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Research Papers

Deformation of Microchannels Embedded in an Elastic Medium

[+] Author and Article Information
Vivek Ramachandran

Integrated Soft Materials Laboratory,
Carnegie Mellon University,
Pittsburgh, PA 15213
e-mail: vivek.ramachandran@epfl.ch

Carmel Majidi

Integrated Soft Materials Laboratory,
Carnegie Mellon University,
Pittsburgh, PA 15213
e-mail: cmajidi@andrew.cmu.edu

1Present address: Laboratory of Intelligent Systems, École Polytechnique Fédérale de Lausanne, Lausanne 1015, Vaud, Switzerland.

2Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 29, 2018; final manuscript received May 30, 2018; published online June 27, 2018. Assoc. Editor: Yihui Zhang.

J. Appl. Mech 85(10), 101004 (Jun 27, 2018) (7 pages) Paper No: JAM-18-1244; doi: 10.1115/1.4040477 History: Received April 29, 2018; Revised May 30, 2018

The deformation of microfluidic channels in a soft elastic medium has a central role in the operation of lab-on-a-chip devices, fluidic soft robots, liquid metal (LM) electronics, and other emerging soft-matter technologies. Understanding the influence of mechanical load on changes in channel cross section is essential for designing systems that either avoid channel collapse or exploit such collapse to control fluid flow and connectivity. In this paper, we examine the deformation of microchannel cross sections under far-field compressive stress and derive a “gauge factor” that relates externally applied pressure with change in cross-sectional area. We treat the surrounding elastomer as a Hookean solid and use two-dimensional plane strain elasticity, which has previously been shown to predict microchannel deformations that are in good agreement with experimental measurements. Numerical solutions to the governing Lamé (Navier) equations are found to match both the analytic solutions obtained from a complex stress function and closed-form algebraic approximations based on linear superposition. The application of this theory to soft microfluidics is demonstrated for several representative channel geometries.

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Figures

Grahic Jump Location
Fig. 1

(a) Examples of soft microfluidics for emerging applications: (i) Quake valve (reproduced with permission from Unger et al. [16] Copyright 2000 by AAAS), (ii) fluidic soft robot (reproduced with permission from Wehner et al. [14] Copyright 2016 by Springer Nature), (iii) LM electronics (reproduced with permission from Ozutemiz et al. [23] Copyright 2018 by Wiley—VCH Verlag GmbH & Co. KGaA), and (iv) tunable nanoparticle filtration (reproduced with permission from Huh et al. [21] Copyright 2007 by Springer Nature). (b) Soft microfluidic channel modeled as a prismatic opening embedded inside of an elastic sheet subject to surface pressure p. (c) Selected channel cross section, which is representative of geometries typically used in soft microfluidics.

Grahic Jump Location
Fig. 3

Gauge factor G versus channel aspect ratio α. (a) Circular cross section with a = 0.1 and varying b: (circular markers) numerical solution, (solid) analytic solution, (dashed) approximate solution. (b) Rectangular cross section with w = 0.1 and varying h: (square markers) numerical solution, (solid) analytic solution, and (dashed) approximate solution. (c) Numerical solutions for triangular (triangle markers) and diamond-shape (diamond markers) also exhibit a similar monotonic dependency; the dashed line corresponds to a fit of G=1.2/α.

Grahic Jump Location
Fig. 4

(a) The solution to the classical Michell problem of a circular opening in an elastic plate under far-field stress is used to estimate the deformation of an elliptical channel with principle dimensions a and b. (b) The deformation of a rectangular channel is estimated by superposing uniform deformation of a homogenous Hookean solid with the opening of a slit under far-field stress.

Grahic Jump Location
Fig. 2

(a) Deformation of an elliptical channel with w = 0.2, h = 0.1, p = 0.05, and ν = 0.49: (gray) initial shape, (markers) numerical solution for deformed shape, (dashed) approximate solution for deformed shape. (b) Cross-sectional area as a function of normalized pressure: (circular markers) numerical solution, (dashed) approximate solution, and (solid) analytic solution. (c) Deformation of a rectangular channel with w = 0.2, h = 0.1, p = 0.05, and ν = 0.49: (gray) initial shape, (markers) numerical solution for deformed shape, (dashed) approximate solution for deformed shape. (d) Cross-sectional area as a function of normalized pressure: (circular markers) numerical solution, (dashed) approximate solution, and (solid) analytic solution.

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