Elastic Constants

For linear, isotropic, homogeneous materials, these constants are related by:

$$E=2G(1+\nu )$$

Young’s Modulus (E)

Young’s modulus, also known as the modulus of elasticity, is a measure of a material’s ability to resist deformation under tensile (or compressive) stress. It is a fundamental mechanical property that describes the stiffness of a material.

It is defined as the ratio of tensile stress ($\sigma$) to tensile strain ($ϵ$):

$$E=\frac{\sigma }{ϵ}$$

• Units: $\text{N}/\text{m}=\text{Pa}$

Shear Modulus of Elasticity (G)

The shear modulus, also known as the modulus of rigidity, measures a material’s ability to resist deformation under shear stress. It describes how a material deforms in shape (without changing volume) when subjected to shear forces.

It’s defined as the ratio of shear stress $\tau$ to shear strain $\gamma$:

$$G=\frac{\tau }{\gamma }$$

• Units: $\text{N}/\text{m}=\text{Pa}$

Poisson’s Ratio ($\nu$)

Poisson’s ratio is a measure of the deformation in the perpendicular directions to the applied load. It describes the ratio of transverse strain to axial strain in a material under uniaxial stress.

It is defined as the negative ratio of transverse strain (${ϵ}_{\text{trans}}$) to axial strain (${ϵ}_{\text{axial}}$):

$$\nu =-\frac{{ϵ}_{\text{trans}}}{{ϵ}_{\text{axial}}}$$

where the transverse strain is perpendicular to the applied load and the axial strain is parallel to the applied load.

• Units: Dimensionless