Green's function method with consideration of temperature dependent material properties for fatigue monitoring of nuclear power plants

In this paper, a method to consider temperature dependent material properties when using the Green's function method is proposed by using a numerical weight function approach. This is verified by using detailed finite element analyses for a pressurizer spray nozzle with various assumed thermal transient load cases. From the results, it is found that the temperature dependent material properties can significantly affect the maximum peak stresses and the proposed method can resolve this problem with the weight function approach. Finally, it is concluded that the temperature dependency of the material properties affects the maximum stress ranges for a fatigue evaluation. Therefore, it is necessary to consider this effect to monitor fatigue damage when using a Green's function method for the real operating conditions in a nuclear power plant.     

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.     

GREEN'S FUNCTION APPROACH TO THREE-DIMENSIONAL HEAT CONDUCTION EQUATION OF FUNCTIONALLY GRADED MATERIALS

A Green's function approach based on the laminate theory is adopted to solve the three-dimensional heat conduction equation of functionally graded materials (FGMs) with one-directionally dependent properties. An approximate solution for each layer is substituted into the governing equation to yield an eigenvalue problem. The eigenvalues and the corresponding eigenfunctions obtained by solving an eigenvalue problem for each layer constitute the Green's function solution for analyzing the three-dimensional transient temperature. The eigenvalues and the corresponding eigenfunctions are determined from the homogeneous boundary conditions at outer sides and from the continuous conditions of temperature and heat flux at the interfaces. A three-dimensional transient temperature solution with a source is formulated by the Green's function. Numerical calculations are carried out for an FGM plate, and the numerical results are shown in tables and figures.     

Green's function for infinite thin plate with elliptic hole under bending heat source and interaction between elliptic hole and crack under uniform bending heat flux

Green's function is derived for the bending problem of an infinite thin plate with an elliptic hole under a bending heat source. Then the interaction problem between an elliptic hole and a crack in a thin plate under uniform bending heat flux is analyzed. First, the complex variable method is developed for the thermoelastic problem of bending. Then an exact solution in explicit form is derived for the Green's function by using the complex variable method. Distributions of temperature moment, heat flux moments, bending moments along the hole edge are shown in figures. For solving the interaction problem, a solution for an infinite thin plate with an adiabatic elliptic hole under uniform bending heat flux, and two Green's functions of the plate under a bending heat source couple and a bending dislocation are given. The interaction problem then reduces into singular integral equations using the Green's functions and the principle of superposition. After the equations are solved numerically, the moment intensity factors at crack tips are presented in the figures.     

Division of frictional heat: The dependence on sliding parameters

This paper investigates the dependence of the division of friction - induced heat on the contact parameters through the identification of the behavior of a characteristic function under a combination of loads and speeds. The rubbing pair is assumed to have established contact over a circular spot. Thus allowing the use of pre-existing Green's function for the temperature rise due to an instantaneous disk source. It is shown that the dependence of heat partition on the contact parameters is reflected only at moderate speeds. At high sliding speeds, however, that dependence is merely a function of the coupling between the local thermal properties (thermal conductivity and thermal capacity).     

Determination of skin temperature distribution and heat flux during simulated fires using Green's functions over finite-length scales

One of the current practices for measuring heat flux during flash fire testing, forest fires, and other industrial cases focuses on the use of semi-infinite models to predict the heat flux during exposure through surface temperature measurements on simulated skin sensors. For short time frames, these models can be shown to have acceptable accuracy. However, when considering longer time exposures at reduced heat fluxes, such as with firefighters in a forest fire, the accuracy of these models could be brought into question. A one-dimensional, finite length scale, transient heat conduction model was developed using a Green's function approach on a rectangular sensor. The model was developed using transient temperature boundary conditions to avoid the use of complicated radiation and convection conditions at each boundary. For comparison, a semi-infinite model utilizing the same boundary condition on the exposed face was solved using both the Laplace transform method and Green's function method. Experimental data was obtained during exposure to a cone calorimeter. All measurements were taken for a minimum duration of 2 min. This temperature data was used to develop appropriate curves for the boundary conditions and validate the analytical models. It was found that the temperature obtained from the one-dimensional transient heat conduction model based on Green's functions agreed well with the experimental results over longer exposure times, and with reduced error when compared with the semi-infinite model. This suggests that modeling the problem on a finite-length scale will produce more accurate or more conservative temperature and heat flux results over extended periods of exposure in high heat load applications.     

GREEN'S FUNCTION FOR A THERMOMECHANICAL MIXED BOUNDARY VALUE PROBLEM OF AN INFINITE PLANE WITH AN ELLIPTIC HOLE

This article derives the Green's function for a thermomechanical mixed boundary value problem of an infinite plane with an elliptic hole under a pair of heat source and sink. To derive the Green's function in closed form, the Cauchy integral method and a basic Green's function for an external force boundary value problem with a pair of heat source and sink are employed. Illustrative numerical results for temperature, heat flux, and stress along the hole edge and stress intensity factors when the hole collapses into a crack are presented graphically.     

On‐line calculation models of temperature and thermal stresses in a 2‐D axis‐symmetric object during heat treating

This paper presents the analytical calculation models of temperature and thermal stresses in a 2‐D axis‐symmetric object during heat treating. Complicated geometries can be taken into account using Green's function method. The models can deal with some nonlinear factors such as temperature‐dependent properties using perturbation method. One case has been given to compare the results calculated by the models deduced with those calculated by FEM (finite element method). It is shown that the models presented in this work have satisfactory accuracy. Another case shows an application of a water quenching process. Combining the presented mathematical models, real‐time controls of quenching and controlled cooling can be accomplished with greatly improved accuracy. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20359     

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.     

ENLARGEMENT OF A THERMALLY DRIVEN PENNY-SHAPED CRACK: STABILITY CONSIDERATIONS

Quasistatic extension of a penny-shaped crack, which occurs prior to its spontaneous propagation, is studied by means of nonlinear fracture mechanics analogous to the methods used to describe plastic deformation in metals. Wnuk's model of final stretch is employed to describe the displacement and stress fields in the vicinity of the crack border. Three kinds of thermal loadings are considered: (a)prescribed heat flux across the surface of the crack, (b) prescribed temperature difference, and (c) given heat extraction rate resulting from the steady-state flow of a cooling fluid circulating in and out of the crack. The latter case is directly applicable to power-generating geothermal systems. All three problems, if treated by linear elastic fracture mechanics methods, lead to singular thermal stresses around the circumference of the crack. Although the first two problems are of a positive K-gradient nature, i.e., inherently unstable, the third one results in a negative K-gradient. Through the approach used in this work the existence of a preliminary phase of stable crack growth, which precedes the catastrophic (cases a and b) or the spontaneous (case c) crack extension, has been demonstrated. The critical parameters involving the thermal load and the terminal dimension of the crack, at which the transition from stable to unstable propagation takes place, are predicted.     

Fatigue crack initiation and propagation as a consequence of thermal striping

An analysis is given of the initiation and subsequent propagation of surface cracks in a stainless steel section subjected to high frequency thermal shocks (thermal striping) at metal temperatures up to 650°C.Estimates of the maximum sustainable metal temperature range allowable to avoid crack nucleation in the surface layers have been made using fatigue endurance data corrected for the thermally induced equibiaxial strain field. Using these methods, it is apparent that a metal temperature difference of 63°C may be allowed. This represents the application of a safety factor of two, in the conventional manner, to the uniaxial fatigue endurance data in order to account for scatter in fatigue data arising from thermal ageing, surface finish and sodium corrosion effects. A discussion of the assessment is given.The extent of crack propagation under thermal striping has been analysed using a linear elastic fracture mechanics approach and it is shown that, in certain circumstances, crack arrest may occur. Methods are outlined which allow the inclusion of multiple surface cracking (and its effect of reducing the peak stress intensity factor) into the analysis.Consideration is also given to the possibility of arrested thermal fatigue cracks being further propagated by creep crack cracking under conditions of high mean stress.     

Steady-state Green's functions for thermal stresses within rectangular region under point heat source

This article presents new steady-state Green's functions for displacements and thermal stresses for plane problem within a rectangular region. These results were derived on the basis of structural formulas for thermoelastic Green's functions which are expressed in terms of Green's functions for Poisson's equation. Structural formulas are formulated in a special theorem, which is proved using the author's developed harmonic integral representation method. Green's functions for thermal stresses within rectangle are obtained in the form of a sum of elementary functions and ordinary series. In the particular cases for a half-strip and strip, ordinary series vanish and Green's functions are presented by elementary functions. These concrete results for Green's functions and respective integration formulas for thermoelastic rectangle, half-strip and strip are presented in another theorem, which is proved on the basis of derived structural formulas. New analytical expressions for thermal stresses to a particular plane problem for a thermoelastic rectangle under a boundary constant temperature gradient also are derived. Analytical solutions were presented in the form of graphics. The fast convergence of the infinite series is demonstrated on a particular thermoelastic boundary value problem (BVP). The proposed technique of constructing thermal stresses Green's functions for a rectangle could be extended to many 3D BVPs for unbounded, semibounded, and bounded parallelepipeds.     

Transient thermal fracture analysis of a penny-shaped magnetic and dielectric crack in a magnetoelectroelastic cylinder

The problem of penny-shaped magnetic and dielectric crack in a magnetoelectroelastic cylinder is investigated under thermal shock load. The problem is reduced to solve three coupled Fredholm integral equations. The field intensity factors are derived. Numerical results of crack opening displacement intensity factors are presented, and the effects of thermal shock time, crack configuration, and magnetoelectrical crack surface conditions on crack propagation and growth are evaluated. Among others, the larger cylinder's radius, the easier to propagate the crack is. For a fixed crack configuration, magnetoelectrical crack surface conditions have different effects on crack propagation as well.     

The singular behavior of the temperature gradient in the vicinity of a macrocrack tip

The power of singularity of the temperature gradient at a macrocrack tip is analyzed in this work. For a crack in an infinite medium. Williams' method of eigenfunction expansions is extended to heat conduction problems with a crack and comparison with the complex function approach is made. The intensity factor of temperature gradient (IFTG) is introduced to quantify the thermal energy cumulated in the neighborhood of a macrocrack tip. As an entirety, the power of singularity of the temperature gradient is analyzed for a crack in both isotropic and orthotropic media, and an interfacial crack between dissimilar materials. It is shown that the power of singularity of the temperature gradient is not affected by the discontinuous jumps of the thermal properties across the material interface, while that for a crack in an orthotropic medium depends on the ratio of thermal conductivities in the principal directions of material orthotropy.     

Effect of point heat source on growth of subsurface crack induced in brittle material machining

Based on the indentation fracture mechanics, a thermal-assisted grinding model is established to investigate the effect of heat source on subsurface crack propagation. Combining the complex function method with the continuous distribution technique of dislocations, the stress intensity factors (SIFs) near the crack tip under various thermal and mechanical loads are calculated numerically. Results show that whether the median crack propagation is inhibited or promoted by the heat source mainly depends on the relative positions between thermal loading and mechanical loading. The stronger heat source in front of the abrasive grain can inhibit the growth of subsurface crack. When the heat source is located behind the abrasive grain, a minimum value of SIF can be obtained if the grinding parameters are well controlled. Especially, with the increase in heat source intensity, the maximum and minimum SIFs occur when the heat source is located at 1.5 times the distance of half crack length behind and in front of the abrasive grain, respectively.     

NORMALIZED STRESS INTENSITY FACTOR RANGE SOLUTIONS OF AN INNER-SURFACE CIRCUMFERENTIAL CRACK IN THIN- TO THICK-WALLED CYLINDERUNDER THERMAL STRIPING BYSEMI-ANALYTICAL NUMERICAL METHOD

The normalized stress intensity factor (SIF) range of an inner-surface circumferential crack in a thin to thick-walled finite-length cylinder under thermal striping was considered in this paper. The edges of the cylinder were rotation-restrained and the outer surface was adiabatically insulated. The inner surface of the cylinder was heated by a fluid with sinusoidal temperature fluctuation. An analytical temperature solution for the problem and our semianalytical numerical SIF evaluation method for the crack were combined and, as a result, it was shown that the transient SIF solution can be expressed in a generalized form by dimensionless parameters such as mean-radius-to-wall-thickness ratio, Biot number, normalized striping frequency, and Fourier number. Finally, normalized SIF ranges for the first cycle and steady state were given for these dimensionless parameters in tables for mean-radius-to-wall-thickness ratio of 10, 5, and 1.     

A Study of the Radiant Heat Exchange within a Controlled-Atmosphere Furnace

Transient three-dimensional heat transfer between a traversing, structured, and rectangular object and an enclosure is studied. This study investigates the heat transfer process that occurs in brazing an aluminum heat exchanger in a controlled-atmosphere furnace. A model's development is discussed with prescribed enclosure temperature boundary conditions. The program determines the radiant heat exchange between gray diffuse surfaces, and solves the three-dimensional conduction equation for a solid with a radiant heat flux boundary condition using an implicit finite-difference method. The structured object's conduction and radiant thermal properties are described by effective values. It was shown that radiative thermal properties of the traversing object and the enclosure's temperature have a strong impact on the object's temperature history. The effective thermal emissivity was found to influence the object's rate of temperature change. The enclosure's temperatures influenced the object's equilibrium temperature. Also, it was shown that the object's position and rotation can alter its temperature distribution, but not as strong as the effect of boundary conditions and thermal properties. In addition to numerical methods, experiments were performed to further understand the process.     

Practical Algorithms for Online Thermal Stress Calculations and Heating Process Control

This article presents two algorithms for online mode thermal stress determination in machinery components based on Green's Function Technique approach and on the artificial neural network, respectively. A modification of the GFT algorithm is proposed involving the use of the component metal temperature rather than the temperature of the flowing fluid. The algorithms developed were used to calculate stresses in rotors and casings of new generation turbines. A possibility was also shown of using the algorithms to solve the inverse problem, i.e., to control the rate of temperature changes while keeping stresses within an allowable range.     

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.     

The shape of a surface crack in a plate based on a given stress intensity factor distribution

A new method is developed for the evaluation of a crack shape based on a given stress intensity factor (SIF) distribution for a surface crack under Mode-I loading conditions. The SIF distribution along the crack front is investigated using a direct simulation technique, in which the effect of crack closure on fatigue crack growth is considered. Then a SIF distribution function is chosen based on the numerical results. Crack shape (and the SIF) is achieved based on the given SIF distribution function using a numerical iterative procedure. Empirical SIF equations for surface cracks in plates subjected to tension and pure bending fatigue load are determined by systematic curve fitting of the numerical results. The depth ratio and the aspect ratio are considered in the ranges of 0.1–0.9 and 0.2–1.2, respectively. The aspect-ratio variation of surface cracks under fatigue loading is predicted. The application of the new method to predict the shape of a surface crack in plates subjected to tension and pure bending and comparisons of the results obtained with the predictions of the empirical equations proposed by Newman and Raju are presented.