  We study the spherical collapse model in the presence of external
gravitational tidal shear fields for different dark energy scenarios and
investigate the impact on the mass function and cluster number counts. While
previous studies of the influence of shear and rotation on $\delta_\mathrm{c}$
have been performed with heuristically motivated models, we try to avoid this
model dependence and sample the external tidal shear values directly from the
statistics of the underlying linearly evolved density field based on first
order Lagrangian perturbation theory. Within this self-consistent approach, in
the sense that we restrict our treatment to scales where linear theory is still
applicable, only fluctuations larger than the scale of the considered objects
are included into the sampling process which naturally introduces a mass
dependence of $\delta_\mathrm{c}$. We find that shear effects are predominant
for smaller objects and at lower redshifts, i. e. the effect on
$\delta_\mathrm{c}$ is at or below the percent level for the $\Lambda$CDM
model. For dark energy models we also find small but noticeable differences,
similar to $\Lambda$CDM. The virial overdensity $\Delta_\mathrm{V}$ is nearly
unaffected by the external shear. The now mass dependent $\delta_c$ is used to
evaluate the mass function for different dark energy scenarios and afterwards
to predict cluster number counts, which indicate that ignoring the shear
contribution can lead to biases of the order of $1\sigma$ in the estimation of
cosmological parameters like $\Omega_\mathrm{m}$, $\sigma_8$ or $w$.
