# Shear Modulus

Written by Jerry Ratzlaff on . Posted in Classical Mechanics Shear modulus, abbreviated as G, also called modulus of rigidity or shear modulus of elasticity, is the ratio of the tangential force per unit area applied to a body or substance to the resulting tangential strain within the elastic limits.

## Shear modulus formulas

 $$\large{ G = \frac { \tau } { \gamma } }$$ $$\large{ G = \frac { \frac{ F }{ w\;l } }{ tan \Delta } }$$ $$\large{ G = \frac { \frac{ F }{ w\;l } }{ \frac{ l_f }{ l } } }$$

### Where:

 Units English Metric $$\large{ G }$$ = shear modulus $$\large{\frac{lbf}{in^2}}$$ $$\large{Pa}$$ $$\large{ F }$$ = applied force $$\large{ lbf }$$ $$\large{N}$$ $$\large{ l }$$ = length $$\large{ in }$$ $$\large{ mm }$$ $$\large{ l_f }$$ = final length $$\large{ in }$$ $$\large{ mm }$$ $$\large{ \gamma }$$  (Greek symbol gamma) = shear strain $$\large{rad}$$ $$\large{rad}$$ $$\large{ \tau }$$  (Greek symbol tau) = shear stress $$\large{\frac{lbf}{in^2}}$$ $$\large{Pa}$$ $$\large{ \Delta }$$ = angle displacement $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ w }$$ = width $$\large{ in }$$ $$\large{ mm }$$

## Related formula

 $$\large{ G = \frac { 8 \; k_s \; n_a \; D^3 } { d^4 } }$$ (Spring)

### Where:

$$\large{ G }$$ = shear modulus

$$\large{ D }$$ = mean coil diameter

$$\large{ n_a }$$ = number of active coils

$$\large{ k_s }$$ = spring constant

$$\large{ d }$$ = wire size 