An FCC crystal is pulled in tension along the [100] direction. (a) Determine the Schmid factor for all slip systems. (b) Identify the slip system(s) that would be activated first. (c) What is the tensile stress under which this crystal will flow plastically? (t. = 50 MPa)

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### Problem Statement: 

#### An FCC crystal is pulled in tension along the [100] direction.

**Tasks:**
(a) Determine the Schmid factor for all slip systems.

(b) Identify the slip system(s) that would be activated first.

(c) What is the tensile stress under which this crystal will flow plastically? 

*(Given τ_c = 50 MPa)*

---

#### Explanation:

1. **Determining the Schmid Factor:**
   - The Schmid factor (m) is calculated as \( m = \cos(\phi) \cos(\lambda) \), where:
     - \( \phi \) is the angle between the direction of the applied stress and the slip direction.
     - \( \lambda \) is the angle between the direction of the applied stress and the normal to the slip plane.
   - In a Face-Centered Cubic (FCC) crystal, the common slip systems are of the form {111}<110>.
   - Detailed step-by-step derivation for each slip system is required to find \( \phi \) and \( \lambda \).

2. **Identifying the First Activated Slip Systems:**
   - The slip system with the highest Schmid factor will be activated first.
   - This involves comparing the Schmid factors calculated in the previous step.

3. **Calculating the Tensile Stress for Plastic Flow:**
   - Using the critical resolved shear stress (\( \tau_c \)), the tensile stress (\( \sigma \)) is calculated using \( \sigma = \frac{\tau_c}{m_{\text{max}}} \), where \( m_{\text{max}} \) is the maximum Schmid factor found.

##### Given:
   - Critical Resolved Shear Stress (\( \tau_c \)) = 50 MPa

This task involves understanding crystallography and the mechanics of materials and will provide insight into the application of theoretical concepts in predicting material behavior under stress.
Transcribed Image Text:### Problem Statement: #### An FCC crystal is pulled in tension along the [100] direction. **Tasks:** (a) Determine the Schmid factor for all slip systems. (b) Identify the slip system(s) that would be activated first. (c) What is the tensile stress under which this crystal will flow plastically? *(Given τ_c = 50 MPa)* --- #### Explanation: 1. **Determining the Schmid Factor:** - The Schmid factor (m) is calculated as \( m = \cos(\phi) \cos(\lambda) \), where: - \( \phi \) is the angle between the direction of the applied stress and the slip direction. - \( \lambda \) is the angle between the direction of the applied stress and the normal to the slip plane. - In a Face-Centered Cubic (FCC) crystal, the common slip systems are of the form {111}<110>. - Detailed step-by-step derivation for each slip system is required to find \( \phi \) and \( \lambda \). 2. **Identifying the First Activated Slip Systems:** - The slip system with the highest Schmid factor will be activated first. - This involves comparing the Schmid factors calculated in the previous step. 3. **Calculating the Tensile Stress for Plastic Flow:** - Using the critical resolved shear stress (\( \tau_c \)), the tensile stress (\( \sigma \)) is calculated using \( \sigma = \frac{\tau_c}{m_{\text{max}}} \), where \( m_{\text{max}} \) is the maximum Schmid factor found. ##### Given: - Critical Resolved Shear Stress (\( \tau_c \)) = 50 MPa This task involves understanding crystallography and the mechanics of materials and will provide insight into the application of theoretical concepts in predicting material behavior under stress.
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