Isotropic Solid Sphere Research Articles

Thermoelastic behavior of an isotropic solid sphere under a non-uniform heat flow according to the MGT thermoelastic model

Moore-Gibson-Thompson (MGT) is an equation which appropriately describes the spread of sound waves in gasses and fluids as well as thermal/mechanical waves in elastic bodies. The objective of this article is to theoretically analyze the generalized thermoelasticity models that have been presented as the development of the Fourier’s law dealing with the paradox of unlimited propagation velocities of thermal waves. For this purpose, a new model is presented which combines the third type of Green and Naghdi model (GN-III) with the generalized theory including the relaxation time based on the MGT equation. The proposed model may be considered as a generalization of previous thermoelastic theories. To examine the introduced approach, the behavior of thermoelastic waves within a homogeneous isotropic sphere in which its surface is exposed to thermal shock with varying heat source is investigated. The variations in different physical fields of a given substance have been computed by means of Laplace transform technique and an efficient numerical technique is implemented in Laplace inversion procedure. The effect of different forms of heat source are also examined and several comparisons for various thermoelasticity approaches are comprehensively conducted.

Diagonalization Method to Hyperbolic Two-Temperature Generalized Thermoelastic Solid Sphere under Mechanical Damage Effect

This study is the first to use the diagonalization method for the new modelling of a homogeneous, thermoelastic, and isotropic solid sphere that has been subjected to mechanical damage. The fundamental equations were derived using the hyperbolic two-temperature generalized thermoelasticity theory with mechanical damage taken into account. The outer surface of the sphere has been assumed to have been shocked thermally without cubical dilatation. The numerical results for the dynamical and conductive temperatures increment, strain, displacement, and average of the principal stresses components have been represented graphically with different values of the hyperbolic two-temperature parameter and mechanical damage parameters. The two-temperature model parameter and the mechanical damage parameter have significant effects. The propagations of the thermomechanical waves take place at finite speeds in the context of the hyperbolic two-temperature theory as well as in the usual context of the Lord–Shulman theory with one-temperature.

The Fractional Strain Influence on a Solid Sphere under Hyperbolic Two-Temperature Generalized Thermoelasticity Theory by Using Diagonalization Method

This article constructs a mathematical model based on fractional-order deformations for a one-dimensional, thermoelastic, homogenous, and isotropic solid sphere. In the context of the hyperbolic two-temperature generalized thermoelasticity theory, the governing equations have been established. Thermally and without deformation, the sphere’s bounding surface is shocked. The singularities of the functions examined at the center of the world were decreased by using L’Hopital’s rule. Numerical results with different parameter fractional-order values, the double temperature function, radial distance, and time have been graphically illustrated. The two-temperature parameter, radial distance, and time have significant effects on all the studied functions, and the fractional-order parameter influences only mechanical functions. In the hyperbolic two-temperature theory as well as in one-temperature theory (the Lord-Shulman model), thermal and mechanical waves spread at low speeds in the thermoelastic organization.

Thermal-stress analysis of a damaged solid sphere using hyperbolic two-temperature generalized thermoelasticity theory

This work aims to study the influence of the rotation on a thermoelastic solid sphere in the context of the hyperbolic two-temperature generalized thermoelasticity theory based on the mechanical damage consideration. Therefore, a mathematical model of thermoelastic, homogenous, and isotropic solid sphere with a rotation based on the mechanical damage definition has been constructed. The governing equations have been written in the context of hyperbolic two-temperature generalized thermoelasticity theory. The bounding surface of the sphere is thermally shocked and without volumetric deformation. The singularities of the studied functions at the center of the sphere have been deleted using L’Hopital’s rule. The numerical results have been represented graphically with various mechanical damage values, two-temperature parameters, and rotation parameter values. The two-temperature parameter has significant effects on all the studied functions. Damage and rotation have a major impact on deformation, displacement, stress, and stress–strain energy, while their effects on conductive and dynamical temperature rise are minimal. The thermal and mechanical waves propagate with finite speeds on the thermoelastic body in the hyperbolic two-temperature theory and the one-temperature theory (Lord-Shulman model).

Solution of a Spherically Symmetric Static Problem of General Relativity for an Elastic Solid Sphere

The paper is concerned with the spherically symmetric static problem of the General Relativity Theory (GRT). The classical interior solution of this problem found in 1916 by K. Schwarzschild for a fluid sphere is generalized for a linear elastic isotropic solid sphere. The GRT equations are supplemented with the equation for the stresses which is similar to the compatibility equation of the theory of elasticity and is derived using the principle of minimum complementary energy for an elastic solid. Numerical analysis of the obtained solution is undertaken.

High frequency signal acquisition using a smartphone in an undergraduate teaching laboratory: Applications in ultrasonic resonance spectra.

A simple and inexpensive approach to acquiring signals in the megahertz frequency range using a smartphone is described. The approach is general, applicable to electromagnetic as well as acoustic measurements, and makes available to undergraduate teaching laboratories experiments that are traditionally inaccessible due to the expensive equipment that are required. This paper focuses on megahertz range ultrasonic resonance spectra in liquids and solids, although there is virtually no upper limit on frequencies measurable using this technique. Acoustic resonance measurements in water and Fluorinert in a one dimensional (1D) resonant cavity were conducted and used to calculate sound speed. The technique is shown to have a precision and accuracy significantly better than one percent in liquid sound speed. Measurements of 3D resonances in an isotropic solid sphere were also made and used to determine the bulk and shear moduli of the sample. The elastic moduli determined from the solid resonance measurements agreed with those determined using a research grade vector network analyzer to better than 0.5%. The apparatus and measurement technique described can thus make research grade measurements using standardly available laboratory equipment for a cost that is two-to-three orders of magnitude less than the traditional measurement equipment used for these measurements.

Generalized Thermoelasticity for an Isotropic Solid Sphere Indual-Phase-Lag of Heat Transfer with Surface Heat Flux

The dual-phase-lag heat transfer model is applied for an isotropic solid sphere. The solution of the problem is carried out when the boundary of the sphere is maintained at constant heat flux and the displacement of the surface is constrained. The analytical solutions of the displacement, temperature, and stresses are determined. Laplace transform technique is used to obtain the solution. The solution of the problem in the physical domain is obtained by using a numerical method for the inversion of the Laplace transforms based on Fourier series expansions. Numerical values of displacement, temperature, and stresses are computed for a particular material and presented graphically. A comparison with other thermoelasticity theories also has been studied.

Free vibration of rotating hollow spheres containing acoustic media

When a vibrating structure is rotated with respect to inertial space, the vibrating pattern rotates at a rate proportional to the inertial rate of rotation. Bryan first observed this effect in 1890. The effect, called Bryan's effect in the sequel, has numerous navigational applications and could be useful in understanding the dynamics of pulsating stars and earthquake series in astrophysics and seismology. Bryan's factor (the coefficient of proportionality between the inertial and vibrating pattern rotation rates) depends on the geometry of the structure and the vibration mode number. The “gyroscopic effects” of a hollow isotropic solid sphere filled with an inviscid acoustic medium are considered here, but the theory is readily adapted to a hollow isotropic solid cylinder filled with an inviscid acoustic medium. A linear theory is developed assuming, among other mild conditions, that the rotation rate is constant and much smaller than the lowest eigenfrequency of the vibrating system. Thus centrifugal forces are considered to be negligible. Before calculating solutions for the displacement of a particle in the isotropic, spherical, distributed body, Bryan's factor is interpreted using a complex function. Here it is demonstrated that neither Bryan's effect nor Bryan's factor is influenced by including light, isotropic, viscous damping in the mathematical model. Hence damping is neglected in the sequel. Two scenarios are then identified. Firstly, we may assume that the acoustic medium is completely involved in the rotation (the spheroidal mode). Secondly, we may assume that the acoustic medium remains static with respect to the inertial reference frame (the torsional mode). We investigate the spheroidal mode using a numerical experiment that compares the rotational angular rate of a sphere (filled with an inviscid acoustic medium) with those of its vibrating patterns at both high and low vibration frequency.

Acoustic resonance scattering from a submerged anisotropic sphere

The resonance scattering theory (RST) is applied to the problem of sound scattering from an elastic transversely isotropic solid sphere suspended in an ideal acoustic fluid medium. The normal mode expansion technique in conjunction with the Frobenius power series solution method is utilized to deal with the material anisotropy. The presented model, which degenerates to the simple isotropic solution in the case of very weak anisotropy, is initially employed to study sensitivity of various resonant modes of vibration to perturbations in elements of the stiffness matrix. Employing a rigid background subtraction, the target’s spectrum of resonances is extracted from the relevant modal backscattering form functions and subsequently traced and discussed through Regge pole trajectory plots. Also, the backscattering form function and resonance spectra, along with the dispersion curves for selected transversely isotropic solid spheres with distinct degrees of material anisotropy, are calculated and discussed. The various modes of propagation associated with the Rayleigh, Whispering Gallery, and fluid-borne Scholte-Stoneley surface waves are identified and examined.

On Stress-Focusing Effect in a Uniformly Heated Solid Sphere

By using the separation of variables technique, the dynamic thermal stress responses in an isotropic solid sphere subjected to uniform temperature rise all over the sphere and a sudden constant pressure at the external surface are performed successfully. The analytical solutions of the radial and hoop dynamic stresses at the center are also obtained. By means of the present method, integral transform can be avoided. Numerical results denote that a very high dynamic stress peak appears periodically at the center of the isotropic solid sphere subjected to uniform temperature rise all over the sphere and a sudden constant pressure at the external surface.

Observation of diffraction-free propagation of surface acoustic waves around a homogeneous isotropic solid sphere

This letter shows an unexpected phenomenon where a surface acoustic wave (Rayleigh wave) excited by a line source with a finite length on a solid sphere propagates along the great circle in a direction perpendicular to the line source without beam spreading due to diffraction. In experiments, a piezoelectric transducer, 1.5 mm in width and 20 mm in length, was glued on a surface of a glass ball, 80 mm in diameter, as a line source with a finite length. A beam of Rayleigh waves with frequencies centered at 1.1 MHz was excited in either direction perpendicular to the transducer length. A receiving transducer with a circular aperture, 2 mm in diameter, was used in direct contact with the surface to detect the distribution of vibration over the surface of the ball. It was observed that the excited Rayleigh wave propagated along a great circle of the ball for at least four roundtrips. The beam was confined within a narrow path around the ball, the width of which was no more than 20 mm.

On Eigenfrequencies of an Anisotropic Sphere

This note presents exact frequency equations of two independent classes of vibrations of a spherically isotropic solid sphere with fixed boundary conditions. Numerical calculations are performed and comparison between two different materials is made. Some useful observations are obtained. [S0021-8936(00)00102-1]

Experimental Study of Stress-Focusing Effect in a Solid Sphere by Using Pulse-Laser.

When an isotropic solid sphere is suddenly subjected to an instantaneous heat source, a stress wave occurs at the moment thermal impact is applied. The stress wave proceeds radially inward to the center of a sphere. The wave may accumulate at the center and give rise to very large stress magnitudes, even though the initial thermal stress is relatively small. This phenomenon is called the stress-focusing effect. In this study, we certify to the stress-focusing effect in a glass solid sphere by using a Nd : YAG pulse laser. We compare the experimental results with the theoretical values and show the mechanism of stress-focusing effect in a solid sphere.

The photoacoustic effect generated by an isotropic solid sphere

In this paper the production of acoustic waves by irradiation of an optically thin, isotropic solid sphere is discussed. Solutions to the wave equation for the displacement potential are obtained which, together with the conservation equations for acoustic waves at an interface, give a frequency domain solution for the photoacoustic pressure. The time domain response of the sphere to a delta function heating pulse is obtained by Fourier transformation of the frequency domain expression. The photoacoustic waveform generated by a short radiation pulse is shown to depend on the radius of the sphere, the density and longitudinal sound speed of the sphere relative to those of the surrounding fluid, and the ratio of the transverse to longitudinal sound speed of the solid. Experiments are reported where a Q-switched laser was used to excite single spheres.

THERMAL STRESS-FOCUSING EFFECT IN A SOLID SPHERE SUBJECTED TO INSTANTANEOUS HEATING

When an isotropic solid sphere is suddently subjected to a uniform temperature rise on the surface of the sphere, a stress wave occurs at the surface the moment thermal impact is applied. The stress wave at the surface proceeds radially inward to the center of the sphere. The wave may accumulate at the center and give rise to very large stress magnitudes, even though the initial thermal stress is relatively small. This phenomenon is called the stress-focusing effect. This paper analyzes, in an exact manner, the effects of these waves using the ray series. The results give a clear indication of the mechanism of stress-focusing effect in a solid sphere.

Stress-focusing effect in a uniformly heated transversely isotropic sphere

The ray theory is applied to the stress-focusing effects in a uniformly heated transversely isotropic solid sphere. The stress-focusing effect is the phenomenon that, under an instantaneous heating, stress waves reflected from the free surface of the sphere result in very high stresses at the center. Using the ray theory, the Fourier transformed solution of stress waves in the sphere is sorted into rays according to the ray path of multiply-reflected waves. The inverse transform of each ray gives rise to the exact solution of the transient response up to the arrival time of the next ray. The results reveal that stresses peak periodically at regular intervals and the stress peak at the center of the sphere depends on the anisotropy of the sphere.

Axisymmetrical Transient Thermal Stress Analysis of a Spherically Isotropic Solid Sphere Due to Partially Distributed Heat Source

本報は,球座標系 (r, θ, φ) における直交異方性材料の非定常熱応力問題を解析したものである.θ-φ 面内での等方性 (球面等方性) を持つ中実球の軸対称問題を扱った.解析にあたっては,球面調和関数の概念を,つり合い方程式に導入し,変位および応力成分を定式化した.また,中実球の頂部 (θ=0゜) を中心とした局部加熱が作用する場合に対し,数値計算を行い,各異方性パラメータが温度,応力に与える影響を吟味した.

Thermal Shock Analysis of Spherical Shapes

A method is described for studying the thermal shock characteristics of a brittle material. An analysis of the thermal stresses developed in a homogeneous isotropic solid sphere has led to the formulation of an equation relating the physical properties of the body to the temperature difference causing failure and time to maximum stress in a single-cycle unsteady-state test. The thermal shock test consisted of plunging a sphere at one uniform temperature into a medium at another temperature. If fracture occurred, the time to fracture was recorded. A large number of tests were run to determine the temperature difference which caused 50% of the spheres to fracture. The thermal shock relationships were tested using a high-alumina body. The physical properties relating to the thermal shock equations were measured, and calculated temperature differences causing failure and times to maximum stress were compared with measured values. Sufficient agreement was found to lend support to the theory. Summary A method has been developed for studying the thermal shock characteristics of a brittle substance. The method consists of a single-cycle test of unsteady-state nature. Two testing conditions have been selected, one having a rather high surface heat transfer coefficient in a liquid bath and the other having a small finite surface heat transfer coefficient in an air bath. These two conditions are at either extremes regarding the thermal shock usually given a substance in practice. The only fair agreement found in the calculated and experimental data indicates that further investigation is necessary. The importance of certain factors, such as time to maximum stress, which was found to be in rather good agreement for the salt bath heat transfer, cannot be overlooked. There are many applications in the high temperature-high stress field in which the concept of time to maximum stress might well be examined. For example, when repeated high-temperature heatings are made on a refractory piece, the cycling might be arranged so that the time to maximum stress was never reached for the particular heating cycle, although the (ΔT) was higher than that necessary to cause failure. Much is to be learned from this type of study which may be applied to actual situations. It must be emphasized, however, that this work has been conducted on one single set of conditions and the factors found here do not necessarily apply to other situations.