I am trying to come up with a formula similar to the product rule for the divergence (see, for example, here) for the tensors appearing in the equations of Linear Elasticity in a manner similar to this question. I am currently struggling with tensor notation (Einstein summation) and fully understanding the types of tensor products involved.
Namely, let $\mathbf{u}, \mathbf{v}$ be two displacement fields, $\varepsilon(\mathbf{u}), \varepsilon(\mathbf{v})$ the associated strain tensors and $\mathcal{C}$ the fourth-order elastic tensor constant throughout the material. I want to prove (although I am not yet certain this is correct)$$\nabla\cdot(\mathcal{C}\varepsilon(\mathbf{v})\mathbf{u})=\nabla \mathbf{u}:\mathcal{C}\varepsilon(\mathbf{v})+\mathbf{u}\cdot(\nabla\cdot\mathcal{C}\varepsilon(\mathbf{v}))$$
where $A:B=A_{ij}B_{ij}$ and the products not marked with any symbol are inner tensor products (please correct me if I am wrong).
So far, I got$$\nabla\cdot(\mathcal{C}\varepsilon(\mathbf{v})\mathbf{u}) = \nabla\cdot(\mathcal{C}_{ijkl}\varepsilon_{kl}\mathbf{u})=\\\nabla\cdot(\mathcal{C}_{ijkl}\varepsilon_{kl}u_j)=(\mathcal{C}_{ijkl}\varepsilon_{kl}u_j)_{,i}=\\\mathcal{C}_{ijkl}\varepsilon_{kl,i}u_j+\mathcal{C}_{ijkl}\varepsilon_{kl}u_{j,i}$$and here I begin to struggle.