**2.2 Evaluation method of creep ductility (***QL***\* parameter)**

The creep crack growth rate (CCGR) is written by *Q* <sup>∗</sup> parameter as shown in the Eq. (2) and (3) [6, 7].

$$\frac{da}{dt} = A^\* \exp\left(\mathbf{Q}^\*\right) = A^\* \sigma\_\mathbf{g}^{m\_\mathbf{g}} \exp\left(-\frac{\Delta H\_\mathbf{g}}{RT}\right) \tag{2}$$

$$Q^\* = -\frac{\Delta H\_\text{g} - m\_\text{g} \ln \sigma\_\text{g}}{RT} \tag{3}$$

By integrating Eq. (2) and (3), the life of creep crack growth is written by Eqns. (4) and (5),

$$t\_f = \int\_0^{t\_f} \frac{da}{A \, ^\ast \sigma\_\text{g}^{m\_\text{g}} \exp\left(-\frac{\Delta H\_\text{g}}{RT}\right)} = \frac{a\_f - a\_i}{A \, ^\ast \sigma\_\text{g}^{m\_\text{g}} \exp\left(-\frac{\Delta H\_\text{g}}{RT}\right)} = \frac{C}{A \, ^\ast \sigma\_\text{g}^{m\_\text{g}} \exp\left(-\frac{\Delta H\_\text{g}}{RT}\right)}\tag{4}$$

$$\frac{1}{t\_f} = A\_\text{g} \sigma\_\text{g}^{m\_\text{g}} \exp\left(-\frac{\Delta H\_\text{g}}{RT}\right) \tag{5}$$

where *tf* is the life of crack growth and *Ag* <sup>¼</sup> *<sup>A</sup>*<sup>∗</sup> *C* . The steady state creep strain rate is given by the Eq. (6).

$$
\dot{\varepsilon}\_s = A\_c \sigma\_g^{m\_c} \exp\left(-\frac{\Delta H\_c}{RT}\right) \tag{6}
$$

Dividing Eq. (6) by Eq. (5), *QL*<sup>∗</sup> parameter is given by Eq. (7) [11, 12].

$$Q L^{\*} = \dot{e}\_{s}^{\;a} \bullet t\_{f} = M \sigma\_{g}^{m} \exp\left(-\frac{\Delta H}{RT}\right) \tag{7}$$

Where *A*<sup>∗</sup> , *Ag* and *Ac* are constants. *σ<sup>g</sup>* is gross stress (MPa), *m*<sup>g</sup> and *m*<sup>c</sup> are exponent of gross stress and Δ*Hg,* Δ*Hc* is activation energy of crack growth and creep strain (J/mol), respectively. *R* is gas constant (J/K・mol),*T* is absolute temperature (K). m ¼ *mc* � *mg*, Δ*H* ¼ Δ*Hc* � Δ*Hg*, *M* is creep ductility. *α* is constant and it nearly equal to unity.

For creep ductile materials, since steady state creep rate well correlates with the inverse value of fracture life, values of *m* and Δ*H* almost equals to zero, respectively, that is, *mc* ¼ *mg* and Δ*Hc* ¼ Δ*Hg*. For such case, Eq. (7) is written by Eq. (8), that is, the relationship between logarithmic values of *ε*\_*<sup>s</sup>* and *tf* is independent of gross stress and temperature.

$$Q L^\* = \dot{\varepsilon}\_s^a \bullet t\_f = M \tag{8}$$

Based on the *QL*<sup>∗</sup> concept, the relationship between logarithmic values of *ε*\_*<sup>s</sup>* and *tf* for various materials are obtained as shown in **Figure 3** [11], that is, the *QL*<sup>∗</sup> map. The data band for each material shows the creep ductility as the value of *QL*<sup>∗</sup> ð Þ <sup>¼</sup> *<sup>M</sup>* .

**Figure 3.** *QL\* map for creep ductile and brittle materials. The relationship between creep fracture life and steady state creep rate.*

*The Quantitative Estimation of Mechanical Performance on the Creep Strength… DOI: http://dx.doi.org/10.5772/intechopen.106419*
