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.