Deterministic assessment of reactor pressure vessel integrity under pressurised thermal shock

Numerical investigations were carried out to assess the integrity of reactor pressure vessels under pressurised thermal shock (PTS). The 4-loop reactor pressure vessel with cladding was subjected to thermo-mechanical loading owing to loss of coolant accident. The loss of coolant accident corresponding to small break as well as hot leg breaks were considered separately, which led to axisymmetric and asymmetric thermal loading conditions respectively. Three different crack configurations, 360° circumferential part through, circumferential semi-elliptical surface and circumferential semi-elliptical under-clad cracks, were postulated in the reactor pressure vessel. Finite element method was used as a tool for transient thermo-elastic analysis. The various fracture parameters such as crack mouth opening displacement (CMOD), stress intensity factor (SIF), nil ductility transition temperature (RT_NDT) etc. were computed for each crack configuration subjected to various type of loading conditions. Finally for each crack a fracture assessment was performed concerning crack initiation based on the fracture toughness curve. The required material RT_NDT was evaluated to avoid crack initiation.     

TWO CRACK GROWTHS IN A FUNCTIONALLY GRADED PLATE UNDER THERMAL SHOCK

This article examines the problem of two thermal cracks under a transient temperature field in a ceramic/metal functionally graded plate. When the functionally graded plate is subjected to thermal shock, multiple cracks often occur on the ceramic surface. It is shown that the crack paths are influenced by interaction between multiple cracks and a compositional profile of the functionally graded plate. Transient thermal stresses are treated as a linear quasi-static thermoelastic problem for a plane strain state. The crack paths of two cracks are obtained using the finite element method with mode I and mode II stress intensity factors.     

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 SHOCK PROBLEMS OF ELASTIC BODIES WITH A CRACK

This paper deals with thermal shock, problems of elastic bodies with a crack. The case considered is that of an infinitely long circular cylinder with an edge crack, and a homogeneous flat plate with an edge crack initially at uniform temperature and suddenly immersed into a medium of lower temperature. The thermal disturbance near the crack tip is assumed to be neglible in the analysis of the temperature field because thermal shocks occur very quickly. We analyze the transient thermal stress problems of elastic solids with a crack and determine the stress intensity factor at the crack tip. The nondimensional maximum transient stress intensity factor is expressed as a function of the Biot number and the nondimensional crack length. Then we propose simplified formulations of the nondimensional maximum transient stress intensity factor as a function of the Biot number and the nondimensional crack length.     

Three-dimensional thermal weight function method for the interface crack problems in bimaterial structures under a transient thermal loading

Different from previous two-dimensional thermal weight function (TWF) method, a three-dimensional (3D) TWF method is proposed for solving elliptical interface crack problems in bimaterial structures under a transient thermal loading. The present 3D TWF method based on the Betti's reciprocal theorem is a powerful tool for dealing with the transient thermal loading due to the stress intensity factors (SIFs) of whole transient process obtained through the static finite element computation. Several representative examples demonstrate that the 3D TWF method can be used to predict the SIFs of elliptical interface crack subjected to transient thermal loading with high accuracy. Moreover, numerical results indicate that the computing efficiency can be enhanced when dealing with transient problems, especially for large amount of time instants.     

Green's function approach for crack propagation problem subjected to high cycle thermal fatigue loading

An efficient numerical approach using Green's function for the analysis of crack propagation under thermal transient load has been presented. The present approach based on multi-Green's functions pre-determined for each stage of the incremental crack growth substantially shortens the calculation time of the stress intensity factor (SIF) ranges. It was shown that the Green's function method (GFM) can be efficiently used to evaluate not only thermal stresses for fatigue analysis but also the SIF for crack propagation analysis. The crack propagation analysis results have been compared with those of the actual observation for the piping structure subjected to thermal striping load in a liquid metal fast breeder reactor. It was shown that the function determined at a fixed temperature can be applied to a relatively wide range of temperatures because of the compensation effect of the material properties, that is, some properties increase while the others decrease as the temperature increases.     

THERMAL STRESSES INTENSITY FACTOR FOR FUNCTIONALLY GRADIENT PLATE WITH AN EDGE CRACK

Thermal stress problems of a functionally gradient plate as one of the advanced high-temperature materials capable of withstanding the extreme temperature environments, with and without an edge crack, are discussed One of the most important problems of the thermal stress in the functionally gradient plate with the crack are how to decrease the thermal stress intensity factor and how to determine the optimally continuous profile of the composition of the plate. The functionally gradient plate is subjected to a cycle of heating and cooling on the ceramics surface of the plate. The material properties of the functionally gradient plate are dependent on the temperature and the position. The optimally continuous profile of the composition of the plate is discussed. The numerical results for thermal stresses and the thermal stress intensity factor are shown for many temperature conditions and for many continuous profiles of the composition of the plate.     

The transient thermo-electro-elastic fields in a piezoelectric plate with a crack

This paper is concerned with a thin plate made by a piezoelectric ceramic material and containing a crack perpendicular to its surfaces. It is assumed that the transient thermal stress is set up by the application of a heat flux as a function of the time and position along the crack edge and the heat flow by convection from the plate surfaces. The plate is also subjected to mechanical and electric loadings. The exact analytical formulae are obtained for transient thermo-electro-elastic fields in the plate. The exact analytical solutions for the stress and electric displacement intensity factors and crack-opening displacement are obtained. Numerical examples show, among others, a dependence of the stress and electric displacement intensity factors on the thermal and elastic, piezoelectric and dielectric constants of the piezoelectric materials.     

Thermoelasticity problem for a multilayer coating/half-space assembly with undercoat crack subjected to convective thermal loading

The transient thermal stress crack problem for a half-space with a multilayer coating under thermal surface loading containing an undercoat crack, perpendicular to the interface, is considered. The problem is solved using the principle of superposition and uncoupled quasi-static thermoelasticity. Transient temperature distribution and corresponding thermal stresses for the uncracked multilayer assembly are obtained in a closed analytical form using the model with generalized thermal boundary conditions of heat exchange of a half-space with ambient media via the coating. The crack problem is formulated as a perturbation mixed boundary value problem, in which the crack surface loading should be equal and opposite to the thermal stresses obtained for the uncracked medium, and is reduced to a singular integral equation and solved numerically. Numerical computations are performed for the analysis of influence of the coating upon thermal stresses and thermal stress intensity factor.     

CRITICAL STRESS INTENSITY FACTORS FOR CRACKED HOLLOW PIPES UNDER TRANSIENT THERMAL LOADS

Finite element analyses of a long hollow cylinder having an axisymmetric circumferential internal edge crack, subjected to convective cooling on the inner surface are performed. The transient thermal stress intensity factor is estimated using a domain version of the J-integral method. The effect of the thickness of the cylinder, crack length, and heat transfer coefficient on the stress intensity factor history are studied. The variations of critical normalized stress intensity factor with crack length-to-thickness ratio for different parameters are presented. The results show that if a small inner surface crack begins to grow, its stress intensity factor will increase with increase in crack length, reach a maximum, and then begin to drop. Based on the results, a fracture-based design methodology for cracked hollow pipes under transient thermal loads is discussed.     

THERMAL STRESS ANALYSIS OF A RECTANGULAR PLATE AND ITS THERMAL STRESS INTENSITY FACTOR FOR COMPRESSIVE STRESS FIELD

Abstract

In this study, a transient thermal stress problem of a rectangular plate due to a nonuniform heat supply is treated theoretically and, thereafter, fracture behaviors of the plate with a crack are examined for compressive stress states. Assuming that a crack located on an arbitrary position, with an arbitrary direction, is sufficiently small and is closed because of the compressive stress field, a temperature field, in a transient state, is analyzed by taking into account the effect of relative heat transfer on both surfaces of the plate. Thereafter, the corresponding thermal stress analysis is developed on the basis of the two-dimensional plane stress problem using Airy's stress function method, and the stress intensity factor is analyzed for the biaxial stress state. As an analytical model, we consider mechanical boundary conditions of prescribed displacement and estimate the stress intensity factor of a crack tip using parameters of the crack configuration such as the location, direction, length, and coefficient of friction. These numerical results are shown in graphical form.     

DETERMINATIONS OF MIXED-MODE STRESS INTENSITY FACTORS FOR CRACKED BODIES SUBJECTED TO THERMAL LOADINGS BY USING THE THERMAL WEIGHT FUNCTION METHOD

In this study, mixed-mode stress intensity factors for cracked bodies subjected to thermal loadings are determined by using the thermal weight function method. The thermal weight function is a universal junction for a given cracked body and can be obtained from any arbitrary mechanical loading system. The thermal weight function may be thought of as the Green's junction for the stress intensity factor of cracked bodies subject to thermal loadings. Once the thermal weight function for a cracked body is determined, the stress intensity factor for any arbitrary distributed temperature can be simply and efficiently evaluated through the integration of the product of the temperature and the correspondent thermal weight function. A numerical boundary element method for the determination of thermal weight functions is used in this study to evaluate mixed-mode stress intensity factors. As a demonstration, some examples of cracked bodies subjected to thermal loadings are solved by using the thermal weight function method, and the results are compared with solutions obtained from other methods.     

EDGE-CRACKED PLATE WITH ONE FREE AND ONE CONSTRAINED BOUNDARY SUBJECTED TO SUDDEN CONVECTIVE COOLING

The transient thermal stress edge crack problem for an elastic strip with free and fully constrained boundaries is considered. The plate is suddenly subjected to convective cooling on the face containing the edge crack while the other face is insulated. The solution of the problem is obtained by using the superposition technique results in a singular integral equation that is solved numerically. The results of the transient temperature and thermal stress distributions in the uncracked strip are presented. Also, numerical results are obtained for the stress-intensity factor in terms of the Fourier number, crack length, and different values of the Biot number.     

Transient Response of a Cracked Piezoelectric Strip Under Thermoelectric Loading

In this study, the theoretical analysis of a transient piezothermoelastic problem is developed for a piezoelectric strip with a parallel crack under static electric loading and thermal shock loading conditions. The crack faces are supposed to be insulated thermally and electrically. By using both the Laplace transform and the Fourier transform, the thermal and electromechanical problems are reduced to a system of singular integral equations, respectively, which are solved numerically. Some numerical results for the temperature change, the stress and electric displacement distributions, and the energy density factor as well as the stress and electric displacement intensity factors in a transient state are shown in figures.     

THERMAL STRESSES IN FUNCTIONALLY GRADED MATERIALS

The thermal stress problems of functionally graded materials (FGMs), as one of the advanced high-temperature materials capable of withstanding the extreme temperature environments, are discussed. The FGMs consist of the continuously changing composi tion of two different materials. For example, one is an engineering ceramic to resist the severe thermal loading from the high-temperature environment, and the other is a light metal to maintain the structural rigidity. When the FGMs are subjected to extremely severe thermal loading, large thermal stresses are produced in the FGMs. Therefore, one of the most important problems of FGMs is how to decrease thermal stresses and how to increase heat resistance. The optimal composition profile problems of the FGMs in decreasing thermal stresses are discussed in detail. When FGMs are subjected to extremely severe thermal loading, the FGMs are damaged. The crack initiates on the ceramic surface and propagates in the FGMs. It is important to discuss the thermal stresses in the FGMs with various types of cracks. The thermal stress intensity factors in the FGMs with various types of crack are treated analytically and numerically. The optimal composition profile problems of the FGMs in decreasing thermal stress intensity factor are studied. Finally, the crack propagation paths due to thermal shock are discussed.     

Investigation Methods for Thermal Shock Crack Problems of Functionally Graded Materials–Part II: Combined Analytical-Numerical Method

Fractures phenomena can be often found in functionally graded materials (FGMs) subjected to thermal shock loadings. This paper aims to develop a set of analytical-numerical methods for analyzing the mixed-mode thermal shock crack problems of a functionally graded plate (FGP). First, a domain-independent interaction energy integral method is developed for obtaining the mixed-mode transient thermal stress intensity factors (TSIFs). A perturbation method is adopted to obtain the transient temperature field. Then an analytical-numerical method combining the interaction energy integral method, a perturbation method, and the finite element method is developed to solve the present crack problem. Particularly, the influences of the materials parameters, crack length, and crack angle on the TSIFs and the crack growth angle are investigated. The results show that the present analytical-numerical method can be used to solve the thermal shock crack problem with high efficiency. The present work will be significant for the fracture mechanics analysis and design of FGM structures.     

Thermal fracture of cracked cylinders associated with nonclassical heat conduction: The effect of material property

The thermoelastic problem of a transversely isotropic hollow cylinder containing a circumferential crack is investigated in the present article based on the non-Fourier heat conduction theory. The temperature and stress fields are obtained by solving the coupled partial differential equations in the Laplace domain, and corresponding thermal axial stress with minus sign is then applied to the crack surface to form a mode I crack problem. Three different kinds of crack are considered, and the singular integral equation method is adopted to solve the fracture problem. Finally, with the definition of stress intensity factor, the effect of material properties, coupling parameter, and crack geometry on the hyperbolic thermal fracture responses of a transversely isotropic hollow cylinder excited by a thermal loading is visualized.     

Pressurized thermal shock analyses of a reactor pressure vessel using critical crack depth diagrams

Evaluated in this study is the pressure vessel integrity under a pressurized thermal shock. Using transient histories such as temperature, pressure and heat transfer coefficient, the stress distribution is calculated and then stress intensity factors are obtained for a wide range of crack sizes. The stress intensity factors are compared with the fracture toughness to check if cracking is expected to occur during the transient. Critical crack depth diagrams are prepared for each transient which is expected to initiate a pressurized thermal shock accident. Plant-specific analyses of the most limiting plant in Korea are performed to assure the structural integrity of the reactor vessel and the results are discussed.     

A Surface Crack in a Graded Coating Bonded to a Homogeneous Substrate under Transient Thermal Loading

The elastodynamic problem of a surface crack in a graded coating bonded to a homogeneous substrate under transient heat flux is considered. The coating is graded along the thickness direction and modelled as a nonhomogeneous medium with an isotropic stress-strain law. The problem is solved under the assumption of plane strain or generalized plane stress conditions. The resulting crack problem is of mode I because the orientations of the crack axis, the material gradient and the heat-flux are all parallel. The equivalent crack surface tractions are first obtained and substituted in the plane elasticity equations which are then converted analytically using appropriate integral transforms into a singular integral equation. The resulting equation is solved numerically using orthogonal Jacobi polynomials to yield the Mode I stress intensity factor. The main objective of the research is to study the effect of the layer thickness and nonhomogeneity parameters on the dynamic crack tip stress intensity factor for the purpose of gaining better understanding on the behavior of graded coatings under transient thermal loading.     

THERMAL SHOCK FRACTURE IN AN EDGE-CRACKED PLATE

In this paper the transient thermal stress problem for an elastic strip with an edge crack is investigated. The elastic medium is assumed to be insulated on one face and cooled by surface convection on the face contaning the edge crack. Using the principle of superposition, the formulation results in a mixed boundary value problem, with the thermal stresses calculated from the thermoelasticity solution for an uncracked strip utilized as the necessary crack surface tractions. The resulting singular integral equation is of a well-known type and is solved numerically. In this paper, inertia effects are assumed negligible and possible temperature dependence of thermoelastic constants is not considered. The numerical results presented, include the stress intensity factor as a function of nondimensional time (Fourier number) and crack length, for various values of the dimensionless Biot number. The temperature distribution and the thermal stresses in the uncracked strip are also included. The time lag, which occurs between the time at which the stress on the surface of the strip is a maximum and the time when a maximum occurs in the stress intensity factor, is clearly shown to be a function of the Biot number for any given ratio of crack length to strip thickness. A result of particular interest is the degree with which the maximum stress intensity factor decreases, as a function of crack length, for decreasing values of the Biot number.