TABLE I.

Nomenclature

i	Tube index, numbered from outer to inner.
s	Arc length measured from the proximal end of tubes.
L	Tube length.
r	Tube radius.
ε	Strain (scalar).
σl, σu	Loading and unloading stress (sealars).
E	Elastic modulus.
G	Shear modulus.
ν	Poisson’s ratio.
I	Cross section area moment of inertia
J	Cross section polar moment of inertia
F0(s)	Cross section reference frame (no twist).
Fi(s)	Cross-section material coordinate frame of ith tube.
	Angle of z-axis rotation from cross section reference
θi(s)	frame to cross section material frame. (θi(0) is kinematic input variable.)
li	Insertion distance of ith tube measured with respect to reference frame F0(0) (kinematic input variable.)
αi(s)	Relative z-axis rotation angle between cross section material frames, θi(s) − θ1(s). If no subscript, i = 2.
β i	Central angle produced by pre-curvature of ith tube.
Rz(α)	Rotation matrix for rotation of α about z-axis.
$u_i^{F_i\left(s\right)}\left(s\right),u_i\left(s\right)$	Three-component curvature vector of ith tube expressed in tube’s material coordinate frame. Also, angular strain rate per unit arc length.
v(s)	Three-component vector of linear strain rate per unit arc length.
${\widehat{u}}_i\left(s\right)$	Initial or pre-curvature vector of ith tube.
mi(s)	Three-component bending moment vector of ith tube.
ni(s)	Three component shear force vector of ith tube.
Ki	Frame-invariant stiffness tensor of ith tube.
kix, kiy, Kiz	Diagonal components of stiffness tensor.
c	Constant given by $(1+\nu )\parallel {\widehat{u}}_1\parallel \parallel {\widehat{u}}_2\parallel$ (scalar).
nd(u|m), cd(u\m)	Jacobi elliptic functions of scalar variables u and m.
K(m)	Complete elliptic integral of the first kind.