A study of different modes of collapse in metallic hemispherical shells resting on flat platen and compressed with hemispherical nosed indenter

The paper presents experimental and analytical studies on axial compression of aluminium spherical shells having Radius/wall thickness (R/t) ratios between 23 and 135. Quasi-static compressive load was applied centrally and with offset through a indenter having diameter of 22 mm. Testing was carried out on an INSTRON machine having 250 T capacity. Shells having different radius and wall thickness were tested, to classify their modes of collapse and their corresponding energy absorption mechanism. In experiments shells of lower R/t values were found to collapse due to formation of an inward dimple associated with a rolling plastic hinge in central as well as in offset loading. On the other hand, shells of higher R/t values were collapsed initially with formation of an axisymmetric inward dimple, but in later stage of compression showed buckling of non-symmetric shape consisting of integral number of lobes and stationary plastic hinges. The stationary hinges were formed between consecutive lobes. Experimental observations are used to propose an analytical model for prediction of load–compression and energy–compression curves. The results obtained from analytical model compared with the experimental results and found match fairly well.     

Effect of thickness inhomogeneities in internally pressurized elastic-plastic spherical shells

The behaviour of elastic-plastic spherical shells under internal pressure is investigated numerically for thickness-to-radius ratios ranging from cases of thin shells to very thick shells. The shells under consideration are made of strain-hardening elastic-plastic material with a smooth yield-surface. Attention is restricted to axisymmetric deformations, and results are presented for initial thickness inhomogeneities in various axisymmetric shapes. For smooth thickness-variations in the shape of the critical bifurcation mode, the reduction in maximum pressure is studied together with the distribution of deformations in the final collapse mode. Also, the possibility of flow localization due to more localized, initially thin regions on a spherical shell is investigated.     

Experimental and numerical investigations into collapse behaviour of thin spherical shells under drop hammer impact

A study of the collapse behaviour of hemi spherical and shallow spherical shells and their modes of deformation under impact loading are presented in this paper. Aluminium spherical shells of various radii and thicknesses were made by spinning. These were subjected to impact loading under a drop hammer and the load histories were obtained in all the cases. Three-dimensional numerical simulations were carried out for all the tested specimen geometries using LS-DYNA^®. Material, geometric and contact nonlinearities were incorporated in the analysis. The uni-axial stress–strain curve for the material was obtained experimentally and was assumed to be piecewise linear in the plastic region. The results from impact experiments are used for the validation of the numerical simulations. Three distinct modes of deformation, namely local flattening, inward dimpling and formation of multiple numbers of lobes were analysed and influence of various parameters on these modes is discussed.     

Ratcheting and wrinkling of tubes due to axial cycling under internal pressure: Part I experiments

Under compression, pressurized tubes thick enough to deform plastically buckle into an axisymmetric wrinkling mode. The wrinkle amplitude is initially small but with persistent compression grows inducing a gradual reduction in axial rigidity and eventually causing a limit load instability, which is followed by collapse. The onset of buckling and collapse can be separated by a strain level of a few percent. This work investigates whether a tube that develops small-amplitude wrinkles can be subsequently collapsed by persistent axial cycling. Part I presents the results from a set of experiments on super-duplex tubes with D/t of 28.5 loaded as follows. A tube is pressurized and then compressed into the plastic range to a level that initiates wrinkling. It is then cycled under stress control about a compressive mean stress while the pressure is kept constant. The combined loads cause simultaneous ratcheting in the hoop and axial directions as well as a gradual growth of the wrinkles. At some stage the amplitude of the wrinkles starts to grow exponentially with the number of cycles N leading to localization and collapse. The rate of ratcheting and the number of cycles to collapse depend on the initial compressive pre-strain, the internal pressure, and the stress cycle parameters all of which were varied sufficiently to generate an adequate data base. Interestingly, in all cases collapse was found to occur when the accumulated average strain reached the value at which the tube localizes under monotonic compression. A shell model coupled to a specially calibrated plasticity model that can reproduce the biaxial ratcheting exhibited in the problem are presented in Part II. The model is first evaluated by comparison to the experiments and then used to study parametrically cyclic loading histories seen in buried pipelines.     

Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

Non-linear vibrations of free-edge shallow spherical shells are investigated, in order to predict the trend of non-linearity (hardening/softening behaviour) for each mode of the shell, as a function of its geometry. The analog for thin shallow shells of von Kármán's theory for large deflection of plates is used. The main difficulty in predicting the trend of non-linearity relies in the truncation used for the analysis of the partial differential equations (PDEs) of motion. Here, non-linear normal modes through real normal form theory are used. This formalism allows deriving the analytical expression of the coefficient governing the trend of non-linearity. The variation of this coefficient with respect to the geometry of the shell (radius of curvature R, thickness h and outer diameter 2a) is then numerically computed, for axisymmetric as well as asymmetric modes. Plates (obtained as R→∞) are known to display a hardening behaviour, whereas shells generally behave in a softening way. The transition between these two types of non-linearity is clearly studied, and the specific role of 2:1 internal resonances in this process is clarified.     

Non-linear vibrations of free-edge thin spherical shells: Experiments on a 1:1:2 internal resonance

This study is devoted to the experimental validation of a theoretical model of large amplitude vibrations of thin spherical shells described in a previous study by the same authors. A modal analysis of the structure is first detailed. Then, a specific mode coupling due to a 1:1:2 internal resonance between an axisymmetric mode and two companion asymmetric modes is especially addressed. The structure is forced with a simple-harmonic signal of frequency close to the natural frequency of the axisymmetric mode. The experimental setup, which allows precise measurements of the vibration amplitudes of the three involved modes, is presented. Experimental frequency response curves showing the amplitude of the modes as functions of the driving frequency are compared to the theoretical ones. A good qualitative agreement is obtained with the predictions given by in the model. Some quantitative discrepancies are observed and discussed, and improvements of the model are proposed.     

The Random Nature of Column Failure

ABSTRACT

An approximate quasi-static theory is developed to predict large plastic deformation and perforation of spherical shells subjected to impact by blunt-ended projectiles at normal incidence. Based on experimental observations for quasi-static load-displacement characteristics, the problem of a spherical shell under normal impact by a flat-ended missile may be analyzed through the solutions to that of an equivalent circular plate struck transversely by the same missile. It is shown that the approximate theoretical predictions with a = 30 (here a is an empirical constant) are in good agreement with experimental data, in terms of maximum permanent transverse displacements and dimple radii. Furthermore, a theoretical formula for ballistic limits of spherical shells under missile impact is presented and the range of applicability of the theory is discussed.     

Influence of fibre orientation and stacking sequence on petalling of glass/polyester composite cylindrical shells under axial compression

The energy absorbing capability of FRP composite cylindrical tubes used as energy absorbers, by destroying itself progressively, depends on the way in which the tube material is crushed i.e., trend of petalling. This paper investigates the influence of fibre orientation and stacking sequence on the petal formation and specific energy absorption (SEA) of four and six-ply, 0°/90° glass/polyester composite cylindrical shells under axial compression. Number of petals formed and the trend of petalling are changed with proportion of axial (0°) and circumferential (90°) fibre content and stacking sequence in the tubes. In the tubes undergo petalling, presence of axial fibres close to inner surface and the proper proportion of circumferential fibres close to outer surface of the tube wall lead to higher energy dissipation. The axial fibres placed nearer to outer surface leads to more number of petal formation, leading to a stable crushing mechanism. The contribution of mode I strain energy release rate (G_Ic) to the energy dissipation in the form of circumferential delamination is also studied with double cantilever beam (DCB) tests. Analytical model which considers petalling is developed, and used to predict the mean crush load and SEA of cylindrical composite shells under axial compression. Results from the analytical model agree well with experimental results and are presented.     

A numerical axisymmetric collapse analysis of viscoplastic cylindrical shells under axial compression

Circularcylindrical shells are frequently used as structural components because of their high strength and their ability to absorb energy during complete structural collapse. Total collapse analyses have mainly been based on experimental work and approaches inspired by this. However, in the last few years, powerful numerical tools have been available and numerical collapse analyses have become more attractive. This paper presents results from an axisymmetric numerical collapse analysis. The analysis is based on a finite rotation shell theory accounting for contact between the shell walls. The strains are assumed to remain small and the shell material is described by an elastic–viscoplastic model. The sensitivity of the collapse behaviour is demonstrated with respect to parameters such as initial imperfections, thickness of the shell, material parameters and rate of deformation. Comparisons between the results numerically obtained and approaches found in the literature are presented. Good agreement was found for the folding length of the developed collapse pattern whereas small differences between the mean crushing loads was observed. Furthermore, it was noted that the developed collapse pattern was strongly dependent on the strain hardening of the material.     

Imperfection sensitivity of an orthotropic spherical shell under external pressure

In the first part of this paper, rib-stiffened thin-walled spherical shells under external hydrostatic pressure are optimized using classical approximate methods and empirical knock-down-factors. In the second part of the paper, the influence of known imperfections is investigated.The thin-walled spherical shells under external pressure are very sensitive to geometrical imperfections. Hoff recognized that for entire isotropic spherical shells the more likely imperfection will be a local circular dent, which for such shells, can always be considered as an axisymmetric one. Hoff's idea has been further investigated by Koga–Hoff, Galletly et al. These results showed that for a given depth of an imperfection a critical size of the corresponding circular dent exists, giving the minimum for the actual load carrying capacity of the shell.This paper suggests to extend Hoff's theory to isogrid and waffle-grid stiffened spherical shells. The issue of these investigations is a set of knock-down-factors plotted versus imperfection amplitude related to the total thickness of the rib-stiffened (isogrid or waffle-grid) shell. These curves fit reasonably with those established for isotropic shells by Hoff et al. or by Koiter, and enable to estimate the jeopardy of measured actual dents.     

Theory of limit equilibrium of bimetallic shells of revolution and circular plates

Bimetallic shells and plates are widely used in technology (see [1, 2]). An investigation into the flexure and stability of thin shells and various types of loading within the limits of elasticity has been carried out in [3]. An investigation into the load-carrying capacity of cylindrical bimetallic shells made of materials which equally resist tension and compression was carried out in [4]. In many cases the materials of the base and plating layers of bimetallic constructions possess substantially different plastic resistance under tension and compression [5]. The given paper is devoted to the investigation of the load-carrying capacity of bimetallic axisymmetric shells which are made of materials that have different resistances to tension and compression; it is also devoted to the assessment of their economy in comparison with homogeneous shells.     

Some improvements on the energy absorbed in axial plastic collapse of hollow cylinders

The control of the plastic flow mechanism during axial collapse of metallic hollow cylinders is of particular interest in the present work for the absorbed energy. Hence, an experimental methodology is developed during which some different tubular structures are loaded under compressive quasi-static strain rate. These structures of various geometrical parameters η = R_m/t and λ = R_m/L (R_m: mean radius, L: initial length and t: thickness of tube) are made either from copper or aluminum considered as an energy dissipating system. At this point, the effects of both parameters on the mean collapse load and absorbed energy are appropriately studied. The role of η ratio, which has been largely investigated previously, is studied again. Moreover, it is found that the λ ratio has a non-negligible influence on the deformation mode for a given η. It is well known that the absorbed energy is influenced by the deformation mechanism, i.e., for the axisymmetric mode, the related absorbed energy becomes more important than that of the diamond fold mechanism for a given cylinder. Accordingly, to maximize the absorbed energy, two different structural solutions, namely fixed-ends and subdivided structure, are developed for encouraging the axisymmetric mode. It is convenient to consider the classical axial collapse situation (noted as free-ends) as a comparison reference. In this work, it is recognized that the subdivided solution is relatively the best solution. As a result, the absorbed energy increases up to 21% in comparison with the free-ends situation for copper tubes.     

A two-dimensional model for linear elastic thick shells

In this paper, a two-dimensional model for linear elastic thick shells is deduced from the three-dimensional problem of a shell thickness 2ε, ε > 0. From different scalings on the tangent and normal components of the displacement u^ε as widely used in recent works, the limit displacement appears to be Kirchhoff–Love displacement of a different type. It contains additional terms to those found in the Reissner–Mindlin model and satisfies more general equations containing the classical terms found in the literature and some new terms related to the third fundamental form. Such terms could not be well handled in the usual framework. Shear stresses across the thickness are also computed. This model appears to be appropriate to handle stiffened shells which, in fact, cannot be considered uniformly as shallow shells. As a by-product it also lays the mathematical background to justify the Reissner–Mindlin model for plates and will probably contribute to a better understanding of the locking phenomenon encountered in computational mechanics.     

Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects

This paper presents an analytical approach to investigate the non-linear axisymmetric response of functionally graded shallow spherical shells subjected to uniform external pressure incorporating the effects of temperature. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and compatibility equations for shallow spherical shells are derived by using the classical shell theory and specialized for axisymmetric deformation with both geometrical non-linearity and initial geometrical imperfection are taken into consideration. One-term deflection mode is assumed and explicit expressions of buckling loads and load-deflection curves are determined due to Galerkin method. Stability analysis for a clamped spherical shell shows the effects of material and geometric parameters, edge restraint and temperature conditions, and imperfection on the behavior of the shells.     

Experimental creep buckling of aluminum cylinders in axial compression

Creep-buckling tests were conducted on aluminum alloy 2024-0 circular cylinders in axial compression at 500° F having nominalR/t values of 90 and 50. Creep-buckling times for a variety of applied creep-stress values were compared with theoretical predictions of Gerard's unified theory of creep buckling of columns, plates and shells. In this theory, creep-buckling solutions are analogous to plastic-buckling solutions, provided that the material parameters used in the theoretical relation are developed from constant-strain-rate stress-strain data derived by a graphical process from compressive-creep data. The theoretical data were evaluated using appropriate classical plastic-buckling theory and previously obtained creep data on the 2024-0 aluminum material at 500° F. End shortening of the cylinders was autographically recorded during the tests and creep-buckling times were obtained from an analysis of the end-shortening record. A comparison of theory and test data indicated that the theory was somewhat conservative in predicting creep-buckling times. The discrepancy may have been due, in part, to the uncertainty in determining the precise time at which the experimental cylinders buckled. The cylinders withR/t～90 buckled in the axisymmetric mode for the lower creep stresses while, forR/t～50, all buckling occurred in the axisymmetric mode.     

On some general variational principles for creep with applications to thin shells

A constitutive relation for a viscous material subject to small strains but finite rotations is postulated and associated variational theorems are formulated. These are similar to the principle of minimum of potential energy and the Hellinger-Reissner theorem of an elastic solid. The derivation of strain-displacement relations for thin shells subject to small strains but moderately large rotations are given. On this basis a mixed variational principle for thin viscous shells is developed. For the problem of creep collapse of long cylindrical shells under external pressure it is demonstrated that the mixed variational principle may be advantageous compared to other variational theorems. A comparison with the creep collapse theory of Hoff et al. is given.     

Bifurcation buckling of shells of revolution including large deflections,plasticity and creep

A summary is first presented of the conceptual difficulties and paradoxes surrounding plastic bifurcation buckling analysis. Briefly discussed are nonconservativeness, loading rate during buckling, and the discrepancy of buckling predictions with use of J₂ flow theory vs J₂ deformation theory. The axisymmetric prebuckling analysis, including large deflections, elastic-plastic material behavior and creep is summarized. Details are given on the analysis of nonsymmetric bifurcation from the deformed axisymmetric state. Both J₂ flow theory and J₂ deformation theory are described. The treatment, based on the finite-difference energy method, applies to layered segmented and branched shells of arbitrary meridional shape composed of a number of different elastic-plastic materials. Numerical results generated with a computer program based on the analysis are presented for an externally pressurized cylinder with conical heads.     

Nonlinear dynamic response and dynamic buckling of shallow spherical shells with circular hole

In this paper, the nonlinear equations of motion for shallow spherical shells with axisymmetric deformation including transverse shear are derived. The nonlinear static and dynamic response and dynamic buckling of shallow spherical shells with circular hole on elastically restrained edge are investigated. By using the orthogonal point collocation method for space and Newmark-β scheme for time, the displacement functions are separated and the nonlinear differential equations are replaced by linear algebraic equations to seek solutions. The numerical results are presented for different cases and compared with available data.     

Study of Near Wall Coherent Flow Structures on Dimpled Surfaces Using Unsteady Pressure Measurements

Unsteady pressure measurements have been performed inside a spherical dimple in a narrow channel for turbulent flow at Re _D ?=?40,000 with the aim to study coherent vortex structures and to get a deep insight into flow physics. Results confirm the formation of asymmetric coherent vortex structures switching between two extreme positions. Analysis of the pressure temporal distributions and correlation functions shows the presence of the anti-phase motion inside the dimple. Typical power laws of the pressure fluctuation energy spectrum ω ^???1 and ω ^???7/3 are reproduced.     

A hybrid detailed level and configuration accounting model for investigating the radiative opacity of gold plasmas with open 4d and 4f shells

A hybrid model combining the detailed level accounting (DLA) and detailed relativistic configuration accounting (DCA) methods is developed to investigate the radiative opacity of gold plasmas with open 4d and 4f shells. Due to the collapse of 4f shells, the configurations with multi-electron excited from 4d and 4f shells are bound and can form a huge number of fine-structure levels and detailed transition lines. A full DLA calculation is time-consuming and intractable and thus a hybrid DLA and DCA method is needed. To obtain accurate radiative opacity, the transitions within the collapsed orbitals and transitions to the relatively lowly excited orbitals are treated by a DLA method, while the transitions to the higher excited orbitals are treated by a DCA method. As an illustrative example, the spectrally resolved, Rosseland and Planck mean opacity of gold plasmas at 100 eV and 0.001 g/cm³ are calculated by using the hybrid model. The present results are compared with those obtained by pure DCA and average atom models, where large discrepancies in the line intensities and positions are found for the strongest 4d–4f transitions due to the collapse of 4f shells indicating the importance of detailed treatment to obtain the accurate opacity.