Determining the Thermoviscoplastic State of Shells of Revolution Subject to Creep Damage

A technique is developed for determining the thermoviscoplastic state of shells of revolution with allowance for creep damage. The technique is based on the hypotheses of rectilinear element and the theory of deformation along paths of small curvature. The equivalent stress appearing in the kinetic equations of damage and creep is determined using a creep-rupture criterion that accounts for the stress mode and the level of irreversible strains. The technique is tested by determining the thermoviscoplastic state and time to failure of tubular specimens under a tensile force and a torque     

Axisymmetric thermoviscoelastoplastic state of thin flexible shells with damages

The paper presents a technique to determine the axisymmetric geometrically nonlinear thermoviscoelastoplastic state of thin shells with damages. The technique is based on the geometrically nonlinear equations that incorporate transverse-shear strains. The equations of thermoelasticity that describe the deformation of the body’s element along paths of small curvature are used as equations of state. The equivalent stress in the kinetic equations of damage and creep is determined from a failure criterion that accounts for the stress mode. As an example, the geometrically nonlinear thermoviscoelastoplastic deformation of a corrugated shell is analyzed and the time to its failure is determined __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 49–60, February 2008.     

Determination of the Axisymmetric Geometrically Nonlinear Thermoviscoelastoplastic State of Laminated Medium-Thickness Shells

A method is developed to determine the axisymmetric geometrically nonlinear thermoelastoviscoplastic stress–strain state of branched laminated medium-thickness shells of revolution. The method is based on the hypotheses of a rectilinear element for the whole set of layers. The shells are subject to loads that cause a meridional stress state and torsion. They can consist of isotropic layers, which deform beyond the elastic limit, and elastic orthotropic layers. The relations of thermoviscoplastic theory, which describe simple processes of loading, are employed as the equations of state for the isotropic layers. The solution of the problem is reduced to numerical integration of systems of differential equations. The geometrically nonlinear elastoplastic state of a two-layer corrugated shell of medium thickness is calculated as an example     

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.      

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．     

Accurate equations for laminated composite deep thick shells

This paper derives accurate equations of elastic deformation for laminated composite deep, thick shells. The equations include shells with a pre-twist and accurate force and moment resultants which are considerably different than those used for plates. This is due to the fact that the stresses over the thickness of the shell have to be integrated on a trapezoidal-like cross-section of a shell element to obtain the stress resultants. Numerical results are obtained and showed that accurate stress resultants are needed for laminated composite deep thick shells, especially if the curvature is not spherical. A consistent set of equations of motion, energy functionals and boundary conditions are also derived. These may be used in obtaining exact solutions or approximate ones like the Ritz or finite element methods.     

On the incremental equations in non-linear elasticity — I. Membrane theory

A new formulation of the equations of membrane theory in non-linear elasticity is described. It is based on the consistent use of certain conjugate variables averaged through the (undeformed) thickness of the thin shell which the membrane approximates. The deformation gradient is taken as the basic measure of deformation, and its average value as the membrane measure of deformation. It is shown that the average elastic strain energy can be regarded as a function of the average deformation gradient to within an error which is of the second order in a certain small parameter. Moreover, to the same order, the average strain energy is a potential function for the average nominal stress. This means that the averages of the conjugate variables (nominal stress and deformation gradient) are also conjugate.In terms of the average conjugate variables, the membrane equilibrium equations are obtained by averaging from the equilibrium equations of the full three-dimensional theory. Discussion of the order of magnitude of the errors involved in the membrane approximation is a feature of the analysis.The corresponding incremental equations are also derived as a prelude to their application in certain bifurcation problems. One such problem is examined in the companion paper (Part II) in which results for thick shells and membranes are compared.     

Determining the Axisymmetric Geometrically Nonlinear Thermoviscoelastoplastic State of Laminated Shells by the Theory of Deformation Along Paths of Small Curvature

A technique is developed for determining the thermoviscoelastoplastic geometrically nonlinear axisymmetric stress–strain state of laminar shells of revolution under loads that induce meridional stress and torsion. The technique is based on the hypotheses of rectilinear element for the whole stack of layers. The relations of the theory of deformations along paths of small curvature are used as equations of state. The solution is reduced to the numerical integration of a system of ordinary differential equations. The technique is tried out by a test example and illustrated by determining the geometrically nonlinear thermoviscoelastoplastic state of a corrugated shell     

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     

Determining the Axisymmetric, Geometrically Nonlinear, Thermoelastoplastic State of Laminated Orthotropic Shells

A technique is developed to determine the axisymmetric, geometrically nonlinear, thermoplastic stress–strain state of laminated ortotropic shells of revolution under loads that cause a meridian stress state and torsion. The technique is based on the rectilinear-element hypotheses for the whole stack of layers. The active elastoplastic deformation of an ortotropic material is described by deformation-type equations that have been derived without resort to the existence conditions for the plastic potential. The scalar functions in the constitutive equations depend on the intensity of shear strains and temperature. The problem is solved through the numerical integration of a system of differential equations. The technique is tried out in designing tubular specimens subjected to axial force and torque. As an example, the elastoplastic state of a corrugated shell is analyzed     

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.     

可渗透圆柱壳流固耦合问题的流场与内力分析

在弹性薄壳的非线性理论和流体力学基本方程的基础上,研究了可渗透圆柱壳的流固耦合问题.假定壳体具有均布孔隙且孔的面积很小,不考虑其阻力,忽略对弯曲刚度和壳体腔内流体微小运动影响,应用相容欧拉--拉格朗日法建立了带孔的圆柱壳在流体中相互作用的基本方程.通过具体算例求解,给出了流场速度与压力的变化、圆柱壳的变形及内力分布,并对相关参数进行了讨论.     

旋转壳的数值传递函数方法

应用数值传递函数方法建立一种用于分析旋转壳静力、动力响应的截锥壳单元,在本方法中,单元的位移在环向展开为Fourier级数的形式,应用薄壳理论可以得到解耦的微分方程,通过Laplace变换可以将方程转化为频域内的常微分方程,将其表示为状态空间形式后,可以应用数值传递函数方法求解,对复杂的系统可以应用与有限元类似的方法,划分多个单元组合求解,文中给出了几种旋转壳的动力、静力问题的数值算例,并与其它方法进行了比较,表明本文方法具有精度高,计算方便等特点。     

A numerical calculation of dynamic buckling of a thin shallow spherical shell under impact

Assuming the deformation of the shell has an axial symmetrical form, we transform Marguerre's equations into difference equations, and use these equations to discuss the buckling of an elastic thin shallow spherical shell subjected to impact loads. The result shows when impact load acts on the shells, a jump of the shell takes place dependent on the values λ and the critical buckling load increases with the enlargement of the loading area.     

The Thermoviscoelastoplastic State of Shells of Revolution Under Axisymmetric Deformation Along Various Flat Paths

The basic results obtained at the S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine in developing a thermoviscoelastoplastic theory of thin-walled shells of revolution subject to arbitrary axisymmetric loading are analyzed. The theory includes the constitutive thermoviscoplastic equations describing the deformation of an isotropic body along arbitrary flat paths with functionals made specific in base experiments and the solutions of boundary-value problems, which indicate the influence of the geometry of strain paths on the stress–strain state of shells     

加权残值法分析轴压圆柱薄壳后屈曲问题

本文首次应用了样条配点法分析了受到轴向压力的圆柱形薄壳的后屈问题,壳体的方程是L.H.Donnell的非线性正交异性圆柱形壳体方程,壳体的挠度试函数及应力函数试函数都是于轴向用了五次B样条函数基函数,周向用余弦函数。计算模型是周向为半个波长的壳块,可适应后屈曲实验变形跳跃现象。非线性代数方程组用了Newton—Rophson迭代法求解。由此所得的理论上的后屈曲曲线与国外近代实验相符。     

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.     

Effects of partial crack–face contact for the bending of thin shell structures

The physical occurrence that crack surfaces are in contact at the compressive edges when a flat or a shell is subjected to a bending load has been recognized. This article presents a theoretical analysis of crack–face contact effect on the stress intensity factor of various shell structures such as spherical shell, cylindrical shell containing an axial crack, cylindrical shell containing a circumferential crack and shell with two non-zero curvatures, under a bending load. The formulation of the problem is based on the shear deformation theory, incorporating crack–face contact by introducing distributed force at the compressive edge. Material orthotropy is concerned in this analysis. Three-dimensional finite element analysis (FEA) is conduced to compare with the theoretical solution. It is found that due to curvature effect crack–face contact behavior in shells differs from that in flat plates, in that partial contact of crack surfaces may occur in shells, depending on the shell curvature and the nature of the bending load. Crack–face contact has significant influence on the stress intensity factor and it increases the membrane component but decreases the bending component.     

Analysis of the stress state of a flattened conical shell of moderate thickness with a structurally weakening hole

Conclusions We determined the relationship between the nature of the stress distribution on the hole surface in a flexible plate as a function of thickness. We observed a great difference between the stress densities in flattened, thin and moderate-thickness conical shells and the stress concentrations near holes in thin cylindrical shells and thin, almost cylindrical, conical shells. The stress distribution near the hole in flattened conical shells of moderate thickness is similar to the stress distribution near the holes in flexible, thick plates. During loading of conical shells by an axial force, the lowest stress concentration factor near the holes is obtained when the axis of the hole is parallel to the shell axis. As the thickness of the shell is increased, the stress concentration factor near the holes increases.Kiev University. Ukrainian Institute of Water Management Engineers, Rovno. Translated from Prikladnaya Mekhanika, Vol. 24, No. 9, pp. 65–70, September, 1988.