Double contraction
= Double contraction
{title2=$A:B=\sum_{ij}A_{ij}B_{ij}$}
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>, $(C:\epsilon)_{ij}=\sum_{kl}C_{ijkl}\epsilon_{kl}$ remains a second-order <tensor>; double contraction need not always give a scalar.