## Bulk vs Shear Modulus

Consider an unknown material is placed under a stress of 40 GPa. After the test is completed, the specimen is observed to have a modulus of elasticity of 200 GPa and a Poisson’s ratio of 0.3.

- What is the material’s shear modulus in GPa?
- Calculate the bulk modulus in GPa.

##
__
__**Hint**

**Hint**

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

where
$$G$$
is the shear modulus or modulus of rigidity,
$$E$$
is the modulus of elasticity, and
$$\nu$$
is Poisson’s ratio.

##
__
__**Hint 2**

**Hint 2**

$$$K=\frac{E}{3(1-2\nu )}$$$

where
$$K$$
is the bulk modulus,
$$E$$
is the modulus of elasticity, and
$$\nu$$
is Poisson’s ratio.

The shear modulus (modulus of rigidity) is a shearing force’s coefficient of elasticity. It is the ratio of shear stress to the displacement per unit length (shear strain).

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

where
$$G$$
is the shear modulus or modulus of rigidity,
$$E$$
is the modulus of elasticity, and
$$\nu$$
is Poisson’s ratio.

$$$G=\frac{200GPa}{2(1+0.3)}=76.9\:GPa$$$

A substance’s bulk modulus describes how resistant the material is to compression. Bulk (Volume) Modulus of Elasticity:

$$$K=\frac{E}{3(1-2\nu )}$$$

where
$$K$$
is the bulk modulus,
$$E$$
is the modulus of elasticity, and
$$\nu$$
is Poisson’s ratio.

$$$K=\frac{200GPa}{3[1-(2)(0.3)]}=\frac{200GPa}{3[1-0.6]}=\frac{200GPa}{3(0.4)}=166.7\:GPa$$$

- 76.9 GPa
- 166.7 GPa