  We present a microscopic derivation, without electrodynamical assumptions, of
$B(x,y,H(t))$, $M(H(t))$, and $J_c(H(t))$, in agreement with experiments on
strongly pinned superconductors, for a range of values of the density and
strength of the pinning sites. We numerically solve the overdamped equations of
motion % dynamics of these flux-gradient-driven vortices which can be
temporarily trapped at pinning centers. The field is increased (decreased) by
the addition (removal) of flux lines at the sample boundary, and complete
hysteresis loops can be achieved by using flux lines with opposite orientation.
The pinning force per unit volume we obtain for strongly-pinned vortices, $J_c
B \sim n_p f_p^{1.6}$, interpolates between the following two extreme
situations: very strongly-pinned independent vortices, where $J_c B \sim n_p
f_p$, and the 2D Larkin-Ovchinikov collective-pinning theory for weakly-pinned
straight vortices, where $J_c B \sim n_p f_p^{2}$. Here, $n_p$ and $f_p$ are
the density and maximum force of the pinning sites.

Materials Science;Superconductivity;Superconductivity