Engineering Transactions, 9, 2, pp. 211-242, 1961

Zmęczenie i Tłumienie w Prętach Pryzmatycznych

J. Madejski
Politechnika Gdańska

The fatigue failure is understood in this paper as a failure without any perceptible total permanent strain resulting from a durable load. A failure occurs after an accumulation of damages n. At the moment of failure we have n = nz. The latter property a is a material function. Instead of n the specific damage, or the defect. D = n/nz < 1 may be used. The defect grows in the course of the process of loading; at the same
time the process of «recovery», consisting in spontaneous suppression of damages, occurs in the material. Denoting by h the rate of that process, Eq. (1.2) is obtained from the balance of the existing damages. The theoretical formula for velocity h is given on page 212; from the scarce experimental data the value AU xJ 6000 cal/mole is obtained for certain types of carbon steel. The injuries, caused by the load per unit time, %, are to be evaluated as the sum of absolute values of significant permanent strains, according to the proposal of the author [3]. The material may be treated a theoretically as a Bingham material with the elastic limit OE and viscosity p. The Eq. (2.5) is the equation of state for such a material in one-dimensional problems. ff the load cycles are short, the value of t may be evaluated from Eq. (2.6). Permanent strains εp depend on the viscosity, which is a characteristic property of fluids. As crystals are not viscous, it is evident that the flow of a solid is caused by the growth of new amorphous phase, which accompanies the crystalline phase. This process may be treated as an endothermic process proceeding at a certain rate. If the latter is commensurable with the rate of growth of the external loads, conspicuous growth of the elastic limit σE may be expected. In this paper it is assumed that the speed of growth of the external loads is small enough (in relation to the velocity of the reaction mentioned) to neglect the changes of σE. The viscosity of the conglomerate of crystalline and amorphous phases is given by Eq. (3.3). The proper value of the activation energy is evaluated from the Hess Law, as the difference between the internal energies of both phases. The process of damping, treated in Sec. 4, is strictly connected with the increase of the temperature of the specimen. Damping is defined as the work of internal friction (the work of the stresses on permanent strains) attached to unit volume and time. Eq. (4.4) follows from the heat and energy balance where 9 is the reduced temperature difference and S the characteristic stress. There are two cases of damping curves (Fig. 3); if the dimensionless number M defined by the Eq. (4.7) exceeds e (the e basis of natural logarithms), damping stabilizes in the course of the loading process. In other cases damping grows infinitely in a finite period of time. To evaluate the defect (Sec. 5), we must take into consideration the surface fibres of the specimen
(the surface layer), because of the influence of micronotches and «self-equilibrant» stresses, caused by machining and heat treatment of various kinds. These influences are taken into account by means
of special function Zp; examples of such functions together with the «damping function» Zw are given in Sec. 7. Thus, the property is defined by Eq. (5.3) whereas Eq. (5.4) describes the dependency of the defect D on time and temperature. This equation should be solved together with Eq. (4.4). The result are the lines D = const and the Wöhler curve (D = 1). Special cases are analysed in Sec. 6; the results enable us to construct the D-(n/N)-diagrams, which are particularly useful for sequential loads. The «self-quilibrated» stresses vary with time in the course of the process of loading. The analysis of this very complicated phenomenon is given in Sec. 8. The results are rather of a qualitative character. Since fatigue and damping processes are very much influenced by heat transfer phenomena, this problem is investigated in Sec. 9. Sec. 10 deals with the confrontation of the theory with the experiments. Fig. 13a gives an experimental S-N diagram with constant of lines damping for mild carbon steel, obtained by LAZAN and Wu (Ref. [20], [21]). Fig. 13 shows the theoretical curves computed by the author. The theory is also confronted with other experiments, which deal with damping properties of various materials. Several important physical data for metals and alloys are obtained on this ground. These investigations show (in agreement with the experiments) that the specific damping capacity depends on the frequency.

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