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.     

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.     

Thermoelectromechanical Interaction Among Multiparallel Cracks in a Piezoelectric Material

This paper investigates the thermoelectromechanical interaction among multi parallel cracks in a piezoelectric material under a uniform heat flow and a uniform mechanical load 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 systems of singular integral equations, respectively. The singular integral equations are solved numerically. Numerical calculations are carried out, and detailed results are presented to illustrate the influence of the thermoelectromechanical interaction on the stress and electric displacement intensity factors.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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 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.     

Modeling Multilayer Piezoelectric Actuators in a Residual Temperature Field

Electromechanical field concentrations near the electrodes in multilayer piezoelectric ceramic actuators in a residual temperature field are examined. The problem is formulated by considering a representative unit in the real multilayer actuators. A singular integral equation, in which the unknown function is the electric potential ahead of the electrode tip, is derived. The solution of the integral equation gives the electroelastic fields near the electrode tip. The thermally induced electric field ahead of the electrode tip is found to be highly singular. On the other hand, the stresses and electric displacements are singular behind the electrode tip. Possible debonding and cracking between the electrodes and the piezoelectric medium is investigated by energy density theory.     

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.     

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.     

Thermoelastic Buckling Analysis of Functionally Graded Circular Plates Integrated with Piezoelectric Layers

Thermal buckling of circular plates made of functionally graded materials with surface-bounded piezoelectric layers are studied. The material properties of the FG plates are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituent materials. The general thermoelastic nonlinear equilibrium and linear stability equations for the piezoelectric FG plate are derived using the variational formulations. Buckling temperatures are derived for solid circular plates under uniform temperature rise, nonlinear and linear temperature variation through the thickness for immovable clamped edge of boundary conditions. The effects of piezo-to-host thickness ratio, applied actuator voltage, boundary condition, and power law index of functionally graded plates on the buckling temperature of plate are investigated. The results are verified with the data in literature.     

Thermal Green's Functions for a Heat Source in a Piezoelectric Solid with a Parabolic Boundary

Using the Stroh formalism combined with the analytical continuation principle of Muskhelishvili, the Green's functions for a line heat source in a piezoelectric solid with a parabolic boundary are obtained in closed form. The obtained Green's functions not only satisfy all the given boundary conditions, but also ensure the displacement and electric potential to be single-valued. As special cases, the solutions for a piezoelectric half-plane are also presented, and they are shown to be consistent with previous works.     

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.     

Transient Piezothermoelasticity of a Two-Layered Composite Hollow Cylinder Constructed of Isotropic Elastic and Piezoelectric Layers Due to Asymmetrical Heating

This article is concerned with the theoretical treatment of transient piezothermoelastic problem involving a two-layered hollow cylinder constructed of isotropic elastic and piezoelectric layers due to asymmetrical heat supply. The transient two-dimensional temperature is analyzed by the method of Laplace transformation. By using the exact solutions for piezoelectric hollow cylinder and isotropic hollow cylinder, the theoretical analysis of transient piezothermoelasticity is developed for a two-layered composite hollow cylinder under the state of plane strain. As an example, numerical calculations are carried out for an isotropic elastic hollow cylinder made of steel, bonded to a piezoelectric layer of cadmium selenide. Some numerical results for the temperature change, the stress and the electric potential distributions in a transient state are shown in figures. Furthermore, the influence of thickness of the piezoelectric layer or the isotropic elastic layer upon the temperature change, stresses and electric potential is investigated.