Turbulent flows of Newtonian fluids have already been compared with non-Newtonian laminar flows. In this paper the analogy between these classes of flows is explored, and a new approach to derive a turbulent model based on a nonlinear constitutive equation is shown. In order to reach this aim, direct numerical simulation databases of turbulent channel flows are used and analyzed in the light of the classical parameters of non-Newtonian constitutive equations. The Reynolds stress tensor is expressed in terms of a set of basis tensors based on a projection of a nonlinear framework. The coefficients of the model are given as functions of the intensity of the mean strain tensor. The apparent turbulent viscosity, the first and second normal stress difference, are presented in function of the shear rate. A turbulent Weissenberg number, based on a characteristic turbulent time ratio of the first normal stress difference to the apparent viscosity is also presented. These material functions, exhibiting a shear-thinning behavior, are fitted with the power law (the Carreau-type) model. The range of the Reynolds number investigated was 180Re τ2000. One of the advantages of the new algebraic nonlinear power law constitutive equation derived in the paper is that its dependence is only on the mean velocity gradient and can be integrated up to the wall.