The Lamé constant (λ) is equal to the shear modulus (G) for specific values of Poisson's ratio (ν):
In a 2D case:
ν = λ / (λ + 2 × G) = 1/3
For a 3D problem:
ν = λ / (2 × (λ + G)) = 0.25
You can check this by plugging in λ = G = 1; the units don't matter since Poisson's ratio is dimensionless. Read more