MAGNETOTHERMOELASTICITY WITH THERMAL RELAXATION IN A CONDUCTING MEDIUM WITH VARIABLE ELECTRICAL AND THERMAL CONDUCTIVITY

The equations of magnetothermoelasticity with one relaxation time and with variable electrical and thermal conductivity for one-dimensional problems including heat sources are cast into matrix form using the state-space and Laplace transform techniques. The resulting formulation is applied to a problem for the whole conducting space with a plane distribution of heat sources. It also is applied to a semispace problem with a traction-free surface and plane distribution of heat sources located inside the conducting medium. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results for the temperature, displacement, and stress distributions are given and illustrated graphically for both problems. A comparison is made with the results obtained in the following cases: (i) the electrical and thermal conductivities have constant values, (ii) the absence of magnetic field, and (iii) the coupled theory in magnetothermoelasticity.     

STATE-SPACE FORMULATION TO GENERALIZED THERMOVISCOELASTICITY WITH THERMAL RELAXATION

The model of the equation of generalized thermoviscoelasticity with one relaxation time is established. The state-space formulation for these equations is introduced. The formulation is valid for problems with or without heat sources. The resulting formulation together with the Laplace transform technique is applied to a variety of problems. The solutions to a thermal shock problem and to a problem of layer media, both without heat sources, are obtained. Also, the effects of a plane distribution of heat sources on the whole and semi-space are studied. A numerical method is employed for the inversion of the Laplace transforms. Numerical results are given and illustrated graphically for each problem. Comparisons are made with the results predicted by the coupled theory.     

Thermal Fracture of an Interpenetrating Phase Composite: Effects of Local Thermal Nonequilibrium

When subjected to thermal shocks, an interpenetrating phase composite may undergo significant, long range temperature difference between the constituent phases due to the interconnected microstructural networks, which facilitate faster heat transfer in the phase of higher thermal diffusivity. This temperature differential may alter the macroscopic temperature field thereby inducing additional thermal stresses in the composite. This work presents a local thermal nonequilibrium (LTNE) thermoelasticity theory for interpenetrating phase composites. In the LTNE thermoelasticity theory, the temperatures of the constituent phases are governed by the LTNE heat conduction equations based on the continuum theory of mixtures. A weighted average of temperatures for the constituents is employed in the thermoelastic constitutive equations of the homogenized composite. The model is subsequently applied to an infinite composite strip with an edge crack subjected to a thermal shock. Asymptotic solutions of temperature, thermal stress, and thermal stress intensity factor are obtained using the Laplace transform technique. The numerical results for an interpenetrating Al₂O₃/Al composite show that the temperature and thermal stress fields of the LTNE theory deviate from those of the classical theory. More importantly, the thermal stress intensity factor is reduced by considering the LTNE effect, which indicates that interpenetrating networks enhance the thermal fracture resistance of ceramic-metal composites.     

State Space Approach for Conducting Magneto-Thermoelastic Medium with Variable Electrical and Thermal Conductivity Subjected to Ramp-Type Heating

The equations of magneto-thermoelasticity with one relaxation time with variable electrical and thermal conductivity for one-dimensional problems are cast into matrix form using the state-space and Laplace transform techniques. The resulting formulation is applied to a half-space subjected to ramp-type heating and traction free. The inversion of the Laplace transform is carried out using a numerical approach. Numerical results for the temperature, the displacement and the stress distributions are given and illustrated graphically.     

DISCONTINUITIES IN GENERALIZED THERMO-VISCOELASTICITY UNDER FOUR THEORIES

A model of the equations of generalized thermoviscoelasticity for isotropic media is given. The formulation is applied to the generalized thermoelasticity theories—Lord–Shulman, Green–Lindsay, and Chandrasekharaiah and Tzou—as well as to the dynamic coupled theory. The state-space approach is adopted for the solution of the one-dimensional problem of plane distribution of heat sources. The Laplace transform technique is used. The expansions of the stress component, the temperature increment, and the displacement, in Laplace transform domain, in power series, and the exact inversions for arbitrary time, are given. The jump discontinuities are calculated for the four theories and the kinematic conditions of compatibility are verified. Numerical results are given and illustrated graphically by employing the numerical method for the inversion of the Laplace transforms. Comparisons are made with the results predicted by the four theories.     

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 fracture of cracked cylinders associated with nonclassical heat conduction: The effect of material property

The thermoelastic problem of a transversely isotropic hollow cylinder containing a circumferential crack is investigated in the present article based on the non-Fourier heat conduction theory. The temperature and stress fields are obtained by solving the coupled partial differential equations in the Laplace domain, and corresponding thermal axial stress with minus sign is then applied to the crack surface to form a mode I crack problem. Three different kinds of crack are considered, and the singular integral equation method is adopted to solve the fracture problem. Finally, with the definition of stress intensity factor, the effect of material properties, coupling parameter, and crack geometry on the hyperbolic thermal fracture responses of a transversely isotropic hollow cylinder excited by a thermal loading is visualized.     

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.     

Influence of moving heat source on skin tissue in the context of two-temperature memory-dependent heat transport law

Abstract

Modeling and understanding heat transport and temperature variations within biological tissues and body organs are key issues in medical thermal therapeutic applications, such as hyperthermia cancer treatment. In the present analysis, the bioheat equation is studied in the context of memory responses. The heat transport equation for this problem involving the memory-dependent derivative (MDD) on a slipping interval in the context of three-phase (3P) lag model under two-temperature theory is formulated and is then used to study the thermal damage within the skin tissue during the thermal therapy. Laplace transform technique is implemented to solve the governing equations. The influences of the MDD and moving heat source velocity on the temperature of skin tissues are precisely investigated. The numerical inversion of the Laplace transform is carried out using Zakian method. The numerical outcomes of temperatures are represented graphically. Excellent predictive capability is demonstrated for identification of an appropriate procedure to select different kernel functions to reach effective heating in hyperthermia treatment. Significant effect of thermal therapy is reported due to the effect of delay time and the velocity of moving heat source as well.     

Thermal stress analysis of eccentric tube receiver using concentrated solar radiation

In the parabolic trough concentrator with tube receiver system, the heat transfer fluid flowing through the tube receiver can induce high thermal stress and deflection. In this study, the eccentric tube receiver is introduced with the aim to reduce the thermal stresses of tube receiver. The ray-thermal-structural sequential coupled numerical analyses are adopted to obtain the concentrated heat flux distributions, temperature distributions and thermal stress fields of both the eccentric and concentric tube receivers. During the sequential coupled numerical analyses, the concentrated heat flux distribution on the bottom half periphery of tube receiver is obtained by Monte-Carlo ray tracing method, and the fitting function method is introduced for the calculated heat flux distribution transformation from the Monte-Carlo ray tracing model to the CFD analysis model. The temperature distributions and thermal stress fields are obtained by the CFD and FEA analyses, respectively. The effects of eccentricity and oriented angle variation on the thermal stresses of eccentric tube receiver are also investigated. It is recommended to adopt the eccentric tube receiver with optimum eccentricity and 90° oriented angle as tube receiver for the parabolic trough concentrator system to reduce the thermal stresses.     

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.     

Dynamic behaviour of one-dimensional flow multistream heat exchangers and their networks

The dynamic behaviour of one-dimensional flow (cocurrent and countercurrent) multistream heat exchangers and their networks is modelled and simulated. The problems can be classified into two types: (1) dynamic responses to arbitrary temperature transients and to sudden flow rate transients from a uniform temperature initial condition or a steady-state condition, which yield a linear mathematical model; (2) dynamic responses to disturbances in thermal flow rates, heat transfer coefficients or flow distributions, which are non-linear problems and should be solved numerically. A linearized model is developed to solve the non-linear problems with small disturbances. The linear model and the linearized model for small disturbances are solved by means of Laplace transform and numerical inverse algorithm. Introducing four matching matrices, the general solution can be applied to various types of one-dimensional flow multistream heat exchangers such as shell-and-tube heat exchangers and plate heat exchangers as well as their networks. The time delays in connecting and bypass pipes are included in the models. The software TAIHE (transient analysis in heat exchangers) is further developed to include the present general solution and is applied to the simulation of fluid temperature responses of multistream heat exchangers. Examples are given to illustrate the procedures in detail.     

Transient Thermoelastic Analysis of a Solid Cylinder Containing a Circumferential Crack Using the C–V Heat Conduction Model

This article presents the transient thermoelastic analysis in a long solid cylinder with a circumferential crack using the C–V heat conduction theory. The outer surface of the cylinder is subjected to a sudden temperature change. The Laplace transform technique is adopted to solve the one-dimensional hyperbolic heat conduction equation, and the axial thermal stress is obtained for the un-cracked cylinder in the Laplace domain. Then this axial thermal stress with a minus sign is applied to the crack surface to form a mixed boundary value problem in the cylindrical coordinate system. A singular integral equation is derived by applying the Fourier and Hankel transforms to solve the mode I crack problem. The transient thermal stress intensity factors are obtained by solving the singular integral equation numerically. The influences of thermal relaxation time, crack geometry, and Biot's number upon transient temperature distributions, axial stress fields, and stress intensity factors are analyzed.     

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.     

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.     

THREE-DIMENSIONAL TRANSIENT THERMAL STRESS ANALYSIS OF A NONHOMOGENEOUS HOLLOW CIRCULAR CYLINDER DUE TO A MOVING HEAT SOURCE IN THE AXIAL DIRECTION

In this paper, a theoretical analysis of a three-dimensional transient thermal stress problem is developed for a nonhomogeneous hollow circular cylinder due to a moving heat source in the axial direction from the inner and /or outer surfaces. Assuming that the hollow circular cylinder has nonhomogeneous thermal and mechanical material properties in the radial direction, the heat conduction problem and the associated thermoelastic behaviors for such nonhomogeneous medium are developed by introducing the theory of laminated composites as one of theoretical approximation. The transient heat conduction problem is treated with the help of the methods of Fourier cosine transformation and Laplace transformation, and the associated thermoelastic field is analyzed making use of the thermoelastic displacement potential, Michell's function, and the Boussinesq's function. Some numerical results for the temperature change and the stress distributions are shown in figures, and the effect of relaxing the thermal stress in the nonhomogeneous hollow circular cylinder and the influence of the velocity of a moving heat source are briefly discussed     

A new numerical method to simulate the non-Fourier heat conduction in a single-phase medium

Many non-equilibrium heat conduction processes can be described by the macroscopic dual-phase lag model (DPL model). In this paper, a numerical method, which combines the dual reciprocity boundary element method (DRBEM) with Laplace transforms, is constructed to solve such mathematical equation. It is used to simulate the non-Fourier phenomenon of heat conduction in a single-phase medium, then numerically predict the differences between the thermal diffusion, the thermal wave and the non-Fourier heat conduction under different boundary conditions including pulse for one- and two-dimensional problems. In order to check this numerical method's reliability, the numerical solutions are still compared with two known analytical solutions.     

Two-dimensional problem for thermoviscoelastic materials with fractional order heat transfer

Abstract

The model of equations of thermo-viscoelasticity with fractional order heat transfer is constructed. Some fundamental theorems on the linear coupled and generalized theories of thermo-viscoelasticity can be easily obtained as special cases. The medium is assumed initially quiescent. Laplace and Fourier integral transforms are utilized. The method of the matrix exponential which constitutes the basis of the state–space approach of modern control theory is applied to the system of two-dimensional equations. The resulting formulation is applied to a thermal shock half-space problem. The inversion process for Fourier and Laplace transforms is carried out using numerical method based on Fourier series expansions. Numerical results are given and illustrated graphically for the problem considered. Comparisons are made with the results predicted by the coupled theory and generalized theory. The effect of the fractional order parameter on all the considered fields is examined.     

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.     

A Problem on Elastic Half Space Under Fractional Order Theory of Thermoelasticity

The present work is concerned with the solution of a problem on fractional order theory of thermoelasticity for an elastic medium. We investigate the thermoelastic interactions inside the medium by employing the fractional order theory of thermoelasticity, recently advocated by Sherief et al. (Int. J. Solids Struct., 47, 269–275, 2010). State space approach together with the Laplace transform technique is used to obtain the general solution of the problem. The general solution is then applied to three specific problems on an elastic half space, whose boundary is subjected to (i) a thermal shock (i.e., a step input in temperature and zero stress), (ii) a normal load (i.e., a step input in stress and zero temperature change) and (iii) a ramp type increase in temperature and zero stress. To observe the variations of displacement, temperature and stress inside the half-space we compute the numerical values of the field variables for a particular material by utilizing a numerical method of Laplace inversion. The effects of fractional order parameter on the variations of different fields inside the medium are analyzed graphically.