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.     

THERMAL STRESSES AROUND AN INSULATED BARRIER

The main interest in this study is to determine the thermal stresses around an insulated barrier due to constant heat flux in a homogeneous semi-infinite medium. Reducing the diffusion equation to a singular integral equation, the temperature distribution around the insulated barrier is obtained by defining an unknown function, the so-called density function, as a series expansion of orthogonal polynomials. After the temperature distribution is obtained then it can be used in equilibrium equations as an input function to find thermal stresses for different thickness parameters. Using the solution of equilibrium equations one can also obtain displacements around an insulated barrier as a secondary interest of the problem.     

TRANSIENT STRESS INTENSITY FACTORS FOR PERIODIC ARRAY OF CRACKS IN A HALF-PLANE DUE TO CONVECTIVE COOLING

The problem of periodic cracks perpendicular to the boundary of a half-plane under transient thermal loading is investigated. The thermal stresses are generated as a result of convective cooling on the boundary of the plane. The problem is solved using the superposition technique. The perturbation problem is formulated using the thermal stresses obtained from the uncracked problem with the opposite sign as the only external load. The formulation results in a singular integral equation of Cauchy type that is solved numerically. Numerical results are obtained for the stress intensity factors as a function of time, crack length, location of the crack, and periodic crack spacing.     

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.     

Thermoelastic analysis of a cylindrical vessel of functionally graded materials

This paper presents a novel method for analyzing steady thermal stresses in a functionally graded hollow cylinder. The thermal and thermoelastic parameters are assumed to arbitrarily vary along the radial direction of the hollow cylinder. The boundary value problem associated with a thermoelastic problem is converted to a Fredholm integral equation. By numerically solving the resulting equation, the distribution of the thermal stresses and radial displacement is obtained. The numerical results obtained are presented graphically and the influence of the gradient variation of the material properties on thermal stresses is investigated. It is found that appropriate gradient can make the distribution of thermal stresses more gentle in the whole structure.     

THERMOELASTIC PROBLEM OF STEADY-STATE HEAT FLOW DISTURBED BY A CRACK PERPENDICULAR TO THE GRADED INTERFACIAL ZONE IN BONDED MATERIALS

The plane thermoelasticity equations are used to investigate the steady-state nonisothermal crack problem for bonded materials with a graded interfacial zone. The interfacial zone is modeled as a nonhomogeneous interlayer having continuously varying thermoelastic moduli in the exponential form between the dissimilar, homogeneous half-planes. A crack is assumed to exist in one of the half-planes oriented perpendicular to the nominal interface, disturbing a uniform heat flow. Based on the method of Fourier integral transform, formulation of the crack problem is reduced to solving two sets of Cauchy-type singular integral equations for temperature and thermal stress fields. The heat-flux intensity factors and the thermally induced mode II stress intensity factors are defined in order to characterize the singular behavior of temperature gradients and thermal stresses, respectively, in the vicinity of the crack tips. In the numerical results, the values of heat-flux and thermal-stress intensity factors are presented for various combinations of material and geometric parameters of the dissimilar media bonded through a thermoelastically graded interfacial zone. The influence of crack-surface partial conductance on the near-tip temperature and thermal stress fields is also addressed.     

Modeling Multilayer Piezoelectric Actuators in a Residual Temperature Field

Electromechanical field concentrations near the electrodes in multilayer piezoelectric ceramic actuators in a residual temperature field are examined. The problem is formulated by considering a representative unit in the real multilayer actuators. A singular integral equation, in which the unknown function is the electric potential ahead of the electrode tip, is derived. The solution of the integral equation gives the electroelastic fields near the electrode tip. The thermally induced electric field ahead of the electrode tip is found to be highly singular. On the other hand, the stresses and electric displacements are singular behind the electrode tip. Possible debonding and cracking between the electrodes and the piezoelectric medium is investigated by energy density theory.     

THERMAL SINGULAR STRESSES NEAR A CIRCUMFERENTIAL EDGE CRACK IN A SPHERICAL CAVITY UNDER UNIFORM AXIAL HEAT FLOW

The linear thermoelastic problem of a spherical cavity with a circumferential edge crack is solved. The thermal stresses are caused by a uniform heat flow disturbed by the presence of the crack and the cavity. The surfaces of the crack and the cavity are assumed to be insulated. Integral transform techniques are used to reduce the problem concerning the temperature and thermoelastic fields to that of solving two singular integral equations of the first kind. The integral equations are solved numerically and the variation of the thermal stress intensity factor with the crack depth and the crack opening displacement are shown graphically.     

The Influence of Graded Coatings on the Thermal Stress Intensity Factors of an Oblique Crack in a Semi-Infinite Substrate Subjected to Local Heating on the Boundary

Thermally induced singular behavior of an arbitrarily oriented crack in a homogeneous substrate overlaid with a functionally graded coating is considered, within the framework of linear plane thermoelasticity. It is assumed that the graded coating/substrate system is subjected to steady-state thermal loading applied over a finite region at the coating surface and the crack in the substrate is thermally insulated, disturbing the prescribed heat flow. Based on the method of Fourier integral transform and the coordinate transformations of basic field variables in thermoelasticity equations, formulation of the crack problem is reduced to two sets of Cauchy-type singular integral equations for temperature and thermal stresses in the coated medium. In the numerical results, the main emphasis is placed on the investigation of influences of loading, geometric, and material parameters of the coated system on the variations of mixed-mode thermal stress intensity factors. Further addressed are the probable cleavage angles for the incipient growth of the original crack and the corresponding values of effective tensile-mode stress intensity factors.     

Effects of Crack Surface Conductance on Intensity Factors for a Functionally Graded Piezoelectric Material Under Thermal Load

Effects of crack surface conductance on intensity factors for a functionally graded piezoelectric material under thermal load are investigated. The heat flux through the crack is assumed to be proportional to the local temperature difference. Moreover, two models for more realistic crack face electric boundary conditions are proposed. By using the Fourier transform, the thermal and electromechanical problems are reduced to a singular integral equation and a system of singular integral equations, respectively, which are solved numerically. Detailed results are presented to illustrate the influence of the thermal and electric conductance on the stress and electric displacement intensity factors.     

THERMAL SHOCK FRACTURE IN A W-Cu DIVERTOR PLATE WITH A FUNCTIONALLY GRADED NONHOMOGENEOUS INTERFACE

The plane elasticity solution is presented in this article for the crack problem of a W-Cu divertor plate under thermal shock. The material is made of a graded layer with exponentially varying thermomechanical properties bonded between a homogeneous substrate and a homogeneous coating and is subjected to a cycle of heating and cooling on the coating surface of the material. The surface layer contains an embedded or a surface crack perpendicular to the boundaries. Using superposition the problem is reduced to a perturbation problem in which the crack surface tractions are only external forces. The dimensions, geometry, and loading conditions of the original problem are such that the perturbation problem may be approximated by a plane strain mode I crack problem for an infinite divertor plate. Fourier transforms are used to formulate the crack problem in terms of a singular integral equation. After giving some sample results regarding the distribution of thermal stresses, stress intensity factors for embedded and surface cracks are presented. Also included are the results for a crack/contact problem in a divertor plate that is under compression near and at the surface and tension in the interior region.     

THERMALLY INDUCED FRACTURE OF A FUNCTIONALLY GRADED PIEZOELECTRIC LAYER

The theoretical analysis of a thermoelectroelasticity problem is developed for a piezoelectric layer due to a thermal load under a uniform electric field and a fixed grip condition. The layer is assumed to be a functionally graded material, meaning that its thermoelectromechanical properties are assumed to be continuous functions of the thickness coordinate. The layer contains an embedded or an edge crack perpendicular to its boundaries. The Fourier transform technique is used to formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gauss–Jacobi integration formula. Numerical calculations are carried out, and the mode I energy density factors are presented for embedded as well as edge cracks for various values of dimensionless parameters representing the size and the location of the crack, the material nonhomogeneity, the surface temperatures, and the loading combinations.     

EDGE CRACK IN A NONHOMOGENEOUS HALF PLANE UNDER THERMAL LOADING

This paper deals with the problem of an edge crack in a semi-infinite nonhomogeneous plate under steady heat flux loading conditions. The objective of the study is to assess the effect of material nonhomogeneity on the thermal stress intensity factor. All material properties are supposed to be exponentially dependent on the distance from the boundary of the plate. By using the Fourier transform, the problem is reduced to a singular integral equation that is solved numerically. The thermal stress intensity factors for various material constants are calculated. The results show that by selecting the material constants appropriately the stress intensity factor can be reduced.     

FRACTURE OF A PIEZOELECTRIC MATERIAL STRIP UNDER STEADY STATE THERMAL LOAD

Thermal effects become important when the piezoelectric material has to be operated in either extremely hot or cold temperature environments. It is essential to know the interaction of mechanical defects with thermal changes. In this article, we examine the piezothermoelastic problem for a Griffith crack that is located in a piezoelectric material strip. The strip is infinite along the x-direction and has finite thickness in the y-direction. The crack plane is parallel to the boundary of the strip. The polarized axis of the piezoelectric material is either normal or parallel to the y-direction. The basic entities are the Fourier transform and singular integral equation techniques. The crack-tip fields are obtained. The variation in crack-tip field intensity factors due to changes of the crack size and location is studied for different poling directions.     

Transient Intensity Factors for a Parallel Crack in a Plate of a Functionally Graded Piezoelectric Material Under Thermal Shock Loading Conditions

A crack in a plate of a functionally graded piezoelectric material is studied under thermal shock loading conditions. The crack faces are supposed to be completely insulated. All material properties are assumed to be exponentially dependent on the distance from the crack line parallel to the boundaries of the plate. By using both the Laplace transform and Fourier transform, the thermal and electromechanical problems are reduced to a singular integral equation and a system of singular integral equations that are solved numerically. The stress and electric displacement intensity factors vs. time for various material constants and geometric parameters are calculated.     

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.     

Thermoelastic-induced vibrations on an elliptical disk with internal heat sources

In this study, integral transform technique is used to investigate the thermally induced vibration of an elliptical disk. The axisymmetric temperature distribution in the disk is determined by conductivity equation and the corresponding initial and boundary conditions using an extended integral transform technique. The problem of thermally induced vibration of the disk with both ends clamped extremes is solved by developing an integral transform for double Laplace differential equation. The thermal moment is derived on the basis of temperature field, whereas maximum normal stresses are derived based on resultant bending moments per unit width. The results are obtained in series form in terms of Mathieu functions, and numerical results are shown in figures.     

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.     

Circumferentially Cracked Bimaterial Hollow Cylinder Under Mechanical and Transient Thermal Loading

The analytical solution for the problem of a circumferential inner surface crack in an elastic, infinitely long composite hollow cylinder, made of two concentric perfectly bonded transversely isotropic cylinders is considered. Uniform axial loading and thermal loading in the form of a sudden cooling on the inner boundary are considered. Out of 10 material parameters involved, two bimaterial parameters and three material parameters for each layer upon which the stress intensity factor depends under uniform loading, are identified. The problem is reduced to a singular integral equation that is solved numerically. Stress intensity factors are presented for various values of material and geometric parameters.     

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.