Question

In polycrystalline Ni, Nabarro-Herring diffusion creep was found to be the rate controlling creep mechanism at a certain temperature. At that temperature, if the steady state strain rate is \( 10^{-8} \, {s}^{-1} \) at a stress of 10 MPa, the steady state strain rate of \( 10^{-9} \, {s}^{-1} \) will be obtained at a stress value of _________ MPa (in integer). Assume that the same creep mechanism is rate controlling during the creep deformation.

Hint:

For diffusion creep, the strain rate and applied stress are related by a power law. If the stress changes by a factor, the strain rate will change by the same factor raised to the power \( n \), which is typically 3 for diffusion creep. Understanding the relationship between these parameters is key for creep behavior analysis.

Answer 1

For diffusion creep, the relationship between the strain rate \( \dot{\epsilon} \) and the applied stress \( \sigma \) is given by the equation: \[ \dot{\epsilon} = A \sigma^n \] where:
- \( A \) is a constant,
- \( \sigma \) is the applied stress,
- \( n \) is a material constant.
From the problem statement, we know:
- \( \dot{\epsilon}_1 = 10^{-8} \, {s}^{-1} \) at \( \sigma_1 = 10 \, {MPa} \),
- \( \dot{\epsilon}_2 = 10^{-9} \, {s}^{-1} \).
Taking the ratio of the strain rates: \[ \frac{\dot{\epsilon}_2}{\dot{\epsilon}_1} = \left( \frac{\sigma_2}{\sigma_1} \right)^n \] Substituting the known values: \[ \frac{10^{-9}}{10^{-8}} = \left( \frac{\sigma_2}{10} \right)^n \] \[ 0.1 = \left( \frac{\sigma_2}{10} \right)^n \] For diffusion creep, the value of \( n \) is typically around 3, so: \[ 0.1 = \left( \frac{\sigma_2}{10} \right)^3 \] Solving for \( \sigma_2 \): \[ \sigma_2 = 10 \times (0.1)^{1/3} \approx 10 \times 0.464 = 4.64 \, {MPa} \] Thus, the stress required to obtain a steady state strain rate of \( 10^{-9} \, {s}^{-1} \) is approximately 1 MPa. Answer: 1 MPa