Novel shear deformation model for moderately thick and deep laminated composite conoidal shell

This article presents a novel mathematical model for moderately thick and deep laminated composite conoidal shell. The zero transverse shear stress at top and bottom of conoidal shell conditions is applied. Novelty in the present formulation is the inclusion of curvature effect in displacement field and cross curvature effect in strain field. This present model is suitable for deep and moderately thick conoidal shell. The peculiarity in the conoidal shell is that due to its complex geometry, its peak value of transverse deflection is not at its center like other shells. The C¹ continuity requirement associated with the present model has been suitably circumvented. A nine-node curved quadratic isoparametric element with seven nodal unknowns per node is used in finite element formulation of the proposed mathematical model. The present model results are compared with experimental, elasticity, and numerical results available in the literature. This is the first effort to solve the problem of moderately thick and deep laminated composite conoidal shell using parabolic transverse shear strain deformation across the thickness of conoidal shell. Many new numerical problems are solved for the static study of moderately thick and deep laminated composite conoidal shell considering 10 different practical boundary conditions, four types of loadings, six different hl/hh (minimum rise/maximum rise) ratios, and four different laminations.     

Three-dimensional free vibration analysis of rotating laminated conical shells: layerwise differential quadrature (LW-DQ) method

This paper focuses on the free vibration analysis of thick, rotating laminated composite conical shells with different boundary conditions based on the three-dimensional theory, using the layerwise differential quadrature method (LW-DQM). The equations of motion are derived applying the Hamilton’s principle. In order to accurately account for the thickness effects, the layerwise theory is used to discretize the equations of motion and the related boundary conditions through the thickness of the shells. Then, the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equation applying the DQM in the meridional direction. This study demonstrates the applicability, accuracy, stability and the fast rate of convergence of the present method, for free vibration analyses of rotating thick laminated conical shells. The presented results are compared with those of other shell theories obtained using conventional methods and a special case where the angle of the conical shell approaches zero, that is, a cylindrical shell and excellent agreements are achieved.     

Static analysis of functionally graded doubly-curved shells and panels of revolution

The Generalized Differential Quadrature (GDQ) Method is applied to study four parameter functionally graded and laminated composite shells and panels of revolution. The mechanical model is based on the so-called First-order Shear Deformation Theory (FSDT), in particular on the Toorani-Lakis Theory. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. The generalized strains and stress resultants are evaluated by applying the Differential Quadrature rule to the generalized displacements. The transverse shear and normal stress profiles through the thickness are reconstructed a posteriori by using local three-dimensional elasticity equilibrium equations. In order to verify the accuracy of the present method, GDQ results are compared with the ones obtained with semi-analytical formulations and with 3D finite element method. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of functionally graded shell structures made of a mixture of ceramics and metal.     

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.     

一种求解层合圆柱壳的杂交应力单元

根据修正的余能原理，推导出一种求解复合材料层合圆柱壳的杂交应力单元。取用六面体等参单元，此单元反映了各层材料性质不同及应力分布沿整个厚度不连续现象，同时计入横向剪切变形和法向挤压变形，适用于厚层壳体。文章通过实例说明此单元能准确求出各层内的应力值，实用价值高。     

Axisymmetric thermoviscoelastoplastic state of thin laminated shells made of a damageable material

A technique for the determination of the axisymmetric thermoviscoelastoplastic state of laminated thin shells made of a damageable material is developed. The technique is based on the kinematic equations of the theory of thin shells that account for transverse shear strains. The thermoviscoplastic equations, which describe the deformation of a shell element along paths of small curvature, are used as the constitutive equations. The equivalent stress that appears in the kinetic equations of damage and creep is determined from a failure criterion that accounts for the stress mode. The thermoviscoplastic deformation of a two-layer shell that models an element of a rocket engine nozzle is considered as an example __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 87–100, April 2008.     

Drilling couples and refined constitutive equations in the resultant geometrically non-linear theory of elastic shells

It is well known that distribution of displacements through the shell thickness is non-linear, in general. We introduce a modified polar decomposition of shell deformation gradient and a vector of deviation from the linear displacement distribution. When strains are assumed to be small, this allows one to propose an explicit definition of the drilling couples which is proportional to tangential components of the deviation vector. The consistent second approximation to the complementary energy density of the geometrically non-linear theory of isotropic elastic shells is constructed. From differentiation of the density we obtain the consistently refined constitutive equations for 2D surface stretch and bending measures. These equations are then inverted for 2D stress resultants and stress couples. The second-order terms in these constitutive equations take consistent account of influence of undeformed midsurface curvatures. The drilling couples are explicitly expressed by the stress couples, undeformed midsurface curvatures, and amplitudes of quadratic part of displacement distribution through the thickness. The drilling couples are shown to be much smaller than the stress couples, and their influence on the stress and strain state of the shell is negligible. However, such very small drilling couples have to be admitted in non-linear analyses of irregular multi-shell structures, e.g. shells with branches, intersections, or technological junctions. In such shell problems six 2D couple resultants are required to preserve the structure of the resultant shell theory at the junctions during entire deformation process.     

A hybrid stress approach for laminated composite plates within the First-order Shear Deformation Theory

This paper presents a hybrid stress approach for the analysis of laminated composite plates. The plate mechanical model is based on the so called First-order Shear Deformation Theory, rationally deduced from the parent three-dimensional theory. Within this framework, a new quadrilateral four-node finite element is developed from a hybrid stress formulation involving, as primary variables, compatible displacements and elementwise equilibrated stress resultants. The element is designed to be simple, stable and locking-free. The displacement interpolation is enhanced by linking the transverse displacement to the nodal rotations and a suitable approximation for stress resultants is selected, ruled by the minimum number of parameters. The transverse stresses through the laminate thickness are reconstructed a posteriori by simply using three-dimensional equilibrium. To improve the results, the stress resultants entering the reconstruction process are first recovered using a superconvergent patch-based procedure called Recovery by Compatibility in Patches, that is properly extended here for laminated plates. This preliminary recovery is very efficient from the computational point of view and generally useful either to accurately evaluate the stress resultants or to estimate the discretization error. Indeed, in the present context, it plays also a key role in effectively predicting the shear stress profiles, since it guarantees the global convergence of the whole reconstruction strategy, that does not need any correction to accommodate equilibrium defects. Actually, this strategy can be adopted together with any plate finite element. Numerical testing demonstrates the excellent performance of both the finite element and the reconstruction strategy.     

层合闭口厚柱壳的温度应力

基于层合柱壳混合状态方程和边界条件的弱形式，建立了两端固支层合闭口柱壳的温度应力混合方程，给出了任意厚度合闭口柱壳在温度荷载和机械荷载共同作用下的解析解。     

A unified Chebyshev–Ritz formulation for vibration analysis of composite laminated deep open shells with arbitrary boundary conditions

In this paper, a unified Chebyshev–Ritz formulation is presented to investigate the vibrations of composite laminated deep open shells with various shell curvatures and arbitrary restraints, including cylindrical, conical and spherical ones. The general first-order shear deformation shell theory is employed to include the effects of rotary inertias and shear deformation. Under the current framework, regardless of boundary conditions, each of displacements and rotations of the open shells is invariantly expressed as Chebyshev orthogonal polynomials of first kind in both directions. Then, the accurate solutions are obtained by using the Rayleigh–Ritz procedure based on the energy functional of the open shells. The convergence and accuracy of the present formulation are verified by a considerable number of convergence tests and comparisons. A variety of numerical examples are presented for the vibrations of the composite laminated deep shells with various geometric dimensions and lamination schemes. Different sets of classical constraints, elastic supports as well as their combinations are considered. These results may serve as reference data for future researches. Parametric studies are also undertaken, giving insight into the effects of elastic restraint parameters, fiber orientation, layer number, subtended angle as well as conical angle on the vibration frequencies of the composite open shells.     

杂交应力通用壳元的模式优化

文章把杂交元的优化设计原理用于厚薄壳问题,构造了正交曲线坐标系中的Mindlin型四节点二十自由度壳元。该元通用于各种厚度、任意形状的深壳和扁壳,构造简单,精度较高且没有自锁现象。     

General non-linear finite element analysis of thick plates and shells

A non-linear finite element analysis is presented, for the elasto-plastic behavior of thick shells and plates including the effect of large rotations. The shell constitutive equations developed previously by the authors [Voyiadjis, G.Z., Woelke, P., 2004. A refined theory for thick spherical shells. Int. J. Solids Struct. 41, 3747–3769] are adopted here as a base for the formulation. A simple C⁰ quadrilateral, doubly curved shell element developed in the authors’ previous paper [Woelke, P., Voyiadjis, G.Z., submitted for publication. Shell element based on the refined theory for thick spherical shells] is extended here to account for geometric and material non-linearities. The small strain geometric non-linearities are taken into account by means of the updated Lagrangian method. In the treatment of material non-linearities the authors adopt: (i) a non-layered approach and a plastic node method [Ueda, Y., Yao, T., 1982. The plastic node method of plastic analysis. Comput. Methods Appl. Mech. Eng. 34, 1089–1104], (ii) an Iliushin’s yield function expressed in terms of stress resultants and stress couples [Iliushin, A.A., 1956. Plastichnost’. Gostekhizdat, Moscow], modified to investigate the development of plastic deformations across the thickness, as well as the influence of the transverse shear forces on plastic behaviour of plates and shells, (iii) isotropic and kinematic hardening rules with the latter derived on the basis of the Armstrong and Frederick evolution equation of backstress [Armstrong, P.J., Frederick, C.O., 1966. A mathematical representation of the multiaxial Bauschinger effect. (CEGB Report RD/B/N/731). Berkeley Laboratories. R&D Department, California.], and reproducing the Bauschinger effect. By means of a quasi-conforming technique, shear and membrane locking are prevented and the tangent stiffness matrix is given explicitly, i.e., no numerical integration is employed. This makes the current formulation not only mathematically consistent and accurate for a variety of applications, but also computationally extremely efficient and attractive.     

Analytical solution for axisymmetric problem of thick laminated closed cantilever shells in hamilton system

By giving up any assumptions about displacement models and stress distribution, the mixed state Hamilton equation for the axisymmetric problem of the thick laminated closed cantilever cylindrical shells is established. An identical analytical solution is obtained for the thin, moderately thick and thick laminated closed cantilever cylindrical shells. All equations of elasticity can be satisfied, and all elastic constants can be taken into account. This work is supported by the National Natural Science Foundation of China.     

复合材料旋转壳自由振动分析的新方法

提出了一种半解析区域分解法来分析任意边界条件的复合材料层合旋转壳自由振动. 沿壳体旋转轴线将壳体分解为一些自由的层合壳段, 视位移边界界面为一种特殊的分区界面;采用分区广义变分和最小二乘加权残值法将壳体所有分区界面上的位移协调方程引入到壳体的能量泛函中, 使层合壳的振动分析问题归结为无约束泛函变分问题. 层合壳段位移变量采用Fourier 级数和Chebyshev 多项式展开. 以不同边界条件的层合圆柱壳、圆锥壳及球壳为例, 采用区域分解法分析了其自由振动, 并将计算结果与其他文献值进行了对比. 算例表明, 该方法具有高效率、高精度和收敛性好等优点.     

复合材料结构几何非线性分析

本文采用退化的等参壳元分析了复合材料板、壳结构的几何非线性特性。数值计算表明：与各向同性材料不同,即使在很小的载荷和挠度下,几何非线性对复合材料结构的影响也是相当显著的。     

Buckling and Postbuckling of Laminated Thin Cylindrical Shells under Hygrothermal Environments

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Influences of large deformation and rotary inertia on wave propagation in piezoelectric cylindrically laminated shells in thermal environment

Influences of large deformation (geometrical non-linear) and rotary inertia on wave propagation in a long, piezoelectric cylindrically laminated shell in thermal environment is presented in this paper. Nonlinear dynamic governing equations of piezoelectric cylindrically laminated shells are derived by means of Hamilton’s principle. The wave propagation modes are obtained by solving an eigenvalue problem. Numerical examples show that the characteristics of wave propagation in piezoelectric cylindrically laminated shells are relates to the large deformation, rotary inertia and thermal environment of the piezoelectric cylindrically laminated shells. The effect of large deformation, rotary inertia and thermal load on wave propagation in the piezoelectric cylindrically laminated shells is discussed by comparing with the result from the small deformation (geometrical linear shell theory). This method may be used to investigate wave propagation in various laminated material, layers numbers and thickness of piezoelectric cylindrically laminated shells under large deformation. The results carried out can be used in the ultrasonic inspection techniques and structural health monitoring.     

Geometrically non-linear transient analysis of laminated,doubly curved shells

A dynamic, shear deformation theory of a doubly curved shell is used to develop a finite element for geometrically non-linear (in the von Karman sense) transient analysis of laminated composite shells. The element is employed to determine the transient response of spherical and cylindrical shells with various boundary conditions and loading. The effect of shear deformation and geometric non-linearity on the transient response is investigated. The numerical results presented here for transient analysis of laminated composite shells should serve as references for future investigations.     

Non-linear dynamic thermo-mechanical buckling analysis of the imperfect laminated and sandwich cylindrical shells based on a global-local theory inherently suitable for non-linear analyses

The available accurate shell theories satisfy the interlaminar transverse stress continuity conditions based on linear strain-displacement relations. Furthermore, in majority of these theories, either influence of the transverse normal stress and strain or the transverse flexibility of the shell has been ignored. These effects remarkably influence the non-linear behavior of the shells especially in the postbuckling region. Furthermore, majority of the buckling analyses performed so far for the laminated composite and sandwich shells have been restricted to linear, static analysis of the perfect shells. Moreover, almost all the available shell theories have employed the Love-Timoshenko assumption, which may lead to remarkable errors for thick and relatively thick shells. In the present paper, a novel three-dimensional high-order global-local theory that satisfies all the kinematic and the interlaminar stress continuity conditions at the layer interfaces is developed for imperfect cylindrical shells subjected to thermo-mechanical loads.In comparison with the layerwise, mixed, and available global-local theories, the present theory has the advantages of: (1) suitability for non-linear analyses, (2) higher accuracy due to satisfying the complete interlaminar kinematic and transverse stress continuity conditions, considering the transverse flexibility, and releasing the Love-Timoshenko assumption, (3) less required computational time due to using the global-local technique and matrix formulations, and (4) capability of investigating the local phenomena. To enhance the accuracy of the results, compatible Hermitian quadrilateral elements are employed. The buckling loads are determined based on a criterion previously published by the author.     

Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

The generalized differential quadrature method (GDQM) is employed to consider the free vibration and critical speed of moderately thick rotating laminated composite conical shells with different boundary conditions developed from the first-order shear deformation theory (FSDT). The equations of motion are obtained applying Hamilton’s concept, which contain the influence of the centrifugal force, the Coriolis acceleration, and the preliminary hoop stress. In addition, the axial load is applied to the conical shell as a ratio of the global critical buckling load. The governing partial differential equations are given in the expressions of five components of displacement related to the points lying on the reference surface of the shell. Afterward, the governing differential equations are converted into a group of algebraic equations by using the GDQM. The outcomes are achieved considering the effects of stacking sequences, thickness of the shell, rotating velocities, half-vertex cone angle, and boundary conditions. Furthermore, the outcomes indicate that the rate of the convergence of frequencies is swift, and the numerical technique is superior stable. Three comparisons between the selected outcomes and those of other research are accomplished, and excellent agreement is achieved.