# 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, 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 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.

## Keywords

Anisotropic mass-growth Axisymmetric mass-growth Elastic-like mass-growth Growth of cylinders and tubes Hyperelastic mass-growth Mass-growth modelling## 1 Introduction

Several types of tube-like hard or soft fibre-reinforced tissue possess natural ability to grow in a manner that preserves their own shape as well as the direction and, hence, the shape of their embedded fibres. Fibres of helical shape are, for instance, commonly met in several different kinds of plant and bone structures (e.g. [1, 2, 3]), as well as in various forms of tube-like soft biological tissue. The latter include arteries and veins (e.g. [4, 5, 6, 7]), muscles (e.g. [8]) and even living creatures of tubular shape [9]. An elephant’s trunk and the arm of an octopus may be referred to as additional examples of tube-like soft tissue containing families of fibres organised along several different orientations (e.g. [10, 11]). In this context, particular mention is made of a set of experimental and theoretical investigations that aim to clarify the role of collagen fibre reinforcement as well as its influence on the mechanical behaviour of arterial wall (see [5, 6, 7] and relevant references therein). These investigations focus attention on the material anisotropy caused in different layers of the arterial vessel by the dispersion and the mean alignment of collagen fibres. It has become, for instance, understood [5] that two families of fibres are present in the intima, media and adventitia of human aortas, and these are helically arranged with respect to the cylinder axis. Often a third and sometimes a fourth fibre family is present in the intima along the respective axial and circumferential directions [5]. However, there exist also artery wall layers, such as the medial layer of human common iliac arteries, which have only a single preferred fibre direction [6, 7].

The present study employs postulates and principles of continuum solid mechanics with the purpose to model and subsequently investigate features associated with the ability of such tube-like types of living and growing fibre-reinforced tissue to control fibre direction. Being a long established vehicle of tissue characterisation, continuum solid mechanics models tissue deformation and movement that is due to either mechanical loading (e.g. [12, 13, 14, 15, 16, 17, 18]) or mass-growth (e.g. [18, 19, 20, 21, 22]) on the basis of kinematical concepts and equilibrium equations that describe and balance deformation of solid media. In this context, it perceives growth of fibres that takes place naturally during mass-growth of fibre-reinforced tissue as a kind of fibre stretch.

It is important to note in this regard, that a variety of inextensible fibres embedded in growing plants, trees and, possibly, other types of living fibre-reinforced tissue resist any kind of stretch that is due to mechanical loading. However, at the same time, they naturally experience growth elongation which, in solid mechanics terms, is perceived as a kind of stretch solely due to, as well as compatible with the implied tissue growth. It follows that the concept of fibre inextensibility is generally not observable when deformation of fibre-reinforced tissue is due purely to mass-growth. More generally, there is a clear distinction between fibre extensibility or stretch due to mechanical loading and its counterpart caused by tissue mass-growth. The term “growth stretch” of a fibre is thus employed in what follows, in order to distinguish the implied fibre growth/elongation from conventional fibre stretch which is due to mechanical loading.

Under these considerations, Sect. 2 considers the most general form of axisymmetric deformation that a circular cylindrical tube may experience, and outlines preliminary details related to its kinematics and equilibrium. It also describes postulates that identify the implied deformation as a mass-growth process, quotes and discusses the boundary conditions of principal interest in this study and provides the relevant equations of dynamic equilibrium in appropriate, cylindrical polar coordinate form. Section 3 derives next conditions that should necessarily hold for a single family of fibres embedded in, and growing with the tube to preserve direction during the axisymmetric deformation of interest. In this context, fibre direction features are initially handled in a general manner. Nevertheless, a comprehensive classification is also presented of all six particular cases emerging when the fibre direction is normal to either one or two of the cylindrical polar coordinate directions.

Most of the information, arguments and concepts presented in Sects. 2 and 3 are valid regardless of whether the axially symmetric deformation of interest is due to externally applied mechanical loading or to mass-growth. Moreover, these are valid regardless of the tube material constitution, as long as the latter is consistent with the local transverse isotropy induced by the presence of a single family of fibres. In this context, Sect. 4 completes the relevant mathematical formulation by quoting basic results of a seemingly elastic-type mass-growth introduced in, hence, [22] and, hence, associating the constitutive behaviour of the growing tube with a relevant concept of “mass-growth hyperelasticity”. Due to the fibre direction restrictions detailed previously in Sect. 3, the final system of equations that govern the described growth model may initially be perceived as overdetermined. However, the additional information provided by those restrictions can be directed towards identification of specific classes of admissible much sought strain energy densities for growth. This observation reveals then that nature can control direction of fibres that grow with and within an elastically growing tube by making use of specific forms of the corresponding strain energy density for growth.

The outlined theoretical developments are applicable to many different particular cases of axisymmetric mass-growth of tube-like tissues, some of which are considered next in Sects. 5–8. In this context, Sect. 5 introduces and discusses a particular case in which hyperelastic mass-growth is of purely dilatational nature. This discussion assists substantially the more general case detailed afterwards in Sect. 6, where dilatational and torsional mass-growth are considered combined. Thus, Sects. 5 and 6 focus on particular axisymmetric mass-growth patterns, while they both keep the fibre direction general. In contrast, Sects. 7 and 8 specify a priori the fibre shape and direction and, in better contact with practical aims of the present growth model, investigate the influence that the preservation of fibre direction exerts on both the mass-growth pattern of interest and the strain energy for growth of the tube material.

In more detail, Sect. 7 considers and discusses the aforementioned case of particular practical interest [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], where fibres grow with and within the tube in a helical manner. A similar kind of interest is shown afterwards for the families of plane fibres considered in Sect. 8 which, in a final application, considers cross-sectional mass growth of a transversely isotropic tube of infinite extent. It is recalled in this connection that the plane strain analysis of an infinitely long transverse isotropic tube offers some analytical simplification and is therefore not rare in conventional hyperelasticity applications (e.g. [23, 24, 25, 26, 27, 28]). Useful observations stemming from the presented mass-growth modelling, analysis and applications are finally summarised in Sect. 9, which also outlines the main conclusions of this investigation as well as relevant future research directions.

## 2 Preliminaries: axially symmetric deformation of a finite tube

*t*denotes time, an undeformed circular cylindrical tube of finite axial length, 2

*H*, and constant mass density, \(\rho \) \(_{0}\), occupies the region

*R*, \(\varTheta \) and

*Z*are standard cylindrical polar coordinate parameters (Fig. 1). The non-negative constant parameters

*A*and

*B*\((0\le A < B)\) represent the inner and the outer radii of the tube, respectively. If

*A*= 0 this representation thus depicts the particular case of a non-hollow (solid) cylinder.

*r*, \(\theta \) and

*z*represent cylindrical polar coordinate parameters in the current, continuously deforming configuration. This general dynamic combination of tube radial and axial expansion with azimuthal and axial shear deformation depends on the form of the functions \(r(R,Z; t),g(R,Z; t)={{\hat{{g}}}}(r, z; t)\) and

*z*(

*R*,

*Z*;

*t*). These are to be determined or specified, subject to the initial conditions

### 2.1 Kinematics

*, are*

**v**### 2.2 Equilibrium

*r*(

*R*,

*Z*;

*t*),

*g*(

*R*,

*Z*;

*t*) and

*z*(

*R*,

*Z*;

*t*) appearing in (2.2). Solution of these differential equations may thus be attempted as soon as the material constitution of the tube is specified, and a relevant set of constitutive equations is thus determined; see, for instance, Sect. 4 below. Nevertheless, solution of some specific relevant boundary value problem usually requires specification of an associated set of boundary conditions.

### 2.3 Boundary conditions

Satisfaction of several of the non-homogeneous traction boundary conditions (2.13) and (2.14) is often required in cases where the tube deformation is due to externally applied loading. However, it will become evident in Sect. 4 below that, if the deformation pattern (2.2) is totally due to mass-growth activity, then the homogeneous version of any of those traction boundary conditions (where the corresponding *q*- and \(\tau \)-quantity is zero) should be given priority and preference. If application of the non-homogeneous form of any of these boundary conditions is found necessary in subsequent sections, its presence will thus be perceived as purely supporting the observed mass-growth process.

It is also noted that, in cases that the outlined geometrical features conform to those of a corresponding non-hollow cylinder (\(A=a\) = 0), the boundary conditions (2.13a) and (2.14a, c) are replaced by the requirement that the deformation and all stress components are finite on the cylinder axis (*r* = 0).

## 3 Growth/deformation patterns that preserve the direction of a single family of fibres

*is generally assumed independent of the circumferential coordinate parameter. In that case, standard transformation rules of continuum mechanics reveal that, in the current tube configuration (\(t > t_{0}\)),*

**A***transforms into the vector*

**A***r*(

*R*,

*Z*;

*t*),

*g*(

*R*,

*Z*;

*t*) and

*z*(

*R*,

*Z*;

*t*), along with the equations of motion (2.12).

The total number (five) of the governing differential equations (2.12) and (3.6) makes the mathematical model seem overdetermined. Alternatively, the additional information provided by (3.5) seems to require identification of further unknown parameters that should accompany the initial set of three unknown functions *r*, *g* and *z*. Such additional unknowns may, however, be relevant to the tube material constitution or to certain privileged fibre directions. It will accordingly become more evident in Sects. 4–8 below that, in cases of hyperelastic mass-growth, (3.6) can lead to identification of privileged classes of strain energy densities that enable a tissue to control their fibre direction.

In this context, (3.6) may alternatively be regarded as a set of two simultaneous first-order linear partial differential equations, whose potential solution might lead to the elimination of two of the three unknown functions *r*, *g* and *z*. The search for a general analytical solutions of (3.6) may, however, meet considerable complications, particularly in cases that the unit vector * A* is dependent on the cylindrical polar coordinate parameters

*R*and

*Z*. Nevertheless, several classes of fibre families met often in nature violate (3.3) and, hence, lead to considerably simplified versions of (3.6) or, equivalently, (3.5). In this regard, the remaining of this section identifies all six different classes of fibre families that violate (3.3) and, for each of those classes, examines afterwards the influence that the corresponding version of (3.5) exerts on the deformation pattern (2.2).

### 3.1 Classification of fibre directions that satisfy \(A_{R} A_{\varTheta } A_{Z} = 0\)

In a similar manner, class (ii\(_{1})\) refers to straight fibres that spread radially on the undeformed tube cross section while, if \(A_{Z}\) is a non-zero constant, class (ii\(_{2})\) specifies straight fibres passing through cylindrical axis, and forming non-zero constant angles with both that axis and the cross-sectional plane of the tube. Nevertheless, class (ii\(_{2})\) enables relevant curved plane fibres to also be considered, if \(A_{Z}\) or, equivalently, \(A_{R}\) depends on the cylindrical polar coordinate parameters.

### 3.2 Restrictions that enable preservation of fibre direction in each of the classes (i)–(iii)

It can easily be verified that (3.7a) satisfies identically all the three restrictions (3.5), thus implying that circumferential fibres always maintain their cross-sectional circumferential profile during the implied axisymmetric tube growth. Hence, in that case

\(\left( {\mathrm{{i}}_1} \right) \) no kinematic restrictions are required.

*Z*and, hence, leads to the relationship

*R*, thus leading to

However, the example applications presented later in Sects. 5 and 6 depart with no restrictions on the fibre direction but, instead, impose certain simplifying assumptions on the axisymmetric deformation pattern (2.2). Sections 7 and 8 specify next a priori the fibre shape and direction and investigate the influence that the latter exerts on the corresponding axisymmetric mass-growth pattern. In this connection, Sect. 7 assumes that the fibres growing with and within the tube material are helically oriented and, hence, a priori specified as (\(\mathrm{i}_{3}\))-class fibres, while Sect. 8 deals with a case that may involve any of the three classes of plane cross-sectional fibres, namely (i\(_{1}\)), (ii\(_{1})\) or (iii). In all four of these examples, consideration of the equilibrium equations (2.12) is enabled in association with the purely hyperelastic type of mass-growth detailed next in Sect. 4.

It is finally worth recalling that in many cases of large axisymmetric tube deformation due to action of externally applied mechanical loading, the observed deformation pattern is a particular case of, and, therefore, simpler than that implied by (2.2). It is similarly very likely that several of the plethora of the different growth mechanisms met in nature can produce axisymmetric deformation patterns which are also simpler than (2.2). In such cases, several of the restriction sets obtained in this section may attain some simplified or modified form, as is demonstrated in Appendix A with an illustrative example.

## 4 Mass-growth hyperelasticity

The information, concepts and arguments presented in Sects. 2 and 3 above are valid regardless of whether the implied axially symmetric deformation is due to mass-growth or to externally applied mechanical loading, as well as regardless of the tube material constitution as long as the latter is consistent with the local transverse isotropy implied by the presence of a single family of fibres. Conventional hyperelasticity may accordingly provide the simplest possible choice of a relevant constitutive law.

*W*represents the strain energy density for growth of the tube material. This excludes the effects of possible plastic mass-growth considered in [22, 29] and, for convenience, is next replaced with its simplest possible form.

It is recalled in this connection (see Sect. 1) that there is a clear distinction between fibre extensibility (or stretch) due to mechanical loading and its counterpart caused by tissue mass-growth. That distinction distinguishes the concept of “growth stretch” of a fibre from the conventional fibre stretch due to mechanical loading. This further implies that, in general, the material response mechanism that accounts for fibre growth stretch is necessarily different from its counterpart that accounts for fibre stretch due to mechanical loading. Indeed, the particular example of mechanically inextensible but otherwise normally growing fibres mentioned in the Introduction makes it obvious that the strain energy density for growth of a fibre-reinforced material is not necessarily identical with its conventional counterpart (e.g. [30]).

Alternatively, these considerations justify the claim that, in general, the strain energy density for growth of a solid tissue differs from its strain energy density due to mechanically caused deformations. In cases that mass-growth as well as some set of mechanical loads act on a solid tissue simultaneously, such a possible pair of different strain energy densities should somehow interact. However, more details of the implied kind of interaction are currently unknown, as is also unknown whether those different strain energy densities present similarities in certain types of tissue or are completely different in others.

This investigation does not aim to explore further or answer any of the numerous questions and/or possible consequences that might follow the outlined claim. In this context, it confines interest only on axisymmetric deformation patterns which are due purely to mass-growth activity. Namely, deformation patterns of the form (2.2), which (i) are due solely to the activity of the continuity condition with growing mass (2.10), and, if/when necessary, (ii) are supported by boundary conditions that do not interfere mechanically with the implied mass-growth process. The latter restriction thus justifies the previously expressed preference on the homogeneous version of the traction boundary conditions (2.13) and (2.14).

### 4.1 Incompressible mass growth. Mass-growth hyperelasticity

In conventional hyperelasticity where \(r_{g} \equiv \) 0, the continuity equation with growing mass (2.10) reduces to the usual mass conservation law. The class of incompressible materials is then naturally associated with the constraint equation \(\rho /{ \rho }_{0} = 1 = \mathrm{{det}}\) * F* or, equivalently, \(\nabla \cdot v=0\). This is accordingly naturally identified as the class of materials which are susceptible to isochoric deformations only (e.g. [31]).

*and, hence, enables (2.10) to convert into the following:*

**F***\(\ne \)1) and, unlike its conventional solid mechanics counterpart, is accordingly neither isochoric nor any other kind of a kinematically constrained deformation.*

**F***s*may either be specified

*a priori*on the basis of existing theoretical/experimental evidence or determined

*a posteriori*by solving some well-posed mass-growth boundary value problem. This parameter is considered independent of time, but not necessarily independent of position and, hence, not necessarily constant; see also [32].

*s*= 1 represents all cases and classes of incompressible mass-growth (4.3), where the ratio coefficient appearing in the last term (4.1) becomes 1. Mass-growth incompressibility enables thus the general constitutive formula (4.1) to take the simpler form

### 4.2 Mass-growth hyperelasticity for a locally transverse isotropic tube

*W*and \(\sigma \) acquire at \(t=t_{0}\) are interconnected as well as influenced by earlier mass-growth stages of the tube material [22, 29]. In this regard, the following set of initial conditions should generally be associated with (4.5):

*\(_{0}\) denotes the prestress tensor. Either of these physical quantities is regarded as a known product of mass-growth activity that took place at \(t \le t_{0}\).*

**T***\(_{0} \ne \)*

**T****0**) may influence the present analysis is analogous to that detailed in [22, 29] for isotropic mass-growth (see also [21]).

*W*, to be a function of the irreducible set of independent invariants (e.g. [30])

## 5 Application 1: purely dilatational mass-growth

*g*to be at most a function of time. It follows that involvement of

*g*in (2.2) represents only a rigid body rotation that includes no mass-growth deformation. One can thus incorporate

*g*into the initial polar angle parameter \(\varTheta \), by setting

*g*= 0. It is then seen that, in this case, the placement boundary condition (2.14a) necessarily acquires its homogeneous form, where \(\alpha \) = 0.

*t*). Validity of (5.1) and (5.4) enables then one to show that the restrictions (3.5) are all satisfied identically in this case, regardless of the fibre shape/direction. Alternatively, the shape and direction of a single family of fibres are always preserved within a tube that grows in the purely dilatational manner (5.5). This kind of purely dilatational mass-growth may well be met in nature. Its features are accordingly detailed and discussed in the remaining of this section, though only in the context of mass-growth incompressibility.

### 5.1 Incompressible mass-growth

*and*

**F***depend only on time, the invariants (4.9) become*

**B***are all independent of cylindrical polar coordinate parameters. It is then convenient for someone to distinguish and discuss next, separately, the case of fibre direction which is also position independent. The more general case, where purely dilatational mass-growth takes place while the unit vector*

**b***depends on position, is considered later in Sect. 5.5.*

**A**### 5.2 Fibre direction independent of position

*are all constant. Hence, because \(\lambda \),*

**A***W*and, hence, the stress components (5.10) depend only on time, the equations of motion (2.12) simplify into the following:

### 5.3 Non-axial fibres with direction independent of position

*W*and, accordingly, enables partial determination of admissible forms of the strain energy for growth. In this context, (5.15) may be converted into a first-order linear partial differential equation (PDE) for

*W*in five different ways, through appropriate use of one of the following choices:

*W*may thus be obtained in a similar manner, by choosing \(\lambda ^{{2}}\) in a manner different from (5.17d) and, hence, converting (5.15) into a first-order PDE different than (5.18). Solution of any such a PDE will provide an additional admissible class of

*W*that, like (5.19), satisfies (5.15) and, hence, enables the constitutive equations (5.10) to attain the simplified form

Attention then naturally turns into the fact that, with appropriate use of (5.17), (5.23) is converted into one of several different versions of a first-order linear PDEs for *W*. This may be achieved in a manner similar to that in which (5.15) is earlier converted into the PDE (5.18). Solution of the pair of simultaneous first-order PDEs generated by such a conversion of (5.15) and (5.23) produces admissible forms of *W* that enable the tube to preserve fibre shape and direction while growing in a completely stress-free manner.

### 5.4 Axial fibres

This stress field satisfies identically the quasi-static equations of motion (5.11a) and (5.12) when (5.13) and thus (5.14) also hold, regardless of the form of, *W*. Hence, (5.26) is a formal representation of the stress state developing within a tube with axial transverse isotropy that grows in accordance with the proportionally dilatational pattern (5.5). However, unless the strain energy for growth satisfies simultaneously both (5.15) and (5.23), the stress distribution (5.26) has to be supported by some externally applied set of appropriate boundary normal tractions.

*W*satisfies (5.15) but violates (5.23), then (5.26) attains again the equi-triaxial form (5.20) and, hence, the tube boundaries must be supported by the set of boundary normal tractions (5.21). Nevertheless, if

*W*satisfies simultaneously both (5.15) and (5.23), then the remaining analysis and, hence, the conclusion detailed in Sect. 5.3 are still fully valid.

### 5.5 Fibre direction dependent on the radial and axial coordinate parameters

Consider finally the case in which the fibre direction vector * A* depends on position in such a manner that its components \(A_{R}\), \(A_{\varTheta }\) and \(A_{Z}\) are still independent of the circumferential coordinate parameter. Because

*may depend on the radial and the axial coordinate parameters only, formation of the axially symmetric growth pattern (2.2) is still possible. The purely dilatational form (5.5) of mass-growth insures that \(\lambda \) and*

**A***W*remain position independent but the stress components (5.10) depend evidently now on this pair of coordinate parameters. Hence, rather than taking a simplified form, the equations of motion (2.12) remain unaltered.

*W*satisfies (5.15). It follows that a fibre-reinforced tube can grow in the incompressible, linearly proportional dilatational manner (5.5) and (5.7) in a completely stress-free manner and, at the same time, preserve the fibre direction, provided that the latter is independent of the circumferential coordinate parameter.

Non-dilatational mass-growth patterns are however also observed in nature, probably on a more regular basis than dilatational ones do (e.g. several types of plants and trees, hair, arteries and veins). In this regard, Sects. 6–8 connect next the analysis presented in this as well as in the preceding sections with certain axisymmetric, non-dilatational mass-growth patterns. These consider mass-growth patterns that, apart from dilatation, they also generate different kinds of shear strains.

## 6 Application 2: combined dilatational and torsional mass-growth—general case (**A** \(_{R}\) **A** \(_{\Theta }\) **A** \(_{Z}\ne \)0)

**A**

**A**

**A**

### 6.1 Kinematics

*g*.

Connection of (6.1) and (6.6) with the incompressible mass-growth form (4.3) of the continuity equation reveals that the mass-growth rate and the fibre growth stretch still relate according to (5.6), regardless of the potential solution of (6.7). Hence, the fibre growth stretch, \(\lambda \), is still given according to (5.7) where \(r_{g}\) should still depend only on time. Both (5.5) and the mass-growth pattern (6.1) with (6.6) are thus triggered by the same class of mass-growth rates and, while preserving fibre shape and direction, they lead to identical predictions of fibre growth stretch. Any of the potential mass-growth solutions sought and/or found next is thus regarded as alternative/additional to the purely dilatational ones developed in Sect. 5.

*t*) should also possess dimensions of (length)\(^{-1}\).

It can be verified that the placement fields (6.6) and (6.9) satisfy the restrictions (3.6) when (3.3) is obeyed and, hence, the fibre direction is general. Moreover, this field satisfies the corresponding restrictions detailed in Sect. 3.2 in five of the six cases that (3.3) is violated, with the class (iii) that refers to plane, cross-sectional spiral fibres being the exception. Nevertheless, a case of cross-sectional spiral fibres that grow in the cross section of a tube of infinite extent is considered and discussed later separately in Sect. 8.3. In this regard, it is fitting to note that Sect. 7.1 is dealt later with radially proportional growth of a tube with embedded helical fibres and makes also use of the azimuthal placement expression (6.9).

Despite that (6.6) coincides with the homogeneous, pure dilatational pattern (5.5), its combination with the exponential form (6.8) of *g* leads to a non-homogeneous form of the mass-growth pattern (6.1); this provides forms of * F*,

*and*

**B***W*that depend not only on time, but also on the radial and the axial coordinate parameters. As a result, the equations of motion (2.12) cannot simplify into some form similar to (5.11).

*g*given by (6.9) is linear in both

*R*and

*Z*and, hence, its association with (6.1) and (6.6) maintains a homogeneous form of the mass-growth pattern. Moreover, this leads to

*makes thus*

**A***W*independent of the axial coordinate parameter, further progress is possible without excessive deviation from the mathematical analysis detailed in Sect. 5.

*g*described by (6.9) is accordingly preferred to (6.8) in the remaining of this section. It is worth noting that, with this choice of

*g*, the present mass-growth pattern associates to the radial and axial components (5.8) of the velocity vector the non-zero azimuthal component

*t*) which is in strict adherence with the rules of quasi-static mass-growth.

### 6.2 Constitutive and equilibrium equations

*is considered constant, the equations of motion (2.12b, c) still simplify into (5.11b, c); and, hence, still lead to (5.12).*

**A***W*the following pair of conditions:

*W*the following additional condition:

*W*that preserve fibre shape and direction should accordingly satisfy simultaneously all three conditions (6.14a,b) and (6.16). Such a class of admissible forms of

*W*that is independent of the strain invariant \(I_{5}\) is found in Appendix D, and is as follows:

*W*, which also depend on \(I_{5}\) or are functions of different combinations of the strain invariants.

*W*into (6.13) provides next corresponding explicit forms of the non-zero stress components. Alternatively, the radial normal stress is obtained in the following form:

*a*= 0) of the cylinder when the single-curved boundary of the latter is kept free of external tractions. Moreover, and, regardless of whether the growing cylinder is hollow or not, a combination of (6.18) with (6.15) yields the hoop stress in the following alternative form:

## 7 Application 3: radially proportional growth of a tube with embedded helical fibres (**A** \(_{R}\) **= 0)**

**A**

*z*and

*g*. Hence, use of (2.5) and (2.6) yields further

*g*, may now be considered unknown implies that this may be determined with use of one of the equations of motion (2.12). The remaining pair of those equations may then be used towards identification of relevant classes of admissible strain energy densities. However,

*g*is generally now a function of both

*R*and

*Z*and, hence, the equations of motion (2.12) cannot attain unconditionally their simplified version (5.14b, c) and (6.15). The experience gained in previous sections suggests however that the analysis may simplify if the form of the last remaining placement component is provided. In such a case, admissible forms of

*W*should satisfy all three equations (2.12).

### 7.1 Kinematics associated with a particular form of the azimuthal placement component

*t*) may still be considered as an arbitrary function that possesses dimensions of (length)\(^{-1}\). The axisymmetric growth patterns (7.1) and (7.5) exhibit thus a specific theoretical connection with its counterpart considered in the preceding section, while its linearity with respect to the axial tube parameter enables the equations of motion (2.12) to attain again their simplified form (5.11b, c) and (6.15).

However, the form of the axial placement (7.5b) is quadratic in the radial coordinate parameter and, hence, the placement fields (7.1) and (7.5) do not anymore represent a homogeneous deformation. Moreover, unlike either (5.5) or (6.6) with (6.9) which predict that (\(b-a)\)/(\(B-A)=h/H\), the present deformation rule enables consideration of growth patterns which, as often happens in nature, do not preserve the ratio of the radial and axial tube dimensions. This is because the azimuthal placement (7.5a) influences now the magnitude of its axial counterpart (7.5b).

*t*) and \(\phi \)(

*t*) may thus still be given according to (C.1) and (C.6), respectively, although the product \({\dot{\lambda }}\phi \) appearing in (7.6c) prevents now a strict adherence with the concept of quasi-static mass-growth.

### 7.2 Constitutive and equilibrium equations

*W*that enable realisation of the mass-growth pattern (7.1) and (7.5) should necessarily obey some set of different restrictions.

*W*that satisfy simultaneously all three differential conditions (7.12) and (7.13) may be sought in the manner described in the preceding sections. Nevertheless, the form

The stress field (7.15) seems then associated with an essentially infinite number of mass-growth patterns of the form (7.1) and (7.5), and this concept of placement arbitrariness is manifested in (7.5) through the involvement of the function \(\phi \)(*t*). Each of those patterns is caused by the same evolution rule of dilatational growth, represented in (7.15) by a certain choice of the function \(\lambda \)(*t*), but involves some different rules of shear strain growth, due to the different possible choices of the function \(\phi \)(*t*). The choice of this pair of otherwise arbitrary functions dictates thus substantially the manner in which shear deformation that is caused through growth of unidirectional helical fibres alters the proportional, single-parameter features of the mass-growth patterns (5.5) and (6.6) considered in Sects. 5 and 6, respectively. The form of (7.5) then reveals that some specific link, similar to that described in Appendix C, may need to be sought and found between \(\lambda \)(*t*) and \(\phi \)(*t*).

*m*and

*n*represent unequal positive integers (\(m \ne n)\); the corresponding equi-triaxial stress field may then easily be calculated with use of (7.15).

## 8 Application 4: Growth of an infinitely long tube with an embedded family of spiral fibres

*ii*\(_{1})\) and (

*iii*), are formed by circumferential, axial and spiral fibres, respectively. Those three families are accordingly considered separately in what follows.

*\(_{0}\) = 0), (8.8) enables the two-dimensional counterpart of the initial conditions (4.7) and (4.8) to yield*

**T***g*= 0 and, therefore, that there is absence of azimuthal shear deformation ( \(\gamma \) = 0) if the fibres are straight radial (\(A_{\varTheta }\) = 0) or concentric circles aligned with the azimuthal direction of the tube cross section (\(A_{R}\) = 0). However, the general case of spiral fibres (\(A_{R}A_{\Theta }\ne \) 0) involves always generation of non-zero azimuthal shear deformation. These three particular cases are accordingly considered and discussed next separately.

### 8.1 Radial fibres (\(A_{\Theta }\) = 0)

*ii*\(_{1})\) in Sect. 3, (8.14) yields \(g = \gamma \) = 0, and the relevant restrictions (3.15) are satisfied identically. It is thus concluded that radial fibres do maintain their direction and shape, which is made alternatively obvious by observing that (8.6) returns

*r*yields the radial normal stress in the following alternative form:

However, if the growing cross section is solid rather than hollow, then the growing long cylinder is indeed free of external tractions. Because *a* = 0 in that case, *r*(*R*; *t*) and \({\hat{{W}}}\) are only required to be such that (8.20) returns a finite value for \( q_{a}(t)\). By virtue of (8.18) and (8.20) that value of \( q_{a}(t)\) will then represent the finite value that the radial normal stress attains at the centre of the cross section.

*r*(

*R*;

*t*) may be determined by inserting (8.17) into the radial equilibrium equation (8.12a), and then solving the resulting differential equation, namely,

### 8.2 Circumferential fibres (\(A_{R}\) = 0)

*g*= \(\gamma \) = 0. The invariants (8.7a, b) attain again the simplified form (8.16a, b), respectively, but the third becomes \(J_3 =\chi _\theta ^2 \). Hence, (8.6) yields

It follows that if (8.22) and (8.27) hold simultaneously under a mass-growth rate that satisfies (8.5), then the cross section of a radially or a circumferentially reinforced infinitely long tube subjected to homogeneous traction boundary conditions (\(q_{a}=q_{b}=\tau _{b\theta } \) = 0) maintains purely radial, inflation-type mass-growth (*g* = \(\gamma \) = 0) in a completely stress-free manner.

### 8.3 Cross-sectional spiral fibres (\(A_{R}A_{\Theta } \ne \) 0)

*is independent of the radial coordinate parameter and, therefore, constant, the differential equation (8.34) admits the general solution*

**A***t*) represents an arbitrary non-dimensional function of time which, by virtue of (8.10b), is still required to conform with the initial condition (8.27b).

*r*(

*R*;

*t*),

*g*(

*R*;

*t*) and \({\hat{{W}}}\left( {J_1 ,J_2 ,J_3 } \right) \). Alternatively, use of (8.14) in connection with (8.36) may convert (8.37) into a single equation for \({\hat{{W}}}\). Potential solution of that highly non-linear integro-differential equation may then provide

*r*(

*R*;

*t*) and

*g*(

*R*;

*t*) through subsequent substitution into (8.35) and (8.14), respectively. However, such a solution will not be pursued here any further. It should be noted that the incompressible mass-growth process of present interest may become possible only if associated with a rate of mass-growth which is obtained by inserting (8.35) into the right- hand side of (8.5).

## 9 Conclusions

Several types of tube-like fibre-reinforced tissue have the ability to grow in a manner that preserves not only their shape, but also the shape and direction of their embedded fibres. Motivated by this observation, this investigation considered the most general axisymmetric mass-growth pattern of a finite tube reinforced by a single family of fibres, and investigated the influence that preservation of the fibre direction exerts on relevant mathematical modelling, as well as on the growth mechanism and the physical behaviour of the tube. Accordingly, relevant sets of necessary conditions that enable axisymmetric mass-growth patterns of a tube to take place were developed not only for fibres that preserve some general direction, but also for all six particular cases in which fibre direction remains normal to either one or two of the associated cylindrical polar coordinates.

Those conditions exert direct influence only on the kinematic characteristics of the tube growth pattern. They are mathematically independent of the tube material features and behaviour, and are thus required to hold in addition to and, therefore, simultaneously with the standard stress equilibrium equations. By enhancing coupling between tube kinematics and constitutive characteristics, they thus reflect the hidden ability of the growing tube to guide its material behaviour in a manner that enables the embedded growing fibres to preserve direction and shape. Those conditions can thus be exploited, and employed in a manner that provides valuable information regarding the constitutional behaviour of a fibre-reinforced tube undergoing axisymmetric mass-growth.

It is accordingly seen with examples that, in cases that validity of (4.3) and (4.5) justify the consideration and the use of the concept of mass-growth hyperelasticity, the additional information provided by the aforementioned conditions enables identification of specific classes of the strain energy density for growth that are suitable and, hence, admissible in different types of axisymmetric tissue mass-growth. In this context, the example applications considered in Sects. 5–8 refer to different tube mass-growth patterns that take place in an incompressible manner and, hence, enable the continuity equation with growing mass to attain the simplified form (4.3).

This form of the continuity equation exerts direct influence on the dilatational features of the mass-growth of interest and leads naturally to (5.6), which relates the fibre growth stretch with the rate of growth of the tube material. The predominantly dilatational nature of the observed mass-growth pattern identifies thus the fibre growth stretch as the principal proportionality parameter that dictates not only the purely dilatational mass-growth pattern discussed in Sect. 5, but also the combined dilatational and shear types of growth detailed afterwards in Sects. 6 and 7. It is emphasised that, due to their different kinematic characteristics, each of the three-dimensional mass-growth patterns studied in Sects. 5–7 is associated with some different class or classes of admissible strain energy density for growth.

However, the example growth patterns studied in Sects. 5 and 6 are both triggered by the same class of mass-growth rates and are applicable regardless of the preserved fibre shape and direction. They lead to identical predictions of fibre growth stretch and, hence, they both preserve during growth the initial ratio between the radial and axial tube dimensions. Mass-growth patterns sought and found in Sect. 6 are accordingly regarded as alternative or additional to their purely dilatational counterparts developed previously in Sect. 5 in the sense that, due to availability and use of a different class of admissible strain energy densities for growth, they superpose on mass-growth dilatation appropriate amounts of torsional deformation.

In contrast, the example application discussed in Sect. 7 focuses attention on a specific shape of fibres. This is the shape of fibres that grow with and within the tube tissue in a helical manner and, as is detailed in the Introduction, met often in nature. This application enables thus the identification of strain energy densities for growth that, as is also observed in nature, allow the tube to grow in a manner that does not preserve the ratio between its radial and axial dimensions.

Several of the outlined observations still hold in the last application considered in Sect. 8 which, however, considered and studied the cross-sectional, plane strain mass-growth of a transversely isotropic tube of infinite extent. An interesting case of stress-free cross-sectional mass-growth has thus been identified when the infinitely long tube of interest is reinforced by a radial (Sect. 8.1) or circumferential family of fibres (Sect. 8.2). Such stress-free mass-growth has the form of a homogeneous axisymmetric deformation and is possible only if the strain energy for growth belongs to a class of functions which, like (8.23) or (8.25), satisfy the differential equation (8.22). Moreover, it fits in an exact mathematical manner the quasi-static equilibrium framework considered, as long as that deformation is radially uniform and linearly proportional to time.

Most interestingly, this kind of homogeneous, stress-free mass-growth is possible in the case of a finite tube regardless of the fibre shape and direction, provided that the strain energy for growth satisfies simultaneously the differential equations (5.15) and (5.23). This is however not possible in cases that the infinitely long tube of interest is reinforced by a cross-sectional family of spiral fibres. Section 8.3 makes thus available a non-homogeneous mass-growth pattern that enables the present analysis to also consider and account for cross-sectional tube mass-growth that preserves the direction of spiral fibres, namely, plane cross-sectional fibres which are neither radial nor circumferential.

There is a large number of additional mass-growth applications that may be considered and studied in a manner similar to that detailed in Sects. 5–8. These may be formed by considering, for instance, several different combinations of non-zero components in the deformation tensors (2.4). Consideration and study of any of those potential axisymmetric mass-growth patterns in association with some particular fibre direction met in nature, like the helical fibre direction considered in Sect. 7, will also be of interest. Moreover, cases of non-axisymmetric mass-growth of tube-like fibre-reinforced tissue may possibly be met in nature, and could be considered and modelled in a similar manner.

Along with the type of the incompressible hyperelastic mass-growth introduced in Sect. 4, there should exist in nature mass-growth processes which are compressible. Rather than (4.3), such types of growth make use of the full form of the continuity equation with growing mass (2.10) and may thus be able to influence directly not only the dilatational, but also the shear deformation features of axisymmetric tube mass-growth. The present hyperelasticity mass-growth model can then be found helpful in directing relevant compressible mass-growth developments [22, 32], where the influence of the divergence of the velocity vector is still accounted for in the corresponding constitutive equation (4.1). Similarly, the presented analysis may assist future developments in which plastic-like mass-growth and/or influence of some non-zero prestress state are further expected to be accounted for [22, 29].

A final comment should refer to relevant tube growth cases that involve more than one family of growing fibres. Most likely, consideration of such cases in association with the postulates and ideas described in the present investigation will increase the number of the kinematic restrictions outlined in Sect. 3.2. As a result, the aforementioned strain energy density classes should be expected to become narrower and, potentially, lead thus to simpler forms of admissible strain energy densities for growth. Such a potential result should not be considered as very restrictive for the mechanical behaviour of a growing tube-like tissue because, as is justified in Sect. 4, the strain energy density for growth and the strain energy density for mechanically caused deformations of the same tissue should, in general, not be the same.

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