Multiscale coupling using a finite element framework at finite temperature

This paper presents the formulation and application of a multiscale methodology that couples three domains using a finite element framework. The proposed method efficiently models atomistic systems by decomposing the system into continuum, bridging, and atomistic domains. The atomistic and bridging domains are solved using a combined finite element–molecular mechanics simulation where the system is discretized into atom/nodal centric elements based on the atomic scale finite element method. Coupling between the atomistic domain and continuum domain is performed through the bridging cells, which contain locally formulated atoms whose displacements are mapped to the nodes of the bridging cell elements. The method implements a temperature‐dependent potential for finite temperature simulations. Validation and demonstration of the methodology are provided through three case studies: displacement in a one‐dimensional chain, stress around nanoscale voids, and fracture. From these studies differences between multiscale and fully atomistic simulations were very small with the simulation time of the proposed methodology being approximately a tenth of the time of the fully atomistic model. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiscale modeling of intergranular fracture in aluminum: constitutive relation for interface debonding

Intergranular fracture is a dominant mode of failure in ultrafine grained materials. In the present study, the atomistic mechanisms of grain-boundary debonding during intergranular fracture in aluminum are modeled using a coupled molecular dynamics—finite element simulation. Using a statistical mechanics approach, a cohesive-zone law in the form of a traction–displacement constitutive relationship, characterizing the load transfer across the plane of a growing edge crack, is extracted from atomistic simulations and then recast in a form suitable for inclusion within a continuum finite element model. The cohesive-zone law derived by the presented technique is free of finite size effects and is statistically representative for describing the interfacial debonding of a grain boundary (GB) interface examined at atomic length scales. By incorporating the cohesive-zone law in cohesive-zone finite elements, the debonding of a GB interface can be simulated in a coupled continuum–atomistic model, in which a crack starts in the continuum environment, smoothly penetrates the continuum–atomistic interface, and continues its propagation in the atomistic environment. This study is a step toward relating atomistically derived decohesion laws to macroscopic predictions of fracture and constructing multiscale models for nanocrystalline and ultrafine grained materials.     

Failure analysis and the optimal toughness design of carbon nanotube-reinforced composites

The combined analysis of the fracture toughness enhancement of carbon nanotube (CNT)-reinforced composites is herein carried out on the basis of atomistic simulation, shear-lag theory and facture mechanics. It is found that neither longer reinforced CNTs nor stronger CNT/matrix interfaces can definitely lead to the better fracture toughness of these composites. In contrast, the optimal interfacial chemical bond density and the optimal CNT length are those making the failure mode just in the transition from CNT pull-out to CNT break. To verify our theory, an atomic/continuum finite element method (FEM) is applied to investigate the fracture behavior of CNT-reinforced composites with different interfacial chemical bond densities. Our analysis shows that the optimal interfacial chemical bond density for (6,6) CNTs is about 5–10% and that increasing the CNT length beyond 100 nm does not further improve fracture toughness, but can easily lead to the self-folding and clustering of the CNTs. The proposed theoretical model is also applicable to short fiber-reinforced composites.     

On propagation of short rolling contact fatigue cracks

Recent accidents involving railway rails have aroused demand for improved and more efficient rail maintenance strategies to reduce the risk of unexpected rail fracture. Numerical tools can aid in generating maintenance strategies: this investigation deals with the numerical modelling and analysis of short crack growth in rails. Factors that influence the fatigue propagation of short surface‐breaking cracks (head checks) in rails are assessed. A proposed numerical procedure incorporates finite element (FE) calculations to predict short crack growth conditions for rolling contact fatigue (RCF) loading. A parameterised FE model for the rolling‐sliding contact of a cylinder on a semi‐infinite half space, with a short surface breaking crack, presented here, is used in linear‐elastic and elastic–plastic FE calculations of short crack propagation, together with fracture mechanics theory. The crack length and orientation, crack face friction, and coefficient of surface friction near the contact load are varied. The FE model is verified for five examples in the literature. Comparison of results from linear‐elastic and elastic–plastic FE calculations, shows that the former cannot describe short RCF crack behaviour properly, in particular 0.1–0.2 mm long (head check) cracks with a shallow angle; elastic–plastic analysis is required instead.     

A robust nano-mechanics approach for tensile and modal analysis using atomistic–continuum mechanics method

Nanoscale engineering has been developing rapidly. However, experimental investigations at the nanoscale level are very difficult to conduct. This research seeks to employ the same model to investigate an atomic-scale structure for tensile and modal analyses, based on atomistic–continuum mechanics (ACM) and a finite element method (FEM). The ACM transfers an originally discrete atomic structure into an equilibrium continuum model using atomistic–continuum transfer elements. All interatomic forces, described by the empirical potential functions, can be transferred into springs to form the atomic structure. The spring network models were also widely utilized in FEM based nano-structure studies. Thus, this paper attempts to explore ACM using three examples including silicon, carbon nanotube, and copper. All of the results are validated by bulk properties or literature.     

The finite element method for the mechanically based model of non‐local continuum

In this paper the finite element method (FEM) for the mechanically based non‐local elastic continuum model is proposed. In such a model each volume element of the domain is considered mutually interacting with the others, beside classical interactions involved by the Cauchy stress field, by means of central body forces that are monotonically decreasing with their inter‐distance and proportional to the product of the interacting volume elements. The constitutive relations of the long‐range interactions involve the product of the relative displacement of the centroids of volume elements by a proper, distance‐decaying function, which accounts for the decrement of the long‐range interactions as long as distance increases. In this study, the elastic problem involving long‐range central interactions for isotropic elastic continuum will be solved with the aid of the FEM. The accuracy of the solution obtained with the proposed FEM code is compared with other solutions obtained with Galerkins' approximation as well as with finite difference method. Moreover, a parametric study regarding the effect of the material length scale in the mechanically based model and in the Kr”oner–Eringen non‐local elasticity has been investigated for a plane elasticity problem. Copyright © 2011 John Wiley & Sons, Ltd.     

A multiscale framework for computational nanomechanics: Application to the modeling of carbon nanotubes

A multiscale computational framework is presented that provides a coupled self‐consistent system of equations involving molecular mechanics at small scales and quasi‐continuum mechanics at large scales. The proposed method permits simultaneous resolution of quasi‐continuum and atomistic length scales and the associated displacement fields in a unified manner. Interatomic interactions are incorporated into the method through a set of analytical equations that contain nanoscale‐based material moduli. These material moduli are defined via internal variables that are functions of the local atomic configuration parameters. Point defects like vacancy defects in nanomaterials perturb the atomic structure locally and generate localized force fields. Formation energy of vacancy is evaluated via interatomic potentials and minimization of this energy leads to nanoscale force fields around defects. These nanoscale force fields are then employed in the multiscale method to solve for the localized displacement fields in the vicinity of vacancies and defects. The finite element method that is developed based on the hierarchical multiscale framework furnishes a two‐level statement of the problem. It concurrently feeds information at the molecular scale, formulated in terms of the nanoscale material moduli, into the quasi‐continuum equations. Representative numerical examples are shown to validate the model and demonstrate its range of applicability. Copyright © 2008 John Wiley & Sons, Ltd.     

Elastic crack propagation model for crystalline solids using a self-consistent coupled atomistic–continuum framework

Deformation and failure processes of crystalline materials are governed by complex phenomena at multiple scales. It is necessary to couple these scales for physics-based modeling of these phenomena, while overcoming limitations of modeling at individual scales. To address this issue, this paper develops self-consistent elastic constitutive and crack propagation relations of crystalline materials containing atomic scale cracks, from observations made in a concurrent multi-scale simulation system coupling atomistic and continuum domain models. The concurrent multi-scale model incorporates a finite temperature atomistic region containing the crack, a continuum region represented by a self-consistent crystal elasticity constitutive model, and a handshaking interphase region. Atomistic modeling is done by the molecular dynamics code LAMMPS, while continuum modeling is conducted by the finite element method. For single crystal nickel a nonlinear and nonlocal crystal elasticity constitutive relation is derived, consistent with the atomic potential function. An efficient, staggered solution scheme with parallel implementation is designed for the coupled problem. The atomistic–continuum coupling is achieved by enforcing geometric compatibility and force equilibrium in the interphase region. Quantitative analyses of the crack propagation process focuses on size dependence, strain energy release rate, crack propagation rate and degradation of the local stiffness. The self-consistent constitutive and crack propagation relations, derived from the concurrent model simulation results are validated by comparing results from the concurrent and full FE models. Excellent accuracy and enhanced efficiency are observed in comparison with pure MD and concurrent model results.     

Non‐reflecting boundary conditions for atomistic,continuum and coupled atomistic/continuum simulations

We present a method to numerically calculate a non‐reflecting boundary condition which is applicable to atomistic, continuum and coupled multiscale atomistic/continuum simulations. The method is based on the assumption that the forces near the domain boundary can be well represented as a linear function of the displacements, and utilizes standard Laplace and Fourier transform techniques to eliminate the unnecessary degrees of freedom. The eliminated degrees of freedom are accounted for in a time‐history kernel that can be calculated for arbitrary crystal lattices and interatomic potentials, or regular finite element meshes using an automated numerical procedure. The new theoretical developments presented in this work allow the application of the method to non‐nearest neighbour atomic interactions; it is also demonstrated that the identical procedure can be used for finite element and mesh‐free simulations. We illustrate the effectiveness of the method on a one‐dimensional model problem, and calculate the time‐history kernel for FCC gold using the embedded atom method (EAM). Copyright © 2005 John Wiley & Sons, Ltd.     

Three Phase Composite Cylinder Assemblage Model for Analyzing the Elastic Behavior of MWCNT-Reinforced Polymers

Evolution of computational modeling and simulation has given more emphasis on the research activities related to carbon nanotube (CNT) reinforced polymer composites recently. This paper presents the composite cylinder assemblage (CCA) approach based on continuum mechanics for investigating the elastic properties of a polymer resin reinforced by multi-walled carbon nanotubes (MWCNTs). A three-phase cylindrical representative volume element (RVE) model is employed based on CCA technique to elucidate the effects of inter layers, chirality, interspacing, volume fraction of MWCNT, interphase properties and temperature conditions on the elastic modulus of the composite. The interface region between CNT and polymer matrix is modeled as the third phase with varying material properties. The constitutive relations for each material system have been derived based on solid mechanics and proper interfacial traction continuity conditions are imposed. The predicted results from the CCA approach are in well agreement with RVE-based finite element model. The outcomes reveal that temperature softening effect becomes more pronounced at higher volume fractions of CNTs.     

A coupled molecular dynamics and extended finite element method for dynamic crack propagation

A multiscale method is presented which couples a molecular dynamics approach for describing fracture at the crack tip with an extended finite element method for discretizing the remainder of the domain. After recalling the basic equations of molecular dynamics and continuum mechanics, the discretization is discussed for the continuum subdomain where the partition‐of‐unity property of finite element shape functions is used, since in this fashion the crack in the wake of its tip is naturally modelled as a traction‐free discontinuity. Next, the zonal coupling method between the atomistic and continuum models is recapitulated. Finally, it is discussed how the stress has been computed in the atomic subdomain, and a two‐dimensional computation is presented of dynamic fracture using the coupled model. The result shows multiple branching, which is reminiscent of recent results from simulations on dynamic fracture using cohesive‐zone models. Copyright © 2009 John Wiley & Sons, Ltd.     

Woven fabric composites—part II: Characterization of macro‐crack initiation loads for global damage analysis

In Part 2, finite element techniques are developed to additionally investigate the global damage of woven fabric composites, focusing on plain weaves. The homogenized elastic properties of the original undamaged and the damaged woven fabric composites from the micromechanical homogenization and the micromechanical progressive damage analysis investigated earlier in Part 1 are subsequently employed in the present global damage analysis. The theory of continuum damage mechanics is utilized in the global damage analysis. The damage variables, the most important material properties of the continuum damage mechanics, are measures of average material degradation at a macro‐mechanics scale. In the present study, these damage variables are calculated numerically using the results from the micromechanical damage analysis in Part 1, instead of being obtained experimentally. Subsequently, a finite element formulation for the global damage analysis of woven fabric composites is developed to predict the initiation loads of macro‐cracks. Copyright © 2001 John Wiley & Sons, Ltd.     

A three‐dimensional surface stress tensor formulation for simulation of adhesive contact in finite deformation

A three‐dimensional surface adhesive contact formulation is proposed to simulate macroscale adhesive contact interaction characterized by the van der Waals interaction between arbitrarily shaped deformable continua under finite deformation. The proposed adhesive contact formulation uses a double‐layer surface integral to replace the conventional double volume integration to compute the adhesive contact force vector. Considering nonlinear finite deformation, we have derived the surface stress tensor and the corresponding tangent stiffness matrix in a Galerkin weak formulation. With the surface stress formulation, the adhesive contact problems are solved in the framework of nonlinear continuum mechanics by using the standard Lagrange finite element method. Surface stress tensors are formulated for both interacting bodies. Numerical examples show that the proposed surface contact algorithm is accurate, efficient, and reliable for three‐dimensional adhesive contact problems of large deformations for both quasi‐static and dynamic simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

An extended finite element/level set method to study surface effects on the mechanical behavior and properties of nanomaterials

We present a new approach based on coupling the extended finite element method (XFEM) and level sets to study surface and interface effects on the mechanical behavior of nanostructures. The coupled XFEM‐level set approach enables a continuum solution to nanomechanical boundary value problems in which discontinuities in both strain and displacement due to surfaces and interfaces are easily handled, while simultaneously accounting for critical nanoscale surface effects, including surface energy, stress, elasticity and interface decohesion. We validate the proposed approach by studying the surface‐stress‐driven relaxation of homogeneous and bi‐layer nanoplates as well as the contribution from the surface elasticity to the effective stiffness of nanobeams. For each case, we compare the numerical results with new analytical solutions that we have derived for these simple problems; for the problem involving the surface‐stress‐driven relaxation of a homogeneous nanoplate, we further validate the proposed approach by comparing the results with those obtained from both fully atomistic simulations and previous multiscale calculations based upon the surface Cauchy–Born model. These numerical results show that the proposed method can be used to gain critical insights into how surface effects impact the mechanical behavior and properties of homogeneous and composite nanobeams under generalized mechanical deformation. Copyright © 2010 John Wiley & Sons, Ltd.     

A semi-continuum finite element approach to evaluate the Young’s modulus of single-walled carbon nanotube reinforced composites

This paper describes a micromechanical finite element approach for the estimation of the effective Young’s modulus of single-walled carbon nanotube reinforced composites. These composite materials consist of aligned carbon nanotubes that are uniformly distributed within the matrix. Based on micromechanical theory, the Young’s modulus of the nanocomposite is estimated by considering a representative cylindrical volume element. Within the representative volume element, the reinforcement is modeled according to its atomistic microstructure while the matrix is modeled as a continuum medium. Spring-based finite elements are employed to simulate the discrete geometric structure and behavior of each single-walled carbon nanotube. The load transfer conditions between the carbon nanotubes and the matrix are modeled using joint elements of changeable stiffness that connect the two materials, simulating the interfacial region. The proposed model has been tested numerically and yields reasonable results for variable stiffness values of the joint elements. The effect of the interface on the performance of the composite is investigated for various volume fractions. The numerical results are compared with experimental and analytical predictions.     

Finite element analysis of membrane wrinkling

New results are presented for the finite element analysis of wrinkling in curved elastic membranes under‐going large deformation. Concise continuum level governing equations are derived in which singularities are eliminated. A simple and efficient algorithm with robust convergence properties is established to find the real strain and stress of the wrinkled membrane for Hookean materials. The continuum theory is implemented into a finite element code. Explicit formulas for the internal forces and the tangent stiffness matrix are derived. Numerical examples are presented that demonstrate the effectiveness of the new theory for predicting wrinkling in membranes undergoing large deformation. Copyright © 2001 John Wiley & Sons, Ltd.     

A bridging domain and strain computation method for coupled atomistic–continuum modelling of solids

We present a multiscale method that couples atomistic models with continuum mechanics. The method is based on an overlapping domain‐decomposition scheme. Constraints are imposed by a Lagrange multiplier method to enforce displacement compatibility in the overlapping subdomain in which atomistic and continuum representations overlap. An efficient version of the method is developed for cases where the continuum can be modelled as a linear elastic material. An iterative scheme is utilized to optimize the coupled configuration. Conditions for the regularity of the constrained matrices are determined. A method for computing strain in atomistic models and handshake domains is formulated based on a moving least‐square approximation which includes both extensional and angle‐bending terms. It is shown that this method exactly computes the linear strain field. Applications to the fracture of defected single‐layer atomic sheets and nanotubes are given. Copyright © 2006 John Wiley & Sons, Ltd.     

Multiscale simulation of nanostructures based on spatial secant model: a discrete hyperelastic approach

The main objective of this paper is to present a coarse-grained material model for the simulation of three-dimensional nanostructures. The developed model is motivated by the recent progress in establishing continuum models for nanomaterials and nanostructures. As there are conceptual differences between the continuum field defined in the classical sense and the nanomaterials consisting of discrete, space-filling atoms, existing continuum measures cannot be directly applied for mapping the nanostructures due to the discreteness at small length scale. In view of the fundamental difficulties associated with the direct application of the continuum approach, we introduce a unique discrete deformation measure called spatial secant and have developed a new hyperelastic model based on this measure. We show that the spatial secant-based model is consistently linked to the underlying atomistic model and provides a geometric exact mapping in the discrete sense. In addition, we outline the corresponding computational framework using the finite element and/or meshfree method. The implementation is within the context of finite deformation. Finally we illustrate the application of the model in studying the mechanics of low-dimensional carbon nanostructures such as carbon nanotubes (CNT). By comparing with full-scale molecular mechanics simulations, we show that the proposed coarse-grained model is robust in that it accurately captures the non-linear mechanical responses of the CNT structures.     

3D modelling of strong discontinuities in elastoplastic solids: fixed and rotating localization formulations

This paper is concerned with the incorporation of displacement discontinuities into a continuum theory of elastoplasticity for the modelling of localization processes such as cracking in brittle materials. Based on the strong discontinuity approach (SDA) (Computational Mechanics 1993; 12: 277–296) mesh objective 2D and 3D finite element formulations are developed using linear and quadratic 2D elements as well as 8‐noded 3D elements. In the formulation of the finite‐element model proposed in the paper, the analogy with standard formulations is emphasized. The parameter defining the amplitude of the displacement jump within the finite element is condensed out at the material level without employing the standard static condensation technique. This approach results in linearized constitutive equations formally identical to continuum models. Therefore, the standard return mapping algorithm is used to solve the non‐linear equations. In analogy to concepts used in continuum smeared crack models, a rotating formulation of the SDA is proposed in addition to the standard concept of fixed discontinuities. It is shown that the rotating localization approach reduces locking effects observed in analyses based on fixed localization directions. The applicability of the proposed SDA finite‐element model as well as its numerical performance is investigated by means of a three‐dimensional ultimate load analysis of a steel anchor embedded in a concrete block subjected to a shear force. Copyright © 2003 John Wiley & Sons, Ltd.