Limit analysis of viscoplastic thick-walled cylinder and spherical shell under internal pressure using a strain gradient plasticity theory

Plastic limit load of viscoplastic thick-walled cylinder and spherical shell subjected to internal pressure is investigated analytically using a strain gradient plasticity theory. As a result, the current solutions can capture the size effect at the micron scale. Numerical results show that the smaller the inner radius of the cylinder or spherical shell, the more significant the scale effects. Results also show that the size effect is more evident with increasing strain or strain-rate sensitivity index. The classical plastic-based solutions of the same problems are shown to be a special case of the present solution.

Thermo-magnetic analysis of thick-walled spherical pressure vessels made of functionally graded materials

This study presents an analytical solution of thermal and mechanical displacements, strains, and stresses for a thick-walled rotating spherical pressure vessel made of functionally graded materials (FGMs). The pressure vessel is subject to axisymmetric mechanical and thermal loadings within a uniform magnetic field. The material properties of the FGM are considered as the power-law distribution along the thickness. Navier’s equation, which is a second-order ordinary differential equation, is derived from the mechanical equilibrium equation with the consideration of the thermal stresses and the Lorentz force resulting from the magnetic field. The distributions of the displacement, strains, and stresses are determined by the exact solution to Navier’s equation. Numerical results clarify the influence of the thermal loading, magnetic field, non-homogeneity constant, internal pressure, and angular velocity on the magneto-thermo-elastic response of the functionally graded spherical vessel. It is observed that these parameters have remarkable effects on the distributions of radial displacement, radial and circumferential strains, and radial and circumferential stresses.

Analysis of Dynamic Failure of Plastic Spherical Shells Under Local Impact

Aim To study the dynamic failure of the plastic spherical shell impacted by a missile. Methods The deformation mode of spherical shells was given by introducing isometric transformation. The governing equation of motion of the rigid plastic spherical shell was given by energy balance. This equation was solved by using Runge Kutta method. Results The relationships between the impact force, dimple radius, central point deflection and time were obtained. The response time initial velocity, the maximal impact force permanent initial velocity, the central point deflection initial velocity and the dimple radius initial velocity characteristics were respectively plotted. Conclusion A comparison made between the theoretical results and the experimental ones indicates that the two groups of results are in conformity with each other.

ANALYSIS OF NONLINEAR PIEZOELECTRIC CIRCULAR SHALLOW SPHERICAL SHELLS BY DIFFERENTIAL QUADRATURE ELEMENT METHOD

The static behavior of piezoelectric circular spherical shallow shells under both electrical and mechanical loads is studied by using the differential quadrature element method (DQEM). Geometrical nonlinearity effect is considered. Detailed formulations and procedures are given for the first time. Several examples are analyzed and accurate results are obtained by the DQEM. Based on the results in this paper, one may conclude that the DQEM is a useful tool for obtaining solutions of structural elements. It can be seen that the shell shape may be theore tically controlled and snap through may occur when the applied voltage reaches a critical value even without mechanical load for certain geometric configurations.

Transient response of a spherical cavity with a partially sealed shell embedded in viscoelastic saturated soil

Based on Biot’s wave equation, this paper discusses the transient response of a spherical cavity with a partially sealed shell embedded in viscoelastic saturated soil. The analytical solution is derived for the transient response to an axisymmetric surface load and fluid pressure in Laplace transform domain. Numerical results are obtained by inverting the Laplace transform presented by Durbin, and are used to analyze the influences of the partial permeable property of boundary and relative rigidity of shell and soil on the transient response of the spherical cavity. It is shown that the influence of these two parameters is remarkable. The available solutions of permeable and impermeable boundary without shell are only two extreme cases of this paper.

Assembly and disassembly mechanics of a spherical snap fit

The snap fit is a common mechanical mechanism.We have studied the spherical snap fit carefully for its physical asymmetry,which is easy to assemble but difficult to disassemble.Because of the complexity of spherical snap fit,it is difficult to get a theoretical formula to describe its physical asymmetry.In this paper,the pushing assembly and pulling disassembly of spherical snap fit are studied by both finite element analysis and experiments.The theoretical formulaes of spherical snap fit have been obtained based on numerical simulations and theoretical results of cylindrical snap fit.

Axial acoustic radiation force on an elastic spherical shell near an impedance boundary for zero-order quasi-Bessel-Gauss beam

Shell structures have increasingly widespread applications in biomedical ultrasound fields such as contrast agents and drug delivery,which requires the precise prediction of the acoustic radiation force under various circumstances to improve the system efficiency.The acoustic radiation force exerted by a zero-order quasi-Bessel-Gauss beam on an elastic spherical shell near an impedance boundary is theoretically and numerically studied in this study.By means of the finite series method and the image theory,a zero-order quasi-Bessel-Gauss beam is expanded in terms of spherical harmonic functions,and the exact solution of the acoustic radiation force is derived based on the acoustic scattering theory.The acoustic radiation force function,which represents the radiation force per unit energy density and per unit cross-sectional surface,is especially investigated.Some simulated results for a polymethyl methacrylate shell and an aluminum shell are provided to illustrate the behavior of acoustic radiation force in this case.The simulated results show the oscillatory property and the negative radiation force caused by the impedance boundary.An appropriate relative thickness of the shell can generate sharp peaks for a polymethyl methacrylate shell.Strong radiation force can be obtained at small half-cone angles and the beam waist only affects the results at high frequencies.Considering that the quasi-Bessel-Gauss beam possesses both the energy focusing property and the non-diffracting advantage,this study is expected to be useful in the development of acoustic tweezers,contrast agent micro-shells,and drug delivery applications.

Static and dynamic snap-through behaviour of an elastic spherical shell

The deformation and snap-through behaviour of a thin-walled elastic spherical shell statically compressed on a flat surface or impacted against a fiat surface are studied the- oretically and numerically in order to estimate the influence of the dynamic effects on the response. A table tennis ball is considered as an example of a thin-walled elastic shell. It is shown that the increase of the impact velocity leads to a variation of the deformed shape thus resulting in larger de- formation energy. The increase of the contact force is caused by both the increased contribution of the inertia forces and contribution of the increased deformation energy. The contact force resulted from deformation/inertia of the ball and the shape of the deformed region are calcu- lated by the proposed theoretical models and compared with the results from both the finite element analysis and some previously obtained experimental data. Good agreement is demonstrated.

VIBRATION AND ACOUSTIC RADIATION FROM SUBMERGED STIFFENED SPHERICAL SHELL WITH DECK-TYPE INTERNAL PLATE

Based on the motion differential equations of vibration and acoustic coupling system for thin elastic spherical shell with an elastic plate attached to its internal surface,in which Dirac-δ functions are employed to introduce the moments and forces applied by the attachment on the surface of shell,by means of expanding field quantities as Legendre series,a semi-analytic solution is derived for the vibration and acoustic radiation from a submerged stiffened spherical shell with a deck-type internal plate,which has a satisfactory computational effectiveness and precision for an arbitrary frequency range.It is easy to analyze the effect of the internal plate on the acoustic radiation field by using the formulas obtained by the method proposed.It is concluded that the internal plate can significantly change the mechanical and acoustic characteristics of shell,and give the coupling system a very rich resonance frequency spectrum.Moreover,the method can be used to study the acoustic radiation mechanism in similar structures as the one studied here.

On Axially Symmetric Vibrations of Fluid Filled Poroelastic Spherical Shells

Employing Biot’s theory of wave propagation in liquid saturated porous media, waves propagating in a hollow poroelastic closed spherical shell filled with fluid are studied. The frequency equation of axially symmetric vibrations for a pervious and an impervious surface is obtained. Free vibrations of a closed spherical shell are studied as a particular case when the fluid is vanished. Frequency as a function of ratio of thickness to inner radius is computed in absence of dissipation for two types of poroelastic materials each for a pervious and an impervious surface. Results of previous works are obtained as a particular case of the present study.

Vibration and Acoustic Radiation from Submerged Spherical Double-Shell

Based on the motion differential equations of vibration and acoustic coupling system for a thin elastic spherical double-shell with several elastic plates attached to the shells, in which Dirac-δ functions are employed to introduce the forces and moments applied by the attachments, and by means of expanding field quantities as the Legendre series, a semi-analytic solution is derived for the solution to the vibration and acoustic radiation from a submerged spherical double-shell. This solution has a satisfying computational effectiveness and precision for arbitrary frequency range excitation. It is concluded that the internal plates attached to shells can change significantly the mechanical and acoustical characteristics of shells, and make the coupling system have a very rich resonance frequency spectrum. Moreover, the present method can be used to study the acoustic radiation mechanism of the type of structure.

THERMOELASTICALLY COUPLED AXISYMMETRIC NONLINEAR VIBRATION OF SHALLOW SPHERICAL AND CONICAL SHELLS

The problem of axisymmetric nonlinear vibration for shallow thin spherical and conical shells when temperature and strain fields are coupled is studied. Based on the large deflection theories of von Krmn and the theory of thermoelasticity, the whole governing equations and their simplified type are derived. The time-spatial variables are separated by Galerkin's technique, thus reducing the governing equations to a system of time-dependent nonlinear ordinary differential equation. By means of regular perturbation method and multiple-scales method, the first-order approximate analytical solution for characteristic relation of frequency vs amplitude parameters along with the decay rate of amplitude are obtained, and the effects of different geometric parameters and coupling factors as well as boundary conditions on thermoelastically coupled nonlinear vibration behaviors are discussed.

SNAP－THROUGH OF LAMINATED ORTHOTROPIC SHALLOW SPHERICAL IMPERFECT SHELLS WITH TRANSVERSE SHEAR

The equations of large deformations of laminated orthotropic spherical shellsare derived. The effects of transverse shear deformation and initial imperfection are considered. on this basis. the semi-analytical solution of the axisymrnetric snap-throughbuckling of laminated orthotropic shallow spherical shells under uniform pressure is obtained using orthogonal collocation method. The effects of material parameters, structuralparameters, initial imperfection and transverse shear deformation are discussed.

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.

Analysis of Error Caused by Replacing Spherical Shell with an Elastic Plate Model in Studying Bending Deformation of the Lithosphere

To study the bending deformation of the lithosphere, the simplification of replacing a spherical shell by a plate model is usually made. Based on the differential equations for the bending of plates and shallow spherical shells, an expression for the error caused by such a simplification is derived in this paper. The effect of model sizes on the error is discussed. It is proved that if we replace the shallow spherical shell by a plate model to solve the bending deformation of lithospheric plate, a large error will be caused. In contrast, if we use a plate on an elastic foundation instead, an approximate solution closer to that of spherical shell can be obtained. In such a way, the error can be reduced effectively and the actual geological condition can be modeled more closely.

DYNAMIC ANALYSIS OF LARGE DEFORMATION OF RIGID-VISCOPLASTIC HARDENING SPHERICAL SHELLS IMPACTED BY A MISSILE

This paper studies the dynamic behavior of large deformation of spherical shells impacted by a flat-nosed missile.By using isometric transformations,the deformation modes are given.On the basis of Perzyna-Symonds viscoplastic constitutive equations,the motion equations of the shells are obtained by rigid- viscoplastic variational principle.A comparison made between the numerical results and experimental ones indicates that the two groups of results are in conformity with each other.

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 Newmarh-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.

Nonlinear stability of sensor elastic element--corrugated shallow spherical shell in coupled multi-field

Nonlinear stability of sensor elastic element- corrugated shallow spherical shell in coupled multi-field is studied. With the equivalent orthotropic parameter obtairled by the author, the corrugated shallow spherical shell is considered as an orthotropic shallow spherical shell, and geometrical nonlinearity and transverse shear deformation are taken into account. Nonlinear governing equations are obtained. The critical load is obtained using a modified iteration method. The effect of temperature variation and shear rigidity variation on stability is analyzed.

TORSIONAL BUCKLING OF SPHERICAL SHELLS UNDER CIRCUMFERENTIAL SHEAR LOADS

By the aid of differential geometry analysis on the initial buckling of shell element, a set of new and exact buckling bifurcation equations of the spherical shells is derived. Making use of Galerkin variational method, the general stability of the hinged spherical shells with the circumferential shear loads is studied. Constructing the buckling mode close to the bifurcation point deformations, the critical eigenvalues, critical load intensities and critical stresses of torsional buckling ranging from the shallow shells to the hemispherical shell are obtained for the first time.

Thermal buckling of axisymmetrically laminated cylindrically orthotropic shallow spherical shells including transverse shear

The nonlinear thermal buckling of symmetrically laminated cylindrically orthotropic shallow spherical shell under temperature field and uniform pressure including transverse shear is studied. Also the analytic formulas for determining the critical buckling loads under different temperature fields are obtained by using the modified iteration method. The effect of transverse shear deformation and different temperature fields on critical buckling load is discussed.