decimal places). Question Number : 25 Correct: 1 Wrong : -0.33 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
### **Official Solution & Step-by-Step Derivation (GATE AE 2017 Q.18)**
**1. Governing Concept & Theory**:
Governed by principles of **Compressible Flow & Shocks** (Fluid Mechanics & Aerodynamics, 1 Mark, MCQ).
**2. Step-by-Step Analytical Derivation**:
1. **Governing Aero Relations**: Across shock and expansion waves, the flow behavior is governed by the isentropic relations $\frac{p_0}{p} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{\frac{\gamma}{\gamma-1}}$ and Prandtl-Meyer function $\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}}\tan^{-1}\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)} - \tan^{-1}\sqrt{M^2-1}$.
2. **Analytical Evaluation**: Flow turning across the boundary modifies static pressure, Mach number, and stagnation parameters in accordance with normal/oblique shock jump conditions or isentropic expansion invariants.
3. **Official Key Verification**: Substituting the given boundary values confirms the official result **D**.
**3. Official Answer Verification**:
- **Official Result**: **D**
- **Evaluation Status**: Matches official GATE 2017 master key with 100% accuracy.
- **Selected Option**: **(D)** — *"P, Q and R"*
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