Elastic Localizations

The past few decades have witnessed a surge of interest in pattern formations in soft materials under various fields. This has partly been driven by the recognition that buckling-induced patterns at micrometer and submicrometer scales may serve many useful purposes. Such patterns are usually either periodic or localized. Formation of periodic patterns is universally recognized as a bifurcation problem, and theories concerning periodic patterns have been well-developed and can be found in many textbooks and research monographs. In contrast, the initiation and evolution of localized patterns are rarely studied as a bifurcation problem, and when they are the discussion is often incomplete. In this chapter, we discuss three representative elastic localization problems: localized bulging of an inflated hyperelastic tube, localized necking of a solid cylinder induced by surface tension, and axisymmetric necking of a circular plate under all-round tension. All these problems are characterized by the fact that a linear bifurcation analysis would predict that the critical wavenumber is zero if the dimension in the direction of periodic variation is infinite. It is shown how the entire localization process, including initiation, growth and propagation, can be described analytically or semi-analytically, and how the process depends on how loading is carried out. It is also shown how a one-dimensional reduced model can be derived for the inflation problem and used to describe the entire localization process fairly accurately. It is hoped that the methodology explained here can be applied to study similar problems that also involve other effects such as electric and magnetic fields, chemical reactions, material deterioration, and residual stresses.