Survey of recent developments in the fields of heat conduction in solids and thermo-elasticity

Boley, Bruno A.

doi:10.1016/0029-5493(72)90109-4

Cited by 50 publications

(11 citation statements)

References 244 publications

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Section: Introductionmentioning

confidence: 99%

“…Various mathematical methods and techniques have been used to study free boundary problems. They have been discussed and summarized in several books [1][2][3][4][5] and many survey papers [6][7][8][9].Free boundary problems have been studied since the nineteenth century by Lame and Clapeyron in 1831, Neumann in the 1860s and Stefan in 1889. However, the only known exact solutions are those of Neumann and some of their extensions and variations.…”

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The Stefan problem with arbitrary initial and boundary conditions

Liu¹

1978

Quart. Appl. Math.

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Abstract.The paper is concerned with the free boundary problem of a semi-infinite body with an arbitrarily prescribed initial condition and an arbitrarily prescribed boundary condition at its face. An analytically exact solution of the problem is established, which is expressed in terms of some functions and polynomials of the similarity variable x/tin and time t. Convergence of the series solution is considered and proved. Hence the solution also serves as an existence proof. Some special initial and boundary conditions are discussed, which include the Neumann problem and the one-phase problem as special cases. Introduction.Diffusive processes with a change of phase of the materials occur in many scientific and engineering problems. They are known as Stefan problems or free boundary problems. Many examples of these problems can be given, e.g. the melting or freezing of an ice-water combination, the crystallization of a binary alloy or dissolution of a gas bubble in a liquid. To find the solutions to this class of problems has been the subject of investigations by many researchers. Because of the presence of a moving boundary between the two phases, the problem is nonlinear. Various mathematical methods and techniques have been used to study free boundary problems. They have been discussed and summarized in several books [1][2][3][4][5] and many survey papers [6][7][8][9].Free boundary problems have been studied since the nineteenth century by Lame and Clapeyron in 1831, Neumann in the 1860s and Stefan in 1889. However, the only known exact solutions are those of Neumann and some of their extensions and variations. These solutions are all expressible in a single similarity variable x/t1'2. This is to say that the partial differential equations of the problem are reducible to a set of ordinary differential equations. In the case of classical free boundary problems, the reductions are possible only when the body is semi-infinite and the boundary and initial conditions are of certain special forms. Neumann considered a semi-infinite body of a constant initial temperature which is suddenly in contact with a different temperature at its face. The exact solutions of the temperature of both phases are then found in the form of the similarity variable x/t1/2. And the solution of the interface location is proportional to t1/2. A generalization of the problem to cover arbitrary initial and boundary conditions is nontrivial, since the reduction to ordinary differential equations is no longer possible. No exact solutions have yet been found for the Neumann problem with arbitrary initial and boundary conditions.It is the purpose of this paper to study the Neumann problem with arbitrary initial and boundary conditions, i.e., the free boundary problem of a semi-infinite body with an

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

The Stefan problem with arbitrary initial and boundary conditions

Liu¹

1978

Quart. Appl. Math.

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“…The thermoelastic coupling terms occurring in the heat conduction equation are usually small and neglected in practice [1][2][3]. Taking them into account, however, has been found important in many applications, such as the modeling of the vibration of resonant microelectromechanical systems (MEMS) [4][5][6], dynamic crack propagation [7][8][9], thermal shocks [10][11][12][13][14], ultrafast laser heating in thermal processing of materials [15][16][17][18], and wave propagation [19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Coupled heat equation in thermoelasticity with temperature dependent moduli

Boussaa

2019

Journal of Thermal Stresses

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New and consistent expressions for the coupled heat equation are developed within the framework of small-strain thermoelasticity for both the Fourier and Cattaneo-Vernotte conduction models. These expressions place no restrictions on the changes in temperature, allow for the temperature dependence of the thermoelastic moduli, and include all the coupling terms as functions of the thermoelastic moduli and their derivatives. As applications, (i) an extended Lord-Shulman-type model is derived that takes into account the temperature dependence of the thermoelastic moduli, and (ii) the equations underpinning the experimental technique of thermoelastic stress analysis are revisited.

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“…These cylindrical problems find application in the rapidly developing field of nuclear technology. The literature, up to 1971, may be found in the comprehensive survey by Boley [5]. Also, for the latest developments on cylindrical geometry, one may refer, for example, to Chen [6] for the end effect, to Huang and Cozzarelli [7] for the cladding effect, to Sugano [8] for transient and anisotropic effects, to Spiga and Lorenzini [9] for the radial neutron distribution, to Amada and Yang [10] for the nonlinear thermal stresses resulting from phase changes in materials, to Thompson [11] for a general study on cylinders of arbitrary crosssection.…”

Section: Introductionmentioning

confidence: 99%