  We present numerical studies of stretching of Hookean dumbbells in a
turbulent Navier-Stokes flow with a linear mean profile, <u_x>=Sy. In addition
to the turbulence features beyond the viscous Kolmogorov scale \eta, the
dynamics at the equilibrium extension of the dumbbells significantly below eta
is well resolved. The variation of the constant shear rate S causes a change of
the turbulent velocity fluctuations on all scales and thus of the intensity of
local stretching rate of the advecting flow. The latter is measured by the
maximum Lyapunov exponent lambda_1 which is found to increase as \lambda_1 ~
S^{3/2}, in agreement with a dimensional argument. The ensemble of up to 2
times 10^6 passively advected dumbbells is advanced by Brownian dynamics
simulations in combination with a pseudospectral integration for the turbulent
shear flow. Anisotropy of stretching is quantified by the statistics of the
azimuthal angle $\phi$ which measures the alignment with the mean flow axis in
the x-y shear plane, and the polar angle theta which determines the orientation
with respect to the shear plane. The asymmetry of the probability density
function (PDF) of phi increases with growing shear rate S. Furthermore, the PDF
becomes increasingly peaked around mean flow direction (phi= 0). In contrast,
the PDF of the polar angle theta is symmetric and less sensitive to changes of
S.
