Subtopic: Conservation Equations of Mass, Momentum & Energy
Question Statement:
The number of independent elastic constants that a fully anisotropic linear elastic
material can have is ______.
Options:
(A) 36
(B) 21
(C) 10
(D) 2
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.16)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to determine the number of independent elastic constants in the generalized Hooke's law for a fully anisotropic (triclinic) linear elastic material.
**2. Step-by-Step Mathematical Derivation**:
1. **Generalized Hooke's Law Formulation**:
In 3D linear elasticity, stress and strain tensors are related via the 4th-order stiffness tensor:
$\sigma_{ij} = C_{ijkl} \varepsilon_{kl} \quad (i, j, k, l \in \{1, 2, 3\})$
The full tensor has $3^4 = 81$ total coefficients.
2. **Minor Symmetries (Stress & Strain Symmetry)**:
Due to equilibrium of moments (no body couples), the Cauchy stress tensor is symmetric: $\sigma_{ij} = \sigma_{ji} \implies C_{ijkl} = C_{jikl}$.
By definition of engineering strain: $\varepsilon_{kl} = \varepsilon_{lk} \implies C_{ijkl} = C_{ijlk}$.
This reduces the $3 \times 3$ tensors to 6 independent stress and 6 independent strain components, reducing the tensor to a $6 \times 6$ matrix with $6 \times 6 = 36$ constants.
3. **Major Symmetry (Existence of Strain Energy Potential)**:
For a hyperelastic material possessing an elastic strain energy potential $U$ such that $\sigma_{ij} = \frac{\partial U}{\partial \varepsilon_{ij}}$:
$C_{ijkl} = \frac{\partial^2 U}{\partial \varepsilon_{ij} \partial \varepsilon_{kl}} = C_{klij}$
The $6 \times 6$ matrix $[C]$ is therefore symmetric ($C_{mn} = C_{nm}$).
The number of independent elements in a symmetric $6 \times 6$ matrix is:
$N = \frac{6(6 + 1)}{2} = \frac{42}{2} = 21$
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 36**: ❌ **INCORRECT**. This is the number before invoking the major symmetry (existence of strain energy function $U$).
- **(B) 21**: ✅ **CORRECT**. Fully anisotropic material has exactly 21 independent elastic stiffness constants.
- **(C) 10**: ❌ **INCORRECT**. Does not correspond to standard crystal symmetry (orthotropic has 9, transversely isotropic has 5).
- **(D) 2**: ❌ **INCORRECT**. Corresponds to an isotropic material (Lamé constants $\lambda, \mu$, or $E, \nu$).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: Memorize the sequence of independent elastic constants: Anisotropic = 21, Orthotropic = 9, Transversely Isotropic = 5, Isotropic = 2.
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