Size effect of the elastic modulus of rectangular nanobeams:Surface elasticity effect

The size-dependent elastic property of rectangular nanobeams （nanowires or nanoplates） induced by the surface elas- ticity effect is investigated by using a developed modified core-shell model. The effect of surface elasticity on the elastic modulus of nanobeams can be characterized by two surface related parameters, i.e., inhomogeneous degree constant and surface layer thickness. The analytical results show that the elastic modulus of the rectangular nanobeam exhibits a distinct size effect when its characteristic size reduces below 1 O0 nm. It is also found that the theoretical results calculated by a mod- ified core-shell model have more obvious advantages than those by other models （core-shell model and core-surface model） by comparing them with relevant experimental measurements and computational results, especially when the dimensions of nanostructures reduce to a few tens of nanometers.

Nano-Contact Problem with Surface Effects on Triangle Distribution Loading

This work presents a theoretical study of contact problem. The Fourier integral transform method based on the surface elasticity theory is adopted to derive the fundamental solution for the contact problem with surface effects, in which both the surface tension and the surface elasticity are considered. As a special case, the deformation induced by a triangle distribution force is discussed in detail. The results are compared with those of the classical contact problem. At nano-scale, the contributions of the surface tension and the surface elasticity to the stress and displacement are not equal at the contact surface. The surface tension plays a major role to the normal stress, whereas the shear stress is mainly affected by the surface elasticity. In addition, the hardness of material depends strongly on the surface effects. This study is helpful to characterize and measure the mechanical properties of soft materials through nanoindentation.

METHOD TO SOLVE ELASTIC CONJUGATE SURFACES IN MULTI-TEETH MESHING

A method to solve the elastic conjugate surfaces in multi-teeth meshing ispresented. In mechanical manufacturing and design, there exist a lot of problems relating toconjugate surfaces, such as three-dimensional engagement, steel rolling and workpiece machining,which cause great effects on the quality of machining and performances of transmission. This methoddescribes relation between conjugate motion and elastic deformation in the process of mesh-in andmesh-out, and can be used to determine the profile of gear tooth by a certain given load sharing.

Photodetachment of H^-near an elastic surface in a magnetic field

This paper investigates the photodetachment of the negative hydrogen ion H- near an elastic wall in a magnetic field. The magnetic field confines the perpendicular motion of the electron, which results in a real three-dimensional well for the detached electron. The analytical formulas for the cross section of the photodetachment in the three-dimensional quantum well are derived based on both the quantum approach and closed-orbit theory. The magnetic field and the elastic surface lead to two completely different modulations to the cross section of the photodetachment. The oscillation amplitude depends on the strength of the magnetic field, the ion-wall distance and the photon polarization as well. Specially, for the circularly polarized photon-induced photodetachment, the cross sections display a suppressed （E - Eth）1/2 threshold law with energy E in the vicinity above Landau energy Eta, contrasting with the （E - Eta）-1/2 threshold law in the presence of only the magnetic field. The semiclassical calculation fits the quantum result quite well, although there are still small deviations. The difference is attributed to the failure of semiclassical mechanics.

Nonlinear Thermal Buoyancy on Ferromagnetic Liquid Stream Over a Radiated Elastic Surface with Non Fourier Heat Flux

The current article discusses the heat transfer characteristics of ferromagnetic liquid over an elastic surface with the thermal radiation and non-Fourier heat flux.In most of the existing studies,the heat flux is considered as constant,but whereas we incorporated the non-Fourier flux to get the exact performance of the flow.Also,we excluded the PWT and PHF cases to control the boundary layer of the flow.The governing equations related to our contemplate are changed into non-linear ordinary differential equations(ODE’s)by utilizing appropriate similarity changes,which are at the point enlightened by Runge–Kutta based shooting approach.The equations are broken down concerning boundary conditions and to be explained prescribed wall temperature(PWT)and prescribed heat flux(PHF)cases.The impacts of diverse non-dimensional physical parameters on velocity and temperature profiles are laid out graphically.Also,the assortment of skin friction and local Nusselt number for both PWT and PHF cases for various assessments of non-dimensional parameters have been sorted out.Towards the wrap-up of the examination,we suspect that the friction factor coefficient is higher in the PWT case compared to the PHF case.This result helps to conclude that the flux conditions are useful for cooling applications.

Approximate Tangent Plane Method for Calculating Surface Deformation in Elastic Contact Problems

A flew numerical method for constructing a pressure distribution to calculate surface elastic deformationcaused by normal contact pressure is developed in this paper. The pressure distribution over one of nonequidistantrectangles is fitted by an approximate tangent plane(ATP), which is formed by five pressure samples. Because thepressure distribution could be expressed as an one order linear polynomial, the iterative expression of elasticdeformation deduced by this method is simple, and the numerical accuracy is higher.

A review on the application of modified continuum models in modeling and simulation of nanostructures

Analysis of the mechanical behavior of nanos- tructures has been very challenging. Surface energy and non- local elasticity of materials have been incorporated into the traditional continuum analysis to create modified continuum mechanics models. This paper reviews recent advancements in the applications of such modified continuum models in nanostructures such as nanotubes, nanowires, nanobeams, graphenes, and nanoplates. A variety of models for these nanostructures under static and dynamic loadings are men- tioned and reviewed. Applications of surface energy and nonlocal elasticity in analysis of piezoelectric nanomateri- als are also mentioned. This paper provides a comprehensive introduction of the development of this area and inspires fur- ther applications of modified continuum models in modeling nanomaterials and nanostructures.

Elastic fields around a nanosized elliptic hole in decagonal quasicrystals

Based on the variational principle, a continuum theory of surface elasticity and new boundary conditions for qua- sicrystals is proposed. The effect of the residual surface stress on a decagonal quasicrystal that is weakened by a nanoscale elliptical hole is considered. The explicit expressions for the hoop stress along the edge of the hole are obtained using the Stroh formalism. The results show that the residual surface stress and the shape of the hole have a significant effect on the elastic state around the hole.

Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching

A model for the mechanics of woven fabrics is developed in the framework of two-dimensional elastic surface theory. Thickness effects are modeled indirectly in terms of appropriate constitutive equations. The model accounts for the strain of the fabric and additional effects associated with the normal bending, geodesic bending, and twisting of the constituent fibers.

Three-dimensionally nonlocal tensile nanobars incorporating surface effect: A self-consistent variational and well-posed model

A naturally discrete nanobar implies that the continuum axiom fails, and the surface-to-volume ratio is very large. The nonlocal theory of elasticity releasing the continuum axiom and the surface theory of elasticity are therefore employed to model tensile nanobars in this work. As commonly believed in the current practice, the axial nonlocal effect is only taken into account to analyze the mechanical behaviors of nanobars, regardless of the three-dimensional inherent atomistic interactions. In this study,a three-dimensional nonlocal constitutive law is developed to model the true nonlocal effect of nanobars, and based on which,a self-consistent variational bar model is proposed. It has been revealed for the first time how both the cross-sectional nonlocal interactions and the axial nonlocality affect the tensile behaviors of nanobars. It is found that the nonlocal influence predicted by the currently axial nonlocal bar model is grossly underestimated. Both the nonlocal cross-sectional and axial interactions become significant when the length-to-height ratio of nanobars is small. If the length-to-height ratio is relatively large(slender bars), the main nonlocal effect stems, however, from the nonlocal cross-sectional effect, rather than the axial nonlocal effect. This work also shows that it is possible to overcome the ill-posed problem of the pure nonlocal integral elasticity by employing both the pure nonlocal integral elasticity and surface elasticity. A well-posed size-dependent governing equation has been established for modeling nanobars under tension, and closed-form solutions are derived for their displacements. Based on the closed-form solutions, the effective elastic modulus is obtained and will be useful for calibrating the physical quantities in the "discretecontinuum" transition region for a span-scale modeling approach. It is shown that the effective elastic modulus may be softening or hardening, depending on the competition between the surface(modulus-hardening) and nonlocal(modulus-softening) effects.

Towards Understanding Why the Thin Membrane Transducer Deforms:Surface Stress-Induced Buckling

This study is directed towards a comprehensive exploration on the deformation mechanism of the thin membrane transducer（TMT） caused by surface stress variation.We stress that the biomolecular interaction has changed the magnitude of the surface stress;and when the surface stress exceeds a critical value the TMT will buckle and deform.Based upon Gurtin＇s theory of surface elasticity and principle of finite deformation,we abstract the TMT as a nanobeam with two clamped ends,and the close-formed governing equation set is derived accordingly.A computer code via the shooting method is developed to solve the presented two-point boundary value problem.In succession,the nanobeam deflection and critical parameters for buckling are quantitatively discussed.This investigation lays the theoretical foundation of TMTs;and it is also beneficial to gain deep insight into characterizing mechanical properties of nanomaterials and engineering nano-devices.

Collisionless tool orientation smoothing above blade stream surface using NURBS envelope

In five-axis machining,tool orientation above a blade stream surface may lead to tool collision and a decrease in workpiece rigidity.Hence,collisionless tool orientation smoothing(TOS)becomes an important issue.On the basis of a constant scallop height tool path,the triangular facets in the faces,vertices format are constructed from cutter contact(CC)using the Voronoi incremental algorithm.The cutter location(CL)points candidate set is represented by an oblique elliptic cone whose vertex lies at CC using NURBS envelope.Whether the CL point is above its CC is judged by the dot product between the normal vector and the point on triangulation nearest to the CL point.The curvatures at CC are obtained by fitting a moving least square(MLS) quadratic patch to the local neighborhood of a vertex and calculating eigenvectors and eigenvalues of the Hessian matrix.Triangular surface elastic energy is employed as the weight in selection from the NURBS envelope.The collision is judged by NURBS surface intersection.TOS can then be expressed by selecting a CL point for each CC point and converted into a numerical control(NC)code automatically according to the postprocessor type of the machine center.The proposed method is verified by finishing of a cryogenic turboexpander impeller of air separation equipment.

Nonlinear Vibration of the Flexoelectric Nanoplate with Surface Elastic Electrodes Under Active Electric Loading

This paper mainly studies the effects of surface elastic electrodes and electric loading on the nonlinear vibration of flexoelectric nanoplate.The governing equations are derived based on the von Karman type strain-displacement relations and the flexoelectricity theory.The nonlinear vibration equation is solved by an approximate method.Numerical results reveal that the surface elastic electrodes and the length-width ratio of flexoelectric nanoplate have a more significant effect on the nonlinear resonant frequency of simply-supported nanoplate than that of the all-edge-clamped nanoplate.Meanwhile,the effect of the graphene electrode on the nonlinear resonant frequency is more notable than that of the A1 electrode.Additionally,the applied electric loading can significantly affect the nonlinear resonant frequency of the flexoelectric nanoplate.For the nanoplates with different boundary conditions,the applied electric loading has different effects on the nonlinear resonant frequency.This study has certain research significance in the structure design and examination of the flexoelectric nanoplate with electrodes.

Elastic ware scattering from rough free surfaces of solids

Elastic wave scattering by a rough free surface of solids is analyzed. The analysis is based on the concept of scattering amplitude (SA) and perturbation approximation. The SA method is very convenient for rough surface scattering problems. By solving the boundary equations, the first and the second order solutions of approximate scattering amplitude are obtained. The general solutions are used for, as an example, the wave scattering by rough surfaces with Gaussian distribution. The mean field and variance are given. Finally, an experiment is designed to verify the theoretical predications.

Propagation of elastic waves in a plate with rough surfaces

The characteristics of Lamb wave propagating in a solid plate with rough surfaces are studied on the basis of small perturbation approximation. The Rayleigh-Lamb frequency equation expressed with SA matrix is presented. The Rayleigh-Lamb frequency equation for a rough surface plate is different from that for a smooth surface plate, resulting in a small perturbation △k on Lamb wave vector k. The imaginary part of △k gives the attenuation caused by wave scattering. An experiment is designed to test our theoretical predications. By using wedge-shape pipes, different Lamb wave modes are excited. The signals at different positions are received and analyzed to get the dispersion curves and attenuations of different modes. The experimental results are compared with the theoretical predications.

Torsional wave in a circular micro-tube with clogging attached to the inner surface

In the present paper, we study the torsional wave propagation along a micro-tube with clog- ging attached to its inner surface. The clogging accumulated on the inner surface of the tube is modeled as an ＂elastic membrane＂ which is described by the so-called surface elasticity. A power-series solution is particularly developed for the lowest order of wave propagation. The dispersion diagram of the lowest-order wave is numerically presented with the surface （clogging） effect.

Axisymmetric loading on nanoscale multilayered media

Multilayered nanoscale structures are used in several applications.Because the effect of surface energy becomes nontrivial at such a small scale,a modified continuum theory is required to accurately predict their mechanical behaviors.A Gurtin–Murdoch continuum model of surface elasticity is implemented to establish a computational scheme for investigating an elastic multilayered system under axisymmetric loads with the incorporation of surface/interface energy.Each layer stiffness matrix is derived based on the general solutions of stresses and displacements obtained in the form of the Hankel integral transform.Numerical solutions to the global equation,which are formulated based on the continuity conditions of tractions and displacements across interfaces between layers,yield the displacements at each layer interface and on the top surface of the multilayered medium.The numerical solutions indicate that the elastic responses of multilayered structures are affected significantly by the surface material properties of both the top surface and interfaces,and that they become size dependent.In addition,the indentation problem of a multilayered nanoscale elastic medium under a rigid frictionless cylindrical punch is investigated to demonstrate the application of the proposed solution scheme.

Analytical Solutions for an Isotropic Elastic Half-Plane with Complete Surface Effects Subjected to a Concentrated/Uniform Surface Load

Within the context of Gurtin-Murdoch surface elasticity theory,closed-form analytical solutions are derived for an isotropic elastic half-plane subjected to a concentrated/uniform surface load.Both the effects of residual surface stress and surface elasticity are included.Airy stress function method and Fourier integral transform technique are used.The solutions are provided in a compact manner that can easily reduce to special situations that take into account either one surface effect or none at all.Numerical results indicate that surface effects generally lower the stress levels and smooth the deformation profiles in the half-plane.Surface elasticity plays a dominant role in the in-plane elastic fields for a tangentially loaded half-plane,while the effect of residual surface stress is fundamentally crucial for the out-of-plane stress and displacement when the half-plane is normally loaded.In the remaining situations,combined effects of surface elasticity and residual surface stress should be considered.The results for a concentrated surface force serve essentially as fundamental solutions of the Flamant and the half-plane Cerruti problems with surface effects.The solutions presented in this work may be helpful for understanding the contact behaviors between solids at the nanoscale.

A Characterization of the Ejiri Torus in S^5

We conjecture that a Willmore torus having Willmore functional between 2π2 and 2π2 √3 is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri＇s torus in S5 is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in S n by reducing them into elastic curves in S3, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S5 attains the minimum 2π2 √3, which indicates our conjecture holds true for Wilhnore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S5. Moreover, similar to Li and Vrancken, we classify all constrained Wilhnore surfaces of tensor product by reducing them with elastic curves in S3. All constrained Willmore tori obtained this way are also shown to bc unstable when the co-dimension is big enough.