Cyclic Deformation, Holdtime Creep and Thermomechanical Fatigue of an ODS Superalloy

Appendix A Fit Functions for Holdtime Creep Data

Region I: This region with negative deformation rates characterizes the unloading situation when the specimen is still under high absolute stresses and stress and strain rate have the same sign (Fig. 4). In Fig. A1 the data of region I are plotted on a logarithmic scale. They can be fitted by a power-law equation of the form:

Fig. A1

The holdtime creep data for negative stresses can be fitted by a power-law function (Eq. (A1)).

where ɛ̇_I is the mean deformation rate, n_I the stress exponent (n_I = 8), A_I a fitparameter (A_I = 1.4 · 10^–34 m²). E is the temperature-dependent Young’s modulus given in [16]:

and D is the volume diffusivity; data for its activation energy Q and pre-exponential factor D₀ were taken for Ni20Cr from [23]; D₀ = 1.6 · 10^–4 m²/s, Q = 285 kJ/mol.

The upper bound of region I data is given by the x-axis in Fig. 4 (ɛ̇ = 0). The stresses at the intersection of the curves with the x-axis do not cause any permanent inelastic deformation which means that the effective stress is zero and the internal stress equals the applied stress. In Fig. A2 the internal stresses are plotted as a function of temperature on a logarithmic scale. This temperature dependence can be described by the following expression (σ_i in MPa and T in K):

Fig. A2

Temperature dependence of the internal stresses σ_i determined from holdtime creep tests. The data can be described by an exponential function of temperature (Eq. (A3)).

Region II: The strain rates in region II are positive (Fig. 4). Only a weak stress dependence exists, and all datapoints except the 1050 °C values approach a straight line which separates region II from region III. For positive stresses the strain rate can here be expressed as a linear function of stress:

For negative stresses the fit functions correspond to straight lines between the intersection of the y-axis (σ = 0 and ɛ̇ = 6.8 · 10^–6 s^–1) and the points on the x-axis which characterize the appropriate internal stress (σ = σ_ί(T), ɛ̇ = 0).

Region IV: The data for positive stresses and strains are plotted in normalized form in Fig. A3. Two regions of different stress dependence exist as can be seen most clearly in the 1050 °C and 750 °C data. For high stresses the stress exponent is n_IV = 3.5 for all temperatures. Assuming that n_IV is not temperature dependent, a power-law fit function was selected which is illustrated in Fig. A3 as parallel lines:

The normalization of the creep rate and the stresses by the diffusivity D and by the temperature dependent Young’s modulus (E(T)), respectively, shows that the temperature dependence of the creep behaviour in this region is not determined by volume diffusion alone (Fig. A3). For an accurate description of the region IV data A_IV(T) is an empirical function of temperature (T in K):

Fig. A3

Normalized plot of holdtime stresses σ_h/E vs the 10 s mean deformation rate ɛ̇/D for positive stresses (region A in Fig. 1) of the holdtime creep experiments. The straight lines are given by Eqs (A5) and (A6).

Region III: The data in the transition between regions II and IV cannot be satisfactorily described by Eqs (A4) and (A5). Therefore the following empirical equation has been applied to describe the behaviour in region III:

The pre-exponent A_III and the stess exponent n_III are functions of temperature and were determined as:

where T has to be expressed in K.

The fit functions of regions III and IV are illustrated in Fig. A4 for the 800 °C and 950 °C data. The full lines represent the mean values of the three fit equations which were applied in order to smoothen the curves.

Fig. A4

Fit procedure for an accurate description of the holdtime creep data. The final fit functions are a mean value of the fit equations for regions III and IV. The result is illustrated for the 800 °C and the 950 °C data.

Appendix B Numerical Calculation of Stress-strain Hysteresis Loops

In the algorithm of the applied programme, the desired cycle with a given time-dependence of strain and temperature and a given strain amplitude was divided into j = 100 points of constant δɛ = (2 · Δɛ)/j, Δt = t/j and δT = (2 · ΔT)/j. Starting with σ₁ = 0, ɛ₁ = 0 and T₁ = (T_max–T_min)/2, time, strain and temperature were increased in a stepwise manner and the stresses σ_j were calculated iteratively in the following way:

The stop-criterion of the iteration procedure was chosen as a deviation better than 0.01 % between the desired strain and the strain recalculated from the appropriate stress σ_j. The functions for the stress and temperature dependences of the strain rates are given in Eqs. (A1) and (A3) to (A9) and the temperature dependence of the Young’s modulus follows Eq. (A2).