A new statistical framework for the determination of safe creep life using the theta projection technique

Abstract

In this article a new estimation framework is put forward for the well-known theta projection technique which enables, for the first time, levels of confidence to be associated with the creep property predictions made using this technique. The predictions made from the resulting model are in the form of distributions, which is a substantial advance on existing life assessment methodologies used in high-temperature applications such as the disks and blades used in aero engines. This additional information should prove invaluable for questions related to the issues of possible life extension. When applied to data on Ti.6.2.4.6, accurate interpolations and extrapolations could be made of the actual distribution of the minimum creep rate, which in combination with the Monkman–Grant relation, also enabled accurate predictions to be made for the time to failure. For example, when extrapolating to a stress of 480 MPa, the time to failure was predicted to follow a non-normal distribution with a median time of 3450 h and a 95% confidence interval of 3250–3800 h. The single experimental data point available at this stress was consistent with such an extrapolation.

Appendix

The modified theta projection method takes the form given by Eq. 6a.

$$ \begin{gathered} \varepsilon_{ij} - \rho \varepsilon_{ij - 1} = {\user1{AF}}_{1} {\kern 1pt} \Uptheta_{1i} (P_{ij} - \rho P_{ij - 1} ) + {\user1{AF}}_{3} {\kern 1pt} \Uptheta_{3i} (T_{ij} - \rho T_{ij - 1} ) + \eta_{i} u_{ij}^{*}\quad {\text{with}}\quad u_{ij}^{*} = {{u_{ij} } \mathord{\left/ {\vphantom {{u_{ij} } {\eta_{i} }}} \right. \kern-\nulldelimiterspace} {\eta_{i} }} \hfill \\ v_{ij \, } = \rho v_{ij - 1} + u_{ij} ,\,P_{ij} = (1 - {\text{e}}^{{ - {\user1{AF}}_{2} \Uptheta_{2i} t_{ij} }} ),\,P_{ij - 1} = (1 - {\text{e}}^{{ - \Uptheta_{2i} t_{ij - 1} }} ),\,T_{ij} = ({\text{e}}^{{{\user1{AF}}_{4} {\kern 1pt} \Uptheta_{4i} t_{ij} }} - 1),\,T_{ij - 1} = ({\text{e}}^{{\Uptheta_{4i} t_{ij - 1} }} - 1) \hfill \\ \end{gathered} $$

(7)

with ρ fixed at unity and with the ${\user1{AF}}_{h}$ defined by Eq. 6b. The probability density function for the change in consecutive strain measurements made on specimen i is given by

$$ f\left( {\varepsilon_{ij} - \rho \varepsilon_{ij - 1} } \right) = \frac{{\lambda^{\lambda - 0.5} }}{{\eta_{i} \Upgamma (\lambda )}}{ \exp }\left[ {\sqrt \lambda \left\{ {u_{ij}^{*} + D} \right\} - \lambda { \exp }\left( {\frac{{u_{ij}^{*} + D}}{\sqrt \lambda }} \right)} \right] $$

(8)

where u ^* _ij is defined by Eq. (7). This type of distribution was first put forward by Bartlett and Kendall [25] and re parameterised by Prentice [26] to improve stability in estimation. Given ρ = 1, the log likelihood for the n _i values of $f\left( {\varepsilon_{ij} - \rho \varepsilon_{ij - 1} } \right)$ made on specimen i, conditional on Θ_1i –Θ_4i and γ₁ to γ₄ is then given by

$$ { \ln }\left( {L_{i} |\Uptheta_{li} - \Uptheta_{4i} ,\gamma_{1} - \gamma_{4} ,\eta_{i} } \right) = \sum\limits_{j = 1}^{{n_{i} }} {\left\{ {(\lambda - 0.5)\ln (\lambda ) - \ln \Upgamma (\lambda ) - \ln (\eta_{i} ) + \sqrt \lambda \left( {u_{ij}^{*} + D} \right) - \lambda e^{{\left( {D + u_{ij}^{*} } \right)/\sqrt \lambda }} } \right\}} $$

(9)

The log likelihood for all i units is then

$$ { \ln }(L) = \sum\limits_{i = 1}^{m} {\ln \left( {L_{i} |\Uptheta_{1i} - \Uptheta_{4i} ,\gamma_{1} - \gamma_{4} ,\eta_{i} } \right)} = \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{{n_{i} }} {\left\{ \begin{gathered} (\lambda - 0.5)\ln (\lambda ) - \ln \Upgamma (\lambda ) - \ln (\eta_{i} ) \hfill \\ + \sqrt \lambda \left( {u_{ij}^{*} + D} \right) - \lambda {\text{e}}^{{\left( {D + u_{ij}^{*} } \right)/\sqrt \lambda }} \hfill \\ \end{gathered} \right\}} } $$

(10)

where there are m test specimens. In principle this log likelihood could be maximised with respect to Θ_1i –Θ_4i , γ₁ to γ₄, and η _i . However, there are two problems with this approach. First, the variance–covariance matrix Σ and the mean matrix μ are absent from this log likelihood and so cannot be estimated in this way. This problem can be overcome by specifying the random nature of Θ_1i to Θ_4i and η _i

$$ \Uptheta_{{ 1 {\text{i}}}} = \mu_{\Uptheta 1} + {\text{z}}_{{ 1 {\text{i}}}} ;\quad \Uptheta_{{ 2 {\text{i}}}} = \mu_{\Uptheta 2} + {\text{z}}_{{ 2 {\text{i}}}} ;\quad \Uptheta_{{ 3 {\text{i}}}} = \mu_{\Uptheta 3} + {\text{z}}_{{ 3 {\text{i}}}} ;\quad \Uptheta_{{ 4 {\text{i}}}} = \mu_{\Uptheta 4} + {\text{z}}_{{ 4 {\text{i}}}} ;\quad \eta_{\text{i}} = \mu_{\eta } + {\text{z}}_{{ 5 {\text{i}}}} $$

(11)

where μ_Θ1 is the average of all the Θ_1i values through to μ_Θ4 which is the average of all the Θ_4i values and μ_η is the average of all the η _i values . Further, z_1i to z_5i are assumed to follow a joint normal distribution with a mean vector of zero and variance–covariance matrix given by Σ, with the restriction that the Θ _h and η are independent of each other. Further, z _1i to z _5i can calculated from standard normal variates in the following way

$$ \begin{gathered} {\text{z}}_{{ 1 {\text{i}}}} = \delta_{\Uptheta 1} {\text{w}}_{{ 1 {\text{i}}}} ;\quad {\text{z}}_{{ 2 {\text{i}}}} = \delta_{\Uptheta 12} {\text{w}}_{{ 1 {\text{i}}}} + \delta_{\Uptheta 2} {\text{w}}_{{ 2 {\text{i}}}} ;\quad {\text{z}}_{{ 3 {\text{i}}}} = \delta_{\Uptheta 13} {\text{w}}_{{ 1 {\text{i}}}} + \delta_{\Uptheta 23} {\text{w}}_{{ 2 {\text{i}}}} + \delta_{\Uptheta 3} {\text{w}}_{{ 3 {\text{i}}}} \hfill \\ {\text{z}}_{{ 4 {\text{i}}}} = \delta_{\Uptheta 14} {\text{w}}_{{ 1 {\text{i}}}} + \delta_{\Uptheta 24} {\text{w}}_{{ 2 {\text{i}}}} + \delta_{\Uptheta 34} {\text{w}}_{{ 3 {\text{i}}}} + \delta_{\Uptheta 4} {\text{w}}_{{ 4 {\text{i}}}} ;\quad {\text{z}}_{{ 5 {\text{i}}}} = \delta_{\eta } {\text{w}}_{{5{\text{i}}}} \quad \left( {{\text{i }} = {\text{ 1 to m}}} \right) \hfill \\ \end{gathered} $$

(12)

where w _1i to w _5i are standard normal variates, i.e., variables that follow independent normal distributions with means of zero and variances of 1. Further, these new delta values are related to the variance–covariance matrix Σ as follows

$$ \Uplambda = \left[ {\begin{array}{*{20}c} {\delta_{\Uptheta 1}^{{}} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {\delta_{\Uptheta 12}^{{}} } & {\delta_{\Uptheta 2}^{{}} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {\delta_{\Uptheta 13}^{{}} } & {\delta_{\Uptheta 23}^{{}} } & {\delta_{\Uptheta 3}^{{}} } & 0 & 0 & 0 & 0 & 0 & 0 \\ {\delta_{\Uptheta 14}^{{}} } & {\delta_{\Uptheta 24}^{{}} } & {\delta_{\Uptheta 34}^{{}} } & {\delta_{\Uptheta 4}^{{}} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\delta_{\eta }^{{}} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\quad {\text{with }}\Uplambda \Uplambda^{T} = \Upsigma $$

(13)

The superscript T in Eq. 13 stands for transpose so that Λ is a Cholesky decomposition of Σ.

The basic steps involved in maximising this resulting log likelihood have been summarised by Greene [27] as follows:

• 1. Obtain at random 5 draws from the standard normal distribution using the well known Box-Muller [28] method. These values then quantify w _1i to w _5i in Eq. 12, for i = 1.

• 2. Repeat steps one above m times to obtain values for w ₁₂ to w ₅₂ through to w _1m to w _5m .

• 3. Choose, at random, starting values for all the parameters in Λ of Eq. 13.

• 4. Insert the values obtained in steps 2 to and 3 into Eq. 12 to obtain values for z _1i to z _5i for i = 1 to m.

• 5. Choose, at random, starting values μ_Θ1 to μ_Θ4, μ_η and all the γ parameters in Eq. 6b. Insert these, together with the z _1i to z _5i values obtained in step 4 above into Eq. 11 to obtain values for Θ_1i to Θ_4i and η _i (for i = 1, m).

• 6. Insert the values obtained for Θ_1i to Θ_4i , η _i and γ₁ to γ₄ from step 5 above into Eq. 7 to calculate values for u ^* _ij (using ρ = 1). Then insert all these u ^* _ij values into Eq. 10 to obtain a numeric value for the log likelihood given a value for λ.

• 7. Then run a standard numerical optimisation procedure to maximise ln(L) with respect to μ_Θ1 to μ_Θ4, μ_η, γ₁ to γ₄ and the parameters in Λ. (Λ can then be converted to Σ using Eq. 13).

• 8. Finally, steps 1–7 are then repeated R times yielding R values for μ_Θ1 to μ_Θ4, μ_η, γ₁ to γ₄ and the parameters in Λ. The average of these R values can be taken as the simulated maximum likelihood estimates for these parameters, and the standard deviation in these R values as an estimate of their standard error. The maximised log likelihood is then given by the average of the R maximised log likelihood values, $\ln (\overline{L} )$

This estimation procedure can be repeated using different values for λ, and the value for λ that maximises $\ln (\overline{L} )$ is chosen as the correct value for λ. This is quite a straightforward procedure in practice because obtaining random draws from a standard normal distribution is easily implemented within popular and commercially available software packages such as Microsoft Excel [29]. In this package the Normsinv (Rand()) function is used to obtain a value at random from this distribution.