Shear modulus

In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:^[1]

${\displaystyle G\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\tau _{xy}}{\gamma _{xy}}}={\frac {F/A}{\Delta x/l}}={\frac {Fl}{A\Delta x}}}$

where

${\displaystyle \tau _{xy}=F/A\,}$ = shear stress

${\displaystyle F}$ is the force which acts

${\displaystyle A}$ is the area on which the force acts

${\displaystyle \gamma _{xy}}$ = shear strain. In engineering ${\displaystyle :=\Delta x/l=\tan \theta }$, elsewhere ${\displaystyle :=\theta }$

${\displaystyle \Delta x}$ is the transverse displacement

${\displaystyle l}$ is the initial length of the area.

The derived SI unit of shear modulus is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousand pounds per square inch (ksi). Its dimensional form is M¹L⁻¹T⁻², replacing force by mass times acceleration.