Master's Degree Thesis

ISRN: BTH-AMT-EX--2014/D07--SE

The Mechanical and Fracture

Property of HDPE-Experiment
Result Combined with Simulation
.

Shaofeng Li
Kang Qi

Department of Mechanical Engineering

Blekinge Institute of Technology
Karlskrona, Sweden
2014

Supervisors: Jian Li, Shanghai Second Polytechnic University

Sharon Kao-Walter, BTH
 The Mechanical and Fracture
Property of HDPE-Experiment
Result Combined with Simulation
Shaofeng Li
Kang Qi
Department of Mechanical Engineering
Blekinge Institute of Technology
Karlskrona, Sweden
2014
Thesis submitted for completion of Master of Science in Mechanical
Engineering with emphasis on Structural Mechanics at the Department of
Mechanical Engineering, Blekinge Institute of Technology, Karlskrona,
Sweden.
Abstract
The aim of this work is the evaluation of different specimen dimensions
and crack length on the mechanical properties of HDPE (high quality
density polyethylene) which it is often used in the packing industry. Two
mother shape specimens are chosen and one is dog-bone shaped
specimen, the other is shear specimen for studying shear damage. A
series of experiments were processed and results show that fracture
initiation in HDPE specimen is sensitive to the size of centre circled and
pre-crack. A corresponding numerical simulation is calculated by
ABAQUS explicit and the corresponding model in numerical simulation
was chosen and result rather good match with shear tests. In addition to
tensile experiment, the microscope analysis was operated to observe the
fracture surface of specimen with scanning electron microscope. In
application section, bottle cap opening study was carried out based on the
numerical and experimental results.
Keywords:
High Density Polyethylene, Crack, Tensile test, Shear damage,
Mechanical properties, Fracture

3
 Acknowledgements
This thesis was carried out at the Department of mechanical Engineering &
electronic, Shanghai Second Polytechnic University, Shanghai, China in
cooperation with Department of Mechanical Engineering, Blekinge Institute
of Technology, Karlskrona, Sweden, under the supervision of Dr. Sharon
Kao-Walter and Dr. Jian Li. The project was started in January 2014 and
finished in June 2014.
We wish to express our sincere gratitude to my supervisor Dr. Sharon Kao-
Walter at the Department of Mechanical Engineering, Blekinge Institute of
Technology, for exposing us to fracture mechanics, for her advice,
encouragement and patience through various aspects of questions. We would
like to thank Dr. Jian Li of Shanghai Second Polytechnic University (SSPU)
for his wonderful and rewarding guidance and insightful comments all the
way along our research and the completion of thesis. Thanks also to PhD
candidate Md. Shafiqul Islam, Blekinge Institute of Technology, Sweden, for
his discussion and advice of ABAQUS technology. Finally, this dissertation is
dedicated to our parents for their endless love, understanding and support.
Shaofeng Li
Kang Qi
Karlskrona, May 2014

4
Contents
Acknowledgements ........................................................................ 4
Contents .......................................................................................... 5
Notation .......................................................................................... 7
1 Introduction ............................................................................ 8
1.1 Background and motivation .............................................................. 8
1.2 Research objectives and method ....................................................... 9
1.3 The scope of the thesis .................................................................... 10
2 Chapter 2.The fundamental knowledge about fracture
mechanics ..................................................................................... 12
2.1 Fundamental knowledge concepts .................................................. 12
2.2 The Linear Elastic Fracture Mechanics (LEFM) mechanics .......... 15
2.3 Elastic Plastic Fracture Mechanics ................................................. 17
2.4 Fracture toughness .......................................................................... 19
2.5 Fundamental theory of mechanical property .................................. 21
3 Chapter 3. Experimental Work........................................... 28
3.1 Introduction ..................................................................................... 28
3.2 Experimental setup.......................................................................... 29
3.2.1The tensile specimen ....................................................................... 29
3.2.2The experiment case ....................................................................... 31
3.2.3The experiment method .................................................................. 31
3.3 The Result and discussion ............................................................... 32
3.3.1The tensile test result ...................................................................... 32
3.3.2The shear test result ........................................................................ 35
3.3.3Shear test with pre-crack condition................................................. 38
3.4 The SEM fracture observation ........................................................ 41
4 Chapter 4 .The numerical simulation ................................. 46
4.1 Introduction ..................................................................................... 46
4.2 The finite element method .............................................................. 46
4.3 ABAQUS Simulation...................................................................... 49
4.4 The material properties for ABAQUS simulation .......................... 50
4.5 The Mechanical property of material and result ............................. 53
4.5.1Dog-bone shape specimen tensile simulation ................................. 53
4.5.2The numerical result of dog-bone shape specimen ......................... 54
4.5.3The numerical result of shear specimen.......................................... 57

5
 4.6 Fracture behavior predicting ........................................................... 65
5 Chapter5. Plastic bottle cap calculation .............................67
5.1 Introduction..................................................................................... 67
5.2 Nonrectangular thread screws cap .................................................. 68
5.3 Parameters used in calculation ........................................................ 70
5.4 The result and discussion ................................................................ 71
5.5 Bottle design suggestion ................................................................. 77
6 Chapter 6. Conclusion and future work .............................79
6.1 Conclusions of HDPE experiment and simulation ......................... 79
6.2 Future work ..................................................................................... 80
References .....................................................................................81
Appendix A Experiment result ...................................................83
Appendix B ABAQUS modelling and relative plasticity data .87
Appendix C Matlab code used in chapter 5 ..............................94

6
Notation
A Area [m2]

E Young’s modulus [MPa]

F Force [N]

L length [m]

KI Stress intensity factor for mode one [MPaξ]

KII Stress intensity factor for mode two [MPaξ]

KIII Stress intensity factor for mode three [MPaξ]

KIc Fracture toughness [MPaξ]

f Friction coefficient

ı Stress [MPa]

Ʌ Angle

İ Strain

Ȳ Thread angle

ȕ Friction angle

Abbreviations

CTOD Crack Tip opening Displacement

HDPE High Density Polyethylene

SEM Scanning Electron Microscope

7
1 Introduction
1.1 Background and motivation
The phenomenon of fracture commonly occurs in our daily lives, for example,
the dish breaks into several pieces when accidently dropped on the floor. In
the package industry, the package of food can be opened by tearing. In case of
fracture occurs, some of these instances are spontaneous and others are
controlled by people and mostly due to the mechanism of the design.
Metal and non-metal material is widely used in our common lives and
especially polymer material has increased significantly in last fifty years. In
this research, high density polyethylene (HDPE) is mainly focused on. HDPE
is widely used in packaging industry.
The pipe industry, HDPE is widely used and has been applied in pressure
piping for many years. Sarkes and Smith (1983) pointed out that the use of
plastic pipes in the gas transmission industry since 1955. Currently, HDPE
pipes are installed more than the other plastic pipes. In the last twenty years,
HDPE pipes have also been used as a protection layer for cables in
construction structure to prevent corrosion [1-2].
In the packaging industry, the history of HDPE used for food packaging can
be traced to 1964. As the development of packaging industry, HDPE used for
packaging has increased because of its low cost, flexibility, durability, and
resistance to many chemicals. As food packaging, HDPE is most used with
milk, oil, and juice bottles and for non-food packaging, it is in cleaning
product containers, supermarket bags, agricultural films and chemical
containers, motor oil containers, paper bag liners, bags, crates, drums, and
pails[2].
In the industry, the products should be test under different types of loading
before it is brought to the market. In the test, when exceeding the load
carrying capacity, materials fail by fracture. Although failure modes may be
different, the failures sequence can often be illustrated into Figure 1.

8
 Figure 1 .Typical failure procedure for plastic[3]
The material displays elasticity which is recoverable when load is removed
and if the load beyond a certain limit, its deformation is unrecoverable which
called plasticity [2]. Furthermore, the plasticity deformation is still stable until
the ultimate tensile strength is reached. If the material continues to be loaded
after the peak loading, the deformation will be unstable and soon leads to
fracture.
In the processing of material fracture, theory and experimental method will be
combined and its result will provide powerful support for the industry.
1.2 Research objectives and method
Many methods have been introduced to describe the phenomenon of fracture
which includes the concepts of stress intensity factor, energy release rate, J-
integral and so on. In this research, the mechanical properties of HDPE like
Young’s modulus, yield strength, and some mechanical variables will be
investigated, such as stress and strain. The tensile tests standard was operated
by following ASTM D638. The specimen is applied under tensile load to find
the yield point during the process.
In fracture part, the pre-crack HDPE specimens will be applied and analysis
the fracture condition in the shear model. The dog-bone shape and shear shape
specimen with different circled will be investigated in this thesis.

9
Typically, with the help of computational power, more complicated problem
can be solved. In this thesis, ABAQUS is chosen to analyze the fracture
situation under the stress. The key processing is building the model and a
successful model depends on an understanding of the physical world upon
which an idealized theory can be proposed. A figure of this learning structure
is illustrated in Figure 2. In this study, an equal emphasis is put on the
following three roles.

Figure 2 .The relationship betwween physical world, abstract theory and

computer simulation [4]

1.3 The scope of the thesis

Following the theoretical part, a combined experimental and numerical
program is carried out to analyze and verify the mechanical and fracture
property under deferent loading condition.
This master thesis is documented in the following format: It starts with
chapter 2 which introduce the general overview of the literature in the field of
fracture and mechanical properties. In this chapter, the reader will build some
fundamental knowledge related to the studies undertaken by the author.
Chapter 3 will present the studies of the experimental result with different
fracture condition and a corresponding SEM observation of fractured surface
is committed.
Chapter4 will show the numerical result in ABAQUS and it will supply the
additional information for the fracture condition of HDPE.
Chapter 5 will study a case of practical issue which calculating how much
force and torque to open a HDPE made bottle cap.

10
Chapter 6 will show the conclusion and recommendation for the further work.
In this chapter key finding will be summarized, the conclusion are derived,
the shortcoming of current study will be shown and the direction and
recommendation of the further research will also be proposed.
Hence, the corresponding chapters are “manuscript-based”, and therefore,
certain material may be repeated in the different chapters. References and
appendix are given at the end of paper.

11
2 Chapter2.The fundamental knowledge
about fracture mechanics
2.1 Fundamental knowledge concepts
There are two types reason for most failures:
1. Negligence during design, construction, or operation of the structure [5].
2. Application of a new design or material, which produces an unexpected
(and undesirable) result [5].
Although polymers are becoming more and more common in structural
applications, it provides lots of advantages over metal, other type fracture
situations will also happen in polymers. The PE material is popular used in
the natural gas transportation system in the world. The great advantage of PE
is that maintenance can be performed on a small branch of the line without
shutting down the whole system; by applying a clamping tool to the PE pipe,
the local area can be shut down, however, although the operation of pinch
clamping save sums of money, this behavior will lead to fracture problem [1].
The Figure 3 and Figure 4 show the typical fractured PE pipe.

12
 Figure 3 .Fracture surface of a PE pipe[1]

Figure 4 .Thumbnail crack produced in a PE pipe [1]

In the fracture approaches, the stress intensity factor (K) and energy criterion
are equivalent in certain circumstances. In this research, we only focus on the
stress intensity factor method.

13
There are three types of loading that a crack can experience, as Figure 5,
Figure 6 and Figure 7 illustrate.

Figure 5.The mode I fracture [6]

Figure 6. The mode II fracture [6]

14
 Figure 7.The model III fracture[6]
Mode I fracture – Opening mode (a tensile stress normal to the plane of the
crack). The load is applied normal to the crack plane, tends to open the crack
Mode II fracture – Sliding mode (a shear stress acting parallel to the plane of
the crack and perpendicular to the crack front), corresponds to in-plane shear
loading and tends to slide one crack face with respect to the other.
Mode III fracture – Tearing mode (shear stress acting parallel to the plane of
the crack and parallel to the crack front). It is associated with a pure shear
condition, typical of a round bar loaded in torsion

2.2 The Linear Elastic Fracture Mechanics (LEFM)

mechanics
The popular method to analysis fracture behavior of a material is Linear
Elastic Fracture Mechanics (LEFM) since the fracture property is mainly
limited to fracture initiation stage. The fracture behavior of a linear elastic
structure can be determined by comparing the applied K (the driving force) to
a critical K.
When the external forces are applied on the crack configurations and
assuming isotropic linear elastic material behavior, the closed form
expressions for stress in body can be obtained. The Figure 8 shows a polar
coordinate axis with the origin at the crack tip.

15
 Figure 8.Polar coordinate axis with the origin at the crack tip[4]
The stress field in any linear elastic cracked body is given by
ౣ
୩ ሺ୫ሻ
ɐ୧୨ ൌ ቀ ቁ ˆ୧୨ ሺɅሻ ൅ σஶ
୫ୀ଴  ୫ ” మ ‰ ୧୨ ሺɅሻ ˄ʹǤʹǤͳ˅
ξ୰

ɐ୧୨ = the stress tensor; r andɅ can be defined in the above figure; k is the
constant; ˆ୧୨ is dimensionless function ofɅin the leading term.
The stress intensity factor can be given to express the mode of loading, so the
stress on the crack tip in an isotropic linear elastic material can be written as:
ሺ୍ሻ ୏౅ ሺ୍ሻ
Ž‹୰՜଴ ɐ୧୨ ൌ ˆ ሺɅሻ Mode I
ξଶ஠୰ ୧୨

ሺ୍୍ሻ ୏౅౅ ሺ୍୍ሻ

Ž‹୰՜଴ ɐ୧୨ ൌ ˆ ሺɅሻ Mode II
ξଶ஠୰ ୧୨

ሺ୍୍୍ሻ ୏౅౅౅ ሺ୍୍୍ሻ

Ž‹୰՜଴ ɐ୧୨ ൌ ˆ୧୨ ሺɅሻ Mode III
ξଶ஠୰

In a mixed-mode problem the principles of linear superposition can be written

as the following format:
ሺ୲୭୲ୟ୪ሻ ሺ୍ሻ ሺ୍୍ሻ ሺ୍୍୍ሻ
ɐ୧୨ ൌ ɐ୧୨ ൅ ɐ୧୨ ൅ ɐ୧୨  ˄ʹǤʹǤʹ˅

16
The Figure 9 shows the relationship between the three critical variables in
fracture mechanics. The stress and flaw size contribute to the driving force,
and the fracture toughness is a measure of the material resistance.

Figure 9 .The relationship between the three critical variables in fracture

mechanics
Although stress intensity solution can be given in different of forms, the stress
intensity factor K can be related to the through crack through the appropriate
correction:
 ሺ୍ǡ୍୍ǡ୍୍୍ሻ ൌ ɐξɎȽ ˄ʹǤʹǤ͵˅
Where the characteristic stress isı; Į is the characteristic crack dimension and
Y is the dimensionless constant which depends on the geometry and the mode
of loading.
For the linear elastic material, due to the linearity, stresses, and the stress
intensity factor will become additive as the loading is consistent. That is
ሺ୲୭୲ୟ୪ሻ ሺ୅ሻ ሺ୆ሻ ሺେሻ
୍ ൌ ୍ ൅ ୍ ൅ ୍  ˄ʹǤʹǤͶ˅

2.3 Elastic Plastic Fracture Mechanics

When material exhibit time independent plastic deformation, then LEFM can
be applied to find material properties. Irwin provides a simple plastic zone
correction to the stress intensity factor, since when plastic zone happens, the
LEFM cannot be operated. The other method which do the correction is
issued by Dugdale and Barenblatt[7].
The Elastic plastic fracture mechanism (EPFM) will be applied to material
that exhibit time-independent, non-linear behavior. In this section, we will
introduce two elastic-plastic elements: The crack tip opening displacement
(CTOD) and J contour integral and both of them can be used to describe the

17
fracture condition. Although there are have limits to the applicability of J and
CTOD, these limits are less restrictive than validity requirement of LEFM. In
this research we mainly focus on the crack tip displacement method.

Crack tip opening displacement

The crack tip displacement theory is introduced by the Well and in his
original paper; Wells [4-5] introduced an approximate analysis which related
CTOD to the stress intensity factor in the limit of small-scale yielding. The
crack with a small plastic area can be shown in the Figure 10. While Irwin [7-
10] introduced an alternative method and CTOD can be solved by solving for
the displacement at the physical crack tip.

Figure 10ˊThe crack with a small plastic area [7]

In Wells theory, the CTOD can be expressed as the following format:

ସ ୏మ౅ ଵ
Ɂൌ ‫כ‬ ‫ כ‬ ˄ʹǤ͵Ǥͳ˅
஠ ஢ౕ౏ ୉

Ɂ is the CTOD
ஔ
As ՜ Ͳˈ the above equation can be expressed as the following format:
஢ౕ౏

୏మ౅
Ɂൌ  ˄ʹǤ͵Ǥʹ˅
୉‫כ‬஢ౕ౏

There are a number of different definitions of CTOD. The two popular

definitions are shown in Figure 11, are the displacement at the original crack

18
tip and the 90°intercept. The two different definitions are equivalent if the
crack blunts in a semi circled.

Figure 11.The typical CTOD figure[7]

The standard method for CTOD is modified hinge modal and the
displacement can be written into the following format:
୏మ౅ ୰౦ ሺ୛ିୟሻ୚౦
Ɂ ൌ Ɂୣ୪ ൅ Ɂ୮ ൌ ൅  ˄ʹǤ͵Ǥ͵˅
୫஢ౕ౏ ୉ᇲ ୰౦ ሺ୛ିୟሻାୟ

Ɂୣ୪  is elastic component and Ɂ୮ is the plastic component; ”୮ is plastic

rotational factor and for typical material the value is 0.44.
2.4 Fracture toughness
The fracture toughness ( ୍ୡ ) as one of the most important properties play an
important role in the material science and describe the ability of a material
containing a crack to resist fracture. [11-17]
The fracture toughness ( ୍ୡ ) is measured by loading a sample containing a
deliberately-introduced contained crack of length 2c or a surface crack of
length c (shown in the Figure 12). Record the tensile stress and bending load
at which the crack suddenly propagates. The fracture toughness ( ୍େ ) can be
calculated into the following format:
 ୍ୡ ൌ ଵ ɐ ‫ כ‬ξɎ… ˄ʹǤͶǤͳ˅
୊
 ୍ୡ ൌ ଶ ‫ כ‬ξɎ… ˄ʹǤͶǤʹ˅
ୠ୵

19
 Figure 12.The fracture toughness calculation relationship[7]

F is the bending load andı is the tensile stress. Y1 and Y2 are geometric
factors. E is Young’s modulus and b and w are the beam thickness and depth,
as in the figure. The fracture toughness can be well obtained followed the
above method for material. For Mullite-fibre composite, the fracture
toughness is about 1.8–3.3 MPa · m1/2[4].
The general relationship between fracture toughness and thickness is shown in
the following figure:

20
 Figure 13. The relationship between fracture toughness and thickness[19]
From Figure 13, the thin parts have a high value of fracture toughness and as
the thickness increase, the Kc value will decrease and this type is called stress-
strain mixed mode. For the thick parts, the entire fracture surface is flat and
the fracture toughness will reach the minimum value of the asymptotic.
Plastic zone sizes at fracture are much larger in thin parts as compared to
thick parts.
2.5 Fundamental theory of mechanical property

Stress-strain curve
Tensile strength as one of the important mechanical property is defined as a
stress which is measured as force per unit area. For some non-homogeneous
materials, it can be viewed as a force per unit width. The cross-sectional area
may change if the material deforms as it is stretched, so the area used in the
calculation is the original un-deformed cross-sectional area Ao
For stress:

21
 Figure 14 .Force in per area

୊୭୰ୡୣ
ɐൌ  ሺʹǤͷǤͳሻ
୅౥

For strain: The strain is a measure of the change in length of the sample
(Figure15). The strain commonly is expressed in one of two ways

22
 Figure15.Length changes in force direction

Elongation:
୐
ɂൌ െ ͳ ሺʹǤͷǤʹሻ
୐౥

Extension ratio:
୐
Ƚൌ  ሺʹǤͷǤ͵ሻ
୐౥

The following figure shows stress - strain curve:

23
 Figure 16 .The typical stress-strain curve[20]
The stress-strain behavior (material deformation) is best characterized in
tension and the degree of deformation depends on the magnitude of an
imposed stress. At low stress levels the strain is directly proportional to the
stress for most materials what was first observed by Hooke in 1678 and
termed Hooke’s law[19].
Many materials are display linear elastic behavior and controlled by linear
stress-strain relationship, that is, a specimen loaded elastically in tension will
elongate, but will return to its original shape and size when unloaded. Beyond
the linear region, the deformations are plastic which the deformed specimen
will not return to its original size and shape when unloaded. In many
applications, the plastic deformation is unacceptable and the engineering
designer should avoid this plastic deformation.

Young’s modulus
Young's modulus is the ratio of stress to strain. It is also called the modulus of
elasticity or the tensile modulus and is a quantity used to characterize
materials. In solid mechanics, the slope of the stress–strain curve at any point
is called the tangent modulus. Stress-strain curves often are not straight-line
plots, indicating that the modulus is changing with the amount of strain. In
this case the initial slope usually is used as the modulus which is illustrated in
the Figure17.

24
 Figure17. The typical figure for calculating Young’s Modulus[20]
In general, rigid materials have a high Young's modulus, fibers have high
Young's modulus values, elastomers have low values.

Deformation of HDPE
An understanding of deformation mechanisms of polymers is important in
order to be able to manage the mechanical characteristics of material.
The deformation of the HDPE depends on many elements, for instance, the
stress mode, speed and magnitude of the applied load. It contains the
rearrangement of the molecules and change in the morphology, which is
elastic (recoverable) at low stress levels. The shape of the stress versus strain
curve shown in the Figure 18, it shows the details of deformation process.

25
 Figure 18. The typical deformation curve for polymer
In the elastic region, the deformation can be regarded as elastic and
homogenous and it can be clearly seen in the Figure 19(stage 1). The yield
point which is the maximum in the stress-strain curve shows heterogeneous
and plastic deformation. In the cold drawing stage, the neck grows along the
specimen and it can be seen in the following figure (stage 2) next stage shows
the segments separate from the lamellae but remain attached to each other by
fiber chains (stage 3). In the hardening stage, it shows the fibrous material
deform uniformly and finally reach break (stage 4).

Figure 19κThe typical deformation for polymer [5]

From the above figure, it shows the processing of rearrangement of HDPE in

tension test. Many elements like shear, compressive and flexural properties of

26
HDPE are controlled by the same morphological characteristics that control
the tensile properties [5]. In the above processing which represented are
reversible and very different to the typical engineering materials such as steel
or concrete.

27
3 Chapter 3. Experimental Work
3.1 Introduction
In this chapter, several experiments including tensile and shear test are
included and the objective of this experimental program is to understand the
mechanical behavior and to determine the material parameters for further
numerical study. The material selected to be experimentally investigated is
HDPE.
HDPE is one of the widely used non-metal materials in industry; its
application includes aircraft structural components, aircraft fittings, hardware,
truck wheel and parts for the transportation industry because of its high
strength and excellent fatigue resistance.
The tensile and shear experiments were conducted using a screw driven 200
KN MTS universal testing machine. Combined numerical-experimental
approach described in this project, required series of experiments and
numerical simulations to obtain desired results.
In the tensile testing experiment, the specimen with crack and without crack
are considered and driven under different velocities (5mm/min, 10mm/min,
15mm/min, 20mm/min, and 25mm/min).
In the shear test, the mechanical property of material and crack initiation and
crack propagation under this loading condition will be obtained.
A parallel numerical study of these test were conducted and reported in
chapter 4 and one of objectives of the numerical simulations are to assess the
obtained material parameter from experiments and to fine the parameters.
Most of the experiment in this master thesis is operated in Blekinge Institute
of Technology(BTH), some are applied in Shanghai Second Polytechnic
University and all test results are presented in Appendix A.

28
3.2 Experimental setup

3.2.1 The tensile specimen

A series of tensile specimens were cut in the longitudinal direction. The

geometry and the dimensions of the specimens used in this line of study are
shown in the Figure 20 and table 3.1:

Figure 20.the geometry of dog-bone specimen

Table 3.1.The dimension of dog-bone specimen

Symbo Name Dimension Symbol Name Dimension

l
L Total length 150mm W Width 20mm

H The distance 115mm d Thickness 4mm

between the
clamp

C The middle 60mm b Width in 10mm

section length middle
section
G0 The gauge 50mm R Radius 60mm
length

For the dog bone shape tensile specimens, ASTM E647-11 test standard is
followed in this experiment.

29
Specimen for shear experiments is shown in Figure 21:

Figure 21. drawing of shear specimen

All the specimens are prepared by the CNC machine and in order to ensure
the precision of specimen, the DFX file from Auto Desk Inventor are used as
input to CNC machine (see Figure 22).

Figure 22 .The CNC machine used to make shear specimen

To make the pre-crack, special scissors are used to make the exact dimension
on specimens.

30
3.2.2 The experiment case

The experimental test has been applied on two shapes of specimen, the tensile
test with dog bone specimen and specially designed shear specimen for shear
test. For investigating the fracture elements and phenomenon, the shear
specimen was further studied under tensile loading by introducing 1mm and
2mm crack length. The different cases are listed in table 3.2.
Table3.2 the test cases table:
Test Case Specimen number Dog bone specimen Shear Specimen

Tensile test 3 Yes Yes

1mm centre 3 Yes No

circled test
2mm centre 3 Yes Yes
circled test
3mm centre 3 No Yes
circled test
1mm both 3 No Yes
sides crack
test
2mm both 3 No Yes
sides crack
test
1mm single 3 No Yes
side crack
test
2mm single 3 No Yes
side crack
test

3.2.3 The experiment method

MTS Test 100 Tensile machine with 2KN load cell will be used as the
experiment device in this master thesis. The tensile test will be operated at
very low loading rates of about 10mm/min as recommend by ASTM D638 to
ensure the tensile test condition. The tensile test machine, MTS, has a pair of

31
pneumatic gripper and it is used for clamping the specimen ends. In the
processing of operating, the lower gripper keep stationary and the upper one
will move with the certain speed. The specimen is placed between the gripper
and it can be avoided the phenomenon of slipping because of the clamp. The
MTS machine is connected with a computer which is installed with the Test
Works software and test data can be collected and shown on the screen. The
testing machine's load and displacement sensors give the load, extension,
stresses and strains as output at each time instant.

Figure 23. MTS Qtest 100 Machine for tensile testing at BTH

3.3 The Result and discussion

3.3.1 The tensile test result

The program includes a set of un-circled and circled dog-bone and shear
shape specimens under axial tension. A material parameter calibration process
is then followed.
Parallel numerical studies of these tests were conducted and are reported in
Chapter 4. The objectives of the numerical simulations are to assess the
obtained material parameter from experiments and to fine tune the parameters.

32
The main aspects to be assessed are the predicted fracture modes and the load-
deflection curves. All test result is shown in Appendix A.
For each case, at least three tests with three specimens are performed.
The simple tensile experiment is applied in BTH and SSPU (Shanghai Second
Polytechnic University) and the loading velocity is 10mm/min. The following
Figure 24 shows the result:

Figure 24.The HDPE tensile test in 10mm/min

In the simple tensile test, for loading along machine direction, larger
elongation has been observed. However, the dog-bone specimen was not
break in the processing of tensile test, the reason is that the HDPE material is
too plastic to break and this can also be seen from the Figure 24. From the
experimental result, the Young’s modulus value can be obtained and the
number is 534Mpa. In the following chapter, the Young’s modulus number
will be used for numerical simulation in ABAQUS.
1mm and 2mm center circled dog-bone specimen test are also considered and
the compared curves are plotted in Figure 25.

33
 Figure 25. HDPE tensile test with different circled condition
Comparing the result from the pure dog-bone specimen and the circled
specimen, displacement-force curve shows the circled specimen has a
profound effect on the initiation of fracture in the upsetting tests. With larger
circled radius, the onset of fracture is significantly increased. With the pre-
circled, the specimen can be easily tended to fracture and less softening
phenomenon happens.
The fracture surfaces are shown in the following chapter. For the circled
specimen, there is a flat area in crack surface and shear lips are on the
opposite side of two separated side (See Figure 26-Figure 27).

Figure 26.1mm center circled dog-bone specimen video

34
 Figure 27.2mm center circled of dog-bone specimen
Observation of the video shows the tensile test processing of a dog-bone
specimen; the stress is mainly focused on the center circled and the fracture
initiates and increased until the specimen break. It can be seen a large portion
of fracture surface is in tension and because of the symmetry of the dog-bone
specimen, two symmetric parts which perpendicular to each other develop the
fracture at the same time. The specimens fractured suddenly in the middle of
the as the increased tension force.

3.3.2 The shear test result

For the shear shape specimen without center circled, the typical example of
force-extension curve is shown in the Figure 28:

35
 Figure 28.The result of shear specimen without center circled
From the above figure, two different curve shapes can be seen and in the
extension scope form origin point to point A, it shows the shear deformation
and from extension scope from point A to point B, the figures shows tensile
deformation combined with shear damage which can be approved from the
video of Figure 29. Furthermore, beyond the point B to the remote area, the
damage model is tensile.

Figure 29.The pure shear specimen tensile video

From the above figures, the shear and tensile deformation can be clearly seen,
and the HDPE material is not broken up to the experiment processing.
The load-deflection curves for the specimen with 2mm and 3mm center
circled are plotted together in the Figure 30.

36
 Figure 30.Comparison of shear specimen with different notch condition
The displacements at fracture of the circled shear specimen show prominent
decline compared with the mother shear specimen. For shear specimen, from
origin point to point A, it shows the shear deformation and from point A to
point B, the figure shows tensile deformation combined with shear damage.
Furthermore, beyond point B to the remote area, the damage model is tensile.
During the strain hardening stage of elongation of the specimen, the load-
displacement curves become linear and well defined.
The reduction in plasticity is due to the higher stress triaxiality, and the
obtained load-deflection curves are different when compared 2mm center
specimen with that 3mm center circled. For the smaller radius 2mm, the load-
displacement curves show that less softening phase exists.
The following figures shows the details of 2mm and 3mm center circled shear
specimens fracture phase.

37
 Figure 31.The tensile video for 2mm center shear specimen

Figure 32.The tensile video for 3mm center shear specimen

Compare above Figure 31 and Figure 32, 2mm diameters center circled shear
specimen is easily tends to fracture than that of 3mm.
3.3.3 Shear test with pre-crack condition

In this part, we consider shear test with different crack length and crack
position. The pre-crack was cut off by using a steel ruler and knife. The
following figure shows the crack cut direction. The Figure 33(a) shows the
both side crack specimen and Figure 33(b) shows the one side crack
specimen. For each crack condition, at least 3 specimens is prepared and
tested on the MTS machine. The result has been plotted in Figure 34.

38
 Figure 33.The shape of shear specimen with two different crack

Figure 34.The load-extension curve for 1mm crack on single side and both
sides of shear specimen

39
The Figure 34 shows that specimen with 1mm crack on both side can be
advantageously tends to fracture than the ones with 1mm crack on single side.
In the case of specimen with 1mm crack on both sides, lower force and
extension to fracture are observed. When the crack occurs on the both sides of
the specimen the fracture behavior tends to less ductile, and the specimen with
1mm single side crack tends to larger strain hardening.

Figure 35. The load-extension curve for 2mm crack on single side and both
sides of shear specimen
The Figure 35 shows the results of experiment on specimens with 2mm crack
on both side and single side. It is clearly shown that with crack on both sides
crack, the specimen tends to be easily fractured with lower force loaded. The
fracture behavior with 2mm crack on the both side of specimen tends to less
ductile fracture.
The distinction between the two specimens fracture is also explained in two
other ways by Griffith, firstly he considered that fracture produces a new
surface area and postulated that for fracture; to occur the increase in energy
required; to produce the new surface must be balanced by a decrease in
elastically spread energy [5].
Secondly, to explain the large discrepancy between the measured strength of
materials and those based on theoretical considerations, he states that the
elastically stored energy is not distributed uniformly throughout the specimen
but is concentrated in the neighborhood of small cracks and those results in

40
fracture because of the spreading of cracks that originate in pre-existing
flaws[5].
Based on the above Griffith theory, the difference between the two curves on
above two figures can be explained reasonably. If the specimen with fewer
cracks tends to fractured, more energy will be needed in the processing. The
following two figures show how crack size influences the result of HDPE
shear specimen fracture.

Figure 36.Comparison of shear specimen with different crack length on single

side and both sides.
From Figure 36, conclusion can be drawn that with larger pre-crack size, the
easier fracture happens on the HDPE shear specimen and that is suitable for
single and both sides specimen. The reason is explained by the Griffith’s
theory and he states that with smaller size pre-crack on the specimen, more
energy or load will be needed in the processing of fracture.
3.4 The SEM fracture observation
Microscopic studies were undertaken to get an insight of the crack surface of
HDPE material. For better observation in SEM, sample with suitable cross-
sectional area were cut off and painted by gold on the surface for better
electrical conductivity [21-22].

41
 Figure 37. The SEM experiment figure
It has been recognized that fracture surface topography reveals inherent
details of deformation and the energy that govern the HDPE fracture
processing. In this program, the fracture surfaces were studied both
qualitatively and quantitatively. In this part of research, scanning electron
microscopy was performed for examination of fractured surfaces of dog-bone
and shear failed specimen. Figure 37 shows the photo of a scanning electron
microscope used to study fracture surface morphologies of HDPE.
For thermoplastic materials HDPE, when put it under a high pressure, it will
produce a certain degree of plastic deformation, this has simulative
densification effect for the pre-crack HDPE material.

42
 Figure 38.SEM figure for fractured HDPE with 1mm pre-crack

Figure 39.SEM magnification figure for fractured HDPE with 1mm pre-crack
The SEM micrographs of a fractured surface of the1mm pre-crack HDPE are
shown in Figure 38. And Figure 38 (a), (b) and (c) show breakage, fractured
surface and pre-crack position respectively.
The Figure 38 (a) and (b) show the crack propagation direction. The
differences between the surface roughness features are rather striking.

43
A higher magnification view of the fracture surface morphology can be seen
in the Figure 38. The micrograph in Figure 38 (b) reveals fiber debonding and
pullout which results in crack front trapping. It can also be noticed that the
crack between micron-size fiber indicating crack strip. As micro-cracks leave
the strip positions, tail lines as well as step patterns in the direction of crack
propagation appear.
Based on the above SEM figures, the shape of fiber matrix has changed
completely, showing splitting and fibrillation and some fibrils surrounding
their original fiber are distributed in the fiber layer.
SEM photomicrographs of tensile fracture surfaces of HDPE specimen with
2mm crack is shown in the figures:

Figure 40.SEM figure of fractured HDPE with 2mm pre-crack

44
 Figure 41. SEM magnification figure for fractured HDPE with 2mm pre-
crack
As shown in the above figure, the micrograph for 2mm pre-crack HDPE
specimen failed specimen shows the fractured cross-section from different
direction.
The microscopic features of fractured surfaces are shown in Figure 40 for
HDPE material. The broken fiber in these micrographs indicates the direction
of crack propagation. Compared the 1mm pre-crack specimen with that of
2mm crack, larger deformation is applied on the 1mm pre-crack specimen and
this phenomenon is matching well with the curve in Figure 36.
The fractured surface of HDPE specimen with 2mm crack specimen is shown
in Figure 41 and reveals a relatively flat smooth and featureless surface
demonstrating its plastic nature with energy expenditure among all the cases.
Figure 41(b) illustrates the fracture surface of 2mm crack dog-bone specimen
case showing in elastically deformed fiber with crack branches on various
fracture planes.
In Figure 41 reveal smaller surface roughness and ruggedness for the 2mm
pre-crack fractured specimens when compared to that with 1mm crack
counterparts and accounts for the higher crack initiation toughness in the
former cases. The reason is with larger crack length, less energy will release
as increasing the load in the processing of experiment.
From the above discussion, the crack propagation direction is clearly defined
and it growths along the fiber of material in the center of the specimen and the
energy release is closely depends on the crack length.
For more study about the shear fractured specimen, SEM of fractured shear
specimen can be further studied in the future study.

45
4 Chapter 4 .The numerical simulation
4.1 Introduction
Computer aid computation has had an enormous influence in all branches
engineering and fracture mechanics is no exception. Numerical modeling has
become an indispensable tool in fracture analysis since analytical method
cannot solve the most of practical problem. In the fracture field, stress-
intensity solution for literally hundreds of configuration have been published,
the majority of which were come from the numerical models [5].
The J integral and crack-tip-opening displacements (CTOD) are commonly
solved by elastic-plastic method and some special problem, ductile crack
growth and dynamic fracture problem is also solved by the advanced
numerical analysis. As the development of computer hardware, more efficient
numerical algorithms have greatly reduced solution times in fracture problem,
such as K and J solutions are easily obtained from the finite element models
with rough meshes.
In this chapter, numerical method theory and numerical analysis in ABAQUS
will be introduced. In ABAQUS analysis section, the shear specimen with
different crack condition will be mainly focused on. In this chapter, several
complicated mathematically derivation is contained and this information is
unavoidable when explaining the basis of the common numerical techniques.
In the processing of fracture analysis, the distribution of stresses and strains in
a body that is subjected to external loads or displacement is often obtained. A
variety of numerical techniques have been used in solving solid mechanics,
such as finite element method [23], finite difference method [24]. The most
popular used method is finite element method in recent years, although finite
difference method is useful in limited cases.
4.2 The finite element method
In the finite element method, the structure body of interest can be subdivided
into many discrete shapes which are called elements. The stiffness finite
element method is usually used in stress analysis problems. This approach is
outlined below for the two-dimensional case.

46
The following figure shows an isoperimetric continuum element for two-
dimensional plane stress or plane strain problems, together with local and
global coordinate axes.
The local coordinates varies from -1 to +1 over the element area which can be
seen from the figure.

Figure 42 .The local coordinates varies from -1 to +1 over the element

area[23]

Consider one of points on the element at (Ɍǡ Ʉ), the global coordinates of this
point can be written as:
š ൌ σ୬୧ୀଵ ୧ ሺɌǡ Ʉሻ (4.2.1)

› ൌ σ୬୧ୀଵ ୧ ሺɌǡ Ʉሻ›୧ (4.2.2)

Where n is the number of nodes in the elements and Ni is the shape functions
corresponding to the node i, whose coordinates are(xi,yi) in the global system
and (Ɍ୧ ǡ Ʉ୧ ) in the parametric system.
The displacements within an element can be written as:
— ൌ σ୬୧ୀଵ ୧ ሺɌǡ Ʉሻ—୧ 



 

47
 ˜ ൌ σ୬୧ୀଵ ୧ ሺɌǡ Ʉሻ˜୧

Where (ui,vi) are the nodal displacement for x and y directions respectively.
The strain matrix at (x,y) can be written as:
ɂ୶
—୧
ɂ
൝ ୷ ൡ ൌ ሾሿ ቄ˜ ቅ (4.2.3)
ɀ୶୷ ୧

Where
μ୧
‫ۓ‬ Ͳ ۗ
ۖ μš ۖ
ۖ μ ۖ୧
ሾሿ ൌ Ͳ
‫۔‬ μ› ۘ
ۖμ୧
ۖ μ୧ ۖ
ۖ
‫ ە‬μ› μš ۙ

And
ப୒౟ ப୒౟
ப୶ பஞ
ቐப୒౟ቑ ൌ ሾ ሿିଵ ቐப୒ ቑ ˄4.2.4˅
౟
ப୷ ப஗

Where J is the Jacobian matrix and can be written as:

μš μ›
‫ۍ‬ ‫ې‬
‫ێ‬ μɌ μɌ ‫ۑ‬
ሾ ሿ ൌ ‫ێ‬
μš μ›‫ۑ‬
‫ێ‬ ‫ۑ‬
‫ۏ‬μɄ μɌ‫ے‬

The stress matrix can be written as:

ሼɐሽ ൌ ሾሿሼɂሽ ˄4.2.5˅

[D] Is the stress-strain matrix.

48
The stresses and strains can be calculated at several Gauss points or
integration points within each element. The global force, displacement, and
stiffness matrices are related and can be written as:
ሾሿሾ—ሿ ൌ ሾ ሿ ˄4.2.6˅

[k] is the elemental stiffness matrix and can be written as:

ଵ
ଵ
ሾሿ ൌ න න ሾሿ୘ ሾሿሾሿ †‡–ȁ ȁ†Ɍ†Ʉ
ିଵ
ିଵ

4.3 ABAQUS Simulation

In this section, ABAQUS 6.10 was chosen as the finite element software to
perform the simulation. Since CAE does not have any fixed units for the input
parameters, it is convenient to set the units by the user. In this study, the
following units are used in the software.
Table 4.1Set of consistent units to be used for the simulation in ABAQUS

Parameter Name Units

Density g/mm3
Force N
Length mm
Stress MPa
Time ms

All material properties and parameters such as elastic, plastic, young’s

modulus and damage values have been obtained from the tensile test of dog-
bone shape HDPE specimen.
The properties have been used to model the shear test under different crack
condition and 3D solid part model is chosen to analysis the specimen. The
following key points have been used in the model:
(1) The material property is considered as isotropic and homogeneous
(2) In the part section, 3D is chosen as the modeling space; deformable and
solid is chosen in the corresponding section.
(3) Young’s modulus, plasticity and damage parameters should be obtained
from the true stress-strain curve.

49
(4) Dynamic, Explicit is chosen as the step procedure type. The units are
consistency in the processing of simulation.
4.4 The material properties for ABAQUS simulation
Density: The volumetric mass density, of a substance is its mass per unit
volume and it can be calculated as [5]:
ƒ••
‡•‹–› ൌ
‘Ž—‡

Young’s Modulus: Young's modulus, E, can be calculated by dividing

the tensile stress by the extensional strain in the elastic (initial, linear) portion
of the stress–strain curve [5]:
‡•‹Ž‡•–”‡••
‘—‰ ᇱ •‘†—Ž—• ൌ 
š–‡•‹‘ƒŽ•–”ƒ‹

Figure 43. The true strain-stress curve[18]

In the ABAQUS simulation model, only values form true stress-strain curve
can be accepted. Some modification for engineering stress-strain curve should
be done to match the numerical simulation requirement.
True stress is the stress determined by the instantaneous load acting on the
instantaneous cross-sectional area true stress is related to engineering stress
[25].Assuming material volume remains constant

50
 ୔ ୔ ୅బ ୔ ୅బ
ɐ୘ ൌ ൌ ‫כ‬ ൌ ‫כ‬ (4.4.1)
୅ ୅ ୅బ ୅బ ୅

୅బ ୐ ஔା୐బ ஔ
ൌ ൌ ൌ ൅ ͳ ൌ ሺͳ ൅ ɂሻ (4.4.2)
୅ ୐బ ୐బ ୐బ

Based on the above equation, the true stress-strain equation can be calculated
as:
୔
ɐ୘ ൌ ሺͳ ൅ ɂሻ ൌ ɐሺͳ ൅ ɂሻ (4.4.3)
୅బ

For the true strain equation:

† 
ɂ୘ ൌ න ൌ Žሺ ሻ
 ଴
୐బ ାο୐
ɂ୘ ൌ Ž ቀ ቁ ൌ Žሺͳ ൅ ɂሻ (4.4.4)
୐బ

Poisson's ratio: is the negative ratio of transverse to axial strain. When a

material is compressed in one direction, it usually tends to expand in the other
two directions perpendicular or parallel to the direction of flow [5].
Plasticity: in physics and engineering, plasticity is the propensity of a
material to undergo permanent deformation under load when compressed. In
ABAQUS model the plasticity value should be obtained from the true stress-
strain curve and followed the formula:
஢౐౫౨౛
ɂ୔୪ୟୱ୲୧ୡ୘୳୰ୣ ൌ ɂ୘ െ ሺ ሻ (4.4.5)
୉

Strain rate: is defined as the change in strain over the change in time
Fracture at strain: The logarithmic strain where the specimen breaks and it
can be obtained from the true stress-strain curve.
Damage Evolution: The evolution of damage variable with the relative
plastic displacement can be specified in tabular, linear, or exponential form
and linear form is chosen in this study [26].
In linear form, the damage variable increases according to the equation 4.4.6
ሶ
౦ౢ ሶ
౦ౢ
୐க തതത
୳
†ሶ ൌ ౦ౢ ൌ ౜౦ౢ (4.4.6)
തതത౜
୳ തതത౜
୳

51
The plastic displacement at failure can be computed as:
ଶୋ౜
—ഥ୤ ୮୪ ൌ (4.4.7)
஢౯బ

Where
୤ is the fracture energy per unit; ɐ୷଴ is the fracture stress; L is the
characteristic length of the element.
Analysis time: The analysis time plays an important role in the ABAQUS
simulation, so the time should be selected properly. In this study, 10ms is
selected as the time in the ABAQUS model.
Mesh control: The mesh quality plays an important role in the accuracy
output values, high quality mesh is needed for obtaining better numerical
result. In ABAQUS 6.12 academic teaching license version, the nodes are
restricted to 20000. So mesh quality should be controlled and the following
pictures shows the finally mesh for dog-bone shape specimen and shear
specimen:

Figure 44. Final Meshing used for tensile specimen

52
 Figure 45 .Final Meshing used for shear specimen with center circled

4.5 The Mechanical property of material and result

4.5.1 Dog-bone shape specimen tensile simulation

Elastic value: the value of Young’s modulus is chosen as 534Mpa and the
poison’s ratio is 0.4
The mass density: In the ABAQUS simulation, 0.0009g/mm3 is chosen as the
mass density
Plastic value: Plasticity value of HDPE was obtained from the pure dog-bone
specimen experimental result. The plasticity value is shown in Appendix B.
Table 4.2 Damage Initiation:

Fracture strain Stress Triaxiality Strain Rate

2.3 -5 0
2.3 5 0

Damage evolution value for HDPE dog-bone shape specimen

53
 Table 4.3.Damage evolution:

Displacement at
failure [mm]
0.1
4.5.2 The numerical result of dog-bone shape specimen

The Figure 46 shows the force-displacement of physical test and numerical

calculation result of dog-bone specimen with 1mm center circled.

Figure 46 .Force-displacement of physical test and numerical calculation

result of dog-bone specimen with 1mm center circled
The plot of HDPE dog-bone specimen with 1mm center circled is obtained
(Figure 47) and it shows the stress concentration inɐଵଵ ,ɐଵଶ , ɐଶଶ .

54
Figure 47.Stress components for dog-bone specimen with 1mm center circled
It can be shown in Figure 47 that the maximum stress mainly concentrated
around the center circled, and the fracture initiates from the center of the
specimen and propagates towards the other directions of the surface.
The Figure 48 shows the force-displacement of physical test and numerical
calculation result of dog-bone specimen with 2mm center circled.

55
Figure 48 .The force-displacement of physical test and numerical calculation
result of dog-bone specimen with 2mm center circled.
The plot of HDPE dog-bone specimen with 2mm center circled is obtained
(Figure 49) and it shows the stress concentration situation.

Figure 49. Stress components for dog-bone specimen with 2mm center circled

56
It can be shown from the Figure 49 that the stress is mainly concentrated
around the 2mm center circled and the fracture initiates and propagates form
the cross direction of the dog-bone shape specimen. Similar crack initiates
and propagates direction condition is obtained from Figure 47 and Figure 49
4.5.3 The numerical result of shear specimen

Table 4.4 Damage Initiation:

Fracture strain Stress Triaxiality Strain Rate

1.9 5 0

Table 4.5 Damage evolution value for HDPE dog-bone shape specimen

Displacement at
failure [mm]
0.1

The Figure 50 shows the force-displacement of physical test and numerical

calculation result of shear specimen with 2mm center circled.

Figure 50. Force-displacement of physical test and numerical calculation

result of shear specimen with 2mm center circled.

57
The magnified plot of HDPE shear specimen with 2mm center circled is
obtained (Figure 51) and it shows the stress concentration inɐଵଵ ,ɐଵଶ , ɐଶଶ .

Figure 51ˊStress components for shear specimen with 2mm center circled
The Figure 52 shows the force-displacement of physical test and numerical
calculation result of shear specimen with 2mm center circled.

Figure 52. Force-displacement of physical test and numerical calculation

result of shear specimen with 3mm center circled.

58
The magnified plot of HDPE shear specimen with 2mm center circled is
obtained (Figure 53) and it shows the stress concentration inɐଵଵ ,ɐଵଶ , ɐଶଶ .

Figure 53. Stress components for dog-bone specimen with 3mm center circled
It is shown in Figure 52 and Figure 50 that the simulation result does not
match the experimental result, and the global deformation is not stable since
there is a significant drop from the maximum force. The reason can be
classified into following aspects: Firstly, the fracture strain and damage
evolution value which used in simulation is not accuracy enough, so it is very
difficult to obtain the numerical results which matches experimental curve.
Secondly, in the simulation model, the shear behavior of specimen is not
considered which resulting little shear damage in simulation results. Finally,
in the simulation processing, the stress which concentrated along the machine
direction does not powerful enough to make the specimen tends to shear
damage significantly and that is the reason why curve are dropping form the
peak point in Figure 52.
The Figure 54 shows the force-displacement of physical test and numerical
calculation result of shear specimen with 2mm single side crack.

59
 Figure 54. force-displacement of physical test and numerical calculation
result of shear specimen with 2mm single side crack
The plot of HDPE shear specimen with 2mm single side crack is obtained
(Figure 55) and it shows the stress concentration inɐଵଵ ,ɐଵଶ , ɐଶଶ .

Figure 55 .Stress components for shear specimen with 2mm single side crack
The Figure 56 shows the force-displacement of physical test and numerical
calculation result of shear specimen with 2mm both side cracks.

60
Figure 56 .The force-displacement of physical test and numerical calculation
result of shear specimen with 2mm both side cracks.

The plot of HDPE shear specimen with 2mm both crack is obtained (Figure
57) and it shows the stress concentration inɐଵଵ ,ɐଵଶ , ɐଶଶ .

61
Figure 57. Stress components for shear specimen with 2mm both side cracks
The Figure 58 shows the force-displacement of physical test and numerical
calculation result of shear specimen with 1mm single side crack.

62
 Figure 58.the force-displacement of physical test and numerical calculation
result of shear specimen with 1mm single side crack.
The plot of HDPE shear specimen with 1mm single crack is obtained (Figure
59) and it shows the stress concentration inɐଵଵ ,ɐଵଶ , ɐଶଶ .

Figure 59.Stress components for shear specimen with 1mm single crack

63
The Figure 60 shows the force-displacement of physical test and numerical
calculation result of shear specimen with 1mm both side cracks.

Figure 60.Force-displacement of physical test and numerical calculation

result of shear specimen with 1mm both side cracks.
The plot of HDPE shear specimen with 1mm both side cracks is obtained
(Figure 61) and it shows the stress concentration inɐଵଵ ,ɐଵଶ , ɐଶଶ .

64
Figure 61 .Stress components for shear specimen with 1mm both side cracks

4.6 Fracture behavior predicting

The HDPE shear specimen behavior with different crack dimensions can be
predicted by using ABAQUS. The prediction is useful in bottle cap opening
problem which will be shown in next chapter.
For better understanding the crack propagation in HDPE shear specimen,
different shear specimens were modeled in ABAQUS.

Figure 62. HDPE shear specimen with different pre-crack dimensions

simulation result

65
In the Figure 62, the results for different pre-crack dimensions are clearly seen,
and the shear specimen with 3mm pre-crack on both sides are easily tends to
fracture compared with others. A maximum load was observed at the yield
point. From the figure, conclusion can be drawn that with larger pre-crack, the
specimens are more easier tends to fracture. Tensile phenomenon will happen
before fracture from the simulation procedure for shear specimen wirh pre-
crack on single side. It is interesting that the shear specimen with 3mm pre-
crack on both sides will be fractured and followed shear fracture model.
In Figure 62, the corresponding fracture toughness value varies tendency can
also be obtained from ABAQUS simulation result. The table shows the value
for each case:
Table 4.6.1 the fracture toughness for each case

3mm pre-crack on 3mm pre-crack 4mm precrack on 5mm precrack

both sides on single side single side on single side

KIC 0.543 Mpaξ݉ 0.89Mpaξ݉ 0.784Mpaξ݉ 0.7088Mpa

ξ݉

The higher fracture toughness value that the case has , the larger CTOD and
engergy relaese rate value that the case will be obtained since the fracture
toughness value shows the same varies tendency with CTOD and engergy
release rate value.
The above table shows the 3mm pre-crack on both sides has less fracture
toughness and with the crack size increasing, the larger facture toughness
value will be obtained. This reault can also be proven in Figure 62.

66
5 Chapter5. Plastic bottle cap calculation
5.1 Introduction
Plastic bottle cap is popular used in packing industry; it can be used to seal the
top opening of a bottle. Caps for plastic bottles are often made of a different
type of plastic than the bottle.
There are many types bottle caps used in industry, for example, the crown
cork, screw cap and sports cap. In this part, the screw cap is mainly focused
on and the figure is shown in the Figure 63.

Figure 63 .The screws cap

Screws on closures are the most common bottle caps. It is easy to apply by
kinds of automated equipment. The application torque of closures must be
controlled properly and applied tight enough to perform good seal. Although
the closure should be resisted back off, it should not be applied too tightly to
remove or open the closure for people. So it is interesting to know the force to
open a bottle cap and in this part, the maximum force and torque for opening
the HDPE made screw cap will be calculated.

67
The screws are popular used for plastic caps and the screws type should be
considered and the nonrectangular thread is chosen as the right type in this
part. Material parameter used in this bottle cap is HDPE which is the material
applied in the project. The following Figure 64 is the typical nonrectangular
thread screws cap which made of HDPE.

Figure 64.The typical HDPE non rectangular thread screws cap

5.2 Nonrectangular thread screws cap

Nonrectangular thread is kind of screws which screw tooth angle is zero.
When the relative rotation between the threaded screw and nut can move at
the same time and this phenomenon can be regards as the wedge body slides
along the wedge cant.

68
 Figure 65.Nonrectangular thread screws[27]

Figure 66.Magnified nonrectangular thread screws[27]

When the plane normal reaction N = Q; When the plane friction Ff = fN = fQ;
Wedge surface normal reaction N / = Q/…‘•Ⱦ; Wedge surface friction Ff/=f N/
=fQ/cosȕ.
f/ =f/cosȕ can be regard as the equivalent friction coefficient. Ff/=f/Q; The
equivalent friction Angle, represented byɏ୴ .
The required horizontal force which makes the body move can be calculated
as the following equations:
The force for ascending period:
୲ ൌ –ƒሺȲ ൅ ɏ୴ ሻ (5.2.1)

The torque for ascending period:

୊౪ ୢమ ୕୲ୟ୬ሺஏା஡౬ ሻୢమ
ൌ ൌ (5.2.2)
ଶ ଶ

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The force for descending period:
୲ ൌ –ƒሺȲ െ ɏ୴ ሻ (5.2.3)

The torque for descending period:

୊౪ ୢమ ୕୲ୟ୬ሺஏି஡౬ ሻୢమ
ൌ ൌ (5.2.4)
ଶ ଶ

Where F is the force, f is the friction coefficient,ɏ୴ is the friction angle, N is

the normal reaction, fN is the friction force.
All equations are treated as the input in Matlab and Appendix C shows the
relative code.
5.3 Parameters used in calculation
The table shows the parameters which will be used in calculating.
Table 5.1 Parameter values

Ȍ ȕ f d2 Joints width Joints thick

10o 30o 0.2 0.028m 0.1mm 0.2mm

Figure 67 shows the shape of bottle cap which applied for calculating and
Figure 68 shows the screw shape which is mentioned in section 5.2.

Figure 67 .The bottle cap model

70
 Figure 68.The bottle cap model

5.4 The result and discussion

The results are summarized in Figure 69 and Figure 70 and Figure 71. The
calculation procedure and MATLAB code is shown in the appendix C.

Figure 69.The true force-strain curve used for calculation

It is interesting to note that the force values increase very fast in small strain
value scope. The true force is the force applied on the bottle cap and it is

71
corresponding to the Q value in equation 5.2.1 and 5.2.2 and it will be used to
calculate the horizontal force and torque for opening the bottle cap.

Figure 70. The torque-strain curve

Figure 71.The horizontal force-true strain curve

From Figure 70 and Figure 71, the torque value and horizontal force for
opening the bottle cap was calculated for different circled joint condition.

72
The table 5.2 shows the results for different circled joints.

Table 5.2 the result for different circled joints

Force[N] Torque[N*m]

Non-circled joint 37.58 0.5261

0.02mm circled 19.09 0.2673

joint
0.04mm circled 17.34 0.2427
joint

From the above table, the circle notched joints shows less force and torque to
open the bottle cap when compared to that non-circled joint. A conclusion can
be drawn that circled joints plays an important role in opening bottle cap and
more efforts can be saved in this processing.
The Figure 72 and Figure 73 shows the calculation result from Inventor
software, it shows the stress distribution along the bottle cap and displacement
values of 0.04mm circled joints respectively.

Figure 72.The stress distribution on cap

From the above figure, the stress distribution is obtained and it mainly
concentrated along the joints section which is reasonable in practice.

73
 Figure 73.The displacement of cap
From the above figure, the maximum displacement value is 0.1027mm which
is enough to open the bottle.
For better understand how joints shape influence the bottle cap opening
question, several calculation has been done based on the experimental and
simulation data in chapter 4. In order to obtain better calculation result, the
shear shape specimen dimension is reduced by 100 times to match the shear
shape joints which are used in this section. By inputting the data into Matlab
codes, the corresponding result can be obtained.
Figure 74, Figure 75, Figure 76 show the result of shear specimen with
different center circled.

74
Figure 74 .The true force-strain curve used for calculation

Figure 75. The torque-strain curve

75
 Figure 76.The horizontal force-true strain curve

From Figure 75 and Figure 76, the torque value and horizontal force for
opening the bottle cap were calculated for different circled joint condition.
The table 5.3 shows the results for shear shape joints with different circled
dimensions.
Table5.3. Results for shear shape joints with different circled dimensions.
Force[N] Torque[N*m]

Non-circled joint 7.268 0.10180

0.02mm circled 6.091 0.08528

joint
0.03mm circled 5.848 0.08187
joint

With the shear shape joints applied, the force and torque to open the HDPE
bottle cap has been decreased greatly. The center circle dimension also has
influence on force and torque values and with larger circle dimension, less
force and torque will be used in bottle cap opening.
In order to study how pre-crack on joints influence the bottle cap opening
processing, several calculations have been done and summarized into the table.

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 Table 5.4 Results for shear shape joints with different pre-crack condition
Force[N] Torque[N*m]

0.01mm single side pre- 7.901 0.11

crack on shear shape
joints
0.01mm both sides pre- 8.279 0.116
crack on shear shape
joints
0.02mm single side pre- 8.126 0.1138
crack on shear shape
joints
0.02 both sides pre-crack 8.384 0.1170
on shear shape joints

0.03mm single side pre- 0.04066 0.0005

crack on shear shape
joints

From the above table, we can clearly note that the joints are more easily tends
to break with 0.03mm single side pre-crack on shear shape joints. From the
calculation, conclusion can be drawn that the shear shape specimen joints are
easier tends to be broken than the rectangular shape ones. With pre-crack or
circled on the shear shape specimen, the bottle cap has great advantages to be
opened.

5.5 Bottle design suggestion

Combined with numerical result, some suggestions can be given to the bottle
manufactures for opening bottle cap easier.
1. In the joints section, some center circled can be done for opening bottle
cap easier. In this case, we suggest the center circle dimension as 0.04mm.
2. For the joints section shape, different shapes can be used. Shear specimen
shape is better than that with rectangular ones since shear specimen shape
are easily tends to fracture.
3. The shear shape specimen with 0.03mm circled or 0.03mm single pre-
crack can be used in the bottle joints for saving effort to open the bottle
cap.

77
4. Try to increase the thread angle of the bottle and cap.

78
6 Chapter 6. Conclusion and future work
6.1 Conclusions of HDPE experiment and simulation
In this project, tensile and shear experiments were applied for studying the
HDPE’s mechanical and fracture properties. The theoretical background of
the HDPE fracture is put forward to incorporate the pre-crack sensitivity and
the circled diameter dependence of ductile fracture.
The project is integrated fracture study with application calculation, a series
of numerical simulation based on a pure HDPE dog-bone shape test were
done to obtain the mechanical properties constants. The initial material
constants were obtained directly from experimental data and error correction
procedure is then followed to get the true stress-strain curve which used to
fine and extrude elastic-plastic material data. The obtained fracture
characterizing material constants are then used to calculate the fracture pattern
of the upsetting tests in ABAQUS.
An important finding is that the for dog-bone shape specimen, the one with
circled tends to fracture easier than that without any circled and the ones with
bigger diameter is easier to break than that with smaller circled diameter.
Crack propagation or fracture growth rate is faster for larger circled or crack
diameter. SEM and video observation are applied in the experimental
processing and it shows the macro and details of crack initiation and
propagate, fracture surface are observed by SEM for better understanding the
fracture type. The shear specimen with different pre-crack has been applied
by ABAQUS and the corresponding stress intensity factor has been calculated.
Form the fracture toughness varies tendency, the specimen with larger pre-
crack will obtain smaller fracture toughness which shows tends to fracture
easier.
A combined application program is further carried out to calculate how much
force and torque to open the HDPE bottle cap based on the experimental and
simulation data. In this section, several circled joints have been applied and
the results shows the joints with larger dimension circled will be broken easier
than that with smaller. For better study how different joints shape will
influence the open bottle question, the shear shape joints with pre-crack and
circled have been calculated. With 0.03mm single pre-crack shear joints, the

79
bottle can be opened easier than the others in this study. The relative
suggestions are given based on the calculation.

6.2 Future work

1. In application part, more accurate numerical test should be operated in the
future.
2. Different specimen dimension will be considered in the processing of
fracture, for example, different thickness.
3. Better method should be applied in ABAQUS numerical simulation to
obtain better result to match tensile experimental result. For example, the
influence of twist should be considered in the ABAQUS modulus.
4. In the shear specimen test, the shear combined with tensile model can be
investigated deeply with experiments and numerical method.
5. In the application part, the numerical simulation can be applied to study
the force and torque which are needed to open the bottle cap. For
obtaining the most optimized pre-crack or circle dimension of the joints,
more simulation can be done in ABAQUS.

80
References
1. Hsuan, Y.G., Zhang, J., (2005), “Stress crack resistance of corrugated
HDPE pipes in different test environments”, Transportation Research
Board (TRB) Annual Meeting, Washington D.C
2. I. Barsoum and J. Faleskog. Rupture mechanisms in combined tension and
shear experiments. International Journal of Solids and Structures,
44(6):1768–1786, 2007.
3. http://www.kazuli.com/UW/4A/ME534/lexan2.htm.
4. Liang Xue.(2007) Ductile Fracture Modelling-Theory, Experimental
Investigation and Numerical verification. Massachusetts Institute of
Technology.
5. Peacock A. J. (2000) Handbook of polyethylene: Structures, properties
and applications. New York, Marcel Dekker Inc.
6. http://en.wikipedia.org .
7. Callister W. D. (2007) Materials science and engineering: An
introduction. New Jersey, John Wiley and Sons.
8. J. M. Hodgkinson (2000), Mechanical Testing of Advanced Fibre
Composites, Cambridge: Woodhead Publishing, Ltd., p. 132–133.
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fracture and toughening mechanisms. Eng Fract Mech 2006;73(16):2375–
98.
10. Kfouri, A.P., 1996. Crack extension under mixed-mode loading in an
anisotropic mode-asymmetric material in respect of resistance to fracture.
Fatigue & Fracture of Engineering Materials & Structures 19 (1), 27–38.
11. J.F. Yau, S.S. Wang, H.T. Corten A mixed-mode crack analysis of
isotropic solids using conservation laws of elasticity ASME Journal of
Applied Mechanics, 47 (1980), pp. 335̢341.
12. Wang, B.L., Mai, Y.W., Noda, N., 2002. Fracture mechanics analysis
model for functionally graded materials with arbitrarily distributed
properties. Int. J. Fract. 116, 161–177.
13. R. Chaouadi, P. D. Meester, and W. Vandermeulen. Damage work as
ductile fracture criterion. International Journal of Fracture, 66(2), 1994.

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14. Xie J, Wu X, Hong Y. Shear bands at the fatigue crack tip of
nanocrystalline nickel. Scripta Mater 2007;57:5–8.
15. Ovid’ko IA. Review on the fracture processes in nanocrystalline materials.
J Mater Sci 2007;42:1694–708.
16. Dalla Torre F, Van Swygenhoven H, Victoria M. Nanocrystalline
electrodeposited Ni: microstructure and tensile properties. Acta
Mater2002;50:3957–70.
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morphology and the mechanical properties of ternary PE/PA6/GF
composites during annealing. Polymer 2007;48(21):6294–303.
18. http://www.fea-optimization.com/ETBX/strainlife_help.html.
19. Y. Bao and T. Wierzbicki. On fracture locus in the equivalent strain and
stress triaxiality space. International Journal of Mechanical Sciences,
46:81–98, 2004.
20. http://faculty.uscupstate.edu/llever/Polymer%20Resources/Mechanical.ht
m.
21. Waddoups, M.E., Eisenmann, J.R., and Kaminski, B.E., “Macroscopic
Fracture Mechanics of Advanced Composite Materials.” Journal of
Composite Materials,Vol. 5, 1971, pp. 446–454.
22. Zhang X, Jia C. The microstructural characteristics of the deformed
nanocrystalline cobalt. Mater Sci Eng A 2006;418:77–80.
23. Lapidus, L. and Pinder, G.F., Numerical Solution of P artial Differential
Equations in Science and Engineering. John Wiley & Sons, New York,
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24. Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method.(4th ed.,)
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27. http://wenku.baidu.com.

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Appendix A Experiment result

The HDPE tensile test for analysising material property

83
The shear specimen with 2mm centred circled test result

The shear specimen with 3mm centred circled test result

84
The shear specimen with 2mm both side pre-crack test result

The shear specimen with 1mm both side pre-crack test result

85
The shear specimen with 1mm single side pre-crack result

The shear specimen with 2mm single side pre-crack result

86
Appendix B ABAQUS modelling and
relative plasticity data
Modelling tensile test in ABAQUS

1. Part

3D –deformable, solid

2. Material behaviours

Elastic: The mechanical properties like young’s modulus, density and

Poisson’s ration have been mentioned in the numerical chapter.

Ductile damage:

Damage Evolution:

87
Plastic:

3. Section

Section-solid-homogenous –assign section

4.Step and output

88
Field output:

89
History output:

5. Interaction

90
6.Boundary condition

91
7.Mesh

92
Seed-edge by number :

8. Amplitude for boundary condition

93
Plasticity Data:
Yield stress Plastic strian
16.4772 0
30.9683 0.2262
33.2119 0.4106
33.9368 0.5662
31.4462 0.7008
36.8256 0.8194
41.5143 0.9255
46.2240 1.0214
50.9773 1.1088
55.7194 1.2637
60.2808 1.3330
64.9015 1.3978
69.4759 1.4586
73.8675 1.5160
78.0546 1.5702
82.0194 1.6217
86.4748 1.6706
90.5749 1.7173
94.4370 1.7618
98.8903 1.8045
102.8762 1.8454
105.7932 1.8847
108.2176 1.9226
106.9642 1.9590
110.6460 1.9942
122.4158 2.0282
129.1826 2.0610
134.9865 2.0928
139.5592 2.1236
143.5579 2.1536
147.0201 2.1826
149.1180 2.1934
153.3560 2.2108
155.8607 2.2383
164.0499 2.2650

Appendix C Matlab code used in chapter 5

clear all;
close all;
clc;

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L=50e-3;
data=load ('tensile.txt');
Force=data(:,1);
extension=data(:,2)*1e-3;
stress=Force/(10e-3*2e-3);
strain=extension/L;
for i=1:length(stress)
true_stress(i)=stress(i)*(L+extension(i))/L;
true_strain(i)=log(1+strain(i));
end
zeta=10;
beta=30;
f=0.2;
true_section=0.1e-3*0.2e-3./true_strain;
true_force=true_stress.*true_section;
max(true_force)
plot(true_strain,true_force,'linewidth',2)
grid on
title('The ture force-strain curve used for calculation')
xlabel('True strain ')
ylabel('Ture Force[N]')
xlim([0 1])
Ft=tand(zeta+atand(f/cos(beta))).*true_force;
T=tand(zeta+atand(f/cos(beta)))*28e-3/2.*true_force;
figure
plot(true_strain,Ft,'linewidth',2)
grid on
title('The force-strain curve')
xlabel('True strain ')
ylabel('Horizontal force applied in opening cap[N]')
figure
plot(true_strain,T,'linewidth',2)
title('The Torque-strain curve')
xlabel('True strain ')
ylabel('Torque applied in opening cap[N*m]')
grid on
max(Ft)
max(T)
grid on

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96
School of Engineering, Department of Mechanical Engineering Telephone: +46 455-38 50 00
Blekinge Institute of Technology E-mail: info@bth.se
SE-371 79 Karlskrona, SWEDEN