A quasicontinuum methodology for multiscale analyses of discrete microstructural models

Abstract

SUMMARYMany studies in different research fields use lattice models to investigate the mechanical behavior of materials. Full lattice calculations are often performed to determine the influence of localized microscale phenomena on large-scale responses but they are usually computationally expensive. In this study the quasicontinuum (QC) method (Phil. Mag. A 1996; 73:1529-1563) is extended towards lattice models that employ discrete elements, such as trusses and beams. The QC method is a multiscale approach that uses a triangulation to interpolate the lattice model in regions with small fluctuations in the deformation field, while in regions of high interest the exact lattice model is obtained by refining the triangulation to the internal spacing of the lattice. Interpolation ensures that the number of unknowns is reduced while summation ensures that only a selective part of the underlying lattice model must be visited to construct the governing equations. As the QC method has so far only been applied to atomic lattice models, the existing summation procedures have been revisited for structural lattice models containing discrete elements. This has led to a new QC method that makes use of the characteristic structure of the considered truss network. The proposed QC method is, to the best of the authors' knowledge, the only QC method that does not need any correction at the interface between the interpolated and the fully resolved region and at the same time gives exact results unlike the cluster QC methods. In its present formulation, the proposed QC method can only be used for lattice models containing nearest neighbor interactions, but with some minor adaptations it can also be used for lattices with next-nearest neighbor interactions such as atomic lattices.

A multiscale quasicontinuum method for dissipative lattice models and discrete networks

“…First, this leads to a large number of degrees of freedom (DOFs) making the system of equations computationally expensive to solve. Secondly, the construction of the system of equations is computationally expensive because all lattice points must be visited for this (Beex et al, 2011).…”

“…The QC method was originally developed for the reduction of atomistic lattice models by Tadmor et al (1996a) and has been widely used to investigate phenomena such as intergranular fracture (Miller et al, 1998) and nanoindentation (Tadmor et al, 1996b;Knap and Ortiz, 2001;Kulkarni et al, 2008). In a previous study (Beex et al, 2011) the applicability of the QC method has been demonstrated for structural lattice models of fibrous materials that employ elastic trusses. An overview of several QC frameworks is given by Miller and Tadmor (2002).…”

“…This is an advantage, since continuum descriptions for fibrous materials tend to be complex to formulate (Peng and Cao, 2005). Another advantage is that a number of QC frameworks (Knap and Ortiz, 2001;Eidel and Stukowski, 2009;Beex et al, 2011Beex et al, , 2012a do not require a handshaking region or coupling procedure between fully resolved domains (in which the exact lattice model is recovered) and coarse domains (in which an approximation is made). Several other multiscale approaches require such a coupling procedure (Shenoy et al, 1999;Xiao and Belytschko, 2004;Fish et al, 2007;Ha-Minh et al, 2011).…”

“…To obtain an accurate solution, all lattice points in the fully resolved regions are sampling points, while in the coarse regions only a small number of points is used (see for instance the studies of Zhang and Gunzburger (2010); Beex et al (2011) and Beex et al (2012a)). …”

“…Also the QC formulation of Beex et al (2011), developed for structural lattice models with elastic interactions only, uses this ansatz. Depending on the application, such an elastic description is adequate (Delincé and Delannay, 2004;Sharma and Sutcliffe, 2004;Gonella and Ruzzene, 2008;Hatami-Marbini and Picu, 2009;Zeman et al, 2011).…”