For an isotropic material, E and ν are often chosen as the two independent engineering constants. There are other elastic consta
nts: the shear modulus G, the bulk modulus K, and the Lames’ constants, μ and λ.
Suppose that G = E/[2(1+v)].
Prove that μ and λ, and K, satisfy the following relations:
μ = G = 3K(1-2v)/[2(1+v)], λ = Ev/ [(1+v)(1-2v)] = 2vG/ (1-2v),
K = E/ 3(1-2v) = λ+(2/3)μ
To prove it, apply pure shear stress τ, and connect it to pure shear strain γ by G, ie. τ = Gγ. Then in the 45° orientation, consider its normal stresses σ, and σ: are related to its normal strains ε, and e, through E and v in Hooke's law. We can easily establish this relation.
