Large amplitude behavior of the Grinfeld instability: a variational approach

TL;DR

This paper introduces a variational method using multi-cycloid curves to analyze the large amplitude behavior of the Grinfeld instability, providing improved accuracy and physical insight over previous amplitude expansion approaches.

Contribution

The authors develop a variational approach with multi-cycloid curves that surpasses previous methods in accuracy and applicability, especially for large amplitudes and gravity effects.

Findings

Analytical calculation with a single cycloid provides qualitative insights.

Including multiple cycloid modes yields quantitative agreement with numerical results.

The method extends to large stresses, amplitudes, and gravity-influenced scenarios.

Abstract

In previous work, we have performed amplitude expansions of the continuum equations for the Grinfeld instability and carried them to high orders. Nevertheless, the approach turned out to be restricted to relatively small amplitudes. In this article, we use a variational approach in terms of multi-cycloid curves instead. Besides its higher precision at given order, the method has the advantages of giving a transparent physical meaning to the appearance of cusp singularities and of not being restricted to interfaces representable as single-valued functions. Using a single cycloid as ansatz function, the entire calculation can be performed analytically, which gives a good qualitative overview of the system. Taking into account several but few cycloid modes, we obtain remarkably good quantitative agreement with previous numerical calculations. With a few more modes taken into consideration, we improve on the accuracy of those calculations. Our approach extends them to situations involving gravity effects. Results on the shape of steady-state solutions are presented at both large stresses and amplitudes. In addition, their stability is investigated.