A tutorial on the electrostatics of deformable materials with a focus on stability and bifurcation analysis

The attached tutorial paper is yet unpublished but I am posting a pre-print since several students I know have found it to be a useful pedagogical resource. You may also access the document on arXiv.

Here is the abstract.

Soft materials, such as liquids, polymers, foams, gels, colloids, granular materials, and most soft biological materials, play an important role in our daily lives. From a mechanical viewpoint, soft materials can easily achieve large deformations due to their low elastic moduli; meanwhile, surface instabilities, including wrinkles, creases, folds, and ridges, among others, are often observed. In particular, soft dielectrics when subjected to electrical stimuli can achieve signicantly large deformations that are often accompanied by instabilities. While instabilities are conventionally thought to cause failures in the engineering context and carry a negative connotation, they can also
be harnessed for various applications such as surface patterning, giant actuation strain, and energy harvesting. In the biological world, instability and bifurcation phenomena often precede important events such as endocytosis, cell fusion, among others. Stability and bifurcation analysis (especially for soft materials) is challenging and often presents a formidable barrier to entry in this important field. A multidisciplinary audience may lack the background in one or more areas that are needed to carry out the requisite modeling or even understand papers in the literature. Furthermore, combining electrostatics together with large deformations brings its own challenges. In this article, we provide a tutorial on the basics of stability and bifurcation analysis in the context of soft electromechanical materials. The aim of the article is to use simple examples and to gently lead a reader, unfamiliar with either stability analysis or electrostatics of deformable media, to develop the ability to understand the pertinent literature that already exists and position them to embark on the state-of-the-art research on this topic.