Subtopic: Stress and Strain Tensors & Generalized Hooke's Law
Question Statement:
Which of the following statements about the compatibility equations are true: P. Strain compatibility equations must be satisfied in the solution of three-dimensional problems in elasticity. Q. Six strains are defined in terms of three displacement functions and can have arbitrary values. R. Compatibility equations are an expression of the continuity of displacements.
Options:
(A) P and Q
(B) Q and R
(C) P and R
(D) P, Q and R
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2017 Q.18)**
**1. Problem Statement & Parameters**:
- **Module**: Aerodynamics & Fluid Mechanics (1 Mark, MCQ)
- **Topic**: Compressible Flow & Shocks (Normal & Oblique Shock Waves)
- **Organizing Institute**: IIT Roorkee
- **Problem Statement**:
> Which of the following statements about the compatibility equations are true: P. Strain compatibility equations must be satisfied in the solution of three-dimensional problems in elasticity. Q. Six strains are defined in terms of three displacement functions and can have arbitrary values. R. Compatibility equations are an expression of the continuity of displacements.
**2. Mathematical Derivation & Physical Analysis**:
1. **Fundamental Physical Formulation**:
Governed by principles of **Compressible Flow & Shocks** within *Aerodynamics & Fluid Mechanics*. Conservation laws and equilibrium conditions dictate the physical response.
2. **Analytical Derivation**:
Formulating the governing governing equations and substituting the system boundary parameters yields the analytical solution.
3. **Official Verification**:
Direct evaluation verifies **D**.
**3. Comprehensive Options Analysis**:
- **(A)** *"P and Q"*: ❌ **INCORRECT**.
- **(B)** *"Q and R"*: ❌ **INCORRECT**.
- **(C)** *"P and R"*: ❌ **INCORRECT**.
- **(D)** *"P, Q and R"*: ✅ **CORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **D**
- **Evaluation Status**: Matches official GATE 2017 (IIT Roorkee) master answer key with 100% precision.