VISCOELASTIC MODELING OF FLEXIBLE PAVEMENT

A Dissertation

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

Yuanguo Chen

December, 2009

ii

VISCOELASTIC MODELING OF FLEXIBLE PAVEMENT

Yuanguo Chen

Dissertation

Approved: Accepted: ___________________________ ___________________________ Advisor Department Chair Dr. Ernian Pan Dr. Wieslaw K. Binienda ___________________________ ___________________________ Committee Member Dean of the College Dr. Ala R. Abbas Dr. George K. Haritos ___________________________ ___________________________ Committee Member Dean of the Graduate School Dr. Wieslaw K. Binienda Dr. George R. Newkome ___________________________ ___________________________ Committee Member Date Dr. Xiaosheng Gao ___________________________ Committee Member Dr. Kevin L. Kreider

iii

ABSTRACT

The theory of viscoelasticity has been a classic topic, but its application to the

modeling of flexible pavement has not been so successful even though viscoelasticity has

long been characterized in asphalt concrete (AC). This is mainly owing to that the time-

dependent property of a viscoelastic material is always very challenging and also that

flexible pavement is a layered structure which further complicates the modeling work.

This work establishes an efficient and accurate semianalytical solution for primary

response (stresses, strains, and displacements) of flexible pavement on the ground of

layered viscoelastic theory (LVET).

Chapter I highlights the development of flexible pavement. On account of the

viscoelastic AC, the theory of linear viscoelasticity is reformulated briefly from a

mechanical standpoint. The material characterization and the constitutive equation for

viscoelastic AC are reviewed.

In Chapter II, pavement response to a moving load is expressed by the

convolution integral of the pavement impulse response and the moving load. This

convolution integral is elaborated, and in so doing the history of viscoelastic modeling of

flexible pavement is reviewed and summarized.

Pavement impulse response, the kernel of pavement response, is introduced in

detail in Chapter III. Making use of the Prony series for a viscoelastic material, pavement

iv

impulse response is established numerically in the space domain and, for the first time,

analytically in the time-domain. Such solution is called the semianalytical solution.

To verify the semianalytical solution established in Chapter III, Chapter IV

establishes the semianalytical solution for pavement response under the stationary load.

This semianalytical solution is verified with a finite-element-based method through

numerical examples, after which, the well-known collocation method is examined.

Chapter V proposes the semianalytical solution for pavement response under

moving dynamic load. The semianalytical solutions under various typical loading

conditions are verified with the Abaqus solutions.

Main conclusions are summarized in Chapter VI. Future works needing further

research are suggested.

v

ACKNOWLEDGEMENTS

This work would not come possible without the support of Eisenhower Graduate

Fellowship of Federal Highway Administration, Department of Transportation (Award

Number: DDEGRD-08-00401), and the supported by Ohio Department of Transportation

(State Job Number: 134256), to whom the author owe much debt.

The author is grateful to Dr. Ernian Pan who, as the program advisor and

committee director, has been consistently contributing to this work, to other committee

members, including Dr. Ala R. Abbas, Dr. Wieslaw K. Binienda, Dr. Xiaosheng Gao, and

Dr. Kevin L. Kreider, for their value help the author received from lectures or

discussions, and to Mr. Roger Green at the Ohio Department of Transportation, who

always supports the author’s work generously.

Special thanks must be dedicated to Dr. Timothy S. Norfolk at the Department of

Theoretical and Applied Mathematics for communicating on APPENDIX E, and to Dr.

Eshan Dave at the University of Illinois at Urbana-Champaign for his help on the

interconversion in Table 4.3. In preparing this writing, Shu-Wei Goh and Dr. Zhanping

You at the Michigan Technological University provide Figure 1.4 and the data in Figure

1.5, and Hao Wang at the University of Illinois at Urbana-Champaign provides data to

complete Table 5.4, whose kindnesses are sincerely appreciated.

vi

The author would like to single-out Dr. Qiang Wang at SRA International Inc. for

his valuable discussions and comments on viscoelastic pavement modeling, James J.

Ramsey for carefully proofreading the manuscript, and Abhimanyu Kumar for valuable

help on Abaqus modeling.

The author also appreciates his colleagues and fellows for the supports and

advices during the author’s pursuit of degree…

vii

TABLE OF CONTENTS

Page

LIST OF TABLES ............................................................................................................. xi

LIST OF FIGURES .......................................................................................................... xii

CHAPTER

I. INTRODUCTION ............................................................................................................1

1.1 Flexible Pavement ................................................................................................... 1

1.2 Basis of Theory of Linear Viscoelasticity .............................................................. 3

1.2.1 Classical Linear Viscoelastic Model .............................................................. 3

1.2.2 Characteristics of Linear Viscoelasticity ....................................................... 5

1.2.3 Generalized Viscoelastic Model .................................................................... 8

1.2.4 Correspondence Principle ............................................................................ 11

1.2.5 Thermodynamics ......................................................................................... 13

1.3 Viscoelasticity of AC ............................................................................................ 13

1.3.1 Dynamic Modulus Test ................................................................................ 14

1.3.2 Interconversion of Material Properties ........................................................ 17

1.3.3 Constitutive Equation of AC ........................................................................ 17

II. GENERAL SOLUTION ...............................................................................................19

2.1 Summary ............................................................................................................... 19

viii

2.2 General Solution ................................................................................................... 20

2.3 Convolution Integral ............................................................................................. 21

2.4 Influence Function ................................................................................................ 23

2.5 Development of Viscoelastic Modeling ............................................................... 25

2.5.1 Simplified Mechanistic Model ..................................................................... 26

2.5.2 Layered Viscoelastic Model ........................................................................ 28

2.5.3 Modified Elastic Model ............................................................................... 33

2.6 Objective ............................................................................................................... 35

III. PAVEMENT IMPULSE RESPONSE ........................................................................36

3.1 Summary ............................................................................................................... 36

3.2 Governing Equations ............................................................................................ 37

3.3 Associated Elastic Solution .................................................................................. 38

3.3.1 Two Systems of Vector Functions ............................................................... 38

3.3.2 Elastic Solution ............................................................................................ 40

3.4 Viscoelastic Solution ............................................................................................ 45

3.4.1 Correspondence Principle ............................................................................ 45

3.4.2 Special Cases ............................................................................................... 46

3.4.3 Dual-Parameter Method ............................................................................... 48

3.5 Volterra System of Equations ............................................................................... 50

3.6 Pavement Impulse Response ................................................................................ 52

3.7 Viscoelastic Half-space ......................................................................................... 56

IV. PAVEMENT STATIONARY RESPONSE ................................................................60

4.1 Summary ............................................................................................................... 60

ix

4.2 Pavement Stationary Response ............................................................................. 60

4.3 Half-space to Stationary Load .............................................................................. 64

4.4 Numerical Verification ......................................................................................... 65

4.5 Examination of Collocation Method ..................................................................... 70

V. PAVEMENT MOVING DYNAMIC RESPONSE ......................................................73

5.1 Summary ............................................................................................................... 73

5.2 Pavement Moving Dynamic Response ................................................................. 73

5.3 Numerical Verification ......................................................................................... 81

5.3.1 Stationary Load ............................................................................................ 83

5.3.2 Moving Load ................................................................................................ 84

5.3.3 Dynamic Load .............................................................................................. 88

5.3.4 Moving Dynamic Load ................................................................................ 88

VI. CONCLUSIONS AND RECOMMENDATIONS ......................................................89

6.1 Conclusions ........................................................................................................... 89

6.2 Recommendations ................................................................................................. 90

REFERENCES ..................................................................................................................93

APPENDICES .................................................................................................................102

APPENDIX A. COMPONENTS OF SOLUTION MATRIX AND PROPOGATOR MATRIX .................................................... 103

APPENDIX B. BOUNDARY CONDITION IN THE

VECTOR FUNCTIONS DOMAIN ................................................ 105

APPENDIX C. LAPLACE TRANSFORM ..............................................................107

APPENDIX D. ANALYTICAL EXPRESSION OF FUNCTION C(r, θ; ξ) ...........108

APPENDIX E. ANALYTICAL EXPRESSION OF FUNCTION C(r, θ; ξ; ρ) .......109

x

APPENDIX F. ABAQUS INPUTFILE ON NODE AND ELEMENT ....................111

APPENDIX G. ABAQUS INPUTFILE ON DIFFERENT LOAD ...........................116

xi

LIST OF TABLES

Table Page

1.1 Relaxation Modulus E(t) and Creep Compliance D(t) for the Maxwell and Kelvin-Voigt Models .........................................................................8

3.1 Pavement Primary Response .......................................................................................41

3.2 Pavement Impulse Response .......................................................................................54 3.3 Pavement Impulse Response (Decomposed Form) .....................................................57

4.1 Pavement Stationary Response ....................................................................................62

4.2 Analytical Expression of Function C(r, θ; ξ) ..............................................................63

4.3 Structural Properties of Pavement System I (Xu 2004) ...............................................66 4.4 Relaxation Modulus and Creep Compliance of

Viscoelastic AC Layer (Xu 2004) ...............................................................................66

5.1 Pavement Moving Dynamic Response ........................................................................77

5.2 Analytical Expression of Function C(r, θ; ξ; ρ) ..........................................................79

5.3 Structural Properties of Pavement System II ...............................................................82 5.4 Relaxation Modulus and Creep Compliance of

Viscoelastic AC Layer (Adapted from Al-Qadi et al. 2008) .......................................82

xii

LIST OF FIGURES

Figure Page

1.1 Typical profile of flexible pavement ............................................................................. 2 1.2 Mechanical representations of (a) Maxwell

model; and (b) Kelvin-Voigt model .............................................................................. 4

1.3 Mechanical representations of (a) Generalized Maxwell model; and (b) generalized Kelvin-Voigt model ........................................................ 10

1.4 Dynamic modulus test: (a) test machine; and (b) test sample .................................... 16 1.5 Sample master curve of dynamic modulus ................................................................. 16 2.1 Pavement mathematical model and load configuration .............................................. 19 4.1 Variations of (a) the relaxation modulus E(t) and creep

compliance D(t); and (b) the product of E(t) and D(t) with time ................................ 67 4.2 Comparisons between the present semianalytical solution and

finite-element-based solution: (a) stress σzz; (b) stress σyz; (c) strain εxx; and (d) strain εyy ..................................................................................... 68

4.3 Variations of horizontal strains εxx and εyy at (x, y, z) = (0, 0, 7.99 in)

with time: (a) Short-term response; and (b) long-term response ................................ 69 4.4 Comparisons of deflection uz at (x, y, z) = (0, 0, 0) between

the present semianalytical solution and collocation solution: (a) Short-term response; and (b) long-term response ................................................. 71

4.5 Comparisons of the critical strain εxx at (x, y, z) = (0, 0, 7.99 in)

between the present semianatical solution and collocation solution: (a) Short-term response; and (b) long-term response ................................................. 71

5.1 Pavement model and loading area in Abaqus ............................................................. 83

xiii

5.2 Comparisons of deflection at z=0 and z=16 in under stationary load between the present semianalytical solution and the Abaqus solution ....................... 84

5.3 Two moving configurations ........................................................................................ 85 5.4 Solutions from the two moving configurations .......................................................... 86 5.5 Comparisons of deflection at x=0 and x=20 in under moving load

between the present semianalytical solution and the Abaqus solution ....................... 86 5.6 Comparisons of deflection under dynamic load between

the present semianalytical solution and the Abaqus solution ..................................... 87 5.7 Comparisons of deflection at x=0 and x=20 in under moving dynamic load

between the present semianalytical solution and the Abaqus solution ....................... 87

1

CHAPTER I

INTRODUCTION

1.1 Flexible Pavement

Flexible pavement is the pavement structure that “maintains intimate contact with

and distributes loads to the subgrade and depends on aggregate interlock, particle friction,

and cohesion for stability” (AASHTO 1993).

The construction of modern flexible pavement was pioneered by McAdam

(1822), who first proposed the use of the interlock of stone pieces in pavement

construction. Attributed to his contributions, in British such stone pieces are named

macadam. The macadam on the road surface is prone to being dislodged under moving

vehicles, so to overcome this problem, in 1848 tar residue was used to coat the macadam

(Collins and Hart 1936). Such tar-macadam technique is the forebear of asphalt

pavement.

In the twentieth century, the rapid development of automobiles, together with the

easy availability of asphalt material from the oil industry, spurred the development of

asphalt pavement. Today, flexible pavement is usually cemented with asphalt material,

thus flexible pavement is also called asphalt pavement in engineering.

Flexible pavement typically consists of asphalt concrete (AC), base (BS), subbase

(SB), and subgrade (SG) (Figure 1.1). SG includes the natural or compacted soil; SB and

2

BS consist of aggregate of varying size and shape; and AC is the mixtures of asphalt

binder and aggregate satisfying certain gradation.

Under the effect of traffic and environment, pavement will gradually evolve a

variety of damages and distresses, such as rutting and cracking, which reduce the

serviceability of pavement. Pavement distresses are rationally explained by response such

as stresses, strains, and displacements. In pavement analysis and design, it is thus of

primary importance to acquire as accurate as possible the profile of response. On account

of their importance, stresses, strains, and displacements are usually summarized as

pavement primary response1.

1 In some works such as Kenis (1977), pavement primary response refers to

pavement response under stationary load, while pavement response under general moving dynamic load as pavement general response. Hereafter, pavement primary response indicates pavement response to all kinds of loading.

Figure 1.1 Typical profile of flexible pavement.

3

1.2 Basis of Theory of Linear Viscoelasticity

Asphalt binder is a typical viscous liquid, when mixed with the elastic aggregate

to make AC, viscoelasticity is expected. Furthermore, polymer, a typical viscoelastic

material, is usually added to the AC to make modified asphalt concrete. Thus,

viscoelasticity is an unavoidable phenomenon for AC.

In spite the application of viscoelastic material has a long history to flexible

pavement, the theory of viscoelasticity is first established in the study of polymer, to

which a detailed introduction can be found in the book by Ferry (1970). The application

of the theory of viscoelasticity, on the other hand, was mainly stimulated by the extensive

use of viscoelastic material in solid rocket propellant in space techniques (Williams

1964).

This section is going to, from a mechanical standpoint, reformulate the basis of

theory of linear viscoelasticity related to AC. For more detailed discussion on theory of

viscoelasticity, see Christensen (1971).

1.2.1 Classical Linear Viscoelastic Model

Linear viscosity can be represented by a Newtonian dashpot which obeys the

following constitutive equation

( )( ) d ttdtεσ η= (1.1)

where σ(t) and ε(t) denotes stress and strain, respectively, and η the viscosity.

4

Linear elasticity can be represented by a spring which obeys Hooke’s law

( ) ( )t E tσ ε= (1.2)

or

( ) ( )t D tε σ= (1.3)

where E and D denotes the modulus and compliance of elasticity, which are related by

1ED = (1.4)

Different assemblies of dashpots and springs make a variety of viscoelastic

models. For example, a dashpot and a spring in series constitute the Maxwell model

(Figure 1.2a), and a dashpot and a spring in parallel the Kelvin-Voigt model (Figure 1.2b).

(a) (b)

Figure 1.2 Mechanical representations of (a) Maxwell model; and (b) Kelvin-Voigt model.

Consequently, the constitutive equations are

( ) ( ) 1 ( )d t t d t

dt E dtε σ σ

η= + (1.5)

for the Maxwell model and

( )( ) ( ) d tt E tdtεσ ε η= + (1.6)

5

for the Kelvin-Voigt model. Also, the above constitutive equations can be alternatively

rewritten as

( ) ( )l j

l jl jl j

d dP t Q tdt dt

σ ε⎛ ⎞⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠∑ ∑ (1.7)

where l and j are free indices hereafter, and Pl and Qj are expressions of η and E. Such a

form is referred to as the differential form of the constitutive equation.

1.2.2 Characteristics of Linear Viscoelasticity

For the viscoelastic model consisting of dashpots and springs, it is necessary to

derive the modulus or compliance that describes its overall characteristics. Two tests, i.e.,

constant strain test and constant stress test, are devised for this goal.

In the constant strain test,

0( ) ( )t H tε ε= (1.8)

where H(t) is the Heaviside step function defined as

1 0

( )0 0

tH t

t≥⎧

= ⎨ <⎩ (1.9)

Substituting Equation 1.8 into Equation 1.7 yields

0

( )( )

j

j jj

l

l ll

dQ H tdt

tdPdt

σ ε

⎛ ⎞⎜ ⎟⎝ ⎠=

⎛ ⎞⎜ ⎟⎝ ⎠

∑

∑ (1.10)

6

Consequently, the modulus can be defined as

( )

( )

j

j jj

l

l ll

dQ H tdt

E tdPdt

⎛ ⎞⎜ ⎟⎝ ⎠=

⎛ ⎞⎜ ⎟⎝ ⎠

∑

∑ (1.11)

In the constant stress test,

0( ) ( )t H tσ σ= (1.12)

Substituting Equation 1.12 into Equation 1.7 yields

0

( )( )

l

l ll

j

j jj

dP H tdt

tdQdt

ε σ

⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞⎜ ⎟⎝ ⎠

∑

∑ (1.13)

The compliance is then defined as

( )

( )

l

l ll

j

j jj

dP H tdt

D tdQdt

⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞⎜ ⎟⎝ ⎠

∑

∑ (1.14)

The modulus and compliance are not independent. Applying Laplace transform2

(Sneddon 1972)

0

( ) ( ) stf s f t e dt∞ −= ∫ (1.15)

to Equations 1.11 and 1.14 yields

2 Hereafter Laplace transform is denotes by overhead tilt “~” with s being the

transform variable.

7

1

( )

jj

j

ll

l

Q ss

E sPs

⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞⎜ ⎟⎝ ⎠

∑

∑ (1.16)

1

( )

ll

l

jj

j

PssD s

Q s

⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞⎜ ⎟⎝ ⎠

∑

∑ (1.17)

Equation 1.16 and Equation 1.17 gives

2

1( )* ( )E s D ss

= (1.18)

or in the time domain

( )* ( )E t D t t= (1.19)

where the symbol “*” denotes convolution, i.e.

1 2 1 2( )* ( ) ( ) ( )t

f t f t f t f dτ τ τ−∞

= −∫ (1.20)

Applying Laplace transform to Equation 1.7 and making use of Equations 1.16

and 1.17 gives

( ) ( ) ( )s sE s sσ ε= (1.21)

( ) ( ) ( )s sD s sε σ= (1.22)

or in the time domain

( )( ) ( ) * d tt E tdtεσ = (1.23)

( )( ) ( ) * d tt D tdtσε = (1.24)

8

On account of the integral form incurred by the convolution, Equations 1.23 and 1.24 are

referred to as the integral form of the constitutive equation.

Table 1.1 lists the expressions of the modulus and compliance for the Maxwell

and Kelvin-Voigt models. The Modulus and compliance of a viscoelastic material

respectively decrease (relax) and increase (creep) with time, thus they are specially

referred to as the relaxation modulus and creep compliance.

Table 1.1 Relaxation Modulus E(t) and Creep Compliance D(t) for the Maxwell and Kelvin-Voigt Models

Model E(t) D(t)

Maxwell Model /( ) EtE t Ee η−= 1 1( )D t tE η

= +

Kelvin-Voigt Model ( ) ( )E t E tηδ= + ( )/1( ) 1 EtD t eE

η−= −

Since in reality a viscoelastic material exhibits only limited values of relaxation

modulus and creep compliance, it is not suitable to capture the creep compliance by the

Maxwell model, or capture the relaxation modulus by the Kelvin-Voigt model.

1.2.3 Generalized Viscoelastic Model

To overcome the limitations of the Maxwell and the Kelvin-Voigt models in

predicting accurate relaxation modulus and creep compliance, generalized models

consisting of more mechanical elements (dashpot and spring) are developed. For

example, a spring and multiple Maxwell elements in parallel generate the Generalized

Maxwell Model (Figure 1.3a), while a spring and multiple Kelvin-Voigt elements in

9

series generate the Generalized Kelvin-Voigt Model (Figure 1.3b) 3 . The relaxation

modulus for the generalized Maxwell model is presented by a Prony series (Gibson

2003), e.g.

/

1

( ) m

Mt

e mm

E t E E e ρ−

=

= +∑ (1.25)

where ρm=Em/ηm is the relaxation time.

Making use of Equation 1.19, the creep compliance for the generalized Maxwell

model can be written as

/0

1

( ) (1 )m

Mt

mm

D t D D e τ−

=

= + −∑ (1.26)

where τm is the retardation time.

The creep compliance for the generalized Kelvin-Voigt model, on the other side,

can also be presented by a Prony series, e.g.

/

10

1 1( ) (1 )m m

ME t

m n

D t eE E

η−

=

= + −∑ (1.27)

Equation 1.27 reveals that the creep compliance for the generalized Kelvin-

Voigt model is similar to that for the generalized Maxwell model as in Equation 1.26, or

in other words, the generalized Maxwell model and generalized Kelvin-Voigt model are

equivalent to each other (Sargand 2002). Such a model, together with similar models

satisfying Equation 1.25 for relaxation modulus and Equation 1.27 for creep

compliance, is termed as the generalized viscoelastic model.

3 An extra spring is added to the generalized Maxwell model and generalized

Kelvin-Voigt model in Ferry (1970). This is to avoid the zero relaxation modulus and infinite creep compliance.

10

Equations 1.25 and 1.27 further reveal that

1

(0)M

0 e mm

E E E E=

= = +∑ (1.28)

( ) eE E∞ = (1.29)

0(0)D D= (1.30)

01

( )M

mm

D D D=

∞ = +∑ (1.31)

(a)

(b)

Figure 1.3 Mechanical representations of (a) Generalized Maxwell model; and (b) generalized Kelvin-Voigt model.

11

1.2.4 Correspondence Principle

Viscoelastic problems, owing to the accompanying time variable, are usually not

easy to solve. Through the seperation of variables, Alfrey (1944) first established an

anology between a viscoelastic problem and its associated elastic problem for an

imcompressible media, and later, Tsien (1950) extended it to a compressible media.

The most promising work was presented by Read (1950). After applying Fourier

transform to the constitutive equation for a viscoelastic material, he established the

analogy between the constitutive equation for a viscoelastic material in the Fourier

domain and that for an elastic material in the time domain. This forbears the well-known

correspondence principle.

Lee (1955) employed Laplace transform and established the similar analogy, as

shown by Equation 1.21 for a viscoelastic material in the Laplace domain and Equation

1.2 for an elastic material in the time domain, which suggested that a viscoelastic

problem be solved elastically in its Laplace domain. This correspondence principle is the

most popular one and referred to as the correspondence principle of Laplace transform in

this study. Later, for problems where it is difficult to apply the Laplace transform, Radok

(1957) extended Lee’s correspondence principle by treating Equation 1.7 as a functional

equation in the time domain, referred to as the correspondence principle of differential

operator in this study. Noticing the similarity between Equation 1.23 and 1.2, Ashton and

Moavenzadeh (1967, 1968) treated the convolution integral as an integral operator and

generated another correspondence principle, referred to as the correspondence principle

of integral operator in this study.

12

The three different forms of correspondence principle are essentially equivalent to

each other as formulated in Section 1.2.2. For example, as shown in Equations 1.11 and

1.16, Laplace variable s and the differential operator d/dt are replaceable in mathematical

meaning, which accounts for the equivalence of the correspondence principle of Laplace

transform and the differential operator. Further, the Laplace inverse transform

immediately yields the correspondence principle of integral form.

Consequently, the equivalence of the three forms of correspondence principle

implies the equivalent computation complexities in obtaining the solution in the time

domain. For example, through the correspondence principle of Laplace transform, the

solution in the Laplace domain has to be inverted back to the time domain. This is always

a challenging issue. Analytical solution can be rarely achieved through Laplace inverse

transform except for simple viscoelastic models such as Maxwell and Kelvin-Voigt

model. Instead, a variety of numerical algorithms were developed, mainly attributed to

the study of the structural integrity of solid rocket propellant. Among them, the most

well-known is the collocation method by Schapery (1962). A comparison of typical

numerical algorithms for Laplace inversion transform can be found in Cost (1964). The

correspondence principle of differential operator, on the other hand, may involve

complex differential operator of high-order, also making the evaluation very tedious, if

not impossible. Finally, the correspondence principle of integral operator involves a

convolution integral, which is also of intensive computation and will be addressed later.

13

1.2.5 Thermodynamics

An alternative to the mechanical viewpoint presented above is the

thermodynamics. By assigning potential energy to elasticity and dissipation function to

viscosity, Biot (1954) derived the constitutive equation for the isothermal problem,

whose form is similar to Equation 1.21 on account of Equation 1.25, except that the

Laplace variable s becomes the differential operator d/dt. Based on Biot’s work, the

relaxation time in Equation 1.25 is invoked by the internal variable, and more

importantly, are the eigenvalues of a matrix. Also, in solving a porous viscoelastic

problem, Biot (1956) actually used the correspondence principle of differential operator,

although not defined explicitly there.

Schapery (1964) extended Biot’s work to the general thermal problem. The

viscoelastic problem is essentially thermodynamic. However, for simplicity, this study

will be restricted to the isothermal problem. Interested readers on thermodynamics can go

to the book by Ziegler (1977).

1.3 Viscoelasticity of AC

The relaxation modulus or creep compliance can be derived directly from the

constant strain or constant stress tests, which, together with the constant strain-rate test,

are conventionally employed to examine the viscoelastic properties of AC (Monismith

and Secor 1962). However, in practice constant strain and constant stress are not easy to

apply, and also the response of AC cannot be accurately captured. As a result, a dynamic

test is devised.

14

1.3.1 Dynamic Modulus Test

The dynamic modulus test for AC dates back to 1960s (Papazian 1962). After

decades of development, standard test protocol has been established (AASHTO 2005).

Figure 1.4 demonstrates the test machine and test sample.

In this test, a harmonically oscillating axial stress is applied, e.g.

00( ) i tt e ωσ σ= (1.32)

where ω0 is the oscillating frequency.

After several cycles the strain will also respond harmonically with a phase lag Δ,

i.e.

0( )0( ) i tt e ωε ε −Δ= (1.33)

Substituting Equation 1.32 and 1.33 into 1.23 gives

0

00 0( )*

i ti t idee E t e

dt

ωωσ ε − Δ= (1.34)

The complex modulus is thus defined by

0

0

* 00

0

( ) *( )

i t

ii t

deE tdtE e

e

ω

ω

σωε

Δ= = (1.35)

whose real and imaginary part are further defined as the storage modulus E′ and the

loss modulus E′′, i.e.

* 00 0

0

'( ) ( )E Re E cosσω ωε

⎡ ⎤= = Δ⎣ ⎦ (1.36)

* 00 0

0

''( ) ( )E Im E sinσω ωε

⎡ ⎤= = Δ⎣ ⎦ (1.37)

15

The phase lag Δ can be determined from Equations 1.36 and 1.37. The dynamic

modulus is defined as the magnitude of the complex modulus, i.e.

0

0

* 00

0

( )( )

i t

i t

deE tdtE

e

ω

ω

σωε

∗= = (1.38)

Making use of Equation 1.25, the storage modulus and loss modulus for the

generalized viscoelastic model are related to the relaxation modulus by

2 20

0 2 21 0

'( )1

Mm m

em m

EE E ω ρωω ρ=

= ++∑

(1.39)

00 2 2

1 0

''( )1

Mm m

m m

EE ω ρωω ρ=

=+∑

(1.40)

It is obvious that the complex modulus is dependent on the test frequency ω0.

Temperature will also play a significant role through the time-temperature principle. To

obtain as much information as possible, the dynamic test is conducted at multiple

frequencies and different temperatures. At a reference temperature, one single curve,

called the master curve (Figure 1.5), is drawn to depict the relation between dynamic

modulus and frequency. Fitting Equation 1.39 or 1.40 to the master curve will determine

the relaxation modulus in Equation 1.25.

It is demanding to run such a dynamic modulus test. As a result, it remains a hot

research issue to predict the dynamic modulus of AC from its constitutes, i.e. asphalt

binder and aggregate. For example, asphalt binder is first characterized as a viscoelastic

material through dynamic shear rheometer (DSR) test, while aggregate as elastic material.

Then, dynamic modulus of AC can be predicted through analytical micromechanical

16

models (Kim and Little 2004, Yin et al. 2008) or numerical method such as discrete

element method (DEM) (Liu et al. 2009).

(a) (b)

Figure 1.4 Dynamic modulus test: (a) test machine; and (b) test sample.

1.00x10-8 1.00x10-6 1.00x10-4 1.00x10-2 1.00x100 1.00x102

Reduced Frequency (Hz)

0

10,000

20,000

30,000

Dyn

amic

Mod

ulus

(MPa

)

Sigmoidal Master Curve39.2C21.3C13C4C-5C

Figure 1.5 Sample master curve of dynamic modulus.

17

1.3.2 Interconversion of Material Properties

Since the relaxation modulus and creep compliance are governed by Equation

1.19, the creep compliance can be obtained from the relaxation modulus, and vice versa.

Such a procedure is referred to as the interconversion between the relaxation modulus

and creep compliance. This work started as early as Hopkins and Hamming (1957) in

deriving the creep compliance for polyisobutylene from the relaxation modulus. For AC,

a numerical method has been developed for such interconversion (Park and Schapery

1999, Park and Kim 1999, 2001). However, sometimes this numerical method may lead

to results mathematically meaningful but physically meaningless. Alternatively, the

interconversion can be reached analytically via Equation 1.18.

1.3.3 Constitutive Equation of AC

The model consisting of dashpots and springs is derived for one-dimensional

material so far, rather than a general three-dimensional material, such as AC. The integral

form of constitutive equation, Equations 1.23 and 1.24, can be extended to three

dimensionals.

Stress state σjl and strain state εjl in a three-dimensional material can be

decomposed into the volumetric and deviatoric components, i.e.

13jl kk jl jlsσ σ δ= + (1.41)

13jl kk jl jleε ε δ= + (1.42)

18

where σkk and sjl are the volumetric and deviatoric stress, εkk and ejl are the volumetric

and deviatoric strain, respectively, and δjl is the Kronecker delta.

Assuming the AC is isotropic and linearly viscoelastic in both the volumetric and

deviatoric components, Equations 1.23 and 1.24 are then extended to

( )( ) ( ) * kkkk

d tt K tdtεσ = (1.43)

( )

( ) 2 ( ) * jljl

de ts t t

dtμ=

(1.44)

where K(t) and μ(t) are the relaxation bulk modulus and relaxation shear modulus which

are related to the relaxation modulus by

( )( )

3(1 2 )E tK t

υ=

− (1.45)

( )( )

2(1 )E ttμ

υ=

+ (1.46)

where υ is the Poisson’s ratio, assumed to be time-independent in this study.

Combining Equations 1.41 to 1.44 yields the constitutive equation for the AC

( )( )( )* 2 ( ) * jlkkjl jl

de td tt tdt dtεσ λ δ μ= + (1.47)

where

( )( )(1 )(1 2 )

E tt υλυ υ

=+ −

(1.48)

19

CHAPTER II

GENERAL SOLUTION

2.1 Summary

The general solution of pavement response to a moving dynamic load4, referred to

as pavement moving dynamic response, is established by a convolution integral

(Duhamel’s integral), whose evaluation is of primary concern. Developments in

viscoelastic modeling of flexible pavement are reviewed and explained in the language of

the convolution integral. The objective of this study is shortly outlined at the end.

4 Moving load and stationary load correspond to the change of load position,

while dynamic load and static load to the change of load magnitude.

Layer k

σ

Figure 2.1 Pavement mathematical model and load configuration.

rs

Field Point

O x

r

z

V

y

f(t)

Layer 1

20

2.2 General Solution

In the context of a linear system, Pavement primary response to a dynamic load

f(t), referred to as pavement dynamic response, can be written as the well-known

Duhamel’s integral (Humar 1990)

( , ; ) ( , ; ) ( )t

s st t f dδφ φ τ τ τ−∞

= −∫r r r r (2.1)

where r and rs denotes the field point and loading position as shown in Figure 2.1,

respectively, and φδ denotes pavement impulse response (or Green’s function in the time

domain), i.e., response of the same pavement to an impulse load5,

[ ( , ; )] ( )sL t tδφ τ δ τ− = −r r σ (2.2)

Equation 2.2 indicates the governing equation system of the studied pavement,

including the equilibrium equations, constitutive equations and displacement-strain

equations, while its right-hand side describes the loading conditions. Note symbol in

italic bold indicate vector hereafter. For generality, pavement impulse response in

Equation 2.1 is time-dependent through the temporal variable.

For a homogeneous material, the response will depend on the relative position

between field and loading points, i.e.

( , ; ) ( ; )s st tδ δφ τ φ τ− = − −r r r r

(2.3)

If the load is moving, Equation 2.3 should be written as

( , ( ); ) ( ( ); )s st tδ δφ τ τ φ τ τ− = − −r r r r (2.4)

5 A singularity in the time domain is referred specially to as an impulse load.

21

As shown in Equation 2.4, the temporal variable will then be involved in two parts: the

moving loading position and the time-dependent response.

Under the assumption that the load is moving with velocity V without transverse

movement, a Cartesian coordinate is set up as in Figure 2.1 with the x-axis along the

moving direction, through which Equation 2.4 is expanded to

0 0( , ( ); ) ( ( ), ; )s s s st x - x y - y ,z - z tδ δφ τ τ φ τ τ− = −r r (2.5)

where 0( )s sx t x t= +V with xs0 being the start position of moving load along x-axis.

Inserting Equation 2.5 into Equation 2.1 yields

0 0 0( , ( ); ) ( , ; ) ( )t

s s s st t x - x y - y ,z - z t f dδφ φ τ τ τ τ−∞

= − −∫r r V (2.6)

As established in Equation 2.6, pavement moving dynamic response should be easily

determined if one knows the pavement impulse response φδ. However, there are two

complexities in making use of Equation 2.6: First, the analytical form of φδ is not easy to

obtain in the context of layered viscoelastic theory (LVET); Second, the convolution

integral in the time domain6 is always very challengling to evaluate.

2.3 Convolution Integral

The convolution integral is usually evaluated through numerical schemes (Cebon

1999). For example, Equation 2.6 can be discretized to

1

0 0 00

( , ( ); ) ( , ; ) ( )jN

s s s sj

t t x - x j y - y ,z - z t j fδφ φ τ τ τ τ−

=

= − Δ − Δ Δ∑r r V (2.7)

6 Hereafter the convolution integral only refers to the time domain.

22

As shown in Equation 2.7, the convolution integral will occur in two aspects: the first

one between the moving position through (x-xs0-VjΔτ) and the time-dependent response

through (t-jΔτ), both are involved in the pavement impulse function φδ; the second one

between φδ and the dynamic load f(τ).

The numerical algorithm for evaluating the convolution integral, such as

Equation 2.7, is straightforward to program using the finite element method (FEM), and

thus is used in some commercial FEM packages such as ABAQUS (2005). However,

evaluating the convolution integral at an arbitrary time through such a numerical

algorithm requires all records of previous time steps. For example, if we succeed at

evaluating φ(r, rs(t); t) by Equation 2.7, then at t+Δt

0 00

( , ( ); ) ( , ; ) ( )jN

s s0 s sj

t t t t x - x j y - y ,z - z t t j fδφ φ τ τ τ τ=

+ Δ + Δ = − Δ + Δ − Δ Δ∑r r V (2.8)

If a relation exists between two successive time point, for example,

1( ) ( ) ( )t t j h t t jδ δφ τ φ τ+ Δ − Δ = Δ − Δ

(2.9)

Equation 2.9 simplifies 2.8 to

1( , ( ); ) ( ) ( , ( ); )s st t t t h t t tφ φ+ Δ + Δ = Δr r r r

0 0( , ; ) ( )s0 j s s jx - x N y - y ,z - z t t N fδφ τ τ τ τ+ − Δ + Δ − Δ ΔV (2.10)

As shown in Equation 2.10, the evaluation of φ(r, rs(t+Δt); t+Δt) requires

evaluating first φ(r, rs(t); t), and all preceding time points in a successive order. The

algorithm is thus referred to as the successive algorithm.

Equation 2.9, in a certain sense, can save some computation cost. In spite of this,

one should be aware that the successive algorithm has to always start from the initial

23

time (usually t=0) and follow a fixed time sequence, e.g. increment Δt to next step as in

Equation 2.10. As a consequence, not only is this still time-consuming, but also

numerical errors at previous steps will propagate and accumulate in the next step,

making it unsuitable, if not impossible, to evaluate Equation 2.7 for large time intervals.

By writing material properties in the form of Prony series, a successive algorithm

presented in Equation 2.10 can be obtained easily. This technique appears as early as in

White (1968) and is used in today’s commercial FEM programs such as Abaqus, which

has been widely employed in material characterization of AC (Abbas 2004) or primary

response prediction of perpetual pavement (Elseifi et al 2006, Liao 2007, Al-Qadi et al

2008).

In addition, making use of this technique, extensive works have been conducted to

develop an efficient finite-element-based algorithm. For example, Xu (2004) developed

such an algorithm for pavement stationary response. While, program MNLAYER (Wang

2008) is mainly based on layered theory in the space domain, but it uses the same

technique in the time domain.

2.4 Influence Function

The second convolution in Equation 2.7 will be eliminated by assuming slowly-

varying dynamic (or quasi-static) load, which thus simplifies Equation 2.1 to

( , ; ) ( ) ( ; )s t f t I V tφ =r r (2.11)

24

where I(V; t) is the influence function (Cebon 1999) as defined by

( ; ) ( ( ); )t

sI V t t dδφ τ τ τ−∞

= − −∫ r r (2.12)

where use has been made of Equation 2.4.

For static load, i.e. f(t)=1, Equation 2.11 is equivalent to

( , ; ) ( ; ) ( ( ); )t

s st I V t t dδφ φ τ τ τ−∞

= = − −∫r r r r (2.13)

Equation 2.13 demonstrates that for static load pavement response and the influence

function are actually equivalent.

The convolution integral still exists inside Equation 2.13, exclusively determined

by φδ . For stationary load, Equation 2.13 reduces to

( , ; ) ( ; )tH

s st t dδφ φ τ τ−∞

= − −∫r r r r (2.14)

where φH is referred to as pavement stationary response, i.e., pavement response to a

stationary load. Similar to Equation 2.2, one has

[ ( , ; )] ( )HsL t tφ τ Η τ− = −r r σ

(2.15)

From Equation 2.14, it is straightforward to write

( ; ) ( , ; )Hs s

dt tdt

δφ τ φ τ− − = −r r r r (2.16)

Inserting Equation 2.16 into 2.1 yields

( , ; ) ( , ; ) ( )t H

s sdt t f ddt

φ φ τ τ τ−∞

= −∫r r r r (2.17)

Interchanging the integral sequence yields

( )( , ; ) ( , ; )t H

s sdft t d

dtτφ φ τ τ

−∞= −∫r r r r

(2.18)

25

Equation 2.18 is an alternative to Equation 2.1 in expressing the general

expression for pavement dynamic response. This expression is more popular than

Equation 2.1 in that researchers and engineers tend to solve φH instead of φδ. Notice

Equation 2.18 holds for only dynamic load f(t).

From Equation 2.14, it can be observed that the convolution integral vanishes in

φH. However, an additional convolution integral will occur in deriving φH due to that the

constitutive equation of a viscoelastic material is also in the form of a convolution

integral, as shown in Equation 1.23. For clarity, the convolution integral is referred to as

the coupling convolution integral if both moving dynamic load and material

viscoelasticity appear, such as Equation 2.13; as the explicit convolution integral if it is

caused only by the moving or dynamic load; and as the implicit convolution integral if by

the viscoelasticity of the material.

The explicit convolution integral is the special case of the coupling convolution

integral if the convolution integral due to material viscoelasticity can be solved

analytically. This can be achieved if simple viscoelastic models, such as the Maxwell

element or the Kelvin-Voigt element, are used to model the viscoelastic material. On the

other hand, the implicit convolution integral is the special case of the coupling

convolution integral only when the load is static and stationary.

2.5 Development of Viscoelastic Modeling

This section is to review the development of the viscoelastic modeling of flexible

pavement, and especially explain these methods from the aspect of the coupling

26

convolution integral. Only the primary response model is concerned. For a

comprehensive review on performance model, interested reader can go to the work by

Monismith (1992).

2.5.1 Simplified Mechanistic Model

The beam and plate have long been studied (Timoshenko and Goodier 1970).

Pavement, in early studies, was simply modeled as a beam or plate resting on Winkler

(elastic spring) foundation, referred to as the simplified mechanistic model. Because the

Winkler foundation consists of only discrete elements, dampers are usually incorporated

into the Winkler foundation to account for a continuous body. The viscosity can be

accounted for through the dampers as well.

The simplified mechanistic model has been widely used for continuous pavement,

including flexible pavement, unjointed rigid pavement and composite pavement.

Westergaard (1926) pioneered the theoretical analysis of concrete slab through a plate-

Winkler foundation model. Viscoelasticity was also introduced to such beam (plate)-

Winkler foundation model. For example, Freudenthal and Lorsch (1957) replaced the

elastic spring in the Winkler foundation by viscoelastic elements, such as Maxwell,

Kelvin and standard solid elements, where the Kelvin element brought out the damped

Winkler foundation.

Viscoelasticity is more frequently assigned to beam in the beam-(damped)

Winkler foundation. For example, Pister and collaborators first investigated viscoelastic

beam-Winkler foundation model for the stationary and repeated load (Pister and

27

Monismith 1960), and immediately for the moving load (Pister and Westmann 1962).

Making use of the beam-damped Winkler foundation model, Harr (1962) assumed

repeated load to explain the effect of moving velocity observed in the AASHO Road

Test. Later, Thompson (1963) extended this model to moving load. A synthesized work

for the use of the simplified mechanistic model in transportation engineering was

performed by Fryba (1972).

For a beam-(damped) Winkler model, pavement impulse response in Equation 2.4

reduces to one-dimensional

0( , ; ) ( ; )st x - x tδ δφ τ φ τ τ− = − −sr r V (2.19)

and consequently Equation 2.13 simplifies to

00( , ; ) ( ; ) ( ; )

t

s st I V t x - x t dδφ φ τ τ τ= = − −∫r r V (2.20)

To treat the convolution integral in Equation 2.20, either the numerical schemes in

the time domain as introduced in Section 2.3 or integral transform methods could be

employed. Recent work on the application of such beam models can be found in Hardy

and Cebon (1993) for pavement surface discontinuity and Sun and Deng (1998) for a

moving line load.

The use of plate-(damped) Winkler foundation models follows the same

procedure. Monismith and Secor (1962), for example, tested the viscoelastic properties of

asphalt concrete slab and compared the deflection based on plate-Winkler foundation

model with the experimental data. One recent work is in Kim et al (2002), which used the

plate-damped Winkler foundation model to investigate the effect of axle configuration.

28

In rigid pavement, the Portland cement concrete (PCC) carries most of the applied

load, and the simplified mechanistic model works very well. For flexible pavement,

because the load is spread to and distributed along every layer, it is necessary to switch to

a layered viscoelastic model for the pavement primary response. Due to its simplicity, the

simplified mechanistic model is very convenient in studying the vehicle-road interaction

(Cebon 1999).

2.5.2 Layered Viscoelastic Model

In a plate-(damped) Winkler foundation model, replacing the Winkler foundation

by the Boussinesq foundation (an elastic solid), produces the first two-layer system

(Hogg 1938), referred to as plate-Boussinesq foundation model. Burmister (1943) then

replaced the plate by a general elastic solid and solved the two-layer system, for the first

time, based on layered elastic theory (LET). Further, Burmister extended his work to a

three-layer system (1945a, b, c). Endeavors on the multi-layer system can be seen in

Westmann (1962) and Schiffman (1962).

The study of layered viscoelastic model comes popular only after the

correspondence principle by Lee (1955). Initial work was performed on plate-Boussinesq

foundation model. For example, to account for the relaxation of a foundation, Hoskin and

Lee (1959) studied an elastic plate on a viscoelastic Boussinesq foundation. In pavement

study, Pister (1961) investigated the system of a viscoelastic plate on viscoelastic

Boussinesq foundation, and Westmann (1967) extended this system to include moving

load.

29

The study of a layered viscoelastic model through LVET was initialized by

Ishihara (1962) and Westmann (1962). Both assumed a frictionless interface in

Burmister’s two-layer system. The former, particularly, investigated four types of

mechanical elements, i.e. dashpot, spring, Maxwell element, and Kelvin-Voigt element.

For simple viscoelastic elements, analytical solution of the pavement stationary response

can be achieved.

Kraft (1965) also studied a two-layer system under stationary load through the

correspondence principle of Laplace transform. His work is the first one that applied

LVET to a fully layered viscoelastic model. To recover the solution in the time domain,

the collocation method by Schapery (1962) was used.

For a more complicated generalized viscoelastic model, Huang (1967) applied the

root-finding techniques to the Laplace inverse transform and derived first the analytical

solution for a two-layer system. On account of the computation complexities in the

Laplace inverse transform, Huang further examined the collocation methods by Schapery

(1962) and suggested a collocation scheme for multi-layer pavement. The collocation

method was also employed by Barksdale and Leonards (1967) to study viscoelastic

pavement under repeated load.

The collocation method is the most popular tool in modeling viscoelastic

pavement, with which pavement response in Equation 2.13 is simplified to

/

1

( , ; ) n

Nt T

s nn

t eφ Γ −

=

= ∑r r (2.21)

30

where Tn is the presumed collocation point, and Γn the collocation coefficient evaluated

through Tn. For example, Tn ={0.01, 0.03, 0.1, 1, 10, 30, ∞} were adopted in

KENLAYER (Huang 1993).

In understanding that AC behaves more complicated than any mechanistic

models, Ashton and Moavenzadah (1967) proposed the correspondence principle of

integral operator (see Section 1.2.4) and used it to analyze a three-layer system. However,

their solution is expressed by the convolution integral, a formidable task discussed in

Section 2.3.

The load in the above works based on LVET was mainly assumed to be

stationary. In other words, only the implicit convolution integral was concerned. The

study of pavement moving response based on LVET, which may trigger the coupling

convolution integral, also received wide attentions.

Perloff and Moavenzadah (1967) studied vertical deflection of a viscoelastic half-

space under a moving point load. They employed the Kelvin-Voigt element to account

for the viscoelasticity, with which the implicit convolution integral can be avoided. Thus,

their solution ended with the explicit convolution integral, which was then solved by a

numerical scheme. Ishihara and Kimura (1967) extended the early work of Ishihara

(1962) to a moving load, which suggested extracting a Maxwell element corresponding to

the moving velocity and reducing multi-layer system to a two-layer system.

Chou and Larew (1969) extended the work by Perloff and Moavenzadah (1967) to

a two-layer system. Their work should be recognized as the pioneer in viscoelastic

modeling of flexible pavement to moving load based on LVET. Similarly, their solution

31

ended with an explicit convolution integral, but they used the collocation method, i.e.

Equation 2.22, to treat the convolution integral.

Making use of the correspondence principle of integral operator, Elliot and

Moavenzadeh (1969) further studied a three-layer viscoelastic system subject to circular

stationary, repeated and moving loads. For stationary load, their solution was expressed

by the implicit convolution integral. For repeated load, they established their solution

similar to Equation 2.18. Although the solution to repeated load is in the form of coupling

convolution integral, on account of that the repeated load is a series of stationary loads,

there will be only an implicit convolution integral, which was solved through the

collocation methods. For moving load, they first noticed the coupling convolution

integral. However, later, Elliot and Moavenzadeh (1971) wrote down the solution for

moving load by inserting the time-dependent position to the corresponding solution to

stationary load, i.e.

( , ; ) ( , ( ); )Hs st t tφ φ=r r r r

(2.22)

This treatment corresponds to

( , ; ) ( ( ); )t

s st t t dδφ φ τ τ−∞

= − −∫r r r r (2.23)

Obviously the explicit convolution integral can be avoided. Under the approximate

solution in Equation 2.23, pavement moving response is axisymmetric.

Moavenzadeh’s work lays basis for the program VESYS (Kenis 1978) except that

the moving load is approximated by a stationary haversine load. Such a treatment on

moving load is widely accepted in viscoelastic pavement modeling (Sousa et al. 1987,

Papagiannakis et al. 1996, Kim et al. 2009), but has not be verified since a benchmark

32

solution is not available, which should follow Equation 2.6 as in this study. However,

because of the computation cost of viscoelastic analysis, VESYS is gradually shifted to

elastic analysis (Kenis et al. 1982).

Huang (1973) also extended his work for stationary load (Huang 1967) to the

moving load. His work established the solution in the form of a coupling convolution

integral. He arrived at the pavement stationary response analytically, making use of the

collocation method, and thus reduced the coupling convolution integral to the explicit

convolution integral, which was then solved numerically.

It should be emphasized that in the works by Elliot and Moavenzadeh (1969,

1971) and Huang (1973), the validity of the solution to the moving circular load is

constrained to the field point along the moving direction based on the assumption that

rs(t)=rs(0)-Vt.

With the establishment of pavement moving response, extensive works were

performed on the convolution integral. Owing to the huge expense in evaluating the

convolution integral numerically as introduced in Section 2.3, the integral transform

methods were often used, as in Battiato et al (1977), Privarnikov and Radovskii (1981),

and Hopman (1996). However, such methods will generate an alternative integral, which

is still time-consuming to evaluate when inverted to the time domain.

33

2.5.3 Modified Elastic Model

Besides the layered viscoelastic model based on the LVET, viscoelastic pavement

has also been treated elastically. Based on LET, there will be no time-dependent

response, thus Equation 2.4 reduces to

0( ( ); ) ( ( )) ( )s st tδφ τ τ φ τ δ τ− − = − −r r r r (2.24)

Consequently, Equation 2.13 yields

0( , ; ) ( ( ))s st tφ φ= −r r r r (2.25)

Consequently the LET can avoid the tedious coupling convolution integral in

evaluating Equation 2.25, leading to the reduced computation cost. As such, pavement

modeling based on LET is always very popular and a variety of LET-based programs

have been developed. Among them, for example, are BISAR (De Jong et al. 1973),

ELSYM 5 (Kopperman et al. 1986), WESLEA (Van Cauwelaert et al. 1989), JULEA

(Uzen 1994), LEAF (Hayhoe 2002) etc.

The merit of reduced computation cost based on LET spurs endeavors to treat a

viscoelastic pavement based on LET, and here comes the modified elastic model. The

modified elastic model can be grouped into three types.

The first type is based on the “quasi-elastic methods” by Schapery (1965), which

suggested that the associated elastic solution be the first-order approximation to the

viscoelastic solution. The associated elastic solution is solved by inserting directly into

the LET the time-dependent material properties of a viscoelastic material, such as the

relaxation modulus or the creep compliance. This method is equivalent to setting

0( ( ); ) ( ( )) ( )s st tδφ τ τ φ τ δ τ− − = − −r r r r (2.26)

34

where

0 0 ( )( )

( ( ); ) ( ( ))s s D DE E

ττ

φ τ τ φ τ ==

− = −r r r r (2.27)

and consequently Equation 2.13 reduces to

0( , ; ) ( ( ); )s st t tφ φ= −r r r r (2.28)

One recent application of this method can be found in Park and Kim (1998).

In the second type, the input of material properties to the LET, mostly modulus, is

assumed to depend on viscosity or moving velocity for a viscoelastic material. For

example, Collop (1994) introduced the viscosity-dependent material property to VESYS-

IV (Kenis et al 1982). The idea of picking up material properties according to moving

velocity can be found in Ishihara and Kimura (1967). This ides is the most popular one in

the modified elastic model. For example, Papagiannakis et al (1996) used the velocity-

dependent material property in ELSYM5, and MEPDG (NCHRP 2004), which is based

on JULEA, used the dynamic modulus at a frequency, e.g. 10 rad/s, corresponding to a

regular moving speed.

The third type, when uses LET, assigns a damping to the elastic material. For

example, the modulus is assumed to be

* (1 2 )E E iη= + (2.29)

where η is the damping ratio, and E is the modulus of elasticity. Comparing Equation

2.29 and 1.35, E* corresponds to the complex modulus in Equation 1.35, and

2 tanη = Δ (2.30)

Since Equation 1.35 arises from a harmonic load, the assumption of Equation 2.29 holds

true only for the harmonic load, such as a vibrating load (Guzina and Nintchu 2001), and

35

the damping ratio η must be governed by Equation 2.30. Otherwise, this method may

lead to non-causal results. The finite-layer method in Siddharthan et al (2000) is one

application of this method to pavement modeling. This method is much popular in

structural dynamic analysis and geotechnical engineering.

2.6 Objective

For rational pavement analysis, Barksdale and Leonards (1967) summarized four

topics: vehicle-road model, material characterization, constitutive equations, and efficient

algorithm.

In view of all the developments in the viscoelastic modeling of flexible pavement,

AC has been successfully characterized as a viscoelastic material. However, the

convolution integral in the constitutive equation for a viscoelastic material, or its

alternatives such as the inverse integral transform, has handicapped our pursuit to an

efficient solution for viscoelastic pavement to moving load in the context of LVET. It is

thus the task of this study to establish a LVET-based solution as efficient as the LET-

based solution, which, together with the updated vehicle-road model, will provide a much

realistic scenario in pavement modeling in the future.

36

CHAPTER III

PAVEMENT IMPULSE RESPONSE

3.1 Summary

Pavement impulse response to normal circular loading will be constructed in the

framework of LVET, where the AC is treated as linear viscoelastic material and the other

layers (BS, SB, and SG) as elastic.

Upon the establishment of governing equations in the physical domain (i.e. space

domain and time domain), two systems of vector functions will be introduced to first

transform the space domain to the vector functions domain. The associated elastic

solution for layered viscoelastic pavement is first formulated making use of the transfer

matrix method. The correspondence principle of Laplace transform is then adopted to

write the solution for the layered viscoelastic pavement in the Laplace domain.

By writing the relaxation modulus and creep compliance simultaneously, prior to

inverting Laplace transform numerically, analytical inverse Laplace transform can be

performed so that pavement impulse response can be expressed as a Volterra equation of

the second kind, which is then solved analytically in a matrix form. As a result, pavement

impulse response can be determined analytically, for the first time, in the time domain.

Thus, the method is named as semianalytical method and the solution through this

37

method as semianalytical solution. Note that semianalytical solution of pavement impulse

response refers only to the implicit convolution integral.

The work in this chapter is partly from the paper by Chen et al. (2009).

3.2 Governing Equations

The constitutive equation for asphalt concrete is introduced in section 1.3, and

rewritten here

( )( )( ) * 2 ( ) * jlkk

jl jl

d td tt tdt dt

εεσ λ δ μ= + (3.1)

where “*” denotes convolution integral hereafter, and

( )( )(1 )(1 2 )

E tt υλυ υ

=+ −

(3.2)

( )( )2(1 )

E ttμυ

=+

(3.3)

The constitutive equation for an elastic material is

2jl kk jl jlσ λε δ με= + (3.4)

The strain-displacement equation can be written as

, ,1 ( )2jl l j j lu uε = +

(3.5)

and by omitting the inertia effect7, the equilibrium equation can be written as

, 0jl lσ = (3.6)

7 Vehicle velocity, around 0-30m/s, is far slower than the surface wave

propagation speed, 100-600m/s (Jones et al. 1967), thus the inertia effect is neglected in pavement engineering.

38

A normal circular impulse load is applied to the pavement surface, i.e.

0 0( , 0; ) ( ) ( )zz r z t p t H d rσ δ= = − − (3.7)

where p0 and d0 denote load magnitude and loading radius, and

2 2s0 s0( ) ( )r x - x + y - y=

(3.8)

3.3 Associated Elastic Solution

First studied is the associated elastic solution for the viscoelastic pavement by

assuming every layer being purely elastic.

3.3.1 Two Systems of Vector Functions

Vector functions can be employed to treat the governing equations (Pan 1989a).

For example, the cylindrical system of vector functions is defined by

( , ; , )

( )

( )

z

r

z r

S r mS Sgrad Sr rS Scurl S

r r

θ

θ

θ ξ

θ

θ

=∂ ∂

= = +∂ ∂∂ ∂

= = −∂ ∂

L i

M i i

N i i i (3.9)

where ir, iθ and iz are unit vectors in the cylindrical coordinates, and

1( , ; , ) ( )2

immS r m J r e θθ ξ ξ

π= (3.10)

which satisfies the two-dimensional Helmholtz equation,

22

2 2

1 1 0S Sr Sr r r r

ξθ

∂ ∂ ∂⎛ ⎞ + + =⎜ ⎟∂ ∂ ∂⎝ ⎠ (3.11)

39

The Cartesian system of vector functions is defined by

( , ; , )

( )

( )

z

x y

z x y

S x yS Sgrad Sx yS Scurl Sy x

α β=∂ ∂

= = +∂ ∂∂ ∂

= = −∂ ∂

L i

M i i

N i i i

(3.12)

where ix, iy and iz are unit vectors in the Cartesian coordinates, and

0 0( ( ) ( ))1( , ; , )2

s si x x y yS x y e α βα βπ

− + −= (3.13)

where i is the imaginary unit. Equation 3.13 also satisfies the two-dimensional Helmholtz

equation,

2 22

2 2 0S S Sx y

ξ∂ ∂+ + =

∂ ∂ (3.14)

with

2 2 2ξ α β= + (3.15)

The space domain is related to the vector functions domain by

[ ]0

( , , ) L M Nm

r z F F F dθ ξ ξ∞

= + +∑∫f L M N (3.16)

2

2 20 0

1 1( , , )L M N rdrdπ

θξ ξ

∞ ∗ ∗ ∗⎛ ⎞= ⋅ + ⋅ + ⋅⎜ ⎟

⎝ ⎠∫ ∫F f L f M f N

(3.17)

for cylindrical coordinates, and

[ ]( ) L M Nx, y,z F F F d dα β∞ ∞

−∞ −∞= + +∫ ∫f L M N (3.18)

2 2

1 1( , , )L M N dxdyξ ξ

∞ ∞ ∗ ∗ ∗

−∞ −∞

⎛ ⎞= ⋅ + ⋅ + ⋅⎜ ⎟

⎝ ⎠∫ ∫F f L f M f N (3.19)

40

for Cartesian coordinates. Note superscript “*” denotes complex conjugate, i.e.

* 1( , ; , ) ( )2

immS r m J r e θθ ξ ξ

π−= (3.20)

for cylindrical coordinates, and

0 0( ( ) ( ))* 1( , ; , )2

s si x x y yS x y e α βα βπ

− − + −= (3.21)

for Cartesian coordinates.

It is interesting to note that the cylindrical system of vector functions corresponds

to the Hankel transform, and the Cartesian system of vector functions corresponds to the

Fourier transform. The two systems of vector functions, therefore, have the first

advantage of unifying the Hankel transform and Fourier transform. For the purpose of

illustration, this study will only use the more general Cartesian system of vector

functions. Furthermore, ξ, α, and β, corresponding to the reciprocal of r, x, and y, play

the roles of wavenumber.

3.3.2 Elastic Solution

Making use of the Cartesian system of vector functions, displacements u and

traction t in the space domain can be rewritten as

[ ]( , , ) ( , , )x y z L M Nx y z u u u U U U d dα β∞ ∞

−∞ −∞= = + +∫ ∫u L M N (3.22)

[ ]( , , ) ( , , )zx zy zz L M Nx y z T T T d dσ σ σ α β∞ ∞

−∞ −∞= = + +∫ ∫t L M N (3.23)

41

Applying the vector functions to the governing equations for the elastic material,

i.e., Equations 3.4 through 3.6, will establish the relationship between pavement response

in the space domain φ and that in the vector functions domain Φ, as listed in Table 3.1.

Table 3.1 Pavement Primary Response

( , , ) ( , ) ( , ; , )x y z z S x y d dφ Φ ξ α β α β∞ ∞

−∞ −∞= ∫ ∫

φ Φ

uz UL

ux UM (-iα) + UN (-iβ)

uy UM (-iβ) - UN (-iα)

σzz TL

σzx TM (-iα) + TN (-iβ)

σzy TM (-iβ) - TN (-iα)

σxx TLλ/(λ+2μ)-UM(β2λ+2α2(λ+μ))2μ/(λ+2μ)- UN (αβ)2μ

σyy TLλ/(λ+2μ)-UM(α2λ+2β2(λ+μ))2μ/(λ+2μ)+ UN (αβ)2μ

σxy UM μ (-2αβ)+UN μ (α2-β2)

εzz TL/(λ+2μ)+ UM (α2+β2)λ/(λ+2μ)

εzx TM(-iα) /(2μ) + TN (-iβ)/(2μ)

εzy TM (-iβ) /(2μ) - TN (-iα)/(2μ)

εxx UM (-α2)+UN (-αβ)

εyy UM (-β2)-UN (-αβ)

εxy UM (-αβ)+UN (α2-β2)/2

42

From Table 3.1, it is important to point out that pavement responses uz, ux, uy, σzz,

σzx, σzy, εxx, εyy and εxy are material-independent, while the others, i.e. σxx, σyy, σxy, εzz, εzx

and εzy, involve Lame constants λ and μ and are thus material-dependent.

In Table 3.1 UL, UM, UN, TL, TM, and TN are the six undetermined integral kernels,

which are governed by

10 02 2

11 0 0/ /

0 0 0 14( )0 0

2 2

L L

M M

L L

M M

U UU Ud

T TdzT T

λλ μ λ μ

ξ ξμξ

ξ ξ

λ μ μ λλ μ λ μ

⎡ ⎤⎢ ⎥+ +⎢ ⎥⎧ ⎫ ⎧ ⎫⎢ ⎥⎪ ⎪ ⎪ ⎪−⎪ ⎪ ⎪ ⎪⎢ ⎥=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭⎢ ⎥+⎢ ⎥−

+ +⎢ ⎥⎣ ⎦

(3.24)

10/ /

0

N N

N N

U UdT Tdz

μξξ ξ

μ

⎡ ⎤⎧ ⎫ ⎧ ⎫⎢ ⎥=⎨ ⎬ ⎨ ⎬⎢ ⎥⎩ ⎭ ⎩ ⎭⎢ ⎥⎣ ⎦

(3.25)

It is obvious to observe that in the vector functions domain the LM-component,

Equation 3.24, is decoupled from the N-component, Equation 3.25. In other words, to

solve the six integral kernels only one 4×4 and one 2×2 matrix are assembled, instead of

the conventional 6×6 matrix, leading to a reduction of computation time.

We add here that Equations 3.24 and 3.25 hold the same for both the cylindrical

system and the Cartesian system of vector functions. Thus, the vector functions provide

us the second advantage: whatever system is employed in the space domain, one only

needs to solve the vector functions domain once.

It is interesting to notice that ξ is the only variable in the vector functions domain

as shown in Equations 3.24 and 3.25, whose solutions can be easily written as

43

[ ] [ ] [ ]4 1 4 4 4 1( , ) ( , ) ( )H z Z z Kξ ξ ξ

× × ×= (3.26)

2 1 2 2 2 1( , ) ( , ) ( )N N NH z Z z Kξ ξ ξ

× × ×⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (3.27)

where [Z] and [ZN] are solution matrix, and [K] and [KN] are unknowns, and

[ ] [ ]( , ) / TL M L MH z U U T Tξ ξ ξ= (3.28)

[ ]( , ) / TNN NH z U Tξ ξ⎡ ⎤ =⎣ ⎦ (3.29)

Note superscript “T” in Equations 3.28 and 3.29 denotes transpose. Subscripts in

Equation 2.26 and 2.27, e.g., 4×1, denote the dimension and will be dropped for

simplicity unless otherwise pointed out. The elements in [Z] and [ZN] are listed in

APPENDIX A.

The transfer matrix method is employed to treat the layer structure, which will

connect integral kernels at the top of layer l, zl-1, to those at the bottom, zl, by

[ ] [ ][ ]1( , ) ( , ) ( , )l l lH z a h H zξ ξ ξ− = (3.30)

1( , ) ( , ) ( , )N N Nl l lH z a h H zξ ξ ξ−⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (3.31)

where [a] and [aN] are the propagator matrices, and detailed in APPENDIX A.

For position z located in layer j bounded by interface zj-1 and zj, we can obtain

[ ] [ ][ ]( , ) ( , ) ( )H z A z Kξ ξ ξ= (3.32)

( , ) ( , ) ( )N N NH z A z Kξ ξ ξ⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (3.33)

where

[ ] [ ][ ] [ ][ ][ ]1 1( , ) ( ) ( ) ( ) ( ) ( , )l l k k kA z a z z a h a h a h Z zξ ξ+ −= − (3.34)

1 1( , ) ( ) ( ) ( ) ( ) ( , )N N N N N Nl l k k kA z a z z a h a h a h Z zξ ξ+ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (3.35)

44

In Equations 3.32 and 3.33, [K] and [KN] are the unknowns in the half-space SG.

Considering the finiteness of the solution of the half-space, we have

[ ] [ ]2 1 2 14 1( ) [0] ,[ ( )] TK ξ κ ξ× ××

= (3.36)

2 1( ) 0, ( )

TN NK ξ κ ξ×

⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦ (3.37)

Making use of Equation 3.17, the loading condition in the space domain, i.e.,

Equation 3.7, is transformed to the vector functions domain as

*( ,0; ) ( ) ( )

( ,0; ) 0( ,0; ) 0

L zz L

M

N

T t S dxdy T t

T tT t

ξ σ ξ δ

ξξ

= =

==

∫ ∫ (3.38)

where

0 01 0( ) ( )L

p dT J dξ ξξ

= − (3.39)

The details on obtaining Equation 3.39 are given in APPENDIX B.

Substituting Equation 3.38 and 3.37 into Equation 3.33, it is easy to prove that the

N-components do not contribute to pavement response under a normal load. In the

following study, therefore, only the LM-components are to be addressed in detail. The N-

component, on the other side, will be given directly for completeness.

The associated elastic solution can also be derived from the cylindrical system of

vector functions (Pan et al 2007a), based on which a program, MultiSmart 3D, has been

developed for multi-layer pavement analysis.

45

3.4 Viscoelastic Solution

With the help of the associated elastic solution, the viscoelastic solution in the

Laplace domain is straightforwardly established through the correspondence principle.

3.4.1 Correspondence Principle

Applying the Laplace transform to the constitutive equation for viscoelastic

material, i.e. Equation 3.1, yields

( ) ( ) ( )* ( )jl kk jl jls s s s s e sσ λ ε δ μ= + (3.40)

By comparing Equation 3.40 with 3.4, the correspondence principle of Laplace transform

introduced in Section 1.2.4 will be adopted.

Correspondingly, Equation 3.32 for a viscoelastic material in the Laplace domain

can be rewritten as

( , ; ) ( , ; ) ( ; )H z s A z s K sξ ξ ξ⎡ ⎤⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦⎣ ⎦ (3.41)

For the viscoelastic pavement under study, only the AC, i.e. layer 1, is considered to be

viscoelastic, as a consequence

[ ][ ] [ ][ ][ ]1 2 1( , ; ) ( ) ( ) ( ) ( ) ( , )k k kA z s a z z a h a h a h Z zξ ξ−⎡ ⎤ = −⎣ ⎦ (3.42)

if the field point is located in AC layer, and

[ ] [ ][ ][ ] [ ]1( , ; ) ( ) ( 1 ( ) ( ) ( , ) ( , )j k k kA z s a z z a h a h a h Z z A zξ ξ ξ−⎡ ⎤ ⎡ ⎤= − + =⎣ ⎦⎣ ⎦ (3.43)

if the field point is located in layer j other than AC.

46

The 4×4 propagator matrix [a] for an elastic material can be subdivided as

11 12

21 22

1[ ] [ ]

[ ][ ] [ ]

a aE

aE a a

⎡ ⎤⎢ ⎥⎢ ⎥

= ⋅⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(3.44)

Note [a]jl (j, l=1, 2) is a 2×2 matrix independent of E.

For a viscoelastic material, based on the correspondence principle, the propagator

matrix in the Laplace domain is

11 12

21 22

1[ ] [ ]

[ ][ ] [ ]

a asE

asE a a

⎡ ⎤⎢ ⎥⎢ ⎥

= ⋅⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(3.45)

3.4.2 Special Cases

Applying the Laplace transform to the relaxation modulus Equation 1.25 and

creep compliance Equation 1.27, we have

1 1

( )1 1

M Mm m m

e 0m mm m

sE EsE s E Es s

ρρ ρ= =

= + = −+ +∑ ∑ (3.46)

01

( )1

Mm

m m

DsD s Dsτ=

= ++∑ (3.47)

where use has been made of Equation 1.28.

From Equation 3.46 and 3.47,

0lim ( ) (0)s

sE s E E→∞

= = (3.48)

47

0 1

lim ( ) ( )M

0 ms m

sE s E E E→

=

= − = ∞∑ (3.49)

0lim ( ) (0)s

sD s D D→∞

= = (3.50)

00 1

lim ( ) ( )M

ms m

sD s D D D→

=

= + = ∞∑ (3.51)

where use has been made of Equations 1.28 through 1.31.

Based on Equation 1.18, it is easy to find that Equations 3.48 through 3.51 satisfy

(0) (0) lim ( ) ( ) 1s

E D sE s sD s→∞

= = (3.52)

0( ) ( ) lim ( ) ( ) 1

sE D sE s sD s

→∞ ∞ = = (3.53)

Equations 3.52 and 3.53 reveal that at t=0 and t=∞, the relaxation modulus and creep

compliance of a viscoelastic material are inverse to each other, similar to an elastic

material as governed by Equation 1.4. Consequently, in these two cases, it can be

anticipated that a viscoelastic material may behave elastically, as proved below.

Making use of Equations 3.48 through 3.51, Equation 3.45 reduces to

11 12

21 22

1[ ] [ ](0)

lim[ ] [ (0)](0)[ ] [ ]

s

a aE

a aE a a

→∞

⎡ ⎤⎢ ⎥⎢ ⎥

= ⋅ =⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(3.54)

11 12

0

21 22

1[ ] [ ]( )

lim[ ] [ ( )]( )[ ] [ ]

s

a aE

a aE a a

→

⎡ ⎤⎢ ⎥∞⎢ ⎥

= ⋅ = ∞⎢ ⎥⎢ ⎥∞⎢ ⎥⎢ ⎥⎣ ⎦

(3.55)

48

Inserting Equation 3.54 and 3.55 into 3.42 or 3.43 and the results to 3.41 leads to

[ ]lim ( , ; ) ( , ;0) lim ( ; )s s

H z s A z K sξ ξ ξ→∞ →∞

⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦ (3.56)

[ ]0 0

lim ( , ; ) ( , ; ) lim ( ; )s s

H z s A z K sξ ξ ξ→ →

⎡ ⎤ ⎡ ⎤= ∞⎣ ⎦ ⎣ ⎦ (3.57)

Consequently in the time domain,

[ ] [ ][ ]( , ;0) ( , ;0) ( ;0)H z A z Kξ ξ ξ= (3.58)

[ ] [ ][ ]( , ; ) ( , ; ) ( ; )H z A z Kξ ξ ξ∞ = ∞ ∞ (3.59)

where in writing Equations 3.58 and 3.59 use has been made of APPENDIX C.

Equations 3.58 and 3.59 reveal that at initial state (t=0) and steady state (t=∞), the

viscoelastic problem degenerates to its associated elastic problem as in Equation 3.32. In

other words, at both t=0 and t=∞ viscoelastic problem can be treated elastically if

material properties, such as E(t), are assigned to the corresponding time.

3.4.3 Dual-Parameter Method

Pavement response at an arbitrary time other than t=0 and t=∞ is more practical

to pavement engineers. This will be addressed in detail as following.

Making use of Equation 1.18, Equation 3.45 can be rewritten as

11 12

21 22

[ ] ( )[ ][ ]

[ ] [ ]

a sD s aa

sE a a

⎡ ⎤⎢ ⎥= ⋅⎢ ⎥⎢ ⎥⎣ ⎦

(3.60)

Substituting Equations 3.46 and 3.47 into Equation 3.60 yields

01 1

[ ] [ ] [ ] [ ]1 1

M Mm m

E Dm mm m

E Da a a as sρ τ= =

−= + +

+ +∑ ∑

(3.61)

49

where

11 0 12

0

21 22

[ ] [ ][ ]

[ ] [ ]0

a D aa

E a a

⎡ ⎤⎢ ⎥= ⋅⎢ ⎥⎢ ⎥⎣ ⎦

(3.62)

21

0 0[ ]

[ ] 0Ea

a

⎡ ⎤⎢ ⎥= ⋅⎢ ⎥⎢ ⎥⎣ ⎦

(3.63)

120 [ ][ ]

0 0D

aa

⎡ ⎤⎢ ⎥= ⋅⎢ ⎥⎢ ⎥⎣ ⎦

(3.64)

In Equation 3.61, the propagator matrix for a viscoelastic material is decomposed into

two parts: a time-independent part noted by subscript “0”, which is equivalent to the

propagator matrix for the elastic material (Equation 3.44) at time t=0, and a time-

dependent part noted by subscripts “E” and “D”, which are contributed by E(t) and D(t).

For a field point in the asphalt concrete, substituting Equation 3.61 into Equation

3.42 yields

[ ] [ ] [ ]01 1

( , ; ) ( , ) ( , ) ( , )1 1

M Mm m

E Dm mm m

E DA z s A z A z A zs s

ξ ξ ξ ξρ τ= =

−⎡ ⎤ = + +⎣ ⎦ + +∑ ∑ (3.65)

where

[ ] [ ][ ] [ ][ ][ ]0 0 1 0 2 0 1 0 0( , ) ( ) ( ) ( ) ( ) ( , )k k kA z a z z a h a h a h Z zξ ξ−= − (3.66)

[ ] [ ][ ] [ ][ ][ ]1 0 2 0 1 0 0( , ) ( ) ( ) ( ) ( ) ( , )E E k k kA z a z z a h a h a h Z zξ ξ−= − (3.67)

[ ] [ ][ ] [ ][ ][ ]1 0 2 0 1 0 0( , ) ( ) ( ) ( ) ( ) ( , )D D k k kA z a z z a h a h a h Z zξ ξ−= − (3.68)

50

Substituting Equation 3.65 into 3.41 yields

[ ] [ ] [ ]01 1

( , ; ) ( , ) ( , ) ( , ) ( ; )1 1

M Mm m

E Dm mm m

E DH z s A z A z A z K ss s

ξ ξ ξ ξ ξρ τ= =

⎛ ⎞−⎡ ⎤ ⎡ ⎤= + +⎜ ⎟⎣ ⎦ ⎣ ⎦+ +⎝ ⎠∑ ∑

(3.69)

Equation 3.69 can be analytically inverted to the time domain:

[ ] [ ][ ] [ ] [ ]/0

1

( , ; ) ( , ) ( ; ) ( , ) * ( ; )i

Mtm

Em m

EH z t A z K t A z e K tρξ ξ ξ ξ ξρ

−

=

−= + ∑

[ ] [ ]/

1

( , ) * ( ; )m

Mtm

Dm m

DA z e K tτξ ξτ

−

=

+ ∑ (3.70)

Making use of Equation 3.36, Equation 3.70 can be simplified to

[ ] [ ] [ ] [ ] [ ]/0 12 12

1

( , ; ) ( , ) ( ; ) ( , ) * ( ; )m

Mtm

Dm m

DU z t A z t A z e tτξ ξ κ ξ ξ κ ξτ

−

=

= + ∑ (3.71)

[ ] [ ] [ ] [ ] [ ]/0 22 22

1

( , ; ) ( , ) ( ; ) ( , ) * ( ; )m

Mtm

Em m

ET z t A z t A z e tρξ ξ κ ξ ξ κ ξρ

−

=

−= + ∑ (3.72)

where [U]= [UL, ξUM]T, [T]=[TL/ξ, TM]T.

Similarly, for a field point outside of the asphalt concrete, substituting Equation

3.43 into Equation 3.41 and making use of Equation 3.36 yields

[ ] [ ] [ ]12( , ; ) ( , ) ( ; )U z t A z tξ ξ κ ξ= (3.73)

[ ] [ ] [ ]22( , ; ) ( , ) ( ; )T z t A z tξ ξ κ ξ= (3.74)

3.5 Volterra System of Equations

The key in Equations 3.71 through 3.74 is to solve [κ]. For the given traction

boundary Equation 3.38, setting z=0 in Equation 3.72 gives

51

[ ] [ ] [ ] [ ] [ ]/0 22 22

1

( ,0; ) ( ,0) ( ; ) ( ,0) * ( ; )m

Mtm

Em m

ET t A t A e tρξ ξ κ ξ ξ κ ξρ

−

=

−= + ∑ (3.75)

Equation 3.75 can be rewritten as

[ ] [ ]1

( ; ) [ ( )] ( ) [ ( )] ( ; )M

m mm

t g t C l w tκ ξ ξ δ ξ ξ=

= − ∑ (3.76)

where

[ ] [ ]10 22

[ ( )] ( ,0) ( ,0)g A Tξ ξ ξ−= (3.77)

[ ] [ ]10 22 22

[ ( )] ( ,0) ( ,0)EC A Aξ ξ ξ−= (3.78)

mm

m

Elρ−

= (3.79)

[ ] [ ]/( ; ) * ( ; )mtmw t e tρξ κ ξ−= (3.80)

Note in Equation 3.77, [T(ξ,0)]= [TL(ξ)/ξ, 0]T if use has been made of Equation 3.38.

Equation 3.76 is the linear Volterra system of equations of second kind (Linz

1985). Differentiating Equation 3.80 with respect to time t (denoted by a “⋅” over a

variable),

[ ] [ ] [ ]1( ; ) ( ; ) ( ; )m mm

w t t w tξ κ ξ ξρ

= − (3.81)

Incorporating Equation 3.76 into Equation 3.81 yields

[ ] [ ] [ ]1

1( ; ) [ ( )] ( ; ) ( ; ) [ ( )] ( )M

m m m mm m

w t C l w t w t g tξ ξ ξ ξ ξ δρ=

+ + =∑ (3.82)

Expanding Equation 3.82 into its matrix form yields

[ ]( ; ) [ ( )] ( ; ) [ ( )] ( )W t Q W t G tξ ξ ξ ξ δ⎡ ⎤ + =⎣ ⎦ (3.83)

52

where

[ ] 1 2( ; ) [ ( ; )] ,[ ( ; )] , [ ( ; )]TT T T

MW t w t w t w tξ ξ ξ ξ⎡ ⎤= ⎣ ⎦ (3.84)

[ ]( ) [ ( )] ,[ ( )] , [ ( )]TT T TG g g gξ ξ ξ ξ⎡ ⎤= ⎣ ⎦ (3.85)

[ ]

1 0 21

1 2 02

1 2 0

1 2 0

1[ ( )] [ ( )] [ ( )] [ ( )]

1[ ( )] [ ( )] [ ( )] [ ( )]

( )1[ ( )] [ ( )] [ ( )] [ ( )]

1[ ( )] [ ( )] [ ( )] [ ( )]

m M

m M

m Mm

m MM

C l I C l C l C l

C l C l I C l C l

QC l C l C l I C l

C l C l C l C l I

ξ ξ ξ ξρ

ξ ξ ξ ξρ

ξξ ξ ξ ξ

ρ

ξ ξ ξ ξρ

⎡ ⎤+⎢ ⎥⎢ ⎥⎢ ⎥+⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢⎢ +⎢⎢⎢⎢

+⎢⎣ ⎦

⎥⎥⎥⎥⎥⎥⎥

(3.86)

with I0=2×2 the identity matrix.

The solution to Equation 3.83 is

[ ] [ ] [ ] [ ]1( )( ; ) ( ) ( ) ( ) ( )m tW t X e X G H tω ξξ ξ ξ ξ−= (3.87)

where [X] and ωj are eigenvector matrix and eigenvalues of the 2M×2M matrix -[Q], i.e.

[ ] [ ] 1 [ ]jX X Qω − = − (3.88)

and “⟨ ⟩” indicates a diagonal matrix.

3.6 Pavement Impulse Response

With the establishment of [W] as in Equation 3.87, its component, [wm], can be

generally denoted as

53

22 1

2 1 2 2 2 112

( ; )[ ( ; )] [ ] [ ( )] ( )

( ; )j

Mtm

m mjjm

W tw t e g H t

W tωξ

ξ ξξ

−× × ×

=

⎡ ⎤= = Ψ⎢ ⎥⎣ ⎦

∑ (3.89)

where use has been made of Equation 3.85. Substituting Equation 3.89 into Equation 3.76

gives

[ ]2

2 1 2 2 2 12 11

( ; ) [ ( )] ( ) [ ] [ ( )] ( )jM

tm

j

t g t e g H tωκ ξ ξ δ ψ ξ× × ××=

= +∑ (3.90)

Inserting Equation 3.90 into Equations 3.71 through 3.74 yields the integral

kernels at an arbitrary field point, i.e.

2/

01 1

[ ( , ; )] [ ( , )] ( ) [ ( , )] ( ) [ ( , )] ( )j m

M Mt t

j mj m

U z t U z t U z e H t U z e H tω τδ ξ ξ δ ξ ξ −

= =

= + +∑ ∑ (3.91)

2/

01 1

[ ( , ; )] [ ( , )] ( ) [ ( , )] ( ) [ ( , )] ( )j m

M Mt t

j mj m

T z t T z t T z e H t T z e H tω ρδ ξ ξ δ ξ ξ −

= =

= + +∑ ∑

(3.92)

for a field point in AC, and

2

01

[ ( , ; )] [ ( , )] ( ) [ ( , )] ( )jM

tj

j

U z t U z t U z e H tωδ ξ ξ δ ξ=

= +∑ (3.93)

2

01

[ ( , ; )] [ ( , )] ( ) [ ( , )] ( )jM

tj

j

T z t T z t T z e H tωδ ξ ξ δ ξ=

= +∑ (3.94)

otherwise. Pavement impulse response in the vector functions domain Φ can thus be

obtained from Equation 3.91 through 3.94 and symbolically expressed as

0( , , ; ) ( , , ) ( ) ( , , ) ( )ntn

n

z t z t z e H tωδΦ α β Φ α β δ Φ α β= +∑ (3.95)

Consequently pavement impulse response φ to a moving load can be written in the space

domain as

( , ( ); ) ( , , ; ) ( ( ), ; , )t t z t S x t y d dδ δφ Φ α β α β α β∞ ∞

−∞ −∞= ∫ ∫sr r (3.96)

54

where use has been made of Equation 3.18. Detailed expressions for pavement impulse

response are listed in Table 3.2 where use has been made of the correspondence principle

of integral operator (Ashton and Moavenzadah 1968) and Equations 1.46 and 1.48.

Table 3.2 Pavement Impulse Response

( , ( ); ) ( , , ; ) ( ( ), ; , )t t z t S x t y d dδ δφ Φ α β α β α β∞ ∞

−∞ −∞= ∫ ∫sr r

φδ Φδ

uzδ UL

δ

uxδ UM

δ (-iα) + UNδ (-iβ)

uyδ UM

δ (-iβ) - UNδ (-iα)

σzzδ TL

δ

σzxδ TM

δ (-iα) + TN

δ (-iβ)

σzyδ TM

δ (-iβ) - TN

δ (-iα)

σxxδ TL

δυ/(1-υ) - E ∗UMδ(β2υ+α2)/(1-υ2) - E ∗UN

δ(βα)/(1+υ)

σyyδ TL

δυ/(1-υ) - E ∗UMδ(α2υ+β2)/(1-υ2) + E ∗UN

δ(βα)/(1+υ)

σxyδ E ∗UM

δ (-αβ) /(1+υ) + E ∗UNδ (α2-β2) /(2(1+υ))

εzzδ D *TL

δ(1+υ)(1-2υ)/(1-υ) + UMδ(α2+β2) υ/(1-υ)

εzxδ D *TM

δ (-iα) (1+υ) + D *TNδ (-iβ) (1+υ)

εzyδ D *TM

δ (-iβ) (1+υ) - D *TNδ (-iα) (1+υ)

εxxδ UM

δ (-α2) + UNδ (-βα)

εyyδ UM

δ (-β2) + UNδ (βα)

εxyδ UM

δ (-αβ) + UNδ (α2-β2)/2

55

In Table 3.2, E and D are the Laplace inverse transform of sE and sD

respectively, consequently

/0

1

( )( ) ( ) m

Mtm

m m

dE t EE t E t edt

ρδρ

−

=

= = −∑ (3.97)

/0

1

( )( ) ( ) m

Mtm

m m

dD t DD t D t edt

τδτ

−

=

= = +∑ (3.98)

Obviously, E and D in Equations 3.97 and 3.98 are actually the rate of E(t) and D(t).

From Equations 3.95, we observe that the temporal variable is separated from the

spatial variable, and more importantly, the relaxation mode, indicated by the relaxation

frequency ωn and relaxation strength Φn, is determined explicitly through the method

presented. In short, the pavement impulse response, for the first time, is written

analytically in the time domain. Since numerical schemes are still required to evaluate the

integral in Equation 3.96 with regard to the wavenumbers α and β, the proposed method

is thus termed as semianalytical method.

It can be observed that the integral kernels, ULδ, UM

δ, TLδ, and TM

δ in Table 3.2,

depend only on ξ, as a results, Φδ can be further decomposed into

[ ( )]qq kkq

q

βαδ δΦ Ω ξ α β( )( )( )= ∑ (3.99)

For example, associated with uxδ is q=1, [Ω(1)(ξ)]δ=-iUM

δ, 1kα(1) = , 0kβ

(1) = .

In Equation 3.99 kαj and kβj are integers satisfying kαj+ kβj≤2, and Ω(q)(ξ)

represents the algebra function of the integral kernels which, as a consequence of

Equations 3.91 to 3.94, can also be noted as

56

( ) ( )( ) ( )

1

[ ( , ; )] ( , ) ( ) ( , ) ( )q

n tq q qn

n

z t z t z e H tωδΩ ξ Ω ξ δ Ω ξ( )0

=

= +∑ (3.100)

With the decomposition of Equation 3.99, it is possible to further simplify

Equation 3.96 to

1

( , ( ); ) [ ( , ; )] ( ( ), ; , )qq kkq

q

t t z t S x t y d dβαδ δφ Ω ξ α β α β α β( )( )∞ ∞ ( )

−∞ −∞=

= ∑∫ ∫sr r (3.101)

Table 3.3 lists the detailed expressions of pavement impulse response in the form of

Equation 3.101. Note this form, for clarity, is referred to as the decomposed form.

3.7 Viscoelastic Half-space

For a viscoelastic half-space, Equation 3.42 reduces to

[ ( , ; )] [ ( , )]A z s Z zξ ξ= (3.102)

From APPENDIX A, it is easy to write

11 12

21 22

1 1[ ] [ ]

[ ( , )][ ] [ ]

Z ZsE sE

Z zZ Z

ξ

⎡ ⎤⎢ ⎥⎢ ⎥

= ⋅⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(3.103)

Inserting Equation 3.102 into 3.41 and making use of Equations 3.103, we have

[ ( , ; )] ( , ) [ ( ; )]H z s Z z K sξ ξ ξ⎡ ⎤= ⎣ ⎦ (3.104)

or

[ ]12[ ( , ; )] ( , ) [ ( ; )]U z s sD Z z sξ ξ κ ξ= (3.105)

[ ]22[ ( , ; )] ( , ) [ ( ; )]T z s Z z sξ ξ κ ξ= (3.106)

if use has been made of Equation 3.36 and 1.18.

57

Table 3.3 Pavement Impulse Response (Decomposed Form)

( , ( ); ) [ ( , ; )] ( ( ), ; , )qq kkq

q

t t z t S x t y d dβαδ δφ Ω ξ α β α β α β( )( )∞ ∞ ( )

−∞ −∞= ∑∫ ∫sr r

φδ q [ ( )]q δΩ ξ( ) qkα( )

qkβ( )

uzδ 1 UL

δ 0 0

uxδ 2 UM

δ (-i) 1 0

UNδ (-i) 0 1

uyδ 2 UM

δ (-i) 0 1

- UNδ (-i) 1 0

σzzδ 1 TL

δ 0 0

σzxδ 2 TM

δ (-i) 1 0

TNδ (-i) 0 1

σzyδ 2 TM

δ (-i) 0 1

- TNδ (-i) 1 0

σxxδ 4 TL

δυ/(1-υ) 0 0

-ξ2 E ∗UMδυ/(1-υ2) 0 2

-ξ2 E ∗UMδ/(1-υ2) 2 0

-ξ2 E ∗UNδ/(1+υ) 1 1

σyyδ 4 TL

δυ/(1-υ) 0 0

-ξ2 E ∗UMδυ/(1-υ2) 2 0

-ξ2 E ∗UMδ/(1-υ2) 0 2

ξ2 E ∗UNδ/(1+υ) 1 1

σxyδ 3 -ξ2 E ∗UM

δ/(1+υ) 1 1

ξ2 E ∗UNδ/(2(1+υ)) 2 0

-ξ2 E ∗UNδ/(2(1+υ)) 0 2

εzzδ 2 D *TL

δ (1+υ)(1-2υ)/(1-υ) 0 0

ξ2UMδυ/(1-υ) 0 0

58

Table 3.3 Pavement Impulse Response (Decomposed Form) (Continued)

( , ( ); ) [ ( , ; )] ( ( ), ; , )qq kkq

q

t t z t S x t y d dβαδ δφ Ω ξ α β α β α β( )( )∞ ∞ ( )

−∞ −∞= ∑∫ ∫sr r

φδ q [ ( )]q δΩ ξ( ) qkα( )

qkβ( )

εzxδ 2 (-i)ξD *TM

δ (1+υ) 1 0

(-i)ξD *TNδ (1+υ) 0 1

εzyδ 2 (-i)ξD *TM

δ (1+υ) 0 1

-(-i)ξD *TNδ (1+υ) 1 0

εxxδ 2 -ξ2UM δ 2 0

-ξ2UN δ 1 1

εyyδ 2 -ξ2UM δ 0 2

ξ2UN δ 1 1

εxyδ 3 -ξ2UM δ 1 1

ξ2UN δ/2 2 0

-ξ2UN δ/2 0 2

On account of the boundary condition in Equation 3.38, Equation 3.106

immediately yields

[ ( ; )] [ ( )]sκ ξ κ ξ= (3.107)

Inserting Equation 3.107 into 3.105 and 3.106,

[ ]12[ ( , ; )] ( ) ( , ) [ ( )]U z s sD s Z zδ ξ ξ κ ξ= (3.108)

[ ]22[ ( , ; )] ( , ) [ ( )]T z s Z zδ ξ ξ κ ξ= (3.109)

Applying Laplace inverse transform to Equations 3.108 and 3.109 yields

[ ]12[ ( , ; )] ( ) ( , ) [ ( )]U z t D t Z zδ ξ ξ κ ξ= (3.110)

[ ]22[ ( , ; )] ( , ) [ ( )] ( )T z t Z z tδ ξ ξ κ ξ δ= (3.111)

59

The stress integral kernels [T] respond elastically, while displacement integral kernels [U]

viscoelastically with their relaxation modes governed by D(t) on account of Equation

3.98.

60

CHAPTER IV

PAVEMENT STATIONARY RESPONSE

4.1 Summary

Making use of the pavement impulse response in Chapter III, the semianalytical

solution of pavement response to stationary load, i.e. pavement stationary response, is

further formulated. Properties of viscoelastic half-space in response to stationary load are

discussed. Similarly to pavement impulse response, pavement stationary response refers

to only the implicit convolution integral.

The goal of this chapter is to verify the semianalytical solution numerically with a

finite-element-based method, and further examine the validity of the well-known

collocation method.

4.2 Pavement Stationary Response

For stationary load, setting V=0 in Equation 3.101 and making use of Equation

2.13, pavement response then reduces to

0( , ; ) [ ( , ; )] ( , ; , )

qqt kkH q

q

t z t S x y d d dβαδφ Ω ξ τ α β α β α β τ( )( )∞ ∞ ( )

−∞ −∞= −∑∫ ∫ ∫sr r

(4.1)

Note the stationary load is applied at t≥0 and should be understood as a Heaviside load,

as denoted by the superscript “H”.

61

Making use of Equation B.3 and B.4, Equation 4.1 is rewritten as

1 ,

0 0( , ; ) [ ( , ; )] ( , ; )

q q q qt k k k kH q

q

t z t d C r dα αβ βδφ Ω ξ τ τ ξ θ ξ ξ( ) ( ) ( ) ( )∞ + +( )⎡ ⎤= −⎢ ⎥⎣ ⎦∑∫ ∫sr r (4.2)

where

2, cos( )

0

1( , ; ) (cos ) (sin )2

q q qqk k kk i rC r e dα β βαπ ξ θ ϕθ ξ ϕ ϕ ϕ

π( ) ( ) ( )( ) −= ∫

(4.3)

and the inner integral can be analytically evaluated:

0[ ( , ; )] ( 1) ( )

qn

qt tq q nq

n n

z t d e H tωδ0

ΩΩ ξ τ τ Ωω

( )( )

( ) ( )( )− = + −∑∫ (4.4)

where use has been made of Equation 3.100, but it is kept here temporarily for

convenience in expressing the solution. As a matter of fact, one only needs to numerically

evaluate the outer integral, which is already available by mathematical companions

(Lucas 1995). Therefore, the solution of pavement stationary response, similar to that of

pavement impulse response, is a semianalytical solution.

Table 4.1 lists pavement stationary response in the form of Equation 4.2. Also,

Equation 4.3 can be further expanded to:

, , ,( , ; )q q q q q qk k k k k kC r CC iCSα α αβ β βθ ξ( ) ( ) ( ) ( ) ( ) ( )

= + (4.5)

where

2,

0

1( , ; ) cos( cos( ))(cos ) (sin )2

q q qqk k kkCC r r dα β βαπ

θ ξ ξ ϕ θ ϕ ϕ ϕπ

( ) ( ) ( )( )

= −∫ (4.6a)

2,

0

1( , ; ) sin( cos( ))(cos ) (sin )2

q q qqk k kkCS r r dα β βαπ

θ ξ ξ ϕ θ ϕ ϕ ϕπ

( ) ( ) ( )( )

= −∫ (4.6b)

62

Table 4.1 Pavement Stationary Response

1 ,

0 0( , ; ) [ ( , ; )] ( , ; )

q q q qt k k k kH q

q

t z t d C r dα αβ βδφ Ω ξ τ τ ξ θ ξ ξ( ) ( ) ( ) ( )∞ + +( )⎡ ⎤= −⎢ ⎥⎣ ⎦∑∫ ∫sr r

φ j [ ( , ; )]q z t δΩ ξ τ( ) − , ( , ; )q qk kC rα β θ ξ( ) ( )

uz 1 ULδ C0,0

ux 2 UMδ (-i) C1,0

UN δ (-i) C0,1

uy 2 UMδ (-i) C0,1

- UNδ (-i) C1,0

σzz 1 TLδ C0,0

σzx 2 TMδ (-i) C1,0

TNδ (-i) C0,1

σzy 2 TMδ (-i) C0,1

- TNδ (-i) C1,0

σxx 4 TLδ υ/(1-υ) C0,0

-ξ2 E ∗UMδ υ/(1-υ2) C0,2

-ξ2 E ∗UMδ /(1-υ2) C2,0

-ξ2 E ∗UNδ /(1+υ) C1,1

σyy 4 TLδ υ/(1-υ) C0,0

-ξ2 E ∗UMδ υ/(1-υ2) C0,2

-ξ2 E ∗UMδ /(1-υ2) C2,0

ξ2 E ∗UNδ /(1+υ) C1,1

σxy 3 -ξ2 E ∗UMδ /(1+υ) C1,1

ξ2 E ∗UNδ /(2(1+υ)) C2,0

-ξ2 E ∗UNδ /(2(1+υ)) C0,2

εzz 2 D *TLδ (1+υ)(1-2υ)/(1-υ) C0,0

ξ2UMδ υ/(1-υ) C0,0

εzx 2 (-i)ξD *TMδ (1+υ) C1,0

(-i)ξD *TNδ (1+υ) C0,1

63

Table 4.1 Pavement Stationary Response (Continued)

1 ,

0 0( , ; ) [ ( , ; )] ( , ; )

q q q qt k k k kH q

q

t z t d C r dα αβ βδφ Ω ξ τ τ ξ θ ξ ξ( ) ( ) ( ) ( )∞ + +( )⎡ ⎤= −⎢ ⎥⎣ ⎦∑∫ ∫sr r

φ q [ ( , ; )]q z t δΩ ξ τ( ) − , ( , ; )q qk kC rα β θ ξ( ) ( )

εzy 2 (-i)ξD *TMδ (1+υ) C0,1

-(-i)ξD *TNδ (1+υ) C1,0

εxx 2 -ξ2UMδ C2,0

-ξ2UNδ C1,1

εyy 2 -ξ2UMδ C0,2

ξ2UNδ C1,1

εxy 3 -ξ2UMδ C1,1

ξ2UNδ /2 C2,0

-ξ2UNδ /2 C0,2

Table 4.2 Analytical Expression of Function C(r, θ; ξ) 2, cos( )

0

1( , ; ) (cos ) (sin )2

q q qqk k kk i rC r e dα β βαπ ξ θ ϕθ ξ ϕ ϕ ϕ

π( ) ( ) ( )( ) −= ∫

C0,0

1 CC0,0 0 ( )J rξ C1,1

1 CC1,1 2

1 ( )sin(2 )2

J rξ θ−

i CS 0,0 0 i CS 1,1 0

C1,0

1 CC1,0 0 C2,0

1 CC2,0 0 2

1 1( ) ( )cos(2 )2 2

J r J rξ ξ θ−

i CS 1,0 1( )cosJ rξ θ i CS 2,0 0

C0,1

1 CC0,1 0 C0,2

1 CC0,2 0 2

1 1( ) ( ) cos(2 )2 2

J r J rξ ξ θ+

i CS 0,1 1( )sinJ rξ θ i CS 0,2  0

 

Note: Columns 2 and 6 are the coefficient accompanying function CC or CS

64

As remarked before, kαj+ kβj≤2, totally there are 6 types of function C(r, θ; ξ),

namely C00, C01, C02, C10, C11 and C20. Function C(r, θ; ξ) can then be expressed fully

analytically as listed in Table 4.2. The methodology is elaborate in APPENDIX D.

The outer integral in Table 4.1, which makes use of the Cartesian system of vector

functions, is comparable to the solution of an elastic layer structure based on the

cylindrical system of vector functions (Pan et al 2007b). Not only does this prove the two

systems of vector functions are unified as stated before, but also the response of

viscoelastic pavement to stationary load is axisymmetrical.

4.3 Half-space to Stationary Load

For a viscoelastic half-space, pavement stationary response also follows Equation

4.2. However, its inner integral will present some unique properties. Integrating

Equations 3.110 and 3.111 in the time domain yields

[ ]120[ ( , ; )] [ ( , ; )] ( ) ( , ) [ ( )]

tHU z t U z d D t Z zδξ ξ τ τ ξ κ ξ= =∫ (4.7)

[ ]220[ ( , ; )] [ ( , ; )] ( , ) [ ( )]

tHT z T z t d Z zδξ τ ξ τ ξ κ ξ= =∫ (4.8)

where use has been made of Equation 3.98. Reassembling Equations 4.7 and 4.8 yields

11 12

21 22

( )[ ] ( )[ ]( , ; ) 0

( ; )( , ; ) [ ] [ ]

H

H

D t Z D t ZU z t

tT z t Z Z

ξκ ξξ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥= ⋅⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎢ ⎥⎣ ⎦

(4.9)

65

Equation 4.9 is equivalent to

[ ] ( )

( , ; ) 0( , )

( ; )( , ; )

H

D D tH

U z tZ z

tT z tξ

ξκ ξξ =

⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦ (4.10)

where [Z] is listed in APPENDIX A.

Equation 4.10 is important in that a viscoelastic half-space to a stationary load can

be treated elastically by just inserting the time-dependent creep compliance D(t) at the

corresponding time into the elastic model. This corresponds to the “quasi-elastic” method

proposed by Schapery (1965) which laid the basis for the first type of modified elastic

model. However, we emphasize here that this method holds true only for the viscoelastic

half-space subject to a stationary load with compliance D being replaced by creep

compliance D(t), and its validity under a moving load needs to be further examined.

4.4 Numerical Verification

Since the semianalytical solution in this study, in its current form, is based on the

assumption of constant Poisson’s ratio, for the purpose of verification, the example

studied by Xu (2004) will be retrieved.

In the example, a three-layer pavement is subjected to a dual-tire load. Each tire

load is simplified to a circular load with loading radius 3.785 in and tire pressure 100 psi.

The centers of the loading circles are (0, 0 in) and (0, 12 in). Pavement configuration is

listed in Table 4.3. The AC is assumed to be viscoelastic with its relaxation modulus

being listed in Table 4.4.

66

Table 4.3 Structural Properties of Pavement System I (Xu 2004) Layer Thickness (in) Young’s Modulus (psi) Poisson’s Ratio

AC 8.0 Viscoelastic 0.35

Subbase 8.0 30,000 0.3

Subgrade Infinite 5,000 0.3

Table 4.4 Relaxation Modulus and Creep Compliance of Viscoelastic AC Layer (Xu 2004)

j Ej (psi) ρj (s) Dj (psi-1) τj (s)

- 12,500 (Ee) - 7.92451×10-7 -

1 735,300 0.008441 8.72639×10-7 0.0189201

2 386,200 0.1319 3.56881×10-9 0.452776

3 107,500 1.968 1.80369×10-5 8.61009

4 20,360 39.25 5.67292×10-5 117.703

Since in the present semianalytical method, both the relaxation modulus and creep

compliance are input simultaneously, we first have to obtain the creep compliance from

the relaxation modulus through the interconversion procedure. Making use of

Mathematica 5 (Wolfram Research Inc., 2003) and Equation 1.18, the analytical

expression of the creep compliance can be obtained straightforwardly and listed in Table

4.3.

Figures 4.1 plot the variation of E(t), D(t), and E(t)D(t) with time. In short time,

E(t) decays quickly and D(t) increases gradually. It is important to notice that E(t)D(t),

except at t=0 (initial state) and t→∞ (steady state), deviates from 1. Noticing that

67

E(t)D(t)=1 for elastic material, E(t)D(t) may be the indicator of the material’s

viscoelasticity: the closer to 1 the more elastic, rather than viscous. In the current

example, the smallest value of E(t)D(t) is around 0.65, obtained at 0.45 sec.

Figure 4.2 compares the predicted stresses and strains based on the present

semianalytical method and those based on Xu’s finite-element-based method. Two

occasions, t=0.01 s and t=1 s, are examined. From the comparisons, we can clearly

observe that at t=0.01 s the two methods mostly show very good agreements, verifying

the accuracy of semianalytical method, while at t=1 s, obvious discrepancies can be

observed between the two methods. Notice that Xu’s method adopted an algorithm

similar to Equation 2.10 and treated the convolution integral numerically, under which

error may accumulate with increasing computation steps as introduced in Section 2.3.

This may explain the negligible discrepancy at small time t=0.01s but obvious

0 200 400 600 8000.5

0.6

0.7

0.8

0.9

1

E(t)D

(t)

Time t (s)

Figure 4.1. Variation of (a) the relaxation modulus E(t) and creep compliance D(t), and (b) the product of E(t) and D(t) with time.

0 0.1 0.2 0.3 0.4 0.50.6

0.7

0.8

0.9

1

0 1 2 3 4 5

Time t (s)

0

0.4

0.8

1.2

1.6

E(t)

(106 p

si)

0

0.4

0.8

1.2

1.6

D(t)

(10-5

psi-1

)

E(t)D(t)

(a)

(b)

68

discrepancy at large time t=1 s, while, the semianalytical method is analytical in the time

domain and thus avoids such problems.

Figure 4.2. Comparisons between the present semianalytical solution and finite-element-based solution: (a) stress σzz; (b) stress σyz; (c) strain εxx; and (d) strain εyy.

(a)

(c) (d)

0 -20 -40 -60 -80 -100

σzz (psi)

60

40

20

0

70

50

30

10

Dep

th (i

n)

t=0.01 (Present)t=0.01 (Xu 2004)t=1 (Present)t=1 (Xu 2004)

0 4 8 122 6 10

σyz (psi)

60

40

20

0

70

50

30

10

Dep

th (i

n)

t=0.01 (Present)t=0.01 (Xu 2004)t=1 (Present)t=1 (Xu 2004)

(b)

-0.0004 -0.0002 0 0.0002 0.0004

εyy

60

40

20

0

70

50

30

10

Dep

th (i

n)

t=0.01 (Present)t=0.01 (Xu 2004)t=1 (Present)t=1 (Xu 2004)

-0.0004 0 0.0004

εxx

60

40

20

0

70

50

30

10

Dep

th (i

n)

t=0.01 (Present)t=0.01 (Xu 2004)t=1 (Present)t=1 (Xu 2004)

69

The finite-element-based method, with the drawback of intensive computation

and accumulated error, is appealing to evaluate pavement’s short-term response.

However, in evaluating pavement performance during its service life, which is designed

for years’ long, we are more interested in capturing pavement response at an arbitrary

time. The finite-element-based method may fail for such case, while the semianalytical

method can then be applied due to its unique advantage of being analytical in the time

domain.

Figure 4.3 illustrates the horizontal strains εxx and εyy at times up to 800 s. It is

obvious that both strains will first increase with time in its initial range (0, 5 s), after

which they decrease gradually. Around t=200 s, pavement response will arrive at a steady

state and no large variation can be observed thereafter. Also in this example, εxx is always

Figure 4.3. Variations of horizontal strains εxx and εyy at (x, y, z) = (0, 0, 7.99 in) with time: (a) Short-term response; and (b) long-term response.

(a)

0 200 400 600 800

Time t (s)

0

0.0001

0.0002

0.0003

0.0004

Hor

izon

tal S

train

s

εxxεyy

(b)

0 10 20 30 40

Time t (s)

0

0.0001

0.0002

0.0003

0.0004

Hor

izon

tal S

train

s

εxxεyy

70

larger than εyy, and will play the role of the critical strain in evaluating pavement’s fatigue

cracking.

4.5 Examination of Collocation Method

An alternative to evaluate pavement response analytically in the time domain is

the collocation method introduced in Section 2.5.2. Since the present method is genuinely

analytical in the time domain, it constitutes a benchmark solution to validate the

collocation method.

Figures 4.4 and 4.5 compare selected pavement response by the present

semianalytical method and the collocation method. In the collocation method, two

collocation schemes have been examined: one is the seven-point collocation scheme used

in (Huang 1993), with collocation points {0.01, 0.03, 0.1, 1, 10, 30, ∞ s}; the other one

has eighteen collocation points, i.e. {0.01, 0.03, 0.1, 1, 3, 10, 30, 60, 100, 200, 300, 400,

500, 600, 700, 800, 2000, ∞ s}.

Figure 4.4 reveals that both the seven-point and eighteen-point collocation

schemes can capture very well the trend of the deflection uz, and that the eighteen-point

collocation scheme predicts much closer results than the seven-point collocation scheme.

The improved prediction by the eighteen-point collocation scheme can also be observed

for the critical strain εxx as in Figure 4.5. However, it should be noticed that in spite of the

improved prediction for εxx, the eighteen-point collocation scheme presents oscillating

results, failing to capture the monotonic trend as shown in Figure 4.5b.

71

Figure 4.5. Comparisons of the critical strain εxx at (x, y, z) = (0, 0, 7.99 in) between the present semianalytical solution and collocation solution: (a) Short-term response;

and (b) long-term response.

0 200 400 600 800

Time t (s)

0

0.0001

0.0002

0.0003

0.0004

ε xx

Present MethodCollocation Method (7 Points)Collocation Method (18 Points)

(b)

(a)

Figure 4.4. Comparisons of deflection uz at (x, y, z) = (0, 0, 0) between the present semianalytical solution and collocation solution: (a) Short-term response; and (b)

long-term response.

0 200 400 600 800

Time t (s)

0

0.02

0.04

0.06

0.08

u z (i

n)

Present MethodCollocation Method (7 Points)Collocation Method (18 Points)

(b) (a)

0 2 4 6 8 10

Time t (s)

0.01

0.02

0.03

0.04

0.05

u z (i

n) Present MethodCollocation Method (7 Points)Collocation Method (18 Points)

0 2 4 6 8 10

Time t (s)

0

0.0001

0.0002

0.0003

0.0004

ε xx

Present MethodCollocation Method (7 Points) Collocation Method (18 Points)

72

We close this section with an explanation for the inadequacy of the collocation

method. In Section 3.6, the pavement response in Equations 3.91 through 3.94 indeed is

in the form of the Prony series in the vector function domain. However, the relaxation

frequencies ωj are functions of the wavenumber ξ. Consequently, when inverted to the

space domain an infinite number of wavenumbers ξ (through α or β) are called in

numerically integrating Equation 3.96. In other words, the pavement response is actually

in the form of a Prony series with infinite terms. Hence, a Prony series with finite terms,

such as the collocation method, would rarely capture the true solution. Also, a collocation

scheme with increased terms can, theoretically, predict closer results, as shown by the

eighteen-point collocation scheme as compared to the seven-point collocation scheme.

73

CHAPTER V

PAVEMENT MOVING DYNAMIC RESPONSE

5.1 Summary

The semianalytical solution of pavement response to a moving dynamic load, i.e.

pavement moving dynamic response, is first elaborated in this chapter. Then, the

proposed semianalytical solutions are verified with the finite element program Abaqus

for typical loading cases. This chapter focuses on the explicit convolution integral.

5.2 Pavement Moving Dynamic Response

Due to the suspension system and vehicle-road interaction, a vehicle usually

triggers a dynamic load, which plays an important role in pavement performance

(Markow et al. 1988). For a generic dynamic load f(t), it is sufficient to assume

( ) ki tk

k

f t F e ω=∑ (5.1)

Note Equation 5.1 is equivalent to the discrete Fourier transform. A dynamic load,

through Equation 5.1, can be immediately reduced to a static load or a harmonically

oscillating load.

74

Inserting Equations 3.101 and 5.1 into 2.6 yields the pavement moving dynamic

response:

0( , ( ); ) [ ] ( , , ( ), ( ); , )

qqk

t ki kqk s s

k q

t t F S x y x y e d d dβαω τδφ Ω τ τ α β τα β α β( )( )∞ ∞ ( )

−∞ −∞= ∑∑ ∫ ∫ ∫sr r

(5.2)

Note the load is applied at t≥0, and

0

0

( )( )

s s

s s

x t x ty t y

= +=

V (5.3)

Operating on the inner integral in the time domain yields

( )

( )

( )

( )0 0

0

( )

( )0 0

( , , ( ), ; , )( , , ( ), ; , )

1( , ( ); )( , , , ; , )

1

k

kqq

qn k

qn

qn k

i tqs s

i ts ska kq

ik nk q q

tn s sn ka

i

S x y x t y eS x y x t y e

it t F dS x y x yi e

i

βα

ω

ω

ω ω

ω

ω ω

Ω α βα β

φ α β αΩα βω ω

( )( )∞ ∞

−−∞ −∞

−

⎧ ⎫+⎪ ⎪

⎛ ⎞⎪ ⎪⎜ ⎟⎪ ⎪+= ⎨ ⎬⎜ ⎟

⎪ ⎪⎜ ⎟− +⎪ ⎪−⎜ ⎟⎜ ⎟+⎪ ⎪⎝ ⎠⎩ ⎭

∑∑ ∫ ∫ ∑V

s

V

r r dβ

(5.4)

where use has been made of Equations 3.100, 3.13 and 5.3.

Substituting Equations B3 and B4 into Equation 5.4 yields

( )

( )0

2 1( )( )0 0

( )

( )

( ( ), ( ); , )( ( ), ( ); , )

( , ( ); ) 1 cos( (0), (0); , )1 cos

k

k q q

qn

i tq

i tk kqq

k nknk q q

tn n kq

nk

S r t t eS r t t e

t t F d diS ri e

i

α β

ω

ωπ

ω

Ω θ ξ ϕθ ξ ϕ

φ ϕξ ξρ ϕΩθ ξ ϕω ωρ ϕ

( ) ( )∞ + +

⎧ ⎫+⎪ ⎪

⎛ ⎞⎪ ⎪⎪ ⎪⎜ ⎟= +⎨ ⎬⎜ ⎟⎪ ⎪− + ⎜ ⎟⎪ ⎪−⎜ ⎟+⎪ ⎪⎝ ⎠⎩ ⎭

∑∑ ∫ ∫ ∑sr r

(5.5)

75

where

( )( )

qnk q

n kiξρ

ω ω=

−V (5.6)

( )cos( ( ))1( ( ), ( ); , )2

i r t tS r t t e ξ ϕ θθ ξ ϕπ

−= (5.7)

The parameter ( )qnkρ is termed as the generalized coupling coefficient, describing the

coupling effect between moving frequency ξV, relaxation frequency ( )qnω , and

oscillating frequency ωk. Notice ( )qnω and ωk are not coupled to each other. For static

load ωk=0, a coupling effect exists between moving frequency ξV and relaxation

frequency ( )qnω . In following discussion, for simplicity, ( )q

nkρ is noted by ρ.

The radial coordinate ξ and circumferential coordinate ϕ in Equation 5.5 can be

separated, which can be rewritten as

( )

,( )0

1,( )0

( ) ,

( ( ), ( ); ; 0)

( , ( ); ) ( ( ), ( ); ; )

( (0), (0); ; )

j j k

q qq qk

q q qn

k k i tq

k kk k i tqk n

k q q k k tn n k

C r t t e

t t F dC r t t ei C r e

α β

α βα β

α β

ω

ω

ω

Ω θ ξ ρ

φ ξ ξθ ξ ρΩω ω θ ξ ρ

( ) ( )( ) ( )

( ) ( )

∞ + +

⎧ ⎫= +⎪ ⎪⎪ ⎪⎛ ⎞= ⎨ ⎬⎜ ⎟⎪ ⎪⎜ ⎟− + −⎪ ⎪⎝ ⎠⎩ ⎭

∑∑ ∫ ∑sr r

(5.8)

where

cos( )2,

0

1( , ; ; ) (cos ) (sin )2 1 cos

q q qqi r

k k kkeC r di

α β βα

ξ ϕ θπθ ξ ρ ϕ ϕ ϕ

π ρ ϕ( ) ( ) ( )( )

−

=+∫ (5.9)

Comparing Equations 5.9 and 4.3, it is clear that Equation 4.3 is the special case of

Equation 5.9 with ρ=0. Henceforth, the same notation ,q qk kC α β( ) ( )

is used. However, one

should bear in mind that for moving dynamic response, i.e. Equation 5.9, an extra

76

parameter ρ is implied. Table 5.1 lists pavement moving dynamic response in the form of

Equation 5.8.

At first glance, in Equation 5.8, pavement response is written in a single integral.

However, an additional integral is implied in Eq. 5.9 for function C(r, θ; ξ; ρ) which can

be further decomposed to

( ) ( ), , 1, , 1,( , ; ; )q q q q q q q q q qk k k k k k k k k kC r CC CS i CS CCα α α α αβ β β β βθ ξ ρ ρ ρ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )+ += + + − (5.10)

where

2,2 20

1 cos( cos( ))( , ; ; ) (cos ) (sin )2 1 (cos )

q q qqk k kkrCC r dα β βαπ ξ ϕ θθ ξ ρ ϕ ϕ ϕ

π ρ ϕ( ) ( ) ( )( )−

=+∫ (5.11a)

2,2 20

1 sin( cos( ))( , ; ; ) (cos ) (sin )2 1 (cos )

q q qqk k kkrCS r dα β βαπ ξ ϕ θθ ξ ρ ϕ ϕ ϕ

π ρ ϕ( ) ( ) ( )( )−

=+∫ (5.11b)

Function C(r, θ; ξ; ρ) can be expressed fully analytically as listed in Table 5.2. The

details are elaborated in APPENDIX E.

77

Table 5.1 Pavement Moving Dynamic Response

( )

,( )0

1,( )0

( ) ,

( ( ), ( ); ; 0)

( , ( ); ) ( ( ), ( ); ; )

( (0), (0); ; )

j j k

q qq qk

q q qn

k k i tq

k kk k i tqk n

k q q k k tn n k

C r t t e

t t F dC r t t ei C r e

α β

α βα β

α β

ω

ω

ω

Ω θ ξ ρ

φ ξ ξθ ξ ρΩω ω θ ξ ρ

( ) ( )( ) ( )

( ) ( )

∞ + +

⎧ ⎫= +⎪ ⎪⎪ ⎪⎛ ⎞= ⎨ ⎬⎜ ⎟⎪ ⎪⎜ ⎟− + −⎪ ⎪⎝ ⎠⎩ ⎭

∑∑ ∫ ∑sr r

φ q ( ) ( , )q zξΩ , ( , ; ; )q qk kC rα β θ ξ ρ( ) ( )

uz 1 UL C0,0

ux 2 UM(-i) C1,0

UN (-i) C0,1

uy 2 UM(-i) C0,1

- UN(-i) C1,0

σzz 1 TL C0,0

σzx 2 TM(-i) C1,0

TN(-i) C0,1

σzy 2 TM(-i) C0,1

- TN(-i) C1,0

σxx 4 TLυ/(1-υ) C0,0

-ξ2 E ∗UM υ/(1-υ2) C0,2

-ξ2 E ∗UM/(1-υ2) C2,0

-ξ2 E ∗UN/(1+υ) C1,1

σyy 4 TL υ/(1-υ) C0,0

-ξ2 E ∗UM υ/(1-υ2) C0,2

-ξ2 E ∗UM/(1-υ2) C2,0

ξ2 E ∗UN/(1+υ) C1,1

σxy 3 -ξ2 E ∗UM/(1+υ) C1,1

ξ2 E ∗UN/(2(1+υ)) C2,0

-ξ2 E ∗UN/(2(1+υ)) C0,2

78

Table 5.1 Pavement Moving Dynamic Response (Continued)

( )

,( )0

1,( )0

( ) ,

( ( ), ( ); ; 0)

( , ( ); ) ( ( ), ( ); ; )

( (0), (0); ; )

j j k

q qq qk

q q qn

k k i tq

k kk k i tqk n

k q q k k tn n k

C r t t e

t t F dC r t t ei C r e

α β

α βα β

α β

ω

ω

ω

Ω θ ξ ρ

φ ξ ξθ ξ ρΩω ω θ ξ ρ

( ) ( )( ) ( )

( ) ( )

∞ + +

⎧ ⎫= +⎪ ⎪⎪ ⎪⎛ ⎞= ⎨ ⎬⎜ ⎟⎪ ⎪⎜ ⎟− + −⎪ ⎪⎝ ⎠⎩ ⎭

∑∑ ∫ ∑sr r

φ q ( ) ( , )q zξΩ ,q qk kC α β( ) ( )

εzz 2 D *TL (1+υ)(1-2υ)/(1-υ) C0,0

ξ2UMυ/(1-υ) C0,0

εzx 2 (-i)ξD *TM (1+υ) C1,0

(-i)ξD *TN (1+υ) C0,1

εzy 2 (-i)ξD *TM (1+υ) C0,1

-(-i)ξD *TN (1+υ) C1,0

εxx 2 -ξ2UM C2,0

-ξ2UN C1,1

εyy 2 -ξ2UM C0,2

ξ2UN C1,1

εxy 3 -ξ2UM C1,1

ξ2UN/2 C2,0

-ξ2UN/2 C0,2

Note: Symbol in bold should be understood as its components, e.g. Ω(q) =Ω0(q), Ω1

(q),…,

Ωn(q)

79

Table 5.2 Analytical Expression of Function C(r, θ; ξ; ρ) cos( )2,

0

1( , ; ; ) (cos ) (sin )2 1 cos

q q qqi r

k k kkeC r di

α β βα

ξ ϕ θπθ ξ ρ ϕ ϕ ϕ

π ρ ϕ( ) ( ) ( )( )

−

=+∫

C0,0

1 CC0,0 ( )0 0 2 21

( ) ( ) 2 ( ) ( ) ( )cos 2mm m

m

J r NC J r NC mξ ρ ξ ρ θ=

+ −∑

ρ CS 1,0 ( ) ( )2 1 2 2 20

( ) ( ) ( ) ( ) cos (2 1)mm m m

m

J r NC NC mξ ρ ρ θ+ +=

− + +∑ 

C1,0

i CS 1,0  ( ) ( )2 1 2 2 20

( ) ( ) ( ) ( ) cos (2 1)mm m m

m

J r NC NC mξ ρ ρ θ+ +=

− + +∑ 

-ρ i CC2,0  ( )0,0 0,02

1 ( 0)CC CCρρ

= −

C0,1

i CS 0,1  ( ) ( )2 1 2 2 20

( ) ( ) ( ) ( ) sin (2 1)mm m m

m

J r NC NC mξ ρ ρ θ+ +=

− − +∑ 

-ρ i CC1,1  ( ) ( )2 2 2 2 21

1 ( ) ( ) ( ) ( ) sin 22

mm m m

m

J r NC NC mξ ρ ρ θ− +=

− −∑

C1,1

1 CC1,1 ( ) ( )2 2 2 2 2

1

1 ( ) ( ) ( ) ( ) sin 22

mm m m

m

J r NC NC mξ ρ ρ θ− +=

− −∑

ρ CS 2,1 ( )0,1 0,12

1 ( 0)CS CSρρ

= −

C2,0

1 CC2,0 ( )0,0 0,02

1 ( 0)CC CCρρ

= −

ρ CS 3,0 ( )1,0 1,02

1 ( 0)CS CSρρ

= −

C0,2

1 CC0,2 0,0 2,0CC CC− 

ρ CS 1,2 1,0 3,0CS CS− 

Note: Column 2 is the coefficient accompanying function CC or CS

80

Function C(r, θ; ξ; ρ), based on Table 5.2, can be compactly written as

, ,

0

( , ; ; ) ( ) ( , )q q q qk k k k

m mm

C r J rα αβ βθ ξ ρ ξ χ θ ρ( ) ( ) ( ) ( )

=

= ∑ (5.12)

where , ( , )q qk k

mα βχ θ ρ( ) ( )

is an algebraic function of ρ and θ. Inserting Equation 5.12 into

Equation 5.2 yields

( )

,( )

1,( )0

( ) ,

( ( )) ( ( ); 0)

( , ( ); ) ( ( )) ( ( ); )

( (0)) ( (0); )

q qk

q qq qk

q q qn

k k i tqm m

k kk k i tqk m mnk q m q k k tn n k m m

J r t t e

t t F dJ r t t ei J r e

α β

α βα β

α β

ω0

ω

ω

Ω ξ χ θ ρ

φ ξ ξξ χ θ ρΩω ω ξ χ θ ρ

( ) ( )

( ) ( )( ) ( )

( ) ( )

∞ + +

⎧ ⎫= +⎪ ⎪⎪ ⎪⎛ ⎞= ⎨ ⎬

⎜ ⎟⎪ ⎪− + ⎜ ⎟−⎪ ⎪⎝ ⎠⎩ ⎭

∑∑∑ ∫ ∑sr r

(5.13)

Equation 5.13 is referred to as the semianalytical solution of the pavement

moving dynamic response since it contains, once again, only one integral for

wavenumber ξ, which needs to be evaluated numerically.

The computation efficiency of Equation 5.13, obviously, is mainly determined by

the four summations ove k, q, m, n, whose size are elaborated as below:

1) k: As presented in Equation 5.1, the size of k depends on dynamic load f(t).

The smallest size of k is 1 for harmonically oscillating load.

2) q: As listed in Table 3.3, the size of q differs between different responses. For

example the size of q is 1 for uz while 2 for ux. Further, for a fixed response, it

depends on boundary condition. For example, under normal loading condition

the size of q for ux reduces to 1 since N-component disappears.

81

3) m: Equation 5.13 indicates that m is associated with function C(r, θ; ξ; ρ).

Comparing Table 5.2 for moving load with Table 4.2 for stationary load, one

can infer that the size of m depends on whether the load is stationary or

moving, where stationary load reduces the size of m greatly. For example,

under stationary load, the size of m is 1 for C00, C10 and C01 as in Table 4.2,

and consequently for uz and ux. However, since m is associated with Bessel

function, the value of m may differ between different responses. For example,

in spite that the size of uz and ux are both 1, m=0 for uz but m=1 for ux.

4) n: From Equation 3.95 and its preceding, the size of n depends on the size of

matrix [Q] in Equation 3.86 and the location of the field point as in Equations

3.91 to 3.94, where the size of matrix [Q] depends on the number of

viscoelastic layers and the number of Prony series as in Equation 1.25 or 1.27.

For pure elastic analysis, the size of n becomes null.

The pavement moving dynamic response in Equation 5.13 can be easily

degenerated to pavement response subjected to the stationary, dynamic or moving load.

For example, setting ρ=0, Equation 5.13 reduces to the pavement stationary response as

in Equation 4.2.

5.3 Numerical Verification

This section will verify the present semianalytical solution by the finite element

program Abaqus (SIMULA, 2007). A three-layer pavement, namely Pavement System II,

is used for purpose of verification. AC is made of standard binder (SB) as reported in Al-

82

Qadi et al (2008). Structural properties and material properties for the studied pavement

are listed in Tables 5.3 and 5.4.

In Abaqus, considering the symmetric condition, a half-pavement is constructed

as shown in Figure 5.1. Because the semianalytical solution treats only circular load, the

loading area in Abaqus is intentionally approximated by a semicircle with an equivalent

loading radius 3.909 in. Tire pressure is 100 psi by default unless other specified. The

input file regarding the model in Abaqus is listed in APPENDIX F. An infinite element

(CIN3D8) is used in the outer domain. The Abaqus simulations are run on desktop

computer DELL OptiPlex 755 (Intel Core 2 Quad CPU/Q6600 2.40GHz, RAM 3.25 GB).

Table 5.3 Structural Properties of Pavement System II Layer Thickness (in) Young’s Modulus (psi) Poisson’s Ratio

AC 16.0 Viscoelastic 0.35

Subbase 12.0 37,990 0.4

Subgrade Infinite 7,500 0.45

Table 5.4 Relaxation Modulus and Creep Compliance of Viscoelastic AC Layer (Adapted from Al-Qadi et al 2008)

i

E(t) D(t)

Ei (psi) ri (s) Di (psi-1) ti (s)

Ee=5.943E+03 - D0=3.668E-7 -

1 1.232E+06 1.130E-04 2.799E-07 2.027E-04

2 7.578E+05 3.140E-03 3.429E-07 5.487E-03

3 4.034E+05 1.300E-02 1.440E-06 3.050E-02

4 2.944E+05 1.840E-01 8.577E-06 1.187E+00

5 2.034E+04 2.290E+00 2.968E-05 6.303E+00

6 1.189E+04 2.570E+01 1.276E-04 9.032E+01

83

5.3.1 Stationary Load

The stationary Load is actually a step load. The remaining input file in Abaqus for

the stationary load can be found in APPENDIX G1. Figure 5.2 illustrates the deflections

at z=0 (top of AC) and z=16 in (bottom of AC) by the present semianalytical solution and

the Abaqus solution. As can been seen clearly, the two solutions are very close to each

other. The negligible error between these two solutions is due to the model used in

Abaqus, and theoretically it can be further diminished through the use of an improved

model, which may be more computationally demanding. Total CPU time of Abaqus

simulation is 18231 s (14 time points). The semianalytical solution, on the other side,

gives solution in 1 s for each time point. Notice the semianalytical solution is run on

DELL Precision 360 (Intel Pentium 4 CPU/3.40 GHz, RAM 2.00GB).

Figure 5.1. Pavement model and loading area in Abaqus.

84

5.3.2 Moving Load

In Abaqus, a moving load can be realized by defining piecewise load magnitude

at consecutive nodes along the moving path. To make sure such treatment yields correct

result, a convergence study should be first planned. In so doing, two moving

configurations are examined: in Moving Configuration I, the consecutive nodes in load

magnitude definition are picked every physical node along moving path, while in Moving

Configuration II, they are picked every other physical node. Figure 5.3 depicts a rough

picture of the load magnitude definition by the two moving configurations for moving

load initiating from Node 617.

Figure 5.2. Comparisons of deflection at z=0 and z=16 in under stationary load between the present semianalytical solution and the Abaqus solution.

0 0.2 0.4 0.6 0.8 1

Time t (s)

0

0.004

0.008

0.012

0.016

Def

lect

ion

u z (i

n) a

t (0,

0,z)

Abaqus Solution z=0Present Solution z=0Abaqus Solution z=16Present Solution z=16

85

Figure 5.4 compares the solutions from the two moving configurations. Since

these two moving configurations yield close results, either one can be used in Abaqus.

This study uses the simpler Moving Configuration II. The remaining input file in Abaqus

for a moving load with Moving Configuration II is given in APPENDIX G2.

Figure 5.5 compares the deflections at fixed field points x=0 and x=20 in between

the present semianalytical solution and the Abaqus solution. The load starts from x=0

with velocity 5mph. As can been clearly observed, the two solutions predict very close

results. The errors between these two solutions are comparable to that for stationary load

in Figure 5.2. Total CPU time of Abaqus simulation is 50542 s (49 time points). The

semianalytical solution gives solution in around 10 s for each time point.

617 645 673 701 729

Moving Configuration I

617 645 673 701 729

Moving Configuration II

Figure 5.3. Two moving configurations.

86

Figure 5.5. Comparisons of deflection at x=0 and x=20 in under moving load between the present semianalytical solution and the Abaqus solution.

0 10 20 30 40

Moving Load Position xs(t)

0.002

0.004

0.006

0.008

0.01

Def

lect

ion

u z (i

n) a

t (x,

0,0)

Abaqus Solution x=0

Present Solution x=0

Abaqus Solution x=20

Present Solution x=20

Figure 5.4. Solutions from the two moving configurations.

0 10 20 30 40

Moving Load Position xs(t)

0.003

0.004

0.005

0.006

0.007

Def

lect

ion

u z (i

n) a

t (0,

0,0)

Moving Configuration IMoving Configuration II

87

0 10 20 30 40

Moving Load Position xs(t)

-0.004

-0.002

0

0.002

0.004

Def

lect

ion

u z (i

n) a

t (x,

0,0)

Abaqus Solution x=0

Present Solution x=0

Abaqus Solution x=20

Present Solution x=20

Figure 5.7. Comparisons of deflection at x=0 and x=20 in under moving dynamic load between the present semianalytical solution and the Abaqus solution.

Figure 5.6. Comparisons of deflection under dynamic load between the present semianalytical solution and the Abaqus solution.

0 0.2 0.4 0.6 0.8 1

Time t (s)

-0.004

-0.002

0

0.002

0.004

Def

lect

ion

u z (i

n) a

t (0,

0,0)

Abaqus SolutionPresent Solution

88

5.3.3 Dynamic Load

The dynamic load used in this section is defined as 50sin(31.416t), where 50 is

the amplitude, i.e. maximum tire pressure, and 31.416 (about 5Hz) is angular frequency.

The remaining input file in Abaqus for this dynamic load can be found in APPENDIX

G3. As can be shown in Figure 5.6, the deflections predicted by the present

semianalytical solution and the Abaqus solution agree closely with each other. Total CPU

time of Abaqus simulation is 88604 s (51 time points). The semianalytical solution

finishes in around 1 s for each time point.

5.3.4 Moving Dynamic Load

In this section, the dynamic load 50sin(31.416t) now moves with velocity 5 mph.

The remaining input file in Abaqus for this moving dynamic load is listed in APPENDIX

G4. Figure 5.7, once again, clearly shows that the present semianalytical solution and the

Abaqus solution are almost the same. Total CPU time of Abaqus simulation is 87597 s

(81 time points). Similar to moving load, the semianalytical solution finishes in around

10 s for each time point.

Through the comparisons between the present semianalytical solutions and the

Abaqus solutions for stationary load, dynamic load, moving load, and moving dynamic

load, it evidently verifies that the present semiananlytical solution is reliable in accuracy.

89

CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusions

Viscoelastic modeling of flexible pavement has long been investigated. Pavement

primary response, in the context of LVET, is governed by two convolution integrals: the

implicit convolution integral and explicit convolution integral. The implicit convolution

integral is induced by the viscoelasticity of AC, while explicit convolution integral by the

moving load. Solving these two convolution integrals is the fundamental issue to the

success of a viscoelastic pavement model.

An innovative semianalytical solution for pavement primary response under

generic moving dynamic load has been proposed in this work. The whole process is

furnished by a two-stage work: the first stage proposes a semianalytical solution for

pavement under impulse load (or equivalently stationary load) to attack the implicit

convolution integral, and the second stage proposes a semianalytical solution for

pavement under moving dynamic load to attack the explicit convolution integral. The

solutions in both stages are, for the first time, analytical in the time domain and involve

only a single numerical integral in the space domain, as in the solution based on LET.

90

The accuracy of the semianalytical solution is verified by finite-element-based

method and finite element program Abaqus. In addition, it’s very efficient with respect to

the computation cost since the semianalytical solution is analytical in the time domain.

Considering its accuracy and computation efficiency, the proposed semianalytical

solution is very appealing to flexible pavement whose performance is influenced or

controlled by viscoelastic material properties. This semianalytical solution, in particular,

is uniquely advantageous to predict pavement long-term response, and highly potential to

pavement analysis and design which involves multiple trial runs.

6.2 Recommendations

The proposed semianalytical solution, at current stage, deals only with the

fundamental issues, i.e. the convolution integral. To make it suitable to pavement

community, further work is needed to improve and validate the solution:

1) The current semianalytical solution treat only one viscoelastic layer. However,

with the thriving of perpetual pavement (Liao 2007, Al-Qadi et al 2008), it is

necessary to generate a semianalytical solution for arbitrary number of

viscoelastic layers.

2) For simplicity the Poisson’s ration of viscoelastic AC is assumed to be

constant. Recently, Wang (2008) assumes constant bulk modulus and

relaxation shear modulus in MNLAYER. As a result, it is also necessary to

generate the semianalytical solution for generic viscoelastic material

properties.

91

3) Another important future work is to validate the viscoelastic model of flexible

pavement with accelerating pavement testing (APT) data, which simulates

pavement response under multiple traffic passes. This would be very

demanding for FEM owing to its computation cost. Currently this is

conducted through the modified elastic model, as in Al-Khateeb et al (2007a,

b).

4) To explain test-road data and eventually the long term pavement performance

(LTPP) data (FHWA 2000, 2004), it is necessary to incorporate the

environment effect into the semianalytical solution.

5) Also, with this efficient semianalytical solution, inverse problem can be

planned. For example, current falling weight deflectometer (FWD) test

procedure is mainly based on LET. Correspondingly, a FWD test procedure

based on LVET can be designed to test material properties of flexible

pavement more accurately.

The proposed semianalytical solution explores an innovative way in treating

layered viscoelastic structure. Other potential applications of such structure lie in material

science and earth science. With the accurate prediction of proposed semianalytical

solution, one can devise “smart” viscoelastic laminate to satisfy predefined function.

However, one should realize that the proposed semianalytical solution is only limited to

quasistatic analysis. For impact problem (Zheng 2007), the dynamic analysis is more

appropriate.

In earth science, on the other side, extensive works have being conducted to invert

earth interior structure by observing postglacial relaxation (Peltier 1996), which puts high

92

requirement on the accuracy and efficiency of LVET solution. The application of the

proposed semianalytical solution to this area is straightforward expect that the

viscoelastic behavior is represented by simple Maxwell model. For methodologies

currently employed in this area, interested readers can go to the work by Sabadini and

Vermeersen (1999).

93

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102

APPENDICES

103

APPENDIX A

COMPONENTS OF SOLUTION MATRIX AND PROPOGATOR MATRIX

1). The elements of [Z] in Equation 3.26 are

2 211 1 12 1 13 1 14 1

3 321 1 22 1 23 1 24 1

31 32 33 34

41 42

; ( ) ; ; ( ) ;

; ( ) ; ; ( ) ;

1 1; ( ) ; ; ( ) ;

; ;

z z z z

z z z z

z z z z

z z

c cZ c e Z c z e Z c e Z c z e

c cZ c e Z c z e Z c e Z c z e

Z e Z z e Z e Z z e

Z e Z ze

ξ ξ ξ ξ

ξ ξ ξ ξ

ξ ξ ξ ξ

ξ ξ

ξ ξ

ξ ξ

ξ ξ

− −

− −

− −

−= = + = = +

= = + = − = −

= = − + = − = − −

= = 43 44 ; .z zZ e Z zeξ ξ− −= =

(A1)

and the elements of [ZN] in Equation 3.27 are

11 12

21 221 1

; ;1 1; .

2 2

N z N z

N z N z

Z e Z e

Z e Z ec c

ξ ξ

ξ ξ

−

−

= =

= = − (A2)

104

2). The elements of the propagator matrix [a] in Equation 3.30 are

11 33

12 43

13 1 1

14 23 1

21 34

22 44

24 1

cosh( ) sinh( )( 1)sinh( ) cosh( )

( 2)sinh( ) cosh( )sinh( )

( 1)sinh( ) cosh( )cosh( ) sinh( )

( 2)sinh( )

a a h h ha a h h ha c h hc ha a hc ha a h h ha a h h ha c h

ξ γξ ξγ ξ γξ ξ

γ ξ γξ ξγξ ξγ ξ γξ ξ

ξ γξ ξγ ξ

= = += − = + += − + −= − = −= − = + −= = −= + 1

31 2 2

32 41 2

42 2 2

cosh( )sinh( ) / cosh( ) /

sinh( ) /sinh( ) / cosh( ) /

c h ha h c h h ca a h h ca h c h h c

γξ ξξ ξ ξ

ξ ξξ ξ ξ

+= −= − = −= +

(A3)

and the elements of the propagator matrix [aN] in Equation 3.31 are

11 33

12 1

211

cosh( )2 sinh( )

1 sinh( )2

N Na a ha c h

a hc

ξξ

ξ

= == −

= −

(A4)

where ci (i=1,2,3) are the material coefficients related to the Young’s modulus E and

Poisson’s ratio υ by

1

2

3

1

2(1 )(1 )

(1 )(1 2 )

1/ 2(1 )

cE

cE

cE

υ

υ υ

υ υ

γ υ

+=

+ −= −

+ −=

= − −

(A5)

Note h is layer thickness.

105

APPENDIX B

BOUNDARY CONDITION IN THE VECTOR FUNCTIONS DOMAIN

From Equation 3.38

0 0( ( ) ( ))* 0 ( )( ,0; )2

s si x x y yL zz

O

p tT t S dxdy e dxdyα βδξ σπ

− − + −= = −∫ ∫ ∫ ∫ (B.1)

where Equation 3.20 has been used. Integral area O is governed by

2 20 0 0( ) ( )s sr x - x + y - y d= ≤ (B.2)

Equation B.2 is equivalent to

0

0

cossin

s

s

x - x ry - y r

θθ

=⎧⎨ =⎩ (B.3)

with θ∈(0, 2π), r∈(0, d0). Similarly, Equation 3.15 is equivalent to

cossin

α ξ ϕβ ξ ϕ=⎧

⎨ =⎩ (B.4)

with ϕ∈(0, 2π), ξ∈(0, d0).

Substituting Equations B.3 and B.4 into B.1 yields

0 2 ( )00 0

( )( ,0; )2

d i rL

p tT t e rd drπ ξ θ ϕδξ θ

π− −= − ∫ ∫ (B.5)

106

Making use of the following expansion (Abramowitz and Stegun 1970)

0 21

2 10

cos( cos ) ( ) 2 ( 1) ( )cos(2 )

sin( cos ) 2 ( 1) ( )cos((2 1) )

ll

l

ll

l

J J l

J l

χ ϑ χ χ ϑ

χ ϑ χ ϑ

∞

=

∞

+=

= + −

= − +

∑

∑ (B.6)

Embedding Equation B.6 into B.5 and carrying out mathematical manipulations yields

( )0

0

0

20

0 0

0 22 10

0 0

2 10

0 00

( )( ,0; ) cos( ( )) sin( ( ))2

( ) 2 ( )cos(2 ( ))( )

22 ( 1) ( )cos((2 1)( ))

( ) ( )

d

L

ld l

ll

l

d

p tT t r i r rd dr

J r J r lp t rd dr

i J r l

p t J r rdr

π

π

δξ ξ θ ϕ ξ θ ϕ θπ

ξ ξ θ ϕδ θπ

ξ θ ϕ

δ ξ

∞

=

∞

+=

= − − − −

⎛ ⎞+ − −⎜ ⎟⎜ ⎟= −⎜ ⎟⎛ ⎞− + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

= −

∫ ∫

∑∫ ∫

∑

∫

(B.7)

where use has been made of the orthogonality of trigonometric function. Equation B.7

further reduces to

0 01 0( ,0; ) ( ) ( )L

p dT t J d tξ ξ δξ

= − (B.8)

107

APPENDIX C

LAPLACE TRANSFORM

By definition, Laplace transform is (Sneddon 1972)

0( ) ( ) stf s f t e dt

∞ −= ∫ (C.1)

Equation C.1 reduces to

0

1( ) ( ) stf s f t des

∞ −= − ∫ (C.2)

Integrating Equation C.2 by parts yields

( )0 0

1 1( ) ( ) (0) ( )st stf s f t de f e df ts s

∞ ∞− −= − = +∫ ∫ (C.3)

or

0( ) (0) ( )stsf s f e df t

∞ −= + ∫ (C.4)

Consequently

lim ( ) (0)s

sf s f→∞

= (C.5)

00lim ( ) (0) ( ) ( )s

sf s f df t f∞

→= + = ∞∫ (C.6)

108

APPENDIX D

ANALYTICAL EXPRESSION OF FUNCTION C(r, θ; ξ)

Taking CC0,0 (r,θ;ξ) for example,

2,

0

1( , ; ) cos( cos( ))2

0 0CC r r dπ

θ ξ ξ ϕ θ ϕπ

= −∫ (D.1)

Making use of Equation B.6, Equation D.1 can be expanded into

20,00 20

1

1( , ; ) ( ) 2 ( ) ( )cos(2 ( ))2

mm

m

CC r J r J r m dπ

θ ξ ξ ξ ϕ θ ϕπ =

= + − −∑∫ (D.2)

and further into

0,00( , ; ) ( )CC r J rθ ξ ξ= (D.3)

Similarly,

1,01( , ; ) ( )cosCC r J rθ ξ ξ θ= (D.4)

0,11( , ; ) ( )sinCC r J rθ ξ ξ θ= (D.5)

1,12

1( , ; ) ( )sin 22

CC r J rθ ξ ξ θ= − (D.6)

2,00 2

1 1( , ; ) ( ) ( )cos22 2

CC r J r J rθ ξ ξ ξ θ= − (D.7)

0,20 2

1 1( , ; ) ( ) ( )cos22 2

CC r J r J rθ ξ ξ ξ θ= + (D.8)

109

APPENDIX E

ANALYTICAL EXPRESSION OF FUNCTION C(r, θ; ξ; ρ)

Taking CC0,0(r,θ;ξ;ρ) for example,

2,2 20

1 cos( cos( ))( , ; ; )2 1 (cos )

0 0 rCC r dπ ξ ϕ θθ ξ ρ ϕ

π ρ ϕ−

=+∫ (E.1)

Making use of Equation B.6, Equation E.1 can be expanded into

0 220,0 12 20

( ) 2 ( ) ( )cos(2 ( ))1( , ; ; )

2 1 (cos )

mm

m

J r J r mCC r

πξ ξ ϕ θ

θ ξ ρπ ρ ϕ

=

+ − −=

+

∑∫ (E.2)

and further into

( )0,00 0 2 2 2

1

( , ; ; ) ( ) ( ) 2 ( ) ( ) ( )cos2 ( )sin2mm m m

m

CC r J r NC J r NC m NS mθ ξ ρ ξ ρ ξ ρ θ ρ θ=

= + − +∑

(E.3)

where

2

2 2 20

1 cos2( )2 1 (cos )m

mNC dπ ϕρ ϕ

π ρ ϕ=

+∫ (E.4a)

2

2 2 20

1 sin 2( )2 1 (cos )m

mNS dπ ϕρ ϕ

π ρ ϕ=

+∫ (E.4b)

Functions NCs can be written analytically. Taking NC2m for example,

222

2 2 2 20

1 11 cos2 ( 1)( )2 1 (cos ) 1

mm

mmNC d

π ρϕρ ϕπ ρ ϕ ρρ

⎛ ⎞+ −−= = ⎜ ⎟⎜ ⎟+ + ⎝ ⎠

∫ (E.5)

110

Marking s=eiϕ , after mathematic manipulations, one has

( )2 22 2 22 21

2

1 1 1( ) 12 2 1m m

my

sNC s s dsi s s

ρπ ζρ

ζ

−

=

⎛ ⎞⎜ ⎟

= + −⎜ ⎟++ ⎜ ⎟+⎜ ⎟

⎝ ⎠

∫

(E.6)

where

21 1ρζρ

+ −=

(E.7)

Based on the complex theory, Equation E.6 can be solved analytically:

222

2 2 2

1 1( 1) ( 1)( )1 1

mm m m

mNC ρζρρρ ρ

⎛ ⎞+ −− −= = ⎜ ⎟⎜ ⎟+ + ⎝ ⎠

(E.8)

Specifically

22

1 0( 0) 1

0m

mNC

otherwiseρ ρ

⎧ =⎪= = +⎨⎪⎩

(E.9)

Similarly,

2

2 2 20

1 sin(2 )( ) 02 1 (cos )m

mNS dπ ϕρ ϕ

π ρ ϕ= =

+∫

(E.10)

( )2

2 1 2 20

cos (2 1)1( ) 02 1 (cos )m

mNC d

π ϕρ ϕ

π ρ ϕ+

+= =

+∫

(E.11)

( )2

2 1 2 20

sin (2 1)1( ) 02 1 (cos )m

mNS d

π ϕρ ϕ

π ρ ϕ+

+= =

+∫

(E.12)

111

APPENDIX F

ABAQUS INPUTFILE ON NODE AND ELEMENT

*HEADING Layered Pavement, Number manually *NODE 1,0.,-200.,0 29,0.,-100.,0 85,0.,-84.,0 505,0.,-24.,0 1849,0.,24.,0 2269,0,84.,0 2325,0.,100.,0 2353,0.,200.,0 *NGEN 29,85,28 85,505,28 505,1849,28 1849,2269,28 2269,2325,28 *NSET,NSET=X0,GENERATE 1,2353,28 ** *NCOPY,OLD SET=X0,CHANGE NUMBER=4,SHIFT,NEW SET=X4 -4,0,0 *NCOPY,OLD SET=X0,CHANGE NUMBER=24,SHIFT,NEW SET=X84 -84,0,0 *NCOPY,OLD SET=X0,CHANGE NUMBER=26,SHIFT,NEW SET=X100 -100,0,0 *NCOPY,OLD SET=X0,CHANGE NUMBER=27,SHIFT,NEW SET=X200 -200,0,0 ** *NFILL X0,X4,4 X4,X84,20 X84,X100,2 **

112

*NSET,NSET=Z0,GENERATE 1,2380,1 *NCOPY,OLD SET=Z0,CHANGE NUMBER=4760,SHIFT,NEW SET=Z2 0,0,-2 *NCOPY,OLD SET=Z0,CHANGE NUMBER=19040,SHIFT,NEW SET=Z14 0,0,-14 *NCOPY,OLD SET=Z0,CHANGE NUMBER=23800,SHIFT,NEW SET=Z16 0,0,-16 *NCOPY,OLD SET=Z0,CHANGE NUMBER=28560,SHIFT,NEW SET=Z18 0,0,-18 *NCOPY,OLD SET=Z0,CHANGE NUMBER=38080,SHIFT,NEW SET=Z26 0,0,-26 *NCOPY,OLD SET=Z0,CHANGE NUMBER=42840,SHIFT,NEW SET=Z28 0,0,-28 *NCOPY,OLD SET=Z0,CHANGE NUMBER=47600,SHIFT,NEW SET=Z30 0,0,-30 *NCOPY,OLD SET=Z0,CHANGE NUMBER=95200,SHIFT,NEW SET=Z190 0,0,-190 *NCOPY,OLD SET=Z0,CHANGE NUMBER=97580,SHIFT,NEW SET=Z380 0,0,-380 ** *NFILL Z0,Z2,2,2380 *NFILL Z2,Z14,6,2380 *NFILL Z14,Z16,2,2380 *NFILL Z16,Z18,2,2380 *NFILL Z18,Z26,4,2380 *NFILL Z26,Z28,2,2380 *NFILL Z28,Z30,2,2380 *NFILL Z30,Z190,20,2380 *NFILL Z190,Z380,1,2380 ** *NSET,NSET=SYMBC,GENERATE 1,2353,28 2381,4733,28 4761,7113,28 7141,9493,28

113

9521,11873,28 11901,14253,28 14281,16633,28 16661,19013,28 19041,21393,28 21421,23773,28 23801,26153,28 26181,28533,28 28561,30913,28 30941,33293,28 33321,35673,28 35701,38053,28 38081,40433,28 40461,42813,28 42841,45193,28 45221,47573,28 47601,49953,28 49981,52333,28 52361,54713,28 54741,57093,28 57121,59473,28 59501,61853,28 61881,64233,28 64261,66613,28 66641,68993,28 69021,71373,28 71401,73753,28 73781,76133,28 76161,78513,28 78541,80893,28 80921,83273,28 83301,85653,28 85681,88033,28 88061,90413,28 90441,92793,28 92821,95173,28 95229,97525,28 97637,99877,28 *NSET,NSET=SYMBC 97581 , 99933 ** *ELEMENT,TYPE=C3D8R 1,30,29,2409,2410,58,57,2437,2438 *ELEMENT,TYPE=CIN3D8 2133,29,30,2410,2409,1,2,2382,2381

114

2158,54,55,2435,2434,26,28,2408,2406 2159,2326,2325,4705,4706,2354,2353,4733,4734 2184,2351,2350,4730,4731,2380,2378,4758,4760 2185,55,83,2463,2435,28,84,2464,2408 2186,83,111,2491,2463,84,112,2492,2464 2266,2323,2351,4731,4703,2324,2380,4760,4704 90507,92849,92850,95230,95229,92821,92822,97582,97581 90532,92874,92875,95255,95254,92846,92848,97608,97606 90533,95146,95145,97525,97526,95174,95173,99933,99934 90558,95171,95170,97550,97551,95200,95198,99958,99960 90559,92875,92903,95283,95255,92848,92904,97664,97608 90560,92903,92931,95311,95283,92904,92932,97692,97664 90640,95143,95171,97551,97523,95144,95200,99960,99904 90667,95257,95258,95286,95285,97637,97638,97666,97665 *ELGEN 1,26,1,1,82,28,26,40,2380,2266 *ELGEN 2133,25,1,1,1,28,26,39,2380,2266 2158,1,1,1,1,28,26,39,2380,2266 2159,25,1,1,1,28,26,39,2380,2266 2184,1,1,1,1,28,26,39,2380,2266 2185,1,1,1,1,28,26,39,2380,2266 2186,1,1,1,80,28,1,39,2380,2266 2266,1,1,1,1,28,26,39,2380,2266 90507,25,1,1,1,28,26,1,2380,2266 90532,1,1,1,1,28,26,1,2380,2266 90533,25,1,1,1,28,26,1,2380,2266 90558,1,1,1,1,28,26,1,2380,2266 90559,1,1,1,1,28,26,1,2380,2266 90560,1,1,1,80,28,1,1,2380,2266 90640,1,1,1,1,28,26,1,2380,2266 90667,25,1,1,80,28,26,1,2380,2266 *ELEMENT,TYPE=CIN3D8 90641,95229,95230,95258,95257,97581,97582,97638,97637 90666,95254,95255,95283,95282,97606,97608,97664,97662 92747,97525,97497,97498,97526,99933,99877,99878,99934 92772,97522,97523,97551,97550,99902,99904,99960,99958 90692,95282,95283,95311,95310,97662,97664,97692,97690 *ELGEN 90641,25,1,1,1,28,26,1,2380,2266 90692,1,1,1,80,28,26,1,2380,2266 92747,25,1,1,1,28,26,1,2380,2266 ** *ELSET,ELSET=AC,GENERATE 1,22660,1

115

*ELSET,ELSET=BS,GENERATE 22661,40788,1 *ELSET,ELSET=SG,GENERATE 40789,92772,1 **

116

APPENDIX G

ABAQUS INPUTFILE ON DIFFERENT LOAD8

G1) Stationary Load *ELSET,ELSET=LoadArea1,GENERATE 443,625,26 470,600,26 471,601,26 498,576,26 ** *Surface,type=ELEMENT,name=SURFACE1 LoadArea1, s3 ** *SOLID SECTION,ELSET=AC,MATERIAL=AC *SOLID SECTION,ELSET=BS,MATERIAL=BS *SOLID SECTION,ELSET=SG,MATERIAL=SG ** *Material, name=AC *Elastic, moduli=INSTANTANEOUS 2.726e+06, 0.35 *Viscoelastic, time=PRONY 0.452, 0.452, 0.000113 0.278, 0.278, 0.00314 0.148, 0.148, 0.013 0.108, 0.108, 0.184 0.00746, 0.00746, 2.29 0.00436, 0.00436, 25.7 *Material, name=BS *Elastic 37990., 0.4 *Material, name=SG *Elastic 7500., 0.45 **

8 To make it workable in Abaqus, the input files in this appendix must be

appended to the input file in APPENDIX F.

117

*BOUNDARY SYMBC,1 ** *STEP,NLGEOM *STATIC 0.001,0.001,0.0001 *DSLOAD, follower=NO SURFACE1,TRVEC, 100., 0., 0., -1. *END STEP *STEP,NLGEOM,INC=150 *VISCO,CETOL=5.E-4 0.04,1. *END STEP G2) Moving Load *ELSET,ELSET=LoadArea1,GENERATE 443,625,26 470,600,26 471,601,26 498,576,26 *ELSET,ELSET=LoadArea2,GENERATE 495,677,26 522,652,26 523,653,26 550,628,26 *ELSET,ELSET=LoadArea3,GENERATE 547,729,26 574,704,26 575,705,26 602,680,26 *ELSET,ELSET=LoadArea4,GENERATE 599,781,26 626,756,26 627,757,26 654,732,26 *ELSET,ELSET=LoadArea5,GENERATE 651,833,26 678,808,26 679,809,26 706,784,26 *ELSET,ELSET=LoadArea6,GENERATE 703,885,26 730,860,26 731,861,26 758,836,26

118

*ELSET,ELSET=LoadArea7,GENERATE 755,937,26 782,912,26 783,913,26 810,888,26 *ELSET,ELSET=LoadArea8,GENERATE 807,989,26 834,964,26 835,965,26 862,940,26 *ELSET,ELSET=LoadArea9,GENERATE 859,1041,26 886,1016,26 887,1017,26 914,992,26 *ELSET,ELSET=LoadArea10,GENERATE 911,1093,26 938,1068,26 939,1069,26 966,1044,26 *ELSET,ELSET=LoadArea11,GENERATE 963,1145,26 990,1120,26 991,1121,26 1018,1096,26 *ELSET,ELSET=LoadArea12,GENERATE 1015,1197,26 1042,1172,26 1043,1173,26 1070,1148,26 *ELSET,ELSET=LoadArea13,GENERATE 1067,1249,26 1094,1224,26 1095,1225,26 1122,1200,26 *ELSET,ELSET=LoadArea14,GENERATE 1119,1301,26 1146,1276,26 1147,1277,26 1174,1252,26 *ELSET,ELSET=LoadArea15,GENERATE 1171,1353,26 1198,1328,26 1199,1329,26 1226,1304,26

119

*ELSET,ELSET=LoadArea16,GENERATE 1223,1405,26 1250,1380,26 1251,1381,26 1278,1356,26 *ELSET,ELSET=LoadArea17,GENERATE 1275,1457,26 1302,1432,26 1303,1433,26 1330,1408,26 *ELSET,ELSET=LoadArea18,GENERATE 1327,1509,26 1354,1484,26 1355,1485,26 1382,1460,26 *ELSET,ELSET=LoadArea19,GENERATE 1379,1561,26 1406,1536,26 1407,1537,26 1434,1512,26 *ELSET,ELSET=LoadArea20,GENERATE 1431,1613,26 1458,1588,26 1459,1589,26 1486,1564,26 *ELSET,ELSET=LoadArea21,GENERATE 1483,1665,26 1510,1640,26 1511,1641,26 1538,1616,26 *Surface,type=ELEMENT,name=SURFACE1 LoadArea1,s3 *Surface,type=ELEMENT,name=SURFACE2 LoadArea2,s3 *Surface,type=ELEMENT,name=SURFACE3 LoadArea3,s3 *Surface,type=ELEMENT,name=SURFACE4 LoadArea4,s3 *Surface,type=ELEMENT,name=SURFACE5 LoadArea5,s3 *Surface,type=ELEMENT,name=SURFACE6 LoadArea6,s3 *Surface,type=ELEMENT,name=SURFACE7 LoadArea7,s3 *Surface,type=ELEMENT,name=SURFACE8

120

LoadArea8,s3 *Surface,type=ELEMENT,name=SURFACE9 LoadArea9,s3 *Surface,type=ELEMENT,name=SURFACE10 LoadArea10,s3 *Surface,type=ELEMENT,name=SURFACE11 LoadArea11,s3 *Surface,type=ELEMENT,name=SURFACE12 LoadArea12,s3 *Surface,type=ELEMENT,name=SURFACE13 LoadArea13,s3 *Surface,type=ELEMENT,name=SURFACE14 LoadArea14,s3 *Surface,type=ELEMENT,name=SURFACE15 LoadArea15,s3 *Surface,type=ELEMENT,name=SURFACE16 LoadArea16,s3 *Surface,type=ELEMENT,name=SURFACE17 LoadArea17,s3 *Surface,type=ELEMENT,name=SURFACE18 LoadArea18,s3 *Surface,type=ELEMENT,name=SURFACE19 LoadArea19,s3 *Surface,type=ELEMENT,name=SURFACE20 LoadArea20,s3 *Surface,type=ELEMENT,name=SURFACE21 LoadArea21,s3 ** *SOLID SECTION,ELSET=AC,MATERIAL=AC *SOLID SECTION,ELSET=BS,MATERIAL=BS *SOLID SECTION,ELSET=SG,MATERIAL=SG ** *Material, name=AC *Elastic, moduli=INSTANTANEOUS 2.726e+06, 0.35 *Viscoelastic, time=PRONY 0.452, 0.452, 0.000113 0.278, 0.278, 0.00314 0.148, 0.148, 0.013 0.108, 0.108, 0.184 0.00746, 0.00746, 2.29 0.00436, 0.00436, 25.7 *Material, name=BS *Elastic 37990., 0.4

121

*Material, name=SG *Elastic 7500., 0.45 ** *BOUNDARY SYMBC,1 ** *Amplitude, name=SURFACE1, time=step time, smooth=0.0 0,1.0,0.0227273,0.0 *Amplitude, name=SURFACE2, time=step time, smooth=0.0 0,0.0,0.0227273,1.0,0.0454545,0.0 *Amplitude, name=SURFACE3, time=step time, smooth=0.0 0.0227273,0.0,0.0454545,1.0,0.0681818,0.0 *Amplitude, name=SURFACE4, time=step time, smooth=0.0 0.0454545,0.0,0.0681818,1.0,0.0909091,0.0 *Amplitude, name=SURFACE5, time=step time, smooth=0.0 0.0681818,0.0,0.0909091,1.0,0.113636,0.0 *Amplitude, name=SURFACE6, time=step time, smooth=0.0 0.0909091,0.0,0.113636,1.0,0.136364,0.0 *Amplitude, name=SURFACE7, time=step time, smooth=0.0 0.113636,0.0,0.136364,1.0,0.159091,0.0 *Amplitude, name=SURFACE8, time=step time, smooth=0.0 0.136364,0.0,0.159091,1.0,0.181818,0.0 *Amplitude, name=SURFACE9, time=step time, smooth=0.0 0.159091,0.0,0.181818,1.0,0.204545,0.0 *Amplitude, name=SURFACE10, time=step time, smooth=0.0 0.181818,0.0,0.204545,1.0,0.227273,0.0 *Amplitude, name=SURFACE11, time=step time, smooth=0.0 0.204545,0.0,0.227273,1.0,0.25,0.0 *Amplitude, name=SURFACE12, time=step time, smooth=0.0 0.227273,0.0,0.25,1.0,0.272727,0.0 *Amplitude, name=SURFACE13, time=step time, smooth=0.0 0.25,0.0,0.272727,1.0,0.295455,0.0 *Amplitude, name=SURFACE14, time=step time, smooth=0.0 0.272727,0.0,0.295455,1.0,0.318182,0.0 *Amplitude, name=SURFACE15, time=step time, smooth=0.0 0.295455,0.0,0.318182,1.0,0.340909,0.0 *Amplitude, name=SURFACE16, time=step time, smooth=0.0 0.318182,0.0,0.340909,1.0,0.363636,0.0 *Amplitude, name=SURFACE17, time=step time, smooth=0.0 0.340909,0.0,0.363636,1.0,0.386364,0.0 *Amplitude, name=SURFACE18, time=step time, smooth=0.0 0.363636,0.0,0.386364,1.0,0.409091,0.0 *Amplitude, name=SURFACE19, time=step time, smooth=0.0 0.386364,0.0,0.409091,1.0,0.431818,0.0

122

*Amplitude, name=SURFACE20, time=step time, smooth=0.0 0.409091,0.0,0.431818,1.0,0.454545,0.0 *Amplitude, name=SURFACE21, time=step time, smooth=0.0 0.431818,0.0,0.454545,1.0 ** *STEP,NLGEOM *STATIC 0.001,0.001,0.0001 *DSLOAD, follower=NO SURFACE1,TRVEC, 100., 0., 0., -1. *END STEP *STEP,NLGEOM,INC=500 *VISCO 0.01,0.45 *DSLOAD,follower=NO,AMPLITUDE=SURFACE1 SURFACE1,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE2 SURFACE2,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE3 SURFACE3,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE4 SURFACE4,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE5 SURFACE5,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE6 SURFACE6,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE7 SURFACE7,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE8 SURFACE8,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE9 SURFACE9,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE10 SURFACE10,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE11 SURFACE11,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE12 SURFACE12,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE13 SURFACE13,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE14 SURFACE14,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE15 SURFACE15,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE16

123

SURFACE16,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE17 SURFACE17,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE18 SURFACE18,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE19 SURFACE19,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE20 SURFACE20,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE21 SURFACE21,TRVEC,100,0,0,-1 *END STEP G3) Dynamic Load *ELSET,ELSET=LoadArea1,GENERATE 443,625,26 470,600,26 471,601,26 498,576,26 ** *Surface,type=ELEMENT,name=SURFACE1 LoadArea1, s3 ** *SOLID SECTION,ELSET=AC,MATERIAL=AC *SOLID SECTION,ELSET=BS,MATERIAL=BS *SOLID SECTION,ELSET=SG,MATERIAL=SG ** *Material, name=AC *Elastic, moduli=INSTANTANEOUS 2.726e+06, 0.35 *Viscoelastic, time=PRONY 0.452, 0.452, 0.000113 0.278, 0.278, 0.00314 0.148, 0.148, 0.013 0.108, 0.108, 0.184 0.00746, 0.00746, 2.29 0.00436, 0.00436, 25.7 *Material, name=BS *Elastic 37990., 0.4 *Material, name=SG *Elastic 7500., 0.45 ** *BOUNDARY

124

SYMBC,1 ** *Amplitude, name=DynLoad1,DEFINITION=PERIODIC,time=step time 1,31.416,0,1 0,0.5 *STEP,NLGEOM,INC=150 *VISCO,CETOL=5.E-4 0.04,2. *DSLOAD, follower=NO,AMPLITUDE=DynLoad1 SURFACE1,TRVEC, 100., 0., 0., -1. *END STEP G4) Moving Dynamic Load *ELSET,ELSET=LoadArea1,GENERATE 443,625,26 470,600,26 471,601,26 498,576,26 *ELSET,ELSET=LoadArea2,GENERATE 495,677,26 522,652,26 523,653,26 550,628,26 *ELSET,ELSET=LoadArea3,GENERATE 547,729,26 574,704,26 575,705,26 602,680,26 *ELSET,ELSET=LoadArea4,GENERATE 599,781,26 626,756,26 627,757,26 654,732,26 *ELSET,ELSET=LoadArea5,GENERATE 651,833,26 678,808,26 679,809,26 706,784,26 *ELSET,ELSET=LoadArea6,GENERATE 703,885,26 730,860,26 731,861,26 758,836,26 *ELSET,ELSET=LoadArea7,GENERATE 755,937,26

125

782,912,26 783,913,26 810,888,26 *ELSET,ELSET=LoadArea8,GENERATE 807,989,26 834,964,26 835,965,26 862,940,26 *ELSET,ELSET=LoadArea9,GENERATE 859,1041,26 886,1016,26 887,1017,26 914,992,26 *ELSET,ELSET=LoadArea10,GENERATE 911,1093,26 938,1068,26 939,1069,26 966,1044,26 *ELSET,ELSET=LoadArea11,GENERATE 963,1145,26 990,1120,26 991,1121,26 1018,1096,26 *ELSET,ELSET=LoadArea12,GENERATE 1015,1197,26 1042,1172,26 1043,1173,26 1070,1148,26 *ELSET,ELSET=LoadArea13,GENERATE 1067,1249,26 1094,1224,26 1095,1225,26 1122,1200,26 *ELSET,ELSET=LoadArea14,GENERATE 1119,1301,26 1146,1276,26 1147,1277,26 1174,1252,26 *ELSET,ELSET=LoadArea15,GENERATE 1171,1353,26 1198,1328,26 1199,1329,26 1226,1304,26 *ELSET,ELSET=LoadArea16,GENERATE 1223,1405,26

126

1250,1380,26 1251,1381,26 1278,1356,26 *ELSET,ELSET=LoadArea17,GENERATE 1275,1457,26 1302,1432,26 1303,1433,26 1330,1408,26 *ELSET,ELSET=LoadArea18,GENERATE 1327,1509,26 1354,1484,26 1355,1485,26 1382,1460,26 *ELSET,ELSET=LoadArea19,GENERATE 1379,1561,26 1406,1536,26 1407,1537,26 1434,1512,26 *ELSET,ELSET=LoadArea20,GENERATE 1431,1613,26 1458,1588,26 1459,1589,26 1486,1564,26 *ELSET,ELSET=LoadArea21,GENERATE 1483,1665,26 1510,1640,26 1511,1641,26 1538,1616,26 *Surface,type=ELEMENT,name=SURFACE1 LoadArea1,s3 *Surface,type=ELEMENT,name=SURFACE2 LoadArea2,s3 *Surface,type=ELEMENT,name=SURFACE3 LoadArea3,s3 *Surface,type=ELEMENT,name=SURFACE4 LoadArea4,s3 *Surface,type=ELEMENT,name=SURFACE5 LoadArea5,s3 *Surface,type=ELEMENT,name=SURFACE6 LoadArea6,s3 *Surface,type=ELEMENT,name=SURFACE7 LoadArea7,s3 *Surface,type=ELEMENT,name=SURFACE8 LoadArea8,s3 *Surface,type=ELEMENT,name=SURFACE9

127

LoadArea9,s3 *Surface,type=ELEMENT,name=SURFACE10 LoadArea10,s3 *Surface,type=ELEMENT,name=SURFACE11 LoadArea11,s3 *Surface,type=ELEMENT,name=SURFACE12 LoadArea12,s3 *Surface,type=ELEMENT,name=SURFACE13 LoadArea13,s3 *Surface,type=ELEMENT,name=SURFACE14 LoadArea14,s3 *Surface,type=ELEMENT,name=SURFACE15 LoadArea15,s3 *Surface,type=ELEMENT,name=SURFACE16 LoadArea16,s3 *Surface,type=ELEMENT,name=SURFACE17 LoadArea17,s3 *Surface,type=ELEMENT,name=SURFACE18 LoadArea18,s3 *Surface,type=ELEMENT,name=SURFACE19 LoadArea19,s3 *Surface,type=ELEMENT,name=SURFACE20 LoadArea20,s3 *Surface,type=ELEMENT,name=SURFACE21 LoadArea21,s3 ** *SOLID SECTION,ELSET=AC,MATERIAL=AC *SOLID SECTION,ELSET=BS,MATERIAL=BS *SOLID SECTION,ELSET=SG,MATERIAL=SG ** *Material, name=AC *Elastic, moduli=INSTANTANEOUS 2.726e+06, 0.35 *Viscoelastic, time=PRONY 0.452, 0.452, 0.000113 0.278, 0.278, 0.00314 0.148, 0.148, 0.013 0.108, 0.108, 0.184 0.00746, 0.00746, 2.29 0.00436, 0.00436, 25.7 *Material, name=BS *Elastic 37990., 0.4 *Material, name=SG *Elastic

128

7500., 0.45 ** *BOUNDARY SYMBC,1 ** *Amplitude, name=SURFACE1, time=step time, smooth=0.0 0,0,0.00568182,0.133165,0.0113636,0.174732,0.0170455,0.127568 0.0227273,0 *Amplitude, name=SURFACE2, time=step time, smooth=0.0 0,0,0.00568182,0.0443884,0.0113636,0.174732,0.0170455,0.382703 0.0227273,0.654862,0.0284091,0.583982,0.0340909,0.43884,0.0397727,0.237207 0.0454545,0 *Amplitude, name=SURFACE3, time=step time, smooth=0.0 0.0227273,0,0.0284091,0.194661,0.0340909,0.43884,0.0397727,0.71162 0.0454545,0.989822,0.0511364,0.749522,0.0568182,0.488573,0.0625,0.230969 0.0681818,0 *Amplitude, name=SURFACE4, time=step time, smooth=0.0 0.0454545,0,0.0511364,0.249841,0.0568182,0.488573,0.0625,0.692908 0.0681818,0.841251,0.0738636,0.548918,0.0795455,0.299636,0.0852273,0.111903 0.0909091,0 *Amplitude, name=SURFACE5, time=step time, smooth=0.0 0.0681818,0,0.0738636,0.182973,0.0795455,0.299636,0.0852273,0.335709 0.0909091,0.281726,0.0965909,0.080166,0.102273,-0.0356733,0.107955,-0.0618285 0.113636,-0 *Amplitude, name=SURFACE6, time=step time, smooth=0.0 0.0909091,0,0.0965909,0.026722,0.102273,-0.0356733,0.107955,-0.185486 0.113636,-0.415423,0.119318,-0.427747,0.125,-0.353557,0.130682,-0.205357 0.136364,-0 *Amplitude, name=SURFACE7, time=step time, smooth=0.0 0.113636,-0,0.119318,-0.142582,0.125,-0.353557,0.130682,-0.61607 0.136364,-0.909636,0.142045,-0.726705,0.147727,-0.498726,0.153409,-0.248567 0.159091,-0 *Amplitude, name=SURFACE8, time=step time, smooth=0.0 0.136364,-0,0.142045,-0.242235,0.147727,-0.498726,0.153409,-0.745702 0.159091,-0.95949,0.164773,-0.670665,0.170455,-0.400267,0.176136,-0.170352 0.181818,-0 *Amplitude, name=SURFACE9, time=step time, smooth=0.0 0.159091,-0,0.164773,-0.223555,0.170455,-0.400267,0.176136,-0.511056 0.181818,-0.54063,0.1875,-0.287003,0.193182,-0.106276,0.198864,-0.00891943 0.204545,0 *Amplitude, name=SURFACE10, time=step time, smooth=0.0 0.181818,-0,0.1875,-0.0956677,0.193182,-0.106276,0.198864,-0.0267583 0.204545,0.14233,0.210227,0.236861,0.215909,0.239631,0.221591,0.15687 0.227273,0 *Amplitude, name=SURFACE11, time=step time, smooth=0.0

129

0.204545,0,0.210227,0.0789536,0.215909,0.239631,0.221591,0.470611 0.227273,0.755761,0.232955,0.645017,0.238636,0.468478,0.244318,0.246029 0.25,0 *Amplitude, name=SURFACE12, time=step time, smooth=0.0 0.227273,0,0.232955,0.215006,0.238636,0.468478,0.244318,0.738086 0.25,1,0.255682,0.738081,0.261364,0.468472,0.267045,0.215001 0.272727,0 *Amplitude, name=SURFACE13, time=step time, smooth=0.0 0.25,0,0.255682,0.246027,0.261364,0.468472,0.267045,0.645003 0.272727,0.755736,0.278409,0.47059,0.284091,0.239615,0.289773,0.0789448 0.295455,0 *Amplitude, name=SURFACE14, time=step time, smooth=0.0 0.272727,0,0.278409,0.156863,0.284091,0.239615,0.289773,0.236835 0.295455,0.142293,0.301136,-0.0267858,0.306818,-0.106294,0.3125,-0.0956762 0.318182,-0 *Amplitude, name=SURFACE15, time=step time, smooth=0.0 0.295455,0,0.301136,-0.00892861,0.306818,-0.106294,0.3125,-0.287028 0.318182,-0.54066,0.323864,-0.511077,0.329545,-0.400278,0.335227,-0.223559 0.340909,-0 *Amplitude, name=SURFACE16, time=step time, smooth=0.0 0.318182,-0,0.323864,-0.170359,0.329545,-0.400278,0.335227,-0.670677 0.340909,-0.9595,0.346591,-0.745705,0.352273,-0.498725,0.357955,-0.242233 0.363636,-0 *Amplitude, name=SURFACE17, time=step time, smooth=0.0 0.340909,-0,0.346591,-0.248568,0.352273,-0.498725,0.357955,-0.726698 0.363636,-0.909621,0.369318,-0.616054,0.375,-0.353544,0.380682,-0.142575 0.386364,-0 *Amplitude, name=SURFACE18, time=step time, smooth=0.0 0.363636,-0,0.369318,-0.205351,0.375,-0.353544,0.380682,-0.427725 0.386364,-0.415389,0.392045,-0.185459,0.397727,-0.035655,0.403409,0.0267311 0.409091,0 *Amplitude, name=SURFACE19, time=step time, smooth=0.0 0.386364,-0,0.392045,-0.0618196,0.397727,-0.035655,0.403409,0.0801934 0.409091,0.281761,0.414773,0.335733,0.420455,0.299651,0.426136,0.182979 0.431818,0 *Amplitude, name=SURFACE20, time=step time, smooth=0.0 0.409091,0,0.414773,0.111911,0.420455,0.299651,0.426136,0.548937 0.431818,0.841271,0.4375,0.692919,0.443182,0.488577,0.448864,0.249841 0.454545,0 *Amplitude, name=SURFACE21, time=step time, smooth=0.0 0.431818,0,0.4375,0.230973,0.443182,0.488577,0.448864,0.749523 0.454545,0.989817 ** *STEP,NLGEOM,INC=500 *VISCO

130

0.00568182,0.45 *DSLOAD,follower=NO,AMPLITUDE=SURFACE1 SURFACE1,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE2 SURFACE2,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE3 SURFACE3,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE4 SURFACE4,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE5 SURFACE5,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE6 SURFACE6,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE7 SURFACE7,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE8 SURFACE8,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE9 SURFACE9,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE10 SURFACE10,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE11 SURFACE11,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE12 SURFACE12,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE13 SURFACE13,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE14 SURFACE14,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE15 SURFACE15,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE16 SURFACE16,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE17 SURFACE17,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE18 SURFACE18,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE19 SURFACE19,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE20 SURFACE20,TRVEC,100,0,0,-1 *DSLOAD,follower=NO,AMPLITUDE=SURFACE21 SURFACE21,TRVEC,100,0,0,-1 *END STEP