The figure shows plots of two yield loci for an isotropic material, where $\sigma$I and $\sigma$II
are the principal stresses, and $\sigma$Y is the yield stress in uniaxial tension. Which of the
following statements is/are correct?
Scientific schematic diagram reproduced from official IIT paper
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 $\sigma$I and $\sigma$II are the principal stresses, and $\sigma$Y 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.
