the stress tensor it
$$\boldsymbol \tau = \left[ \begin{array}{ccc}
\sigma_\text{xx} & \tau_\text{xy} & \tau_\text{xz}\\
\tau_\text{yx} & \sigma_\text{yy} & \tau_\text{yz}\\
\tau_\text{zx} & \tau_\text{zy} & \sigma_\text{zz}\\
\end{array} \right]$$
with the principal diagonal elements the normal stress and the off diagonal elements the shear stress. The shear stresses can be defined by many number of laws, simplest of which is the law from Newton.
$$\tau = \mu \nabla v$$
in tensor form, shear stress can be expressed as the function of the viscosity $\mu$ and the gradient of the $i$th velocity in the $j$th direction, as given below.
$$\tau_{ij} = \mu \left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right)$$
that is the most general form of the viscosity equation. For more detailed info, please refer
Truckenbrodt, E.
Fluidmechanik 1: Grundlagen und elementare
Stromungsvorgange dichtebestandiger Fluide
. Springer, Berlin, 2 Auflage,
1996 (in german) or
Munson, B. R., Young, D. F., and Okiishi, T. H.:
Fundamentals of Fluid Mechanics
.
John Wiley and Sons Inc., 2006
For Newtonian fluids (such as water and air), the viscous stress tensor, $T_{ij}$, is proportional to the rate of deformation tensor, $D_{ij}$:
$$D_{ij} = \frac{1}{2}\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right)$$
$$T_{ij} = \lambda\Delta\delta_{ij} + 2\mu D_{ij}$$
where $\Delta \equiv D_{11} + D_{22} + D_{33}$. The Navier-Stokes equation for Newtonian fluids can then be written as:
$$\rho\left(\frac{\partial v_i}{\partial t} + v_j\frac{\partial v_i}{\partial x_j}\right) = -\frac{\partial p}{\partial x_i} + \rho B_i + \frac{\partial T_{ij}}{\partial x_j}$$
The Navier-Stokes equation above governs both laminar and turbulent flow using the same stress tensor. This shows that the definition of shear rate is the same in both laminar and turbulent flows, however, their values will be very different.
For non-Newtonian fluids, the same is true. Instead of the stress tensor defined above, replace it with a non-Newtonian stress tensor. Still the same governing equation applies to laminar and turbulent flows so the definition of shear rate is the same for both regimes.
As you mention, turbulent flow does not have nice, orderly layers. As a result, there can be acute stress localizations.
Best Answer
I didn't catch this the first time I read this thread, but your equation for the rate of deformation tensor D is incorrect; it should not have the dilatation terms along the diagonal. The definition of the rate of deformation tensor is "the symmetric part of the velocity gradient tensor":
$$\mathbf{D}=\frac{(\nabla \mathbf{v})+(\nabla \mathbf{v})^T}{2}$$
Reiner and Rivlin derived a general constitutive equation for a non-viscoelastic non-linear fluid by expressing the stress tensor $\boldsymbol{\tau}$ as a Taylor series in D, and applying the Cauley Hamilton theorem to obtain: $$\boldsymbol{\tau}=a+b\mathbf{D}+c\mathbf{D^2}$$where the scalar material parameters a, b, and c are functions of the three invariants of D. The linearized version of this is a Newtonian fluid, with "a" being a function only of the dilatation (first invariant), b being a constant, and c being zero.