Tear Energy of Natural Rubber Under Dynamic Loading

Home , Natural rubber

TEAR ENERGY OF NATURAL RUBBER UNDER DYNAMIC LOADING

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

Linling Chen

December, 2008 TEAR ENERGY OF NATURAL RUBBER UNDER DYNAMIC LOADING

Linling Chen

Thesis

Approved: Accepted:

______Advisor Dean of the College Michelle S. Hoo Fatt George K. Haritos

______Faculty Reader Dean of the Graduate School T. S. Srivatsan George R. Newkome

______Department Chair Date Celal Batur

ii ABSTRACT

This study focuses on the fracture behavior of natural rubber under dynamic loading. Experiments were performed to determine the tear energy of rubber using trouser and pure shear specimens. The sample strain rates of the pure shear specimen were allowed to range from 0.01 to 311 s -1 by performing the tests with an MTS servo- hydraulic machine for sample strain rates 0.01 to 10 s-1 and a Charpy tensile impact

apparatus for sample strain rates exceeding 10 s-1. Finite element analysis was used to

define the pure shear region of the pure shear specimen. It was found that the tear energy

varied with sample strain rates and specimen geometry and therefore, it may not be a

proper parameter to predict the fracture of natural rubber. Finite element analysis of the

fractured specimen indicated that comparing the strain energy density and the maximum

principal stress at the crack tip to the toughness and tensile strength, respectively, may be

more valuable in predicting crack initiation.

iii ACKNOWLEDGEMENTS

I would like to acknowledge the National Science Foundation, and I would like to thank the Department of Mechanical Engineering at The University of Akron for providing funding and facilities during my graduate studies and research work.

I express my gratitude to my advisor, Dr. Michelle S. Hoo Fatt, for her constant guidance, continuous advice and encouragement that led me throughout this research. I would like to thank Dr. T. S. Srivatsan and Dr. Xiaosheng Gao for agreeing to be on the defense committee and for their valuable suggestions.

Also, I thank Amy Blake and Glenn Jansen for their assistance in the pure shear fracture tests. I also appreciate the helpful advice from Min Liu.

I dedicate this thesis to my parents and grandparents who have been a constant source of inspiration.

iv TABLE OF CONTENTS

Page

LIST OF TABLES ...... vii

LIST OF FIGURES ...... viii

CHAPTER

I. INTRODUCTION ...... 1

II. LITERATURE REVIEW ...... 4

2.1 Introduction of Fracture Mechanics ...... 4

2.2 Historical Overview of Fracture Mechanics ...... 5

2.2.1 Griffith’s energy relation (1920) ...... 5

2.2.2 Linear-elastic fracture mechanics - Irwin's modification (1956) ...... 6

2.2.3 Elastic-plastic fracture mechanics - J.R. Rice (1960) ...... 8

2.2.4 Fully plastic fracture mechanics ...... 8

2.3 Rubber Fracture Mechanics ...... 9

III. EXPERIMENTS ...... 15

3.1 Materials ...... 15

3.2 Trouser Test ...... 16

3.2.1 Specimen preparation...... 16

3.2.2 Test procedure ...... 17

3.2.3 Data processing ...... 17 v 3.2.4 Trouser test results ...... 19

3.3 Pure Shear Test ...... 21

3.3.1 Specimen preparation...... 21

3.3.2 Test procedure ...... 22

3.3.3 The Charpy impact test ...... 23

3.3.4 Pure shear region of pure shear specimen...... 37

IV. FINITE ELEMENT ANALYSIS ...... 40

4.1 Mesh Convergence Study ...... 40

4.2 Mesh Study around the Crack Tip ...... 45

4.3 Element Type Study ...... 47

4.4 Pure Shear Region...... 49

4.4.1 Pure shear specimen without crack ...... 49

4.4.2 Defining the pure shear region ...... 50

4.4.3 Results ...... 55

V. RESULTS AND DISCUSSION ...... 56

5.1 Results of Pure Shear Fracture Tests ...... 56

5.2 Comparison with Tensile Strip ...... 58

5.3 Other Significant Fracture Parameters ...... 61

VI. CONCLUSION AND FUTURE WORK ...... 65

REFERENCES ...... 67

vi LIST OF TABLES

Table Page

3.1 Formulation of NR0 and NR25...... 16

3.2 Tear strength for trouser tests...... 20

3.3 Drop heights converted to strain rates...... 32

4.1 Hyperelastic constants for NR0 and NR25...... 41

4.2 Convergence study from 3500 elements to 56000 elements...... 42

4.3 Tear energy of NR0 and NR25 from trouser tests and pure shear tests...... 55

vii LIST OF FIGURES

Figure Page

2.1 Tay bridge disasters (taken Ref. [3])...... 5

2.2 Simplified family tree of fracture mechanics ...... 9

2.3 Stress-strain curve ...... 11

2.4 Three fracture tests: (a) Tensile strip, (b) Trousers and (c) Pure shear ...... 12

2.5 Crack growth with a distance of ...... 13

3.1 Trousers test pieces: (a) Plan view and (b) Side view and force orientation ...... 16

3.2 ASTM standard trouser test ...... 18

3.3 Load vs. load separation in NR0 trouser test ...... 19

3.4 Load vs. grip separation in NR25 trouser test...... 20

3.5 Geometry of pure shear specimen...... 21

3.6 Razor used to create the crack ...... 22

3.7 Side view of Charpy impact pendulum ...... 24

3.8 Schematic of Charpy impact pure shear fracture apparatus ...... 25

3.9 Free body diagram of the reaction grip ...... 26

3.10 Effect of inertial resistance and friction on the total specimen force of NR0 at 40 in drop height ...... 29

3.11 Force vs. time from piezoelectric load cells ...... 30

viii 3.12 Strain vs. time of NR0 pure shear specimen at 1m drop height ...... 31

3.13 Snapshots of unfilled NR pure shear specimen at 229 s -1 sample rate: (a) Start of test, t=0 s, (b) Blunting, t=0.001s, (c) Fracture onset, t=0.003s, (d) Intermediate crack propagation, t=0.0115s, (e) Tearing and necking, t=0.0135s, and (f) Final fracture, t=0.0175s ...... 33

3.14 Gage length vs. time from high-speed camera ...... 34

3.15 Synchronize force vs. time ...... 34

3.16 Stress-strain curve of NR0 at 40in drop height ...... 35

3.17 Engineering stress-strain curves for NR0 ...... 36

3.18 Engineering stress-strain curves for NR25 ...... 36

3.19 Determing crack speeds in the pure shear test...... 37

3.20 Schematic diagram of pure shear test piece ...... 38

4.1 A whole pure shear specimen with a sharp crack ...... 41

4.2 Convergence of maximum principal stress at the crack tip ...... 42

4.3 Convergence of strain energy density at the crack tip ...... 43

4.4 Result of the mesh around the crack tip after convergence study...... 44

4.5 Bias mesh around the crack tip ...... 44

4.6 Mesh study with radius around the crack tip: (a) Bias mesh at the crack tip, (b) System bias mesh at the crack tip, (c) Features assigned bias mesh at the crack tip and (d) Result of features assigned bias mesh around the crack tip ...... 46

4.7 Comparison of integration points: (a) C3D8 2x2x2 integration point and (b) C3D8R 1x1x1 integration point ...... 48

4.8 FEA comparisons of strain energy density with: (a) C3D8 elements and (b) C3D8R elements ...... 48

4.9 Pure shear specimen without crack ...... 49

4.10 Normal stress distributions in center pure shear specimen without crack ...... 49

ix 4.11 FEA mesh of upper half of pure shear fracture specimen ...... 50

4.12 Comparison of ALLAE with ALLSE ...... 51

4.13 Pure shear region in NR25 pure shear specimen ...... 52

4.14 Pure shear region comparison of viscous vs. hyperelastic: (a) t=15ms, (b) t=35ms and (c) t=55ms ...... 54

5.1 Variation of tear energy with far-field sample strain rate from pure shear fracture specimen ...... 57

5.2 Variation of crack speed with far-field sample strain rate from tensile strip specimen ...... 58

5.3 Comparison of the variation of tear energy with far-field sample strain rate from both tensile strip specimen and pure shear specimen: (a) Unfilled NR and (b) 25 phr carbon black-filled NR ...... 59

5.4 Comparison of for unfilled NR data with Lake et al. (2000) for tensile and pure shear tests ...... 61

5.5 Contour plot of the strain energy density at the onset of crack propagation in the NR25 pure shear specimen at 10 s -1 ...... 62

5.6 Contour plot of the maximum principal stress at the onset of crack propagation in the NR25 pure shear specimen at 10 s -1 ...... 62

5.7 Variation of strain energy density with rate of first invariant of left Cauchy Green deformation tensor for 25 phr carbon black-filled NR ...... 63

5.8 Variation of maximum principal stress at the crack tip with rate of first invariant of left Cauchy Green deformation tensor for 25 phr carbon black-filled NR ...... 64

x CHAPTER I

INTRODUCTION

Natural and synthetic rubbers, because of their large extensibility and good damping capability, are now widely used in many products. Ranging from household to industrial products, these elastomeric materials enter the production stream as either intermediate parts or as final products. Rubber is used in door and window profiles, hoses, belts, matting, flooring and damping elements for automotive industry. Rubber gloves are also widely used in the medical field. However, tires and tubes make up the largest consumption of rubber.

Virtually, all rubber materials contain defects, such as flaws, porosity, fatigue cracks, or stress corrosion cracks. Defects cannot be avoided and will always occur.

These defects arise either during production or during service. Fracture or catastrophic failure of rubber components will eventually occur due to the defects. To make better use of rubber products, engineers are now obliged to investigate and gain more and deeper understanding of the behavior of rubbers.

The objective of this research is to determine fracture characteristics of rubbers under dynamic loading. The fracture toughness is a parameter that is often used to characterize a material resistance to fracture when defects are present. It is a material

1 property independent of specimen geometries and loading. In rubber, the fracture toughness is referred to as the fracture or tear energy, G, as will be discussed in Chapter

II. The maximum principal stress is another parameter that could describe the driving force to open a crack. The crack will propagate in the direction perpendicular to the maximum stress around the crack tip. The critical strain energy density yet another important material failure parameter to describe how much strain energy can be stored in the material before it breaks. This parameter can also be used to investigate crack propagation in rubber. However, tear energy (G) is the most popular of all the above mentioned material properties.

Conventional quasi-static strain rates are generally below 1 s -1, but rubber tires

will experience higher strain rate than this when they are in use. Hussain [1] calculated

that when an automobile is traveling between 35-125 mph, the strain rates in the tire foot

print will vary from 155-416 s -1. To simulate real life conditions, this fracture study will

generate data for strain rates ranging from 0.1 to 300s -1, for which the vehicle speed is around 90 mph. In this thesis, fracture experiments will be conducted on unfilled and carbon black-filled natural rubber, two compounds that were characterized under high strain rate by Hussain [1].

Natural rubber, because of its ability to undergo strain-induced crystallization, provides an outstanding tensile strength while being used. However, natural rubber (NR) is very expensive. To reduce cost, people often use synthetic rubbers, such as Styrene

Butadiene Rubber (SBR). Styrene Butadiene Rubber cannot crystallize and are about

90% weaker when compared to NR. Carbon black and silica are used as filler material to

2 improve the stiffness and abrasion resistance of rubbers. For example, fillers can increase the strength of SBR more than tenfold [2].

Hussain [1] found that at quasi-static rates of loading, small amounts of carbon black increased the tensile strength by increasing strain-induced crystallization of NR, whereas too much carbon black impeded strain-induced crystallization. Natural rubber with 25 phr N550 carbon black was considered as the optimum amount of filler to give the highest tensile strength. In this study, two types of fracture test will be performed using unfilled NR and NR with 25 phr N550 carbon black specimens. The two fracture tests include a pure shear test and trouser test.

Quasi-static fracture tests will be conducted on an INSTRON electro-mechanical machine and dynamic tensile tests will be accomplished using an MTS servohydraulic machine and a Charpy tensile impact apparatus. Finite element analysis using ABAQUS explicit will be done to identify the pure shear region in the pure shear test specimen.

The pure shear region is part of the whole specimen and it is the only region that stores the strain energy.

A critical literature review on fracture mechanics of rubber will be given in

Chapter II; how to calculate G, the strain energy for the three models will also be included in this chapter. Chapter III will introduce the Charpy tensile impact apparatus, experimental procedures, specimen preparation and experimental results. Finite element analysis of a pure shear specimen under dynamic loading will be covered in Chapter IV.

Finally, the conclusions and a discussion of future work will be given in Chapter V.

3 CHAPTER II

LITERATURE REVIEW

2.1 Introduction of Fracture Mechanics

Fracture mechanics is the study of fracture in materials that contains defects or flaws. It includes methods to calculate the driving force of a crack and characterize the material's resistance to fracture. Fracture mechanics is an important tool to improve the mechanical performance of materials and components. It encompasses the physics and the relationship of stress and strain by using the theories of elasticity and plasticity of materials.

In many cases, fracture failure of structures can be catastrophic; one example is the Tay Rail Bridge [3] shown in Figure 2.1. Disasters like this occur because engineering structures contain defects, such as weld flaws, porosity, forging laps, fatigue cracks, or stress corrosion cracks [4]. Some defects will occur either during production or during service. These cracks will grow and their growth will lower the strength of the structure. This became the reason for evaluating the fracture strength of pre-cracked structures in fracture mechanics.

Figure 2.1 Tay bridge disasters (taken Ref. [3]).

2.2 Historical Overview of Fracture Mechanics

Fracture mechanics is the study of mechanics concerned with the breaking of structures which contain material defects.

2.2.1 Griffith’s energy relation (1920)

Fracture Mechanics was first developed by an English aeronautical engineer, A.

A. Griffith, during World War I. Griffith suggested that “the reason for observed strength of glass being much less then that expected on the basis of the theoretical forces between molecules was the presence of small cracks, which would create stress concentrations at their tips, thus reducing the measured strength” [5]. Later Griffith developed a thermodynamic approach. He assumed that growth of a crack requires creation of surface energy, which is supplied by the loss of strain energy accompanying the relaxation of local stresses as the crack advances. Failure occurs when the loss of 5 strain energy is sufficient to provide the increase in surface energy. Griffith’s approach is described as an energy balance written for the whole specimen. For the simple case of a thin rectangular plate with a crack perpendicular to the load Griffith’s theory becomes

πσ 2a G = E (2-1)

where σ is the applied stress, a is half the crack length, and E is the Young’s modulus.

The strain energy release rate can otherwise be understood as the rate at which energy is absorbed by growth of the crack.

2.2.2 Linear-elastic fracture mechanics - Irwin's modification (1956)

Griffith’s work was ignored for over twenty years until a group under G.R. Irwin

[6, 7] at the U.S. Naval Research Laboratory (NRL) brought it up during World War II.

Back then the naval materials were not perfectly elastic but underwent plastic deformation at the tip of a crack. Researchers were obliged to investigate and make assumption to develop a theory for linear-elastic fracture mechanics.

Based on Griffith's approach, Irwin and his colleagues developed a modified form of energy balance. Their work defined a new material property, the fracture toughness K, which is now universally accepted as the defining property of fracture mechanics.

Irwin’s [7] modification of Griffith’s theory emerged from this work: a term called the stress intensity factor replaced strain energy release rate and a term called the fracture toughness replaced surface energy. Both of these terms are simply related to the energy terms that Griffith used as follows:

= σ π K I a (2-2)

6 and

= K C EG (plane stress) (2-3)

EG K = (plane strain) (2-4) IC 1−υ 2

ν where K I is the stress intensity, K C or K IC is the fracture toughness, and is Poisson’s ratio. It is important to recognize the fact that fracture toughness has different values when measured under plane stress and plane strain.

≥ Fracture occurs when K I K C . For the special case of plane strain deformation,

K C becomes K IC and is considered a material property. The subscript I arises because of the different ways of loading a material to enable a crack to propagate.

There are three ways of applying a force to enable a crack to propagate:

Mode I crack – Opening mode.

Mode II crack – In-plane shear mode.

Mode III crack – Out-plane or tearing mode.

Note that the expression for K I in Equation (2-2) will be different for different

geometries, and it is necessary to introduce a dimensionless correction factor, Y, in order

to characterize the geometry. We thus have

K Yσ πa I = (2-5) where Y is a function of the crack length and width of sheet given by

a πa Y ( ) = sec( ) W W (2-6) for a sheet of finite width W containing a through-thickness crack of length 2a, or

7 a 41.0 a 18 7. a Y ( ) = 12.1 - + ( ) 2 ...- π π W W W (2-7) for a sheet of finite width W containing a through-thickness edge crack of length a.

2.2.3 Elastic-plastic fracture mechanics - J.R. Rice (1960)

In the mid-1960s, Rice [8, 9] developed a new toughness measure to express phenomenon where there the deformation is sufficient at the crack tip that no longer follows the linear-elastic approximation. Rice's assumption of non-linear elastic deformation ahead of the crack tip is designated by the J-integral. The limit of this analysis occurs to the situations where plastic deformation at the crack tip does not extend to the furthest edge of the loaded part. It commands that the assumed non-linear elastic behavior of the material has a rational approximation in shape and the magnitude

to the real material's load response. The elastic-plastic failure parameter is designated J IC

and it could be conventionally converted to K IC by

EJ IC K IC = 2 -1 ν (2-8)

The J-integral approach reduces to the Griffith theory for linear-elastic behavior.

2.2.4 Fully plastic fracture mechanics

If the material is so tough that the yielded region ahead of the crack extends to the far edge of the specimen before fracture, the crack is no longer an effective stress concentrator. Instead, the presence of the crack merely serves to reduce the load-bearing

8 area. In this case, the failure stress is conventionally assumed to be the average of the yield and ultimate strengths of the material. Neither G nor J would apply in fully plastic fracture mechanics.

2.3 Rubber Fracture Mechanics

Rubber-like materials exhibit highly nonlinear elastic and hyperelastic behavior.

Rubbers have large deformation and can be stretched 300-700% before break. Fracture properties of rubber are also rate-dependant. Hence, the above-mentioned methods are invalid for rubber. Figure 2.2 [4] shows a general view of fracture mechanics studies.

Our rubber fracture belongs to time-dependent material.

Linear Linear Elastic Time-Independent Fracture Materials Mechanics

Elastic-Plastic Nonlinear Fracture Time-Independent Mechanics Materials

Dynamic Visco-elastic Visco-plastic Time-Dependent Fracture Fracture Fracture Materials Mechanics Mechanics Mechanics

Figure 2.2 Simplified family tree of fracture mechanics .

9 The study of dynamic fracture analysis which deals with nonlinear and time- dependent material behavior is a very new field. With rapidly advancing measuring devices, such as high speed cameras, data acquisition systems, and super computers, it is now possible to study the process of fracture under high loading rates. Because of the rate sensitivity of rubber, even strain rate above 0.01 s-1 may be considered as high loading rates. We will study the fracture of rubber under impact and dynamic in the attempt of modeling the most severe misuse which materials can be subjected in the real life.

Rivlin and Thomas [10, 11] adapted Griffith’s approach to the tear behavior for rubber by defining the elastic strain energy released rate as a crack grows in rubber.

Later, this approach was verified experimentally by Lake [12-15], who compared the behavior of test pieces of different shapes and showed that crack growth was independent of the test piece geometry, so that critical G c became a true fracture property of a material.

In this research, we will use the energy approach to determine the energy release rate of crack growth right before the crack propagation in the material. As what has been discussed earlier, the main theory from the energy balance for crack growth is expressed by

∆ ∆ ∆ ∆ WS + WP = G( A) + WE (2-9)

∆ where WS is the net release of stored strain energy or the so-called potential energy

stored in the specimen, ∆WP is the energy gained or the work done directly by test

∆ machine, A is the cross section of fracture plane, and ∆WE is the new surface strained energy. On the left side of this equation is the energy supplied to the crack front and the

10 right side is the energy expended during crack growth. Here, G, the tear energy is due to the combination of breaking bonds across fracture plane and the hysteretic losses that are approximately independent of test geometry but depend on exposure temperature and the

applied load. The stored strain energy density, U b , at which crack propagates is considered the area that under the stress-strain curve shown in Figure 2.3.

Stress σ b

U b

Strain Figure 2.3 Stress-strain curve .

When the stress-strain curve is linear

U = σ 2 2/ E b b (2-10)

σ where E is the Young’s modulus, and b is the stress at break.

Three commonly used rubber fracture test pieces are shown in Figures 2.4 (a)-(c).

For the tensile and pure shear test pieces, we have both ∆WP and ∆WE equal to zero.

This is because the test machine does no work to open the crack and there are no new surfaces strained before the crack propagates.

F F t

c t ∆ c l0

F F (a) Tensile strip. (b) Trousers specimen.

F t

∆ h a c 0

Figure 2.4 Three fracture tests: (a) Tensile strip, (b) Trousers specimen and (c) Pure shear specimen. 12 From Equation (2-9), we have for the tensile and pure shear

∆ = ∆ WS G( A) (2-11)

∆ = ∆ ∆ where A t c and WS is different for the different test geometries.

For the simple tensile strip

∆ Ws = U b (V f -Vi ) (2-12)

and the initial volume of material Vi is

V = kc 2t i (2-13)

where the final volume of material V f is

V = k(c + ∆c) 2 t = k(c 2 + 2c∆c + ∆c 2 )t f (2-14)

∆ 2 = Here c 0 . The change in volume, V f -Vi , is the volume of material that becomes

unstressed as a result of crack growth of a distance of ∆c , as shown in Figure 2.5. The shaded region is the energy lost when the crack propagates.

Figure 2.5 Crack growth with a distance of ∆c.

By substituting Equations (2-13) and (2-14) into Equation (2-12) and satisfying

Equation (2-11), one gets

2 σ b kc G = E (2-15) where k = π 1( + ε ) − 2/1 is small if the stress-strain curve is linear and big otherwise.

For the pure shear test,

∆W = U (hct ) s b (2-16)

Substituting Equation (2-16) into Equation (2-11) gives

G = U h b (2-17)

∆ For the trousers testWS = 0 . However ∆WP is not zero because work is being done by the test machine. This work is given as

∆W = F 2( c + 2e) p (2-18)

where c is the crack length, e is the strained length due to the increase strain in one leg as

the crack grows. The factor of 2 in Equation (2-18) indicates that the trousers test has two

legs. The new surface strain energy ∆WE is given by

(F / wt ) 2 ∆W = 2( cwt ) (2-19) E 2E

(F /ωt) 2 The entire term is the stored strain energy density in legs and 2cωt is 2E

the newly strained volume in legs. Satisfying Equation (2-9) for the trouser test gives

2F G = t (2-20)

14 CHAPTER III

EXPERIMENTS

Trouser and pure shear fracture tests on both unfilled NR and NR with 25 phr

N550 carbon black fillers were performed using two types of test apparatus. An MTS servo-hydraulic machine was used to execute quasi-static and intermediate rate tests, while a Charpy tensile impact apparatus was used for tests at high rates. Only quasi- static tests were done on the trouser test pieces. This chapter covers the experimental setup, procedures, and the results from these tests.

3.1 Materials

The unfilled NR (NR0) and NR with 25 phr N550 carbon black fillers (NR25) were prepared according to standard ISO 2393 methods [16]. The formulations, cure time, cure temperature and cure pressure for each material are listed in Table 3.1. A

152.4mm x 152.4mm x 2.54mm thick (6in x 6in x 0.1in thick) mold was used to cure the rubber in sheet form.

15 Table 3.1 Formulation of NR0 and NR25.

NR0 NR25 Ingredients Amount (phr*) Amount (phr*) NR 100 100 Process Oil 2 5 Stearic Acid 2 2 Zinc Oxide 5 5 6 PPD 1 1.5 Sulfur 2.75 2.75 MBTS 1 1 TMTD 0.1 0.1 Carbon Black N 550 0 25 *Parts by weight per 100 parts by weight of rubber. NR0: Cure for 11.2 minutes at 150˚C (300˚F) and 2.07 MPa (300 psi). NR25: Cure for 9 minutes at 150˚C (300˚F) and 2.07 MPa (300 psi).

3.2 Trouser Test

The trouser fracture test follows ASTM standard: D624-00 [18]. The geometry and dimension will be shown as in Figure 3.1 (a) and (b).

3.2.1 Specimen preparation

The trouser fracture test piece was cut from the 152.4mm x 152.4mm x 2.54mm thick rubber sheets with a razor and hammer.

15 mm 7.5 mm

F F 40mm F 2mm

150 mm

(a) Plan view. (b) Side view and force orientation.

Figure 3.1 Trousers test pieces: (a) Plan view and (b) Side view and force orientation.

16 3.2.2 Test procedure

The trouser fracture tests were conducted using standard procedures from ASTM

D624-54 [17]. These procedures are as follows:

1. Place the test pieces in the grips of the MTS machine.

2. Adjust the clamped test pieces in the grips so that the whole specimen would

be strained uniformly, and avoid slippage.

3. Operate the MTS machine in displacement control mode with the rate of jaw

separation equal to 50 mm/min.

4. Strain the test piece until it is completely ruptured.

5. Record displacement and force.

3.2.3 Data processing

The output from the system included the time, force and grip separation distance.

Figure 3.2 shows the load vs. grip specimen graph from one of the tests. The curve

illustrates a characteristic tear, commonly called “knotty tear,” which means a large

magnitude transient increase in tearing force followed by a precipitous decrease. For this

type of tear, the increase and decrease process repeated in a cyclic fashion. Each

increasing force stage eventually produced a rapid tear rupture, which relieved

concentrated stress and increased the torn length [18].

17 14 X 12 X

10 O 8 X X

6 Load (N) O O 4

0 0 50 100 150 200 Grip separation (mm)

Figure 3.2 ASTM standard trouser test.

As discussed in Section 2.3, the tear strength for trousers specimen was calculated

2F G = t where F can be the peak (denoted by “ X ”in Figure 2), valley (denoted by “ O ”in Figure

2), mean or median force obtained from the force record. In this project, the mean value was selected for F. When processing the data, the mean force was computed from the load values of the peak and the valley, summing them together, and then dividing by the total number of peaks and valleys.

18 3.2.4 Trouser test results

According to ASTM D624-54 [17], a proper test requires at least three tests to be conducted at each strain rate and the median value would then be taken. If one sample displayed an outlier, which means it was outside a 20% range of the median, two more tests should be performed. The median would then be taken from five tests.

Three tests, including the median and two others inside the 20% range of the

median, were processed for both NR0 and NR25, as shown in Figures 3.3 and 3.4,

respectively. It was found that all the curves followed the same pattern: tear forces

increase and decrease in a cyclic fashion or knotty tear.

10 Test 1 Test 2 8 Test 3

Load(N) 6

0 0 50 100 150 200 Grip separation (mm)

Figure 3.3 Load vs. load separation in NR0 trouser test.

19 20 18 16 14

12 Test 1 10 Test 2

Load(N) 8 Test 3 6 4 2 0 0 50 100 150 200 Grip separation (mm)

Figure 3.4 Load vs. grip separation in NR25 trouser test.

By compiling the raw data from the tests, the tear energy for each test was calculated, and they are listed in Table 3.2. The tear energy calculated from the median value are G =8.64 kJ/m 2 for NR0 and G = 10.87 kJ/m 2 for NR25.

Table 3.2 Tear strength for trouser tests.

NR0 NR25 ID Test 1 Test 2 Test 3 Test 1 Test 2 Test 3

G (kJ/m 2) 9.4 8.6 7.3 9.8 11.5 10.9

Standard 9.0 0.0 15.5 9.8 6.1 0.0 Deviation (%)

20 3.3 Pure Shear Test

There is no ASTM standard for the pure shear fracture test. The geometry and dimension of pure shear test piece is shown in Figure 2.4(b).

3.3.1 Specimen preparation

The design for the pure shear fracture specimen was inspired by a fatigue specimen from the ASTM D4482-99 method [19]. To prevent slipping and tearing of the rubber piece in the grips, the ends of the specimen were beaded. For a state of pure shear, the sample would need to break under constrained tension and to achieve this, the ratio of width to height was required to be 10:1. A special mold was designed for producing the pure shear fracture specimen with the 10:1 aspect ratio and beaded ends, as shown in Figure 3.5. The crack length for the specimen was determined from prior testing with crack lengths between 50.8 mm to76.2 mm. A standard 63.5 mm crack length was deemed the appropriate initial crack length whereby the tear energy would be independent of the initial crack length. The tear energy determined with cracks length less than this value was found to vary with the crack length.

diam=6.35 mm 19 mm

63.5 mm 260 mm

Thickness = 2 mm Figure 3.5 Geometry of pure shear specimen.

21 To produce a long crack, a special 152 mm (6 in) long razor blade was used to create the crack incision. Also, a razor holder was designed as shown in Figure 3.6. The crack was produced by hammering the top of the holder, so that a specified distance of the razor would provide a blunt crack in the specimen. The thickness of this razor will effect the FEA simulation, as will be discussed later in Chapter IV.

. F F 0.93 mm

Holder 2.74 mm

152 mm Razor

Figure 3.6 Razor used to create the crack.

3.3.2 Test procedure

The test procedure for the pure shear fracture test using the MTS machine and

Charp device are as follows:

1. Place the beads at each edge of the test piece into the grips.

2. Operate the MTS machine in displacement control mode at constant strain

rate (10 s-1, 1 s-1, 0.1 s-1, 0.01 s-1).

3. Use the Charpy tensile impact apparatus for higher rates. (This will be

discussed in more details later.)

4. Strain the test piece until it has completely ruptured.

5. Record grip specimen distance and force.

6. Record crack tip position using high-speed camera. 22 3.3.3 The Charpy impact test

The conventional Charpy impact test is a standardized high strain-rate test that could determine the amount of energy absorbed by a material (usually steel) during its fracture. The absorbed energy, transferred from potential energy to kinetic energy and eventually strain energy, could be used to measure the toughness of different materials.

For our pure shear fracture test on rubbers, the Charpy impact pendulum was modified as part of the loading mechanism. The general layout of Charpy impact test is shown in

Figures 3.7 and 3.8.

The procedure of the Charpy impact test involved four general steps:

Step 1: The apparatus consisted of a pendulum swing. The pendulum would be first raised to a certain height, which helped to set up the potential energy for the system, and then it would be released to hit a brass slider bar.

Step 2: The brass slider bar, which could travel freely along the steel rail, was connected to one end of the copper/steel cables by a pulley, and the other end of the cables was connected to an eyelet on a movable grip. The potential energy of the pendulum mass at a certain height was transferred to kinetic energy in this step.

Step 3: The movable grip was mounted on linear bearings that slide along rails. It would pull one end of the specimen when the pendulum impacted the brass slider bar.

The other end of the specimen was fixed by a reaction grip, which was also mounted on linear bearings to the rails. The reaction forces from the reaction grip were transmitted to a force link load cell connected to a shaft. The acceleration of the reaction grip was also measured by an accelerometer mounted on it.

23 Step 4: The extension of specimen and position of the crick tip were measured by a high-speed camera, which was triggered on/off by a laser triggering device (see Figures

3.7 and 3.8). Loads, extensions and accelerations were measured as the specimen extended to break. The speed of the tests was controlled by the drop height of the pendulum.

Pendulum Brass Slider Bar

Camera Input Laser Trigge r

Data Acquisition Input Grooved Cables

Figure 3.7 Side view of Charpy impact pendulum.

Laser Trigger

Charpy Impact Pendulum Brass Slider Bar Pulley

Cable

Rails

Eyelet

Specimen Movable Grips

Load Cell Reaction Grips

Power Supply Accelerometer & Signal Conditioner Angle Iron

Data High-Speed Computer Computer Acquisition Camera

Figure 3.8 Schematic of Charpy impact pure shear fracture apparatus.

25 Tensile forces in specimen

The reaction grip on the far end of the base plate was mounted to a rod connected to the load cell. The load recorded by the load cell was lower than the load applied to the specimen. There were two main reasons for this:

1. Friction loss in the linear bearing.

2. Inertial force of the reaction grip.

To improve the accuracy of the system, an accelerometer was attached to the reaction grip; the accelerometer measured the accelerations of the reaction grip. The total force in the specimen could then be determined as the combination of the inertial forces, amount of friction, and the dynamic forces measured by the load cell. A force balance for the specimen is shown in Figure 3.9, where F SP represents the tensile force in the specimen.

FSP X

F m 2 x f && FLC

Figure 3.9 Free body diagram of the reaction grip.

26 The tensile force is calculated from the equation of motion for the reaction grip as follows:

FSP = m2 x&& + FLC + F f (3-1)

where m 2 is the mass of the reaction grip, x&& 2 is the acceleration of the reaction grip, F LC is

the dynamic forces measures by the load cell, and Ff is the friction force.

Load cell and accelerometer

A piezoelectric load cell was used to measure the dynamic forces in the experiments. The amount of impact force in the pure shear fracture experiments was anticipated to be large. In this case, the force link was chosen because it allowed for a high load range. The piezoelectric force sensor, Model 221B03, was chosen; it had a linear range in tension of 0- 500 lb. The load cell was fitted between threaded rods connecting from the reaction grip to a rigidly-supported angle iron. Attached to the reaction grip, an accelerometer measured the accelerations that the grip would see in the dynamic equation of motion. The accelerometer was an ICP accelerometer Model

353B15. An 8-Channel PCB 482A22 Signal Conditioner was used with the force sensor and accelerometer. The force-time data and acceleration-time data were recorded with a

DATAQ DI-720-USB data acquisition system, which had a data acquisition rate of

250,000 data per second and allowed up to 4 channels to collect data at the same time.

Camera with laser triggering device

The Photron Ultima APX FASTCAM monochrome high-speed video camera was employed to record the displacement of the specimen. The videos recorded by the 27 camera provided the extension and the crack speed of the specimens. Data from the high- speed camera was synchronized with the load cell and accelerometer readings electronically.

The laser triggering device shown in Figure 3.7 prompted the high-speed

FASTCAM-Ultima to record data. When the Charpy pendulum hit the slider bar, the

slider bar would cut the laser beam and signal the camera to start recording. Both the

camera and data acquisition were set to capture 50,000 fps and 50,000 data per second,

respectively. These acquisition rates ensured that displacement and force would be

measured every 0.02 ms in the experiments, which usually lasted under 30 milliseconds.

Pure shear data processing

The data processing for the pure shear tests was more complicated than that for the trouser tests. Since the objective of this project was to define the tear energy, we needed to set up a first step of finding the engineering stress-strain curves under different strain rates.

The data acquisition would provide time, load, displacement, and acceleration.

From Section 3.3.3, we know that the total force was determined by the inertial forces, amount of friction, and the dynamic forces. Here, the inertia force would be the acceleration multiplied by 1.8163 kg, which was the mass of the reaction grip. The friction force would be the weight of grip multiplied by the coefficient of friction which was 0.125. Equation (3-1) could then be specified as

FSP = 1.8163 × x&& 2 + FLC + .1 8163 × 81.9 × .0 125 (3-2)

28 Figure 3.10 shows how much the acceleration and friction on the reaction grip would affect the total force of NR0 at 1 m (40 in) drop height.

270 Total force

220 Force from load cell 170

Sum of inertial 120 resistance and friction

Total Force (N) 70

0.2 0.204 0.208 0.212 0.216 0.22 0.224 -30 Times (s)

Figure 3.10 Effect of inertial resistance and friction on the total specimen force of NR0 at 40 in drop height.

The raw data of the total force was displayed with noise, shown as the blue curve in Figure 3.11. This was because the data acquisition system recorded the data point every 0.02 ms, and it was considered an over-saturation of data. To obtain the main signal of the total force, the raw data needed to be filtered and would be smoothed by plotting force - time curve. OriginPro Version 7.5 [20], a commercial scientific graphing and analysis program was used for FFT filtering. It examined the plot and eliminated certain points up to a certain frequency setting. To eliminate unnecessary filter, this procedure was performed for adjusting the total force only. Figure 3.11 shows the force- time plots for both raw and the smooth data. 29 250

200

150 Raw Data 100 Smooth Data Total Force (N) Force Total

0 0.2 0.204 0.208 0.212 0.216 0.22 0.224 Times (s)

Figure 3.11 Force vs. time from piezoelectric load cells.

From Figure 3.11, the fracture point was found at the load drop. However, the start time of the test was not obvious from the force-time plot. This might be because of the material shear lag and inertial effects. The start of test was found from data synchronization with the grip displacement from the high-speed camera. The ProAnalyst

Professional Edition Image Analysis Software [21] was used to track the displacement and velocity of certain points in the video images. The start of the test was when the specimen began to stretch.

Before any further discussion, we need to provide how the drop height was related to the strain rate. As mentioned above, the camera tracked points, which led to the crack tip position and grip separation distance. The strain could then be determined from the original gage length and the current gage length. The strain is the ratio of the specimen’s

∆∆∆ elongation to the original length Lo : 30 ∆ ε = (3-3) Lo

For the pure shear specimen L0 =19 mm.

After plotting the strain-time curve, the slope of the curve would then be

considered as the strain rate of the specific test. Figure 3.12 is an example of the

variation of strain with time for NR0 and a drop height at 1 m (40 in).

1.6 y = 212.23x - 44.131 R2 = 0.999 1.2 Strain 0.8

0.4

0 0.195 0.2 0.205 0.21 0.215 0.22 Time (s)

Figure 3.12 Strain vs. time of NR0 pure shear specimen at 1m drop height.

Table 3.3 is a compilation of various drop heights and the strain rates for all the high strain rate tests. The last two drop heights (40 in and 45 in) were achieved by switching from copper to steel cables. Note the transition in strain rates between the 35 in and 40 in drop height.

31 Table 3.3 Drop heights converted to strain rates.

0.51 m 0.76 m 0.89 m 1.00 m 1.14 m

Drop Height (20 in) (30 in) (35 in) (40 in) (45 in)

NR0 (s-1 ) 132 188 270 229 311

NR25 (s-1 ) 92 123 226 168 277

Snapshots of the experiment for unfilled Natural Rubber specimen strain from 40 in drop height, or about 229 s-1strain rate, are shown in Figures 3.13 (a)-(f). There was

significant blunting at the crack tip before the crack propagated at 0.003 s. In order to

keep track of the crack tip a white dot was painted on to the crack tip with Liquid Paper.

(a) Start of test, t=0 s.

(b) Blunting, t= 0.001 s.

(d) Intermediate crack propagation, t=0.0115 s.

(e) Tearing and necking, t= 0.0135 s.

(f) Final fracture, t= 0.0175 s. Figure 3.13 Snapshots of unfilled NR pure shear specimen at 229 s -1 sample rate: (a) Start of test, t=0 s, (b) Blunting, t=0.001 s, (c) Fracture onset, t=0.003 s, (d) Intermediate crack propagation, t=0.0115 s, (e) Tearing and necking, t=0.0135 s and (f) Final fracture, t=0.0175 s. 33 A plot of the specimen gage length-time curve for the specimen is shown in

Figure 3.14 as well. The start time and the end time of the test are indicated in Figure

3.14. Figure 3.15 shows the corresponding synchronized force-time curve.

30 Start of const. rate Start 0.008 s

Gage length Gage (mm) 15 0.0135 s

0 0.195 0.2 0.205 0.21 0.215 0.22 Time (s)

Figure3.14 Gage length vs. time from high-speed camera.

250 Synchronized Data

200

150

100

(N) Force Total 50

0 0.2 0.204 0.208 0.212 0.216 0.22 0.224 Times (s)

Figure 3.15 Synchronize force vs. time. 34 With the synchronized data of total force, the stress could then be determined.

The engineering stress-strain curve for the specimen is shown in Figure 3.16.

0.7

0.6

0.5

0.4 0.3

0.2 y = 0.1534x 4 - 0.0052x 3 - 0.8182x 2 + 1.2279x + 0.0205 0.1 2 R = 0.9975 Engineering Stress (Mpa) 0 0 0.2 0.4 0.6 0.8 1 1.2 Engineering Strain

Figure 3.16 Stress-strain curve of NR0 at 40in drop height.

For each sample and drop height, 3-5 tests were done and the median was chosen.

Engineering stress-strain curves for NR0 and NR25 are shown in Figures 3.17 and 3.18,

respectively. In Figures 3.17 and 3.18, the curves at 0.1 s-1, 1 s -1 and 10 s -1 were obtained from the MTS machine, and all curves with strain rates higher than 10 s -1 were obtained from the Charpy impact apparatus.

35 1 0.01 1/s

0.1 1/s 0.8 1 1/s

0.6 10 1/s 132 1/s

0.4 188 1/s

229 1/s 0.2 270 1/s Engineering Stress (MPa) Stress Engineering 311 1/s 0 0 0.5 1 1.5 2 2.5

Engineering Strain

Figure 3.17 Engineering stress-strain curves for NR0.

1.8 0.01 1/s 1.6 0.1 1/s 1.4 1 1/s 1.2 10 1/s 1 92 1/s 0.8 123 1/s 0.6 168 1/s 0.4 226 1/s 0.2 Engineering Stress (MPa) Stress Engineering 277 1/s 0 0 0.5 1 1.5 Engineering Strain

Figure 3.18 Engineering stress-strain curves for NR25.

36 Crack speed

We also used the high speed video ProAnalysis software program [20] to track the

crack speed. The position of the trick tip as it varied in time is shown in Figure 3.19. A

trend line for determining the crack speed is indicated by a black line in Figure 3.19.

After fitting this line to a linear equation, we define the gradient of the line as the crack

speed in the test piece.

120

100 y = 20038x - 4331.3 R2 = 0.994 80

Position tip of (mm) crack 20

0 0.215 0.216 0.217 0.218 0.219 0.22 0.221 0.222 0.223 Time (s)

Figure 3.19 Determing crack speeds in the pure shear test.

3.3.4 Pure shear region of pure shear specimen

For a pure shear test, the specimen could be divided into four regions as shown in

Figure 3.20: Region A is an unstrained region, Region B is the pure shear region, Region

C is a complicated strain region and Region D is a region with edge effects. The tear

37 energy is stored only in the pure shear region. During crack propagation, Region C simply shifts into Region B. The energy release rate from which the tear energy is calculated, is therefore found only in Region B of the pure shear region.

C B D

A h

Figure 3.20 Schematic diagram of pure shear test piece.

In order to identify where and how big the pure shear region is, we modified

Equation (2-17) as

= = G U sh h0 KUh 0 (3-4) where U sh is the strain energy density in pure shear region, U is the strain energy density in whole specimen, and K is a factor defined by

∆ ∫ Fsh d ∆ w F d∆ U A h A0 ∫ Fsh d ∫ sh K = sh = sh 0 = = 0 U ∆ ∆ ∆ (3-5) ∫ Fd Ash ∫ Fd wsh ∫ Fd

A0h0 where w0 is the whole width of the test pieces, wsh is the width in the pure shear region,

Fsh is the total reaction force in the pure shear region and F is the total reaction force in

38 the whole specimen. To define the factor K, finite element analysis (FEA) was carried out and this will be discussed in the next chapter.

39 CHAPTER IV

FINITE ELEMENT ANALYSIS

This chapter focuses on defining the pure shear region in the pure shear specimen.

Using ABAQUS Explicit, a mesh convergence study on the pure shear specimen will be done first using hyperelastic material property for rubbers. After finding the best mesh, a hyper-viscoelastic material model will be used to find the pure shear region in the specimen.

4.1 Mesh Convergence Study

In finite element modeling, a finer mesh will typically provide a more accurate solution. However, for different FEA commercial software there is always a limit on the number of elements. In addition to this, computation time and disk storage increase as a mesh is made finer. How to get a mesh that satisfactorily balances the accuracy and computing resources is answered by a mesh convergence study. The procedures of a mesh convergence study are to create a mesh, analyze the terms of interest, increase the mesh density, re-analyzing the same terms of interest and repeat the processes until the terms of interest converge to fixed values. The goal of our convergence study is to estimate the maximum number of elements that allows the program and at the same time

40 find the mesh that is sufficiently dense around the region that we are interested in without overly-demanding computing resources.

First, the entire pure shear specimen with a sharp crack was modeled in FEA, as

shown in Figure 4.1. Hyperelastic material property for NR0 (seen Table 4.1), was

obtained from Al-Quraishi [23] and used for our convergence study. Continuum C3D8 elements (see details in Section 4.3) were chosen as the element type. The specimen was clamped at the lower end and pulled at the upper end as in the test. The test at 10 s-1

specimen strain rate was simulated up to the point of crack propagation. The terms of

interest in the FEA are the maximum principal stress and strain energy density at the

crack tip.

Figure 4.1 A whole pure shear specimen with a sharp crack.

Table 4.1 Hyperelastic constants for NR0 and NR25.

Material C10 (MPa) C20 (MPa) C30 (MPa) Unfilled NR 0.18269 -0.00043 0.00004 Filled NR 0.37282 0.00111 0.00017

For our specific case, 16B=56,000 elements is the maximum allowable number of

elements that can be run in ABAQUS Explicit 6.6. Table 4.2 shows that when the

element number varies from 1B to 16B, both of the maximum principal stress and the

strain energy density at the crack tip increase in value. Figures 4.2 and 4.3 show that

both of the maximum principal stress and strain energy density around the crack tip 41 increase until the maximum element number that the program allows to run.

Convergence with this mesh appears to be unlikely.

Table 4.2 Convergence study from 3500 elements to 56000 elements.

Element Element Number Max. Principal Strain Energy CPU Disk Number through Stress Density Time Space (B=3500) Thickness (MPa) (KJ/m^3) (Min) (GB) B 1 4.8914 46 2 0.25 2B 1 6.2880 59 4 0.49 4B 2 6.2360 56 9 0.94 8B 2 7.9155 72 24 1.88 16B 2 10.0878 90 71 3.75 Half 16B 2 10.0811 90 36 1.88

Figure 4.2 Convergence of maximum principal stress at the crack tip.

Figure 4.3 Convergence of strain energy density at the crack tip.

At the same time, increasing element number consumes more disk space and requires more run time. To have optimum use of computing resources, we set up a half model, which has the same material property, mesh size, element type but a symmetry boundary condition. The actual pure shear fracture test in this project was not symmetric, because only the top of the specimen was pulled. We take a half-model to represent the whole specimen as an approximation. As shown in Table 4.2, 16B compared very well with half 16B. Obviously, half 16B kept the accuracy of our terms of interest, but decreased the CPU time and the disk space to one half of the original full test model.

From here on, a half model will be used for all future FEA simulation.

The mesh that was used in this convergence was not adequate for finding local crack tip parameters such as energy density and maximum principal stress. This is because the mesh near the crack tip is always coarse when the specimen is pulled, as shown in Figure 4.4.

Figure 4.4 Result of the mesh around the crack tip after convergence study.

Another try was to have a bias mesh around the crack tip, as shown in Figure 4.5.

By using a bias mesh, the region around the crack tip would have a very fine or dense mesh. However, no matter how dense we made the mesh, the elements around the crack always had extremely large deformation, similar to what was shown in Figure 4.4. This new mesh definitely would not provide accurate results. In the next study, an improved mesh is introduced.

Figure 4.5 Bias mesh around the crack tip.

44 4.2 Mesh Study around the Crack Tip

To have the FEA model provide more and accurate information around the crack

tip, the mesh around the crack tip would to be finer. The procedures to achieve this

objective are shown in Figures 4.6 (a)-(d).

A radius, which is half of the thickness of the blade, as mentioned in Section

3.2.1, was assigned, as shown in Figure 4.6 (a). Several features, including datum planes

and partition cells were created to achieve a very fine mesh around the crack tip. Without

these features, the program would generate the disorderly mesh shown in Figure 4.6 (b).

The final mesh provided a much finer diffusion of elements after the deformation, as

shown in Figures 4.6 (c) and (d).

Bias mesh at the crack tip.

(b) System bias mesh at the crack tip. (c) Features assigned bias mesh at the crack tip.

(d) Result of features assigned bias mesh around the crack tip.

Figure 4.6 Mesh study with radius around the crack tip: (a) Bias mesh at the crack tip, (b) System bias mesh at the crack tip, (c) Features assigned bias mesh at the crack tip and (d) Result of features assigned bias mesh around the crack tip.

46 4.3 Element Type Study

In the prior convergence study, C3D8, a continuum eight-node solid element with

full integration, as shown in Figure 4.7 (a), was used as the element type. Unfortunately,

the program aborted when C3D8 was used. The deformation ended with an irregular

form and the program only accomplished 59 out of 100 frames. One way to fix the

"overly-distorted" element problem was to make a finer mesh. However, the mesh we

had was the densest we could afford from a computational time standpoint. Already this

program had run for several days. Fortunately, ABAQUS Explicit Version 6.7 provided

C3D8R, a continuum eight-node solid element with reduced integration. The shape

functions of C3D8R are the same as for the C3D8 element. The node numbering 1-8 and

the integration point (solid black circle) for C3D8R are shown in Figure 4.7 (b).

Element C3D8R was chosen to use instead of C3D8. However, the C3D8R element requires hourglass control, which means that for C3D8R elements the correct solution is superposed with arbitrarily large displacements corresponding to the zero energy modes. Also, hourglassing could create an artificial stress field on the top of the real stress field. To reduce the artificial strain energy, a finer mesh would have to be used. The way to check for hourglassing is to look at the artificial strain energy and compare it to strain energy. Figure 4.8 shows the FEA comparison results of C3D8 full integration elements with C3D8R reduced integration elements.

47 1

(a) C3D8 2x2x2 integration point. (b) C3D8R 1x1x1 integration point.

Figure 4.7 Comparison of integration points: (a) C3D8 2x2x2 integration point and (b) C3D8R 1x1x1 integration point.

(a) C3D8 elements. (b) C3D8R elements.

Figure 4.8 FEA comparisons of strain energy density with: (a) C3D8 elements and (b) C3D8R elements.

48 4.4 Pure Shear Region

As mentioned in Section 3.3.4, for the pure shear specimen, tear energy only stored in the pure shear region.

4.4.1 Pure shear specimen without crack

A criterion for finding the pure shear region will be defined by analyzing the stress field in an un-cracked pure shear specimen. Another FEA model was set up to accomplish the objective. Figure 4.9 shows the pure shear specimen without a crack.

The specimen material is described as hyperelastic with NR0 properties (see Table 4.1) and it is being pulled at a sample stain rate at 10 s -1 up to the same amount of deformation that would cause it to break in a pure shear fracture test.

Figure 4.9 Pure shear specimen without crack.

1.6

1.4

1.2 t=2 ms 1 t=4 ms 0.8 0.6 t=6 ms

S22Stress (MPa) 0.4 t=8 ms 0.2 t=10 ms 0 0 50 100 150 200 250 Distance from left edge( mm)

Figure 4.10 Normal stress distributions in center pure shear specimen without crack.

49 Figure 4.10 shows the distribution of stress in 2-direction or normal direction

along the center of the specimen. The stresses are uniformly distributed at different

times, except at the edges of the specimen. Thus, a criterion for finding the pure shear

region in the pure shear fracture specimen could be the region of uniform stress S 22

ahead of the crack tip.

4.4.2 Defining the pure shear region

The pure shear fracture specimen was modeled using ABAQUS Explicit, Version

6.7. An upper half model of the pure shear fracture specimen is shown in Figure 4.11.

Hyper-viscoelastic material properties for NR25 was incorporated in a user-defined

material subroutine (VUMAT) program obtained from Al-Quraishi [23]. All degrees of

freedom except displacement in the 2-direction were constrained at the bottom of

specimen, and displacement ( ∆∆∆ ) was applied on the top of the specimen. The displacements control for NR25 pure shear specimen was the test at 10 s -1 sample strain rate. Continuum C3D8R was chosen as the element type. The total number of elements used in this model was 38,728; there were eight elements through the thickness.

∆

a=0.0635 mm

Y-SYMMETRIC

Figure 4.11 FEA mesh of upper half of pure shear fracture specimen .

50 Before we make any conclusion, we need to check for hourglassing because

C3D8R has one integration point. As was mentioned above, the artificial strain energy

(ALLAE) should be small compared to the strain energy (ALLSE). Figure 4.12 shows that compared with the strain energy (ALLSE); ALLAE is less than 1% and negligible.

ALLAE

ALLSE 2 Energy (J)

0 0 20 40 60 Time (ms)

Figure 4.12 Comparison of ALLAE with ALLSE.

Now we are in a position to use the criterion that we found from the pure shear

specimen without a crack. From the output of our FEA model, we plot the normal stress

(S22) versus the distance from the crack tip at various times in Figure 4.13. Near the

crack tip the stress is non-uniform. This is the so-called complicated strain region, or

Region C as defined in Figure 3.20 . Beyond this complicated strain region, the stresses decrease to a uniform value until it decreased due to the edge effect. The uniform stress

51 region between the complicated strain region and the edge effect region is the pure shear region. Figures 4.13 shows a way to define the pure shear region.

A procedure for defining the pure shear region is as follows:

1. plot the normal stress ahead of the crack tip at different time.

2. Superimpose a horizontal line indicating constant or uniform stress (red lines in

Figure 4.13).

3. Draw two vertical lines intersecting points where the red lines deviate from the normal stress.

The region between the vertical lines is where the stress has uniform normal stress or the pure shear region.

t=15 ms 18 t=25 ms 15 t=35 ms

12 t=45 ms

9 t=55 ms

Stress S22 (MPa) 6

0 -5 25 55 85 115 145 175 205 Displacement from the crack tip (mm)

Figure 4.13 Pure shear region in NR25 pure shear specimen.

52 Based on the previous study and the information that we obtained from FEA model, we calculate K, which is about 1.61 for NR25. To confirm this K value, similar studies for NR0 and NR25 with hyperelastic material properties were executed. Figures

4.14 (a), (b) and (c) show a comparison study of the pure shear region for different materials and at different times. Although the pure shear region varies slightly for the different materials, the K value will not vary too much. Here, we conclude that the K value for pure shear fracture test is 1.61.

53 t=15ms

6 Hyper-viscoelastic NR25

5 Hyperelastic NR25 Hyperelastic NR0 4

2 Stress S22 (MPa) S22 Stress

0 0 40 80 120 160 200 Distance (mm)

(a) t=15ms

t=35ms

6 Hyper-viscoelastic NR25

5 Hyperelastic NR25 Hyperelastic NR0 4

2 Stress S22 (MPa) S22 Stress

0 0 40 80 120 160 200 Distance (mm)

(a) t=35ms

t=55ms

6 Hyper-viscoelastic NR25

5 Hyperelastic NR25 Hyperelastic NR0 4

2 Stress S22 (MPa) S22 Stress

0 0 40 80 120 160 200 Distance (mm)

Figure 4.14 Pure shear region comparison of viscous vs. hyperelastic: (a) t=15ms; (b) t=35ms and (c) t=55ms. 54 4.4.3 Results

After we define the K value, the tear energy, G, for the pure shear tests can be calculated from Equations (3-4) and (3-5). The tear energy for NR0 and NR25 for the both the trouser tests and the pure shear tests are listed in Table 4.3.

Table 4.3 Tear energy of NR0 and NR25 from trouser tests and pure shear tests.

NR0 NR25 Strain Rate G (kJ/m2) Strain Rate G (kJ/m2) (1/s) (1/s) Trousers test 8.63 10.88 0.01 22.16 0.01 16.79 0.1 29.47 0.1 12.62 1 17.22 1 10.51 10 13.17 10 11.6 Pure shear test 132 17.62 92 25.45 188 17.67 123 21.17 270 29.57 168 35.32 229 23.94 226 24.06 311 36.72 277 45.16

From Table 4.3, we can see that for the pure shear fracture tests, different strain rates will result in different values of G. When comparing the pure shear test strain rate at 0.01 s -1 with trouser tests, the G values are also different. Results from this project indicate that tear strength G for rubbers is not a constant material value. It varies with strain rates and test methods or specimen geometries. More use with the finite element analysis (FEA) will be given in Chapter V.

CHAPTER V

RESULTS AND DISCUSSION

5.1 Results of Pure Shear Fracture Tests

The variation of the tear energy with respect to sample strain rate for the unfilled

NR and carbon black-filled NR pure shear specimens is shown Figure 5.1. The unfilled

NR pure shear specimen varies substantially with strain rate and is higher than the tear energy found from the ASTM Standard trouser test (=8.64 kJ/m 2 for the unfilled NR).

The tear energy at 0.1 s -1 strain rate is almost double what it is at 0.01, 1, 10 and 132 s -1

strain rate. On the other hand, the tear energy from the 25 phr carbon black-filled pure

shear specimen is roughly constant from 0.01 to 10 s -1 strain rate and approximately equal to the value found from the ASTM standard trouser test (=10.87 kJ/m 2 for 25 phr carbon black-filled NR). In general, the tear energy from the unfilled and carbon black- filled NR pure shear specimen increases with sample strain rate when the strain rates are greater than 10 s -1. Thus tear energy varies with sample strain rate.

56 50 45 40 35 25 phr carbon black-filled NR 30 25 Unfilled NR 20 15 10 Tear Energy G (KJ/m^2) 5 0 0.001 0.01 0.1 1 10 100 1000 Sample Strain Rate (1/s)

Figure 5.1 Variation of tear energy with far-field sample strain rate from pure shear fracture specimen.

The corresponding variation of crack speed with sample strain rates for the two compounds in pure shear is shown in Figure 5.2. Very good repeatability in the crack speeds for both unfilled NR and carbon black-filled NR pure shear specimen are noted at

0.1, 1 and 10 s -1. However, there is a wider scatter in the crack speed data for the unfilled

NR pure shear specimen compared to the carbon black-filled NR pure shear specimen when the strain rates are less than 10 s -1. For sample strain rates above 100 s -1, the crack

speed increases with increasing strain rate.

20 25 phr carbon 15 black-filled NR Unfilled NR 10 Crack Speed (m/s) 5

0 0.001 0.01 0.1 1 10 100 1000 10000 Sample Strain Rate (1/s)

Figure 5.2 Variation of crack speed with far-field sample strain rate from tensile strip specimen.

5.2 Comparison with Tensile Strip

Ali-Quraishi [23] has performed fracture tests with the NR0 and NR25 using

tensile strip specimens. The variation of the tear energy with far-field sample strain rate

from tensile strip and our pure shear specimens are plotted together for the unfilled and

carbon black-filled NR in Figures 5.3 (a) and (b), respectively. Clearly there are

inconsistencies in the tear energy between sample strain rates 0.01 to 1 s -1. Above a sample strain rate of 10 s-1, the tear energy calculated from either specimen are comparable or at least within the margin of experimental error.

58 50

30 Tensile strip (Al-Quarashi, 25 2007) Pure shear 20

Tear Energy G (KJ/m^2) 10

0 0.001 0.01 0.1 1 10 100 1000 Sample Strain Rate (1/s)

(a) unfilled NR.

30 Tensile strip 25 (Al-Quarashi, 2007) 20 Pure shear

10 Tear (KJ/m^2) Energy Tear G 5

0 0.001 0.01 0.1 1 10 100 1000 Sample Strain Rate (1/s)

(b) 25 phr carbon black-filled NR.

Figure 5.3 Comparison of the variation of tear energy with far-field sample strain rate from both tensile strip specimen and pure shear specimen: (a) Unfilled NR and (b) 25 phr carbon black-filled NR . 59 The differences in the tear energy derived from the tensile strip and pure shear

specimens may be attributed to differences in the local crack tip stress state. The pure

shear fracture specimen is in a state of constrained tension due to the 10:1 width to height

ratio specimen geometry, while the tensile strip is a state of relatively unconstrained

tension. Secondly, variations in the tear energy between sample strain rates 0.01 to 1 s -1 could be due to strain-induced crystallization. The load duration in tensile strip experiments for specimens at 10 s -1 is about 56 ms and the load duration in the pure shear

experiments at 10 s -1 is about 71 ms. Since strain induced crystallization has a characteristic time of 50-60 ms, experiments with strain rates greater than 10 s -1, will not

show any effects of it. Finally, the crack speeds found from the tensile strip and pure

shear specimens are very different. Lake et al . [12] have proposed that the tear energy of

NR varies with crack speed.

We plot the crack speed versus the tear energy for the unfilled NR, and compare it with the data from Lake et al. [12] in Figure 5.4. The comparison will testify to the accuracy of our results and indicate any variation of the tear energy with crack speed.

The results shows that the crack speeds of pure shear tests are at the same level, but the tear energy are lower than results from Lake et al. [12]. The unfilled NR in their experiments is not identical to our NR0. A similar trend in the variation of crack speed and tear energy is observed in our unfilled NR tensile strip and pure shear fracture tests when compared to their results. However, note that there is a small range of crack speeds in our tensile strip experiments, 1.4 m/s – 4.2 m/s, and that Lake et al . [12] was able to get high crack speeds in the tensile strip by first stretching the specimen and then introducing the crack.

60 Here, we conclude that the tear energy might not be a good or constant material

property to anticipate the fracture of rubbers or rubber like materials. With the accurate

FEA model we have set up, we explore other parameters, such as strain energy density

and the maximum principal stress at the crack tip for predicting fracture of rubbers.

100

Tensile Strip, Lake et al. (2000)

Pure Shear, Lake et al. 10 (2000)

Pure Shear Crack Speed (m/s) Crack

Tensile Strip, Al-Quraishi (2007)

1 1 10 100 1000 Tear Energy (KJ/m^2)

Figure 5.4 Comparison of for unfilled NR data with Lake et al. (2000) for tensile and pure shear tests.

5.3 Other Significant Fracture Parameters

The NR25 pure shear specimen experiment at strain rate 10 s -1 was simulated in

FEA. The FEA contour plot of the strain energy density and maximum principal stress in the pure shear specimen at the onset of crack propagation are shown in Figures 5.5 and

5.6, respectively. It is obvious from the plots that the strain energy density and the maximum principal stress are highest near the crack tip.

Figure 5.5 Contour plot of the strain energy density at the onset of crack propagation in the NR25 pure shear specimen at 10 s -1.

Figure 5.6 Contour plot of the maximum principal stress at the onset of crack propagation in the NR25 pure shear specimen at 10 s -1.

62 The toughness and maximum principal stress at the crack tip for 25 phr carbon

black-filled NR was plotted as a function of the time rate of first invariant of the left

I I Cauchy Green deformation tensor at the material break point, &1 . Here &1 is defined as the value of I 1 at fracture onset divided by the time to reach this value. The strain energy

densities and maximum principal stresses at the crack tip from the FEA fall within the

material test data [23] as shown in Figures 5.7 and 5.8 with a solid circle. The same

procedure was used on the NR25 tensile strip pulled at 10 s -1 and the FEA results are

presented as a solid square in Figures 5.7 and 5.8. Thus comparing the strain energy

density to the material toughness and maximum principal stress to the true tensile

strength of the natural rubber would give relatively accurate predictions for the onset of

crack propagation.

60 Toughness, Al-Quraishi 50 (2007) Tensile Strip 10 1/s, Al- 40 Quraishi (2007) Pure shear 10 1/s 30

20 Strain Energy Density (MPa)

0 0.1 1 10 100 1000 10000

Rate of First Invariant of Left Cauchy Green tensor, I B_dot

Figure 5.7 Variation of strain energy density with rate of first invariant of left Cauchy Green deformation tensor for 25 phr carbon black-filled NR .

210

180

150 Tensile Strength, Al-Quraishi (2007) 120 Tensile Strip 10 1/s, Al- Quraishi (2007) 90 Pure shear 10 1/s

60 Max.principal stress(Mpa)

0 0.1 1 10 100 1000 10000

Rate of First Invariant of Left Cauchy Green tensor, I B_dot Figure 5.8 Variation of maximum principal stress at the crack tip with rate of first invariant of left Cauchy Green deformation tensor for 25 phr carbon black-filled NR .

In all, Figures 5.7 and 5.8 indicate that simple fracture parameters such as the critical strain energy density and the maximum principal stress at the crack tip may be used to predict the onset of fracture at the crack tip at different strain rates.

64 CHAPTER VI

CONCLUSION AND FUTURE WORK

An experimental study was conducted to characterize the fracture behavior of

unfilled and 25 phr carbon black-filled natural rubber with varying loading rates. Quasi-

static and dynamic tensile fracture tests were performed with a servo-hydraulic MTS

machine and Charpy tensile apparatus on pure shear specimens with far-field sample

strain rate in the range 0.01 to 311 s -1. A high-speed video camera operating at 50,000

frames per second was used to measure elongations and crack speeds.

In order to find the pure shear region of the pure shear specimen, finite element analysis (FEA) was done with hyperviscoelastic and hyperelstic material properties of the

25 phr carbon black-filled natural rubber. The FEA procedures included a convergence and mesh design study to get the best FEA model. The finite element results from both hyperviscoelastic and hyperelstic material properties of rubbers allowed us to find a simple correction factor to evaluate the strain energy density of the entire pure shear specimen. Applying this factor to the experiment data, we obtained the tear energy of the pure shear specimens under different strain rates.

65 From the results of pure shear fracture tests, we found that the tear energy varied with sample strain rates, i.e., the tear energy are rate-dependent. Furthermore, on comparing the pure shear fracture tests with the tensile strip and trouser fracture tests, it was found that the tear energy was different for different specimen geometries even when the tests were done at the same sample strain rate. For these two reasons, we conclude that the tear energy may not be a good parameter to predict the fracture of rubbers.

Our finite element analysis indicated that a critical strain energy density and the maximum principal stress at the crack tip could be better fracture parameters for predicting crack growth. The strain energy density and maximum principal strain at the crack tip at the point of fracture onset were approximately equal to the toughness and tensile strength of the rubber at the same average time rate of the first invariant of the left

Cauchy-Green deformation tensor. Thus, these two parameters may be used to predict the onset of fracture at the crack tip at different strain rates. Future work in this research could involve the use of a critical strain energy density or the maximum principal stress to predict crack growth. Such a study would also be very important in setting up criteria fatigue damage of rubbers.

66 REFERENCES

[1] Hussain, S., “Effect of Carbon Black Fillers on High Strain Rate Properties of Natural Rubber,” Master Thesis, The University of Akron, December, 2005.

[2] Bateman, L., “The Chemistry and Physics of Rubber-like Substances”, MacLaren and Sons Ltd., London, 1963.

[3] http://en.wikipedia.org/wiki/Fracture_mechanics.

[4] Anderson, T.L., "Fracture Mechanics: Fundamentals and Applications", 3 rd Edition, Boca Raton, Florida, 2004.

[5] Griffith, A.A., “The Theory of Rupture,” in the Proceedings of the 1 st International Congress on Applied Mechanics, edited by C.B. Bienzo and J.M. Burgers, Delft, pp. 55-63, 1925.

[6] Thomas, A.G., “The Development of Fracture Mechanics for Elastomers,” Goodyear Medal Paper presented to the American Society Rubber Division Meeting Chicago, April 1994, Rubber Chemistry and Technology, Vol. 67, No. 3, G50-G60, 1994.

[7] Irwin, G.R., “Onset of Fast Crack Propagation in High Strength Steel and Aluminum Alloys,” Sagamore Research Conference Proceedings, Vol. 2, pp. 289-305, 1956.

[8] Rice, J.R., “Mathematical Analysis in the Mechanics of Fracture,” in Fracture: An Advanced Treatise, edited by H. Liebowitz . Vol. II, Academic Press, New York, 1968, pp. 191-311, 1968.

[9] Rice, J.R. and Rosengren, G.F., “Plane Strain Deformation Near a Crack Tip in a Power Law Hardening Materials,” Journal of the Mechanics and Physics of Solids, Vol. 16, pp. 1-12, 1968.

[10] Thomas, A.G., “The Development of Fracture Mechanics for Elastomers,” Goodyear Medal Paper presented to the American Society Rubber Division Meeting Chicago, April 1994, Rubber Chemistry and Technology, Vol. 67, No. 3, G50-G60, 1994.

67 [11] Rivlin, R.S. and Thomas, A.G., “Rupture of Rubber. I. Characteristic Energy for Tearing,” Journal of Polymer Science, Vol. X, pp. 291-318, 1953.

[12] Lake, G.J., Lawrence, C.C. and Thomas, A.G., “High-Speed Fracture of Elastomers: Part I,” Rubber Chemistry and Technology, Vol. 73, No. 5, pp. 801-817, 2000.

[13] Lake, G.J., “Fatigue and Fracture of Elastomers,” Rubber Chemistry and Technology, Vol. 68, No. 3, pp. 435-460, 1995.

[14] Lake, G.J., “Fracture Mechanics and its Application to Failure in Rubber Articles,” Rubber Chemistry and Technology, Vol. 76, No. 3, pp.567-591, 2003.

[15] Lake, G.J. and Thomas, A.G., “Strength,” in Engineering with rubber, How to design rubber components, 2 nd edition, A.N. Gent (editor), Hanser Gardner Publications, Cincinnati, 2001.

[16] ISO 2393: Rubber Test Mixes – Preparation, Mixing and Vulcanization – Equipment and Procedures, 2 nd Edition, 1994.

[17] ASTM D 624-54, Standard Methods of Test for Tear Resistance of Vulcanized Rubber, 1954.

[18] ASTM D 624-00, Standard Test Method for Tear Strength of Conventional Vulcanized Rubber and Thermoplastic Elastomers, 2000.

[19] ASTM D 4482-99, Standard Test Method for Rubber Property-Extension Cycling Fatigue, 1999.

[20] OriginLab. 2003. Originpro Version 7.5, www.OriginLab.com.

[21] ProAnalyst Professional Edition Image Analysis Software, Version 1.5.0.1, Xcitex Inc., August 2005.

[22] A.G. Thomas, “The Development of Fracture Mechanics for Elastomers”, Rubber Chemistry and Technology, Vol. 67, No. 3, pp. 50-60, 1994.

[23] Al-Quraishi, A.A., “The Deformation and Fracture Energy of Natural Rubber under High Strain Rates,” Doctoral dissertation, The University of Akron, August, 2007.