Numerical stability for a dynamic Naghdi shell

We consider an elastic model for a shell incorporating shear, membrane, bending and dynamic effects. We make use of the theory proposed by Arnold and Brezzi [1] based on a locking free non-standard mixed variational formulation. This method is implemented in terms of the displacement and rotation variables as the minimization of an altered energy functional. We extend this theory to the shell vibrations problem and establish optimal error estimates independent of the thickness, thereby proving that shear and membrane locking is avoided. We study the numerical stability both in static and dynamic regimes. The approximation schemes are tested on specific examples and the numerical results confirm the estimates obtained from theory.     

Dispersion models for annular climbing film flow

The mathematical theory of dispersion in annular climbing film flow is developed. Starting with dispersion in a uniform film, the theory is extended to incorporate successively the effects of a viscous sublayer, disturbance waves and interchange of material with entrained droplet. These effects are considered independently but their combined influence on the overall dispersion characteristics of the system is shown to be capable of analysis in terms of an interchange dispersion model (IDM). A solution method for this interchange model is given which may be used to obtain values for the dispersion parameter, P_f, and an ion fractionation coefficient, _f, by non-linear regression on experimental concentration distributions. Values for the dispersion parameter so obtained can be used to give an induction of the viscous layer thickness as well as other film characteristics.     

Theory of vibration-damping polymeric coatings

Conclusions 1. An approximate method for calculating the characteristics of vibration-damping polymeric coatings of complex construction, based on equations of the theory of multilayer plates and shells, has been developed.2. The dimensionless parameters of the theory of multilayer plates and shells, g and h/r, are the basic quantities which determine the effectiveness of a damping coating.3. An increase in number of layers in the coating, while preserving the overall thickness of the packet and an assigned ratio between the thicknesses of the polymeric and rigid layers, exerts only a small effect on the effectiveness of the coating.Moscow Power Institute. Translated from Mekhanika Polimerov, No. 2, pp. 269–276, March–April, 1978.     

变厚度夹层截顶扁锥壳的非线性稳定性分析

对具有变厚度夹层截顶扁锥壳的非线性稳定问题进行了研究。利用变分原理导出表层为等厚度而夹心为变厚度的夹层截顶扁锥壳的非线性稳定问题的控制方程和边界条件,采用修正迭代法求得了具有双曲型变厚度夹层截顶扁锥壳的非线性稳定性问题的解析解,得到了内边缘与一刚性中心固结而外边缘为可移夹紧固支的变厚度夹层截顶扁锥壳临界屈曲载荷的解析表达式,讨论了几何参数和物理参数对壳体屈曲行为的影响。     

Errors for a shear model in studying stress concentrations

The possibility is studied of using revised theories for calculating stress concentration in transtropic and isotropic plates. It has been shown that in studying stress concentration the error for revised theory is of the same order as that for Kirchhoff theory. The first gives results with an excess, but the second gives results with a deficit. It has been noted that for transtropic plates of average thickness acceptable accuracy for engineering may be obtained by an energy-symptotic method. However, the nonlinear nature of stress distribution through the thickness obtained by three-dimensional theory cannot be revealed by the energy-asymptotic method.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 20–25, 1987.     

Bending analysis of size-dependent functionally graded annular sector microplates based on the modified couple stress theory

In the present paper, a non-classical model for functionally graded annular sector microplates under distributed transverse loading is developed based on the modified couple stress theory and the first-order shear deformation plate theory. The model contains a single material length scale parameter which can capture the size effect. The material properties are graded through the thickness of plates according to a power-law distribution of the volume fraction of the constituents. The equilibrium equations and boundary conditions are simultaneously derived from the principle of minimum total potential energy. The system of equilibrium equations is then solved using the generalized differential quadrature method. The effects of length scale parameter, power-law index and geometrical parameters on the bending response of annular sector plates subjected to distributed transverse loading are investigated.     

Thermoelasticity of a regularly inhomogeneous curved layer with wavy surfaces

A curved inhomogeneous anisotropic layer of variable thickness is considered that has wavy surfaces. It is assumed that the elastic, thermo-physical characteristics of the layer material and the shape of its upper and lower surfaces are periodic in structure with a single periodicity cell (PC). The period of the structure is here comparable in magnitude with the layer thickness, which is assumed to be much less than the other linear dimensions of the body and the radius of curvature of its middle surface.On the basis of a general scheme for taking the average of processes in periodic media /1, 2/, a method is developed which enables a transition to be made from a spatial quasistatic thermoelasticity problem to a system of thermoelasticity equations for an average shell whose effective and thermophysical coefficients are determined from the solution of local problems in a PC. Results obtained for the static theory of elasticity in /3/ are used. The heat conduction problem is averaged to determine the temperature components occurring in the equation of motion.The model constructed enables thermoelastic strains, stresses and the temperature distribution to be obtained in shells and plates of composite or porous materials with a different kind of reinforcement of the periodic structure (waffle, ribbed, corrugated) in reinforced and grid-like shells and plates. In the limiting case of “smooth” surfaces and a homogeneous material, the thermoelasticity equations are obtained for shallow anisotropic shells and the heat conduction equations of anisotropic shells assuming a linear temperature distribution law over the thickness.     

The Axisymmetric Deformation of a Thin,or Moderately Thick,Elastic Spherical Cap

A refined shell theory is developed for the elastostatics of a moderately thick spherical cap in axisymmetric deformation. This is a two-term asymptotic theory, valid as the dimensionless shell thickness tends to zero.The theory is more accurate than “thin shell” theory, but is still much more tractable than the full three-dimensional theory. A fundamental difficulty encountered in the formulation of shell (and plate) theories is the determination of correct two-dimensional boundary conditions, applicable to the shell solution, from edge data prescribed for the three-dimensional problem. A major contribution of this article is the derivation of such boundary conditions for our refined theory of the spherical cap. These conditions are more difficult to obtain than those already known for the semi-infinite cylindrical shell, since they depend on the cap angle as well as the dimensionless thickness. For the stress boundary value problem, we find that a Saint-Venant-type principle does not apply in the refined theory, although it does hold in thin shell theory. We also obtain correct boundary conditions for pure displacement and mixed boundary data. In these cases, conventional formulations do not generally provide even the first approximation solution correctly. As an illustration of the refined theory, we obtain two-term asymptotic solutions to two problems, (i) a complete spherical shell subjected to a normally directed equatorial line loading and (ii) an unloaded spherical cap rotating about its axis of symmetry.     

A finite element approach for modelling hip joint contact

In this contribution a finite element model for the three dimensional investigation of hip joint contact is described. A shell-like interface element with variable thickness is developed for modelling fluid flow in the synovial gap. For this purpose the Taylor-Hood element is extended in order to take a spatial thickness distribution and local thickness changes into account. The interaction between the synovial fluid and the cartilage layers is solved by a staggered iteration using an artificial compressibility method. Cartilage is modelled using the theory of porous media and three dimensional geometries are reconstructed from medical imaging data. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new approach to the cascade theory

The cascade theory of cosmic rays is investigated from a new point of view; we deal with the number of particles produced in a certain thickness of matter, the energy of each particle being greater than E at the point of its production. This new approach leads to an elegant asymptotic theory for large thicknesses and it is better adapted to the interpretation of experiments in nuclear emulsions.     

An analytic method of studying the nonlinearly elastic equilibrium of a rectangular plate of variable thickness under bending

We propose an approximate analytic method of solving three-dimensional boundary-value problems of the physically nonlinear theory of elasticity for thick rectangular plates of variable thickness subject to a transverse load. The method is used to seek a solution of this problem in the form of double power series in small dimensionless parameters. In arbitrary approximation the original problem is reduced to a sequence of linear inhomogeneous boundary-value problems for plates of constant thickness. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 36–40     

中厚度复合材料夹芯层板变分渐近精细模型

为准确预测对中厚度复合材料夹芯层板分层开裂至关重要的沿厚向应力/应变分布,利用板固有小参数将原三维板分析严格拆分为沿厚向的一维分析和二维板非线性分析,并将原三维能量渐近扩展为系列二维近似能量泛函;通过对近似能量泛函中主导变分项(含翘曲项)的渐近修正,得到与原三维模型尽可能接近的近似能量,从而构建无需任何场变量假设的精细模型,并转换为工程常用的Reissner模型形式．通过4层复合材料夹芯板柱形弯曲算例表明:基于所构建模型重构的三维场精度较一阶剪切变形理论和经典层合理论更好,与精确解基本一致;由于所构建的变分渐近模型为等效单层板模型,在保证足够精度的前提下,相比三维有限元计算可减少2~3阶计算量,在精确性和有效性间取得较好的折衷．     

Numerical Solution of the Radiative Transfer Equation in Spherical Shells

A method of solving numerically problems of radiative transferin shells with spherical symmetry is proposed. The differenceequations are derived by discrete ordinate methods and solvedusing algorithms due to Grant & Hunt (1968, 1969). We haveshown that the method is stable, and will yield non-negativesolutions, provided the optical thickness of an elementary shelland the ratio of its thickness to the radius satisfy certainrestrictions. Two simple problems are presented to illustratethe theory.     

Minimal energy configurations of strained multi-layers

We derive an effective plate theory for internally stressed thin elastic layers as are used, e.g., in the fabrication of nano- and microscrolls. The shape of the energy minimizers of the effective energy functional is investigated without a priori assumptions on the geometry. For configurations in two dimensions (corresponding to Euler-Bernoulli theory) we also take into account a non-interpenetration condition for films of small but non-vanishing thickness.      

Analysis of functionally graded plates using higher order shear deformation theory

This work addresses a static analysis of functionally graded material (FGM) plates using higher order shear deformation theory. In the theory the transverse shear stresses are represented as quadratic through the thickness and hence it requires no shear correction factor. The material property gradient is assumed to vary in the thickness direction. Mori and Tanaka theory (1973) [1] is used to represent the material property of FGM plate at any point. The thermal gradient across the plate thickness is represented accurately by utilizing the thermal properties of the constituent materials. Results have been obtained by employing a C° continuous isoparametric Lagrangian finite element with seven degrees of freedom for each node. The convergence and comparison studies are presented and effects of the different material composition and the plate geometry (side-thickness, side–side) on deflection and temperature are investigated. Effect of skew angle on deflection and axial stress of the plate is also studied. Effects of material constant n on deflection and the temperature distribution are also discussed in detail.     

A minimalist two-level foundation for constructive mathematics

We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin.One level is given by an intensional type theory, called Minimal type theory. This theory extends a previous version with collections.The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model.This two-level theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the “proofs-as-programs” paradigm and acts as a programming language.     

Surface waves supported by thin-film/substrate interactions

A systematic approximation to the linear equations for small-amplitudesurface waves in an elastic half space, interacting with a residuallystressed thin film of different material bonded to its planeboundary, is developed in powers of the film thickness, assumingthe latter to be small compared to the wavelength of the disturbance.The theory is illustrated by calculating asymptotic expansionsof the wave speeds for Love and Rayleigh waves valid to secondorder in the dimensionless film thickness for a transverselyisotropic film bonded to an isotropic substrate.     

Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity

In this paper, to consider all surface effects including surface elasticity, surface stress, and surface density, on the nonlinear free vibration analysis of simply-supported functionally graded Euler–Bernoulli nanobeams using nonlocal elasticity theory, the balance conditions between FG nanobeam bulk and its surfaces are considered to be satisfied assuming a cubic variation for the component of the normal stress through the FG nanobeam thickness. The nonlinear governing equation includes the von Kármán geometric nonlinearity and the material properties change continuously through the thickness of the FG nanobeam according to a power-law distribution of the volume fraction of the constituents. The multiple scale method is employed as an analytical solution for the nonlinear governing equation to obtain the nonlinear natural frequencies of FG nanobeams. The effect of the gradient index, the nanobeam length, thickness to length ratio, mode number, amplitude of deflection to radius of gyration ratio and nonlocal parameter on the frequency ratios of FG nanobeams is investigated.     

Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory

Closed-form solutions for free vibration analysis of orthotropic plates are obtained in this paper based on two variable refined plate theory. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the natural frequency of orthotropic plates are investigated and discussed in detail.