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.     

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.     

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.     

Thermal Intensity Factors for a Parallel Crack in a Functionally Graded Piezoelectric Strip

In this paper, the mixed-mode thermoelectromechanical fracture problem for a functionally graded piezoelectric material (FGPM) strip is considered. 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. The problem is formulated in terms of a system of singular integral equations. The stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.     

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.     

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.     

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.     

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.     

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.     

THERMAL STRESS IN AN ORTHOTROPIC PLATE CONTAINING A PAIR OF COPLANAR CENTRAL CRACKS

Abstract

The thermal stress problem for a pair of coplanar central cracks contained in an orthotropic plate is investigated using the techniques of Fourier transforms and finite Hilbert transforms. The crack surfaces are subjected to symmetrical thermal loadings. Exact expressions for the temperature field, the crack shapes, and the thermal stresses in the crack plane are obtained for the case of constant temperature. The opening-mode stress intensity factors at the inner and outer crack tips are also obtained in closed forms in terms of the material properties and the distance between the cracks. The properties of the stress intensity factors are shown to be different from the shear-mode stress intensity factors because of the disturbance of a uniform heat flow by a pair of central cracks.     

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.     

Thermoelectromechanical Response of a Piezoelectric Strip with Two Parallel Cracks of Different Lengths

This work is concerned with the thermoelectromechanical fracture behavior of two parallel cracks of different lengths in a piezoelectric material strip under thermal loading. The crack faces are assumed to be insulated thermally and electrically. Fourier transform techniques are used to reduce the mixed boundary value problems to two systems of singular integral equations. Numerical calculations are carried out, and detailed results are presented to illustrate the influence of the geometric parameters on the thermal stress and electric displacement intensity factors.     

Two Parallel Cracks in Arbitrary Positions of a Piezoelectric Material Strip Under Thermo-Electric Loadings

In this article, thermo-electro-elastic fracture behavior of two parallel cracks in arbitrary positions of a piezoelectric material strip under thermo-electric loadings is considered. The crack faces are assumed to be insulated thermally and electrically. Fourier transform techniques are used to reduce the mixed boundary value problems to two systems of singular integral equations. Numerical calculations are carried out, and detailed results are presented to illustrate the influence of the geometric parameters on the stress and electric displacement intensity factors. The results for the temperature and electro-elastic fields are also included.     

CIRCUMFERENTIAL CRACK PROBLEM FOR AN FGM CYLINDER UNDER THERMAL STRESSES

The main objective of this study is to determine the stress intensity factors associated with a circumferential crack in a thin-walled cylinder subjected to quasi-static thermal loading. The cylinder is assumed to be a functionally graded material. In order to make the problem analytically tractable, the thin-walled cylinder is modeled as a layer on an elastic foundation whose thermal and mechanical properties are exponential functions of the thickness coordinate. Hence a plane strain crack problem is obtained. First temperature and thermal stress distributions for a crack-free layer are determined. Then using these solutions, the crack problem is reduced to a local perturbation problem where the only nonzero loads are the crack surface tractions. Both internal and edge cracks are considered. Stress intensity factors are computed as functions of crack geometry, material properties, and time.     

Further applications of variable singularity boundary elements in the study of neighbouring singularities

The application of two new variable singularity boundary elements (BEs) in the solution of interaction of neighbouring singularities is presented. The first element can model the variable order strain and traction singularities at one end of the element. The second element can simulate the variable strain and traction singularities at both ends of the element simultaneously. These elements are employed for the computation of stress intensity factors in crack–crack interaction problems. To improve the accuracy of such computations a modified crack closure integral (MCCI) method is adopted. To illustrate the usefulness of these elements in the study of neighbouring singularities, a number of examples are presented including, Z-shaped crack, multiple kinked surface crack, multiple edge cracks, interaction of a microcrack with a cross-shaped crack at a bimaterial strip.     

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.     

Analysis of Mode-III fracture problem with multiple cracks by boundary element alternating method

An efficient boundary element alternating method was developed in this present study for the analysis of a Mode-III fracture problem with single or multiple cracks in a finite sheet. Firstly, an analytical solution for a crack in an infinite domain, subjected to an arbitrary longitudinal shear loading across the crack surface, is developed herein. The solution can then be obtained through the iterative superposition of this analytical solution and the boundary element of a finite uncracked sheet. Several Mode-III fracture problems in a finite sheet, with single and multiple cracks under various boundary conditions, are discussed for confirming the validity of this work. Excellent agreement can be observed. The interaction effects among cracks and influence of the boundaries are also considered.     

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.     

Transient thermoelectroelastic response of a piezoelectric material strip with two parallel cracks in arbitrary positions

This work is concerned with the thermoelectromechanical fracture behavior of two parallel cracks in arbitrary positions of a piezoelectric material strip under thermal shock loading. 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 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. The intensity factors versus time for various geometric parameters are calculated and presented in graphical forms. The temperature, stress and electric displacement distributions in a transient state are also included.     

Transient Thermoelectromechanical Response of a Piezoelectric Strip with Two Parallel Cracks of Different Lengths

This work is concerned with the thermoelectromechanical fracture behavior of two parallel cracks of different lengths in a piezoelectric material strip under thermal shock loading. 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 two systems of singular integral equations, respectively, which are solved numerically. A numerical method is employed to obtain the time dependent solutions by way of a Laplace inversion technique. The intensity factors versus time for various geometric parameters are calculated and presented in graphical forms. Temperature change, the stress and electric displacement distributions in a transient state are also included.