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.     

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.     

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 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.     

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.     

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.     

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.     

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.     

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.     

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.     

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 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.     

Ligament yielding of a plate with semi-elliptical surface cracks under uniform tension

The ligament yielding of a plate with a semi-elliptical surface crack under tension loading is calculated using the Dugdale model. The stress intensity factor for the strip load is obtained using the weight-function method. The necessary crack opening displacement is calculated with an approach proposed by Petroski and Achenbach.     

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.     

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.     

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.     

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.     

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.