NEW APPROACH IN CHARACTERIZING ACCESSORY DRIVE · PDF filenew approach in characterizing...

NEW APPROACH IN CHARACTERIZING ACCESSORY DRIVE BELTS FOR

FINITE ELEMENT APPLICATIONS

By

FARBOD NASSIRI

A thesis submitted in conformity with the requirements

for the degree of

MASTER OF APPLIED SCIENCE

Department of Mechanical and Industrial Engineering University of Toronto

©Copyright by Farbod Nassiri 2010

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NEW APPROACH IN CHARACTERIZING ACCESSORY DRIVE BELTS FOR FINITE ELEMENT APPLICATIONS

Farbod Nassiri

Master of Applied Science

Department of Mechanical and Industrial Engineering University of Toronto

2010

ABSTRACT

Multi-ribbed serpentine belt is the core of the automotive accessory drive system, which

distributes the engine power to other auxiliary systems of the car. Development of a belt life

model is of a significant importance to the accessory drive system manufacturers, in order to

prevent any premature failures of these belts. However, any numerical analysis on the belt life is

heavily dependent on gaining an understanding of stress distribution in the belts under the

operational loading conditions.

The presented work demonstrates a new systematic approach for determining the hyperelastic

material parameters of rubber with specific application in Finite Element Analysis (FEA) of

serpentine accessory drive belts. This new approach can be used as a stand-alone tool by

manufacturers to determine the stress distribution in the belt under operational conditions; the

results of which can be applied to assess the life of accessory drive belts, in a relatively short

time.

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ACKNOWLEDGEMENTS

This thesis would not have been possible, without the love, mentorship, assistance, and

guidance of many dear people around me.

First, and foremost, I would like to thank my parents, brother, and grandparents for their

continuous love, inspiration, and encouragement. Without my parents’ nourishment, devotion,

and sacrifice, I certainly would not have been here. The love for learning and hard work that

they have seeded in me during my childhood, and making the tough decision to come to Canada

for their children’s better education, is far beyond appreciation.

I would like to thank my supervisor, Professor J. W. Zu, for her continuous mentorship,

patience, and support during my graduate studies at University of Toronto. I am very grateful to

Dr. Zu, for her kindness, believing in me during challenging times, and making my experience as

a graduate student to be such a joy.

I want to acknowledge the staff of Litens Automotive Group, namely Mr. Mike Clark, Dr.

Seevaratnam Karunendiran, and Dr. Steven Jia for initiating and facilitating this research project

and sharing their technical knowledge of accessory drive belts. All the experimental work

presented in this thesis was conducted at Litens Automotive Group’s facility at Concord, ON,

Canada. I would also want to thank Maby Boado, my friend and colleague, for her help with this

project.

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I am very thankful to Professor Hani Naguib for sharing his knowledge, and devoting his time

for my questions. Lastly, I would also want to express my appreciation to the personnel of the

graduate office, for their assistance.

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TABLE OF CONTENTS

ABSTRACT .................................................................................................................................. II

ACKNOWLEDGEMENTS ....................................................................................................... III

TABLE OF CONTENTS ............................................................................................................ V

LIST OF TABLES .................................................................................................................... VII

LIST OF FIGURES ................................................................................................................. VIII

LIST OF APPENDICES ............................................................................................................ XI

NOMENCLATURE .................................................................................................................. XII

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

1.1 Background ...................................................................................................................... 1

1.2 Research Motivation ........................................................................................................ 3

1.3 Objective and Contributions ............................................................................................. 4

1.4 Thesis Outline .................................................................................................................. 5

2 LITERATURE REVIEW ..................................................................................................... 7

2.1 Belt Mechanics and the Existing Material Models .......................................................... 7

2.2 Conventional Material Characterization Tests ............................................................... 10

2.2.1 Uniaxial Tensile Testing ......................................................................................... 11

2.2.2 Equi-biaxial Tensile and Uniaxial Compression Testing ....................................... 16

2.2.3 Planar Tensile and Pure Shear Testing ................................................................... 19

2.3 Micro-Mechanical Testing ............................................................................................. 24

2.4 Shortcomings of the Conventional and Micro-Mechanical Test Methods ................... 28

3 NEW PROCEDURE FOR MECHANICAL TESTING OF ACCESSORY DRIVE BELTS .......................................................................................................................................... 31

3.1 Mechanical Testing ........................................................................................................ 31

3.2 Uniaxial Tensile Testing of the Whole Belt and the Rubber Rib ................................... 32

3.3 Uniaxial Compression Testing of the Belt and the Rubber Rib ..................................... 36

3.4 Pure Shear Testing of the Belt and the Rubber Rib ....................................................... 40

3.5 Three-Point-Bending Test of the Belt ............................................................................ 42

3.6 Conclusion ...................................................................................................................... 43

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4 DETERMINATION OF THE BELT CONSTITUENTS’ PARAMETERS .................. 44

4.1 Determination of the Cord Region Elastic Modulus ...................................................... 44

4.2 Conversion of the Force-Displacement Test Results into the Stress-Strain Field for the Rubber Region ........................................................................................................................... 46

4.3 Determination of the Mooney-Rivlin Constants ............................................................ 51

4.4 Conclusion ...................................................................................................................... 56

5 FINITE ELEMENT ANALYSIS VERIFICATION ........................................................ 57

5.1 Finite Element Modeling ................................................................................................ 57

5.2 Single-Mode of Deformation Models ............................................................................ 58

5.3 Multi-Mode of Deformation Models .............................................................................. 61

5.4 Comparison between the FEA and the Experimental Results ........................................ 63

5.5 Conclusion ...................................................................................................................... 68

6 APPLICATION IN BELT LIFE PREDICTION ............................................................. 69

6.1 Fatigue in Accessory Drive Belts ................................................................................... 69

6.2 Stress Distribution in Accessory Drive Belts ................................................................. 72

6.3 Determination of the Equivalent Uniaxial Stresses at the Belt Rib Tip ......................... 73

6.4 Correlation between the Stress Distribution and Life in Accessory Drive Belts ........... 75

6.5 Conclusion ...................................................................................................................... 76

7 SUMMARY & CONCLUSIONS ....................................................................................... 77

7.1 Summary ........................................................................................................................ 77

7.2 Recommendation for Future Work ................................................................................ 80

REFERENCES ............................................................................................................................ 84

APPENDICES ............................................................................................................................. 90

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LIST OF TABLES

Table 1: Dimensions of standard dies for cutting dumbbell shape samples as per ASTM D412 . 14

Table 2: Dimensions of standard dies for cutting ring samples as per ASTM D412 [24] ............ 14

Table 3: Sample sizes as per ASTM D954 [36] ........................................................................... 24

Table 4: Summary of the approximated Mooney-Rivlin parameters under various modes of

deformation using Hyperfit ............................................................................................... 55

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LIST OF FIGURES

Figure 1: Accessory drive system (courtesy of General Motors) ................................................... 1

Figure 2: Constitutive components of a typical accessory drive belt ............................................. 2

Figure 3: (a) Cross sectional view of a 5-ribbed accessory drive belt, and (b) its CAD model ..... 8

Figure 4: Basic modes of deformation for rubber [21] ................................................................. 11

Figure 5: Uniaxial tensile testing of rubber [22] ........................................................................... 12

Figure 6: Standard dies for cutting dumbbell specimens as per ASTM D412 [24] ...................... 15

Figure 7: Rubber equi-biaxial tension testing at NASA space center [9] ..................................... 16

Figure 8: A bi-axial extension experiment with Axel Products' setup [21] .................................. 17

Figure 9: Equivalent Testing Modes [20] ..................................................................................... 18

Figure 10: Compression testing of rubber [22] ............................................................................. 19

Figure 11: Schematic representation of planar tension [22][35] .................................................. 20

Figure 12: Planar tension test on a rubber specimen glued to metallic brackets [6] .................... 21

Figure 13: Planar tension test grips, Axel Products [21] .............................................................. 21

Figure 14: Planar tension test, Axel products [21] ....................................................................... 22

Figure 15: Advanced Yerzley Oscillograph [36] .......................................................................... 23

Figure 16: Sample mounted on the nano tensile tester (NanoBionix, MTS, USA) [39] .............. 26

Figure 17: Flexure-based testing machine [41] ............................................................................ 27

Figure 18: Schematic illustrating the methodology proposed in [37] ........................................... 28

Figure 19: Rubber rib and belt uniaxial tensile tests ..................................................................... 33

Figure 20: Removal of the ribs at the ends of the belt sample for the uniaxial tensile test .......... 33

Figure 21: The cord uniaxial tensile test result ............................................................................. 34

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Figure 22: The rubber rib uniaxial tensile test result .................................................................... 35

Figure 23: The belt uniaxial tensile test result .............................................................................. 35

Figure 24: Rubber rib and belt uniaxial compression tests ........................................................... 37

Figure 25: (a) Groove-on-flat and (b) Groove-on-groove surface interactions ............................ 37

Figure 26: The rubber rib uniaxial compression test result .......................................................... 38

Figure 27: The belt groove-on-flat compression test result .......................................................... 38

Figure 28: The belt groove-on-groove compression test result .................................................... 39

Figure 29: (a) Shear test setup, (b) Belt and moving block interaction ........................................ 41

Figure 30: Force-displacement result of the shear test – 61 kg dead weight ................................ 41

Figure 31: The three-point-bending test of the belt ...................................................................... 42

Figure 32: The three-point-bending test results ............................................................................ 43

Figure 33: Cross-sectional snapshot of (a) the belt and (b) the rubber rib ................................... 45

Figure 34: Compression of the rubber rib ..................................................................................... 47

Figure 35: Stress-strain behavior of rubber in uniaxial tensile ..................................................... 49

Figure 36: Stress-strain behavior of rubber in uniaxial compression ........................................... 50

Figure 37: Stress-strain behavior of rubber in shear ..................................................................... 50

Figure 38: Curve fitting the Mooney-Rivlin equation to the uniaxial tensile data ....................... 52

Figure 39: Curve fitting the Mooney-Rivlin equation to the uniaxial compression data ............. 52

Figure 40: Curve fitting the Mooney-Rivlin equation to the pure shear data using Hyperfit ....... 53

Figure 41: FE simulation of the rubber rib compression test (0.25mm displacement) ................. 59

Figure 42: FE simulation of the rubber rib tensile test (5.5mm displacement) ............................ 60

Figure 43: FE simulation of the belt groove-on-flat compression test (0.27mm displacement) .. 61

Figure 44: FE simulation of the belt groove-on-groove compression test (0.1mm displacement)62

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Figure 45: FE simulation of the belt bending test (3.98mm deflection) ....................................... 63

Figure 46: The rubber rib uniaxial tensile test - experimental and numerical force-displacement

comparison ................................................................................................................... 65

Figure 47: The rubber rib uniaxial compression test - experimental and numerical force-

displacement comparison ............................................................................................. 65

Figure 48: The belt groove-on-flat uniaxial compression test - experimental and numerical force-

displacement comparison ............................................................................................. 66

Figure 49: The belt groove-on-groove uniaxial compression test - experimental and numerical

force-displacement comparison .................................................................................... 66

Figure 50: Belt bending – cord side up - experimental and numerical force-displacement

comparison ................................................................................................................... 67

Figure 51: Initiation of cracks at the rubber rib tips [5] ................................................................ 70

Figure 52: Stress variation due to tension ..................................................................................... 72

Figure 53: State of stress at the rubber rib tip of accessory drive belts ........................................ 73

Figure 54: Snapshot of the cross-ribbed accessory drive belt during testing [5] .......................... 82

Figure 55: Snapshot of the belt cross section ................................................................................ 90

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LIST OF APPENDICES

Appendix A: Calculation of the Cord Region Elastic Modulus ................................................... 90

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NOMENCLATURE

Roman Characters A Cross-sectional area

b Fatigue strength exponent

C01,C10 2-term Mooney-Rivlin material constants

d Incompressibility factor

Dr Diameter of the driver pulley

E Modulus of Elasticity

hf Hours to failure

I1, I2, I3 Right Cuchy-Grean tensor strain invariants

K Material stiffness

L Total belt length

Nf Number of reversals to failure

RPM Driver RPM

W Strain energy density function

Y0,Y1 Rubber rib top and bottom base lengths

Yinstantaneous The instantaneous base length of the rubber rib when compressed by

length L

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Greek Characters λi Principle stretch ratio in direction i

εT True strain

εE Engineering strain

εi Principle strain

ν Poison ratio

σm Total mean stress

σf Fatigue strength coefficient

σa Total alternating stress

σqa Equivalent alternating stress

σa1, σa2, σa3 Principle alternating stresses

σqm Equivalent mean stress

σmx, σmy , σmz Mean normal stresses in any three mutually perpendicular directions

σm1…. σmi Equivalent mean stress at pulley 1 to i

σa1…. σai Equivalent alternating stress at pulley 1 to i

σt Tight side uniaxial tension stress

σs Slack side uniaxial tension stress

σb Bending stress

σz Transverse stress

σq Stress in the circumferential direction

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σE Engineering stress

σT True stress

σx Longitudinal tensile stress

μs Coefficient of static friction

Chapter 1: Introduction

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1 INTRODUCTION

1.1 Background

The multi-ribbed serpentine belt is considered as an essential part of power transmission

between the engine crankshaft and other accessory drive components of a car (Figure 1). These

belts are mainly fabricated with composite materials that combine rubber, cords and either an

elastomeric or fabric backing (Figure 2) [1]. With each cycle, they experience static and dynamic

stresses from belt pre-tension and bending, crankshaft torque fluctuation, and component

loading. They also experience friction wear, age hardening from temperature variations, ozone

attack and abrasion from dust and rubber particles. As a result, cracks are formed at the rubber

rib tips and the internal cords lose their resilience, finally resulting in failure of the belt.

Therefore, accurate prediction of the belt life under various drive conditions is deemed crucial

for accessory drive manufacturers.

Figure 1 Accessory drive system (courtesy of General Motors)


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Figure 2 Constitutive components of a typical accessory drive belt

Experimental belt wear and fatigue tests are common among the accessory drive belt

manufacturers. However, the variety of factors impacting the fatigue life of the belt makes these

tests complex in nature and time consuming. This necessitates the development of a simple, yet

versatile fatigue life model of the belt under various loading conditions. Several researchers have

investigated fatigue of transmission V-belts and have attempted to develop prediction models.

Marco, Starkey and Hornung were one of the first researchers to provide a design method for the

selection of belts based on a horsepower-life relationship [2]. Oliver, Johnson and Breig

highlighted the importance of including cord stress ranges into the V-belt fatigue formulas since

cord is the load carrying element [3]. Fritzon developed “the parametric belt segment model” to

examine the stress-strain state at any location in the belt, while simulating a running belt [4].

Karunendiran utilized a stress-life approach to develop the belt life equation, where serpentine

belt rib stresses were used as the damage parameter [5].

All of the above belt fatigue life models were derived on the basis of results obtained from

vast experimental belt testing. Moreover, most of these models have one common requirement,

which is the necessity of obtaining maximum stress levels of the belt during a full cycle.

Therefore, the first step towards developing a fatigue life model is to determine the stress state of


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the belt, during its interaction with the pulley system. Finite Element Analysis (FEA) is

considered as one of the prevailing stress analysis methods.

Material modeling and characterization is one of the essential steps in FEA. Mechanical

properties of each material must be accurately determined, so that the FE simulation represents

that material as close as possible under the given loads and boundary conditions.

The cord section of the belt acts as the main load-carrying component. Due to its high

stiffness, it shows a linear-elastic behaviour under the operation loads for automotive

applications. The ribbed section of the belt is made of rubber to create the required grip between

the belt and pulley during power transfer. It is common to consider rubber as a hyperelastic

material. Such a material deforms in a non-linear fashion with the ability of sustaining reversible

strains of several hundreds of percent.

1.2 Research Motivation

The state of stress in the belt, when it interacts with a pulley, is very complex and there are no

simple empirical formulas for the analysis of this type of problem [5]. Therefore, a Finite

Element Model (FEM) is the only available option for representing and analyzing the mechanics

behind belt-pulley interactions. Various examples of FEA of accessory drive belts can be found

in literature [5]-[8]; however, in all such attempts, the belt properties are whether taken directly

from literature, or received from a manufacturer. Such luxury is often far from reality. The

accessory drive manufacturers mostly purchase the belts from variety of vendors, who do not

give out the specific properties of the belt, which is required for FE modeling.


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Experimental data obtained from tests are useful to calibrate constitutive models in order to

accurately reproduce mechanical behavior of the materials in FE simulations [6]. In order to use

any of the existing hyperelastic material models for rubber, certain ASTM tests are

recommended. However, the rubber specimens that can be extracted from the belt are much

smaller than the required size, and therefore, no standard test on rubber is feasible. Therefore, a

new technique for parameterizing the accessory drive belt is much needed.

1.3 Objective and Contributions

The presented research work introduces a new methodology to characterize the constitutive

components of accessory drive belts. The major objective of this research is to devise a technique

to determine all the mechanical parameters of a multi-ribbed automotive accessory drive belt,

required for simulating it in a commercial FEA package. This objective consists of the following

tasks:

1. Designing a set of mechanical testing methods to obtain force-displacement behaviour

of the belt constituents, under various modes of deformation. While some of these

tests are already conventional (tensile testing of the belt), some of the suggested

experiments (such as the belt compression and shear tests) are new for the purpose of

characterizing the belt for FEA applications.

2. Utilizing the force-displacement results to determine the stress-strain behaviour of the

belt’s elastomeric region for the conducted tests.

3. Extracting the hyperelastic material properties of the belt from the stress-strain graphs.

4. Applying the determined mechanical properties of the belt constituent’s in a series of

FE simulations, and comparing the numerical and experimental results.


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The proposed methodology offers a relatively fast, cost effective and efficient solution for

determining the stress distribution in accessory drive belts, under variety of operational loading

conditions. Most importantly, this technique is completely independent of the belt manufacturers,

and would only utilize the finished belt, for extracting the necessary properties. Therefore, this

new approach can be used as a stand-alone tool by manufacturers to assess and compare the

stress distribution of different accessory drive belts, in a relatively short time.

1.4 Thesis Outline

The following is a brief overview of each chapter of this thesis, which illustrates the sequence

of tasks required for this approach to parameterize the accessory drive belt:

Chapter 1 presents some background information on automotive accessory drive belts, and

gives a brief introduction on the work that has been done thus far on the belt fatigue life models.

This will lead into the motivation behind this research, the major objectives that are

accomplished, and the application of these outcomes for the potential users. Lastly, the thesis

outline is illustrated in this chapter.

Chapter 2 gives a detailed literature review on FEA of accessory drive belts, and more

specifically, discusses the conventional methods for parameterizing elastomeric materials. This

chapter discusses all the shortcomings of these conventional methods and builds up the structure

and expectations for a new technique.


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Chapter 3 provides the details of the newly proposed mechanical test procedures for

characterizing the belt constituents’ mechanical properties. As a result, series of force-

displacement graphs are generated that describe the behaviour of the belt components under

various modes of deformation.

Chapter 4 fully describes how the required belt constituent’s material parameters are

extracted from the mechanical test results. Most importantly, the force-displacement graphs of

the mechanical tests are converted to stress-strain field, for the rubber section of the belt, from

which the rubber hyperelastic parameters are extracted.

Chapter 5 is devoted to verification of the discussed parameterization methodology. In this

chapter, series of finite element simulations are presented, which replicate each of the previously

conducted mechanical tests. These models verify the correlation between the experimental and

numerical results.

Chapter 6 presents the application of this research work in belt life prediction. This chapter

briefly discusses the relationship between the FE stress distribution and the existing fatigue life

models, and the contribution of this research work on predicting the life of the accessory drive

belts.

Chapter 7 concludes this thesis, and summarizes the objectives and accomplishments of this

research work. Furthermore, it proposes additional novel ideas for future research on the similar

grounds.

Chapter 2: Literature Review

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2 LITERATURE REVIEW

This chapter is a thorough literature review on the subject of parameterizing elastomers for

finite element application, previously conducted by other researchers. First, the prevailing rubber

material models are introduced. The Mooney-Rivlin hyperelastic material model is described and

it is discussed why this model is suitable to represent the rubber region of the belt during finite

element analysis. Thereafter, the conventional mechanical tests for extracting the required

hyperelastic parameters are discussed. Lastly, the feasibility of theses conventional techniques

for accessory drive belt modeling is assessed.

2.1 Belt Mechanics and the Existing Material Models

The first step in any material modeling technique is to determine the parameters, which

explain the behavior of that material under the given stress-strain state. These parameters depend

on the constitutive model representing the material.

The cord section of the belt acts as a linear elastic material in the accessory drive belt, and

carries most of the tensile load during the power transfer (Figure 3). For such a material,

Young’s Modulus is the most important parameter that needs to be determined, in order to

predict the behavior of the cord in the elastic region. The ribbed section of the belt is made of

rubber to create the required grip between the belt and pulley during power transfer (Figure 3).


8

Figure 3 (a) Cross sectional view of a 5-ribbed accessory drive belt, and (b) its CAD model

Although stress-strain behavior of rubber shows great dependency on time, most models

proposed for rubber are equilibrium based. Such models correspond to equilibrium material

response at infinitely slow loading condition [8]. A common equilibrium-based approach is to

model rubber as a hyperelastic-elastic material. Such a material deforms in a non-linear fashion

with the ability of sustaining reversible strains of several hundreds of percent; thus more

parameters are required for representing rubber [9]. The most preferred method to determine

these parameters is through experiments [11]. The objective of these tests is to put the rubber

into well-defined, “pure” strain fields, such that the stress-strain curve represents the elastomer’s

behavior only in the desired state. By combining different modes of stress-strain data, a general

and transferable material model can be obtained [12]. Various constitutive material models exist

which explain the non-linear deformation of rubber based on the strain energy density function

W [13]. There are two classes of such constitutive models [14]. The first class assumes that the

strain energy density is a separable function of the three principal stretch ratios λi. The principle


9

stretch ratios are related to the principle strains, εi by:

ii ελ += 1 3,2,1=i (1)

Ogden [15] and Peng -Landel [16] material models are examples within this class. In the

second class, it is assumed that the material is isotropic and the strain energy can be defined as a

function of the right Cuchy-Grean Tensor strain invariants I1, I2 and I3:

),,( 321 IIIfW = (2)

I1, I2 and I3 can be written as functions of the principle stretch ratios. When incompressibility

is assumed, this model is commonly referred to as the Rivlin material [17], and If only first-order

terms are used, the model is referred to as the Mooney-Rivlin material [18].

When choosing any of these models, the range of accuracy and computational efficiency are

important factors to consider, all of which depends on the application. Mooney-Rivlin is a

commonly used constitutive material model, capable of predicting the behavior of elastomers of

up to 200% stretch, with sound accuracy [19]. The general strain energy potential for the two-

term Mooney-Rivlin is as follows [17]:

2201110 )1(1)3()3(−

+−+−=Jd

ICICW (3)


10

C01 and C10 are the material constants and d is the incompressibility factor, associated with the

Mooney-Rivlin material model. For the purpose of this research, the above formulation has been

considered as the constitutive material model of the choice. This is because the range of strains

imposed on the rubber component of the accessory drive belts are under 10%, during regular

operation [5]. Furthermore, the computational efficiency of this material model is much higher

than other more precise, but complex ones, such as Ogden or Arruda-Boyce.

In order to characterize the elastomeric component of the belt, the Mooney-Rivlin constants,

C10 and C01 must be determined. Since rubber is almost incompressible, the third term in (3)

disappears. In order to determine these parameters, series of conventional mechanical tests are

recommended in literature [20]. Each test puts the material in a particular mode of deformation

(pure strain) in order to study the behavior of rubber in that desired state. By superimposing the

stress-strain results of simple deformation modes, the behavior of rubber under mixed, more

complex modes of strain can be determined.

2.2 Conventional Material Characterization Tests

Rubber can deform under four basic modes of deformation: uniaxial, equi-biaxial, planar and

volumetric (the former three modes are illustrated in Figure 4) [9]. Furthermore, each of these

modes can be done in tensile or compression states. For an incompressible elastomer, uniaxial

tension, equi-biaxial tension, and planar tension are considered as standard material

characterization tests [21].


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Figure 4 Basic modes of deformation for rubber [21]

2.2.1 Uniaxial Tensile Testing

The first step in any material modeling technique is to determine the parameters, which

explain the behavior of that material under the given stress-strain state. These parameters depend

on the constitutive model representing the material.

The tensile testing of rubber has become a standard procedure under ASTMD 412 [9][24].

ISO 37:2005 also provides requirements for uniaxial compression and tension test on rubber

[6][25]. The most significant requirement for this test is that the test sample must be much longer

in the direction of stretching than in width and thickness directions (length: width ratio > 10).

This is necessary to eliminate any lateral constraint to specimen thinning. Typically for this test,


12

dumbbell shaped specimens are cut and used in standard uniaxial tension machine and video

extensometer is used in order to capture strain in the area of the specimen with a uniform

distribution of deformation [21]. Figure 5 illustrates a tension experiment using a video

extensometer.

Figure 5 Uniaxial tensile testing of rubber [22]

Sasso et al. has performed the tension test as follows [6]:

“A standard uniaxial stretching test was performed on dumbbell specimens with a thickness of

1.7mm. The “useful” part of the specimen, that is the part with maximum and uniform

deformation, was delimited by two white markers. This zone has a length of 40mm and a width

of 9mm. The test rig consists of an electro-mechanical testing machine, model Zwick Z050.

Load is measured by a 5KN load cell, the displacement, which is proportional to the engineering

strain, is measured by the crosshead LVDT, while the true strain through the gauge part of the


13

specimen is measured by a high resolution video extensometer by comparing distances between

markers along time.”

As per ASTM D412, the uniaxial tensile test samples can be dumbbell or cut ring shape. For

dumbbell shape samples, the following must be noted:

“Whenever, possible, the test specimens shall be injection molded or cut from a flat sheet not

less than 1.3 mm (0.05in) nor more than 3.3 mm (0.13 in) thick and of a size which will permit

cutting a specimen by one of the standard methods (see Practice D3182). Sheets may be prepared

directly by processing or from finished articles by cutting and buffing. If obtained from a

manufactured article, the specimen shall be free of surface roughness, fabric layers, etc. in

accordance with the procedure described in Practice D3183. All specimens shall be cut so that

the lengthwise portion of the specimens is parallel to the grain unless, otherwise specified. In the

case of sheets prepared in accordance with Practice D3182, the specimen shall be 2.0 ± 0.2 mm

(0.08 ± 0.008 in) thick died out in the direction of the grain. Use Die C, (Unless otherwise noted)

to cut the specimens from the sheet with a single impact stoke (hand or machine) to ensure

smooth cut surfaces.”

Table 1 illustrates the standard die sizes for cutting the dumbbell shape samples. Figure 6,

illustrates the standard dies for cutting dumbbell specimens. ASTM D3182 & D3183 are the

standard practices for preparing and cutting rubber specimens [26][27]. For cut ring samples, two

types are allowed by ASTM D412 as specified in Table 1. Unless otherwise specified, the Type 1

ring specimen shall be used.


14

Table 1 Dimensions of standard dies for cutting dumbbell shape samples as per ASTM D412

Table 2 Dimensions of standard dies for cutting ring samples as per ASTM D412 [24]

Type 1 mm in.

Circumference (inside) 50.0 ±0.01 2.0±0.004

Diameter (inside) 15.92±0.003 0.637±0.001

Radial width 1.0±0.01 0.040±0.0004

Thickness, minimum 1.0 0.040

Maximum 3.3 0.13

Type 2 mm in.

Circumference mean 100.0±0.2 4.0±0.0004

Diameter (inside) 29.8±0.06 1.19±0.0001

Radial width 2.0±0.02 0.08±0.0008

Thickness, minimum 1.0 0.04

Maximum 3.3 0.13


15

Figure 6 Standard dies for cutting dumbbell specimens as per ASTM D412 [24]


16

2.2.2 Equi-biaxial Tensile and Uniaxial Compression Testing

The use of only the uniaxial tensile test for the characterization of the material properties of

rubber is insufficient, and further tests are necessary to capture the behaviour of rubber in other

modes of deformation [28]-[33]. Figure 7, highlights the rubber equi-biaxial stretching on an

Iwawoto machine. The cost and complexity of the procedure has resulted in development of

other test rigs, such as the jig, Ford Scientific Labs made at Ann Arbor Axel Products, in

Michigan (Figure 8). The Axel Products’ tester stretches a round sheet of rubber with cables

passing by pulleys, while a laser monitors deformation of grids [9].

Figure 7 Rubber equi-biaxial tension testing at NASA space center [9]


17

Figure 8 A bi-axial extension experiment with Axel Products' setup [21]

As can be notices, bi-axial stretching of rubber is a complex procedure that might show non-

repeatable results and it is still a research and development topic [9]. Fortunately, since rubber is

almost incompressible, upon addition of hydro-static stresses, equi-biaxial tension and

compression tests become identical to uniaxial tension and compression (Figure 9). Therefore,

the equi-biaxial tensile test can be replaced by uniaxial compression of rubber, which is well

documented under ASTM D575 [34]. Figure 10 illustrates uniaxial compressive loading of

rubber.


18

Figure 9 Equivalent Testing Modes [20]

As per ASTM D575 the test specimens must be of the following dimensions:

“Standard test specimens shall be 28.6 ± 0.1 mm (1.129 ± 0.005 in) in diameter (650 mm2

(1.000 in2) in area) and 12.5 ±0.5 mm (0.49 ±0.02 in) in thickness, from which all molded

surface layers have been removed. “


19

Figure 10 Compression testing of rubber [22]

2.2.3 Planar Tensile and Pure Shear Testing

The purpose for planar tensioning of rubber is to make the transverse stress (normal to the

sheet) zero. Although at the first glance this strain mode might be confused with the uniaxial

tensile test, due to the high width to length ratio, all the thinning occurs in one direction [35]

(through thickness) as illustrated in Figure 11 (Figure 12-Figure 14 illustrate the planar

tensioning test setup). For this purpose, short, yet very wide rubber strips must be used. Sasso et

al. describes the planar tensioning test procedure as follows [6]:


20

“The test rig is very simple because a strain measurement system is not required. A flat rubber

specimen is glued to metal brackets in such a way that the effective undeformed length is 15 mm

and the width of 60 mm is fixed on the boundaries throughout the test. The thickness of the

specimen is 1.7 mm. The brackets have a “L” shape in order to match the clamping of the tensile

machine; with this kind of fixture the load can be applied avoiding any shrinking of the specimen

between the clamps. Taking these cautions on boundary conditions, it is possible to put fixed

thickness and width on the lower and upper edges of the specimen into the FEM model, without

introducing excessive approximation. Displacement is applied at 2 mm/min and crosshead

position and load are recorded during the test. This data will be compared with numerical data

from FEM to validate the previously fitted constitutive models of the rubber.”

Figure 11 Schematic representation of planar tension [22][35]


21

Figure 12 Planar tension test on a rubber specimen glued to metallic brackets [6]

Figure 13 Planar tension test grips, Axel Products [21]


22

Figure 14 Planar tension test, Axel products [21]

When incompressibility is assumed, the planar tension test and pure shear tests are

replaceable as well. As a matter of fact, one method to perform pure shear test is exactly the

same as the above planar test. Since the material is almost incompressible, a state of pure shear

exists in the specimen at a 45-degree angle to the stretching direction [21]. The most significant

aspect of the specimen is that it is much shorter in the direction of stretching than the width.

Another method of performing pure shear test is via the Mechanical Oscillograph (Figure 15),

as per ASTM D945 [36]. ASTMD 945 is the standard practice for determining “rubber

properties in compression and shear”. Table 3 details the specimen sizes for the shear test under


23

ASTMD 945. For measurements in compression, the test specimen is prepared from solid rubber

as follows:

“For solid rubber specimens: At least two specimens shall be tested, except that at least

three shall be required if measurement of creep is to be included. “

Figure 15 Advanced Yerzley Oscillograph [36]

“As per ASTMD945 for solid rubber samples, the test specimens for measurements in

compression shall be right circular cylinders chosen from the following alternatives:”


24

Table 3 Sample sizes as per ASTM D954 [36]

Shape Factor Primary Practice Diameter

0.390 SI Units 19.5±0.13mm

0.375 Inch-pound Units 0.75±0.005in

Shape Factor Height Reference Area of Nominal Circle

0.390 12.5±0.25mm 300mm2

0.375 0.5±0.010in 0.442in2

2.3 Micro-Mechanical Testing

Another alternative stream of tests for the purpose of characterizing the rubber component of

the accessory drive belts is micro-mechanical testing. These tests are generally designed for the

specimen of orders of magnitude smaller dimensions, and consequently much lower load levels.

The following gives a thorough overview of the most prevailing of such tests.

When the available rubber samples are very small, the traditional macro-scale mechanical

tests cannot be utilized, as the specimen for those tests requires a minimum regulated size. Also,

traditional test methods are difficult to apply to micro-scale ones due to the fragility of the

samples [37]. Micro-mechanical testing is a viable solution for those cases. These micro tests are

usually the same as those used in traditional mechanical testing; however, fixtures and samples

are much smaller than the standard ones. Most of these equipments require very low forces, but

very high resolution, as the deformations obtained are also very small. The micro-mechanical

tests found in literature are mostly micro-tensile, hardness and three-point bending tests. Planar


25

tension and uniaxial compression tests have not been greatly explored.

Micro testing standards for rubber are limited to hardness test, which is included in ISO48.

This test is essentially similar to the conventional hardness test, but receives specimens of 2mm

thickness and uses a 0.0395mm diameter ball with 153.mN total force and an appropriately

reduced foot for the equipment [38]. This procedure points out that even through the micro and

conventional hardness tests are the same in principle; results from these tests are poorly

correlated. The main reason for the discrepancies is that friction effects are much more

significant in micro testing. Furthermore, ASTM D1708 outlines the standard test method for

tensile properties of plastics in micro-scale, which is applicable to elastomers as well, since

rubbers and thermoplastics have similar structures [38]. No direct standard for micro-tensile

testing of elastomers is available. The ASTM D1708 standard is superseded by D882 (Test

method for tensile properties of thin plastic sheets) or D638 (Test method for tensile properties

of plastics) when applicable. However, yield, tensile and ultimate stresses are the only properties

that can be extracted from this test. Therefore, it cannot be used to calculate elastic modulus.

An in-depth literature review shows that there have been some research efforts towards

development of micro-tensile techniques. A research at the National University of Singapur lead

to development of tensile testing of a single-strand polycaprolactone electrospun ultrafine fiber

using a commercial nano-tensile testing system (see Figure 16) [39]. They successfully obtained

stress-strain curves for PCL microfibers of various diameters ranging within 1.4 +/- 0.3 mm.

They also used an existing atomic force microscope (AFM)-based nano-indentation system for

the tensile testing of these microscale fibers [40]. During the process for which the system is


26

intended for, the depth of the indent relative to the sample surface as well as the indentation load

are continuously recorded. A plot of indentation load against displacement during the loading

and unloading of the nano-indenter is then obtained and mechanical properties such as hardness

and Young’s modulus of elasticity are derived. An alternative use of this system would be to

reverse the loading direction of the nano-indenter thereby converting it into a nano-tensile tester.

Figure 16 Sample mounted on the nano tensile tester (NanoBionix, MTS, USA) [39]

Gudlavalleti, Gearing and Anand [41] described the design of two flexure-based

micromechanical testers capable of measuring forces with an accuracy of 25 μN over a range of

1-30 N, and specimen extensions with an accuracy of 20 nm over a range of 1-5 mm (Figure 17).

The novelty of their test machine is the use of two compound flexures in an integrated

monolithic frame: one flexure working as a precision guide for actuation, and the other flexure as


27

a linear spring for force measurement. Stress-strain curves can be obtained from the force-

displacement data recorded.

Figure 17 Flexure-based testing machine [41]

Another possible technique to obtain material parameters of a micro sample would be to use a

surface profilometer as explained by Hopcroft [37]. A contact surface profilometer stylus was

swept along the length of a suspended cantilever beam of the material under analysis (Figure 18).

Since the stylus contacts the sample with a constant force, the deflection downwards was

recorded as the instrument traces along the beam. This data set was then analyzed in combination

with the geometry of the test structure to determine mechanical and materials properties of the

specimen. This method is insensitive to potential errors in test structure fabrication, such as

undercut of the cantilever root or residual stress in the material under test, which are all potential


28

flaws during fabrications of microscale samples. Tensile, compression and bending tests can also

be performed by changing loading and specimen fixtures. The load resolution of the test rig was

10 μN and the displacement resolution was 5 nm.

Figure 18 Schematic illustrating the methodology proposed in [37]

2.4 Shortcomings of the Conventional and Micro-Mechanical Test Methods

The discussed mechanical testing procedures are not suitable for characterizing accessory

drive belts for the following reasons:

1. There are no rubber samples available from the belt manufacturers that can be used as

mechanical test specimens, mainly due to the nature of the accessory drive belt

manufacturing. The rubber component of the belt is cured while being molded around the


29

cord section. Therefore, the raw rubber that is used in the process does not have the same

properties as the rubber section of the finished belt. Therefore, the belt manufacturers have

to custom-mold and cure the rubber samples of the desired sizes and shapes, separately for

the test purposes, which is not a feasible option.

2. The ASTM D412, D575 and D945 thoroughly describe the uniaxial tensile, uniaxial

compression and shear test procedures for elastomers respectively. However, all these

procedures require the use of rubber specimens of certain dimensions that are not possible

to extract from a finished belt. The elastomeric portion of the belt is too thin, and narrow

to be used in any of the standard tests.

3. Some of the conventional mechanical tests, such as the biaxial compression and tension

are complex in nature, and hardly repeatable. Furthermore, these tests involve complex

and expensive machinery, which are mainly custom-made. One of the main objectives of

this research project is to use a material characterization procedure that is simple, and

inexpensive.

4. The majority of the micro-mechanical tests are not regulated under any industry standard,

and they are still the topic of research and development. Therefore, the reliability and

repeatability of these tests are questionable. Furthermore, the load levels that these tests

deal with are mostly too low for the accessory belt application.


30

The above reasons necessitate a customized methodology of testing the accessory drive belt,

in order to extract its mechanical properties. The main criteria for this methodology are high

computational efficiency, low cost, and complete independency from the belt manufacturer for

extracting the required belt parameters.

Chapter 3: New Procedure for Mechanical Testing of Accessory Drive Belts

31

3 NEW PROCEDURE FOR MECHANICAL TESTING OF

ACCESSORY DRIVE BELTS

Chapter 2 addresses the common practices used for determining the mechanical properties of

accessory drive belts, most importantly for extracting the parameters of the belt’s elastomeric

region. The ribbed region of the belt is made of rubber. For FE applications, this region is

considered as a hyperelastic material, since rubber shows a non-linear reversible deformation

behavior under load. Chapter 2 concludes that the conventional test methods are not suitable for

determining the hyperelastic properties of rubber in accessory drive belts, mainly due to the

complexity of some these techniques, and the specimen size requirements by ASTM standards.

Chapter 3 introduces new mechanical tests that are specifically customized for parameterizing

accessory drive belts. These tests are conducted once on the whole belt, and once on an extracted

strip of the rubber rib. They are designed to impose single and multi-modes of strain on the

rubber region of the belt. The details and results of each test are fully described in this chapter.

3.1 Mechanical Testing

In order to resolve the hyperelastic characterization problem in accessory drive belts, series of

customized tests were devised. Although these tests do not follow the conventional rubber

parameterization standards discussed earlier, they are designed to impose simple strain on the

rubber portion of the belt, in the desired directions. Each test is conducted once on the whole

belt (5-ribbed specimen), and once on an extracted rib of the belt (hereafter called the rubber rib).

The output of each test is a force-displacement graph, which will be converted to stress-strain


32

field. All these tests were conducted with a low strain rate and the material response was

measured at equilibrium.

3.2 Uniaxial Tensile Testing of the Whole Belt and the Rubber Rib

Belt, cord and rubber rib samples of 400mm, 156mm, and 56mm long were cut and prepared

for the uniaxial tensile tests in MTS-ET64 machine as illustrated in Figure 19. The tensile test

chamber was controlled at the room temperature during the whole experiment. In order to create

a tight grip between the belt and the tensile tester jaws, the rubber ribs of both ends of the sample

belt were removed as illustrated in Figure 20.

The force displacement attributes of the belt in tensile can be related to its cord region

stiffness, since almost the entire tensile load is carried by the cords and not by the rubber ribs.

The results of the cord and rubber rib tests were used alongside to compare against the belt

tensile results, and to verify the stiffness of the cord region. Figures Figure 21- Figure 23

illustrate the force-displacement behavior of the cord, rubber rib and the belt respectively.


33

Figure 19 Rubber rib and belt uniaxial tensile tests

Figure 20 Removal of the ribs at the ends of the belt sample for the uniaxial tensile test


34

Figure 21 The cord uniaxial tensile test result


35

Figure 22 The rubber rib uniaxial tensile test result

Figure 23 The belt uniaxial tensile test result


36

3.3 Uniaxial Compression Testing of the Belt and the Rubber Rib

Two 100mm belt and rubber rib samples were cut and prepared for the uniaxial compression

tests in MTS ET65 machine as illustrated in Figure 24. The assumption in the belt compression

test is that the cord region is much stiffer than the rubber and therefore, the rubber ribs will be

the only components to deform under compression.

This test was once performed by compressing the ribbed side of the belt against a flat surface

(hereafter called groove-on-flat test), and once by compressing the ribbed region against a

grooved surface, replicating the pulley grooves (hereafter called groove-on-groove test) (Figure

25). The groove-on-flat test is used to generate the required stress-strain curves for

parameterizing the rubber in uniaxial compression deformation mode, because it is assumed that

the much stiffer cord region’s deformation is minimal in comparison to the rubber ribs’

compression. Therefore, it is assumed that the rubber ribs are the only components that are

taking the compression loads. Since they are being squeezed against a flat surface, the only

active mode of strain is uniaxial compression.

On the other hand, the groove-on-groove compression test, is used as a point of reference to

verify the FEA results. This is since the groove-on-groove compression test imposes multi modes

of deformation on the rubber region: uniaxial compression and shear (between the belt and

pulley ribs). Therefore, this test is not suitable for generating stress-strain curves of rubber under

simple strain direction.


37

Figure 24 Rubber rib and belt uniaxial compression tests

Figure 25 (a) Groove-on-flat and (b) Groove-on-groove surface interactions

The force-displacement results of the rubber rib, belt groove-on-flat and the belt groove-on

groove tests are illustrated in Figure 26 - Figure 28.


38

Figure 26 The rubber rib uniaxial compression test result

Figure 27 The belt groove-on-flat compression test result


39

Figure 28 The belt groove-on-groove compression test result

As can be noticed in the above figures, although the compression tests were performed in

low strain rates, hysteresis is noticeable between loading and unloading phases, due to

damping effects of rubber. The rubber hysteresis is not of an importance when characterizing a

hyperelastic material. For the purpose of extracting the material parameters, either of the

loading, unloading, or the average force-displacement curves can be used. Ultimately, the

force-displacement results of the FE simulation must match that of the experiment; if the

loading portion of the compression curve was considered for generating the parameters to

begin with, the FE force-displacement results of the compression test must match that of the

loading portion of the experiment, and vice versa. For the purpose of this research work, the

loading portions of the compression load-displacement curves were considered. This is


40

because from the engineering perspective, the loading portion generates higher stresses in the

rubber, and is considered to be the worst case scenario; a factor to design for in this process.

3.4 Pure Shear Testing of the Belt and the Rubber Rib

Two 156mm belt and rubber rib samples were cut and prepared for the shear tests in MTS

ET65 machine. The test setup is illustrated in Figure 29. The belt is clamped to the base on its

flat side, while its ribbed side is free to slide against the moving block’s grooves. The moving

block is pulled in the illustrated direction, while the force and displacement is measured. As the

force increases, the block starts sliding against the belt, converting it to a friction test. Therefore,

the portion of the force-displacement curve before the sliding of the moving block (Figure 30)

can be used for the purpose of shear testing, while the sliding portion of the graph can be used to

determine the friction coefficient between the rubber and the pulley.

This test was conducted with three sets of dead weights keeping the pressure between the

moving block and the belt: 17.7 kg, 43.2 kg and 61 kg. The coefficient of static friction between

the rubber and moving block grooves, μs, is measure to be 0.6, which showed to be consistent for

all three sets of dead weights.


41

Figure 29 (a) Shear test setup, (b) Belt and moving block interaction

Figure 30 Force-displacement result of the shear test – 61 kg dead weight


42

3.5 Three-Point-Bending Test of the Belt

A 100mm belt sample was prepared for the three point bending test. The actuator loads the

cord side of the belt as illustrated in Figure 31, while the displacement of the center of the belt is

measured. The force-displacement behavior of the belt in bending is illustrated in figure. The

bending stiffness of the belt is measured to be 0.19 KN/m.

The belt bending test involves various modes of strain such as axial tensile and compression,

and shear (contrary to the uniaxial tensile testing of the rubber rib, or the groove-on-flat uniaxial

compression testing of the belt, which involve single modes of deformation). Therefore, this test

will not be directly used to determine the hyperelastic parameters of the rubber, but will be

applied as a control experiment to be compared against the three-point-bending FE simulation.

Figure 31 The three-point-bending test of the belt


43

Figure 32 The three-point-bending test results

3.6 Conclusion

In this chapter, a series of customized mechanical tests are introduced. The first series of these

tests are designed to put the rubber region of the belt under simple state of deformation. These

experiments are once conducted on the whole belt and once on an extracted rubber strip of the

belt, and are namely uniaxial tensile, uniaxial compression, and shear tests. The results of these

experiments are utilized in the proceeding chapters to determine the hyperelastic parameters of

the rubber region. The second set of tests, namely groove-on-groove compression of the belt

(which imposes downward compression on the belt against a grooved surface) and the three-

point bending of the belt, put the rubber region under multi-modes of deformation. The results of

these tests are used in Chapter 5 to verify that of the FE simulations.

Chapter 4: Determination of the Belt Constituents’ Parameters

44

4 DETERMINATION OF THE BELT CONSTITUENTS’

PARAMETERS

Chapter 3 describes a series of customized mechanical testing procedures to determine the

mechanical properties of the belt’s components. The focus of the majority of these tests is to put

the hyperelastic rubber section of the belt under various modes of deformation. As a result, the

force-displacement behavior of rubber under uniaxial tensile, uniaxial compression and shear

modes of deformation is determined.

In Chapter 4 the mechanical properties of the belt components are determined based on the

conducted mechanical tests. First, the elastic modulus of the cord region is determined.

Furthermore, the force-displacement behavior of the rubber region is converted to the stress-

strain field. Lastly, the hyperelastic parameters of the rubber region are extracted from these

stress-strain curves.

4.1 Determination of the Cord Region Elastic Modulus

Figure 33 illustrates the cross-sectional snapshots of the belt and the rubber rib respectively.

The cord region of the belt is considered as a continuous fiber-reinforced hybrid material. The

tensile cords and the rubber replicate the fiber and matrix constituents respectively. The elastic

modulus of the belt can be calculated once by extracting it from the belt tensile test implicitly,

and another time by explicitly approximating it from the cord and rubber rib tensile stiffness

values, via composite formulation.


45

Figure 33 Cross-sectional snapshot of (a) the belt and (b) the rubber rib

The cord is a linear elastic material, much stiffer than the rubber matrix. Since the cord fibers

dominate the tensile behavior of the cord region, this section can be treated as a linear elastic

material. Furthermore, the rubber matrix is exposed to small amount of strain in this region;

therefore, its tensile behavior falls under the linear elastic section. For a linear elastic material,

once the stiffness K is determined for geometry of length L and cross-sectional area A, the elastic

modulus E can be determined as follows:

AKLE = (4)

In the first approach, the stiffness of the cord region was measured from the belt tensile test.


46

Consequently, the cord region elastic modulus was calculated as 7.9GPa, using (4).

In the second approach, the stiffness values of the rubber and the cord were measured from

the rubber rib and the cord tensile tests. The cord and rubber elastic moduli were measured as

18.6GPa and 7.7MPa respectively. In order to approximate the cord region’s elastic modulus

Etotal from its constituents, the composite material formulation is used, as follows [23]

cordrubber

rubberrubbercordcordtotaol AA

AEAEE

++

= (5)

The elastic modulus of the cord region was calculated as 7.8GPa using (5). The correlation

between both approaches in approximating the elastic modulus of the cord region proves that

neglecting the involvement of the rubber ribs in bearing any tensile load in belt tensile test has

been a sound assumption.

4.2 Conversion of the Force-Displacement Test Results into the Stress-Strain Field for the Rubber Region

In the next step, the resultant force-displacement graphs were converted to stress-strain field.

Series of assumptions were made in order to carry out this conversion with reasonable accuracy.

This is necessary since the non-uniform cross-sectional area of the test specimens, make this

conversion unconventional.

As illustrated in Figure 34, the rubber rib has a trapezoidal shape, with sides Y0 and Y1. When

the rib is compressed by length L, the base length Y0 increases to Yinstantaneous. When converting


47

the applied loads to real stresses in the compression test, the average of Y0 and Yinstantaneous is

calculated for each compression length L to take the instantaneous cross section area into

account. Furthermore, during the shear tests, it was assumed that the cross-sectional area does

not change due to shear, and it solely distorts.

Figure 34 Compression of the rubber rib

The resultant stress-strain behavior of the rubber material under uniaxial tensile, uniaxial

compression and pure shear tests are illustrated in Figure 35- Figure 37. As can be seen, the

whole belt, and the rubber rib engineering stress-strain results of the shear test correlated closely.

When dealing with hyperelastic materials, commercial software packages such as ABAQUS,

or ANSYS require the engineering stress-strain data to be inserted for parameterization purposes.

The software will try to estimate the true stress-strain values based on the provided material data,

geometry and the generic conversions formulas, as per following [42]:


48

)1( EET εσσ += (6)

)1ln( ET εε += (7)

In equations (6) & (7), σT, σE, εT and εE represent the true and engineering stresses and strains

respectively. When comparing the engineering stress-strain behavior of the rubber rib and the

belt groove-on-flat compression tests, the results did not correlate. However, after taking the

assumption, discussed earlier with regards to the uniaxial compression force-displacement

conversion procedure, and using (6) and (7), the generated true uniaxial compression stress-strain

curves of the rubber rib and the belt, correlated closely (Figure 36). This phenomena is somewhat

expected, as the engineering stress-strain curve for the compression case, cannot be an accurate

measure of comparison between the rubber rib and the belt stress-strain behavior under

compression. This is because the engineering stress-strain is blind to the actual situation that

happens when the belt is compressed radially, due to the rib’s non-uniform trapezoidal cross-

section, as discussed earlier. The true stress-strain conversion procedure is a more practical

measure of comparing the two results, and therefore it is deemed to be more reliable. In spite of

this fact, the engineering stress-strain curves of both the rubber rib and the belt groove-on flat

compression tests were used to extract the required Mooney-Rivlin parameters. That is why there

will be two sets of Mooney-Rivlin parameters, one for the rubber rib compression, and one for

the belt groove-on flat compression.

Overall, the presented results in Figure 35- Figure 37 indicate that the method applied for

generating the stress-strain curves was reasonable, as both the rubber rib and whole belt tests

resulted in similar stress-strain curves. The results further verify the assumptions taken in


49

conducting the force-displacement to stress-strain conversions.

Figure 35 Stress-strain behavior of rubber in uniaxial tensile


50

Figure 36 Stress-strain behavior of rubber in uniaxial compression

Figure 37 Stress-strain behavior of rubber in shear


51

4.3 Determination of the Mooney-Rivlin Constants

The generated stress-strain curves must now be used to extract the Mooney-Rivlin constants,

which represent rubber’s behavior under various modes of strain. When defining a hyperelastic

material in ABAQUS, the software gives an option of inputting the material model constants

directly, or inserting the stress-strain curves of the uniaxial tensile, uniaxial compression, planar

tensile, and volumetric tests for rubber, so that these parameters are automatically calculated by

the software. When the latter procedure was used, the estimated Mooney-Rivlin constants made

the material unstable (and thus the generated FE simulations never converged). Therefore, the

Mooney-Rivlin parameters must be extracted separately before being used by ABAQUS.

In order to extract the required parameters, equation (3) has to be curve fitted to the stress-

strain curves of the rubber, through the method of least squares. For this purpose, commercial

software, Hyperfit, was used, which conducts this curve fitting conveniently. The software goes

through thousands of iterations to find the most optimized values for C01 and C10 constants,

which fit (3) to the stress-strain curves.

The Mooney-Rivlin parameters, representing the hyperelastic rubber rib in a FE simulation

fully depend on the imposed mode of deformation. For instance, when conducting FE simulation

of the rubber rib tensile test in ABAQUS, only the uniaxial tensile stress-strain curve should be

used in the curve-fitting process. Similarly, for simulating the rubber rib and the belt groove-on-

flat compression tests, only the uniaxial compression stress-strain curve is used for extracting the

Mooney-Rivlin constants. Figure 38 - Figure 40 illustrate the results of this curve fitting process.


52

Figure 38: Curve fitting the Mooney-Rivlin equation to the uniaxial tensile data using Hyperfit

Figure 39: Curve fitting the Mooney-Rivlin equation to the uniaxial compression data using Hyperfit


53

Figure 40: Curve fitting the Mooney-Rivlin equation to the pure shear data using Hyperfit

When simulating a single-axis mode of deformation scenario, the relevant stress-strain curve

of that mode is used to extract the required parameters, as described above. However, when

dealing with more complex modes of deformation, such as the groove-on-groove compression

and the three-point-bending tests, the stress-strain behavior of all the interacting modes must be

considered in the curve-fitting process.

The groove-on-groove compression process mainly involves two modes of deformation: the

radial compression, and the shear between the belt and the pulley ribs. In order to take both of

these modes into account, the Mooney-Rivlin equation (3) must be fitted to both of these curves

simultaneously. Similarly, the three-point-bending involves three modes of deformation: axial

tensile, axial compression, and shear. For the purpose of this curve-fitting process, both of the


54

uniaxial tensile and shear stress-strain curves are inserted. The axial compression strain mode

(the line of action of which runs along the belt) is fundamentally different from the radial

compression mode (the action line of which runs through the belt thickness, in the radial

direction). Therefore, the rubber compression stress-strain curve is not considered for this case.

The longitudinal compression is treated as the inverse of the longitudinal tensile by ABAQUS,

therefore, it is safe to only consider the uniaxial tensile for both modes of strain in longitudinal

direction. Table 4 summarizes the approximated Mooney-Rivlin parameters determined by

Hyperfit for all the discussed deformation modes.

The method of extracting the Mooney-Rivlin constants from the stress-strain curves can be

further improved, for the cases which involve multi-modes of deformation (bending and groove-

on-groove compression). This can be done by varying the weight of each simple strain mode

involved in that deformation based on the significance of that strain mode in the overall

deformation. In another words, the role of each mode of strain (tensile, compression, shear) in

the whole deformation can be controlled by assigning specific weights, during curve fitting

procedure. Perhaps a script can be written in C or MATLAB to have more control over this

process.


55

Table 4 Summary of the approximated Mooney-Rivlin parameters under various modes of deformation using Hyperfit

Test Modes of Deformation

Approximated Mooney‐Rivlin Constants

Belt and Rubber Rib Tensile

Tensile C01: 2.175

C10: ‐0.140

Rubber Rib Compression

Compression C01: 6.52

C10: ‐1.054

Belt Groove‐on‐Flat

Compression

Compression C01: 22.739

C10: ‐3.757

Belt Groove‐on‐Groove Compression

Radial Compression, Shear

C01: 22.555

C10: ‐3.725

Belt Bending Tensile and Axial Compression,

Shear

C01: ‐0.184

C10: 0.252


56

4.4 Conclusion

In this chapter, the resultant force-displacement graphs of the mechanical tests are converted

to stress-strain field. This conversion is not conventional, as the cross-sectional area of each

rubber rib is trapezoidal shape and hence non-uniform. Series of assumptions are made to

simplify this process. When converting the applied loads to real stresses in the compression test,

the average of the basal lengths of each trapezoidal rubber rib is considered for each compression

length, in order to take the instantaneous cross-sectional area into account. Furthermore, during

the shear tests, it is assumed that the cross-sectional area does not change due to shear, and it

solely distorts.

The belt cord’s elastic modulus is calculated once directly from the belt tensile test result, and

once via composite formulation, using the elastic modulus of the rubber rib and the cord. The

results correlate closely. Furthermore, the stress-strain behavior of the whole belt and the rubber

rib under uniaxial tensile, uniaxial compression and shear strains are compared, which also

correlate closely. This indicates that the assumptions made for performing this conversion are

sound.

Chapter 5: Finite Element Analysis Verification

57

5 FINITE ELEMENT ANALYSIS VERIFICATION

In Chapter 4, the belt constituent’s mechanical properties, namely the 2-term Mooney-Rivlin

hyperelastic parameters are determined, for various modes of deformation. The extracted

Mooney-Rivlin parameters are specific to the applied mode of strain on rubber. These results are

summarized in Table 4.

In chapter 5, the determined mechanical parameters of the belt components are inserted in

series of FE simulations. These simulations replicate the previously conducted mechanical tests.

The intention of this chapter is to compare the experimental and FEA results, and to verify the

reliability of presented material characterization procedure.

5.1 Finite Element Modeling

In order to verify the parameterization process and the approximated Mooney-Rivlin

constants, series of finite element models were developed in ABAQUS, each replicating one of

the mechanical tests, previously discussed. In each simulation, the model is exposed to a range of

displacements as boundary conditions, and the resultant reaction forces were compared against

the experimental measurements.

The optimized mesh density of each FE model was finalized after performing convergence

analysis. The element type of the choice for all the presented models was C3D20 (3D 20-node

quadratic element). This type of element is capable of capturing the deformation of hyperelastic

materials accurately. The conducted simulations can be categorized in two sections: the ones


58

involving single-axis mode of deformation, and the models with multi-axis modes of

deformation.

In all the discussed scenarios in this chapter, the cord is modeled as a linear elastic material

with Poisson ratio ν, and elastic modulus, Ecord, of 0.3 and 7.8 GPa respectively, as discussed in

section 4.1 of Chapter 4. In all the compression simulations, the flat, or grooved surfaces, against

which the rubber rib or belt were compressed, are assumed to be rigid bodies. The rubber

portions were modeled as 2-term Mooney-Rivlin hyperelastic materials. The appropriate

Mooney-Rivlin parameters for each simulation were directly taken from Table 4 , as calculated

and fully discussed in Chapter 4.

5.2 Single-Mode of Deformation Models

The first set of FE models, the rubber rib tensile, the rubber rib uniaxial compression, and the

belt groove-on-flat compression simulations, all represent simple modes of deformation. These 3

models were preliminarily used to extract the Mooney-Rivlin constants of the belt under tensile

and compressive loads. Therefore if the applied finite element modeling technique is sound, it is

somewhat expected from these models to generate similar force-displacement results as those of

the experiments. In Chapter 4, the force-displacement results of these three experiments were

used to generate the FE material parameters, and now the reverse operation is conducted, in

which the FE simulations are applied to determine whether the original experimental force-

displacement curves are achievable.

Figure 41 is a snapshot of the rubber rib compression simulation. For computational

efficiency, only 1/100th of the rubber rib length (1mm) is modeled. There are two boundary


59

conditions imposed on this model: the flat surface is fixed, while the top surface is displaced

radially (downward). Two types of contact were defined: normal, and tangential, with static

coefficient of friction of 0.6.

During the post-processing operation, the bottom face of the flat rigid body was selected, and

all the nodal reaction forces of that face were added together. The resultant value was multiplied

by 200 (to make up for the real length of the belt, 100mm, and the two ribs, which were being

tested simultaneously as per Figure 24).

Figure 41: FE simulation of the rubber rib compression test (0.25mm displacement)


60

Figure 42 is a snapshot of the rubber rib tensile FE simulation. There are two boundary

conditions imposed on this model: one end of the rib is set to be displaced axially, while

symmetry in the axial direction is imposed on the other end. During the post-processing

operation, the nodal reaction forces of the fixed side of the rubber rib were probed and added

together, to estimate the total reaction forces due to the imposed displacement.

Figure 42 FE simulation of the rubber rib tensile test (5.5mm displacement)

Figure 43 is a snapshot of the belt groove-on-flat compression simulation. For computational

efficiency, only 1/100th of the belt length (1mm) is modeled. There are two boundary conditions

imposed on this model: the bottom of the flat surface is fixed, while the top surface of the cord

section is displaced radially (downward). The cord and rubber sections are connected together

via the node-to-node tied condition. The rubber to flat surface contact parameters, and the post-


61

processing method to estimate the reaction forces are similar to the rubber rib compression

simulation, discussed earlier.

Figure 43 FE simulation of the belt groove-on-flat compression test (0.27mm displacement)

5.3 Multi-Mode of Deformation Models

The second and final set of FE models, the belt groove-on groove compression, and the belt

three-point-bending simulations, represent multi modes of deformation scenarios. The results of

these two experiments were not used earlier to extract the rubber stress-strain behavior, since

each test engages more than one mode of strain. The groove-on-groove test involves both the

radial compression and shear strain modes, while the stress-strain behavior of the three-point-

bending test is a combination of axial tensile, axial compression and shear modes of deformation.

The results of these two simulations are of a special importance. It can be claimed that the


62

reliability of the discussed material characterization methodology as a whole is concluded in

these two models. If these two FE models are capable of predicting the experimental force-

displacement results closely, the credibility of most of the assumptions discussed in this research

project would be more apparent.

Figure 44 illustrates a snapshot of the groove-on-groove compression simulation. The details

of this FE modeling process are very similar to the groove-on-flat model, discussed in section

5.1.

Figure 44 FE simulation of the belt groove-on-groove compression test (0.1mm displacement)

Figure 45 illustrates a snapshot of the three-point-bending simulation. In order to keep this

model simple, and to increase its computational efficiency, the probe and two pivot knives, that


63

were part of the actual experiment (Figure 31) were not directly modeled; though their impact

was included as a set of boundary conditions. The belt length of 100mm (which was the distance

between the pivot points in the actual experiment) is modeled in this simulation. Both pivot

edges of the belt are restricted to move radially (downward), while the center node of the belt, at

the top surface of the cord section, is being displaced downward by a range of prescribed values.

The cord and the rubber sections are connected by the node-to-node tied condition. During the

post-processing operation, the reaction force seen by that central node was probed.

Figure 45 FE simulation of the belt bending test (3.98mm deflection)

5.4 Comparison between the FEA and the Experimental Results


64

Figure 46 - Figure 50 compare the force-displacement results of the FE simulations discussed

earlier in sections 5.1 and 5.2 and that of the experiments discussed in chapter 3. As can be seen

all of the numerical and experimental results correlate closely.

It is noticeable that the experimental and numerical results start to tail off and diverge for

larger displacement values. However, it is important to notice that the aim of this project has

been to facilitate the FEA of the accessory drive belts within the operational loading range. As

discussed earlier in section 2.1 of chapter 2, the maximum strain seen by the belt during

operation is 10%. This amount of strain in the rubber rib uniaxial tensile, rubber rib uniaxial

compression and the belt groove-on-flat compression experiments would translate into 5.6mm,

0.15mm and 0.23 mm of displacement receptively. At 10% strain, the percentage error between

the numerical predictions and experimental results of the rubber rib uniaxial tensile, rubber rib

uniaxial compression and the belt groove-on-flat compression tests are 9%, 1% and 6%

respectively. These values are well accepted for engineering design and analysis purposes within

FEA capabilities.


65

Figure 46 The rubber rib uniaxial tensile test - experimental and numerical force-displacement

comparison

Figure 47 The rubber rib uniaxial compression test - experimental and numerical force-displacement

comparison


66

Figure 48 The belt groove-on-flat uniaxial compression test - experimental and numerical force-

displacement comparison

Figure 49 The belt groove-on-groove uniaxial compression test - experimental and numerical force-

displacement comparison


67

Figure 50 Belt bending – cord side up - experimental and numerical force-displacement comparison

The presented results conclude that the discussed mechanical tests, the assumptions made to

generate the stress-strain curves, and the method used to approximate the Mooney-Rivlin

constants for the rubber, although not conventional, work best for characterizing accessory drive

belts. In practice, any mode of deformation that the accessory drive belt is exposed to would be a

combination of tensile, compression and bending. Therefore, knowing the range of each mode,

the stress state of the belt under the combined loading can be determined by superimposing the

impact of individual FEA models.


68

5.5 Conclusion

In this chapter, series of FE simulations are devised, each replicating the previously conducted

mechanical tests. The extracted hyperelastic parameters, estimated from chapter 4 are inserted as

material properties of rubber for each of these simulations. These parameters are specific to the

existing mode of deformation in each test. The force-displacement results of the FE simulations

are compared to that of the experiments. The results correlate closely, for strains of under 10%

(which is the maximum strain level that the belt is exposed to during operation). This verifies

that the new methodology utilized for characterizing accessory drive belt components has been

reliable.

Chapter 6: Application in Belt Life Prediction

69

6 APPLICATION IN BELT LIFE PREDICTION

In Chapter 5, the extracted mechanical properties of the accessory drive belt constituents are

inserted in series of FE simulations. These simulations replicate each of the previously conducted

mechanical experiments. This chapter concludes that the force-displacement results of the

simulations correlated closely with that of the experiments; therefore, the soundness of the

demonstrated parameterization procedure is verified.

Chapter 6 describes the role of FEA on fatigue life prediction of the accessory drive belts.

This chapter presents the prevailing fatigue life prediction model in accessory drive belts, and

how it is related to the stress distribution in the belt.

6.1 Fatigue in Accessory Drive Belts

The most important application of the accessory drive belt FEA is in its fatigue life study.

Experimental research has shown that initiation of transverse cracks on the rubber rib tips is a

dominant cause of fatigue failure in accessory drive belts (Figure 51) [5]. Therefore, the stress

distribution and its magnitude at the tip, is of a special interest.


70

Figure 51 Initiation of cracks at the rubber rib tips [5]

Fatigue failure in belts occurs in two phases: First as a crack nucleation, and second as a crack

growth [43] [44]. These two phases occur due to cyclic stresses, tensile stresses and plastic

strains. Although fatigue life models for rubber materials are available, since the tensile cords in

the belt limit the strain magnitude of the rubber component to the linear region, metal like fatigue

methods are used for fatigue in rubber portion of the serpentine belt [5][45].

In general, fatigue can be divided into high cycle and low cycle. The high cycle is associated

with lower loads and longer lives, with the strains associated to the elastic range, while the low

cycle is associated with high loads and shorter lives, with plastic strains occurring during each

cycle. For the purpose of the belt life problem, the former assumption (elastic strain controlled

approach) is made. The Basquin relation can describe the stress-life relationship as follows [46]:

σa σf 2Nf( )b⋅ (8)


71

σa, σf, b, and 2Nf are stress amplitude, fatigue strength, fatigue strength coefficient, and

number of reversal to failure respectively. Although laboratory tests are based on fully reversed

loading, in reality, besides the stress amplitude, mean stresses developed in material are not

always zero and they have to be taken into account as follows [46]:

σa σf σm−( ) 2Nf( )b⋅ (9)

The fatigue failures however can be more complex and occur due to multi-axial state of stress.

In order to deal with such a situation, equivalent uniaxial mean and amplitude stresses must be

developed via strain energy or Sines methods [47]. Both of these approaches result in the

following equivalent alternating (σqa) and mean (σqm) stresses as follows [5]:

σqaσa1 σa2−( )2 σa2 σa3−( )2+ σa3 σa1−( )2+

2 (10)

σqm σmx σmy+ σmz+ (11)

In the above formulas, σai represents the principal alternating stresses in direction i, and σmi,

represent mean normal stresses in any three mutually perpendicular directions. Consequently, by

determining these equivalent mean and alternating stress values, the fatigue formals can be used

to derive the belt life.


72

6.2 Stress Distribution in Accessory Drive Belts

During operation, the accessory drive belt is exposed to series of loading scenarios. The belt is

constantly under pre-tension, the amount of which is pre-set for various systems and drive

conditions, by the belt manufacturer. This pre-tension generates tensile stresses in the belt.

However, in a belt drive system, the belt transfers the crankshaft power to the pulleys via torque,

which is created through the tangential force. This tangential force results in variation in belt

tension at points of contact to the pulley, between the tight side T1 (last point of contact) and the

slack side T2 (first point of contact) (Figure 52). Consequently, the torque transfer generates

longitudinal tensile stresses in the belt, varying from σs (tensile stress on the slack side) to σt

(tensile stress on the tight side) [5].

Figure 52 Stress variation due to tension

Depending on which side of the belt is wrapped around the pulley, the belt bending around the

pulley generates further tensile stresses in the belt in longitudinal direction. For instance, when

the backside is wrapped around the pulley, longitudinal tensile stresses are developed in the

rubber side of the belt σb, maximum of which occurs at the rib tip.


73

Furthermore, when the grooved side is travelling around the pulley, compression between the

belt rib and the pulley groove will cause the belt rib tip to bulge, since the rubber is

incompressible. Due to this bulging, transverse tensile stresses are developed at the rib tip σz.

Since the rib tip never contacts the pulley, and therefore has a free surface, no radial stresses are

developed at the rib tip [5] (Figure 53).

Figure 53 State of stress at the rubber rib tip of accessory drive belts

6.3 Determination of the Equivalent Uniaxial Stresses at the Belt Rib Tip

As discussed above, the overall stress mode at the rib tip can be summarized in the

longitudinal and transverse directions, and therefore the stress developed at the rib tip is a bi-

axial stress case. Therefore, this biaxial stress mode must be transformed into an equivalent

uniaxial stress mode to be used for fatigue life predictions.


74

Furthermore, each stress mode consists of mean and alternating portions. The static load on

the belt (due to pre-tension and wrapping around pulleys) causes the mean portion. The belt

tension fluctuations can be due to variety of reasons: torsional vibrations in the system,

acceleration and deceleration and variable RPM, and the moment of inertia of the pulleys in the

system (which imposes counter torque on the belt). These dynamic loadings impose alternating

stresses on the belt. The longitudinal mean and alternating tensile stresses due to torque transfer

between belt and pulley (σx.m, σx.a) can be determined as follows [5]:

σx.mσt σs+

2 σx.a

± σt σs−( )⋅

2 (12)

The longitudinal mean and alternating tensile stresses due to bending (σb.m, σb.a) can be

determined as follows:

σb.mσb2

σb.a±σb

2 (13)

Therefore, the total longitudinal mean and alternating stresses, when taking both the bending

and tensile modes of deformation into account can be calculated as follows:

σl.mσt σs+ σb+

2 σl.a

± σt σs− σb+( )⋅

2 (14)


75

Similarly, the mean and alternating stresses in transverse direction can be calculated as

follows:

σz.mσz2

σz.a± σz( )⋅

2 (15)

Lastly, the effect of longitudinal and transverse stresses can be combined as per Sines method,

to determine the total mean and alternating stresses (σm, σa) at the rib tip of the drive belt, at each

pulley:

mzmlm σσσ += (16)

2

2)(2)(2)( azalazala

σσσσσ

++−= (17)

6.4 Correlation between the Stress Distribution and Life in Accessory Drive Belts

By conducting a FEA of the belt under the range of the operational loading conditions, the

stress state of the tip can be determined, and all the discussed alternating and mean longitudinal

and transverse stresses can be estimated using (12)-(15). Thereafter, the total alternating and

mean stresses at the belt rib tip can be calculated using (16) & (17). These values can be used in

the fatigue life model of the belt and they can be applied to predict the belt life under various

driving conditions as follows [5]:


76

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

=

b

mf

ab

mf

ab

mf

a

RPMDr

BeltLengthL

fh

1

3

321

11

2

221

11

1

121

1)....60(

σσ

σ

σσ

σ

σσ

σ

π

(18)

Equation (18) approximates the belt life for a 3-pulley accessory drive system, with a constant

RPM, while neglecting any frictional drag from bearings. In this equation, hf is the time to

failure in hours, σai and σmi are total equivalent uniaxial alternating and mean stresses at pulley i,

and b is the fatigue strength component.

As can be noticed, determining the stress distribution at the tip of the belt rib via FEA plays a

major role in assessing the life of the accessory drive belts. Parameterizing the belt constituents

is a crucial and inevitable part of this FEA.

6.5 Conclusion

In this chapter, the role of stress distribution in fatigue life prediction of accessory drive belt is

investigated. It is experimentally known that the majority of catastrophic ruptures of the belt

initiate from the tip of the rubber ribs, which makes this spot of a special interest. The biaxial

state of stress at the tip can be converted to the equivalent mean and alternating uniaxial stresses.

The equivalent stress values can be used in the existing empirical belt life formulas to predict the

life of the belt. This concludes that conducting stress analysis on accessory drive belts is an

essential part of the belt life prediction.

Chapter 7: Summary & Conclusions

77

7 SUMMARY & CONCLUSIONS

7.1 Summary

The presented research work has elaborated on a new methodology to parameterize the

constituents of accessory drive belts for FEA applications. The importance of this research is its

applicability for accessory drive system manufacturers (which are usually not the belt

manufacturer themselves), as a design and optimization tool. More specifically, this technique is

devised to be relatively quick, cost efficient and accurate and enables the customer to extract the

required parameters form the finished product (the accessory drive belt), without any knowledge

of each belt constituent’s material.

Almost all accessory drive belts consist of cord and elastomeric ribbed regions to take the

tensile loads, and to create the required grip between the ribs and the pulley respectively. In order

to perform any analysis on these belts, the properties of each region must be fully determined, in

order to be inserted into its representative FE model. The cord region can be assumed as a linear

elastic component, due to its high stiffness and elastic response. On the other hand, the

elastomeric ribbed region is modeled as a hyperelastic material. This is mainly due to low, non-

linear stiffness for this portion of the belt. More specifically, Mooney-Rivlin was chosen as the

hyperelastic material model of the choice, since this model is computationally efficient, and yet

contains the acceptable level of accuracy for the range of strains that the belt is exposed to. An

accessory belt will see a maximum of 10% strain during its life-time in vehicles.


78

The belt tensile property is mostly attributed to its cords. Therefore, tensile testing of the belt

determines the elastic modulus of the cord region. However, the hyperelastic rubber ribs of the

belt require more parameters to describe its behavior. Conventionally, elastomeric materials are

cut to ASTM prescribed sample sizes, and are loaded in “simple strain” directions – mainly

uniaxial tensile, uniaxial compression, and pure shear.

After reviewing these conventional mechanical test procedures, it was concluded that these

tests were not feasible for accessory drive belts. This is partly due to complexity, and

unrepeatability of some of these methods. One of the major objectives of this research is to make

the parameterization methodology simple, cost efficient and relatively quick, which is not

feasible via the conventional testing techniques. Furthermore, the test samples are not available

in the required shape and sizes as specified by the ASTM standards. The only available source of

extracting the test samples is the belt itself, which is much thinner and narrower than required.

In the next step, series of customized tests were devised that impose the similar modes of

simple strain (as required by the conventional tests) on the rubber region of the belt. These tests

were namely uniaxial tensile, uniaxial compression and pure shear, and they were conducted

once on an extracted rubber rib, and once on the whole belt. The compression testing of the belt

was once performed by pressing the ribbed side of the belt against a flat rigid surface (groove-

on-flat test), and another time against a grooved rigid surface (groove-on-groove test), replicating

the belt and pulley interaction. Furthermore, three-point bending of the belt was performed to

determine the bending stiffness of the belt.


79

As a result, the force-displacement curves of the rubber rib and the whole belt were measured

under variety of loading scenarios, which the belt is mostly exposed to during operation. The

elastic modulus of the cord region was measured once directly using the belt tensile test stiffness

measurements, and once through the tensile stiffness values of the cord and rubber rib, via

composite material formulation. As both approximated elastic moduli correlated closely, it is

concluded that rubber ribs do not play a noticeable role in bearing any load during tensile testing

of the belt.

The force-displacement graphs of the rubber rib uniaxial tensile, rubber rib and belt groove-

on-flat uniaxial compression, and rubber rib and belt pure shear tests were converted to stress-

strain field. The groove-on-groove and belt bending force-displacement graphs were not used in

this step, since these experiments put the belt under multi-modes of deformation, and therefore

their representative stress-strain curves will not be of any use. These conversions were conducted

while taking series of assumptions. This is necessary since the non-uniform cross-sectional area

of the test specimens, makes this conversion unconventional. During the shear tests, it was

assumed that the cross sectional area of the rubber rib region does not change due to shear and it

solely distorts. Furthermore, since the rubber rib has a trapezoidal shape, when converting the

applied loads to stresses in the compression test, the average of both bases of each rib is

calculated for each compression length, in order to take the instantaneous cross section area into

account.

Consequently, the stress-strain behavior of the rubber under various modes of deformation

was measured using the rubber rib and the whole belt force-displacement data separately. Both


80

approaches predicted similar behavior for the uniaxial tensile, uniaxial compression and shear

modes. This verifies the methodology and assumptions used to perform the force-displacement

to stress-strain conversion. The proposed methodology of the devised conventional tests,

combined with the procedure for converting the test results to stress-strain field, satisfies the

objective of parameterizing the accessory drive belt components independent of its manufacturer.

The stress-strain relation of the rubber was used to determine the Mooney-Rivlin parameters,

through least square curve fitting technique. These parameters were used for creating series of

FE models, which replicated each of the previously conducted experiments (rubber rib tensile,

rubber rib compression, flat-on-groove and groove-on-groove compression of the whole belt, and

belt bending). In each simulation, similar ranges of displacements to the actual tests were

applied, and the resultant displacements were compared to that of the experiments. The force-

displacement results of the FEA simulations showed close correlation to the experimental

measurements. Although this was some-what expected from the simple strain models (tensile

and groove-on-flat compression simulations), the models with multi modes of strain (groove-on-

groove compression and bending) also compared closely to the experiments. This concludes that

the procedure applied to generate the Mooney-Rivlin constants and the method of combining

them for cases involving multi modes of strain is accurate for small strains.

7.2 Recommendation for Future Work

In the current project, the groove-on-groove compression and the three-point bending tests

have been considered as the control experiments of the choice, in order to validate the described

methodology under multi-axis modes of loading. These two experiments can be combined into a


81

single test that puts the belt under both loading scenarios; one that perhaps exposes the belt to

more realistic situation as the belt sees during operation. One possible experiment is to equip the

belt with strain gauges in longitudinal direction, while wrapping it around two pulleys. The

pulleys can be actuated with a very low, constant RPM, in order to replicate the desired quasi-

static loading condition. The strain level, read from the gauges can be directly compared to a

similar FE simulation. The effectiveness of the presented methodology can be determined much

clearer this way, when more practical scenarios are simulated.

All the discussed characterization tests in this research project were performed at room

temperature. However, temperature is a factor that plays an important role in stress distribution

and consequently the life of the accessory drive belts. Therefore, this research can be extended to

include temperature as a parameter during the experimental and numerical works. For this

purpose, all the discussed tests must be conducted for a range of temperatures that the belt is

exposed to during operation, and the hyperelastic parameters are extracted for each temperature.

The method of extracting the Mooney-Rivlin constants from the stress strain curves can be

further improved, especially for the cases which involve multi-modes of deformation (bending

and groove-on-groove compression). This can be done by varying the weight of each simple

strain mode involved in that deformation. In another words, the role of each mode of strain

(tensile, compression, shear) in the whole deformation can be controlled by assigning specific

weights, during curve fitting procedure. Perhaps a script can be written in C or MATLAB to

have more control over this process.


82

This research work can be extended to other types of accessory drive belts, namely the cross

ribbed accessory drive belts (Figure 54). These belts are designed to prevent the accumulation of

dust and debris in between the ribs. The stress distribution and mechanical properties of these

belts are completely different from that of the serpentine multi-ribbed belts. Therefore,

development of a characterization method and an FE model for these belts can be a valuable

future research topic.

Figure 54: Snapshot of the cross-ribbed accessory drive belt during testing [5]

The methodology proposed for parameterizing the constituents of the accessory drive belts

can have variety of applications involving composite products with elastomeric matrixes.

Therefore, the proposed method can be generalized for products for which the test specimens can


83

neither be provided by the manufacturer, nor be extracted from the product itself. Perhaps

certain revisions have to be done to customize the test methods for the specific application.

Timing belts, bushings and dampeners, and bio materials (such as synthetic skin), are few areas

in which this new methodology can be applied.

Lastly, for improving the accuracy of the FE models, application of higher order Mooney-

Rivlin, or other hyperelastic material models, such as Arruda Boyce and Ogden can be

investigated. Although these models will not have the computational efficiency of the 2-term

Mooney-Rivlin, they will be suitable for strain levels of above 10%.


84

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Chapter 8: Appendix

90

APPENDIX A: CALCULATION OF THE CORD REGION

ELASTIC MODULUS

The following is the detailed calculation of the cord region elastic modulus (Figure 55), using

MathCad. The modulus was once calculated directly from the whole belt tensile test, and once

indirectly approximated, using the tensile test results of the rubber rib (an extracted rubber strip

of the ribbed region) and the cord (tensile cord strand), via the composite material formulation.

For the latter method, it was assumed that the cord region behaves as a hybrid material.

Figure 55: Snapshot of the belt cross section

8.1 Direct Calculation Using the Belt Uniaxial Tensile Test Result

Chapter 8: Appendix

91

8.2 Calculation Using the Composite Material Formulation, via the Rubber rib

and Cord Tensile Test Results

Chapter 8: Appendix

92

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