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 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.     

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.     

Thermal Stresses Around a Crack in a Non-Homogeneous Diffusion Layer Between a Coating Plate and an Elastic Half-Plane

ABSTRACT

This article proposes a method for determining the thermal stress field around a crack in a thin non homogeneous layer located between a ceramic plate and a metallic half-plane. For these calculations, the crack surfaces are assumed to be insulated and a uniform heat flux flows perpendicular to the crack. The material properties of the layer are assumed to vary continuously from those of the ceramic plate to those of the half-plane. The Fourier transform technique is employed to transform the problem into a set of integral equations. These equations are solved by expanding the differences in the crack surface temperature and the crack surface displacements in a series of functions that are automatically zero outside the crack. The Schmidt method is then used to determine the unknown coefficients in the series. Using this procedure, the stress intensity factors are calculated numerically for several ceramic plate thickness values.     

The Crack Problem in Piezoelectric Strip Under Thermoelectric Loading

This paper investigates the thermoelectromechanical fracture behavior of a parallel crack in a piezoelectric strip under thermoelectric loading. The crack faces are supposed to be insulated thermally and electrically. By using the Fourier transform, the thermal and electromechanical problems are reduced to a system of singular integral equations, respectively, which are solved numerically. Numerical calculations are carried out, and the energy density factor as well as the stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the size and the location of the crack and the magnitude of the electric loading.     

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.     

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.     

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.     

Transient thermal response of a functionally graded piezoelectric laminate with a crack normal to the bimaterial interface

In this article, the problem of a functionally graded piezoelectric material strip (FGPM strip) containing a crack perpendicular to the interface between the FGPM strip and a homogeneous layer is analyzed under transient thermal loading condition. The crack faces are supposed to be completely insulated. Material properties are assumed to be exponentially dependent on the distance from the interface. Using the Laplace and Fourier transforms, the thermoelectromechanical problem is reduced to a singular integral equation, which is solved numerically. The stress intensity factors of embedded and edge cracks are computed. The results for the crack contact problem are also included.     

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.     

A generalized thermoelastic problem due to nonlocal effect in presence of mode I crack

Abstract

This article constructs a new model of nonlocal thermoelasticity which resolves a dynamical problem of a homogeneous, isotropic infinite space weakened by a finite linear mode I crack. The boundary of the crack is being subjected to a prescribed temperature distribution and stress. In the context of three-phase lag model of generalized thermoelasticity, the governing equations have been solved employing the Laplace and the Fourier transforms, which reduces to four dual integral equations, the solution of which is equivalent to solving the Fredholm’s integral equation of the first kind. These integral equations have been solved employing the Maple software package, while the numerical inversion of the Laplace transform is carried out with the help of Bellman method. Numerical computations for a copper material are performed and demonstrated graphically. The results provide a motivation to further investigate the problem and draw concluding remarks due to the influence of nonlocality also.     

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.     

Thermoelectromechanical Response of a Center Crack in a Symmetrical Functionally Graded Piezoelectric Strip

The thermoelectromechanical fracture problem for a symmetrical functionally graded piezoelectric strip containing a center crack parallel to the free boundaries is considered in this study. It is assumed that the thermoelectroelastic properties of the medium vary continuously in the thickness direction, and that the strip is under thermomechanical loadings. The crack faces are supposed to be insulated thermally and electrically. By using the Fourier transform, the thermal and electromechanical problems are reduced to singular integral equations, respectively, which are solved numerically. Numerical calculations are carried out, and detailed results are presented to illustrate the influence of the crack length and the material nonhomogeneity on the temperature-stress distributions and the stress intensity factor.     

THERMAL STRESSES IN A CIRCULAR CYLINDER WITH A CIRCUMFERENTIAL EDGE CRACK UNDER UNIFORM HEAT FLOW

The linear thermoelastic problem of an infinitely long circular cylinder with a circumferential edge crack is solved. The thermal stresses are caused by a uniform heat flow disturbed by the presence of the crack. The crack surfaces and the cylindrical surface are assumed to be insulated. Integral transform techniques are used to reduce the problem to that of solving two singular integral equations of the first kind. The equations are solved numerically, and the variation of the stress intensity factor with the crack depth is shown graphically.     

Infinite Row of Parallel Cracks in a Functionally Graded Piezoelectric Material Strip Under Mechanical and Transient Thermal Loading Conditions

In this paper, the problem of an infinite row of parallel cracks in a functionally graded piezoelectric material strip (FGPM strip) is analyzed under static mechanical and transient thermal loading conditions. The crack faces are supposed to be completely insulated. Material properties are assumed to be exponentially dependent on the distance from the bottom surface. By using the Laplace and Fourier transforms, the thermoelectromechanical problem is reduced to a singular integral equation, which is solved numerically. The stress intensity factors for both the embedded and edge cracks are computed. The results for the crack contact problem are also included.     

The effect of a stiffener on a cracked plate under bending

In this study the problem of a stiffened plate containing a through-crack under uniform bending load is analyzed. The problem is formulated for a specially orthotropic material by using Reissner's plate theory. By using the Fourier integral transform technique the problem is reduced to a singular integral equation. This singular integral equation is then solved numerically by using Gȧuss-Chebyshev and Gauss-Jacobi quadrature formulas. The special case of the problem in which the crack tip terminates at the stiffener is also analyzed in order to assess the crack arrest effectiveness of the stiffener. The asymptotic stress state near the crack tip terminating at the stiffener is examined, and normalized Mode I stress intensity factors are tabulated. The results also include the effect of Poisson's ratio, stiffness constants and material orthotropy for specially orthotropic materials on the stress intensity factors.     

Thermal Stress Analysis in Magneto-Electro-Thermo-Elasticity with a Penny-Shaped Crack Under Uniform Heat Flow

ABSTRACT

The magnetoelectroelastic material possesses the dual feature that the application of magnetic field induces electric polarization and electric field induces magnetization. Piezoelectric-piezomagnetic materials exhibit magneto-electric effect. When magneto-electro-elastic materials are subjected to thermal flow, they can fracture prematurely due to their brittle behavior. Hence, it should be important to know the fracture behavior of magneto-electro- elastic materials. The penny-shaped crack problem in a medium possessing coupled electro-magneto-thermo-elastic is considered in this paper. It is assumed that the crack is isothermal. The analysis is an exact treatment of penny-shaped crack in a magneto-electroelastic solid subjected to uniform heat flow far away from the crack region. The governing equations of temperature, elastic displacements and electric potential as well as magnetic potential for an anisotropic magneto-electro-elastic medium are partial differential equations of second order, which are solved by means of the Hankel transform technique. Expressions for elastic displacements, thermal stresses, electric displacements and magnetic inductions are determined from the dual integral equation method. Exact thermal stress intensity factor of the problem is obtained, and the near crack tip solutions are provided.     

Thermal Stress Intensity Factors for a Normal Crack in a Piezoelectric Material Strip

This paper investigates the electromechanical fracture behavior of a normal crack in a piezoelectric material strip subjected to a uniform heat flow far away from the crack region. The crack faces are supposed to be insulated thermally and electrically. By using the Fourier transform, the thermal and electromechanical problems are reduced to singular integral equations, respectively, which are solved numerically. Both the cases of an internal crack and an edge crack are studied. Numerical calculations are carried out, and detailed results are presented to illustrate the influence of the crack location and length on the temperature distribution and the stress intensity factors.     

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.     

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.