2007 / Accepted: 20 September 2007 © Springer-Verlag 2007 Abstract In this paper, a cohesive finite element method (FEM) is proposed for a quasi-continuum (QC), i.e. a contin- uum model that utilizes the information of underlying atomis- tic microstructures. Most cohesive laws used in conventional cohesive FEMs are based on either empirical or idealized constitutive models that do not accurately reflect the actual lattice structures. The cohesive quasi-continuum finite element method, or cohesive QC-FEM in short, is a step forward in the sense that: (1) the cohesive relation between interface traction and displacement opening is now obtained based on atomistic potentials along the interface, rather than empirical assumptions; (2) it allows the local QC method to simulate certain inhomogeneous deformation patterns. To this end, we introduce an interface or discontinuous Cauchy– Born rule so the interfacial cohesive laws are consistent with the surface separation kinematics as well as the atomistically enriched hyperelasticity of the solid. Therefore, one can sim- ulate inhomogeneous or discontinuous displacement fields by using a simple local QC model. A numerical example of a screw dislocation propagation has been carried out to dem- onstrate the validity, efficiency, and versatility of the method. Keywords Cohesive laws · Dislocation · Finite element method · Nano-mechanics · Quasi-continuum 1 Introduction The numerical simulation of strong and weak discontinuities has been one of the major focuses of computational fail- ure mechanics and engineering reliability analysis in recent years. Several finite element methods (FEMs) have been X. Liu · S. Li (B ) · N. Sheng Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720, USA e-mail: shaofan@berkeley.edu proposed. Two of the most successful methods are the cohesive FEM proposed by [17], and the so-called extended finite element method (X-FEM) proposed by Belytschko et al. [1]. However, most of these methods adopt phenomenolog- ical constitutive relations, which may not be suitable for computational nanomechanics. For example, when it comes to simulations of individual dislocation motions, we are not only interested in how a dislocation moves but also how it affects the motion of neighboring atoms. The traditional FE based methods usually have difficulties in capturing those details. An alternative method is molecular dynamics (MD), which has been successfully applied to simulations of the crack growth. However MD simulates the motion of every single atom in the domain, the computational cost can be enormous if the size of the spatial domain of the simulation is up or beyond nanoscale. Today most MD simulations of fracture are only limited in nano- or sub-nanoscales. To build a cost-effective simulation tool, a class of so-called coarse-grained methods has been proposed. These methods exploit the information at the atomic level but retain some basic features of continuum mechanics. One of the pop- ular coarse-grained methods is the Quasi-continuum (QC) method, e.g. [16]. A comprehensive review can be found in [12]. There are two versions of the QC method: the local QC and the non-local QC. The local version of QC method adopts the Cauchy–Born rule, and hence it can only apply to where the local deformation is uniform; while the non- local version was designed to simulate inhomogeneous local deformations. To achieve that, it needs atomic resolution, and hence it is not really a coarse grain model. Therefore, its computational cost is more expensive and comparable to that of MD simulations. 1 1 In the rest of the paper, when we use the term “QC method,” we refer to the local version of QC method, unless indicated otherwise. 123