  This paper is devoted to classical variational problems for planar elastic
curves of clamped endpoints, so-called Euler's elastica problem. We investigate
a straightening limit that means enlarging the distance of the endpoints, and
obtain several new results concerning properties of least energy solutions. In
particular we reach a first uniqueness result that assumes no symmetry. As a
key ingredient we develop a foundational singular perturbation theory for the
modified total squared curvature energy. It turns out that our energy has
almost the same variational structure as a phase transition energy of
Modica-Mortola type at the level of a first order singular limit.
