Cauchy Born Research Articles

A Reduced-Order General Continuum Method for Dynamic Simulations of Carbon Nanotube

In this paper, a reduced-order general continuum method for simulating the three-dimensional transient mechanical behaviors of carbon nanotube (CNT) is presented. The method builds the potential energy density for carbon nanotubes by applying the macroscopic deformation gradient to the atomistic energy potential based on a modified Cauchy-Born rule. The minimum energy principle in finite element methods can be used to obtain the equilibrium solution of the resulting equations. To compute the dynamic behaviors of carbon nanotubes, an explicit Newmark time integration scheme is applied to augment the static formulation. Details of the proposed formulation to study CNT dynamics are described. The results of simulation cases are then presented. The cases include a two-dimensional carbon atomic ring interacting with carbon substrate, static deformation of carbon nanotube, such as elongation, buckling, and twisting, and dynamic deformation of CNT-AFM probe with point force loading. Where possible, the results are compared with those obtained by molecular dynamics and atomic based finite element methods. The comparison shows that the current method can capture unique nonlinear characteristics of CNTs and provide predictions of the CNT’s transient behaviors.

A nonlocal continuum model based on atomistic model at zero temperature

In traditional continuum theories, constitutive relations such as generalized Hooke's law and the Fourier law for heat conduction, are representations of macroscopic phenomenons. These theories have severe limitations as the length scale of the structure becomes close to the atomistic dimensions. The failure of the traditional continuum theories at atomistic scale can be attribute to ignoring of the length scale induced by intrinsic non-locality of atomistic interactions. The primary objective of this work is to construct a continuum model strictly based on the atomistic model bypassing any empirical constitutive law and any local homogeneous assumption such as Cauchy-Born rule. The intrinsic non-locality of the atomistic interactions are explicitly built into the model to make the model applicable to cases of strongly nonuniform deformation at the atomistic scales. When the deformation is homogeneous, the model reduces to that derived from Cauchy-Born rule. The present continuum model is expected to work well at atomistic scales. It is also consistent with the traditional continuum as the length scale approach to macro scale.

Analysis of the elastostatic specific stiffness of 2D stretching-dominated lattice materials

This paper presents a matrix-based procedure to characterize the specific stiffness properties of 2D lattice materials with any arbitrary cell topology. Unlike previous works, the current study automates the analysis process to include lattice materials whose unit cell has elements extending between adjacent cells and thus intersecting their envelopes. The main challenge in the analysis of this periodic lattice structures is that the unit cell does not have the full information concerning its nodal kinematic and static periodicity. For this reason, we introduce the Dummy Node Scheme, which enables the analysis of lattice material with any cell topology. The lattice material is modelled here as a pin-jointed infinite micro-truss structure. The results of the determinacy analysis are used to distinguish between the bending-dominated and the stretching-dominated behaviours of the material. The Cauchy–Born hypothesis is used to homogenize the lattice material properties by formulating the microscopic lattice nodal deformations in terms of the material macroscopic strain field. This formulation, in turn, is used to express the microscopic element deformations in terms of the macroscopic strain field, from which the material macroscopic stiffness properties are derived. In this process, the Dummy Node Scheme is a necessary step to construct the nodal periodicity within the unit cell, which is used to apply the Cauchy–Born kinematic boundary condition to the nodal deformation wave functions. The procedure introduced in this paper is applied to 10 lattice topologies, five of which have unit cells with a square Bravais lattice symmetry and the other five have unit cells with a hexagonal Bravais lattice symmetry. Finally, charts representing the relative elastic moduli of the lattice material versus its relative density are developed. These charts assist the selection of the best topology of a stretching-dominated lattice material for a given application that requires a material with specific stiffness properties.

On (Andersen–)Parrinello–Rahman Molecular Dynamics, the Related Metadynamics, and the Use of the Cauchy–Born Rule

It is shown that the Parrinello–Rahman Lagrangian is obtained by assuming that (i) the cell inertia tensor is spherical and constant in time, and that (ii) the cell fluctuation motions are irrotational. A slightly different Lagrangian is suggested, arrived at by dropping the sphericity assumption in (i). A related metadynamics is also proposed, based on replacing by (ii) the various no-cell-rotation assumptions that are customarily used. Finally, a zero-temperature, scale-bridging relation is proposed between the intermolecular potential and the macroscopic stored-energy mapping.

A New Extension of Cauchy–Born Rule for Monolayer Crystal Films

By combining with the physical concept of inscribed surface, the standard Cauchy–Born rule (CBR) is straightly extended to have a rigorous and accurate atomistic continuum theory for the monolayer crystal films. Resorting to using Tersoff–Brenner potential, the present theory to graphite sheet and single-walled carbon nanotubes (SWCNTs) is applied to evaluate the mechanical properties. The results are validated by the comparison with previously reported studies.

Poroelasticity-I: Governing Equations of the Mechanics of Fluid-Saturated Porous Materials

We present here the approach to the theory of fluid-filled poroelastics based on consideration of poroelastics as a continuum of “macropoints” (representative elementary volumes), which “internal” states can be described by as a set of internal parameters, such as local relative velocity of fluid and solid, density of fluid, internal strain tensor, specific area, and position of the center of mass of porous space. We use the generalized Cauchy–Born hypothesis and suggest that there is a system of (structural) relationships between external parameters, describing the deformation of the continuum and internal parameters, characterizing the state of representative elementary volumes. We show that in nonhomogenous (and, particularly, nonlinear) poroelastics, an interaction force between solid and fluid appears. Because this force is proportional to the gradient of porosity, absent in homogeneous poroelastics, and one can neglect with dynamics of internal degrees of freedom, this force is equivalent to the interaction force, introduced earlier by Nikolaevskiy from phenomenological reasons. At last, we show that developed theory naturally incorporates three mechanisms of energy absorption: visco-inertial Darcy mechanism, “squirt flow” attenuation, and skeleton attenuation.

On the generalized virial theorem and Eshelby tensors

The formal relationships between the scalar and tensorial virials and Eshelby tensors have been presently investigated. The key idea is to evaluate the Eshelby stress from discrete or atomistic simulations for a structured body, conceived as a numerical homogenization method to reconstitute the macroscopic continuum behavior in multiscale modelling approaches. Extending first the writing of the scalar virial to a material format, it is shown that the average of the elaborated scalar material virial is the trace of the (material) Eshelby stress. The spatial and material virials are further related to eachother in the framework of hyperelasticity, and a tensorial extension of the material virial is provided. Interpretation of those results from the microscopic point of view shows that Eshelby stress may be identified and calculated at the discrete level from the average of the virial tensor. Consideration of the material version of the virial theorem further leads to express Eshelby stress versus the average of the internal tensorial material virial and of the kinetic energy. The average scalar virial is further identified to the grand potential in a thermodynamic context. A definition of the material scalar virial for a second order continuum is lastly proposed, based on the identification of a second order Eshelby stress and in line with the second order Cauchy–Born rule.

Inhomogenous Dislocation Nucleation Based on Atom Potential in Hexagonal Noncentrosymmetric Crystal Sheet

By introducing internal degree, the deformation of hexagonal noncentrosymmetric crystal sheet can be described by the revised Cauchy–Born rule based on atomic potential. The instability criterion is deduced to investigate the inhomogeneous dislocation nucleation behavior of the crystal sheet under simple loading. The anisotropic characters of dislocation nucleation under uniaxial tension are studied by using the continuum method associated with the instability criterion. The results show a strong relationship between yield stress and crystal sheet chirality. The results also indicate that the instability criterion has sufficient ability to capture the dislocation nucleation site and expansion. To observe the internal dislocation phenomenon, the prediction of the dislocation nucleation site and expansion domain is illustrated by MD simulations. The developed method is another way to explain the dislocation nucleation phenomenon.

Strain effect analysis on phonon thermal conductivity of two-dimensional nanocomposites

In this paper, we present a model that combines lattice dynamics and the phonon Boltzmann transport equation (BTE) to analyze strain effect on the cross-plane phonon thermal conductivity of silicon wire-germanium host nanocomposites. For a given strain condition, mechanical strain is translated to crystal lattice deformation by using the Cauchy–Born rule. Strain-dependent phonon thermal properties of Si and Ge obtained from lattice dynamics with Tersoff empirical interatomic potential are then incorporated into the BTE, in which ballistic transport within one material and diffuse scattering between Si–Ge interface are employed. The strain-dependent BTE is solved numerically on an unstructured triangular mesh by using a finite volume method. Nanocomposites with different Si nanowire cross sections are also investigated. The results show that the phonon thermal conductivity of the nanocomposites can be significantly decreased (or increased) by a tensile (or compressive) strain. With the same length change, hydrostatic strain produces a larger variation in phonon thermal conductivity than uniaxial strain. In addition, it is shown that with the same atomic percentage, the cross-sectional shape makes little difference to the thermal conductivity except at very small characteristic lengths of the Si nanowire.

A multiscale cohesive zone model and simulations of fractures

In this work, a novel multiscale cohesive zone model is proposed, in which the bulk material is modeled as a local quasi-continuum medium that obeys the Cauchy–Born rule while the cohesive force and displacement relations inside the cohesive zone are governed by a coarse grained depletion potential. The interface depletion potential is constructed based on the Derjaguin approximation of nonlocal colloidal interactions. By doing so, the interface constitutive descriptions are made genetically consistent with the bulk constitutive relations that are enriched from underneath atomistic structure. The method provides an effective means to describe properties of material inhomogeneities such as grain boundaries, bi-material interfaces, slip lines, and inclusions, etc. We have developed and implemented the proposed multiscale cohesive zone model in a cohesive finite element weak Galerkin formulation, and we have applied it to simulate dynamic fracture problems in solids. The numerical simulation results have shown that the method has successfully captured the phenomenon of spall fracture during simulations of impacts and penetrations.

Higher Order Cauchy-Born Rule Based Study of Chiral Single-Walled Carbon Nanotubes

Higher Order Cauchy-Born Rule Based Study of Chiral Single-Walled Carbon Nanotubes

On the applicability of the Cauchy-Born rule

Numerical computations are presented confirming the possible breakdown of the Cauchy-Born rule for an infinite planar square lattice whose vertices are occupied by interacting particles, in agreement with the theoretical predictions of Friesecke and Theil (2002). Energy minimization yields solution branches exhibiting a non-zero deviatoric inner displacement for shearing and extensional deformation. A statically square lattice can become unstable when subjected to sufficiently large deformation. Numerical solutions of a boundary-value problem illustrate the development of irregular lattice configurations under supercritical conditions.

Graphene Nano-Ribbons Under Tension

The mechanical response of graphene nano-ribbon under tensile loading has been investigated using atomistic simulation. Lattice symmetry dependence of elastic properties are found, which fits prediction from Cauchy-Born rule well. Concurrent brittle and ductile behaviors are observed in the failure process at elastic limit, which dominates at low and high temperature respectively. In addition, the free edges of finite width ribbon help to activate bond-flip events and initialize ductile behavior.

Quantifying the size-dependent effect of the residual surface stress on the resonantfrequencies of silicon nanowires if finite deformation kinematics are considered

There are two major objectives to the present work. The first objective is to demonstratethat, in contrast to predictions from linear surface elastic theory, when nonlinear, finitedeformation kinematics are considered, the residual surface stress does impact the resonantfrequencies of silicon nanowires. The second objective of this work is to delineate,as a function of nanowire size, the relative contributions of both the residual(strain-independent) and the surface elastic (strain-dependent) parts of the surface stress tothe nanowire resonant frequencies. Both goals are accomplished by using the recentlydeveloped surface Cauchy–Born model, which accounts for nanoscale surface stressesthrough a nonlinear, finite deformation continuum mechanics model that leads to thesolution of a standard finite element eigenvalue problem for the nanowire resonantfrequencies. In addition to demonstrating that the residual surface stress does impact theresonant frequencies of silicon nanowires, we further show that there is a strong sizedependence to its effect; in particular, we find that consideration of the residual surfacestress alone leads to significant errors in predictions of the nanowire resonant frequency,with an increase in error with decreasing nanowire size. Correspondingly, thestrain-dependent part of the surface stress is found to have an increasingly importanteffect on the resonant frequencies of the nanowires with decreasing nanowire size.

Stability and size-dependency of Cauchy–Born hypothesis in three-dimensional applications

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 gold metal to consider the real metallic behavior in numerical simulations. Two failure criteria are introduced in the strain and stress domains; the validity surfaces are derived for the Cauchy–Born hypothesis; and the size effect of specimens is investigated on the convergency of results. It is shown that the gold crystal deforms homogeneously inside the validity surface, in which the material is elastic and the CB has remained valid. It is observed that although the deformation is inhomogeneous and the CB is invalid outside the validity surface, the crystalline structure may exhibit elastic or plastic behavior in this region. Moreover, it is numerically shown that the size-dependency of validity surface decreases with the increase of the size of specimens. These observations are meticulously investigated by loading and unloading several cubic specimens using molecular dynamics simulation.

Energy landscape for martensitic phase transformation in shape memory NiTi

First-principles calculations are presented for parent B2 phase and martensitic B19 and B19′ phases in NiTi. The results indicate that both B19 and B19′ are energetically more stable than the parent B2 phase. By means of ab initio density functional theory, the complete distortion–shuffle energy landscape associated with B2 → B19 transformation in NiTi is then determined. In addition to accounting for the Bain-type deformation through the Cauchy–Born rule, the study explicitly accounts for the shuffle displacements experienced by the internal ions in NiTi. The energy landscape allows the energy barrier associated with the B2 → B19 transformation pathway to be identified. The results indicate that a barrier of 0.48 mRyd atom −1 (relative to the B2 phase) must be overcome to transform the parent B2 NiTi to orthorhombic B19 martensite.

On the derivation of linear elasticity from atomistic models

We derive linear elastic energy functionals from atomistic models as a -limit when the number of atoms tends to infinity, respec tively, when the interatomic distances tend to zero. Our approach generalizes a recent result of Braides, Solci and Vitali (2). In particular, we study mass spring models with full nearest and next-to-nearest pair interactions. We also consider boundary value problems where a part of the boundary is free. 1. Introduction. The passage from discrete atomic models to continuum theories is an active area of current research in continuum mechanics. For elastic systems one usually refers to the Cauchy-Born rule to obtain macroscopic energy densities from atomistic interaction functionals. The Cauchy-Born rule states that - roughly speaking - each individual atom follows the macroscopic deformation gradient and in particular does not take into account fine scale oscillations on the microscopic scale. For a two-dimensional mass spring model, the validity of the Cauchy-Born rule for deformations close to a rigid motion has been proved by Friesecke and Theil in (8). Their result has been generalized to arbitrary dimensions by Conti, Dolzmann, Kirchheim and Muller in (3). If the deformation gradients are very close to SO(d), the set of orientation pre- serving rigid motions, then we expect linear elasticity theory to apply. This relation has been made rigorous by Dal Maso, Negri and Percivale who derive the energy functional of linear elasticity as a -limit of nonlinear el asticity for small displace- ments in (5). (See also the author's article (10) for a strong convergence result for the associated minimum problems.) Recently it has been noted that one can derive linear elasticity functionals directly from certain atomistic pair potentials: For a special class of pair interaction models Braides, Solci and Vitali prove -convergence of the discre te energy functionals to the energy functional of an associated continuum linear elasticity energy functional (see (2)). In this set-up one has to deal with two small parameters e and δ measuring the typical interatomic distance and the local distance of the deformations from the set of rigid motions, respectively. The aim of the present article is to extend these results in three directions. Firstly, we will drop the assumption that atoms are allowed to interact only along

ENERGY OF ARMCHAIR NANOTUBE USING THE MODIFIED CAUCHY-BORN RULE

Due to the difference of nanotube diameters, the single-walled carbon nanotubes (SWCNTs) show the different energy and mechanical properties. In order to take the effect of the curvature of nanotubes into account in the modeling of those structures, the present paper proposes an atomistic based continuum model with using a type of modified Cauchy-Born to link the continuum strain energy to the interatomic potential. This modified Cauchy-Born is developed by incorporating the concept of differential mean value theorem into the standard Cauchy-Born rule. The present model not only can bridge the microscopic and macroscopic length scales, but also can investigate the curvature effect of a single layer film on the continuum level. Application of the current model to armchair carbon nanotubes and graphite shows an excellent prediction of the size dependent strain energy which are compared in a good agreement with the existing experimental and theoretical results.

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

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.

Instabilities in shear and simple shear deformations of gold crystals

We use the tight-binding potential and molecular mechanics simulations to study local and global instabilities in shear and simple shear deformations of three initially defect-free finite cubes of gold single crystal containing 3480, 7813, and 58,825 atoms. Displacements on all bounding surfaces are prescribed while studying simple shear deformations, but displacements on only two opposite surfaces are assigned during simulations of shear deformations with the remaining four surfaces kept free of external forces. The criteria used to delineate local instabilities in the system include the following: (i) a component of the second-order spatial gradients of the displacement field having large values relative to its average value in the body, (ii) the minimum eigenvalue of the Hessian of the energy of an atom becoming non-positive, and (iii) structural changes represented by a high value of the common neighborhood parameter. It is found that these criteria are met essentially simultaneously at the same atomic position. Effects of free surfaces are evidenced by different deformation patterns for the same specimen deformed in shear and simple shear. The shear strength of a specimen deformed in simple shear is more than three times that of the same specimen deformed in shear. It is found that for each cubic specimen deformed in simple shear the evolution with the shear strain of the average shear stress, prior to the onset of instabilities, is almost identical to that in an equivalent hyperelastic material with strain energy density derived from the tight-binding potential and the assumption that it obeys the Cauchy–Born rule. Even though the material response of the hyperelastic body predicted from the strain energy density is stable over the range of the shear strain simulated in this work, the molecular mechanics simulations predict local and global instabilities in the three specimens.