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

Soldatos, Konstantinos (2018) On the preservation of fibre direction during axisymmetric hyperelastic mass-growth of a finite fibre-reinforced tube. Journal of Engineering Mathematics, 109 (1). pp. 173-210. ISSN 1573-2703

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Several types of tube-like fibre-reinforced tissue, including arteries and veins, different kinds of muscle, 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, three-dimensional, 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 the fibres grow normal to either one or two of the cylindrical polar coordinate directions. The implied conditions are of kinematic character but are 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 enable 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.

Item Type: Article
RIS ID: https://nottingham-repository.worktribe.com/output/930148
Keywords: Axisymmetric mass-growth, Elastic-like mass-growth, Growth of cylinders and tubes, Hyperelastic mass-growth, Anisotropic mass-growth, Mass-growth modelling.
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Identification Number: https://doi.org/10.1007/s10665-017-9942-6
Depositing User: Soldatos, Konstantinos
Date Deposited: 30 Aug 2017 08:45
Last Modified: 04 May 2020 19:34
URI: https://eprints.nottingham.ac.uk/id/eprint/45204

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