Transient swelling of cylindrical hydrogels under coupled extension-torsion: Analytical and 3D FEM solutions

Abstract

To design more accurate soft sensors and actuators, there is a significant requirement for the application of complicated deformation conditions. Coupled extension-torsion is a type of deformation which may be utilized in the property characterization of materials with complex behavior such as soft hydrogels. Hydrogels with coupled diffusion and large deformation behavior have intricate kinetics which should be studied in detail. In this work, a robust semi-analytical procedure is proposed to capture the transient behavior of cylindrical hydrogels in combined loading of extension and torsion in different conditions. Moreover, the whole problem with the same conditions is simulated in COMSOL-Multiphysics as a verification framework of the proposed semi-analytical procedure, where comparing the results shows a good agreement. The results demonstrate that the transient behavior of the cylindrical hydrogel under coupled extension-torsion is significantly dependent on the rate of the deformation and material properties. Besides, since the hydrogel undergoes a highly nonlinear swelling in this deformation regime, the simulations predict a complicated reaction force and moment. Visualizing the effect of different parameters engaged in the material shows the robust behavior of the proposed semi-analytical procedure, which can be employed in constitutive models’ calibration as well as in the design and optimization of hydrogels structures.

Keywords

Coupled extension-torsion transient kinetics diffusion large deformation hydrogels

1. Introduction

Hydrogels may imbibe solvent molecules in their polymeric chains and then swell upon diffusion of the solvents. This swelling phenomenon brings interesting applications in microfluidic systems (Amiri and Mazaheri, 2020) such as drug delivery systems (Gharehnazifam et al., 2021; Hoare and Kohane, 2008), micro-valves (Ghasemkhani et al., 2021; Mazaheri et al., 2018), micro-pumps (Richter et al., 2009), actuators (Nourian et al., 2020), self-folding devices (Guan et al., 2005), grippers (Yoon, 2019), biosensors (Dolatabadi et al., 2022; Goh et al., 2017, 2019), and cell immobilization (Jen et al., 1996). Depending on different features, hydrogels can be divided into various categories. For instance, depending on the response to the external stimuli, they may be classified into temperature-sensitive (Khanjani et al., 2020b; Mazaheri et al., 2016), pH-sensitive (Khanjani et al., 2020a; Marcombe et al., 2010), pressure-sensitive (Baït et al., 2011), salt concentration sensitive (Zheng and Liu, 2019), and sensitive to magnetism (Liu et al., 2017). There are also a lot of hydrogels that have no reaction to the external stimulus. These hydrogels are conventionally called neutral hydrogels (Mauck et al., 2003). They just need time to swell in a wet environment due to the osmotic pressure. Neutral hydrogels with high compatibility with body organisms may be used in artificial tissues (Lee and Mooney, 2001) for those who suffer from organ loss. Such as women who have lost their breasts because of breast cancer or people who lose their tissue in an accident, in these cases, a neutral hydrogel can be a great help. As another example, a recent work by Goh et al. (2018), shows the great potential of utilizing hydrogels as dialysis membranes for patients who suffer from chronic kidney disease. Wang and Yang (2021), employed a finite deformation theory to analyze the inhomogeneous deformation of a spherical hydrogel under chemo-mechanical loading.

The behavior of hydrogels may be considered in equilibrium states where the transient kinetics of the material is not included in formulations (Brannon-Peppas and Peppas, 1991). This type of study ignores some important features of the material behavior since the diffusion phenomenon as the main occurring phenomenon in the hydrogel is time-dependent and requires considering the kinetics of the problem. However, this means investigating the behavior of the hydrogel in transient mode of swelling, which makes the governing equations more complex (Bayat et al., 2020). Furthermore, the diffusion and deformation are influenced by each other, which results in a fully coupled multiphysics problem.

Some studies analytically examine the deformation of the hydrogel materials at an equilibrium state of swelling (Niroumandi et al., 2021; Shojaeifard et al., 2022). Abdolahi et al. (2016), presented an analytical solution for swelling-induced bending of temperature-sensitive hydrogel bilayers. The same group did an analytical and numerical study on the bending of pH-sensitive hydrogel bilayers (Arbabi et al., 2017). Dai and Song (2011) considered three different cases of free-swelling, nearly free-swelling, and general inhomogeneous swelling of the hydrogel, at an equilibrium state. More recently, Shojaeifard et al. (2019) studied the swelling-induced finite bending of functionally graded (FG) pH-sensitive hydrogels via a semi-analytical approach. The same group did similar research for FG temperature-sensitive hydrogels, again both numerically and analytically. Furthermore, Zhou et al. (2019) proposed an equilibrium constitutive model for fiber-reinforced hydrogels and studied different deformation modes of these materials analytically. All these works have one feature in common, which they do not consider the time-dependent effect of diffusion in their formulation and just observe the behavior of the hydrogel at an equilibrium state. This leads to ignoring the important complex behavior of the hydrogels in real conditions. Specifically, in small scales (less than millimeters) at which the diffusion phenomenon is fast, this main assumption leads to unrealistic solutions.

There are several standard deformation regimes used for material characterization (Pashazadeh et al., 2021; Xolin et al., 2020), and one of the most important is coupled extension and torsion. This is due to the fact that this deformation regime involves both the tensile and shear modes of deformation simultaneously. Hence, developing analytical or semi-analytical solutions to investigate the nonlinear behavior of hydrogels, has attracted the researchers’ attention. Treloar (1972) did the first work on a swollen cylinder under torsion. In this work, he utilized Flory-Huggins’ theory for the swelling of the crosslinked rubber. He proposed a solution for the large torsional strains using a computational iterative solution method. Treloar’s theory was only applicable to the equilibrium behavior of the swollen rubber. Gandhi et al. (1989) considered the gradients in the fluid density throughout the mixture theory. They compared the results for volume changes due to the torsion of the swollen cylinder with experimental results, where a relatively good agreement was observed.

Recently, Shojaeifard and Baghani (2020) proposed a semi-analytical solution for the swelling of a temperature-sensitive hydrogel cylinder under combined extension and torsion in an equilibrium state. Employing a UHYPER subroutine, developed in the finite element method (FEM) framework, they showed that their proposed solution has an excellent agreement with FEM results. In another work, Shojaeifard et al. (2021) proposed a coupled solution for the same problem, while the temperature distribution was solved simultaneously. Again, their solution was only acceptable for the equilibrium behavior of the hydrogel. Mollica et al. (2012) studied the combined extension and torsion of a swollen hydrogel in unsteady conditions. Applying a continuum theory of mixture and adopting the Flory-Rehner theory to describe the Helmholtz free energy of the hydrogel, they examined the problem for a cylindrical hydrogel which initially is at a fully swollen state. More recently, Namdar and Mazaheri (2020) investigated the kinetics of a cylindrical temperature-sensitive hydrogel in the absence of any external deformation.

In the present work, we propose a semi-analytical framework using nonlinear solid mechanics theories for predicting the transient behavior of the cylindrical hydrogels under coupled extension and torsion deformation. It is worth mentioning that most of the previous works, do not consider the transient behavior of the material, and this is the first semi-analytical solution for the extension and torsion of the hydrogel which includes the time-dependent swelling phenomenon of the material from a fully shrunk to a fully swollen state. We also compare all semi-analytical results with those extracted from 3D FEM to verify the validity of the proposed solution.

This paper is organized as follows: in section 2, first, we briefly introduce the material model and governing constitutive equations of the hydrogel. Then, we illustrate the kinematics of the coupled extension and torsion deformation for a compressible cylinder and derive the related deformation gradient tensor. Next, applying balance equations of momentum and mass, we propose a semi-analytical solution for extension-torsion of a cylindrical hydrogel using the coupled theory of large deformation and diffusion. In the last part of this section, we introduce the boundary conditions of the problem and give some notes on the normalization procedure used in this study. In section 3, we report results and will have a detailed discussion. Finally, we bring a summary and present a conclusion in section 4.

2. Coupled extension-torsion

In this section, first, a material model is presented to describe the transient coupled large deformation and diffusion behavior of the hydrogel, where, the constitutive equations of the stress and chemical potential are reported. In the next step, the kinematics of the extension and torsion of the cylindrical hydrogels are described. Thereafter, employing the balance equations of momentum and mass, a transient semi-analytical solution is proposed for the coupled extension-torsion problem. Then, we discuss the boundary conditions.

2.1. Material model and constitutive equations

The coupled diffusion and large deformation theory introduced by Hong et al. (2008) is used in this work as the constitutive equations for the hydrogel. This model considers two energy functions, one for the stretch of polymer chains and the other one for the mixing of solvent molecules and long polymers:

W (F, C) = W_{s} (F) + W_{m} (C),

(1)

where $W_{s} (F)$ , as a function of the deformation gradient $F$ stands for the contribution of stretching in polymer chains, and $W_{m} (C)$ as a function of solvent concentration $C$ , is the contribution of mixing. One may use an invariant based neo-Hookean form for the stretching energy function:

W_{s} (F) = \frac{1}{2} NkT (I_{1} - 3 - \log I_{3}),

(2)

in which, $N$ is the number of polymer chains per unit volume of the dry state, $k$ is the Boltzmann’s constant, and $T$ denotes the absolute temperature. The first and third invariants of the left Green Cauchy deformation tensor, $B = F F^{T}$ , are denoted by $I_{1}$ and $I_{3}$ , respectively. According to Huggins (1941) and Flory (1942), the energy function of mixing can be assumed as:

W_{m} (C) = \frac{- kT}{ν} (ν C \log (1 + \frac{1}{ν C}) + \frac{χ}{1 + ν C}),

(3)

in which, $ν$ is the volume of one solvent molecule and the dimensionless parameter $χ$ represents the enthalpy change within the hydrogel. Using free energy functions of equations (2) and (3), one may derive two sets of constitutive equations for Cauchy stresses $σ$ and chemical potential $μ$ . For Cauchy stress, we have (Holzapfel, 2000):

σ = 2 J^{- 1} [I_{3} \frac{\partial W}{\partial I_{3}} I + \frac{\partial W}{\partial I_{1}} B] - Π I,

(4)

in which, the Lagrange multiplier $Π$ is responsible for the osmotic pressure occurring within the hydrogel and $I$ is identity tensor.

The second constitutive equation of chemical potential is derived as

μ = \frac{\partial W}{\partial C} + Π ν

(5)

The conservation of solvent molecules within the hydrogel can be expressed as

\frac{\partial C}{\partial t} = - \frac{\partial j_{K}}{\partial X_{K}},

(6)

in which $j$ is the solvent flux and is defined as

j_{K} = - M_{KL} \frac{\partial μ}{\partial X_{L}},

(7)

and $X_{L}$ are the components of reference coordinate system and $M_{KL}$ is the mobility tensor with the following definition

M_{KL} = \frac{D}{ν kT} H_{iK} H_{iL} (J - 1),

(8)

where D is the diffusion coefficient which is assumed to be isotropic and independent of the deformation gradient $F$ . While tensor $H_{ik}$ is the transpose of the inverse of the deformation gradient $F$ and J = det F.

2.2. Kinematics of coupled extension and torsion

Describing the undeformed configuration of the hydrogel in reference cylindrical coordinate system $(R, Θ, Z)$ , we have:

0 \leq R \leq R_{out}, 0 \leq Θ \leq 2 π, 0 \leq Z \leq L,

(9)

As shown in Figure 1, one may express the coupled extension-torsion problem in current coordinate system of $(r, θ, z)$ .

Figure 1.

Reference and current coordinate systems, describing the coupled extension and torsion deformation of a cylindrical hydrogel.

The hydrogel is subjected to a uniformly distributed axial extension $γ$ and a torsion $τ$ defined as twist angle per unit length of the stretched hydrogel. Thus, having these parameters in hand, the deformation field can be expressed as (Kirkinis and Ogden, 2002)

r = r (R, t), θ = Θ + γ (t) τ (t) Z, z = γ (t) Z,

(10)

Since the diffusion in hydrogel has a transient characteristic, the current radius is a function of both the reference radius and time. For the sake of brevity in equations, we use the kinematics parameters without their arguments ( $t$ and $R$ ).

The deformation gradient $F$ related to this deformation field is computed as:

F = (\begin{matrix} r' & 0 & 0 \\ 0 & r / R & γ τ r \\ 0 & 0 & γ \end{matrix}),

(11)

which is expressed with respect to the reference and current cylindrical coordinate systems, and the prime shows the differentiation with respect to $R$ . The left Cauchy-Green deformation tensor can be presented by:

B = (\begin{matrix} {r'}^{2} & 0 & 0 \\ 0 & {(r / R)}^{2} + r^{2} γ^{2} τ^{2} & r γ^{2} τ \\ 0 & r γ^{2} τ & γ^{2} \end{matrix}) .

(12)

Therefore, the principal invariants of $I_{1}, I_{2}, I_{3}$ are found as;

\begin{matrix} I_{1} = {r'}^{2} + {(r / R)}^{2} + γ^{2} + r^{2} γ^{2} τ^{2}, \\ I_{2} = {r'}^{2} {(r / R)}^{2} + γ^{2} {(r / R)}^{2} + {r'}^{2} γ^{2} + r^{2} τ^{2} {r'}^{2} γ^{2}, \\ I_{3} = {r'}^{2} {(r / R)}^{2} γ^{2} . \end{matrix}

(13)

2.3. Transient semi-analytical solution

Using the deformation invariants expressed in equation (13), the stress constitutive equations of equation (4), are recast as:

\begin{matrix} σ_{rr} = \frac{RNkT}{r' r γ} ({r'}^{2} - 1) - Π, \\ σ_{θ θ} = \frac{NkT}{Rr' r γ} (r^{2} τ^{2} γ^{2} R^{2} + r^{2} - R^{2}) - Π, \\ σ_{zz} = \frac{RNkT}{r' r γ} (γ^{2} - 1) - Π, \\ σ_{θ z} = \frac{RNkT τ γ}{r'} . \end{matrix}

(14)

The only non-trivial equilibrium equation in cylindrical coordinate is:

\frac{1}{r'} \frac{\partial σ_{rr}}{\partial R} + \frac{1}{r} (σ_{rr} - σ_{θ θ}) = 0 .

(15)

The other trivial equilibrium equations give that $\frac{\partial Π}{\partial θ} = \frac{\partial Π}{\partial z} = 0$ . Thus, one may conclude that the Lagrange multiplier $Π = Π (R)$ is a function of only $R$ . Recasting equation (15) for $\frac{\partial Π}{\partial R}$ , we have:

\frac{\partial Π}{\partial R} = - \frac{NTk}{γ {r'}^{2} r^{2} R} (- R^{2} r r ″ ({r'}^{2} + 1) + r' (- R r {r'}^{2} + ((γ^{2} R^{2} τ^{2} + 1) r^{2} - R^{2}) r' + Rr))

(16)

Rewriting equation (8) for radial direction gives:

M_{rr} = \frac{D}{ν kT} [\frac{r' r γ - R}{{r'}^{2} R}] .

(17)

The chemical potential in equation (5) is also recast as:

μ = kT [\log \frac{ν C}{1 + ν C} + \frac{1}{1 + ν C} + \frac{χ}{{(1 + ν C)}^{2}}] + Π ν .

(18)

It is common to assume that the polymer chains and solvent molecules are incompressible, that is,

J = 1 + ν C .

(19)

Substituting the concentration $C$ from equation (19), into equation (18) to find the chemical potential $μ$ , and then the result into equation (7), the radial solvent flux will be found in terms of $R, r, r'$ and $r ″$

j_{r} = A_{1} + \frac{A_{2}}{r'} + \frac{A_{3}}{{r'}^{2}} + \frac{A_{4}}{{r'}^{3}} + \frac{A_{5}}{{r'}^{4}} + \frac{A_{6}}{{r'}^{5}},

(20)

where

\begin{matrix} A_{1} = - \frac{DN}{R}, \\ A_{2} = \frac{DN}{r γ R^{2}} ((γ^{3} r^{2} τ^{2} - γ r r ″ - γ + 1) R^{2} + γ r^{2}), \\ A_{3} = - \frac{D}{R r^{2} γ ν} ((γ^{2} N ν r^{2} τ^{2} - N ν r r ″ - N ν - 2 χ + 1) R^{2} - N r^{2} ν (γ - 1)), \\ A_{4} = - \frac{D}{ν r^{3} γ^{2}} (2 χ R^{2} + r^{2} γ (γ N ν r r ″ + N ν + 2 χ - 1)), \\ A_{5} = \frac{DR}{r^{2} ν γ^{2}} (γ r ″ (N ν + 2 χ - 1) r + 2 χ), \\ A_{6} = - \frac{2 D R^{2} χ r ″}{r^{2} ν γ^{2}} . \end{matrix}

(21)

Moreover, it should be noted that equation (16) as the first derivation of $Π (R)$ with respect to $R$ was used in equation (7).

Substituting equation (20) into equation (6), we arrive at a partial differential equation (PDE) which is first order in terms of time and third order with respect to $R$ . Accordingly, we subdivide the time interval of interest [0, t] in sub-increments and solve the evolution equation (6) over the generic interval [t_n, t_n_+ 1] with t_n_+ 1 > t_n, to have a third order ordinary differential equation (ODE) with respect to $R$

\frac{C_{n + 1} - C_{n}}{Δ t} + \frac{\partial j_{r}}{\partial R} = 0 .

(22)

The final form of equation (22) is expressed as

B_{1} r' + B_{2} + \frac{B_{3}}{r'} + \frac{B_{4}}{{r'}^{2}} + \frac{B_{5}}{{r'}^{3}} + \frac{B_{6}}{{r'}^{4}} + \frac{B_{7}}{{r'}^{5}} + \frac{B_{8}}{{r'}^{6}} = 0,

(23)

in which

\begin{matrix} B_{1} = \frac{γ r}{R ν Δ t}, \\ B_{2} = \frac{1}{r^{2} γ R^{2} ν Δ t} ((γ (γ^{2} ND ν Δ t τ^{2} - C_{n} ν - 1) r^{2} + N Δ tD ν (γ - 1)) R^{2} + 2 N r^{2} Δ t γ D ν), \\ B_{3} = - \frac{D}{r^{3} γ R^{3} ν} ((N ν r r ″ + 2 N ν + 4 χ - 2) R^{4} + Nr ″' r^{3} γ ν R^{3} + 2 N γ ν r^{4}), \\ B_{4} = \frac{D}{R^{2} ν r^{4} γ^{2}} (r^{2} γ (- N r ″ τ^{2} γ^{3} ν r^{3} + N γ ν (- γ τ^{2} + {r ″}^{2}) r^{2} + Nr ″ γ ν r + 2 N ν + 4 χ - 2) R^{2}) \\ + \frac{D}{R^{2} ν r^{4} γ^{2}} (6 R^{4} χ + Nr ″' r^{3} γ ν R^{3} - N r^{4} γ ν (γ r r ″ + γ - 1)), \\ B_{5} = - \frac{2 D}{r^{3} R ν γ^{2}} (\begin{matrix} (- Nr ″ τ^{2} γ^{3} ν r^{3} + N {r ″}^{2} r^{2} γ ν + \frac{3 r ″ γ (N ν + 2 χ - 1) r}{2} + 4 χ) R^{2} \\ + \frac{Nr ″' R r^{3} γ^{2} ν}{2} + Nr ″ r^{3} γ ν (γ - 1) \end{matrix}), \\ B_{6} = \frac{3 D}{r^{3} ν γ^{2}} (\frac{10 r ″ χ R^{2}}{3} + \frac{r ″' r^{2} γ (N ν + 2 χ - 1) R}{3} + r (N {r ″}^{2} r^{2} γ^{2} ν + \frac{4 r ″ γ (N ν + 2 χ - 1) r}{3} + \frac{2 χ}{3})), \\ B_{7} = - \frac{4 RD}{r^{2} ν γ^{2}} (\frac{r ″' χ R}{2} + r ″ (r ″ γ (N ν + 2 χ - 1) r + 3 χ)), \\ B_{8} = \frac{10 R^{2} χ D {r ″}^{2}}{r^{2} ν γ^{2}} . \end{matrix}

(24)

The resultant axial force F and torque M required to maintain the mentioned coupled extension-torsion deformation are calculated as

F = \int \int σ_{zz} dA = 2 π \int_{0}^{r_{out}} r σ_{zz} dr,

(25)

and

M = \int \int r σ_{θ z} dA = 2 π \int_{0}^{r_{out}} r^{2} σ_{θ z} dr,

(26)

where A is cross sectional area of the cylinder.

2.4. Boundary conditions

There are four boundary conditions (BCs) in this problem and two of them are dependent. It means they create a set of three independent BCs for solving the third-order ODE in equation (22). The first BC comes from the symmetry condition of the cylinder which leads to $r (0) = 0$ at any time. Also, the symmetry condition causes a zero flux at the center of the cylinder which means $\frac{\partial μ}{\partial R} |_{R = 0} = 0$ . The two dependent BCs occur at the outer surface of the cylinder, saying that the radial stress $σ_{rr}$ and chemical potential $μ$ are zero at the outer radius ( $R = R_{out}$ ). The Lagrange multiplier $Π$ which depends on both the stress and chemical potential, is also unknown. Since this term appears in both the stress and chemical potential relations, one may relate the stress to chemical potential by eliminating of $Π$ . Hence, the two former boundary conditions at the outer radius would generate the third boundary condition for the mentioned third order ODE of equation (22). One may express this boundary condition as

σ_{rr} = \frac{RNkT}{r' r γ} ({r'}^{2} - 1) - {μ - \frac{kT}{ν} [\log \frac{ν C}{1 + ν C} + \frac{1}{1 + ν C} + \frac{χ}{{(1 + ν C)}^{2}}]},

(27)

As one may notice, in this equation the unknown parameter $Π$ has been eliminated.

For better results visualization and comparison, the components of stress are normalized by $kT / ν$ and chemical potential $μ$ are normalized by $kT$ . Time $t$ is also normalized by $Dt / {R^{2}}_{out}$ to reach the dimensionless form of equation (22).

Volume of a single water molecule has the value of $ν = 3 \times 10^{- 29} m^{3}$ . Also, at room temperature, $kT = 4 \times 10^{- 21} J$ and $kT / ν = \frac{4}{3} \times 10^{8} Pa$ . The dimensionless parameter $χ$ which strongly affects the final swelling of the hydrogel in this study is $χ = 0.6$ , and $N ν$ which expresses the strength of the hydrogel is $N ν = 0.0015$ . Finally, the diffusion coefficient is assumed as $D = 10^{- 7} m^{2} / s$ .

3. Results and discussion

Equation (22) subject to the mentioned BC’s has been solved using finite difference method (FDM), by MATLAB function bvp4c. This function can solve nonlinear set of ODEs with proper boundary values via FDM. We discretize the problem in 2500 time-intervals of $Δ t = 0.04 s$ . Since in this study, the hydrogel is subjected to the coupled extension and torsion and starts to swell from a fully shrunk state, simultaneously, the proper initial value should be considered. Also, the radial domain of $R$ is divided to 1000 points; however, the solution is very fast, which is an advantage of the proposed semi-analytical solution.

The finite element method is used here to verify the proposed semi-analytical solutions. Implementing the mentioned coupled theory of diffusion and large deformation of this work in COMSOL 5.5 enables us to simulate the kinetics of coupled extension and torsion loading on cylindrical hydrogels. Despite wide flexibility for simulating various problems in Multiphysics tools of COMSOL, it is a relatively time-consuming process compared to analytical and semi-analytical procedures. The simulations of this study have been done for a cylinder with a height and radius of 1 and 0.25 cm, respectively. After performing a mesh-independency study, 19,141 tetrahedron elements were used to reach an optimized number of elements. In the following, an axial stretch of $γ = 1.8$ and a twist of $τ = \frac{π}{3.6} rad / cm$ with rate of $0.001 s^{- 1}$ , are applied to the cylindrical hydrogel.

Figure 2 depicts the radial displacement of the hydrogel along the radial direction at different times. As one may observe from Figure 2, passing time leads to larger displacement within the hydrogel stemming from the diffusion and swelling of the hydrogel. As the applied deformation stops at $tD / {R^{2}}_{out} \approx 15$ , the nonlinear form of the radial displacement gets closer gradually to a linear form at later times. That is because the hydrogel reaches an equilibrium state as time passes and the displacement distribution becomes homogenized. Finally, at $tD / {R^{2}}_{out} \approx 80$ which the hydrogel is fully swollen at an equilibrium state, the radial displacement takes an exact linear form.

Figure 2.

Normalized radial displacement along normalized radial direction at different time frames for $γ = 1.8$ and $τ = \frac{π}{3.6} rad / cm$ .

The normalized chemical potential distribution is depicted in Figure 3. As shown, at any time frame the chemical potential of the outer radius is in balance with the chemical potential of the external solution which is assumed to be zero. It is noted that, at the beginning where the hydrogel is at its shrunk state, there is a large difference in the chemical potential between the center and outer surface of the cylinder. It means that the hydrogel is not at its equilibrium state, and gradually over time and by diffusing more solvent molecules into the hydrogel, $μ$ reaches a uniform distribution and would be in balance with the chemical potential of the external solvent.

Figure 3.

Normalized chemical potential along the normalized radial direction at different times for $γ = 1.8$ and $τ = \frac{π}{3.6} rad / cm$ .

Figure 4 reports good evidence for the swelling within the hydrogel. As shown, each curve can be a representative illustration of an increase in the hydrogel volume. As illustrated in Figure 4, over time the outer surface of the cylinder gets a higher concentration which arises from its direct contact with the external solvent. However, it gradually finds a uniform distribution as the hydrogel reaches an equilibrated state.

Figure 4.

Normalized chemical potential of the hydrogel along normalized radial direction at different times for $γ = 1.8$ and $τ = \frac{π}{3.6} rad / cm$ .

Parameter $λ_{θ} = r (R) / R$ interpreted as the circumferential stretch is depicted in Figure 5 at different times. In the early stages, where the hydrogel does not undergo a significant swelling, a relatively uniform behavior for $λ_{θ}$ is observed in the diagram. As the complex deformation of extension-torsion is applied and the swelling occurs within the hydrogel, the curves take more nonuniform shapes at later times. Since $λ_{θ} = r (R) / R$ in Figure 5, takes the values smaller and greater than one at some time frames, both tensile and compression modes of stretching occur along the radial direction of the hydrogel.

Figure 5.

Circumferential stretch of $r (R) / R$ along normalized radial direction at different time frames for $γ = 1.8$ and $τ = \frac{π}{3.6} rad / cm$ .

Now we examine the axial reaction force and torque developed in the structure, presented in equations (25) and (26). Since the reactions are highly dependent on the rate of deformation, the reaction force during the extension-torsion on the hydrogel is depicted in Figure 6(a) for different extension rates of $\overset{\cdot}{γ} = 0.0010, 0.0013, 0.0020$ and $0.0040$ .

Figure 6.

Normalized reaction (a) force and (b) torque, of the cylinder due to the coupled extension and torsion deformation versus time for different deformation rates of extension and torsion.

Similarly, Figure 6(b) plots the reaction torque for different torsion rates of $\overset{\cdot}{τ} = 0.0010, 0.0013, 0.0020$ and $0.0040$ . It is worth mentioning that in Figure 6(a) for slower extension rates of $\overset{\cdot}{γ} = 0.0010$ and $0.0013$ , at the early steps of the process, the diffusion phenomenon as the faster mechanism compared to the applied axial deformation, creates a compressive force. It means at these two extension rates, the stretching due to the swelling, happens faster than the stretching from the applied external deformation $γ$ . On the other hand, at higher extension rates, no compressive reaction force appears in the diagram, since the external stretching in these cases is faster than that of the swelling. However, the steady-state solutions of the reaction force for any stretching rates are the same. At higher extension rates, the material undergoes a larger maximum reaction force due to the applied deformation as the governing mode of deformation. The same line of reasoning can be followed for the developed torque in the cylinder shown in Figure 6(b).

Finally, the effect of the dimensionless parameter $χ$ on the swelling response of the hydrogel is depicted in Figure 7. This parameter characterizes the difference in interaction energy of a solvent molecule immersed in pure polymer compared to the one in pure solvent (Nagy, 2018). If $χ > 0$ , the solvent and polymer dislike each other, so decreasing its value leads to an increase in final equilibrium swelling volume. As one may observe from Figure 7, the material with $χ = 0.7$ would reach an equilibrium state after about $\frac{tD}{R_{out}^{2}} = 15$ with a smaller value of final volume relative to the materials with smaller value of $χ$ . On the other hand, the swelling ratio for $χ = 0.5$ does not meet a steady state, even after $\frac{tD}{R_{out}^{2}} = 80$ . Moreover, it experiences a higher swelling ratio relative to the material with higher values of $χ$ .

Figure 7.

Swelling of the cylinder for different values of $χ$ for $γ = 1.8$ and $τ = π / 3.6 rad / cm$ .

Figure 8 illustrates the three-dimensional deformation of the cylindrical hydrogel at different time frames including a nondimensionalized von Mises stress contour obtained from 3D FEM analysis. It should be noted that this figure represents the deformation of the cylindrical hydrogel with $χ = 0.6$ . Referring to Figure 7, one can observe a high swelling gradient around $tD / R_{out}^{2} = 5$ , which leads to a relatively nonuniform stress distribution within the material as presented in Figure 8(c).

Figure 8.

Three-dimensional deformation and normalized von Mises stress contour for the cylindrical hydrogel at different time frames of $tD / R_{out}^{2}$ approximately equal to (a) 0, (b) 1, (c) 5, (d) 15, (e) 30, (f) 80, for $γ = 1.8$ and $τ = π / 3.6 rad / cm$ .

4. Conclusion

In this paper, a semi-analytical approach was proposed to describe the coupled large deformation-diffusion kinetics behavior of hydrogel cylinders under combined loading of extension and torsion. The hydrogel cylinder starts to swell while being in a wet environment. The semi-analytical approach proposed in this paper employs relations of nonlinear solid mechanics to properly represent the mechanical behavior of such hydrogel cylinders under extension and torsion. The transient behavior of hydrogels needs to consider several parameters to be properly used in sensors and actuators. Using the free energy function of the hydrogel in the mass and momentum balance equations and utilizing an appropriate kinetics law, the proposed semi-analytical approach may estimate the behavior of the cylindrical hydrogel very well. The final PDE equation was solved by applying a numerical scheme. To verify the proposed approach, the results of the semi-analytical method were compared with those of 3D FEM where a great agreement was observed.

From the results, one may conclude that the transient behavior of cylindrical hydrogels under coupled extension torsion is extremely dependent on the deformation rate, time, and material properties. Changing in any of these may affect the system response significantly. For instance, the reaction axial force in low rates of deformation is compressive due to the swelling phenomenon which is faster than the loading rate at the early stages of the process. Or circumferential stretch around the cylinder center becomes compressive after a long time due to the coupling of the swelling and extension-torsion deformation.

Some parameters such as time duration of applying deformation, Flory-Huggins’ interaction parameter, the elastic modulus of the polymer chains, and value of stretching were selected for parametric study. The behavior of the cylindrical hydrogel by changing these parameters was illustrated in diagrams and in all cases, there was a perfect agreement between the results of the proposed method and FEM. The proposed solution can be used as an efficient tool to calibrate constitutive models and to study the effects of changing any of the material or geometrical parameters on smart structures containing hydrogel components undergoing extension torsion for their design and optimization which require a large number of simulations.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Majid Baniassadi

Mostafa Baghani

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