The relationship between the elastic constants, derived from the general constitutive equations, establishes that Young's modulus ( $E$ ) and Poisson's ratio ( $\nu$ ) can be fully expressed by the Bulk modulus ( $K$ ) and the Shear modulus ( $G$ ) for an isotropic material. This derivation fundamentally relies on separating stress and strain into volumetric (governed by $K$ ) and deviatoric (governed by $G$ ) components. The key intermediate result is the relationship $E= 2 G(1+\nu)$, which connects the stiffness ($E$) to the resistance to shear ($G$) and lateral contraction ($\nu$). The final expressions, $E=\frac{9 K G}{3 K+G}$ and $\nu=\frac{3 K-2 G}{6 K+2 G}$, show how the material's resistance to volume change ( $K$ ) and resistance to shape change ( $G$ ) combine to define its overall elastic behavior.
<aside> 🧄
$\complement\cdots$Counselor
</aside>
Interrelation of Elastic Constants: The fundamental takeaway is that all four common elastic constants ( $E, \nu, K$, and $G$ ) are interdependent. In an isotropic material, knowing any two allows you to determine the other two.
Role of Trace Operations: The derivation relies on decomposing the stress and strain tensors into their volumetric (dilatational) and deviatoric (shear) components:
Physical Meaning of the Constants:
Key Intermediate Relationship: The relationship derived from the pure shear condition is crucial: $E=2 G(1+\nu)$. This is one of the most common and essential relationships between the constants.
Incompressibility Limit ( $\nu \rightarrow 1 / 2$ ): The final equation for $\nu$ is: $\nu=\frac{3 K-2 G}{6 K+2 G}$
For an incompressible material (like rubber), $\nu \approx 0.5$ (or 1 / 2 ). Substituting this value into the equation shows that the Bulk Modulus ( $K$ ) must approach infinity ( $K \rightarrow \infty$ ), which physically means an infinite pressure is required to change the material's volume.
‣
<aside> 🧄
</aside>