Analytical solution for transient temperature and thermal stresses within convective multilayer disks with time-dependent internal heat generation,Part II: Applications

This work investigates the analytical solution for transient temperature and thermal stresses within three circular geometries. First, the transient temperature and thermal stresses within a composite disk are addressed. Then, two examples regarding transient temperature and thermal stresses throughout circular heaters are analyzed. Pulsed and sinusoidal internal heat generations are incorporated into the second and third examples, respectively. For the composite hollow-disk example, merely the separation of variables method (SVM) is used to overcome the energy partial differential equation. For the other two examples, the combination of the SVM and Duhamel's theorem are adopted to solve the partial differential equations. Accordingly, assuming plane stress formulation, the transient thermal stresses within structures are obtained.     

Two-Dimensional Green's Function for Thermal Stresses in a Semi-Layer Under a Point Heat Source

We present specific new expressions for thermal stresses as Green's functions for a plane boundary value problem of steady-state thermoelasticity for a semi-layer. We also obtain new integration formulas of Green's type, which determine the thermal stresses in the form of integrals of the products of the given distributed internal heat source, boundary temperature, and heat flux and derived kernels. Elementary functions results obtained are formulated in a theorem, which is proved using the harmonic integral representations method to derive thermal stresses Green's functions, which are written in terms of Green's functions for Poisson's equation. A new solution to particular two-dimensional boundary value problem for a semi-layer under a boundary constant temperature gradient is obtained in explicit form. Graphical presentations for thermal stresses Green's functions created by a unit heat source (line load in out-of-plane direction) and by a temperature gradient are also included.     

Monitoring of transient 3D temperature distribution and thermal stress in pressure elements based on the wall temperature measurement

Abstract

Two methods for monitoring the thermal stresses in pressure components of thermal power plants are presented. In the first method, the transient temperature distribution in the pressure component is determined by measuring the transient wall temperature at several points located on the outer insulated surface of the component. The transient temperature distribution in the pressure component, including the temperature of the inner surface is determined from the solution of the inverse heat conduction problem (IHCP). In the first method, there is no need to know the temperature of the fluid and the heat transfer coefficient. In the second method, thermal stresses in a pressure component with a complicated shape are computed using the finite element method (FEM) based on experimentally estimated fluid temperature and known heat transfer coefficient. A new thermometer with good dynamic properties has been developed and applied in practice, providing a much more accurate measurement of the temperature of the flowing fluid in comparison with standard thermometers. The heat transfer coefficient on the inner surface of a pressure element can be determined from the empirical relationships available in the literature. A numerical-experimental method of determination of the transient heat transfer coefficient based on the solution of the 3D-inverse heat conduction problem has also been proposed. The heat transfer coefficient on the internal surface of a pressure element is determined based on an experimentally determined local transient temperature distribution on the external surface of the element or the basis of wall temperature measurement at six points located near the internal surface if fluid temperature changes are fast. Examples of determining thermal and pressure stresses in the thick-walled horizontal superheater header and the horizontal header of the steam cooler in a power boiler with the use of real measurement data are presented.     

DYNAMIC SHAPE CONTROL OF SOLIDS AND STRUCTURES BY THERMAL EXPANSION STRAINS

The present article is concerned with the dynamic shape control of solids and structures by means of thermal expansion strains or, equivalently, by means of thermal expansion stresses. We study transient disturbances produced by imposed forces, and we wish to identify a transient temperature distribution that, when superimposed, leads to zero total displacements of the body. The derivation is based on the theorem of work expended and on Graffi's theorem. Using the anisotropic constitutive equations of linear thermoelasticity, we arrive at a dynamic extension of the principle of virtual forces and at a dynamic extension of Maysel's formula of thermoelasticity. Comparing these two convolution statements, it is found that, in order to make the total displacement zero everywhere in the body, the thermal expansion stress in the thermal problem should be the quasi-static stress minus the thermal expansion stress in the force problem. Any quasi-static stress due to a thermal loading may be added to the thermal expansion stress without changing the validity of this theorem. The practical application of this result is facilitated when the production of heat due to deformation may be neglected and when the applied forces are separable in space and time. The validity of the analytical solution is checked, by means of finite element computations.     

Solution in Elementary Functions to a BVP of Thermoelasticity: Green's Functions and Green's-Type Integral Formula for Thermal Stresses within a Half-Strip

This article presents new elementary Green's functions for displacements and stresses created by a unit heat source applied in an arbitrary interior point of a half-strip. We also obtain the corresponding new integration formulas of Green's and Poisson's types which directly determine the thermal stresses in the form of integrals of the products of internal distributed heat source, temperature, or heat flux prescribed on boundary and derived thermoelastic influence functions (kernels). All these results are presented in terms of elementary functions in the form of a theorem. Based on this theorem and on derived early by author general Green's type integral formula, we obtain a new solution to one particular boundary value problem of thermoelasticity for half-strip. The graphical presentation of thermal stresses created by a unit point heat source and of thermal stresses for one particular boundary value problem of thermoelasticity for half-strip is also included. The proposed method of constructing thermoelastic Green's functions and integration formulas are applicable not only for a half-strip but also for many other two- and three-dimensional canonical domains of Cartesian system of coordinates.     

Transient Thermoelastic Analysis of a Solid Cylinder Containing a Circumferential Crack Using the C–V Heat Conduction Model

This article presents the transient thermoelastic analysis in a long solid cylinder with a circumferential crack using the C–V heat conduction theory. The outer surface of the cylinder is subjected to a sudden temperature change. The Laplace transform technique is adopted to solve the one-dimensional hyperbolic heat conduction equation, and the axial thermal stress is obtained for the un-cracked cylinder in the Laplace domain. Then this axial thermal stress with a minus sign is applied to the crack surface to form a mixed boundary value problem in the cylindrical coordinate system. A singular integral equation is derived by applying the Fourier and Hankel transforms to solve the mode I crack problem. The transient thermal stress intensity factors are obtained by solving the singular integral equation numerically. The influences of thermal relaxation time, crack geometry, and Biot's number upon transient temperature distributions, axial stress fields, and stress intensity factors are analyzed.     

THERMAL STRESSES IN ANNULAR FINS WITH TEMPERATURE-DEPENDENT CONDUCTIVITY UNDER PERIODIC BOUNDARY CONDITION

A stress-field in an annular fin of temperature-dependent conductivity under a periodic heat transfer boundary condition is analyzed by the Adomian's decomposition method. The distribution of the transient thermal stress is obtained by direct integration of the temperature distribution. The heat transfer process is governed by the parameters of the convectional fin parameter N, the thermal conductivity parameter k , the frequency parameter B, and the amplitude parameter s. For k < 0 , the mean temperature is decreased, which in turn increases the thermal stress and enhances the oscillation of the timewise thermal stresses. The opposite effect occurs for k > 0 . The maximum radial stress appears at R = 1.3 , and the maximum tangential stress occurs at the inner base of the fins. Detailed results showing the effects of various parameters on temperature and thermal stresses are presented and discussed.     

TRANSIENT THERMOELASTIC PROBLEM FOR AN INFINITE PLATE CONTAINING A PENNY-SHAPED CRACK AT AN ARBITRARY POSITION

This article deals with the transient thermoelastic problem for an infinite plate containing a penny-shaped crack that is parallel to the surfaces of the plate but at an arbitrary position of the plate. The transient thermal stresses are set up by the heat generation on the surfaces and the sudden heat exchange on the surfaces. By using the finite difference method for the time variable, the analytical solution for spatial variables can be obtained. The numerical results for the temperature and stress intensity factor are obtained, and results are shown in graphs.     

ANALYSIS OF TRANSIENT THERMAL BENDING MOMENTS AND STRESSES OF THE WORKPIECE DURING SURFACE GRINDING

Geometrical inaccuracy is often induced by heat generated during grinding. Furthermore, the transient thermal process is the main cause for the residual stresses on theground surface. The objective of this article is to investigate the three-dimensional transient temperature distribution of the workpiece using the finite difference method,and based on the acquired temperature and beam theory, the thermal moment and thermoelastic stress as calculated using Simpson's multiple numerical integral method. The energypartition is the key factor in accurately predicting the temperature distribution, on which the solution of the thermal moment and stress rely. As the heat conductivity of the workpiece decreases, the stress and moment increase near the wheel-workpiece contact zone and the peaks move closer to the contact position. A smaller thickness results in higher thermal stress and lower thermal moment. Enhancing cooling in grinding effectively reduces temperature and the induced stress.     

Thermoelastic study of a functionally graded annular fin with variable thermal parameters using semiexact solution

Abstract

In this paper, the thermoelastic behavior of a functionally graded material (FGM) annular fin is investigated. The material properties of the annular fin are assumed to vary radially. The heat transfer coefficient and internal heat generation are considered to be functions of temperature. A closed form solution of nonlinear heat transfer equation for the FGM fin is obtained using the homotopy perturbation method (HPM) which leads to nonuniform temperature distributions within the fin. The temperature field is then coupled with the classical theory of elasticity and the associated thermal stresses are derived analytically. For the correctness of the present closed form solution for the stress field, the results are compared with the ANSYS-based finite element method (FEM) solution. The present HPM-based closed form solution of the stress field exhibits a good agreement with the FEM results. The effect of various thermal parameters such as the thermogeometric parameter, conduction-radiation parameter, internal heat generation parameter, coefficient of variation of thermal conductivity, and the coefficient of thermal expansion on the thermal stresses are discussed. The results are presented in both nondimensional and dimensional form. The dimensional stress analysis discloses the suitability of FGM as the fin material in practical applications.     

THREE-DIMENSIONAL TRANSIENT THERMAL STRESS ANALYSIS OF A NONHOMOGENEOUS HOLLOW CIRCULAR CYLINDER DUE TO A MOVING HEAT SOURCE IN THE AXIAL DIRECTION

In this paper, a theoretical analysis of a three-dimensional transient thermal stress problem is developed for a nonhomogeneous hollow circular cylinder due to a moving heat source in the axial direction from the inner and /or outer surfaces. Assuming that the hollow circular cylinder has nonhomogeneous thermal and mechanical material properties in the radial direction, the heat conduction problem and the associated thermoelastic behaviors for such nonhomogeneous medium are developed by introducing the theory of laminated composites as one of theoretical approximation. The transient heat conduction problem is treated with the help of the methods of Fourier cosine transformation and Laplace transformation, and the associated thermoelastic field is analyzed making use of the thermoelastic displacement potential, Michell's function, and the Boussinesq's function. Some numerical results for the temperature change and the stress distributions are shown in figures, and the effect of relaxing the thermal stress in the nonhomogeneous hollow circular cylinder and the influence of the velocity of a moving heat source are briefly discussed     

NEW FORMULAE FOR DYNAMICAL THERMAL STRESSES

A generalization of the function of influence of a unit heat source to the displacements is suggested for the boundary value problems in the dynamical uncoupled thermoelasticity. This generalization is a convolution over time and bulk of two influence functions. One of them is a Green's function for the heat conduction problem. The other is a function of influence of unit concentrated forces onto bulk dilatation. Broad possibilities are shown in constructing these influence functions. In particular, the theorem on dilatation constructing is proved. To calculate the convolutions successfully the following properties of the introduced function are found to be useful. (1) In coordinates of the point of observation, the function satisfies the equations used to find the Green's functions in the problem of heat conduction, with the unit heat source being replaced by the influence function of concentrated force onto dilatation; and (2) in coordinates of the point of heat source application, it satisfies the boundary value problem used to find Green's matrix, with the unit concentrated forces being replaced by derivatives of Green's function in the problem of heat conduction. Based on the introduced influence function, some new integral formulae for displacements and stresses are obtained, which are a generalization of Mysel's formula in the theory of dynamical thermal stresses. The proposed formulae have certain advantages allowing us to unite the two-staged process of finding the solutions for boundary value problems in thermoelasticity in a single stage. It is established that, based on the obtained results it becomes possible to compile a whole handbook on the influence functions and integral solutions for boundary value problems in dynamical thermoelasticity. As examples, the solutions for two boundary value problems in the theory of dynamical thermal stresses for the half-space and quarter-space are presented.     

TRANSIENT THERMAL STRESS ANALYSIS AND BENDING BEHAVIOR OF AN ANGLE-PLY LAMINATED SLAB

Abstract

This paper is concerned with a theoretical treatment of thermal stress and bending behavior in a transient state of a multilayered, nonisotropic, laminated slab. As an analytical model, we consider an infinitely long, laminated slab, which consists of obliquely directed layers with orthotropic material properties; the model corresponds to the so-called angle-ply laminate. We solve the thermoelastic problem for the slab under the condition of uniformly distributed heat supply from its one surface. Introducing a method of Laplace transforms to the temperature field, we obtain the temperature solution using the residue theorem, and we evaluate the thermal stresses in a transient state by using the elementary plate theory. As an example, we carry out numerical calculations for the five-layered angle-ply laminate, evaluate the thermal stress distributions and the bending behavior, and examine the influence of the ply angle on the thermal stress distribution.     

Transient stress analysis of sandwich plate and shell panels with functionally graded material core under thermal shock

A finite element formulation for stress analysis of functionally graded material (FGM) sandwich plates and shell panels under thermal shock is presented in this work. A higher-order layerwise theory in conjunction with Sanders’ approximation for shells is used to develop the finite element formulation for transient stress analysis of FGM sandwich panels. The top and the bottom surfaces of FGM sandwich panels are made of pure ceramic and metal, respectively, and core of the sandwich is assumed to be made of FGM. The temperature profile in the thickness direction of the panels is considered to be varying as per the Fourier’s law of heat conduction equation for unsteady state. The heat conduction equations are solved using the central difference method in conjunction with the Crank–Nicolson approach. Transient thermal displacements of the sandwich panels are obtained using Newmark average acceleration method and the transient thermal stresses are obtained using stress–strain relations, subsequently. Results obtained from the present layerwise finite element formulations are first validated with available solutions in literature. Parametric studies are taken up to study the effects of volume fraction index, temperature dependency of material properties, core thickness, panel configuration, geometric and thermal boundary conditions on transient thermal stresses of FGM sandwich plates and shells.     

Thermo-Elastic Analysis of a Cracked Half-Plane Under a Thermal Shock Impact Using the Hyperbolic Heat Conduction Theory

The transient thermal stresses around a crack in a thermo-elastic half-plane are obtained under a thermal shock using the hyperbolic heat conduction theory. Fourier, Laplace transforms and singular integral equations are applied to solve the temperature and thermal stress fields consecutively. The integral equations are solved numerically and the asymptotic fields around the crack tip are obtained. Numerical results show that the hyperbolic heat conduction have significant influence on the dynamic temperature and stress field. It is suggested that to design materials and structures against fracture under thermal loading, the hyperbolic model is more appropriate than the Fourier heat conduction model.     

Transverse temperature distribution and heat generation rate in composite superconductors subjected to constant thermal disturbance

Analytical solution of one-dimensional, transient heat conduction with a distributed heat source is obtained to predict the transverse temperature distribution and heat generation rate per unit volume of the composite superconductor. The solution indicates that temperature distribution and heat generation rate depend on three dimensionless parameters: the dimensionless external disturbance w₀, the dimensionless interface temperature θ¹, and the dimensionless parameter φ that is dependent on the thickness and the thermal conductivity of the superconductor. Results of transient and steady-state solutions are presented. It is shown that the heat generation rate per unit volume of the composite, Q/Qc, is directly proportional to the current in the stabilizer.     

A simplified approach for the thermoelastic large deflection in the thin clamped annular sector plate

The present article is concerned with analysis of large deflection of a heated thin annular sector plate with clamped edges under transient temperature distribution using Berger’s approximate methods. The prescribed surface temperature is at the top face of the plate whereas the bottom face is kept at zero temperature. In this study, the Laplace transform as well as the classical method have been used for the solution of heat conduction equation. The thermal moment is derived on the basis of temperature distribution, and its stresses are obtained using resultant bending moment and resultant forces per unit length. The calculations are obtained for the aluminium plate in the form of an infinite series involving Bessel functions, and the numerical results for temperature, deflection, resultant bending moments, and thermal stresses have been illustrated by graphs.     

Thermal and mechanical stresses in a functionally graded thick sphere

In this paper, a general solution for the one-dimensional steady-state thermal and mechanical stresses in a hollow thick sphere made of functionally graded material is presented. The temperature distribution is assumed to be a function of radius, with general thermal and mechanical boundary conditions on the inside and outside surfaces of the sphere. The material properties, except Poisson's ratio, are assumed to vary along the radius r according to a power law function. The analytical solution of the heat conduction equation and the Navier equation lead to the temperature profile, radial displacement, radial stress, and hoop stress as a function of radial direction.     

Quasi-Static Thermal Stresses Due to Heat Generation in a Thin Hollow Circular Disk

The present paper deals with the determination of displacement and thermal stresses in a thin hollow circular disk defined by a ≤ r ≤ b due to internal heat generation within it. Time dependent heat flux Q(t) is applied at the outer circular boundary (r = b), whereas inner circular boundary (r = a) is at zero heat flux. Also, initially the circular disk is at arbitrary temperature F(r). The governing heat conduction equation has been solved by the method of integral transform technique. The radial stress function σ_rr is zero at inner and outer circular boundaries (r = a and r = b). The results are obtained in a series form in terms of Bessel's functions. The results for displacement and stresses have been computed numerically and illustrated graphically.     

ANALYSIS ON TRANSIENT THERMAL STRESSES IN AN ANNULAR FIN

Transient thermal stresses are an important consideration in production processes involving large temperature changes. Recently, thermal stresses have also become significant in design problems related to microelectronic devices through their effects on material properties and system parameters. To calculate the thermal stresses, three kinds of methods are available. The first is the analytical method, in which the elastic theory is used to find the exact solution. The second approach consists of some kind of approximate technique, such as a perturbation procedure. The third method is the use of a numerical process, such as a finite-difference or a finite-element method.

This article investigates the transient thermal stresses in an annular fin with its base subjected to a heat flux of a decayed exponential function of time. In order to obtain the solution of the governing equation, which is a partial differential equation, the following procedures of analysis are used.

1. Normalize the governing partial differential equation subject to appropriate initial and boundary conditions.

2. Take the Laplace transform of the resulting equation with respect to time.

3. Utilize the exponential-like solutions introduced by Keller and Keller to solve the transformed system.

4. Achieve the inverse Laplace transform by means of complex contour integration and the residue theorem.

5. Substitute the temperature distribution function into the governing equation of thermal stresses. Then use Simpson's rule to obtain the thermal stress distribution as a function of time and position of the fin.