[Paper Review] Finite wavelength surface-tension driven instabilities in soft solids

Xuan Chen, John S. Biggins|arXiv (Cornell University)|Dec 16, 2015

Geomagnetism and Paleomagnetism Studies3 citations

TL;DR

This study uses linear stability analysis to investigate surface-tension-driven instabilities in soft solids, revealing that finite-wavelength instabilities are generic in cylindrical geometries—contrary to the classical infinite-wavelength case. The key finding is that surface tension thresholds for instability depend on geometry and material contrast, with finite wavelengths emerging when elastic and capillary forces balance at specific $λ \propto R_0$ and $\gamma \propto \mu R_0$. The results show that infinite wavelength instabilities are the exception, not the rule, in elasto-capillary systems.

ABSTRACT

We deploy linear stability analysis to find the threshold wavelength ($\lambda$) and surface tension ($\gamma$) of Rayleigh-Plateau type peristaltic instabilities in incompressible neo-Hookean solids in a range of cylindrical geometries with radius $R_0$. First we consider a solid cylinder, and recover the well-known, infinite wavelength instability for $\gamma\ge6 \mu R_0$, where $\mu$ is the solid's shear modulus. Second, we consider a volume-conserving (e.g. fluid filled) cylindrical cavity through an infinite solid, and demonstrate infinite wavelength instability, but for $\gamma\ge 2 \mu R_0$. Third, we consider a solid cylinder embedded in a different infinite solid, and find a finite wavelength instability with $\lambda\propto R_0$, at surface tension $\gamma \propto \mu R_0$, where the constants depend on the two solids' modulus ratio. Finally, we consider an empty cylindrical cavity through an infinite solid, and find an instability with finite wavelength, $\lambda \approx2 R_0$, for $\gamma\ge 2.543... \mu R_0$. We argue that such finite wavelengths are generic for elasto-capillary instabilities, with the simple cylinder's infinite wavelength being the exception rather than the rule.

Motivation & Objective

• To understand the conditions under which surface-tension-driven instabilities occur in soft, incompressible neo-Hookean solids.
• To determine whether finite-wavelength instabilities arise in various cylindrical geometries beyond the classical infinite-wavelength case.
• To quantify the threshold surface tension and critical wavelength for instability in different configurations involving solid cylinders, cavities, and embedded solids.
• To clarify the role of material contrast and geometry in determining instability characteristics in soft elastic materials.

Proposed method

• Linear stability analysis is applied to model Rayleigh-Plateau type instabilities in incompressible neo-Hookean solids.
• The analysis considers four distinct geometries: a solid cylinder, a fluid-filled cylindrical cavity in an infinite solid, a solid cylinder embedded in a different infinite solid, and an empty cylindrical cavity in an infinite solid.
• The stability of small perturbations is evaluated by solving the eigenvalue problem derived from the linearized elasticity and surface tension equations.
• The critical surface tension $\gamma$ and critical wavelength $\lambda$ are determined by solving the resulting dispersion relation for each geometry.
• The analysis accounts for incompressibility and uses the shear modulus $\mu$ as a key material parameter.
• The dependence of $\lambda$ and $\gamma$ on the radius $R_0$ and modulus ratio between materials is systematically evaluated.

Experimental results

Research questions

• RQ1 What is the threshold surface tension $\gamma$ for instability in a solid cylinder, and does it support infinite-wavelength modes?
• RQ2 How does the presence of a fluid-filled cylindrical cavity in an infinite solid affect the instability threshold and wavelength?
• RQ3 What determines the emergence of finite-wavelength instabilities in a solid cylinder embedded in a different infinite solid?
• RQ4 Does an empty cylindrical cavity in an infinite solid exhibit finite-wavelength instabilities, and if so, what is the critical $\gamma$ and $\lambda$?
• RQ5 Is the infinite-wavelength instability a generic outcome, or is it restricted to specific geometries?

Key findings

• For a solid cylinder, the instability threshold is $\gamma \geq 6\mu R_0$, with infinite wavelength, consistent with classical results.
• For a fluid-filled cylindrical cavity in an infinite solid, the instability threshold is $\gamma \geq 2\mu R_0$, also with infinite wavelength.
• In the case of a solid cylinder embedded in a different infinite solid, a finite-wavelength instability emerges with $\lambda \propto R_0$ and $\gamma \propto \mu R_0$, where the proportionality constants depend on the modulus ratio.
• For an empty cylindrical cavity in an infinite solid, the instability has a finite wavelength $\lambda \approx 2R_0$ and threshold $\gamma \geq 2.543...\mu R_0$, indicating a non-trivial geometric and elastic dependence.
• The study concludes that finite-wavelength instabilities are generic in elasto-capillary systems, with the infinite-wavelength case being the exception rather than the rule.