## 5 Life Prediction Models
### Multiaxial LCF--Non Proportional Loading  
Many structural components, such as thrust chambers, experience complex mechanical and thermal loads during their normal working conditions. Such loads can generate multiaxial stress-strain histories and also changes in the principal stress directions. Therefore,  
Figure 10: Steady stress vs. damage time [16].  
fatigue criteria, taking into account both multi-axiality and non-proportional loading, have to be adopted in these cases [18,19,30,31]. Most of the fatigue criteria propose to estimate the value of a uniaxial equivalent strain, which can be considered as being equivalent to the multiaxial strain field. The relationship between this uniaxial equivalent strain and the number of cycles to failure can be obtained through the Basquin-Manson-Coffin relationship [30], represented through the following equation:  
\[\epsilon_{a,eq}=\frac{\sigma_{f}^{\prime}-2\sigma_{n,mean}}{E}\left(2N_{f} \right)^{b}+\epsilon_{f}^{\prime}(2N_{f})^{c} \tag{14}\]  
where the fatigue strength coefficient \(\sigma_{f}^{\prime}\) and exponent \(b\) are parameters of Basquin law, whereas fatigue ductility coefficient \(\epsilon_{f}^{\prime}\) and exponent \(c\) are parameters of Coffin-Manson curve, \(\sigma_{n,mean}\) represents the mean stress normal to the maximum shear plane, and \(2N_{f}\) is the number of reversals to failure.  
The Wang-Brown criterion is based on a critical plane approach. As demonstrated through numerous experiments [14], fatigue life is usually dominated by crack-growth along either shear planes or tensile planes. This criterion is then based on the idea that shear strains help to nucleate cracks, while normal strain contributes to their growth. The amplitude of the equivalent strain is evaluated according to the following equation:  
\[\epsilon_{a,eq}=\frac{\gamma_{max}+S\delta\epsilon_{n}}{1+v^{\prime}+S(1-v^{ \prime})} \tag{15}\]  
where \(\gamma_{max}\) is the maximum shear strain amplitude on the critical plane, \(\delta\epsilon_{n}\) is the normal strain excursion, and \(v^{\prime}\) is the effective Poisson's ratio. The experimental data regarding fatigue tests on CuCrZr coupons are reported in Figure 11 [16]. An analytical relationship between the equivalent strain amplitude and the number of cycles to failure was evaluated by adopting a multiple linear regression model to fit the CuCrZr experimental fatigue tests. Then, the equivalent strain amplitude \(\epsilon_{a,eq}\), estimated according to the Wang-Brown method, was used to estimate the fatigue life according to the following relationship:  
\[\epsilon_{a,eq}=D+H\bigg{(}\frac{N_{f}}{10,000}\bigg{)}^{-\frac{0.25T}{1000}}+ F\frac{T}{1000} \tag{16}\]  
where \(D\), \(H\), and \(F\) are constants set as equal to 0.1684, 0.4580, and \(-0.5987\), respectively. The corresponding coefficient of determination, \(R^{2}\), is equal to 0.96.  
Figure 11: Strain range vs. number of cycles to failure [16].  
Finally, a usage factor \(UF\) was defined in order to sum the effects of all the described thermal-mechanical phenomena:  
\[UF=\frac{1}{N} \tag{17}\]  
whereas the total number of cycles prior to failure \(N_{tot}\) was evaluated as:  
\[N_{tot}=\frac{1}{UF_{tot}}=\frac{1}{UF_{i}+UF_{c}+UF_{f}}=\frac{1}{\frac{1}{N_{ i}}+\frac{1}{N_{c}}+\frac{1}{N_{f}}} \tag{18}\]  
where \(UF_{i}\), \(UF_{c}\), and \(UF_{f}\) are, respectively, the usage factors associated with plastic instability, creep, and low-cycle fatigue.