Elastic Spherical Shell Research Articles

Deflagration inside an elastic spherical shell: Fluid-structure interaction effects

We examine the problem of deflagration of a flammable gaseous mixture contained within an elastic spherical shell and ignited at its centre. We predict analytically the pressure wave radiated by an expanding spherical deflagration front and we study its interaction with the elastic shell prior to failure. We predict the waves reflected at the inner wall of the shell and radiated in the space outside the shell, which we assume initially filled with air at atmospheric pressure, and we calculate the magnitude of these waves as a function of density, thickness and elastic modulus of the shell. The findings of this model are used to critically assess laboratory setups used in controlled deflagration experiments, in which a flammable mixture is initially contained by a soft shell. The interpretation of data from these tests currently assumes that the gaseous mixture is effectively unconfined during the deflagration, due to the low mass, low stiffness and early failure of the containing shell. We show that this assumption is not adequate and it leads to errors in the interpretation of the measurements; the proposed model is used to quantify these errors.

Acoustic Radiation Force on an Arbitrarily Placed Elastic Spherical Shell near an Impedance Boundary for Bessel Waves

Acoustic Radiation Force on an Arbitrarily Placed Elastic Spherical Shell near an Impedance Boundary for Bessel Waves

Exploring the interplay of liquid crystal orientation and spherical elastic shell deformation in spatial confinement.

The application of liquid crystal technology typically relies on the precise control of molecular orientation at a surface or interface. This control can be achieved through a combination of morphological and chemical methods. Consequently, variations in constrained boundary flexibility can result in a diverse range of phase behaviors. In this study, we delve into the self-assembly of liquid crystals within elastic spatial confinement by using the Gay-Berne model with the aid of molecular dynamics simulations. Our findings reveal that a spherical elastic shell promotes a more regular and orderly alignment of liquid crystals compared to a hard shell. Moreover, during the cooling process, the hard-shell confined system undergoes an isotropic-smectic phase transition. In contrast, the phase behavior within the spherical elastic shell closely mirrors the isotropic-nematic-smectic phase transition observed in bulk systems. This indicates that the orientational arrangement of liquid crystals and the deformations induced by a flexible interface engage in a competitive interplay during the self-assembly process. Importantly, we found that phase behavior could be manipulated by altering the flexibility of the confined boundaries. This insight offers a fresh perspective for the design of innovative materials, particularly in the realm of liquid crystal/polymer composites.

Non-spherical axisymmetric deformations of hyperelastic shells

Spherical elastic shells commonly appear both in nature and man-made devices. Often, their functionality is governed by an incoming- or outgoing flux of fluid. The transient traction that the fluid exerts in the process causes the shell to depart from sphericity. Here, we develop a framework for determining non-spherical axisymmetric deformations, by combining tools from nonlinear continuum mechanics, structural mechanics, and asymptotic analysis. We apply our framework to analyze an exemplary problem of a Mooney–Rivlin shell that is filled by viscous fluid. Collectively, our framework and the insights gained from its application, promote the understanding of the mechanics of such fluid-filled deformable membranes and shells.

Directivity of spherical acoustic scattering based on COMSOL

The acoustic scattering characteristics of spheres are the basis for studying the sound scattering law of the target. The directivity of acoustic scattering is one of the most important characteristics of the target sound scattering law. In this paper, the acoustic scattering characteristics of rigid spheres and elastic spherical shells are analyzed from the two typical basic models of the rigid sphere and elastic spherical shell. The sound scattering directivity corresponding to different ka values under different models is analyzed by COMSOL finite element simulation analysis. The change law of spheroid scattering directivity of spheres under different elastic backgrounds of the rigid sphere and elastic spherical shell is compared and studied. It is found that the scattering directivity of both rigid spheres and elastic spherical shells will change with the change of ka value. The larger the ka, the stronger the forward directivity, and the more side lobes for rigid spheres. The backward directivity is greater than the forward directivity at ka≤1, while the forward directivity of the elastic spherical shell is always greater than the backward direction. And it is not difficult to find that the directivity of rigid spheres is relatively uniform in backward scattering when ka=1. When ka>1, the directivity increases with ka, the forward directivity becomes stronger and stronger, and the side lobes gradually increase. The elastic spherical shell uses air as the internal filler, and its directivity has always been forward than backward. The forward directivity also increases with the increase of ka, and the side lobes also increase with the increase of frequency.

Vibrations of a thin, prestressed, fluid-loaded spherical shell forced by a point sound source

This paper investigates vibrations of and sound scattering by a thin elastic spherical shell when static pressures are different in the fluids inside and outside the shell. The work is motivated by stratospheric balloons being employed to carry acoustic sensors and by the proposed use of large, encapsulated gas bubbles for passive suppression of underwater sound [O. A. Godin and A. B. Baynes, J. Acoust. Soc. Am., 143, EL67–EL73 (2018)]. Linearized equations of motion of thin, prestressed shells are derived from first principles. Differences in the fluid-loading terms from previously proposed ad hoc models are identified and their significance is analyzed. Analytic solutions are derived for vibrations of a spherical shell excited by an incident spherical sound wave and for acoustic fields in the internal and external fluids. The results are verified against exact solutions in several particular cases. The mathematical model of the shell vibrations is applied to characterize the influence of the shell’s material properties on passive suppression of underwater noise and to quantify the effect of wave scattering by the balloon on performance of balloon-borne infrasound sensors in the atmosphere. [Work supported by ONR.]

Lithospheric Stress Due to Mantle Convection and Mantle Plume over East Africa from GOCE and Seismic Data

The Afar and Ethiopian plateaus are in a dynamic uplift due to the mantle plume, therefore, considering the plume effect is necessary for any geophysical investigation including the estimation of lithospheric stress in this area. The Earth gravity models of the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) and lithospheric structure models can be applied to estimate the stress tensor inside the Ethiopian lithosphere. To do so, the boundary-value problem of elasticity is solved to derive a general solution for the displacement field in a thin elastic spherical shell representing the lithosphere. After that, general solutions for the elements of the strain tensor are derived from the displacement field, and finally the stress tensor from the strain tensor. The horizontal shear stresses due to mantle convection and the vertical stress due to the mantle plume are taken as the lower boundary value at the base of the lithosphere, and no stress at the upper boundary value of the lithospheric shell. The stress tensor and maximum stress directions are computed at the Moho boundary in three scenarios: considering horizontal shear stresses due to mantle convection, vertical stresses due to mantle plume, and their combination. The estimated maximum horizontal shear stresses’ locations are consistent with tectonics and seismic activities in the study area. In addition, the maximum shear stress directions are highly correlated with the World Stress Map 2016, especially when the effect of the mantle plume is solely considered, indicating the stress in the study area mainly comes from the plume.

Acoustic radiation force of elastic spherical shell with eccentric droplet in plane wave acoustic field

Based on the application of acoustic waves in cell manipulation, a model consisting of an elastic spherical shell and eccentric droplet is established to simulate a eukaryotic cell and analyze the acoustic radiation force (ARF) on the cell. In this work, we derive an exact expression for the ARF on the liquid-filled spherical shell. The influence of eccentric distance, radius of the eccentric droplet and impedance of the medium inside the liquid-filled spherical shell on the ARF are analyzed numerically. The results show that the ARF is very sensitive to the position and size of the eccentric droplet. As the eccentricity of the eccentric droplet increases, the ARF becomes greater. In a low frequency region (<i>ka</i><3) the resonance peak point increases, and the position of the curve ventral point shifts to the high frequency region (<i>ka</i>>3) with the increase of the radius of the eccentric droplet. The effect of the position variation on the ARF is more significant than that of the radius change, and both of their effects will be superimposed on each other. The ARF, as a function of <i>ka,</i> is mainly affected by the variation of the nucleus characteristic impedance. The ARF amplitude around <i>ka</i> = 5 increases and the position of the ventral point tends to shift rightwards with the enlargement of the nucleus impedance. Therefore, the radiation response at a certain frequency or in a cell size range can be enhanced when the nucleus impedance increases. The results of this study provide theoretical basis for the cell sorting and targeted therapy.

ON THE STRESS WAVE IN THE EARTH'S BODY DUE TO DIFFERENTIAL ROTATION OF THE MANTLE AND CORE

Seismic energy generated as a result of elastic deformation of tectonic plates or blocks of rocks is realized by unloading at deformations exceeding the ultimate strength of rocks. The currently generally accepted earthquake source models are essentially local and are not able to establish a connection between seismic activity and some geophysical phenomena on a planetary scale. One of the features of seismicity is the recurrence of strong earthquakes in one place after a certain time interval. The purpose of this work is to study the influence of differential rotation of the solid core and mantle on seismic processes. It has now been established that the rate of rotation of the solid core exceeds the rate of rotation of the mantle. In relative motion, the solid core makes one revolution presumably in 200–400 years. Earlier, based on this, an explanation was proposed for long-period variations in the length of the day. In this paper, on a simple mechanical model, it is shown that the differential rotation of the elastic mantle and the solid core of an ellipsoidal shape generates a stress wave in the Earth's body. The idea of a stress wave is qualitatively new in the physical description of a seismic process. To demonstrate this effect, a solution is given to the problem of deformation of an elastic spherical shell interacting with a dumbbell-shaped solid body rotating in its cavity. When solving the problem, the apparatus of spherical vectors was used. The stress distribution of the spherical layer inside the shell is constructed and the areas of greatest risk are identified. It is shown that taking into account the differential rotation leads to the migration of the zones of greatest risk with a period equal to half the period of a complete rotation of the dumbbell relative to the shell (100–200 years). The results of the work can be useful in studying the relationship between seismicity and planetary rotation regimes.

Dynamic modeling of the motions of variable-shape wave energy converters

In the recently introduced Variable-Shape heaving wave energy converters, the buoy changes its shape actively in response to changing incident waves. In this study, a Lagrangian approach for the dynamic modeling of a spherical Variable-Shape Wave Energy Converter is described. The classical bending theory is used to write the stress–strain equations for the flexible body using Love’s approximation. The elastic spherical shell is assumed to have an axisymmetric vibrational behavior. The Rayleigh–Ritz discretization method is adopted to find an approximate solution for the vibration model of the spherical shell. A novel equation of motion is presented that serves as a substitute for Cummins equation for flexible buoys. Also, novel hydrodynamic coefficients that account for the buoy mode shapes are proposed. The developed dynamic model is coupled with the open-source boundary element method software NEMOH. Two-way and one-way Fluid–Structure Interaction simulations are performed using MATLAB to study the effect of using a flexible shape buoy in the wave energy converter on its trajectory and power production. Finally, the variable shape buoy was able to harvest more energy for all the tested wave conditions.

Numerical Analysis of Deformation Characteristics of Elastic Inhomogeneous Rotational Shells at Arbitrary Displacements and Rotation Angles

Adequate mathematical models and computational algorithms are developed in this study to investigate specific features of the deformation processes of elastic rotational shells at large displacements and arbitrary rotation angles of the normal line. A finite difference method (FDM) is used to discretize the original continuum problem in spatial variables, replacing the differential operators with a second-order finite difference approximation. The computational algorithm for solving the nonlinear boundary value problem is based on a quasi-dynamic form of the ascertainment method with the construction of an explicit two-layer time-difference scheme of second-order accuracy. The influence of physical and mechanical characteristics of isotropic and composite materials on the deformation features of elastic spherical shells under the action of surface loading of “tracking” type is investigated. The results of the studies conducted have shown that the physical and mechanical characteristics of isotropic and composite materials significantly affect the nature of the deformation of the clamped spherical shell in both the subcritical and post-critical domains. The developed mathematical models and computational algorithms can be applied in the future to study shells of rotation made of hyperelastic (non-linearly elastic) materials and soft shells.

Acoustic radiation force dependence on properties of elastic spherical shells in standing waves

In this study, we theoretically investigate the acoustic radiation force acting on elastic spherical shells in standing waves in the dimensionless frequency range of 0<ka<0.7 (k is the wave number in the host fluid and a is the outer radius of the spherical shells), we show that in the low-frequency range (0<ka<0.2), the acoustic radiation force on core–shell spheres can vary from positive to negative by decreasing the effective density and effective bulk modulus of the spherical shells, while the effective density and bulk modulus are dependent on the hollow ratio of the core–shell spheres, the density and bulk modulus ratios of the shell to the core. Thus, the core–shell spheres cannot only be trapped from the pressure node to the pressure antinode by increasing the hollow ratio or decreasing the shell’s density or longitudinal or shear velocity, but are also unresponsive to ultrasound waves with an optimal hollow ratio, density, or acoustic velocity of the shell. We further show that in the relatively low-frequency range of0<ka<0.7, the acoustic radiation force on core–shell spheres can be significantly enhanced around resonant frequencies. The acoustic radiation force could be enhanced at lower frequencies by increasing the hollow ratio or decreasing the shell’s density or shear wave velocity. Our results pave the way for optimising the design of core–shell particles for efficient ultrasound-mediated drug delivery.

Large strains of cylindrical and spherical elastic shells with distributed dislocations

Based on the 6-parameter nonlinear theory of elastic shells, the influence of continuously distributed dislocations on the deformation of a thin circular cylinder loaded with hydrostatic pressure, longitudinal force, and torque and a closed spherical shell loaded with tension and a distributed couple load is studied. For these structures, exact solutions of the equilibrium equations are constructed valid for large strains and rotations. The relative simplicity of the obtained solutions allows using them in experimental studies of the mechanical behavior of two-dimensional materials with defects.

Computational method of underwater acoustic radiation from a spherical shell coupled with nonlinear systems

During the continuous research and practical application of nonlinear vibration isolation systems, new problems arise for the vibro-acoustic calculation of underwater targets. The present frequency-domain analysis method can only deal with linear problems, so the acoustic radiation calculation of underwater targets coupled with nonlinear systems can only be performed in the time domain. When it comes to the real-time calculation, the huge computation cost will definitely bring challenges to the practical application. To solve this problem, a new calculation method is established for the underwater vibro-acoustic problems of structures coupled with nonlinear systems, taking elastic spherical shells as research objects in this paper. This method realizes the time-frequency domain conversion of the fluid-structure coupled vibration equation of an elastic spherical shell through the Fourier transform. The explicit difference method is further used to calculate the vibration response of the spherical shell, while the Newmark method is used to calculate the vibration response of the internal nonlinear system. An efficient integrated calculation of the two coupling parts is then achieved by considering boundary conditions at the connection point. Without the time-domain acoustic field solution, this method is able to calculate the vibration response of both the internal nonlinear system and the spherical shell, as well as the radiated acoustic power generated by the spherical shell vibration in real time. The calculation complexity and efficiency are radically improved. The nonlinear system discussed in this paper is a nonlinear mass-spring oscillator system installed at the bottom of a spherical shell. Finally, the correctness of the method described in this paper is verified through specific numerical examples, and a series of calculations and analyzes are carried out to explore the influence of nonlinear effects on the acoustic radiation of underwater structures. The calculation method described in this article is not limited to spherical shells, and is still applicable to general underwater targets.

Quantifying the efficiency of the acoustic radiation force from two-wave beating for structural resonance excitation of a thin spherical shell

The structural echoes obtained from low-frequency (∼kHz) excitation of resonant underwater targets can serve as a basis for automated target recognition. Acoustic radiation force (ARF) can provide a means to excite those structural echoes with compact transducing mechanisms as it relies on high-frequency transducers to generate locally low-frequency excitation via the classical beating effect. However, a key question is quantifying the efficiency of ARF excitation mechanisms to excite those structural echoes when compared to using direct low-frequency pulses. Here, we develop a time-domain finite element framework that compares the ARF and direct low-frequency excitation methods to excite classical resonances scenarios such as thickness resonances of a thin plate or the so-called “mid-frequency enhancement echo” for elastic spherical shells. These canonical target objects provide a straightforward procedure to investigate the ARF excitation efficiency and establishes the numerical framework to investigate more complex targets with a wider range of environmental parameters.

Analytical solution for a vibrating rigid sphere with an elastic shell in an infinite linear elastic medium

The analytical solution is given for a vibrating rigid core sphere, oscillating up and down without volume change, situated at the center of an elastic material spherical shell, which in turn is situated inside an infinite (possible different) elastic medium. The solution is based on symmetry considerations and the continuity of the displacement both at the core and the shell-outer medium boundaries as well as the continuity of the stress at the outer edge of the shell. Furthermore, a separation into longitudinal and transverse waves is used. Analysis of the solution shows that a surprisingly complex range of physical phenomena can be observed when the frequency is changed while keeping the material parameters the same, especially when compared to the case of a core without any shell. With a careful choice of materials, shell thickness and vibration frequency, it is possible to filter out most of the longitudinal waves and generate pure tangential waves in the infinite domain (and vice-versa, we can filter out the tangential waves and generate longitudinal waves). When the solution is applied to different frequencies and with the help of a fast Fourier transform (FFT), a pulsed vibration is shown to exhibit the separation of the longitudinal (L) and transverse (T) waves (often called P- and S-waves in earthquake terminology).

Thermal and mechanical stabilities of Core-shell microparticles containing a liquid core

Thorough understanding of the behaviour of core–shell microparticles with a liquid core is essential for determining their performance in applications under different operation conditions. This paper reports the behaviour of core–shell particles with a liquid core under thermal and mechanical loads. First, we formulated an analytical model for the heating process of a core–shell microparticle with a liquid core. Next, we utilised an axisymmetric model of an elastic spherical shell upon compression to describe the deformation of a core–shell microparticle. Finally, we conducted experiments to validate these models. Both thermal and mechanical models agree well with the experimental data. The maximum temperature a core–shell microparticle can withstand depends on the liquid, the geometry, and the material of the shell. The critical compression force before rupture of a core–shell microparticle depends on the Poisson’s ratio of the shell material and the shell thickness relative to the outer shell radius. The rupture force and rupture temperature increase with increasing shell thickness.

Krylov subspace iterative methods for time domain equivalent sources method based nearfield acoustical holography

This paper studies the acoustic field reconstruction on an arbitrarily shaped vibrating surface using the time-domain equivalent source method (ESM) based near-field acoustic holography (NAH) from measurements of the pressure field on a nearby conformal surface. For the classical frequency-dependent ESM, we represent the acoustic field by a set of frequency-dependent monopole sources strategically distributed over a close and conformal surface to the vibrating structure. We demonstrated that in this setup, the ESM representation approximates the classical integral representations, which justifies the ESM as an extension to the classical integral theory. For time-domain ESM in a similar manner, we demonstrate that a set of time-dependent monopole sources approximates the single layer retarded potential and then characterize the acoustic field. The discretization of time-domain ESM produces a matrix system with a particular sparse structure that can be efficiently solved using Krylov subspace iterative methods. The iterative methods that will be of our interest are Conjugate Gradients Normal equation Residual, Conjugate Gradients Normal equation Error and Least Squares QR. To reduce the amplification of measurement in the numerical reconstructions, we combine the proposed iterative methods with heuristic stopping rules like L-curve, Quasi-Optimality, Hanke–Raus, Brezinski–Rodriguez–Seatzu and a posteriori stopping rule. To determine the best combination of reconstruction technique, iteration and stopping rule, we develop a statistical study that uses multiple samples of numerically generated data by Gaussian pulses over a sphere. We produce these samples using a random source position and different noise levels. Finally, we validate our proposed methodology using scattering data from an elastic spherical shell.

Dehydration-induced mechanical instabilities in active elastic spherical shells

Active elastic instabilities are common phenomena in the natural world, where they have the character of sudden mechanical morphings. Frequently, the driving force of the instability mechanisms has a chemo-mechanical nature, which makes the instabilities very different from the standard elastic instabilities. In this paper, we describe and study the active elastic instability occurring in a swollen spherical closed shell, confining a water-filled cavity, during a dehydration process. We set up a few numerical experiments based on a stress-diffusion model to give an insight into the phenomenon. Then, we present a study that looks at the chemo-mechanical problem and, through a few simplifying assumptions, allows us to derive a semi-analytical model of the phenomenon. It takes into account both the stress state and the water concentration in the walls of the shell at the onset of the instability. Moreover, it considers the invariance of the cavity volume at the onset of instability, which is due to the impossibility of instantaneously changing the cavity volume filled with water. Eventually, it is shown that the semi-analytic model matches very well the outcomes of the numerical experiments far from the initial regime; the ranges of validity of the approximated analytical model are also discussed.

Delayed buckling of spherical shells due to viscoelastic knockdown of the critical load

We performed dynamic pressure buckling experiments on defect-seeded spherical shells made of a common silicone elastomer. Unlike in quasi-static experiments, shells buckled at ostensibly subcritical pressures, i.e. below the experimentally determined critical load at which buckling occurs elastically, often following a significant delay period from the time of load application. While emphasizing the close connections to elastic shell buckling, we rely on viscoelasticity to explain our observations. In particular, we demonstrate that the lower critical load may be determined from the material properties, which is rationalized by a simple analogy to elastic spherical shell buckling. We then introduce a model centred on empirical quantities to show that viscoelastic creep deformation lowers the critical load in the same predictable, quantifiable way that a growing defect would in an elastic shell. This allows us to capture how both the deflection at instability and the time delay depend on the applied pressure, material properties and defect geometry. These quantities are straightforward to measure in experiments. Thus, our work not only provides intuition for viscoelastic behaviour from an elastic shell buckling perspective but also offers an accessible pathway to introduce tunable, time-controlled actuation to existing mechanical actuators, e.g. pneumatic grippers.