Large-strain constitutive laws for the cylindrical deformation of shells

Expressions are developed for the strain energy and stress resultants in the case of large strain-large bending cylindrical deformations of shells of non-linear materials. The deformation variables are the extension and curvature of the material reference surface. The Mooney material is used as an example where appropriate.The influence of the large deformations on the constitutive laws is evaluated. Of special interest is the effect of cross terms and of the ability to express the stress resultants as partial derivatives of the strain energy with respect to appropriate deformation variables.     

A general nonlinear theory of elastic shells

This paper presents a general nonlinear theory of elastic shells for large deflections and finite strains in reference to a certain natural state. By expanding the displacement components into power series in the coordinate θ³ normal to the undeformed middle surface of shells, the expansions of the Cauchy-Green strain tensors are expressed in terms of these expanded displacement components. Through the modified Hellinger-Reissner variational principle for a three-dimensional elastic continuum, a set of the fundamental shell equations is derived in terms of the expanded Cauchy-Green strain tensors and Kirchhoff stress resultants. The Love-Kirchhoff hypothesis is not assumed and higher order stretching and bending are taken into consideration. For elastic shells of isotropic materials, assuming the strain-energy to be an analytic function of the strain measures, general nonlinear constitutive equations are then derived. Thus, a complete and consistent two-dimensional shell theory incorporating the geometrical and physical nonlinearities is established. The classical theories of shells are directly derivable from the present results by proper truncations of the series.     

基于SHPB装置的膨胀圆柱管实验技术

采用改进的SHPB实验装置对45钢薄壁金属圆柱管进行了膨胀断裂加载,完成了不同变形程度 (覆盖圆柱管整个变形及断裂过程)圆柱管的冻结回收实验。回收样品可用于断裂机理研究分析,通过数值模 拟辅助分析和实验数据拟合得到了圆柱管凸起处的径向应变、应变率和环向拉伸应力。通过在圆柱管端粘贴 应变片监测断裂信号的方式,准确判断了圆柱管凸起处发生断裂的时间,以及径向断裂应变、应变率和环向拉 伸断裂应力,在102~104s-1应变率范围内,SHPB实验装置可用于研究金属圆柱管膨胀断裂。     

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.     

Fracture behavior of explosively loaded spherical molded steel shells

Experimental and numerical works are made to study the fracture of explosively loaded spherical molded steel shells. The first series of experiments included three sawdust recovery shots to save fragments for examination. In this series, detonation was initiated from the center of the sphere. Results of the experiments show that two types of fractures are observed in spherical shells: radial and shear as in cylindrical shells. Spall fracture is also observed in spherical shells. To assist understanding of the experimental results, a computer simulation of expanding shells is performed to provide information on the stress, strain, strain rate and position of each element of the shell wall as a function of time after detonation. For t=7.5 μs after detonation, triaxial non-uniform strain prevails in the middle of the thickness of the wall. The peak of the stress equals to 6.5 GPa and exceeds the spall strength of carbon steel. In the second series of experiments, spall fracture is suppressed by displacing the point of detonation initiation from the center to periphery of HE charge.     

Non-linear static-dynamic finite element formulation for composite shells

Three non-linear finite element formulations for a composite shell are discussed. They are the simplified large rotation (SLR), the large displacement large rotation (LDLR), and the Jaumann analysis of general shells (JAGS). The SLR and the LDLR theories are based on total Lagrangian approach, and the JAGS is based on a co-rotational approach. Both the SLR and LDLR theories represent the in-plane strains exactly the same as Green's strain-displacement relations, whereas, only linear displacement terms are used to represent the transverse shear strain. However, a higher order kinematic through the thickness assumption is used in the SLR theory, which leads to parabolic transverse shear stress distribution compared to a first order kinematic through the thickness relationship used in the LDLR theory that leads to linear transverse shear stress distribution. Furthermore, the LDLR theory uses an Euler-like angle in the kinematics to account for the large displacement and rotation. The JAGS theory decomposes the deformation into stretches and rigid body rotations, where an orthogonal coordinate system translates and rotates with the deformed infinitesimal volume element. The Jaumann stresses and strains are used. Layer-wise stretching and shear warping through the thickness functions are used to model the three-dimensional behavior of the shell, where displacement and stress continuities are enforced along the ply interfaces. The kinematic behavior is related to the original undeformed coordinate system using the global displacements and their derivatives. Numerical analyses of composite shells are performed to compare the three theories. The commercial code ABAQUS is also used in this investigation as a comparison.     

Physically Nonlinear Deformation of a Long Orthotropic Cylindrical Shell with Elliptic Cross-Section*

International Applied Mechanics - The stress–strain state of long cylindrical shells with elliptical cross-section is studied. The shells are made of a nonlinear elastic orthotropic organic...     

Buckling analysis of laminated cylindrical shells by 3-D multilayer hybrid element

A 3-D multilayer hybrid element is developed for the analysis of thick laminated plates and shells. The stresses are assumed independently in each sublayer element and the stress continuity between adjacent sublayers is applied to form the stress pattern of the multilayer element. Both interlaminar stress concentration and global structure response can be adequately predicted by the element model. The buckling analysis of orthotropic cylindrical shells under the external pressure is performed and the results show that the plane strain assumption is not applicable to the buckling of long orthotropic cylindrical shells.     

Some Approaches to the Solution of Problems on Thin Shells with Variable Geometrical and Mechanical Parameters

A number of approaches to the solution of stress problems for anisotropic inhomogeneous shells in the classical formulation are discussed. A review is made of approaches to the solution of one- and two-dimensional static problems for thin shells with variable parameters and to the solution of stress–strain problems for anisotropic shells of revolution under axisymmetric and non-axisymmetric loading, shallow convexo-convex shells, noncircular cylindrical shells, plates of various shapes, and shells of complex geometry     

三种壳体在脉冲电子束辐射下动力学响应的实验研究

在闪光二号电子束装置上 ,对材料或形状有异的两端固支的三种壳体进行了辐射结构响应的实验研究。实验采用在壳体内表面粘贴一个特制长引线应变片并组成直流桥路的测量原理。结果表明 :当壳体受辐射方位角为 0～ 180时 ,应变峰值随方位角的增大而减小 ;编织材料壳体中的应变峰值和热激波应力峰值都比硬铝壳体中的小 ;当电子束能注量为 15 0～ 2 10J/cm2 、壳体半径厚度比为 13～ 18 5时 ,壳体变形处于弹塑性或全塑性状态。     

旋转壳非线性蠕变分析

基于现有的蠕变理论,本文推导出复杂应力状态下的蠕变本构方程,以内力与转角为基本未知量,得到旋转壳在轴对称载荷作用下的非线性蠕变控制方程.对基本未知量在空间上采用样条配点法离散,在时域上采用初应力法离散,求出壳体在不同时域的蠕变应变和应力,而最终可求出蠕变后的稳态应力重新分布.计算结果表明同,本文分析方法是有效的.     

基于一种简化Green应变的薄壳大挠度方程

将正交曲线坐标下的格林应变张量引入到薄壳大变形分析,通过建立恰当的基本假设,直接导出了用格林应变张量表示的壳体变形几何方程,将该方程代入到本构方程,由能量原理得到了小应变非线性变形平衡方程、内力方程和边界条件,在此基础上提出了大应变变形的简化分析方法．文中导得的方程涵盖薄板壳大、小变形的全部方程,推导过程简捷、系统,所得结果规则、清晰,与此前有关分析方法的结果完全吻合．     

The nonlinear strain components of thin shells

The nonlinear strain components of thin shells are the foundations of nonlinear shell mechanics. They are needed in the investigation of various thin shell stability and large displacement problems. Due to the geometrical variety of this shells we have not seen in existing literatures a complete set of general formulae expressing nonlinearity of shell strain components. In this paper we have derived six of them expressed in Lamé coefficients and orthogonal curvilinear coordinates, including both linear and nonlinear parts, three of them are tensile strain components, the other three are shear strain components.     

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.     

Stress Analysis of Biconvex Laminated Orthotropic Shells that are Shallow to a Variable Degree

The stress–strain state of biconvex laminated orthotropic shells is analyzed against the degree of shallowness and the parameters of orthotropy. Numerical values of deflections and stresses are obtained by solving two-dimensional boundary-value problems using spline-functions and the discrete-orthogonalization method. The effect of the rise of shells on the displacement and stress fields is analyzed for different parameters of orthotropy     

Solving Stress Problems for Elastic Systems of Anisotropic Shells of Revolution with Regard for Transverse Shear and Reduction

An approach is proposed for refined solution of stress problems for elastic systems consisting of coaxial shells of revolution. Transverse shear and reduction are taken into account. Multivariant calculations made for orthotropic cylindrical shells with elliptical end-plates allow us to analyze the influence of the semiaxis ratio and intermediate supports on the stress–strain state of the shell systems under consideration     

A nonlinear composite shell theory

A general nonlinear theory for the dynamics of elastic anisotropic circular cylindrical shells undergoing small strains and moderate-rotation vibrations is presented. The theory fully accounts for extensionality and geometric nonlinearities by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. Moreover, the linear part of the theory contains, as special cases, most of the classical linear theories when appropriate stress resultants and couples are defined. Parabolic distributions of the transverse shear strains are accounted for by using a third-order theory and hence shear correction factors are not required. Five third-order nonlinear partial differential equations describing the extension, bending, and shear vibrations of shells are obtained using the principle of virtual work and an asymptotic analysis. These equations show that laminated shells display linear elastic and nonlinear geometric couplings among all motions.     

On the influence of prebuckling deformations on the buckling of thin elastic shells

A buckling theory valid for finite prebuckling deformations is presented for thin homogeneous, isotropic and elastic shells. It is subject to the restriction of the Kirchhoff hypothesis. A set of stability equations is derived by decomposing strain and stress components into four classes according to their characteristics.The influence of the prebuckling deformations on the buckling of thin circular cylindrical shells under lateral pressure is investigated with the aid of the basic equations derived above and the results are compared with the solutions of the Flügge equations and those obtained by Yamaki.     

Generalized stress/strain rate relations for creep of thin shells; a comparison of “exact” and approximate relations

A set of approximate generalized stress/strain rate relations which has been used for the stationary creep analysis of thin shells is compared with the corresponding ‘exact’ relations. The comparison is made by computing the functions from which the relations are derived and plotting the corresponding surfaces. Results are included for a limiting condition in which the stress/strain rate relations become those for a rigid-plastic material obeying the von Mises yield condition and associated flow rule. Although the comparison is made only for conditions valid in a cylindrical shell under rotationally symmetric loading, it indicates the errors which are likely to occur when the approximate relations are used in stationary creep analysis.