An axisymmetric crack in a functionally graded thermal barrier coating bonded to a homogeneous elastic substrate under transient thermal loading

Abstract

In this paper, the fracture problem of an axisymmetric crack in a functionally graded thermal barrier coating (FGTBC) bonded to a homogeneous substrate is considered. The problem is solved for the laminate that is suddenly heated from the upper surface of the FGTBC. The bottom surface of the homogeneous substrate is maintained at the initial temperature. The crack faces are supposed to be completely insulated. Material properties are assumed to be exponentially dependent on the distance from the interface. By using both the Laplace and Hankel transforms, the thermo-mechanical fracture problem is reduced to a singular integral equation and a system of singular integral equations which are solved numerically. The stress intensity factors of the crack are computed and presented as functions of the normalized time for various values of the nonhomogeneous and geometric parameters.     

A MODE-I CRACK PROBLEM FOR AN INFINITE SPACE IN GENERALIZED THERMOELASTICITY

ABSTRACT

In this work, we solve a dynamical problem of an infinite space with a finite linear crack inside the medium. The Fourier and Laplace transform techniques are used. The problem is reduced to the solution of a system of four dual integral equations. The solution of these equations is shown to be equivalent to the solution of a Fredholm integral equation of the first kind. This integral equation is solved numerically using the method of regularization. The inverse Laplace transforms are obtained numerically using a method based on Fourier expansion techniques. Numerical values for the temperature, stress, displacement, and the stress intensity factor are obtained and represented graphically.     

Thermo-Elastic Analysis of a Cracked Half-Plane Under a Thermal Shock Impact Using the Hyperbolic Heat Conduction Theory

The transient thermal stresses around a crack in a thermo-elastic half-plane are obtained under a thermal shock using the hyperbolic heat conduction theory. Fourier, Laplace transforms and singular integral equations are applied to solve the temperature and thermal stress fields consecutively. The integral equations are solved numerically and the asymptotic fields around the crack tip are obtained. Numerical results show that the hyperbolic heat conduction have significant influence on the dynamic temperature and stress field. It is suggested that to design materials and structures against fracture under thermal loading, the hyperbolic model is more appropriate than the Fourier heat conduction model.     

Three-dimensional thermoelastic problem in orthotropic medium

Abstract

The present article is concerned with a thermoelastic boundary-value problem with a time-dependent thermal shock on traction-free half-space for a homogeneous orthotropic heat-conducting solid. The governing equations of the three-dimensional problem of generalized thermoelasticity in orthotropic medium are obtained as a vector-matrix differential equation form by employing normal mode analysis to the considered equations which is then solved by the eigenfunction expansion method. The distribution of thermal stress, displacements, and temperature are presented graphically and compared with other thermoelastic models.     

GENERALIZED THERMOELASTICITY OF AN INFINITE BODY WITH A CYLINDRICAL CAVITY AND VARIABLE MATERIAL PROPERTIES

ABSTRACT

The equations of generalized thermoelasticity with one relaxation time in an isotropic elastic medium with temperature-dependent mechanical and thermal properties are established. The modulus of elasticity and the thermal conductivity are taken as linear function of temperature. A problem of an infinite body with a cylindrical cavity has been solved by using Laplace transform techniques. The interior surface of the cavity is subjected to thermal and mechanical shocks. The inverse of the Laplace transform is done numerically using a method based on Fourier expansion techniques. The temperature, the displacement, and the stress distributions are represented graphically. A comparison was made with the results obtained in the case of temperature-independent mechanical and thermal properties.     

The Influence of Graded Coatings on the Thermal Stress Intensity Factors of an Oblique Crack in a Semi-Infinite Substrate Subjected to Local Heating on the Boundary

Thermally induced singular behavior of an arbitrarily oriented crack in a homogeneous substrate overlaid with a functionally graded coating is considered, within the framework of linear plane thermoelasticity. It is assumed that the graded coating/substrate system is subjected to steady-state thermal loading applied over a finite region at the coating surface and the crack in the substrate is thermally insulated, disturbing the prescribed heat flow. Based on the method of Fourier integral transform and the coordinate transformations of basic field variables in thermoelasticity equations, formulation of the crack problem is reduced to two sets of Cauchy-type singular integral equations for temperature and thermal stresses in the coated medium. In the numerical results, the main emphasis is placed on the investigation of influences of loading, geometric, and material parameters of the coated system on the variations of mixed-mode thermal stress intensity factors. Further addressed are the probable cleavage angles for the incipient growth of the original crack and the corresponding values of effective tensile-mode stress intensity factors.     

IMPORTANCE OF INERTIA TERM IN DYNAMIC CRACK PROBLEMS CONSIDERING LORD–SHULMAN THEORY OF THERMOELASTICITY

A numerical technique is presented for the accurate calculation of stress intensity factors as a function of time for generalized coupled thermoelastic problems. In this task, the effect of the inertia term is investigated, considering different theories of thermoelasticity, and its importance is shown.

A boundary element method using the Laplace transform in time domain is developed for the analysis of fracture mechanics; dynamic coupled thermoelasticity problems with relaxation time are considered in the two-dimensional finite domain. The Laplace transform method is applied to the time domain and the resulting equations in the transformed field are discretized using the boundary element method. Actual physical quantities in the time domain are obtained using the numerical inversion of the Laplace transform method.

The singular behavior of the temperature and displacement fields in the vicinity of the crack tip is modeled by quarter-point elements. The thermal dynamic stress intensity factor for mode I is evaluated using the J-integral method. The accuracy of the method is investigated through comparison of the results with the data available in literature.

The J integral, which represents the dynamic energy release rate for propagating cracks, contains a boundary integral and a domain integral. The boundary integral contains strain energy, tractions, and strains whereas the domain integral contains inertia and strains. The J-integral method allows these two terms to be calculated separately. In this way, the importance of each term may be investigated by considering different theories of dynamic thermoelasticity.     

Memory response for thermal distributions moving over a magneto-thermoelastic rod under Eringen’s nonlocal theory

Abstract

Enlightened by the Caputo fractional derivative, this study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of magnetic field and moving heat source in a rod in the context of Lord–Shulman (LS) theory of thermoelasticity based on Eringen’s nonlocal elasticity. Both ends of the rod are fixed and heat insulated. Employing Laplace transform as a tool, the problem has been transformed into the space domain and solved analytically. Finally, solutions in the real-time domain are obtained by applying the inverse Laplace transform. Numerical calculation for stress, displacement, and temperature within the rod is carried out and displayed graphically. The effects of moving heat source speed, time instance, memory-dependent derivative, magnetic field and nonlocality on temperature, stress, and temperature are studied.     

Generalized Coupled Thermoelasticity of a Layer

The generalized thermoelasticity based on the Lord-Shulman (LS), Green-Lindsay (GL), and Green-Naghdi (GN) theories admit the second sound effect. By introducing some parameters all these theories are combined and a unified set of equations is rendered. These equations are then solved for a layer of isotropic and homogeneous material to study the thermal and mechanical wave propagations. The disturbances are generated by a sudden application of temperature to the boundary. The non-dimensionalized form of the governing equations are solved utilizing the Laplace transform method in time domain. Closed form solutions are obtained for the layer in Laplace transform domain, and a numerical inverse Laplace transform method is used to obtain the temperature, displacement, and stress fields in the physical time domain. The thermo-mechanical wave propagations and reflections from the layer boundaries are investigated.     

AN INTERNAL PENNY-SHAPED CRACK IN AN INFINITE THERMOELASTIC SOLID

In this work, we solve a dynamical problem for an infinite thermoelastic solid with an internal penny-shaped crack, which is subjected to prescribed temperature and stress distributions. The problem is solved using the Laplace and Hankel transforms. The boundary conditions of the problem give a set of four dual integral equations. The operators of fractional calculus are used to transform the dual integral equations into a Fredholm integral equation of the second kind, which is solved numerically. The inverse Hankel and Laplace transforms are obtained using a numerical technique. Numerical results for the temperature, stress, and displacement distributions, as well as for the stress intensity factor, are shown graphically.     

Waves in generalized thermo-viscoelastic infinite continuum with cylindrical cavity due to three-phase-lag time-nonlocal heat transfer

Abstract

Generalized thermoelastic interactions due to three-phase-lag time-nonlocal heat transfer in a Kelvin-Voigt type infinitely extended visco-thermoelastic continuum with cylindrical cavity has been investigated. The two-temperature generalized thermoelasticity theory has also been taken into account. The problem has been solved in the domain of Laplace on the assumption that the surface of the cavity is free from traction and is subjected to a smooth and time-dependent-heating effect. Laplace inversion of the transformed solutions has been carried out numerically. The obtained numerical data for different considerations are plotted in graphs to study the effects of time-nonlocal parameter, two-temperature parameter and visco-thermoelastic relaxation parameter on different thermoelastic quantities of physical interest.     

Fundamental solution for a line source of heat in the fractional order theory of thermoelasticity using the new Caputo definition

Abstract

The new Caputo Fabrizio fractional differential operator is used to investigate a problem in the fractional order theory of thermoelasticity. The problem concerns an infinite elastic space under the effect of a continuous line source of heat. The problem is solved using asymptotic expansions valid for short times. Laplace and Hankel transforms are used to solve the problem. A brief study to the nature of propagation of waves is introduced. Graphical results are presented and discussed.     

An Embedded Crack in a Graded Coating Bonded to a Homogeneous Substrate Under Thermo-Mechanical Loading

ABSTRACT

The problem of an embedded partially insulated crack in a graded coating bonded to a homogeneous substrate under thermal and mechanical loading is considered. The heat conduction and the plane elasticity equations are converted into singular integral equations which are solved to yield the temperature and the displacement fields in the medium as well as the crack tip stress intensity factors. A crack-closure algorithm is applied to avoid interpenetration. The main objective of the paper is to study the effect of the coating nonhomogeneity parameters, partial insulation of the crack surfaces and crack-closure on the crack tip stress intensity factors for the purpose of gaining better understanding of the thermo-mechanical behavior of graded coatings.     

EIGENVALUE APPROACH TO THERMOELASTICITY

The general problem of the one-dimensional linearized simultaneous equations of thermoelasticity has been solved in the Laplace transform domain by following the algebraic eigenvalue approach. This is an alternative which retains the original structure of the problem compared to the recent state space methodology of Bahar and Hetnarski [1, 2]. The use of thermoelastic potential is thus avoided. The results obtained can be used to a broad class of problems in thermoelasticity or other coupled fields. Applications to problems pertaining to a half space and a layer are presented.     

Hybrid Numerical Method Applied to Multilayered Hollow Cylinder with Time-Dependent Boundary Conditions

ABSTRACT

This article deals with one-dimensional axisymmetric quasi-static coupled thermoelastic problems with time-dependent boundary conditions. Laplace transform and finite difference methods are used to analyze the problems. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in transform domain. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. We obtain solutions for the temperature and thermal deformation distributions for a transient and steady state. It is demonstrated that the computational procedures established in this article are capable of solving the generalized thermoelasticity problem of a hollow cylinder with nonhomogeneous layers.     

ON THE AXISYMMETRIC PROBLEMS OF GENERALIZED ANISOTROPIC THERMOELASTICITY

In this article a two-dimensional axisymmetric problem in a homogeneous transversely isotropic medium has been studied by employing the eigenvalue approach after applying the technique of Laplace and Hankel transforms. An example of infinite space with concentrated force at the origin has been presented to illustrate the application of the approach. The results for coupled thermoelasticity and in the case of a homogeneous isotropic medium have also been deduced. The results obtained can be used for a broad class of problems in generalized thermoelasticity. The integral transforms have been inverted by using a, numerical technique to obtain the displacements, temperature, and stresses in the physical domain. The results for these quantities are given and illustrated graphically.     

Investigation of the thermal-elastic problem in cracked semi-infinite FGM under thermal shock using hyperbolic heat conduction theory

In this paper, a thermoelastic analytical model is established for a functionally graded half-plane containing a crack under a thermal shock in the framework of hyperbolic heat conduction theory. The moduli of functionally graded materials (FGMs) are assumed to vary exponentially with the coordinates. By employing the Fourier transform and Laplace transform, coupled with singular integral equations, the governing partial differential equations under mixed, thermo-mechanical boundary conditions are solved numerically. For both the temperature distribution and transient stress intensity factors (SIFs) in FGMs, the results of hyperbolic heat conduction model are significantly different than those of Fourier’s Law, which should be considered carefully in designing FGMs.     

Thermoelastic Analysis of an Annulus Using the Green-Naghdi Model

ABSTRACT

The linear theory of thermoelasticity of Green-Naghdi (GN) types II and III for homogeneous and isotropic materials are employed to study the thermal and mechanical waves in an annulus domain. The disturbances are generated by sudden application of temperature to the boundary. The nondimensional form of the governing equations are solved utilizing the Laplace transform method. Locally transversal linearization (LTL) technique, and a numerical inverse Laplace transform method are used to obtain the temperature, displacement, and stress fields in the physical time domain. The thermomechanical wave propagation and reflection from the boundary are investigated and the influence of the damping parameter on temperature, displacement, and stress fields in the Green-Naghdi type III is discussed.     

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.     

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.