Large Deformation of Hydrogels Coupled with Solvent Diffusionruihuang/talks/hydrogel2015.pdf ·...
Transcript of Large Deformation of Hydrogels Coupled with Solvent Diffusionruihuang/talks/hydrogel2015.pdf ·...
Large Deformation of Hydrogels Coupled with Solvent Diffusion
Rui Huang
Center for Mechanics of Solids, Structures and MaterialsDepartment of Aerospace Engineering and Engineering Mechanics
The University of Texas at Austin
July 2015
Ambrosio 2010
Crosslinkedpolymer
Solvent molecules
• Large and reversible deformation facilitated by solvent migration• Response to mechanical/chemical/electrical stimuli• Applications: biomedical, soft machines/robots
Hydrogel = polymer + water (solvent)
Transient behaviors of hydrogelsTanaka et al, 1987
• Swelling• Indentation• Fracture• …
Bonn et al, 1998
Hu et al, 2011
Linear Poroelasticity
3 material parameters: G, ν, k/η
i
j
j
iij x
uxu
21
0
ij
ij bx
kk xxpk
t
2
kk x
pkJ
0 kk
ijijkkijij pG
212
Biot, 1941; Scherer, 1989; Suo et al., 2010.
Linear constitutive relation
Linear kinematics Linear kinetics Inertia ignored
A Nonlinear Continuum Theory for Gels),( CU F
iKiK F
Us
0
iK
iK BXsCU
Diffusion kinetics:L
KLLk
L
k
K
BK X
MXx
XxX
TkDCJ
LKL
KK
K
XM
XXJ
tC Conservation of solvent
molecules:
Nonlinear constitutive relations
Nonlinear kinematics Geometrically nonlinear
kinetics Inertia ignored
Free energy density function:
Nominal stress:
Chemical potential:
Mechanical equilibrium:
Hong, Zhao, Zhou, and Suo, 2008.
A Material Model)()(),( CUUCU me FFFlory‐Rehner (1943) free energy
density function:
)det(ln2321 FF iKiKBe FFTNkU
CC
CCCTkCU B
m 11ln)(
F)FF det(1)()(),( CCUUCU me
Flory‐Huggins polymer solution theory:
Neo‐Hookean rubber elasticity:
C1)det(F
4 material parameters: NkBT, NΩ, χ, D
Volume of gel::
Linearization near a swollen state
Isotropic swollen state: 0321
1
50
20
30
20
2)1(
122
1
NN
0TNkG B
3
0060
30
30
300 1111
ln
N
TkB
ijijkkijij G
0
212
kkBk x
MxTk
Dcj
00
30
30
01
TkDMB
300 1 c 0cckk
kkk
k
xxM
xj
tc
2
0
0p c 30
30 1
TkDk
B
The linearized equations are identical to the theory of linear poroelasticity.
The 3 parameters in the linearized equations depend on the swelling ratio.
Bouklas and Huang, Soft Matter 2012.
Poisson’s ratioYoon et al., Soft Matter 6, 6004‐6012 (2010).
1
50
20
30
20
2)1(
122
1
NN
Instantaneously incompressible (ν0 = 0.5) The equilibrium Poisson’s ratio depends on
the initial state (λ0 or μ0) as well as the intrinsic material properties (χ and NΩ).
1 1.5 2 2.5 3 3.50.2
0.25
0.3
0.35
0.4
0.45
0.5
0
Poi
sson
's ra
tio,
N = 0.001
= 0.1-0.6
0 0.2 0.4 0.6 0.8 10.2
0.25
0.3
0.35
0.4
0.45
0.5
Poi
sson
's ra
tio
N = 10-4, 10-3, 10-2, 10-1
Bouklas and Huang, Soft Matter 2012.
poor solvent
good solvent
A nonlinear, transient finite element methodWeak form for the coupled fields:
0 0 0 0
, ,t
iJ i J K K i i i iV V S S
s u C tJ dV b u r t C dV T u dS i t dS
Discretizationu nu N un N μ
u nδu N δun N δμ
For numerical stability, the 2D Taylor‐Hood elements (8u4p) are implemented.
Bouklas et al., JMPS 2015CCUU -) (F,) (F, ˆLegendre transform
),()()(),( CFUCUUCU cme FF
A slightly different material model: 2
21)det(),( CKCUc
FF
Numerical stability for mixed finite element method• Nearly incompressible
behavior at the instantaneous response limit ⟶0 for K >> NkBT
• Discontinuity of the boundary condition at 0 with respect to the initial conditions
Choice of appropriate interpolations:LBB condition(Ladyzhenskaya‐Babuska‐Brezzi)
UNSTABLEEqual order8u8p
STABLETaylor Hood8u4p
• Incompressible linear elasticity• Stokes flow• Linear poroelasticity
Constrained swelling of hydrogel layersBC’sIC’s
0, 0 u X X X
0, 0 X
1 : 0S
2 2: , 0S J u 03 1 1: 0, 0S u J
UNSTABLEEqual order8u8μ element
μ/KBT
time
Oscillations in the early stage
Evolution of chemical potential oscillations
0
3
1.4, 0.001
0.4, 10 B
N
K Nk T
0 0.5 1-0.06
-0.04
-0.02
0
X2/H
/(k
BT)
0 0.5 1-0.06
-0.04
-0.02
0
X2/H
/(k
BT)
0 0.5 1-0.04
-0.03
-0.02
-0.01
0
X2/H
/(k
BT)
0 0.5 1-0.04
-0.03
-0.02
-0.01
0
X2/H
/(k
BT)
t/ = 10-3
t/ = 10-1
t/ = 10-4
t/ = 10-2
UNSTABLE8u8μ
2HD
Effect of compressibility
0 1.4, 0.0010.4
N
4/ 10t The oscillations decrease with increasing compressibility (K decreasing)
0 0.5 1-0.06
-0.04
-0.02
0
X2/H
/(k
BT)
0 0.5 1-0.06
-0.04
-0.02
0
X2/H
/(k
BT)
0 0.5 1-0.06
-0.04
-0.02
0
X2/H
/(k
BT)
0 0.5 1-0.06
-0.04
-0.02
0
X2/H
/(k
BT)
K =10NkBTK = 102NkBT
K = 103NkBTK = 104NkBTUNSTABLE8u8μ
Use of stable interpolationSTABLETaylor Hood8u4μ element
0.5 0.6 0.7 0.8 0.9 11-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
X2/H
/(K
BT)
t/ = 10-4
t/ = 10-3
t/ = 10-2
t/ = 10-1
K = 103NkBT
0.5 0.6 0.7 0.8 0.9 11-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
/(K
BT)
X2/H
K = 104 NkBT
K = 103 NkBT
K = 102 NkBT
K = 10 NkBT
t/ = 10-4
• Oscillations are significantly reduced and do not depend oncompressibility.
• The remaining oscillations are due to discontinuity of the boundary condition at 0 .
Bouklas et al., JMPS 2015
0
3
1.4, 0.001
0.4, 10 B
N
K Nk T
Swelling induced surface instability
10-3 10-2 10-1 100 101 102 103 1041
1.5
2
2.5
3
3.5
4
4.5
h/H
t/
ne = 50ne = 30ne = 20ne = 4
= 0.4
= 0.6
Swell induced compressive stress could lead to surface instability For a poor solvent (χ = 0.6), the surface remains flat and stable. For a good solvent (χ = 0.4), the homogeneous hydrogel layer is
instantaneously unstable upon swelling; numerically, divergence occurs at a finite time that depends on the mesh size.
Bouklas et al., JMPS 2015
Swell induced surface wrinkling
s11
10-2 10-1 100 101 102 103 1040
0.01
0.02
0.03
0.04
0.05
0.06
t/
A/h
0
• Regularization of the surface instability by using a stiff skin layer.
• Wrinkle amplitude grows over time, with a selected wrinkle wavelength.
Bouklas et al., JMPS 2015
Two types of surface instability
A B
Wu, Bouklas and Huang, IJSS 50, 578‐587 (2013).
Type A: soft-on-hard bilayer, critical condition at the short wave limit, forming surface creases;
Type B: hard-on-soft bilayer, critical condition at a finite wavelength, forming surface wrinkles first.
A soft‐on‐hard gel bilayer (Type A)
A hard‐on‐soft gel bilayer (Type B)
Flat Punch Indentation
S2x1
x2S5 hS3
S1
F (δ)
S4aa
b
2
D
10-2 100 102 104 1060
0.2
0.4
0.6
0.8
1
t/
(F(t)
-F(
)) /
(F(0
)-F(
))
/h = 10-3
/h = 10-2
/h = 0.1/h = 0.2fitting
F ( ( ) ( )) / ( (0) ( ))t F t F F F
σ/NkBT
σ/NkBT
σ/NkBT
σ/NkBT µ/kBT
µ/kBT
µ/kBT
µ/kBT
Fracture Properties of Hydrogels
Sun et al., 2012.Kwon et al., 2011.
• Reported values of fracture toughness range from 1 to 10000 J/m2
• Fracture mechanism may vary over different types of hydrogels• Potential rate dependence may relate to viscoelasticity or solvent diffusion
• Experimental methods to measure fracture properties of hydrogels are not well developed.
A center crack model
• Consider a center crack in a hydrogel plate under remote strain• A nonlinear, transient finite element method is employed• Sharp and notched crack geometries are used for small to large
deformationBouklas et al., JAM 2015
Immersed and not‐immersed cases
0t t Hydrogel is immersed in solvent Assume solvent‐permeable boundaries
and crack faces
Hydrogel is not immersed in solvent Assume solvent‐impermeable
boundaries and crack faces Overall volume is conserved, but
solvent redistribution within the gel may lead to local volume change.
1
2 3 2 50 0 0 0
1 1 22 2 ( 1)
N N
0 0.5
0t t
LEFM Predictions
0TNkG B
jiIij
Iij Tf
rKr 112
,
Crack‐tip stress field:
1
2G
aKI T
Crack opening: 21
212 2ˆ xaxu
Stress would relax over time
Crack opening would remain constant, independent of modulus or Poisson’s ratio
Crack‐tip stress fields 310
00.5
t
0.2415t
• The square‐root singularity is maintained at both the instantaneous and equilibrium states, but the stress intensity factor is lower in the equilibrium state due to poroelastic relaxation.
Evolution of crack opening
Immersed Not immersed
10-4 100 104 108
2
2.5
3
x 10-3
/a
t/
immersednot immersedLEFM
= 10-3 h/a
100
20
10
Opening at the center:
Bouklas et al., JAM 2015
Solvent diffusion around crack tip
Instantaneous limit:Homogeneous solvent concentration, but inhomogeneous chemical potential.
Equilibrium state:Homogeneous chemical potential, but inhomogeneous solvent concentration.
Bouklas et al., JAM 2015
Evolution of Solvent Concentration Immersed
A singular concentration field is approached in the equilibrium state for both the immersed and not‐immersed cases.
10-3 10-2 10-1
10-4
10-2
100
(C-C
0)
r/a
t/ = 10-1
t/ = 1t/ = 102
t/ = 104 = 10-3
Not‐immersed
10-3 10-2 10-1
10-4
10-2
100
(C-C
0) r/a
t/ = 10-1
t/ = 1t/ = 102
t/=104 = 10-3
Bouklas et al., JAM 2015
Energy release rate Rate of change of potential energy:
Total energy:
Rate of total energy change due to crack extension
Rate of dissipation due to diffusion:
Conserved throughout the transient evolution (without crack growth).
0 0 0 0 0
i ii iV V S V S
dx dxd dU dV b dV T dS rdV idSdt dt dt dt
0KV
K
d J dVdt X
0ddt
0 0
iK Ki KV S
K
dxdI dId dU dV T N dSda da X da da da
A modified J‐integral
• J* is path independent.• The second form is more convenient for numerical calculations.• A domain integral method is used to calculate J*.
1 0
*1
1 1
iiJ JS V
x CJ UN s N dS dVX X
1 0
*1
1 1
ˆ iiJ JS V
xJ UN s N dS CdVX X
* dJda
Define a nominal energy release rate: (per unit length of crack in the reference state)
Bouklas et al., JAM 2015
Evolution of J‐integral
10-4 10-2 100 102 104 106 1080
1
2
3
4
5
6
7
8 x 10-5
J* /(aN
k BT)
t/
not immersedimmersedLEFM ( = 0.5)LEFM ( = 0.2415)
h/a = 10, 20, 100
= 10-3
aTNktJ B2
0* 40 aTNktJ B
20*
12
LEFM predictions at the instantaneous and equilibrium limits:
Dependence on material parameters
10-6 10-3 100 103 1060.9
1
1.1
1.2
1.3
1.4
1.5
J* /(4G
2
02 a
)
t/*
N = 10-2
N = 10-3
N = 10-4
= 0.2 = 10-3
10-6 10-3 100 103 1060.9
1
1.1
1.2
1.3
1.4
1.5
J* /(4G
2
02 a
)
t/*
= 0.2 = 0.4 = 0.6
N = 10-2
= 10-3
The J* is well predicted by LEFM at the instantaneous limit (t → 0) The equilibrium J* depends sensitively on χ (good/poor solvent) The time scale is captured effectively by the poroelastic diffusivity Delayed fracture possible.
Bouklas et al., JAM 2015
Under moderately large remote strain
10-4 10-3 10-2 10-110-1
100
101
102
22/(N
k BT)
r/a
t/ = 10-4
t/ = 105 = 0.1
~ r-1/2
~ r-1
• Similar behavior for crack opening (and J* too, not shown)• Stress distributions show transition of singularity: with a stronger 1/r
singularity due to hyperelasticity and a weaker singularity closer to the crack tip due to solvent diffusion; the typical square‐root singularity is approached at the equilibrium limit.
1.0
Bouklas et al., JAM 2015
Under large remote strainImmersed Not‐immersed
5.0
• The 1/r singularity becomes more prominent in both cases• The singularity weakens over time for the immersed case
Bouklas et al., JAM 2015
J‐integral
10-4 100 104 1080
0.1
0.2
0.3
0.4
0.5
0.6
0.7
t/
J* /(4G
2
02 a)
immersednot-immersed = 0.5
• The J* is considerably lower than the LEFM prediction, while the evolution trends are similar to the small‐strain behaviors.
Bouklas et al., JAM 2015
Fracture Criterion
?**cJtJ
The right hand side of the fracture criterion has to be determined by well‐designed experiments.
Could the right‐hand side depend on time due to the evolving, inhomogeneous field of solvent concentration around the crack tip?
A nonlinear, transient finite element method was implemented for large deformation of hydrogels coupled with solvent diffusion.
The transient processes of constrained swelling (with surface instability) and flat punch indentation were studied in some details.
Effects of solvent diffusion on the crack‐tip fields and driving force for fracture of hydrogels were studied. A path‐independent, modified J‐integral was derived as the transient energy release rate of crack growth in hydrogels, although a complete fracture criterion is yet to be established.
Summary
Acknowledgments
• Min‐Kyoo Kang• Nikolas Bouklas• Zhigen Wu• Chad Landis
• Funding by NSF