THERMALLY INDUCED VIBRATIONS IN AN ELASTIC BODY WITH A SPHERICAL CAVITY

An elastic heat-conducting body with a cavity can be set in vibration if the temperature of the cavity varies cyclically with time. A simple closed solution can be derived if the geometry is spherical and if the temperature oscillates harmonically. The amplitude of the stresses increases with increasing inertia. If the velocities of temperature waves and elastic waves coincide, no resonancelike amplification of the stresses occurs.     

ON THE EXISTENCE OF GENERALIZED SOLUTIONS FOR THERMALLY INDUCED VIBRATIONS IN A NONHOMOGENEOUS AND NONLINEAR PLATE

The existence of generalized solutions of a nonlinear problem concerning thermally induced vibrations in a nonhomogeneous plate is considered. The boundary conditions are for convenience, assumed to be homogeneous. This corresponds to the physical situation of a clamped plate. The initial conditions are imposed on one function only. The method of dealing with the problem is analogous to that for a homogeneous plate without the influence of temperature. Some lemmas and a theorem are formulated; the theorem asserts that under certain conditions on the temperature there exists at least one generalized solution to the problem.     

EFFECTS OF THERMAL RELAXATIONS ON THERMOVISCOELASTIC INTERACTIONS IN AN UNBOUNDED BODY WITH A SPHERICAL CAVITY SUBJECTED TO A PERIODIC LOADING ON THE BOUNDARY

Thermoviscoelastic interactions in an infinite homogeneous viscoelastic medium with a spherical cavity are studied. The cavity surface is subjected to a periodic loading and zero temperature change. The classical dynamical theory of thermoelasticity as well as the generalized theories of thermoelasticity are applied to consider the thermoelastic coupling. The analytical expressions for the closed-form solutions of displacement, temperature, and stresses are obtained; and the thermal relaxation effects on the interactions are studied to compare the three theories. The numerical values of the physical quantities are computed for a suitable material. The results are presented graphically to illustrate the problem.     

THERMALLY INDUCED VIBRATIONS OF COMPOSITE BEAMS WITH INTERLAYER SLIP

Thermally excited vibrations of composite viscoelastic two-layer beams with interfacial slip are analyzed. Geometrically linearized conditions are considered, and the Bernoulli-Euler hypothesis is applied to each layer. At the interface a linear viscoelastic slip law is assigned. The resulting sixth-order initial boundary value problem of the deflection is solved in the time domain by separating the dynamic response in a quasistatic and a complementary dynamic portions. The quasistatic solution is determined in closed form, and the remaining complementary dynamic part is approximated by a truncated modal series that exhibits accelerated convergence. Numerical results are obtained for single-span composite beams with interlayer slip by means of a time-stepping procedure based on the linear interpolation of the driving terms within the time intervals.     

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.     

THERMAL STRESSES FOR AN INFINITE BODY WITH A SPHERICAL CAVITY WITH TWO RELAXATION TIMES

Thermoelastic interactions caused in a homogeneous and isotropic infinite body with a spherical cavity are considered for the two different theories of generalized thermoelasticity, that is, Lord, and Shulman's theory and Green and Lindsay's theory. Analytical expressions for the temperature, displacement, and thermal stress fields are obtained; and the results are compared with the classical dynamical coupled theory.     

RELAXATION EFFECTS ON GENERALIZED THERMOELASTIC WAVES

Green and Lindsay's theory ofthermoelasticity is used to study the effects of the two relaxation times t₁ and t₂ on thermoelastic waves. A perturbation technique is used to obtain asymptotic solutions. Also, complete results obtained by direct numerical solution of the frequency equation for various ratios(t₁/t₂) are presented. The wave speeds and attenuation coefficient results obtained can be applied directly to the numerical solution of complete dynamic boundary value problems in generalized thermoelasticity.     

A SHORT TIME SOLUTION FOR A PROBLEM IN THERMOELASTICITY OF AN INFINITE MEDIUM WITH A SPHERICAL CAVITY

The problem of a thermoelastic infinite medium with a spherical cavity is considered within the context of the theory of thermoelasticity with two relaxation times. The surface of the cavity is stress free and suddenly subjected to a time-dependent thermal shock. Laplace transform techniques are used. The inverse Laplace transforms are obtained analytically using asymptotic expansions valid for small values of time. Numerical computations for the temperature, the displacement, and stress distribution are carried out and represented graphically.     

THERMALLY INDUCED LOCAL EFFECTS IN LAMINATED COMPOSITES

The steady-state thermally induced local effects in laminated composite materials are investigated from the micro- and macromechanical points of view. The micro-mechanical model is represented by the alternating layers of a perfectly bonded matrix-reinforcement system while the macromechanical counterpart is characterized by the effective thermal and mechanical properties of the constituent layers. The solutions are obtained within the framework of linear plane thermoelasticity via the Fourier transform technique and the flexibility/stiffness matrix method. As a numerical illustration, the micro- and macromechanical temperature and thermal stress fields due to given surface temperature distributions are presented to measure the degree of correspondence between the two models. Considerable discrepancies are observed to exist between the micro- and macromechanical thermal stress fields, especially, in regions involving high temperature gradients.     

VIBRATIONS IN A THERMALLY STRESSED THIN PLATE

Eigenfrequencies of originally flat thin plates are considered when in-plane thermal stresses are applied in the absence of lateral pressure. First pre-buckling and later postbuckling vibrations are envisaged. A method of superposition of partial solutions is used throughout.     

SHEAR DEFORMATION EFFECTS ON THERMALLY INDUCED INSTABILITY OF LAMINATED PLATES

A theoretical investigation of dynamics for linear elastic moderately thick structures due to uniform space- and time-dependent temperature fields is presented. The structures aredescribed by partial differential equations including a transverse inertia term for a general deformation state with interlaminar shear strains. A viscous model of external damping with a constant proportionality coefficient is assumed to describe a dissipation of the structural energy in the transverse motion. The particular problem of dynamic cylindrical bending due to the temperature with Gaussian and harmonic distributions is analyzed in detail. Dynamic thermal buckling of both symmetric alternating cross-ply composite plates and antisymmetric angle-ply composite laminates is investigated. The influence of neglecting the shear effects on dynamic stability regions is shown.     

EXISTENCE OF A GENERALIZED SOLUTION IN THERMOELASTICITY WITH ONE RELAXATION TIME

The existence of a generalized solution to a natural stress-flux initial boundary-value problem of thermoelasticity with one relaxation time is established. It is shown that the solution is the limit of a sequence of classical solutions belonging to a functional space of the Sobolev type.     

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.     

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.     

THERMALLY INDUCED STRESSES IN A TRILAYERED SYSTEM

The objective of this investigation is to conduct a parametric study of thermally induced stresses in trilayered media. In particular, the resulting shear and normal stresses along the media interfaces is analyzed. Because this investigation is concerned only with the failure at the bonding surfaces, the results of bending stresses, although calculated, are not presented. A finite element model utilizing a recently developed element that provides for both axial and lateral displacement continuity is employed. First, the effect of the material property, that is, coefficient of thermal expansion, is examined. Then the effect of geometric configuration, that is, the length and thickness dimensions of the midlayer, is analyzed. Finally, a study of the effect of a nonuniform temperature field on the trimaterial medium is conducted.     

ELECTROMAGNETO-THERMOELASTIC PLANE WAVES WITH THERMAL RELAXATION IN A MEDIUM OF PERFECT CONDUCTIVITY

The model of the two-dimensional equations of generalized magneto-thermoelasticity with one relaxation time in a perfectly conducting medium is established. The normal mode analysis is used to obtain the exact expressions for the temperature distribution, thermal stresses, and the displacement components. The resulting formulation is applied to three different concrete problems. The first deals with a thick plate of perfect conductivity subjected to a time-dependent heat source on each face; the second concerns the case of a heated punch moving across the surface of a semi-infinite thermoelastic half-space of perfect conductivity subject to appropriate boundary conditions; and the third problem deals with a plate with thermo-isolated surfaces subjected to time-dependent compression. Numerical results are given and illustrated graphically for each problem. Comparisons are made with the results predicted by the coupled theory and with the theory of generalized thermoelasticity with one relaxation time.     

BUOYANT CONVECTION IN A SQUARE CAVITY PARTIALLY FILLED WITH A HEAT-GENERATING POROUS MEDIUM

Steady-state buoyant convection in a rectangular cavity, partially filled with a fluid-saturated porous medium with spatially uniform internal heat generation, is considered. The Brinkman-extended Darcy model in the porous region is adopted. The overall Rayleigh number is large to render a boundary-layer-type global flow pattern. Scale analysis is performed to obtain a rudimentary understanding of the flow characteristics. In parallel with the theoretical endeavors, numerical solutions are secured over broad ranges of nondimensional parameters. The results indicate that there exists an asymptotic convection regime where the flow is nearly independent of the permeability and conductivity of the porous medium. The effect of the thermal conductivity of porous material is appreciable in the intermediate regime. In the conduction-dominant regime, the porous region acts like a heat-generating solid block. The numerical study gives credence to the reliability of the theoretical arguments.     

EXISTENCE OF A GENERALIZED SOLUTION IN THERMOELASTICITY WITH TWO RELAXATION TIMES: PART I

Abstract

Existence of a generalized solution to a central problem of thermoelasticity with two relaxation times described by a pair (φ, θ), where φ is a scalar thermoelastic displacement potential and θ is a temperature, is established. The solution is shown to be a limit of the fundamental sequence in a functional space of the Sobolev type.     

THERMALLY INDUCED STRESS WAVES IN FUNCTIONALLY GRADED MATERIALS WITH TEMPERATURE-DEPENDENT MATERIAL PROPERTIES

This article is concerned with the dynamic treatment of thermally induced stress waves in an infinite elastic plate subjected to impulsive electromagnetic radiation. The plate is assumed to be a functionally graded material (FGM), meaning that the material is composed of multiconstituents in ceramics and metals, the volume fractions of which distribute continuously inside the material. The mathematical problem is one of wave propagation in a typical nonhomogeneous material The radiation absorption is assumed to occur at a constant rate for the duration of the pulse and to diminish exponentially with distance from the surface of the plate, assuming negligible heat conduction. In treating problems, the nature of the stress-wave buildup in the plate is studied for the case of a temperature-dependent solid, that is, when material properties vary with temperature. The numerical procedure employs the characteristic method based on the integration of the governing equations along the characteristics. Numerical calculations are carried out for ceramic-metal FGM plates showing the influences of the temperature-dependent material properties and the volume fractions of the phases composing the FGM on the magnitude of the dynamic thermal stresses.