ELASTOMERLERİN SAYISAL YÖNTEMLERLE ANALİZİ VE 

ŞEKİLSEL DÖKÜM YÖNTEMİ İLE DARBELERE DAYANIKLI 

ROBOT GELİŞTİRİLMESİ 

 

 
 

NUMERICAL ANALYSIS AND MANUFACTURING OF 

ELASTOMERS UTILIZING SHAPE DEPOSITION 

MANUFACTURING METHOD TO DESIGN A RUGGED 

ROBOT 

 

 

 

BEHRANG SHAMSADINLO 

 

 

 

DR. ÖZGÜR ÜNVER 

Supervisor 

 

 

 

 

Submited to the Institute of Science of Hacettepe University 

As a Partial Fulfillment to the Requirements  

For the Award of the Degree of Master of Science  

In Mechanics 

 

 

2013



This work named “ Numerical Analysis and Manufacturing of Elastomers  

Utilizing Shape Deposition Manufacturing Method to Design a Rugged Robot” 

by BEHRANG SHAMSADINLO has been approved as a thesis for the degree of 

Master of Science in Mechanic by below mentioned Examining Committee 

Members. 

 

Asst. Prof. Benat KOÇKAR         ................................................................... 

Head 

 

Dr. Özgür ÜNVER    ................................................................... 

Supervisor 

 

Asst. Prof. Tuncay KARAÇAY                  .................................................................. 

Member 

 

 

 

This thesis has been approved as a thesis for the Degree of MASTER OF SCIENCE 

IN MECHANICS by Board of Directors of the Institute for Graduate Studies in Science 

and Engineering.  

 
 
 
 
 
 
 
 
 
 

Prof. Dr. Fatma SEVİN DÜZ  

Director of the Institute of Graduate Studies in Science 

 

 



 
 

 

 

 

 

 

 

 

 

 

 

 

To my family 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 
 

ETHICS 

 

In this thesis study, prepared in accordance with the spelling rules of Institute of 

Graduate Studies in Science of Hacettepe University,  

 

I declare that  

 all the information and documents have been obtained in the base of the 

academic rules  

 all audio-visual and written information and results have been presented 

according to the rules of scientific ethics 

 in case of using others Works, related studies have been cited in accordance 

with the scientific standards 

 all cited studies have been fully referenced  

 I did not do any distortion in the data set  

 and any part of this thesis has not been presented as another thesis study at 

this or any other university. 

 

 

 

 

 

 

16/12/2013 
 

BEHRANG SHAMSADINLO 

 

 

 

 

 



i 

 

ÖZET 

 

 

ELASTOMERLERİN SAYISAL YÖNTEMLERLE ANALİZİ VE 

ŞEKİLSEL DÖKÜM YÖNTEMİ İLE DARBELERE DAYANIKLI ROBOT 

GELİŞTİRİLMESİ 

 

 

Behrang SHAMSADINLO 

Yüksek Lisans, Makina Mühendisliği Bölümü 

Tez danışmanı: Dr. Özgür Ünver 

Aralık, 2013 

 

 

Günümüzde kullanımları eksponansiyel olarak artan mekatronik uygulamalar ve 
robotlar her geçen gün daha hızlı, akıllı ve kullanışlı olmaktadırlar. Yeni nesil robotlar 
gövdelerinde bir çok kamera, algılayıcı ve eyleyici gibi yüksek ivme ve gerilim altında 
kırılgan olabilecek parçalarla çalışmaktadır. Bu tezin amacı; hali hazırda tedarik 
edilen algılayıcı ve eyleyicilerin elastomerik malzemelerin içine gömülmesi ile 
robotların çok daha dayanıklı hale getirilmesidir. Bu proje kapsamında hızlı 
prototipleme yöntemlerinden biri olan Şekilsel Döküm ve Üretim Yöntemi (ŞDÜ) 
kullanılmıştır.  

Bu tez kapsamında; yüksek aşınma direnci, yırtılma direnci, kimyasal direnç, 
mikrobiyal direnç ve ayarlanabilir Shore sertliği nedenleri ile hızlı hareketlerde 
viskoelatik özelliği, yüksek deformasyonda ise hiperelastik özellik gösteren 
poliüretanın kullanılmasına karar verilmiştir. 

Bu tezin hedefi; malzemelerin viskoelastik ve hiperelastik özelliklerini kullanarak 
darbelere karşı dayanıklı bir robot geliştirme algoritmasının bulunmasıdır. Bu amaç 
doğrultusunda; seçilen elastomerik malzemeler Sonlu Elemanlar Yöntemi ve Yapay 
Sinir Ağı gibi saysal yöntemler kullanarak analiz edilmiştir. Sonlu elemanlar modelini 
oluşturmak için tek eksenli çekme, kesme, iki eksenli çekme gibi test sonuç bilgilerine 
ihtiyaç vardır. Ancak kesme ve iki eksenli çekme testleri çok nadir bulunan ve pahalı 
test cihazları tarafından gerçekleştirilebilmektedir. Bu yüzden, bu testler fiziksel olarak 



ii 

 

gerçekleştirilmemiş, yerine Valanis-Landel yöntemi ile çekme ve basma test verileri 
kullanarak yaklaşık sonuçlar bulunmuştur. 

Elde edilen tüm sonuçların test edilmesi ve modellerle karşılaştırılması amacıyla bu 
tez kapsamında düşme test cihazı tasarlanmış ve üretilmiştir. Bu test düzeneği ile 
farklı alan ve kalınlıklara sahip malzemeler farklı değişik hız ve ağırlıklarla test 
edilmiş, böylece tüm bu değişkenlerin malzeme üzerinde oluşturduğu azami ivme ve 
gerilme değerleri üzerine olan etkileri belirlenmiştir. 

 
 
Anahtar kelımeler: Poliüretan Elastomer, Sonlu Elemanlar Yöntemi, Hiperelastik ve 
Viskoelastik Malzemeler, Şekilsel Döküm ve Üretim. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 



iii 

 

ABSTRACT 

 

 

NUMERICAL ANALYSIS AND MANUFACTURING OF ELASTOMERS 

UTILIZING SHAPE DEPOSITION MANUFACTURING METHOD TO 

DESIGN A RUGGED ROBOT 

 

 

Behrang SHAMSADINLO 

Master of Science, Department of Mechanical Engineering 

Supervisor: Dr. Özgür ÜNVER 

December, 2013 

 

 

The robotic and its evolution in recent years is one of the most transformative 
inventions of man. Over years innovations have made robots faster, more intelligent 
and smart. Modern robots utilize increasing numbers of cameras, sensors, actuators 
and etc. Some of these mechatronic devices are sensitive to applied acceleration 
and/or stress.  More electronic devices are expected to provide more functionality 
whilst remaining safe when using in harsh environment. In this thesis, the ultimate 
goal is to design a rugged robot which provides maximum level of protection for 
sensitive components. This performance would be satisfied by embedding the 
electronic devices into a host objects made by rubber-like materials; which are widely 
used in decreasing intensity of impact loads or acceleration. Polyurethane elastomers 
are viscoelastic rubber-like materials with very wide shore hardness which provide 
different dynamic properties. These materials are selected due to their superior 
properties such as; viscoelastic behavior under high strain rates and elastic behavior 
under low strain-rates. 

Embedding of the components in the host material requires special manufacturing 
method to manufacture the robot in 3-D and monolithic way. We used a special rapid 
prototyping method named Shape Deposition Manufacturing (SDM) to manufacture 
the robot.  



iv 

 

The main objective of this thesis is to design a rugged robot by analyzing viscoelastic 
and hyperelastic behaviors of the material. Numerical methods such as Finite 
Element Method (FEM) and Neural network are implemented to analyze the behavior 
of material under different loading conditions. FEM requires data-set results of 
materials from multiple test set-ups (uniaxial tension, pure shear, biaxial). Shear and 
biaxial tests require special mechanical devices that are scarce and expensive, 
therefore; an alternative Valanis-Landen method was used instead of performing 
these tests. 

To analyze the realistic impact, a test device is designed to test the specimens with 
different area and thickness under different testing conditions of drop height and drop 
mass. Finally, the best design parameters are selected to ensure protection of the 
rugged robot. 

 
 
Keywords: Polyurethane Elastomer, Finite Element Method, Hyperelastic and 
Viscoelastic Analysis, Shape Deposition Manufacturing.  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



v 

 

TEŞEKKÜR 

 
 

Yüksek lisans eğitimime ilk başladığım andan itibaren özveriyle bilgisini, sabrını ve insani 

ilgisini esirgemeyen tez danışmanım Dr. Özgür Ünver’e, yeni çıkacağım bu yolda sonsuz 

desteği için teşekkürler.  

Ayrıca, başta Sayın Yrd. Doç. Dr. Benat Koçkar ve Prof. Bora Yıldırım ve olmak üzere, 

yüksek lisans eğitimim süresince üzerimde emeği geçen Hacettepe Üniversitesi Makine 

Mühendisliği Bölümü’ndeki bütün akademisyenlere teşekkür ederim. Aynı zamanda, tez 

komitesinde jüri üyeliğinde bulunan Sayın Yrd. Doç. Dr. Tuncay Karaçay’a tezime 

gösterdiği ilgi için çok teşekkür ederim.  

Yüksek lisansım süresince çalıştığım Hacettepe Makine Mühendisliği bölümündeki 

Senstech Laboratuarındaki çalışma arkadaşlarıma göstermiş oldukları yardımları ve 

dostlukları için çok teşekkür ederim. Yüksek lisansta her zaman yanımda olan Peyman 

Ansari ve Morteza Dusti’ye teşekkürler. Ayrıca yüksek lisans süresinde beraber bir evde 

yaşadığımız Rashad Qafarov ve kardeş gibi herzaman yanımda olan Mehdi 

Mazaheri’den çok teşekkür ederim. 

Hayatımda her zaman yanımda olan ve benden maddi ve manevi hiçbir desteği 

esirgemeyen babama ve anneme çok teşekkür ederim. Yüksek lisansım süresince 

manevi desteklerinden dolayı abim Ali’ye de teşekkür ederim. 

 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 



vi 

 

CONTENTS 

                     Page 

ÖZET ............................................................................................................................. i 

ABSTRACT ................................................................................................................. iii 

TEŞEKKÜR .................................................................................................................. v 

CONTENTS ................................................................................................................. vi 

LIST OF TABLES ...................................................................................................... viii 

LIST OF FIGURES .................................................................................................... viii 

NOMENCLATURE ....................................................................................................... x 

1. INTRODUCTION .................................................................................................. 1 

1.1. Problem Statement ............................................................................................... 1 

1.2. Research Goal and Objectives ............................................................................. 1 

1.3. Challenges ........................................................................................................... 2 

1.4. Outline .................................................................................................................. 3 

2. LITERATURE SURVEY .......................................................................................... 4 

2. 1. Definition of the Critical Issues ............................................................................ 4 

2. 2. Analyzing the Hyperelasticity and Viscoelasticity of Materials ............................. 5 

2.2.1. Hyperelastic Material Models .......................................................................... 10 

2.2.1.1. Mooney-Rivlin ............................................................................................... 11 

2.2.1.2. Ogden ........................................................................................................... 12 

2.2.1.3. Neo-Hookean ............................................................................................... 12 

2.2.1.4. Gent .............................................................................................................. 13 

2.2.2. Experimental set-up ........................................................................................ 14 

2.2.2.1. Details of the Material ................................................................................... 15 

2.2.2.2. Uniaxial Tension Test ................................................................................... 16 

2.2.2.3. Biaxial Test ................................................................................................... 18 

2.2.2.4. Shear Test .................................................................................................... 20 

2.2.2.5. Drop Test Device .......................................................................................... 21 

2.3. Manufacturing Process ....................................................................................... 25 

2.3.1. Shape Deposition Manufacturing (SDM) ......................................................... 25 



vii 

 

2.3.2. CAD Model Creation........................................................................................ 26 

2.3.3. Materials in SDM ............................................................................................. 27 

2.3.3.1. Part material ................................................................................................. 28 

2.3.3.2. Support Material ........................................................................................... 29 

2.3.3.3. Temporary Material ...................................................................................... 29 

2.3.3.4. Sacrificial Material ........................................................................................ 29 

2.3.3.5. Embedded Components ............................................................................... 29 

2.3.3.6. General Considerations in Material Selection ............................................... 31 

2.3.3.7. Effective Parameters on Changing Mechanical Properties of Elastomers .... 33 

2.3.3.7.1. Cure time ................................................................................................... 33 

2.3.3.7.2. Mixing time ................................................................................................ 34 

2.3.3.7.3. Annealing .................................................................................................. 34 

2.3.3.7.4. Creep ......................................................................................................... 36 

2.3.3.7.5. Stress softening (Mullins effect) ................................................................ 36 

2.3.4. Different Manufacturing Process of SDM ........................................................ 37 

3.3.5. Fixturing Challenges for Flexible Components ................................................ 38 

3. RESULTS AND DISCUSSIONS ............................................................................ 40 

3.1. Operations on Material ....................................................................................... 40 

3.2. Increasing SDM Performance ............................................................................ 41 

3.3. Finding constants of Hyperelastic Materials ....................................................... 43 

3.3.1. Constants of Mooney-Rivlin ............................................................................. 43 

3.3.2. Constants of Ogden ........................................................................................ 44 

3.3.3. Constants of Neo-Hookean ............................................................................. 44 

3.3.4. Constants of Gent ........................................................................................... 44 

3.4. Results of FEA ................................................................................................... 45 

3.4.1. Data-set Combination Effect on Different Material Models in Uniaxial Tension 

Test ........................................................................................................................... 46 

3.4.2. Data-set Combination Effect on Different Material Models in Shear Test ........ 48 

3.5. Comparing Different Material Model Results ...................................................... 50 

3.6. Impact Modeling ................................................................................................. 53 

3.6.1. Modeling of PD ................................................................................................ 54 



viii 

 

3.6.1.1. Analytical Model of PD ................................................................................. 54 

3.6.1.2. Empirical Modeling (NN method) .................................................................. 57 

3.6.1.3. Comparison of Analytical and Empirical Modeling ........................................ 60 

3.6.2. Stress Analysis ................................................................................................ 61 

3.6.2.1. Differential Equation ..................................................................................... 61 

3.6.2.2. Finite Element Method .................................................................................. 62 

3.6.2.3. Comparison of FEM and Differential Equation ............................................. 65 

4. CONCLUSIONS .................................................................................................... 66 

4.1. Improving SDM ................................................................................................... 66 

4.2. Analyzing a Hyperelastic Material using FEM .................................................... 66 

4.3. Impact Analyzes of Viscoelastic Material............................................................ 67 

4.4. Future Works ...................................................................................................... 67 

REFERENCES .......................................................................................................... 68 

APPENDIX ................................................................................................................ 71 

CURRICULUM VITAE ............................................................................................... 75 

 

LIST OF TABLES 

Table 2.1. Mechanical properties of the V10 polyurethane elastomer ....................... 15 

Table 2.2. Specimens manufactured for testing ........................................................ 23 

Table 2.3. Sample variables and PD results of specimens ....................................... 24 

Table 2.4. Examples of mechanical properties of elastomeric polyurethanes. .......... 28 

Table 3.1. Tension test on specimens with different contact area ............................. 42 

Table 3.2. 2 and 9-parameter Mooney-Rivlin constants ............................................ 44 

Table 3.3. RMS error of material models for each test models .................................. 52 

Table 3.4. The artifical NN structure .......................................................................... 58 

Table 3.5. Specimens manufactured for testing ........................................................ 60 

Table 3.6. RMS error of each model ......................................................................... 60 

 

 
 

LIST OF FIGURES 

 

Figure 2.1. Schematic diagram of the behavior of the elastomer in simple shear [7] .. 6 

Figure 2.2. Middle cross section of entire structure [10] .............................................. 7 

file:///C:/Users/behrang/Desktop/savunma%2012.docx%23_Toc375999472


ix 

 

Figure 2.3. The multi-layered buffering system with embedded SIP [4] ...................... 8 

Figure 2.4. Nonlinear impact model [12] ...................................................................... 9 

Figure 2.5. A photo of the uniaxial tension test machine during testing in 

approximately 300% of elongation (left) and the drawing of the specimen (right) ..... 17 

Figure 2.6. Estimated results of biaxial tension test using the equivalent equation and 

VL function ................................................................................................................ 19 

Figure 2.7. A photo of the uniaxial compression test (left) and the drawing of the 

specimen (right) ......................................................................................................... 20 

Figure 2.8. Comparison of pure shear test and corresponding VL function results. .. 21 

Figure 2.9. A photo of the designed drop test; cylindrical bearing (1), impact rig (2), 

adjustable mass (3), Specimen (4), and accelerometer (5) ....................................... 22 

Figure 2.10. A sample plot of the impact deceleration of a specimen, where; A=28.26 

cm2, T=5.19 mm, M=1.866 kg and H= 5 cm versus time .......................................... 23 

Figure 2.11. The manufacturing process of the SDM ................................................ 25 

Figure 2.12. The SDM cyclic process involving material addition and removal and 

component embedding .............................................................................................. 26 

Figure 2.13. An example of SDM manufacturing process: Designing (A), Embedding 

process after machining the mold (B), Deposit part material (C), Machining (D), and 

Releasing from mold (E). [30] .................................................................................... 27 

Figure 2.14. Sacrificial material (yellow), support material (blue) and embedded 

components [30] ........................................................................................................ 30 

Figure 2.15. Example of embedded components Double-Jointed compliant leg 

(fabricated by SDM) [30]............................................................................................ 31 

Figure 2.16. Polyurethane used as a part material .................................................... 33 

Figure 2.17. Effect of annealing on arrangement of domains [40] ............................. 35 

Figure 2.18. Annealing device ................................................................................... 35 

Figure 2.19. Creep effect [40] .................................................................................... 36 

Figure 2.20. Mullins effect [18] .................................................................................. 37 

Figure 3.1. Plot of elongation vs. time (days) ............................................................ 40 

Figure 3.2. Plot of load/thickness vs. time (days) ...................................................... 41 

Figure 3.3. Bonding of two different materials ........................................................... 42 

Figure 3.4. The block diagram of ANSYS analysis .................................................... 46 

Figure 3.5. FEA of the uniaxial tension test using Mooney-Rivlin (9-parameter) model

 .................................................................................................................................. 47 

Figure 3.6. FEA of the uniaxial tension test using Ogden 1rd model ......................... 47 

Figure 3.7. FEA of the uniaxial tension test using Neo-Hookean model .................... 48 

Figure 3.8. FEA of the uniaxial tension test using Gent model .................................. 48 

Figure 3.9. FEA of the pure shear test using Mooney-Rivlin (9-parameter) model .... 49 

Figure 3.10. FEA of the pure shear test using Ogden (1-rd) model ........................... 49 

Figure 3.11. FEA of the pure shear test using Gent model ....................................... 50 

Figure 3.12. FEA of the pure shear test using Neo-Hookean model ......................... 50 



x 

 

Figure 3.13. Difference of material models in analyzing uniaxial tension test............ 51 

Figure 3.14. Difference of material models in analyzing shear test ........................... 51 

Figure 3.15. An example plot of PD vs. thickness with different drop height ............. 54 

Figure 3.16. A best fit plot of deceleration ratio vs. thickness .................................... 55 

Figure 3.17. A best fit plot of deceleration ratio vs. Area ........................................... 56 

Figure 3.18. A best fit plot of deceleration ratio vs. mass .......................................... 56 

Figure 3.19. A best fit plot of PD vs. height ............................................................... 57 

Figure 3.20. Regression analysis of MPL network. ................................................... 59 

Figure 3.21. Performance of the NN. ......................................................................... 59 

Figure 3.22. 3-D analyzing the drop test in ANSYS ................................................... 64 

Figure 3.23. Von-misses stresses over time during an impact .................................. 64 

Figure 3.24. A plot of comparison of calculated stresses. ......................................... 65 

Figure 3.25. A plot of stress vs. mass ....................................................................... 65 

 
 

 

NOMENCLATURE 

 

  Principal stretch ratio 

  Engineering strain  

  Engineering stress 

T True stress 

t Thickness of specimen 

  Gage Length of specimen 

  Cross section area of specimen 

D Diameter of specimen 

                Constants of Mooney-Rivlin material model 

  ̅   ̅ First and second deviatoric strain invariant 

d Material incompressibility  

J Determinant of the elastic deformation gradient 



xi 

 

w Strain energy density function value 

  Initial shear modulus 

   Ogden material model constant 

   Ogden material model constant 

  Gent material model constant 

   Gent material model constant 

T(t) Thickness function 

A(a) Area function 

M(m) Mass function 

H(h) Drop height function 

   Static stress 

 

 
 

 

 

 

 

 

 



1. INTRODUCTION 

1.1. Problem Statement 

Nowadays many robots are manufactured for various applications. Some of these 

robots work in real environment conditions which increased risk of damage. Most of 

robots are sensitive and have potential of receiving large damages in very small 

impacts, falling from height, rolling over or collisions. In this thesis we are trying to find 

a way to overcome these problems. Sometimes increasing material or using another 

soft material (as a protector) around the sensitive object considered to increase 

robustness. However, there is an alternative method which enables us to embed 

different mechanic and electronic parts of the robot in a soft viscoelastic material 

using shape deposition manufacturing (SDM) method. 

 

Protecting of products plays significant role in engineering and almost in every 

industry. It leads to generate the science of packaging. In this study protection of 

sensitive robot components is considered. Some of the sensors are sensitive to 

applied acceleration. In this thesis, the Peak deceleration (PD) of the applied impact 

load is modeled empirically, analytically and compared with experimental data. Also 

the maximum stress on the material is calculated using the Finite Element Method 

(FEM). 

 

1.2. Research Goal and Objectives 

The main objective of this study is to model viscoelastic material behavior under 

Impact loads to design a rugged robot which could endure external impact loads. This 

thesis presents the methods of manufacturing rugged robots. Moreover, all dynamic 

and physical conditions are explored and documented in this work. Following 

questions will be answer in this study: 

 

 What type of materials should be used to provide maximum protection for 

sensitive components? 



2 

 

 How to enhance the mechanical properties of a material? 

 Which parameters of the material are more important to reach the 

robustness? 

 How to manufacture integrated components without using screws, washers, 

and nuts? 

 How to protect sensitive components from impacts? 

 How to manufacture objects with less than a millimeter-scale? 

 How to interface two different elastomers? 

 How to model viscoelastic material? 

 How do thickness and area of the specimen affect the peak deceleration? 

 How do the robot weight and speed affect the peak deceleration? 

 How to model the dynamic behavior of the material? 

 How to reach the stress on the material from the peak deceleration? 

 How to model the Peak Deceleration (PD) analytically and numerically? 

 How to analyze these materials in ANSYS? 

 Which material model must be used for modeling the hyperelastic material 

using Finite Element Analysis (FEA)? 

 How to predict the stress-strain of the material under shear and biaxial test 

without  perform a test  

 Which combination of a data-set could solve the model more precisely? 

 Which combination of a data-set has the best results? 

 

1.3. Challenges 

There are lots of problems in numerically analyzing the material behavior of the 

elastomeric polyurethane in ANSYS. Numerical modeling of the specimen using finite 

element method requires adequate experimental data-sets in three different test set-

ups. Three experimental set-ups are: uniaxial tension, uniaxial simple shear and 

biaxial tension tests. Unfortunately, shear and biaxial tests need special test setups 

that are rarely found and expensive. In order to compensate the lack of experimental 

data, an estimation method could be used to predict experimental data set of shear 



3 

 

and biaxial without practically performing these tests. For this purpose the Valanis-

Landel method is used. Shear and biaxial data-sets are predicted from uniaxial 

tension and uniaxial compression tests which are mostly available in all laboratories. 

 

Another problem is modeling the dynamic behavior of the elastomers. Modeling of the 

viscoelastic material as a spring and damper is really hard and is not very applicable. 

On the other hand, the specimen spring constant and damping coefficient are fully 

related to the specimen size and testing conditions. Since the peak deceleration is 

needed to model in this work; a special test device designed and manufactured. By 

using this test device, the Peak Deceleration (PD) and Stress is modeled according to 

specimen size and testing conditions. The biggest problem in PD of the specimens in 

drop test is the maximum measurable acceleration of the accelerometer that affects 

the measurable impact range of constitutive material modeling. 

 
1.4. Outline 

In this study, next section is devoted to the literature review. Then, section 3 is 

allocated for results of analytical and numerical solution of PD and stress of 

polyurethane elastomers; afterwards, characterization of different material models 

and different combinations are presented. Finally, section 4 is dedicated to the 

conclusion. 

 

 

 

 

 

 

 

 



4 

 

2. LITERATURE SURVEY 

2. 1. Definition of the Critical Issues 

Many researchers have been used the viscoelastic materials to protect embedded 

electronic systems under impact loads [1-4]. In [4] for instance; a numerical model of 

an electronic system, which is embedded in multiple viscoelastic materials under 

impact loads, was studied. According to the work, a design methodology is defined for 

buffer layers of viscoelastic material which significantly increased the protection of 

embedded electronic systems. The effect of the thickness of different viscoelastic 

buffer layers on decreasing the stress on the electronic circuits was investigated using 

Finite Element Method (FEM). However, authors compared two thicknesses only. It is 

numerically proven that the thickness of the viscoelastic materials would dramatically 

affect the maximum stress on the material [4].  

 

Since we do not have chance to test all the materials with all variables at any time, we 

need a model to find the impact forces and peak decelerations (PD) in colliding 

bodies. It is difficult to determine the impact forces using elastic stress wave theory 

[1]. Dan K. Song [5] has worked on modeling the impact phenomena and predicting 

the impact force when a mass experiences a collision to a contact pad. The 

elastomeric contact pad is modeled as a linear and nonlinear spring and nonlinear a 

damper. In another work [6], the viscoelastic contact pad is modeled as linear and/or 

nonlinear contact stiffness. These studies [5][7] model the impact force when an ideal 

mass with a constant area experiences a collision at different heights only. Few 

studies have been investigated to examine the effects of physical parameters such 

as; specimen size, temperature, drop mass, density and etc. on impact modeling of 

the viscoelastic materials so far. J. A. Sherwood, et al [7] have been worked on 

modeling the compressive stress-strain response of polyurethane foam under 

compressive impact loads considering the effects of density, temperature and strain 

rate as separable functions.  

 



5 

 

Modeling the behavior of the viscoelastic material would enable us to reach desired 

protection. Further, it could help us to design the protective material size. There are 

many investigations on modeling the behavior of the material under low strain rates 

which known as hyperelastic material modeling [8-11] and high strain rates which 

calls viscoelasticity. In this study, different material models behavior is analyzed in 

details.  

 

Finding a best material, modeling its dynamic behavior, finding the ideal specimen 

size to protect the embedded components of the robot and defining the special 

manufacturing process (SDM) are the purpose of this study. Each goal is introduced 

briefly in the next sections.  

 

SDM process is a useful method for manufacturing 3-D models with extremely 

complex geometries in very high accuracy [12]. The SDM is appropriate 

manufacturing process for pourable materials such as polyurethane elastomer which 

is used in this study.  

 

2. 2. Analyzing the Hyperelasticity and Viscoelasticity of Materials 

The polyurethane elastomers are widely used materials in many areas such as; 

aerospace, automotive, civil engineering and medical field due to their superior 

properties such as; viscoelastic behavior under high strain rates, elastic behavior 

under large deformations and abrasion, cut, microbial, chemical resistance and etc. 

When the load is applied to a material, most of them show an elastic behavior in a 

limited region where the stress and strain have a linear behavior. Elastic behavior of 

rubber-like materials under large deformations is known as hyperelasticity. These 

materials have a large elastic region with nonlinear relationship between stress and 

strain [8]. Linear elastic models do not accurately simulate the nonlinear behavior of 

hyperelastic materials; therefore, a special material model is needed. The motivation 

and developments in formulating the material models have discussed by Gent [9]. 

Rubber-like materials show hyperelastic behavior under static loads; while they show 



6 

 

viscoelasticity in dynamic mechanical load. Their behavior is fully related to the 

deformation speed. Elastomeric polyurethane enables fast, continuous and large 

deformations. These properties could guarantee protect of sensitive electrical parts. 

 

Lee. S. J [10] has worked on analyzing the hyperelastic behavior of a composite 

material to design a shear actuator. In this work, a novel active material for 

controllable applications was designed and tested. The active material used is a lens 

shaped element to implement pure shear motion using gas pressure actuation. The 

lens elements were embedded in a polyurethane matrix. He tested the polyurethane 

elastomer in different test set-ups and derived their experimental data-set and then 

numerically analyzed them using Finite Element Method. The aim of the paper was to 

design the best lens shaped element to control shear deformation on the hyperelastic 

polyurethane. During the numerical analysis, the Ogden material model [11] was 

selected as the best model for impact test. The conclusion of the paper is: the lens 

element actuators approach a full circle when pressurized and causes shear 

deformation. Different hyperelastic material models are used to analyze deformation 

of the material due to shear stress. Schematic diagram of the system presented in 

Figure 2.1.  

 

 

 
Figure 2.1. Schematic diagram of the behavior of the elastomer in simple shear [7] 

 

In another work, Alsakraneh [13] presented the use of rubber buffer layer to protect 

embedded System-In-a-Package (SIP) device from high mechanical forces. The aim 



7 

 

of the paper is to reduce the transmitted stress to the embedded SIP device. The 

system was numerically analyzed the stress-strain on the embedded SIP using 

different testing conditions. Finally it is demonstrated that the strain on the SIP could 

decrease up to 84% with relatively thin layer of buffer material. Further, different 

variables such as the effect of thickness of rubber and epoxy, rubber brand and static 

force on minimizing the strain on the SIP were presented. The structure is presented 

in Figure 2.2. 

 

 
 

 

Examples of using the hyperelastic behavior of rubber-like materials are presented 

above. Many researches performed on the viscoelastic behavior of the materials. The 

viscoelastic materials are particularly applicable to the protection of sensitive 

components from impact loads. They can be used as bumpers in automobiles, 

cushion in vibration systems, packaging, impact isolators and etc. 

 

The numerical modeling of an electronic system embedded in multiple viscoelastic 

materials under impact loads was studied in [4]. According to the work, a design 

methodology defined for buffer layers which could significantly increase the protection 

of embedded electronic systems. The effect of thickness of different viscoelastic 

 

Figure 2.2. Middle cross section of entire structure [13] 



8 

 

materials on decreasing the stress on the electronic circuit was presented in the work. 

The author used a ball structure shown in Figure 2.3 which consists of different layers 

of viscoelastic materials. 

 

 
 

Figure 2.1. The multi-layered buffering system with embedded SIP [4] 

 

In another work, dynamic model of polyurethane foam cushion is established [14]. 

The static compression testing and dynamic compression testing is performed on the 

material to build the nonlinear viscoelastic plastic deformation. Sung, D. K [5] 

modeled the nonlinear elastomer impact problem when a mass experience a collision 

with a contact pad which is mounted on the underlying linear dynamic structure. The 

viscoelastic contact pad modeled as linear and nonlinear stiffness and nonlinear 

contact damper. The underlying dynamic structure is calculated by Fourier spectral 

analyzer. The impact force used in this study developed when an ideal mass 

experiences a collision with the elastomeric contact pad. However the model does not 

include the specimen size or the dropped mass area and weight. The test structure is 

presented in Figure 2.4. The model analyzed just for special contact pad in specific 

test condition.  



9 

 

 
 

Figure 2.2. Nonlinear impact model [5] 

 

We are trying to model the elastomeric contact pad in a way that the model includes 

the specimen size and testing conditions. For this purpose, the PD is modeled 

analytically by defining an analytic model and empirically using Neural Networks (NN). 

Then, PD is converted to the stress by using a derived differential equation. Further, 

the ANSYS software is used to numerically analyzing the stress on the specimen. 

ANSYS uses material models to simulate the material behavior. As mentioned before, 

linear isotropic material models could not accurately simulate behavior of hyperelastic 

materials. Then, nonlinear elastic material models are used. Each material model has 

a special Strain Energy Density (SED) function. There are many material models; 

however, in this study four material models are analyzed. 

For numeric analysis of these materials, we first prepared the experimental data-set 

and then used them to evaluate different hyperelastic material models by using 

ANSYS. Experimental data of shear, uniaxial tension and biaxial tests are needed to 

obtain an acceptable Finite Element (FE) result. However, measurement of shear and 

biaxial tests require special mechanical devices. To compensate the lack of test 

setups, Valanis-Landen method is used to estimate the data-set of these tests. The 

FE implementation of specimens under uniaxial tension and pure shear tests is 



10 

 

evaluated using four separate material models in four different experimental data-set 

combinations.  

 
In this study, data-sets of shear and biaxial tests are predicted from uniaxial tension 

and uniaxial compression tests which are mostly available in all laboratories. These 

data-sets are used to evaluate four different material models. Beomkeun Kim 

compared three different material models using experimental data-sets of 

Chloroprene Rubber [15]. According to the results; Ogden 3rd was shown to have a 

better convergence with the experimental data-sets, while Mooney-Rivlin and Neo-

Hookean were not in an acceptable agreement range. However, the Gent model 

which is suitable for rubber modeling was not investigated. Jang et al worked on 

weatherstrip which is made of EMPM and TPE. However, he considered the 

combination of biaxial + uniaxial tension tests data-sets only to calculate the 

constants of material models [16]. 

 
2.2.1. Hyperelastic Material Models 

As mentioned before, special material models are selected to define the nonlinear 

behavior of the material. The major difficulties of using FEM are: finding out the 

behavior of different material models, and determination of constants of relevant 

material model. In this thesis four material models are used to model the hyperelastic 

materials. These models are:Mooney-Rivlin, Ogden, Neo-Hookean and Gent. These 

material models have special SED functions. It should be noticed that, the constants 

of each material model and the constants of related SED functions are the same 

values. 

 

SED functions are used for numerical modeling of the behavior of rubber-like 

materials. Constants of these SED functions are found using curve-fitting tool of 

ANSYS utilizing experimental data. To make a decent polynomial fit to the entire data-

set and to find out the material model constants, data-set of multiple types of tests 

(uniaxial simple tension, planar shear and biaxial tension) are needed. Unfortunately, 

shear and biaxial tests need special test setups that are rarely found and expensive. 



11 

 

In order to compensate the lack of experimental data, known success of the VL 

function in characterizing the behavior of rubber-like materials is used. The 

effectiveness of the VL function in estimating the data-set of shear and biaxial tests 

from uniaxial tension and uniaxial compression tests has already been shown in [17-

20]. This paper however presents a detailed description of numerical analysis of the 

hyperelastic materials utilizing experimental and estimated data-sets. 

 

There are many approaches to estimate the experimental tests data-sets [17-20] and 

to analyze the hyperelastic material models [15-16] separately. However, to the best 

of our knowledge, numerical analysis of hyperelastic elastomers, based on predicted 

test results, has not been reported yet. Experimental set-up and estimation method is 

described in section (2.2.2). 

 

2.2.1.1. Mooney-Rivlin 

Mooney-Rivlin method is a hyperelastic material model introduced by Mooney and 

Rivlin which has options of 2, 3, 5 or 9-parameters. In general, increasing the order 

gives more accurate results. It is known that, material models having higher orders 

can be used for varied structural components of up to 200% deformation. Also, 2-

parameter Mooney-Rivlin option is suitable for stains of up to 100% in tension and 

30% in compression [21].SED function of the material model is composed of linear 

combination of two invariants of the left Cauchy-Green deformation tensor as given 

below: 

 

 (  ̅   ̅)     (  ̅   )     (  ̅   )  
 

 
(   )  (1) 

 

where,     and     are the material constants characterizing the deviatoric 

deformation of the material,   ̅ and   ̅ are the first and second deviatoric strain 

invariant, and d is the material incompressibility parameter. j is the determinant of the 

elastic deformation gradient which is equal to 1 for incompressible materials. 



12 

 

Therefore, for the last segment of the equation becomes zero [22].The general 

equation of SED function for infinite series is as follows: 

 

  ∑    

   

     

(  ̅   ) (  ̅   )  (2) 

 

2.2.1.2. Ogden 

An Ogden model is a hyperelastic material model. The SED function of this model is 

expressed by principal stretch ratios. Ogden model is proven to be a good 

hyperelastic constitutive model for large strains of incompressible materials [11]. The 

SED function of Ogden model is a special case of the VL function that has a capability 

of expanding experimental data-set of uniaxial tension to the other two test formats. It 

can be expressed as a nonlinear function of principle stretch ratios which is 

expressed as: 

 

 (         )   ∑
  

  
(  

     
     

     )

   

   

 (3) 

 

where,   (j=1, 2, 3),   ,    are the principal stretch ratios, and material constants 

respectively. Ogden model shows a good agreement with the experimental data even 

in high deformations of up to 700%, therefore; it is used most commonly to analyze 

the rubber-like components [21]. 

 

2.2.1.3. Neo-Hookean 

This model uses the Hook’s law which is applicable for materials having linear 

behavior at the first regime. A Neo-Hookean hyperelastic material model could 

analyze various structural components in small strains of up to 30% [21].  Mooney-

Rivlin SED function is a series of powers which are usually truncated in the first term. 

Neo-Hookean model only depends on the first term of the Mooney- Rivlin SED 

function. Therefore, the SED function of this material model becomes: 



13 

 

 

 (  ̅)     (  ̅   ) (4) 

 

the     constant could be expressed as: 

 

    
 

 
 (5) 

 

where   is the initial shear modulus [23]. 

 

2.2.1.4. Gent 

The Gent model is a phenomenological model for hyperelastic materials which can be 

used to analyze the stress-strain behavior of isotropic incompressible hyperelastic 

materials. The Gent model, which is based on the theory of limiting chain extensibility, 

has a logarithmic SED function [24]. The SED function of the Gent model depends 

only on the first invariant (  ̅) of the left Cauchy-Green strain tensor, which is a 

logarithmic function of   ̅ and two material parameters   and    which are initial shear 

modulus and limiting value of   ̅   . SED function of the Gent model can be shown 

as: 

 

   
   
 

  (  
  ̅   

  
)  

 

 
( 
    

 
   ( ) ) (6) 

 

where, d is the material incompressibility. Therefore, the last segment of the function 

tends to be zero.  

 

For rubber-like materials, the relationship between the applied stress and the strain is 

fully nonlinear; therefore, the material models with nonlinear formula of SED function 

such as Ogden and Gent are expected to have better convergence with the 

experimental data set.  

 



14 

 

2.2.2. Experimental set-up 

Two main experimental set-ups are used in this work to perform static and dynamic 

(impact) tests. The static tests are performed to define the constants of material 

models in order to use for numerical analysis of the material. The dynamic tests are 

performed to analyze the PD and stress during an impact. 

 

As previously mentioned, Experimental results of only simple uniaxial tension test 

would not be sufficient. Therefore, finding the constants of a SED function (material 

model) requires adequate number of experimental data-sets. In this work, we use one 

practical (uniaxial tension), and two estimated (shear and biaxial) data-sets as an 

input. Due to lack of shear and biaxial test set-ups the results of these tests estimated 

using VL function. In order to explore the reliability of VL function, few shear tests are 

performed and the results are compared with the predicted shear test results. During 

the shear tests, the specimens fail around the clamp edges after about 120% of 

elongation. Therefore, the comparison can be done only for a range of limited strain. 

However, we could not make the biaxial tests due to the lack of experimental set-up in 

our laboratory. 

 

During the static and dynamic tests, following conditions must be considered: 

 

 Humidity and temperature dramatically affect the structural properties of the 

elastomers. 

 High strain rate increases the damping force which consequently increases the 

value of stress-strain gradient and usually decreases the maximum endurable 

strain in simple tension test [8]. 

 According to the experimental experiences, the mechanical properties of the 

specimens saturate after about 30 days of manufacturing due to the 

vaporization.  

 



15 

 

By considering the arguments given above; all specimens should be manufactured 

and stored under the same conditions. Testing of hyperelastic materials should be 

performed in low strain rates (i.e. 5mm/min), and they should be tested after 30 days 

of manufacturing. Finally, if a test needs to be repeated; at least 30 minutes should be 

passed over the previous one due to the stress softening. When the rubber-like 

materials are deformed for the first time, their mechanical properties change. 

Therefore, after manufacturing the material, it should be stretched many times to 

saturate the mechanical properties before gathering data sets. This phenomenon is 

known as Mullins effect and the recovery time differs from material to material [25]. 

 

2.2.2.1. Details of the Material 

The material used in this study is Vytaflex-10 which is a polyurethane elastomer with 

a shore hardness of A10, manufactured by Smooth-on. The material is commercially 

available in two separate containers that must be mixed in 1:1 proportions. The 

mechanical properties of Vytaflex-10 provided by the manufacturer are presented in 

Table 2.1. According to the manufacturer, a specimen can reach up to 1000% of 

elongation when tested 7 days after manufactured. However, due to the evaporation, 

the specimen reaches to its stable condition in 30 days and the maximum elongation 

decreases down to 800%. 

 

Table 2.1. Mechanical properties of the V10 polyurethane elastomer 

 

Material Vytaflex-10 

Shore A hardness [ASTM D-2240] A10 

Tensile strength [ASTM D-412]Mpa (psi) 1.37 (200) 

Elongation at break [ASTM D-412] % 1000% 

100% modulus [ASTM D-412] 25 



16 

 

Poisson’s ratio 0.499 

Density kg/m3 990 

 

2.2.2.2. Uniaxial Tension Test 

Uniaxial tensile and compression tests are performed with a machine manufactured 

by U-test designed for uniaxial tests. For the tension test, the probe dimensions are 

selected in accordance with ASTM D-412 standard which is suitable for volcanized 

rubber and thermoplastic elastomers. According to ASTM D-412, the specimen has to 

have a hollow dumbbell shape. The reason of manufacturing the specimens in 

dumbbell shape is to prevent the specimen failure at the clamps. 

 
Uniaxial tension tests are performed on 20 specimens utilizing U-test machine. The 

variation of the test results about the mean are calculated to be around 10%. This 

variation could be mainly due to the small defects in the manufacturing process. 

 

In this work, deformation only applies in one direction and the principal stretch ratio 

can be calculated using the equation given below: 

 

     
 

  
             

 

√ 
 (7) 

 
where,    (i=1, 2, 3),    ,  , L,   are principle stretch ratio, nominal tensile strain, 

stretch along uniaxial loading direction, gage length of the specimen during and 

before deformation respectively. 

 



17 

 

 

 

Figure 2.3. A photo of the uniaxial tension test machine during testing in 
approximately 300% of elongation (left) and the drawing of the specimen (right) 

 

As shown in Figure 2.5,   (i=1, 2) is the length and W is the width of the specimen. In 

addition, engineering stress can be calculated by dividing the force measurements to 

the un-deformed cross-section area of the specimen. Principle stress states can be 

defined as: 

 

     
 

  
         (8) 

 
where, F,   ,  (i=1,2,3) are the applied force, cross-sectional area and axial stress, 

respectively. According to Kearsley and Zapas, VL function can be calculated utilizing 

uniaxial tension and compression tests. [20] Following equation can be used to obtain 

 ́( ) of the VL function: 

 

   
   

∑  ( (  ⁄ )
 

)   (  (  ⁄ )(  ⁄ )
 

) 

   

   

   ́( ) (9) 



18 

 

where, t represents the true stress. According to this equation, there is an infinite 

number of true stresses and the first and the second terms are in opposite side of the 

un-deformed state (   ). Therefore, both uniaxial tension and compression test 

results are needed to solve the equation for  ́( ) of the desired points. The equation 

of the uniaxial engineering stresses can be written in terms of VL function as given 

below: 

 

   ́( )    (  ⁄ ) ́(  (  ⁄ )) (10) 

 

2.2.2.3. Biaxial Test 

Biaxial test requires special fixtures and high cost equipment, therefore; to minimize 

the total cost of the procedure and speed up the process, an alternative method 

should be used. For incompressible materials, uniaxial tension is equivalent to biaxial 

compression, uniaxial compression is equivalent to biaxial tension, and planar tension 

is equivalent to planar compression [8]. Biaxial test results can be estimated by using 

two methods; first, using an equivalent equation, and second, using a VL function. 

Using a simple mathematical conversion, biaxial stress-strain can be written as a 

function of uniaxial compression as follows: 

 

                       
 

 ⁄  (11) 

 
In another approach, biaxial stress-strain relationship can be expressed in terms of 

VL function which can be given as: 

 

          ́( )  
 

  
 ́(

 

  
) (12) 

 
where,   represents the principle stretch ratio. Comparison of these two approaches 

is shown in Figure 2.6. According to the plot, both models generate almost the same 

results. 



19 

 

 

 

Figure 2.4. Estimated results of biaxial tension test using the equivalent equation and 
VL function 

 

In uniaxial compression test, top and bottom surfaces of the specimen is lubricated in 

order to minimize the shear deformation. As shown in Figure 2.7, D and t represent 

the diameter and the thickness of the specimen which are 40 mm and 12 mm in 

dimensions respectively. Deformation and stress state of the uniaxial compression 

tests can be defined as: 

 

     
 

  
       

 

  
 (13) 

and 

     
 

  
          (14) 

 
respectively where,    (i=1,2,3) is the principal stretch ratio and    (i=1,2,3) is the axial 

stress.  



20 

 

 

 

Figure 2.5. A photo of the uniaxial compression test (left) and the drawing of the 
specimen (right) 

 

2.2.2.4. Shear Test 

The stress state of the planar shear test is similar to the pure shear test. Shear test is 

also known as uniaxial tension test with a very wide specimen [26]. The most 

important feature of the specimen is that the width should be much larger than the 

direction of stretching to make thinning in thickness direction only.  Experimental 

studies have shown that, the width of the specimen must be at least 10 times of the 

length. Only in this condition, the specimen is considered to be tested in the condition 

of the plane strain rather than the plain stress known as uniaxial tension test [11].  

 
W ≥ 10L (15) 

 
where, W and L represent the width and length of the specimen respectively. For the 

tests, we manufactured 5 specimens which are 200 15 0.9 mm’s in dimensions. 

During the tests, these specimens were all ruptured at the clamps after approximately 

120% of deformation. Strain and stress states of planar shear test can respectively be 

defined as: 

 

     
 

  
             (16) 

and 

               (17) 



21 

 

where,    (i=1,2,3) and    (i=1,2,3) represent principal stretch ratio and axial stress 

respectively. In addition, i = 1, 2, 3, corresponds to the length, width and thickness. 

Stress-strain relationship of the planar shear in the form of VL function can be 

expressed as: 

 

        ́( )  
 

  
 ́(

 

 
) (18) 

 
Comparison of the VL function and the experimental test result of pure shear tests is 

shown in Figure 2.8. 

 

 

 

Figure 2.6. Comparison of pure shear test and corresponding VL function results. 

 

2.2.2.5. Drop Test Device 

In order to perform a dynamic compression test, a test device is designed (Figure 2.9) 

similar to [27]. The mass and the height of the impact rig are adjustable. The 

minimum mass of the impact rig is 1.866 kg and can be increased by adding mass. 

Specimens having a top area of up to 254 cm2 can be tested in the setup. All 

specimens are manufactured by pouring the elastomer in cylindrical molds made of 

Plexiglas. The accelerometer used in this study is 3256A1 manufactured by Dytran 

which detects shocks of up to 300 g and working frequency range of 1 to 10 KHz.  

 



22 

 

 

 

Figure 2.7. A photo of the designed drop test; cylindrical bearing (1), impact rig (2), 
adjustable mass (3), Specimen (4), and accelerometer (5) 

 

Deceleration data of the collision is measured by the accelerometer and transferred to 

data acquisition card via a signal conditioner. LABVIEW software is used to record the 

data. A sample test result of the specimen with an area of 28.26 cm2, a thickness of 

5.19 mm, a drop height of 5 cm and a drop mass of 1.866 kg  is shown in Figure 2.10. 

As seen in this figure (vertical upward direction is positive), there are many local PDs, 

however, only the first PD wave is considered for modeling due to its highest value. 

The subsequent peaks of decelerations are caused by rebounding of the impact rig. 

Other tests were performed for drop heights of between 5 to 60 cm (or velocities from 

1 to 2.62 m/s) and drop masses of between 1.866 to 4.363 kg. In this study, seven 

specimens with constant thicknesses and varying areas between 3.14 to 200.96 cm2; 



23 

 

nine specimens with constant areas and varying thicknesses from 3.7 to 27 mm are 

tested. To model the material behavior 16 specimens are manufactured and tested; 

specimens presented in Table 2.2. 

 

 

 

Figure 2.8. A sample plot of the impact deceleration of a specimen, where; A=28.26 
cm2, T=5.19 mm, M=1.866 kg and H= 5 cm versus time 

 

Table 2.3 presents the results of some experiments made in this work. Note that the 

experiments given in Table 2.3 are a very small portion of the whole tests done in this 

work. 

 
Table 2.2. Specimens manufactured for testing 

 

Area 
28.26 
Cm2 

The specimen number 
Thickness (mm) 

1 
3.7 

2 
5.2 

3 
7.75 

4 
8.33 

5 
10.88 

6 
11.63 

7 
16.7 

8 
22.3 

9 
27 

Thickness 
8.5 
mm 

The specimen number 
Area (cm2) 

10 
3.14 

11 
12.56 

12 
28.26 

13 
50.24 

14 
78.5 

15 
113 

16 
153.8 

 

 



24 

 

Table 2.3. Sample variables and PD results of specimens 

 

Target 
variable 

Area 
A 

(cm2) 

Thickness 
T 

(mm) 

Drop 
height 

H 
(cm) 

Drop 
mass 

M 
(kg) 

Average peak 
deceleration (g) 

Thickness 
T 

28.26 

5.19 

5 

1.866 

138.26 

10 182.15 

15 255.75 

16.7 

5 

1.866 

25.75 

15 79.38 

25 141.1 

Drop 
height 

H 
28.26 11.63 

5 

1.866 

32.25 

15 108.85 

25 197.75 

35 294.98 

Area 
A 

12.56 

8.3 

5 

1.866 

41.3 

15 109.06 

25 192.12 

78.5 

5 

1.866 

105.91 

10 203.65 

15 245.65 

Drop mass 
M 

28.26 11.63 15 

1.866 109.01 

2.479 91.11 

2.965 83.41 

3.431 79.23 

3.897 78.93 

 

Each test is repeated four times under the same conditions and the average PD 

values are presented here only. The PD may increase when the tests are repeated 

while keeping the other parameters constant due to the viscoelastic relaxation of the 

material. This phenomenon could be eliminated by waiting for a while depending on 

the material type between the tests [28].  

 

Using table 2.3, following conclusions can be drawn: The PD increases with 

increasing drop height and area and decreases with increasing thickness and mass 

individually, when the other constants are kept constant. 

 



25 

 

2.3. Manufacturing Process 

In this thesis, we tried to find a method to manufacture a rugged robot. For this 

purpose, a unique manufacturing method called Shape Deposition Manufacturing 

(SDM) is used. As it comes from its name, this method includes shaping and 

deposition in order to manufacture desired models. The rugged robot must endure 

external impacts. Vulnerable components of the robot must be embedded in a host 

object which reduces the impact force considerably. There are some materials that 

satisfy this mechanical property. In the next section, properties of the host material 

are going to be explained which covers the robot. In this section, manufacturing 

process of the SDM is described in detail. 

 
2.3.1. Shape Deposition Manufacturing (SDM) 

This manufacturing method enables us to manufacture mechanical 4 bars linkages 

monolithically without needing a screw and a bolt. One of the properties that make 

SDM interesting is its repetitive material addition, and material removal that allow 

making models layer by layer with desired thicknesses. This property enables to 

manufacture 3-D shapes that conclude making design more simple and monolithic. 

Other feature that makes it more interesting is embedding mechanical and electrical 

components between the materials. It can be anything like a motor, spring or a wire 

and etc. Figure 2.11 presents the process of SDM: 

 

 

 
 

Figure 2.9. The manufacturing process of the SDM 

 

design 
material 
selection 

machining 
the mold 

place 
sacrificial 
materials 

deposit part 
materials 

Degasing & 
cure 

machining  extract part 



26 

 

In order to embed components and manufacture different 3-D shapes, it is required to 

make repetitive cycle as it is shown in Figure 2.12. 

 

 
 
 

Figure 2.10. The SDM cyclic process involving material addition and removal and 

component embedding 

 

This method needs a mold (made of gypsum or wax), pour-able materials (for 

example polyurethane elastomer), material for making it easy to extract poured 

material from mold (for example Vernik or Glycerol) and if requires, embedding 

components. The mold and the part material are machined using the same method, 

typically a CNC mill. It should be noticed that some of the materials are not machine-

able. The mold uses a sacrificial material. This means that the mold can be used 

again or ejected after being used. The polyurethane which is known as part material 

will be pour after the machining the mold [26-27]. 

 

2.3.2. CAD Model Creation 

Using CATIA as a CAD would enable us to design desired models. However, 

important issue is the general strategies considered to build order. 

machining 

embedding 

deposit 

degassing 
& cure 



27 

 

 

 
 

Figure 2.11. An example of SDM manufacturing process: Designing (A), Embedding 

process after machining the mold (B), Depositing the part material (C), Machining (D), 

and Releasing from mold (E). [29] 

 

The embedded components are designed, selected and prepared for embedding. 

This process involves the fixtures, machining the part and sacrificial material and 

even pre-assemblies. It is important to notice the tolerances of mold. Applying proper 

fixtures for the embedded components depend on considering appropriate tolerances 

in design process. 

   
2.3.3. Materials in SDM 

The materials involved in SDM are classified into 5 categories: part material, support 

material, temporary material, sacrificial material (dam material) and embedded 

components. 

 



28 

 

2.3.3.1. Part material 

Polyurethane, rubbers, elastomers, ceramics and metals can be used as the part 

material in SDM [30-31]. In this work, the part material is chosen among polyurethane 

elastomers. Selection of the part material needs some consideration factors such as; 

 

 Shore hardness 

 Shrinkage 

 Chemical stability 

 Gel time 

 Machinability 

 Viscosity 

 Curing time 

  

In some models like flexible joints, two different materials are required. In this 

condition, binding, gel time and chemical reaction between two materials must be 

tested. Table 2.4 presents some of polyurethane elastomers with different mechanical 

properties. 

 
Table 2.4. Examples of mechanical properties of elastomeric polyurethanes 

 

Hardness 
Cost 

 
Company Tensile 

Cure 
time 

Elongation 
Specific 
gravity 

Gel time 

(shore A) tl/kg 
 

psi hour 
 

gr/cm3 min 

18 171 C3 140 24 150% 1.25 20 

24 428 C3 140 24 150% 1.25 20 

34 130 C3 140 24 150% 1.25 20 

40 81 C3 140 24 150% 1.25 20 

50 149 C3 140 24 150% 1.25 20 

60 73 C3 140 24 150% 1.25 20 

87 330 C3 140 24 150% 1.25 20 

20 100 vyta flex 200 16 1000% 1 30  

40 100 vyta flex 522 16 660% 1.02 30  

60 100 vyta flex 880 16 480% 1.04 60  

33 30 Era 290 18 435% 1.2 N/A 

83 30 Era 551 18 450% 1.08 N/A 



29 

 

95 30 Era 1885 18 375% 1.13 N/A 

75 
shore D 

40 Era 6526 22 N/A 1.19 N/A 

 

2.3.3.2. Support Material 

The support material is typically chosen from different waxes but we decided to use 

gypsum as a support material. A proper candidate for the support material should 

have high chemical stability, good machinability, low shrinkage, and ease to remove 

from the part. All of these properties can be found in gypsum. Gypsum has perfect 

chemical stability, machinability, no shrinkage and could be easily removed from the 

mold. 

 
2.3.3.3. Temporary Material 

The temporary material is used to make an internal geometry such as bearing houses 

or tube connectors. The typical temporary materials are water solvable or low melting 

point waxes. Some chemicals such as PEG can be very easily dissolved in water. 

Various chemical solutions have also been used to be dissolved in special types of 

polymers. It should be easily removed or dissolved after the manufacturing process to 

create the internal void in the part assembly. The temporary material is especially 

useful for embedding mobile components where the joint should have a certain 

clearance. 

 
2.3.3.4. Sacrificial Material 

Sacrificial material is known as dam material. It is used as a dam during casting to 

hold the liquid state polymer material. It is usual to use modeling clay to build the dam 

walls; however, other materials also can be used. It should be removed before 

urethane is machined. The clay and it’s usage is shown in Figure 2.14; the yellow 

material is the temporary material which is used for making internal voids. 

 
2.3.3.5. Embedded Components 

As previously mentioned, one of the abilities of SDM is embedding the components 



30 

 

between part materials. During the designing, it should be noticed that;  

 

 Robust binding: components should have a robust binding with the part 

materials;  

 There are some problems like weak bonding of soft and hard polyurethane 

because of bubbling and chemical reaction between two materials.  

 Embedded components should not be damaged in the machining process.  

 The components should keep required clearance from the part material 

where a free motion is required.  

 Shrinkage must be considered. 

 

It is important to minimize the bubbling using various techniques such as pre-

vacuuming the material before molding or molding in vacuum environment, and 

improving the surface finish of the embedded components. The typical embedded 

components include motors, pistons, air cylinders, chips, and strain gage sensors. A 

collection of embedded components are shown in Figure 2.13 [29]. 

 

 
 

Figure 2.12. Sacrificial material (yellow), support material (blue) and embedded 
components [29] 



31 

 

 

 
Figure 2.13. Example of embedded components Double-Jointed compliant leg 

(fabricated by SDM) [29] 

 

2.3.3.6. General Considerations in Material Selection 

In selecting the part material general considerations are: 

 

 Low cost 

 Long shelf-lives, long pot-lives, low toxicity 

 Strong layer to layer binding 

 Ability to pour even in low temperatures 

 Material shrinkage should be minimized for the dimensional accuracy of the 

SDM process 

 Part materials should exhibit good machinability 

 Good mechanical properties and chemical stability for part material 

 Part materials should solidify without the formation of internal voids 

 Part and support materials should have physical properties such as low 

viscosity and good machinability that favor the process control of SDM 

 Part and support materials must be physically and chemically compatible 

 Part material should be easily taken out of the support material after the 



32 

 

molding process completed 

 Good binding strength between soft and hard polymeric materials. 

 

other considerations specified for biomimetic robot fabrication; 

 

 Good inter layer shear strength between the embedded components and 

the surrounding polymer material 

 Good binding property between polymeric materials and metals (e.g. 

embedded components) 

 The temporary material should be easy to remove in order to make internal 

cavities such as pistons 

 The embedded sensors, tubes and actuators should lead to minimum 

trapped air in the polymer material after casting. 

 

By considering the notes above; we decided to use Vytaflex polyurethane with 

different shore hardness. The Era and C3 companies were not selected because they 

need special manufacturing process like pre-heating the mold before molding and 

curing in specific temperature. Polyurethane which is used as a part material is shown 

in Figure 2.16. It consists of two materials (resin and hardener) which has a mixing 

ratio of 1:1. 

 
Interaction between materials plays an important role in bonding and embedding of 

different component in materials. Understanding the chemical reaction in combination 

of two materials, strength of binding, flexibility of the joints and considering the 

feasibility of connections are important factors which should be considered. For 

example, it is difficult to connect soft material to hard material because they do not 

machine very well. For solving this problem, orderly manufacturing methodology is 

used. For example, deposit hard polyurethane first and then machine it for making 

cavities to deposit the soft material. 

 



33 

 

 
 

Figure 2.14. Polyurethane used as a part material 

 
2.3.3.7. Effective Parameters on Changing Mechanical Properties of Elastomers 

These parameters could change the mechanical properties: 

 

 Cure time 

 Mixing time 

 Annealing 

 Creep 

 Degassing 

 Stress softening (Mullins effect) 

 

Each effect is described below in detail.  

 
2.3.3.7.1. Cure time 

After molding the polyurethane, it takes 1 day to fully cure and ready to take it out 

from the mold. As time passes, the gasses in the specimen vaporizes and it`s color 

gets darker. At the same time, its mechanical properties such as stiffness (K), 

damping coefficient (C), and maximum stress and strain durability change. The 

results are presented in section 3. 



34 

 

 

2.3.3.7.2. Mixing time 

Polyurethane elastomers are manufactured by mixing two liquid in one by one ratio 

for four minutes. In short mixing time polyurethane chains will not have enough 

chance to bond well and this generates a negative effect on mechanical properties of 

polymer.  

 

2.3.3.7.3. Annealing 

In order to obtain ultimate dynamic properties of a polyurethane elastomer, the 

material need to be annealed.  Most effective temperature for annealing is generally 

just below the melting point (  ) of the material. Mechanical properties such as 

resilience and flexibility in low temperature can be improved by annealing as well. 

These improvements occur because of reduction in the amount of interphase 

between the hard and soft block domains. Since the interphase is the mixture of hard 

and soft block domains, the material able to rearrange and/or separate, becoming 

either part of an increasingly well-structured hard phase or an amorphous soft phase 

as shown in Figure 2.17. The annealing is performed using the device shown in 

Figure 2.18.  

 



35 

 

 

 

Figure 2.15. Effect of annealing on arrangement of domains [32] 

 

 
 

Figure 2.16. Annealing device 

 



36 

 

2.3.3.7.4. Creep 

Creep in polymers happens in every temperature. Creep rate (velocity of creep) 

increases by increasing temperature and stress. 

 

 

 
Figure 2.17. Creep effect [32] 

 

2.3.3.7.5. Stress softening (Mullins effect) 

Under repeated tensile strain, most of the polymers exhibit a reduction in stress after 

the initial extension; this phenomenon is known as the Mullins Effect [16]. The Mullins 

effect is “Phenomenon observed in rubber-like materials where the equilibrium stress-

strain response softens with the strain history”. The Mullins effect has these features: 

 

 The cycled material has a more compliant response at strains smaller than 

the previous maximum strain 

 When the Mullins effect occurs, the microstructure breaks down with 

increasing deformation, and the initial structure cannot rebuilt 

 



37 

 

 
 

Figure 2.18. Mullins effect [16] 

 

2.3.4. Different Manufacturing Process of SDM 

The SDM process includes two main manufacturing process; the material addition 

and removal. Each of these two processes is subdivided into bulk (uncontrolled) and 

selective (controlled) methods. 

 

Selective deposition means controlled material addition. In this process, the materials 

are deposited into a designed geometry using a special 3-D rapid prototyping 

machine. The selective material deposition requires Fused Material Deposition (FDM) 

or stereo lithography which are expensive. 

 

Bulk deposition process known as uncontrolled material addition. In this case material 

is free to fill the mold. Example of this type is explained before in figure 2.14. After the 

molded material is completely cured, the material gets ready for machining. Removing 

the undesired parts from the material is known as selective removal. 

 



38 

 

Selective removal is the controlled material removing process. The undesired 

sections of models are removed using a CNC. 

 

Bulk removal is uncontrolled material removal. Hot water jet, melting, chemical 

etching are example of this method. Whole sacrificial material is removed in bulk 

removal method; consequently this method is not very accurate. Four main 

manufacturing process of SDM is described in detail in Appendix 1. Furthermore, an 

example of manufacturing a spring-loaded hinge is presented in Appendix 2. 

 

3.3.5. Fixturing Challenges for Flexible Components 

Embedding flexible components in the part material is very challenging issue 

because; the flexible components do not keep its shape constant in designed place 

without fixtures. To solve this problem, three main methods are described: 

 

Fluid adhesive fixture method: This method is not very accurate. This is a simple and 

valid method when the required positioning accuracy is not very tight and it is not 

need to be in tension. However, it is a rather unreliable skill-dependent method since 

performance consistency cannot be expected. 

 

Pre-encapsulation: This method describes pre-embed flexible component in a 

polymer. Having a layer of polymer, the flexible component keeps safe from chemical 

reactions. Embedded components should have sufficient rigidity and density to keep 

its shape and displacement by flotation. This method has less spatial limitations 

during embedding process and also fixture can be removed from the component 

before its incorporation into the mechanism in production. The best way to define 

geometry of a flexible component is to apply tension. For example fiber reinforcement 

elastomers can be produced by holding fibers under tension condition by using 

anchors and string alignment nut placed in the mold under tension condition and 

pouring polymer to encapsulate fibers. The pre-encapsulation fixture is described in 

Appendix 3. 



39 

 

 

Suspending fixture method: This method is used to create a specific harness for 

flexible components to keep their shape in cavity. The harness keeps flexible 

components in three or more locations. These multiple securing parts would be in one 

rigid piece until poured polymer fully got rigid. For releasing polymers from cavity top 

part of the support block removed by CNC. Since flexible polymers can be damaged 

in this method and complexity of designing the support block, it is not very common 

method. The pre-encapsulation method is proper for simple geometries and also it 

can be applied in small sizes as well.  But this method has its own advantages like 

enabling to achieve more complex geometries. The suspending fixture is presented in 

Appendix 4. 

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 



40 

 

3. RESULTS AND DISCUSSIONS 

3.1. Operations on Material 

Effective parameters on mechanical properties of elastomeric materials are described 

in previous section. In this section, results of some of these operations are presented. 

As mentioned before the mechanical properties of elastomeric materials could change 

over time. This happens because of vaporization of chemical gasses trapped in the 

material. Temperature and humidity have large effects on vaporization of the gasses; 

therefore, all specimens are stored in room conditions. Figure 3.1 shows elongation at 

break over time. Five specimens are manufactured and tested each day under the 

same conditions. 

 

 

 

Figure 3.1. Plot of elongation vs. time (days) 

 

As it is seen in the plot, the elongation decreases rapidly in the first days but the 

graph saturates after 30 days. As it seen in Figure 3.1, the specimens tested in day 

19 have large errors mainly due to the manufacturing problems. The maximum force 



41 

 

that specimen endures during uniaxial tension test is shown in Figure 3.2. It should be 

noticed that the width and the length of the specimens are kept constant. . 

 

 

 

Figure 3.2. Plot of load/thickness vs. time (days) 

    

As previously mentioned, the polyurethane elastomer is made by mixing two materials 

(resin and hardener). According to manufacturer the elastomer should be mixed for 

three minutes. We have found that mixing time below three minutes dramatically 

decreases the maximum endurable stress of the material. Further, the annealing 

could be applied to the material by heat the specimen  to150°F (65°C) for 4 to 8 hours 

after overnight cure; the maximum endurable stress increases up to 30% however the 

maximum strain decreases 5%. 

 

3.2. Increasing SDM Performance 

Conventional SDM process uses flat surfaces as contact surfaces each layer; 

however, we have found that increasing the cross section of bonding would increase 

the maximum bonding between layers. For this purpose, we manufactured and tested 



42 

 

different cross sectional bonding. Different cross sections and the results of uniaxial 

tension are presented in Table 3.1. Example of two different materials bonded is 

presented in Figure 3.3. 

 

 

 

Figure 3.3. Bonding of two different materials 

 

Table 3.1. Tension test on specimens with different contact area 

 

Bonding type 
Average of 
maximum 
force(N) 

Average of 
thickness 

(mm) 

Force(N)/ 
thickness 

(mm) 

 
 

 

735 

 

7.38 

 

99.5 

 

 

652 

 

7.7 

 

84.6 

 

 

647 

 

6.1 

 

106 



43 

 

 

 

562 

 

7.4 

 

75.9 

 

 

441 

 

7.3 

 

60.4 

 

 

646 

 

6.1 

 

105.9 

 

 

702 

 

7.2 

 

97.5 

 

 

650 

 

7.4 

 

87.8 

 
 

3.3. Finding constants of Hyperelastic Materials 

3.3.1. Constants of Mooney-Rivlin 

Constants of SED function having 2 and 9-parameters of Mooney-Rivlin models 

derived from using the experimental uniaxial tension test and two estimated shear 

and biaxial test results are listed in Table 3.2. 

 
 
 
 
 
 



44 

 

Table 3.2. 2 and 9-parameter Mooney-Rivlin constants 

 

Constants 
Mooney-Rivlin options 

2-parameter 9-parameter 

C10 4.5383E5 2.8203E5 

C01 42559 2.422E5 

C20  14067 

C11  -12031 

C02  -4124 

C30  -1104 

C21  1538.6 

C12  44.44 

C03  -5.748 

 

3.3.2. Constants of Ogden 

Constants of the hyperelastic material model according to the experimental uniaxial 

tension test and two estimated shear and biaxial test results are calculated as: 

 
  =1.2313E+006 

  =1.8068 

 
3.3.3. Constants of Neo-Hookean 

According to the experimental uniaxial tension test and two estimated shear and 

biaxial test results, the constant of the Ogden material model is calculated as [23]: 

 
   =1.0322E6 

 
3.3.4. Constants of Gent 

Constants of this hyperelastic material model utilizing uniaxial tension test and two 

estimated shear and biaxial test results are calculated as: 



45 

 

 
  = 1.0322e6 

  = -1.7989e12 

 

3.4. Results of FEA 

Numerical analysis is applied to the specimen under uniaxial tension and pure shear 

using ANSYS. Four material models (Mooney-Rivlin, Ogden, Neo-Hookean, Gent) 

were numerically tested in four test combinations (uniaxial tension + biaxial, uniaxial 

tension+ shear, shear + biaxial, uniaxial tension + Shear + Biaxial) using 

Experimental data-set of uniaxial tension, and two estimated data-set of shear and 

biaxial tests. 

 

The work started by developing a FE model using ANSYS 11. The element type was 

selected as solid 20node186 for meshing the specimen; the element has quadratic 

displacement behavior and characterized by 20nodes having three degrees of 

freedom on each node: translations in the nodal x, y, z directions. Moreover, because 

of its large deformation capabilities, it is used for meshing hyperelastic materials. 

Constants of the SED function of the selected material model were calculated using 

ANSYS curve fitting tool. As boundary conditions; one end of the specimen is fixed 

and the displacement is applied to the other end. The detailed algorithm of the 

numerical analysis of hyperelastic materials using ANSYS is described as a block 

diagram as given in Figure 3.4. 

 



46 

 

ANSYS general processor 

operation (select element 

type, modeling, meshing)

Select data-set combination 
(i)

Select material model
(j)

Fit by least-square-method

Error<ε

Define boundary conditions

No

Yes

Solve (Current LS)

Solution
Done?

Export results

End

Yes

No

i=i+1

i > 4

No
j=j+1

i=1

i=1; j=1

Yes

 

 

Figure 3.4. The block diagram of ANSYS analysis 

 

3.4.1. Data-set Combination Effect on Different Material Models in Uniaxial 
Tension Test 

In this section, comparison of numerical analysis of a specimen under uniaxial tension 

test using different inputted data-sets combinations are considered. For this purpose, 

four material models were evaluated and the results were presented in Figure 3.5-3.8. 

As shown in these figures, combination of uniaxial tension + biaxial and biaxial + 



47 

 

shear data-sets presented the best and worst results respectively, in all material 

models. As shown in Figure 3.5 Mooney-Rivlin cannot numerically solve the problem 

for large strains using the combination of uniaxial tension + shear or biaxial + shear 

data-sets.  

 

 

 

Figure 3.5. FEA of the uniaxial tension test using Mooney-Rivlin (9-parameter) model 

 

 

Figure 3.6. FEA of the uniaxial tension test using Ogden 1rd model 



48 

 

 

 

Figure 3.7. FEA of the uniaxial tension test using Neo-Hookean model 

 

 

 

Figure 19. FEA of the uniaxial tension test using Gent model 

 

3.4.2. Data-set Combination Effect on Different Material Models in Shear Test 

In this section, numerical analysis of the specimen under pure shear test was 

performed using different data-sets combinations and material models. As shown in 

Figure. 3.9-3.12, biaxial + shear data-sets combination presented the best 

convergence in all material models. Moreover, Mooney-Rivlin could solve the model 



49 

 

using biaxial + shear data-sets combination only. In addition, Ogden model could not 

solve the model using combination of all three types of test data-sets. 

 

 

 

Figure 3.9. FEA of the pure shear test using Mooney-Rivlin (9-parameter) model 

 

 

Figure 3.10. FEA of the pure shear test using Ogden (1-rd) model 



50 

 

 

Figure 3.11. FEA of the pure shear test using Gent model 

 

Figure 3.12. FEA of the pure shear test using Neo-Hookean model 

 

3.5. Comparing Different Material Model Results 

The best combination of inputted data-sets for uniaxial tension and pure shear test 

was found in the previous section. In this section, the best data-sets combination of 

each model is compared. 

 
Figure 3.13 shows the comparison of aforementioned material models, using uniaxial 

tension + biaxial data-sets combination which is proven to be the best combination in 

previous section for uniaxial tension. In order to study the effects of orders in material 

models, 2-parameter Mooney-Rivlin is also evaluated. As shown in Figure 3.13, 9-

parameter Mooney-Rivlin is more precise than 2-parameter model. However, none of 



51 

 

them are in good compliance with the experimental data-set. It is shown that the Gent 

model has the best convergence. 

 
Figure 3.14 shows the comparison of different material models using combination of 

biaxial + shear data-sets which was already proven to be the best combination for 

pure shear test in the previous section. Neo-Hookean model demonstrates the best 

convergence with the estimated pure shear test. 

 

 

Figure 3.13. Difference of material models in analyzing uniaxial tension test 

 

 
Figure 3.14. Difference of material models in analyzing shear test 

 

 



52 

 

To compare FE results with the experimental results, the average Root-Mean-Square 

(RMS) error was utilized for different strain intervals. The average RMS error can 

calculated as: 

                   
 

 
√∑|

        

    
|

 

   

 (19) 

 
where, M is the number of points at which the stress is calculated by FE Analysis, 

     is the stress of experimental data and     is the stress calculated from FE 

analysis. Results are presented in table 3.3. This Table shows the average RMS error 

of different material models calculated for analyzing uniaxial tension and pure shear 

tests. 

 
Table 3.3. RMS error of material models for each test models 

 

Uniaxial tension test 

 
Mooney-

Rivlin 
9-parameter 

Mooney-
Rivlin 

2-parameter 

Ogden 
1-rd 

Neo-
Hookean 

Gent 

RMS 

0   1 
0.160 0.166 0.198 0.192 0.25 

RMS 

1   2 
0.191 0.226 0.201 0.193 0.104 

RMS 

1   2.5 
0.306 0.367 0.335 0.316 0.117 

RMS 

0   2.5 
0.1466 0.1883 0.162 0.1542 0.100 

Pure shear test 

 
Mooney-Rivlin 
9-parameter 

Ogden 
1-rd 

Neo-
Hookean 

Gent 

RMS 

0   1 
0.380 0.320 0.318 0.381 

RMS 

1   2 
0.273 0.213 0.201 0.281 

RMS 

1   2.5 
0.239 0.164 0.143 0.284 

RMS 

0   2.5 
0.164 0.128 0.121 0.182 

 



53 

 

3.6. Impact Modeling 

In this work, we developed an analytical and empirical models of a peak deceleration 

(PD) using the results of experimental drop tests. In the analytical model; thickness, 

area, drop mass and drop height are considered as separate functions. In the 

empirical model; Neural Network (NN) is used. Further, stresses on the materials are 

calculated using differential equation and Finite Element Method (FEM). Finally, all 

methods are compared and the best approaches are found.   

 
Mechatronic applications are increasing exponentially and taking over the pure 

mechanisms recently. However, these applications utilize cameras, sensors, 

actuators and etc. which are sensitive to applied acceleration and stress. Therefore, 

many mechatronics devices cannot be used in many real-world applications. In this 

work, we focused on the ways of minimizing acceleration and stress by embedding 

the electronic devices in a host object made by rubber-like materials; which are widely 

used in minimizing the intensity of impact loads or accelerations. Polyurethane 

elastomers are viscoelastic rubber-like materials with very wide shore hardness which 

provide different dynamic properties. They can be used as bumpers in automobiles, 

cushion in vibration systems, packaging material, impact isolators and etc. 

 
Since, the aim of this study is to protect the vulnerable components of a mechatronics 

device from elevated accelerations; in this study we developed an analytical and 

empirical (using Neural Networks) model of PD. These models could predict the PD of 

hypothetical specimens under different impact loadings. To the best of our 

knowledge, there has not been any work done to model the PD on variable specimen 

size and testing conditions. In both analytical and empirical models; the area, 

thickness, collision velocity (drop height) and collision mass are considered as 

separable functions. The stress in the material is also analyzed both analytically using 

a differential equation by converting PD to the stress and numerically using FEM. 

Finally, these models are compared to each other.   

 
In this study, the elastomeric contact pad (specimen) is assumed to be the cover body 

of the robot experiencing the collision which is mimicked by a test device 



54 

 

manufactured in this work. This work continues with the details of the experimental 

set-up, modeling of the PD, stress analysis, discussion and conclusion. 

 

3.6.1. Modeling of PD 

3.6.1.1. Analytical Model of PD 

The analytical model of deceleration proposed in this work considers area, thickness, 

drop height and drop mass as variables and assumes that these parameters are not 

functions of each other’s. 

 

PD = A (a)   T (t)   H (h)   M (m) (20) 

 

where, A, T, H and M are functions of area, thickness, drop height and drop mass 

respectively.  

 

 

 

Figure 3.14. An example plot of PD vs. thickness with different drop height 

 
Figure 3.15 shows the experimental results of deceleration of specimens with different 

thicknesses for different drop heights. According to the results, deceleration trends 

look similar for different drop heights keeping the area and drop mass constant but 

changing the thickness. Utilizing the plot; thickness of 8.33 mm, area of 28.26 cm2 



55 

 

and drop mass of 1.866 kg are chosen as reference points. The coefficient of 

thickness is found utilizing a best fit by keeping all parameters fixed but changing the 

thickness with a reference point of 8.33 mm. The coefficients of area and drop mass 

can be found similarly as well. The area function is calculated at an area of 28.26 cm2 

as the reference point where thickness, drop hight and mass are constants and fixed 

to 8.5 mm, 15 cm and 1.866 kg respectively. The mass function is calculated at a 

mass of 1.866 kg as the reference point where thickness, area and drop hight are 

constants and fixed to 8.33 mm, 28.26 cm2, 15 cm respectively. In this work, all the 

test results are best fitted at an arbitrarily chosen drop height of 15 cm using 

MATLAB. The coefficients can be calculated using the best fit curves given below; 

 

 ( )        (       )              (           ) (21) 

 ( )           (     √   
⁄ ) (22) 

 ( )         (       )              (           ) (23) 

 

where, t (mm), a (cm2), m (kg) are the variables of thickness, area and mass 

respectively.  

 

 

 

Figure 3.15. A best fit plot of deceleration ratio vs. thickness 



56 

 

 

 

Figure 3.1620. A best fit plot of deceleration ratio vs. Area 
 

 

 

Figure 21. A best fit plot of deceleration ratio vs. mass 
 

Although thickness, area and mass are converted into dimensionless number, the 

function of drop height is left with a unit of g (m/sec2). This function is modeled up to 

35 cm height due to accelerometer limitations where the reference specimen`s area, 

thickness and drop mass are constants and fixed to 28.26 cm2, 8.33 mm and 1.866 

kg respectively.  



57 

 

 

 

Figure 228. A best fit plot of PD vs. height 

 

The best fit of the drop height function is; 

 

 ( )               (24) 

 

where, h is the drop height of the impact rig. Using the coefficients of each parameter 

and the drop height function, PD of any V10 specimen can be predicted. 

 

3.6.1.2. Empirical Modeling (NN method) 

In the previous section, the effect of four parameters including, area, thickness, drop 

weight, and drop height is modelled analytically. However, in this section the NN is 

applied to the experimental data.  Designed NN could predict the PD of any 

combination of the four parameters. The perceptron algorithm is implemented for this 

problem. Two kinds of perceptron algorithms are compared in this work. One of them 

has two layers and the other has one layer. It is seen that the two-layer perceptron 

have smaller Minimum Squared Error (MSE). Then, the two-layer perceptron is 

decided to be used. 

 



58 

 

Tansig and Purelin transfer functions are used for the first and second layer, 

respectively. The designed NN uses Levenberg-Marquardt algorithm as the 

instruction function of training. Input data are divided randomly to the three sets which 

are as follows: 70% for training, 15% for validating and 15% used for testing. The 

regression analysis of different data-set using multilayer perceptron (MLP) 

implemented in this work is shown in Figure 3.20. The regression analysis verifies the 

suitability of the designed MLP [33]. Figure 3.21 shows the performance diagram of 

the selected algorithm. The MSE of validation graph has a descending trend and the 

indicated circle shows the best point for training the algorithm. The overall NN 

structure is provided in table 3.4. The number of neurons in each layer is determined 

by trial and error. 

 
Table 3.4. The artificial NN structure 

 

Structure Multilayer perceptron 

Number of hidden layers 2 

Number of neurons on each hidden layers 
First layer: 5; 

Second layer: 4 

Hidden layers activation function Purelin, Tansig 

Number of training sample 2569 

Learning method Levenberg-Marquardt algorithm 

 
 



59 

 

 

Figure 3.19. Regression analysis of MPL network. 

 

 

 

Figure 230. Performance of the NN. 



60 

 

3.6.1.3. Comparison of Analytical and Empirical Modeling 

Some specimens are manufactured to analyze the accuracy of the analytical and 

empirical models. Table 3.5 represents the specimen size and testing conditions of 

these specimens. To compare the analytical and empirical results with the 

experimental results, the average Root-Mean-Square (RMS) error is utilized. The 

average RMS error can be calculated as: 

 

                   
 

 
√∑|

             

     
|

 

   

 (25) 

 

where, M is the number of points at which the PD is calculated analytically or 

empirically,       is the result of experimental data and         is calculated via 

analytical or empirical analysis. Results are presented in table 3.6. 

 
Table 3.5. Specimens manufactured for testing 

 

specimen No. 1 No. 2 No. 3 No. 4 No. 5 

Area 
(cm2) 

16 12.56 8 19.26 9.07 

Thickness 
(mm) 

16.5 8.3 9.6 11.5 40 

Drop mass 
(kg) 

2.479 1.866 4.363 3.431 2.965 

Drop height 
(cm) 

57 5 27 13 40 

 
Table 3.6. RMS error of each model 

 

specimen No. 1 No. 2 No. 3 No. 4 No. 5 RMS 

Experimental 
(m/s2) 

164.2 41.34 98.2 77.8 81.2 - 

Analytical 
(m/s2) 

180.9 44.99 88.56 64.1 94.62 0.158 



61 

 

Empirical 
(m/s2) 

167.2 41.2 91.8 79.4 78.5 0.075 

 

3.6.2. Stress Analysis 

PD of the material is modeled using analytical and numerical models in the previous 

sections. Besides PD, the stress level of the material is also very important for long 

term mechatronics applications. Therefore; the stress on the materials should also be 

studied. To analyze the stresses on the materials, two methods can be utilized. These 

methods are; a differential equation and FEM. The differential equation converts PD 

to the stress assuming a uniform stress distribution on the specimen. In other words, 

it neglects the shear stress at the corners. To solve this problem, numerical software, 

i.e. ANSYS 14.5, is utilized in this work. 

 
3.6.2.1. Differential Equation 

In this section, the differential equation of dynamic deceleration is derived to convert 

deceleration to the stress. Modeling of the viscoelastic elastomer as a spring and a 

damper is difficult due to its nonlinearities. Further, its mechanical properties change 

with the specimen size. Therefore, instead of dynamically analyzing the material, we 

used a methodology to convert the measured acceleration to the stress. In general, 

impact force of a viscoelastic material could be modeled as; 

 

    ̇     (26) 

 

where, k and c represent the stiffness and damping coefficient respectively. According 

to [34], the force on the material can be calculated as; 

 

    ̈     (27) 

 



62 

 

where m is the mass of the impact rig, g is the acceleration of the gravity and  ̈ is the 

deceleration measured by the accelerometer. If the equation is divided into the area 

of the specimen, the stress equation can be derived as; 

 

   
  

 
 ̈     (28) 

 

where,     
  

 ⁄  is the static stress. The initial conditions of the equation of motion 

are:  

 

 ( )     ̇( )  √     ̈( )    (29) 

 

where, H is the drop height. In addition, if the stresses are known, it can also be 

converted into the acceleration as well using the equation below; 

 

 ̈   
 

  
(    ) 

(30) 

 

 

3.6.2.2. Finite Element Method 

In this section, the process of FEM using ANSYS is described in detail. The same 

element type is used to mesh all asymmetric 3-D specimens. The element type is 

selected as solid 20node186 for meshing the specimen and the impact rig; the 

element has a quadratic displacement behavior and characterized by 20nodes having 

three degrees of freedom on each node: translations in the nodal x, y, z directions. 

Moreover, because of the large deformation capabilities of the elements, it can be 

used for meshing hyperelastic and viscoelastic materials. For the sake of simplicity, 

other element type of 8node183 is used for symmetric specimens. The element has a 

quadratic displacement behavior and is defined by 8 nodes having two degrees of 

freedom at each node: translation in x and y directions. This element is suitable for 



63 

 

axisymmetric models. Further, it has a capability of simulating the nearly 

incompressible or fully incompressible hyperelastic materials. 

 
In order to neglect the contact stiffness, Lagrange multiplier method was used to 

model the contact. After modeling and meshing, the contact elements could be 

selected automatically in ANSYS 14.5 by selecting the target line/surface and the 

contact line/surface. It should be notice that the impact rig is the moving object and 

must be selected as the contact surface and the specim