An Annular Crack in a Functionally Graded Piezoelectric Strip Under Thermoelectric Loadings

A thermoelectroelastic problem of a functionally graded piezoelectric material (FGPM) strip containing an annular crack is solved. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under the thermoelectric loadings. The crack faces are supposed to be insulated thermally and electrically. Using integral transform techniques, the problem is reduced to that of solving a singular integral equation and a system of singular integral equations. Numerical calculations are carried out, and the variations of the stress and electric displacement intensity factors are plotted against the geometric parameters for some values of the material non-homogeneity parameters.     

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.     

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.     

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.     

Transient Thermoelectroelastic Response of a Functionally Graded Piezoelectric Strip with Two Parallel Axisymmetric Cracks

In this article, the problem of two parallel axisymmetric cracks in a plate of a functionally graded piezoelectric material (FGPM) strip is analyzed under transient thermal loading conditions. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the crack faces are supposed to be insulated thermally and electrically. By using both the Laplace and Hankel transforms, the thermal and electromechanical problems are reduced to two systems of singular integral equations. The singular integral equations are solved numerically, and a numerical method is then employed to obtain the time dependent solutions by way of a Laplace inversion technique. Systematic numerical calculations are carried out, and the field intensity factors versus time are presented for various values of dimensionless parameters representing the crack geometry and the material non-homogeneity.     

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 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.     

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.     

THERMAL STRESS ANALYSIS IN THERMOPIEZOELASTIC STRIP WITH AN EDGE CRACK

The analysis of thermal stresses becomes important when the piezoelectric material has to be operated in either extremely cold or hot temperature environments. Hence, it is essential to know the interaction of mechanical defects with temperature changes. This investigation is concerned with a strip problem of transversely isotropic thermopiezoelastic material containing an edge crack under partial thermal and electric loading conditions. Thermopiezoelastic stresses are analyzed by introducing potential functions and Fourier transforms. The problem reduces to solving a singular integral equation, and the singular integral equation is solved. Numerical calculations of the thermal stress intensity factors are carried out for a cadmium selenide material.     

Thermal fracture model for a functionally graded material with general thermomechanical properties and collinear cracks

In this article, a fracture mechanics model for functionally graded materials (FGMs) with general thermomechanical properties and collinear cracks under thermal loading is proposed. Assuming the thermomechanical properties of FGM strip to be general continuous functions of the coordinate in the thickness direction, the FGM strip is divided into a multilayered medium with the thermomechanical properties varying exponentially in each layer. Using the superposition method, the problem is reduced to a perturbation problem in which the crack surface tractions are the only external forces. Finally, the crack problem is reduced to integral equations with generalized Cauchy kernel and solved numerically. Some typical examples are discussed and the thermal stress intensity factors (TSIFs) for the collinear cracks are presented. The influences of the geometry parameters and the interaction between both collinear cracks on the TSIFs are discussed. Some important conclusions are drawn.     

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.     

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.     

Elasto-plastic analysis of a cracked ductile cylindrical pressure vessel

The deformation of a long cylindrical pressure vessel made of strain hardening material with a long longitudinal crack is analyzed under internal pressure by an FEM code. Stress distributions, the shape of the crack opening, the plastic zone, the J-integral and the CTOD are calculated.The $\text{J}\text{CTOD}$ ratio is found to be practically independent of the amount of pressure load, as well as of the crack height.Comparisons are made with a fully plastic strip yield model which seems to give a reasonable estimate of crack opening. This model suggests that the CTOD is linearly dependent on the radius of cylinder. This implies that the size of the pressure vessel is an important parameter when considering the tolerance of flaws.     

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.     

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.     

A crack in a finite medium under transient non-Fourier heat conduction

This paper investigates the problem of a finite crack in a material layer under the theory of non-Fourier heat conduction. The concept of thermal flow intensity factor is introduced to show the singularity of the thermal flow at the crack tip. Dependence of the crack tip thermal flow field on the thermal flow intensity factor is established in closed-form. Time-varying crack tip thermal flow intensity factors are obtained with sufficient accuracy. In addition to the single crack problem, solution technique and numerical results for the problem of two collinear cracks are given. Effects of crack length and layer thickness on the thermal flow intensity factors are discussed in detail.     

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.     

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.     

EDGE CRACK PROBLEMS IN HOMOGENOUS AND FUNCTIONALLY GRADED MATERIAL THERMAL BARRIER COATINGS UNDER UNIFORM THERMAL LOADING

In this study the axisymmetric crack problem for thermal barrier coatings under a uniform temperature change is considered. Modes I and II stress intensity factors and the strain energy release rate are calculated for various sizes and locations of the crack. The main variables in the problem are the material inhomogeneity parameter of the functionally graded material coating, the size and the location of the crack, and the relative dimensions of the specimen. The effect of the temperature dependence of the material properties on the stress intensity factors and the strain energy release rate is also investigated. The finite element method is used to solve the problem. The material property grading is accounted for by developing special inhomogeneous elements and the stress intensity factors are calculated by using enriched crack tip elements.