  Based on previous work for the static problem, in this paper we first derive
one form of dynamic finite-strain shell equations for incompressible
hyperelastic materials that involve three shell constitutive relations. In
order to single out the bending effect as well as to reduce the number of shell
constitutive relations, a further refinement is performed, which leads to a
refined dynamic finite-strain shell theory with only two shell constitutive
relations (deducible from the given three-dimensional (3D) strain energy
function) and some new insights are also deduced. By using the weak formulation
of the shell equations and the variation of the 3D Lagrange functional,
boundary conditions and the two-dimensional (2D) shell virtual work principle
are derived. As a benchmark problem, we consider the extension and inflation of
an arterial segment. The good agreement between the asymptotic solution based
on the shell equations and that from the 3D exact one gives verification of the
former. The refined shell theory is also applied to study the plane-strain
vibrations of a pressurized artery, and the effects of the axial pre-stretch,
pressure and fibre angle on the vibration frequencies are investigated in
detail.
