Stability of thin spherical shells under dynamic loading

Moscow Aviation Institute. Translated from Prikladnaya Mekhanika, Vol. 29, No. 1, pp. 41–48, January, 1993.     

Buckling of thin spherical shells

Summary This paper deals with the buckling of a shallow spherical cap subjected to uniform edge moment and a clamped deep spherical shell under uniform pressure. The first problem is formulated in integral equations which are solved by an iterative procedure. The buckling moments are determined for a wide range of the shell geometrical parameter. The second problem is based on the concept that the highly deformed region around the apex is treated as a shallow spherical cap elastically supported by the rest of the shell. The stability of a thin sphere is treated as a special case. The results obtained in both problems are compared with existing solutions.

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Übersicht Es wird das Beulen sowohl für eine flache Kugelkalotte mit gleichförmigem Randmoment, als auch für eine tiefe Kugelschale unter gleichförmigem Druck untersucht. Die erstgenannte Aufgabe wird auf Integral-gleichungen zurückgeführt, die durch Iteration gelöst werden. Die Beulmomente werden für einen weiten Bereich der geometrischen Parameter der Schale bestimmt. Für die Lösung der zweiten Aufgabe wird angenommen, daß der stark verformte Teil der Schale in der Umgebung des zentralen Punktes als eine flache Kugelschale aufgefaßt werden kann, die elastisch von dem Rest der Schale getragen wird. Die Stabilität einer dünnen Kugel wird als Spezialfall betrachtet. In beiden Fällen werden die Ergebnisse mit vorhandenen Lösungen verglichen.

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The first problem in the analysis was sponsored by the National Research Council of Canada. The author is very grateful to Professor K. N. Tong for his illuminating suggestions regarding the second problem.     

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.     

Buckling behaviour of imperfect spherical shells

For the design of spherical shells under external pressure relatively few information can be found in corresponding codes and recommendations, e.g. not at all in the new draft of Eurocode 3 ENV 1993-1-6. Under this aspect, new design rules for these shells were developed, which take into account relevant details like boundary conditions, material properties, and imperfections. They are usually based on a large number of systematic numerical simulations to obtain results describing the load carrying behaviour and imperfection sensitivity of thin spherical shells. In addition, previous theoretical and experimental results are discussed. Based on the results, diagrams and design rules have been developed which might be used for new recommendations in the design concept of the Eurocode.     

PERFORATION OF PLASTIC SPHERICAL SHELLS UNDER IMPACT BY CYLINDRICAL PROJECTILES

The objective is to study the perforation of a plastic spherical shell impacted by a cylindrical projectile. First, the deformation modes of the shell were given by introducing an isometric transformation. Then, the perforation mechanism of the shell was analyzed and an analytical model was advanced. Based on Hamilton principle, the governing equation was obtained and solved using Runge-Kuta method. Finally, some important theoretical predictions were given to describe the perforation mechanism of the shell. The results will play an important role in understanding the perforation mechanism of spherical shells impacted by a projectile.     

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.     

Elastic instabilities for strain-stiffening rubber-like spherical and cylindrical thin shells under inflation

This paper is concerned with investigation of the effects of strain-stiffening on the classical limit point instability that is well-known to occur in the inflation of internally pressurized rubber-like spherical thin shells (balloons) and circular cylindrical thin tubes composed of incompressible isotropic non-linearly elastic materials. For a variety of specific strain-energy densities that give rise to strain-stiffening in the stress-stretch response, the inflation pressure versus stretch relations are given explicitly and the non-monotonic character of the inflation curves is examined. While such results are known for constitutive models that exhibit a gradual stiffening (e.g. exponential and power-law models), our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two phenomenological constitutive models that reflect limiting chain extensibility at the molecular level. It is shown that for materials with sufficiently low extensibility no limit point instability occurs and so stable inflation is then predicted for such materials. Potential applications of the results to the biomechanics of soft tissues are indicated.     

Experimental and theoretical/numerical investigations of thin films bonding strength

A laser spallation facility has been developed to measure the strength of planar interfaces between a substrate and a thin coating. This quantity is a central requirement in contemporary thin film and protective coatings technology and its successful measurement should improve the scientific/technological potential for the design of advanced composites, protective coatings of composites that operate in hostile environments, and in joining of dissimilar materials. The technique involves impinging a laser pulse of ultra short duration on the rear surface of the substrate, which is coated by a thin layer of energy absorbing metal such as Sn and Pb. The explosive evaporation of the metallic layer, confined between a fused quartz crystal and the substrate, induces a compressive shock wave, which propagates through the substrate toward the material interface. Upon reflection from the free surface of the coating, the pressure pulse is converted into a tensile wave which, under certain conditions, can lead to spallation at the interface. It is shown by mathematical simulation that atomic bond rupture is the mechanism of separation in this experiment. Since the interaction of laser energy with matter is a complicated, highly non-linear process, our investigations, at first, were based on measurement of the pressure pulse generated by the threshold flux level that leads to spallation, by using a micro-electronics device with a piezo-electric crystal, and on computation of the tensile stress experienced at the material interface, by numerical simulation of the induced stress wave propagation. Several substrate/coating (ceramic/ceramic and ceramic/metal) systems have been investigated such as, 1–15 μm SiC by CVD, 1–4 μm TiC and TiN by PVD coatings on sapphire substrates, as well as 1–2 μm Au, Sn and Ag coatings by sputtering on sapphire, fused quartz and glass substrates. For identically prepared specimens, the measured threshold energy levels are reproducible, thus leading to reproducible bond strength values, while the spall size, as expected, is dependent on the laser pulse energy level. Finally, the bond strength values obtained are in very good agreement with similar data derived by direct experimental techniques based on Laser-Doppler-Interferometry.     

大药片落锤撞击感度研究

设计建立了一种炸药大药片撞击感度试验方法,落锤质量为20 kg、落高度为0~15 m,对规格20 mm5 mm、重约2.8 g的大药片进行撞击感度测试。试验测试了两种典型炸药Tetryl和JOB-9003炸药的落锤撞击感度,落锤撞击Tetryl炸药和JOB-9003炸药的爆炸阈值落高分别约3.5 m和6.5 m。对落锤撞击JOB-9003炸药样品的过程进行了数值计算,计算结果与试验值相符。试验结果表明,该试验方法可以测量炸药的落锤撞击感度。     

Experimental study of thin cylindrical shells under local axial loadings

A thin cylindrical shell was subjected to local axial loads. These loads were transferred to the shell through uniformly spaced pads mounted inside the cylinder. The stress distribution in the vicinity of these pads was determined experimentally, and the results were then compared with theoretical findings. Parameters examined included the number and size of pads and the eccentricity of the loads applied to the pads with respect to the middle surface of the cylindrical skin.     

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.     

Experimental and numerical investigations of pressure drop in a rectangular duct with modified power law fluids

Numerical solutions are presented for fully developed laminar flow for a modified power law fluid (MPL) in a rectangular duct. The solutions are applicable to pseudoplastic fluids over a wide shear rate range from Newtonian behavior at low shear rates, through a transition region, to power law behavior at higher shear rates. The analysis identified a dimensionless shear rate parameter which, for a given set of operating conditions, specifies where in the shear rate range a particular system is operating, i.e. in the Newtonian, transition, or power law regions. The numerical results of the friction factor times Reynolds number for the Newtonian and power law region are compared with previously published results showing agreement within 0.05% in the Newtonian region, and 0.9% and 5.1% in the power law region. Rheological flow curves were measured for three CMC-7H4 solutions and were found to be well represented by the MPL constitutive equation. The friction factor times Reynolds number values were measured in the transition region for which previous measurements were unavailable. Good agreement was found between experiment and calculation thus confirming the validity of the analysis.     

Experimental and numerical investigations into flow features in an intake duct for the waterjet propulsion under mooring conditions

Acta Mechanica Sinica - The waterjet propulsion is widely applied in the marine vessels over 30 knots, and the intake duct is considered as an essential component that strongly relates to the...     

Experimental and numerical investigations of the deformation of soft materials under tangential loading

The surface ‘tensile test’, in which tangential loads are applied through surface mounted adhesive tapes, is a viable method for the assessment of mechanical properties of soft materials, particularly biological soft materials in vivo. In the present work the deformation pattern and force–displacement relationship in the surface tensile test were experimentally investigated using surface displacement analysis (SDA) and numerically simulated using finite element modelling. The experimental and FE results showed close agreement using silicone rubber as a model material. The force–displacement relationship was found to be dependent on the tape separations. SDA measurements and FE simulation showed that the displacement and strain fields were not uniform and the distribution pattern varies with tape separation. A combined experimental–numerical approach to inversely extract material properties using multiple tests with different length scales is proposed and assessed using a model material.     

撞击载荷作用下单层球面网壳动力响应模型实验研究

通过单层KIEWITT8型网壳模型在落锤撞击作用下的实验,研究单层网壳在撞击作用下的动力稳定性。利用动态应变仪和力传感器,获取了落锤撞击网壳时撞击力时程曲线、杆件轴力时程曲线和稳定临界力。利用高速摄影机拍摄了撞击历程、撞击失稳模态及破坏形态。结果表明：撞击作用为三角脉冲荷载形式,其最大幅值和脉宽与撞击冲量和网壳所处变形阶段的刚度性能相关;撞击荷载持续作用的时间为3.00~22.36 ms;撞击接触时间的突然增大对应着网壳的失稳;杆件开始响应的时刻比撞击力开始作用的时刻滞后0.2~0.4 ms;对失稳前的撞击,落锤回弹速度较大;对失稳时的撞击,落锤回弹速度很小;模型具有较大的后屈曲抗撞击能力,在顶点垂直撞击下没有发生连续断裂。