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
Poland

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.

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