Which of the following statements regarding the elasticity matrix and elastic constants of linear isotropic and anisotropic aerospace materials are **CORRECT**?

(A) A general anisotropic linear elastic material has 21 independent elastic constants in its compliance/stiffness matrix

(B) An orthotropic material (such as a unidirectional composite ply) has 9 independent elastic constants

(C) A transversely isotropic material (symmetric about an axis of fiber alignment) has 5 independent elastic constants

(D) An isotropic homogeneous material has only 2 independent elastic constants (e.g. Young's modulus $E$ and Poisson's ratio $\nu$)

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles Classification of material elastic symmetry in 3D linear elasticity: 1. **General Anisotropic:** Begins with $6 \times 6 = 36$ constants. Existence of a strain energy potential function makes the stiffness matrix symmetric ($C_{ij} = C_{ji}$), reducing the number of independent constants to $\frac{6 \times 7}{2} = 21$. 2. **Orthotropic:** Three mutually orthogonal planes of material symmetry eliminate shear-extension coupling terms, reducing the number of independent constants to **9** ($E_1, E_2, E_3, G_{12}, G_{23}, G_{31}, \nu_{12}, \nu_{23}, \nu_{31}$). 3. **Transversely Isotropic:** One axis of symmetry with isotropic behavior in the transverse plane (unidirectional fiber composite), reducing the independent constants to **5**. 4. **Isotropic:** Infinite planes of symmetry, reducing the independent constants to strictly **2** (e.g., $E$ and $\nu$, or Lamé parameters $\lambda$ and $\mu$). ### 2. Statement Evaluation - **All statements A, B, C, D are universally CORRECT.** ### 3. Exam Trap & Standard Reference - **Trap Alert:** Memorize the classic hierarchy: Anisotropic (21) $\to$ Monoclinic (13) $\to$ Orthotropic (9) $\to$ Transversely Isotropic (5) $\to$ Isotropic (2). - **Standard Reference:** Jones, R.M., *Mechanics of Composite Materials*, Chapter 2.

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