in linear elasticity we have, for an isotropic material, $$C[E]=2 \mu E + \lambda \operatorname{tr}(E)I$$ where $\mu,\lambda$ are called Lamè moduli and $E=\frac{\nabla u + \nabla{u}^T}{2}$

I've seen (https://en.wikiversity.org/wiki/Introduction_to_Elasticity/Constitutive_relations#Isotropic_elasticity) that we can write $$C=\lambda I \otimes I + 2 \mu \mathbf{I}$$ where $\mathbf{I}$ is the fourth order identity tensor (since $C$ is a fourth order tensor). My question is: **how can one derive the latter expression for $C$, starting from the one for $C[E]$?

In linear elasticity we have, for an isotropic material, $$C[E]=2 \mu E + \lambda \operatorname{tr}(E)I$$ where $\mu,\lambda$ are called Lamè moduli and $E=\frac{\nabla u + \nabla{u}^T}{2}$

I've seen that we can write $$C=\lambda I \otimes I + 2 \mu \mathbf{I}$$ where $\mathbf{I}$ is the fourth order identity tensor (since $C$ is a fourth order tensor). My question is: how can one derive the latter expression for $C$, starting from the one for $C[E]$?