Mathematical modelling of phononic nanoplate and its size-dependent dispersion and topological properties

A new model with analysis for the propagation of flexural waves in a phononic plate at nanoscale is developed. The Gurtin-Murdoch theory for surface elasticity is adopted to model the surface heterogeneity. The Mindlin (or first-order) plate theory is further generalized to establish the governing equations for flexural waves in a phononic plate with surface effect, for which the plane wave expansion method is applied to derive the dispersion relation. A numerical model is developed using the finite element method and very good consistency between theory and numerical solution is observed. It is found that the surface density and the surface residual stress play the main role that affects the band structures. The surface effect can be approximately regarded as the competition between frequency decrease due to surface density and frequency increase caused by surface residual stress, which effectively increases the low-frequency bands but decreases the high-frequency bands. The quantum spin Hall effect is observed in the phononic plate at nanoscale, and the surface effect is studied numerically. By applying the k.p perturbation method, a theoretical framework is established to calculate the spin Chern number, which is an important topological invariant that determines the quantum spin Hall effect. Based on the topological analysis, an efficient waveguide with a zig-zag path is designed, in which a topologically protected wave in the interface state can robustly propagate along the path against disorders. The theory and numerical study developed in this paper will help better understand the size-dependent quantum spin Hall effect in nanostructures and it may also provide guidance for the design of topological wave devices at nanoscale.     

The stressed state in a plate caused by the distortion tensor

The improved plate theory, which makes it possible to determine all the components of the stress tensor, is generalized to the case when the stressed state of a plate is caused by the distortion tensor. The equations obtained can be useful in the study of residual stresses in thinwalled elements, in applications of the method of distortion in the mechanics of failure and elsewhere.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 53–57.     

The stressed state caused by a residual strain field in plates with stress concentrators

We obtain an analytic solution of the problem of determining the stressed state caused by a given residual strain field (while taking account of three-dimensional effects) in a round plate with a concentric foreign inclusion. We study the influence of the geometric parameters of the given system, the nonlinearity of the distribution of the given residual strains over the thickness of the plate, and a possible jump in distortion at the surface of contact on the stressed state of the system. We discover an internal boundary-layer effect that is significant in the case when the given residual strain field is strongly gradient.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 60–66.     

First Integrals and the Residual Solution for Orthotropic Plates in Plane Strain or Axisymmetric Deformations

Two classes of exact solutions are derived for the equations of three dimensional linear orthotropic elasticity theory governing flat (plate) bodies in plane strain or axisymmetric deformations. One of these is the analogue of the Lévy solution for plane strain deformations of isotropic plates and is designated as the interior solutions. The other complementary class correspond to the Papkovich-Fadle Eigenfunction solutions for isotropic rectangular strips and is designated as the residual solutions. For sufficiently thin plates, the latter exhibits rapid exponential decay away from the plate edges. A set of first integrals of the elasticity equations is also derived. These first integrals are then transformed into a set of exact necessary conditions for the elastostatic state of the body to be a residual state. The results effectively remove the asymptoticity restriction of rapid exponential decay of the residual state inherent in the corresponding necessary conditions for isotropic plate problems. The requirement of rapid exponential decay effectively limits their applicability to thin plates. The result of the present paper extend the known results to thick plate problems and to orthotropic plate problems. They enable us to formulate the correct edge conditions for two-dimensional orthotropic thick plate theories with stress or mixed edge data.     

Inverse Design of a Displacement Body with a discrete Adjoint Method

Wind tunnel experiments at German Aerospace Center (DLR) for the analysis of laminar turbulent transition showed that a nearly linearly decreasing pressure distribution at a plate is able to suppress Tollmien-Schlichting boundary layer instabilities. Nevertheless, this pressure distribution leads to the growth of nonlinear instabilities in cross flow direction. Within up-ucoming research projects at DLR it is intended to further investigate this cross flow boundary layer instabilities. The experimental setup consists of a flat plate, a displacement body mounted above the plate and an additional supporting wing. The supporting wing prevents flow separations. The displacement body imposes the particular pressure distribution upon the upper side of the plate. However, contour and position of the displacement body is not known by detail. Therefore, the given pressure distribution is to be recreated by an inverse design procedure. The task can be seen as a design optimisation to a given pressure function. To provide a fast evaluation the discrete adjoint method of the TAU-Code is used. The problem's cost function is regarded as the residual of the current and the target pressure. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

双模量复合材料正交迭层矩形厚板的热弯曲问题

本文给出了双模量复合材料迭层板热弯曲的加权残数解。各层都假定为弹性和热弹性的双模量各向异性材料。该模型是建立在Whitney-Pagano迭层板理论和热弹性模型基础上,考虑了沿板厚的剪切应变。所得挠度和中性面位置的结果和精确解非常吻合。     

A posteriori error estimator for a mixed finite element method for Reissner-Mindlin plate

We present an a posteriori error estimator for a mixed finite element method for the Reissner-Mindlin plate model. The finite element method we deal with, was analyzed by Durán and Liberman in 1992 and can also be seen as a particular example of the general family analyzed by Brezzi, Fortin, and Stenberg in 1991. The estimator is based on the evaluation of the residual of the finite element solution. We show that the estimator yields local lower and global upper bounds of the error in the numerical solution in a natural norm for the problem, which includes the norms of the terms corresponding to the deflection and the rotation and a dual norm for the shearing force. The estimates are valid uniformly with respect to the plate thickness.

     

The elastoplastic stressed state of a half-plane under local nonstationary heating

We solve the thermoplastic problem for a semi-infinite plate under local nonstationary heating by heat sources. The physical equations are taken to be the relations of the nonisothermic theory of plastic flow associated with the Mises fluidity condition. The solution of the problem is constructed by the method of integral equations and the self-correcting method of sequential loading, where time is taken as the loading parameter. We carry out numerical computations of the stresses in the case of heating a plate with heat output by normal-circular heat sources. We study the problem of optimization of heating regimes in order to introduce favorable residual compressive stresses (from the point of view of hardness) in a given region of a half-plane. Two figures.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 29–34.     

Strongly-coupled modelling and analysis of energy harvesting devices

A monolithic approach is proposed that provides simultaneous modelling and analysis of the harvester, which involves surface-coupled fluid-structure interaction, volume-coupled electro- mechanics and a controlling energy harvesting circuit for applications in energy harvesting. A space-time finite element approximation is used for numerical solution of the weighted residual form of the governing equations of the flow-driven piezoelectric energy harvesting device. This method enables time-domain investigation of different types of structures (plate, shells) subject to exterior/interior flow with varying cross sections, material compositions, and attached electrical circuits with respect to the electrical power output generated. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second-order two-scale method for bending behavior analysis of composite plate with 3-D periodic configuration and its approximation

This paper considers the bending behaviors of composite plate with 3-D periodic configuration.A second-order two-scale(SOTS)computational method is designed by means of construction way.First,by 3-D elastic composite plate model,the cell functions which are defined on the reference cell are constructed.Then the effective homogenization parameters of composites are calculated,and the homogenized plate problem on original domain is defined.Based on the Reissner-Mindlin deformation pattern,the homogenization solution is obtained.And then the SOTS’s approximate solution is obtained by the cell functions and the homogenization solution.Second,the approximation of the SOTS’s solution in energy norm is analyzed and the residual of SOTS’s solution for 3-D original in the pointwise sense is investigated.Finally,the procedure of SOTS’s method is given.A set of numerical results are demonstrated for predicting the effective parameters and the displacement and strains of composite plate.It shows that SOTS’s method can capture the 3-D local behaviors caused by3-D micro-structures well.     

Evaluation of mechanical parameters of an elastic thin film system by modeling and numerical simulation of telephone cord buckles

The morphology of telephone cord buckles of elastic thin films can be used to evaluate the initial residual stress and interface toughness of the film-substrate system. The maximum out-of-plane displacement δ, the wavelength λ and amplitude A of the wave buckles can be measured in physical experiments. Through δ, λ, and A, the buckle morphology is obtained by an annular sector model established using the von Karman plate equations in polar coordinates for the elastic thin film. The mode-mix fracture criterion is applied to determine the shape and scale parameters. A numerical algorithm combining the Newmark-β scheme and the Chebyshev collocation method is adopted to numerically solve the problem in a quasi-dynamic process. Numerical experiments show that the numerical results agree well with physical experiments.     

Spatial Regression Models,Response Surfaces,and Process Optimization

Abstract

Spatial regression models are developed as a complementary alternative to second-order polynomial response surfaces in the context of process optimization. These models provide estimates of design variable effects and smooth, data-faithful approximations to the unknown response function over the design space. The predicted response surfaces are driven by the covariance structures of the models. Several structures, isotropic and anisotropic, are considered and connections with thin plate splines are reviewed. Estimation of covariance parameters is achieved via maximum likelihood and residual maximum likelihood. A feature of the spatial regression approach is the visually appealing graphical summaries that are produced. These allow rapid and intuitive identification of process windows on the design space for which the response achieves target performance. Relevant design issues are briefly discussed and spatial designs, such as the packing designs available in Gosset, are suggested as a suitable design complement. The spatial regression models also perform well with no global design, for example with data obtained from series of designs on the same space of design variables. The approach is illustrated with an example involving the optimization of components in a DNA amplification assay. A Monte Carlo comparison of the spatial models with both thin plate splines and second-order polynomial response surfaces for a scenario motivated by the example is also given. This shows superior performance of the spatial models to the second-order polynomials with respect to both prediction over the complete design space and for cross-validation prediction error in the region of the optimum. An anisotropic spatial regression model performs best for a high noise case and both this model and the thin plate spline for a low noise case. Spatial regression is recommended for construction of response surfaces in all process optimization applications.     

An infinite element theory

A technique for solution of plane problems in mathematical physics or mechanics is presented. The method was developed initially for elastic plate problems with transverse load concentrations but is here extended to other two-dimensional problems. Draw-down problems with a number of wells in a porous medium, heat-flow problems in a plate with a distributed source, and with discrete sinks, or membrane displacements under concentrated forces, are considered in particular.The technique is to treat the two-dimensional medium as infinite: each source or sink is treated separately and equilibrated by an auxiliary source function. Superposition allows the effects of the separate sources and sinks to be added. The residual effect of the auxiliary source, after superposition, is negligible so that the summed effects of the individual sources and sinks gives the required solution. The method is approximate but the error may be made arbitrarily small. It is simple and well suited to the class of problems above, and to either manual or computer based analyses. Examples of its use are given.     

Thermal and surface effects on the pull-in characteristics of circular nanoplate NEMS actuator based on nonlocal elasticity theory

This paper aims to investigate the coupling influences of thermal loading and surface effects on pull-in instability of electrically actuated circular nanoplate based on Eringen's nonlocal elasticity theory, where the electrostatic force and thermally corrected Casimir force are considered. By utilizing the Kirchhoff plate theory, the nonlinear equilibrium equation of axisymmetric circular nanoplate with variable coefficients and clamped boundary conditions is derived and analytically solved. The results describe the influences of surface effect and thermal loading on pull-in displacements and pull-in voltages of nanoplate under thermal corrected Casimir force. It is seen that the surface effect becomes significant at the pull-in state with the decrease of nanoplate thicknesses, and the residual surface tension exerts a greater influence on the pull-in behavior compared to the surface elastic modulus. In addition, it is found that temperature change plays a great role in the pull-in phenomenon; when the temperature change grows, the circular nanoplate without applied voltage is also led to collapse.     

Investigation of the Thermoelastoplastic State of Locally Heated Shallow Shells of Double Curvature

We develop a numerical-analytic method for calculating the thermoelastoplastic state of shallow shells of arbitrary curvature under conditions of heating by localized heat sources. For solving auxiliary problems that arise in applying the method of additional deformations, we use the Fourier integral transformation and integral representations of a solution. Detailed investigations were performed for shells of cylindrical, spherical, and ellipsoidal shapes. We establish the dependence of the distribution of thermal and residual stresses on the curvatures of a shell and on the mode of heating. We consider the limiting case where the construction has the shape of a plate.     

Dynamic stability of a porous rectangular plate

The study is devoted to a axial compressed porous-cellular rectangular plate. Mechanical properties of the plate vary across is its thickness which is defined by the non-linear function with dimensionless variable and coefficient of porosity. The material model used in the current paper is as described by Magnucki, Stasiewicz papers. The middle plane of the plate is the symmetry plane. First of all, a displacement field of any cross section of the plane was defined. The geometric and physical (according to Hook's law) relationships are linear. Afterwards, the components of strain and stress states in the plate were found. The Hamilton's principle to the problem of dynamic stability is used. This principle was allowed to formulate a system of five differential equations of dynamic stability of the plate satisfying boundary conditions. This basic system of differential equations was approximately solved with the use of Galerkin's method. The forms of unknown functions were assumed and the system of equations was reduced to a single ordinary differential equation of motion. The critical load determined used numerically processed was solved. Results of solution shown in the Figures for a family of isotropic porous-cellular plates. The effect of porosity on the critical loads is presented. In the particular case of a rectangular plate made of an isotropic homogeneous material, the elasticity coefficients do not depend on the coordinate (thickness direction), giving a classical plate. The results obtained for porous plates are compared to a homogeneous isotropic rectangular plate. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The effect of overloads on the residual strength and life of laminated GRP

The effect of a short duration cyclic overload on the residual life and strength of laminated glass-fiber reinforced polyester is studied. A uniaxial tensile fatigue loading with the stress ratio 0.1 is considered. The residual life of the composite decreases due to the overload, while the residual strength is almost unaffected. A reasonable agreement of experimental data with the prediction by a residual strength model and by Miner's rule is observed.Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 5, pp. 701–706, May–June, 1999.     

A Study on the Propagation of Plane Stress Waves across the Thickness of a Plate by the Method of Analytic Continuation in Time

The interaction of plane tension/compression waves propagating within a plate perpendicularly to its surface is considered. The analytic solution is obtained by a modified method of characteristics for the one-dimensional wave equation used in problems on an impact of a rigid body on the surface of a plate. The displacements, velocities, and stresses in the plate are determined by the edge disturbance caused by the initial velocity and the stationary force field of masses of the striker and the plate. The method of analytic continuation in time put forward allows a stress analysis for an arbitrary time interval by using finite expressions. Contrary to a stress analysis in the frequency domain, which is commonly used in harmonic expansion of disturbances, the approach advanced allows one to analyze the solution in the case of discontinuous first derivatives of displacements without calculating jumps in summing series. A generalized closed-form solution is obtained for stresses in an arbitrary cycle n(t), which is determined by the multiplicity of the time of wave travel across the double thickness of the plate. A method of recurrent solution based on calculating the convolution of repeated integrals of the initial form of disturbance at t = 0 is elaborated. The procedure can be used for evaluating the maximum stress and the contact time in a plane impact on the surface of a plate.     

On the stress distribution in a rectangular thick plate of a composite material with curved structure under forced vibration

The stress distribution in a rectangular plate of a multilayer composite material with a periodically curved structure under forced vibration is studied. It is assumed that the plate is hinge supported at opposite sides. The investigation is carried out within the exact three-dimensional linear theory of elasticity. The mechanical relationships of the plate material are described by the continuum theory of Akbarov and Guz'. The numerical results obtained by the finite element method show that even in low-frequency dynamic loading of the plate the extreme values of stresses, which appear as a result of the curving in the plate structure, considerably exceed those in the corresponding static loading.Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan, Baku, Azerbaijan. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 447–454, July–August, 1999.     

On the three-dimensional undulation instability of a rectangular viscoelastic composite plate in biaxial compression

Within the frame work of the second version of small precritical deformation in the three-dimensional linearized theory of stability of deformable bodies (TDLTSDB), the undulation instability problem for a simply supported rectangular plate made of a viscoelastic composite material is investigated in biaxial compression in the plate plane. The corresponding boundary-value problem is solved by employing the Laplace transformation and the principle of correspondence. For comparison and estimation of the accuracy of results given by the TDLTSDB, the same problem is also solved by using various approximate plate theories. The viscoelasticity properties of the plate material are described by the Rabotnov fractional-exponential operator. The numerical results and their discussion are presented for the case where the plate is made of a multilayered viscoelastic composite material. In particular, the variation range of problem parameters is established for which it is necessary to investigate the undulation instability of the viscoelastic composite plate by using the TDLTSDB. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 1, pp. 93–108, January–February, 2009.