On the preservation of fibre direction during axisymmetric hyperelastic mass-growth of a finite fibre-reinforced tube

Abstract

Several types of tube-like fibre-reinforced tissue, including arteries and veins, different kinds of muscles, biological tubes as well as plants and trees, grow in an axially symmetric manner that preserves their own shape as well as the direction and, hence, the shape of their embedded fibres. This study considers the general, threedimensional, axisymmetric mass-growth pattern of a finite tube reinforced by a single family of fibres growing with and within the tube, and investigates the influence that the preservation of fibre direction exerts on relevant mathematical modelling, as well on the physical behaviour of the tube. Accordingly, complete sets of necessary conditions that enable such axisymmetric tube patterns to take place are initially developed, not only for fibres preserving a general direction, but also for all six particular cases in which fibres grow normal to either one or two of the cylindrical polar coordinate directions. The implied conditions are of kinematic character but independent of the constitutive behaviour of the growing tube material. Because they hold in addition to and simultaneously with standard kinematic relations and equilibrium equations, they describe growth by an overdetermined system of equations. In cases of hyperelastic mass-growth, the additional information they thus provide enables identification of specific classes of strain energy densities for growth that are admissible and, therefore, suitable for the implied type of axisymmetric tube mass-growth to take place. The presented analysis is applicable to many different particular cases of axisymmetric mass-growth of tube-like tissue, though admissible classes of relevant strain energy densities for growth are identified only for a few example applications. These consider and discuss cases of relevant hyperelastic mass-growth which (i) is of purely dilatational nature, (ii) combines dilatational and torsional deformation, (iii) enables preservation of shape and direction of helically growing fibres, as well as (iv) plane fibres growing on the cross section of an infinitely long fibre-reinforced tube. The analysis can be extended towards mass-growth modelling of tube-like tissue that contains two or more families of fibres. Potential combination of the outlined theoretical process with suitable data obtained from relevant experimental observations could lead to realistic forms of much sought strain energy functions for growth.

A complete set of independent and physically meaningful invariants in the mechanics of solids reinforced by two families of fibres

“…As is mentioned in the Introduction, this short communication stemmed from the author's interest to extend the hyperelastic mass-growth modelling presented in [1] towards mass-growth of tube-like tissue that preserves the shape and direction of two families of unidirectional fibres. In particular, tubes reinforced by two families of continuously distributed helical fibres wound symmetrically in opposing directions are met very often in several different kinds of plant and bone structures (e.g., [10,11]), as well as in various forms of tube-like soft or hard biological tissue.…”

“…Mass-growth of a fibre-reinforced circular cylindrical tube can naturally be modelled with the use of known principles employed in the theory of hyperelastcity. It is thus recently shown [1] that, if mass-growth preserves the direction and shape of a single family of fibres embedded in, and growing with and within the tube, then the corresponding set of hyperelasticity type equations appears overdetermined. However, the extra equations emerging in the model enable formation of one or more linear relationships between the strain energy density for growth, W, and its derivatives with respect to the principal deformation invariants of the growing system.…”

“…However, if the number, n say, of fibre families embedded in a hyperelastic material is bigger than one, then the classical deformation invariants involved in the strain energy density are not any more independent (e.g., [2,3]). It follows that relationships between W and its derivatives that, in analogy with [1] would reflect mass-growth ability of the material to preserve fibre direction, can be considered as partial differential equations for W only after: (i) the exact number, m say, of independent invariants is identified; (ii) a manner is found for a complete basis of precisely m independent invariants to be formed; and (iii) W is considered as function of those m independent invariants only.…”

“…This short communication is motivated by its author's interest to extend the analysis detailed in [1] towards hyperelastic mass-growth modelling of tissue reinforced by two or more families of fibres. It accordingly considers the particular but practically important case of tissue reinforced by two unidirectional families of fibres (n = 2) and, based on the aforementioned useful result [2,3], demonstrates a manner in which a complete set of six independent strain invariants can conveniently be identified without much deviation from wellknown features that characterise their classical counterparts.…”

“…where, R, Θ and Z are standard cylindrical polar coordinates, and the non-negative constants A and B (0 ≤ A < B) represent its inner and the outer radii (e.g., [1]). In that cylindrical polar co-ordinate system the unit vectors implied in (4) acquire the form…”