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The ultrasonic frequency radial vibration of an elastic thin spherical shell composed of isotropic materials is studied, the frequency equation of its vibration is deduced, and the equivalent circuit is obtained. The finite element software COMSOL is used to verify the obtained analytical theory, and the calculated results are basically consistent with the theoretical calculated values. The relevant conclusions provide references for the application of spherical shell structure in related fields.

We present a hydrodynamic model for a thin spherical shell of active nematic liquid crystal with an arbitrary configuration of defects. The active flows generated by defects in the director lead to the formation of stable vortices, analogous to those seen in confined systems in flat geometries, which generate effective dynamics for four +1/2 defects that reproduces the tetrahedral to planar oscillations observed in experiments. As the activity is increased and two counterrotating vortices dominate the flow, the defects are drawn more tightly into pairs, rotating about antipodal points. We extend this situation to also describe the dynamics of other configurations of defects. For example, two +1 defects are found to attract or repel according to the local geometric character of the director field around them and the extensile or contractile nature of the material, while additional pairs of opposite charge defects can give rise to flow states containing more than two vortices. Finally, we describe the generic relationship between defects in the orientation and singular points of the flow, and suggest implications for the three-dimensional nature of the flow and deformation in the shape of the shell.

Can uncorrelated surrounding sound sources be used to generate extended diffuse sound fields? By definition, targets are a constant sound pressure level, a vanishing active sound intensity, and uncorrelated sound waves arriving isotropically from all directions. Are there ideal source layouts to synthesize a maximum diffuse sound field within? As methods, we employ numeric simulations and undertake a series of considerations based on uncorrelated source layouts at a finite radius. Statistically expected active sound intensity and sound energy density are insightful and highlight the relation of active sound intensity to potential theory. Correspondingly, both Gauß’ divergence and Newton’s spherical shell theorem apply, and they provide valuable insights. In a circular layout, uncorrelated elementary point-source fields decaying by 1/√r ideally compose an extended sound field of vanishing active sound intensity; in spherical layouts this is the case with a 1/r decay. None of the layouts synthesizes a perfectly constant sound energy density inside. Theory and simulation offer a broad basis for understanding the synthesis of diffuse sound fields with uncorrelated sources in the free sound field.

This paper employs the Higher-Order Shear and Normal Deformation Theory (HOSNDT) to a static analysis of spherical composite shells. By using a third-order parabolic variation of shear strains to account for transverse shear deformation effects, the theory does away with the necessity for shear correction factors and precisely captures deformation of the shell along the thickness. The concept of virtual work is used to systematically develop the governing equilibrium equations and associated boundary conditions. Under static transverse loads, the mathematical model is solved using Navier’s analytical method, which works well for simply supported doubly curved laminated shells. The solution provides non-dimensional displacements and stress components that allow for direct comparison with benchmark solutions. The validity and applicability of the developed theory are confirmed by the current numerical results, which show excellent agreement with the body of existing literature.

Dielectric elastomers (DEs) have attracted significant attention in many engineering fields due to their excellent deformation capabilities and mechanical properties. In this work, the nonlinear vibrations of the DE spherical shell characterized by the third-order Ogden model are investigated. The governing equation describing the radially symmetric motions of the shell is derived by the incompressibility constraint and the variational method. Through the qualitative and quantitative analyses, the nonlinear dynamical behaviors are discussed, along with the parameter analyses of the structural damping, pressure, direct current (DC) voltage and alternating current (AC) voltage. It is shown that with the increasing pressure and voltage, the vibrations transition from periodic vibrations to chaotic vibrations via the period-doubling bifurcation. Particularly, the self-similar structure of the system is found.

Here we present general multivariate mixed Ostrowski type inequalities over spherical shells and balls. We cover the radial and not necessarily radial cases. The proofs derive by the use of some estimates coming out of some new trigonometric and hyperbolic Taylor's formulae ([2]) and reducing the multivariate problem to a univariate one via general polar coordinates.

Gravity forward modeling as a basic tool has been widely used for topography correction and 3D density inversion. The source region is usually discretized into tesseroids (i.e., spherical prisms) to consider the influence of the curvature of planets in global or large-scale problems. Traditional gravity forward modeling methods in spherical coordinates, including the Taylor expansion and Gaussian–Legendre quadrature, are all based on spatial domains, which mostly have low computational efficiency. This study proposes a high-efficiency forward modeling method of gravitational fields in the spherical harmonic domain, in which the gravity anomalies and gradient tensors can be expressed as spherical harmonic synthesis forms of spherical harmonic coefficients of 3D density distribution. A homogeneous spherical shell model is used to test its effectiveness compared with traditional spatial domain methods. It demonstrates that the computational efficiency of the proposed spherical harmonic domain method is improved by four orders of magnitude with a similar level of computational accuracy compared with the optimized 3D GLQ method. The test also shows that the computational time of the proposed method is not affected by the observation height. Finally, the proposed forward method is applied to the topography correction of the Moon. The results show that the gravity response of the topography obtained with our method is close to that of the optimized 3D GLQ method and is also consistent with previous results.

The spherical shell and spherical zonal band are two elemental geometries that are often used as benchmarks for gravity field modeling. When applying the spherical shell and spherical zonal band discretized into tesseroids, the errors may be reduced or cancelled for the superposition of the tesseroids due to the spherical symmetry of the spherical shell and spherical zonal band. In previous studies, this superposition error elimination effect (SEEE) of the spherical shell and spherical zonal band has not been taken seriously, and it needs to be investigated carefully. In this contribution, the analytical formulas of the signal of derivatives of the gravitational potential up to third order (e.g., V, Vz, Vzz, Vxx, Vyy, Vzzz, Vxxz, and Vyyz) of a tesseroid are derived when the computation point is situated on the polar axis. In comparison with prior research, simpler analytical expressions of the gravitational effects of a spherical zonal band are derived from these novel expressions of a tesseroid. In the numerical experiments, the relative errors of the gravitational effects of the individual tesseroid are compared to those of the spherical zonal band and spherical shell not only with different 3D Gauss–Legendre quadrature orders ranging from (1,1,1) to (7,7,7) but also with different grid sizes (i.e., 5∘×5∘, 2∘×2∘, 1∘×1∘, 30′×30′, and 15′×15′) at a satellite altitude of 260 km. Numerical results reveal that the SEEE does not occur for the gravitational components V, Vz, Vzz, and Vzzz of a spherical zonal band discretized into tesseroids. The SEEE can be found for the Vxx and Vyy, whereas the superposition error effect exists for the Vxxz and Vyyz of a spherical zonal band discretized into tesseroids on the overall average. In most instances, the SEEE occurs for a spherical shell discretized into tesseroids. In summary, numerical experiments demonstrate the existence of the SEEE of a spherical zonal band and a spherical shell, and the analytical solutions for a tesseroid can benefit the investigation of the SEEE. The single tesseroid benchmark can be proposed in comparison to the spherical shell and spherical zonal band benchmarks in gravity field modeling based on these new analytical formulas of a tesseroid.

The problem of elastic and limiting equilibrium of a shallow spherical shell weakened by a pair of rectilinear cuts along the meridian is considered in a two-dimensional statement. One is considered to be a contact crack capable of closing, while the other one — a narrow slit with free faces. The contact interaction of the crack faces under bending load is described using a contact model along a line in the front surface of the shell. The numerical solution to the problem is developed by the method of singular integral equations using the mechanical quadrature algorithm. The regularities of the contact reaction distribution on the closed crack faces are studied. The effect of the shell curvature and the relative location of the defects on the values of intensity coefficients of forces and moments and on the value of the fracture load is studied.

A thin-walled spherical shell made of pure iron material is a key part of precision physical experiments. Tool wear characteristics of freeform surface parts are significantly different from those of single-point turning due to the movement of the contact point between the tool and the workpiece, which affects the form accuracy and surface integrity of workpiece. However, there is lack of a comprehensive understanding of the tool wear characteristics in machining pure iron materials. Therefore, we proposed the mathematical model to investigate the wear characteristics of cemented carbide tool further when spherical shell turning pure iron materials. The results show that uniform flank wear land and notable notch wear occur when turning end face, but notch wear disappears and only flank wear land exists when turning spherical shell. Based on major notch position and minor notch position, a mathematical model is developed to explain formation mechanisms of flank wear land during turning spherical shell of pure iron materials. Theoretical and experimental results show that flank wear land results from the major and minor notch movement. Spherical shell turning and end face turning have the same wear mechanisms, mainly composed of adhesive wear, diffusion wear, and oxidation wear.