EFFECT OF HYDRATION ON THE MECHANICAL PROPERTIES OF A PVA 
DUAL-CROSSLINKED HYDROGEL 
 
 
 
 
 
 
 
A Thesis 
Presented to the Faculty of the Graduate School 
of Cornell University 
In Partial Fulfillment of the Requirements for the Degree of 
Master of Science  
 
 
 
 
 
by 
Fan Cui 
August 2021 
 
 
 
 
 
 
 
 
 
 
 
 
© 2021 Fan Cui 
 
ABSTRACT 
 
The effect of drying on the constitutive behavior of a dual cross-linked poly(vinyl 
alcohol) (PVA) hydrogel is studied. This gel contains about 90% water when fully 
hydrated. The mass-volume relationship of the gel is measured using a microbalance 
with a density kit. Our results show that as the gel dries the volume is linearly 
proportional to the mass. The impact of drying on the gel’s mechanical properties is 
measured in uniaxial tension tests, which include loading-unloading tests at three 
different constant stretch rates, a complex loading history test and a stress-relaxation 
test. Data from specimens with different hydration levels can be described by a 
constitutive model of the gel. The results show that the model parameters are strongly 
dependent on hydration level and that as the gels dry, the gels become much stiffer than 
in the fully hydrated state. Fracture tests, including constant rate loading test and creep 
test, are also done for this hydrogel at different hydration levels. The results show that 
the breaking stress is much larger as the gels dry. Furthermore,  experimental and 
numerical studies are done on the effect of compression in torsion-rheometry tests. The 
results show the compression will make us overestimate the dynamic modulus and give 
a shift factor for different compression. 
BIOGRAPHICAL SKETCH 
 
 
Fan Cui was born in Dalian, a coastal city in Northeastern China. In 2014, after 
graduating from Dalian Yuming High School, he was admitted to Tsinghua University 
to pursue a bachelor's degree in Engineering Mechanics at the School of Aerospace 
Engineering and graduated in 2018. In 2019, he began his Master of Science program 
in Sibley School of Mechanical and Aerospace Engineering at Cornell University. His 
study focuses on Solid Mechanics and Materials.  
iii   
 
 
 
 
 
 
 
 
 
 
 
To my families 
iv   
ACKNOWLEDGMENTS 
 
This work would not have been possible without the help of many people. 
First, I would like to thank my advisor, Prof. Alan Zehnder. He is the most phenomenal 
adviser that I have ever met. During my M.S. program in Cornell University, I got a lot 
of valuable guidance and support from him. Whenever I met some trouble in my 
research or in my course, he was always ready to help me. As an international student, 
my expression in English is not always as good as native speakers. When I worked on 
my paper and thesis, he helped me a lot on the language. I also remember that when we 
had a video meeting, he always encouraged me to express my opinions. He is also very 
nice in person and always treats me with kindness. 
I would also give my thanks to Prof. Herbert Hui for serving as my special committee 
members. We worked in the same group on the hydrogel project. We had a lot of 
inspiring discussions on this project. He suggested me a lot in the area of computational 
solid mechanics. He always works with energy and passion, which inspire me a lot.  
Then I would like to thank people who help me with my research, Mincong Liu, Jingyi 
Guo and Jikun Wang. Mincong and Jingyi did most of the previous work and they 
helped me start my research. Jikun is my co-worker and we always discussed together 
when we met any problem. 
Finally, I’d like to express my special gratitude to my parents. Their unconditional 
support is the best thing I could have ever had. 
 
 
v   
TABLE OF CONTENTS 
BIOGRAPHICAL SKETCH ························································ iii 
DEDICATION ········································································ iv 
ACKNOWLEDGEMENTS ·························································· v 
TABLE OF CONTENTS ···························································· vi 
1 INTRODUCTION 1 
2 EXPERIMENTAL SETUP 6 
2.1 Introduction ······································································· 6 
2.2 Material Preparation ····························································· 6 
2.3 Tensile Tester ···································································· 10 
3 THE EFFECT OF HYDRATION ON THE CONSTITUTIVE RESPONSE 
OF THE MATERIAL 15 
3.1 Introduction ······································································ 15 
3.2 Experiment Overview ··························································· 15 
3.2.1  Mass and Volume Measurement ········································ 15 
3.2.2  Uniaxial Tests ······························································ 18 
3.3 Results ············································································ 19 
3.3.1  Mass-Volume Relationship ·············································· 19 
3.3.2  Uniaxial Tests Results ···················································· 20 
3.4 Data Analysis ···································································· 22 
3.4.1  Review of the Constitutive Model ······································ 22 
3.4.2  Determination of Material Parameters ·································· 25 
3.5 Discussion ········································································ 32 
3.5.1  Effect of Dehydration ····················································· 32 
3.5.2  Limitation of the Model ·················································· 35 
3.6 Conclusion ······································································· 36 
4 STUDY ON THE EFFECT OF HYDRATION: FRACTURE TESTS 38 
4.1 Introduction ······································································ 38 
4.2 Experiment ······································································· 38 
4.3 Results ············································································ 41 
4.3.1  Constant Rate Loading Fracture Test ··································· 41 
4.3.2  Creep Tests ································································· 42 
4.4 Discussion ········································································ 44 
4.5 Conclusion ······································································· 48 
5 NUMERICAL AND EXPERIMENTAL STUDY ON THE EFFECT OF 
COMPRESSION IN THE RHEOLOGY TEST 50 
5.1 Introduction ······································································ 50 
5.2 Finite Element Analysis ························································ 50 
5.2.1  Methodology ······························································· 50 
vi   
5.2.2  FEM Calculation ·························································· 51 
5.3 Experiment ······································································· 54 
5.3.1  Experiment setup ·························································· 54 
5.3.2  Torsion Rheometry Experiments ········································ 55 
5.4 Results ············································································ 55 
5.4.1  FEM Results ······························································· 55 
5.4.2  Experimental Results ····················································· 59 
5.5 Discussion ········································································ 60 
5.5.1  FEA Calculation ··························································· 60 
5.5.2  Experiments ································································ 63 
5.6 Conclusion ······································································· 66 
A APPENDIX 68 
A.1 Results of Recovery Tests ······················································ 68 
A.2 Long-time Stress in Relaxation Test ·········································· 69 
A.3 Feedback Load Control ························································· 70 
A.4 UMAT for PVA Model ························································· 72
vii  
CHAPTER 1 
INTRODUCTION 
 
A hydrogel is essentially a network of polymer chains swollen in water. With 
desirable properties, such as high compliance, low friction and good compatibility 
with biological tissues, hydrogels are widely used in many areas, especially in 
biomedical engineering. People used hydrogels as cell scaffolds in tissue engineering 
[1][2], as vehicles for drug delivery [3] and as artificial cartilage [4]. In most of 
applications, good load-carrying capabilities, such as high stretchability, high 
toughness, and degree of self-healing, are required. Conventional hydrogels, usually 
single crosslinked, may not meet the requirements in use. In the last decades, a wide 
range of hydrogels have been developed 
[5][6][7][8][9][10][11][12][13][14][15][16][17] with intriguing properties such as the 
ability to self-heal, to undergo extremely large deformations and to show elastic or 
viscoelastic behaviors. One of the methods to enhance the toughness of hydrogel is to 
build highly stretchable polymer network by introducing a non-organic component to 
the network. Haraguchi et al. [18] developed a nanocomposite hydrogel by polymer 
intercalating into disk-shaped nano-clay sheets. This gel exhibited high structural 
homogeneity, superior elongation with near-complete recovery. Another idea used by 
many researchers is introducing a second network as a sacrificial network. One 
example is the double-network (DN) hydrogel developed by Gong et al.[19]. In this 
DN gel, the gel contains two types of network. The highly cross-linked first network 
1 
has a relatively high Young’s modulus but is rather brittle. The loosely cross-linked 
second network is highly extensible and prevents the formation of macroscopic cracks. 
The toughness of this hydrogel can match that of synthetic rubber (approximately 
10,000 𝐽/𝑚2). However, this gel cannot fully recover to its original state after damage 
since the first sacrificial network is chemically cross-linked and cannot reattach after 
breaking [20]. Nevertheless, these DN gels still have good fatigue resistance to crack 
propagation [21]. Several groups of researchers also have synthesized highly 
stretchable and tough hydrogels by introducing non-covalent (physical) transient 
bonds as a sacrificial network [8][13][15][16][22][23]. These physically cross-linked 
networks are reversibly broken during loading, but they can reform and recover to 
their initial state after unloading and resting. For example, J. Sun et al. [8] synthesized 
tough hydrogels by mixing ionically crosslinked alginate and covalently crosslinked 
polyacrylamide. The covalent crosslinks preserve the memory of the initial state, 
allowing the material to recover upon removal of the load. The ionic crosslinks unzip, 
causing internal hysteresis, dissipating energy, but they heal by re-zipping. The 
resulting gel can be stretched 20 times their initial length, and have a fracture energy 
of about 9,000 𝐽/𝑚2, despite its high water content of about 90%. T. Sun et al.[12] 
reported the synthesis of a tough polyampholytes (PA) hydrogel which has a fracture 
energy of about 4,000 𝐽/𝑚2 with only ionic bonds. This PA gel has a phase-separated 
structure composed of soft and stiff phases, with a structure length of 100 nm 
[24][25][26][27]. The ionic bonds have a wide distribution of strength, and it can be 
divided into two types: strong bonds and weak bonds. These bonds serve as sacrificial 
2 
networks and enhance the fatigue [28] , fracture resistance and self-healing behavior 
[29] of PA gels.  
Being comprised mostly of water, hydrogels are prone to drying, leading to shrinkage 
and to changes of mechanical properties. The effect of hydration on hydrogels 
including drying and shrinkage has been studied by several researchers 
[30][31][32][33][34][35][36][37][38][39]. Some researchers focused on the behavior 
of the hydrogel during the dehydration-rehydration cycle [30][31]. Lee et al.[30]  
reported the shape stability of PVA hydrogels during the drying-rewetting cycle. 
Thomas et al.[31]  found that a poly(vinyl alcohol) and poly(vinyl pyrrolidone) 
(PVA/PVP) gel shrinks in a dehydration-rehydration cycle and that  the instantaneous 
compressive modulus increases. The more water it loses during the dehydration, the 
more the gel shrinks, and the more the modulus increases. This result is broadly 
consistent with available data. For example, Lee et al.’s work on PVA hydrogel [32] 
and Truong et al.’s work on rec1-resilin gel [33] both reported that the stiffness of 
their gel decreases as the hydration level increases. Gao and Gu [34] developed a 
relationship between the water content  and elastic modulus  for hydrated porous 
materials, showing that the modulus decreases with increasing hydration.  Smyth et 
al.[35] reported that the stiffness of  hydrogel films becomes lower when soaking in 
water. Zhang et al.[36] reported that for solvent treated VM-3-10-1 hydrogel, the 
modulus decreases as the gel becomes hydrated until the water content reaches 75% 
and then changes little as it reaches full hydration. There are some exceptions, e.g., Li 
et al.[37] found that for a bacterial cellulose and polyvinyl alcohol (BC/PVA) 
hydrogel, the compressive modulus of samples with PVA content less than 20% 
3 
decreases compared to fully hydrated samples.  The effect of hydration on the tensile 
strength of  hydrogel films have been studied by Olivas et al.[38] and Pereira et 
al.[39]. They reported the tensile strength (breaking stress) of the film is lower when 
surface humidity is higher. Zhang et al.[36] reported that for the solvent treated VM-3-
10-1 hydrogel the strain to failure increases as the water content increases. 
While stiffness and tensile strength are important mechanical properties, they do not 
fully characterize the complex, time dependent behavior of viscoelastic gels. Thus, the 
overall goal of this study is to understand the rate-dependent, large deformation 
mechanical response of hydrogels at different hydration levels, with more focus on the 
experimental aspects. In this study, we focused on a specific type of hydrogel, the 
poly(vinyl alcohol) (PVA) dual-crosslinked hydrogel developed by Mayumi et al.[14]. 
Dual cross-linked means that the crosslinks are a mixture of physical (ionic) and 
chemical (covalent) bonds. The chemical crosslinks form a permanent network. The 
physical crosslinks form a temporary network which can break and reattach. About 
10% of the cross-links are chemical with the balance physical bonds. For a fully 
hydrated gel, Long et al.[40] and Guo et al.[41] have established a constitutive  model 
which accurately predicts its behavior in uniaxial tension [40][41] and rheology tests 
[42]. A numerical scheme developed by Guo et al.[43] allows the application of this 
constitutive model in  finite element (FE) to simulate tests with more complicated 
geometry and loading history. The accuracy of this numerical application has been 
checked by comparing the simulation results with experimental results [43][44].  Liu 
at el.[45] demonstrated that the mechanical properties of this PVA hydrogel over a 
range of temperatures can also described by this constitutive model. Meacham at 
4 
el.[46] performed a preliminary study of the effect of hydration on the tensile response 
of this PVA hydrogel. Both the constitutive model and the numerical scheme provide 
effective tools to further study this material. 
This thesis is organized as follows. In Chapter 2, the details of the experimental 
methodology are provided, including hydrogel synthesis and experimental setup. In 
Chapter 3, we investigate the effects of drying on the volume and time dependent 
constitutive response of the PVA hydrogel. We find that the constitutive model used 
for the fully hydrated gel can be used to quantify our experimental results and that 
changes in the model parameters provide insight into the role of hydration on the 
chemical and physical bonds. This work has been accepted by Mechanics of Materials 
[47]. Basic study on how hydration levels influence the fracture response of the PVA 
hydrogel is contained in Chapter 4. There have been a lot of studies on the fracture of 
hydrogels[48][49][50][51][52][53][54][55], but few has focused on the effect of 
hydration. We compare the fracture response of fully hydrated hydrogels with that of 
drier hydrogels. However, more work is needed in the future study on this topic. In 
Chapter 5, we report some of the study on the rheology tests. We use both numerical 
and experimental method to study the effect of pre-compression in the rheology test. 
The shift parameters for different degree of compression are given. However, more 
experiments are needed to verify the FEA results.
5 
CHAPTER 2 
EXPERIMENTAL SETUP 
 
2.1 Introduction 
In this work, we use experimental methods to study the mechanical properties of PVA 
dual-crosslinked hydrogel, especially on the dehydrated hydrogels. In this chapter, we 
will introduce the basic experimental setups of our experiments. These setups are the 
fundamentals for the studies presented in this thesis that most of the tests are done 
with them.  
2.2 Material Preparation 
The synthesis process of this dual-crosslinked PVA hydrogel was first introduced by 
Mayumi et al. in [14]. Basically, it was synthesized by incorporating borate ions in a 
chemically crosslinked PVA gel. The detail of the synthesis process is explained in the 
following paragraphs. 
PVA Solution 
The first step of the synthesis process is making 16%wt PVA solution. It was made by 
dissolving PVA powder in distilled water at a high temperature (>90 ℃). The PVA 
powder used here was from Sigma-Aldrich (Mw 89,000-98,000, 98%+ hydrolyzed, 
341584-500G). We used a water bath here to control the temperature. First, we added 
ice to the water bath to achieve a low temperature (<5 ℃). Then we put the container 
6 
in the water bath and added distilled water and PVA powder to the container with 
active stirring. The weight ratio of distilled water and PVA powder is 21:4 since we 
want a target of 16% wt PVA solution. The low temperature environment helps make 
the PVA powder distribute evenly in the water. Then we covered the lid of the 
container, heated the water bath to 90 ℃ and kept the temperature. After another 2 
hours stirring, we could let the solution cool at the room temperature. The solution can 
be kept at room temperature for several weeks before we use it. 
Chemically Crosslinked Hydrogel 
Then the next step is to add chemical crosslinks. In this step, we mixed PVA solution 
and glutaraldehyde (GA) solution at pH 1.7. The target concentration of the PVA is 
12%. The molar ratio of GA crosslinker to PVA monomer is 1:500. To make 40 g of 
such solution, we need 30 g of PVA 16% solution, 0.872 g of GA 2.5% solution, 0.8 
mL of hydrochloric acid (HCl) (1 mol/L) and 8.328 g of distilled water. The GA 2.5% 
solution is always freshly made by diluting GA 25% solution (Sigma-Aldrich, Grade 
Ⅱ, G6357-10ML). We first added PVA 16% solution to the beaker and kept stirring. 
Then we added distilled water and GA 2.5% solution to the beaker. After 10 minutes, 
we added HCl to the solution and kept stirring for another 10 minutes to make the 
solution be mixed evenly. Then the solution was injected into a custom-built mold 
using a syringe. A schematic view of the mold is shown in Figure 2.1. It was made by 
acrylic plates spaced by silicone rubber with PET films between the rubber and acrylic 
plates. The PET film could prevent the chemically crosslinked gel from sticking to the 
plate. The solution was left sitting in the mold for at least 24 hours with a good seal. 
After 24 hours, the gel needed to be washed to neutral pH. We peeled the chemical gel 
7 
from the mold and place it into a container with distilled water. Water was changed 
regularly during a 24-hour period.  
 
Figure 2.1: Acrylic mold used for shaping chemical PVA gel. 
Physical Crosslinking 
After washing the chemically crosslinked gel, we soaked the gel in a freshly made 
Borax/NaCl solution to add the physical crosslinks and reduce the degree of swelling. 
The concentration of borax and NaCl in the solution was 1 mM/L and 90 mM/L 
respectively. Borax and NaCl used were from Sigma-Aldrich (Borax: Sodium 
tetraborate decahydrate, ACS reagent, ≥ 99.5%, S9640-500G; NaCl: Sodium chloride, 
puriss. pa., ≥ 99.5%, 71380-1 KG). In this step, the weight of the solution is 20 times 
the weight of the chemical gel to make the process consistent. The gel was soaked in 
the solution with a sealed container for at least three days to reach equilibrium. The 
dual-crosslinked PVA hydrogels could be stored in the solution for about 2 weeks. 
Specimen Preparation 
For tensile relaxation tests, the specimen we used was rectangular. The size of the test 
part was about 30 mm × 10 mm × 2 mm. The specimen was cut from the as-prepared 
dual-crosslinked hydrogel sheets (about 50 mm width) using a 60 mm × 10 mm 
rectangular cutter (see Figure 2.2). After cutting, we used a digital caliper to measure 
8 
the width of the specimen to make sure that the width was accurate. The thickness of 
the specimen was hard to accurately measure with a caliper since the material is too 
soft. So we built a measurement tool with a Newport translational stage and a 
GAERTNER modular microscope. The measurement tool is shown in Figure 2.3. We 
first aligned the center of the microscope crosshair with one edge of the specimen. The 
we moved the microscope with the translational stage until the crosshair center was 
aligned with the other edge of the specimen. Then the displacement was the thickness 
of the specimen. 
 
Figure 2.2: Specimen preparation for uniaxial tests. 
9 
 
Figure 2.3: Thickness measurement tool. 
2.3 Tensile Tester 
For the uniaxial tests that we performed in this study, we used a custom-built tensile 
tester [55], as shown in Figure 2.4. In the following sections, we will give more details 
of the setup. 
Translational Motion 
The translational motion was provided by a Zaber X-LSM200A-E03 translational 
stage. The stage was mounted on a Newport optical stand and connected to the 
hydrogel specimen via a load cell, an aluminum rod and aluminum grips. And it was 
controlled by the laptop in the lab through Zaber Console Software. 
10 
 
Figure 2.4: Custom-built tensile tester. 
Gripping 
During the tests, the hydrogel samples were held by two aluminum grips. On each 
individual grip, the specimen was held by two parallel aluminum plates which were 
connected by the screws. On both plates, we glued sandpaper sheets to prevent the 
specimen from slipping. Figure 2.5 gives a close-up view of the sample with the grips. 
11 
Because the gel is soft and brittle, the screw should not be tightened too much. When 
the screw was being tightened, the sample would be squeezed out a bit and there 
would be a residual stress in the specimen. So before the test, we needed to stretch the 
sample a bit (about 1 mm) to cancel out the effect of residual stress. 
 
Figure 2.5: A close-up view of the specimen. 
Oil Container 
For a certain specimen, one set of uniaxial tests could cost about 2 hours. Because 
90% of the hydrogel is water, the hydrogel sample was prone to dry in the air. So for 
most experiments, the hydrogel specimen was immersed in mineral oil to prevent 
drying. The custom-built oil container, as shown in Figure 2.4, has a removable front 
12 
door which is attached to the container using screws. Between the door and the body, 
we placed rubber foam to improve sealing. When we did the experiments, we held the 
sample first and then closed the door and filled the container with mineral oil. One 
thing should be noticed is that the oil exerts a buoyant force on the grips and this force 
will change during the loading or unloading. So calculation is needed to eliminate the 
systematic error. 
Data Collection 
In the uniaxial tests, the data we are interested in are the stress, the strain, and the time. 
In order to calculate the nominal stress, we needed to know the force and the cross-
sectional area of the specimen. The force was measured by an Interface SMT 1-1.1 
load cell (1.1 lbf capacity). The force signal we obtained from the load cell was passed 
to a CALEX 163MK signal conditioner to be amplified and filtered. Then it was sent 
to a Keithley Model 2701 data logger. The signal we obtained is in voltage unit, so we 
needed to convert it into force. We hung dead weights (from 0 g to 20 g, which is 
consistent with the applied load range in the tests) on the grips and recorded the 
corresponding voltage. We could perform a linear fitting on the weight and the 
voltage. The slope was obtained as the conversion factor (unit: N/V). The force can be 
calculated by multiplying the voltage by the conversion factor. To calculate the strain, 
we needed to measure the displacement, using an OMEGA LD 620 LVDT. The signal 
of the LVDT was passed directly to the data logger. And the conversion factor of the 
LVDT was provided as 20 mm/V. We could also obtain the factor using the same 
method as the force measurement. The Keithley data logger can record data at a 
maximum speed of 25 data pairs per second when sampling the data at two channels. 
13 
In our experiments, we usually recorded the data every 0.1 second. We used a 
Keithley Kickstart Software to control the data logger on laptop. Figure 2.6 shows the 
connection of the load cell, LVDT and data logger. 
 
Figure 2.6: Connection of the data collection devices.
14 
 
CHAPTER 3 
THE EFFECT OF HYDRATION ON THE 
CONSTITUTIVE RESPONSE OF THE MATERIAL 
 
3.1 Introduction 
For hydrogels with transient physical crosslinks, hydration can have significant effects 
on the mechanical response. Varying the hydration level changes the breaking and 
reattaching rates of the physical crosslinks, which is equivalent to varying the loading 
rates. Understanding the effects of hydration on the mechanical response will facilitate 
the practical application of such materials under different environments. The effect of 
hydration on hydrogels including drying and shrinkage has been studied by several 
researchers [30][31][32][33][34][35][36][37][38][39]. However, these studies do not 
fully characterize the complex, time dependent behavior of viscoelastic gels. Based on 
the constitutive model developed by Guo et al.[41], the material parameters at 
different hydration levels can be obtained by fitting the model prediction to the 
experimental results. Analyzing those material parameters provides quantitative 
understanding of the hydration effects and the inherent relation between the material 
response during uniaxial tension tests. 
3.2 Experiment Overview 
3.2.1 Mass and Volume Measurement 
15 
 
The hydration levels of the hydrogels were characterized by the amount of mass loss 
during drying. In this section, we want to find out how much the gel shrank as it 
dehydrated. So we measured what we called “mass-volume relationship” for the PVA 
dual-crosslinked hydrogel at different hydration levels. In other words, we measured 
the volume percentage (the volume of the sample after drying divided by the initial 
volume of the sample) for a given mass percentage (the mass of the sample after 
drying divided by the initial mass of the sample). This “mass-volume relationship” 
will also help us calculate the nominal stress for the tests done using drying gels. More 
details can be found in section 3.2.2. The following paragraphs will explain the details 
of the measuring. 
We cut the fully hydrated hydrogel sheet into many different 10 𝑚𝑚 × 10 𝑚𝑚 
samples. For each sample, the mass and volume of were measured initially when it 
was fully hydrated. We then measured its mass every 5-10 minutes until it reached 
95% of the initial mass. And then we measured its volume. Then we used new samples 
to repeat this procedure at the 90%, 85%, 80%, 75%, 70%, 65% and 60% of their 
initial mass. The mass was measured by a Mettler Toledo AG285 microbalance with 
an accuracy of 0.01 mg. The volume was measured by weighing the sample in air and 
in mineral oil using a density determination kit (DDK) on the microbalance. A 
schematic of the DDK is shown in Figure 3.1. The red parts are connected to the 
microbalance plate so that we can measure the mass of the sample on the plate or in 
the basket. The black parts hold up the beaker and are connected to the base of the 
balance, not to the balance plate itself. 
16 
 
 
Figure 3.1: Schematics of the density determination kit. 
First, we measured the weight of the gel 𝑊𝑎 on the plate in air. The mass of the sample 
in oil, 𝑊𝑜, was measured by placing it on the basket which was immersed in oil. By 
Archimedes principle, the buoyancy equals the volume of the sample (𝑉𝑠𝑎𝑚𝑝𝑙𝑒) 
multiplied by the density of the oil, 𝜌𝑜𝑖𝑙, times the gravitational acceleration, 𝑔. The 
buoyancy is 
𝐹𝑏 = 𝑊𝑎 − 𝑊𝑜 (3.1) 
 
From 
𝑊𝑎 − 𝑊𝑜 = 𝜌𝑜𝑖𝑙𝑔𝑉𝑠𝑎𝑚𝑝𝑙𝑒 (3.2) 
 
We obtain 
𝑊𝑎 − 𝑊𝑜
𝑉𝑠𝑎𝑚𝑝𝑙𝑒 = (3.3) 𝜌𝑜𝑖𝑙𝑔
17 
 
The density of the oil was measured by using a pipette to inject 1 mL of oil into a 
container on the microbalance. The reported density value is the average of three 
measurements. 
3.2.2 Uniaxial Tests 
We performed uniaxial tests on the hydrogel specimens at different loading histories 
and different hydration levels. The tests were performed with a custom-built tensile 
tester. All the tests were done in oil to prevent the specimen from further drying. The 
details of the setup and the geometry of the specimen can be found in Chapter 2. 
At each hydration level (100, 90, 80, 70 and 60%), six tests were performed. Tests 
were performed sequentially using the same sample for consistency. Between each 
test, the sample was allowed to relax for about 12 minutes to fully recover to its initial 
state. We carried out three different types of tests. For the fully hydrated gels (100% of 
mass), 90% and 80% mass gels, we first carried out cyclic tests where the sample was 
loaded to a stretch of 𝜆 = 1.3 at stretch rates of ?̇? = 0.003/𝑠, 0.01/𝑠, 0.03/𝑠 and then 
unloaded to 𝜆 = 1 at the same rate. Second, we carried out a complex loading history 
in which the sample was loaded to a stretch of 𝜆 = 1.15 at a rate of ?̇? = 0.003/𝑠, held 
for 1 minute and then loaded to 𝜆 = 1.3 at ?̇? = 0.03/𝑠, held for another 1 minute and 
then unloaded to 𝜆 = 1 at a rate of ?̇? = 0.01/𝑠. Then the sample was loaded to 𝜆 =
1.3 at  ?̇? = 0.1/𝑠 and unloaded to 𝜆 = 1 at ?̇? = 0.001/𝑠. This test is followed by a 
tensile-relaxation test at a maximum stretch of 𝜆 = 1.3 with an initial stretch rate of 
?̇? = 0.5/𝑠. For the 70% and 60% mass gels, the tensile relaxation test is performed 
18 
 
first; followed by the complex loading history test and loading-unloading tests at 
different stretch rates. 
Results are presented in terms of the nominal stress, 𝜎, i.e. applied force divided by 
the initial cross-sectional area of the sample. For the fully hydrated gel, the cross-
sectional area was 2 𝑚𝑚 × 10 𝑚𝑚 . For the drying gels, we assume that the gel 
shrinks uniformly in all three dimensions, thus the cross-sectional area is calculated 
using 2 𝑚𝑚 × 10 𝑚𝑚 × (𝑉%)2/3, where 𝑉% is the volume percentage (volume after 
drying divided by initial volume at full hydration). The value of 𝑉% is found from the 
mass-volume relationship described below. 
3.3 Results 
3.3.1 Mass-Volume Relationship 
The results of mass-volume experiments are shown in Figure 3.2. The volume 
percentage (volume after drying divided by initial volume) decreases linearly with the 
mass percentage (mass after drying divided by initial mass). The data in Figure 3.2 can 
be fit using 𝑉% = 1.03𝑀% − 3.04%, where 𝑉% is the volume percentage and 𝑀% is 
the mass percentage. This equation allows us to calculate the undeformed cross-
sectional area of the gel at different hydration levels as described above. 
19 
 
 
Figure 3.2: Mass-Volume relationship of PVA hydrogels during drying. 
3.3.2 Uniaxial Tests Results 
Figures 3.3a-d show the stress-strain curves for PVA hydrogels at different hydration 
levels for the cyclic test at different loading-unloading rates. Figure 3.3e shows the 
stress-strain curves for PVA hydrogels at different hydration levels for the complex-
loading-history loading test. Figure 3.3f shows the stress-time curves for PVA 
hydrogels at different hydration levels in tensile-relaxation tests. 
20 
 
 
Figure 3.3: a) Nominal stress vs. stretch at five hydration states.  Loading-
unloading at stretch rate of ?̇? = 0.003/𝑠. b) Nominal stress vs. stretch at five 
hydration states. Loading-unloading at stretch rate of ?̇? = 0.01/𝑠. c) Nominal 
stress vs. stretch at five hydration states. Loading-unloading at stretch rate of 
?̇? = 0.03/𝑠. d) Nominal stress vs. stretch at five hydration states. Loading at 
stretch rate of ?̇? = 0.1/𝑠  and unloading at stretch rate of ?̇? = 0.001/𝑠 . e) 
Nominal stress vs. stretch at five hydration states. A complex loading history 
(see text for description). f) Nominal stress vs. time in log scale for five 
hydration states in relaxation tests. Samples are loaded at ?̇? = 0.5/𝑠  to a 
maximum stretch of 1.3. 
 
 
These figures show that for all stretch rates the stress is significantly higher as the gel 
dries.  Figures 3.3a-e show that stress increases slowly at the initial stages of loading 
21 
 
for the 70 and 60% hydration levels. Except for this initial dip, the maximum nominal 
stresses of the drier gels are higher, and across all tests we see that the gel stiffens as it 
dehydrates. It appears that that gel samples with mass percentages less than or equal to 
70% did not had sufficient time to recover after the previous experiment. Indeed, we 
observed that these gels did not fully recover to their initial state after unloading to a 
stretch 𝜆 = 1 after the tensile-relaxation test, which indicates that in the drier gels the 
network may undergo permanent change during the loading. For fully hydrated 
hydrogels and 70% mass hydration level hydrogels, we did a recovery test. In this test, 
we first loaded the sample to stretch 𝜆 = 1.3 at stretch rate ?̇? = 0.5/𝑠, then we held 
the sample at 𝜆 = 1.3 to perform a relaxation test. After 30 minutes of relaxation, we 
unloaded the sample to 𝜆 = 1 at rate ?̇? = 0.001/𝑠. Then the sample was held for 4 
hours to recover. We found that the 100% hydrated gels recovered fully after 700 s. 
However, the 70% gel did not fully recover even after 4 hours. The results of the  
recovery test are given in the Appendix. 
3.4 Data Analysis 
3.4.1 Review of the Constitutive Model 
The constitutive model of this PVA dual-crosslinked hydrogel was first developed by  
Long et al. [40] and later modified by Guo et al. [41]. For fully hydrated PVA dual-
crosslink hydrogel at room temperature, it has been shown that the constitutive model 
can accurately predict the stress vs. stretch behaviors of uniaxial tension tests. Here we 
briefly review the model. 
The key assumptions in this model are: 
22 
 
1. The elastic strands in the polymer network are connected by both chemical 
crosslinks and physical crosslinks. The chemical crosslinks behave as the 
permanent network during the deformation that they don’t break. The physical 
crosslinks can break and reform during the loading and show the viscoelastic 
behavior. 
2. The total strain energy in the network is equal to the sum of the strain energy 
carried by each individual polymer chains. 
3. Macroscopically, the hydrogel is assumed to be incompressible and isotropic. 
The chains between physical and chemical networks deform in the same way when 
they are subject to the same stress. Thus, the same strain energy model applies to 
both permanent and transient strands. The strain energy of the undamaged network, 
𝑊0, is assumed to depend only on the first invariant of the right Cauchy-Green 
tensor, 𝐼1 = 𝑡𝑟[(𝑭
0→𝑡)𝑇𝑭0→𝑡. 
4. The breaking and reforming of the physical crosslinks reaches a dynamic 
equilibrium state before the hydrogel is subject to any loading, which implies that 
the breaking and reforming rates are equal and independent of time. Also the 
breaking and reforming rates are independent of the strain history. This rate is 
denoted by ?̅?∞. 
5. The stress in the strand that attaches to the physical crosslinks is instantaneously 
relaxed when the crosslinks break. The strand does not carry any strain energy at 
this time. When the physical crosslinks reform at time 𝜏, the deformation of the 
23 
 
corresponding chain is governed by the deformation gradient tensor 𝑭𝜏→𝑡, where t 
is the current time. 
With all these assumptions, the nominal stress tensor P can be related to the 
deformation gradient tensor 𝑭𝜏→𝑡 by 
𝑑𝑊
𝑷 = −𝑝(𝑭0→𝑡)−𝑇
0
+ 2[𝜌 + 𝑛(𝑡)] | × 𝑭0→𝑡 +
𝑑𝐼1 𝐼1=𝐻(0,𝑡)
𝑡  𝑡 − 𝜏 𝑑𝑊0
2?̅?∞ ∫ 𝜙𝐵 ( ) | × 𝑭
𝜏→𝑡(𝑭0→𝜏)−𝑇𝑑𝜏 (3.4)
0 𝑡𝐵 𝑑𝐼1 𝐼1=𝐻(𝜏,𝑡)
 
Where 
• 𝑝 is the Lagrange multiplier that enforces incompressibility. 
• 𝜌 is the molar fraction of the chemical crosslinks. 
• 𝑊0 is the strain energy. In this study, we assume that all chains are Gaussian. 
Then the energy is given by the neo-Hookean model: 
𝜇
𝑊0 = (𝐼1 − 3) (3.5) 2
where 𝜇 is the small strain shear modulus of neo-Hookean model. 
• ?̅?∞ is the steady state reattachment rate of the physical chains, i.e., molar 
fraction of the physical chains reattached per unit time, hence ?̅?∞𝑑𝜏 is the 
number of newly reattached physical chains. 
• The integrand 
1
𝑡 − 𝜏 𝑡 1−𝛼𝐵
(?̅?∞𝑑𝜏)𝜙𝐵 ( ) = ?̅?∞ [(1 + (𝛼𝐵 − 1) ) ] 𝑑𝜏 (3.6) 𝑡𝐵 𝑡𝐵
24 
 
is the number of chains that healed between 𝜏 and 𝜏 + 𝑑𝜏 and survives until 𝑡, 
where 𝑡𝐵 is the characteristic time for breaking; 1 < 𝛼𝐵 < 2 is a material 
constant that specifies the rate of decay of 𝜙𝐵. 
• 𝑛(𝑡) is the molar fraction of physical bonds at 𝑡 = 0 and still attached at time 
𝑡; it is given by 
2−𝛼
0 𝐵𝑡 − 𝜏 𝑡𝐵 𝑡 1−𝛼𝐵
𝑛(𝑡) = ?̅?∞ ∫ 𝜙𝐵 ( ) 𝑑𝜏 = ?̅?∞ (1 + (𝛼𝐵 − 1) ) (3.7) 
−∞ 𝑡𝐵 2 − 𝛼𝐵 𝑡𝐵
For uniaxial case, the equation can be simplified to the following form: 
1 𝑡 𝑡 − 𝜏 𝜆(𝑡) 𝜆(𝜏)
𝜎 = 𝜇(𝜌 + 𝑛(𝑡)) [𝜆(𝑡) − ] + 𝜇?̅? × ∫ 𝜙 ( ) [ − ] 𝑑𝜏 (3.8) 
𝜆2(𝑡) ∞ 𝐵 2 20 𝑡𝐵 𝜆 (𝜏) 𝜆 (𝑡)
where 𝜎 is the nominal stress and 𝜆 is the stretch. 
In this model, the constitutive relationship is completely specified by four independent 
material parameters: 𝜇𝜌, 𝜇?̅?∞, 𝛼𝐵 and 𝑡𝐵. For physical understanding, : 𝜇𝜌  
corresponds to the shear modulus of the network crosslinked only by the chemical 
crosslinks. 𝜇?̅?∞ can be roughly thought of as a parameter measuring how stresses 
increase with the number of reattached physical chains per unit time. 1 < 𝛼𝐵 < 2  
controls the average survival time of a newly attached transient bond and 𝑡𝐵 is the 
characteristic time for breaking. 
3.4.2 Determination of Material Parameters 
In our previous work, we determined the four material parameters mainly using the 
stress relaxation test parameters [41]. We then used these parameters and equation (2c) 
to predict the stress-stretch curve for cyclic test. These predicted stress strain curves 
are then compared with experiments and manually adjusted to provide a good overall 
25 
 
fit. Here we accelerated the fitting process using a machine learning algorithm (MLA) 
to determine these material parameters. Details of this method can be found in our 
recent paper [56] and a brief summary of the method is given below. 
Based on our constitutive model [41], for any fixed strain history, we can calculate the 
stress history for different sets of parameters. If we randomly pick several points from 
our parameter space and calculate the stress history and then put them together, we can 
get a stress matrix and apply singular value decomposition on it to get the basis and 
principal components of each stress history. Then we use the principal components of 
all those stress histories to train a Gaussian process. After training, under the same 
strain history, this Gaussian process can make predictions about the principal 
components of the stress history of any parameters it has not seen. In this specific 
problem, we can train 1000 points in the parameter space with 6 different strain 
history which corresponds to the six uniaxial experiments we have performed. And 
finally it will go through 1 million points in the parameter space to find the best fitting 
results. The workflow of this method is shown in Figure 3.4. 
26 
 
 
Figure 3.4: Workflow of Gaussian Process to identify constitutive model. (a) 1000 
stress history curves calculated directly from the constitutive model are 
decomposed into basis and projections on this basis (i.e. principal components) 
using SVD; (b) the principal components of the 1000 parameter sets are taken as 
training set. After training, the GP can make predictions about the principal 
components of any parameter set not in the training set; (c) with the predicted 
principal components, the stress history of any parameter set can be predicted by 
applying SVD inverse transform to the predicted principal component without 
calculating the constitutive model; (d) the best parameters for the experimental 
data can be obtained by comparing the experimental stress history with the stress 
histories calculated for a large number of parameter sets. 
 
 
We use all the six experiments to fit the data of 100%, 90%, 80% mass hydration 
levels.  We use only the first 200 seconds of the relaxation tests to fit the data for the 
70% and 60% mass hydration level hydrogels since the initial dip in stress that occurs 
in the cyclic loading tests cannot be fit by the model. The parameters obtained using 
our MLA are shown in Table 3.1. 
27 
 
𝝁?̅?∞𝒕𝑩
Hydration 𝝁𝝆(𝒌𝑷𝒂) 𝜶𝑩 𝒕𝑩(𝒔) 𝝁?̅?∞(𝒌𝑷𝒂/𝒔) (𝒌𝑷𝒂) 𝟐 − 𝜶𝑩
100% mass 2.347 1.634 0.4618 19.91 25.15 
90% mass 2.567 1.633 0.4335 26.72 31.53 
80% mass 2.640 1.664 0.4250 31.62 40.05 
70% mass 2.994 1.674 0.3383 48.76 50.62 
60% mass 3.439 1.687 0.2338 101.2 75.55 
 
Table 3.1: Data fitting results for PVA hydrogels at different hydration levels. 
 
 
Comparisons of model and experiments are shown in Figures 3.5-3.9. Figures 3.5-3.7 
show that there is excellent agreement between experiments with different loading 
histories and constitutive model as long as drying is not so severe that permanent 
deformation is experienced by the samples. For the 60% and 70% hydrated gels, our 
model fails to predict the cyclic tests and the complex loading test. As shown in 
section 3.3.2, for samples at 70% and 60% mass, the stress increases slowly at the 
initial part of loading for the loading-unloading tests. This is likely due to permanent 
deformation remaining after the relaxation test. For these samples, our constitutive 
model does not work well. Hence, we use only the first 200 seconds of relaxation test 
to compare with the model (recall that the relaxation test was performed on a virgin 
sample at 70% and 60% mass hydration level). 
28 
 
 
Figure 3.5: Comparison between model and experiments for fully hydrated (100% 
mass) PVA hydrogel. Nominal Stress vs. Stretch for (a) Loading-unloading to 
stretch 𝜆 = 1.3 at stretch rate 0.003/s. (b) Loading-unloading to stretch 𝜆 = 1.3 
at stretch rate 0.01/s. (c) Loading-unloading to stretch 𝜆 = 1.3 at stretch rate 
0.03/s. (d) Loading to stretch 𝜆 = 1.3 at stretch rate 0.1/s and unloading at stretch 
rate 0.001/s. (e) A complex loading history which was mentioned above. (f) 
Nominal Stress vs. Time for loading at stretch rate 0.5/s to a stretch of 1.3 and 
then holding as the stress relaxes. 
 
29 
 
 
Figure 3.6: Comparison between model and experiments for 90% mass PVA 
hydrogel. Nominal Stress vs. Stretch for (a) Loading-unloading to stretch 𝜆 = 1.3 
at stretch rate 0.003/s. (b) Loading-unloading to stretch 𝜆 = 1.3 at stretch rate 
0.01/s. (c) Loading-unloading to stretch 𝜆 = 1.3 at stretch rate 0.03/s. (d) Loading 
to stretch 𝜆 = 1.3 at stretch rate 0.1/s and unloading at stretch rate 0.001/s. (e) A 
complex loading history which was mentioned above. (f) Nominal Stress vs. Time 
for loading at stretch rate 0.5/s to a stretch of 1.3 and then holding as the stress 
relaxes. 
30 
 
 
Figure 3.7: Comparison between model and experiments for 80% mass PVA 
hydrogel. Nominal Stress vs. Stretch for (a) Loading-unloading to stretch 𝜆 = 1.3 
at stretch rate 0.003/s. (b) Loading-unloading to stretch 𝜆 = 1.3 at stretch rate 
0.01/s. (c) Loading-unloading to stretch 𝜆 = 1.3 at stretch rate 0.03/s. (d) Loading 
to stretch 𝜆 = 1.3 at stretch rate 0.1/s and unloading at stretch rate 0.001/s. (e) A 
complex loading history which was mentioned above. (f) Nominal Stress vs. Time 
for loading at stretch rate 0.5/s to a stretch of 1.3 and then holding as the stress 
relaxes. 
31 
 
 
Figure 3.8: Comparison between model and experiments for 70% mass PVA 
hydrogel. (a) Nominal Stress vs. Stretch for loading at stretch rate 0.5/s to a stretch 
of 1.3 and then holding as the stress relaxes. (b) Nominal Stress vs. Time for loading 
at stretch rate 0.5/s to a stretch of 1.3 and then holding as the stress relaxes. 
 
 
 
Figure 3.9: Comparison between model and experiments for 60% mass PVA 
hydrogel. (a) Nominal Stress vs. Stretch for loading at stretch rate 0.5/s to a stretch 
of 1.3 and then holding as the stress relaxes. (b) Nominal Stress vs. Time for loading 
at stretch rate 0.5/s to a stretch of 1.3 and then holding as the stress relaxes. 
 
3.5 Discussion 
3.5.1 Effect of Dehydration 
Volume Change 
In section 3.3.1 and Figure 3.2, we showed that the volume percentage (volume after 
drying divided by initial volume) decreases linearly with the mass percentage (mass 
32 
 
after drying divided by initial mass). The relationship can be described by equation 
𝑉% = 1.03𝑀% − 3.04%, where 𝑉% is the volume percentage and 𝑀% is the mass 
percentage. The equation can also be written as, 
𝑉𝐷 𝑀𝐷
= 1.03 × − 0.0304 (3.9) 
𝑉𝐼 𝑀𝐼
where 𝑉𝐼 is the volume after drying, 𝑀𝐷 is the initial volume, 𝑀𝐼 is the mass after 
drying and 𝑉𝐷 is the initial mass. Rounding the above, equation can be written as 
𝑉𝐷 𝑀𝐷
− 1 = 1.03 × ( − 1) (3.10) 
𝑉𝐼 𝑀𝐼
or 
𝑀𝐼 𝑀𝐼 − 𝑀𝐷
= 1.03 × (3.11) 
𝑉𝐼 𝑉𝐼 − 𝑉𝐷
The left side of the equation is the density of the initial sample which is fully hydrated. 
In our experiment, the measured density of the fully hydrated PVA hydrogel is 1.006 
𝑔/𝑐𝑚3. So we can obtain that (𝑀𝐼 − 𝑀𝐷)/(𝑉 − 𝑉 ) = 0.977 𝑔/𝑐𝑚
3
𝐼 𝐷 . We know that 
𝑀𝐼 − 𝑀𝐷 is the mass loss of water during drying. If we consider the water and 
polymer network as two separate parts, then (𝑀𝐼 − 𝑀𝐷)/(𝑉𝐼 − 𝑉𝐷) should be exactly 
the density of the water, which is 1 𝑔/𝑐𝑚3. Therefore, the results show that while the 
volume loss of the PVA gel during drying is mainly caused by water loss, additional 
shrinkage occurs due to other mechanisms. 
Mechanical Properties 
According to the constitutive model, at long time (when the physical network does not 
carry any load), the behavior of the nominal stress 𝜎 ≡ 𝜎∞ in a relaxation test of 
stretch 𝜆0 is 
33 
 
1
𝜎∞ = 𝜇𝜌 (𝜆0 − (2) 3.12) 𝜆0
Thus, the parameter 𝜇𝜌 is the equilibrium modulus which is governed by the 
permanent (chemical) network. 
The initial slope of the stress versus time curve in a tensile relaxation test is 
𝜎 𝜇?̅?∞𝑡𝐵
lim = 3?̇? (𝜇𝜌 + ) (3.13) 
𝑡→0 𝑡 2 − 𝛼𝐵
The parameter ?̅?∞𝑡𝐵/(2 − 𝛼𝐵) represents the molar fraction of connected physical 
chains and hence 𝜇?̅?∞𝑡𝐵/(2 − 𝛼𝐵) can be interpreted as the instantaneous modulus 
associated with the physical (transient) network. Figure 3.10 shows the change of both 
𝜇𝜌 and 𝜇?̅?∞𝑡𝐵/(2 − 𝛼𝐵) with respect to hydration level. 
 
Figure 3.10: The change of 𝜇𝜌 and 𝜇?̅?∞𝑡𝐵/(2 − 𝛼𝐵) with respect to the hydration 
level. 
 
 
From Table 1 or Figure 3.10,  𝜇𝜌 increases by about 47% when the PVA gel dries 
from 100% to 60% mass. This means the chemical network is stiffer and carries more 
34 
 
load when the gel dries. Table 1 shows that 𝛼𝐵 changes little when the gel dehydrates 
while 𝑡𝐵  decreases and 𝜇?̅?∞ increases. This suggests physical bonds break and 
reattach faster when the gel loses water.  From the last column in Table 1, and Figure 
3.10, 𝜇?̅?∞𝑡𝐵/(2 − 𝛼𝐵), increases by a factor or  three when the PVA gel dries from 
100% to 60% mass. Thus, the short time modulus of the gel, which involves both the 
physical bonds and chemical network, increases strongly with drying. From Table 1 
and Figure 3.10, we see that the drier PVA gels are stiffer and hat the water content 
has a much more significant influence on the physical crosslinks than the chemical 
crosslinks as evidenced by the far greater increase in 𝜇?̅?∞𝑡𝐵/(2 − 𝛼𝐵) relative to 𝜇𝜌. 
Physically, the loss of water will enhance the ion concentration inside the hydrogel, 
reduce screening and making the ionic interaction more active and stronger. Hence, 
the hydration level should have a more significant influence on the physical 
interaction than the permanent chemical network. This is consistent with our 
experimental results. 
3.5.2 Limitations of the Model 
For the 70% and 60% mass PVA dual-crosslinked hydrogels, our constitutive model 
cannot totally explain the experimental results. These gels do not fully recover to their 
initial state after the tensile-relaxation test, which indicates that the network may have 
suffered permanent change during the test. In the Appendix, we compare the results of 
the recovery test for fully hydrated hydrogels and 70% mass hydration level 
hydrogels. Our result shows that the 70% mass gels have a large residual stress after 
unloading to their initial length and that this residual stress persists even after the gel is 
35 
 
rested for several hours. Figure 3.3f shows that the nominal stress in tensile-relaxation 
test after 1000 second drops rapidly for the 70% and 60% mass gels and the stress 
eventually becomes smaller than the stress of gels at higher hydration levels (please 
see the zoom-in Figure for the long-time stresses for these cases in the Appendix). 
This result contradicts our model which predicts that at long times, physical bonds do 
not carry load in a relaxation test and the stress should plateau at 𝜇𝜌(𝜆 −20 − 𝜆0 ), where 
𝜆0 is the imposed stretch. Since 𝜇𝜌 is the modulus associated with the chemical 
crosslinks, which increases as the gel dries (see Table 1), the plateau stress should be 
higher as the gels dry instead of lower. Likewise, the stress of 60% mass hydration 
level hydrogel did not reach a plateau within our time of observation. Recall that we 
fit the relaxation experiment using first 200 seconds of data and it fits well for the 
loading part and the start of the relaxation part. These results, i.e., the rapid drop in 
stress and the lower long-time stress for gels at 70% and 60% mass hydration level 
suggest some permanent change to the chemical network during deformation of the 
60% and 70% mass gels. Thus, our constitutive model does not totally explain the last 
part of the relaxation test as well as the tension tests performed following the 
relaxation tests. This is a subject for future work. 
3.6 Conclusion 
In this Chapter, we present experimental results on how drying changes the 
mechanical properties of a PVA dual-crosslinked hydrogel. We performed six 
experiments on this hydrogel with different strain histories under uniaxial stress state 
and use a new fitting method which is based on Gaussian Process Machine learning to 
36 
 
obtain the material parameters for our constitutive model. For gels that are moderately 
hydrated, our constitutive model is shown to be highly accurate and able to predict 
complex loading behavior. Our model shows both the physical and chemical network 
stiffens as the gel dries. However, drying stiffens the physical network much more 
than the chemical network.
37 
 
CHAPTER 4 
STUDY ON THE EFFECT OF HYDRATION: 
FRACTURE TESTS  
 
4.1 Introduction 
To facilitate applications of hydrogels, their fracture behavior needs to be better 
understood. Understanding of the fracture mechanism of the material can provide 
insight into approaches to designing hydrogel systems with higher toughness. 
Furthermore, results on the fracture behavior of hydrogel at different hydration level 
can help us understand the change in hydrogel when dehydrating. 
There have been a lot of studies on the fracture of different kinds of hydrogels, such as 
studies on energy dissipating [48][49][50], and studies on the fatigue fracture 
[52][53][54]. In this study, we use a crack tip stress based model developed by 
Mincong [55] to study the fracture behavior of PVA dual-crosslinked hydrogel at 
different hydration levels. 
4.2 Experiment 
Experiment Overview 
At each hydration level (100, 90, 80, 70, and 60% of initial mass), a constant-rate 
loading fracture test was performed with a specimen which had a crack. Figure 4.1 
shows the sketch of the geometry of the specimen. The loading rate is the same, 
0.003/s, for all hydration level. We stretched the sample until it broke. Stress history 
38 
 
and breaking time were recorded. Then we performed several creep fracture tests at 
different hydration level under different creep stress with the sample in same 
geometry. For fully hydrated hydrogels, the stress level was 2.5 kPa, 3 kPa, and 4 kPa. 
For hydrogels at 80% mass hydration level, the stress level was 2.5 kPa, 3kPa, 4kPa 
and 5 kPa. 
Experiment Setup 
The experiments were performed using the custom-built setup introduced in Chapter 
2. For the constant rate loading tests, we still used the Zaber Console Software to 
control the translation stage to provide a stretch. And we used the Kickstart Software 
to control the data logger to record the force and displacement. But this open-loop 
control, the translation stage controls the motion of the top grip which stretches the 
specimen and thus induces force with the specimen, can not keep the force constant in 
a creep test. Then a closed-loop force control scheme is required for the creep test.  
To control the applied force in the creep test, we used a feedback load control, the PID 
control, through MATLAB code. For each discrete time step, the current force signal 
𝑉𝑐𝑢𝑟𝑟𝑒𝑛𝑡 was read from the Keithley data logger into MATLAB. We compared this 
signal with the target force 𝑉𝑡𝑎𝑟𝑔𝑒𝑡 and calculated the difference 𝑉𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 =
𝑉𝑡𝑎𝑟𝑔𝑒𝑡 − 𝑉𝑐𝑢𝑟𝑟𝑒𝑛𝑡. Then the difference 𝑉𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 is incorporated in a PID algorithm 
to compute a velocity correction term. Then the velocity correction term was sent to 
Zaber translation stage to adjust the loading rate. More details about this feedback 
control will be given in Appendix. 
Specimen Preparation 
39 
 
Same as Section 2.2, the specimen was cut from a PVA sheet using the 
60 𝑚𝑚 × 10 𝑚𝑚 cutter. Usually the test part was 30 𝑚𝑚 × 10 𝑚𝑚 and the thickness 
of the specimen was 2 mm. Since we were doing a crack test, a 3 mm crack was made 
on the edge of the specimen using a X-ACTO knife, as shown in Figure 4.1. 
 
Figure 4.1: Sketch of the geometry of a specimen for crack test. 
 
Recall that we did the crack test not only for the fully hydrated hydrogels, but also for 
drying gels. As the gel dried, it lost water and shrank. The way for preparing a 
specimen corresponding to its hydration level is shown below. 
1. We cut a 10 mm width sample from the fully hydrated hydrogel sheet and weighted 
it. 
2. Every 10-15 minutes, we weighted the sample until it reached the target weight, for 
example, 80% of its initial weight. 
3. In section 3.3.1, we obtained the mass-volume relationship for the hydrogel. Then 
we calculated the volume percentage V% using equation 𝑉% = 1.03𝑀% − 3.04%. 
40 
 
Similar with the calculation of cross-sectional area, we assumed that the hydrogel 
sample shrank uniformly in all 3 directions. Then the width of the sample was 
calculated as 𝑑 = 10 𝑚𝑚 × (𝑉%)1/3. This was also verified by the measurement 
using a digital caliper. 
4. We cut a 0.3d long crack on the edge of the specimen and then gripped the 
specimen to the setup. 
Same as the uniaxial tests, all experiments were performed in oil and the buoyancy 
was subtracted through calculation to eliminate the system error. 
4.3 Results 
4.3.1 Constant Rate Loading Fracture Test 
Figure 4.2 shows the stress vs. stretch curve for PVA dual-crosslinked hydrogels at 
different hydration levels. As the gel dried, the stress increased faster, which was 
consistent with results in section 3.3.2 that the hydrogel became stiffer as it dried. The 
breaking stress also increased as the gel lost water. However, the breaking stretch for 
the hydrogel at different hydration levels did not show any regular pattern. Since all 
loading rates were the same, the breaking time did not show any regular pattern, too. 
41 
 
 
Figure 4.2: Stress vs. stretch for hydrogels at different hydration levels in crack test. 
4.3.2 Creep Tests 
The results for the constant stress tests of fully hydrated hydrogel and 80% mass 
hydration level hydrogel are shown in Figure 4.3 and 4.4. As creep occurs the stretch 
increases with time. This behavior is because the physical crosslinks break and reform 
during loading. When a physical crosslink breaks, the polymer chain that attaches to it 
is relaxed and it does not carry any stress. Due to this relaxation mechanism, the 
specimen stretches even while the nominal stress is held constant. For all the nominal 
stress levels and hydration levels tested in this study, the specimen eventually 
42 
 
fractured. The fracture time increases as the nominal stress decreases.
 
Figure 4.3: Stretch vs. time under constant applied nominal stress, fully hydrated 
hydrogels. 
 
Figure 4.4: Stretch vs. time under constant applied nominal stress, 80% mass 
hydration level hydrogels. 
 
43 
 
4.4 Discussion 
From Figure 4.2, as the gels dry, they become stiffer and their breaking stress becomes 
higher. The breaking stress told us the proper stresses that we used in creep test. And it 
implied that under same applied nominal stress, the fully hydrated hydrogels would 
creep faster than the drying gels. The breaking time for fully hydrated hydrogel would 
also be shorter. Figure 4.5 and Figure 4.6 show the comparison between the fully 
hydrated hydrogel sample and 80% of initial mass hydrogel sample at the same 
applied nominal stress. It is consistent with our derivation.  
 
Figure 4.5: Comparison between fully hydrated hydrogel and 80% mass hydration 
level hydrogel in creep test with constant nominal stress 4 kPa. 
 
44 
 
 
Figure 4.6: Comparison between fully hydrated hydrogel and 80% mass hydration 
level hydrogel in creep test with constant nominal stress 3 kPa. 
 
Several works have been done on the relationship between the time to failure and the 
applied nominal stress in uniaxial creep test [50][51][55][57]. A thermally activated 
fracture process has been discovered. Under such condition, the time to failure is 
governed by the classic durability equation [58]: 
𝑈 − 𝛾𝜎
𝑡𝑓 = 𝜏0 exp ( ) (4.1) 𝑘𝑇
where 𝑡 −13𝑓 is the time to failure, 𝜏0 ≈ 10  s is the characteristic oscillation time of 
atoms in a solid, 𝑈 is the activation energy of fracture, 𝛾 is a material parameter, 
characterizing the activation volume of bond rupture, 𝜎 is the applied stress, 𝑘 is the 
Boltzmann constant (1.38 × 10−23𝐽 ∙ 𝐾−1), and 𝑇 is the absolute temperature.  
Based on equation 4.1, replacing the applied stress 𝜎 by the crack tip stress value 
𝐶𝐵
𝜎𝑐 = 𝑝 (4.2) 
√𝑟𝑐
45 
 
Mincong gave the governing equation for a creep test with a crack in [55]: 
𝛾 𝐶𝐵 𝑈
ln 𝑡𝑓 = − 𝑝 + + ln 𝜏 (4.3) 𝑘𝑇 √𝑟 𝑘𝑇
0
𝑐
where 𝐶𝐵 is the scale coefficient which was assumed to be constant, 𝛾 is the activation 
volume of fracture, 𝑟𝑐 is a distance used to compute the crack tip stresses, 𝛾/√𝑟𝑐 is a 
material parameter, and 𝑝 is the applied nominal stress. From equation 4.3, we can see 
that for a certain material, the log of time to failure will be linearly proportional to the 
applied far-field nominal stress. This linear relationship for fully hydrated hydrogel 
and 80% mass hydration level hydrogel is shown in Figure 4.7 and Figure 4.8. 
 
Figure 4.7: Time to failure vs. applied stress for fully hydrated hydrogels. 
46 
 
 
Figure 4.8: Time to failure vs. applied stress for 80% mass hydration level hydrogels. 
 
For fully hydrated hydrogel, linear fitting is good. However, for 80% mass hydration 
level hydrogel, the results of creep at 5 kPa is not in the linear relationship. But if we 
looked at the stress-time curve of this test, as shown in figure 4.9, the stress actually 
didn’t reach 5 kPa at all. The fracture occurred in the loading part. Then in an ideal 
creep test, the time to failure will be much shorter than the true experiment. 
47 
 
 
Figure 4.9: Stress vs. time in creep test under 5 kPa for 80% mass hydrogels. 
Although linear relationship was found between the time to failure and the applied 
nominal stress, there was not enough data to determine the material parameters using 
crack tests. If we want to find how hydration effect the fracture behavior, we need 
more experimental results. At each hydration level, we need to do a full set of crack 
tests, including constant loading rate tests at different rates and creep tests under more 
stress levels. That is the work for the further research. 
4.5 Conclusion 
In this chapter, we present the results of some basic crack tests on fully hydrated 
hydrogels as well as dehydrated hydrogels. The gels become stiffer, and the breaking 
stress increases as the gels dry. Constant applied nominal stress tests show that both 
fully hydrated hydrogel and 80% mass hydration level hydrogel follow the linear 
relationship between the log of the time to failure and the applied nominal stress. More 
48 
 
experiments are needed to determine the material parameters for crack test at different 
hydration levels.
49 
 
CHAPTER 5 
NUMERICAL AND EXPERIMENTAL STUDY ON 
THE EFFECT OF COMPRESSION IN THE 
RHEOLOGY TEST 
 
5.1 Introduction 
Small strain rheology test is often used to obtain the viscoelasticity parameter of the 
material, such as storage modulus, loss modulus, and tangent delta. One of the most 
often used geometry of rheology test is the parallel plate. In order to prevent specimen 
slippage during the experiment, the hydrogel specimen was pre-compressed. However, 
this pre-compression may affect the measurement of the rheology test. Sullivan et 
al.[59] developed a nonlinear viscoelastic model for viscoelastic behavior at large 
static deformation. Lee et al.[60] gave a new practical modulus characterization 
method and compared the model with the experiment. In our previous study [42][45], 
we multiplied the dynamics modulus by a shifting factor to correct the dynamic 
modulus. This shifting parameter was usually found by practice. In this study, we use 
both numerical and experimental method to characterize the effect of pre-compression 
in the rheology test for PVA dual-crosslinked hydrogel. 
5.2 Finite Element Analysis 
5.2.1 Methodology 
50 
 
To find how much the compression would affect the results in rheology test, we first 
used finite element analysis (FEA) method. We used the ABAQUS to do the 
calculation with a user material (UMAT) to satisfy the PVA model. The constitutive 
model used in FEA is based on the model introduced in section 3.4.1, developed by 
Long et al. and later modified by Guo et al. The UMAT is described in [43] by Jingyi 
Guo.  and details of the are given in Appendix. We first performed a FE simulation of 
an ideal torsion-rheometry test on a cylinder sample with 20 mm diameter and 2 mm 
thickness. Then a torsion-rheometry test under compression was simulated on a 
cylinder sample with 20 mm diameter and 2.1 mm thickness. We first compressed the 
sample from 2.1 mm to 2 mm thickness and then simulated the torsion-rheometry test. 
Both  simulations were performed at frequencies of 𝜔 = 0.01, 0.1, 1 𝑎𝑛𝑑 10. Then we 
compare the difference between the steady state torque on both samples.  
After that, we compared the steady state torque at the same frequency 𝜔 = 0.1 but 
under different compression (from 0% to 30%). 
5.2.2 FEM Calculation 
Geometry 
The basic geometry of the sample was a cylinder with 20 mm diameter and different 
thickness (from 2 mm to 2.857 mm) which was determined by the compression we 
want. Figure 5.1 shows the geometry in the ABAQUS. The partitions help us to 
control the mesh. 
51 
 
 
Figure 5.1: The geometry of the simulation. Black lines indicate partitions used to 
facilitate uniform meshing. 
Mesh 
In the calculation, we used the element type of 20-node quadratic brick for 3D stress 
(C3D20), using hybrid element to satisfy incompressible conditions. Along the 
cylindrical sample, at least 4 elements were needed for small compression and more 
elements were needed when the compression is larger than 10%. Figure 5.2 shows the 
mesh in ABAQUS for the 4 elements along the generatrix of the cylindrical case. 
52 
 
 
Figure 5.2: The mesh of the simulation. 
Boundary Conditions 
For the simulation of the torsion-rheometry tests without compression, we applied the 
following boundary conditions. On bottom surface, we applied a fixed boundary 
condition. All displacement of nodes on the bottom surface were all set to zero. On top 
surface, we applied a time periodic displacement boundary condition that the surface 
rotated by 
𝜃 = 𝜃0 sin 𝜔𝑡 (5.1) 
The way we applied this boundary condition in ABAQUS was that we first 
constrained the surface with its center node and then applied rotations to that center 
node. Here 𝜃0 = 0.0006 will give a maximum strain of 0.003, and 𝜔 is the frequency 
which will be chosen from 0.01 to 10. On the side surface, we applied traction free 
boundary conditions.  
53 
 
For the simulation with compression, same boundary conditions were applied to the 
bottom surface and the side surface. On top surface, we first applied the compression 
in the first step. All the nodes on top surface moved a same displacement along in-
plane direction. Then we let the sample relax for about 10 minutes. Then the rotation 
was applied to these nodes. For the rotation that applied on top surface, sine wave was 
not easy to apply in ABAQUS. One approximation was that we divided one cycle to 
many small parts and used linear piecewise functions. Here we used 24 time steps to 
approximate the 𝜃 in each cycle of rotation. As observed, the relaxation time of the 
material is about 10-15 seconds, which means that it usually took 10-15s to reach the 
steady state. Thus, for the frequency 𝜔 = 0.01 and 𝜔 = 0.1, we used 5 cycles. For the 
frequency 𝜔 = 1, 15 cycles of rotation were simulated. However for the frequency 
𝜔 = 10, 100-150 cycles are too large for calculation on my laptop. We calculated only 
70 cycles of rotation for the sample with and without compression. So at this 
frequency, the torque that we calculated for the last cycle may not be the steady state 
torque. 
5.3 Experiment 
5.3.1 Experiment setup 
The torsion-rheometry tests performed in this chapter were done using TA Instruments 
DHR-3 Rheometer. Figure 5.3 shows the structure of the DHR-3 Rheometer. It is a 
stress-controlled shear rheometer with a range of measurement options. It measures 
various properties including viscosity, shear stress, storage and loss modulus, strain, 
phase angle, etc. It measures the specimen in different geometries: various diameter 
54 
 
parallel plate and cone plate, cup and rotor with options of a vaned or conical rotor, 
and a number of tribological accessories. In this Chapter, we used the geometry of 20 
mm diameter parallel plate.  
 
Figure 5.3: DHR-3 Rheometer. 
 
5.3.2 Torsion Rheometry Experiments 
For the fully hydrated hydrogels, we performed torsion-rheometry tests to find the 
change of storage and loss modulus with respect to the frequency at different 
compressions: we compressed the specimen from 2.0 mm to 1.9 mm, 1.8 mm, and 1.7 
mm.  
5.4 Results 
55 
 
5.4.1 FEM Results 
For the compression of 4.76% (from 2.1 mm to 2 mm), Figures 5.4-5.7 show the total 
torque on the top surface of the sample with and without compression at different 
frequencies. From Figures 5.4-5.7, we can find that when the cyclic rotation started, 
the torque of the samples without compression was always larger than that of the 
samples with compression for the first 5 seconds. However, after 5 seconds, the torque 
of the samples without compression dropped more quickly. And finally when the 
sample reached steady state, the torque of the samples with compression were always 
larger, which is consistent with other works and experiments. Table 5.1 shows the 
peak torque in the last cycle of the simulation for the samples with and without 
compression. 
 
Frequency, Hz 𝟎. 𝟎𝟏 𝟎. 𝟏 𝟏 𝟏𝟎 
Steady state torque for 
samples with compression, 49.97 101.34 120.34 122.67 
𝒎𝑵 × 𝒎𝒎 
Steady state torque for 
samples without 46.69 94.51 112.53 117.84 
compression, 𝒎𝑵 × 𝒎𝒎 
 
Table 5.1: Steady state torque for samples with and without compression at 
different frequencies. 
56 
 
 
Figure 5.4: Torque (𝑚𝑁 × 𝑚𝑚) vs. time (s) for sample with and without compression 
at frequency 0.01Hz. 
 
Figure 5.5: Torque (𝑚𝑁 × 𝑚𝑚) vs. time (s) for sample with and without compression 
at frequency 0.1 Hz. 
57 
 
 
Figure 5.6: Torque (𝑚𝑁 × 𝑚𝑚) vs. time (s) for sample with and without compression 
at frequency 1 Hz. 
 
Figure 5.7: Torque (𝑚𝑁 × 𝑚𝑚) vs. time (s) for sample with and without compression 
at frequency 10 Hz. 
 
58 
 
At frequency 𝜔 = 0.1, the final steady state torque of samples with different 
compression is shown in Table 5.2. Also the results of the same calculation for hyper-
elastic material are shown to compare. 
viscoelastic material (PVA)  hyper-elastic material 
𝑑 = 20 𝑚𝑚 compression torque (𝑚𝑁 × 𝑚𝑚) torque 
difference (𝑚𝑁 × difference 
4 edge 8 edge 12 edge 
𝑚𝑚) 
elements elements elements 
ℎ = 2 𝑚𝑚 0 94.51 94.45  0 9.24 0 
ℎ: 2.1
4.76% 101.34   6.74%   
→ 2 𝑚𝑚 
ℎ: 2.105
5% 101.65   7.02%   
→ 2 𝑚𝑚 
ℎ: 2.222
10% 108.16 107.07  12.62% 9.26 0.22% 
→ 2 𝑚𝑚 
ℎ: 2.353
15% 114.7 111.38  15.15%   
→ 2 𝑚𝑚 
ℎ: 2.500
20%  114.02 114.03 17.11% 9.22 -0.22% 
→ 2 𝑚𝑚 
ℎ: 2.667
25%  117.53 116.49 18.90%   
→ 2 𝑚𝑚 
ℎ: 2.857
30%  120.75 118.23 20.06% 8.9 -3.82% 
→ 2 𝑚𝑚 
 
Table 5.2: FE calculation results for different compression at frequency 0.1. 
From Table 5.2, we can find that for the viscoelastic material, as the compression 
increased, more elements were needed to converge, and the difference of the torque 
also increased. The increasing of the difference in torque is slower than the increasing 
of compression. And this is a property only for the viscoelastic materials since the 
hyper-elastic material was not influenced by the compression when the compression is 
not big. For 30% of compression, the torque of the hyper-elastic material decreases 
mainly because of the change of the sample in shape. 
5.4.2 Experimental Results 
59 
 
Figure 5.8 shows the results of storage modulus and loss modulus vs. frequency under 
different compression. From the figure we can see that as we compressed more, the 
storage modulus measured by the rheometer would increase. 
 
Figure 5.8: Storage modulus/Loss modulus vs. frequency under different compression. 
 
5.5 Discussion 
5.5.1 FEA Calculation 
Torque and Dynamic Modulus 
In the simulation, the final results we got was the total torque on the top surface. We 
needed to find the connection between the torque and the storage and loss modulus. 
Basically, in a torsion-rheometry test, if we give the shear strain 
𝛾(𝑡) = 𝛾0𝑒
𝑖𝜔𝑡 , (5.2𝑎) 
60 
 
then the steady state shear stress can be written as 
𝜏(𝑡) = 𝛾 𝑖𝜔𝑡0[𝐺1(𝜔) + 𝑖𝐺2(𝜔)]𝑒 (5.2𝑏) 
where 𝛾0 is the given maximum strain, 𝐺1 is the storage modulus, 𝐺2 is the loss 
modulus, and  𝜔 is the frequency. 
In our simulation, we applied a sine wave rotation 𝜃 = 𝜃0 sin 𝜔𝑡. Then the shear strain 
on the top surface is 
𝛾0𝑟
𝛾(𝑟, 𝑡) = 𝛾(𝑟)sin𝜔𝑡 = sin𝜔𝑡 (5.3) 
𝑟0
where 𝑟 is the position in polar coordinate and 𝑟0 is the radius of the sample.  
Then the shear stress at steady state can be calculated as 
𝜏(𝑟, 𝑡) =  𝛾(𝑟)[𝐺1(𝜔) sin 𝜔𝑡 + 𝐺2(𝜔) cos 𝜔𝑡] =
𝛾0𝑟 1 𝜋  
𝐺1(𝜔)√1 + sin (𝜔𝑡 + − 𝛿) (5.4)𝑟 20 tan 𝛿 2
And the torque on the surface (steady state) is 
𝛾0 1 𝜋
𝑇(𝑡) = ∬ 𝜏(𝑟, 𝑡) ∙ 𝑟𝑑𝐴 = 𝐺1(𝜔)√1 + sin (𝜔𝑡 + − 𝛿) ∬ 𝑟
2𝑑𝐴 (5.5) 
𝑟0 tan
2 𝛿 2
Thus, 
𝛾0𝐼𝜌 1
𝑇𝑚𝑎𝑥 = 𝐺1(𝜔)√1 + (5.6) 𝑟0 tan
2 𝛿
where 𝛾0 is the given maximum shear strain, 𝐼𝜌 is the polar moment of inertia 
determined by the geometry, and 𝑟0 is the radius of the sample. 
Parameter Shifting 
From Table 5.1, after 5 cycles, for frequency 𝜔 = 0.01, the peak stress in the last 
cycle of the simulation without compression is 6.55% lower than that of the simulation 
61 
 
with compression. For frequency 𝜔 = 0.1, the peak stress in the last cycle of the 
simulation without compression is 6.74% lower than that of the simulation with 
compression. After 15 cycles, for frequency 𝜔 = 1, the peak stress in the last cycle of 
the simulation without compression is 6.50% lower than that of the simulation with 
compression. After 70 cycles, for frequency 𝜔 = 10, the peak stress in the last cycle 
of the simulation without compression is 3.94% lower than that of the simulation with 
compression. How ever, from the Figure 5.4-5.6, we can find that for frequency 𝜔 =
0.01, 0.1 𝑎𝑛𝑑 1, it takes about 15 seconds for the sample to reach the steady state. So 
for the frequency 𝜔 = 10, 70 cycles mean just 7 seconds and it may not reach the 
steady state. Also Figure 5.7 tells us that we need more cycles to reach the steady 
state. Thus at least for frequency 0.01 to 1, the difference between the torque on the 
top surface of a sample with or without compression is almost the same. This helps us 
shift the parameters. 
Another usable information that we can get from the figure is that we can clear see 
that the phases shift of the curve at all frequency, with or without compression, are the 
same. Then it means that the tangent delta of the material does not change as the 
compression condition changes. Then from equation 5.6, we can know that the 
maximum torque on the surface is propotional to the storage modulus. Then in real 
experiments, the rheometer obtain the torque-time curve and than calculate the 
dynamic modulus using this curve. If the experiment was done idealy, then the curve 
would be 6.74% lower than the curve got from the rheometer (When the compression 
is 4.76%, or from 2.1 mm to 2 mm). So the real modulus should be 6.74% lower than 
the results given by rheometer. Then we need to multiply the experimental storage 
62 
 
modulus by 0.9326. Since tangent delta does not change with the compression, the 
loss modulus should also be multiplied by the same factor 0.9326. 
The results given in Table 5.2 tell us the shift factor for other cases of compression, as 
shown in Figure 5.9. 
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
0 10 20 30 40
Compression, %
 
Figure 5.9: Shift parameters for different compression. 
However, for the hyper-elastic material results given in Table 5.2, when the 
compression reached 30%, then the torque is no longer same as the torque without 
compression. This indicates that the geometry of the sample may change a lot under 
such compression and the torque will also be influenced by the geometry. So in real 
experiments, it’s best not to give a compression larger than 20%. 
5.5.2 Experiments 
Dynamic Modulus from Constitutive Model 
63 
Shift Parameter
 
From [42], based on the constitutive model (equation 3.4) in Section 3.4.1, the 
relationship between storage modulus 𝐺′ and loss modulus 𝐺” follows the equation: 
𝐺′(𝜔) = 𝜇𝜌 + 𝜇𝜌𝐾(𝑇)𝐼1(𝜔𝑡𝐵 , 𝛼𝐵) (5.7𝑎) 
𝐺"(𝜔) = 𝜇𝜌𝐾(𝑇)𝐼2(𝜔𝑡𝐵, 𝛼𝐵) (5.7𝑏) 
where 
2 − 𝛼 ∞ 𝜔𝑡 1𝐵 𝐵 −
𝐼1(𝜔𝑡𝐵 , 𝛼𝐵) = 1 − ∫ cos ( 𝑥) (1 + 𝑥) 𝛼𝐵−1𝑑𝑥 (5.8𝑎) 𝛼𝐵 − 1 0 𝛼𝐵 − 1
2 − 𝛼 ∞𝐵 𝜔𝑡 1𝐵 −
𝐼2(𝜔𝑡𝐵, 𝛼𝐵) = ∫ sin ( 𝑥) (1 + 𝑥) 𝛼𝐵−1𝑑𝑥 (5.8𝑏) 𝛼𝐵 − 1 0 𝛼𝐵 − 1
(1 − 𝜌)/𝜌 ?̅?∞𝑡𝐵
𝐾(𝑇) = = (5.8𝑐) 
1 + (2 − 𝛼𝐵)(𝑡𝐻/𝑡𝐵) (2 − 𝛼𝐵)𝜌
where 𝜔 is the frequency in radians/s. 𝑡𝐻 is the characteristic healing time in old 
constitutive model and has the relationship 
1 − 𝜌
?̅?∞ = (5.9)𝑡𝐵  𝑡𝐻 + 2 − 𝛼𝐵
For this sheet of PVA sample, characteristic tests had been performed first and the 
fitting results is shown in Table 5.3 and Figure 5.10. 
𝝁𝝆, 𝒌𝑷𝒂 𝜶𝑩 𝒕𝑩, 𝒔 𝝁?̅?∞, 𝒌𝑷𝒂/𝒔 
4.0 1.645 0.30 30.50 
 
Table 5.3: Fitting parameters for PVA dual-crosslinked hydrogel. 
64 
 
 
Figure 5.10: Comparison between experiment result and model prediction. 
We calculated the dynamic modulus using these parameters and equation 5.7, 5.8. 
Then we shifted the experiment results according to Table 5.2 and then compared 
them to the model prediction. Figures 5.11 and 5.12 show the comparison of dynamic 
modulus in experiment result and model prediction. 
 
Figure 5.11: Comparison of model and experiment in storage modulus. 
65 
 
 
Figure 5.12: Comparison of model and experiment in loss modulus. 
From Figure 5.11 and Figure 5.12, we can see that some of the curves fit the theory 
well, but some do not. The reason is that there are also other factors influencing the 
results in a real experiment. In FE calculation, we assumed that the bottom surface 
was fixed, and the top surface was rotated by a given displacement. But in the actual 
experiments, when the compression was small, the pressure between the plate and the 
material may not provide sufficient friction to prevent slipping which would result in 
the e torque being be smaller than expected. However, if the compression was large, 
then more material was squeezed out. There would be slipping in the radial direction 
during the compression step. The real diameter of the specimen is larger, and the 
torque will also be larger than expected. One way to do an ideal experiment is to glue 
the top and bottom surface to the plate of the rheometer to prevent slipping. However, 
because of some limitation, such experiments could not be performed on this 
rheometer. Further experiments are needed to verify the FEA calculation results in the 
future. 
66 
 
5.6 Conclusion 
In this Chapter, we present the results of the study on the effect of compression in 
torsion-rheometry tests for the viscoelastic material, PVA dual-crosslinked hydrogel, 
using both experiments and finite element simulations. The finite element simulation 
results show that for the same geometry and same compression, the effect of 
compression on the steady state torque is almost the same for different frequencies of 
oscillation. And we gave the shifting factor for different compression according to the 
calculation. From experimental results, it is shown that for some of the tests, this 
shifting factor is fine. However, because in real experiments, the condition is not ideal, 
other factors will also influence the results. It’s not always accurate if we shift all  
experimental results using only this factor. Also, more accurate experiments are 
needed to verify the FE calculation.
67 
 
APPENDIX A 
APPENDIX 
 
A.1 Results of Recovery Tests 
The stress-time curve of the recovery test of fully hydrated hydrogels and 70% mass 
hydration level hydrogels are shown in Figure A.1. We can see that for the fully 
hydrated hydrogels, the stress recovered to almost zero after about 700s of recovery. 
So when we perform the uniaxial tests, 12 minutes (720 seconds) is a reasonable 
recovery time between tests. However, for 70% mass hydration level hydrogels, the 
stress could not recover to zero after even 4 hours. 
 
(a) 
68 
 
 
(b) 
Figure A.1: (a) Nominal stress vs. time for fully hydrated hydrogels. (b) Nominal 
stress vs. time for 70% mass hydration level hydrogels. 
 
A.2 Long-time Stress in Relaxation Test 
A zoom-in figure of the long-time stress for samples at different hydration levels is 
given in Figure A.2. We can see that for samples at 60% mass hydration level, the 
stress does not plateau but drop quickly at long time.  
69 
 
 
Figure A.2: The long-time stress for PVA gel at different hydration levels. 
 
A.3 Feedback Load Control 
The pseudo code for PID load control is shown below. 
connect to Zaber and Keithley 
set system parameters  % convert displacement and velocity in motor; convert 
force and displacement to voltage signal in Keithley 
set sample geometry 
set 𝑜𝑖𝑙𝑓𝑎𝑐𝑡𝑜𝑟  % eliminate the effect of buoyancy 
set target force 𝐹𝑡𝑎𝑟𝑔𝑒𝑡 
compute target force signal 𝑉𝑡𝑎𝑟𝑔𝑒𝑡 
set PID parameters 𝐾𝑃, 𝐾𝐼 , 𝐾𝐷 
set ending condition  % finish the loading 
set 𝑒𝑟𝑟𝑜𝑟(1) = 0, 𝑒𝑟𝑟𝑜𝑟𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑙 = 0, 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑇𝑖𝑚𝑒(1) = 0 
for 𝑖 = 2: 𝑛 do 
 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑇𝑖𝑚𝑒(𝑖) = 𝑟𝑒𝑎𝑑𝐶𝑢𝑟𝑟𝑒𝑛𝑡𝑇𝑖𝑚𝑒() 
 𝑑𝑡 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑇𝑖𝑚𝑒(𝑖) − 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑇𝑖𝑚𝑒(𝑖 − 1) 
 𝑉𝑐𝑢𝑟𝑟𝑒𝑛𝑡(𝑖) = 𝑟𝑒𝑎𝑑𝐹𝑜𝑟𝑐𝑒𝑆𝑖𝑔𝑛𝑎𝑙() 
 𝑑𝑐𝑢𝑟𝑟𝑒𝑛𝑡(𝑖) = 𝑟𝑒𝑎𝑑𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡𝑆𝑖𝑔𝑛𝑎𝑙() 
𝑉𝑐𝑢𝑟𝑟𝑒𝑛𝑡(𝑖) = 𝑉𝑐𝑢𝑟𝑟𝑒𝑛𝑡(𝑖) − 𝑜𝑖𝑙𝑓𝑎𝑐𝑡𝑜𝑟 ∗ 𝑑𝑐𝑢𝑟𝑟𝑒𝑛𝑡(𝑖)  % eliminate the 
effect of buoyancy 
70 
 
 𝑒𝑟𝑟𝑜𝑟(𝑖) = 𝑉_𝑡𝑎𝑟𝑔𝑒𝑡 − 𝑉_𝑐𝑢𝑟𝑟𝑒𝑛𝑡(𝑖) 
 𝑑𝑢𝑃 = 𝐾𝑃 ∗ 𝑒𝑟𝑟𝑜𝑟(𝑖)  % velocity correction with P 
 𝑒𝑟𝑟𝑜𝑟𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑙 = 𝑒𝑟𝑟𝑜𝑟𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑙 + 0.5 ∗ (𝑒𝑟𝑟𝑜𝑟(𝑖) + 𝑒𝑟𝑟𝑜𝑟(𝑖 − 1)) ∗ 𝑑𝑡 
 𝑑𝑢𝐼 = 𝐾𝐼 ∗ 𝑒𝑟𝑟𝑜𝑟𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑙  % velocity correction with I 
 𝑒𝑟𝑟𝑜𝑟𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 = (𝑒𝑟𝑟𝑜𝑟(𝑖) − 𝑒𝑟𝑟𝑜𝑟(𝑖 − 1))/𝑑𝑡 
 𝑑𝑢𝐷 = 𝐾𝐷 ∗ 𝑒𝑟𝑟𝑜𝑟𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙  % velocity correction with D 
 𝑑𝑢 = 𝑑𝑢𝑃 + 𝑑𝑢𝐼 + 𝑑𝑢𝐷 
 if 𝑎𝑏𝑠(𝑑𝑢) > 2𝑚𝑚/𝑠 
  𝑑𝑢 = 𝑑𝑢/(𝑎𝑏𝑠(𝑑𝑢)) ∗ 2𝑚𝑚/𝑠 % upper bound for 𝑑𝑢 
 else if 𝑎𝑏𝑠(𝑒𝑟𝑟𝑜𝑟(𝑖))/𝑉_𝑡𝑎𝑟𝑔𝑒𝑡 < 0.01 
  𝑑𝑢 = 0.001𝑚𝑚/𝑠 %bottom limit for 𝑑𝑢 
 set ZaberSpeed(𝑑𝑢) 
end 
 
The system parameters for Zaber motor are given by: 
1𝑚𝑚 =  20997 in displacement unit in zaber motor; 
1𝑚𝑚/𝑠 =  34402 in speed unit in zaber motor. 
The way to obtain system parameters for Keithley data logger is given in section 2.3. 
The 𝑜𝑖𝑙𝑓𝑎𝑐𝑡𝑜𝑟 can be calculated as below: 
We measured the length and width of the oil container, 𝑎 𝑚𝑚 × 𝑏 𝑚𝑚. Then the 
cross-sectional area is 𝐴1 = 𝑎𝑏 𝑚𝑚
2. We measured the diameter of the cylinder 
which connected the grip and the load cell, 𝑑 𝑚𝑚. Then the cross-sectional area for 
𝜋𝑑2
the cylinder is 𝐴 =  𝑚𝑚22 . 4
If the grip went up for a displacement 𝑥+ 𝑚𝑚, then the oil level dropped down for 
𝐴 𝑥+
𝑥− = 2  𝑚𝑚. The change in buoyancy is  
𝐴1−𝐴2
𝐴1𝐴2
𝐹 = 𝜌 + − +𝑜𝑖𝑙𝑔𝑉 = 𝜌𝑜𝑙𝑖𝑔𝐴2(𝑥 + 𝑥 ) = 𝜌𝑜𝑖𝑙𝑔 𝑥 (𝐴. 1) 𝐴1 − 𝐴2
71 
 
𝐴 𝐴
Then the 𝑜𝑖𝑙𝑓𝑎𝑐𝑡𝑜𝑟 is given by 𝜌 1 2𝑜𝑖𝑙𝑔 ∙ 𝐶, where 𝜌𝑜𝑖𝑙 is the density of the oil, 𝑔 is 𝐴1−𝐴2
the gravitational acceleration, and 𝐶 is calculated using the convert factor that convert 
force and displacement to voltage. 
A.4 UMAT for PVA Model 
Since no current models in ABAQUS library can approach our PVA model accurately, 
we need to write our own user material in Fortran to do calculation with our model. 
In UMAT, the following quantities are available: 
– Stress, strain, and SDVs at the start of the increment 
– Strain increment, rotation increment, and deformation gradient at the start and end of 
the increment 
– Total and incremental values of time, temperature, and user-defined field variables 
– Material constants, material point position, and a characteristic element length 
– Element, integration point, and composite layer number (for shells and layered 
solids) 
– Current step and increment numbers 
And the following quantities must be defined: 
– Stress, SDV(Solution dependent state variable)s, and material Jacobian 
So to define our constitutive equation properly, we need at least three steps. 
1. Updating stresses. 
2. Updating Jacobian (required for ABAQUS/Standard UMAT) to check the 
convergence. 
3. Updating SDVs. 
72 
 
For large deformation, we need to calculate the true stress for each increment. Since 
the model is an incompressible non-linear visco-elastic model, the stress will also 
depend on the deformation history, which means it’s necessary to store the entire 
deformation history at each node. Then it will be difficult to do the calculation because 
it need a very large memory. Here, Jingyi Guo has developed an integration technique 
that resolves the difficulty with data storage while preserving the bond breaking and 
reforming physics of the constitutive model [43]. 
Then for the incompressible model, one common method is to first model the model as 
a compressible material with a very big bulk modulus 𝜅 (104 𝑡𝑜 106  times its shear 
modulus) and then to use hybrid elements in simulation to ensure incompressibility. 
From problem background, the constitutive equation(the relationship between the 
nominal stress tensor P and the deformation gradient tensor 𝐅𝜏→𝑡) is given as: 
𝑑𝑊0
𝐏 = −𝑝(𝐅0→𝑡)−𝑇 + 2[𝑛(𝑡) + 𝜌] | 𝐅0→𝑡 +
𝑑𝐼 𝐼=𝐼(𝒙,𝑡)
𝑡  𝑡 − 𝜏 𝑑𝑊0
2?̅? ∫ 𝜙 ( ) | 𝐅𝜏→𝑡(𝐅0→𝑡)−𝑇∞ 𝐵 𝑑𝜏 (𝐴. 2)
0 𝑡𝐵 𝑑𝐼 𝐼=𝐻(𝒙,𝜏,𝑡)
 
Where 
• 𝑝 is the Lagrange multiplier that enforces incompressibility. 
• 𝜌 is the molar fraction of the chemical crosslinks. 
• 𝑊0 is the strain energy.  
• ?̅?∞ is the steady state reattachment rate of the physical chains, i.e., molar 
fraction of the physical chains reattached per unit time, hence ?̅?∞𝑑𝜏 is the 
number of newly reattached physical chains. 
73 
 
• The integrand 
1
𝑡 − 𝜏 𝑡 1−𝛼𝐵
(?̅?∞𝑑𝜏)𝜙𝐵 ( ) = ?̅?∞ [(1 + (𝛼𝐵 − 1) ) ] 𝑑𝜏 (𝐴. 3) 𝑡𝐵 𝑡𝐵
is the number of chains that healed between 𝜏 and 𝜏 + 𝑑𝜏 and survives until 𝑡, 
where 𝑡𝐵 is the characteristic time for breaking; 1 < 𝛼𝐵 < 2 is a material 
constant that specifies the rate of decay of 𝜙𝐵. 
• 𝑛(𝑡) is the molar fraction of physical bonds at 𝑡 = 0 and still attached at time 
𝑡; it is given by 
2−𝛼
0 𝐵𝑡 − 𝜏 𝑡𝐵 𝑡 1−𝛼𝐵
𝑛(𝑡) = ?̅?  ∞ ∫ 𝜙𝐵 ( ) 𝑑𝜏 = ?̅?∞ (1 + (𝛼𝐵 − 1) ) (𝐴. 4)
−∞ 𝑡𝐵 2 − 𝛼𝐵 𝑡𝐵
 
Using a simple compressible Neo-Hookean work function: 
𝜇 𝜅
𝑊0(𝑡) = (𝐼1̅ − 3) + (𝐽 − 1)
2 (𝐴. 5) 
2 2
 
Where 𝐼1̅ ≡ 𝐼1/𝐽
2⁄3. Since the material is almost incompressible with 𝐽 ≈ 1, we can 
make an approximation of 𝐼1̅ ≈ 𝐼1 to simplify the calculation. Then the compressible 
constitutive equation which relate the true stress 𝝈 (Cauchy stress) with the 
deformation gradient tensor 𝐅𝜏→𝑡 becomes: 
𝜇 𝜇𝛾 𝑡∞ 𝑡 − 𝜏
𝝈(𝑡) = [𝜌 + 𝑛(𝑡)] [ 𝐁0→𝑡 + 𝜅(𝐽(𝑡) − 1)𝐈] + ∫ 𝜙𝐵 ( ) 𝐁
𝜏→𝑡𝑑𝜏
𝐽(𝑡) 𝐽(𝑡) 0 𝑡𝐵
 
𝜇𝛾 𝑡∞ 𝑡 − 𝜏 𝐽(𝑡) 𝐽(𝑡)
+ ∫ 𝜙𝐵 ( ) [ − 1] 𝑑𝜏 𝐈 (𝐴. 6)𝐽(𝑡) 0 𝑡𝐵 𝐽(𝜏) 𝐽(𝜏)
 
74 
 
Where 𝐁 = 𝐅 ∙ 𝐅T is the left Cauchy-Green tensor. 
What we need to do is to calculate the Cauchy stress after a increment ∆𝑡 (𝝈(𝑡 + ∆𝑡)) 
with the current information(we know everything at time t) and the increment of 
deformation gradient tensor ∆𝐅 = 𝐅𝑡→𝑡+∆𝑡 
Here, Jingyi introduced a method:  
Using prony series to approach the power law function inside the integral: 
1 𝑁
𝑡 − 𝜏 𝑡 1−𝛼 𝑡𝐵 −
𝜙𝐵 ( ) = (1 + (𝛼𝐵 − 1) ) → ∑ 𝑔𝑖𝑒 𝜏𝑖 (𝐴. 7) 𝑡𝐵 𝑡𝐵
𝑖=1
And 𝑁 = 4 is accurate enough by practice. 
Then the three terms of 𝝈(𝑡 + ∆𝑡) can be calculated as below: 
𝜇
[𝜌 + 𝑛(𝑡 + ∆𝑡)] [ 𝐁0→𝑡+∆𝑡 + 𝜅(𝐽(𝑡 + ∆𝑡) − 1)𝐈] =
𝐽(𝑡 + ∆𝑡)
𝜇  
[𝜌 + 𝑛(𝑡 + ∆𝑡)] [ ∆𝐅 ∙ 𝐁0→𝑡 ∙ (∆𝐅)T + 𝜅(𝐽(𝑡)|∆𝐅| − 1)𝐈] (𝐴. 8𝑎)
𝐽(𝑡)|∆𝐅|
 
Where  
0 𝑡 − 𝜏
𝑛(𝑡) = ?̅?∞ ∫ 𝜙𝐵 ( ) 𝑑𝜏 (𝐴. 8𝑏) 
−∞ 𝑡𝐵
Can be approached by prony series: 
𝑁
𝑡
−
𝑛(𝑡) = ?̅?∞ ∑ 𝑔𝑖𝜏𝑖𝑒 𝜏𝑖 (𝐴. 8𝑐) 
𝑖=1
𝜇𝛾 𝑡+∆𝑡
𝑁
∞ 𝑡 + ∆𝑡 − 𝜏 𝜇𝛾
∫ 𝜙 ( ) 𝐁𝜏→𝑡+∆𝑡
∞
𝑑𝜏 = ∑ 𝑔 (𝑖)
𝐽(𝑡 + ∆𝑡) 𝐵 𝑡 𝐽(𝑡)|∆𝐅| 𝑖
𝐓 (𝑡 + ∆𝑡) (𝐴. 9𝑎) 
0 𝐵 𝑖=1
 
Where 
75 
 
𝑡 𝑡−𝜏
𝐓(𝑖)
−
(𝑡) = ∫ 𝑒 𝜏𝑖 𝐁𝜏→𝑡𝑑𝜏 (𝐴. 9𝑏) 
0
𝑡+∆𝑡 𝑡+∆𝑡−𝜏
𝐓(𝑖)
−
(𝑡 + ∆𝑡) = ∫ 𝑒 𝜏𝑖 𝐁𝜏→𝑡+∆𝑡𝑑𝜏
0
𝑡 𝑡+∆𝑡−𝜏 𝑡+∆𝑡 𝑡+∆𝑡−𝜏
− −
= ∫ 𝑒 𝜏𝑖 𝐁𝜏→𝑡+∆𝑡𝑑𝜏 + ∫ 𝑒 𝜏𝑖 𝐁𝜏→𝑡+∆𝑡𝑑𝜏  
0 𝑡
∆𝑡 ∆𝑡 ∆𝑡− −
= 𝑒 𝜏𝑖 ∆𝐅 ∙ 𝐓(𝑖)(𝑡) ∙ (∆𝐅)T + [𝑒 𝜏𝑖 ∆𝐅 ∙ (∆𝐅)T + 𝐈] (𝐴. 9𝑐)
2
𝜇𝛾 𝑡+∆𝑡∞ 𝑡 + ∆𝑡 − 𝜏 𝐽(𝑡 + ∆𝑡) 𝐽(𝑡 + ∆𝑡)
∫ 𝜙 ( ) [ − 1] 𝑑𝜏
𝐽(𝑡 + ∆𝑡) 𝐵0 𝑡𝐵 𝐽(𝜏) 𝐽(𝜏)
𝑁
𝜇𝛾 𝑡+∆𝑡 𝑡+∆𝑡−𝜏 2∞ − 𝐽 (𝑡 + ∆𝑡)
= ∑ 𝑔 ∫ 𝑒 𝜏𝑖 𝑑𝜏 −
𝐽(𝑡)|∆𝐅| 𝑖 0 𝐽
2(𝜏)  
𝑖=1
𝑁
𝜇𝛾 𝑡+∆𝑡 𝑡+∆𝑡−𝜏∞ − 𝐽(𝑡 + ∆𝑡)
∑ 𝑔 ∫ 𝑒 𝜏𝑖 𝑑𝜏 (𝐴. 10𝑎)
𝐽(𝑡)|∆𝐅| 𝑖 𝐽(𝜏)
𝑖=1 0
Where 
𝑡+∆𝑡 𝑡+∆𝑡−𝜏
− 𝐽2(𝑡 + ∆𝑡)𝜏 ( )∫ 𝑒 𝑖 𝑑𝜏 = 𝑠 𝑖
𝐽2(𝜏) 1
(𝑡 + ∆𝑡)
0
𝑡 𝑡+∆𝑡−𝜏 𝐽2− (𝑡 + ∆𝑡)
𝑡+∆𝑡 𝑡+∆𝑡−𝜏 𝐽2− (𝑡 + ∆𝑡)
= ∫ 𝑒 𝜏𝑖 𝑑𝜏 + ∫ 𝑒 𝜏𝑖 𝑑𝜏  
𝐽20 (𝜏) 𝑡 𝐽
2(𝜏)
∆𝑡 ∆𝑡
−𝜏 | |2 (𝑖)
∆𝑡 −
= 𝑒 𝑖 ∆𝐅 𝑠1 (𝑡) + [𝑒
𝜏𝑖 |∆𝐅|2 + 1] (𝐴. 10𝑏)
2
𝑡+∆𝑡 𝑡+∆𝑡−𝜏
− 𝐽(𝑡 + ∆𝑡)𝜏 (𝑖)∫ 𝑒 𝑖 𝑑𝜏 = 𝑠 (𝑡 + ∆𝑡)
0 𝐽(𝜏)
2
𝑡 𝑡+∆𝑡−𝜏
− 𝐽(𝑡 + ∆𝑡)
𝑡+∆𝑡 𝑡+∆𝑡−𝜏
𝜏 −
𝐽(𝑡 + ∆𝑡)
= ∫ 𝑒 𝑖 𝑑𝜏 + ∫ 𝑒 𝜏𝑖 𝑑𝜏  
2
0 𝐽 (𝜏)
2
𝑡 𝐽 (𝜏)
∆𝑡 ∆𝑡
− ( ) ∆𝑡 −
= 𝑒 𝜏𝑖 |∆𝐅|𝑠 𝑖 (𝑡) + [𝑒 𝜏𝑖1 |∆𝐅| + 1] (𝐴. 10𝑐)2
 
Then we finish the update of the Cauchy stress. 
The next step is to calculate the Jacobian, which is defined as 
76 
 
1 𝜕∆(𝐽𝜎) 1 𝜕(𝐽𝜎) 𝜕(𝐽𝜎)
𝐂 = = ( ∙ 𝐅𝑇 + 𝐅 ∙ ) (𝐴. 11𝑎) 
𝐽 𝜕∆𝜖 2𝐽 𝜕𝐅 𝜕𝐅𝑇
In components form: 
1 𝜕(𝐽𝜎𝑖𝑗) 𝜕(𝐽𝜎𝑖𝑗)
𝐶𝑖𝑗𝑘𝑙 = ( 𝐹𝑙𝑚 + 𝐹𝑘𝑚 ) (𝐴. 11𝑏) 2𝐽 𝜕𝐹𝑘𝑚 𝐹𝑙𝑚
Subsitute 𝜎𝑖𝑗 in, we can get,  
𝑁
𝑡
−
𝐶𝑖𝑗𝑘𝑙(𝑡) = (𝜌 + ?̅?∞ ∑ 𝑔𝑚𝜏𝑚𝑒 𝜏𝑚)
𝑚=1
𝜇
{ (𝐵0→𝑡𝑖𝑘 𝛿𝑗𝑙 + 𝐵
0→𝑡 0→𝑡 0→𝑡
𝑖𝑙 𝛿𝑗𝑘 + 𝐵𝑗𝑘 𝛿𝑖𝑙 + 𝐵𝑗𝑙 𝛿𝑖𝑘) + 𝜅[2𝐽(𝑡) − 1]𝛿 𝛿2𝐽(𝑡) 𝑖𝑗 𝑘𝑙
}
𝑁  
𝜇?̅?∞ (𝑚)( ) (𝑚)( ) ( ) (+ ∑ 𝑔 [𝑇 𝑡 𝛿 + 𝑇 𝑡 𝛿 + 𝑇 𝑚 (𝑡)𝛿 + 𝑇 𝑚
)(𝑡)𝛿 ]
2𝐽(𝑡) 𝑚 𝑖𝑘 𝑗𝑙 𝑖𝑙 𝑗𝑘 𝑗𝑘 𝑖𝑙 𝑗𝑙 𝑖𝑘
𝑚=1
𝑁
𝜇?̅?∞ ( ) ( )
+ ∑ 𝑔𝑚[2𝑠
𝑚 𝑚
𝐽(𝑡) 1
(𝑡) − 𝑠2 (𝑡)] 𝛿𝑖𝑗𝛿𝑘𝑙 (𝐴. 11𝑐)
𝑚=1
 
(𝑚)
Where 𝑇𝑖𝑗 (𝑡)
(𝑚) (𝑚)
, 𝑠1 (𝑡) and 𝑠2 (𝑡) is given in stress section. 
Then we can update the Jacobian. 
 
 
 
 
 
 
 
 
77 
 
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