Effect of Intrinsic Twist and Orthotropy on Extension–Twist–Inflation Coupling in Compressible Circular Tubes

Abstract

We present effects of intrinsic twist and material orthotropy on extension–twist–inflation coupling in circular tubes about their stress-free state. Simple analytical expressions for coupling stiffnesses corresponding to extension–twist, twist–inflation and extension–inflation couplings are obtained. We show that the sign of the extension–twist coupling stiffness, which governs initial overwinding/unwinding in tubes during their extension, is not just dependent on the tube’s intrinsic twist but also on two other parameters: ratio of the Young’s moduli in the lateral surface of orthotropic tube and the excess of the Poisson’s ratio from an isotropy condition. By tuning these two parameters, one can generate the counter-intuitive overwinding as reported earlier in the case of DNA. Similarly, we show that even with positive Poisson’s ratio, an intrinsically twisted tube could inflate on being stretched. We also present a scheme to obtain all the relevant stiffnesses of chiral single-walled carbon nanotubes from a “one-atom unit cell” calculation. These stiffnesses, when plotted vs. the nanotube’s chirality, exhibit interesting periodicity which has its origin in the 6-fold symmetry of graphene. This trend is captured in the continuum model for a nanotube when it is assumed to be comprised of three families of parallel fibers on its lateral surface.

Appendices

Appendix A: Saint-Venant’s Orthotropic Constitutive Relations

We show below the expression for Saint-Venant’s orthotropic constitutive relations as presented in Itskov and Aksel [24]:

$$ W \approx W\textsuperscript{(2)}(\mathbf{E}) = \frac{1}{2} \sum_{i,j=1} ^{3} a_{ij} \, \mathit{tr}( \mathbf{E}\mathbf{L}_{i})\,\mathit{tr}(\mathbf{E}\mathbf{L}_{j}) + \sum _{i,j \ne i}^{3} G_{ij} \,\mathit{tr}(\mathbf{E} \mathbf{L}_{i}\mathbf{E} \mathbf{L}_{j}). $$

(44)

For small deformations, $\mathit{tr}(\mathbf{E}\mathbf{L}_{i})$ denotes infinitesimal normal strain ($\epsilon_{ii}^{\ast }$) along the $i$th principal direction while $\mathit{tr}(\mathbf{E}\mathbf{L}_{i}\mathbf{E} \mathbf{L}_{j})$ denotes the square of shear strain ($\epsilon_{ij} ^{\ast }$) in the plane formed by $(i,j)$ principal material directions. Accordingly, the Cauchy stress vs. the infinitesimal strain relation in the coordinate system of principal material directions appear as

$$ \begin{bmatrix} \sigma_{11}^{\ast } \\ \sigma_{22}^{\ast } \\ \sigma_{33}^{\ast } \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31}&a_{32}&a_{33} \end{bmatrix} \begin{bmatrix} \epsilon_{11}^{\ast } \\ \epsilon_{22}^{\ast } \\ \epsilon_{33}^{\ast } \end{bmatrix} ,\quad \sigma_{ij}^{\ast }=2G_{ij} \epsilon_{ij}^{\ast }\ (i\ne j). $$

(45)

The meaning of other terms in (44) and (45) are as follows:

$$ \begin{aligned} \mathbf{L}_{j} &=\mathbf{i}_{j} \otimes \mathbf{i}_{j}\quad (j\mathrm{th}\ \text{principal material tensor},\ j=1,2,3 ), \\ a_{ii} &=E_{i}\frac{1-\nu_{jk}\nu_{kj}}{\Delta }, \qquad a_{ij}=a_{ji}=E _{i}\frac{\nu_{ij}+\nu_{kj}\nu_{ik}}{\Delta }, \quad i \ne j \ne k \ne i\ (i,j=1,2,3), \\ \Delta &=1-\nu_{12}\nu_{21}-\nu_{23} \nu_{32}-\nu_{31}\nu_{13}-2\nu _{21} \nu_{32}\nu_{13}, \\ E_{i} &= \text{Young's moduli} \quad (i = 1,2,3), \\ G_{ij} &=G_{ji}= \text{Shear moduli}\quad (i \ne j = 1,2,3), \\ \nu_{ij} &=\nu_{ji}\frac{E_{i}}{E_{j}} = \text{Poisson's ratios} \quad (i \ne j = 1,2,3). \end{aligned} $$

(46)

The orthotropic material constants must be such that they lead to positive definiteness of the quadratic form $W$ in (44) with respect to infinitesimal strains. Accordingly, the orthotropic constants must satisfy the following inequality constraints:

$$ \begin{aligned} &E_{1},E_{2},E_{3},G_{12},G_{23},G_{13}>0,\qquad 1-\nu_{12}\nu_{21}>0, \\ &1- \nu_{23} \nu_{32}>0,\qquad 1-\nu_{13}\nu_{31}>0,\qquad \Delta >0. \end{aligned} $$

(47)

In case of thin tubes, only four constants ($E_{2},E_{3},G_{23},\nu _{32}$) are relevant. Accordingly, only the following constraints need to be satisfied:

$$ E_{2},E_{3},G_{23}>0, \qquad 1- \frac{E_{2}}{E_{3}}\nu_{32}^{2}>0. $$

(48)

Appendix B: System of ODEs and Boundary Conditions for Linearized In-Plane Warping Functions

We now present the system of ODEs and boundary conditions for linearized in-plane warping functions. We first present them in Table 4 for the case when no inflation strain is imposed. These linear ODEs are obtained by substituting the Taylor-expanded warping function (8a) in (12a) and then collecting the coefficients of $\epsilon$ and $\kappa$ separately.

Table 4 System of linear ODEs and boundary conditions for ($\hat{f},\hat{g}$)

The system of ODEs and boundary conditions for the case in which inflation strain is also imposed are obtained similarly. Here, the kinematic boundary condition at inner radius implies:

$$ \begin{aligned} \tilde{b}(r_{1}) &=b_{1} \\ \Rightarrow \epsilon \tilde{f}(r_{1})+\kappa \tilde{g}(r_{1})+b_{1} \tilde{h}(r_{1}) &= b_{1}\quad \bigl(\text{using (8c)}\bigr) \\ \Rightarrow \tilde{f}(r_{1})=\tilde{g}(r_{1})=0,\quad \tilde{h}(r_{1}) &=1. \end{aligned} $$

(49)

The ODEs and the above boundary conditions are presented in Table 5 for readers’ convenience.

Table 5 System of linear ODEs and boundary conditions for ($\tilde{f},\tilde{g},\tilde{h}$)

Appendix C: Thin Tube Limits

Upon Taylor-expanding the radial stress $\sigma_{rr}$ about inner radius, we obtain:

$$ \sigma_{rr}(r)=\sigma_{rr}(r_{1})+ \frac{\mathrm{d}\sigma_{rr}}{\mathrm{d}r}(r_{1})\cdot (r-r_{1})+ \frac{1}{2}\frac{\mathrm{d}^{2}\sigma_{rr}}{\mathrm{d}r^{2}}(r_{1})\cdot (r-r_{1})^{2}+ \cdots $$

(50)

As both ($\sigma_{rr}(r_{1}),\sigma_{rr}(r_{2})$) vanish due to traction free boundary conditions for special Cosserat rod model, evaluating (50) at $r=r_{2}$ leads to

$$ \begin{aligned} 0 &=\frac{\mathrm{d}\sigma_{rr}}{\mathrm{d}r}(r_{1})\cdot T+ \frac{1}{2}\frac{ \mathrm{d}^{2}\sigma_{rr}}{\mathrm{d}r^{2}}(r_{1})\cdot T^{2}+ \cdots \\ \Rightarrow 0 &=\frac{\mathrm{d}\sigma_{rr}}{\mathrm{d}r}(r_{1})+ \frac{1}{2} \frac{\mathrm{d}^{2}\sigma_{rr}}{\mathrm{d}r^{2}}(r_{1})\cdot T+\cdots \quad \bigl(\text{on dividing by thickness}(T)\bigr). \end{aligned} $$

(51)

In the thin tube limit ($T\to 0$), we then obtain $\frac{\mathrm{d} \sigma_{rr}}{\mathrm{d}r}(r_{1})=0$. This results in (17), i.e.,

$$\begin{aligned} \sigma_{rr}(r_{1})=0,\qquad \frac{\mathrm{d}\sigma_{rr}}{\mathrm{d} r}(r_{1})=0. \end{aligned}$$

(52)

In case inflation strain is also prescribed, $\sigma_{rr}(r_{1}) \ne 0$ but $\sigma_{rr}(r_{2})=0$. Substituting them in (50) for $r=r_{2}$, dividing by thickness and further taking thin-tube limit, we obtain (22), i.e.,

$$ \frac{d\sigma_{rr}}{dr}(r_{1})=-\sigma_{rr}(r_{1})/T. $$

(53)

Appendix D: Stiffness Expressions for Isotropic Thick Tubes

The stiffness expressions simplify a lot here due to isotropy. In case no inflation is imposed, we have:

$$ \hat{\mathcal{B}} = \mathit{GJ}, \qquad \hat{\mathcal{D}}=\mathit{EA}, \qquad \hat{\mathcal{E}}=0. $$

(54)

Here, $A$ and $J$ are the cross-sectional area and the polar moment of area respectively. The stiffnesses for the case when inflation is also imposed are presented in Table 6. An interesting observation to note here is that the extensional and inflation stiffnesses are not the same although they match in thin tube limit. We can also note that the ratio of $\tilde{\mathcal{G}}$ and $\tilde{\mathcal{I}}$ is $\nu$ implying that an isotropic tube always exhibits Poisson effect as dictated by its Poisson’s ratio.

Table 6 Stiffnesses for isotropic thick tubes