Coupled transient thermoelastic contact problems for axial cracks in hollow cylinders

A coupled transient thermoelastic behaviour of an axial-cracked hollow circular cylinder subjected to a sudden heating is investigated in this study. It is shown that surface heating may induce the compressive thermal stress near the inner surface of the cylinder which in turn may force the cracked surfaces to close together. Assuming that the existence of the crack does not alter the temperature distribution, we can divide this problem into two parts and solve it by the principle of superposition. First, the temperature and transient thermal stress distributions along the axisymmetric surface of the imaginary cylinder without crack are obtained by finite element implicit time integration method Secondly, the opposite sense of the stress distributions along the cracked surfaces, which is obtained previously, is treated as the traction boundary conditions; the contact length and contact pressure of the real cracked cylinder are obtained by modified elimination finite element scheme. Finally, we also obtained the normalized stress intensity factor for the crack tip of the cylinder. It is concluded that the effect due to thermoelastic coupling term on stress intensity factor becomes more important for higher coupling coefficient, and this coupling term also results in a small time lag in temperature, thermal stress and stress intensity factor.     

Transient thermal contact problem for axial crack in a hollow circular cylinder

The transient behavior of an axial-cracked hollow circular cylinder subjected to a sudden heating is investigated. It is shown that surface heating may induce compressive thermal stress near the inner surface of the cylinder which in turn may force the cracked surfaces to close together. Assuming that the existence of the crack does not alter the temperature distribution, this problem can be divided into two parts and solved by the principle of superposition. First, the temperature and transient thermal stress distributions along the axisymmetric surface of the imaginary cylinder without a crack are obtained by finite element/implicit time integration method. The calculated temperature and thermal stress distributions are in good agreement with the values predicted by the analytical method. Secondly, the opposite senses of the stress distributions along the cracked surfaces, which are obtained previously, are treated as the traction boundary conditions, and the contact length and contact pressure of the real cracked cylinder are obtained by a modified elimination finite element scheme. In this scheme, the concepts of contact-node-pairs' penetration, contact-double-forces and compliance matrix are introduced. The calculated results indicate that the contact length ratio becomes smaller when the crack length ratio increases, and becomes larger as the radius ratio increases. Finally, the normalized stress intensity factor for the crack tip of the cylinder is obtained. It is shown that the larger the crack length ratio the higher the stress intensity factor.     

Effect of axial heat flux on the temperature field of a hollow cylinder

Solutions are given for two-dimensional problems of heat conduction in a hollow cylinder with quasi-steady heating (constant heating rate) for a boundary condition of the second kind at the inner surface of the cylinder, and boundary conditions of the third or first kind at the outer surface. The effect of axial heat flux on the temperature field of the hollow cylinder is estimated.     

功能梯度压电压磁圆柱轴对称的静力学响应

针对无限长功能梯度压电压磁空心圆柱,研究轴对称空心圆柱电磁弹耦合静力学问题。在柱坐标系下,假定材料参数沿径向为幂函数分布,推导出在外激励作用下空心圆柱体中位移、应力、电势以及磁势等物理场的解析解。数值讨论中,针对BaTiO₃-CoFeO₄复合材料空心柱体,在不同边界条件下得到了不同梯度参数及不同厚度下空心圆柱应力、电势和磁势的分布规律。结果表明,对于功能梯度压电压磁空心圆柱,材料的失效主要是由于环向应力引起的,且圆柱厚度对环向应力有着显著影响。此外,梯度参数强烈影响功能梯度压电压磁传感器/制动器的电/磁输出和力学性能。结果对压电压磁圆柱型传感器/制动器的设计和分析具有一定的指导作用。     

Stable crack growth in a surface compression-strengthened hollow cylinder

In this paper the static fatigue problem for a circumferentially cracked hollow cylinder is examined. For this particular configuration, stable crack growth, in the absense of any external forces, is determined for cylinders with axial components of residual stress which are compressive on the inner and outer radial surfaces and tensile in the cylinder wall. An initial surface crack which is deep enough to penetrate the compression strengthened surface region and enters the tensile zone may propagate in a stable manner until either sudden spontaneous failure occurs or the crack arrests. Since a portion of the crack near the cylinder surface will be closed because of the compressive residual stress field, an additional unknown in the problem is the extent of the crack surface contact. This crack surface contact length is determined by iteration on the integral equation which arises in the mathematical derivation for an embedded circumferential crack in a hollow cylinder. As an illustration of stable crack growth for this geometry with a realistic residual stress distribution, numerical results are presented for a hollow, soda-lime glass cylinder, based on crack growth rates in soda-lime glass exposed to water at 25‡ C. Using the fracture toughness and slow crack growth characteristics for soda-lime glass, the conditions for no crack propagation, crack propagation leading to crack arrest, and catastrophic failure are established.     

Magnetohydrodynamic mixed convection flow in vertical concentric annuli with time periodic boundary condition: Steady periodic regime

This work reports an analytical solution for fully developed mixed convection flow of viscous, incompressible, electrically conducting fluid in vertical concentric annuli under the influence of a transverse magnetic field, where the outer surface of inner cylinder is heated sinusoidally and the inner surface of outer cylinder is kept at a constant temperature. The analysis is carried out for fully developed parallel flow and steady-periodic regime. The governing dimensionless momentum and energy equations are separated into steady and periodic parts and solved analytically. Closed form solutions are expressed in terms of modified Bessel function of first and second kind. The influence of each governing parameters such as magnetic field parameter, Prandtl number and the dimensionless frequency of heating on flow formation and thermal behaviour are discussed with the aid of graphs. During the course of investigation, it is found that the oscillation amplitude of the friction factor is maximized at a resonance frequency near the surface of the concentric annuli where there is periodic heating. Furthermore, increasing transverse magnetic field decreases the oscillation amplitude of the friction factor.     

Material tailoring for functionally graded hollow cylinders and spheres

We present a technique to tailor materials for functionally graded (FG) linear elastic hollow cylinders and spheres to attain through-the-thickness either a constant hoop (or circumferential) stress or a constant in-plane shear stress. The volume fractions of two phases of a FG material (FGM) are assumed to vary only with the radius and the effective material properties are estimated by using either the rule of mixtures or the Mori-Tanaka scheme; the analysis is applicable to other homogenization methods. For a FG cylinder we find the required radial variation of the volume fractions of constituents to make a linear combination of the radial and the hoop stresses uniform throughout the thickness. The through-the-thickness uniformity of the hoop stress automatically eliminates the stress concentration near the inner surface of a very thick cylinder. The through-the-thickness variations of Young’s moduli obtained with and without considering the variation of Poisson’s ratio are very close to each other for a moderately thick hollow cylinder but are quite different in a very thick hollow cylinder. For an FG sphere the required radial variation of the volume fractions of the two phases to get a constant circumferential stress is similar to that in an FG cylinder. The material tailoring results presented here should help structural engineers and material scientists optimally design hollow cylinders and spheres comprised of inhomogeneous materials.     

Non-axial-symmetric thermal stresses in circular discs or cylinders

A solution is presented for the computation of the transient thermoelastic stresses in a hollow cylinder with temperature boundary conditions given as a circumferential variation of surface heat transfer coefficient. The temperature distribution is solved explicitly. The problem is set up using the Airy stress function which leads to the biharmonic equation. This approach requires the satisfaction of three Michell integrals at the inner boundary in order to ensure single-valued displacements and rotation. An iterative method is described in which these integrals are all simultaneously satisfied and thus provide the necessary non-zero boundary conditions for the solution of the biharmonic equation which is rapidly solved by Gaussian elimination. Results are presented for the general case where the temperature is a function of r and θ. The computer program is checked by assuming a constant value of the surface heat transfer coefficients. In this case a closed form solution is obtained.     

Estimation of temperature distributions and thermal stresses in a functionally graded hollow cylinder simultaneously subjected to inner-and-outer boundary heat fluxes

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to simultaneously estimate the unknown time-dependent inner and outer boundary heat fluxes in a functionally graded hollow circular cylinder from the knowledge of temperature measurements taken within the cylinder. Subsequently, the distributions of temperature and thermal stresses in the cylinder can be determined as well. It is assumed that no prior information is available on the functional forms of the unknown heat fluxes; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors and locations in these measurements upon the precision of the estimated results is also considered. Results show that an excellent estimation on the time-dependent heat fluxes, temperature distributions, and thermal stresses can be obtained for the test case considered in this study.     

On Thermally Induced Multiple Cracking of a Surface: An Experimental Study

An experimental study of the thermally induced surface multiple cracking, on the example of steam superheater collector was carried out. The structure’s perforated area has 3 rows of holes, to which the connection sleeves are welded. On the first stage, the cracks initiate on the inner surface of the perforated area of the collector. These cracks are due to the thermal fatigue, caused by temperature changes at start-stops and by temperature fluctuations during operation. The histograms and cumulative distribution functions of crack lengths and the inclination angles with respect to the axes of fastening pipes holes were obtained. The length of most cracks was less than 2 mm. Also, an important localized damage on the collector inner surface in the form of a partly circumferential crack with a length of 149 mm and with the largest depth of 37.8 mm was observed.     

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.     

低温风洞电动推杆热防护结构设计及传热分析

为解决低温风洞中电动推杆的热防护问题,设计了使其能够在低温风洞高温工况(323 K)和低温工况(110 K)下正常工作的热防护结构。采用数值模拟方法校核了热防护结构强度及刚度,分析了电机发热功率,冷却气体流量和加热片加热功率等因素对推杆元件温度的影响。结果表明:低温风洞内压力达到极值0.35 MPa时,由厚度5 mm,材料S30408不锈钢制成的圆筒形热防护结构最大变形量为0.397 mm,最大应力为160.62 MPa;冷却气体流量大于等于0.005 kg/s时,高温和低温工况下电机最高温度均不大于418 K的允许工作温度;当加热功率达到500 W时,缸杆端部各考察截面温度均高于263 K;在高温工况和加热功率为500 W的低温工况下,冷却气体流量为0.005 5kg/s时,缸体、缸杆均能维持在263—313 K的工作温度,且高温工况最大温升与温差分别为3.62、2.81 K,低温工况最大温降与温差分别为4.94、6.82 K,满足温度稳定性与均匀性要求。     

Asymmetric temperature field of an unbounded cylinder with a moving heating line

The temperature distribution in an unbounded hollow cylinder, a portion of whose inner surface is asymmetrically heated, is obtained under the assumption that the heating line moves at a certain speed toward the cylinder axis.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 15, No. 6, pp. 1080–1085, December, 1968.     

Analysis of shear bands in a dynamically loaded viscoplastic cylinder containing two rigid inclusions

Summary We study plane strain thermomechanical deformations of a hollow circular cylinder containing two rigid non-heat-conducting ellipsoidal inclusions placed on a radial line symmetrically with respect to the center. These inclusions can be viewed as precipitates or second phase particles in an alloy. The material of the cylinder is presumed to exhibit thermal softening, but strain and strain-rate hardening. The impact load applied on the inner surface of the cylinder is modeled by prescribing a radial velocity and zero tangential tractions at material particles situated on the inner surface. Rigid body motion of the inclusion is considered and no slip condition between the inclusion and the cylinder material is imposed.It is found that shear bands initiate from points adjacent to inclusion tips near the inner surface of the cylinder and propagate toward this surface. At inclusion tips near the outer surface of the cylinder, the maximum principal logarithmic strain and the temperature are high and the effective stress is low, but severe deformations there do not propagate outward.     

Closed‐form thermal stress intensity factors for an internal circumferential crack in a thick‐walled cylinder

In this paper the method of weight functions is employed to calculate the stress intensity factors for an internal circumferential crack in a thick‐walled cylinder. The pressurized cylinder is also subjected to convection cooling on the inner surface. Finite element method is used to determine an accurate weight function for the crack and a closed‐form thermal stress intensity factor with the aid of the weight function method is extracted. The influence of crack parameter and the heat transfer coefficient on the stress intensity factors are determined. Comparison of the results in the special cases with those cited in the literature and the finite element data shows that the results are in very good agreement.     

Optimum heating of pressure components of large steam boilers

The method of determining the optimum time changes of fluid temperature during heating process of boiler structural pressure elements is presented. The stresses are kept at the maximum allowed levels. The optimization procedure is based on the discrete form of the Duhamel’s integral. The optimum temperature changes of the fluid were determined for the cylindrical and spherical components, and Y-branch. The heating of the elements with and without the internal pressure was analyzed. The examples of the optimum heating processes of the boiler pressure elements are also presented as the illustration of the application of the developed method.     

Temperature and thermal stress evaluation in complex-shaped boiler components

A fast way of temperature and thermal stress evaluation in complex-shaped pressurized elements is presented. Transient temperature and thermal stresses in the point of concentration were calculated using the discrete form of the Duhamel integral. The time changes of temperature and maximal stresses $u(\vec r,t)$ were calculated for unitary step change of the fluid temperature using the Finite Element Method. The Duhamel integral was then approximated numerically and the temperature and thermal stresses were then evaluated for arbitrary time changes of fluid temperature. Practical application of the presented method was illustrated by determining the circumferential stresses on the edge of the boiler drum-downcomer joint.     

On the optimum design of rotating two-layered composite tubes subject to internal heating or pressure

Subject of this analytical investigation is a rotating two-layered hollow cylinder under generalized plane strain subject to an elevated temperature at the inner surface or to internal pressure. It is presupposed that the inner cylindrical layer consists of the heavier material, whereas the outer layer is made of a material with lower density, like for example in a steel/aluminum tube. Criterion for the maximum permissible stress is the yield criterion by von Mises, and the device is optimized with respect to its weight. It is found that plasticization may start at different radii, and the study provides not only a comprehensive overview of the elastic limits of composite tubes of the above type but also a straightforward procedure for determining the optimum composition.     

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.     

Axisymmetric stress in an electrostrictive hollow cylinder under electric loading

Axisymmetric problems for an electrostrictive hollow cylinder under electric loading are studied based on the potential function method. First, in the Cartesian coordinate system, the general solutions for the displacement are presented. Then, explicit results of displacement and stress are derived. For two special cases of in-plane and off-plane electric loading, the stresses around the inner and outer boundaries of the cylinder are given in an analytic form and the effects of Maxwell stress are discussed. Numerical results are also presented and graphically shown. It is found that for a thick-walled hollow cylinder, the electrostrictive stress in the internal boundary of the cylinder would be significantly large in magnitude when in-plane electrical loading is applied.