PSI - Issue 80

496 Vaclav Sklenicka et al. / Procedia Structural Integrity 80 (2026) 493–500 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 Uniaxial constant-stress creep tests under tension were carried out over a range of applied stresses from 10 to 350 MPa and temperature intervals of 300 to 700°C. Under these loading conditions , the minimum creep rate, ̇ m , is from 3 x 10 -8 s -1 to 1.7 x 10 -3 s -1 . Different stresses were used at each testing temperature. Representative time dependences of the creep strain (standard creep curves) are shown in Fig. 1. As shown in Fig. 1, a typical creep curve consists of four different creep stages: an instantaneous strain, ε 0 , which is composed of elastic, anelastic, and plastic components, and appears after sudden loading of the creep specimens, and the primary, secondary creep deformation, during which the creep rate decreases due to the generation of dislocations. In the secondary creep stage, the creep rate is constant due to the balance between the hardening and recovery rates. The secondary creep stage is the minimum strain rate of the creep curve. In the tertiary creep stage, the creep rate continuously increases with time, and the stage concludes with the fracture of the specimen. As demonstrated in Fig. 1, the shapes of standard curves differ considerably due to different loading conditions (applied stress, temperature). The effect of the applied stress is evident in Fig. 1(a, b), and the impact of testing temperature is depicted in Fig. 1 (d). The reason for creep test interruption was a technical option of the constant-stress creep machine (the strain ε > 40%, see 2.2.) and/or the prediction of extremely long-term creep exposure (Fig.1(c)). However, in all creep tests, the value of the minimum creep rate was achieved using replotting the standard creep curves in the modified form of the strain rate, ̇ , vs. the time of creep exposure, t, (Sklenicka et al. (2023)). 3.2. Stress dependences of the main creep parameters In this work, the main creep parameters are the minimum creep rate, ̇ m , the time to fracture, t f , and the strain to fracture, ε f . The stress dependence of the minimum creep rate is frequently expressed by the constitutive relations based on simplified physical models (Mukherjee et al. (1969), Čadek (1988), Kassner (2009). For the applied stress, σ , and temperature, T , the dependence of the minimum creep rate, ̇ m , can usually be described by a simple Norton power-law constitutive relation as follows (Mukherjee et al. (1969)): ̇ m = A (σ ) n exp(-Q c /kT), (1) where A is the structural parameter related to the alloy, n = (∂ln ̇ m /∂ ln σ) T is the stress exponent of the minimum creep rate, Q c = [∂ln ̇ m /∂ ( -1/kT)] σ is the activation energy for creep , k is Boltzmann constant, and T is the absolute temperature. Analogous as in Eq. (1), the constitutive relationship for the stress dependence of the time to fracture (creep life) , t f , can be expressed as (Maruyama (2008)): (2) where B is a material constant, m = - (∂ln t f /∂ln σ) T is the stress exponent of the time to fracture, and Q f is the activation energy for the time to fracture. The stress dependences of the minimum creep rate, ̇ m , are shown in Fig. 2(a) for different testing temperatures, using double logarithmic axes. It can be seen from Fig. 2(a) that these stress dependencies may not be linear, which is clearly demonstrated by the characteristic shape of the relevant curve. The slopes in Fig.2(a) and, hence, the stress exponents of the minimum creep rate n strongly depend on applied stress and, especially, testing temperature. Generally, the value of the stress exponent n increases with increasing stress and decreases with increasing testing temperature. For instance, while the values n within the range 4 - 7 were observed for temperatures 500 and 700°C and lower applied stresses, σ ≤ 100 MPa, at exceptionally high applied stresses ( σ ≥ 200 MPa) associated with lower temperatures ( T ≤ 400°C), the values of the stress exponent exhibits extremely high values and the stress dependences nearly straight lines. It should be noted that temperature s from 300 to 400°C are typical during water reactor operations. Similarly, the stress dependences on the time to fracture, t f, or the time to interrupt the creep tests, t exp, are shown in Fig. 2(b). As evident in the figure, the lines of the stress dependences of the times to t f or t exp , are not linear, and the different values of the stress exponents m are also the result of acting stress and temperature and are close to the determined values of n . 4 t f = B (σ) -m exp(Q f /kT),