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.     

基于SE(3)群局部标架的5/6 Dofs CB壳单元

基于李群局部标架(local frame of Lie group,LFLG)的多柔体系统动力学建模方法可自然消除刚体运动带来的几何非线性,使系统的广义弹性力、广义惯性力及其雅可比矩阵满足刚体运动的不变性.本文融合李群局部标架思想和基于连续体(continuum-based,CB)的壳理论,提出基于SE(3)群局部标架...     

功能梯度截顶圆锥壳的热弹性弯曲精确解

研究了功能梯度材料截顶圆锥壳在横向机械载荷与非均匀热载荷同时作用下的变形问题. 基于经典线性壳体理论推导出了以横向剪力和中面转角为基本未知量的功能梯度薄圆锥壳轴对称变形的混合型控制方程. 假设功能梯度圆锥壳的材料性质为沿厚度方向按照幂函数连续变化的形式. 然后采用解析方法求解,得到了问题的精确解. 分别就两端简支和两端固支边界条件,给出了圆锥壳的变形随其载荷、材料参数等变化的特征关系曲线,重点分析和讨论了载荷参数与材料梯度变化参数对变形的影响.      

GENERALIZATION OF KIRCHHOFF KINETIC ANALOGY TO THIN ELASTIC SHELLS 1)

将弹性细杆的"Kirchhoff动力学比拟"方法推广到弹性薄壳,使弹性薄壳的变形在物理概念上和刚体的运动对应, 在数学表述上等同,从而可以用刚体动力学的理论和方法研究弹性薄壳的变形,为连续的弹性薄壳提供新的离散化方法. 在直法线假设下,在弹性中面上构筑空间正交轴系, 此轴系沿坐标线"运动"的角速度构成两自变量的弯扭度. 沿两个坐标线的弯扭度表达了弹性薄壳的变形和位形,证明了弯扭度之间以及弯扭度与中面切矢间的相容关系. 用Euler角和Lam$\acute{e}$系数表达了非完整约束和中面位形的微分方程,用弯扭度和Lam$\acute{e}$系数表达了应变和应力以及内力及其本构方程.导出了用分布内力集度表达的弹性薄壳在变形后位形上的平衡偏微分方程组,方程的形式与刚体动力学的Euler方程和弹性细杆的Kirchhoff方程具有相似性,实现了Kirchhoff动力学比拟对弹性薄壳的推广.总结了弹性薄壳静力学和刚体动力学以及弹性细杆静力学在概念上的比拟关系.最后给出了一个算例. 为研究弹性薄壳的变形和运动提供新的建模方法和研究思路.也可进一步推广到弹性薄壳动力学.     

Kirchhoff动力学比拟对弹性薄壳的推广

将弹性细杆的"Kirchhoff动力学比拟"方法推广到弹性薄壳,使弹性薄壳的变形在物理概念上和刚体的运动对应,在数学表述上等同,从而可以用刚体动力学的理论和方法研究弹性薄壳的变形,为连续的弹性薄壳提供新的离散化方法.在直法线假设下,在弹性中面上构筑空间正交轴系,此轴系沿坐标线"运动"的角速度构成两自变量的弯扭度.沿两个...     

Phenomenological invariants and their application to geometrically nonlinear formulation of triangular finite elements of shear deformable shells

A phenomenological definition of classical invariants of strain and stress tensors is considered. Based on this definition, the strain and stress invariants of a shell obeying the assumptions of the Reissner–Mindlin plate theory are determined using only three normal components of the corresponding tensors associated with three independent directions at the shell middle surface. The relations obtained for the invariants are employed to formulate a 15-dof curved triangular finite element for geometrically nonlinear analysis of thin and moderately thick elastic transversely isotropic shells undergoing arbitrarily large displacements and rotations. The question of improving nonlinear capabilities of the finite element without increasing the number of degrees of freedom is solved by assuming that the element sides are extensible planar nearly circular arcs. The shear locking is eliminated by approximating the curvature changes and transverse shear strains based on the solution of the Timoshenko beam equations. The performance of the finite element is studied using geometrically linear and nonlinear benchmark problems of plates and shells.     

Determining the axisymmetric elastoplastic state of thin shells with allowance for the third invariant of the stress deviator

A technique to determine the axisymmetric elastoplastic state of thin shells with allowance for the third invariant of the stress deviator is developed. The technique is based on the theory of thin shells that takes into account transverse shear and torsional strains. Plastic equations that relate the components of the stress tensor in Eulerian coordinates with the linear components of the finite-strain tensor are used as constitutive equations. The nonlinear scalar functions in the constitutive equations are found from base tests on tubular specimens under proportional loading for different stress modes. The boundary-value problem is solved by numerically integrating a system of ordinary differential equations     

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.     

Buckling of thick cylindrical shells under external pressure: A new analytical expression for the critical load and comparison with elasticity solutions

In this paper a set of stability equations for thick cylindrical shells is derived and solved analytically. The set is obtained by integration of the differential stability equations across the thickness of the shell. The effects of transverse shear and the non-linear variation of the stresses and displacements are accounted for with the aid of the higher order shell theory proposed by [Voyiadjis, G.Z. and Shi, G., 1991, A refined two-dimensional theory for thick cylindrical shells, International Journal of Solids and Structures, 27(3), 261–282.]. For a thick shell under external hydrostatic pressure, the stability equations are solved analytically and yield an improved expression for the buckling load. Reference solutions are also obtained by solving numerically the differential stability equations. Both the full set that contains strains and rotations as well as the simplified set that contains rotations only were solved numerically. The relative magnitude of shear strain and rotation was examined and the effect of thickness was quantified. Differences between the benchmark solutions and the analytic expressions based on the refined theory and the classical shell theory are analysed and discussed. It is shown that the new analytic expression provides significantly improved predictions compared to the formula based on thin shell theory.     

Effects of imperfections on the buckling response of compression-loaded composite shells

The results of an experimental and analytical study of the effects of initial imperfections on the buckling and postbuckling response of three unstiffened thin-walled compression-loaded graphite-epoxy cylindrical shells with different orthotropic and quasi-isotropic shell-wall laminates are presented. The results identify the effects of traditional and non-traditional initial imperfections on the non-linear response and buckling loads of the shells. The traditional imperfections include the geometric shell-wall mid-surface imperfections that are commonly discussed in the literature on thin shell buckling. The non-traditional imperfections include shell-wall thickness variations, local shell-wall ply-gaps associated with the fabrication process, shell-end geometric imperfections, non-uniform applied end loads, and variations in the boundary conditions including the effects of elastic boundary conditions. A high-fidelity non-linear shell analysis procedure that accurately accounts for the effects of these traditional and non-traditional imperfections on the non-linear responses and buckling loads of the shells is described. The analysis procedure includes a non-linear static analysis that predicts stable response characteristics of the shells and a non-linear transient analysis that predicts unstable response characteristics.     

Determination of the deformed position of a thin shell from surface strains and height function

This paper focuses on the following problem: given the strain tensor of a deformed reference surface of a thin shell and the distances of the points on this surface from some arbitrarily fixed reference plane (the so-called height function) find the position of this reference surface. Two alternative procedures supplying the solution are developed. The first one follows from the ideas developed by Darboux (Leçons sur la Théorie Général des Surfaces, Troisiéme Partie, Gauthier-Villars, Paris, 1894), whereas the second one is based on the polar decomposition theorem and techniques developed in continuum mechanics. These procedures are purely kinematic, valid for arbitrary surface geometry and for unrestricted surface strains. Szwabowicz (Deformable surfaces and almost inextensional deflections of thin shells, Habilitation Thesis, 1999) proposed a relatively simple non-linear boundary-value problem (BVP) for thin elastic shells, which was expressed in three surface strains and the height function as basic independent field variables. The results of this paper suggest that this approach to the non-linear problems of thin shells may be an attractive alternative to other BVPs developed in the literature.     

Asymptotic Justification of the Kirchhoff–Love Assumptions for a Linearly Elastic Clamped Shell

The displacement vector of a linearly elastic shell can be computed by using the two-dimensional Koiter's model, based on the a priori Kirchhoff–Love assumptions. These hypotheses imply that the displacement of any point of the shell is an affine function of the transverse variable x ₃. The term independent of x ₃ of this approximation is equal to the displacement vector of the two-dimensional Koiter's model. The term linear in x ₃ depends on the infinitesimal rotation vector of the normal. After an appropriate scaling, we estimate here the difference between the three-dimensional displacement and this affine vector field in the case of shells clamped along their entire lateral face. Besides, in the case of shells with uniformly elliptic middle surface, taking into account the term depending of the rotation of the normal allows to improve the asymptotic estimate between the three-dimensionnal displacement and Koiter's bidimensional displacement. This revised version was published online in August 2006 with corrections to the Cover Date.     

A new kind of mixed elements for shells of revolution subjected to arbitrary loads

In this paper a generalized variational principle with two-field variables is derived from the Reissner principle of elasticity in the curvilinear coordinates of a revolution shell, based on which, a new kind of mixed elements with independent transverse rotations is formulated for revolution shells subjected to harmonic external loads. The resultant-stress interpolations are carefully selected so that the shear part of the element stiffness contains the Kirchhoff hypothesis for thin shells and element stiffness matrices have correct ranks. The elements are free from shear locking and spurious kinematic modes. Numerical examples show that the new elements have good generality and high accuracy for thin and moderately-thick revolution shells.     

载流圆锥薄壳的磁弹性应力与变形分析

在建立旋转壳体的非线性磁弹性运动方程的基础上,研究了电磁场和机械载荷联合作用下载流圆锥薄壳的磁弹性效应．通过算例,得到了载流圆锥薄壳的位移及应力与通电电流强度之间的关系．解决了圆锥薄壳顶点处的奇异性问题,给出了轴对称条件下的数值解．计算结果表明：改变通电电流强度,可以改变载流圆锥薄壳的应力与变形状态,达到控制圆锥薄壳的受力与变形的目的．     

FINITE ELEMENT DISPLACEMENT PERTURBATION METHOD FOR GEOMETRIC NONLINEAR BEHAVIORS OF SHELLS OF REVOLUTION OVERALL BENDING IN A MERIDIONAL PLANE AND APPLICATION TO BELLOWS (Ⅰ)

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Geometrically non-linear analysis of arbitrary elastic supported plates and shells using an element-based Lagrangian shell element

In the present paper, the ELF (element-based Lagrangian formulation) 9-node ANS (assumed natural strain) shell element was combined with the spring element for geometrically non-linear analysis of plates and shells sustained by arbitrary elastic edge supports that are subjected to variation in loading.This particular spring element serves as tool for modeling an arbitrary elastic edge support with 6 DOF (degrees of freedom). The elastic edge support was modeled by combining different spring models. The ANS method was used to overcome shear and membrane locking problems inherent in some thin plate and shell problems. In the formulation of the ELF characteristic arrays, the expression of element strains was adopted in the framework of the element natural coordinates. The non-linear analysis results of idealized edge supports were validated against the reference solutions available in the literature. As a result of the numerical test, the combination of the ELF 9-node shell element and spring element shows an exceptional performance for non-linear analysis of plates and shells under elastic edge supports.     

FINITE ELEMENT DISPLACEMENT PERTURBATION METHOD FOR GEOMETRIC NONLINEAR BEHAVIORS OF SHELLS OF REVOLUTION OVERALL BENDING IN A MERIDIONAL PLANE AND APPLICATION TO BELLOWS (Ⅰ)

In order to analyze bellows effectively and practically, the finite-element-displacement-perturbation method (FEDPM) is proposed for the geometric nonlinear behaviors of shells of revolution subjected to pure bending moments or lateral forces in one of their meridional planes. The formulations are mainly based upon the idea of perturba-tion that the nodal displacement vector and the nodal force vector of each finite element are expanded by taking root-mean-square value of circumferential strains of the shells as a perturbation parameter. The load steps and the iteration times are not as arbitrary and unpredictable as in usual nonlinear analysis. Instead, there are certain relations between the load steps and the displacement increments, and no need of iteration for each load step. Besides, in the formulations, the shell is idealized into a series of conical frusta for the convenience of practice, Sander’s nonlinear geometric equations of moderate small rotation are used, and the shell made of more than one material ply is also considered.     

Fracture mechanics of plates and shells applied to fail-safe analysis of fuselage Part I: Theory

In this paper, a general theory on the asymptotic field near the crack tip for plates and shells with and without shear deformation effect is established. It is found that four stress intensity factors, two for symmetrical and antisymmetrical stretching and two for symmetrical and antisymmetrical bending, are required to describe arbitrary asymptotic fields near the crack tip for plates without shear deformation. An additional stress intensity factor is required for the transverse shearing force induced by antisymmetrical bending when the shear deformation is included in the analysis. It is also proven by means of the complex variable technique that for problems of plates with shear deformation, there exist similarities in the asymptotic expressions of moments and membrane forces and also in the asymptotic expressions of in-plane displacements and rotations of the mid-surface. The energy release rate associated with crack growth in the direction of the crack line can be expressed in terms of stress intensity factors by means of Irwin's method of work and energy associated with a virtual crack extension. A combined stress intensity factor can be defined through the total energy release rate. The theory of the fracture of plates is generalized and applied to the study of problems in the fracture of shells. An example of an infinitely long cylindrical shell with a circumferential crack subjected to remote axial tension is given to demonstrate the application of the theory and to test the accuracy of the numerical analysis used for the problem.     

Analysis of a strengthened weakly conical shell using a complete system of orthogonal functions on an arbitrary contour of its transverse cross-section

Thin-walled weakly conical and cylindrical shells with arbitrary open, simply or multiply closed contour of transverse cross-sections strengthened by longitudinal elements (such as stringers and longerons) are used in the design of wings, fuselages, and ship hulls. To avoid significant deformations of the contour, such structures are stiffened by transverse elements (such as ribs and frames). Various computational models and methods are used to analyze the stress-strain states of such compound structures. In particular, the ground stress-strain states in bending, transverse shear, and twisting of elongated structures are often analyzed with the use of the theory of thin-walled beams [1] based on the hypothesis of free (unconstrained) warping and bending of the contour of transverse cross-sections. In general, the computations with the contour warping and bending constraints caused by the variable load distribution, transverse stiffening elements, and the difference in the geometric and rigidity parameters of the shell cells are usually performed by the finite element method or the superelement (substructure) method [2, 3]. In several special cases (mainly for separate cells of cylindrical and weakly conical shells located between transverse stiffening elements, with the use of some additional simplifying assumptions), efficient variation methods for computations in displacements [4–8] and in stresses [9] were developed, so that they reduce the problem to a system of ordinary differential equations. In the one-and two-term approximations, these methods permit obtaining analytic solutions, which are convenient in practical computations. But for shells with multiply closed contours of transverse cross-sections and in the case of exact computations by using the Vlasov variational method [4], difficulties are encountered in choosing the functions representing the warping and bending of the contour of transverse cross-sections. In [10], in computations of a cylindrical shell with simply closed undeformed contour of the transverse section, warping was represented in the form of expansions in the eigenfunctions orthogonal on the contour, which were determined by the method of separation of variables from a special integro-differential equation. In [11], a second-order ordinary differential equation of Sturm-Liouville type was obtained; its solutions form a complete system of orthogonal functions with orthogonal derivatives on an arbitrary open simply or multiply closed contour of a membrane cylindrical shell stiffened by longitudinal elements. The analysis of such a shell with expansion of the displacements in these functions leads to ordinary differential equations that are not coupled with each other. In the present paper, by using the method of separation of variables, we obtain differential and the corresponding variational equations for numerically determining complete systems of eigenfunctions on an arbitrary contour of a discretely stiffened membrane weakly conical shell and a weakly conical shell with undeformed contour. The obtained systems of eigenfunctions are used to reduce the problem of deformation of shells of these two types to uncoupled differential equations, which can be solved exactly.     

一类新型受任意载荷的杂交旋转壳单元

本文从弹性力学Reissner变分原理出发推导旋转壳曲线坐标系下内分，位移的二类变量广义变分原理，依据这个原理推导一类旋转壳坐标系中具有独立横向转角的受谐和外载荷下的杂交旋转壳单元，内力模式的选用使刚度矩阵的剪切部份在薄壳情况下能反映Kirchhoff假设，并使单元刚度矩阵满秩，从而保证单元无剪切自锁和零能模式，数例证明这类单元对中厚和薄旋转壳具有良好的通用性和较高的精度。