Engineering Transactions, 10, 2, pp. 281-305, 1962

### Obliczanie Wytężenia Materiału w Stanach Podkrytycznych

M. Życzkowski
Politechnika Krakowska
Poland

This is a continuation of Ref. [16], where a general measure of the «exertion» of material in sub- critical states is proposed (degree in which the physical state of the material approaches the dangerous state). This measure, expressed by the Eqs. (1.2) and (1.3), depends on the distribution of the density of probability p of reaching the limit surface in each particular direction. In the case of the proportional loading process (the point probability) it becomes identical with the «reduced stress». Some examples of computation of the exertion are given in the case of a two-dimensional space of exertion factors. Closed-form solutions are obtained in the case where the limit curve is composed of rectilinear segments (Figs. 2 and 3) or is an elliptic or parabolic curve (Figs. 5 and 7 respectively) with a step-like probability distribution. These are the most frequent cases of limit curves. In the latter two cases the results are expressed by means of elliptic integrals. A parabola may correspond to a simultaneous action of stress and temperature. The influence of the distribution of the probability density p assumed on the results obtained is analysed and illustrated by Tables
4-7 and Figs. 8-11. With other forms of the limit curve (limit surface) or with other distributions of probability density p and, especially, with a higher number of dimensions of the space of exertion factors it will, as a rule, be necessary to use approximate integration methods. These methods are discussed briefly in Sec. 2.

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