Viscoelasticity of biomimetic scale beams from trapped complex fluids

TL;DR

This paper develops an energy-based analytical model for a biomimetic scale-covered beam with fluid trapped between scales, capturing how substrate elasticity, scale geometry, and shear-dependent complex fluids jointly produce nonlinear viscoelastic dissipation. Using a Carreau fluid description and a coupled elastic-kinematic framework, the authors derive a moment-curvature relation and define the relative energy dissipation (RED) as the ratio of dissipated work to total work, showing how RED depends on lubrication gap $\delta_L$, lubrication area $\alpha_L$, overlap ratio $\eta$, initial inclination $\theta_0$, and fluid rheology parameter $m$ under ramp and oscillatory loads. Key findings include a regime-differentiated dissipation with a robust scaling $\delta_L \alpha_L^{0.87} = \text{constant}$, stronger dissipation for shear-thickening fluids, and nonmonotonic effects of $\theta_0$ on RED, highlighting geometry- and rheology-driven tunability of damping. The work provides design insights for soft robotics and smart damping systems, offering a first-principles framework to engineer fluid-mediated viscoelastic responses in biomimetic scale architectures.

Abstract

We investigate the nonlinear viscoelastic behavior of a biomimetic scale-covered beam in which shear-dependent complex fluids are trapped between overlapping scales under bending loads. These fluids mimic biological mucus and slime layers commonly enveloping the skins found in nature. An energy-based analytical model is developed to quantify the interplay between substrate elasticity, scale geometry, and fluid rheology at multiple length scales. Constant strain rate and oscillatory bending are examined for Newtonian, shear-thinning, and shear-thickening fluids. The analysis reveals unique, geometry- and rate-dependent viscoelastic response, distinct from classical mechanisms such as material dissipation, frictional resistance, or air drag. Energy dissipation is shown to emerge from a nonlinear coupling of tribological parameters, fluid rheology, and system kinematics, exhibiting distinct regime-differentiated characteristics. The model captures the competitions and cooperations between elastic and geometrical parameters to influence the viscoelastic behavior and lead to geometry and rheology scaling laws for relative energy dissipation. The pronounced nonlinearity in the moment-curvature relationships, along with the geometry-controlled regimes of performance, highlights the potential for using tailored and engineered complex inks for soft robotics and smart damping systems.

Figures (7)

• Figure 1: (a) Natural examples of slime-covered organisms, including fish (image reproduced with permission from © Reefs.com Reefs2020), snail (© Sarah Swanson Swanson2020), salamander (© Jake Scott, iNaturalist Scott2024), and slug (© David Shetlar, The Ohio State University Extension OSUExtension2024), demonstrating the role of biological slime in protection, lubrication, and load-bearing functionality. (b) Schematic illustration of the multiscale-multiphysics nature of the problem - from the structure level to RVE (Representative Volume Element) to the lubrication length scale from left to right. (c) Detailed geometric relationships showing scale articulation, sliding kinematics, normal and dissipative contact forces under curvature-induced deformation (Figure 1(c) is adopted from ghosh2017non with slight modifications).
• Figure 2: Variation of the normalized relative sliding velocity $\dot{\bar{r}}/\dot{\psi}$ with $\bar{\psi}$ ($\psi/\pi$) for different values of $\eta$. Here, results are plotted for constant, $\dot\psi$ = 1 rad/s.
• Figure 3: (a) Variation of the normalized moment, $\bar{M}$, with $\bar{\psi}$ ($\psi/\pi$), for different values of $m$ and $\dot{\psi}$ ($\eta$ = 5). (b) Variation of the normalized moment, $\bar{M}$, with $\bar{\psi}$, for different values of $m$ and $\eta$ ($\dot{\psi} = 1 \, \text{rad/s}$). (c) Time-dependent variation of $\bar{M}$ for different frequency ratios, $\Omega/\Omega_n$. The inset highlights the variation of the bending curvature over time for different values of $\Omega/\Omega_n$ ($\eta$ = 5). (d) Time-dependent variation of the normalized moment, $\bar{M}$, for different $m$ values at $\Omega/\Omega_n = 1$ ($\eta$ = 5). The inset highlights the variation of the bending curvature over time for $\Omega/\Omega_n$ = 1. Here, results are plotted for $\delta_L = 2 \times 10^{-4}$ and $\alpha_L = 0.01$, and $T_n$ in Fig. 3(c)-(d) is the time period corresponding to the natural frequency of the unscaled beam.
• Figure 4: Log--log plots of the Relative Energy Dissipation (RED) factor with respect to tribological parameters: (a-b) RED versus the lubrication gap $\delta_L$ (with $\alpha_L = 0.01$), for two cases: (a) varying overlap ratio $\eta$, and (b) varying initial inclination angle $\theta_0$; (c-d) RED versus the lubrication area ratio $\alpha_L$ (with $\delta_L = 2 \times 10^{-4}$), for two cases: (c) varying $\eta$, and (d) varying $\theta_0$. The results correspond to $\bar{E}_B = 3.4 \times 10^{-4}$, $m = 1$, and a constant curvature rate $\dot{\psi} = 1 \, \text{rad/s}$.
• Figure 5: Log--log plots of the RED factor with the lubrication gap parameter $\delta_L$ varying: (a) $m$ ($\dot{\psi} = 1\, \mathrm{rad/s}$), and (b) $\dot{\psi}$ ($m = 1$). The results correspond to $\bar{E}_B = 3.4 \times 10^{-4}$, $\eta = 5$, $\theta_0 = 0^\circ$, and $\alpha_L = 0.01$.