An Embedded Crack in a Graded Coating Bonded to a Homogeneous Substrate Under Thermo-Mechanical Loading

ABSTRACT

The problem of an embedded partially insulated crack in a graded coating bonded to a homogeneous substrate under thermal and mechanical loading is considered. The heat conduction and the plane elasticity equations are converted into singular integral equations which are solved to yield the temperature and the displacement fields in the medium as well as the crack tip stress intensity factors. A crack-closure algorithm is applied to avoid interpenetration. The main objective of the paper is to study the effect of the coating nonhomogeneity parameters, partial insulation of the crack surfaces and crack-closure on the crack tip stress intensity factors for the purpose of gaining better understanding of the thermo-mechanical behavior of graded coatings.     

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

The elastostatic problem of a surface crack in a graded coating bonded to a homogeneous substrate under steady-state heat flux is considered. The coating is graded along the thickness direction and modeled 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 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 article is to study the effect of the layer thickness and nonhomogeneity parameters on the crack tip stress intensity factor for the purpose of gaining better understanding on the behavior of graded coatings under thermal loading.     

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 Stresses Around Two Collinear Interface Cracks between a Nonhomogeneous Bonding Layer and One of the Two Dissimilar Elastic Half-Planes Under Uniform Heat Flux

In composite materials, in which two dissimilar elastic half-planes are bonded by a nonhomogeneous elastic layer, two collinear cracks are situated at the interface between the nonhomogeneous elastic layer and one of the two dissimilar half-planes. The stress intensity factors are solved under uniform heat flux normal to the cracks. The material properties of the bonding layer vary continuously from the lower half-plane to the upper half-plane. The boundary conditions are reduced to dual integral equations using the Fourier transform technique. In order to satisfy the boundary conditions outside the cracks, the differences in temperature and displacements at the crack surfaces are expanded in a series of functions that vanish outside the cracks. The unknown coefficients in each series are evaluated using the Schmidt method. The stress intensity factors were calculated numerically for selected crack configurations.     

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.     

An axisymmetric crack in a functionally graded thermal barrier coating bonded to a homogeneous elastic substrate under transient thermal loading

Abstract

In this paper, the fracture problem of an axisymmetric crack in a functionally graded thermal barrier coating (FGTBC) bonded to a homogeneous substrate is considered. The problem is solved for the laminate that is suddenly heated from the upper surface of the FGTBC. The bottom surface of the homogeneous substrate is maintained at the initial temperature. The crack faces are supposed to be completely insulated. Material properties are assumed to be exponentially dependent on the distance from the interface. By using both the Laplace and Hankel transforms, the thermo-mechanical fracture problem is reduced to a singular integral equation and a system of singular integral equations which are solved numerically. The stress intensity factors of the crack are computed and presented as functions of the normalized time for various values of the nonhomogeneous and geometric parameters.     

Thermal Stresses Around Two Parallel Interface Cracks Between a Nonhomogeneous Bonding Layer and Two Dissimilar Elastic Half-Planes Under Uniform Heat Flux

Composite materials consisting of two dissimilar elastic half-planes bonded by a nonhomogeneous elastic layer contain two interface cracks; one is situated at the lower interface between the layer and the lower half-plane, while the other is situated at the upper interface between the layer and the upper dissimilar half-plane. The stress intensity factors are solved under uniform heat flux normal to the cracks. The material properties of the bonding layer vary continuously from the lower half-plane to the upper half-plane. The boundary conditions are reduced to dual integral equations using the Fourier transform technique, and they are satisfied outside the cracks by expanding the differences in temperature and displacements at the crack surfaces using a series of functions that vanish outside the cracks. The unknown coefficients in each series are evaluated using the Schmidt method. The stress intensity factors were calculated numerically for selected crack configurations.     

Thermal Stress Analysis of an Embedded Crack in a Graded Orthotropic Coating-Substrate Structure

An analysis of a coupled plane thermoelastic problem for a graded orthotropic coating-substrate structure is performed under thermomechanical loading conditions. The crack direction is parallel to the free surface. Applying the superposition principle and Fourier integral transform, the heat conduction and plane elasticity equations lends themselves to the derivation of two sets of Cauchy-type singular integral equations. The thermal stress intensity factors are defined and evaluated. In the numerical results, the effects of the orthotropy parameters, thermoelastic non-homogeneity parameters, and dimensionless thermal resistance on the temperature distribution and the thermal stress intensity factors (TSIFs) are studied. The obtained results can be used to design graded orthotropic coating-substrate structures under thermomechanical loading.     

The Axisymmetric Problem of a Partially Insulated Mixed-Mode Crack Embedded in a Functionally Graded Pyro Magneto-Electro-Elastic Infinite Medium Subjected to Thermal Loading

This article describes our investigation of the influence of an axisymmetric partially insulated mixed-mode crack on the coupled response of a functionally graded magneto-electro-elastic material (FGMEEM) subjected to thermal loading. The crack is embedded at the center of an infinite medium, and the material is graded in the direction orthogonal to the crack plane and is modeled as a nonhomogeneous medium with anisotropic constitutive laws. The heat conduction equation is first solved using the Hankel transform to yield the temperature field in the medium. Using the same integral transform, the magneto-electro-elasticity equations are converted analytically into a system of four singular integral equations that are solved numerically to yield the crack-tip mode I and II stress intensity factors, the electric displacement intensity factor and the magnetic induction intensity factor. The main objective of this research is to study the influence of material nonhomogeneity on the fields’ intensity factors for the purpose of gaining better understanding on the behavior of graded pyro magneto-electro-elastic materials.     

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.     

Transient Thermal Stress Analysis of Orthotropic Functionally Graded Materials with a Crack

The transient thermal fracture problem of a crack (perpendicular to the gradient direction) in a graded orthotropic strip is investigated. Most of the materials properties are assumed to vary as an exponential function of thickness direction. The transient two-dimensional temperature problem is analyzed by the methods of Laplace and Fourier transformations. A system of singular integral equations are obtained and solved numerically. Numerical results are figured out to show the variation of the temperature on the crack faces and extended line and stress intensity factors for different material parameters with dimensionless time.     

A NEW MODEL OF FUNCTIONALLY GRADED COATINGS WITH A CRACK UNDER THERMAL LOADING

A new model for fracture analysis oaf functionally graded materials (FGMs) with arbitrarily varying material properties under thermal loading is developed. The FGM is modeled as a multilayered medium and in each layer both shear modulus and thermal conductivity are assumed to be a linear function of the depth and are continuous on the subinterfaces. To make the crack problem tractable, thermal expansion and conductivity of the FGMs are supposed to have the same form. With this new model, the crack problem of a functionally graded coating bonded to a homogeneous substrate under steady-state thermal loading is investigated. Employment of the Fourier integral transform technique reduces the problem to a system of Cauchy singular integral equations that are solved numerically. Thermal stress intensity factors (TSIFs) are obtained for various forms of thermal conductivity or expansion. The results reveal that the present model is very efficient and in the frame of the present model both the form of thermal conductivity/expansion and that of its derivative can influence the TSIFs significantly.     

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 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.     

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.     

Contact Zone Approach for an Electrically Impermeable Crack in Piezoelectric Materials Under Thermal-Mechanical Loading

A partial contact zone model is developed for the stress and electric displacement fields due to the obstruction of a uniform heat flux by an electrically impermeable crack in piezoelectric materials. Green's function method is used to reduce the problem to a set of singular integral equations that are solved in closed form. When the crack is assumed to be traction free, the crack opening displacement is found to be negative over one-half of the crack unless a sufficiently large far field tensile stress is superposed. The problem is reformulated assuming a contact zone at one crack tip. The extent of this zone, the stress and electric displacement intensity factors at each crack tip are obtained as functions of the applied mechanical stress and heat flux.     

MULTIPLE SURFACE CRACKING IN GLASS-FIBER-REINFORCED PLASTICS DUE TO A THERMAL SHOCK AT A LOW TEMPERATURE

We consider the transient thermal singular stress problem of multiple surface cracking in glass-fiber-reinforced plastics due to a thermal shock at a low temperature. The layered composite is made of a layer bonded between two layers of different physical properties, and it is suddenly cooled on the surfaces. The surface layers contain parallel arrays of the embedded or edge cracks perpendicular to the boundaries. The thermal and elastic properties of the material are dependent on the temperature. For the case of the crack that ends at the interface between orthotropic elastic materials, the order of stress singularity around the tip of the crack is obtained. Finite element calculations are carried out, and the transient thermal stress intensity factors are shown graphically.     

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 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.     

Investigation of the thermal-elastic problem in cracked semi-infinite FGM under thermal shock using hyperbolic heat conduction theory

In this paper, a thermoelastic analytical model is established for a functionally graded half-plane containing a crack under a thermal shock in the framework of hyperbolic heat conduction theory. The moduli of functionally graded materials (FGMs) are assumed to vary exponentially with the coordinates. By employing the Fourier transform and Laplace transform, coupled with singular integral equations, the governing partial differential equations under mixed, thermo-mechanical boundary conditions are solved numerically. For both the temperature distribution and transient stress intensity factors (SIFs) in FGMs, the results of hyperbolic heat conduction model are significantly different than those of Fourier’s Law, which should be considered carefully in designing FGMs.