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Research Papers

A Variational Approach and Finite Element Implementation for Swelling of Polymeric Hydrogels Under Geometric Constraints

[+] Author and Article Information
Min Kyoo Kang

Department of Aerospace Engineering and Engineering Mechanics, University of Texas, Austin, TX 78712

Rui Huang

Department of Aerospace Engineering and Engineering Mechanics, University of Texas, Austin, TX 78712ruihuang@mail.utexas.edu

J. Appl. Mech 77(6), 061004 (Aug 17, 2010) (12 pages) doi:10.1115/1.4001715 History: Received August 06, 2009; Revised February 07, 2010; Posted May 05, 2010; Published August 17, 2010; Online August 17, 2010

A hydrogel consists of a cross-linked polymer network and solvent molecules. Depending on its chemical and mechanical environment, the polymer network may undergo enormous volume change. The present work develops a general formulation based on a variational approach, which leads to a set of governing equations coupling mechanical and chemical equilibrium conditions along with proper boundary conditions. A specific material model is employed in a finite element implementation, for which the nonlinear constitutive behavior is derived from a free energy function, with explicit formula for the true stress and tangent modulus at the current state of deformation and chemical potential. Such implementation enables numerical simulations of hydrogels swelling under various constraints. Several examples are presented, with both homogeneous and inhomogeneous swelling deformation. In particular, the effect of geometric constraint is emphasized for the inhomogeneous swelling of surface-attached hydrogel lines of rectangular cross sections, which depends on the width-to-height aspect ratio of the line. The present numerical simulations show that, beyond a critical aspect ratio, creaselike surface instability occurs upon swelling.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic illustration of the reference state (dry) and the equilibrium state (swollen) of a hydrogel, along with an auxiliary initial state used in numerical simulations

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Figure 2

Comparison between numerical results and analytical solution for free, isotropic swelling of a hydrogel

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Figure 3

Anisotropic swelling of a hydrogel film under lateral constraint: (a) the swelling ratio in the thickness direction and (b) the swelling-induced true stress in the lateral direction. Numerical results from two different implementations (UMAT and UHYPER ) are compared with the analytical solution in Eqs. 56,57. Note that the results from UHYPER correspond to a different analytical solution with an isotropic initial swelling (29).

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Figure 4

Anisotropic swelling of a hydrogel line under longitudinal constraint: (a) the swelling ratio in the lateral direction and (b) the swelling-induced true stress in the longitudinal direction

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Figure 5

Numerical steps to simulate inhomogeneous swelling of a hydrogel line (W/H=1) attached to a rigid substrate: (a) the dry state, (b) the initial state, (c) deformation after releasing the side pressure in (b), and (d) equilibrium swelling at μ=0, with the dashed box as the scaled dry state

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Figure 6

Inhomogeneous swelling of surface-attached hydrogel lines: (a) average longitudinal stress and (b) volume ratio of swelling. The solid and dashed lines are analytical solutions for the homogeneous limits with W/H→∞ and W/H→0, respectively.

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Figure 7

Simulated swelling deformation and longitudinal stress distribution in surface-attached hydrogel lines of different aspect ratios: (a) W/H=1, (b) W/H=5, and (c) W/H=10. The rectangular boxes outline the cross sections at the dry state.

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Figure 8

Equilibrium volume ratio as a function of the dry-state width-to-height aspect ratio for inhomogeneous swelling of surface-attached hydrogel lines

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Figure 9

Formation of surface creases in a surface-attached hydrogel line with W/H=12 as the chemical potential increases: (a) μ¯=−0.00075, (b) μ¯=−0.0003, and (c) μ¯=0

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