## 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.
1. What is the material’s shear modulus in GPa?
2. Calculate the bulk modulus in GPa.

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 $$K=\frac{E}{3(1-2\nu )}$$$
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. $$K=\frac{200GPa}{3[1-(2)(0.3)]}=\frac{200GPa}{3[1-0.6]}=\frac{200GPa}{3(0.4)}=166.7\:GPa$$$
1. 76.9 GPa
2. 166.7 GPa