GREEN'S FUNCTION FOR A THERMOMECHANICAL MIXED BOUNDARY VALUE PROBLEM OF AN INFINITE PLANE WITH AN ELLIPTIC HOLE

This article derives the Green's function for a thermomechanical mixed boundary value problem of an infinite plane with an elliptic hole under a pair of heat source and sink. To derive the Green's function in closed form, the Cauchy integral method and a basic Green's function for an external force boundary value problem with a pair of heat source and sink are employed. Illustrative numerical results for temperature, heat flux, and stress along the hole edge and stress intensity factors when the hole collapses into a crack are presented graphically.     

Effect of uniform heat flux on stress distribution around a triangular hole in anisotropic infinite plates

The presence of a hole in an anisotropic plate under uniform heat flux causes thermal stress around the hole. In this study, on the basis of two-dimensional thermoelastic theory and using Lekhnitskii’s complex variable technique, the stress analysis of an anisotropic infinite plate with a circular hole under a uniform heat flux is developed to the plate containing a triangular hole. For this purpose, an infinite plate containing a triangular hole is mapped to the outside of a unit circle using a conformal mapping function. Stress and displacement distributions around the triangular holes in an anisotropic infinite plate are investigated in thermal steady-state condition. The plate is under uniform heat flux at infinity and Neumann boundary conditions and thermal-insulated condition on the hole boundary are considered. The rotation angle of the hole, fiber angle, the angle of heat flux, bluntness, and the aspect ratio of hole size are investigated in the present study. The accuracy of the analytical results is also confirmed by finite element analysis.     

Study of the effective parameters on stress distribution around triangular hole in metallic plates subjected to uniform heat flux

On the basis of the steady-state two-dimensional theory of thermoelasticity, stress field around a triangular hole in an infinite isotropic plate is discussed. A metallic plate subjected to uniform heat flux and thermal-insulated condition along the hole boundary is assumed. The method used for this study is the expansion of Goodier and Florence's method. They used the complex variable method for stress analysis of infinite isotropic plates with an elliptical or circular hole. The rotation angle of the hole, bluntness, aspect ratio of hole size, and angle of heat flux are important parameters considered in this paper.     

Thermal stress analysis of infinite anisotropic plate with elliptical hole under uniform heat flux

Lekhnitskii’s complex variable method was developed to investigate the effect of uniform heat flux on perforated anisotropic plate with elliptical hole. The Cauchy’s integral formula was simplified by conformal mapping, and infinite area external to the hole was represented by the area outside the unit circle. In this article, Neumann boundary conditions and thermal-insulated condition along with the hole boundary were considered. Important parameters affecting stress distribution and displacement were those of rotation angle of hole, aspect ratio of hole size, and fiber angle. Results determined in this article were verified by finite element analysis.     

Analyses of Thermal Conduction and Stress Induced by Electric Current in an Infinite Thin Plate with an Elliptical Hole

Uniform electric current at infinity is applied to a thin infinite conductor with an elliptical hole disturbing the electric current, which gives rise to Joule heat, temperature increase and heat flux. Joule heat produces uniform and uneven temperature fields which in turn initiate thermal stress. These electrical current, Joule heat, temperature, heat flux and thermal stress analyses are carried out and their closed form solutions are obtained. The heat conduction problem is solved as a temperature boundary value problem. Figures of distribution of Joule heat, temperature, heat flux and stress are shown. A dislocation and a rotation terms for thermal stress analysis appear, which makes problem complex. Solutions of Joule heat, temperature, heat flux and thermal stress are nonlinear for the direction of electric current. For an infinite plate with a circular hole, stress components do not occur on the whole plate. As a special case, a crack problem is analyzed and intensities at the crack tip of each problem are investigated. Relations between melting temperature and electric current density, and between fracture toughness value and electric current density are investigated for some crack lengths for steel.     

Contact Zone Approach for an Electrically Impermeable Crack in Piezoelectric Materials Under Thermal-Mechanical Loading

A partial contact zone model is developed for the stress and electric displacement fields due to the obstruction of a uniform heat flux by an electrically impermeable crack in piezoelectric materials. Green's function method is used to reduce the problem to a set of singular integral equations that are solved in closed form. When the crack is assumed to be traction free, the crack opening displacement is found to be negative over one-half of the crack unless a sufficiently large far field tensile stress is superposed. The problem is reformulated assuming a contact zone at one crack tip. The extent of this zone, the stress and electric displacement intensity factors at each crack tip are obtained as functions of the applied mechanical stress and heat flux.     

The effect of a stiffener on a cracked plate under bending

In this study the problem of a stiffened plate containing a through-crack under uniform bending load is analyzed. The problem is formulated for a specially orthotropic material by using Reissner's plate theory. By using the Fourier integral transform technique the problem is reduced to a singular integral equation. This singular integral equation is then solved numerically by using Gȧuss-Chebyshev and Gauss-Jacobi quadrature formulas. The special case of the problem in which the crack tip terminates at the stiffener is also analyzed in order to assess the crack arrest effectiveness of the stiffener. The asymptotic stress state near the crack tip terminating at the stiffener is examined, and normalized Mode I stress intensity factors are tabulated. The results also include the effect of Poisson's ratio, stiffness constants and material orthotropy for specially orthotropic materials on the stress intensity factors.     

GREEN'S FUNCTION APPROACH TO THREE-DIMENSIONAL HEAT CONDUCTION EQUATION OF FUNCTIONALLY GRADED MATERIALS

A Green's function approach based on the laminate theory is adopted to solve the three-dimensional heat conduction equation of functionally graded materials (FGMs) with one-directionally dependent properties. An approximate solution for each layer is substituted into the governing equation to yield an eigenvalue problem. The eigenvalues and the corresponding eigenfunctions obtained by solving an eigenvalue problem for each layer constitute the Green's function solution for analyzing the three-dimensional transient temperature. The eigenvalues and the corresponding eigenfunctions are determined from the homogeneous boundary conditions at outer sides and from the continuous conditions of temperature and heat flux at the interfaces. A three-dimensional transient temperature solution with a source is formulated by the Green's function. Numerical calculations are carried out for an FGM plate, and the numerical results are shown in tables and figures.     

SOLUTION OF DISPLACEMENT BOUNDARY VALUE PROBLEM UNDER UNIFORM HEAT FLUX

The general solution of the displacement boundary value problem is obtained for an infinite plate with an arbitrary shaped hole under uniform heat flux in any direction. The complex stress functions, the dislocation method, and a rational mapping function are used and the closed solution is obtained. An infinite plate with a circular hole and a slit is analyzed under the condition of the constrained displacements. The singularity at the tip of the slit of the constrained displacement is investigated     

Hygrothermoelastic response in the bending analysis of elliptic plate due to hygrothermal loading

Abstract

This study investigates the theoretical outline to couple both classical Fourier’s and Fick’s laws to frame a new model of two-temperature hygrothermoelastic diffusion theory for a non-simple rigid material. Based on hygrothermoelasticity method, a system of linearly coupled partial differential equations for the thermal and moisture diffusion for the case of a non-simple medium is established. The transient response using the decoupled technique of a multilayered elliptic plate perpendicular to the axial axis, subjected to hygrothermal loading is considered, to derive closed-form expressions for temperature, moisture, deflection, bending moments, and hygrothermal stresses. The solutions to the governing coupled equations and its boundary conditions are solved by employing a new integral transform technique. The small deflection equation is found and utilized to preserve the intensities of bending moments and stresses, involving the Mathieu functions and its derivatives. Moreover, the elliptical region can be degenerated into a circular part by applying limitations. Numerical results of the transient response of hygrothermoelastic fields are established graphically for the better understanding the underlying elliptic structure, improved understanding of its relationship to circular profile, and better estimates of the effect of the associated hygrothermoelastic responses.     

A simplified approach for the thermoelastic large deflection in the thin clamped annular sector plate

The present article is concerned with analysis of large deflection of a heated thin annular sector plate with clamped edges under transient temperature distribution using Berger’s approximate methods. The prescribed surface temperature is at the top face of the plate whereas the bottom face is kept at zero temperature. In this study, the Laplace transform as well as the classical method have been used for the solution of heat conduction equation. The thermal moment is derived on the basis of temperature distribution, and its stresses are obtained using resultant bending moment and resultant forces per unit length. The calculations are obtained for the aluminium plate in the form of an infinite series involving Bessel functions, and the numerical results for temperature, deflection, resultant bending moments, and thermal stresses have been illustrated by graphs.     

THERMAL STRESS PROBLEM FOR MIXED HEAT CONDUCTION BOUNDARY AROUND AN ARBITRARILY SHAPED HOLE WITH CRACK UNDER UNIFORM HEAT FLUX

This article deals with a thermal stress problem for thermal conduction around an arbitrarily shaped hole with a crack under uniform heat flux. Two cases for the hole edge and the crack faces are assumed: adiabatic and isothermal conditions or vice versa (isothermal and adiabatic). A closed-form solution is obtained using conformal mapping, dislocation functions, and the complex variable method. Results of temperature, heat flux, stress, and stress intensity factor are illustrated.     

GREEN'S FUNCTION FOR THERMAL STRESS MIXED BOUNDARY VALUE PROBLEM OF AN INFINITE PLANE WITH AN ARBITRARY HOLE UNDER A POINT HEAT SOURCE

Green's function of a point heat source is derived for a mechanical mixed boundary value problem of an infinite plane with an arbitrary hole, for which zero-displacement and traction-free boundary conditions are prescribed to its boundary. As the thermal boundary condition on the hole, either an adiabatic or isothermal condition is considered. By employing the mapping technique and complex variable method, an explicit solution including a hypergeometrical function is obtained. Stress distributions are shown in illustrative examples for a square hole.     

Stress Analysis of an Infinite Plate with a Coated Elliptic Hole Under a Remote Uniform Heat Flow

A thermoelastic solution to a coated elliptic hole embedded in an infinite matrix subjected to a remote uniform heat flow is provided in this article. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, the general expressions of the temperature and stresses in the coated layer and the matrix are derived explicitly in a series form. Some numerical results are provided to investigate the effects of the material combinations and geometric configurations on the interfacial stresses. It is found that a coated layer has a strong effect on thermal stresses of the problem with an elliptic hole embedded in an infinite plate.     

SOLUTION OF AN ELLIPTICAL RIGID INCLUSION WITH DEBONDINGS IN AN INFINITE PLANE UNDER THE UNIFORM HEAT FLUX

A problem of an elliptical rigid inclusion in an infinite plane subjected to uniform heat flux in an arbitrary direction and with a number of debondings on the interface of the elliptical rigid inclusion and the elastic matrix is solved. The rotation of the inclusion under the uniform heat flux is considered. The complex variable method is used, and the closed-form solution is obtained. The stress intensity of debonding at the debonding tips is calculated and the extension of debonding is investigated. Examples of stress distributions and resultant moments on the inclusion are shown for the inclusion with two debondings.     

NATURAL CONVECTION BETWEEN PARALLEL PLATES WITH CONJUGATE CONDUCTIVE EFFECTS

This article deals with the conjugate conductive and convective heat transfer in a vertical channel with finite thickness conductive parallel plates. The investigation is carried out numerically by solving the full elliptic Navier-Stokes and energy equations with the finite volume method in a composite I-shaped domain. The results are reported as functions of the main geometrical (B/b and L/b) and thermal (K and Ra*) parameters, for a Prandtl number of 0.71 (i.e., air). The results show that the thinner the plate the more uniform the heat flux distribution along the plate. As the thickness of the plate increases, the heat flux distribution is less uniform and at the inlet corner the temperatures present much higher values than at the infinitely thin plate. Lower plate thermal conductivity implies a more uniform heat flux distribution and lower temperature increments than the infinitely thin plate. The dependence of the heat flux and temperature distributions along the solid-fluid interface is stronger than the lower the Ra* value.     

Steady-state Green's functions for thermal stresses within rectangular region under point heat source

This article presents new steady-state Green's functions for displacements and thermal stresses for plane problem within a rectangular region. These results were derived on the basis of structural formulas for thermoelastic Green's functions which are expressed in terms of Green's functions for Poisson's equation. Structural formulas are formulated in a special theorem, which is proved using the author's developed harmonic integral representation method. Green's functions for thermal stresses within rectangle are obtained in the form of a sum of elementary functions and ordinary series. In the particular cases for a half-strip and strip, ordinary series vanish and Green's functions are presented by elementary functions. These concrete results for Green's functions and respective integration formulas for thermoelastic rectangle, half-strip and strip are presented in another theorem, which is proved on the basis of derived structural formulas. New analytical expressions for thermal stresses to a particular plane problem for a thermoelastic rectangle under a boundary constant temperature gradient also are derived. Analytical solutions were presented in the form of graphics. The fast convergence of the infinite series is demonstrated on a particular thermoelastic boundary value problem (BVP). The proposed technique of constructing thermal stresses Green's functions for a rectangle could be extended to many 3D BVPs for unbounded, semibounded, and bounded parallelepipeds.     

The interaction of two parallel non-coplanar identical surface cracks under tension and bending

In this paper, the interaction between two identical, non-coplanar, semi-elliptical surface cracks is investigated. The interacting cracks are assumed to be in an infinite plate subjected to remote tension or to pure bending loads. The stress intensity factors (SIFs) for these cracks are calculated using three-dimensional linear finite element analysis. A parametric study involving the relative horizontal and vertical separation distance between the two surface cracks is carried out for a specific crack shape and crack depth to plate thickness ratios of 0.3 and 0.2, respectively. An empirical formula is derived that relates the effects of the relative positions of these cracks to their SIFs.     

Plastic collapse moment of 90° long radius elbows with internal circumferential surface crack at intrados under in-plane bending

Piping elbows under in-plane bending moment are vulnerable to cracking. The crack initiates at the surface and eventually reaches through the thickness and may lead to failure. The structural integrity assessment requires knowledge of the limit load. Limit load solutions for elbows with through-wall crack configurations are available in the open literature. But solutions for surface crack are not available. This paper presents a closed form expression for the plastic collapse moment (PCM) of 90°, long radius elbows with circumferential surface cracks at the intrados, under in-plane bending moment. The expression is derived, based on the results of non-linear (geometric and material) FE analyses covering a wide range of geometries and crack sizes. These plastic collapse moments evaluated herein will help in structural integrity assessment.     

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.