Superposition Method in Thermal-Stress Problems for Rectangular Plates

Abstract

The classical two-dimensional biharmonic problem for a rectangular domain is considered. Some historical aspects of the problem are elucidated. The superposition method turns out to be efficient in solving thermoelastic-equilibrium problems in a rectangle. The relationship between the mathematical and engineering approaches to these problems is studied. A few typical examples are given Keywords: biharmonic problem, thermoelastic stresses in rectangle, infinite systems of linear algebraic equations Introduction. Nowadays, the needs of technology provoke interest in thermal-stress problems for various structures. Thermal stresses in structural members were intensively studied in the mid-20th century because of important practical problems that arose in designing new steam and gas turbines, jet and rocket engines, and nuclear reactors [1,6]. Members of such structures are subject to nonuniform and often nonstationary heating that causes temperature gradients and unequal thermal expansion of isolated components. Such nonuniform thermal expansion cannot proceed freely in an elastic body and induces thermal stresses. The magnitude and behavior of thermal stresses must be known to allow comprehensive strength analysis of structures.Thermoelasticity became an independent division of elasticity theory quite a long time ago-linear equations incorporating the thermal effect were derived independently by Duhamel [47] and Neumann [68]. In these fundamental works, they not only set up the governing equations of thermoelasticity, but also solved elementary one-dimensional problems for thermal stresses. The general formulation of the stress problem was discussed in the recent publications [42,43]. Thermoelastic problems were addressed in fundamental courses on elasticity theory due to S. P. 15, 16, and drills therein] and in fundamental treatises due to A. E. H. Love [23, Sect. 74], N. I. Muskhelishvili [31, Sect. 62], and A. I. Lur'e [ 22, Ch. 1, Sect. 13, Ch. 3, Sect. 7, Ch. 4, Sect. 4]. Moreover, these problems were discussed in monographs of Soviet and foreign authors [1,6,12,13,21,24,26,32,34], which include detailed bibliographies on the subject. In this variety of publications, A. D. Kovalenko's textbooks [14,15,17] stand out due to their concise format, clear formulations of problems and methods of their solution, wide coverage of almost all divisions of static and dynamic thermoelasticity, and in-deep analysis of the possibilities for practical application of results. It is not surprising that the first textbook was translated into English [54].It is often possible to analyze the thermostressed state of an elastic body under plane-strain conditions: the plane strain of an extended body of constant cross section where thermal processes do not depend on the longitudinal coordinate and (generalized) plane stress state of a thin, simply connected, isotropic plate of constant thickness. Two-dimensional quasistatic problems of thermoelasticity for a known temperature field and no surface mechanical loads on the boundary of t...

Thermal Stresses in an Elastic Rectangle

“…This method was used in previous papers [31,32] to evaluate the stress fields in the two-dimensional biharmonic problems of stresses in an elastic rectangle and the bending of a clamped rectangular plate; the papers were accepted by Don Carlson, the Editor-in-Chief of the Journal of Elasticity. Previous studies of similar thermoelastic problems [33][34][35] were restricted to prescribed temperature fields that exhibited various degrees of symmetry.…”

Stress state of a cross-base prism under torsion

“…The applications of the method are reviewed in [4,7,9,17]. The use of the limitants method is difficult because of the necessity of solving a great number of finite systems of linear algebraic equations.…”

Evolution of the planar vibrations of a rectangular piezoceramic plate as its aspect ratio is changed

“…When a square plate undergoes longitudinal vibrations, its expansion along one axis is accompanied by contraction along the perpendicular axis. In the case of radial vibrations, expansion (or compression) occurs in phase and simultaneously in all directions, as in a thin circular disk.According to [14,22,23], when a square plate becomes rectangular, the radial mode splits into several modes, giving rise to longitudinal and transverse resonances. Such modes are distinguished by complex behavior of the strain state and cannot be described in terms of simple harmonic functions.…”

“…In the general case where the plate cannot be considered narrow (aspect ratio less than five), the deformation is a very complicated process so that the radial, edge, and even bending resonances appear between the fundamental longitudinal mode L1 and its overtones L3 and L5 [2,16,[22][23][24].…”

“…According to [14,22,23], when a square plate becomes rectangular, the radial mode splits into several modes, giving rise to longitudinal and transverse resonances. Such modes are distinguished by complex behavior of the strain state and cannot be described in terms of simple harmonic functions.…”