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.     

Kinematic hardening rules in uncoupled modeling for multiaxial ratcheting simulation

An earlier paper by the authors evaluated the performance of several coupled models in simulating a series of uniaxial and biaxial ratcheting responses. This paper evaluates the performance of various kinematic hardening rules in an uncoupled model for the same set of ratcheting responses. A modified version of the Dafalias–Popov uncoupled model has been demonstrated to perform well for uniaxial ratcheting simulation. However, its performance in multiaxial ratcheting simulation is significantly influenced by the kinematic hardening rules employed in the model. Performances of eight different kinematic hardening rules, when engaged with the modified Dafalias–Popov model, are evaluated against a series of rate-independent multiaxial ratcheting responses of cyclically stabilized carbon steels. The kinematic hardening rules proposed by Armstrong–Frederick, Voyiadjis–Sivakumar, Phillips, Tseng–Lee, Kaneko, Xia–Ellyin, Chaboche and Ohno–Wang are examined. The Armstrong–Frederick rule performs reasonably for one type of the biaxial ratcheting response, but fails in others. The Voyiadjis–Sivakumar rule and its constituents, the Phillips and the Tseng–Lee rules, can not simulate the biaxial ratcheting responses. The Kaneko rule, composed of the Ziegler and the prestress directions, and the Xia–Ellyin rule, composed of the Ziegler and Mroz directions, also fail to simulate the biaxial ratcheting responses. The Chaboche rule, with three decomposed Armstrong–Frederick rules, performs the best for the whole set of ratcheting responses. The Ohno–Wang rule performs well for the data set, except for one biaxial response where it predicts shakedown with subsequent reversal of ratcheting.     

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.     

Effect of the geometry on the non-linear vibration of circular cylindrical shells

The non-linear vibration of simply supported, circular cylindrical shells is analysed. Geometric non-linearities due to finite-amplitude shell motion are considered by using Donnell's non-linear shallow-shell theory; the effect of viscous structural damping is taken into account. A discretization method based on a series expansion of an unlimited number of linear modes, including axisymmetric and asymmetric modes, following the Galerkin procedure, is developed. Both driven and companion modes are included, allowing for travelling-wave response of the shell. Axisymmetric modes are included because they are essential in simulating the inward mean deflection of the oscillation with respect to the equilibrium position. The fundamental role of the axisymmetric modes is confirmed and the role of higher order asymmetric modes is clarified in order to obtain the correct character of the circular cylindrical shell non-linearity. The effect of the geometric shell characteristics, i.e., radius, length and thickness, on the non-linear behaviour is analysed: very short or thick shells display a hardening non-linearity; conversely, a softening type non-linearity is found in a wide range of shell geometries.     

Modeling multi-axial deformation of planar anisotropic elasto-plastic materials,part I: Theory

In sheet metal forming processes local material points can experience multi-axial and multi-path loadings. Under such loading conditions, conventional phenomenological material formulations are not capable to predict the deformation behavior within satisfying accuracy. While micro-mechanical models have significantly improved the understanding of the deformation processes under such conditions, these models require large sets of material data to describe the micromechanical evolution and quite enormous computation expenses for industrial applications. To reduce the drawbacks of phenomenological material models under the multi-path loadings a new anisotropic elasto-plastic material formulation is suggested. The model enables the anisotropic yield surface to grow (isotropic hardening), translate (kinematic hardening) and rotate (rotation of the anisotropy axes) with respect to the deformation, while the shape of the yield surface remains essentially unchanged.Essentially, the model is formulated on the basis of an Armstrong–Frederick type kinematic hardening, the plastic spin theory for the reorientation of the symmetry axes of the anisotropic yield function, and additional terms coupling these expressions. The capability of the model is illustrated with multi-path loading simulations in ‘tension-shear’ and ‘reverse-shear’ to assess its performance with ‘cross’ hardening and ‘Bauschinger’ effects.     

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.     

Global structural behaviour of thin and moderately thick monoclinic spherical shells using a mixed shear deformation model

Summary The static and dynamic responses of anisotropic spherical shells under a uniformly distributed transverse load are investigated. Analytical solutions using the mixed variational formulation are presented for spherical shells subjected to various boundary conditions. Numerical results of a refined mixed first-order shear deformation theory for natural frequencies, critical buckling, center deflections and stresses are compared with those obtained using the classical shell theory. A variety of simply-supported and clamped boundary conditions are considered and comparisons with the existing literature are made. The sample numerical results presented herein for global structural behaviour of monoclinic spherical shells should serve as references for future comparisons.     

广义协调元在薄壳自由振动分析中的应用

本文采用２０个自由度的矩形平板型壳单元分析薄筒壳自由振动问题。在文献（１）的基础上，将广义协调元的应用范围扩大到了壳体分析中，目的是找出简单实用的方法分析板壳这类具有特殊性质的结构。该单元由平面应力单元和平板弯曲单元经简单叠加而成，是一种性能良好的单元。沿用常规作法非常便利，程序容易实现。经算例验证，该单元自由度少，精度较好，实用方便，适合于工程应用。     

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。     

Stress–strain state of nonthin orthotropic spherical shells of variable thickness

The stress–strain state of an orthotropic spherical shell with thickness varying in two coordinate directions is analyzed. Different boundary conditions are considered, and a refined problem statement is used. A numerical analytic method based on spline-approximation and discrete orthogonalization is developed. The stress–strain state of spherical orthotropic shells with variable thickness is studied     

轴向运动层合圆柱壳体的振动特性

研究了轴向运动层合圆柱壳体的振动特性．基于Donnell壳体理论,建立了轴向运动层合圆柱壳体的横向振动方程,使用Galerkin方法求解该振动方程,得到其固有频率,通过与有限元结果对比说明方法的有效性．分析了轴向速度、纤维方向角、长径比和厚径比对壳体振动特性的影响．研究表明：当纤维方向角为 (15?/-15?)s时,轴向运动柱壳前3阶固有频率达到最大值．     

On the stability analysis of plates and shells using a quadrilateral, 16-degrees of freedom plat shell element DKQ16

The linear buckling problems of plates and shells were analysed using a recently developped quadrilateral, 16-degrees of freedom flat shell element ( called DKQ16 ). The geometrical stiffness matrix was established. Comparison of the numerical results for several typical problems shows that the DKQ16 element has a very good precision for the linear buckling problems of plates and shells.     

含孔边裂纹球壳的断裂分析

在依据Reissner理论得出的球壳裂纹尖端应力应变场展开式基础上,采用局部—整体分析法和权函数方法分别计算承受内压的含孔边裂纹球壳的应力强度因子.在有限元的模式中考虑剪切变形的影响,并对奇异元模式的应力应变场展开式的项数选择、奇异元最佳尺寸的选取进行了分析.本文计算和分析了在不同几何尺寸,不同开孔大小以及不同剪切参量条件下承受内压的含孔边裂纹球壳的应力强度因子及其变化规律.     

一种薄壳厚壳通用的轴对称曲壳单元

有限元法分析轴对称壳体时，常需区分薄壳和厚壳并选用不同的单元，给计算带来不便．为此，通过对现有的轴对称壳体单元的研究，基于加权残值法，将广义协调条件引入剪应变场，构造了一种挠度和转角各自独立的新型轴对称曲壳单元．算例结果表明，新单元具有很高的精度，既可用于厚壳，也可用于薄壳．该单元可用于轴对称壳体结构的计算分析．     

Slow Motion of an Assemblage of Porous Spherical Shells Relative to a Fluid

The body-force-driven motion of a homogeneous distribution of spherically symmetric porous shells in an incompressible Newtonian fluid and the fluid flow through a bed of these shell particles are investigated analytically. The effect of the hydrodynamic interaction among the porous shell particles is taken into account by employing a cell-model representation. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field around a single particle in a unit cell, and the drag force acting on the particle by the fluid is obtained in closed forms. For a suspension of porous spherical shells, the mobility of the particles decreases or the hydrodynamic interaction among the particles increases monotonically with a decrease in the permeability of the porous shells. The effect of particle interactions on the creeping motion of porous spherical shells relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solution describing the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres. The particle-interaction behavior for a suspension of porous spherical shells with a relatively low permeability may be approximated by that of permeable spheres when the porous shells are sufficiently thick.     

Plastic deformation modeling of AL-6XN stainless steel at low and high strain rates and temperatures using a combination of bcc and fcc mechanisms of metals

Combination of physically based constitutive models for body centered cubic (bcc) and face centered cubic (fcc) metals developed recently by the authors [Voyiadjis, G.Z., Abed, F.H., 2005. Microstructural based models for bcc and fcc metals with temperature and strain rate dependency. Mech. Mater. 37, 355–378] are used in modeling the plastic deformation of AL-6XN stainless steel over a wide range of strain rates between 0.001 and 8300 s⁻¹ at temperatures from 77 to 1000 K. The concept of thermal activation analysis as well as the dislocation interaction mechanism is used in developing the plastic flow model for both the isothermal and adiabatic plastic deformation. In addition, the experimental observations of AL-6XN conducted by Nemat-Nasser et al. [Nemat-Nasser, S., Guo, W., Kihl, D., 2001. Thermomechanical response of AL-6XN stainless steel over a wide range of strain rates and temperatures, J. Mech. Phys. Solids 49, 1823–1846] are utilized in understanding the underlying deformation mechanisms. The plastic flow is considered in the range of temperatures and strain rates where diffusion and creep are not dominant, i.e., the plastic deformation is attributed to the motion of dislocations only. The modeling of the true stress–true strain curves for AL-6XN stainless steel is achieved using the classical secant modulus for the case of unidirectional deformation. The model parameters are obtained using the experimental results of three strain rates (0.001, 0.1, and 3500 s⁻¹). Good agreement is obtained between the experimental results and the model predictions. Moreover, the independency of the present model to the experiments used in the modeling is verified by comparing the theoretical results to an independent set of experimental data at the strain rate of 8300 s⁻¹ and various initial temperatures. Good correlation is observed between the model predictions and the experimental observations.     

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.     

Model identification and FE simulations: Effect of different yield loci and hardening laws in sheet forming

The bi-axial experimental equipment [Flores, P., Rondia, E., Habraken, A.M., 2005a. Development of an experimental equipment for the identification of constitutive laws (Special Issue). International Journal of Forming Processes] developed by Flores enables to perform Bauschinger shear tests and successive or simultaneous simple shear tests and plane strain tests. Flores investigates the material behavior with the help of classical tensile tests and the ones performed in his bi-axial machine in order to identify the yield locus and the hardening model. With tests performed on one steel grade, the methods applied to identify classical yield surfaces such as [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic materials. Proceedings of the Royal Society of London A 193, 281–297; Hosford, W.F., 1979. On yield loci of anisotropic cubic metals. In: Proceedings of the 7th North American Metalworking Conf. (NMRC), SME, Dearborn, MI, pp. 191–197] ones as well as isotropic Swift type hardening, kinematic Armstrong–Frederick or Teodosiu and Hu hardening models are explained. Comparison with the Taylor–Bishop–Hill yield locus is also provided. The effect of both yield locus and hardening model choices is presented for two applications: plane strain tensile test and Single Point Incremental Forming (SPIF).     

A robust shell element in meshfree SPH method

With the incorporation of total Lagrangian smoothed particle hydrodynamics(SPH) method equation and moving least square(MLS) function,the traditional SPH method is improved regarding the stability and consistency.Based on Mindlin-Ressiner plate theory,the SPH method simulating dynamic behavior via one layer of particles is applied to plate’s mid-plane,i.e.,a SPH shell model is constructed.Finally,through comparative analyses on the dynamic response of square,stiffened shells and cylindrical shells under various strong impact loads with common finite element software,the feasibility,validity and numerical accuracy of the SPH shell method are verified.Consequently,further researches on SPH shell may well pave the way towards solving problems involving dynamic plastic damage,tearing or even crushing.     

Modeling fracture in large scale shell structures

We consider a problem of modeling fracture and failure preceded by large scale yielding of ductile shells from the point of view of large-scale structural analysis. We place a special emphasis on the computational efficiency of the constitutive formulation. In this context, we seek the formulation embedded in the shell mechanics framework, which is both theoretically sound and easily implementable into a large-scale explicit dynamic finite element code without precluding vectorization or parallelization. This is achieved through the elasto-plastic damage constitutive model for finite-element analysis of plates and shells. The proposed damage model is purely phenomenological with a scalar damage parameter, which has no physical interpretation, except that it represents on a global scale the micromechanical changes the material undergoes during the process of necking and fracture. The localization leading to softening and fracture is represented by the damage calibration function with exponential damage growth after the onset of necking. The proposed phenomenological damage model uses a general plasticity and shell mechanics frameworks which makes it general and easily implementable into existing finite element codes. The proposed formulation has been implemented into the explicit dynamic finite element software code EPSA (Atkatsh et al., 1980, Atkatsh et al., 1983).