层合板壳脱层曲面的有限元分析

通过建立一类新的参考面有限单元,得到适应于分析层合板壳脱层屈曲问题的有限元方法,指出了利用Ｍｉｎｄｌｉｎ假设意义下的变形协调条件,可以将大多数胜任层合板壳分析的一般板壳单元改造为相应的参考面单元,这一方法确保了位移场的合理性和协调条件满足,为验证参考而单元的有效性和协调还对壳体脱层屈曲的几个算例作了数值分析。     

Micro-vibrations of thin cylindrical shells with uniperiodic structure

Micro-vibrations of thin linear-elastic Kirchhoff-Love cylindrical shells, having a periodic structure along one direction tangent to the shell midsurface, are investigated using the new averaged non-asymptotic model of such shells proposed by Tomczyk (2008). This model describes the effect of microstructure size on the overall shell behaviour and makes it possible to analyze the shell's micro-dynamics independently of its macro-dynamics. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Routes to chaos in continuous mechanical systems. Part 1: Mathematical models and solution methods

In this work chaotic dynamics of continuous mechanical systems such as flexible plates and shallow shells is studied. Namely, a wide class of the mentioned objects is analyzed including flexible plates and cylinder-like panels of infinite length, rectangular spherical and cylindrical shells, closed cylindrical shells, axially symmetric plates, as well as spherical and conical shells. The considered problems are solved by the Bubnov–Galerkin and higher approximation Ritz methods. Convergence and validation of those methods are studied. The Cauchy problems are solved mainly by the fourth Runge-Kutta method, although all variants of the Runge-Kutta methods are considered. New scenarios of transition from regular to chaotic orbits are detected, analyzed and discussed.First part of the paper is devoted to the validation of results obtained. This is why the same infinite length problem is reduced to that of a finite dimension through the FDM (Finite Difference Method) with the approximation order of O(c²), BGM (Bubnov–Galerkin Method) or RM (Ritz Method) with higher approximations. We pay attention not only to convergence of the mentioned methods regarding the number of partitions of the interval [0, 1] in the FDM or regarding the number of terms in the series applied either in the BGM or RM methods, but we also compare the results obtained via the mentioned different approaches. Furthermore, a so called practical convergence of different Runge-Kutta type methods are tested starting from the second and ending with the eighth order.Second part of the work is devoted to a study of routes to chaos in the so far mentioned mechanical objects. For this purpose the so-called “dynamical charts” are constructed versus control parameters {q₀, ω_p}, where q₀ denotes the loading amplitude, and ω_p is the loading frequency. The charts are constructed through analyses of frequency power spectra and the largest Lyapunov exponent (LE). Analysis of the mentioned charts indicates clearly that different routes to chaos exist and allow us to control the objects being investigated. In some cases we detect the classical Feigenbaum scenario and we compute also the Feigenbaum constant. This scenario accompanied all problems which we studied. In addition, we detect and illustrate novel scenarios of transition from regularity into chaos including the Ruelle–Takens–Newhouse–Feigenbaum scenario, and the so called modified Pomeau–Manneville scenario.Third part of the paper is devoted to analysis of the Lyapunov exponents. Namely, while investigating evolutions of vibration regimes of a shell associated with an increase of excitation amplitude q₀ phase transitions chaos–hyper chaos as well as chaos-hyper chaos–hyper–hyper chaos dynamics are illustrated and studied. Furthermore, for all investigated plates and shells the Sharkovskiy windows of periodicity are detected. In particular, a space-temporal chaos/turbulence is studied.     

Geometrically non-linear equations in the theory of momentless shells with applications to problems on the non-classical forms of loss of stability of a cylinder

Geometrically non-linear and linearized equations in the theory of momentless shells are set up based on the kinematic relations in [Paimushin VN, Shalashilin VI. Relations of the theory of deformations in the quadratic approximation and problems of constructing refined versions of the geometrically non-linear theory of laminar structural components. Prikl Mat Mekh 2005; 69(5): 861–81]. The use of these equations, unlike in the case of the well-known equations, enables one to avoid the occurrence of spurious bifurcation points in solving real problems. Non-classical problems of the stability of cylindrical shells under an external pressure, axial compression and torsion are considered, which can be formulated on the basis of the derived equations of the theory of momentless shells. Their exact analytical solutions are found and enable one to estimate the quality of the previously obtained relations [Paimushin VN, Shalashilin VI. Relations of the theory of deformations in the quadratic approximation and problems of constructing refined versions of the geometrically non-linear theory of laminar structural components. Prikl Mat Mekh 2005; 69(5): 861–81] and the richness of content of the equations which have been constructed compared with well-known equations in the mechanics of thin shells. It is established that the majority of the new forms of loss of stability of cylindrical shells which are revealed relate to a number of shear forms, the onset of which is possible before the flexural forms which have been well studied up to now, in the case of small values of the shear modulus of a shell material with a very highly pronounced anisotropy in its properties.     

Perturbation Sensitivity and Limit Loads of Shells

The perturbation sensitivity and its influence on the limit loads of shells are widely discussed phenomena. Both phenomena may be classified with respect to the type of perturbation. As perturbations influence the stability of shells, the identification of unfavourable perturbations is essential. The perturbation energy concept enables to identify unfavourable non–initial perturbation loads and to evaluate the perturbation sensitivity of fundamental states by the perturbation energy. This measure is also the basis for a load–level–specific optimisation of the perturbation sensitivity. Hence, the present paper discusses primarily the perturbation sensitivity of axially loaded cylindrical shells with different boundary conditions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New variational-asymptotic formulations for 3-D stress analyses of laminated composite shells with a circular hole

A new approach for three-dimensional stress analyses in composite cylindrical shells is presented. The method of composite expansions along with Hellinger-Reissner variational formulation is employed to derive the interior and edge layer problems for high order approximations. Classical assumptions have been justified and new approximations have been established. These formulations are directed especially towards, new high integrity mixed-hybrid finite element schemes. The expository examples chosen are of cross-ply and angle-ply laminated shells. The circumferential location of the delamination failure initiation, for angle-ply laminates containing a circular hole, is within a sector located symmetrically around the perpendicular direction to the applied load.     

Decomposition of shell deformations – Asymptotic behavior of the Green–St Venant strain tensor

This Note deals with a new method, based on a decomposition of the deformations, to study thin shells. In particular, we give the asymptotic behavior of the Green–St Venant's strain tensor. To cite this article: D. Blanchard, G. Griso, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

一种新的叠层板壳高阶理论

本文提出了一种新的叠层板壳高阶理论,然后又研究了正交对称叠层板,反对称叠层板,圆柱弯曲和球壳弯曲问题.为了检验理论的准确性,文中计算了几个特殊例子,数值结果和精确解吻合得相当好,说明本理论具有较高的准确度,且表现出未知数较少,解题方便的优点.     

Nonlinear free vibration analysis of SSMFG cylindrical shells resting on nonlinear viscoelastic foundation in thermal environment

In this paper, a semi-analytical method for the free vibration behavior of spiral stiffened multilayer functionally graded (SSMFG) cylindrical shells under the thermal environment is investigated. The distribution of linear and uniform temperature along the direction of thickness is assumed. The structure is embedded within a generalized nonlinear viscoelastic foundation, which is composed of a two-parameter Winkler-Pasternak foundation augmented by a Kelvin-Voigt viscoelastic model with a nonlinear cubic stiffness. The cylindrical shell has three layers consist of ceramic, FGM, and metal in two cases. In the first model i.e. Ceramic-FGM-Metal (CFM), the exterior layer of the cylindrical shell is rich ceramic while the interior layer is rich metal and the functionally graded material is located between these layers and the material distribution is in reverse order in the second model i.e. Metal-FGM-Ceramic (MFC). The material constitutive of the stiffeners is continuously changed through the thickness. Using the Galerkin method based on the von Kármán equations and the smeared stiffeners technique, the problem of nonlinear vibration has been solved. In order to find the nonlinear vibration responses, the fourth order Runge–Kutta method is utilized. The results show that the different angles of stiffeners and nonlinear elastic foundation parameters have a strong effect on the vibration behaviors of the SSMFG cylindrical shells. Also, the results illustrate that the vibration amplitude and the natural frequency for CFM and MFC shells with the first longitudinal and third transversal modes (m = 1, n = 3) with the stiffeners angle θ = 30°, β = 60° and θ = β = 30° is less than and more than others, respectively.     

Numerical simulation of crack–propagation in shells

We develop a finite element method for the simulation of fragmentation of thin shells. The method is valid for completely non–linear problems, but is restricted to through–the–thickness cracks, which are normal to the midsurface. The methodology is based on the extended finite element method (X–FEM). The use of X–FEM allows arbitrary crack path evolution and does not require a priori knowledge of the crack zone or remeshing. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermoelastoplastic deformation of complexly reinforced shells

The problem on the elastoplastic deformation of reinforced shells of variable thickness under thermal and force loadings is formulated. A qualitative analysis of the problem is carried out and its linearization is indicated. Calculations of isotropic and metal composite cylindrical shells have shown that the load-carrying capacity of shell structures under elastoplastic deformations is several times (sometimes by an order of magnitude) higher than under purely elastic ones; the heating of shells with certain patterns of reinforcement sharply reduces their resistance to elastic deformations, but only slightly affects their resistance to elastoplastic ones; not always does the reinforcement in the directions of principal stresses and strains provide the greatest load-carrying capacity of a shell; there are reinforcement schemes that ensure practically the same resistance of shells at different types of their fastening. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 6, pp. 707–728, November–December, 2006.     

A variational principle in a geometrically nonlinear theory of heterogeneous viscoelastic shells

The use of the hereditary theory for shells heterogeneous across their thickness is considered. A variational method is formulated for calculating thin anisotropic shells made of a material whose deformation behavior can be described by relations of the linear theory of viscoelasticity. In order to transform the corresponding functional into a form suitable for shells, some assumptions related to concepts of the theory of thin shells are introduced. In the capacity of Euler equations, physical relations, nonlinear equilibrium equations, and nonlinear boundary conditions are derived. The state equations are deduced for a multilayered shell. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 231–240, March–April, 2009.     

Numerical analysis method for studying local forms of stability loss of bearing layers of three-layered shells using mixed forms

A revised formulation of linearized stability problems of three-layered shells with a sofi filler has been presented. The form of stability loss of the rigid layers is mixed in the shells when the moment precritical stress-strain state (SSS) is reached and is localized in the principal moment SSS zones. If the filler thickness is much greater than the thickness of the rigid layers, the size of the bulges and thickness of the filler have the same order of magnitude. Thus, a very fine grid must be used for a numerical solution of the stability loss equations, which poses considerable computational difficulties. A numerical analysis method is proposed for the local forms of mixed mode stability loss of the rigid layers of a three-layered shell. Using this method, the solution of equations for the precritical SSS by the finite element scheme is found but an analytical solution of reduced stability loss equations is presented for estimating the critical load. This solution is an asymptotic approximation for local modes of stability loss implemented into design.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 1, pp. 88–100, January–February, 1995.     

主动约束阻尼开口柱壳的NLMS反馈减振控制

为缩减开口柱壳结构的振动,给出了一种局部主动约束阻尼(ALCD)敷设结构,并结合Lagrange方程和Sanders薄壳理论构建了压电耦合开口柱壳的动力学模型,根据推得的系统状态空间形式,应用归一化最小均方差自适应滤波算法(NLMS)和线性二次规划算法(LQR)设计了一种自适应反馈控制器,通过数值仿真研究了控制参数对开...     

Computations on thin non‐inhibited hyperbolic elastic shells. Bench‐marks for membrane locking

We study the bending limit problem of shells in relation to the membrane locking, encountered in finite element computation of non‐inhibited very thin shells. Using a new approach of the theory of inextensional displacements (or infinitesimal bendings) we solve the bending limit problem in the case of a clamped hyperbolic paraboloid. We then use this solution to validate computations which can be used as bench‐marks for the membrane locking. Such configuration, non‐inhibited hyperbolic very thin shells, usually lacks numerical ‘validation’. Copyright © 1999 John Wiley & Sons, Ltd.     

High order explicit Runge–Kutta Nyström pairs

Explicit Runge–Kutta Nyström pairs provide an efficient way to find numerical solutions to second-order initial value problems when the derivative is cheap to evaluate. We present new optimal pairs of orders ten and twelve from existing families of pairs that are intended for accurate integrations in double precision arithmetic. We also present a summary of numerical comparisons between the new pairs on a set of eight problems which includes realistic models of the Solar System. Our searching for new order twelve pairs shows that there is often not quantitative agreement between the size of the principal error coefficients and the efficiency of the pairs for the tolerances we are interested in. Our numerical comparisons, as well as establishing the efficiency of the new pairs, show that the order ten pairs are more efficient than the order twelve pairs on some problems, even at limiting precision in double precision.     

Optimized composite finite difference schemes for atmospheric flow modeling

In this paper, we use some finite difference methods in order to solve an atmospheric flow problem described by an advection–diffusion equation. This flow problem was solved by Clancy using forward‐time central space (FTCS) scheme and is challenging to simulate due to large errors in phase and amplitude which are generated especially over long propagation times. Clancy also derived stability limits for FTCS scheme. We use Von Neumann stability analysis and the approach of Hindmarsch et al. which is an improved technique over that of Clancy in order to obtain the region of stability of some methods such as FTCS, Lax–Wendroff (LW), Crank–Nicolson. We also construct a nonstandard finite difference (NSFD) scheme. Properties like stability and consistency are studied. To improve the results due to significant numerical dispersion or numerical dissipation, we derive a new composite scheme consisting of three applications of LW followed by one application of NSFD. The latter acts like a filter to remove the dispersive oscillations from LW. We further improve the composite scheme by computing the optimal temporal step size at a given spatial step size using two techniques namely; by minimizing the square of dispersion error and by minimizing the sum of squares of dispersion and dissipation errors.     

A Nonlinear Sandwich Shell Theory Accounting for Transverse Core Compressibility

The present study is concerned with an advanced theory of sandwich shells with transversely compressible core. The model is based on the standard Kirchhoff‐Love hypothesis for the face sheets and a third‐order displacement expansion for the core. Consistent equations of motion and boundary conditions are derived by means of Hamilton's variational principle. The model is applied to a postbuckling analysis of cylindrical shells under axial compression.     

Deformation of a system of cylindrical shells under the action of an internal nonstationary pressure wave

We study the problem of nonstationary deformation of a system of imbedded cylindrical shells under the action of an internal compression wave. The rigorous method of solution we apply makes it possible to obtain a system of Volterra integral equations with retarded arguments in the deflections of the shells. We give the results of computations for the case of a linear source and a source of finite size with limiting hardness properties. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 60–66.     

A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells

Nonlinear vibration analysis of circular cylindrical shells has received considerable attention from researchers for many decades. Analytical approaches developed to solve such problem, even not involved simplifying assumptions, are still far from sufficiency, and an efficient numerical scheme capable of solving the problem is worthy of development. The present article aims at devising a novel numerical solution strategy to describe the nonlinear free and forced vibrations of cylindrical shells. For this purpose, the energy functional of the structure is derived based on the first-order shear deformation theory and the von–Kármán geometric nonlinearity. The governing equations are discretized employing the generalized differential quadrature (GDQ) method and periodic differential operators along axial and circumferential directions, respectively. Then, based on Hamilton's principle and by the use of variational differential quadrature (VDQ) method, the discretized nonlinear governing equations are obtained. Finally, a time periodic discretization is performed and the frequency response of the cylindrical shell with different boundary conditions is determined by applying the pseudo-arc length continuation method. After revealing the efficiency and accuracy of the proposed numerical approach, comprehensive results are presented to study the influences of the model parameters such as thickness-to-radius, length-to-radius ratios and boundary conditions on the nonlinear vibration behavior of the cylindrical shells. The results indicate that variation of fundamental vibrational mode shape significantly affects frequency response curves of cylindrical shells.