Question Statement:
For a homogeneous, isotropic material, the relation between the shear modulus
(G), Young's modulus (E), and Poisson's ratio (𝜈) is ____.
Options:
-
(A) G = 2E(1 + 𝜈)
-
(B) 2G = E(1 + 𝜈)
-
(C) E = 2G(1 + 𝜈)
-
(D) 2E = G(1 + 𝜈)
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.26)**
**1. Intuitive Physical Setup & Given Parameters**:
For a homogeneous, isotropic, linear elastic material, we are asked to identify the fundamental constitutive relation connecting:
- Shear modulus ($G$)
- Young's modulus ($E$)
- Poisson's ratio ($\nu$)
**2. Step-by-Step Continuum Mechanics Derivation**:
1. **Constitutive Stress-Strain Formulation**:
- Consider a square element subjected to pure shear stress $\tau_{xy} = \tau$ with all normal stresses zero ($\sigma_x = \sigma_y = 0$).
- The shear strain is:
$\gamma_{xy} = \frac{\tau}{G}$
2. **Analysis Along Principal Axes**:
- Under pure shear $\tau$, rotating the coordinate system by $45^\circ$ yields principal stresses:
$\sigma_1 = +\tau, \quad \sigma_2 = -\tau$
- By generalized Hooke's law, the principal tensile strain along direction 1 is:
$\varepsilon_1 = \frac{\sigma_1}{E} - \nu \frac{\sigma_2}{E} = \frac{\tau}{E} - \nu \left(-\frac{\tau}{E}\right) = \frac{\tau}{E}(1 + \nu)$
3. **Kinematic Strain Transformation**:
- From Mohr's circle of strain, the maximum engineering shear strain $\gamma_{xy}$ is related to the principal strains by:
$\gamma_{xy} = \varepsilon_1 - \varepsilon_2 = \varepsilon_1 - (-\varepsilon_1) = 2\varepsilon_1$
- Substituting $\varepsilon_1$:
$\gamma_{xy} = 2 \left[ \frac{\tau}{E}(1 + \nu) \right] = \frac{2\tau(1 + \nu)}{E}$
4. **Equating Shear Strains**:
$\frac{\tau}{G} = \frac{2\tau(1 + \nu)}{E} \implies \frac{1}{G} = \frac{2(1 + \nu)}{E}$
$E = 2G(1 + \nu)$
**3. Comprehensive Options & Distractor Analysis**:
- **(A) G = 2E(1 + 𝜈)**: ❌ **INCORRECT**. Inverts the elastic moduli positions.
- **(B) 2G = E(1 + 𝜈)**: ❌ **INCORRECT**. Divides by 2 on the wrong side.
- **(C) E = 2G(1 + 𝜈)**: ✅ **CORRECT**. The classic isotropic elasticity relation.
- **(D) 2E = G(1 + 𝜈)**: ❌ **INCORRECT**. Inverts the factor of 2.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: The twin isotropic relations are: $E = 2G(1 + \nu)$ and $E = 3K(1 - 2\nu)$, relating Young's modulus to shear and bulk moduli.
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