GENERALIZED COUPLED THERMOELASTICITY OF DISKS BASED ON THE LORD–SHULMAN MODEL

A one-dimensional generalized thermoelasticity model of a disk based on the Lord–Shulman theory is presented. The dynamic thermoelastic response of the disk under axisymmetric thermal shock loading is studied. The effects of the relaxation time and coupling coefficient are studied. The Laplace transform method is used to transform the coupled governing equations into the space domain, where the Galerkin finite element method is employed to solve the resulting equations in the transformed domain. The dimensionless temperature, displacement, and stresses in the transformed domain are inverted to obtain the actual physical quantities using the numerical inversion of the Laplace transform method.     

Thermally nonlinear generalized thermoelasticity of a layer

This article addresses a clarification study on the thermally nonlinear Fourier/ non-Fourier dynamic coupled (generalized) thermoelasticity. Based on the Maxwell-Cattaneo’s heat conduction law and the infinitesimal theory of thermoelasticity, governing equations for the thermally nonlinear small deformation type of generalized thermoelasticity are derived. The Bubnov–Galerkin scheme is implemented for spatial discretization. The spatially discretized equations are directly discretized in time domain using the fully damped Newmark method. The Newton–Raphson procedure is used to linearize the finite element equations. The layers are exposed to a thermal shock, so that the displacement, temperature, and stress waves propagate in layers. The effects of the time evolution, thermoelastic coupling, and thermal relaxation time on the response of the layers are investigated. Results reveal the significance of the thermally nonlinear analysis of generalized thermoelasticity for the conditions where large temperature elevations exist.     

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.     

Thermal-shock problem for a hollow cylinder via a multi-dual-phase-lag theory

Abstract

An exact solution of a thermal shock for a circular cylinder is presented. A refined multi-dual-phase-lag generalized thermoelasticity model is proposed. The application of initial conditions without using Laplace transform is effected. The exact solutions of main physical fields are obtained analytically in the radial direction using the normal mode technique. For the case where the mechanical and thermal loads are applied on the inner and outer surfaces of the cylinder. Numerical results for the distributions of radial displacement, temperature, radial, hoop, and axial stresses are illustrated graphically. Extensive results are tabulated to show the accuracy of the present model. The results will also be used as benchmarks for forthcoming comparisons with other investigations. The results indicate that the effects of internal and external pressures and time are very pronounced.     

Transient responses of generalized magnetothermoelasto-diffusive problems with rotation using Laplace transform–finite element method

This article is mainly devoted to the transient response analysis of generalized magnetothermoelasto-diffusive problems with rotation in the context of the generalized thermoelastic diffusion theory. Due to the complexity of governing equations, Laplace transform–finite element method is applied to solve them. As a numerical example, a thermally and electrically conducting rotating half-space whose surface is subjected to a zonal time-dependent thermal and chemical shock is investigated. The transient responses, i.e., dimensionless temperature, chemical potential, displacement as well as stresses, are obtained and illustrated graphically. The parametric studies are performed to evaluate the rotation effect on the transient magnetothermoelasto-diffusive responses.     

Generalized coupled transient response of 3-D multilayered hollow cylinder

This paper deals three-dimensional axisymmetric quasi-static coupled thermoelastic problems. Laplace transform and finite difference methods are used to analyze 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 in a transient and steady state. Moreover, the computational procedures established in this thesis, can solve the generalized thermoelasticity problem for different length hollow cylinder with nonhomogeneous materials.     

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.     

Analysis of Thermoelastic Waves in Functionally Graded Hollow Spheres Based on the Green-Lindsay Theory

In this article, the Green–Lindsay theory of thermoelasticity is employed to study the thermoelastic response of functionally graded hollow spheres. This generalized coupled thermoelasticity theory admits the second sound phenomena and depicts a finite speed for temperature wave propagation. The materials of the hollow sphere are assumed to be graded through its thickness in the radial direction while a symmetric thermal shock load is applied to its boundary. The Galerkin finite element method via the Laplace transformation is used to solve the coupled form of governing equations. A numerical inversion of the Laplace transform is employed to obtain the results in time domain. Using the obtained solution, the temperature, displacement, radial stress, and hoop stress waves propagation are studied. Also the material distribution effects on temperature, displacement and stresses are investigated. Finally, the obtained results for the Green–Lindsay theory are compared with the results of classical thermoelasticity theory.     

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.     

Two-dimensional thermal shock problem of generalized thermoelastic multilayered structures considering interfacial conditions

A two-dimensional model of the generalized thermoelasticity with one relaxation time is established. The resulting nondimensional coupled equations together with the Laplace and Fourier transform techniques are applied to a specific problem of multilayered structures considering thermal resistance subjected to thermal shock and traction-free surface. The solutions in the transformed domain are obtained by a direct approach. Numerical inversion techniques are used to obtain the inverse double transform. Numerical results are represented graphically to estimate the effects of the thermal resistance and thermal conductivities on the temperature, displacement, and stress distributions.     

Analytical solution of generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads

In this article, a fully analytical solution of the generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads, based on Lord–Shulman model, is presented. The general forms of axisymmetric thermal and mechanical boundary conditions as arbitrary time-dependent heat transfer and traction, respectively, are considered at the inner and outer radii of the disk. The governing equations are solved analytically using the principle of superposition and the Fourier–Bessel transform. The general closed form solutions are presented for temperature and displacement fields. To validate the solutions, the results of this study are compared with the numerical results available in the literature, which show good agreement. For the temperature, displacement and stresses, radial distributions, and time histories are plotted and discussed. The propagation of thermoelastic waves and their reflection from the boundary of the disk are clearly shown. Moreover, effects of relaxation time and angular velocity on temperature, displacement, and stress fields are investigated.     

The thermal and mechanical behaviour of structural steel piping systems

The temperature, the deformation and the stress field in thermo-mechanical problems play a very important role in engineering applications. This paper presents a finite element algorithm developed to perform the thermal and mechanical analysis of structural steel piping systems subjected to elevated temperatures. The new pipe element with 22 degrees of freedom has a displacement field that results from the superposition of a beam displacement, with the displacement field associated with the section distortion. Having determined the temperature field, the consequent thermal displacement produced in the piping systems due to the thermal variation can be calculated. The temperature rise produces thermal expansion and a consequent increase of pipe length in the structural elements. For small values of the ratio of the pipe thickness to mean radius, the thermal behaviour can be calculated with adequate precision using a one-dimensional mesh approach, with thermal boundary conditions of an axisymmetric type across the pipe section. With this condition, several case studies of piping systems subjected to elevated temperatures and mechanical loads are presented and compared with corresponding results from commercial finite element codes. The main advantage of this formulation is associated with reduced time for mesh generation with a low number of elements and nodes. Considerable computational effort may be saved with the use of this finite pipe element.     

An investigation on strain and temperature rate-dependent thermoelasticity and its infinite speed behavior

Abstract

The present work is aimed at a mathematical analysis of the newly proposed strain and temperature rate-dependent thermoelasticity theory, also called a modified Green–Lindsay model (MGL) theory, given by Yu et al. (2018). This model is also an attempt to remove the discontinuity in the displacement field observed under temperature rate-dependent thermoelasticity theory proposed by Green and Lindsay. We study thermoelastic interactions in an infinite homogeneous, isotropic elastic medium with a cylindrical cavity based on this model when the surface of the cavity is subjected to thermal shock. The solutions for the distribution of displacement, temperature, and stress components are obtained by using the Laplace transform technique. The inversion of the Laplace transform is carried out by short-time approximation. A detailed comparison of the analytical results predicted by the MGL model with the corresponding predictions by the Lord–Shulman model and the Green–Lindsay model is performed. It is observed that strain rate terms in the constitutive equation avoid the prediction of discontinuity in the displacement field and other significant effects are noted. However, the new theory predicts the infinite speed of disturbance like the classical theory. Variations of field variables at different time are graphically displayed for different models and compared by using a numerical method.     

THERMOELASTIC INTERACTIONS WITHOUT ENERGY DISSIPATION IN AN UNBOUNDED MEDIUM WITH A SPHERICAL CAVITY DUE TO A THERMAL SHOCK AT THE BOUNDARY

Thermoelastic interactions without energy dissipation in an unbounded elastic medium with a spherical cavity have been investigated. The cavity surface is assumed to be stress free and is subjected to a thermal shock. The solutions for displacement, temperature, and stresses are obtained using the Laplace transform procedure. The discontinuities of the distributions of the physical quantities are determined and compared with earlier findings. The inversions are also carried out with a numerical method based on Fourier series expansions of functions. The results are compared with the corresponding results obtained in cases of conventional thermoelasticity theory and the generalized theories of thermoelasticity with thermal relaxation time parameters.     

Development of new analytical solutions for elastic thermal stress components in a hollow cylinder under sinusoidal transient thermal loading

For the analysis of high-cycle thermal fatigue due to striping (such as has been observed due to turbulence at mixing tees of class 1–2–3 piping of nuclear power reactors) it can be necessary to consider the time-dependent temperature gradient within the pipe wall thickness rather than just at the surface. To address this, a set of analytical solutions with several new features has been developed for the temperature field and the associated elastic thermal stress distributions for a hollow circular cylinder subjected to sinusoidal transient thermal loading at the inner surface. The approach uses a finite Hankel transform and some properties of Bessel functions. The analytical predictions have been successfully benchmarked by comparison with results from finite element analysis, and also with some results of independent studies.     

THERMOELASTIC TRANSIENT RESPONSE OF MULTILAYERED HOLLOW CYLINDER WITH INITIAL INTERFACE PRESSURE

This article deals with the transient response of one-dimensional axisymmetric quasi-static coupled thermoelastic problems with initial interface pressure. The initial interface pressure in a multilayered cylinder caused by the heat-assembling method is considered as an initial condition for the thermoelastic equilibrium problem. The Laplace transform and finite difference methods are used to analyze problems. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in the transform domain. The solution is obtained using the matrix similarity transformation and inverse Laplace transform. We obtained solutions for the temperature and thermal stress distributions in a transient state. Moreover, the computational procedures established in this article can solve the generalized thermoelasticity problem for a multilayered hollow cylinder.     

Analysis of Thermoelastic Crack Problems Using Green–Lindsay Theory

A boundary element method using the Laplace transform in time domain is presented for the analysis of fracture mechanics under thermal shock using the Green and Lindsay (GL) theory of thermoelasticity. The dynamic thermoelastic model of Green and Lindsay is selected to show the effect of thermal wave propagation at finite speed on crack tip stress intensity factor evaluation. The singular behavior of the temperature and displacement fields in the vicinity of the crack tip is modeled by the quarter-point elements. Thermal dynamic stress intensity factor for mode I is evaluated from computed nodal values, using the well-known displacement and traction formulas. The accuracy of the method is investigated through comparison of the results with the available data in literature. Condition where the inertia term plays important role is discussed and variations of dynamic stress intensity factor is investigated. Different relaxation times are chosen to briefly show their effects on stress intensity factor in the Green and Lindsay theory.     

Transient stress analysis of sandwich plate and shell panels with functionally graded material core under thermal shock

A finite element formulation for stress analysis of functionally graded material (FGM) sandwich plates and shell panels under thermal shock is presented in this work. A higher-order layerwise theory in conjunction with Sanders’ approximation for shells is used to develop the finite element formulation for transient stress analysis of FGM sandwich panels. The top and the bottom surfaces of FGM sandwich panels are made of pure ceramic and metal, respectively, and core of the sandwich is assumed to be made of FGM. The temperature profile in the thickness direction of the panels is considered to be varying as per the Fourier’s law of heat conduction equation for unsteady state. The heat conduction equations are solved using the central difference method in conjunction with the Crank–Nicolson approach. Transient thermal displacements of the sandwich panels are obtained using Newmark average acceleration method and the transient thermal stresses are obtained using stress–strain relations, subsequently. Results obtained from the present layerwise finite element formulations are first validated with available solutions in literature. Parametric studies are taken up to study the effects of volume fraction index, temperature dependency of material properties, core thickness, panel configuration, geometric and thermal boundary conditions on transient thermal stresses of FGM sandwich plates and shells.     

Two temperature fractional order thermoelasticity theory in a spherical domain

This article is an application of fractional thermoelasticity in association with two-temperature theory. The fractional heat conduction model has been proposed to investigate the thermal variations within the bounded spherical region. The corresponding heat conduction equation has been derived in the context of the generalized two-temperature theory of fractional thermoelasticity. The analytical solutions of thermal variations have been obtained in the Laplace domain, which are inverted using the Gaver–Stehfest algorithm in the time domain. Kuznetsov convergence criterion has been discussed for the bounded variations and stability of the problem. The delay time translations used in the heat flux vector and the temperature gradient result in the finite speed of thermal wave propagation. As a special case of time fractional derivative, the classical and generalized thermoelasticity theories have been recovered.