Kachanov represented continuum damage as an effective loss in mate­rial cross-section due to the formation and growth of internal voids. Consequently the internal stress corresponding to a nominal externally applied load increases with increasing damage. Kachanov assumed that damage could be represented by a quantity which he called the ‘continuity.’ The continuity is essentially the ratio of the remaining effective area A to the original area A0 . With accumulation of damage, the resulting internal stress (o) increases from initial value o0 to a value given by

°t = °o A [3.54]

The continuity term was later modified by Rabotnov and was called the damage parameter a, where

[3.55]

By assuming a power-law dependence of stress, the creep rate at constant temperature was described as

where m and p are material parameters. At time t = 0, a = 0 and the above equation assumes the power-law form. As a increases, the creep rate increases and when it achieves a critical value, the creep rate tends towards infinity and failure follows.

In order to describe the evolution of damage, Kachanov assumed that dam­age is a function of the initial stress a0. This was later generalized by Rabotnov who assumed that the damage is instead a function of the instantaneous stress and described the rate of change of damage through the following:

dm _ BO

~dt ~ (1 — m)r. [3.57]

Solving the above two equations gives the creep strain in the following form:

[3.58]

where ec is the instantaneous creep strain, eR is the rupture strain, t is the time and tR is the time to rupture. The shape of the creep curve described by Equation [3.58] is as shown in Fig. 3.23.

The damage tolerance parameter X is given by the following equation:

Я

1 + r — p

The material fails in the steady-state creep regime when X = 1. Ashby and Dyson97 have demonstrated that each damage micromechanism has a char­acteristic X and a characteristic shape of the creep curve. This implies that

the creep curve would assume different shapes for different values of X. Phaniraj et al.9S have established a correlation between the ratio of time to Monkman-Grant ductility (tMGD) and time to rupture (tR) and the damage tolerance parameter as given by

Figure 3.24 is based on this Equation [3.60] and shows that tMGD/tR is essen­tially constant for X > 4. The fMGD was suggested as time for onset of true tertiary creep damage and was considered to be an important parameter in identifying the useful creep life of a material. It also describes the time for which minimum creep ductility is ensured. Hence Phaniraj et al. contend that the stress to cause tMGD in 105 h can be used as a useful design criterion for creep of elevated temperature components.

Before concluding we present a few examples where the concepts dis­cussed in the previous sections may not be directly applied. Rather subtle modifications to the models are necessary in order to simulate the actual behavior of the material.

3.4 Case studies illustrating the role of other factors

In the following section, the effects of impurities, second phases and multi­axial loadings on creep of materials are discussed with examples taken from various classes of materials including ionic solids.