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The Cauchy-Born (CB) hypothesis has been widely used in multi-scale modeling of crystalline nano-structures. The violation of CB hypothesis in stress space and the transition to plasticity, which is equivalent to the violation of CB hypothesis in strain space, are generally confused and it becomes crucial to differentiate between the two distinct phenomena; the violation of the former usually occurs at high values of stress and at regions where the surface effects are manifest while the violation of the latter occurs at low stresses when the material loses its strength to tolerate the applied loading. In this paper, a novel technique is developed to investigate the validity of CB hypothesis... 

The Cauchy–Born (CB) hypothesis has been widely used in multi-scale modeling of crystalline nano-structures. The violation of CB hypothesis in stress space and the transition to plasticity, which is equivalent to the violation of CB hypothesis in strain space, are generally confused and it becomes crucial to differentiate between the two distinct phenomena; the violation of the former usually occurs at high values of stress and at regions where the surface effects are manifest while the violation of the latter occurs at low stresses when the material loses its strength to tolerate the applied loading. In this paper, a novel technique is developed to investigate the validity of CB hypothesis... 

The band gap offset is an effect of coordination numbers (CNs) of atom reduction at the edge of transversal cross-section of Silicon nanowires (SiNWs). In this paper, a hierarchical multi-scale technique is developed to model the edge effect on the band gap shift of SiNWs since the geometric effect is dominant in the energy gap due to the appearance of strain in the self-equilibrium state. The multi-scale model is performed based on the molecular dynamics approach and finite element method for the micro- (atomistic) and macro-scale levels, respectively. The Cauchy-Born (CB) hypothesis is used to relate the atomic positions to the continuum field through the deformation gradient. Finally, the... 

One of the most popular constitutive rules that correlate the continuum and atomic properties in multi-scale models is the Cauchy-Born (CB) hypothesis. Based on this constitutive law of continuum media, it assumes that all atoms follow the deformation subjected to the boundary of crystal. In this paper, the validity and failure of CB hypothesis are investigated for the silicon nano-structure by comparison of the continuum and atomic properties. In the atomistic level, the stresses and position of atoms are calculated using the molecular dynamics (MD) simulation based on the Tersoff inter-atomic potential. The stresses and strains are compared between the atomistic and continuous media to... 

In this paper, a new multi-scale approach is presented based on the modified boundary Cauchy-Born (MBCB) technique to model the surface effects of nano-structures. The salient point of the MBCB model is the definition of radial quadrature used in the surface elements which is an indicator of material behavior. The characteristics of quadrature are derived by interpolating data from atoms laid in a circular support around the quadrature, in a least-square scene. The total-Lagrangian formulation is derived for the equivalent continua by employing the Cauchy-Born hypothesis for calculating the strain energy density function of the continua. The numerical results of the proposed method are... 

In continuum mechanics, the constitutive models are usually based on the Cauchy-Born (CB) hypothesis which seeks the intrinsic characteristics of the material via the atomistic information and it is valid in small deformation. The main purpose of this paper is to investigate the temperature effect on the stability and size-dependency of Cauchy-Born hypothesis. Three-dimensional temperature-related Cauchy-Born formulations are developed for crystalline structure and the stability and size-dependency of temperature-related Cauchy-Born hypothesis are investigated by means of direct comparison between atomistic and continuous mediums. In order to control the temperature effect, the Nose-Hoover... 

In this paper, a novel multi-scale approach is developed for modeling of the surface effect in crystalline nano-structures. The technique is based on the Cauchy-Born hypothesis in which the strain energy density of the equivalent continua is calculated by means of inter-atomic potentials. The notion of introducing the surface effect in the finite element method is based on the intrinsic function of quadratures, called as an indicator of material behavior. The information of quadratures is derived by interpolating the data from probable representative atoms in their proximity. The technique is implemented by the definition of reference boundary CB elements, which enable to capture not only... 

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... 

The Cauchy-Born hypothesis (CB) provides a hierarchical approach in the molecular theory of crystal elasticity to relate the continuum and atomic deformations. This kinematic theory has been extensively used as the constitutive law of continuum regions in multi-scale models. In these models, the fine scale is proposed to describe the real behavior of crystalline structure wherever the continuum description fails. The main objective of this article is to investigate the stability and size-dependency of CB hypothesis in three-dimensional applications by direct comparison of information between atomistic and continuous description of a medium. The Sutton-Chen many-body potential is used for the... 

The band gap offset is an effect of coordination numbers (CNs) of atom reduction at the edge of transversal cross-section of Silicon nanowires (SiNWs). In this paper, a hierarchical multi-scale technique is developed to model the edge effect on the band gap shift of SiNWs since the geometric effect is dominant in the energy gap due to the appearance of strain in the self-equilibrium state. The multi-scale model is performed based on the molecular dynamics approach and finite element method for the micro- (atomistic) and macro-scale levels, respectively. The Cauchy–Born (CB) hypothesis is used to relate the atomic positions to the continuum field through the deformation gradient. Finally, the... 

One of the most popular constitutive rules that correlate the continuum and atomic properties in multi-scale models is the Cauchy–Born (CB) hypothesis. Based on this constitutive law of continuum media, it assumes that all atoms follow the deformation subjected to the boundary of crystal. In this paper, the validity and failure of CB hypothesis are investigated for the silicon nano-structure by comparison of the continuum and atomic properties. In the atomistic level, the stresses and position of atoms are calculated using the molecular dynamics (MD) simulation based on the Tersoff inter-atomic potential. The stresses and strains are compared between the atomistic and continuous media to... 

In this paper, a novel temperature-dependent multi-scale method is developed to investigate the role of temperature on surface effects in the analysis of nano-scale materials. In order to evaluate the temperature effect in the micro-scale (atomic) level, the temperature related Cauchy-Born hypothesis is implemented by employing the Helmholtz free energy, as the energy density of equivalent continua relating to the inter-atomic potential. The multi-scale technique is applied in atomistic level (nano-scale) to exhibit the temperature related characteristics. The first Piola-Kirchhoff stress and tangential stiffness tensor are computed, as the first and second derivatives of the free energy... 

In this paper, an atomistic-based higher-order continuum model is developed in the framework of nonlinear finite element method to present the geometrically nonlinear behavior of nano-structures. In order to model the inhomogeneous deformation within the Cauchy-Born hypothesis, the higher-order CB hypothesis is presented based on a hierarchical multi-scale technique, in which the constitutive model of higher-order continuum is obtained using the derivatives of strain energy density. In order to avoid the use of C1–continuity element, as an alternative procedure, the mixed-type element is utilized employing the nodal deformation gradient as additional degrees of freedom. The relation between... 

A large amount of research in computational mechanics has biased toward atomistic simulations. This trend, on one hand, is due to the increased demand to perform computations in nanoscale and, on the other hand, is due to the rather simple applications of pairwise potentials in modeling the interactions between atoms of a given crystal. The Cauchy–Born (CB) hypothesis has been used effectively to model the behavior of crystals under different loading conditions, in which the comparison with molecular dynamics simulations presents desirable coincidence between the results. A number of research works have been devoted to the validity of CB hypothesis and its application in post-elastic limit....