Deformation of an Elastic Beam due to Viscous Flow in an Embedded Parallel Channel Network

TL;DR

This paper models how viscous fluid flow within an embedded channel network causes elastic deformation of a slender beam, providing a mathematical framework for designing complex deformation patterns in soft-robotic applications.

Contribution

It introduces a fourth-order integro-differential equation linking viscous flow and elastic deflection, enabling the design of dynamic deformation patterns in embedded channel networks.

Findings

Derived a governing equation for beam deflection due to viscous flow.

Identified conditions for inertia-like wave solutions in beam deformation.

Provided a basis for designing complex soft-robotic deformation patterns.

Abstract

Elastic deformation due to embedded fluidic networks is currently studied in the context of soft-actuators and soft-robotic applications. In this work, we analyze interaction between the elastic deflection of a slender beam and viscous flow within a long serpentine channel, embedded in the elastic beam. The channel is positioned asymmetrically with regard to the midplane of the beam, and thus pressure within the channel creates a local moment deforming the beam. We focus on creeping flows and small deflections of the elastic beam and obtain, in leading order, a fourth-order partial integro-differential equation governing the time-dependent deflection field. This relation enables the design of complex time-dependent deformation patterns of beams with embedded channel networks, including inertia-like standing and moving wave solutions in configurations with negligible inertia.