Introduction

Modulus

The most commonly used modulus are:

• Bulk modulus

• Shear modulus

• Young’s modulus

Bulk modulus

\[K = -V\frac{\Delta P}{\Delta V}\]

It quantifies the change of material volume in response to stress change. The material volume is also proportional to the material density. Therefore, the equation could also be written as:

\[k = \rho\frac{\Delta P}{\Delta \rho}\]

Note

The negative sign is removed in \(\rho\) expresson because decreased volume \(\Delta V\) will lead to increased \(\Delta \rho\)

Shear modulus

A schematic plot showing the shear strain (By C.lingg - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=4354169)

The shear modulus quantifies the strain under shear stress, it is defined as:

\[G = \frac{\tau_{xy}}{\gamma_{xy}}\]

where \(\gamma_{xy}\) is defined as \(\frac{\Delta x}{L}\).