Cauchy Born Research Articles

Higher-Order Elasticity Constants of Carbon Nanotubes

In this paper, the higher-order elasticity constants are evaluated in the theoretical scheme of higher-order continuum. A single-walled carbon nanotube is treated as a higher-order continuum cylindrical tube with a thin wall, and the representative cell is chosen as a triangle unit that contains four carbon atoms. The Brenner potential is employed to describe the C-C atomic interaction, and the higher-order constitutive relationship is derived by virtue of the higher-order Cauchy-Born rule. The higher-order elasticity constants of carbon nanotubes are evaluated based on the derived higher-order constitutive model, which can provide a foundation for the further analysis of the mechanical properties of carbon nanotubes in the theoretical scheme of higher-order continuum.

Multi-scale modeling of edge effect on band gap offset in polygonal cross-section Silicon nanowires

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 applicability of proposed multi-scale model is illustrated in numerical simulations of four SiNWs cross-sections, i.e. the circular, hexagonal, rectangular and triangular, and the results are compared with the fully atomistic model, experimental data and analytical solution.

Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues

The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to continuum methods based on partial differential equations, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born rule (CBR). Although the CBR does not hold exactly for noncrystalline materials, it may still be used as a first-order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal is to investigate numerically the applicability of the CBR to two-dimensional cellular-scale models by assessing the mechanical behavior of model biological tissues, including crystalline (honeycomb) and noncrystalline reference states. The numerical procedure involves applying an affine deformation to the boundary cells and computing the quasistatic position of internal cells. The position of internal cells is then compared with the prediction of the CBR and an average deviation is calculated in the strain domain. For center-based cell models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free configuration is defined by a honeycomb lattice. We show further that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration. By contrast, for vertex-based cell models, a similar analysis reveals that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications of these results for concurrent discrete and continuous modeling, adaptation of atom-to-continuum techniques to biological tissues, and model classification.

Homogenization of a graphene sheet

A continuum model for a graphene sheet undergoing infinitesimal in-plane deformations is derived by applying the arguments of homogenization theory. The model turns out to coincide with that found by various authors with different methods, but it avoids anticipations on the validity of any properly adjusted or generalized Cauchy–Born rule. The constitutive equation for stress and the effective Young’s modulus and Poisson ratio are explicitly given in terms of the bond constants.

A thickness-independent nonlocal shell model for describing the stability behavior of carbon nanotubes under compression

In this article, a nonlocal Flügge shell model incorporating interatomic potentials is developed to study the buckling behavior of an axially loaded single-walled carbon nanotube (SWCNT). The theory incorporates the relations resulting from establishing a linkage between the strain energy induced in the continuum and the potential energy stored in the atomic bonds, using the so-called Cauchy–Born rule, into the constitutive relations of Eringen’s nonlocal elasticity theory. An exact solution is implemented to solve the set of coupled field equations. In comparison to classical models, the present model provides a much better fit to molecular dynamics (MDs) simulations results and proposes the appropriate value of nonlocal parameter for SWCNTs with simply-supported end conditions. Furthermore, the model developed herein is independent of the nanotube wall thickness and Young’s modulus whose values are scattered in the literature.

Strength of graphene in biaxial tension

A simple analytical study of a single-atom-thick sheet of graphene under biaxial tension is presented. It is based on the combination of the approaches of continuum and molecular mechanics. On the molecular level the Tersoff-Brenner potential with a modified cut-off function is used as an example. Transition to a continuum description is achieved by employing the Cauchy–Born rule. In this analysis the graphene sheet is considered as a crystal composed of two simple Bravais lattices and the mutual atomic relaxation between these lattices is taken into account. Following this approach a critical failure surface is produced for strains in biaxial tension. The adopted methodology is discussed in the context of the alternative approaches. ► Analytical study of a single-atom-thick sheet of graphene under biaxial tension is presented. ► Approaches of continuum and molecular mechanics are combined. ► The graphene sheet is considered as a crystal composed of two simple Bravais lattices. ► The mutual atomic relaxation between these lattices is taken into account. ► A critical failure surface is produced for strains in biaxial tension.

MOLECULAR DYNAMICS/XFEM COUPLING BY A THREE-DIMENSIONAL EXTENDED BRIDGING DOMAIN WITH APPLICATIONS TO DYNAMIC BRITTLE FRACTURE

We propose a method to couple a three-dimensional continuum domain to a molecular dynamics domain to simulate propagating cracks in dynamics. The continuum domain is treated by an extended finite element method to handle the discontinuities. The coupling is based on the bridging domain method, which blends the continuum and atomistic energies. The Lennard-Jones potential is used to model the interactions in the atomistic domain, and the Cauchy-Born rule is used to compute the material behavior in the continuum domain. To our knowledge, it is the first time that a three dimensional extended bridging domain method is reported. To show the suitability of the proposed method, a three-dimensional crack problem with an atomistic region around the crack front is solved. The results show that the method is capable of handling crack propagation and dislocation nucleation.

On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth

We derive continuum limits of atomistic models in the realm of nonlinear elasticity theory rigorously as the interatomic distances tend to zero. In particular we obtain an integral functional acting on the deformation gradient in the continuum theory which depends on the underlying atomistic interaction potentials and the lattice geometry. The interaction potentials to which our theory applies are general finite range models on multilattices which in particular can also account for multi-pole interactions and bond-angle dependent contributions. Furthermore, we discuss the applicability of the Cauchy-Born rule. Our class of limiting energy densities consists of general quasiconvex functions and the class of linearized limiting energies consistent with the Cauchy-Born rule consists of general quadratic forms not restricted by the Cauchy relations.

About an analytical verification of quasi-continuum methods with Γ-convergence techniques

ABSTRACTQuasi-continuum (QC) methods are computational techniques, which reduce the complexity of atomistic simulations in a static setting while keeping information on small-scale structures and effects. The main idea is to couple atomistic and continuum models and thus to obtain quite detailed but still not too expensive numerical simulations.We aim at a mathematically rigorous verification of QC methods by means of discrete to continuum limits. In this article we present our first results for the so-called quasi-nonlocal QC method in the context of fracture mechanics. To this end we start from a one-dimensional chain of atoms with nearest and next-to-nearest neighbour interactions of Lennard-Jones type. This is considered as a fully atomistic model of which the Γ-limits (of zeroth and first order) for an infinite number of atoms are known [7].The QC models we construct are equal to this fully atomistic model in the atomistic region; in the continuum regime we approximate the next-to-nearest neighbour interactions by some nearest neighbour potential which is related to the so-called Cauchy-Born rule. Further we choose certain representative atoms in order to coarsen the mesh in the continuum region. It turns out that the selection of the representative atoms is crucial and influences the Γ-limits.We regard a QC model as good if the Γ-limits of zeroth and first order or at least their minimal values and minimizers are the same as those of the fully atomistic model. Our analysis shows that, while in an elastic regime only the size of the atomistic region matters, in the case of fracture a proper choice of the representative atoms is an essential ingredient.

FULL COUPLING RESPONSE OF SINGLE-WALLED CARBON NANOTUBES

Using a nanoscale continuum theory established via the higher order Cauchy-Born rule, the full coupling response among axial extension, radial deformation and torsion is investigated for single-walled carbon nanotubes (SWCNTs). The axial-strain-induced torsion and the radial strain-induced torsion are both limited to chiral nanotubes with asymmetric atomic structures. The maximum coupling response is obtained in SWCNTs with the chiral angle of 15°. Furthermore, the influences of the diameter and the chirality of SWCNTs on these coupling response are also investigated.

Quasi-Continuum Model for the Finite Deformation of Single-Layer Graphene Sheets Based on the Temperature-Related Higher Order Cauchy-Born Rule

Quasi-Continuum Model for the Finite Deformation of Single-Layer Graphene Sheets Based on the Temperature-Related Higher Order Cauchy-Born Rule

A three‐dimensional atomistic‐based process zone model simulation of fragmentation in polycrystalline solids

SUMMARYA three‐dimensional atomistic‐based process zone model (APZM) is used to simulate high‐speed impact induced dynamic fracture process such as fragmentation and spall fracture. This multiscale simulation model combines the Cauchy–Born rule, colloidal crystal process model, and micromechanics homogenization technique to construct constitutive relations in both grains and grain boundary at mesoscale. The proposed APZM has some inherent advantages to describe mechanical behaviors of polycrystalline solids. First, in contrast to macroscale phenomenological constitutive models, the APZM takes into account atomistic binding energy and atomistic lattice structure. In particular, the electron density related embedded atom method (EAM) potential has been adopted to describe interatomistic interactions of metallic polycrystalline solids in bulk elements; second, a mixed type of EAM potential and colloidal crystal depletion potential is constructed to describe heterogeneous microstructure in the process zone; third, the atomistic potential in both bulk material and process zone has the same atomistic origin, and hence, the bulk and process potentials are self‐consistent. The simulation of dynamic fracture process of a cylinder made of aluminum powder metallurgy (P/M) alloy during high‐speed impact/penetration is carried out, and numerical results demonstrate that APZM finite element method has remarkable ability to accurately capture complex three‐dimensional fragmentation formation and damage morphology. Copyright © 2012 John Wiley & Sons, Ltd.

Justification of the Cauchy–Born Approximation of Elastodynamics

We present sharp convergence results for the Cauchy—Born approximation of general classical atomistic interactions, for static problems with small data and for dynamic problems on a macroscopic time interval.

Finite Element Analysis of Cauchy–Born Approximations to Atomistic Models

This paper is devoted to a new finite element consistency analysis of Cauchy–Born approximations to atomistic models of crystalline materials in two and three space dimensions. Through this approach new “atomistic Cauchy–Born” models are introduced and analyzed. These intermediate models can be seen as first level atomistic/quasicontinuum approximations in the sense that they involve only short-range interactions. The analysis and the models developed herein are expected to be useful in the design of coupled atomistic/continuum methods in more than one dimension. Taking full advantage of the symmetries of the atomistic lattice, we show that the consistency error of the models considered both in energies and in dual W 1,p type norms is \({\mathcal{O}(\varepsilon^2)}\), where \({\varepsilon}\) denotes the interatomic distance in the lattice.

A computational framework for transverse compression of microtubules based on a higher-order Cauchy–Born rule

This work presents a computational framework for the transverse compression of microtubules using the Cauchy–Born rule. Atomistic-continuum simulation and mesh-free method are employed in the computation for the theoretical scheme of the higher-order gradient continuum. Elastic properties and mechanical responses of microtubules under transverse compression are intensively studied. Without tracing each atom in large protein structures, a homogenization technique is proposed to evaluate interatomic energy stored among macromolecules. The concept of fictitious bonds is proposed for microtubules to bridge the gap between the atomistic simulation and the continuum approach, which is of great significance for application of the continuum approach to macromolecular structures. To reflect the inhomogeneous deformations of a cylindrical structure, the higher-order Cauchy–Born rule is employed to calculate the fictitious bond vectors emanating from a given evaluation point during the deformation process. By selecting a representative unit, a higher-order gradient continuum constitutive relationship is established to take atomistic interactions into consideration. Elastic modulus and transverse mechanics, including critical hydrostatic pressure and transverse compression-induced structure transitions, are numerically simulated. Example problems are carefully selected and the obtained results are discussed in detail.

Stability and the continuum limit of the spin-polarized Thomas-Fermi-Dirac-von Weizsäcker model

The continuum limit of the spin-polarized Thomas-Fermi-Dirac-von Weizsäcker model in an external magnetic field is studied. An extension of the classical Cauchy-Born rule for crystal lattices is established for the electronic structure under sharp stability conditions on charge density and spin density waves. A Landau-Lifshitz type of micromagnetic energy functional is derived.

ELASTIC LIMIT OF SQUARE LATTICES WITH THREE-POINT INTERACTIONS

We derive the equivalent energy of a square lattice that either deforms into the three- dimensional Euclidean space or remains planar. Interactions are not restricted to pairs of points and take into account changes of angles. Under some relationships between the local energies associated with the four vertices of an elementary square, we show that the limit energy can be obtained by mere quasiconvexification of the elementary cell energy and that the limit process does not involve any relaxation at the atomic scale. In this case, it can be said that the Cauchy–Born rule holds true. Our results apply to classical models of mechanical trusses that include torques between adjacent bars and to atomistic models.

A MLS-BASED LATTICE SPRING MODEL FOR SIMULATING ELASTICITY OF MATERIALS

A MLS-based lattice spring model is presented for numerical modeling of elasticity of materials. In the model, shear springs between particles are introduced in addition to normal springs. However, the unknowns contain only particle displacements but no particle rotations. The novelty of the model lies in that the deformations of shear springs are computed by using the local strain obtained by the moving least squares (MLS) approximation rather than using the particle displacements directly. By doing so, the proposed lattice spring model can represent the diversity of Poisson's ratio without violating the requirement of rotational invariance. Relationships between micro spring parameters and macro material constants are derived from the Cauchy-born rules and the hyperelastic theory. Numerical examples show that the proposed model is able to reproduce elastic solutions obtained by finite element methods for problems without fractures. Therefore, it is capable of simulating solid materials which are initially continuous, but eventually fracture when critical stress and/or displacement levels are reached. A demonstrating example is presented.

An analysis of the boundary layer in the 1D surface Cauchy–Born model

The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is O(1) in the mesh size; however, we are able to identify an alternative approximation parameter – the stiffness of the interaction potential – with respect to which the relative error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary. In this case we even obtain pointwise error estimates for the strain.

Prediction of the biaxial buckling and vibration behavior of graphene via a nonlocal atomistic-based plate theory

The present article is concerned with the applicability of an elastic plate theory incorporating the interatomic potentials for biaxial buckling and vibration analysis of single-layer graphene sheets (SLGSs) and accounting for the small scale effects. For this purpose, the relations based on the interatomic potential and Eringen’s nonlocal equation are incorporated into the classical plate theory. The former relations are obtained through constructing a linkage between the strain energy induced in the continuum and the potential energy stored in the atomic bonds using the Cauchy–Born rule. The nonlocal governing equations of motion for buckling and vibration of the SLGSs with simply-supported boundary conditions are exactly solved and explicit formulae for the frequencies and critical buckling load are derived. The results generated from the present model are compared with those of molecular dynamic (MD) simulations and the other previously reported ones and a good agreement is achieved. The model developed herein is independent of Young’s modulus which is of an ambiguous definition in the literature. It is found that the small scale effect on buckling and vibrational response of the SLGSs is profound and it becomes more prominent when the side length is low.