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APPLICATION OF FRACTURE MECHANICS TO THE
IMPACT BEHAVIOUR OF POLYMERS
by
EVANGELIA PLATI
B.Sc.(Eng.), M.Sc., D.I.C.
A thesis submitted for the degree of
Doctor of Philosophy .
of the
University of London
Department of Mechanical Engineering
July 1975 Imperial College of Science and Technology London SW7 2BX
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ABSTRACT
In conventional types of impact tests the impact strength is reported
in terms of the energy to fracture W divided by the ligament area A. It
is well known that such an analysis of the data is not satisfactory,.
mainly due to the fadt that w/A has a strong geometrical dependence.
The research work described in this thesis dealt with the
examination of these geometrical effects for polymers in the Charpy and
Izod loading situations, and, by employing the fracture mechanics concepts,,
the critical strain energy release rate, Gc, was deduced directly from
the energy measurements.
The success of this approach to the field of impact testing has been
clearly indicated throughout the thesis, since the same Gc value was
obtained for both Charpy and Izod tests.
The effect of temperature on the impact behaviour of polymers was
also examined. The concept of the plane strain fracture toughness Gcl
and the plane stress fracture toughness Gc2 with yield stress changes
gave a good picture of variations with temperature and specimen thickness.
Finally, the analysis of blunt notch data showed that the fracture
mechanics idea of a plastic zone provided a method of describing blunt
notch impact data in terms of the sharp notch result Gc and the plane
strain elastic energy gyp?.
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ACKNOWLEDGEMENTS
The author is grateful for the encouragement and invaluable help
received from her supervisor, Professor J.G. Williams, during the course
of this study.
The generous financial support of BP Chemicals (UK) Limited for the
full duration of this study is gratefully acknowledged.
The author also wishes to thank Mr L.H. Coutts for his valuable
assistance on the technical aspects of the experimental work.
For their assistance and advice throughout this project, the author
expresses her gratitude to Dr G.P. Marshall, Mr P.D. Ewing and
Mr M.W. Birch. -
Special thanks are also given to Miss E.A. Quin for accomplishing
the considerable task of typing the manuscript.
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To my beloved parents
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CONTENTS
Page
Abstract 2
Acknowledgements 3
Contents 5
Notation 11
Abbreviations 14
Introduction 15
CHAPTER 1: LITERATURE SURVEY 17
1.1 HISTORICAL INTRODUCTION TO IMPACT TESTING 17
1\,, 1.2 IMPACT TESTING OF PLASTICS 18
1.3 SPECIFIC IMPACT TESTS 19
1.3.1 Limiting Energy Impact Testing Methods 20
1.3.2 Excess Energy Impact Testing Methods 20
1.4 TENSILE IMPACT TEST 22
1.5 CHARPY AND HOD TESTS 22
1.5.1 The Effect of Notch Tip Radius on the Impact
Strength 24
1.5.2 Notch Stress Distribution for Charpy and
Izod Tests 25
1.5.3 Effect of Clamping Pressure for the Izod
Test 26
1.6 BRITTLE AND DUCTILE IMPACT FAILURES 27
1.7 IMPACT STRENGTH - ENERGY TO FRACTURE 28
1.7.1 Energy to Initiate and to Propagate Fracture 28
1.7.2 Energy Lost in Plastic Deformation 31
1.7.3 Kinetic Energy of the Broken Half 32
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1.7.4 Energy Lost in the Apparatus 35
1.8 EFFECTS OF TEMPERATURE ON IMPACT STRENGTH 36
1.9 THERMAL STABILITY - MECHANICAL LOSSES OF POLYMERS 38
1.9.1 Dynamic Mechanical Losses and Impact Strength
of Polymers 39
1.10 FRACTURE MECHANICS APPROACH TO IMPACT 43
1.10:1 The Griffith Approach 44
1.10.2 Strain Energy Release Rate 46
1.10.3 Stress Intensity Approach 46
1.10.4 The Relationship Between Fracture Toughness
and Absorbed Energy for the Charpy Impact
Test 49
1.10.5 Plastic Zone Size 54
1.10.6 Fracture Toughness and Specimen Thickness 55
1.11 INSTRUMENTED IMPACT 56
1.11.1 The Fracture Mechanics Approach to the
Instrumented Impact Test 57
CHAPTER 2: CALIBRATION FACTORS (I) 59
2.1 INTRODUCTION 59
2.2 COMPUTATION OF THE CALIBRATION FACTOR 4 FROM THE Y
POLYNOMIAL FOR THE CHARPY TEST 60
2.3 THE FACTOR AND THE COMPLIANCE RELATIONSHIP 60
2.4 EXPERIMENTAL CALIBRATION OF (1) FOR THE IZOD TEST 62
2.4.1 Specimens and Test Procedure 62
2.4.2 Experimental Results - Discussion 63
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2.5 CORRELATION BETWEEN COMPUTED AND EXPERIMENTALLY
DETERMINED CHARPY CALIBRATION FACTORS (1) 64
2.6 DERIVATION OF (I) FROM THEORETICAL COMPLIANCE BY
APPROXIMATION TO VERY SMALL CRACK LENGTHS 65
2.6.1 The Charpy Case 65
2.6.2 The Izod Case 67
2.7 COMPARISON BETWEEN THEORETICAL AND EXPERIMENTAL
CALIBRATION FACTOR 4) 68
2.7.1 (1) for the Charpy Test 68
2.7.2 for the Izod Test 68
CHAPTER 3: IMPACT MACHINE 72
3.1 INTRODUCTION 72
3.2 DESCRIPTION OF THE APPARATUS 73
3.3 ZERO AND VICE OFFSET - WINDAGE AND FRICTION LOSSES 75
3.3.1 The Zero Offset 75
3.3.2 The Vice Offset 75
3.3.3 Windage/Friction Losses 76
3.4 EFFECTIVE RELEASE POINT OF THE TUP 77
3.5 POTENTIAL ENERGY OF THE TUP 78
3.6 ENERGY TO FRACTURE - CALIBRATION TABLES 79
3.7 SOME CHECKS OF PERFORMANCE OF THE MACHINE 80
CHAPTER 4: CHARPY AND IZOD IMPACT FRACTURE TOUGHNESS OF POLYMERS 82
4.1 INTRODUCTION 82
4.2 MATERIALS 83
4.3 THE CHARPY TEST - EXPERIMENTAL PROCEDURE 84
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4.3.1 Test Conditions and Apparatus 84
4.3.2 Specimens and Notching Technique 85
4.3.3 Testing Procedure 86
4.4 ANALYSIS OF EXPERIMENTAL DATA 87
4.5 EXPERIMENTAL RESULTS- DISCUSSION 88
4.5.1 Low Impact Fracture Toughness Polymers 89
4.5.2 Medium Impact Fracture Toughness Polymers 89
4.5.3 High Impact Fracture Toughness Polymers 91
4.6 ANALYSIS FOR HIGH TOUGHNESS POLYMERS 91
4.6.1 The Effective Crack Length Approach 91
4.6.2 The Rice's Contour Integral Approach 92
4.7 EXPERIMENTAL RESULTS FOR HIGH TOUGHNESS MATERIALS 94
4.8 THE IZOD TEST - EXPERIMENTAL PROCEDURE 95
4.8.1 Specimens and Notching 95
4.8.2 Testing Procedure 95
4.9 ANALYSIS OF THE 'HOD TEST DATA 96
4.10 IZOD TEST RESULTS - DISCUSSION 96
4.11 CONCLUSION ON THE CHARPY AND IZOD IMPACT TESTS OF
POLYMERS 98
4.12 SOME FACTORS AFFECTING THE IMPACT FRACTURE TOUGHNESS
OF POLYMERS 99
4.12.1 Effect of Molecular Weight on the Impact
Fracture Toughness of PMMA 99
4.12.2 Materials Tested 100
4.12.3 Molecular Weight and Relative Viscosity
Relationship 100
4.12.4 Experimental Procedure 101
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4.12.5 Experimental Results - Discussion 101
4.13 EFFECT OF MOISTURE CONTENT ON THE IMPACT FRACTURE
TOUGHNESS OF NYLON 66 102
4.13.1 Experimental Results - Discussion 103
CHAPTER 5: EFFECT OF TEMPERATURE ON THE IMPACT FRACTURE TOUGHNESS
OF POLYMERS 105
5.1 INTRODUCTION 105
5.2 SPECIMENS AND TEST PROCEDURE 106
5.2.1 Materials 106
5.2.2 Test Conditions and Apparatus 106
5.2.3 Specimens and Notching 107
5.3 EXPERIMENTAL RESULTS 108
5.4 THICKNESS EFFECT - THEORETICAL ANALYSIS 109
5.4.1 Plane Stress Elastic Work to Yielding and Gc
Relationship 112
5.5 YIELD STRESS AND TEMPERATURE - TEST PROCEDURE 114
5.6 EXPERIMENTAL RESULTS - DISCUSSION 114
5.7 CONCLUSIONS 116
.CHAPTER 6: EFFECT OF NOTCH RADIUS ON THE IMPACT FRACTURE
TOUGHNESS OF POLYMERS 117
6.1 INTRODUCTION 117
6.2 THEORETICAL ANALYSIS 118
6.2.1 Relation Between the Plane Strain Elastic
Work W and the Plane Stress Elastic Work pl
p2 121
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6.3 SPECIMENS AND TEST PROCEDURE 123
6.3.1 Materials 123
6.3.2 Specimens and Notching Technique 123
6.3.3 Test Conditions 124
6.4 EXPERIMENTAL RESULTS 124
6.5 CONCLUSION 125
CHAPTER 7: CONCLUSIONS 127
Tables 129
Figures 143
APPENDIX I 245
I.1 THE RELATIONSHIP BETWEEN FRACTURE TOUGHNESS AND 245
ABSORBED ENERGY FOR THE IZOD IMPACT TEST 245
APPENDIX II: STRESS CONCENTRATIONS AND BLUNT CRACKS 247
II.1 INTRODUCTION 247
11.2 STRESSES AROUND AN ELLIPTICAL HOLE 247
11.3 STRESSES AROUND A BLUNT CRACK 249
Figure for Appendix II 250
References 251
Paper 1
Paper 2
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NOTATION
a Crack length. In infinite plate, half crack length.
of
Crack and plastic zone length. •
A Ligament area.
B Specimen thickness.
cr
• . Count recorded.
cw/f :
Count lost due to windage and friction.
Co
Compliance for a zero crack length specimen.
Ca
Compliance for a cracked specimen of notch length a.
C : Total experimental compliance.
CT
Total theoretical compliance.
D Specimen width.
E • . Young's modulus.
G • . Strain energy release rate.
Gc
• . Critical value of G. Sharp crack fracture toughness.
Gel • . Plane strain fracture toughness.
Gc2
• . Plane stress fracture toughness.
GB
• . Blunt notch fracture toughness.
g • • Constant of gravity.
j Subscripts of tensor notation.
I Moment of inertia.
J Rice's contour integral.
Jc
Critical value of J.
K Stress intensity factor.
c Critical value of K for sharp cracks.
Kcl
Plane strain critical stress intensity factor.
c
• 2
Plane stress critical stress intensity factor.
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KB
: Blunt notch critical stress intensity factor.
L : Half span for three point bending and cantilever bending.
M : Average molecular weight.
: Mass of tup.
n. : Number of counts recorded.
P : Applied load.
✓ : Radial distance from the crack tip.
rp : Radius of Irwin plastic zone.
r p2 Plastic zone size under plane stress conditions.
T Temperature.
U Elastic strain energy per unit thickness.
u Displacement.
Elastic strain energy to fracture.
W' Kinetic energy of the fractured specimen (Charpy or Izod).
rat • • Energy to give first yielding.
/ • • Plane strain elastic energy to yielding.
WP2 • Plane stress elastic energy to yielding.
Coordinate.
Crosshead speed.
Coordinate.
Finite plate correction factor.
a, s : Constants.
: Surface energy.
Ab
: Bending deflection of a beam.
A : Shear deflection of a beam.
A : Total deflection of a beam (= Ab+ As).
• : Engineer's strain.
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: Relative viscosity.
e Angular measure.
: Poisson's ratio.
p : Notch tip radius.
a : Applied stress.
a : Yield stress. y
ac
: Stress at the tip of a blunt notch.
of : Stress at fracture.
Calibration faces for Charpy and Izod tests.
Angular measure.
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ABBREVIATIONS
ABS Acry)onitrile-butadiene-styrene
GPPS : General purpose polystyrene
HIPS : High impact polystyrene
HDPE : High density polyethylene
LDPE : Low density polyethylene
PC : Polycarbonate
PE : Polyethylene
PP : Polypropylene
PS : Polystyrene
PMMA : Poly(methyl methacrylate)
PTFE : Polytetrafluoroethylene
PVC : Polyvinyl chloride
SCF : Stress concentration factor
ZO : Zero offset
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INTRODUCTION
Impact strength is widely acknowledged to be one of the most
important properties of materials. It is considered as a major criterion
in the specification of the mechanical usefulness of any material, plastic
or metal. The importance of the impact test lies in the fact that it
provides a method of quality control, mainly for plastics, and also
provides design information for research and development. In quality
control it is used to determine the uniformity of production of a given
material. By design information is meant prediction of the relative
toughness of a material under practical conditions. Unfortunately,
although impact -testing is very popular and often discussed, it is
seldom fully understood. To quote Westover (1958) "... Out of the chaos
of two centuries of investigations of impact on metals and three decades
of impact applications to plastics, we can find little ground for
agreement among present day investigations. Notched and unnotched
specimens have been made in various shapes and sizes and have been
subjected to tensile, compressive, torsion and bending impacts.
Materials have been thrown, dropped and subjected to blows from hammers,
bullets, falling weights, pendulums, falling balls, horizontally moving
balls and projections from flywheels."
The impact strength of a material is assumed to be equivalent to the
loss in kinetic energy resulting from the momentum exchange between a
moving mass and the test specimen. In conventional types of impact tests
the impact strength is reported in terms of the energy, w, absorbed by
the specimen when it is struck and fails under the impact, and this is
generally divided by the ligament area A to give an apparent surface
energy W/4.
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It is well known that such an analysis of the data is not
satisfactory, particularly since the parameter has a strong geometrical
dependence.
The main aim of the present research work is to examine the nature
of these geometrical effects for polymers in the Charpy and Izod loading
situations, and, by employing the concepts of fracture mechanics, to
deduce the critical strain energy release rate, Gc, directly from the
absorbed energy measurements. The work began with an attempt to
determine impact fracture toughness values using the Charpy and Izod
tests for various polymers at room temperature. Good correlation
between Charpy and Izod impact fracture toughness values would provide
a basis for studying the effects of temperature and notch tip radius on
the impact behaviour of polymers.
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CHAPTER 1
LITERATURE SURVEY
1.1 HISTORICAL INTRODUCTION TO IMPACT TESTING
Studies on the subject of impact testing can be traced back over
two centuries. A monumental report on impact testing up to 1948 was
presented by Lethersich (1948) in which more than 200 references were
quoted. Historically, impact testing originated when it was realised
that metals which appeared satisfactory when tested by the usual methods
sometimes failed when subjected to shock conditions. The impact test
was able to discriminate between good and faulty steels, and specification
of a minimum impact strength fora given size of specimen was sufficient
to provide a rational quality evaluation.
The history of impact testing for metals goes back to 1734 when a
German metallurgist, Swedenborg, tested iron bars by throwing them
against a sharp edge. In America in 1824 T. Tredgold theoretically
examined the resistance of cast iron beams to impulsive forces, and in
1874 R.H. Thurston computed the impact resistance to single and repeated
blows from the area under the static stress-strain diagram. Events in
Europe are described by Charpy (1901) at the Conference of Testing
Materials. In 1901 he designed his pendulum machine, which could be
used on specimens with three point bend or cantilever-type loading.
He then discarded the cantilever support because it was thought that the
clamping pressure would affect the results, and the three point bend
support was used and named after him. In Britain Izod (1903) developed
his pendulum machine in which one end of the notched specimen was clamped
in a vice and the other end struck by a hammer so that the notch was
opened. The Izod support is, of course, of the cantilever form and
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although named after him, it was originally designed by Charpy.
Perhaps the most significant contribution to the subject of impact
testing of metals was made in 1923, when P. Ludwik discovered that two
types of fracture could be distinguished. If failure occurs in shear,
ductile fracture results, but if the cohesion between the molecules is
broken, brittle fracture results. In 1925, Moser (1925) was the first
to measure the volume of the plastic deformation around the notch, and
he found it to be proportional to the impact strength and independent
of the specimen size. So, for the first time, the impact strengths of
specimens of different sizes could be compared.
Until 1926, impact testing was confined to metals and particularly
to steel, but development of plastics (mainly for electrical insulation)
led to the application of impact testing to these materials.
In this manner, the controversial subject of impact testing of
plastics was introduced into the world of science and engineering.
1.2 IMPACT TESTING OF PLASTICS
One of the many properties of a plastic material which influence its
choice for a particular article or application is its ability to resist
the inevitable impacts met in day to day use. Impact tests attempt to
rank materials in terms of their resistance to breakage. Impact testing
• of plastics over the last 50 years has assumed great practical importance
due to the greatly increased use of these materials in everyday life and
in many engineering applications; however, argument and confusion among
the various investigators has grown proportionally.
The complexity of impact testing results from a number of factors.
There is a remarkably large number of different impact testing machines
and test methods. All the various types of tests measure different
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quantities, some of which are not clearly defined or understood. Tests
are made on specimens of various sizes and shapes. The specimens are
broken under different kinds of stress distributions and under different
impact velocities. Variations in the specimens themselves make it
difficult to obtain reproducible results. For example, specimens may
have varying degrees of molecular orientation, which may be parallel or
perpendicular to the stresses developed during the impact test. Plastics
are considered to be notch sensitive materials (some more than others),
so that small variations in the notch tip radius can cause wide
divergence of the impact strength values obtained in the test. Humidity
and temperature control of the laboratory where the impact test is
performed is also important, as plastics are sensitive to environmental
conditions, any variation of which may result in a different impact
strength value. A typical example is nylon which tends to absorb.
moisture from the environment. This can have a remarkable effect on
the impact behaviour of the material. (The effect of moisture on the
impact behaviour of Nylon 66 is discussed in section 4.13.)
1.3 SPECIFIC IMPACT TESTS
Many methods of measuring impact strength are in use in the plastics
industry. These methods can be broadly divided into two categories:
1. Excess energy methods
2. Limiting energy methods
The essential characteristic of the excess energy method is that the
kinetic energy of the striker is much greater than the fracture energy
of the specimens, so that the velocity of the striker can be assumed to
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be constant during impact. The energy absorbed is determined from the
loss of kinetic energy or the decrease in angular velocity of a flywheel
striker. In limiting energy methods, the kinetic energy of the striker
is adjusted to the point at which only a fraction of the specimen,
usually a half, is broken. A simple form of limiting energy tests
consists of dropping an article from a range of heights. In
conventional tests it is more common to alter the mass of the striker
rather than the impact velocity.
1.3.1 Limiting Energy Impact Testing Methods
The falling weight test or drop dart test falls into this category.
It is usually carried out on fabricated or semi-fabricated articles such
as sheet or piping. In the British Standard version of the test
(BS 2782:1970) the specimens are 24" discs cut from sheet, and are freely
supported on a hollow steel cylinder of internal diameter 2". The
striker consists of a 1-" diameter steel ball attached to a weight carrier,
falling freely between guides from a height of 24". • Repeated trials of
differently weighted strikers are made until the minimum weight to
produce penetration is obtained. An alternative method of repeated
trials is sometimes used in which the same weight drops from increasing
heights. An obvious disadvantage of the falling weight impact test is
that a large number of trials and samples are needed for a proper
material assessment.
1.3.2 Excess Energy Impact Testing Methods
There are three main types of pendulum impact tests that are
considered to be excess energy methods:
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1. The Charpy test, *which employs specimens supported as a
three point bend bar.
2. The Izod test, which employs specimens supported as a
cantilever.
3. The tensile impact test, which employs dumbell specimens
loaded in uniaxial tension.
The Izod and the.Charpy pendulum tests were the earliest impact tests to
be standardised for plastics and they are still the most widely quoted.
This is not surprising, as these tests were originally derived from
traditional tests for metals and in any application where plastics were
to replace metals, comparative test data was required. However, the
Izod and the Charpy tests suffer from a number of disadvantages. Both
are very sensitive to errors in forming the notch, and any small
variation of the notch tip radius could affect the result. Test values
must be obtained from a standard specimen geometry and can be compared
on the basis of that standard specimen only. The complexity of the
stress distribution around the notch is another factor that creates many
difficulties in analysing the data, since very little theoretical work
has been done on the bending of plastic materials under impact loads.
Lee (1940) showed that the deflection curve of a beam under impact deviates
widely from the static deflection curve. He stated: "... A material
test carried out at high speeds may be markedly influenced by plastic
wave propagation effects. In such a case a variation of strain occurs
along the test specimen, and the stress-strain relation cannot be
determined from measurements made on the specimen as a whole."
Finally, the broken half error, or the so-called "toss-factor", has
to be considered in the Charpy and Izod tests. In the case of the Izod
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test, the broken portion of the specimen is thrown forward by the
pendulum and in the case of the Charpy test the broken halves of the
specimen are ejected after impact. Thus, both tests involve some form
of energy loss dissipated as kinetic energy by the broken specimen.
This portion of energy loss is included in the result, so the actual
energy to failure should be less than the total energy recorded.
Since Izod and Charpy test originated for metals, metallurgists paid
little attention to this error, as it appeared to be small in comparison
to the high impact strength of these materials. For plastics, however,
because their impact strength is relatively low, the broken half error
can be considerable. Many investigators have favoured the tensile
impact test. Evans (1960), Maxwell (1952), Bragaw (1956), Westover
(1958) and (1961) argued that the tensile test is a more meaningful test,
giving results that are easier to analyse than those of the Charpy or of
the Izod test. The main attraction is that the stress system is simple,
and the strain rates are known.
1.4 TENSILE IMPACT TEST
The tensile impact test is a simple modification of the Izod test
(ASTM D1822-1964). The modification consists of replacing the Izod vice
with one that holds the fixed end of a dumbell specimen and attaching a
free metal grip to the other end. The pendulum is adapted so that it
strikes the metal grip on swinging and therefore breaks the specimen in
simple tension. If the effective gauge length of the specimen is known,
the approximate strain rate may be calculated from the pendulum velocity.
1.5 CHARPY AND IZOD TESTS
The Charpy and the Izod tests are excess energy tests in which a bar
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is broken in flexure by a blow from a pendulum type striker. A scale
records the reduction in the amplitude of the pendulum swing and hence
the energy to break the specimen. In the Charpy test the notched
specimen is supported (horizontally) and is hit at the centre behind
the tip of the notch by the pendulum striker, so that fracture takes place
by three point bending.
In the Izod test, one end of the notched specimen is firmly clamped
in the vertical position in a vice and the pendulum striker hits the other
end horizontally.
Both test methods employ a range of pendulum heads with different
masses so that various plastics having a range of impact strength can be
tested. The most commonly used apparatus for the Charpy testing of
plastics is the Hounsfield impact tester of which Vincent (1971) gives
a brief description.
Some tests are carried out on unnotched specimens, but most specimens
are notched centrally. The main purpose of the notch for both methods
of testing is to•concentrate stress at its tip and hence locate the point
of fracture initiation. The Izod test is performed in two slightly
different forms, as a British Standard (BS 2782:1970) and as an American
Standard (ASTM D256-56:1961).
Table 1.1 gives specimen dimensions for the Izod test for both the
British and American specifications, as well as the dimensions for the
standard Charpy specimen. The main difference between the BS and ASTM
Izod specimens is that of the notch radius. A notch radius of 0.040"
is specified for BS specimens whereas a notch of 0.01" characterises the
ASTM specimens. Since the ASTM notch is sharper, it is more likely to
produce plane strain conditions in the specimen, so that crack initiation
energies are lower for some plastics which are very sensitive to notch
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tip radius. Thus, a significant increase in the impact strength can be
expected when tested to British Standard specifications. Horsley (1962)
compared the BS and ASTM Izod impact strengths of a number of plastics.
He found that the BS test gives a higher impact strength for all the
plastics tested. However, he observed that the increase was more
noticeable for some Plastics than for others.
1.5.1 The Effect of Notch Tip Radius on the Impact Strength
When plastics with a high degree of notch sensitivity are used, care
should be taken in design to avoid any points of stress concentration,
whereas using plastics with a low degree of notch sensitivity such
factors are not so critical. Stephenson (1957) examined the effect of
notch tip radius on the impact strength of PMMA by testing specimens with
keyhole notches of various notch radii, and compared the data with those
obtained from ASTM and BS specifications. The data indicated an
approximately linear increase in the impact strength with notch tip
radius. The effect of notch tip radius on the impact behaviour of
plastics has also been examined. Vincent (1971), Reid and Horsley
(1959), Hulse and Taylor (1957), Adams et al (1956).
Lethersich (1948) attributed the increase in impact strength with
notch tip radius for a given specimen size and notch depth to two factors;
the greater stress concentration that arises with sharper notches, and
the increase in the spatial stress ratio* as the radius of the notch
decreases. The latter increases the probability of brittle failure.
Petrenko (1925) found experimentally that the impact strength I and the
* The ratio of the triaxial tensile stress to the shear stress.
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notch radius p could be related by the empirical equation:
I = aDpVT f SD B2
where D and B are the width and the thickness of the specimen and a and
are constants.
Inglis (1913) showed that the tensile stress at the root of the
notch is given by:
ac
= a (14- C Va/p)
(1.2)
where a is the applied stress, a is the notch depth and p is the notch
tip radius.
The constant C was found to be nearly 2. Equation (1.2) indicates
that any increase in the notch tip radius should reduce the impact.
strength. The ratio of stress at the root of the notch to the applied
stress (a /a) is defined as the "stress concentration factor" (SCF).
1.5.2 Notch Stress Distribution for Charpy and Izod Tests
Although the notch serves the same function for both tests, the
stress distribution round the notch varies considerably. Coker (1957)
examined photoelastically the general characteristics of the stress
distribution round the notch tip for both tests, and observed a
dissimilar distribution. An aslymetrical stress distribution was
observed for the Izod test. This asiymetry was believed to be due to
the applied clamping pressure since it resulted in additional stress
round the notch tip area. It must be concluded that the Charpy and the
Izod impact strengths as defined by conventional methods, WA, cannot be
directly compared since they do not measure exactly the same quantity.
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This is a very important point in the author's opinion, and it can be
considered responsible for the inconsistency between the Charpy and Izod
test data. This is the main reason for the introduction of the fracture
mechanics approach into the field of impact testing, since a single
parameter, the "fracture toughness", Gc, can be deduced from any test
(Charpy or Izod) and it is characteristic of the material and independent
of the loading configuration. The fracture mechanics approach will be
discussed in section 1.10.4.
1.5.3 Effect of Clamping Pressure for the Izod Test
The main complication of the Izod test over the Charpy test is the
effect of clamping pressure on the results. BS and ASTM do not specify
the clamping pressure to be applied to the test specimen.
Stephenson (1957) performed a series of tests in which the clamping
pressure was varied. The results indicated that there is a linear
decrease of impact strength with increasing clamping pressure, which
could be represented to a fair approximation by the formula:
Impact Strength (ft.lb/in of notch) = 0.362 - 0.000024 P'
(P!= clamping pressure in lb/in2)
Adams et al (1951) examined the effect of clamping pressure on the
impact strength of several plastics and found that some plastics are more
sensitive to clamping pressure variations than others. He noted that
styrene showed a consistent decrease in impact strength with increasing
the clamping pressure. Therefore, co-operating laboratories should
agree on a means of standardising the gripping force, for instance by
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using a torque wrench on the screw of the specimen vice.
1.6 BRITTLE AND DUCTILE IMPACT FAILURES
Generally all plastics under impact conditions fail in a ductile
(tough) or brittle manner. Horsley (1962) relates each type of fracture
failure to the stress level at crack initiation with the yield stress of
the material. In the case of a ductile type of failure the material in
the fracture area yields and flows, whereas in the brittle case, only
small elastic deformations take place prior to fracture. The existence
of a ductile or a brittle type of failure depends upon whether under
specific experimental conditions, the specimen yields prior to crack
initiation or whether the crack initiates before the yield stress is
reached. A brittle type failure occurs if the stress at crack
initiation is lower than the yield stress, as the elastic energy stored
in the sample at the moment of crack initiation is usually sufficient to
propagate the crack. Conversely, if the crack initiation stress is
higher than the yield stress, a ductile failure results. So if, during
any impact test, the load/deflection curve were recorded, a material
which failed in a brittle manner would give a straight line relationship
with fracture occurring at the maximum recorded load as shown in
Figure 1.1, whereas for a ductile failure a curve would be obtained with
fracture occurring at some point after the maximum load had been recorded.
The area under each centre could give a measure of the impact strength.
Therefore, any factor that affects the yield strength, the crack initiation
stress or both can have an influence on the type of failure and the
consequent impact strength value obtained. Such factors may be either
structural changes (e.g. preferred orientation or surface imperfections)
or changes in the experimental conditioning of the test to which the
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-28-
specimen is subjected (e.g. humidity and temperature variations or
some environmental changes such as the effect of various chemicals).
The effect of temperature on the impact strength of plastics, and the
brittle-ductile transition type of failure are discussed in Chapter 4.
1.7 IMPACT STRENGTH - ENERGY TO FRACTURE
In the pendulum type of impact tests the energy absorbed in
fracturing the specimen is measured by the excess swing of the pendulum.
Telfair and Nason (1943) defined the "energy to break" the specimen as
the sum of energies consumed by several mechanisms taking place during
the test. They summarised these several forms of energy as:
1. Energy to initiate fracture of the specimen.
2. Energy to propagate the fracture across the specimen.
3. Energy to deform the specimen plastically.
4. Energy to eject the broken ends of the test specimen.
5. Energy lost through vibration of the apparatus and at
its base through friction.
1.7.1 Energy to Initiate and to Propagate Fracture
Lethersich (1948) discussed the opinions of various workers who
considered that the energy required to fracture a specimen is made up of
two parts: the energy to initiate the fracture and the energy to
propagate the fracture. He stated that opinion is divided as to whether
only the first part is required or both. About a century ago F. Kick
first showed that the energy required to initiate fracture was
proportional to the volume of the specimen. Some years later Charpy
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-29-
suggested that the energy required to separate the two halves of the
specimen is proportional to cross-sectional area. From the above
considerations, the total energy to fracture would he given by:
I =a1/4- S S (1.3)
where V and S are the volume and cross-section of the specimen, and a
and s are constants. It was shown experimentally that for brittle
materials 13 was zero which suggests that the propagation energy is
negligible in this case, whereas for ductile materials the constant a
becomes small.
Stephenson -(1961) showed that the above equation is true with a
slight modification in the initiation energy term aV. He showed
experimentally that the elastic energy stored in the specimen at the
time of breakage is available to propagate fracture before imparting
kinetic energy. If, however, there is not sufficient elastic energy
available, then for complete fracture, extra energy will be absorbed
from the pendulum. The energy that will then be measured will be the
sum of the energy to propagate the crack and that part of the stored
elastic energy which has been lost. The energy required for crack
propagation will be proportional to the area of the q-ew surfaces
formed, i.e. the cross-sectional area of the specimen. The stored-up
elastic energy is proportional to the volume of the specimen, therefore
if it is assumed that the energy lost is proportional to the stored-up
energy it will also be proportional to the volume. By this mechanism
the measured impact energy is given as the sum of the energy required for
crack propagation and the stored-up energy which has been lost.
Using this hypothesis, the product aV gives the amount of the crack
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- 30 -
initiation energy which has been lost. Therefore, the crack initiation
energy is considered to be a criterion for impact failure. With
notched specimens which differ only in the notch radius, the energy for
crack propagation will be the same for every notch radius, whereas the
crack initiation energy will decrease with decrease in the notch radius.
Therefore, there will be a critical radius above which the measured value
will increase with notch radius because crack initiation is measured, and
below which there will be a very little dependence on the radius because
crack propagation is measured. Therefore, meaningful data on crack
initiation are obtained only if the notch radius is above a critical value.
For a PMMA Izod specimen of standard dimensions, the critical radius was
found to be aboilt that of the ASTM notch (i.e. 0.010").
Vincent (1971) states that there are clearly at least two different
physical properties (i.e. crack initiation and crack propagation energy)
underlying the impact behaviour. He considered the results of tests
which were performed on samples of rigid polyvinylchloride (PVC) and
acrolonitrile-butadiene-styrene (ABS) at room temperature with sharp
notches (tip radius 0.25 mm) and with blunt notches (tip radius 2 mm).
When the sharp notched specimens were tested it was found that ABS had a
much higher impact strength, but when the blunt notched specimens were
tested PVC had a higher impact strength. He explained these results
assuming that both crack initiation energy and crack propagation energy
can contribute to the measured impact strength. By this interpretation,
PVC must have a relatively high crack initiation energy to account for
its good behaviour with blunt notches but a low crack propagation energy
to account for its poor behaviour with sharp notches. Conversely, ABS
has a relatively high propagation energy but a low crack initiation energy.
It is clear that Vincent's explanation coincides with Stephenson's,
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- 31 -
that is to say that crack initiation energy is the predominant factor in
the blunt notch case. Vincent states that when a material has a low
crack propagation energy, the impact strength measures only the crack
initiation energy; once the crack has initiated, the stored elastic
energy is sufficient to propagate the crack completely across the specimen
without absorbing further energy. The crack propagation energy is more
difficult to measure. It can be estimated in the special case of a
sharply notched specimen only partly broken in the test. In this case
the ratio of the energy lost by the weight of the pendulum to the area of
new surface created, provides an upper limiting estimate of the crack
propagation energy.
1.7.2 Energy Lost in Plastic Deformation
Although the notch in the Charpy and Izod test has mainly the -purpose
of concentrating the stress and preventing plastic deformation, it is
quite usual for plastic deformations to take place during the impact
process resulting in a ductile type failure. In this case the specimen
may break (or may not break completely - hinge failure) with obvious signs
of permanent macroscopic deformations at the fracture surface. A
whitened region observed in the fracture surface indicates that some
plastic deformation has taken place. The amount of whitening can be
varied considerably by varying some experimental conditions such as the
temperature. Generally, the impact energy increases as the amount of
whitening increases, resulting in very high impact strength values.
Vincent (1971) in his monograph discusses various types of fracture in
which plastic deformations have taken place during various stages of the
fracture process. These types can be summarised as:
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- 32-
1. The specimen yields at first round the crack tip region,
a whitened region is formed and the crack continues to
propagate within this region. The specimen may break or
may not, depending on the material's resistance to crack
propagation.
2. The case in which the crack initiates and propagates in a
brittle manner. No whitening is observed in the fracture
surface but suddenly the material yields and crack
propagation stops, the ligament forming a flexible hinge.
In this case the material is significantly more resistant
to crack propagation than to crack initiation.
3. Finally, the case in which the specimen yields at first but
then the crack propagates throughout the entire fracture
area in a brittle manner, resulting in a brittle type of
fracture with a small whitened region round the crack tip.
In this case the material is more resistant to crack
initiation than to crack propagation. •
If any one of these types of fracture occurs in an impact test a high
impact energy value can be expected because of the yielding process.
Energy lost in plastic deformation is included in the measured impact
energy value.
1.7.3 Kinetic Energy of the Broken Half
Kinetic energy of the broken half, in the case of the Izod test, or
of the two broken halves in the case of the Charpy test, is one of the
most important factors assumed to contribute to the measured impact
strength. In the Izod test the broken portion of the specimen is thrown
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-33-
forwards by the pendulum, taking"some energy from it. A similar energy
loss occurs in the Charpy test; in this case both broken halves of the
test specimen are ejected after impact. This energy should not be
included in the impact strength value and contributes to what is referred
to as the "broken half" error or sometimes as the "toss factor". In
order to correct the Izod impact strength value for tossing of the broken
half, the broken half of the specimen is replaced and struck again.
The energy to re-toss the broken half is considered to be the tossing
error. This method was first introduced by Zinzow (1938). The main (ICS —
advantage of this method is that the actual tossing velocity for a specimen
usually differs from the velocity of a previously broken sample. Another
point to be considered is that the above method of correction does not
include rotational energy. Lethersich (1948) noted that rotational
kinetic energies as high as / the value of the linear kinetic energy have
been reported in the Izod test during the breaking stroke.
Callendar (1942) estimated the broken half error for the Izod test
by replacing the broken piece and finding the energy required to throw it
the same distance as it flew in the test. He observed, however, that the
broken half of an ebonite specimen went further in the test that it was
knocked when it was replaced. He attributed this difference to the
kinetic energy derived from the stored-up elastic energy which should not
be regarded as an error. He stated that there is always some stored
elastic energy in a stressed specimen and there is the possibility that
some kinetic energy is derived from it. Stephenson (1961) aimed to find
the energy which would just be enough to crack the specimen. In this
case there would be no broken half error. He performed the standard Izod
test for poly(methyl methacrylate) (PMMA) in which the available energy
was varied by changing the starting height and therefore the impact
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-34-
velocity. He found that when the specimen was broken, the broken half
was projected forward with some velocity, even in the case when the
available energy was just sufficient to break it. This indicates that
part of the kinetic energy comes from the stored elastic energy. His
results indicated that the height reached by the broken half is
approximately the same, irrespective of the impact velocity, whereas the
distance travelled increases with the impact velocity. This behaviour
is consistent with the assumption that the broken half always leaves the
pendulum with the same velocity relative to it. He calculated the
final velocity of the pendulum from its final energy and so obtained
corrected values for the distance travelled.
He found a -characteristic value for the height and distance to which
a broken half will go, if allowance is made for the horizontal component
of velocity of the pendulum. Stephenson's results indicate that the
energy in the broken half of the Izod specimen consists of two components.
One component of energy which is characteristic of the specimen, and
should not be considered as an error when included in impact energy, and
a second component of energy imparted by the pendulum, which should be
considered as an error. Therefore, the correction will be overestimated
if the total kinetic energy of the broken half is used.
Maxwell and Rahm (1949) presented a method of impact testing which
eliminates the toss factor in the Izod test. The standard Izod-type
specimen is attached to the periphery of a flywheel that can provide a
wide range of loading rates. An anvil obstructs the path of the free
end of the specimen and fractures it and the energy removed from the
flywheel is determined. Since the specimen is in motion prior to the
impact, it contains the kinetic energy necessary to eject itself after
fracture, and thus there is no energy lost from the flywheel for the
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- 35 -
toss factor. Burns (1954) introduced the Dozi (Izod spelt backwards)
impact testing machine which differed from the Izod tester in that the
Dozi-type specimen is clamped in the pendulum. This gives results
similar to the Maxwell's machine as it eliminates the toss factor.
1.7.4 Energy Lost in the Apparatus
Energy losses due to vibration of the apparatus may be large in
testing metals but are apparently negligible for plastic materials. This
assumption is based on a statement given by Westover (1958) who pointed
out that the energy lost by the pendulum during an impact test is shared
by the specimen and the machine in an inverse ratio of their elastic
moduli. That 1s to say, the greater the modulus of elasticity of the
specimen (as in the case of metals) the greater will be the proportion
of the energy absorbed by the machine, and the smaller the modulus of
elasticity of the specimen (as in the case of plastics) the smaller will
be the proportion of the energy absorbed by the machine. Friction losses
are largely eliminated by careful design and operation of the apparatus.
For example, if was pointed out that it is important, in pendulum type
machines, that the centre of percussion of the pendulum coincides with the
point of impact. Itthis condition is not fulfilled, then energy is lost
from shock in the bearings at the top of the pendulum. Lethersich (1948)
suggested that losses due to friction at the bearings of the pendulum, due
to friction at the idle pointer, and losses due to windage can usually be
estimated by performing a blank test, i.e. a test in which the specimen is
omitted. The measured energy loss gives the magnitude of these errors.
Bluhm (1955) assumed a model in which the force acting on the pendulum
was the same as that acting on the specimen. He showed that the
discrepancies in the measurement of energy absorption from one machine to
-
-36-
another may be attributed to:the flexibility of the impact machine and
that flexibilities can give rise to the differential behaviour of high-
and low-strength specimens having the same toughness. He concluded that
to ensure adequate design the stiffness of the pendulum should exceed a
certain minimum value.
1.8 EFFECTS OF TEMPERATURE ON IMPACT STRENGTH
The physical properties of polymers in general show a strong
dependence on temperature. Impact strength is not an exception.
Although it is obvious that in the majority of practical applications the
impact strength at room temperature is more important, graphs of impact
strength against temperature could be very useful since they could give
a much better understanding and appreciation of the polymer impact
behaviour than a single temperature value could do.
Vincent (1971) considered the temperature effect on the Charpy
impact strength of three polymers (PMMA, polypropylene (PP), and rigid PVC).
He tested unnotched and notched specimens with two different notch tip
radii, p = 0.25 mm and p = 2 mm. His results indicated that although a
wide variation in numeric values was immediately apparent, there were
nevertheless some marked similarities in the behaviour of the polymers
tested. Their impact strength showed a low value at low temperatures
(less than -20°C) with the value staying almost constant with further
decrease in temperature. With the exception of PMMA, when tested with
sharp notches (p = 0.25 mm), all polymers showed a very sudden increase
in the impact strength, during a quite small increase in temperature.
The temperature at which this increase occurred was different for the
three polymers and it also differed from one notch radius to another for
the same polymer. He examined in detail the behaviour of the polymer
-
-37-
within this relatively narrow range of temperature, and the appearance of
the specimens after testing showed that their behaviour changed from
brittle at lower temperatures to ductile at higher temperatures. He
proposed that this important temperature region might be called the
"tough-brittle" transition region. Reid and Horsley (1959) compared the
Charpy notched impact strength with the falling weight impact strength
of various polymers tested in the temperature range from -40°C to +60°C.
They found that the variation of the Charpy notched impact strength with
temperature was very different from that of the sheet in the falling
weight test. Although a good correlation was observed with both types
of test at low temperatures (the impact strength values were low and
fairly constant), the temperature at which the sudden rise in the impact
strength occurred was different for both tests. However, they reported
that three polymers (cellulose nitrate, styrene-acrylonitrile rubber and
high impact polystyrene (HIPS)) showed a very similar impact behaviour
with temperature for both tests. They stated that this agreement was due
to the fact that these three polymers were identified as insensitive to
notch radius. It would seem, therefore, that notch sensitivity is
responsible for changes in the material properties in the falling weight
test. Horsley (1962) reported a tough-brittle transition region for
unplasticised PVC at about 10°C. Below this temperature a significant
drop in the impact strength was observed. He pointed out that as the
transition from tough to brittle type failures is accompanied by a marked
reduction in the impact strength of the material, the major purpose of
impact testing should be to ascertain the conditions under which such a
transition occurs, so that brittle type failure can be avoided in practice
if possible.
One generalisation frequently made, Turley (1968), is that any polymer
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38 -
at a temperature above or near its glass transition temperature is ductile
(i.e. it has a high impact strength), whereas any polymer at a temperature
well below its glass transition temperature is brittle (i.e. it has a low
impact strength). This assumption created doubts when it was realised
that some polymers behaved in a more complicated manner and that the above
generalisation could not be true, Boyer (1968). It was then recognised
that most polymers had transitions and relaxations lying below the glass
transition temperature and that those secondary transitions appeared quite
important in polymers which were ductile below their glass transition
temperature.
1 .9 THERMAL STABILITY - MECHANICAL LOSSES OF POLYMERS
Over the years the thermal stability of polymers, the temperature
transitions and the relaxation processes have been well examined and
discussed in detail by various investigators in many texts and in a
large number of published articles. The aim of the present review is not
to consider molecular mechanisms and relaxation processes in polymers, but
rather to refer to some views on how the impact behaviour of various
polymers could be related to the relaxation processes and damping peaks.
Figure 1.2 shows a schematic representation of three relaxation spectra
of the same polymer, measured by three different test methods at three
different frequencies (1 Hz, 1000 Hz and 107 Hz), all as a function of
temperature (after Boyer (1968)). Generally, the same energy absorption
peaks are shown up by all three methods, moving to higher temperatures as
the frequency is increased. The low frequency dynamic mechanical test
illustrates the following characteristics of the absorption spectra:
1. The melting point, TM, is the highest observed transition
-
- 39 -
referred to as the primary transition.
2. The glass transition temperature, TG, frequently referred
to as the a-relaxation. It is observed at a considerably
lower temperature than the melting point. This type of
relaxation corresponds to the motion of a large number of
carbOn atoms about the main polymer chain.
3. The strong T < TG relaxation is a second order relaxation
frequently referred to as the (3 or y relaxation. This type
usually involves motion of a small number (4 to 8) of
carbon atoms about the main polymer chain. It is believed
that this is the relaxation process related with the high
impact strength of some polymers at temperatures below TG.
Many investigators have considered the possibility that
there is some relation between the short-term toughness of
polymers as defined by their impact strengths and their
moduli and mechanical losses determined by dynamic
mechanical experiments. An excellent historical review
on the dependence of mechanical properties on the molecular
motion in polymers is given by Boyer (1968). In this
article over one hundred references are quoted.
1.9.1 Dynamic Mechanical Losses and Im act Strength of Polymers
Heijboer (1968) reported the impact strength of various polymers as a
function of temperature and investigated the possibility of a
relationship between the impact strength and the damping peaks. For PMMA
he observed two damping peaks at -80°C and +10°C, respectively, for 1 Hz
frequency. The damping peak at +10°C is probably the well known (3-peak
-
-40-
for PMMA which starts as low as -50°C, Jenkins (1972). The impact
strength of PMMA was found to increase slightly in the -80°C temperature
region, whereas in the +10oC temperature region no change in the impact
strength was observed. The impact strength for polycarbonate (PC)
showed a broad damping peak at about -110°C and the impact strength
transition was observed at about -130°C. Therefore, for PC the impact
strength transition could be well related to the damping peak. For
polyoxymethylene a good correlation between the impact strength transition
and the damping peak was observed at about -70°C, whereas for high density
polyethylene (HDPE) the damping peak at -120°C was not accompanied by an
increase in the impact strength; the impact strength transition was
observed at a somewhat lower temperature instead. From the behaviour of
these four polymers it might be concluded that the molecular movements
may have an influence on the impact behaviour. However, the exact
location of the impact strength transition cannot be predicted from the
location of the damping peak. A point that has to be emphasised is that
the damping values have been reported for 1 Hz frequency and, since impact
failure occurs in a shorter time, one might expect a better correlation
at higher frequencies.
Oberst (1963) studies the correlation between impact strength and
dynamic mechanical properties for PVC at 1000 Hz frequency. He reported
that the high impact strength for PVC at room temperature arises from the
s-relaxation. It had been previously shown that PVC shows a low, broad
s-peak at about -30°C to -50°C. This rapidly moves to higher
temperatures with increasing frequency and disappears on the addition of
plasticiser. From the last statement it follows that if the high impact
strength of PVC at room temperature is directly related to the damping
peak, one should expect the impact strength to drop on adding small amounts
-
-41 -
plasticiser. Bohn (1963) reported a drop in the impact strength of PVC
from 3 (ft.lbs) to 0:3 (ft.lbs) on adding up to 10% of plasticiser
(dioctylphthaiate). Vincent (1960) also studied the impact strength of
PVC as a function of plasticiser content and observed a minimum at about
10% plasticiser.
Special attention has been given to the impact behaviour as a
function of temperature for two phase polymers, such as the rubber
modified polystyrene. The addition of rubber to the polystyrene phase
markedly improves the impact behaviour of the polymer. It has been
observed that rubber modified polystyrene has several times the impact
strength of crystal polystyrene with the degree of improvement dependent
on three variables. The amount of rubber, its type, and the method of
its addition. Boyer (1968) represented schematically the mechanical
loss curves for unmodified and rubber modified polystyrene as a function
of temperature. The point of interest is the fact that for rubber
modified polystyrene an additional peak was observed (rubber peak) at
about -50°C in addition to the f3-peak for polystyrene at the much higher
temperature of about +50°C. It is believed that the rubber peak is
related to the high impact strength of rubber modified polystyrene at
quite low temperatures. However, the mechanism of rubber reinforcement
of impact strength in polystyrene is controversial (Schmitt and Keskula
(1960), Arrends (1966)) and is not within the scope of this review.
Bucknall and Smith (1965) commented on the temperature dependence of the
impact strength of rubber modified polystyrene and he identified three
regions as a function of temperature:
a) Below -30°C the impact strength is very low and almost
constant. The specimens are brittle.
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-42-
b) From -30°C to +40°C a small but increasing amount of
stress-whitening is observed near the notch and the
impact strength rises steadily.
c) Above +40°C a dense stress-whitening occurs at the fracture
surface and both the impact strength and the extent of
whitening increase rapidly with temperature. The
immediate cause of the impact strength increase is
believed to be the second order transition in the rubber
(rubber peak), whereas the continuous increase at
temperatures well above this transition region was
expected to be due to the activated nature of crazes.
Vincent (1974) considered how far the impact strength and the
damping peaks could be related in polymers, and he presented some evidence
relating damping peaks in brittle and impact strength to relaxation
processes. He stated that careful selection of the notch tip radius
may be needed to demonstrate peaks in the Charpy impact strength of
polymers associated with peaks in the dynamic losses. He explained that
if the notch is too blunt, the specimens become tough in the region of
dynamic loss and the peak in the impact strength appears as a slight bump
on the low temperature side of the steeply rising impact strength curve.
. If the notch is too sharp, the peak in impact strength may not appear.
To justify the last statement he tested polycarbonate with sharp notches
and with i mm radius notches and looked for any relation between mechanical
losses and impact strength as a function of temperature. The mechanical
loss curve showed a peak at about -70°C. The impact strength with very
sharp notches was found to be constant between -100°C and +60°C and was
apparently unaffected by the f3-process. In constrast, the impact
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-43-
strength with 4 mm radius notches was nearly doubled between -100°C and
-40°C, presumably because of the presence of the R-peak. Between -40°C
and 0°C the impact strength increased even more rapidly towards the very
high impact strength at +20°C.
Vincent (1974) also tested polyoxymethylene with 4 mm radius notches,
and in this case the damping peak at -50°C did not coincide with the
impact strength peak observed at a somewhat much lower temperature. His
results on PTFE, tested with very sharp notches, showed a great similarity
between damping peaks and impact strength peaks as a function of
temperature.
From the above review on the relation between impact strength and
mechanical losses in polymers, the author concludes that low temperature
loss peaks in polymers are neither a necessary nor a sufficient condition
to guarantee peaks in their impact strength.
1.10 FRACTURE MECHANICS APPROACH TO IMPACT
The results from conventional impact testing are expressed in terms
of the specific fracture energy WA, where W is the energy absorbed to
break the notched specimen and A is the cross-sectional area of the
fractured ligament. It has been previously discussed that such an
analysis of the data is not satisfactory due to the fact that the
parameter riV4 is very dependent on the dimensions of the test specimen,
the notch length and the type of impact test used. This classical
method of analysis provides no correlation between Charpy and Izod
impact strengths for the same materials. Some recent publications
(Marshall et al (1973), and Brown (1973)) showed that assuming linear
deformations, the linear fracture mechanics theory can be extended to
impact data and Gc, the fracture toughness parameter, can be deduced
-
- 44 -
directly from the absorbed energy measured.
A full literature review on the theories of fracture mechanics will
not be presented here, since many good reviews are available
(e.g. Liebowitz (1968), Turner (1972), Hayes (1970)). The purpose of
this section is to give a short summary of the derivations of parameters
which are used in the main part of the thesis to describe impact failure
in polymers from a fracture mechanics point of view.
1.10.1 The Griffith Approach
The fundamental concepts of fracture mechanics were proposed in the
early 1920's by A.A. Griffith (1921) who explained why materials fail at
stress levels well below those that could be predicted theoretically from
considerations of atomic structure. He carried out several studies of
brittle fracture using glass as a model material and he suggested that all
real materials were permeated with small crack-like flaws which act as
localised stress raisers. He argued that at the tips of these flaws
stresses could be raised to'such an extent that the material's
theoretical strength would be reached and failure would result. Thus,
Griffith considered fracture to be dependent on the local conditions at the
tip of a flaw. He formulated the problem in energy terms and proposed
that crack growth under plane stress conditions will occur if:
d (_ 62 IT a2
4ay) = 0 da (1.4)
where the first term inside the parentheses represents the elastic energy
loss of a plate of unit thickness under a stress, a, measured far away
from the crack; if a crack of length 2a was suddenly cut into the plate
at right angles to the direction of a. The second term represents the
-
- 45 -
energy gain of the plate due to the creation of the new surface having a
surface tension y. This is illustrated in Figure 1.3 which is a
schematic representation of the two energy terms and their sum as a
function of the crack length. When the elastic energy release due to
an increment of crack growth, da, outstrips the demand for surface energy
for the same crack growth, the crack will become unstable. A critical
fracture stress could be defined from this instability condition for a
centrally notched plate of infinite dimensions, shown in Figure 1.4 as:
af ✓2Ey/Ira (1.5)
which has been shown in the form afI = constant to hold quite well for
brittle and semi-brittle metals. (af
the critical stress at fracture).
In 1944, Zener and Hollomon (1944) converted the Griffith crack
propagation concept with the brittle fracture of metallic materials for
the first time. Orowan (1945) referred to X-ray work which showed
extensive plastic deformation on the fracture surfaces of materials which
failed in a "brittle" fashion. Irwin (1948) pointed out that the
Griffith-type energy balance must be between the strain energy stored in
the specimen and the surface energy plus the work done in plastic
deformation. He also recognised that for relatively ductile materials
the work done against the surface tension is generally not significant
in comparison to the work done against plastic deformation.
Irwin and Orowan (1949) suggested a modification to Griffith's theory
to account for a limited amount of plastic deformation. Their approach
was simply to add a plastic work factor P to the surface tension y in
equation (1.5). Orowan (1955) noted that the plastic work term was
approximately three orders of magnitude greater than the surface energy
-
46 -
term and hence would dominate fracture behaviour. Both Irwin and
Orowan argued that, provided the zone where plastic deformation takes
place is small in comparison with crack length and specimen thickness,
the energy released by crack extension could still be calculated from
elastic analysis. Under this restriction all the analyses that were
available for Griffith's theory applied to situations where limited
plasticity took place prior to fracture, provided yp replaced y
(where yp = y f P).
1.10.2 Strain Energy Release Rate
Irwin (1948) generalised the Griffith criterion by proposing that
crack propagatiOn occurs when the strain energy release rate (W/3(2)
reaches a critical value. He named the energy release rate G (after
Griffith) and the critical value at fracture, Gc , is known as the
"fracture toughness". Because two new surfaces are formed at fracture -
each requiring surface works- the relation between F7 and yp is given
by:
Gc = 2yp (1.6)
1.10.3 Stress Intensity Approach
Linear elasticity theory provides unique and single-valued
relationships between stress, strain and energy. Therefore, a fracture
criterion expressed in terms of an energy concept has its equivalent
stress and strain criteria. Irwin (1957) produced a fracture criterion
via an analysis of the stress field in the vicinity of the crack. He
considered that fracture can also take place when critical conditions are
attained in the material at the tip of the crack. Using the solution
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47 -
for an elastic cracked sheet obtained by Westergaard (1939), Irwin derived
the solution for the stresses in the vicinity of the crack tip of a
centrally notched plate (Figure 1.4) as:
a = K (27r r) 2. f.. (e) '74 Ij
(1.7)
where r and e are polar co-ordinates with an origin at the crack tip.
Equation (1.7) indicates that identical stress fields are obtained for
identical K values. The parameter K is called the "stress intensity
factor" and is a function of the applied stress and of the crack
geometry. For a crack length 2a in an infinite plate the stress
intensity factor is given by:
K = a (7r a) (1.8)
If the critical stress system under which failure occurs is characterised
by a stress intensity factor, Kc , which is in itself a material
characteristic and is referred to as the "critical stress intensity
factor" or fracture toughness, then a Griffith-type relationship results
without consideration of any energy-dissipation process involved. Kc,
in the same way as Gc, is a material property, but like most material
constants, it is influenced by temperature, strain rate and some other
testing variables. Irwin also identified a simple relationship between
K and G as:
G = K2/E' (1.9)
where E' is the reduced Young's modulus, E for plane stress, and E//-v2
for plane strain (v is the Poisson's ratio). Strictly speaking,
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equation (1.8) is only applicable for a line crack in an infinite plate
and to linear elastic materials exhibiting no more than small scale
yielding, i.e. when the crack length is very much greater than the plastic
zone size or when the ratio of the applied stress to the yield stress is
about 0.7 (Liu, 1965). To apply the Kc concept to a practical test
specimen geometry some modification has to be applied to equation (1.8) to
take into account the finite width of the test specimen. The factor
(TO i ' in equation (1.8) was replaced by Brown and Srawley (1966) by a
correction factor "Y" and the general form of equation (1.8) becomes:
K = a Y
(1.10)
The factor Y depends on the geometry and on the loading configuration of
the specimen in question. For example, for a single-edge notched .(SEN)
plate in tension Y is given by:
Y = 1.99 - 0.41 (a/D) + 18.70 (a/D)2 - 38.48 (a/D)3 4. 53.85 (a/D)4 (1.11)
For single-edge notched bend specimens the correction factor Y is
represented by fourth degree polynomials of the following form:
Y = Ao + Al (a/D) + A
2 (a/D)2 + A
3 (2/D)3 A
4 (a/D)4
(1.12)
For a three point bend test (which is the loading configuration for the
Charpy impact test specimen) the coefficients of the polynomial depend on
the span to depth ratio (2L/D) of the specimen. Brown and Srawley (1966)
derived numerical values for the coefficients for 2L/D = 4 and for
2L/D =.8.
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For:2L/D = 4:
Y = 1.93 - 3.07 (a/D) 14.53 (a/D)2 - 25.11 (a/D)3 25.80 (a/D)4 (1.13)
For 2L/D = 8:
Y = 1.96 - 2.75 (a/D) 13.66 (a/D)2 - 23.98 (a/D)3 4- 25.22 (a/D)4 (1.14)
1.10.4 The Relationship Between Fracture Toughness and Absorbed
Energy for the Charpy Impact Test
Since the conventional types of impact tests record the energy to
failure, an attempt was made by Marshall et al (1973) to develop a
relationship between the recorded impact fracture energy, W, and the
fracture toughness, G,, in polymers. They considered the Charpy impact
test because it appeared to be easier to analyse than the Izod test.
The loading pattern of the Charpy test specimen is identical to the
three point bend bar. In the following analysis, the same relationships
between bending moment, load and stress are assumed to hold as the ones
described by classical bending theory. The strain energy, U, per unit
thickness absorbed in deflecting a cracked elastic test specimen of
thickness B is given by:
U = PA/2B (1.15)
where P is the load and A is the deflection of its point of application.
If the crack a is extended by an amount da, the strain energy release rate,
G, per unit thickness is:
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50 -
du dA G E (2) + cl-.)/2B (1.16)
The compliance is given by:
C = A/P (1.17)
and differentiating with respect to crack length gives:
At constant load:
dC _ 1 dA _ A dP da - P • da TP2- ' dai
dC do P = da ay
(1.18)
(1.19)
Substituting equation (1.19) in equation (1.16) gives.:
p2 c i
2B da (1.20)
Substituting equations (1.9) and (1.10) in equation (1.20) gives:
y2a2a P2 dC
- 28 (da ) (1.21)
The factor Y is given from equations (1.13) and (1.14) depending on the
(2L/D) value. From three point bend theory (Timoshenko (1951)) the
nominal stress, 0-, is given as:
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a = 6P(2L)/4BD2
(1.22)
Combining equations (1.21) and (1.22) and integrating, the compliance C
can be obtained as:
C 9(2E)2 r
j Yea da f C 2BDIIE"
(1.23)
where Co
is the compliance for zero crack length. From the conventional
theory of three point bending:
= (2L)3/4EBD3 (1.24)
Thus, if the only energy absorbed, W, were the elastic strain energy, UB,
then from equations (1.15) and (1.17):
P2 U =2 c
2B
(1.25)
then by substituting for C from equations (1.23) and (1.24), and expressing
P in terms of a from equation (1.22), equation (1.25) gives:
W = GB [f Y2a da f (2L) Tyza (1.26)
= GBD4,
(1.27)
where = [f Y2x dx + (18LD)1 /y203
(1.28)
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where x = a/D the non-dimensional crack length, referred to as the "crack
depth": From equation (1.28) it is clear that the quantity (I) is a
function of the non-dimensional crack length (a/D) as well as of (2L/D).
Marshall et al (1973) developed curves of (I) against (a/D) for
2L/D = 4, 6 and 8.
At fracture G = G and equation (1.27) become:
G = W/PD(I) (1.29)
The above equation gives a powerful relationship between the fracture
toughness Gc and the energy to fracture W. They used PMMA as a model
material and theY tested a number of sharply notched specimens with various
crack lengths in the Charpy mode of failure. The results of W versus BD4)
followed a predominantly linear pattern as expected from equation (1.29).
Contrary to expectation, however, the line did not pass through the origin,
a least square fit to the data showing that there was a positive
intercept w' on the energy axis, implying that there is some additional
form of energy to be considered. Nonetheless, Marshall et al (1973)
showed the slopes of the lines for different specimen geometries were
very consistent, thereby implying a constant value of Gc independent of
both notch length and specimen size. They considered the positive
intercept W' to be interpreted as the kinetic energy loss term. They
estimated the kinetic energy loss term from classical mechanics and
argued that it will depend on the relative sizes of the specimen and
pendulum.
From classical mechanics, a mass M (the pendulum) striking, with
velocity V, a mass m (the test specimen) at rest, will impart to it a
velocity v where:
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= 53 -
v = V (n m) (1 e)
(1.30)
e is the coefficient of restitution (e = 0.58) (Paper I). Thus, the
positive intercept iv' on the energy axis can easily be evaluated from
the kinetic energy equation as:
W' = z m v2 = z m V2 (in )2 (1 4. e)2
(1.31)
Marshall et al (1975) evaluated W' for various specimen dimensions.
(Charpy data in this thesis have W' = 0.01 Joule1). Fraser and Ward
(1974) followed a slightly different approach to calculate the kinetic
energy of a bend specimen (four point bend). They assumed that at
fracture the specimen halves are thin bars rotating about their outer
support points, with the inner (striking points) moving with the same
velocity (v) as the striking pendulum. They considered an element of
thickness dy from the half broken specimen, a distance y from the outer
support. The velocity of this element is V A and its mass is given by
BD dy E where B is the thickness and D is the width of the specimen.
£ is the density of the material. The kinetic energy of the element in
this case is: Vt
15 BD dy E (1J,L)2 (1.32)
and the kinetic energy of the whole specimen is:
V2 W / = BD e f y2 dy
2,2 —x (1.33)
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(t+g) is half of the span for the three point bend case, i.e. t f g = L.
Brown (1973) also applied the fracture mechanics theory to the
ure-Mt- Charpy impact energy data for polymers. He tested math.a.Rel-, polycarbonate,
amorphous polyetheleneterephthalate (PET), high molecular weight PET and
ABS and attempted to determine their fracture toughness. The data
obtained for all the polymers tested (except ABS) when plotted followed
a predominantly linear pattern and thus the fracture toughness was easily
defined. However, for ABS the plot was not linear, making the
determination of G impossible in this case. The deviation from
linearity is due to the fact that the theory assumes linear elastic
behaviour. At this point it must be emphasised that ABS and some other
ductile polymers (e.g. high impact polystyrene (HIPS)) undergo
considerable plastic deformation even at these high impact speeds and
thus some correction has to be considered to account for small scale
yielding as it will be discussed in section 4.6.
1.10.5 Plastic Zone Size
Linear fracture mechanics provides a method of measuring the "brittle"
strength of a material by using the linear elasticity solution for a
mathematically sharp crack tip (equation (1.7)) (i.e. radius of
curvature of the crack tip is "zero"). In reality, however, it is
impossible for a mathematically sharp crack tip to be achieved and thus
some plastic yielding certainly takes place during loading and the stress
level always remains finite. If plasticity phenomena are negligible in
relation to the phenomena occurring in the elastically stressed region,
the error will be negligible. As circumstances develop which increase the
ratio of volume subjected to plastic flow to volume under elastic
conditions the error will increase. Thus it is necessary to ensure that
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- 55 -
the errors introduced by plastic yielding are very small or adequately
corrected for.
Irwin (1960) proposed a plastic zone correction factor, r , to take
into account small-scale plastic yielding at the crack tip. In this
case, the stress field can be adequately described by linear elasticity
theory and the approximate plastic zone size can be obtained from
equation (1.7) by the simple yield criterion that Gyy = ay and since
fYY (0) = 1 for o.= 0, then:
1K 2 r = p 2 IT
(1.34)
Irwin then suggested that the crack length should be adjusted to include
this plastic zone estimate, and that the new crack length should be r
longer than the original crack length.
At the onset of fracture where K = K, the error introduced by
plastic yielding could be estimated from the ratio (r Az) = (1/27ra)(K/a )2
which is equal to 2(G,A1 Ys)2, where ccf. is the gross fracture stress.
From this, it is evident that fracture mechanics is a good mathematical
model as long as the gross fracture stress is small compared to the yield
stress of the material. Irwin proposed that stresses up to 0.7 a could
be dealt with. A fracture mode change, from plane stress to plane strain,
may be accompanied by a drastic change in plastic zone size and a fracture
mechanics analysis may well apply to the plane strain condition but not
to the plane stress condition. (Irwin et al (1958) and Irwin (1960)).
1.10.6 Fracture Toughness and Specimen Thickness
A fracture mode change can he caused by a change in the thickness of
the test specimen. A plane strain situation exists when B » r , where
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-56-
B i$ the specimen thickness. Irwin (1960), Bluhm (1961), Repko et al
(1962), Bluhm (1962) postulated that Kc (or Gc ) is strongly dependent on
the specimen thickness and only after a certain thickness had been
exceeded could K (or G) be regarded as a material property dependent
only on the testing environment. It has been shown that KC and Gc
increase as the specimen thickness decreases and the fracture mode
changes from plane strain to plane stress. Under plane strain conditions
the fracture toughness has its minimum value denoted by kic or G . IC
To compare the fracture mode transition behaviour of various materials
Irwin (1964) considered it convenient to express the specimen thickness
in terms of a non-dimensional parameter a where:
K a = —B
A , c*2
y
(1.35)
From equations (1.30) and (1.31) the ratio of plastic zone size to
specimen thickness is given as a/7. He showed experimentally, for a
large variety of high strength metals, that when the plastic zone size
was less than the specimen thickness, i.e. a < 7) most of the specimens
showed less than 50% shear. When the plastic zone size was greater than
twice the specimen thickness, a > 2', the shear lips occupied nearly 100%
of the specimen thickness. Irwin proposed that the fracture mode
transition from flat fracture (plane strain) to shear fracture (plane
stress) occurs at the region around a . 2.4.
1.11 INSTRUMENTED IMPACT
In conventional types of impact tests the impact strength is reported
in terms of the energy absorbed by the specimen when it is struck and fails
under impact. It has been argued that this conventional impact strength
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-57-
energy could be much greater than the actual energy to failure. This
discrepancy probably arises because after the test specimen has reached
the elastic limit it does not break but it starts to yield and thus some
form of energy could be absorbed during this plastic drawing process.
It was mainly the consideration of this rather complicated yielding
process in the impact behaviour that led to the development of impact
testing equipment that would show the load time relationship of the
specimen during impact. This type of impact test is referred to as the
"instrumented impact test". Wolstenholme (1962) gave a description for
such an instrumented impact tester of the Izod type.
The equipment consists of a strain gauge transducer connected to the
specimen with an oscillascope to display the transducer output and a
camera to record the oscillascope trace. The oscilloscope y-axis
deflection is calibrated directly in load units, and the calibrated x-axis
provides the time base. It can be seen from this brief description that
most types of impact testing equipment could be modified in a similar
manner to provide the dynamic stress-time data. Wolstenholme reported
that three general types of impulse curves could be recorded for various
materials according to the degree of ductility. Schematic diagrams for
these three types are illustrated in Figure 1.5.
1.11.1 The Fracture Mechanics Approach to the Instrumented Impact Test
In recent years the fracture mechanics approach has been further
extended to the instrumented impact test. The main advantage of applying
fracture mechanics concepts to instrumented impact test data rather than
to