On the Global Stability of Compressible Elastic Cylinders in Tension

Abstract Consider a three-dimensional, homogeneous, compressible, hyperelastic body that occupies a cylindrical domain in its reference configuration. We identify a variety of hypotheses on the structure of the stored-energy function under which there exists an axisymmetric, homogeneous deformation that globally minimizes the energy. For certain classes of energy functions the uniqueness of this minimizer is also established. The primary boundary condition considered is the extension of the cylinder via the prescription of its deformed axial length, but the sliding contact of the curved surface is also briefly considered. In particular, the results contained in this paper give conditions on the stored-energy function under which material instabilities, such as necking or the formation of shear bands, do not occur. Mathematics Subject Classifications (2010): 74B20, 35J50, 49J40, 74G65