• id: JME-08-N09
• title: Inter-Relations of Y, G, B, and Poisson's Ratio
• tags: elastic-constants, relations, derivation

Three key relationships connect the four elastic constants:

1. $Y = 2G(1 + \nu)$
2. $Y = 3B(1 - 2\nu)$
3. $Y = \frac{9BG}{3B + G}$

From (1): $\nu = Y/(2G) - 1$. From (2): $\nu = (1 - Y/(3B))/2$. Equating gives relation (3). These are derived using the theory of isotropic linear elasticity. Only 2 of the 4 constants ($Y$, $G$, $B$, $\nu$) are independent — any two determine the others. For $\nu = 0.5$: $Y = 3G$ and $B \to \infty$ (incompressible). For $\nu = 0$: $Y = 2G = 3B$.