Elastic Spherical Shell Research Articles

Acoustic field effect on encapsulated spherical drop located in liquid filled cylindrical vessel

The acoustical problem dealing with the interaction of a plane wave propagating along the liquid-filled cylindrical cavity with an elastic spherical shell filled, in turn, with the compressible liquid is solved. An axisymmetric case is studied. The boundary problem for simultaneous solution of the wave equations describing the motion in the fluid filling the vessel as well as in the fluid contained within the shell together with equations of motion of the thin elastic shell is formulated. The problem statement contains the boundary conditions for both fluid–shell interfaces, the condition at the vessel surface and far-field radiation conditions. The incident wave propagating along the vessel axis is expanded into a normal mode series. In order to meet all boundary conditions, the variable separation method along with the spherical wave function representations in terms of the cylindrical ones and vice versa are used. This approach yields exact solution of the boundary problem and enables one to analyze the scattered fields in the low and medium frequency ranges. The problem is reduced to an infinite system of algebraic equations that is solved by the truncation method. The goal of the study is to determine the interaction features between the shell and the vessel surfaces. Obtained numerical results describe scattered field of pressure and velocity in the neighborhood of the shell as functions of frequency, the internal liquid characteristics, and the elastic characteristics of the shell. It is ascertained, that peculiarities of the resonance behavior of the liquid-filled shell can be essentially amplified by the acoustical interaction with the vessel surface.

A Semi-Analytical Method for the 3D Elastic Structural-Acoustic Radiation in Shallow Water

A hybrid computational method is proposed for three-dimensional structural-acoustic radiation in shallow water, which combines the wave superposition method (WSM) with the adjustable Green function and the multi-physical-coupled finite element method (FEM). A low-frequency acoustic model in the near field is established by the FEM to discretize the continuous and arbitrary-shaped radiator, and the normal vibration velocity on the wetted surface is used as the input in the WSM to calculate the strengths of simple sources enclosed within the radiator. Finally, the sound propagation in the far field of the elastic structure is efficiently computed by superimposing the sound field of each simple source. The numerical examples of the pulsating sphere and elastic spherical shell in the shallow water are developed to verify the high accuracy and efficiency of the presented method, respectively. Moreover, the regularization algorithms are employed as the complementary approaches to improve the stability of this method in engineering practices, it is found that the algorithms can significantly hold the calculation accuracy in case the velocity data contain noise, and the least-squares QR is an efficient and accurate method with a few iteration numbers and better stability effect as compared with other algorithms.

Global Acoustic Impedance for the Analysis of Acoustic Scattering From Elastic Structures

The interaction between the sound field outside an elastic structure and its boundary is of paramount importance in the field of acoustics. This paper describes the global acoustic impedance in the form of impedance integral operator and impedance matrix for the analysis of elastic structures. In general, the excitation applied locally to the structure can induce vibration on the entire structure surface and hence the global acoustic impedance is introduced to characterize the relationship between the resultant sound pressure and acoustic particle velocity. The main advantage of the global acoustic impedance over the local acoustic impedance is that it facilitates improved physical insights into the sound field due to the presence of excitation sources. Furthermore, the acoustic impedance of an elastic spherical shell in matrix form is solved and analyzed by solving the finite element equation of structural dynamics with programming techniques. The impedance matrix is subsequently taken as the boundary condition to investigate the acoustic scattering from the elastic shell structure using the acoustic boundary element method. Explanations for the difference between the acoustic scatterings calculated based on the global acoustic impedance and the local acoustic impedance are offered.

РЕЛАКСАЦИЯ ОСТАТОЧНЫХ НАПРЯЖЕНИЙ В ДЕКАЭДРИЧЕСКИХ ЧАСТИЦАХ ПУТЕМ ОБРАЗОВАНИЯ ЦЕНТРАЛЬНОЙ СФЕРИЧЕСКОЙ ПОРЫ

Small metal particles with a body-centered crystal lattice (BCC) often take the form of polyhedrons with fifth-order symmetry axes such as the icosahedron, decahedron, and pentagonal prism. The quintic symmetry axes, forbidden by the traditional crystallography laws, cause inhomogeneous elastic stress and strain in these particles. Under certain conditions, these stress and strain could relax through the change in the particle structure: the formation of partial and perfect dislocations, misfit layers, and the nucleation of cracks and voids. Within the quasi-equilibrium energy approach, the authors proposed a theoretical model of residual stress relaxation in decahedral particles due to the formation of a central spherical void. The explicit analytical expressions for energies of solid and hollow decahedral particles are found. The elastic energy of a hollow decahedral particle is defined as the work spent on the nucleation of a positive wedge disclination with the power ω≈0.0163 rad (≈7°20') in the elastic spherical shell under its own stress field. The authors determined the change in the surface energy due to the formation of a void considering the influence of the relaxation effect of the first coordination sphere surrounding the vacancy on the particle volume change. The energy change of decahedral particles during the formation of a spherical void is calculated and the optimal and critical parameters of this process are determined. The study shows that there some critical radius of a particle, if reached the formation of the central spherical void becomes energetically favorable. Moreover, the study shows that a pore germ will grow until it reaches a certain optimal size corresponding to the greatest change in the system energy. The numerical calculations correspond with experimental observations of unstable voids in the rather small silver and gold decahedral particles with the diameter of 30–40 nm and stable voids in relatively large copper decahedral particles with the diameter of ~1 μm.

Stress-stain state in elastic spherical shells under the action some asymmetrical of circular loads

The paper presents the basic differential equations for calculating shells of rotation. It is considered that the material of the shell works beyond the elastic limit. The conclusion of the resolving system of differential equations is based on the linear shell theory and the Kirchhoff-Love hypothesis. The physical correlations are based on the small elastic-plastic deformation theory. With the construction of the solution of the load function, also unknown resolving functions expand to the Fourier series along the circumferential coordinate. In the present work, the action of the asymmetrical distribution along the parallel loads is considered. The solution is approached by the elastic decisions method. The study of stress-stain state in the isotropic spherical shells is presented in the work. The results of the calculation are presented in the example of a spherical shell. The work of the material beyond the elastic limit is taken into account using assuming linear hardening of the material. The shell is loaded with ring distributed loads, and they are applied on a certain segment in a parallel circle. The diagrams of bending moments and areas of the appearance of plastic deformations along the thickness of the shell are presented. The areas of plastic deformations for ring loads distributed along the parallel relative to the zero meridian in the range of 45, 90,135 degrees are shown.

A New Analytical Method for Spherical Thin Shells’ Axisymmetric Vibrations

In order to improve the spherical thin shells’ vibrations analysis, we introduce a new analytical method. In this method, we take into consideration the terms of the inertial couples in the stress couples’ differential equations of motion. These inertial couples are omitted in the theories provided by Naghdi–Kalnins and Kunieda. The results show that the current method can solve the axisymmetric vibrations’ equations of elastic thin spherical shells. In this paper, we focus on verifying the current method, particularly for free vibrations with free edge and clamped edge boundary conditions. To check the validity and accuracy of the current analytical method, the natural frequencies determined by this method are compared with those available in the literature and those obtained by a finite element calculation.

Axial acoustic radiation force on an elastic spherical shell near an impedance boundary for zero-order quasi-Bessel–Gauss beam* *Project supported by the National Natural Science Foundation of China (Grant Nos. 81527901, 11604361, and 91630309).

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.

Analysis of Two Colliding Thin Spherical Shells

In the present paper, the collision of two elastic spherical shells is investigated using the wave theory of impact. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. Since the local bearing of the materials of the colliding elastic shells is taken into account, then the solution in the contact domain is found via the Hertz contact theory.

Buckling of an Epithelium Growing under Spherical Confinement.

SummaryMany organs are formed through folding of an epithelium. This change in shape is usually attributed to tissue heterogeneities, for example, local apical contraction. In contrast, compressive stresses have been proposed to fold a homogeneous epithelium by buckling. While buckling is an appealing mechanism, demonstrating that it underlies folding requires measurement of the stress field and the material properties of the tissue, which are currently inaccessible in vivo. Here, we show that monolayers of identical cells proliferating on the inner surface of elastic spherical shells can spontaneously fold. By measuring the elastic deformation of the shell, we infer the forces acting within the monolayer and its elastic modulus. Using analytical and numerical theories linking forces to shape, we find that buckling quantitatively accounts for the shape changes of our monolayers. Our study shows that forces arising from epithelial growth in three-dimensional confinement are sufficient to drive folding by buckling.

A mixed analytical-numerical method for the acoustic radiation of a double elastic spherical shell in finite-depth water

A mixed analytical-numerical method is proposed to evaluate the acoustic radiation of a double elastic spherical shell in finite-depth water by combining the analytical method with the numerical Wave Based Method (WBM) [or Wave Superposition Method, WSM]. The effects of reflection from both the free surface and the seabed are considered. The correctness of this method is verified through several numerical examples. Theoretically, this method can be extended to address the acoustic radiation problem of a double spherical shell in more complex hydroacoustic waveguide environment, because only a change of Green's function used in the WBM is needed. The method presented in this work can provide a benchmark, which is quite scarce at present, for the fluid-structure coupled vibration, acoustic radiation and sound propagation of three-dimensional bodies of an arbitrary shape in the hydroacoustic waveguide environment. Furthermore, the influence of free surface and seabed boundaries on the behaviors of structural vibration, acoustic radiation and sound propagation is presented. Some physical images are also included to illustrate the mechanism more clearly.

Dynamic response and damage assessment of spherical robot GFRP spherical shell under low velocity impact

Abstract In the normal operation of a spherical robot, its spherical shell structure is often accompanied by low velocity impact. At the same time, the spherical shell structure should take into account its protective ability against high strength impact of small probability. Glass fiber reinforced polymer (GFRP), which has high specific strength, specific stiffness, corrosion resistance and an impact energy dissipation coefficient, can be used as the ideal spherical shell material for spherical robots. In this study, the low velocity impact damage of GFRP spherical shell is studied based on the background that a thin-walled shell structure of spherical robot may suffer from large deformation and dynamic load. This study is divided into three aspects: experiment, simulation and calculation. The dynamic response and residual bearing capacity of the GFRP spherical shell is obtained through experiments; the progressive damage model of the composite structure, which demonstrates expounded stress distribution, a structural deformation mode and an energy dissipation mechanism under impact, is established based on the Hashin criterion. Low velocity impact-penetrating failure of the elastic brittle thin-walled GFRP spherical shell is calculated according to the geometrical principle and energy method. In this paper, the material dynamic behavior and impact damage of a GFRP spherical shell are systematically studied. This is of great significance for the development of high-performance spherical robots and the realization of accurate control.

Lithospheric stress, strain and displacement changes from GRACE-FO time-variable gravity: case study for Sar-e-Pol Zahab Earthquake 2018

SUMMARY Temporal variations in the Earth's gravity field can be used for monitoring of lithospheric deformations. The network of continuously operating gravity stations is required for this purpose but a global coverage by such network is currently extremely sparse. Temporal variations in long-wavelength part of the Earth's gravity field have been, however, observed by two satellite missions, namely the Gravity Recovery And Climate Experiment (GRACE) and the GRACE Follow-On (GRACE-FO). These satellite gravity observations can be used to study long-wavelength deformations of the lithosphere. Consequently, considering the lithosphere as a spherical elastic shell and solving the partial differential equation of elasticity for it, the stress, strain and displacement inside the lithosphere can be estimated. The lower boundary of this shell is assumed to be stressed by mantle convection, which has a direct relation to the Earth's gravity field according to Runcorn's theory. Changes in gravity field lead to changes in the sublithospheric stress and the stress propagated throughout the lithosphere. In this study, we develop mathematical models in spherical coordinates for describing the stress propagation from the sublithosphere through the lithosphere. We then organize a system of observation equations for finding a special solution to the boundary-value problem of elasticity in the way that provides a stable solution. In contrast, models presented in previously published studies are ill-posed. Furthermore, we use constants of the solution determined from the boundary stresses to determine the strain and displacements leading to these stresses, while in previous studies only the stress has been considered according to rheological properties of the lithosphere. We demonstrate a practical applicability of this theoretical model to estimate the stress–strain redistribution caused by the Sar-e-Pol Zahab 2018 earthquake in Iran by using the GRACE-FO monthly solutions.

Numerical Analysis on Target Strength of the Filled Spherical Elastic Shell

In this paper, the acoustic target strength of spherical shells in water with internal fillers is calculated by using the normal series expansion method, and the effects of metal and non-spherical shell materials, size factors and different fillers on the target strength of spherical shells are analyzed. The results show that for different elastic shell parameters, the influence of the acoustic characteristics of the internal filler on the target strength is quite different. For non-metallic sound transmission spherical shells filled with fluid, only when the sound velocity difference between the filler and the water medium is large, the target strength is greatly improved compared with the corresponding rigid sphere, and the shell size parameters have little effect on the target strength. However, for metal spherical shells with internal fillers, the material and size factors of the shell have a greater impact on its target strength, and the sensitivity to sound velocity mismatch inside and outside the shell is relatively small.

Buckling of pressurized spherical shells containing a through-thickness defect

We present a study on the pressure buckling of thin, elastic spherical shells containing a thickness defect. Methodologically, we combine precision model experiments, finite element simulations, and a reduced axisymmetric shell model. We observe qualitatively different buckling behavior by varying the geometry of the defect: either one buckling event or two events comprising local buckling at the defect and global buckling of the entire shell. We systematically analyze the loading path for the imperfect shell under prescribed pressure or volume change and identify three buckling regimes. We then explore a wide parameter space to study the dependence of the buckling regimes on the defect geometry, thus obtaining a phase diagram with quantitative relationships between critical buckling pressures and defect geometry. We find that the global buckling becomes insensitive to the defect beyond a critical value of its amplitude, and we demonstrate that the buckling regimes are governed by the three geometric parameters of the defect, namely its width, amplitude and the width of the transition region across the edge of the defect.

The influence of the thickness of the elastic spherical shell with liquid on its stress-strain state

Investigations of the dynamics of the system consisting of elastic spherical shell filled with ideal compressible fluid and gas cavity in the center of the system are presented in this paper. The excitation pulse-modulated source is introduced into the gas cavity in the center of the system. The effect of the shell thickness on its dynamics and the stress-state during the pulsations is investigated. The results for radial displacements changes of the middle surface, the thickness of the fluid separation from the shell, the stress intensity in the shell during its free pulsations are obtained. The comparison of calculations for the separation thickness in cases of free and partially fixed shell is carried out.

Buckling Thresholds for Pre-Loaded Spherical Shells Subject to Localized Blasts

Abstract This paper investigates the robustness against localized impacts of elastic spherical shells pre-loaded under uniform external pressure. We subjected a pre-loaded spherical shell that is clamped at its equator to axisymmetric blast-like impacts applied to its polar region. The resulting axisymmetric dynamic response is computed for increasing amplitudes of the blast. Both perfect shells and shells with axisymmetric geometric imperfections are analyzed. The impact energy threshold causing buckling is identified and compared with the energy barrier that exists between the buckled and unbuckled static equilibrium states of the energy landscape associated with the pre-loaded pressure. The extent to which the impact energy of the threshold blast exceeds the energy barrier depends on the details of its shape and width. Targeted blasts that approximately replicate the size and shape of the energy barrier buckling mode defined in the paper have an energy threshold that is only modestly larger than the energy barrier. An extensive study is carried out for more realistic Gaussian-shaped blasts revealing that the buckling threshold energy for these blasts is typically in the range of at least 10–40% above the energy barrier, depending on the pressure pre-load and the blast width. The energy discrepancy between the buckling threshold and energy barrier is due to elastic waves spreading outward from the impact and dissipation associated with the numerical integration scheme. Buckling is confined to the vicinity of the pole such that, if the shell is not shallow, the buckling thresholds are not strongly dependent on the location of the clamping boundary, as illustrated for a shell clamped halfway between the pole and the equator.

Partitioned Coupling for Structural Acoustics

Abstract We expand the second-order fluid–structure coupling scheme of Farhat et al. (1998, “Load and Motion Transfer Algorithms for 19 Fluid/Structure Interaction Problems With Non-Matching Discrete Interfaces: Momentum and Energy Conservation, Optimal Discretization and Application to Aeroelasticity,” Comput. Methods Appl. Mech. Eng., 157(1–2), pp. 95–114; 2006, “Provably Second-Order Time-Accurate Loosely-Coupled Solution Algorithms for Transient Nonlinear Computational Aeroelasticity,” Comput. Methods Appl. Mech. Eng., 195(17), pp. 1973–2001) to structural acoustics. The staggered structural acoustics solution method is demonstrated to be second-order accurate in time, and numerical results are compared to a monolithically coupled system. The partitioned coupling method is implemented in the Sierra Mechanics software suite, allowing for the loose coupling of time domain acoustics in sierra/sd to structural dynamics (sierra/sd) or solid mechanics (sierra/sm). The coupling is demonstrated to work for nonconforming meshes. Results are verified for a one-dimensional piston, and the staggered and monolithic results are compared to an exact solution. Huang, H. (1969, “Transient Interaction of Plane Acoustic Waves With a Spherical Elastic Shell,” J. Acoust. Soc. Am., 45(3), pp. 661–670) sphere scattering problem with a spherically spreading acoustic load demonstrates parallel capability on a complex problem. Our numerical results compare well for a bronze plate submerged in water and sinusoidally excited (Fahnline and Shepherd, 2017, “Transient Finite Element/Equivalent Sources Using Direct Coupling and Treating the Acoustic Coupling Matrix as Sparse,” J. Acoust. Soc. Am., 142(2), pp. 1011–1024).

Interference structure in the form function for bistatic scattering from elastic spherical shells in water

Although the monostatic scattering from a sphere is independent of viewing angle, the bistatic frequency response contains rich structures a function of scattering angle. The exact solution for the far-field scattering from underwater elastic spherical shells is examined as a function of scattering angle. The high-frequency scattering from spherical shells includes contributions from leaky Lamb-type waves that circumnavigate the shell while continuously radiating into the fluid. The form function in scattering angle contains multiple interference patterns between the specular reflection and the different modes of Lamb waves. This interference structure is similar to the case for solid spheres and cylinders recently studied [A.M. Gunderson, A.L. Espana, and P.L. Marston, J. Acoust. Soc. Am. 142, 110–115 (2017)], where the interference patterns between the specular reflection and Rayleigh waves were examined. For the elastic shell case, the greater number of Lamb wavemodes and their highly dispersive nature creates richer interference structure in the form function from that of the solid sphere. The interference loci between the specular reflection and the two lowest mode Lamb waves are calculated and compared with the bistatic form function. [Work supported by the Office of Naval Research.]

Localization in spherical shell buckling

This paper addresses localization of the deformation due to buckling that occurs immediately following the onset of bifurcation in the axisymmetric buckling of a perfect spherical elastic shell subject to external pressure. The localization process is so abrupt that the buckling mode of the classical eigenvalue analysis, which undulates over the entire shell, becomes modified immediately after bifurcation transitioning to an isolated dimple surrounded by an unbuckled expanse of the shell. The paper begins by revisiting earlier attempts to analyze the initial post-buckling behavior of the spherical shell, illustrating their severely limited range of validity. The unsuccessful attempts are followed by an approximate Rayleigh-Ritz solution which captures the essence of the localization process. The approximate solution reveals the pathway that begins at bifurcation from the classical mode shape to the localized dimple buckle. The second part of the paper presents an exact asymptotic expansion of the initial post-buckling behavior which accounts for localization and which further exposes the analytic details of the abruptness of the transition.

Scattering and radiation force dependence on properties of empty elastic spherical shells: Low-frequency phase-shift derivation.

It is helpful to evaluate scattering and acoustic radiation forces on spheres for idealized cases in which the effects of energy dissipation are ignorable. Let x denote the product of the acoustic wave number and the sphere's radius. Previously expansions were obtained for fluid and solid spheres involving powers of x and algebraic expressions containing material properties. The present analysis concerns the case of empty elastic shells and reveals how expansion coefficients also depend on shell thickness. Incident waves considered are plane traveling and standing waves, though relevance to Bessel wave-fields is also noted. The expansions give leading-order corrections to the usual Rayleigh scattering approximation.