3. A novel damage constitutive model based on TTC relations
In recent investigation28, the intrinsic relations between threshold stress, tensile properties, and creep behavior were established. These relations, called TTC relations, can be expressed as
where , , and are constants, is the threshold stress, is the ultimate tensile strength, is the minimum creep rate, is the gas constant (), is rupture time, and is the apparent activation energy of TTC relations which can be calculated at constant . Rearranging Eq. and Eq. to obtain and as follows
TTC relations are established on the basis of assumption that and as , while and as according to threshold stress theory38–40. Combining the TTC relations with continuum damage mechanics (CDM), a novel damage constitutive model, called TTC CDM-based model can be derived as
where is creep strain rate, is damage (,) is damage rate, and are parameters which depend on stress and temperature. To ensure when failure occurs, the constant is also defined as
where is the final creep rate, which can be determined directly from experimental data. Introducing constant into Eq., the experimental damage takes the form
Integration of damage evolution Eq. with assumption of initial time and initial damage , the analytical damage takes the form
when , . The constant controls the curvature of damage evolution curve, which can be determined by least square fitting of experimental damage Eq. and analytical damage Eq..