An embedded atom hyperelastic constitutive model and multiscale cohesive finite element method

Based on embedded atom method (EAM), an embedded atom hyperelastic (EAH) constitutive model is developed. The proposed EAH constitutive model provides a multiscale formalism to determine mesoscale or macroscale material behavior by atomistic information. By combining the EAH with cohesive zone model (CZM), a multiscale embedded atom cohesive finite element model (EA-cohesive FEM) is developed for simulating failure of materials at mesoscale and macroscale, e.g. fracture and crack propagation etc. Based on EAH, the EA-cohesive FEM applies the Cauchy-Born rule to calculate mesoscale or macroscale material response for bulk elements. Within the cohesive zone, a generalized Cauchy-Born rule is applied to find the effective normal and tangential traction-separation cohesive laws of EAH material. Since the EAM is a realistic semi-empirical interatomic potential formalism, the EAH constitutive model and the EA-cohesive FEM are physically meaningful when it is compared with experimental data. The proposed EA-cohesive FEM is validated by comparing the simulation results with the results of large scale molecular dynamics simulation. Simulation result of dynamic crack propagation is presented to demonstrate the capacity of EA-cohesive FEM in capturing the dynamic fracture.     

Real‐Time Monitoring of Colloidal Crystallization in Electrostatically‐Levitated Drops

Colloidal crystallization is analogous to the crystallization in bulk atomic systems in various aspects, which has been explored as a model system. However, a real‐time probing of the phenomenon still remains challenging. Here, a levitation system for a study of colloidal crystallization is demonstrated. Colloidal particles in a levitated droplet are gradually concentrated by isotropic evaporation of water from the surface of the droplet, resulting in crystallization. The structural change of the colloidal array during crystallization is investigated by simultaneously measuring the volume and reflectance spectra of the droplet. The crystal nucleates from the surface of the droplet at which the volume fraction exceeds the threshold and then the growth proceeds. The crystal growth behavior depends on the initial concentrations of colloidal particles and salts which determine the overall direction of crystal growth and interparticle spacing, respectively. The results show that a levitating bulk droplet has a great potential as a tool for in situ investigation of colloidal crystallization.     

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.     

Finite element methods for the non‐linear mechanics of crystalline sheets and nanotubes

The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy–Born rule. The constitutive model for a two‐dimensional continuum deforming in three dimensions (a surface) is written explicitly in terms of the underlying atomistic model. The resulting hyper‐elastic potential depends on the stretch and the curvature of the surface, as well as on internal elastic variables describing the rearrangements of the crystal within the unit cell. Coarse grained calculations of carbon nanotubes (CNTs) are performed by discretizing this continuum mechanics theory by finite elements. A smooth discrete representation of the surface is required, and subdivision finite elements, proposed for thin‐shell analysis, are used. A detailed set of numerical experiments, in which the continuum/finite element solutions are compared to the corresponding full atomistic calculations of CNTs, involving very large deformations and geometric instabilities, demonstrates the accuracy of the proposed approach. Simulations for large multi‐million systems illustrate the computational savings which can be achieved. Copyright © 2003 John Wiley & Sons, Ltd.     

Aspects of rapid crack propagation in silicon

Fracture experiments with silicon specimens in recent years have shown the need for a new approach to the analysis of rapidly propagating cracks in single crystals. Behaviour and phenomena have been revealed that fracture in these materials is rather different from the fracture of both amorphous and polycrystalline materials. We show that continuum mechanics is insufficient for analyzing crack propagation in single crystals since it is unable to consider atomistic‐scale phenomena. Accordingly, we describe basic phenomena associated with rapid crack propagation in silicon : (i) anisotropic velocity‐dependent R‐curve behaviour, as a key phenomenon dictating atomistic scale behaviour, (ii) crack deflection from one cleavage plane to another as a mesoscopic scale phenomenon in single‐crystal fracture, (iii) the Rayleigh surface wave speed as the limiting crack tip velocity is re‐examined, (vi) the lowest crack velocity in brittle crystals is examined, and finally (v) the interaction between crack path and preferred cleavage planes in single crystals is depicted.     

A full‐discontinuous Galerkin formulation of nonlinear Kirchhoff–Love shells: elasto‐plastic finite deformations,parallel computation,and fracture applications

Because of its ability to take into account discontinuities, the discontinuous Galerkin (DG) method presents some advantages for modeling cracks initiation and propagation. This concept has been recently applied to three‐dimensional simulations and to elastic thin bodies. In this last case, the assumption of small elastic deformations before cracks initiation or propagation reduces drastically the applicability of the framework to a reduced number of materials. To remove this limitation, a full‐DG formulation of nonlinear Kirchhoff–Love shells is presented and is used in combination with an elasto‐plastic finite deformations model. The results obtained by this new formulation are in agreement with other continuum elasto‐plastic shell formulations. Then, this full‐DG formulation of Kirchhoff–Love shells is coupled with the cohesive zone model to perform thin body fracture simulations. As this method considers elasto‐plastic constitutive laws in combination with the cohesive model, accurate results compared with the experiments are found. In particular, the crack path and propagation rate of a blasted cylinder are shown to match experimental results. One of the main advantages of this framework is its ability to run in parallel with a high speed‐up factor, allowing the simulation of ultra fine meshes. Copyright © 2012 John Wiley & Sons, Ltd.     

A two‐dimensional ordinary,state‐based peridynamic model for linearly elastic solids

Peridynamics is a non‐local mechanics theory that uses integral equations to include discontinuities directly in the constitutive equations. A three‐dimensional, state‐based peridynamics model has been developed previously for linearly elastic solids with a customizable Poisson's ratio. For plane stress and plane strain conditions, however, a two‐dimensional model is more efficient computationally. Here, such a two‐dimensional state‐based peridynamics model is presented. For verification, a 2D rectangular plate with a round hole in the middle is simulated under constant tensile stress. Dynamic relaxation and energy minimization methods are used to find the steady‐state solution. The model shows m‐convergence and δ‐convergence behaviors when m increases and δ decreases. Simulation results show a close quantitative matching of the displacement and stress obtained from the 2D peridynamics and a finite element model used for comparison. Copyright © 2014 John Wiley & Sons, Ltd.     

Experimental Investigations of Fracture Process in Concrete by Means of X‐ray Micro‐computed Tomography

The paper describes investigation results on fracture in notched concrete beams under quasi‐static three‐point bending by the X‐ray micro‐computed tomography. The two‐dimensional (2D) and three‐dimensional image procedures were used. Attention was paid to width, length, height and shape of cracks along beam depth. In addition, the displacements on the surface of concrete beams during the deformation process were measured with the 2D digital image correlation technique in order to detect strain localisation before a discrete crack occurred. The 2D fracture patterns in beams were numerically simulated with the finite‐element method using an isotropic damage constitutive model enhanced by a characteristic length of micro‐structure. Concrete was modelled as a random heterogeneous four‐phase material composed of aggregate, cement matrix, interfacial transitional zones and air voids. The advantages of the X‐ray micro‐computed tomography were outlined.     

Colloidal Self‐Assembly Meets Nanofabrication: From Two‐Dimensional Colloidal Crystals to Nanostructure Arrays

Self‐assembly of colloidal microspheres or nanospheres is an effective strategy for fabrication of ordered nanostructures. By combination of colloidal self‐assembly with nanofabrication techniques, two‐dimensional (2D) colloidal crystals have been employed as masks or templates for evaporation, deposition, etching, and imprinting, etc. These methods are defined as “colloidal lithography”, which is now recognized as a facile, inexpensive, and repeatable nanofabrication technique. This paper presents an overview of 2D colloidal crystals and nanostructure arrays fabricated by colloidal lithography. First, different methods for fabricating self‐assembled 2D colloidal crystals and complex 2D colloidal crystal structures are summarized. After that, according to the nanofabrication strategy employed in colloidal lithography, related works are reviewed as colloidal‐crystal‐assisted evaporation, deposition, etching, imprinting, and dewetting, respectively.     

A 3D anisotropic elastoplastic‐damage model using discontinuous displacement fields

This paper is concerned with the development of constitutive equations for finite element formulations based on discontinuous displacement fields. For this purpose, an elastoplastic continuum model (stress–strain relation) as well as an anisotropic damage model (stress–strain relation) are projected onto a surface leading to traction separation laws. The coupling of both continuum models and, subsequently, the derivation of the corresponding constitutive interface law are described in detail. For a simple calibration of the proposed model, the fracture energy resulting from the coupled elastoplastic‐damage traction separation law is computed. By this, the softening evolution is linearly dependent on the fracture energy. The second part of the present paper deals with the numerical implementation. Based on a local and incompatible additive split of the displacement field into a continuous and a discontinuous part, the parameters specifying the jump of the displacement field are condensed out at the material level without employing the standard static condensation technique. To reduce locking effects, a rotating localization zone formulation is applied. The applicability and the performance of the proposed numerical implementation is investigated by means of a re‐analysis of a two‐dimensional L‐shaped slab as well as by means of a three‐dimensional ultimate load analysis of a steel anchor embedded in a concrete block. Copyright © 2004 John Wiley & Sons, Ltd.     

The fracture process in quasi‐brittle materials simulated using a lattice dynamical model

The damage process in quasi‐brittle materials is characterized by the evolution of a micro‐crack field, followed by the joining of micro‐cracks, stress localization and crack instability. In network models, masses are lumped at nodal points which are interconnected by one‐dimensional elements with a bilinear constitutive relation, considering the energy consistency during the simulated process. In order to replicate the material imperfections, to render a realistic behaviour in damage localization, the model has not only random elastic and rupture properties, but also a geometric perturbation. In the present paper 2D plates with different levels of brittleness are simulated. The numerical results are presented in terms of global stress vs strain diagram, final network configuration, energy balance during the process and as geometric damage evolution. Therefore, the predictive potential of the lattice discrete element model to capture fracture processes in quasi‐brittle materials is demonstrated.     

Higher order Cauchy–Born rule based multiscale cohesive zone model and prediction of fracture toughness of silicon thin films

In this work, we extend the multiscale cohesive zone model (MCZM) (Zeng and Li in Comput Methods Appl Mech Eng 199:547–556, 2010), in which interatomic potential is embedded into constitutive relation to express cohesive law in fracture process zone, to include the hierarchical Cauchy–Born rule in the process zone and to simulate three dimensional fracture in silicon thin films. The model has been applied to simulate fracture stress and fracture toughness of single-crystal silicon thin film by using the Tersoff potential. In this study, a new approach has been developed to capture inhomogeneous deformation inside the cohesive zone. For this purpose, we introduce higher order Cauchy–Born rules to construct constitutive relations for corresponding higher order process zone elements, and we introduce a sigmoidal function supported bubble mode in finite element shape function of those higher order cohesive zone elements to capture the nonlinear inhomogeneous deformation inside the cohesive zone elements. Benchmark tests with simple 3D models have confirmed that the present method can predict the fracture toughness of silicon thin films. Interestingly, this is accomplished without increasing of computational cost, because the present model does not require quadratic elements to represent heterogeneous deformation, which is the inherent weakness of the previous MCZM model. Quantitative comparisons with experimental results are performed by computing crack propagation in non-notched and initially notched silicon thin films, and it is found that our model can reproduce essential material properties, such as Young’s modulus, fracture stress, and fracture toughness of single-crystal silicon thin films.     

A simple dynamical scale‐coupling method for concurrent simulation of hybridized atomistic/coarse‐grained‐particle system

We propose a simple method for dynamical coupling of two sub‐systems with different characteristic scales described with different theoretical models, such as the fine‐scale sub‐system with the atomistic model (AM) such as the empirical inter‐atomic potential and the coarse‐scale sub‐system with the coarse‐grained particle (CGP) method, in a concurrent hybrid simulation scheme. Naive coupling of the different‐scale sub‐systems results in reflection of high wavenumber waves at the interface because of the differences in the phonon Brillouin‐zone and in the dispersion relation. To solve the problem, the present scale‐coupling method introduces (virtual) extra atoms and particles for the AM and the CGP sub‐systems, respectively, beyond the atom–particle interface, and uses the extra atoms and the particles to mutually transfer information of the waves between the two sub‐systems and to suppress the artificial reflection of the incident wave in the whole wavenumber range. As the algorithm in the present scale‐coupling method is local in time and space, it is applicable to hybrid systems with any interface shape at low computation and memory requirement. Accuracy of the present scale‐coupling method is compared with that of the existing methods for a simple model system. The hybrid AM‐CGP simulation of indentation of a graphene nano‐drum using the present scale‐coupling method is performed to demonstrate its accuracy and usefulness through its comparison with the fully atomistic results. Copyright © 2010 John Wiley & Sons, Ltd.     

Electron Mobility of 24 cm2 V−1 s−1 in PbSe Colloidal‐Quantum‐Dot Superlattices

Colloidal quantum dots (CQDs) are nanoscale building blocks for bottom‐up fabrication of semiconducting solids with tailorable properties beyond the possibilities of bulk materials. Achieving ordered, macroscopic crystal‐like assemblies has been in the focus of researchers for years, since it would allow exploitation of the quantum‐confinement‐based electronic properties with tunable dimensionality. Lead‐chalcogenide CQDs show especially strong tendencies to self‐organize into 2D superlattices with micrometer‐scale order, making the array fabrication fairly simple. However, most studies concentrate on the fundamentals of the assembly process, and none have investigated the electronic properties and their dependence on the nanoscale structure induced by different ligands. Here, it is discussed how different chemical treatments on the initial superlattices affect the nanostructure, the optical, and the electronic‐transport properties. Transistors with average two‐terminal electron mobilities of 13 cm² V^?1 s^?1 and contactless mobility of 24 cm² V^?1 s^?1 are obtained for small‐area superlattice field‐effect transistors. Such mobility values are the highest reported for CQD devices wherein the quantum confinement is substantially present and are comparable to those reported for heavy sintering. The considerable mobility with the simultaneous preservation of the optical bandgap displays the vast potential of colloidal QD superlattices for optoelectronic applications.     

Monitoring Nanocrystal Self‐Assembly in Real Time Using In Situ Small‐Angle X‐Ray Scattering

Self‐assembled nanocrystal superlattices have attracted large scientific attention due to their potential technological applications. However, the nucleation and growth mechanisms of superlattice assemblies remain largely unresolved due to experimental difficulties to monitor intermediate states. Here, the self‐assembly of colloidal PbS nanocrystals is studied in real time by a combination of controlled solvent evaporation from the bulk solution and in situ small‐angle X‐ray scattering (SAXS) in transmission geometry. For the first time for the investigated system a hexagonal closed‐packed (hcp) superlattice formed in a solvent vapor saturated atmosphere is observed during slow solvent evaporation from a colloidal suspension. The highly ordered hcp superlattice is followed by a transition into the final body‐centered cubic superlattice upon complete drying. Additionally, X‐ray cross‐correlation analysis of Bragg reflections is applied to access information on precursor structures in the assembly process, which is not evident from conventional SAXS analysis. The detailed evolution of the crystal structure with time provides key results for understanding the assembly mechanism and the role of ligand–solvent interactions, which is important both for fundamental research and for fabrication of superlattices with desired properties.     

A surface Cauchy–Born model for nanoscale materials

We present an energy‐based continuum model for the analysis of nanoscale materials where surface effects are expected to contribute significantly to the mechanical response. The approach adopts principles utilized in Cauchy–Born constitutive modelling in that the strain energy density of the continuum is derived from an underlying crystal structure and interatomic potential. The key to the success of the proposed method lies in decomposing the potential energy of the material into bulk (volumetric) and surface area components. In doing so, the method naturally satisfies a variational formulation in which the bulk volume and surface area contribute independently to the overall system energy. Because the surface area to volume ratio increases as the length scale of a body decreases, the variational form naturally allows the surface energy to become important at small length scales; this feature allows the accurate representation of size and surface effects on the mechanical response. Finite element simulations utilizing the proposed approach are compared against fully atomistic simulations for verification and validation. Copyright © 2006 John Wiley & Sons, Ltd.     

Nanoscale void nucleation and growth and crack tip stress evolution ahead of a growing crack in a single crystal

A constrained three-dimensional atomistic model of a cracked aluminum single crystal has been employed to investigate the growth behavior of a nanoscale crack in a single crystal using molecular dynamics simulations with the EAM potential. This study is focused on the stress field around the crack tip and its evolution during fast crack growth. Simulation results of the observed nanoscale fracture behavior are presented in terms of atomistic stresses. Major findings from the simulation results are the following: (a) crack growth is in the form of void nucleation, growth and coalescence ahead of the crack tip, thus resembling that of ductile fracture at the continuum scale; (b) void nucleation occurs at a certain distance ahead of the current crack tip or the forward edge of the leading void ahead of the crack tip; (c) just before void nucleation the mean atomic stress (or equivalently its ratio to the von Mises effective stress, which is called the stress constraint or triaxiality) has a high concentration at the site of void nucleation; and (d) the stress field ahead of the current crack tip or the forward edge of the leading void is more or less self-similar (so that the forward edge of the leading void can be viewed as the effective crack tip).     

Mixed mode failure analysis of bonded joints with rate‐dependent interface models

The recent developments in joining technologies and the increasing use of composites materials in structural design justify the wide interest of structural mechanics researchers in bonded joints. Joints often represent the weakness zone of the structure and appropriate and rigorous mechanical models are required in order to describe deformation, durability and failure. The present work is devoted to the theoretical formulation and numerical implementation of an interface model suitable to simulate the time‐dependent behaviour of bonded joints. The interface laws are formulated in the framework of viscoplasticity for generalized standard materials and describe the softening response of the joint along its decohesion process in presence of shear and tensile normal tractions. These laws are derived in a thermodynamic consistent manner and take into account the rate dependency modifications of the fracture process zone making use of a sort of non‐local instantaneous dissipation. The interface constitutive laws are expressed both in rate and discrete incremental form for the purpose of numerical implementation. The consistent tangent matrix is derived. Finally, the problem of model parameters identification is approached making use of the finite element method for the experiments simulation and of an evolution strategy to solve the constrained optimization problem which mathematically represents the parameter identification inverse problem. Copyright © 2006 John Wiley & Sons, Ltd.     

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.