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.     

Surface stress effects on the free vibration behavior of nanoplates

A new frontier of research in the area of computational nanomechanics is to study the behavior of structures at very small length scales. As the dimensions of a structure approach the nanoscale, the classical continuum theories may fail to accurately predict the mechanical behavior of nanostructures. Among these nanostructures, nanobeams are attracting more and more attention due to their great potential engineering applications. One of the most important factors that influence the behavior of such submicron-sized structures is surface stress effect because of their high surface to volume ratio. In this paper, a non-classical solution is proposed to analyze bending and buckling responses of nanobeams including surface stress effects. Explicit formulas are proposed relevant to each type of beam theory to evaluate the surface stress effects on the displacement profile and critical buckling load of the nanobeams. Numerical results are presented to demonstrate the difference between the behaviors of the nanobeam predicted by the classical and non-classical solutions which depends on the magnitudes of the surface elastic constants.     

A surface energy model and application to mechanical behavior analysis of single crystals at sub-micron scale

Size effects of mechanical behaviors of materials are referred to the variation of the mechanical behavior due to the sample sizes changing from macroscale to micro-/nanoscales. At the micro-/nanoscale, since sample has a relatively high specific surface area (SSA) (ratio of surface area to volume), the surface energy effect, although it is often neglected at the macroscale, becomes prominent in governing the mechanical behavior. In the present research, a continuum model considering the surface energy effect is developed through introducing the surface energy to total potential energy. Simultaneously, a corresponding finite element method is developed. The model is used to analyze the axial equilibrium strain problem for a Cu nanowire at the external loading-free state. As another application of the model, from dimensional analysis, the size effects of uniform compression tests on the microscale cylinder specimens for Ni and Au single crystals are analyzed and compared with experiments in literatures.     

Evaluation of a physical length scale for granular materials

Classical continuum mechanics considers the interaction of microstructural units of the material through stresses and displacements of material points. Therefore, conventional continuum mechanics approaches can not incorporate any intrinsic material length scale. However in reality interaction of grains may include rotations and the corresponding couple stresses as well, and real materials have a number of important length scales (e.g., grains, particles, fibers, etc.). An equation for determining the length scale is proposed. The proposed length scale equations include the effect of plastic deformation (microrotation), the effect of normal stress and contact area. The proposed length scale is implemented into elastic–elastoplastic Cosserat formulation. The effect of length scale on the finite element simulation and yield surface was evaluated by using the proposed length scale equation. The importance of length scale on the constitutive modeling of granular materials is analyzed in numerical simulations.     

Electric Field Gradient Theory with Surface Effect for Nano-Dielectrics

The electric field gradient effect is very strong for nanoscale dielectrics. In addition, neither the surface effect nor electrostatic force can be ignored. In this paper, the electric Gibbs free energy variational principle for nanosized dielectrics is established with the strain/electric field gradient effects, as well as the effects of surface and electrostatic force. As regards the surface effects both the surface stress and surface polarization are considered. From this variational principle, the governing equations and the generalized electromechanical Young-Laplace equations, which take into account the effects of strain/electric field gradient, surface and electrostatic force, are derived. The generalized bulk and surface electrostatic stress are obtained from the variational principle naturally. The form are different from those derived from the flexoelectric theory. Based on the present theory, the size-dependent electromechanical phenomenon in nano-dielectrics can be predicted.     

Crack patterns obtained by unidirectional drying of a colloidal suspension in a capillary tube: experiments and numerical simulations using a two-dimensional variational approach

Basalt columns, septarias, and mud cracks possess beautiful and intriguing crack patterns that are hard to predict because of the presence of cracks intersections and branches. The variational approach to brittle fracture provides a mathematically sound model based on minimization of the sum of bulk and fracture energies. It does not require any a priori assumption on fracture patterns and can therefore deal naturally with complex geometries. Here, we consider shrinkage cracks obtained during unidirectional drying of a colloidal suspension confined in a capillary tube. We focus on a portion of the tube where the cross-sectional shape cracks does not change as they propagate. We apply the variational approach to fracture to a tube cross-section and look for two-dimensional crack configurations minimizing the energy for a given loading level. We achieve qualitative and quantitative agreement between experiments and numerical simulations using a regularized energy (without any assumption on the cracks shape) or solutions obtained with traditional techniques (fixing the overall crack shape a priori). The results prove the efficiency of the variational approach when dealing with crack intersections and its ability to predict complex crack morphologies without any a priori assumption on their shape.     

A finite element‐based level set method for fluid–elastic solid interaction with surface tension

A numerical method for simulating fluid–elastic solid interaction with surface tension is presented. A level set method is used to capture the interface between the solid bodies and the incompressible surrounding fluid, within an Eulerian approach. The mixed velocity–pressure variational formulation is established for the global coupled mechanical problem and discretized using a continuous linear approximation in both velocity and pressure. Three ways are investigated to reduce the spurious oscillations of the pressure that appear at the fluid–solid interface. First, two stabilized finite element methods are used: the MINI‐element and the algebraic subgrid method. Second, the surface integral corresponding to the surface tension term is treated either by the continuum surface force technique or by a surface local reconstruction algorithm. Finally, besides the direct evaluation method proposed by Bruchon et al., an alternative method is proposed to avoid the explicit computation of the surface curvature, which may be a source of difficulty. These different issues are addressed through various numerical examples, such as the two incompressible fluid flow, the elastic inclusion embedded into a Newtonian fluid, or the study of a granular packing. Copyright © 2012 John Wiley & Sons, Ltd.     

Interphase layer theory and application in the mechanics of composite materials

In the present research work the interphase layer model is developed as a continuum media with local cohesion and adhesion effects. By the model it was found that these effects can help to understand/predict macro/micro mechanics of the material, if the boundary conditions and phase effects are modeled across the length scales. This paper describes the kinematics of continuum media, the formulation of governing equations (fundamentals) and the statement of boundary conditions for multi-scale modeling of the material. An approach and the model has been validated to predict some basic mechanical properties of a polymeric matrix reinforced with nanoscale particles/fibres/tubes (including carbon nanotubes) as a function of size and also dispersion of nanoparticles. Presented mathematical model of an interphase layer allows estimating an interaction around and nearby interfaces of nanoparticle and material matrix. Using these approaches the prediction methodology and modeling tools have been developed by numerical simulations and analysis of the mechanical properties across the length scales. Results of the work will provide a platform for the development and understanding of nanoparticle-reinforced materials that are light-weight, vibration and shock resistant.     

Variational principles in dissipative electro‐magneto‐mechanics: A framework for the macro‐modeling of functional materials

This paper presents a general framework for the macroscopic, continuum‐based formulation and numerical implementation of dissipative functional materials with electro‐magneto‐mechanical couplings based on incremental variational principles. We focus on quasi‐static problems, where mechanical inertia effects and time‐dependent electro‐magnetic couplings are a priori neglected and a time‐dependence enters the formulation only through a possible rate‐dependent dissipative material response. The underlying variational structure of non‐reversible coupled processes is related to a canonical constitutive modeling approach, often addressed to so‐called standard dissipative materials. It is shown to have enormous consequences with respect to all aspects of the continuum‐based modeling in macroscopic electro‐magneto‐mechanics. At first, the local constitutive modeling of the coupled dissipative response, i.e. stress, electric and magnetic fields versus strain, electric displacement and magnetic induction, is shown to be variational based, governed by incremental minimization and saddle‐point principles. Next, the implications on the formulation of boundary‐value problems are addressed, which appear in energy‐based formulations as minimization principles and in enthalpy‐based formulations in the form of saddle‐point principles. Furthermore, the material stability of dissipative electro‐magneto‐mechanics on the macroscopic level is defined based on the convexity/concavity of incremental potentials. We provide a comprehensive outline of alternative variational structures and discuss details of their computational implementation, such as formulation of constitutive update algorithms and finite element solvers. From the viewpoint of constitutive modeling, including the understanding of the stability in coupled electro‐magneto‐mechanics, an energy‐based formulation is shown to be the canonical setting. From the viewpoint of the computational convenience, an enthalpy‐based formulation is the most convenient setting. A numerical investigation of a multiferroic composite demonstrates perspectives of the proposed framework with regard to the future design of new functional materials. Copyright © 2011 John Wiley & Sons, Ltd.     

Variational methods in irreversible thermoelasticity: theoretical developments and minimum principles for the discrete form

A variational approach for fully coupled dynamic irreversible thermoelasticity is developed for continua, which considers both the conservative and dissipative character in terms of mixed variables. By introducing a consistent variational scheme for the spatial and temporal discretization of the governing equations, a mixed continuum element is established under the Hamiltonian-Lagrangian formalism. The proposed method leads to the development of minimum principles in discrete form with the proper selection of state variables and temporal action sum operators. Consequently, this novel mixed variational formulation can provide the basis for a class of optimization-based methods for irreversible thermomechanics. Several applications are considered to demonstrate the robustness of the proposed variational approach, including transient dynamic response of thermoelastic media due to surface heating caused by ramp- and step-type heat fluxes, and a sequence of laser pulses.     

Superductile,Wavy Silica Nanostructures Inspired by Diatom Algae

Biology implements intriguing structural design principles that allow for attractive mechanical properties—such as high strength, toughness, and extensibility despite being made of weak and brittle constituents, as observed in biomineralized structures. For example, diatom algae contain nanoporous hierarchical silicified shells, called frustules, which provide mechanical protection from predators and virus penetration. These frustules generally have a morphology resembling honeycombs within honeycombs, meshes, or wavy shapes, and are surprisingly tough when compared to bulk silica, which is one of the most brittle materials known. However, the reason for its extreme extensibility has not been explained from a molecular level upwards. By carrying out a series of molecular dynamics simulations with the first principles‐based reactive force field ReaxFF, the mechanical response of the structures is elucidated and correlated with underlying deformation mechanisms. Specifically, it is shown that for wavy silica, unfolding mechanisms are achieved for increasing amplitude and allow for greater ductility of up to 270% strain. This mechanism is reminiscent to the uncoiling of hidden length from proteins that allows for enhanced energy dissipation capacity and, as a result, toughness. We report the development of an analytical continuum model that captures the results from atomistic simulations and can be used in multiscale models to bridge to larger scales. Our results demonstrate that tuning the geometric parameters of amplitude and width in wavy silica nanostructures are beneficial in improving the mechanical properties, including enhanced deformability, effectively overcoming the intrinsic shortcomings of the base material that features extreme brittleness.     

Non-equilibrium thermodynamics and variational principles for fully coupled thermal–mechanical–chemical processes

In this paper, the chemical Gibbs function variational principle, the Helmholtz function variational principle and the internal energy variational principle based on irreversible thermodynamics are proposed for the thermal–chemical–mechanical fully coupling problems. The complete fully coupling governing equations, including the heat conduction, mass diffusion and chemical reactions, are derived from the variational principles. The convective effect can also be derived in the diffusion and energy equations from the variational principles naturally. Moreover, the concentrations and entropy jump conditions on the moving interface between the products due to chemical reactions and the matrix can be derived from the variational principles naturally. This work provides the basis for the analyses and computations of thermochemomechanical coupling problems.     

A variational framework for fiber‐reinforced viscoelastic soft tissues

The mechanical properties of soft biological tissues vary depending on how the internal structure is organized. Classical examples of tissues are ligaments, tendons, skin, arteries, and annulus fibrous. The main element of such tissues is the fibers which are responsible for the tissue resistance and the main mechanical characteristic is their viscoelastic anisotropic behavior. The objective of this paper is to extend an existing model for isotropic viscoelastic materials in order to include anisotropy provided by fiber reinforcement. The incorporation of the fiber allows the mechanical behavior of these tissues to be simulated. The model is based on a variational framework in which its mechanical behavior is described by a free energy incremental potential whose local minimization provides the constraints for the internal variable updates for each load increment. The main advantage of this variational approach is the ability to represent different material models depending on the choice of suitable potential functions. Finally, the model is implemented in a finite‐element code in order to perform numerical tests to show the ability of the proposed model to represent fiber‐reinforced materials. The material parameters used in the tests were obtained through parameter identification using experimental data available in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

On instabilities and autocatalytic effects associated with the nucleation of martensite bands in polycrystalline SMA flat tensile specimens

The mechanical response in the pseudoelastic range of polycrystalline NiTi SMA specimens in displacement controlled tensile tests exhibits a load drop associated with the nucleation of martensitic bands, a phenomenon which is similar to the formation of Lüders bands in mild steel. A continuum mechanical energy approach allows to calculate the nominal stress–strain curve incorporating the characteristic load drop. The influence of the geometrical aspect ratio, i.e., thickness-to-width ratio of the flat tensile test specimens on the load drop is discussed and compared to the experimental data.     

Elastic mechanical behavior of nano-scaled FGM films incorporating surface energies

A general, global theory is developed for nano-scaled functionally graded films considering surface effects. In addition to the Kirchhoff hypothesis of the classical thin plate theory, the surface layers of the film are modeled by the continuum theory of surface elasticity. Bulk stresses on the surfaces are required to satisfy the surface balance conditions involving surface stresses. Unlike the classical plate theory, the bulk transverse normal stress is preserved here. By incorporating the surface energies into the principle of minimum potential energy, a series of non-classical governing differential equations which include intrinsic length scales are derived. To illustrate application of the theory, a simply supported nano-scaled film in cylindrical bending is investigated. Numerical examples are presented to clarify the effects of surface energies on the bending behavior of FGM films, whose effective elastic moduli are predicted using the Mori–Tanaka method. Finally, the nature of intrinsic length scales, and the effects of gradient index and aspect ratio on the displacements are discussed.     

TiNi基形状记忆合金薄膜研究进展

TiNi基形状记忆合金薄膜除具有和体材料相同的形状记忆特性外,还因表面积大、散热能力好,改善了体材料响应速度较慢的缺点,在微电子机械系统领域是一种具有广阔应用前景的材料.综述了近年来TiNi基形状记忆合金薄膜研究现状,并对其应用进行了评述.     

Influence of different sources of microstructural heterogeneity on the degradation of asphalt mixtures

The response and degradation of the hot mix asphalt (HMA) materials used in pavement structures are affected by their inherent heterogeneity. The objective of this work is to study the impact of two different sources of HMA heterogeneity in the uncertainty of the mechanical moisture degradation of HMA. The first source of heterogeneity is the spatial variability of the properties of the bulk fine aggregate matrix (FAM) of the mixture, and the second is the location and shape of the coarse aggregate particles. The heterogeneity of the bulk FAM phase was modelled using a random field technique, while that of the coarse aggregates was accounted for by randomly generating realistic probable sets of aggregate particles. Thus, ‘computational replicates’ of HMA microstructures were generated and subjected to moisture diffusion and mechanical loading using a finite element approach. In the mechanical simulations, a non-linear viscoelastic moisture damage constitutive relationship based on continuum damage mechanics theory was selected to characterise the response of the bulk FAM phase. The results show that conducting computational simulations with realistic HMA microstructures that properly capture the heterogeneity of the material is useful to quantify the mean values and dispersion (i.e. uncertainty) associated with the response and degradation of the mixture. This information, which cannot be easily obtained in the field or in the laboratory due to the difficulty of acquiring a sufficient amount of data, is useful to conduct structural reliability analysis and to predict the life cycle behaviour of the material.     

Review of the role of the interphase in the control of composite performance on micro- and nano-length scales

In fiber reinforced composites (FRCs), exhibiting heterogeneous structure at multiple length scales, the interphase phenomena at various length scales were shown to be of pivotal importance for the control of the performance and reliability of such structures. Various models based on continuum mechanics were used to describe effects of the macro- and meso-scale interphase on the mechanical response of laminates and large FRC parts, satisfactorilly. At the micro-scale, the interphase is considered a 3D continuum with ascribed average properties. Number of continuum mechanics models was derived over the last 50 years to describe the stress transfer between matrix and individual fiber with realtively good success. In these models, the interphase was characterized by some average shear strength, τ _a, and elastic modulus, E _a. On the other hand, models for tranforming the properties of the micro-scale interphase around individual fiber into the mechanical response of macroscopic multifiber composite have not been generally successfull. The anisotropy of these composite structures are the main reasons causing the failure of these models. The strong thickness dependence of the elastic modulus of the micro-scale interphase suggested the presence of its underlying sub-structure. On the nano-scale, the discrete molecular structure of the polymer has to be considered. The term interphase, originally introduced for continuum matter, has to be re-defined to include the discrete nature of the matter at this length scale. The segmental immobilization resulting in retarded reptation of chains caused by interactions with solid surface seems to be the primary phenomenon which can be used to re-define term interphase on the nano-scale. Thus, the Rubinstein reptation model and a simple percolation model were used to describe immobilization of chains near solid nano-particles and to explain the peculiarities in the viscoleastic response of nano-scale “interphase.” It has also been shown that below 5 nm, Bernoulli–Euler continuum elasticity becomes not valid and higher-order elasticity along with the proposed reptation dynamics approach can provide suitable means for bridging the gap in modeling the transition between the mechanics of continuum matter at the micro-scale and mechanics of discrete matter at the nano-scale.     

Multi‐scale constitutive modelling of heterogeneous materials with a gradient‐enhanced computational homogenization scheme

A gradient‐enhanced computational homogenization procedure, that allows for the modelling of microstructural size effects, is proposed within a general non‐linear framework. In this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element (RVE). This enables us to incorporate the microstructural size and to account for non‐uniform macroscopic deformation fields within the microstructural cell. Every microstructural constituent is modelled as a classical continuum and the RVE problem is formulated in terms of standard equilibrium and boundary conditions. From the solution of the microstructural boundary value problem, the macroscopic stress tensor and the higher‐order stress tensor are derived based on an extension of the Hill–Mandel condition. This automatically delivers the microstructurally based constitutive response of the higher‐order macro continuum and deals with the microstructural size in a natural way. Several examples illustrate the approach, particularly the microstructural size effects. Copyright © 2002 John Wiley & Sons, Ltd.     

Torsional deformation behavior of cracked gold nano-wires

The presence of in-homogeneity or defects in materials cannot be ignored. There is great need to understand the influence of defects on the mechanical response of nano-materials. In this study, atomistic simulations have been used to investigate the mechanical response of gold nano-wires under twisting. Simulations show that nano-wires have different elastic properties when defects are present. Embedded cracks of different sizes have been created in nano-wires to quantify in-homogeneity. The inter-atomic interactions are represented by employing an embedded-atom potential. The effect of different sizes of crack on potential energy, torque and stresses for investigating the mechanical response of a nano-wire is part of the whole investigation. It is predicted from our simulation that the presence of a crack and dimensions of the crack control the torsional response of gold nano-wires. Deviation in the behavior of gold nano-wires from continuum expectations is also discussed. The comparison of results of atomistic simulations is made with a linear elasticity model (of healthy and cracked nano-wires) to get deep insights into the nano-scale behavior of nano-wires.