Nonlinear poroviscoelastic behavior of gelatin-based hydrogel

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Abstract

The mechanical properties of hydrogels involve solvent diffusion as well as viscous dissipation in the network, with the coupling called poroviscoelasticity. To explore the poroviscoelastic behavior, we performed tests on a gelatin-based hydrogel in which poroelastic swelling/drying occurred at controlled humidity levels while simultaneously developing creep deformation under a uniaxial load. These experimental results allowed the decomposition of the deformation into volumetric and isochoric parts, and the inference that volumetric deformation was dominated by poroelastic effects while isochoric deformation was influenced by both viscoelastic and poroelastic effects. We developed a nonlinear model based on an extension of the Flory–Rehner theory of the ternary system and the fractional derivative model of viscoelasticity to capture nonlinear poroviscoelastic behavior. The model was implemented into FEniCS finite element software. Comparisons of numerical simulations based on this model to experimental measurements were used to calibrate material parameters.

Introduction

Hydrogels are polymer networks embedded in a solvent that is predominantly water. They are able to absorb and retain a large amount of water and yet, retain a solid-like phase; they have a high degree of flexibility and excellent biocompatibility, making them appealing in a wide range of applications, including drug delivery (Qiu and Park, 2001, Hoare and Kohane, 2008), soft robots (Zheng et al., 2015), contact lenses (Hyon et al., 1994), tissue engineering (Lee and Mooney, 2001), agriculture industry (Ullah et al., 2015) and so on. Therefore, it is essential to understand the mechanical properties of hydrogels completely.

The deformation of hydrogels is time-dependent due to two fundamentally different mechanisms. First, the diffusive motion of the solvent molecules through the network leads to swelling or shrinking behavior of the gel macroscopically; this is the poroelastic mechanism. The linear and nonlinear theories of poroelasticity were developed to account for the solvent migration (e.g. Biot, 1941, Flory, 1942, Huggins, 1941, Hong et al., 2008, Hong et al., 2009, Chester and Anand, 2010, Bouklas and Huang, 2012, Bouklas et al., 2015). Second, the rearrangement of long flexible polymer chains results in creep or relaxation in the microscale. The theory of viscoelasticity (or viscoplasticity) takes such processes into account; there are a considerable number of mathematical models that have been developed, incorporating both linear and nonlinear responses (e.g. Christensen, 1982, Lakes, 1998, Ferry, 1980, Bergström and Boyce, 1998). The behavior of hydrogels involves these two mechanisms acting together in a coupled manner; the resulting behavior is called poroviscoelastic behavior (linear or nonlinear).

Since poroviscoelastic properties also exist in soil, rocks, articular cartilage and other materials, it has already gained attention from researchers in the last century. Biot, 1941, Biot, 1973 established the equations of deformation of an anisotropic linear viscoelastic porous solid containing a viscous fluid, and then extended to the semilinear case in 1973. Mak (1986) proposed a biphasic poroviscoelastic model for saturated articular cartilage in two versions under the assumption of elastic and viscoelastic bulk deformation respectively. There are also models that have been developed for soils and rocks (Abousleiman et al., 1996, Vgenopoulou and Beskos, 1992). In addition, many test methods have been developed to measure the poroviscoelastic properties, like the confined compression test, the indentation test and the hydrostatic compression test (Setton et al., 1993, DiSilvestro and Suh, 2001, Makhnenko and Podladchikov, 2018). Recently, to capture poroviscoelastic behavior of hydrogels, the theoretical framework based on nonequilibrium thermodynamics was established and developed by a number of researchers (e.g. Hu and Suo, 2012, Wang and Hong, 2012, Chester and Anand, 2010, Caccavo and Lamberti, 2017, He and Hu, 2020). Experimental methods have been proposed to measure poroelasticity and viscoelasticity separately (Strange et al., 2013, Wang et al., 2014); separation is possible since poroelastic behavior is size-dependent but viscoelastic behavior is not (Hu et al., 2011). Indentation and compression relaxation tests are commonly performed to measure separate poroelastic and viscoelastic properties (Chan et al., 2012, Gentile et al., 2013, Shapiro and Oyen, 2014, Caccavo et al., 2017).

Despite all the results available for hydrogels, there are still some questions that remain unanswered: (1) Do poroelastic and viscoelastic behavior influence each other? If yes, how? It is known that viscoelasticity can be influenced by temperature, polymer concentration and other environment impact (Janáček and Ferry, 1969, Ferry, 1980). Besides, Bosnjak et al. (2020) showed that fully swollen VHB specimen does not exhibit significant viscous deformation while the dry one does. (2) How does the coupling enter a constitutive model? (3) How to analyze the experiments when poroelastic and viscoelastic behavior are exhibited simultaneously so that a proper constitutive model can be calibrated?

The principal aim of this manuscript is to consider the poroviscoelastic coupling and material calibration. It is organized as follows: the experimental method and characterization of poroviscoelastic behavior are addressed first in Sections 2 Materials and methods, 3 Results. The formulation of the nonlinear poroviscoelastic model based on thermodynamics and the two-potential framework are described in Section 4. The specific model used for our gelatin-based hydrogel is presented in Section 5. The governing equations are normalized, and the general initial and boundary conditions are discussed in Section 6. The weak form and the numerical solution procedure are described in Section 7. The calibration of material parameters from comparison between numerical and experimental results is also demonstrated in Section 7.

Section snippets

Specimen preparation

The gelatin-based hydrogel was prepared by dissolving 15% weight fraction of Fisher gelatin type A powder (G8-500, derived from porcine skin) into a solvent, which is a mixture of 51% by weight glycerol and 34% by weight water. This corresponds to the composition A used in our previous work (Chen et al., 2020). The solution was heated up to 80°C and stirred continuously to completely dissolve the gelatin and obtain a homogeneous mixture. In order to remove dissolved gases and the bubbles that

Principal stretches

Since no shear deformation can be sustained through swelling/drying and creeping, the deformation gradient tensor is diagonal with principal values λx,λy,λz. Here, we use the average horizontal and vertical stretches λx and λy of the front surface calculated based on strain components obtained from DIC: λx=1+ɛxx, λy=1+ɛyy. We plot the average principal stretches versus time of two specimens for three representative cases as shown in Fig. 2. In the case of 73% humidity, as shown in Fig. 2(a),

Balance laws and thermodynamics

We now turn to the formulation of the nonlinear poroviscoelastic problem with the aim of using the experimental data to extract a calibration of the material properties.

Specific material constitutive model

We will now consider a specific form of the constitutive model that is applicable to the three-component gelatin based gel, including a power-law creep form of the viscoelastic behavior.

Normalized governing equations, and initial and boundary conditions

We summarize all the governing equation developed above, in a nondimensionalized form.

Weak form

We write the weak-form of the governing equations by considering test functions v, q and Cvtest for the displacement vector field u, the chemical potential scalar field μ, and the viscous deformation tensor field Cv, respectively. Multiplying the local force balance equation (Eq. (33)) by v, integrating over the volume of body B, and using the divergence theorem yields: BS:vXdV=BbvdV+BtTvdSSimilarly, multiplying the local mass balance equation (Eq. (35)) by q, and using implicit

Conclusion

We analyze concurrent or coupled poroelastic and viscoelastic behavior of gelatin-based gels through experiments and numerical simulations. Experiments are performed in a humidity-controlled chamber to explore poroelastic swelling as well as drying with and without mechanical loading, and viscoelastic creep with or without solvent diffusion. These experimental results are interpreted in terms of a nonlinear poroviscoelastic model. We begin with the two-potential formulation for representing the

CRediT authorship contribution statement

Si Chen: Conceptualization, Methodology, Data analysis, Data interpretation, Writing – original draft, Writing – review & editing. Krishnaswamy Ravi-Chandar: Conceptualization, Methodology, Data interpretation, Writing – original draft, Writing – review & editing, Supervision, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research was performed during the course of an investigation into the behavior of hydrogels funded by the National Science Foundation, USA through a grant CMMI-1538658; this support is gratefully acknowledged. The authors also thank Dr. Rui Huang at the University of Texas at Austin for many valuable discussions on the behavior of hydrogels and for comments on this manuscript.

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