An investigation on the validity of Cauchy–Born hypothesis using Sutton-Chen many-body potential

Abstract

The Cauchy–Born hypothesis has been used to concurrently bridge atomistic information to continuum model. It has been a prevalent assumption in computational nano-mechanics during the past decade. This kinematic assumption relates the deformation of the continuum to the deformation of its underlying crystalline structure. The main objective of this paper is to investigate the validity of this hypothesis by means of direct atomistic simulations and the continuum mechanic calculations. In fact, we intend to determine under which strain or stress state the crystalline structure undergoes inhomogeneous deformation due to a small perturbation of the homogeneously deformed system. Two failure criteria are produced in principal strain and stress domains using different deformation paths, to illustrate the validity of Cauchy–Born hypothesis. It has been shown that the continuum model essentially obeys the hyperelastic model in the inner area of the produced curves. Obviously, the validity of Cauchy–Born hypothesis is analogous to the yield surface used in the theory of plasticity.

Introduction

Tremendous advances in nanoscience during the past two decades have drawn a new horizon for the future of science. Many structural elements such as nanotubes, nanowires, thin films and quantum dots are studied carefully. These studies span from experimental aspects by means of high-tech microscopes such as STM and AFM, to numerical aspects due to availability and increase in computational power of personal computers. Among several methodologies for numerical simulation of nanostructures, molecular dynamics (MD) is the one which has reached an exceptional prosperity in computational nanoscience [1], [2]. In addition to the modeling of nanostructures, molecular dynamics has shown promising results in modeling of physical phenomena which are direct results of microscopic organization and deformation of atomic lattices [3]. The objective of this branch of studies is to determine why and how material fails [4]. Despite this popularity, since MD works at the length scale of an atom and the time scale in the order of femtoseconds, running a simulation in large scales is absolutely prohibitive due to excessive computational expense [5]. Therefore, the notion of multiscale modeling to achieve atomistic information without spending too much computational expense has been triggered in mind [6].

The multiscale methods, such as quasicontinuum [7], [8], bridging scale [9] and bridging domain [10] are among the notable efforts to bridge the atomistic information to the continuum description, concurrently. While these methods use continuum description in the region of nearly homogeneous deformation, they retain atomistic configuration in the inhomogenously deformed regions. The quasicontinuum method is based on an adaptive procedure, in which the required area of atomistic data can be specified as the simulation proceeds [11]. The absence of automatic adaption for the second method is explicitly mentioned by Liu et al. [12]. In order to define a criterion for automatic adaption, it must be clearly specified that under which strain or stress state the continuum description fails and the atomistic resolution necessitates. The continuum model which is already used in these multiscale methods, is based on Cauchy–Born hypothesis.

Cauchy–Born (CB) is a homogenization hypothesis which is capable of bridging information from the atomistic scale to macro-scale. It stems from Cauchy who derived expressions for the linear elastic moduli from atomistic pair potentials [13]. This hypothesis postulates that when the boundaries of a single crystalline body are subjected to a deformation, all atoms of that body will follow this deformation [14]. Due to its simplicity, different researchers have tried to modify CB to consider different effects at the nano scale. Arroyo and Belytschko [15] proposed the exponential CB to model curved single layer films. Xiao and Yang [16] proposed the temperature-related CB which considers the thermal effect. In an effort to insert the effect of surface to CB, Park et al. [17] proposed the surface Cauchy–Born model. As it has been observed from the original definition, CB is valid in the elastic phase and when the atomistic deformation becomes inhomogeneous, this hypothesis fails.

The modern investigation of the issue starts with the treatise of Born and Huang [18]. Braides et al. [19] studied atomistic models using the concept of Γ-convergence, and proved that certain discrete functionals with pair-wise interaction converge to a continuum model. Friesecke and Theil [20] analytically studied the validity of CB in a cubic lattice via harmonic spring between the nearest and diagonal neighbors, using the notion of poly-convexity of harmonic lattices. They concluded that CB fails for large displacements and unfavorable values of spring. Steinmann et al. [21] investigated the validity of CB using direct atomistic simulation and compared their results with the acoustic tensor of continuum tensor [22]. They used Lenard–Jones potential which cannot model metallic behaviors. Their work merely checked the failure of CB in tension and shear deformation paths and they did not propose a domain of validity in their two-dimensional investigations.

In the present paper, the validity of Cauchy–Born hypothesis is studied by direct implementation of atomistic simulations, using Sutton-Chen many-body potential. The failure criterion is produced in both strain and stress spaces using different deformation paths. The failure criterion, which is analogous to the yield surface in plasticity, is defined to present the validity and invalidity of Cauchy–Born hypothesis. The effect of material anisotropy at nano scale, known as the orientation effect on the shape of failure criterion, is investigated. The paper is organized in the following sections. In Section 2, a concise definition of CB is presented along with the continuum based on Cauchy–Born hypothesis. The definition of the stress tensor and explanation of relaxation concept in a continuum based on CB are given in this section. In Section 3, the backgrounds of quasi-static atomistic simulation are represented and subsequently details of MD experiments are illustrated. The factors for instability determination of CB in both the strain and stress domains are defined in Section 4. Several numerical results along with validity curves are also presented in this section. Finally, some concluding remarks are given in Section 5.