- Email: [email protected]
Analytical and numerical analysis of swelling-induced large bending of thermally-activated hydrogel bilayers
Accepted Manuscript
Analytical and numerical analysis of swelling-induced large bending of thermally-activated hydrogel bilayers Jalal Abdolahi , Mostafa Baghani , Nasser Arbabi , Hashem Mazaheri PII: DOI: Reference:
S0020-7683(16)30234-7 10.1016/j.ijsolstr.2016.08.017 SAS 9277
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
11 May 2016 25 July 2016 22 August 2016
Please cite this article as: Jalal Abdolahi , Mostafa Baghani , Nasser Arbabi , Hashem Mazaheri , Analytical and numerical analysis of swelling-induced large bending of thermally-activated hydrogel bilayers, International Journal of Solids and Structures (2016), doi: 10.1016/j.ijsolstr.2016.08.017
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
CR IP T
Analytical and numerical analysis of swelling-induced large bending of thermally-activated hydrogel bilayers Jalal Abdolahia, Mostafa Baghani*a, Nasser Arbabia and Hashem Mazaherib a
School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran. b
Department of Mechanical Engineering, Bu-ali Sina University, Hamedan, Iran.
AN US
*Corresponding Author Email: [email protected]
Abstract
Temperature-sensitive hydrogels have recently been implemented vastly for biomedical and microfluidic sensors and actuators. The accurate and efficient design of the bilayer sensors and actuators made of
M
temperature sensitive hydrogels are of crucial importance. In this work, we develop an analytical
ED
method to solve the swelling induced bending of temperature responsive hydrogel bilayer under plane strain condition. The bilayer consists of a neutral incompressible elastomer layer attached to a
PT
temperature sensitive hydrogel layer. An analytical approach is developed to predict the thermomechanical response of these bilayers. At the other hand, the finite bending of the bilayer is
CE
simulated applying the finite element method. Several cases are solved to demonstrate the validity and performance of the proposed analytical method. The deformation and the stresses inside the layers are
AC
presented for various material parameters employing both the developed analytical formulation as well as the finite element method. A good correspondence between the presented method and the finite element method is observed. Finally, the effect of material and geometrical parameters on curvature are also investigated.
ACCEPTED MANUSCRIPT
Keywords: Bilayers, Thermally-activated hydrogels, Finite bending, Analytical solution, Finite element method.
1 Introduction
CR IP T
Smart hydrogels are lightly cross-linked polymer chains that have capability to undergo large deformation in response to various external stimuli such as temperature (Cai and Suo, 2011; Mazaheri et al., 2015; Morimoto and Ashida, 2015), pH (Drozdov, 2015; He et al., 2012; Hong et al., 2009; Marcombe et al., 2010; Mazaheri et al., 2016; Yan et al., 2014), light (Ahn et al., 2008; Toh et al., 2014), electric field
AN US
(Lai et al., 2010; Li, 2009; Zhou et al., 2002) and etc. This feature makes smart hydrogels an ideal material for fabrication of both sensors and actuators (Ionov, 2013; Richter, 2009; Shin et al., 2010; Yu et al., 2001). Smart hydrogel sensors and actuators have a promising future for application in microfluidics (Bäcker et al., 2012; Domachuk et al., 2010; Richter et al., 2009; Zhang et al., 2007), biomedical devices
M
(Guenther et al., 2013; Jones et al., 2008) and soft robotics (Kwon et al., 2008).
ED
Bilayer cantilever beams have been served as sensors and actuators for many decades. Classical bilayer sensors mostly were made of two strips of metals (bi-metal beam) with different thermal
PT
expansion coefficients (Timoshenko, 1925). Bi-metal sensors and actuators are not biocompatible due to their weak corrosion resistance in the aqueous environment. An alternative is a bilayer composed of
CE
smart hydrogels. A strip made of smart hydrogels can be attached to one layer of neutral elastomer or
AC
hydrogels with different swelling characteristics to form a highly biocompatible bilayer sensor or actuator (Agrawal et al., 2014; Swann and Ryan, 2009; Zhang et al., 2012). Hydrogel bilayer beams are able to deform noticeably in comparison with classic bi-metals, results in higher sensitivity and accuracy, and also makes them suitable to be implemented in self-folding structures. These structures have various applications in drug-delivery systems (Fernandes and Gracias, 2012; He et al., 2006; Randall et
ACCEPTED MANUSCRIPT
al., 2012). Bilayer hydrogel beams also are used as micro-valve and micro-pump for biomedical applications (Kwon et al., 2011; Yu et al., 2001). There are several theories describing the swelling behavior of thermally sensitive hydrogels. Chester and Anand (2011) developed a coupled mechanical deformation, fluid permeation and heat transfer
CR IP T
theory describing the swelling behavior of temperature sensitive hydrogels. Considering the gel to be isotropic, they developed a model based on Flory-Huggins mixing energy. Cai and Suo (2011) proposed a theory for temperature sensitive N-isopropylacrylamide hydrogels (PNIPAM). Adopting Flory-Rehner model (Flory and Rehner Jr, 1943) and defining the interaction parameter as a function of temperature
AN US
and polymer volume fraction, they developed mathematical framework describing phase transition phenomenon in PNIPAM hydrogels. Aforementioned models showed snap-through instability and multiple solutions near to the Phase Transition Temperature (PTT), which is a restriction for these
M
models to be implemented into finite element (FE) frameworks. Mazaheri et al. (2015) solved the socalled discontinuity and instability problems through modifying the mixing energy for PNIPAM
ED
hydrogels. This theory predicted the experimental result accurately without showing serious discontinuity in the vicinity of PTT.
PT
The advantages of the analytical solutions have encouraged the researchers to develop mathematical
CE
methods to solve the finite bending of the polymeric bilayers including the hydrogel bilayers. Roccabianca et al. (2010) analytically solved the finite bending of a multilayered elastic incompressible
AC
thick plate under plane strain condition. They also performed experiments to confirm the theoretical results. Lucantonio et al. (2014) studied the swelling-induced bending of layered gel beams, based on the assumption of splitting the swelling-induced deformation gradient into a homogeneous swelling component and a pure bending component. They have determined stretch and bending curvature of beams, depending on the material and geometrical characteristics. Morimoto and Ashida (2015) proposed a new approach to investigate the finite plane strain bending of a bilayer gel made of two gel
ACCEPTED MANUSCRIPT
layers with different swelling properties. They assumed that swelling-layer first undergoes a virtual onedimensional homogeneous swelling. Then, the non-swelling homogeneous pure bending balance all forces and moments on the gel bilayer. They compared the method predictions with the linear model based on the elementary beam theory. Recently, Nardinocchi and Puntel (2016) presented a method to
CR IP T
solve swelling-induced bending of layered gels based on the basic assumption of multiplicative decomposition of the deformation gradient tensor. Based on this assumption, the deformation gradient tensor split into two components of elastic deformation and elastic distortion. They also compared outcomes of their method with numerical results in the literature.
AN US
In this paper, a new analytical approach is presented to investigate the finite bending of bilayer gel beams under plane strain condition. In particular, we study bilayers composed of a neutral incompressible elastomeric layer and a PNIPAM hydrogel layer. The bilayer is assumed flat in the
M
reference (initial) state and experiences a swelling induced bending due to the temperature changes in the responsive layer. We employ a deformation gradient tensor which maps the initial state to the
ED
deformed configuration without assuming any virtual intermediate state. For describing the swelling behavior of the PNIPAM hydrogel, a modified model of the Helmholtz free energy density that proposed
PT
by Mazaheri et al. (2015) is used. Also, a UHYPER code is developed and implemented in ABAQUS for
CE
Finite Element (FE) modeling of the bilayer to numerically validate the proposed analytical model predictions. Finally, we studied the effects of some geometrical and material parameters on the bilayer
AC
configuration and the stresses distribution inside the layers. The article is organized as follow: firstly, the proposed analytical method for the plane strain
swelling-induced bending of the bilayer beam is fully explained in section 2. Then, a brief discussion on the swelling theory and the Helmholtz free energy density is presented. Thereafter, based on the swelling theory the governing equations of the system are derived. Later in section 3, we set up some case studies in order to demonstrate the accuracy and robustness of the presented method. The results
ACCEPTED MANUSCRIPT
are validated through comparing them with FEM results. Finally, we present a summary and draw conclusions in section 4.
2 Bending deformation of thermally activated hydrogel bilayers
CR IP T
In this work, a bilayer beam made of two attached layers is studied. One layer is made of a neutral incompressible elastomer (indexed by n=1) and the other is made of temperature sensitive poly-(N-isopropylacrylamide) (PNIPAM) hydrogel (indexed by n=2). Schematics of the bi-strip beam both the
reference
(flat)
and
current
(curved)
states
are
shown
in
ED
M
AN US
in
PT
Fig. 1. Finite bending of the bilayer is triggered only by the temperature variation due to an increase in the swelling of the hydrogel layer. The out of plane dimension is assumed to be infinite and the
CE
problem is formulated as plane strain. Moreover, it is assumed that the bilayer cross sections remain plane after the deformation, and the end-effects are neglected in both analytic and FE analyses.
AC
Introducing X (X1 ,X2 ,X3 ) as a reference Cartesian coordinates for the bilayer, the position of a
generic point in the reference state is given by: X X1 e1 X2 e2 X3 e3 ,
with
(1)
ACCEPTED MANUSCRIPT
L L X1 [ , ], X 2 [0, H (1) H (2) ], X3 ( , ). 2 2
(2)
where ei (i 1, 2,3) are the axes of the Cartesian coordinate, H ( n) and L are initial thicknesses and length of each layer in the undeformed configuration, respectively. It is evident that the total initial thickness of
CR IP T
the bilayer is H H (1) H (2) . In addition, the deformed configuration in the current state is considered as a portion of a cylindrical tube with the semi-angle of . Introducing x (r, , z) as current polar coordinates for each layer, the position of the generic point x( n) in the deformed state is given by:
AN US
x( n) r (n) er ( n) eθ z ( n) ez ,
with
r (1) [ri(1) , ri(1) h(1) ], r (2) [ri(2) , ri(2) h(2) ], ( n) [ , ],
(4)
M
z ( n) ( , ).
(3)
in which ei (i r, , z) are the axes of the polar coordinate, h( n) and ri ( n) are the thickness and the inner
ED
radius of each layer in the deformed configuration, while ri(2) ri(1) h(1) .
PT
Assuming r (n) r ( n) (X2 ) , (n) ( n) (X1 ) and z ( n) X3 , the deformation gradient can be found as
AC
CE
(Roccabianca et al., 2010)
F( n)
dr ( n) (X 2 ) dX 2
er e2 r ( n)
d ( n) dX1
e e1 e z e3 ,
(5)
in which is the dyadic operator. Eq. (5) also defines the principal stretches for each layer to be:
r( n)
dr ( n) (X 2 ) ( n) d ( n) ( n) , r ( n) , z 1. dX 2 dX1
(6)
ACCEPTED MANUSCRIPT
Assuming a linear distribution form for the tangential component of the position vector in the polar coordinates as (n) 2 X1 / L , the principal stretch in tangential direction for each layer is:
( n)
2 ( n) r ( X 2 ). L
(7)
r(1)
L 2 r ( X 2 ) (1)
.
CR IP T
In light of the incompressibility condition for the elastomeric layer, its radial principal stretch is recast as:
(8)
AN US
The radial stretch for the elastomeric layer that is found by the deformation gradient (Eq. (6)) should be equal to Eq. (8) (dr (1) (X2 ) / dX2 L / 2 r (1) (X2 )) . This forms a first order Ordinary Differential Equation (ODE) with a boundary condition (r (1) (0) ri(1) ) at X2 0 . Solving this ODE, the distribution of r (1) (X2 ) is
M
calculated as:
L
X 2 (ri(1) )2 ,
(9)
ED
r (1) ( X 2 )
where and ri(1) are kinematic unknowns.
PT
For modeling the swelling behavior of the temperature-sensitive PNIPAM hydrogel, a modified model
CE
of the Helmholtz free energy density recently proposed by Mazaheri et al. (2015) is employed. This modified energy is more stable in the vicinity of the phase transition temperature (PTT), which results in
AC
numerical instability elimination besides the accurate prediction of the hydrogel swelling behavior. This modified model of the free energy is convex, which made it applicable to the FEA. The modified free energy density is (Mazaheri et al., 2015):
ACCEPTED MANUSCRIPT
W (2) (1(2) , 2(2) , 3(2) ; T )
G (2) (2) 1 2
2
(2) 2
2
(2) 2 3
3 2 ln J (2)
1 k BT (2) 1 1 ( ; T ) ( J 1) (2) , (2) 2 (2) 3 v 2( J ) 3( J ) J (2) J
(10)
in which W (2) is free energy density for the hydrogel. The first term on the right-hand side of Eq. (10) is
CR IP T
the Neo-Hookean hyperelastic energy that describes the elastic deformation of the polymer network and the second term stands for the mixing part of the free energy density. G (2) is the shear modulus of elasticity of the hydrogel and is equal to G(2) N kBT , in which k B is Boltzmann’s constant is the density of the polymer chains. In Eq. (10), i (n) (i 1, 2,3) are
AN US
(kB 1.38 1023 J / K ) and N
principal stretches and J (n) 1(n) (2n) 3( n) is the determinant of the deformation gradient tensor, which is related to the volume changes for each layer. Also, v and T are the volume of a single solvent molecule and the absolute temperature, respectively. The dimensionless mixing parameter ( ; T ) is determined
M
via the following relations (Afroze et al., 2000):
(11)
ED
( ; T ) 0 1 , 0 A0 B0T , 1 A1 B1T ,
PT
where 0 and 1 are functions of the temperature and other experimental constants ( A0 , B0 , A1 and B1 ) , is the volume fraction of the polymer in the swollen state. Assuming that the swelling of the hydrogel
CE
only originates from the incompressible fluid diffusion through the network, can be found as (Hong et
AC
al., 2008):
1 C
1
J (2) ,
(12)
where C is the nominal concentration of the solvent. In addition, the behavior of the elastomeric layer is also modeled by a Neo-Hookean constitutive law. Free energy density for the elastomeric part is:
ACCEPTED MANUSCRIPT
W (1) (1(1) , 2(1) , 3(1) )
G (1) (1) 1 2
2
(1) 2
2
(1) 2 3
3 2 ln J (1) (ln( J (1) ))2 ,
(13)
where G (1) is the shear modulus of elasticity of the elastomer and is a Lagrangian multiplier
CR IP T
representing hydrostatic pressure. The principal true stresses for the elastomeric layer (n 1) as an incompressible material ( J (1) 1) and for the hydrogel layer (n 2) can be found from the free energy density as below (Cai and Suo, 2011):
i (2)
W (1) i(1)
(i 1, 2,3),
AN US
i(1) i(1)
i (2) W (2) (i 1, 2,3). J (2) i (2)
(14)
(15)
M
Assuming T0 as the reference temperature, the hydrogel layer has a free-swelling deformation from
ED
(2) the dry state to this reference state with 1(2) (2) 2 3 0 . The reference stretches for the hydrogel
layer can be calculated from the equilibrium of the principal true stresses as: (T ) 1 1 1 N k BT 2 0 2 0 1 k BT 0 2 0 3 0 6 0 9 0 (T ) 1 3 0 3 1 2 2 3 0 (T ) (T ) k BT 03 1 0 4 0 7 0 10 0 (T ) 1 3 0 4 1 7 1 0, T T0 . 0 0
(16)
CE
PT
AC
We consider that the hydrogel and the elastomer layers are attached to each other at the reference temperature and the bilayer beam experiences no bending at this reference point. We study the beam deformation and bending from this reference point at which the bilayer is straight and free-stress. Thus, we should rewrite the Helmholtz free energy density according to this reference state. In this regard, we substitute i by 0 i in the free energy statement of Eq. (10) and divide it by 30 . The altered free
ACCEPTED MANUSCRIPT
energy density is used to furnish a UHYPER code implemented into ABAQUS, a commercial finite element package. Due to symmetry, there is no variations in tangential direction ( r ( n) / 0 ), thus, the equilibrium
r( n ) X 2 r ( n ) (X 2 )
r( n ) ( n ) r ( n ) (X 2 )
0.
X 2
CR IP T
equation for each layer in the polar coordinates is given by:
(17)
AN US
For the elastomeric layer, substituting Eq. (8) into the second term of the balance equation (Eq. (17)) and by integrating this term with respect to X 2 , the principal true stress in the radial direction is computed as a function of X 2 .
r(1) (1) dr (1) (X 2 ) r ( n) (X 2 )
dX 2
M
r(1) ( X 2 )
dX 2 ,
(18)
ED
where is the integration constant that is identified through satisfying the stress-free condition at the inner radius of the elastomeric layer ( r (1) 0 at r (1) (0) ri(1) ) . Thereafter, substituting r(1) (X2 ) into the
PT
equilibrium equation, the principal true stresses of the elastomer are determined as:
AC
CE
r(1) (X 2 )
(1) (X 2 )
1G 8
(1)
32(r
1G 8
(1) 6 5 i )
(1)
16(r
(1) 4 4 i )
L 2
16 LX 2 (ri(1) )2 3 L4 X 2
(ri(1) )2
LX 2 (ri(1) )2
80 LX 2 (ri(1) ) 4 4 48L2 X 22 (ri(1) ) 2 3 2 L4 (ri(1) ) 2 L5 X 2
2
(ri(1) )2
LX 2 L
2
(ri(1) )2
(19)
,
.
(20)
Furthermore, for the hydrogel layer, after substituting the principal stretches (Eq. (6)) into the true principal stresses (Eq. (15)), the balance equation (Eq. (17)) would result into a second order ODE. The appropriate boundary conditions for this ODE are: the stress-free condition at the outer radius of the
ACCEPTED MANUSCRIPT
bilayer ( r (2) 0 at X2 H (1) H (2) ) , and the stress field continuity in the radial direction at the interface of two layers ( r (1) r (2) at X2 H (1) ) . The net force and the moment components in the tangential directions for any cross section of the
H (1)
(1)
0
dr (1) (X 2 ) (X 2 ) dX 2 dX 2
H (1) H (2)
H (1)
H (1)
r
(1)
(X 2 )
0
(1)
dr (2) (X 2 ) dX 2 0, dX 2
AN US
and,
(2) (X 2 )
CR IP T
bilayer must be zero due to its free deformation. Thus, we have:
dr (1) (X 2 ) (X 2 ) dX 2 dX 2
H (1) H (2)
H
(1)
r (2) (X 2 ) (2) (X 2 )
dr (2) (X 2 ) dX 2 0. dX 2
(21)
(22)
Now the governing equations are constructed and should be solved employing an appropriate method.
M
The problem-solving procedure for swelling-induced bending of the bilayer at a given temperature is:
ED
firstly, we guess unknown parameters based on the previous relevant values ( and ri(1) ). Then, the second order ODE (Eq.(17)) is numerically solved in order to find the distribution of r (2) (X2 ) . Finally,
PT
substituting r (2) (X2 ) and r (1) (X2 ) from Eq. (9) into Eqs. (21) and (22), new values of and ri(1) are calculated applying the Newton-Raphson method to Eqs. (21) and (22). These calculated values are used
CE
again as initial guesses for solving the ODE to find the new distribution of r (2) (X2 ) .These iterations
AC
continue until a suitable convergence tolerance for the values of and ri(1) is reached.
3 Results and Discussion In this section, to demonstrate the accuracy and strength of the presented method, we study finite bending of bi-layer beams in some special cases. The experimental constants of the dimensionless parameter ( ; T ) are A0 12.947, B 0 0.04496 K 1, A1 17.92, B1 0.0569 K 1 (Afroze et al., 2000). The
ACCEPTED MANUSCRIPT
volume per water molecule is 3 1029 m3 (Morimoto and Ashida, 2015) and the shear modulus of the elastomeric layer is assumed to be G(1) 2 MPa . The reference stretch ( 0 ) is calculated for various values of the dimensionless crosslink density N at the reference temperature of T0 =320 K . It should be
CR IP T
noted that however the effect of the thickness ratio ( H (2) / H (1) ) on bending of the bilayer beams is noticeable (Nardinocchi and Puntel, 2016), our case studies are limited only to bilayers that have an equal thickness of layers ( H (2) H (1) H / 2) . Since there is no variation in the tangential direction, all results except the semi-angle ( ) are independent of the length-thickness (aspect) ratio L / H
AN US
(Morimoto and Ashida, 2015).
At first, we investigated the relation between the free-swelling of the PNIPAM hydrogels versus the temperature variation. We considered a cube made of the hydrogel which can swell freely in all
M
(2) (2) directions (1(2) (2) 2 3 ) from the reference state to the current state. Applying the stress-free
condition, we can calculate the free-swelling stretch ( (2) ) using Eq. (15). The swelling ratio
2, the hydrogel has a PTT near to T=305 K , and swells dramatically as the
PT
observed form Fig.
ED
( J (2) ( (2) )3 ) for different values of N are calculated as a function of temperature. As it can be
temperature drops below this point. Additionally, the hydrogels with lower values of N have a higher
CE
swelling ratio at the temperatures below PTT. The free-swelling behavior of the gels are widely discussed in the literature (Cai and Suo, 2011; Mazaheri et al., 2015), which are recommended to
AC
readers who seeks for detailed discussions. Fig. 3 shows the distribution of r ( n) (X2 ) calculated from the analytical (ANL) solution and the Finite
Element Method (FEM) for N 0.01,0.005 and 0.001 at T=288K , in which the swelling ratio and the deformation are large with respect to the reference state. Since r ( n) (X2 ) is independent of the aspect ratio, the magnitude of X 2 is only normalized by the thickness ( H ) of bilayers. In the hydrogel layer,
ACCEPTED MANUSCRIPT
the slope of r (2) (X2 ) increases as the N decreases, since the swelling ratio is higher at lower crosslink densities. In the elastomeric layer, the slope of r (1) (X2 ) is independent of the crosslink density of the gel, while the magnitude of r (1) (X2 ) decreases with decreasing of N . This is because of large isochoric
CR IP T
deformation of the elastomeric layer in lower values of N . As shown in Fig. 3, for both the FEM and the presented analytical method, a discontinuity in slope of r ( n) (X2 ) is observed at the interface of the layers for all values of N , due to discontinuity in the bilayer material at this point. At the other hand, the presented method is in a very good agreement with the FEM results.
AN US
For a more precise study, the normalized radial and hoop stresses versus the dimensionless radius (r ( n) ri(1) ) / (h(1) h(2) ) are calculated for N 0.01, 0.005 and 0.001 at T=288K and depicted in Fig. 4.
Radial and hoop stresses are also normalized by kBT / . In the hydrogel layer, the magnitude of the radial and the hoop stresses are low for small values of N . The reason is that a lowly cross-linked
M
network has a higher volume expansion, which leads to a more stress relaxation. Furthermore, the
ED
maximum of the radial stress inside the gel occurs at the layers interface due to confinement of both layers at this point and is shifted to the left as the bending curvature increases. As one may observe
PT
from Fig. 4(a), the extreme bending of the bilayer with low N induces high radial stresses inside the elastomeric layer, while in the hydrogel layer the stress level is small due to soft nature of the low cross-
CE
linked network. Increasing N results in an increase in the stress level in the hydrogel layer and a decrease in the elastomeric one. As it is obvious from Fig. 4(b), there are two neutral axes that each one
AC
appears on one layer. The neutral axis of the gel layer is approximately at (r (2) ri(1) ) / (h(1) h(2) ) 0.7 for all values of N . The number and locations of the neutral axes might be influenced by the geometry or the stiffness ratio of two layers (Morimoto and Ashida, 2015). The hoop stress has opposite maximums near to the interface between each layer. For a more detailed study, the normalized radial and the hoop stress distributions in both layers are calculated and shown in Fig. 5 on the deformed shape, for both
ACCEPTED MANUSCRIPT
the analytical method and the FEM for N 0.005 at T=288K . In this case study, the swelling ratio and the deformation are large with respect to the reference state. As can be seen, despite the large deformations, the analytic method shows a very good agreement with FEM results which validates the presented method successfully.
CR IP T
In the next step, the effect of the temperature changes on the displacement and stress fields are investigated and shown in Fig. 6. In Fig. 6(a), the distribution of r (n) (X2 / H ) are illustrated at T=308K (above PTT), 300 K and 288K (below PTT) for N 0.005 . The bilayer is stress-free without any
AN US
deformation at the reference temperature. As the temperature decreases, the inner and the interfacial radii of the bilayer decrease continuously, while the outer radius decreases and then increases, because of the high swelling ratio of the hydrogel at temperatures below PTT. The normalized radial and hoop stresses versus the dimensionless radius are also depicted in Fig. 6(b) and Fig. 6(c), respectively. For
ED
swelling ratio of the hydrogel.
M
both directions, the magnitude of stresses grows by lowering the temperature due to an increase in the
The distribution of the swelling ratio in the cross-sections of a bilayer with different values of N at
PT
T=288K are computed by the proposed analytical solution and are shown in Fig. 7 . The initial length of
the bilayer and its aspect ratio are L 0.03m and L / H 3 , respectively. As it can be found from Fig.
CE
6(b, c), the bending configuration creates an inhomogeneous stress field inside the bilayer thickness. Since the mechanical stress can affect the swelling, the swelling ratio has different values in the
AC
cross-section. The swelling ratio of the elastomer is equal to unity ( J (1) 1) because of the incompressibility condition for the elastomeric layer. Lower values of N results in higher values of swelling ratio in the hydrogel layer. At the interface, the swelling ratio of the hydrogel ( J (2) ) is minimum since the high magnitude of the radial stress in this area prevents the hydrogel from further swelling. At the outer radius, which the stress-free condition is applied, the swelling ratio has its maximum value.
ACCEPTED MANUSCRIPT
The swelling ratio also grows continuously in the hydrogel layer since the hydrogel become less constrained as we get away from the inner radius to the outer radius. In conclusion, there is a reverse relation between the amount of the swelling ratio and the magnitude of the stress field inside the hydrogel so that the swelling ratio is low in areas with high stress magnitudes and vice versa.
CR IP T
Additionally, the magnitude of the free-swelling (see Fig. 2) is totally higher than the coupled swelling ratio of the hydrogel in the bilayer. The reason is that the constraints that are applied by both the elastomeric layer and the plane strain condition prevent the hydrogel from further swelling.
AN US
An important parameter in the bilayer design is its curvature and the semi-angle ( ) which has a direct impact on its deflections. Thus, the bending curvature ( 1/ ri(1) ) and the semi-angle of bilayers, with the different stiffness of the gel layer, are demonstrated in Fig. 8 versus the temperature changes. As shown, starting from the reference point (T0 =320 K) , as the temperature decreases, the bending
M
curvature and semi-angle start to grow. In the vicinity of the PTT ( 305 K) bilayers experience a sharp phase transition which leads to relatively large curvatures and semi-angles at temperatures below PTT.
ED
The softer gels (low N ) have relatively large curvatures and semi-angles, in comparison to the stiffer
PT
ones, as a result of high swelling ratio. As mentioned earlier, the aspect ratio has almost no influence on the curvatures, however, it affects the bending semi-angle directly and as a result, Fig. 8(a) is applicable
CE
for all values of L / H , while Fig. 8(b) is only valid for the aspect ratio of L / H 3 . Thus, for investigating the effect of the bilayer aspect ratio, the bending semi-angle for N 0.005 at
AC
aspect ratios of L / H 1, 2, 3 and 4 are calculated by the proposed analytical method and shown in Fig. 9(a). As shown, at higher values of L / H , the bending semi-angle is longer. For L / H 4 , the bilayer bends excessively at low temperatures; therefore, the two ends of the bilayer pass each other theoretically which is a key parameter in the bilayer design procedure. However, this is a virtual state since, in reality, the beam ends contact each other, resulting in smaller curvature. Additionally, the
ACCEPTED MANUSCRIPT
increasing of the bending semi-angle by L / H is approximately linear at a special temperature and N . For instance, the distribution of the bending semi-angle at different L / H for N .01, 0.005 and 0.001 at T=288 K are shown in Fig. 9(b). Both of Increasing L / H and decreasing in N result in increasing the bending semi-angle. Similar to the previous statement, at high values of L / H , the bending semi-
CR IP T
angle is greater than , where the two ends of the bilayer pass each other theoretically.
4 Summary and Conclusions
In this work, an analytical method was developed to investigate the finite bending of a bilayer
AN US
composed of a temperature-responsive gel under plane-strain condition. The bilayer is made of two attached layers: one layer is made of a neutral incompressible elastomer and the other layer is made of a temperature sensitive hydrogel. The bilayer bends in response to the environment temperature variations due to the existence of the responsive hydrogel layer. A novel analytical method is presented
M
to investigate the bilayer bending problem. Considering the free energy density and the balance
ED
equations, a second order ODE coupled with two algebraic equations is reached. The governing equations were solved employing iterative numerical techniques, where the unknown parameters were
PT
determined in terms of the temperature changes. On the other hand, a UHYPER code was developed to be implemented in ABAQUS in order to simulate the finite bending problem by FEM and validate the
CE
presented analytical proposed method. Due to a very good agreement between the presented method
AC
results and the FEM results the validity of the method was confirmed. Influence of the crosslink density on the distribution of the displacements and the stresses fields was
studied. Lowering the temperature, the radial and the hoop stresses started to grow. Decreasing the crosslink density leads to relatively large curvatures and semi-angles. At higher values of aspect ratio, larger bending semi-angles were predicted. Based on the perfect agreement of the analytical and the FEM results, it was concluded that the presented method is capable of predicting the swelling induced
ACCEPTED MANUSCRIPT
finite bending of bilayers even at below the PTT, where the gel layer experiences extreme volume changes. This method can be useful for analysis and design of bilayer sensors and actuators in various applications such as thermal sensors, self-folding structures and drug delivery systems.
CR IP T
References
AC
CE
PT
ED
M
AN US
Afroze, F., Nies, E., Berghmans, H., 2000. Phase transitions in the system poly(Nisopropylacrylamide)/water and swelling behaviour of the corresponding networks. J. Mol. Struct. 554, 55-68. Agrawal, A., Yun, T., Pesek, S.L., Chapman, W.G., Verduzco, R., 2014. Shape-responsive liquid crystal elastomer bilayers. Soft Matter 10, 1411-1415. Ahn, S.-k., Kasi, R.M., Kim, S.-C., Sharma, N., Zhou, Y., 2008. Stimuli-responsive polymer gels. Soft Matter 4, 1151-1157. Bäcker, M., Raue, M., Schusser, S., Jeitner, C., Breuer, L., Wagner, P., Poghossian, A., Förster, A., Mang, T., Schöning, M.J., 2012. Microfluidic chip with integrated microvalves based on temperature- and pH-responsive hydrogel thin films. Phys. Status Solidi A 209, 839-845. Cai, S., Suo, Z., 2011. Mechanics and chemical thermodynamics of phase transition in temperature-sensitive hydrogels. J Mech Phys Solids 59, 2259-2278. Chester, S.A., Anand, L., 2011. A thermo-mechanically coupled theory for fluid permeation in elastomeric materials: application to thermally responsive gels. J. Mech. Ph. Solids 59, 1978-2006. Domachuk, P., Tsioris, K., Omenetto, F.G., Kaplan, D.L., 2010. Bio‐microfluidics: Biomaterials and Biomimetic Designs. Adv. Mater. 22, 249-260. Drozdov, A.D., 2015. Swelling of pH-responsive cationic gels: Constitutive modeling and structure–property relations. Int. J. Solids Struct. 64–65, 176-190. Fernandes, R., Gracias, D.H., 2012. Self-folding polymeric containers for encapsulation and delivery of drugs. Adv. Drug Delivery Rev. 64, 1579-1589. Flory, P.J., Rehner Jr, J., 1943. Statistical mechanics of cross-linked polymer networks II. Swelling. J. Chem. Phys. 11, 521-526. Guenther, M., Gerlach, G., Wallmersperger, T., Avula, M.N., Cho, S.H., Xie, X., Devener, B., Solzbacher, F., Tathireddy, P., Magda, J.J., 2013. Smart hydrogel-based biochemical microsensor array for medical diagnostics, Adv. Sci. Tech. Trans Tech Publ, pp. 47-52. He, H., Guan, J., Lee, J.L., 2006. An oral delivery device based on self-folding hydrogels. J. Controlled Release 110, 339-346. He, T., Li, M., Zhou, J., 2012. Modeling deformation and contacts of pH sensitive hydrogels for microfluidic flow control. Soft Matter 8, 3083. Hong, W., Liu, Z., Suo, Z., 2009. Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load. Int. J. Solids Struct. 46, 3282-3289. Hong, W., Zhao, X., Zhou, J., Suo, Z., 2008. A theory of coupled diffusion and large deformation in polymeric gels. J. Mech. Ph. Solids 56, 1779-1793. Ionov, L., 2013. Biomimetic Hydrogel-Based Actuating Systems. Adv. Funct. Mater. 23, 4555-4570.
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Jones, D.S., Lorimer, C.P., McCoy, C.P., Gorman, S.P., 2008. Characterization of the physicochemical, antimicrobial, and drug release properties of thermoresponsive hydrogel copolymers designed for medical device applications. J. Biomed. Mater. Res. Part B 85, 417426. Kwon, G.H., Jeong, G.S., Park, J.Y., Moon, J.H., Lee, S.-H., 2011. A low-energy-consumption electroactive valveless hydrogel micropump for long-term biomedical applications. Lab Chip 11, 2910-2915. Kwon, G.H., Park, J.Y., Kim, J.Y., Frisk, M.L., Beebe, D.J., Lee, S.H., 2008. Biomimetic soft multifunctional miniature aquabots. J Small. 4, 2148-2153. Lai, F., Li, H., Luo, R., 2010. Chemo-electro-mechanical modeling of ionic-strength-sensitive hydrogel: Influence of Young’s modulus. Int. J. Solids Struct. 47, 3141-3149. Li, H., 2009. Kinetics of smart hydrogels responding to electric field: A transient deformation analysis. Int. J. Solids Struct. 46, 1326-1333. Lucantonio, A., Nardinocchi, P., Pezzulla, M., 2014. Swelling-induced and controlled curving in layered gel beams, P. Roy. Soc. A-Math. Phy. The Royal Society, p. 20140467. Marcombe, R., Cai, S., Hong, W., 2010. A theory of constrained swelling of a pH-sensitive hydrogel Soft Matter 6, 784-793. Mazaheri, H., Baghani, M., Naghdabadi, R., 2015. Inhomogeneous and homogeneous swelling behavior of temperature-sensitive poly-(N-isopropylacrylamide) hydrogels. J. Intel. Mat. Syst. Str., 1045389X15571381. Mazaheri, H., Naghdabadi, R., Baghani, M., sohrabpour, S., 2016. pH/temperature sensitive hydrogels: Coupling behavior of the pH/temperature sensitive hydrogels for the inhomogeneous and homogeneous swelling. under revision in "Smart Mater. Struct.". Morimoto, T., Ashida, F., 2015. Temperature-responsive bending of a bilayer gel. Int. J. Solids Struct. Nardinocchi, P., Puntel, E., 2016. Finite bending solutions for layered gel beams. Int. J. Solids Struct. Randall, C.L., Gultepe, E., Gracias, D.H., 2012. Self-folding devices and materials for biomedical applications. Trends Biotechnol. 30, 138-146. Richter, A., 2009. Hydrogels for actuators, Hydrogel sensors and actuators. Springer, pp. 221-248. Richter, A., Klatt, S., Paschew, G., Klenke, C., 2009. Micropumps operated by swelling and shrinking of temperature-sensitive hydrogels. Lab Chip 9, 613-618. Roccabianca, S., Gei, M., Bigoni, D., 2010. Plane strain bifurcations of elastic layered structures subject to finite bending: theory versus experiments. IMA J. Appl. Math., hxq020. Shin, J., Braun, P.V., Lee, W., 2010. Fast response photonic crystal pH sensor based on templated photo-polymerized hydrogel inverse opal. "Sens. Actuators, B " 150, 183-190. Swann, J.M., Ryan, A.J., 2009. Chemical actuation in responsive hydrogels. Polym. Int. 58, 285-289. Timoshenko, S., 1925. Analysis of bi-metal thermostats. JOSA 11, 233-255. Toh, W., Ng, T.Y., Hu, J., Liu, Z., 2014. Mechanics of inhomogeneous large deformation of photo-thermal sensitive hydrogels. Int. J. Solids Struct. 51, 4440-4451. Yan, H., Jin, B., Gao, S., Chen, L., 2014. Equilibrium swelling and electrochemistry of polyampholytic pH-sensitive hydrogel. Int. J. Solids Struct. 51, 4149-4156. Yu, Q., Bauer, J.M., Moore, J.S., Beebe, D.J., 2001. Responsive biomimetic hydrogel valve for microfluidics. Appl. Phys. Lett. 78, 2589.
ACCEPTED MANUSCRIPT
AN US
CR IP T
Zhang, C., Xing, D., Li, Y., 2007. Micropumps, microvalves, and micromixers within PCR microfluidic chips: Advances and trends. Biotechnol Adv 25, 483-514. Zhang, J., Wu, J., Sun, J., Zhou, Q., 2012. Temperature-sensitive bending of bigel strip bonded by macroscopic molecular recognition. Soft Matter 8, 5750-5752. Zhou, X., Hon, Y., Sun, S., Mak, A., 2002. Numerical simulation of the steady-state deformation of a smart hydrogel under an external electric field. Smart Mater. Struct. 11, 459.
AC
CE
PT
ED
M
Fig. 1. Hydrogel bilayer geometry schematics: a) the initial configuration and b) the deformed configuration.
Fig. 2. The free-swelling ratio of the Hydrogel with N 0.01,0.005 and 0.001 versus temperature.
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
Fig. 3. FEM and analytical (ANL) results for the radius distribution for N 0.01,0.005 and 0.001 at T=288K .
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
Fig. 4. Normalized a) radial stress and b) hoop stress distributions calculated from the analytical (ANL) and FEM solution for N 0.01, 0.005 and 0.001 .
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
Fig. 5. The radial and the hoop stress distributions on the deformed bilayer at T=288 K with N 0.005. a) FEM radial stress, b) Analytical radial stress, c) FEM hoop stress and d) Analytical hoop stress.
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
ACCEPTED MANUSCRIPT
AN US
CR IP T
Fig. 6. a) Radius, b) normalized radial stress and, b) normalized hoop stress distributions at T=308, 300 and 288K for N 0.005 computed by the proposed analytical method.
AC
CE
PT
ED
M
Fig. 7. Distribution of the swelling ratio in the cross-section for the bilayer with different values of N at T 288K .
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Fig. 8. a) Bending curvature and b) bending semi-angle of bilayers with N 0.01,0.005 and 0.001 and L / H 3 computed by the proposed analytical method.
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Fig. 9. a) Bending semi-angle of bilayers with L / H 1, 2, 3 and 4 for N 0.005 .b) Variation of the bending semi-angle of bilayers at different L / H for N 0.01, 0.005 and 0.001 at T=288K .
Analytical and numerical analysis of swelling-induced large bending of thermally-activated hydrogel bilayers Jalal Abdolahi , Mostafa Baghani , Nasser Arbabi , Hashem Mazaheri PII: DOI: Reference:
S0020-7683(16)30234-7 10.1016/j.ijsolstr.2016.08.017 SAS 9277
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
11 May 2016 25 July 2016 22 August 2016
Please cite this article as: Jalal Abdolahi , Mostafa Baghani , Nasser Arbabi , Hashem Mazaheri , Analytical and numerical analysis of swelling-induced large bending of thermally-activated hydrogel bilayers, International Journal of Solids and Structures (2016), doi: 10.1016/j.ijsolstr.2016.08.017
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
CR IP T
Analytical and numerical analysis of swelling-induced large bending of thermally-activated hydrogel bilayers Jalal Abdolahia, Mostafa Baghani*a, Nasser Arbabia and Hashem Mazaherib a
School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran. b
Department of Mechanical Engineering, Bu-ali Sina University, Hamedan, Iran.
AN US
*Corresponding Author Email: [email protected]
Abstract
Temperature-sensitive hydrogels have recently been implemented vastly for biomedical and microfluidic sensors and actuators. The accurate and efficient design of the bilayer sensors and actuators made of
M
temperature sensitive hydrogels are of crucial importance. In this work, we develop an analytical
ED
method to solve the swelling induced bending of temperature responsive hydrogel bilayer under plane strain condition. The bilayer consists of a neutral incompressible elastomer layer attached to a
PT
temperature sensitive hydrogel layer. An analytical approach is developed to predict the thermomechanical response of these bilayers. At the other hand, the finite bending of the bilayer is
CE
simulated applying the finite element method. Several cases are solved to demonstrate the validity and performance of the proposed analytical method. The deformation and the stresses inside the layers are
AC
presented for various material parameters employing both the developed analytical formulation as well as the finite element method. A good correspondence between the presented method and the finite element method is observed. Finally, the effect of material and geometrical parameters on curvature are also investigated.
ACCEPTED MANUSCRIPT
Keywords: Bilayers, Thermally-activated hydrogels, Finite bending, Analytical solution, Finite element method.
1 Introduction
CR IP T
Smart hydrogels are lightly cross-linked polymer chains that have capability to undergo large deformation in response to various external stimuli such as temperature (Cai and Suo, 2011; Mazaheri et al., 2015; Morimoto and Ashida, 2015), pH (Drozdov, 2015; He et al., 2012; Hong et al., 2009; Marcombe et al., 2010; Mazaheri et al., 2016; Yan et al., 2014), light (Ahn et al., 2008; Toh et al., 2014), electric field
AN US
(Lai et al., 2010; Li, 2009; Zhou et al., 2002) and etc. This feature makes smart hydrogels an ideal material for fabrication of both sensors and actuators (Ionov, 2013; Richter, 2009; Shin et al., 2010; Yu et al., 2001). Smart hydrogel sensors and actuators have a promising future for application in microfluidics (Bäcker et al., 2012; Domachuk et al., 2010; Richter et al., 2009; Zhang et al., 2007), biomedical devices
M
(Guenther et al., 2013; Jones et al., 2008) and soft robotics (Kwon et al., 2008).
ED
Bilayer cantilever beams have been served as sensors and actuators for many decades. Classical bilayer sensors mostly were made of two strips of metals (bi-metal beam) with different thermal
PT
expansion coefficients (Timoshenko, 1925). Bi-metal sensors and actuators are not biocompatible due to their weak corrosion resistance in the aqueous environment. An alternative is a bilayer composed of
CE
smart hydrogels. A strip made of smart hydrogels can be attached to one layer of neutral elastomer or
AC
hydrogels with different swelling characteristics to form a highly biocompatible bilayer sensor or actuator (Agrawal et al., 2014; Swann and Ryan, 2009; Zhang et al., 2012). Hydrogel bilayer beams are able to deform noticeably in comparison with classic bi-metals, results in higher sensitivity and accuracy, and also makes them suitable to be implemented in self-folding structures. These structures have various applications in drug-delivery systems (Fernandes and Gracias, 2012; He et al., 2006; Randall et
ACCEPTED MANUSCRIPT
al., 2012). Bilayer hydrogel beams also are used as micro-valve and micro-pump for biomedical applications (Kwon et al., 2011; Yu et al., 2001). There are several theories describing the swelling behavior of thermally sensitive hydrogels. Chester and Anand (2011) developed a coupled mechanical deformation, fluid permeation and heat transfer
CR IP T
theory describing the swelling behavior of temperature sensitive hydrogels. Considering the gel to be isotropic, they developed a model based on Flory-Huggins mixing energy. Cai and Suo (2011) proposed a theory for temperature sensitive N-isopropylacrylamide hydrogels (PNIPAM). Adopting Flory-Rehner model (Flory and Rehner Jr, 1943) and defining the interaction parameter as a function of temperature
AN US
and polymer volume fraction, they developed mathematical framework describing phase transition phenomenon in PNIPAM hydrogels. Aforementioned models showed snap-through instability and multiple solutions near to the Phase Transition Temperature (PTT), which is a restriction for these
M
models to be implemented into finite element (FE) frameworks. Mazaheri et al. (2015) solved the socalled discontinuity and instability problems through modifying the mixing energy for PNIPAM
ED
hydrogels. This theory predicted the experimental result accurately without showing serious discontinuity in the vicinity of PTT.
PT
The advantages of the analytical solutions have encouraged the researchers to develop mathematical
CE
methods to solve the finite bending of the polymeric bilayers including the hydrogel bilayers. Roccabianca et al. (2010) analytically solved the finite bending of a multilayered elastic incompressible
AC
thick plate under plane strain condition. They also performed experiments to confirm the theoretical results. Lucantonio et al. (2014) studied the swelling-induced bending of layered gel beams, based on the assumption of splitting the swelling-induced deformation gradient into a homogeneous swelling component and a pure bending component. They have determined stretch and bending curvature of beams, depending on the material and geometrical characteristics. Morimoto and Ashida (2015) proposed a new approach to investigate the finite plane strain bending of a bilayer gel made of two gel
ACCEPTED MANUSCRIPT
layers with different swelling properties. They assumed that swelling-layer first undergoes a virtual onedimensional homogeneous swelling. Then, the non-swelling homogeneous pure bending balance all forces and moments on the gel bilayer. They compared the method predictions with the linear model based on the elementary beam theory. Recently, Nardinocchi and Puntel (2016) presented a method to
CR IP T
solve swelling-induced bending of layered gels based on the basic assumption of multiplicative decomposition of the deformation gradient tensor. Based on this assumption, the deformation gradient tensor split into two components of elastic deformation and elastic distortion. They also compared outcomes of their method with numerical results in the literature.
AN US
In this paper, a new analytical approach is presented to investigate the finite bending of bilayer gel beams under plane strain condition. In particular, we study bilayers composed of a neutral incompressible elastomeric layer and a PNIPAM hydrogel layer. The bilayer is assumed flat in the
M
reference (initial) state and experiences a swelling induced bending due to the temperature changes in the responsive layer. We employ a deformation gradient tensor which maps the initial state to the
ED
deformed configuration without assuming any virtual intermediate state. For describing the swelling behavior of the PNIPAM hydrogel, a modified model of the Helmholtz free energy density that proposed
PT
by Mazaheri et al. (2015) is used. Also, a UHYPER code is developed and implemented in ABAQUS for
CE
Finite Element (FE) modeling of the bilayer to numerically validate the proposed analytical model predictions. Finally, we studied the effects of some geometrical and material parameters on the bilayer
AC
configuration and the stresses distribution inside the layers. The article is organized as follow: firstly, the proposed analytical method for the plane strain
swelling-induced bending of the bilayer beam is fully explained in section 2. Then, a brief discussion on the swelling theory and the Helmholtz free energy density is presented. Thereafter, based on the swelling theory the governing equations of the system are derived. Later in section 3, we set up some case studies in order to demonstrate the accuracy and robustness of the presented method. The results
ACCEPTED MANUSCRIPT
are validated through comparing them with FEM results. Finally, we present a summary and draw conclusions in section 4.
2 Bending deformation of thermally activated hydrogel bilayers
CR IP T
In this work, a bilayer beam made of two attached layers is studied. One layer is made of a neutral incompressible elastomer (indexed by n=1) and the other is made of temperature sensitive poly-(N-isopropylacrylamide) (PNIPAM) hydrogel (indexed by n=2). Schematics of the bi-strip beam both the
reference
(flat)
and
current
(curved)
states
are
shown
in
ED
M
AN US
in
PT
Fig. 1. Finite bending of the bilayer is triggered only by the temperature variation due to an increase in the swelling of the hydrogel layer. The out of plane dimension is assumed to be infinite and the
CE
problem is formulated as plane strain. Moreover, it is assumed that the bilayer cross sections remain plane after the deformation, and the end-effects are neglected in both analytic and FE analyses.
AC
Introducing X (X1 ,X2 ,X3 ) as a reference Cartesian coordinates for the bilayer, the position of a
generic point in the reference state is given by: X X1 e1 X2 e2 X3 e3 ,
with
(1)
ACCEPTED MANUSCRIPT
L L X1 [ , ], X 2 [0, H (1) H (2) ], X3 ( , ). 2 2
(2)
where ei (i 1, 2,3) are the axes of the Cartesian coordinate, H ( n) and L are initial thicknesses and length of each layer in the undeformed configuration, respectively. It is evident that the total initial thickness of
CR IP T
the bilayer is H H (1) H (2) . In addition, the deformed configuration in the current state is considered as a portion of a cylindrical tube with the semi-angle of . Introducing x (r, , z) as current polar coordinates for each layer, the position of the generic point x( n) in the deformed state is given by:
AN US
x( n) r (n) er ( n) eθ z ( n) ez ,
with
r (1) [ri(1) , ri(1) h(1) ], r (2) [ri(2) , ri(2) h(2) ], ( n) [ , ],
(4)
M
z ( n) ( , ).
(3)
in which ei (i r, , z) are the axes of the polar coordinate, h( n) and ri ( n) are the thickness and the inner
ED
radius of each layer in the deformed configuration, while ri(2) ri(1) h(1) .
PT
Assuming r (n) r ( n) (X2 ) , (n) ( n) (X1 ) and z ( n) X3 , the deformation gradient can be found as
AC
CE
(Roccabianca et al., 2010)
F( n)
dr ( n) (X 2 ) dX 2
er e2 r ( n)
d ( n) dX1
e e1 e z e3 ,
(5)
in which is the dyadic operator. Eq. (5) also defines the principal stretches for each layer to be:
r( n)
dr ( n) (X 2 ) ( n) d ( n) ( n) , r ( n) , z 1. dX 2 dX1
(6)
ACCEPTED MANUSCRIPT
Assuming a linear distribution form for the tangential component of the position vector in the polar coordinates as (n) 2 X1 / L , the principal stretch in tangential direction for each layer is:
( n)
2 ( n) r ( X 2 ). L
(7)
r(1)
L 2 r ( X 2 ) (1)
.
CR IP T
In light of the incompressibility condition for the elastomeric layer, its radial principal stretch is recast as:
(8)
AN US
The radial stretch for the elastomeric layer that is found by the deformation gradient (Eq. (6)) should be equal to Eq. (8) (dr (1) (X2 ) / dX2 L / 2 r (1) (X2 )) . This forms a first order Ordinary Differential Equation (ODE) with a boundary condition (r (1) (0) ri(1) ) at X2 0 . Solving this ODE, the distribution of r (1) (X2 ) is
M
calculated as:
L
X 2 (ri(1) )2 ,
(9)
ED
r (1) ( X 2 )
where and ri(1) are kinematic unknowns.
PT
For modeling the swelling behavior of the temperature-sensitive PNIPAM hydrogel, a modified model
CE
of the Helmholtz free energy density recently proposed by Mazaheri et al. (2015) is employed. This modified energy is more stable in the vicinity of the phase transition temperature (PTT), which results in
AC
numerical instability elimination besides the accurate prediction of the hydrogel swelling behavior. This modified model of the free energy is convex, which made it applicable to the FEA. The modified free energy density is (Mazaheri et al., 2015):
ACCEPTED MANUSCRIPT
W (2) (1(2) , 2(2) , 3(2) ; T )
G (2) (2) 1 2
2
(2) 2
2
(2) 2 3
3 2 ln J (2)
1 k BT (2) 1 1 ( ; T ) ( J 1) (2) , (2) 2 (2) 3 v 2( J ) 3( J ) J (2) J
(10)
in which W (2) is free energy density for the hydrogel. The first term on the right-hand side of Eq. (10) is
CR IP T
the Neo-Hookean hyperelastic energy that describes the elastic deformation of the polymer network and the second term stands for the mixing part of the free energy density. G (2) is the shear modulus of elasticity of the hydrogel and is equal to G(2) N kBT , in which k B is Boltzmann’s constant is the density of the polymer chains. In Eq. (10), i (n) (i 1, 2,3) are
AN US
(kB 1.38 1023 J / K ) and N
principal stretches and J (n) 1(n) (2n) 3( n) is the determinant of the deformation gradient tensor, which is related to the volume changes for each layer. Also, v and T are the volume of a single solvent molecule and the absolute temperature, respectively. The dimensionless mixing parameter ( ; T ) is determined
M
via the following relations (Afroze et al., 2000):
(11)
ED
( ; T ) 0 1 , 0 A0 B0T , 1 A1 B1T ,
PT
where 0 and 1 are functions of the temperature and other experimental constants ( A0 , B0 , A1 and B1 ) , is the volume fraction of the polymer in the swollen state. Assuming that the swelling of the hydrogel
CE
only originates from the incompressible fluid diffusion through the network, can be found as (Hong et
AC
al., 2008):
1 C
1
J (2) ,
(12)
where C is the nominal concentration of the solvent. In addition, the behavior of the elastomeric layer is also modeled by a Neo-Hookean constitutive law. Free energy density for the elastomeric part is:
ACCEPTED MANUSCRIPT
W (1) (1(1) , 2(1) , 3(1) )
G (1) (1) 1 2
2
(1) 2
2
(1) 2 3
3 2 ln J (1) (ln( J (1) ))2 ,
(13)
where G (1) is the shear modulus of elasticity of the elastomer and is a Lagrangian multiplier
CR IP T
representing hydrostatic pressure. The principal true stresses for the elastomeric layer (n 1) as an incompressible material ( J (1) 1) and for the hydrogel layer (n 2) can be found from the free energy density as below (Cai and Suo, 2011):
i (2)
W (1) i(1)
(i 1, 2,3),
AN US
i(1) i(1)
i (2) W (2) (i 1, 2,3). J (2) i (2)
(14)
(15)
M
Assuming T0 as the reference temperature, the hydrogel layer has a free-swelling deformation from
ED
(2) the dry state to this reference state with 1(2) (2) 2 3 0 . The reference stretches for the hydrogel
layer can be calculated from the equilibrium of the principal true stresses as: (T ) 1 1 1 N k BT 2 0 2 0 1 k BT 0 2 0 3 0 6 0 9 0 (T ) 1 3 0 3 1 2 2 3 0 (T ) (T ) k BT 03 1 0 4 0 7 0 10 0 (T ) 1 3 0 4 1 7 1 0, T T0 . 0 0
(16)
CE
PT
AC
We consider that the hydrogel and the elastomer layers are attached to each other at the reference temperature and the bilayer beam experiences no bending at this reference point. We study the beam deformation and bending from this reference point at which the bilayer is straight and free-stress. Thus, we should rewrite the Helmholtz free energy density according to this reference state. In this regard, we substitute i by 0 i in the free energy statement of Eq. (10) and divide it by 30 . The altered free
ACCEPTED MANUSCRIPT
energy density is used to furnish a UHYPER code implemented into ABAQUS, a commercial finite element package. Due to symmetry, there is no variations in tangential direction ( r ( n) / 0 ), thus, the equilibrium
r( n ) X 2 r ( n ) (X 2 )
r( n ) ( n ) r ( n ) (X 2 )
0.
X 2
CR IP T
equation for each layer in the polar coordinates is given by:
(17)
AN US
For the elastomeric layer, substituting Eq. (8) into the second term of the balance equation (Eq. (17)) and by integrating this term with respect to X 2 , the principal true stress in the radial direction is computed as a function of X 2 .
r(1) (1) dr (1) (X 2 ) r ( n) (X 2 )
dX 2
M
r(1) ( X 2 )
dX 2 ,
(18)
ED
where is the integration constant that is identified through satisfying the stress-free condition at the inner radius of the elastomeric layer ( r (1) 0 at r (1) (0) ri(1) ) . Thereafter, substituting r(1) (X2 ) into the
PT
equilibrium equation, the principal true stresses of the elastomer are determined as:
AC
CE
r(1) (X 2 )
(1) (X 2 )
1G 8
(1)
32(r
1G 8
(1) 6 5 i )
(1)
16(r
(1) 4 4 i )
L 2
16 LX 2 (ri(1) )2 3 L4 X 2
(ri(1) )2
LX 2 (ri(1) )2
80 LX 2 (ri(1) ) 4 4 48L2 X 22 (ri(1) ) 2 3 2 L4 (ri(1) ) 2 L5 X 2
2
(ri(1) )2
LX 2 L
2
(ri(1) )2
(19)
,
.
(20)
Furthermore, for the hydrogel layer, after substituting the principal stretches (Eq. (6)) into the true principal stresses (Eq. (15)), the balance equation (Eq. (17)) would result into a second order ODE. The appropriate boundary conditions for this ODE are: the stress-free condition at the outer radius of the
ACCEPTED MANUSCRIPT
bilayer ( r (2) 0 at X2 H (1) H (2) ) , and the stress field continuity in the radial direction at the interface of two layers ( r (1) r (2) at X2 H (1) ) . The net force and the moment components in the tangential directions for any cross section of the
H (1)
(1)
0
dr (1) (X 2 ) (X 2 ) dX 2 dX 2
H (1) H (2)
H (1)
H (1)
r
(1)
(X 2 )
0
(1)
dr (2) (X 2 ) dX 2 0, dX 2
AN US
and,
(2) (X 2 )
CR IP T
bilayer must be zero due to its free deformation. Thus, we have:
dr (1) (X 2 ) (X 2 ) dX 2 dX 2
H (1) H (2)
H
(1)
r (2) (X 2 ) (2) (X 2 )
dr (2) (X 2 ) dX 2 0. dX 2
(21)
(22)
Now the governing equations are constructed and should be solved employing an appropriate method.
M
The problem-solving procedure for swelling-induced bending of the bilayer at a given temperature is:
ED
firstly, we guess unknown parameters based on the previous relevant values ( and ri(1) ). Then, the second order ODE (Eq.(17)) is numerically solved in order to find the distribution of r (2) (X2 ) . Finally,
PT
substituting r (2) (X2 ) and r (1) (X2 ) from Eq. (9) into Eqs. (21) and (22), new values of and ri(1) are calculated applying the Newton-Raphson method to Eqs. (21) and (22). These calculated values are used
CE
again as initial guesses for solving the ODE to find the new distribution of r (2) (X2 ) .These iterations
AC
continue until a suitable convergence tolerance for the values of and ri(1) is reached.
3 Results and Discussion In this section, to demonstrate the accuracy and strength of the presented method, we study finite bending of bi-layer beams in some special cases. The experimental constants of the dimensionless parameter ( ; T ) are A0 12.947, B 0 0.04496 K 1, A1 17.92, B1 0.0569 K 1 (Afroze et al., 2000). The
ACCEPTED MANUSCRIPT
volume per water molecule is 3 1029 m3 (Morimoto and Ashida, 2015) and the shear modulus of the elastomeric layer is assumed to be G(1) 2 MPa . The reference stretch ( 0 ) is calculated for various values of the dimensionless crosslink density N at the reference temperature of T0 =320 K . It should be
CR IP T
noted that however the effect of the thickness ratio ( H (2) / H (1) ) on bending of the bilayer beams is noticeable (Nardinocchi and Puntel, 2016), our case studies are limited only to bilayers that have an equal thickness of layers ( H (2) H (1) H / 2) . Since there is no variation in the tangential direction, all results except the semi-angle ( ) are independent of the length-thickness (aspect) ratio L / H
AN US
(Morimoto and Ashida, 2015).
At first, we investigated the relation between the free-swelling of the PNIPAM hydrogels versus the temperature variation. We considered a cube made of the hydrogel which can swell freely in all
M
(2) (2) directions (1(2) (2) 2 3 ) from the reference state to the current state. Applying the stress-free
condition, we can calculate the free-swelling stretch ( (2) ) using Eq. (15). The swelling ratio
2, the hydrogel has a PTT near to T=305 K , and swells dramatically as the
PT
observed form Fig.
ED
( J (2) ( (2) )3 ) for different values of N are calculated as a function of temperature. As it can be
temperature drops below this point. Additionally, the hydrogels with lower values of N have a higher
CE
swelling ratio at the temperatures below PTT. The free-swelling behavior of the gels are widely discussed in the literature (Cai and Suo, 2011; Mazaheri et al., 2015), which are recommended to
AC
readers who seeks for detailed discussions. Fig. 3 shows the distribution of r ( n) (X2 ) calculated from the analytical (ANL) solution and the Finite
Element Method (FEM) for N 0.01,0.005 and 0.001 at T=288K , in which the swelling ratio and the deformation are large with respect to the reference state. Since r ( n) (X2 ) is independent of the aspect ratio, the magnitude of X 2 is only normalized by the thickness ( H ) of bilayers. In the hydrogel layer,
ACCEPTED MANUSCRIPT
the slope of r (2) (X2 ) increases as the N decreases, since the swelling ratio is higher at lower crosslink densities. In the elastomeric layer, the slope of r (1) (X2 ) is independent of the crosslink density of the gel, while the magnitude of r (1) (X2 ) decreases with decreasing of N . This is because of large isochoric
CR IP T
deformation of the elastomeric layer in lower values of N . As shown in Fig. 3, for both the FEM and the presented analytical method, a discontinuity in slope of r ( n) (X2 ) is observed at the interface of the layers for all values of N , due to discontinuity in the bilayer material at this point. At the other hand, the presented method is in a very good agreement with the FEM results.
AN US
For a more precise study, the normalized radial and hoop stresses versus the dimensionless radius (r ( n) ri(1) ) / (h(1) h(2) ) are calculated for N 0.01, 0.005 and 0.001 at T=288K and depicted in Fig. 4.
Radial and hoop stresses are also normalized by kBT / . In the hydrogel layer, the magnitude of the radial and the hoop stresses are low for small values of N . The reason is that a lowly cross-linked
M
network has a higher volume expansion, which leads to a more stress relaxation. Furthermore, the
ED
maximum of the radial stress inside the gel occurs at the layers interface due to confinement of both layers at this point and is shifted to the left as the bending curvature increases. As one may observe
PT
from Fig. 4(a), the extreme bending of the bilayer with low N induces high radial stresses inside the elastomeric layer, while in the hydrogel layer the stress level is small due to soft nature of the low cross-
CE
linked network. Increasing N results in an increase in the stress level in the hydrogel layer and a decrease in the elastomeric one. As it is obvious from Fig. 4(b), there are two neutral axes that each one
AC
appears on one layer. The neutral axis of the gel layer is approximately at (r (2) ri(1) ) / (h(1) h(2) ) 0.7 for all values of N . The number and locations of the neutral axes might be influenced by the geometry or the stiffness ratio of two layers (Morimoto and Ashida, 2015). The hoop stress has opposite maximums near to the interface between each layer. For a more detailed study, the normalized radial and the hoop stress distributions in both layers are calculated and shown in Fig. 5 on the deformed shape, for both
ACCEPTED MANUSCRIPT
the analytical method and the FEM for N 0.005 at T=288K . In this case study, the swelling ratio and the deformation are large with respect to the reference state. As can be seen, despite the large deformations, the analytic method shows a very good agreement with FEM results which validates the presented method successfully.
CR IP T
In the next step, the effect of the temperature changes on the displacement and stress fields are investigated and shown in Fig. 6. In Fig. 6(a), the distribution of r (n) (X2 / H ) are illustrated at T=308K (above PTT), 300 K and 288K (below PTT) for N 0.005 . The bilayer is stress-free without any
AN US
deformation at the reference temperature. As the temperature decreases, the inner and the interfacial radii of the bilayer decrease continuously, while the outer radius decreases and then increases, because of the high swelling ratio of the hydrogel at temperatures below PTT. The normalized radial and hoop stresses versus the dimensionless radius are also depicted in Fig. 6(b) and Fig. 6(c), respectively. For
ED
swelling ratio of the hydrogel.
M
both directions, the magnitude of stresses grows by lowering the temperature due to an increase in the
The distribution of the swelling ratio in the cross-sections of a bilayer with different values of N at
PT
T=288K are computed by the proposed analytical solution and are shown in Fig. 7 . The initial length of
the bilayer and its aspect ratio are L 0.03m and L / H 3 , respectively. As it can be found from Fig.
CE
6(b, c), the bending configuration creates an inhomogeneous stress field inside the bilayer thickness. Since the mechanical stress can affect the swelling, the swelling ratio has different values in the
AC
cross-section. The swelling ratio of the elastomer is equal to unity ( J (1) 1) because of the incompressibility condition for the elastomeric layer. Lower values of N results in higher values of swelling ratio in the hydrogel layer. At the interface, the swelling ratio of the hydrogel ( J (2) ) is minimum since the high magnitude of the radial stress in this area prevents the hydrogel from further swelling. At the outer radius, which the stress-free condition is applied, the swelling ratio has its maximum value.
ACCEPTED MANUSCRIPT
The swelling ratio also grows continuously in the hydrogel layer since the hydrogel become less constrained as we get away from the inner radius to the outer radius. In conclusion, there is a reverse relation between the amount of the swelling ratio and the magnitude of the stress field inside the hydrogel so that the swelling ratio is low in areas with high stress magnitudes and vice versa.
CR IP T
Additionally, the magnitude of the free-swelling (see Fig. 2) is totally higher than the coupled swelling ratio of the hydrogel in the bilayer. The reason is that the constraints that are applied by both the elastomeric layer and the plane strain condition prevent the hydrogel from further swelling.
AN US
An important parameter in the bilayer design is its curvature and the semi-angle ( ) which has a direct impact on its deflections. Thus, the bending curvature ( 1/ ri(1) ) and the semi-angle of bilayers, with the different stiffness of the gel layer, are demonstrated in Fig. 8 versus the temperature changes. As shown, starting from the reference point (T0 =320 K) , as the temperature decreases, the bending
M
curvature and semi-angle start to grow. In the vicinity of the PTT ( 305 K) bilayers experience a sharp phase transition which leads to relatively large curvatures and semi-angles at temperatures below PTT.
ED
The softer gels (low N ) have relatively large curvatures and semi-angles, in comparison to the stiffer
PT
ones, as a result of high swelling ratio. As mentioned earlier, the aspect ratio has almost no influence on the curvatures, however, it affects the bending semi-angle directly and as a result, Fig. 8(a) is applicable
CE
for all values of L / H , while Fig. 8(b) is only valid for the aspect ratio of L / H 3 . Thus, for investigating the effect of the bilayer aspect ratio, the bending semi-angle for N 0.005 at
AC
aspect ratios of L / H 1, 2, 3 and 4 are calculated by the proposed analytical method and shown in Fig. 9(a). As shown, at higher values of L / H , the bending semi-angle is longer. For L / H 4 , the bilayer bends excessively at low temperatures; therefore, the two ends of the bilayer pass each other theoretically which is a key parameter in the bilayer design procedure. However, this is a virtual state since, in reality, the beam ends contact each other, resulting in smaller curvature. Additionally, the
ACCEPTED MANUSCRIPT
increasing of the bending semi-angle by L / H is approximately linear at a special temperature and N . For instance, the distribution of the bending semi-angle at different L / H for N .01, 0.005 and 0.001 at T=288 K are shown in Fig. 9(b). Both of Increasing L / H and decreasing in N result in increasing the bending semi-angle. Similar to the previous statement, at high values of L / H , the bending semi-
CR IP T
angle is greater than , where the two ends of the bilayer pass each other theoretically.
4 Summary and Conclusions
In this work, an analytical method was developed to investigate the finite bending of a bilayer
AN US
composed of a temperature-responsive gel under plane-strain condition. The bilayer is made of two attached layers: one layer is made of a neutral incompressible elastomer and the other layer is made of a temperature sensitive hydrogel. The bilayer bends in response to the environment temperature variations due to the existence of the responsive hydrogel layer. A novel analytical method is presented
M
to investigate the bilayer bending problem. Considering the free energy density and the balance
ED
equations, a second order ODE coupled with two algebraic equations is reached. The governing equations were solved employing iterative numerical techniques, where the unknown parameters were
PT
determined in terms of the temperature changes. On the other hand, a UHYPER code was developed to be implemented in ABAQUS in order to simulate the finite bending problem by FEM and validate the
CE
presented analytical proposed method. Due to a very good agreement between the presented method
AC
results and the FEM results the validity of the method was confirmed. Influence of the crosslink density on the distribution of the displacements and the stresses fields was
studied. Lowering the temperature, the radial and the hoop stresses started to grow. Decreasing the crosslink density leads to relatively large curvatures and semi-angles. At higher values of aspect ratio, larger bending semi-angles were predicted. Based on the perfect agreement of the analytical and the FEM results, it was concluded that the presented method is capable of predicting the swelling induced
ACCEPTED MANUSCRIPT
finite bending of bilayers even at below the PTT, where the gel layer experiences extreme volume changes. This method can be useful for analysis and design of bilayer sensors and actuators in various applications such as thermal sensors, self-folding structures and drug delivery systems.
CR IP T
References
AC
CE
PT
ED
M
AN US
Afroze, F., Nies, E., Berghmans, H., 2000. Phase transitions in the system poly(Nisopropylacrylamide)/water and swelling behaviour of the corresponding networks. J. Mol. Struct. 554, 55-68. Agrawal, A., Yun, T., Pesek, S.L., Chapman, W.G., Verduzco, R., 2014. Shape-responsive liquid crystal elastomer bilayers. Soft Matter 10, 1411-1415. Ahn, S.-k., Kasi, R.M., Kim, S.-C., Sharma, N., Zhou, Y., 2008. Stimuli-responsive polymer gels. Soft Matter 4, 1151-1157. Bäcker, M., Raue, M., Schusser, S., Jeitner, C., Breuer, L., Wagner, P., Poghossian, A., Förster, A., Mang, T., Schöning, M.J., 2012. Microfluidic chip with integrated microvalves based on temperature- and pH-responsive hydrogel thin films. Phys. Status Solidi A 209, 839-845. Cai, S., Suo, Z., 2011. Mechanics and chemical thermodynamics of phase transition in temperature-sensitive hydrogels. J Mech Phys Solids 59, 2259-2278. Chester, S.A., Anand, L., 2011. A thermo-mechanically coupled theory for fluid permeation in elastomeric materials: application to thermally responsive gels. J. Mech. Ph. Solids 59, 1978-2006. Domachuk, P., Tsioris, K., Omenetto, F.G., Kaplan, D.L., 2010. Bio‐microfluidics: Biomaterials and Biomimetic Designs. Adv. Mater. 22, 249-260. Drozdov, A.D., 2015. Swelling of pH-responsive cationic gels: Constitutive modeling and structure–property relations. Int. J. Solids Struct. 64–65, 176-190. Fernandes, R., Gracias, D.H., 2012. Self-folding polymeric containers for encapsulation and delivery of drugs. Adv. Drug Delivery Rev. 64, 1579-1589. Flory, P.J., Rehner Jr, J., 1943. Statistical mechanics of cross-linked polymer networks II. Swelling. J. Chem. Phys. 11, 521-526. Guenther, M., Gerlach, G., Wallmersperger, T., Avula, M.N., Cho, S.H., Xie, X., Devener, B., Solzbacher, F., Tathireddy, P., Magda, J.J., 2013. Smart hydrogel-based biochemical microsensor array for medical diagnostics, Adv. Sci. Tech. Trans Tech Publ, pp. 47-52. He, H., Guan, J., Lee, J.L., 2006. An oral delivery device based on self-folding hydrogels. J. Controlled Release 110, 339-346. He, T., Li, M., Zhou, J., 2012. Modeling deformation and contacts of pH sensitive hydrogels for microfluidic flow control. Soft Matter 8, 3083. Hong, W., Liu, Z., Suo, Z., 2009. Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load. Int. J. Solids Struct. 46, 3282-3289. Hong, W., Zhao, X., Zhou, J., Suo, Z., 2008. A theory of coupled diffusion and large deformation in polymeric gels. J. Mech. Ph. Solids 56, 1779-1793. Ionov, L., 2013. Biomimetic Hydrogel-Based Actuating Systems. Adv. Funct. Mater. 23, 4555-4570.
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Jones, D.S., Lorimer, C.P., McCoy, C.P., Gorman, S.P., 2008. Characterization of the physicochemical, antimicrobial, and drug release properties of thermoresponsive hydrogel copolymers designed for medical device applications. J. Biomed. Mater. Res. Part B 85, 417426. Kwon, G.H., Jeong, G.S., Park, J.Y., Moon, J.H., Lee, S.-H., 2011. A low-energy-consumption electroactive valveless hydrogel micropump for long-term biomedical applications. Lab Chip 11, 2910-2915. Kwon, G.H., Park, J.Y., Kim, J.Y., Frisk, M.L., Beebe, D.J., Lee, S.H., 2008. Biomimetic soft multifunctional miniature aquabots. J Small. 4, 2148-2153. Lai, F., Li, H., Luo, R., 2010. Chemo-electro-mechanical modeling of ionic-strength-sensitive hydrogel: Influence of Young’s modulus. Int. J. Solids Struct. 47, 3141-3149. Li, H., 2009. Kinetics of smart hydrogels responding to electric field: A transient deformation analysis. Int. J. Solids Struct. 46, 1326-1333. Lucantonio, A., Nardinocchi, P., Pezzulla, M., 2014. Swelling-induced and controlled curving in layered gel beams, P. Roy. Soc. A-Math. Phy. The Royal Society, p. 20140467. Marcombe, R., Cai, S., Hong, W., 2010. A theory of constrained swelling of a pH-sensitive hydrogel Soft Matter 6, 784-793. Mazaheri, H., Baghani, M., Naghdabadi, R., 2015. Inhomogeneous and homogeneous swelling behavior of temperature-sensitive poly-(N-isopropylacrylamide) hydrogels. J. Intel. Mat. Syst. Str., 1045389X15571381. Mazaheri, H., Naghdabadi, R., Baghani, M., sohrabpour, S., 2016. pH/temperature sensitive hydrogels: Coupling behavior of the pH/temperature sensitive hydrogels for the inhomogeneous and homogeneous swelling. under revision in "Smart Mater. Struct.". Morimoto, T., Ashida, F., 2015. Temperature-responsive bending of a bilayer gel. Int. J. Solids Struct. Nardinocchi, P., Puntel, E., 2016. Finite bending solutions for layered gel beams. Int. J. Solids Struct. Randall, C.L., Gultepe, E., Gracias, D.H., 2012. Self-folding devices and materials for biomedical applications. Trends Biotechnol. 30, 138-146. Richter, A., 2009. Hydrogels for actuators, Hydrogel sensors and actuators. Springer, pp. 221-248. Richter, A., Klatt, S., Paschew, G., Klenke, C., 2009. Micropumps operated by swelling and shrinking of temperature-sensitive hydrogels. Lab Chip 9, 613-618. Roccabianca, S., Gei, M., Bigoni, D., 2010. Plane strain bifurcations of elastic layered structures subject to finite bending: theory versus experiments. IMA J. Appl. Math., hxq020. Shin, J., Braun, P.V., Lee, W., 2010. Fast response photonic crystal pH sensor based on templated photo-polymerized hydrogel inverse opal. "Sens. Actuators, B " 150, 183-190. Swann, J.M., Ryan, A.J., 2009. Chemical actuation in responsive hydrogels. Polym. Int. 58, 285-289. Timoshenko, S., 1925. Analysis of bi-metal thermostats. JOSA 11, 233-255. Toh, W., Ng, T.Y., Hu, J., Liu, Z., 2014. Mechanics of inhomogeneous large deformation of photo-thermal sensitive hydrogels. Int. J. Solids Struct. 51, 4440-4451. Yan, H., Jin, B., Gao, S., Chen, L., 2014. Equilibrium swelling and electrochemistry of polyampholytic pH-sensitive hydrogel. Int. J. Solids Struct. 51, 4149-4156. Yu, Q., Bauer, J.M., Moore, J.S., Beebe, D.J., 2001. Responsive biomimetic hydrogel valve for microfluidics. Appl. Phys. Lett. 78, 2589.
ACCEPTED MANUSCRIPT
AN US
CR IP T
Zhang, C., Xing, D., Li, Y., 2007. Micropumps, microvalves, and micromixers within PCR microfluidic chips: Advances and trends. Biotechnol Adv 25, 483-514. Zhang, J., Wu, J., Sun, J., Zhou, Q., 2012. Temperature-sensitive bending of bigel strip bonded by macroscopic molecular recognition. Soft Matter 8, 5750-5752. Zhou, X., Hon, Y., Sun, S., Mak, A., 2002. Numerical simulation of the steady-state deformation of a smart hydrogel under an external electric field. Smart Mater. Struct. 11, 459.
AC
CE
PT
ED
M
Fig. 1. Hydrogel bilayer geometry schematics: a) the initial configuration and b) the deformed configuration.
Fig. 2. The free-swelling ratio of the Hydrogel with N 0.01,0.005 and 0.001 versus temperature.
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
Fig. 3. FEM and analytical (ANL) results for the radius distribution for N 0.01,0.005 and 0.001 at T=288K .
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
Fig. 4. Normalized a) radial stress and b) hoop stress distributions calculated from the analytical (ANL) and FEM solution for N 0.01, 0.005 and 0.001 .
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
Fig. 5. The radial and the hoop stress distributions on the deformed bilayer at T=288 K with N 0.005. a) FEM radial stress, b) Analytical radial stress, c) FEM hoop stress and d) Analytical hoop stress.
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
ACCEPTED MANUSCRIPT
AN US
CR IP T
Fig. 6. a) Radius, b) normalized radial stress and, b) normalized hoop stress distributions at T=308, 300 and 288K for N 0.005 computed by the proposed analytical method.
AC
CE
PT
ED
M
Fig. 7. Distribution of the swelling ratio in the cross-section for the bilayer with different values of N at T 288K .
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Fig. 8. a) Bending curvature and b) bending semi-angle of bilayers with N 0.01,0.005 and 0.001 and L / H 3 computed by the proposed analytical method.
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Fig. 9. a) Bending semi-angle of bilayers with L / H 1, 2, 3 and 4 for N 0.005 .b) Variation of the bending semi-angle of bilayers at different L / H for N 0.01, 0.005 and 0.001 at T=288K .