Interaction Between an Oscillating Sphere and a Thin Elastic Cylindrical Shell Filled with a Compressible Liquid. Internal Axisymmetric Problem

The problem on the interaction between a spherical body that oscillates in a prescribed manner and a thin elastic cylindrical shell filled with an ideal compressible liquid is formulated. It is assumed that the geometrical center of the sphere is located on the cylinder axis. The problem is solved based on the possibility of representing a partial solution of the Helmholtz equation written in cylindrical coordinates in terms of partial solutions in spherical coordinates, and vice versa. By satisfying the boundary conditions on the surfaces of the sphere and the shell, we obtain an infinite system of linear algebraic equations to determine the coefficients of expansion of the liquid-velocity potential into a Fourier series in terms of Legendre polynomials. The hydrodynamic characteristics of the liquid filling the cylindrical shell are determined and compared with the cases where a sphere oscillates in an infinite liquid and in a rigid cylindrical vessel     

Stability of nonlinear local axisymmetrical strains of shells of revolution

A brief review of the results of investigation of the stability of the axisymmetrical strains of elastic shells of revolution is contained in [1, 2]. In [3] the problem was formulated and solved for a round shell, uniformly loaded along its hinged edge by a radial compressive force. Below, this problem is formulated for an arbitrary shell of revolution with a uniformly compressed hinged edge. Results of its solution are given for conical and spherical shells.     

Spherical-symmetric steady-state response of piezoelectric spherical shell under external excitation

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｔｈｅｓｃｉｅｎｃｅａｎｄｔｅｃｈｎｏｌｏｇｙ ,ｍｏｒｅａｎｄｍｏｒｅｃｏｍｐｌｅｘｍｅｃｈａｎｉｓｍａｎｄｓｔｒｕｃｔｕｒｅｓａｒｅｐｕｔｉｎｔｏｕｓｅ,ｓｕｃｈａｓｆｌｅｘｉｂｌｅｒｏｂｏｔ,ｆｌｅｘｉｂｌｅａｒｍ ,ａｉｒｃｒａｆｔａｎｄｓｐａｃｅｓｔａｔｉｏｎ .Ｒｅｓｅａｒｃｈｅｒｓｐａｙｍｏｒｅａｔｔｅｎｔｉｏｎｔｏｔｈｅｐｒｏｂｌｅｍｏｆｅｆｆｅｃｔｉｖｅｄｙｎａｍｉｃｄｅｔｅｃｔｉｏｎａｎｄｃｏｎｔｒｏｌｏｆｓｕ…     

Investigation of changes in the spherical shell shape under the action of pulsed loading due to contact interaction with a rigid block

An axisymmetric problem of high strains in a spherical lead shell enclosed into an aluminum “spacesuit” under the action of pulsed loading is considered. The shell straining is described with the use of equations of mechanics of elastoviscoplastic media in Lagrangian variables, and the kinematic relations are determined in the current state metrics. Equations of state are taken in the form of equations of the flow theory with isotropic hardening. The problem is solved numerically by using the variational difference method and the “cross” explicit scheme of integration with respect to time. The influence of the yield stress as a function of the strain rate on changes in the shell shape is studied for different values of loading. The calculated final shape and residual strains are demonstrated to be in good agreement with experimental data.     

Relationships of the Timoshenko-type theory of thin shells with arbitrary displacements and strains

A new modified version of the Timoshenko theory of thin shells is proposed to describe the process of deformation of thin shells with arbitrary displacements and strains. The new version is based on introducing an unknown function in the form of a rotation vector whose components in the basis fitted to the deformed mid-surface of the shell are the components of the transverse shear vector and the extensibility in the transverse direction according to Chernykh. For the case with the shell mid-surface fitted to an arbitrary non-orthogonal system of curvilinear coordinates, relationships based on the use of true stresses and true strains in accordance with Novozhilov are obtained for internal forces and moments. Based on these relationships, a problem of static instability of an isotropic spherical shell experiencing internal pressure is solved. The shell is considered to be made either of a linear elastic material or of an elastomer (rubber), which is described by Chernykh’s relationships.     

Elastoplastic state of flexible spherical shells with a reinforced elliptic hole

The elastoplastic state of a thin spherical shell weakened by an elliptic hole is analyzed. Finite deflections are considered. The hole is reinforced with a thin ring. The shell is made of an isotropic homogeneous material. The load is internal pressure. A relevant problem is formulated and solved numerically with allowance for physical and geometrical nonlinearities. The distribution of stresses, strains, and displacements along the elliptic boundary and in the zone of their concentration is studied. The stress–strain state of the shell near the hole is analyzed Translated from Prikladnaya Mekhanika, Vol. 44, No. 12, pp. 93–101, December 2008.     

Axisymmetric oscillations of two spherical shells in a compressible fluid

To find the interaction between spherical shells at the frequency of their free oscillations in a fluid, we examine the problem of axisymmetric oscillations of two identical spherical shells under the assumption that the shell centers of curvature do not coincide. The solution is found for the cases of a compressible and an incompressible fluid by the series method with reduction to an infinite system of linear equations. A mathematical justification of the method used is presented.     

Numerical Stress–Strain Analysis of Shells Including the Nonlinear and Shear Properties of Composites

A presentation is made of the numerical results obtained in a stress–strain analysis of thin and nonthin orthotropic shells with due regard for the physical nonlinearity and small and nonsmall shear stiffness of composites. A spherical shell with a circular hole is used as an example to analyze how the above-mentioned factors affect the distribution of stresses and strains depending on the shell thickness for adopted deformation models (the Kirchhoff–Love and Timoshenko hypotheses). Generalized conclusions are drawn from which it is possible to decide which of the composite properties and shell models should be given more priority.     

Dynamic Elastoplastic Interaction between an Impactor and a Spherical Shell

Dynamic axisymmetric elastoplastic interaction between a massive body and a simply supported, circular segment of a spherical shell is studied. The problem of determining the contactinteraction force is formulated for the case of spherical and conical bodies. A nonlinear integral equation is derived for various models of local plastic compression using the equations of equilibrium of a membrane spherical shell written in terms of radial displacement of the shell. Numerical results are presented graphically.     

Transient response of a moving spherical shell in an acoustic medium

A ring-stiffened spherical shell is submerged in an acoustic medium. The shell is thin and elastic. The acoustic medium is inviscid, irrotational and compressible. The center of mass of the shell is subjected to a translational acceleration which is an arbitrary function of time. The absolute displacements of the shell are expressed in terms of the relative displacements and the displacement of the base of the shell, base being defined as the rigid ring placed at the equator. The motion of the acoustic medium is governed by the wave equation. The transient response of the shell is investigated numerically. The results are compared with the results of the in-vacuo response. The effects of the plane wave approximation and the base velocity on the transient response of the shell are studied. The numerical results show that the plane wave approximation accurately predicts the response of the shell in the acoustic medium for short times after excitation. The displacements of the shell in fluid are larger than those in vacuo. But when the base of the shell is restrained from translating, the displacements in fluid are smaller than those in vacuo. Therefore, base translation has a very significant effect on the transient response of the shells submerged in an acoustic medium.     

Linear and weakly nonlinear variable viscosity convection in spherical shells

A theoretical study of linear and weakly nonlinear thermal convection in a spherical shell is performed. The Boussinesq fluid is of infinite Prandtl number and its viscosity is temperature dependent. The linear stability eigenvalue problem is derived and solved by a shooting method assuming isothermal, stress-free boundaries, a self-gravitating fluid, and corresponding to two heating models. The first is heating from below, and the second is a model of combined heating from below and within, such that convection is described by a self-adjoint linear stability formulation. In addition, nonlinear, hemispherical, axisymmetric convection is computed by a finite volume technique for a shell with 0.5 aspect ratio. It is shown that 2-cell convection occurs as transcritical bifurcation for a viscosity constrast across the shell up to about 150. Motions with four cells are also possible. As expected, the subcritical range is found to increase with increasing viscosity contrast, even when the linear operator is self-adjoint.This research was supported by the AT&T Foundation.     

Analysis of a spherical tank under a local action

A spherical tank, being perfect as far as weight is concerned, is used in spacecraft, where the thin-walled elements (shells) are united by frames. Obviously, local actions on the shell and hence the stress concentration in the shell cannot be avoided. Attempts to make weight structure of the spacecraft perfect inevitably decrease the safetymargin of the components, which is possible only if the stress-strain state of the components is determined with a controlled error. A mathematical model of shell deformation mechanics is proposed for this purpose, and its linear differential equations are obtained with an error that does not exceed the error of Kirchhoff assumptions in the theory of shells. The algorithm for solving these equations contains procedures for estimating the convergence of the Fourier series and the series of the hypergeometric function with a prescribed error, and the problem can be solved analytically.     

Quasi-Green＇s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green＇s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi- Green＇s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the prob- lem. The mode shape differential equations of the free vibration problem of a simply- supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equa- tion, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green＇s function method.     

An Exact Analysis for Free Vibration of a Composite Shell Structure-Hermetic Capsule

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｍｐｏｓｉｔｅｓｔｒｕｃｔｕｒｅｓｃｏｎｓｉｓｔｉｎｇｏｆｓｈｅｌｌｓｏｆｒｅｖｏｌｕｔｉｏｎｈａｖｅｗｉｄｅａｐｐｌｉｃａｔｉｏｎｓｉｎｖａｒｉｏｕｓｅｎｇｉｎｅｅｒｉｎｇｆｉｅｌｄｓｓｕｃｈａｓａｅｒｏｓｐａｃｅ ,ｃｈｅｍｉｃａｌ,ｃｉｖｉｌ,ｍｅｃｈａｎｉｃａｌ,ｍａｒｉｎｅｅｎｇｉｎｅｅｒｉｎｇ .Ｄｕｅｔｏｔｈｅｍａｔｈｅｍａｔｉｃａｌｃｏｍｐｌｅｘｉｔｙｏｆｓｈｅｌｌｅｑｕａｔｉｏｎｓａｎｄｔｈｅｄｉｆｆｉｃｕｌｔｙｔｏｍａｔｃｈｃｏｎｄｉｔｉｏｎｓｏｆｔｈｅ…     

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.     

Stokes flow past an assemblage of axisymmetric porous spherical shell-in-cell models: effect of stress jump condition

The quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous concentric spherical shell-in-cell model is studied. Boundary conditions on the cell surface that correspond to the Happel, Kuwabara, Kvashnin and Cunningham/Mehta-Morse models are considered. At the fluid-porous interfaces, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid are used. The hydrodynamic drag force acting on the porous shell by the external fluid in each of the four boundary conditions on the cell surface is evaluated. It is found that the normalized mobility of the particles (the hydrodynamic interaction among the porous shell particles) depends not only on the permeability of the porous shells and volume fraction of the porous shell particles, but also on the stress jump coefficient. As a limiting case, 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 with jump.     

Canonical equations in the geometrically nonlinear theory of thin anisotropic shells

The functional in the principle of minimum potential energy of layered anisotropic shells with a nonlinear relationship between strains and displacements is transformed into a canonical integral that coincides with the functional in the Reissner principle. Partial forms of the functional are derived for problem formulations where the dimension can be reduced with respect to one of the coordinates. The canonical system of equations is linearized and then normalized. The boundary-value problem is solved by the numerical discrete-orthogonalization method. An anisotropic spherical shell under external compression is analyzed for stability as an example     

Free vibration analysis of a finite circular cylindrical shell in contact with unbounded external fluid

The free flexural vibration of a finite cylindrical shell in contact with external fluid is investigated. The fluid is assumed to be inviscid and irrotational. The cylindrical shell is modeled by using the Rayleigh–Ritz method based on the Donnell–Mushtari shell theory. The fluid is modeled based on the baffled shell model, which is applied to fluid–structure interaction problems. The kinetic energy of the fluid is derived by solving the boundary-value problem. The natural vibration characteristics of the submerged cylindrical shell are discussed with respect to the added virtual mass approach. In this study, the nondimensionalized added virtual mass incremental factor for the submerged finite shell is derived. This factor can be readily used to estimate the change in the natural frequency of the shell due to the presence of the external fluid. Numerical results showed the efficacy of the proposed method, and comparison with previous results showed the validity of the theoretical results.     

Analytical solution of cracked shell resting on elastic foundation

The elastostatic problem for cracked shallow spherical shell resting on linear elastic foundation is considered. The problem is formulated for a homogeneous isotropic material within the confines of a linearized shallow shell theory. By making use of integral transforms and asymptotic analysis, the problem is reduced to the solution of a pair of singular integral equations. The stress distribution obtained, around the crack tip, is similar to that of the elasticity solutions. The numerical results obtained agree well with those of previous work, where the elastic supports were neglected. The influences of the shell curvature and the modulus of subgrade reaction on the stress intensity factor are given.     

Axisymmetric spherical shell with variable wall thickness

This paper is engaged in research of the problem of axisymmetric spherical shell with variable wall thickness. The solutions for the problem are given for the spherical shell segment which does not contain the pole of sphere and the point of zero wall thickness. First Received July 24, 1982