2. The existing creep damage constitutive models
2.1. Kachanov-Rabotnov damage model
Continuum damage mechanics (CDM) was first developed by Kachanov7, and then extended by Rabotnov8. The Kachanov-Rabotnov damage model, also known as K-R model can be expressed as:
where is creep strain rate, is the damage ranging from 0 to 1, is creep damage rate, is the applied stress, is the stress exponent, and ,,and are the material constants. When , failure occurs. The K-R model assumes that the damage mainly occurs in the tertiary stage, i.e. damage is zero before entering the tertiary stage. Hence when Eq. can be written as:
where is the minimum creep strain rate. It can be seen that the K-R model is essentially based on the power-law equation. There are five material constants are required to define in the K-R model, and several constant determination methods have been proposed by different researchers30–32.
2.2. Liu-Murakami damage model
Liu and Murakami pointed out that significant damage localization and mesh-dependence of K-R model could be found when used in FE analysis33. This can be attributed to the damage term in the denominator. As the damage approaches to unity, the strain rate and damage rate will approach to infinity, due to the terms and in Eq. and Eq., respectively. Therefore, the exponential form of damage constitutive model was proposed by Liu and Murakami33:
where is the stress exponent, and, , and are the material constants. Research results of copper showed that the great improvements in damage localization and mesh-dependence were achieved by the Liu-Murakami model34.
2.3. Sinh damage model
Both the K-R model and the Liu-Murakami model are established on the basis of the power-law equation. The characteristics of the stress exponent changing with stress and temperature limit their applications. To overcome the limitation of power-law equation, Haque and Stewart proposed the Sinh damage constitutive model20:
where and are the secondary creep stage constants which can be obtained from the minimum creep rate equation , , , and are material constants. To ensure when failure occurs, constant is defined as , where is the final creep rate when specimen fracture, which can be determined directly from the experimental data.
It can be observed that the above damage models are essentially based on the power-law equation and Sinh equation. The undetermined stress and temperature dependence of related parameters make it difficult to achieve reliable extrapolation. Therefore, most of the published literatures aimed to model the creep behavior within a certain stress range and at a single temperature17,35–37. This means that it is very challenging to extrapolate the short-term creep data obtained in the laboratory to the long-term practical service. It is necessary to develop a novel damage constitutive model to achieve reliable extrapolation, in which all parameters have clear stress and temperature dependence.