  This paper proposes a novel way to solve transient linear, and non-linear
solid dynamics for compressible, nearly incompressible, and incompressible
material in the updated Lagrangian framework for tetrahedral unstructured
finite elements. It consists of a mixed formulation in both displacement and
pressure, where the momentum equation of the continuum is complemented with a
pressure equation that handles incompresibility inherently. It is obtained
through the deviatoric and volumetric split of the stress, that enables us to
solve the problem in the incompressible limit. The Varitaional Multi-Scale
method (VMS) is developed based on the orthogonal decomposition of the
variables, which damps out spurious pressure fields for piece wise linear
tetrahedral elements. Various numerical examples are presented to assess the
robustness, accuracy and capabilities of our scheme in bending dominated
problems, and for complex geometries.
