One-dimensional structured phase transformations under prescribed loads (Q1067820)

A bounded one-dimensional elastic body identified with the interval [0,L] has stored energy \(\hat W.\) The form of \(\hat W\) is as follows: \(\hat W=W(\gamma)+\epsilon (\gamma^ 1)^ 2. W(\gamma)\) is a \(C^ 2\)- function of strain \(\gamma\), and \(\epsilon\) is a ''small'' positive constant. The solution of the minimum problem \(\min_{\gamma (x)}\int^{L}_{0}[W(\gamma)-f\gamma +\epsilon^ 2(\gamma ')^ 2]dx\) is given over the class of absolutely continuous functions \(\gamma\) on [0,L], when the load f is constant. It is proved that all non-constant solutions \(\gamma\) of the minimum problem above mentioned are unstable.