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Investigating size effects of complex nanostructures through Young-Laplace equation and finite element analysis
Journal of Applied Physics
Abstract
Analytical studies on the size effects of a simply-shaped beam fixed at both ends have successfully explained the sudden changes of effective Young's modulus as its diameter decreases below 100 nm. Yet they are invalid for complex nanostructures ubiquitously existing in nature. In accordance with a generalized Young-Laplace equation, one of the representative size effects is transferred to non-uniformly distributed pressure against an external surface due to the imbalance of inward and outward loads. Because the magnitude of pressure depends on the principal curvatures, iterative steps have to be adopted to gradually stabilize the structure in finite element analysis. Computational results are in good agreement with both experiment data and theoretical prediction. Furthermore, the investigation on strengthened and softened Young's modulus for two complex nanostructures demonstrates that the proposed computational method provides a general and effective approach to analyze the size effects for nanostructures in arbitrary shape.
... For analyzing the effects of the residual surface stress and surface elastic modulus in different regular polygonal prism structures, three typical cases [30,41] Figure 4 shows the effective modulus of different regular polygonal prism structures. From Fig. 4a and Fig. 3, as the surface elastic modulus increases from 50 N/m to 100 N/m, the effective modulus increases obviously. ...
... However, the difference in the effective modulus for different surface elastic moduli can be investigated deeply. In Fig. 5, the difference in the effective modulus is analyzed in the following cases: (i) E S = 50 N/m and E S = 25 N/m, (ii) E S = 100 N/m and E S = 50 N/m [30,41]. For different ligament sizes, the black and the red curves represent cases (i) and (ii), respectively, as shown in Fig. 5a. ...
Nanoporous materials exhibit remarkable mechanical properties, which are influenced by their surface effects and microstructures. Here, we theoretically analyze Young’s modulus of various regular polygonal prism structures, which represent the first-level (conventional) nanoporous material and higher-level hierarchical nanoporous material. The effects of surface elastic modulus, porosity, residual surface stress, and side number on Young’s modulus of a regular polygonal prism unit cell are investigated in detail. The results reveal that Young’s modulus is controlled by the surface elastic modulus, but it is not sensitive to the residual surface stress. This trend is in good consistency with the existing work. Moreover, the effective modulus strongly depends upon the surface elastic modulus. Under the certain range of high porosity, maximum Young’s modulus always relies on a critical side number of 6, regardless of the surface effect and hierarchical structure. Interestingly, for different regular polygonal prism structures, the difference in effective modulus tends to decrease with the increasing porosity. The current findings provide insights to improve the mechanical properties of nanoporous materials via tuning microstructure, porosity, and surface elastic modulus.
... It shall be mentioned that such a concept can be generalized to more ''complex" nanostructures. For example, Lu et al. (2015) used the finite element analysis to solve the Laplace-Young equation interacting with deformable structures. ...
The surface energy has been one of the topics of atomistic research for nanoparticles in the last decades. However, the physical role of surface stress and its quantification have been a lot less an object of research. Assumptions for the surface stress, going back to the thermodynamic basis of continua, have been popular. As an example the surface stress (state) follows as derivatives of the surface energy with a rather “classical” evolution equation for the deformation energy. The current concept introduces a combination of atomistic modelling and continuum mechanics for a core-shell system. Considering crystalline and amorphous gold nanoparticles with radii in the range of 1 nm to 12 nm, we are finally able to independently calculate the values of surface stress and surface energy, both slightly decreasing with the increasing particle radius. Surprisingly large values of surface stress are predicted for the case of amorphous nanoparticles.
... The aforementioned theoretical studies are based on the models of simple beams and regular beam systems, which are theoretically viable but geometrically dissimilar to nanostructures. Later, an iterative finite-element method was proposed to directly simulate the deformation of complex nanostructures under the curvature-dependent pressure resulting from the size effects [22]. Though this computational approach can be used to validate the experimental data for randomly shaped nanoporous structures, it is inapplicable to nanostructures less than 10 nm due to the low efficiency and convergence difficulty at such a small scale. ...
The size effects that reveal the dramatic changes of mechanical behaviour at nanoscales have traditionally been analysed for regular beam systems. Here, the method of using finite-element analysis is explored with the intention of evaluating the size effects for complex nanostructures. The surface elasticity theory and generalized Young-Laplace equation are integrated into a beam element to account for the size effects in classical Euler- Bernoulli and Timoshenko beam theories. Computational results match well with the theoretical predictions on the size effect for a cantilever beam and a cubic unit cell containing 24 horizontal/vertical ligaments. For a simply supported nanowire, it is found that the results are very close to the experimental data. With the assumption that nanoporous gold is composed of many randomly connected beams, for the first time, the size effect of such a complex structure is numerically determined.
Ceramics have some of the highest strength- and stiffness-to-weight ratios of any material but are suboptimal for use as structural
materials because of their brittleness and sensitivity to flaws. We demonstrate the creation of structural metamaterials composed
of nanoscale ceramics that are simultaneously ultralight, strong, and energy-absorbing and can recover their original shape
after compressions in excess of 50% strain. Hollow-tube alumina nanolattices were fabricated using two-photon lithography,
atomic layer deposition, and oxygen plasma etching. Structures were made with wall thicknesses of 5 to 60 nanometers and densities
of 6.3 to 258 kilograms per cubic meter. Compression experiments revealed that optimizing the wall thickness-to-radius ratio
of the tubes can suppress brittle fracture in the constituent solid in favor of elastic shell buckling, resulting in ductile-like
deformation and recoverability.
IntroductionProblem Statement and Material Interpolation SchemeSensitivity Analysis and Sensitivity NumberExamplesConclusion
Appendix 4.1References
T cells that accompany allogeneic hematopoietic grafts for treating leukemia enhance engraftment and mediate the graft-versus-leukemia
effect. Unfortunately, alloreactive T cells also cause graft-versus-host disease (GVHD). T cell depletion prevents GVHD but
increases the risk of graft rejection and leukemic relapse. In human transplants, we show that donor-versus-recipient natural
killer (NK)–cell alloreactivity could eliminate leukemia relapse and graft rejection and protect patients against GVHD. In
mice, the pretransplant infusion of alloreactive NK cells obviated the need for high-intensity conditioning and reduced GVHD.
NK cell alloreactivity may thus provide a powerful tool for enhancing the efficacy and safety of allogeneic hematopoietic
transplantation.
Several studies of the surface effect on bending properties of a nanowire (NW) have been conducted. However, these analyses are mainly based on theoretical predictions, and there is seldom integration study in combination between theoretical predictions and simulation results. Thus, based on the molecular dynamics (MD) simulation and different modified beam theories, a comprehensive theoretical and numerical study for bending properties of nanowires considering surface/intrinsic stress effects and axial extension effect is conducted in this work. The discussion begins from the Euler-Bernoulli beam theory and Timoshenko beam theory augmented with surface effect. It is found that when the NW possesses a relatively small cross-sectional size, these two theories cannot accurately interpret the true surface effect. The incorporation of axial extension effect into Euler-Bernoulli beam theory provides a nonlinear solution that agrees with the nonlinear-elastic experimental and MD results. However, it is still found inaccurate when the NW cross-sectional size is relatively small. Such inaccuracy is also observed for the Euler-Bernoulli beam theory augmented with both contributions from surface effect and axial extension effect. A comprehensive model for completely considering influences from surface stress, intrinsic stress, and axial extension is then proposed, which leads to good agreement with MD simulation results. It is thus concluded that, for NWs with a relatively small cross-sectional size, a simple consideration of surface stress effect is inappropriate, and a comprehensive consideration of the intrinsic stress effect is required.
Silver nanowires with different diameters were synthesized by a hydrothermal chemical method. The elastic properties of the nanowires with outer diameters ranging from 20 to 140 nm were measured using contact atomic force microscopy. The apparent Young modulus of the nanowires is found to decrease with the increase of the diameter. When the diameter of the silver nanowires is larger than 100 nm, the Young modulus approaches a constant value. The size dependence of the apparent Young modulus of the silver nanowires is attributed to the surface effect, which includes the effects of the surface stress, the oxidation layer, and the surface roughness. Thus, a theoretical analysis is presented to explain the size dependence. This analysis is different from the previous models in that both the surface stress and the surface moduli are included in it. We also show that the apparent surface modulus and the surface stress of the silver nanowires can be experimentally determined.
The tensile strengths of individual multiwalled carbon nanotubes (MWCNTs) were measured with a “nanostressing stage” located
within a scanning electron microscope. The tensile-loading experiment was prepared and observed entirely within the microscope
and was recorded on video. The MWCNTs broke in the outermost layer (“sword-in-sheath” failure), and the tensile strength of
this layer ranged from 11 to 63 gigapascals for the set of 19 MWCNTs that were loaded. Analysis of the stress-strain curves
for individual MWCNTs indicated that the Young's modulus E of the outermost layer varied from 270 to 950 gigapascals. Transmission electron microscopic examination of the broken nanotube
fragments revealed a variety of structures, such as a nanotube ribbon, a wave pattern, and partial radial collapse.
The results of two sets of experiments to measure the elastic–plastic behaviour of gold at the nanometre length scale are reported. One set of experiments was on free-standing nanoscale single crystals of gold, and the other was on free-standing nanoscale specimens of open-celled nanoporous gold. Both types of specimens were fabricated from commercially available leaf which was either pure Au or a Au/Ag alloy following by dealloying of the Ag. Mechanical testing specimens of a 'dog-bone' shape were fabricated from the leaf using standard lithographic procedures after the leaf had been glued onto a silicon wafer. The thickness of the gauge portion of the specimens was about 100 nm, the width between 250 nm and 300 nm and the length 7 µm. The specimens were mechanically loaded with a nanoindenter (MTS) at the approximate midpoint of the gauge length. The resulting force–displacement curve of the single crystal gold was serrated and it was evident that slip localization occurred on individual slip systems; however, the early stages of the plastic deformation occurred in a non-localized manner. The results of detailed finite element analyses of the specimen suggest that the critical resolved shear stress of the gold single crystal was as high as 135 MPa which would lead to a maximum uniaxial stress of about 500 MPa after several per cent strain. The behaviour of the nanoporous gold was substantially different. It exhibited an apparent elastic behaviour until the point where it failed in an apparently brittle manner, although it is assumed that plastic deformation occurred in the ligaments prior to failure. The average elastic stiffness of three specimens was measured to be Enp = 8.8 GPa and the stress at ultimate failure averaged 190 MPa for the three specimens tested. Scaling arguments suggest that the stress in the individual ligaments could approach the theoretical shear strength.
Surface stress was incorporated into the finite element absolute nodal coordinate formulation in order to model elastic bending
of nanowires in large deformation. The absolute nodal coordinate formulation is a numerical method to model bending structures
in large deformation. The generalized Young-Laplace equation was employed to model the surface stress effect on bending nanowires.
Effects from surface stress and large deformation on static bending nanowires are presented and discussed. The results calculated
with the absolute nodal coordinate formulation incorporated with surface stress show that the surface stress effect makes
the bending nanowires behave like softer or stiffer materials depending on the boundary condition. The surface stress effect
diminishes as the dimensions of the bending structures increase beyond the nanoscale. The developed algorithm is consistent
with the classical absolute nodal coordinate formulation at the macroscale.
A mathematical framework is developed to study the mechanical behavior of material surfaces. The tensorial nature of surface stress is established using the force and moment balance laws. Bodies whose boundaries are material surfaces are discussed and the relation between surface and body stress examined. Elastic surfaces are defined and a linear theory with non-vanishing residual stress derived. The free-surface problem is posed within the linear theory and uniqueness of solution demonstrated. Predictions of the linear theory are noted and compared with the corresponding classical results. A note on frame-indifference and symmetry for material surfaces is appended.
In a nanostructured material, the interface-to- volume ratio is so high that the interface energy, which is usually negligible
with respect to the bulk energy in solid mechanics, can no longer be neglected. The interfaces in a number of nanomaterials
can be appropriately characterized by the coherent interface model. According to the latter, the displacement vector field
is continuous across an interface in a medium while the traction vector field across the same interface is discontinuous and
must satisfy the Laplace–Young equation. The present work aims to elaborate an efficient numerical approach to dealing with
the interface effects described by the coherent interface model and to determining the size-dependent effective elastic moduli
of nanocomposites. To achieve this twofold objective, a computational technique combining the level set method and the extended
finite element method is developed and implemented. The numerical results obtained by the developed computational technique
in the two-dimensional (2D) context are compared and discussed with respect to the relevant exact analytical solutions used
as benchmarks. The computational technique elaborated in the present work is expected to be an efficient tool for evaluating
the overall size-dependent elastic behaviour of nanomaterials and nano-sized structures.
Surface and interface stresses in solids are defined and their role in the thermodynamics of solids is presented. A discussion concerning the physical meaning of these quantities is given, along with a review of selected theoretical calculations and experimental measurements. It is shown that for a solid phase with one or more of its dimensions smaller than about 10 nm, the surface and interface stresses can be principal factors in determining the equilibrium structure and behavior of the solid. In particular, the effects of surface and interface stresses on thin films are reviered along with the related topic of surface reconstructions in metals.
Nanomech. bending behavior and elastic modulus of silver nanowires (65-140 nm[nullset]) suspended across silicon microchannels were investigated using digital pulsed force mode (DPFM) at. force microscopy through coincident imaging and force profiling. Deflection profiles analyzed off-line demonstrate the role of bending nanowire shape and symmetry in exptl. detg. boundary conditions, eliminating the need to rely on isolated midpoint bending measurements and the usual assumptions for supported-end behavior. Elastic moduli for as-prepd. silver nanowires ranged from 80.4 ± 5.3 to 96.4 ± 12.8 GPa, which met or exceeded the literature values for bulk silver. The calcd. moduli were based on classic modeling, both with one-dimensional anal. solns. and three-dimensional finite element anal. Modeling results indicate that the classic models are accurate as long as the boundary conditions are not arbitrarily assumed but directly confirmed by data anal. DPFM also facilitated the exptl. detn. of sample gauge lengths from images and bending profiles. [on SciFinder(R)]
Mechanical behaviour analysis plays an important role in the design of micro/nano-electromechanical system (MEMS/NEMS) devices for reliability. In this paper, the size-dependent mechanical properties of nanostructures are numerically studied with the finite element method (FEM) by developing a kind of surface element to take into account the surface elastic effect. This method is then applied to the investigation of the interaction between two pressurized nanovoids and the effective moduli of two-dimensional nanoporous material. The numerical results indicate that surface elasticity can significantly alter the nature of interaction forms and the effective moduli by inducing a strong size dependence in conventional results.
The surface effect from surface stress and surface elasticity on the elastic behavior of nanowires in static bending is incorporated into Euler-Bernoulli beam theory via the Young-Laplace equation. Explicit solutions are presented to study the dependence of the surface effect on the overall Young's modulus of nanowires for three different boundary conditions: cantilever, simply supported, and fixed-fixed. The solutions indicate that the cantilever nanowires behave as softer materials when deflected while the other structures behave like stiffer materials as the nanowire cross-sectional size decreases for positive surface stresses. These solutions agree with size dependent nanowire overall Young's moduli observed from static bending tests by other researchers. This study also discusses possible reasons for variations of nanowire overall Young's moduli observed.
The significant rise in the strength and stiffness of porous materials at nanoscale cannot be described by conventional scaling laws. This letter investigates the effective Young's modulus of such materials by taking into account surface effect in a microcellular architecture designed for an ultralight material whose stiffness is an order of magnitude higher than most porous materials. We find that by considering the surface effects the predicted stiffness using Euler-Bernoulli beam theory compares well to experimental data for spongelike nanoporous gold with random microstructures. Analytical results show that, of the two factors influencing the effective Young's modulus, the residual stress is more important than the surface stiffness.
The paper describes theoretical and computational studies associated with the interface elastic properties of noncoherent metallic bicrystals. Analytical forms of interface energy, interface stresses, and interface elastic constants are derived in terms of interatomic potential functions. Embedded-atom method potentials are then incorporated into the model to compute these excess thermodynamics variables, using energy minimization in a parallel computing environment. The proposed model is validated by calculating surface thermodynamic variables and comparing them with preexisting data. Next, the interface elastic properties of several fcc-fcc bicrystals are computed. The excess energies and stresses of interfaces are smaller than those on free surfaces of the same crystal orientations. In addition, no negative values of interface stresses are observed. Current results can be applied to various heterogeneous materials where interfaces assume a prominent role in the systems' mechanical behavior.
There are three basic finite element formulations,which are used in multibody dynamics. These are the floating frame of reference approach, the incremental method and the large rotation vector approach. In the floating frame of reference and incremental formulations, the slopes are assumed small in order to define infinitestimal rotations that can be treated and transformed as vectors. This description, however, limits the use of some important elements such as beams and plates in a wide range of large displacement applications. As demonstrated in some recent publications, if infinitesimal rotations are used as nodal coordinates, the use of the finite element incremental formulation in the large reference displacement analysis doss nor lend to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. In this paper, a simple non-incremental finite element procedure that employs the mathematical definition of the slope and rises it to define the element coordinates instead of the infinitestimal and finite rotations is developed for large rotation and deformation problems. By using this description and by defining the element coordinates in the global system, nor only the need for performing coordinate transformation is avoided, bur also a simple expression for the inertia forces is obtained The resulting mass matrix is constant and it is the same matrix that appears in linen, structural dynamics. It is demonstrated in this paper that this coordinate description leads to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. Nonetheless, the stiffness matrix becomes nonlinear function even in the case of small displacements. The method presented in this paper differs from previous large rotation vector formulations in the sense that the inertia forces, the kinetic energy, and the strain energy are not expressed in terms of any orientation coordinates and therefore, the method does not require interpolation of finite rotations. While the use of the formulation is demonstrated using a simple planar beam element, the generalization of the method to other element types and to the three dimensional case is straightforward. Using the finite element procedure presented in this paper; beams and plates can be treated as isoparametric elements.
We present a Surface Cauchy-Born approach to modeling non-centrosymmetric, semiconducting nanostructures such as silicon that exist in a diamond cubic lattice structure. The model is based on an extension to the standard Cauchy-Born theory in which a surface energy term that is obtained from the underlying crystal structure and governing interatomic potential is used to augment the bulk energy. The incorporation of the surface energy leads naturally to the existence of surface stresses, which are key to capturing the size-dependent mechanical behavior and properties of nanomaterials. We present the approach in detail, then demonstrate its capabilities by calculating the minimum energy configurations of silicon nanowires due to surface stresses as compared to full scale atomistic calculations.
Elastic properties of crystal surfaces are useful in understanding mechanical properties of nanostructures. This paper presents a fully nonlinear treatment of surface stress and surface elastic constants. A method for the determination of surface elastic properties from atomistic simulations is developed. This method is illustrated with examples of several crystal faces of some fcc metals modeled with embedded atom potentials. The key finding in this study is the importance of accounting for the additional relaxations of atoms at the crystal surface due to strain. Although these relaxations do not affect the values of surface stress (as had been determined in previous works), they have a profound effect on the surface elastic constants. Failure to account for these relaxations can lead to values of elastic constants that are incorrect not only in magnitude but also in sign. A possible method for the experimental determination of the surface elastic constants is outlined.
Atomistic simulations with modified embedded atom method (MEAM), embedded atom method (EAM) and surface embedded atom method (SEAM) potentials reveal that, at certain sizes, a face centered cubic (fcc) gold ⟨100⟩ nanowire reorients into an fcc ⟨110⟩ nanowire. In MEAM simulations, the reorientation consists of two successive processes. First, surface stress and thermal vibrations cause the fcc ⟨100⟩ nanowire to transform into a body centered tetragonal (bct) nanowire. Second, the bct nanowire becomes unstable with respect to shear and transforms into an fcc ⟨110⟩ nanowire. In EAM and SEAM simulations a different reorientation mechanism exists. The surface stress in the fcc ⟨100⟩ nanowire induces slip on a {111}⟨112⟩ system. Progressive slip on adjacent {111} planes changes the stacking sequence of these {111} planes from ABCABC to ACBACB, and the nanowire reorients into an fcc ⟨110⟩ nanowire. The difference in reorientation mechanisms is rooted in the differences in the unstable stacking fault energy and orientation dependence of electron density in the potentials. In spite of these differences, the final structures of the reoriented nanowires are the same, which helps to explain why uniform fcc ⟨110⟩ nanowires are observed much more often in experiments than nanowires of other orientations.
The fracture strain, strength, and flexibility of ZnO nanowires (NWs) with a large range of diameters (85-542 nm) are investigated at a quantitative level. Large strains up to 4%-7% have been obtained before the final elastic fracture, corresponding to fracture strengths close to the theoretical strength. The flexibility of a NW is discussed quantitatively in terms of the diameter and fracture strain. The fundamental mechanisms responsible for the observed exceptional properties are discussed. (c) 2007 American Institute of Physics.
Nanoindentation behaviors of zinc oxide (ZnO) nanofilms with different film thicknesses are studied by using both molecular mechanics (MM) simulations and continuum analyses. It is found that there is a significant size effect on the indentation modulus obtained from MM simulations, which is absent in the continuum studies. The indentation modulus increases with the film thickness, and it also increases with the indentation depth; the trend of such a variation also depends on the film thickness. The contributions of the contact size effect, film thickness size effect, and microstructural size effect (surface effect) are elucidated and their couplings are explored. In addition, the substrate effect and nonlinear hyperelastic effect are incorporated to explain the size dependence of elastic indentation behaviors of ZnO nanofilms.
We investigate the effective elastic properties of nanoporous materials with hierarchical structures, which exhibit a distinct dependence on the characteristic sizes of their microstructure. A core shell model is first used to account for the effects of both surface tension and surface elasticity. We derive the effective Young's modulus of porous materials with one level of nanosized open or closed cells with surface effects. Then hierarchically structured nanoporous materials consisting of nanosized cells nested in another level of microsized lattice structure are considered to correlate their effective properties with the hierarchical structure. Particular attention is paid to nanoporous gold with multimodal ligament size distributions. (C) 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
The mechanical properties of a nanoporous material depend not only on its porosity but also on its characteristic sizes of microstructure, e.g., the average sizes of ligaments. Classical continuum mechanics models cannot interpret this type of size dependence. We here present a unit-cell micromechanics model to predict the effective Young's modulus of open-cell nanoporous materials. The theory of surface elasticity is adopted to incorporate the effects of surface energy and residual surface stress on the effective elastic property of nanoporous materials. This model can reasonably elucidate the relevant experimental results.
Effective stiffness properties (D) of nanosized structural elements such as plates and beams differ from those predicted by standard continuum mechanics (Dc). These differences (D-Dc)/Dc depend on the size of the structural element. A simple model is constructed to predict this size dependence of the effective properties. The important length scale in the problem is identified to be the ratio of the surface elastic modulus to the elastic modulus of the bulk. In general, the non-dimensional difference in the elastic properties from continuum predictions (D-Dc)/Dc is found to scale as αS/Eh, where α is a constant which depends on the geometry of the structural element considered, S is a surface elastic constant, E is a bulk elastic modulus and h a length defining the size of the structural element. Thus, the quantity S/E is identified as a material length scale for elasticity of nanosized structures. The model is compared with direct atomistic simulations of nanoscale structures using the embedded atom method for FCC Al and the Stillinger-Weber model of Si. Excellent agreement between the simulations and the model is found.
A comprehensive study on the relationship between yield strength, relative density and ligament sizes is presented for nanoporous Au foams. Depth-sensing nanoindentation tests were performed on nanoporous foams ranging from 20% to 42% relative density with ligament sizes ranging from 10 to 900 nm. The Gibson and Ashby yield strength equation for open-cell macrocellular foams is modified in order to incorporate ligament size effects. This study demonstrates that, at the nanoscale, foam strength is governed by ligament size, in addition to relative density. Furthermore, we present the ligament length scale as a new parameter to tailor foam properties and achieve high strength at low densities.
In nanoscaled solids, the mathematical behavior of a curved interface between two different phases with interface stress effects can be described by the generalized Young-Laplace equations T. Young, Philos. Trans. R. Soc. London 95, 65 1805; P. S. Laplace, Traite de Mechanique Celeste Gauthier-Villars, Paris, 1805, Vol. 4, Supplements au Livre X. Here we present a geometric illustration to prove the equations. By considering a small element of the curved thin interface, we model the interface stresses as in-plane stresses acting along its edges, while on the top and bottom faces of the interface the tractions are contributed from its three-dimensional bulk neighborhood. With this schematic illustration, simple force balance considerations will give the Young-Laplace equations across the interface. Similar procedures can be applied to conduction phenomena. This will allow us to reconstruct one type of imperfect interfaces, referred to as highly conducting interfaces. © 2006 American Institute of Physics.
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.
Surfaces and interfaces in solids may behave differently from their bulk counterparts, particularly when the geometry is on
the nanoscale. Our objective in this work is to assess the overall behavior of composites containing cylindrical inclusions
with surface effects prevailing along the interfaces. In the formulation, we first decompose the loadings into three different
deformation modes: the axisymmetric loadings, the transverse shear and the antiplane shear. For each deformation mode, we
derive the energy potential incorporating the surface effects. Using a variational approach, we construct the Euler-Lagrange
equation together with the natural transition (jump) conditions. The surface effects are represented by an interface of a
membrane type, with in-plane moduli different from those of either phase. The overall elastic behavior of the composite is
characterized by five constants. Four of them, except the transverse shear modulus, are derived in simple closed forms using
an approach of neutral inclusion. For the transverse shear, we derive the value based on the generalized self-consistent method.
The effect of surface energies, strains, and stresses on the size-dependent elastic state of embedded inhomogeneities are investigated. At nanolength scales, due to the increasing surface-to-volume ratio, surface effects become important and induce a size dependency in the otherwise size-independent classical elasticity solutions. In this letter, closed-form expressions are derived for the elastic state of eigenstrained spherical inhomogeneities with surface effects using a variational formulation. Our results indicate that surface elasticity can significantly alter the fundamental nature of stress state at nanometer length scales. Additional applications of our work on nanostructures such as quantum dots, composites, etc. are implied. © 2003 American Institute of Physics.
We present in this paper the development of a high-resolution projection micro-stereolithography (PμSL) process by using the Digital Micromirror Device (DMD™, Texas Instruments) as a dynamic mask. This unique technology provides a parallel fabrication of complex three-dimensional (3D) microstructures used for micro electro-mechanical systems (MEMS). Based on the understanding of underlying mechanisms, a process model has been developed with all critical parameters obtained from the experimental measurement. By coupling the experimental measurement and the process model, the photon-induced curing behavior of the resin has been quantitatively studied. The role of UV doping has been thereafter justified, as it can effectively reduce the curing depth without compromising the chemical property of the resin. The fabrication of complex 3D microstructures, such as matrix, and micro-spring array, with the smallest feature of 0.6 μm, has been demonstrated.
In this paper, we study triply periodic surfaces with minimal surface area
under a constraint in the volume fraction of the regions (phases) that the
surface separates. Using a variational level set method formulation, we present
a theoretical characterization of and a numerical algorithm for computing these
surfaces. We use our theoretical and computational formulation to study the
optimality of the Schwartz P, Schwartz D, and Schoen G surfaces when the volume
fractions of the two phases are equal and explore the properties of optimal
structures when the volume fractions of the two phases not equal. Due to the
computational cost of the fully, three-dimensional shape optimization problem,
we implement our numerical simulations using a parallel level set method
software package.
We devise new numerical algorithms, called PSC algorithms, for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front also can be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand sides, by using techniques from hyperbolic conservation laws. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be also used for more general Hamilton-Jacobi-type problems. We demonstrate our algorithms by computing the solution to a variety of surface motion problems.
The excellent mechanical properties of carbon nanotubes are being exploited in a growing number of applications from ballistic armour to nanoelectronics. However, measurements of these properties have not achieved the values predicted by theory due to a combination of artifacts introduced during sample preparation and inadequate measurements. Here we report multiwalled carbon nanotubes with a mean fracture strength >100 GPa, which exceeds earlier observations by a factor of approximately three. These results are in excellent agreement with quantum-mechanical estimates for nanotubes containing only an occasional vacancy defect, and are approximately 80% of the values expected for defect-free tubes. This performance is made possible by omitting chemical treatments from the sample preparation process, thus avoiding the formation of defects. High-resolution imaging was used to directly determine the number of fractured shells and the chirality of the outer shell. Electron irradiation at 200 keV for 10, 100 and 1,800 s led to improvements in the maximum sustainable loads by factors of 2.4, 7.9 and 11.6 compared with non-irradiated samples of similar diameter. This effect is attributed to crosslinking between the shells. Computer simulations also illustrate the effects of various irradiation-induced crosslinking defects on load sharing between the shells.
We report a size dependence of Young's modulus in [0001] oriented ZnO nanowires (NWs) with diameters ranging from 17 to 550 nm for the first time. The measured modulus for NWs with diameters smaller than about 120 nm is increasing dramatically with the decreasing diameters, and is significantly higher than that of the larger ones whose modulus tends to that of bulk ZnO. A core-shell composite NW model in terms of the surface stiffening effect correlated with significant bond length contractions occurred near the {1010} free surfaces (which extend several layers deep into the bulk and fade off slowly) is proposed to explore the origin of the size dependence, and present experimental result is well explained. Furthermore, it is possible to estimate the size-related elastic properties of GaN nanotubes and relative nanostructures by using this model.
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