A material experiences transitions in mechanisms when the applied stress or the test temperature is varied.

3.5.1 The Bird-Mukherjee-Dorn equation4

As discussed earlier, for the same temperature and stress combinations, a material can creep via different mechanisms if the grain size is different. In fact, as Equation [3.25] would suggest, a material creeps with higher strain rates for smaller grain sizes. For relatively smaller grain sizes, creep could occur by diffusion of vacancies through the grain boundaries. But larger grain sized materials, under the same stress and temperature conditions, could creep by dislocation-based processes or by lattice diffusion processes.

In order to illustrate the effect of stress and temperature on transitions in mechanisms it is necessary to suitably modify the creep equation. Sherby analyzed steady-state creep-rate results using strain rate compensated by diffusivity versus stress normalized by temperature-dependent modulus of elasticity16

[3.42]

so that different materials can be compared with each other. While this equation seems to work well, it would be more appropriate to use dimen­sionless strain rate as well, and Dorn and co-workers4 proposed a dimen­sionless equation that can appropriately describe the effect of changes in stress, temperature and microstructure on mechanisms of creep. This equa­tion known as the Bird-Mukherjee-Dorn (BMD) equation is given by

[3.43]

As shown in Fig. 3.16a, changes in stress and temperature for a given con­stant microstructure of the material can reveal changes in the stress expo­nent value.60 At low normalized stress values, the deformation mechanism appears to proceed with a stress exponent value of 1. At intermediate stress values a stress exponent value of 2 corresponding to GBS is obtained. At the highest normalized stress values, the mechanism of deformation oper­ates with a stress exponent value of 5 corresponding to power-law creep. The diffusivity value utilized for constructing the plot corresponds to the lattice diffusion activation energy of titanium, and thus data at different temperatures follow different curves in the GBS and viscous creep regimes where the appropriate activation energy is that for grain boundary diffu­sion. On the contrary, if one chooses to use the activation energy for grain boundary diffusion, the data at high stresses will lead to different lines for different temperatures. The BMD plot thus allows an easy understanding of the transitions in creep mechanisms following changes in stress and temper­ature. Such an analysis was found to be very useful in delineating various creep mechanisms in Zr-based alloys as depicted in Fig. 3.16b.73 Moreover, such plots made for different materials would show the material behaviors at equivalent loading conditions.4