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.     

Asymptotic limits and wrinkling patterns in a pressurised shallow spherical cap

The asymmetric bifurcation problem for a shallow spherical cap is examined. The applied pressure can act either external or internal to the cap and both cases are treated here. Assuming a non-linear axisymmetric basic state, the linearised bifurcation equations for the pressurised shell are investigated in the limit when the thickness of the cap is much less than the maximum rise of the shell mid-surface. Within this regime the wrinkling patterns in both cases are confined to a narrow zone near the edge of the shell, making it possible to solve asymptotically the corresponding equations and derive analytical predictions for both the critical pressure and the corresponding number of wrinkles. Some comparisons with direct numerical simulations are included as well.     

Strain gradient plasticity solution for an internally pressurized thick-walled spherical shell of an elastic–plastic material

An analytical solution is presented for an internally pressurized thick-walled spherical shell of an elastic strain-hardening plastic material. A strain gradient plasticity theory is used to describe the constitutive behavior of the material undergoing plastic deformations, whereas the generalized Hooke’s law is invoked to represent the material response in the elastic region. The solution gives explicit expressions for the stress, strain and displacement components. The inner radius of the shell enters these expressions not only in non-dimensional forms but also with its own dimensional identity, unlike classical plasticity-based solutions. As a result, the current solution can capture the size effect. The classical plasticity-based solution of the same problem is shown to be a special case of the present solution. Numerical results for the maximum effective stress in the shell wall are also provided to illustrate applications of the newly derived solution. The new solution can be used to construct improved expanding cavity models in indentation mechanics that incorporate both the strain-hardening and indentation size effects.     

Bifurcation phenomena of a biphasic compressible hyperelastic spherical continuum

A basic linear bifurcation problem is solved for a representative biphasic composite spherical cell. Setting is that of an internal inclusion that expends with an external shell made of a different material. Exact rate equations are derived in the framework of hyperelastic continuum mechanics. Sample examples are solved numerically revealing sensitivity of critical strain levels and bifurcated mode pattern to strength differential between both phases. Mathematical framework can be adapted to other types of constitutive responses.     

Generalization of the Lamé problem for three-stage decelerated corrosion process of an elastic hollow sphere

The paper presents an analytical solution to Lamé's problem for a hollow sphere with unknown evolving boundaries. The double-sided uniform corrosion of a linearly elastic thick-walled spherical shell under internal and external pressure is considered. It is assumed that the corrosion rates are piecewise linear functions of the maximum principal stress on the related surface, and exponentially decaying with time. The corrosion process is supposed to be divided into three successive stages: constant rate double-sided corrosive wear, a stage of corrosion accelerated on only one of the surfaces of the shell, and a double-sided mechanochemical corrosion. Closed-formed expressions for all the consecutive stages are obtained with their junction points (corresponding to stress corrosion thresholds) being taken into account.     

Uniqueness and stability of rigid-plastic cylinders under internal pressure and axial tension

Uniqueness of deformation and stability of the equilibrium configuration of a long closed-ended cylinder of rigid-plastic material, obeying the von Mises yield criterion, are examined under internal pressure and axial tensile load. Sufficient conditions are derived for uniqueness of the current state of the finitely deformed cylinder. By considering a material model of the Ramberg-Osgood type, it is shown that uniqueness is guaranteed up to a stage when either of the loads (or both) attains a maximum. For such a material model, “pressure-tension interaction curves” are obtained for some values of the wall-ratio and the strain-hardening index. Under internal pressure and small tension, however, the possibility of a bifurcation preceding a stability loss is shown to exist for certain cylinder geometry and material hardening properties.     

Torsional buckling of spherical shells under circumferential shear loads

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Snap-through buckling of eccentrically stiffened shallow spherical caps

The problem of snap-through buckling of a clamped, eccentrically stiffened shallow spherical cap is considered under quasi-statically applied uniform pressure and a special case of dynamically applied uniform pressure. This dynamic case is the constant load infinite duration case (step time-function) and it represents an extreme case of blast loading-large decay time, small decay rate.The analysis is based on the nonlinear shallow shell equations under the assumption of axisymmetric deformations and linear stress-strain laws. The eccentric stiff eners are disposed orthogonally along directions of principal curvature in such a way that the smeared mass, and extensional and flexural stiffnesses are constant. The stiffeners are also taken to be one-sided with constant eccentricity, and the stiffener-shell connection is assumed to be monolithic.The method developed in an earlier paper is employed. In this method, critical pressures are associated with characteristics of the total potential surface in the configuration space of the generalized coordinates.In addition, buckling of the complete thin eccentrically stiffened spherical shell under uniform quasi-statically applied pressure is considered, and these results are used to check the numerical answers. The complete spherical shell is stiffened in the same manner as the shallow cap.The results are presented in graphical form as load parameter vs initial rise parameter. Geometric configurations corresponding to isotropic, lightly stiffened, moderately stiffened and heavily stiffened geometries are considered. By lightly stiffened geometry one means that most of the extensional stiffness is provided by the thin shell. A computer program was written to solve for critical pressures. The Georgia Tech Univac 1108 high speed digital computer was used for this purpose.     

Mechanics of sheet metal formed by hydraulic pressure into axisymmetrical shells

A new approach is made to the process known as the bulge test by deducing, from the membrane-stress formula, the relationship between the geometry and the stresses in a thin shell under internal pressure. A quantitative measure, called prolateness, is introduced for the deviation of the local geometry from the perfect spherical shape, and it is shown that prolateness is a geometrical property distinct from curvature. Surfaces of constant prolateness are examined both for their shapes and for the stresses in thin shells of such shapes. The special experimental techniques used in this project are explained and the experimental results presented. It is found that the metal shell formed is perfectly spherical only at the pole and along a circle of constant latitude. Inside this circle, the shell is prolate (or more pointed than a sphere) and outside it, it is oblate (or more flattened than a sphere). As the forming progresses, this circle of perfect spherical surface expands till the process becomes unstable and the whole surface becomes prolate. The stress distributions at various stages of the forming process are also shown and discussed.     

Behaviour of an incrementally elastic thick walled sphere under internal and external pressure

The behaviour of a thick walled sphere underinternal and external pressure is considered. The material of the sphere is assumed to obey an incrementally elastic constitutive law. There is no restriction on the size of the deformation and a solution is given in terms of special functions associated with the non-linear differential equations of the problem.As a numerical example the behaviour of a spherical shell, subjected to internal pressure, is described. It is shown that at a certain critical pressure instability of the second kind (inflation) is obtained.     

电活性聚合物球壳的力电不稳定性

应用连续介质力学有限变形理论,分析了不可压电活性聚合物球壳在外加电场及内压作用下发生非对称变形的力电不稳定性问题。文中给出了不同外加电场下球壳的变形曲线和应力分布曲线, 结果表明对壁厚小于临界壁厚值的薄壁球壳,当内压大于临界内压值时,球壳可以产生不稳定的非对称变形。文中求得了球壳发生不稳定变形的临界壁厚及临界内压,探讨了外加电场对两个临界值的影响规律,同时讨论了外加电场对球壳中应力分布的影响。     

Lebesgue measure for creep

The Lebesgue strain measure for creep has been developed by taking the Lebesgue integral of the function of a weighted function instead of the power of a weighted function as was taken in Seth's strain measure. The problem of a spherical shell under internal pressure is considered and it is shown that the results obtained by using Seth's concept of measure can be derived from the more general analysis presented herein.     

Stability and limit states of elastoplastic spherical shells under static and dynamic loading

The problem of elastoplastic deformation, buckling, and postcritical behavior of spherical shells is solved using a finite element method and a cross-type explicit scheme of time integration. Stability problems for hemispherical shells under external pressure and compression between rigid plates are considered. The influence of holes and boundary conditions on shell deformation is investigated. It is shown that the calculation results are in good agreement with experimental data.     

Two new expanding cavity models for indentation deformations of elastic strain-hardening materials

Two expanding cavity models (ECMs) are developed for describing indentation deformations of elastic power-law hardening and elastic linear-hardening materials. The derivations are based on two elastic–plastic solutions for internally pressurized thick-walled spherical shells of strain-hardening materials. Closed-form formulas are provided for both conical and spherical indentations, which explicitly show that for a given indenter geometry indentation hardness depends on Young’s modulus, yield stress and strain-hardening index of the indented material. The two new models reduce to Johnson’s ECM for elastic-perfectly plastic materials when the strain-hardening effect is not considered. The sample numerical results obtained using the two newly developed models reveal that the indentation hardness increases with the Young’s modulus and strain-hardening level of the indented material. For conical indentations the values of the indentation hardness are found to depend on the sharpness of the indenter: the sharper the indenter, the larger the hardness. For spherical indentations it is shown that the hardness is significantly affected by the strain-hardening level when the indented material is stiff (i.e., with a large ratio of Young’s modulus to yield stress) and/or the indentation depth is large. When the indentation depth is small such that little or no plastic deformation is induced by the spherical indenter, the hardness appears to be independent of the strain-hardening level. These predicted trends for spherical indentations are in fairly good agreement with the recent finite element results of Park and Pharr.     

Instability and failure of internally pressurized ductile metal cylinders

Forlong, ductile, thick-walled tubes under internal pressure instabilities and final failure modes are studied experimentally and theoretically. The test specimens are closed-end cylinders made of an aluminum alloy and of pure copper and the experiments have been carried out for a number of different initial external radius to internal radius ratios. The experiments show necking on one side of the tubes at a stage somewhat beyond the maximum internal pressure. All tubes, except for one aluminum alloy tube, failed by shear fracture under decreasing pressure. The aluminum alloy tubes exhibited localized shear deformations in the neck region prior to fracture and also occasionally surface wave instabilities. The numerical investigation is based on an elastic-plastic material model for a solid that develops a vertex on the yield surface, using representations of the uniaxial stress-strain curves found experimentally. In contrast to the simplest flow theory of plasticity this material model predicts shear band instabilities at a realistic level of strain. A rather sharp vertex is used in the material model for the aluminum alloy, while a more blunt vertex is used to characterize copper. The theoretically predicted bifurcation into a necking mode, the cross-sectional shape of the neck, and finally the initiation and growth of shear bands from the highly strained internal surface in the neck region are in good agreement with the experimental observations.     

Radial oscillations of nonhomogeneous,thick-walled cylindrical and spherical shells subjected to finite deformations

The infinitesimal breathing motions of long cylindrical tubes and hollow spherical shells of arbitrary wall thickness subjected to a finite deformation field caused by uniform internal and/or external pressures are investigated. A neo-Hookean material with a material constant varying continuously along the radial direction is used. The shell is first subjected to finite static deformations and is then exposed to a secondary dynamic displacement field. Based on the theory of small deformations superposed on large deformations, closed form expressions are obtained for the frequency of small oscillations about the highly prestressed state. Frequency versus initial deformation parameter curves are given for several nohomogeneity functions and for various wall thicknesses.     

棘轮安定曲线在不锈钢压力容器应变强化技术中的应用

应变强化是不锈钢压力容器结构实现轻型化的重要途径,而应变强化内压的确定则是应变强化技术的核心。为了能够更准确有效地达到结构应变强化的目的,对循环加载的应变强化方式进行了研究。通过304不锈钢材料的室温单轴棘轮试验,建立了应力比R0条件下的棘轮安定曲线。根据Mises等效原理,利用该曲线通过一次弹塑性有限元分析直接获得结构在循环载荷作用下的强化内压和产生的塑性应变,与试验结果吻合较好,说明运用循环加载的应变强化方式可以有效地达到应变强化的目的。在达到相同的应变强化程度要求下,该方法降低了强化内压,因此可以减小过载加压的风险。     

Localized bulging in an inflated cylindrical tube of arbitrary thickness – the effect of bending stiffness

We study localized bulging of a cylindrical hyperelastic tube of arbitrary thickness when it is subjected to the combined action of inflation and axial extension. It is shown that with the internal pressure P and resultant axial force F viewed as functions of the azimuthal stretch on the inner surface and the axial stretch, the bifurcation condition for the initiation of a localized bulge is that the Jacobian of the vector function $(P,F)$ should vanish. This is established using the dynamical systems theory by first computing the eigenvalues of a certain eigenvalue problem governing incremental deformations, and then deriving the bifurcation condition explicitly. The bifurcation condition is valid for all loading conditions, and in the special case of fixed resultant axial force it gives the expected result that the initiation pressure for localized bulging is precisely the maximum pressure in uniform inflation. It is shown that even if localized bulging cannot take place when the axial force is fixed, it is still possible if the axial stretch is fixed instead. The explicit bifurcation condition also provides a means to quantify precisely the effect of bending stiffness on the initiation pressure. It is shown that the (approximate) membrane theory gives good predictions for the initiation pressure, with a relative error less than 5%, for thickness/radius ratios up to 0.67. A two-term asymptotic bifurcation condition for localized bulging that incorporates the effect of bending stiffness is proposed, and is shown to be capable of giving extremely accurate predictions for the initiation pressure for thickness/radius ratios up to as large as 1.2.     

Effect of elastomer slight compressibility

This paper is concerned with the constitutive equation for slightly compressible elastic material under finite deformations. We show that material slight compressibility can be effectively taken into account in the case of high hydrostatic pressure or highly confined material. In all other situations the application of the incompressible and nearly incompressible material theories gives practically the same results. Therefore it is of interest to consider the problem in which allowing for material slight compressibility leads to results essentially different from those obtained with help of the incompressible material model. In the present paper this difference has been demonstrated for the problem of high hydrostatic pressure causing an increase of the ‘bulk’ and ‘shear’ material moduli. The behavior of a long hollow cylinder of real material under finite deformations is analyzed. The cylinder is subjected to internal pressure and axial and circular displacements at the outer surface. This problem has been solved analytically using the small parameter method. The solution obtained predicts a decrease of the axial and circular displacements of the outer surface under fixed shear (axial and circular) forces when the internal pressure is applied.     

Identification of mechanical properties of inhomogeneous materials

In the present paper, the stress-strain state of tubes made of inhomogeneous elastic materials is considered. We discuss what causes the onset of inhomogeneity and solve a problem for a tube consisting of an inhomogeneous and a homogeneous layer. It is shown how the variations in the thickness ratio of the homogeneous and inhomogeneous material layers affect the values of the longitudinal and circular deformations on the external surface of the tube under the action of constant internal pressure; it is noted that this effect can be used to monitor the pipeline state and to ensure its safe operation. A method for identifying mechanical properties of deformable inhomogeneous materials is proposed; this method is based on the use of thick-walled tubular specimens in calibration tests, which is especially convenient when analyzing the action of aggressive media or radiation on the properties of deformable materials.