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APPROVED: Jaehyung Ju, Major Professor Xun Yu, Committee Member Sheldon Shi, Committee Member Yong Tao, Chair of the Department of
Mechanical and Energy Engineering Costas Tsatsoulis, Dean of the College of
Engineering Mark Wardell, Dean of the Toulouse Graduate
School
CHARACTERIZATION OF VISCOELASTIC PROPERTIES OF A MATERIAL
USED FOR AN ADDITIVE MANUFACTURING METHOD
Shaheer Iqbal
Thesis Prepared for the Degree of
MASTER OF SCIENCE
UNIVERSITY OF NORTH TEXAS
December 2013
Iqbal, Shaheer. Characterization of Viscoelastic Properties of a Material Used for an
Additive Manufacturing Method. Master of Science (Mechanical & Energy Engineering),
December 2013, 58 pp., 8 tables, 34 figures, references, 25 titles.
Recent development of additive manufacturing technologies has led to lack of
information on the base materials being used. A need arises to know the mechanical behaviors
of these base materials so that it can be linked with macroscopic mechanical behaviors of 3D
network structures manufactured from the 3D printer. The main objectives of my research are
to characterize properties of a material for an additive manufacturing method (commonly
referred to as 3D printing). Also, to model viscoelastic properties of Procast material that is
obtained from 3D printer. For this purpose, a 3D CAD model is made using ProE and 3D printed
using Projet HD3500. Series of uniaxial tensile tests, creep tests, and dynamic mechanical
analysis are carried out to obtained viscoelastic behavior of Procast. Test data is fitted using
various linear and nonlinear viscoelastic models. Validation of model is also carried out using
tensile test data and frequency sweep data. Various other mechanical characterization have
also been carried out in order to find density, melting temperature, glass transition
temperature, and strain rate dependent elastic modulus of Procast material. It can be
concluded that melting temperature of Procast material is around 337°C, the elastic modulus is
around 0.7-0.8 GPa, and yield stress is around 16-19 MPa.
ACKNOWLEDGEMENTS
I would like to express the deepest appreciation to my advisor, Dr. Jaehyung Ju, for his
constructive guidance, understanding and support. I feel really obliged to have worked with
him on his research. Without his guidance and persistent help this thesis would not have been
possible.
I would also like to thank my committee member Dr. Sheldon Shi and Dr. Xun Yu, for
granting me permission to use their equipments. Also, I would like to thanks Dr. Nandika Anne
D’souza for allowing me to use her lab and equipments.
I want to express my thanks to all other faculty members and my friends in the
Mechanical and Energy Engineering Department for their assistance and excellent help which
have helped me on my thesis.
iii
TABLE OF CONTENTS Page
ACKNOWLEDGEMENTS ................................................................................................................... iii
LIST OF TABLES ................................................................................................................................ vi
LIST OF FIGURES ............................................................................................................................. vii
CHAPTER 1 INTRODUCTION ........................................................................................................... 1
1.1 Motivation ........................................................................................................................ 2
1.2 Objectives ......................................................................................................................... 2
CHAPTER 2 LITERATURE REVIEW .................................................................................................... 3
2.1 Additive Manufacturing ........................................................................................................ 3
2.2 Mechanical Characterization ........................................................................................... 3
2.2.1 Tensile Test ............................................................................................................... 3
2.2.2 Creep Test ................................................................................................................. 4
2.2.3 Dynamic Mechanical Analysis (DMA) ....................................................................... 4
2.3 Viscoelastic Models .......................................................................................................... 5
2.3.1 Maxwell Model ......................................................................................................... 5
2.3.2 Generalized Maxwell Model ..................................................................................... 6
2.3.3 Voigt Model ............................................................................................................... 7
2.3.4 Voigt-Kelvin Model.................................................................................................... 8
2.3.5 Prony Series .............................................................................................................. 9
2.3.6 Schapery Model ...................................................................................................... 14
CHAPTER 3 MECHANICAL TESTING ............................................................................................... 16
3.1 Tensile Testing ................................................................................................................ 16
3.2 Creep Test ...................................................................................................................... 18
3.3 Dynamic Mechanical Analysis Test ................................................................................ 19
3.4 Differential Scanning Calorimetry (DSC) ........................................................................ 23
3.5 Density ............................................................................................................................ 24
CHAPTER 4 NONLINEAR REGRESSION .......................................................................................... 26
iv
CHAPTER 5 LINEAR VISCOELASTIC MODELS ................................................................................. 28
5.1 Maxwell Model ................................................................................................................... 28
5.2 Voigt-Kelvin Model ......................................................................................................... 30
5.3 Prony Series .................................................................................................................... 34
CHAPTER 6 NON-LINEAR VISCOELASTIC MODEL .......................................................................... 39
CHAPTER 7 VALIDATION OF MODEL ............................................................................................. 46
CHAPTER 8 CONCLUSIONS AND FUTURE WORK .......................................................................... 55
REFERENCES .................................................................................................................................. 56
v
LIST OF TABLES
Table 1 : Different strain rates and speed of crosshead ............................................................... 17
Table 2 : Young's Modulus and Yield Strength as a function of strain rate .................................. 18
Table 3 : Density of Procast material ............................................................................................ 25
Table 4 : Material Parameters for Maxwell Model ....................................................................... 28
Table 5: Material Parameters for Voigt-Kelvin Model .................................................................. 31
Table 6 : Creep Material Parameters for Prony Series ................................................................ 36
Table 7 : Material Parameters for Schapery Model ..................................................................... 42
Table 8 : Polynomial Constants for Nonlinear Material Paramters ............................................. 44
vi
LIST OF FIGURES
Fig. 1 : Projet HD 3500 3D Printer ................................................................................................... 1
Fig. 2 : Some 3D Printed Samples .................................................................................................. 2
Fig. 3 : Maxwell Model .................................................................................................................... 6
Fig. 4 : Generalized Maxwell Model ............................................................................................... 7
Fig. 5 : Voigt Model ......................................................................................................................... 7
Fig. 6 : Voigt-Kelvin Model ............................................................................................................. 8
Fig. 7 : Shimadzu AGS-X Series Universal Testing Machine ......................................................... 16
Fig. 8 : Stress-Strain Behavior of Procast at different strain rates................................................ 17
Fig. 9 : Creep Test Results from 1-10 MPa .................................................................................... 19
Fig. 10 : Storage Modulus and Loss Modulus plot over temperature of Procast at 1 Hz ............ 21
Fig. 11 Storage Modulus and Tan Delta plot over temperature of Procast at 1 Hz .................... 22
Fig. 12 : Frequency Sweep Response of Loss Modulus at Room Temperature ............................ 22
Fig. 13 : Frequency Sweep Response of Storage Modulus at Room Temperature ...................... 23
Fig. 14 : DSC Graph for Procast ..................................................................................................... 24
Fig. 15 : Creep Strain for Maxwell model for 1 -3 MPa stress ...................................................... 29
Fig. 16 : Creep Strain for Maxwell model for 1-8 MPa stress ....................................................... 29
Fig. 17 : Creep Strain for Voigt-Klevin Model for 1 -3 MPa stress ................................................ 32
Fig. 18 : Creep Strain for Voigt-Kelvin Model for 1-8 MPa stress ................................................. 33
Fig. 19: Creep Strain prediction for Prony Series .......................................................................... 36
Fig. 20 : Schapery Model Prediction of Creep Test ...................................................................... 43
Fig. 21 : Nonlinear Parameters for Schapery Model at Various Stress Levels .............................. 43
vii
Fig. 22 : Strain-Rate Dependent Tensile Test Data Including Yielding Region .............................. 46
Fig 23 : Validation for Maxwell Model at Various Strain Rates .................................................... 47
Fig. 24 : Validation for Voigt-Kelvin Model at Various Strain Rates ............................................. 47
Fig. 25 : Validation for Prony Series at Various Strain Rates ........................................................ 48
Fig. 26 : Validation for Schapery Model at Various Strain Rates .................................................. 49
Fig. 27 : Loss Modulus response for Maxwell model ................................................................... 50
Fig. 28 : Loss Modulus response for Voigt-Kelvin model ............................................................. 50
Fig. 29 : Loss Modulus response for Prony model ....................................................................... 51
Fig. 30 : Loss Modulus response for Schapery Model .................................................................. 51
Fig. 31 : Storage Modulus response for Maxwell model .............................................................. 52
Fig. 32 : Storage Modulus response for Voigt-Kelvin model ....................................................... 53
Fig. 33 : Storage Modulus response for Prony Series ................................................................... 53
Fig. 34 : Storage Modulus response for Schapery model ............................................................. 54
viii
CHAPTER 1
INTRODUCTION
Recent development of additive manufacturing technologies has led to a lack of
information on the base materials being used. A need arises to know the mechanical behaviors
of these base materials so that it can be linked with macroscopic mechanical behaviors of 3D
network structures manufactured from the 3D printer. The base material used by our 3D
printer is Procast. The objective of my research is to investigate non-linear stress-strain
behaviors of the Procast obtained from the 3D printer.
Fig. 1 : Projet HD 3500 3D Printer
For this purpose, different mechanical tests are conducted on Procast such as tensile
test, and creep test. Some dynamic mechanical analysis is also being carried out. For this
purpose, a 3D CAD model is made using ProE and 3D printed using Projet HD3500 plus Fig. 1.
For post-processing, a Projet finisher oven is used to melt the support material, and digital
ultrasonic cleaner is used for cleaning the remaining support material. Tensile and creep tests
1
are performed using Shimadzu AGS-X series Universal Testing Machine following ASTM
standards [1]-[5][8][9]. Rheometric Scientific equipment is used to carry out dynamic
mechanical analysis. Experimental data is fitted using different viscoelastic models both in time
domain and frequency domain. Validation of the data is carried out using experimental test
data and viscoelastic model data.
Fig. 2 : Some 3D Printed Samples
1.1 Motivation
Additive manufacturing (3D printing) is a relatively new technique and the material used
in this method is unknown. 3D printer can make complex cellular structures which were not
possible the conventional manufacturing technique.
When considering a cellular structure, its geometry and material gives the material
behavior. In my research I am working on the material aspect and will be providing research
community the information regarding the base material (which in our case is called Procast)
and its properties.
1.2 Objectives
The main objectives of my research are to characterize properties of a material for an
additive manufacturing method (commonly referred to as 3D printing). Also, to model
viscoelastic properties of Procast material that is obtained from 3D printer.
2
CHAPTER 2
LITERATURE REVIEW
2.1 Additive Manufacturing
Additive manufacturing is a method in which 3D solid objects are printed using a 3D
printer. In common terminology, additive manufacturing is also termed as 3D printing. The
general principle of additive manufacturing is layer by layer deposition of base material to make
3D parts. Additive manufacturing uses a 3D CAD model as a soft copy for 3D part. The
corresponding CAD file is converted into STL format which is the required format for the 3D
printer [1]. Upon receiving the STL file, the 3D printer starts printing the 3D part layer by layer.
There are differing types of printing process available [2]; the following two processes
are used by our Projet HD 3500 plus, 3D Systems printer.
a) Fused deposition modeling (FDM): In this method, layer by layer deposition of base
material is made by the 3D printer.
b) Stereolithography (SLA): This process uses a laser to solidify the base material as it is
being printed by FDM method.
2.2 Mechanical Characterization
Since 3D printing is a new technique, and base materials being used for it are relatively
new as well, a need arises to conduct different mechanical tests in order to find their
mechanical properties.
2.2.1 Tensile Test
Tensile tests were carried out using ASTM Standards D412 [1] and D638 [5]. For our
Procast material, dumbbell shaped specimen Fig. 2 was printed using Projet HD3500 Fig. 1. The
3
same dimensions were considered for Procast as Die C in ASTM standard D412 [1]. Once the
sample is printed, it is placed in the Projet finisher oven to melt the support material; in our
case it is wax. The Procast sample is further cleaned using Digital Ultrasonic Cleaner, and the
material obtained is dried by placing it in a desiccator.
For the tensile test, Shimadzu AGS-X series Universal Testing Machine Fig. 7 is used with
a load cell of 5kN. The test is conducted at room temperature 23 ± 2°C as per ASTM standards.
For different strain rates, the cross-head speed of the machine is adjusted to obtain the desired
strain rate.
2.2.2 Creep Test
Creep tests were carried out using Shimadzu AGS-X series Universal Testing Machine. In
a creep test, stress is held constant for some time duration at a specified temperature. We
conducted the test using different stress levels at room temperature 23 ± 2°C for 1800 seconds.
ASTM Standard D2990 [3] was followed in preparing the sample. The sample dimension was in
standing with Type 1 in ASTM Standard D638 [5].
2.2.3 Dynamic Mechanical Analysis (DMA)
Dynamic mechanical properties refer to the response of the material when it is
subjected to dynamic loading. These properties may be expressed in terms of a storage
modulus, loss modulus and a damping term.
Temperature sweep test is conducted where we study the response of the material at
fixed frequency over a wide range of temperature. This temperature sweep allows in studying
the transitions in the material including the glass transition temperature (𝑇𝑇𝑔𝑔).
4
A Frequency sweep test is conducted where we study the response of the material at
fixed temperature but over a wide range of frequencies, we use this data for time temperature
superposition. We find the material properties by the following equations
𝑆𝑡𝑜𝑟𝑎𝑔𝑒 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 (E’) = 𝜎𝑜Є𝑜𝑐𝑜𝑠𝛿 (2. 1)
𝐿𝑜𝑠𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 (E’’) = 𝜎𝑜Є𝑜𝑠𝑖𝑛𝛿 (2. 2)
𝑃ℎ𝑎𝑠𝑒 𝑎𝑛𝑔𝑙𝑒 ∶ 𝑇𝑇𝑎𝑛(𝑑𝑒𝑙𝑡𝑎) = E’’E’
(2. 3)
2.3 Viscoelastic Models
Viscoelasticity is made up of two words: viscous and elastic. If a material exhibits both
elastic and viscous nature, then it is termed as viscoelasticity, and such material exhibits a time-
dependent strain. A viscoelastic material has an elastic component and a viscous component.
This viscous component gives a time-dependent strain. The elastic component is modeled as
spring (𝜎 = 𝐸𝜀), while the viscous component is modeled as dashpot (𝜎 = 𝜂 𝑑𝜀𝑑𝑡
)
There are various viscoelastic material models that give a good time dependent
response of the material. Few viscoelastic material models are studied here.
2.3.1 Maxwell Model
Maxwell model is the simplest model that gives linear viscoelastic material response
[11] [13]. In this model, spring is connected in a series with a dashpot. Here, strain in both the
elements is different while stress remains same 𝜎𝑇𝑂𝑇𝐴𝐿 = 𝜎𝐷 = 𝜎𝑆
Total strain is the sum of strains in spring and dashpot 𝜀𝑇𝑂𝑇𝐴𝐿 = 𝜀𝐷 + 𝜀𝑆
5
Fig. 3 : Maxwell Model
To obtain strain rate,
𝑑𝜀𝑇𝑂𝑇𝐴𝐿𝑑𝑡
=𝑑𝜀𝐷𝑑𝑡
+𝑑𝜀𝑆𝑑𝑡
=𝜎𝜂
+1𝐸𝑑𝜎𝑑𝑡
(2. 4)
𝜀̇ =𝜎𝜂
+�̇�𝐸
(2. 5)
For Stress Relaxation test, the above equations becomes
0 =𝜎𝜂
+�̇�𝐸
(2. 6)
�̇�𝐸
= −𝜎𝜂
(2. 7)
𝑑𝜎𝜎
= −𝐸𝜂𝑑𝑡 (2. 8)
𝜎 = 𝜎𝑜𝑒−𝑡 𝜏� (2. 9)
Here, τ is the relaxation time given by 𝜏 = 𝜂𝐸�
And Relaxation Modulus 𝐸(𝑡) is given by
𝐸(𝑡) = 𝐸𝑒−𝑡 𝜏� (2. 10)
And for Creep Compliance,
𝐷(𝑡) = 𝐷(1 + 𝑡 𝜏� ) (2. 11)
2.3.2 Generalized Maxwell Model
Generalized Maxwell model is obtained by adding parallel combinations of n Maxwell
units. This model is used to model relaxation behavior [12]-[13].
𝐸
𝜂
6
Fig. 4 : Generalized Maxwell Model
The relation for stress relaxation for the generalized Maxwell model is given by
𝐸(𝑡) = �𝐸𝑖𝑒−𝑡 𝜏𝑖�
𝑛
𝑖=1
(2. 12)
The equation for Generalized Maxwell model in frequency domain is given by
(𝑆𝑡𝑜𝑟𝑎𝑔𝑒 𝑀𝑜𝑑𝑢𝑙𝑢𝑠)𝐸′(𝜔) = ��𝐸𝑖𝜏𝑖2𝜔2
1 + 𝜏𝑖2𝜔2�𝑛
𝑖=1
(2. 13)
(𝐿𝑜𝑠𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠)𝐸′′(𝜔) = ��𝐸𝑖𝜏𝑖𝜔
1 + 𝜏𝑖2𝜔2�𝑛
𝑖=1
(2. 14)
2.3.3 Voigt Model
Voigt model is another type of linear viscoelastic material model in which spring and
dashpot are connected in parallel.
Fig. 5 : Voigt Model
Here, stress is different in both the elements while strain remains same: 𝜀𝑇𝑂𝑇𝐴𝐿 = 𝜀𝐷 = 𝜀𝑆
𝐸 𝜂
7
Total stress is sum of stresses in spring and dashpot: 𝜎𝑇𝑂𝑇𝐴𝐿 = 𝜎𝐷 + 𝜎𝑆
𝜎(𝑡) = 𝐸 𝜀(𝑡) + 𝜂𝑑𝜀(𝑡)𝑑𝑡
(2. 15)
For creep test,
𝜎𝑜 = 𝐸 𝜀(𝑡) + 𝜂𝑑𝜀(𝑡)𝑑𝑡
(2. 16)
𝜀(𝑡) =𝜎𝑜𝐸�1 − 𝑒−𝑡 𝜏� � (2. 17)
Here, τ is the relaxation time given by 𝜏 = 𝜂𝐸�
Thus, creep compliance is given by
𝐷(𝑡) = 𝐷 �1 − 𝑒−𝑡 𝜏� � (2. 18)
2.3.4 Voigt-Kelvin Model
The Voigt-Kelvin model is a generalization of the Voigt model, in which Voigt elements
are added in series [11] [12]. This model gives good response for creep behavior, but it is
relatively poor for relaxation behavior:
Fig. 6 : Voigt-Kelvin Model
8
So, for n number of Voigt elements, the creep compliance is given by
𝐷(𝑡) = �𝐷𝑖 �1 − 𝑒−𝑡 𝜏𝑖� �
𝑛
𝑖=1
(2. 19)
Applying Fourier Transformation,
(𝑆𝑡𝑜𝑟𝑎𝑔𝑒 𝑀𝑜𝑑𝑢𝑙𝑢𝑠) 𝐷′(𝜔) = ��𝐷𝑖
1 + 𝜏𝑖2𝜔2�𝑛
𝑖=1
(2. 20)
(𝐿𝑜𝑠𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠) 𝐷′′(𝜔) = �𝐷𝑖 �𝜏𝑖𝜔
1 + 𝜏𝑖2𝜔2�𝑛
𝑖=1
(2. 21)
2.3.5 Prony Series
A common form for the linear viscoelastic response is given by Prony Series by the
following equation [10]:
�𝛼𝑖𝑒−𝑡 𝜏𝑖�
𝑁
𝑖=1
(2. 22)
Here, 𝜏𝑖 are the time constants and 𝛼𝑖 are the exponential coefficients.
In order to determine material parameters, creep and relaxation tests are used mostly.
There are various approaches to determine Prony coefficients; we have used a nonlinear
regression approach to determine the coefficients.
We know that for a relaxation test, its constitutive equation is given by the following equation:
𝜎(𝑡) = 𝑌(𝑡)Є0 (2. 23)
Here, 𝑌(𝑡) is the relaxation function, and its response under Prony Series is given by
𝑌(𝑡) = 𝐸0.�1 −�𝑝𝑖(1 − 𝑒−𝑡 𝜏𝑖�
𝑛
𝑖=1
� (2. 24)
Here,
9
𝑝𝑖 is the ith Prony constant 𝑖 = (1,2,3, … )
𝜏𝑖 is the ith Prony Retardation time constant 𝑖 = (1,2,3, … )
𝐸0 is the instantaneous modulus
When time 𝑡 = 0, 𝑌(0) = 𝐸0
And when time 𝑡 = ∞, 𝑌(∞) = 𝐸∞(1 − ∑𝑝𝑖)
To determine the stress state at a particular time, the deformation history must be
taken into account. For linear viscoelastic materials, a superposition of hereditary integrals
gives a time dependent response. In the case for stress relaxation, the specimen is under no
strain level prior to time 𝑡 = 0, at which a strain is applied and its corresponding stress
response for time 𝑡 > 0 is given by
𝜎(𝑡) = Є0𝑌(𝑡) + � 𝑌(𝑡 − 𝜉)𝑑Є(𝜉)𝑑𝜉
𝑑𝜉𝑡
0 (2. 25)
Here, 𝑌(𝑡) is the relaxation function and 𝑑Є(𝜉)𝑑𝜉
is the strain rate.
The process in general relaxation test is divided into 2 segments [10]: loading response
(increasing strain rate) and constant strain response (zero strain rate). Their functions are given
below:
Є(𝑡) = �Є1𝑡
(𝑡1 − 𝑡0)� ; 𝑡0 < 𝑡 < 𝑡1
Є1 ; 𝑡1 < 𝑡 < 𝑡2�
𝑑Є𝑑𝑡
= �Є1
(𝑡1 − 𝑡0)� ; 𝑡0 < 𝑡 < 𝑡1
0 ; 𝑡1 < 𝑡 < 𝑡2�
Here, Є0 = 0, Є1 is the strain level at which the strain is kept constant, and 𝑡0 = 0
10
Using the above strain response, the stress function for the loading response can be given as
[10]
Step 1(𝒕𝟎 < 𝑡 ≤ 𝒕𝟏)
𝜎1(𝑡) = Є0𝑌(𝑡) + � 𝑌(𝑡 − 𝜉)𝑑Є(𝜉)𝑑𝜉
𝑑𝜉𝑡
0 (2. 26)
𝜎1(𝑡) = 0 + � 𝐸0. �1 −�𝑝𝑖(1 − 𝑒−(𝑡−𝜉)
𝜏𝑖�𝑛
𝑖=1
�Є1𝑡1
𝑑𝜉𝑡
0 (2. 27)
𝜎1(𝑡) = 𝐸0Є1𝑡1
� �1 −�𝑝𝑖(1 − 𝑒−(𝑡−𝜉)
𝜏𝑖�𝑛
𝑖=1
� 𝑑𝜉𝑡
0 (2. 28)
𝜎1(𝑡) = 𝐸0Є1𝑡1
� �1 −�𝑝𝑖
𝑛
𝑖=1
+ �𝑝𝑖𝑒−(𝑡−𝜉)
𝜏𝑖�𝑛
𝑖=1
� 𝑑𝜉𝑡
0 (2. 29)
𝜎1(𝑡) = 𝐸0Є1𝑡1
�𝜉 −�𝑝𝑖𝜉𝑛
𝑖=1
+ �𝜏𝑖𝑝𝑖𝑒−(𝑡−𝜉)
𝜏𝑖�𝑛
𝑖=1
�0
𝑡
(2. 30)
𝜎1(𝑡) = 𝐸0Є1𝑡1
�𝑡 −�𝑝𝑖𝑡 +�𝑝𝑖𝜏𝑖 −�𝑝𝑖𝜏𝑖𝑒−𝑡 𝜏𝑖� � (2. 31)
Here, n is the number of terms in the Prony Series.
Step 2(𝒕𝟏 < 𝑡 ≤ 𝒕𝟐)
In this step, the strain is kept constant.
𝜎2(𝑡) = Є0𝑌(𝑡) + � 𝑌(𝑡 − 𝜉)𝑑Є(𝜉)𝑑𝜉
𝑑𝜉𝑡1−
0+ � 𝑌(𝑡 − 𝜉)
𝑑Є(𝜉)𝑑𝜉
𝑑𝜉𝑡
𝑡1+ (2. 32)
𝜎2(𝑡) = 0 +𝐸0Є1𝑡1
�𝜉 −�𝑝𝑖𝜉𝑛
𝑖=1
+ �𝜏𝑖𝑝𝑖𝑒−(𝑡−𝜉)
𝜏𝑖�𝑛
𝑖=1
�0
𝑡1
+ 0 (2. 33)
𝜎2(𝑡) = 𝐸0Є1𝑡1
�𝑡1 −�𝑝𝑖𝑡1 +�𝑝𝑖𝜏𝑖𝑒−(𝑡−𝑡1)
𝜏𝑖� −�𝑝𝑖𝜏𝑖𝑒−𝑡 𝜏𝑖� � (2. 34)
11
Using a non-linear regression technique [10], the above stress function can be determined by a
stress relaxation test.
To get material response in frequency domain for Prony Series, we apply Fourier
transformation. Prony Series is represented in terms of shear relaxation modulus by the
following expression [20]:
𝑔𝑅(𝑡) = 1 −�𝑔𝑖 �1 − 𝑒−𝑡 𝜏𝑖� �
𝑁
𝑖=1
(2. 35)
Here, 𝑔𝑖 and 𝜏𝑖 are material parameters and 𝑔𝑅(𝑡) is the dimensionless relaxation modulus
given by
𝑔𝑅(𝑡) =𝐺𝑅(𝑡)𝐺0
(2. 36)
Apply Fourier Transformation:
(𝑆𝑡𝑜𝑟𝑎𝑔𝑒 𝑀𝑜𝑑𝑢𝑙𝑢𝑠) 𝐺′(𝜔) = 𝐺0 �1 −�𝑔𝑖
𝑁
𝑖=1
� + 𝐺0�𝑔𝑖
𝑁
𝑖=1
�𝜏𝑖2𝜔2
1 + 𝜏𝑖2𝜔2� (2. 37)
(𝐿𝑜𝑠𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠) 𝐺′′(𝜔) = 𝐺0�𝑔𝑖
𝑁
𝑖=1
�𝜏𝑖𝜔
1 + 𝜏𝑖2𝜔2� (2. 38)
We know that for creep compliance [10],
Є(𝑡) = 𝐽(𝑡)𝜎0 (2. 39)
Here, 𝐽(𝑡) is the creep compliance function and its response under Prony Series is given by
𝐽(𝑡) = 𝐽0.�1 −�𝑝𝑖𝑒−𝑡 𝜏𝑖�
𝑛
𝑖=1
� (2. 40)
Є(𝑡) = 𝜎0𝐽(𝑡) + � 𝐽(𝑡 − 𝜉)𝑑𝜎(𝜉)𝑑𝜉
𝑑𝜉𝑡
0 (2. 41)
12
𝜎(𝑡) = �𝜎1𝑡
(𝑡1 − 𝑡0)� ; 𝑡0 < 𝑡 < 𝑡1
𝜎1 ; 𝑡1 < 𝑡 < 𝑡2�
𝑑𝜎𝑑𝑡
= �𝜎1
(𝑡1 − 𝑡0)� ; 𝑡0 < 𝑡 < 𝑡1 0 ; 𝑡1 < 𝑡 < 𝑡2
�
Here, 𝜎0 = 0, 𝜎1 is the stress level at which the stress is kept constant, and 𝑡0 = 0
Step 1(𝒕𝟎 < 𝒕 ≤ 𝒕𝟏)
Є1(𝑡) = 𝜎0𝐽(𝑡) + � 𝐽(𝑡 − 𝜉)𝑑𝜎(𝜉)𝑑𝜉
𝑑𝜉𝑡
0 (2. 42)
Є1(𝑡) = 0 + � 𝐽0.�1 −�𝑝𝑖𝑒−(𝑡−𝜉)
𝜏𝑖�𝑛
𝑖=1
�𝜎1𝑡1
𝑑𝜉𝑡
0 (2. 43)
Є1(𝑡) = 0 + � 𝐽0.�1 −�𝑝𝑖𝑒−(𝑡−𝜉)
𝜏𝑖�𝑛
𝑖=1
�𝜎1𝑡1
𝑑𝜉𝑡
0 (2. 44)
Є1(𝑡) = 𝐽0𝜎1𝑡1
�𝜉 −�𝜏𝑖𝑝𝑖𝑒−(𝑡−𝜉)
𝜏𝑖�𝑛
𝑖=1
�0
𝑡
(2. 45)
Є1(𝑡) = 𝐽0𝜎1𝑡1
�𝑡 −�𝑝𝑖𝜏𝑖 +�𝑝𝑖𝜏𝑖𝑒−𝑡 𝜏𝑖� � (2. 46)
Step 2(𝒕𝟏 < 𝒕 ≤ 𝒕𝟐)
In this step, the stress is kept constant.
Є2(𝑡) = 𝜎0𝐽(𝑡) + � 𝐽(𝑡 − 𝜉)𝑑𝜎(𝜉)𝑑𝜉
𝑑𝜉𝑡1−
0+ � 𝐽(𝑡 − 𝜉)
𝑑𝜎(𝜉)𝑑𝜉
𝑑𝜉𝑡
𝑡1+ (2. 47)
Є2(𝑡) = 0 +𝐽0𝜎1𝑡1
�𝜉 −�𝜏𝑖𝑝𝑖𝑒−(𝑡−𝜉)
𝜏𝑖�𝑛
𝑖=1
�0
𝑡1
+ 0 (2. 48)
Є2(𝑡) = 𝐽0𝜎1𝑡1
�𝑡1 −�𝑝𝑖𝜏𝑖𝑒−(𝑡−𝑡1)
𝜏𝑖� + �𝑝𝑖𝜏𝑖𝑒−𝑡 𝜏𝑖� � (2. 49)
13
2.3.6 Schapery Model
Schapery model is used to model the behavior of non-linear viscoelastic materials. We
know that for creep test,
Є𝜎
= 𝐷(𝑡) (2. 50)
Here, 𝐷(𝑡) is the creep compliance. For this creep test, the stress-strain equation is given by
[18]- [19]:
Є = 𝐷0𝜎 + 𝛥𝐷(𝑡)𝜎 (2. 51)
Here, 𝐷0 is the initial value of compliance and
𝛥𝐷(𝑡) = 𝐷(𝑡) − 𝐷0 (2. 52)
Similar response can be obtained for stress relaxation test.
When we have creep test data, i.e. 𝐷(𝑡) is known, we can calculate strain by using Boltzmann
superposition principal which is given by,
Є = 𝐷0𝜎 + � 𝛥𝐷(𝑡 − 𝜏)𝑡
0
𝑑𝜎𝑑𝜏
𝑑𝜏 (1) (2. 53)
This was the linear response of the material [14]-[19], slight changes are to made in the above
equation to obtain nonlinear constitutive equation, which is given below
Є(𝑡) = 𝑔0𝐷0𝜎 + 𝑔1 � 𝛥𝐷[𝜓(𝑡) − 𝜓′(𝜏)]𝑡
0
𝑑𝑔2𝜎𝑑𝜏
𝑑𝜏 (2) (2. 54)
The above equation gives 1-D representation for Schapery model. Here, 𝐷0 and 𝛥𝐷(𝜓) are the
instantaneous and transient linear viscoelastic creep compliance components which have been
defined previously, and 𝜓 is reduced-time given by
14
𝜓 = �𝑑𝑡′
𝑎𝜎[𝜎(𝑡′)]
𝑡
0
(2. 55)
And
𝜓′ = 𝜓(𝜏) = �𝑑𝑡′
𝑎𝜎[𝜎(𝑡′)]
𝜏
0
(2. 56)
By comparing equations (2. 53 and (2. 54 and 1 we see that 𝑔0,𝑔1,𝑔2,𝑎𝜎 = 1 when the stress is
sufficiently small. For case of creep test, constant stress 𝜎 is used and the strain rate 𝑑𝑔𝑔2𝜎𝑑𝜏
𝑑𝜏
goes to zero. So the equation modifies to [14]-[19]
𝐷𝑛 =Є𝜎
= 𝑔0𝐷0 + 𝑔1𝑔2𝛥𝐷 �𝑡𝑎𝜎� (2. 57)
If we have 𝑔0,𝑔1,𝑔2,𝑎𝜎 = 1 then above equation becomes
𝐷𝑛 =Є𝜎
= 𝐷0 + 𝛥𝐷(𝑡) (2. 58)
This is the response for linear Schapery model. The transient tensile component 𝛥𝐷(𝜓) is
expressed in terms of Prony series [17]
𝛥𝐷(𝜓) = �𝐷𝑛[1 − exp (−𝜆𝑛𝜓)]𝑁
1
(2. 59)
Here, 𝐷𝑛 and 𝜆𝑛 are Prony constants which can be determined by tensile creep compliance
data and non-linear material parameter 𝑔0,𝑔1,𝑔2,𝑎𝜎 can be determined by creep-recovery
tests.
15
CHAPTER 3
MECHANICAL TESTING
In this chapter we found mechanical properties of 3D printed material Procast obtained
from a 3D printer. Procast is a new 3D material, and its properties are unknown. Tensile tests
are conducted at 4 different strain rates to study the effect of elastic modulus. Stress relaxation
and creep test are also conducted to study the viscoelastic response of the material.
3.1 Tensile Testing
For Uniaxial Testing Shimadzu AGS-X Series universal Testing Machine Fig. 7 is used with
a load cell of 5kN. The maximum speed to elongate with the current universal testing machine
was 1000 mm/sec.
Fig. 7 : Shimadzu AGS-X Series Universal Testing Machine
A uniaxial tension test was performed with sample dimensions as per ASTM D412
standards [1]. Testing speed was taken from 1.5 mm/sec to 75 mm/sec, which corresponds to a
strain rate of 0.001 sec−1 to 0.05 sec−1.
16
Tests were performed at 4 different strain rates.
Table 1 : Different strain rates and speed of crosshead
Strain Rate Є̇ Velocity of crosshead
0.001 sec−1 1.5 mm/min
0.005 sec−1 7.5 mm/min
0.01 sec−1 15 mm/min
0.05 sec−1 75 mm/min
The quasi-static stress-strain behavior of the Procast material is shown in Fig. 8. The material
was tested at four different strain rates, and as the graph indicates, the material becomes
stiffer as we increase the strain rate.
Fig. 8 : Stress-Strain Behavior of Procast at different strain rates
0
10
20
30
40
50
60
0.00% 5.00% 10.00% 15.00% 20.00%
Stre
ss (M
Pa)
Strain (%)
Stress-Strain
Strain Rate = 0.001
Strain Rate = 0.005
Strain Rate = 0.01
Strain Rate = 0.05
17
The material response can also be studied from Table 2, where Young’s modulus and yield
strength are listed at various strain rates.
Table 2 : Young's Modulus and Yield Strength as a function of strain rate
Strain Rate
(𝒔𝒆𝒄−𝟏)
Young’s Modulus
(𝑮𝐏𝐚)
Yield Strength
(𝐌𝐏𝐚)
Yield Force
(𝐍)
0.001 0.727±0.051 16.33±3 184.66±34.7
0.005 0.77±0.05 16.66±2.62 188.66±29.17
0.01 0.788±0.056 18±2.9 203.33±33.29
0.05 0.831±0.059 19±1.63 222±17.57
From Fig. 8, we can see that initially the material experiences a linear response. After some
time, it undergoes non-linear deformation and fails around 10-12% strain. As we increase the
strain rate the material becomes more stiff and its Young’s modulus increases which is given in
Table 2.
3.2 Creep Test
Creep tests were carried out Shimadzu AGS-X Series Universal Testing Machine Fig. 7.
Procast sample was hold at different stress levels (1-10 MPa) for 1800 seconds.
Their responses are given below:
18
Fig. 9 : Creep Test Results from 1-10 MPa
Fig. 9 shows strain versus time response for creep test. It can be seen from these figures that as
the stress values are held constant, strain increases exponentially for that period.
3.3 Dynamic Mechanical Analysis Test
The DMA test was conducted using a Rheometric Scientific Equipment in bending mode.
The sample was placed on a clamping fixture and the strain amplitude was applied with by a
movable clamp at the center of the sample. Distance between clamps was 25 mm. The
response from the sample was measured as stress. The dynamic modulus (E’), dynamic loss
modulus (E”) was determined by phase angle, the strain applied and the measured stress and
their formula is given below [7]-[8].
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
0 500 1000 1500 2000
Stra
in
Time (sec)
Strain at 1-10 MPa
1 MPa2 MPa3 MPa4 MPa5 MPa6 MPa7 MPa8 MPa9 MPa10 MPa
19
𝑆𝑡𝑜𝑟𝑎𝑔𝑒 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 (E’) = 𝜎𝑜Є𝑜𝑐𝑜𝑠𝛿 (3. 1)
𝐿𝑜𝑠𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 (E’’) = 𝜎𝑜Є𝑜𝑠𝑖𝑛𝛿 (3. 2)
𝑃ℎ𝑎𝑠𝑒 𝑎𝑛𝑔𝑙𝑒 ∶ 𝑇𝑇𝑎𝑛(𝑑𝑒𝑙𝑡𝑎) = E’’E’
(3. 3)
Initially a strain sweep test was conducted in order to find the strain amplitude that can
be used for the material to follow Hook’s Law. Strain amplitude used for this polyurethane was
0.005%. Once strain amplitude was found a dynamic temperature sweep test was conducted at
a frequency of 1 Hz over a temperature range of -100 °C to 150 °C with a scanning rate of
3°C 𝑚𝑖𝑛−1. A sinusoidal strain was applied and material response was measured as stress. By
approximating the applied sinusoidal strain wave with a triangular strain wave, the average
strain was calculated as ∈̇=2 ∈ 𝜔.
Fig. 10 gives temperature sweep response for Procast at 1 Hz. Maximum value of E’ is
observed in the region of -65°C till -35°C which is in the order of 13 GPa. The values of E’ is 9.4
GPa at around 25°C. It can be clearly seen that E’ decreases around 25°C and this steep
decrease is observed till 100°C. From 100°C till 150°C material maintains almost a constant
value E’ of 0.1-0.2 GPa so this can be taken In account where material application comes in.
20
Fig. 10 : Storage Modulus and Loss Modulus plot over temperature of Procast at 1 Hz
Fig. 11 gives response for Tan Delta curve response for Procast at 1 Hz. Since, there is an
inverse relation between Storage Modulus (E’) and Tan δ given by Tan δ = Eʹ𝐸ʹʹ
, so Tan δ peak is
observed corresponding to decrease in E’ in similar temperature range Fig. 11. So, peak in Tan
delta region gives material glass transition 𝑇𝑇𝑔𝑔 value, which in for Procast is around 81°C.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
1
10
100
-150 -100 -50 0 50 100 150 200
E" (G
Pa)
E' (G
Pa)
Temp (C)
Temperature Sweep @ 1 Hz
E'(Storage Modulus)
E''(Loss Modulus)
21
Fig. 11 Storage Modulus and Tan Delta plot over temperature of Procast at 1 Hz
A frequency sweep test 0.01 Hz to 80 Hz on Procast material was also conducted at
room temperature (25°C) using same material dimension as stated in ASTM Standard.
Fig. 12 : Frequency Sweep Response of Loss Modulus at Room Temperature
0.01
0.1
1
0.1
1
10
100
-150 -100 -50 0 50 100 150 200
Tan_
Delta
E' (G
Pa)
Temp (C)
Temperature Sweep @ 1 Hz E'(Storage Modulus)Tan_Delta
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.00E+00
1.00E+08
2.00E+08
3.00E+08
4.00E+08
5.00E+08
6.00E+08
7.00E+08
8.00E+08
0 20 40 60 80 100
Tan_
Delta
E'' (
Pa)
Freq (Hz)
Loss Modulus
Tan_Delta
22
Fig. 12 gives frequency sweep response for Loss Modulus of Procast material at room
temperature. In this figure, we can see that Loss modulus values slightly increase from 0.2 GPa
at around 0.01 Hz to 0.7GPa at around 80 Hz.
Fig. 13 : Frequency Sweep Response of Storage Modulus at Room Temperature
Fig. 13 gives frequency sweep response for Storage Modulus of Procast material at room
temperature. In this figure, we can see that Storage modulus values slightly increase from 3.6
GPa at around 0.01 Hz to 5.1 GPa at around 80 Hz.
3.4 Differential Scanning Calorimetry (DSC)
To determine the melting temperature point of Procast material, a differential scanning
calorimetry (DSC) is used. In a typical DSC experiment, the difference of power required to heat
a reference pan and a sample pan is measured over wide temperature range. Typically DSC
pans are made up of Aluminium, while in certain specific experiments pans of gold, platinum
and stainless steel are also used.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.00E+00
1.00E+09
2.00E+09
3.00E+09
4.00E+09
5.00E+09
6.00E+09
0 20 40 60 80 100
Tan_
Delta
E' (P
a)
Freq (Hz)
Storage Modulus
Tan_Delta
23
In our experiment we have used a DSC 6 Perkin Elmer machine. Procast sample was
prepared in Aluminium pans (30µL), weighed and crimped. For reference a black Aluminium
pan was used. Peak observed in the following graph indicated Procast melting temperature
which is 337.5 °C.
Fig. 14 : DSC Graph for Procast
3.5 Density
We also measured density of Procast sample. For this purpose we printed out 5 by 5 by 5
rectangular samples from 3D printer at high definition, ultra high definition and extreme
ultra high definition. Three samples for each resolution level were measured to obtain the
density whose result is given in Table 3. The densities of the three samples are
1125.797±1.356 kg/m3, 1155.503±0.984 kg/m3, and 1162.283±0.835 kg/m3 for High
Definition, Ultra High-Definition, and Extreme Ultra High-Definition resolution respectively.
0
5
10
15
20
25
30
0 50 100 150 200 250 300 350 400 450
Heat
Flo
w (m
W)
Temperature (oC)
DSC Graph for Procast
𝑇𝑇𝑔𝑔 = 97.591 oC
𝑇𝑇𝑚𝑚 = 337.105 oC
24
Table 3 : Density of Procast material
High Definition Ultra High-Definition Extreme Ulta High-Definition
Sample
1
Sample
2
Sample
3
Sample
1
Sample
2
Sample
3
Sample
1
Sample
2
Sample
3
Volume
mm3 124.99 124.248 125.482 124.744 124.744 124.744 124.998 124.747 124.998
Mass
mg 140.67 139.71 141.5 144.15 143.99 144.29 145.17 145.13 145.26
Density
kg/m3 1125.3 1124.44 1127.65 1155.56 1154.27 1156.68 1161.37 1163.39 1162.09
25
CHAPTER 4
NONLINEAR REGRESSION
Nonlinear regression analysis is a technique in which experimental data is modeled by
function which is a combination of material/model parameters. The experimental data is fitted
by using an algorithm or a fitting approach to best predict the behavior. Curve fitting technique
is used to describe experimental data with mathematical equations.
Suppose there is a function y=f(x). Where x is the independent variable and y is dependent
variable, which is measured; and f is the function which uses one or more model parameters to
describe y. Better prediction of these material parameters by any algorithm, the more accurate
the function describes the data. There are many approaches and software available to find out
the best model parameters that gives a good fit. I have used Excel, to obtain my material
parameters. Excel contains the SOLVER function, which comes with every MS office package. It
uses an iterative approach to fit the data with non-linear functions [22][25].
The method used for this approach is called iterative non-linear least square fitting. In a
linear regression (least square approach), we try to minimize the value of squared sum of the
difference between experimental value and predicted/fitted value.
𝑆𝑆 = ��𝑦 − 𝑦𝑓𝑖𝑡�2
𝑛
𝑖=1
(4. 1)
Here, y is the experimental value, 𝑦𝑓𝑖𝑡 is the predicted/fitted value, and SS is the sum of the
squares. The difference between linear regression and non-linear regression is that, in the later
we use iterations to get this SS value to minimum.
26
Starting point for this method is to assume or predict good initial parameters for the
function. This good starting value provides less iteration to compute the function and obtain
best result.
In the first iteration, once we have given some initial starting value, the algorithm runs
the function and obtains some SS value. In second iteration, SOLVER makes small changes in the
initial parameters values and recalculates value of SS.
This method is repeated many times to ensure we have the smallest possible value of
SS. Several different algorithms can be used for non-linear regression, such as the Guass-
Newton, the Marquardt-Levenberg, and the Nelder-Mead. However, Excel SOLVER uses
another iterative approach called GRG (generalized reduced gradient) method. A detailed
description about this code can be found elsewhere [23]-[25].
All of these algorithms have similar properties. Each of them requires the user to input
initial parameters, and based on that it predicts or gives the best fit that function.
27
CHAPTER 5
LINEAR VISCOELASTIC MODELS
In this chapter we will study linear viscoelastic models and try to determine which
model is better for our 3D printed material Procast. We have used 3 linear viscoelastic models:
Maxwell Model, Generalized Voigt-Kelvin Model, and Prony Series. Material parameters are
determined using creep test previous chapter.
5.1 Maxwell Model
The following equation gives the response for creep strain in Maxwell model
𝜖(𝑡) = 𝜎𝑜�𝐷�1 + 𝑡 𝜏� �� (5. 1)
To find material parameters 𝐷 and 𝜏 we use Creep Test data and optimizing technique to fit our
model with the experimental data. After doing non-linear regression, we got the following
material parameters that give a good response for creep behavior for low stress levels.
Table 4 : Material Parameters for Maxwell Model
𝐷 1.5 × 10−9
𝜏 2 × 104
Using the above equation we obtain the creep strain response for Procast at different creep
stress levels which is given by the following figures
28
Fig. 15 : Creep Strain for Maxwell model for 1 -3 MPa stress
Fig. 15 shows creep strain prediction for Maxwell model at low stress levels. It can be seen from the
figure that at low stress levels the response is a good fit. For
Fig. 16 we can see that Maxwell model fails to capture the response for higher stress values.
Fig. 16 : Creep Strain for Maxwell model for 1-8 MPa stress
0.00%
0.50%
1.00%
1.50%
2.00%
0 500 1000 1500 2000
Stra
in
Time (sec)
Maxwell Model
3 MPa experiment2 MPa experiment1 MPa experiment3 MPa prediction2 MPa prediction1 MPa prediction
0.00%
0.50%
1.00%
1.50%
2.00%
0 500 1000 1500 2000
Stra
in
Time (sec)
Maxwell Model 8 MPa experiment7 MPa experiment6 MPa experiment5 MPa experiment4 MPa experiment3 MPa experiment2 MPa experiment1 MPa experiment8 MPa prediction7 MPa prediction6 MPa prediction5 MPa prediction4 MPa prediction3 MPa prediction2 MPa prediction
29
To obtain material response from time domain to frequency domain, Fourier transformation is
used on the relaxation modulus equation for Maxwell model.
𝐸(𝑡) = 𝐸𝑒−𝑡 𝜏� (5. 2)
Apply Fourier Transformation,
𝐸(𝜔) = 𝐸 �1
1 + 𝑗𝜏𝜔� (5. 3)
Here, j is the imaginary number with a value of √−1
𝐸(𝜔) = 𝐸 �𝜏𝜔
𝜏𝜔 + 𝑗� (5. 4)
Multiply and divide by conjugate:
𝐸(𝜔) = 𝐸 �𝜏𝜔
𝜏𝜔 + 𝑗� �𝜏𝜔 − 𝑗𝜏𝜔 − 𝑗
� (5. 5)
𝐸(𝜔) = 𝐸 �𝜏2𝜔2 − 𝑗𝜏𝜔
1 + 𝜏2𝜔2 � (5. 6)
Separating real and imaginary parts, we get
𝑆𝑡𝑜𝑟𝑎𝑔𝑒 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 𝐸′(𝜔) = �𝐸𝜏2𝜔2
1 + 𝜏2𝜔2� (5. 7)
𝐿𝑜𝑠𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 𝐸′′(𝜔) = �𝐸𝜏𝜔
1 + 𝜏2𝜔2� (5. 8)
5.2 Voigt-Kelvin Model
The Voigt-Kelvin model is a generalization of Voigt models in which Voigt elements are
connected in series. The Voigt-Kelvin model gives a better viscoelastic response than the simple
Voigt model.
30
The following equation gives the relation for creep compliance for this Voigt-Kelvin model
[11]-[13].
𝐷(𝑡) = �𝐷𝑖 �1 − 𝑒−𝑡 𝜏𝑖� �
𝑧
𝑖=1
(5. 9)
For our model, we considered two Voigt elements connected in series and so the above
equation becomes
𝐷(𝑡) = 𝐷1 �1 − 𝑒−𝑡 𝜏1� �+ 𝐷2 �1 − 𝑒
−𝑡 𝜏2� � (5. 10)
To find material parameters 𝐷1,𝐷2, 𝜏1, and 𝜏2 we use Creep Test data and optimizing technique
to fit our model with the experimental data.
After doing non-linear regression, we got the following material parameters that give a good
response for creep behavior
Table 5: Material Parameters for Voigt-Kelvin Model
𝐷1 1.12 × 10−10
𝐷2 1.5 × 10−9
𝜏1 399
𝜏2 3.406
Using the above equation we obtain the creep strain response for Procast at different creep
stress levels which is given by the following figures
31
Fig. 17 : Creep Strain for Voigt-Klevin Model for 1 -3 MPa stress
Fig. 17 shows creep strain prediction for Voigt-Kelvin model at low stress levels. It can be seen
from the figure that at low stress levels the response is a good fit. For Fig. 18 we can see that
Voigt-Kelvin model fails to capture the response for higher stress values.
0.00%
0.50%
1.00%
1.50%
2.00%
0 500 1000 1500 2000
Stra
in
Time (sec)
Voight-Kelvin Model
3 MPa experiment
2 MPa experiment
1 MPa experiment
3 MPa prediction
2 MPa prediction
1 MPa prediction
32
Fig. 18 : Creep Strain for Voigt-Kelvin Model for 1-8 MPa stress
Above figures give material response in time domain; to see material response in frequency
domain, we apply Fourier transform following equation.
𝐷(𝑡) = �𝐷𝑖 �1 − 𝑒−𝑡 𝜏𝑖� �
𝑛
𝑖=1
(5. 11)
Applying Fourier Transformation,
𝐷(𝜔) = �𝐷𝑖
𝑛
𝑖=1
�1 − �1
1 + 𝑗𝜏𝑖𝜔
�� (5. 12)
Here, j is the imaginary number with a value of √−1
𝐷(𝜔) = �𝐷𝑖
𝑛
𝑖=1
�1 − �𝜏𝑖𝜔
𝜏𝑖𝜔 + 𝑗�� (5. 13)
0.00%
0.50%
1.00%
1.50%
2.00%
0 500 1000 1500 2000
Stra
in
Time (sec)
Voight-Kelvin Model 8 MPa experiment7 MPa experiment6 MPa experiment5 MPa experiment4 MPa experiment3 MPa experiment2 MPa experiment1 MPa experiment8 MPa prediction7 MPa prediction6 MPa prediction5 MPa prediction4 MPa prediction3 MPa prediction2 MPa prediction1 MPa prediction
33
Multiply and divide by conjugate
𝐷(𝜔) = �𝐷𝑖
𝑛
𝑖=1
�1 − �𝜏𝑖𝜔
𝜏𝑖𝜔 + 𝑗� �𝜏𝑖𝜔 − 𝑗𝜏𝑖𝜔 − 𝑗
�� (5. 14)
𝐷(𝜔) = �𝐷𝑖 �1 − �𝜏𝑖2𝜔2 − 𝑗𝜏𝑖𝜔𝜏𝑖2𝜔2 − 𝑗2
��𝑛
𝑖=1
(5. 15)
𝐷(𝜔) = �𝐷𝑖 �1 − �𝜏𝑖2𝜔2 − 𝑗𝜏𝑖𝜔𝜏𝑖2𝜔2 + 1
��𝑛
𝑖=1
(5. 16)
𝐷(𝜔) = �𝐷𝑖 �1 + 𝜏𝑖2𝜔2 − 𝜏𝑖2𝜔2 + 𝑗𝜏𝑖𝜔
𝜏𝑖2𝜔2 + 1�
𝑛
𝑖=1
(5. 17)
𝐷(𝜔) = �𝐷𝑖 �1 + 𝑗𝜏𝑖𝜔
1 + 𝜏𝑖2𝜔2�𝑛
𝑖=1
(5. 18)
Separating real and imaginary parts, we get
(𝑆𝑡𝑜𝑟𝑎𝑔𝑒 𝑀𝑜𝑑𝑢𝑙𝑢𝑠) 𝐷′(𝜔) = ��𝐷𝑖
1 + 𝜏𝑖2𝜔2�𝑛
𝑖=1
(5. 19)
(𝐿𝑜𝑠𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠) 𝐷′′(𝜔) = �𝐷𝑖 �𝜏𝑖𝜔
1 + 𝜏𝑖2𝜔2�𝑛
𝑖=1
(5. 20)
5.3 Prony Series
A common form for the linear viscoelastic response is given by Prony Series by the
following equation [10]-[20]:
�𝛼𝑖𝑒−𝑡 𝜏𝑖�
𝑁
𝑖=1
(5. 21)
Here, 𝜏𝑖 are the time constants and 𝛼𝑖 are the exponential coefficients.
For creep test we use the creep compliance equation [10]-[20]
34
Є(𝑡) = 𝐽(𝑡)𝜎0 (5. 22)
Here, 𝐽(𝑡) is the creep compliance function and its response under Prony Series is given by
𝐽(𝑡) = 𝐽0.�1 −�𝑝𝑖𝑒−𝑡 𝜏𝑖�
𝑛
𝑖=1
� (5. 23)
Є(𝑡) = 𝜎0𝐽(𝑡) + � 𝐽(𝑡 − 𝜉)𝑑𝜎(𝜉)𝑑𝜉
𝑑𝜉𝑡
0 (5. 24)
𝜎(𝑡) = �𝜎1𝑡
(𝑡1 − 𝑡0)� ; 𝑡0 < 𝑡 < 𝑡1
𝜎1 ; 𝑡1 < 𝑡 < 𝑡2�
𝑑𝜎𝑑𝑡
= �𝜎1
(𝑡1 − 𝑡0)� ; 𝑡0 < 𝑡 < 𝑡1 0 ; 𝑡1 < 𝑡 < 𝑡2
�
Here, 𝜎0 = 0, 𝜎1 is the stress level at which the stress is kept constant, and 𝑡0 = 0
Step 1(𝒕𝟎 < 𝒕 ≤ 𝒕𝟏)
Є1(𝑡) = 𝜎0𝐽(𝑡) + � 𝐽(𝑡 − 𝜉)𝑑𝜎(𝜉)𝑑𝜉
𝑑𝜉𝑡
0 (5. 25)
Є1(𝑡) = 𝐽0𝜎1𝑡1
�𝑡 −�𝑝𝑖𝜏𝑖 +�𝑝𝑖𝜏𝑖𝑒−𝑡 𝜏𝑖� � (5. 26)
Є1(𝑡) = 𝐽0𝜎1𝑡1
�𝑡 − 𝑃1𝜏1 − 𝑃2𝜏2 + 𝑃1𝜏1𝑒−𝑡 𝜏1� + 𝑃2𝜏2𝑒
−𝑡 𝜏2� � (5. 27)
We obtained a two-term Prony Series
Step 2(𝒕𝟏 < 𝒕 ≤ 𝒕𝟐)
In this step, the stress is kept constant.
Є2(𝑡) = 𝐽0𝜎1𝑡1
�𝑡1 −�𝑝𝑖𝜏𝑖𝑒−(𝑡−𝑡1)
𝜏𝑖� + �𝑝𝑖𝜏𝑖𝑒−𝑡 𝜏𝑖� � (5. 28)
Є2(𝑡) = 𝐽0𝜎1𝑡1
�𝑡1 − 𝑃1𝜏1𝑒−(𝑡−𝑡1)
𝜏1� − 𝑃2𝜏2𝑒−(𝑡−𝑡1)
𝜏2� + 𝑃1𝜏1𝑒−𝑡 𝜏1� + 𝑃2𝜏2𝑒
−𝑡 𝜏2� � (5. 29)
35
We used the following material parameters to predict our creep response
Table 6 : Creep Material Parameters for Prony Series
𝐽 1.75 × 10−9
𝑃1 0.12
𝑃2 0.2
𝜏1 25
𝜏2 1000
Fig. 19 gives creep strain prediction using Prony Series. We can see that it gives a perfect
response till 5 MPa stress, and at higher stress levels it does not show a good response. It gives
better response than previous two viscoelastic models as it catches the trend of the creep
strain and shows a better fit.
Fig. 19: Creep Strain prediction for Prony Series
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
0 500 1000 1500 2000
Cree
p St
rain
Time (sec)
Prony Series 10 MPa9 MPa8 MPa7 MPa6 MPa5 MPa4 MPa3 MPa2 MPa1 MPa10 MPa prediction9 MPa prediction8 MPa prediction7 MPa prediction6 MPa prediction5 MPa prediction4 MPa prediction3 MPa prediction2 MPa prediction1 MPa prediction
36
To get material response in frequency domain for Prony Series, we apply Fourier
transformation. Prony Series is represented in terms of shear relaxation modulus by the
following expression [20]:
𝑔𝑅(𝑡) = 1 −�𝑔𝑖 �1 − 𝑒−𝑡 𝜏𝑖� �
𝑁
𝑖=1
(5. 30)
Here, 𝑔𝑖 and 𝜏𝑖 are material parameters and 𝑔𝑅(𝑡) is the dimensionless relaxation modulus
given by
𝑔𝑅(𝑡) =𝐺𝑅(𝑡)𝐺0
(5. 31)
𝐺𝑅(𝑡)𝐺0
= 1 −�𝑔𝑖 �1 − 𝑒−𝑡 𝜏𝑖� �
𝑁
𝑖=1
(5. 32)
𝐺𝑅(𝑡)𝐺0
= 1 −�𝑔𝑖
𝑁
𝑖=1
+ �𝑔𝑖
𝑁
𝑖=1
𝑒−𝑡 𝜏𝑖� (5. 33)
𝐺𝑅(𝑡) = 𝐺0 �1 −�𝑔𝑖
𝑁
𝑖=1
+ �𝑔𝑖
𝑁
𝑖=1
𝑒−𝑡 𝜏𝑖� � (5. 34)
𝐺𝑅(𝑡) = 𝐺0 �1 −�𝑔𝑖
𝑁
𝑖=1
� + 𝐺0�𝑔𝑖
𝑁
𝑖=1
𝑒−𝑡 𝜏𝑖� (5. 35)
Apply Fourier Transformation:
𝐺(𝜔) = 𝐺0 �1 −�𝑔𝑖
𝑁
𝑖=1
� + 𝐺0�𝑔𝑖
𝑁
𝑖=1
1
1 + 𝑗𝜏𝑖𝜔�
(5. 36)
Here, j is the imaginary number with a value of √−1
37
𝐺(𝜔) = 𝐺0 �1 −�𝑔𝑖
𝑁
𝑖=1
� + 𝐺0�𝑔𝑖
𝑁
𝑖=1
1
1 + 𝑗𝜏𝑖𝜔�
(5. 37)
𝐺(𝜔) = 𝐺0 �1 −�𝑔𝑖
𝑁
𝑖=1
� + 𝐺0�𝑔𝑖
𝑁
𝑖=1
𝜏𝑖𝜔𝜏𝑖𝜔 + 𝑗
(5. 38)
Multiply and divide by conjugate:
𝐺(𝜔) = 𝐺0 �1 −�𝑔𝑖
𝑁
𝑖=1
� + 𝐺0�𝑔𝑖
𝑁
𝑖=1
�𝜏𝑖𝜔
𝜏𝑖𝜔 + 𝑗� �𝜏𝑖𝜔 − 𝑗𝜏𝑖𝜔 − 𝑗
� (5. 39)
𝐺(𝜔) = 𝐺0 �1 −�𝑔𝑖
𝑁
𝑖=1
� + 𝐺0�𝑔𝑖
𝑁
𝑖=1
�𝜏𝑖2𝜔2 − 𝜏𝑖𝜔𝑗𝜏𝑖2𝜔2 − 𝑗2
� (5. 40)
𝐺(𝜔) = 𝐺0 �1 −�𝑔𝑖
𝑁
𝑖=1
� + 𝐺0�𝑔𝑖
𝑁
𝑖=1
�𝜏𝑖2𝜔2
𝜏𝑖2𝜔2 + 1� − 𝑗𝐺0�𝑔𝑖
𝑁
𝑖=1
�𝜏𝑖𝜔
𝜏𝑖2𝜔2 + 1� (5. 41)
We know that
𝐺(𝜔) = 𝐺′(𝜔) + 𝑗𝐺′′(𝜔) (5. 42)
Separating real and imaginary parts, we get
(𝑆𝑡𝑜𝑟𝑎𝑔𝑒 𝑀𝑜𝑑𝑢𝑙𝑢𝑠) 𝐺′(𝜔) = 𝐺0 �1 −�𝑔𝑖
𝑁
𝑖=1
� + 𝐺0�𝑔𝑖
𝑁
𝑖=1
�𝜏𝑖2𝜔2
1 + 𝜏𝑖2𝜔2� (5. 43)
(𝐿𝑜𝑠𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠) 𝐺′′(𝜔) = 𝐺0�𝑔𝑖
𝑁
𝑖=1
�𝜏𝑖𝜔
1 + 𝜏𝑖2𝜔2� (5. 44)
38
CHAPTER 6
NON-LINEAR VISCOELASTIC MODEL
In this chapter we will discuss methods of characterization non-linear viscoelastic
materials. We will be using constitutive equations from Chapter 1 and using experimental data
(creep test data) to obtain material parameters. In small deformation materials, the total strain
is the sum of viscoelastic strain and viscoplastic strain [14]-[19].
Є(𝑡) = Є𝑣𝑒(𝑡) + Є𝑣𝑝(𝑡) (6. 1)
While, the incremental strain is given by
𝛥Є(𝑡) = 𝛥Є𝑣𝑒(𝑡) + 𝛥Є𝑣𝑝(𝑡) At time t>0 (6. 2)
Here, Є𝑣𝑒refers to viscoelastic strain and Є𝑣𝑝 refers to viscoplastic strain and 𝛥Є𝑣𝑒 gives the
incremental viscoelastic strain while 𝛥Є𝑣𝑝 gives the incremental viscoplastic strain. In our
model we are only concerned with elastic deformation and have omitted the viscoplastic part.
Schapery equation in 1-D form is given by the following equation [14]-[19],
Є(𝑡) = 𝑔0𝐷0𝜎 + 𝑔1 � 𝛥𝐷[𝜓(𝑡) − 𝜓′(𝜏)]𝑡
0
𝑑𝑔2𝜎𝑑𝜏
𝑑𝜏 (1) (6. 3)
Here, 𝐷0 and 𝛥𝐷(𝜓) are the instantaneous and transient linear viscoelastic creep compliance
components which have been defined previously, and 𝜓 is reduced-time given by
𝜓 = �𝑑𝑡′
𝑎𝜎[𝜎(𝑡′)]
𝑡
0
(6. 4)
And
𝜓′ = 𝜓(𝜏) = �𝑑𝑡′
𝑎𝜎[𝜎(𝑡′)]
𝜏
0
(6. 5)
39
In a 3-D representation of Schapery model, stress and strain are given by its deviatoric and
hydrostatic components.
𝑆𝑖𝑗 = 𝜎𝑖𝑗 −13𝛿𝑖𝑗𝜎𝑘𝑘 (6. 6)
𝑑𝑖𝑗 = Є𝑖𝑗 −13𝛿𝑖𝑗Є𝑘𝑘 (6. 7)
Here, 𝛿𝑖𝑗 is Kronecker Delta and 13𝜎𝑘𝑘 and 1
3Є𝑘𝑘 are the hydrostatic stress and strain
respectively.
For an isotropic linear elastic material, the relation between stress and strain can be expressed
as [14]-[19]
Є𝑖𝑗 =12𝐽𝑆𝑖𝑗 +
19𝐵𝛿𝑖𝑗𝜎𝑘𝑘 (6. 8)
Here, 𝐽 is the shear compliance and 𝐵 is the bulk compliance.
So, using our 1-D Schapery equation, we can get 3-D non-linear viscoelastic model as
Є𝑖𝑗(𝑡) =12𝑔0𝐽0𝑆𝑖𝑗(𝑡) +
12𝑔1 � 𝛥𝐽[𝜓(𝑡) − 𝜓′(𝜏)]
𝑡
0
𝑑𝑔2𝑆𝑖𝑗𝑑𝜏
𝑑𝜏
+19𝑔0𝐵0𝛿𝑖𝑗𝜎𝑘𝑘(𝑡) +
19𝑔1𝛿𝑖𝑗 � 𝛥𝐵[𝜓(𝑡) − 𝜓′(𝜏)]
𝑡
0
𝑑𝑔2𝜎𝑘𝑘𝑑𝜏
𝑑𝜏
(6. 9)
The instantaneous shear compliance 𝐽0 and instantaneous bulk compliance 𝐵0 can be given by
the following equations, where ν is the Poison’s ratio [14]-[19].
𝐽0 = 2(1 + 𝝂)𝛥𝐷(𝜓) (6. 10)
𝐵0 = 3(1 − 2𝝂)𝛥𝐷(𝜓) (6. 11)
The transient tensile component 𝛥𝐷(𝜓) is expressed in terms of Prony series [17]
40
𝛥𝐷(𝜓) = �𝐷𝑛[1 − exp (−𝜆𝑛𝜓)]𝑁
1
(6. 12)
Here, 𝐷𝑛 and 𝜆𝑛 are Prony constants which can be determined by tensile creep compliance
data.
Using the relation between shear compliance and tensile compliance we can write the following
equation as;
𝛥𝐽(𝜓) = �𝐽𝑛[1 − exp(−𝜆𝑛𝜓)]𝑁
1
(6. 13)
Similarly,
𝛥𝐵(𝜓) = �𝐵𝑛[1 − exp(−𝜆𝑛𝜓)]𝑁
1
(6. 14)
Here,
𝐽𝑛 = 2(1 + 𝝂)𝐷𝑛 (6. 15)
𝐵 = 3(1 − 2𝝂)𝐷𝑛 (6. 16)
The Prony constants 𝐷𝑛 and 𝜆𝑛 can be obtained creep test data by using curve fitting
approach, and by using the relation between shear compliance and tensile compliance we can
obtain the remaining terms. At first linear response of the material is considered (at low value
of stress) and non-linear material parameters 𝑔0,𝑔1,𝑔2, and 𝑎𝜎 is considered as 1.
Transient creep compliance can be given by the following by using simple power law [17]-[19]
𝛥𝐷(𝜓) = 𝐶𝜓𝑚𝑚 (6. 17)
Here, 𝐶 and 𝑚 are material constants. In the case of creep test where σ is held constant for
time> 0, equation (6. 3 becomes
41
Є𝑐(𝑡) = 𝑔0𝐷0𝜎 +𝑔1𝑔2𝐶𝑎𝜎𝑚𝑚
𝑡𝑚𝑚 (6. 18)
Taking low stress region into account, in our case creep test at 1 MPa, where the response of
the material is almost linear and non-linear material parameters 𝑔0,𝑔1,𝑔2, and 𝑎𝜎 is
considered as 1, above equation deduced into [17][19]
Є𝑐(𝑡) = 𝐷0𝜎 + 𝐶𝑡𝑚𝑚 (6. 19)
Using curve fitting technique to find out the constants 𝐷0, 𝐶 and 𝑚. Using the above
equations, creep strain response is predicted by using the following equation.
Є𝑐(𝑡) = �𝑔0𝐷0 + 𝑔1𝑔2𝛥𝐷�𝑡 𝑎𝜎� �� 𝜎 (6. 20)
This is the required equation we will be using in order to predict our response. We use the
following material parameters to get the response, while 𝑔0,𝑔1,𝑔2,𝑎𝜎 = 1 is used for linear
visco-elastic response. Here 𝐷0 = 1.43 × 10−3 MPa−1.
Table 7 : Material Parameters for Schapery Model
𝑵 𝑫𝒏(𝑴𝑷𝒂−𝟏) 𝝀𝒏(𝒔−𝟏)
1 1.11 × 10−5 1
2 1 × 10−5 0.1
3 1 × 10−5 0.01
4 4.6 × 10−5 0.001
5 7.5 × 10−4 0.0001
6 9 × 10−4 0.00001
42
Fig. 20 : Schapery Model Prediction of Creep Test
Fig. 20 gives creep strain prediction of Procast using Schapery model. We can see that it gives a
better material response than Prony series, even at higher stress values.
Fig. 21 : Nonlinear Parameters for Schapery Model at Various Stress Levels
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
0 500 1000 1500 2000
Cree
p St
rain
Time (sec)
Schapery Model 10 MPa9 MPa8 MPa7 MPa6 MPa5 MPa4 MPa3 MPa2 MPa1 MPa10 MPa prediction9 MPa prediction8 MPa prediction7 MPa Prediction6 MPa Prediction5 MPa Prediction4 MPa prediction3 MPa Prediction2MPa Prediction1MPa Prediction
0.5
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12
Non
linea
r vis
coel
astic
par
amet
ers
Stress (MPa)
g0
g1
g2
a
43
Nonlinear viscoelastic parameters can be obtained for other stress levels by using the constants
listed in Table 8 for polynomial equations, where subscript of α denotes exponent of stress for
example;
𝑓(𝑔0,𝑔1,𝑔2,𝑎) = 𝛼𝑛𝜎𝑛 + 𝛼𝑛−1𝜎𝑛−1 + ⋯+ 𝛼0𝜎0
Table 8 : Polynomial Constants for Nonlinear Material Paramters
𝜶𝟒 𝜶𝟑 𝜶𝟐 𝜶𝟏 𝜶𝟎
𝑔0 - - 0.0032 -0.0111 1.0037
𝑔1 0.0005 -0.0107 0.0719 -0.1091 1.0119
𝑔2 0.0004 -0.0085 0.057 -0.0866 1.0095
𝑎 0.0001 -0.0026 0.0171 -0.0254 1.0026
Equation (6. 20 gives material response in time domain, to obtain material response in
frequency domain we apply Fourier transformation.
Є𝑐(𝑡)𝜎
= �𝑔0𝐷0 + 𝑔1𝑔2𝛥𝐷�𝑡 𝑎𝜎� �� (6. 21)
Using equation (6. 12, we get
𝐷(𝑡) = 𝑔0𝐷0 + 𝑔1𝑔2�𝐷𝑛�1 − exp �−𝜆𝑛 𝑡 𝑎𝜎� ��𝑁
1
(6. 22)
Applying Fourier Transformation,
𝐷(𝜔) = 𝑔0𝐷0 + 𝑔1𝑔2𝐷𝑛 �1 − �1
1 + 𝑗𝜔 �
𝜆𝑛𝑎 �
�� (6. 23)
44
𝐷(𝜔) = 𝑔0𝐷0 + 𝑔1𝑔2𝐷𝑛 �1 − �𝜔
𝜔 + 𝑗 �𝜆𝑛𝑎 ��� (6. 24)
Multiply and divide by conjugate
𝐷(𝜔) = 𝑔0𝐷0 + 𝑔1𝑔2𝐷𝑛 �1 − �𝜔
𝜔 + 𝑗 �𝜆𝑛𝑎 ���
𝜔 − 𝑗 �𝜆𝑛𝑎 �
𝜔 − 𝑗 �𝜆𝑛𝑎 ��� (6. 25)
𝐷(𝜔) = 𝑔0𝐷0 + 𝑔1𝑔2𝐷𝑛
⎣⎢⎢⎡1 −
⎝
⎛𝜔2 − 𝑗𝜔 �𝜆𝑛𝑎 �
𝜔2 + �𝜆𝑛𝑎 �2
⎠
⎞
⎦⎥⎥⎤ (6. 26)
𝐷(𝜔) = 𝑔0𝐷0 + 𝑔1𝑔2𝐷𝑛
⎝
⎛𝜔2 + �𝜆𝑛𝑎 �
2− 𝜔2 + 𝑗𝜔 �𝜆𝑛𝑎 �
𝜔2 + �𝜆𝑛𝑎 �2
⎠
⎞ (6. 27)
𝐷(𝜔) = 𝑔0𝐷0 + 𝑔1𝑔2𝐷𝑛
⎝
⎛�𝜆𝑛𝑎 �
2+ 𝑗𝜔 �𝜆𝑛𝑎 �
𝜔2 + �𝜆𝑛𝑎 �2
⎠
⎞ (6. 28)
Separating real and imaginary parts, we get
(𝑆𝑡𝑜𝑟𝑎𝑔𝑒 𝑀𝑜𝑑𝑢𝑙𝑢𝑠) 𝐷′(𝜔) = 𝑔0𝐷0 + 𝑔1𝑔2𝐷𝑛
⎝
⎛�𝜆𝑛𝑎 �
2
𝜔2 + �𝜆𝑛𝑎 �2
⎠
⎞ (6. 29)
(𝐿𝑜𝑠𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠) 𝐷′(𝜔) = 𝑔1𝑔2𝐷𝑛
⎝
⎛𝜔 �𝜆𝑛𝑎 �
𝜔2 + �𝜆𝑛𝑎 �2
⎠
⎞ (6. 30)
45
CHAPTER 7
VALIDATION OF MODEL
In this chapter we will try to validate our models with the uniaxial tensile tests we
performed. For this purpose we have taken tensile test data in Fig. 8 and plotted its response
for various models. We have considered only the elastic region which in our case is under 2.5%
strain, which can be seen by following figure.
Fig. 22 : Strain-Rate Dependent Tensile Test Data Including Yielding Region
The yielding region is considered to be under 2.5 % strain. This is due to the fact that we are
considering the elastic response, rather than the plastic deformation. We will try to fit the
validation of different material model under 2.5 % strain.
0
10
20
30
40
50
60
0.00% 5.00% 10.00% 15.00% 20.00%
Stre
ss (M
Pa)
Strain (%)
Stress-Strain
Strain Rate = 0.001
Strain Rate = 0.005
Strain Rate = 0.01
Strain Rate = 0.05
Yield Point
46
Fig 23 : Validation for Maxwell Model at Various Strain Rates
Fig 23 shows response of Maxwell model at different strain rates. It can be seen that current
Maxwell model is unable to capture complete deformation at various strain rates. This is due to
the fact of very high time constant.
Fig. 24 : Validation for Voigt-Kelvin Model at Various Strain Rates
0
5
10
15
20
25
0.00% 0.50% 1.00% 1.50% 2.00% 2.50%
Stre
ss (M
Pa)
Strain
Maxwell Model
Model 0.05
Model 0.01
Model 0.005
Model 0.001
Experiment 0.05
Experiment 0.01
Experiment 0.005
Experiment 0.001
0
10
20
30
40
50
60
70
0.00% 0.50% 1.00% 1.50% 2.00% 2.50%
Stre
ss (M
Pa)
Strain
Voigt-Kelvin Model
Model 0.05
Model 0.01
Model 0.005
Model 0.001
Experiment 0.05
Experiment 0.01
Experiment 0.005
Experiment 0.001
47
Fig. 24 shows response of Voigt-Kelvin model at different strain rates. It can be seen that at low
strain rates the response matches but at higher strain rates it starts deviating. The main reason
for this is due to high values of time constants.
Fig. 25 : Validation for Prony Series at Various Strain Rates
Fig. 25 shows response of Prony series at different strain rates. It can be seen that Prony series
captures very well the deformation at various strain rates.
0
5
10
15
20
25
0.00% 0.50% 1.00% 1.50% 2.00% 2.50%
Stre
ss (P
a)
Strain
Prony Series
Model 0.05
Model 0.01
Model 0.005
Model 0.001
Experiment 0.05
Experiment 0.01
Experiment 0.005
Experiment 0.001
48
Fig. 26 : Validation for Schapery Model at Various Strain Rates
Fig. 26 shows response of Schapery model at different strain rates. It can be seen that Schapery
model also captures the deformation very well at various strain rates.
Using the equations for dynamic response, we try to predict the loss modulus and
storage modulus values of frequency sweep of Procast data. Fig. 27Fig. 30 gives loss modulus
response for all viscoelastic models. We can see that all models do not capture a complete
response for loss modulus behavior.
0
5
10
15
20
25
0.00% 0.50% 1.00% 1.50% 2.00% 2.50%
Stre
ss (M
Pa)
Strain
Schapery Model
Model 0.05
Model 0.01
Model 0.005
Model 0.001
Experiment 0.05
Experiment 0.01
Experiment 0.005
Experiment 0.001
49
Fig. 27 : Loss Modulus response for Maxwell model
Fig. 28 : Loss Modulus response for Voigt-Kelvin model
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
0 20 40 60 80 100
Loss
Mod
ulus
(Pa)
Frequency (Hz)
Maxwell Model
Experiment
Maxwell Model
1.00E+001.00E+011.00E+021.00E+031.00E+041.00E+051.00E+061.00E+071.00E+081.00E+091.00E+101.00E+111.00E+12
0 20 40 60 80 100
Loss
Mod
ulus
(Pa)
Frequency (Hz)
Voigt-Kelvin Model
Experiment
Voigt-Kelvin Model
50
Fig. 29 : Loss Modulus response for Prony model
Fig. 30 : Loss Modulus response for Schapery Model
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
0 20 40 60 80 100
Loss
Mod
ulus
(Pa)
Frequency (Hz)
Prony Series
Experiment
Prony Series
1.00E+06
1.00E+07
1.00E+08
1.00E+09
1.00E+10
1.00E+11
1.00E+12
1.00E+13
0 20 40 60 80 100
Loss
Mod
ulus
(Pa)
Frequency (Hz)
Schapery Model
Experiment
Schapery
51
Using the equations for dynamic response (storage modulus), we try to predict the
storage modulus values of frequency sweep of Procast data. Fig. 30 Fig. 31Fig. 34 gives storage
modulus response for all viscoelastic models. We can see that Maxwell and Voigt-Kelvin models
do not capture the complete dynamic response while Prony series and Schapery model gives a
better prediction in terms of storage modulus.
Fig. 31 : Storage Modulus response for Maxwell model
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
1.00E+10
1.00E+11
1.00E+12
0 20 40 60 80 100
Stor
age
Mod
ulus
(Pa)
Frequency (Hz)
Maxwell Model
Experiment
Maxwell Model
52
Fig. 32 : Storage Modulus response for Voigt-Kelvin model
Fig. 33 : Storage Modulus response for Prony Series
1.00E+00
1.00E+02
1.00E+04
1.00E+06
1.00E+08
1.00E+10
1.00E+12
1.00E+14
0 20 40 60 80 100
Stor
age
Mod
ulus
(Pa)
Frequency (Hz)
Voigt-Kelvin Model
Experiment
Voigt-Kelvin
1.00E+06
1.00E+07
1.00E+08
1.00E+09
1.00E+10
0 20 40 60 80 100
Stor
age
Mod
ulus
(Pa)
Frequency (Hz)
Prony Series
Experiment
Prony Series
53
Fig. 34 : Storage Modulus response for Schapery model
1.00E+06
1.00E+07
1.00E+08
1.00E+09
1.00E+10
0 20 40 60 80 100
Stor
age
Mod
ulus
(Pa)
Frequency (Hz)
Schapery Model
Experiment Schapery
54
CHAPTER 8
CONCLUSIONS AND FUTURE WORK
Additive manufacturing is a relatively new technique and mechanical response of the base
material used for printing is also unknown. In my thesis I have conducted various tests to
determine the base properties for Procast material. My major findings include
(i) Glass transition temperature of Procast, which is in the range of 81°C
(ii) Melting temperature of Procast which is around 337.1°C
(iii) Density of Procast material can be seen from Table 3 which is around 1125-1162
kg/m3
(iv) Strain rate dependent elastic modulus which can be seen from Table 2 which is
around 0.7-0.8 GPa
(v) Procast yield stress which is around 16-19 MPa
(vi) Linear viscoelastic stress level for Procast which is around 3 MPa
(vii) Prony Series and Schapery Model show good validation response at various strain
rates.
These finding from my thesis can benefit the research community. It can also be seen from my
viscoelastic models that Prony Series show better fit for linear viscoelastic response and
Schapery model shows good fit for all stress levels for non-linear viscoelastic region.
So far, I have studied the material response for Procast material, in the future we can also look
at complex cellular geometry behavior of Procast material.
55
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New York : Springer, 2010
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Creep and Creep-Rupture, ASTM D2990-09
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56
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57
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