PSI - Issue 14

30 L.R. Botvina / Procedia Structural Integrity 14 (2019) 26–33 L.R.Botvina / Structural Integrity Procedia 00 (2018) 000 – 000 5 where f s is the volume fraction of the solid phase: ̇ is the shear rate of the melt. The phase transition during crystallization in which the first derivatives of some physical quantities change in a jumpwise manner and heat is evolved is traditionally classified as the first-order phase transition. In fact, at the critical point for cyclic loading, we observe an abrupt increase in the temperature of the specimen surface, the crack growth rate, and the acceleration of the crack growth, i.e. the first and the second derivatives of the crack length. In order cases of failure, no increase in the temperature of the specimen surface is observed at the critical stress. On the other hand, according to Botvina et al. (2010), a sharp increase in the heat capacity of the steel associated with the presence of a crack in specimen was observed. As is well known, an abrupt increase in the heat capacity is a sign of a second-order phase transition. These observations lead to the conclusion that the phase transitions in the case of fracture have features characteristic of phase transitions of both the first and the second kind. Fig. 2c shows the dependence of the rate of crystallization on the product of the current viscosity of the material, η , characterizing the crystallization process, and the shear rate ̇ , which is the main factor affecting the intensity of the crystallization process. In this case, the middle section of the crystallization diagram is described by a power-law relation with a power exponent for a given alloy m = 2.6: / ~( ̇√ ) (5) It follows that curves similar to those shown in Fig. 2d can be described by power relations of the form: / ~( √ ) , (6) where K and dK /d τ are the characteristic and rate of the process development; P is a loading parameter. The curves in Fig. 2c are similar to each other, therefore in normalized coordinates they lie on a single universal curve (Fig. 2 d), which indicates the self-similarity of the crystallization process and does not differ in form from the kinetic diagram of fatigue fracture (Fig. 1b) described by the Paris equation (1). It was shown by Botvina and Oparina (1992) that the same power-law dependence describes the process of accumulation of defects during multiple fracture of material before the macrocrack formation: / ~( √ ) , (7) where / is the rate of increase in damage; σ is the applied stress. This means that the same power relations can be detected by analyzing the change in the physical properties of materials with defects. The approach used and the proposed estimate of the critical exponents for fracture and crystallization make it possible to compare the characteristics of phase transitions in different media. To plot kinetic diagrams of ultrasonic attenuation reflecting changes in the structure or material damage, it is necessary to obtain a family of curves of changing the attenuation coefficient corresponding to a various structure or damage of material in function of frequency, and to select a governing parameter reflecting the self-similarity of the process of accumulation of damages noted above. The kinetic diagram of ultrasonic attenuation was plotted by Botvina et al. (1995, 2000) for a material with different grain sizes, described by the relation: / = ( √ / ) , (8) where / is the rate of change of the ultrasonic attenuation coefficient with frequency ( f ), D is the grain size in the steel, λ is the wavelength of ultrasound at a given frequency. Analysis of the ultrasonic wave attenuation carried out by Botvina et al. (1995) on the base of the similarity theory approaches and the hypothesis of incomplete self-similarity of Barenblatt (1987), and the studies of 3. Kinetic diagrams of ultrasonic attenuation