Application of a Discrete Model for the Concurrent Simulation of Nanostructures

Due to their outstanding mechanical properties, developing robust modeling and simulation methods for nanomaterials is becoming increasingly important for a wide range of applications. Nanomaterials are distinguished by their near-perfect molecular structures at the nanometer scale and outstanding mechanical properties defined at the macroscopic level. Correspondingly, continuum modeling approach has the unique advantage in its ability to describe the structure-property relations at relatively large scale. A critical concern in the use of the existing continuum model is the kinematics based on continuum assumption, which will cause inconsistency if the discrete nature of the nanoscale system is considered. To remove such an inconsistency, a constitutive model based on the concept of spatial secant is outlined. Although the model is similar in its form to the crystal elasticity model, it is a discrete model due to the deformation measure that is introduced. Following the presentation of the model, the robustness of this model and its link to the atomistic model are discussed and shown through benchmark problems. I. Introduction t is well-known that the outstanding mechanical and physical properties of the nanomaterials are closely related to their almost defect-free molecular structures. To study the structure-property relationship, two major categories of approaches exist: continuum and atomistic approaches. In the atomistic approach, potential models that govern the interactions among the atoms are developed based on either first-principle calculations or experimental measurements. Such interaction models can be expressed in either empirical form or calculated from a particular first-principle approach. Although the empirical model can handle up to billions of atoms, there is a gap between the scales that can be treated by modeling and simulation and the real-life applications. On the other hand, continuum approach offers the advantage of extending the scale of the atomistic simulations. The continuum models that have been developed so far include both phenomenological theories such as linear elasticity and mechanism-based theories such as crystal elasticity. A key difference between the atomistic model and continuum model is the type of systems they treat. Atomistic model directly treats systems of atoms which occupies space in a discrete fashion. If a continuum model needs to be established for the same system, the atomic system must be first converted to a “continuum” by introducing certain principle of equivalence. Since the continuum model is established based on the kinematics of mapping of infinitesimal points, inconsistency arises if the length scale of the inter-atomic interaction becomes important. The main objective of this paper is to present a discrete model that aims at removing this inconsistency by developing a geometric exact approach. The developed measure for enforcing the exact kinematical mapping is called spatial secant. Correspondingly, a constitutive model is developed based on the introduced spatial secant. Although this model is similar in its form to the crystal elasticity model, it is a discrete model because of the underlying cluster representation. The definitions of the spatial secant and the material model are introduced in the next section. The implementation of the model is outlined in section III. The robustness of the model is illustrated in section IV through example problems and final conclusions are made in section V.