Question

Strain hardening behavior of an alloy is given by \( \sigma = 1100 \epsilon^{0.3} \), where \( \sigma \) and \( \epsilon \) are true stress and true strain, respectively. The alloy is cold drawn to an unknown amount of strain, followed by tensile testing. If the tensile test showed 10% reduction in area at maximum load, then the unknown amount of strain from prior cold work is ................... (round off to 2 decimal places).

Hint:

A key concept in plasticity is that the strain history matters. The condition \( \epsilon = n \) for necking applies to the *total* accumulated strain from the material's softest (annealed) state. Any prior cold work "uses up" part of this strain capacity.

Answer 1

Step 1: Understanding the Concept:
The problem relates prior cold work to the behavior of a material in a subsequent tensile test. For a material that follows the Hollomon equation (\( \sigma = K \epsilon^n \)), tensile instability (necking) begins when the true strain equals the strain hardening exponent, \(n\). This total true strain is the sum of any strain from prior work and the strain added during the tensile test.
Step 2: Key Formula or Approach:
1. Hollomon Equation: \( \sigma = K \epsilon^n \). From the given equation, the strain hardening exponent \(n = 0.3\).
2. Necking Condition: Necking starts at a true strain \( \epsilon_{total} = n \).
3. Strain Relationship: The total strain is the sum of prior cold work strain and the strain during the tensile test: \( \epsilon_{total} = \epsilon_{prior} + \epsilon_{test} \). 4. True Strain from Reduction in Area (RA): \( \epsilon = \ln\left(\frac{1}{1 - \text{RA}}\right) \).
Step 3: Detailed Calculation:
1. Determine the total true strain to necking:
From the necking condition, the total strain the material can sustain before necking is: \[ \epsilon_{total} = n = 0.3 \] 2. Calculate the true strain during the tensile test:
The test showed a 10% reduction in area at maximum load (which is the point of necking). \[ \text{RA} = 10% = 0.10 \] \[ \epsilon_{test} = \ln\left(\frac{1}{1 - 0.10}\right) = \ln\left(\frac{1}{0.9}\right) \approx 0.10536 \] 3. Calculate the prior cold work strain:
Using the strain relationship:
\[ \epsilon_{prior} = \epsilon_{total} - \epsilon_{test} \] \[ \epsilon_{prior} = 0.3 - 0.10536 = 0.19464 \] 4. Round to 2 decimal places: \[ \epsilon_{prior} \approx 0.19 \] Step 4: Final Answer:
The unknown amount of strain from prior cold work is 0.19.