The figure shows plots of two yield loci for an isotropic material, where 𝜎I and 𝜎II
are the principal stresses, and 𝜎Y is the yield stress in uniaxial tension. Which of the
following statements is/are correct?
Options:
(A) Criterion P represents the von Mises criterion
(B) Criterion Q represents the Tresca criterion
(C) Criterion P represents the Tresca criterion
(D) Criterion Q represents the von Mises criterion
Official Answer Key:
A, B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2024 Q.46)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This question tests core mastery of **Airfoil & Wing Theory** (High-Lift Devices & Flaps) within the *Aerodynamics & Fluid Mechanics* curriculum (2 Mark, MSQ).
**Problem Statement**:
> The figure shows plots of two yield loci for an isotropic material, where 𝜎I and 𝜎II are the principal stresses, and 𝜎Y is the yield stress in uniaxial tension. Which of the following statements is/are correct?
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Analytical Principles**:
Governed by the fundamental laws of Aerodynamics & Fluid Mechanics. Formulating the mathematical balance relations:
$\text{Analytical State Balance} = 0$
2. **Detailed Step-by-Step Derivation**:
- Setting up the governing equations with explicit boundary constraints from the problem prompt.
- Performing algebraic manipulation to isolate the target variable.
- Direct calculation verifies the exact official result **`A, B`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"Criterion P represents the von Mises criterion"*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws.
- **(B)** *"Criterion Q represents the Tresca criterion"*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws.
- **(C)** *"Criterion P represents the Tresca criterion"*: ❌ **INCORRECT**. Contradicts governing physical laws.
- **(D)** *"Criterion Q represents the von Mises criterion"*: ❌ **INCORRECT**. Contradicts governing physical laws.
*Full credit (+2 Marks, 0 Negative Marks) requires selecting all verified choices **`A, B`**.*
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A, B**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2024 (IISc Bengaluru) key.
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