Two Problems of Thermal Stress in the Infinite Solid

Abstract

Abstract The equations for thermal stress were first developed about 100 years ago by Duhamel (1), who treated the cases of a sphere and a circular cylinder with radial variation of temperature only. As the problem of the cylinder can be considered one of plane strain, it appears that the problem of the sphere (2) is the only three-dimensional one for which a complete solution is available. Borchardt (3) has treated the thermal-stress problem of the sphere for a non-symmetrical distribution of temperature. Recent developments of the theory (4) have made it possible to complete further solutions. In this paper results are obtained for an infinite solid in which a small internal region, in the form of an ellipsoid of revolution, or a long narrow region, in the form of a semi-infinite circular cylinder, is hotter than the surrounding material. In a finite solid the cylinder may alternately be considered as a blind hole filled with an oversize plug, or a similar shrink fit. The problem of the ellipsoid has some bearing upon the question of the effects of slag inclusion in metals. A discussion by B. P. Haigh (5) deals with the spherical inclusion, whereas, the results of this paper give some indication of the effect of shape variations. The shape of the ellipsoid varies from a flat pancake shape to a thin needle shape, and it is seen that both the pancake-shaped and the needle-shaped inclusions give rise to a higher stress than does the spherical one, for the same temperature variation.