# Problem Restatement A steel is given **10% cold work**. Original material properties are: - Yield strength, \( S_y = 34.0 \) kpsi - Ultimate strength, \( S_u = 52.0 \) kpsi - True fracture strength, \( S_f = 95.5 \) kpsi - Strain hardening exponent, \( m = 0.26 \) **Find:** The new values for yield strength and ultimate strength after 10% cold work. --- # Step 1: Understand Cold Work (%CW) **Percent cold work** is defined as: \[ \%CW = \frac{A_0 - A_f}{A_0} \times 100\% \] Given \(\%CW = 10\%\), so: \[ \frac{A_0 - A_f}{A_0} = 0.10 \] \[ \frac{A_f}{A_0} = 1 - 0.10 = 0.90 \] --- # Step 2: Relationship Between Strength and Cold Work For metals: \[ \text{True strain}~(\epsilon) = \ln \left(\frac{A_0}{A_f}\right) \] \[ \Rightarrow \epsilon = \ln \left(\frac{1}{0.90}\right) = \ln(1.1111) \approx 0.1054 \] The **true stress at a given true strain** is: \[ \sigma = K \epsilon^m \] But for cold working, yield and ultimate strengths increase with cold work. The new yield strength (\(S_{y, \text{new}}\)) and ultimate strength (\(S_{u, \text{new}}\)) can be estimated as: \[ S_{\text{new}} = S_{\text{orig}} \times \left( \frac{A_0}{A_f} \right)^m \] or \[ S_{\text{new}} = S_{\text{orig}} \times e^{m \cdot \epsilon} \] --- # Step 3: Calculate New Strengths ## a. Calculate the Multiplier \[ e^{m \cdot \epsilon} = e^{0.26 \times 0.1054} = e^{0.027404} \approx 1.0278 \] ## b. New Yield Strength \[ S_{y, \text{new}} = S_y \times 1.0278 = 34.0~\text{kpsi} \times 1.0278 \approx 34.95~\text{kpsi} \] ## c. New Ultimate Strength \[ S_{u, \text{new}} = S_u \times 1.0278 = 52.0~\text{kpsi} \times 1.0278 \approx 53.45~\text{kpsi} \] --- # Step 4: Final Answers - **New yield strength:** \(\boxed{34.95~\text{kpsi}}\) - **New ultimate strength:** \(\boxed{53.45~\text{kpsi}}\) --- ## Summary Table | Property | Original (kpsi) | After 10% CW (kpsi) | |-------------------|-----------------|---------------------| | Yield strength | 34.0 | 34.95 | | Ultimate strength | 52.0 | 53.45 | --- ## Explanation - Cold working increases the strength of steel by introducing strain hardening. - The increase is calculated based on the strain hardening exponent (\(m\)) and the true strain from reduction in area. - The formula \(S_\text{new} = S_\text{orig} \times e^{m \epsilon}\) gives the new strength after cold work. - For 10% cold work, both yield and ultimate strengths increase by about 2.78%.