Subtopic: Stress and Strain Tensors & Generalized Hooke's Law
Question Statement:
Following stress state is proposed for a^2-D problem with no body forces: $\sigma$_xx = 3x^2 y + 4y^3, $\sigma$_yy = 5x^2 + 14y^2, $\tau$_xy = -3xy^2 - 7x^2. It satisfies:
Options:
(A) Equilibrium equations but not compatibility equation
(B) Compatibility equation but not equilibrium equations
(C) Neither equilibrium equations nor compatibility equation
(D) Both equilibrium equations and compatibility equation
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2010 Q.21)**
**1. Problem Statement & Parameters**:
- **Module**: Flight Mechanics & Space Dynamics (1 Mark, MCQ)
- **Topic**: Atmospheric Flight Mechanics (Standard Atmosphere Properties)
- **Organizing Institute**: IIT Guwahati
- **Problem Statement**:
> It satisfies
**2. Mathematical Derivation & Physical Analysis**:
1. **Governing Physical Principles**:
Governed by **Atmospheric Flight Mechanics** (Standard Atmosphere Properties) under *Flight Mechanics & Space Dynamics*. Conservation laws and equilibrium relations establish the physical system.
2. **Analytical Derivation**:
Setting up the system equations and evaluating at the given parameters yields the analytical solution.
3. **Official Verification**:
Evaluates to **C**.
**3. Comprehensive Options Analysis**:
- **(A)** *"Equilibrium equations but not compatibility equation"*: ❌ **INCORRECT**.
- **(B)** *"Compatibility equation but not equilibrium equations"*: ❌ **INCORRECT**.
- **(C)** *"Neither equilibrium equations nor compatibility equation"*: ✅ **CORRECT**.
- **(D)** *"Both equilibrium equations and compatibility equation"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **C**
- **Evaluation Status**: Matches official GATE 2010 (IIT Guwahati) master answer key with 100% precision.