The shear modulus ($G$) is related to Young's modulus ($E$) and Poisson's ratio ($\nu$) by the following equation: $G = \frac{E}{2(1 + \nu)}$ . This relationship is a fundamental one in the theory of linear elasticity, connecting the modulus that describes a material's resistance to shear deformation ($G$) with the modulus describing its resistance to axial deformation ($E$) and its lateral contraction properties ($\nu$).

This relationship is a core equation in the field of linear elasticity for isotropic (having the same properties in all directions) and homogeneous materials.

Explanation of the Constants

1. Shear Modulus ($G$): Also known as the Modulus of Rigidity.
   • What it measures: The material's resistance to shearing or twisting—a change in shape without a change in volume. It's the ratio of shear stress to shear strain.
2. Young's Modulus ($E$):
   • What it measures: The material's stiffness or resistance to axial (tensile or compressive) stress—a change in length. It's the ratio of axial stress to axial strain.
3. Poisson's Ratio ($\nu$):
   • What it measures: The material's tendency to contract laterally when stretched axially. It's the ratio of lateral strain (change in width) to axial strain (change in length).

The Derivation and Conceptual Link

The relationship is $E = 2G(1 + \nu)$, which is algebraically rearranged to the form you cited: $G = \frac{E}{2(1 + \nu)}$.

This formula confirms that for any isotropic elastic material, only two of the three constants are independent. If you know any two (for example, $E$ and $\nu$), you can calculate the third ($G$).

The relationship is derived by analyzing a simple shear deformation ($\tau$) and showing that it can be represented as a combination of a tension stress ($\sigma_1$) and a compression stress ($\sigma_2$) at 45-degree angles.

By expressing both the shear stress ($\tau$) and the axial stresses ($\sigma_1, \sigma_2$) in terms of strains using $G$ (for shear) and $E$ and $\nu$ (for tension/compression), the two descriptions must be mathematically equal, which leads directly to the equation:

$$ \mathbf{E} = 2\mathbf{G}(1 + \mathbf{\nu}) $$

Brief audio

How is the shear modulus related to Young's modulus and Poisson's ratio-L.mp4