Evaluation of Healing and Constitutive Modeling of …...EVALUATION OF HEALING AND CONSTITUTIVE MODELING OF ASPHALT CONCRETE BY MEANS OF THE THEORY OF NONLINEAR VISCOELASTICITY AND

EVALUATION OF HEALING AND CONSTITUTIVE MODELING OF

ASPHALT CONCRETE BY MEANS OF THE THEORY OF

NONLINEAR VISCOELASTICITY AND DAMAGE MECHANICS

by

YOUNGSOO R. KIM

and

DALLAS N. LITTLE

Final Report

National Science Foundation

Grant No. ECE-8511852

September 1988

ABSTRACT

It has been proved by many researchers that existing fatigue

failure criteria based on constant amplitude loading tests

underpredict the fatigue life of asphalt concrete pavements.

Unrealistic loading conditions in laboratory testing are the major

sources of this discrepancy. Two major differences between

laboratory and field loading conditions were addressed in this study:

the existence of rest periods and the random sequence of load

applications of varying magnitudes.

Based on an extensive literature review, three mechanisms were

identified as influencing the behavior of asphalt concrete subjected

to multi-level repetitive loads interrupted by various durations of

rest periods. They are: fatigue as a damage accumulation process,

relaxation due to the viscoelastic nature of asphalt concrete, and

chemical healing across crack faces during rest periods. Visual

evidence of healing was achieved in this research by means of a

Scanning Electron Microscope analysis of fracture faces from Izod

impact tests on samples of various asphalt grades and sources.

The effort to evaluate the mechanism of chemical healing in the

microcrack process zone is confounded by the concomitant occurrence

of viscoelastic relaxation. Schapery's correspondence principle of

nonlinear viscoelastic media was successfully used to separate

viscoelastic relaxation from chemical healing. Application of the

procedure of separating out the viscoelastic relaxation yields a

method by which to quantify chemical healing in a damaged asphalt

concrete body.

iii

Chemical healing as a function of the duration of rest periods

is quantified using a healing index based on pseudo energy density.

This healing index is presented for three asphalts of varied

composition.

As a result of the techniques applied to separate the

relaxation and healing mechanisms, a uniaxial constitutive model was

developed by employing the correspondence principle in concert with

damage mechanics. The verification of this equation was successfully

accomplished under realistic loading conditions, such as multi-level

loading with various lengths of rest periods.

iv

TABLE OF CONTENTS

CHAPTER

I INTRODUCTION

II TERMINOLOGY

I I I LITERATURE REVIEW'

1. Effects of Rest Periods 2. Healing Mechanism ... 3. Fatigue Characterization

IV MATERIALS AND TESTING PROGRAMS

v

VI

1.

2.

Materials . . . . . . . . SEM Study . . . . ... Three Point Bend and Uniaxial Testing

Sample Fabrication and Testing Methods Izod Impact Testing . . . Three Point Bend Testing Uniaxial Testing

MICROSCOPIC EVALUATION OF HEALING

1. Results ..

2. Discussion

THEORY OF VISCOELASTICITY AND DAMAGE MECHANICS

1. Theory of Linear Viscoelasticity

2. Theory of Nonlinear Viscoelasticity Correspondence Principle I Correspondence Principle II . Correspondence Principle III.

3. Constitutive Modeling of Asphalt Concrete Damage of Asphalt Concrete Damage Parameter

VII THREE POINT BEND TESTING

1. Derivation of Pseudo Displacement for Constant Strain Rate Testing

2. Results and Discussion

Page

1

6

8

8 10 15

20

21 21 22

23 23 28 30

34

34

40

47

48

so 51 51 52

52 57 60

64

64

67

v

TABLE OF CONTENTS (Continued)

CHAPTER

VIII UNIAXIAL TESTING - EVALUATION OF HEALING

1. Method of Analysis

2. Results and Discussion Relaxation Testing Constant Strain Rate Simple Loading Tests with Rest Periods . . . . .

IX UNIAXIAL TESTING - CONSTI~IVE MODELING

1. Study of Rate-Dependence

2. Determination of Damage Parameter

3. Constant Strain Rate Monotonic Loading Tests

4. Constant Strain Rate Simple Loading Tests . .

5. Verification of Constitutive Equation (IX.ll)

X CONCLUSIONS AND RECOMMENDATIONS

1. Conclusions . .

2. Recommendations

REFERENCES

APPENDIX A - DEVELOPMENT OF PSEUDO QUANTITIES

APPENDIX B - GENERALIZED J- INTEGRAL THEORY

APPENDIX C - VERIFICATION TEST RESULTS (CONSTANT STRAIN RATE SIMPLE LOADING TEST YITH VARIOUS LENGTHS OF REST PERIODS) . .

Page

78

78

81 81

85

96

96

97

102

105

110

142

142

143

146

155

160

168

vi

TABLE

1

2

3

USTOFTIW~

Corbett analyses on the three asphalt cements used in testing . . . . . . . . . . . . . . .

Penetration information of different binders . . . .

Governing equations for linear elastic and linear viscoelastic materials . ; . . . . . . . . . .

vii

Page

24

43

49

FIGURE

1

2

3

4

5

6

7

8

9

10

11

12

LIST OF FIGURES

Schematic illustration of various loading conditions.

Gradation plot of granite fines and the AASHTO specification of fine aggregate for bituminous paving mixtures. . ........... .

Izod impact test and sample configuration used in preparation of fracture surfaces for SEM analysis.

Configuration of three point bend testing sample with a chevron notch. . . . . . .

Picture and schematic presentation of uniaxial testing apparatus. . . . . . .....

(a) Microscopic video camera with testing apparatus. (b) Image of cracking area pictured from TV monitor.

Fracture surfaces of Izod samples with different asphalts: (a) AC-5, (b) AC-20, and (c) SBR latex-modified AC-5. . ....... .

Fracture surfaces of Izod samples with AC-20 and SBR latex-modified AC-5 at higher magnifications: (a) 150x magnification of AC-20, (b) 450x magnifi cation of AC-20, (c) 150x magnification of latexmodified AC-5, and (d) 450x magnification of latex-modified AC-5. . . . . . . . . . . ...

Fracture surfaces of Izod samples with AC-5: (a) control fracture surface and fracture surfaces after healing periods of (b) 5 minutes, (c) 10 minutes, and (d) 20 minutes. . ...... .

Fracture surfaces of Izod samples with AC-20: (a) control fracture surface and fracture surfaces after healing periods of (b) 20 minutes, (c) 40 minutes, and (d) 60 minutes. . ........ .

Comparison between (a) control fracture surface and (b) fracture surface after healing period of 20 minutes from Izod samples with AC-20. . ...

Fracture surfaces of Izod samples with SBR latexmodified AC-5: (a) control fracture surface and fracture surfaces after healing periods of (b) 10 minutes, and (c) and (d) 20 minutes. . .....

viii

Page

7

25

26

29

32

33

35

36

38

39

41

42

FIGURE

13

14

15

16

17

LIST OF FIGURES (Continued)

Illustration of rate-dependency in asphalt concrete (stress-strain curves of constant strain rate monotonic loading tests). . ...... .

The effect of the maximum strain during the past strain history on (a) stress-strain behavior and (b) stress-pseudo strain relationship after the application of correspondence principle.

Illustration of damage accumulation under the large strain amplitude: (a) stress-strain behavior and (b) stress-pseudo strain relationship after the application of correspondence principle.

Load versus displacement curves of uncracked samples without prefatigue.

Isochronal curves constructed from Figure 16.

18 Shift factor versus time of uncracked samples without prefatigue.

19

20

21

22

23

24

Load versus pseudo displacement curves of uncracked samples without prefatigue.

Load versus displacement curves of uncracked samples with prefatigue.

Load versus pseudo displacement curves of uncracked samples with prefatigue.

Load versus displacement curves of cracked samples with prefatigue.

Load versus pseudo displacement curves of cracked samples with prefatigue.

Strain history for tests "b" and "c".

Page

54

55

58

68

69

70

72

73

74

76

77

80

25 Relaxation data for the mixtures with Witco AR-4000. 82

26 Relaxation data for the mixtures with Fina AC-20. 83

27 Relaxation data for the mixtures with Shamrock AC-20. 84

28 Stress versus pseudo strain of initial 10 cycles with negligible damage (Shamrock AC-20). 87

ix


FIGURE

29 Stress versus pseudo strain before and after 40-minute rest period with negligible damage (Shamrock

Page

AC- 20). . . . . . . . . . . . . . . . . 88

30 Stress versus pseudo strain of initial 20 cycles with strain amplitude of 0.0092 in./in. (Witco AR-4000).. 89

31 Stress versus pseudo strain before and after 40-minute rest period with strain amplitude of 0.0092 in./in. (Witco AR-4000). . . . . . . 90

32 Illustration of pseudo energy densities before and after rest period. . . . . . . . . . . . . 93

33 Healing potential of different binders as a function of the duration of rest period. . . . . . 95

34 Strain history for the study of rate-dependency. 97

35 Stress versus pseudo strain for the first cycles at different strain rates shown in Figure 34. . . . 99

36 Stress versus pseudo strain for different rates (constant strain rate monotonic loading). 100

37 Damage parameter versus time for monotonic loading.. 104

38 Damage parameter versus time for constant strain rate simple loading (20 cycles). . . . . . . 105

39 Back-calculated F versus fR/e~ for constant strain rate simple loading. 108

40 Damage coefficient versus damage parameter (after Schapery ( 44)) . . . . . . . . . . 109

41 Back-calculated damage coefficient (G) versus damage parameter for constant strain rate simple loading. 110

42 Stress-strain curves for constant strain rate monotonic loading. . . . . . . . 112

43 Stress-strain curves for a constant strain rate simple loading test (strain amplitude = 0.00184 in./in.). . . . . . . . . 113

44 Stress-strain curves for a constant strain rate simple loading test (strain amplitude =

0.00369 in./in.). . . . . . . . . . . . . 114

X


FIGURE Page

45 Strain history of a multi-level loading verification test with 30-second rest periods. 116

46 Strain history of a multi-level loading verification test with random durations of rest periods. 117

47 Stress-strain curves of initial 20 cycles for the constant strain rate simple loading verification test shown in Figure 24 (strain amplitude 0.00276 in./in.). 118

48 Stress-strain curves after the 1st 5-minute rest period of the constant strain rate simple loading verification test. 119

49 Stress-strain curves after the 3rd 40-minute rest period of the constant strain rate simple loading verification test. 120

50 Stress-strain curves of Group 1 loading of the multilevel loading verification test shown in Figure 45.. 122








58 Stress-strain curves of Group 9 loading of the multilevel loading verification test shown in Figure 45 .. 130


xi


FIGURE Page










xii

CHAPTER I

INTRODUCTION

Failure criteria associated with the fracture and fatigue of

asphalt concrete layers have been developed based on mathematical

models or phenomenological relationships. Perhaps the most commonly

used fatigue failure criterion was presented by Epps and Monismith

(1) in the form:

where

1 1 or a

Nf - the total number of constant amplitude load

repetitions,

K1 to K4 regression constants,

E = the initial value of the bending strain induced per

load application, and

a = the repeated stress level per load application.

This phenomenological relationship based on constant amplitude

loading, which results in fatigue failure, has been used in a variety

of layered elastic pavement design and/or analysis schemes.

A number of researchers have shown that this classic fatigue

failure relationship grossly underpredicts field fatigue life by as

much as 100 times. Finn, et al. (2) actually demonstrated that the

laboratory-derived phenomenological fatigue relationships for the

The format of this dissertation follows the style of the

Transportation Research Board's Transportation Research Record.

l

asphalt concrete used at the AASHTO Road Test required a shift of 13

to match actual fatigue cracking data derived from AASHTO field

sections. This difference between laboratory and field fatigue

curves may be attributed to loading differences between the

laboratory and the field.

Continuous cycles of loadings at a constant strain or stress

amplitude, generally applied in laboratory tests, do not

realistically simulate the compound-loading conditions to which a

paving material is subjected under actual traffic conditions. Major

differences between the laboratory and the field loading conditions

are due to:

a. rest periods which occur in the field but not (normally) in

the laboratory,

b. the sequence of the load applications of varying magnitude,

and

c. reactions or frictional forces encountered in the field

between the asphalt concrete surface and the base layer.

When an asphalt concrete pavement is subjected to repetitive

applications of multi-level vehicular loads with various durations of

rest periods, three major mechanisms take place: fatigue, which can

be regarded as damage accumulation during loading; relaxation of

stresses in the system due to the viscoelastic nature of asphalt

concrete; and chemical healing across microcrack and macrocrack faces

during rest periods. The fatigue damage mechanism degrades pavement

performance, while relaxation and healing mechanisms enhance the

fatigue life of asphalt concrete pavement. A realistic fatigue model

should be able to account for these mechanisms.

2

The difficulty of evaluating these mechanisms arises from the

fact that they occur simultaneously in an asphalt concrete pavement.

For example, the degree of fatigue damage sustained under loading

depends on how well the material relaxes, but healing as well as

relaxation take place simultaneously in a damaged pavement.

The objectives of this research were to: (a) verify, through

literature review and experimentation, that healing does indeed occur

as a result of rest periods introduced in the cyclic fatigue testing

of asphalt concrete; (b) identify the magnitude of this healing

phenomenon; and (c) identify the mechanism(s) through literature

review and experimentation, by which microcrack healing occurs.

Two reports have resulted from this research. This report deals

with identification of the magnitude of healing which occurs. A

companion report addresses the mechanisms which support this healing

phenomenon.

In order to quantitatively evaluate healing, it was neccessary

to develop a procedure to separate the hereditary viscoelastic

effects from the healing effects. The correspondence principle of

the theory of nonlinear viscoelasticity developed by Schapery (3) was

applied to accomplish this. The information from the mechanical

evaluations of healing discussed in this report provided the support

data for the study of the mechanisms influencing chemical healing

(4).

As a result of the techniques applied to differentiate

relaxation and healing, a uniaxial constitutive relationship was

developed by employing the correspondence principle in concert with

damage mechanics. This constitutive equation successfully predicts

3

the behavior of asphalt concrete under realistic loading conditions

(i.e. multi-level repetitive loading with various lengths of rest

periods).

Microscopic studies were performed as a part of this research to

verify the healing phenomenon of asphalt concrete. The Scanning

Electron Microscope w~s utilized, and effects of the duration of rest

periods and type and grade of asphalt cement binder were studied.

Following this introductor~ chapter, a chapter entitled

"Terminology" describes the various types of loading used in the

testing phases of this research. Literature review and the

description of materials and testing plans are presented in Chapters

III and IV, respectively. The microscopic verification of healing is

presented in Chapter V. Chapter VI establishes the background

theories which are used to separate the relaxation and healing and to

model a constitutive relationship. The applicability of these

theories to asphalt concrete is proved by means of three point bend

testing in Chapter VII and uniaxial tensile testing described in

Chapter VIII. Based on the methodology discussed in Chapters VII and

VIII, uniaxial repetitive loading tests with rest periods were

performed on notched samples to evaluate the healing potentials of

different asphalts. The procedures used and results are presented in

Chapter VIII. In Chapter IX, a uniaxial constitutive equation is

developed based on the theories presented in Chapter VI. The

experimental approach to obtain coefficients and exponents for this

equation is presented in Chapter IX. Also, the constitutive model is

verified in Chapter IX under various loading conditions. Finally,

conclusions and recommendations for future research are presented in

4

5

Chapter X.

CHAPTKR II

TERMINOLOGY

In this section terminology is defined to aid the reader's

understanding and to avoid lengthy descriptions within the text.

Five types of loading will be discussed in this text: constant

strain-rate monotonic loading, simple loading, constant-strain-rate

simple loading, pulsed loading and multi-level loading. Schematic

illustration of these loading types is presented in Figure 1.

Constant-strain-rate monotonic loading is continuous loading

during which strain is increasing throughout testing at a constant

rate. Simple loading is defined as continuous, repetitive loading of

a single wave form at a constant amplitude of strain. When simple

loading is composed of a "saw-tooth" strain wave (i.e. constant

strain-rate), with symmetric loading and unloading segments, it is

called constant-strain-rate simple loading. Pulsed loading is the

same as simple loading except that a rest period is introduced after

each loading application. Multi-level loading is repetitive loading

with various levels of strain amplitude. Multi-level loading can be

continuous (i.e. no rest period) or discontinuous (with rest

periods). When different lengths of rest periods are introduced

randomly among the applications of multi-level loading, this

represents a loading condition which is most similar to actual

conditions.

6

Time

(a) Constant strain rate monotonic loading.

(c) Pulsed loading.

(d) Multi-level loading.

Time

(b) Constant. strain rate simple loading.

Time

FIGURE 1 Schematic illustration of various loading conditions.

7

CHAPTER III

LITERATURE REVIEW

1. Effects of Rest Periods

The significance of rest periods between load applications has

been recognized by several researchers. Monismith et al. (5) varied

rest time from 1.9 seconds to 19 seconds on beam samples tested by a

repeated-flexure apparatus. No significant change in fatigue

performance was observed. This result may be partially explained by

the specific testing configurations, such as the deflection measuring

point and the elastic response from the spring base. Deacon and

Monismith (6) used pulsed loading instead of simple loading to

simulate the recovery of asphalt concrete pavement due to the

viscoelastic nature of the material. Raithby and Sterling (7)

performed uniaxial tensile cyclic tests on beam samples sawed from a

rolled carpet of asphalt concrete. Pulsed loading with rest periods

of up to 3 times longer than the loading cycle was applied until

failure occurred. It was observed that the strain recovery during

the rest periods resulted in longer fatigue life by a factor of five

or more than the life under simple loading. Francken (8) developed a

new expression for the cumulative cycle damage ratio in Miner's law

by accounting for effects of rest periods.

McElvaney and Pell (9) performed rotating bending fatigue tests

on a typical English base course mix and concluded that rest periods

have a beneficial effect on the fatigue life depending on the damage

accumulated during loading ~eriods. Other researchers (10-13) have

8

also reported beneficial effects of rest periods on the fatigue

performance of different asphalt concrete mixes. The testing mode,

frequency, temperature, duration of rest periods, and resulting

beneficial effects of these factors are well-summarized by Bonnaure

et al. (14). Bonnaure et al. (14) investigated the effects of rest

periods on a typical Dutch asphalt concrete by means of a three point

bending apparatus. They concluded that higher test temperatures and

softer binders result in a more beneficial effect from rest periods.

At Texas A&M University, efforts (15,16) have been made recently

to evaluate the increase in work done after rest periods from

displacement-controlled cyclic testing. Balbissi (15) studied the

effects of rest periods on the fatigue life of plastiGized sulfur

binders used in asphalt-like mixtures. A mathematical expression for

the shift between laboratory and field fatigue lives was developed.

A slightly modified version of Balbissi's shift factor is currently

used in the Florida DOT flexible pavement performance model (17), and

is as follows:

where

SF= [.--1--

1-po t-m

SF the shift factor,

P0 the percent of stress under maximum load which

remains as residual stress (-0.2 < P0 < 0.2),

K2 the fatigue constant,

m the slope of the log of creep compliance versus log·

time of loading curve,

t 1 the rest period between maximum loads, and

nr the number of rest periods between maximum loads in

the traffic stream.

9

Little, et al. (16) reported an increase in work done after rest

periods in controlled-displacement crack growth testing in asphalt

concrete mixes modified with various additives. They evaluated the

effectiveness of additives on fatigue performance which was

influenced not only by crack growth rate but also by healing

potential.

2. Healing Mechanism

Even though a considerable volume of work exists discussing the

effects of rest periods on asphalt concrete pavements, only one paper

was found which treated the chemical healing potential of asphalt

concrete. Bazin and Saunier (18) introduced rest periods to asphalt

concrete beam samples which were previously failed under uniaxial

tensile testing. Then the same testing was performed with a rest

period, and the healing ratio, ratio of tensile strength after the

rest period to that before the rest period, was plotted against the

duration of the rest period. It was reported that an ordinary dense

mix could recover 90 percent of its initial resistance with only 3

days of rest at 77°F, and that the healing seemed to become complete

after one month at that temperature. The same procedure with cyclic

fatigue testing was performed before and after rest periods. The

life ratio, the ratio of the number of cycles to failure after the

rest period to that before the rest period, was evaluated. The ratio

was over 50 percent after a 1 day rest period with 0.213 psi pressure

pressing the cracked faces together. This research showed clear

evidence of healing in asphalt concrete, but the durations of rest

periods were too long (1 to 100 days) to realistically mimic field

10

loading conditions. Also it has been concluded that the pressure

applied at contact faces has a great influence on healing.

Therefore, in order to realistically evaluate the effect of healing

in asphalt concrete pavement which usually contains many microcracks,

one needs to consider healing of partially cracked samples rather

than that of fully ruptured samples.

Whereas only limited research in the area of asphalt concrete

healing has been reported, the mechanism of the healing within

polymeric materials has been intensely studied. The healing

mechanism of polymers is well described by Prager and Tirrel (19) as

follows:

"When two pieces of the same amorphous polymeric material

are brought into contact at a temperature above the glass

transition, the junction surface gradually develops

increasing mechanical strength until, at long enough

contact times, the full fracture strength of the virgin

material is reached. At this point the junction surface

has in all respects become indistinguishable from any other

surface that might be located within the bulk material:

we say the junction has healed."

Jud, et al. (20) identified three different concepts for the

time-dependent build-up of joint-strength between two polymer

surfaces: (a) polymer-polymer interdiffusion (21-23); (b) adhesion

between rough surfaces (24-26); and (c) jointing by flow of molten

material (27,28).

In the diffusion model, Wool and O'Connor (22) identified the

following stages of healing which influence mechanical and

11

spectroscopic measurements: (a) surface rearrangement, (b) surface

approach, (c) wetting, (d) diffusion, and (e) randomization. Kim and

Wool (23) introduced the concept of minor chains and described the

diffusion model as follows:

"By the end of the wetting stage, potential barriers

associated with the inhomogeneities at the interface

disappear, and the stages of diffusion and randomization

are the most important ones because chains are free to move

across the interface and the characteristic strength of a

polymer material appears in these stages."

The reptation model proposed by de Gennes (29) explains these

microscopic sequences very well. The term "reptation" was defined

(30) as a chain travelling in a snake-like fashion, due to thermal

fluctuation, through a tube-like region created by the presence of

neighboring chains in a three-dimensional network. De Gennes (29)

explained that the wriggling motions occur rapidly, that their

magnitudes are small, and that in a time scale greater than that of

the wriggling motions, a chain, on average, moves coherently back and

forth along the center line of the tube with a certain diffusion

constant, keeping its arc length constant.

Macromechanically, the most common technique to describe the

healing properties of polymers is to measure fracture mechanics

parameters of a healed specimen, such as energy release rate, G1 ;

stress intensity factor, K1 ; fracture stress, af; and fracture

strain, fr· These parameters are dependent on the duration of the

healing period, temperature, molecular weight, and pressure applied

during the healing period. Four models based on the reptation model

12

have been proposed to theoretically describe the fracture mechanics

parameters in terms of these variables: (a) Prager and Tirrell's

model, (b) de Gennes' model, (c) Jud, Kausch, and Williams' model,

and (d) Kim and Wool's model.

Kim and Wool (23) introduced the concept of minor chains and

assumed that the chain interpenetration distance is the controlling

factor. Minor chains can be defined as the portion of a chain that

escapes from the tube-like region. Their model predicted that

G - to.s M-o.s IC

where G1 c the critical energy release rate in an opening mode,

t the duration of healing period, and

M molecular weight.

They also proposed the following experimental relationship:

where afh the fracture healing strength, and

a00 the original strength.

While the square-root-time-dependence of the energy release rate has

been agreed upon by other models and proved experimentally (21, 31),

there is a disagreement on the value of the exponent of the molecular

weight.

Temperature dependence of healing mechanisms has been reported

by many researchers (20, 21, 26, 32). An increase in the healing

temperature shifts the recovery response to shorter times. Wool (26)

has constructed master healing curves by time-temperature

superposition.

In the adhesion model (24-26), surface irregularities are

reduced by local flow of polymer material under the action of

13

adhesive forces. This model suggests that facial healing occurs by

restoration of secondary bonding between chains or microstructural

components and that van der Waals forces or London dispersion forces

play a very important role in healing (26). Briscoe (25) concluded

that surface forces, such as van der Waals forces, electrostatic

forces, and hydrogen bonds are responsible for adhesion. He also

pointed out that the interaction of adhesive forces and the bulk

viscoelastic properties of the "hinterland" adjacent to the interface

are the most important factors in the adhesion of elastomers.

The flow model (27, 28) suggests that the orientation and

interpenetration of the flowing material influences the strength of

the joint. Bucknall, et al. (28) experimentally found that these

factors are dependent on healing temperature, contact period, and

extent of melt displacement.

In order to understand the healing mechanism of asphalt

concrete, the chemistry of asphalt cement must be studied with the

healing models of polymers in mind. Petersen (33) claims that the

association fore~ (secondary bond) is the main factor controlling the

physical properties of asphalt. That is, the higher the polarity,

the stronger the association force, and the more viscous is the

fraction, even if molecular weights are relatively low. He also

illuminated the effect of degree of peptization on the flow

properties of asphalt as follows:

"Consider what happens when a highly polar asphaltene fraction

having a strong tendency to self-associate is added to a

petrolene fraction having a relatively poor solvent power for

the asphaltenes. Intermolecular agglomeration will result,

14

producing large, interacting, viscosity-building networks.

Conversely, when an asphaltene fraction is added to a petrolene

fraction having relatively high solvent power for the

asphaltenes, molecular agglomerates are broken up or dispersed

to form smaller associated species with less interassociation;

thus, the viscosity-building effect of the asphaltenes is

reduced."

Traxler (34) also suggested that the degree of dispersion of the

asphalt components is inversely related to the complex (non

Newtonian) flow properties of asphalt.

Ensley et al. (35) and Thompson (36) ascribe to the view that

asphalt cement consists of aggregations of micelles. These micelles

consist of two or more molecules of asphaltenes and associated (if

present) peptizing lower molecular weight materials. These peptizing

materials grade upward in size (from outside to inside the micelle)

from napthenes and paraffins to resins and polar compounds coating

the asphaltenes (36). The interactions of these micelles among

themselves and with aggregates largely determine cohesion and bond

strengths, respec.tively.

3. Fatigue Characterization

Since the AASHO Road Test results were reported in 1962 (37), it

has been generally accepted that fatigue is a process of cumulative

damage and one of the major causes of cracking in asphalt concrete

pavements. In order to model the fatigue life of asphalt concrete

pavements, different configurations of repetitive testing have been

performed (6, 18, 38). These tests proved that the number of cycles

15

16

to failure (Nr) can be predicted from a simple power form of initial

bending strain or stress. Epps and Monismith (1) summarized studies

which had shown that this power form is valid for different mixes

under continuous, constant amplitude loading. Other fatigue failure

criteria, such as a modified power form of Nf versus bending strain

(8) and the failure criterion based on total dissipated energy during

the fatigue test (38) have been reported as being successful in

predicting the fatigue life of asphalt concrete samples.

Meanwhile, it has been found (2) from the comparison of field

data with laboratory results (from continuous flexural fatigue

testing at a constant stress amplitude) that laboratory data

underpredict the fatigue life of asphalt concrete pavements. It has

been reasoned that the discrepancy comes from the complexity of

loading magnitude and sequence and rest periods between load

applications (5-8, 39).

In order to account for the effect of multi-level loading with

random sequence, Miner's law or the modified form of Miner's law has

been successfully used. Miner (40), in 1945, suggested a linear

summation of cycle ratios hypothesis (cumulative damage hypothesis)

from his research on the fatigue of aircraft metals. This

hypothesis, so-called Miner's law, states that fatigue failure will

occur when

n h ni

= 1 i=l Ni

where n 1 number of applications of stress level i, and

number of applications of stress level i required to

cause failure under simple loading.

This hypothesis was applied to asphalt concrete mixes, and some

modified versions (6, 8, 39) were reported. Francken (8) used a

modified power law of Nf versus bending strain and a generalized

Miner's law which accounts for the effect of rest periods and showed

that his cumulative cycle ratio at failure was much closer to 1 than

others reported in the literature, References (6) and (39).

Whether or not the power fatigue law and Miner's rule, modified

or unmodified, have contributed significantly to the fatigue study of

asphalt concrete, they are empirical. Furthermore, flexural fatigue

testing is time-consuming and usually results in large data

variation.

In 1973, Fitzgerald and Vakili (41) developed a nonlinear

stress-strain relationship of sand-asphalt concrete by means of the

maximum strain in the loading history and a weighted average of the

strain history. Their model was verified successfully for different

histories of strain input. It was also concluded that linear

viscoelasticity did not seem to be an applicable theory for

characterizing materials with asphalt binder under repeated loads.

Another rational constitutive model was developed by Perl, et

al. (42), which predicted the uniaxial stress-strain behavior of

asphalt concrete under realistic repetitive loading. The total

strain was separated into four components; elastic, plastic,

viscoelastic, and viscoplastic. The explicit dependence of the

strain components on stress level, time, and number of load

repetitions was evaluated. The final form of each strain component

was somewhat complex, but the results showed satisfactory agreement

between the measured and predicted values.

17

The key to effective constitutive modeling is the ability to

characterize and predict inelastic response of a given material.

Response of many materials to mechanical and environmental

disturbances is significantly influenced by widespread local

structural changes such as initiation and growth of cracks in the

opening and shearing modes, holes, crazes, and shear bands (43).

Schapery (44) used the term "damage" for these-changes and explain

them as follows:

"The changes are affected as much by constituent properties as

by mechanical and possibly chemical interactions among

constituents; for example, particles and fibers may initiate

matrix cracks through stress concentrations and also serve as

barriers to subsequent crack growth. These changes in the

microstructure are not necessarily deleterious to the

composite's behavior as they often increase the overall

toughness or resistance to global fracture. Quantities in the

global constitutive equations which reflect these changes are

called damage parameters."

The need for accurate prediction of damage in the context of

continuum mechanics is well recognized, and there have been

remarkable advances in this area based on empirical or theoretical

concepts (43-57). Some damage models have been developed for civil

engineering materials, such as clay (51), soft marine sediment (52),

concrete (58-60), rock (61), and polymers (62).

In order to model the damage process for a given material, one

needs to understand the major microstructural damage mechanism. From

the microscopic study of asphalt concrete under the repeated loading

18

of wheel tracking tests, Van Dijk (38) concluded that the fatigue

process could be classified into three stages associated with the

development of hairline cracks, real cracks, and failure of the mix.

Hence, the microcrack growth is considered to be a major fatigue

mechanism of asphalt concrete under repetitive loading.

Schapery (44) developed a one-dimensional constitutive equation

of particle-reinforced rubber by means of damage parameters based on

the law of microcrack growth. The basic form of his theory has

proven to work successfully for fiber-reinforced plastics (43),

metals (50), and soils (52). Considering that particle-reinforced

rubber is a highly-filled and very nonlinear, viscoelastic material,

Schapery's damage parameter is regarded as an appropriate.tool by

which to model the cumulative damage process occurring in asphalt

concrete. The detailed theoretical concepts behind this parameter

are reviewed and discussed in Chapter VI.

This extensive literature review suggests that constitutive

modeling with an appropriate failure criterion can provide better

and more meaningful mechanistic fatigue characterization.

19

CHAPTER IV

MATERIALS AND TESTING PROGRAMS

Three types of testing were performed in this research, each

with a specific purpose. They are:

a. Izod impact testing,

b. three point bend testing, and

c. uniaxial tensile testing.

Izod impact loads were applied to Sharpy specimens to provide

fracture faces for visual evaluation. These faces were studied

before and following the introduction of rest periods using a

Scanning Electron Microscope (SEM). The purpose of these experiments

was to determine whether or not visual evidence of healing exists and

to aid understanding of the healing mechanism of asphalt concrete.

Three point bend testing was used to verify the correspondence

principle of the theory of nonlinear viscoelasticity .. Beam samples

were tested at various rates in a vertical displacement-controlled

mode. Isochronal curves were constructed from load-displacement

curves, and the exponent of the power law between relaxation modulus

and time was predicted from these curves. Then uniaxial tensile

relaxation testing was performed to measure the exponent of

relaxation modulus versus time relationship. The measured exponent

from uniaxial relaxation testing and the predicted exponent from bend

testing were compared for purposes of verification of the

application of the theory of nonlinear viscoelasticity to asphalt

concrete. Also, a limited amount of verification work on notched

samples was performed using this technique.

20

All data for the evaluation of healing and construction of a

constitutive equation were generated using uniaxial tensile testing.

Verification of the nonlinear viscoelastic correspondence principle

occurred prior to healing tests using simple loading with various

lengths of rest periods introduced. The strain amplitude used in

this verification stage was small enough so as not to induce damage

growth in the sample. After verification, the healing potentials of

three different asphalts were measured through simple loading tests

with rest periods. In these tests, notched beam samples were loaded

up to the strain amplitudes which can produce macrocrack growth. In

constructing the constitutive law, two types of uniaxial tests were

performed: constant-strain-rate monotonic loading tests at various

strain rates and simple loading tests with different levels of strain

amplitude. These tests provided sufficient information to construct

a constitutive model based on the nonlinear viscoelastic

correspondence principle and damage mechanics.

Load, displacement, and Krak gage data were acquired through a

Hewlett-Packard data acquisition unit 3497A and stored in a

microcomputer. Data reduction and plotting programs were used to

quickly generate plots for visual data analysis. This computerized

procedure made the time-consuming calculations possible and

eliminated the potential for algebraic mistakes.

1. Materials

SEM Study

21

Three types of binder.s were studied: AC-5, AC-20, and styrene

butadiene latex rubber modified AC-5. The AC-5 and AC-20 grades were

from the Texaco refinery at Port Neches, Texas, which produced a

blend of crude oils from East Texas, Mexico, South America and

Wyoming. The SBR latex was obtained from Textile Rubber and

Chemical Company and is identified by the trade name of Ultrapave

70. It is an anionic emulsion and contains approximately 70 percent

solids; the supplier has not provided any other information on the

composition.

In the production of SBR modified AC-5, 5 percent by weight of

SBR solids were blended for 5 minutes with AC-5 and Ottawa sand at

275°F. The SBR appeared to be only partially soluble in the Texaco

AC-5 asphalt. An aggregate was mixed to provide a "carrier" for the

binder in thin film. Ottawa sand was selected as it is a uniformly

graded, clean aggregate which minimizes irregularities in the SEM

evaluation.

Three Point Bend and Uniaxial Testing

The sources and grades of asphalt cement used in three point

bend and uniaxial testing were as follows: Witco AR-4000, Fina AC-20,

and Diamond Shamrock AC-20. Viscosities for all asphalts at 140°F

were approximately 2000 poises. The Witco asphalt was from a

California refinery which processes crude oil from the San Joaquin

Valley. Fina asphalt was from the refinery in Big Spring, Texas,

which uses 100 percent domestic Permian basin crude. The source of

the Shamrock asphalt was the refinery in Sunray, Texas, which uses

22

100 percent domestic crude. Corbett analyses of these asphalts are

presented in Table 1.

For the three point bend testing and the study of constitutive

modeling, only Witco AR-4000 was used. All three asphalts were used

in the healing study. A syenitic granite aggregate (crusher fines)

was used for three point bend and uniaxial testing. The gradation is

illustrated in Figure 2. Fracture within this mixture resulted in

uniform crack surfaces without the irregular crack growth pattern

typical of mixtures employing larger and more well-graded

aggregates.

2. Sample Fabrication and Testing Methods

Izod Impact Testing

Izod impact experiments (American Society for Testing and

Materials (ASTM) E23) were conducted to provide fracture surfaces

produced at a controlled loading rate. Several binder types were

investigated in the experiment. The sample and the test apparatus

configurations are shown in Figure 3.

Ottawa sand was mixed with 13 percent asphalt cement by weight

of dry sand at 300°F, and samples were compacted at 275°F. The

mixing and compaction temperatures were determined based on viscosity

versus temperature data. Two blows of a 10-pound, Proctor-type

hammer were applied to the sample to provide compaction. Compacted

samples were stored at 68oF in a curing room for one day before

impact testing. Four samples were fabricated for each type of

binder. Samples were fractured by the Izod impact test machine (tmi,

23

TABLE 1 CORBETT ANALYSES ON THE THREE ASPHALT CEMENTS USED IN TESTING

Binder Saturates (%)

Witco AR-4000 11.22

Fina AC-20 13.95

Shamrock AC-20 4.92

Napthenic Aromatics

(%)

32.49

30.02

39.12

Polar Aromatics

(%)

51.14

42.37

51.67

Aspha1tenes (%)

S.15

13.66

4.29

N .p..

100 y-

/ /

rf 80 I

I I

bC

I ~ -M UJ UJ 60 co I P-.

~ I I ~ <l) I I (.)

H

I I <l)

40 p...

M I I co

I I ~ 0

t-< I I • • Granite Fines

I - -o--o-- AASHTQ M 29-33 20 / I (Grading reauirements of

/ / I fine aggregate for bitumi-

,./ nous paving mixtures)

--.J 0

200 100 50 30 16 8 4

Sieve Number

FIGURE 2 Gradation plot of granite fines and the AASHTO specification of fine aggregate for bituminous paving mixtures.

25

Striking edge

<~ Impact loading

Specimen

f-- 2.165 in. ___,

EJ ~0.079 in.

I A I ~45°\~ -t ~~-394 in.

0.394 in.

FIGURE 3 Izod impact test and sample configuration used in preparation of fracture surfaces for SEM analysis.

26

Testing Machine Inc.) with 1-pound impact hammer. One sample was

used to produce a replica of the fracture surface (control) for SEM

evaluation. The two fractured surfaces of the other samples were

brought back into contact with each other. Then these samples were

placed vertically and undisturbed for various periods of time at

68°F. Following these healing periods, the samples were again

fractured by the Izod impact test machine, and replicas of the healed

surfaces were immediately prepared for SEM inspection.

The use of the surface replication procedure for SEM

investigation was unavoidable because there was concern among staff

members in the Electron Microscope center about possible damage to

the SEM due to the evaporation of hydrocarbons from the asphalt

cement under the electronic beam.

The replication technique used in this study has been used very

successfully by anthropologists for many years. Detailed

information about the materials and procedure was reported by Rose

(63)~ The first stage of replication required the mixing of 6.0 ml

of Zantoprene Blue (silicone-based material) with 0.26 ml of

hardener. The mixture was then squeezed onto the surface by a

syringe which facilitated the flow of Zantoprene into tiny cracks on

the surface being replicated and prevented air bubbles from forming

on the surface of the mold. After six minutes, the mold was

carefully removed from the sample, and a wall of the mold was

constructed which was made of 10 ml of Optosil (silicone-based

impression putty) with 0.09 ml of hardener. After one hour of

hardening, the cast epoxy mixture of 20 parts of Epo-tek #301 and 5

parts of hardener, was gently poured into the mold. The epoxy was

27

further hardened overnight, and the mold and wall were removed.

These replicas were sputter-coated for one minute and 15 seconds with

125 R of gold-palladium before examination under a JEOL JSM-25

electron microscope.

Three Point Bend Testing

Three point bend testing was performed in accordance with ASTM

E813 (Figure 4). Granite fines with 9 percent of Witco AR-4000

asphalt by weight of dry aggregate were mixed at 300°F and compacted

at 275°F. An asphalt content of 9 percent was selected as one which

provided adequate specimen stability during testing, yet which

promoted uniform crack growth during fracture. Two-inch wide, three

inch high, and thirteen-inch long beam samples were fabricated using

a Cox kneading compactor.

Compacted samples were stored at 73oF for 24 hours, and their

bulk specific gravities were measured. Then, the samples were moved

to a 50°F curing room and cured between seven and fourteen days prior

to testing.

The compactive effort used during fabrication was as follows:

Layer No. Pressure Applied (psi) No. of Tamps

1 100 5

2 100 20

200 20

400 40

500 50

This compactive effort was designed to provide uniform density

throughout the specimen and to avoid a density (air void) gradient

28

LOAD

L w

Fracture Surface

A-A

-A

Enlargement Detail

W = 3 in. L=l3in.

ao :: 1 . 5 in. t = 2 in.

FIGURE 4 Configuration of three point bend testing sample with a chevron notch.

29

within the beam (16). The resulting air void content of all beam

specimens was in the range of 17 ± 0.5 percent without a noticeable

air void gradient. This high air void content is a function of the

uniformity and size (fine) of the aggregate. Although the high air

void content is not representative of dense mixes, the purpose of

these experiments was to study the relative degree of healing and not

to predict specific levels of healing in densely-graded mixtures.

Three different types of bend testing were performed on:

a. unnotched samples without prefatigue,

b. unnotched samples with prefatigue, and

c. notched samples with prefatigue.

The magnitude of the sinusoidal prefatigue load was 1.5 lbs. with the

frequency of 1 second/cycle. Selected samples were notched before

testing by cutting a chevron notch, 1.5-inch in length, with a

carbide tip blade with a 45° angle tip. The crack tip was then

sharpened using a razor blade. Prefatigue loading was applied until

the crack tip passed the chevron and the crack length reached 1.55-

inches, which took about 2800 cycles. For unnotched samples with

prefatigue loading, 2800 cycles were applied prior to testing.

For fracture testing (with notched samples), the crack length

was monitored by means of a Krak gage. The Krak gage, a thin metal

foil, was glued onto the side of a sample using a very thin layer of

epoxy. All tests were controlled by an MTS servo-controlled,

electro-hydraulic system.

Uniaxial Testing

30

The same sample fabrication technique for bend tesing specimen

fabrication was used for uniaxial specimens except that a straight

notch with l-inch crack length was fabricated in notched samples.

Uniaxial testing was performed using a device fabricated for this

study (Figure 5) With this machine, the samples were subjected to

a controlled horizontal movement of the base plate. Bending due to

the weight of the asphalt concrete samples was eliminated using this

testing configuration. The possibility of misalignment was minimized

as the pulling direction was g~ided by a linear track.

The crack length was monitored through a microscope video camera

(Figure 6). Chartpak pattern film graduated at 50 lines per inch was

attached beside the anticipated crack path and was used as a guide by

which to monitor crack growth using the microscope. The crack

information from the microscope was stored on videotape and was

studied following each test.

All tests were performed in a displacement-controlled mode at

73°F. The strain was calculated from the. movement of the hydraulic

ram and the original sample length. This calculated strain was very

close to the strain measured using two linear variable differential

transformers (L.V.D.T. 's) in the middle of the sample with gage

lengths of one inch.

31

3 /2 /1

~ 5

1. Beam epoxied to metal end support. 2. Sharp-tipped notch (introduced in some beams). 3. Metal end support. 4. Fixed platen.

I I

'I

'' I I I I

I I I I

I' :I 1 I

3 /

Horizontal movement introduced (servohydraulically controlled)

5. Moving platen. 6. L.V.D.T. (connected to M.T.S. controller) 7. Load cell. 8. Microscopic video camera.

FIGURE 5 Picture and schematic presentation of uniaxial testing

apparatus.

32

l. Beam sample with a shar?-tipped notch. 2. ~icroscopic video camera. 3. r: monitor. 4. Chartpak pattern film (0 02 in. between lines). 5. ~acrocrack. 6. ~icrocracks.

FIG~RE 6 (a) Microscopic ~ideo camera with testing apparatus. (b) Image of cracking area pictured from TV monitor.

33

CHAPTER V

MICROSCOPIC EVALUATION OF HEALING

1. Results

The technique used to replicate the fracture surface of the

asphalt mixture for SEM evaluation proved quite satisfactory. Since

the preparation and treatment procedure was relatively short and

simple, details of the fracture surface were not lost due to the flow

of the asphalt cement during the procedure. In order to produce

satisfactory resolution and viewing of the fracture surfaces using

the SEM, 5kV of accelerating voltage and 48 mm of working distance

from the bottom pole piece of the objective lens to the sample

surface were required.

In order to compare the fracture surface patterns of different

binders and to investigate the effects of healing time and binder

type, magnifications of 45, 150, and 450 were used. Higher

magnifications did not yield any additional information due to the

reproduction limitations of the replication technique.

Figure 7 shows the fracture patterns of AC-5, AC-20 and latex

modified AC-5. While the AC-5 fracture surface looks dull and

ductile, the fracture surfaces of AC-20 and latex-modified AC-5

reveal a sharper and more brittle pattern.

Higher magnifications of the fracture surfaces of AC-20 and

latex-modified AC-5 in Figure 8 reveal the different fracture

patterns of these asphalts. In these figures, the areas marked by

S's are the surfaces of sand aggregate. The AC-20 fracture surface

34

FIGURE 7 Fracture surfaces of Izod samples with different asphalts: (a) AC-5, (b) AC-20, and (c) SBR latex-modified AC-5.

35

FIGURE 8 Fracture surfaces of Izod samples wich AC-20 and SBR La~exmodified AC-5 at higher magnifications: (a) l50x magnification of AC-20, (b) 450x magnification of AC-20, (c) l50x magnification of Latexmodified AC-5, and (d) 450x magnification of latex-modified AC-5.

36

presents sharp and long lips (marked A) of the binder, while the

latex-modified AC-5 fracture surface is composed of two distinct

fracture patterns: cleavage type fracture (marked B) and ductile

fracture (marked C). The direction of the impact loading (arrow D)

can be predicted from the orientation of AC-20 lips in Figure 8-a.

The effect of the healing period on the fracture surface of the

AC-5 binder is presented in Figure 9. Figure 9-a is the control

fracture surface, and Figures 9-b, 9-c, and 9-d are the fracture

surfaces of AC-5 after healing periods of 5, 10, and 20 minutes,

respectively. Actually the fracture surfaces identified as

"following healing" were re-fractured using the Izod impact device

following the identified period of contact healing. The philosophy

of evaluation is that when a fracture surface "following healing" is

identical to the control (no healing), then the rest period has

produced total healing based or1 the visual criterion. The fracture

surface following a healing period of 5 minutes shows a more ductile

pattern with dull lips (marked B) than the control or the surface

after a 10-minute healing period. The fracture surface following a

20-minute period of healing shows essentially the same fracture

pattern with long and sharp lips (marked A) as does the control.

Figure 10 presents the AC-20 fracture surfaces with and without

healing. The fracture surfaces of AC-20 were allowed to heal for 20,

40 and 60 minutes. This experiment revealed that 20-, 40-, and 60-

minute healing periods were required with AC-20 to yield the visually

determined level of healing achieved in AC-5 following 5-, 10-, and

20-minute healing periods, respectively. That is, after a 20-minute

healing period, the AC-20 fracture surface demonstrated a ductile

37

FIGURE 9 Fracture surfaces of Izod samples with AC-5: (a) control fracture surface and fracture surfaces after healing periods of (b) 5 minutes, (c) lO minutes. and (d) 20 minutes.

38

FIGURE 10 Fracture s~rfaces of Izod samples with AC-20: (a) con=~Jl fracture surface and frac=ure surfaces after healing periods of (bJ

20 minutes, (c) 40 mi2u=es, and (d) 60 minutes.

39

fracture with smooth lips (marked B). As the healing time increased

toward 60 minutes, sharp and long lips (marked A) were observed more

frequently.

Figure 11 illustrates the difference between the control

fracture surface (Figure 11-a) and the fracture surface following a

20-minute healing period (Figure 11-b) for AC-20. The lips (marked

B) of the fracture surface following a 20-minute healing period were

smoother than the lips (marked A) of the control fracture surface.

Point healing (marked C) was observed in the area with less asphalt

cement.

The fracture surfaces following various periods of healing for

latex-modified AC-5 are presented in Figure 12. Figure 12-a shows

the control fracture surface, Figure 12-b shows the fracture surface

following a 10-minute period, and Figures 12-c and 12-d show the

fracture surfaces following a healing period of 20 minutes. From

Figures 12-a, 12-b, and 12-c, it can be seen that the brittle

fracture patterns (marked A) change very little. The area marked B

shows the ductile fracture demonstrated only by AC-5 binder.

2. Discussion

As far as fracture of asphalt concrete pavement is concerned,

the viscosity of binder at the time of fracture plays an important

role. In addition, if flow is considered as a part of the healing

mechanism, one can argue that the viscosity of the binder controls

not only the fracture but also the healing phenomenon of the asphalt

concrete. The penetration values of three binders were obtained

from Little, et al., (16) and tabulated in Table 2.

40

·~ ., / •.. /"

....,.,/' ""*J· ,.- - ..

"'¢ ..... ........ ~ ....

, ··" l ,;·

FIGURE 11 Comparison between (a) control fracture surface and (b) fracture surface after heal ittg period of 20 minutes from Izod samples with AC-20.

./> ,_.

FIGURE 12 Fracture surfaces of Izod samples with SBR latex-modified AC-5: (a) control fracture surface and fracture surfaces after healing periods of (b) 10 minutes, and (c) and (d) 20 minutes.

42

TABLE 2 PENETRATION INFORMATION OF DIFFERENT BINDERS AT 77oF

Type Penetration1 at 77°F, 100 g, 5 sec. (units of O.lmm)

Texaco AC-5 186

Styrene-butadiene rubber latex-modified AC-5

Texaco AC-20

114

75

---------- -------------------------

1. In accordance with the American Association of State Highway and Transportation Officials (AASHTO) 1'49

~ w

The sharper, more brittle looking surfaces of AC-20 and latex

modified AC-5 samples in Figures 7-b and 7-c compared to the fracture

pattern of AC-5 (Figure 7-a) is expected because the viscosities of

AC-20 and latex-modified AC-5 are higher than is the viscosity of AC-

5 at the test temperature. However, it has been found from studying

higher SEM magnifications (Figures 8-c and 8-d) that the

incompatibility of latex with Texaco AC-5 contributes to the brittle

fracture. That is, latex which is a solid at room temperature

results in a clevage type of fracture, while AC-5 produces ductile

fracture. An asphalt which is incompatible with the polymer will

result in a two-phase system in lieu of a homogeneous mass.

Figures 9 and 10 suggest that two stages are involved in the

healing mechanism. One is interpenetration, and the other is

bonding. When asphalt cement from two surfaces is brought into

contact, the interface will disappear as a function of time. Then,

the bonding energy develops also as a function of time and

contributes the major structural effect to the healed asphalt cement.

After 5-minute healing period, the interpenetration stage for the AC-

5 specimen is essentially complete, but the surface has not regained

its structural capability. The result is a dull looking surface.

After a 20-minute healing period the surface regains its original

strength, and the result is a fracture surface similar to the control

(Figure 9).

In contrast, it takes about 60 minutes of healing for AC-20

samples to regain the visual appearance of the control fracture

surface (Figure 10). This can be explained by the higher viscosity

44

of AC-20. That is, AC-20 needs a longer time to flow and

interpenetrate across the interface.

From Figure 12, it is apparent that the latex phase does not

change its shape significantly as healing time increases. Therefore,

it is apparent that at 68°F, AC-5 is the phase which contributes to

healing of latex-modified AC-5 over the range of healing times

studied here, not the latex phase. Latex is a solid material at

68°F.

Based on the observations discussed in the preceding paragraphs,

it is suggested that the appropriate healing model should represent

both initial surface penetration and the development of structural

bonding. Perhaps the viscosity of the binder determines the rate of

initial interprenetration and the level of structural bonding. A

binder with low viscosity will result in a higher rate of initial

interpenetration but a lower level of structural bonding after

complete healing.

This microscopic study could not clarify which phenomenon

contributes most to the healing mechanism for asphalt concrete. In a

global sense, the flow of the asphalt cement controls the healing

phenomenon. If the flow is governed by association forces (secondary

bonds) as reported by Petersen (33), secondary bonds among micelles

are the important factors in a healing mechanism.

In addition to the association force, it is suggested that

rearrangement of chain-like molecules can contribute to the time

dependent healing mechanism. When asphalt concrete is fractured, the

fracture surfaces are in a non-equilibrium stage. When the fracture

surfaces are in contact under pressure, initial interpenetration

45

occurs and chain-like molecules try to return to an equilibrium

stage. In amorphous materials, equilibrium can be obtained when

chains are in random order. Even though the average chain length of

asphalt cement is much shorter than for polymers', the degree and

rate of reentanglement of chain-like molecules can govern the healing

mechanism, as can association forces.

46

GHAPTER. VI

THEORY OF VISCOELASTICITY AND DAKAGE MECHANICS

Asphalt concrete pavement is subjected to different amplitudes

of repetitive loading and different durations of rest periods.

Modeling of asphalt concrete under this compound loading condition

can be quite difficult due to the history-dependent nature of the

material. That is, the material response is not only determined by

the current state of stress, but is also determined by all past

states of stress. Therefore, during loading and unloading paths, the

inelastic response of the material can be due to damage accumulation

processes and/or the viscoelastic nature of the material. Relaxation

and healing also occur at the same time during rest periods.

For many viscoelastic materials, the theory of viscoelasticity

has been successfully used to describe the history-dependent be

havior. In the theory of viscoelasticity, the influence of loading

history is usually assessed through a convolution integral. The

system is linear when the following two conditions are met:

a. superposition:

R{ 11 } + R{ 12 }

b. homogeneity:

where 11 and 12 different input histories,

R response, and

C constant.

The symbol{} represents the functional form. That is, R{I 1 } should

be read "response due to a function of the input 11 history".

47

A careful manipulation of the constitutive relationships of

viscoelastic (linear or nonlinear) materials can result in the so

called elastic-viscoelastic correspondence principle. The

correspondence principle reveals that the vast catalog of static

elastic solutions can be converted to quasi-static viscoelastic

solutions. The entire procedure involves replacing elastic moduli by

the appropriate transformation of the viscoelastic properties,

reinterpreting elastic field variables as transformed viscoelastic

field variables, and then inverting (64). A different way of

interpreting the correspondence principle is that one can reduce a

viscoelastic problem to an elastic problem merely by working within

an appropriately transformed domain with the substitution of elastic

moduli.

In this chapter, correspondence principles of linear

viscoelastic and nonlinear viscoelastic media are reviewed. The

correspondence principle of nonlinear viscoelastic media is used in

this study to separately evaluate healing and damage from history

dependence. Also, the internal state variable formulation of

constitutive modeling is demonstrated with the review of the damage

parameter developed by Schapery (43).

1. Theory of Linear Viscoelasticity

Before introducing the theory of nonlinear viscoelasticity and

its application, the theory of linear viscoelasticity will be briefly

reviewed.

As shown in Table 3, all the field equations and boundary

conditions for nonaging linear viscoelastic media are identical to

48

TABLE 3 GOVERNING EQUATIONS FOR LINEAR ELASTIC AND LINEAR VISCOELASTIC MATERIALS

Field Equations

Conservation of Linear Momentum

Kinematic Equation

Compatibility Equation

Constitutive Equation

Boundary Conditions

Linear Elastic

a i J . J ~ 0

£ij = l/2 (ui. J

V2 £ = o

aiJ - cijkl£kl

o 1 JnJ = T 1 011 ST

u 1 = u 1 on Su

Linear Viscoelastic

a i J • J - 0

+ uj . i ) £ij ~ l/2 (ui. J + uJ . i )

'V2 £ - 0

aiJ - It ( 8£kl o c i j k l t - ., ) --a-., - d.,

o 1 JnJ = T 1 on S1

u 1 = u 1 on Su

+'\.0

those of the linear elastic case except that the constitutive

equation of the linear viscoelastic material is in a convolution

form. However, if one takes the Laplace transform of both

constitutive equations, they will be reduced as follows:

Linear elastic case:

0 i j = ci j k 1 10 k 1

Linear viscoelastic case:

where f a Laplace transform of f and

fa sf= Carson transform of f.

Therefore, taking the Laplace transform of the governing field and

boundary equations of viscoelastic problems with respect to time

reduces them so that they are mathematically equivalent to those for

elasticity problems with the substitution of elastic moduli. This

correspondence principle allows one to reduce the linear viscoelastic

problem to the linear elastic problem merely by working in the

Laplace-transformed domain with Carson-transformed elastic moduli.

It is noted that when elastic moduli are constant, the Carson

transformed elastic moduli are elastic moduli themselves.

2. Theory of Nonlinear Viscoelasticity

Schapery (3) suggested that the constitutive equations for

nonaging nonlinear viscoelastic media are identical to those for the

nonlinear elastic case, but the stresses and displacements are not

necessarily physical quantities in the viscoelastic body. Instead,

they are "Pseudo stresses" and "Pseudo displacements" which are in

the form of convolution integrals such that

50

R u

i

R C7 ij

1 J: Ea

Ea J:

E(t-r) aui dr 8r

D(t-T) 8a ij 'dT

8r

where E(t) and D(t) relaxation modulus and creep compliance,

respectively, and

the reference modulus which is an

arb~trary constant.

The theoretical development of pseudo parameters is shown in Appendix

A.

Three different correspondence principles were introduced (3)

for different time-dependent boundary conditions.

Correspondence Principle I

The viscoelastic solution is

1 r . R

E(t-r) a a i j dr and ER o ar

J: D(t-r) R

ER aui dr ---

ar

where and u~ satisfy equations of the reference elastic problem.

This correspondence principle is valid for time-independent boundary

conditions.

Correspondence Principle II

This correspondence principle is proper when applied to the ;ase

of a growing traction boundary surface (dST/dt ~ 0), such as crack

growth problems. Here, ST is the surface of the traction boundary.

The solution of the viscoelastic case is

51

52

R and aij = aij

~I: R

ui D(t-r) aui dr ar

R R where a 1 j and u1 satisfy equations of the reference elastic problem.

Correspondence Principle III

When the surface of a traction boundary decreases with time,

correspondence principle II is no longer valid. Contact problems

and crack healing problems are represented by the cases when dST/dt <

0. The viscoelastic solution for these cases is

1 J: E(t-r) "::j dr and

where a~j and u~ satisfy equations of reference elastic problem.

In this research, only correspondence principle II is

considered, since in most of the tests performed, dST/dt ~ 0. Again,

correspondence principle II states that using physical stresses with

pseudo displacements one can reduce the nonlinear viscoelastic

problem to a nonlinear elastic case. The explicit form of the

constitutive equation between stresses and pseudo displacements is

dependent on material type, sample geometry, and loading geometry.

3. Constitutive Modeling of Asphalt Concrete

Asphalt concrete is a rate-dependent, history-dependent

composite material. In order to model the behavior of asphalt

concrete under complicated, realistic loading, one needs to account

for stress-induced damage along with these characteristics. Examples

of sources of damage in asphalt concrete are microcracking,

macrocracking, shear yielding, permanent deformation, and healing at

crack faces. Some of the structural changes are advantageous to the

overall behavior of asphalt concrete even though we classify them

here by the term "damage".

The rate-dependence of asphalt concrete is presented in Figure

13. When the strain rate is increased, the stress at the same

magnitude of strain increases. The history-dependence of asphalt

concrete is shown in Figure 14. Not only are the stresses at the

same strain different on the first loading and unloading paths, but

stresses also drop as cycling continues. The behavior shown in

Figure 14 will be studied in detail in the next section. The data

shown in Figures 13 and 14 are the actual data collected from

uniaxial tensile testing under constant strain rate monotonic loading

and constant strain rate simple loading, respectively.

In order to mathematically model this complicated behavior of

asphalt concrete, internal state variable formulation was used. That

is, by investigating the behavior of asphalt concrete under loading,

one can establish a functional form of the stress-strain

relationship, and discrepancies from the real response will be

accounted for using a sufficient number of internal state variables.

We propose that the form of the constitutive relationship for asphalt

concrete is as follows:

a i j ( tk 1 , t, T,

where aij stresses in a body,

aT a~

(VI. l)

53

"" ..... (f)

0... .../

(f) (f)

01 L

+-> tf)

so~~~~~~--,-~~~~~~~~~--,-~~~

40 ·-

30 I .----- E = 0.0368 in./in./min .

20 ·- - ( = 0.0184 in./in./min.

10 ·- £ = 0.0092 in./in./min.

___.----------------- '""'€ 0.0046 in./in./min.

E 0.0023 in./in./min.

QL-----~--~_.~--~--._~~~~~~~----~~~_.--~--~----~----J

0 . 001 • 002 . 003 . 004 • 005 . 006

Strain (in. /in.)

FIGURE 13 Illustration of rate-dependency in asphalt concrete (stress-strain curves of constant strain rate monotonic loading tests).

V1 -1:-

Input

Time

Application of correspondence principle ~

en en Q)• !-< ~ (/)

Strain

15~--------~--------~----------r---------~---------,

<ll 1st loading 10

5 ./ .//

0~------------------~~----------------------------_,

-5 (b)

-10~--------~--------~----------~--------~--------~ -0.001 -0.0005 0 0.0005 0.001

Pseudo Strain

FIGURE 14 The effect of the maximum strain during the past strain history on (a) stress-strain behavior and (b) stress-pseudo strain relationship after the application of correspondence principle.

55

t

T

aT

a"Xn

strains in a body,

time elapsed from the first application of loading,

temperature,

spatial temperature gradients in a body,

am = internal state variables,

i, j, k, 1, n = 1, 2, 3, and

m = 1, 2, 3, . . . , M.

Assuming that the temperature is constant spatially, Equation (VI.l)

reduces to

(VI. 2)

The nonlinear viscoelastic correspondence principle suggests

that the constitutive form of Equation (VI.2) can be reduced to :

(VI. 3)

The effect of temperature on the stress-strain relationship can be

included in the convolution integral using reduced time, e. Assuming

that the asphalt concrete is a tbermorheologically simple material

which has been proven (65) to be a good approximation, one can

construct a master curve of the relaxation modulus versus the reduced

time e. Since the material is thermorheologically simple, the

reduced time can be determined from:

loge= log t - log ar(T)

where ar(T) =the shift factor which can be obtained from the

horizontal translations of the curves at different temperatures to a

single master curve. This change of variable scheme results in

E(t,T) = ~(0 .

56

57

If the temperature does not vary during the testing period, the

pseudo strains can be determined from:

1

Ie' 0 ~(~-()

With the above expression for Equation (VI.3) can be

reduced to:

In this research, only the uniaxial case with constant

temperature is considered; therefore,

(VI. 4)

where E(t-r) dr .

Damage of Asphalt Concrete

In order to investigate the damage of asphalt concrete under

compound loading, two uniaxial tests with simple loading were

performed (Figures 14 and 15). For these tests Witco AR-4000 asphalt

was used. The strain amplitudes used in Figures 14 and 15 were 6.13

x 10- 4 in./in. and 1.84 x 10- 3 in./in., respectively. The strain

amplitude of 6.13 x 10- 4 in./in. was small enough for damage growth

to be negligible. As a contrast, the strain amplitude of 1.84 x 10- 3

in.jin. induced additional damage after each cycle.

From the first loading and unloading curves shown in Figure 14

(a), there are two different stresses (aL and au) for one strain

value (E0). On the loading path, the stress is a function of current

strain only, while on the unloading path, the stress is a function of

current strain and the maximum strain during the strain history up to

,...... ·rl Ul 0.

Ul Ul QJ

~ ~

Cf)

20

10

0

-10 -

Input

Time

Application of correspondence

(b)

-0.002

<

0

Pseudo Strain

Strain

~

Change in stress as the number of cycle increases

0.002 0.004

FIGURE 15 Illustration of damage accumulation under the large strain amplitude: (a) stress-strain behavior and (b) stress-pseudo strain relationship after the application of correspondence principle.

58

that time. This concept is also well illustrated in Figure 14 (b)

with the application of the correspondence principle. The first

loading path is different from the rest of the loading and unloading

paths. The reason for this is that the largest pseudo strain in the

loading history, E~, for the first loading is different from that in

remaining cycles. That is, €~ for the first loading is the current

pseudo strain, while €~ from the first unloading path to the end of

the strain history is a constant which is the largest pseudo strain

during the first loading. Furthermore, negligible damage growth

during the loading-unloading cycles results in practically the same

loading and unloading paths after the first loading.

R From the above observation, it is proposed that EL can be used

as the first internal state variable in the constitutive equation

(VI.4). Now Equation (VI.4) becomes

~VI. 5)

where the maximum pseudo strain in the past history and

m = 2, 3, ... , M.

However, from Figure 15, it is found that €~ is not sufficient

to accurately model the damage accumulation process of asphalt

concrete under repetitive loading. That is, even though E~ is

constant after the first loading, the stress at a specific level of

pseudo strain decreases in Figure 15 (b) as the number of cycle

increases. This behavior suggests another variable in the

constitutive equation which should represent the damage growth in an

asphalt concrete body. The "damage parameter" as it is termed by

Schapery (43) will be used in this research as the last internal

state variable in the constitutive model.

59

60

Damage Parameter

Schapery has developed (43) damage parameters based on a

microcrack growth law which has been predicted using a generalized J

integral theory for viscoelastic materials (3). In the remainder of

this subsection, the major steps of development of the damage

parameter are reviewed.

In 1975, Schapery published a series of papers (66) on the

theoretical analysis of the crack growth in a linear viscoelastic

media using the correspondence principle based on Laplace

transformation. He derived the power form of the relationship

between the crack growth rate and the stress intensity factor, which

is the so-called Paris law. The form of Paris law is:

da (VI.6) dt

where a = crack length,

~ax the amplitude of the oscillating stress intensity

factor, and

A and n = regression coefficients.

Schapery showed the theoretical relationship among the power law

constants A and n and the creep parameters and material properties of

viscoelastic media (66). In fact, for a linear viscoelastic

material, he proved the exponent in the crack growth law to be:

n 2 (1 + _!_) m

where m the exponent of the power law between creep compliance

and time, i.e. D(t) = D1 tm.

The major concepts of this work have been extended to develop a

generalized J integral for large deformation (3). In addition,

Schapery derived an implicit form of crack speed as a function of the

viscoelastic J-integral, Jv:

da

dt

Summarized reviews of J-integral theory and the crack growth law

shown above are presented in Appendix B.

Based on his J-integral theory, Schapery (67) derived the

explicit form of crack speed versus Jv relationship as follows:

da

dt (VI.7)

This form is valid for a failure zone in a continuum obeying power

law relationships between stress and strain and between creep

compliance and time i.e. a-I£RIN and D(t) = D1 tm. The exponent k

was shown to be a simple function of both exponents, m and N, as

follows:

a. if the maximum stress in the failure zone and the fracture

energy are constant, then

k = 1 + 1

m

b. if the failure zone size and the fracture energy are

constant, then

k 1

m

c. if the failure zone size and crack-opening displacement are

constant, then

k 1

(l+N)m

61

It has been shown (43) that for nonlinear power law materials,

Jv - lall+l/N f 2 , where f 2 is independent of stress, a, but in

general varies with time through the effects of varying crack

geometry. For an isolated penny-shaped crack, f 2 - f 3 a, where f 3 is

a constant depending on the crack dimensions. Integrating Equation

(VI.7) in this case yields

As can be observed from this equation, the time dependence of the

crack length comes only from the integral S term which is defined as

s = dt . (VI. 8)

Schapery (43) generalized this analysis to microcrack growth from the

fact that most of the failure time is consumed before the crack grows

appreciably. The integral S in Equation (VI.8) was defined (43) as

the damage parameter based on stress.

The same type of approach can be applied to the Paris law with

linear materials in which the stress intensity factor can be

presented as (68):

K = Y a J;;i

where Y depends on the crack geometry. Substituting the above

expression for K into the Paris law in Equation (VI.6) yields:

a = n d 1-n/2 a t + a 0

J 2/(Z-n)

where C is a constant in time and is a function of Paris' law

constants (A and n), Y and a 0 . From the analysis of crack growth in

linear viscoelastic materials (66), n = 2 (1+1/m). Therefore, the

time dependence of the crack length in linear viscoelastic materials

comes from the following integral S':

62

63

S' - J: 2(1+1/m) d 0 t .

With the assumption that f 1-l, the integral S' is a linear

viscoelastic representation of the integral S in Equation (VI.8)

because N - 1 in the linear case and k - 1 + 1/m assuming constant

maximum stress in the failure zone and constant fracture energy.

Since o-leRIN, the damage parameter based on pseudo strain can

be obtained from Equation (VI.8) as follows:

Assuming that f 1 - 1, the damage parameter SP was developed (43) as a

Lebesgue norm of eR. The Lebesgue norm of a function f is defined

as

Finally, the damage parameter was developed as

(VI. 9)

where P = (l + N) k. (VI.lO)

An important property of SP is that Sp ~ leRimax asp~ ro (43).

Therefore, the maximum pseudo strain occurring during loading

history, which was earlier identified as a necessary variable in the

constitutive model, is merely a damage parameter at an infinitive p

value.

Now the implicit form of constitutive model is

0 (VI.ll)

The explicit form of this model was investigated through a series of

uniaxial tests. The results of this study are presented in Chapter

IX.

CHAPTER VII

THREE POINT BEND TESTING

In this chapter, the correspondence principle is applied to.

three point bend testing. The exponent of the power law relationship

between the relaxation modulus and time is predicted from a series of

three point bend tests at various strain rates. This exponent was

verified successfully by uniaxial tensile relaxation testing.

A limited amount of fracture mechanics testing was performed in

accordance with ASTM E813, and the crack length was measured by means

of Krak gages. The results were used to develop a power law

relationship between the Jv integral and crack speed. However, it

was found that the crack length measured from the Krak gages was much

longer than the visual crack length. Since the Krak gage was much

more brittle than the mixture of granite fines and AR-4000 grade

asphalt cement at 73oF, it could not resist the small movement in the

damage zone ahead of the crack tip which can be regarded as a

deformation in the mixture rather than a separation of the material.

Consequently, the results were not satisfactory.

1. Derivation of Pseudo Displacement for Constant Strain Rate

Testing.

Pseudo displacement for a uniaxial case is determined from:

J: E(t-r) du

dr dr .

For a constant displacement rate test dujdt = C where C is a

constant. Then,

64

Let x t - r, then

R u E(t-r) dr .

UR _ ~- J0

t - '-'R E(x) dx .

Since u = Ct,

R u

where E'(t) = ~ Jt E(x) dx . t a

u E' (t)

(VII.l)

The term E'(t) is the relaxation modulus in a time average sense.

The same procedure can be used to derive the pseudo

displacement in a three point bend testing configuration, except that

the relaxation modulus in the convolution integral, E(t), should be

replaced by the relaxation modulus derived from bending tests. Thus,

the pseudo displacement in constant displacement rate bend testing is

determined from:

where E" (t)

R u

E" ( t) u

- E8 (x) dx and 1 It t a

(VII. 2)

(VII. 3)

EB (t) relaxation modulus in the bending configuration.

In order to derive E8 (t), beam theory was applied. For a

simply supported beam with a concentrated load in the middle of the

span, maximum vertical displacement at the center is:

u (VII.4) 48EI

where p = the magnitude of the concentrated load,

L = the span length,

65

E Young's modulus, and

I moment of inertia.

The relaxation modulus can be defined as a load response from

the unit step function of the displacement input. Using this

definition with Equation (VII.4) yields

1 E L3 B

48EI

i.e. EB 48EI - LJ

From the correspondence principle, the relaxation modulus derived

from the three point bending test is

48E(t)I

LJ

For the geometry and size of the sample used in this research,

EB(t) = 0.147E(t) . (VII. 5)

The relaxation modulus, E(t), is often represented in a power form,

(VII. 6)

Substituting Equations (VII.S) and (VII.6) for E"(t) in Equation

(VII. 3) yields

0.147E1 -m E"(t) - t . (VII. 7)

1-m

Two different approaches can be used to verify the validity of

the correspondence principle. Knowing the uniaxial relaxation

modulus as a function of time, one can run a series of constant

strain rate bending tests at various rates and calculate the bending

pseudo displacement based on Equations (VII.2) and (VII.7). The load

versus pseudo displacement curves at different loading rates should

fall onto the same line for negligible damage because the rate-

66

dependence of the material has been accounted for using the

correspondence principle.

The second method is to observe that the pseudo displacement is

a product form of the physical displacement and E"(t)/E1t· One can

then shift the physical displacements at constant times and obtain

pseudo displacements. The magnitude of shift as a function of time

can be predicted by a power relationship. The exponent of this power

relationship should be identical to the exponent of the relaxation

modulus, m, in Equation (VII.7). In this research, the latter method

was used simply because the relaxation data were not available prior

to three point bend testing.


As mentioned in Chapter IV, Section 2, bend testing on three

types of samples was performed:

a. unnotched samples without prefatigue,

b. unnotched samples with prefatigue, and

c. notched samples with prefatigue.

Load versus displacement curves of Test "a" with three different

displacement rates are shown in Figure 16. As expected, the

material was rate-dependent and the faster the rate, the higher the

stiffness. A series of isochronal curves were constructed at

different times from Figure 16 and are presented in Figure 17. The

isochronal curve at 80 seconds was selected as a reference curve, and

other curves were shifted horizontally to the reference curve. The

shift factor versus time was plotted in Figure 18 on a log-log scale.

67

" _ci ...--. '-J

"'0 0 0

_j

20~~----~~--~~----~~--~~~--~~----~~----~--r-~~-----,

15 ·-

10 ·-

5 ·-

-4 u = 6.667 x 10 in./sec.

.... ,-........

_,.,"' / ... ,"'" .~·-

/ "" __,- . ,---- -.......... . r ~,-- u=

y"' ~

//

-4 u = 5.000 x 10 in./sec. ~~ ~-- / ,_,.

,_,

-4 3.333 x 10 in./sec.

0~~~--._~--~~~--~~--~~~--._----~~----~----~~----~-J 0 .02 .04 .06 .08 • 1 • 12

Displacement (in.)

FIGURE 16 Load versus displacement curves of uncracked samples without prefatigue.

(J\

CXl

/'""'

,.0 rl

'0 C1l 0

,....:l

20

10

0 0.005 0.01 0.05

Displacement (in.)

FIGURE 17 Isochronal curves constructed from Figure 16.

0.1

0\ \0

,-..

,-.. ~

......,I r:.:: = w w .......,

1-< 0 ~

u C1l ~

~ 4-< •rl ...c: CfJ

5

1

8

8

8

0

10 so 100

Time (sec.)

FIGURE 18 Shift factor versus time of uncracked samples without prefatigue.

-.J 0

The relationship between log (shift factor) and log (time) was

determined from the regression analysis as follows:

shift factor- 18.535 t-0.666 (VII.8)

with R2 = 0.999. Therefore, the time exponent of the shift factor

was -0.666, which will be compared later with the time exponent of

relaxation modulus from uniaxial tensile relaxation testing. The

shift factors were calculated for each data point from Equation

(VII.8) and multiplied by the physical displacements to determine

pseudo displacements. Pseudo displacements are plotted against load

for different loading rates in Figure 19. As predicted, the rate

dependency of the material disappeared using the pseudo

displacements.

The same procedure was applied to data from unnotched samples

with prefatigue (Figures 20 and 21). The shift factors calculated

from Equation (VII.8) were used to obtain pseudo displacements. The

rate-dependency was again satisfactorily eliminated as demonstrated

in Figure 21. This suggests that the application of prefatigue

loading did not influence the viscoelastic properties of the

material. However, from the comparison of Figures 19 and 21, it

should be noted that the stiffness of the samples with prefatigue is

higher than that of the samples without prefatigue. Possibly,

residual stresses developed in the sample upon bending after

prefatigue loading. In an effort to produce the same magnitude of

bending displacement in the sample after prefatigue loading as for

the sample without prefatigue loading, a greater load was required

due to upward recovery action after prefatigue loading.

71

,......

_ci ...... '-"

-u a 0

_j

20 I .

u --4 6.667 x 10 in./sec. -4 ---- 5.000 x 10 in./sec.

151--4 -·- 3.333 x 10 in./sec.

10

~!'/ 5t- ---~-

/.

,_-< /' "

,""/' / ""'///'

. 02 . 04 . 06

/' /

/ ,/./

Pseudo Displacement

... "

,-I ./

. 08 . 1 . 12

FIGURE 19 Load versus pseudo displacement curves of uncracked samples without prefatigue.

...... N

;--..

.d ....... ..__,

\J 0 0

_J

40~~~~~~~,-~~~r-~~-.~~--~~----

30 ..

20 /

/ ,.. /"

/' -,"' ....

101- ,/' -~-~ I ,/ -~-~ "-.·

,'// u

v ~-

o~ 0 . 02 . 04

-4 u = 6.667 x 10 in./sec.

_,. /'

/ .... -- ~-..,./_ ~· ,... u =

-4 5.000 x 10 in./sec.

-4 3.333 x 10 in./sec.

. 06 • 08 • 1

Displacement (in.)

FIGURE 20 Load versus displacement curves of uncracked samples with prefatigue.

• 12

-....)

w

,.... _ci ...... '-'

u a 0

_..J

40

I

301-

20 ·-

10 ·-

------·-

. u -

-4 6.667 x 10 in./sec. -4 5.000 x 10 in./sec. -4 3.333 x 10 in./sec.

~.'/ /

. / /

~/'· /

,.- _./../..- / /

/~~ ............ /' ....

. 02 . 04

'I'

'l

. 06

r

.-'

.J

rr

Pseudo Displacement

I

• 08 • 1 • 12

FIGURE 21 Load versus pseudo displacement curves of uncracked samples with prefatigue.

-...J .p-

The analysis of data from the notched samples was identical to

the analysis for the unnotched samples until the plots of load versus

pseudo displacement were obtained (Figures 22 and 23). The same

shift factors derived from Equation (VII.8) were satisfactorily used

to eliminate the rate-dependency, which implies that the introduction

of a crack to the sample does not affect the viscoelastic properties

of the material significantly. The reduction of the stiffness of the

notched samples with prefatigue compared to the unnotched samples

with prefatigue is due to the decrease in the ligament area.

Uniaxial tensile relaxation testing was performed on beam

samples to compare the time exponent in Equation (VII.8) with that

measured from relaxation testing. The detailed testing plan will be

described in Chapter VIII; however, the regression equation

representing the relationship between E(t) and time is:

E(t) = 5007.55 t- 0 · 661 (VII.9)

with R2 = 0.996. As a result, the exponents in Equations (VII.8) and

(VII.9) are essentially the same. This proves that the

correspondence principle developed by Schapery (3) can be

successfully applied to bend testing of asphalt concrete mixtures.

75

......,

.0 ...... ......,

"'0 0 0

_J

25r-~~~~~~~-a~--~--~~~~-a~~~~

20 ·-

/-- --,

-4 x 10 in. I sec. -

15 ,-;-<

/-- ~ -4 = 5.000 x 10 in.lsec.-

10 ·-

-" "'

--r-

_/

,. "'"' ,....,J'-'~ / ~·

.......-

/~ ~ . -4 I "' ~· ~u = 3.333 x 10 in. sec.

/" ,.-/'' // _.,/'

I . ./.

;/ ( .

sl· 'Y

o~~--~--._~~_.--~--~~--~--~--._~--~--~--~~--~--~----~ 0 .02 .04 . 06 • 08 • 1

Displacement (in.)

FIGURE 22 Load versus displacement curves of cracked samples with prefatigue.

-....J Q'\

"' ..!) ...... '--'

-o 0 0

__J

25

I

201----------

I 15

I

10 ·-

5 ·-

. u -

-4 6.667 x 10 in./sec. -4 5.000 x 10 in./sec. -4 3.333 x 10 in./sec.

~-- I ~- ................

/

J

r:

""

,.. t'

ol,L~;~,----' 0 I I ~1

----......1 1 ~~ _._____.__

1 1 j I I I I , ( I I I

. 02 • 04 . 06 • 08 • 1

Pseudo Displacement

FIGURE 23 Load versus pseudo displacement curves of cracked samples with prefatigue.

'-1 '-1

CHAPTER. VIII

UNIAXIAL TESTING - EVAIJJATION OF I:IF.ALING

1. Method of Analysis

When a viscoelastic material is subjected to cyclic loading, a

hysteresis loop is usually observed in the stress-strain diagram.

According to correspondence principle theory, the hysteresis loop

will disappear in the stress-ps~udo strain plot if damage growth is

negligible during the loading history. That is, the relationship

between stress and pseudo strain is a single-valued function.

Furthermore, even when the loading paths before and after rest

periods are compared, this elastic-like behavior will be maintained,

if negligible damage has occurred. This is because the relaxation

during the rest period is taken into account by integrating the

convolution integral from the initial loading time to the present

time.

When the damage is large, the stress will decrease, in

displacement-controlled testing, as the number of cycles increases.

The difference in the stress at the same pseudo strain level is due

to the damage growth in the sample. If rest periods are introduced

in the loading history and relaxation is the only phenomenon

occurring during the rest period, the stress after the rest period

should be equal to or less than the stress before the rest period for

the same pseudo strain, based on correspondence principle theory.

If the stress after ·the rest period is larger than the stress before

the rest period, at the same pseudo strain level, the increase in

78

stress must logically be attributed to some chemical healing

mechanism.

The concept, outlined in the preceding paragraphs, was used to

evaluate the healing potential of three different asphalt cements.

In this evaluation, the following three types of uniaxial tests were

performed:

a. relaxation tests,

b. constant-strain-rate simple loading tests with rest periods

(Figure 24). The magnitudes of displacements were within

the linear viscoelastic range of the material (negligible

permanent damage), and

c. constant-strain-rate simple loading tests with rest periods

(Figure 24). The magnitudes of displacements resulted in

crack growth.

Beam samples with an edge crack were used only for Test Series "c".

All other samples were beam samples without fabricated edge (sharp

tipped) starter cracks.

Test Series "b" was designed exclusively for the verification of

the applicability of the correspondence principle based on the

relaxation moduli obtained from Test Series "a". Since the maximum

strain level for Test series "b" is in the linear viscoelastic range

which results in negligible damage, the stress-pseudo strain plot

must be linear; and no stress drop should be observed at the same

pseudo strain as the cycle number increases.

The loading history for Test series "b" and "c" is shown in

Figure 24. The numbers of initial simple loading cycles for Tests

"b" and "c" were 10 and 20, respectively. Then a set of four rest

79

c ·rl r: ).... ~

cr.

~ A A n A I

~ ~ 1--- Initial londin~ + Rest

rwriod

~ ~ ~ A ~

I ~ V V V + 5 cycles +

of lo:1ding

FIGURE 24 Strain history for tests "b" and "c".

Rest period

~ I

+- 5 cycles -+of loading

I

I V Rest

period ---1

~ Time

00 0

periods of 5, 10, 20, or 40 minutes duration was introduced in a

random sequence. Five cycles of simple loading were applied after

each rest period, and this loading pattern (rest period followed by

five cycles of simple loading) was repeated until 3 repetitions for

each rest period length were achieved.

It was experimentally found that a strain level of 6.13 x 10- 4

in.jin. was small enough for all the mixtures investigated in this

research to produce linear behavior. Two strain levels, 6.74 x 10- 3

in./in. and 9.20 x 10- 3 in./in., were used in Test Series "c" and

were large enough to propagate the crack in the middle of the sample.


Relaxation Testing

Uniaxial tensile relaxation testing was performed for three

different mixtures, and the results are plotted in Figures 25

through 27. Various strain levels were used, and the loading time to

peak strain was three seconds. The strain level dependency could not

be identified for the range of strain levels evaluated because the

variation among samples was substantially greater than the strain

level sensitivity. Theoretically, strain in the form of a step

function should be applied in relaxation testing; however, due to

load cell range limitations, a three-second initial ramp was

unavoidable.

Usually, the relaxation data are represented in the form of a

pure power law:

81

""' ..... (/) 0.

'"--'

(/) :J

....... :J -o 0

::L

r: 0 ..• •J n X u ~· 01 0::

105

~ ............... ~~-Strain amplitude Symbol

0.0023 in./in. • 0.0023 in./in. X

0.0038 in./in. 0

104 ·-

0.0046 in./in. +

E(t)=40.303+5457.956t-0. 7 l6

103

E(t)=ll7.044+281.066t-O.l62

• • • 102

1 10 ~-~1_.--~~~~~0--~_.~~~~~1--~~~~~~~2~~_.~~~~~3~~~_.~~~ 4

10 10 10 10 10 10

Time (seconds)

FIGURE 25 Relaxation data for the mixtures with Witco AR-4000.

00 N

,.., ..... (/) Q_

'--'

(/) ::J ._.... ::J -u 0 ~

r: 0 .....

.j.) tJ X 0

..-< aJ

n::

105~~~~~ Strain amplitude Symbol

0.0023 in./in. • 0.0023 in./in. x 0.0038 in./in. 0

104 L 0

0 o~E(t)=-41.518+5320.255t-0.553

103

0 E(t)=56.307+3207.465t-0.477

102

I 10 ~-~.~--._~~~~0~~~~~~~~.~~~~~~~~2~~~~~~~~3--~~~~~~ 4

10 10 10 10 10 10

Time <seconds)

FIGURE 26 Relaxation data for the mixtures with Fina AC-20.

00 w

,..... ...... (II CL ..__,

(II ::>

....... ::>

u 0

2::

r: 0 ...... +J u X ll

....... OJ

0::

105

~~~~~ Strain amplitude Symbol

0.0015 in./in. • 0.0023 in./in. x 0.0023 in./in. 0

104

E(t)=-160.052+8405.693t-0. 486

103

0 E(t)=l54.072+11622.0lt-0.669

102

I 10 L-_~,~--._~~~~0~~_.~~~~~,--~~_.~~~~2~~~_.~~~-3~~~_.~~~ .c

10 10 10 10 10 10

Time <seconds)

FIGURE 27 Relaxation data for the mixtures with Shamrock AC-20.

00 .p.

or by a generalized power law:

E(t) - E0 + E2 t-n

where E(t) the relaxation modulus and

regression constants.

The best way of fitting the data in Figures 25 through 27 was to

divide the whole curve into two sections, and to fit each curve using

a generalized power form. The regression constants in the

generalized power law were obtained by a trial and error method.

That is, assuming the value of n, a linear regression analysis was

performed between E(t) and t-n and the sum of squares of error was

calculated. Repeating this procedure for a series of n's, then

which resulted in the smallest sum of squares of error was

determined. This technique yielded the predicted curves whose

regression constants are shown in Figures 25 through 27.

Constant Strain Rate Simple Loading Tests with Rest Periods

The calculation of pseudo strains for repetitive loading is more

involved than Equation (VII.l) for constant strain rate monotonic

loading. During the first loading path of constant strain rate

simple loading (Figure 24), the sample does not know whether the test

type is monotonic or cyclic. Therefore, pseudo strains can be

calculated from Equation (VII.l) by substituting displacement (u)

with strain (e). Thus, for the first loading path,

eR = ~ J: E(x) dx (VIII.l)

where C = de/dt, i.e. constant strain rate.

During the first unloading, the same practice yields:

85

ER ~ [ fo 1 E(x) dx - J:

1

E(x) dx ]

where t 1 - the time of peak loading.

Similarly, the pseudo strain at any time can be calculated

analytically as long as the loading history and the relaxation

modulus as a function of time are known, regardless of the existence

of rest periods.

The results of the Test "b" series demonstrated the validity of

the correspondence principle and its applicability to this research.

Furthermore, the Test "b" series demonstrated that the relaxation

moduli derived from the Test "a" series were satisfactory

measurements. The stress-pseudo strain plots of the first ten cycles

and of the cycles before and after the 40-minute rest period for Test

"b" are presented in Figures 28 and 29, respectively. The asphalt

cement studied in these figures was Shamrock AC-20 asphalt. As shown

in Figure 28, the loading and unloading paths of the first ten cycles

practically fall on the same line. Also in Figure 29, the stress

pseudo strain ·curves before and after the 40-minute rest period are

practically the same. The results for the other binders and for

different lengths of rest periods verified the success of this

procedure.

The results of the Test "c" series with Witco AR-4000 asphalt

are shown in Figures 30 and 31. In Figure 30, the loading and

unloading paths of the initial 20 cycles are plotted, and the stress

pseudo strain behavior before and after the 40-minute rest period is

presented in Figure 31.

86

;--.. ·.-4

(Jl 0...

--.>

(Jl (/) 01 L .u U1

15r-----------~------------.-------------r-----------~----------~

G) 1st loading

10 ·-~

5

0

-5 ..

-10~----------~------------~-------------L------------~------------~ -.001 -.0005 0 .0005 .001 . 0015

Pseudo Strain

FIGURE 28 Stress versus pseudo strain of initial 10 cycles with negligible damage (Shamrock AC-20).

00 -.....!

,... ...... (/) 0...

'-"

(/) (/) OJ L

.l-l (J)

15~----------~------------~----------~~-----------r----------~

10 ·-

5

0

-5 ·-+ Before 40-rnin. rest period ~ After 40-rnin. rest period

-lOL-----------~------------~------------~------------~----------~ -.oot ~ooo5 o .ooo5 .oot • 0015

Pseudo Strain

FIGURE 29 Stress versus pseudo strain before and after 40-minute rest period with negligible damage (Shamrock AC-20).

co co

,..._ ..... (f) Q_ .._,

(f) (f) (j)

L .a (f)

30

I @--- 1st loading path

- -{D-- 1st unloading path 20 1- llo. 2nd loading path

- -dr- - 2nd unloading path

20th loading path

101- --+--20th unloading path

ol~ f/ r /~/~---~--

- . -~.,-, ,-, _,J •• --~--j<1!f/ --- I -10 e~,;! a."="'~:.:e;.:: ~ ~~-

-20~----~------~------~----~~----~------~------._----~~----~ -.004 -.002 0 .002 .004

Pseudo Strain

FIGURE 30 Stress versus pseudo sirain of initial 20 cycles with strain amplitude of 0.0092 in./in. (Witco AR-4000).

CXl \0

r-..... (/) (}_ ...._,

(/) (/) 01 L

.1-l U1

20r---.---~----~--~---r----.---.---~--~

Loading path before ----+---- 40-min. rest period

Unloading path before --r-- 40-min. rest period

10 ~ Loading path after --*-- 40-min. rest period

Unloading path after --~-- 40-min. rest perlod

01----------------~----~-----

/// 7

~~ -1 .--' ~ ;/ 0~ _ __... _,__...._._-- __.._--~-·~

. _ _,.._. ,...... . ..-.. -----...---./-¥ ,)It K

/

-20~----_.------~------~------~----~------~------._----~~----~ -.004 -.002 0 .002 .004

Psgudo Strain

FIGURE 31 Stress versus pseudo strain before and after 40-minute rest period with strain amplitude of 0.0092 in./in. (Witco AR-4000).

"' 0

The first point to note in Figure 30 is that loading and

unloading paths form a hysteresis loop which disappeared after

applying the correspondence principle to the Test "b" series results.

However, under the conditions of Test "c" (i.e. damage accumulation

through crack growth), visual dissipation of energy is due to the

damage growth in the sample. Since the test was performed in a

controlled strain mode with a constant strain amplitude, the stress

at a selected pseudo strain level became smaller as c.ycling

continued.

In Figures 28 and 30, the first loading path is different from

the rest of the loading paths. The reason for this is that the

largest pseudo strain in the loading history, e~, for .the first

loading is different from that in the remaining cycles (41,44). That

is, R eL for the first loading is the current pseudo strain, while

R eL for the following cycles is a constant which is the largest

pseudo strain during the first loading. The effect of this e~ on a

constitutive equation modeling the stress-pseudo strain relationship

will be discussed in the next chapter.

From the comparison of the recorded crack length and the stress-

pseudo strain behavior, it was found that the stiffness increase

during the last part of loading (Figure 30) was due to an additional

growth in the crack length from the prior loading cycle. The crack

was growing faster for the Witco asphalt mixture for the same maximum

strain level than for Fina and Shamrock mixtures.

As discussed previously, the stress-pseudo strain curve after

the rest period should be positioned ·somewhat lower than the curve

before the rest period if the relaxation is the only mechanism

91

influencing behavior during the rest period. However, as shown in

Figure 31, there was a significant increase in the stress for the

same pseudo strain. This behavior was not observed in Figure 29

where damage level was almost negligible. Therefore, it is concluded

that during rest periods in a damaged asphalt concrete body, there

exists a mechanism other than relaxation which provides a beneficial

structural change. Assuming that cracking, regardless of the size of

cracks, is the major cause of damage in these asphalt concrete

samples, this advantageous structural alteration is attributed to the

healing mechanism within the asphalt cement.

Since four different rest periods were randomly applied to each

sample until three repetitions for each rest period were introduced,

each repetition induced a different crack length and damage level.

A methodology which can normalize the difference in damage level is

necessary to evaluate the healing which occurs at different rest

periods with different damage levels.

In this study, pseudo energy density and a healing index were

used to represent the healing capacity of a specific binder as a

function of rest time. The pseudo energy density, ~R, in a uniaxial

case, is defined as:

~R J a deR .

The pseudo energy densities before and after a rest period are

illustrated in Figure 32. As shown, only the tension part (positive

stress) is used to calculate the pseudo energy density. From the

observation that the pseudo energy density after the rest period is a

unique material function which can be related to the specified damage

92

"' (/) CL

'-'

(/) (/) (jj

L .w U1

20r-----~-------.------~----~.-----~-------r------T-----~-------,

10 ·-

0

-10 ·-

~

~

Pseudo energy density before rest period

Pseudo energy density after rest period

-20 ° -.004 -.002 0 .Q02

PsQudo Strain

. 004

FIGURE 32 Illustration of pseudo energy densities before and after rest period.

\0 w

94

level, the healing index was used to represent the healing potential

of different binders at different rest times. The healing index, H,

is defined as:

H ~R ~RB ~A - ~

where the pseudo energy density after the rest period, and

t/JR the pseudo energy density before the rest period. B

The healing indices at four rest periods of three mixtures are

plotted in Figure 33. This index successfully normalizes the

difference in the damage level for Fina and Shamrock asphalt, while

the discrepancy becomes a little larger for Witco asphalt.

As a result, the mixture using Witco asphalt shows the lowest

level of healing, while the mixtures with Fina and Shamrock

demonstrate higher levels of healing than for the Witco asphalt. The

chemical nature of the asphalt which influence these results has

been studied in parallel research as a part of this National Science

Foundation project (4).

Regression analysis between the healing index and the duration

of the rest period resulted in the time exponent range of 0.13 -

0.22. This can be compared to the time exponent of the strength

ratio, 0.25, determined by Kim and Wool (23) for polymers. The

higher exponent of Witco asphalt indicates that, even if the initial

healing is not as effective as for the other asphalt, the increase in

healing potential as a function of rest time is greater for Witco

asphalt.

X 01

'U c ..... m c .....

..-< 0 OJ

I

.5~------~~~~~~~~~~----------~------------.------------Strain am=de (in./in.) . 0.009~ 0.00674

W1tco 8 • Fina 8. A

• 4 I Shamrock 0 !____/. : -----. ----------

& ~lt -\ Shamrock AC-20: :=0.251(RP)0.138

• 31- #..,.. ~ · I Fina AC- 20: H=O. 24 7 (RP) 0. 138

• 2

• 1

0 0

f

Q 8

0 ~ 8

~. 0 -~

- I : _ Witco

10 20

§

• • •

AR-4000: H=0.077(RP) 0 ' 217

30 40

Rest Period (min.)

50

FIGURE 33 Healing potential of different binders as a function of the duration of rest period.

"' \.J1

96

CHAPTER. IX

UNIAXIAL TESTING - CONSTITUTIVE MODELING

As discussed in Chapter VI, the internal state variable

formulation resulted in a constitutive equation of the form:

(VI.ll)

In order to find the explicit form of Equation (VI.ll), two types of

tests were performed in a displacement-controlled mode. They are:

a. constant strain rate monotonic loading tests with various

rates and

b. constant strain rate simple loading tests.

For Test Series "b", two levels of strain amplitude were applied;

1.84 x 10- 3 in./in. and 3.687 x 10- 3 in./in.

1. Study of Rate-Dependence

Before the data from Test Series "a" and "b" were used for

constitutive modeling, rate-dependence of the asphalt materials was

studied using the ·correspondence principle. A beam sample was

subjected to simple loading with rest periods (Figure 34) in the

uniaxial testing apparatus. The strain rate was changed after each

rest period. The length of rest periods was fixed at 30 seconds,

and the strain amplitude was 6.13 x 10- 4 in./in. which produced

negligible damage in the testing stage the purpose of which was to

verify the applicability of the correspondence principle (Chapter

VIII, Section 2).

g ~ IVV\M MMA AA6AA N0M !iNJ!1 Time

FIGURE 34 Strain history for the study of rate-dependency.

I.D -.1

Pseudo strains were calculated for the applied strain history

(Figure 34). In order to present the rate-independence of the

stress-pseudo strain relationship clearly, stress versus pseudo

strain curves at the first cycles after each rest period are shown in

Figure 35. Regardless of the rate, stress versus pseudo strain

curves converge onto the same line.

Another study of rate-dependence was performed using constant

strain rate monotonic loading tests with different loading rates.

Five beam samples were loaded in tension at five rates, ranging from

0.0023 in./in./min. to 0.0368 in./in./min. The responses are

plotted in Figure 13. Pseudo strains were calculated based on

Equation (VIII.l) and plotted against stresses as shown in Figure

36. The curves representing different loading rates fall on the same

line until higher levels of stress or strain are induced. Then the

variance in the agreement of stress-pseudo strain curves for the

different loading rates becomes greater.

The two studies discussed above suggest that the correspondence

principle can successfully eliminate the rate-dependency of the

material when the damage is negligible. However, if a significant

level of damage is induced in the sample, additional variables should

be employed in a constitutive equation. In the following section,

the damage parameter developed by Schapery (43), discussed in Chapter

VI, Section 3, will be determined for the strain histories of Test

Series "a" and "b".

2. Determination of Damage Parameter

98

r-. ..... (/) 0... .....,

(/) (/) OJ L

.1-) Ul

20

15

+

~ X X

- + X+

10

5

&&Q&

../.,..A

Q fl +

0

-5

?/. - +>t

as a a +*

-10 +*

-15 -.002 -.001 0 • 001 • 002

Pseudo Strain

FIGURE 35 Stress versus pseudo strain for the first cycles at different strain rates shown in Figure 34.

1.0 1.0

'""" ..... 11)

0... ../

11) 11)

OJ L

.j.) (f)

so~~~--~~~--~~~--~~~--~~~~~~--~~~--r-~~--~~~

40 ·-

30

20 ·-

10 ..

"' d

'"' "'

. 001

-:;.---

. 002 . 003 . 004 . 005

Pseudo Strain

FIGURE 36 Stress versus pseudo strain for different rates (constant strain rate monotonic loading).

....... 0 0

'-

The damage parameter developed by Schapery (43) is a Lebesgue

norm of fR presented in Equation (VI.9), i.e.

(VI. 9)

For the monotonic loading condition, this integral can be integrated

analytically (44) assuming a power law,

(IX.l)

where A and a are positive constants. Substituting Equation (IX.l)

into Equation (VI.9) and integrating yields

[

t ]1/p SP = lfRI -- . pa+l

(IX.2)

Knowing p and a, one can calculate SP for the time of interest from

Equation (IX. 2).

When repetitive loading is applied, Equation (IX.2) is not

valid. Numerical integration can be used to obtain Sp assuming that

d€R/dt is constant within the range of the experimental data points.

Letting dt

lfRip dt +t-tj-1

-r-'Yj-1

1

c

'Yp+1_'Y·-1p+1

p + 1 (IX. 3)

An advantage of Equation (IX.3) over Equation (IX.2) is that the

damage parameter can be calculated without knowing a. However, p

must still be known to calculate SP. There are four ways to obtain

the value of p for a given material. The first method is to use the

101

definition of p shown in Equation (VI.lO). The problem with this

method is that the crack growth law, Equation (VI.7), must be known a

priori. The second method was presented by Schapery (44) in a

graphical way. He performed constant strain rate monotonic loading

tests at different rates and assumed that the constitutive equation

for this loading could be represented by the product of functions of

eR and SP. Then several values of p were selected, and the log a

versus log SP relationship for a set of constant values of eR was

plotted. If the assumption of the product form is reasonable, the

curves should be superposed by translations along the log a axis.

The value of p was selected which provided the best fit after

superposition.

The third method was proposed by Fitzgerald and Vakili (41).

They performed uniaxial relaxation tests and constant strain rate

monotonic loading tests. These tests resulted in relaxation moduli

and stresses from the monotonic testing as power forms of time. The

coefficients and exponents from these two power laws were compared to

determine p.

The fourth method to determine p is to assume that the material

is linearly viscoelastic. If so, the value of p is equal to 2(1 +

1/m) where m is the exponent of creep compliance versus time in the

power law relationship.

The second and the third methods were tried using relaxation

data and constant strain rate monotonic loading test results

·(Figures 25 and 13). The range of the p value in the second method

was from 0.1 to 20. It was difficult to determine a single value of

p because, after translations, the degrees of fit were very close for

102

103

various values of p. The third method results in p values ranging

from 10 to 18 depending on the strain rate.

It was decided in this research to use the linear viscoelastic

expression of p (i.e. the fourth method). Since m obtained from

relaxation testing is 0.661, the p value is approximately 5. This

linear viscoelastic assumption is acceptable because of the linearity

observed in Figures 28, 29, and 35 with negligible damage. That is,

the material itself is essentially linear, but the nonlinearity is

due to the damage mechanism.

In this research, Equation (IX.3) with p = 5 was used to

calculate SP regardless of the loading conditions applied. Typical

shapes of SP versus time for monotonic loading and cyclic loading

tests are presented in Figures 37 and 38. During unloading paths

presented in Figure 38, the damage parameter was kept constant

because, in principle, there is no damage growth during unloading.

3. Constant Strain Rate Monotonic Loading Tests

The implicit form of the constitutive Equation (VI.ll) can be

simplified for monotonic loading as follows:

(IX.4)

R R because e1 - e for monotonic loading. According to Equation (IX.4),

SP is responsible for the stress deviation shown in Figure 36. Since

the superposition technique used to determine p provided a

satisfactory fit after translation, the explicit form of Equation

(IX.4) was a product of functions of fR and SP.

After a detailed study of data, the following function produced

the best fit for all of the rates applied:

0... lfl

.006r-~--~--~--~~--~--~--~-T---r--~~~-T--~--T-~~~--~--~~

. 004 •.

. 002 ..

0o~~--~--~--~----~--._--~~---5~~--~--~--~_.--~--~~~----~10

Time (seconds)

FIGURE 37 Damage parameter versus time for monotonic loading.

,__.. 0 -1'-

0.. U1

.005~~~-.~~~.-~~~~~~,---~~~----~~~--

. 004 ·-

. 003 ·-

. 002

. 001

0o 20 40 60 80 100 120 140

Time (seconds)

FIGURE 38 Damage parameter versus time for constant strain rate simple loading (20 cycles).

....... 0 Vl

(IX.5)

0.995. The parameter I is the stiffness pseudo modulus

(i.e. I u/ER) at the early linear part of a stress-pseudo strain

curve. This stiffness pseudo modulus was necessary to reduce the

effect of sample-to-sample variability on constitutive modeling.

Substituting Sp in Equation (IX.5) with the expression of SP in

Equation (IX.2) yields

a _ ( ER) 1 . a 4 1 t- a . a 6 9 .

Therefore, when damage is small (i.e. time elapsed is small), u- ER.

This is the behavior response of linear viscoelastic materials. This

again verifies the linear behavior shown in Figures 23, 29, and 35

for negligible damage.

4. Constant Strain Rate Simple Loading Tests

In order to model the behavior of the material under cyclic

loading, the role off~ in Equation (VI.ll) should be considered.

R The concept of EL has been introduced in References (41) and (44) in

h f f R/ R t e orm o E. E1 . The same approach was used in this research.

Therefore, Equation (VI.ll) becomes

R This equation must satisfy Equation (IX.S) when EL

form of Equation (IX.6) is assumed to be:

u = F( ) I (ER)l.386(SP)-a.345.

(IX. 6)

ER. Thus, the

(IX. 7)

Comparing Equations (IX.5) and (IX. 7) yields that F(l) 1.31.

106

In order to determine the function F(fR/€~), measured stresses

from constant strain rate 20 cycle simple loading tests were divided

Figure 39, a strong inverse power relationship was observed between

calculated F and (fR/e~), regardless of the strain amplitude applied.

The study of this function produced a form of F(ER/f~) as follows:

F = 0.47 + 0.6553 [ :; rl. 024 when

ER < 0.9

R €L

(IX.8)

when ER

;::: 0.9. R F = 1.2

€L

It is noted that F(l) = 1.2 which is close to 1. 31 in montonic

loading. The constitutive equation is now

a = I F( (IX.9)

where F is given in Equation (IX.8).

However, the predicted stresses of Test Series "b" using

Equation (IX.9) were higher than the measured stresses at small

values of SP. This behavior was observed and predicted theoretically

(44) using a micromechanics model (69). As shown in Figure 40, the

damage coefficient which is a function of SP was proven to approach a

specific value when SP goes to zero. The damage coefficient in

Equation (IX.9), G(SP), is (S )-0.345. p When sp approaches zero, the

damage coefficient goes to infinity and results in a higher value of

predicted stress. In order to obtain a precise form of G(SP),

measured stresses were divided by IF (€R) 1 · 386 and plotted against

SP for Test Series B. The result is plotted in Figure 41. The form

of G(SP) is found from the polynomial regression to be:

107

"" "0 (U 4J 0 ..... :J u ..... 0 u I

X u 0 ro ........

u.

15~~------~~--~--~~---r--~~~~--~--~~------r-----~--~~

10 ·-

5 ·-

+ + +

a+ + + +

l +

0

0~ 't\

+

+

\0 tb

+

0

Strain amplitude Symbol

0.00184 in./in. +

0.003687 in./in. 0

~~~--~~~~~ 0 o+-o

0~_. ______ ~_.--~~------~~--~--~~--~--~~--~~------~~ 0 . 2 . 4 .6 • 8

Pseudo Strain/Max. Pseudo Strain

FIGURE 39 Back-calculated F versus fR/f~ for constant strain rate simple loading.

...... 0 (X)

1.2

~ 0.8 aJ ~ (J

~ '+-< '+-< aJ 0 u aJ 00 t1l e ~ 0.4

0

0 20

From theory (69) with one crack.

From theory (69) with two cracks.

40 60

Damage Parameter

FIGURE 40 Damage coefficient versus damage parameter (after Schapery (44)).

109

,.-... '""Cl (!) w (I)

rl ;:l

11 I I I i 9 f I I i I I i I I j 9 I I i I I i i B i I ' i ' ' I I i I 9 II i i I I I i I • I I I I I i I

10

+

0 '\. ~ G = (S )-0.345

p

9 ·-~ 0

~ + '0

'-+

u ~ 8 ·-u I

.X u (I)

p:)

'-'> 7 ·-

Strain amp1i.tude Symbol

6 ·- 0.003687 in./in. + 0.00184 in./in. 0

5~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

0 • 001 • 002 • 003 • 004 • 005

Sp

FIGURE 41 Back-calculated damage coefficient (G) versus damage parameter for constant strain rate simple loading.

......

...... 0

G- 11.073 - 1019.782(Sp) - 180623.458(Sp) 2 + 40615298.665(SP) 3

when SP < 0.0033, and

G = (Sp)- 0 · 345 when SP ~ 0.0033 (IX.lO)

Finally, the uniaxial constitutive equation of the mixture of granite

fines and Witco AR-4000 asphalt is

a = I F( (IX.ll)

where F and G are given in ~quations (IX.8) and (IX.lO),

respectively.

5. Verification of Constitutive Equation (IX.ll)

Before additional verification testing was performed, the

stress-strain behavior of the tests used in developing Equation

(IX.11) (Test series A and B) was checked by Equation (IX.ll). They

are presented in Figures 42-44. The overall agreement between the

experimental stresses and the predicted stresses was satisfactory.

For the monotonic loading tests (Figure 42), however, as the

strain rate becomes lower, the agreement between the predicted

stresses and the experimentally measured stresses at small strain

values is not as good. This discrepancy comes from the mathematical

form of the damage coefficient, G(Sp). When the inverse power form

of G was used as in Equation (IX.S), the agreement at the lower

strain rates was better than the case using the expression for G in

Equation (IX.10). The observation probably demonstrates the rate-

dependency of the damage coefficient.

111

,.... ...... (/) a.. ......;

(/) (/) Cll L +) Ul

so~~T-~~~~~~~~~~~~~~~~~~~--~~T-~~~--~~~

40

30 ·-

20

++++ Experimental

Predicted

+ +

+

0.0368 in./in./min.

+ + + + + + + + + + + + + + + +

~£ = 0.0184 in./in./min.

'--.....£ 0.0092 in./in./min.

±++d..&.+ • t dr I I I • - + 1 1 1 I I I I ~ I I~ 4 ~~~~~~J ~+±++±++++++ . _ ...... ~.:Li;ii;{~in./in./min. £ = 0.0023 in./in./min .

• 001 .• 002 • 003 • 004 • 005

Strain (in./in.)

FIGURE 42 Stress-strain curves for constant strain rate monotonic loading.

• 006

,..... ,..... N

,..., ...... (J)

0.. '-'

(J)

Ul OJ L +) lJ)

20,_~~~~-.~~~~~.-~~~~-&~~~~~-

***** Experimental

Predicted 15

0

-10

-15~----~--~~--~--~~--~--~~------~----~--._----~--~--~~

0 . 0005 . 00 1 . 00 15

Strain (in. /i11.)

FIGURE 43 Stress-strain curves for a coustaut strain rate simple loading test (strain amplitude - 0.00184 in./in.).

. 002

...... ...... w

....... ..... (}) 0 . ......,

(}) (}) ClJ l. .. ,

lf}

20~~~~~~~~~~~~~~~~~~--~---

***** Experimental

Predicted 15

10 ·-

5

0

-5

-10

-15L-~--~--~~--~--~~--~--~~--~--~~~~--~--~~--------~ 0 . 001 . 002 . 003 . 004

Strain (in. lin.)

FIGURE 44 Stress-strain curves for a constant strain rate simple loading test (strain amplitude- 0.00369 in./in.).

1-' 1-' ..,..

For verification purposes, three types of uniaxial testing were

performed; constant strain rate simple loading with various lengths

of rest periods (Figure 24), multi-level loading with 30 second rest

periods (Figure 45), and multi-level loading with various lengths of

rest periods (Figure 46). The strain amplitude of the simple loading

test was 2.76 x 10- 3 in./in. The loading history of the simple

loading test was the same as that described for the healing test in

Chapter VIII. The lengths of rest periods applied in simple loading

tests were 5, 10, 20, and 40 minutes. In simple loading, three sets

of four rest periods of 5, 10, 20, and 40 minutes duration were

applied. The sequence of rest periods in each set was random. The

number of initial simple loading cycles was 20. Five cycles of

simple loading were applied after each rest period.

Constitutive Equation (IX.ll) was used to predict stresses from

the simple loading test history (Figure 24). The stress-strain

curves of three representative groups of loading were presented in

Figures 47-49: stress-strain curves for the initial 20 cycles in

Figure 47, stress-strain curves after the first introduction of 5-

minute rest period in Figure 48, and stress-strain curves after the

third introduction of 40-minute rest period in Figure 49. The

stress-strain curves for the rest of the simple loading test were

presented in Appendix C. In general, the predictions made by the

constitutive Equation (IX.ll) are very satisfactory. Constitutive

equation (IX.ll) was able to predict the decrease in stress drop at a

selected strain level as the number of cycle increases. Also,

different amounts of the increase in stress after various durations

of rest periods were successfully predicted.

115

,... . c .....

" . r: ..... '-../

r: ..... 0 L ~.l (f)

• 003 ' I

Group 7

-. 002

Grou 1 4 Group 8

Group 2 Grou) 5 Group 9

• 001

~roup

I Group 3 Group 6

. oo 100 200 300 400 500 600

Time (seconds)

FIGURE 45 Strain history of a multi-level loading verification test with 30-second rest

periods. ...... ...... 0\

,-..

c ...... ........ . c .....

'--"

c ..... c L .v Ul

.003--~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

. 0021-

( ) indicates the number of cycles in each Group.

Group 4 ( 10)

I Group 2 (5) I

. 001

~~~Group H ( 5)

Group II 6(5)

Gr~u~~ Gro~p I Group I ~II 5

(5) < 5) I (5)

00 1000

Time (seconds)

Group 9 (5)

I Group 10 (3)

2000 3000

FIGURE 46 Strain history of a multi-level loading verification test with random durations of

rest periods. ...... ...... -...!

'"' ...... (/) ()_

'-"

(/) 1.1)

OJ L

+-> (.f)

20~~~~~~~~--~~~~~~~~--~~~~~~~~----~~~--~~

***** 15

Experimental

Predicted

-15~~~~--~~~--~~~~--~~----~~~~~~~----~._~~--_.~

0 . 00 I . 002 . 003

Strain (in./in.)

FIGURE 47 Stress-strain curves of initial 20 cycles for the constant strain rate simple loading verification test shown in Figure 24 (strain amplitude= 0.00276 in./in.).

...... ,...... CXl

,.... (/) 0..

"-./

(/) (/)

OJ L -iJ Ul

20r-~~~~~~~--~~~~~~~--~~--~~~~~--~~--~--~---,

. 15 ·-

10 ·-

***** Experimental

Predicted

-lOr-- *

-15~--~~~~~~~~--._~~--~~~--~--~~--~~~--~--~~--~

0 . 00 I • 002 • 003

Strain (in. /in.)

FIGURE 48 Stress-strain curves after the 1st 5-minute rest period of the constant strain rate

simple loading verification test.

......

...... '-0

,.... ..... Ill 0...

Ill Ill Ql L .jJ Ul

20~~~~~~~~--~~~~~~~~--~~~~~~~~--~~~~--~~

15 ·-

10 ·-

-10 ·-

***** Experimental

Predicted

*

. 001 -150~~~----------~~~--------~--~~----~--~~__J

. 003 . 002

Strain (in. /in.)

FIGURE 49 Stress-strain curves after the 3rd 40-minute rest period of the constant strain rate simple loading verification test.

,_. N 0

The minimum and maximum strain amplitudes used in multi-level

loading tests were 6.13 x 10- 4 in./in. and 2.76 x 10- 3 in./in.,

respectively. The sequence of strain amplitudes was random in the

multi-level loading verification test with 30-second rest periods.

In the multi-level loading verification test with various lengths of·

rest periods, both the sequences of strain amplitudes and rest

periods were random. The strain histories in these multi-level

loading verification tests were composed of 9 and 10 groups of strain

applications as shown in Figures 45 and 46, respectively. The strain

amplitude used for each group was held constant.

Figures 50 through 58 show the accuracy of predicted stresses

from Equation (IX.ll) under multi-level loading with 30-second rest

periods. It has been observed that the increase in damage due to

repetitive strain applications becomes smaller when the amplitude of

the preceding strain application is larger than that of the current

one. In other words, a light truck following a heavy truck will

induce less damage than the same light truck following a car. This

behavior can be observed by comparing Figures 50 and 52, Figures 51

and 54, or Fi.gures 53 and 57. The constitutive equation (IX.ll)

predicted this behavior very successfully. For example, in Figures

53 and 57, the stress drop from the second to the fifth cycle was

much larger in Figure 53 than that in Figure 57.

The multi-level loading test with various lengths of rest

periods represents the most randomized, realistic loading history

used in this research (Figure 46). The measured and predicted

stresses were plotted against strain in Figures 59-68. The

predictions made by Equation (IX.ll) are satisfactory. The stress

121

,..... ·.-< (/) ()_

'J

(/) (/)

OJ L

.j.) lf)

15~~~~~--~~~~~~~~~,-~~~~~~

***** Experimental

Predicted

-10

-15~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

0 . 000 1 . 0002 . 0003 . 0004 • 0005 . 0006 . 0007

Strain Cin. /in.)

FIGURE 50 Stress-strain curves of Group 1 loading of the multi-level loading verification test shown in Figure 45.

,_. N N

"' 'M ({) 0... ......,

({) (/)

OJ L +) U)

15~--~~~~~~~~~~~~~~~~~~~~--~--~--~~~~~~

10 ·-

5 ·-

0

-5

***** Experimental

Predicted

~ -==--11 N

-10 * *

*

-15~~~~----~~~~~--~~~~~~--~~~~~--~~~~--~~--J

0 . 0005 . 001 . 0015

Strain (in. /in.)


...... N w

,...... ..... (/) Q_

'-./

(/) (/) QJ L +l U)

15~~~~~~~~~,-~~~~r-~~~~~~--~~~~~~~~~~~

10 ·-

***** Experimental

Predicted

-5~ • ~ -10 ·-

*

-15~~~~~~~~~~~~~~~~~~~~--~~~~~~~~~~~~

0 . 000 1 . 0002 • 0003 . 0004 • 0005 • 0006 . 0007

Strain (in. /in.)


,_.. N P-

,... ...... (f)

0.... v

(f) (f) 01 L +> (f)

15~~--~~--~--~~--~--~~--~~~~--~~--~--~~--~--~~

10--

5·-

0

-5

***** Experimental

Predicted

* ---~

-1 o L •-===--------. -

-150 .001

Strain Cin. /in.)

. 002


,_. N U1

,...... ·.-4

(f)

0... v

(f) (f) (])

L +> lf)

15~~~~~~~~~~~~~~.-~~~~~~~

***** Experimental

Predicted

10 ·--"- u~~~~~

0

-10

-150 .0005 . 001 . 0015

Strain (in. /in.)

FIGURE 54 Stress-strain curves of Group 5 loading of the multi-level loading verification test

shown in Figure 45.

...... N 0\

" ....... (/) ()_

'-./

(/) (/)

OJ L

.jJ lf)

15~--~--------~~~~--~~~~~~~~~~~--~--------~~

10 ·-

-10 ·-

***** Experimental

Predicted

-15~~~~~~~~~~~~~~~~~~~~~~~~~~~--~~~~~

0 . 000 1 . 0002 . 0003 . 0004 . 0005 . 0005 • 0007

Strain (in. /in.)


,_. N -...)

r-.. ...... Ul 0...

v

Ul (/)

OJ L +)

U1

15 ___,

10 ·-

5 ..

0

-5

***** Experimental

Predicted

-lOl • • 7 ""--~-

-15~~~~~~~~~~~~~~~~~~--~~~~~~~~~--~~~~

0 .001 .002 .003

Strain (in./in.)


...... N 00

/"'. ..... (/)

0... v

(/) (/) OJ L

+1 lJ)

15~~--~~--~--~~--~--r-~--~--~~--~~--~--~----~----~

-10·--

***** Experimental

Predicted

-15~~----~~~--~~--~--~~--~--~----~~------~~--~--~--J 0 . 001 • 002

Strain (in./in.)


...... N 1.0

,...., ...... ({) o._

'-/

({) ({) QJ L +) lf)

15~--~~~~~~~~-r~~~~~~~~~~~~--~~~~~~~~

10 ·-

-5

-10

***** Experimental

Predicted

-15L-~~--~~--~~--~~~~~~~~~~~~~--~~~--~~--~~

0 .0005 .001 . 0015

Strain Cin. /in.)


,_... w 0

,..... ..... Ul a..

"-./

Ul Ul QJ L +) (J)

15~~~~~~~~~~~~~~~--~~~~--~~~~~~~~~~~_,

***** Experimental

Predicted 10 ·-

*

' I

-10 ·-

-1s~~~_._.~~~~~~._~~~--~~~~~~~._._~--~~~~~--J

0 .0001 .0002 .0003 .0004 .0005 . 0006 . 0007

Strain (in. /in.)


...... w ......

,--.. ..... (}) (L

'-./

(}) (}) 01 L

-+) (J)

15~--~~~~~~~~~~~--~--~~~~---

***** Experimental

Predicted

10

5 ·-

0

-5

-10 • t

-l 5oL-~~~~~~_.~~-.0-0~0-5~~~~~~~~---.~00-1~~~_.--~--~._~_~0015

Strain (in./in.)


....... w N

"' ..... Ul (L .._,.

Ul Ul Cll L

.j..) (f)

15~~~~~~~~-,~~~--~~~~~~~~

***** Experimental

Predicted 10 •.

* * *

4

-10 ·-

-15L-~~_.~~~~~~~~~~~~~~~~~~~~._~--~~----_._J

0 .0001 .0002 .0003 .0004 .0005 . 0006 . 0007

Strain (in./in.)


,_... w w

,...... ..... (/) 0... ...._,

(/) (/) 01 L +> Ul

15~~--~~~~--~~--~--~~--~--~~--~~~~--~~--~--~~

10 ·-

5 ·-

0

-5~

-10

***** Experimental

Predicted

;----.~

• ~

-t5L-~--~~--~--~~--~~--~~--~--~~--~~--~--~~--~~ 0 • 001 • 002

Strain (in. /in.)


....... w .j::--

"'"' ...... (/) CL ......,.

(/) (/)

01 L

+> U')

15~~~~~~~~~~~~~~~~~-T~~~~~~~~~~~~~-T_,

10 ·-

-10--

***** Experimental

Predicted

'

* * *

-15~--~~~----~~~~----~~~~~----~~~~----~~~--~ 0 .0001 .0002 .0003 .0004 .0005 . 0006 • 0007

Strain (in. /in.>


,...... w Vl

,..... ..... (J) 0..

'--'

(J) (J) (]I L +I (f)

15~~~~~~~~~~~~~~~~~~~-T~~~~~~~~~~~~~

10 ·-

-5

-10 ·-

***** Experimental

Predicted

* * ..

*

-150 . 0005

* *

I

. 001 • 0015

Strain (in./in.)


,..... (.;..)

0\

"' ..... Ul 0..

-.../

Ul Ul ClJ L .u

(f)

15~~~~~~~~~--~--~--~~~~~~~~--~~--~~~~~~-,

10 ·-

5 ·-

0

-5

***** Experimental

Predicted

-lOL •---=

-150 .001

*

. 002 . 003

Strain (in./in.)


..... w -...J

,.... ...... Ul n. ...._,

Ul (J) ()j L +J (f)

15~~--~--~~--~--~~--~--~~--~--~~~~--~--~~--~--~_,

***** Experimental

Predicted 10

* •

-10

-15L-~--~~--~~~~--~~--~~--~--~~--~----~--~----~~

• 001 • 002 0

Strain (in./in.)


....... w 00

,.... ..... (/) 0... ....., (/) (/) 01 L +> (J)

15~~~--~~~--~~--~--~~~--~~--~~~~~~~----~~~~_,

10 ·-

***** Experimental

Predicted

-5~- •• ~ * -10 ·-

-150 • 0005

Strain (in./in.)

. 001 • 0015


....... w \0

" ..... (/)

a.. .....,

(II (II Ql L +) Ul

15~~--~-T~~~~~~~~~~~~--~~~~~~~~~~~~~---,

10

0

-10 ·-

***** Experimental

Predicted

•

*

-15~~~~~~~~~~~~~~~~~~~_.~~~~~~._~~~~--~

0 .0001 .0002 .0003 .0004 .0005 . 0006 • 0007

Strain (in./in.)


....... p. 0

increases after various lengths of rest periods were predicted

successfully, as was the effect of the sequence of the load

applications of varying magnitudes.

The strain histories for multi-level loading tests with a 30-

second rest period and with random durations of rest period (Figures

45 and 46) were designed to be able to investigate the effect of the

healing mechanism on constitutive modeling. That is, in Figures 45

and 46, the strain histories until the first cycle of Group 4 loading

are identical except for the lengths of rest periods between loading

groups. Then, for the multi-level loading test with random durations

of rest period, ten cycles of loading followed by 600-second rest

period were applied, while five cycles of loading followed by 30-

second rest period were applied in the multi-level loading test with

a constant-length rest period. More cycles will, of course, induce

more damage, and longer rest periods after greater damage will

produce greater effects of healing.

With these characteristics of testing histories in mind, Figures

59-68 are compared with Figures 50-58. From the comparison of peak

stresses during the first cycle in Figures 50 and 59, Figures 51 and

60, Figures 52 and 61, and Figures 53 and 62, it is apparent that

there is an additional stress increase after the longer rest period,

which constitutive Equation (IX.ll) can not predict. This

observation is highlighted by the comparison of Figures 54 and 63.

The discrepancy between the experimental and the predicted stresses

at the strain level of 0.000613 in./in. (which is the strain

amplitude in Figure 63) is greater in Figure 63 than in Figure 54.

Even though the discrepancy after the short rest period is almost

141

negligible, the accuracy of constitutive modeling can be slightly

improved by including another internal state variable which

represents the beneficial effect from the healing mechanism.

Especially, when the effect of healing becomes significant, that is,

at an elevated temperature or through the modification of asphalt

cement, an appropriate constitutive model should be able to predict

the beneficial effect of healing as well as relaxation and

detrimental damage accumulation.

142

CHAPTER X

CONCLUSIONS AND RECOMMENDATIONS

1. Conclusions

The influence of rest periods on laboratory fatigue testing has

been documented by a large number of authors around the world. The

overwhelming consensus is that tbe rest periods enhance fatigue life

due to healing and relaxation mechanisms. Understanding such

mechanisms will be an overpoweringly important contribution to the

understanding of fatigue performance and may allow us to select

asphalts with greater healing potential.

The work of polymer researchers has provided insight to the

polymer healing phenomenon which may be applicable to asphalt. The

visual study of fracture surfaces after various lengths of rest

periods using an SEM has proved the existence of chemical healing in

asphalt concrete. The literature review of healing in polymeric

materials and this visual study suggests that the appropriate healing

model must represent both initial surface penetration and the

development of structural bonding.

A procedure was developed in this study by which the chemical

healing can be quantified by separating out the concomitantly

occurring viscoelastic effect of relaxation. The process

demonstrates that a quantifiable chemical healing does indeed occur

in asphalt concrete. The amount of this healing varies among the

asphalt cements tested.

143

In addition to the evaluation of chemical healing, accumulated

damage under repetitive loading was modeled using the correspondence

principle of nonlinear viscoelastic media and damage mechanics. A

uniaxial constitutive equation was developed from tests on mixtures

of asphalt concrete composed of crushed granite fines and asphalt

cement. In a verification stage, a random sequence of multi-level

loading with random duration of rest periods was applied. The

constitutive equation satisfactorily predicted the effect of the

sequence of the load applications of varying magnitudes as well as

the beneficial effect of rest periods. It should be possible to

improve the accuracy of predicting stresses after long rest periods

by adding an internal state variable which accounts for the healing

mechanism.

A constitutive model of this nature is an indispensible link

required in the fatigue performance prediction of asphalt concrete

pavements. This type of model will allow the researcher or

practicing engineer to predict fracture fatigue potential while

realistically accounting for the effects of the loading spectra.

2. Recommendations

Even though it has been proved that chemical healing does indeed

occur, the microstructural, governing mechanism has not been

identified. The use of a replication procedure with an SEM limited

the magnitude of magnification, and therefore, microstructural

information. Probably, a chemical study of the structure of asphalt

cement should accompany the visual study of healing in order to

144

identify the nature of the chemical healing of asphalt cement. This

is addressed to some extent in the companion report (4).

It has been found (16) that some polymeric additives can enhance

the fatigue performance of asphalt concrete pavement significantly.

However, this beneficial effect of additives can result either from

the change of viscoelastic properties or improved healing

characteristics. The procedure presented in this research will

separate the effects of these two mechanisms and enable the evaluator

to suggest which additive can be most beneficial to the fatigue

performance of asphalt concrete pavement over a range of conditions.

The uniaxial constitutive equation developed in this research

showed great potential as a means of predicting the stress-strain

behavior of asphalt concrete under realistic cyclic loading. In

order for this model to be used in a fatigue failure criterion, a~

appropriate failure criterion should be identified. This can be done

using a critical stress or strain concept or possibly identifying the

critical value of the damage parameter.

More research is required to extend the one-dimensional

constitutive law to a multiaxial case. Also, environmental effects,

such as temperature and humidity should be accounted for in the

equation. There is a need for a parameter representing the

beneficial effect of healing to be studied at elevated temperatures.

145

REFERENCES

1. J. A. Epps and C. L. Monismith. Fatigue of Asphalt Concrete

Mixtures - Summary of Existing Information. STP 508, ASTM,

Philadelphia, Pennsylvania, 1972.

2. F. Finn, C. Saraf, R. Kulkarni, K. Nair, W. Smith, and A.

Abdullah. The Use of Prediction Subsystems for the Design of

Pavement Structures. Proc., Fourth International Conference on

Structural Design of Asphalt Pavements, 1977, pp. 3-38.

3. R. A. Schapery. Correspondence Principles and a Generalized J

Integral for Large Deformation and Fracture Analysis of

Viscoelastic Media. International Journal of Fracture, Vol. 25,

1984, pp. 195-223.

4. F. C. Benson. Correlation among Asphalt Concrete Mechanical

Healing, Asphalt Cement Dispersion, FT-IR Chemical Functional

Groups and NMR Chemical Structure. Ph.D. Dissertation, Texas

A&M University, College Station, Texas, 1988.

5. C. L. Monismith, K. E. Secor, and E. W. Blackmer. Asphalt

Mixture Behavior in Repeated Flexure. Proc., Association of

Asphalt Paving Technologists, Vol. 30, 1961, pp. 188-215.

6. J. A. Deacon and C. L. Monismith. Laboratory Flexural-Fatigue

Testing of Asphalt-Concrete with Emphasis on Compound-Loading

Tests. In Highway Research Record 158, HRB, 1967, pp. 1-32.

7. K. D. Raithby and A. B. Sterling. The Effect of Rest Periods on

the Fatigue Performance of a Hot-Rolled Asphalt under Reversed

Axial Loading, Proc., Association of Asphalt Paving

Technologists, Vol. 39, 1970, pp. 134-147.

146

8. L. Francken. Fatigue Performance of a Bituminous Road Mix under

Realistic Test Conditions. In Transportation Research Record

712, TRB, 1979, pp. 30-37.

9. J. McElvaney and P. S. Pell. Fatigue Damage of Asphalt - Effect

of Rest Periods. Highways and Road Construction, Vol. 41, No.

1766, October 1973, pp. 16-20.

10. J. Verstraeten. Aspects Divers de la Fatigue des Melanges

Bitumineux. Report No. 170/Jv/(1976), Centre de Recherche

Routieres, Bruxelles, 1976.

11. T. H. Doan. Les Etudes de Fatigue des Enrobes Bitumineux au

LCPC. Bulletin de Liaison des Laboratoires des Ponts et

Chaussees, numero special V. Bitumes et Enrobes Bitumineux.

12. W. van Dijk and W. Visser. The Energy Approach to Fatigue for

Pavement Design. Proc., Association of Asphalt Paving

Technologists, Vol. 46, 1977, pp. 1-40.

13. P. Uge, A. Gravois, and J. N. Lemaire. Le Comportement en

Fatigue des Bitumineux: Influence du Liant. Revue Generale des

Routes et des Aerodromes, No. 521, June 1976.

14. F. P. Bonnaure, A. H. J. J. Huibers, and A. Boonders. A

Laboratory Investigation of the Influence of Rest Periods on the

Fatigue Characteristics of Bituminous Mixes. Proc., Association

of Asphalt Paving Technologists, Vol. 51, 1982, pp. 104-128.

15. A. H. A1-Balbissi. A Comparative Analysis of the Fracture and

Fatigue Properties of Asphalt Concrete and Sulph1ex. Ph.D.

Dissertation, Texas A&M University, College Station, Texas,

1983.

147

16. J. W. Button, D. N. Little, Y. Kim, and J. Ahmed. Mechanistic

Evaluation of Selected Asphalt Additives. Proc., Association of

Asphalt Paving Technologists, Vol. 56, 1987, pp. 62-90.

17. K. H. Tseng and R. L. Lytton. Documentation of Flexible

Pavement Performance Analysis Program. Final Report for Florida

DOT, Texas Transportation Institute, College Station, Texas,

1988.

18. P. Bazin and J. B. Saunier. Deformability, Fatigue and Healing

Properties of Asphalt Mixes. Proc., Second International

Conference on the Structural Design of Asphalt Pavements, Ann

Arbor, Michigan, 1967, pp. 553-569.

19. S. Prager and M. Tirrell. The Healing Prcess at Polymer-Polymer

Interfaces. Journal of Chemical Physics, Vol. 75, No. 10,

November 1981, pp. 5194-5198.

20. K. Jud, H. H. Kausch, and J. G. Williams. Fracture Mechanics

Studies of Crack Healing and Welding of Polymers. Journal of

Materials Science, Vol. 16, 1981, pp. 204-210.

21. K. Jud and H. H Kausch. Load Transfer through Chain Molecules

after Interpenetration at Interfaces. Polymer Bulletin 1, 1979,

pp. 697-707.

22. R. P. Wool and K. M. O'Connor. A Theory of Crack Healing in

Polymers. Journal of Applied Physics, Vol. 52, No. 10, 1981,

pp. 5953-5963.

23. Y. Kim and R. P. Wool. A Theory of Healing at a Polymer-Polymer

Interface. Macromolecules, Vol. 16, 1983, pp. 1115-1120.

24. H. Krupp. Advances in Colloid and Interface Science. Vol. 1,

No. 2, Elsevier, Amsterdam, 1976, p. 111.

148

25. B. J. Briscoe. Some Aspects of the Autoadhesion of Elastomers.

In: Polymer Surfaces, D. T. Clark, and W. J. Feast (eds.), Wiley

Interscience Pub., New York, 1978, pp. 25-46.

26. R. P. Wool. Crack Healing in Semicrystalline Polymers, Block

Copolymers and Filled Elastomers. In: Adhesion and Adsorption

of Polymers, Polymer Science and Technology, Vol. 12A, Plenum

Press, New York, 1979, pp. 341-362.

27. K. DeZeeuw and H. Potente. The Determination of Welding

Parameters in the Hot Plate Welding of Large-Diameter Pipes of

Rigid Polyethylene. Society of Plastics Engineers, Technical

Paper, Vol. 23, 1977, pp. 55-80.

28. C. B. Bucknall, I. C. Drinkwater, and G. R. Smith. Hot Plate

Welding of Plastics: Factors Affecting Weld Strength. Polymer

Engineering and Science, Vol. 20, No. 6, April 1980, pp. 432-

440.

29. P. G. de Gennes. Reptation of a Polymer Chain .in the Presence

of Fixed Obstacles. The Journal of Chemical Physics, Vol. 55,

No. 2, July 1971, pp. 572-579.

30. P. G. de Geenes. Scaling Concepts in Polymer Physics. Cornell

University Press. Ithaca and London, 1979.

31. R. P. Wool and K. M. O'Connor. Time Dependence of Crack

Healing. Journal of Polymer Science - Polymer Letters Edition,

Vol. 20, No. 1, 1982, .pp. 7-16.

32. K. M. O'Connor and R. P. Wool. Optical Studies of Void

Formation and Healing in Styrene-Isoprene-Styrene Block

Copolymers. Journal of Applied Physics, Vol. 51, No. 10,

October 1980, pp. 5075-5079.

149

33. J. C. Petersen. Chemical Composition of Asphalt as Related to

Asphalt Durability: State of the Art. In Transportation

Research Record 999, TRB, 1984, pp. 13-30.

34. R. N. Traxler. Relation Between Hardening and Composition of

Asphalt. Preprints, Division of Petroleum Chemistry, American

Chemistry Society, Vol. 5, No. 4, 1960, pp. A71-A77.

35. E. Ensley, H. Plancher, R. Robertson, and J. Petersen. Asphalt

- We Hardly Know You. Journal of Chemical Education, Vol. 55,

No. 10, October 1978, pp. 656-658.

36. E. A. Thompson. Chemical Aspects of Asphalt Pavement

Rejuvenation. Proc., Twenty Sixth Annual Conference of Canadian

Technical Asphalt Association, Vol. 26, November 1981, pp. 84-

96.

37. Highway Research Board. The AASHO Road Test; Report 5.

Pavement Research. Special Report 61E, 1962.

38. W. van Dijk. Practical Fatigue Characterization of Bituminous

Mixes. Proc., Association of Asphalt Paving Technologists, Vol.

44, 1975, pp. 38-74.

39. J. McElvaney and P. S. Pell. Fatigue Damage of Asphalt under

Compound-Loading. Transportation Engineering Journal, ASCE, Vol.

100, No. TE3, 1974, pp. 701-718.

40. M. A. Miner. Cumulative Damage in Fatigue. Journal of Applied

Mechanics, Transactions ASME, Vol. 66, 1945, pp. Al59-Al64.

41. J.E. Fitzgerald and J. Vakili. Nonlinear Characterization of

Sand-Asphalt Concrete by Means of Permanent Memory Norms.

Experimental Mechanics, Vol. 13, No. 12, December 1973, pp. 504-

510.

150

42. M. Perl, J. Uzan, and A. Sides. Visco-Elasto-Plastic

Constitutive Law for a Bituminous Mixture under Repeated

Loading. In Transportation Research Record 911, TRB, 1984,

pp.20-27.

43. R. A. Schapery. On Viscoelastic Deformation and Failure

Behavior of Composite Materials with Distributed Flaws.

Advances in Aerospace Structures and Materials, AD-Vol. 01, S.

S. Wang and W. J. Renton (eds.), ASME, 1981, pp. 5-20.

44. R. A. Schapery. Models for Damage Growth and Fracture in

Nonlinear Viscoelastic Particulate Composites. Proc., Ninth

U.S. National Congress of Applied Mechanics, ASME, 1982, pp.

237-245.

45. D. C. Stouffer, K. Krempl, and J. E. Fitzgerald (Steering

Committee). Workshop on Continuum Mechanics Approach to Damage

and Life Predition. National Science Foundation, May 1980.

46. H. Kraus. Creep Analysis. John Wiley and Sons, 1980, pp. 104-

123.

47. R. W. Hertzberg and J. A. Manson. Fatigue of Engineering

Plastics, Academic Press, 1980.

48. R. J. Farris. The Stress-Strain Behavior of Mechanically

Degradable Polymers. Polymer Networks: Structural and

Mechanical Properties. A. J. Chompff and S. Newman (eds.),

Plenum Press, 1971, pp. 341-394.

49. B. D. Agarwal and L. J. Broutman. Analysis and Performance of

Fiber Composites, John Wiley & Sons, 1980.

151

SO. F. A. Leckie and D. R. Hayhurst. Creep Rupture of Structures.

Proc., Royal Society of London, Series A, Vol. 340, 1974, pp.

323-347.

51. J. K. Mitchell. Fundamentals of Soil Behavior, John Wiley &

Sons, 1976.

52. R. A. Schapery and M. Riggins. Development of Cyclic Nonlinear

Viscoelastic Constitutive Equations for marine Sediment.

International Symposium on Numerical Models in Geomechanics,

Zurich, September 1982, pp. 172-182.

53. R. A. Schapery. Deformation and Fracture Characterization of

Inelastic Composite Materials Using Potentials. Polymer

Engineering and Science, January 1987, Vol. 28, No. 1, pp. 63-

76.

54. R. A. Schapery. A Theory of Mechanical Behavior of Elastic

Media with Growing Damage and Other Changes in Structure.

Report No. MM 5762-88-1, March 1988, Texas A&M University,

College Station, Texas.

55.

56.

Constitutive Modeling for Nontraditional

Materials. The Applied Mechanics Division, ASME, V. K. Stokes

and D. Krajcinovic (eds.), AMD-Vol. 85, December 1987.

Mechanics of Damage and Fatigue. International

Union of Theoretical and Applied Mechanics, Symposium on

Mechanics of Damage and Fatigue. S. R. Bodner and Z. Hashin

(eds.), Haifa and Tel Aviv, Israel, July 1985, Engineering

Fracture Mechanics, Vol. 25.

152

57. Damage Mechanics in Composites. Presented at the

Winter Annual Meeting of the American Society of Mechanical

Engineers, AD-Vol. 12, A. S. D. Wang and G. K. Haritos (eds.).

Boston, Massachusetts, December 1987.

58. J. Mazars. A Description of Micro- and Macroscale Damage of

Concrete Strctures. Engineering Fracture Mechanics, Vol. 25,

Nos. 5/6, 1986, pp. 729-737.

59. D. Krajcinovic and D. Fanella. A Micromechanical Damage Model

for Concrete. Engineering Fracture Mechanics, Vol. 25, Nos.

5/6, 1986, pp. 585-596.

60. Z. P. Bazant. Plastic Fracture for Damage in Geo-Materials

Including Concrete. In Workshop on a Continum Mechanics

Approach to Damage and Life Prediction. National Science

Foundation, May 1980, pp. 110-118.

61. W. Wawersik. Direct and Indirect Observation of the Evolution

of Damage in Some Rocks. In Workshop on a Continum Mechanics

Approach to Damage and Life Predition. National Science

Foundation, May 1980.

62. R. P. Wool. Material Damage in Polymers. In Workshop on a

Continum Mechanics Approach to Damage and Life Prediction.

National Science Foundation, May 1980, pp. 28-35.

63. J. J. Rose. A Replication Technique for Scanning Electron

Microscopy: Applications for Anthropologists. American Journal

of Physical Anthropology, Vol. 62, 1983, pp. 255-261.

64. R. M. Christensen. Theory of Viscoelasticity - An Introduction.

2nd Edition, Academic Press, 1982.

153

65. C. L. Monismith, R. L. Alexander and K. E. Secor. Rheologic

Behavior of Asphalt Concrete. Proc., Association of Asphalt

Paving Technologists, Vol. 35, 1966, pp. 400-450.

66. R. A. Schapery. A Theory of Crack Initiation and Growth in

Viscoelastic Media. International Journal of Fracture, Vol. 11,

August 1975, Part I. Theoretical Development, pp. 141-149;

Part II. Approximate Methods of Analysis, pp. 369-388; Part

III. Analysis of Continuous Growth, pp. 549-562.

67. R. A. Schapery. Time-Dependent Fracture: Continum Aspects of

Crack Growth. In: Encyclopedia of Materials Science and

Engineering, M. B. Bever (ed.), Pergamon Press Ltd., 1986, pp.

5043-5053.

68. D. Broek. Elementary Engineering Fracture Mechanics. Third

Revised Edition, Martinus Nijhoff Publishers, 1982.

69. R. A. Schapery. A Micromechanical Model for Nonlinear

Viscoelastic Behavior of Particle-Reinforced Rubber with

Distributed Damage. Engineering Fracture Mechanics, Vol. 25,

Nos. 5/6, 1986, pp. 845-867.

70. J. D. Eshelby. Calculation of Energy Release Rate. In:

Prospects of Fracture Mechanics, Sih, Van Elst, and Broek

(eds.), Noordhoff, 1974, pp. 69-84.

71. J. R. Rice. A Path Independent Integral and the Approximate

Analysis of Strain Concentration by Notches and Cracks. Journal

of Applied Mechanics, Vol. 35, Series E, No. 2, June 1968, pp.

379-386.

154

155

APPENDIX A

DEVELOPMENT OF PSEUDO QUANTITIES

APPENDIX A

DKVKLOPKENT OF PSEUDO QUANTITIES

For a linear isotropic material, the inversion of Hooke's law

gives the following tensor equation:

where E

1

E

Young's modulus,

v =Poisson's ratio, and

o1 j Kronecker delta.

In the following derivation, only the axial strain, f 11 , and the

shear strain, f 12 , are considered. Other strains can be worked in

the same manner. For a linear isotropic material.

Let S = 1

E , then

1

E

1 + v

E

For general stress history,

S(t)o11 (t) - S(t)v(t)[o22 (t) + o33 (t)] and

S(t)[l + v(t)]o12 (t) .

/

Now, for a linear viscoelastic, isotropic, nonaging material, f 11 and

f 12 can be determined from:

Jt ao11 Jt a

f 11 (t) = 0

S(t-r) ar dr -0

S(t-r) v(t-r) ar (o22 +o33 ) dr (A.l)

156

and Jt aa12 elz(t)-

0

S(t-r) [1 + v(t-r)] dr. ar

(A.2)

In order to identify S(t) and v(t), consider a uniaxial tensile

creep test where a 11 = a 0 H(t) and a 22 = a 33 = a 12

The creep compliance, D(t), is defined as:

From Equation (A.l), e11 (t) = S(t) a 0 • Hence,

D(t) = S(t) .

Also, Poisson's ratio for the creep test, v0(t) is defined as:

From Equation (A.l), e11 (t)

-S(t)v(t)a0 • Therefore,

S(t) a 0 , and similarly, e 22 (t) =

-S(t)v(t)a0

S(t)a0

v(t) .

The results (A.3) and (A.4) state that S(t) and v(t) in the

(A.3)

(A.4)

constitutive Equations (A.l) and (A.2) are the creep compliance and

Poisson's ratio of the creep test, respectively. Now Equations (A.l)

and (A.2) can be rewritten as:

and e12 (t) = Jt D(t-r)[l+v0(t-r)] aal2 dr .

0 ar (A.6)

Taking the Laplace transformation of Equations (A.S) and (A.6) yields

157

158

where f t{f} - J: f(t)e-stdt (i.e. Laplace transform of f(t)) and

s - a real or complex constant.

Assuming that vc is a constant (which is usually a good

approximation) ,

where f sf

Carson transform of f(t).

Since DE = 1,

Taking the Laplace transform inversion results in

J: E(t-r) 8~;11 dr 8r

J: E(t-r) 8~;12 dr

8r

To make these equations look like elastic ones, introduce a reference

modulus, Ea. which is an arbitrary constant as follows:

Ea [ 1

J: E(t-r) 8~;11 dr l ER 8r

Ea [ 1

I: E(t-r) 8~;12 dr l

Ea 8r

Let "pseudo strain" be defined as:

Then,

These equations are identical to an elastic form except that ER and

R Eij replace E and fij. For a nonlinear viscoelastic, isotropic,

nonaging material, this analogy suggests that the use of ~ and

pseudo parameters can reduce the viscoelastic problem to an elastic

one. Since there is not a general constitutive law for a nonlinear

viscoelastic media, an appropriate function between stresses and

pseudo strains must be obtained empirically.

159

160

APPENDIX B

GENERALIZED J-INTEGRAL THEORY

161

APPENDIX .8

GENERALIZED J- INTEGRAL THEORY

Eshelby (70) has defined a number of contour integrals which are

path independent by virtue of the theorem of energy conservation.

The two dimensional form of one of these integrals can be written as

I au J = r (Wdy - T ds)

ax

where W is the strain energy per unit volume, r is a closed contour

followed counter-clockwise in a stressed solid, T is the traction

vector perpendicular to r in the outside direction, u is the

displacement in the X direction, and ds is an element of f.

Rice (71) applied this integral to cracked body problems and

showed that appropriate integration path choices permitted direct

evaluation of the J-integral as follows:

J = -av a a

where V is the potential energy and a is the crack length.

Schapery (3), in 1984, generalized the J-integral for a three-

dimensional case with large deformation. The conservation of linear

momentum states that

R aai j .....R 0 + l'i = (B.l)

axj

and the relation between surface traction ~ and stresses are given

by

T~ 1

(B.2)

Multiplying Equation (B.l) by the weighing factor _aui and axl

integrating over the volume results in a volume integral such that

L [- R R aai j aui

axj axl ) dv - 0 .

Assuming that potentials ~ and ~F exist with the properties that

a~ R

R where R aui

aij R ui•J

a(u. . ) axj 1. J

and ~ a~F R '

aui

the above volume integral can be reduced into the following surface

integral by virtue of the divergence theorem;

I L [ The failure zone with possible high density microcracks or

microvoids is introduced at the crack tip. Then the surface S can be

divided into three subsurfaces (Figure B.l); surface surrounding the

tip of the failure zone (Stip), surface between the continuum and the

failure zone (Sf), and the balance of S (S1 ). Assuming that the

integral over the surface Stip can be neglected, it is shown that

[ (~+~F )nl R

]ds If ~ J -~ aui -£3 J f (B.4) 1

sf axl

where i 3 is the length of the crack edge and Jf is defined as

Jf = J: R

R at.ui d~ Ti

a~

Here, o: is the size of the failure R is the normal stress, and zone, Tz

162

s, _______ _/ __ I ' ( ~

I ~ I 1--a1 I ' I I

---------'-----~- f I

l

....... ______ .Lf._ stip I

/

a

I I I

\ I ' I \ / '-----------

FIGURE B.l Crack tip model by Schapery (3).

163

164

R R r 1 and r 3 are the shearing stresses in the x1 and x2 directions along

the interface between the failure zone and continuum.

A surface-independent integral, Jv, is defined as

and it can be shown from Equations (B.3) and (B.4) that

It should be noted that the above result allows us to evaluate the

integral over the surface surrounding the failure zone by evaluating

the integral over the far field surface. The integral Jv can be

reduced to Rice's path-independent J-integral by omitting the body

force and assuming two-dimensional deformations.

In order to show that Jv is in fact an energy release rate,

change in crack length oa is allowed and self-similar crack growth is

imposed. Through the virtual work expression, it is finally reduced

to the familiar result,

Jv = Jf = - 8Pv a A

where Pv Jv (.P-~u~) dv - J sr ~u~ ds and

A crack surface area.

Applying the correspondence principle II to the surface integral

form results in the J-integral for a nonlinear viscoelastic case as

follows:

1

R where T1 is the physical traction on the body, ui is the pseudo

displacements, and .P and .p, are potentials defined such that

165

and R a (ui. j)

with physical stresses a 1 J.

Another choice of measuring the nonlinear viscoelastic J-

integral is to view the J-integral as an energy release rate; that

is,

This expression is much easier than the form of the J-integral from

the path independent integral for an experimentalist to deal with.

Knowing the potential energy change with respect to change in crack

length, one can measure the J-integral. According to the

correspondence principle II, the potential energy for nonlinear

viscoelastic material can be obtained merely using the pseudo

displacement instead of physical displacement. This procedure

eliminates the time-dependence of viscoelastic material.

Schapery (3) has defined Wf to be the work per unit undeformed

area input by the continuum to a given material element in the

failure zone from the time the crack tip arrives at the element, ta,

to the time the left end of the failure zone arrives, t~; that is,

dt at

where the quantities r 1 and 6u1 are Piola stresses and relative

displacements along the interface, respectively. Comparing the Jf

for the elastic problem with Wf, one should note that Jf is for a

fixed time whereas Wf is for a fixed value of x 1 . Thus, Jf does not

in general reflect the deformation of a given material element. As a

166

result, Wr is believed to be a more basic parameter for defining

material failure (3).

Finally, Schapery (3) expressed the work input to the crack tip

until the initiation time t 1 in terms of the far field parameter Jv.

It was shown to be

Based on the earlier work for the linear viscoelastic media (66),

Schapery was able to reduce the above equation for a local steady-

state condition, i.e. for constant a, as follows:

where ta is the effective time parameter which represents an

equivalent time to give the same compliance for viscoelastic rather

than elastic behavior of the material. Recognizing that the time

taken for the elastic crack tip to move a distance a (failure zone

size) was equal to a/a, the effective time for the viscoelastic case

could be obtained from

a

where k is the correction factor used to express the effective time

of viscoelastic crack growth. In most cases, k is approximately 1/3.

Knowing that the energy required for initiation of the crack should

be equal to the energy available at the crack tip, Schapery (3)

introduced the "fracture initiation energy", r, which is defined like

a surface energy. It is claimed that if a is sufficiently small and

nonlinear behavior of ¢ in far field is well behaved, r is dependent

at most on a and Jv. Then Equation (5) becomes

2r(a, Jv) - ~ D( ~ , t) Jv = 0. a.

This results in an implicit form of crack speed,

167

APPENDIX C

VERIFICATION TEST RESULTS

(CONSTANT STRAIN RATE SIMPLE LOADING TEST

WITH VARIOUS LENGTHS OF REST PERIODS)

168

...., Ul 0... .._.

Ul Ul OJ L .jJ lf)

20~~~--~~~~~~~~~~-.~--~~--~1

15

10 ·-

-10--

***** Experimental

Predicted

*

* *

*

-tsL-~~~~~~~--~~~~~~~~--~--~~~~~~----~~--~~~

0 . 00 l . 002 . 003

Strain Cin./in.)

FIGURE C.l Stress-strain curves after the 1st 20-minute rest period of the constant strain rate simple loading verification test.

....... (J\

1.0

,.. ...... U) Q._

'-"

U) U)

ru L +' Ul

20~~~~~~~--~~~~~~~~--~~~~~~~~--~~--~~~~~

15 ·-

10 ·-

-10--

***** Experimental

Predicted

·~ =-=; ; ; ; 1

-15~*-~~~~~~----*-~~~--~~~~~--~~--~------------~--~ 0 . 001 . 002 . 003

Strain (in. /in.)

FIGURE C.2 Stress-strain curves after the 1st 10-minute rest period of the constant strain rate simple loading verification test.

....... --....J 0

,..... ..... (/)

0... ...., (/) (/)

Ill L

.j.) lfl

20~~~--~~~~~~~~~--~~~~~~--~1

Experimental

Predicted *****

15 ·-

-10 ·-

-150 0 • 001 . 002 . 003

Strain (in./in.)

FIGURE C.3 Stress-strain curves after the 1st 40-minute rest period of the .constant strain rate simple loading verification test.

....... -...J .......

,... ..... Ill a.. "" Ill Ul 01 L .j.J Ul

20~~~--~~~.-~~~~~~~~~~--~~--

15

10 ·-

***** Experimental

Predicted

-sf " • . • -10

-15 . 0 . 00 1 . 002 . 003

Strain (in. /in.)

FIGURE C.4 Stress-strain curves after the 2nd 10-minute rest period of the constant strain rate simple loading verification test.

..... -...1 N

,...... ..... CJ) 0..

'-"

CJ) CJ) 01 L i.J

lJ)

20r-~~~~~--~~~~r-~~~~~~--~~~~~--~~~~~~~-,

15 ·-

10 ·-

***** Experimental

Predicted

-5

L . I • -

-10 ..

-15~----~~~~~--~~~~--~~~~~~~~~--~~._~--~~--~

• 002 0 • 001 • 003

Strain (in. /in.)

FIGURE C.S Stress-strain curves after the 2nd 5-minute rest period of the constant strain rate simple loading verification test.

........ -...J w

,.... .... Ul 0... .....,

Ul Ul Ql L ...,

Ul

20r-~~--~~~----~~~--~~~----~~~--~~~----~~~----~_,

***** Experimental

151- Predicted

10 ·-

-10 ·-

-JSL-._~~----~~--~~~~--~~~--~~~~~--~--~~~._~~--~

0 . 00 1 . 002 . 003

Strain (in./in.)


.......

...... ~

,..... ...... w D.. .._,

w w 01 L .j.) Ul

20--~~--~~~~~~~~----~----~~--~~

***** Experimental

Predicted 15 ·-

0

-sl . ,_ -·

-10

-JSL---~~~--~~~~--~~----~~~~----~~--~~~--._._~~---J 0 . 001 . 002 . 003

Strain (in. lin.)


...... -...J V1

" ..... Ul a.

'-/

Ul Ul 01 L +) Ul

20~--~~~~--~~--~~~~~~--~~~~~~~~~~~~--~~_,

15 ·-10

5

0

-5

-10 ·-

***** Experimental

Predicted

-15~~._--~~----~--~----~._~--~~--~~--~--~~._----~--~ 0 . 00 1 . 002 . 003

Strain (in./in.)

FIGURE C.8 Stress-strain curves after the 3rd 5-minute rest period of the constant strain rate simple loading verification test.

,..... '-I 0\

""' ..... (/)

a.. ~

(/) (/)

01 L +) (J)

20r---~~~~--~~--~--~--~~~~--~~~~------~~--~--~-,

15

10 ·-

***** Experimental

Predicted

-5L• -====-=· ~· ~ L • , -10 ·-

-15~--~----~~--------._----~~~------------~_.~----._------~~ 0 .001 .0~ .003

Strain (in./in.)

FIGURE C.9 Stress-strain curves after the 3rd 10-minute rest period of the constant strain rate simple loading verification test.

...... -...J -...J

,... ...... Ul 0... .......

Ul Ul (II L +> U)

20~~~~--~~~~~~T-~--~~~~~~~--~--~~~----~--~~~

15 ·-

10 r

f

-5

-10 ·-

***** Experimental

Predicted

* ;------; *

-15L-._._ __ ~~~--~--~~~._._~--~------~--~~~~--~~--~-J 0 . 00 1 . 002 • 003

Strain (in. /in.)

FIGURE C.lO Stress-strain curves after the 3rd 20-minute rest period of the constant strain rate simple loading verification test.

,..... '-J 00

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