engleski

Atomistic-to-continuum modelling in solid mechanics

An increased competition in consumer electronics has pushed the boundaries of technological development towards miniaturisation. Ever increasing weight/size and power demand limitations resulted in the rise of nano-materials. We focus primarily on the conceptually new class of materials that are only one atom thick, called by common name "graphene". More precisely, we consider single-atomic layer of carbon atoms tightly packed into a two-dimensional, honeycomb lattice. The molecular mechanics of the chamical bonds is determined by the Morse empirical interatomic potential. The experimental measurement of the mechanical properties of graphene is still considered a difficult task which requires tests to be performed at the nano-scale. Thus, there is not yet a large number of existing works on experimental evaluation of the mechanical properties. Consequently, quantifying the mechanical properties by the numerical simulations becomes of even greater importance. However, simulation of this kind ought to start at nano-scale to properly consider the material, i.e. lattice structure. We use here molecular mechanics based on the assumption that atoms are the smallest unit needed to be modelled. This enables, furthermore to study the discrete atomic structure as a multi-particle system. Due to the lack of computational power, performing a fully atomistic simulation of practical carbon nanosystems is not always possible. Thus, we seek to find an alternative, more effective modelling strategy. At first we concern the substitute, continuum modelling of pristine, defect-free graphene in the small and large strain regime. This procedure is often called hierarchical multiscale modelling. In the case of the small strain deformation, the homogenised continuum model boils down to the isotropic linear elastic model. However, in the available literature on the subject a large scatter of the material constants is observed. We review principal mechanisms causing the scatter and develop stiffness bounds related to the type of the imposed boundary conditions, namely force or displacement. This proves to be yet another reason that may cause the discrepancy between the reported results. In order to have an effective design tool for novel applications of graphene the large strain regime is equally important. We developed a homogenised constitutive model written in terms of strain energy potential as a function of principal stretches, that fits well in the large deformation membrane theory. Having a well defined surrogate continuum model of pristine graphene, we turn to concurrent multiscale methodology which limits atomic model to a small cluster of atoms near the hot spot, i.e. defect in graphene lattice. The proposed methodology is based on the overlapping domain decomposition scheme and coupling of discreet, atomic and continuum models, called the bridging domain or Arlequin method. The latter enables to have efficient continuum model, preserving at the same time the accuracy of atomistic model. This methodology is implemented in MATLAB and tested first on a simple chain-like model. We present brief discussion about the spurious effects (termed ghost forces) that may arise in and near the coupling domain depending on the different coupling options. Furthermore, we give an overview of salient features of the main multiscale families with a special attention towards the role of model adaptivity. The quasicontinuum method uses an adaptive coarse graining approach rather than classical coupling, and is, thus, used as a reference for adaptive strategy. Moreover, we brought the two mentioned mainstream multiscale methods to bear on the chosen model problem. In the process, either method is further advanced from its standard implementation which shows the possibility of unique formulation. The two-dimensional L2 and H1 coupling formulation for the defected graphene is present at the end. The numerical efficiency of the derived algorithms is demonstrated by a number of illustrative numerical examples.

graphene; molecular mechanics; multiscale; bridging domain; Arlequin; quasicontinuum

Rad je izrađen u okviru dvojnog doktorata prema Ugovoru o dvojnom doktoratu potpisanom između Sveučilišta u Zagrebu i Ecole Normale Superioure de Cachan, Francuska.