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.