  We present a detailed analysis of the local evolution of 206 Lagrangian
Volumes (LVs) selected at high redshift around galaxy seeds, identified in a
large-volume $\Lambda$ cold dark matter ($\Lambda$CDM) hydrodynamical
simulation. The LVs have a mass range of $1 - 1500 \times 10^{10} M_\odot$. We
follow the dynamical evolution of the density field inside these initially
spherical LVs from $z=10$ up to $z_{\rm low} = 0.05$, witnessing highly
non-linear, anisotropic mass rearrangements within them, leading to the
emergence of the local cosmic web (CW). These mass arrangements have been
analysed in terms of the reduced inertia tensor $I_{ij}^r$, focusing on the
evolution of the principal axes of inertia and their corresponding
eigendirections, and paying particular attention to the times when the
evolution of these two structural elements declines. In addition, mass and
component effects along this process have also been investigated. We have found
that deformations are led by dark matter dynamics and they transform most of
the initially spherical LVs into prolate shapes, i.e. filamentary structures.
An analysis of the individual freezing-out time distributions for shapes and
eigendirections shows that first most of the LVs fix their three axes of
symmetry (like a skeleton) early on, while accretion flows towards them still
continue. Very remarkably, we have found that more massive LVs fix their
skeleton earlier on than less massive ones. We briefly discuss the
astrophysical implications our findings could have, including the galaxy
mass-morphology relation and the effects on the galaxy-galaxy merger parameter
space, among others.
