Exploiting Functional Relationships Between MPC Omega, Theta, and Sinh-Hyperbolic Continuum Damage Mechanics Model
Authors: Haque Mohammad Shafinul , Stewart Calvin M. 
Published: December 2016

The MPC Omega and Theta models for creep deformation and life prediction have become popular in recent years. Both models offer better prediction than classical constitutive models such as Norton Power law, Bailey-Norton law, and Norton-Soderberg law to name a few. The Omega model uses a strain hardening approach and requires two material constants for creep deformation and life prediction. An analytical solution to the constants are available and it is easy to manipulate and implement numerically. However, the analytical damage of the Omega model predicts an unrealistic linear damage evolution. The Theta model uses a time-hardening approach, and requires four constant that are a function of stress and temperature. For materials under isothermal conditions, with tertiary creep dominant deformation, the Theta model constants can be determined using only two constants. Life prediction using the Theta and Omega models depends on the final creep strain. The final creep strain observed in an experiment is stochastic; dependent on the material, testing conditions, and operator. The statistics of final creep strain must be investigated before the Theta or Omega models can be applied. In literature, some authors add a nonlinear damage variable to the Theta model; however, critical damage at rupture is not unity violating the assumptions of continuum damage mechanics. There is a superior Sin-hyperbolic continuum damage model available in the literature that can be used to overcome these problems. It is hypothesized that a functional relationships exist between the three models and these relationships can be exploited to achieve more accurate and easy to implement creep deformation and life predictions. In this study, the relationships between the constants of MPC Omega, Theta, and a Sin-hyperbolic CDM models are determined analytically. The sin-hyperbolic model incorporates a continuum damage variable in the creep strain rate equation. The damage function exhibits a more realistic elliptical path and is constructed such that damage is always unity at rupture. This function facilitates conversion of one models’ constants to the constants of the other two. The relationships between the constants are identified, while maintaining dimensional homogeneity. Using the derived relationships, the three models can be easily compared and the disadvantages of each respective model can be avoided. Experimental data at four different configurations of stress (6.3 to 36.5 ksi) and temperature (1200 to 1800°F) (sixteen data sets) for Hastelloy X is used to compare the models. Creep rupture data at seven temperature levels (600 to 1000°C) and a wide stress range (5 to 500 MPa) is used to analyze life prediction. The constants for each model are determined. Using one models’ constants and the derived relationships; the predictions of the other two models are generated. It is observed that the relationship generated curves agree with experimental data.
Finally, it is demonstrated that using the derived relationships, the most useful aspects of each model can combined. An elliptical damage evolution curve is obtained for the Omega model. The final creep strain rate dependency problem of the Theta model can be avoided. It is observed that the Sinh model becomes more flexible and easy to implement.