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.     

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.     

GENERALIZED THERMOELASTICITY IN AN INFINITE SOLID WITH A HOLE

This paper deals with one-dimensional generalized thermoelasticity based on the theories of Lord and Shulman and of Green and Lindsay. A formulation of generalized thermoelasticity that combines both generalized theories is derived. The generalized thermoelastic problems for an infinite solid with a cylindrical hole and an infinite solid with a spherical hole are analyzed by means of the Laplace transform technique. Numerical calculations for temperature, displacement, and stresses under the generalized formulation are carried out and compared with those of classical dynamic coupled theory.     

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.     

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.     

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.     

Thermal stresses in chiral plates

This article is concerned with the linear theory of chiral Cosserat thermoelastic bodies. We investigate the deformation of chiral plates. First, we present the basic equations which govern the deformation of thin plates. Then, we present reciprocity and uniqueness results. In the next section, we establish the instability of solutions whenever the internal energy is negative. We use a semigroup approach to prove the existence of a solution. The deformation of an infinite plate with a circular hole is investigated.     

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.     

Closed Forms of Goursat Functions in Presence of Heat for Curvilinear Holes

In this paper, we consider a general conformal mapping function with complex constant coefficients, to solve the elasticity problems for an infinite plate weakened by a curvilinear hole. Conformal was used outside and inside of a unit circle in the presence of an initial heat flowing perpendicular to the plate. The use of the complex variable method gives convenient expressions of Goursat functions in applications, it also achieves the objective rapidly. Several previous works are considered as special cases of this work. The hole takes different shapes that make this study applicable to many cases, like tunnels, caves, excavations in soil or rock, etc. Stress and strain components have been obtained and plotted to investigate their physical meanings. With the aid of a computer, shapes of holes were received, and distribution of stresses obtained.     

Dynamic problem of fractional order thermoelasticity for a thick circular plate with finite wave speeds

This article is aimed at determining the thermoelastic displacement, stress, and temperature in a thick circular plate of finite thickness and infinite extent whose lower and upper surfaces are traction free, subjected to a given axisymmetric temperature distribution. The problem is formulated in the context of fractional order thermoelasticity theory with finite wave speeds. Integral transform technique is used to obtain the general solution in Laplace transform domain. Inversion of the Laplace transforms is done using a numerical scheme. A mathematical model is prepared for a copper material plate. Thermoelastic stresses, temperature and displacement are shown graphically and the effects of fractional-order parameters are discussed.     

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.     

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.     

Analytical solution for an infinite elastic power-law hardening plate containing an elastic circular inhomogeneity and subjected to equi-biaxial tension

This paper presents an clasto-plastic analytical solution for the plane stress inclusion problem of an elastic power-law hardening plate containing an elastic circular inhomogeneity and subjected to equi-biaxial far-field tension. Hencky's deformation theory (for compressible materials) and von Mises' yield criterion are applied, and infinitesimal deformations are assumed. The solution is derived by using a stress formulation and with the help of a modified Nadai's auxiliary-variable method and the extended Michell theorem. All expressions for the stress, strain and displacement components are derived in explicit forms in terms of an auxiliary variable and four constant parameters, which are determined from the given boundary conditions. Three specific solutions of practical interest are presented as limiting cases, one of which is the closed-form solution for the plate containing a traction-free circular hole. Numerical results are also provided to demonstrate quantitatively applications of the solution in the opening and reinforcement design of spherical pressure vessels.     

Hydrodynamical study of flow in a permeable channel: Application to flat plate dialyzer

This article investigates the problem of creeping motion of a Newtonian fluid in a permeable slit. The seepage velocity at the slit walls is assumed to obey the Darcy's law whereas the tangential velocity is taken in accordance with the Beaver's and Joseph's slip boundary condition. Exact solution of the hydrodynamical equations is obtained using the separation of variables technique. Expressions describing various quantities of interests are also derived. The influence of slip parameter is observed on the velocity and pressure field. The obtained results are applied to the physiological problem of blood flow in a flat plate dialyzer. Using the available data in the literature, theoretical values of filtration coefficient and the mean pressure drop in a flat plate dialyzer are computed and found to be in good agreement with the corresponding available experimental values.     

On plane stress state and stress free deformation of thick plate with FGM interface under thermal loading

This paper demonstrates the plane stress state and the stress free thermo-elastic deformation of FGM thick plate under thermal loading. First, the Sneddon-Lockett theorem on the plane stress state in an isotropic infinite thick plate is generalized for a case of FGM problem in which all thermo-mechanical properties are optional functions of depth co-ordinate. The proof is based on application of the Iljushin thermo-elastic potential to displacement type system of equations that reduces it to the plane stress state problem. Then an existence of the purely thermal deformation is proved in two ways: first, it is shown that the unique solution fulfils conditions of simultaneous constant temperature and linear gradation of thermal expansion coefficient, second, proof is based directly on stress type system of equations which straightforwardly reduces to compatibility equations for purely thermal deformation if only stress field is homogeneous in domain and at boundary. Finally, couple examples of application to an engineering problem are presented.     

TRANSIENT THERMOELASTIC PROBLEM FOR AN INFINITE PLATE CONTAINING A PENNY-SHAPED CRACK AT AN ARBITRARY POSITION

This article deals with the transient thermoelastic problem for an infinite plate containing a penny-shaped crack that is parallel to the surfaces of the plate but at an arbitrary position of the plate. The transient thermal stresses are set up by the heat generation on the surfaces and the sudden heat exchange on the surfaces. By using the finite difference method for the time variable, the analytical solution for spatial variables can be obtained. The numerical results for the temperature and stress intensity factor are obtained, and results are shown in graphs.     

THERMOELASTIC PROBLEM OF ARC CRACK WITH HEAT SOURCE OR DOUBLET

The problem of a circular arc-shaped crack in an infinite plate under the action of a heat source or doublet is solved by using the complex variable function method. The original problem can be considered a superposition of three particular problems. The first problem relates to a heat source applied at the origin point (z = 0) of an infinite continuous medium. In the second problem, the heat flow passes through the arc crack face. In the third problem, tractions are applied on the arc crack face. All three problems can be solved in a closed form. When heat sources are present, the behavior of the complex potentials is studied in detail. Finally, the solution for the original problem is obtained.     

STRESS-TEMPERATURE EQUATIONS OF MOTION OF IGNACZAK TYPE FOR A THREE-DIMENSIONAL PROBLEM OF MICROPOLAR THERMOELASTICITY

A stress-temperature initial boundary value problem (STIBVP) of Ignaczak type for the coupled dynamic three-dimensional micropolar thermoelasticity proposed by Eringen?Nowacki is discussed. First, a completeness of the problem is proved. Then, a closed-form singular solution, representing the displacement, rotation, stress, and temperature under the action of harmonic concentrated mass forces and moments, and heat source, for an infinite solid is obtained. Finally, conclusions resulting from the stress-temperature formulation are listed.     

Three-dimensional steady-state general solution for isotropic hygrothermoelastic media

The potential theory method was utilized to derive the steady-state general solution for three-dimensional (3D) hygrothermoelastic media. Two displacement functions are introduced to simplify the governing equations, with which the elastic, moisture, and temperature fields are thus simplified. Using the differential operator theory and superposition principle, all the physical quantities can be expressed in terms of two functions, one of which satisfies a harmonic equation and the other satisfies an eight-order partial differential equation. With the aid of generalized Almansi’s theorem, all the physical quantities like displacements, moisture, and temperature are expressed in terms of five quasi-harmonic functions for various cases of material characteristic roots. The obtained general solutions are in simple form and thus they may bring more convenience to certain boundary problems. As an example, the fundamental solutions for a point moisture source combined with heat source in the interior of infinite hygrothermoelastic body are presented by virtue of the obtained general solution. A planar crack of arbitrary shape in an infinite medium subjected to mechanical, moisture, and temperature loads is investigated to illustrate the application of the solution in boundary value problems. Specifically, for a penny-shaped crack under uniform combined loads, the complete, exact solutions are presented.     

Refined theories based on non-polynomial kinematics for the thermoelastic analysis of functionally graded plates

This article presents an analytical solution for the thermoelastic analysis of simply supported functionally graded sandwich plates using the Carrera unified formulation, which allows the automatic implementation of various structural theories. The governing equations for plates under thermal loads are obtained using the principal of virtual displacement and solved using the Navier method. Linear and nonlinear temperature fields through the thickness are taken into account. Particular attention is focused on plate theories with nonpolynomial refined kinematics. The results of the present displacement fields are compared with the classical polynomial ones, proposed by Carrera, for several orders of expansion.