Double contraction ($A:B={\sum }_{ij}{A}_{ij}{B}_{ij}$)

A tensor contraction over two paired indices. For real second-order tensors, it gives the displayed Frobenius inner product. For a fourth-order elasticity tensor and a second-order infinitesimal strain tensor, $(C:ϵ{)}_{ij}={\sum }_{kl}{C}_{ijkl}{ϵ}_{kl}$ remains a second-order tensor; double contraction need not always give a scalar.