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.     

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.     

The transient thermo-electro-elastic fields in a piezoelectric plate with a crack

This paper is concerned with a thin plate made by a piezoelectric ceramic material and containing a crack perpendicular to its surfaces. It is assumed that the transient thermal stress is set up by the application of a heat flux as a function of the time and position along the crack edge and the heat flow by convection from the plate surfaces. The plate is also subjected to mechanical and electric loadings. The exact analytical formulae are obtained for transient thermo-electro-elastic fields in the plate. The exact analytical solutions for the stress and electric displacement intensity factors and crack-opening displacement are obtained. Numerical examples show, among others, a dependence of the stress and electric displacement intensity factors on the thermal and elastic, piezoelectric and dielectric constants of the piezoelectric materials.     

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.     

Thermal Stress Intensity Factor for an Opening Crack in Thermomagnetoelectroelastic Solids

An opening crack model is proposed by considering the effects of various physical properties inside a crack of thermomagnetoelectroelastic solids in the present article. The thermal flux, electric displacement, and magnetic induction at the crack surfaces are assumed to be dependent on crack opening displacement. Applying the Fourier transform technique, the thermal stress intensity factor for an opening Griffith crack in thermomagnetoelectroelastic solids is given in the explicit form. The effects of applied magnetoelectromechanical loadings and the physical properties of crack interior on the normalized thermal stress intensity factor are investigated. The obtained results reveal that the thermal stress intensity factor depends not only on applied thermal loadings but also on applied magnetoelectromechanical loadings and various physical properties of crack interior for a crack filled with a medium.     

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.     

Transient Thermal Fracture Analysis of Transversely Isotropic Magneto-Electro-Elastic Materials

Modern materials such as magneto-electro-elastic materials are used in the development of smart structures. The magneto-electro-elastic materials possess the dual features that the application of electric field induces magnetization and magnetic field induces electric polarization. The theory of linear magneto-electro-elasticity is applied to solve transient thermal fracture in magneto-electro-elastic cylinder under sudden heating on its outer surface. The equilibrium equations are obtained from the constitutive equations. The governing partial differential equations are deduced by using equilibrium equations of elastic, electric and magnetic fields. The heat conduction equation is solved by separation of variable technique. Hankel transform is applied to solve elastic displacements, electric potential and magnetic potential. The problem is reduced into integral equation involving Bessel functions which is treated exactly using Abel's integral equation. Transient distributions of temperature, stress, displacement and magnetic inductions are derived for magneto-electro-elastic cylinder. Thermal stress, electric displacement and magnetic induction-intensity factors are obtained. The solutions are valid for both impermeable and permeable crack models. The studies are valuable for such material analysis and design.     

The Axisymmetric Problem of a Partially Insulated Mixed-Mode Crack Embedded in a Functionally Graded Pyro Magneto-Electro-Elastic Infinite Medium Subjected to Thermal Loading

This article describes our investigation of the influence of an axisymmetric partially insulated mixed-mode crack on the coupled response of a functionally graded magneto-electro-elastic material (FGMEEM) subjected to thermal loading. The crack is embedded at the center of an infinite medium, and the material is graded in the direction orthogonal to the crack plane and is modeled as a nonhomogeneous medium with anisotropic constitutive laws. The heat conduction equation is first solved using the Hankel transform to yield the temperature field in the medium. Using the same integral transform, the magneto-electro-elasticity equations are converted analytically into a system of four singular integral equations that are solved numerically to yield the crack-tip mode I and II stress intensity factors, the electric displacement intensity factor and the magnetic induction intensity factor. The main objective of this research is to study the influence of material nonhomogeneity on the fields’ intensity factors for the purpose of gaining better understanding on the behavior of graded pyro magneto-electro-elastic materials.     

Deterministic assessment of reactor pressure vessel integrity under pressurised thermal shock

Numerical investigations were carried out to assess the integrity of reactor pressure vessels under pressurised thermal shock (PTS). The 4-loop reactor pressure vessel with cladding was subjected to thermo-mechanical loading owing to loss of coolant accident. The loss of coolant accident corresponding to small break as well as hot leg breaks were considered separately, which led to axisymmetric and asymmetric thermal loading conditions respectively. Three different crack configurations, 360° circumferential part through, circumferential semi-elliptical surface and circumferential semi-elliptical under-clad cracks, were postulated in the reactor pressure vessel. Finite element method was used as a tool for transient thermo-elastic analysis. The various fracture parameters such as crack mouth opening displacement (CMOD), stress intensity factor (SIF), nil ductility transition temperature (RT_NDT) etc. were computed for each crack configuration subjected to various type of loading conditions. Finally for each crack a fracture assessment was performed concerning crack initiation based on the fracture toughness curve. The required material RT_NDT was evaluated to avoid crack initiation.     

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 Thermoelastic Analysis of a Solid Cylinder Containing a Circumferential Crack Using the C–V Heat Conduction Model

This article presents the transient thermoelastic analysis in a long solid cylinder with a circumferential crack using the C–V heat conduction theory. The outer surface of the cylinder is subjected to a sudden temperature change. The Laplace transform technique is adopted to solve the one-dimensional hyperbolic heat conduction equation, and the axial thermal stress is obtained for the un-cracked cylinder in the Laplace domain. Then this axial thermal stress with a minus sign is applied to the crack surface to form a mixed boundary value problem in the cylindrical coordinate system. A singular integral equation is derived by applying the Fourier and Hankel transforms to solve the mode I crack problem. The transient thermal stress intensity factors are obtained by solving the singular integral equation numerically. The influences of thermal relaxation time, crack geometry, and Biot's number upon transient temperature distributions, axial stress fields, and stress intensity factors are analyzed.     

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.     

The limited electrically permeable crack model in linear piezoelasticity

The problem of a crack in a piezoelectric material under electromechanical loading is considered. The exact solution, obtained in this work, includes the unknown a priori normal component of the electric displacement inside the crack. Several different physical assumptions associated with limited electric permeability of the crack are utilized to determine those unknown electric fluxes through the crack boundaries. Analytical formulae for the stress and electric intensity factors are derived. The effects of electric boundary conditions (limited permeability) at the crack surface on the basic parameters of fracture mechanics are analyzed and some features of the solution are discussed. If the permeability of the medium inside the crack tends to zero or is very large the extreme results i.e. impermeable or permeable crack solution are obtained.     

Three-dimensional thermal weight function method for the interface crack problems in bimaterial structures under a transient thermal loading

Different from previous two-dimensional thermal weight function (TWF) method, a three-dimensional (3D) TWF method is proposed for solving elliptical interface crack problems in bimaterial structures under a transient thermal loading. The present 3D TWF method based on the Betti's reciprocal theorem is a powerful tool for dealing with the transient thermal loading due to the stress intensity factors (SIFs) of whole transient process obtained through the static finite element computation. Several representative examples demonstrate that the 3D TWF method can be used to predict the SIFs of elliptical interface crack subjected to transient thermal loading with high accuracy. Moreover, numerical results indicate that the computing efficiency can be enhanced when dealing with transient problems, especially for large amount of time instants.     

Analysis of arbitrarily shaped planar cracks in three-dimensional isotropic hygrothermoelastic media

In the present article, a planar crack of arbitrary shape embedded in three-dimensional isotropic hygrothermoelastic media is investigated. Based on the general solutions and Hankel transform technique, the fundamental solutions for unit-point and extended displacement discontinuities (EDD; including the displacement discontinuities, moisture concentration discontinuity, and the temperature discontinuity) are derived. The EDD boundary integral equations for an arbitrarily shaped, planar crack in the hygrothermoelastic medium are established in terms of the EDD. Utilizing the boundary integral equation method, the singularities of near-crack front fields are analyzed, and the stress, moisture flux, and heat flux intensity factors are all derived in terms of the EDD. As a special case, the analytical solution for a penny-shaped crack under uniform combined loadings is presented. The EDD boundary element method is proposed for numerical simulation. The numerical result for a penny-shaped crack subjected to uniform mechanical–moisture–thermal loading is compared with the analytical solution to verify the correctness of the proposed method. Two coplanar elliptical cracks subjected to combined loadings are simulated as an application, and the influences of applied loadings and the ellipticity ratio are discussed.     

An explicit integral expression for the stress intensity factor of a semi-elliptic surface crack subjected to thermal transient loading

An explicit integral expression for the stress intensity factor of a semi-elliptic surface crack in a plate subjected to thermal transient loading was developed. The stress intensity factor of a semi-elliptic surface crack in a plate, which is exposed to a step change of fluid temperature, was calculated on the basis of the weight function method. The change of the stress intensity factor for a semi-elliptic surface crack subjected to an arbitrary change of the boundary fluid temperature was obtained by Duhamel integration for the product of the step function result and the time varying fluid temperature. The result obtained by the present method has shown good agreement with those obtained by the influence function method. As a practical application, a parametric analysis was performed for the crack behavior during the emergency cool down of reactor coolant in the reactor pressure vessel. Also, the present expression can be effectively applied to the simulation of fatigue crack growth of a semi-elliptic surface crack subjected to various thermal transient loading.     

THERMAL STRESS ANALYSIS OF A RECTANGULAR PLATE AND ITS THERMAL STRESS INTENSITY FACTOR FOR COMPRESSIVE STRESS FIELD

Abstract

In this study, a transient thermal stress problem of a rectangular plate due to a nonuniform heat supply is treated theoretically and, thereafter, fracture behaviors of the plate with a crack are examined for compressive stress states. Assuming that a crack located on an arbitrary position, with an arbitrary direction, is sufficiently small and is closed because of the compressive stress field, a temperature field, in a transient state, is analyzed by taking into account the effect of relative heat transfer on both surfaces of the plate. Thereafter, the corresponding thermal stress analysis is developed on the basis of the two-dimensional plane stress problem using Airy's stress function method, and the stress intensity factor is analyzed for the biaxial stress state. As an analytical model, we consider mechanical boundary conditions of prescribed displacement and estimate the stress intensity factor of a crack tip using parameters of the crack configuration such as the location, direction, length, and coefficient of friction. These numerical results are shown in graphical form.     

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.