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.

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.

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.

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.

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.

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.

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.

Nonlinear free vibration of reticulated shallow spherical shells taking into account transverse shear deformation

This paper deals with nonlinear free vibration of reticulated shallow spherical shells taking into account the effect of transverse shear deformation. The shell is formed by beam members placed in two orthogonal directions. The nondimensional fundamental governing equations in terms of the deflection, rotational angle, and force function are presented, and the solution for the nonlinear free frequency is derived by using the asymptotic iteration method. The asymptotic solution can be used readily to perform the parameter analysis of such space structures with numerous geometrical and material parameters. Numerical examples are given to illustrate the characteristic amplitudefrequency relation and softening and hardening nonlinear behaviors as well as the effect of transverse shear on the linear and nonlinear frequencies of reticulated shells and plates.

Nonlinear stability of double-deck reticulated circular shallow spherical shell

Based on the variational equation of the nonlinear bending theory of doubledeck reticulated shallow shells, equations of large deflection and boundary conditions for a double-deck reticulated circular shallow spherical shell under a uniformly distributed pressure are derived by using coordinate transformation means and the principle of stationary complementary energy. The characteristic relationship and critical buckling pressure for the shell with two types of boundary conditions are obtained by taking the modified iteration method. Effects of geometrical parameters on the buckling behavior are also discussed.

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.

A GENERAL SOLUTION OF AXISYMMETRIC PROBLEM OF ARBITRARY THICK SPHERICAL SHELL AND SOLID SPHERE

In this paper, the axisymmetric problems of arbitrary thick spherical shell and solid sphere are studied directly from equilibrium equations of three-dimensional problem, and the general solutions informs of Legendre serifs for thick spherical shell and solid sphere are given by using the method of weighted residuals.

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.

REFINED DIFFERENTIAL EQUATIONS OF DEFLECTIONS IN AXIAL SYMMETRICAL BENDING PROBLEMS OF SPHERICAL SHELL AND THEIR SINGULAR PERTURBATION SOLUTIONS

This paper deals with the research of accuracy of differential equations of deflections. The basic idea is as follows. Firstly, considering the boundary effect the meridian midsurface displacement u=0, thus we derive the deflection differential equations; secondly we accurately prove that by use of the deflection differential equations or the original differential equations the same inner forces solutions are obtained; finally, we accurately prove that considering the boundary effect the meridian surface displacement u = 0 is an exact solution. In this paper we give the singular perturbation solution of the deflection differential equations. Finally we check the equilibrium condition and prove the inner forces solved by perturbation method and the outer load are fully equilibrated. It shows that perturbation solution is accurate. On the other hand, it shows again that the deflection differential equation is an exact equation.The features of the new differential equations are as follows:1. The accuracies of the new differential equations and the original differential e-quations are the same.2. The new differential equations can satisfy the boundary conditions simply.3. It is advantageous to use perturbation method with the new differential equations.4 We may obtain the deflection expression(w)and slope expression (dw/da) by using the new differential equations.The new differential equations greatly simplify the calculation of spherical shell. The notation adopted in this paper is the same as that in Ref. [1]

ASYMMETRIC DYNAMIC INSTABILITY OF AXISYMMETRIC POLAR DIMPLING OF THIN SHALLOW SPHERICAL SHELLS

If the parameter , which measures the thickness-to-rise of the sliell, is small, the axismnnetrie polar dimpling oj .shallow .spherical .shell due to quadratic pressure distribution i.s dynamic instability, i.e., a small perturbation can change il to an asymmetric polar dimple mode. In two cases, the problem can be reduced to an eigenvalue problem where T can approximately be reduced to a Sturm-Liouvi/le operator if The existence of at least one real eigenvalue of T, which means that the axisyntmetric polar dimpling is dynamically unstable, i.s proved by spectral theorem or Hilbert theorem. Furthermore, an eigenfunction, which represents one of the asymmetric modes of the unstable dimple shell, belonging to an eigenvalue of T, is found.

THE SINGULAR PERTURBATION METHOD APPLIED TO THE NONLINEAR STABILITY PROBLEM OF A SHALLOW SPHERICAL SHELL(Ⅱ)

In this paper we consider the nonlinear stability of a thin elastic circular shallow spherical shell under the action of uniform normal pressure with a clamped edge. When the geometrical parameter k is large, the uniformly valid asymptotic solutions are obtained by means of the singular perturbation method. In addition, we give the analytic formula for determining the centre deflection and the critical load, and the stability curve is also derived. This paper is a continuation of the author's previous paper[11]

EXACT STATIC ANALYSIS OF A ROTATING PIEZOELECTRIC SPHERICAL SHELL

An exact analysis of a rotating piezoelectric spherical shell with arbitrary thickness is given. Three displacement functions are introduced to simplify the basic equations of a spherically isotropic, piezoelectric medium. By expanding the displacement functions as well as the electric potential in terms of spherical harmonics, the basic equations of equilibrium are converted to an uncoupled Euler type, second order ordinary differential equation and a coupled system of three second order ordinary differential equations. A general solution to the homogeneous equations of equilibrium is then derived. The static analysis of a rotating spherical shell is performed and the numerical example is presented. (Edited author abstract) 13 Refs.

GREEN QUASIFUNCTION METHOD FOR FREE VIBRATION OF SIMPLY-SUPPORTED TRAPEZOIDAL SHALLOW SPHERICAL SHELL ON WINKLER FOUNDATION

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

Shielding Effectiveness of Superconductor in the Shape of Spherical Shell

The Shielding coefficient of superconductor in the shape of the spherical shell is derived on the basis of Maxwell's equations and London's two-liquid model. Some cases of superconductor shielding have also been discussed in this paper.

Non-linear acoustic radiation by an underwater vibrating spherical shell

This paper deals with the non-linear propagation of sound waves radiated by a thin spherical shell vibrating in ideal lluid. Non-linear sound field is obtained by the method of renormalization proposed by Kelly and Nayfeh. Considering the acoustic load, we use Junger and his cooperators' spherical function expan8ion to obtain the structural response of the spherical shell. By combining both of the above methods, the structural response, the waveshape before the formation of shock wave and the first two order waves (including the 1st and 2nd harmonics, sum and difference frequency waves) are calculated respectively. They vary with frequency, distance and directions. Through the comparison of the above results with linear theoretical olles1 we analyse the rules of generation and propagation of non-linear waves.

Time-domain analysis for vibration and sound radiation of submerged spherical shell excited by force

The vibration and sound radiation of a submerged spherical shell are studied in the time-domain by Laplace transform method, where a CW pulse force acts at the apex of the shell. The numerical results for the case of long pulse show that the different vibrational modes and the resonant or beat radiated sound are excited for different carrier-frequencies, but litle sound is radiated for some vibrational modes. For the case of short pulse the waveforms of the pulse become widened and deformed, when the pulse propagates between apexes of the shell. Then, the Doubly Asymptotic Approximations (DAA2) and Kirchhoff's Retarded Potential Formulate (KRPF)are used to solve the same problem. It is shown that the results of DAA2 and KRPF method have a good agreement with the results of Laplace transform method.