A geometrically exact viscoplastic membrane-shell with viscoelastic transverse shear resistance avoiding degeneracy in the thin-shell limit.

Abstract.

We reduce a viscoelastic finite-strain continuum model to a two-dimensional membrane-plate. The reduction is based on assumed kinematics, analytical integration through the thickness and physically motivated simplifications. The resulting formulation is observer-invariant and accounts for thickness stretch and finite rotations.

The membrane energy is a quadratic, uniformly Legendre-Hadamard elliptic, first order energy in contrast to classical membrane models and the corresponding system of balance equations remains of second order. An evolution equation for some independent rotation is appended (already in the bulk-model) introducing viscoelastic transverse shear resistance. It can be shown that this reduced membrane formulation is locally well-posed. Use is made of a dimensionally reduced version of an extended Korn’s first inequality.

In the equilibrium relaxation limit an intrinsic membrane-plate formulation is obtained similar to the proposal of Fox/Simo, which is, however, non-elliptic. Nevertheless, the linearization of this last equilibrium model coincides with the classical linear membrane-plate model. In this sense, the new viscoelastic membrane-plate model regularizes the occurring loss of ellipticity in classical finite-strain membrane models.