Elastic Cylindrical Shell Research Articles

Scattering of Acoustic Waves by an Inhomogeneous Elastic Cylindrical Shell of Finite Length in a Half-Space

An analytical solution to the problem of diffraction of a plane acoustic wave on a radially inhomogeneous thick-walled elastic cylindrical shell of finite length is obtained. The cylindrical shell is located in an acoustic half-space filled with an ideal liquid. The boundary of the half-space is an acoustically rigid or acoustically soft surface. The results of calculations of the acoustic field in the far zone are presented.

Mathematical modelling of stability problems for thin transversally graded cylindrical shells

The objects of consideration are thin linearly elastic Kirchhoff–Love-type open circular cylindrical shells having a functionally graded macrostructure and a tolerance-periodic microstructure in circumferential direction. The first aim of this contribution is to formulate and discuss a new mathematical averaged non-asymptotic model for the analysis of selected stability problems for such shells. As a tool of modelling we shall apply the tolerance averaging technique. The second aim is to derive and discuss a new mathematical averaged asymptotic model. This model will be formulated using the consistent asymptotic modelling procedure. The starting equations are the well-known governing equations of linear Kirchhoff–Love second-order theory of thin elastic cylindrical shells. For the functionally graded shells under consideration, the starting equations have highly oscillating, non-continuous and tolerance-periodic coefficients in circumferential direction, whereas equations of the proposed models have continuous and slowly-varying coefficients. Moreover, some of coefficients of the tolerance model equations depend on a microstructure size. It will be shown that in the framework of the tolerance model not only the fundamental cell-independent, but also the new additional cell-dependent critical forces can be derived and analysed.

Uncertainty quantification for locally resonant coated plates and shells

The effect of uncertainty in design parameters on the acoustic performance of coated plates and coated shells for maritime applications is presented. The locally resonant coatings are designed using a viscoelastic material with an impedance similar to water and embedded with layers of inclusions. Both voids and hard inclusions, which respectively exhibit monopole and dipole resonance scattering, are considered. Effective medium approximation theory is employed to characterise the layers of inclusions as homogenised layers with effective material and geometric properties. The coating is then modelled as a multilayered equivalent fluid composed of alternating layers of the homogeneous viscoelastic material and the homogenised layers of inclusions. The coating is bonded to a rigid backing plate and the acoustic response is calculated using the transfer matrix method. The coating is also externally applied to an elastic cylindrical shell. The acoustic response is calculated by expressing the shell displacements and acoustic pressures in the coating and exterior domain in terms of Fourier series expansions, and applying continuity equations at each interface between the shell surface, coating layers and the surrounding water. Efficient stochastic models based on the non-intrusive polynomial chaos expansion (PCE) method are developed by transforming the analytical models for the coated plates and shells into computationally efficient surrogate models using point collocation. Uncertainty in dominant design parameters associated with the geometry of the inclusions and material properties of the coating is examined. The influence of the design parameters for inclusions tuned to different resonance frequencies and for multiple layers of inclusions is also reported. Strong acoustic performance of coating designs occurs in a broad frequency range around local resonance of the inclusions. For all coating models, uncertainty in parameters which predominantly influence the local resonance of the inclusions were observed to yield the greatest variation in the acoustic responses of the coated plates and shells.

A revisit to the plane problem for low-frequency acoustic scattering by an elastic cylindrical shell

The proposed revisit to a classical problem in fluid–structure interaction is due to an interest in the analysis of the narrow resonances corresponding to a low-frequency fluid-borne wave, inspired by modeling and design of metamaterials. In this case, numerical implementations would greatly benefit from preliminary asymptotic predictions. The normal incidence of an acoustic wave is studied for a circular cylindrical shell governed by plane strain equations in elasticity. A novel high-order asymptotic procedure is established considering for the first time all the peculiarities of the low-frequency behavior of a thin fluid-loaded cylinder. The obtained results are exposed in the form suggested by the Resonance Scattering Theory. It is shown that the pressure scattered by rigid cylinder is the best choice for a background component. Simple explicit formulae for resonant frequencies, amplitudes, and widths are presented. They support various important observations, including comparison between widths and the error of the asymptotic expansion for frequencies.

Propagation of Waves in a Fluid in a Thin Elastic Cylindrical Shell

The oscillatory process of a viscoelastic shell of a cylindrical tipe filled with a liquid is considered. Unlike other works, this paper focuses on the viscoelastic properties of a cylindrical shell and a liquid. Differential equations for joint vibrations of a shell and liquid are obtained by the equations of a thin shell that satisfies the Kirchhoff–Love hypotheses, and the equations of motion of a viscous liquid obey the Navier–Stokes equation. After simple transformations, the integro-differential equations are reduced to ordinary differential equations and solved using Godunov's orthogonal run method combined with Muller's method. Based on the developed algorithm, natural frequencies and corresponding vibration modes were obtained. For steady-state oscillations, all eigenvalues and eigenmodes turned out to be complex. For the first time, it was found that the damping coefficient branches out after certain values of wave numbers. It was found that the motion in a cylindrical shell is localized on the surface of the shell. At slow localization, starting from a certain wave number, the natural oscillations become aperiodic.

Odd elastic stability of cylindrical shells

Biologically active shells typically exhibit complex mechanical behaviors, with their morphogenesis involving rich symmetry breaking events and non-conservative biological forces. In this article, an accurate buckling solution model of active cylindrical shells is established based on the odd elasticity theory. The tension buckling, non-reciprocal torsional buckling and stress-free active buckling of odd elastic cylindrical shells are reported, which are an anomalous instability caused by odd elastic effects. The result shows that the anomalous buckling of cylindrical shells can only occur in the form of chiral deformation, because the energy required for instability can be obtained based on the odd elastic effects.

Chiral Standing Spin Waves and Unidirectional Waves of Odd Elastic Cylindrical Shells

Abstract Rotating waves can be observed in structures with periodic conditions, such as cylinders and spheres. Compared with traveling waves and standing waves, rotating waves have received less attention. In this paper, an odd elastic dynamic model of the cylindrical shells is established, and the dispersion relation, traveling waves, and standing waves are investigated. The non-Hermitian rotating waves and single-handedness chiral standing spin waves are reported, which are novel dynamic phenomenon caused by odd elastic effects. Waves generally cannot propagate in passive materials with vanishingly small elastic modulus. However, a unidirectional wave with the highest cut off frequency can occur in an odd elastic cylindrical shell with vanishingly small elastic modulus. For incompletely restrained end displacements, the odd elastic cylindrical shell can also generate a hybrid mode combining standing spin waves with unidirectional waves.

EQUILIBRIUM OF A NORMALLY LOADED ELASTIC SHALLOW CYLINDRICAL SHELL WITH DISLOCATIONS AND DISCLINATIONS

We consider the problem of the equilibrium of a normally loaded elastic shallow rectangular circular cylindrical shell, which contains sources of internal stress in the form of fields of continuously distributed edge dislocations and wedge disclinations or other sources, for example, thermal ones. Based on the modified system of Karman equations for the equilibrium of an elastic plate with dislocations and disclinations, a system of nonlinear equilibrium equations for the shell was constructed. The resulting system of equations differs from the Karman system of equilibrium equations for an elastic shallow cylindrical shell under the action of a normal load by the presence on the right side of the continuity equation of a nonzero function, called the incompatibility function, which is expressed through the densities of edge dislocations and wedge disclinations. The edges of the shell are freely clamped or hinged. In the absence of internal stress sources, this system transforms into a system of equilibrium equations for a shallow shell, and in the case of an infinitely large radius of curvature (and the presence of internal stress sources) into a modified system of Karman equilibrium equations for a flexible elastic plate with dislocations and disclinations. To solve the system of nonlinear shell equilibrium equations, the Newton – Kantorovich method in combination with the difference method is proposed. For the case of small values of the normal load and small values of the incompatibility function, linearization is carried out with respect to the deflection and stress function. As a result, a linear boundary value problem was obtained in the form of a system of differential equations for the deflection and the stress function, which is solved by the difference method. The solution of the linearized system is used as an initial approximation in the implementation of the Newton – Kantorovich method. Examples of numerical calculations of deflection and stress function for given values of normal load and incompatibility function are given, and corresponding graphs are constructed.

THERMOMECHANICAL DEFORMATION OF CARBON NANOTUBES IN POLYMER MATRIXES

Annotation: A theoretical study of the effects of nucleation of intrastructural thermal stresses in polymer materials reinforced with carbon nanotubes, caused by thermomechanical incompatibility of materials, was performed. Differential equations of its thermoelastic interaction with epoxy, polyamide, phenol-formaldehyde, polyester, polycarbonate, polystyrene, and polypropylene media in uniform and non-uniform temperature fields were constructed on the basis of modeling of a nanotube with an elastic cylindrical shell with aggregated values of thickness, modulus of elasticity, Poisson's ratio, and coefficient of linear thermal expansion. The solutions of these equations are found in closed form. The distribution functions of thermal stresses in the matrix medium and thermal forces in the nanotube wall are constructed. It was demonstrated that the thermal stresses in the matrix medium have the form of concentrators on the interface surface between the phases of the system and decrease inversely proportional to the square of the distance to the axis of the nanotube. It is shown that the intensities of these thermal stresses increase with increasing incompatibility of the thermomechanical parameters of the system phases.

Отражение звука от упругой конечной цилиндрической оболочки различной относительной длины

A calculation of the sound pressure frequency dependences scattered by a finite elastic cylindrical shell placed in a liquid medium is presented. The shell has hemispherical ends and is considered either hollow or filled with gas or liquid. The scattered sound pressure under conditions of hydroelastic contact on the shell surfaces is found by jointly using the Kirchhoff integral and the integral equation for the elastic medium displacement vector, obeying the Lamé equation. Boundary conditions regarding stresses and displacements are formulated for each of the shell contact surfaces with the external and internal environments. Considerating approach is based on the numerical transformation of continuous integral equations into a system of linear algebraic equations using curvilinear isoparametric boundary elements. In this case, the elements geometry and the main variables (displacements and stresses) are specified using the same interpolating relations (shape functions). The scattered sound pressure frequency dependences are calculated and analyzed for various ratios of the length and shell diameter.

Behaviour of circumferential and helical guided waves in two-layered composite cylindrical shell

A model of the acoustic scattering of a plane wave incident obliquely upon an infinite elastic bi-layered cylindrical shell is developed based on normal mode expansion technique. The far field backscattering form function for the composite copper/stainless steel cylindrical shell is calculated for reduced frequency range of k⊥a1=0–70 and for different values of incident angles α (k⊥=k1cos⁡α, k1 is the wave number in the fluid surrounding the shell and a1 is the outer radius of the shell). Numerical calculations show the behaviour of the resonances of circumferential and helical guided waves as the angle of incidence increases. The obtained results are compared with those of copper and stainless steel one-layered cylindrical shells. The use of frequency-angle representation depicts the helical propagation of surface waves along the axis of the shell. This representation, along with the identification plan, enables to describe the evolution of acoustic scattering as a function of incidence angle and vibration mode n. Furthermore, a comparative analysis of the resonance trajectory evolution between mono-layered and composite cylindrical shells is performed. The evolution of the helical guided waves, the bifurcation of the A0 wave into the A0− and A0+ waves, and a region with high backscattering are investigated. The presented results show that the extension of the region with high backscattering depends on the thickness and physical properties of the bi-layered cylindrical shell.

Acoustic radiation from a baffled finite shell in an underwater waveguide

Analytical modelling of vibroacoustic systems can help us to understand the physical phenomena involved in more complex problems, and it provides a benchmark solution and reference upon which more complex systems can be built. In the present work, the system of interest consists of a finite elastic cylindrical shell inserted in infinitely rigid cylindrical baffles and immersed in an underwater acoustic waveguide. The latter consists of a finite fluid layer bounded by an upper free surface and a lower rigid floor. In such a fluid domain, the acoustic waves radiated from the excited shell will exhibit reflections off the boundaries. This phenomenon is modelled by the image-source theory and embedded in the fluid loading term, which intervenes in the shell equations. Investigations into the influence on the finiteness of the elastic shell, types of supports (i.e., simply supported, clamped, free, and combinations of these), and depth of the waveguide on the shell’s acoustic radiation are presented.

A new combined asymptotic-tolerance model of thermoelasticity problems for thin uniperiodic cylindrical shells

AbstractThe objects of consideration are thin linearly thermoelastic Kirchhoff–Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential direction (uniperiodic shells). The aim of this contribution is to formulate and discuss a new averaged mathematical model for the analysis of selecteddynamic thermoelasticity problemsfor the shells under consideration. This so-called combined asymptotic-tolerance model is derived by applying the combined modelling including the consistent asymptotic and the tolerance non-asymptotic modelling techniques, which are conjugated with themselves into a newprocedure. The starting equations are the well-known governing equations of linear Kirchhoff–Love theory of thin elastic cylindrical shells combined with Duhamel–Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation. For the periodic shells, the starting equations have highly oscillating, non-continuous and periodic coefficients, whereas equations of the proposed model have constant coefficients dependent also on a cell size.

On the scattering of cylindrical elastic shell having trifurcation and structural variations at interfaces

The present research focuses on the study of how acoustic radiation modes behave in an elastic shell that has trifurcated junctions and structural variations. To express the vibration of a thin, flexible shell, the Donnell–Mushtari equations are utilized. These modes possess non-orthogonal characteristics and form a system that is linearly dependent. To analyze the scattering phenomenon in an elastic shell with a finite length that is connected to an extended trifurcated inlet and outlet with different ring conditions and step discontinuities, the mode-matching solution is developed. This approach uses continuity conditions for pressure and velocity modes, as well as generalized orthogonality conditions, to transform the differential systems into linear algebraic systems which are numerically solved after truncation. To ensure the accuracy of the algebraic calculations and the convergence of truncated coefficients, an energy flux identity based on the conservation law is developed and verified through continuity conditions. The results of numerical experiments conducted using the truncated solution offer valuable insights for designing effective noise reduction strategies in various industrial and engineering applications.

Use of interpolation methods for modeling the stress-strain state of operated oil storage tanks

The aim of the research is the comparison of two approaches for computer modeling of the stress-strain state of thin-walled shells of engineering structures, considering the imperfections of the geometric shapes arising due to their operation. The object of the study is the operated steel vertical cylindrical reservoir with imperfections of the geometric shape intended for storage of petroleum products. The first, so-called classical, approach provides geometric modeling of the surface of the tank's shell with the subsequent import of the geometric model into one of the systems of finite element analysis to calculate the stress-strain state of the structure and determine its technical condition, and the possibility of further operation. The geometric modeling of the shell surface with imperfections was performed using a two-dimensional interpolation method based on the 1st order smoothness outlines implemented in the point calculus. The calculation of the stress-strain state of the shell was carried out in the SCAD Office computer complex, taking into account geometric and structural non-linearity on the basis of the octahedral tangential stress theory. The second approach assumes modeling of an array of functions of vertical deflection of the tank wall by means of interpolation, solution of an array of differential equations of the elastic cylindrical shell under axisymmetric loading, improved by introduction of vertical deflection functions of the wall, followed by two-dimensional interpolation and analysis of the deformed state of the shell based on displacements arising in the tank wall from the hydrostatic load. As a result of the effective use of two-dimensional interpolation in the process of implementing the second approach, it was possible to achieve a significant increase in the speed of the numerical solution while maintaining sufficient accuracy for engineering calculations.

Wave Propagation in the Viscoelastic Functionally Graded Cylindrical Shell Based on the First-Order Shear Deformation Theory.

Based on the first-order shear deformation theory (FSDT) and Kelvin-Voigt viscoelastic model, one derives a wave equation of longitudinal guide waves in viscoelastic orthotropic cylindrical shells, which analytically solves the wave equation and explains the intrinsic meaning of the wave propagation. In the numerical examples, the velocity curves of the first few modes for the elastic cylindrical shell are first calculated, and the results of the available literature are compared to verify the derivation and programming. Furthermore, the phase velocity curves and attenuation coefficient curves of the guide waves for a functionally graded (FG) shell are calculated, and the effects of viscoelastic parameters, material gradient patterns, material volume fractions, and size ratios on the phase velocity curves and attenuation curves are studied. This study can be widely used to analytically model the wave propagating in inhomogeneous viscoelastic composite structures and present a theoretical basis for the excellent service performance of composite structures and ultrasonic devices.

Low-Frequency Fiber Optic Hydrophone Based on Ultra-Weak Fiber Bragg Grating

A winding low-frequency hydrophone based on ultra-weak fiber Bragg grating was studied. Through analyzing the hydrophone principle and the sensitivity factors, such as the material, radius, and thickness of the elastic cylindrical shell, the probe structure was optimized. In addition, an adaptive filtering technique based on the least mean square algorithm was introduced to improve the signal-to-noise ratio of the system. A low-frequency hydrophone with a working depth of 100 m and a diameter of 15 mm was developed and validated using the moving-water-column method. Results showed that the average acoustic pressure sensitivity of the hydrophone in the range of 1-100 Hz was –144.836 dB (re rad/μPa), the acoustic pressure sensitivity at 1 Hz was up to –130.85 (dB re rad/μPa). After using LMS filtering algorithm, the maximum signal-to-noise ratio of hydrophone can be increased by 4.62 dB, and the minimum detection pressure is 1.69×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">–4</sup> Pa/Hz <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> . This hydrophone with high sensitivity and signal-to-noise ratio provides a reference for low-frequency underwater detection.

Transient Deformation of an Anisotropic Cylindrical Shell with Structural Features

In this work, a new transient function of normal displacements is obtained in a thin elastic anisotropic cylindrical shell of constant thickness with structural features. Structural features form of two sequences in angular direction of weld spots or rivets. Structural features are mathematically described by point boundary conditions of rigid pinching or point boundary conditions of hinge support, accordingly. A load with time variable amplitude and impact boundaries influences on the outer surface of the shell along the normal. Such movement of the shell is considered in the cylindrical coordinate system as connected with the shell axis. Kirchhoff–Love hypothesis are confirmed as a shell model. The solution of this problem can be formed with the help of Green function method and lift compensation method. The required function of transient normal displacements is represented as the sum of integral operators of Green convolution for an infinite shell with functions of current transient load and compensating loads out of point boundary conditions. Compensating loads satisfying to boundary conditions can be found based on the solution of Volterra integral equations of the first kind with the difference kernel. Green function is such kernel. The system solution can be executed including preliminary discretization of compensating time loads. The convolution integrals can be taken numerically with the help of quadrature formula by the method of rectangles. The verification of the method is represented by comparing the solution of specific case with the known solution for a hinged isotropic cylindrical shell.

Asymptotic corrections to the low-frequency theory for a cylindrical elastic shell

The general scaling underlying the asymptotic derivation of 2D theory for thin shells from the original equations of motion in 3D elasticity fails for cylindrical shells due to the cancellation of the leading-order terms in the geometric relations for the mid-surface deformations corresponding to shear and circumferential extension. As a consequence, a cylindrical shell as an elastic waveguide supports a small cut-off frequency for each circumferential mode. The value of this cut-off tends to zero at the thin shell limit. In this case, the near-cut-off behaviour is strongly affected by the presence of two small parameters associated with the relative thickness and wavenumber. It is not obvious whether it can be treated within the 2D theory. For the first time, a novel special scaling is introduced, in order to derive an asymptotically consistent formulation for a cylindrical shell starting from 3D framework. Comparisons with the previous results obtained using the popular 2D Sanders–Koiter shell theory are made. Asymptotic corrections are deduced for the fourth-order equation of low-frequency motion and some of other relations, including the formulae for tangential shear stress resultants.

A new combined asymptotic-tolerance model of thermoelasticity problems for thin biperiodic cylindrical shells

The objects of consideration are thin linearly thermoelastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to formulate and discuss a new averaged mathematical model for the analysis of selected dynamic thermoelasticity problems for the shells under consideration. This so-called combined asymptotic-tolerance model is derived by applying the combined modelling including the consistent asymptotic and the tolerance non-asymptotic modelling techniques, which are conjugated with themselves into a new procedure. The starting equations are the well-known governing equations of linear Kirchhoff-Love theory of thin elastic cylindrical shells combined with Duhamel-Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation. For the periodic shells, the starting equations have highly oscillating, non-continuous and periodic coefficients, whereas equations of the proposed model have constant coefficients dependent also on a cell size.