The constitutive relationships identified in the previous sections are appli­cable for a wide variety of metals and alloys. However the strain rates of deformation might assume values different from model predictions even while the parametric dependencies remain the same. This was discussed in case study (3.7.2) and was attributed to the presence of impurities. On the other hand, there are cases where the strain rates of deformation as well as parametric dependencies can turn out to be different. Such instances are encountered while dealing with two — or multi-phase alloys that exhibit pre­cipitation or dispersion strengthening. Strain compatibility issues as well as differences in deformation rates of individual phases contribute to discrep­ancies in experimental observations and traditional creep model predic­tions. To this end, analytical models have been proposed to understand the creep behavior of multi-phase alloys.103-105 Here we present a case where the second phase is rigid and is added to enhance the overall strength of the alloy. For particle strengthened alloys, stress exponents higher than those predicted by established creep models106 and/or anomalous varia­tion of stress exponent with stress are observed.107 This is rationalized by the introduction of a friction or resisting stress also known as back stress

as demonstrated by Li et al. im As noted in Fig. 3.27a, the stress exponent decreases with increasing stress.

By introducing a friction stress (t0) the creep behavior of this alloy could be described by the following equation:

where у is the strain rate of deformation, т is the applied stress, t0 is the threshold stress and the rest of the terms are as described previously. Following the introduction of the threshold stress, the creep strain rates, Y1/n, are plotted on a linear scale against the applied stress. Here n is chosen as that value where a linear correlation between jlln and т is obtained. Extrapolation of the straight line to the x-axis provides a value for the threshold stress (Fig. 3.27b). The threshold stress is attributed to the pres­ence of the second phase which acts as an obstacle to dislocation motion. Such threshold stress based models have been considered in 60 s in analyzing the creep behavior of precipitation and dispersion strengthened alloys.108