Engineering Transactions, 16, 1, pp. 33-67, 1968

Zagadnienia termosprężystości w obszarach ograniczonych powierzchniami kulistymi i stożkowymi

Z.F. Baczyński
Zakład Mechaniki Ośrodków Ciągłych Instytutu Podstawowych Problemów Techniki PAN

One of the subject matters of the present paper is a method of homogeneous solutions of boundary-value problems of heat conduction, stress and displacement for a certain class of elastic and conical surfaces. This method is illustrated by solving the three-dimensional axially symmetric problem of a thick-walled open spherical shell.
The general homogeneous solutions of hest conduction problems given in Sec. 1 enable us to determine the temperature field in any simply connected region of a body bounded by concentric spherical surfaces and the surface of a circular cone of any apex angle.
The conditions on the spherical surfaces and the conical surface are arbitrary. The general solution of the problem of heating of the body at the spherical surfaces, the conical surface being kept at zero temperature is obtained in the form of the series (1.8), which is (are) a Fourier-Legendre series constructed by means of orthogonal functions and the roots of the transcendental Legendre equation resulting from the homogeneous condition for the temperature at the conical surface. It is shown that the series (1.8) are absolutely and uniformly convergent in the regions considered. This convergence is slow, however, so that the substitution of a few terms results in an error of more than then percent and the substitution of several tens of terms - in an error of a few percent. In particular cases we must solve in addition the transcendental Legendre equation to obtain the sequence of real roots. The solution of such an equation is impossible in the general case.
The problem in which the body is heated in any manner on its conical surface the spherical surfaces being kept at zero temperature leads to a solution in the form of the series (1.14). This is a Fourier-Mehler series constructed by means of orthogonal sets of functions and complex roots of the transcendental equation which is derived from the homogeneous conditions for the temperature at the spherical surfaces. The complex coefficients of the series (1.14) are determined directly by solving the infinite set of equations (1.20). It should be stressed that the solution (1.14), by contrast with (1.8), does not require the solution of the (a) transcendental equation for each case of the body considered. Each particular region of the body is characterized by the coefficients of the series (1.14) through the values of Mehler's functions with constant arguments [in the solution (1.8) different regions are characterized by different indices of the Legendre functions with constant arguments].
In Sec. 2 is solved the problem of thermal stress and displacement produced by any axially symmetric temperature field in a simply connected spherical sector. The state of stress and displacement is the sum of a potential state and two additional states corresponding to auxiliary solutions. The potential state [cf. the Eqs. (2.12) and (2.13)] corresponds to a particular solution of the equations of the field (7.9) and is determined by a potential of thermoelastic displacement. In this state, on the surfaces bounding the region under consideration there occur, in general, normal and shear stresses different from zero. The additional state of stress σ_(ij  )^((1)), σ_(j  )^((1)) [Eqs. (2.20) and (2,21)] corresponds to the general solution of the first problem of the equations of the theory of elasticity, in which the stresses σ_(ij  )^((1)) take prescribed values on the spherical surfaces. The additional state of stress σ_(ij  )^((2)), u_(i  )^((1))corresponds to the general solution of the second problem of the equations of elasticity in which the stresses σ_(ij  )^((2)) take prescribed values on the conical surface.
The states of stress and strain σ_(ij  )^((0)), σ_(ij  )^((1)), u_(ij  )^((0)),u_(i  )^((1))are described by Legendre-Fourier series constructed by means of a set of orthogonal functions and roots of the same transcendental equation as for the solution (1.8).
From the analysis of the state of stress and the numerical computation it follows that the potential states of stress σ_(ij  )^1and σ_(ij  )^((1))  lay a dominating role. The additional state of stress σ_(ij  )^((2)) has the character of residual stress which may be rejected in many practical cases. It is possible to estimate the error thus committed.

Full Text: PDF
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).


J. M. C. DUHAMEL, Second mémoire sur le phénomènes thermomécanigues, Journal de l'École Polytechnique, 25, 15 (1837), 1-57.

J. M. C. DUHAMEL, Mémoire sur le calcul des actions moléculaires developpés par les changements de temperature dans les corps solides, Mémoires par divers savans, j. de l'École Polytechnique, 5 (1838), 440 498.

, C. W. BORCHARDT, Untersuchungen über die Elasticität fester isotroper Körper unter Berücksichtigung der Wärme, Monatsber, Akad. Wiss., Berlin, Germany, 9 (1873).

[in Russian]

J.N. GOODIER, On the Integration of the Thermo-Elastic Equations, Philosophical Magazine, 23 (1937), 1017.

R. D. MINDLIN, D.H. CHENG, Thermoelastic stress in the seminfinite elastic solid, J. Appl. Phys., 21 (1950), 931.

B. SHARMA, Stresses due to a nucleus of thermoelastic strain (i) in an infinite elastic solid with spherical cavity and (ii) in a solid elastic sphere, ZAMP, 2, 8 (1957), 142-150.

E. L. Mc DOWELL, E. STERNBERG, Axisymmetrical thermal stresses in a spherical shell of arbitrary thicknees, J. Appl. Mech., 24 (1957), p. 376.

J. IGNACZAK, The axially symmetric boundary-value problem of thermoelasticity for a hemispherical shell of any thickness, Arch. Mech. Stos., 12 (1960), 415.

W. E. WARREN, A transient axisymmetric thermoelastic problem for the hollow sphere., AIAA Journal, 1, 11 (1963), 2569-2576.

W. E. WARREN, A note on the transient axisymmetric thermoelastic problem for the solid sphere, J. Appl. Mech., 2, 31, (E) (1964), 348-350.

G. EICHELBERG, Temperaturverlauf und Wärmespannungen in Verbrennungsmotoren, VDI- Forschungsheft 263, Berlin 1923.

S. P. TIMOSHENKO, Theory of Plates and Shells, McGraw-Hill, New-York 1940.

[in Russian]

[in Russian]

H. PARKUS, Wärmespannungen in Rotationsschalen bei drechsymmetrischer Temperaturver teilung, Sitzungsber. Österr. Akad. Wiss., Abt. IIa, 160, 1 (1951).

E. MELAN, Wärmespannungen bei der Abkühlung einer Kugel, Acta Physica Austriaca, 1--2, 10 (1956), 6-81.

W. NOWACKI, Thermoelasticity, Pergamon Press, London 1962, 19. Z. F. BACZYŃSKI, Axisymmetric steady-state conduction in an open spherical shell of arbitrary thickness, IASS, Symposium on Non-Classical Shell Problems, Warsaw 1963; (Summary in Bull. Acad., Polon. Sci., Série Sci. Techn., 11a, 12 (1964), 7-8.

Z. F. BACZYŃSKI, The steady-state axisymmetric thermoelastic problem for an open spherical shell of arbitrary thickness, Bull. Acad. Polon. Sci., Série Sci. Techn., 12 (1964), 581-587.

Z. F. BACZYŃSKI, Naprężenia termiczne w cienkiej otwartej powłoce kulistej, Arch. Bud. Maszyn, 3, 13 (1966).

Z. F. BACZYŃSKI, The thermoelastic steady-state stresses and strains for and strains for an open spherical shell of arbitrary thickness, Arch. Mech. Stos., 7, 18 (1966).

Z. F. BACZYŃSKI, Jednorodne rozwiązania termosprężystości dla obszarów kulistych, referat na konferencji naukowej ZMOC, IPPT, PAN, Augustów 1965.

Z. F. BACZYNSKI, Naprężenia termiczne w grubych otwartych powłokach kulistych (rozprawa doktorska), IPPT, PAN, Warszawa 1966.

[in Russian]

[in Russian]

E. STERNBERG, R. A. EUBANKS, M. A. SADOWSKY, On axisymmetric problem of elasticity theory for a region bounded by two concentric spheres, Proc. First Nat. Congr. Appl. Mech, A.S.M.E., New York 1951.

B. A. BOLEY, J. H. WEINER, Theory of Thermal Stresses, New York-London 1962.

[in Russian]

[in Russian]