Transient piezothermoelastic analysis for a functionally graded thermopiezoelectric hollow sphere

This paper is concerned with the theoretical treatment of transient piezothermoelastic problem involving a functionally graded thermopiezoelectric hollow sphere due to uniform heat supply. The transient one-dimensional temperature is analyzed by the method of Laplace transformation. The thermal, thermoelastic and piezoelectric constants of the hollow sphere are expressed as power functions of the radial coordinate. The one-dimensional solution for the temperature change in a transient state, and piezothermoelastic response of a functionally graded thermopiezoelectric hollow sphere is obtained herein. Some numerical results for the temperature change, displacement, stress and electric potential distributions are shown. Furthermore, the influence of the nonhomogeneity of the material upon the temperature change, displacement, stresses and electric potential is investigated.     

Transient thermoelastic analysis for a multilayered hollow sphere with piecewise power law nonhomogeneity

This paper is concerned with the theoretical treatment of transient thermoelastic problem involving a multilayered hollow sphere with piecewise power law nonhomogeneity due to uniform heat supply. The thermal and thermoelastic constants of each layer are expressed as power functions of the radial coordinate, and their values continue on the interfaces. We obtain the exact solution for the one-dimensional temperature change in a transient state, and thermoelastic response. Some numerical results for the temperature change, the displacement and the stress distributions are shown in figures.     

功能梯度压电压磁球对称问题的静力响应

研究径向载荷作用下功能梯度压电压磁空心球壳的空间球对称电磁弹耦合静力学问题。假设压电压磁空心球壳的材料参数沿球厚度方向呈幂函数分布, 在球坐标系下, 由材料的参数方程、本构方程、几何方程和平衡方程导出在外激励作用下空心球壳体的应力、电势、磁势等物理量的解析解。结构内表面材料设为常用BaTiO₃-CoFeO₄复合材料, 分别对球壳内表面受力以及内外表面存在电势差或磁势差的情况进行数值讨论, 分别给出不同梯度参数下结构内部径向应力、环向应力、电势和磁势的分布。结果表明梯度参数的选取对功能梯度压电压磁球壳的性能有很大的影响。     

Thermoelastic dynamic solution of a multilayered spherically isotropic hollow sphere for spherically symmetric problems

Summary. The thermoelastic dynamic solution of a multilayered spherically isotropic hollow sphere in the state of spherical symmetry is obtained. By the method of superposition, the displacement is divided into two parts: one is quasi-static and the other is dynamic. The quasi-static solution is first derived in an explicit form by using the transfer matrix method. Then by introducing a new dependent variable, the governing equations, boundary conditions as well as the initial conditions for the dynamic solution are rewritten, and the dynamic solution is obtained by the separation of variables method coupled with the initial parameter method as well as the orthogonal expansion technique. The present method is suitable for a multilayered spherically isotropic hollow sphere consisting of arbitrary layers and subjected to arbitrary spherically symmetric thermal loads. Numerical results are finally presented and discussed.     

Coupled thermal stress problem in a hollow sphere under a partial heating

In a precise analysis of transient thermal stress problem, this requires the corrected heat equation taking into account of the thermo-mechanical coupling term. In this report, we deal with the transient thermal stress problem in a hollow sphere accompanied by the simultaneous determination of stress and temperature distributions. A new technique is developed in this report and numerical works are carried out. From our results, it should be concluded that the small lag can be found in case of ordinal practical materials.     

Stressed state of a conducting sphere under the electromagnetic action with pulsed modulating signal

A dynamic centrally symmetric problem of thermomechanics is posed for a hollow conducting sphere subjected to a uniform nonstationary electromagnetic action. By using the cubic approximations of the azimuthal component of the magnetic-field strength and the radial component of the stress tensor in the radial coordinate, we obtain the solution of the problem and perform the numerical analysis of the thermal stressed state and the load-carrying capacity of nonferromagnetic spheres under the electromagnetic action in the mode with pulsed modulating signal.     

Numerical Simulation of the Mechanical Properties of Sintered and Bonded Perforated Hollow Sphere Structures (PHSS)

This paper investigated the uniaxial mechanical properties of a new type of hollow sphere structures. For this new type, the sphere shell was perforated by several holes in order to open the inner sphere volume and surface. The mechanical properties, i.e. elastic properties and initial yield stress of perforated hollow sphere structures (PHSS) in a primitive cubic arrangement were numerically evaluated for different hole diameters and different joining techniques of the hollow spheres. The results are compa...     

Vibration analysis of a poroelastic composite hollow sphere

Vibrations in a poroelastic composite hollow sphere are investigated employing Biot’s theory of wave propagation in poroelastic media. A composite hollow poroelastic sphere consists of two concentric poroelastic spherical layers both of which are made of different poroelastic materials with each poroelastic material being homogeneous and isotropic. The boundaries of the composite hollow poroelastic sphere are free from stress. The frequency equations of both radial and rotatory vibrations are obtained each for pervious and impervious surfaces. The frequency equation of vibrations of a poroelastic composite hollow sphere with rigid core is derived as a particular case. The non-dimensional frequency for propagating modes is computed as a function of ratio of thickness to inner radius of core. The results are presented graphically for two types of poroelastic composite spheres and then discussed.     

Stress intensity factors of two diametrically opposed edge cracks in a thick-walled functionally graded material cylinder

A method is developed to evaluate stress intensity factors for two diametrically-opposed edge cracks emanating from the inner surface of a thick-walled functionally graded material (FGM) cylinder. The crack and the cylinder inner surfaces are subjected to an internal pressure. The thermal eigenstrain induced in the cylinder material due to nonuniform coefficient of thermal expansion after cooling from the sintering temperature is taken into account. First, the FGM cylinder is homogenized by simulating its nonhomogeneous material properties by an equivalent eigenstrain, whereby the problem is reduced to the solution of a cracked homogenized cylinder with an induced thermal and an equivalent eigenstrains and under an internal pressure. Then, representing the cracks by a continuous distribution of edge dislocations and using their complex potential functions, generalized formulations are developed to calculate stress intensity factors for the cracks in the homogenized cylinder. The stress intensity factors calculated for the cracks in homogenized cylinder represents the stress intensity factors for the same cracks in the FGM cylinder. The application of the formulations are demonstrated for a thick-walled TiC/Al₂O₃ FGM cylinder and some numerical results of stress intensity factors are presented for different profiles of material distribution in the FGM cylinder.     

Collinear cracks in a layered half – plane with a graded nonhomogeneous interfacial zone – Part II: Thermal shock response

In Part I of the paper, the problem of collinear cracks in a layered half-plane with a graded nonhomogeneous interfacial zone was investigated under mechanical loading and the cracking behavior was addressed by evaluating the stress intensity factors as functions of various geometric and material parameters. In Part II, the solution framework is extended to the problem of thermal shock on the basis of uncoupled, quasi-static thermoelasticity. The interfacial zone, in this case, is assumed to have the graded thermoelastic properties. Using the principle of superposition, a system of singular integral equations is solved subjected to equivalent crack surface tractions obtained form the transient thermoelasticity solution for a uncracked medium. Main results presented are the transient thermal stress intensity factors of collinear cracks to illustrate the parametric effects of geometric and material combinations of the layered medium with the thermoelastically graded interfacial zone. This revised version was published online in July 2006 with corrections to the Cover Date.     

Edge cracks in a transversely isotropic hollow cylinder

The analytical solution for the linear elastic, axisymmetric problem of inner and outer edge cracks in a transversely isotropic infinitely long hollow cylinder is considered. The z = 0 plane on which the crack lies is a plane of symmetry. The loading is uniform crack surface pressure. The mixed boundary value problem is reduced to a singular integral equation where the unknown is the derivative of the crack surface displacement. An asymptotic analysis is done to derive the generalized Cauchy kernel associated with edge cracks. It is shown that the stress intensity factor is a function of three material parameters. The singular integral equation is solved numerically. Stress intensity factors are presented for various values of material and geometric parameters.     

Finite element simulation of the thermal conductivity of perforated hollow sphere structures (PHSS): Parametric study

This paper investigates the thermal properties of a new type of hollow sphere structures. For this new type, the sphere shell is perforated by several holes in order to open the inner sphere volume and surface. The effective thermal conductivity of perforated sphere structures in a primitive cubic arrangement is numerically evaluated for different hole diameters and different dimensions of the joining elements.     

Non-uniform temperature gradients and thermal stresses produced by a moving heat flux applied on a hollow sphere

This work presents numerical analyses of transient temperature and thermally-induced stress distributions in a hollow steel sphere heated by a moving uniform heat source applied on a certain zenithal segment (the heated zenithal segment, Θ _H ) of its outer surface (the processed surface) under stagnant ambient conditions. Along the process, the moving heat source (MHS) moves angularly from the first zenithal segment to the last zenithal segment on the processed surface with a constant angular speed, ω, and then returns backward to the first zenithal segment with the same speed. It is assumed that the inner surface is heat-isolated and that the outer surface except the heated segment is under stagnant ambient conditions. The numerical calculations are performed individually for a wide range of thermal conductivity, λ, of steel and for the different Θ _H s. The maximum effective thermal stress ratio calculated as per the heat flux intensity (q ₀) can be reduced in considerable amounts. By increasing λ(∼75%) and ω(∼63%) the maximum effective thermal stress ratio calculated can be significantly reduced.     

Thermal stresses near a penny-shaped crack in an elastic sphere embedded in an infinite elastic space

This paper gives an analysis of the distribution of thermal stresses in a sphere which is bonded to an infinite elastic medium. The thermal and the elastic properties of the sphere and the elastic infinite medium are assumed to be different. The penny-shaped crack lies on the diametral plane of the sphere and the centre of the crack is the centre of the sphere. By making a suitable representation of the temperature function, the heat conduction problem is reduced to the solution of a Fredholm integral equation of the second kind. Using suitable solution of the thermoelastic displacement differential equation, the problem is then reduced to the solution of a Fredholm integral equation, in which the solution of the earlier integral equation arising from heat conduction problem occurs as a known function. Numerical solutions of these two Fredholm integral equations are obtained. These solutions are used to evaluate numerical values for the stress intensity factors. These values are displayed graphically.     

The influence of thermomechanical loading history on the stress level in WWER NPP reactor pressure vessels under thermal shock

We present the results of calculations of the kinetics of stress-strain state and stress intensity factors for surface and under-the-cladding circumferential cracks in modeling the emergency core cooldown conditions for the WWER-1000 reactor. The calculation procedure is based on a mixed finite-element method statement which provides stability of numerical solution and a high accuracy of results for both the displacements as well as stresses and strains. The authors analyze the influence of the density of the finite-element discretization of the crack-tip area for the surface and under-the-cladding circumferential cracks on the accuracy and convergence of computation of fracture-mechanics parameters in the modeling of thermal shock conditions. The results of calculation of kinetics of stress intensity factors allowing for the thermomechanical loading history and residual process-induced stress fields are reported. It is demonstrated that if the elastoplastic deformation history and residual process-induced stress fields are disregarded in the calculations of stress intensity factors for under-the-cladding cracks the reactor pressure vessel strength and lifetime may turn out to be overestimated.     

Stress intensity factors for axial cracks in hollow cylinders subjected to thermal shock

Stress intensity factors are determined for one and two, internal and external axial cracks in hollow cylinders subjected to stress gradients arising from a thermal shock. A closedform weight function formula was used in the calculation. Results covering a wide range of cylinder geometries are presented in graphical form.     

Mechanical and thermal stresses at shot particles during fatigue of Kaowool aluminum composites at 20°C

During fatigue of Kaowool fiber reinforced aluminum composites at 20°C, cracks are initiated at hollow Kaowool particles. The stress concentrations associated with these particles arise from two sources: (i) residual stresses due to differential thermal contraction of the Kaowool and aluminum and (ii) the applied cyclic fatigue stress. These stresses are calculated from a finite element model which incorporates plasticity of the aluminum matrix. In general, the mechanical stresses are considerably larger than the thermal stresses. The total stress, in both the aluminum matrix and the Kaowool particle, increases with decreasing particle wall thickness and the proximity of the particle to the surface. In general, the stress concentrations in the aluminum matrix are more critical than those in the Kaowool particles, and are predicted to exceed locally the yield strength of 339 aluminum for all values of wall thickness. The particles observed experimentally at the fatigue fracture origins are thin walled and close to the surface, in quantitative agreement with the predictions of the finite element model.     

Exact solution of a plastic hollow sphere with a Mises–Schleicher matrix

In this paper we establish the exact solution for a hollow sphere with a rigid-plastic pressure-sensitive matrix and subjected to hydrostatic tension or compression. The matrix is assumed to obey to a parabolic Mises–Schleicher criterion. The closed-form expressions of the velocity field and of the stress field are provided. These exact solutions, expressed by means of the Lambert W function, allow to assess and discuss existing results.     

Thermoelastic state of a two-component hollow cylinder with edge radial cracks

Stationary two-dimensional axisymmetric problems of thermal conductivity and thermoelasticity for a hollow two-component cylinder with cracks are studied by the method of singular integral equations. The cross section of the cylinder has the form of a circular concentric ring with a layer of another material that also has the form of a concentric ring and contains edge radial cracks. The surfaces of the cylinder are free of stresses. Thermal processes on these surfaces are characterized by temperature conditions of the third kind. Conditions of ideal thermal and mechanical contact are satisfied on the interface of the two media. A numerical solution is obtained for the case where the inner and outer cylindrical surfaces are kept at different constant temperatures. Stress intensity factors near the tip of one or two edge cracks were found for various values of thermal and mechanical characteristics of the cylinder.Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 30, No. 4, pp. 76–80, July – August, 1994.     

Transient thermoelastic analysis for a multilayered hollow cylinder with piecewise power law nonhomogeneity due to asymmetric surface heating

This paper is concerned with the theoretical treatment of transient thermoelastic problems involving a multilayered hollow cylinder with piecewise power law nonhomogeneity due to the asymmetrical heating of its surfaces. The thermal and thermoelastic constants of each layer are expressed as power functions of the radial coordinate, and their values continue on the interfaces. The exact solution for the two-dimensional temperature change in a transient state is obtained using the Laplace transformation and separation-of-variables method. The exact solution for the thermoelastic response of a multilayered hollow cylinder under the state of generalized plane strain, where the strain is not bound, is obtained herein. Some numerical results for the temperature change and the stress distributions are presented in figures. The influence of the functional grading on the thermal stresses is investigated. Furthermore, the influence on the axial stress of the restraint condition in the axial direction is investigated.