A one-dimensional model for axisymmetric deformations of an inflated hyperelastic tube of finite wall thickness

Abstract

We derive a one-dimensional (1d) model for the analysis of bulging or necking in an inflated hyperelastic tube of finite wall thickness from the three-dimensional (3d) finite elasticity theory by applying the dimension reduction methodology proposed by Audoly and Hutchinson (2016). The 1d model makes it much easier to characterize fully nonlinear axisymmetric deformations of a thick-walled tube using simple numerical schemes such as the finite difference method. The new model recovers the diffuse interface model for analyzing bulging in a membrane tube and the 1d model for investigating necking in a stretched solid cylinder as two limiting cases. It is consistent with, but significantly refines, the exact linear and weakly nonlinear bifurcation analyses. Comparisons with finite element simulations show that for the bulging problem, the 1d model is capable of describing the entire bulging process accurately, from initiation, growth, to propagation. The 1d model provides a stepping stone from which similar 1d models can be derived and used to study other effects such as anisotropy and electric loading, and other phenomena such as rupture.

Introduction

Hyperelastic tubes are commonly found in various applications ranging from soft robotics (Ma et al., 2015, Lu et al., 2015, Lu et al., 2020, Stano and Percoco, 2021) to energy harvesting (Lu and Suo, 2012, Bucchi and Hearn, 2013, Smith, 2016, Collins et al., 2021, Bastola and Hossain, 2021). They are also used to model human arteries in order to understand pathological conditions such as aneurysms (Fu et al., 2012, Alhayani et al., 2014, Demirkoparan and Merodio, 2017, Varatharajan and DasGupta, 2017, Hejazi et al., 2021). Inflation of a hyperelastic tube is one of the few boundary value problems in nonlinear elasticity that have closed-form solutions, and it provides the simplest setup to explain bifurcation, localization, loss of convexity, and “two-phase” deformations. Thus, understanding this problem is not only important for applications, but may also shed light on other more complicated stability and bifurcation problems.

Simple inflation experiments with commercially available latex rubber tubes show that localized bulging is the dominant deformation form. For almost all realistic constitutive models for rubber, the pressure versus volume curve has an up-down-up shape under the condition of fixed resultant axial force (Green and Adkins, 1960). This led Yin (1977) and Chater and Hutchinson (1984) to analyze the final observable configuration as that corresponding to a “two-phase” deformation. The subsequent experimental studies carried out by Kyriakides and Chang, 1990, Kyriakides and Chang, 1991, Pamplona et al. (2006) and Goncalves et al. (2008) have provided a clear picture on how a localized bulge initiates, grows and then propagates under fixed axial force or fixed-ends conditions.

When the membrane assumption is made, the governing equations for tube inflation can be viewed as a finite-dimensional spatial dynamical system that has two conservation laws/integrals (Pipkin, 1968). This realization enabled Fu et al. (2008) to demonstrate explicitly how a localized solution initiates as a zero-wave-number mode from the uniform deformation and how it evolves into a “two-phase” state. The stability of bulging solutions and their sensitivity to imperfections have been studied under the same framework (Pearce and Fu, 2010, Fu and Xie, 2010, Fu and Il’ichev, 2015). Fresh analytical insight into the case of fixed ends has also been obtained. It is shown that the bifurcation condition for this case corresponds to the axial force reaching a maximum at a fixed pressure (Fu and Il’ichev, 2015); in other words, as pressure is increased, the critical pressure is the value of pressure at which the axial force reaches a maximum when viewed as a function of the axial stretch. Also, in contrast with the case of fixed axial force where the measured pressure approaches a constant value (the propagation pressure), the measured pressure in the case of fixed ends has an up-down-up shape where the right branch approaches a master curve that is independent of the pre-axial-stretch or the tube length (Guo et al., 2022).

In some practical applications, however, the tube wall may be of moderate or even large thickness and the membrane model no longer applies. For example, in the context of aneurysm formation, a human artery can be as thick as a quarter of its outer radius (Müller et al., 2008), and fiber-reinforcement also seems to reduce the range of validity of the membrane assumption (Wang and Fu, 2018). Thus, recent studies have begun to consider hyperelastic tubes of finite wall thickness. Fu et al. (2016) showed that the associated bifurcation condition for localized bulging corresponds to the vanishing of the Jacobian determinant of the internal pressure and the resultant axial force as functions of the azimuthal stretch on the inner surface and the axial stretch; see also Yu and Fu (2022) for an alternative derivation. This provides a framework under which additional effects such as rotation (Wang et al., 2017), double-fiber-reinforcement (Wang and Fu, 2018), bi-laying (Liu et al., 2019, Ye et al., 2019), torsion (Althobaiti, 2022), and surface tension (Emery and Fu, 2021a, Emery and Fu, 2021b, Emery and Fu, 2021c, Emery, 2023) can be assessed in a systematic manner. Ye et al. (2020) conducted a weakly non-linear analysis and derived the bulging solution explicitly. The analytic predictions were corroborated by numerical simulations (Lin et al., 2020) and experiments (Wang et al., 2019).

For tubes of finite wall thickness, the equations that govern their axisymmetric deformations are coupled nonlinear partial differential equations. Although analytic solutions can be obtained in the near-critical regime using asymptotic methods (Ye et al., 2020), the complexity of the governing equations forbids any further analytic attempts to understand the bulging evolution further away from the bifurcation point. The post-bifurcation behavior in the fully nonlinear regime has so far only been investigated by resorting to Abaqus simulations (Wang et al., 2019, Lin et al., 2020). This is not satisfactory since the insight provided by full-scale simulations tends to be limited and there are situations where repeated calculations of the bulging profile are required (e.g. in the assessment of the rupture potential (Hejazi et al., 2021)).

A recent series of studies by Audoly and coworkers has opened the possibility that a 1d reduced model can be derived to describe the fully nonlinear evolution of bulging or necking. In the first of this series, Audoly and Hutchinson (2016), the authors derived a 1d model for tensile necking localization in a 3d prismatic solid of arbitrary cross-section. The key idea of their derivation is a dimension reduction assuming slow variation in the axial direction that respects self-consistency. In terms of the language of perturbation analysis, the leading-order solution is almost correct and higher-order terms are only added to restore self-consistency. The method was later applied by Lestrigant and Audoly to obtain a diffuse interface model for the characterization of propagating bulges in membrane tubes (Lestringant and Audoly, 2018) and a 1d model for predicting surface tension-driven necking in soft elastic cylinders (Lestringant and Audoly, 2020b). It has also been used recently to derive a 1d model for elastic ribbons (Audoly and Neukirch, 2021) and for tape springs (Kumar et al., 2022). The systematic reduction method for deriving 1d strain-gradient models for nonlinear slender structures was further generalized by Lestringant and Audoly (2020a). It is worth pointing out that although the 1d models are built on the assumption that localized solutions vary slowly in the longitudinal direction, it is surprisingly accurate, even in the region where the localization is well developed. This is illustrated by the numeric examples in the aforementioned work and in the comparative studies by Wang and Fu (2021) and Fu et al. (2021).

This work aims to extend the diffuse interface model of Lestringant and Audoly (2018) for membrane tubes to tubes of finite wall thickness, in a similar spirit as the previous work Fu et al. (2016) and Ye et al. (2020) that extend the bifurcation condition and the weakly nonlinear analysis from membrane tubes to thick-walled tubes. In contrast with the case under the membrane assumption where the original governing equations are already one-dimensional, the governing equations for the current case are two-dimensional, and the uniformly inflated deformation is no longer homogeneous since the uniform solution depends on the radial variable. It will be shown that a 1d reduced model can still be derived with the associated energy functional simplified to the form

$${\mathcal{E}}_{\text{1d}}\left[a\right]={\int }_{-L}^L\left(G(a,\lambda \left(a\right))+\frac{1}{2}D\left(a\right)a^{\prime }{\left(Z\right)}^2\right)dZ+C\left(a\right)a^{\prime }\left(Z\right){|}_{-L}^L,$$

where $L$ is the initial half length, $Z$ is the axial coordinate, $a\left(Z\right)$ is the azimuthal stretch on the inner surface (a constant multiple of the deformed inner radius) and the expressions for $G(a,\lambda \left(a\right))$, $D\left(a\right)$ and $C\left(a\right)$ are given in (3.10), (4.22), (4.23), respectively. The first term $G$ in (1.1) corresponds to the energy of the uniform deformation, which determines the amplitudes of the two phases in the bulge propagation stage; the second term accounts for the contribution of the strain gradient to the total energy, which describes how the two phases are connected. The Euler–Lagrange equation associated with the energy functional (1.1) is a second-order nonlinear ordinary differential equation for $a\left(Z\right)$, which is a drastic simplification from the original nonlinear partial differential equations. This 1d model is validated by comparison with finite element simulations, showing excellent agreement with numerical results even for the propagation stage.

The rest of this paper is as follows. In Section 2, we formulate the 3d axisymmetric finite-strain model for a tube of finite wall thickness under inflation and axial stretching. In Section 3, we summarize solutions corresponding to uniform inflation of the tube, making preparation for the subsequent dimension reduction. In Section 4, we carry out the dimension reduction and derive the aforementioned 1d strain-gradient model. The connection of the 1d model with prior work is given in Section 5. In Section 6, we validate the 1d model by making comparison with finite element simulations. Finally, concluding remarks are given in Section 7.