GENERALIZED ONE-DIMENSIONAL THERMAL-SHOCK PROBLEM FOR SMALL TIMES

This paper is concerned with the one-dimensional problem of distribution of thermal stresses and temperature in the generalized thermoelastic semispace subjected 10 sudden heating with constant temperature on the bounding plane. The Laplace transform technique is used to solve the problem. The inverse transforms are obtained in an approximate manner as proposed by Paria |19| and Hetnarski |5|. Numerical computations for two particular cases are carried out for small values of lime, and the results are compared with the corresponding solution of the coupled problem obtained by Hetnarski |5|.     

THEORY OF GENERALIZED MICROPOLAR THERMOELASTICITY AND AN AXISYMMETRIC HALF-SPACE PROBLEM

ABSTRACT

The general equations of motion and constitutive equations are derived for a general homogeneous anisotropic medium with a microstructure, taking into account the effects of heat and allowing for second sound effects. A uniqueness theorem is also derived. As an illustration of the obtained equations, we solve a problem for a half-space whose boundary is rigidly fixed and subjected to an axisymmetric thermal shock. There are no body forces, body couples, or heat sources affecting the medium. Laplace and Hankel transform techniques are used. Numerical results are obtained and graphically illustrated.     

A PROBLEM OF GENERALIZED MAGNETOTHERMOELASTICITY FOR AN INFINITELY LONG,PERFECTLY CONDUCTING CYLINDER

This article concerns the investigation of the stress, temperature, and magnetic field in a transversely isotropic, elastic cylinder of infinite length and perfectly conducting material placed in a primary constant magnetic field when the curved surface of the cylinder is subjected to periodic loading. The analysis encompasses Lord and Shulman and Green and Lindsay theories of generalized thermoelasticity to account for the finite velocity of heat equation. The analysis of the numerical results for stress, temperature, and numerical values of the perturbed magnetic field in the free space is carried out at various points of the cylindrical medium. It is found that the effect of the applied magnetic field is an increase in the elastic wave velocity or, in other words, the increase of the solidity of the body. Furthermore, it has been shown graphically that the stress and perturbed magnetic field are modified due to the thermal relaxation time effect. In the absence of the magnetic field or relaxation times, our results reduce to those of generalized thermoelasticity and classical dynamical thermoelasticity, respectively.     

MECHANICAL AND THERMAL SOURCES IN A GENERALIZED THERMOELASTIC HALF-SPACE

The disturbance due to mechanical point loads and thermal sources acting on the boundary of a homogeneous isotropic thermoelastic half-space has been investigated upon applying the Laplace and Hankel transforms in the context of generalized theories of thermoelasticity. The integral transforms have been inverted using a numerical technique to obtain the displacements, temperature, and stresses in the physical domain. The numerical technique expresses the integrand as a Fourier series representation with respect to the Laplace transform parameter and evaluates the inverse Hankel transform integral via Romberg integration with an adaptive stepsize after using the results from successive refinements of the extended trapezoidal rule followed by extrapolating the results to the limit when the stepsize tends to zero. The results for various physical quantities are computed and presented graphically. A comparison of the results for different generalized theories of thermoelasticity are also presented.     

SHORT TIME SOLUTION FOR A PROBLEM IN MAGNETOTHERMOELASTICITY WITH THERMAL RELAXATION

In this work, we study a problem in electromagnetothermoelasticity with thermal relaxation for a half-space whose surface is subjected to a thermal shock and is laid on a rigid foundation. Laplace transform techniques are used to obtain the solution by a direct approach. Wave propagation in the elastic medium and in the free space, bounding it, is investigated. The solution of the problem is obtained analytically using asymptotic expansions valid for short times. The temperature, displacement, stress, and the induced magnetic and electric field distributions are obtained analytically. The numerical values of these functions are represented graphically.     

STATE-SPACE APPROACH TO GENERALIZED MAGNETOTHERMOELASTICITY WITH THERMAL RELAXATION IN A MEDIUM OF PERFECT CONDUCTIVITY

The model of the two-dimensional equations of generalized magnetothermoelasticity with one relaxation time in a perfectly conducting medium is established. The method of the matrix exponential [Bahar and Hetnarski, J. Thermal Stresses, vol. 1, pp. 135-145, 1978] which constitutes the basis of the state-space approach of modern theory, is applied to the nondimensional equations. Laplace and Fourier integral transforms are used. The resulting formulation is applied to a problem of a thick plate subject to heating on parts of the upper and lower surfaces of the plate that varies exponentially with time. Numerical results are given and illustrated graphically for the problem considered. A comparison was made with the results obtained in the absence of a magnetic field.     

TWO-DIMENSIONAL PROBLEM IN GENERALIZED THERMOELASTICITY WITH HEAT SOURCES

In this work, a two-dimensional problem for a half-space is solved. The problem is in the context of the theory of generalized thermoelasticity with one relaxation time. The surface of the half-space is taken to be traction free and the temperature on it is specified. Heat sources permeate the medium. Laplace and exponential Fourier transform techniques are used. The solution in the transformed domain is obtained by a direct approach. The inverse double transform is evaluated numerically. Numerical results are obtained and represented graphically.     

GENERATION OF GENERALIZED MAGNETOTHERMOELASTIC WAVES BY THERMAL SHOCK IN A PERFECTLY CONDUCTING HALF-SPACE

The problem of distribution of thermal stresses and temperature is considered in a perfectly conducting half-space, in contact with a vacuum, permeated by an initial magnetic field when the bounding plane is suddenly heated to a constant temperature. The problem is in the context of generalized magnetothermoelasticity with one relaxation time. The solution is obtained using the method of potentials. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using asymptotic expansions valid for small values of time. Numerical computations for the temperature and stress distributions are carried out and represented graphically. A comparison is made with the results obtained in the absence of a magnetic field.     

TWO-DIMENSIONAL PROBLEM OF A MOVING HEATED PUNCH IN GENERALIZED THERMOELASTICITY

The two-dimensional equations of generalized thermoelasticity are solved for the case of a heated punch moving across the surface of a semi-infinite thermoelastic half-space subject to appropriate boundary conditions. The exponential Fourier transform with respect to one space variable in a coordinate system moving with the punch is applied. The resulting equations are solved and numerical results are given. The results are compared with those obtained by Roberts for the coupled thermoelastic case.     

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.     

REFLECTION OF GENERALIZED THERMOELASTIC WAVES FROM THE BOUNDARY OF A HALF-SPACE

We investigated the problem of thermoelastic wave reflection from the insulated and isothermal stress-free as well as rigidly fixed boundaries of homogeneous isotropic solid half-spaces in the context of various linear theories of thermoelasticity, namely, Lord-Shulman, Green-Lindsay, Green-Nagdhi, coupled thermoelasticity, and uncoupled thermoelasticity. The ratios of reflection coefficients to that of incident coefficients are obtained for P- and SV-wave incidence cases. The results for partition of the energy for various values of the angle of incidence are computed numerically and presented graphically for aluminum-epoxy composite material in case of incident P- and SV-waves from the stress-free and rigidly fixed thermally insulated boundaries. The results obtained are discussed and compared in various models of thermoelasticity.     

THERMAL SHOCK IN A VISCOELASTIC HALF-SPACE

Abstract

Thermal shock due to sudden heating of the bounding surface of a viscoelastic half-space is discussed. Coupling of the temperature and strain fields has been taken into account, and the effects of thermal relaxation on the shock propagation have also been considered. The linear standard model has been used to represent the viscoelastic behavior of the solid in shear which is elastic in hulk deformation. The analysis, although valid for small times, shows that the stress and temperature fields have discontinuities at two points at a given instant of lime. It is further noted that attenuation takes place more rapidly than in the corresponding case of an elastic continuum. The stress and temperature fields are the result of superposition of two shock waves: one is mechanical in origin, the other is due to thermal relaxation. An estimate of the finite speed of the heat signal is also obtained for a specific case.     

NONLINEAR MAGNETOTHERMOELASTICITY OF ANISOTROPIC PLATES IMMERSED IN A MAGNETIC FIELD

A geometrically nonlinear theory of magnetothermoelasticity of electroconductive anisotropic plates in a magnetic field is developed. In this context, the Kirchhoff hypothesis is adopted for the plate modeling and the geometrical nonlinearities are considered in the von Kármán sense. In addition, the assumptions related to the distribution of electric and magnetic field disturbances through the plate thickness as proposed by Ambartsumyan and his collaborators are adopted. Based on the electromagnetic equations (i.e., the ones by Faraday, Ampère, Ohm, Maxwell, and Lorentz), on the modified Fourier law of heat conduction, and elastokinetic field equations, the three-dimensional coupled problem is reduced to an equivalent two-dimensional one appropriate to the theory of plates. The theory developed herein enables one to investigate the interacting effects among the magnetic, thermal, and elastic fields in orthotropic thin plates. As a special case, the problem of the free vibration of simply supported plate strips immersed in a transversal magnetic field is considered. Effects of the directionality property of the constituent material, magnetic and temperature fields, and electric conductivity, as well as thermal expansion coefficients, on the characteristics of vibrational behavior of the plate strips are investigated.     

A PROBLEM IN GENERALIZED THERMOELASTICITY FOR AN INFINITELY LONG ANNULAR CYLINDER COMPOSED OF TWO DIFFERENT MATERIALS

The problem of heat conduction of a thermoelaslic cylindrical medium composed of two different materials is considered. The problem is solved in the framework of the generalized thermoelasticity theory with one relaxation time. The solution is obtained in the Laplace transform domain by using the potential function approach. Numerical inversion formula is used to obtain the corresponding solutions in the physical domain     

EIGENVALUE APPROACH TO MAGNETOTHERMOELASTICITY

In this paper, the equations representing the one-dimensional, linearized, simultaneous, coupled problem of magnetothermoelasticity are arranged in the form of a matrix differential equation in the Laplace transform domain. The problem is then converted to an algebraic eigenvalue problem and solved in the same domain. The results obtained can be used to solve a broad class of problems in magnetothermoelasticity. Applications to problems pertaining to a half-space and a layer are presented. The results obtained for some special cases are compared to the results in the literature.     

STATE SPACE APPROACH TO MAGNETOTHERMOELASTICITY

The general problem of the one-dimensional linearized simultaneous equations of magnetothermoelasticity has been solved in the Laplace transform domain by following the state space approach of modern control theory. The results obtained can be used to generate solutions in the Laplace transform domain to a broad class of problems in magnetothermoelasticity. Application to a problem pertaining to a conducting half-space is presented, and the result is in complete agreement with the existing literature of magnetothermoelasticity.     

GENERALIZED THERMOELASTIC PROBLEM FOR AN INFINITELY LONG HOLLOW CYLINDER FOR SHORT TIMES

In this work, we consider the one-dimensional problem for an infinitely long hollow cylinder in the context of the theory of generalized thermoelasticity with one relaxation time. The Laplace transform technique is used to solve the problem. The solution in the transformed domain is obtained by a direct approach. The inverse transforms are obtained in an approximate analytical manner using asymptotic expansions valid for small values of time. The temperature, displacement, and stress are computed and represented graphically.     

STUDY OF THERMOELASTIC WAVE PROPAGATION IN A HALF-SPACE USING GN THEORY

Thermoelastic interactions in an isotropic homogeneous elastic half-space are studied using mathematical methods under the purview of the GN model of linear theory of generalized thermoelasticity. Of specific concern here is the propagation of waves owing to ramp-type heating of the bounding surface, which is either (i) traction-free or (ii) held rigidly fixed whereby the displacement is constrained. Using the integral transform approach, approximate solutions valid in the small time range are obtained for the situation when Fourier conductivity is dominant. Also exact solutions for temperature, displacement, and stress fields are obtained for the same problem for cases where conductivity is ignorable. Numerical estimates are also obtained for a particular example.     

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.     

THE COUPLED THERMOELASTICITY PROBLEM OF THE TRANSIENT MOTION OF A LINE HEAT/MECHANICAL SOURCE OVER A HALF-SPACE

Abstract

A procedure is developed for obtaining fundamental thermoelastic two-dimensional solutions for thermal and/or mechanical loadings moving unsteadily over the surface of a half-space. These solutions are within the bounds of the transient coupled thermoelastodynamic theory of M. A. Biot. The concentrated line loadings (sources) are suddenly applied on the surface of the half-space and then move in a fixed direction with nonuniform speed. The problem is of basic interest in contact mechanics and tribology, and it is especially related to the well-known heat checking problem (thermo-mechanical cracking in an unflawed half-space material from high-speed asperity excitations). Here, an exact and general formulation is considered and explicit results are given for some special cases. These results are obtained by generating asymptotic expressions from one- and two-sided Laplace transforms and then performing the inversions in an exact manner.