Surface Constrained Elastic Rods with Application to the Sphere

Abstract

The unilateral contact between an elastic rod and a rigid surface is encountered in numerous biological and engineering applications. Along continuous contacts, the centerline of the rod reflects the geometry of the constraining surface. This restriction of the rod axis to surface bound configurations enters its local equilibrium through a reaction pressure which ensures the compatibility between the deformation and the restraint geometry. The classic equations governing the static equilibrium of elastic rods are particularized to surface bound configurations by (i) specifying the location of the rod axis in terms of its coordinates in the parameter space associated with the constraining surface parameterization, and (ii) characterizing the orientation of its material frame through its rotation with respect to the surface normal. This formulation, which emphasizes the relations between the rod configuration and the geometry of the constraint, also leads to an expression for the reaction pressure. This approach is then validated on a spherical surface by comparing with known solutions for elastic curves, i.e., inextensible and twist free rods.

Appendix: Closed Elastica on the Sphere

1.1 A.1 Components of $\boldsymbol{\varsigma} (\xi )$ in the Cylindrical Coordinate System

In the cylindrical coordinate system, the scaled position vector and the unit tangent to the elastica axis read $\boldsymbol{\varsigma} (\xi )=\varrho \boldsymbol{e}_{\varrho}+\zeta \boldsymbol {e}_{\zeta}$ and $\boldsymbol{\varsigma}' (\xi )=\varrho' \boldsymbol {e}_{\varrho}+\varrho \vartheta' \boldsymbol{e}_{\vartheta}+\zeta' \boldsymbol{e}_{\zeta}$, respectively. The constant vector $\boldsymbol{\mathcal{I}}=\mathcal {I} \boldsymbol{e}_{\zeta}$, aligned with the $\zeta$-axis, may alternatively be decomposed in the Darboux frame $\{ \boldsymbol{d}_{3},\boldsymbol{N}\times \boldsymbol{d}_{3},\boldsymbol{N} \}$ as

$$\begin{aligned} \boldsymbol{\mathcal{I}} & = (\mathcal{M}_{3}+\mathcal{F}_{g} )\boldsymbol{d}_{3}- (\mathcal{F}_{3}+1 )\boldsymbol{N}\times \boldsymbol{d}_{3}+\varkappa_{g}\boldsymbol{N}, \end{aligned}$$

(65)

since $\boldsymbol{N}\times\boldsymbol{\mathcal{F}}=\mathcal {F}_{3}\boldsymbol{N}\times\boldsymbol{d}_{3}-\mathcal{F}_{g}\boldsymbol {d}_{3}$. Hence, projecting this vector on the scaled position vector $\boldsymbol {\varsigma} (\xi )$ leads to

$$\begin{aligned} \boldsymbol{\mathcal{I}}\cdot\boldsymbol{\varsigma} & =-\varkappa _{g}, \end{aligned}$$

(66)

$$\begin{aligned} & =\mathcal{I} \zeta, \end{aligned}$$

(67)

which reduces to the relation (56.a). Alternatively, the vector product between $\boldsymbol{\mathcal{I}}$ and $\boldsymbol{\varsigma}'$ reads $\boldsymbol{\mathcal{I}}\times \boldsymbol{\varsigma}'=-\mathcal{I} (\varrho \vartheta' \boldsymbol{e}_{\varrho}-\varrho' \boldsymbol{e}_{\vartheta} )$, in the cylindrical coordinate system, or $\boldsymbol{\mathcal{I}}\times \boldsymbol{\varsigma}'=\varkappa_{g}\boldsymbol{N}\times\boldsymbol {d}_{3}+ (\mathcal{F}_{3}+1 )\boldsymbol{N}$, in the Darboux frame. Therefore, taking the dot product of both expressions with the scaled position vector yields

$$\begin{aligned} \bigl(\boldsymbol{\mathcal{I}}\times\boldsymbol{\varsigma}' \bigr) \cdot \boldsymbol{\varsigma} & =-\mathcal{I} \varrho^{2} \vartheta', \end{aligned}$$

(68)

$$\begin{aligned} & =- (\mathcal{F}_{3}+1 ), \end{aligned}$$

(69)

which, taking into account expression (49.c), reduces to the relation (56.b).

1.2 A.2 Magnitude of the Constant Vector $\boldsymbol{\mathcal {I}}$

Equation (50) may be integrated to obtain

$$\begin{aligned} \bigl(\varkappa_{g}' \bigr)^{2} & =C_{1}-\frac{1}{4} \bigl(\varkappa _{g}^{4}-2 \sigma \varkappa_{g}^{2} \bigr), \end{aligned}$$

(70)

where $C_{1}$ is a constant of integration. As a global maximum point, $\xi_{m}$, of the squared geodesic curvature $\varkappa_{g}^{2}$ is a zero of $\varkappa_{g}'$, this constant reads

$$\begin{aligned} C_{1} & =\frac{1}{4} \bigl(\varkappa_{m}^{4}-2 \sigma \varkappa _{m}^{2} \bigr), \end{aligned}$$

(71)

such that expression (70) reduces to

$$\begin{aligned} \bigl(\varkappa_{g}' \bigr)^{2} & = \frac{1}{4} \bigl(\varkappa _{m}^{2}- \varkappa_{g}^{2} \bigr) \bigl(\varkappa_{g}^{2}+ \varkappa _{m}^{2}-2 \sigma \bigr). \end{aligned}$$

(72)

As the term $(\varkappa_{m}^{2}-\varkappa_{g}^{2} )$ is always non-negative, the term $(\varkappa_{g}^{2}+\varkappa _{m}^{2}-2 \sigma )$ has also to be non-negative for any $\varkappa_{g}^{2}$, that is $\varkappa_{m}^{2}\geq\sigma$. Computing the magnitude of the constant vector $\boldsymbol{\mathcal{I}}=\boldsymbol{\mathcal{M}}-\boldsymbol {N}\times\boldsymbol{\mathcal{F}}$ is then straightforward

$$\begin{aligned} \boldsymbol{\mathcal{I}}\cdot\boldsymbol{\mathcal{I}} & = \bigl(\varkappa _{g}' \bigr)^{2}+\frac{1}{4} \bigl[ \varkappa_{g}^{2} \bigl(\varkappa _{g}^{2}-2 \sigma \bigr)+ (2+\sigma )^{2} \bigr], \end{aligned}$$

(73)

which, accounting for expression (72), reduces to (57).