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.     

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.     

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.     

THERMAL STRESSES AROUND TWO PARALLEL CRACKS IN AN INFINITE ORTHOTROPIC PLATE UNDER UNIFORM HEAT FLOW

Stress intensity factors around two parallel cracks in an infinite orthotropic plate have been determined. Uniform heat flows normally to the cracks. Thermal insulation is assumed on the surfaces of the cracks. The mixed boundary value conditions are reduced to dual integral equations using the Fourier transform technique. In order to solve the equations, the temperatures and displacements at the two cracks are expanded in a series of functions that are zero outside the cracks. The unknown coefficients in each series are solved by the Schmidt method. The stress intensity factors are calculated numerically for steel and ceramic-fiber-reinforced ceramics.     

Thermal Stresses Around a Crack in a Non-Homogeneous Diffusion Layer Between a Coating Plate and an Elastic Half-Plane

ABSTRACT

This article proposes a method for determining the thermal stress field around a crack in a thin non homogeneous layer located between a ceramic plate and a metallic half-plane. For these calculations, the crack surfaces are assumed to be insulated and a uniform heat flux flows perpendicular to the crack. The material properties of the layer are assumed to vary continuously from those of the ceramic plate to those of the half-plane. The Fourier transform technique is employed to transform the problem into a set of integral equations. These equations are solved by expanding the differences in the crack surface temperature and the crack surface displacements in a series of functions that are automatically zero outside the crack. The Schmidt method is then used to determine the unknown coefficients in the series. Using this procedure, the stress intensity factors are calculated numerically for several ceramic plate thickness values.     

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.     

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.     

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.     

Three-dimensional thermal boundary layer corrections for circular heat flux gauges mounted in a flat plate with a surface temperature discontinuity

Three-dimensional Navier–Stokes computational fluid dynamics (CFD) analysis has been performed in an effort to determine thermal boundary layer correction factors for circular convective heat flux gauges (such as Schmidt–Boelter and plug type) mounted flush in a flat plate subjected to a stepwise surface temperature discontinuity. Turbulent flow solutions with temperature-dependent properties are obtained for a freestream Reynolds number of 1E6, and freestream Mach numbers of 2 and 4. The effect of gauge diameter and the plate surface temperature have been investigated. The 3D CFD results for the heat flux correction factors are compared to quasi-2D results deduced from constant property integral solutions and also 2D CFD analysis with both constant and variable properties. The role of three-dimensionality and of property variations on the heat flux correction factors has been demonstrated.     

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.     

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.     

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.     

Thermal Stress Analysis of an Embedded Crack in a Graded Orthotropic Coating-Substrate Structure

An analysis of a coupled plane thermoelastic problem for a graded orthotropic coating-substrate structure is performed under thermomechanical loading conditions. The crack direction is parallel to the free surface. Applying the superposition principle and Fourier integral transform, the heat conduction and plane elasticity equations lends themselves to the derivation of two sets of Cauchy-type singular integral equations. The thermal stress intensity factors are defined and evaluated. In the numerical results, the effects of the orthotropy parameters, thermoelastic non-homogeneity parameters, and dimensionless thermal resistance on the temperature distribution and the thermal stress intensity factors (TSIFs) are studied. The obtained results can be used to design graded orthotropic coating-substrate structures under thermomechanical loading.     

Analysis of an interface crack of arbitrary shape in a three-dimensional transversely isotropic magnetoelectrothermoelastic bimaterial—part 1: Theoretical solution

The extended displacement discontinuity boundary integral–differential equation method is developed for the analysis of an interface crack of arbitrary shape in a three-dimensional (3D), transversely isotropic magnetoelectrothermoelastic bimaterial. The extended displacement discontinuities (EDDs) include conventional displacement discontinuities, electric potential discontinuity, magnetic potential discontinuity as well as temperature discontinuity across the interface crack faces, correspondingly, while the extended stresses represent conventional mechanical stresses, electric displacement, magnetic induction, heat flux, etc. By virtue of the potential functions and Hankel transformation technique, the fundamental solutions for unit-point EDDs on the interface in a 3D transversely isotropic magnetoelectrothermoelastic bimaterial are derived, then the extended displacements and stresses are all obtained in terms of EDDs. An analysis method is proposed based on the analogy with the solution in an isotropic thermoelastic bimaterial. The singular indices and the singular behaviors of the near crack-tip fields are studied. The combined extended stress intensity factors for three new fracture modes are derived in terms of the EDDs and are compared with those in magnetoelectroelastic bimaterials.     

Dynamic analysis of cracks under thermal shock considering thermoelasticity without energy dissipation

This article reports a study of a cracked finite isotropic medium under nonclassic thermal shock based on thermoelasticity without energy dissipation. The time history of stress intensity factors as well as the temperature distribution around the crack tip is analyzed thoroughly. The fully coupled governing equations are discretized in the space by employing the extended finite-element method. The Newmark method is used as the time integration scheme to solve discretized equations. The stress intensity factors, which are extracted using the interaction integral method, are compared with other theories of thermoelasticity. The results of a cracked plate under temperature shock demonstrate that the stress intensity factors based on thermoelasticity without energy dissipation are significantly greater than those based on classic and Lord–Shulman models, whereas the peaks of stress intensity factors under heat flux shock are nearly equal for various theories of thermoelasticity. Furthermore, a mobile cold region is created along slanted crack in the temperature distribution, in which the temperature is less than the applied thermal boundary condition.     

Heat and mass diffusion fluxes on a permeable wall with foreign-gas blowing

In the present paper, we consider the variation of heat and mass diffusion fluxes on a permeable plate with blowing of a foreign gas into the boundary layer, the fluxes being considered as functions of the permeability parameter, varied through variation of blowing intensity, free-stream velocity, or longitudinal coordinate. It is shown that at a fixed distance from the leading edge of the plate one can, varying the value of blowing intensity while preserving the uniformity of the blowing over the plate length, obtain a non-monotonic variation of the wall heat and mass diffusion fluxes. In contrast to the heat and mass diffusion fluxes, the shear stress always monotonically decreases with increasing the blowing intensity. Similar to the shear stress, on increase of permeability parameter achieved through changing either the free-stream velocity or the longitudinal coordinate the heat flux and the mass diffusion flux both show a monotonic reduction. Using the integral relations of boundary-layer theory, we have derived simple analytical expressions allowing determination of the maximum values of the heat and mass diffusion fluxes in laminar and turbulent flow regimes. The obtained analytical relations were verified by performed numerical simulations.     

Thermoelastic Models of Continua

Axially symmetric thermal stresses in an elastic pipe weakened by two cylindrical cracks are provided. The surfaces of the cracks are assumed to be thermally insulated. The outer surface of the pipe is heated to maintain a constant temperature T _d , and the inner surface of the pipe is cooled to maintain a constant temperature T _b . As a first step, the boundary conditions related to the temperature field are reduced to dual integral equations using the Fourier transform technique. To satisfy the boundary conditions outside the cracks, the temperature difference at the crack surfaces is expanded into a series of functions that diminish to zero outside the cracks. The unknown coefficients in the series are determined by the Schmidt method so as to satisfy the thermal insulation inside the cracks. Next, the boundary conditions related to the stress field are reduced to dual integral equations. To solve the equations, the differences in the displacements at the crack surfaces are again expanded in a series of functions that diminish to zero outside the cracks. The Schmidt method is also used to solve the unknown coefficients in the series so as to satisfy the stress-free conditions inside the cracks. The stress intensity factors are defined and calculated numerically for several configurations of the pipe.     

Natural convection heat transfer of non-Newtonian fluids in porous media from a vertical cone under mixed thermal boundary conditions

This work studies the problem of the steady natural convection boundary layer flow over a downward-pointing vertical cone in porous media saturated with non-Newtonian power-law fluids under mixed thermal boundary conditions. A similarity analysis is performed, and the obtained similar equations are solved by cubic spline collocation method. The effects of the power-law viscosity index and the similarity exponent on the heat transfer characteristics under mixed thermal boundary conditions have been studied. Under mixed thermal boundary conditions, both the surface heat flux and the surface temperature are found to decrease when the power-law viscosity index of the non-Newtonian power-law fluid in porous media is increased. Moreover, an increase in the similarity exponent tends to increase the boundary layer thickness and thus decreases the surface heat flux under mixed thermal conditions. The generalized governing equations derived in this work can be applied to the cases of prescribed surface temperature and prescribed heat flux.     

TRANSIENT STRESS INTENSITY FACTORS FOR PERIODIC ARRAY OF CRACKS IN A HALF-PLANE DUE TO CONVECTIVE COOLING

The problem of periodic cracks perpendicular to the boundary of a half-plane under transient thermal loading is investigated. The thermal stresses are generated as a result of convective cooling on the boundary of the plane. The problem is solved using the superposition technique. The perturbation problem is formulated using the thermal stresses obtained from the uncracked problem with the opposite sign as the only external load. The formulation results in a singular integral equation of Cauchy type that is solved numerically. Numerical results are obtained for the stress intensity factors as a function of time, crack length, location of the crack, and periodic crack spacing.     

Green's function for infinite thin plate with elliptic hole under bending heat source and interaction between elliptic hole and crack under uniform bending heat flux

Green's function is derived for the bending problem of an infinite thin plate with an elliptic hole under a bending heat source. Then the interaction problem between an elliptic hole and a crack in a thin plate under uniform bending heat flux is analyzed. First, the complex variable method is developed for the thermoelastic problem of bending. Then an exact solution in explicit form is derived for the Green's function by using the complex variable method. Distributions of temperature moment, heat flux moments, bending moments along the hole edge are shown in figures. For solving the interaction problem, a solution for an infinite thin plate with an adiabatic elliptic hole under uniform bending heat flux, and two Green's functions of the plate under a bending heat source couple and a bending dislocation are given. The interaction problem then reduces into singular integral equations using the Green's functions and the principle of superposition. After the equations are solved numerically, the moment intensity factors at crack tips are presented in the figures.