Mass-growth of a finite tube reinforced by a pair of helical fibres

Abstract

Several types of tube-like fibre-reinforced tissue, including the layers of arteries and veins, different kinds of muscle, biological tubes as well as plants and trees, are reinforced by a pair of helical fibres wound symmetrically around the tube axis in opposite directions. In many cases, this kind of biological structures grow in an axially symmetric manner that preserves their own shape as well as the direction and shape of their embedded pair of helical fibres. This study considers and investigates the influence that preservation of fibre direction exerts on pseudo-elastic (elastic-like) mass-growth modelling of the described fibrereinforced structure. Complete sets of necessary conditions that enable the implied axisymmetric tube massgrowth to take place are sought, found and presented. These hold in addition to, and simultaneously with standard kinematic relations and equilibrium equations met in conventional hyperelasticity. They thus render this mass-growth mathematical model the properties of an apparently overdetermined boundary value problem. However, the additional information they provide leads to identification of admissible classes of strain energy densities for growth that enable realisation of the implied type of tube mass-growth. The analysis is applicable to several different types of mass-growth of tube-like tissue reinforced by a pair of symmetrically wound helical fibres. This is demonstrated with an application which considers that massgrowth of the fibre-reinforced tube takes place in an incompressible manner; namely, in a non-isochoric manner that along with fibre direction, it also preserves the material density of the growing tube.

The number of independent invariants for m unit vectors and n symmetric second order tensors is 2m+ 6n-3

“…The construction of rotational invariants of sets containing vectors and tensors in continuum mechanics, especially for anisotropic materials [1,16,17,20,21], has been active for around 70 years. The "classical" invariants and minimal integrity bases constructed in Spencer [22], published in 1971, have been extensively used in the literature.…”

“…We also give relations (not necessarily syzygies) between classical invariants in the corresponding minimal integrity basis. Knowing the number of independent invariants is crucial in modelling [20,21] and in a rigorous construction of a constitutive equation of a particular material, where it is determined by doing tests that hold all, except one, of the independent invariants constant so that the dependence in the one invariant can be identified [7,8,12].…”