Let's break down the solution step by step: --- ### **Step 1: Understanding the Problem** - **Material:** 3105 Aluminum - **Initial thickness (\( t_0 \))**: 3.0 in - **Final thickness (\( t_f \))**: 0.3 in - **Tensile strength required:** ≥ 25,000 psi - **% Elongation required:** ≥ 5% - **Maximum cold work per step:** 80% reduction in thickness --- ### **Step 2: Calculate Total % Reduction Required** \[ \text{Total % Reduction} = \frac{t_0 - t_f}{t_0} \times 100\% \] \[ = \frac{3.0 - 0.3}{3.0} \times 100\% = \frac{2.7}{3.0} \times 100\% = 90\% \] --- ### **Step 3: Maximum Cold Work per Step** Maximum reduction per cold working step = 80%. Let’s check if we can do this in one step: - After one 80% reduction: \[ t = t_0 \times (1 - 0.8) = 3.0 \times 0.2 = 0.6 \text{ in} \] - But we need 0.3 in, so **one step isn't enough**. #### **Number of Steps Needed** Let’s denote the thickness after the first step as \( t_1 \): 1. **First Step:** \[ t_1 = t_0 \times (1 - 0.8) = 3.0 \times 0.2 = 0.6 \text{ in} \] 2. **Second Step:** \[ t_2 = t_1 \times (1 - 0.8) = 0.6 \times 0.2 = 0.12 \text{ in} \] This is less than 0.3 in, so we cannot do two full 80% reductions. #### **Better Approach: Two Steps, Second with Less Reduction** - **Step 1:** 80% reduction: \( 3.0 \to 0.6 \) in - **Step 2:** Reduction from 0.6 in to 0.3 in: \[ \text{Reduction} = \frac{0.6 - 0.3}{0.6} \times 100\% = 50\% \] Which is less than 80%, so **feasible**. --- ### **Step 4: Cold-Working and Annealing Procedure** - **Step 1:** Cold work from 3.0 in to 0.6 in (80% reduction) - Likely, after this much cold work, the material will be heavily strain hardened and brittle, with low ductility. - **Anneal** after this step to restore ductility. - **Step 2:** Cold work from 0.6 in to 0.3 in (50% reduction) - After this, the material will again be strain hardened. - For the required properties (tensile strength ≥ 25,000 psi, elongation ≥ 5%), **do not anneal after this step**, as annealing would lower strength. --- ### **Step 5: Check Properties after Cold Working** - **Cold Work:** Increases strength, reduces ductility. - **Annealing:** Recovers ductility, reduces strength. - After the final reduction, we want the material to be in a strain-hardened (cold-worked) condition to have high strength, but not too much so it still maintains at least 5% elongation. #### **Typical Properties of 3105 Aluminum:** - As-rolled (cold-worked) 3105-H18: - Tensile strength: ~28,000 psi - Elongation: ~2-5% - 3105-O (annealed): - Tensile strength: ~16,000 psi - Elongation: ~20% **So, with a 50% cold reduction in the final step, you are likely to meet the minimum required properties.** --- ### **Step 6: Process Summary** 1. **Step 1:** Cold work 3.0 in → 0.6 in (80% reduction) 2. **Step 2:** Anneal at appropriate temperature for 3105 Al (e.g., ~350–400°C) 3. **Step 3:** Cold work 0.6 in → 0.3 in (50% reduction) 4. **Step 4:** Do **not** anneal after final step to retain strength. --- ### **Step 7: Hot Working Alternative** If **hot working** is possible for the initial deformation, you could: - **Hot work** the plate from 3.0 in to 0.6 in (or even directly to 0.3 in if possible). - Hot working does not strain harden the material as much, so it remains ductile and easier to work. - **Cold work** only the final pass (0.6 in → 0.3 in, 50% reduction) to achieve the desired strength. - This makes the process easier and reduces the need to anneal between steps. --- ## **Final Answer (Summary Table)** | Step | Operation | Thickness (in) | % Reduction | Anneal? | |------|---------------------|----------------|-------------|-----------| | 1 | Cold work | 3.0 → 0.6 | 80% | No | | 2 | Anneal | 0.6 | — | Yes | | 3 | Cold work | 0.6 → 0.3 | 50% | No | - This sequence achieves the desired thickness and mechanical properties (≥25,000 psi strength, ≥5% elongation). **If hot working is possible for the first reduction:** - Hot work 3.0 in → 0.6 in (or 0.3 in) - Cold work only the final step to 0.3 in - No intermediate anneal needed --- ### **Conclusion** - **Cold work–anneal–cold work** is the best route if constrained to cold working. - **Hot working** is more efficient for large thickness reductions, followed by a final cold work for property control. Let me know if you need more detail on the calculations or diagrams!