1. Introduction
Due to the increasing requirements of low carbon emission and high efficiency, creep failure has become one of the most important failure modes for many engineering components, such as gas turbine, aero-engine, and power generation system, etc. Therefore, the ability to accurately evaluate and predict the creep behavior is of great importance for the integrity of high temperature structure. The typical creep deformation curve can be defined in three distinct stages, i.e. primary, secondary, and tertiary stages1. In the primary stage, with the increase of time-dependence creep deformation, the work hardening occurs gradually, the mobile dislocation density and the creep strain rate decrease with time2. When the creep rate reaches to the minimum value, creep enters the secondary stage. With further development of deformation, the onset of tertiary creep occurs. Dislocations gradually pill up at defects, such as grain boundaries and second-phase particles, which lead to stress concentration. And the cavities gradually nucleate at the location of stress concentration, then grow up and coalesce into macro-cracks3–5. Finally, it results in the fracture of the material.
It is obvious that the creep deformation is always accompanied by the damage evolution process, Dyson6 identified three categories of creep damage: (1) strain-induced damage(including grain boundary cavitation, dynamic subgrain coarsening, and mobile dislocation multiplication); (2) thermal-induced damage(including coarsening of particles, and solute depletion); (3) environment-induced damage (including oxidation, sulphuration, and carbonization). In order to study the damage process, continuum damage mechanics (CDM) was developed by Kachanov7 who introduced a continuity factor to reflect the degree of material degradation. Based on the creative work of Kachanov, Rabotnov8 established the damage variable and defined the concept of effective stress. After that, Lemaitre9, Chaboche10, Krajcinovic11, and Murakami12, etc. further developed the continuum damage mechanics theory.
Continuum damage mechanics is a branch of damage mechanics, which is used to study the mechanical process of damage. And it can be divided into physically based damage constitutive model and empirically based damage constitutive model13. The former usually develops multiple damage variables to distinguish the effect of different damage mechanisms, and defines a damage variable separately for each damage mechanism. On the contrary, the latter defines a single empirical damage variable to quantify the material degradation caused by different damage mechanisms. In general, physically based damage constitutive model contains a large number of material parameters, which leads to many difficulties and inconveniences in engineering application. Because of the simplicity of the empirically based damage constitutive model, such as Kachanov-Rabotnov damage model, Liu-Murakami damage model and Sinh damage model, it has been widely used in practical applications. Hyde studied the creep behavior of notched P91 steel specimens by using Kachanov-Rabotnov damage model and Liu-Murakami damage model14. Guo investigated damage accumulation of manifold component and modified Liu-Murakami damage model15. Saberi utilized Liu-Murakami damage model to assess the life of blade-disk attachments16. Praveen17 and Wang18 used Kachanov-Rabotnov damage model to predict the creep damage behavior of 316LN steel and UNS N1003 alloy, respectively. Recently, a Sinh damage model was proposed by Haque19, and some comparative analysis of Sinh and Kachanov-Rabotnov damage models can also be found in available published literatures20–22.
However, it is worth noting that the empirically based damage constitutive model is generally established on the basis of power-law equation and Sinh equation. It is difficult to reliably extrapolate the short-term creep data obtained under high stress and high temperature in laboratory condition to the long-term practical condition of engineering components, due to the lack of explicit stress and temperature dependence of the constants in above damage constitutive models. In order to achieve reasonable extrapolation, Cano23tried to combine the Wilshire equations with the damage constitutive model. However, the adopted region-splitting method was contrary to the original intention of Wilshire and made the model more complex24–27.
Our latest research results28 showed that intrinsic relations exist between threshold stress, tensile properties and creep behavior, and the TTC relations were proposed, which were better at predicting the creep behavior than Wilshire equations. Therefore, to ensure the accuracy of extrapolation of long-term creep behavior, a novel damage constitutive model was derived which combines the TTC relations with continuum damage mechanics. Post-audit validation was also conducted to verify the predictive ability of the novel model, and it was determined that the model displays satisfactory predictive results. Interestingly, the nonlinear creep damage accumulation effect was revealed by the novel model, which was consistent with our previous experimental results29. And this phenomenon further showed the accuracy of the damage constitutive model from the aspect of damage accumulation process.