## Abstract

Electro-active polymers (EAPs) for large actuations are nowadays well-known and promising candidates for producing sensors, actuators and generators. In general, polymeric materials are sensitive to differential temperature histories. During experimental characterizations of EAPs under electro-mechanically coupled loads, it is difficult to maintain constant temperature not only because of an external differential temperature history but also because of the changes in internal temperature caused by the application of high electric loads. In this contribution, a thermo-electro-mechanically coupled constitutive framework is proposed based on the total energy approach. Departing from relevant laws of thermodynamics, thermodynamically consistent constitutive equations are formulated. To demonstrate the performance of the proposed thermo-electro-mechanically coupled framework, a frequently used non-homogeneous boundary-value problem, i.e. the extension and inflation of a cylindrical tube, is solved analytically. The results illustrate the influence of various thermo-electro-mechanical couplings.

## 1. Introduction

In the last couple of decades, the so-called electro-active polymers (EAPs) have been gaining popularity within the smart materials community. The core mechanism of EAPs is an electro-mechanical coupling behaviour that converts the energy of an electric field into mechanical energy. In contrast to traditional piezoelectric materials of small strain actuation mechanisms, the excitation on EAPs by an external electric field results in a large deformation. It may also result in a change in their material behaviour, e.g. the polarization response. EAPs can be divided into two major categories depending on the mechanism of deformation: electronic electro-active polymers (EEAPs) and ionic electro-active polymers (IEAPs). In general, two electro-mechanical forces are mainly responsible for the deformation of the polymer. Firstly, the Maxwell stress that originates from the elastostatic forces between electric charges. Secondly, the electrostriction that is due to intramolecular forces of the material. EAP materials have many potential applications, e.g. as dielectric elastomers (DEs) in artificial muscles and as sensors and generators [1–5]. EEAP as a DE is used in the form of a thin film that is coated with two compliant electrodes and sandwiched in between them. The dielectric elastomer’s thickness has to be thin-filmed for a large activation output with a moderate electric potential difference. When the thin film is squeezed in the direction of the applied electric field, the material expands in the transverse planar direction as the underlying polymeric material is idealized as a nearly incompressible material [6–11].

Experimental studies of EAPs under either purely mechanical or electro-mechanical loading are very rare in the literature [12,13]. Diaconu *et al*. [12] studied the electro-mechanical properties of a synthesized polyurethane (PU) elastomer film-based polyester. They verified that the deformation of a PU elastomer varies quadratically according to the applied electric field. Qiang *et al.* [13] presented an experimental study on the dielectric properties of a polyacrylate DE. Ma & Cross [14] conducted an experimental investigation of the electro-mechanical response in a dielectric acrylic elastomer in various frequency ranges at room temperature. Michel *et al.* [15] performed a comparison between silicone and acrylic elastomers that can potentially be used as dielectric materials in EAP actuators. Moreover, natural rubber and styrene-based elastomers have recently been shown to demonstrate very good electro-mechanical properties (even better than silicone and acrylic elastomers) for the development of DE transducers (e.g. [16,17]).

Hossain *et al.* [18] initially carried out a comprehensive mechanical characterization of an acrylic elastomer (VHB 4910; 3M) using different standard experiments such as single-step relaxation tests, multi-step relaxation tests and loading–unloading cyclic tests. In modelling the mechanical behaviour, a modified version of the micro-mechanically motivated Bergström–Boyce viscoelastic model [19,20] was used along with a finite linear evolution law. Later on, Hossain *et al.* [21] conducted experiments with electro-mechanically coupled loads with standard tests already used for a viscoelastic polymeric material characterization in the purely mechanical case. In all experimental cases, the polymer samples were pre-stretched by up to several hundred per cent to achieve an initial thickness reduction in order to increase the effect of the applied electro-mechanically coupled load. Subsequently, the pre-stretched samples were subjected to various amounts of mechanical as well as coupled deformations at different strain rates. The data produced from several loading–unloading tests, single-step relaxation tests and multi-step relaxation tests show that electric loading has a profound effect on the time-dependent behaviour of the VHB 4910 elastomer.

Recently, Ask *et al.* [22,23] modelled the electrostrictive behaviour of viscoelastic polymers, particularly electrostrictive PU elastomer, using a phenomenologically motivated constitutive model. After proposing a thermodynamically consistent electro-mechanical coupled finite strain viscoelastic model based on a multiplicative decomposition of the deformation gradient, they used data from Johlitz *et al.* [24] to identify the purely mechanical parameters of the model and data from Diaconu *et al.* [12] to identify the coupled parameters as well as for their model validation. Büschel and co-workers [25] proposed an electro-viscoelastic model using a multiplicative decomposition of the deformation gradient into an elastic part and a viscous part for the mechanical deformation where the mechanical free energy basically stems from an Ogden-type energy function [26]. Another interesting electro-viscoelastically coupled model was proposed by Vogel *et al*. [27,28]. Following the classical approach of the deformation gradient decomposition, Saxena *et al.* [29,30] proposed an electro-viscoelastic model that takes into account an additive decomposition of the vector-valued electric field variable into an equilibrium part and a non-equilibrium part. Compared with Vogel *et al.*, Saxena and co-workers formulated their governing equations following the idea of McMeeking & Landis [31]. However, none of the earlier constitutive models takes a thermo-electro-mechanically coupled approach into account.

Vertechy *et al*. [32,33] presented a finite-deformation thermo-electro-elastically coupled continuum model for electrostrictive elastomers. In the framework, they assumed that EAPs are isotropic modified-entropic hyperelastic dielectrics that can deform under the combined actions of electrical, thermal and mechanical stimuli. They have demonstrated the thermodynamical consistency of the formulation. Moreover, the formulation does not require the postulation of any force or stress tensor of electrical origin (cf. [32]). In order to identify the material parameters appearing in the model, they conducted a thermo-electro-elastically experimental study on a lozenge-shaped linear actuator which is optimally designed for DEs (cf. [33]). They were able to demonstrate an excellent correlation of the proposed thermo-electro-elastic model with the experimental data. However, the model was applied to the study of a uniaxial mechanical load case and the temperature distribution was assumed to be homogeneous for the selected geometry. Therefore, the solution of any real thermo-electro-mechanical boundary value problem is absent here.

A coupled theory for damage and time dependence, e.g. viscoelasticity or ageing of nonlinear thermo-electro-mechanical and thermo-magneto-mechanical problems, is developed in Chen [34,35]. Therein, the total energy includes the contribution of the electric field as a functional of the histories of stress, temperature, temperature gradient and electric field while an internal state variable is introduced in the case of damage modelling. The framework is developed in a thermodynamically consistent way. A superposition principle of time, ageing, temperature, stress and electric field is proposed for materials with memory on an intrinsic time scale so that the long-term property functions may be represented with horizontal and vertical shifting of the momentary master curves. However, none of the purely theoretical contributions demonstrate solutions of any boundary-value problem for either thermo-electro-viscoelasticity or thermo-magneto-viscoelasticity.

In this contribution, we present a general constitutive framework for a thermo-electro-mechanical case departing from the second of law of thermodynamics and neglecting any time-dependent phenomena. Starting with the standard definition of the heat capacity, we devise a generalized formulation for the thermo-electro-mechanical free energy function in an additive form where the electro-mechanically coupled effect is linearly scaled with the temperature. Moreover, three different forms of electro-mechanical coupled energy functions are proposed. For the numerical case study, we analytically solve the extension and inflation of a cylindrical tube that works under a thermo-electro-mechanical load where, in addition to electro-mechanical loads, heat flow also occurs. To the best of the authors’ knowledge, this widely used benchmark problem is not solved yet in the literature for the thermo-electro-mechanical case. Polymeric materials are typically viscoelastic and non-perfect electrical insulators where temperature changes can also occur as a result of dissipation. In addition, there are various other sources for temperature changes, e.g. external heat flux or prescribed temperature boundary conditions. The current contribution proposes a general framework for electro-thermo-mechanical coupling whereby temperature changes due to viscoelastic dissipation and current leakage are not included for the sake of simplicity. Consideration of all sources of dissipations and the resulting temperature variations are important and will be addressed in our forthcoming contributions.

The paper is organized as follows: §2 will briefly review the nonlinear kinematics and will elaborate basic equations in nonlinear electro-mechanics. In §3, the main mathematical foundation that leads to a thermo-electro-mechanical formulation for electro-active polymers is discussed in detail. It includes coupled constitutive equations, the total energy function and the modified heat equation of the thermo-electro-mechanically coupled problem. A non-homogeneous boundary problem is solved analytically with the proposed formulation which is elaborated in §4. Section 5 concludes the paper with a summary and an outlook for future works.

## 2. Basics of nonlinear electro-mechanics

### (a) Kinematics

In the case of large deformations, we distinguish between the reference configuration ** X**. In the deformed configuration

**which is connected to**

*x***through the nonlinear deformation map**

*X***=**

*x***(**

*χ***). The deformation gradient**

*X***is defined as the gradient of the deformation map with respect to the material coordinates**

*F**J*is the Jacobian determinant of the deformation gradient. Using the deformation gradient, we can introduce the left and right Cauchy–Green tensors

**and**

*b***, respectively,**

*C*### (b) Balance laws in electrostatics

#### (i) Spatial configuration

The electric displacement *ε*_{0}
_{1} is satisfied exactly by an electric field that is derived from a scalar potential [27,36–38]. Therefore, *φ* is the gradient of the electric potential with respect to the spatial coordinates. The mechanical behaviour of the material is governed by the balance of linear momentum
*b*_{t} and the total Cauchy-type symmetric stress tensor *σ*^{tot} [9] that consists of a non-symmetric mechanical Cauchy stress ** σ** and the ponderomotive stress

*σ*^{pon}[8]

*σ*^{pol}[28,39] and the symmetric Maxwell stress

**is the second-order identity tensor in the spatial configuration. Outside of matter the polarization**

*i***and the polarization stress**

*σ*

*σ*^{pol}vanish but the Maxwell stress and the electric displacement

**are prescribed**

*χ***on the surface of**

*n*^{out}−•

^{in}]. The jump conditions for the electrical quantities are defined as

#### (ii) Material configuration

Various electric quantities described in the previous section are expressed in the spatial configuration. We now want to transform them in the undeformed configuration ** X** in

*σ*^{tot}can be transformed into a material counterpart, the total Piola stress tensor

*P*^{tot},

**and the ponderomotive part**

*P*

*P*^{pon}

*P*^{pol}and the Maxwell stress

*P*^{max}

*dA*to the spatial configuration with the respective area element

*da*.

## 3. Basics of nonlinear thermo-electro-mechanics

### (a) Constitutive equations

The energy density *Ψ* per unit volume in *θ* is the absolute temperature. The second law of thermodynamics in the form of the Clausius–Duhem inequality leads to [40–44]
*H* is the entropy and ** Q** is a heat flux vector defined in the material configuration that can be transformed to the spatial form via

*J*

**=**

*q*

*F***. Taking the time derivative of**

*Q**Ω*according to

*p*is a Lagrange multiplier associated with the incompressibility constraint [7] and

**is the second-order identity tensor in the current configuration. After applying the Coleman–Noll argumentation [40] to equation (3.5), the reduced conductive dissipation power density reads**

*i*### (b) Energy function

To derive a thermo-electro-mechanically coupled free energy function, we start with the heat capacity *θ*_{0} is the reference temperature
*c*_{0} is also related to the negative second derivative of the energy *Ψ* with respect to the absolute temperature *θ* as
*θ*_{0} to an arbitrary temperature *θ* leads to
*C*_{1} may depend on the deformation gradient ** F** and the electric field

*θ*

_{0}to an arbitrary temperature

*θ*brings us to

*W*depends on the electro-mechanical coupled invariants, i.e.

*I*

_{1}–

*I*

_{6}as

*I*

_{1},

*I*

_{2},

*I*

_{4},

*I*

_{5},

*I*

_{6}) are defined as a combination of the right Cauchy–Green tensor

**and the electric field**

*C**C*

_{1}contains electro-mechanical contributions which we decompose additively into a purely mechanical part

*M*(

**) and an electro-mechanically coupled part**

*F**M*(

**), i.e.**

*F**α*and

*κ*are the thermal expansion and bulk modulus coefficients at the reference temperature and

*γ*is a dimensionless constant material parameter. Note that, since

*M*(

**) only accounts for volumetric changes, it vanishes for incompressible materials. For the electro-mechanically coupled part**

*F*If we do not consider any thermo-electro-mechanical coupling, the electro-mechanically coupled part simply vanishes, i.e.
*π*_{0} and the deformation is neglected, the formulation reads
*I*_{4} is the fourth invariant defined in equation (3.15). In the third case, thermo-electro-mechanical effects are considered, thus we assume
*I*_{5} is the fifth invariant defined in equation (3.15). This will give a pyroelectric stress in a form analogous to the Maxwell stress as

### (c) Heat equation

Considering the first law of thermodynamics, the governing equation used to describe the thermal field can be written in entropy form as
** Q** in the material configuration. By using the constitutive relation

*H*=−∂

*Ψ*/∂

*θ*with the help of the chain rule and the definition of the specific heat capacity

*c*

_{0}, we obtain

*et al.*[32] for a similar expression.

## 4. Non-homogeneous boundary-value problems

In this section, the above thermo-electro-mechanical framework is applied to a controllable non-homogeneous deformation, e.g. the extension and inflation of an incompressible cylindrical tube with a cylindrical symmetry (cf. [50–55]). Therefore, for the considered tube problem, it is reasonable to work in cylindrical coordinates. Expressing the divergence of the electric displacement *ϕ*=∂(•)/∂*z*=0, equations (4.1) and (4.2) reduce to

### (a) Extension and inflation of a tube

We consider a thick-walled tube in cylindrical coordinates (figure 1). The geometry in the spatial configuration is described by
*z*-direction (*r*,*Φ*,*z*) are defined as
** σ**=

**can be formulated from equation (4.4) as**

*o**a*

*b*

*c*

With the constitutive relation (3.8), for incompressibility the total Cauchy stress can be calculated as
*σ*^{tot}

#### (i) Temperature function

In order to allow for an analytical solution available in the literature [56,57], we neglect the thermo-mechanical and thermo-electrical cooling/heating effects. As a result, the heat equation from equation (3.23) simply yields
** q**=−

*κ*grad

*θ*and Δ is a Laplacian operator. In this case, a constant value for the spatial thermal conductivity

*κ*has been assumed. Note that the considered expression is a special case which, in the case of any dependency on both the deformation and electric field, needs to be extended to incorporate more complex coupling effects by choosing a different constitutive equation. For more details, we refer, for example, to the works of Eringen & Maugin [58]. In the (quasi) static case, the heat equation is reduced to the Laplace equation [56], which is also known as the stationary heat equation. In the cylindrical coordinates (

*r*,

*ϕ*,

*z*), the equation for an axial symmetric problem, e.g. a cylindrical hollow tube, can be written as

*r*is the actual radius of the tube. An analytical solution for the equation can be found in Bland [56] and also in Rajagopal & Huang [57]

*k*

_{1}and

*k*

_{2}can be determined by the boundary conditions at the inner and outer surfaces of the tube, respectively. For an internal radius

*a*

_{i}and an external radius

*a*

_{e}with the corresponding actual temperatures

*θ*(

*a*

_{i}) and

*θ*(

*a*

_{e}), respectively, we find

#### (ii) Axially applied electric field

Now, an energy function *W* at the reference temperature is required. As an example, we assume an incompressible neo-Hookean-like material as in [37,38] that depends on the purely mechanical invariant *I*_{1}, the shear modulus *μ* that depends on the purely electrical invariant *I*_{4}, the coupled invariant *I*_{5} and the constants *c*_{1}, *c*_{2}. Other advanced forms of energy functions for rubber-like materials can be coupled with the electric part of the energy to improve the modelling (cf. [19,59–61]). Without considering the surrounding free space, the energy function is formulated as
*I*_{1},*I*_{2},*I*_{4},*I*_{5},*I*_{6}) are defined in equation (3.15). Note that this type of free energy function incorporating electro-mechanical coupling is frequently used in the literature [6,7,18,27,28] and is motivated by neo-Hookean-type pure mechanical response modelling of rubber-like materials. It should be noted that the large deformations considered here lead to a strong polarization of the material. Owing to the small value of the vacuum permittivity the free space term in the total energy formulation (3.4) can be neglected as in [37] and it holds that *μ*(*I*_{4})=*g*_{0}+*g*_{1}*I*_{4}, where *g*_{0},*g*_{1} are material parameters. Therefore, the derivatives ∂*Ω*_{i}/∂*I*_{i}=:*Ω*_{i} with respect to the invariants are
*c* is an integration constant which can be determined from the boundary conditions for the stress. If the outer surface of the tube is free of mechanical loads we find
*P*,
*k*_{1} from the temperature equation (4.16) does not depend on the actual radius *r*. Hence, as is demonstrated in appendix A, the first part of the pressure *P*_{1} can be expressed as
_{i}=*a*_{i}/*A*_{i}, λ_{e}=*a*_{e}/*A*_{e}. The second integral contains an expression that is very difficult to calculate with analytical integration methods. Therefore, it is solved using the computing environment Maple 18 [62]. Detailed derivations are given in appendix A.

Another important term for the demonstration of the results for this tube example is the normal force *σ*_{zz} denotes the axial component of the mechanical traction that is applied. With the stress definitions (4.11), the definition of the axial component of the Maxwell stress is *c*_{2}=*ε*_{0}/2. Each term of the integral on the r.h.s. of equation (4.28) is evaluated separately (see appendix A). In the isothermal case, the normal force can be expressed as
*ζ*=*A*_{e}/*A*_{i}. The introduction of a temperature gradient adds a number of terms to the formulation of the normal force and leads to an expression that is significantly larger than the equivalent expression for the pressure presented above. Therefore, the final expression is presented only in appendix A. The constants used in the subsequent calculations are summarized in table 1. The values of *g*_{0} and *g*_{1} are taken from Bustamante [6].

#### (iii) Results and discussions

In order to illustrate the general behaviour of the example, we initially focus on the electro-mechanical load case for a cylindrical tube with an initial internal radius of 10 mm. Thus, we prescribe an axial stretch λ_{z}, a radial inflation or compression characterized by λ_{i}, the ratio of the internal radius after the deformation to the initial radius and a purely axial electric field. In figure 2, the radial pressure *P* is plotted for selected values of the electric field *ζ*, i.e. the ratio of the initial external to the initial internal radii, depending on the parameters λ_{i} and λ_{z} without the influence of a temperature gradient.

Figure 2*a* shows that, for the inflation of the tube (λ_{i}>1), a positive pressure has to be applied on the internal surface. This pressure increases with increasing values of λ_{i}. In the case of a radial compression (λ_{i}≤1), we find that the pressure is negative and that it decreases for smaller values of λ_{i}. Furthermore, it can be seen from figure 2*a* that the material softens due to the applied electric field as the magnitude of the pressure decreases. In contrast to an axial compression λ_{z}<1, the applied pressure at the inner surface is positive and it is reducing for increasing values of λ_{z} (figure 2*b*). An increase in the wall thickness of the tube characterized by larger values of *ζ* leads to a higher pressure level that is needed both for the inflation and for the axial stretch of the tube. In both plots of figure 2, it can be seen that the influence of the electric field on the applied pressure increases for larger values of *ζ*, leading to a more distinctive reduction of the pressure.

Now we also incorporate a temperature gradient along the radial axis. To achieve this, the temperature at the outer surface of the tube *θ*_{e} is varied but the temperature at the internal surface is kept at the reference value *θ*_{0}. An increase in the absolute temperature *θ*_{e} (*θ*_{e}>*θ*_{0}) leads to a pressure increase. By cooling the external tube surface and keeping the external temperature less than the reference one (*θ*_{e}<*θ*_{0}) the pressure decreases (figure 3). This behaviour is typical for a rubber-like material in which the stresses increase with increasing temperature, as presented for example in [63].

Next we investigate the variation of the pressure with respect to the thickness ratio *ζ* and the electric field strength *E*_{0} in the axial direction.

It is obvious that, for a larger value of *ζ*, we find an increase in the magnitude of the pressure (figure 4). Furthermore, with a larger value of *ζ* the absolute difference between the pressure in the isothermal case and that in the cases with heating or cooling becomes larger but the ratio of the pressure in these two cases decreases. The latter shows a decreased influence of the temperature change due to the increased wall thickness. This is explained in more detail in the later examples.

We will now focus on the behaviour of the scaled normal force _{i}>1) at a constant volume leads to a contraction in the axial direction for a prescribed value of λ_{z}=1, there has to be a positive normal force acting in the axial direction to prevent any contraction. For an increased tube thickness, the level of the normal force increases as well. This behaviour is plotted in figure 5*a*. The applied electric field causes the material to contract in the axial direction. Simultaneously, the electric field leads to a softening of the material due to the dependency of the shear modulus on *I*_{4}. As the softening effect has a stronger influence than the instigated contraction, the applied load *b* shows the response of the normal force _{z}. For increased values of the axial stretch characterized by λ_{z}, the normalized force

At this stage, we take into account a radial temperature gradient for the case of normal force variation. As before, this is achieved by a temperature change at the external surface of the tube while the temperature at the internal tube surface is kept constant at the reference temperature *θ*_{0}. A plot of this case shows that the normal force needed to maintain the axial stretch increases if the temperature at the external surface is increased. By contrast, the cooling of the external surface leads to a decrease in the normal force. This effect strongly depends also on the thickness of the tube (cf. figure 6).

For increasing values of λ_{i} and λ_{z} the relative influence of the temperature gradient slightly decreases. In figure 7, the normal force is plotted with respect to *ζ* and the electric field strength *E*_{0}.

Figure 7 shows that the difference between the values for the isothermal case and the values for the cases with temperature gradient increases. Similar to the previously presented behaviour of the pressure, it can be shown that the ratio of the normal force in both cases decreases, which means a decreased influence of the temperature due to a smaller temperature gradient. Furthermore, figure 7*b* confirms that a temperature gradient has only a slight influence on the normal force in the case of compression but a stronger influence in the case of tube inflation.

Finally, we focus solely on the thermal behaviour of the material. Therefore, both the absolute pressure and the pressure ratio with respect to the pressure at the reference temperature are plotted in figure 8 against the temperature at the external tube surface. The temperature at the internal surface is kept constant and is equal to the reference temperature *θ*_{0}=293 *K*. We vary the value of λ_{i} to evaluate radial compression (λ_{i}<1), inflation (λ_{i}>1) and a case without mechanical deformation (λ_{i}=1)

In the case of compression, the absolute influence of the temperature change on the pressure is more distinctive than in the case of inflation but the relative change is smaller due to the increased tube thickness in the case of compression (figure 8). Next we vary the value of the axial stretch λ_{z} (figure 9). We evaluate the cases of axial compression (λ_{z}<1), elongation (λ_{z}>1) and again without mechanical deformation (λ_{z}=1).

Again the absolute influence of a temperature gradient on the pressure is the strongest for the largest thickness of the tube, i.e. for the smallest values of λ_{z}, but here the change in the pressure ratio is not as distinctive as in the previous example due to the smaller influence of λ_{z} on the wall thickness of the tube. Finally, the influence of the initial tube thickness is analysed (figure 10). The tube has to remain thick-walled (*ζ*>1) as for the thin-walled case the temperature function is not defined.

In this case, the ratio of the pressure shows clearly that a temperature change has the strongest influence on the pressure for the smallest tube thickness as this leads to a larger temperature gradient (figure 10).

## 5. Conclusion and outlook

Experimental studies of EAPs under isothermal conditions are rather difficult to perform. Moreover, constraining temperature variation during tests is far from reality since polymeric materials are prone to temperature generation during experimental studies. Therefore, the formulation of a thermo-electro-mechanically coupled modelling framework is necessary to capture realistic experimental results. In this contribution, we propose a thermo-electro-mechanically coupled constitutive framework that obeys relevant laws of thermodynamics. To make the model plausible, a widely used non-homogeneous boundary-value problem, i.e. a cylindrical tube subject to an electro-mechanical load in addition to a thermal load, is performed. As a first step towards a comprehensive thermo-electro-mechanical approach, the time dependence, e.g. inelasticity of the underlying polymeric materials, is discarded. Therefore, the proposed model needs to be extended for viscoelastic dissipative behaviour. Moreover, in order to find an analytical solution several simplifications are made. For example, a constant heat coefficient *κ* are assumed. The implementation of more general forms of these terms needs to be addressed in future contributions. In order to simulate real-life boundary-value problems and complex geometries, the framework needs to be implemented into a coupled finite-element framework. In a forthcoming contribution, a detailed finite-element implementation of the thermo-electro-mechanically coupled formulations will be elaborated. In the future, there are plans to conduct some experimental studies in order to identify constitutive material parameters and to validate the model with real experimental data.

## Authors' contributions

M.M. and M.H. jointly wrote the initial draft of the paper, while M.M. performed the numerical calculations in the results section. P.S. improved the initial draft significantly. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

The authors acknowledge funding within DFG project no. STE 544/52-1. M.H. and P.S. would like to express their sincere gratitude for the funding by the ERC within the Advanced Grant project MOCOPOLY.

## Appendix A

**(a) Derivation of pressure P**

*Ω*_{1}=(*θ*(*r*)/2*θ*_{0})[*g*_{0}+*g*_{1}*I*_{4}], *Ω*_{2}=0, _{i}=*a*_{i}/*A*_{i}, λ_{e}=*a*_{e}/*A*_{e} it follows that
*P*_{1} and *P*_{2} are evaluated separately. With *P*_{1}
*P*_{2} can be transformed,

**(b) Derivation of axial force N**

With the axial component of the mechanical tractions at the end of the tube, we obtain
*Ω*_{5}=(*θ*(*r*)/*θ*_{0})*c*_{2}=(*θ*(*r*)/*θ*_{0})(*ϵ*_{0}/2). The terms

- Received March 5, 2016.
- Accepted May 10, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.