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The part of an elastic rod's elastic energy that penalizes departure of its bending strain of a rod $u$ from the intrinsic value $\hat{u}$. Here $B$ is the bending stiffness and $S$ is reference arc length. For an inextensible rod, the same formula can be written using spatial curvature and current arc length.

An elastic rod is a slender elastic body represented by a centreline and an attached orthonormal director frame. Its elastic energy penalizes stretching, bending and twisting relative to a preferred configuration. The director kinematics of a rod describe changes of the frame along the centreline. An unshearable elastic rod has its third director parallel to the centreline tangent vector.

An inextensible rod preserves material arc length, so its stretch ratio is one. In an unshearable elastic rod, $d\mathbf{r}/dS={\mathbf{d}}_3$ for reference arc length $S$.

A right-handed orthonormal frame attached to the cross-sections of an elastic rod. For an unshearable elastic rod, ${\mathbf{d}}_3$ is the unit tangent vector to its centreline, while ${\mathbf{d}}_1$ and ${\mathbf{d}}_2$ follow the orientation of its material cross-section. Unlike a frame constructed only from the centreline curvature, the material frame records the twist of the material about the tangent.

The material frame of a rod obeys the director kinematics of a rodIn the stress-free reference configuration, the constitutive equation gives $u_1=0$, $u_2=\hat{u}$, and $u_3=0$, since all three stiffnesses are positive. Thus $\mathbf{u}=\hat{u}{\mathbf{d}}_2$. Its cross product with ${\mathbf{d}}_2$ vanishes, soConsequently this director is a fixed vector in space, even though the other two directors rotate along the elastic rod.

The first term is the bending energy of rod 1, whose preferred bending strain of a rod is zero. The second is the bending energy of rod 2 relative to its preferred intrinsic curvature of an elastic filament $\hat{u}$. The third is its stretching energy, with stretching stiffness $k{B}_2$ and stress-free stretch ratio ${\alpha }^{+}=1$. Each term is an energy per unit reference arc length before integration.

Substitute $u^{+}=u$ and ${\alpha }^{+}-1=-au$. The resulting quadratic function of $u$ has densityThere are no derivatives of $u$ or imposed end constraints in this minimum potential energy principle in elasticity. Sincethe unique pointwise minimizer is also the constant global minimizer:For example, completing the square giveswhich also proves uniqueness, up to equality almost everywhere, without merely assuming a uniform curvature.

With $k$, $a$ and $\hat{u}$ fixed, the two limits areIn the first limit, rod 1 dominates the bending energy and suppresses the bend. In the second, rod 1 offers negligible bending resistance, but following rod 2's preferred bending strain of a rod still shortens rod 2 through the bonded geometry. The competition between its bending energy and stretching energy leaves $u_{\ast }<\hat{u}$ when $\hat{u}>0$ and $k{a}^2>0$. Only when that stretching penalty also vanishes does $u_{\ast }\to \hat{u}$.

These formulae describe an admissible bonded elastic rod provided $1-a{u}_{\ast }>0$. If parameters instead force $a{u}_{\ast }\ge 1$, the unconstrained quadratic minimizer is outside the kinematic regime of the model; the given energy does not itself enforce positive stretch ratio.

Because the two elastic rods each have thickness $a$, their centrelines are separated by $a$ along the common director ${\mathbf{d}}_1$. The bonded geometry isDifferentiate with respect to reference arc length. The first rod is an inextensible rod, and the printed director kinematics of a rod giveFor the physically admissible orientation-preserving configuration $1-au>0$, the rods have the same unit tangent ${\mathbf{d}}_3^{+}={\mathbf{d}}_3$ and the same planar material frame of a rod. ThereforeIn particular, the second rod is shortened on the inside of a bend with $u>0$.

The bending strain of a rod is defined here by differentiation with respect to reference arc length. Since the reference labels match,The distinction from spatial curvature matters: the second centreline hasThus $u/(1-au)$ is its signed spatial curvature, rather than the $u^{+}$ appearing in the given elastic energy.

Before contact with both walls, the elastic rod can grow freely, so its elastic stretch ratio is $\alpha =1$ and its length is its stress-free grown length,The first contact occurs when $\ell (t_c)=L_1$. For the intended growth case $c>0$,The condition $L_1>L_0$ ensures $t_c>0$. If $c=0$, contact never occurs; negative $c$ describes shrinkage rather than the stated continuing growth.

The ratio of current to reference lengths of an infinitesimal material line element. Along an elastic rod, it is $\alpha =ds/dS=|d\mathbf{r}/dS|>0$. A principal stretch is a stretch ratio along a principal direction of the stretch tensor.