Cauchy Born Research Articles

Surface stress effects on the resonant properties of metal nanowires: The importance of finite deformation kinematics and the impact of the residual surface stress

We utilize the recently developed surface Cauchy–Born model, which extends the standard Cauchy–Born theory to account for surface stresses due to undercoordinated surface atoms, to study the coupled influence of boundary conditions and surface stresses on the resonant properties of 〈 1 0 0 〉 gold nanowires with { 1 0 0 } surfaces. There are two major purposes to the present work. First, we quantify, for the first time, variations in the nanowire resonant frequencies due to surface stresses as compared to the corresponding bulk material which does not observe surface effects within a finite deformation framework depending on whether fixed/free or fixed/fixed boundary conditions are utilized. We find that while the resonant frequencies of fixed/fixed nanowires are elevated as compared to the corresponding bulk material, the resonant frequencies of fixed/free nanowires are reduced as a result of compressive strain caused by the surface stresses. Furthermore, we find that for a diverse range of nanowire geometries, the variation in resonant frequencies for both boundary conditions due to surface stresses is a geometric effect that is characterized by the nanowire aspect ratio. The present results are found to agree well with existing experimental data for both types of boundary conditions. The second major goal of this work is to quantify, for the first time, how both the residual (strain-independent) and surface elastic (strain-dependent) parts of the surface stress impact the resonant frequencies of metal nanowires within the framework of nonlinear, finite deformation kinematics. We find that if finite deformation kinematics are considered, the strain-independent surface stress substantially alters the resonant frequencies of the nanowires; however, we also find that the strain-dependent surface stress has a significant effect, one that can be comparable to or even larger than the effect of the strain-independent surface stress depending on the boundary condition, in shifting the resonant frequencies of the nanowires as compared to the bulk material.

Strain sensing through the resonant properties of deformed metal nanowires

In this article, we study the potential of gold nanowires as resonant nanoscale strain sensors. The sensing ability of the nanowires is determined by calculating the variations in resonant frequency that occur due to applied uniaxial tensile and compressive strain. The resonant frequencies are obtained using the surface Cauchy–Born model, which captures surface stress effects on the nanowires through a nonlinear continuum mechanics framework; due to the continuum formulation, the strain-dependent nanowire resonant frequencies are calculated through the solution of a standard finite element eigenvalue problem, where the coupled effects of the applied uniaxial strain and surface stress are naturally included through the finite element stiffness matrix. The nanowires are found to be more sensitive to compressive than tensile strain, with resonant frequency shifts around 200–400 MHz with the application of 1% tensile and compressive strain. In general, the strain sensitivity of the nanowires is found to increase with decreasing cross-sectional size, with additional dependencies on their aspect ratio.

Surface stress effects on the resonant properties of silicon nanowires

The purpose of the present work is to quantify the coupled effects of surface stresses and boundary conditions on the resonant properties of silicon nanowires. We accomplish this by using the surface Cauchy–Born model, which is a nonlinear, finite deformation continuum mechanics model that enables the determination of the nanowire resonant frequencies including surface stress effects through solution of a standard finite element eigenvalue problem. By calculating the resonant frequencies of both fixed/fixed and fixed/free ⟨100⟩ silicon nanowires with unreconstructed {100} surfaces using two formulations, one that accounts for surface stresses and one that does not, it is quantified how surface stresses cause variations in nanowire resonant frequencies from those expected from continuum beam theory. We find that surface stresses significantly reduce the resonant frequencies of fixed/fixed nanowires as compared to continuum beam theory predictions, while small increases in resonant frequency with respect to continuum beam theory are found for fixed/free nanowires. It is also found that the nanowire aspect ratio, and not the surface area to volume ratio, is the key parameter that correlates deviations in nanowire resonant frequencies due to surface stresses from continuum beam theory.

On the Cauchy—Born Rule

To relate continuum theories of elastic phenomena to typical atomic or molecular theories, one needs some way of correlating changes in positions of the entities in the latter with descriptions of deformation used in the former. For crystals, the most commonly used bridge used for this is the Cauchy—Born rule. In recent years, there has been a flurry of activity aimed at better understanding it and in trying to generalize it. I will describe and comment on important ideas that have been generated, point out some successes and failures of the rule and indicate kinds of applications that have motivated such work.

Study of the Damage-induced Anisotropy of Quasi-brittle Materials using the Component Assembling Model

Damage-induced anisotropy of quasi-brittle materials is investigated using component assembling model in this study. Damage-induced anisotropy is one significant character of quasi-brittle materials coupled with nonlinearity and strain softening. Formulation of such complicated phenomena is a difficult problem till now. The present model is based on the component assembling concept, where constitutive equations of materials are formed by means of assembling two kinds of components' response functions. These two kinds of components, orientational and volumetric ones, are abstracted based on pair-functional potentials and the Cauchy—Born rule. Moreover, macroscopic damage of quasi-brittle materials can be reflected by stiffness changing of orientational components, which represent grouped atomic bonds along discrete directions. Simultaneously, anisotropic characters are captured by the naturally directional property of the orientational component. Initial damage surface in the axial-shear stress space is calculated and analyzed. Furthermore, the anisotropic quasi-brittle damage behaviors of concrete under uniaxial, proportional, and nonproportional combined loading are analyzed to elucidate the utility and limitations of the present damage model. The numerical results show good agreement with the experimental data and predicted results of the classical anisotropic damage models.

Continuum limits of discrete thin films with superlinear growth densities

We provide a rigorous derivation by Γ-convergence of an effective theory of thin films in hyperelastic regime in the so-called discrete to continuous framework. By considering a discrete thin film obtained piling up at microscopic distance a finite number M of copies of a discrete monolayer ω, we provide a continuum description analogous to that in the dimension-reduction theories for continuum thin films. Our energetic description of the continuum limit model accounts for microscopic effects and in particular depends in a non-trivial way on M. We also consider the problem of homogenization and discuss several cases of interest when an explicit formula for the homogenized energy density can be obtained, with an interpretation in terms of the Cauchy–Born rule.

The buckling of single-walled carbon nanotubes upon bending: The higher order gradient continuum and mesh-free method

The bending buckling of single-walled carbon nanotubes (SWCNTs) is studied in the theoretical scheme of the higher order gradient continuum. The deformation of the underlying lattice vectors is approximated with an extended Cauchy–Born rule in which the effect of the second order deformation gradient is considered, and the continuum constitutive responses are determined by minimizing the energy of the representative cell. A mesh-free method is developed to implement the numerical modeling of SWCNTs, and their bending buckling behavior is numerically simulated with the developed method. The results are compared with those obtained with a full atomistic simulation, and it is revealed that the developed mesh-free method can accurately exhibit the bending deformation of SWCNTs. Different types of carbon nanotubes (CNTs) are studied, and the buckling mechanism is investigated.

A multiscale, finite deformation formulation for surface stress effects on the coupled thermomechanical behavior of nanomaterials

We present a multiscale, finite deformation formulation that accounts for surface stress effects on the coupled thermomechanical behavior and properties of nanomaterials. The foundation of the work lies in the development of a multiscale surface Helmholtz free energy, which is constructed through utilization of the surface Cauchy–Born hypothesis. By doing so, temperature-dependent surface stress measures as well as a novel form of the heat equation are obtained directly from the surface free energy. The development of temperature-dependent surface stresses distinguishes the present approach, as the method can be utilized to study the behavior of nanomaterials by capturing the size-dependent variations in the thermoelastic properties with decreasing nanostructure size. The coupled heat and momentum equations are solved in 1D using a fully implicit, monolithic scheme, and show the importance of capturing surface stress effects in accurately modeling the thermomechanical behavior of nanoscale materials.

A finite element formulation for nanoscale resonant mass sensing using the surface Cauchy–Born model

The purpose of this work is to develop the theoretical basis needed to study nanoscale resonant mass sensing with finite elements using the surface Cauchy–Born (SCB) model. The theory is developed in 1D, where it is identified that the primary modeling issue lies in capturing inhomogeneous surface stresses arising from adsorbate/substrate interactions. By utilizing internal degrees of freedom within the SCB framework, we show that the SCB model can represent the bonding energies, and thus the inhomogeneous surface stress that arises due to interactions by atoms of dissimilar materials. A key outcome of this is that it is shown that a finite element solution using the SCB model is able to simultaneously capture both mass and stiffness variations due to adsorbate/substrate interactions, and their effects on the nanostructure resonant properties. We first verify that the SCB model accurately captures the resonant properties of monatomic 1D atomic chains, then demonstrate the approach by studying the resonant properties of 1D atomic chains that interact with adsorbates. Importantly, we demonstrate that a finite element solution using the SCB model can predict the distinct shifts in resonant frequency that occur due to the adsorption of different materials on the 1D monatomic chain.

Application of the higher‐order Cauchy–Born rule in mesh‐free continuum and multiscale simulation of carbon nanotubes

AbstractThis paper investigates the application of a recently proposed higher‐order Cauchy–Born rule in the continuum simulation and multiscale analysis of carbon nanotubes (CNTs). A mesh‐free computational framework is developed to implement the numerical computation of the hyper‐elastic constitutive model that is derived from the higher‐order Cauchy–Born rule. The numerical computation reveals that the buckling pattern of a single‐walled carbon nanotube (SWCNT) can be accurately displayed by taking into consideration the second‐order deformation gradient, and fewer mesh‐free nodes can provide a good simulation of homogeneous deformation. The bridging domain method is employed to couple the developed mesh‐free method and the atomistic simulation. The coupling method is used to simulate the bending buckling of an SWCNT and the tensile failure of an SWCNT with a single‐atom vacancy defect, and good computational results are obtained. Copyright © 2008 John Wiley & Sons, Ltd.

A finite temperature continuum theory based on interatomic potential in crystalline solids

A finite temperature continuum theory of crystalline solid based on an approximate Helmholtz free energy expression is proposed. The free energy expression is specifically derived for simple implementation in atomistic-based continuum methods (i.e. quasicontinuum method) via the Cauchy–Born rule at finite temperature. It is obtained by the method of statistical moments via the quasi-harmonic approximation together with Taylor series expansion of a given interatomic potential. The phonons are assumed to follow the Bose–Einstein distribution so that the quantum effects at low temperature are accounted for. The resulting free energy is in terms of a given interatomic potential and a simple function of displacement that accounts for thermal expansion. It is employed to formulate two finite temperature continuum methods via Cauchy–Born rule and via the virtual atomic cluster (VAC). It is validated through comparison with experimental results of various thermodynamic quantities. In the case of fcc metals, the proposed free energy expression is shown to be valid for a wide range of temperatures above 50 K.

A multilattice quasicontinuum for phase transforming materials: Cascading Cauchy Born kinematics

The quasicontinuum (QC) method is applied to materials possessing a multilattice crystal structure. Cauchy-Born (CB) kinematics, which accounts for the shifts of the crystal basis, is used in continuum regions to relate atomic motions to continuum deformation gradients. To avoid failures of the CB kinematics, QC is augmented with a phonon stability analysis that detects lattice period extensions and identifies the minimum required periodic cell size. This augmented approach is referred to as Cascading Cauchy-Born kinematics. The method is analyzed for both first- and second-order phase transformations, and demonstrated numerically on a one-dimensional test problem.

The Elastic Continuum Limit of the Tight Binding Model*

The authors consider the simplest quantum mechanics model of solids, the tight binding model, and prove that in the continuum limit, the energy of tight binding model converges to that of the continuum elasticity model obtained using Cauchy-Born rule. The technique in this paper is based mainly on spectral perturbation theory for large matrices.

Mesh-free simulation of single-walled carbon nanotubes using higher order Cauchy–Born rule

A mesh-free computational framework is developed to study the deformation behavior of single-walled carbon nanotubes (SWCNTs) by considering the effect of the second-order deformation gradient. The analysis is based on a hyper-elastic constitutive model derived from the higher order Cauchy–Born rule, in which the atomic-scale deformed lattice vectors are calculated with both the first- and second-order deformation gradients. Within the theoretical scheme of the higher order Cauchy–Born rule, the structural properties of SWCNTs and the constitutive response of the system are determined by minimizing the energy of the representative cell. The compression and torsion tests of SWCNTs are numerically simulated with the developed method. The numerical computations reveal that a less amount of mesh-free nodes can provide a good simulation for the homogeneous deformation stage, and the buckling pattern can be truly displayed with the application of the increasing amount of nodes.

Cauchy-Born Rule and the Stability of Crystalline Solids: Dynamic Problems

We study continuum and atomistic models for the elastodynamics of crystalline solids at zero temperature. We establish sharp criterion for the regime of validity of the nonlinear elastic wave equations derived from the well-known Cauchy-Born rule.

Ab initio calculations of strain fields and failure patterns in silicon nitride intergranular glassy films

Theoretical experiments were performed on silicon nitride intergranular glassy film (IGF) models subjected to tensile loading using an accurate ab initio method. The results were used to investigate the strain fields within IGF models. The Green–Lagrange strain fields were calculated from the displacement gradients of the IGF models under various loads. Significant deviations from the first-order Cauchy–Born rule were observed for IGF models even under small load. The strain fields were also analysed to understand atomic-scale mechanisms that lead to intergranular and intragranular failures.

Extension of the temperature-related Cauchy–Born rule: Material stability analysis and thermo-mechanical coupling

In this paper, we first study the material stability of nanostructured materials via the continuum linearized stability analysis technique with the temperature-related Cauchy–Born (TCB) rule. As a temperature-related homogenization technique, the TCB rule considers the free energy instead of the potential so that temperature effects on material stability can be investigated. In addition, we develop a thermo-mechanical coupling model through implementing the thermal diffusion equation into nanoscale continuum approximation. Crack propagation at a nanoplate is studied as an example. Since the nanoscale phenomenon of bond breaking is involved when crack propagates, temperature increasing around the crack tip due to the released potential is considered in our thermo-mechanical coupling model.

Studies of Energy and Mechanical Properties of Single-Walled Carbon Nanotubes via Higher Order Cauchy Born Rule

Based on the higher order Cauchy-Born rule, a nanoscale finite deformation continuum theory, which links interatomic potentials and atomic microstructure of carbon nanotubes to a constitutive model, is presented for analysis of the mechanics of carbon nanotubes. By using of Tersoff-Brenner potential with two sets of parameters, the energy and Young’s modulus of graphite sheet and single-walled carbon nanotubes are studied based on the theory presented. The findings are in good agreement with the existing experimental and theoretical results.

A contact mechanics model for quasi‐continua

AbstractA computational multiscale contact mechanics model is proposed to describe the interaction between deformable solids based on the interaction of individual atoms or molecules belonging to the solids. The contact model, formulated in the framework of large deformation continuum mechanics, is derived from coarsening the molecular dynamics (MD) description of a large assembly of individual atoms, and it thus bears some of the characteristics of the underlying atomic structure. The multiscale contact model distinguishes between atoms interacting within a small neighbourhood within the solids and atoms interacting over large distances between remote regions of the solids. The former furnishes a constitutive relation for the continuum, like the Cauchy–Born Rule, while the latter is used to model the interaction between distinct bodies. The proposed contact model is formulated as a variational weak form and implemented within an updated Lagrangian finite element method. It is shown that, as the problem size increases, the description of the model can be simplified to yield more efficient computational algorithms. In this respect, the proposed multiscale formulation leads to a smooth transition from MD to continuum contact mechanics. The general behaviour of the contact model is studied, and some numerical examples are given. Copyright © 2007 John Wiley & Sons, Ltd.

The continuum limit and QM-continuum approximation of quantum mechanical models of solids

We consider the continuum limit for models of solids that arise in density functional theory and the QM-continuum approximation of such models. Two different versions of QM- continuum approximation are proposed, depending on the level at which the Cauchy-Born rule is used, one at the level of electron density and one at the level of energy. Consistency at the interface between the smooth and the non-smooth regions is analyzed. We show that if the Cauchy-Born rule is used at the level of electron density, then the resulting QM-continuum model is free of the so-called “ghost force” at the interface. We also present dynamic models that bridge naturally the Car-Parrinello method and the QM-continuum approximation.