In the Abaqus help file, the matrix is represented in bold, such as S, which represents the matrix. What does the "s: s" part in the formula stand for?
Two-point multiplication, see the textbook of arbitrary tensor analysis for details.
Written as a component:
A little multiplication: S*S is equivalent to Sij*Sjk, and the result is a second-order tensor.
Two-point multiplication: S:S is equivalent to Sij*Sji, and the result is a scalar.