= Solution
{c}
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 $kB_2$ 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 density
$$
q(u)=\frac12(B_1+B_2+ka^2B_2)u^2-B_2\hat u\,u+\frac12B_2\hat u^2.
$$
There are no derivatives of $u$ or imposed end constraints in this <minimum potential energy principle in elasticity>. Since
$$
q'(u)=(B_1+B_2+ka^2B_2)u-B_2\hat u,\qquad
q''(u)=B_1+B_2+ka^2B_2>0,
$$
the unique pointwise minimizer is also the constant global minimizer:
$$
\boxed{u_* =\frac{B_2\hat u}{B_1+B_2+ka^2B_2}
=\frac{\hat u}{B_1/B_2+1+ka^2}.}
$$
For example, <completing the square> gives
$$
\mathcal E[u]-\mathcal E[u_*]
=\frac{B_1+B_2+ka^2B_2}{2}\int_0^L(u-u_*)^2\,dS\ge0,
$$
which also proves uniqueness, up to equality <almost everywhere>, without merely assuming a uniform <curvature>.
With $k$, $a$ and $\hat u$ fixed, the two limits are
$$
\boxed{\begin{aligned}
B_1/B_2\to\infty &: \quad u_*\to0,\\
B_1/B_2\to0 &: \quad u_*\to\frac{\hat u}{1+ka^2}.
\end{aligned}}
$$
In 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 $u_*<\hat u$ when $\hat u>0$ and $ka^2>0$. Only when that stretching penalty also vanishes does $u_*\to\hat u$.
These formulae describe an admissible bonded <elastic rod> provided $1-au_*>0$. If parameters instead force $au_*\ge1$, the unconstrained quadratic minimizer is outside the kinematic regime of the model; the given energy does not itself enforce positive <stretch ratio>.
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