Research Papers

A Micromechanically Based Constitutive Model for the Inelastic and Swelling Behaviors in Double Network Hydrogels

[+] Author and Article Information
Yin Liu

State Key Laboratory of Structural Analysis for
Industrial Equipment,
Department of Engineering Mechanics,
Faculty of Vehicle Engineering and Mechanics,
Dalian University of Technology,
Dalian 116024, China

Hongwu Zhang

State Key Laboratory of Structural
Analysis for Industrial Equipment,
Department of Engineering Mechanics,
Faculty of Vehicle Engineering and Mechanics,
Dalian University of Technology,
Dalian 116024, China

Yonggang Zheng

State Key Laboratory of Structural
Analysis for Industrial Equipment,
Department of Engineering Mechanics,
Faculty of Vehicle Engineering and Mechanics,
Dalian University of Technology,
Dalian 116024, China
e-mail: zhengyg@dlut.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 8, 2015; final manuscript received October 26, 2015; published online November 16, 2015. Assoc. Editor: Thomas Siegmund.

J. Appl. Mech 83(2), 021008 (Nov 16, 2015) (14 pages) Paper No: JAM-15-1486; doi: 10.1115/1.4031897 History: Received September 08, 2015; Revised October 26, 2015

This paper presents a micromechanically based constitutive model within the framework of the continuum mechanics to characterize the inelastic elastomeric and swelling behaviors of double network (DN) hydrogels, such as the stress-softening, necking instability, hardening, and stretch-induced anisotropy. The strain-energy density function of the material is decomposed into two independent contributions from the tight and brittle first network and the soft and loose second network, each of which is obtained by integrating the strain energy of one-dimensional (1D) polymer chains in each direction of a unit sphere. The damage process is derived from the irreversible breakages of sacrificial chains in the first network and characterized by the directional stretch-dependent evolution laws for the equivalent modulus and the locking stretch in the non-Gauss statistical model of a single polymer chain. The constitutive model with the optimized-material evolution law predicts stress–stretch curves in a good agreement with the experimental results during loading, unloading, and reloading paths for both ionic and covalent DN hydrogels. The deformation-induced anisotropy is investigated and demonstrated by the constitutive model for the free swelling of damaged specimen. The constitutive model is embedded into the finite-element (FE) procedure and proved to be efficient to model the necking and neck propagation in the plane-strain uniaxial elongation. Based on the procedure, the effects of imperfection and boundary conditions on the loading path and the material evolution during different stages of deformation are investigated.

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Fig. 1

Experimental stress–stretch curves for loading, unloading, and reloading of a DN hydrogel specimen, which include three typical stages, i.e., prenecking, necking, and hardening (Reproduced with permission from Nakajima et al. [8] based on Fig. 4. Copyright 2013 by The Royal Society of Chemistry).

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Fig. 2

(a) Three-dimensional network repartition in a unit sphere and (b) the positions of integration points on half of the unit sphere. The number i in (b) denotes the location of the direction vector di.

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Fig. 3

Parameter effects for the evolution of the effective modulus and the nominal locking stretch for the first network

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Fig. 4

Comparison of (a) the stress–stretch curves and (b) the initial tangent moduli between the experimental data and the constitutive model. (a) The experimental data for fitting (circles) and for prediction (squares) can be found in Fig. 1 and the fitting and prediction curves are calculated according to Eq. (9) and material constants in Table 1.

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Fig. 5

Evolution of (a) the equivalent shear modulus G1,1 and the stretch factor ξ1/n1,1 in the tensile direction d1 and (b) the tensile stress P11 and P12 distributed in the first and the second network, respectively. Note that ξ1 is equal to λ1 according to Eq. (4) and Fig. 2(b).

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Fig. 6

The effects of G1p, p = 2,3 and n13 on the evolution of stress–stretch curves with respect to the first loading path

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Fig. 7

The evolution of the swelling ratios and the bulk modulus, Gp, for damaged specimens with maximal uniaxial stretch λm. The theoretical swelling ratios, λ1s and λ2s, are calculated by Eq. (25). The experimental data λ̃1s and λ̃2s are obtained from the work of Nakajima et al. [8], in which λ̃2s is the average value of the swelling ratios in the isotropic plane.

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Fig. 8

Comparison of the stress–stretch curves between the constitutive model and experiments [9] for cyclic loading on an ionic DN hydrogel

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Fig. 9

The tensile stress–stretch curves for the DN hydrogel with material constants given in Table 3. The theoretical result is calculated by Eq. (32) and the numerical result is obtained from the FE procedure with one Q4 element.

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Fig. 10

The Maxwell line for (a) a nonlinear elastic solid and (b) a typical DN hydrogel in quasi-static, steady-state neck propagation

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Fig. 11

The stress–stretch curves for uniaxial loading and unloading of the hydrogel block and several representative configurations at different deformation stages

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Fig. 12

The computed tensile stress–stretch curves with different imperfection parameters ap = 1×10−8, 1×10−6, and 1×10−3

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Fig. 13

The evolution of the material parameters in the tensile direction, G1,1 and n1,1, at material points A, B, and C. The distributions of G1 and n1 are at point A before and after the neck passes through it (λ1=1.90 and 2.25) are also shown.

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Fig. 14

Boundary effects on the computed stress–stretch curves for the three specimens with a different length lAB



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