### 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 MPaThis 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.

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)

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)

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

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Concept explainers

The Space Lattice and Unit Cells

Every matter consists of atoms, molecules, and ions. If these atoms, molecules, and ions (micro-particles) are represented in a three-dimensional space-occupying certain specific points in space, then these arrangements are termed as space lattices and the specific points which are being occupied by atoms, molecules and ions are termed as lattice points. The atoms, molecules, and ions at the lattice points are generally grouped in a regular fashion and repeat periodically at the three-dimensional space.

Question

### 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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