THERMAL STRESSES IN JOINED SEMI-INFINITE SOLIDS

The elastic stress field caused by an ellipsoidal inclusion with uniform dilatational eigenstrains in one of two perfectly bonded semi-infinite solids is investigated. The thermal stress in domain z induced by the temperature variation j T could be simulated effectively by pure dilatational eigenstrain. The solutions are obtained using the method of dilatation centers. The potential functions for the problem solved are the harmonic potential functions of attracting matter filling the element of volume and expressed in terms of derivatives of elliptic integrals. Numerical examples are given to show the normal and tangential stresses along the boundary of the ellipsoidal inclusion and the maximum principle stresses along the major and minor axes in the inclusion that are important to the fracture problems. The effects of the inclusion depth from the interface, the ratio of elastic moduli of joined semi-infinite solids, the shapes of the inclusion and the rotation angle of the ellipsoidal inclusion are also studied. They are of great significance in physical applications pertaining to thermal stress problems.     

HYPERBOLIC STEFAN PROBLEM WITH APPLIED SURFACE HEAT FLUX AND TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY

Abstract

The hyperbolic Stefan problem with an applied surface heat flux and temperature-dependent thermal conductivity is solved numerically for a semi-infinite slab using Mac-Cormack's predictor-corrector method. Solutions are presented for cases where the melt temperature is both below and above the instantaneous jump in surface temperature at time t = O⁺. The interface condition, surface temperature, and internal temperatures are presented for different Stefan numbers and melt temperatures, as well as thermal conductivity both increasing and decreasing with temperature. The results obtained from the hyperbolic solution are compared with those obtained from the parabolic solution.     

Solutions of Fourier’s equation appropriate for experiments using thermochromic liquid crystal

In transient heat-transfer experiments, the time to activate the thermochromic liquid crystal (TLC) can be used to evaluate h, the heat transfer coefficient. Most experimenters use the solution of Fourier’s equation for a semi-infinite substrate with a step-change in the temperature of the fluid to determine h. The ‘semi-infinite solution’ can also be used to determine T_ad, the adiabatic surface temperature, but this is an error-prone method suitable only for experiments with relatively large values of Bi, the Biot number. For Bi > 2, which covers most practical cases, more accurate results could be achieved using a composite substrate of two materials. Using TLC to determine the temperature–time history of the surface of the composite substrate, h and T_ad could be computed from the numerical solution of Fourier’s equation. Alternatively, h and T_ad could be determined analytically from a combination of the semi-infinite and steady-state solutions.     

STRESS INTENSITY FACTOR FOR TRANSIENT THERMAL STRESSES IN AN INFINITE ELASTIC BODY WITH AN EXTERNAL CRACK

Abstract

This paper deals with a transient thermal stress problem in an infinite body with an external crack. The elastic medium is cooled by time- and position-dependent temperature on the external crack. It is very difficult to obtain the analytical expression for the temperature, so the finite-difference method is used with respect to a time variable. Thus, the analytical expression for the temperature with respect to the spatial variables may be obtained. The temperature solution reduces to a dual-integral equation for spatial variables by use of the finite-difference method for a time variable. The numerical results for stress intensity factor are obtained.     

Optimization and application of sensitivity-based spatial discrete method for numerical solution of transient flow and heat transfer

Abstract

In this article, we start with the spatial dispersion difficulty encountered in the solution of physical phenomena controlled by partial differential equations. The one-dimensional diffusion phenomenon is taken as the research object, and the optimal precision that can be achieved under the theoretical grid number is obtained by programing. And a sensitivity analysis method for spatial discrete optimization with no analytical solution is proposed and evaluated. At the same time, this paper takes the non-adiabatic single-pore cavity of typical components in the research of aero engine air system network as an example, and applies the above method to the calculation of strong nonlinear fluid. We find that the optimal spatial discrete method obtained by applying the sensitivity method explored in this paper is at least seven times more efficient than the experience-based meshing method.     

Analytical technique for thermoelectroelastic field in piezoelectric bodies with D∞ symmetry

A general solution technique for thermoelectroelastic problems in bodies with D_∞ symmetry is constructed. The displacement and electric field are expressed in terms of the respective potential functions, and the thermoelectroelastic field quantities are expressed in terms of the elastic and piezoelastic potential functions, each of which satisfies the Laplace equation with respect to the appropriately transformed spatial coordinates, combined with the two thermoelastic displacement potential functions. As an application of the technique, the theoretical analysis of a semi-infinite body subjected to a temperature distribution is performed, and the significance of the thermoelectroelastic analyses is demonstrated.     

Stationary Theory of Heat-Conductivity for an Axi-Symmetrical Piece Homogeneous Space with Circular Inclusion

This paper discusses the theory of thermo-conductivity due steady-state heat flow of semi-infinite homogeneous solids containing a perfectly rigid circular inclusion located on the plane where the two half-solids are joint. The half-spaces have different thermo-physical characteristics and a steady heat flow is applied to the solids. The effects of selected spatial heat sources are also investigated. Mathematical model and general solution is presented along with selected particular cases and simulations, followed by pertinent discussions and conclusions.     

THERMOELASTIC PROPERTIES OF COMPOSITES CONTAINING ELLIPSOIDAL INHOMOGENEITIES

This article studies the thermal stresses and the effective thermoelastic properties of composites containing ellipsoidal inhomogeneities. The cluster scheme developed recently by A. Molinari and M. El Mouden in The Problem of Elastic Inclusions at Finite Concentration, Int. J. Solids Struct, vol. 33, pp. 3131 - 3150, 1996, for the case of elastic inclusions embedded in an isotropic elastic matrix, is generalized to the case of ellipsoidal thermoelastic inclusions embedded in an anisotropic thermoelastic matrix. The shape, spatial distribution, and orientation of the inhomogeneities are taken into account in our scheme. The theoretical results for a composite of S_iO₂ particles in a Kerimid matrix are in good agreement with experimental measurements.     

Assessment of different spatial/angular agglomeration multigrid schemes for the acceleration of FVM radiative heat transfer computations

ABSTRACT

The development and comparison of different parallel spatial/angular agglomeration multigrid schemes to accelerate the finite volume method, for the prediction of radiative heat transfer, are reported in this study. The proposed multigrid methodologies are based on the solution of radiative transfer equation with the full approximation scheme coupled with the full multigrid method, considering different types of sequentially coarser spatial and angular resolutions as well as different V-cycle types. The encountered numerical tests, involving highly scattering media and reflecting boundaries, reveal the superiority of the nested scheme along with the V(2,0)-cycle-type strategy, while they highlight the significant contribution of the angular extension of the multigrid technique.     

General solutions for elasticity of transversely isotropic materials with thermal and other effects: A review

Abstract

The well-known general solution for uncoupled thermoelasticity of isotropic bodies was proposed by Goodier, which has been utilized extensively since its birth in 1937. When the steady-state response is considered, the temperature field satisfies Laplace’s equation, and the corresponding elastic field can be expressed in terms of a harmonic function, giving rise to the Williams solution. However, for anisotropic bodies, there are no steady-state thermoelastic general solutions that are expressed in terms of (quasi)harmonic functions until 2000. This article presents a short review of the harmonic general solutions for uncoupled elasticity of transversely isotropic materials with thermal and other effects. These solutions are obtained by simply but forcedly combining the heat conduction equation with other governing equations (e.g., the Navier equations). We will show that the Williams solution as well as some other solutions all can be deduced as the special cases. Two application scenarios of the general solutions are also highlighted to demonstrate their elegance and versatility.     

Enhanced convergence of eigenfunction expansions in convection-diffusion with multiscale space variable coefficients

ABSTRACT

A convergence enhancement technique known as the integral balance approach is employed in combination with the Generalized Integral Transform Technique (GITT) for solving diffusion or convection-diffusion problems in physical domains with subregions of markedly different materials properties and/or spatial scales. GITT is employed in the solution of the differential eigenvalue problem with space variable coefficients, by adopting simpler auxiliary eigenproblems for the eigenfunction representation. The examples provided deal with heat conduction in heterogeneous media and forced convection in a microchannel embedded in a substrate. The convergence characteristics of the proposed novel solution are critically compared against the conventional approach through integral transforms without the integral balance enhancement, with the aid of fully converged results from the available exact solutions.     

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.     

The Second Law of Thermodynamics for a Two-Temperature Model of Heat Transport in Metal Films

ABSTRACT

The second law of thermodynamics asserts that heat will always flow “downhill”, i.e., from an object having a higher temperature to one having a lower temperature. For a parabolic rigid heat conductor with a single temperature T and a single heat-flux q this amounts to the statement that the inner product of q and ?T must be non-positive for every point x of the conductor and for every non-negative time t. For a homogeneous and isotropic body in which classical Fourier law with a heat conductivity coefficient k is postulated, the second law is satisfied if k is a positive parameter. For ultra-fast pulse-laser heating on metal films, a parabolic two-temperature model coupling an electron temperature T_e with a metal lattice temperature T_l has been proposed by several authors. For such a model, at a given point of space x and a given time t there are two different temperatures T_e and T_l as well as two different heat-fluxes q _e and q _l related to the gradients of T_e and T_l, respectively, through classical Fourier law. As a result, for a homogeneous and isotropic model the positive definiteness of the heat conductivity coefficients k_e and k_l corresponding to T_e and T_l, respectively, implies that the second law of thermodynamics is satisfied for each of the pairs (T_e, q _e) and (T_l, q _l), separately. Also, the positive definiteness of k_e and k_l, and of the corresponding heat capacities c_e and c_l as well as of a coupling factor G imply that a temperature initial-boundary value problem for the two-temperature model has unique solution. In the present paper, an alternative form of the second law of thermodynamics for the two-temperature model with k_l = 0 and q _l =  0 is obtained from which it follows that in a one-dimensional case the electron heat-flux q_e(x, t) has direction that is opposite not only to that of ?T_e(x, t)/?x but also to that of ?T_l(x, t + τ_T)/?x, where τ_T is an intrinsic small time of the model. Also, for a general two-temperature rigid heat conductor in which k_e, k_l, c_e, c_l, and G are positive, an inequality of the second law of thermodynamics type involving a pair (T_e ? T_l, q _e ?  q _l) is postulated to prove that a two-heat-flux initial-boundary value problem of the two-temperature model has a unique solution. For a one-dimensional case, the semi-infinite sectors of the plane ( q _l, q _e) over which uniqueness does not hold true are also revealed.     

APPLICATION OF THE METHOD OF INTEGRAL RELATIONS TO TRANSIENT HEAT CONDUCTION

Abstract

This paper presents a general integral method of solving the one-dimensional transient heat conduction equation in a semi-infinite region with constant or variable thermal conductivity. The method is based upon the general framework of the orthonormal method of integral relations of Fletcher and Holt, with a modified procedure for obtaining the integral relations. In addition, a technique for obtaining the necessary initial conditions for the integral relations is proposed instead of the usual practice of using similarity solutions. The main advantages of the present method over most existing integral methods are its generality and controlled accuracy, as demonstrated by the numerical results obtained herein. Comparisons with available exact solutions are generally good.     

Coupled 1-D stress and thermal waves in temperature dependent nonlinear elastic and viscoelastic media*

Abstract

A general analysis is formulated for the closed loop coupled thermal and displacement viscoelastic 1-D wave problem. The proper inclusion of the highly temperature sensitive viscoelastic material properties renders the problem nonlinear, even though the displacements and material properties are considered to obey linear relations. In the present article. the previous analysis is enlarged and reformulated by (a) the inclusion of nonlinear elastic and viscoelastic constitutive relations as formulated in Hilton, (b) the addition of thermal waves to the displacement waves, and by (c) temperature dependent material density and viscoelastic moduli and compliances. The wave problem studied here is of significant importance in modeling, material characterization, determination of instantaneous moduli, nonlinear analytical solution protocols and the nonlinear interaction of temperature, material properties, and wave motions. Analytical and numerical solution protocols are presented and evaluated.     

A characteristic ADI finite difference method for spatial-fractional convection-dominated diffusion equation

Abstract

Based on characteristic method and shifted Grünwald fractional difference method, a characteristic finite difference method is proposed for solving the one/two/three-dimension spatial-fractional convection-dominated diffusion equation. The resulting schemes are first-order accuracy in time and second-order accuracy in space. For high-dimensional problems, alternating direction implicit (ADI) schemes are further proposed. The stability and convergence properties of these schemes are discussed. Numerical experiments are carried out to support the theoretical analysis, and some comparisons with the implicit upwind finite difference scheme are presented to show the effectiveness of the proposed method.     

Identifying heat conductivity and source functions for a nonlinear convective-diffusive equation by energetic boundary functional methods

Abstract

In the article, we solve the inverse problems to recover unknown space-time dependent functions of heat conductivity and heat source for a nonlinear convective-diffusive equation, without needing of initial temperature, final time temperature, and internal temperature data. After adopting a homogenization technique, a set of spatial boundary functions are derived, which satisfy the homogeneous boundary conditions. The homogeneous boundary functions and zero element constitute a linear space, and then a new energetic functional is derived in the linear space, which preserves the time-dependent energy. The linear systems and iterative algorithms to recover the unknown parameters with energetic boundary functions as the bases are developed, which are convergent fast at each time marching step. The data required for the recovery of unknown functions are parsimonious, including the boundary data of temperatures and heat fluxes and the boundary data of unknown functions to be recovered. The accuracy and robustness of present methods are confirmed by comparing the exact solutions with the identified results, which are obtained under large noisy disturbance.     

A closed form solution for temperature rise inside solid substrate due to time exponentially varying pulse

Modeling of laser heating process minimizes the experimental cost and enables to optimize the process parameters for improved end product quality. In the present study, an analytical solution for laser conduction limited heating due to time exponentially varying pulse is presented. The governing equation of heat diffusion is solved analytically using a Laplace transformation method. The closed form solution is validated by a solution for a step input pulse intensity presented in the previous study as well as numerical predictions. Temperature rise inside the substrate material is computed for steel. It is found that the present solution reduces to previous solution once the pulse parameter (β=0) are set to zero. Temperatures obtained from the closed form solution agree well with the numerical predictions. Moreover, temperature rises rapidly in the surface vicinity due to time exponentially varying pulse. The pulse parameter (β∗/γ∗) has a significant effect on the temperature rise. In this case, low value of (β∗/γ∗) results in high temperature rise in the surface vicinity of the substrate material.     

TRANSIENT EFFECTS ON HEAT CONDUCTION IN SLIDING BODIES

ABSTRACT

The finite-element method is used to study transient heat conduction in two bodies sliding over one another with frictional heat generated at the contact interface. Temperature profiles and heat partition distributions are determined for three cases: two-dimensional conduction between two semi-infinite sliding bodies in contact over an infinite strip, three-dimensional conduction between two semi-infinite sliding bodies in contact over a square area, and two-dimensional conduction between a sliding railcar wheel and the rail. For Peclet numbers greater than 10, the two-dimensional model and three-dimensional models were found to give similar results for centerline temperatures and heat partition distributions.     

Hygrothermal effects on static stability of embedded single-layer graphene sheets based on nonlocal strain gradient elasticity theory

Abstract

This article develops a nonlocal strain gradient plate model for buckling analysis of graphene sheets under hygrothermal environments. For more accurate analysis of graphene sheets, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. Graphene sheet is modeled via a two-variable shear deformation plate theory needless of shear correction factors. Governing equations of a nonlocal strain gradient graphene sheet on elastic substrate are derived via Hamilton’s principle. Galerkin’s method is implemented to solve the governing equations for different boundary conditions. Effects of different factors such as moisture concentration rise, temperature rise, nonlocal parameter, length scale parameter, elastic foundation and geometrical parameters on buckling characteristics a graphene sheets are examined.