If two pieces of material A and B have the same bulk modulus, but the value of modulus of elasticity for B is 1% greater than that for A, find the value of rigidity for the material B in terms of modulus of elasticity and modulus of rigidity for materials A.

Given: $For material A, E_A=E (modulus of elasticity)$

$G_A=G(modulus of rigidity)$

$K_A=K(bulk modulus)$

$Material B= E_B=1.01E_A=1.01E$

$G_B=?$

$K_B=K_A=K$

For material, the relation between elastic content is,

$E=\frac{9GK}{3K+G}$

$3EK+EG=9GK$

$EG=9GK-3EK$

$EG=(9G-3E)K$

$i.e K=\frac{EG}{9G-3E}$

$Now, K_A=K_B is given$

$\frac{EG}{9G-3E}=1.01\frac{EG_B}{9G_B-3*1.01E}$

$9GG_B-3.03EG_A=9.09GG_B-3.03EG_B$

$9.09GG_B-3.03EG_B-9GG_B=-3.03EG$

$0.09GG_B-3.03EG_B=-3.03EG$

$G_B(0.09G-3.03E)=-3.03EG$

$G_B=\frac{- 3.03EG}{0.09G-3.03E}$