Summary: We address some properties of a scalar two-dimensional model that has been proposed to describe microstructue in martinsitic phase transformations, consisting of minimizing the bulk energy \[ I[u]= \int^{l_x}_0 \int_0^{l_y} u^2_x +\sigma|u_{yy}| \] where \(|u_y|=1\) a.e. and \(u(0,\cdot) =0\). R. V. Kohn and S. Müller [Commun. Pure Appl. Math. 47, 405-435 (1994; Zbl 0803.49007)] proved the existence of a minimizer for \(\sigma >0\) and obtained bounds on the total energy that suggested self-similarity of the minimizer. Building upon their work, we derive a local upper bound on the energy and on the minimizer itslef and show that the minimizer \(u\) is asymptotically self-similar in the sense that the sequence \[ u^j(x,y)= \theta^{-2j/3} u(\theta^jx, \theta^{2j/3}y) \] \((0<\theta <1)\) has a strongly converging subsequence in \(W^{1,2}\).